Nonlinear modes in a complex parabolic potential - APS Link Manager

PHYSICAL REVIEW A 81, 013606 (2010)

Nonlinear modes in a complex parabolic potential Dmitry A. Zezyulin,1,* Georgy L. Alfimov,1,† and Vladimir V. Konotop2,‡ 1

Moscow Institute of Electronic Engineering, Zelenograd, Moscow RU-124498, Russia Centro de F´ısica Te´orica e Computacional, Universidade de Lisboa, Complexo Interdisciplinar, Avenida Professor Gama Pinto 2, Lisboa P-1649-003, Portugal, and Departamento de F´ısica, Faculdade de Ciˆencias, Universidade de Lisboa, Campo Grande, Edif´ıcio C8, Piso 6, Lisboa P-1749-016, Portugal (Received 16 October 2009; published 11 January 2010) 2

We report on analysis of the mode structure of a Bose-Einstein condensate loaded in a complex parabolic potential and subjected to a constant pump. Stationary solutions for the positive and negative scattering lengths are addressed. In the case of a positive scattering length and large number of atoms the ground state is described by the Thomas-Fermi distribution, whose properties in the presence of the dissipation are very different from its conservative counterpart. It is shown that for a positive scattering length only the ground state appears to be stable. DOI: 10.1103/PhysRevA.81.013606

PACS number(s): 03.75.Lm, 05.45.Yv, 42.65.Tg

I. INTRODUCTION

Progress in producing and managing Bose-Einstein condensates (BECs) is intimately related to possibilities of cooling atoms and of their confinement in trapping potentials. The latter phenomenon is based on interaction of atoms with the external electromagnetic field (see, e.g., [1,2]). For the applications related to trapping of the atoms the frequencies of the field are usually chosen far enough from the atomic resonances. In such a situation interactions of atoms with the trap can be safely considered to be elastic and respective potentials to be conservative. At the same time losses in the atom-field interactions, even though negligible in the leading order, necessarily appear in higher orders of a more accurate description of the system. Moreover, the losses can be enhanced, say, by changing either the field frequency or the intensity. Then, in the mean-field approximation the inelastic interactions of atoms with an external field can be described by the complex part of the respective trap potential, leading to decay of the number of trapped atoms. If, at the same time, one is interested in maintaining nonlinear density profiles unchanged, the induced losses must be compensated by some gain, say, by the linear pump. In the BEC theory such a gain can be originated by the condensation of atoms from the thermal cloud [3] or created artificially by properly manipulating the trap [4] (in this last case the linear density grows, while the total number of atoms decreases). Various aspects of the effect of the linear pump on the evolution of BECs with losses [4,5] and on BECs in inverted parabolic traps [6] have been addressed in a number of earlier publications. In a general situation the properties of dissipative systems are very different from the properties of their conservative counterparts [7]. Accordingly, even small (but non-negligible) dissipation may result in dramatic changes of the condensate dynamics. In particular, combined with a pump it may lead

*

[email protected] [email protected] ‡ [email protected] †

1050-2947/2010/81(1)/013606(7)

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to enhanced stability of some of the density patterns (in mathematical terms they become attractors), making all other states unstable. This selective stability, in turn, may have practical applications, justifying the relevance of a study of possible density patterns in complex potentials. The aim of the present work is to describe the atomic distribution of a quasi-one-dimensional BEC loaded in a complex parabolic trap, where the interaction of the atoms with the external field are not purely elastic. The system is described by the equation of a complex nonlinear oscillator. This is the simplest geometry of a trap used in numerous experimental settings. In particular, a typical example of such a situation occurs when atoms are loaded in an optical trap, whose atomic polarizability is not purely real but contains a complex imaginary part [2], which must be taken into account when the laser field intensity is large enough. In the conservative case the nonlinear modes of the parabolic trap have been studied in numerous publications (see, e.g., [8,9] and references therein) and presently are fairly well understood for both attractive and repulsive condensates. A number of particles in a mode, or alternatively the chemical potential, can be considered as a quantity parametrizing the families of the modes bifurcating from the respective solutions for the linear oscillator. Later in this article we show that the situation is essentially different when the dissipative processes are taken into account. This is reflected in the structure of the modes, their stability, and especially in the control parameters. In the dissipative case neither the number of particles nor the chemical potential of a mode is a free parameter any more; instead they both are defined by the balance between the dissipation and the pump in the system. The organization of this article is as follows. We start with the formulation of the model in Sec. II, subsequently addressing its linear limit, as well as some general mathematical properties expressed in terms of the relations among the parameters. In Sec. III we obtain the Thomas-Fermi limit of the model and discuss its physical properties. Numerical investigation of the structure of the modes is given in Sec. IV. The stability of the modes is addressed in Sec. V. In Sec. VI, we summarize the outcomes of the article and outline perspectives. ©2010 The American Physical Society

ZEZYULIN, ALFIMOV, AND KONOTOP


Next, taking into account the inequality ∞ √ x 2 |ψ|2 dx = 2 N Xψx 2 ψ22 2ψx 2

II. THE MODEL AND SOME GENERAL RELATIONS A. The model

Let us consider the one-dimensional Gross-Pitaevskii (GP) equation it = −xx + ν 2 x 2 + σ ||2 + i,

(1)

where is the dimensionless order parameter, σ = sgn(as ) where as is the scattering length, and we introduced the linear pump [5] describing loading atoms in the potential and characterized by the constant strength . The linear oscillator “frequency” ν = νr − iνi is now complex. In what follows we also use the notations ν 2 = α1 − iα2 (i.e., α1 = νr2 − νi2 and α2 = 2νr νi ), and we assume that all introduced constants are positive, that is, νr,i > 0, αr,i > 0, and > 0. Spatially localized solutions we are interested in correspond to the ansatz (t, x) = e−iµt ψ(x), where µ is a real constant, subject to the zero boundary conditions: lim|x|→∞ |ψ(x)| = 0. Then the introduced complex-valued function ψ(x) solves the stationary equation −ψxx + (−µ + i)ψ + (α1 − iα2 )x 2 ψ + σ |ψ|2 ψ = 0.

(2)

Taking into account the total amount of the parameters (they are four), we first observe that one of them can be scaled out. Nevertheless, we prefer to keep all of them, for the sake of a simpler analysis of physically different limits. The characteristics of the complex potential αr,i (or alternatively νr,i ) and the pump are considered as the control parameters: they are supposed to be given in an experiment, whereas the chemical potential µ is a parameter determined by the stationary state of the system. As the first step we observe that the following equalities hold: (3) |ψx |2 dx − µN + α1 N X2 + σ |ψ|4 dx = 0 and X=

[it is obtained using the Cauchy √ inequality as well as definition (5)], one verifies that N 2Xψx 2 . Then by taking the square of both parts of the last relation and recalling the link (4), one readily obtains N

4 ψx 22 . α2

(9)

The obtained result acquires transparent physical meaning after one introduces the Fourier transform of the wave function ˆ ψ(k) = √12π eikx ψ(x)dx and takes into account that the mean-square spectral width K of a mode can be defined as

2 ˆ k 2 |ψ(k)| dk ψx 22 K = = . 2 dk ˆ N |ψ(k)| 2

(10)

Then (9) gives the lower bound for the spectral width: K √ α2 /4 (as it is clear now the “uncertainty” relation takes the form KX 1/2). From the estimate (9) combined with the Sobolev inequality we obtain other useful relations: ψ44 A2 N 2ψx 2 N 3/2 16

3/2 3/2 α2

ψx 42 ,

(4)

we have introduced the total number of particles N = where ∞ 2 |ψ| dx and the mean-square width of the mode −∞ ∞ 1 x 2 |ψ|2 dx. (5) X=√ N −∞ Thus we arrive at the result that the mean-square width of the mode does not depend on the number of particles and is uniquely determined by the relation between dissipation and pump. Belowwe use also the standard notations for the Lp norm: p ψp = |ψ|p dx. As it is evident in these notations N = ψ22 and using Eq. (4) one can rewrite Eq. (3) in the form MN = ψx 22 + σ ψ44

(6)

where we have introduced the quantity M =µ−

α1 α2

whose physical meaning is clarified below.

(7)

(11)

where we have introduced the amplitude A of the mode: A = supx (|ψ|). B. The linear limit

Let us now consider the linear case, which is formally obtained by setting σ = 0. The respective linear eigenvalue problem − ψ˜ xx + i ψ˜ + (α1 − iα2 )x 2 ψ˜ = µ˜ ψ˜

/α2 ,

(8)

−∞

(12)

(hereafter a tilde is introduced to distinguish the linear solution) possesses a set of eigenvalues µ˜ n = (2n + 1)νr ,

(13)

where n = 0, 1, . . . , provided the dissipation and pump are balanced, that is, provided the pump is chosen as = ˜ n , where ˜ n = (2n + 1)νi

(14)

(for the linear eigenmodes of the inverted harmonic oscillator, that is, the oscillator with purely imaginary frequencies which is also referred to as the expulsive trap see [10] and references therein). The corresponding eigenfunctions read as √ 2 (15) ψ˜ n (x) = Hn ( νx)e−νx /2 , where H√ n (z) is the nth Hermite polynomial and the branch of the root ν is chosen to have positive real part. We also observe that now M = M˜ n , where 1 νr2 + νi2 M˜ n = n + > 0. (16) 2 νr

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NONLINEAR MODES IN A COMPLEX PARABOLIC POTENTIAL

For the conservative case, νi = 0, the obtained quantity M˜ n is nothing but the energy of the linear quantum oscillator. Let us now fix n. The nth linear mode ψ˜ n (x)e−i µ˜ n t is a solution of Eq. (1) with σ = 0 and = n . Simple calculations show that this mode is stable only if, for each k = 0, 1, . . . , the difference ˜ n − ˜ k = 2(n − k)νi is nonpositive. This takes place if n = 0 and νi > 0. There are two important conclusions following from the obtained result. First, in the linear case only the ground state is stable; that is, the ground state is the only attractor (corresponding to a nontrivial solution). Second, the potential must be dissipative (i.e., there are no stable modes in a potential with gain even in the presence of linear dissipation). This last property justifies the signs fixed above for the parameters of the problem. On the other hand, in terms of the original nonlinear system, the linear case can also be seen as the limit N → 0. Hence, when the balance (14) is verified the stable state of the system corresponds to the absence of any particle. If, however, = ˜ n , nonzero stable solutions are possible. Their study is performed in the rest of the present article. Bearing in mind that in the linear case only the ground state is stable, we pay special attention to the solutions branching from the zero mode (n = 0) for repulsive (σ = 1) and attractive (σ = −1) nonlinearities. For comparison we also briefly discuss the solutions from the next branch (n = 1). C. Some general relations

Now we turn to the nonlinear problem in the repulsive case (positive scattering length, σ = 1). From (6) it follows that M > 0, that is, the chemical potential µ is positive, and that ψx 22 MN . Further, the last inequality is compatible with (9) only when √ α1 α2 + α1 , (17) 4 α2 thus giving the lower (however, not necessarily the best) bound for the chemical potential. Next we obtain from (11) and (6) the relation 3/2 1 + 16 3/2 ψx 22 ψx 22 − MN 0. (18) α2 µ µ0 =

Recalling that M > 0 and MN ψx 22 , we finally get 1 + 16MN (/α2 )3/2 − 1 ψx 22 M. (19) M 3/2 32MN (/α2 ) N The last √ result allows one to attribute the physical meaning for M: M defines the order of the spectral width of the solution. Indeed, recalling (10) and assuming that the factor 16MN(/α2 )3/2 is of the order of 1 we obtain from (19) √ K ∼ M. (20) Passing to the attractive case (σ = −1) we first observe that for sufficiently small amplitudes of the nonlinear modes one has M > 0 [since this is the case of the linear limit, see (16)]; that is, the chemical potential is positive. An increase of the amplitude of the nonlinear mode is accompanied by a decrease of M, which becomes negative for sufficiently large


amplitudes. In the point M = 0 for the amplitude the estimate √ A α2 /4 is valid. Then for M < 0, from (6) and (11) we obtain 3/2

α2 ψx 22 4N 3 . (21) 16 3/2 From this relation and from (4) one concludes that, in the attractive case at a constant dissipation α2 , a decrease of the pump leads to an increase of the number of particles N in the stationary modes and to a decrease of its width X. III. THOMAS-FERMI DISTRIBUTION IN THE REPULSIVE CASE A. Analytical form of the Thomas-Fermi distribution

Now we turn to a more detailed study of the mode branching from the linear ground state, for which ˜ 0 = νi and µ˜ 0 = νr . We concentrate on the case of a large number of particles, N → ∞, and consider the case of a positive scattering length, σ = 1. As it is well known, in the conservative case, that is, at = α2 = 0, the respective solution is described by the ThomasFermi (TF) distribution [1,2]. We hold this terminology also for the dissipative problem. The first immediate conclusion is that contrary to the conservative case, where the TF solution is real (or more precisely can have only a constant phase), the solution of Eq. (2) must be complex (that is, have a phase depending on the spatial coordinate). Indeed, let us rewrite Eq. (2) in terms of real amplitude ρ(x) and phase ϕ(x) of the wave function ψ(x) = ρ(x)eiϕ(x) : ρxx (22a) + (µ − α1 x 2 ) − ϕx2 − ρ 2 = 0, ρ 2ρx ϕx + ϕxx = 0. (22b) (− + α2 x 2 ) + ρ Let us also introduce a slow variable y = εx as well as the renormalized phase (y) = εa ϕ(x) and amplitude R(y) = εb ρ(x). Hence, ε 1 is a small parameter whose physical meaning is clarified later in this article. The exponents a and b are to be found from the respective scaling of Eqs. (22), subject to the conditions R ∼ Ry ∼ y ∼ 1, which now take the form Ryy V2 y2 R2 + µ − α1 2 − 2a−2 − 2b = 0, (23a) ε2 R ε ε ε 2

2V Ry Vy y + − + α2 + a−2 = 0. (23b) ε2 Rεa−2 ε Here we have introduced the renormalized superfluid velocity V (y) = y (y). It is obvious, that in Eq. (23b) one cannot neglect either of the last two terms. Notice that neglecting V would mean that the phase is constant. Neither one can neglect the complex potential ∼ α2 , since this would result in unbounded solutions. Therefore, one has to require a = 4. Thus, introducing γ = ε2 , we rewrite Eq. (23b) as 2V Ry + Vy − γ + α2 y 2 = 0. (24) R Turning to Eq. (23a) one readily concludes that the first term can be neglected compared with the last two terms. This

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result is natural for a slowly varying TF distribution. Another rather surprising result comes from the fact that, for a = 4, the term defined by the velocity V appears to be of order ε−6 , while the parabolic trap is only of the order of ε−2 . Thus unlike in the conservative case the spatial domain of the TF distribution is determined by the superfluid velocity and hence by the balance between the dissipative part of the potential and the linear pump. Finally, we have to choose b = 3. The small parameter ε is not defined yet. To do this we recall that for the TF approximation in the conservative case ρ(0) = √ µ. Expecting a similar relation for the conservative case, that is, expecting that in the case at hand the TF approximation corresponds to sufficiently large µ, and taking into account that the above scaling gives R = ε3 ρ, we arrive at the requirement ε = µ−1/6 ,

(25)

defining the meaning of the formal, so far, small parameter. Now Eq. (23a) can be rewritten in a simple form, R 2 + V 2 = 1.

(26)

The system of equations defining the amplitude and the phase, that is, (26) and (24), can be solved explicitly. To this end we introduce the new dependent variable (y) according to the relations R(y) = cos[ (y)],

V (y) = sin[ (y)].

(27)

Since R(y) is positive and V (y) is real (by their definitions) we are interested in real functions (y) ∈ [−π/2, π/2]. Substituting the ansatz (27) in Eq. (24) it is straightforward to obtain 2 − 3 cos y = α2 y 2 − γ . (28) cos Next we recall that we are dealing with the ground state solution whose density distribution is symmetric with respect to the center of the trap; that is, R is an even function and the velocity in the center is zero: V (0) = 0. This allows us to integrate (28) and compute an implicit formula for : α2 1 + sin − 3 sin = y 3 − γ y. 2 ln (29) cos 3 The obtained expression (29) can also be viewed as an √ implicit link between the density ρ(x) = µ cos[ (µ−1/6 x)] √ and the superfluid velocity v(x) ≡ ϕx = µ sin[ (µ−1/6 x)]. In the original physical variables this link acquires the form √ µ+v α2 √ − 3v = x 3 − x. (30) 2 µ ln ρ 3 B. Discussion of the formal solution

Let us now pass to the discussion of the obtained results. First of all we observe that, in the center of the trap, the √ condensate density is given by ρ(0) = µ, as expected by analogy with the conservative case. However, now, in the presence of the dissipation and gain, the density does not have a finite support determined by the Fermi radius, but is nonzero in the whole space: ρ(x) = 0 for all |x| = ∞. Taking into

account the zero limits for the density we obtain from (30) in the leading order the following decay rate of the density: √ √ 3 (31) ρ ∼ µe−α2 x /(6 µ) , x → ∞. The curvature of the density in the vicinity of x = 0 is determined by the dissipative terms ρ=

√

µ−

1 2x 2 √ + O(x 3 ), 2 µ

x → 0,

(32)

that is, in the leading order is independent of the conservative part of the parabolic potential. Moreover, it follows from (24) that in the vicinity of the center of the potential there exists an outflow (vx (0) = ). This occurs because of the dominating pump which expels the condensate from the center of the trap. The absolute value of the superfluid velocity is a growing function of |x|, with √ the maximal velocity achieved at infinity: vm = µ (this of course does not mean an increase of the current density, which goes to zero because of exponential decay of the density). So far, however, we still did not compute the chemical potential of the TF distribution. Whereas in the conservative case it is defined by the total number of atoms, now µ is determined by the balance between the pump and the dissipation. For the TF approximation one can compute µ by requiring the (physically meaningful) solution to be continuous everywhere in space. In other words, we obtain the chemical potential from the requirement for (y) to be a continuous function. √ To this end we introduce the parameter y∗ = γ /α2 and √ quantity ∗ = arccos( 2/3). Then it follows from Eq. (28) ˜ = ∗ at any y = y˜ = y∗ , then (y) has a that, if (y) discontinuity at y = y˜ [this results from the fact that in this point y( ) has a minimum, for y˜ > y∗ , a maximum, for y˜ < y∗ ]. Hence one has to require y˜ = y∗ ; that is, (y∗ ) = ∗ . This allows one to compute the chemical potential of the TF distribution: µTF = µ∗

3 , α2

µ∗ =

√

4

9[ 3 − ln(2 +

√

3)]2

≈ 2.58.

(33)

IV. FAMILIES OF THE NONLINEAR MODES A. Numerical approach

Let us consider a solution ψ(x) of Eq. (2) vanishing as x → +∞. Then for x large enough the nonlinear term in (2) is negligible and one concludes that ψ(x) obeys the following asymptotics: µ + i 1 2 ln x e−νx /2 , x → +∞, ψ ∼ C exp − 2ν 2 (34) where C is a complex constant. This asymptotics can be used to estimate ψ(x∞ ) and ψx (x∞ ) for some point x∞ 1 and arbitrary complex C. Having done this one can find ψ(x) for x in the range from x∞ to 0, using, for example, the Runge-Kutta method. Such a simple idea was used in [9] to obtain nonlinear modes in the conservative parabolic potential. Let us note that Eq. (7) is invariant with respect to the transformation ψ(x) → ψ(x)eiϕ , ϕ ∈ R. The important consequence is that it is sufficient to regard real C only. Indeed, let

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us consider a vanishing solution ψ(x) of (2) which obeys (34) with some complex C = |C|ei arg C . Then ψ(x)e−i arg C is also a vanishing solution for (2) but it corresponds to real C in (34). Thus, in what follows we suppose that C is real. For numerical purposes it is convenient to use the following substitution, ψ(x) = Cu(x), and introduce a new real parameter k defined as k = σ C 2 . Then from (2) we obtain the equation uxx + (µ + i − ν 2 x 2 )u + k|u|2 u = 0

Γ

(35)

0

Pn=0 -14

-4 µ

6

-14

-4

Pn=0 Pn=16

30 25

(36)

(37)

B. Numerical results

Using the approach described above we have constructed numerically the lowest families of the stationary nonlinear modes for both repulsive and attractive cases. The results for the two families (n = 0 and n = 1) are shown in Fig. 1. The upper panel of Fig. 1 shows the dependence of the pump on the chemical potential. Let us note that, instead of the µ() representation which is physically more natural, in the panel the families of nonlinear modes are shown as the curves (µ). Such manner of visualization is chosen to make comparison easier between the upper panel of Fig. 1 and the lower one where the nonlinear modes are shown in perspective, more typical for the conservative case, that is, in the form N (µ). For a fixed n the dependence (µ) appears to be a monotonously increasing function. Since there are no stable solutions in a system with dissipation but without pump (i.e., when α2 > 0 and < 0), all the curves belong to the half-plane > 0. In the limit → 0 we obtain µ ∼ M → −∞.

20

N

Let us fix one parameter, for example, µ. Then we have two equations (36) with respect to two real parameters, and k. If some solution of system (36) has been found then one can restore the corresponding solution ψ(x) using definitions of k and u(x). To solve system (36) we start with the linear limit σ = k = 0. It is known that u(x) ≡ 0 is the solution for (35) and corresponding values of and µ are defined by (13) and (14) with n = 0, 1, . . . . Then varying µ one can use the Newton method to modify and k. Let us introduce the parameter plane (, µ). According to the above arguments it is natural to expect that there are several branches in this plane. These branches bifurcate from the linear limit (13), (14) and are continuous and that is why they can be numbered by the index n corresponding to their linear limit. Respectively, each point of a branch represents a nonlinear mode ψn (x) which is defined up to the factor eiϕ with ϕ ∈ R. The extension to the case of odd solutions is straightforward. In this case the conditions (36) should be replaced by Re u(0; µ, , k) = Im u(0; µ, , k) = 0.

1

0.5

to be explored. There are three real parameters in Eq. (35) to be determined: µ, , and k. According to the imposed boundary conditions u ≡ u(x; µ, , k) is a solution of (35) vanishing as x → +∞, corresponding to particular values of these parameters. We concentrate on even solutions of (35). Respectively, we look for solutions u(x; µ, , k) satisfying the conditions Re ux (0; µ, , k) = Im ux (0; µ, , k) = 0.

Pn=1

1.5

15 10 5 0

µ

FIG. 1. Families of the nonlinear modes for Eq. (2) on the (, µ) plane (upper panel) and on the (N, µ) plane (lower panel) for α1 = α2 = 1. To make comparison easier between both panels the families of nonlinear modes are shown as curves (µ). Solid and dashed fragments correspond to the attractive and repulsive cases, respectively. In the boxes spatial profiles |ψn (x)| are schematically shown. Points Pn=0,1 indicate the linear limit for n = 0 and n = 1, respectively (i.e., the points Pn correspond to = ˜ n , µ = µ˜ n , and N = 0).

The lowest family n = 0 corresponds to the ground state. Like in the conservative case, the modulus ρ(x) of the ground state is nodeless. For the solutions from the family n = 1, the distribution ρ(x) has exactly one node at x = 0. In the repulsive case for large values of N the lowest family tends to the TF distribution which was studied in detail in Sec. III. Regarding the attractive case we observe that like in the case of the real potential (when α2 = 0) the spatial profile ρ(x) = |ψ(x)| of the ground state is well approximated by the nodeless solution of the equation ρxx + µρ + ρ 3 = 0, or in other words by the function ρ(x) =

2|µ|sech( |µ|x)

(38)

(we, however, emphasize that, unlike in the conservative case, now µ is not a free parameter, but is uniquely determined by the balance between dissipation and pump). Thus, for large N the ground state is strongly localized at the center of the potential, what corroborates with the exact analytical estimate (21). These facts are illustrated in Fig. 2 where a numerically obtained profile is compared with the approximation (38).

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10 40

5

0

0

0.5

1 x

1.5

20 10 0 −6

2

FIG. 2. The ground state profile |ψ0 (x)|2 for ≈ 0.14 and µ ≈ −5.00 (solid line) and ρ 2 (x) obtained from (38) for the same value of µ (dashed line).

V. STABILITY OF MODES

In order to study the dynamics and the stability of the nonlinear modes ψn (x) we solve numerically the time-dependent equation (1) starting with the initial profile (0, x) obtained by the numerical approach described previously. For this purpose the numerical scheme from [11] was adopted for the dissipative equation (1). The results can be summarized as follows: Repulsive case. First we address the stability of the ground state. Obtained with the shooting method, solutions ψ0 (x) were perturbed with small-amplitude noise and the resulting profiles were used as the initial data (0, x) for Eq. (1). Performed computations show that the ground state is stable, which naturally corroborates with the arguments presented previously for the linear limit (see Sec. II B). Moreover, we observed that for fixed > ˜ 0 = νi initial data (0, x) that had no instilled semblance to the ground state shape evolve to the corresponding ground state mode with the same value of . This fact is illustrated by Fig. 3 which describes the evolution 2 of the initial data of the form (0, x) = (1 + i)e−0.5(x−1) (i.e., a shifted Gauss profile) and ≈ 0.97. As one can see in Fig. 3 the initial profile recovers the symmetry fast enough and evolves to the ground state corresponding to the given value of . Unlike in the conservative parabolic potential, in the case at hand no oscillatory motion is observed. Turning to the family n = 1, we found it to be unstable. In a general situation even a small perturbation leads to evolution of the mode ψ1 (x) to the ground state ψ0 (x) with the same value of . A typical evolution for the nonlinear mode with n = 1 is illustrated in Fig. 4. In this figure the initial profile (0, x) = ψ1 (x) belongs to the family with n = 1 and ≈ 1.73. The profile ψ1 (x) was obtained with

20 5 10 0 −6

−3

t 0 x

3

6 0

FIG. 3. The evolution |(t, x)|2 for the initial data of the form 2 (0, x) = (1 + i)e−0.5(x−1) and ≈ 0.97 and σ = 1 (i.e., for the repulsive case). For t large enough the initial profile |(0, x)|2 evolves to the ground state profile |ψ0 (x)|2 with the same value of .

20 t −3

0

3 x

6

0

FIG. 4. The evolution |(t, x)|2 of the nonlinear mode ψ1 (x) for ≈ 1.73 and σ = 1 (i.e., for the repulsive case). For t large enough the initial profile |ψ1 (x)|2 evolves to the ground state profile |ψ0 (x)|2 with the same value of .

the numerical shooting method described previously and no additional perturbation was introduced into it. The profile persists for small values of t. However at some t it abruptly transforms into a nodeless profile corresponding to the ground state with the same value of . Computations also show that nonlinear modes from the family n = 1 are unstable with respect to even perturbations only. In numerical simulations even perturbations can be suppressed if the following boundary is chosen: (t, 0) = (t, +∞) = 0. Then solutions from the family n = 1 demonstrate stable dynamics. Attractive case. As in the repulsive case, numerical simulations show that nonground modes are unstable. The situation for the ground state is more delicate. Our numerical simulations indicate that the ground state behaves as a stable entity, if the number of particles is not too large (i.e., the ground state mode is close to the linear limit). For the ground states with large numbers of particles numerical study revealed an instability. In order to study the origin of this instability and exclude the effects of numerical approximation one needs more advanced analysis. However, a preliminary study allows one to suppose that the ground state in attractive case is unstable in general, but the increment of instability falls when approaching the linear limit. The analysis of this interesting issue is postponed for a future study. VI. CONCLUSION

To conclude, we have considered nonlinear modes of a Bose-Einstein condensate embedded in a complex parabolic potential. We have found that stationary modes exist only for dissipative potentials and therefore must be supported by the linear pump. In the case of a positive scattering length, the only stable mode is the ground state, which appears to be an attractor to which higher modes subjected to small perturbations evolve. So far, our numerics were not conclusive about the stability of the modes in the attractive case, which is left for further comprehensive study. In the repulsive case, we have constructed the TF distribution valid in the limit of large numbers of particles. This distribution appears to be a complex function and is determined by the balance of the dissipation and the pump, rather than by the real part of the parabolic trap, as happens in the conservative case.

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The strong stability of the ground state (as this is typical of attractors of dissipative systems) allows one to suggest that, by changing the intensity of the pump, the mode can be “moved” along the lowest branch. As another immediate consequence of the stability properties we have that, due to inelastic collisions with the potential, loading atoms by means of the linear pump can be done only in the ground state.

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PHYSICAL REVIEW A 81, 013606 (2010) ACKNOWLEDGMENTS

The authors would like to thank I. Pustobaev and E. Ustinova, Moscow Institute of Electronic Engineering, for help in numerical calculations. The research of VVK was partially supported by the European Commission within the 7th European Community Framework Programme under Grant PIIF-GA-2009-236099 (NOMATOS).

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