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Physical Review A

Mixed excitation- and deexcitation-operator coherent states for the SU(1,1) group

G. Satya Prakash, G. S. Agarwal

Published Nov 1, 1994• 32 citations
Publisher

Abstract

Squeezed coherent states associated with the SU(1,1) group defined by (\ensuremath{\mu}${\mathit{a}}^{2}$+\ensuremath{\nu}${\mathit{a}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}2}$)\ensuremath{\Vert}\ensuremath{\psi}〉=\ensuremath{\lambda}\ensuremath{\Vert}\ensuremath{\psi}〉, \ensuremath{\Vert}\ensuremath{\mu}${\mathrm{\ensuremath{\Vert}}}^{2}$-\ensuremath{\Vert}\ensuremath{\nu}${\mathrm{\ensuremath{\Vert}}}^{2}$=1 are studied in detail. An explicit expression for the quasidistribution Q(\ensuremath{\alpha},${\mathrm{\ensuremath{\alpha}}}^{\mathrm{*}}$) in terms of the parabolic cylindrical functions is derived. The nonclassical properties of these states are discussed. The possibility of producing SU(1,1) squeezed coherent states using the down-conversion process is also demonstrated.

DOI
10.1103/physreva.50.4258

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Squeezed coherent states associated with the SU(1,1) group defined by (\ensuremath{\mu}a2{\mathit{a}}^{2}a2+\ensuremath{\nu}a\ifmmode∘\else\textdegree\fi2{\mathit{a}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}2}a\ifmmode∘\else\textdegree\fi2)\ensuremath{\Vert}\ensuremath{\psi}〉=\ensuremath{\lambda}\ensuremath{\Vert}\ensuremath{\psi}〉, \ensuremath{\Vert}\ensuremath{\mu}\ensuremath∥2{\mathrm{\ensuremath{\Vert}}}^{2}\ensuremath∥2-\ensuremath{\Vert}\ensuremath{\nu}\ensuremath∥2{\mathrm{\ensuremath{\Vert}}}^{2}\ensuremath∥2=1 are studied in detail. An explicit expression for the quasidistribution Q(\ensuremath{\alpha},\ensuremathα∗{\mathrm{\ensuremath{\alpha}}}^{\mathrm{*}}\ensuremathα∗) in terms of the parabolic cylindrical functions is derived. The nonclassical properties of these states are discussed. The possibility of producing SU(1,1) squeezed coherent states using the down-conversion process is also demonstrated.
In this paper, after a brief review on the entangled coherent state, a general form of the entangled nonlinear coherent state has been introduced based on the nonlinear coherent states approach. By choosing particular f-deformation functions, theoretical schemes for generation of two distinct classes of such states associated with SU(1,1) and SU(2) groups are presented. The entanglement characteristic of the corresponding entangled coherent states has been studied by evaluating the concurrence. In the continuation, other nonclassical behaviors of the produced states have been analyzed by investigating sub-Poissonian photon statistics, the oscillatory photon number distribution, one- and two-mode quadrature squeezing, and sum and difference squeezing. Accordingly, in addition to their entanglement property, other nonclassicality features of the introduced states are well established in detail.
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.
Two new types of coherent states associated with the Cλ-extended oscillator, where Cλ is the cyclic group of order λ, are introduced. The first ones include as special cases both the Barut-Girardello and the Perelomov su(1,1) coherent states for λ = 2, as well as the annihilation-operator coherent states of the Cλ-extended oscillator spectrum generating algebra for higher λ values. The second ones, which are eigenstates of the Cλ-extended oscillator annihilation operator, extend to higher λ values the paraboson coherent states, to which they reduce for λ = 2. All these states satisfy a unity resolution relation in the Cλ-extended oscillator Fock space (or in some subspace thereof). They give rise to Bargmann representations of the latter, wherein the generators of the Cλ-extended oscillator algebra are realized as differential-operatorvalued matrices (or differential operators). The statistical and squeezing properties of the new coherent states are investigated over a wide range of parameters and some interesting nonclassical features are exhibited. 1 1
We introduce states defined by \ensuremath{\Vert}\ensuremath{\alpha},m〉=${\mathit{a}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}\mathit{m}}$\ensuremath{\Vert}\ensuremath{\alpha}〉 up to a normalization constant, where \ensuremath{\Vert}\ensuremath{\alpha}〉 is a coherent state and m an integer. We study the mathematical and physical properties of such states. We demonstrate phase squeezing and the sub-Poissonian character of the fields in such states. We study in detail the quasiprobability distributions and the distribution of the field quadrature. We also show how such states can be produced in nonlinear processes in cavities.
The idea of construction of the nonlinear coherent states based on the hypergeometric- type operators associated to the Weyl-Heisenberg group [J:P hys:A 45(2012) 095304], are generalized to the similar states for the arbitrary Lie group SU(1, 1). By using of a discrete, unitary and irreducible representation of the Lie algebra su(1, 1) wide range of generalized nonlinear coherent states(GNCS) have been introduced, which admit a resolution of the identity through positive definite measures on the complex plane. We have shown that realization of these states for different values of the deformation pa- rameters r = 1 and 2 lead to the well-known Klauder-Perelomov and Barut-Girardello coherent states associated to the Irreps of the Lie algebra su(1, 1), respectively. It is worth to mention that, like the canonical coherent states, GNCS possess the temporal stability property. Finally, studying some statistical characters implies that they have indeed nonclassical features such as squeezing, anti-bunching effect and sub-Poissonian statistics, too.
The properties of correlated two-mode SU(1, 1) coherent states are studied. These states are a generalization of the two-mode squeezed vacuum states and are generated by applying the two-mode squeeze operator to a state in which there are initially q photons in one mode, |q, 0〉, the squeezed vacuum (q = 0) obviously being a special case. For these states the photon-number distribution and the nonclassical properties of photon antibunching, enhanced phase fluctuations, violations of the Cauchy–Schwarz inequality, two-mode squeezing, and sum squeezing are studied. A Q-function quasi-probability distribution for these states is also presented.