F65

Electromagnetically induced transparency(EIT)

Preface

Imagine you can make a black wall transparent just by shining a certain light onto that wall.In principle, this is what happens to an atomic medium when it experiences electromagneticallyinduced transparency (EIT). A light beam which is completely absorbed by an atomic mediumcan suddenly go through just by shining in another certain light beam. A feature that raises theidea of an optical transistor, which has recently been implemented with EIT for just one atom[1].Furthermore, the phenomenon of EIT is closely related to slow light, i.e. the reduction of thespeed of light in an atomic system. A light speed reduction down to 17 m

s was achieved this wayin an ultracold atomic gas [2]. Pursuing this method further, light can even be stopped, openingup the possibility of storing light in an atomic medium. If a quantum bit is encoded in thatlight, the atomic system can this way be made a quantum memory, as has been realized withsingle atoms not long ago [3]. EIT is the starting point for these developments that pave theway towards quantum communication as well as quantum computing.The subject of this lab course is the observation and the quantitative analysis of EIT in a sampleof Rubidium atoms. The preparation of EIT requires a Doppler-free saturation spectroscopy onRubidium as well as the frequency modulation of laser beams, which is realized with acousto-optical modulators. As an application, the EIT-signal will be employed as a magnetometer tomeasure Bohr’s magneton.On the practical side, the lab course includes the alignment and preparation of laser beams withopto-mechanical components as well as polarization optics, the handling of electronic devices andcomputer-aided analysis. On the theoretical side, the course aims at deepening the understandingof atom-light interaction, reviewing concepts like the atomic two-level system, Rabi oscillationsor Zeeman splitting.

Errors, suggestions, criticism or whatever kind of feedback on this manuscript or the lab courseis very welcome at EIT@matterwave.de. 1

1Last updated February 17, 2014

2

Contents

1. Introduction 5

2. Theory of EIT 72.1. The two-level atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2. Static description of EIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3. Dynamic description of EIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4. Experimental realization of an EIT-system . . . . . . . . . . . . . . . . . . . . . . 14

3. EIT-related phenomena 173.1. Slow light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2. Light storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4. Experimental setup 214.1. Laser beam generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2. Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.3. Frequency generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.4. EIT-cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5. Devices in detail 255.1. λ

2 -plate and λ4 -plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.2. Polarizing Beamsplitter (PBS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.3. Glan-Taylor polarizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.4. DFB laser diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.5. Optical isolator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.6. Anamorphic prism pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.7. Doppler-free saturation spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 275.8. Acoustooptical modulator (AOM) . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.9. Optical fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6. Course manual 31

A. Appendix 33A.1. Rubidium 85 D1 transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33A.2. Rubidium 87 D1 transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34A.3. Data sheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

B. Bibliography 37

3

1. Introduction

Consider a three level system as indicated in figure 1.1, with three eigenstates |1〉 , |2〉 and |3〉.Transitions |1〉 ↔ |3〉 and |2〉 ↔ |3〉 shall be allowed while the transition |1〉 ↔ |2〉 shall bedipole-forbidden. Due to its shape, this system is called Λ-System. It is irradiated by two laserbeams named probe beam and couple beam with angular frequencies ωp and ωc, respectively.

|1>

|2>

|3>

ωp

ωcprobe beam couple beam

dipole forbidden

Figure 1.1.: Schematic of a general Λ-System. The couple beam with angular frequency ωccouples eigenstates |2〉 and |3〉 while the probe beam with angular frequency ωpprobes the transition |1〉 ↔ |3〉.

If the couple beam is switched off while the frequency of the probe beam is varied one wouldexpect an absorption profile around the resonance frequency, as illustrated on the top of figure1.2.

atomic sample

PD

probe beam

atomic sample

PD

couple beam

probe beam

Figure 1.2.: First observation of EIT in Strontium vapor [4]. When the couple beam is switchedon, a tiny transparency window occurs in the absorption valley of the otherwiseopaque atomic medium.

5

If the couple beam is turned on a peculiar phenomenon occurs: A tiny transparency window opensaround the resonance frequency, reaching a transmission of nearly 50% in an otherwise opaquesurrounding (bottom of figure 1.2). This effect is called electromagnetically induced transparency(EIT) and was first observed in Strontium vapor by [4], whose results are shown in figure 1.2.Simply speaking, this means that a medium can be made transparent for a certain frequency byswitching on a couple beam.In the next chapter an attempt will be made to provide a theoretical understanding of this notvery intuitive EIT-phenomenon. On top of that, the equations for calculating the EIT-featuresare deduced.

6

2. Theory of EIT

To obtain a theoretical understanding of EIT, a quantum mechanical treatment of the three-level system is necessary. Such a treatment builts on the theory of the two-level system, whichshould have been subject of the quantum mechanics lecture. For memorization a brief recallingof the two-level system will be given in the following section, where the underlying conceptsare introduced. Based on that, the theory of EIT will first be described in a static way onlylooking at the eigenstates of the three-level system. For obtaining the dynamic features of EIT asobserved in the experimental course, the three-level system will then be treated dynamically.

2.1. The two-level atom

A detailed derivation of the two-level system can be found in for example [5, 6, 7, 8, 9].The approach presented here mainly follows [5] and [9].Consider an atom that has only a ground state |g〉 and an excited state |e〉. These are eigenstatesof the atomic Hamiltonian H0, i.e.

H0 |e〉 = ~ωe |e〉H0 |g〉 = ~ωg |g〉

The atom shall now be illuminated with laser light of the angular frequency ωl corresponding toa detuning δ = ωl − ω with ω = ωe − ωg.

|g>

|e> ћωe

ћωg

ωωl

Figure 2.1.: Scheme of the idealized two-level atom with ground state |g〉 and excited state |e〉that have energies ~ωg and ~ωe respectively. The two-level system is irradiated bylight with angular frequency ωl.

Assuming a classical light field, the time-evolution of the system can be described with theSchrödinger equation. However, including spontaneous emission requires to describe the systemby the density matrix

ρ =

(ρee ρegρge ρgg

)(2.1)

7

where ρee and ρgg are the probabilities to find the system in state |e〉 and |g〉, respectively. Theoff-diagonals ρeg and ρge determine the coherence between the two states.The time evolution of the density and thereby the time evolution of the state populations isobtained by the von Neuman equation

i~ρ = [H, ρ] (2.2)

where H = H0 + H′ denotes the Hamilton operator which is composed of the atomic Hamil-tonian H0 and the Hamiltonian H′ that describes the atom–light interaction. In the dipoleapproximation this Hamiltonian can be expressed as

H′ = −d ·E (2.3)

where d is the electric dipole operator and E the electric field produced by the laser at the locationof the atom. Evaluating the differential equation (2.2) and introducing the Rabi-frequency

Ω :=d ·E~

(2.4)

yields the so-called Optical Bloch Equations (OBE)

ρgg = γρee + i2 (Ω∗ρeg − Ωρge)

ρee = −γρee + i2 (Ωρge − Ω∗ρeg)

˙ρge = −γ2ρge + i

2 [Ω∗(ρee − ρgg)− 2δρge]

˙ρeg = −γ2ρeg + i

2 [Ω(ρgg − ρee) + 2δρeg]

(2.5)

where the first summand has been added to account for spontaneous emission1, that occurs atthe spontaneous decay rate γ. Furthermore, ρge = ρgee

iδt and ρeg = ρegeiδt have been introduced

and ∗ denotes the complex conjugate.The spontaneous decay rate γ is obtained from a fully quantized theory [5], where also theelectromagnetic field is treated quantum mechanically2. It is given by

γ =ω3e2d2

3πε0~c3(2.6)

Using conservation of populationρgg + ρee = 1 (2.7)

and optical coherenceρeg = ρ∗ge (2.8)

these equations can further be reduced to

ρeg = −(γ

2− iδ

)ρeg +

iωΩ

2w = −γw − i

(Ωρ∗eg − Ω∗ρeg

)+ γ

where w := ρgg − ρee denotes the difference in population.1this is a somewhat sloppy way of introducing the spontaneous emission. However, a more rigourous derivation[10] leads to the same result, but would exceed the scope of this manual.

2The correct expression for the spontaneous decay rate was already found by a phenomenological theory ofEinstein [5].

8

The temporal behaviour of the system can be found by direct numerical integration of the opticalBloch equations. The result for different detunings and Rabi-frequencies is shown in figure 2.2.

0 5 10 150

0.2

0.4

0.6

0.8

1

t in [1/Ω]

ρ ee

0 5 10 150

0.2

0.4

0.6

0.8

1

t in [1/Ω]

ρ ee

Ω = 3 γΩ = γ

δ = 3 Ωδ = 0

Figure 2.2.: Rabi oscillations for different detunings without spontaneous emission (left) andincluding spontaneous emission for different Rabi-frequencies (right).

Without spontaneous emission (i.e. γ = 0) the two-level system performs oscillations between theground and the excited state. In case of resonance (δ = 0) these oscillations are fully modulatedand occur at the Rabi-frequency Ω, whereas for δ 6= 0 the modulation of the oscillations decreasesand the frequency increases to

Ω′ =√δ2 + Ω2 . (2.9)

If spontaneous emission is included (i.e. γ 6= 0) one can distinguish between two different timedomains:

• For t 1γ the damping caused by spontaneous emission is negligible.

• For t 1γ a steady-state will be reached due to damping caused by spontaneous emission.

The steady-state population of the excited state ρee in the second case can be obtained byrequiring

ρeg = w = 0 (2.10)

which yields

ρee =s/2

1 + s+ 4δ2

γ2

(2.11)

where the saturation parameter

s :=2Ω2

γ2=

2|d|2|E|2

~γ2=

I

Is(2.12)

has been defined. Furthermore, the saturation intensity

Is :=~cε0γ2

4d2

(2.6)=

πhcγ

3λ3(2.13)

is introduced, while using that the intensity of the light field is related to the electric field via

I =1

2ε0c|E|2 . (2.14)

For the case of low saturation with s 1, the population is predominantly in the ground state,whereas in the case of high saturation, the population is equally distributed between ground and

9

excited state.From (2.11) the total scattering rate γs of light from the laser field is given by

γs = γρee =γ

2

s

1 + s+ 4δ2

γ2

(2.15)

which saturates for large s to the maximum scattering rate γ/2.Plotting the scattering rate as a function of detuning δ (figure 2.3) yields a Lorentzian-profilewith a FWHM of

γ′ = γ√

1 + s (2.16)

which is referred to as the power-broadened linewidth.

−10 −5 0 5 100

0.1

0.2

0.3

0.4

0.5

δ in [γ]

γ s in [γ

]

s=0.1s= 1s= 10s=100

Figure 2.3.: The scattering rate γs as a function of detuning δ for different saturation parameters is shown. For s > 1 the line profiles are substantially broadened.

10

2.2. Static description of EIT

Having worked through the two-level system, the three-level system can be approached. For atheoretical description of EIT, the simple lambda system in figure 1.1 shall be elaborated a bitfurther. To that objective the relevant parameters are added in figure 2.4.

|1>

|2>

|3>

ωcΩc

ωpΩp

∆c∆p

γ21

γ31

γ32probe beam couple beam

dipole forbidden

Figure 2.4.: Schematic of a Λ-System with Rabi frequencies Ωp and Ωc driven by probe and couplebeam respectively, spontaneous decay rates γij for the decay from level |i〉 to |j〉 andthe detunings ∆p and ∆c of the beams relative to the corresponding transition.

Ωp and Ωc denote the Rabi frequencies of the transitions driven by the probe-beam and thecouple-beam respectively, which are related to the corresponding beam intensities Ip and Ic by

Ω2p =

3γ31λ3

4π2~c· Ip Ω2

c =3γ32λ

3

4π2~c· Ic . (2.17)

The detunings of the probe and the couple frequency relative to the corresponding transitionfrequencies are

∆p = ω31 − ωp ∆c = ω32 − ωc (2.18)

where ω31 = ω3 − ω1 and ω32 = ω3 − ω2 denote the angular frequency differences between thecorresponding eigenstates. The spontaneous decay rates γ31 and γ32 account for the spontaneousdecay from state |3〉 to |1〉 and |2〉 respectively. Although the transition from |2〉 to |1〉 is dipoleforbidden there still might be a tiny probability for a spontaneous decay accounted for by γ21.The crucial point for EIT is that the presence of the probe and couple beam changes the Hamil-tonian, such that an interaction Hamiltonian Hint describing the interaction between the atomand the incoming light has to be added to the mere atomic Hamiltonian H0:

H = H0 +Hint (2.19)

The eigenstates ofH0, |1〉, |2〉 and |3〉, are in general no longer eigenstates of the total HamiltonianH. After having introduced the dipole approximation as well as the rotating wave approximation,the interaction Hamiltonian can be represented in a rotating frame by

Hint = −~2

0 0 Ωp

0 −2(∆p −∆c) Ωc

Ωp Ωc −2∆p

(2.20)

11

The eigenvalues of the interaction Hamiltonian Hint can be calculated to

~ω+ =~2

(∆p +

√∆2p + Ω2

p + Ω2c

)(2.21)

~ω− =~2

(∆p −

√∆2p + Ω2

p + Ω2c

)(2.22)

~ω0 = 0 . (2.23)

It is straightforward to verify that the corresponding eigenstates of the interaction HamiltonianHint are

|a+〉 = sin θ sinφ |1〉+ cos θ sinφ |2〉+ cosφ |3〉 (2.24)|a−〉 = sin θ cosφ |1〉+ cos θ cosφ |2〉 − sinφ |3〉 (2.25)|a0〉 = cos θ |1〉 − sin θ |2〉 (2.26)

where the mixing angles θ and φ with

tan θ =Ωp

Ωctanφ =

√Ω2p + Ω2

c√Ω2p + Ω2

c + ∆2p + ∆p

(2.27)

have been introduced. Now the state |a0〉 shall be considered in detail, especially the question isasked: how does |a0〉 couple to state |3〉 or in other words, what is the transition probability togo from state |a0〉 to state |3〉 when Hint is applied?

〈3|Hint |a0〉 = 0 (2.28)

This means, that |a0〉 does not interact with the light or in other words, a transition from |a0〉 to|3〉 is not possible. This is why |a0〉 is called a dark state. This provides a formal understanding ofthe electromagnetically-induced transparency described in the previous section: the couple-beamtransfers the atom into a dark state from which an interaction with the probe light is no longerpossible. Therefore, it appears to be transparent.Now consider the special case that Ωp Ωc, i.e. the couple beam is much stronger than theprobe beam. In this case

sin θ → 0 cos θ → 1 |a0〉 → |1〉 (2.29)

So the original state |1〉 becomes the dark state.

12

2.3. Dynamic description of EIT

With the static description above, the basic phenomenon of EIT could be explained. In order toderive the experimentally observed absorption profile, a dynamic description is needed that alsoincludes spontaneous emission (note that spontaneous emission has been neglected in the previoustreatment). As for the two-level system, this can be done with the density matrix formalism.Again, ρ denotes the density matrix, whose diagonal elements ρii represent the measurementprobability of state i, while the off-diagonal elements ρij are referred to as coherences, since theyprovide information about the relative phase of state i and j.The time evolution of the density matrix is given by

i~ρ = [Hint, ρ] . (2.30)

This equation does not account for spontaneous emission yet. Therefore decay terms have tobe added which occur at rates γ31 and γ32 for spontaneous emissions |3〉 → |1〉 and |3〉 → |2〉,respectively. This procedure yields a set of differential equations that can be solved for eachelement of the density matrix ρ.The final objective is to derive an expression for the transmission T which is given by Lambert-Beers Law

T (∆p) = e−a(∆p)L (2.31)

where a denotes the absorption coefficient and L the length of the atomic sample. The absorptioncoefficient can directly be calculated from ρ13 which in turn is obtained by solving the abovedifferential equation. For a sufficiently weak probe field (Ωp γ31) and ∆c = 0, the absorptioncoefficient can be approximated by [11]

a(∆p) =nωpµ

213

ε0c~·

2γ21(γ21γ31 + Ω2c) + 8∆2

pγ31

(4∆2p − γ21γ31 − Ω2

c)2 + 4∆2

p(γ21 + γ31)2absorption coefficient (2.32)

where n denotes the atomic number density and µ13 the transition matrix dipole element.Assume that a probe beam shines through an atomic sample while scanning its frequency, i.e. itsdetuning ∆p relative to the resonance frequency ω31. The simplification shall be made, that allatoms have zero velocity, such that the doppler broadening can be neglected. In this case theresulting transmission, i.e. the signal one would expect on a photodiode behind the atomic samplecan be calculated with the above formula (2.32). If the couple beam is turned off, i.e. Ωc = 0,and if γ21 = 0, i.e. there is no ’leakage’ from state |2〉 to state |1〉, then (2.32) reduces to

a(∆p) =nωpµ

213

ε0c~· 2γ31

4∆2p + γ2

31

(2.33)

which is the well-known natural line profile with a full-width-at-half-maximum (FWHM) of γ31.The resulting transmission T (∆p) is plotted on the left in figure 2.3. If now the couple beam isswitched on, i.e. Ωc 6= 0 then a narrow transparency window emerges at ∆p = 0 (right hand sideof figure 2.3), which very much resembles the experimental result shown in the introduction. In[12] it is shown, that for Ωp,Ωc γ31, γ32 this profile is approximately the difference of theabove broad Lorentzian and a narrow Lorentzian with a FWHM of

γEIT =Ω2p + Ω2

c

γ31 + γ32. (2.34)

This implies the astonishing fact, that the linewidth of the EIT-signal can in principle be madearbitrarily small, far below the natural linewidth, just by decreasing the intensity of the probeand the couple beam.

13

−4 −2 0 2 40.4

0.6

0.8

1

probe detuning/γ31

tran

smis

sion

Ωc=0

−4 −2 0 2 40.4

0.6

0.8

1

probe detuning/γ31

tran

smis

sion

Ωc=0.5 ⋅γ

31

Figure 2.5.: Transmission calculated with formula (2.32) with and without couple beam switchedon. γ21 = 0.02 · γ31 for both plots. With the applied couple beam a narrow trans-mission window emerges at resonance frequency (right), which is the EIT signal.

For Ωc γ13 the transparency window becomes very large and two Lorentz-Profiles emerge thatare split by the Rabi-frequency Ωc. This phenomenon is referred to as Autler-Townes splittingand is conceptually different from EIT [13].As mentioned, the theoretical description was so far only valid for atoms with zero velocity, whichis nearly the case in an ultracold atomic gas. In a hot gas, as used in this experiment, Doppler-broadening has to be accounted for. However, it turns out that for co-propagating couple andprobe beam, narrow EIT-resonances can also be obtained in a hot gas [14].

2.4. Experimental realization of an EIT-system

So far the description of EIT was made for a general Λ-system, which for an experimentalrealization needs to be prepared in an atomic system. In this experiment the D1-line of 87Rb isemployed as indicated in figure 2.4.

87Rb has a nuclear spin of I = 3/2, therefore the lower 52S1/2 and the upper 52P1/2 each splitinto hyperfine-levels with F = I ± J = 1, 2. Each of these levels consists of 2F + 1 Zeeman-sublevels (which are degenerate in the absence of a magnetic field), as illustrated in figure 2.4for the case of the lower F = 2 and the upper F = 1 level.The question is, how a Λ-system can be realized in that particular atomic setting. One Λ-systemis for example realized by the three states marked in red. Since selection rules require

∆MF = ±1 or 0 (2.35)

for the magnetic quantum number, the |1〉 ↔ |2〉 transition is dipole forbidden, thus fulfilling theprerequisite for a Λ-system. Without the presence of a magnetic field, couple and probe beamhave the same frequency, but different polarizations recalling that ∆MF = +1 transitions areinduced by σ+ light while ∆MF = −1 transitions are induced by σ− light. When a magnetic

14

|1>

|3>

|2> |1>

|3>

|2>

∆ωz

B=0 B≠0

Figure 2.6.: The D1-line of 87Rb. On the left the hyperfine splitting is illustrated. The Zeeman-sublevels of the 52S1/2(F = 2) and the 52P1/2(F = 1) are degenerated for a vanishingmagnetic field (center) and split when a magnetic field is present (right). A Λ-systemcan for example be realized with the indicated Zeeman-sublevels.

field of strength B is applied, the magnetic sublevels split, i.e. the energy of a magnetic substatewith quantum number MF is shifted by the frequency

∆ωZ = MF gFµB/~B (2.36)

as indicated on the right in figure 2.4. Due to this relation, EIT can serve as a magnetic fieldsensor and in this experiment, Bohrs magneton will be measured this way.In the following some optical transition properties of the 87RB D1-line are listed:

wavelength λ 794.979 nmdecay rate /Natural line width (FWHM) γ = γ31 = γ32 2π · 5.746(8) MHz

transition dipole moment µ 2.537 · 10−29 C ·m

These values can be inserted into expression 2.32 using typical densities ranging from n =1016...1022 atoms

m3 . It is recommended to plot the resulting transmission for different values of Ωc,γ12 and n to get a feeling for the effect of the different parameters. Since the transition |1〉 ⇔ |2〉is dipole forbidden γ12 γ31 can be chosen or γ12 may even be set to zero.

15

3. EIT-related phenomena

EIT is the starting point for many further physics. As will be revealed during the practicalcourse, the narrow EIT-signal can be used as a magnetometer for measuring low magnetic fieldswith high precision ([15, 16, 17]). The property of having the control to let a probe beam passor not by switching on a couple beam raises the idea of an optical transistor. This techniquehas been scaled down by Muecke et al. ([1]) to just one single atom in a cavity acting as aquantum-optical transistor for single photons. In this section a brief glance is shed on how theEIT-resonance can be used for realizing slow light and light storage.

3.1. Slow light

In the theory section the focus laid on the absorptive features of EIT and therefore an expressionfor the absorption coefficient a was aimed at. It is related to the imaginary part of the refractiveindex n by

a = I(n) . (3.1)

The real part of the refractive index instead describes the dispersive properties, i.e. how lightpropagates through the medium. The group velocity of the light traveling through the medium

1

0 w0w

Re( w))n(

Im( w))n(Re

al-a

nd

ima

gin

ary

pa

rto

f th

ere

fra

ctiv

ein

de

x[a

.u.]

Figure 3.1.: Real and imaginary part for the refractive index as a function of angular frequency.At resonance where the absorption is strongest, the derivative of the real part of therefractive index also obtains its maximum, which leads to a small group velocity vgin the medium.

17

is related to the real part of the refractive index R(n) by:

vg ∝(dR(n)

dω

)−1

(3.2)

i.e. the group velocity is inversely proportional to the derivative of R(n) with respect to theangular frequency ω. This relation implies that the steeper the dispersion curve in a medium,the slower the light will propagate. Figure 3.1 shows the real and the imaginary part of therefractive index at an EIT-resonance. The steepness of the dispersion curve is related to thewidth of the EIT signal, i.e. the smaller the EIT-width the steeper the dispersion curve and theslower the light. More general it can be shown that

vg ∝Ω2c

N(3.3)

which implies that slower velocities can be achieved by either decreasing the intensity of thecouple beam (i.e. reducing the EIT-linewidth) or by increasing the atomic density N . Exactlythis is how Hau et al. [2] achieved to reduce the speed of light down to 17 m

s (figure 3.2), by goingto high atomic densities in an ultracold atomic gas, while inducing EIT with a couple beam oflow intensity. They observed a time delay in a light pulse propagating through an atomic sodiumsample, which at lowest temperature corresponds to a group velocity of vg ∼17 m

s .

Figure 3.2.: Slow light in an ultracold gas of sodium atoms observed by Hau et al. [2]. Theingoing reference pulse (open circles) is delayed by ∼7µs while traveling through the∼229µm long atomic cloud. The corresponding light speed is 32.5 m

s .

18

3.2. Light storage

The phenomenon of slow light raises the question, whether it is possible to make the lightstop entirely. Recalling equation (3.3) for the group velocity it results, that upon adiabaticallyramping down the couple beam power while the probe pulse is inside the atomic sample, the groupvelocity can actually be brought down to zero. The probe pulse remains spatially compressedinside the atomic cloud until the couple beam is switched on again. This effect was first observedby Liu et al. [18], whose results are shown in figure 3.3.

Figure 3.3.: Light storage observed by Liu et al. [18]. In a) the reference pulse (open circles)enters the atomic sample while the couple beam (dashed curve) is switched on. Adelay of the outgoing pulse is observed as explained in the previous section. In b) thecouple beam is adiabatically switched off when the incomig probe pulse is entirelyenclosed in the atomic sample. Upon switching on the couple beam the light pulsecomes out again. Storage times of more then a second were achieved this way.

Specht et al. even achieved to realize this method for coherently storinga single photon in a single atom [3]. This way they were able toimplement a single-atom quantum memory, by performing write andread operations of arbitrary polarization states of light into and out ofa single atom trapped inside an optical cavity.

19

4. Experimental setup

Figure 4.1.: Impression of the experimental setup in the laboratory

The aim of this advanced lab course is the observation and the quantitative analysis of EIT. TheΛ-system is generated in the D1-line of 87Rb, as described in section 2.4. In the following theexperimental setup shown in figure 4.2 will be explained. The employed devices are describedin detail in the following chapter. The experimental setup can be divided into the following 4parts:

optical2f breOsingle2mode2and2

polarization-maintainingM

Faraday2isolator

beam2shaper2

Rb-spectroscopycell

Iris

lensphoto2diode21

PBS

λ/2

DFB-Diode Laser

PBS

λ/2

λ/2

λ/2λ/2

PBS

PBS

λ/4

λ/4

λ/4

iris

iris

lens

lens

iris

iris

EIT-cellλ/4 λ/4

Glan-TaylorPolarizer

AOMs

curved2mirror

mirror

LASER RADIATION

photo2diode22

Figure 4.2.: Schematic of the Experimental setup for the observation of EIT

21

4.1. Laser beam generation

A DFB-diode laser provides a light beam at 795 nm which passes an optical isolator, that protectsthe laser from back-reflection which can cause frequency instabilities and in the worst casedamage. With the subsequent beam shaper the light beam, which is leaving the laser with anelliptical shape, is made circular. The beam shaper is a polarization dependent device, which iswhy a λ/2-plate is placed in front for aligning the polarization. This part of the experiment isalready aligned and must not be touched.

4.2. Spectroscopy

As described in section 2.4, the D1-line of 87Rb will be employed for EIT. Therefore the laserhas to be adjusted to the frequency of that line. In order to find it, Doppler-free saturationspectroscopy will be performed. For that reason, a fraction of the laser beam is diverted by theλ/2-plate and the PBS, together acting as a power splitter. The laser beam which has passedthe Rubidium sample twice than goes undeflected through the PBS, since it went through theλ/4-plate twice. Behind the PBS it is diverted onto a photodiode. By Scanning the laserfrequency (for example by scanning the lasercurrent) a Doppler-free spectrum of the D1-line canbe observed and the desired frequency positioned.

4.3. Frequency generation

For EIT a probe and a couple beam are required. As explained in section 2.4, they need to haveopposite polarizations and in the non-degenerated case (i.e. when a magnetic field is applied)also their frequencies are different. Furthermore, the frequency of the probe beam has to bescanned in order to observe the desired transparency profile.A convenient optical device standardly used to shift the frequency of a laser beam is an acousto-optical-modulator (AOM). Here, AOM’s in double-pass configuration are used to adjust thefrequencies of couple and probe beam. The frequency-tuned beams with orthogonal polarizationare subsequently combined in a polarizing beam splitter and finally coupled into an optical fiber.Optimal coupling into the fiber guarantees a perfect overlap of couple and probe beam as wellas a Gaussian beam leaving the other end of the fiber.

4.4. EIT-cell

The overlapping probe and couple beam, that are leaving the fiber have to be guided throughthe EIT-cell. The EIT-cell is similar to the spectroscopy cell, but optimized for optimal EITperformance. In order to increase the atomic density the Rubidium (which to 72% consists of85Rb) is enriched with 87Rb. Furthermore, the cell walls can be heated such that the rubidiumatoms do not condense there, which would lower the atomic density. Suppression of disturbingmagnetic fields is achieved by several layers of Mu-metal (a metal with high permeability) ascan be seen in figure 4.3. The Mu-metal can be demagnetized by a degaussing-technique withcoils that are wound around the Mu-metal layers. Controlled magnetic fields can be applied tothe cell via a coil, which is placed inside of the magnetic shielding. A λ/4-plate in front of thecell changes the linear polarization of the probe and the couple beam to circular polarization.

22

Behind the cell the polarization is again changed back to linear polarization such that the probebeam passes through the adjacent PBS while the couple beam is deflected. For a very efficientseparation of probe and couple beam, a Glan-Taylor polarizer is used. The probe beam is thenfocused onto a photodiode, where the EIT- signal can be observed.

Figure 4.3.: Opened EIT-cell. A glass cell very similar to the spectroscopy cell is filled withenriched 87Rb. The cell can be heated with resistance wires in order to increase theatomic density. The magnetic field is generated with a coil of 98mm length, 41mmdiameter and 167 windings. The cell is enclosed in several layers of Mu-metal formagnetic shielding. This shielding has to be demagnetized from time to time byapplying degaussing-currents to wires wound around the Mu-metal layers.

23

5. Devices in detail

In this section the devices employed in the experimental setup are described in more detail.However, only the properties relevant for this experiment will be explained. For a more detaileddiscussion for example [19] or [20] may be consulted.

5.1. λ2-plate and λ

4-plate

Linearly Polarized Light

Left HandedCircularly Polarized Light

optical axis

Lambda/4-plate

Lambda/2-plateA λ2 -plate is a birefringent crystal, which is cut such that the

optical (also called extraordinary or fast) axis is parallel to theentrance surface. Light polarized along this axis travels fasterthan light polarized along the perpendicular axis in the plane ofthe surface (slow or ordinary axis). Depending on the thickness ofthe crystal, light with polarization components along both axeswill emerge in a different polarization state. If the thickness ischosen, such that the crystal causes a phase shift between the twocomponents of λ/2, the linear polarization is turned by 90(topof right hand figure).If the thickness of the crystal is chosen, such that it causes aphase shift of λ/4, linearly polarized light is changed to ellipticallypolarized light and vice versa. In the special case that the anglebetween optical axis and incoming linear polarization is 45, linearpolarization is changed to circular polarization (bottom of righthand figure).

5.2. Polarizing Beamsplitter (PBS)

A polarizing beamsplitter (PBS) is used to separate or combine light with per-pendicular polarization components. The most common type consists of twoprism that are glued together with a dielectric coating in between, that hasa high reflectivity for vertically (or s-) polarized light and a low reflectivity forhorizontally (or p-) polarized light. Typically the transmission of the s-polarizedlight Ts is suppressed by a factor of ∼1000 with respect to the transmission ofthe p-polarized light Tp (i.e. Tp/Ts ∼ 1000), whereas for the reflected beam itsonly Ts/Tp ∼ 100. Due to the very thin layer of dielectric coating and the cubicdesign, the PBS only causes a negligible shift to the path of the transmitted beam.A PBS together with a λ/2-plate in front is standardly used as a powersplitter.

25

5.3. Glan-Taylor polarizer

air-gap

optical axes

ordinary ray

extraordinary ray

If a purer separation of the polarization is needed a Glan-Taylorpolarizer can be employed, which features an extinction ratio ofTp/Ts ∼ 100.000 for the transmitted beam (the reflected beam isnot that well polarized). In opposite to the PBS described above,the separation of the polarization components is the result of bire-fringency and total internal reflection. The Glan-Taylor polarizerconsists of two birefringent prisms that are separated by a tiny airslit (∼ 50µm). The optical axes of the two prisms lie horizontallyin the plane of the front surface. Hence, the p-polarized (extraor-dinary) beam and the s-polarized (ordinary) beam are travelingalong the same path, but with different velocities due the differ-ent refractive indices the two beams experience. The cutting angle of the two prisms is chosenaccording to the different refractive indices for s- and p-polarization, such that the s-polarizedlight experiences total internal reflection, while for the p-polarized component the angle is closeto the Brewster angle. The ordinary ray is therefore reflected, while the extraordinary ray istransmitted. Since the air gap is very tiny, the path of the transmitted beam remains nearlyunshifted. The reflected beam instead leaves the cube with an angle typically around 70.

5.4. DFB laser diode

cladding layer

cladding layer

cladding layer

cladding layer

polished

facet

polished

facet

Active layer

N

P

emitted

laserlight

drive current

n-electrode

p-electrode

cladding layer

cladding layer

cladding layer

cladding layer

AR-coatingHR-coating

Active layer

Bragg-grating

N

P

emitted

laserlight

drive current

n-electrode

p-electrode

Fabry-Perot

laser

Distributed-feedback

laser

In this experiment a DFB (Distributed feedback) laserdiode at 795nm is used. In usual Fabry-Perot diodes(top of right hand figure) the polished front surfaces ofthe diode build a cavity for frequency selection. Itslinewidth is typically on the order of several nm. ADFB diode instead has a diffraction grating within the ac-tive region as a frequency selective element, with whicha linewidth of only 1MHz to 5MHz is achieved whilebeing able to scan over a large frequency range (sev-eral 100 GHz). The frequency of the DFB-laser can bevaried by either changing the temperature or the cur-rent. In order to keep the frequency constant, thetemperature of the diode is stabilized by a peltier-element.

5.5. Optical isolator

The optical isolator is used to prevent backreflections into the laser,which can cause frequency instabilities and in the worst case damage.It consists of a polarizer (for example a PBS) that only lets througha certain polarization followed by a faraday rotator that changes thepolarization by 45 through the faraday effect. The subsequent polar-izer is also rotated by 45, such that light traveling in this direction

26

remains unaffected. However, light entering backwards will be polar-ized by the analyzer, turned to horizontal polarization by the Faradayrotator and then blocked by the polarizer. Hence, light can only passthe optical isolator in one direction.

5.6. Anamorphic prism pair

An elliptical beam can be shaped with an anamorphic prism pair,i.e. one axis can be expanded or compressed while the other axisstays unchanged. The prisms are usually oriented, such that theyare near Brewster condition to ensure maximum transmission.Therefore a λ/2-plate for aligning the polarization is placed infront.

5.7. Doppler-free saturation spectroscopy

If a laser is sent through an atomic sample with its laser frequency scanned around the atomicresonance frequency, the transmission signal behind is a Doppler-profile, which is caused by thevelocity distribution of the atoms (top of figure 5.1). The Doppler-width is typically on theorder of ∼ 1GHz which is much larger than the natural linewidth (typically on the order of∼ 10MHz). In order to resolve the natural lineprofile, Doppler-free saturation spectroscopy canbe performed.

0

1

PDspectroscopy cell

0

1

frequency

spectroscopy cell

mirrorλ/4

PD f0

Lamb dip

Figure 5.1.: Doppler-free saturation spectroscopy. In the setup shown on the top only theDoppler-broadened absorption profile can be observed on a photodiode. By passingthe spectroscopy twice with counterpropagating laser beams, as illustrated on thebottom, the natural line profile can be resolved.

Its idea is illustrated in figure 5.2. Without excitation, the atoms are populated in the groundstate according to a Maxwell-Boltzmann velocity distribution which in one dimension is a Gaus-sian curve. An atom with velocity ~v encountering a laser beam with wavevector ~k experiences a

27

laser frequency shifted by the Doppler-shift

∆fD = − 1

2π~k · ~v (5.1)

If the laser frequency fl is detuned from the atomic resonance frequency f0 by ∆f = fl − f0,then the frequency experienced by the atom is

f0 + ∆f −∆fD . (5.2)

If the detuning matches the Doppler-shift, i.e. if

∆f = − 1

2π~k · ~v (5.3)

then the atom is in resonance. If the detuning is negative, an atom traveling antiparallel to thelaser beam (i.e. ~k · ~v < 0) with a certain velocity |~v| is in resonance and will thus be transferredto the excited state. If the laser beam is reflected by a mirror as in figure 5.1, on the way backthe laser beam will be in resonance with atoms of opposite velocity −~v, since ~k has turned to −~k.If the laser beam is on resonance, both the incoming laser beam (also called saturation beam)and the reflected beam (probe beam) are in resonance with the same atoms of zero velocity. Dueto that, the probe beam experiences less absorption which becomes apparent as a little dip inthe Doppler-profile called Lamb-Dip (bottom of figure 5.1), which corresponds to the naturallineprofile (only power and pressure-broadened).

0

atoms in excited state

0velocity

atoms in ground state

incomingsaturation

beam

returningprobebeam

0

saturationbeam

probebeam

vk -k-vatom atom

atoms in excited state

atoms in ground state

0velocity

Figure 5.2.: Illustration of Doppler-free laser spectroscopy. On the left atoms with differentvelocities are addressed, which results in a strong absorption signal. On the rightthe same atoms with zero velocity are addressed, which results in a lower absorptionsignal, because the atoms are already excited by the incoming saturation beam.

Cross-over resonancesA special feature occurs for atomic systems with not just one but two excited states, whose energydifference is smaller than the Doppler-width. Performing Doppler-free saturation spectroscopyon such a system, one would expect an extended absorption-profile with 2 Lamb-dips (left offigure 5.3). Curiously, a third peak exactly in the middle of the two Lamb-Dips occurs.The mechanism leading to such a cross-over resonance is illustrated on the right in figure 5.3.

A saturation beam with frequency fl = f1 + f2−f12 will excite those atoms to state 1, that are

traveling in the same direction and have a velocity v that fulfills f2−f12 = − 1

2π~k ·~v. The returning

28

frequency

cross-over

absorption spectrum

f1

f2

f1+f

2

2

vatom

k

hfl

E2

E1

E0

h∆fD

-k

hfl

E2

E1

E0

h∆fD

saturation beam probe beam

Figure 5.3.: Cross-over resonance. Left) When performing Doppler-free saturation spectroscopyon an atomic system with two excited states, an additional Lamb-Dip appears in themiddle of the two atomic resonances. Right) Illustration of the mechanism leadingto a cross-over resonance.

probe beam is now resonant with the second transition of the same atoms. Since these atomsalready absorb photons from the saturation beam, the absorption of the probe beam is reduced,which results in an additional lamb dip, referred to as cross-over resonance.

5.8. Acoustooptical modulator (AOM)

Δf

Absorber

Piezo

fin

fin

+∆f

fin

Θ

Θ

In atom-optics an AOM is a standard device for imposing afrequency shift ∆f on a given laser beam of frequency fin. Itconsists of a piezo-element tapping against a crystal with thedriving frequency ∆f , inducing a sound wave in the crystalwhich acts as a diffractive grating. If the light enters withthe Bragg angle

sin(θB) =c ·∆f

2vsound · fin

where c is the speed of light and vsound the speed of sound in the crystal, then part of the light isdeflected. Regarding the sound wave in the crystal as phonons, that interact with the incomingphotons, energy conservation requires that the frequency of the outgoing diffracted laser beamis

fout = fin + ∆f .

As shown above, the angle of the deflected beam depends on the piezo-frequency ∆f . This impliesthat if a diffracted beam is used for subsequent beam alignment, ∆f cannot be changed anymorewithout destroying it. This problem is overcome by an AOM used in double-pass configuration.In this configuration a retroreflector is placed behind the AOM. This way, all beams leaving theAOM are sent back to where the came from independent of the diffraction angle (i.e. independentof ∆f). On their way back through the AOM they are again diffracted, such that they obtaintwice the frequency shift ∆f . Such a retroreflector can for example be realized by a curvedmirror which is placed at the distance of its radius behind the AOM as shown in figure 5.4. Sincethe active region of the AOM (in this experiment 1mm) is often smaller than the laser beam (inthis experiment 2mm), the beam is focused into the AOM by a lens. In order to obtain a lowdivergence of the beam at the entrance of the AOM, the focal length of the lens is usually chosenrather high (in this experiment 30 cm).

29

Figure 5.4.: Schematic of an AOM in double-pass configuration

5.9. Optical fiber

collimated

beam

lens

cladding

core

numerical aperture

of fiber

cladding

core

acceptance angle

α

multimode fibre (step-index)

singlemode fibre

An optical fiber serves as a waveguide for light. A multi-mode fiber consists of a core of typically 50 µm to more than1000 µm surrounded by a cladding whose refractive index n2

is smaller than the refractive index n1 of the core. Rays thatenter the fiber core with an angle below the acceptance an-gle are totally reflected within the core, whereas rays witha steeper angle are refracted into the surrounding cladding.Due to the large core, several modes can propagate throughthe fiber which leads to a wide and structured output beamprofile, depending on the input (top of right figure)1.A single-mode fiber has a much smaller core then a mul-timode fiber of a few µm, which is typically only a fewtimes bigger than the wavelength. Light can only propagatethrough the fiber in a single transverse mode, which leads toa nearly Gaussian beam profile at the output.Often a collimated laser beam has to be coupled into a fiber,which is typically done with a lens, that focuses the beamonto the core. The best coupling is achieved, if the conver-gence of the focused beam matches the acceptance angle and if the focused spot size matchesthe core-diameter of the fibre2. Therefore, the correct lens has to be chosen for each laser beamdiameter. If the laser beam is too large for the lens, it will enter the fibre with an angle largerthan its acceptance angle. If it is too small, the focused spot size will be bigger than the fibrecore.In this experiment a single-mode fiber is used to guarantee a perfect overlap of the probe- andcouple-beam and to obtain a Gaussian beam profile.

1in opposite to the step-index multimode fiber shown here, a gradient-index multimode fiber has a gradual profileof the refractive index to avoid modal dispersion.

2more precisely, the focused spot size has to match the mode field diameter(MFD) of the fibre, which is usuallyslightly bigger than the core.

30

6. Course manual

This section shall serve as a rough guideline through the experiment. Main practical information,explanation and help will be given by the tutor.

The different steps of the practical course go along with the different parts of the experiment asdescribed in the experimental setup.

1. Spectroscopy

The aim of this part is to obtain a spectroscopy of the D1-line of Rubidium, i.e. a graphabsorption vs. frequency where the absorption lines are visible. For that reason a Doppler-free saturation spectroscopy as described in section 5.7 has to be set up. The evaluationconsists in

• creating a comprehensive graph of the spectrum with a calibrated frequency axis

• addressing different spectroscopy features to the corresponding transitions

• assigning values of the laser-current to the different transitions and a rough calibrationlaser frequency vs. laser current

• relation (2.16) between the broadened linewidth and the intensity of the laser beamshall be measured. From the resulting fit, the saturation intensity can be deducedand compared with the literature value.

2. Double-pass AOM

A double-pass AOM is needed for the couple beam as well as for the probe beam, to scanthe frequency of the probe beam around the resonance frequency. The one for the couplebeam is already aligned, while the one for probe has to be set up.The AOM has a maximum diffraction efficiency of ∼90%. Thus, for the double-pass AOMa maximum efficiency of ∼80% can be achieved. Efficiencies around 65% are typicallyachieved in the current setup due to non-perfect laser beam properties and reflection lossesat each optical device. The final efficiencies of the alignment for probe and couple shouldbe recorded.

3. Fibre coupling

The couple beam is already coupled into the fibre, while the probe beam still needs to.With a well collimated Gaussian laser beam with a diameter that fits to the lens of thefibre coupler, more than 80% of the laser power can be transported through the fibre. Inthe current setup efficiencies of only 50% are typically achieved again due to non-perfectlaser beam properties. The final efficiencies of the alignment for probe and couple shouldbe recorded.

31

4. Observation of EIT

Once the previous parts have been accomplished, the main aligning work is done and itcan be proceeded to the actual experiment. Probe and couple beam are leaving the fibrecoupler superposed but with orthogonal linear polarization and need to be guided throughthe EIT cell. Behind the cell, the two beams have to be separated again with the Glan-Taylor Prism. The transmitted probe beam is imaged onto a photodiode. Optionally, thedeflected couple beam can also be monitored with another photodiode. The cell needs tobe heated to ∼60C for obtaining a reasonable EIT-signal.Some preparations have to be done:

• The λ/4 waveplates need to be aligned, such that the transmission of the probe beamthrough the Glan Taylor prism is maximal.

• The frequency of the laser has to be adjusted to the right frequency by the lasercurrent.

• The frequency scan of the probe beam needs to be set at the AOM-driver.

If the frequencies of the AOM are set correctly and if the double-pass AOM has beenadjusted to the first order, a beating signal should be seen on the photodiode due to theinterference of the probe beam and the remaining contribution of the couple beam. Byslightly varying the laser frequency and the power balance between probe and couple, anEIT signal (i.e. a transparency window in the transmission) should appear on the photo-diode. As described in the theory section, center frequency and width of the EIT-signaldepend on magnetic field and couple power, respectively. These two relations shall bemeasured. The evaluation should include the following:

• From relation (2.36) between center frequency and magnetic field strength Bohrsmagneton and offset-magnetic field can be determined quite precisely.

• It is quite hard to determine the width of the EIT-signal with the oscilloscope. Formore precise analysis the signals from the oscilloscope should be transferred to aUSB-stick and evaluated with a computer using relation (2.34).

32

A. Appendix

A.1. Rubidium 85 D1 transition

Figure A.1.: Hyperfine structure of D1 transition in Rubidium 85, with frequency splittings be-tween the hyperfine energy levels (taken from [21])

33

A.2. Rubidium 87 D1 transition

Figure A.2.: Hyperfine structure of D1 transition in Rubidium 87, with frequency splittings be-tween the hyperfine energy levels (taken from [22])

34

A.3. Data sheets

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35

B. Bibliography

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