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arXiv:quant-ph/0407260

On the Dynamics of Generalized Coherent States. I. Exact and Stable Evolution

Blagovest A. Nikolov, D. A. Trifonov

Published Jul 30, 2004• 2 citations
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Abstract

The exact and stable evolutions of generalized coherent states (GCS) for quantum systems are considered by making use of the time-dependent integrals of motion method and of the Klauder approach to the relationship between quantum and classical mechanics. It is shown that one can construct for any quantum system overcomplete family of states (OFS), related to the unitary representations of the Lie group G by means of integral of motion generators, and the possibility of using this group as a dynamical symmetry group is pointed out. The relation of the OFS with quantum measurement theory is also established.

DOI
10.48550/arxiv.quant-ph/0407260

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The exact and stable evolutions of generalized coherent states (GCS) for quantum systems are considered by making use of the time-dependent integrals of motion method and of the Klauder approach to the relationship between quantum and classical mechanics. It is shown that one can construct for any quantum system overcomplete family of states (OFS), related to the unitary representations of the Lie group G by means of integral of motion generators, and the possibility of using this group as a dynamical symmetry group is pointed out. The relation of the OFS with quantum measurement theory is also established.
Using the Klauder approach the stable evolution of generalized coherent states (GCS) for some groups (SU(2), SU(1,1) and SU(N)) is considered and it is shown that one and the same classical solution z(t) can correctly characterize the quantum evolution of many different (in general nonequivalent) systems. As examples some concrete systems are treated in greater detail: it is obtained that the nonstationary systems of the singular oscillator, of the particle motion in a magnetic field, and of the oscillator with friction all have stable SU(1,1) GCS whose quantum evolution is determined by one and the same classical function z(t). The physical properties of the constructed SU(1,1) GCS are discussed and it is shown particularly that in the case of discrete series D_k^+ they are those states for which the quantum mean values coincide with the statistical ones for an oscillator in a thermostat.
The topic of this Bachelor's thesis is the quantum-to-classical crossover using generalized coherent states (GCS). In other words, the thesis consists of a mathematically rigorous consideration of the transition between quantum theory and classical theory, and how this can be carried out using the theory of GCS. The necessary mathematical background for this is also reviewed. For the theory of generalized coherent states, Lie algebras and Lie groups are first reviewed. Lie groups are groups with a structure of a differentiable manifold. For the Lie groups we consider, there exist coresponding Lie algebras, whose elements generate the Lie groups using the exponential map. Generalized coherent states are a generalization of Glauber states, which are continuous quantum states used in describing the harmonic oscillator. The generalization is carried through using some Lie group other than the Heisenberg-Weyl group used in defining the Glauber states. A special property of GCS is their semiclassical structure: a structure resembling the classical phase space may be defined in the differentiable manifold used in generating the GCS. An example of GCS are the spin coherent states, generated using the Lie group SU(2). For the coherent states to be applicable in the quantum-to-classical crossover, one has to also find some symmetry so that the system under consideration may be described using a single GCS. In the reference literature it is assumed that a global symmetry enables the description of a system of N elements by a coherent state corresponding to a coherent state of a single element. In the spin chain example however, the symmetry can be explicitly applied on the system and it turns out that this assumption is valid: using suitable symmetries, a spin chain may be represented by a single spin coherent state. The quantum-to-classical crossover can be carried out by defining some classical operators whose expectation value with respect to the GCS corresponds to the functions of classical mechanics. In addition to this, the coherent states in question have to fulfill some requirements. In the case of spin coherent states these requirements are fulfilled and the quantum-to-classical crossover may be carried through.
The review is devoted to an analysis of definite overcomplete non-orthogonal state systems that are connected with irreducible representations of Lie groups–the so called systems of generalized coherent states. These systems, which the author is the first to propose, are generalizations of Glauber's coherentstate system and arise in natural fashion in physical problems that have dynamic symmetry. They permit a considerable simplification of the solution of the quantum problem by reducing it to a simpler classical problem. The review deals with the properties of generalized-coherent-state systems connected with the simplest Lie groups.
We present a detailed discussion of a general theory of phase-space distributions, introduced recently by the authors [J. Phys. A 31, L9 (1998)]. This theory provides a unified phase-space formulation of quantum mechanics for physical systems possessing Lie-group symmetries. The concept of generalized coherent states and the method of harmonic analysis are used to construct explicitly a family of phase-space functions which are postulated to satisfy the Stratonovich-Weyl correspondence with a generalized tracing condition. The symbol calculus for the phase-space functions is given by means of the generalized twisted product. The phase-space formalism is used to study the problem of the reconstruction of quantum states. In particular, we consider the reconstruction method based on measurements of displaced projectors, which comprises a number of recently proposed quantum-optical schemes and is also related to the standard methods of signal processing. A general group-theoretic description of this method is developed using the technique of harmonic expansions on the phase space.
In this review, a general algorithm for constructing coherent states of dynamical groups for a given quantum physical system is presented. The result is that, for a given dynamical group, the coherent states are isomorphic to a coset space of group geometrical space. Thus the topological and algebraic structure of the coherent states as well as the associated dynamical system can be extensively discussed. In addition, a quantum-mechanical phase-space representation is constructed via the coherent-state theory. Several useful methods for employing the coherent states to study the physical phenomena of quantum-dynamic systems, such as the path integral, variational principle, classical limit, and thermodynamic limit of quantum mechanics, are described.
This paper discusses an application to the study of dynamics of the typical overcomplete, non-independent sets of unit vectors that characterize continuous-representation theory. It is shown in particular that the conventional, classical Hamiltonian dynamical formalism arises from an analysis of quantum dynamics restricted to an overcomplete, nonindependent set of vectors which lie in one-to-one correspondence with, and are labeled by, points in phase space. A generalized ``classical'' mechanics is then defined by the extremal of the quantum-mechanical action functional with respect to a restricted set of unit vectors whose c-number labels become the dynamical variables. This kind of ``classical'' formalism is discussed in some generality, and is applied not only to simple single-particle problems, but also to finite-spin degrees of freedom and to fermion field oscillators. These latter cases are examples of an important class of problems called exact, for which a study of the classical dynamics alone is sufficient to infer the correct quantum dynamics.