A dipolar droplet bound in a trapped Bose-Einstein condensate

A dipolar droplet bound in a trapped Bose-Einstein condensate

Abstract

We study the statics and dynamics of a dipolar Bose-Einstein condensate (BEC) droplet bound by inter-species contact interaction in a trapped non-dipolar BEC. Our findings are demonstrated in terms of stability plots of a dipolar Dy droplet bound in a trapped non-dipolar Rb BEC with a variable number of Dy atoms and the inter-species scattering length. A trapped non-dipolar BEC of a fixed number of atoms can only bind a dipolar droplet containing atoms less than a critical number for the inter-species scattering length between two critical values. The shape and size (statics) as well as the small breathing oscillation (dynamics) of the dipolar BEC droplet are studied using the numerical and variational solutions of a mean-field model. We also suggest an experimental procedure for achieving such a Dy droplet by relaxing the trap on the Dy BEC in a trapped binary Rb-Dy mixture.

pacs:
03.75.Hh, 03.75.Mn, 03.75.Kk

I Introduction

The experimental observation of a dipolar Bose-Einstein condensate (BEC) of Cr (4); (2); (3); (1); (5), Dy (7); (6) and Er (8) atoms with large magnetic dipole moments has opened new directions of research in cold atoms in the quest of novel and interesting features related to the anisotropic long-range dipolar interaction. Polar molecules, with much larger (electric) dipole moment, are also being considered (9) for BEC experiments. The atomic interaction in a dilute BEC of alkali-metal and other types of atoms (with negligible dipole moment) is represented by an S-wave contact (delta-function) potential. However, the non-local anisotropic long-range dipolar interaction acts in all partial waves and is attractive in certain directions and repulsive in others.

An untrapped three-dimensional (3D) BEC with attractive interaction does not exist in nature due to collapse instability (10). However, the collapse of an untrapped BEC can be avoided due to interspecies contact interaction in a binary mixture with a trapped BEC. The shape of such a stable droplet bound in a trapped BEC is controlled by the inter-species interactions, whereas that of a trapped BEC is determined by the underlying trap. Here we consider such a dipolar droplet bound in a trapped nondipolar BEC. The effect of the underlying atomic interaction, specially that of the anisotropic dipolar interaction, will easily manifest in such a dipolar droplet. These droplets will be termed quasi-free as they can easily move around inside the larger trapped BEC responsible for their binding. By taking the trapped BEC to be nondipolar, one can easily study the effect of intra-species dipolar interaction on the quasi-free droplet in the absence of any inter-species dipolar interaction.

Dipolar BECs are immediately distinguishable from those with purely contact interactions by their strong shape and stability dependence (11) on trapping geometry. Here, we are proposing a new way to trap a dipolar BEC in the form of a quasi-free dipolar droplet, using an attractive inter-species mean-field potential, which introduces a unique trapping geometry that results in further interesting shape and stability characteristics of dipolar BECs. We study the statics and dynamics of a quasi-free dipolar droplet using numerical solution and variational approximation of a mean-field model (12); (13). For this purpose we consider a binary mixture of non-dipolar Rb87 and dipolar Dy164 where the trapped non-dipolar Rb BEC could be in a cigar- or disk-shape. Among the available dipolar BECs (4); (7); (8), Dy atoms have the largest (magnetic) dipolar interaction strength (1); (6). The existence of stable 3D Dy droplets is illustrated by stability plots involving the number of atoms of the two species and the inter-species scattering length. For a fixed number (Rb) of Rb atoms the dipolar droplet could be bound below a critical number (Dy) of Dy atoms between two limiting values of inter-species attraction. If the inter-species attraction is too small, the Dy atoms in the droplet cannot be bound and, for a very large inter-species attraction, the droplet is destroyed by collapse instability. Usually, the dipolar droplet has the same shape as the trap acting on the non-dipolar BEC. However, for a large number of Dy atoms, as the inter-species attraction approaches the collapse instability, the dipolar droplet always is of cigar shape even if the external trap is disk shaped and eventually the dipolar droplet collapses on the axial axis from the cylindrical configuration. For a small number of atoms, the dipolar droplet collapses to the center maintaining its shape rather than on the axial axis.

The variational approximation to the sizes and chemical potentials of the stationary droplets is compared with the numerical solution of the mean-field model. The numerical study of breathing oscillation of the stable dipolar droplet is found to be in reasonable agreement with a time-dependent variational model calculation. We also demonstrate a viable experimental way of creating a Dy droplet bound in a trapped Rb BEC, e.g., by slowly removing the trap on a trapped binary Dy-Rb mixture, while the trapped Dy BEC evolves into a quasi-free droplet.

In Sec. II the mean-field model for a dipolar BEC droplet bound in a trapped non-dipolar BEC is developed. A time-dependent, analytic, Euler-Lagrange Gaussian variational approximation of the model is also presented. The results of numerical calculation are shown in Sec. III. Finally, in Sec. IV we present a brief summary of our findings.

Ii Mean-field model for a quasi-free dipolar droplet

We consider a binary BEC, where one of the species is dipolar and the other non-dipolar, interacting via inter- and intra-species interactions with the mass, number, magnetic /colorred dipole moment, and scattering length for the two species denoted by , respectively. The first species (Rb) is taken to be non-dipolar () and trapped while the second species (Dy) is dipolar () and polarized along axial direction. The angular frequencies for the axially-symmetric trap on Rb along , and directions are taken as and . The inter- () and intra-species () interactions for two atoms at positions and are taken as

(1)
(2)

where is the permeability of free space, is the angle made by the vector with the polarization direction, and is the reduced mass of the two species of atoms. With these interactions, the coupled Gross-Pitaevskii (GP) equations for the binary dipolar BEC can be written as (13)

(3)
(4)
(5)

To compare the dipolar and contact interactions, the intra-species dipolar interaction is expressed in terms of the length scale , defined by We express the strength of the dipolar interaction in Eq. (II) by this length scale and transform Eqs. (3) and (II) into the following dimensionless form (13)

(6)
(7)

where In Eqs. (6) and (7), length is expressed in units of oscillator length of the first species , energy in units of oscillator energy , density in units of , and time in units of .

Convenient analytic variational approximation to Eqs. (6) and (7) can be obtained with the following ansatz for the wave functions (14); (15); (16)

(8)

where , and are the widths and and are additional variational parameters. The effective Lagrangian for the binary system is

(9)
(10)

where , and . In these equations the overhead dot denotes time derivative. The Euler-Lagrange variational equations for the widths for the effective Lagrangian (II), obtained in usual fashion (16), are:

(11)
(12)
(13)
(14)
(15)
(16)

The solution of the time-dependent Eqs. (11) (14) gives the dynamics of the variational approximation.

If is the chemical potential with which the stationary wave function propagates in time, e.g. , then the variational estimate for is:

(17)
(18)

The energy of the system is given by

(19)

The widths of the stationary state can be obtained from the solution of Eqs. (11) (14) setting the time derivatives of the widths equal to zero. This procedure is equivalent to a minimization of the energy (II), provided the stationary state is stable and corresponds to a energy minimum.

Iii Numerical Results

We solve Eqs. (6) and (7) by split-step Crank-Nicolson discretization scheme using a space step of 0.1 and the time step 0.001 (15); (17). The contribution of the dipolar interaction is calculated in momentum space by Fourier transformation (15). For both species of atoms Rb and Dy we take the intra-species scattering length as , and the strength of dipolar interaction as (10); (7). The yet unknown inter-species scattering length is taken as a variable. The variation of can be achieved experimentally by the Feshbach resonance technique (18). We consider the trap frequency Hz, so that the length scale m and time scale ms.

Figure 1: (Color online) (Dy) stability plot showing the domain of stable Dy droplet in a binary Rb-Dy mixture for (a) (Rb) = 2000, (b) (Rb) = 10000, and (c) (Rb) = 50000, and for and 4. The intra-species scattering lengths are taken as .

We find that a quasi-free Dy droplet is achievable for a moderately attractive inter-species attraction (negative ) and for appropriate values of the number of atoms of the two species and . We illustrate the domain of existence of a stable Dy droplet in terms of stability plots in Figs. 1 (a), (b) and (c) for (Rb) and 50000, and for (disk-shaped trap) and 0.25 (cigar-shaped trap). We consider cigar- and disk-shaped Dy droplets in this study, where the effect of dipolar interaction is expected to be more prominent, and not spherically symmetric ones where the effect of dipolar interaction is expected to be a minimum (1); (5). In all plots of Figs. 1, for a fixed (Rb), the Dy droplet can be bound for a maximum of (Dy). This maximum of (Dy) increases with (Rb), all other parameters remaining fixed. For a fixed (Rb), the disk-shaped trap can accommodate a larger maximum number of Dy atoms than the cigar-shaped trap. For (Dy) smaller than this maximum number, the Dy droplet can be bound for between two limiting values. For above the upper limit, there is too much inter-species attraction on the the Dy droplet leading to its collapse. For below the lower limit, there is not enough attraction and the Dy droplet cannot be bound. The lower limit of is small and tends to zero as (Rb) tends to infinity. For fixed values of (Rb) and (Dy) and for small near the lower limit, the disk-shaped configuration is favored and it can bind the Dy droplet for smaller values of . However, for a fixed (Rb) and for a large near the upper limit, the disk-shaped configuration is favored only for a large (Dy) allowing a Dy droplet for larger , whereas the cigar-shaped configuration is favored for a small (Dy) allowing a Dy droplet for larger

Figure 2: (Color online) Variational (v) and numerical (n) results for chemical potential and rms sizes for the trapped Rb BEC of 10000 atoms and the bound Dy droplet versus the number of Dy atoms for (a) and (b) . The inter-species scattering length is .

In Figs. 2 we illustrate the numerical and variational results for chemical potentials and root-mean-square (rms) sizes and of the trapped Rb BEC of 10000 atoms and of the bound Dy droplet versus (Dy) for (a) and (b) 0.25. Considering that the Dy droplet may not have a Gaussian shape as assumed in the variational approximation, the agreement between the numerical and variational results is quite satisfactory.

(Dy)
approx -40 1000 4 2.7597 0.7106 1.0755 0.4847
var -40 1000 4 2.7475 0.7091 1.0703 0.4834
approx -80 1000 0.25 2.2780 8.9525 0.6468 2.5799
var -80 1000 0.25 2.2549 8.8558 0.6378 2.5525
approx -60 500 4 2.7597 0.7106 0.8478 0.3768
var -60 500 4 2.7492 0.7090 0.8443 0.3757
approx -100 100 0.25 2.2780 8.9525 0.6384 1.5803
var -100 100 0.25 2.2752 8.9392 0.6373 1.5779
Table 1: Root-mean-square sizes of the binary Rb-Dy system of 50000 Rb atoms from variational approximation (11) (14) (var), and an approximate energy minimization (approx).

The existence of the quasi-free Dy droplet can be studied qualitatively by a minimization of the energy (II). However, the widths of the Rb BEC for a fixed (Rb) do not vary much as (Dy) or is varied. Hence for a qualitative understanding of the existence of a stable Dy droplet for a fixed (Rb), we can make further approximation in the energy (II) and take the widths of the trapped Rb BEC to be the same as the widths of the Rb BEC in the absence of Dy atoms under otherwise identical conditions. The widths of the Dy droplet so obtained for (Rb) = 50000 are compared in Table I for several values of , (Dy), and with the widths obtained from exact energy minimization. We find from Table I that the sizes of the Rb for a fixed (Rb) remains reasonably constant with respect to the variation of (Dy) and and that the approximate energy minimization with fixed widths for Rb BEC leads to a good approximation to the widths of the Dy droplet. In the numerical solution of the binary GP equations (6) and (7) we also verified that the shape and size of the trapped Rb BEC are fairly independent of the presence or absence of the Dy droplet. Hence in the study of the shape, size and dynamics of the dipolar Dy droplet bound in the trapped Rb BEC, the trapped BEC will only have a passive role and we shall highlight only the shape, size and dynamics of the Dy droplet in the following.

Figure 3: (Color online) (a) The isodensity contour of a disk-shaped () Rb BEC of 10000 atoms and that of the bound Dy droplet of (b) 1000 atoms (), (c) 1000 atoms (), (d) 4000 atoms (). The isodensity contour of the Rb BEC is practically the same in all three cases. The density on the contour in all cases is 0.001 and length is measured in units of (= 1 m).

We consider the shape of a stable Dy droplet in the stability plots of Figs. 1 and study its change as the collapse instability is approached. In Fig. 3 (a) we show the isodensity contour plot (density ) of a disk-shaped () Rb BEC of 10000 atoms and that (density ) of the bound Dy droplet in Figs. 3 (b) for (Dy) = 1000 (), (c) for (Dy) = 1000 (), and in (d) for (Dy) = 4000 (). The isodensity of the Rb BEC is practically the same in all three cases. Of the isodensities of the Dy droplet, the one in Fig. 3 (b) is deep inside the stability region far away from collapse instability. The parameters of Figs. 3 (c) and (d) are close to the region of collapse instability. Of these two, Fig. 3 (c) corresponds to a medium number of Dy atoms and Fig. 3 (d) to a large number. Independent of the initial shape, a Dy droplet with a medium number of Dy atoms always collapses towards the center with shrinking size. However, a Dy droplet containing a large number of atoms, independent of the associated trap symmetry, always first takes a cigar shape and then collapses on the axial direction. A strong dipolar interaction prohibits a collapse to center of a large Dy droplet and favors a cigar shape. In Figs. 3 the trap is disk-shaped. The medium-sized Dy droplet of Fig. 3 (c) collapses to center maintaining the disk shape as the net dipolar interaction is smaller in this case. The large Dy droplet in Fig. 3 (d) has changed its shape from the supporting disk-shaped trap to a cigar shape. If the attraction is further increased by increasing , the Dy droplet of Fig. 3 (d) would collapse from the cigar shape on the axial axis.

Figure 4: (Color online)(a) The isodensity contour of a cigar-shaped () Rb BEC of 10000 atoms and that of the bound Dy droplet of (b) 1000 atoms (), (c) 1000 atoms (), (d) 3000 atoms (). The isodensity contour of the Rb BEC is practically the same in all three cases. The density on the contour in all cases is 0.001 and length is measured in units of (= 1 m).
Figure 5: (Color online) The rms sizes (a) and (b) from numerical simulation (num) and variational approximation (var) during breathing oscillation initiated by the sudden change versus time in units of (= 1.38 ms) of a Dy droplet of 100 atoms bound in a trapped Rb BEC of 2000 atoms for and .

In Fig. 4 (a) we illustrate the isodensity contour of a cigar-shaped () Rb BEC of 10000 atoms and the same of the Dy droplet are shown in Figs. 4 (b) for (Dy) = 1000 (), (c) for (Dy) = 1000 (), and in (d) for (Dy) = 3000 (). The density of the Rb BEC is virtually the same in all three cases. Of the three Dy droplets, the one in Fig. 4 (b) is deep inside the stability region far away from collapse instability. The parameters for Figs. 4 (c) and (d) are close to the region of collapse instability. In all cases the Dy droplet maintains the cigar shape of the accompanying trap acting on the Rb BEC. The size of the Dy droplet in Fig. 4 (b) is larger than that of Fig. 4 (c) with the same number of atoms due to a strong inter-species attraction acting on the latter. The Dy droplet of Fig. 4 (d) is larger than those in Figs. 4 (b) and (c) due to a larger number of Dy atoms in it.

Figure 6: (Color online) The rms sizes (a) and (b) from numerical simulation (num) and variational approximation (var) during breathing oscillation initiated by the sudden change versus time in units of (= 1.38 ms) of a Dy droplet of 1000 atoms bound in a trapped Rb BEC of 10000 atoms for and .

We investigate the dynamics of a bound Dy droplet in a trapped Rb BEC. First, to test the present scheme we consider a small Dy droplet of 100 atoms bound in a disk-shaped () Rb BEC of 2000 atoms with . This corresponds to the stable region of Fig. 1 (a). In real-time evolution of the system, the scattering length is suddenly changed to thus starting the breathing oscillation. In Figs. 5 we show the resultant oscillation of the rms sizes (a) and (b) during time evolution as obtained from numerical solution of Eqs. (6) and (7) and variational approximation (11) (14).

After obtaining a satisfactory result of dynamics with a small system, we venture with a larger system of experimental interest, e. g., a stable Dy droplet of 1000 atoms bound in a cigar-shaped () Rb BEC of 10000 atoms with corresponding to the stable region of Fig. 1 (b). In real-time evolution of the system, the scattering length is again suddenly changed to thus starting the breathing oscillation. In Figs. 6 we show the resultant oscillation of the rms sizes (a) and (b) during time evolution as obtained from a numerical solution of Eqs. (6) and (7) and variational approximation (11) (14). The agreement between the numerical simulation and variational approximation in both cases of breathing oscillation illustrated in Figs. 5 and 6 for disk- and cigar-shapes is quite satisfactory considering the facts that (i) during oscillation the bound Dy droplet might have a density distribution different from a Gaussian distribution assumed in the variational approximation and (ii) the large non-linearity in the Rb BEC will also make its density distribution non-Gaussian. The stable oscillation under small perturbation as shown in Figs. 5 and 6 confirms the dynamical stability of the Dy droplet for both disk and cigar type environments.

Figure 7: (Color online) (a) The rms sizes and of the Dy and Rb BECs versus time during the passage of the trapped Dy BEC in the binary Rb-Dy mixture with (Rb) = 10000, (Dy) = 1000, and as the confining trap on the Dy BEC is relaxed for exponentially as , so that this trap is practically zero for . The linear 1D densities (b) and (c) at times (in units of ) during the expansion together with the numerically calculated density of the stationary (sta) Dy droplet.

The present quasi-free dipolar droplet is not just of theoretical interest, but can be realized experimentally by initially preparing a binary Rb-Dy mixture where both the components are harmonically trapped. The trap on Dy can then be ramped to zero exponentially during a few milli seconds. The Dy cloud will initially expand and finally form the quasi-free Dy droplet. To illustrate numerically the viability of this procedure, we consider an initial Rb-Dy binary mixture with following parameters: (Rb) = 10000, (Dy) = 1000, . We take the same axial trap with acting on both components. Then we perform real-time propagation of the binary GP equation with this initial state and ramp down the Dy trap by (Dy) for time . The trap on Dy is practically zero for times . The trapped Dy BEC first expands for and eventually it emerges as the quasi-free Dy droplet. The time evolution of the rms sizes of the Dy droplet is shown in Fig. 7 (a). The widths of the Dy droplet first increase and eventually execute small oscillation illustrating the stable Dy droplet. To investigate the shape of the Dy droplet, we plot in Figs. 7 (b) and (c) the one-dimensional (1D) densities and at times and 100 along with the corresponding numerical densities of the stationary droplet calculated from the solution of Eqs. (6) and (7). The 1D numerical densities of the stationary Dy droplet are in agreement with the densities obtained from dynamical simulation of the passage of the trapped Dy BEC to a quasi-free droplet. In Figs. 7 (b) and (c) one can easily identify the small oscillation of the evolving dynamical droplet around its stationary shape.

Iv Summary and Discussion

Using variational approximation and numerical solution of a set of coupled mean-field GP equations, we demonstrate the existence of a stable dipolar Dy droplet bound by inter-species attraction in a trapped non-dipolar BEC of Rb atoms. The domain of stability of the Dy droplet is highlighted in stability plots of number of Dy atoms and inter-species scattering length for both cigar- and disk-shaped traps acting on the Rb BEC. Results of variational approximation and numerical solution for the statics (sizes and chemical potentials) and dynamics (breathing oscillation) of the Dy droplet are found to be in satisfactory agreement with each other. We also demonstrate numerically that such droplets can be obtained experimentally by considering a trapped binary Rb-Dy BEC and then removing the trap on Dy. The Dy BEC then expands and transforms into a bound dipolar droplet.

Dipolar interaction among atoms is quite different from normal short-range atomic interaction and manifests in different ways in a trapped dipolar BEC. However, in a trapped dipolar BEC, the confining constraints could be too strong and could make the effect of dipolar interaction difficult to observe. Special experimental set up and theoretical formulation might be necessary to study the effect of dipolar interaction. However, it will be much easier to see the effect of dipolar interaction in the present binary mixture of non-dipolar Rb and dipolar Dy. Strong dipolar interaction in the axial polarization direction should elongate the dipolar BEC along this direction thus transforming it to a cigar shape. However, it will be difficult to observe this change in the presence of a harmonic trap along . The effect of dipolar interaction is clearly seen in the present quasi-free Dy droplet of Fig. 3 (d) where due to a strong dipolar interaction the shape of the Dy droplet has changed from the disk shape the shape of the Rb BEC of Fig. 3 (a) responsible for its binding to the cigar shape. The thin disk-shaped Rb BEC stays only near the central plane of the Dy droplet. Most of the Dy atoms lying outside the Rb BEC are bound due to the intra-species dipolar interaction. In the dipolar Dy droplet of Fig. 3 (d) the dipolar interaction is playing a more important role in its binding and shape.

The present quasi-free dipolar droplet may also find other interesting applications in the BEC phenomenology. Among the interesting features in a trapped dipolar BEC, one can mention its peculiar shape and stability properties (11), and D-wave collapse (19), anisotropic soliton, vortex soliton (20) and vortex lattice (21), anisotropic shock and sound wave propagation (22), anisotropic Landau critical velocity (23), stable checkerboard, stripe, and star configurations in a two-dimensional (2D) optical lattice as stable Mott insulator (24) as well as super-fluid soliton (25) states. It would be of great interest to find out how these features and properties of a trapped dipolar BEC would manifest in a 3D quasi-free dipolar droplet. For example, a quasi-free dipolar droplet could be used in the experimental study of anisotropic sound and shock-wave propagation (22), collapse dynamics (19), anisotropic Landau critical velocity (23), formation of vortex dipoles and vortex lattice (21) etc in a different setting of confinement which will facilitate the observation of the effect of anisotropic dipolar interaction.

Acknowledgements.
We thank FAPESP and CNPq (Brazil) for partial support.

Footnotes

  1. lyoung@ift.unesp.br
  2. adhikari@ift.unesp.br; URL: http://www.ift.unesp.br/users/adhikari

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