Kapitza stabilization of a repulsive Bose-Einstein condensatein an oscillating optical lattice

J. Martin,1 B. Georgeot,2 D. Guery-Odelin,3 and D. L. Shepelyansky2

1Institut de Physique Nucleaire, Atomique et de Spectroscopie,CESAM, Universite de Liege, Batiment B15, B - 4000 Liege, Belgium

2Laboratoire de Physique Theorique, IRSAMC, Universite de Toulouse, CNRS, UPS, France3Laboratoire Collisions, Agregats, Reactivite, IRSAMC, Universite de Toulouse, CNRS, UPS, France

(Dated: March 7, 2018)

We show that the Kapitza stabilization can occur in the context of nonlinear quantum fields.Through this phenomenon, an amplitude-modulated lattice can stabilize a Bose-Einstein condensatewith repulsive interactions and prevent the spreading for long times. We present a classical andquantum analysis in the framework of Gross-Pitaevskii equation, specifying the parameter regionwhere stabilization occurs. Effects of nonlinearity lead to a significant increase of the stabilitydomain compared with the classical case. Our proposal can be experimentally implemented withcurrent cold atom settings.

I. INTRODUCTION

The striking example of the Kapitza pendulum showsthat an oscillating force with zero average can lead tothe phenomenon of Kapitza stabilization, with transfor-mation of an unstable fixed point into a stable one [1, 2].The theory of this nonlinear system is well establishedin a classical context [3]. Some applications to quantumsystems have been proposed, including optical molasses[4], stability of optical resonators [5], trapping by laserfields [6, 7], cold atoms with oscillating interactions [8],the periodically driven sine-Gordon model [9], and polari-ton Rabi oscillations [10]. However, the emergence of thisphenomenon for nonlinear quantum fields of repulsive in-teractions has not been analyzed. In this paper, we showthat a similar effect appears for a repulsive Bose-Einsteincondensate (BEC) in an oscillating optical lattice. Forthis system, the oscillating lattice enables the localiza-tion of a wave packet of repulsive atoms through Kapitzastabilization: thus, while in the absence of the lattice theatoms spread over the system, they remain trapped ina localized wave packet in the presence of the oscillat-ing force with zero mean, an effect due to the interplaybetween dynamical renormalization of the potential andatom-atom interactions. The evolution is described bythe Gross-Pitaevskii Equation [11] (GPE), with the re-pulsive nonlinear interaction creating the unstable fixedpoint in the vicinity of the maximum of the wave packet.In contrast with the standard classical Kapitza pendu-lum, where the potential is fixed in the vicinity of theunstable fixed point, the present GPE setting creates amore complex situation where the potential varies withthe shape of the wave function. In the following we de-scribe the physics of this remarkable phenomenon andpresent realistic parameter values for an experimental re-alization with a BEC. We note that the general problemof stabilization by oscillating fields finds various impor-tant applications; e.g., Paul traps for charged particles[12].

II. CLASSICAL SYSTEM DYNAMICS

We first analyze a classical inverted harmonic oscillatorin one dimension in an oscillating periodic potential, withthe Hamiltonian

H =p2

2m− 1

2mω2

i x2 + V0(x) cos(ω`t), (1)

with V0(x) = U0 cos (2πx/d) where U0 is the potentialamplitude. Here m is the particle mass, x and p areposition and momentum, ωi characterizes the unstablefixed point and the periodic potential has a spatial pe-riod d and an amplitude oscillation of frequency ω`. Wedefine a characteristic momentum p0 = 4

√mU0 and os-

cillation frequency ω0 = 2π√U0/(md2), leading to the

dimensionless variables

X = 2πx

d, P =

p

p0, T =

ω`t

2π(2)

and the frequency ratios

Ri0 =ωiω0, R0` =

ω0

ω`. (3)

Following the standard methods of dynamical systems[13], we describe the dynamics through the Poincaresection, with typical phase-space structures shown inFig. 1. The bottom-left panel shows the regime wherethe Kapitza stabilization is too weak and the pointX = P = 0 remains unstable. The top left panel showsthe regime of Kapitza stabilization with a stability islandaround X = P = 0; the island is surrounded by a chaoticcomponent where the trajectories can escape to infinity.The bottom right panel corresponds to a very weak valueof Ri0 and relatively strong driving, with overlapping res-onances leading to onset of chaos as determined by theChirikov criterion [14]. To determine numerically the sta-bility diagram, we follow trajectories with random initialconditions for sufficiently long time ∆T . A trajectoryof initial conditions (X(0), P (0)) = (0, P (0)) is consid-ered unstable if |X(T ) − 0| > π for some T ∈ [0 : 1000].

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FIG. 1. Poincare sections formed by a few thousand trajec-tories with random initial conditions (X(0), P (0)) ∈ [−2ξ :2ξ] × [−0.1 : 0.1] (ξ = 1 or π) propagated during a times-pan ∆T = 400 for the frequency ratios Ri0 = 0.075 andR0` = 0.45 (top left), Ri0 = 0.15 and R0` = 0.2 (bottom left),and Ri0 = 0.02 and R0` = 0.7 (bottom right). (top right)Stability region in the parameter space of frequency ratios.Color shows the largest initial momentum Pmax(0) for whichtrajectories with initial conditions (X(0), P (0)) = (0, P (0))remain stable for time T ∈ [0 : 1000]. Crosses mark parame-ters of the Poincare sections; dashed black lines show theory(4), upper red/grey dashed line shows refined theory takinginto account secondary resonances (see text).

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FIG. 2. Poincare sections for R0` = 0.65 (left panels) andR0` = 0.7 (right panels) at Ri0 = 0 showing the loss of sta-bility of the fixed point at P = X = 0.

The top-right panel shows a density plot of the largestinitial momentum P (0) giving rise to a stable trajectoryas a function of Ri0 and R0`. It highlights the parame-ter region where the Kapitza phenomenon stabilizes theunstable fixed point. The specific shape of this regiondepends on two main borders, the lower one determinedthrough Kapitza’s original argument [1–3], and the upperone through the Chirikov criterion [14], yielding

R0` >√

2Ri0, R0` < 0.45 (4)

The derivation of the first relation directly followsthe approach of Kapitza pendulum [3]: the effectiveaverage potential created by the oscillating force isUeff = 〈p2〉/(2mω2

` ) = (π2U20 /(md

2ω2` )) sin2(2πx/d),

which combined with the inverted harmonic potential,gives for small oscillations the squared effective frequencyω2

eff = (ω20/ω`)

2/2 − ω2i . Thus, x = 0 is stable if

ω2eff > 0, leading to the first inequality of Eq. (4).

The second inequality follows from the Chirikov crite-rion [14]. The nonlinear resonances are located at posi-tions p± = ±ω`d/2π. The resonant term of the Hamil-tonian is reduced to a pendulum Hamiltonian Hrs =p2/2m + U0/2 cos θ with the phase θ = 2πx/d ± ω`tconjugate to p. According to the standard results fora pendulum [14], the frequency width of the pendulumseparatrix is ∆ω = 2ω0, and the frequency distancebetween resonances is δω = 2ω`. The parameter ofChirikov resonance overlap criterion is S = ∆ω/δω andthe chaotic transitions between two resonances take placeat K ≈ 2.5S2 > 1, leading to Eq. (4) (the coefficient 2.5takes into account the effect of secondary resonances) [14](see also [15]).

These two theoretical borders of Eq. (4) are shownby straight black dashed lines in Fig. 1. We note thatthe lower border is in excellent agreement with the nu-merical data. The upper border is lower than the sta-bility region centred around R0` ≈ 0.5. The reason isthat the Chirikov overlap criterion gives the border for achaotic transition between resonances while the destruc-tion of resonances takes place at higher values (e.g. in theChirikov standard map, the primary resonance becomesunstable at K ≈ 2.5S2 = 4 while the last invariant curveis destroyed at K ≈ 0.9716). In our case with two pri-mary resonances, the secondary resonance at P = X = 0becomes unstable at R0` ≈ 0.7 and Ri0 = 0, as it isshown in the Poincare sections in Fig. 2 for R0` = 0.65(fixed point P = X = 0 is stable) and R0` = 0.7 (fixedpoint P = X = 0 is unstable) at Ri0 = 0. The refinedupper stability border R0` = 0.7 is shown in top rightpanel of Fig. 1 by the upper horizontal red dashed line.

III. QUANTUM EVOLUTION WITH GPE

We now turn to the quantum case, and set U0 = sEL/2where s is a dimensionless parameter characterizing thelattice depth and EL = 2π2~2/(md2) is a lattice char-acteristic energy [16]. When the dimensionless posi-

3

tion and momentum are turned into operators, X = Xand P = (~eff/i)∂X , the canonical commutation rela-tion [x, p] = i~ leads to an effective Planck’s constant~eff = 1/(2

√s).

In this work, we use the one-dimensional (1D) GPE tostudy the dynamics of a BEC with N atoms subjectedto the driving potential V0(x) cos(ω`t). This equationis valid in the weakly interacting regime. More quan-titatively, this regime is obtained from an anisotropicthree-dimensional (3D) confinement tightly confined inthe radial direction, i.e., when the confinement energy,~ω⊥, greatly exceeds the mean-field interaction energy.As a result, at sufficiently low temperature the radialmotion is frozen and therefore governed by the groundstate wave function of the radial harmonic oscillator.The strength of the interaction in the GPE equation isthen renormalized by averaging the 3D interaction overthe radial density profile: g = 2~2as/m`

2⊥, where as is

the 3D scattering length and `⊥ = (~/mω⊥)1/2 the har-monic oscillator length associated with the radial har-monic confinement. In 1D, the criterion of the weaklyinteracting regime reads γ = mg/~2n � 1 where n isthe 1D atomic density [17, 18]. In the weakly interactingregime, the BEC wave function (normalized according to∫|ψ|2dx =

∫|Ψ|2dX = 1 where Ψ ≡

√d/2π ψ) is thus

governed by the nonlinear equation

i

4π∂TΨ = R0`

(−~eff∂

2X +

cos(2πT ) cosX

8~eff+g|Ψ|2

2

)Ψ

(5)where g = 2πNg1D/(~ω0d) = 4πNω⊥as/(ω0d) [19].

By expanding the nonlinear potential for a Gaussianwave packet |ψ(x)|2 = exp[−x2/(2σ2)]/(σ

√2π) of rms

width σ around its maximum, we obtain an effective in-verted harmonic potential with rescaled frequency

Ri0,eff ≡ωi,eff

ω0= 23/4π−1/4

√~eff g

σ3(6)

where σ = 2πσ/d. From the expression for Ueff , it followsthat all points at X = mπ with integer m become stablefor Ri0,eff < R0`/

√2.

In our simulations, we use the Strang-Marchukoperator-splitting method [20] to approximate the evolu-tion operator corresponding to Eq. (5). We take Ns = 216

basis states for the wave function, a range of X valuescorresponding to Qtot = 64 periods of the driven opti-cal lattice [which leads to a numerical grid with δX =(2πQtot)/Ns ≈ 0.006 and δP = ~eff/Qtot ≈ 5.5 × 10−4

for s = 200], and a time step δT ∈ [0.0002 : 0.001]. If Qdenotes the total number of initially populated potentialwells of the static potential −V0(X), then the effects ofinteractions in our simulations only depend on the ratiog/Q since the nonlinear potential is given by g|Ψ|2. In thecase of one localized packet (Q = 1), the initial state isthe ground state (without atom-atom interactions) of thepotential well centered at X = 0 of the static potential−V0(X), which for large enough s corresponds to a Gaus-

sian wave packet |Ψ(X)|2 = exp[−X2/(2σ2)]/(σ√

2π) of

0.75

0.5

0.25

0

PS(T

)

1

0.75

0.5

0.25

0

0.75

0.5

0.25

050403020100

T

2π0−2π

1

0.5

0

X

1

0.5

0

|Ψ(X

,T)|2

1.5

1

0.5

0

FIG. 3. Left panels show the probability density |Ψ(X,T )|2vs position X at T = 0 (single Gaussian wave packet of

rms width σ = 200−1/4 ≈ 0.27, dashed curve delimitinggray shaded area), T = 1 (green solid curve with triangles),T = 2 (blue solid curve with circles), T = 3 (red solid curvewith squares) for R0` = 0.6 and Q = 1. Right panels show

PS(T ) =∫ +π/2

−π/2 |Ψ(X,T )|2dX. Top row shows without driv-

ing (s = 0) and with g/Q = 0.4. Middle row shows with driv-ing given by Ri0,eff ≈ 0.16 at s = 200 and g/Q = 0.008. Bot-tom row shows with driving corresponding to Ri0,eff ≈ 1.09at s = 200 and g/Q = 0.4 (left cross in Fig. 7).

rms width σ = 1/s1/4 and zero average momentum. Forsuch initial states, we have Ri0,eff = (2/π)1/4s1/8√g. Wealso consider as initial state a chain of wave packets pe-riodically repeated in potential wells (Q = Qtot), whichwe describe as the ground state (with atom-atom inter-actions) of GPE for the static potential −V0(X). Suchstates can be easily prepared experimentally by switchinga static optical lattice with a formation of BEC in eachpotential minimum. For such an initial state one shouldreplace g by g/Qtot in Eq. (6).

IV. KAPITZA STABILIZATION OF QUANTUMSTATES

The time evolution of a single wave packet and its fullwidth at half maximum (FWHM) are shown in Figs. 3and 4, respectively. Without the oscillating potential, thewave packet spreads over the whole lattice, leading to amonotonic drop of the probability inside the initial poten-

tial well, PS(T ) =∫ +π/2

−π/2 |Ψ(X,T )|2dX, and a linear in-

crease of the FWHM with time. In contrast, in the pres-ence of the oscillating potential, the Kapitza stabilizationleads to conservation of a large part of the probability inthe initial well. The larger the interaction strength g, thebetter the stabilization. Interestingly enough, quantumstabilization exists inside the classical stability domain(see middle panels), but also at Ri0,eff ≈ 1.09, signifi-

4

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20151050

5

4

3

2

1

0

FIG. 4. Full width at half maximum (FWHM) of |Ψ(X,T )|2as a function of time T for an initial Gaussian wave packet ofrms width σ = 200−1/4 ≈ 0.27 for R0` = 0.6 and g/Q = 0.4(with Q = 1). Red dashed curve is without driving ; greenthick solid curve is without driving but with static potential−V0(X) at s = 200 ; blue solid curve is with driving corre-sponding to Ri0,eff ≈ 1.09 at s = 200.

1

0.75

0.5

0.25

0

PP(T

)

0.75

0.5

0.25

050403020100

T

PP(T

)

2π0−2π

0.02

0.01

0

X

|Ψ(X

,T)|2

0.02

0.01

0

|Ψ(X

,T)|2

FIG. 5. Left panels show probability density |Ψ(X,T )|2 vsposition X at T = 0 (periodic ground-state wave functionof GPE with static potential −V0(X), dashed curve delim-iting the gray shaded area, here σ ≈ 0.40), T = 1 (greensolid curve with triangles), T = 2 (blue solid curve with cir-cles), T = 3 (red solid curve with squares) for R0` = 0.6and g/Q = 0.4 with Q = 64. Right panels show PP (T ) =∫{X:V0(X)>0} |Ψ(X,T )|2dX. Top row shows without driv-

ing (s = 0). Bottom row shows with driving given byRi0,eff ≈ 0.85 and s = 200.

cantly above the classical stability border Ri0,eff = 0.21from Eq. (4). We attribute this quantum enhancement ofKapitza stabilization to the presence of oscillations in thewidth of the wave packet (see Fig. 4), which we discussbelow.

Another possible initial state is given by a chain ofBEC wave packets corresponding to the ground state of

FIG. 6. Density plot of probability density as a functionof position X and time T for R0` = 0.6, g/Q = 0.6, ands = 200. Top panel shows that the initial state is a singleGaussian wave packet of rms width σ = 200−1/4 ≈ 0.27 cen-tered on X = 0 (Q = 1), with Ri0,eff ≈ 1.34 (see right cross inFig. 7). Bottom panel shows that the initial state is the peri-odic ground-state wave function of GPE with static potential−V0(X) (Q = 64) and Ri0,eff ≈ 1.03.

GPE at each potential minimum of the lattice. Thisstate is obtained numerically by the standard methodof imaginary time propagation of GPE. The time evolu-tion is shown in Fig. 5. Without the oscillating potential,the periodic peak structure becomes less pronounced, de-creasing with time. Thus the probability escapes fromthe vicinity of the unstable fixed points as measured byPP (T ) =

∫{X:V0(X)>0} |Ψ(X,T )|2dX, which oscillates in

time between 0 and 1. In contrast, in the presence ofthe oscillating potential, the probability remains in thevicinity of the unstable fixed points, even if the param-eter Ri0,eff ≈ 0.85 is significantly beyond the classicalstability border of Eq. (4).

The origin of the quantum enhancement of the Kapitzastabilization seen in Figs. 3 and 5 can be understood fromthe typical evolution of the wave function shown in Fig. 6.Indeed, the width σ of the wave packet oscillates in timeby a factor f ≈ 2 (see Fig. 4), which renormalizes σ.Since Ri0,eff ∝ σ−3/2, this gives a reduction factor f−3/2

of the values of Ri0,eff in Fig. 3 from Ri0,eff = 1.09 toRi0,eff = 0.39, significantly closer to the theoretical classi-cal border of Eq. (4) at Ri0,eff = 0.21. Moreover, the timeoscillation of |Ψ(X,T )|2 creates a supplementary oscillat-

5

FIG. 7. Top left panel shows classical stability map as inFig. 1. Top right and bottom panels show density plot of PS =110

∫ 3Tsp+10

3TspPS(T )dT , with PS(T ) =

∫ +π/2

−π/2 |Ψ(X,T )|2dX and

where Tsp is a characteristic spreading time of the initialwave packet in the absence of driving (s = 0) determinedby P s=0

S (Tsp) = 0.75, as a function of the frequency ratioR0` and the effective frequency ratio Ri0,eff of Eq. (6) (multi-plied by a scaling coefficient α = 0.2) for s = 200 (top right),s = 100 (bottom left) and s = 50 (bottom right). Initial state

is a single Gaussian wave packet of rms width σ = 1/s1/4

centered around X = 0 (Q = 1). The two crosses correspondto parameters g/Q = 0.4 (left cross) and g/Q = 0.6 (rightcross) for R0` = 0.6.

ing potential which can generate an additional Kapitzastabilization.

The region of quantum Kapitza stabilization in thisGPE system is shown in Fig. 7, which displays the time-averaged probability to stay in the vicinity of the unsta-ble fixed point X = 0 as a function of Ri0,eff and R0`

together with the classical stability diagram of Fig. 1.In the regime of small effective Planck’s constant cor-responding to large s values, a large stability region isvisible, with a shape similar to the classical stability do-main. In Fig. 7, the values of Ri0,eff are rescaled by a mul-tiplicative factor α = 0.2 corresponding to the fact thatquantum stabilization exists at Ri0,eff values significantlylarger than in the classical case, given by Eq. (4). We at-tribute the presence of this factor to the quantum fluctu-ations as discussed above. For decreasing values of s, thequantum stability region becomes less pronounced. Weexplain this by the fact that the effective ~ becomes com-parable to the phase-space area of the classical Kapitzastability island (see Fig. 1, top left panel). In this case,

quantum tunneling from the island becomes importantand leads to the destruction of the Kapitza phenomenon.

We note that in the absence of interaction, quantumtunneling will always induce a decay of probability in-side the stability island induced by Kapitza stabilization.However, this tunneling can be affected in a nontrivialway by the nonlinearities. Indeed, nonlinearities can pro-duce solitonic solutions which remain stable for all times.Therefore, a rigorous answer to this interesting questionrequires a careful mathematical analysis. All time stablesolutions may appear when the stability region has verylarge depth and sufficiently large width.

V. PROPOSED EXPERIMENTALREALIZATION

The experimental implementation can be carried outby loading adiabatically a BEC into a deep static hori-zontal 1D optical lattice (s ∼ 50) realized with far off-resonant lasers. As a result, we obtain a chain of smallBEC at the bottom of the potential wells. To place themat the top of the potential hills of the lattice, we haveto shift suddenly by half the spatial period the opti-cal lattice as in Ref. [16]. The amplitude of the latticeshall be subsequently modulated to ensure the Kapitzastabilization. In practice, the control of the lattice pa-rameters (phase, amplitude) can be performed by usingphase-locked synthesizers that imprint their signals onlight through acousto-optic modulators (AOMs) placedon each lattice beam before they interfere to produce thelattice. The range of interaction strengths that we pro-pose is readily achievable with a standard rubidium-87BEC placed in an optical lattice made of two counter-propagating lasers at 1064 nm. With ω⊥ ≈ 2π × 200 Hzand a lattice spacing d ≈ 532 nm, g ≈ 0.003N/

√s, we

have g ≈ 21 for N = 105 and a depth s = 200, andg/Q ' 0.55 for a BEC of typical size 20 µm. Interest-ingly, the enhancement of interactions through Feshbachresonances is not necessary to observe the dynamical sta-bilization phenomenon. We note that Kapitza stabiliza-tion of cold atoms in optical lattices starts to attract theinterest of experimental groups [21].

VI. CONCLUSION

We have shown that the Kapitza phenomenon can sta-bilize a BEC with repulsive interaction by means of anoscillating force with zero average. This represents anapplication of the Kapitza effect in the context of non-linear quantum fields. Our theoretical proposal can beexperimentally realized with current cold atom technol-ogy. Besides its fundamental interest, it should providenew tools for the long-time manipulation of BECs.

6

ACKNOWLEDGMENTS

This work was supported in part by the Programme In-vestissements d’Avenir ANR-11-IDEX-0002-02, referenceANR-10-LABX-0037-NEXT (projects THETRACOM

and TRAFIC). Computational resources were providedby the Consortium des Equipements de Calcul Intensif(CECI), funded by the Fonds de la Recherche Scientifiquede Belgique (F.R.S.-FNRS) under Grant No. 2.5020.11and by CAlcul en MIdi-Pyrenees (CALMIP).

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