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Physical Review A

Asymmetric bidrectional Kibble-Zurek mechanism in a spin-orbit-coupled Bose-Einstein condensate across first-order quantum phase transitions

Jun-Hang Ren, Sheng Liu, Yong-Sheng Zhang

Published Nov 12, 2025• 1 citation
Publisher

Abstract

Quench dynamics of first-order phase transitions in a homogeneous Bose-Einstein condensate with Raman-type spin-orbit coupling is investigated. By adjusting the coupling strength, the system can be driven across the critical point between the plane-wave phase and stripe phase in two opposite directions, and we find the scaling-law behaviors for both processes of phase transitions. This is different from previous studies, which only focus on the quench dynamics in the unidirectional transitions from a higher symmetric phase to a lower symmetric phase. Subsequently, we obtain the power law of the domain numbers (correlation length) and formation time (correlation time) of these phase transitions and extract the critical exponents. The results reveal the effect of the directionality of the phase transition on the dynamic scaling law.

DOI
10.1103/w1n4-d3lf

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Quench dynamics of first-order phase transitions in a homogeneous Bose-Einstein condensate with Raman-type spin-orbit coupling is investigated. By adjusting the coupling strength, the system can be driven across the critical point between the plane-wave phase and stripe phase in two opposite directions, and we find the scaling-law behaviors for both processes of phase transitions. This is different from previous studies, which only focus on the quench dynamics in the unidirectional transitions from a higher symmetric phase to a lower symmetric phase. Subsequently, we obtain the power law of the domain numbers (correlation length) and formation time (correlation time) of these phase transitions and extract the critical exponents. The results reveal the effect of the directionality of the phase transition on the dynamic scaling law.
The famous Kibble-Zurek mechanism offers us a significant clue to study quantum phase transitions out of equilibrium. Here, we investigate an intriguing phenomenon of a spin-orbit coupled Bose-Einstein condensate by quenching the Raman coupling strength from a high-symmetry phase (nonmagnetic phase) to a low-symmetry phase (magnetic phase). When crossing the critical point, the fluctuation of momentum distribution leads to delayed bifurcation structures. Simultaneously, the domain information emerges in momentum space. Moreover, the universal scalings of spatiotemporal dynamics are extracted from the fluctuations and domains, which manifests homogeneous and inhomogeneous Kibble-Zurek power laws at different timescales. Our work demonstrates a paradigmatic study on the inhomogeneous Kibble-Zurek mechanism.
Abstract 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.
Continuous phase transitions occur in a wide range of physical systems, and provide a context for the study of non-equilibrium dynamics and the formation of topological defects.The Kibble-Zurek (KZ) mechanism predicts the scaling of the resulting density of defects as a function of the quench rate through a critical point, and this can provide an estimate of the critical exponents of a phase transition.In this work we extend our previous study of the miscible-immiscible phase transition of a binary Bose-Einstein condensate (BEC) composed of two hyperfine states in which the spin dynamics are confined to one dimension [J.Sabbatini et al., Phys.Rev. Lett.107, 230402 (2011)].The transition is engineered by controlling a Hamiltonian quench of the coupling amplitude of the two hyperfine states, and results in the formation of a random pattern of spatial domains.Using the numerical truncated Wigner phase space method, we show that in a ring BEC the number of domains formed in the phase transitions scales as predicted by the KZ theory.We also consider the same experiment performed with a harmonically trapped BEC, and investigate how the density inhomogeneity modifies the dynamics of the phase transition and the KZ scaling law for the number of domains.We then make use of the symmetry between inhomogeneous phase transitions in anisotropic systems, and an inhomogeneous quench in a homogeneous system, to engineer coupling quenches that allow us to quantify several aspects of inhomogeneous phase transitions.In particular, we quantify the effect of causality in the propagation of the phase transition front on the resulting formation of domain walls, and find indications that the density of defects is determined during the impulse to adiabatic transition after the crossing of the critical point.
Continuous phase transitions occur in a wide range of physical systems, and provide a context for the study of non-equilibrium dynamics and the formation of topological defects. The Kibble–Zurek (KZ) mechanism predicts the scaling of the resulting density of defects as a function of the quench rate through a critical point, and this can provide an estimate of the critical exponents of a phase transition. In this work, we extend our previous study of the miscible–immiscible phase transition of a binary Bose–Einstein condensate (BEC) composed of two hyperfine states in which the spin dynamics are confined to one dimension (Sabbatini et al 2011 Phys. Rev. Lett. 107 230402). The transition is engineered by controlling a Hamiltonian quench of the coupling amplitude of the two hyperfine states and results in the formation of a random pattern of spatial domains. Using the numerical truncated Wigner phase-space method, we show that in a ring BEC the number of domains formed in the phase transitions scales as predicted by the KZ theory. We also consider the same experiment performed with a harmonically trapped BEC, and investigate how the density inhomogeneity modifies the dynamics of the phase transition and the KZ scaling law for the number of domains. We then make use of the symmetry between inhomogeneous phase transitions in anisotropic systems, and an inhomogeneous quench in a homogeneous system, to engineer coupling quenches that allow us to quantify several aspects of inhomogeneous phase transitions. In particular, we quantify the effect of causality in the propagation of the phase transition front on the resulting formation of domain walls and find indications that the density of defects is determined during the impulse to adiabatic transition after the crossing of the critical point.
We investigate the quench dynamics in a one-dimensional spin-orbit-coupled Bose-Einstein condensate (SOC-BEC) across the phase transition from plane-wave (PW) to stripe (ST), incorporating phenomenological damping. In the dissipation-free case, a state stagnation phenomenon emerges during the PW-ST quench: for slow quenches, the system remains trapped in the PW phase due to the energy gap induced by critical slowing down, which prevents spontaneous relaxation to the stripe ground state. To explore this phenomenon and examine the universal scaling predicted by the Kibble-Zurek mechanism (KZM) in open systems, we introduce a dissipative Gross-Pitaevskii equation with a phenomenological damping term. Numerical simulations reveal that weak dissipation preserves the expected KZM power-law scaling for the freeze-out time and defect density, whereas strong dissipation or long quench times lead to significant deviations. Our results demonstrate that the KZM remains applicable in dissipative quantum systems under appropriate conditions, providing insights into nonequilibrium dynamics in open quantum systems.