Stable and mobile excited two-dimensional dipolar Bose-Einstein condensate solitons
We demonstrate robust, stable, mobile excited states of quasi-two-dimensional (quasi-2D) dipolar Bose-Einstein condensate (BEC) solitons for repulsive contact interaction with a harmonic trap along the direction perpendicular to the polarization direction . Such a soliton can freely move in the plane. A rich variety of such excitations is considered: one quanta of excitation for movement along (i) axis or (ii) axis or (iii) both. A proposal for creating these excited solitonic states in a laboratory by phase imprinting is also discussed. We also consider excited states of quasi-2D dipolar BEC soliton where the sign of the dipolar interaction is reversed by a rotating orienting field. In this sign-changed case the soliton moves freely in the plane under the action of a harmonic trap in the direction. At medium velocity the head-on collision of two such solitons is found to be quasi elastic with practically no deformation. The findings are illustrated using numerical simulation in three and two spatial dimensions employing realistic interaction parameters for a dipolar Dy BEC.
pacs:03.75.Hh, 03.75.Mn, 03.75.Kk, 03.75.Lm
A bright soliton is a localized intensity peak that maintains its shape, while traveling at a constant velocity in one dimension (1D), due to a cancellation of nonlinear attraction and dispersive effects . Solitons have been observed in water waves, nonlinear optics, and Bose-Einstein condensate (BEC) etc. among others. In physical three-dimensional (3D) world, quasi-one-dimensional (quasi-1D) solitons are observed where a reduced (integrated) 1D density exhibit soliton-like property . Experimentally, bright matter-wave solitons and soliton trains were created in a BEC of Li  and Rb atoms . However, due to collapse instability, 3D BEC bright solitons are fragile and can accommodate only a small number of atoms .
Recent observation of BECs of Dy [6, 5], Er  and Cr [8, 9] atoms with large magnetic dipole moments has opened new directions of research in BEC solitons. In a dipolar BEC, in addition to the quasi-1D solitons  it is possible to have a quasi-two-dimensional (quasi-2D) soliton  free to move in a plane with a constant velocity and trapped in the perpendicular direction. Although, such a quasi-2D dipolar BEC soliton has not been experimentally observed, this seems to be the only simplest example of an experimentally realizable quasi-2D soliton. Dipolar BEC solitons can be stabilized either by a harmonic or an optical-lattice trap in quasi-1D  and quasi-2D  configurations. More interestingly, one can have dipolar BEC solitons for fully repulsive contact interaction . Hence these dipolar solitons bound by long-range dipolar interaction could be robust and less vulnerable to collapse for a large number of atoms due to the repulsive contact interaction. The contact repulsion gives stability against collapse and dipolar attraction prevents the escape of the atoms from the soliton.
The possibility of realizing a trapped BEC in an excited (non-ground) state has been a topic of intense research . Here we demonstrate the existence of stable excited quasi-2D dipolar BEC solitons harmonically trapped in the direction capable of moving in the plane with a constant velocity with the polarization direction. They are stable and stationary excitations of the quasi-2D bright solitons. We consider three types of excitations of the solitons: lowest excitation for dynamics along (i) axis, (ii) axis, and (iii) both and axes. The wave function for the lowest () excitation is antisymmetric in () with a zero in the () plane. The symmetries of these states are quite similar to the excited states of a 3D trapped harmonic oscillator, the only difference is that in the case of the quasi-2D dipolar solitons there is no harmonic trap along and axes and the confinement in these directions is achieved solely by dipolar attraction.
In addition to the normal excited quasi-2D dipolar BEC soliton, we also considered a set-up where the sign of the dipolar interaction is reversed by a rotating orienting field . In this sign changed configuration, the dipolar interaction is repulsive along axis and attractive in the plane and an excited quasi-2D dipolar BEC soliton can be realized by applying a harmonic trap along the axis. The stationary excited quasi-2D bright solitons were obtained by a imaginary-time propagation of the mean-field Gross-Pitaevskii (GP) equation in 3D. These stationary excited solitons can also be termed quasi-2D dipolar dark-in-bright solitons bearing resemblance with the quasi-1D dipolar dark-in-bright solitons studied recently . The quasi-1D dipolar dark-in-bright solitons were also established to be excitations of dipolar bright solitons [16, 17].
We also studied the dynamics of these quasi-2D solitons by real-time simulation using an effective 2D mean-field model. The stability of the excited quasi-2D dipolar solitons was established by studying the breathing oscillation of the system upon small perturbation over long time in real-time propagation. The head-on collision between two excited quasi-2D solitons is found to be quasi elastic at medium velocities of few mm/s. In such a collision, two excited solitons pass through each other without significant deformation. However, at lower velocities the collision becomes inelastic, as only the strictly 1D integrable solitons can have elastic collision at all velocities . We also demonstrate by real-time simulation in 2D the possibility of the creation of the excited quasi-2D solitons by phase imprinting [18, 19] a normal quasi-2D bright soliton with identical parameters with, for example, a phase difference of between the two parts of the BEC wave function situated at and .
In Sec. II the time-dependent 3D mean-field model for a quasi-2D dipolar BEC soliton is presented. A reduced 2D model appropriate for the present study is also considered. The numerical investigation of the quasi-2D solitons is considered in Sec. III. The domain of the appearance of the ground and excited states of the quasi-2D solitons is illustrated in a phase plot with realistic values of contact and dipolar interactions of Dy atoms exhibiting the maximum number of atoms in the quasi-2D soliton versus the scattering length. The evolution of collision between two excited quasi-2D solitons is considered by real-time evolution. The dynamical simulation of the creation of an excited quasi-2D soliton from a phase-imprinted ground state of a quasi-2D soliton is also demonstrated. The possibility of creating a dark soliton by phase imprinting a BEC by means of a detuned laser has been illustrated experimentally . Finally, in Sec. IV we present a brief summary and concluding remarks.
2 Mean-field Gross-Pitaevskii equations
We consider a dipolar BEC soliton, with the mass, number of atoms, magnetic dipole moment, and scattering length given by . The interaction between two atoms at and is 
where is the permeability of free space, is the angle made by the vector with the polarization direction. The parameter () can be tuned by a rapidly rotating magnetic field allowing the change of the sign of dipolar interaction . We will consider two cases in this study: (i) corresponding to the normal dipolar interaction and (ii) corresponding to a sign-changed dipolar interaction.
The first possibility (i) leads to a fully asymmetric quasi-2D excited solitonic state in the presence of a harmonic trap along axis. The dimensionless GP equation in this case can be written as 
where . In (3), length is expressed in units of oscillator length , where is the circular frequency of the harmonic trap acting along axis. The energy is in units of oscillator energy , probability density in units of , and time in units of .
The second possibility (ii) leads to an axially-symmetric excited quasi-2D dipolar soliton in the presence of a harmonic trap along axis. The dimensionless GP equation in this case is 
where now is the circular frequency of the harmonic trap along axis. In the case of (3) the dipolar interaction is attractive along axis and repulsive in the plane; the opposite is true for (4). The anisotropic dipolar interaction is circularly symmetric in the plane in both cases. Hence, (3) is fully anisotropic and leads to fully-anisotropic quasi-2D solitons, whereas (4) is axially symmetric and hence leads to quasi-2D solitons with this symmetry.
In place of (4), a quasi-2D model appropriate for the quasi-2D excited solitons of large spatial extension is very economic and convenient from a computational point of view, specially for real-time dynamics. The system is assumed to be in the ground state of the axial trap and the wave function can be written as , where is an effective wave function in the plane. Using this ansatz in (4), the dependence can be integrated out to obtain the following effective 2D equation 
where , and the dipolar term is written in Fourier momentum space.
For this study we consider Dy atoms of magnetic moment  with the Bohr magneton leading to the dipolar length Dy, with the Bohr radius. We consider here m corresponding to a radial angular trap frequency Hz corresponding to ms.
3 Numerical Results
We solve the 3D equations (3) and (4) or the 2D equation (2) by the split-step Crank-Nicolson discretization scheme using both real- and imaginary-time propagation in 3D or 2D Cartesian coordinates, respectively, using a space step of 0.1 0.2 and a time step of in the imaginary-time simulation, and of in real-time simulation . A smaller time step is employed in the real-time propagation to obtain reliable and accurate results. The dipolar potential term is treated by a Fourier transformation to the momentum space using a convolution rule .
The stationary profile of the excited quasi-2D solitons can be obtained by imaginary-time simulation. A symmetric initial state in the harmonic oscillator problem converges to the ground state in imaginary-time propagation, whereas an antisymmetric initial state converges to the lowest excited state. The imaginary-time routine preserves the symmetry of the initial state. The anisotropic quasi-2D soliton of (3) lies in the plane and we consider three types of excited solitons: excitation in (i) , (ii) , and in both (iii) and . In imaginary-time simulation these stationary excitations can be obtained with an initial antisymmetric trial function (as in the harmonic oscillator problem), for example, , where (antisymmetric in ), (antisymmetric in ), and (antisymmetric in both and ) for excitations (i), (ii), and (iii), respectively. In case of the circularly symmetric quasi-2D soliton of (4) in plane the excitation could be (i) in (antisymmetric in ), and in both (ii) and (antisymmetric in both ) which can be obtained in imaginary-time propagation with the initial state , where and , respectively. Hence, while using the effective 2D equation (2), the excited quasi-2D solitons of (4) can be obtained in imaginary-time propagation with the initial state , where or for excitations in or in both and , respectively. For fast convergence the constants and in these initial guesses are taken to be small denoting large spatial extension of the excited quasi-2D soliton.
In this study we consider excited quasi-2D dipolar solitons in a BEC of Dy atoms with magnetic moment . The reason for considering Dy atoms is that these atoms have the largest magnetic moment among those used in dipolar BEC experiments and a large dipole moment is fundamental for achieving excited quasi-2D dipolar solitons with a large number of atoms. The dipolar length in this case is . In the case of a sign-changed dipolar interaction by a rotating orienting field  we take and . We take the harmonic trap length m, corresponding to a harmonic trap frequency of Hz and time scale ms.
First we study the domain of the appearance of the nonexcited quasi-2D solitons (ground states) of (3) and of (4). These solitons for a specific value of scattering length can exist for the number of atoms below a critical number beyond which the system collapses . In figure 1 we plot this critical number versus from numerical simulation for and . We find that a quasi-2D soliton is possible for the number of atoms below this critical number . In the collapse region, the soliton collapses due to an excess of dipolar attraction. In the stable region there is a balance between attraction and repulsion and a stable soliton can be formed. In  we considered the critical number of Dy atoms in the ground and excited states of quasi-1D solitons with a strong trap in and directions and no trap in the direction. The critical number of quasi-1D ground-state solitons of  is also plotted in figure 1 for a comparison. These two critical numbers are quite similar and the critical number for quasi-2D solitons is larger than that for quasi-1D solitons. This last finding is not unexpected in view of the stronger trap in the quasi-1D case, as compared to the quasi-2D case, facilitating the collapse in a dipolar BEC, thus leading to a smaller critical number in the quasi-1D case. A similar result is expected in the case of the excited quasi-2D solitons when contrasted with the quasi-1D solitons.
The excited quasi-2D solitons to be considered next have a larger spatial extension and can accommodate a larger number of atoms. However, due to a larger spatial extension, an accurate calculation of the critical number of atoms in excited quasi-2D solitons necessitates a huge amount of RAM and CPU time and the critical number of these excited solitons will not be considered here. The plots of figure 1 represent a lower limit for the critical numbers for the excited quasi-2D solitons. From figure 1 we see that the number of atoms in the quasi-2D solitons in ground and excited states could be quite large and will be of experimental interest. The size of quasi-1D nondipolar solitons is usually quite small and these solitons can accomodate only a small number of atoms. A quasi-2D soliton cannot be realized in a nondipolar BEC as the long-range dipolar interaction plays a crucial role in their formation and stability.
Next we present the profile of the quasi-2D solitons as obtained from (3) by imaginary-time propagation for 1000 Dy atoms with the scattering length adjusted to by the Feshbach resonance technique  and with the dipolar length . A 3D Gaussian input wave function converges to the unexcited quasi-2D soliton illustrated in figure 2 (a). An input of 3D Gaussian times , with a node at representing an excitation in , converges to the excited quasi-2D soliton of figure 2 (b). Similarly figure 2 (c) shows a quasi-2D soliton with the lowest excitation in and figure 2 (d) shows the same with an excitation in both and . From figure 2 we find that the excitation of the quasi-2D soliton increases the spatial extension significantly for the same number (1000) of Dy atoms. Moreover, the excited quasi-2D solitons can accommodate a larger number of atoms compared to the unexcited quasi-2D solitons implying a larger critical number shown in figure 1. The quasi-2D solitons of Figs. 2 (b) and (c) with one quantum of excitation in or are larger in size compared to the ground state shown in figure 2 (a) and the quasi-2D soliton of figure 2 (d) with two quanta of excitation is larger those of Figs. (b) and (c) with one quantum of excitation each. The size of the excited quasi-2D solitons of figure 2 is about 120 m and it increases rapidly with the scattering length . The size tends to as .
Similar to the Feshbach resonance technique for manipulating the scattering length by magnetic  and optical  means, it is possible to manipulate the dipolar interaction by a rotating orienting field . Now we present results for quasi-2D solitons with a sign-changed dipolar length of for Dy atoms with an axial trap along the polarization direction satisfying (4). Here we consider the lowest-energy quasi-2D soliton of 1000 Dy atoms for the scattering length . The 3D isodensity contour is illustrated in figure 3 (a) as obtained by imaginary-time propagation of (4). As (4) is axially symmetric, profile of the quasi-2D soliton with one quantum of excitation in is the same as the one with one quantum of excitation in and we present the one with one quantum of excitation in in figure 3 (b). Finally, in figure 3 (c) we present a quasi-2D soliton with one quantum of excitation each in and . As in figure 2, the excited quasi-2D solitons of figure 3 containing the same number of atoms have larger spatial extension. The size of the quasi-2D soliton of figure 3 (c)
The excited quasi-2D solitons presented in Figs. 2 and 3 are stable and robust as tested under real-time propagation in 3D with a reasonable perturbation in the parameters. The robustness comes from the large contact repulsion for a reasonably large scattering length which strongly inhibits collapse. Also, an appropriate combination of the harmonic trap and long-range dipolar interaction provides confinement of the quasi-2D soliton and prevents leakage of the atoms to infinity. In a nondipolar BEC soliton, the contact attraction alone provides the binding and there is no repulsion to stop the collapse. Consequently, a nondipolar BEC soliton is usually fragile against collapse and auto-destruction. A stringent test of the robustness of these excited solitons is provided in their behavior under head-on collision. Like the quasi-1D dipolar solitons, the collision of quasi-2D solitons are expected to be quasi elastic with the solitons emerging with little deformation at medium velocities. However, at low velocities the collision is expected to be inelastic. Only the collision between two integrable 1D solitons is known to be perfectly elastic at all velocities.
To demonstrate the robustness of the solitons we consider a head-on collision between two excited quasi-2D solitons moving in opposite directions. However, because of the large spatial extension of the excited solitons, from a consideration of the required RAM and CPU time, it is prohibitive to carry on these studies in 3D. Hence for the study of the dynamics of the excited quasi-2D solitons we consider the reduced 2D equation (2) in place of the 3D equation (4) with the sign changed dipolar interaction. The reduced 2D equation (2) should give a reasonable account of the quasi-2D solitons for medium values of contact and dipolar nonlinearity parameters. Before the study of this collision dynamics, we first compare the densities of a quasi-2D soliton calculated using imaginary-time propagation of the 3D and 2D equations (4) and (2), respectively. In Figs. 4 (a), (b), and (c) we plot the effective 2D density in the plane of the quasi-2D excited solitons shown in Figs. 3 (a), (b), and (c), respectively, from a numerical solution of the 3D GP equation (4). In Figs. 4 (d), (e), and (f) we plot the same as obtained from the imaginary-time solution of the reduced 2D (2). The agreement between the two sets of densities presented in figure 4 justifies the use of (5) for the study of the dynamics. However, it should be noted that nonetheless the collision dynamics can differ slightly in the 2D and 3D descriptions, because in the 3D case energy can be transferred in the additional degree of freedom in the direction which is eliminated in the 2D case .
We consider the collision dynamics between two identical excited quasi-2D solitons each of 1000 Dy atoms as illustrated in figure 3 (b). The collision dynamics of two such solitons as generated from a real-time simulation of (2) is shown in figure 5. The initial profiles of the two colliding solitons are obtained by a imaginary-time simulation of the same. The initial velocities of the two solitons placed at are attributed by multiplying the initial wave functions of the two solitons by phase factors . The parameter is chosen by trial. The two excited quasi-2D solitons are initially placed at and the real-time simulation started. The two solitons move in opposite directions and suffer a head-on collision. The collision dynamics is best illustrated by plotting effective linear density along direction versus and as shown in figure 5. The velocity of each soliton is about mm/s using m and ms. The smooth density profiles of the dynamics presented in figure 5 illustrates the quasi elastic nature of the dynamics.
As the excited quasi-2D solitons are stable and robust, they can be prepared by phase imprinting  a bright soliton. In experiment a homogeneous potential generated by the dipole potential of a far detuned laser beam is applied on one half of the bright soliton () for an interval of time so as to imprint an extra phase of on the wave function for . The thus phase-imprinted quasi-2D bright soliton is propagated in real-time, while it slowly transforms into a excited quasi-2D soliton. This simulation is done with no axial trap. In actual experiment a very weak axial trap can be kept during generating the excited quasi-2D soliton, which can be eventually removed. The simulation is illustrated in figure 6, where we plot the linear axial density versus time. It is demonstrated that at large times the linear density tends towards that of the stable excited quasi-2D soliton.
We demonstrated the possibility of creating mobile, stable, excited quasi-2D solitons in dipolar BEC with a notch in the central plane and capable of moving in a plane with a constant velocity. These solitons are stationary solutions of the mean-field GP equation. The head-on collision between two such solitons with a relative velocity of about 5 mm/s is quasi elastic with the solitons passing through each other with practically no deformation. A possible way of preparing these excited quasi-2D solitons by phase imprinting a bright soliton is demonstrated using real-time propagation in a mean-field model. The results and conclusions of this paper can be tested in experiments with present-day know-how and technology and should lead to interesting future investigations.
We thank FAPESP and CNPq (Brazil) for partial support.
- Adhikari@ift.unesp.br; URL: http://www.ift.unesp.br/users/Adhikari
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