Eur. Phys. J. B 70, 193–199 (2009) DOI: 10.1140/epjb/e2009-00216-2

THE EUROPEAN PHYSICAL JOURNAL B

Regular Article

Metastable state and macroscopic quantum tunneling of binary mixtures C.F. Liu1 and Y. Tang1,2,a 1 2

Department of Physics, Xiangtan University, Xiangtan 411105, Hunan, P.R. China Institute of Modern Physics, Xiangtan University, Xiangtan 411105, Hunan, P.R. China Received 4 December 2008 / Received in final form 19 March 2009 c EDP Sciences, Societ` Published online 23 June 2009 –  a Italiana di Fisica, Springer-Verlag 2009 Abstract. We provide a mixing model to explore the metastable state and the macroscopic quantum tunnel√ ing (MQT) of binary mixtures. For g12 > g1 g2 , we first observe the two condensates form the symmetrybreaking state (SBS) and then suddenly transfer to the symmetry-preserving state (SPS) through the MQT. The SBS is shown to be the metastable state in our system. We find the MQT does not spontaneously arise. The inducement mechanism is the damping but not the excitations. The damping mechanism can also control the lifetime and the tunneling decay rate of the SBS. Finally, we further present the origin of these phenomena by examining the energy of the system. PACS. 03.75.Mn Multicomponent condensates; spinor condensates – 03.75.Lm Tunneling, Josephson effect, Bose-Einstein condensates in periodic potentials, solitons, vortices, and topological excitations – 05.30.Jp Boson systems – 67.85.De Dynamic properties of condensates; excitations, and superfluid flow

1 Introduction Binary mixtures of Bose-Einstein condensates (BECs) [1–23] have been shown to possess fascinating macroscopic quantum phenomena, such as the complex spatial structure [7–14], metastable states [15–17], and symmetry breaking instabilities [18–23]. Most previous theoretical studies about the binary mixtures concentrate on the static properties. To determine the density profile, Ho and Shenoy have first presented a simple algorithm within the Thomas-Fermi approximation (TFA) [7]. More accurately, the Hartree-Fock theory is provided to achieve the rich spatial structures [8–10]. Generally, the interspecies interaction coefficient g12 plays an important role in determining the structure of the ground state. √ When the inequality g12  g1 g2 is satisfied, the two √ condensates are miscible; and when g12 > g1 g2 , the trapped BECs are immiscible due to the strong interspecies repulsion. In addition, the spatial patterns of the ground state of the phase-separated mixtures have two basic configurations: the symmetry-breaking state (SBS) and the symmetry-preserving state (SPS). Which configuration the ground state would take depends not only on the interspecies interaction but also on the particle number, the intraspecies interaction, and the shape of the trapping potential [23]. Furthermore, the quantum a

e-mail: tang [email protected]

phase transition of the ground state between the two configurations has also been investigated [18–23]. The metastability of the binary mixtures with repulsive interactions has been reported in references [15–17]. The core-shell structure in reference [17] is a typical case to display the metastability according to the spatial distribution. In fact, the metastability of the binary mixtures mainly comes from the competition between intra- and interspecies interactions. The quantum phase transition of the ground state implies that one of the configurations may act as a metastable state of the other one. Theoretically, Kasamatsu et al. have shown the existence of the metastable configuration in the binary mixtures with repulsive interactions [24]. They have also shown the possibility that the transition from the metastable state to the ground state through the macroscopic quantum tunneling (MQT) [24]. Now we know the MQT only originates from the spatial structures of the quantum system itself. Other studies about the MQT of the binary mixtures mostly refer to the spin regime [25,26] or depend on the spatial separated external potentials [27]. Can we straight establish the metastable state? How to induce the MQT? Kasamatsu et al. only show the possibility of the existence of those phenomena [24], but the practical situation which induces the MQT is not systematically considered. In general, the metastable state should be maintained in a closed system. If the state of the system has a change through the MQT, some inducement

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mechanism may be needed. In this study, we concentrate on a direct mixing process of two initially separated condensates with different components. One may believe the mixing process would exhibit the similar properties as that of the static binary mixtures. However, the initial spatial separation would cause a non-equilibrium state. Meanwhile, one condensate can not wrap around the other one to form the core-shell configuration especially for the √ 1D system with g12 > g1 g2 . They have to firstly form the SBS that would be a metastable state for a big system. Can the system be kept on the SBS? If the mixtures continue to evolve to the SPS, what would induce the tunneling? We consider the mixing process is performed along with the damping mechanism. The center of mass coordinates are used to characterize the whole evolution pro√ cess. For g12  g1 g2 , the two miscible condensates go to the √ ground state with damping oscillation. While for g12 > g1 g2 , the two immiscible condensates first form the SBS, which is confirmed to be the metastable state of the system. The SBS can hold for some time and then it suddenly transfer to the SPS through the MQT. We give a visual explanation to the tunneling by analyzing the configuration of the system. Importantly, we find the thermal excitation may be a main obstacle for the tunneling. Meanwhile, the MQT would not spontaneously arise but it may be induced by the damping. The damping mechanism can also effectively control the lifetime and the tunneling decay rate of the SBS. Finally, we calculate the energy of both the SBS and the SPS. The origin of these phenomena is further clarified.

2 Model and basic equations Usually, the two-body interaction is changed by the effect of tight confinement of the trapping potential [28,29]. One would consider the binary mixtures in a three dimensional trap potential which can be described by i Vext (r) =

1 1 2 mi ωi2 x2 + mi ωi⊥ (y 2 + z 2 ), 2 2

i = 1, 2

(1)

where mi is the atomic mass, and ωi and ωi⊥ are the longitudinal and transverse trapping frequencies, respectively. If the longitudinal trapping frequency is much smaller than the transverse trapping frequency, the trapping potential is cigar shaped. In addition, ωi does not affect the transverse component of the wave functions when the twobody interaction energy is much smaller than ω⊥ . These features allow us to investigate the mixture problem in the one-dimensional space. Here, we consider the two condensates with different components are initially separated and trapped in each of the symmetric double well, respectively. The middle barrier is unloaded with a time scale t0 , then the final harmonic well is achieved and the two initially separated condensates are put together. The external potential can

be written as: i (x, t) = Vext     ⎧ m ω 2 x2 t x2 ⎪ ⎨ i i + αi 1 − exp − 2 t  t0 2 t0 ω0i , 2 2 ⎪ m ω x i ⎩ i t > t0 2

i = 1, 2 (2)

where αi is the height of the middle barrier. mi and ωi are the mass of the species i and the trap frequency of the species i, respectively. ω0i is used to tune the width of the barrier. The system can be described by the coupled Gross-Pitaevskii (GP) equations [7–14]:  ∂ 2 ∇2 1 i ψ1 (x, t) = − + Vext (x, t) + g1 |ψ1 (x, t)|2 ∂t 2m1 2

+ g12 |ψ2 (x, t)|

ψ1 (x, t),

(3)

 2 ∇2 ∂ 2 2 + Vext (x, t) + g12 |ψ1 (x, t)| i ψ2 (x, t) = − ∂t 2m2 + g2 |ψ2 (x, t)|2

ψ2 (x, t),

(4)

ψ1 , ψ2 stand for the macroscopic wave function of the two condensates respectively. The intraspecies interaction co2 2 a1 a2 efficients are g1 = 4π and g2 = 4π m1 m2 , where a1 , a2 are the scattering length of the species 1 and the species 2 respectively; and the interspecies interaction coefficient is 2 a12 g12 = m1 m2π , where a12 is the scattering length 2 /(m1 +m2 ) between the species 1 and the species 2. In addition, each wave

function is normalized by the number of particles Ni as dx |ψi (x)|2 = Ni . Another point we must note is the damping mechanism which is a simplified version of the cooling methods used by several groups [30–33]. This mechanism has been applied to help them to cool the condensates. In a realistic system, the condensates are prone to damping due to the small thermal cloud. Thus the damping mechanism will be phenomenologically considered to damp out the excitations in our model. We introduce a phenomenological damping term γ∂ψ/∂t on the left-hand side of equations (3, 4). Then we get  ∂ 2 ∇2 2 1 i(1 + iγ) ψ1 (x, t) = − +Vext (x, t)+g1 |ψ1 (x, t)| ∂t 2m1 2

+ g12 |ψ2 (x, t)|

ψ1 (x, t),

(5)

 2 ∇2 ∂ 2 +Vext (x, t)+g12 |ψ1 (x, t)|2 i(1 + iγ) ψ2 (x, t) = − ∂t 2m2 2

+ g2 |ψ2 (x, t)|

ψ2 (x, t).

(6)

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Fig. 1. (Color online) The time evolution of the two condensates. (a), (b) and (c) show the density evolution of the condensate 1, √ the condensate 2 and the combination for g12  g1 g2 respectively. The parameters are g1 = 1, g2 = 0.6 and g12 = 0.4. (d), √ (e) and (f) are that for g12 > g1 g2 , respectively. The parameters are g1 = 1, g2 = 0.6 and g12 = 1.6. The dissipation rate γ is  0.03 in the simulations. The time unit is ω −1 and the length unit is ξ = /(mω). The corresponding units in the following pictures are the same as in this picture.

The inclusion of this phenomenological damping has been considered in references [30,31]. The damping mechanism will cause a dissipation of energy from the system, such that it transforms towards the ground state. Any excitations in the system will become damped with the rate being controlled by the value of the dimensionless parameter γ.

3 Numerical simulations, results and discussions In the following, we apply the Crank-Nicholson method to demonstrate the dynamical mixing process. At t = 0, we take the density profile approximated by the Thomas-Fermi (TF) solution as initial condition, i (x))/gi μ  V (x) (μi − Vext , i = 1, 2 (7) ni (x) = 0 else μi is the chemical potential of the species i. For the sake of simplicity, we assume the confining potentials for the two 1 2 components are equal, i.e., Vext (x) = Vext (x) . In practical

experiments, the number of atoms can reach 106 . There√ fore, we can set α1 = α2 = α = 2.25, ω1 = ω2 = ω = 1002 and ω01 = ω02 = 15 for the potentials. In addition, we choose μ1 = 1 and μ2 = 0.76125 to get N1 = N2 = N0 . We set  = 1 and the atomic mass m1 = m2 = m = 1. In this situation, the spatial extent  of the system is characterized by the healing length ξ = /(mω), and the time unit is ω −1 . For these parameters, the middle barrier is so high and broad that the two BECs can be completely separated. Furthermore, we unload the middle barrier with t0 = 1000ω −1. The middle barrier is not suddenly removed in order to avoid too much density fluctuation caused by the collision. In fact, the interspecies interaction plays an important role in determining the evolution of the two condensates. We must note the two condensates consist of different components, and g1 = g2 . Based on previous investigations, we can naturally divide the evolution into √ √ two cases: g12  g1 g2 and g12 > g1 g2 . Figures 1a–1c show the density evolution of the miscible condensates. Figures 1d–1f illustrate that of the immiscible condensates. The two condensates first form the SBS. After holding for some time, the condensate 1 suddenly splits into

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Fig. 3. (Color online) The time evolution of the center of mass coordinates of the two condensates with various g12 . Here, g1 = 1.0 and g2 = 0.6; μ1 = 1 and μ2 = 0.76125; the dissipation coefficient γ is 0.03.

Fig. 2. (Color online) The scheme of the MQT in our model. This picture corresponds to Figure 1f. The red lines, the blue lines and the black lines denote the condensate 1, the condensate 2 and the potential, respectively.

two parts and the condensate 2 starts to locate in the center of the trap. Why is there a holding process? Why does the condensate 1 transmit to the other side of the condensate 2? In reference [24], Kasamatsu et al. have provided a description to these phenomena by using the MQT. They have analyzed the transition from the SBS to the SPS within the WKB approximation. After their investigation, there are not other experiments to further make sure these phenomena and reveal the necessary inducement mechanism. Let us first examine the transition in detail. Figure 2 shows some snapshots about the MQT in Figure 1f from t = 4500ω −1 to 7000ω −1. The red line and the blue line denote the density of the condensate 1 and that of the condensate 2, respectively. The black line stands for the external potential. Obviously, the condensate 1 has tunneled through the condensate 2, and the condensate 2 starts to become the ‘core’ of the system. Furthermore, this process is unlike that for the miscible condensates, which can completely penetrate with each other. Therefore, this phenomenon provides a more stringent test about the quantum mechanics than that in the one particle case. Here we analyze the MQT only according to the structure of the two condensates. In big system, the condensates are more likely to act as liquid, the kinetic energy

of the wave function is much smaller than the potential energy. We know the configuration of the SBS is not symmetric. The shape of the condensate 2 is sharper than that of the condensate 1 (Fig. 2). For the condensate 1, the potential at the right edge of the condensate 2 is much lower than that at the left edge of the condensate 1. Thus they form a non-equilibrium state. As a quantum system, the condensate 1 tunnel to the right hand of the condensate 2. Till a symmetric structure is formed, the system achieves the equilibrium state. Therefore, the SBS and the SPS may correspond to the metastable state and the ground state of the system, respectively. In order to further explore the mixing process of the two condensates, we now examine the evolution process by calculating the center of mass coordinates of the two condensates. Figure 3 shows the results of the evolution of the two condensates with various g12 . We keep all other conditions to be fixed. The two condensates in each case are presented by the lines of the same color. The line and the dot line correspond to the condensate 1 and the condensate 2, respectively. At t = 0, the initial two condensates are separated by the middle barrier. However, the two condensates are compelled to meet with √ each other when the barrier is being unloaded. For g12 < g1 g2 , the interspecies interaction is weak. In many studies, one observes that the two condensates are miscible. Here, the two condensates form a damping oscillation in the mix√ ing process. For the critical value g12 = g1 g2 , the center of mass coordinates of the two condensates directly approaches to the center of the trap (the blue lines). While √ for g12 > g1 g2 , the two condensates are immiscible due to the strong interspecies repulsion. The spatial separation of the two condensates often is typified by the core-shell ground-state structure in 3D space. In our 1D system, one of the condensates can not wrap around the other one to form the core-shell structure. Thus, it is not surprising the center of mass coordinates of the two condensates keeps

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Fig. 5. (Color online) The energy of the SBS and the SPS as a function of g12 . Here g1 = 1 and g2 = 0.6. The number of the atoms of each component is N1 = N2 = N0 ; The inset shows the energy difference between the two states. The energy is in unit of ω.

Fig. 4. (Color online) (a) The time-dependence of the center of mass coordinates of the condensates with the various damping coefficient γ. The parameters are g1 = 1.0, g2 = 0.6 and g12 = 1.6; μ1 = 1 and μ2 = 0.76125. (b) The amplificatory picture of (a).

parallel. However, after holding for some time, the center of mass coordinates start to come together. This means the two condensates perform the transition from the SBS to the SPS. In addition, the larger the interspecies interaction coefficient g12 is, the longer is the lifetime of the SBS. The most obvious parameter that determines the lifetime of the SBS is the damping coefficient γ. Here, we only change the damping coefficient γ and keep all other parameters fixed. The evolution process would help us to realize the two states and the MQT. Figure 4a shows the evolution process with γ = 0, 0.01, 0.02, 0.03, 0.05 and 0.1 respectively. The damping coefficient can indeed affect the lifetime of the SBS. When the damping coefficient γ increases, the lifetime of the SBS becomes shorter, and the transition process is much faster. This means the tunneling decay rate of the SBS is determined by the damping. For γ = 0, even we prolong the simulation time, the MQT does not occur at all. Therefore, we can infer that the damping mechanism induces the MQT. For clarity, we further plot an amplificatory picture about Figure 4a (see Fig. 4b). The lifetime of the SBS can be easily identified in Figure 4b. We also can find the SBS is not completely static. The two condensates oscillate with damping around a metastable equilibrium

position. Another important point is that the oscillation has decayed to zero before the tunneling, i.e., the MQT may arise only after the excitations are damped out. Usually, one may believe the excitations would induce the tunneling. However, we obtain a different outcome in this experiment. Now, we examine the energy of both the SBS and the SPS to further confirm the above views. In our system, the two condensates are trapped in the harmonic well when the middle barrier is unloaded. Thus we can use the following one-dimensional coupled Gross-Pitaevskii energy function   2 

 2  ∂ψi  1 2 2 2 E[ψ1 , ψ2 ] = dx + mi ωi x |ψi | 2mi  ∂x  2 i=1,2  1 1 4 4 2 2 + g1 |ψ1 | + g2 |ψ2 | + g12 |ψ1 | |ψ2 | . 2 2 (8) In this calculation, we use the imaginary time method based on the Crank-Nicholson scheme. Figure 5 shows the energy of the systems as a function of the coefficient g12 . All other conditions are fixed. The black line shows the energies of the SPS and the red line is that of the SBS. The energies of the regimes increase with increasing the value of g12 . But the increase of the energy of the SPS obviously √ becomes slow when g12 > g1 g2 . It means the properties of the system have an essential change. Meanwhile, the energy of the SBS can be worked out. The energy difference between the two states is shown in the inset. Obviously, the energy difference between the two states decreases with g12 increasing. Comparing with the above numerical simulations, we can now give a summary about the emergence of the MQT. The SBS is the metastable state of the system. The energy difference between the two states determines the stability of the SBS. With a big energy difference between the two states, the MQT can arise easily.

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Fig. 6. (Color online) The evolution of the condensates with the stop of the damping mechanism after some given time t. g1 = 1, g2 = 0.6 and g12 = 1.6; the dissipation rate γ is 0.03 before the given time t. (a) The time evolution of energy of the system; (b) the time evolution of the center of mass coordinates of the condensates.

To further check whether the transition from the SBS to the SPS relies on the damping mechanism, we can stop the damping mechanism during the tunneling process. If the transition pauses with the absence of the damping, it shows the transition should depend on the damping. Here, we take Figures 1d–1f for instance. Figure 6 shows evolution of the energy of the whole system and the corresponding center of mass coordinates of the condensates. Note that we switch off the damping at t = 4000ω −1, 5000ω −1, 5500ω −1, 6000ω −1 and 8000ω −1, respectively. It is clear that the transition does not continue when the damping mechanism is switched off at t = 4000ω −1, 5000ω −1, 5500ω −1 and 6000ω −1, respectively. In addition, the energy of the system does not continue to decrease. These results show the transition relies on the damping mechanism to reduce the energy of the system and to induce the tunneling. The oscillation of the center of mass coordinates means the collective oscillation of the system, i.e., the state of the system is unstable. However, the oscillation does not appear in SBS (t = 4000ω −1) and the SPS (t = 8000ω −1). This property also indicates the SBS and the SPS are the metastable state and ground state of the system respectively. We have first carried out the MQT in the numerical experiments. In fact, the MQT is a decay process for the system to bounce to the ground state. In reference [24], the tunneling decay rate of the metastable state has been detailedly analyzed by using the collective coordinate method under the WKB approximation. The thermal effects on the tunneling decay rate are also estimated. According to their analysis, the MQT may derive from the excitation that overcomes some latent energy barrier. Here, we ascribe the origin of these phenomena to the damping. The latent energy barrier is not shown in the tunneling process. Our experiments also indicate the excitation is an obstacle for the tunneling. We believe

the emergence of the MQT needs the collaboration of the damping mechanism. The stability of the metastable state and the tunneling decay rate also depend on the damping mechanism. We have only considered the damping mechanism to induce the MQT and the interspecies interaction coefficient to determine the essential properties of the metastable state. In fact, these phenomena are also dependent on other factors such as the trapping potential, the number of the two species and the intraspecies interaction [23,24]. Although this work preliminarily focuses on the formation of the metastable state and the emergence of the MQT, it has clearly demonstrated the properties of those phenomena. In addition, the asymmetric structure would be the ground state if the number of the two species is very small. This means the mixing model can not essentially achieve the MQT. Furthermore, the metastable state would be very hard to transfer to the ground state when the energy difference between the two states is small enough. It implies the possibility to achieve some robust metastable state. Thus, in real experiment, the MQT might be easily observed by tuning the inter√ species scattering length a12 to approach a1 a2 , increasing the difference between a1 and a2 , and increasing the number of the two species.

4 Summary A direct mixing model is provided to numerically demonstrate the metastable state and the MQT of the binary √ mixtures. For the immiscible regime, i.e., g12 > g1 g2 , the two condensates firstly form the SBS, which is confirmed to be the metastable state. The SBS can hold for some time and then suddenly evolves to the SPS through the MQT. We give a visual explanation to the MQT by analyzing the structure of the system. The evolution processes are also characterized by calculating the center of mass coordinates of the condensates. We find the MQT is induced by the damping but not the excitations. In addition, the damping is necessary for the mixtures to carry out the MQT, and it can also effectively control the lifetime and the tunneling decay rate of the SBS. Finally, we calculate the energy of both the SBS and the SPS and apply a damp-controlling experiment to further present the origin of these phenomena. This work was supported by the Key Project of Hunan Provincial Educational Department of China under Grant No. 04A058, and the General Project of Hunan Provincial Educational Department of China under Grant No. 07C754.

References 1. D.S. Hall, M.R. Matthews, J.R. Ensher, C.E. Wieman, E.A. Cornell, Phys. Rev. Lett. 81, 1539 (1998) 2. J. Stenger, S. Inouye, D.M. Stamper-Kurn, H.-J. Miesner, A.P. Chikkatur, W. Ketterle, Nature (London) 396, 345 (1998)

C.F. Liu and Y. Tang: Metastable state and macroscopic quantum tunneling of binary mixtures 3. C.J. Myatt, E.A. Burt, R.W. Ghrist, E.A. Cornell, C.E. Wieman, Phys. Rev. Lett. 78, 586 (1997) 4. I. Kourakis, P.K. Shukla, M. Marklund, L. Stenflo, Eur. Phys. J. B 46, 381 (2005) 5. B. Apagyi, D. Schumayer, Eur. Phys. J. B 45, 55 (2005) 6. P. Ao, S.T. Chui, J. Phys. B: At. Mol. Opt. Phys. 33, 535 (2000) 7. T.-L. Ho, V.B. Shenoy, Phys. Rev. Lett. 77, 3276 (1996) 8. B.D. Esry, C.H. Greene, J.P. Burke, Jr., J.L. Bohn, Phys. Rev. Lett. 78, 3594 (1997) 9. E. Timmermans, Phys. Rev. Lett. 81, 5718 (1998) ¨ 10. P. Ohberg, S. Stenholm, Phys. Rev. A 57, 1272 (1998) 11. H. Pu, N.P. Bigelow, Phys. Rev. Lett. 80, 1130 (1998) 12. R. Graham, D. Walls, Phys. Rev. A 57, 484 (1998) 13. S.T. Chui, P. Ao, Phys. Rev. A 59, 1473 (1999) 14. P. Ao, S.T. Chui, Phys. Rev. A 58, 4836 (1998) 15. H.-J. Miesner, D.M. Stamper-Kurn, J. Stenger, S. Inouye, A.P. Chikkatur, W. Ketterle, Phys. Rev. Lett. 82, 2228 (1999) 16. D.M. Stamper-Kurn, H.-J. Miesner, A.P. Chikkatur, S. Inouye, J. Stenger, W. Ketterle, Phys. Rev. Lett. 83, 661 (1999) 17. H. Pu, N.P. Bigelow, Phys. Rev. Lett. 80, 1134 (1998) 18. A.A. Svidzinsky, S.T. Chui, Phys. Rev. A 67, 053608 (2003)

19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.

199

D. Gordon, C.M. Savage, Phys. Rev. A 58, 1440 (1998) B.D. Esry, C.H. Greene, Phys. Rev. A 59, 1457 (1999) F. Riboli, M. Modugno, Phys. Rev. A 65, 063614 (2002) D.M. Jezek, P. Capuzzi, Phys. Rev. A 66, 015602 (2002) ¨ P. Ohberg, Phys. Rev. A 59, 634 (1999) K. Kasamatsu, Y. Yasui, M. Tsubota, Phys. Rev. A 64, 053605 (2001) C. Lee, W. Hai, X. Luo, L. Shi, K. Gao, Phys. Rev. A 68, 053614 (2003) L. Yang, Y. Zhang, Phys. Rev. A 74, 043604 (2006) H.T. Ng, C.K. Law, P.T. Leung, Phys. Rev. A 68, 013604 (2003) M. Olshanii, Phys. Rev. Lett. 81, 938 (1998) D.S. Petrov, M. Holzmann, G.V. Shlyapnikov, Phys. Rev. Lett. 84, 2551 (2000) K. Kasamatsu, M. Tsubota, Phys. Rev. Lett. 93, 100402 (2004) S. Choi, S.A. Morgan, K. Burnett, Phys. Rev. A 57, 4057 (1998) A.A. Penckwitt, R.J. Ballagh, C.W. Gardiner, Phys. Rev. Lett. 89, 260402 (2002) K.M. Mertes, J.W. Merrill, R. Carretero-Gonz´ alez, D.J. Frantzeskakis, P.G. Kevrekidis, D.S. Hall, Phys. Rev. Lett. 99, 190402 (2007)