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International Journal of Modern Physics B

STEADY-SQUEEZING STATES

A. A. Bakasov

Published Jul 10, 1993• 0 citations
Publisher

Abstract

Wave functions have been determined which realize constant squeezing (steady-squeezing states) for a quadrature component of the oscillator with a quadratic self-coupling at an á priori given initial complex amplitude of the oscillator. The set of these wave functions forms a kernel of a special generating operator and is divided into two mutually orthogonal subsets: the subset of superpositions of the states with an even number of quanta of fluctuations around the classical value of the oscillator energy and the subset of analogous states but with odd number of such quanta. The value of the steady squeezing is not a monotonic function of the self-coupling constant and possesses an unexpected minimum. Steady-squeezing states do not minimize the uncertainty product and are not the Gaussian states at the same time.

DOI
10.1142/s0217979293003048

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Wave functions have been determined which realize constant squeezing (steady-squeezing states) for a quadrature component of the oscillator with a quadratic self-coupling at an á priori given initial complex amplitude of the oscillator. The set of these wave functions forms a kernel of a special generating operator and is divided into two mutually orthogonal subsets: the subset of superpositions of the states with an even number of quanta of fluctuations around the classical value of the oscillator energy and the subset of analogous states but with odd number of such quanta. The value of the steady squeezing is not a monotonic function of the self-coupling constant and possesses an unexpected minimum. Steady-squeezing states do not minimize the uncertainty product and are not the Gaussian states at the same time.
Some of the most enduring questions in physics--including the quantum measurement problem and the quantization of gravity--involve the interaction of a quantum system with a classical environment. Two linearly coupled harmonic oscillators provide a simple, exactly soluble model for exploring such interaction. Even the ground state of a pair of identical oscillators exhibits effects on the quantum nature of one oscillator, e.g., a diminution of position uncertainty, and an increase in momentum uncertainty and uncertainty product, from their unperturbed values. Interaction between quantum and classical oscillators is simulated by constructing a quantum state with one oscillator initially in its ground state, the other in a coherent or Glauber state. The subsequent wave function for this state is calculated exactly, both for identical and distinct oscillators. The reduced probability distribution for the quantum oscillator, and its position and momentum expectation values and uncertainties, are obtained from this wave function. The oscillator acquires an oscillation amplitude corresponding to a beating between the normal modes of the system; the behavior of the position and momentum uncertainties can become quite complicated. For oscillators with equal unperturbed frequencies, i.e., at resonance, the uncertainties exhibit a time-dependent quantum squeezing which can be extreme.
Certain superposition (linear combination) of energy eigenfunctions of linear harmonic oscillators are called coherent states because they are of significance in optics for the representation of coherent light waves. A hallmark of coherent states is that the variances (squares of the uncertainties) of x and px are constant in time and further their product becomes the minimum value allowed by the Heisenberg uncertainty principle. On the other hand, some linear combinations of harmonic oscillator energy functions give rise to squeezed states . A curious property of squeezed states is that the variances of x and px oscillate in time 180o out of phase with one another with the frequency twice of the oscillator. The wave packet of a coherent state possesses a minimum uncertainty. 〈x〉 and 〈px〉 have the same oscillatory forms as in the classical case. The coherent states have also been called minimum uncertainty coherent states, the Schrödinger coherent states or the Glauber coherent states [1-6]. The squeezed states are sometimes referred to in the literature as two-photon coherent states and generalized coherent states.
We obtained the uncertainty relation in squeezed states for a time-dependent oscillator. The uncertainty relation in coherent states is same as that of the number states with n=0. However, the uncertainty relation in squeezed states does not satisfy this property and depends on squeezing parameter c. For instance, the uncertainty relation is ℏ/2 which is the minimum value as far as quantum mechanics permits for c=1, same as that in coherent state for c=±∞, and infinity for c=-1. If the time-dependency of the Hamiltonian for the system vanishes, the uncertainty relation in squeezed states will no longer depend on c and becomes the same as that in number state with n=0, like the uncertainty relation in coherent states.
A displacement operator for a subset of the supercoherent states is found. It is also shown that a diagonal version of the squeezing operator reduces the fluctuations of one of the quadratures, but the reduction is not enough to go below the standard quantum noise level. New states are found, which are minimum uncertainty for the new operators X and P, and lead to coherence and squeezing, in the sense of equal fluctuations in the two quadrature and reduction in one of them, respectively. Moreover, these states are associated with the model of Jaynes and Cummings (1963) in the strong-coupling limit.