Nonlinear atom-optical delta-kicked harmonic oscillator using a Bose-Einstein condensate

a r X i v :c o n d -m a t /0406545v 1 [c o n d -m a t .s o f t ] 23 J u n 2004

Nonlinear atom-optical delta-kicked harmonic oscillator using a Bose-Einstein

condensate

G.J.Du?y,A.S.Mellish,K.J.Challis,and A.C.Wilson

Department of Physics,University of Otago,

P.O.Box 56,Dunedin,New Zealand

(Dated:February 2,2008)

We experimentally investigate the atom-optical delta-kicked harmonic oscillator for the case of nonlinearity due to collisional interactions present in a Bose-Einstein condensate.A Bose con-densate of rubidium atoms tightly con?ned in a static harmonic magnetic trap is exposed to a one-dimensional optical standing-wave potential that is pulsed on periodically.We focus on the quantum anti-resonance case for which the classical periodic behavior is simple and well under-stood.We show that after a small number of kicks the dynamics is dominated by dephasing of matter wave interference due to the ?nite width of the condensate’s initial momentum distribution.In addition,we demonstrate that the nonlinear mean-?eld interaction in a typical harmonically con?ned Bose condensate is not su?cient to give rise to chaotic behavior.

PACS numbers:05.45.Mt,03.75.Lm,32.80.Qk,03.65.Yz

The delta-kicked rotor is an extensively investigated system in the ?eld of classical chaos theory.During the last decade great progress has been achieved in under-standing quantum dynamics of a classically chaotic sys-tem using atom-optical techniques and cold atoms.From an experimental point of view,cold atoms in optical po-tentials [1,2,3,4,5]provide an ideal environment to ex-plore quantum dynamics.To date,all experimental work has focused on linear atomic systems,(see,for example,[6,7,8,9]and references therein)where the quantum dy-namics is stable due to the linearity of the Schr¨o dinger equation.In stark contrast to the chaotic behavior of classical dynamics,the linear quantum systems exhibit anti-resonance (periodic motion),dynamical localization (quasi-periodic motion)or resonant dynamics [10,11].Recently,theoretical investigations have considered how the nonlinearity due to many-body (collisional)in-teractions in a Bose-Einstein condensate modi?es the be-havior of the atom-optical kicked rotor system,provid-ing a route to chaotic dynamics.Gardiner et al.devel-oped a theoretical formalism to treat the one-dimensional nonlinear kicked harmonic oscillator (a particular mani-festation of the generic delta-kicked rotor)using Gross-Pitaevskii and Liouville-type equations to describe the dynamics of a Bose-Einstein condensate,and estimated the growth rate in the number of non-condensate parti-cles [12].Zhang et al.investigated the generalized quan-tum kicked rotor by considering a periodically kicked Bose condensate con?ned in a ring potential for the case of quantum anti-resonance [13].As opposed to the famil-iar periodic behavior exhibited by a corresponding linear system,they predicted quasi-periodic variation in energy for a weak interaction strength and chaotic behavior for strong interactions.

In this work we investigate the nonlinear delta-kicked harmonic oscillator by performing experiments on Bose-Einstein condensates in a harmonic potential.A Bose condensate of rubidium atoms tightly con?ned in a static

harmonic magnetic trap is exposed to a periodically pulsed one-dimensional optical standing-wave potential.Our focus is on the particular case of quantum anti-resonance for which the linear behavior is simple and well understood [14].The ?nite width of the initial conden-sate momentum distribution is shown to have a profound e?ect on the dynamics.After a small number of kicks the behavior is dominated by dephasing of matter wave in-terference.We present numerical solutions of the Gross-Pitaevskii equation which match the observed behavior and con?rm our interpretation.

In the atom-optical kicked harmonic oscillator,the ef-fective Planck’s constant ˉk can be adjusted to,in a sense,make the system “more”or “less”quantum mechanical.At speci?c values of ˉk -in particular,where ˉk is a rational multiple of 2π-quite remarkable phenomena can occur in the form of so-called quantum resonances and anti-resonances [6,15,16,17,18,19,20].In this work we fo-cus our attention on the case of the ˉk =2πanti-resonance at which the energy of a linear system exhibits simple pe-riodic behavior.This anti-resonance requires a particular initial momentum state which evolves in such a way that during the period of free evolution in between kicks,the di?erent components of the state vector of the system experience a phase shift that alternates in sign from one momentum component to the next,so that the system returns identically to its initial state after every second kick.The underlying physics of linear atom-optical kick-ing at anti-resonance has already been neatly described,albeit in a di?erent context [14].In the short pulse (thin grating)limit the ?rst kick imprints a sinusoidal phase pro?le onto the plane matter wave thereby populating a number of momentum states (di?raction orders),and the phase evolution of the n th state is proportional to n 2so that after free evolution (between kicks)corresponding to half the Talbot time (T T =h/4E r ,where the recoil energy E r =(ˉh k )2/2m ,k is the wave-vector and m is the atomic mass)the second pulse cancels the spatial

2 variation induced by the?rst.For multiple pulses this

process repeats so that the initial plane wave state is re-

constructed after every second pulse.

Bose condensate evolution in an optical standing wave,

or lattice,has previously been well described by the

Gross-Pitaevskii equation(GPE)(see,for example,[21]),

and condensate behavior in a kicked harmonic potential

can be described in this formalism using the one dimen-

sional GPE along the direction of the kicking beams,

i ?ψ(x,t)

?x2

?

κˉk

4

x2+C|ψ(x,t)|2 ψ(x,t),(1)

whereψ(x,t)is the condensate wave function andκ= E r?Tτp/ˉh is the classical stochasticity parameter(or kick strength)for the e?ective Rabi frequency?.Here f(t?nT)represents a square pulse,such that f(t?nT)=1for0

?x2

?

κˉk

4x2+

1

FIG.2:Momentum distribution versus kick number for the numerical simulation in Fig.1

M o m e n

t u m

K i c k n u m b e r

the momentum distribution of the condensate.The cen-tral (zero momentum)region of the initial condensate momentum distribution couples to the higher-order mo-mentum states at a slower rate than the non-zero wings of the condensate wave function.This causes,for exam-ple,the development of the double-peaked structure in the ?rst order di?raction components.As time evolves the cycling between momentum states for di?erent com-ponents of the initial distribution become progressively

4

lower kick strength.We repeated our measurements for a kick strengthκ=4.125and observed similar features to those presented in Fig.1,with the main di?erence being a smaller amplitude of the energy oscillations.We esti-mate that we would have to reduce our kick strength by a factor of100to enter the chaotic regime predicted by Zhang et al.[13].While it may seem straightforward to simply further reduce the intensity of the kicking beams, this reduces the energy of the system to the point where shot-to-shot variations exceed the predicted signal.For a kick strength lower thanκ≈4the signal to noise is compromised and the energy of our system becomes im-measurable.

In summary,we experimentally investigated the possi-bility of using nonlinear collisional interactions in a typi-cal Bose-Einstein condensate to observe chaotic dynamics in the quantum-kicked harmonic oscillator system.We applied a pulsed,far-detuned,optical standing wave to a rubidium Bose condensate,and measured the system energy as a function of kick number for the case of the quantum anti-resonance condition atˉk=2π.We found that,even in the presence of nonlinear interactions,our system exhibits the well-known periodic behavior associ-ated with the linear https://www.sodocs.net/doc/6013745267.html,ing numerical solutions to the Gross-Pitaevskii equation,we showed that observed dephasing of the oscillations is due to the?nite width of the condensate’s initial momentum distribution.This severely limits the possibility of observing an extended period of chaotic behavior in the energy of the system.

Acknowledgments

The authors acknowledge the support of the Mars-den Fund of the Royal Society of New Zealand(grant 02UOO080)and KJC acknowledges the support of FRST Top Achiever Doctoral Scholarship(TAD884).We thank Simon Gardiner and Scott Parkins for helpful dis-cussions.

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