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Journal of Experimental and Theoretical Physics Letters

Entangled photon states in consecutive nonlinear optical interactions

A. V. Rodionov, A. S. Chirkin

Published Mar 1, 2004• 35 citations
Publisher

Abstract

Quantum theory of two consecutive light-wave parametric interactions with aliquant frequencies produced by a common pump wave in a crystal is developed. Using the differentiation method, the unitary evolution operator of the system is reduced to an ordered form that allows the calculation of the field state and the statistical characteristics of interacting waves. It is shown that, for the initial vacuum field state, the created photons obey the super-Poisson statistics at the interacting frequencies and are in a multiparticle entangled state.

DOI
10.1134/1.1759404

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Quantum theory of two consecutive light-wave parametric interactions with aliquant frequencies produced by a common pump wave in a crystal is developed. Using the differentiation method, the unitary evolution operator of the system is reduced to an ordered form that allows the calculation of the field state and the statistical characteristics of interacting waves. It is shown that, for the initial vacuum field state, the created photons obey the super-Poisson statistics at the interacting frequencies and are in a multiparticle entangled state.
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The results of quantum analysis of the light generated by consecutive and simultaneous quasi-phase-matched (QPM) wave interactions in periodically poled nonlinear crystals (PPNCs) are presented. In the case of the consecutive interactions of waves with multiple frequencies ω, 2ω and 3ω, the parametric amplification at low-frequency pumping is investigated. Generation of the quadrature-squeezed light at frequencies ω and 3ω in the 2ω frequency pumping field, the photon statistics and correlation as well as the entanglement properties for photons produced at different frequencies are studied. In the case of simultaneous QPM spontaneous parametric down-conversion processes in single PPNC, the main attention is paid to obtaining the polarization-entangled states at the collinear geometry of the wave interaction.
Quasi-phase-matched consecutive three-frequency interactions between waves with nonmultiple frequencies are considered. The quantum theory of parametric amplification is developed in the approximation of an undepleted low-frequency pump field in the Heisenberg representation. It is shown that, with a certain choice of quasi-phase-matching orders, the regime of exponential increase in the mean number of photons of interacting waves can be realized. In this case, the photon statistics of generated waves is super-Poissonian one. The correlation of fluctuations of numbers of photons with different frequencies is also considered.
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.
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.