Interacting Rydberg atoms - Heidelberg University

Interactions in an ultracold gasof

Rydberg atoms

Inaugural-Dissertationzur

Erlangung des Doktorgrades

der Fakultat fur Mathematik und Physikder Albert-Ludwigs-UniversitatFreiburg im Breisgau, Germany

vorgelegt von

Dipl.-Phys. Kilian Talo Theodor Singer

aus Uberlingen am Bodensee

im Oktober 2004

Dekan: Professor Dr. Josef HonerkampLeiter der Arbeit: Professor Dr. Matthias WeidemullerReferent: Professor Dr. Matthias WeidemullerKorreferent: Professor Dr. Hanspeter HelmTag der Verkundigungdes Prufungsergebnisses: 26.11.2004

2

Publications

Part of the work presented in this thesis is based on the following publications:

• M. Weidemuller, K. Singer, M. Reetz-Lamour, T. Amthor and L. G. MarcasseUltralong-Range Interactions and Blockade of Excitation in a ColdRydberg Gasto appear in Atomic Physics XIX (Proceedings of ICAP 2004) and in Braz. J.Phys. (2004)

• K. Singer, J. Stanojevic, M. Weidemuller, and R. CoteLong range interaction potentials for the ns-ns, np-np and nd-ndasymptotes for rubidium Rydberg atom pairsJ. Phys. B in press (2004)

• K. Singer, M. Reetz-Lamour, T. Amthor, S. Folling, M. Tscherneck, and M.WeidemullerSpectroscopy of an ultracold Rydberg gas and signatures of Rydberg-Rydberg interactionsJ. Phys. B in press (2004)

• K. Singer, M. Reetz-Lamour, T. Amthor, L. G. Marcassa, and M. WeidemullerSpectral Broadening and Suppression of Excitation Induced byUltralong-Range Interactions in a Cold Gas of Rydberg AtomsPhys. Rev. Lett. 93, 163001 (2004)

also selected for the October 25 issue ofVirtual Journal of Nanoscale Science & Technology (2004)

• K. Singer, M. Tscherneck, M. Eichhorn, M. Reetz-Lamour, and M. WeidemullerMethod and apparatus for the coherent addition of laser beams fromdistinct laser sourcesGerman Patent DE 102 43 367 (2004)

• K. Singer, M. Tscherneck, M. Eichhorn, M. Reetz-Lamour, S. Folling, and M.WeidemullerPhase-coherent addition of laser beams with identical spectral prop-ertiesOptics Communications 218, 371 (2003)

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• K. Singer, M. Reetz-Lamour, M. Tscherneck, S. Folling, and M. WeidemullerTowards an ultracold dense gas of Rydberg atomsin: Interaction in Ultracold Gases: From Atoms to Molecules (Wiley, New York2003)

• K. Singer, S. Jochim, M. Mudrich, A. Mosk, and M. WeidemullerLow-cost mechanical shutter for light beamsRev. Sci. Instrum. 73, 4402 (2002)

In addition the author contributed to the following publications:

• M. Mudrich, S. Kraft, K. Singer, A. Mosk, and M. WeidemullerThermodynamics in an ultracold mixture of optically trapped atomicgasesin: Interaction in Ultracold Gases: From Atoms to Molecules (Wiley, New York2003)

• G. Fahsold, K. Singer, and A. PucciIn-situ IR-transmission study of vibrational and electronic propertiesduring the formation of thin-film β-FeSi2Journal of Applied Physics 91, 145 (2002)

• M. Mudrich, S. Kraft, K. Singer, R. Grimm, A. Mosk, and M.WeidemullerSympathetic Cooling with Two Atomic Species in an Optical TrapPhys. Rev. Lett. 88, 253001-1 (2002)

• S. Kraft, M. Mudrich, K. Singer, R. Grimm, A. Mosk, and M. WeidemullerSympathetic cooling of lithium by laser-cooled cesiumin: Laser Spectroscopy XV, Proceedings of the International Conference onLaser Spectroscopy (ICOLS01), 341-344 (2002)

• A. Mosk, M. Mudrich, S. Kraft, K. Singer, W. Wohlleben, R. Grimm, and M.WeidemullerMixture of ultracold lithium and cesium atoms in an optical dipoletrapAppl. Phys. B 79, 791 (2001)

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Abstract: This thesis presents observations of ultralong range interac-tions in a frozen Rydberg gas. The observed signatures are interaction-induced line broadenings and a suppression of resonant Rydberg excita-tion. The latter effect can be interpreted as the onset of a local dipoleblockade and can be exploited to perform quantum information process-ing with mesoscopic ensembles. The dominating interaction between apair of Rydberg atoms separated as far as 100 000 Bohr radii is thevan der Waals interaction. The strength of this interaction is calculatedquantitatively in a perturbative approach. We describe in detail ourexperimental apparatus which employs narrow-bandwidth continuous-wave (cw) laser excitation in a two-photon excitation process to producea frozen Rydberg gas from magneto-optically trapped 87Rb atoms. As aparticular feature of the apparatus we have implemented a novel schemefor the coherent addition of laser intensities and we have realized 3D de-generate Raman sideband cooling resulting in atom cloud temperaturesin the sub-µK range. Systematic studies of the Rydberg spectra asa function of excitation-laser intensity, density of excitable atoms andexcitation time are presented.

Zusammenfassung: In dieser Doktorarbeit werden die langreichweiti-gen Wechselwirkungen in einem ultrakalten Gas aus Rydbergatomen un-tersucht. Die beobachteten Signaturen sind wechselwirkungsverursachteLinienverbreiterungen und eine Anregungsunterdruckung bei resonan-ter Rydberganregung. Der letztere Effekt kann als Beginn einer lokalenDipolblockade interpretiert werden, welche zur Realisierung von Quan-teninformationsverarbeitung mit mesoskopischen Gesamtheiten aus-genutzt werden kann. Die dominierende Wechselwirkung zwischen zweiRydbergatomen in einem Abstand von 100 000 Bohrradien ist dievan-der-Waals-Wechselwirkung. Die Starke der Wechselwirkung wurdequantitativ in einem storungstheoretischen Ansatz berechnet. Der ex-perimentelle Aufbau wird detailliert beschrieben. Wir setzen schmal-bandige kontinuierliche Laseranregung in einem Zweiphotonenprozesszur Rydberganregung aus magneto-optisch gefangenen 87Rb Atomenein. In unserem Aufbau kommt ein neuartiges Verfahren zur koharentenAddition von Laserintensitaten zum Einsatz. Des Weiteren wurde 3Dentartete Ramanseitenbandkuhlung implementiert und Atomwolken-temperaturen im unteren µK Bereich erreicht. Systematische Studiender Rydbergspektren als Funktion der Anregungslaserintensitaten, derDichte der anregbaren Atomen und der Anregungszeit wurden durchge-fuhrt.

5

Contents

1 Introduction 9

2 Creation of an ultracold gas of atoms 132.1 Laser cooling and trapping . . . . . . . . . . . . . . . . . . . . . . 132.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3 3D degenerate Raman sideband cooling . . . . . . . . . . . . . . . 27

3 Rydberg atoms 333.1 Background on Rydberg atoms . . . . . . . . . . . . . . . . . . . 333.2 Excitation and detection of Rydberg atoms . . . . . . . . . . . . . 413.3 Spectra of Rydberg atoms . . . . . . . . . . . . . . . . . . . . . . 48

4 Interactions in a frozen gas of Rydberg atoms 554.1 Qualitative theoretical description . . . . . . . . . . . . . . . . . . 564.2 Quantitative perturbative approach . . . . . . . . . . . . . . . . . 604.3 Interaction-induced line broadening of Rydberg resonances . . . . 72

5 Blockade of Rydberg excitation on resonance 855.1 Quantum information processing with Rydberg atoms . . . . . . . 855.2 Interaction-induced inhibition of excitation . . . . . . . . . . . . . 905.3 Excitation of dipole-forbidden states . . . . . . . . . . . . . . . . 97

6 Conclusion and perspectives 101

A Experimental control system 105

B Detailed expressions for the interaction potentials 115

Bibliography 123

Acknowledgements 131

7

CONTENTS

8

Chapter 1

Introduction

Rydberg atoms are atoms in highly excited electronic states at energies close tothe ionization limit. The orbital radius of the electron extends over more thanthousands of bohr radii. These exaggerated properties lead to very strong polariz-abilities of Rydberg atoms. They are very fragile and can easily be perturbed e.g.by electric fields. Rydberg atoms at room temperatures have been extensivelystudied over many decades since the end of the 19th century [Gallagher, 1994].With the advances in laser cooling and trapping, the creation of an ultracold gasof Rydberg atoms became realizable and totally new perspectives for the inves-tigation of Rydberg states have opened since then. With modern laser coolingtechniques, Rydberg atoms can now be excited out of a gas of trapped alkaliatoms at densities exceeding 1010cm−3, and at temperatures in the micro-Kelvinrange. An important feature of Rydberg atoms excited out of laser-cooled atomsis that they do not move significantly during their radiative lifetime (“frozenRydberg gas”). The situation can be compared to an amorphous solid. In thefrozen Rydberg gas, resonant excitation exchange (Forster process) was reported, leading to unexpected effects such as the many-body diffusion of excitation[Mourachko et al., 1998, Anderson et al., 1998]. Other interesting effects are thepopulation of long-living high angular-momentum states through free charges[Dutta et al., 2001] and the spontaneous formation and recombination of ultra-cold plasmas [Robinson et al., 2000, Gallagher et al., 2003].

The interaction strength between Rydberg atoms can easily be tuned overseveral orders of magnitude e.g. by changing the density or by exciting themto different principal quantum numbers n. Adjacent Rydberg atoms interactmainly by the long-range van der Waals interaction whose strength scales withn11. The interaction declines with internuclear distance (R) as R−6 but is stillstrong enough that Rydberg atoms more than a thousand Bohr radii apart arepredicted to form a bound molecular state at temperatures reachable by modernlaser cooling techniques [Boisseau et al., 2002]. Furthermore molecules consistingof a ground state atom and a Rydberg atom with very high dipole moments arepredicted to form[Greene et al., 2000]. In this molecules the ground state atom

9

CHAPTER 1. INTRODUCTION

Figure 1.1: Schematic explanation of the dipole blockade. (a) Energy lev-els for a pair of atoms. |g〉 and |r〉 denote the ground and Rydberg staterespectively. The ultralong range dipole interaction splits the Rydberg pairstate and thus suppresses excitation of a Rydberg pair by a resonant laserfield, as indicated by the vertical arrows for small internuclear distances. (b)An excitation laser beam is overlapped with a cloud of cold atoms. Rydbergexcitation out of the gas is suppressed in the vicinity of a Rydberg atom bythe interaction, resulting in many domains within which only one Rydbergatom is excited.

is trapped in one of the nodes of the Rydberg electron wave function.

Observed signatures of the interaction between Rydberg atoms are density-dependent line broadening of resonances [Raimond et al., 1981], modification ofcollisional processes [de Oliveira et al., 2003], and molecular crossover resonancesdue to avoided crossings [Farooqi et al., 2003]. Apart from tunable interaction,Rydberg atoms offer the unique possibility of tuning the energy level of the Ryd-berg state with small electric fields by exploiting the dc-Stark effect. This canbe done by applying only small fields of several V/cm as Rydberg states are veryfield sensitive. If Rydberg levels are tuned exactly between two other levels towhich dipole transitions are allowed, then the interaction between the Rydbergatoms is strongly increased due to resonant dipole-dipole interaction scaling asR−3.

This strong dipole-dipole interaction between Rydberg atoms has been pro-posed to be used in the implementation of a fast phase gate for quantum in-formation processing [Jaksch et al., 2000]. In contrast to already implementedsystems with ions [Schmidt-Kaler et al., 2003] where the Coulomb interaction isthe basic coupling mechanism used for entanglement, [Jaksch et al., 2000] pro-posed to use long-lived metastable ground states of neutral atoms as storagestates and entangle states by exciting for a short period of time into Rydbergstates which strongly interact. This approach would have the great advantagethat the atoms are decoupled from the environment during storage time, whichavoids decoherence effects. The proposal of [Jaksch et al., 2000] was extendedto a version using mesoscopic clouds of atoms avoiding the need for single atom

10

control [Lukin et al., 2001]. The underlying mechanism in the latter approachis the “dipole blockade”, which is the interaction-induced excitation inhibition ofmultiple Rydberg excitations within a mesoscopic cloud of atoms (see Figure 1.1).

This thesis is dedicated to the quest for signatures of the “dipole blockade”.Systematic density-dependent studies were performed on an ultracold gas ofRydberg atoms, showing that an excitation inhibition of Rydberg atoms canindeed be observed. This is interpreted as a local blockade of Rydberg exci-tation [Tong et al., 2004, Singer et al., 2004a]. The findings are supported byspectroscopic measurements of density-dependent line broadening and quantita-tive theoretical calculations of the van der Waals interaction potentials betweenRydberg atoms. The thesis is structured in five parts. In Chapter 2, we presentthe experimental setup for the creation of an ultracold gas of rubidium atoms.An overview of the special features of Rydberg atoms together with details aboutthe excitation into Rydberg states including the Rydberg detection and the cali-bration procedures is presented in Chapter 3. Chapter 4 is dedicated to processesinvolved in the interaction between Rydberg atoms. Starting with a quantitativetheoretical picture of the various types of dipole-dipole interaction between Ryd-berg atoms, we present quantitative perturbative calculations of the long-rangevan der Waals interaction potentials. Experimental results on interaction-inducedline broadening are presented. In Chapter 5, we present our central results onthe inhibition of Rydberg excitation due to long range Rydberg-Rydberg interac-tions. Finally a conclusion and outlook on future perspectives of the experimentare given in Chapter 6.

11

CHAPTER 1. INTRODUCTION

12

Chapter 2

Creation of an ultracold gas ofatoms

Laser-cooled alkali atoms can be generated at densities exceeding 1010cm−3 andtemperatures below 1mK. They are the starting point for many exciting exper-iments leading to key breakthroughs in basic research like the first realizationof a Bose-Einstein condensate in a dilute gas of atoms [Anderson et al., 1995,Bradley et al., 1995, Davis et al., 1995]. In our experiment, laser-cooled rubid-ium atoms are used as starting point for Rydberg excitation.

In this chapter, we show how to create an ultracold gas of rubidium atoms.The magneto-optical trap (MOT) [Raab et al., 1987], the working horse foratomic physicists, is presented in Section 2.1. The experimental apparatus insidewhich the MOT is located, is presented in Section 2.2.1. Details on the lasersystem, including the patented laser addition system, are shown in Section 2.2.2.In Section 2.3, we present the experimental realization and the results of Ramansideband cooling reaching temperatures in the sub-µK range.

2.1 Laser cooling and trapping

2.1.1 Rubidium

In our experiments, we use 87Rb, which is a nonstable isotope of rubidium (theonly stable isotope being 85Rb) which has a nuclear lifetime of 4.8810 and decaysby β− decay to 87Sr. The overall physical properties of 87Rb are given in Table 2.1.It has a nuclear spin of I = 3/2 and, like all alkali metals, it has only one unpairedvalence electron. As a result the electron spin is S = 1/2. Therefore, 87Rbobeys bosonic quantum statistics. The optical properties of the D2 transitionare summarized in Table 2.2, and the level scheme is depicted in Figure 2.1[Steck, 2001]. Due to the existence of two closed transitions 52S1/2(F = 1) →52P3/2(F

′ = 0) and 52S1/2(F = 2) → 52P3/2(F′ = 3), the D2 transition is perfectly

13

CHAPTER 2. CREATION OF AN ULTRACOLD GAS OF ATOMS

Table 2.1: Physical Properties of 87Rb (taken from [Steck, 2001]).

Atomic Numbers Z 37Total Nucleons Z +N 87Relative Natural Abundance η(87Rb) 27.83(2)%Nuclear Lifetime τn 4.88× 1010 yrAtomic Mass m 86.909 180 520(15) u

1.443 160 60(11) ×10−25 kgDensity at 25C ρm 1.53 g/cm3

Melting Point TM 39.31 CBoiling Point TB 688 CSpecific Heat Capacity cp 0.363 J/g·KMolar Heat Capacity Cp 31.060 J/mol·KVapor Pressure at 25C PV 3.0×10−7 torrNuclear Spin I 3/2Ionization Limit EI 33 690.8048(2) cm−1

4.177 127 0(2) eV

suited for laser cooling applications. The D1 transition does not possess a closedtransition and is therefore not taken into consideration.

2.1.2 Radiation pressure force

A comprehensive descriptions of the magneto-optical trap can be found in themonograph on laser cooling and trapping [Metcalf and van der Straten, 1999].The principle of the MOT is based on the radiation pressure force actingon an atom as it interacts with near-resonant light. For a simple two-levelsystem, consisting of a single ground state g and a single excited state e,one finds the following semiclassical expression for the radiation pressure force[Cohen-Tannoudji et al., 1988]:

Frad = ~kΓ1

2

I/IS

1 + I/IS + 4 (δ/Γ)2 (2.1)

where δ = ω − ω0 is the detuning from resonance and IS = ~Γω30/12πc2 denotes

the saturation intensity.

Equation 2.1 can be interpreted in a very intuitive way: ~k is the momentumof a photon from the laser wave. This momentum is transferred by the atom whenabsorbing the photon. Subsequently, a photon is emitted by spontaneous emis-sion, but on average no momentum is transferred to the atom because 〈~k′〉 = 0with ~k′ denoting the momentum of the spontaneously emitted photon. There-fore, the force on the atom is the momentum transfer multiplied by the photon

14

2.1. LASER COOLING AND TRAPPING

193.7408(46) MHz

229.8518(56) MHz

302.0738(88) MHz

72.9113(32) MHz 266.650(9) MHz

156.947(7) MHz

72.218(4) MHz

F = 3

F = 2

F = 1

F = 0

5 P3/22

F = 2

F = 1

2.563 005 979 089 11(4) GHz

4.271 676 631 815 19(6) GHz

6.834 682 610 904 29(9) GHz

g = 1/2(0.70 MHz/G)

F

g = -1/2(-0.70 MHz/G)

F

5 S1/22

g = 2/3(0.93 MHz/G)

F

g = 2/3(0.93 MHz/G)

F

g = 2/3(0.93 MHz/G)

F

780.246 291 629(11) nm 384.227 981 877 3(55) THz12 816.465 912 47(18) cm-1

Figure 2.1: The 87Rb D2 transition with hyperfine splitting (taken from[Steck, 2001]). For each level, the approximate Lande gf -factor together withthe Zeeman splitting between adjacent magnetic sublevels is given.

15


Table 2.2: Optical properties of the 87Rb D2(52S1/2 → 52P3/2) transition

(taken from [Steck, 2001]).

Frequency ω0 2π · 384.227981877 THzWavelength (Vacuum) λ 780.246 291 629(11) nmWavelength (Air) λair 780.037 08 nmWave Number (Vacuum) kL/2π 12 816.465 912 47(18) cm−1

Lifetime τ 26.24(4) nsDecay Rate Γ 38.11(6)×106 s−1

Natural Line Width (FWHM) Γ 2π · 6.065(9) MHzAbsorption oscillator strength f 0.6956(15)Recoil Velocity vr 5.8845 mm/sRecoil Energy ωr 2π · 3.7709 kHzRecoil Temperature Tr 361.95 nKDoppler Shift ∆ωd(vatom = vr) 2π · 7.5418 kHzDoppler Temperature TD 146 µK

scattering rate Γsc. The scattering rate is the product of the spontaneous emis-sion rate Γ and the population of the excited state which, for a two-level atom,is given by the last term of Equation 2.1. Although the momentum transfer byabsorption of one single photon is rather small, the acceleration caused by the ra-diation pressure reaches very high values of 106− 107g due to the high scatteringrate Γsc ∼ 107 s−1.

From Equation 2.1 it follows that the radiation pressure force critically de-pends on the detuning of the laser light from resonance. An atom moving with avelocity v “sees” a laser frequency that is shifted by an amount ∆ω = ±kv due tothe Doppler effect. The effective detuning of the laser from resonance is thereforegiven by

δDoppler = ω − ω0 ∓ k · v . (2.2)

If an atom moves towards the laser beam, the laser frequency is shifted towardshigher frequencies. Consequently, when the laser is detuned below the atomicresonance, a counter-moving atom will be shifted towards resonance and thusexperience a stronger force. If an atom is exposed to laser radiation from oppo-site directions at a frequency below the atomic resonance, the atom will alwaysabsorb more photons from the counter-propagating laser field than from the co-propagating one, which gives rise to a friction force according to Eq. (2.2). Themotion of an atom illuminated by laser beams from all six spatial directions isthus strongly damped. The term optical molasses has been coined for pairs ofcounter-propagating laser beams detuned below the atomic resonance, in whichthe atoms are cooled by the friction force exerted by the radiation pressure.

16

2.1. LASER COOLING AND TRAPPING

Figure 2.2: Magneto-optical trap. (a) Schematic drawing of the setup. Thecurrents induce an axially symmetric quadrupole magnetic field B. The laserbeams are circularly polarized. (b) Energy level diagram for a ground statewith total angular momentum Fg=0 and an excited state with Fe=1 in aquadrupole magnetic field. The laser frequency ω is tuned below the atomicresonance frequency at zero field. At positions right from the center, the laserbeam driving the σ− transition is tuned closer to resonance by the Zeemaneffect. The atom therefore absorbs more photons from the σ− beam thanfrom the σ+ beam. The atom is thus pushed towards the center which is thepoint of stable equilibrium for the radiation pressure of all beams.

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2.1.3 Magneto-optical force

The friction force only provides compression in velocity space. To additionallyconfine the atoms to a small volume, a spatially dependent force is needed. Sucha force can be created by combining a magnetic quadrupole field with an appro-priate polarization of the light fields, as indicated in Figure 2.2(a). Due to theZeeman effect in the inhomogeneous field B(r), the detuning becomes a functionof position

δZeeman = ω − ω0 ± µ′B(r) (2.3)

where µ′ = (geme − ggmg)µB is the effective magnetic moment of the transitionused (gg,e = g-factor the ground and excited state, mg,e = magnetic quantumnumber, µB = Bohr’s magneton). The laser frequency is chosen below the atomicresonance, and the polarizations are adjusted so that an atom away from thecenter always absorbs more photons from the wave pushing it towards the centerthan from the opposite wave, as schematically shown in Figure 2.2(b). Thus,the atoms experience a spatially dependent restoring force and the point of zeromagnetic field constitutes the center of the trap.

In this way, the magneto-optical trap provides two nice things at the sametime: cooling through the Doppler effect and confinement via the Zeeman effect[Raab et al., 1987]. When both the Zeeman and the Doppler shift are small incomparison to the laser detuning ω − ω0, the force can be expanded in velocityand position:

FMOT ' −βfricv − κconfr . (2.4)

Thus, the atomic ensemble is cooled and compressed into the center of the mag-netic quadrupole field.

Cooling in the MOT is even more effective due to the presence of polarization-gradient cooling mechanisms which rely on selective optical pumping betweenZeeman sublevels of the ground state in fields with spatially varying polarizations[Dalibard and Cohen-Tannoudji, 1989]. The final velocity of the trapped atomscorresponds to only a few photon recoils ~k/m, the natural limit of laser cooling(“recoil limit”). 87Rb with its closed transition 5S1/2(F = 2) → 5P3/2(F

′ = 3)(see Section 2.1.1) is perfectly suited for the application of polarization-gradientcooling mechanisms. Achievable temperatures in a MOT are around 100 µK.

2.2 Experimental setup

2.2.1 Vacuum system

The MOT is located inside a vacuum chamber. Figure 2.3 shows the whole vac-uum setup including the ion getter pump and titanium sublimation pump whichmaintain a base pressure below 10−10mbar monitored by an ionization gauge

18

2.2. EXPERIMENTAL SETUP

Figure 2.3: Vacuum system: Ion getter and titanium sublimation pumpmaintain UHV base pressure below 10−10mbar, which is monitored by theionization gage pressure sensor. The UHV valve is connected to a turbomolecular pump which is only used during evacuation. The main physicshappens in the stainless steel cube where the MOT is located. The compen-sation coils are in Helmholtz configuration in order to compensate the earthmagnetic field.

photo diode

fieldplates

iondetection

trappinglaser

absorptionlaser

CCD camera forabsorption imaging

Rb-dispenser

excitation laser

Figure 2.4: Schematic of the experiment.

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Figure 2.5: (a) Rendered image of the CAD-document used to designthe chamber. The individual elements are: 1) stainless steel cube (length123mm), 2) magnetic coils used to generate the quadrupole field for theMOT, 3) offset coil to generate an offset field for magnetic trapping, 4) ru-bidium dispenser glowing red behind heat shield, 5) atom cloud, 6) microchannel plates (MCP) for ion and electron detection, 7) field grids to shieldfield from MCP, 8)central field grids used for field ionization of the Rydbergatoms.

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Figure 2.6: rubidium dispensers as atom source. Heat shield removed inboth pictures.

pressure sensor. During evacuation, an additional turbo molecular pump is con-nected to the UHV valve. The main physics happens in the stainless steel cubewhich is schematically shown in Figure 2.4(a). A cloud of cold 87Rb atoms is pre-pared in a magneto-optical trap (MOT) [Raab et al., 1987]. The magnetic coilsneeded for the MOT are placed inside the stainless steel cube with a side lengthof 124mm (Kimball Physics), providing optical access along the three mutuallyorthogonal main axes and along the four diagonals. The earth magnetic fieldis compensated by three compensation coils in Helmholtz configuration (yellowcoils in Figure 2.3). Figure 2.5(a) shows the rendered image of the CAD docu-ment which was used for the design of the whole setup. The view through oneof the main view ports with window removed is depicted in Figure 2.5. The sixmain windows are used for the trapping and cooling laser beams, while two diag-onals serve as electrical feed-throughs. The Rydberg excitation laser beam passesthrough another diagonal, and the fourth diagonal is used for fluorescence moni-toring. The atom cloud is loaded from rubidium vapor released from a dispenser(SAES Getters) situated close to the center of the vacuum cube (see Figure 2.6)[Fortagh et al., 1998]. A spark eroded heat shield (removed at Figure 2.6) is putin front of the dispensers. This heat shield has two slits designed in such a waythat rubidium atoms are still captured by the MOT beams but direct exposure ofthe atom cloud is avoided in order to reduce exposure to thermal radiation. Thisis an important issue when dealing with Rydberg atoms, as black body radiationcan significantly reduce the lifetime of Rydberg states.

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Figure 2.7: Schematic of the laser system for cooling, trapping and Rydbergexcitation.

2.2.2 Laser system

General setup of the laser system The schematic of the laser sys-tem for driving the MOT is depicted in Figure 2.7. The major laserbeams used to drive the MOT are emitted by a novel patented laser system[Singer et al., 2004d, Singer et al., 2003] which coherently adds two laser beams.It is described in Section 2.2.2. An external cavity diode laser (ECDL) elec-tronically locked to an atomic transition with a Doppler-free dichroism spec-troscopy [Corwin et al., 1998] serves as the primary frequency reference (see[Tscherneck, 2002, Folling, 2003]). The laser beam is frequency-shifted with anacousto-optical modulator (AOM) to a detuning of 1.5 Γ (natural line widthΓ/2π = 6.1 MHz) below the transition 5S1/2(F=2) → 5P3/2(F=3) (see Figure 2.8)for atom cooling and trapping in the MOT. The beam seeds two laser-diodeswhich are coherently added to produce a single laser beam with an intensitywhich is equal to the sum of the intensities of the two sources. The resultingbeam is guided through a single mode fiber for perfect overlap and mode clean-ing, providing 70mW of laser power at the fiber output. The fiber output beam issplit into six beams which are passing quarter-wave plates and are sent throughthe six main windows of the vacuum chamber with a background pressure of5× 10−11mbar to provide the cooling and trapping light for the MOT.

The cooling and trapping beams are superimposed with beams from a secondECDL (see Figure 2.7) driving the transition 5S1/2(F=1) → 5P3/2(F=2) in orderto repump atoms that have been off-resonantly excited out of the cooling andtrapping cycle (see Figure 2.8). This laser is locked to an atomic transitionusing Doppler-free frequency-modulation spectroscopy [Bjorklund, 1980]. A smallportion of the frequency-reference ECDL is frequency-shifted with an AOM and

22


Figure 2.8: Energy levels of 87Rb with the relevant transitions addressedby transitions driven by the trapping laser, the repumping, the depumpinglasers and the lasers used for absorption imaging.

is used to intentionally depump atoms out of the cooling and trapping cycle. Thisallows for systematic measurements of density-dependent effects as described indetail in Section 4.3.2. All laser beams can be switched or attenuated by means ofAOMs in combination with mechanical shutters [Singer et al., 2002] to allow forcontrolled manipulation and excitation of internal states. The timing is controlledby a dedicated microcontroller with integrated RISC processor for precise timingprocessing (see Appendix A for further details).

The frequency-reference ECDL laser also provides a beam for absorption im-ages of the atomic cloud in the MOT, which are detected by a CCD camera. Sizeand density of the cloud are determined by analyzing absorption images. Tem-perature measurements are performed by monitoring the ballistic expansion ofthe cloud after turning off all trapping lasers. The number of atoms in the MOTis monitored by collecting the fluorescence light on a photodiode. The measuredfluorescence signal is calibrated with absorption images to yield absolute atomnumbers.

The atom cloud can be approximated by a Gaussian spatial distribution withan 1/e diameter of 1.2(0.3) mm, resulting in peak densities of 1.1(3)× 1010 cm−3

and a total amount of atoms of 1.2(3)×107, as determined by absorption imagingwith a resonant probe laser beam ∗. Loading times of the MOT are roughly 5 s(see Figure 2.9). The steady state temperature of the MOT is about 100µK.The temperature can be further reduced to 4.5µK [Lett et al., 1988] in pulsedoperation by means of optical molasses down or below 1µK by means of 3Ddegenerate Raman sideband cooling (see Section 2.3) .

∗We estimate a factor of 2 as the systematic error for the determination of the density andnumber of atoms.

23


0 5 10 15 20

0

1x107

2x107

3x107

4x107

5x107

6x107

7x107

8x107

Fluorescence level

Fit to A(1-e -t/T), A=7.6 10 7, T=3.9s

Num

ber

of A

tom

s

Time[s]

Figure 2.9: Loading curve of the MOT (Dispenser current 4.7A, MOT-coilcurrent 1.7 A, cooling laser detuning 17 MHz). The curve is well fitted by anexponential saturation curve.

Phase-coherent addition of laser beams Sufficient laser power is crucial foroptimal trapping conditions in the MOT. However, the maximum output powerof laser sources is technically limited. Phase-coherent addition of laser beamsfrom separate sources offers a way to overcome this limitation. The ultimate goalis to attain an output beam which is indistinguishable from a beam originat-ing from a single laser source. For this purpose several, schemes were realized.A lossless beam combiner for laser sources with slightly different frequencies isdescribed in [Haubrich et al., 2000]. For sources with equal frequencies, addi-tion can be achieved by injection locking the laser sources to a master oscillatorand combining the laser beams interferometrically by means of a simple beamsplitter [Tempus et al., 1993, Kerr and Hough, 1989, Musha et al., 2001]. Alter-native combination methods for phase-locked laser beams with equal frequenciesare realized by means of a binary phase grating [Leger et al., 1986] or a birefrin-gent element [Menard et al., 1996]. An electro-optical beam combiner with anintegrated phase modulator based on a Ti : LiNbO3 waveguide was realized by[Buhl and Alferness, 1987].

In our setup, we use a novel patented method for the coherent combina-tion of two laser beams in order to obtain a laser beam with added intensityat a single frequency [Singer et al., 2003, Singer et al., 2004d]. To obtain anoutput beam that is indistinguishable from the laser beam emitted by a sin-gle laser source, the two initial laser beams should have exactly the same fre-

24


S1

S2

M1M2

PBC1

HW1

HW2

QW1

piezodiff

stabilization

PBC2

optical

isolator O1

master beamoutput beam

HW3

AP1

PD1

PD2

optical

isolator O2

single

mode

fiber

Figure 2.10: Detailed view of the coherent laser beam combiner.

quency with a constant phase relation and well-matched phase fronts. In or-der to satisfy the first two conditions, we use the method of injection lock-ing [Siegmann, 1986, Bouyer and Breant, 1992, Gertsvolf and Rosenbluh, 1999,Mercier and McCall, 1997, Ringot et al., 1999]. Two laser diodes (slave lasers)are seeded by a reference laser (master laser) in order to force the slave diodes toemit laser light with spectral properties that are identical to those of the referencelaser. The phase fronts are matched by coupling the beams into an optical fiber.

The basic setup is shown in Figure 2.10. The master beam enters the slavediode setup through an optical isolator. The master beam is split into two com-ponents at a polarizing beam splitter cube (PBC) in order to simultaneously seedtwo slave lasers. The beams of the injection-locked slave lasers are superposedon the PBC and the resulting beam passes through the optical isolator in re-verse direction. The output beam emanates from the second port of the opticalisolator. If the relative phase is equal to zero, linear polarization is attained af-ter the PBC. To maximize the output of the optical isolator, the polarization ofthe superposed laser beam is adjusted by means of the half-wave plate (HWP).Thermal expansion and acoustic noise can lead to drifts and fluctuations of therelative phase which in turn cause fluctuations in the polarization of the super-posed beam. Therefore, a monitor beam emanating from the second port of thePBC is used as input for a stabilization scheme which controls the phase of slavelaser S2 with the aid of a phase-shifting element. We achieve an output beamwith low intensity noise and a power equivalent to the sum of the powers of theindividual slave diodes. The combined beam is coupled into a single-mode polar-ization maintaining optical fiber with the same efficiency as a beam originatingfrom a single laser source (74 %).

The injection lock is best characterized by the locking range, i.e. the frequency

25


Figure 2.11: (a) Locking range for the combined slave laser as a function ofthe injected power. The dashed curve is a square root function fitted to thedata. (b) Error signal (black) and the corresponding output power (red) asa function of the piezo voltage. The vertical line shows the piezo voltage atwhich the system is stabilized for maximum output power.

range of the master laser over which the slave diodes follow the master laser. Thelocking range depends on the overlap and mode matching between the beamsof master and slave. It increases with the intensity of the master laser which iscoupled into the slave diode. Under optimal conditions, the minimum master-beam intensity for stable injection locking of both slaves was as low as 20 µW. The

range is given by ∆νlock = γe

π

√Pmaster

Pslave[Siegmann, 1986], with Pmaster and Pslave

denoting the powers of master and slave respectively, and γe the energy loss rateof the resonator. One therefore expects a square root functional dependence ofthe locking range on the injected power. Figure 2.11(a) depicts the measuredlocking range as a function of the master intensity monitored on a Fabry-Perotcavity into which both slave lasers are coupled. The locking range of the combinedsetup is equal to the locking range of one laser diodes. The data were fitted with∆ν = A

√Pmaster and A = (1090± 30) MHz/

√mW (dashed line).

In order to characterize the quality of the injection lock, we measured thefrequency spectrum of the output beam by recording the beat note between themaster beam and the slave laser with a fast photodiode. The master beam wasshifted by about 140 MHz with an acousto-optical modulator (AOM). The mea-sured width of the beat signal was less than 1 kHz and was thus completelydetermined by the resolution bandwidth of the spectrum analyzer. The centerfrequency of the beat note drifted over a range of about 1 kHz on a time scaleof several seconds which reflects a drift of the voltage controlled oscillator in thedriving electronics of the AOM. There is no indication of incoherent contributionsto the beat signal.

Exact matching of both slave beams after the polarizing beam splitter cubePBC1 is crucial for producing an added beam that is indistinguishable from a

26

2.3. 3D DEGENERATE RAMAN SIDEBAND COOLING

beam coming from a single laser source. Therefore, the output beam is coupledinto a single-mode polarization-maintaining optical fiber in order to filter thebeam spatially and attain perfect mode matching. We achieve a transmission of74% for the combined laser beam. A second optical isolator O2 is inserted beforethe fiber as the phase stabilization is extremely sensitive to feedback of light fromthe fiber. Figure 2.11(b) shows a measurement of the power transmitted throughthe fiber while the piezo voltage is scanned periodically. From the visibility ofthe interference pattern one can deduce the degree of coherence γ = 0.975(1) (fordetails see [Singer et al., 2003]).

2.3 3D degenerate Raman sideband cooling

The temperatures in a MOT can be further lowered below 1µK with the helpof Raman sideband cooling. At these temperatures, 87Rb atoms have thermalvelocities around 17µm/ms. In a typical MOT, the average nearest neighbor dis-tance is about 5µm, while Rydberg lifetimes can be as long as several hundredmicroseconds. Temperatures in the sub-µK regime are therefore ideal to preparefrozen Rydberg gases with long interaction times. Degenerate 3D Raman side-band cooling [Kerman et al., 2000] provides these temperatures by cooling thevibrational atomic motion in an optical lattice. It has been implemented success-fully for cesium, reaching temperatures as low as 200nK [Treutlein et al., 2001].Apart from lower temperatures, the use of an optical lattice offers an additionalfeature that is interesting for the preparation of frozen Rydberg gases: Opticallattices show long-range order even for filling factors far below 1 (as typical for aMOT) [Weidemuller et al., 1998], which may alter the effects of ultralong-rangeinteractions. For filling factors of 1 (typical in the Mott insulator state of aBEC [Greiner et al., 2002] and if unity excitation probability can be achieved),the Rydberg gas actually constitutes a crystal. In this case, the electrons mightexhibit phase transitions into a metal-like state.

2.3.1 Setup for Raman sideband cooling

Our setup corresponds to the setup reported in [Treutlein et al., 2001] (see Fig-ure 2.12). It is described in detail in the diploma thesis of Simon Folling[Folling, 2003]. In brief, the atoms in their ground state

∣∣5S1/2, F = 1⟩

aretrapped in an optical lattice in the Lamb-Dicke regime. This assures thatscattered photons do not change the vibrational quantum number. Whileall mF -sub-states feel the same optical potential, a magnetic field shifts theenergy of adjacent Zeeman states by exactly one quantum of the vibra-tional motion. If a photon from a pumping beam (σ+- and π-polarizedlight resonant with

∣∣5S1/2, F = 1⟩→

∣∣5P3/2, F′ = 0

⟩) is absorbed by the∣∣5S1/2, F = 1,mF = −1, ν

⟩(∣∣5S1/2, F = 1,mF = 0, ν

⟩) state, the atom preferably

27


a) b)

Figure 2.12: (a) Principle of 3D degenerate Raman sideband cooling asexplained in the text. (b) Our optical setup following [Treutlein et al., 2001]

Figure 2.13: Relevant energy levels of 87Rb the Raman lasers at 780 nm.

decays into∣∣5S1/2, F = 1,mF = 1, ν

⟩(see level scheme in Figure 2.13). How-

ever, since the atom keeps its vibrational quantum number, energy of up one ortwo times the Zeeman splitting is removed (equal to one or two vibrational en-ergy quanta). Two-photon Raman coupling transfers atoms back to the degener-ate

∣∣5S1/2, F = 1,mF = 0, ν − 1⟩

and∣∣5S1/2, F = 1,mF = −1, ν − 2

⟩states, and

the cycle can start again. The atoms finally end up in the dark overall groundstate

∣∣5S1/2, F = 1,mF = 1, ν = 0⟩. The atomic sample is simultaneously spin-

polarized by the cooling cycles.

The optical lattice is created by a standing wave along the x-direction and tworunning waves along +y and +z (see Figure 2.12(b)). These beams are providedby a Ti:Sapphire laser (Coherent MBR 110 pumped by a Coherent Innova 90 Ar+

ion laser) delivering 85mW after passing through a single mode polarization-maintaining optical fiber. The frequency was set 15 GHz below the transition5S1/2(F = 1) → 5P3/2(F

′ = 0). The 1/e-radius of the beams is 1.1mm with

28


equal power for all 4 beams. All polarizations are chosen to be linear and in theyz-plane, in order to ensure maximum lattice depth. The two counterpropagatingbeams along x are polarized with an inclination relative to the xy-plane. Theinclination angles determine the two-photon Raman coupling and are optimizedexperimentally. In the presented measurements, both inclination angles are setclose to 45. The calculated lattice depths (in units of the Boltzmann constant)are 18µK along y/z and 38µK along x, resulting in vibrational frequencies of27 kHz and 59kHz respectively. In order to suppress heating during release fromthe lattice, the lattice laser power was lowered adiabatically with an acousto-optical modulator [Kastberg et al., 1995].

The pumping beam for closing the Raman cycle is provided by an ECDLlocked to a vapor cell close to the 5S1/2(F = 1) → 5P3/2(F

′ = 0) resonance. Op-timal cooling is achieved for intensities around 60µW/cm2. While the pointingof this beam is fixed (the beam is propagating in the xy-plane with an angle of∼ 10 relative to the y-axis), the direction of the coupling magnetic field can bechosen freely and is optimized experimentally. Optimal cooling is achieved withcircularly polarized light with a magnetic field pointing almost parallel to thepumping beam. This ensures that the pump beam contains mainly σ+, little πand no σ− polarization components [Kerman et al., 2000]. As the pumping laseroff-resonantly scatters atoms into the 5S1/2(F = 2) states, an additional ECDL onthe transition 5S1/2(F = 2) → 5P3/2(F

′ = 2) pumps atoms back into the coolingcycle. Since Raman sideband cooling relies on well-defined magnetic fields, straymagnetic offset fields (e.g. earth fields, vacuum pumps) must be compensated.We determined the zero of the magnetic field by means of dark states (mechanicalHanle effect [Kaiser et al., 1991]): since the pumping laser runs on a closed tran-sition (

∣∣5S1/2, F = 1⟩→

∣∣5P3/2, F′ = 0

⟩), it pushes the atoms out of the MOT

region if the lattice is not switched on. However, for σ+(π)-polarized light, themF = 0, −1 (mF = ±1)-states do not couple to the laser light and the atoms willnot be pushed by the pumping beam. In order to cancel the magnetic field, themagneto-optical trap is switched off and the pumping beam is turned on for 13ms.Afterwards, the MOT-light is switched on again and the amount of recapturedatoms is measured. This rate is much larger for well-defined σ+(π)-polarizationthan for mixed polarizations. Since the pumping beam travels (almost) along+y, right-handed circularly polarized light has σ+ polarization if and only if themagnetic field points along ±y. Therefore, the magnetic field along x/z can easilybe canceled by maximizing the recaptured atom numbers. The same holds forlight which is linearly polarized along x (the magnetic field must be zero alongy/z in this case).

2.3.2 Experimental results of Raman cooling

For the measurements of Raman sideband cooling, a MOT is loaded with 1.4×107

atoms. Subsequently, the magnetic quadrupole field is switched off, and the

29


0 20 40 60 80 100 120 1400

1

2

3

Zeeman shift (kHz)

Tem

pera

ture

(

K)

0

10

20

30

40

50

59 kHz27 kHz

Cap

ture

effic

ienc

y (%

)

Figure 2.14: Magnetic field dependence of capture efficiency () and tem-perature (Tx •, Ty ). The minimum temperature was determined with afit on the ballistic expansion of the atomic cloud after adiabatic release fromthe optical lattice, giving 0.70± 0.01µK along x and 1.07± 0.02µK along y.The other temperatures were calculated from the cloud width measured 60msafter release. The calculated vibrational frequencies of 27kHz and 59kHz areindicated by the vertical lines.

30


0 20 40 60 80 100 120 140

0,8

1,0

1,2

1,4

1,6

1,8

2,0

2,2

Clo

ud W

idth

[mm

]

Time [ms]

x Width fit to x Width, T

x= 720nK

y Width fit to y Width, T

y=1120nK

Figure 2.15: Ballistic expansion of an atom cloud of 5×106. The tempera-tures are obtained by fits to the expansion curve: 0.71(1)µK along the x-axisand 1.12(3)µK along the y-axis.

31


trapping laser is detuned by −133MHz to the cooling transition for a molassesphase of 5ms. This pre-cools the atomic cloud to 4.5µK. Afterwards, the coolinglaser is switched off, and the pumping beam, the repumping beam and the latticeare turned on. After 10ms the pumping and repumping light is turned off andthe lattice is ramped down adiabatically. After 60ms of ballistic an absorptionimage of the atom cloud is taken. This cycle is repeated for several values anddirections of the magnetic field. While the measured number of atoms reflectsthe capture efficiency of the lattice, the cloud size after a fixed expansion timedefines the temperature. Figure 2.14 shows a magnetic field scan. The lowesttemperature is achieved for magnetic fields that correspond to a Zeeman shift of40kHz, which corresponds to the vibrational frequencies of 27kHz along the y/zand 59kHz along the x direction. At these field strengths, 45% of the atoms arecaptured. Higher capture rates (∼ 50%) are achieved for higher magnetic fields,but at the cost of a slightly higher temperature. The minimum temperaturewas determined by means of ballistic expansion measurements and is as low as0.70± 0.01µK along x and 1.07± 0.02µK along y (see Figure 2.15). The fittinginterval is started at 45ms as hot atoms not captured by the lattice would havelead to an overestimation of the initial cloud width.

Raman sideband cooling in combination with Rydberg excitation requires apulsed operation of the atom trap, characterized by a cycle of loading and mo-lasses pre-cooling followed by Raman sideband cooling and subsequent excitationto Rydberg states. For the sake of short experimental cycling times and in orderto circumvent long-term drifts of the excitation laser, the measurements in Sec-tion 4 and Section 5 were performed on a steady-state MOT without additionalRaman cooling stages. However, Raman sideband cooling is an integral part ofour experimental apparatus and provides new prospects for future experimentson frozen Rydberg gases. In future experiments long-term drifts of the Rydbergexcitation laser will be eliminated by locking it to a very stable cavity (see Sec-tion 6), so that Raman sideband cooled atoms will be used as a starting pointfor Rydberg experiments.

32

Chapter 3

Rydberg atoms

In this chapter, we describe how a “frozen gas” of Rydberg atoms[Mourachko et al., 1998, Anderson et al., 1998] is created out of an ultracoldgas of 87Rb atoms (see Section 2). The exaggerated properties of Rydbergatoms are discussed in Section 3.1.1. These properties are mainly determinedby the behavior of the electron wave function. In Section 3.1.2, we showthat wave functions can be easily obtained due to the hydrogen like nature ofthese systems. In Section 3.2, we show how we create a frozen gas of Ryd-berg atom out of rubidium atoms trapped in a MOT and how we detect them[Singer et al., 2004b, Weidemuller et al., 2004]. Calibration issues of the detectorare discussed at the end of this section. Rydberg spectra at n ∼ 60 and n ∼ 80are presented in Section 3.3. Field dependent Rydberg spectra (stark maps) arepresented. They are used to compensate stray fields.

3.1 Background on Rydberg atoms

3.1.1 General properties of Rydberg atoms

The first spectroscopic signatures of Rydberg atoms were already reported atthe end of the 19th century, when Balmer related the line positions of atomichydrogen to the famous formula named after him [Gallagher, 1994]

λ =bn2

n2 − 4, (3.1)

with b = 3645.6 A. Hartley [Hartley, 1883] expressed Balmer’s formula in termsof wave numbers instead of wavelength

ν =

(1

4b

) (1

4− 1

n2

). (3.2)

From this expression it was obvious that the series were caused by energy dif-ferences between n′ = 2 and high lying Rydberg states. Rydberg was able to

33

CHAPTER 3. RYDBERG ATOMS

express the wavenumbers of different series of alkali atoms (classified into sharp(s), principal (p) and diffuse (d) series) [Rydberg, 1890] with a single formula

ν` = ν∞` −RRyd

(n− δ`)2for ` = s, p, d, (3.3)

where ν∞` is the series limit, δ` is the quantum defect (which will be introduced inthe next section) and RRyd = 109721.6 cm−1 is the universal Rydberg constant.Another great achievement of Rydberg was the discovery that the lines connectingseries can be obtained by setting ν∞` = RRyd/(n− δ`′)

2.The physical meaning of the principal quantum number n and the Rydberg

constant became clear in 1913 when Bohr proposed his model for the hydrogenatom. He was able to relate RRyd to

RRyd =Z2e4mel

2(4πε0~)2. (3.4)

Rydberg atoms can be generated by charge exchange, when a positive ion collideswith a ground state atom. The ion captures one electron from the ground stateatom and is left in a Rydberg state. Electron impact can be used to generateRydberg atoms with very high ` states. Here, an electron beam hits groundstate atoms and excites them to Rydberg states. Optical excitation becamefeasible with the advent of lasers. This method has one big advantage overcharge exchange and electron impact namely that individual Rydberg states canbe accurately accessed through tuning of the photon energy. Investigation ofRydberg atoms entered a new realm unveiling exaggerated properties of Rydbergatoms when compared to ground state atoms.

The properties are listed in Table 3.1. As an example, the orbital radius forn = 60 is about 5000 Bohr radii. As a consequence, Rydberg atoms have verylarge dipole moments and polarizabilities which leads to a pronounced Stark effect(see Section 3.3.2) and van der Waals interaction between pairs of Rydberg atoms(see Section 4). Neighboring states with different principal quantum numbers areas close as 30 GHz for n = 60, so that transition between Rydberg states canbe driven by microwaves. Furthermore, the radiative lifetime (215µs for n=60)can be so large that other processes like black body radiation lead to excitationredistribution and even ionization, and significantly shorten the total lifetime ofthe Rydberg state. For n=60, the electronic state is only several meV below theionization limit which explains the fragility of these states. Even small electricfields can already lead to field ionization (see Section 3.2).

3.1.2 Theoretical description of Rydberg atoms

Schrodinger equation of the hydrogen atom To make quantitative theo-retical predictions of the properties of Rydberg, the wave functions for Rydberg

34

3.1. BACKGROUND ON RYDBERG ATOMS

Table 3.1: Selected properties of Rydberg atoms and their depen-dence on the effective principal quantum number n∗ = n − δl adaptedto rubidium from [Gallagher, 1994]. Some formulas are taken out of[Bransden and Joachain, 2003]. The formula for the radiative lifetime isobtained by fitting to theoretical calculations [Gallagher, 1994]. The ba-sic formula for the radiative lifetime is obtained by summing the transitionstrength to lower levels: ( e2

3~c3πε0

∑l=l′±1n<n′

`max

2`′+1ω3| 〈n′l′| r |nl〉 |2)−1. Fine struc-

ture splitting can be found in [Liberman and Pinard, 1979] for p states andin [Chang and Larijani, 1980] for d states.

Property Formula (n∗)x Rb(60d)

Binding energy En −RRyd

(n∗)2(n∗)−2 3.96 meV

Energy spacing En − En−1 (n∗)−3 33.5 GHzOrbital radius 〈r〉 ' 1

2(3(n∗)2 − `(`+ 1)) (n∗)2 5156 a0

Geo. cross section π 〈r〉2 (n∗)4 8.35× 107 a20

Dipole moment 〈nd | er | nf〉 (n∗)2 138.3 ea0

Polarizability 2e2∑

n6=n′,l,m|〈nlm|z|n′l′m′〉|2Enlm−En′l′m′

(n∗)7 191 MHz(V/cm)2

Radiative lifetime 2.09 ns (n∗)2.85 (n∗)3 215 µs

Black body transition 1τbbnl

= 4α3kT3(n∗)2

(n∗)−2 1 kHz

Fine structure 4.8× 106 MHz 52(n∗)−3 (n∗)−3 59 MHz

35


Table 3.2: Atomic units [Gallagher, 1994].

Quantity Definition in atomic units Value

Length Radius of the first Bohr orbit 0.529177 AMass Electron mass 9.10939× 10−28gVelocity Velocity of electron at first Bohr orbit 2.18769× 108cm/sCharge Electron charge 1.60218× 10−19 CEnergy Twice the ionization energy of hydrogen 27.2112 eVElectric field Field at the fist Bohr orbit 5.14221× 109 V/cm

states are needed. To this aim, we start with the Schrodinger equation for theelectron of a hydrogen atom, which is written in atomic units as(

−∆2

2µ− Z

r

)ψ = Wψ, (3.5)

whereW = −1/(2n2) is the energy of the state under consideration, r the distanceof the electron to the core, Z = 1 the charge of the atom core and µ = Mat/(Mat+Mel) the reduced mass of the electron and atom in atomic units. Atomic units aredefined in such a way that all relevant parameters of the ground state of hydrogenhave magnitude one (see Table 3.2). ∆2 is the Laplacian operator defined as

∆2 =∂

∂r2+

2

r

∂

∂r+

1

r2 sin θ

∂

∂θ

(sin θ

∂

∂θ

)+

1

r2 sin2 θ

∂2

∂φ2(3.6)

If we write the wave function as a separable product of radial and angular functionψ = Y (θ, φ)R(r), the Schrodinger equation takes the following form[

1

2µ

(∂2R

∂r2+

2

r

∂R

∂r

)+

(W +

1

r

)R

]Y+ (3.7)

1

2µ

[1

r2 sin θ

∂

∂θ

(sin θ

∂Y

∂θ

)+

1

r2 sin2 θ

∂2Y

∂φ2

]R = 0.

If we divide this equation by RY/r2, we obtain

r2

R

[1

2µ

(∂2R

∂r2+

2

r

∂R

∂R

)+

(W +

Z

r

)R

]+ (3.8)

1

2µY

[1

sin θ

∂

∂θ

(sin θ

∂Y

∂θ

)+

1

sin2 θ

∂2Y

∂φ2

]= 0.

The first and second term are independent of each other and must be separatelyequal to a constant ±λ

1

sin θ

∂

∂θ

(sin θ

∂Y

∂θ

)+

1

sin2θ

∂2Y

∂φ2= −2λµY. (3.9)

36


The solutions are the spherical harmonics Y`m(θ, φ) defined in term of Pm` (cos θ),

the unnormalized associated Legendre polynomials

Y`m(θ, φ) =

√(`−m)!

(`+m)!

2`+ 1

4πPm

` (cos θ)eimφ, (3.10)

where ` is a nonnegative integer and m takes integral values ranging between −`and `. The angular equation also requires that 2λµ = `(`+ 1), so that the radialequation takes the form

1

2µ

(∂2R

∂r2+

2

r

∂R

∂R

)+

(W +

Z

r− `(`+ 1)

2µr2

)R = 0. (3.11)

This radial equation can be further simplified by substituting R(r) = U(r)/rleading to the following equation(

− 1

2µ

d2

dr2− Z

r+l(l + 1)

2r2µ

)U(r) = WU(r), (3.12)

the first term is the kinetic energy, the second the Coulomb potential, and thethird the centrifugal potential. The solutions can be expressed analytically interms of associated Laguerre polynomials [Sakurai, 1994]. We are interested inthe wave functions for alkali atoms where approximate solutions are only numer-ically obtainable due to the effect of the core electrons on the valence electron.This effect is accounted for by introducing a quantum defect.

Quantum defect The main difference between hydrogen atoms and alkaliatoms is that the core electrons have an influence on the valence electron. Whenthe electron is far from the core, it sees a net positive charge of 1. However, if itcomes closer to the core, which is especially important for states with low angularmomentum, it sees a higher charge, and then the core can be polarized. So we geta depression of low ` states due to the core polarization and the penetration. Thiscan be accounted for by replacing n by an effective principal quantum number

n∗ = n− δn,j,` (3.13)

with the quantum defect δn,j,` [Seaton, 1983]. The quantum defect itself can be ex-pressed with the extended Rydberg-Rietz formula [Lorenzen and Niemax, 1983]

δn,j,` = a+b

(n− a)2+

c

(n− a)4+

d

(n− a)6+

e

(n− a)8+ ... (3.14)

where the coefficients are obtained from the table in Figure 3.1. To calculatethe line positions of Rydberg states excited from the ground state, the followingformula has to be applied

En,j,l = Ei −RRyd

(n− δn,j,`)2, (3.15)

where Ei is the ionization energy.

37


Term RRyd Ei,cm-1

a b c d e n E, Ref.

series cm-1

mKa

n2S1/2 0.399 468 0.030 233 - 0.002 8 0.011 5 _ 2 ±30 [a]

n2 0.047 263 -0.026 18 0.022 1 -0.068 3 2 ±30 [a]

n2DJ 0.002 129 -0.014 91 0.175 9 -0.850 7 — 3 ±30 [a]

7Li

n2FJ

10 9728.64 43 487.15(3)

- 0.000 077 0.021 856 -0.421 1 2.389 1 — 4 ±30 [a]

n2S1/2 1.347 964 0.060 673 0.023 3 -0.008 5 — 3 ±30 [b,c]

n2P1/2 0.855 380 0.113 63 0.038 4 0.141 2 — 3 ±30 [b,c]

n2P3/2 0.854 565 0.114 195 0.035 2 0.153 3 — 3 ±30 [b,c]

n2DJ 0.015 543 -0.085 35 0.795 8 -4.0513 — 3 ±30 [b,c]

23 Na

n2FJ

10 9734.69 41 449.44(3)

0.001 453 0.017 312 -0.780 9 7.021 — 4 ±30 [b,c]

n2S1/2 35 009.8138(7) 2.180 198 5(150) 0.135 58(300) 0.075 9 0.117 - 0.206 4 ±0.5 [d]

n2P1/2 35 009.8153(30) 1.713 892(30) 0.233 294(5000) 0.16137 0.534 5 - 0.234 4 ±3 [e]

n2P3/2 35 009.8151(30) 1.710 848(30) 0.235 437(6000) 0.11551 1.101 5 -2.0356 4 ±3 [e]

n2D3/2 35 009.8140(7) 0.276 9700(60) -1.024 911(1000) -0.709 174 11.839 -26.689 3 ± 0.5 [d]

n2D5/2 35 009.8141(7) 0.277 1580(60) -1.025 635(2000) -0.592 01 10.005 3 -19.0244 3 ±0.5 [d]

39K

n2FJ

10 9735.774

35 009.814(15) 0.010 098 -0.100 224 1.563 34 -12.685 1 — 4 ±15 [b]

n2S1/2 33 690.7989(5) 3.131 180 4(10) 0.178 4(6) -1.8 — — 14 ±0.5 [f]

n2P1/2 33 690.799(3) 2.654 884 9(10) 0.290 0(6) - 7.904 0 116.437 3 -405.907 11 ±3 [g,g,e,e,e]

n2P3/2 33 690.797(3) 2.641 673 7(10) 0.295 0(7) -0.974 95 14.600 1 -44.7265 13 ±3 [g,g,e,e,e]

n2D3/2 33 690.7978(30) 1.348 091 71(40) -0.602 86(26) -1.505 17 -2.420 6 19.736 4 ±3 [g,g,e,e,e]

n2D5/2 33 690.7978(30) 1.346 465 72(30) -0.596 00(18) -1.505 17 -2.420 6 19.736 4 ±3 [g,g,e,e,e]

85Rb

n2FJ

10 9736.605

33 690.799(10) 0.016 312 - 0.064 007 -0.360 05 3.239 0 — 4 ±10 [h]

n2S1/2 31 406.4710(7) 4.049 352 1(380) 0.238 322(7000) 0.240 44 0.121 77 -0.0105 6 ±0.5 [e]

n2P1/2 31 406.490(3) 3.591 610(500) 0.363 662(10000) 0.342 84 1.239 86 — 6 ±30 [i]

n2P3/2 31 406.474(30) 3.558 960(700) 0.331 248 0.280 13 1.576 31 — 6 ±30 [i]

n2D3/2 31 406.4710(7) 2.475 454(20) 0.009 -0.433 24 -0.965 55 -16.9464 5 ±0.5 [e]

n2D5/2 31 406.4710(7) 2.466 308(30) 0.015 256 (6000) - 0.436 74 -0.744 42 -17.5100 5 ±0.5 [e]

133Cs

n2F5/2

10 9736.86

31 406.4710(30) 0.033 587 -0.213 732 0.700 25 -3.662 16 — 4 ±3 [j]

Figure 3.1: Rydberg-Rietz coefficients for the calculation of the quantumdefect of Rydberg alkali atoms. The values are valid for Rydberg stateswith a principal quantum number larger than or equal to the numberslisted in the n ≥ column. All listed data refer to the centers of grav-ity of the hyperfine-splitted ground and excited states. Data of termseries labeled n2XJ refer to the center of gravity of the fine-structuredoublets. The error of the calculated energies is denoted in the ∆Ecolumn. The data is collected out of Reference e and g. The detailed ref-erences are: [a]=[Johansson, 1958], [b]=[Risberg, 1956], [c]=[Martin, 1980],[d]=[Lorenzen et al., 1981], [e]=[Lorenzen and Niemax, 1983],[f]=[Lee et al., 1978], [g]=[Li et al., 2003],[h]=[Johansson, 1961],[i]=[Lorenzen and Niemax, 1979], [j]=[Eriksson and Wenaker, 1970].

38


a) b)

200 400 600 800 1000r @a0D

-0.1

-0.05

0

0.05

0.1

UHrL

0 500 1000 1500r @a0D

-0.0125

-0.01

-0.0075

-0.005

-0.0025

0

0.0025

0.005

Ene

rgy@a

.u.D

Figure 3.2: (a) Radial wave functions for the electron of a hydrogen atom(n = 18, ` = 1). For small radii, it starts to oscillate with increasing frequency.(b) Corresponding effective potential. Zero energy refers to the ionizationlimit of hydrogen.

Numerical integration of the Schrodinger equation The bound radialwave functions U(r) are calculated by numerically integrating the radial time-independent Schrodinger equation 3.12. A result for hydrogen and n = 18, ` =1 can be seen in Figure 3.2, together with the corresponding effective radialpotential (zero refers to the ionization limit).

The integration of the radial Schrodinger equation for bound states is per-formed by beginning at large radii with the starting condition that ∂U(r)

∂r= 0,

so that the first two values of the wave function have a small value ε 1. Theresult must be normalized and can be seen in Figure 3.2(a). The oscillation of thewave function increases with decreasing radii. In order to avoid the accumulationof numerical errors, especially at small radii one has to continuously adjust theintegration step sizes, which makes the implementation of the integration routinequite lengthy and slow. Another method is to rescale the Schrodinger equationin such a way that the solutions oscillate with a nearly constant period. This canbe accomplished by using the variable substitution v =

√r and the scaled radial

wave function [Bhatti et al., 1981]

Ξ(v) = v−12U(v2). (3.16)

The scaled radial Schrodinger equation now has the following form

d2

dv2Ξ(v) =

((1

2+ 2l)(3

2+ 2l)

v2+

4v2µ

n∗2− 8Zµ

)Ξ(v). (3.17)

The resulting scaled radial wave functions are integrated with the Numerovmethod [Blatt, 1967] with step size h from large v for which we set Ξ(v − h) :=Ξ(v) := ε (ε << 1) to smaller values of v. The advantage of the Numerov methodis that the integration error scales as O(h6). The resulting rescaled wave functionare depicted in Figure 3.3(a) for hydrogen (n = 18, l = 1 red curve) together with

39


a) b)

0 10 20 30 40!!!

r @!!!!!!a0 D

0

10

20

30

gH!!! rL

Figure 3.3: (a) Scaled radial wave functions (see Equation 3.16) for theelectron of a hydrogen atom with n = 18, ` = 1 (red line) and of a rubidiumatom with n = 20, ` = 1 (black line). In contrast to the unscaled wave func-tion depicted in Figure 3.2, the scaled version oscillates with nearly constantfrequency. (b) Effective potential for the scaled radial wave function.

a) b)

0 10 20 30 40!!!

r @!!!!!!a0 D

-0.04

-0.02

0

0.02

0.04

H!!! rL-

12UHrL

0 10 20 30 40!!!

r @!!!!!!a0 D

-30

-20

-10

0

10

gH!!! rL

Figure 3.4: (a) Scaled radial wave functions for the electron of a rubidiumatom with n = 20, ` = 1 (black line). Core polarization and penetration isincluded in the model potential. (b) Corresponding model potential includingcore polarization and penetration.

40

3.2. EXCITATION AND DETECTION OF RYDBERG ATOMS

the effective rescaled radial potential which is depicted in Figure 3.3(b) (zerorefers to the energy of the the state for hydrogen with n = 18, l = 1). If non-hydrogenic wave functions as those for rubidium (n = 20, l = 1 black curve) arecalculated, then the wave functions should exponentially decay to zero for smallvalues of v in the classically forbidden range. Instead they diverge because theeffect of the core is only included by the quantum defect. We therefore truncatewave functions if they diverge within the classically forbidden region or start tooscillate faster than the integration grid size. The error introduced thereby isnegligible when dealing with Rydberg states. As an example, the matrix elementof 〈n = 30, ` = 0 | r | n = 10, ` = 1〉 calculated with the truncated wave functionsdiffers by only 0.01% from the same matrix element calculated with wave func-tions using model potentials [Marinescu et al., 1994] (see Figure 3.4(b)) wherethe influence of the core is taken into account. Figure 3.4(a) shows a rescaledradial wave function when the effect of the core is taken into account. The fastoscillations near the core are suppressed by R−2 when overlap integrals betweenwave functions are calculated. The major error introduced by truncation stemsfrom the difference in the normalization of the wave function. When only overlapintegrals between Rydberg states are of interest, the errors introduced by trun-cation can be neglected in view of the large spacial extends of Rydberg states.

3.2 Excitation and detection of Rydberg atoms

3.2.1 Excitation into Rydberg states

The laser light needed to drive the transition to Rydberg states 5P3/2(F=3) → n`jis provided by a commercial laser system (Toptica, TA-SHG 110) consisting of anextended cavity diode laser (ECDL) at 960 nm, which is subsequently amplifiedto 1W and frequency doubled to 479 nm and has a line width <2MHz. Thewavelength of this laser can be tuned by±1 nm to address Rydberg levels, startingfrom n' 30 up to the ionization threshold (see Figure 3.5). The output beam thatcan be switched with an acousto-optical modulator (AOM) is guided along oneof the diagonal axes of the vacuum chamber and focused to a waist of 80(10)µmat the center of the atom cloud. The overlap between the atom cloud and theblue laser beam defines an effective Rydberg excitation volume of Vexc = 1.1 ×10−2 mm3.

The two-step excitation into Rydberg states is schematically shown in Fig-ure 3.5. At the beginning of an excitation cycle, the trapping laser is tuned intoresonance with the 5S1/2(F = 2) → 5P3/2(F

′ = 3) transition. Subsequently, theblue Rydberg excitation, tuned to the Rydberg manifold, is switched on for avariable time (typically 20µs). The delay between changing the MOT laser de-tuning and switching the blue laser can be precisely controlled on the 100-ns levelby switching an acousto-optical modulator (AOM). Rydberg atoms are accumu-

41


Figure 3.5: Energy levels of 87Rb. The trapping laser is tuned into resonancein order to excite resonantly into the launch state. Then the blue excitationlaser at 479 nm excites into the Rydberg manifold.

42


lated in the excitation volume during this time, as the lifetime of the Rydbergstates exceeds 100µs [Gallagher, 1994, Theodosiou, 2000] for states of interest(n=60...80). After the Rydberg excitation laser is blocked, the trapping laser isreset to the MOT detuning of −1.5 Γ. In order to apply well-defined electric fieldsfor Rydberg atom manipulation and ionization, the MOT is placed between twometal grids, with an optical transmission of 95%, spaced 13.2mm apart, throughwhich the MOT laser beams pass almost undisturbed.

3.2.2 Detection of Rydberg atoms

Rydberg atoms are detected by means of field ionization. The voltage needed forionizing Rydberg states can be obtained quite accurately by adding the Coulombpotential to the potential generated by the field plates

V = −1

r+ Ez. (3.18)

If tunneling and the shift of energy levels with an electric field is neglected, theelectric field necessary for ionization can be calculated classically by finding thesaddle point

zmax = −√E (3.19)

V (zmax) = −2√E. (3.20)

With V = −1/(2n2), one obtains in atomic units

E =1

16n4. (3.21)

Ions and electrons are accelerated by electric fields onto oppositely placed micro-channel plates (MCP) (see Figure 2.5-1) which amplify the charge signal. Theschematic of the MCP readout for electrons and ions is depicted in Figure 3.6. Thesignal obtained at the anode of the MCP is amplified by a fast preamplifier andthen integrated and discriminated with a boxcar integrator. The resulting signalis recorded by a digital oscilloscope with eight bit single channel resolution. Tofurther increase the data-acquisition resolution, two measures are taken. In orderto also resolve small peaks near big signals, we recorded two channels of the scopesimultaneously. One channel was set at high, the other at low sensitivity. Afterdata-acquisition, the recorded signal is merged into a single spectrum. In orderto increase the resolution of high dynamic signals, we additionally took about20 to 200 samples per data point and binned them. The bin size is determinedby calculating the autocorrelation function of the recorded data. The phase ofthe bin is then calculated by convoluting the original data with a comb functionhaving the period of the bin size.

43


Figure 3.6: Schematic of the MCP read-out.

During Rydberg excitation, residual electrical fields are mainly caused by theMCP potential adjusted to −1.9 kV for ion detection. This effect is minimized bygrounding the grids in front of the ion MCP (see Figure 2.5-7). Field componentsperpendicular to the field plates are compensated by applying a small voltageto the central field plates during excitation, while parallel field components aremeasured to be less than 0.16 V/cm. To avoid the accumulation of ions in theintervals between consecutive excitation cycles, we maintain the potential at thecentral field plates (see Figure 2.5-8) at the voltage necessary for field ionization(typically 40 V).

The excitation cycle is typically repeated every 20 ms while a continuouslyloaded steady-state MOT is maintained. To take an excitation spectrum, thefrequency of the blue laser is scanned at a rate of usually 300 MHz/s.

3.2.3 Calibration of the Rydberg spectrometer

The amplification of the MCP depends exponentially on the voltage applied overthe device. But with higher voltages the response becomes increasingly nonlinear.In order to optimize the sensitivity we used the MCP in the nonlinear mode (1900Volt) but corrected for nonlinearity by comparing the signals with identical scansat lower MCP voltages (1700 Volt). To this aim, we varied the excitation timeon resonance of the 82S peak. As can be seen in Figure 3.8(a), the scan in which1700 Volt is applied to the MCP saturates slightly. Looking at the scan where1900 Volt is applied to the MCP (see Figure 3.8(b)), we can see a much strongersaturation. The saturation parameter is determined by fitting with a heuristicsaturation function (see text inside Figure 3.8(a) and (b)). Having obtained the

44


z[a0 ] (x

²+y²

)[a

1/2

0]

Energ

y[R

Ryd]

Figure 3.7: Potential curve of the Rydberg electron exposed to a uniformelectrical field.

a) b)

10 20 30 40

Excitation time t s

0.5

1

1.5

2

2.5

3

PC

Ml

an

giS

Stl

oV

Sp t

1 t

tsat

Fitresults:

Estimate Asymptotic SE

p 0.111603 0.00483858

xsat 87.752 14.2351

limt

S p tsat 9.79335 1.64443

10 20 30 40

Excitation time t s

2

4

6

8

10

PC

Ml

an

giS

Stl

oV

Sp t

1 t

tsat

Fitresults:

Estimate Asymptotic SE

p 0.657928 0.0266617

xsat 27.9834 2.19878

limt

S p tsat 18.4111 1.6277

Figure 3.8: Excitation time scans of the 82S resonance at different MCPvoltages: (a) 1700 Volt; (b) 1900 Volt. The excitation time is varied from 2µsto 40µs. Saturation values are obtained by fitting with a heuristic saturationfunctions (see inset in plots).

45


a) b)

Figure 3.9: The blue excitation laser is flashed on for 20µs with a repetitionrate of 50Hz while it is scanned over a Rydberg resonance. Signals taken atdifferent resonances are compared. (a) 82S Resonance; lower Graph: Theblack line shows the number of atoms in the MOT determined by monitoringthe fluorescence. The red line depicts the atom lost from the MOT. It isobtained by subtracting the MOT reloading curve from the black line. Thegreen line represents the integrated linearized MCP signal. The normalizationis fitted; it determines the absolute calibration of the MCP signal. UpperGraph: The black line represents the original MCP signal. The red lineshows the number of Rydberg atoms obtained by the linearized and calibratedMCP signal. (b) 60D resonance; lower Graph: The black line shows thenumber of atoms in the MOT determined by monitoring the fluorescence. Thered line depicts the atom lost from the MOT. It is obtained by subtractingthe MOT reloading curve from the black line. The green line representsthe integrated linearized MCP signal. The normalization is taken out ofthe fit of the 82S line. The dashed green line shows the discrepancy if nolinearization is performed. Upper Graph: The black line shows the originalMCP signal. The red line represents the number of Rydberg atoms obtainedby the linearized and calibrated MCP signal. The amount of Rydberg atomsexcited on resonance is about 50 % of the total amount of atoms available inthe laser focus.

46


0 0.02 0.04 0.06 0.08Fluoloss

0

0.02

0.04

0.06

0.08

Inte

grat

edM

CP

Sig

nal

Figure 3.10: Integrated desaturated MCP signal versus fluorescence loss.The solid line is a linear fit to the points. The tips of the lower error barsdenote the integrated original MCP Signal. The tips of the upper error barsdenote the integrated MCP signal which was desaturated under the assump-tion that the highest MCP signal measured has already fully saturated theMCP.

47


fit parameters, we applied the inverse of the saturation function to the MCPsignals in order to linearize it. But if we totally linearized both scans, thensaturation values below the observed MCP signals are obtained. Therefore, wecan give a lower limit for the MCP saturation obtained by assuming that thesaturation seen in the first scan (1700 Volt) is caused by the interaction betweenthe Rydberg atoms and not by the MCP. The upper limit is given by meansof the highest MCP signals measured. The MCP linearity can also be verifiedby varying the linearization parameters and comparing the integrated linearizedMCP signal of several scans with the loss of atoms out of the MOT determined byfluorescence loss signals (see Figure 3.10). The fluorescence signal is monitoredwith a photodiode and calibrated by means of absorption images. The MCPsaturation obtained by this method gives a result close to the lower limit obtainedby the time scans. This also gives the absolute calibration of the MCP signalwith an uncertainty factor of 2 due to the uncertainties of the determination ofabsolute atoms numbers by absorption images. In Figure 3.9(a), lower graph, afluorescence loss signal (black line) is depicted. The excitation laser was scannedover the resonance of the 82S line and flashed into the atom cloud for 20µs at arepetition rate of 50 Hz. The black line is corrected for MOT reloading resultingin the red line. The green line shows the integral of the linearized MCP signal (redline in upper graph of Figure 3.9(a)). The normalization of the integral is fittedto follow the red curve. In Figure 3.9(b), the same procedure is repeated with the60D resonance where much more atoms are excited. The green line representsthe integrated linearized MCP signal. The normalization is taken out of the 82Sscans. We see a good agreement between the green and the red line. On the otherhand, if no linearization is performed, the result is the dashed green line, whichshows the discrepancy. The absolute number of Rydberg atoms excited at the60D line is about 50 % of the atoms available in the laser focus. In the following,we present linearized data obtained under the assumption that the MCP at 1700Volt is linear. We add upper error bars indicating maximum MCP saturation,which is determined by taking the maximum signal measured as saturation value.

Rydberg densities are obtained by dividing the Rydberg atom number by theeffective Rydberg excitation volume. The latter is derived by considering theoverlap region of the atom cloud (1.2mm 1/e-diameter) with the excitation laserbeam (80(10)µm waist), which define a cylindrical Rydberg excitation volume of0.011mm3.

3.3 Spectra of Rydberg atoms

3.3.1 Spectra without electric field

Figure 3.11 and Figure 3.12 show spectra around the principal quantum numbersn=60 and n=80. The power of the excitation laser was 350 Watt/cm3 and 100

48

3.3. SPECTRA OF RYDBERG ATOMS

624828.5 624829.0 624829.5 624830.0

0

100

200

300

61P3/2

61P1/2

Ryd

berg

ato

m n

umbe

r

Laser frequency (GHz)

624844.5 624844.8 624845.10

5

10

15

62S

Ryd

berg

ato

m n

umbe

r (10

3 )

Laser frequency (GHz)624837.6 624838.4 624839.2

0

5

10

15

ghost 5P3/2 (F=2)

60D

Ryd

berg

ato

m n

umbe

r (10

3 )


624830 624835 624840 6248450

5

10

15

Ryd

berg

ato

m n

umbe

r (10

3 )


Figure 3.11: Rydberg spectra at n'60. The x-axis is the frequency of theblue excitation laser providing the second step in the two-photon excitationscheme. Besides the dipole-allowed nS and nD states, residual electric fieldsalso allow for the excitation of nP states. The small peak near the 60D stateis a hyperfine ghost excited from the 5P3/2(F = 2) state.

49


625266 625267 625268 625269 625270

0

100

200

300D*

79h

Ryd

berg

ato

m n

umbe

r

Laser frequency (GHz)625258.2 625258.4

0

100

200

300

400

500

625259.6 625259.8

81P3/2

81P1/2

Ryd

berg

ato

m n

umbe

r


( )*

625265.2 625265.4 625265.60.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.082S

Ryd

berg

ato

m n

umbe

r (10

3 )

Laser frequency (GHz)625262 625263

0

1

2

3

4

5

6

ghost 5P3/2 (F=2)

80D

Ryd

berg

ato

m n

umbe

r (10

3 )


625260 625270 6252800

1

2

3

4

5

6

D*80h79h

83S

81D

82S

82P81P

80D

Ryd

berg

ato

m n

umbe

r (10

3 )


( )*( )*

Figure 3.12: Rydberg spectra at n'80. The x-axis is the frequency of theblue excitation laser providing the second step in the two-photon excitationscheme. Apart from the dipole-allowed nS and nD states, residual electricfields also allow for the excitation of nP and the hydrogen-like nh (n`, ` ≥ 3)states. The lines marked with (*) have not been assigned so far. The smallpeak near the 80D state is a hyperfine ghost excited from the 5P3/2(F = 2)state.

50


Table 3.3: Measured frequencies compared to calculated values. The finestructure doublet of the D resonances cannot be resolved.

Assignment Measured peak position Calculated value81P1/2 625258.382 625258.325581P3/2 625258.531 625258.446280D3/2 625262.479 625262.464680D5/2 625262.479 625262.486682S1/2 625265.420 625265.417182P1/2 625271.694 625271.750582P3/2 625271.846 625271.864581D3/2 625275.763 625275.734081D5/2 625275.763 625275.755283S1/2 625278.584 625278.5781

Watt/cm3 respectively. Only excitations of nS and nD states are dipole-allowed inthe two-photon excitation scheme employed in our experiments (see Figure 3.5).However, small residual electric fields can break the selection rules and thereforeallow for the excitation of nP and n` states (` ≥ 3). We determined a smallresidual electrical field of less than 0.2 V/cm (see Section 3.3.2). The states with` ≥ 3 are are called hydrogen-like states because of their negligible quantumdefect. Because the polarizability strongly increases with higher n, we can onlysee the hydrogen-like states at n'80. For the dipole-allowed states, the excitationprobability decreases with higher n as it scales with n−3 [Gallagher, 1994]. Thelinewidths of the n ∼ 80 spectra are of the order of 10MHz and mainly determinedby the linewidth of the saturated first excitation step. On this level of precision,the maximum influence of the magnetic quadrupole field of the MOT is less than5 MHz and can thus be neglected.

The absolute frequency of the resonances was measured with a commercialwavemeter (Burleigh) with an accuracy of 500MHz. Table 3.3 gives a comparisonbetween measured frequencies and calculated values of the resonance. The mea-sured spectral positions of all major resonances in the spectrum coincide with thecalculated values, taking into account the corresponding quantum defects of thelevels. To the blue side of the major resonances, there are hyperfine ghosts whichare excited from 5P3/2(F = 2). However, the lines in the n=80 spectrum markedwith (*) could not be related to atomic resonances even when taking other hy-perfine levels of the 5S1/2 ground and 5P3/2 intermediate state into account (e.g.off-resonant two-photon excitation from the 5S1/2(F = 1) state, a so-called hy-perfine ghost, see Figure 3.5). Further field-dependent studies show that the D*line has D character. Remarkably, these lines do not appear at n'60 althoughthe excitation rate is higher there. Non-atomic resonances have been found to becaused by two-particle interactions leading to level crossings [Farooqi et al., 2003].

51


In order to find possible explanations of the unknown lines, we qualitatively cal-culated molecular potential curves for the long-range van der Waals interactionsin Section 4.2. In Section 6, we give a possible explanation of the D* line in termsof a molecular resonance due to an avoided crossing.

3.3.2 Field-dependent spectra

As Rydberg states exhibit large polarizabilities, Rydberg excitation spectra de-pend on static electric fields [Luc-Koenig et al., 1991]. The dependence of thespectrum at n'80 is shown in Figure 3.13. The line positions of the observed res-onances can be compared to quantum mechanical calculations for Rydberg atomsin electric fields [Zimmerman et al., 1979]. It thus allows for a calibration of theelectric field at the position of the atoms. In particular, the field componentperpendicular to the field plates can be precisely compensated in this way.

The excitation rate of dipole-forbidden transitions (as for nP states) scalesquadratically with the electric field. This is due to the fact that the electricfield enters in first order perturbation theory and leads to a linear dependenceof admixtures of dipole allowed states as a function of the electric field. As thetransition probability is proportional to the square of transition matrix elements,the quadratic dependence of the excitation probability of dipole forbidden statesexposed to electric fields is obtained. By measuring the peak area of the 82P linefor varying electric fields, we determined the residual field to be 0.16V/cm (com-ponent parallel to the grids). An alternative calibration method is to monitor thesplitting of the hydrogen-like states which grow linearly with increasing electricfield. This provides a similar value of 0.12 V/cm for the residual field. The resid-ual field is mainly a result of the high voltages at the MCPs and inhomogeneitiesinduced by the field grids. While field components perpendicular to the grids canbe compensated by a small bias field, parallel components cannot be cancelled.

The calculation of the Stark spectra is performed as described in[Zimmerman et al., 1979] by diagonalizing the total Hamiltonian

H = Ho + Fz +Hfs, (3.22)

where H0 is the Hamiltonian of the unperturbed system (see Equation 3.5), Fthe electric field in z direction, and Hfs the energy shift of the states due tofinestructure, which has to be included in the calculations for heavier alkali atomslike rubidium.

Hfs = −µ ·B = −1

r

dV

drL · S (3.23)

As a basis, one can use the wave functions of H0 which are obtained as describedin Section 3.1.2. If the quantumdefect described in Section 3.1.2 is used, thenthe finestructure for the diagonal elements is already included in H0. We as-sume that the finestructure is excluded from H0 and that the energy of the basis

52


-0,2 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4

-17,9

-17,8

-17,7

-17,6

-17,5

-17,4

-17,3

-17,2

-17,1

( )

( )

( )

*

( )

*

*

*

81P

80D

82S

79h

82P

81D

83S

80h

Energ

y (

cm

-1)

Electric field (V/cm)

Figure 3.13: Electric field dependence of the excitation spectra for n'80(Stark map). The experimental data (black) is overlapped with the result ofperturbation theory (shown as gray lines).

53


states is taken to be the center of gravity of the finestructure multiplets. Thediagonal terms of Hfs are obtained by the finestructure splitting obtained by[Liberman and Pinard, 1979] for p states and by [Chang and Larijani, 1980] ford states

〈n, ` = 1|Hfs |n, ` = 1〉 = 8.70× 10−3 3

2

1

(n− δ)3(3.24)

〈n, ` = 2|Hfs |n, ` = 2〉 = 7.28× 10−4 5

2

1

(n− δ)3(3.25)

The off-diagonal terms can be obtained under consideration of the fact thatthe finestructure interaction depends only on the behavior of the wave functionnear the origin due to the r−3 dependence of Hfs. For a given `, the wave functionat the origin varies only with the normalization of the wave function, so that offdiagonal terms 〈n, l|Hfs |n′, l〉 are obtained by [Zimmerman et al., 1979]

〈n, l|Hfs |n′, l〉 =√〈n, l|Hfs |n, l〉〈n′, l|Hfs |n′, l〉 (3.26)

The matrix elements of Fz are obtained by

〈n, l, j,mj|Fz∣∣n′, l′, j′,m′

j

⟩= δ(mj,m

′j)δ(`, `

′ ± 1) 〈n, l| r |n′, l′〉F

×∑

ml=mj±1

⟨`,

1

2,ml,mj −ml | j,mj

⟩ ⟨`′,

1

2,ml,mj −ml | j′,mj

⟩× 〈l,ml| cos θ |l′,ml〉 , (3.27)

where the terms in the second line are Clebsch-Gordan coefficients and

〈l,ml| cos θ |l − 1,ml〉 =

(`2 −m2

(2`+ 1)(2`− 1)

)1/2

(3.28)

〈l,ml| cos θ |l + 1,ml〉 =

((`+ 1)2 −m2

(2`+ 3)(2`+ 1)

)1/2

. (3.29)

The stark map is obtained by diagonalizing the total Hamiltonian H for eachvalue of F .

54

Chapter 4

Interactions in a frozen gas ofRydberg atoms

Rydberg atoms in a pure state do not have a permanent dipole moment. How-ever, they do possess rather strong polarizabilities, as shown in Section 3.1.1 andthey interact strongly via the van der Waals interaction. Classically, this can beunderstood in terms of the following process. Atom A has a temporary dipolemoment µA due to fluctuations in the electron distribution. This dipole momentgenerates an electrical field proportional to R−3 which in turn induces a dipolemoment in Atom B: µB ∼ αbE, where αB is the polarizability of atom B. Thedipole-dipole interaction potential has the following form

Vdd =µA · µB

R3− 3(µA ·R)(µB ·R)

R5. (4.1)

Two dipoles therefore have an interaction potential proportional to µaµb/R3.

Plugging in the expressions for µb, we get an overall R−6 behavior of the van derWaals interaction. If a small electric field is applied, states of different paritymix and permanent dipoles are induced in the Rydberg atoms. These perma-nent dipoles lead to an interaction proportional to R−3. The same scaling isachieved if the dipoles interact via a resonant dipole-dipole interaction (Forsterprocess [Forster, 1996]). The resonance condition can be tuned by applying asmall electric field thereby shifting the states via the dc-Stark effect.

In the following section a qualitative quantum mechanical model of the dipole-dipole interaction is presented, giving an intuitive picture of the van der Waals in-teraction and the transition to resonant dipole-dipole interaction. In Section 4.2,we show how explicit values for the long-range van der Waals interaction betweenRydberg atoms can be calculated following a perturbative approach. Experi-mental signatures of interaction-induced spectral line broadening are presentedin Section 4.3.

55

CHAPTER 4. INTERACTIONS IN A FROZEN GAS OF RYDBERG ATOMS

Figure 4.1: (a) Single-atom energy levels. ∆/2 is the detuning of the |p〉state to the average value of the |s〉 and the |s′〉 state and is much smallerthan other energy differences. (b) Two-atom energy levels. ∆ is the energydifference between the |p〉 ⊗ |p〉 and the |s〉 ⊗ |s′〉/|s′〉 ⊗ |s〉 state.

4.1 Qualitative theoretical description

Model Hamiltonian The van der Waals interaction can be described quali-tatively with a simple quantum mechanical model. The model consists of threestates interacting via dipole-dipole interaction and an additional state which isused to shift the state of interest into resonance via the Stark effect in orderto make the transition from non-resonant to resonant dipole-dipole interaction.An analytic expression for the potential of the interaction between two Rydbergatoms in the |p〉 (` = 1) state is derived. Energetically, the |p〉 state should havetwo nearby states |s〉 and |s′〉 (` = 0). The energies of the single particle statesare depicted in Figure 4.1. ∆/2 denotes the energy difference between the |p〉state and the average energy of |s〉 and |s′〉. The dipole matrix elements between|p〉 and |s〉 and between |p〉 and |s′〉 are nonvanishing, whereas the one between|s〉 and |s′〉 is vanishing:

µ1 = 〈p | r | s〉 6= 0 , µ2 = 〈p | r | s′ 〉 6= 0 , 〈s | r | s′ 〉 = 0 (4.2)

We further make the simplifying assumption that the state |d〉 is decoupled fromthe system as long as no external electric field is present and that |d〉 and |p〉 arethe only states which are coupled by the Stark effect. Γ denotes the energy differ-ence between the |d〉 state and the average energy of |s〉 and |s′〉. The interactionenergy between two |p〉 states is obtained by writing down the Hamiltonian forthe two-particle state |a〉 ⊗ |b〉. Symmetry issues are not considered here, theyare treated in detail in Section 4.2. The only two-particle states we do considerare the ones that have a non-vanishing dipole-dipole interaction with the |p〉⊗|p〉state and are closest in energy (see Figure 4.1) and result in four states |s〉 ⊗ |s′〉

56

4.1. QUALITATIVE THEORETICAL DESCRIPTION

Figure 4.2: (a) For ∆ > 0 we obtain an attractive interaction potential forthe |p〉 ⊗ |p〉 state. (b) Repulsive interaction potential for the |p〉 ⊗ |p〉 stateif ∆ > 0.

,|s′〉 ⊗ |s〉 ,|d〉 ⊗ |p〉 and |p〉 ⊗ |d〉. As the electric field is assumed to couple onlyto the |p〉 and |d〉 states, we only have to add the two-particle states |p〉⊗ |d〉 and|d〉 ⊗ |p〉. In the basis |s〉 ⊗ |s′〉, |p〉 ⊗ |p〉, |s′〉 ⊗ |s〉, |p〉 ⊗ |d〉 and |d〉 ⊗ |p〉, thetotal Hamiltonian takes the following form

H = H0 +HvdW +HStark (4.3)

with

H0 =

0 0 0 0 00 ∆ 0 0 00 0 0 Γ 00 0 0 0 Γ

, (4.4)

HvdW =

0 µ1µ2

R3 0 0 0µ1µ2

R3 0 µ1µ2

R3 0 00 µ1µ2

R3 0 0 00 0 0 0 00 0 0 0 0

(4.5)

and

HStark = −

0 0 0 0 00 0 0 µ3E µ3E0 0 0 0 00 µ3E 0 0 00 µ3E 0 0 0

, (4.6)

with µ3 = e 〈p| z |d〉 if the electric field is assumed to be along the z-axis.

Analytic solution for the field-free case If no external electrical field ispresent (E = 0), the total Hamiltonian simplifies to a 3× 3 matrix and thus can

57


Figure 4.3: Resonant dipole-dipole interaction for ∆ = 0. The potentialscales as R−3.

be diagonalized analytically, leading to the following unnormalized two-particleeigenstates

|1〉 = |s〉 ⊗ |s′〉 − |s′〉 ⊗ |s〉 (4.7)

|2〉 = |s〉 ⊗ |s′〉+ |s′〉 ⊗ |s〉+R3 ∆−

√R6 ∆2 + 8µ1

2 µ22

2µ1 µ2

|p〉 ⊗ |p〉 (4.8)

|3〉 = |s〉 ⊗ |s′〉+ |s′〉 ⊗ |s〉+R3 ∆ +

√R6 ∆2 + 8µ1

2 µ22

2µ1 µ2

|p〉 ⊗ |p〉 (4.9)

with eigenenergies

H |1〉 = 0 (4.10)

H |2〉 =R6 ∆−R3

√R6 ∆2 + 8µ1

2 µ22

2R6|2〉 ∼ C6

R6if ∆ 6= 0 (4.11)

H |3〉 =R6 ∆ +R3

√R6 ∆2 + 8µ1

2 µ22

2R6|3〉 ∼ C0 −

C6

R6if ∆ 6= 0 (4.12)

We are following the common definition of the C-coefficients in defining theinteraction energy as VvdW = −

∑nCn/R

n. The expansion in powers of 1/Rclearly shows the 1/R6 behavior if ∆ 6= 0. In Figure 4.2, the resulting interactionpotentials for ∆ < 0 and ∆ > 0 are plotted. Figure 4.2(b) uses realistic valuesfor the energies and dipole moments for rubidium excited to the 33P3/2 statewith neighboring 33S1/2 and 34S1/2 states. For ∆ < 0 we obtain attractive p-p interaction potentials. For large distances, the corresponding state is |3〉 =|p〉 ⊗ |p〉. It gets additional admixtures of the s states for closer distances. Forlarge R, the two states |1〉 and |2〉 are pure |s〉 states and are degenerate. Thedegeneracy is lifted at closer distances where the |1〉 state is totally decoupledand the |2〉 state gets additional admixtures of the |p〉 state and becomes shiftedin energy. If ∆ = 0, the scaling with R changes drastically. Even for large R,the |2〉 and the |3〉 two-particle states always have admixtures of both |s〉 and |p〉

58

4.1. QUALITATIVE THEORETICAL DESCRIPTION

Figure 4.4: Interaction potentials for various fields: (a) 0 V/cm, (b) 3V/cm,(c) 6.7 V/cm, (d) 10 V/cm

character, which leads to a permanent dipole moment of the atom. This is calledresonant dipole-dipole interaction and has a 1/R3 interaction scaling, as can beseen from Equation 4.10. The potential curves are depicted in Figure 4.3. Thissituation corresponds to rubidium excited to 38P3/2. The resonance conditioncan be induced with Rydberg atoms as the dc-Stark effect (see Section 3.3.2) canbe used to tune energy levels into resonance.

Tuning of energy levels with an electric field Tuning the energy levels bymeans of an electric field causes some admixtures to the initial zero field state. Weuse the simplification that only the |d〉 state is coupled to the |p〉 state throughthe electric field. The 5 × 5 matrix for the total Hamiltonian can be furtherreduced to a 3 × 3 matrix by skipping antisymmetric states that are decoupled(|1〉 and (|p〉 ⊗ |d〉 − |d〉 ⊗ |p〉)). The resulting matrix can only be diagonalizednumerically for each radius. The result can be seen in Figure 4.2. Realistic valuesfor the energies and dipole moments of rubidium excited to the 33P3/2 state withneighboring 33S1/2 and 34S1/2 state are used. The |d〉 state is chosen in sucha way that a realistic quadratic stark effect for the |p〉 state is obtained. Theresulting potentials can be seen in Figure 4.4. We can see that quite moderatefields of around 7 V/cm are enough to shift the p state into resonance. We geta resonant dipole-dipole interaction which, in this case, scales as 1/R3.5. Thedeviation from 1/R3 is due to the occurrence of an avoided crossing between the

59


states coupled by the electric field. As long as an avoided crossing is far away,which is usually the case at large distances, the field-free potential curves withadequately shifted energy levels can be used to describe the interaction potentials.However, one has to keep in mind that at small distances, avoided crossings occurbetween states that are coupled by the electric field. This leads to a modificationof the potential behavior.

4.2 Quantitative perturbative approach

The approach presented in the preceding section cannot be used to calculatequantitative values for the interaction potentials as the matrix grows quadrati-cally with the amount of states involved. This is why we use perturbation theoryto calculate the van der Waals interaction potentials. We present explicit expres-sions for long-range interaction potentials between Rydberg atoms in s, p and dangular momentum states. We focus on homonuclear dimers where both atomsare in the same state. Rydberg atoms are the ideal candidates as they have verylarge polarizabilities leading to pronounced van der Waals interactions betweenmutually induced dipoles. We evaluate the three leading C-dispersion coefficientsof the asymptotic interaction potential V (R) given as an expansion of the inter-atomic distance R, i.e. V (R) ' −C6/R

6 − C8/R8 − C10/R

10 (ns-ns asymptote),−C5/R

5 − C6/R6 − C8/R

8 (np-np asymptote), and −C5/R5 − C6/R

6 − C7/R7

(nd-nd asymptote). The calculations presented are applied to rubidium atoms.Calculations for all alkali atoms can be found in [Singer et al., 2004c]. For casesnot reported there, refer to our online potential calculator∗. As an example,the potential curves for rubidium around n = 80 are plotted in Figure 4.5. Wegive the values of the C-coefficients as a function of n in terms of simple fittingparameters.

Theoretical concepts: The interaction energy between two atoms separatedby large internuclear distances R can be expanded as an infinite sum of powersof 1/R [Marinescu, 1997, Dalgarno and Davison, 1966]

V (r1, r2) = −∞∑

n=1

Cn

Rn=

∞∑`,L=1

V`L(r1, r2)

R`+L+1(4.13)

where r1 and r2 are the relative positions of each electron relative to the atomcore, and

V`L(r1, r2) =(−1)L4π√

(2`+ 1)(2L+ 1)

∑m

√(`+ L

`+m

) (`+ L

L+m

)r`1r

L2 Y`m(r1)YL−m(r2) ,

(4.14)

∗http://quantendynamik.physik.uni-freiburg.de/potcalc.htm

60

4.2. QUANTITATIVE PERTURBATIVE APPROACH

0.0 5.0x104 1.0x105-1.0

-0.5

0.0

0.5

1.0

1 +g,

3 +u

81P-81P

80S-80S

Pai

r int

erac

tion

ener

gy [c

m-1]

Internuclear distance [a.u.]

80D-80D

RLR

Figure 4.5: Potential curves for the different symmetries of 81p-81p (solidlines), 81s-81s (dashed line) and 80d-80d (dotted lines) of rubidium. RLR

denotes the Le Roy radius.

61


where(

nk

)= n!/(k!(n− k)!) is the binomial coefficient and Y`m(r) are spherical

harmonics. In order for equations (4.13) and (4.14) to be valid, the electron wavefunctions of the two atoms must not overlap so that exchange and charge overlapinteractions can be neglected. This is the case if R is larger than the Le Royradius RLR [Roy, 1974]:

RLR = 2(〈n1`1| r2 |n1`1〉1/2 + 〈n2`2| r2 |n2`2〉1/2) , (4.15)

where 〈n1`1| r2 |n1`1〉 are the matrix elements of r2 between the radial wave func-tions belonging to the valence electron of an alkali atom. Then, the energy shiftcaused by the long-range interaction can be calculated using perturbation theory,taking the interaction potential (4.13) as a perturbation to the Hamiltonian forthe noninteracting asymptotic case.

Symmetries All geometric symmetry operations of homonuclear diatomicmolecules form the point group D∞h. The symmetry elements that we usu-ally consider are rotations about the internuclear axis, reflections (σν) througha plane containing the rotation axis, and the inversion i of the spatial coordi-nates at the center point between both atom cores [Herzberg, 1950]. The wholepoint group is genereated by multiple applications of these symmetry operations.The group symmetry gives the classification of molecular states and some goodquantum numbers. The molecular wave functions are naturally expressed in themolecule-fixed coordinate system. The projection of the total angular momen-tum M = m1 + m2 onto the molecular axis is conserved as a consequence ofthe rotational symmetry. If M 6= 0, reflections through a plane containing themolecular axis change the sign of the projection of the angular momentum on theaxis. For M 6= 0, the reflected molecular state has the same energy as the initialone. Consequently, the (anti)symmetrization |M±〉 = (1 ± σν)/

√2 |M〉 of the

M 6= 0 states does not break the degeneracy between the states with the sameabsolute value of M . Only for Σ-states (i.e. M = 0), the (anti)symmetrization|Σ±〉 = (1 ± σν)/

√2 |Σ〉 can give non-degenerate states. Thus, the symme-

try property under reflections can be used to distinguish different molecularpotentials. It turns out that the representation of the reflection operator σν

in the molecule-fixed coordinate system is not unique because the position ofthe symmetry axis of a linear molecule is determined only by two Euler angles[Brown and Carrington, 2003]. The absence of off-axis nuclei impedes a uniquedefinition of the way how the third Euler angle is transformed under the space-fixed inversion giving an additional phase factor in the representation of σν . Thisfactor is fixed by an additional convention [Brown and Carrington, 2003].

In case M = m1 = m2 = 0 for the n`-n` asymptotes, only Σ+ states casnexist. This follows from Eq. (4.16) below. The inversion operation i inverts thesingle electronic state relative to the respective atom core and then translates thestate to the other atom core, so that the first electron is located near the second

62


atom core and vice versa. As i× i = 1, the eigenvalues p of i must be either +1 or−1. Molecular states which do not change the sign of the spatial wave functionunder this symmetry operation are labeled gerade states and those that changesign are labeled ungerade states. To fulfill the condition that the total electronicwave function is antisymmetric under exchange of both electrons, the spatial partof the total wave function must be symmetric for the antisymmetric singlet spinstate, and antisymmetric for the symmetric triplet spin states.

Defining σ = (−1)S with the total spin S, the proper symmetrized spatial partof the electronic wave function of the homonuclear atom pair has the followingasymptotic form:∣∣n1l1m1

n2l2m2,M ;σ; p

⟩(4.16)

'[|n1l1m1; R1〉 |n2l2m2; R2〉+ σp(−1)l1+l2 |n2l2m2; R1〉 |n1l1m1; R2〉

]+(−1)l1+l2p

[|n1l1m1; R2〉 |n2l2m2; R1〉+ σp(−1)l1+l2 |n2l2m2; R2〉 |n1l1m1; R1〉

],

where n1 and n2 are the principal quantum numbers, l1 and l2 the angular mo-mentum quantum numbers, and m1 and m2 the separate projections of the an-gular momentum of each atom onto the molecular axis, satisfying the constraintM = m1 +m2. In our notation, the ket |nilimi;Rk〉 |njljmj;Rk′〉 means that thefirst electron is in state n = ni, l = li and m = mi while the second electron isin state n = nj, l = lj and m = mj. Rk and Rk′ are the positions of the twonuclei k and k′ (with k, k′ = 1, 2). It is further understood in the following wayby letting the bra 〈r1| 〈r2| act from the left:

〈r1 |nilimi; Rk〉〈 r2 |njljmj; Rk′〉 = Φnilimi(r1 −Rk) Φnj ljmj

(r2 −Rk′) ,(4.17)

where r1 and r2 denote the absolute position vectors to the first and secondelectron and Φnlm (r) = Rnl(r)Ylm(r) is the spatial wave function for the electronof a single atom, where Rnl(r) represents the electron’s radial wave function andthe spherical harmonic Ylm(r) its angular wave function.

Expressions for C coefficients In order to obtain the expressions for the C-coefficients, perturbation theory is applied. The interaction potential V (r1, r2)in Eq. (4.13) is taken as a perturbation to the Hamiltonian of a noninteractingatom pair, H0. The total Hamiltonian H has the following form

H = H0 + V (r1, r2) . (4.18)

The asymptotic expressions (4.16) of the preceding sections are used as unper-turbed zero order molecular wave functions. The first order energy shift for non-degenerate states is obtained by calculating the expectation values of V definedin Eq. (4.13) with respect to the selected state

∆E(1) =⟨

n1`1,m1

n2`2,m2,M ;σ; p∣∣∣V ∣∣∣n1`1,m1

n2`2,m2,M ;σ; p

⟩. (4.19)

63


Symmetry considerations show that M, σ and p are good quantum numbers whichare not mixed by the Hamiltonian (4.18) even when we look at higher order per-turbation terms. If R > RLR, the electron wave functions for each atom do notoverlap, so that mutual terms between the first square bracket and the secondsquare bracket of Eq. (4.16) are zero. Additionally, perturbation terms evaluatedfor the first square bracket are the same as for the second square bracket since theonly difference in the states is a permutation of the position vectors of the twonuclei. As a consequence we can further simplify the expression for the unper-turbed zero order wave function (4.16) by reducing it to the proper normalizedfirst square bracket. This greatly simplifies the calculations and therefore Eq.(4.19) can be rewritten as

1

2

[⟨n1`1m1n2`2m2

∣∣ + σp(−1)`1+`2⟨

n2`2m2n1`1m1

∣∣]V [∣∣n1`1m1n2`2m2

⟩+ σp(−1)`1+`2

∣∣n2`2m2n1`1m1

⟩], (4.20)

with M = m1 +m2 and the notation∣∣n1`1m1n2`2m2

⟩:= |n1`1m1〉 |n2`2m2〉 . (4.21)

From Equation 4.20, we can see a twofold degeneracy between singlet gerade andtriplet ungerade states and between singlet ungerade and triplet gerade states asthe Hamiltonian (4.18) is spin-independent and no matrix elements depend on σand p separately, but only on the product σp. This degeneracy is removed at closerdistances as the mutual contributions of the two square brackets of Equation 4.16are not negligible and the matrix elements of (4.18) start to depend on p and pσ.Under the condition R > RLR, we can use Equation 4.20 in our calculation, sothat we have to sum matrix elements of the following form

〈n′1`′1m′

1

n′2`′2m′2| V`L |n1`1m1

n2`2m2〉 =

(−1)L4π√(2`+ 1)(2L+ 1)

×∑m

√(`+ L

`+m

) (`+ L

L+m

)〈n′1`′1| r` |n1`1〉〈n′2`′2| rL |n2`2〉

× 〈`′1m′1|Y`m |`1m1〉〈`′2m′

2|YL−m |`2m2〉 . (4.22)

where 〈n′`′| rk |n`〉 is the matrix element of rk

〈n′`′| rk |n`〉 =

∫ ∞

0

Rn′`′(r)rkRn`(r)r

2dr, (4.23)

and 〈`′m′|YLM |`m〉 is the matrix element of the spherical harmonics which canbe expressed as [Marinescu, 1997]

〈`′m′|YLM |`m〉 =

∫Y ∗

`′m′(r)YLM(r)Y`m(r)dΩ

= (−1)m′

√(2`′ + 1)(2L+ 1)(2`+ 1)

4π

(`′ L `0 0 0

) (`′ L `−m′ M m

), (4.24)

64


where the two terms in brackets are the Wigner 3j-symbols defined in[Messiah, 1962, Shore and Menzel, 1968], which can be calculated by using theRacah formula [Racah, 1942].

For degenerate zero order molecular states∣∣M (i)

⟩, degenerate perturbation

theory has to be applied. In order to find the correct eigenvectors the followingmatrix must be diagonalized 〈M (1) | V |M (1)〉〈M (1) | V |M (2)〉 . . .

〈M (2) | V |M (1)〉〈M (2) | V |M (2)〉 . . ....

.... . .

.

The eigenvalues are the first order energy shifts and the eigenvectors are thezero order basis states. We obtain molecular states of the same symmetry andasymptotic energies but with different interaction potentials.

The energy correction in the second order perturbation has the form

∆E(2) =∑φi

⟨n1`1m1n2`2m2

,M ;σ; p∣∣V |φi〉〈φi|V

∣∣n1`1m1n2`2m2

,M ;σ; p⟩

EM − EMi

, (4.25)

where the sum is over a complete orthogonal basis set. Like in the first ordercase, we have to sum matrix elements of the following form

∑niliminjljmj

⟨n′1l′1m′

1

n′2l′2m′2

∣∣∣V ∣∣∣niliminj ljmj

⟩ ⟨nj ljmj

nilimi

∣∣∣V ∣∣n1l1m1n2l2m2

⟩(En1l1 + En2l2)− (Enili + Enj lj)

(4.26)

=∞∑

l,L,l′,L′=1

1

Rl+L+l′+L′+2×

∑niliminjljmj

⟨n′1l′1m′

1

n′2l′2m′2

∣∣∣Vl′L′

∣∣∣niliminj ljmj

⟩ ⟨niliminj ljmj

∣∣∣VlL

∣∣n1l1m1n2l2m2


and the sum is over all possible intermediate states. To simplify the notation, wedefine the following matrix

⟨n′1l′1m′

1

n′2l′2m′2

∣∣∣W l′L′

lL

∣∣n1l1m1n2l2m2

⟩=

∑niliminjljmj

⟨n′1l′1m′

1

n′2l′2m′2

∣∣∣Vl′L′

∣∣∣niliminj ljmj

⟩ ⟨niliminj ljmj

∣∣∣VlL

∣∣n1l1m1n2l2m2


. (4.27)

The calculational effort is simplified by the following symmetry properties:⟨n′1`′1m′

1

n′2`′2m′2

∣∣∣V`L

∣∣n1`1m1n2`2m2

⟩= (−1)`+L

⟨n′2`′2m′

2

n′1`′1m′1

∣∣∣VL`

∣∣n2`2m2n1`1m1

⟩(4.28)

=⟨

n′1`′1−m′1

n′2`′2−m′2

∣∣∣V`L

∣∣n1`1−m1n2`2−m2

⟩=

⟨n1`1m1n2`2m2

∣∣V`L

∣∣∣n′1`′1m′1

n′2`′2m′2

⟩.

65


Also,⟨

n′1`′1m′1

n′2`′2m′2

∣∣∣V`L

∣∣n1`1m1n2`2m2

⟩= 0 if at least one of the following conditions is true:

`′1 + `1 + ` = (odd), `′2 + `2 + L = (odd) (4.29)

m′1 +m′

2 6= m1 +m2,

` < |`′1 − `1| , ` > `′1 + `1, L < |`′2 − `2| , L > `′2 + `2.

The symmetry properties of⟨

n′1`′1m′1

n′2`′2m′2

∣∣∣W `′L′

`L

∣∣n1`1m1n2`2m2

⟩are:⟨

n′1`′1m′1

n′2`′2m′2

∣∣∣W `′L′

`L

∣∣n1`1m1n2`2m2

⟩= (−1)`+`′+L+L′

⟨n′2`′2m′

2

n′1`′1m′1

∣∣∣WL′`′

L`

∣∣n2`2m2n1`1m1

⟩(4.30)

=⟨

n′1`′1−m′1

n′2`′2−m′2

∣∣∣W `′L′

`L

∣∣n1`1−m1n2`2−m2

⟩=

⟨n1`1m1n2`2m2

∣∣W `L`′L′

∣∣∣n′1`′1m′1

n′2`′2m′2

⟩.

Again,⟨

n′1`′1m′1

n′2`′2m′2

∣∣∣W `′L′

`L

∣∣n1`1m1n2`2m2

⟩= 0 if at least one of the following conditions is

true:

`1 + `′1 + `+ `′ = (odd), `2 + `′2 + L+ L′ = (odd) (4.31)

m′1 +m′

2 6= m1 +m2

|`′1 − `′| > `1 + `, |`1 − `| > `′1 + `′

|`′2 − L′| > `2 + L, |`2 − L| > `′2 + L′.

Evaluation of potentials In what follows, the interaction potentials for Ryd-berg states correlated to the ns-ns, np-np and nd-nd asymptotes of alkali atomsare obtained by taking Equation 4.13 as a perturbation. In our approach, thefine structure is neglected, and we choose the center of gravity as the energy ofn` states. The np-np case has already been extensively discussed and analyzedin [Boisseau et al., 2002, Marinescu, 1997]. The nd-nd symmetries also have aC7 coefficient arising from the first order correction to the energy. The determi-nation of the C5 and C7 coefficients, as well as the higher order corrections, isfacing additional difficulties from the asymptotic degenerate states for which theC7 does not vanish. In these cases, even the zeroth order wave functions dependon the internuclear distance R. In order to get C5 and C7 for those states, oneneeds to diagonalize the matrix of the first-order correction to the energy andthen expand its eigenvalues in powers of 1/R. The matrices are of the followingtype

R−5(M0 +R−2M1), (4.32)

where Eq. (4.32) represents the right hand side of Eq. (4.13) in the set of thedegenerate asymptotic states. In these cases C5 and C7 are

C5 = 〈ni|M0 |ni〉 , (4.33)

C7 = 〈ni|M1 |ni〉 ,

66


and |ni〉 are the eigenstates of M0. Note that the R−5 and R−7 interaction termswith the C5 and C7 given by (4.33) might be significantly less accurate than theeigenvalues of (4.32) themselves at shorter distances when R reaches the Le Royradius RLR.

Following the classification of molecular states by Wigner and Witmer[Wigner and Witmer, 1928], the molecular states are denoted by capital Greekletters (Σ, Π, ∆, Φ, . . . for M = 0, 1, 2, 3, . . . ). 1M and 3M represent singlet andtriplet states respectively and Mg and Mu represent gerade and ungerade statesrespectively. The values for the C-coefficients are extracted by collecting termswith equal power of R. The explicit expressions of the perturbation terms arequite lengthy and are supplied in Appendix B. In table 4.1 we give a list of theunperturbed molecular states for the ns-ns, np-np and nd-nd asymptotes. Thefirst column denotes the projection M of the total angular momentum on theinternuclear axis. In the second column we give the molecular symmetry, and inthe next column, the state itself.

In table 4.2, we give the C6, C8, and C10 coefficients for the ns-ns asymptotes,the C5, C6, and C8 coefficients for the np-np asymptotes and the C5, C6, and C7

coefficients for the nd-nd asymptotes of high Rydberg states of all alkali atoms.The C5, C6, C7, C8, and C10 coefficients are scaled by their major dependenceon n, which is n8, n11, n12, n15, and n22, respectively. For 30 ≤ n ≤ 95 theresidual dependence on n is fitted to a polynomial of the form a+ bn+ cn2 + · · · .The fitting parameters are presented in table 4.2. As an example, in Figure 4.6,we have plotted the C5, C6 and C7 coefficient for the 1Γg ,3Γu symmetry. Thedivergence near n = 35 is due to a resonance in the interaction potentials, whichleads to a resonant dipole-dipole interaction where our perturbative approachfails. A specific example is given in Figure 4.5, where the calculated interactionpotentials for Rb-Rb pairs near the n ∼ 80 manifold are shown.

In a few cases, for some symmetries and some n, the two-atom levels are veryclose to each other and the second order correction (4.25) gives large values. Inthese cases perturbation theory fails. We have added a resonance term of theform c−1/(n− n0) to the polynomial in order to simplify the presentation of thedata away from the resonance by the fitted polynomial. To check the validity ofthe potentials at certain interatomic distances R, we have calculated the Le Royradius RLR for different effective quantum numbers and different alkali atoms.For all alkali atoms, the Le Roy radius RLR is approximately RLR ∼ 5n2.

We have tested the numerical accuracy of our radial wave functions with re-spect to their truncation at small distance by comparing them with wave functionsobtained using model potentials where the core is taken into account. We alsotested the convergence of our results as a function of the number of intermediatestates (n′) included, and found that convergence to the 5 significant digits wasattained by taking n− 20 ≤ n′ ≤ n+ 20 (n being the principal quantum numberof the state under consideration).

The results presented here are not corrected for spin-orbit coupling (fine struc-

67


Table 4.1: Unperturbed molecular states for different symmetries accordingto Wigner and Witmer [Wigner and Witmer, 1928]. States with equal |M |and symmetry are asymptotically degenerate and are obtained by degenerateperturbation theory.|M | Symmetry Representation for the ns-ns asymptote0 1Σ+

g , 3Σ+u |n00

n00〉|M | Symmetry Representation for the np-np asymptote2 1∆g,

3∆u |n11n11〉

1 1Πu,3Πg

1√2(|n11

n10〉 − |n10n11〉)

1 1Πg,3Πu

1√2(|n11

n10〉+ |n10n11〉)

0 1Σ−u , 3Σ−

g1√2

(∣∣n11n1−1

⟩−

∣∣n1−1n11

⟩)0 1Σ+

g , 3Σ+u

√23|n10n10〉+ 1√

6

∣∣n11n1−1

⟩+ 1√

6

∣∣n1−1n11

⟩0 1Σ+

g , 3Σ+u − 1√

3|n10n10〉+ 1√

3

∣∣n11n1−1

⟩+ 1√

3

∣∣n1−1n11

⟩|M | Symmetry Representation for the nd-nd asymptote4 1Γg,

3Γu |n22n22〉

3 1Φu,3Φg

1√2(|n22

n21〉 − |n21n22〉)

3 1Φg,3Φu

1√2(|n22

n21〉+ |n21n22〉)

2 1∆u,3∆g

1√2(|n22

n20〉 − |n20n22〉)

2 1∆g,3∆u

√819|n22n20〉+

√819|n20n22〉+

√319|n21n21〉

2 1∆g,3∆u −

√338|n22n20〉 −

√338|n20n22〉+ 4√

19|n21n21〉

1 1Πu,3Πg

√14

+ 74√

55(|n21

n20〉 − |n20n21〉) +

√14− 7

4√

55

(∣∣n22n2−1

⟩−

∣∣n2−1n22

⟩)1 1Πu,

3Πg

√14− 7

4√

55(− |n21

n20〉+ |n20n21〉) +

√14

+ 74√

55

(∣∣n22n2−1

⟩−

∣∣n2−1n22

⟩)1 1Πg,

3Πu

√14− 5

4√

79(− |n21

n20〉 − |n20n21〉) +

√14

+ 54√

79

(∣∣n22n2−1

⟩+

∣∣n2−1n22

⟩)1 1Πg,

3Πu

√14

+ 54√

79(|n21

n20〉+ |n20n21〉) +

√14− 5

4√

79

(∣∣n22n2−1

⟩+

∣∣n2−1n22

⟩)0 1Σ−

u , 3Σ−g

1√5+

√5

(∣∣n21n2−1

⟩−

∣∣n2−1n21

⟩)+ 1√

5−√

5

(∣∣n22n2−2

⟩−

∣∣n2−2n22

⟩)0 1Σ−

u , 3Σ−g

1√5−

√5

(−

∣∣n21n2−1

⟩+

∣∣n2−1n21

⟩)+ 1√

5+√

5

(∣∣n22n2−2

⟩−

∣∣n2−2n22

⟩)0 1Σ+

g , 3Σ+u 0.4121

(∣∣n21n2−1

⟩+

∣∣n2−1n21

⟩)+ 0.5204

(∣∣n22n2−2

⟩+

∣∣n2−2n22

⟩)+ 0.3445 |n20

n20〉0 1Σ+

g , 3Σ+u −0.1316

(∣∣n21n2−1

⟩+

∣∣n2−1n21

⟩)− 0.2064

(∣∣n22n2−2

⟩+

∣∣n2−2n22

⟩)+ 0.9382 |n20

n20〉0 1Σ+

g , 3Σ+u −0.5593

(∣∣n21n2−1

⟩+

∣∣n2−1n21

⟩)+ 0.4320

(∣∣n22n2−2

⟩+

∣∣n2−2n22

⟩)+ 0.0331 |n20

n20〉

68


Figure 4.6: The C5, C6 and C7 coefficient for the 1Γg ,3Γu symmetry. Thedivergence near n = 35 is due to a resonance in the interaction potentials lead-ing to a resonant dipole-dipole interaction where our perturbative approachfails.

69


ture). These effects appear as difference in the quantum defect of p1/2 and p3/2,or d3/2 and d5/2, and may play a role especially for Rb and Cs. For these twospecies, we used the center-of-gravity for the energy level entering the second or-der perturbation expressions, and the quantum defect of j = `−1/2 for the radialmatrix elements of the various multipole moments (which are not very sensitive tothe j component used). As an estimate of the error introduced by neglecting thespin-orbit interaction, we performed calculations replacing the center-of-gravityenergy by the energy of one of the fine structure components. At n = 60 forrubidium, some C6 dispersion coefficients obtained by second order perturbationtheory deviate by a factor of 3 to 5. It should also be noted that the resonanceis shifted from n = 35 to n = 38.

We have found that in most cases, the dispersion coefficients follow simplescaling behaviors. The variation from this scaling behavior is well fitted by simplepolynomials. However, in few instances, we found divergent coefficients nearsome accidental degeneracies of energy levels. This is especially noticeable forthe C6 coefficient of Rb near n = 35. Even though in those cases perturbationtheory fails, we have added a resonance term to simplify our fitting procedure forcoefficients far from those resonances where our calculation is valid.

70


Table 4.2: Dispersion coefficients of high Rydberg states for the ns-ns,np-np and nd-nd asymptotes of Rb-Rb. The coefficients are scaled by theirmajor dependence on n and the residual dependence is fitted using threefitting parameters as indicated in the table. All nd-nd C6 coefficients showresonances. They have to be calculated differently as indicated by the formulain the table with n0 = 35.14.ns-ns C6 = n11(c0 + c1n + c2n2) C8 = n15(c0 + c1n + c2n2) C10 = n22(c0 + c1n + c2n2)

c0 c1 c2 c0 c1 c2 c0 c1 c2

symmetry (×101) (×10−1) (×10−3) (×100) (×10−1) (×10−3) (×10−4) (×10−6) (×10−9)

1Σ+g ,3Σ+

u 1.197 -8.486 3.385 -7.303 8.068 -3.792 -5.546 5.242 -3.154

np-np C5 = n8(c0 + c1n + c2n2) C6 = n11(c0 + c1n + c2n2) C8 = n15(c0 + c1n + c2n2)

c0 c1 c2 c0 c1 c2 c0 c1 c2

symmetry (×100) (×10−2) (×10−2) (×10−1) (×10−1) (×10−4) (×101) (×100) (×10−2)

1∆g ,3∆u -0.231 -1.976 0.010 3.620 -0.579 2.778 1.199 -0.624 0.250

1Πu ,3Πg vanishes 6.070 -1.273 6.157 1.173 0.010 -0.069

1Πg ,3Πu 0.922 7.903 -0.041 3.575 -0.183 0.816 2.973 -2.281 0.990

1Σ−u ,3Σ−

g vanishes 2.373 -0.034 0.107 2.176 -1.711 0.747

1Σ+g ,3Σ+

u -1.383 -11.850 0.061 43.010 3.575 -1.714a 5.359 -3.984 1.729

1Σ+g ,3Σ+

u vanishes 5.461 -1.133 5.476 0.712 0.244 -0.162

nd-nd C5 = n8(c0 + c1n + c2n2) C6 = n11(c0 + c1n +c−1

n−n0) C7 = n11(c0 + c1n + c2n2)

c0 c1 c2 c0 c1 c−1 c0 c1 c2

symmetry (×100) (×10−2) (×10−4) (×101) (×10−2) (×100) (×101) (×10−1) (×10−1)

1Γg ,3Γu -1.445 -2.731 1.477 2.603 1.454 66.310 0.235 0.920 -0.487

1Φu ,3Φg -0.722 -1.366 0.738 4.124 2.475 105.000 -0.059 -0.230 0.122

1Φg ,3Φu 2.167 4.097 -2.215 0.643 -0.273 16.570 -0.997 -3.911 2.069

1∆u ,3∆g 1.686 3.186 -1.723 1.730 0.525 44.200 0.352 1.380 -0.730

1∆g ,3∆u 1.565 2.959 -1.600 0.571 -0.584 14.650 1.260 4.940 -2.614

1∆g ,3∆u -0.722 -1.366 0.738 5.157 3.032 131.200 0.031 0.121 -0.064

1Πu ,3Πg -1.013 -1.915 1.036 5.157 2.864 131.200 -0.499 -1.958 1.036

1Πu ,3Πg 0.773 1.460 -0.789 1.789 0.563 45.640 -0.381 -1.493 0.790

1Πg ,3Πu 1.191 2.250 -1.217 2.217 0.916 56.570 0.151 0.594 -0.314

1Πg ,3Πu -0.950 -1.795 0.971 0.907 -0.779 22.990 -0.093 -0.364 0.193

1Σ−u ,3Σ−

g -2.338 -4.419 2.389 0.274 -0.870 6.928 0.849 3.330 -1.762

1Σ−u ,3Σ−

g 0.893 1.688 -0.913 2.676 1.229 68.210 0.324 1.272 -0.673

1Σ+g ,3Σ+

u -2.748 -5.195 2.809 0.298 -1.100 7.455 0.264 1.037 -0.548

1Σ+g ,3Σ+

u -1.271 -2.403 1.299 4.755 2.449 120.900 -1.118 -4.386 2.320

1Σ+g ,3Σ+

u 0.408 0.771 -0.417 2.942 1.436 74.940 -0.319 -1.252 0.663

a The following resonance term has to be added to the polynomial:

C6 = n11(c0 + c1n + c2n2 + 6.931× 102/(n− 29.5))

71


Figure 4.7: (a) Rydberg spectrum around n = 40 taken at Rydberg densitiesof 1010 cm−3 and excitation intensities of 200 W/10−4 cm2, observed linewidth ∼ 7 GHz. (b) Same Rydberg spectrum at Rydberg densities of 2×1011

cm−3 and excitation intensities of 5 kW/10−4 cm2, observed line width 30−40GHz. (taken from [Raimond et al., 1981])

4.3 Interaction-induced line broadening of Ryd-

berg resonances

The first spectroscopic signs of interaction-induced effects in Rydberg gases wereseen in atomic beam experiments [Raimond et al., 1981]. In these experiments,atoms at densities of ' 1013cm−3 were excited into Rydberg states (n∼40) usingbroad-bandwidth pulsed lasers. With increasing laser power (and thus increasingRydberg atom densities) the excitation lines broadened, which was explained bythe dipole-dipole interaction in dense Rydberg gases (see Figure 4.7). Our setupallows for much higher resolution and leads to similar findings at launch statedensities which are about three orders of magnitude smaller (' 1010cm−3). Thespectrum presented here is recorded at n ∼ 80 and a high excitation intensity ofabout 650 W/cm2. In this regime, the internuclear distances are small enoughfor the interaction potentials (see Figure 4.5) to play a role. The spectrum inFigure 4.8 is strikingly different from the spectra presented in Section 3.3 (whichare taken at lower n around 60 and at higher n around 80 but with low excitation

72

4.3. INTERACTION-INDUCED LINE BROADENING OF RYDBERGRESONANCES

625266 625267 625268 625269 6252700.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

ghost 5P3/2 (F=2)

D*

79h

Ryd

berg

ato

m n

umbe

r [10

3 ]

Laser frequency (GHz)625258.2 625258.4

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

625259.6 625259.8

81P3/2

81P1/2

Ryd

berg

ato

m n

umbe

r [1

03 ]


( )*

625265.2 625265.4 625265.60

1

2

3

4

5

6

ghost 5P3/2 (F=2)

82S

Ryd

berg

ato

m n

umbe

r (10

3 )

Laser frequency (GHz)625262 625263

0

2

4

6

8

ghost 5P3/2 (F=2)

80D

Ryd

berg

ato

m n

umbe

r (10

3 )


625260 625270 6252800

2

4

6

8

D*

80h79h

83S

81D

82S

82P81P

80D

Ryd

berg

ato

m n

umbe

r (10

3 )


( )*( )*

Figure 4.8: Rydberg spectrum at n'80. The x-axis is the frequency of theblue excitation laser providing the second step in the two-photon excitationscheme. Besides the dipole-allowed nS and nD states, residual electric fieldsalso allow for the excitation of nP states. The small peak near the 80D stateis a hyperfine ghost excited from the 5P3/2(F = 2) state. The spectrum istaken at an excitation intensity of 650 W/cm2. When compared to spectrashown in Section 3.3 two striking differences are observed: The lines areasymmetrically broadened to the red and small peaks appear inside the redwings.

73


Figure 4.9: (a) Rydberg atom pairs at closer internuclear distances areshifted in energy due to interaction potentials. They can only be excited bya red/blue detuned two-photon excitation process. (b) Rydberg atom pairsexcited with the excitation laser.

power). We can see a strong asymmetric broadening to the red, and a hugeamount of small peaks appear inside the red wings.

There are several possible explanations for the observed line broadening whichare also discussed in [Raimond et al., 1981]. Pure saturation broadening causedby the intensity of the the blue laser can be ruled out for three reasons. First,only about 10% of the atoms are actually excited into Rydberg states (roughly60,000 atoms in the interaction volume are available in the 5P3/2 intermediatestate), which can only explain a much smaller broadening than the one observed.Second, the line is asymmetrically broadened (see inset of Figure 4.10), and third,extensive density-dependent studies show that only the combination of high laserpower with high densities results in the observed broadening [Singer et al., 2004a].Stark broadening due to electric fields caused by ions in the vicinity of the Ryd-berg atoms cannot not account for the observed broadening as we estimate anionization rate due to collisions and black-body radiation of the order of 3 kHz[Gallagher, 1994] and therefore do not expect an appreciable number of ions toaccumulate during the time of excitation.

The line broadening can be caused by off-resonant excitation of atom pairsinto a two-body interacting state with the energy shift being caused either due tolong-range dipole-dipole forces or van der Waals interaction (see Figure 4.9). Theobserved broadening thus directly reflects the interaction energy between pairsof Rydberg atoms. Note that under the conditions of our experiment, the inter-atomic spacing is still one order of magnitude larger than the extension of theelectron wave function (nearest-neighbor distances are in the order of 5µm; as acomparison the extension of the wave function is ∼ 802aBohr ' 0.3µm ). Follow-ing this interpretation, it is tempting to attribute the pronounced structure on thered wing of the resonances, the structure never appear on the blue side of the res-onance (independent of the scanning direction), to the excitation of bound statesin the attractive interatomic Rydberg-Rydberg potential [Boisseau et al., 2002].

74


-200 0 200 400-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

-200 0 2000

1

2

3

4

Num

ber o

f Ryd

berg

atom

s (1

03 )

(MHz)

636 W/cm2 528 W/cm2 229 W/cm2 127 W/cm2 57 W/cm2 25 W/cm2 13 W/cm2

Figure 4.10: Excitation spectrum of the 82S resonance for different inten-sities of the blue laser driving the second excitation step in the two-photonexcitation scheme (see Figure 3.5). The first excitation step is saturated. Theinset shows a high resolution spectrum of the same transition.

Full resolution of these structures has so far been precluded by slow drifts of theblue excitation laser. As these structures seem to be very narrow we are currentlysetting up a stabilization scheme for the excitation laser by locking it to a stablereference cavity in order to achieve the needed stability in laser frequency (seeSection 6).

To further understand the spectra, we carried out systematic studies of theline shape as a function of laser power (Section 4.3.1), launch state density (Sec-tion 4.3.2) and excitation time (Section 4.3.3).

75


-0.1 -0.05 0 0.05 0.1Detuning @GHzD

0.5

1

1.5

2

2.5

3

3.5

4

inte

nsity

scal

ing

expo

nent

Figure 4.11: Intensity scaling of the Rydberg excitation as a function of thedetuning of the excitation laser. For each detuning, we have plotted the fittedpotential dependence α of the amount of Rydberg atoms N on the intensityI of the excitation laser: N ∝ Iα.

76


-0.5 0.0 0.5 1.0 1.50.00

0.05

0.10

0.15

0.20

0.25

0.0

0.5

launch state peak density [1010cm

-3]

fluor

esce

nce

[mV

]

time [ms]

trap laser tunedto resonance

1.0

1.0

Figure 4.12: Typical offset corrected fluorescence signal during depopula-tion. At t = 0 the repumping laser is switched off and a depumping laser isswitched on. For Rydberg excitation the trap laser is tuned into resonanceand the blue excitation laser is switched on. Corresponding peak density ofthe launch state is indicated on the right axis.

4.3.1 Intensity-dependent studies

In the following, we study how the spectrum evolves from the low-power to thehigh-power regime. Figure 4.10 shows the result of an intensity scan of the 82Sresonance. With increasing laser intensity, the resonance lines broaden signifi-cantly. The inset in Figure 4.10 shows a high-resolution scan of the same tran-sition, revealing that the line broadening is asymmetric and that the red wingof the resonance contains some pronounced structure. A shift of the resonancelines could not be seen in our experiment, as we do not have absolute values forthe frequency axis. The traces in Figure 4.10 are all centered around ∆ν = 0at their maximum value. Intensity scans give essential information of the exci-tation process. If a two-photon excitation process (Figure 4.9) is involved, thena quadratic scaling with intensity is expected. Figure 4.11 shows a plot of theintensity scaling exponent versus detuning. The exponent α was determined foreach detuning by fitting the function Iα to the plot of the amount of generatedRydberg atoms versus intensity I. With increasing detuning, the exponent growsand is consistent with a two-photon process for larger detunings. The satura-tion near resonance is partly caused by power saturation and interaction effectsbetween Rydberg atoms. It is discussed in Section 5.2.1.

77


4.3.2 Density-dependent studies of line shapes

The best way to obtain a clear signature for density-dependent effects is changingthe density itself. The density of Rydberg atoms can be controlled by varyingthe population in the 5S1/2(F = 2) launch state by means of optical pumpinginto the 5S1/2(F = 1) state. For this purpose, the repumping laser is attenuatedby an AOM while an additional depumping laser resonant with the transition5S1/2(F=2) → 5P3/2(F=2) (see Figure 3.5) is switched on. Within 1 ms, thedensity of atoms in the launch state is lowered by one order of magnitude, as ismeasured by recording the fluorescence on the closed 5S1/2(F = 2) - 5P3/2(F = 3)transition with a photodiode (see Figure 4.12). Note that no atoms are lost fromthe MOT capture volume during this short period of time, and that the excitationvolume Vexc remains unaltered. By delaying the two-step Rydberg excitationscheme with respect to the start of the depumping process, one can thus modifythe density of atoms in the launch state without changing the total density ofatoms in the MOT.

Figure 4.13 shows the density dependence of the dipole-forbidden fine struc-ture doublet 81P1/2 and 81P3/2 (left and right peak in Figure 4.13(a), respec-tively), together with theoretical calculations of the pair potentials ∗. With in-creasing density, the 81P3/2 peak develops a wing on the red side of the resonance,again with the fluctuating resonances, while the 81P1/2 peak is hardly broadened.As shown in Figure 4.13(b), due to a coincidental near-resonant enhancement inthe C6 coefficient, the attractive 81P3/2 interaction asymptote is much steeperthan the 81P1/2 asymptote. This explains why broadening is only observed forthe red wing of the 81P3/2 line, and not for the 81P1/2 line. The average near-est neighbor distances derived from the launch state densities are depicted bythe dashed lines in Figure 4.13(b). The red broadening starts when the averagenearest neighbor distance becomes comparable to interatomic distances at whichthe attractive interaction potential starts to bend, and observed broadening andinteraction energy are within the same order of magnitude.

Figure 4.14 shows excitation spectra of the 82S1/2 state recorded for differentRydberg densities. The density is changed either by varying the power of theexcitation laser or, as described above, by changing the density ng of atoms in the5S1/2(F = 2) launch state. Figs. 4.14(a1) and 4.14(a2) are taken at low intensity

of the blue excitation laser (I = 6 W/cm2). In this “low power regime”, the lineshows no broadening even at highest densities. The behavior changes drasticallyat higher intensities of the blue excitation laser. Figs. 4.14(a3) and (a4) showthe resonance line at an excitation intensity of about 500 W/cm2. In the “high-

∗The interaction potentials are calculated without considering the effect of the electric fieldas no avoided crossings with other states occur in the region of interest (see Section 4.1). Thefine structure is only taken into account by using different energies levels for the perturbationcalculation presented in Section 4.2. This is inaccurate as Hund’s coupling case (c) should beused instead of (a). We intent to tackle the problem in future work.

78


Figure 4.13: (a) Density dependence of the resonance line for the fine struc-ture doublet 81P1/2 (left resonance) and 81P3/2 (right resonance) [launch statedensities from left to right: 1.0, 0.83, 0.55, 0.25 ×1010 cm−3]. (b) Calculatedvan der Waals pair potentials for the 81P1/2 + 81P1/2 and 81P3/2 + 81P3/2

asymptote versus the interatomic distance. The dashed lines indicate theaverage nearest neighbor distances for the corresponding spectra.

79


Figure 4.14: (a) Density dependence of the resonance line for the 82S1/2 line

[from left to right: 6 W/cm2 @ 1.6× 109 cm−3, 6 W/cm2 @ 1.0× 1010 cm−3,500 W/cm2 @ 1.5 × 109 cm−3, 500 W/cm2 @ 1.0 × 1010 cm−3]. (b) Calcu-lated van der Waals pair potentials for the 82S1/2 + 82S1/2 asymptote versusinteratomic distance. The dashed lines indicate the average nearest neighbordistances for both density limits.

80


-0.1 -0.05 0 0.05 0.1Detuning @GHzD

1

2

3

4

5

6

laun

chst

ate

dens

itysc

alin

gex

pone

nt

Figure 4.15: Density scaling of the Rydberg excitation as function of thedetuning of the excitation laser. For each detuning we, have plotted the fittedpotential dependence α of the amount of Rydberg atoms N on the launchstate density n: N ∝ nα.

81


-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

-6

-4

-20

24

6

810

1214

16

18

2 s8 s

14 s20 s26 s32 sN

umbe

r of R

ydbe

rg a

tom

s (1

03 )

(GHz)

38 s

Figure 4.16: Excitation spectrum of the 82S resonance for different dura-tions of the blue exciation laser. For long durations of excitation an additionalfeature appears at ∆ν ≈ −0.25GHz whose origin has not yet been clarified.

power” regime characterized by Rydberg densities in the 108 cm−3 range, againan asymmetric broadening of the line and the spectral features in the red wingof the resonance line are observed. They both become more pronounced withincreasing density. The origin of the asymmetry to the red cannot be explainedby the van der Waals interaction potentials as only a single repulsive interactionpotential exists for the 82S-82S asymptote, which can be seen in Figure 4.14(b).In Figure 4.11, the density-scaling exponent is plotted versus detuning. Theexponent α was determined for each detuning by fitting the function nα to theplot of generated Rydberg atoms versus the launch state density n. For detuningsof 100 MHz, the amount of generated Rydberg atoms scales quadratically withthe launch state density. This is an indication that a two-particle process isinvolved. Again, near resonance we see a saturation of the excitation which willbe discussed in Section 5.2.2.

4.3.3 Time-dependent spectrum

To gain more insight into the dynamics of the Rydberg excitation process inthe regime of Rydberg-Rydberg interactions, we studied the number of Rydberg

82


-0.05 0 0.05 0.1Detuning @GHzD

0.5

1

1.5

2

2.5

3

Exc

itatio

ntim

esc

alin

gex

pone

nt

Figure 4.17: Excitation time scaling of the Rydberg excitation as functionof the detuning of the excitation laser. For each detuning we have plottedthe fitted potential dependence α of the amount of Rydberg atoms N on theexcitation time t of the blue laser: N ∝ tα.

83


atoms as a function of the time for which the excitation laser is switched on. Theintensity of the excitation laser is kept constant at 650 W/cm2. The resultingspectra of 82S resonance are shown in Figure 4.16. We observe asymmetric broad-ening which occurs already at short excitation times. Even though the minimumexcitation time is a factor of 10 smaller than in the density scans and the max-imum excitation time is a factor of 2 larger, the peak shape seems to be ratherunaffected. This becomes even more obvious if we plot the time-scaling exponentversus detuning. The exponent α was determined for each detuning by fitting thefunction tα to the plot of generated Rydberg atoms versus the excitation time t.When compared to Figure 4.11 or Figure 4.15, the exponents in Figure 4.17 aremuch closer to linear scaling, especially for zero detuning (see Section 5.2.3). Apossible explanation for this different behavior is that the initial excitation intoRydberg states gets redistributed in states which couple less to the initial state.So only during the initial phase of excitation the energy levels are perturbed bysimultaneously excited Rydberg atoms. Due to the redistribution of excitation,Rydberg atoms excited later do not see the earlier Rydberg atoms. The validityof this interpretation must await further state-selective measurements which arecurrently prepared.

84

Chapter 5

Blockade of Rydberg excitationon resonance

In the first two sections of this chapter, we show how quantum computing canbe realized on the basis of Rydberg atoms. We describe two possible realizationsof a phase gate: the two-body (Section 5.1.1) and the mesoscopic many-bodyimplementation (Section 5.1.2). The latter exploits an interaction-induced exci-tation inhibition of multiple Rydberg excitations. These proposals motivate ourexperimental studies of on resonance excitation inhibition in a gas of Rydbergatoms which are presented in Section 5.2.

5.1 Quantum information processing with Ryd-

berg atoms

Possible implementations of a computer that operates according to the principlesof quantum mechanics are currently investigated: Systems exploiting nuclearmagnetic resonance, trapped dipolar molecules, trapped ions, solid state devices,systems based on cavity quantum electro dynamics and trapped neutral atoms.The latter are a promising candidate as the interaction between neutral atomscan be controlled through its electronic state. Neutral atoms in the ground statehardly interact with the environment and can therefore be used as robust storagestates. However, when excited into Rydberg states, they can strongly interactvia dipole-dipole interaction, which can be exploited for the implementation ofquantum operations.

5.1.1 A phase gate with Rydberg atoms

Quantum computing with neutral atoms exploiting the strong dipole-dipole in-teraction of Rydberg states to obtain entanglement and perform gate oper-ations was first proposed by [Jaksch et al., 2000]. In our setup, the pair of

85

CHAPTER 5. BLOCKADE OF RYDBERG EXCITATION ON RESONANCE

Figure 5.1: Relevant levels for implementation of a phase gate with twoRydberg atoms. Depicted is the situation where atom 1 is already excited toa Rydberg state. The same laser frequency which was resonant for atom 1now applied to atom 2 does not lead to excitation because this state is shiftedin energy due to dipole-dipole interactions between Rydberg atoms.

metastable sublevels of the groundstate manifold of 87Rb |g〉 =∣∣52S1/2, F = 1

⟩and |e〉 =

∣∣52S1/2, F = 2⟩

would serve as long-living storage states. This hasthe advantage that the states hardly interact during storage time, which leadsto very long coherence times. Let us consider two neighboring trapped atoms.The gate operations are performed by exciting the atoms to Rydberg states |r〉letting them interact via the strong dipole-dipole interaction of strength u (seeFigure 5.1). This interaction can be caused by field induced dipoles, by res-onant dipoles or even by van der Waals forces. The only requirement is thatthe interaction is strong enough that the gate operations can be performed on amuch faster time scale than typical trapping oscillation times in order to avoidentanglement of the motional degrees of freedom with the internal states of theatoms. [Jaksch et al., 2000] describe several possible implementations of phasegates witch Rydberg atoms. We focus on one implementation which can be ex-tended easily to a version working on mesoscopic ensembles of atoms as we willsee in the next section. This implementation assumes that the Rabi frequency Ωis much smaller than the interaction u/~, and that both atoms can be addressedseparately∗. The Rabi frequency is defined as ~Ω = µE0 with µ the dipole mo-ment of the upper hyperfine ground state |e〉 to the Rydberg state |r〉 and E0 thefield amplitude of the excitation laser on the transition from |e〉 → |r〉. The gate

∗Individual addressing can be avoided by using an adiabatic version of the phase gate. Thisis accomplished by using time varying detunings and intensities of the excitation laser, therebyadiabatically eliminating the Rydberg state [Jaksch et al., 2000].

86

5.1. QUANTUM INFORMATION PROCESSING WITH RYDBERG ATOMS

operation is now performed in three steps: (i) The gate operation is initiated byapplying a π pulse to the first atom

|e〉 ⊗ |e〉 → i |r〉 ⊗ |e〉 (5.1)

|e〉 ⊗ |g〉 → i |r〉 ⊗ |g〉|g〉 ⊗ |e〉 → |g〉 ⊗ |e〉|g〉 ⊗ |g〉 → |g〉 ⊗ |g〉 .

(ii) Then a 2π pulse is applied to the second atom. If the first atom is in state|r〉 and the second atom is in state |e〉, then the second atom sees a detunedpulse by δ = u/~ due to the interaction energy u between Rydberg atoms (seeFigure 5.1). If the laser bandwidth ∆ν is smaller than δ/2π, then the secondatom is not excited. It only accumulates a small phase which is obtained by theRabi formula for the probability of the second atom being excited to a Rydbergstate

P (t) =Ω2

Ω2 + δ2sin2

[√Ω2 + δ2

t

2

]. (5.2)

If δ = 0, the excitation probability can reach the value one and we have accu-mulated a phase φ = π/2. In our case δ = u/~, and as u >> ~Ω, the excitationprobability oscillates rapidly, having a maximum value of (Ω/δ)2, which leads toa maximum accumulated phase of φ = πΩ/(2δ) which is nearly zero. Thereforethe states are affected as follows

i |r〉 ⊗ |e〉 → i |r〉 ⊗ |e〉 (5.3)

i |r〉 ⊗ |g〉 → i |r〉 ⊗ |g〉|g〉 ⊗ |e〉 → − |g〉 ⊗ |e〉|g〉 ⊗ |g〉 → |g〉 ⊗ |g〉 .

(iii) Finally, a second π pulse is applied to the first atom yielding the followingoverall truth table

|e〉 ⊗ |e〉 → − |e〉 ⊗ |e〉 (5.4)

|e〉 ⊗ |g〉 → − |e〉 ⊗ |g〉|g〉 ⊗ |e〉 → − |g〉 ⊗ |e〉|g〉 ⊗ |g〉 → |g〉 ⊗ |g〉 .

If a two-photon excitation scheme is used, and if the intermediate state is off-resonant, then the transition can be described by an effective two-photon Rabifrequency [Teo et al., 2003].

The presented implementation of a phase gate is based on an excitation inhi-bition of the second atom due to interaction induced level shifts. However, it isquite difficult to control single atoms which have to be trapped and excited intoRydberg states. In the next section, an approach is presented using basically thesame mechanism but without the need for single atom control.

87


Figure 5.2: Schematic explanation of the dipole blockade. (a) Energylevels for a pair of atoms. |g〉 and |r〉 denote the ground and Rydberg staterespectively. The ultralong range dipole interaction splits the Rydberg pairstate and thus suppresses excitation of a Rydberg pair by a resonant laserfield, as indicated by the vertical arrows for small internuclear distances. (b)An excitation laser beam is overlapped with a cloud of cold atoms. Rydbergexcitation out of the gas is suppressed in the vicinity of a Rydberg atom bythe interaction, resulting in many domains within which only one Rydbergatom is excited.

5.1.2 Dipole blockade in mesoscopic atomic ensembles

If we expose a cloud of atoms to laser light with band width ∆ν resonant to aRydberg transition, we will get mesoscopic domains of single Rydberg excitationsinside which the other atoms cannot be excited (see Figure 5.2(b)) due to theprocess described in the previous section. Figure 5.2(a) illustrates this excitationinhibition caused by interaction induced level shifts δ = u/~. The radius of thesedomains rD is defined by the condition that the laser bandwidth must be smallerthan the energy shift

u > h∆νlaser. (5.5)

[Lukin et al., 2001] have termed this excitation blocking ”dipole blockade”, re-sembling the term ”Coulomb blockade”, which is a similar process occurring insolid state devices like the single electron transistor [Altshuler et al., 1991]. The”Coulomb blockade” is the inhibition of electron tunneling in mesoscopic metalclusters if this cluster is already charged by one electron. The clusters must besub-micron sized, so that the capacitance C to the tunneling gates is so smallthat the one electron charging energy e2/C is larger than thermal energies.

[Lukin et al., 2001] have proposed to exploit the ”dipole blockade” to entangletwo mesoscopic domains in order to implement a phase gate. This has the bigadvantage that, unlike in the previous section, no single atom control is requiredwhich would be technologically quite difficult. [Lukin et al., 2001] propose usingcollective excitations inside a mesoscopic ensemble of atoms as storage states.Again, a pair of metastable states of the ground state serve as storage states |g〉

88

5.1. QUANTUM INFORMATION PROCESSING WITH RYDBERG ATOMS

and |e〉 and a Rydberg state |r〉 serves as an interacting state, but now collectivestates are considered. The collective ground state of N atoms is represented by∣∣gN

⟩= |g〉⊗ . . .

N⊗ |g〉 . (5.6)

The multiparticle state with n atoms in the state |e〉 and N − n in the groundstate |g〉 is represented by the properly symmetrized sum over possible singleexcitations∣∣en gN−n

⟩=

√(N − n)!/(N !n!)

∑permutations

|e〉⊗ . . .n⊗ |e〉 ⊗ |g〉⊗ . . .

N − n⊗ |g〉 (5.7)

A weak light field tuned to the |g〉 → |r〉 transition can excite the transitionfrom the collective ground state

∣∣gN⟩

to the state with one excited Rydbergatom

∣∣r1gN−1⟩. The interaction energy u between two Rydberg states shifts the

energy of multiple Rydberg excitations. If this shift is large enough, then multipleRydberg excitations are inhibited.

This process can be used to realize a quantum register, i.e., to excite ex-act numbers of atoms in the storage states. By applying to the

∣∣gN⟩

state amultiparticle π pulse tuned to the transition |g〉 → |r〉, we obtain a single Ryd-berg excitation

∣∣r1gN−1⟩. Note that double-excited Rydberg states

∣∣r2gN−2⟩

areshifted due to the interaction between Rydberg states and cannot be excited. Byapplying a second π pulse tuned to the transition |r〉 → |e〉, we can obtain stateswhere only one atom is excited in the second storage state

∣∣e1gN−1⟩. This can be

applied subsequently to increase the number of atoms in the |e〉 storage state.A phase gate can be implemented by means of two atom clouds separated

by a few µm. Using the same procedure as described in the section above theentanglement between these two mesoscopic ensembles can be established dueto the fact that a collective Rydberg state excited in one cloud can inhibit theexcitation of the second cloud. Two mesoscopic clouds are then described bye.g. the product state

∣∣gN⟩⊗

∣∣e1gN−1⟩. By using the same pulse sequences as

described in the preceding section, the following truth table is obtained∣∣e1gN−1⟩⊗

∣∣e1gN−1⟩→ −

∣∣e1gN−1⟩⊗

∣∣e1gN−1⟩

(5.8)∣∣e1gN−1⟩⊗

∣∣gN⟩→ −

∣∣e1gN−1⟩⊗

∣∣gN⟩∣∣gN

⟩⊗

∣∣e1gN−1⟩→ −

∣∣gN⟩⊗

∣∣e1gN−1⟩∣∣gN

⟩⊗

∣∣gN⟩→

∣∣gN⟩⊗

∣∣gN⟩.

Note that the truth table is equivalent to table 5.4. This can be seen by meansof the following replacements: |g〉 →

∣∣gN⟩, |e〉 →

∣∣e1gN−1⟩, (|r〉 →

∣∣r1gN−1⟩.

The implementation of [Lukin et al., 2001] uses Rydberg states which are cou-pled by resonant dipole-dipole interaction. This has the advantage that the in-teraction energy u scales with the radius proportional to R−3 (see Section 4.1).Other interaction types like the van der Waals interaction (u ∝ R−6) can be used

89


0 100 200 300 400 500 600 7000

1

2

3

4

5

6

Num

ber o

f Ryd

berg

ato

ms

[103 ]

Intensity [W/cm2]

Figure 5.3: On-resonance excitation of the 82S Rydberg state as a func-tion of excitation intensity. A strong saturation of on-resonance excitation isobserved. However, power saturation and depletion of atoms are entangledwith interaction-induced excitation inhibition.

in the same manner. The central process in realizing a phase gate with meso-scopic ensembles is the interaction induced excitation inhibition which we havestudied in the following sections.

5.2 Interaction-induced inhibition of excitation

First evidence for the ”dipole blockade” has recently been found by Tong et al.[Tong et al., 2004] as a function of laser intensity and density for pulsed laser exci-tation, and by our group [Singer et al., 2004a, Singer et al., 2004b] as a functionof intensity, density and excitation time for cw excitation. These observationsare the first step towards the realization of a quantum phase gate using Rydbergatoms.

5.2.1 Intensity dependence of excitation

We have performed studies of the intensity dependence of the 82S state (seeFigure 5.3) [Singer et al., 2004b]. The on-resonance excitation is extracted outof the intensity-dependent spectra presented in Section 4.3.1. A saturation ofexcitation is indeed observed. However, a significant portion of atoms in theexcitation volume is excited into Rydberg states. Therefore, saturation of thenumber of Rydberg atoms partly stems from depletion of atoms in the launch stateand partly from an interaction-induced excitation blockade. These two effectscannot be disentangled easily from the data plotted in Figure 5.3. Furthermore,

90

5.2. INTERACTION-INDUCED INHIBITION OF EXCITATION

Figure 5.4: MOT excitation fraction as a function of scaled irradiancefor different principal quantum numbers (measured by the group in Storrs[Tong et al., 2004]). The irradiance is scaled to account for the fact that theexcitation probability decreases with the effective principal quantum num-ber as (n∗)−3. Rydberg states are excited by a pulsed single photon process.The strong saturation especially at higher principal quantum numbers is at-tributed to interaction induced excitation inhibition. Power saturation anddepletion is excluded.

91


intensity scans have an upper limit for the excitation power applied to the atomsas power saturation has to be avoided. Therefore, we resort to density-dependentmeasurements for a more detailed investigation of excitation inhibition.

Similar saturation behavior is observed by [Tong et al., 2004]. As can beseen in Figure 5.4, with increasing effective principal quantum number n∗, theRydberg excitation saturates even though the excitation probability decreaseswith increasing n∗. The irradiance is scaled by (n∗)−3 to account for the decreasein excitation probability. The laser power was chosen weak enough in order toavoid any power saturation. In the isolated atom limit at low irradiances, allcurves are identical. However, at higher irradiances excitation to higher principalquantum numbers strongly saturates as the atomic transition is shifted out ofresonance due to the interaction. The saturation behavior is well described by amean field model using Bloch-like equations for the time-dependent amplitudesfor atoms in the ground state and for atoms in the Rydberg state. The interactionis included by shifting the energy level for the Rydberg state, which dependson the amount and distances of neighboring Rydberg atoms. These findingssupport the interpretation of our data as a local blockade caused by van derWaals interaction.

5.2.2 Density-dependence of excitation

Density-dependent measurements [Singer et al., 2004a] of the on-resonance ex-citation are performed by changing the amount of atoms in the launch state(5S1/2(F = 2)), as described in Section 4.3.2. These measurements have theadvantage that other saturation effects like power saturation or depletion of theatom cloud do not lead to a density-dependent saturation. Thus any observedexcitation inhibition must come from interaction effects. However, not only Ryd-berg atoms can interact with other Rydberg atoms. Ions also strongly interactwith Rydberg atoms. However, under our experimental conditions, the ion for-mation rate for n ∼ 80 states through black body radiation [Gallagher, 1994]is estimated to be approximately at 3kHz. Ionization rates due to collisions ofcold Rydberg atoms with residual hot Rydberg atoms are of the same order ofmagnitude under the assumption that an upper limit of 108 cm−3 for the den-sities of hot Rydberg atoms. This upper limit is obtained by taking the MOTdensity of 1010 cm−3 as an unreachable upper limit for the background atomswhile taking into account the fact that the excitation of these atoms are dopplerbroadened. We therefore do not expect significant effects of spontaneous creationof ions and avalanche processes during the timescales of excitation (∼ 20 µs)[Robinson et al., 2000]. Up to know, we have not yet studied at which rate ionsare produced by the interaction of cold Rydberg atoms through processes likePenning ionization.

Apart from the discussed line broadening in Section 4.3.2, we find suppressionof excitation on resonance. In Figure 5.5(b) we have plotted the peak density of

92


(a)

0 0.5x1010 1x10100.0

5.0x108

1.0x109

1.5x109

2.0x109

2.5x109

3.0x109

3.5x109

82S

Ryd

berg

pea

k de

nsity

[cm

-3]

Launch state peak density [cm-3]

62S

109 1010

107

108

109

(a3)

(a4)

(a2)

low power 82S

Ryd

berg

pea

k de

nsity

[cm

-3]

Launch state peak density [cm-3]

high power 82S

(a1)

(a4)

(a3)

(a2)

(a1)

(c)

(b)

-0.3 0.0 0.30.0

0.1

0.2

[GHz]

0.0

0.5

0

2

4

0

5

10

Num

ber o

f Ryd

berg

ato

ms

[103 ]

Figure 5.5: (a) Excitation spectra of the 82S1/2 state at different intensities

of the blue excitation laser [6 W/cm2 for (a1) and (a2), 500 W/cm2 for (a3)and (a4)] and launch state densities [(a1) 1.6×109 cm−3, (a2) 1.0×1010 cm−3,(a3) 1.5×109 cm−3, (a4) 1.0×1010 cm−3]. (b) Rydberg peak densities on the82S1/2 resonance versus launch state density for 6 W/cm2 () and 500 W/cm2

(♦). The dashed line is a linear extrapolation from the origin through thefirst data point. Data points corresponding to the spectra (a1) to (a4) aremarked. (c) Comparison of the density-dependent suppression of Rydbergexcitation for the 82S1/2 (same data as in (b)) and 62S1/2 (©) resonances.

The 62S1/2 data was taken at 350 W/cm2. The solid lines in (b) and (c)show fitted saturation functions (see text). The upper error bars are causedby assuming that the maximum measured signals fully saturate the MCP.We see that the 82S saturation is nearly not affected. For a more detaileddiscussion of the error bars see Section 3.2.3.

93


Rydberg atoms excited on resonance versus peak density ng of 5S1/2(F = 2)atoms for both the “low-power” and the “high-power” regimes. If no interactionis present, the peak density of Rydberg atoms scales linearly with ng. This isrealized in the “low power” regime (6 W/cm2), where a fit to the data yields

nRyd = p82,low n0.99(2)g with a probability p82,low = 0.006(1) for an atom to become

excited in a Rydberg state during the 20-µs excitation interval. In the “high-power” regime (500 W/cm2), however, the increase of Rydberg density scales lessthan linear with ng clearly shows a suppression of excitation. At high excitationrate and density, we detect only 104 Rydberg atoms, which is a factor of ∼ 2.7 lessthan expected from simple linear density scaling (see dashed line in Figure 5.5(b)).The data is fitted with a heuristic saturation function of the form

nRyd = p82,high ng/(1 +ng

nsat

) (5.9)

giving p82,high = 0.31(2) and nsat = 4.9(4) × 109 cm−3 ∗. The deviation ofp82,high/p82,low = 56 from the ratio of the blue excitation laser power of 83 in-dicates a slight power saturation of the Rydberg excitation. It is important tonote that power saturation can explain neither the asymmetric broadening of theexcitation lines nor the density-dependent saturation of the Rydberg excitation.Saturation caused by the MCP detection is accounted for in the calibration andlinearization procedure (see Section 3.2.3). Uncertainties are indicated by errorbars. Therefore, we attribute the suppression of the Rydberg excitation to inter-atomic interactions resulting in the onset of a dipole blockade caused by van derWaals interactions. The expected domain size is estimated at 15 µm from the vander Waals interaction potentials, assuming a line width of 10 MHz as obtainedfrom low power scans.

To further test this interpretation of a van der Waals blockade, we have com-pared the saturation for different principal quantum numbers n. The van derWaals interaction potential between two Rydberg atoms strongly increases withn. Therefore, one expects weaker suppression of Rydberg excitation for lineswith lower principal quantum number, although the transition strength of theselines is much higher as it scales with (n∗)−3 where n∗ denotes the principal quan-tum number corrected for the corresponding quantum defect [Gallagher, 1994].Figure 5.5(c) shows a comparison of resonantly excited Rydberg atoms on the82S1/2 line and the 62S1/2 line. The solid lines show fits of the saturationfunction (Equation 5.9). For the 62S1/2 line we find p62,high = 0.37(2) andnsat = 1.7(3) × 1010 cm−3 †. The saturation density of the 62S1/2 state is largerfactor 3.4 than the saturation density of the 82S1/2 state. Additionally, theasymptotic Rydberg density (prydnsat for ng nsat)) for 62S1/2 relative to 82S1/2

∗The value for nsat in [Singer et al., 2004a] is 4.1(4) × 109 cm−3. Revision of the MCPcalibration procedure led to a slight correction of this value.

†The value for nsat in [Singer et al., 2004a] is 9(1) × 109 cm−3. Revision of the MCP cali-bration procedure led to a slight correction of this value.

94


-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

-6-4-202468

1012141618

0 10 20 30 400

10

20

(b)

2 s8 s

14 s20 s26 s32 s

Ryd

berg

ato

ms

(103 )

(GHz)

(a)

38 s

Ryd

berg

ato

ms

(103 )

Excitation time ( s)

Figure 5.6: (a) Excitation spectrum of the 82S line for different durationsof the second excitation step driven by the blue laser. The graphs are offsetfor clarity, with the duration time indicated on the right of each spectrum.The dashed line indicates the peak resonance at which the values are takenfor the graph to the right. (b) Peak atom number for 82S as a function ofexcitation time. The curves show theoretical curves including induced andspontaneous emission. The solid line uses a fitted excitation rate, the dashedline uses an excitation rate extrapolated from low-power measurements. (Fora discussion of the error bars, see Section 3.2.3.)

is larger by a factor of 4. Both findings support the interpretation in terms ofan interaction-induced blockade. While the spectral broadening of the resonancelines (see Section 4.3.2) can only be understood as an “instantaneous” excitationof Rydberg pairs, it is not clear whether the density-dependent saturation occursin a similar way or whether it is induced by a process based on the successivecreation of Rydberg atoms. We have therefore performed time-dependent studiesof on-resonance Rydberg excitation.

5.2.3 Time dependence of excitation

Besides the smaller laser band width, cw excitation offers an additional advantageover pulsed excitation, namely the possibility of varying the time during whichthe atoms are exposed to the excitation laser light. Figure 5.9(b) show the peak

95


Figure 5.7: Effective three-level scheme for modelling the excitation process(for details, see text).

height of the 82S as a function of excitation time.

As can be seen from Figure 5.9(b), the saturation of the number of atoms inthe 82S state with increasing excitation time much weaker than in the previousintensity and density scans [Singer et al., 2004b]. This is even more amazing asthe excitation for the highest signals is a factor of two longer than in the previousscans. This can be explained by a redistribution of Rydberg excitation intostates that interact less with the initial state, as described in Section 4.3.3. Inorder to describe the slight saturation of Rydberg excitation through depletionof launch state atoms, we model the excitation process in the following way:The level structure of the atoms is reduced to an effective three-level system(|g〉, |r〉, |d〉) with the intermediate 5P3/2 state |g〉 coupled to the Rydberg state|r〉 which slowly decays into a state |d〉 (see Figure 5.7). The correspondingpopulations are Ng, Nr, and Nd. Different degeneracies of the states are neglectedhere. The rate of the excitation into the Rydberg state is given by Rgr Ng, andstimulated emission occurs at a rate Rrg Nr, where the single atom coupling rateRrg = Rgr = R is proportional to the laser intensity. Spontaneous emission from|r〉 occurs mainly to states energetically close to |r〉 (see [Gallagher, 1994]) whichtogether are represented by |d〉. Decay of |d〉 to |g〉 happens on a much longertimescale and is neglected here. We assume the first excitation step on the closedtransition 5S1/2(F = 2) → 5P3/2(F = 3) to be saturated. Then the dynamics of

96

5.3. EXCITATION OF DIPOLE-FORBIDDEN STATES

this system are described by the following set of equations

Ng =1

2(Ntotal −Nr −Nd) (5.10)

Nr = +RNg −RNr −γNr (5.11)

Nd = +γNr (5.12)

where Nr(t = 0) = Nd(t = 0) = 0 and γ representing the decay rate of the 82Sstate. Its value is taken from the literature as γ = 1/560µs+1/290µs = 1/191µs,the two terms in the sum representing spontaneous decay and black-body-induceddecay respectively[Gallagher, 1994]. The population of |g〉, corresponding to5P3/2, is assumed to be permanently refilled from 5S1/2 to 50% of the number ofatoms which are not state |r〉 or |d〉. Ntotal ' 105 is the total number of atoms inthe excitation volume. The only free parameter is the Rydberg excitation rate R.The solid line in Figure 5.9(b) shows a fit to the data. For comparison, the dottedline in Figure 5.9(b) shows the solution of the above differential equations usingthe excitation rate which we extrapolated from measurements at low laser inten-sity.∗ The discrepancy between the low-power extrapolation and the model fit tothe data may indicate that additional processes have to be included to describethe saturation of the Rydberg excitation. Indeed, we believe this to be another,albeit weak sign for Rydberg-Rydberg interactions resulting in an inhibition ofRydberg excitation.

5.3 Excitation of dipole-forbidden states

Dipole-forbidden states are excited due to a small residual electric field and showa totally different behavior of on-resonant excitation [Singer et al., 2004a]. Con-trary to the dipole-allowed S and D lines, the dipole-forbidden 81P1/2 peak growsmuch faster than linearly with the launch state density ng, as shown in Fig-

ure 5.8(b). A fit to the data of the 81P1/2 resonance yields a scaling with n2.4(1)g .

The behavior of the 81P3/2 peak is more complicated, but shows saturation rel-ative to the 81P1/2 peak, which may again be caused by an interaction-inducedblockade effect.

We have seen similar results during time scans with the dipole-forbidden 79`states with ` ≥3. The spectrum of these states becomes more prominent withincreasing excitation time (see Figure 5.9(a)) [Singer et al., 2004b]. The dashedlines indicates the frequencies of the atom number measurements depicted inFigure 5.9(b), which shows the excitation probability of the normally dipole-forbidden transition to the hydrogen-like 79` (` ≥ 3) states as a function of

∗From these low-power measurements (excitation rate 550Hz, laser intensity 6W/cm2, spec-ified laser-bandwidth 2 MHz), we extract the transition matrix element µgr = 〈5P3/2|ez|82S1/2〉to be 3.4× 10−4a.u.

97


(b)

-0.3 0.0 0.3

0

1

2

[GHz]

0

1

2

(a4)

0

1

2

(a)

(a3)

(a2)

(a1)0 0.5x10

101x10

10

0

1x108

2x108

3x108

81P3/2

81P1/2

Launch state peak density [cm-3

]

Ryd

be

rg p

ea

k d

en

sity

[cm

-3]

(a4)

(a2)

(a1)

(a3)

0

1

2

Nu

mb

er

of

Ryd

be

rg a

tom

s [

10

3]

Figure 5.8: (a) Resonance line for the fine structure doublet 81P1/2 (left)

and 81P3/2 (right) at an excitation intensity of about 500 W/cm2 and differentlaunch state densities [(a1) 2.5 × 109 cm−3, (a2) 5.5 × 109 cm−3, (a3) 8.3 ×109 cm−3, (a4) 1.0× 1010 cm−3]. (b) Resonance peak densities of the 81P1/2

and the 81P3/2 line as a function of the launch state peak density. The solidline is a polynomial fit to the data. The data points corresponding to thegraphs depicted in (a) are marked.

98

5.3. EXCITATION OF DIPOLE-FORBIDDEN STATES

1 2 3-7

-6

-5

-4

-3

-2-1

0

1

2

3

4

0 10 20 30 400.0

0.5

1.0

1.5

(b)(a)

2 s

8 s

14 s

20 s

26 s

32 s

Ryd

berg

ato

ms

(103 )

(GHz)

(a)

38 s

Ryd

berg

ato

ms

(103 )

Excitation time ( s)

Figure 5.9: (a) Excitation spectrum of the hydrogen-like 79` (` ≥3) statesfor different durations of the second excitation step driven by the blue laser.The graphs are offset for clarity with the duration time indicated on the left ofeach spectrum. The dashed line indicates the frequencies of the atom numbermeasurements depicted in (b). (b) Peak atom number for 79` as a function ofexcitation time. The solid line shows a fit of a linear plus quadratic function.The dashed line depicts the linear term.

99


excitation time. If no interactions were present, this dipole-forbidden line shouldincrease linearly with excitation duration. As shown in Figure 5.9(c), the linecenter grows much faster than linearly. The data points up to 30µs are fitted bya linear plus quadratic curve, the linear part is indicated by the dashed line, andis related to excitation due to residual external electric fields. The strong increasein excitation for times longer than 30µs has not yet been clarified and may be anexperimental artefact or caused by additional processes like Penning ionization.

We attribute the quadratic growth of the Rydberg excitation in both densityand time scan to an increasing admixture of dipole-allowed states into the atomicstates due to the interaction between the Rydberg atoms. This quadratic scalingwith launch state density and time could indicate that permanent dipole-dipoleinteraction (∝ R−3) is involved. This perturbation would lead to an admixtureof dipole-allowed states scaling linearly to the perturbation, which means ∝ R−3.The square of the transition matrix elements is proportional to the excitationprobability which is therefore proportional to R−6. For small excitation proba-bilities, the density of Rydberg atoms nRyd ∝ R−3 should be a linear functionof the launch state density and excitation time, leading to a quadratic scaling ofthe excitation probability with launch state density and excitation time. In thisway, the excitation rate increases with growing number of Rydberg atoms, whilethe excitation due to the residual field (< 0.16V/cm) acts as a seed for this pro-cess and is responsible for linear growth of the signal (dashed line in Figure 5.9).Further experiments including state-selective detection of the Rydberg atoms areneeded to clarify the origin of the quadratic growth of dipole-forbidden lines inthe presence of a small electric field.

100

Chapter 6

Conclusion and perspectives

We have described a compact experimental setup for the study of frozen Ryd-berg gases which combines state-of-the-art laser cooling and trapping techniqueswith two-photon cw excitation of high-lying Rydberg states. By qualitativelyanalyzing the dependence of the excitation spectra on the intensity, launch statedensity and the duration of the second excitation step, we find indications forlong-range interactions between Rydberg atoms. Two signatures of interactionsin a cold Rydberg gas are observed: a broadening of Rydberg resonance lines withadditional spectral features in the red wings and a suppression of on-resonanceexcitation which is interpreted as the onset of a van der Waals blockade. The lineshapes and dependencies of the excitation rates on the different parameters arenot yet quantitatively understood. We therefore require an appropriate model forthe line shapes based on the actual interaction potentials which we have calcu-lated for the asymptotic limit neglecting fine structure in a perturbative approach.We have also seen first indications of bound molecular states as there are manypeaks observed inside the red wings of Rydberg resonances which become moreprominent with increasing density. However, the positions of these resonancescannot yet be reproduced between subsequent scans, which might be due to thefact that the excitation laser is not yet stabilized.

Technological improvements One of the first technological improvements tobe implemented in our experiment is the stabilization of the excitation laser bylocking it to an ultra-precise reference cavity built out of ULE, a material witha very low thermal expansion coefficient, and located inside a vacuum tube tooptimize frequency stability. The tunability will be realized by means of a verywide range tunable acousto-optical modulator. This setup will enable measure-ments with much higher frequency accuracy. We hope to be able to reproduce thepositions of the small resonance peaks inside the red wings in order to performsystematic density-dependent studies. The analysis of the peak positions couldgive important information if molecular states are involved.

101

CHAPTER 6. CONCLUSION AND PERSPECTIVES

Rydberg excitation out of a 3D-optical lattice Once the excitation laseris stabilized, excitation of Rydberg states out of an optical lattice of 3D degener-ate Raman sideband cooled atoms becomes feasible. This could inhibit collisionprocesses between Rydberg atoms, because a minimal distance is guaranteed bythe optical lattice. Rydberg atoms localized in a 3D lattice could show interest-ing effects like the Mott insulator transition, as the delocalization of the Rydbergelectron can be controlled by the excitation to different electronic states.

Long range Rydberg-Rydberg molecular resonances The calculated longrange van der Waals interaction potentials for Rydberg atom pairs can give apossible explanation for the D* line in the n ∼ 80 spectra of Figure 3.12. We finda crossing between the 1Σ+

g -3Σ+u states of the dipole-allowed 82S-82S, 81D-81D

and 82S-81D interaction potentials, which perfectly matches the position of theresonance (see Figure 6.1). Because the 82S-82S, the 81D-81D and the 82S-81Dhave states with equal symmetry an avoided crossing can occur. This avoidedcrossing leads to a flattening of the potential curves at the crossing point. Theconsequence is that a larger range of internuclear distances can be accessed withthe same laser frequency.

Anisotropy of permanent dipole-dipole interaction Similar to experi-ments with resonant dipole-dipole interaction [Carroll et al., 2004], a small elec-tric field could be applied to induce permanent dipole moments in the Rydbergatoms. If the excitation laser is focused strong enough, it should be able toobserve the anisotropy of the interaction between two permanent dipoles. Thiswould be reflected in a more asymmetric line shape with decreasing waist of theexcitation laser.

Collective Rabi oscillations Currently, the density is not high enough for areal mesoscopic amount of atoms to be within an excitation-inhibited domain.The density can be improved by loading the atoms into a focused dipole trap,which can lead to a density increase of up to two orders of magnitude, therebyproceeding from the two-body limit to the full many-body regime of the dipoleblockade [Lukin et al., 2001]. One interesting aspect of the many-body regimeis the possibility to drive collective Rabi oscillations Ωcoll =

√NΩ with N the

amount of atoms involved and Ω the single atom Rabi frequency. Apart from highdensities, homogenous excitation conditions are required for the observation ofcollective Rabi oscillations. This means that the excitation volume has to containatoms at constant density and that it has to be exposed to spatially homogenousexcitation light. This is best realized using diffraction elements which generate atop-hat shaped beam profile. Rabi cycles with a two-photon excitation processcan be driven by means of two crossed beams where laser frequencies are detunedfrom the intermediate state [Teo et al., 2003].

102

Figure 6.1: Potential curves for the 1Σ+g -3Σ+

u states of the 82S-82S, 81D-81Dand 82S-81D asymptotes. We find a resonance of the Rydberg spectra (blacklines plotted with frequency as y-axis and Rydberg atom number as x-axis)exactly at the crossing point of the three interaction potentials (red circles).Neighboring resonances can be explained in a similar manner. Because allthree interaction potentials belong to the same symmetry, they can coupleand develop an avoided crossing. Due to the flatness of the potential curvesat the crossing point, this leads to an increase in excitation probability.

103

CHAPTER 6. CONCLUSION AND PERSPECTIVES

Full blockade If the excitation volume is small enough, then it might be pos-sible to observe a single domain where only one Rydberg atom is excitable. Bygoing to higher principal quantum numbers and minimizing the line width witha detuned two-photon excitation process, the domain size of single Rydberg ex-citation can be maximized.

Entangling two mesoscopic clouds Two mesoscopic clouds can be trappedinside a CO2 standing wave trap. The atom clouds is then about ∼ 5µm apart.The typical domain is larger than the spacing between two clouds, so that theblockading mechanism is expected to work between the two sub-ensembles.

104

Appendix A

Experimental control system

Each individual experiment is divided into a sequence of steps that are executedat a pre-programmed timing which is controlled by the timing computer. Themeasured values are obtained from absorption images taken during the sequenceor from other sensors controlled by the timing computer, like the MOT fluores-cence or control signals like the laser output power or the background pressure(see also [Folling, 2003]).

Timing computer “Brain332”

The timing computer system is a custom-made microcontroller design based onthe Motorola 68332 processor and a modified VME Bus (“BrainBus”) as back-plane. The MC68332 was used because of its internal Time Processor Unit (TPU)module which allows the generation of very flexible, high precision timings. Thecomplete system currently consists of four modules: The microcontroller board,an output board with 16 digital outputs and an analog output board with 8outputs. An input board provides two digital inputs and two trigger lines withSchmitt-trigger type comparator inputs.

Architecture

As underlying architecture, a VME bus system is chosen for its simplicity andavailability. A single-channel (96-pin, 16 Bit) VME backplane is used as BrainBus(see Figure A.1), the bus connecting the individual modules. The synchronousprotocol used for data transactions on the bus is a subset of the Motorola 68k-family processor bus protocol. This is much simpler than the complex VMEprotocol and has a far lower latency for individual accesses, which is a crucialfactor for timing applications. Furthermore, no bus arbitration is implemented -the controller board is the only bus master.

The rough structure of the pin layout on the Bus is unchanged with respectto the VME specification. The power, interrupt, data and address bus lines are

105

APPENDIX A. EXPERIMENTAL CONTROL SYSTEM

Pin Assignment for the VMEbus-Compatible P1/J1 Connector

Pin Row a Row b Row c

1 D00 TP8 D08

2 D01 TP7 D09

3 D02 TP6 D10

4 D03 BG0IN* D11

5 D04 BG0OUT* D12

6 D05 BG1IN* D13

7 D06 BG1OUT* D14

8 D07 BG2IN* D15

9 GND BG2OUT* GND

10 SYSCLK BG3IN* CS2*

11 GND BG3OUT* CS3*

12 DS1* TP5 SYSRESET*

13 DS0* TP4 CS4*

14 WRITE* TP3 CS5*

15 GND TP2 A23 / CS10*

16 DTACK* TP1 A22 / CS9*

17 GND TP0 A21 / CS8*

18 AS* CS0* A20 / CS7*

19 GND CS1* A19 / CS6*

20 TP10 GND A18

21 IACKIN* SERA A17

22 IACKOUT* SERB A16

23 TP9 GND A15

24 A07 IRQ7* A14

25 A06 IRQ6* A13

26 A05 IRQ5* A12

27 A04 IRQ4* A11

28 A03 IRQ3* A10

29 A02 IRQ2* A09

30 A01 IRQ1* A08

31 -12 VDC +5VSTBY +12 VDC

32 +5 VDC +5 VDC +5 VDC

Figure A.1: ExpBUS Connector Pin Assignment. ExpBUS is a modifiedVME-32 Bus using 68332 signals on a standard VME backplane. Note: (*):indicates active low signal. Shaded regions indicate new signals defined orredefined under ExpBUS, as compared to VME32.

106

12

34

56

78

ABCD

87

65

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D C B A

Title

Num

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Size A3

Dat

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Sep-

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_3.d

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raw

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:

BERR

70

BKPT

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LK56

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RXD

53

T2CL

K12

8

TSC

57

V D

DSY

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V S

TBY

19

XFC

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AD

DR0

90

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DR1

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031

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132

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DR1

233

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DR1

335

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DR1

436

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DR1

537

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DR1

638

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DR1

741

AD

DR1

842

AD

DR2

21

AD

DR2

3/CS

1012

5

AD

DR3

22

AD

DR4

23

AD

DR5

24

AD

DR6

25

AD

DR7

26

AD

DR8

27

AD

DR9

30

AS

82

BG/C

S111

4

CLK

OU

T66

CSBO

OT

112

D0

111

D1

110

D10

98

D11

97

D12

94

D13

93

D14

92

D15

91

D2

109

D3

108

D4

105

D5

104

D6

103

D7

102

D8

100

D9

99

FREE

ZE/ Q

UO

T58

IPIP

E/ D

SO54

PC0/

FC0/

CS3

118

PC1/

FC1/

CS4

119

PC2/

FC2/

CS5

120

PC3/

AD

DR1

9/CS

612

1

PC4/

AD

DR2

0/CS

712

2

PC5/

AD

DR2

1/CS

812

3

PC6/

AD

DR2

2/CS

912

4

R/ W

79

XTA

L60

BGA

CK/C

S211

5

BR/C

S011

3

HA

LT69

IFET

CH/ D

SI55

PE0/

DSA

CK0

89

PE1/

DSA

CK1

88

PE2/

AV

EC87

PE3/

RMC

86

PE5/

DS

85

PE6/

SIZ

081

PE7/

SIZ

180

PF0/

MO

DCL

K78

PF1/

IRQ

177

PF2/

IRQ

276

PF3/

IRQ

375

PF4/

IRQ

474

PF5/

IRQ

573

PF6/

IRQ

672

PF7/

IRQ

771

PQS0

/MIS

O43

PQS1

/MO

SI44

PQS2

/SCK

45

PQS3

/PCS

0/SS

46

PQS4

/PCS

147

PQS5

/PCS

248

PQS6

/PCS

349

PQS7

/ TX

D52

RESE

T68

TPU

CH0

16

TPU

CH1

15

TPU

CH10

4

TPU

CH11

3

TPU

CH12

132

TPU

CH13

131

TPU

CH14

130

TPU

CH15

129

TPU

CH2

14

TPU

CH3

13

TPU

CH4

12

TPU

CH5

11

TPU

CH6

10

TPU

CH7

9

TPU

CH8

6

TPU

CH9

5V DD18

V DD28

V DD39

V DD50

V DD63

V DD65

V DD84

V DD96

V DD107

V DD116

V SS 106

VDD7

VDD1

VDD126

V SS 29

V SS 34

V SS 40

V SS 51

V SS 59

V SS 83

V SS 95

V SS 101

V SS 67

V SS 17

V SS 2

V SS 8

V SS 117

V SS 127

U24

MC6

8332

ACF

C16(

132)

A23

A18

A17

A16

A15

A14

A13

A12

A11

A10

A9

A8

A7

A6

A5

A4

A3

A2

A1

A0

A22

A21

A20

A19

D0

D1

D2

D3

D4

D5

D6

D7

D8

D9

D10

D11

D12

D13

D14

D15

/IRQ

7/IR

Q6

/IRQ

5/IR

Q4

/IRQ

3/IR

Q2

/IRQ

1

/CS1

/CS3

/CS4

/CS5

/CS2

/CS0

TP1

TP2

TP3

TP4

TP5

TP6

TP7

TP8

TP9

TP10

TP11

TP12

TP13

TP14

TP15

EXTA

L

XTA

L

R3 10M

C6 22pF

C7 22pF

R4 330k

EXTA

L

XTA

L

CLK

OU

T

/AV

EC/R

MC

/DS

SIZ0

MO

DCL

K

R/W

/AS

12

34

56

78

910

ST2

STEC

K10

/BK

PT/B

ERR

RXD

/CSB

OO

T

FREE

ZE/IP

IPE

T2CL

KTS

CV

DD

SYN

VST

BYX

FC/H

ALT

/IFET

CH

/DSA

CK1

SIZ1

Refe

renc

e Cl

ock

Osc

illat

or (3

2.78

6 H

z)

Rese

t But

ton

Refe

renc

e Cl

ock

Pow

er a

nd P

LL F

ilter

BERG

-Inte

rface

VCC

/RES

ET

/MIS

O/M

OSI

/PCS

0

/DS

/RES

ET

/BER

R/B

KPT

FREE

ZE/IF

ETCH

/IPIP

EV

CC

VCC

VD

DSY

N

XFC

/RES

ET

C4 1uF

TXD

/PCS

3/P

CS2

/PCS

1

SCK

VCC

C3 1uF

C5 1uF

C21u

F

C11u

F

VCC

TXD

RXD

RS23

2-RX

D

RS23

2-TX

DRS23

2 In

terfa

ce

R1 IN

13

R2 IN

8

T1 IN

11

T2 IN

10

GND 15V+2

V-6

VCC16

R1 O

UT

12

R2 O

UT

9

T1 O

UT

14

T2 O

UT

7

C1+

1

C1 -

3C2

+4

C2 -

5

U29

MA

X23

2ACP

E(16

)

Y1

32.7

68 k

Hz

C9 100n

F

C8 100n

F

TP0

1 2

J1 RESE

T

GN

D

VCC

12

34

56

78

910

ST1

Ser.-

inte

rface

D00

1A

D01

2A

D02

3A

D03

4A

D04

5A

D05

6A

D06

7A

D07

8A

SYSC

LK10

A

/DS1

12A

/DS0

13A

/WRI

TE14

A

/DTA

CK16

A

/AS

18A

tp10

20A

/IACK

IN21

A

/IACK

OU

T22

A

tp9

23A

A07

24A

A06

25A

A05

26A

A04

27A

A03

28A

A02

29A

A01

30A

-12V

31A

+5V

32A

tp8

1B

tp7

2B

tp6

3B

/BG

0IN

4B

/BG

0OU

T5B

/BG

1IN

6B

/BG

1OU

T7B

/BG

2IN

8B

/BG

2OU

T9B

/BG

3IN

10B

/BG

3OU

T11

B

tp5

12B

tp4

13B

tp3

14B

tp2

15B

tp1

16B

tp0

17B

/cs0

18B

/cs1

19B

sera

21B

serb

22B

/IRQ

724

B/IR

Q6

25B

/IRQ

526

B/IR

Q4

27B

/IRQ

328

B/IR

Q2

29B

/IRQ

130

B

+5V

(Bat

t.)31

B+5

V32

B

D08

1C

D09

2C

D10

3C

D11

4C

D12

5C

D13

6C

D14

7C

D15

8C

/cs2

10C

/cs3

11C

/SY

SRES

ET12

C

/cs4

13C

/cs5

14C

A23

15C

A22

16C

A21

17C

A20

18C

A19

19C

A18

20C

A17

21C

A16

22C

A15

23C

A14

24C

A13

25C

A12

26C

A11

27C

A10

28C

A09

29C

A08

30C

-12V

31C

+5V

32C

VM

E2

VM

E96-

BRA

IN33

2

G1

1G

219

A1

2

Y1

18

A2

3

Y2

17

A3

4

Y3

16

A4

5

Y4

15

A5

6

Y5

14

A6

7

Y6

13

A7

8

Y7

12

A8

9

Y8

11

U16

SN74

LS54

1

A1

A2

A3

A4

A5

A6

A7

A8

A9

A10

A11

A12

A13

A14

A15

A16

A17

A18

A19

A20

A21

A22

A23

D0

D1

D2

D3

D4

D5

D6

D7

D8

D9

D10

D11

D12

D13

D14

D15

VD

0V

D1

VD

2V

D3

VD

4V

D5

VD

6V

D7

VD

8V

D9

VD

10V

D11

VD

12V

D13

VD

14V

D15

VD

0V

D1

VD

2V

D3

VD

4V

D5

VD

6V

D7

VD

8V

D9

VD

10V

D11

VD

12V

D13

VD

14V

D15

/IRQ

1/IR

Q2

/IRQ

3/IR

Q4

/IRQ

5/IR

Q6

/IRQ

7

VA

1V

A2

VA

3V

A4

VA

5V

A6

VA

7V

A8

VA

9V

A10

VA

11V

A12

VA

13V

A14

VA

15V

A16

VA

1V

A2

VA

3V

A4

VA

5V

A6

VA

7V

A8

VA

9V

A10

VA

11V

A12

VA

13V

A14

VA

15V

A16

VA

17V

A18

VA

19V

A20

VA

21V

A22

VA

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Figure A.2: Microcontroller board with the 68332.

107


identical to the VME standard. All signal lines used in the VME protocol for busarbitration and special transaction modes have been used to accommodate 10internal timing reference channels controlled by the TPU, a serial interface con-trolled by the CPU and chip-select lines which eliminate the need for individualchip-select logic circuits on the I/O-boards.

CPU and controller board

The Motorola MC68332 processor is an integrated microcontroller based on theMotorola 68020 32-bit CPU core (see Figure A.2). It allows for completely trans-parent 32-bit programming even though the external data bus is only 16 bitswide. The address bus is up to 24 bits wide, allowing for direct addressing of 16megabytes. The internal address space is 32 Bits. Integrated into the controlleris a two-channel serial interface, a background debugging interface, a timing pro-cessor unit (TPU) and a complete, configurable chip-select logic module. Theinternal chip-select logic allows for simple, direct connection of periphery compo-nents to the processor’s bus, without the need for additional circuitry for addressdecoding and implementing the bus protocol and timing. Therefore, extremelysimple designs are possible. The controller board is a complete computer systemcomprising the microcontroller as CPU, several megabytes of RAM and two se-rial interfaces. To connect it to the BrainBus, additional bus drivers are required.These are necessary because the backplane is terminated passively on both endsof the bus to avoid signal reflection. Each bus line has an electric load of 170Ω,requiring a bus driver capable of delivering 30mA. The microcontroller cannotdrive such high loads, so all signals to the bus have to be buffered.

A prototype of the controller board, which is still used in the experiment,uses a commercial evaluation board (Nohau Emul16/300-PC/BDM) containingan RS232 serial interface and 128kB of 8 bit wide RAM. It is mounted on a secondboard which contains the bus interface. In this configuration, the processor isoperated at 16 MHz.

In the next version of the controller board, which is already designed andawaits construction, the internal memory is addressed with the full data bus widthof 16 bits and the processor clock is increased to 20 MHz. This will increase thespeed of the system by almost 100%.

Pattern generation

To generate arbitrary bit patterns with precise timing, the internal TPU of themicrocontroller is used. This is a processor module specially designed for timingneeds. It has up to 16 timing channels which can be configured both as inputsand as outputs. In the output mode, the TPU channel is programmed to generatepulses at specified times relative to the CPU clock, which in turn is generated

108

Figure A.3: Host interface to PC. Note that diode D1 is added to theoriginal schematic proposed by Motorola to avoid reset problems because acoaxial cable to the computer acts like a capacitance and puts the processorin a dead loop.

by a quartz oscillator. The resolution is two clock cycles, so the maximal timingresolution is 0.25µs in the present setup, with quartz precision.

Timing pulses of one of these channels are used to trigger the output boards.Each output has an internal memory into which the next output value is storedbefore the board is triggered. This internal memory is accessed by the controllerboard via the BrainBus backplane. At the specified time, the TPU generatesa trigger pulse, causing the output boards to change their output state to thestored value.

This defines the maximal pattern generation rate: Before a new output pat-tern can be generated, all internal memories of the channels that are to be changedhave to be pre-loaded with the new output values. With the current controllerboard and the current software, this takes 30-50 µs, which is the time that mustpass between two changes of the output signals. The new controller board andfurther optimization of the software should bring this time down to 10-20 µs ifrequired. Even faster timing updates for digital signals could be achieved bydirectly using the TPU output channels.

User interface and host computer

The timing computer is controlled by a host computer (PC) which directly writesto the memory of the microcontroller via the background debugging interface ofthe MC68332, connected to the parallel port of the PC (see Figure A.3). Beforeeach individual run of the experiment, the host computer generates a timingtable which is sent to the controller and is executed by its “operating system”.

109


Figure A.4: Digital output board.

This operating software is stored into the microcontroller RAM before its firstuse after power-on (“booting”). By adding non-volatile memory, booting will bemade unnecessary. This will be the case with the new revision of the controllerboard.

The tables containing the output patterns and the timings are specified in agraphical user interface programmed in LabView (National Instruments). Theyare converted into the controller’s format and sent to the parallel interface byan external DLL programmed in C++. The current version of the user interfacecan address up to 1000 digital and analog channels, and can automatically scanthrough a parameter space of up to eight parameters.

External triggers

In order to be able to react to external events, the execution can be influencedby waiting for external triggers. For this, one of the free TPU channels is pro-grammed as an input channel. The TPU is programmed in such a way that thegeneration of the next output trigger pulse is linked to this input channel beingtriggered by an external signal. The link is created by programmable logic inthe TPU, so the only delay between the arrival of the external trigger and thereaction of the output is caused by the gate delays of this internal logic and thegate delays of the output boards. All of these are much below 1µs, and have sofar been negligible in all applications.

Output boards

The output channels of the system are on special output boards which are con-nected to the central bus. Two kinds of output boards have been designed: A

110

Figure A.5: Analog out board.

Figure A.6: Digital input board.

111


1 2 3 4 5 6

A

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654321

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Title

Number RevisionSize

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Date: 9-Sep-2004 Sheet of File: C:\_d\eigene dateien ki\electronic\Protel\Brain332main\68332triggerv1.0.ddbDrawn By:

D00 1AD01 2AD02 3AD03 4AD04 5AD05 6AD06 7AD07 8A

SYSCLK10A /DS112A /DS013A /WRITE14A /DTACK16A /AS18A

tp1020A

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tp923A

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A04 27A

A03 28A

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A01 30A

-15V31A

+5V 32A

tp81B tp72B tp63B

/BG0IN4B /BG0OUT5B /BG1IN6B /BG1OUT7B /BG2IN8B /BG2OUT9B /BG3IN10B /BG3OUT11B

tp512B tp413B tp314B tp215B tp116B tp017B

/cs0 18B

/cs1 19B

sera21B serb22B

/IRQ724B

/IRQ625B

/IRQ526B

/IRQ427B

/IRQ328B

/IRQ229B

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+5V(Batt.)31B

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D08 1CD09 2CD10 3CD11 4CD12 5CD13 6CD14 7CD15 8C

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Figure A.7: Trigger board.

112

digital board (see Figure A.4) which can generate two sets of 16 digital signalseach. Presently in our experiment, 16 channels are used by the experiment. Eachoutput uses a two-stage latch: The first is used as a buffer memory and is trig-gered by the write cycle on the BrainBus. It stores the value that was writtento it and forwards it to the input of the second latch, which is triggered by thetiming pulse from the TPU to change its output state to the new values anddrives the outputs.

The second board is an analog output board (Figure A.5) with eight channels.Each channel has an output range of ±10V and a resolution of 16 bits, resultingin a voltage resolution of 0.3mV. As with the digital board, if the output value isto be updated, the new value is first written into a buffer in the digital-to-analogconverter (BB DAC 712P). Only when the trigger pulse from the TPU arrives,is the output signal actually updated.

Since the minimal time between channel updates is 30µs, the maximum up-date frequency for analog channels is 20kHz. It was therefore not very usefulas signal generator. To generate shaped pulses with a high bandwidth, an ex-ternal arbitrary function generator is pre-programmed with the pulse shape andtriggered at the desired time by a pulse from one of the digital output channels.

Trigger inputs

An additional board provides input channels for the computer (see Figure A.7).This board has so far mainly been used for triggering the timing sequence whena measured signal crossed a threshold. Usually, the fluorescence light from theMOT is monitored and the timing sequence is stopped during MOT loading, tobe continued when a certain fluorescence level is reached. This ensures that thenumber of atoms in subsequent experiments is approximately constant, allowingfor comparable measurements.

To this avail, the trigger board has two channels with two analog inputs each.The two inputs are compared with a Schmitt-trigger type comparator, and theresult of the comparison is converted to a TTL signal which drives a TPU inputchannel. Typically, one of the comparator inputs is connected to an analog outputchannel, which makes the threshold value programmable.

113


114

Appendix B

Detailed expressions for theinteraction potentials

C coefficients for the ns-ns asymptote

The energy terms from first order perturbation are all zero. The second orderterms give for the 1Σ+

g and 3Σ+u state of the ns-ns asymptote (see Eq. (9,11,15)

of chapter 2.2 for definitions)

C6 = −⟨

n00n00

∣∣W 1111

∣∣n00n00

⟩(B.1)

C8 = −2⟨

n00n00

∣∣W 1212

∣∣n00n00

⟩(B.2)

C10 = −2⟨

n00n00

∣∣W 1313

∣∣n00n00

⟩−

⟨n00n00

∣∣W 2222

∣∣n00n00

⟩(B.3)

C coefficients for the np-np asymptote

First order perturbation theory reveals information about the C5 coefficients. TheC6 and C8 terms are obtained from second order perturbation.

• Case M=2: The first order term for the 1∆g and 3∆u state of the np-npasymptote gives

C5 = − 6

25〈n1| r2 |n1〉2 . (B.4)

The second order terms are

C6 = −⟨

n11n11

∣∣W 1111

∣∣n11n11

⟩(B.5)

C8 = −4⟨

n11n11

∣∣W 1113

∣∣n11n11

⟩− 2

⟨n11n11

∣∣W 1212

∣∣n11n11

⟩(B.6)

• Case M=1: For the 1Πu and 3Πg state of the np-np asymptote for whichσp = −1 the first order terms are all zero

C5 = 0. (B.7)

115

APPENDIX B. DETAILED EXPRESSIONS FOR THE INTERACTIONPOTENTIALS

For the 1Πg and 3Πu state of the np-np assymptote for which σp = 1 thefirst order term gives

C5 =24

25〈n1| r2 |n1〉2 . (B.8)

The second order terms give

C6 = −⟨

n11n10

∣∣W 1111

∣∣n11n10

⟩− σp

⟨n11n10

∣∣W 1111

∣∣n10n11

⟩(B.9)

C8 = −2⟨

n11n10

∣∣W 1113

∣∣n11n10

⟩− 2

⟨n11n10

∣∣W 1131

∣∣n11n10

⟩−

⟨n11n10

∣∣W 1212

∣∣n11n10

⟩(B.10)

−⟨

n11n10

∣∣W 2121

∣∣n11n10

⟩−2σp

(⟨n11n10

∣∣W 1113

∣∣n10n11

⟩+

⟨n11n10

∣∣W 1131

∣∣n10n11

⟩+

⟨n11n10

∣∣W 1212

∣∣n10n11

⟩)• Case M=0: For the 1Σ−

u and 3Σ−g state of the np-np asymptote the first

order terms are all zero and consequently

C5 = 0. (B.11)


C6 = −⟨

n11n1−1

∣∣W 1111

∣∣n11n1−1

⟩+

⟨n11n1−1

∣∣W 1111

∣∣n1−1n11

⟩(B.12)

C8 = −4⟨

n11n1−1

∣∣W 1113

∣∣n11n1−1

⟩− 2

⟨n11n1−1

∣∣W 1212

∣∣n11n1−1

⟩+ 4

⟨n11n1−1

∣∣W 1113

∣∣n1−1n11

⟩+2

⟨n11n1−1

∣∣W 1212

∣∣n1−1n11

⟩(B.13)

Both 1Σ+g and 3Σ+

u states of the np-np asymptote have two different degen-erate asymptotic representations which are degenerate so that the matrixelements of V (r1, r1) has to be diagonalized:

1

R5

12

25〈n2| r2 |n2〉2

(1

√2√

2 2

)(B.14)

We obtain the new eigenstates (see Table 1). For the first degenerate statethe first order perturbation term gives

C5 = −36

25〈n1| r2 |n1〉 . (B.15)


C6 = −4

3

⟨n11n1−1

∣∣W 1111

∣∣n10n10

⟩− 2

3

⟨n10n10

∣∣W 1111

∣∣n10n10

⟩− 1

3

⟨n11n1−1

∣∣W 1111

∣∣n1−1n11

⟩−1

3

⟨n11n1−1

∣∣W 1111

∣∣n11n1−1

⟩(B.16)

C8 =8

3

(−

⟨n10n10

∣∣W 1113

∣∣n10n10

⟩−

⟨n11n1−1

∣∣W 1311

∣∣n10n10

⟩−

⟨n11n1−1

∣∣W 1212

∣∣n10n10

⟩(B.17)

−⟨

n11n1−1

∣∣W 1113

∣∣n10n10

⟩)+

4

3

(−

⟨n10n10

∣∣W 1212

∣∣n10n10

⟩−

⟨n11n1−1

∣∣W 1113

∣∣n1−1n11

⟩−

⟨n11n1−1

∣∣W 1113

∣∣n11n1−1

⟩)+

2

3

(−

⟨n11n1−1

∣∣W 1212

∣∣n1−1n11

⟩−

⟨n11n1−1

∣∣W 1212

∣∣n11n1−1

⟩)116

For the second degenerate state the first order perturbation term is zero

C5 = 0. (B.18)


C6 = +4

3

⟨n11n1−1

∣∣W 1111

∣∣n10n10

⟩− 2

3

⟨n11n1−1

∣∣W 1111

∣∣n1−1n11

⟩− 2

3

⟨n11n1−1

∣∣W 1111

∣∣n11n1−1

⟩−1

3

⟨n10n10

∣∣W 1111

∣∣n10n10

⟩(B.19)

C8 =8

3

(⟨n11n1−1

∣∣W 1311

∣∣n10n10

⟩+

⟨n11n1−1

∣∣W 1212

∣∣n10n10

⟩+

⟨n11n1−1

∣∣W 1113

∣∣n10n10

⟩(B.20)

−⟨

n11n1−1

∣∣W 1113

∣∣n1−1n11

⟩−

⟨n11n1−1

∣∣W 1113

∣∣n11n1−1

⟩)+

4

3

(−

⟨n10n10

∣∣W 1113

∣∣n10n10

⟩−

⟨n11n1−1

∣∣W 1212

∣∣n1−1n11

⟩−

⟨n11n1−1

∣∣W 1212

∣∣n11n1−1

⟩)+

2

3

(−

⟨n10n10

∣∣W 1212

∣∣n10n10

⟩)C coefficients for the nd-nd asymptote

The first non vanishing energy terms from first order perturbation is the C5 andC7 term. The C6 terms come from second order perturbation.

• Case M=4: The first order terms for the 1Γg and 3Γu state of the nd-ndasymptote give

C5 = −24

49〈n2| r2 |n2〉2 , (B.21)

C7 =20

49〈n2| r2 |n2〉〈n2| r4 |n2〉 . (B.22)

The second order term gives

C6 = −⟨

n22n22

∣∣W 1111

∣∣n22n22

⟩(B.23)

• Case M=3: The first order terms for the 1Φu and 3Φg state of the nd-ndasymptote for which σp = −1 give

C5 = −12

49〈n2| r2 |n2〉2 (B.24)

C7 = − 5

49〈n2| r2 |n2〉〈n2| r4 |n2〉 . (B.25)

The second order term gives:

C6 =⟨

n21n22

∣∣W 1111

∣∣n22n21

⟩−

⟨n21n22

∣∣W 1111

∣∣n21n22

⟩(B.26)

117


The first order terms for the 1Φg and 3Φu state of the nd-nd asymptote forwhich σp = 1 give

C5 =36

49〈n2| r2 |n2〉2 , (B.27)

C7 = −85

49〈n2| r2 |n2〉〈n2| r4 |n2〉 . (B.28)

The second order terms is

C6 = −⟨

n22n21

∣∣W 1111

∣∣n22n21

⟩+

⟨n22n21

∣∣W 1111

∣∣n21n22

⟩(B.29)

• Case M=2: The first order terms for the 1∆u and 3∆g state of the nd-ndasymptote for which σp = −1 give

C5 =4

7〈n2| r2 |n2〉2 , (B.30)

C7 =30

49〈n2| r2 |n2〉〈n2| r4 |n2〉 . (B.31)

The second order terms is:

C6 = −⟨

n22n20

∣∣W 1111

∣∣n22n20

⟩+

⟨n22n20

∣∣W 1111

∣∣n20n22

⟩(B.32)

Both 1∆g and 3∆u states of the nd-nd asymptote have two different degen-erate asymptotic representations which are degenerate so that the matrixof V (r1, r1) has to be diagonalized:

1

R5

1

49〈n1| r2 |n1〉2

(20 8

√3

8√

3 −6

)(B.33)

We obtain the new eigenstates (see Table 1). For the first degenerate statethe first order perturbation terms give

C5 =26

49〈n2| r2 |n2〉2 , (B.34)

C7 =2040

931〈n2| r2 |n2〉〈n2| r4 |n2〉 . (B.35)


C6 = −8√

6

19

⟨n22n20

∣∣W 1111

∣∣n21n21

⟩+

16

19

(−

⟨n22n20

∣∣W 1111

∣∣n20n22

⟩−

⟨n22n20

∣∣W 1111

∣∣n22n20

⟩)− 3

19

⟨n21n21

∣∣W 1111

∣∣n21n21

⟩(B.36)

118

For the second degenerate state the first order perturbation terms give

C5 = −12

49〈n2| r2 |n2〉2 , (B.37)

C7 =50

931〈n2| r2 |n2〉〈n2| r4 |n2〉 . (B.38)


C6 =8√

6

19

⟨n22n20

∣∣W 1111

∣∣n21n21

⟩− 16

19

⟨n21n21

∣∣W 1111

∣∣n21n21

⟩+

3

19

(−

⟨n22n20

∣∣W 1111

∣∣n20n22

⟩−

⟨n22n20

∣∣W 1111

∣∣n22n20

⟩)(B.39)

• Case M = 1: Both 1Πu and 3Πg states of the nd-nd asymptote for whichσp = −1 have two different degenerate asymptotic representations whichare degenerate so that the matrix of V (r1, r1) has to be diagonalized:

1

R5

1

49〈n1| r2 |n1〉2

(−16 −2

√6

−2√

6 12

)(B.40)


C5 = − 2

49

(1 +

√55

)〈n2| r2 |n2〉2 , (B.41)

C7 = −15(55 +√

55)

1078〈n2| r2 |n2〉〈n2| r4 |n2〉 . (B.42)


C6 =55 + 7

√55

110

(⟨n21n20

∣∣W 1111

∣∣n20n21

⟩−

⟨n21n20

∣∣W 1111

∣∣n21n20

⟩)(B.43)

+

√6

55

(⟨n21n20

∣∣W 1111

∣∣n2−1n22

⟩−

⟨n21n20

∣∣W 1111

∣∣n22n2−1

⟩)+

55− 7√

55

110

(⟨n22n2−1

∣∣W 1111

∣∣n2−1n22

⟩−

⟨n22n2−1

∣∣W 1111

∣∣n22n2−1

⟩)For the second degenerate state the first order perturbation terms give

C5 =2

49

(−1 +

√55

)〈n2| r2 |n2〉2 , (B.44)

C7 =15(−55 +

√55)

1078〈n2| r2 |n2〉〈n2| r4 |n2〉 . (B.45)

119



C6 =55 + 7

√55

110

(⟨n22n2−1

∣∣W 1111

∣∣n2−1n22

⟩−

⟨n22n2−1

∣∣W 1111

∣∣n22n2−1

⟩)(B.46)

+

√6

55

(−

⟨n21n20

∣∣W 1111

∣∣n2−1n22

⟩+

⟨n21n20

∣∣W 1111

∣∣n22n2−1

⟩)+

55− 7√

55

110

(⟨n21n20

∣∣W 1111

∣∣n20n21

⟩−

⟨n21n20

∣∣W 1111

∣∣n21n20

⟩)Both 1Πg and 3Πu states of the nd-nd asymptote for which σp = 1 havetwo different degenerate asymptotic representations which are degenerateso that the matrix of V (r1, r1) has to be diagonalized:

1

R5

1

49〈n1| r2 |n1〉2

(−8 −6

√6

−6√

6 12

)(B.47)


C5 =2

49

(1 +

√79

)〈n2| r2 |n2〉2 , (B.48)

C7 =5(79 + 37

√79)

7742〈n2| r2 |n2〉〈n2| r4 |n2〉 . (B.49)


C6 = 3

√6

79

(⟨n21n20

∣∣W 1111

∣∣n2−1n22

⟩+

⟨n21n20

∣∣W 1111

∣∣n22n2−1

⟩)(B.50)

+79 + 5

√79

158

(−

⟨n22n2−1

∣∣W 1111

∣∣n2−1n22

⟩−

⟨n22n2−1

∣∣W 1111

∣∣n22n2−1

⟩)+

79− 5√

79

158

(−

⟨n21n20

∣∣W 1111

∣∣n20n21

⟩−

⟨n21n20

∣∣W 1111

∣∣n21n20


C5 = −2− 2√

79

49〈n2| r2 |n2〉2 . (B.51)

C7 =5

98− 185

98√

79〈n2| r2 |n2〉〈n2| r4 |n2〉 . (B.52)

120


C6 = 3

√6

79

(−

⟨n21n20

∣∣W 1111

∣∣n2−1n22

⟩−

⟨n21n20

∣∣W 1111

∣∣n22n2−1

⟩)(B.53)

+79 + 5

√79

158

(−

⟨n21n20

∣∣W 1111

∣∣n20n21

⟩−

⟨n21n20

∣∣W 1111

∣∣n21n20

⟩)+

79− 5√

79

158

(−

⟨n22n2−1

∣∣W 1111

∣∣n2−1n22

⟩−

⟨n22n2−1

∣∣W 1111

∣∣n22n2−1

⟩)• Case M=0: Both 1Σ−

u and 3Σ−g states of the nd-nd asymptote for which

σp = −1 have two different degenerate asymptotic representations whichare degenerate so that the matrix of V (r1, r1) has to be diagonalized:

1

R5

1

49〈n1| r2 |n1〉2

(0 −24−24 −24

)(B.54)


C5 = −12

49

(1 +

√5)〈n2| r2 |n2〉2 , (B.55)

C7 =10

49(5 +

√5) 〈n2| r2 |n2〉〈n2| r4 |n2〉 . (B.56)


C6 =2√5

(⟨n21n2−1

∣∣W 1111

∣∣n2−2n22

⟩−

⟨n21n2−1

∣∣W 1111

∣∣n22n2−2

⟩)(B.57)

+2

5−√

5

(−

⟨n22n2−2

∣∣W 1111

∣∣n22n2−2

⟩+

⟨n22n2−2

∣∣W 1111

∣∣n2−2n22

⟩)+

5−√

5

10

(⟨n21n2−1

∣∣W 1111

∣∣n2−1n21

⟩−

⟨n21n2−1

∣∣W 1111

∣∣n21n2−1


C5 =12

49

(−1 +

√5)〈n2| r2 |n2〉2 , (B.58)

C7 = −10

49(−5 +

√5) 〈n2| r2 |n2〉〈n2| r4 |n2〉 . (B.59)


C6 =2√5

(−

⟨n21n2−1

∣∣W 1111

∣∣n2−2n22

⟩+

⟨n21n2−1

∣∣W 1111

∣∣n22n2−2

⟩)(B.60)

+3 +

√5

5 +√

5

(−

⟨n21n2−1

∣∣W 1111

∣∣n21n2−1

⟩+

⟨n21n2−1

∣∣W 1111

∣∣n2−1n21

⟩)+

5−√

5

10

(−

⟨n22n2−2

∣∣W 1111

∣∣n22n2−2

⟩+

⟨n22n2−2

∣∣W 1111

∣∣n2−2n22

⟩)121


Both 1Σ−g and 3Σ−

u states of the nd-nd asymptote for which σp = 1 havethree different degenerate asymptotic representations which are degenerateso that the matrix of V (r1, r1) has to be diagonalized:

1

R5

4

49〈n1| r2 |n1〉2

−3 −6 −√

2

−6 −6 −√

2

−√

2 −√

2 −6

For this matrix we only got numerical solutions (see Table 1). For the firstdegenerate state the first order perturbation terms give

C5 = −0.931688 〈n2| r2 |n2〉2 , (B.61)

C7 = 0.459786 〈n2| r2 |n2〉〈n2| r4 |n2〉 . (B.62)


C6 = 0.857832(−

⟨n21n2−1

∣∣W 1111

∣∣n2−2n22

⟩−

⟨n21n2−1

∣∣W 1111

∣∣n22n2−2

⟩)(B.63)

−0.717193⟨

n22n2−2

∣∣W 1111

∣∣n20n20

⟩− 0.567915

⟨n21n2−1

∣∣W 1111

∣∣n20n20

⟩+0.541658

(−

⟨n22n2−2

∣∣W 1111

∣∣n2−2n22

⟩−

⟨n22n2−2

∣∣W 1111

∣∣n22n2−2

⟩)+0.33964

(−

⟨n21n2−1

∣∣W 1111

∣∣n2−1n21

⟩−

⟨n21n2−1

∣∣W 1111

∣∣n21n2−1

⟩)−0.118702

⟨n20n20

∣∣W 1111

∣∣n20n20

⟩For the second degenerate state the first order perturbation terms give

C5 = −0.430986 〈n2| r2 |n2〉2 , (B.64)

C7 = −1.94514 〈n2| r2 |n2〉〈n2| r4 |n2〉 . (B.65)


C6 = −0.880201⟨

n20n20

∣∣W 1111

∣∣n20n20

⟩+ 0.774416

⟨n22n2−2

∣∣W 1111

∣∣n20n20

⟩(B.66)

+0.493816⟨

n21n2−1

∣∣W 1111

∣∣n20n20

⟩+ 0.108617

(−

⟨n21n2−1

∣∣W 1111

∣∣n2−2n22

⟩−

⟨n21n2−1

∣∣W 1111

∣∣n22n2−2

⟩)+ 0.0851681

(−

⟨n22n2−2

∣∣W 1111

∣∣n2−2n22

⟩−

⟨n22n2−2

∣∣W 1111

∣∣n22n2−2

⟩)+ 0.0346305

(−

⟨n21n2−1

∣∣W 1111

∣∣n2−1n21

⟩−

⟨n21n2−1

∣∣W 1111

∣∣n21n2−1

⟩)For the third degenerate state the first order perturbation terms give

C5 = 0.138184 〈n2| r2 |n2〉2 , (B.67)

C7 = −1.94514 〈n2| r2 |n2〉〈n2| r4 |n2〉 . (B.68)

The second order term gives:

C6 = 0.966449(⟨

n21n2−1

∣∣W 1111

∣∣n2−2n22

⟩+

⟨n21n2−1

∣∣W 1111

∣∣n22n2−2

⟩)(B.69)

+0.625729(−

⟨n21n2−1

∣∣W 1111

∣∣n2−1n21

⟩−

⟨n21n2−1

∣∣W 1111

∣∣n21n2−1

⟩)+0.373174

(−

⟨n22n2−2

∣∣W 1111

∣∣n2−2n22

⟩−

⟨n22n2−2

∣∣W 1111

∣∣n22n2−2

⟩)+0.0740983

⟨n21n2−1

∣∣W 1111

∣∣n20n20

⟩− 0.057223

⟨n22n2−2

∣∣W 1111

∣∣n20n20

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⟩122

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Acknowledgements

The work presented would have been impossible to realize and daily life wouldnot have been so enjoyable without the help and the friendship of the followingpeople to whom I want to express my gratitude.

My greatest thanks go to my advisor Professor Matthias Weidemuller. Iam very grateful that he gave me the possibility to work in his group. I greatlyappreciate all the support I received from him. He is a very innovative scientistand I learned a lot from him in various discussions. I consider him one of the fewpeople who can inspire others with his visions and ideas of physics, the universeand everything.

Big thanks for all the help go to my coworker Markus Reetz-Lamour withwhom I worked together nearly from the beginning of our project. Together, wesolved lots of problems and mastered difficult situations. In brief: He is a greatanalyst, a professional problem solver and a good friend.

I want to thank Thomas Amthor who joined our team last year. Thanksfor all the work especially with data analysis and paper writing. He is a realmulti talent being gifted in areas so different as physics, languages, music,electronics and so on. Thanks also for the fun time.

The other two people who worked on the project during their diploma thesiswere Michaela Tscherneck and Simon Folling. Thanks for the great workand the nice time, as well as for keeping in touch.

For the time at the Max-Planck-Institute in Heidelberg I want to thankProfessor Dirk Schwalm, our research director. I want to thank him for hisgenerous support and the nice atmosphere he created.

I want to thank Allard Mosk for revealing me the real secret of science:peace of mind. Whenever I want to solve problems and face difficulties, Iremember his words.

131

Special thanks go to the Storrs group whom I visited during my three monthDAAD research scholarship: I want to thank Professor Phil Gould for thegenerous invitation, for the inspiring discussions and for finding an appartmentfor me especially in a situation when housing was rare. My thanks go to ProfessorEd Eyler for fruitful discussions especially about important technical details.My thanks got to Professor Robin Cote and his coworker Jovica Stanojevicfor the discussions of theoretical issues and for the joint effort on the publicationon interaction potentials between Rydberg atoms.

For my nice stay in Brazil at the ICAP and Sao Carlos I want to thankProfessor Luis Gustavo Marcassa and Professor Vanderlei Bagnato. Mythanks go also to his coworker Gustavo Telles for showing me the Brazilianway of life.

My special thanks for solving all non physical problem and keeping overviewover all travel refunds and other complicated constellations go to the heart ofour group Helga Muller.

Without the support of all the people of the mechanical and electronicsworkshop and the staff in Heidelberg and Freiburg not a single step of theexperiment could have been realized. My special thanks go to Herr Mallinger, Oliver Koschorreck and Helga Krieger (Heidelberg) and Ulrich Personand Hartmut Gotz (Freiburg).

I want to thank Roland Wester for the collaboration and especially forbeing the major coordinator for the Heidelberg-Freiburg move.

For the nice cooperation and good atmosphere here in Freiburg I want tothank our group members: Johannes Deiglmayr, Judith Eng, JurgenEurisch, Ulrike Fruhling, Christian Giese, Stephan Kraft, Jorg Lange,Torsten Losekamm, Jochen Mikosch, Ulrich Poschinger, HelmutRathgen, Wenzel Salzmann, Peter Staanum, Christoph Strauß, ...

For the nice days in Heidelberg I want to thank former group members:Annabelle Blum, Achim Dahlbokum, Marc Eichhorn, Udo Eisen-barth, Bjorn Eike, Patrick Friedmann, Sandro Hannemann, DominickNiedenzu, Selim Jochim, Henning Moritz, Marcel Mudrich, MatthiasStaudt, ...

For the friendly reception and nice days in Storrs I want to thank: ShahidFarooqi, Phil Gee, Hey Won Clara Kim and Joe, Sulabha Krishnan,Mab Owen, Dave Tong, Matt Wright, Yanpeng Zhang, ...

For financial support I want to thank the Max-Planck-Institute in Heidelberg.The work was also supported in part by the Landesstiftung Baden-Wurttembergin the framework of the ”Quantum Information Processing” program and theDAAD-PROBRAL and DAAD-USA programs.

Thanks to Ernst Florin from the European Molecular Biology Laboratoryfor the fruitful collaboration.

Thanks to Helmut Singer for sending me the 68332 prototype board andall disclosed documentations.

Thanks to Juliane Schmitt for all the proofreading and for being a contextsensitive dictionary.

Thanks to my friends Kai Nickel and Selim Jochim for listening patientlywhen I tell them about my newest ideas and thanks also to Marcel Mudrichand Marc Eichhorn for inspiring me with crazy inventions.

My deepest gratitude goes to my parents Yoshiko and Wolfgang for theirsupport and belief in me. I thank my two sisters Julika and Marika forsupporting me during my thesis by enduring my musical creations.

And my deepest devotion goes to my girlfriend Aliki Bellou mon bijou ex-traordinaire. Thanks for supporting me and for staying with me numerous week-ends at the Max-Planck-Institute. And thanks for being quite patient when Ispent all day in the lab even during weekends and when I was not really talkativeon the phone. You are my special girl:)

Interacting Rydberg atoms - Heidelberg University

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