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.
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.
Barut–Girardello coherent states corresponding to the (l−m)- and (l+m)-integer discrete irreducible representations of su(1,1) Lie algebra are calculated by the spherical harmonics Ylm(θ,ϕ). Their explicit compact forms and also, to realize the resolution of the identity, their corresponding positive definite measures on the complex plane are obtained in terms of the known functions. It is also shown that coherent states of both positive and negative representations separately lead us to construct new representation bases for su(1,1) Lie algebra. Then, it is shown that the su(1,1)-Barut–Girardello coherent states corresponding to two particles containing the spatial parity symmetries of a bipartite quantum system can be entangled in ten different ways.
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.
We study the violations of the Bell–Clauser–Horne–Shimony–Holt inequality for entangled SU(1,1) Barut–Girardello coherent states. As in a previous paper where violations of the inequality were studied for entangled SU(1,1) coherent states of the Perelomov form [Phys. Rev. A.93, 042104 (2016)PLRAAN1050-294710.1103/PhysRevA.93.042104], we choose as our observable the SU(1,1) parity operator, though here we discuss the physical meaning of the operator for single-mode and two-mode bosonic realizations of the su(1,1) Lie algebra. We show that the SU(1,1) parity operator is not the same as the usual photon number parity operator but rather is a form of higher-order photon number parity. Distant observers Alice and Bob perform on each of their subsystems non-compact SU(1,1) transformations characterized by hyperbolic angles, followed by measurements of the SU(1,1) parity operators. We find strong violations of the inequality over a wide range of parameters, and these violations are stronger than those that can be obtained by the entangled Perelomov SU(1,1) coherent state.
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.
We start with the Heisenberg–Weyl algebra and after the definitions of the Fock states we give the definition of the coherent state of this group. This is followed by the exposition of the SU(2) and SU(1, 1) algebras and their coherent states. From there we go on describing the binomial state and its extensions as realizations of the SU(2) group. This is followed by considering the negative binomial states, and some squeezed states as realizations of the SU(1, 1) group. Generation schemes based on physical systems are mentioned for some of these states.
In this paper, we first suggest a scheme for the generation of a particular class of Gilmore–Perelomov-type SU(1,1) coherent states, which may be established as nonlinear coherent states. The proposal employs a two-level atom that interacts with a single-mode quantized cavity field (by using an intensity-dependent Jaynes–Cummings model) and at the same time a strong external classical field. The time evolution of the system first leads to the generation of a superposition of SU(1,1) coherent states. Depending on the initial states of the atom and the field which may be appropriately prepared, and also under the conditions in which the atom is detected (in the excited or ground state) after the occurrence of the interaction, the field will be collapsed to arbitrary combinations or a single class of Gilmore–Perelomov-type SU(1,1) coherent states. Then, it is shown that, following a similar procedure, our proposed scheme can successfully generate various superpositions and, in particular, a single class of SU(2) coherent states, too.