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Journal of Optics B Quantum and Semiclassical Optics

Squeezing evolution with non-dissipative and dissipativeSU(2) systems

Saad S. M. Hassan, M. Sebawe Abdalla, G. M. Abd Al-Kader, L.A-M. Hanna

Published Apr 5, 2002• 5 citations
Publisher

Abstract

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.

DOI
10.1088/1464-4266/4/3/376

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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.
In the present communication, we consider the problem of two quantum systems with the Kerr-like medium nonlinearity. The system is cast form of an interaction between two operators of the form su(1,1) Lie algebra and su(2) Lie algebra. We obtain the wave function via the evolution operator where we use the Heisenberg equations of motion to derive the constants of motion. We discuss the atomic inversion. It is found that the Kerr-like medium decreases the amplitude and increases the fluctuations. Also we consider different types of squeezing, it is shown that the entropy squeezing is pronounced in the second quadrature, but it shows a small amount in the first quadrature. For the variance squeezing, a small amount occurs in the presence of the Kerr-like medium. However, the normal squeezing occurs in the first quadrature where the squeezing is sensitive to both the Kerr-like medium parameter and the initial state. Furthermore, the degree of entanglement is examined through the linear entropy. It is shown that the function decreases besides rapid fluctuations. The correlation function displays nonclassical behavior in addition to an increase in the amplitude of the fluctuations.
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.
A Hamiltonian, that describes the interaction between a two-level atom (su(2) algebra) and a system governed by su(1,1) Lie algebra besides two external interaction, is considered. Two canonical transformations are used, which results into removing the external terms and changing the frequencies of the interacting systems. The solution of the equations of motion of the operators is obtained and used to discuss the atomic inversion, entanglement, squeezing and correlation functions of the present system. Initially the atom is considered to be in the excited state while the other systems is in the Perelomov coherent state. Effects of the variations in the coupling parameters to the external systems are considered. They are found to be sensitive to changing entanglement, variance and entropy squeezing.
The relaxation process of a single two-level atom driven by an intense resonant coherent field is studied in the presence of a broadband squeezed vacuum field. Generalized forms for the self-field operator and the field-dependent damping coefficients are derived. In the steady state, positive atomic inversion (~5%) is shown for some range of the phase of the squeezed vacuum field. The squeezing-induced enhanced and asymmetric coherence (i.e.?non-zero dispersive atomic polarization) induces profound asymmetry in the side-bands of the Mollow fluorescent spectrum and the absorptive?dispersive spectra near the Rabi side-band frequencies. The case of two and three cooperative atoms, instead of a single atom, shows (i)?positive atomic inversion in the steady state for larger interval of the squeeze phase parameter and more enhanced extrema for the atomic polarization components and (ii) the extra side peaks in the fluorescent spectrum become asymmetric and of dispersive-like profile. Dressed-state analysis in the single-atom case with the field-dependent decay process in the presence of the squeezed vacuum field is presented and shows that the inequality of the (field-dependent) decay rates of the dressed states leads to more positive inversion in the dressed states as compared with the normal vacuum case.