Investigations into Quantum Resonance and Anti-Resonance of Cold Rb87 Atoms Interacting with an Optical Moving Standing Wave

Transcription

1 Investigations into Quantum Resonance and Anti-Resonance of Cold Rb87 Atoms Interacting with an Optical Moving Standing Wave Jean Anne Currivan In collaboration with Maarten Hoogerland, Arif Ullah and Rémi Blinder July 14, 2008 Abstract Context. By creating a small frequency difference between the two counter-propagating lasers in our optical trap, 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 can explore the quantum resonances and anti-resonances of the δ-kicked rotor, which is an important example of quantum chaos. Aims. The goal of this paper is to exhibit the quantum resonances and anti-resonances of our Rubidium 87 atoms when they interact with a moving standing wave and to compare these data results with the theoretical expectations of the experiment, for different initial parameters. Methods. We generated a kicking pulse sequence that interacts with our cold cloud of Rb87 atoms. We then varied the number of kicks, the frequency difference between the counter-propagating waves that make up the pulse sequence and the free period between kicks in the pulse sequence to observe quantum resonances and anti-resonances for these different parameters. Results. We detected quantum resonances and anti-resonances for both 2 and 4 kicks, for a frequency difference from 0kHz to 30.16kHz, and for a free period difference of 33.15µs, 66.3µs, and 99.45µs. We compared these results with our theoretical predictions. 1

2 Contents 1 Introduction 2 2 Theory The Classical Standard Mapping Standard Mapping in Quantum Regime for a Strong Perturbation Applying to Our Experimental Setup: Dilute Atomic Gas in Pulsed Optical Standing Wave The Atomic Hamiltonian The Laser Field and its Interaction with the Atom Quantum Resonance and Anti-Resonance Calculating Energy Experimental Setup Block Diagram and Alignment Generating Pulse Sequence Observing Overlapping Pulses Running the Experiment Experimental Results Contour Plots Energy vs. Initial Momentum β No. of Atoms vs. Momentum for Each Recoil Momentum Comparison to Theoretical Results from M. Saunders, et al Conclusions and Future Prospects 16 6 Acknowledgments 16 7 References 16 8 Appendix Contour Plots for n=4 Kicks List of Figures 1 Basic block diagram Pulse sequence on scope Block diagram for pulse detection Single pulse shown on scope Pulses at center of MOT from the counter-propagating lasers Quantum resonance and anti-resonance contour plots for n=2 kicks Energy vs. initial recoil momentum, n= Energy vs. initial recoil momentum, n= No. of Atoms vs. momentum for each recoil momenta β, n= No. of Atoms vs. momentum, n=2 compared with M. Saunders et al Quantum resonance and anti-resonance contour plots for n=4 kicks

3 1 Introduction The Bose-Einstein Condensate (BEC) created in our lab interacts with a standing wave generated from two counter-propagating lasers with equal amplitude, frequency and phase. If we instead set a small frequency difference between the lasers, i.e. ω << ω, we will generate a moving standing wave. With the atoms in the BEC moving relative to the laser trap, we can perform diffraction experiments. In the canonical diffraction experiment, light encounters a matter grating; conversely, here the atoms encounter an optical grating, and we can observe the resulting quantum interference effects. In this paper we explore one major quantum interference effect: the quantum resonance and antiresonance of cold atoms when interacting with a pulsed laser field. This is an experimental realization of the δ-kicked rotor. As the quantum analog of the standard mapping of dynamical chaos, the δ-kicked rotor is an important area for understanding how classically chaotic systems act in the quantum regime. By looking at its classical behavior in the regime of very strong kicking over a longer time versus its quantum behavior when the kicks are less strong and in the time just after kicking, we can better understand the transition between classical and quantum regimes. The resonances and anti-resonances of this δ-kicked rotor are a specifically quantum feature that can further our understanding of this transition, and of the behavior of classically chaotic systems when in the quantum regime. In the 1990s, with new advancements in atom-cooling techniques, many kicked rotor experiments were performed [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 kicks needed for the system to reach the classical regime. The realization of a BEC in 1995, with a well-defined initial atomic state and much colder temperatures [6], lead to a new range of experiments that were able to focus on the specifically quantum nature of the δ-kicked rotor soon after the kicks were applied. It was found 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 we seek to lay out the quantum resonance and anti-resonance phenomena for a variety of initial parameters, so as to give a fuller analysis of how these resonances depend on the initial momentum of the atoms and the period between kicks. Part two of this paper lays out the theory of quantum resonance, from which we develop theoretical simulations to compare with the data results. Part three outlines the experimental setup, including the programming of the arbitrary waveform generator, the process of overlapping the two laser beams and running the experiment itself. Then in part four we discuss the data analysis and experimental results and in 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]. A quantum resonance occurs if, when kicked n times with strength ɛ 0, the system acts as if it was kicked only once with strength nɛ 0. Additionally, while usually in time the energy of the system increases linearly as the atom cloud expands, at a quantum resonance the energy increases quadratically. An anti-resonance occurs when 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-kicked rotor. 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 kicked gravity field. In general, the Hamiltonian is given as Ĥ general = H o (p) + V (θ)f(t), withf(t + T ) = f(t). (1) 2

4 Here H o (p) is the unperturbed Hamiltonian and V (θ)f(t) is the nonlinear perturbation, periodic with period T between kicks. We can think of the external kicking field as a periodic delta function, and thus the Hamiltonian can be written as δ T (t) = t= δ(t tt ), (2) H = p2 2I + ɛ 0 cos θ δ T (t) (3) where p is the angular momentum, ɛ 0 is the kicking strength, θ is the angular displacement and I is the moment of inertia of the pendulum. The standard mapping is obtained by getting the equations of motion from H and then integrating over one period T. Thus we find the standard mapping to be P t+1 = P t + K sin θ t ; Θ t+1 = Θ t + P t+1 (4) where P pt, K = ɛ 0 T. This standard mapping is used to gain understanding of dynamical chaos. It can be used to describe the change in time of a generic Hamiltonian system, in the case that there is one isolated, 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 perturbation is in the regime of K >> 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 the pendulum kicked in a gravity field that was outlined in the previous section. We can translate that classical Hamiltonian into the quantum regime: Thus the time-independent Schrödinger equation is We assume instantaneous, periodic kicks such that Ĥ = ˆp2 2m ɛ 0 cos ˆx δ T (t). (5) i ψ t = ˆp2 2m ψ ɛ 0 cos ˆx δ T (t)ψ. (6) ψ(x, t + T ) = Ûψ(x, t), (7) where Û is a unitary operator. From the time-independent Schrödinger equation, we find [13] Û = e iɛ0 e i τ 2 2 x 2, τ T m. (8) Thus, to get ψ just after an infinitesimal kick, we apply Û to ψ. The parameter ɛ 0 can be thought of as the perturbation strength, and τ is related to the period between kicks. The first term in Û is the kicking term, and the second term is the free expansion. 3

5 2.3 Applying to Our Experimental Setup: Dilute Atomic Gas in Pulsed Optical Standing Wave We can now turn to looking at how this time evolution operator Û acts in the specified case of a dilute atomic 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 we can focus on a singe-particle Hamiltonian, similar to the discussion in the previous section. Additionally, since the gas is non-interacting, the ˆx, ŷ and ẑ directions are separable and we can just look at ψ(x) along the axis of the standing wave. Recall equation that describes the Hamiltonian of a δ-kicked rotor. We seek to achieve this same Hamiltonian 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 The Atomic Hamiltonian We will treat the atom as having two states e and g separated by E = ω 0. This gives the Hamiltonian Ĥ atom = ˆp2 2m ω 0σ z (9) where m is the mass of the atom, ω 0 is the resonant frequency between the two energy levels, and the Pauli matrix σ z is given as σ z = e e g g 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 the same. 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 to the 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 to the original frequency, but it can be shown that this frequency difference can be translated as the phase difference needed for a moving standing wave. Because we design the difference in frequency to be small compared to the original laser frequency, this frequency difference can indeed be viewed as a phase difference between the two lasers, as needed in the above theory. With no frequency difference, the counter-propagating waves are defined as y 1 = A sin(kx ωt) (10) y 2 = A sin(kx + ωt). (11) When a small frequency difference is introduced, after time T pulse y 2 becomes We know that the group velocity c of the wave is dω dk, and thus So, y 2 can be written as y 2 = A sin[(k + dk)x + (ω + dω)t pulse ]. (12) y 2 = A sin[(k + dω/c)x + (ω + dω)t pulse ]. (13) y 2 = A sin[kx + ωt + dω(x/c + T pulse) ]. (14) and therefore is the original y 2 offset by a phase of dω(x/c + T pulse ) which depends on the position and the size of the pulse. 4

6 With this moving standing wave we get the Hamiltonian Ĥ atom field int = 1 2 Ω(ei(k l ˆx ω l t+φ 1 e g ) Ω(e i(k l ˆx+ω l t φ 2 e g ). (15) Here k l is the laser wavevector magnitude, defined as k l = 2π λ where λ is the laser wavelength, in our experiment equal to 780nm. The parameter ω l is the frequency of the two laser beams and φ 1 and φ 2 are the respective laser phase differences. In our experiment ω l is set to about Hz and our change in frequency, as shown, corresponds to the phase φ. The parameter Ω is the Rabi frequency of the two laser beams, which gives the strength of coupling between the E&M field and ω 0, the frequency between the two energy levels. Our two beams are at the same intensity and thus have the same Ω, defined as Ω 2 = I power Γ 2 I saturated 2. (16) I power is the intensity of the laser for a given power, defined as I power = P ower πr, where r is the radius of 2 the laser beam. I sat is the saturation intensity, found to be 16 W/m 2 [12], and the natural linewidth of transition Γ is found to be 2π 5.9 MHz [12]. Γ is the spread in frequency about ω 0. Thus, the total Hamiltonian for both the atom and the field is found to be Ĥ total = Ĥatom + Ĥatom field int (17) Ĥ total = ˆp2 2m ω 0σ z Ω(ei(k l ˆx ω l t+φ 1 e g ) Ω(e i(k l ˆx+ω l t φ 2 e g ). (18) It is useful to manipulate this Hamiltonian by multiplying by two unitary operators: After applying these, we get e e + g g Û 1 = exp[i(ω l e e ω 0 t)], (19) 2 Û 2 = exp( iω 2 g g t ). (20) 8(ω 0 ω l ) Ĥ δ = ˆp2 2m ɛ 0 cos(k ˆx)δ T (t) (21) which is the same form as that of a δ-kicked rotor in equation Here m is the Rb87 atomic mass, K 2k l and the kicking strength ɛ 0 = Ω2 T pulse 8(ω 0 ω l ). 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 antiresonances. Recall in equation we found that the time evolution of a δ-kicked rotor is governed by Û = e iɛ0 cos ˆx e ˆp2 T 2m (22) where T is the time between pulses. For the δ-kicked rotor in our experimental setup, ˆx and ˆp are continuous, and thus can be separated into discrete and continuous parts. We let ˆx = 1 K (2πˆl + ˆθ) and ˆp = K(ˆk + ˆβ) where ˆl and ˆk are integer values and ˆθ and ˆβ are continuous: ˆθ [ π, π], ˆβ [0, 2]. Thus Û can be re-written as Û = e iɛ0 cos ˆθe i K2 T 2m (ˆk+ β 2 )2. (23) 5

7 Here ˆβ is replaced by its eigenvalue β/2 (the 2 is for convenience) since β is a conserved quantity. β is referred to as the quasimomentum. We usually choose to describe β in units of recoils, where one recoil is the change in momentum of the atom when it absorbs one photon. β can be thought of as the initial momentum of the atoms, in units of recoils. This can be translated into units of frequency. We can show how an offset in frequency between the two counter-propagating lasers leads to changing the momentum of the atom by an amount determined by the doppler shift. When absorbing a single photon, an atom changes its velocity by v 1 recoil = k l (24) 2m since the momentum of one photon is k l, where k l = 2π λ laser. But, since with the small frequency difference we create a moving standing wave with which the atom interacts, the atom moving relative to the field observes a Doppler shift of ω doppler shift = k l v. (25) If the velocity of the atom is v 1 recoil then this corresponds to a Doppler shift of k2 l 2m. We know the free propagation velocity of the atom is 2 k 2 l 2m, and so we find that KE atom = ω doppler shift. (26) This shows that our introduced frequency difference between the two counter-propagating beams does lead to a change in the initial kinetic energy of the atoms, causing them to be kicked into different momentum states. We can calculate this recoil frequency by E = ω recoil = 2 k 2 l 2m. (27) In our experiment, with λ laser =780nm, one recoil frequency corresponds to f recoil =3.77kHz. In equation for Û, 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 two counter-propagating laser beams is in units of initial momentum β, and thus the phase between the two beams is determined by β. To simplify equation we let the period between pulses, T, be defined as T = 2πm l, (28) K2 where l is a positive integer. For our Rubidium 87 atoms and with λ laser =780nm, the coefficient 2πm K = µs. Thus, to exhibit quantum resonance and anti-resonance we construct a Matlab program that, for a given β and l, applies Û to ψ for each kick up to the total number of kicks, n. For example, if β=0 then Û reduces to Û(β = 0) = e ilπˆk 2 iɛ0 cos ˆθ e (29) The eigenvalues of ˆk are integers, so e ilπˆk 2 = e ilπ(±ˆk), and therefore Û(β = 0) = e ±ilπˆke iɛ0 cos ˆθ (30) Here we can see that the time evolution of ψ according to Û depends greatly on whether l is even or odd. Let us assume we apply the kicking pulse twice, i.e. n=2. Then Û n=2 (β = 0) = e ±ilπˆke iɛ0 cos ˆθ Û n=2 (β = 0) = e ilπˆke iɛ0 cos ˆθe +ilπˆke iɛ0 cos ˆθ (31) (32) 6

8 One can find that [ˆθ, ˆk] = i [11], and thus Û n=2 (β = 0) = e iɛ0 cos(ˆθ lπ) iɛ0 cos ˆθ e Û n=2 (β = 0) = e iɛ0[1+( 1)l ] cos ˆθ If l is even, for example l=2 and thus T =66.3µs, we get (33) (34) Û n=2 (β = 0) = e i2ɛ0 cos ˆθ. (35) This is a quantum resonance: instead of acting like two kicks of strength ɛ 0 separated by T, it can be seen as acting as if a single kick was applied with a greater strength of 2ɛ 0. If l is odd, for example l=1 and thus T =33.15µs or l=3 and T =99.45µs, we get Û n=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 ψ after the kicking sequence. ψ(θ, t) can be written as an expansion of coefficients ψ(θ, t) = 1 2π P p = P A p (t)e ip θ (37) where A p (t) 2 is the probability distribution in momentum space after t number of kicks. Thus the energy can be defined as p E(t) = p 2 A p (t) 2 p A, (38) p (t) 2 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 0 to 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 energy and 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-resonance of 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 we are essentially dealing with a cloud of very cold atoms at around 50 nk. We have about atoms in the 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 kicks are 300ns, compared to the free evolution time T which varies between 33.15µs and 99.45µs, and thus the kick is very small compared to T and our setup can approximate a δ-kicked rotor. But, the difference in our setup is that the momentum space is continuous, while for the ideal rotor the momentum space is discrete. 7

9 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-optical modulator to control the intensity and frequency of the laser beams that also pass into the modulator. The laser beams, originating from the 780nm diode kick laser, then continue along a fiber and into the vacuum chamber from opposite sides. The mirrors are adjusted so that the lasers completely overlap, such that the laser 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 from channel 1 and channel 2 of the waveform generator. The 80MHz is the frequency shift we put into the AOM to shift the laser frequency. We first adjust the different mirrors so that we detect on a power meter the light coming out of both fibers. We then make a crude alignment by observing the laser light on an infrared sensor card and adjusting the mirrors until the light overlaps as much as possible. For a finer alignment, we place the detector on the entering end of one of the fibers and tune the mirrors until laser light coming from the other channel is detected exiting the fiber. Once the two laser beams are correctly overlapping, we can input our desired pulse sequences into the waveform generator. 3.2 Generating Pulse Sequence The Tektronix arbitrary waveform generator comes with the program ArbExpress to create an arbitrary waveform. We use this program to develop a pulse sequence of the form sin(2πft) (n pulse sequence with T pulse, T free ) (39) where T pulse =300ns and T free =33.15µs, 66.3µs or 99.45µs, corresponding to l=1,2 or 3. The amplitude of the pulses is set to 3.5Vpp. The following is an example equation used in the program to develop a pulse sequence, with f=80mhz, T pulse =300ns, T free =66.3µs and n=2. range(0,133.2us) k1=80e6 sin(2*pi*k1*t) 8

10 range(0,300ns) v*1.75 range(300ns,66.6us) v*0 range(66.6us,66.9us) v*1.75 range(66.9us,133.2us) v*0 When 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: Sampling Rate. (40) T otal P eriod For example, with a sampling rate of 250MSamples/second for the above total period of 133.2µs, we must set the total number of samples to be An example pulse sequence generated is in Fig. 2. This is for the same parameters as in the above example, except here n=4. N o. of Samples = Figure 2: Pulse sequence on scope, Tpulse =300ns, Tf ree =66.3µs, n=4. Each pulse is made up of a rapidly oscillating sine wave. Compared to the free period Tf ree, which ranges between 33.15µs and 99.45µs, the 300ns pulse Tpulse is very short in time. So, this is indeed a representation of a delta kick and can be used to observe quantum resonances and anti-resonances. We use this ArbExpress equation editor to create a series of pulse sequences for each Tf ree. We set the first laser pulse sequence to have a frequency of 80MHz and vary the second pulse sequence from a frequency of 80MHz kHz up to 80MHz kHz, to correspond with the frecoil derived from equation This 80MHz plus an offset is what we put into an acoustic-optical modulator to shift the laser frequency of about 1015 Hz by the desired amount. For each Tf ree, we run over a range of frequency differences between the two laser beams, which correspond to different quasimomentum values β from zero to two recoils. In this way we can observe the quantum resonances and anti-resonances of the Rb atoms, for both n=2 and n=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 generator does indeed create the correct pulsed laser sequence. An example block diagram is in Fig. 3 to check the 9

11 pulse sequence that is output by channel 1 of the waveform generator. Figure 3: Block diagram for pulse detection, Tpulse =300ns, Tf ree =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 the focus. In this way we can observe on the scope both the pulse produced by the waveform generator and the pulse 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 pulse at 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 can confirm 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 and when it is observed by the photo-detector, but this delay will not be important as long as it is the same delay for both counter-propagating lasers, such that the pulses overlap at the same time in the MOT. 10

12 We observe this pulse for both counter-propagating beams, as shown in Fig. 5, but unfortunately the time 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 pulse sequences are sent through, since the two amplifiers used are not the same model. The offset is only about 140ns 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 solution we add 140ns to the beginning of the channel 2 pulse sequence to cancel the offset. We do not need to know the 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 experiment itself is straight-forward. We program into the Arbitrary Waveform Generator the two pulse sequences with our desired frequency difference β for a set value of l. Then using the computer interface that controls the experimental setup, we run the sequence. First the CO 2 laser trap is Lorentzian ramped from 100µW down to a final power of between 60 and 70µW, allowing the atoms to fall into a BEC. Next a shutter is opened, allowing the kick laser, which is in the desired pulse sequence, to interact with the atoms. We wait a set expansion time of 5ms and then turn on the probe laser for only 0.1ms. The CCD camera detects a shadow image showing the absorption of the probe laser by the atoms. We repeat the process for our range of β, recording multiple trials for each data set to generate the error in the data. We in turn repeat the process for l=1, 2 and 3. 11

13 4 Experimental Results 4.1 Contour Plots The contour plots in Fig. 6 show definite quantum resonances and anti-resonances as predicted by our theory. For example (compared to Section 2.4), we can see that for n=2 kicks, the β=0 first plot on the left for each is a resonance for l=2 and an anti-resonance for l=1 and l=3. For the resonance the atoms are kicked out into many momentum states, while for the anti-resonance almost all of the atoms are in the center 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=1,2 and 3. For each plot, running through β from 0 to 2 recoils. This corresponds to a frequency difference between 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 energy of ψ for each β, both in units of recoils, and compare it to our predicted energy curve for each value of n and l. To obtain the energy for a specific data set, we input the data set into a Matlab code that fits a curve to the data set peaks. The program then uses that fitted curve to generate the energy as defined by equation The error for the data points is generated by finding the standard deviation in the energy for each trial 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 the theoretical black curve. for l=3, 99.45us the data points are slightly outside the theoretical curve, but the resonance/anti-resonance pattern is still very visible. Particularly for l=3 the β=0 anti-resonance state seems to 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 the theory. We were not able to take l=3 data for 4 kicks because it required too many sampling points. In the l=2 66.3us graph, the resonance of the center peak is offset to slightly lower values of β. This may have 12

14 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. 13

15 to do with an assymetry between the kicking of the atoms into the momentum states to the left and right of the zero momentum state. Experimental effects such as vibrations in the path of the laser beam or the assymetries in our optical trap may cause the atoms to be kicked farther in one direction than in the opposite direction from the zero state. This effect can similarly be seen in the contour plots, where sometimes the momentum 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 probability density A p (t) 2 gives the number of atoms present in a certain momentum state p. We include in Fig. 9 the resultant curves for n=2, l=1, 2 and 3. The y axis includes both the number of atoms and the recoil momentum 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 14

16 completely kicked back into the zero momentum state. Note that the data and the theory did not have the same normalization, so we rescaled the theoretical curves to have a center peak that is the average of the β=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 and anti-resonances [11]. We can compare our data to their simulations as well, to cross-reference the validity of our 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.. Red solid 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 when we do not have a center peak in the data. 15

17 5 Conclusions and Future Prospects In this paper we have performed an experiment to exhibit the quantum resonance of a delta-kicked cloud of cold 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 a standing wave into a moving standing wave and have explained the experimental setup we used to achieve our results. Our atoms did indeed exhibit quantum resonance and anti-resonance, as compared to our theoretical simulations. This resonance shows a fundamental behavior of the delta-kicked rotor in the field of quantum chaos. This experiment used short 300ns pulses to kick the atoms so quickly that it was basically an instantaneous kick, allowing us to display quantum resonance. The next step in our lab is to build on this method toward the creation of an atom laser. If we instead use a longer and weaker pulse in our pulse sequence and a larger frequency difference between counter-propagating beams, the atoms will have time to significantly change their position, leading toward a coherent directed flow of atoms. The weaker interaction strength will generate the Bragg scattering needed for an atom laser [6]. The atom laser, first developed 1997 [14], has applications in precision measurements of fundamental constants and in atomic beam deposition, but in our lab we will use the atom laser to accurately retrieve one-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 start work in the exciting field of laser physics. I also acknowledge Professor Reiner Leonhardt for the effort he put 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 the lab equipment and explaining details about the experiment. 7 References [1] H. Ammann, R. Gray, I. Shvarchuck, and N. Christensen. Quantum delta-kicked rotor: Experimental observation of decoherence, Phys. Rev. Lett (1998). [2] M. B. d Arcy, R. M. Godun, M. K. Oberthaler, D. Cassettari, and G.S. Summy. Quantum enhancement of momentum diffusion in the delta-kicked rotor, Phys. Rev. Lett (2001). [3] Mark G. Raizen. Quantum chaos with cold atoms. Advances in Atomic, Molecular and Optical Physics 41 (1999), [4] J. Ringot, P. Szriftgiser, J.C. Garreau, and D. Delande. Experimental evidence of dynamical localization and delocalization in a quasiperiodic driven system. Phys. Rev. Lett (2000). [5] M. E. K. Williams, M. P. Sadgrove, A. J. Daley, R. N. C. Gray, S. M. Tan, A. S. Parkins, Christensen N., and R. Leonhardt. Measurements of diffusion resonances for the atom optics quantum kicked rotor. J. Opt. B: Quantum Semiclass. Opt. 6 (2004), [6] Fabienne, Catherine Haupert. Diffraction of a Bose-Einstein Condensate and the Path to an Atom Laser. Master s thesis, University of Auckland, [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. E 60, 3881 (1999). 16

18 [10] F.M. Israilev. Simple Models of Quantum Chaos: Spectrum and Eigenfunctions. Institute of Nuclear Physics. Physics Reports 196, 5-6 (1990) [11] M. Saunders, P.L. Halkyard, K.J. Challis, and S.A. Gardiner. Manifestation of Quantum Resonances and Antiresonances in a Finite-Temperature Dilute Atomic Gas. Phys. Rev. A 76, (2007). [12] Wayper, Stephanie. The Delta-Kicked Rotor and Construction Towards a Dipole Trap. Master s thesis, University of Auckland, [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: Heidelberg, 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. 17