Physical Review

Configuration-Space Photon Number Operators in Quantum Optics

L. Mandeļ

Published Apr 29, 1966• 138 citations

Abstract

Some properties of the operator ${\stackrel{^}{n}}_{V,t}$ representing the number of photons localized in a finite volume $V$ at time $t$ are investigated. To some extent these properties reflect the well-known difficulty of localizing photons in space-time. However, it is shown that, when the linear dimensions of $V$ are large compared with the wave-length of any occupied mode of the field, the ${\stackrel{^}{n}}_{V,t}$ operator acquires some simple properties. The commutator of ${\stackrel{^}{n}}_{V,t}$ and the detection operator $\stackrel{^}{\mathrm{A}}(\mathrm{x}, t)$ is expressible in an interesting form. The commutator and relations derived from it become particularly simple for certain space-time regions which we label conjoint and disjoint. An orthogonal set of eigenstates of ${\stackrel{^}{n}}_{V,t}$ is found, together with the corresponding eigenvalues, and it is shown that an arbitrary state is expressible in terms of these eigenstates. Some $N\mathrm{th}$-order correlations of the ${\stackrel{^}{n}}_{V,t}$ operators are evaluated, and the results are used to calculate the probability distribution of eigenvalues of ${\stackrel{^}{n}}_{V,t}$ for an arbitrary state of the field.

DOI: 10.1103/physrev.144.1071

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Antinormally Ordered Correlations and Quantum Counters

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Physical ReviewDec 2, 196685 citations

Photon position eigenstates in configuration space

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quant-pharXiv:2605.25857Physical Review AMay 26, 20260 citations

Physical meaning of the photon wave function

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Physical Review AMar 1, 199810 citations

Phase Space Representations of the Density Operator

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Apr 19, 20160 citations

Coherent and Incoherent States of the Radiation Field

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Multi-mode, multi-photon coherent states

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Quantum Optics Journal of the European Optical Society Part BDec 1, 19893 citations

Physical Review

Configuration-Space Photon Number Operators in Quantum Optics

L. Mandeļ

Published Apr 29, 1966• 138 citations

Publisher

Abstract

Some properties of the operator

{\stackrel{^}{n}}_{V,t}

representing the number of photons localized in a finite volume

V

at time

t

are investigated. To some extent these properties reflect the well-known difficulty of localizing photons in space-time. However, it is shown that, when the linear dimensions of

V

are large compared with the wave-length of any occupied mode of the field, the

{\stackrel{^}{n}}_{V,t}

operator acquires some simple properties. The commutator of

{\stackrel{^}{n}}_{V,t}

and the detection operator

\stackrel{^}{\mathrm{A}}(\mathrm{x}, t)

is expressible in an interesting form. The commutator and relations derived from it become particularly simple for certain space-time regions which we label conjoint and disjoint. An orthogonal set of eigenstates of

{\stackrel{^}{n}}_{V,t}

is found, together with the corresponding eigenvalues, and it is shown that an arbitrary state is expressible in terms of these eigenstates. Some

N\mathrm{th}

-order correlations of the

{\stackrel{^}{n}}_{V,t}

operators are evaluated, and the results are used to calculate the probability distribution of eigenvalues of

{\stackrel{^}{n}}_{V,t}

for an arbitrary state of the field.

DOI: 10.1103/physrev.144.1071

Related papers

Antinormally Ordered Correlations and Quantum Counters

L. Mandeļ

The properties of a photon counter functioning by stimulated emission---rather than by absorption---of photons in an external field are examined. A possible scheme for such a quantum counter is described, and it is shown that correlation measurements performed with a number of quantum counters correspond to antinormally ordered products of field operators. These correlations, unlike the normally ordered ones, are always positive definite and depend explicitly on the number of radiation modes per unit volume to which the counter is coupled. It is shown that the antinormally ordered correlations carry useful information about the field only when the average photon occupation number per mode is large. A general expression for the probability that the quantum counter registers $n$ counts in a certain time interval is derived and is shown to be related in an interesting way to the corresponding expression for the photoelectric detector. The variance of the probability distribution is evaluated for some simple states of the field.The properties of a photon counter functioning by stimulated emission---rather than by absorption---of photons in an external field are examined. A possible scheme for such a quantum counter is described, and it is shown that correlation measurements performed with a number of quantum counters correspond to antinormally ordered products of field operators. These correlations, unlike the normally ordered ones, are always positive definite and depend explicitly on the number of radiation modes per unit volume to which the counter is coupled. It is shown that the antinormally ordered correlations carry useful information about the field only when the average photon occupation number per mode is large. A general expression for the probability that the quantum counter registers $n$ counts in a certain time interval is derived and is shown to be related in an interesting way to the corresponding expression for the photoelectric detector. The variance of the probability distribution is evaluated for some simple states of the field.

Physical ReviewDec 2, 196685 citations

Photon position eigenstates in configuration space

Artemio González-López, Luis Martínez Alonso

The expressions of the eigenfunctions of the Hawton photon position operator in configuration space are derived for several classes of wave function, including the Riemann--Silberstein and Landau--Peierls cases. Although these eigenfunctions have a simple form in momentum space, the explicit characterization of their representations in configuration space is rather more involved. We provide closed expressions of these eigenfunctions in terms of linear combinations of the complete elliptic integrals $K(κ)$ and $E(κ)$ with modulus $κ$ depending on trigonometric functions of the polar angle. We show that they diverge not only at the value $\mathbf q$ of the position eigenvalue, but also on a plane containing $\mathbf q$ and that they decay as inverse powers of the distance from $\mathbf q$ .The expressions of the eigenfunctions of the Hawton photon position operator in configuration space are derived for several classes of wave function, including the Riemann--Silberstein and Landau--Peierls cases. Although these eigenfunctions have a simple form in momentum space, the explicit characterization of their representations in configuration space is rather more involved. We provide closed expressions of these eigenfunctions in terms of linear combinations of the complete elliptic integrals $K(κ)$ and $E(κ)$ with modulus $κ$ depending on trigonometric functions of the polar angle. We show that they diverge not only at the value $\mathbf q$ of the position eigenvalue, but also on a plane containing $\mathbf q$ and that they decay as inverse powers of the distance from $\mathbf q$.

quant-pharXiv:2605.25857Physical Review AMay 26, 20260 citations

Physical meaning of the photon wave function

Toshio Inagaki

Recently, the author [Phys. Rev. A 49, 2839 (1994)] proposed a quantum-mechanical theory of a photon, in which negative energy states can be dismissed from physical photon states without causing any difficulties. In this Brief Report the physical meaning of a photon wave function is investigated more precisely. The interpretation of a photon wave function as a probability amplitude is guaranteed by requiring the coarse-grained condition that the linear dimensions of a volume or a surface concerned in the configuration space are large compared to the photon wavelengths. The calculation made here is essentially based on that presented by Mandel [Phys. Rev. 144, 1071 (1966)] in his analysis of the photon number operator. But our calculation includes that of Mandel as the zeroth-order approximation. As a result, it is shown that although the position operator ◯ in the ordinary quantum mechanics be considered a correct photon operator strictly cannot, it becomes meaningful to some extent under the coarse-grained condition. In connection with ◯, we also examine the velocity operator $[\mathbf{◯},\ifmmode \hat{H}\else \^{H}\fi{}]/i\ensuremath{\Elzxh}$ ($\ifmmode \hat{H}\else \^{H}\fi{}$ is the photon Hamiltonian), and show that the probability current density is definable under the coarse-grained condition.Recently, the author [Phys. Rev. A 49, 2839 (1994)] proposed a quantum-mechanical theory of a photon, in which negative energy states can be dismissed from physical photon states without causing any difficulties. In this Brief Report the physical meaning of a photon wave function is investigated more precisely. The interpretation of a photon wave function as a probability amplitude is guaranteed by requiring the coarse-grained condition that the linear dimensions of a volume or a surface concerned in the configuration space are large compared to the photon wavelengths. The calculation made here is essentially based on that presented by Mandel [Phys. Rev. 144, 1071 (1966)] in his analysis of the photon number operator. But our calculation includes that of Mandel as the zeroth-order approximation. As a result, it is shown that although the position operator ◯ in the ordinary quantum mechanics be considered a correct photon operator strictly cannot, it becomes meaningful to some extent under the coarse-grained condition. In connection with ◯, we also examine the velocity operator $[\mathbf{◯},\ifmmode \hat{H}\else \^{H}\fi{}]/i\ensuremath{\Elzxh}$ ($\ifmmode \hat{H}\else \^{H}\fi{}$ is the photon Hamiltonian), and show that the probability current density is definable under the coarse-grained condition.

Physical Review AMar 1, 199810 citations

Phase Space Representations of the Density Operator

Z. Ficek, Mohamed Ridza Wahiddin

In Chapter 4, we introduced the concept of the photon number representation of the electromagnetic (EM) ﬁeld that is based on the number states, the eigenstates of the Hamiltonian of the EM ﬁeld. However, this is not the only possible representation, in fact not always the most convenient. In this chapter, we introduce the concept of phase space representations of a quantum system which is in a state described by the density operator ρˆ. The density operator of a given system encodes classical as well as non-classical (quantum) properties of the system. How to distinguish these two kinds of properties is of basic importance in quantum optics. Therefore, we shall address this issue by using representations of the density operator that are based on the parametric space of complex eigenvalues of the annihilation operator aˆ in a coherent state |α〉. We ﬁrst determine what we mean by the density operator of a quantum system and discuss its basic properties. Next, we introduce diﬀerent representations of the density operator in terms of coherent state projectors |α〉〈α|. The representations show that a state of the EM ﬁeld or of an arbitrary quantum system may be regarded as a mixture of coherent states. Along the way, we will learn how to ﬁnd the P , Q and Wigner representations of the density operator, what are their properties, how to interpret the properties and how to calculate relations between the diﬀerent representations. Most importantly, certain ﬁeld states exhibit non- classical features and these non-classical features can be manifested in the phase space representations. Accordingly, we will learn how the representations are very convenient tools to describe quantum states of simple systems.In Chapter 4, we introduced the concept of the photon number representation of the electromagnetic (EM) ﬁeld that is based on the number states, the eigenstates of the Hamiltonian of the EM ﬁeld. However, this is not the only possible representation, in fact not always the most convenient. In this chapter, we introduce the concept of phase space representations of a quantum system which is in a state described by the density operator ρˆ. The density operator of a given system encodes classical as well as non-classical (quantum) properties of the system. How to distinguish these two kinds of properties is of basic importance in quantum optics. Therefore, we shall address this issue by using representations of the density operator that are based on the parametric space of complex eigenvalues of the annihilation operator aˆ in a coherent state |α〉. We ﬁrst determine what we mean by the density operator of a quantum system and discuss its basic properties. Next, we introduce diﬀerent representations of the density operator in terms of coherent state projectors |α〉〈α|. The representations show that a state of the EM ﬁeld or of an arbitrary quantum system may be regarded as a mixture of coherent states. Along the way, we will learn how to ﬁnd the P , Q and Wigner representations of the density operator, what are their properties, how to interpret the properties and how to calculate relations between the diﬀerent representations. Most importantly, certain ﬁeld states exhibit non- classical features and these non-classical features can be manifested in the phase space representations. Accordingly, we will learn how the representations are very convenient tools to describe quantum states of simple systems.

Apr 19, 20160 citations

Coherent and Incoherent States of the Radiation Field

Roy J. Glauber

Methods are developed for discussing the photon statistics of arbitrary fields in fully quantum-mechanical terms. In order to keep the classical limit of quantum electrodynamics plainly in view, extensive use is made of the coherent states of the field. These states, which reduce the field correlation functions to factorized forms, are shown to offer a convenient basis for the description of fields of all types. Although they are not orthogonal to one another, the coherent states form a complete set. It is shown that any quantum state of the field may be expanded in terms of them in a unique way. Expansions are also developed for arbitrary operators in terms of products of the coherent state vectors. These expansions are discussed as a general method of representing the density operator for the field. A particular form is exhibited for the density operator which makes it possible to carry out many quantum-mechanical calculations by methods resembling those of classical theory. This representation permits clear insights into the essential distinction between the quantum and classical descriptions of the field. It leads, in addition, to a simple formulation of a superposition law for photon fields. Detailed discussions are given of the incoherent fields which are generated by superposing the outputs of many stationary sources. These fields are all shown to have intimately related properties, some of which have been known for the particular case of blackbody radiation.Methods are developed for discussing the photon statistics of arbitrary fields in fully quantum-mechanical terms. In order to keep the classical limit of quantum electrodynamics plainly in view, extensive use is made of the coherent states of the field. These states, which reduce the field correlation functions to factorized forms, are shown to offer a convenient basis for the description of fields of all types. Although they are not orthogonal to one another, the coherent states form a complete set. It is shown that any quantum state of the field may be expanded in terms of them in a unique way. Expansions are also developed for arbitrary operators in terms of products of the coherent state vectors. These expansions are discussed as a general method of representing the density operator for the field. A particular form is exhibited for the density operator which makes it possible to carry out many quantum-mechanical calculations by methods resembling those of classical theory. This representation permits clear insights into the essential distinction between the quantum and classical descriptions of the field. It leads, in addition, to a simple formulation of a superposition law for photon fields. Detailed discussions are given of the incoherent fields which are generated by superposing the outputs of many stationary sources. These fields are all shown to have intimately related properties, some of which have been known for the particular case of blackbody radiation.

Physical ReviewSep 15, 19636,794 citations

Multi-mode, multi-photon coherent states

Jacob Katriel, A. I. Solomon

A two-mode photon operator satisfying the canonical boson commutation relations is used to define two-mode coherent states for the Weyl, SU(2) and SU(1,l) groups. In all these cases squeezing is determined by an appropriate correlation, which we evaluate. The generalisation to multi-mode, many-photon states is indicated. In this letter we introduce a formalism for describing coherent states, tailored to the multi-mode case. We shall present the two-mode case for definiteness, although the generalisation to several modes will be obvious. Similarly, we show at the end of this note how to accommodate multi-photon states with little extra effort. States associ- ated with the Weyl group (two-mode coherent states), and the groups SU.(2) and SU(1,l) will be given; and we show that (for a specific representation) the last case reduces to the two-mode sqaeezed states of Caves and Schumaker (1986). In the conventional quantum optics approach, a mode of the electromagnetic field is represented by a creation (annihilation) operator a(at), with commutation relation (a, a') = 1. Our starting point in considering the two-mode case is to introduce a two-mode operator A analogous to the single-mode operator a. We do this by defining (Katriel and Hummer 1981) A = (max(Al, A2) + 1)'2ala2 , (1) Here al, a2 are the (annihilation) operators referring to each mode; they obey the basic commutation relation (ai, ait) = 6, but commute with each other. The number operators AI, A2 are defined in the usual way, Ai = aitai and functions of Ai will be evaluated in simultaneous eigenstates of rii, where of course they equal the functions of the corresponding eigenvalues. Now, we may show easily that (A, A') = 1. We note that AA = AtA = min(A1, h2). (The generalisation of (1) to three (or more) modes, which we shall not pursue, is immediate on replacing one of the two single-mode operators by a two-mode one.) Note that all the states defined here give variance-free single-mode operators; since At always creates equal numbers of 1 and 2 modesA two-mode photon operator satisfying the canonical boson commutation relations is used to define two-mode coherent states for the Weyl, SU(2) and SU(1,l) groups. In all these cases squeezing is determined by an appropriate correlation, which we evaluate. The generalisation to multi-mode, many-photon states is indicated. In this letter we introduce a formalism for describing coherent states, tailored to the multi-mode case. We shall present the two-mode case for definiteness, although the generalisation to several modes will be obvious. Similarly, we show at the end of this note how to accommodate multi-photon states with little extra effort. States associ- ated with the Weyl group (two-mode coherent states), and the groups SU.(2) and SU(1,l) will be given; and we show that (for a specific representation) the last case reduces to the two-mode sqaeezed states of Caves and Schumaker (1986). In the conventional quantum optics approach, a mode of the electromagnetic field is represented by a creation (annihilation) operator a(at), with commutation relation (a, a') = 1. Our starting point in considering the two-mode case is to introduce a two-mode operator A analogous to the single-mode operator a. We do this by defining (Katriel and Hummer 1981) A = (max(Al, A2) + 1)'2ala2 , (1) Here al, a2 are the (annihilation) operators referring to each mode; they obey the basic commutation relation (ai, ait) = 6, but commute with each other. The number operators AI, A2 are defined in the usual way, Ai = aitai and functions of Ai will be evaluated in simultaneous eigenstates of rii, where of course they equal the functions of the corresponding eigenvalues. Now, we may show easily that (A, A') = 1. We note that AA = AtA = min(A1, h2). (The generalisation of (1) to three (or more) modes, which we shall not pursue, is immediate on replacing one of the two single-mode operators by a two-mode one.) Note that all the states defined here give variance-free single-mode operators; since At always creates equal numbers of 1 and 2 modes

Quantum Optics Journal of the European Optical Society Part BDec 1, 19893 citations

Some properties of the operator

{\stackrel{^}{n}}_{V,t}

representing the number of photons localized in a finite volume

V

at time

t

are investigated. To some extent these properties reflect the well-known difficulty of localizing photons in space-time. However, it is shown that, when the linear dimensions of

V

are large compared with the wave-length of any occupied mode of the field, the

{\stackrel{^}{n}}_{V,t}

operator acquires some simple properties. The commutator of

{\stackrel{^}{n}}_{V,t}

and the detection operator

\stackrel{^}{\mathrm{A}}(\mathrm{x}, t)

{\stackrel{^}{n}}_{V,t}

is found, together with the corresponding eigenvalues, and it is shown that an arbitrary state is expressible in terms of these eigenstates. Some

N\mathrm{th}

-order correlations of the

{\stackrel{^}{n}}_{V,t}

operators are evaluated, and the results are used to calculate the probability distribution of eigenvalues of

{\stackrel{^}{n}}_{V,t}

for an arbitrary state of the field.

n

counts in a certain time interval is derived and is shown to be related in an interesting way to the corresponding expression for the photoelectric detector. The variance of the probability distribution is evaluated for some simple states of the field.

K(κ)

and

E(κ)

with modulus

κ

depending on trigonometric functions of the polar angle. We show that they diverge not only at the value

\mathbf q

of the position eigenvalue, but also on a plane containing

\mathbf q

and that they decay as inverse powers of the distance from

\mathbf q