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arXiv:1009.1646 · Physical Review A

Prethermalization in quenched spinor condensates

Ryan Barnett, Anatoli Polkovnikov, Mukund Vengalattore

Published Aug 5, 2011• 87 citations
arXivHTMLPDF

Abstract

Motivated by recent experiments, we consider the dynamics of spin-1 spinor condensates after a quantum quench from the polar to ferromagnetic state from varying the quadratic Zeeman field $q$. We apply the truncated Wigner approximation (TWA) to the spinor system, including all spatial and spin degrees of freedom. For short times, we find full agreement with the linearized Bogoliubov analysis. For longer times, where the Bogoliubov theory fails, we find that the system reaches a quasisteady prethermalized state. We compute the Bogoliubov spectrum about the ferromagnetic state with general $q$ and show that the resulting finite-temperature correlation functions grossly disagree with the full TWA results, thus indicating that the system does not thermalize even over very long time scales. Finally, we show that the absence of thermalization over realistic time scales is consistent with calculations of Landau damping rates of excitations in the finite-temperature condensate.

DOI
10.1103/physreva.84.023606

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Motivated by recent experiments, we consider the dynamics of spin-1 spinor condensates after a quantum quench from the polar to ferromagnetic state from varying the quadratic Zeeman field qqq. We apply the truncated Wigner approximation (TWA) to the spinor system, including all spatial and spin degrees of freedom. For short times, we find full agreement with the linearized Bogoliubov analysis. For longer times, where the Bogoliubov theory fails, we find that the system reaches a quasisteady prethermalized state. We compute the Bogoliubov spectrum about the ferromagnetic state with general qqq and show that the resulting finite-temperature correlation functions grossly disagree with the full TWA results, thus indicating that the system does not thermalize even over very long time scales. Finally, we show that the absence of thermalization over realistic time scales is consistent with calculations of Landau damping rates of excitations in the finite-temperature condensate.
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Recent experimental studies of 87Rb^{87}\text{R}\text{b}87Rb spinor Bose Einstein condensates have shown the existence of a magnetic field driven quantum phase transition accompanied by structure formation on the ferromagnetic side of the transition. In this theoretical study we examine the dynamics of the unstable modes following the transition within the framework of the semiclassical truncated Wigner approximation. In the process we present a systematic study of the effects of the trap, nonlinearities, finite temperature, and dipole-dipole interactions. Starting from an initial state which includes quantum fluctuations, we attempt to make quantitative comparisons with recent experimental data on this system by combining results presented here with those presented previously [S. Leslie, J. Guzman, M. Vengalattore, J. D. Sau, M. L. Cohen, and D. M. Stamper-Kurn, Phys. Rev. A 79, 043631 (2009)]. In the process we estimate the contribution of quantum zero-point fluctuations to the domain formation with quantitative accuracy and find discrepancies between the calculations and experiments at the quantitative level, despite the qualitative agreement between theory and experiment. We discuss the possible origins of these discrepancies. Finally, using the strong anisotropy of the trap, we propose ways to observe directly the effects of dipole-dipole interactions on the spinor condensate dynamics.
Thermalization of isolated quantum systems is a long-standing fundamental problem where different mechanisms are proposed over time. We contribute to this discussion by classifying the diverse quench-dynamical behaviors of spin-1 Bose-Einstein condensates, which includes well-defined quantum collapse and revivals, thermalization, and certain special cases. These special cases are either nonthermal equilibration with no revival but a collapse even though the system has finite degrees of freedom or no equilibration with no collapse and revival. Given that some integrable systems are already shown to demonstrate the weak form of eigenstate thermalization hypothesis (ETH), we determine the regions where ETH holds and fails in this integrable isolated quantum system. The reason behind both thermalizing and nonthermalizing behaviors in the same model under different initial conditions is linked to the discussion of ``rare'' nonthermal states existing in the spectrum. We also propose a method to predict the collapse and revival time scales and find how they scale with the number of particles in the condensate. We use a sudden quench to drive the system to nonequilibrium and hence the theoretical predictions given in this paper can be probed in experiments.
We discuss the ordering of a spin-1 condensate when quenched from its paramagnetic phase to its ferromagnetic phase by reducing the magnetic field. We first elucidate the nature of the equilibrium quantum phase transition. Quenching rapidly through this transition reveals XY ordering either at a specific wave vector, or the "light-cone" correlations familiar from relativistic theories, depending on the end point of the quench. For a quench proceeding at a finite rate the ordering scale is governed by the Kibble-Zurek mechanism. The creation of vortices through growth of the magnetization fluctuations is also discussed. The long-time dynamics again depends on the end point, conserving the order parameter in a zero field, but not at a finite field, with differing exponents for the coarsening of magnetic order. The results are discussed in the light of a recent experiment by Sadler et al.
We examine the dynamics of a quasi-two-dimensional spin-1 condensate in which the quadratic Zeeman energy qqq is suddenly quenched to a value where the system has a ferromagnetic ground state. There are two distinct types of ferromagnetic phases, i.e., a range of qqq values where the magnetization prefers to be in the direction of the external field (easy axis) and a range of qqq values where it prefers to be transverse to the field (easy plane). We study the quench dynamics for a variety of qqq values and show that there is a single dynamic critical exponent to characterize the scale-invariant domain growth for each ferromagnetic phase. For both quenches we give simple analytic models that capture the essential scale-invariant dynamics and correctly predict the exponents. Because the order parameter for each phase is different, the natures of the domains and the relevant topological defects in each type of coarsening are also different. To explore these differences we characterize the fractal dimension of the domain walls and the relationship of polar-core spin vortices to the domains in the easy-plane phase. Finally, we consider how the energy liberated from the quench thermalizes in the easy-axis quench. We show that local equilibrium is established in the spin waves on moderate time scales, but continues to evolve as the domains anneal.
We show that the early time dynamics of easy-axis magnetic domain formation in a spinor condensate is described by percolation theory. These dynamics could be initialized using a quench of the spin-dependent interaction parameter. We propose a scheme to observe the same dynamics by quenching the quadratic Zeeman energy and applying a generalized spin rotation to a ferromagnetic spin-1 condensate. Using simulations we investigate the finite-size scaling behavior to extract the correlation length critical exponent and the transition point. We analyze the sensitivity of our results to the early-time dynamics of the system, the quadratic Zeeman energy, and the threshold condition used to define the positive (percolating) domains.