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arXiv:2312.16555 · Communications Physics

Dynamics of a nonequilibrium discontinuous quantum phase transition in a spinor Bose–Einstein condensate

Matthew T. Wheeler, Hayder Salman, Magnus O. Borgh

Published Apr 11, 2025• 5 citations
arXivHTMLPDF

Abstract

Abstract Symmetry-breaking quantum phase transitions lead to the production of topological defects or domain walls in a wide range of physical systems. In second-order transitions, these exhibit universal scaling laws described by the Kibble–Zurek mechanism, but for first-order transitions a similarly universal approach is still lacking. Here, we propose a spinor Bose–Einstein condensate as a testbed system where critical scaling behaviour in a first-order quantum phase transition can be understood from generic properties. We demonstrate the applicability of the Kibble–Zurek mechanism for this transition to determine the critical exponents for: (1) the onset of the decay of the metastable state on short times scales, and (2) the number of resulting phase-separated ferromagnetic domains at longer times, as a one-dimensional spin-1 condensate is ramped across a first-order quantum phase transition. The predictions are in excellent agreement with mean-field numerical simulations and provide a paradigm for studying the decay of metastable states in experimentally accessible systems.

DOI
10.1038/s42005-025-02048-7

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Abstract Symmetry-breaking quantum phase transitions lead to the production of topological defects or domain walls in a wide range of physical systems. In second-order transitions, these exhibit universal scaling laws described by the Kibble–Zurek mechanism, but for first-order transitions a similarly universal approach is still lacking. Here, we propose a spinor Bose–Einstein condensate as a testbed system where critical scaling behaviour in a first-order quantum phase transition can be understood from generic properties. We demonstrate the applicability of the Kibble–Zurek mechanism for this transition to determine the critical exponents for: (1) the onset of the decay of the metastable state on short times scales, and (2) the number of resulting phase-separated ferromagnetic domains at longer times, as a one-dimensional spin-1 condensate is ramped across a first-order quantum phase transition. The predictions are in excellent agreement with mean-field numerical simulations and provide a paradigm for studying the decay of metastable states in experimentally accessible systems.
The Kibble-Zurek mechanism provides a unified theory to describe the universal scaling laws in the dynamics when a system is driven through a second-order quantum phase transition. However, for first-order quantum phase transitions, the Kibble-Zurek mechanism is usually not applicable. Here, we experimentally demonstrate and theoretically analyze a power-law scaling in the dynamics of a spin-1 condensate across a first-order quantum phase transition when a system is slowly driven from a polar phase to an antiferromagnetic phase. We show that this power-law scaling can be described by a generalized Kibble-Zurek mechanism. Furthermore, by experimentally measuring the spin population, we show the power-law scaling of the temporal onset of spin excitations with respect to the quench rate, which agrees well with our numerical simulation results. Our results open the door for further exploring the generalized Kibble-Zurek mechanism to understand the dynamics across first-order quantum phase transitions.
We study the equilibrium and dynamical properties of a ferromagnetic spinor atomic Bose-Einstein condensate. In the vicinity of the critical point for a continuous quantum phase transition, universal behaviors are observed both in the equilibrium state and in the dynamics when the quadratic Zeeman shift is swept linearly. Three distinct dynamical regions are identified for different sweeping time scales ($\ensuremath{\tau}$) when compared to the time scale ${\ensuremath{\tau}}_{\mathrm{KZ}}\ensuremath{\sim}{N}^{(1+\ensuremath{\nu}z)/\ensuremath{\nu}d}$ decided by external driving in a system with finite size NNN ($\ensuremath{\nu},z$ are critical exponents and ddd the dimensionality). They are manifested by the excitation probability P\mathcal{P}P and the excess heat density Q\mathcal{Q}Q. The adiabatic region of $\mathcal{P}\ensuremath{\sim}\mathcal{Q}\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}2}$ follows from the adiabatic perturbation theory when $\ensuremath{\tau}>{\ensuremath{\tau}}_{\mathrm{KZ}}$, while the nonadiabatic universal region of $\mathcal{P}\ensuremath{\sim}\mathcal{Q}\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}1}$ in the thermodynamic limit is described by the Kibble-Zurek mechanism when ${\ensuremath{\tau}}_{\mathrm{KZ}}>\ensuremath{\tau}>$ the time scale given by initial gap. The Kibble-Zurek scaling hypothesis is augmented with finite-size scaling in the latter region and several experimentally falsifiable features for the finite system we consider are predicted. The region of the fastest sweeping is found to be nonuniversal and far from equilibrium, with P\mathcal{P}P and Q\mathcal{Q}Q essentially being constants independent of $\ensuremath{\tau}$.
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.
The dynamics of a quantum phase transition are explored using slow quenches from the polar to the broken-axisymmetry phases in a small spin-1 ferromagnetic Bose-Einstein condensate. Measurements of the evolution of the spin populations reveal a power-law scaling of the temporal onset of excitations versus quench speed as predicted from quantum extensions of the Kibble-Zurek mechanism. The satisfactory agreement of the measured scaling exponent with the analytical theory and numerical simulations provides experimental confirmation of the quantum Kibble-Zurek model.
We consider a phase transition from an antiferromagnetic to a phase separated ground state in a spin-1 Bose-Einstein condensate of ultracold atoms. We demonstrate the occurrence of two scaling laws, for the number of spin domain seeds just after the phase transition, and for the number of spin domains in the final, stable configuration. Only the first scaling can be explained by the standard Kibble-Żurek mechanism. We explain the occurrence of two scaling laws by a model including postselection of spin domains due to the conservation of condensate magnetization.