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Physical Review A

Coherent states for four-mode systems in quantum optics

Bindu A. Bambah, G. S. Agarwal

Published Jun 1, 1995• 10 citations
Publisher

Abstract

We introduce and study the properties of a class of coherent states for the group SU(1,1)\ifmmode\times\else\texttimes\fi{}SU(1,1) and derive explicit expressions for these using the Clebsch-Gordan algebra for the SU(1,1) group. We also derive a phase space representation based on pair coherent states rather than the standard harmonic-oscillator coherent states. We discuss the utility of the resulting ``bi-pair coherent states'' in the context of four-mode interactions in quantum optics.

DOI
10.1103/physreva.51.4918

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We introduce and study the properties of a class of coherent states for the group SU(1,1)\ifmmode\times\else\texttimes\fi{}SU(1,1) and derive explicit expressions for these using the Clebsch-Gordan algebra for the SU(1,1) group. We also derive a phase space representation based on pair coherent states rather than the standard harmonic-oscillator coherent states. We discuss the utility of the resulting ``bi-pair coherent states'' in the context of four-mode interactions in quantum optics.
We introduce and study the properties of a class of coherent states for the group SU(1,1) X SU(1,1) and derive explicit expressions for these using the Clebsch-Gordan algebra for the SU(1,1) group. We restrict ourselves to the discrete series representations of SU(1,1). These are the generalization of the `Barut Girardello' coherent states to the Kronecker Product of two non-compact groups.The resolution of the identity and the analytic phase space representation of these states is presented. This phase space representation is based on the basis of products of `pair coherent states' rather than the standard number state canonical basis. We discuss the utility of the resulting `bi-pair coherent states' in the context of four-mode interactions in quantum optics.
We study two types of coherent states for two-mode realizations of the direct product group $\mathrm{SU}(1,1)\ensuremath{\bigotimes}\mathrm{SU}(1,1)$ [which is locally isomorphic to SO(2,2)] constructed from the coupling of two single-mode realizations of SU(1,1). The basis states for the relevant representations of $\mathrm{SU}(1,1)\ensuremath{\bigotimes}\mathrm{SU}(1,1)$ are constructed from the SU(1,1) Clebsch-Gordon coefficients. From these, Perelomov and Barut-Giradello coherent states are constructed. Various properties of the states are discussed, and methods for generating them are proposed. Some of the states can be generated by the operation of beam splitters with two-mode squeezed vacuum states or pair coherent states as inputs. We show that a competitive two-channel two-photon process gives rise to states of the Barut-Girardello type as the steady-state solutions of the associated master equation for appropriate initial conditions.
Unitary operator coherent states, as defined by Klauder (1963), Perelomov (1972) and Gilmore (1974), are considered for the harmonic series irreducible representations of U(p,q). Various properties of these states are reviewed, including their reproducing kernel and measure explicit forms, as well as the representation they lead to for the U(p,q) generators. Starting from a contraction of the su(p,q) Lie algebra, both Dyson and Holstein-Primakoff boson realisations of u(p,q) are obtained in terms of pq pairs of boson creation, annihilation operators and of the generators of a U(p)*U(q) intrinsic group. Such boson realisations are applied to determine the matrix elements of the U(p,q) generators between discrete bases classified according to the chain U(p,q) contains/implies U(p)*U(q). Finally, the isomorphism between so(4,2) and su(2,2) is employed to derive the corresponding properties of the SO(4,2) unitary operator coherent states for application in atomic physics.
Entangled SU (2) and SU (1,1) coherent states are developed as superpositions of multiparticle SU (2) and SU (1,1) coherent states. As a special example of entangled SU (2) coherent states, entangled binomial states are introduced and these entangled binomial states enable the contraction from entangled SU (2) coherent states to entangled harmonic oscillator coherent states. Entangled SU (2) coherent states are discussed in the context of pairs of qubits. We also introduce the entangled negative binomial states and entangled squeezed states as examples of entangled SU (1,1) coherent states. A method for generating the entangled SU (2) and SU (1,1) coherent states is discussed and degrees of entanglement calculated. Two types of SU (1,1) coherent states are discussed in each case: Perelomov coherent states and Barut-Girardello coherent states.
Starting from the symplectic Lie groups Sp (2 n , ) of an n -mode system in classical and quantum mechanics we discuss, in particular, Sp (2, )~ SO (2,1) for a single mode and Sp (4, )~ SO (3,2) with some of its important subgroups such as SU (1,1)~ Sp (2, ) and SU (2). We then look at their applications in quantum optics (squeezing, phase states, beamsplitter and polarization). We explicitly construct the root scheme of the Lie algebra sp (4, ) in two different bases as it is an important means for the investigation of the group properties and the construction of the irreducible representations. Apart from the well known realizations of Sp (2, ) by single-mode and two-mode systems, we discuss some other realizations of this group by fractal combinations of boson operators.