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arXiv:quant-ph/0407261

On the Dynamics of Generalized Coherent States. II. Classical Equations of Motions

Blagovest A. Nikolov, D. A. Trifonov

Published Jul 30, 2004• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/0407261

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arXiv:1212.6888•Dec 31, 2012•0 citations
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 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.
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.
We consider an example of a quantum system with a continuous and unbounded spectrum—a particle in the parabolic potential barrier. Conditionally, we call this system the inverse oscillator. Following the method used in our previous publication (Bagrov et al 2012 J. Phys. A: Math. Theor. 45 125306), we construct different families of generalized coherent states (GCS). We discuss properties of the constructed GCS and their relation to what is already known in the literature about semiclassical states for the inverse oscillator. Then in the same manner, we construct families of GCS for the normal oscillator. Using special variables, we succeeded in constructing GCS for the normal oscillator that admit different limiting cases: free particle GCS, the usual Schrödinger, Glauber CS and GCS of the inverse oscillator.
The idea of coherent states has proved very useful in the study of the quantum nature of electromagnetic radiation. Coherent states are also interesting because their evolution in time comes closest to classical dynamics. So they are the best approximation in quantum mechanics to classical behaviour. In recent years, attempts have been made to generalize the concept of coherent states to systems other than those made up of the simple harmonic oscillator (such as light). In one such attempt, Klauder introduces a set of physical criteria that generalized coherent states should have (Klauder J R 1996 J. Phys. A: Math. Gen. 29 L293). These criteria have been applied to the hydrogen atom and several other systems. In this paper, we apply Klauder's criteria to identify generalized coherent states for the quadratic Hamiltonian H = (f1a†a + f*2a2 + f2a†2). We check if two-photon coherent states of light satisfy these criteria. We find that they do not, which is expected: their non-classical properties are well known. We then use Klauder's criteria to find a set of states that are associated with the quadratic Hamiltonian. We find that these states too display non-classical behaviour even for large values of the quantum numbers. This non-classical behaviour is revealed in the form of sub-Poisson photon statistics. We will also briefly consider the one-photon Hamiltonian, apply Klauder's criteria to it and comment on its statistics.
The idea of construction of the nonlinear coherent states based on the hypergeometric- type operators associated to the Weyl-Heisenberg group [J:P hys:A 45(2012) 095304], are generalized to the similar states for the arbitrary Lie group SU(1, 1). By using of a discrete, unitary and irreducible representation of the Lie algebra su(1, 1) wide range of generalized nonlinear coherent states(GNCS) have been introduced, which admit a resolution of the identity through positive definite measures on the complex plane. We have shown that realization of these states for different values of the deformation pa- rameters r = 1 and 2 lead to the well-known Klauder-Perelomov and Barut-Girardello coherent states associated to the Irreps of the Lie algebra su(1, 1), respectively. It is worth to mention that, like the canonical coherent states, GNCS possess the temporal stability property. Finally, studying some statistical characters implies that they have indeed nonclassical features such as squeezing, anti-bunching effect and sub-Poissonian statistics, too.