Zero-momentum Phase Research Articles

Elementary excitations in a spin–orbit-coupled Floquet spinor Bose–Einstein condensate

We study elementary excitations in a Floquet-engineered spin-1 BEC with spin–orbit coupling, which can be realized by periodically driving the quadratic Zeeman shift. We calculate the excitation spectrum of the plane-wave phase, the zero-momentum phase, and the stripe phase, as well as the corresponding response function and static structure factor. In special, we find that two roton modes appear in the lowest two excitation bands of the stripe phase, which are tunable by changing the driving parameters. The phase transition between different phases can be identified by the sound velocity of these phases. We also investigate spin dynamics in such a system, which strongly depend on the driving parameter.

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.

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.

Excitation spectrum of tunable spin-orbit coupled Bose-Einstein condensates and its effective regulation

<sec>In a recent experiment, the excitation spectrum of spin-orbit (SO) coupled Bose-Einstein condensates (BECs) of <inline-formula><tex-math id="M1">\begin{document}$^{87}{\rm{Rb}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222306_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222306_M1.png"/></alternatives></inline-formula> atoms was studied by using Bragg spectroscopy, and the roton-maxon structure was found to exist in the excitation spectrum of magnetized phase. In addition, the roton-mode and its softening phenomenon are obtained by using various artificial SO couplings such as Rashba SO coupling and spin-orbital-angular-momentum coupling. However, the SO coupling strength in previous studies could not be controlled in real time, which limits the further study and precise regulation of the excitation spectrum of condensate. Thus, it is still an important topic to study how to regulate the SO coupling strength of the system through an external driving field, and further regulate the excitation spectrum of SO coupled BECs.</sec> <sec>In this work, the excitation spectrum of a tunable SO coupled BECs in free space is studied by using Bogoliubov theory. The time-independent effective Floquet Hamiltonian with two-body interaction is obtained in the high frequency approximation, and then a tunable SO coupling and an effective two-body interaction that can be regulated by the periodic driving of Raman coupling are obtained. Based on the effective Floquet Hamiltonian of the system, the dispersion relation of the BECs with interactions is numerically calculated. It is found that the periodic driving can effectively regulate the structure of the dispersion relation, which indicates that the periodic driving can regulate the phase transition between the zero-momentum phase and the plane wave phase. Then, the Bogoliubov-de-Gennes (BdG) equation of the system is obtained by using Bogoliubov theory. Moreover, the excitation spectrum of the BECs in the zero momentum phase and the plane wave phase are studied, respectively. Only the phonon excitation exists in the excitation spectrum of the zero momentum phase, and the excitation spectrum behaves as a Bessel function with the increase of the periodic driving strength. The phonon and roton excitations exist in the excitation spectrum of the plane wave phase, and the roton mode gradually softens with the increase of periodically driving strength. Therefore, the phonon and roton excitations in the excitation spectrum of SO coupled BECs can be regulated in real time by periodically driving Raman coupling.</sec>

Superfluidity of a Raman spin-orbit-coupled Bose gas at finite temperature

We investigate the superfluidity of a three-dimensional weakly interacting Bose gas with a one-dimensional Raman-type spin-orbit coupling at both zero and finite temperatures. Using the imaginary-time Green's function within the Bogoliubov approximation, we explicitly derive analytic expressions of the current-current response functions in the plane-wave and zero-momentum phases, from which we extract the superfluid density in the limits of long wavelength and zero frequency. At zero temperature, we check that the resultant superfluid density agrees exactly with our previous analytic prediction obtained from a phase-twist approach. Both results also satisfy a generalized Josephson relation in the presence of spin-orbit coupling. At finite temperature, we find a significant non-monotonic temperature dependence of superfluid density near the transition from the plane-wave phase to the zero-momentum phase. We show that this non-trivial behavior might be understood from the sound velocity, which has a similar temperature dependence. The non-monotonic temperature dependence is also shared by Landau critical velocity, above which the spin-orbit-coupled Bose gas loses its superfluidity. Our results would be useful for further theoretical and experimental studies of superfluidity in exotic spin-orbit coupled quantum gases.

Faraday patterns in spin-orbit-coupled Bose-Einstein condensates

We study the Faraday patterns generated by spin-orbit-coupling induced parametric resonance in a spinor Bose-Einstein condensate with repulsive interaction. The collective elementary excitations of the Bose-Einstein condensate, including density waves and spin waves, are coupled as the result of the Raman-induced spin-orbit coupling and a quench of the relative phase of two Raman lasers without the modulation of any of the system's parameters. We observed several higher parametric resonance tongues at integer multiples of the driving frequency and investigated the interplay between Faraday instabilities and modulation instabilities when we quench the spin-orbit-coupled Bose-Einstein condensate from zero-momentum phase to plane-wave phase. If the detuning is equal to zero, the wave number of combination resonance barely changes as the strength of spin-orbit coupling increases. If the detuning is not equal to zero after a quench, a single combination resonance tongue will split into two parts.

The Bloch oscillations of spin-orbit coupled Bose-Einstein condensates in deep optical lattices

We study the ground state and the Bloch oscillations of the spin-orbit coupled Bose-Einstein condensates in one-dimensional deep optical lattices. The phase transition between the zero-momentum phase and the plane-wave phase in untilted deep optical lattices is studied analytically. The Bloch oscillations in tilted lattices started in different ground state phases are analyzed. It is found that the Bloch dynamics is harmonic (anharmonic) in the zero-momentum phase (plane-wave phase). Furthermore, the modification of the Bloch oscillations induced by spin-orbit coupling, tunnelling coefficient and the external force (the inclination of the tilt) is revealed. In addition, the analytical condition for emerging the long-lived Bloch oscillations is obtained and the corresponding diagram in parameter space is provided. Particularly, we find that the spin-orbit coupling can extend the duration of the Bloch oscillations in the plane-wave phase. We provide an effective method to manipulate the Bloch dynamics in the deep optical lattices.

Anderson localization of a spin–orbit coupled Bose–Einstein condensate in disorder potential

We present numerical results of a one-dimensional spin–orbit coupled Bose–Einstein condensate expanding in a speckle disorder potential by employing the Gross–Pitaevskii equation. Localization properties of a spin–orbit coupled Bose–Einstein condensate in zero-momentum phase, magnetic phase and stripe phase are studied. It is found that the localizing behavior in the zero-momentum phase is similar to the normal Bose–Einstein condensate. Moreover, in both magnetic phase and stripe phase, the localization length changes non-monotonically as the fitting interval increases. In magnetic phases, the Bose–Einstein condensate will experience spin relaxation in disorder potential.

Symmetry breaking and phase transitions in Bose-Einstein condensates with spin–orbital-angular-momentum coupling

Theoretical study is presented for a spinor Bose-Einstein condensate, whose two components are coupled by copropagating Raman beams with different orbital angular momenta. The investigation is focused on the behavior of the ground state of this condensate, depending on the atom-light coupling strength. By analyzing the ground state, we have identified a number of quantum phases, which reflect the symmetries of the effective Hamiltonian and are characterized by the specific structure of the wave function. In addition to the well-known stripe, polarized and zero-momentum phases, our results show that the system can support phases, whose wave function contains a complex vortex molecule. Such molecule plays an important role in the continuous phase transitions of the system. The predicted behavior of vortex-molecule phases can be examined in cold-atom experiments using currently existing techniques.

Stability and quantum escape dynamics of spin-orbit-coupled Bose-Einstein condensates in the shallow trap.

The Bose-Einstein condensates in a finite depth potential well provide an ideal platform to study the quantum escape dynamics. In this paper, the ground state, tunneling, and diffusion dynamics of the spin-orbit coupling (SOC) of Bose-Einstein condensates with two pseudospin components in a shallow trap are studied analytically and numerically. The phase transition between the plane-wave phase and zero-momentum phase of the ground state is obtained. Furthermore, the stability of the ground state is discussed, and the stability diagram in the parameter space is provided. The bound state (in which condensates are stably trapped in the potential well), the quasibound state (in which condensates tunnel through the well), and the unstable state (in which diffusion occurs) are revealed. We find that the finite depth potential well has an important effect on the phase transition of the ground state, and, interestingly, SOC can stabilize the system against the diffusion and manipulate the tunneling and diffusion dynamics. In particular, spatial anisotropic tunneling and diffusion dynamics of the two pseudospin components induced by SOC in quasibound and unstable states are observed. We provide an effective model and method to study and control the quantum tunneling and diffusion dynamics.

Bound states of spin–orbit coupled cold atoms in a Dirac delta-function potential

Dirac delta-function potential is widely studied in quantum mechanics because it usually can be exactly solved and at the same time is useful in modeling various physical systems. Here we study a system of delta-potential trapped spin–orbit coupled cold atoms. The spin–orbit coupled atomic matter wave has two kinds of evanescent modes, one of which has a pure imaginary wavevector and is an ordinary evanescent wave; while the other with a complex number wave vector is recognized as an oscillating evanescent wave. We identified the eigenenergy spectra and the existence of bound states in this system. The bound states can be constructed analytically using the two kinds of evanescent modes and we found that they exhibit typical features of stripe phase, separated phase or zero-momentum phase. In addition to that, the properties of semi-bound states are also discussed, which is a localized wave packet on a plane wave background.

Two-fluid theory for a superfluid system with anisotropic effective masses

In this work, we generalize the two-fluid theory to a superfluid system with anisotropic effective masses along different principal axis directions. As a specific example, such a theory can be applied to spin-orbit coupled Bose-Einstein condensate (BEC) at low temperature. The normal density from phonon excitations and the second sound velocity are obtained analytically. Near the phase transition from the plane wave to zero-momentum phases, due to the effective mass divergence, the normal density from phonon excitation increases greatly, while the second sound velocity is suppressed significantly. With quantum hydrodynamic formalism, we give unified derivations for suppressed superfluid density and Josephson relation. At last, the momentum distribution function and fluctuation of phase for the long wave length are also discussed.

The effect of a finite two-photon detuning on the quantum tricriticality and phase transitions in a spin-orbit coupled Bose gas

In the current experiments, realizing the synthetic spin-orbital coupling (SOC) with ultracold atoms, a finite two-photon detuning (TPD) is present. However, in most relevant theoretical studies so far, the TPD has been assumed as zero. Here, we investigate how a finite detuning affects the ground phase diagram in a Bose gas with equal-weight Rashba and Dresselhuas SOC. We show that the single-particle dispersion relation in the presence of a finite TPD can be significantly modified compared to the case with zero detuning, which gives rise to experimentally observable effects on quantum tricriticality and phase transitions. In particular, the zero-momentum phase that can arise for a zero TPD becomes unstable against a finite TPD and disappears. Our results are relevant to the current cold atom experiments using the Raman laser-induced gauge field.

Quantum depletion and superfluid density of a supersolid in Raman spin-orbit-coupled Bose gases

We theoretically investigate a three-dimensional weakly interacting Bose gas with one-dimensional Raman-type spin-orbit coupling at zero temperature. By employing an improved ansatz, including high-order harmonics in the stripe phase, we show that the critical transition from the stripe to the plane-wave phases is shifted to a relatively larger Rabi frequency compared to the prediction by previous work [Li $\textit{et al.}$, Phys. Rev. Lett. $\textbf{108}$, 225301 (2012)] using a first-order stripe ansatz. We also determine the quantum depletion and superfluid density over a large range of Rabi frequency in different phases. The depletion exhibits an intriguing behavior with a discontinuous jump at the transition between the stripe and plane-wave phases, and a maximum at the transition between the plane-wave and zero-momentum phases. The superfluid density is derived through a phase-twist method. In the plane-wave and zero-momentum phases, it is significantly suppressed along the spin-orbit-coupling direction and vanishes at the transition, consistent with a recent work [Zhang $\textit{et al.}$, Phys. Rev. A $\textbf{94}$, 033635 (2016)], while in the stripe phase, it smoothly decreases with increasing Rabi frequency. Our predictions would be useful for further theoretical and experimental studies of the exotic supersolid stripe phase.

Universal dynamics of zero-momentum to plane-wave transition in spin–orbit coupled Bose–Einstein condensates

We investigate the universal spatiotemporal dynamics in spin–orbit coupled Bose–Einstein condensates which are driven from the zero-momentum phase to the plane-wave phase. The excitation spectrum reveals that, at the critical point, the Landau critical velocity vanishes and the correlation length diverges. Therefore, according to the Kibble–Zurek mechanism, spatial domains will spontaneously appear in such a quench through the critical point. By simulating the real-time dynamics, we numerically extract the static correlation length critical exponent and the dynamic critical exponent z from the scalings of the temporal bifurcation delay and the spatial domain number. The numerical scalings consist well with the analytical ones obtained by analyzing the excitation spectrum.

Sound Wave of Spin–Orbit Coupled Bose–Einstein Condensates in Optical Lattice**Supported by the National Natural Science Foundation of China under Grant Nos 11305132, 11274255 and 11475027, and the Scientific Research Project of Gansu Higher Education under Grant No 2016A-005.

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.

Quantum and thermal fluctuations in a Raman spin-orbit-coupled Bose gas

We theoretically study a three-dimensional weakly-interacting Bose gas with Raman-induced spin-orbit coupling at finite temperature. By employing a generalized Hartree-Fock-Bogoliubov theory with Popov approximation, we determine a complete finite-temperature phase diagram of three exotic condensation phases (i.e., the stripe, plane-wave and zero-momentum phases), against both quantum and thermal fluctuations. We find that the plane-wave phase is significantly broadened by thermal fluctuations. The phonon mode and sound velocity at the transition from the plane-wave phase to the zero-momentum phase are thoughtfully analyzed. At zero temperature, we find that quantum fluctuations open an unexpected gap in sound velocity at the phase transition, in stark contrast to the previous theoretical prediction of a vanishing sound velocity. At finite temperature, thermal fluctuations continue to significantly enlarge the gap, and simultaneously shift the critical minimum. For a Bose gas of $^{87}$Rb atoms at the typical experimental temperature, $T=0.3T_{0}$, where $T_{0}$ is the critical temperature of an ideal Bose gas without spin-orbit coupling, our results of gap opening and critical minimum shifting in the sound velocity, are qualitatively consistent with the recent experimental observation {[}S.-C. Ji \textit{et al.}, Phys. Rev. Lett. \textbf{114}, 105301 (2015){]}.

Landau critical velocity of spin-orbit-coupled Bose-Einstein condensate across quantum phase transition

An impurity immersed in a superfluid can move without friction when its velocity is below a critical value. This phenomenon can be explained by the famous Landau criterion, according to which, the critical velocity is determined by the elementary excitation spectrum of the superfluid. Landau critical velocity has been measured in the isotropic superfluid, such as the liquid He-Ⅱ and the Bose-Einstein condensates of dilute atomic gases, where the onset of dissipation is due to the creation of roton and phonon, respectively. The recent realization of synthetic spin-orbit coupling in quantum gas opens up possibilities for the study of novel superfluidity with ultracold atoms. To date, a specific type of spin-orbit coupling, which is generated by a pair of Raman laser beams, has been achieved in a Bose-Einstein condensate of 87Rb experimentally. Remarkably, the excitation spectrum of this system is anisotropic and can be feasibly tuned by the external laser field. While the anisotropic dynamics has been observed experimentally, the critical velocity has not been measured so far. It is a conventional wisdom that in an anisotropic superfluid, the critical velocity is determined by the excitation spectrum in the moving direction of the impurity. However, this is not always the case. In this work, we investigate the motion of a point-like impurity in a spin-orbit-coupled condensate with the spin-dependent interatomic interaction. In the vicinity of the quantum phase transition between the plane-wave (PW) phase and the zero-momentum (ZM) phase, the onset of the dissipation is due to the emission of a phonon, and the Landau critical velocity vc depends on the anisotropic sound velocity. While the sound velocity varies smoothly across the PW-ZM phase transition, the critical velocity in the direction perpendicular to the axis of spin-orbit coupling exhibits a sudden jump at the phase boundary. The value of vc on the PW phase side of the transition is generally smaller than the one on the ZM phase side, and the jump amplitude of vc is an increasing function of the spin-dependent interaction strength. Beyond the critical velocity, the energy dissipation rate of the impurity is explicitly calculated via a perturbation approach. The discontinuity of vc at the phase boundary can be clearly seen from the dissipation curves, which can be measured through the heating of the condensate. Our prediction can be tested in the current experiments with ultracold atoms.

Phase transitions and elementary excitations in spin-1 Bose gases with Raman-induced spin-orbit coupling

We study the ground state phase diagram and the quantum phase transitions in spin-1 Bose gases with Raman induced spin-orbit coupling. In addition to the Bose-Einstein condensates with uniform density, three types of stripe condensation phases that simultaneously break the U(1) symmetry and the translation symmetry are identified. The transitions between these phases are investigated, and the occurrences of the various tricritical points are predicted. The excitation spectra in the plane-wave phase and the zero-momentum phase show rich roton-maxon structures, and their instabilities indicate the tendency to develop the crystalline order. We propose the atomic gas of $^{23}$Na could be a candidate for observing the stripe condensate with high contrast fringes.

Spin–orbit coupled Bose–Einstein condensates with Rydberg-dressing interaction*Project supported by the National Basic Research Program of China (Grant No. 2011CB921504) and the National Natural Science Foundation of China (Grant No. 11104292).

Interaction between Rydberg atoms can be used to control the properties of interatomic interaction in ultracold gases by weakly dressing the atoms with a Rydberg state. Here we investigate the effect of the Rydberg-dressing interaction on the ground-state properties of a Bose–Einstein condensate imposed by Raman-induced spin–orbit coupling. We find that, in the case of SU(2)-invariant s-wave interactions, the gas is only in the plane-wave phase and the zero-momentum phase is absent. In particular, we also predict an unexpected magnetic stripe phase composed of two plane-wave components with unequal weight when s-wave interactions are non-symmetric, which originates from the Rydberg-dressing interaction.