The minimum-uncertainty squeezed states for atoms and photons in a cavity

Kryuchkov, Sergey; Suslov, Sergeĭ K.; Vega-Guzmán, José

doi:10.1088/0953-4075/46/10/104007

Cited by 25 publications

(48 citation statements)

References 181 publications

(424 reference statements)

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“…After completion, see , U ( t ) would satisfy the time‐dependent Schrödinger equation with a quadratic Hamiltonian for a certain class of arbitrary initial data. In general, the evolution operator in and should be applied for our initial “multiparameter squeezed number states” given by

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

(see also Marhic and Kryuchkov et al). The traditional squeeze operator corresponds to the special initial data α (0) = θ (0) = ϕ (0) = 0 and

τ false(0 false) = ((), 1 false/ 2) \ln ((), β^{2} ((), 0) false/ ω)

.…”

Section: Resultsmentioning

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%

“…Example Special cases include the degenerate parametric amplifiers, which provide a natural mechanism of creation of the multiparameter squeezed states of light (see also Marhic and Kryuchkov and the references therein). For the Hamiltonian of the form

Ĥ false(t false) = \frac{ω}{2} ((), \hat{a} {\overset{a}{true}}^{†} + {\overset{a}{true}}^{†} \hat{a}) + \frac{λ}{2} ((), e^{2 i ω t} {(\overset{a}{true})}^{2} - e^{- 2 i ω t} {((), {\overset{a}{true}}^{†})}^{2}),

another presentation of this Hamiltonian of the form is given by

\begin{matrix} alignleft \\ align-1 Ĥ (t) & align-2 = \frac{1}{2} (1 + \frac{λ}{ω} \cos (2 ω t)) {\hat{p}}^{2} \\ align-1 & align-2 + \frac{ω^{2}}{2} (1 - \frac{λ}{ω} \cos (2 ω t)) {\hat{q}}^{2} \\ align-1 & align-2 + \frac{λ}{2} \sin (2 ω t) (\hat{p} \hat{q} + \hat{q} \hat{p}) . \end{matrix}

The form is obtained using the standard annihilation and creation operators ; our solutions of Heisenberg's equations are given by

\begin{matrix} alignleft \\ align-1 \overset{a}{true} (t) & align-2 = {boldU}^{- 1} (t) a \end{matrix}

…”

Section: Solving Heisenberg's Equations Of Motionmentioning

confidence: 99%

“…In some cases, the complex parametrization and quasi‐invariants of the Ermakov‐type system may help to simplify the arguments of the hypergeometric functions. Examples are discussed in Acosta‐Humanez et al and Kryuchkov et al…”

Section: On Variable Photon Statisticsmentioning

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

(see also Marhic and Kryuchkov et al). The traditional squeeze operator corresponds to the special initial data α (0) = θ (0) = ϕ (0) = 0 and

τ false(0 false) = ((), 1 false/ 2) \ln ((), β^{2} ((), 0) false/ ω)

.…”

Section: Resultsmentioning

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%

Ĥ false(t false) = \frac{ω}{2} ((), \hat{a} {\overset{a}{true}}^{†} + {\overset{a}{true}}^{†} \hat{a}) + \frac{λ}{2} ((), e^{2 i ω t} {(\overset{a}{true})}^{2} - e^{- 2 i ω t} {((), {\overset{a}{true}}^{†})}^{2}),

another presentation of this Hamiltonian of the form is given by

\begin{matrix} alignleft \\ align-1 Ĥ (t) & align-2 = \frac{1}{2} (1 + \frac{λ}{ω} \cos (2 ω t)) {\hat{p}}^{2} \\ align-1 & align-2 + \frac{ω^{2}}{2} (1 - \frac{λ}{ω} \cos (2 ω t)) {\hat{q}}^{2} \\ align-1 & align-2 + \frac{λ}{2} \sin (2 ω t) (\hat{p} \hat{q} + \hat{q} \hat{p}) . \end{matrix}

The form is obtained using the standard annihilation and creation operators ; our solutions of Heisenberg's equations are given by

\begin{matrix} alignleft \\ align-1 \overset{a}{true} (t) & align-2 = {boldU}^{- 1} (t) a \end{matrix}

…”

Section: Solving Heisenberg's Equations Of Motionmentioning

confidence: 99%

Section: On Variable Photon Statisticsmentioning

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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“…Among other quantum mechanical analogs, the minimum-uncertainty squeezed states for atoms and photons in a cavity are reviewed in [28]. (See also [6,10,19,29] and the references therein for extensions to nonlinear geometrical optics; an optoacoustic experiment is proposed in [30].)…”

Section: × Expmentioning

confidence: 99%

Spiral laser beams in inhomogeneous media

2013

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Explicit solutions of the inhomogeneous paraxial wave equation in a linear and quadratic approximation are applied to wave fields with invariant features, such as oscillating laser beams in a parabolic waveguide and spiral light beams in varying media. A similar effect of superfocusing of particle beams in a thin monocrystal film, harmonic oscillations of cold trapped atoms, and motion in magnetic field are also mentioned. Green function and generalized Fresnel integrals. In the context of quantum mechanics, a one-dimensional (1D) linear Schrödinger equation for generalized driven harmonic oscillators,(a, b, c, d, f , and g are suitable real-valued functions of time t only), can be solved by the integral superposition principle:Gx; y; tψy; 0dy;where Gx; y; t 2πμ 0 t −1∕2× expiα 0 tx 2 β 0 txy γ 0 ty 2 δ 0 tx ε 0 ty κ 0 t; (3) for certain initial data ψx; 0 φx (see [1][2][3][4] and the references therein for more details). The intrinsic connection between Hamiltonian mechanics and the process of wave propagation is anything but a new idea [5,6]. Yet, in paraxial optics, when the time variable t represents the coordinate, say s, in the direction of wave propagation, Eqs. (2) and (3) can be thought of as a generalization of the Fresnel integral [7][8][9][10].In the paraxial approximation, a 2D coherent light field in a parabolic inhomogeneous medium with coordinates r; s x; y; s is described by the following equation for the complex field amplitude: iA s −aA xx A yy bx 2 y 2 A

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