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Physical Review A

Two-mode coherent states for SU(1,1)⊗SU(1,1)

Christopher C. Gerry, Adil Benmoussa

Published Aug 15, 2000• 36 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.62.033812

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We study two types of coherent states for two-mode realizations of the direct product group SU(1,1)\ensuremath⨂SU(1,1)\mathrm{SU}(1,1)\ensuremath{\bigotimes}\mathrm{SU}(1,1)SU(1,1)\ensuremath⨂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 SU(1,1)\ensuremath⨂SU(1,1)\mathrm{SU}(1,1)\ensuremath{\bigotimes}\mathrm{SU}(1,1)SU(1,1)\ensuremath⨂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.
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.
The two types of SU (1, 1) coherent states of Barut and Girardello and of Perelomov are dual in a sense that the operators in the eigenvalue equation and in the exponentials which create these types of coherent states from the lowest eigenstate (vacuum) form an asymmetric Heisenberg–Weyl algebra. A new type of SU (1, 1) coherent states which takes on an intermediate position between the two dual types of SU (1, 1) coherent states already mentioned is established and investigated. In this new type, the duality of the operators becomes a self-duality and the corresponding operators form a usual Heisenberg–Weyl algebra. Properties of the different SU (1, 1) coherent states are investigated for the realization of SU (1, 1) by one-dimensional quantum-mechanical potential problems leading to a quadratic law of energy-level spacing. Coherent SU (1, 1) phase states are discussed for this realization of SU (1, 1). It is shown that no wavepackets corresponding to any of the SU (1, 1) coherent states can exactly preserve their shape during time evolution.
The Clebsch–Gordan problems of the Barut–Girardello, and Perelomov coherent states of SL(2,R) are studied using the associated Hilbert spaces as the respective carrier spaces of the representations of the group. For the Barut–Girardello coherent states this Hilbert space is a subspace of the Bargmann–Segal Hilbert space B(C2) called the ‘‘reduced Bargmann space.’’ The generators of the group in this realization are essentially the boson operators of Holman and Biedenharn which provide a convenient starting point of the problem. For the Perelomov coherent states the associated Hilbert space turns out to be Bargmann’s canonical carrier space for the realization of the discrete series of representations, namely, the Hilbert space of functions analytic inside the open unit disc. The scalar product, the principal vector, and a complete orthonormal set in these Hilbert spaces are constructed and used for the explicit evaluation of the Clebsch–Gordan coefficients. For each of the coherent state systems the product state turns out to be the principal vector and, therefore, the coupled state itself is the Clebsch–Gordan coefficient. For the Barut–Girardello coherent states this is, apart from normalization, the product of a Bessel function and d-function. For the Perelomov coherent states, on the other hand, this closely resembles the Clebsch–Gordan coefficient of the SU(2) coherent states.
Abstract A new type of pair quantum states is introduced. Which can be considered as pair Barut-Girardello coherent states. It is an eigenstate of the operators K − ab = K − a K − b and K 3 a − K 3 b . We construct these eigenstates and generation scheme is proposed in terms of two mode state described in terms of su(1, 1) Lie algebra , We employ the second-order correlation function to discuss some non-classical properties,and violations of Cauchy-Schwarz inequalities. The phenomenon of squeezing is examined , squeezing is clear and Q-functions support that. Finally the phase distribution in the framework of an appropriate Pegg and Barnett formalism is considered and discussed.
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.
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.