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Physical Review A

Correlation of photon pairs from the double Raman amplifier: Generalized analytical quantum Langevin theory

C. H. Raymond Ooi, Qingqing Sun, Muhammad Suhail Zubairy, Marlan O. Scully

Published Jan 31, 2007• 58 citations
Publisher

Abstract

We present a largely analytical theory for two-photon correlations ${G}^{(2)}$ between Stokes $(s)$ and anti-Stokes $(a)$ photon pairs from an extended medium (amplifier) composed of double-$\ensuremath{\Lambda}$ atoms in counterpropagating geometry. We generalize the parametric coupled equations with quantum Langevin noise given in a beautiful experimental paper of Balic et al. [Phys. Rev. Lett. 94, 183601 (2005)] beyond adiabatic approximation and valid for arbitrary strength and detuning of laser fields. We derive an analytical formula for cross correlation ${G}_{as}^{(2)}=⟨{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{E}}_{s}^{\ifmmode\dagger\else\textdagger\fi{}}(L){\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{E}}_{a}^{\ifmmode\dagger\else\textdagger\fi{}}(0,\ensuremath{\tau}){\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{E}}_{a}(0,\ensuremath{\tau}){\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{E}}_{s}(L)⟩$ and use it to obtain results that are in good quantitative agreement with the experimental data. Results for ${G}_{as}^{(2)}$ obtained using our coupled equations are in good quantitative agreement with the results using the equations of Balic et al., while perfect agreement is obtained for sufficiently large detuning. We also compute the reverse correlation ${G}_{sa}^{(2)}$ which turns out to be negligibly small and remains classical while the cross correlation violates the Cauchy-Schwartz inequality by a factor of more than a hundred.

DOI
10.1103/physreva.75.013820

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Publisher's Note: Correlation of photon pairs from the double Raman amplifier: Generalized analytical quantum Langevin theory [Phys. Rev. A75, 013820 (2007)]

C. H. Raymond Ooi, Qingqing Sun, Muhammad Suhail Zubairy +1 more

Received 1 February 2007DOI:https://doi.org/10.1103/PhysRevA.75.029902©2007 American Physical Society

Physical Review A•Feb 6, 2007•0 citations
We present a largely analytical theory for two-photon correlations G(2){G}^{(2)}G(2) between Stokes (s)(s)(s) and anti-Stokes (a)(a)(a) photon pairs from an extended medium (amplifier) composed of double-\ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ atoms in counterpropagating geometry. We generalize the parametric coupled equations with quantum Langevin noise given in a beautiful experimental paper of Balic et al. [Phys. Rev. Lett. 94, 183601 (2005)] beyond adiabatic approximation and valid for arbitrary strength and detuning of laser fields. We derive an analytical formula for cross correlation Gas(2)=⟨E\ifmmode^\elseˆ\fis\ifmmode†\else\textdagger\fi(L)E\ifmmode^\elseˆ\fia\ifmmode†\else\textdagger\fi(0,\ensuremathτ)E\ifmmode^\elseˆ\fia(0,\ensuremathτ)E\ifmmode^\elseˆ\fis(L)⟩{G}_{as}^{(2)}=⟨{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{E}}_{s}^{\ifmmode\dagger\else\textdagger\fi{}}(L){\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{E}}_{a}^{\ifmmode\dagger\else\textdagger\fi{}}(0,\ensuremath{\tau}){\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{E}}_{a}(0,\ensuremath{\tau}){\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{E}}_{s}(L)⟩Gas(2)​=⟨E\ifmmode^\elseˆ\fi​s\ifmmode†\else\textdagger\fi​(L)E\ifmmode^\elseˆ\fi​a\ifmmode†\else\textdagger\fi​(0,\ensuremathτ)E\ifmmode^\elseˆ\fi​a​(0,\ensuremathτ)E\ifmmode^\elseˆ\fi​s​(L)⟩ and use it to obtain results that are in good quantitative agreement with the experimental data. Results for Gas(2){G}_{as}^{(2)}Gas(2)​ obtained using our coupled equations are in good quantitative agreement with the results using the equations of Balic et al., while perfect agreement is obtained for sufficiently large detuning. We also compute the reverse correlation Gsa(2){G}_{sa}^{(2)}Gsa(2)​ which turns out to be negligibly small and remains classical while the cross correlation violates the Cauchy-Schwartz inequality by a factor of more than a hundred.
An extended medium driven in a double Raman configuration generates Stokes and anti-Stokes fields that are described by coupled parametric oscillator equations with solutions that depend on input boundary operators and quantum noise operators. We identify the conditions where the boundary operators can be the substitute to the noise operators for describing two-photon cross correlation in forward and backward geometries. These conditions include short sample and small decoherence between ground states ${\ensuremath{\gamma}}_{bc}$, and they are fulfilled by the spontaneous Raman electromagnetic induced transparency scheme (weak pump with large detuning). We verify the correspondence between the results from boundary and noise operators analytically and show that the correlation due to the boundary operators is typically smaller than that due to the noise operators. In general the noise operators are required to obtain the correct correlation, especially when the control laser field is weak and ${\ensuremath{\gamma}}_{bc}$ is finite. Explanations for the findings are given based on the physics represented by the boundary operators and noise operators. Similar conclusions are obtained for the Stokes and anti-Stokes intensities.
We present a comprehensive study of co- and counterpropagating Stokes and anti-Stokes quantum fields in double Raman four-wave mixing parametric oscillators using full quantum Heisenberg-Langevin framework with noise operators. General analytical solutions of the fields operators at any point in the Raman medium are obtained for four cases: two possible copropagating (forward) and two counterpropagating (backward) Stokes and anti-Stokes. We analyze the symmetrical properties of the complex linear and nonlinear susceptibilities spectra of the quantum fields, nonclassicality of two-photon correlation functions, spatial variations of the quantum fields, and the two-mode relative intensity squeezing. We compare the results of forward and backward cases for several limiting double Raman schemes. We find interesting resonant effects of medium length, laser detunings and laser field strengths (Rabi frequencies) for backward-propagating geometries. Analysis of the solutions provide insights on the resonant conditions while computation over multiple variables enables us to identify the values of laser detuning, field strength, and propagation length that give enhanced nonclassical intensity squeezing and persistent correlations. The present work sets the crucial foundations for optimization of the nonclassicality of photons in double Raman systems and would be useful for quantum information storage, quantum nonlinear optics, and quantum spectroscopy.
We study several aspects of two-photon correlation of an optically driven extended medium (amplifier) with parametric down-conversion (PDC) scheme and three-level cascade scheme. The correlation for the PDC scheme is modeled by coupled parametric equations with constant self-coupling and cross coupling coefficients. This provides simple physical insights to how the correlation profile depends on the sign and magnitude of the coefficients in the presence of propagation group delay between signal and idler photons. The results with constant coefficients are related to the results of the driven three-level cascade scheme obtained from quantum Langevin-Maxwell's equations. The correlation transforms from bunching to antibunching as the pump field increases. Cauchy-Schwartz inequality is violated for all time delay in the case of off-resonant pump, showing nonclassical correlation. We verify that the correlation obtained without noise operators is qualitatively correct regardless of the optical density, especially for large detuning. We also discuss the interesting physics behind antibunching and oscillations found in the reverse correlation.
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.
We use a Langevin approach to analyze the quantum noise in coherent anti-Stokes Raman spectroscopy in several experimental scenarios: with continuous-wave input fields acting simultaneously and with fast sequential pulsed lasers where one field scatters off the coherence generated by other fields and for interactions within a cavity and in free space. In all cases, the signal and quantum noise due to spontaneous decay and decoherence in the medium are shown to be described by the same general expression. Our theory in particular shows that for short interaction times, the medium noise is not important and the efficiency is limited only by the intrinsic quantum nature of the photon. We obtain fully analytic results without making an adiabatic approximation; the fluctuations of the medium and the fields are solved self-consistently.