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

Characteristic States of the Electromagnetic Radiation Field

Marvin M. Miller, Eli A. Mishkin

Published Dec 23, 1966• 24 citations
Publisher

Abstract

It has been argued that the positive-frequency part of the quantized electromagnetic field is the "observable" that one would most naturally associate with field measurements using quantum photodetectors. However, since it is possible in principle to make field measurements via the process of stimulated emission, the question of the possible solutions of the characteristic-value equation for the creation operator $a\ifmmode\dagger\else\textdagger\fi{}$ is examined. Various proofs are given to demonstrate that the characteristic kets of $a\ifmmode\dagger\else\textdagger\fi{}$ are not physically admissible states of the radiation field. The possible existence of other useful basis states besides $|n〉$, $|\ensuremath{\alpha}〉$, and states generated from these by unitary transformations is then considered. It is shown that when certain restrictions are placed on the correspondence between Hermitian combinations of the arbitrary non-normal operators $b$ and $b\ifmmode\dagger\else\textdagger\fi{}$ and the harmonic-oscillator variables $x$, $p$, and $H$, then the only possible basis states are the coherent states $|\ensuremath{\alpha}〉$ and the number states $|n〉$. A $\ensuremath{\lambda}$-dependent variation on the photon annihilation operator $a$ is also considered. Its characteristic states for $\ensuremath{-}1\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}1$ are derived, and shown to form a complete set.

DOI
10.1103/physrev.152.1110

Related papers

Quantum Radiation Field: Spontaneous Emission and Entangled Photons

Duncan G. Steel

Abstract One of the greatest successes in quantum theory, and certainly one of the more important parts for application to devices and applications is the prediction of the emission of light through the quantization of an electromagnetic field. Broadly, this is the field of quantum electrodynamics. In this chapter, we develop the Hamiltonian for the classical electromagnetic field. It is seen that the Hamiltonian for each mode (identified by the k-vector and polarization of the field) of the plane wave electromagnetic field is identical to that of the harmonic oscillator. One unit of energy, ℏω, in a mode is a called a photon. The eigenkets for the system are number states (Fock states). We then consider a two-level system described by a Hamiltonian which couples the two-level quantum system to the quantized electromagnetic field. Using the Weisskopf–Wigner formalism developed in Chapter 14, we solve the equations of motion for the time dependent Schrödinger equation assuming the system starts in the excited state with no radiation present in the vacuum field. The results show the creation of one unit of energy in an electromagnetic mode corresponding to the emission of a photon. The excited state probability decays exponentially with the emission of this photon. We consider the important and special case of such a two-level system but in a cavity restricting the radiation field to a single mode. The Jaynes–Cummings Hamiltonian shows that this system, if started in the excited state, Rabi oscillates with no radiation incident on the system.

•Apr 30, 2021•0 citations

Coherent and squeezed states of the radiation field

Marlan O. Scully, Muhammad Suhail Zubairy

Following the development of the quantum theory of radiation and with the advent of the laser, the states of the field that most nearly describe a classical electromagnetic field were widely studied. In order to realize such ‘classical’ states, we will consider the field generated by a classical monochromatic current, and find that the quantum state thus generated has many interesting properties and deserves to be called a coherent state . An important consequence of the quantization of the radiation field is the associated uncertainty relation for the conjugate field variables. It therefore appears reasonable to propose that the wave function which corresponds most closely to the classical field must have minimum uncertainty for all times subject to the appropriate simple harmonic potential. In this chapter we show that a displaced simple harmonic oscillator ground state wave function satisfies this property and the wave packet oscillates sinusoidally in the oscillator potential without changing shape as shown in Fig. 2.1. This coherent wave packet always has minimum uncertainty, and resembles the classical field as nearly as quantum mechanics permits. The corresponding state vector is the coherent state |α〉, which is the eigenstate of the positive frequency part of the electric field operator, or, equivalently, the eigenstate of the destruction operator of the field. Classically an electromagnetic field consists of waves with welldefined amplitude and phase. Such is not the case when we treat the field quantum mechanically.

Cambridge University Press eBooks•Sep 4, 1997•3 citations

General Description and Quantization of EM Fields

Z. Ficek, Mohamed Ridza Wahiddin

We will begin our journey through the background of quantum optics with an elementary, but quantitative, classical theory of radiative fields.Wewill first briefly outline the electromagnetic (EM) theory of radiation, and describe how the EM radiation may be understood as a wavewhich can be represented by a set of harmonic oscillators. We shall describe how the free or non-interacting EM field may be understood as a collection of harmonic oscillators which is quantized in the standard manner, and whose energy cannot be zero as a consequence of the basic non-commutability of the canonical field variables. This chapter discusses the properties of plane EM waves and normalization of the EM field in one dimension. This is followed by a description of the Hamiltonian and the amplitudes of the EM field in terms of the annihilation and creation operators. Based on this chapter, it is possible to considerably simplify the formulation of the physical basis for the mathematical description of the major problems of quantum optics.

•Apr 19, 2016•0 citations

The quantum electromagnetic field in the absence of sources

G. Compagno, Roberto Passante, Franco Persico

Introduction . The purpose of this chapter is to present the quantum theory of the electromagnetic field in the absence of charges and currents. Thus the classical field discussed in the previous chapter is subjected to canonical quantization in Section 2.1, where creation and annihilation operators for plane-wave and spherical wave modes, as well as their commutation relations, are derived along with various field-field commutators related to field propagators. In the next section we introduce the concept of the photon as an elementary excitation of the electromagnetic field. The attention is focused on the ground state of the quantized electromagnetic field in the absence of sources, which is the photon vacuum. The amplitude fluctuations, or zero-point fluctuations of this vacuum, are evaluated. Excited states of the field are examined in the next sections. In particular, Section 2.3 is concerned with number states and coherent states, the latter being obtained by a Glauber transformation of the vacuum, and with their statistical properties. Squeezed states of the field are introduced in Section 2.4 by a unitary transformation leading from the normal to the squeezed vacuum, whose statistical properties are compared with those of a coherent state. Section 2.5 is devoted to a brief discussion of thermal states. The final section of this chapter is dedicated to a discussion of the nonlocalizability of the photon.

Cambridge University Press eBooks•Aug 3, 1995•0 citations

Minimum-Uncertainty States |γ〉 and the States |n〉γ of the Electromagnetic Field

Paul Bertrand, K. Moy, Eli A. Mishkin

Two vector bases suitable for the representations of states of the electromagnetic field are introduced. The minimum-uncertainty states $|\ensuremath{\gamma}〉$ form an overcomplete nonorthogonal set. The ${|n〉}_{\ensuremath{\gamma}}$ states form a complete orthogonal set. The various relations they satisfy are derived via an oscillator frequency transformation. The corresponding similarity transformation of the non-normal operator changes the physical character of its characteristic states. The photon statistics of the states $|\ensuremath{\gamma}〉$ and ${|n〉}_{\ensuremath{\gamma}}$ are shown to display anticorrelation effects.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Sep 15, 1971•9 citations

Field quantization by means of a single harmonic oscillator

Marek Czachor

A new scheme of field quantization is proposed. Instead of associating with different frequencies different oscillators we begin with a single oscillator that can exist in a superposition of different frequencies. The idea is applied to the electromagnetic radiation field. Using the standard Dirac-type mode-quantization of the electromagnetic field we obtain several standard properties such as coherent states or spontaneous and stimulated emission. As opposed to the standard approach the vacuum energy is finite and does not have to be removed by any ad hoc procedure.

arXiv:quant-ph/9912102•Dec 22, 1999•1 citation
It has been argued that the positive-frequency part of the quantized electromagnetic field is the "observable" that one would most naturally associate with field measurements using quantum photodetectors. However, since it is possible in principle to make field measurements via the process of stimulated emission, the question of the possible solutions of the characteristic-value equation for the creation operator a\ifmmode†\else\textdagger\fia\ifmmode\dagger\else\textdagger\fi{}a\ifmmode†\else\textdagger\fi is examined. Various proofs are given to demonstrate that the characteristic kets of a\ifmmode†\else\textdagger\fia\ifmmode\dagger\else\textdagger\fi{}a\ifmmode†\else\textdagger\fi are not physically admissible states of the radiation field. The possible existence of other useful basis states besides ∣n〉|n〉∣n〉, ∣\ensuremathα〉|\ensuremath{\alpha}〉∣\ensuremathα〉, and states generated from these by unitary transformations is then considered. It is shown that when certain restrictions are placed on the correspondence between Hermitian combinations of the arbitrary non-normal operators bbb and b\ifmmode†\else\textdagger\fib\ifmmode\dagger\else\textdagger\fi{}b\ifmmode†\else\textdagger\fi and the harmonic-oscillator variables xxx, ppp, and HHH, then the only possible basis states are the coherent states ∣\ensuremathα〉|\ensuremath{\alpha}〉∣\ensuremathα〉 and the number states ∣n〉|n〉∣n〉. A \ensuremathλ\ensuremath{\lambda}\ensuremathλ-dependent variation on the photon annihilation operator aaa is also considered. Its characteristic states for \ensuremath−1\ensuremath≤\ensuremathλ\ensuremath≤1\ensuremath{-}1\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}1\ensuremath−1\ensuremath≤\ensuremathλ\ensuremath≤1 are derived, and shown to form a complete set.
Abstract One of the greatest successes in quantum theory, and certainly one of the more important parts for application to devices and applications is the prediction of the emission of light through the quantization of an electromagnetic field. Broadly, this is the field of quantum electrodynamics. In this chapter, we develop the Hamiltonian for the classical electromagnetic field. It is seen that the Hamiltonian for each mode (identified by the k-vector and polarization of the field) of the plane wave electromagnetic field is identical to that of the harmonic oscillator. One unit of energy, ℏω, in a mode is a called a photon. The eigenkets for the system are number states (Fock states). We then consider a two-level system described by a Hamiltonian which couples the two-level quantum system to the quantized electromagnetic field. Using the Weisskopf–Wigner formalism developed in Chapter 14, we solve the equations of motion for the time dependent Schrödinger equation assuming the system starts in the excited state with no radiation present in the vacuum field. The results show the creation of one unit of energy in an electromagnetic mode corresponding to the emission of a photon. The excited state probability decays exponentially with the emission of this photon. We consider the important and special case of such a two-level system but in a cavity restricting the radiation field to a single mode. The Jaynes–Cummings Hamiltonian shows that this system, if started in the excited state, Rabi oscillates with no radiation incident on the system.
Following the development of the quantum theory of radiation and with the advent of the laser, the states of the field that most nearly describe a classical electromagnetic field were widely studied. In order to realize such ‘classical’ states, we will consider the field generated by a classical monochromatic current, and find that the quantum state thus generated has many interesting properties and deserves to be called a coherent state . An important consequence of the quantization of the radiation field is the associated uncertainty relation for the conjugate field variables. It therefore appears reasonable to propose that the wave function which corresponds most closely to the classical field must have minimum uncertainty for all times subject to the appropriate simple harmonic potential. In this chapter we show that a displaced simple harmonic oscillator ground state wave function satisfies this property and the wave packet oscillates sinusoidally in the oscillator potential without changing shape as shown in Fig. 2.1. This coherent wave packet always has minimum uncertainty, and resembles the classical field as nearly as quantum mechanics permits. The corresponding state vector is the coherent state |α〉, which is the eigenstate of the positive frequency part of the electric field operator, or, equivalently, the eigenstate of the destruction operator of the field. Classically an electromagnetic field consists of waves with welldefined amplitude and phase. Such is not the case when we treat the field quantum mechanically.
We will begin our journey through the background of quantum optics with an elementary, but quantitative, classical theory of radiative fields.Wewill first briefly outline the electromagnetic (EM) theory of radiation, and describe how the EM radiation may be understood as a wavewhich can be represented by a set of harmonic oscillators. We shall describe how the free or non-interacting EM field may be understood as a collection of harmonic oscillators which is quantized in the standard manner, and whose energy cannot be zero as a consequence of the basic non-commutability of the canonical field variables. This chapter discusses the properties of plane EM waves and normalization of the EM field in one dimension. This is followed by a description of the Hamiltonian and the amplitudes of the EM field in terms of the annihilation and creation operators. Based on this chapter, it is possible to considerably simplify the formulation of the physical basis for the mathematical description of the major problems of quantum optics.
Introduction . The purpose of this chapter is to present the quantum theory of the electromagnetic field in the absence of charges and currents. Thus the classical field discussed in the previous chapter is subjected to canonical quantization in Section 2.1, where creation and annihilation operators for plane-wave and spherical wave modes, as well as their commutation relations, are derived along with various field-field commutators related to field propagators. In the next section we introduce the concept of the photon as an elementary excitation of the electromagnetic field. The attention is focused on the ground state of the quantized electromagnetic field in the absence of sources, which is the photon vacuum. The amplitude fluctuations, or zero-point fluctuations of this vacuum, are evaluated. Excited states of the field are examined in the next sections. In particular, Section 2.3 is concerned with number states and coherent states, the latter being obtained by a Glauber transformation of the vacuum, and with their statistical properties. Squeezed states of the field are introduced in Section 2.4 by a unitary transformation leading from the normal to the squeezed vacuum, whose statistical properties are compared with those of a coherent state. Section 2.5 is devoted to a brief discussion of thermal states. The final section of this chapter is dedicated to a discussion of the nonlocalizability of the photon.
Two vector bases suitable for the representations of states of the electromagnetic field are introduced. The minimum-uncertainty states $|\ensuremath{\gamma}〉$ form an overcomplete nonorthogonal set. The ${|n〉}_{\ensuremath{\gamma}}$ states form a complete orthogonal set. The various relations they satisfy are derived via an oscillator frequency transformation. The corresponding similarity transformation of the non-normal operator changes the physical character of its characteristic states. The photon statistics of the states $|\ensuremath{\gamma}〉$ and ${|n〉}_{\ensuremath{\gamma}}$ are shown to display anticorrelation effects.
A new scheme of field quantization is proposed. Instead of associating with different frequencies different oscillators we begin with a single oscillator that can exist in a superposition of different frequencies. The idea is applied to the electromagnetic radiation field. Using the standard Dirac-type mode-quantization of the electromagnetic field we obtain several standard properties such as coherent states or spontaneous and stimulated emission. As opposed to the standard approach the vacuum energy is finite and does not have to be removed by any ad hoc procedure.