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Physical Review A

Quantum parametric double Raman oscillators with co- and counterpropagating fields: Relative intensity squeezing and spatial photon correlations

C. H. Raymond Ooi, Konstantin E. Dorfman

Published Nov 7, 2022• 3 citations

Abstract

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.

DOI: 10.1103/physreva.106.053705

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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.

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 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.

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)}

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.

Statistical properties of optical fields in nonlinear couplers composed of two waveguides in which Raman or Brillouin processes (with classical pumping) are in operation and which are mutually connected through the mutual Stokes and/or anti-Stokes linear interactions are investigated within the framework of a generalized superposition of coherent fields and quantum noise. Heisenberg equations describing the couplers are solved both analytically under special conditions and numerically in general cases. Regimes for nonclassical properties of optical fields, such as sub-Poissonian photon-number statistics, negative reduced moments of integrated intensity and squeezing of quadrature fluctuations are discussed for the cases of single and compound fields. General results are compared with those from the short-length approximation.

The squeezing properties of the Raman scattering process which takes place when the lower output frequency from a non-degenerate optical parametric interaction approaches a resonance of the nonlinear medium placed inside an optical resonant cavity are presented. The linear stability analysis is performed and the spectrum for the output fields is given in terms of the quadrature phase components in the Wigner representation. Perfect squeezing for the amplitude difference between the output Stokes and anti-Stokes modes at the Hopf bifurcation point is obtained.

Nonclassical light is now available in a number oflaboratories, being generated by Frank—Hertz experiment, optical parametric processes, four-wave mixing including optical fibres, by optical system with feedback, etc. Therefore it is senseful to reconsider various nonlinear optical interactions involving initially nonclassical light ,originally solved with coherent (laser) light. Adopting tools of quasidistributions related to antinormal ordering, the photocount distributions, its factorial moments and characteristics of squeezing of vacuum fluctuations can be investigated assuming initially nonclassical (squeezed) light in the framework of the Heisenberg—Langevin of Schrödinger approaches. Photon statistics of stimulated Raman scattering of squeezed light can be derived assuming phonon system in chaotic, coherent or squeezed states and including additive noise. The influence of nonlinear dynamics can be established on initially inclassical light and it can be shown that the initially nonclassical effects are encresed by the nonlinear dynamics provided that the Stokes mode is less amplified than the anti-Stokes mode is damped, whereas in the opposite case the nonclassical features are smoothed out. Also frequency detuning can be optimized to have maximum nonclassical effects, which are missing with increasing phonon number, even if the phonon system is squeezed. Small initial squeezing can by increased by the nonlinear dynamics, whereas strong squeezing can only be diminished.