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arXiv:quant-ph/9802048 · Journal of Physics A Mathematical and General

General approach to functional forms for the exponential quadratic operators in coordinate-momentum space

Xiang‐Bin Wang, C. H. Oh, L. C. Kwek

Published May 8, 1998• 14 citations
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Abstract

In a recent paper (Nieto M M 1996 Quantum Semiclass. Opt. 8 1061, quant-ph/9605032), the one-dimensional squeezed and harmonic oscillator time-displacement operators were reordered in coordinate-momentum space. In this paper, we give a general approach for reordering the multidimensional exponential quadratic operator (EQO) in coordinate-momentum space. An explicit computational formula is provided and applied to the single-mode and double-mode EQO through the squeezed operator and the time-displacement operator of the harmonic oscillator.

DOI
10.1088/0305-4470/31/18/020

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In a recent paper (Nieto M M 1996 Quantum Semiclass. Opt. 8 1061, quant-ph/9605032), the one-dimensional squeezed and harmonic oscillator time-displacement operators were reordered in coordinate-momentum space. In this paper, we give a general approach for reordering the multidimensional exponential quadratic operator (EQO) in coordinate-momentum space. An explicit computational formula is provided and applied to the single-mode and double-mode EQO through the squeezed operator and the time-displacement operator of the harmonic oscillator.
Among the three methods of defining the squeezed states of the harmonic oscillator, only two, i.e. the minimum uncertainty and ladder-operator methods were extended to investigate the squeezed states of the general symmetry systems. The only exception is the displacement-operator formalism. In this letter we propose a displacement-operator approach to the squeezed states for general systems. We show that the ladder-operator squeezed states for general symmetry systems can be obtained by applying an exponential displacement-operator on the ``vacuum'' state. The general formalism is presented and it is illustrated with su(1,1) as an explicit example. As a byproduct, the squeezed states of the q-deformed oscillator are obtained and, in particular, expressed in terms of the {\\it usual exponential} displacement-operator acting the ``vacuum'' state.
The family of displacement operators , a central concept in the theory of coherent states of a quantum mechanical harmonic oscillator, has been successfully generalized to systems of quantized, cyclic or finite position coordinates. However, out of the plethora of mutually equivalent expressions for the displacement operators valid in the continuous case, only few are directly applicable in the other systems of interest. The aim of this paper is to strengthen the analogy between the different cases by identifying the root cause of the issues accompanying the straightforward generalization of certain important expressions and, more importantly, offering alternative ones of general validity. Ultimately we arrive at an algorithm allowing one to express any displacement operator as an exponential of a pure imaginary multiple of a generalized 'quadrature' observable that is not obtained by a linear combination of position and momentum observables but rather by a shear transform of one of them in the system's phase space.
We propose a displacement-operator approach to some aspects of squeezed states for general multiphoton systems. The explicit displacement-operators of the squeezed vacuum and the coherent states are achieved and expresses as the ordinary exponential form. As a byproduct the coherent states of the qqq-oscillator are obtained by the {\it usual exponential} displacement-operator.
Time evolution of squeezed coherent states for a quantum parametric oscillator with the most general self-adjoint quadratic Hamiltonian is found explicitly. For this, we use the unitary displacement and squeeze operators in coordinate representation and the evolution operator obtained by the Wei-Norman Lie algebraic approach. Then, we analyze squeezing properties of the wave packets according to the complex parameter of the squeeze operator and the time-variable parameters of the Hamiltonian. As an application, we construct all exactly solvable generalized quantum oscillator models classically corresponding to a driven simple harmonic oscillator. For each model, defined according to the frequency modification in position space, we describe explicitly the squeezing and displacement properties of the wave packets. This allows us to see the exact influence of all parameters and make a basic comparison between the different models.