Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1108.4233 · Physical Review Letters

Spin-Orbit Coupled Weakly Interacting Bose-Einstein Condensates in Harmonic Traps

Hui Hu, B. Ramachandhran, Han Pu, Xia-Ji Liu

Published Jan 4, 2012• 311 citations
arXivHTMLPDF

Abstract

We investigate theoretically the phase diagram of a spin-orbit coupled Bose gas in two-dimensional harmonic traps. We show that at strong spin-orbit coupling the single-particle spectrum decomposes into different manifolds separated by ℏω{⊥}, where ω{⊥} is the trapping frequency. For a weakly interacting gas, quantum states with Skyrmion lattice patterns emerge spontaneously and preserve either parity symmetry or combined parity-time-reversal symmetry. These phases can be readily observed in a spin-orbit coupled gas of ^{87}Rb atoms in a highly oblate trap.

DOI
10.1103/physrevlett.108.010402

Related papers

Half-quantum vortex state in a spin-orbit-coupled Bose-Einstein condensate

B. Ramachandhran, Bogdan Opanchuk, Xia-Ji Liu +3 more

We theoretically investigate the condensate state and collective excitations of a two-component Bose gas in a two-dimensional harmonic trap subject to isotropic Rashba spin-orbit coupling. In the weakly interacting regime when the interspecies interaction is larger than the intraspecies interaction (${g}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}>g$), we find that the condensate ground state has a half-quantum angular momentum vortex configuration with spatial rotational symmetry and skyrmion-type spin texture. Upon increasing the interatomic interaction beyond a threshold ${g}_{c}$, the ground state starts to involve higher-order angular momentum components and thus breaks rotational symmetry. In the case of ${g}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}<g$, the condensate becomes unstable toward the superposition of two degenerate half-quantum vortex states. Both instabilities (at $g>{g}_{c}$ and ${g}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}<g$) can be determined by solving the Bogoliubov equations for collective density oscillations of the half-quantum vortex state and by analyzing the softening of mode frequencies. We obtain the phase diagram as a function of the interatomic interactions and the spin-orbit coupling. In addition, we directly simulate the time-dependent Gross-Pitaevskii equation to examine the dynamical properties of the system. Finally, we investigate the stability of the half-quantum vortex state against both trap anisotropy and anisotropy in the spin-orbit coupling.

arXiv:1201.1471•Physical Review A•Feb 6, 2012•163 citations

Symmetry-protected skyrmions in three-dimensional spin-orbit-coupled Bose gases

Guanjun Chen, Tiantian Li, Yunbo Zhang

We present a variational study of pseudo-spin-$1/2$ Bose gases in a harmonic trap with weak three-dimensional (3D) spin-orbit coupling of $\mathbf{\ensuremath{\sigma}}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ type. This spin-orbit coupling mixes states with different parities, which inspires us to approximate the single-particle state with the eigenstates of the total angular momentum, i.e., superposition of harmonic $s$-wave and $p$-wave states. As the time-reversal symmetry is protected by two-body interaction, we set the variational order parameter as the combination of two mutually time-reversal symmetric eigenstates of the total angular momentum. The variational results essentially reproduce the 3D skyrmionlike ground state recently identified by Kawakami et al. [T. Kawakami, T. Mizushima, M. Nitta, and K. Machida, Phys. Rev. Lett. 109, 015301 (2012)]. We show that these skyrmionlike ground states emerging in this model are primarily caused by the $p$-wave spatial mode involved in the variational order parameter that drives two spin components spatially separated. We find the ground state of this system falls into two phases with different density distribution symmetries depending on the relative magnitude of intraspecies and interspecies interaction: phase I has parity symmetric and axisymmetric density distributions, while phase II is featured with special joint symmetries of discrete rotational and time-reversal symmetry. With the increasing interaction strength the transition occurs between two phases with distinct density distributions, while the topological 3D skyrmionlike spin texture is symmetry protected.

arXiv:1503.00128•Physical Review A•May 26, 2015•14 citations

Magnetic and nematic phases in a Weyl type spin–orbit-coupled spin-1 Bose gas

Guanjun Chen, Li Chen, Yunbo Zhang

We present a variational study of the spin-1 Bose gases in a harmonic trap with three-dimensional spin–orbit (SO) coupling of Weyl type. For weak SO coupling, we treat the single-particle ground states as the form of perturbational harmonic oscillator states in the lowest total angular momentum manifold with j = 1, m j = 1, 0, −1. When the two-body interaction is considered, we set the trail order parameter as the superposition of three degenerate single-particle ground-states and the weight coefficients are determined by minimizing the energy functional. Two ground state phases, namely the magnetic and the nematic phases, are identified depending on the spin-independent and the spin-dependent interactions. Unlike the non-SO-coupled spin-1 Bose–Einstein condensate for which the phase boundary between the magnetic and the nematic phase lies exactly at zero spin-dependent interaction, the boundary is modified by the SO-coupling. We find the magnetic phase is featured with phase-separated density distributions, 3D skyrmion-like spin textures and competing magnetic and biaxial nematic orders, while the nematic phase is featured with miscible density distributions, zero magnetization and spatially modulated uniaxial nematic order. The emergence of higher spin order creates new opportunities for exploring spin-tensor-related physics in SO coupled superfluid.

arXiv:1512.06394•New Journal of Physics•Jun 8, 2016•6 citations

Lattice configurations in spin-1 Bose–Einstein condensates with the SU(3) spin–orbit coupling*

Ji-Guo Wang, Yueqing Li, Yu-Fei Dong

We consider the SU(3) spin–orbit coupled spin-1 Bose–Einstein condensates in a two-dimensional harmonic trap. The competition between the SU(3) spin–orbit coupling and the spin-exchange interaction results in a rich variety of lattice configurations. The ground-state phase diagram spanned by the isotropic SU(3) spin–orbit coupling and the spin–spin interaction is presented. Five ground-state phases can be identified on the phase diagram, including the plane wave phase, the stripe phase, the kagome lattice phase, the stripe-honeycomb lattice phase, and the honeycomb hexagonal lattice phase. The system undergoes a sequence of phase transitions from the rectangular lattice phase to the honeycomb hexagonal lattice phase, and to the triangular lattice phase in spin-1 Bose–Einstein condensates with anisotrpic SU(3) spin–orbit coupling.

Chinese Physics B•Aug 1, 2020•8 citations

Spin-orbit-coupled spinor gap solitons in Bose-Einstein condensates

Jing Yang, Yongping Zhang

Spin-1 spin-orbit-coupled spinor Bose-Einstein condensates have been realized in experiment. We study spin-orbit-coupled spinor gap solitons in this experimentally realizable system with an optical lattice. The spin-dependent parity symmetry of the spin-orbit coupling plays an important role in the properties of gap solitons. Two families of solitons with opposite spin-dependent parity are found. Using an approximate model by replacing the optical lattice with a harmonic trap, we demonstrate the physical origin of the two families. For the zero effective quadratic Zeeman shift, we also find a type of gap soliton that spontaneously breaks the spin-dependent parity symmetry.

arXiv:2209.10190•Physical Review A•Feb 22, 2023•21 citations

Emergence of topological and strongly correlated ground states in trapped Rashba spin-orbit-coupled Bose gases

B. Ramachandhran, Hui Hu, Han Pu

We theoretically study an interacting few-body system of Rashba spin-orbit-coupled two-component Bose gases confined in a harmonic trapping potential. We solve the interacting Hamiltonian at large Rashba coupling strengths using an exact-diagonalization scheme, and obtain the ground-state phase diagram for a range of interatomic interactions and particle numbers. At small particle numbers, we observe that the bosons condense to an array of topological states with $n+1/2$ quantum angular momentum vortex configurations, where $n=0,1,2,3,...$. At large particle numbers, we observe two distinct regimes: at weaker-interaction strengths, we obtain ground states with topological and symmetry properties that are consistent with mean-field theory computations; at stronger-interaction strengths, we report the emergence of strongly correlated ground states.

arXiv:1301.0800•Physical Review A•Mar 26, 2013•21 citations
We investigate theoretically the phase diagram of a spin-orbit coupled Bose gas in two-dimensional harmonic traps. We show that at strong spin-orbit coupling the single-particle spectrum decomposes into different manifolds separated by ℏω{⊥}, where ω{⊥} is the trapping frequency. For a weakly interacting gas, quantum states with Skyrmion lattice patterns emerge spontaneously and preserve either parity symmetry or combined parity-time-reversal symmetry. These phases can be readily observed in a spin-orbit coupled gas of ^{87}Rb atoms in a highly oblate trap.
We theoretically investigate the condensate state and collective excitations of a two-component Bose gas in a two-dimensional harmonic trap subject to isotropic Rashba spin-orbit coupling. In the weakly interacting regime when the interspecies interaction is larger than the intraspecies interaction (${g}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}>g$), we find that the condensate ground state has a half-quantum angular momentum vortex configuration with spatial rotational symmetry and skyrmion-type spin texture. Upon increasing the interatomic interaction beyond a threshold gc{g}_{c}gc​, the ground state starts to involve higher-order angular momentum components and thus breaks rotational symmetry. In the case of ${g}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}<g$, the condensate becomes unstable toward the superposition of two degenerate half-quantum vortex states. Both instabilities (at g>gcg>{g}_{c}g>gc​ and ${g}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}<g$) can be determined by solving the Bogoliubov equations for collective density oscillations of the half-quantum vortex state and by analyzing the softening of mode frequencies. We obtain the phase diagram as a function of the interatomic interactions and the spin-orbit coupling. In addition, we directly simulate the time-dependent Gross-Pitaevskii equation to examine the dynamical properties of the system. Finally, we investigate the stability of the half-quantum vortex state against both trap anisotropy and anisotropy in the spin-orbit coupling.
We present a variational study of pseudo-spin-1/21/21/2 Bose gases in a harmonic trap with weak three-dimensional (3D) spin-orbit coupling of $\mathbf{\ensuremath{\sigma}}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ type. This spin-orbit coupling mixes states with different parities, which inspires us to approximate the single-particle state with the eigenstates of the total angular momentum, i.e., superposition of harmonic sss-wave and ppp-wave states. As the time-reversal symmetry is protected by two-body interaction, we set the variational order parameter as the combination of two mutually time-reversal symmetric eigenstates of the total angular momentum. The variational results essentially reproduce the 3D skyrmionlike ground state recently identified by Kawakami et al. [T. Kawakami, T. Mizushima, M. Nitta, and K. Machida, Phys. Rev. Lett. 109, 015301 (2012)]. We show that these skyrmionlike ground states emerging in this model are primarily caused by the ppp-wave spatial mode involved in the variational order parameter that drives two spin components spatially separated. We find the ground state of this system falls into two phases with different density distribution symmetries depending on the relative magnitude of intraspecies and interspecies interaction: phase I has parity symmetric and axisymmetric density distributions, while phase II is featured with special joint symmetries of discrete rotational and time-reversal symmetry. With the increasing interaction strength the transition occurs between two phases with distinct density distributions, while the topological 3D skyrmionlike spin texture is symmetry protected.
We present a variational study of the spin-1 Bose gases in a harmonic trap with three-dimensional spin–orbit (SO) coupling of Weyl type. For weak SO coupling, we treat the single-particle ground states as the form of perturbational harmonic oscillator states in the lowest total angular momentum manifold with j = 1, m j = 1, 0, −1. When the two-body interaction is considered, we set the trail order parameter as the superposition of three degenerate single-particle ground-states and the weight coefficients are determined by minimizing the energy functional. Two ground state phases, namely the magnetic and the nematic phases, are identified depending on the spin-independent and the spin-dependent interactions. Unlike the non-SO-coupled spin-1 Bose–Einstein condensate for which the phase boundary between the magnetic and the nematic phase lies exactly at zero spin-dependent interaction, the boundary is modified by the SO-coupling. We find the magnetic phase is featured with phase-separated density distributions, 3D skyrmion-like spin textures and competing magnetic and biaxial nematic orders, while the nematic phase is featured with miscible density distributions, zero magnetization and spatially modulated uniaxial nematic order. The emergence of higher spin order creates new opportunities for exploring spin-tensor-related physics in SO coupled superfluid.
We consider the SU(3) spin–orbit coupled spin-1 Bose–Einstein condensates in a two-dimensional harmonic trap. The competition between the SU(3) spin–orbit coupling and the spin-exchange interaction results in a rich variety of lattice configurations. The ground-state phase diagram spanned by the isotropic SU(3) spin–orbit coupling and the spin–spin interaction is presented. Five ground-state phases can be identified on the phase diagram, including the plane wave phase, the stripe phase, the kagome lattice phase, the stripe-honeycomb lattice phase, and the honeycomb hexagonal lattice phase. The system undergoes a sequence of phase transitions from the rectangular lattice phase to the honeycomb hexagonal lattice phase, and to the triangular lattice phase in spin-1 Bose–Einstein condensates with anisotrpic SU(3) spin–orbit coupling.
Spin-1 spin-orbit-coupled spinor Bose-Einstein condensates have been realized in experiment. We study spin-orbit-coupled spinor gap solitons in this experimentally realizable system with an optical lattice. The spin-dependent parity symmetry of the spin-orbit coupling plays an important role in the properties of gap solitons. Two families of solitons with opposite spin-dependent parity are found. Using an approximate model by replacing the optical lattice with a harmonic trap, we demonstrate the physical origin of the two families. For the zero effective quadratic Zeeman shift, we also find a type of gap soliton that spontaneously breaks the spin-dependent parity symmetry.
We theoretically study an interacting few-body system of Rashba spin-orbit-coupled two-component Bose gases confined in a harmonic trapping potential. We solve the interacting Hamiltonian at large Rashba coupling strengths using an exact-diagonalization scheme, and obtain the ground-state phase diagram for a range of interatomic interactions and particle numbers. At small particle numbers, we observe that the bosons condense to an array of topological states with n+1/2n+1/2n+1/2 quantum angular momentum vortex configurations, where n=0,1,2,3,...n=0,1,2,3,...n=0,1,2,3,.... At large particle numbers, we observe two distinct regimes: at weaker-interaction strengths, we obtain ground states with topological and symmetry properties that are consistent with mean-field theory computations; at stronger-interaction strengths, we report the emergence of strongly correlated ground states.