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Journal of Modern Optics

Time Evolution of Stable Squeezed States

Xin Ma

Published Aug 1, 1989• 4 citations
Publisher

Abstract

It is shown using a straightforward approach that, for a single mode of an electromagnetic field, only quadratic Hamiltonians can preserve squeezed states for all time. The time-evolution equations for the parameters of a stable squeezed state are given for arbitrary quadratic Hamiltonians. It is also pointed out that there is no single Hamiltonian that preserves minimum uncertainty for all squeezed minimum-uncertainty states.

DOI
10.1080/09500348914551091

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Journal of Physics A Mathematical and General•Mar 11, 1984•102 citations
It is shown using a straightforward approach that, for a single mode of an electromagnetic field, only quadratic Hamiltonians can preserve squeezed states for all time. The time-evolution equations for the parameters of a stable squeezed state are given for arbitrary quadratic Hamiltonians. It is also pointed out that there is no single Hamiltonian that preserves minimum uncertainty for all squeezed minimum-uncertainty states.
The amplitude of the electric field of a mode of the electromagnetic field is not a fixed quantity: there are always quantum mechanical fluctuations. The amplitude, having both a magnitude and a phase, is a complex number and is described by the mode annihilation operator a. It is also possible to characterize the amplitude by its real and imaginary parts which correspond to the Hermitian and anti-Hermitian parts of a, X sub 1 = 1/2(a(sup +) + a) and X sub 2 = i/2(a(sup +) - a), respectively. These operators do not commute and, as a result, obey the uncertainty relation (h = 1) delta X sub 1(delta X sub 2) greater than or = 1/4. From this relation we see that the amplitude fluctuates within an 'error box' in the complex plane whose area is at least 1/4. Coherent states, among them the vacuum state, are minimum uncertainty states with delta X sub 1 = delta X sub 2 = 1/2. A squeezed state, squeezed in the X sub 1 direction, has the property that delta X sub 1 is less than 1/2. A squeezed state need not be a minimum uncertainty state, but those that are can be obtained by applying the squeeze operator.