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arXiv:0909.5164 · Journal of Physics A Mathematical and Theoretical

Nonlinearity and constrained quantum motion

Dorje C. Brody, Anna C T Gustavsson, Lane P. Hughston

Published Feb 8, 2010• 10 citations
arXivHTMLPDF

Abstract

The dynamical equation satisfied by the density matrix when a quantum system is subjected to one or more constraints arising from conserved quantities is derived. The resulting nonlinear evolution of the density matrix has the property that it is independent of the specific composition of the pure-state mixture generating the initial state of the system.

DOI
10.1088/1751-8113/43/8/082003

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The dynamical equation satisfied by the density matrix when a quantum system is subjected to one or more constraints arising from conserved quantities is derived. The resulting nonlinear evolution of the density matrix has the property that it is independent of the specific composition of the pure-state mixture generating the initial state of the system.
Hamiltonian formulation of quantum dynamics and nonlinear constraints are used to derive dynamical equations of a hybrid classical-quantum system. Starting with a compound quantum system in the Hamiltonian formulation, conditions for classical behavior are imposed on one of its subsystems and the corresponding hybrid dynamical equations are derived. The dynamical equations for hybrid systems in pure and in mixed states indicate that the hybrid systems have properties that are not exhausted by those of quantum and classical systems.
Ordinary quantal conservation laws are associated with null operators for time rate of change and are valid for causal evolution of the system through either pure or mixed states. There is, however, a larger class of quantal constants of the motion, some of which are conserved only for pure state evolution. After an analysis of the theoretical origins of these exotic conserved quantities, several illustrations are presented with empirical interpretations based on the quorum theory methods used in Paper I.
We investigate the conserved quantities for a closed, N -state quantum system. A simple derivation from the equation of motion for the density matrix shows that there is an infinite number of such conserved quantities. However, only the first N of these constants of the motion are independent. These constants uniquely determine the degree of mixing in the system, which is itself conserved. We discuss the relationship between these results, the properties of Hamiltonians under unitary transformation and van Vleck's principle of spectroscopic stability.
The dynamics of Gaussian states for systems with Hamiltonians, quadratic in the position and momentum operators, gives rise to the definition of a system of nonlinear coupled differential equations for the density matrix parameters of the system states. In this work, we show that, using the Gaussian-state covariance-matrix evolution, it is possible to solve the system of these nonlinear equations. Some examples of applications of the method developed are given for one- and two-dimensional quadratic-system Gaussian states. It is noted that this formalism can also be used to find new constants of motion related to the covariance matrices of quadratic systems.
The vast majority of the literature dealing with quantum dynamics is concerned with linear evolution of the wave function or the density matrix. A complete dynamical description requires a full understanding of the evolution of measured quantum systems, necessary to explain actual experimental results. The dynamics of such systems is intrinsically nonlinear even at the level of distribution functions, both classically as well as quantum mechanically. Aside from being physically more complete, this treatment reveals the existence of dynamical regimes, such as chaos, that have no counterpart in the linear case. Here, we present a short introductory review of some of these aspects, with a few illustrative results and examples.
High Resolution Image Download MS PowerPoint Slide A practical approach is put forward for a compact representation of the time evolving density matrix of the forced Morse oscillator. This approach uses the factorized product form of the unitary time evolution operator, à la Wei-Norman. This product form casts the time evolution operator in the basis of operators that form a closed Lie algebra. The further requirement that the Hamiltonian of the system be closed within this Lie algebra is satisfied by restricting the dynamics to its sudden limit. One is thereby able to propagate in time both pure and mixed quantum states. As an example, for a thermal initial state, the time-evolved density matrix of maximum entropy is derived, and it is compacted to be described by only three explicit constraints: one time-dependent constraint, which is a dynamical symmetry, and two constants of the motion, with corresponding time-independent coefficients. This representation is a significant reduction from O ( j 2 ) constraints down to just three, where j is the number of bound states of the Morse oscillator.