We extend recent results on expectation values of coherent oscillator states and SU(2) coherent states to the case of the discrete representations of su(1,1). Systematic semiclassical expansions of products of arbitrary operators are derived. In particular, the leading order of the energy uncertainty of an arbitrary Hamiltonian is found to be given purely in terms of the time dependence of the classical variables. The coherent states considered here include the Perelomov-Gilmore (PG) coherent states. As an important application we discuss the pseudoharmonic oscillator and compare the PG states with the states introduced by Barut and Girardello. The latter ones turn out to be closer to the classical limit as their relative energy variance decays with the inverse square root of energy, while in the former case a constant is approached.
Abstract We extend recent results on expectation values of coherent oscillator states and SU(2) coherent states to the case of the discrete representations of su(1,1). Systematic semiclassical expansions of products of arbitrary operators are derived. In particular, the leading order of the energy uncertainty of an arbitrary Hamiltonian is found to be given purely in terms of the time dependence of the classical variables. The coherent states considered here include the Perelomov–Gilmore (PG) coherent states. As an important application we discuss the pseudoharmonic oscillator and compare the PG states with the states introduced by Barut and Girardello. The latter ones turn out to be closer to the classical limit as their relative energy variance decays with the inverse square root of energy, while in the former case a constant is approached.
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 quantum dynamics of arbitrary
N-level systems, including dissipative systems, are modeled exactly here by the dynamics of classical coupled oscillators. A one-to-one correspondence is established between the quantum states and the positions of the oscillators. Quantum coherence, expectation values, and measurement probabilities for system observables can therefore be realized from the corresponding classical states. Although the well-known equivalence [SU(2), SO(3) homomorphism] of two-level quantum dynamics to a rotation in real, physical space cannot be generalized to arbitrary
N-level systems, the representation of quantum dynamics by a system of coupled harmonic oscillators in one physical dimension
is general for any
N. The time evolution of an
N-level system [generated by a complex element of the SU(
N) group], is first represented as the rotation of a real state vector in (unphysical) hyperspace, as previously known for density matrix states and also extended here to include Schr\"odinger states. The resulting rotor in
n Euclidean dimensions [the rotation group SO(
n)] is then mapped directly to
n oscillators in one physical dimension, which significantly reduces the level of abstraction required to visualize quantum dynamics compared to vector models or generalized Bloch spheres in higher dimensions. The number of such oscillators needed to represent
N-level systems scales as
N2 for the density matrix formalism but increases only linearly with
N for Schr\"odinger states. Values for the classical coupling constants are readily derived from the system Hamiltonian, allowing construction of classical mechanical systems that provide insight into the dynamics of abstract quantum systems (new dynamical invariants) as well as a metric for characterizing the interface between quantum and classical mechanics. A distinctive attribute of the quantum-classical connection as presented here is the necessity for both positive and negative couplings and, in the case of dissipative systems, antisymmetric couplings.
Abstract Using coherent states as initial states, we investigate the quantum dynamics of the Lipkin–Meshkov–Glick (LMG) and Dicke models in the semi-classical limit. They are representative models of bounded systems with one- and two-degrees of freedom, respectively. The first model is integrable, while the second one has both regular and chaotic regimes. Our analysis is based on the survival probability. Within the regular regime, the energy distribution of the initial coherent states consists of quasi-harmonic sub-sequences of energies with Gaussian weights. This allows for the derivation of analytical expressions that accurately describe the entire evolution of the survival probability, from t = 0 to the saturation of the dynamics. The evolution shows decaying oscillations with a rate that depends on the anharmonicity of the spectrum and, in the case of the Dicke model, on interference terms coming from the simultaneous excitation of its two-degrees of freedom. As we move away from the regular regime, the complexity of the survival probability is shown to be closely connected with the properties of the corresponding classical phase space. Our approach has broad applicability, since its central assumptions are not particular of the studied models.