Spin-orbit Coupled Bose-Einstein Condensate Research Articles

Effective spin dynamics of spin-orbit coupled matter-wave solitons in optical lattices

We consider matter–wave solitons in spin–orbit coupled Bose–Einstein condensates embedded in an optical lattice and study the dynamics of the soliton within the framework of Gross–Pitaevskii equations. We express spin components of the soliton pair in terms of nonlinear Bloch equations and investigate the effective spin dynamics. It is seen that the effective magnetic field that appears in the Bloch equation is affected by optical lattices, and thus the optical lattice influences the precessional frequency of the spin components. We make use of numerical approaches to investigate the dynamical behavior of density profiles and center-of-mass of the soliton pair in the presence of the optical lattice. It is shown that the spin density is periodically varying due to flipping of spinors between the two states. The amplitude of spin-flipping oscillation increases with lattice strength. We find that the system can also exhibit interesting nonlinear behavior for chosen values of parameters. We present a fixed point analysis to study the effects of optical lattices on the nonlinear dynamics of the spin components. It is seen that the optical lattice can act as a control parameter to change the dynamical behavior of the spin components from periodic to chaotic.

Transfer of solitons and half-vortex solitons via adiabatic passage

We show that the transfer of matter-wave solitons and half-vortex solitons in a spin-orbit coupled Bose-Einstein condensate between two (or more) arbitrarily chosen sites of an optical lattice can be implemented using the adiabatic passage. The underlying linear Hamiltonian has a flat band in its spectrum, so that even sufficiently weak interatomic interactions can sustain well-localized Wannier solitons which are involved in the transfer process. The adiabatic passage is assisted by properly chosen spatial and temporal modulations of the Rabi frequency. Within the framework of a few-mode approximation, the mechanism is enabled by a dark state created by coupling the initial and target low-energy solitons with a high-energy extended Bloch state, like in the conventional stimulated Raman adiabatic passage used for the coherent control of quantum states. In real space, however, the atomic transfer between initial and target states is sustained by the current carried by the extended Bloch state, which remains populated during the whole process. The full description of the transfer is provided by the Gross-Pitaevskii equation. Protocols for the adiabatic passage are described for one- and two-dimensional optical lattices, as well as for splitting and subsequent transfer of an initial wave packet simultaneously to two different target locations. Published by the American Physical Society 2024

Theory-guided neural network for studying the ground state of 2D spin-orbit coupled Bose–Einstein condensates

Machine learning has been a powerful tool to study various models in many fields, which requires plenty of data to ensure accuracy. It is a challenge to reduce the dependence on the amount of data and ensure that the predictions are both accurate and physical. We develop a theory-guided neural network (TgNN) to explore the ground states of two-component spin–orbit coupled Bose–Einstein condensation in two-dimension. Only randomly selected few data from the twelve ground states as training data, TgNN performs higher accuracy, broader generalization ability and stronger robustness than deep neural network (DNN). Moreover, in the neighborhood of the phase transition where the predictions of DNN have obvious deviations, while TgNN still keeps relative accuracy. TgNN is less dependent on data and exhibits potential for high-dimensional and complicated models.

Gap solitons of spin–orbit-coupled Bose–Einstein condensates in a Jacobian elliptic sine potential

Gap solitons of quasi-one-dimensional Bose–Einstein condensate loaded in a Jacobian elliptic sine potential with spin–orbit-coupled are investigated theoretically. Under the mean-field approximation, the dynamical behaviors of such a system are described using the Gross–Pitaevskii equation (GPE). Firstly, we linearize the GPE to obtain the band-gap structures. Secondly, the Newton-Conjugate-Gradient (NCG) method is used to search for gap solitons. Thirdly, the dynamic stability of the gap solitons obtained by the NCG method was investigated using linear stability analysis and nonlinear dynamic evolution methods. It is found that in this nonlinear system, there exist both stable gap solitons and structurally rich unstable gap solitons. The external potential modulus and the strength of the spin–orbit coupling are important parameters that influence the stability of the gap solitons.

Stable stripe and vortex solitons in two-dimensional spin-orbit coupled Bose–Einstein condensates

We present a flexible manipulation and control of solitons via Bose–Einstein condensates. In the presence of Rashba spin–orbit coupling and repulsive interactions within a harmonic potential, our investigation reveals the numerical local solutions within the system. By manipulating the strength of repulsive interactions and adjusting spin–orbit coupling while maintaining a zero-frequency rotation, diverse soliton structures emerge within the system. These include plane-wave solitons, two distinct types of stripe solitons, and odd petal solitons with both single and double layers. The stability of these solitons is intricately dependent on the varying strength of spin–orbit coupling. Specifically, stripe solitons can maintain a stable existence within regions characterized by enhanced spin–orbit coupling while petal solitons are unable to sustain a stable existence under similar conditions. When rotational frequency is introduced to the system, solitons undergo a transition from stripe solitons to a vortex array characterized by a sustained rotation. The rotational directions of clockwise and counterclockwise are non-equivalent owing to spin–orbit coupling. As a result, the properties of vortex solitons exhibit significant variation and are capable of maintaining a stable existence in the presence of repulsive interactions.

Anderson localization and quenched induced dynamics of spin orbit coupled Bose–Einstein condensate in a random potential

We consider the localization and dynamical properties of a one dimensional spin orbit coupled Bose–Einstein condensate trapped by a disordered speckle potential. We numerically solve coupled Gross–Pitaevskii equation to obtain ground sate solutions. The effects of spin–orbit coupling and detuning parameter on localization are investigated. It is found that the increase of spin–orbit coupling delocalizes the condensate while the increase of detuning favors localization. After achieving the numerical ground state solutions, we examine the quench induced dynamics of the condensate by the complete cessation of the spin–orbit coupling. We show that at parameters where the ground state is not localized, the dynamics of the system is chaotic.

Kármán vortex street in a spin–orbit-coupled Bose–Einstein condensate with PT symmetry

The dynamics of spin–orbit-coupled Bose–Einstein condensate with parity-time symmetry through a moving obstacle potential is simulated numerically. In the miscible two-component condensate, the formation of the Kármán vortex street is observed in one component, while ‘the half-quantum vortex street’ is observed in the other component. Other patterns of vortex shedding, such as oblique vortex dipoles, V-shaped vortex pairs, irregular turbulence, and combined modes of various wakes, can also be found. The ratio of inter-vortex spacing in one row to the distance between vortex rows is approximately 0.18, which is less than the stability condition 0.28 of classical fluid. The drag force acting on the obstacle potential is simulated. The parametric regions of Kármán vortex street and other vortex patterns are calculated. The range of Kármán vortex street is surrounded by the region of combined modes. In addition, spin–orbit coupling disrupts the symmetry of the system and the gain-loss affects the local particle distribution of the system, which leads to the local symmetry breaking of the system, and finally influences the stability of the Kármán vortex street. Finally, we propose an experimental protocol to realize the Kármán vortex street in a system.

Compressing the atomic cloud in a matter-wave stripe soliton

We consider an attractive quasi-one dimensional spin-orbit coupled Bose–Einstein condensate (SOC BEC) confined in a periodic potential produced by the combination of linear and nonlinear optical lattices, and study the effects of squeezing a stripe soliton by varying the inter-atomic interaction in the nonlinear lattice. It is observed that the nodes on the soliton arising entirely due to the effect of spin–orbit coupling tend to disappear as we increase the squeezing effect to finally get a stable fundamental soliton. This leads us to conclude that external pressure can reduce the effect of spin–orbit coupling in the SOC BEC and even convert the system to a traditional BEC without spin–orbit coupling. We make use of an information theoretic measure to visualize how does the atomic density distribution in the condensate respond to continual reduction in the spin–orbit coupling effect.

The dynamical stripes in spin–orbit coupled Bose–Einstein condensates with Josephson junctions

The Josephson dynamics of the Bose–Einstein condensation with Raman-induced spin–orbit coupling is investigated. A quasi-1D trap is divided into two reservoirs by an optical barrier. Before the tunneling between the reservoirs is turned on, the system stays in its equilibrium ground state, which displays a rich phase diagram for different spin–orbit coupling parameters and interaction strengths. In this work we focus on the plane wave phase and the stripe phase. Our calculation shows that, when the tunneling is turned on, the plane wave phase evolves into a dynamical stripe phase, that is, the density of the particle changes from uniform to periodically modulated. Basically, this stripe is described by a sine function and the wave length, the amplitude and the initial phase of the function are all varying with time. If the system stays in stripe phase initially, the stripes become “sliding” when the tunneling is turned on, which reflects the running of one of the phases of the wave function.

Emergence of a Spin Microemulsion in Spin-Orbit Coupled Bose-Einstein Condensates.

We report the first numerical prediction of a "spin microemulsion"-a phase with undulating spin domains resembling classical bicontinuous oil-water-surfactant emulsions-in two-dimensional systems of spinor Bose-Einstein condensates with isotropic Rashba spin-orbit coupling. Using field-theoretic numerical simulations, we investigated the melting of a low-temperature stripe phase with supersolid character and find that the stripes lose their superfluidity at elevated temperature and undergo a Kosterlitz-Thouless-like transition into a spin microemulsion. Momentum distribution calculations highlight a thermally broadened occupation of the Rashba circle of low-energy states with macroscopic and isotropic occupation around the ring. We provide a finite-temperature phase diagram that positions the emulsion as an intermediate, structured isotropic phase with residual quantum character before transitioning at higher temperature into a structureless normal fluid.

Multi-stability in cavity QED with spin–orbit coupled Bose–Einstein condensate

Multi-stability in cavity QED with spin–orbit coupled Bose–Einstein condensate

Patterning by dynamically unstable spin–orbit-coupled Bose–Einstein condensates

In a two-dimensional atomic Bose–Einstein condensate, we demonstrate Rashba spin–orbit coupling can always introduce dynamical instability into specific zero-quasimomentum states in all parameter regimes. During the evolution of the zero-quasimomentum states, such spin–orbit-coupling-induced instability can fragment the states and lead to a dynamically patterning process. The features of formed patterns are identified from the symmetries of the Bogoliubov–de Gennes Hamiltonian. We show that spin–orbit-coupled Bose–Einstein condensates provide an interesting platform for the investigation of pattern formations.

Induced position-dependent spin–orbit coupling in atomic Bose–Einstein condensate

In this paper, we study-induced spin–orbit coupling (SOC) in one-dimensional ultracold quantum gases composed of atoms of different species. One species is subjected to an equal combination of Rashba and Dresselhaus SOC generated by Raman transition. The two species interact with each other through spin-independent and spin-exchange contact interactions. The spin-exchange interaction introduces a locking pattern of the momentum and spin degrees of freedom for the species without direct SOC. For small spin-independent interactions, the two species overlap and the induced SOC is effective. For large spin-independent interactions, however, the two species keep separate from each other and the induced SOC is negligible. We propose to generate inhomogeneous SOC by exploiting density engineering technique applied to the directly spin–orbit coupled Bose–Einstein condensate. When the effective potential is of Gaussian type, the single-particle eigenenergies and eigenstates are calculated by exact diagonalization method. For general cases, mean-field ground states are obtained by numerically searching for the minimum of energy functional. With the density engineering technique, it is possible to produce hybrid structures of plane wave, stripe and/or zero-momentum phases.

Metastable supersolid in spin-orbit-coupled Bose-Einstein condensates

Supersolid is a special state of matter with both superfluid properties and spontaneous modulation of particle density. In this paper, we focus on the supersolid stripe phase realized in a spin-orbit coupled Bose-Einstein condensate and explore the properties of a class of metastable supersolids. In particular, we study a one-dimensional supersolid whose characteristic wave number $k$ (magnitude of wave vector) deviates from $k_{m}$, i.e., the one at ground state. In other words, the period of density modulation is shorter or longer than the one at ground state. We find that this class of supersolids can still be stable if their wave numbers fall in the range $k_{c1}<k<k_{c2}$, with two thresholds $k_{c1}$ and $k_{c2}$. Stripes with $k$ outside this range suffer from dynamical instability with complex Bogoliubov excitation spectrum at long wavelength. Experimentally, these stripes with $k$ away from $k_m$ are accessible by exciting the longitudinal spin dipole mode, resulting in temporal oscillation of stripe period as well as $k$. Within the mean-field Gross-Pitaevskii theory, we numerically confirm that for a large enough amplitude of spin dipole oscillation, the stripe states become unstable through breaking periodicity, in qualitative agreement with the existence of thresholds of $k$ for stable stripes. Our work extends the concept of supersolid and uncovers a new class of metastable supersolids to explore.

Fundamental and multipole gap solitons in spin-orbit-coupled Bose-Einstein condensates with parity-time-symmetric Zeeman lattices

Our results demonstrate that fundamental and tripole matter-wave gap solitons can be stable in spin-orbit-coupled Bose-Einstein condensates (SOC-BECs) in one-dimensional (1D) parity-time (PT)-symmetric Zeeman lattices with attractive interactions. The solitons are found to be in a semi-infinite gap and both the components are complex. The real and imaginary parts of two components of these solitons respectively are even and odd functions. The peak numbers of the second component are twice that of the first component. We analyze the linear stability for the matter-wave gap solitons and the correctness is confirmed by the soliton evolutions. Moreover, the strength of spin-orbit (SO) coupling cannot change the critical threshold of the PT-symmetric Zeeman lattices.

Bright–dark Peregrine solutions in helicoidal spin–orbit coupled Bose–Einstein condensates

Vector Gross–Pitaevskii system in helicoidal spin–orbit coupled Bose–Einstein condensates can be reduced into vector nonlinear Schrödinger system. Employing the nonrecursive Darboux method, we find bright–dark Peregrine solutions, such as first-order and second-order vector Peregrine solutions, including vector Peregrine “three sisters” solutions. These results will provide a deeper understanding of nonlinear wave phenomena in helicoidal spin–orbit coupled Bose–Einstein condensates.

Anharmonicity-induced phase transition of spin–orbit coupled Bose–Einstein condensates

In the mean-field framework, using variational analysis and numerical simulation, we investigate the effect of anharmonic trap and atomic interaction on the ground-state phases of spin-orbit (SO) coupled Bose–Einstein condensates (BECs) in the harmonic plus quartic potential. Then, the Gaussian wave function is selected to predict the analytical conditions of the phase transition boundary of the SO coupled BECs by using the variational method. We found that the anharmonicity of the external potential induces the SO coupled BECs to undergo a phase transition between the zero-momentum phase and plane-wave phase, which is more pronounced in the cases of weak harmonic potential or strong interspecies interaction. Since the potential energy of the system modified by anharmonicity competes with other energies of the system, the anharmonicity changes the critical SO coupling strength and Raman coupling strength when the phase transition occurs. At the same time, the critical anharmonic coefficients are also affected by interspecies interaction and harmonic potential. Finally, the correctness of the theoretical results is verified by numerical simulation of the Gross-Pitaevskii equation.

Adiabatic elimination theory of multi-level system in spin-orbit coupled Bose-Einstein condensate

In quantum optics, adiabatic elimination simplifies multi-level quantum system by eliminating the fast oscillatory degree of freedom and preserving the slow-varying dynamics, thus obtaining an efficient description of the system. Adiabatic elimination has important applications in quantum simulation and quantum precision measurement. For example, spin-orbit coupling has been realized in ultracold atoms by using three-level Raman coupling and adiabatic elimination. In this paper, we investigate the theoretical method and generalize the adiabatic elimination in three-level non-Hermitian systems and multi-level systems on the basis of standard elimination scheme. These can provide theoretical guidance for realizing the interdiscipline of non-Hermitian physics and spin-orbit coupling effects and their potential applications. We mainly discuss the influences of dissipative effect on the population dynamics of the system, the validity and accuracy of the adiabatic elimination theory under different parameters for both non-Hermitian and two types of five-level systems. Specifically, the dynamics satisfying the large detuning condition gives very accurate results for quite a long evolution time with the adiabatic elimination theory, but when the two-photon detuning &lt;i&gt;δ&lt;/i&gt; and the Rabi frequency &lt;inline-formula&gt;&lt;tex-math id="Z-20231017181120"&gt;\begin{document}$\varOmega $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20231052_Z-20231017181120.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20231052_Z-20231017181120.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; gradually increase, leading to the violation of the large detuning condition &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \varOmega,\gamma, \delta \ll \Delta$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20231052_M1.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20231052_M1.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, the effective two-level model can no longer describe the fast-varying dynamics of the system even in a short evolution time. Thus the choice of system parameters affects the effectiveness of adiabatic elimination of the excited levels. In a non-Hermitian system, the population in the ground state oscillates with gain periodically at the beginning, while that in the ground state oscillates with loss and decreases with time, with the total population decreasing with oscillation. For long-time evolution the gain in the system causes the population to diverge, and the adiabatic elimination of the effective two-energy level system describes this behavior accurately. The effect of the non-Hermitian parameters on the dynamics of the system in the resonance case is manifested in the case that the total population remains conserved, while the total population tends to diverge for finite two-photon detuning. We find that with the increase of detuning, the divergence appears earlier and the total number of particles can be kept constant by choosing the ratio of gain to loss appropriately. This study provides a theoretical basis for state preparation and dynamical manipulation in dissipative multi-energy quantum systems.

Quantum droplets with topological textures in a dipolar condensate

We investigate a spin–orbit coupled Bose–Einstein condensate containing both dipolar and non-dipolar components. Combined with Rashba spin–orbit coupling, the competition between contact and dipolar interaction reveals four distinctive exotic phases: droplet lattice phase, stripe phase, oblique stripe phase, and labyrinthine stripe phase. In particular, we adopt a condensate order parameter ansatz to characterize the superstripe phase in a harmonic trap through minimizing the analytical total energy. Furthermore, we find that the energy analysis of the dipolar condensate in the trap uncovers a higher order transition from lattice to stripe phases when contact interaction increases. Our results demonstrate the possibility for physicists to realize a wealth of controllable configurations in spin–orbit coupled dipolar condensates, including droplet lattice with topology and superstripe adjusted by changing the strength of contact and dipolar interactions. The topological phases we predict in this work can be realized by using state-of-the-art techniques in ultracold atomic experiments of chromium, erbium, or dysprosium condensates.

Spin Josephson effects of spin–orbit-coupled Bose–Einstein condensates in a non-Hermitian double well

In this paper, we investigate the spin and tunneling dynamics of a spin–orbit-coupled noninteracting Bose–Einstein condensate in a periodically driven non-Hermitian double-well potential. Under high-frequency driving, we obtain the effective time-averaged Hamiltonian by using the standard time-averaging method, and analytically calculate the Floquet quasienergies, revealing that the parity-time ()-breaking phase transition appears even for arbitrarily small non-Hermitian parameters when the spin–orbit coupling strength takes half-integer value, irrespective of the values of other parameters used. When the system is -symmetric with balanced gain and loss, we find numerically and analytically that in the broken -symmetric regions, there will exist the net spin current together with a vanishing atomic current, if we drop the contribution of the exponential growth of the norm to the current behaviors. When the system is non--symmetric, though the quasienergies are partial complex, a stable net spin current can be generated by controlling the periodic driving field, which is accompanied by a spatial localization of the condensate in the well with gain. The results deepen the understanding of non-Hermitian physics and could be useful for engineering a variety of devices for spintronics.