Cent. Eur. J. Phys. • 10(2) • 2012 • 335-341DOI: 10.2478/s11534-012-0006-2

Central European Journal of Physics

Oscillations in a parametrically excited Bose-Einsteincondensate in combined harmonic and optical latticetrap

Research Article

Priyanka Verma1, Aranya B. Bhattacherjee2∗, Man Mohan1

1 University of Delhi, Department of Physics and Astrophysics,Delhi-110007, India

2 University of Delhi (South Campus), Department of Physics, ARSD College,New Delhi-110021, India

Received 16 August 2011; accepted 18 January 2012

Abstract: In this work, we study parametric excitations in an elongated cigar-shaped BEC in a combined harmonictrap and a time dependent optical lattice by using numerical techniques. We show that there exists a relativecompetition between the harmonic trap which tries to spatially localize the BEC and the time varying opticallattice which tries to delocalize the BEC. This competition gives rise to parametric excitations (oscillationsof the BEC width). Regular oscillations disappear when one of the competing factors, i.e. the strength ofharmonic trap or the strength of optical lattice, dominates. Parametric instabilities (chaotic dynamics) arisefor large variations in the strength of the optical lattice.

PACS (2008): 03.75.-b,03.75.Kk,03.75.Lm

Keywords: Bose Einstein condensate • optical lattice • parametric excitation© Versita Sp. z o.o.

1. Introduction

A periodic variation of a system parameter leads tothe exponential growth of certain modes of the system.This phenomenon is known as parametric resonance andoccurs in a wide variety of systems such as classicaloscillators, nonlinear optics, system governed by non-linear Schrödinger equations and in Hamiltonian chaoticsystems. Parametric excitations in the form of Faraday∗E-mail: bhattach@arsd.du.ac.in

waves, were observed in a cigar-shaped Bose-Einsteincondensate (BEC) by periodically modulating the radialtrap frequency [1]. The modulation of the radial trapfrequency leads to a periodic modulation of the density ofthe cloud in time, which in turn leads to a periodic changein the nonlinear interactions. This leads to the parametricexcitation of longitudinal sound-like waves (Faradaywaves) in the direction of weak confinement. From thetheoretical side, parametric excitations have already beeninvestigated by numerous authors [2–12]. A BEC in anoptical lattice potential exhibits parametric excitationswhen the depth of the optical lattice potential is peri-odically modulated in time. These parametric excitations335

Oscillations in a parametrically excited Bose-Einstein condensate in combined harmonic and optical lattice trap

correspond to an exponential growth of the populationof counter-propagating Bogoliubov excitations [13, 14].When the frequency of oscillation of the scattering lengthis an even multiple of the radial or axial natural oscilla-tion frequency, respectively, the radial or axial oscillationof the condensate exhibits resonance [5]. Parametricresonances in a BEC under periodic oscillations of theposition of a quadratic plus quartic trap have also beenpredicted [7]. Experimentally, Stöferle et al. [19, 20]measured the excitation spectrum of the condensate inthe superfluid regime by time modulating the opticallattice depth. The width of the expanding cloud after itis released from the trap is taken as a measure of theexcitation energy. The observed resonances in these ex-periments were explained by the generation of parametricamplification of Bogoliubov states and the subsequentnonlinear dynamics, which leads to the broadening of themomentum distribution [13, 14]. Parametric resonancesof trapped condensates under the action of periodic mod-ulation of the trap’s strength have been studied earlier[15–18]. Various dynamical effects in periodically time-modulated optical lattice potentials were studied in [16].In this work, we study parametric excitations in an elon-gated cigar-shaped BEC in a combined harmonic trapand a time dependent optical lattice by using numericaltechniques. In order to distinguish the present work fromparametric excitations studied earlier, we always considerthe case when the frequency of oscillation of the opticallattice depth is not equal to the axial or radial naturaloscillation frequency. We show that there exists a relativecompetition between the harmonic trap which tries tospatially localize the BEC and the time varying opticallattice which tries to delocalize the BEC. This competitiongives rise to parametric excitations (oscillations of theBEC width). Regular oscillations disappear when oneof the competing factors i.e strength of harmonic trap orthe strength of optical lattice dominates. On one hand,parametric instabilities (chaotic dynamics) arise for largevariations in the strength of the optical lattice and on theother hand, stable oscillations of the condensate widthoccurs when the harmonic trap strength dominates.2. Model

We consider an elongated cigar shaped Bose-Einsteincondensate(BEC) of N two-level 87Rb atoms in the |F =1〉 state with mass m and frequency ωa of the |F = 1〉 →|F ′ = 2〉 transition of the D2 line of 87Rb in combined har-monic and optical lattice potential. In order to create anelongated BEC, the frequency of the harmonic trap alongthe transverse direction should be much larger than one in

the axial (along the direction of the optical lattice) direc-tion. This system is described by the standard mean-fieldmodel for the BEC, the GP equation in 3D which isi~∂ψ(r, t)

∂t = [− ~22m∇2 + V ′(x, y, z, t)+NU0|ψ(r, t)|2]ψ(r, t), (1)

where ψ(r, t) is the condensate wavefunction, m is themass of an atom, N represents the total number of theatoms in the condensate, a is the atomic scattering length,which can be tuned by the Feshbach resonance technique,and U0 = 4π~2a/m is the three dimensional inter-atomicinteraction. The normalization condition of the order pa-rameter is ∫ |ψ(r, t)|2dr = 1. The confining potential forthe BEC is given byV ′(x, y, z, t) = γ2m (ω2

xx2 + ω2yy2 + ω2

zz2) + V OL1D (t), (2)where ωx = νω,ωy = κω, and ωz = λω represent thetrapping frequencies in the respective directions. We haveassumed that harmonic potential in the x-direction is su-perimposed by a 1D OL potential of the form

V OL1D (t) = V sin2 (2πxλl

) (1 + α sin(Ωt)) , (3)where V is the amplitude of the OL potential. We havechosen the 1D OL potential to make the numerics simplerbut experimentally it is possible to create a 1D OLpotential inside a cavity [21, 22]. In a cavity, 1D standingwave can be formed which acts as an 1D optical lattice.K = 2π

λland λl is the wavelength of the laser beam usedto create the 1D OL potential. γ is the parameter rep-resenting the strength of harmonic confinement such that0〈γ〈1. In all of our calculations γ ∼= 0.001 shows veryweak and γ = 1 shows very strong harmonic confinement.Ω is the frequency of the external perturbation at anytime t in the time dependent 1D OL potential with theparameter α < 1. We are interested in studying theeffect of a time dependent potential on the dynamics ofthe cigar-shaped BEC. Taking the y and z directionsto be transverse and x as longitudinal we assume theconfinement in the transverse directions to be sufficientlystrong (ωz , ωy ωx ) so that the transverse degrees offreedom (y, z) freeze out and the condensate motion getsconfined to the longitudinal direction only and the BECis quasi-1D. Note that we are studying localized states

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which are low-lying states of the system with the givenoptical potential [23]. They are distinct from gap solitonswhich have finite spatial extension and are excited statesof the system without a linear counterpart.To solve the above problem numerically, we first make Eqn.1 dimensionless through a set of linear transformationswhich we will use throughout the paper: t → τω , x →

Xl , y → Y l , z → Zl , l = √~mω , and ψ(x, y, z, t) =

φ(X,Y ,Z ,τ)√l3 . After these transformations Eqn. 1 becomes

i ∂φ(X, Y , Z , τ)∂τ = [− 12

(∂2∂X 2 + ∂2

∂Y 2 + ∂2∂Z 2

)+γ2 (ν2X 2 + κ2Y 2 + λ2Z 2)+ V OL1D (τ)

~ω +η3D |φ(X, Y , Z , τ)|2]φ(X, Y , Z , τ), (4)

where η3D is the non-linearity in the GP equation. Toreduce the above Eqn to a quasi-1D form it is assumedthat the system remains confined to the ground state inthe transverse direction. By integrating the above Eqnwith respect to the ground state transverse wavefunctionsusing the method of separation of variables, and settingν=1, we obtain the quasi-1D form as

i ∂φ(X, τ)∂τ = [− 12 ∂2

∂X 2 + (γ2X 2 + V OL1D (τ)~ω

)+η1D |φ(X, τ)|2]φ(X, τ), (5)

where η1D = 2aN√λκ/l with the normalization∫ +∞−∞ φ(X, τ)dX = 1 and where we assume l to be theharmonic oscillator’s length in the longitudinal direction.Also, VOL1D (τ)

~ω = AT (τ) sin2 ( 2πXλl

), where A = Vo 2π2λ2l ,

T (τ) = (1+α sin(ωrτ)) , Vo = V /ER , λl = λll , ωr = Ω/ωx ,and ER = ~2K 22m .Therefore we can rewrite the above Eqn. 5 asi ∂φ(X, τ)

∂τ =[− 12 ∂2

∂X 2 + (γ2X 2 + AT (τ) sin2 (2πXλl

))+η1D |φ(X, τ)|2]φ(X, τ). (6)

3. Numerical results

In the numerical simulations, we have used the split-stepCrank-Nicolson method to solve the GP equation in onespace variable. We have used the imagetime1D.F andrealtime1D.F codes [24]. The ground state solution isfound by imagetime1D.F code with NPAS = 100000,NRUN = 20000 with space step dX = 0.01, time stepdτ = 0.0002, and the initial Gaussian wavefunctionφ(X, τ = 0) = e−X2/2/√π. Then, to study the time evolu-tion of the GP equation, we have used the realtime1D.Fcode with dτ = 0.001. The results of the numericalsimulations are now presented below.To perform a systematic numerical study of the aboveEqn. 6, we take the values λl = 4.0, Vo = 10.0, andωr = 0.5. We have taken the maximum depth of theoptical lattice as 10 times the recoil energy. From thetheoretical point of view, the full numerical simulation ofthe non-linear Schrödinger equation takes into accountall of the bands. From the experimental point of view[25], it was demonstrated that for a 1D optical latticethe potential depth of the optical lattice depth needed toobserve the superfluid to Mott transition is substantiallylower than that in 3D. In our case, in order for the resultsto be valid, the density (the number of atoms per latticesite ) should be less than unity. This basically meanswe are ignoring the Mott state and that we are alwaysin the superfluid regime. For λ = κ = 100 and a longoptical lattice (we have used X = 1000), density 〈1 canbe achieved. Note that we are always in this regime withthe maximum density not exceeding 0.8. In the figures,we only show a truncated X scale. We have checked ourresults for much lower values of the optical lattice also(5 recoil energy) and have found that the results are thesame except that the onset of chaotic dynamics startsat a later time. To perform the simulations we have firstprepared the ground state wavefunction of BEC using theimaginary time propagation method with the V (X, τ) atτ = 0, and then this ground state is propagated in thereal time with V (X, τ) = γX22 + AT (τ) sin2 ( 2πX

λl

). Wehave taken η1D = 17.0 for the numerical simulation. Whenthe frequency of oscillation of the scattering length isan even multiple of the radial or axial natural oscillationfrequency, respectively, the radial or axial oscillationof the condensate exhibits resonance [5]. In order todistinguish the present work from parametric excitationsstudied earlier [5], we always consider the case whenthe frequency of oscillation of the optical lattice depthis not equal to the axial or radial natural oscillationfrequency. The natural frequency of axial oscillations is2ω, so in order to stay away from resonance we havetaken ωr = 0.5 as mentioned above.337

Oscillations in a parametrically excited Bose-Einstein condensate in combined harmonic and optical lattice trap

Fig. 1 shows the time evolution of the rms value ofthe condensate axial width 〈Xrms〉 in a weak harmonicpotential γ = 0.02 for four different values of the pertur-bation strength: α = 0.01 (plot a1), α = 0.1 (plot a2),α = 0.2 (plot a3), and α = 0.5 (plot a4). We clearly seethe competing effect of the harmonic potential and theperiodic modulation of the intensity of the optical lattice.The harmonic potential tries to localize the condensateat the center of the trap while the periodic modulation ofthe lattice generates parametric excitations which growwith time (chaotic dynamics) and tend to delocalize thecondensate. In plot a1, the periodically varying partof the lattice, α sinωrτ , is weak and varies betweenzero and α . The oscillations seen in plot a1 are aresult of competition between the harmonic potential andα sinωrτ . Due to the weak value of the perturbing factorα , the harmonic potential does not allow the condensateto become unstable and the balance between both thecompeting factors induces continuous oscillations in theamplitude of rms value of the condensate width. Nowas we increase the perturbing strength α to 0.1 (plota2), amplification of the parametric excitations begins toappear after a certain time interval. After a certain time,the harmonic potential is no longer able to stabilize andlocalize the condensate and there is an unprecedentedgrowth of the parametric excitations and the condensatebecomes unstable (the condensate width increases indef-initely like that of an impact oscillator [1]). On increasingα further to 0.2 (a3) and 0.5 (a4), the parametricamplification and hence instability of the condensate ismore pronounced and the growth of the unstable modesstarts almost immediately and the harmonic potential iscompletely unable to stabilize the condensate.In Fig. 2, we now show the dynamics of the rms valueof the condensate width in slightly stronger harmonicpotential γ = 0.1 for same values of α , α = 0.01 (plotb1), α = 0.1 (plot b2), α = 0.2 (plot b3), and α = 0.5(plot b4). Now since in Fig. 2, the harmonic potentialis stronger than that in Fig. 1, naturally the value of αneeded to make the condensate unstable and amplify theunstable modes is higher. At α = 0.1, we observe thatthe width still shows the typical growth and decay cyclesas compared to the previous case of Fig. 1, where atα = 0.1, the condensate becomes parametrically unstableafter a certain time. In this case now, the condensateis completely unstable only when α ≥ 0.2. Anotherimportant point to note is that the rms width of thecondensate in the relatively stronger harmonic potentialof Fig. 2 is much smaller as compared to that in Fig. 1.The parametric instability observed in Fig. 1 and Fig. 2for large value of α is a consequence of the exponentialgrowth of parametric excitations of the system induced

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Figure 1. Parametric excitations in the width of the BEC vs dimen-sionless time τ in a combined harmonic trap and a timemodulated optical lattice. The parameters are, γ = 0.02,(a1): α = 0.01, (a2): α = 0.1, (a3): α = 0.2, (a4): α = 0.5.

by the periodic variation of the optical lattice depth. Inorder for these excitations to be amplified, they mustpreexist at t = 0. This means that the initial state is nota pure ground state. Indeed the system is certainly notin its ground state at t = 0 due to excitations produced338

Priyanka Verma et al.

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Figure 2. Parametric excitations in the width of the BEC vs dimen-sionless time τ in a combined harmonic trap and a timemodulated optical lattice. The parameters are, γ = 0.1,(b1): α = 0.01, (b2): α = 0.1, (b3): α = 0.2, (b4): α = 0.5.

when loading the condensate into the lattice. Some nu-merical noise is always present and yields a very smalloccupation of the excitations. The exponential growth ofthe parametric instability with time is due to the nonlin-ear mean-field dynamics of the condensate governed by

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Figure 3. Density distribution |φ(X, τ)|2 at τ = 1000 vs X corre-sponding to the cases considered in Fig.1. The param-eters are γ = 0.001, L(a1): α = 0.01, L(a2): α = 0.1,L(a3): α = 0.2, L(a4): α = 0.5.

the GP equation.This nonlinear dynamics is triggered bythe growth of the parametrically unstable modes. Theoccurrence of this type of chaotic dynamics associatedwith parametric excitations in Bose-Einstein condensatesoriginates from the existence of undamped collective ex-339

Oscillations in a parametrically excited Bose-Einstein condensate in combined harmonic and optical lattice trap

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Figure 4. Density distribution |φ(X, τ)|2 at τ = 1000 vs X corre-sponding to the cases considered in Fig.1. The param-eters are γ = 0.1, L(b1): α = 0.01, L(b2): α = 0.1, L(b3):α = 0.2, L(b4): α = 0.5.

citations which grows with time. The transition to thechaotic dynamics simply is a manifestation of the generalbehaviour of driven nonlinear systems under the action ofa periodic driving force.

The growth of the rms width with time due to the periodicvariation of the optical lattice also motivated us to ana-lyze how this periodic modulation influences the spatialextension of the condensate density. Fig. 3 and Fig. 4show the density plots corresponding to Fig. 1 and Fig. 2respectively. Clearly as the strength of the perturbationα increases, the wavefunction also gets progressively de-localized and starts extending over more lattice sites. Onincreasing the strength of the harmonic potential (Fig. 4),the influence of α in delocalizing the condensate wave-function is less effective, since a stronger harmonic poten-tial now localizes the condensate to a greater extent andacts against the perturbation. The periodic modulation re-distributes the atoms in the lattice wells in a random man-ner and hence the nonuniform density distribution acrossthe lattice wells for large α is seen in Fig. 3 and Fig. 4.4. ConclusionIn this work we have explored the parametric excitations inan elongated Bose-Einstein condensate subject to a com-bined harmonic potential and 1D optical lattice whoseintensity is periodically modulated in time. On one handthe periodic modulation generates parametric excitationswhich grows with time and tends to increase the conden-sate size and hence delocalize it and on the other handthe harmonic trap tries to localize the condensate at thecenter of the trap and act against the instability createdby the parametric excitations. The ensuing nonlinear dy-namics leads to oscillations of the condensate width. Theparametric excitations and the chaotic dynamics are nicelyaccounted for by the numerical calculations.5. AcknowledgementsOne of the authors, Priyanka Verma, thanks the Univer-sity Grants Commission, New Delhi for a Junior ResearchFellowship. The authors thank P. Murugandam for dis-cussions on the numerical simulations.References

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