Investigations into Quantum Resonance andAnti-Resonance of Cold Rb87 Atoms Interacting with

an Optical Moving Standing Wave

Jean Anne CurrivanIn collaboration with Maarten Hoogerland, Arif Ullah and Remi Blinder

July 14, 2008

Abstract

Context. By creating a small frequency difference between the two counter-propagating lasers in our opticaltrap, we can transform our experiment from an interaction between the Bose-Einstein Condensate (BEC)atoms and a standing wave to an interaction with a moving standing wave. With this new setup we canexplore the quantum resonances and anti-resonances of the δ-kicked rotor, which is an important exampleof quantum chaos.Aims. The goal of this paper is to exhibit the quantum resonances and anti-resonances of our Rubidium 87atoms when they interact with a moving standing wave and to compare these data results with the theoreticalexpectations of the experiment, for different initial parameters.Methods. We generated a kicking pulse sequence that interacts with our cold cloud of Rb87 atoms. We thenvaried the number of kicks, the frequency difference between the counter-propagating waves that make upthe pulse sequence and the free period between kicks in the pulse sequence to observe quantum resonancesand anti-resonances for these different parameters.Results. We detected quantum resonances and anti-resonances for both 2 and 4 kicks, for a frequencydifference from 0kHz to 30.16kHz, and for a free period difference of 33.15µs, 66.3µs, and 99.45µs. Wecompared these results with our theoretical predictions.

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Contents

1 Introduction 2

2 Theory 22.1 The Classical Standard Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Standard Mapping in Quantum Regime for a Strong Perturbation . . . . . . . . . . . . . . . 32.3 Applying to Our Experimental Setup: Dilute Atomic Gas in Pulsed Optical Standing Wave . 4

2.3.1 The Atomic Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3.2 The Laser Field and its Interaction with the Atom . . . . . . . . . . . . . . . . . . . . 4

2.4 Quantum Resonance and Anti-Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.5 Calculating Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Experimental Setup 73.1 Block Diagram and Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Generating Pulse Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Observing Overlapping Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.4 Running the Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Experimental Results 124.1 Contour Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 Energy vs. Initial Momentum β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.3 No. of Atoms vs. Momentum for Each Recoil Momentum . . . . . . . . . . . . . . . . . . . . 144.4 Comparison to Theoretical Results from M. Saunders, et al. . . . . . . . . . . . . . . . . . . . 15

5 Conclusions and Future Prospects 16

6 Acknowledgments 16

7 References 16

8 Appendix 178.1 Contour Plots for n=4 Kicks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

List of Figures

1 Basic block diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Pulse sequence on scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Block diagram for pulse detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Single pulse shown on scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Pulses at center of MOT from the counter-propagating lasers . . . . . . . . . . . . . . . . . . 116 Quantum resonance and anti-resonance contour plots for n=2 kicks . . . . . . . . . . . . . . . 127 Energy vs. initial recoil momentum, n=2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Energy vs. initial recoil momentum, n=4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 No. of Atoms vs. momentum for each recoil momenta β, n=2 . . . . . . . . . . . . . . . . . . 1410 No. of Atoms vs. momentum, n=2 compared with M. Saunders et al. . . . . . . . . . . . . . 1511 Quantum resonance and anti-resonance contour plots for n=4 kicks . . . . . . . . . . . . . . . 17

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1 Introduction

The Bose-Einstein Condensate (BEC) created in our lab interacts with a standing wave generated from twocounter-propagating lasers with equal amplitude, frequency and phase. If we instead set a small frequencydifference between the lasers, i.e. ∆ω << ω, we will generate a moving standing wave. With the atoms in theBEC moving relative to the laser trap, we can perform diffraction experiments. In the canonical diffractionexperiment, light encounters a matter grating; conversely, here the atoms encounter an optical grating, andwe can observe the resulting quantum interference effects.

In this paper we explore one major quantum interference effect: the quantum resonance and anti-resonance of cold atoms when interacting with a pulsed laser field. This is an experimental realizationof the δ-kicked rotor. As the quantum analog of the standard mapping of dynamical chaos, the δ-kickedrotor is an important area for understanding how classically chaotic systems act in the quantum regime. Bylooking at its classical behavior in the regime of very strong kicking over a longer time versus its quantumbehavior when the kicks are less strong and in the time just after kicking, we can better understand thetransition between classical and quantum regimes. The resonances and anti-resonances of this δ-kicked rotorare a specifically quantum feature that can further our understanding of this transition, and of the behaviorof classically chaotic systems when in the quantum regime.

In the 1990s, with new advancements in atom-cooling techniques, many kicked rotor experiments wereperformed [1-5] to understand the transition of the δ-kicked rotor between the classical and quantum regimes.Most of these experiments focused on characterizing the long time and sufficiently large number of kicksneeded for the system to reach the classical regime. The realization of a BEC in 1995, with a well-definedinitial atomic state and much colder temperatures [6], lead to a new range of experiments that were ableto focus on the specifically quantum nature of the δ-kicked rotor soon after the kicks were applied. It wasfound that a BEC interacting with an external field could accurately represent a δ-kicked rotor [2, 3, 7].Since this time, other groups [2, 5, 8, 9] have worked on observing quantum resonances. In this paper weseek to lay out the quantum resonance and anti-resonance phenomena for a variety of initial parameters, soas to give a fuller analysis of how these resonances depend on the initial momentum of the atoms and theperiod between kicks.

Part two of this paper lays out the theory of quantum resonance, from which we develop theoreticalsimulations to compare with the data results. Part three outlines the experimental setup, including theprogramming of the arbitrary waveform generator, the process of overlapping the two laser beams andrunning the experiment itself. Then in part four we discuss the data analysis and experimental results andin part five state our conclusions and future prospects.

2 Theory

The Standard Mapping is the main model used for studying dynamical chaos in the classical regime [10]. Aquantum resonance occurs if, when kicked n times with strength ε0, the system acts as if it was kicked onlyonce with strength nε0. Additionally, while usually in time the energy of the system increases linearly as theatom cloud expands, at a quantum resonance the energy increases quadratically. An anti-resonance occurswhen the kicking causes the system to fall back into its initial state.

This theory section briefly outlines the classical model for dynamical chaos and relates it to the δ-kickedrotor. We then describe the time evolution of the δ-kicked rotor, as predicted by our theoretical simulations.We show how this time evolution gives forth quantum resonances and anti-resonances.

2.1 The Classical Standard Mapping

In the classical regime, the model system used to obtain the standard mapping is a pendulum in a kickedgravity field. In general, the Hamiltonian is given as

Hgeneral = Ho(p) + V (θ)f(t), withf(t+ T ) = f(t). (1)

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Here Ho(p) is the unperturbed Hamiltonian and V (θ)f(t) is the nonlinear perturbation, periodic with periodT between kicks. We can think of the external kicking field as a periodic delta function,

δT (t) =∞∑

t=−∞

δ(t− tT ), (2)

and thus the Hamiltonian can be written as

H =p2

2I+ ε0 cos θ δT (t) (3)

where p is the angular momentum, ε0 is the kicking strength, θ is the angular displacement and I is themoment of inertia of the pendulum.

The standard mapping is obtained by getting the equations of motion from H and then integrating overone period T . Thus we find the standard mapping to be

Pt+1 = Pt +K sin θt; Θt+1 = Θt + Pt+1 (4)

where P ≡ pT, K = ε0T . This standard mapping is used to gain understanding of dynamical chaos. Itcan be used to describe the change in time of a generic Hamiltonian system, in the case that there is oneisolated, nonlinear resonance, and all other resonances can be considered as a perturbation [10].

In equation 4 we can see that increasing K gives increasing perturbations. Thus a strong perturbationis in the regime of K >> 1.

2.2 Standard Mapping in Quantum Regime for a Strong Perturbation

The δ-kicked rotor involves atoms periodically kicked by an external field, and thus is analogous to thependulum kicked in a gravity field that was outlined in the previous section. We can translate that classicalHamiltonian into the quantum regime:

H =p2

2m− ε0 cos x δT (t). (5)

Thus the time-independent Schrodinger equation is

i~∂ψ

∂t=

p2

2mψ − ε0 cos x δT (t)ψ. (6)

We assume instantaneous, periodic kicks such that

ψ(x, t+ T ) = Uψ(x, t), (7)

where U is a unitary operator. From the time-independent Schrodinger equation, we find [13]

U = eiε0e−i τ2

∂2

∂x2 , τ ≡ ~Tm. (8)

Thus, to get ψ just after an infinitesimal kick, we apply U to ψ. The parameter ε0 can be thought of asthe perturbation strength, and τ is related to the period between kicks. The first term in U is the kickingterm, and the second term is the free expansion.

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2.3 Applying to Our Experimental Setup: Dilute Atomic Gas in Pulsed OpticalStanding Wave

We can now turn to looking at how this time evolution operator U acts in the specified case of a diluteatomic gas interacting with a pulsed optical standing wave [11].

In our setup we deal with a very dilute gas and so it is reasonable to neglect atomic collisions. Thus wecan focus on a singe-particle Hamiltonian, similar to the discussion in the previous section. Additionally,since the gas is non-interacting, the x, y and z directions are separable and we can just look at ψ(x) alongthe axis of the standing wave.

Recall equation 2.2.5 that describes the Hamiltonian of a δ-kicked rotor. We seek to achieve this sameHamiltonian in our experimental setup. The Hamiltonian in our experiment will be made up of two parts:that of the atom and that of the interaction between the atom and the field.

2.3.1 The Atomic Hamiltonian

We will treat the atom as having two states |e〉 and |g〉 separated by ∆E = ~ω0. This gives the Hamiltonian

Hatom =p2

2m+

12

~ω0σz (9)

where m is the mass of the atom, ω0 is the resonant frequency between the two energy levels, and the Paulimatrix σz is given as σz = |e〉 〈e| − |g〉 〈g|.

2.3.2 The Laser Field and its Interaction with the Atom

We create the optical standing wave with which the atoms interact by using two counter-propagating,overlapping laser beams. Both beams come from the same diode laser, and thus their amplitudes are thesame. If they additionally have the same phase then at the point they overlap a standing wave is generated.But, to observe interference effects between the atoms and the field, the atoms must be moving relative tothe field. Thus we must introduce a phase difference between the two counter-propagating beams.

In our experiment we instead introduce a small frequency difference between the beams compared tothe original frequency, but it can be shown that this frequency difference can be translated as the phasedifference needed for a moving standing wave. Because we design the difference in frequency to be smallcompared to the original laser frequency, this frequency difference can indeed be viewed as a phase differencebetween the two lasers, as needed in the above theory.

With no frequency difference, the counter-propagating waves are defined as

y1 = A sin(kx− ωt) (10)

y2 = A sin(kx+ ωt). (11)

When a small frequency difference is introduced, after time Tpulse y2 becomes

y2 = A sin[(k + dk)x+ (ω + dω)Tpulse]. (12)

We know that the group velocity c of the wave is dωdk , and thus

y2 = A sin[(k + dω/c)x+ (ω + dω)Tpulse]. (13)

So, y2 can be written asy2 = A sin[kx+ ωt+ dω(x/c+ Tpulse)]. (14)

and therefore is the original y2 offset by a phase of dω(x/c+ Tpulse) which depends on the position and thesize of the pulse.

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With this moving standing wave we get the Hamiltonian

Hatom−field int =12

~Ω(ei(klx−ωlt+φ1 |e〉 〈g|) +12

~Ω(e−i(klx+ωlt−φ2 |e〉 〈g|). (15)

Here kl is the laser wavevector magnitude, defined as kl = 2πλ where λ is the laser wavelength, in our

experiment equal to 780nm. The parameter ωl is the frequency of the two laser beams and φ1 and φ2 are therespective laser phase differences. In our experiment ωl is set to about 1015Hz and our change in frequency,as shown, corresponds to the phase φ. The parameter Ω is the Rabi frequency of the two laser beams, whichgives the strength of coupling between the E&M field and ω0, the frequency between the two energy levels.Our two beams are at the same intensity and thus have the same Ω, defined as

Ω2 =Ipower

Isaturated

Γ2

2. (16)

Ipower is the intensity of the laser for a given power, defined as Ipower = Powerπr2 , where r is the radius of

the laser beam. Isat is the saturation intensity, found to be 16 W/m2 [12], and the natural linewidth oftransition Γ is found to be 2π ∗ 5.9 MHz [12]. Γ is the spread in frequency about ω0.

Thus, the total Hamiltonian for both the atom and the field is found to be

Htotal = Hatom + Hatom−field int (17)

Htotal =p2

2m+

12

~ω0σz +12

~Ω(ei(klx−ωlt+φ1 |e〉 〈g|) +12

~Ω(e−i(klx+ωlt−φ2 |e〉 〈g|). (18)

It is useful to manipulate this Hamiltonian by multiplying by two unitary operators:

U1 = exp[i(ωl |e〉 〈e| − ω0|e〉 〈e|+ |g〉 〈g|

2t)], (19)

U2 = exp(−iΩ2 |g〉 〈g| t

8(ω0 − ωl)). (20)

After applying these, we get

Hδ =p2

2m− ε0 cos(Kx)δT (t) (21)

which is the same form as that of a δ-kicked rotor in equation 2.2.5. Here m is the Rb87 atomic mass,K ≡ 2kl and the kicking strength ε0 = Ω2Tpulse

8(ω0−ωl).

Thus our experimental setup does indeed exhibit a quantum kicked rotor.

2.4 Quantum Resonance and Anti-Resonance

Given this Hamiltonian, we can show how our periodic kicks give rise to quantum resonances and anti-resonances. Recall in equation 2.2.8 we found that the time evolution of a δ-kicked rotor is governed by

U = eiε0 cos xe−p2T2m (22)

where T is the time between pulses.For the δ-kicked rotor in our experimental setup, x and p are continuous, and thus can be separated into

discrete and continuous parts. We let x = 1K (2πl + θ) and p = ~K(k + β) where l and k are integer values

and θ and β are continuous: θ ∈ [−π, π], β ∈ [0, 2].Thus U can be re-written as

U = eiε0 cos θe−i~K2T

2m (k+ β2 )2 . (23)

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Here β is replaced by its eigenvalue β/2 (the 2 is for convenience) since β is a conserved quantity. β isreferred to as the ”‘quasimomentum.”’ We usually choose to describe β in units of recoils, where one recoilis the change in momentum of the atom when it absorbs one photon. β can be thought of as the initialmomentum of the atoms, in units of recoils. This can be translated into units of frequency. We can showhow an offset in frequency between the two counter-propagating lasers leads to changing the momentum ofthe atom by an amount determined by the doppler shift. When absorbing a single photon, an atom changesits velocity by

v1 recoil =~kl

2m(24)

since the momentum of one photon is ~kl, where kl = 2πλlaser

. But, since with the small frequency differencewe create a moving standing wave with which the atom interacts, the atom moving relative to the fieldobserves a Doppler shift of

ωdoppler shift = −~kl • ~v. (25)

If the velocity of the atom is v1 recoil then this corresponds to a Doppler shift of ~k2l

2m . We know the free

propagation velocity of the atom is ~2k2l

2m , and so we find that

KEatom = ~ ∗ ωdoppler shift. (26)

This shows that our introduced frequency difference between the two counter-propagating beams does leadto a change in the initial kinetic energy of the atoms, causing them to be kicked into different momentumstates. We can calculate this recoil frequency by

E = ~ωrecoil =~2k2

l

2m. (27)

In our experiment, with λlaser=780nm, one recoil frequency corresponds to frecoil=3.77kHz.In equation 2.4.23 for U , β is in units of quarter-recoils, e.g. β=0.25 corresponds to 1 recoil of 3.77kHz

and β=2 corresponds to 8 recoils of 30.16kHz. The frequency difference we introduce between our twocounter-propagating laser beams is in units of initial momentum β, and thus the phase between the twobeams is determined by β.

To simplify equation 2.4.23 we let the period between pulses, T , be defined as

T =2πm~K2

l, (28)

where l is a positive integer. For our Rubidium 87 atoms and with λlaser=780nm, the coefficient 2πm~K2 =

33.15µs.Thus, to exhibit quantum resonance and anti-resonance we construct a Matlab program that, for a given

β and l, applies U to ψ for each kick up to the total number of kicks, n.For example, if β=0 then U reduces to

U(β = 0) = e−ilπk2eiε0 cos θ (29)

The eigenvalues of k are integers, so e−ilπk2= e−ilπ(±k), and therefore

U(β = 0) = e±ilπkeiε0 cos θ (30)

Here we can see that the time evolution of ψ according to U depends greatly on whether l is even or odd.Let us assume we apply the kicking pulse twice, i.e. n=2. Then

Un=2(β = 0) = e±ilπkeiε0 cos θ (31)

Un=2(β = 0) = e−ilπkeiε0 cos θe+ilπkeiε0 cos θ (32)

6

One can find that [θ, k] = i [11], and thus

Un=2(β = 0) = eiε0 cos(θ−lπ)eiε0 cos θ (33)

Un=2(β = 0) = eiε0[1+(−1)l] cos θ (34)

If l is even, for example l=2 and thus T=66.3µs, we get

Un=2(β = 0) = ei2ε0 cos θ. (35)

This is a quantum resonance: instead of acting like two kicks of strength ε0 separated by T , it can be seenas acting as if a single kick was applied with a greater strength of 2ε0.

If l is odd, for example l=1 and thus T=33.15µs or l=3 and T=99.45µs, we get

Un=2(β = 0) = 1. (36)

This is an anti-resonance: after the two kicks ψ falls back into its initial state.

2.5 Calculating Energy

One useful way to quantify the resonances and anti-resonances is by looking at the energy of the state ψafter the kicking sequence. ψ(θ, t) can be written as an expansion of coefficients

ψ(θ, t) =1√2π

P∑p′=−P

Ap′(t)eip′θ (37)

where |Ap′(t)|2 is the probability distribution in momentum space after t number of kicks. Thus the energycan be defined as

E(t) =

∑p′ p

′2 |Ap′(t)|2∑p′ |Ap′(t)|2

, (38)

with p′ running -P to P where P is some maximum momentum.In our Matlab code, we can simulate this kicking process for any β, l and n. We chose to do experiments

for both n=2 and n=4 number of kicks. We run through β from 0 to 2 in increments of quarter recoils, i.e.from 0 to 8 recoils. Equivalently this is varying the frequency difference between the two laser beams from 0to 30.16kHz in increments of 3.77kHz. We collected data for l=1,2, and 3, which corresponds to T=33.15µs,66.3µs and 99.45µs. We can then compare the kicked ψ in our code to our data. We look at both the energyand the |ψ|2 values for each resultant wavefunction.

3 Experimental Setup

We will now outline the experimental setup that is used to exhibit the quantum resonance and anti-resonanceof our Rb87 atoms.

In our lab we create a BEC using Rb atoms and then let the BEC expand for about 0.5ms, and thus weare essentially dealing with a cloud of very cold atoms at around 50 nK. We have about 2 ∗ 104 atoms inthe BEC [6]. Since it is not possible to create the instantaneous kicks described in the δ-kicked rotor model,our setup approaches the δ-kicked rotor regime as the kicks become infinitesimally short in time. Our kicksare 300ns, compared to the free evolution time T which varies between 33.15µs and 99.45µs, and thus thekick is very small compared to T and our setup can approximate a δ-kicked rotor. But, the difference in oursetup is that the momentum space is continuous, while for the ideal rotor the momentum space is discrete.

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Figure 1: Block diagram. For description see text.

3.1 Block Diagram and Alignment

See Fig. 1 for a simplified block diagram of the experimental setup.We use the Tektronix AFG 3252 Arbitrary Waveform Generator to produce two kick-laser pulse sequences

from channel 1 and channel 2. Both pulses travel through an amplifier and through an acoustic-opticalmodulator to control the intensity and frequency of the laser beams that also pass into the modulator. Thelaser beams, originating from the 780nm diode kick laser, then continue along a fiber and into the vacuumchamber from opposite sides. The mirrors are adjusted so that the lasers completely overlap, such that thelaser from one side enters back into the fiber on the other side.

To align the setup, we set two 80MHz (plus desired offset frequency), 3.5Vpp sine waves exiting fromchannel 1 and channel 2 of the waveform generator. The 80MHz is the frequency shift we put into the AOMto shift the laser frequency. We first adjust the different mirrors so that we detect on a power meter thelight coming out of both fibers. We then make a crude alignment by observing the laser light on an infraredsensor card and adjusting the mirrors until the light overlaps as much as possible. For a finer alignment, weplace the detector on the entering end of one of the fibers and tune the mirrors until laser light coming fromthe other channel is detected exiting the fiber.

Once the two laser beams are correctly overlapping, we can input our desired pulse sequences into thewaveform generator.

3.2 Generating Pulse Sequence

The Tektronix arbitrary waveform generator comes with the program ArbExpress to create an arbitrarywaveform. We use this program to develop a pulse sequence of the form

sin(2πft) ∗ (n pulse sequence with Tpulse, Tfree) (39)

where Tpulse=300ns and Tfree=33.15µs, 66.3µs or 99.45µs, corresponding to l=1,2 or 3. The amplitude ofthe pulses is set to 3.5Vpp.

The following is an example equation used in the program to develop a pulse sequence, with f=80MHz,Tpulse=300ns, Tfree=66.3µs and n=2.

range(0,133.2us)k1=80e6sin(2*pi*k1*t)

8

range(0,300ns)v*1.75range(300ns,66.6us)v*0range(66.6us,66.9us)v*1.75range(66.9us,133.2us)v*0When creating a pulse sequence like the one above, we must set the number of samples for a given

sampling rate to achieve the correct total period:

No. ofSamples =Sampling Rate

Total Period. (40)

For example, with a sampling rate of 250MSamples/second for the above total period of 133.2µs, we mustset the total number of samples to be 33300.

An example pulse sequence generated is in Fig. 2. This is for the same parameters as in the aboveexample, except here n=4.

Figure 2: Pulse sequence on scope, Tpulse=300ns, Tfree=66.3µs, n=4. Each pulse is made up of a rapidlyoscillating sine wave.

Compared to the free period Tfree, which ranges between 33.15µs and 99.45µs, the 300ns pulse Tpulse isvery short in time. So, this is indeed a representation of a delta kick and can be used to observe quantumresonances and anti-resonances.

We use this ArbExpress equation editor to create a series of pulse sequences for each Tfree. We set thefirst laser pulse sequence to have a frequency of 80MHz and vary the second pulse sequence from a frequencyof 80MHz + 3.77kHz up to 80MHz + 30.16kHz, to correspond with the frecoil derived from equation 2.4.27.This 80MHz plus an offset is what we put into an acoustic-optical modulator to shift the laser frequency ofabout 1015Hz by the desired amount. For each Tfree, we run over a range of frequency differences betweenthe two laser beams, which correspond to different quasimomentum values β from zero to two recoils. Inthis way we can observe the quantum resonances and anti-resonances of the Rb atoms, for both n=2 andn=4 number of pulses.

3.3 Observing Overlapping Pulses

We use a photo-detector to make sure the pulse sequence programmed into the arbitrary waveform generatordoes indeed create the correct pulsed laser sequence. An example block diagram is in Fig. 3 to check the

9

pulse sequence that is output by channel 1 of the waveform generator.

Figure 3: Block diagram for pulse detection, Tpulse=300ns, Tfree=66.3µs and n=4.

The equipment is set up such that the focus of the laser beam is at about the center of the MOT;therefore, we place the photo-detector about the same distance away from the mirror so that it is at thefocus. In this way we can observe on the scope both the pulse produced by the waveform generator and thepulse at the center of the MOT, as shown in Fig. 4.

Figure 4: Pulse shown on scope. Top graph is pulse from arbitrary waveform generator, lower graph is pulseat center of MOT.

The channel 1 graph shown in Fig. 4 is the 3.5Vpp, 300ns pulse produced by the waveform generator.The lower channel 2 graph is the 300ns pulse observed by the photo-detector of about 1mV. So, we canconfirm that the pulse generated by the waveform generator does travel through to the center of the MOT.As seen above, there is a time delay between when the pulse is produced by the waveform generator andwhen it is observed by the photo-detector, but this delay will not be important as long as it is the samedelay for both counter-propagating lasers, such that the pulses overlap at the same time in the MOT.

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We observe this pulse for both counter-propagating beams, as shown in Fig. 5, but unfortunately thetime delay is not the same for the two different beams.

Figure 5: Pulses at center of MOT from the counter-propagating lasers.

This difference in the time delay is most likely due to differences in the two amplifiers that the pulsesequences are sent through, since the two amplifiers used are not the same model. The offset is only about140ns between the two pulses, but compared to our 300ns pulse it is much too large of a difference to ignore.The long-term solution to this problem is to make a new amplifier to match one of the two existing amplifiers,but since we find a constant offset of about 140ns between channel 1 and channel 2, for a short-term solutionwe add 140ns to the beginning of the channel 2 pulse sequence to cancel the offset. We do not need to knowthe offset to a higher precision because differences of 5ns or so will not substantially affect the pulse overlap.

3.4 Running the Experiment

Once the pulse sequences of the two counter-propagating beams are set correctly, running the experimentitself is straight-forward. We program into the Arbitrary Waveform Generator the two pulse sequences withour desired frequency difference β for a set value of l. Then using the computer interface that controls theexperimental setup, we run the sequence.

First the CO2 laser trap is Lorentzian ramped from 100µW down to a final power of between 60 and70µW, allowing the atoms to fall into a BEC. Next a shutter is opened, allowing the kick laser, which is inthe desired pulse sequence, to interact with the atoms. We wait a set expansion time of 5ms and then turnon the probe laser for only 0.1ms. The CCD camera detects a shadow image showing the absorption of theprobe laser by the atoms. We repeat the process for our range of β, recording multiple trials for each dataset to generate the error in the data. We in turn repeat the process for l=1, 2 and 3.

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4 Experimental Results

4.1 Contour Plots

The contour plots in Fig. 6 show definite quantum resonances and anti-resonances as predicted by ourtheory. For example (compared to Section 2.4), we can see that for n=2 kicks, the β=0 first plot on theleft for each is a resonance for l=2 and an anti-resonance for l=1 and l=3. For the resonance the atomsare kicked out into many momentum states, while for the anti-resonance almost all of the atoms are in thecenter momentum state. Please see the Appendix section for additional n=4 contour plots.

Figure 6: Quantum resonance and anti-resonance contour plots for n=2 kicks. Starting top left, for l=1,2and 3. For each plot, running through β from 0 to 2 recoils. This corresponds to a frequency differencebetween the counter-propagating laser beams of 0*3.77kHz to 8*3.77kHz.

4.2 Energy vs. Initial Momentum β

To better quantify these quantum resonances that we observe in the contour plots, we calculate the energyof ψ for each β, both in units of recoils, and compare it to our predicted energy curve for each value of n andl. To obtain the energy for a specific data set, we input the data set into a Matlab code that fits a curve tothe data set peaks. The program then uses that fitted curve to generate the energy as defined by equation2.5.38. The error for the data points is generated by finding the standard deviation in the energy for eachtrial of a certain data set, compared to the energy of the average of all three data sets.

We first look at the n=2 kick data. Fig. 7 shows a close correlation between the data in red and thetheoretical black curve. for l=3, 99.45us the data points are slightly outside the theoretical curve, but theresonance/anti-resonance pattern is still very visible. Particularly for l=3 the β=0 anti-resonance state seemsto not completely fall back into the center state, as can also be noted in the contour plot.

For n=4 kicks in Fig. 8, we can see that the resonance/anti-resonance pattern here too agrees with thetheory. We were not able to take l=3 data for 4 kicks because it required too many sampling points. Inthe l=2 66.3us graph, the resonance of the center peak is offset to slightly lower values of β. This may have

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Figure 7: Energy in units of recoils vs. β in recoil momentum, for both the data and simulations, n=2.

Figure 8: Energy in units of recoils vs. β in recoil momentum, for both the data and simulations, n=4.

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to do with an assymetry between the kicking of the atoms into the momentum states to the left and rightof the zero momentum state. Experimental effects such as vibrations in the path of the laser beam or theassymetries in our optical trap may cause the atoms to be kicked farther in one direction than in the oppositedirection from the zero state. This effect can similarly be seen in the contour plots, where sometimes themomentum state to one side is slightly farther away than the momentum state to the other side.

4.3 No. of Atoms vs. Momentum for Each Recoil Momentum

A second way to quantify these results is by calculating the number of atoms vs. momentum. The probabilitydensity |Ap′(t)|2 gives the number of atoms present in a certain momentum state p’. We include in Fig. 9the resultant curves for n=2, l=1, 2 and 3. The y axis includes both the number of atoms and the recoilmomentum β, and the x axis is in units of momentum, with the zeroith momentum in the center.

Figure 9: No. of Atoms vs. momentum for each recoil momenta β, n=2. Red solid curve is experimental,black dotted curve is theoretical. Starting from top left, for l=1,2 and 3.

The data and theory agree reasonably well, although sometimes for the anti-resonance the data is not

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completely kicked back into the zero momentum state. Note that the data and the theory did not have thesame normalization, so we rescaled the theoretical curves to have a center peak that is the average of theβ=0 and β=2 center peaks of the data.

4.4 Comparison to Theoretical Results from M. Saunders, et al.

M. Saunder’s group at Durham University, U.K. has done theoretical simulations on quantum resonances andanti-resonances [11]. We can compare our data to their simulations as well, to cross-reference the validity ofour results. Fig. 10 shows the number of atoms curves for two kicks compared to M. Saunders’ simulations.

Figure 10: No. of Atoms vs. momentum for each recoil momenta β compared with M. Saunders et al.. Redsolid curve is experimental, black dotted curve is theoretical. Starting from top left, for l=1,2 and 3.

A very similar patter is seen, although sometimes the M. Saunder’s theory predicts a center peak whenwe do not have a center peak in the data.

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5 Conclusions and Future Prospects

In this paper we have performed an experiment to exhibit the quantum resonance of a delta-kicked cloud ofcold Rubidium atoms and shown how our experimental results agree with the outlined theoretical predictions.To achieve this end we have explained the process of transforming our counter-propagating beams from astanding wave into a moving standing wave and have explained the experimental setup we used to achieveour results.

Our atoms did indeed exhibit quantum resonance and anti-resonance, as compared to our theoreticalsimulations. This resonance shows a fundamental behavior of the delta-kicked rotor in the field of quantumchaos.

This experiment used short 300ns pulses to kick the atoms so quickly that it was basically an instantaneouskick, allowing us to display quantum resonance. The next step in our lab is to build on this method towardthe creation of an atom laser. If we instead use a longer and weaker pulse in our pulse sequence and alarger frequency difference between counter-propagating beams, the atoms will have time to significantlychange their position, leading toward a coherent directed flow of atoms. The weaker interaction strengthwill generate the Bragg scattering needed for an atom laser [6].

The atom laser, first developed 1997 [14], has applications in precision measurements of fundamentalconstants and in atomic beam deposition, but in our lab we will use the atom laser to accurately retrieveone-by-one counting statistics of our atoms.

6 Acknowledgments

I would like to thank my supervisor, Professor Maarten Hoogerland, for allowing me this chance to startwork in the exciting field of laser physics. I also acknowledge Professor Reiner Leonhardt for the effort heput into developing a special course for me and for putting up with my worrying about it.

Additionally, Arif Ullah, a Ph.D. student in the lab, was especially kind in helping me get used to thelab equipment and explaining details about the experiment.

7 References

[1] H. Ammann, R. Gray, I. Shvarchuck, and N. Christensen. Quantum delta-kicked rotor: Experimentalobservation of decoherence, Phys. Rev. Lett. 80 4111-4115 (1998).[2] M. B. d’Arcy, R. M. Godun, M. K. Oberthaler, D. Cassettari, and G.S. Summy. Quantum enhancementof momentum diffusion in the delta-kicked rotor, Phys. Rev. Lett. 87 074102 (2001).[3] Mark G. Raizen. Quantum chaos with cold atoms. Advances in Atomic, Molecular and Optical Physics41 (1999), 43-81.[4] J. Ringot, P. Szriftgiser, J.C. Garreau, and D. Delande. Experimental evidence of dynamical localizationand delocalization in a quasiperiodic driven system. Phys. Rev. Lett. 85 2741-2744 (2000).[5] M. E. K. Williams, M. P. Sadgrove, A. J. Daley, R. N. C. Gray, S. M. Tan, A. S. Parkins, ChristensenN., and R. Leonhardt. Measurements of diffusion resonances for the atom optics quantum kicked rotor. J.Opt. B: Quantum Semiclass. Opt. 6 (2004), 28-33.[6] Fabienne, Catherine Haupert. Diffraction of a Bose-Einstein Condensate and the Path to an Atom Laser.Master’s thesis, University of Auckland, 2007.[7] F. L. Moore, J. C. Robinson, C. F. Bharucha, B. Sundaram, and M. G. Raizen. Phys. Rev. Lett. 75,4598 (1995).[8] W. H. Oskay, D. A. Steck, V. Milner, B. G. Klappauf, and M. G. Raizen. Opt. Commun. 179, 137(2000).[9] C. F. Bharucha, J. C. Robinson, F. L. Moore, B. Sundaram, Q. Niu, and M. G. Raizen. Phys. Rev. E60, 3881 (1999).

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[10] F.M. Israilev. Simple Models of Quantum Chaos: Spectrum and Eigenfunctions. Institute of NuclearPhysics. Physics Reports 196, 5-6 (1990). 299-392.[11] M. Saunders, P.L. Halkyard, K.J. Challis, and S.A. Gardiner. Manifestation of Quantum Resonancesand Antiresonances in a Finite-Temperature Dilute Atomic Gas. Phys. Rev. A 76, 043415 (2007).[12] Wayper, Stephanie. The Delta-Kicked Rotor and Construction Towards a Dipole Trap. Master’s thesis,University of Auckland, 2005.[13] G. Casati, B. V. Chirikov, J. Ford and F. M. Izrailev, Lect. Notes Phys. 93 (1979) 334.[14] H. J. Metcalf and Peter van der Straten. Laser Cooling and Trapping. Springer-Verlag. Berlin: Heidel-berg, 1999.

8 Appendix

8.1 Contour Plots for n=4 Kicks

Here are additional contour plots for 4 kicks.

Figure 11: Quantum resonance and anti-resonance contour plots for n=4 kicks, l=1 and 2. For each plot,running through β from 0 to 2 recoils.

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