Soliton response to transient trapvariations

S. Sree Ranjani†1, Utpal Roy‡2, P. K. Panigrahi∗3 and A. K. Kapoor†4,† School of Physics, University of Hyderabad, Hyderabad 500 061, India.‡ Dipartimento di Fisica, Universita di Camerino, I - 62032 Camerino, Italy.∗ Indian Institute of Science Education and Research (IISER) Kolkata, Salt

Lake, Kolkata 700 106, India.

Abstract

The response of bright and dark solitons to rapid variations in an ex-pulsive longitudinal trap is investigated. We concentrate on the effect oftransient changes in the trap frequency in the form of temporal delta kicksand the hyperbolic cotangent functions. Exact expressions are obtainedfor the soliton profiles. This is accomplished using the fact that a suitablelinear Schrodinger stationary state solution in time can be effectively com-bined with the solutions of non-linear Schrodinger equation, for obtainingsolutions of the Gross-Pitaevskii equation with time dependent scatteringlength in a harmonic trap. Interestingly, there is rapid pulse amplificationin certain scenarios.

PACS number(s) : 03.75-b, 05.45.Yv

I Introduction

Bose-Einstein condensates (BECs) [1] - [3] can be confined to one and twodimensions in suitable traps [4] - [6], which has opened up the possibility for theirtechnological applications. BEC on a chip is a two dimensional configuration[7], whereas cigar shaped BECs refer to quasi one dimensional scenario [8] -[10]. In the latter case, the transverse confinement is taken to be strong, withthe longitudinal trap frequency having comparatively much smaller value. Inthis condition, dark [11] - [13] and bright [14] - [19] solitons have been observedin repulsive and attractive coupling regimes respectively. In the context of theBEC, bright and dark solitons respectively correspond to density lumps andrarefactions compared to the background. The corresponding mean field Gross-Pitaevskii (GP) equation being the familiar non-linear Schrodinger equation(NLSE) in quasi-one dimensions [20], the observation of localized solitons hasdemonstrated the ubiquity of solitons in diverse branches of physics, which sharecommon non-linear behavior.

Very interestingly, in the cigar shaped BEC, a number of parameters like thescattering length and trap frequencies can be changed in a controlled manner,leading to the possibility of coherent control of solitons. The former can have a

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temporal variation due to Feshbach resonance, a scenario which has been exten-sively studied in the literature [21] - [23]. In particular, the discovery of brightsoliton and soliton trains has led to a systematic investigation of the impact ofthe change in the scattering length on the BEC profile. Recently, the temporalmodulation of the non-linearity in one dimension arising due to the correspond-ing variations in the transverse trap frequency have been systematically studiedin the context of generation of Faraday modes [24], [25].

In a recent paper [26], a method to obtain analytical solutions of the GPequation for a range of temporal variations of parameters like the scatteringlength, the longitudinal trap frequency etc. was demonstrated. It was shownthat in the process of reducing the GP equation to a known non-linear equa-tion, having Jacobi elliptic functions as solutions, the consistency conditionsobtained mapped onto the linear Schrodinger equation. This mapping enablesus to obtain exact solutions of the GP equation for a wide range of temporalmodulations of the control parameters. These solutions exhibit soliton trains,localized solitons (both bright and dark) for different regimes of the couplingparameter. Interestingly, exact solutions of the GP equation have been obtainedfor other kind of trap potentials, for example linear traps [27] using Darbouxtransformations. The effect of the variations in the time dependent coupling onthe soliton profiles has also received a lot of attention [27] - [29].

The goal of the present paper is to investigate the soliton dynamics in aBEC in an expulsive harmonic trap, whose frequency is time dependent. Theexpulsive trap accelerates the soliton. Very interestingly it is found that forcertain conditions it can be compressed, particularly under a sudden transientvariation in the trap frequency. As we have mentioned earlier, a temporal mod-ulation in the transverse frequency leads to the generation of Faraday modes inthe longitudinal direction. Hence, it is of interest to study the effect of transientvariations in the longitudinal trap on the BEC profiles. We concentrate hereon transient variations and for explicitness we use the delta functions and thehyperbolic cotangent functions to model rapid temporal variations in the trapfrequency. We obtain exact solutions and study the spatio-temporal dynamicsof the solitons and their response to the changes in the trap frequency. We pointout that harmonic traps decorated with delta functions and a dimple potentialhave been recently studied numerically [30] and the properties like atomic den-sity, chemical potential, critical temperature etc. of the condensate have beenobtained.

In the following section, we obtain the exact solutions of the GP equationand illustrate how we can modulate the expulsive trap frequency using suitablequantum mechanical models [26]. In section III, the trap frequency is tran-siently changed by hyperbolic cotangent functions. In section IV, we study theresponse of the solitons to sudden changes in the trap frequency in the formof delta kicks. For all the cases studied we obtain profiles of both dark andbright soliton trains and localized soliton solutions. The spatio-temporal dy-namics of the soliton solutions and the behavior of the non-linearity with timein the attractive and repulsive coupling regimes are plotted. It is found thatfor certain examples there is an amplification in the soliton amplitude resulting

2

in its compression while in other examples the solitons spread with a decayingamplitude.

II The Dynamics of the Bose-Einstein condensate

For a BEC at zero temperature, confined in cylindrical harmonic trap V0(x, y) =m′ω2

⊥(x2 + y2)/2 and time dependent harmonic confinement along the z direc-tion V1(z, t) = m′ω2

0(t)z2/2, the dynamics is governed by the GP equation

ih∂Ψ(r, t)∂t

=(− h2

2m′∇2 + U |Ψ(r, t)|2 + V

)Ψ(r, t), (1)

where V = V0+V1, m′ is the mass of the particle and U = 4πh2as(t)/m′ gives thenon-linear coupling with as(t) denoting the scattering length. The nonlinearitycan be either attractive (as < 0) or repulsive (as > 0). In order to reduce the3D GP equation to quasi-one dimensions, it is assumed that the interactionenergy of atoms is much less than the kinetic energy in the transverse direction[9]. Substituting the following trial wave function in dimensionless units,

Ψ(r, t) =1√

2πaBa⊥ψ

(z

a⊥, ω⊥t

)exp

(−iω⊥t−

x2 + y2

2a2⊥

), (2)

we obtain the quasi-1D NLSE

ı∂tψ = −12∂zzψ + γ(t)|ψ|2ψ +

12M(t)z2ψ. (3)

Here, γ(t) = 2as(t)/aB , M(t) = ω20(t)/ω2

⊥, a⊥ = (h/m′ω⊥)1/2 with aB beingthe Bohr’s radius. We have kept M(t), related to the trap frequency, timedependent. This enables us to modulate the trap frequency, allowing us tostudy the effect of different temporal profiles of the trapping potential on thesolitons. The specific case of M(t) being a constant, describes the scenariowhere the oscillator potential is either confining (M > 0) or repulsive (M < 0).Analytic solutions to the above NLSE are obtained using the ansatz solution

ψ(z, t) =√A(t)F{A(t)[z − l(t)]} exp[ıΦ(z, t)], (4)

where A(t) describes the amplitude and l(t) =∫ t0v(t′)dt′ determines the location

of the center of mass. We assume the phase to be of the form

Φ(z, t) = a(t)− 12c(t)z2. (5)

Substitution of ψ(z, t) in the NLSE gives us the consistency conditions

A(t) = A0 exp(∫ t

0

c(t′)dt′),dl(t)dt

+ c(t)l(t) = 0 (6)

and

γ(t) = γ0A(t)A0

, (7)

3

where A0, γ0 and l0 are constants. We also obtain a(t) = a0 + λ−12

∫ t0A2(t′)dt′

along with the Riccati equation for c(t),

dc(t)dt− c2(t) = M(t). (8)

From the above equations, it can be seen that A(t), γ(t) and l(t) are all non-trivially related to the phase component c(t). Added to this, the fact that theRiccati equation can be mapped onto the linear Schrodinger eigenvalue problemwith potential V (t),

− φ′′(t)−M(t)φ(t) = 0 (9)

with M(t) = M0 − V (t), using the change of variable,

c(t) = − d

dtln[φ(t)] (10)

gives us control over the dynamics of the BEC. Hence, knowing c(t) or thesolution φ(t) of (9), allows us to obtain analytical expressions of the controlparameters through equations (10) and (6)-(7). It is a well known fact that formany potential models, the Schrodinger equation can be solved exactly. In thenext sections, we exploit this and the connection between c(t) and φ(t) to studythe dynamics of the BEC analytically, when the oscillator frequency undergoesa rapid change.

The use of the above consistency conditions along with the ansatz solutionin the NLSE gives us the following differential equation for F , in terms of thenew variable T = A(t)[z − l(t)],

F ′′(T )− λF (T ) + 2κF 3(T ) = 0 (11)

with κ = − γ0A0

and the differentiation is with respect to T . The general solutionof this differential equation can be written in terms of the 12 Jacobi ellipticfunctions sn (T,m) , cn (T,m) , dn (T,m) etc where m is the elliptic modulus[31]. The solitons will arise in the limit m→ 1, where dn (T,m) = cn (T,m)→sech (T ) and sn (T,m)→ tanh(T ).

Thus for γ0 < 0, bright soliton trains of the form

ψ(z, t) =√A(t) cn (T/τ0,m) exp[iΦ(z, t)] (12)

exist, where τ20 = −mA0/γ0 and λ = (2m−1)/τ2

0 (these coefficients are obtainedusing equations (6) -(9). In the limiting case m = 1, equation (12) correspondsto a bright soliton solution. It has been seen that for γ0 > 0, there exist darksoliton trains of the form

ψ(z, t) =√A(t) sn(T/τ0,m) exp[iΦ(z, t)], (13)

with τ20 = mA0/γ0, λ = −(m+1)/τ2

0 and in the limit m→ 1, the above solutioncorresponds to a dark soliton.

4

In the following sections, we obtain the profiles of bright and dark solitontrains and solitons in a BEC in an expulsive oscillator potential, with experi-mentally achievable temporal variations of the trap frequency. As seen in (8),the oscillator frequency M(t) depends on c(t). We first concentrate on the re-sponse of the soliton profiles to the variations in the trap frequency when c(t)is equal to the hyperbolic cotangent functions.

III Dynamics of the BEC with c(t) = ±B coth (t)

In this section, we look at the examples where the trap frequency of theconfining trap is modulated using c(t) = ±Bcoth (t) , B > 0. Using the consis-tency conditions in (6) and (7) we obtain the analytical solutions of the NLSEas shown below.

case (a)We take c(t) = Bcoth (t), B > 0 in (8) and obtain

M(t) = −B2 −B(B + 1)cosech 2(t). (14)

Since M0 = −B2, we are looking at an expulsive oscillator scenario. We pointout that V (x) = B(B+1)cosech 2(x) corresponds to the supersymmetric Rosen-Morse potential with superpotential W = B coth (x) [32]. The function V (t)has a singularity at t = 0 which causes a sudden change in the trap frequency.Substituting for c(t) in equations (6) and (7) one obtains

a(t) = a0 −(λ− 1)A2

0

2(sinh (t)− 4t) (15)

A(t) = A0

(sinh (t)sinh (t0)

)B, γ(t) = γ0

(sinh (t)sinh (t0)

)B, l(t) = l0

(sinh (t0)sinh (t)

)B,

(16)which when substituted in (12) and (13), gives the following solutions describingthe bright and dark soliton trains

ψ(z, t) =√A0sinh (t) cn

(A0sinh (t)

[z − l0cosech (t)]τ0

,m

)exp(iΦ(z, t)), (17)

ψ(z, t) =√A0sinh (t) sn

(A0sinh (t)

[z − l0cosech (t)]τ0

,m

)exp(iΦ(z, t)) (18)

respectively. The dynamics of these solutions are plotted in fig 1(a) and 1(b),where the matter wave density |ψ(z, t)|2 is plotted for varying t and z. We cansee that the soliton trains get amplified with time as we move away from the ori-gin. In the limit m→ 1, from (17) and (18) we get the bright and dark solitons,plotted in fig 1(c) and 1(d) respectively. These solitons are highly localized andare seen to be diverging from the t = 0 line in the t − z plane. By changingthe initial time to some t0, instead of t = 0 in the consistency conditions andvarying l0, the location of the center of mass can be changed. The non-linearity

5

γ(t) is plotted against varying t, for the attractive (γ0 < 0) and the repulsive(γ0 > 0) coupling regimes in fig 1(e) and 1(f) respectively and we see that fora given γ0, the non-linearity does not change sign implying we are either in anattractive or repulsive coupling regimes. By changing the constants B, A0, γ0,l0 and t0 the amplification, location of the solitons can be changed.

Case (b)For c(t) = −B coth (t), B > 0, M(t) = −B2 −B(B − 1)cosech 2(t). Here againwe are looking at the expulsive oscillator scenario and the potential B(B −1)cosech 2(x) corresponds to the supersymmetric Poschl-Teller potential [32].Moreover, this case is interesting because for B = 1, this potential correspondsto the free particle problem which has been studied in [26]. Substituting c(t) inequations (6) and (7), we obtain

A(t) = A0

(sinh (t0)sinh (t)

)B, γ(t) = γ0

(sinh (t0)sinh (t)

)B, l(t) = l0

(sinh (t)sinh (t0)

)B.

(19)As in the above case, we obtain the profiles of both bright and dark solitontrains and localized solitons which are plotted in figure 2(a) - 2(f). Althoughthe function V (t) has a singularity at t = 0, like in case (a), the response of thesoliton solutions is entirely different. We see that the soliton trains have a veryhigh amplitude around t = 0 and they decay as we move away from the origini.e. the soliton profile spreads with time. The amplitude and the decay timecan be controlled by changing A0. The solitons plotted in figures 2(c) and 2(d)are again highly localized, but in this case converge towards the t = 0 line inthe t−z plane. From figures 2(e) and 2(f), we see that though the non-linearitydoes not change sign for a given γ0, it does increase rapidly in magnitude aroundt = 0, unlike the previous case were γ(t) varied slowly with time.

IV Response of BEC to delta kicks in the trap frequency

In this example, we vary the trap frequency with an attractive delta potentiallocated at the origin and see how the solitons respond to a sudden kick. Here,

V (t) = −V0 δ(t), V0 > 0. (20)

and the solutions of (9) are

φ1(t) =

√V0

2exp(−V0

2t) (t > 0)

=

√V0

2exp(

V0

2t) (t < 0) (21)

φ2(t) =

√V0

2exp(

V0

2t) (t > 0)

= −√V0

2exp(−V0

2t) (t < 0) (22)

6

with M0 = −V202 . In order to obtain φ2(t), we have made use of the fact that

the second solution of a second order differential equation can be obtained fromthe first solution using the relation φ2(t) = φ1(t)

∫ t[dt′(φ1(t′))−2]. We can writec(t) in terms of φ(t) using (10) and obtain the consistency conditions in termsof φ(t) as

A(t) = A0φ(0)φ(t)

; γ(t) = γ0φ(0)φ(t)

, l(t) = l0φ(t)φ(0)

. (23)

Now we proceed to study the soliton profiles using the two solutions given in(21) and (22) separately.

Case (a)First we consider the solutions in (21) and from equation (10) we obtain c(t) =V0/2 for t > 0 and c(t) = −V0/2 for t < 0. From (23), we obtain

A(t) = A0 exp(V0

2|t|) , γ(t) = γ0 exp(

V0

2|t|) , l(t) = l0 exp(−V0

2|t|) (24)

and also find that

a(t) = a0 +(λ− 1)A2

0

2V0[exp(V0|t|)− 1] . (25)

Substituting A(t), γ(t) and l(t) in (12), we obtain the solutions describingbright and dark soliton profiles, which are plotted in figure 3 along with theplots showing the behavior of the nonlinearity. From the figures we can see thatthe sudden temporal kick to the soliton profile results in rapid amplification ofthe soliton trains over time. Similar to the case (a) of the previous section, thesolitons are highly localized and diverge from the origin. By varying V0, we cancreate a soliton with the required amplitude. The non-linearity γ(t) for thiscase turns out to be an exponential function. Thus the scattering length as(t)turns out to be an exponential in time [18]. It is known that the dynamics ofa BEC in an expulsive parabolic potential are governed by an NLSE with anexponentially varying scattering length. The conditions in which these systemsare one dimensional and the range of the parameter values for which the con-densate is stable are discussed in [16].

Case (b)Here, we repeat the above analysis with the solution given in (22). For thissolution, the nonlinearity γ(t) changes sign when we go from the negative timedomain to the positive time domain. Here, we study the soliton dynamics fort > 0 and the corresponding expressions are

c(t) = −V0

2, a(t) = a0 −

(λ− 1)A20

2V0[exp(−V0t)− 1] , (26)

A(t) = A0 exp(−V0

2t) , γ(t) = γ0 exp(−V0

2t) , l(t) = l0 exp(−V0

2t). (27)

7

The soliton profiles are plotted in figure 4. We observe that the soliton trainamplitude decays rapidly as t increases unlike the previous case. For smallervalues of V0, the trains exist for longer time. Thus the decay of the solutionscan be controlled by varying V0.

In conclusion, we have analytically treated the effect of variations of thelongitudinal trap frequency on the BEC profile. We have looked at the caseswhere the trap frequency is modulated using hyperbolic cotangent functions anddelta kicks. It is observed that in the gain/loss less scenario, these variations inthe trap frequency resulted in the soliton profiles either amplifying or spreading.We have also shown that we can manipulate the amplification and the locationof the solitons by varying the control parameters.

The fact that a stationary Schrodinger eigenvalue problem can be combinedwith solutions of non-linear Schrodinger equation to obtain the solutions ofGross-Pitaevskii equation with time dependent parameters, enabled us to ob-tain the exact dynamical behavior. As mentioned in the introduction, there areother methods of obtaining exact solutions of the GP equation. The methodused in this paper is simpler and the fact that the Schrodinger equation can besolved exactly for many potential models, offers us a huge choice of temporalvariations of the trap frequency from the class of exactly solvable models. Wecan choose the models which can be realized experimentally. Along with thiswide choice, the other advantage is that we have total control over the solitondynamics. It would be interesting to see how solitons evolve in space and timefor other choices of variations in the trap frequency. Another interesting prob-lem will be the study of two soliton dynamics under similar variations in thetrap frequency [29].

AcknowledgmentsS. S. R. acknowledges financial support provided by Council of Scientific andIndustrial Research (CSIR), Government of India.

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(a) Bright soliton train, m = 0.5 ,τ0 = 0.375

(b) Dark soliton train, m = 0.5, τ0 =0.375

(c) Bright soliton, m = 1, τ0 = 1 (d) Dark soliton, m = 1, τ0 = 1

(e) γ(t) versus t, γ0 = −0.5 (f) γ(t) versus t, γ0 = 0.5

Figure 1: c(t) = coth(t) in the expulsive oscillator regime. Plots of thebright [(a),(c)] and dark [(b),(d)] soliton solutions, ψ(z, t) =

√A(t)F{(A(t)[z−

l(t)])/τ0,m} exp[ıa(t) − ı2c(t)z

2] with F = cn (T/τ0,m) and F = sn(T/τ0,m)respectively, are shown. The plots (a)-(d) show |ψ(z, t)|2 versus t and z and theplots (e) and (f) show the variation of the nonlinearity γ(t) with t. The valuesof the constant parameters are A0 = 0.5, l0 = 5, t0 = 0.9 and B = 2.

11

(a) Bright soliton train, m = 0.5, τ0 =0.375

(b) Dark soliton train, m = 0.5, τ0 = 0.375

(c) Bright soliton, m = 1, τ0 = 1 (d) Dark soliton, m = 1, τ0 = 1

(e) γ(t) versus t, γ0 = −0.5 (f) γ(t) versus t, γ0 = 0.5

Figure 2: c(t) = −coth(t) in the expulsive oscillator regime. Plots of thebright [(a),(c)] and dark [(b),(d)] soliton solutions, ψ(z, t) =

√A(t)F{(A(t)[z−

l(t)])/τ0,m} exp[ıa(t) − ı2c(t)z

2] with F = cn (T/τ0,m) and F = sn(T/τ0,m)respectively, are shown. The plots (a)-(d) show |ψ(z, t)|2 versus t and z and theplots (e) and (f) show the variation of the nonlinearity γ(t) with t. The valuesof the constant parameters are A0 = 0.5, l0 = 5, t0 = 0.8812 and B = 2.

12

(a) Bright soliton train, m = 0.5,τ0 = 0.375

(b) Dark soliton train, m = 0.5, τ0 =0.375

(c) Bright soliton, m = 1, τ0 = 1 (d) Dark soliton, m = 1, τ0 = 1

(e) γ(t) versus t, γ0 = −0.5 (f) γ(t) versus t, γ0 = 0.5

Figure 3: Attractive δ potential in the expulsive oscillator regime, φ(t) =√V0/2 exp(−V0|t|/2). Plots of the bright [(a),(c)] and dark [(b),(d)] soliton

solutions, ψ(z, t) =√A(t)F{(A(t)[z − l(t)])/τ0,m} exp[ıa(t) − ı

2c(t)z2] with

F = cn (T/τ0,m) and F = sn(T/τ0,m) respectively, are shown. The plots (a)-(d) show |ψ(z, t)|2 versus t and z and the plots (e) and (f) show the variation ofthe nonlinearity γ(t) with t. The values of the constant parameters are V0 = 10,A0 = 0.5, l0 = 5.

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(a) Bright soliton train, m = 0.5,τ0 = 0.375

(b) Dark soliton train, m = 0.5, τ0 =0.375

(c) Bright soliton, m = 1, τ0 = 1 (d) Dark soliton, m = 1, τ0 = 1

(e) γ(t) versus t, γ0 = −0.5 (f) γ(t) versus t, γ0 = 0.5

Figure 4: Attractive δ potential in the expulsive oscillator regime, φ(t) =√2/V0 exp(V0t/2). Plots of the bright [(a),(c)] and dark [(b),(d)] soliton

solutions, ψ(z, t) =√A(t)F{(A(t)[z − l(t)])/τ0,m} exp[ıa(t) − ı

2c(t)z2] with

F = cn (T/τ0,m) and F = sn(T/τ0,m) respectively, are shown. The plots (a)-(d) show |ψ(z, t)|2 versus t and z and the plots (e) and (f) show the variation ofthe nonlinearity γ(t) with t. The values of the constant parameters are V0 = 10,A0 = 0.5, l0 = 5.

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