Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Coherent states in a finite-dimensional Hilbert space

Adam Miranowicz, K. Pia̧tek, R. Tanaś

Published Oct 1, 1994• 50 citations
Publisher

Abstract

We construct, according to Glauber's definition, the coherent states in a univocally specified finite-dimensional Hilbert space. Their analytical form is obtained as an analytical solution of the problem raised by Bu\ifmmode \check{z}\else \v{z}\fi{}ek et al. [Phys. Rev. A 45, 8079 (1992)]. The phase properties of the coherent states are studied within the Pegg-Barnett formalism.

DOI
10.1103/physreva.50.3423

Related papers

Pegg–Barnett coherent states

I. Ramos-Prieto, Luis A Andrade-Morales, Francisco Soto +1 more

In this paper, we show that the Pegg–Barnett formalism accepts coherent states constructed as eigenstates of the annihilation operator, considering both the number and the phase. These operators are defined within a ( s + 1 ) -dimensional Hilbert space H s and with periodic conditions. The coherent states that we find are determined by the eigenvalue of the annihilation operator, which leads to a discrete spectrum. This approach allows calculation of the discrete-finite counterpart of the Wigner function in a phase space defined by the variables of number and phase.

Journal of the Optical Society of America B•Dec 12, 2019•4 citations

Coherent states in finite-dimensional Hilbert space and their Wigner representation

Tomáš Opatrný, Adam Miranowicz, J. Bajer

We consider two definitions of coherent states in a finite-dimensional Hilbert space based on (i) truncation of the usual coherent state expansion and (ii) generalization of the displacement operator acting on vacuum. The number-phase Wigner function is computed for such states. Analytical results and numerically computed graphs are presented. Special attention is paid to two-level states and to their Stokes parameter representations.

Journal of Modern Optics•Feb 1, 1996•23 citations

Physical realizations and Wigner representation of coherent states of finite-dimensional Hilbert spaces (Analytical Study of Quantum Information and Related Fields)

Adam Miranowicz, Şahin Kaya Özdemir, Yu-xi Liu +2 more

Two essentially different mathematical approaches to define coherent states of finite-dimensional Hilbert spaces are analyzed.Physical realizations of the s0-called quantum scissors to generate the finite-dimensional states via optical truncation are discussed including the linear-0ptical scheme of Pegg, Phillips and Barnett [Phys.Rev. Lett.81, 1604 (1998)] and nonlinear one of Leonski and Tanas [Phys.Rev. A49, R20 (1994)].Distinct properties of the generated states are studied by applying adiscrete Wigner function.

•May 1, 2002•0 citations

Even and Odd Phase Coherent States Associated with the Pegg-Barnett Hermitian Phase Theory

Jiu-Yun Zhu, Le‐Man Kuang

Even and odd phase coherent states associated with the Pegg-Barnett Hermitian phase theory are introduced. Even and odd coherent-state formalism of the Pegg-Barnett phase operator is given in terms of the projection operator in the even and odd coherent-state space. The number-phase uncertainty relation is investigated for these states.

Chinese Physics Letters•Apr 1, 1996•0 citations

Coherent states in a finite-dimensional basis: Their phase properties and relationship to coherent states of light

Vladimír Bužek, A. D. Wilson‐Gordon, P. L. Knight +1 more

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.

Physical Review A•Jun 1, 1992•89 citations

Phase operators on Hilbert space

John A. Vaccaro

A simple formalism is introduced for describing the quantum phase of a single-mode field (or, equivalently, a harmonic oscillator) using a particular infinite-dimensional Hilbert space ${\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}$. This space is spanned by vectors which represent states of the single field mode containing non-negative numbers of photons. It has a symmetry in the sense that it contains vectors in the neighborhood of both the vacuum state and the infinite photon number state. A striking property of ${\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}$ is that it supports Hermitian and unitary phase operators. Indeed the Pegg-Barnett unitary phase operators are shown to converge strongly on the Hilbert space ${\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}$ and so the correponding limit points are ordinary operators on ${\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}$. This can be compared with the situation for the infinite-dimensional Hilbert space H conventionally used in quantum optics on which the Pegg-Barnett unitary phase operators converge weakly, in general.

Physical Review A•Apr 1, 1995•25 citations
We construct, according to Glauber's definition, the coherent states in a univocally specified finite-dimensional Hilbert space. Their analytical form is obtained as an analytical solution of the problem raised by Bu\ifmmode \check{z}\else \v{z}\fi{}ek et al. [Phys. Rev. A 45, 8079 (1992)]. The phase properties of the coherent states are studied within the Pegg-Barnett formalism.
In this paper, we show that the Pegg–Barnett formalism accepts coherent states constructed as eigenstates of the annihilation operator, considering both the number and the phase. These operators are defined within a ( s + 1 ) -dimensional Hilbert space H s and with periodic conditions. The coherent states that we find are determined by the eigenvalue of the annihilation operator, which leads to a discrete spectrum. This approach allows calculation of the discrete-finite counterpart of the Wigner function in a phase space defined by the variables of number and phase.
Two essentially different mathematical approaches to define coherent states of finite-dimensional Hilbert spaces are analyzed.Physical realizations of the s0-called quantum scissors to generate the finite-dimensional states via optical truncation are discussed including the linear-0ptical scheme of Pegg, Phillips and Barnett [Phys.Rev. Lett.81, 1604 (1998)] and nonlinear one of Leonski and Tanas [Phys.Rev. A49, R20 (1994)].Distinct properties of the generated states are studied by applying adiscrete Wigner function.
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.
A simple formalism is introduced for describing the quantum phase of a single-mode field (or, equivalently, a harmonic oscillator) using a particular infinite-dimensional Hilbert space Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​. This space is spanned by vectors which represent states of the single field mode containing non-negative numbers of photons. It has a symmetry in the sense that it contains vectors in the neighborhood of both the vacuum state and the infinite photon number state. A striking property of Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​ is that it supports Hermitian and unitary phase operators. Indeed the Pegg-Barnett unitary phase operators are shown to converge strongly on the Hilbert space Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​ and so the correponding limit points are ordinary operators on Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​. This can be compared with the situation for the infinite-dimensional Hilbert space H conventionally used in quantum optics on which the Pegg-Barnett unitary phase operators converge weakly, in general.