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Journal of the Optical Society of America B

Mixed-state SU(2) squeezing and its occurrence in two quantum optical models

P. K. Aravind

Published Dec 1, 1986• 17 citations
Publisher

Abstract

The atomic pseudospin operators describing a two-level atom, and obeying the SU(2) Lie algebra, can exhibit reduced quantum fluctuations (“squeezing”) in certain states of the atom. These states include both pure and mixed states of the atom. Parameterizing the atom by the spherical polar components S, θ, and ϕ of its pseudospin vector S, we determine the parameter ranges in which squeezing can occur and the magnitude of this squeezing. We illustrate our considerations by examining two models, namely, the coherent-state Jaynes–Cummings model and the optical Bloch equations, in which mixed-state SU(2) squeezing arises dynamically.

DOI
10.1364/josab.3.001712

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The semiclassical Dicke model and the ideal parametric amplifier are described by Hamiltonians linear in the SU(2) and SU( 1,1) group generators, respectively. We use the usual pseudospin vector as well as its SU(1,1) analog1 to give a unified description of the dynamics of the coherent states in these models. The pseudospin dynamics is governed by a precession (or pseudoprecession) equation, hence simple geometrical arguments can be used to infer the evolution of the coherent state parameters. To study fluctuations and squeezing we introduce a correlation tensor which is a higher-order generalization of the pseudospin vector. The correlation tensor obeys a generalized precession equation which can be factored into scalar, vector, and quadrupole parts and solved exactly by application of the Cayley-Hamilton theorem. Our approach to the calculation of squeezing and fluctuations provides an alternative to the more traditional approach employing disentangling theorems. It also provides an attractive and closely parallel geometrical description of coherent state evolution and squeezing in the two models.

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The atomic pseudospin operators describing a two-level atom, and obeying the SU(2) Lie algebra, can exhibit reduced quantum fluctuations (“squeezing”) in certain states of the atom. These states include both pure and mixed states of the atom. Parameterizing the atom by the spherical polar components S, θ, and ϕ of its pseudospin vector S, we determine the parameter ranges in which squeezing can occur and the magnitude of this squeezing. We illustrate our considerations by examining two models, namely, the coherent-state Jaynes–Cummings model and the optical Bloch equations, in which mixed-state SU(2) squeezing arises dynamically.
The theory of generalized coherent states associated with the SU(2) and SU(1,1) Lie algebras is discussed and applied in an investigation of possible reductions of fluctuations. In the framework of a system of N two-level atoms the squeezing of angular-momentum [SU(2)] fluctuations is exhibited for optical coherent transients involving the photon echo. The SU(1,1) fluctuations are discussed and established for general two-photon processes involving dynamical variables different from the creation and annihilation operators.
We examine the time-dependent squeezing of fluctuations associated with the atomic coherent state of the SU (2) Lie algebra. Based on the exact solutions for a non-dissipative SU (2) system, regions of squeezing and its symmetric/asymmetric patterns are investigated in the cases of weak and strong coupling and for the resonance condition. For a spontaneously damped two-level atom, transient squeezing occurs in the atomic polarization components with initial atomic coherent state |θ,ϕ⟩, with asymmetric structure with respect to the state-of-zero atomic inversion. The rapidly vanishing squeezed regions for <θ<π are accompanied by entropy reduction.
We study interaction between a pair of indistinguishable two-level atoms and the single-mode cavity field. It is supposed that the pair of two-level atoms is laser cooled and trapped into the ground vibrational state, in which vibrational quantum number ⟨nv⟩=0\langle n_v\rangle =0⟨nv​⟩=0 . Two Jaynes–Cummings models are investigated. One is the Jaynes–Cummings model with intensity-dependent coupling and the another is the two-photon Jaynes–Cummings model of a pair of indistinguishable two-level atoms. It should be noted that in the present model, at initial moment t=0t=0t=0 , in intensity-dependent Jaynes–Cummings model the cavity field is prepared in Holstein–Primakoff SU(1,1) CS, while in two-photon Jaynes–Cummings model it is prepared in the squeezed vacuum state. Moreover, at initial moment t=0t=0t=0 , pair of atoms is supposed to be in the first excited state ∣e1⟩\vert e_1\rangle ∣e1​⟩ in both models. By using exact analytical solutions for state-vectors of the coupled atom-field systems amplitude-squared squeezing of the quantized cavity field is examined as a function of the ∣ξ∣\vert \xi \vert ∣ξ∣ parameter. In this situation, in both models higher-order squeezing has the tendency towards oscillations, but exact periodicity of these oscillations is violated by the analogy with the second-order squeezing.
In this paper, the behavior of SU(2) squeezing of the mixed-state in two-level atom are studied by means of the time-evolution operator method. We have obtained new results that the mixed-state atomic squeezing under interacting of the light field is deeper than the general pure-state atomic SU(2) squeezing. Furthermore, the squeezing value of the atom in pseudo-ground state is more greater than that in pseudo-excited state. The relation between the statistical probability of atomic level population and the squeezing is studied by numerical calculations.
The semi-classical Dicke model and the ideal parametric amplifier are described by Hamiltonians linear in the SU(2) and SU(1, 1) group generators, respectively. We use the usual pseudospin vector as well as its SU(1, 1) analog to give a unified description of the dynamics of the coherent states in these models. The pseudospin vector obeys a precession (or pseudoprecession) equation; hence simple geometrical arguments can be used to relate its evolution to that of the coherent-state parameters. To study fluctuations and squeezing we introduce the pseudospin tensor, which is a higher-order generalization of the pseudospin vector. The pseudospin tensor obeys a generalized precession equation that can be factored into scalar, vector, and quadrupole parts and solved exactly by application of the Cayley–Hamilton theorem. Squeezing is related directly to the behavior of the. pseudospin tensor and can therefore be calculated without invoking the group-theoretical disentangling formulas. Our approach is thus elementary in character. It also provides an attractive and closely parallel geometrical account of coherent-state evolution and squeezing in the two models.
The semiclassical Dicke model and the ideal parametric amplifier are described by Hamiltonians linear in the SU(2) and SU( 1,1) group generators, respectively. We use the usual pseudospin vector as well as its SU(1,1) analog1 to give a unified description of the dynamics of the coherent states in these models. The pseudospin dynamics is governed by a precession (or pseudoprecession) equation, hence simple geometrical arguments can be used to infer the evolution of the coherent state parameters. To study fluctuations and squeezing we introduce a correlation tensor which is a higher-order generalization of the pseudospin vector. The correlation tensor obeys a generalized precession equation which can be factored into scalar, vector, and quadrupole parts and solved exactly by application of the Cayley-Hamilton theorem. Our approach to the calculation of squeezing and fluctuations provides an alternative to the more traditional approach employing disentangling theorems. It also provides an attractive and closely parallel geometrical description of coherent state evolution and squeezing in the two models.