PHYSICAL REVIEW A 83, 013603 (2011)

Suppression of Faraday waves in a Bose-Einstein condensate in the presence of an optical lattice

Pablo Capuzzi,1,2 Mario Gattobigio,3 and Patrizia Vignolo3

1Departamento de Fisica, FCEN Universidad de Buenos Aires, Ciudad Universitaria, Pab. I C1428EGA Buenos Aires, Argentina2Instituto de Fısica de Buenos Aires–CONICET, Argentina

3Universite de Nice–Sophia Antipolis, Institut non Lineaire de Nice, CNRS, 1361 route des Lucioles, F-06560 Valbonne, France(Received 8 October 2010; published 10 January 2011)

We study the formation of Faraday waves in an elongated Bose-Einstein condensate in the presence of aone-dimensional optical lattice. The waves are parametrically excited by modulating the radial confinement ofthe condensate close to a transverse breathing mode of the system. For very shallow optical lattices, phonons witha well-defined wave vector propagate along the condensate, as in the absence of the lattice, and we observe theformation of a Faraday pattern. We find that by increasing the potential depth the local sound velocity decreases,and when it equals the condensate local phase velocity, the condensate develops an incoherent superposition ofseveral modes and the parametric excitation of Faraday waves is suppressed.

DOI: 10.1103/PhysRevA.83.013603 PACS number(s): 03.75.Kk, 03.75.Lm

I. INTRODUCTION

The term Faraday waves refers to a surface densitymodulation generated by the interference of excitations createdby an oscillatory motion of a nonlinear medium [1]. Theformation of Faraday waves has been studied in severalphysical contexts [2], including in convective fluids, nematicliquid crystals, nonlinear optics, and biology, and, recently,in Bose-Einstein condensates (BECs) [3]. In the contextof ultracold gases, Faraday patterns can be excited by themodulation of the scattering length [4] or by varying thetransverse trap confinement [3,5–9], the main ingredient beingthe modulation of the nonlinear interaction. The study of theformation of spatial structures in a quantum fluid confined ina smooth potential, such as the cigar-shaped harmonic trapused in a BEC experiment [3], can give access to informationabout the excitation spectrum (and thus about the equation ofstate of the system) through the relation between the excitationfrequency and the measured wave vector of the Faraday wave.This is somehow similar to Bragg spectroscopy on ultracoldgases confined in an optical lattice, where the excitationspectrum is inferred by exposing the system to a periodic latticemodulation and measuring the energy absorption [10–14].Proceeding along this line, one may expect to probe thespectrum of an ultracold gas, even in the presence of anoptical lattice [15], by observing the formation (or the lack offormation) of a Faraday pattern for any lattice depth providedthat the system remains compressible (i.e., the lattice does notinduce a superfluid-to-Mott transition).

In this paper we show that the presence of an opticallattice may dramatically alter the parametric excitation ofFaraday waves. In particular, we show that the formation of theFaraday waves can be suppressed not only by the appearanceof a gap in the spectrum of collective excitations but alsowhen the local flow of the condensate exceeds the Landaucritical velocity [16]. We focus our study on a cigar-shapedcondensate where phonons are parametrically excited bymodulating the tight radial confinement at a frequency �

close to a transverse breathing mode. The parametricallyexcited phonons have an energy h�/2 and a Faraday wavevector qF = q(�/2) determined by the energy spectrum of thesystem. We add a one-dimensional (1D) optical lattice along

the axial direction, and, by increasing the lattice potential depthV0, we observe the suppression of the parametric excitationof phonons with wave vector qF , even if h�/2 is still anallowed energy of the collective modes. The sound modeat qF , excited by the parametric instability, is submerged byseveral other modes excited by the dynamical instability. Thisinstability can be related to the previously studied instabilitiesin bosonic superfluid currents moving with respect to anoptical lattice potential. In these setups the current remainsstable if the superflow momentum does not exceed half therecoil momentum [17–19]. At variance, as in our system thecondensate center of mass is at rest, the dynamical instabilitywashing out the Faraday pattern is due to a local supersonicflow through the periodic potential induced by the continuousdriving of the radial breathing mode. Indeed, we observe thatthe critical value V0,crit at which Faraday waves are suppressedcorresponds to the critical value of the local BEC flow [16].

The paper is organized as follows. In Sec. II we intro-duce the nonlinear Schrodinger equations that describe thecondensate in the elongated geometry. We make use of thethree-dimensional (3D) Gross-Pitaevskii equation (GP) forthe evaluation of the low-lying energy spectrum (Sec. III)and of a time-dependent nonpolynomial nonlinear Schrodingerequation (NPSE) to numerically study the spatial Faradaypattern formation (Sec. IV). In Sec. IV we compare the resultsfor a cylindrical and a cigar-shaped trap and show that thesuppression of the parametric excitations of Faraday wavesoccurs when the local condensate flow reaches the criticalvalue associated with the Landau criterion. Our concludingremarks are given in Sec. V.

II. THEORETICAL FRAMEWORK

We study a 3D Bose condensate trapped into a cigar-shapedpotential, possibly with a 1D periodic potential superimposedalong the axial direction. The full description of the system inthe limit of zero temperature and low density is given by the3D GP equation

ih∂

∂tψ(r,t) =

[− h2

2m∇2 + U (r) + gN |ψ(r,t)|2

]ψ(r,t),

(1)

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PABLO CAPUZZI, MARIO GATTOBIGIO, AND PATRIZIA VIGNOLO PHYSICAL REVIEW A 83, 013603 (2011)

with ψ(r,t) the wave function of the condensate, m the massof the atoms comprising the condensate, N the number ofparticles, and g = 4πh2as/m the interaction coupling constantwith as the s-wave scattering length between particles. In oursetup, the trapping potential U (r) is the sum of a cigar-shapedharmonic trap plus an optical potential V (z):

U (r) = 12m ω2

⊥(x2 + y2) + 12m ω2

zz2 + V (z), (2)

with ω⊥ the trapping frequency in the perpendicular direction,ωz the trapping frequency in the longitudinal direction, andωz � ω⊥. The additional potential V (z) is either zero or equalto a periodic potential V (z) = V0 sin2(qBz), with periodicityd = π/qB .

Given this trap geometry, it has been shown in Ref. [20]that a reliable description of the condensate dynamics is givenby an effective 1D time-dependent NPSE. This equation isderived by a variational Ansatz for the wave function

ψ(r,t) = φ(x,y,t,σ (z,t)) f (z,t), (3)

where the transverse wave function φ is modeled by a Gaussianfunction

φ(x,y,t,σ (z,t)) = 1√πσ (z,t)

e−(x2+y2)/2σ (z,t)2, (4)

with a time- and longitudinal-dependent variance σ (z,t). Thevalidity of this description is based on the assumption that thetransverse wave function φ is slowly varying in such a waythat the radial velocity can be neglected. Furthermore, it hasbeen shown that neglecting the transverse excitations does notaffect the prediction for energy and dynamical instabilities [21]or the onset of Faraday waves for a parametrically excitedcondensate [8]. Both the longitudinal wave function f (z,t) andthe variance σ (z,t) are determined by the variational principlefor the energy, and the result is that the longitudinal wavefunction is governed by the NPSE

ih∂

∂tf =

[− h2

2m

∂2

∂z2+ 1

2m ω2

zz2 + V (z)

+ hω⊥1 + 3asN |f |2√1 + 2asN |f |2

]f, (5)

while the variance is algebraically determined by f (z,t),

σ 2(z,t) = h

m ω⊥

√1 + 2asN |f (z,t)|2. (6)

The 3D density profile and velocity field can be obtained as

ρ(r) = ρ(z)e−r2/σ 2

πσ 2,

(7)

v(r) = v(r) z = h

2mi

f ′∗(z)f (z) − f ′(z)f ∗(z)

ρz,

with ρ(z) = |f |2 the integrated 1D density. In the followingwe will consider the case of a radial confinement ω⊥,0/2π =200 Hz and lattice spacing d = 4680 nm; this gives a recoilenergy Er = h2 q2

B/2m � 0.14hω⊥,0. The potential depth V0

will be used as a parameter. We will also consider a condensatewith a large number of particles per site as we are interestedin the effect of a weak optical lattice on top of the elongatedcondensate. Furthermore, the choice of a large lattice spacing,

which could be achieved with two counterpropagating laserbeams tilted by an angle, would provide an increased resolutionof the spatial patterns with respect to setups with parallelcounterpropagating beams typically providing lattice spacingan order of magnitude smaller.

III. THE LOW-LYING ENERGY SPECTRUMWITH PERIODIC POTENTIAL

We calculate the low-lying energy spectrum of the conden-sate subject to a cylindrical confinement (ωz = 0) with periodicboundary conditions (PBC) for different lattice potentialdepths. Writing Eq. (1) in terms of the particle density ρ andvelocity v and considering the hydrodynamic limit of the 3DGP [22], we calculate the collective modes spectrum ω(q)solving the eigenvalue equation

−mω2δρ = g∇ · (ρ0∇δρ), (8)

ρ0 being the equilibrium density in the Thomas-Fermiapproximation, and δρ(r) = δρ(r⊥,z)eiqzbeing the densityperturbation with zero angular momentum propagating alongthe z axis. We discretize Eq. (8) in a rectangular domain(r,z), and we impose PBC in z, i.e., δρ(r⊥,z + d) = δρ(r⊥,z).Figure 1 shows the low-lying energy spectrum for the case ofN0 = 3.2 × 105 particles per site and several values of V0. Asexpected, by increasing the lattice depth, the lowest-energyband gets narrower and an energy gap opens at q = qB .

IV. FARADAY WAVE EXCITATIONS

In this section we present our numerical results fordynamics of the condensate in the presence of an opticallattice. To excite Faraday waves we proceed as follows: Firstwe calculate the ground state f0(z) for a static potential,setting ω⊥ = ω⊥,0, then we switch on the trap modulationω⊥(t) = ω⊥,0(1 + ε cos �t) at the frequency � and amplitudeε. We fix ε = 0.1 and choose � = 2ω⊥,0 in order to be closeenough to the natural breathing mode in the absence of thelattice.

0 0.2 0.4 0.6 0.8 1

1

2

q/qB

ω/ω

⊥,0

FIG. 1. Low-lying energy spectrum in the hydrodynamic approx-imation for the infinite lattice for V0/Er = 0 (continuous line),50 (dashed line), 130 (dot-dashed line), 190 (circles and continuousline), 250 (circles and dashed line), and 350 (circles and dot-dashedline).

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We focus on the dynamics of the density and velocity fieldof the atoms for increasing values of the lattice depth, andwe compare the case of an infinite cylinder, i.e., ωz = 0,and the case of a cigar-shaped BEC, for the same numberN0 of particles in the central site. The NPSE is numericallysolved by using a split-step method and spatial fast Fouriertransforms (FFT) in a finite domain with PBC. In our numericalcalculations, we found that it is enough to consider a spatialgrid with 1024 points and a time step δt � 10−4/ω⊥,0 to followthe dynamics for long times up to several hundreds of thetransverse frequency period.

A. The infinite cylinder

To analyze the formation of a pattern on top of thetime-modulated density we proceed as follows. At each timewe separate the part with the spatial periodicity of the opticallattice from the total density profile. The former is taken betime dependent since, even in the absence of pattern generationand due to the oscillation of the transverse frequency, thecondensate wave function evolves in time. For shallow latticesthis evolution can be shown to be a simple rescaling of time andposition as in a harmonic trap [23]. In our case we explicitlywrite ρ(z,t) = ρ0(z,t) + δρ, where

ρ0(z,t) = A(t) + B(t) cos[2qB z + ϕ(t)], (9)

with A,B, and ϕ fitting parameters that characterize therenormalized trap, and δρ the remaining fluctuation. TheFaraday pattern will be analyzed in terms of the behavior ofthis fluctuation.

In Fig. 2 we show the Fourier transform of central densityρ(z = 0) for a lattice depth V0/Er = 20. At ω⊥,0 t � 300 theoscillation starts departing from a simple scaling solution anda different mode gets populated. This is seen clearly in thetemporal FFT of the central density ρ(z = 0) depicted in Fig. 2,where the frequency ω = �/2 = ω⊥,0 shows up. Shortly after,a well-defined Faraday pattern is superimposed on the lattice.This is shown in Fig. 3, where the linear density ρ(z) andthe spatial FFT of its fluctuation δρ(z) are plotted at a time

ρ(z

=0,t)

ω/ω⊥,0

FIG. 2. Temporal FFT of the central density from ω⊥,0t = 0 to530 with a lattice depth V0/Er = 20.

ρ

qB z

δρ

q/qB

FIG. 3. Faraday pattern in a lattice with depth V0/Er = 20. Thetop panel shows the density profile δρ(z) as a function of qBz

at ω⊥,0t = 530. The bottom panel shows the spatial FFT of thefluctuation δρ(z) at the same time.

ω⊥,0t = 530. It is worthwhile noticing that in the spatial FFTof δρ the peaks at q = ±2qB are never present because ofthe choice of the fitting function ρ0 in Eq. (9). The peaks inthe bottom panel at q = ±0.49qB are the wave vectors of thecounterpropagating phonons giving rise to the Faraday pattern,as expected from Fig. 1. For longer times many more modes arepopulated and a pure Faraday excitation cannot be observed.

The growth of the Faraday pattern can also be studied interms of the evolution of the amplitude A of the Fourier com-ponent of δρ at the Faraday wave vector qF , A = δρ(q = qF ),and the visibility of the pattern defined as

C = ρe max − ρe min

ρe max + ρe min, (10)

where ρe max and ρe min are, respectively, the maximum andminimum of the envelope of the density ρ as shown in Fig. 3.Both A and C have been averaged over a transverse period,2π/ω⊥,0, to remove the time dependence directly associatedwith the modulation. In Fig. 4 we show the results for thecase of V0/Er = 20 with qF = 0.49qB ; it can be seen that the

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PABLO CAPUZZI, MARIO GATTOBIGIO, AND PATRIZIA VIGNOLO PHYSICAL REVIEW A 83, 013603 (2011)

0 100 200 300 400 500 6000

1

2

3

4

5

ω⊥, 0 t

C[%

]

A [ar

b. u

nits

]

FIG. 4. Comparison between the time-averaged visibility C andthe Fourier amplitude A (in arbitrary units) at qF = 0.49qB , for alattice depth V0/Er = 20. The circles correspond to C and the squaresto A.

behaviors of C and A are the same, showing a sharp increaseat ω⊥,0t ≈ 500.

For deep lattices (V0/Er >∼ 230), the frequency �/2 lieswithin the energy gap and no Faraday pattern can be formed.However, as � can intersect an energy band, we may expectfluctuations associated with the wavelength of the correspond-ing collective excitation. In particular, at V0/Er = 350, modesat �/2 are no longer excited, while modes at ω = � are stillpresent. In the spatial domain (see Fig. 5) two main features canbe detected: (i) contributions from higher Brillouin zones atq = ±2qB n, stemming from the periodicity of the GS densityprofile, which is not contained in the weak-V0 limit ansatzfor ρ0(z,t) in Eq. (9), and (ii) a weak mode in the vicinityof qB , originating from the collective mode at ω = � (cf. thecorresponding upper band in the Fig. 1).

Although for intermediate values of V0, well before ap-proaching the edge of the first Brillouin zone, one naturallyexpects to excite the pattern, we found that, for V0/Er > 50,the mode at ω = �/2 is not seen to increase substantially,and therefore it is not possible to identify a Faraday pattern.Indeed, we observe several excited wave vectors. In orderto quantify the spreading in q space as a function of V0 wecalculated the minimum of the average of |q|, 〈|q|〉min, over theexcitation δρ during a given time interval. This is displayedis Fig. 6 together with its dispersion �q = (〈q2〉 − 〈|q|〉2)1/2

(schematized as error bars in the same figure) evaluated atthe same time as 〈|q|〉min. For V0/Er > 50 the dispersion�q is of the same order of qB ; more precisely, roughly allq vectors greater than 0.2–0.3 qB are excited. In Fig. 6 wecompare these results with the expected qF value (solid line),as deduced by the low-lying energy spectrum (see Fig. 1) bysetting qF = q(�/2). As already anticipated, we can concludethat a clear Faraday pattern cannot be identified if V0/Er > 50,rather before the appearance of the energy gap.

Aiming at understanding the mechanism of suppression ofthe Faraday wave excitation, we compare the local condensatevelocity v(z) with the local sound velocity cloc

s = √gρ/m.

At the maxima of the lattice potential, the central densityρ = ρ/(πσ 2) is minimum, and we expect the ratio v(z)/cloc

s (z)to take its maximum value. Figure 7 shows the behavior of the

ρ

qB z

δρ

q/qB

FIG. 5. The same as in Fig. 3, but for V0/Er = 350 andω⊥,0t = 150. In this case we observe the contribution from higherBrillouin zones.

maximum value of the ratio v(z)/clocs (z) as a function of time,

during the parametric excitation, for the case of a shallow(top panel) and a deep (bottom panel) lattice. In the first case,

q F/q

B

V0/Er

FIG. 6. Main wave vector 〈|q|〉min (stars) identified in the densityfluctuation of the lattice during a time interval ω⊥,0t � 1500 andits dispersion �q (schematized as error bars) compared with theprediction of a Faraday pattern with a wavelength qF (solid line),as deduced by the energy spectrum evaluated in the hydrodynamicapproach.

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SUPPRESSION OF FARADAY WAVES IN A BOSE- . . . PHYSICAL REVIEW A 83, 013603 (2011)m

ax{v

(z)/cl

oc

s(z

)}

ω⊥t

max{v

(z)/cl

oc

s(z

)}

ω⊥t

FIG. 7. Maximum of the ratio of the local velocity and the localsound velocity for a homogeneous BEC attained over the lattice asfunctions of time for V0/Er = 20 (top panel) and 100 (bottom panel).

where v(z) is always lower than clocs (z), the increase of v(z)

corresponds to the onset of the parametric excitation of Faradaywaves, while in the second plot, as soon as v(z) becomesgreater than cloc

s (z) we cannot identify a well-defined Faradaypattern but see several unstable spatial modes. These findingsare qualitatively reproduced if one compares the maximumlocal velocity to the sound velocity obtained from the slope ofthe dispersion relationship ω(q).

Since the local velocity v(z) increases with the modulationamplitude ε as shown above, one expects to excite theparametric instability without breaking the superfluid only if ε

is sufficiently small. However, this may provoke a considerabledelay for the onset of the Faraday pattern, in both numericaland real experiments. Indeed, we have investigated this case forε = 0.03 and found that for V0/Er � 100 we do not observethe excitation of the Faraday mode during the modulation upto ω⊥,0t = 4.5 × 104, corresponding to roughly 30 s. Sincethe lifetime of a condensate in most experiments does notexceed a few seconds, a sizable value of ε such as that chosenin our first numerical study would be needed to observe theFaraday mode. In turn, this would give rise to a rapid increase

of the local flow velocity. On the other hand, for the trappingparameters in our setup the Faraday pattern could be readilyseen in the column density after 2–3 s of modulation in thecase of the shallow lattice and thus it could be experimentallyobserved.

B. Excitations in a confined lattice

To allow for a more direct application to current exper-iments carried out in optical lattices, hereafter we considerthe effects of the longitudinal harmonic confinement. For thatgoal we take ωz/(2π ) = 20 Hz and a total number of atomsN = 4.7 × 106, parameters that ensure the same number ofatoms, N0, in the central lattice site as in the cylindricalconfinement.

In Fig. 8 we show results for V0/Er = 1. The continuousline, which corresponds to ρ0 + 5δρ, the factor of 5 havingbeen chosen to zoom the density fluctuation, shows thesetup of the Faraday pattern at ω⊥,0t = 450 compared to

ρ

z/d

δρ

q/qB

×10−4

FIG. 8. Faraday pattern in an optical lattice with V0/Er = 1confined in an elongated trap. The top panel shows the “zoomed”density profile ρ0 + 5δρ at ω⊥,0t = 450 (solid line) together with theground-state density profile (dashed line). Th bottom panel shows thespatial FFT of the fluctuation δρ at the same time.

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PABLO CAPUZZI, MARIO GATTOBIGIO, AND PATRIZIA VIGNOLO PHYSICAL REVIEW A 83, 013603 (2011)

the ground state at t = 0 (dashed line).The presence of theFaraday excitation can be confirmed by the inspection of thespatial FFT of the density fluctuation at z = 0. The lowest|q| peaks correspond to qF = q(�/2) (see Fig. 1), while thesecond main peaks correspond to |q| = qF + 2qB , followingthe lattice periodicity. The broadening of the peaks with respectto the spatial FFT in the cylindrical confinement is due to theinhomogeneous density profile.

Our analysis for the case of a cigar-shaped confinement isin agreement with our study for the cylindrical trap; namely,the axial confinement, in the limit where ωz/ω⊥,0 � 1, doesnot affect the results shown in the previous section.

V. FINAL REMARKS

We studied the existence of Faraday waves in an elongatedcondensate with a superimposed optical lattice and an oscillat-ing transverse confinement. We focused on condensates witha large number of particles for which the speed of sound isenhanced. This choice has in principle two advantages: (i) itincreases the spacing of the Faraday pattern, and thus thedetection resolution, and (ii) it avoids Landau instabilities forperturbation amplitudes larger than those in other experimentalconditions. In the superfluid regime where the average welloccupation is large, the parametric excitation of eigenmodesand thus the observation of Faraday waves is expected tobe possible if the excited mode does not lie within theBogoliubov energy gaps of the superfluid [15]. In this workwe point out a mechanism of suppression of Faraday wavesin an energetically allowed region. During the parametricexcitation, there is a local superfluid flow through the latticebarriers. We found that the presence of these barriers does

not hinder the growth of phononic modes with a well-definedwave vector while the flow velocity remains below the localsound velocity [16]. By increasing the potential depth, thiscondition fails to be satisfied: The density at the potentialmaxima decreases and thus the local sound velocity decreasesas well. We found that the superfluid becomes unstable with thesimultaneous appearance of several incoherent modes and, asa consequence, the spectrum of the system cannot be inferredby the parametric excitation. These results are verified both inan infinitely long cylindrical trap and in a cigar-shaped trapwith a weak confinement along the axial direction.

It would be interesting to extend these investigationsto atomic samples subject to very deep lattices forming astrongly correlated gas, where there are few particles per well.However, since a deep lattice potential induces a quantumphase transition toward the incompressible Mott state, theformation of Faraday patterns spanning several wells couldbe precluded. Indeed, it has been shown that even in the Mottregime parametric excitations induced by the modulation ofthe lattice depth can be used as a tool to detect featuresin the density of states of the system [24] and they couldlead to interesting out-of-equilibrium phenomena. Moreover,the one-dimensional optical lattice also gives the possibilityof exploring the interplay between gap solitons at the bandedge [25] and the Faraday waves dynamics discussed here.

ACKNOWLEDGMENTS

This work was supported by the CNRS-CONICET interna-tional cooperation Grant No. 22966. P.C. also acknowledgespartial financial support from ANPCyT, Argentina, throughGrant No. PICT-2008-0682.

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