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arXiv:1602.05724 · Physical review. B./Physical review. B

Scaling behavior of quantum critical relaxation dynamics of a system in a heat bath

Shuai Yin, Chung-Yu Lo, Pochung Chen

Published May 2, 2016• 25 citations
arXivHTMLPDF

Abstract

We study the scaling behavior of the relaxation dynamics to thermal equilibrium when a quantum system is near the quantum critical point. In particular, we investigate systems whose relaxation dynamics is described by a Lindblad master equation. We find that the universal scaling behavior not only appears in the equilibrium stage at the long-time limit but also manifests in the nonequilibrium relaxation process. While the critical behavior is dictated by the low-lying energy levels of the Hamiltonian, the dissipative part in the Lindblad equation also plays important roles in two aspects: First, the dissipative part makes the high-energy levels decay fast, after which the universal behavior controlled by the low-lying modes emerges. Second, the dissipation rate gives rise to a time scale that affects the scaling behavior. We confirm our theory by solving the Lindblad equation for the one-dimensional transverse-field Ising model.

DOI
10.1103/physrevb.93.184301

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Cambridge University Press eBooks•Jan 28, 2015•0 citations
We study the scaling behavior of the relaxation dynamics to thermal equilibrium when a quantum system is near the quantum critical point. In particular, we investigate systems whose relaxation dynamics is described by a Lindblad master equation. We find that the universal scaling behavior not only appears in the equilibrium stage at the long-time limit but also manifests in the nonequilibrium relaxation process. While the critical behavior is dictated by the low-lying energy levels of the Hamiltonian, the dissipative part in the Lindblad equation also plays important roles in two aspects: First, the dissipative part makes the high-energy levels decay fast, after which the universal behavior controlled by the low-lying modes emerges. Second, the dissipation rate gives rise to a time scale that affects the scaling behavior. We confirm our theory by solving the Lindblad equation for the one-dimensional transverse-field Ising model.
We propose that nonequilibrium quantum criticality in open systems under the Born-Markov approximation can be described by a master equation of the Lindblad form. This master equation is derived from a system coupling weakly to a heat bath microscopically and is suggested to provide an approach to study dynamic quantum critical behavior of the system at finite temperatures. We find that the dissipation rate in the equation representing the coupling must be included in the scaling forms as an indispensable additional scaling variable in order to correctly describe the nonequilibrium quantum critical behavior, yet the equilibrium fixed point determines the nonequilibrium critical behavior in the weak coupling limit. Through numerically solving the Lindblad equation for the quantum Ising chain, we affirm these propositions by finite-time scaling forms with the dissipation rate. Nonequilibrium dynamic critical behavior of spontaneous emissions in dissipative open systems at zero temperature near their quantum critical points is discovered and is also described well by the scaling forms.
We study the critical behavior of the nonequilibrium dynamics and of the steady states emerging from the competition between coherent and dissipative dynamics close to quantum phase transitions. The latter is induced by the coupling of the system with a Markovian bath, such that the evolution of the system's density matrix can be effectively described by a Lindblad master equation. We devise general scaling behaviors for the out-of-equilibrium evolution and the stationary states emerging in the large-time limit for generic initial conditions in terms of the parameters of the Hamiltonian providing the coherent driving and those associated with the dissipative interactions with the environment. Our framework is supported by numerical results for the dynamics of a one-dimensional lattice fermion gas undergoing a quantum Ising transition in the presence of dissipative mechanisms which include local pumping and decay of particles.
We investigate the effects of dissipation on the quantum dynamics of many-body systems at quantum transitions, especially considering those of the first order. This issue is studied within the paradigmatic one-dimensional quantum Ising model. We analyze the out-of-equilibrium dynamics arising from quenches of the Hamiltonian parameters and dissipative mechanisms modeled by a Lindblad master equation, with either local or global spin operators acting as dissipative operators. Analogously to what happens at continuous quantum transitions, we observe a regime where the system develops a nontrivial dynamic scaling behavior, which is realized when the dissipation parameter uuu (globally controlling the decay rate of the dissipation within the Lindblad framework) scales as the energy difference $\mathrm{\ensuremath{\Delta}}$ of the lowest levels of the Hamiltonian, i.e., $u\ensuremath{\sim}\mathrm{\ensuremath{\Delta}}$. However, unlike continuous quantum transitions where $\mathrm{\ensuremath{\Delta}}$ is power-law suppressed, at first-order quantum transitions $\mathrm{\ensuremath{\Delta}}$ is exponentially suppressed with increasing the system size (provided the boundary conditions do not favor any particular phase).
We address the out-of-equilibrium dynamics arising from quantum-quench (QQ) protocols (instantaneous changes of the Hamiltonian parameters) in many-body systems within their quantum critical regime and in contact with (homogeneously coupled) thermal baths. We consider two classes of QQ protocols. In one of them, the thermal bath is used to prepare the initial finite-temperature Gibbs state; then, after quenching, the thermal bath is removed and the dynamics of the system is unitary. We also address a more complex QQ protocol where the thermal bath is not removed after quenching, thus the quantum evolution is also driven by the interaction with the bath, which may be described by appropriate master equations for the density matrix of the system, where a further relevant time scale, or inverse decay rate, characterizes the system-bath coupling. Under these QQ protocols, the critical system develops out-of-equilibrium scaling behaviors, which extend those for isolated critical systems, by introducing further scaling variables proportional to the temperature of the thermal bath and the decay rate of the system-bath interactions. These out-of-equilibrium scaling behaviors are checked by analyzing QQ protocols within fermionic Kitaev wires, or equivalently quantum Ising chains, supplemented with a particular modelization of thermal bath that guarantees the asymptotic thermalization within the Lindblad master equation for the dynamics of open systems.
Near continuous phase transitions, universal power-law scaling, characterized by critical exponents, emerges. This behavior reflects the singular responses of physical systems to continuous control parameters like temperature or external fields. Universal scaling extends to non-equilibrium dynamics in isolated quantum systems after a quench, where time takes the role of the control parameter. Our research unveils critical scaling in time also during the relaxation dynamics of an open quantum system. Here we experimentally realize such a system by the spin of individual Cesium atoms dissipatively coupled through spin-exchange processes to a bath of ultracold Rubidium atoms. Through a finite-size scaling analysis of the entropy dynamics via numerical simulations, we identify a critical point in time in the thermodynamic limit. This critical point is accompanied by the divergence of a characteristic length, which is described by critical exponents that turn out to be unaffected by system specifics.
A critical point is associated with a diverging relaxation time which always makes the dynamics across a critical point fascinating. For example, when the temperature of a classical system undergoing a classical phase transition (CPT) at a finite temperature T c is suddenly changed from a value higher than the critical temperature to a lower value across the critical point, the system does not equilibrate instantaneously. Domains of ordered regions are formed which grow following a phase ordering dynamics which leads to a dynamical scaling [104]. In this section, we will discuss the recent studies of non-equilibrium dynamics of quantum systems driven across QCPs where the dynamics is unitary unlike the dynamics across a finite temperature critical point. The non-equilibrium dynamics of a transverse XY spin chain was first investigated in a series of papers [48, 46, 47] where the time evolution of the model was studied in the presence of various time-dependent magnetic fields and the nonergodic behavior of the magnetization was pointed out. A similar result was also obtained in [505]. There is a recent upsurge in studies of non-equilibrium dynamics of a quantum system swept across a QCP. These studies are important for exploring the universality associated with quantum critical dynamics. Moreover, recent experiments with ultracold atomic gases [343, 663, 478, 87] have stimulated numerous theoretical studies. The main properties of these atomic gases are low dissipation rates and phase coherence over a long time so that the dynamics is well described by the usual quantum evolution of a closed system. In the subsequent sections, we shall discuss that when a quantum system initially prepared in its ground state is driven across a QCP, the dynamics fails to be adiabatic however slow the rate of change in the parameters of the Hamiltonian may be. This is due to the divergence of the characteristic time scale of the quantum system, namely, the relaxation time close to the QCP. This non-adiabaticity results in the occurrence of defects in the final state of the quantum Hamiltonian.