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Generalized coherent states and the quadratic Hamiltonian

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 = ℏ(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.