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arXiv:1508.06949 · Physical Review A

Higher-order two-mode and multimode entanglement in Raman processes

Sandip Kumar Giri, Biswajit Sen, Anirban Pathak, Paresh Chandra Jana

Published Jan 25, 2016• 23 citations
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Abstract

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.

DOI
10.1103/physreva.93.012340

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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.
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.
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 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.
The Hamiltonian and the equations of motion for various modes involving spontaneous and stimulated Raman processes are derived. The coupled differential equations involving these modes are solved analytically by using an intuitive approach (IA). The solution of the present paper does not require any short time approximation. As a matter of fact, the present solution for field operators and hence the squeezing for various modes are valid for all interaction time t. In this way, the present paper is more general compared to the earlier investigations where short-time approximations were found a must. By exploiting the solutions for the field operators, we obtain the squeezing effects of input coherent light for pure and for mixed field modes. The IA employs the perturbation theory and hence we came across the secular terms in the solution. Of course, for small coupling constants these secular terms could be summed for all orders. To establish our claim, we remove the secular terms at least for one occasion.
Higher-order nonclassical properties of fields propagating through a codirectional asymmetric nonlinear optical coupler which is prepared by combining a linear waveguide and a nonlinear (quadratic) waveguide operated by second harmonic generation are studied. A completely quantum mechanical description is used here to describe the system. Closed form analytic solutions of Heisenberg's equations of motion for various modes are used to show the existence of higher-order antibunching, higher-order squeezing, and higher-order two-mode and multimode entanglement in the asymmetric nonlinear optical coupler. It is also shown that nonclassical properties of light can transfer from a nonlinear waveguide to a linear waveguide.
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.
We use the dressed-state representation to generate the full quantum-mechanical treatment of stimulated Raman scattering (SRS). Equations of motion and propagation are obtained from the Heisenberg equations of motion in the plane-wave limit and under conditions such that anti-Stokes and higher-order Stokes processes can be neglected. Equations for a stochastic model, based upon antinormal ordering and a quantum-classical correspondence are developed from them, suitable for numerical ensemble calculations for the study of amplified quantum fluctuations and spontaneously generated phase waves and solitons. Several alternative approaches, which have more general validity but present greater difficulty in numerical implementation, are also developed by introduction of quasiprobability functions and their associated Fokker-Planck equations from which corresponding stochastic equations of motion and propagation are derived. The Q representation, whose moments correspond to expectation values of products of antinormally ordered fields, and normally ordered atomic operators yield stochastic equations identical to the equations for the stochastic model developed directly from the Heisenberg equations for the same initial and boundary conditions, giving strong credence to the stochastic model. The Wigner representation, whose moments correspond to expectation values of symmetrically ordered operators, leads to a set of equations that are valid where the stochastic model fails, namely, where conditions are such that quantum fluctuations and nonlinearity are simultaneously important. The stochastic model, which is shown to be, by far, the simplest to implement in numerical calculations, is shown also to be valid for the conditions of SRS.