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Approaching quantum dynamics from a classical perspective: dynamical flow, trajectory ensembles and state-space distributions for closed and open quantum systems

Amro Dodin, Adam P. Willard

Published May 24, 2018• 0 citations
OpenAlex

Abstract

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.

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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.
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 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.
The classical and quantum evolution of a generic probability distribution is analyzed. To that end, a formalism based on the decomposition of the distribution in terms of its statistical moments is used, which makes explicit the differences between the classical and quantum dynamics. In particular, there are two different sources of quantum effects. Distributional effects, which are also present in the classical evolution of an extended distribution, are due to the fact that all moments cannot be vanishing because of the Heisenberg uncertainty principle. In addition, the noncommutativity of the basic quantum operators add some terms to the quantum equations of motion that explicitly depend on the Planck constant and are not present in the classical setting. These are thus purely quantum effects. Some particular Hamiltonians are analyzed that have very special properties regarding the evolution they generate in the classical and quantum sector. In addition, a large class of inequalities obeyed by high-order statistical moments, and in particular uncertainty relations that bound the information that is possible to obtain from a quantum system, are derived.
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.
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.
The long term behavior of non-integrable quantum systems is investigated to find a classical counterpart that describes the statistical properties of dynamical fluctuation of quantum observables. On the basis of the quantum-classical correspondence in the integrable system, we introduce the ensemble of classical trajectories (ECT) , which is defined as a set of weighted classical trajectories whose mean density in phase space is one per Planck cell. In the non-integrable system, due to the rapid breakdown of quantum-classical correspondence of the wavepacket dynamics, the ECT is unable to yield the correct temporal behavior of quantum observables. However, numerical experiments reveal that it gives a good approximation of various statistical properties of the quantum dynamical fluctuation. This result strongly suggests that, as far as the statistical properties are concerned, the dominant part of the quantum dynamical fluctuation is determined by classical dynamics, i.e. a set of classical trajectories. The meaning and interpretation of the ECT in relation to the coherent subsystem is also discussed.