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Journal of Experimental and Theoretical Physics

Simultaneous parametric generation and up-conversion of entangled optical images

M. Yu. Saygin, A. S. Chirkin

Published Jul 1, 2010• 12 citations
Publisher

Abstract

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.

DOI
10.1134/s1063776110070022

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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.
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.
We develop for the first time a quantum theory of simultaneous amplification and frequency up-conversion of optical images in two coupled three-wave processes. The theory developed takes into account the diffraction effect and the difference of the group velocities of the interacting waves. Due to the degenerate parametric process involved in the nonlinear optical interactions, the up-conversion also possesses phase sensitivity. As a result, the statistical characteristics of the amplified image and the converted one depend on the phase relation between the signal and the pump waves. It is shown that at the output of the nonlinear crystal the signal-to-noise ratios for the amplified image and the converted one may be same as those for the initial image, i.e. transformation of the optical image may be 'noiseless'.
We develop quantum theory of process of the parametric optical image amplification at high-frequency pumping which is accompanied by up - conversion one. Such interactions can be implemented by two coupled three-frequency optical processes. The theory developed takes into account the diffraction effect and the difference of group velocities of interacting waves. The expressions for mean photon numbers and noise figures of the interacting waves are obtained and the dependence of these parameters on the interaction length are examined.
We develop for the first time the quantum theory of parametric image amplification at low-frequency pumping. Such a process can be implemented by coupled consecutive three-frequency optical interactions. As a result, besides the amplified optical image on the signal frequency, there are two other images with frequencies which are below the pumping frequency. The theory developed takes into account the diffraction effect and group mismatch of interacting waves. For the interacting waves the expressions for mean photon numbers and signal-to-noise ratios are deduced and the dependence of these parameters on the interaction length and coordinate at the image plane are studied.
The teleportation of entangled optical images is for the first time considered. The scheme offered is based on process of four-partite state generation in coupled parametric interactions realizing in an aperiodic nonlinear photonic crystal. The coupled nonlinear optical process consists of one parametric down-conversion process which is followed by two up-conversion ones. We study nonclassical properties of the generated fields by analyzing the EPR correlations for different pairs of modes and carry out the comparison of the EPR correlations of the input and images output images and also calculate fidelity of the teleportation exploiting the covariance matrices of the images. We take into account the difference of spatial frequency bandwidths of the entangled images and fields generated in the coupled process. It has been established that under certain parameters of the scheme one can be obtain rather high fidelity of the teleportation and good preservation of the quantum correlations of the input images for all details of the images.
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.