Dynamical localization of matter-wave solitons in managed barrier potentials

Dynamical localization of matter-wave solitons in managed barrier potentials

Fatkhulla Kh. Abdullaev1and Josselin Garnier2

1

Physical-Technical Institute of the Uzbekistan Academy of Sciences, 700084, Tashkent-84, G.Mavlyanov str., 2-b, Uzbekistan and Instituto de Física Teorica, UNESP, Rua Pamplona, 145, São Paulo, Brazil

2

Laboratoire de Probabilités et Modèles Aléatoires & Laboratoire Jacques-Louis Lions, Université Paris VII, 2 Place Jussieu, 75251 Paris Cedex 5, France

共Received 29 September 2006; revised manuscript received 1 December 2006; published 8 March 2007兲

The bright matter-wave soliton propagation through a barrier with a rapidly oscillating position is investi-gated. The averaged-over rapid oscillations Gross-Pitaevskii equation is derived, where the effective potential has the form of a finite well. Dynamical trapping and quantum tunneling of the soliton in the effective finite well are investigated. The analytical predictions for the effective soliton dynamics is confirmed by numerical simulations of the full Gross-Pitaevskii equation.

DOI:10.1103/PhysRevA.75.033603 PACS number共s兲: 03.75.Lm, 02.30.Jr, 05.45.Yv, 42.65.Tg

I. INTRODUCTION

The dynamical localization and tunneling of quantum par-ticles in oscillating in time potentials is one of the problems of quantum mechanics that has attracted a lot of attention recently关1,2兴. In particular, it is interesting to study the lo-calization and transmission phenomena under action of time-dependent barriers. Indeed, experiments with dynamical tun-neling of cold atoms in time-dependent barriers of the multiplicative formf共t兲V共x兲show the possibility for the tun-neling period control关3,4兴. The theory is developed for lin-ear wave packets in Ref. 关5兴. Soliton propagation through such a barrier is investigated in Ref.关6兴. Another interesting problem is to consider a barrier with a rapidly oscillating position, i.e., a potential of the formV关x−f共t兲兴. In the linear regime of propagation resonant effects are exhibited in the transmission process of matter wave packets through an os-cillating barrier关7,8兴. The split of repulsive condensate under a trap with an oscillating position has been investigated in Ref. 关9兴. The dynamical localization of optical wave in the array of periodically curved optical waveguides has been demonstrated recently in Refs. 关10,11兴. This system is de-scribed by a nonlinear Schrödinger- 共NLS兲 type equation with a periodic potential with an oscillating position. It is therefore relevant to investigate the propagation ofa nonlin-ear wave packetand, in particular, the dynamical localization of wave packets in barriers with time-dependent parameters. We consider this problem for a bright matter-wave soliton in an attractive Bose-Einstein condensate共BEC兲in the presence of a barrier potential with an oscillating position. Such a barrier can be achieved by using a laser beam with a blue-detuned far-off-resonant frequency. This generates a repul-sive potential and the laser sheet plays the role of a mirror for cold atoms. Such a laser sheet has been used to study the bouncing of a BEC cloud off a mirror关12兴. By moving the position of the mirror we obtain an oscillating barrier.

In this paper we are mainly interested in the study of the dynamical trapping of soliton in BEC with a rapidly oscillat-ing barrier potential. The averaged-over rapid oscillations Gross-Pitaevskii共GP兲equation is derived in Sec. II, and we show that the effective potential has the form of a finite well. Applying a variational approach to this equation, we analyze

the conditions for soliton trapping by the effective potential in Sec. III. The analytical predictions are checked by numeri-cal simulations of the full GP equation in Sec. IV. We ana-lyze carefully the case of broad solitons in Sec. V. In Sec. VI we make the important remark that the management tech-nique based on rapid oscillations of a barrier potential is very flexible and allows the generation of asymmetric traps. As a consequence, this technique can be used to investigate soli-ton tunneling with sub-barrier energies as we now explain. In Refs. 关13,14兴 the topological soliton tunneling in an asym-metric well with a finite barrier was investigated. In this paper we analyze a similar problem for the NLS soliton case and show that the tunneling in our case has the form of continuous shedding of tunneling radiation by the soliton

共due mainly to the fact that the NLS soliton is the dynamical soliton兲 and formation in the effective well ofbreather-type

solutions. This phenomenon represents a quantum evapora-tion of the soliton from the dynamically generated potential.

II. AVERAGED GROSS-PITAEVSKII EQUATION

The BEC wave function␺in a quasi-one-dimensional ge-ometry is described by the GP equation,

iប␺T+ ប2

2m␺XX−Vp关X−fp共T兲,T兴␺−g1D兩␺兩

2_{␺= 0,}

whereV_p共X,T兲 is a barrier potential. Here g_1D= 2បa_s␻_⬜ is the one-dimensional mean-field nonlinearity constant, a_s is the atomic scattering length,␻_⬜ is the transverse oscillator period, and a_⬜=

冑

ប/共m␻_⬜兲 is the transverse oscillator length. Below we consider the case of BEC with attractive interactions, i.e., as⬍0. To avoid collapse in the attractive

BEC, the condition兩as兩N/a_⬜⬍0.67 should be satisfied, with

Nthe number of atoms. The barrier potential has an oscillat-ing position described by the zero-mean time-periodic func-tion fp共T兲 with period Tp. We consider the case of a

fast-moving potential in the sense that ␧ª␻⬜TpⰆ1. We also

x=

冑

2X

a_⬜ , t=␻⬜T, u=

冑

兩as兩␺, V= V_p

ប␻_⬜, we obtain the GP equation with a fast-moving potential

iu_t+u_xx−V

冋

x−f

冉

t

␧

冊

,t

册

u+ 2兩u兩

2_u

= 0, 共1兲

where

f共t兲=

冑

2

a_⬜fp共Tpt兲

is a periodic function with period 1 and mean 0. The small parameter ␧ describes the fast oscillation period in the di-mensionless variables. We look for the solution in the form

u共x,t兲=u共0兲

冉

x,t,t

␧

冊

+␧u

共1兲

冉

_x,t,t

␧

冊

+␧

2_u共2兲

冉

_x,t,t

␧

冊

+ ¯ , 共2兲

where u共0兲, u共1兲,…are periodic in the argument ␶=t/␧. We substitute this ansatz into Eq.共1兲and collect the terms with the same powers of␧. We obtain the hierarchy of equations,

iu_␶共0兲= 0, 共3兲

iu_t共0兲+u_xx共0兲−V关x−f共␶兲,t兴u共0兲+ 2兩u共0兲兩2_u共0兲_{= −iu} ␶ 共1兲

, 共4兲

iu_t共1兲+u_xx共1兲−V关x−f共␶兲,t兴u共1兲+ 4共u共0兲兲2_u共1兲

+ 2兩u共0兲兩2_u共1兲

= −iu_␶共2兲. 共5兲

The first equation共3兲 shows that the leading-order termu共0兲

does not depend on ␶. The second equation 共3兲 gives the compatibility equation for the existence of the expansion共2兲,

iu_t共0兲+u_xx共0兲−具V关x−f共·兲,t兴典u共0兲+ 2兩u共0兲兩2_u共0兲_{= 0,} where具·典stands for an average in␶. Denoting

Veff共x兲=

冕

0 1

V关x−f共␶兲,t兴d␶, 共6兲

we obtain the averaged GP equation

iut+uxx−Veff共x,t兲u+ 2兩u兩2u= 0. 共7兲 Equation共5兲allows us to compute the first-order correction, whose amplitude is of order␧.

If, for example, f共␶兲=asin共2␲␶兲, then a straightforward calculation shows that

Veff共x,t兲=KeffⴱV共x,t兲, 共8兲 whereⴱstands for the convolution product共inx兲andKeffis given by

Keff共x兲= 1 ␲

1

冑

a2−x21共−a,a兲共x兲. 共9兲

If, moreover, V共x,t兲 is a deltalike potential centered at a positionx0共t兲that moves slowly in time, then

V_eff共x,t兲=K_eff关x−x₀共t兲兴

is a moving double-barrier potential. By this way, one can generate a moving trap potential to manage the soliton posi-tion.

III. VARIATIONAL APPROACH FOR THE SOLITON MOTION

In the absence of effective potentialVeff= 0 the NLS equa-tion共7兲supports soliton solutions of the form

u₀共x,t兲= 2␯e

i2␮关x−x_s共t兲兴+4i共␮2+␯2兲t

cosh兵2␯关x−xs共t兲兴其,

withxs共t兲=xs共0兲+ 4␮t. In the presence of a stationary poten-tialVeff共x兲, the soliton dynamics can be studied by perturba-tion techniques. Applying the first-order perturbed inverse scattering transform共IST兲theory关17兴, we obtain the system of equations for the soliton amplitude and velocity

d␯

dt = 0, d␮

dt = −

1 4␯W␯

⬘

共xs兲,

dx_s

dt = 4␮,

where the prime stands for a derivative with respect toxand

W_␯ has the form

W_␯共x兲=␯

冕

−⬁

⬁ 1

cosh2共z兲Veff

冉

x−

z

2␯

冊

dz=K␯ⴱVeff共x兲,

共10兲

K␯共x兲= 2␯2

cosh2_共2␯x兲. 共11兲

Therefore, we can write the effective equation describing the dynamics of the soliton center as a quasiparticle moving in the effective potentialW_␯

0,

␯0

d2xs

dt2 = −W␯

⬘

0共xs兲, 共12兲

where 4␯0 is the mass共number of atoms兲of the initial soli-ton. This system has the integral of motion

␯0 2

冉

dxs

dt

冊

2

+W␯₀共xs兲= 8␯0␮0 2

, 共13兲

where 4␮0is the velocity of the initial soliton. Note that the first-order perturbed IST takes into account the nonlinear wave nature of the soliton, but neglects radiations phenom-ena by considering that the matter wave remains in the form of a soliton with time-dependent coefficients共velocity, cen-ter, and phase兲. This approach gives the same result as the adiabatic perturbation theory for solitons, which is a first-order method as well. This adiabatic perturbation theory was originally introduced for optical solitons关18,19兴 and it was recently applied to matter-wave solitons关20兴.

W_␯共x兲=共K_␯ⴱK_eff兲ⴱV共x兲.

In the case where f共␶兲=asin共2␲␶兲, the kernelKeffis given by Eq. 共9兲 and the Fourier transform of the quasiparticle potential has the explicit form

Wˆ␯共k兲=

␲kJ0共ka兲

2 sinh

冉

␲k 4␯

冊

Vˆ共k兲,

whereVˆ is the Fourier transform of the reference potentialV. The effective quasiparticle potential is plotted in Fig.1for a Gaussian barrier potential V. The potential W␯ has a local minimum atx= 0 between two global maxima that are close tox= ±a. The trap amplitude is

⌬W_␯= max

x

W_␯共x兲−W_␯共0兲.

Let us consider an input soliton at x= 0 with parameters

共␯0,␮0兲. If the initial soliton velocity is large enough, then the soliton escapes the trap. There is a critical value␮cfor

the initial soliton velocity parameter␮0defined by

␮_c2=

⌬W_␯

0 8␯0

, 共14兲

which determines the type of motion.

共1兲 If 兩␮0兩⬍␮c then the soliton is trapped. Its motion is

oscillatory between the positions ±x_f defined by

W_␯

0共xf兲−W␯0共0兲= 8␯0␮0 2

. 共15兲

共2兲If 兩␮0兩⬎␮c, then the soliton motion is unbounded. It

escapes the trap and reaches the asymptotic velocity param-eter␮a given by

␮_a2=

W␯₀共0兲 8␯0

+␮₀2, 共16兲

which shows that the transmitted soliton velocity is larger than the initial soliton velocity.

As we shall see in the numerical simulations, if the initial soliton parameters are close to the critical case 兩␮0兩 ⬃␮c,

then radiation effects become non-negligible. The construc-tion of an efficient trap requires one to generate a barrier potentialVthat is high enough so that ⌬W␯₀ is significantly larger than 8␯₀2␮0.

Finally, if the potential V is not stationary but has an explicit time-dependence, such as a drift, then the adiabatic equations derived in this section still hold true if the time-dependence is slow enough.

IV. NUMERICAL SIMULATIONS FOR DYNAMICAL LOCALIZATION

In this section we show the motion of a soliton with initial parameters␯0=␮0= 0.25 共the soliton velocity is 1兲. The ref-erence barrier potential isV共x兲= 10 exp共−x2/ 4兲. Note that in this case, ⌬W_␯

0⯝0.323 and ␮c⯝0.4. Therefore, the quasi-particle approach predicts trapping.

In Fig.2the modulated potential isV关x− 10 sin共50t兲兴 共 sta-tionary trap兲. Here ␧= 2␲/ 50⯝0.126 and we check by full numerical simulations of the GP equation共1兲that␧ is small enough to ensure the validity of the theoretical predictions based on the averaged GP equation 共7兲. We plot the soliton profile and the soliton mass, which shows that radiative ef-fects are very small. We also compare the soliton motion obtained by the numerical simulations with the one predicted by the quasiparticle approach, which shows very good agree-ment.

In Fig. 3 the modulated potential is V关x− 10 sin共50t兲

− 0.25t兴 共moving trap, with velocity 0.25兲. We plot the soli-ton profile and the solisoli-ton mass, which shows that radiative effects are very small.

In Fig. 4 the modulated potential is V关x− 10 sin共50t兲

+ 0.25t兴 共moving trap, with velocity −0.25兲. Radiation effects are small, but not completely negligible. It can be seen that some radiation is transmitted through the edges of the effec-tive double-barrier trap. We then double the potential ampli-tude and consider 2V关x− 10 sin共50t兲+ 0.25t兴. The new simu-lation is shown in Fig.5, where radiation effects are seen to be negligible.

In Fig. 6 we show the motion of a soliton with initial parameters␯0= 0.25 and␮0= 0.5 共the soliton velocity is 2兲. Since we have⌬W_␯

0⯝0.323 and ␮c= 0.4, the quasiparticle approach predicts unbounded motion. The results of the nu-merical simulations confirm this prediction. We can observe that half the initial soliton mass is transmitted through the effective barrier during the first interaction, and the transmit-ted soliton has a velocity that is larger than 4␮0.

Let us estimate the typical values of the parameters for an experimental configuration. We consider the case of a con-densate with 7Li atoms in the a cigar-shaped trap potential with␻_⬜= 2␲⫻1 kHz. In this case,␧= 2␲/ 50 corresponds to

−200 −10 0 10 20

0.5 1 1.5 2 2.5

Veff

x

−200 −10 0 10 20

0.2 0.4 0.6 0.8 1

∆W ν

W

ν

x

1 /Tp= 50 kHz. The barrier height isV0= 10ប␻⬜. The number of atoms can be taken as N⬃103 _{for the scattering} length as= −0.2 nm. The transverse oscillator length is

a_⬜= 1.2␮m, so the barrier width is d⯝2␮m and the oscillations amplitude of the barrier position isL⯝10␮m. The velocity is normalized by the sound velocity, which is

c⯝4 mm/ s, so the soliton velocity parameter␮0= 0.25 cor-responds to a velocity equal toc.

V. BROAD SOLITON DYNAMICS

If the soliton is broad and its width becomes comparable to 共or larger than兲 the width of the trap generated by the effective potential Veff, in the sense that 2␯a⬃1, then the quasiparticle potential W_␯ does not present any local mini-mum共see Fig.7兲. Indeed, by Eq.共10兲the quasiparticle po-tential W␯ is the convolution of the effective potential with x

t

−50 0 50

50

100

150

200

0 50 100 150 200

0.96 0.98 1 1.02 1.04

t

mass

0 50 100 150 200

−5 0 5

soliton

center

t

FIG. 2. Soliton profile in a nar-row trap V关x− 10 sin共50t兲兴 共top left兲. Mass decay共top right兲. Bot-tom: Soliton motion predicted

by the quasiparticle approach

共dashed兲 and observed in the nu-merical simulations共solid兲.

x

t

−50 0 50

0

50

100

150

200

0 50 100 150 200

0.99 0.995 1 1.005

t

mass

FIG. 3. Soliton profile in a narrow moving trap

V关x− 10 sin共50t兲− 0.25t兴 共top兲. Mass decay共bottom兲.

x

t

−50 0 50

0

50

100

150

200

0 50 100 150 200

0.8 0.85 0.9 0.95 1

t

mass

the convolution kernel K␯, which is a sech 关2兴 function scaled by␯. As a result, for large␯, i.e., for narrow soliton, the quasiparticle potentialW_␯共x兲 resembles the effective po-tentialVeff共x兲. For small␯, i.e., for broad soliton, the convo-lution with K_␯ smooths the trap, the local minimum is washed out, and the quasiparticle potential has the form of a broad Gaussian barrier centered at 0. In this case, a soliton centered at 0 with a nonzero velocity escapes the trap. A soliton centered at 0 with initial zero velocity ␮0= 0 has a different dynamics. The quasiparticle approach predicts an absence of motion. However, for long times, the soliton emits radiation and then becomes a Gaussian-type breather whose width periodically oscillates in the trap Veff. After a long time of the order of a few thousands, these oscillations are damped and the solution converges to a stationary Gaussian-type wave function. The periodic oscillations and the asymptotic Gaussian wave function can be predicted ana-lytically. The effective potentialVeffcan be fitted by the qua-dratic trap

Veff共x兲=V0+V1x2, 共17兲 where

V1= 1

2Veff

⬙

共0兲= − 1 4␲

冕

V

ˆ_共k兲Kˆ

eff共k兲k2dk

= − 1 4␲

冕

V

ˆ_共k兲J

0共ka兲k2dk.

In the linear approximation the width of a stationary Gauss-ian wave functionu共x兲=u_⬁exp共−x2_/x

⬁ 2_兲

in the quadratic trap

共17兲is

x_⬁=

冑

2V₁−1/4. 共18兲

This result is in good agreement with the numerically ob-served asymptotic wave function in Fig.8, wherex⬁⯝5. It is possible to take into account both the cubic nonlinearity and the quadratic trap by using the variational approach with a chirped Gaussian ansatz关21兴,

u共x,t兲=A共t兲exp

冋

− x 2

w共t兲2+ib共t兲x 2

册

.

Substituting this ansatz into the average Lagrangian of Eq.

共7兲 with Eq. 共17兲 and equating to zero its variations, we arrive at the following variational equation for the widthw:

x

t

−50 0 50

0

50

100

150

200

0 50 100 150 200

0.99 0.995 1 1.005

t

mass

FIG. 5. Soliton profile in the moving trap 2V关x− 10 sin共50t兲 + 0.25t兴 共top兲. Mass decay共bottom兲.

x

t

−100 −50 0 50 100

0

10

20

30

40

0 10 20 30 40

0 0.2 0.4 0.6 0.8 1

t

mass

FIG. 6. Soliton profile in the stationary trapV关x− 10 sin共50t兲兴 共top兲. Mass decay共bottom兲. Here the initial velocity parameter is ␮₀= 0.5.

−200 −10 0 10 20

0.5 1 1.5 2 2.5

W

ν

(x

)/(

2

ν

)

x

ν=0.05

ν=0.1

ν=0.2

ν=0.4

wtt=

16

w3−

8E

冑

␲w2− 4V1w, 共19兲

where the conserved quantity is the number of atoms

E_=兰兩u兩2_dx

=

冑

␲/ 2兩A兩2_{共t兲w共t兲}

. This second-order ordinary dif-ferential equation describes the periodic oscillations of the wave function observed in the numerical simulations of Fig. 8. It can be used also to predict the width of the stationary Gaussian wave functionu共x兲=u_⬁exp共−x2/x_⬁2兲in the nonlin-ear regime,

x⬁= 25/4共u⬁ 2

+

冑

u_⬁4+ 8V1兲−1/2,

which is very close to the value共18兲obtained in the linear approximation.

VI. ASYMMETRIC TRAPS AND QUANTUM TUNNELING It is possible to generate asymmetric potentials by the management technique discussed in the paper. For instance, if the barrier positionf共␶兲is of the form

f共␶兲=a−a

2关1 + sin共2␲␶兲兴 2

,

then the convolution kernel in Eq.共8兲is given by

Keff共x兲= 1 2␲

1

共a−x兲3/4_共

冑

2a−

冑

a−x兲1/21共−a,a兲共x兲. The corresponding effective potential Veff and quasiparticle potentialW_␯共x兲for a Gaussian barrier potentialVare plotted in Fig.9, which shows the existence of an asymmetric trap. The dynamics of a soliton in such a trap is very similar to the one in a symmetric trap. In particular, if the initial soliton is centered at 0 with zero velocity, then we do not observe any soliton tunneling. As shown in Fig.10the soliton emits radiation that escapes the effective trapVeffand transforms into a breather whose center corresponds to the minimum of

the effective trap Veff. The breather slowly converges to a stationary Gaussian-type wave function.

It is relevant to compare these results with recent works on soliton tunneling with sub-barrier energies 关13,14兴. The motion of topological soliton in an asymmetric potential well with a finite barrier has been investigated. When the width of the well is larger than the soliton width and the width or height of the barrier is large, the soliton dynamics is the one of a classical particle. When the soliton width is comparable x

t

−50 0 50 0

200

400

600

800

1000

0 1000 2000 3000 4000 0.1

0.12 0.14 0.16 0.18 0.2

soliton

amplitude

−20 0 20 0.005

0.01 0.015 0.02

x

soliton

profile

FIG. 8. Top left: Soliton profile in the case ␯0= 0.05, ␮0= 0, V共x兲= 10 exp共−x2/ 4兲, and f共␶兲= 10 sin共2␲␶兲. Top right: Soliton amplitude vs time. Bottom: the input profile 兩u共x, 0兲兩2 _is plotted in dashed lines, the output profile兩u共x,t兲兩2 is plotted in solid lines at timet= 4000, and the effective potential Veff 共up to a multiplicative constant兲is also represented in dotted lines.

−200 −10 0 10 20

1 2 3 4

V eff

x

−200 −10 0 10 20

1 2 3 4

W

ν

(x

)/(

2

ν

)

x

ν=0.05

ν=0.1

ν=0.2

ν=0.4

with the well width, soliton tunneling with sub-barrier ki-netic energies can be observed. Our observations show that, if the NLS soliton is narrow, then soliton tunneling is absent, which is analogous to the topological soliton dynamics case

关13兴. Increasing the soliton width we observe the continuous shedding of a small amount of radiation from the soliton, which leads to the formation of a breather-type solution that exists for long times in the dynamically generated potential.

VII. CONCLUSION

In conclusion we have studied the propagation of a bright matter-wave soliton through a barrier potential with a rapidly oscillating position. We have shown that the soliton can be dynamically trapped by such a potential if its width is smaller than the trap width and if its initial velocity is below a critical value predicted by the perturbed IST theory. If the soliton width is comparable to the trap width and its initial

velocity is zero, then tunneling radiation emission leads to the generation of a breather in the effective trap.

It would be interesting to extend this model to the case of a standing well and an oscillating barrier with a large oscil-lation amplitude. In this case the effective potential has the form of a well between two opaque barriers. Applying the results obtained in Ref.关15兴we can expect for the transmis-sion of the nonlinear matter waves such phenomena as the instability and quantum turbulence on the time scale of the quantum dwell time in the well. Another physically relevant system is an oscillating barrier enclosed in a broad well. The effective potential in this case simulates a three-well, two-barrier heterostructure and effective nonlinearities can ex-hibit chaotic dynamics in the tunneling process关16兴. These problems require a separate investigation.

ACKNOWLEDGMENT

F.Kh.A. is grateful to FAPESP for the partial support of his work.

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