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Journal of Modern Optics

Three boson interaction process: spectra and coherent states

Goce Chadzitaskos, C. Daskaloyannis, Jan Smotlacha

Published Mar 1, 2013• 0 citations
Publisher

Abstract

We use the methods of constructions of and deformed coherent states in order to construct the coherent states for down conversion processes. The down conversion process can be understood as a quasi-exactly solvable model of quantum mechanics. After the reduction of the Hamiltonian, we use the Turbiner polynomials approach, and the eigenvalues of the Hamiltonian for low number of photons are calculated and the approximation formula is found out. After the discussion on the time evolution and the entanglement, the coherent states are constructed as the eigenstates of the reduced annihilation operator.

DOI
10.1080/09500340.2013.793415

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We use the methods of constructions of and deformed coherent states in order to construct the coherent states for down conversion processes. The down conversion process can be understood as a quasi-exactly solvable model of quantum mechanics. After the reduction of the Hamiltonian, we use the Turbiner polynomials approach, and the eigenvalues of the Hamiltonian for low number of photons are calculated and the approximation formula is found out. After the discussion on the time evolution and the entanglement, the coherent states are constructed as the eigenstates of the reduced annihilation operator.
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A general procedure for constructing coherent states, which are eigenstates of annihilation operators, related to quantum mechanical potential problems, is presented. These coherent states, by construction are not potential specific and rely on the properties of the orthogonal polynomials, for their derivation. The information about a given quantum mechanical potential enters into these states, through the orthogonal polynomials associated with it and also through its ground state wave function. The time evolution of some of these states exhibit fractional revivals, having relevance to the factorization problem.
We introduce a set of coherent states which are associated with quantum systems governed by a trilinear boson Hamiltonian. These states are produced by the action of a nonunitary displacement operator on a reference state, and can be equivalently defined by some eigenvalue equations. The system prepared initially in the reference state will evolve into the coherent state during the first instants of the interaction process. Some properties of the coherent states are discussed. In particular, the resolution of the identity is derived and the related analytic representation in the complex plane is developed. It is shown that this analytic representation coincides with a double representation based on the Glauber coherent states of the pump mode and on the SU(1,1) Perelomov coherent states of the signal-idler system. Entanglement between the field modes and photon statistics of the coherent states are studied. Connections between the coherent states and the long-time evolution induced by the trilinear Hamiltonian are considered. \textcopyright{} 1996 The American Physical Society.
Making use of the recently introduced inverses of boson creation and annihilation operators, the authors show that the 'photon-added coherent states' are eigenstates of the operator (a Dagger m aa Dagger -m ) and also of the operator (a-ma Dagger -1 ) with eigenvalue alpha . The authors discuss completeness of these states. They obtain another family of photon-added coherent states which are proportional to a -m mod alpha ), where m is a positive integer and mod alpha ) is a coherent state. They show that these are the eigenstates of the operator (a -m a m+1 ) with eigenvalue alpha . Analogously, the authors introduce 'photon-depleted coherent states', a Dagger -m mod alpha ), and show that these are the eigenstates of the operator (a+ma Dagger -1 ) with eigenvalue alpha . The conditions for squeezing and antibunching of photons have also been discussed.
In order to construct time-dependent pseudo-bosonic coherent states, first, we extend the non-Hermitian integrals of motion method to cases where the quantum systems are described by time-dependent non-Hermitian Hamiltonians; second, we introduce a pseudo-bosonic annihilation operator defined as a time-dependent non-Hermitian linear invariant. The pseudo-bosons operators are a pseudo-Hermitian extension of usual boson operators. In fact, they are obtained from the modification of usual boson commutation relations where the annihilation and creation operators are related to their adjoint operators via the bounded Hermitian invertible operator or metric operator. Thus, the pseudo-bosonic coherent states are just obtained as eigenstates of the pseudo-bosonic annihilation operator. As an illustration, we study the time-dependent non-Hermitian Swanson Hamiltonian and we compare the obtained results with those in the literature (Swanson model but time-independent).