On the Problem of Electromagnetic-Field Quantization

Krattenthaler, Christian; Kryuchkov, Sergey; Mahalov, Alex; Suslov, Sergeĭ K.

doi:10.1007/s10773-013-1764-3

Cited by 10 publications

(39 citation statements)

References 91 publications

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“…Under proper boundary conditions, solution of the spatial equations provides a complete set of eigenfunctions and eigenvalues for the corresponding field expansions. In the time domain, we follow the mathematical technique of the field quantization for a variable quadratic system in an abstract (Fock‐)Hilbert space recently discussed in Krattenthaler et al, primarily concentrating on a single mode of the radiation field. In this picture, the electric and magnetic fields are represented by certain time‐independent operators and the time evolution is governed by the Schrödinger equation for the field state vector

(|⟩, ψ false(t false))

i \frac{d}{d t} (|⟩, ψ false(t false)) = Ĥ false(t false) (|⟩, ψ false(t false)),

with a variable quadratic Hamiltonian of the form

Ĥ false(t false) = a false(t false) {(\overset{p}{true})}^{2} + b false(t false) {(\overset{q}{true})}^{2} + c false(t false) (\overset{q}{true}) (\overset{p}{true}) - i d false(t false) - f false(t false) (\overset{q}{true}) - g false(t false) (\overset{p}{true}),

where a , b , c , d , f , and g are suitable real‐valued functions of time only and the time‐independent operators

\overset{q}{true}

and

\overset{p}{true}

obey the canonical commutation rule

([], (\overset{q}{true}), (\overset{p}{true})) = i

(in the units of

ℏ false)

.…”

Section: Dynamical Invariants For Quantization Of Radiation Fieldmentioning

confidence: 99%

“…The time‐dependent annihilation

\overset{b}{true} false(t false)

and creation

{\overset{b}{true}}^{†} false(t false)

operators (linear integrals of motion) are described by theorem 1 of Krattenthaler et al,

\begin{matrix} alignleft \\ align-1 & align-2 \hat{b} (t) = \frac{e^{- 2 i γ (t)}}{\sqrt{2}} (β (t) \hat{q} + ϵ (t) + i \frac{\hat{p} - 2 α (t) \hat{q} - δ (t)}{β (t)}), \\ align-1 & align-2 {\hat{b}}^{†} (t) = \frac{e^{2 i γ (t)}}{\sqrt{2}} (β (t) \hat{q} + ϵ (t) - i \frac{\hat{p} - 2 α (t) \hat{q} - δ (t)}{β (t)}), \end{matrix}

in terms of solutions of the Ermakov‐type system

\frac{d α}{d t} + b + 2 c α + 4 a α^{2} = a β^{4},

\frac{d β}{d t} + ((), c + 4 a α) β = 0,

\frac{d γ}{d t}

…”

Section: Dynamical Invariants For Quantization Of Radiation Fieldmentioning

confidence: 99%

“…These dynamical invariants obey the canonical commutation rule

\overset{b}{true} false(t false) {\overset{b}{true}}^{†} false(t false) - {\overset{b}{true}}^{†} false(t false) \overset{b}{true} false(t false) = 1

at all times. The corresponding evolution (Heisenberg‐type) equations take the form:

i \frac{d}{d t} \overset{b}{true} false(t false) + ([], \overset{b}{true} false(t false), Ĥ false(t false)) = 0, i \frac{d}{d t} {\overset{b}{true}}^{†} false(t false) + ([], {\overset{b}{true}}^{†}, Ĥ false(t false)) = 0 .

…”

Section: Dynamical Invariants For Quantization Of Radiation Fieldmentioning

confidence: 99%

“…Different forms of solutions of the Ermakov‐type system have been found in Krattenthaler et al Explicit special cases are discussed in Acosta‐Humanez et al and Marhic …”

Section: Dynamical Invariants For Quantization Of Radiation Fieldmentioning

confidence: 99%

“…As a result, one may conclude that the minimum‐uncertainty squeezed states, which are important in most applications, occur when

α ((), t_{\min}) = 0

and n = 0 (see, for example, equation 5.5 of Krattenthaler et al). Indeed, only at these instances, by the following conditions hold,

θ ((), t_{\min}) = ϕ ((), t_{\min}) = α ((), t_{\min}) = 0

, and a traditional definition of single‐mode squeeze operator from previous studies can be used, say “stroboscopically.” In general, the evolution operator in and should be applied for our initial “multiparameter squeezed number states” given by

(|⟩, ψ_{n} ((), 0)) = boldU false(0 false) (|⟩, n)

(see also Marhic and Kryuchkov et al).…”

Section: The Canonical Transformation and Evolution Operatormentioning

confidence: 99%

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Time‐dependent photon statistics in variable media

Kryuchkov

Suazo

Suslov

2018

Math Methods in App Sciences

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We find explicit solutions of the Heisenberg equations of motion for a quadratic Hamiltonian, which describes a generic model of variable media in the case of multiparameter squeezed input photon configuration. The corresponding probability amplitudes and photon statistics are also derived in the Schrödinger picture in an abstract operator setting of the quantum electrodynamics; a comparison discussion is made in Heisenberg's picture as well. The unitary transformation and an extension of the squeeze/evolution operator are introduced formally. The time‐dependent photon probability amplitudes with respect to the Fock basis are indeed derived in an operator form. Further, explicit expressions for the matrix elements of the displacement and squeeze operators are derived in terms of hypergeometric functions and solutions of a certain Ermakov‐type system. In the Supporting Information, we provide a computer algebra verification of the derivation of the Ermakov‐system and of the solutions of the Heisenberg equations.

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(|⟩, ψ false(t false))

i \frac{d}{d t} (|⟩, ψ false(t false)) = Ĥ false(t false) (|⟩, ψ false(t false)),

with a variable quadratic Hamiltonian of the form

Ĥ false(t false) = a false(t false) {(\overset{p}{true})}^{2} + b false(t false) {(\overset{q}{true})}^{2} + c false(t false) (\overset{q}{true}) (\overset{p}{true}) - i d false(t false) - f false(t false) (\overset{q}{true}) - g false(t false) (\overset{p}{true}),

where a , b , c , d , f , and g are suitable real‐valued functions of time only and the time‐independent operators

\overset{q}{true}

and

\overset{p}{true}

obey the canonical commutation rule

([], (\overset{q}{true}), (\overset{p}{true})) = i

(in the units of

ℏ false)

.…”

Section: Dynamical Invariants For Quantization Of Radiation Fieldmentioning

confidence: 99%

“…The time‐dependent annihilation

\overset{b}{true} false(t false)

and creation

{\overset{b}{true}}^{†} false(t false)

operators (linear integrals of motion) are described by theorem 1 of Krattenthaler et al,

\begin{matrix} alignleft \\ align-1 & align-2 \hat{b} (t) = \frac{e^{- 2 i γ (t)}}{\sqrt{2}} (β (t) \hat{q} + ϵ (t) + i \frac{\hat{p} - 2 α (t) \hat{q} - δ (t)}{β (t)}), \\ align-1 & align-2 {\hat{b}}^{†} (t) = \frac{e^{2 i γ (t)}}{\sqrt{2}} (β (t) \hat{q} + ϵ (t) - i \frac{\hat{p} - 2 α (t) \hat{q} - δ (t)}{β (t)}), \end{matrix}

in terms of solutions of the Ermakov‐type system

\frac{d α}{d t} + b + 2 c α + 4 a α^{2} = a β^{4},

\frac{d β}{d t} + ((), c + 4 a α) β = 0,

\frac{d γ}{d t}

…”

Section: Dynamical Invariants For Quantization Of Radiation Fieldmentioning

confidence: 99%

“…These dynamical invariants obey the canonical commutation rule

\overset{b}{true} false(t false) {\overset{b}{true}}^{†} false(t false) - {\overset{b}{true}}^{†} false(t false) \overset{b}{true} false(t false) = 1

at all times. The corresponding evolution (Heisenberg‐type) equations take the form:

i \frac{d}{d t} \overset{b}{true} false(t false) + ([], \overset{b}{true} false(t false), Ĥ false(t false)) = 0, i \frac{d}{d t} {\overset{b}{true}}^{†} false(t false) + ([], {\overset{b}{true}}^{†}, Ĥ false(t false)) = 0 .

…”

Section: Dynamical Invariants For Quantization Of Radiation Fieldmentioning

confidence: 99%

“…Different forms of solutions of the Ermakov‐type system have been found in Krattenthaler et al Explicit special cases are discussed in Acosta‐Humanez et al and Marhic …”

Section: Dynamical Invariants For Quantization Of Radiation Fieldmentioning

confidence: 99%

“…As a result, one may conclude that the minimum‐uncertainty squeezed states, which are important in most applications, occur when

α ((), t_{\min}) = 0

and n = 0 (see, for example, equation 5.5 of Krattenthaler et al). Indeed, only at these instances, by the following conditions hold,

θ ((), t_{\min}) = ϕ ((), t_{\min}) = α ((), t_{\min}) = 0

(|⟩, ψ_{n} ((), 0)) = boldU false(0 false) (|⟩, n)

(see also Marhic and Kryuchkov et al).…”

Section: The Canonical Transformation and Evolution Operatormentioning

confidence: 99%

See 3 more Smart Citations

Time‐dependent photon statistics in variable media

Kryuchkov

Suazo

Suslov

2018

Math Methods in App Sciences

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show abstract

An Introduction to Special Functions with Some Applications to Quantum Mechanics

Suslov

Vega-Guzmán

Barley

2020

Orthogonal Polynomials

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Complex Form of Classical and Quantum Electrodynamics

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2017

Springer Proceedings in Mathematics &Amp; Statistics

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Physical laws should have mathematical beauty. " -P. A. M. DiracAbstract. We consider a complex covariant form of the macroscopic Maxwell equations, in a moving medium or at rest, following the original ideas of Minkowski. A compact, Lorentz invariant, derivation of the energy-momentum tensor and the corresponding differential balance equations are given. Conservation laws and quantization of the electromagnetic field will be discussed in this covariant approach elsewhere.

show abstract

On the Problem of Electromagnetic-Field Quantization

Cited by 10 publications

References 91 publications

Time‐dependent photon statistics in variable media

Time‐dependent photon statistics in variable media

An Introduction to Special Functions with Some Applications to Quantum Mechanics

Complex Form of Classical and Quantum Electrodynamics

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