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arXiv:1108.4717 · New Journal of Physics

Qutrit squeezing via semiclassical evolution

Andrei B Klimov, Hossein T. Dinani, Zachari E D Medendorp, Hubert de Guise

Published Nov 23, 2011• 10 citations
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Abstract

We introduce a concept of squeezing in collective qutrit systems through a geometrical picture connected to the deformation of the isotropic fluctuations of su (3) operators when evaluated in a coherent state. This kind of squeezing can be generated by Hamiltonians nonlinear in the generators of su (3) algebra. A simplest model of such a nonlinear evolution is analyzed in terms of semiclassical evolution of the SU (3) Wigner function.

DOI
10.1088/1367-2630/13/11/113033

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We introduce a concept of squeezing in collective qutrit systems through a geometrical picture connected to the deformation of the isotropic fluctuations of su (3) operators when evaluated in a coherent state. This kind of squeezing can be generated by Hamiltonians nonlinear in the generators of su (3) algebra. A simplest model of such a nonlinear evolution is analyzed in terms of semiclassical evolution of the SU (3) Wigner function.
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We study the temporal evolution of a coherent state under the action of a parametric oscillator and a nonlinear Kerr-like medium. We make use of the interaction-picture representation and use an exact time-evolution operator for the time-independent part of the Hamiltonian. We approximate the interaction-picture Hamiltonian in such a way as to make it a member of a Lie algebra. The corresponding time-evolution operator behaves like a squeezing operator due to the temporal dependence of the oscillator’s frequency. We analyze the probability amplitude, the auto-correlation function, and the Husimi distribution function for different Hamiltonian parameters. We find very good agreement between our approximate results and converged 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.
Abstract SU(1, 1) coherent and squeezed states have been widely studied in optical, hybrid atom–light, and spinor Bose–Einstein condensate experiments. Traditionally, the definition of SU(1, 1) squeezing refers to the uncertainty relationship, which implies that the SU(1, 1) coherent state is already squeezed if it is placed in an appropriate system of coordinates. In this report we investigate the basic concept of SU(1, 1) squeezing and discuss the principle for the generation of squeezed states. We establish a new definition of the squeezing criterion for the SU(1, 1) states. We study the dynamical properties in an atom–molecule system and show that the SU(1, 1) squeezed state can be generated with a nonlinear atomic interaction, which is similar to the one-axis twisting mechanism in spin squeezing. We also discuss the effect of the atomic number on squeezing, and show that the optimal squeezing and optimal squeezing angle exhibit a sudden transition at a fixed atomic number value.