Raman Spectroscopy of Degenerate Fermions

David Shirokoff

October 15, 2005

Contents

1 Theoretical Background 61.1 Fermi Statistics and the Semiclassical Limit . . . . . . . . . . . . 61.2 Calculations of Fermi Distributions . . . . . . . . . . . . . . . . . 81.3 Basic Theory of Raman Spectroscopy . . . . . . . . . . . . . . . 101.4 Raman Process: Calculation of Atomic Momentum Change . . . 11

2 The Raman Laser Apparatus 132.1 Design Parameters and Calculations . . . . . . . . . . . . . . . . 13

2.1.1 Laser Intensity: I . . . . . . . . . . . . . . . . . . . . . . . 132.1.2 The Laser Frequency: ωL . . . . . . . . . . . . . . . . . . 142.1.3 The Offset Frequency: ε . . . . . . . . . . . . . . . . . . . 15

2.2 The (Actual!) Laser Apparatus . . . . . . . . . . . . . . . . . . . 152.2.1 Efficiency, Power and Alignment . . . . . . . . . . . . . . 152.2.2 Output Polarizations . . . . . . . . . . . . . . . . . . . . . 172.2.3 Optical Transience . . . . . . . . . . . . . . . . . . . . . . 182.2.4 Modulation Frequency Response . . . . . . . . . . . . . . 18

3 The AOM Driver Circuits 243.1 Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2 Inputs and Outputs: . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.1 RF Power, Gain and Coupling . . . . . . . . . . . . . . . 243.2.2 DC Power Inputs . . . . . . . . . . . . . . . . . . . . . . . 253.2.3 Adwin and TTL: Voltage Controls from Command Central 25

3.3 The Driver Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3.1 RF Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3.2 DC Control of the RF Circuit . . . . . . . . . . . . . . . . 27

3.4 Start Up/Shut Down Procedure - What Not to Do! . . . . . . . . 27

1

List of Figures

1.1 Phase space density of states . . . . . . . . . . . . . . . . . . . . 71.2 Momentum distribution of cold Fermions . . . . . . . . . . . . . 91.3 Raman atomic energy transitions . . . . . . . . . . . . . . . . . . 11

2.1 Raman optics setup . . . . . . . . . . . . . . . . . . . . . . . . . 162.2 Raman optics apparatus . . . . . . . . . . . . . . . . . . . . . . . 172.3 Raman optics apparatus . . . . . . . . . . . . . . . . . . . . . . . 182.4 Polarization and energy transitions . . . . . . . . . . . . . . . . . 192.5 Laser response to 35 µs pulse . . . . . . . . . . . . . . . . . . . . 202.6 Laser response to 20 µs pulse . . . . . . . . . . . . . . . . . . . . 202.7 Laser response to 10 µs pulse . . . . . . . . . . . . . . . . . . . . 212.8 Laser pulse transience . . . . . . . . . . . . . . . . . . . . . . . . 212.9 Laser response to periodic pulse . . . . . . . . . . . . . . . . . . . 222.10 Frequency dependence of the AOMs . . . . . . . . . . . . . . . . 222.11 Frequency dependence of Raman apparatus . . . . . . . . . . . . 23

3.1 RF Input panel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 The AOM outputs . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3 The input/output panel . . . . . . . . . . . . . . . . . . . . . . . 273.4 AOM RF driver circuit . . . . . . . . . . . . . . . . . . . . . . . . 283.5 AOM RF driver circuit . . . . . . . . . . . . . . . . . . . . . . . . 293.6 RF Circuit diagram . . . . . . . . . . . . . . . . . . . . . . . . . 293.7 Output RF drive power vs. coupling . . . . . . . . . . . . . . . . 303.8 Output RF Drive voltage vs. coupling . . . . . . . . . . . . . . . 303.9 RF Output transience . . . . . . . . . . . . . . . . . . . . . . . . 313.10 TTL Switching transience . . . . . . . . . . . . . . . . . . . . . . 313.11 TTL Switching transience . . . . . . . . . . . . . . . . . . . . . . 323.12 DC Adwin control circuit . . . . . . . . . . . . . . . . . . . . . . 323.13 Variable voltage attenuator control circuit - transience . . . . . . 33

2

Acknowledgements

I would like to thank Professor Joseph Thywissen for supervising my work thissummer. I am indebted to Seth Aubin and Allan Stummer for their continuous,cheerful help and their feedback on the difficulties of the real world. In addi-tion, my summer work and learning experience would not have been so fruitfulwithout Stefan Myrskog, Marcius Extavour, Lindsay LeBlanc and Dave McKay.Lastly, thank you to the National Science and Research Council (NSERC) forfunding this work.

3

Introduction

There are two types of particles in our universe: Bosons and Fermions. Bothparticles appear to us in our daily lives. For example, photons, or particlesof light, are Bosons while electrons, protons and neutrons are Fermions. Thedistinction between the two forms of particles arises from quantum mechanicalsymmetry. The important point here is that for a system of Fermions, themany-body wavefunction must be antisymmetric under the swapping of twoparticles, i.e. Suppose the wavefunction ψ(x1, x2) describes a pair of Fermions,then one requires: ψ(x1, x2) = −ψ(x2, x1). Stated alternatively, the symmetryrestriction, known as the Pauli Exclusion Principle, prohibits two Fermions fromoccupying the same quantum state. In contrast, the other fundamental particle,Bosons, have a symmetric wavefunction under particle exchange and thereforedo not satisfy the exclusion principle.

At the University of Toronto’s Ultra-Cold Atoms Lab, current research con-sists of cooling a collection of Fermions to very low energies. Upon cooling, thebehaviour of such a collection gradually undergoes a transition from a classical,thermal distribution to a quantum mechanical, Fermi-Dirac one. As the nameof the research group implies, the Fermions we work with are not the com-monly thought of electron or proton, but rather composite Fermionic particles:potassium-40 atoms or 40K. A collection of Fermion atoms exhibiting quantumstatistics is known as a degenerate Fermi gas or DFG. In contrast to electrons,entire atoms have a larger mass and therefore a smaller DeBroglie wavelength.Consequently, to construct a DFG one requires extremely low temperatures.Despite the long standing theoretical prediction of such a gas [2, 1]1 and itsBosonic cousin [4, 3], a Bose-Einstein condensate, only recently [5, 6] have therebeen physical manifestations in the lab.

As the temperature of a gas of fermions decreases, the statistical distributionof energy changes from a classical Gaussian to a quantum mechanical one. Theheart of the deviation from classical physics lies in the Pauli exclusion principle.As the energy of a collection of atoms decreases, one may think semiclassically,that no two atoms may occupy the same quantum state. Hence, the collectionof atoms must start to fill in the ladder of allowable energy states. Specifically,as T → 0K the atoms fill all energy levels below some characteristic energy, theFermi energy: Ef .

1References are for Fermi-Dirac statistics

4

One may gather information about a system of Fermions, such as the tem-perature and subsequently the degree to which they exhibit quantum degener-acy, by measuring the momentum distribution. By targeting individual atomswith a pre-chosen momentum, Raman spectroscopy offers a precise method formeasuring the momentum distribution of such a gas. My summer researchproject consists of the design and construction of a Raman spectroscopy tool.Moreover, I incorporate theoretical considerations of the design parameters andRaman operation. The report is composed as follows: I introduce a theoreticalbackground, specifically on the statistics of a Fermi gas. Secondly, I include adescription of the Raman laser apparatus and conclude with a description ofthe complementary electrical components.

5

Chapter 1

Theoretical Background

1.1 Fermi Statistics and the Semiclassical Limit

The Pauli exclusion principle states that no two Fermions may occupy the samequantum state. Application of the principle, made independently by Fermi [2]and Dirac [1] to a collection of Fermions satisfying Boltzmann statistics resultsin a new probability function:

F(ε) =1

1 + eε−µkT

(1.1)

where µ(T,N) is the chemical potential and N is the number of Fermions in thesystem. Often µ is replaced in (1.1) with fugacity, defined by: ξ = e

µKT .

For systems with a large number of particles, we wish to throw out themanybody wavefunction and implement a statistical treatment. In our system,we have spin-polarized atoms in a magnetic trap1. The standard treatment isto introduce a density of states function, g(ε), that represents the density ofenergy states at each energy2. We now seek to change the focus from energy, ε,to observable quantities, momentum ~p and position ~r. In this semiclassical limit,one converts the density of energy states to a phase space density D(~r, ~p). Here,one interprets D(~r, ~p)dr3dp3 as the number of quantum states in a phase spacevolume dr3dp3 at location (~r, ~p). Miraculously, for sufficiently large energies,D(~r, ~p) = ( 1

2πh )3, which is completely independent of the Hamiltonian potentialand density of energy states g(ε).

To convert from a density of energy states g(ε) to a phase space densityof states D(~r, ~p), first consider some arbitrary attractive potential V (~r). Inphase space, surfaces of constant energy form closed shells3. One may think ofsmearing the number of quantum states between ε and ε + dε (i.e. g(ε)) into

1All atoms have the same spin, hence we have no degeneracy in energy states due to variousspins.

2g(ε) is the growth in the number of energy states, n : dndε

= g(ε)3particles are in a bounded potential V , thus ~r or ~p cannot →∞

6

Phase Space

Figure 1.1: Phase space plot with trajectories at energies ε and ε + dε. Thephase space density of states at D(~r0, ~p0), with H(~r0, ~p0) = ε, is the number ofstates, g(ε), divided throughout the differential phase space volume as shown.In the large energy limit, D = (2πh)−3, a constant independent of energy.

the volume of phase space found between the closed surfaces of energy at ε andε+ dε (see Fig. 1.1). The ratio of the two quantities is the phase space densityof states D(~r0, ~p0) = (2πh)−3 4. The ratio assumes that the locus of phase spacestates (~r, ~p) satisfying H(~r, ~p) = ε are equally likely. Here, H(~r, ~p) = ~p2

2m + V(~r)is the classical Hamiltonian, while g(ε) is the density of states for the analogousquantum operator.

The number of particles we expect to find at each point in phase space, thephase space density5 is therefore [7]:

W(~r, ~p) =1

(2πh)3F(~r, ~p) =

1(2πh)3

1

1 + eH(~r,~p)−µ(T,N)

kT

(1.2)

4D = Constant, implies that the volume in phase space grows at the same rate as thedensity of states g(ε). Therefore, integrating volume in phase space provides an efficientmethod for calculating the density of states and approximating the eigenvalues of the quantummechanical operator H(~R, ~P )

5Not to be confused with the number of possible quantum states at each point in phasespace: D

7

1.2 Calculations of Fermi Distributions

Given (1.2) and the Hamiltonian:

H(~r, ~p) =1

2m(p2

x + p2y + p2

z) +mω2

2(x2 + y2 + α2z2) (1.3)

where in our case m = 40 amu = 6.64 × 10−26kg is the mass of a 40K atom,ω = (2π) 760 Hz are the short axis trapping frequency, while α = 3/76 = 0.04is the ratio of the trapping frequencies. In Raman spectroscopy, detuned lasersprobe a sample of Fermion atoms along one axis in the harmonic potentialdescribed by (1.3). The one dimensional probe implies that we measure aprojection of the phase space density onto one of the momentum axis. Themomentum components, pi for i = x, y, z, enter symmetrically into the Hamil-tonian and hence the phase space projection is independent of momentum axis.One obtains the projection by integrating out the three space and two mo-mentum dimensions. The harmonic potential is nice since every variable entersas a square. To perform the integration, first re-scale each variable appropri-ately (i.e. pi →

√(2m)pi) so that each ri, pi for i = x, y, z, enter as perfect

squares. One now makes the transformation to hyper-spherical coordinates,letting ρ = p2

x + p2y + x2 + y2 + z2. The momentum distribution is:

Π(pz) =(kThω

)3 83πα

√2mkT

∫ ∞

0

ρ4dρ

1 + eρ2+p2z−µ/kT

(1.4)

= −(kThω

)3 1α√π2mkT

Li5/2(−e1

kT (p2

z2m−µ)) (1.5)

where Li is the polylogarithmic function We know all parameters here exceptthe chemical potential µ. One determines µ(T,N) implicitly via integration overall occupied energy states: g(ε)F(ε). For the harmonic trap in question:

g(ε) =ε2

2α(hω)3(1.6)

Hence, we have

N =∫ ∞

0

g(ε)F(ε)dε (1.7)

N = − 1α

(kThω

)3

Li3(−eµ(T,N)

kT ) (1.8)

Equation (1.8) defines the chemical potential or fugacity implicitly. Onecommonly eliminates the number of atoms, N, by recasting temperature intodimensionless units: τ = T/Tf , where the Fermi Temperature Tf (N) = Ef/kis defined with respect to the characteristic Fermi energy, Ef

6 All meaningful

6At absolute zero, Fermions stack into all sequential energy states. The Fermi energy isthe least upper bound of the occupied energy states.

8

physics is characterized by τ . For my plots I chose N = 40000, our expectedatom number. At several temperatures, I solve equation (1.8) numerically andobtain the chemical potential. Upon determination of µ(T,N), I insert ourphysical parameters into (1.5) and generate plots for the expected momentumdistribution Π(pz). The plots are converted from the variable pz to a measurablefrequency. Specifically, we are interested in comparing the Fermi degeneratemomentum distribution to a thermal one. Hence, I plot the correspondingthermal distribution, normalized to N atoms on top of the quantum mechanicalone. One may easily check that (1.5) does approach a thermal distributionat large temperatures since Lin(−x) ≈ −x + x2/2n, for small x. Here, theargument of Li quickly → 0 as T increases.

Momentum Distributions

Figure 1.2: The graphs show the momentum density distribution over pz. Start-ing at the top left and moving from l-r, for N = 4 × 104, (T/Tf , T nK) =(1, 500), (.5, 250), (.2, 100), (.1, 50). The blue curve represents a classical Gaus-sian distribution, while the red indicates a Fermi-Dirac one. The vertical axisis the number of atoms. The horizontal axis is the frequency (Hz), δ/2π, cor-responding to atom momentum, pz, by equation (1.9). These plots refer toa Raman scheme, which requires a two photon-atom interaction. Hence, thefrequency axis is reduced by a factor of two, as required by equation (1.9).

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1.3 Basic Theory of Raman Spectroscopy

We wish to use Raman Spectroscopy as a tool to image the momentum dis-tribution of a collection of cold Fermion atoms. The Fermion atoms have anexpected momentum as described by the numerical calculations. The first steptowards measuring such a distribution is to somehow probe the distribution atsome given momentum. That is, for each momentum p, we wish to measurethe number of atoms that have momentum between p and p + dp, where theresolution of our equipment determines dp. Raman spectroscopy is appealingbecause it offers a method for targeting atoms with a predetermined momentumsimply by choosing a controlling frequency. To outline the general process, weconsider an atom in a light field. The light field consists of two counter prop-agating laser beams with frequencies ωL + ε/2 and ωL − ε/2. In our case, wefix ωL to approximately target the fine transition in potassium: 4S1/2 → 4P3/2.By varying the frequency ε, we systematically pick out all atoms with a givenmomentum. To understand the relation between the momentum of an atom andits response frequency ε, consider a basic momentum and energy balance. Theatom absorbs a photon with frequency ωL + ε/2 from one beam, followed by astimulated decay induced by a photon with frequency ωL − ε/2. In this case,a portion of the energy from the frequency ε goes into changing the potentialenergy of the atom. The remaining portion δ is converted to kinetic energy.Equating the absorbed photon energy with the kinetic energy:

δ = 2~k · ~pAtom/m+ 2h~k2/m (1.9)

Here ~k is the wavevector of the absorbed beam, while m is the mass of the atom.Equation (1.9) nicely illustrates the velocity dependence of the process. The ~k ·~pterm on the right hand side corresponds to a Doppler shift observed in the atomsreference frame. In the atoms reference frame, the two photon beams undergo afrequency shift. The absorption and emission of the now Doppler shifted beamsmust correspond simply to the energy deposited by the two photons. Asidefrom solely targeting atoms with momentum patom, the result of the doublephoton interaction is two fold. Firstly, the atoms final momentum changes:~pf = ~pi + 2h~k. Secondly, the internal energy state of the atom changes fromF = 9/2 to F = 7/2.

The previous description does not provide insight into the physical origin ofthe Raman process. Consider again the beloved problem of atomic physicists:an atom in a photon field. The standard approach to the problem is to introducea time-dependant perturbation into the Hamiltonian, representing the electricfield potential. Grinding out the math and physics yields several importantresults and parameters [8]. Specifically, the atom may absorb a photon fromone laser beam and emit into the second one. Mathematically, time dependentprobability amplitudes for the elevated and ground states describe the atom.The amplitudes oscillate in time with a characteristic frequency: the Rabi fre-quency. For a more detailed treatment of the problem, one may quantize photon

10

Raman Atomic Energy Transitions

Figure 1.3: The plot shows the internal energy states of an atom - two photonRaman interaction. Note the omission of Zeeman energy levels.

field [9]7. In the appropriate limits, the quantized field model yields the sameresults as the perturbed Hamiltonian model.

1.4 Raman Process: Calculation of Atomic Mo-mentum Change

In the case of a BEC, Bragg spectroscopy, the cousin of Raman spectroscopy,is a useful way of tagging atoms with a predetermined momentum. In a BEC,however, atoms move slowly compared to those in a degenerate Fermi gas. Asa result, the momentum kick provided by two photons in the Bragg scheme isenough to separate out the atoms in question. Unfortunately the same is nottrue in a Fermi gas: calculations show the two photon kick will not provide asubstantial velocity change8. For a harmonic potential, the characteristic energyof a Fermion cloud is:

Ef = 2π 6h(N × ωxωyωz)1/3 (1.10)

= 6(1.054× 10−34)(40000× 30× 760× 760)1/3 (1.11)= 5.6× 10−30 J (1.12)

The corresponding characteristic velocity is:

vF =√

2Ef/m (1.13)

=√

2× 5.6× 10−30/6.64× 10−26 (1.14)7Although some would argue such a treatment is overkill.8Although experimentally there is nothing stopping us from trying!

11

= 0.013 m/s (1.15)

In contrast, the change in velocity due to the interaction with two photons is:

∆v = 2hk/m (1.16)

=2(1.054× 10−34)

(767× 10−9)(6.64× 10−26)(1.17)

= 0.004 m/s (1.18)

The change in velocity is only approximately 1/3 of the characteristic velocityof the atom cloud. The ballistic expansion of the thermal cloud is comparableto the post collision velocity of the targeted atoms, especially when targetingslow atoms. A possible scheme, suited for Bragg spectroscopy, would be to usethree pulses: each pulse targeting atoms with momentum p, p+2hk and p+4hkrespectively. If each pulse is a length corresponding to a π pulse, then it may bepossible to give the system three sequential boosts. The atoms we are interestedin, at the original momentum p, will appear as a miniature cloud on the extremeright of left of the imaging.

12

Chapter 2

The Raman LaserApparatus

2.1 Design Parameters and Calculations

To image our Fermi gas with Raman spectroscopy, we must carefully controlthe frequencies of our laser apparatus. When targeting the Fermi gas, we usetwo counter propagating laser beams with almost identical wavelengths. Therequired mismatch in wavelength between the two beams is introduced to induceRaman transitions. Starting with one beam at a frequency ωL, we divide the ray(via a polarizing beam splitter cube) into two and then apply small frequencyshifts to each. The frequency shifts are of the same magnitude, but not in thesame direction: one beam increases by ε/2 while the second decreases by ε/2 .The result is to have two lasers with frequencies ωL + ε/2 and ωL − ε/2 . TheRaman transition only depends on the difference in frequency between the twobeams: ε and hence is insensible to variations in the initial laser frequency ωL.We use a 767nm diode laser as our initial beam with frequency ωL and retroinject to increase the precision on the wavelength (approximately 1 part in amillion). In the current setup, we control the frequency of the initial Ramanbeam using the repump laser for injection.

As with any propagating laser, the defining characteristics are intensity,frequency and polarization. We control polarization with passive optical com-ponents such as waveplates. In contrast, we use active components to controlthe intensity and frequencies. The frequencies we consider are ωL + ε/2 andωL − ε/2. Each of the frequency parts ωL and δ contribute to target specificatomic energy transitions.

2.1.1 Laser Intensity: I

The final optical intensity delivered to the Fermions depends on the outputof the 767nm diode laser and the inefficiencies in the Raman apparatus. The

13

intensity does have an effect on the atom-photon behaviour. Higher opticalintensity translates into a greater photon density. Atoms that see more photonswill be more likely to interact with them, thus increasing the frequency theyflop between an excited internal energy state and a lower one. This floppingfrequency, the rate at which the probability amplitude of the atom sitting in anelevated eigenstate oscillates, is known as the Rabi frequency. We calculate themaximum single photon Rabi frequency based on the parameters of our setup:

Ω1 = (C.G.)Γ√I/2Is

= (C.G.)2π × 6√

10/2(1.1) rad ·MHz

= (C.G.)80.3 rad ·MHz

Here C.G. is the Clebsh-Gordon coefficient for the specific transition involved.Γ = 2π × 6 MHz is the linewidth of the transition. Is = 1.1 mW/cm2 is thesaturation intensity of the repump transition in 40K. The saturation intensityis given by: Is = πhcΓ/3λ3 = (6.6×10−34)(3×108)π(2π×6×106)

3(767×10−9)3 = 1.1 mW/cm2.The intensity I is the maximum laser intensity that the atoms will see. Inthe laser apparatus, the maximum throughput power per beam is 1 mW. Fora beam diameter of 3 mm we have: Imax = 1/π(.15)2 ≈ 10 mW/cm2. Sincethe total atom-photon transition is a two photon interaction, we really careabout the frequency of the two photon oscillations (not just the one photonRabi frequency). The two photon (Rabi) frequency, Ω2, depends on both theone photon Rabi frequency and on the frequency of the laser: ωL.

2.1.2 The Laser Frequency: ωL

As previously stated, ωL does not effect the final Raman transition since onlythe difference in frequency of the counter propagating beams matters. Thefrequency ωL does, however effect the type of two photon-atom interaction aswell as the rate at which such an interaction occurs. Firstly, an atom in anexcited internal state will decay to a lower state by spontaneously emitting aphoton, or undergoing a stimulated emission provoked by an external photon.During spectroscopy, we wish to induce a stimulated decay while avoid heatingthe sample through spontaneous ones. The trick is in detuning the frequencyωL away from the fine structure resonance transition. For large detunings ∆,the rate of spontaneous decay is:

Γspont =Ω2

1Γ2∆2

(2.1)

On the other hand, the rate of stimulated emission, the two-photon Rabi fre-quency, is:

Ω2 =Ω2

1

2∆(2.2)

We exploit our current setup with the repump as an injection laser. The injectionlaser provides us with the required fine structure transition plus a detuning

14

∆ = ωHF /2 = 2π × 643 MHz. The detuning is large enough to avoid heatingsince the ratio Γspont/Ω2 = Γ/2∆ ≈ 0.005 1. Using equation (2.2), we havea maximum Ωmax

2 = 2π × (C.G.)264 kHz. We are primarily concerned with πpulses: a pulse that will invert the atomic population. Taking the C.G. ≈ 0.5,the maximum Robbie frequency will require the shortest pulse time: T = 30 µs1.

2.1.3 The Offset Frequency: ε

The frequency ε = ωHF +ωZ +δ, where ωHF = 2π×1285.8 MHz corresponds tothe hyperfine transition 2S

1/2, F = 9/2 to F = 7/2, ωZ corresponds to the smallfrequency for the Zeeman transitions 2, while δ is the off resonance detuningfor the hyperfine transition. From the theoretical plots in chapter 1, we requireε to range from −2π × 50 kHz to 2π × 50 kHz in order to scan through themomentum distribution. For the design of the Raman apparatus, we doublepass two acousto optical modulators (AOM’s). Each pass through the modulatorshifts the frequency of the laser by some fixed amount. Thus, each AOM mustimpose a frequency shift of ε/4 = ωHF /4+δ/4 ≈ 2π×320MHz+δ/4. Moreover,to resolve the momentum distributions in the theoretical plots, we would liketo control δ/4 to at least a hundred Hz in precision. Each of the AOM’s usedhas a center frequency of 320 MHz. That is, we drive the AOM with an RFvoltage signal at 320 MHz + δ/4. During testing of the final Raman apparatus,the AOM’s revealed an FWHM of 2π× 12.5 MHz which is orders of magnitudegreater than the required range of 2π× 15 kHz for δ/4. Standard AOM driversdo not, however provide the required precision for the AOM signal: one partper million. To drive the AOM’s, we built our own circuit that uses a digitalsynthesizer as a stable RF signal source. We do better than one part per million,and can resolve the AOM driver signal of 320 MHz to ≈ 1 Hz. I outline thedesign of the AOM driver circuit in chapter 3.

2.2 The (Actual!) Laser Apparatus

2.2.1 Efficiency, Power and Alignment

The laser loses some power at each optical device. The optical isolator transmits92% while each AOM has approximately 68% efficiency per pass when fullypowered3. The fiber optic cable transmits approximately 42% of the powerfor each of the two AOM beams. Figure 2.1 shows a diagram of the Ramanlaser apparatus. Should the laser become misaligned, the following tips mayprove helpful in the realignment process. First, the efficiencies stated in theprevious paragraph are taken when the laser diode is powered by 29.9 mA.

1The pulse time for a π pulse varies linearly with laser intensity2The Zeeman transition frequency is dependent on B field3To fully power the AOM, apply a 10 V signal to the AOM input controller, virtually no

current is required

15

Raman Optics Setup

Figure 2.1: The Raman laser apparatus. The laser is divided into two beams.Each beam double passes an AOM (ε/4)thus acquiring a total change ±ε/2 infrequency. The beams are recombined and coupled into a fiber optic cable.

After passing through the isolator, the laser emits 10.4 mW of power. Thedivision of the laser into separate beams is performed by the half waveplate -beamsplitter cube combination prior to the AOM passes. Turn the waveplate,so all 10.4 mW travels into AOM 1. I found that the efficiency of individualpasses through the AOM varied. Particularly, for AOM 1, the backpass wasmore efficient than the forward pass. Use mirrors M1 and M2 to maximize thetransmission through the AO. Maximizing the backpass should yield a powerof around 4.8 mW. If after maximization, the transmission is still low, try:rotating the quarter waveplate in the AOM pass, or varying the drive powerto the AOM. Increasing the drive power to the AO increases the first orderdiffraction laser power. Alignment adjustment to AOM 2 are more difficult.The maximum output power after double passing the AO is also approximately4.8 mW. A beam splitter cube and not a mirror control the coarse alignment.Do everything in your ability to maximize the power without unscrewing thebeam splitter cube. Try adjusting the fine knobs on the cube mount, and themirror M4 to control the backpass of the AO. Also, try increasing the drivepower to AOM 2 to increase the efficiency. Only serious misalignments shouldrequire unscrewing the beamsplitter cube and coarsely rotating it. The twobeams recombine through a beamsplitter cube located between mirrors M6 andM7. One should independently maximize the coupling of each beam into thefiber cable. Using the half waveplate before the AOM passes, transfer all the

16

power to pass through AOM 1. Use mirrors M7 and M8 to maximize couplinginto the fiber cable. At optimal coupling, there should be approximately 2.01mW transferred through the fiber. Repeat the process for AOM 2 using mirrorsM5 and M6 to maximize the coupling into the fiber. Again, one should observeapproximately 2.01 mW through the fiber.

Raman Optics Setup

Figure 2.2: Raman optics apparatus

2.2.2 Output Polarizations

In the current set up, beams with frequencies ωL + ε/2 and ωL− ε/2 propagatewith a linear polarization orthogonal to each other. By inserting a half waveplate- beamsplitter cube - half waveplate combination, one may align the linearpolarizations with the price of a 50% loss. On the exiting end of the fiber,the linear polarization must be converted into a circular one. In addition tojumping hyperfine levels in the Raman transition, we also induce Zeeman leveltransitions. The Zeeman levels are small in comparison, approximately 1 MHz.For the Zeeman jumps to occur, light must deliver both optical energy andangular momentum. Thus, one has surprising control over the possible allowableatomic transitions. For two counter propagating beams, we have two possiblecases: the two beams may travel with either the same or opposite polarizationhandedness. If the beams have the same handedness, the two photon process will

17

Raman Optics Setup

Figure 2.3: Raman optics apparatus

result in a transfer of angular momentum to the atom: ∆mf = 2. In comparison,two beams with opposite handedness will not change the end angular momentumof the atom: ∆mf = 0. One should know a priori which transitions are targetedand account for the required shift in Zeeman energy.

2.2.3 Optical Transience

The electrical circuits have a response of 3µs, and so AOM transition speed limitsthe response of the laser. The AOMs trade off response time for high efficiency.Thus, to increase operation speed, one should refocus of the laser and realignthe AOs. The characteristic time constant for the response is approximately 5µs. Figures 2.5, 2.6, 2.7, 2.8 and 2.9 show the laser response to various inputsignals.

2.2.4 Modulation Frequency Response

The laser power depends on the imposed frequency shift ε. The maximumlaser output from the AOM’s occurs at a shift of approximately 320 MHz, as

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Polarization and Energy Transitions

Figure 2.4: The energy levels are in 4S1/2 for potassium 40. The left andright diagrams contrast the possible angular momentum exchange and theircorresponding Zeeman transitions

shown in figure 2.10. The plot shows the characteristic FWHM4on the frequencyattenuation curve of approximately 16.5 MHz. In the current setup, the gaincurves of the two AOM’s have peak frequencies of 318.2 and 322.9 MHz. Figure2.11 shows the frequency dependence of the laser intensity emitted from theoptical fiber. For the test, each of the two beams had equal intensity at 320MHz. Due to the mismatch in center frequencies, the transfer curve is actuallyan unequal mixing of power with frequencies ωL + ε/2 and ωL − ε/2. One canshift the center frequency of an AOM, however this requires adjusting the AOMalignment. Note that testing revealed the FWHM of the AOM is due to thephysical limitations of the device and not current optical alignment.

4Frequency width half max

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35 µs Laser Pulse and Response

Figure 2.5: Laser response to 35 µs pulse

20 µs Laser Pulse and Response

Figure 2.6: Laser response to 20 µs pulse

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10 µs Laser Pulse and Response

Figure 2.7: Laser response to 10 µs pulse

Laser Pulse Transience

Figure 2.8: The output laser transient (bottom) has a time constant of τ = 5µs.The propagation delay of the acoustic wave in the AO accounts for about 4 ofthe initial 6 µs delay.

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Repeated 10 µs Laser Pulse

Figure 2.9: Laser response to periodic 10 µs pulses. The response indicatesstrong attenuation.

Frequency Dependence of the AOMs

Figure 2.10: Vertical axis shows the double pass efficiency of the AOM’s. Centerfrequencies are 318.2 and 322.9 MHz for AOMs 1 and 2 respectively. The FWHMis 16.5 MHz.

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Frequency Dependence of Raman Apparatus

Figure 2.11: Vertical axis shows the fractional power transferred through theentire Raman apparatus. The horizontal axis is the frequency range of theAOM’s. To convert the arb. units to meaningful ones, the peak of the graphoccurs at an efficiency of 0.652 × 0.42 = 0.18, where 0.65 is the fractional lossthrough each AOM pass while 0.42 is the fiber loss. The center frequency andFWHM are 320.8 and 12.5 MHz respectively.

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Chapter 3

The AOM Driver Circuits

3.1 Requirements

The AOM requires a radio frequency (RF) signal with an average power of upto 1.5 W. Moreover, our AOMs must shift the optical frequency by 320 MHz+ε, where we control ε over a 0 to 15 kHz range. The difficulty arises in ourrequired precision. We eventually wish to measure the momentum distributionof cold atoms. Our precision in the momentum distribution translates directlyfrom the precision in ε. Current AOM drivers on the market do not meet ourrequirement. Rather, we use a digital synthesis (DDS) to control the stabilityand precision of the RF signal. The digital synthesis has an incredible precisionin frequency: approximately 1 part in 108. Unfortunately the DDS outputs a-10 dBm signal, whereas we require a 1.5 W or approximately 32 dBm output.We use a preamp to boost the -10 dBm signal to 12 dBm. Thus, the AOMdriver circuit must provide a 20 dB gain and operates at a center frequency of320 MHz1. Moreover, to control the output amplitude, we must have variablevoltage control on the gain of the circuit. Lastly, the apparatus shares the DDSsignal with the RF evaporative cooling scheme. Thus, the circuit requires twoRF inputs, one for the DDS high precision RF signal and the other as a coarseRF input required to keep the AOMs thermally stable. To share the DDS signal,we simply switch between the two RF inputs when the apparatus is not in use.Note: do not exceed 12 dBm input (0.9 V) on either of the RF inputs.

3.2 Inputs and Outputs:

3.2.1 RF Power, Gain and Coupling

The amplifier circuit accepts an RF input signal at 12 dBm, provides a variablegain, and emits an output to the AOMs between 0 and 32 dBm. The variable

1The bandwidth on all components is much larger than the required 1 MHz. I have onlyincluded the bandwidth for completion.

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gain and switching will be described in the next section. There are two radiofrequency inputs as seen in figure (3.1). Only one signal is chosen by the TTLto be amplified and sent to the AOs. Due to the design of the apparatus, usershave independent amplitude control to each of the two AOM outputs. There aretwo coupling outputs for each of the two AOMs. Users can monitor the amountof power being sent to each AOM and the amount that is being reflected back.Figure (3.3) shows the location of the coupling outputs.

RF Inputs

Figure 3.1: RF Input Panel. When off (0V), the TTL switches the left input(RF input 1), when on (3V) the right input (RF input 2) is chosen.

3.2.2 DC Power Inputs

The DC power inputs are located on the left hand side on the input/outputpanel, as seen in figure 3.3. The two ground connections are connected internallybehind the panel so that the two power supplies share a common ground. AllDC voltages are label on the input panel.

3.2.3 Adwin and TTL: Voltage Controls from CommandCentral

There are three input control signals. Two analogue signals (0 to 10 V) controlthe output amplitude to the AOMs. Essentially, they are the analogue turn on -turn off switches. The inputs are BNC connections located in the middle of theinput panel (Figure 3.3). To operated both AOMs simultaneously, connect thetwo inputs together. The third input signal is the TTL. The TTL is essentially

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AOM Output

Figure 3.2: The outputs to the AOMs are located behind the input panel. Theoutputs are SMA connections.

a voltage controlled switch: 0 V is off while 3 V is on. The TTL input is locatedat the bottom right of the front input panel. The signal chooses one of the twoRF inputs that will will be amplified and sent to the AOMs. Figure 3.1 showsthe correspondence of the TTL signal with the two RF inputs.

3.3 The Driver Circuit

3.3.1 RF Circuit

The RF circuit accepts one of two inputs signals at 12 dBm and amplifies thesignal to a maximum 31.8 dBm. Figure (3.6) shows a schematic of the RFcircuit. Upon accepting one of the two RF inputs, the driver circuit dividesthe signal into two: one signal to power each of the two AOs. Each of the twosignal amplitudes is controlled independently by variable attenuators. Thus, onehas individual control over each of the two AOM signals. The amplification isperformed by high powered amplifiers. For protection, 2 dB power attenuatorssit after the amplifiers, before the signals propagate to the AOs. The lastcomponent in the circuit couples a percentage of the output power to input-output panel. The coupling allows one to monitor the RF power sent to theAOs. Figures (3.7) and (3.8) show the relationship between the coupling signaland the actual output to the AOMs. I used a 400 MHz scope to monitor thepower sent to the AOs. As a result, due to internal attenuation of the scope,the voltage - power transfer curve is slightly different for the scope than figure(3.7). For the calibration curve, see D. Shirokoff Lab book pg 187.

The RF circuit outputs a signal in the range 320 MHz. Figures (3.9 3.103.11) show the response of the RF envelope when the drive signal is turned onand off. The two methods of signal amplitude control, the TTL switch andvariable attenuator, have internal response times of ≈ 3µs.

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Input and Output Panel

Figure 3.3: The input panel localizes almost all the inputs and outputs. Theonly inputs not included are the two RF sources. The two AOM outputs do notcouple into the panel but rather go directly to the AOMs.

3.3.2 DC Control of the RF Circuit

The DC control circuit takes in an analogue voltage from the Adwin and booststhe power to 0-13.8 V at 30mA. The purpose is to convert an analogue logicsignal into a power signal that controls the variable attenuators. There areactually two circuits, as seen in figure (3.12), one for each AOM. Thus, by usingtwo Adwin signals, one may individually control the output of the AOMs.

Figure (3.13) shows the output response of the DC control circuit. Theresponse is extremely fast compared to the AOM acoustic and RF componentresponses (≈ .25µs).

3.4 Start Up/Shut Down Procedure - What Notto Do!

DO NOT EXCEED 12 dBm or .9 V on the RF input to the circuit. At 12dBm input and maximum circuit gain, the AOMs receiver approximately 31.8dBm (just over 1.5 W). Increasing the 12 dBm input may directly damage theAOMs. The lack of protection arises from the fact that the large amplifiersare not yet at their 3dB compression point. When the driver circuit is runningat maximum gain, the large amplifiers emit approximately 34 dBm. The largeamps, however, are able to handle 35.5 dBm. Therefore, increasing the originalinput above 12dBm will cause the amplifiers to exceed an output of 34 dBm. If

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Figure 3.4: AOM RF driver circuit. The two DC power supplies are located onthe right.

instead one terminates the output to the AOMs with 50 Ω, then the maximumRF input allowed is 18 dBm. At 18 dBm, the large amplifiers receive theirmaximum allowable power input. Exceeding this amount could damage theamplifiers. Before turning anything on, ensure that the output to the AOMs iseither connected to the AOM or terminated with 50 Ω. As a safety precaution,activate all DC power supplies before administering RF to the driver circuit.Even if RF from the DDS is sent into the driver circuit, attenuators will diminishthe signal before reaching the inactive amplifiers. Input RF may be applied oncethe DC power supplies are on. The system is now in an active mode. With theinput RF signal and DC supplies on, a 0 to 10 V signal from the Adwin willcontrol the output amplification sent to the AOMs. To turn off, reverse theactivation procedure. Turn off the Adwin control voltage signal and ensure thetermination of all RF inputs. Turn off DC power supplies. Note, the laser canbe turned on or off at any time.

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Figure 3.5: AOM RF driver circuit

RF Circuit Diagram

Figure 3.6: The diagram shows the relevant RF inputs (12dBm) and outputs.The circuit has a variable gain with a maximum 20 dB.

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Output RF Drive Power vs. Coupling Voltage

Figure 3.7: We can monitor the actual RF drive power sent to the AOM bytapping into the signal. A fraction of the power sent to the AOMs is divertedby the coupler. The plot shows the relation between the amplitude of the couplerRMS voltage and the actual RMS voltage amplitude. On the 400 MHz scopecalibration curve (see D. Shirokoff lab book pg. 187), extended data pointsfurther justify the V 2 extrapolation dependence on power.

Output RF Drive Voltage vs. Coupling Voltage

Figure 3.8: The plot is analogous to 3.7, however; we plot the output voltageto the AOMs. The two sets of data points are taken from the two individualAOMs. A linear extrapolation extends to the maximum recommended AOMdrive power of 1.5 W.

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RF Output Transient

Figure 3.9: Top signal: Adwin input. Bottom signal: Measurement of the drivercircuit response, AOM RF voltage envelope. The voltage controlled attenuatorslimit the transience to 3µs. The circuit has a delay of approximately 2.5 µs.

RF Output Response to Periodic Input

Figure 3.10: The bottom signal is the RF envelope response to the top, TTL,signal.

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TTL Switching Transience

Figure 3.11: The bottom signal is the RF envelope response to the top TTLsignal.

DC Voltage Control Circuit

Figure 3.12: DC Voltage control circuit takes an analogue voltage from theAdwin (0-10V, very little current) and boosts the power to 0-13.8V at 30 mA.The output voltage controls the variable attenuators. Hence, the voltage actsas a tap to a running hose of water.

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Variable Voltage Control Circuit Transience

Figure 3.13: The bottom signal is the voltage response to the DC voltage controlcircuit when responding to an input from the top signal (Adwin). The circuitis slew rate limited as evidenced by the linear response.

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Bibliography

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[4] S.N. Bose, Z. Phys. 26, 178 (1924).

[5] B. DeMarco and D.S. Jin, Science 285, 1703 (1999).

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[7] D.A. Butts, D.S. Rokhsar, Phys. Rev. A 55, 4346 (1997)

[8] H. Metcalf, P. Straten, Laser Cooling and Trapping, (1999)

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