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

Chinese Physics Letters

Sound Wave of Spin–Orbit Coupled Bose–Einstein Condensates in Optical Lattice

Xudan Chai, Zi-Fa Yu, Ai‐Xia Zhang, Ju-Kui Xue

Published Aug 1, 2017• 5 citations
Publisher

Abstract

We study the phonon mode excitation of spin–orbit (SO) coupled Bose–Einstein condensates trapped in a one-dimensional optical lattice. The sound speed of the system is obtained analytically. Softening of the phonon mode, i.e., the vanishing of sound speed, in the optical lattice is revealed. When the lattice is absent, the softening of phonon mode occurs only at the phase transition point, which is not influenced by the atomic interaction and Raman coupling when the SO coupling is strong. However, when the lattice is present, the softening of phonon modes can take place in a regime near the phase transition point. Particularly, the regime is widened as lattice strength and SO coupling increase or atomic interaction decreases. The suppression of sound speed by the lattice strongly depends on atomic interaction, Raman coupling, and SO coupling. Furthermore, we find that the sound speed in plane wave phase regime and zero-momentum phase regime behaves with very different characteristics as Raman coupling and SO coupling change. In zero-momentum phase regime, sound speed monotonically increases/decreases with Raman coupling/SO coupling, while in plane wave phase regime, sound speed can either increase or decrease with Raman coupling and SO coupling, which depends on atomic interaction.

DOI
10.1088/0256-307x/34/9/090301

Related papers

Elementary excitation of spin-orbit coupled Bose-Einstein condensates in optical lattice

Xudan Chai, Ju-Kui Xue

The phonon mode excitations of spin-orbit (SO) coupled Bose-Einstein condensates (BECs) trapped in an one-dimensional optical lattice are studied and the sound speed of the system is obtained analytically.

Journal of Physics Conference Series•Jul 1, 2017•0 citations

Plane- and stripe-wave phases of a spin-orbit-coupled Bose-Einstein condensate in an optical lattice with a Zeeman field

Kristian Mæland, Andreas T. G. Janssønn, Jonas H. Rygh +1 more

A weakly interacting, spin-orbit-coupled, ultracold, dilute Bose gas on a two-dimensional square lattice with an external Zeeman field is studied. We explore the plane- and stripe-wave phases of the system involving nonzero condensate momenta, which occur when the Zeeman field is below a critical value. Their excitation spectra are found using Bogoliubov theory and by two different routes. The validity of each method to obtain the excitation spectrum is discussed, and it is found that projection on the lowest single-particle band is an excellent approximation in the plane-wave phase, while it is a poor approximation in the stripe-wave phase. While the plane-wave phase has a phonon minimum at its single condensate momentum, revealing a nonzero sound velocity of the excitations, the stripe-wave phase has quadratic minima at its two condensate momenta showing zero sound velocity of the excitations. We discuss how the presence of more than one condensate momentum is essential for these differences between the two phases. Additionally, it is emphasized that the zero sound velocity in the stripe-wave phase is a lattice effect, since continuum studies of the same phase have shown nonzero sound velocity.

arXiv:2009.04538•Physical Review A•Nov 19, 2020•11 citations

Spin-orbit-coupling–induced anharmonic collective modes in a Bose-Einstein condensate

Zi-Fa Yu, Ju-Kui Xue

The collective excitations of a spin-orbit–coupled Bose-Einstein condensate are investigated by using a variational method and numerical simulation. The collective modes are depicted in zero-momentum and plane-wave phases, while the mechanism for inducing the anharmonic collective oscillations is revealed analytically. The coupling among spin-orbit coupling, Raman coupling, and atomic interaction results in multiple external collective modes, which can lead to the collective dynamics being the superposition of the multiple harmonic oscillations. When one of the external frequencies is close to the inherent frequency of the conventional condensate and the amplitudes of other external frequencies vanish, the collective oscillations are harmonic, otherwise the collective oscillations are anharmonic. Thus the collective dynamics can undergo a harmonicity-anharmonicity transition. This can provide a possible way for elaborating distinct collective oscillation modes. Moreover, the collective excitations exhibit distinct properties in different phases, while the softening of collective modes is detected at the phase transition point, which can be used to map out the phase boundaries in experiment.

Europhysics Letters (EPL)•Jan 1, 2018•3 citations

Anisotropic dynamics of a spin-orbit-coupled Bose-Einstein condensate

Giovanni Italo Martone, Yun Li, Лев П. Питаевский +1 more

By calculating the density response function we identify the excitation spectrum of a Bose-Einstein condensate with equal Rashba and Dresselhaus spin-orbit coupling. We find that the velocity of sound along the direction of spin-orbit coupling is deeply quenched and vanishes when one approaches the second-order phase transition between the plane-wave and the zero momentum quantum phases. We also point out the emergence of a roton minimum in the excitation spectrum for small values of the Raman coupling, providing the onset of the transition to the stripe phase. Our findings point out the occurrence of a strong anisotropy in the dynamic behavior of the gas. A hydrodynamic description accounting for the collective oscillations in both uniform and harmonically trapped gases is also derived.

arXiv:1207.6804•Physical Review A•Dec 19, 2012•161 citations

Collective modes of a spin-orbit-coupled Bose-Einstein condensate: A hydrodynamic approach

Wei Zheng, Zhibing Li

We studied the collective modes of a Bose-Einstein condensate (BEC) with spin-orbit coupling. We developed the hydrodynamic equations for spin-orbit coupled BECs and used them to study collective modes in the plane-wave phase and large Rabi coupling regime for both a uniform BEC and a BEC in a harmonic trap. In the homogeneous situation, we obtained energy spectra of elementary excitations and found that the spin-orbit coupling can increase the effective mass of the atoms, which will suppress the sound velocity. The spin-orbit coupling can also change the spin mixing, which will modify the interaction energy, and may lead to an enhancement of sound velocity. The competition between these two effects gives the behavior of sound velocity. In a harmonic trap, we found that the dipole mode and the breathing mode are coupled together in the plane-wave phase, and these two modes have a $\ensuremath{\pi}/2$ phase difference, because the spin-orbit coupling and the interaction are not invariant under spin rotation. However, in the large Rabi coupling regime, the dipole mode and the breathing mode are decoupled due to the symmetry restriction.

Physical Review A•May 9, 2012•44 citations

Sound speed of a Bose-Einstein condensate in an optical lattice

Zhaoxin Liang, Xi Dong, Zhang Zhi-dong +1 more

The speed of sound of a Bose-Einstein condensate in an optical lattice is studied both analytically and numerically in all spatial three dimensions. Our investigation shows that the sound speed depends strongly on the strength of the lattice. In the one-dimensional case, the speed of sound falls monotonically with increasing lattice strength. The dependence on lattice strength becomes much richer in two and three dimensions. In the two-dimensional case, when the interaction is weak, the sound speed first increases and then decreases as the lattice strength increases. For the three-dimensional lattice, the sound speed can even oscillate with the lattice strength. These rich behaviors can be understood in terms of compressibility and effective mass. Our analytical results in the limit of weak lattices also offer an interesting perspective to help with our understanding: they show that the lattice component perpendicular to the sound propagation increases the sound speed while the lattice component parallel to the propagation decreases the sound speed. The various dependences of the sound speed on the lattice strength are the result of this competition.

arXiv:0705.4322•Physical Review A•Aug 18, 2008•34 citations
We study the phonon mode excitation of spin–orbit (SO) coupled Bose–Einstein condensates trapped in a one-dimensional optical lattice. The sound speed of the system is obtained analytically. Softening of the phonon mode, i.e., the vanishing of sound speed, in the optical lattice is revealed. When the lattice is absent, the softening of phonon mode occurs only at the phase transition point, which is not influenced by the atomic interaction and Raman coupling when the SO coupling is strong. However, when the lattice is present, the softening of phonon modes can take place in a regime near the phase transition point. Particularly, the regime is widened as lattice strength and SO coupling increase or atomic interaction decreases. The suppression of sound speed by the lattice strongly depends on atomic interaction, Raman coupling, and SO coupling. Furthermore, we find that the sound speed in plane wave phase regime and zero-momentum phase regime behaves with very different characteristics as Raman coupling and SO coupling change. In zero-momentum phase regime, sound speed monotonically increases/decreases with Raman coupling/SO coupling, while in plane wave phase regime, sound speed can either increase or decrease with Raman coupling and SO coupling, which depends on atomic interaction.
A weakly interacting, spin-orbit-coupled, ultracold, dilute Bose gas on a two-dimensional square lattice with an external Zeeman field is studied. We explore the plane- and stripe-wave phases of the system involving nonzero condensate momenta, which occur when the Zeeman field is below a critical value. Their excitation spectra are found using Bogoliubov theory and by two different routes. The validity of each method to obtain the excitation spectrum is discussed, and it is found that projection on the lowest single-particle band is an excellent approximation in the plane-wave phase, while it is a poor approximation in the stripe-wave phase. While the plane-wave phase has a phonon minimum at its single condensate momentum, revealing a nonzero sound velocity of the excitations, the stripe-wave phase has quadratic minima at its two condensate momenta showing zero sound velocity of the excitations. We discuss how the presence of more than one condensate momentum is essential for these differences between the two phases. Additionally, it is emphasized that the zero sound velocity in the stripe-wave phase is a lattice effect, since continuum studies of the same phase have shown nonzero sound velocity.
The collective excitations of a spin-orbit–coupled Bose-Einstein condensate are investigated by using a variational method and numerical simulation. The collective modes are depicted in zero-momentum and plane-wave phases, while the mechanism for inducing the anharmonic collective oscillations is revealed analytically. The coupling among spin-orbit coupling, Raman coupling, and atomic interaction results in multiple external collective modes, which can lead to the collective dynamics being the superposition of the multiple harmonic oscillations. When one of the external frequencies is close to the inherent frequency of the conventional condensate and the amplitudes of other external frequencies vanish, the collective oscillations are harmonic, otherwise the collective oscillations are anharmonic. Thus the collective dynamics can undergo a harmonicity-anharmonicity transition. This can provide a possible way for elaborating distinct collective oscillation modes. Moreover, the collective excitations exhibit distinct properties in different phases, while the softening of collective modes is detected at the phase transition point, which can be used to map out the phase boundaries in experiment.
By calculating the density response function we identify the excitation spectrum of a Bose-Einstein condensate with equal Rashba and Dresselhaus spin-orbit coupling. We find that the velocity of sound along the direction of spin-orbit coupling is deeply quenched and vanishes when one approaches the second-order phase transition between the plane-wave and the zero momentum quantum phases. We also point out the emergence of a roton minimum in the excitation spectrum for small values of the Raman coupling, providing the onset of the transition to the stripe phase. Our findings point out the occurrence of a strong anisotropy in the dynamic behavior of the gas. A hydrodynamic description accounting for the collective oscillations in both uniform and harmonically trapped gases is also derived.
We studied the collective modes of a Bose-Einstein condensate (BEC) with spin-orbit coupling. We developed the hydrodynamic equations for spin-orbit coupled BECs and used them to study collective modes in the plane-wave phase and large Rabi coupling regime for both a uniform BEC and a BEC in a harmonic trap. In the homogeneous situation, we obtained energy spectra of elementary excitations and found that the spin-orbit coupling can increase the effective mass of the atoms, which will suppress the sound velocity. The spin-orbit coupling can also change the spin mixing, which will modify the interaction energy, and may lead to an enhancement of sound velocity. The competition between these two effects gives the behavior of sound velocity. In a harmonic trap, we found that the dipole mode and the breathing mode are coupled together in the plane-wave phase, and these two modes have a $\ensuremath{\pi}/2$ phase difference, because the spin-orbit coupling and the interaction are not invariant under spin rotation. However, in the large Rabi coupling regime, the dipole mode and the breathing mode are decoupled due to the symmetry restriction.
The speed of sound of a Bose-Einstein condensate in an optical lattice is studied both analytically and numerically in all spatial three dimensions. Our investigation shows that the sound speed depends strongly on the strength of the lattice. In the one-dimensional case, the speed of sound falls monotonically with increasing lattice strength. The dependence on lattice strength becomes much richer in two and three dimensions. In the two-dimensional case, when the interaction is weak, the sound speed first increases and then decreases as the lattice strength increases. For the three-dimensional lattice, the sound speed can even oscillate with the lattice strength. These rich behaviors can be understood in terms of compressibility and effective mass. Our analytical results in the limit of weak lattices also offer an interesting perspective to help with our understanding: they show that the lattice component perpendicular to the sound propagation increases the sound speed while the lattice component parallel to the propagation decreases the sound speed. The various dependences of the sound speed on the lattice strength are the result of this competition.