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Physical Review A

Quantum statistics of parametric oscillation

K. J. McNeil, Crispin W. Gardiner

Published Sep 1, 1983• 103 citations
Publisher

Abstract

We present a quantum-statistical analysis of the steady-state light fields in driven parametric oscillation in a cavity. Using the solution of a Fokker-Planck equation for the generalized $P$ representation of the signal and idler modes, we calculate the mean photon numbers, second-order correlation functions, and intermode correlation function as functions of the driving field. The generalized $P$ distribution describes the signal and idler modes' statistics over the whole range of driving-field strengths except in the region far below the oscillation threshold. The second-order correlation functions are found to violate the Cauchy-Schwartz inequality, a violation allowed because the $P$ distribution is complex. Squeezing is also found in a linear combination of the signal and idler fields.

DOI
10.1103/physreva.28.1560

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We present a quantum-statistical analysis of the steady-state light fields in driven parametric oscillation in a cavity. Using the solution of a Fokker-Planck equation for the generalized PPP representation of the signal and idler modes, we calculate the mean photon numbers, second-order correlation functions, and intermode correlation function as functions of the driving field. The generalized PPP distribution describes the signal and idler modes' statistics over the whole range of driving-field strengths except in the region far below the oscillation threshold. The second-order correlation functions are found to violate the Cauchy-Schwartz inequality, a violation allowed because the PPP distribution is complex. Squeezing is also found in a linear combination of the signal and idler fields.
In this thesis, we investigate the quantum optical properties of a single cavity mode interacting with an external classical coherent driving field inside a closed cavity, which is coupled to a thermal reservoir via a single-port mirror. We establish a mathematically comprehensive framework utilizing both the Master Equation under the density operator formalism and the Quantum Langevin Equation under the Heisenberg picture. By analyzing the system with an arbitrary ordering of reservoir noise operators, we derive the steady-state P-representation of the quasi-probability distribution. Employing this quasi-probability distribution, we analytically compute the mean photon number, photon number variance, photon number distribution, and the second-order intensity correlation function. Our findings establish that the interaction of the cavity mode with the coherent drive in the presence of thermal fluctuations exhibits strictly super-Poissonian statistics. The mean photon number and photon number variance scale monotonically with both the strength of the external coherent drive and the average thermal photon occupation of the reservoir. Furthermore, we prove that the produced cavity field does not exhibit quadrature squeezing, confirming that the presence of thermal noise effectively destroys quantum squeezing in this configuration. Finally, the input-output is evaluated to verify the preservation of the commutation relations of the formalism output field. Keywords: Quantum Optics, Master Equation, Fokker-Planck, Quantum Langevin Equation, P- function, Photon Statistics, Super-Poissonian, Squeezing.
Employing the quantum Hamiltonian describing the interaction of a two-mode light (signal–idler modes) generated by a non-degenerate parametric oscillator (NDPO) with two uncorrelated squeezed vacuum reservoirs (USVR), we derive the master and the Fokker–Planck equations. The corresponding Fokker–Planck equation for the Q-function is then solved employing a propagator method developed by K. Fesseha [J. Math. Phys. 33 2179 (1992)]. Making use of this Q-function, we calculate the quadrature fluctuations of the optical system. From these results we infer that the signal–idler modes are in squeezed states. When the NDPO operates below threshold we show that, for a large squeezing parameter, a squeezing amounting to a noise suppression approaching 100% below the vacuum level in one of the quadratures can be achieved.
In this thesis, we investigate the quantum optical properties of a single cavity mode interacting with an external classical coherent driving field inside a closed cavity, which is coupled to a thermal reservoir via a single-port mirror. We establish a mathematically comprehensive framework utilizing both the Master Equation under the density operator formalism and the Quantum Langevin Equation under the Heisenberg picture. By analyzing the system with an arbitrary ordering of reservoir noise operators, we derive the steady-state P-representation of the quasi-probability distribution. Employing this quasi-probability distribution, we analytically compute the mean photon number, photon number variance, photon number distribution, and the second-order intensity correlation function. Our findings establish that the interaction of the cavity mode with the coherent drive in the presence of thermal fluctuations exhibits strictly super-Poissonian statistics. The mean photon number and photon number variance scale monotonically with both the strength of the external coherent drive and the average thermal photon occupation of the reservoir. Furthermore, we prove that the produced cavity field does not exhibit quadrature squeezing, confirming that the presence of thermal noise effectively destroys quantum squeezing in this configuration. Finally, the input-output is evaluated to verify the preservation of the commutation relations of the formalism output field. Keywords: Quantum Optics, Master Equation, Fokker-Planck, Quantum Langevin Equation, P- function, Photon Statistics, Super-Poissonian, Squeezing.
The effect of squeezing on the degenerate parametric oscillator is studied. It is shown that when the ordinary-vacuum reservoir is replaced by a squeezed-vacuum reservoir, the maximum 50% limit of the second-order squeezing inside the cavity can be crossed. It is also shown that the mean number of photons increases considerably. An exact steady-state solution for the Fokker-Planck equation for the field mode is also presented. The photon distribution for the intracavity field and the higher-order variances have been obtained by using the solution of this equation.
The optical parametric oscillator is one of the most well-characterized devices in nonlinear optics, with many useful applications, especially as a frequency converter and as a wide-band amplifier. Novel discoveries made with these devices in the quantum domain include demonstrations of large amounts of both single-mode and two-mode squeezing, and, in the two-mode case, significant quantum intensity correlations between the two modes and quadrature correlations, which provided the first experimental demonstration of the original Einstein–Podolsky–Rosen (EPR) paradox. In this chapter, a systematic theory is developed of quantum fluctuations in the degenerate parametric oscillator. The quantum statistical effects produced by the degenerate parametric amplifier and oscillator have been intensely studied. As we have seen, the amount of transient squeezing produced in an ideal lossless parametric amplifier was calculated using asymptotic methods by Crouch and Braunstein. It was found to scale inversely with the square root of the initial pump photon number. When the system is put into a cavity, and driving and dissipation are introduced, we find that the steady state of a driven degenerate parametric oscillator bifurcates. There is a single steady state at low driving strengths, but, as the driving field increases, a threshold value is reached and the single state splits and forms two stable states. Below the threshold, a linearized analysis of squeezing in the quantum quadratures is possible; while above this threshold, or critical point, the tunneling time between the two states due to quantum fluctuations can be calculated.
A full quantum treatment about the process of parametric down-conversion with frequency degenerate but polarization non-degenerate in an optical parametric oscillator (OPO) cavity is presented. Using the linearized Langevin equations and spectral matrix, we calculated the squeezing spectra of the coupled mode in the output field. The squeezing as a function of driving field and detection frequency is obtained. The results obtained, which are compared with those found semiclassically by Reynaud et al. , indicate that it is possible to generate a two-mode coherent squeezed state with large amplitude. The quantum correlation between the signal and the idler modes is also discussed. It is shown that there is an inseparable relationship between the two-mode squeezing and the intermode quantum correlation.