Evolution Of Quantum System Research Articles

Stochastic evolution in quantum systems

Starting from the standard Lindblad form of a dissipative master equation, we identify a stochastic parameter in the quantum time evolution. This lacks dynamic evolution but its value is affected by the dissipative part of the master equation. An unravelling of the master equation in terms of quantum trajectories then shows this parameter performing the erratic variation characteristic of a stochastic variable. When the two evolutions separate, the stochastic Markov process can be extracted and solved independently and imposed on the parameter of the quantum evolution. This is the traditional form of treatment of stochastic variables in quantum mechanics. The relations to earlier works are established, and the approach is illustrated by simple model examples.

New derivation of the geometric phase

Using the usual distance function (the Fubini-Study metric) and a new distance function on the projective Hilbert space, the geometric phase is defined for non-cyclic, non-unitary and non-Schrödinger evolutions of quantum systems.

Quantum jumps, atomic shelving and Monte Carlo fluorescence spectra

We study the resonance fluorescence dynamics of a driven three-level atom in a “V”-configuration in which one transition is to a strongly fluorescing level and the other transition is to a metastable shelving state. This is the configuration in which quantum jumps, periods of bright fluorescence randomly interrupted by periods of darkness, have been studied experimentally using single three-level trapped ions. We use a Monte Carlo wavefunction simulation method to describe the evolution of single quantum systems. We concentrate on mean values for populations and intensities and also correlation functions for sequential detection of decay photons. For a single ensemble member the population of the state |;1〉 undergoes rapid oscillations which are interrupted by the detection of a photon. This resets the system in the ground state |;0〉. Very infrequently the oscillations die out as the system evolves into the metastable level |;2〉; this is the period of darkness which is terminated (in direct detection) by a quantum jump into the ground state. The ensemble averaged population shares features of the second order correlation function g (2) which we calculate using a recently-developed Monte-Carlo wave function method. We present results for the first and second order correlation functions and the spectra.

Gauge-invariant reference section and geometric phase

We use a gauge-invariant 'reference section' and define the geometric phase for all quantum evolutions in a closed form. This geometric phase is obtained by integrating the inner product of the 'reference section' and its path derivative along the evolution curve which is valid for non-cyclic, non-unitary and non-Schrodinger evolutions of quantum systems. Two non-trivial examples are studied to realize our new expression.

High-order adiabatic approximation for non-Hermitian quantum system and complexification of Berry's phase

In this paper the evolution of a quantum system driven by a non-Hermitian Hamiltonian depending on slowly-changing parameters is studied by building an universal high-order adiabatic approximation (HOAA) method with Berry's phase, which is valid for either the Hermitian or the non-Hermitian cases. This method can be regarded as a non-trivial generalization of the HOAA method for closed quantum system presented by this author before. In a general situation, the probabilities of adiabatic decay and non-adiabatic transitions are explicitly obtained for the evolution of the non-Hermitian quantum system. It is also shown that the non-Hermitian analog of the Berry's phase factor for the non-Hermitian case just enjoys the holonomy structure of the dual linear bundle over the parameter manifold. The non-Hermitian evolution of the generalized force harmonic oscillator is discussed as an illustrative example.

Complex scaling and Lyapunov converters

AbstractIt is proved that for a certain class of potentials dilation operators used in the method of complex scaling (CSM) represent a special case of Lyapunov converters. The time evolution of quantum systems is investigated from the viewpoint of this result, with special emphasis on the time evolution of resonant functions. © 1993 John Wiley & Sons, Inc.

Time evolution of chaotic quantum systems

We consider a quantum system which is open in the sense that a number of metastable states are coupled to a set of decay channels. A solution of the time dependent Schro¨dinger equation is developed that allows us to describe both the excitation and the decay process. Assuming the excitation to be practically instantaneous, the decay is studied in detail for systems which have a random Hamiltonian or—what is closely related—behave chaotically. The decay function is worked out in a form amenable to numerical evaluation with standard techniques. For several instructive special cases, it is evaluated in closed analytical form. It turns out to be generally neither exponential nor independent of the way by which the system was excited. It is exponential and independent of the excitation only if a large number of decay channels are present. If only one channel is open, the decay function is asymptotically proportional to t −3/2.

Time evolution of chaotic quantum systems: H. L. Harney and F.-M. Dittes. Zentralinstitut für Kernforschung Rossendorf, Insitut für subatomare Physik, O-8051 Dresden, Germany; A. Müller. Max-Planck-Institut für Kernphysik, W-6900 Heidelberg, Germany

Time evolution of chaotic quantum systems: H. L. Harney and F.-M. Dittes. Zentralinstitut für Kernforschung Rossendorf, Insitut für subatomare Physik, O-8051 Dresden, Germany; A. Müller. Max-Planck-Institut für Kernphysik, W-6900 Heidelberg, Germany

Dissipative quantum dynamics: solution of the generalized von Neumann equation for the damped harmonic oscillator

A recently proposed generalized von Neumann equation describing dissipative time evolution of quantum systems is applied to a damped and driven harmonic oscillator. Utilizing Lie algebraic methods the nonlinear operator equation is solved and from this solution the behaviour of the mean value (x) is extracted, including both amplitude and phaseshift relative to the driving force. A differential equation for (x) is derived and the damping constant is identified.

Minimum relative entropies of low-dimensional spin systems

Abstract Work of Band and Park in the mid-1970's in which they proposed — on information-theoretic grounds — an alternative to the von Neumann entropy measure, S(ϱ) = -Tr ϱ ln ϱ, of a density matrix (ϱ) has apparently not been further applied. In this paper, however, specific measures are generated of the information-theoretic entropy of spin - 1 2 , spin-1 and two-photon mixed states. For this purpose, the minimum relative entropies of arbitrary mixed states with respect to uniform prior distributions over the pure states are determined. Though Band and Park did not specifically discuss this approach, it is contended that it is harmonious with their position. The duality theory of convex programming is employed to interrelate the von Neumann entropy and the minimum relative entropy measure adopted. Some concluding remarks are made on the possible use of such relative entropy indices in modeling the nonunitary (irreversible) evolution of quantum systems.

Comment concerning the optimum control of transformations in an unbounded quantum system

We discuss the achievability of optimal control of the evolution of an unbounded quantum system by the action of external fields. Previous work has established that the evolution of a quantum system with a nondegenerate discrete and bounded spectrum of states can, in principle, be fully controlled, i.e., that the system wave function can be guided by external fields to approach arbitrarily closely a selected target state wave function. The optimal control of the evolution of a quantum system with a discrete and bounded spectrum of states has also been studied, using a method of analysis that depends on the localized character of bound state wave functions and the fact that the spectrum of states is bounded. In this paper, we examine whether it is possible to control, partially or fully, the evolution of an unbounded quantum system. We show that optimal control of the evolution of an unbounded quantum system is possible, in the sense that it is possible to minimize the difference between the product function formed at time t0 from a localizing function and a continuum wave function and a similarly defined target function at time tf. We have not been able to establish that such optimal control is equivalent to full control, i.e., that the difference between the initial and target functions can be made arbitrarily small, which would require showing that the set of control functions is complete with respect to the function space of the system. Our analysis establishes the existence of external fields that can optimally guide the unbounded system evolution in the absence of other constraints than the Schrödinger equation of motion, but does not provide an algorithm for the construction of such fields. The relationship between optimal control of the evolution of a quantum system and the existence of chaotic dynamics in the continuum domain is briefly discussed. We argue that the optimally controlled system cannot have chaotic dynamics even if the bare system does. As illustrations of the results obtained, we discuss briefly the optimal control of evolution in the subset of discrete states from a spectrum with both discrete and continuum states, the optimal control of evolution in resonant states, and the inversion of the optimally controlled localized product continuum target function to yield the corresponding system wave function.

Gauge invariant asymptotic expansion of schrödinger propagators on manifolds

A semiclassical technique is developed for approximating the time evolution of quantum systems on semi-Riemannian manifolds in the presence of external electromagnetic fields. In particular, a geometrically covariant and gauge covariant formal asymptotic expansion of the Schrödinger propagator is derived. The theory is applicable not only to spinless nonrelativistic quantum systems with constraints, but also to general-relativistic scalar field theory through the Schwinger-DeWitt equation. The basic classical objects in this approach are the geodesics of the manifold. The expansion—in the derivatives of the gravitational and electromagnetic fields—is valid when both propagator arguments lie in a geodesically convex region. The zeroth order approximation consists of the wkb propagator for free quantum evolution on the manifold, modified by a gauge-integral phase factor which carries all the gauge structure of the exact propagator. Higher order expansion coefficients are gauge invariant solutions of a one-dimensional integral recurrence relation derived from geodesic transport. By using a Jacobi field and a Green function associated with the geodesic deviation equation, the expansion coefficients can be put into an explicit form. The first one is computed in detail and manifestly displays the effects of gravity and electromagnetism, both separately and in combination. The expansion derived here generalizes the well-known short time expansion in several ways.

Non-Hermitian evolution of two-level quantum systems.

In this paper we discuss in detail the dynamics of a two-state system ruled by a non-Hermitian Hamiltonian. The relevance of the results to multiphoton ionization is also discussed and a comparison to earlier works is presented.

A static ensemble approximation for stochastically modulated quantum systems

A short-time approximation for the evolution of quantum systems governed by a Hamiltonian with stochastic time dependence is derived. The evolution operator of the system is replaced by an ensemble of evolution operators with time-independent Hamiltonians, weighted by a distribution function related to the ‘‘line shape function’’ of a randomly modulated harmonic oscillator. The approximation conserves the trace of the density matrix and converges to the exact solution in the case of very slow and in the case of very rapid stochastic modulation. The approximation is applied to hyperfine-induced singlet–triplet transitions of a biradical-like system with fluctuating exchange interaction.

Some aspects of approach to equilibrium in classical and quantum systems

AbstractInvestigations of general properties of time evolution of both classical and quantum systems show that the evolution (on a microscopic level) is reversible, and, if there is an approach to equilibrium, it exists only in the sense of weak convergence and necessarily in both directions of time. In this connection attempts to derive irreversibility and the approach to equilibrium in the sense of strong convergence and without any loss of information (using the so‐called Lyapunov converters) are discussed. After demonstrating their drawbacks, an alternative approach is proposed based on the spectral properties of the generator of the group. The relevant spectral criteria for the approach of both classical and quantum ensembles to equilibrium are given, and the decisive role of loss of information in obtaining stronger variants of convergence is emphasized. The asymmetry between initial and any other state —consisting of different levels of information needed to prepare the states—is proposed as an adequate explanation of irreversibility. The importance of the method of complex scaling in this context is discussed, and its application in proving some spectral properties in the quantal Hamiltonian formulation is stressed.

Diffusion processes with singular drift fields

A class of stochastic differential equations with highly singular drift fields is considered. Using a purely probabilistic approach, we can show the unattainability of the nodal set. Moreover, a global existence and uniqueness theorem for diffusion processes with singular drift fields is established. The finite action condition of Carlen and Zheng can be modified. We relate our results to the diffusions which describe the time evolution of quantum systems in stochastic mechanics.

N-level quantum systems with SU(2) dynamic symmetry

We present an analytic solution of the coherent evolution of a laser-driven N-level quantum system that possesses an SU(2) dynamic symmetry.

The time evolution of quantum systems with discrete energy spectra

Using the method of matrix functions, we describe the time evolution of quantum systems with discrete energy spectra. The time evolution of the state vectors and expressions for the perturbed eigenfunctions and their associated projectors are obtained for an atom with n discrete levels. The particular cases n = 2 and n = 3 are presented; level crossing and nonexponential decay of coupled states are emphasized.

Evolution of quantum systems driven by a Hamiltonian written in terms of the SU(1,1) group generators.

In this paper we exploit the Wei-Norman algebraic disentangling procedure to study the evolution of quantum states driven by a time-dependent Hamiltonian linear combination of the generators of the SU(1,1) group.

Finite-difference equation and quasi-diagonal form in quantum-statistical mechanics

We consider a finite-difference Liouville-von Neumann equation for the density operator ϱ which shows some peculiar new features of the evolution of a quantum system. In particular this equation automatically leads to the quasi-diagonal form in the energy representation. Furthermore, the new formulation proposed by us seems to be of interest for a consistent quantum theory of measurement.