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International Conference Laser Optics (ICLO)

Multipartite entangled quantum states in coupled optical parametric interactions: Density matrix and entropy characterization

A. S. Chirkin, M. Yu. Saygin, Timur Tlyachev, M. Yu. Saygin

Published Jun 1, 2014• 0 citations
Publisher

Abstract

We describe the normal ordering method (NOM) as a convinient way to obtaining explicit forms of the state vectors and density matrices of quantum states, usually generated in multiple coupled parametric optical processes. As a specific example, our method is demonstrated for the states, generated in three second-order parametric optical processes - one parametric down-conversion accompanied by two up-conversion processes. The application of the method allowed us to calculate the entropic characteristics of the multipartite quantum system.

DOI
10.1109/lo.2014.6886440

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We describe the normal ordering method (NOM) as a convinient way to obtaining explicit forms of the state vectors and density matrices of quantum states, usually generated in multiple coupled parametric optical processes. As a specific example, our method is demonstrated for the states, generated in three second-order parametric optical processes - one parametric down-conversion accompanied by two up-conversion processes. The application of the method allowed us to calculate the entropic characteristics of the multipartite quantum system.
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.
Abstract The possibility to realize multiple nonlinear optical processes in a single crystal as means to produce multicolor quantum states favours stability and compactness of optical settings. Hence, this approach can be advantageous compared to the traditional one based on cascaded arrangement of optical elements. However, it comes with an obstacle—the class of accessible quantum states is narrower than that of the cascade counterpart. In this letter, we study this task using an example of three coupled nonlinear optical processes, namely, one parametric down-conversion and two of sum-frequency generation. To this end, the singular value decomposition has been applied to find the cascade representation of the compound field evolution. We have found the link between the parameters of the multiwave processes and the relevant cascade parameters—beam-splitting and squeezing parameters, by means of which the generated quantum states have been characterized. The relation between the squeezing parameters that has been found in the course of this work shows that the squeezing resource, produced in the parametric down-conversion, is shared among the modes involved in the compound interactions. Moreover, we have shown that the degree of two-mode entanglement carried by the up-converted frequencies cannot exceed that of the down-converted frequencies.
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 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.
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 report the creation of a wide range of quantum states with controllable degrees of entanglement and entropy using an optical two-qubit source based on spontaneous parametric down-conversion. The states are characterized using measures of entanglement and entropy determined from tomographically determined density matrices. The tangle-entropy plane is introduced as a graphical representation of these states, and the theoretic upper bound for the maximum amount of entanglement possible for a given entropy is presented. Such a combination of general quantum state creation and accurate characterization is an essential prerequisite for quantum device development.