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Physica Scripta

Canonical transformations and multipartite coupled parametric processes

Timur Tlyachev, Alexander Mikhailovich Chebotarev, A. S. Chirkin

Published Apr 1, 2014• 9 citations
Publisher

Abstract

A general approach is considered to find wave function and density operator evolution of the systems including the arbitrary number of coupled optical parametric interactions described by the quadratic Hamiltonian. This approach is based on the time-dependent canonical transformations that define the evolution of the system in the Heisenberg picture or in the interaction picture. An application is illustrated with four-frequency entangled light fields, which can be produced at the nonlinear optical interactions in vapors and solid states. The process includes one non-degenerate parametric down-conversion followed by two parametric up-conversions occurring in the field of the same classical pumping wave.

DOI
10.1088/0031-8949/2014/t160/014041

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A general approach is considered to find wave function and density operator evolution of the systems including the arbitrary number of coupled optical parametric interactions described by the quadratic Hamiltonian. This approach is based on the time-dependent canonical transformations that define the evolution of the system in the Heisenberg picture or in the interaction picture. An application is illustrated with four-frequency entangled light fields, which can be produced at the nonlinear optical interactions in vapors and solid states. The process includes one non-degenerate parametric down-conversion followed by two parametric up-conversions occurring in the field of the same classical pumping wave.
A quantum theory of parametric amplification and frequency conversion of an optical image in coupled nonlinear optical processes that include one parametric amplification process at high-frequency pumping and two up-conversion processes in the same pump field is developed. The field momentum operator that takes into account the diffraction and group velocities of the waves is used to derive the quantum equations related to the spatial dynamics of the images during the interaction. An optical scheme for the amplification and conversion of a close image is considered. The mean photon number density and signal-to-noise ratio are calculated in the fixed-pump-field approximation for images at various frequencies. It has been established that the signal-to-noise ratio decreases with increasing interaction length in the amplified image and increases in the images at the generated frequencies, tending to asymptotic values for all interacting waves. The variance of the difference of the numbers of photons is calculated for various pairs of frequencies. The quantum entanglement of the optical images formed in a high-frequency pump field is shown to be converted to higher frequencies during the generation of sum frequencies. Thus, two pairs of entangled optical images are produced in the process considered.
The dynamics of a generalized non-degenerate optical parametric down-conversion interaction whose Hamiltonian includes an arbitrary time-dependent driving part and a two-mode coupled part is studied by adopting the Lewis–Riesenfeld invariant theory. The closed formulae for the evolution of the quantum states and the evolution operators of the system are obtained. It is shown that various generalized squeezed states arise naturally in the process, and the two-mode squeezed effect is independent of the driving part. An explicitly analytical solution of the Schrödinger equation is further derived as the classical generalized force acting on each mode and the coupling of the two modes both have harmonic time dependences. This solution is found to be in agreement with previous research in special cases.
Extending the scheme developed for a single mode of the electromagnetic field in the preceding paper [F. Dell'Anno, S. De Siena, and F. Illuminati, Phys. Rev. A 69, 033812 (2004)], we introduce two-mode nonlinear canonical transformations depending on two heterodyne mixing angles. They are defined in terms of Hermitian nonlinear functions that realize heterodyne superpositions of conjugate quadratures of bipartite systems. The canonical transformations diagonalize a class of Hamiltonians describing nondegenerate and degenerate multiphoton processes. We determine the coherent states associated with the canonical transformations, which generalize the nondegenerate two-photon squeezed states. Such heterodyne multiphoton squeezed states are defined as the simultaneous eigenstates of the transformed, coupled annihilation operators. They are generated by nonlinear unitary evolutions acting on two-mode squeezed states. They are non-Gaussian, highly nonclassical, entangled states. For a quadratic nonlinearity the heterodyne multiphoton squeezed states define two-mode cubic phase states. The statistical properties of these states can be widely adjusted by tuning the heterodyne mixing angles, the phases of the nonlinear couplings, as well as the strength of the nonlinearity. For quadratic nonlinearity, we study the higher-order contributions to the susceptibility in nonlinear media and we suggest possible experimental realizations of multiphoton conversion processes generating the cubic-phase heterodyne squeezed states.
Quantum properties of parametric amplification and frequency conversion of an optical image in coupled parametric interactions are analyzed for the case of a remote object. The coupled-wave interaction consists of a traditional process of parametric amplification in the high-frequency pump field accompanied by the frequency mixing of generated and pump waves. The photon number density and signal-to-noise ratio at the interacting frequencies are studied in the image plane at the output of the nonlinear frequency converter in the fixed pump field approximation. The degree of entanglement of optical images at frequencies above and below the pump frequency is investigated.
By properly selecting the time-dependent unitary transformation for the linear combination of the number operators, we construct a time-dependent invariant and derive the corresponding auxiliary equations for the degenerate and non-degenerate coupled parametric down-conversion system with driving term. By means of this invariant and the Lewis–Riesenfeld quantum invariant theory, we obtain closed formulae of the quantum state and the evolution operator of the system. We show that the time evolution of the quantum system directly leads to production of various generalized one- and two-mode combination squeezed states, and the squeezed effect is independent of the driving term of the Hamiltonian. In some special cases, the current solution can reduce to the results of the previous works.