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arXiv:1805.09756 · The Journal of Chemical Physics

State space distribution and dynamical flow for closed and open quantum systems

Amro Dodin, Adam P. Willard

Published Aug 9, 2019• 6 citations
arXivHTMLPDF

Abstract

We present a general formalism for studying the effects of heterogeneity in open quantum systems. We develop this formalism in the state space of density operators, on which ensembles of quantum states can be conveniently represented by probability distributions. We describe how this representation reduces ambiguity in the definition of quantum ensembles by providing the ability to explicitly separate classical and quantum sources of probabilistic uncertainty. We then derive explicit equations of motion for state space distributions of both open and closed quantum systems and demonstrate that resulting dynamics take a fluid mechanical form analogous to a classical probability fluid on Hamiltonian phase space, thus enabling a straightforward quantum generalization of Liouville’s theorem. We illustrate the utility of our formalism by analyzing the dynamics of an open two-level system using the state-space formalism that is shown to be consistent with the derived analytical results.

DOI
10.1063/1.5100736

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We present a general formalism for studying the effects of heterogeneity in open quantum systems. We develop this formalism in the state space of density operators, on which ensembles of quantum states can be conveniently represented by probability distributions. We describe how this representation reduces ambiguity in the definition of quantum ensembles by providing the ability to explicitly separate classical and quantum sources of probabilistic uncertainty. We then derive explicit equations of motion for state space distributions of both open and closed quantum systems and demonstrate that resulting dynamics take a fluid mechanical form analogous to a classical probability fluid on Hamiltonian phase space, thus enabling a straightforward quantum generalization of Liouville’s theorem. We illustrate the utility of our formalism by analyzing the dynamics of an open two-level system using the state-space formalism that is shown to be consistent with the derived analytical results.
We present a general formalism for studying the effects of dynamical heterogeneity in opens quantum systems. We develop this formalism in the state-space of density operators, on which ensembles of quantum states can be conveniently represented by probability distributions. We describe how this representation reduces ambiguity in the definition of quantum ensembles by providing the ability to explicitly separate classical and quantum sources of probabilistic uncertainty. We then derive explicit equations of motion for state-space distributions of both open and closed quantum systems and demonstrate that resulting dynamics take a fluid mechanical form analogous to a classical probability fluid on Hamiltonian phase-space, thus enabling a straightforward quantum generalization of Liouville's theorem. We illustrate the utility of our formalism by analyzing the dynamics of an open two-level system using the state-space formalism that are shown to be consistent with the derived analytical results.
We demonstrate that if the dynamics of quantum systems are expressed in the state-space of den- sity operators, then they exhibit dynamical flow properties that are analogous to classical Hamilto- nian systems. We show that individual quantum states can be formulated equivalently as different distributions on this state-space and that ambiguity in this formulation can be leveraged to gain information about how various sources of uncertainty influence system dynamics. We derive explicit equations of motion for populations on this state-space for both closed and open quantum systems. We argue that trajectory ensembles expressed in this state-space can be analyzed and interpreted using the tools of classical statistical mechanics and along these lines we present a quantum gener- alization of transition path sampling. Finally, as an example, we explore trajectory ensembles of a model open quantum system.
A quantum mechanical representation suitable for studying the time evolution of quantum densities in phase space is proposed and examined in detail. This representation on ℒ2 (2) phase space is based on definitions of the operators P̂ and Q̂ in phase space that satisfy various correspondences for the Liouville equation in classical and quantum phase space, as well as quantum position and momentum ℒ2 (1) spaces. The definitions presented here, P̂=p/2−iℏ∂/∂q and Q̂=q/2+iℏ∂/∂p, are related to definitions that have been recently proposed [J. Chem. Phys. 93, 8862 (1990)]. The resulting quantum phase space representation shares many of the mathematical properties of usual representations in coordinate and momentum spaces. Within this representation, time evolution equations for complex-valued functions (wave functions) and their square magnitudes (distribution functions) are derived, and it is shown that the coordinate and momentum space time evolution equations can be recovered by a simple Fourier projection. The phase space quantum probability conservation equation obtained is a good illustration of the quantization rule that requires one to replace the classical Poisson bracket between the Hamiltonian and the probability density with the quantum commutator between the corresponding operators. The possible classical analogs to quantum probabilities densities are also considered and some of the present results are illustrated for the dynamics of the coherent state.
The starting point of the classical description of motion is the Newton equations that yield a phase space trajectory (rN (t), pN (t)) for a given initial condition (rN (0), pN (0)). Alternatively one may describe classical motion in the framework of the Liouville equation (Section (1.2.2)) that describes the time evolution of the phase space probability density f (rN , pN ; t). For a closed system fully described in terms of a well specified initial condition, the two descriptions are completely equivalent. Probabilistic treatment becomes essential in reduced descriptions that focus on parts of an overall system, as was demonstrated in Sections 5.1–5.3 for equilibrium systems, and in Chapters 7 and 8 that focus on the time evolution of classical systems that interact with their thermal environments. This chapter deals with the analogous quantum mechanical problem. Within the limitations imposed by its nature as expressed, for example, by Heisenbergtype uncertainty principles, the Schrödinger equation is deterministic. Obviously it describes a deterministic evolution of the quantum mechanical wavefunction. The analog of the phase space probability density f (rN , pN ; t) is now the quantum mechanical density operator (often referred to as the “density matrix”), whose time evolution is determined by the quantum Liouville equation. Again, when the system is fully described in terms of a well specified initial wavefunction, the two descriptions are equivalent. The density operator formalism can, however, be carried over to situations where the initial state of the system is not well characterized and/or a reduced description of part of the overall system is desired. Such situations are considered later in this chapter.
A formulation of quantum statistical ensembles in terms of probability distributions on a projective Hilbert space is developed. The combination of statistically independent systems and the reduction of a system to one of its subsystems are described by means of a tensor product of probability distributions and a reduction formula for the reduced probability distribution. Within this framework the dynamics of open quantum systems is investigated starting form a microscopic system-plus-reservoir model. Employing the Markov approximation of the classical theory of stochastic processes the short time behavior of the conditional transition probability is derived and shown to yield a differential Chapman-Kolmogorov equation. The latter has the form of a Liouville master equation for the reduced probability distribution of the open quantum system. Thus it is shown that the time-dependent wave function of an open quantum system represents a well-defined and unique stochastic process in the space of rays of the underlying Hilbert space. This stochastic process consists of a continuous time evolution generated by a nonlinear Schr\"odinger equation and a discontinuous jump process; the realizations of the processes correspond to those of the Monte Carlo wave function simulation method and to those of the jump-type evolution of the quantum trajectory method. The quantum master equation for the statistical operator is derived as the equation of motion for the two-point correlation function of the stochastic process.
General statistical ensembles in the Hamiltonian formulation of hybrid quantum-classical systems are analyzed. It is argued that arbitrary probability densities on the hybrid phase space must be considered as the class of possible physically distinguishable statistical ensembles of hybrid systems. Nevertheless, statistical operators associated with the hybrid system and with the quantum subsystem can be consistently defined. Dynamical equations for the statistical operators representing the mixed states of the hybrid system and its quantum subsystem are derived and analyzed. In particular, these equations irreducibly depend on the total probability density on the hybrid phase space.