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

中国科学通报：英文版

Phase Properties of a Quantized Field Interacting With an Atom

周鹏

Published Jan 1, 1993• 0 citations
OpenAlex

Abstract

The study of the quantized field phase dates back to Diracs definition of the phase operator which came from a polar decomposition of the annihilation operator following a simple analogy with the phase of a classical harmonic oscillator, according to the usual procedure of quantization. Unfortunately, this approach causes inconsistency. Some people attempted to investigate the phase properties from various angles after Dirac.

Related papers

Quantum phase and phase dependent measurements

W. P. Schleich, Stephen Mark Barnett

The correct description of the phase variable in quantum mechanics is a question rooted in its earliest formulations. The atom mechanics [1] of Bohr and Sommerfeld—the precursor of modern quantum mechanics—ascribes a central role to action-angle variables. However, Heisenberg's matrix mechanics and Schrödinger's wave mechanics are formulated [2] in terms of the canonical variables representing cartesian coordinates and momenta. Stimulated by this work London [3] attempted a reformulation of these theories in the previously favored action-angle variables. However, this attempt failed due to the difficulty in ascribing quantum operators to the angle variables of classical theory. Despite these difficulties London found [4] an operator representation of the complex exponential of these angle variables. The quantum phase of light made its first appearance in Dirac's classic paper [5] on the quantization of the radiation field. In contrast to modern methods, he constructed the annihilation and creation operators for each field mode from the corresponding amplitude and phase operators. Phase and its quantum nature acquired new significance with the development of lasers in the early sixties: theoretical investigations highlighted significant problems with Dirac's original proposal for the phase operator. A particularly elegant illustration of the difficulty was given by Louisell [6] in 1963. The advent of phase sensitive quantum noise as demonstrated experimentally in the production and detection of squeezed light [7] has created a new wave of interest in the nature of quantum optical phase leading to the discovery of the hermitian optical phase operator [8]. There has followed an explosion of theoretical activity in this area stimulating fresh experimental investigations. In preparing this special issue we have attempted to present the current state of this active and rapidly moving field. We have arranged the papers into what we hope is a coherent representation of the field. The first papers give a historical perspective and overview of current thinking. The two recent experimental investigations which follow are intimately connected to the phase space description of quantum mechanics based on quasi-probability distributions. The representa tion of phase via phase space and its connection with phase-dependent measurements and the phase operator are addressed in the next section. Some more formal considerations pertinent to phase are presented in the following section. Gravitational wave detection and optical communication have motivated the study of the limits of phase noise. Some recent investigations on such optimal phase states are presented. The issue concludes with two papers discussing the significance of phase in light-matter interactions. In concluding we express our gratitude to the authors of the papers in this volume for their efforts in preparing their high quality presentations.

Physica Scripta•Jan 1, 1993•74 citations

Roots of the phase operators

George Jorjadze, I. T. Sarishvili

Abstract. The kth root taken of the bosonic phase operator is considered. This leads to the extension of the Hilbert space. In the case k = 2 an ordinary fermionic extension arises. Particles whose statistics depends on k are introduced for other values of k. 1. Phase operators The problem of polar decomposition of the creation and annihilation operators (a ∗ , a) of a harmonic oscillator goes back to Dirac [1]. In this section some aspects of this problem will be reviewed [2]. Let h be the oscillator hamiltonian h = (p 2 + q 2)/2 (1.1) on the phase plane Γ with coordinates p, q and Poisson brackets (PB) If the polar angle ϕ is introduced by then for the complex variables a and a ∗ {q, p} = 1. (1.2) p = √ 2h sin ϕ, q = √ 2h cos ϕ, (1.3) a = q + ip

Georgian Mathematical Journal•Jan 1, 1995•0 citations

Phase operators of the harmonic oscillator and suppression of a number state

C V Sukumar

The phase properties of the quantized electromagnetic field have been studied using operators ${P}^{\ifmmode\pm\else\textpm\fi{}}$ of the harmonic oscillator whose classical analogs are exp(\ensuremath{\mp}i\ensuremath{\theta}), \ensuremath{\theta} being the classical phase. ${P}^{+}$ and ${P}^{\mathrm{\ensuremath{-}}}$ are used to construct a unitary operator ${U}_{p}$(\ensuremath{\alpha})=exp(\ensuremath{\alpha}${P}^{+}$-${\ensuremath{\alpha}}^{\mathrm{*}}$${P}^{\mathrm{\ensuremath{-}}}$) and the properties of the state generated by ${U}_{p}$ from a number state are studied. It is shown that by tuning \ensuremath{\alpha} it is possible to produce a pure state ${\ensuremath{\psi}}_{0}$=${U}_{p}$(\ensuremath{\alpha})\ensuremath{\Vert}0〉 in which a specific number-state component is suppressed.

Physical Review A•Nov 1, 1989•1 citation

Phase properties of the quantized single-mode electromagnetic field

David Thomas Pegg, Stephen Mark Barnett

The usual mathematical model of the single-mode electromagnetic field is the harmonic oscillator with an infinite-dimensional state space, which unfortunately cannot accommodate the existence of a Hermitian phase operator. Recently we indicated that this difficulty may be circumvented by using an alternative, and physically indistinguishable, mathematical model of the single-mode field involving a finite but arbitrarily large state space, the dimension of which is allowed to tend to infinity after physically measurable results, such as expectation values, are calculated. In this paper we investigate the properties of a Hermitian phase operator which follows directly and uniquely from the form of the phase states in this space and find them to be well behaved. The phase-number commutator is not subject to the difficulties inherent in Dirac's original commutator, but still preserves the commutator--Poisson-bracket correspondence for physical field states. In the quantum regime of small field strengths, the phase operator predicts phase properties substantially different from those obtained using the conventional Susskind-Glogower operators. In particular, our results are consistent with the vacuum being a state of random phase and the phases of two vacuum fields being uncorrelated. For higher-intensity fields, the quantum phase properties agree with those previously obtained by phenomenological and semiclassical approaches, where such approximations are valid. We illustrate the properties of the phase with a discussion of partial phase states. The Hermitian phase operator also allows us to construct a unitary number-shift operator and phase-moment generating functions. We conclude that the alternative mathematical description of the single-mode field presented here provides a valid, and potentially useful, quantum-mechanical approach for calculating the phase properties of the electromagnetic field.

Physical Review A•Feb 1, 1989•873 citations

Phase operators and phase relations for photon states

Howard C. Volkin

For a quantized mode of the radiation field, the operator whose classical analog is the ordinary phase factor of the mode amplitudes has been shown to be nonunitary. A mathematically rigorous formulation of the phase P is given on the basis of the canonical factorization theorem. All other phase quantities are defined in terms of P and P†. Many of the seemingly complex features of phase operators are found to be simple direct consequences of the general mathematical theory. It is easily seen that P is a partial isometry but not a unitary operator. In contrast to the amplitude operator, it is found that P is not a spectral operator and the set of phase eigenstates is not complete. Mathematically precise operator relations, including the rigorous statement of commutation rules, are developed. For each of the phase operators, a complete spectral analysis is given, with the nature of the spectrum, spectral decomposition (if any), and eigenstates shown explicitly. The values of the various phase operators are compared between phase states and coherent states.

Journal of Mathematical Physics•Dec 1, 1973•15 citations

Comprehensive theory of the relative phase in atom-field interactions

J. Delgado, E. C. Yustas, Luis Lorenzo Sánchez-Soto +1 more

We explore the role played by the quantum relative phase in a well-known model of atom-field interaction, namely, the Dicke model. We introduce an appropriate polar decomposition of the atom-field relative amplitudes that leads to a truly Hermitian relative-phase operator, whose eigenstates correctly describe the phase properties, as we demonstrate by studying the positive operator-valued measure derived from it. We find the probability distribution for this relative phase and, by resorting to a numerical procedure, we study its time evolution.

arXiv:quant-ph/0102026•Physical Review A•May 4, 2001•8 citations
The study of the quantized field phase dates back to Diracs definition of the phase operator which came from a polar decomposition of the annihilation operator following a simple analogy with the phase of a classical harmonic oscillator, according to the usual procedure of quantization. Unfortunately, this approach causes inconsistency. Some people attempted to investigate the phase properties from various angles after Dirac.
The correct description of the phase variable in quantum mechanics is a question rooted in its earliest formulations. The atom mechanics [1] of Bohr and Sommerfeld—the precursor of modern quantum mechanics—ascribes a central role to action-angle variables. However, Heisenberg's matrix mechanics and Schrödinger's wave mechanics are formulated [2] in terms of the canonical variables representing cartesian coordinates and momenta. Stimulated by this work London [3] attempted a reformulation of these theories in the previously favored action-angle variables. However, this attempt failed due to the difficulty in ascribing quantum operators to the angle variables of classical theory. Despite these difficulties London found [4] an operator representation of the complex exponential of these angle variables. The quantum phase of light made its first appearance in Dirac's classic paper [5] on the quantization of the radiation field. In contrast to modern methods, he constructed the annihilation and creation operators for each field mode from the corresponding amplitude and phase operators. Phase and its quantum nature acquired new significance with the development of lasers in the early sixties: theoretical investigations highlighted significant problems with Dirac's original proposal for the phase operator. A particularly elegant illustration of the difficulty was given by Louisell [6] in 1963. The advent of phase sensitive quantum noise as demonstrated experimentally in the production and detection of squeezed light [7] has created a new wave of interest in the nature of quantum optical phase leading to the discovery of the hermitian optical phase operator [8]. There has followed an explosion of theoretical activity in this area stimulating fresh experimental investigations. In preparing this special issue we have attempted to present the current state of this active and rapidly moving field. We have arranged the papers into what we hope is a coherent representation of the field. The first papers give a historical perspective and overview of current thinking. The two recent experimental investigations which follow are intimately connected to the phase space description of quantum mechanics based on quasi-probability distributions. The representa tion of phase via phase space and its connection with phase-dependent measurements and the phase operator are addressed in the next section. Some more formal considerations pertinent to phase are presented in the following section. Gravitational wave detection and optical communication have motivated the study of the limits of phase noise. Some recent investigations on such optimal phase states are presented. The issue concludes with two papers discussing the significance of phase in light-matter interactions. In concluding we express our gratitude to the authors of the papers in this volume for their efforts in preparing their high quality presentations.
Abstract. The kth root taken of the bosonic phase operator is considered. This leads to the extension of the Hilbert space. In the case k = 2 an ordinary fermionic extension arises. Particles whose statistics depends on k are introduced for other values of k. 1. Phase operators The problem of polar decomposition of the creation and annihilation operators (a ∗ , a) of a harmonic oscillator goes back to Dirac [1]. In this section some aspects of this problem will be reviewed [2]. Let h be the oscillator hamiltonian h = (p 2 + q 2)/2 (1.1) on the phase plane Γ with coordinates p, q and Poisson brackets (PB) If the polar angle ϕ is introduced by then for the complex variables a and a ∗ {q, p} = 1. (1.2) p = √ 2h sin ϕ, q = √ 2h cos ϕ, (1.3) a = q + ip
The phase properties of the quantized electromagnetic field have been studied using operators ${P}^{\ifmmode\pm\else\textpm\fi{}}$ of the harmonic oscillator whose classical analogs are exp(\ensuremath{\mp}i\ensuremath{\theta}), \ensuremath{\theta} being the classical phase. P+{P}^{+}P+ and ${P}^{\mathrm{\ensuremath{-}}}$ are used to construct a unitary operator Up{U}_{p}Up​(\ensuremath{\alpha})=exp(\ensuremath{\alpha}P+{P}^{+}P+-${\ensuremath{\alpha}}^{\mathrm{*}}$${P}^{\mathrm{\ensuremath{-}}}$) and the properties of the state generated by Up{U}_{p}Up​ from a number state are studied. It is shown that by tuning \ensuremath{\alpha} it is possible to produce a pure state ${\ensuremath{\psi}}_{0}$=Up{U}_{p}Up​(\ensuremath{\alpha})\ensuremath{\Vert}0〉 in which a specific number-state component is suppressed.
The usual mathematical model of the single-mode electromagnetic field is the harmonic oscillator with an infinite-dimensional state space, which unfortunately cannot accommodate the existence of a Hermitian phase operator. Recently we indicated that this difficulty may be circumvented by using an alternative, and physically indistinguishable, mathematical model of the single-mode field involving a finite but arbitrarily large state space, the dimension of which is allowed to tend to infinity after physically measurable results, such as expectation values, are calculated. In this paper we investigate the properties of a Hermitian phase operator which follows directly and uniquely from the form of the phase states in this space and find them to be well behaved. The phase-number commutator is not subject to the difficulties inherent in Dirac's original commutator, but still preserves the commutator--Poisson-bracket correspondence for physical field states. In the quantum regime of small field strengths, the phase operator predicts phase properties substantially different from those obtained using the conventional Susskind-Glogower operators. In particular, our results are consistent with the vacuum being a state of random phase and the phases of two vacuum fields being uncorrelated. For higher-intensity fields, the quantum phase properties agree with those previously obtained by phenomenological and semiclassical approaches, where such approximations are valid. We illustrate the properties of the phase with a discussion of partial phase states. The Hermitian phase operator also allows us to construct a unitary number-shift operator and phase-moment generating functions. We conclude that the alternative mathematical description of the single-mode field presented here provides a valid, and potentially useful, quantum-mechanical approach for calculating the phase properties of the electromagnetic field.
For a quantized mode of the radiation field, the operator whose classical analog is the ordinary phase factor of the mode amplitudes has been shown to be nonunitary. A mathematically rigorous formulation of the phase P is given on the basis of the canonical factorization theorem. All other phase quantities are defined in terms of P and P†. Many of the seemingly complex features of phase operators are found to be simple direct consequences of the general mathematical theory. It is easily seen that P is a partial isometry but not a unitary operator. In contrast to the amplitude operator, it is found that P is not a spectral operator and the set of phase eigenstates is not complete. Mathematically precise operator relations, including the rigorous statement of commutation rules, are developed. For each of the phase operators, a complete spectral analysis is given, with the nature of the spectrum, spectral decomposition (if any), and eigenstates shown explicitly. The values of the various phase operators are compared between phase states and coherent states.
We explore the role played by the quantum relative phase in a well-known model of atom-field interaction, namely, the Dicke model. We introduce an appropriate polar decomposition of the atom-field relative amplitudes that leads to a truly Hermitian relative-phase operator, whose eigenstates correctly describe the phase properties, as we demonstrate by studying the positive operator-valued measure derived from it. We find the probability distribution for this relative phase and, by resorting to a numerical procedure, we study its time evolution.