# Dynamical generation of dark solitons in spin-orbit-coupled Bose-Einstein condensates

###### Abstract

We numerically investigate the ground state, the Raman-driving dynamics and the nonlinear excitations of a realized spin-orbit-coupled Bose-Einstein condensate in a one-dimensional harmonic trap. Depending on the Raman coupling and the interatomic interactions, three ground-state phases are identified: stripe, plane wave and zero-momentum phases. A narrow parameter regime with coexistence of stripe and zero-momentum or plane wave phases in real space is found. Several sweep progresses across different phases by driving the Raman coupling linearly in time is simulated and the non-equilibrium dynamics of the system in these sweeps are studied. We find kinds of nonlinear excitations, with the particular dark solitons excited in the sweep from the stripe phase to the plane wave or zero-momentum phase within the trap. Moreover, the number and the stability of the dark solitons can be controlled in the driving, which provide a direct and easy way to generate dark solitons and study their dynamics and interaction properties.

###### pacs:

(020.0020) Atomic and molecular physics; (020.1475) Bose-Einstein condensates; (190.6135) Spatial solitons.## I Introduction

Spin-orbit (SO) coupling plays an important role in many fundamental quantum phenomena, ranging from the atomic fine structure to the newly discovered topological insulators Hasan (); Qi (). Recently, a synthetic SO coupling has been successfully engineered in Bose-Einstein condensations (BECs) and degenerate Fermi gases Lin2011 (); Williams (); Zhang2012 (); Qu (); LeBlanc (); Olson (); Zhang2014 (); Wang1 (); Cheuk (); Wang2 (). These cold atomic systems with adjustable SO coupling provide an ideal platform for investigating a wide range of interesting physics in many SO-coupled systems, such as atomic spin Hall effects Zhu2006 (); Beeler (), Anderson localization of relativistic particles Zhu2009 (), fractional Fermi number Zhu2012 () and topological superfluid with Majorana fermions Zhu2011 (). The cold atoms with synthetic SO-coupling also have potential applications in designing atomic interferometry Anderson2011 ().

Since the newly realized SO-coupled BEC has no direct analog in solids, great attention has been paid to this unique many-body system. Especially, the ground state of SO-coupled BECs has been studied extensively in theory Wang (); Ho (); Wu (); Li (); Jian (); Hu (); Xu (); Ozawa (); Sinha (); Zhou (); Radic (); Xu2 (); Yu (); Ramachandhran (); He (); Ozawa2 (); Zhou2 (); Lewenstein (); Zhu2013 (); Zhao2014 (). For a two-dimensional homogeneous bosonic gas with Rashba SO coupling, depending on the interatomic interactions, the ground state of the system exhibits two unconventional phases: a plane wave phase with condensate in a single momentum state and a stripe phase with condensate in two opposite momenta Wang (); Ho (); Wu (). While for the realized SO-coupled BEC with equal contributions of Rashba and Dresselhaus SO couplings Lin2011 (); Williams (); Zhang2012 (); Qu (); LeBlanc (); Olson (); Zhang2014 (), it is found that the ground state belong to the two phases (the stripe phase and the plane wave phase) or the conventional zero-momentum phase, and thus an interesting tri-critical point was predicted in the phase diagram Li (). Taking the finite temperature Jian (), a harmonic trap Hu (); Xu (); Ozawa () or a rotating trap Sinha (); Zhou (); Radic (); Xu2 () into account, more exotic phases were revealed, such as half vortices and Skyrmion lattices. Loading the SO-coupled bosons into optical lattices, many novel magnetic ground states were also found Lewenstein (); Zhu2013 (); Zhao2014 ().

The dynamics of SO-coupled BECs has also been studied in different contexts Zhang2012 (); Duine (); Tokatly (); Natu (); Josephson (); Zhang (); KT (). For example, they have been demonstrated to exhibit unconventional collective dipole oscillations Zhang2012 (); Duine () and spin dynamics Tokatly (); Natu (), interesting spin Josephson effects Josephson () and relativistic dynamics with analogs of Zitterbewegung Zhang () and Klein tunneling KT () under certain conditions. For the realized Rashba-Dresselhaus SO-coupled BEC, if the Raman coupling is driven in time, the system will pass through phase transition points. However, such Raman-driving dynamics in this system is yet to be explored.

On the other hand, the interatomic interactions in BECs lead to inherent nonlinearity under the Gross-Pitaevskii mean-field description. Thus the dynamics of a BEC is governed by a nonlinear Schrödinger equation, where the interaction strength could be tuned by Feshbach resonances Chin (). With the tunable nonlinearity, the dark solitons Fialko (); Achilleos1 (), bright solitons Achilleos2 (); yongXu (), and gap solitons Kartashov () in SO-coupled BECs are also investigated recently. Interestingly, the dark solitons can be excited by other methods. One can engineer a phase difference in a condensate to create a dark soliton at the interface between the phase domains Burger () or merge two coherent condensates to create multiple dark solitons Weller (). Other schemes of generating dark solitons involve driving the system away from equilibrium Zurek (); Damski (), such as using nonlinearity-assisted quantum tunneling Lee (). Thus, a natural question is whether we can dynamically generate the solitons in the realized SO-coupled BEC by the Raman-driving.

In this paper, we numerically investigate the Raman-driving dynamics and the nonlinear excitations of the realized SO-coupled BEC in a one-dimensional (1D) harmonic trap. By using imaginary time evolution method, we first obtain the ground state of the system and identify the three quantum phases depending on the Raman coupling strength and the interatomic interactions. Within the external trapping potential, we find a narrow parameter regime with coexistence of stripe and zero-momentum or plane wave phases in real space, which is absent in the homogenous case Li (). With the operator-splitting procedure, we then simulate four different sweeps across the ground-state phases by driving the Raman coupling linearly in time. In these sweeps, we study the non-equilibrium dynamics of the system and obtain different nonlinear excitations. Interestingly, we find that dark solitons can be excited within the trap in the sweep from the stripe phase to the plane wave or zero-momentum phase. Moreover, we show that the number and the stability of the dark solitons can be controlled in the dynamical progress, providing a direct and easy way to create dark solitons and study their dynamics and interaction properties.

The remaining part of this paper is organized as follows. We introduce the model for a SO-coupled BEC in a harmonic trap in Sec. II. The ground-state properties of the system are numerically studied in Sec. III. Then we investigate the Raman-driving dynamics and excitations in the sweeps across different ground-state phases in Sec. IV. Finally, a short summary is given in Sec. V.

## Ii The model

We consider a SO-coupled BEC confined in a quasi-1D harmonic trap with equal strength of Rashba and Dresselhaus SO couplings, which has been realized with Rb atoms Lin2011 (); Williams (); Zhang2012 (); Qu (); LeBlanc (); Olson (); Zhang2014 (). In the mean-field approach, the energy functional of the system is , with the energy density

(1) |

where with and being the two (pseudo-) spin wave functions of the BEC, and the effective 1D interaction constants are defined by the s-wave scattering lengths with , the particle number and the oscillator length associated with a harmonic vertical confinement . Hereafter we assume and for simplicity. The single particle Hamiltonian in Eq. (1) is given by Lin2011 ()

(2) |

where kg is Rb atomic mass, is the momentum operator, is the wave number of the Raman lasers for coupling the two hyperfine states, are Pauli matrices, is the Raman coupling strength, is an effective Zeeman field for the spin states, and is the external trapping potential with frequency . We assume the typical trapping frequency Hz, such that m. We further assume , , and the total number of atoms . Then the interaction strengths and are on the range from several tens to several hundreds of for typical scattering lengths.

Measuring the length in units of , time in units of , energy in units of , we derive the dimensionless equations of motion for wave functions Achilleos1 ():

(3) | |||||

(4) | |||||

Here we have used , , and , thus now .

In the following, we numerically minimize the energy functional to obtain the ground-state wave functions of the system described by Eq. (1) and Eqs. (3,4) for varying Raman coupling strength using the imaginary time evolution method. Then in the next section, we simulate the Raman-driving dynamics across different ground-state phases by integrating the equations of motion with the well-developed operator-splitting procedure. In the following numerical calculations, we will mainly consider for simplicity.

In simulations, we fix and mainly consider two typical groups of interaction parameters: and as the first case, and as the second case. The reason for our choice is that they respectively give rise to the three and two ground-state phases and meanwhile within the typical experimental parameter range. However, we note that the ground-state properties and the Raman-driving dynamics with excited dark solitons, which will be discussed in the following two sections, mostly remain for different choices of interaction parameters. In experiments, the values of and may be directly changed by varying the transverse trapping strength, but the ratio is fixed in this way.

## Iii Ground states

The system in the uniform case () has been well studied in Ref. Li (), and it is analytically found that if the SO-coupling dominates and the condition is satisfied with , there are three different phases with interesting condensate states for varying Raman coupling strength Li (). However, if the atomic interactions dominate and the condition is not satisfied, there are two condensate states Li (). Even though the boundary condition can not be solved analytically in the presence of a trapping potential, our simulations demonstrate that the two cases also appear. The numerical results of the ground states are shown in Fig. 1 and Fig. 2, corresponding to the two groups of interaction parameters, respectively. Here we plot the total density for the ground state wave functions and the difference density between two spin components (spin polarization) in real space as a function of Raman coupling strength, with

(5) |

We also plot the total density in the momentum space , where the wave function . We will see that , and together characterize different condensate states, with the central momentum of the condensate and the spin polarization being the order parameters Li ().

We first consider the case that the SO-coupling dominates with , and . As shown in Fig. 1, three ground-state phases are respectively denoted by phases I, II and III with increasing , and their boundaries are marked by the red solid lines. Phase I is the stripe phase for relatively small , characterized by fringes in the density in Fig. 1(a). In this phase, the atoms condense in an equal-probability superposition of two plane waves with opposite wave vectors as shown in Fig. 1(c). Besides, the spin polarization identically vanishes, leading to in Fig. 1(b). It is interesting to note that if , the spin symmetry will be broken and then the phase separation of the ground-state BEC can appear in the regime of phase I, which has been observed in experiments Lin2011 (); Zhang2012 (). We have numerically confirmed the appearance of phase separation for finite and above a critical Raman coupling strength in the regime of Phase I, which is not shown in Fig. 1 for the case.

Phase II is the plane wave phase for intermediate , characterized by a single non-zero momentum peak in and finite spin polarization. Phase III is the zero-momentum phase for relatively large , where the central momentum of the condensate is zero and the spin polarization also vanishes. The phase transition between I and II belongs to the first order phase transition with a jump in the central momentum as shown in the insert figure in Fig. 1(c). While the one between II and III belongs to the second order phase transition. In all of the three phases, the overall density configuration exhibits a Gaussian distribution due to the external harmonic trap, which also demonstrates that the local density approximation works well here.

For the case that the interactions dominate with , and , the results of the ground states are shown in Fig. 2. In this case, we can see that there are only phases I and III. The phase transition between I and III belongs to the first order phase transition. Interestingly, we find the coexistence of the two phases in a small region near the critical Raman coupling strength, characterized by Gaussian density distribution in the trap center with stripe density in the edge as shown in Fig. 2(a). This phenomenon is due to the inhomogeneous interaction effect by the trapping potential, which is absent in the homogenous case Li (). In the local density approximation, the density distribution of atoms is inhomogeneous and decrease away from the center of the trap, leading to the decreasing interaction strength from the center to the edge. Since the interactions play an crucial role in this case and the critical Raman coupling strength between phases I and III depends on the interaction strength, the central part of the condensate will enter into the phase III earlier than the edge when increasing Raman coupling.

One may also expect a coexistence of stripe and plane wave phases in real space from the same mechanism, which is absent for the case in Fig. 1 due to the weak interactions. Actually, for appropriately intermediate interaction strengths, the I-II coexistence phase is indeed found in our numerical simulations. An example is shown in Fig. 2(c) for and , where the I-II coexistence phase is characterized by Gaussian density distribution at the edge with stripe density in the center. This result further demonstrates that the two coexistence phases are not always present in the whole interaction parameter regime and actually from the competitions between repulsive interactions and external trapping.

## Iv Raman-driven dynamics and generation of dark solitons

In this section, we consider the quantum dynamics and excitations of the SO-coupled BEC in the sweeps between three ground-state phases. To do this, the system is initially prepared in one of its ground states, and is then driven across the phase transition points by the Raman-driving, which can be achieved by changing the intensity of the Raman lasers in experiments. We consider that the Raman coupling strength varies linearly in time :

(6) |

where is the Raman coupling strength at and is the Raman-driving rate. The ground states depend on , and , with the phase boundaries shown in Figs. 1 and 2. In the following, we numerically simulate these sweeping progresses by integrating the equations of motion (3) and (4).

### iv.1 Dark solitons in I-II and I-III sweeps

We first consider the sweep from phase I to phase II with typical interaction strengths and . At the outset the Raman coupling , and the atoms condense into the stripe ground state within the region of phase I, which is obtained numerically by the imaginary time evolution method. Then the Raman coupling strength is increased as the form given by Eq. (6). The numerical results of the Raman-driving dynamics for some typical sweeping rates are shown in Fig. 3, where we plot the density evolution of the SO-coupled BEC . Here the two horizontal red lines denote the phase boundaries. For slow sweep with as shown in Fig. 3(a), the BEC begins to oscillate after passing through the first phase transition point between I and II and then gradually exhibits breathing oscillation. The amplitude and frequency of the breathing oscillation characterize the propagation of excitation modes, which are different from the sound modes in this system when subjected to a perturbation driving Li (). We have also checked that in the very slow sweeping limit, the system will evolve adiabatically without collective excitations (which is not shown here).

When the sweep is fast enough with , we find that a pair of dark solitons exhibit in the BEC after passing through the phase transition point as shown in Fig. 3(b). The dark solitons are generated in the center because the nonlinear interactions are stronger there within the external harmonic trap. Due to the harmonic force, the solitons are reflected from the edges of the trap and collide in the center again and again, giving rise to the oscillation behavior of the dark solitons and interference pattern in Fig. 3(b). When increasing the sweep rate with as shown in Figs. 3(c,d), more and more dark soltions are excited in the system with similar oscillation dynamics and interference pattern.

We then consider the sweep from phase I to phase III with typical interaction strengths and , which is also the first order phase transition. We also set the system initially in the ground state with and the Raman coupling is increased linearly in time. The numerical results of the evolution of atomic density are shown in Fig. 4 for different sweep rates. For very slow sweep with as shown in Fig. 4(a), neither dark solitons nor breathing modes are excited in the BEC in this case. While for in Fig. 4(b), we find that more than one pair of dark solitons are generated compared to the I-II sweep with the same sweeping rate in Fig. 3(a). This is due to the facts that the stronger atomic interaction in this case leads to more density dips for the dynamical generation of dark solitons. In this case, also more and more dark solitons are excited with similar oscillation and interference dynamics when increasing the sweep rate with as shown in Figs. 4(c,d). In a word, the dark solitons are more easily excited in the I-III sweep than in the I-II sweep.

We should note that the boundaries among the three ground-state phases and the corresponding spin polarization will be modified for the case , which corresponds to broken spin symmetry Li (). However, we have numerically confirmed that the dark solitons can also be created in these non-adiabatic sweeps for . Moreover, this way of generation of dark solitons is not specific to the particular choices of interaction and SO-coupling strengths as those in Figs. 3 and 4. As long as the sweeps are from the stripe ground state and fast enough, the dark solitons will be excited and exhibit oscillatory motion and collision in the trap. In contract, the existence of solitons in SO-coupled BECs in previous work closely relies on certain particular conditions, such as ring-shape trapping potentials Fialko (), stationary points in the dispersion relation Achilleos1 (), attractive interactions Achilleos2 (); yongXu () and spatially periodic Zeeman fields Kartashov (). Therefore, our work demonstrated a general method of dynamically generating solitons in SO-coupled BECs, which is easy to be implemented in current experiments without carefully designing the Raman lasers and atomic interactions.

### iv.2 Formation, stability and dynamics of the excited dark solitons

In this part, we further discuss the formation, stability and dynamics of the excited dark solitons in the sweeps. In experiments, one could use the phase imprinting and density engineering methods to create dark solitons in an ordinary BEC Burger (); Weller (). However, the different mechanism of dynamically generating dark solitons here is by sweeping the BECs away from equilibrium stripe ground states and across phase boundaries Zurek (); Damski (). In a harmonically trapped condensate with repulsive interatomic interactions, the nodes of the excited nonlinear eigenstates can evolve into the dark solitons Lee (); Kivshar ().

For the stripe condensate, there are many dips in its density distribution to support the formation of dark solitons when the system is driven fast enough from phase I to phase II, as shown in Figs. 3(b-d). To see this point more clearly, we first plot the spin-up-component density and the phase of the stripe ground state in Fig. 5(a). Even though the phase is relatively smooth in space, the maximum phase variation locates at the density dips, which is favorable to form dark solitons. The situations of the spin-down component are similar. We then plot the spin density and the phase of the excited dark solitons [when in Fig. 3(b)] in Fig. 5(b), where the density dips and the accompanied phase jumps of the two dark solitons can be seen clearly.

To verify the stability of the generated dark solitons in Fig. 6, we calculate the long-time evolution of atomic density after the driving time , corresponding to and the same parameters in Fig. 3(b). Figures 6(a) and 6(b) correspond to the cases of keeping on and tuning off the trap when the driving is stopped at , respectively. It is clear from Fig. 6(a) that the dark solitons remain periodically oscillating in the trap after stopping driving, without significant decay for a long time. Therefore we can conclude that the dark solitons in this system are stable within the trapping potential. If the trap is also turned off with the driving, the dark solitons will propagate and spread out, which is shown in Fig. 6(b).

For an equilibrium SO-coupled BEC without driving, the analytical single-dark-soliton solution of Eqs. (3,4) can be approximately obtained as the forms in Ref. Achilleos1 (). However, for the non-equilibrium system with multiple interacting dark solitons here, it is too complicated to get the analytical forms and derive an effective model to exactly describe their dynamics as shown in Figs. 3 and Fig. 4. In Ref. Achilleos1 (), the size of a single dark soliton is characterized to be proportional to an effective healing length, which is determined by the atomic interaction strengths and the stationary points in the dispersion relation. In numerical simulations for our case, we can not find a clear and direct connection between the size of excited multiple dark solitons and the (effective) healing length. For the simplest two-soliton case in Figs. 3(b) and 6(a), the dark solitons clearly show particle-like behaviors with oscillatory motion and mutually interactions within the external trap, which has been observed in ordinary BECs Weller (). For well-separated dark solitons, their center approximately satisfies the equation of motion in the form , where is the effective mass of the dark solitons Achilleos1 () and is the oscillation frequency determined by the harmonic trap Burger (); Weller (). Due to the interactions between dark solitons, which can be treated by additional potentials, the oscillation frequencies of interacting are modified Weller (). In addition, as the non-adiabatic driving does, the motion and collision of multiple dark solitons gradually generate excitation modes in the SO-coupled BEC, resulting in the complicated breathing-like behavior in the background as shown in Figs. 3(c,d) and 4(c,d). Clearly, the oscillation amplitude of the multiple dark solitons does not remain constant in this case.

### iv.3 Dynamics in sweeps between phases II and III

Now we turn to consider the sweeps between phase II and phase III. The first case we considered is the sweep from phases II to III, which belongs to the second order phase transition. To do this, we set the system initially in phase II with and choose the same interaction parameters as those in Fig. 3: and . Then the Raman coupling is also increased linearly in time. In this case, the collective center-of-mass (COM) motion in and the periodical oscillation of the spin polarization in can emerge, with two examples being shown in Figs. 7(a,d) and 7(b,e) for and , respectively. It is clear to see from Fig. 7 that for very slow sweep (), the system almost follows its ground state in the beginning and then exhibits the dynamical oscillation once passing through the phase transition point. In addition, the period and the amplitude of the oscillation are both increased by increasing the Raman driving rate. However, we find that the dark solitons can not be excited by the Raman-driving even in the fast sweep, such as in Figs. 7(c,f), which just exhibit similar oscillating COM motion and spin dynamics. We have also checked that no dark soliton can be excited in a much faster sweep in this case. This is due to the absence of density dips in the initial plane-wave state to evolve into dark-soliton excitations. Since the dark solitons can be excited only from the stripe state, the appearance of dark solitons in the Raman driving is an indication of the existence of stripe state, which has not directly been imaged in experiments Lin2011 (); Zhang2012 ().

To further study the COM motion of the system, we calculate the corresponding population evolution of the two spin components as and the COM evolution as

(7) |

As shown in Fig. 8(a,b), the atoms initially condense mainly in one of two spins components, then their populations tend to be equal by the Raman driving and finally exhibit periodical oscillation after passing through the II-III phase transition point. However, both of the period and the amplitude of the spin-population oscillation decrease in time. As for the COM motion shown in Fig. 8(c,d), the oscillation appears in the beginning even with very slow sweep [such as in Fig. 8(c)], but its amplitude is very small because the system almost follows its ground state. We note that the oscillation in this case is similar to the well-known Zitterbewegung, which has been observed in the SO-coupled BECs Qu (); LeBlanc (). After passing through the transition point, the oscillating COM motion recovers and the oscillation amplitude is greatly enlarged and can be further increased by increasing the Raman-driving rate.

At the end of this part, we consider the inverse sweep, i.e., the sweep from phase III to phase II. The calculated density evolutions are shown in Fig. 9, where the initial state of the system is in the phase III regime with and the Raman coupling is decreased linearly in time with the rates [Figs. 9(a,d)], [Figs. 9(b,e)] and [Figs. 9(c,f)]. In these cases, the system can follow its ground state with zero spin polarization at the beginning. As shown in Fig. 9, the longitudinal spin polarization of the system becomes finite and finally the excitations appear in the spin density with periodic oscillation after the system passes through the III-II phase boundary, and the excitations increase and the spin oscillation becomes more complicated when speeding up the driving. In this sweep, there is also no excited dark soliton for the same reason.

## V Conclusions

In conclusion, we have numerically studied the ground states of a 1D SO-coupled BEC in a harmonic trap and the Raman-driving dynamics across different ground-state phases. We found the coexistence of stripe and zero-momentum or plane wave phases in real space due to the the competitions between repulsive interactions and external trapping. We also showed a new method of dynamical generation of dark solitons in the Raman-driving by sweeping the BEC through the phase transition points. Both of the number and the stability of the excited dark solitons can be controlled in this direct and convenient way. Therefore, this system may provide a controllable platform for creating dark solitons and studying their dynamics and interaction properties. In view of the fact that the investigated SO-coupled BEC with adjustable Raman coupling strength has been realized by several experimental groups Lin2011 (); Williams (); Zhang2012 (); Qu (); LeBlanc (); Olson (); Zhang2014 (), it is anticipated that our predictions in this work can be tested in an experiment in the near future.

###### Acknowledgements.

We thank Profs. Shi-Liang Zhu, Chaohong Lee and Biao Wu for many helpful discussions. This work was supported by the NSF (Grant No. 11404108), the PCSIRT and the SRFGS of SCNU. D. W. Z. acknowledges support from the postdoctoral fellowship of HKU.## References

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