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arXiv:1502.03996 · Physical Review A

Quantum decoherence and thermalization at finite temperature within the canonical-thermal-state ensemble

M. A. Novotny, Fengping Jin, Shengjun Yuan, Seiji Miyashita, Hans A. De Raedt, Kristel F. L. Michielsen

Published Mar 7, 2016• 12 citations
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Abstract

We study measures of decoherence and thermalization of a quantum system $S$ in the presence of a quantum environment (bath) $E$. The entirety $S+E$ is prepared in a canonical-thermal state at a finite temperature; that is, the entirety is in a steady state. Both our numerical results and theoretical predictions show that measures of the decoherence and the thermalization of $S$ are generally finite, even in the thermodynamic limit, when the entirety $S+E$ is at finite temperature. Notably, applying perturbation theory with respect to the system-environment coupling strength, we find that under common Hamiltonian symmetries, up to first order in the coupling strength it is sufficient to consider $S$ uncoupled from $E$, but entangled with $E$, to predict decoherence and thermalization measures of $S$. This decoupling allows closed-form expressions for perturbative expansions for the measures of decoherence and thermalization in terms of the free energies of $S$ and of $E$. Large-scale numerical results for both coupled and uncoupled entireties with up to 40 quantum spins support these findings.

DOI
10.1103/physreva.93.032110

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We study measures of decoherence and thermalization of a quantum system SSS in the presence of a quantum environment (bath) EEE. The entirety S+ES+ES+E is prepared in a canonical-thermal state at a finite temperature; that is, the entirety is in a steady state. Both our numerical results and theoretical predictions show that measures of the decoherence and the thermalization of SSS are generally finite, even in the thermodynamic limit, when the entirety S+ES+ES+E is at finite temperature. Notably, applying perturbation theory with respect to the system-environment coupling strength, we find that under common Hamiltonian symmetries, up to first order in the coupling strength it is sufficient to consider SSS uncoupled from EEE, but entangled with EEE, to predict decoherence and thermalization measures of SSS. This decoupling allows closed-form expressions for perturbative expansions for the measures of decoherence and thermalization in terms of the free energies of SSS and of EEE. Large-scale numerical results for both coupled and uncoupled entireties with up to 40 quantum spins support these findings.
In this review, we discuss the decoherence and thermalization of a quantum spin system interacting with a spin bath environment, by numerically solving the time-dependent Schrödinger equation of the whole system. The effects of the topologic structure and the initial state of the environment on the decoherence of the two-spin and many-spin system are discussed. The role of different spin-spin coupling is considered. We show under which conditions the environment drives the reduced density matrix of the system to a fully decoherent state, and how the diagonal elements of the reduced density matrix approach those expected for the system in the microcanonical or canonical ensemble, depending on the character of the additional integrals of motion. Our demonstration does not rely on time-averaging of observables nor does it assume that the coupling between system and bath is weak. Our findings show that the microcanonical distribution (in each eigenenergy subspace) and canonical ensemble are states that may result from pure quantum dynamics, suggesting that quantum mechanics may be regarded as the foundation of quantum statistical mechanics. Furthermore, our numerical results show that a fully decoherent quantum system prefers to stay in an equilibrium state with a maximum entropy, indicating the validity of the second law of thermodynamics in the decoherence process.
The rather new field of quantum thermodynamics intends to derive thermodynamics from quantum mechanics, as well as to study thermodynamical behavior of quantum systems. It turns out that, indeed, thermalization can be obtained even for very small quantum systems due to the interaction with an appropriate environment. The consideration of an ensemble, as in statistical mechanics, is not necessary. However, the total system - consisting of system and environment - in quantum thermodynamics is usually considered to be closed. In particular, there exists no interaction with an external observer. Since quantum measurements typically influence the measured system, one may wonder which influence such an observation will have in the quantum thermodynamical setting. In this thesis, we study this influence of measurements in quantum thermodynamical systems. It turns out that various concepts known from statistical mechanics arise from such a "quantum thermodynamics under observation", although they do not have any meaning in the isolated quantum thermodynamical setting. The model discussed in the first part consists of a two-level system coupled to a large spin environment, the magnetization of which is measured periodically. This leads to quasi-classical trajectories for the system and - as in statistical mechanics - thermalization and temperature only have a meaning in terms of an ensemble average or a long-time average. Calculating these averages, we prove the ergodicity of the system. Interestingly, the entropy of the obtained attractor state depends on the time between the measurements and can be shown to be connected with the subjective lack of knowledge of the observer. In case of optimal measurements, the ensemble averaged state of the system under periodic measurements equals the state of the system in the quantum thermodynamical model without measurements. Introducing a detuning between system and environment, we find that the system can be prepared in almost arbitrary thermal states due to the periodic measurements. Even the creation of inversion becomes possible. As shown in numerical simulations, our results, which were analytically obtained for large environments, are even applicable to describe the behavior of the system coupled to small environments consisting of only a few spins. Besides measurements of the environment, we also study direct projective measurements of the system, as well as measurements with finite duration, obtaining that the dynamics of the system is almost independent of the concrete realization of the measurements. In the second part of the thesis, we discuss the meaning of temperature fluctuations in quantum thermodynamical systems. In statistical mechanics, there exist different approaches to the problem of temperature fluctuations, which lead to the well-known temperature fluctuation formula. The nature and operational accessibility of these fluctuations, however, remains rather unclear. We argue that according to quantum thermodynamics a system relaxes into a quasi-stationary thermal state and therefore the temperature does not fluctuate. However, since temperature is not an observable, it can only be determined indirectly. For modular systems, we show that the fluctuations of the temperature estimate obtained from a single energy measurement, indeed, are given by the temperature fluctuation formula derived in statistical mechanics, but only within a certain temperature range. The validity range of this formula increases with increasing size of the system. Thus, one can conclude that these "temperature fluctuations" can be interpreted as representing a fundamental limit for the instantaneous determination of temperature.
We consider quenches of a quantum system that is prepared in a canonical equilibrium state of one Hamiltonian and then evolves unitarily in time under a different Hamiltonian. Technically, our main result is a systematic expansion of the pre- and postquench canonical ensembles in the quench strength. We first demonstrate how this can be used to predict the system's temperature after the quench from equilibrium properties at the prequench temperature. For a thermalizing postquench system, it furthermore allows us to calculate equilibrium observable expectation values. Finally, in the presence of additional conserved quantities besides the Hamiltonian, we obtain a hierarchy of necessary conditions for thermalization towards the (postquench) canonical ensemble. At first order, these thermalization conditions have a nice geometric interpretation in operator space with the canonical covariance as a semi-inner product: The quench operator (difference between post- and prequench Hamiltonians) and the conserved quantity must be orthogonal in the orthogonal complement of the postquench Hamiltonian. We illustrate the results numerically for a variety of setups involving integrable and nonintegrable models.
The role of the system-bath coupling is studied for the generalized canonical thermalization of a system S surrounded by a heat bath B, which was most recently shown [S. Popescu, et al., Nature Physics 2,754(2006) and S. Goldstein et al., Phys. Rev. Lett. 96, 050403(2006)] as a kinematic reduction to a canonical equilibrium state from almost all the pure states of the universe formed by S plus B. This kinematic process is revisited by explicitly considering the energy shell deformation due to the interaction between S and B with an exactly solvable model. By calculating the state numbers of the universe and its subsystems S and B in the deformed energy shells, it is found that, for the overwhelming majority of the universe states (they are entangled at least), the diagonal canonical typicality remains robust with respect to finite interactions between S and B. Particularly, the kinematic decoherence is utilized here to account for the vanishing of the off-diagonal elements of the reduced density matrix of S. It is pointed out that the non-vanishing off-diagonal elements due to the finiteness of bath and the stronger system-bath interaction might offer novelties of the quantum thermalization.
We present an exactly solvable model to study the role of the system-bath coupling for the generalized canonical thermalization, which reduces almost all the pure states of the ``universe'' (formed by a system SSS plus its surrounding heat bath BBB) to a canonical equilibrium state of SSS. It is found that, for the overwhelming majority of the ``universe'' states (they are entangled at least), the diagonal canonical typicality remains robust with respect to finite interactions between SSS and BBB. Particularly, a decoherence mechanism is utilized here to account for the vanishing of the off-diagonal elements of the reduced density matrix of SSS. The nonvanishing off-diagonal elements due to the finite size of the bath and the stronger system-bath interaction might offer more to quantum thermalization.