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Physical Review A

Intermodal entanglement in Raman processes

Biswajit Sen, Sandip Kumar Giri, Swapan Mandal, C. H. Raymond Ooi, Anirban Pathak

Published Feb 19, 2013• 24 citations
Publisher

Abstract

The operator solution of a completely quantum mechanical Hamiltonian of the Raman processes is used here to investigate the possibility of obtaining intermodal entanglement between different modes involved in the Raman processes [e.g., pump mode, Stokes mode, vibration (phonon) mode and anti-Stokes mode]. Intermodal entanglement is reported between (a) pump mode and anti-Stokes mode, (b) pump mode and vibration (phonon) mode, (c) Stokes mode and vibration phonon mode, and (d) Stokes mode and anti-Stokes mode in the stimulated Raman processes for variation of the phase angle of complex eigenvalue ${\ensuremath{\alpha}}_{1}$ of pump mode $a$. Some incidents of intermodal entanglement in the spontaneous and the partially spontaneous Raman processes are also reported. Further, it is shown that the specific choice of coupling constants may produce genuine entanglement among Stokes mode, anti-Stokes mode, and vibration-phonon mode. It is also shown that the two-mode entanglement not identified by Duan's criterion may be identified by Hillery-Zubairy criteria. It is further shown that intermodal entanglement, intermodal antibunching, and intermodal squeezing are independent phenomena.

DOI
10.1103/physreva.87.022325

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The operator solution of a completely quantum mechanical Hamiltonian of the Raman processes is used here to investigate the possibility of obtaining intermodal entanglement between different modes involved in the Raman processes [e.g., pump mode, Stokes mode, vibration (phonon) mode and anti-Stokes mode]. Intermodal entanglement is reported between (a) pump mode and anti-Stokes mode, (b) pump mode and vibration (phonon) mode, (c) Stokes mode and vibration phonon mode, and (d) Stokes mode and anti-Stokes mode in the stimulated Raman processes for variation of the phase angle of complex eigenvalue \ensuremathα1{\ensuremath{\alpha}}_{1}\ensuremathα1​ of pump mode aaa. Some incidents of intermodal entanglement in the spontaneous and the partially spontaneous Raman processes are also reported. Further, it is shown that the specific choice of coupling constants may produce genuine entanglement among Stokes mode, anti-Stokes mode, and vibration-phonon mode. It is also shown that the two-mode entanglement not identified by Duan's criterion may be identified by Hillery-Zubairy criteria. It is further shown that intermodal entanglement, intermodal antibunching, and intermodal squeezing are independent phenomena.
A perturbative analytic operator solution of a completely quantum mechanical Hamiltonian of multi-photon pump non-degenerate hyper-Raman process is obtained. It is shown that the obtained solution is general in nature as the solutions of non-degenerate hyper-Raman and stimulated Raman processes can be obtained as special cases of the present solution. The analytic solutions obtained here are used to investigate the nonclassical properties of the different modes in the stimulated, spontaneous and partially spontaneous multi-photon pump non-degenerate hyper-Raman processes. The nonclassical nature of these processes is witnessed by means of single mode and intermodal quadrature squeezing, intermodal entanglement of different orders, lower order and higher order photon antibunching. Interestingly, manifesting the multiphoton nature of the pump modes, a bunch of nonclassicality involving them are observed due to self-interaction of various pump modes.
A quantum model of the Raman process with the independent Stokes and anti-Stokes nonlinear interactions is developed to study nonclassical correlations between the photons in the Stokes and anti-Stokes fields. The role of the laser pump amplitude, the ratio of the Stokes and anti-Stokes coupling constants, and the population and losses of the vibrational mode in forming the correlations is elucidated. The g(2){g}^{(2)}g(2) intensity cross-correlation function, noise-reduction factor, two-mode principal squeezing variance, logarithmic negativity, nonclassicality depth, steering parameter, and the Bell parameter are analyzed side by side to shed light to the correlations between the Stokes and anti-Stokes fields. Conditions for having the Stokes and anti-Stokes fields composed of only photon pairs, similarly as it occurs in twin beams in parametric down-conversion, are revealed. They allow for nonzero mean thermal phonon numbers.
The light initially prepared in a coherent state runs the stimulated and spontaneous Raman processes in a second-order nonlinear medium. Using analytical expressions of field operators for the light modes, we determine the quantum statistical properties of the modes in the stimulated and spontaneous Raman processes. By a choice of phase, we obtain a boson-pair property correlating the pump and vibration modes and such a property correlating the pump and anti-Stokes modes. The Stokes and vibration modes exhibit the boson-pair property for all initial phase differences between the pump mode and the other ones. In the spontaneous Raman process only the Stokes and vibration modes show such a property.
Coherent light coupled to a third-order nonlinear medium is responsible for spontaneous and for stimulated Raman processes. The analytical solutions up to second order in coupling constants of various field modes for spontaneous Raman and for stimulated Raman processes are utilized to obtain the amplitude-squared and amplitude-cubed squeezing of the input coherent light. The amplitude-squared and amplitude-cubed squeezing are completely ruled out for all modes for spontaneous Raman processes. For stimulated Raman processes, we observe that the squeezing for amplitude-squared and amplitude-cubed are possible for pump and for vibrational phonon modes. It is found that the percentage of higher ordered amplitude squeezing decreases with the increase of the orders. The squeezing in Stokes and in anti-Stokes modes are ruled out even for stimulated Raman processes.
The existence of higher-order entanglement in the stimulated and spontaneous Raman processes is established using the perturbative solutions of the Heisenberg equations of motion for various field modes that are obtained using the Sen-Mandal technique and a fully quantum mechanical Hamiltonian that describes the stimulated and spontaneous Raman processes. Specifically, the perturbative Sen-Mandal solutions are exploited here to show the signature of the higher-order two-mode and multimode entanglement. In some special cases, we have also observed higher-order entanglement in the partially spontaneous Raman processes. Further, it is shown that the depth of the nonclassicality indicators (parameters) can be manipulated by the specific choice of coupling constants, and it is observed that the depth of nonclassicality parameters increases with the order.