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arXiv:quant-ph/0108080 · John Wiley & Sons, Inc. eBooks

Quantum-Optical States in Finite-Dimensional Hilbert Space. I. General Formalism

Adam Miranowicz, W. Leoński, Nobuyuki Imoto

Published Apr 2, 2003• 37 citations
arXivHTMLPDF

Abstract

This chapter presents two essentially different constructions of harmonic oscillator states in a finite-dimensional (FD) Hilbert space. The authors propose some new definitions of the states and find their explicit forms in the Fock representation. For the convenience of the reader, they bring together several known FD quantum-optical states, including FD coherent states, FD phase coherent states, FD displaced number states, FD Schrödinger cats, and FD squeezed vacuums. The discrete Wigner function is applied in order to demonstrate intriguing properties of the various states.

DOI
10.1002/0471231479.ch3

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Journal of Modern Optics•Mar 1, 1994•5 citations
This chapter presents two essentially different constructions of harmonic oscillator states in a finite-dimensional (FD) Hilbert space. The authors propose some new definitions of the states and find their explicit forms in the Fock representation. For the convenience of the reader, they bring together several known FD quantum-optical states, including FD coherent states, FD phase coherent states, FD displaced number states, FD Schrödinger cats, and FD squeezed vacuums. The discrete Wigner function is applied in order to demonstrate intriguing properties of the various states.
The authors begin by summarizing some schemes of generating finite-dimensional (FD) quantum-optical states. Although it is possible to generate n-photon Fock states and then to construct a desired state from these states, the focus of the chapter is on generation schemes that lead directly to the FD coherent states and FD squeezed vacuum. The method described is based on the quantum nonlinear oscillator evolution. The oscillator is assumed to be driven by external excitation. Depending on the character of the excitation, various FD states can be produced.
The Pegg-Barnett phase-operator formalism utilizes a finite basis set to represent operators of the harmonic oscillator; this enables the phase to be represented by a Hermitian operator, but rests on taking the dimensionality of the basis set to infinity for observable quantities. Simultaneously, in their approach Pegg and Barnett consider quantum states of a harmonic oscillator which are normalized in the Fock space, i.e., the dimensionality of the basis set in which the states of the harmonic oscillator are defined is supposed to be infinite, while the phase operator is defined in the finite-dimensional basis set. In this paper we address the problem of a consistent definition of a coherent state within a finite state basis. We employ displacement operators to define such coherent states and numerically evaluate observables as a function of the size of the basis set. We investigate phase properties of these coherent states. We find that if the dimensionality of the state space is much larger than the mean occupation number of the coherent states, then the results obtained in the finite-dimensional basis are applicable in the case of a ordinary quantum-mechanical harmonic oscillator. These coherent states are minimum uncertainty states with respect to quadrature operators (i.e., the position and momentum operators) and do not exhibit quadrature squeezing. A weakly excited (compared with the dimensionality of the state space) coherent state in finite-dimensional basis is not strictly speaking a minimum uncertainty state with respect to the number and phase operators. We give definitions of amplitude and phase squeezing and show that weakly excited coherent states can be amplitude squeezed. In the high-intensity limit (again compared with the dimensionality of the state space) these states exhibit phase squeezing.
New coherent states of a harmonic oscillator in a finite-dimensional Fock space are introduced. Some properties of these coherent states are discussed. The second-order squeezing of these coherent states with respect to the quadrature operators is studied in detail. In particular, for a two-state system the arbitrary higher-order squeezing of these states is investigated. It is shown that these coherent states exhibit much richer squeezing properties than the coherent states of a usual harmonic oscillator in an infinite-dimensional Fock space. It is found that these coherent states have not only second-order squeezing but also higher-order squeezing with respect to the quadrature operators of the field under consideration.
Using the Pegg-Barnett phase operator formalism we have introduced the coherent phase state and the displaced phase state in a finite dimension. Using the definition of phase creation and annihilation operators we have constructed a displacement operator which yields coherent phase states and displaced phase states within a finite-dimensional basis and numerically evaluated observables as a function of the size of the basis set. We have shown that, when the dimension of the state space is much larger than the average excitation of phase quanta, then these states are the coherent phase state and displaced phase state of a harmonic oscillator which has an infinite number of eigenstates and exhibits phase squeezing up to 100%. The coherent phase state is a minimum-uncertainty state with respect to the phase quadrature operators but not the minimum-uncertainty state with respect to number and phase operators. The properties of coherent phase states for a two-state system are also investigated. It is interesting to realize that these coherent phase states and displaced phase states are exactly the same as the phase vacuum state and different phase states respectively for the single-mode light field case.