Degenerate Parametric Oscillator Research Articles

Isotropic and squeezed colored pump noise effects on the degenerate parametric oscillator

The degenerate parametric oscillator above threshold is studied with phenomenological stochastic colored ~nonwhite! pump noise for arbitrary pump to subharmonic relaxation rate. The current experimental limits of large intracavity threshold photon numbers ~very small quantum noise! are considered, allowing for a semiclassical treatment of the system dynamics. A comparison between the effects of isotropic and squeezed pump noise on the internal transient and steady-state fluctuations is presented by simulation of the nonlinear semiclassical stochastic Langevin equations in the Wigner quadrature representation. It is found that the transient squeezing for the system starting in the unstable steady state ~the vacuum! is not degraded by stochastic pump noise. A damped oscillatory behavior of the noise levels ~periodic exchange of fluctuations between the squeezed quadrature of the signal and the pump! is observed for large damping of the signal in the turn on of pump depletion. Finally, it is shown that the limited squeezing above threshold in the steady state (50%) due to pump depletion can be enhanced if squeezed stochastic noise with sufficient significant spectral components ~broadband squeezing! is fed to the pump. The above-threshold steady-state squeezing has been calculated analytically from the linearized stochastic equations and the effects of the time scale associated to the relaxation of the pump noise ~the noise correlation time! compared to the dissipative time scale of the system and the pump to subharmonic loss ratio are presented. @S1050-2947~96!03609-8#

Fractional revivals in optical parametric interactions

A study is presented of the long-time dynamical behaviour of the degenerate parametric oscillator in the semiclassical and quantum regimes for various initial conditions. It is shown that the eigenstates of the trilinear Hamiltonian lead to the collapse and revival phenomena as well as the novel phenomena of fractional revivals in the photon statistics of the field modes. It is also demonstrated that fractional revivals become more prominent if the pump mode possesses sub-Poissonian statistics.

Photon-counting statistics of the degenerate parametric oscillator in the transient regime: The factorial moments method

Photon-counting statistics of a subthreshold degenerate parametric oscillator (DPO) are studied in the transient regime. The field produced by the parametric oscillator is expressed in terms of two independent real Gaussian variables. We use this propery of the field variables to derive analytically the generating function for th photoelectron statistics. From this generating function one general expression for the factorial moments is obtained, and expressions for the first and second factorial moments and variance are presented. Using the Mandel-Q parameter we show that the statistics is Super-Poissonian for the DPO in the transient regime. Some curves are presented to illustrate that behavior of the variance and the Mandel-Q parameter.

Transient macroscopic quantum superposition states in degenerate parametric oscillation using squeezed reservoir fields.

We predict the effect on the intracavity quadrature-phase-amplitude probability distribution of coupling the signal mode of a degenerate parametric oscillator to a squeezed bath. Solutions in a transient regime reveal interference fringes which are indicative of the formation of a ``Schr\odinger cat'' state. The use of the squeezed reservoir allows fringes to be predicted for smaller nonlinearity-to-loss ratios.

Using feedback to eliminate back-action in quantum measurements.

A quantum-nondemolition (QND) measurement is one which does not disturb the quantity it measures. Any measurement can be regarded as a QND measurement followed by an additional back-action which may disturb that quantity. By feeding back the measurement result to control the system dynamics, this back-action can always be eliminated in principle. A practical device based on homodyne measurement of the x quadrature of a cavity mode is investigated. First, a χ(2) crystal is inserted in the cavity so that it becomes a degenerate parametric oscillator at threshold. Then the photocurrent is used to control coherent driving of the cavity, so as to give positive feedback with unit gain. This feedback, combined with the nonlinearity, turns the homodyne measurement into a QND measurement of x. The device can also be used to measure the x quadrature of a traveling wave, and gives near-perfect QND correlations over a large bandwidth. However, the quality of the measurement is badly degraded by even slightly inefficient photodetectors.

Transient macroscopic quantum superposition states in degenerate parametric oscillation: Calculations in the large-quantum-noise limit using the positive P representation.

Recent work by Wolinsky and Carmichael $[---OSA Annual Meeting, 1989 Technical Digest Series Vol. 18 (Optical Society of America, Washington, DC, 1989)] suggests that a superposition of two macroscopically distinct coherent states (a ``Schr\"odinger-cat'' state) may be produced in degenerate parametric oscillation in a transient regime. We investigate this possibility by performing numerical simulations of quantum stochastic equations derived using the positive P representation. The equations allow for finite signal cavity loss. Interference fringes which indicate the existence of the macroscopic superposition states are indeed predicted in regimes where the parametric nonlinearity is sufficiently large compared to the signal cavity losses. Experimental criteria needed for the generation of macroscopic quantum superposition states using parametric oscillation are thus established.

Decay of a single mode electromagnetic field inside a cavity in the presence of an even coherent field

An equation of motion for the reduced density matrix, describing the decay of a single mode field inside a cavity in the presence of an even coherent field, has been derived. With the help of this equation, it is predicted that such an even coherent field can play an important role in the enhancement of steady state squeezing for the degenerate parametric oscillator.

Quantum fluctuations of a non-linear parametric oscillator

The theory of a non-linear degenerate parametric oscillator including the quantification of pump and signal mode is presented in the framework of quantum electrodynamics. The expressions for the dispersions of the signal mode quadratures are obtained at arbitrary temperature and dampings of the pump and signal mode.

Single-atom fluorescence with nonclassical light.

The statistical properties of fluorescent light from a single two-level atom driven by nonclassical light beams that exhibit antibunching and squeezing are discussed in the weak-field limit. The fields produced in intracavity second-harmonic generation and those produced by superposing the light from a degenerate parametric oscillator with a coherent field are considered to be the models of nonclassical light beams. These fields can be described in terms of real Gaussian random processes with nonzero mean coherent components. Analytical expressions for the fluorescent light intensity, the second-order intensity correlation function, and the variance of the photon number in the steady state are obtained. The effects of driving-field correlations, detuning, and other parameters characterizing the atom-field interaction on the statistics of scattered light are investigated. It is found that for a certain range of parameters the second-order intensity correlation function may vanish at nonzero times. Under certain other circumstances the antibunching of the fluorescent photons is enhanced. These and other interesting features that appear in the scattered light are discussed.

Spatial structure of a squeezed vacuum.

We analyze the quantum properties of the signal near field emitted by a degenerate parametric oscillator below threshold, including diffraction and its effects on the threshold for signal generation. The field is probed by a local oscillator field of arbitrary spatial configuration. The results are expressed in terms of an appropriate spectrum which describes how the level of squeezing varies with the angle from the direction of propagation. The results hold for cavities both with plane and with spherical mirrors.

Homodyne photon statistics of the subthreshold degenerate parametric oscillator.

Homodyne statistics when the light from a degenerate parametric oscillator (DPO) is mixed with coherent light from a local oscillator are discussed. By using dynamical models of these beams we discuss the photon sequences underlying the superposed beam in terms of photoelectric pulse sequences recorded by a detector. The field produced by the parametric oscillator is expressed in terms of two independent real Gaussian random variables. Using this property of the field variables, we derive the generating function for the photoelectron statistics analytically. From this generating function, expressions for the photoelectron-counting distribution, factorial moments, and the waiting-time distribution are derived. These quantities are directly measurable in photon-counting experiments. The results reported here are applicable to arbitrary strengths of the signal and the local-oscillator fields. We also show that, depending on the relative strength and relative phase between coherent light from the local oscillator and squeezed light from the DPO, the superposed light beam may exhibit many interesting nonclassical features. These and other effects are described and curves are presented to illustrate the behavior.

Effect of bistability and superpositions on quantum statistics in degenerate parametric oscillation.

The degenerate parametric oscillator operating above threshold exhibits bistability, and recent work suggests that it might also exist as a quantum-superposition state. We calculate the effect of the bistability on the two-time correlation function and the transmitted-field spectrum. The measurable steady-state probability distribution at the output of a homodyne detector is also calculated in a particular limit, in order to investigate whether one can experimentally distinguish between quantum-superposition and classical states.

Generation of nonclassical states by nonlinear optical systems

We review the main results obtained recently by our group in the field of quantum noise reduction in nonlinear optical systems. We illustrate the possibility of obtaining significant effects in degenerate parametric oscillators, intracavity second-harmonic generation, nondegenerate four-wave mixing, and one- and two-photon optical bistability.

S-order nondiagonal quasiprobabilities.

This paper presents a correspondence between the annihilation a^ and creation a${\mathrm{^}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$ operators and two independent complex variables \ensuremath{\alpha} and \ensuremath{\beta}, which makes possible the definition of positive nondiagonal quasiprobabilities for any Cahill-Glauber s order. A generalized version of the Cahill and Glauber s-ordered displacement operator displacing the annihilation and creation operators by \ensuremath{\alpha} and \ensuremath{\beta} is defined. Corresponding to this n/Inondiagonal displacement, a nondiagonal ordering operator is introduced so that a map of s-ordered operators into c numbers a^\ensuremath{\rightarrow}\ensuremath{\alpha}, a${\mathrm{^}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$\ensuremath{\rightarrow}\ensuremath{\beta} can be defined. The Drummond-Gardiner projector and the Cahill-Glauber diagonal ordering operator are obtained as particular cases. In order to use the nondiagonal correspondence in the determination of quantum expectation values of observables, several families of quasiprobabilities are defined, generalizing Drummond-Gardiner (normal order and nondiagonal) positive-P-function and Cahill-Glauber (s-order and diagonal) quasiprobabilities. It is demonstrated that these nondiagonal quasiprobabilities exist as well-behaved functions where at least one is non-negative, for any state and any order. The time evolution of these quasiprobabilities is discussed both in the Schr\odinger and in the Heisenberg pictures. In the Heisenberg picture, a method for obtaining c-number stochastic differential equations (SDE's) directly from operator equations using the s-order nondiagonal correspondence is described. The main difference between this method and the Langevin approach is that in the latter a diagonal correspondence is used, leading eventually to wrong results. The use of these SDE's to solve quantum optical problems is discussed, and an application to the nonlinearly damped degenerate parametric oscillator is made. In order to obtain the SDE, various (inequivalent) truncation schemes are necessary.

Gravity-wave detection via an optical parametric oscillator.

An active gravity-wave detector based on a two-arm optical parametric oscillator (OPO) is proposed and analyzed. The signal and idler waves of a nearly degenerate parametric oscillator are internally separated such that the signal beam propagates along one arm of the OPO and the idler beam along the orthogonal arm. By monitoring the signal-idler beat frequency, a sensitive measure is obtained of the differential path displacement in the two arms of this OPO interferometer. The phase noise of the output beat frequency is calculated and is shown to exhibit time-dependent squeezing. The detection sensitivity is limited by the signal-idler phase diffusion and photon shot noise, and is the same as that of the usual passive Michelson interferometer or active two-laser interferometer for a given apparatus size and detected optical power. Squeezed vacuum can, in principle, be used to improve the detection sensitivity.

Effect of squeezing on the degenerate parametric oscillator.

Effect of squeezing on the degenerate parametric oscillator.

Amplitude-squared squeezing inside and outside a cavity.

Fluctuations of the square of the field amplitude inside and outside a degenerate parametric oscillator are investigated. It is shown that a large reduction of the noise of the square of the field amplitude can be achieved outside the cavity while the maximum amplitude-squared squeezing inside the cavity is only by 50%. This situation is similar to that for normal squeezing.

A comparison between degenerate paramagnetic oscillators and two-photon correlated-spontaneous-emission lasers

The degenerate parametric oscillator (DPO) is analyzed with the nonlinear material treated quantum mechanically. The source of squeezing in the DPO is the same as in the two-photon correlated-spontaneous-emission laser (CEL) — the level coherence. In this sense, they are similar. The amount of squeezing in DPO is the same as in the two-photon CEL at two-photon resonance, but the gain in DPO is usually much smaller than that in the two-photon CEL at two-photon resonance.

Berry phase analogies in nonlinear optics

We consider two simple models which describe degenerate four-wave mixing and the degenerate parametric oscillator, respectively. In both cases we point out the existence of a Berry-like phase induced by the variation of the time.

Microscopic source of squeezing in degenerate parametric oscillation

The degenerate parametric oscillator (DPO) is treated quantum mechanically for both the signal field and the nonlinear material. The squeezing of the signal field in the DPO is due to the full coherence between the energy levels of the material, which is produced by the pumping field. The gain of the signal field is also due to the coherence. Therefore, the DPO is a kind of two-photon correlated spontaneous emission laser operating far off resonances.