Exact transmission state of a Bose–Einstein condensate in a near-harmonic trap

Physics Letters A 373 (2009) 1560–1564

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Physics Letters A www.elsevier.com/locate/pla

Exact transmission state of a Bose–Einstein condensate in a near-harmonic trap Jianwen Song a,b , Wenhua Hai a,∗ , Xiaobing Luo c a b c

Key Laboratory of Low-dimensional Quantum Structures and Quantum Control of Ministry of Education, and Department of Physics, Hunan Normal University, Changsha 410081, China College of Photo-Electric Science and Engineering, National University of Defense Technology, Changsha 410073, China Department of Physics, Jinggangshan University, Ji’an 343009, China

a r t i c l e

i n f o

Article history: Received 1 December 2008 Received in revised form 25 February 2009 Accepted 2 March 2009 Available online 6 March 2009 Communicated by A.R. Bishop

a b s t r a c t We introduce a new confining potential which simulates preferably the realistic near-harmonic trap for a quasi-one-dimensional (1D) Bose–Einstein condensate (BEC). An exact transmission state of the BEC system is found and the corresponding spatial configurations, metastability, superfluidity and the transport properties are analyzed. Resonant transmission through the potential is predicted from the exact solution. © 2009 Elsevier B.V. All rights reserved.

PACS: 03.75.Lm 03.75.Hh 03.65.Ge 68.65.Jp Keywords: Realistic confining potential Exact soliton solution Metastability Transmission state Superfluidity

1. Introduction Bose–Einstein condensates (BECs) of ultracold gases provide an ideal test ground for the nonlinear physics of many-body system. The zero-temperature BEC consisting of N atoms can be described by the Gross–Pitaevskii equation (GPE) [1]. Assuming that the transversal confinement is tight and the parameters do not enter the Tonks–Girardeau regime [2], the transverse wave function is approximately in ground state of a harmonic oscillator of radial frequency ωr . Given the one-dimensional (1D) time-independent potential V (x), the BEC has the longitudinal stationary state ψ(x) governed by the dimensionless quasi-1D GPE [3–5] 1

− ψxx + g 1D |ψ|2 ψ + V (x)ψ = μψ, 2

(1)

where μ is the chemical potential and g 1D represents the 1D atomic interaction strength [4]. In Eq. (1), we have taken fixed con√ stants 1/σ0 and σ0 with σ0 = 2π /(550 nm) as the units of the dimensionless spatial coordinate and wave function respectively,

*

Corresponding author. Tel.: +867318856890; fax: +867314478098. E-mail address: [email protected] (W. Hai).

0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.03.004

so the external potential V (x) and chemical potential μ are normalized by two times of the recoil energy E r = h¯ 2 σ02 /(2m) with m being the atomic mass. In such units, the interaction strength reads [4] g 1D = 2as /(σ0 lr2 ) with as being the s-wave scattering  length and lr = h¯ /(mωr ) the radial length of harmonic oscillator. The g 1D may be positive or negative, which represents repulsive or attractive interactions. The repulsive interaction means the usual case in experiments etc. In the mean-field description, it is principal to solve the GPE for studying physical properties of the condensed atoms. However, we notice that for most of the practical potentials the GPE is not exactly solvable. Therefore, constructing exact special solution of GPE is a challenging job. It is well known that the free GPE without external potential is an integrable system and the integrability could be easily broken by the potentials of different forms [6]. For this reason, the previous analytical works concerning exact solutions dealt with mainly 1D stationary systems with some simple potentials, such as the infinite square-well [7,8], finite square-well [5,9, 10], step-potentials [11], δ potentials [12,13], δ comb potentials [14, 15], linear ramp potential [16,17] and the optical lattice potentials [18–20]. In the treatments to the BEC system with cosine-shaped potential, the balance conditions between the external potentials and interatomic interaction are established [21–23]. Resonance and

J. Song et al. / Physics Letters A 373 (2009) 1560–1564

transport properties of the BEC have also received substantial attentions in recent years [24,25]. In this Letter, we suggest a new 1D model potential which simulates the realistic trap in experiment, and investigate the physical properties of BEC in this trap by constructing an exact solution of GPE. Based on the exact solution, we show the effect of the external potential and internal atom-atom interaction on the flow density and flow velocity. We find that for some conditions the considered external potential suppresses the flow density and the many-body interaction enhances the atomic transport. More interesting thing is that the BEC in our system is metastable superfluid, that leads to the atomic resonant transmission to the obstacle potential. The number of condensed atoms is also estimated, which agrees with the result of experiment [26]. 2. Model potential

V (x) =

Vb

(η − sech2 σ x)2

Fig. 1. Plots of the potential in Eq. (2) used to simulate the realistic confining potential (ring line). Here the parameters obey relations V  = V b /η2 + V c , V b < 0 and η > 1. Dotted line fits a harmonic-oscillator potential.

R xx =

In many experiments, BECs are confined in the quasi-1D cigarshaped traps and the different confining potentials share two common criterions: Firstly, any confining potential consists with a harmonic potential in a certain spatial range. Secondly, with the increase of |x|, it tends to a constant. Obviously, the harmonic potential dose not meet the second criterion, because of its infinite value at x = ±∞. Due to the above considerations, some new potential functions were introduced to model the real traps [27–30]. For example, R. Dum et al. suggested a potential function in the form V (x) = 12 Ωx2 x2 for V (x) < V c and V = V c for V (x)  V c , where Ωx and V c are two constants [27]. S.J. Wang et al. used function V (x) = V a tanh2 (σ x) with constants V a and σ to simulate the realistic confining potential [28]. S.K. Adhikari considered the 2 decayed harmonic potential V (r ) = r 2 e −cr with constant c [29]. In H. Ott and his co-worker’s paper [30], the i realistic trap was numerically simulated by a polynomial ai x with several nonzero coefficients ai . Now we are going to demonstrate that the new model potential

+ Vc

(2)

satisfies the mentioned experimental standards and a BEC held in such a potential has an exact solution. Here η , V b , V c and σ are some constants. The constant η is dimensionless, σ is in units of σ0 = 2π /(550 nm), and V b , V c are normalized by two times of the recoil energy E r . The positive constant 1/σ denotes the width of well (or barrier), which is in order of the longitudinal length ofa harmonic oscillator model with frequency ωx , namely 1/σ ∼ h¯ /(mωx ). The potential depth is associated with the constants η , V b and V c . When η > 1 is taken, Eq. (2) represents an attractive well or a repulsive barrier for the case V b < 0 or V b > 0. In order to fit the zero point of a harmonic potential we select V c = − V b /(η − 1)2 such that V (0) = 0. The attractive well of Eq. (2) is illustrated by the curve with rings in Fig. 1, where the potential is rescaled by V  = V b /η2 + V c and the parameters are taken as η = 1.8, V b = −2, V c = 3.12 and σ = 0.25. For comparison with harmonic potential we plot the function 0.8x2 as the curve with dots. From Fig. 1, we see that the potential of Eq. (2) coincides with the harmonic potential for small |x| values, whereas for great |x| values the potential approaches a finite constant. Therefore, the function of Eq. (2) can simulate the realistic 1D confining potential preferably. 3. Exact solution of transmission state The complex solution of Eq. (1) has the exponential form ψ(x) = R (x)e i θ(x) with norm R (x) and phase θ(x) being real functions. Inserting this form into Eq. (1) and separating the real and imaginary parts, one has the two real equations

1561

θxx +

  + 2g 1D R 3 + 2 V (x) − μ R ,

J2

R3 2θx R x R

(3)

= 0.

(4)

Here J = θx (x0 ) R 2 (x0 ) is the first integration constant of Eq. (4), denoting the flow density which describes the transport property of BEC atoms. The flow density is determined by the derivative θx (x0 ) and atomic number density R 2 (x0 ) at the boundary point x0 . When the external potential V (x) is absent, Eq. (3) becomes R 0,xx =

J 02 R 30

+ 2g 1D R 30 − 2μ0 R 0 .

(5)

This is an integrable equation, whose exact general solution can be constructed easily by using the Jacobi elliptic function. When the modulus of the elliptic function is equal to 1, we arrive at the non-propagated soliton solution as [31]







R 0 (x) = f x, g 1D , μ0 , J 02 = A 0 sech2

 1/2 − A 0 g 1D x + B 0

(6)

which is determined by the boundary conditions [32]. Substituting Eq. (6) into Eq. (5), one has the relationships of constants A 0 , B 0 and J 0 , μ0 as A 0 = −3B 0 + 2μ0 / g 1D , J 02 = −2g 1D B 30 + 2μ0 B 20 . It is well known that Eq. (6) contains the famous homoclinic solution with g 1D < 0, μ0 < 0, J 0 = B 0 = 0 and heteroclinic solution with g 1D > 0, μ0 > 0, J 0 = 0, B 0 = − A 0 = μ0 / g 1D > 0. Although for most potentials Eq. (3) cannot be solved exactly, we here prove that the potential of Eq. (2) can lead Eq. (3) to exactly solvable equation. To do this, we compare Eq. (2) with Eq. (6) to exhibit that the potential of Eq. (2) has the form V (x) = α0 R 0−4 (x) + β0 with constants α0 and β0 . This implies the relation V (x) = α R −4 (x) + β

(7)

between the exact solution R (x) of Eq. (3) and V (x), where α and β are undetermined constants. Introducing the relation of Eq. (7) into Eq. (3) yields a new GPE without external potential R xx =

J 2 + 2α R3

+ 2g 1D R 3 − 2(μ − β) R .

(8)

The form of Eq. (8) is analogous with Eq. (5). The difference potenbetween them is that the flow density J 0 and chemical  tial μ0 are replaced with the effective ones J E = J 2 + 2α and μ E = μ − β respectively. Consequently, Eq. (8) has the exact special solution in the form of Eq. (6),



R (x) = f x, g 1D , μ E , J 2E



   1/ 2 = A sech2 − Ag 1D x + B .

(9)

The substitution of Eqs. (9) and (2) into Eq. (7) produces a group of equations which determine the undetermined constants α , β , A and B as

1562

J. Song et al. / Physics Letters A 373 (2009) 1560–1564

Fig. 2. Images of the spatial configuration of transmission state BEC for

J 2E = −2g 1D B 3 + 2μ E B 2 ;

A = −3B + 2μ E / g 1D , 1

Vb = α

A2 B

 σ = − Ag1D ,

,

η=− ,

β = Vc = −

A

Vb

(η − 1)2

.

(10)

Solving above equations one gets A=−

α=

σ2 g 1D

Vbσ 4 2 g 1D

ησ 2

,

B=

,

β =−

g 1D

, Vb

(η − 1)2

.

(11)

Inserting the constants A, B into Eq. (9), we obtain the exact solution of Eq. (3) and the corresponding density function



 R (x) =

2

A sech

ρ (x) = R 2 (x) =

σx+ B =

 σ2  η − sech2 σ x ,

g 1D

 σ2  η − sech2 σ x ,

g 1D

integrable equation. It is immediate for us to write the general solution of Eq. (8) in terms of the Jacobi elliptic function [31]. Then by substituting the general solution into Eq. (7) we find the required potential which must be in the form of Eq. (2) with ellipticshaped function instead of the sech-shaped one. The elliptic-cosine solution, of course, can be reduced to the sech-shaped soliton solution when its modulus is equal to 1. According to the uniqueness theorem, the different forms of solutions are determined by the boundary conditions uniquely. The gray soliton solutions are associated with constant boundary values of [ψ(x0 ), ψx (x0 )] and the bright solitons may be corresponded to the zero boundary values. The usual periodic boundary conditions are required for the elliptic-shaped periodic solution with modulus being less than 1. If we adopt the elliptic-cosine function to replace the sech-shaped function in Eqs. (2) and (9), the parameter relationships in Eqs. (10) and (11) will be remained such that the chemical potential keeps its form of Eq. (13). Differing from the soliton solution, the ellipticshaped periodic solution describe periodic structure of the BEC generated in a cloud of cold atoms, which can be detected by the Bragg scattering of an optical probe beam [33].

(12)

where B = ησ 2 / g 1D > 0 is the value of atomic density at the infinite boundary x0 = ∞, because of sech2 ∞ = 0. Sometimes the constant B is called the background of BEC cloud. The constant | A | represents altitude of the matter wave-packet (or depth of the matter wave-trough) compared to the background density. Noticing the expression μ E = μ − β and combining the first equation of Eq. (10) with Eq. (11), one has the positive chemical potential

μ = −0.5σ 2 + 1.5ησ 2 − V b /(η − 1)2 .

η = 1.01 and g1D = 1. The gray area shows the background of BEC cloud.

(13)

The spatial configuration of BEC matter wave depends on sign of the constants A. The positive A corresponds to matter wavepacket and negative one to matter wave-trough. To avoid singularity of the potential and solution in Eqs. (2) and (9), the inequalities B > 0 and η > 1 are required. Given the two inequalities, from Eq. (11) we know that g 1D must be positive and A must be negative. Therefore the density function of Eq. (12) describes a matter wave-trough at the center of confining potential, which is illustrated as in Fig. 2. This wave-trough could be called nonpropagated “gray soliton”, which has the same width with the trapping potential. When the parameter η is taken as 1, the density vanishes at the wave-trough center and the “gray soliton” becomes “black soliton”. The depth | A | of wave-trough is related to the system parameters σ and g 1D , through Eq. (11). If g 1D tends to infinity, Eqs. (11) and (12) give A → 0 and the constant density ρ (x) = B → 0 such that the wave-trough will disappear and the BEC cloud will spread. The above-mentioned results show that for any potential of Eq. (7) the GPE (3) can be reduced to the free GPE (8) which is an

4. Metastability and size estimating For the purpose of applications, stabilities of the BECs have attracted much interest. The used research methods concerned the linear stability analysis [34–36], Gaussian variational approach [37], and the numerical simulations to the partial differential GPEs [19, 34]. For the zero flow case of Eq. (3) with J = 0, we have established a criterion for analyzing the stability of a specific solution [35]. In general case of Eq. (3) with nonzero flow, it is difficult to solve the linearized equation [19,34] for analyzing the stability of exact solution (12). However, it is well known that the condensate is metastable [3], corresponding to local minima of the energy functional for different parameters. By the metastability we mean the stability against stretching and shrinking of the BEC, which is relatively easy to demonstrate. The metastability of an attractive BEC has been investigated in Ref. [3], that leads to a critical value N cr of atomic number. In this section, for simplicity we analyze the metastability of the exact solution (12) by investigating the local minima of energy per atom. The total energy functional E [ρ ] of BEC consists of three parts: kinetic energy E kin , potential energy E ext and interaction energy E int respectively [3], namely E [ρ ] = E kin + E ext + E int .

(14)

By direct integration to GPE (1) one has the useful expression

μ N = E kin + E ext + 2E int which implies the energy per particle

(15)

J. Song et al. / Physics Letters A 373 (2009) 1560–1564

1563

Fig. 5. (a) Image of the dependence of flow density J on depth | V b | of the potential √ −1 2 well, where U = η2 σ 2 (η − 1) and J  = g 1D σ U . (b) Image of the relation between the flow density J and nonlinear interaction g 1D B. Here parameters V b = −0.5, σ 2 = 0.3 and g1D = 1 are adopted. Fig. 3. (a) Images of the configuration of condensed atoms in realistic experiment; (b) Modification of the atomic configuration for our exact solution.



l=

12η − 5 + 7.8 0.2 − 0.9η + η2 2η(7η − 3)σi

(20)

,

where l is simply proportional to the width 1/σi of trapping potential. Inserting Eq. (20) into Eq. (17), we get the number N of condensed atoms, N=

Fig. 4. Images of the energy per atom vs width of the trapping potential for and the different values of l.

E / N = μ − E int / N .

η = 1.2

(16)

So far the realistic experiments have been carried out with a finite atomic number, therefore the size of BEC is limited as in Fig. 3(a), and infinite boundary is not required. Here we assume the boundary coordinates x = ±l as shown in Fig. 3(b). Regarding ±l as spacial integration limits and using the exact solution (12), we calculate the number N of condensed atoms and the interaction energy E int , respectively,

l N=

ρ dx = −l

E int =

g 1D 2

2σ g 1D

(ηlσ − 1),

ρ 2 dx = −l

η2l g 1D

σ4 −

3g 1D

σ 3.

(18)

3ησ 2 − σ 2 2

−

3η2 lσ 3 − (6η − 2)σ 2 6ηlσ − 6

2(7η − 3)

g 1D

+ C,

μ of Eq. (13)

(19)

here C =| V b | /(η − 1)2 . In Ref. [3], the local minima of energy per atom are illustrated for different values of parameter N. We here plot the energy E (1/σ ) as a function of 1/σ for several values of parameter l, as in Fig. 4. One clearly sees that the local minima of energy appear at different extreme points 1/σi for i = 1, 2, 3 and different l. This demonstrates that there exists a metastable state for each parameter pair (l, σi ), which is consistent with the previous result: “dark soliton” in quasi-1D BEC to be metastable for T = 0 [38,39]. We now consider that the system is in the metastable state with parameter pair (l, σi ) determined by the local minima of energy. From Eq. (19) taking the first derivative of function E (1/σ ) as zero at the extreme points 1/σi , one finds the relation between l and σi as

−1 .

(21)

In the transmission state, we have the constant flow density J = 0. If there exists a BEC resource and a receiver, the constant atomic flow can be transported across the obstacle (in our system, it is the confining potential) from resource to receiver, which is analogous with a stable direct current driven by a power supply. During a time of τ , atoms of quantity J τ transport from one side of the obstacle to another. Considering the first two equations of Eq. (10) and J 2E = J 2 + 2α , we have the flow density

σ2



g 1D

η2 σ 2 (η − 1) − 2| V b |

(22)

and the flow velocity field [3] v (x) = θx (x) =

Inserting the above N, E int and the chemical potential into Eq. (16), one has the energy per atom E /N =



12η − 5 + 7.8 0.2 − 0.9η + η2

5. Flow and superfluidity

(17)

6η − 2



In the experiments [26], the used  parameters read ωx = 19 Hz and ωr = 250 Hz so that 1/σi ≈ h¯ /(mωx ) ≈ 11.9 μm and 1/lr =  h¯ /mωr ≈ 3.3 μm for 23 Na. The typical value for s-wave length a s is 5–10 nm [40], which leads g 1D to 0.011–0.022. Setting η = 1.01, we obtain the number of condensed atoms N = (3–6) × 106 , which is consistent with the experimental value 5 × 106 [26].

J=

l

2σi

J

ρ (x)

.

(23)

If system parameters are given, J is fixed. When ρ (x) takes minimal value at the center of external potential, the flow velocity will be maximum v max = v (0) = J /ρ (0). At the boundary, ρ (x) takes maximal value and the flow velocity tends to be minimum v min = θx (∞) = J /ρ (∞). From Eq. (22) we know that for the positive g 1D the value of flow density decreases with the increase of the well depth | V b |, as shown in Fig. 5(a). When | V b | increases to η2 σ 2 (η − 1)/2, the flow density J vanishes, that means the external confining potential suppresses the atomic flow. In the studies of nonlinear quantum transport, many attentions focused on how the nonlinearity affects the transport of particles [41]. For our system with an obstacle, the effect of nonlinearity on the transport is expressed by the relation between the flow density J and interaction “potential” g 1D |ψ(x)|2 . From Eq. (12) one knows |ψ(x)|2 equals B at the boundary x → ±∞. As a result, g 1D B can reflect the intensity of nonlinearity at the boundary. Noticing Eq. (11), we rewrite Eq. (22) as J=







B 2 g 1D B − σ 2 − 2α .

(24)

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J. Song et al. / Physics Letters A 373 (2009) 1560–1564

From Eq. (24) and considering B > 0, for the given potential parameters V b and σ , the flow density J increases with the boundary interaction energy | g 1D B |, as shown in Fig. 5(b). Noticing the relations J = ρ v and ρ = B at the boundary,  Eqs. (11), (23) and (24) give the boundary flow velocity v = g 1D B − σ 2 + 2V b /η2 . This also shows that the increase of nonlinearity strength can raise the transport velocity of atoms in our system. Based on the exact solution, we now seek the relation between the superfluidity and transport property of the nonlinear system. For a BEC in superfluid state, the presence of obstacle may destroy the superfluidity. N. Pavlof et al. [42] have studied the effect of obstacle potentials on superfluidity, concerning the δ potential, square well and Gaussian potential. A drag F d was defined to describe the degree of the effect, dx ρ (x)

dV (x) dx

−∞

.

(25)

Its physical meaning is the mean interactional force between the BEC atoms and the obstacle potential. A finite drag implies the breakdown of superfluidity, whereas the zero drag corresponds to the superfluid flow. According to the result of N. Pavlof, there exists a critical velocity below which the atom flow is a superfluid. Besides, for some penetrable obstacles, it is shown that the superfluidity is recovered at large beam velocity. In the considered system, applying Eq. (7) to Eq. (25) and considering the density function of Eq. (12) produce the drag

+∞ Fd =

dx ρ (x)

d[αρ (x)−2 + β] dx

−∞

=

Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant Nos. 10575034 and 10875039. References

+∞ Fd =

obstacle potential, which describes the BEC superfluid in the exact resonant transmission state. It is worth noting that we can use an elliptic-cosine function instead of the sech-shaped function in Eqs. (2) and (12). Thus the exact results are easily extended to the BEC system with the periodic potential, where the periodic boundary conditions should be required. Based on the capacity of current experiments, the results can be tested experimentally.

ρ (x)

ρ (+∞) 2α

= 0.

(26)

ρ (−∞)

This means the BEC flow in our system being a stable superfluid. T. Paul and co-workers [43] used the drag to measure the proximity of the scattering state to a resonant state when BEC atoms propagate through a double barrier potential in a magnetic waveguide. A vanishing drag leads to a perfect resonant transmission. According to their definition, the resonant transmission is also revealed in our system and the exact solution R (x) in Eq. (12) is just the resonant state. 6. Conclusions We have investigated the physical properties of the quasi-1D BEC held in the confining potential (2) by using the exact solution (12). It is shown that the spatial configuration of atomic density shapes as the matter wave-trough which is called the nonpropagated “gray soliton”. The analysis of metastability demonstrates that the soliton state BEC is perfectly metastable for some width values of the trapping potential. The number of condensed atoms described by our exact solution is consistent with the experimental result. The transport properties of the system are studied and the results show that the flow density decreases with the increase of the intensity of external potential. On the contrary, the flow density and transport velocity raise with the increase of the nonlinear interaction strength. More interesting thing is that our exact solution is associated with the zero drag for the considered

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