Quantum Statistics of Stimulated Raman and Hyper-Raman Scattering of Squeezed Light with Pump Depletion

Abstract

Using the approximation of small fluctuations around a stationary solution, photon statistics and quadrature variances are derived in stimulated Raman and hyper-Raman scattering including the coupling of pumping, Stokes, anti-Stokes and phonon modes. The depletion of pump light is included. Compared to Raman scattering, additional regimes to generate anti-bunched, sub-Poisson and squeezed light were found caused by self-interaction of the pump mode. It was possible to include effects of nonlinear dynamics, initial squeezing of radiation and phonon system, external noise and losses.     

Amplitude-squared and amplitude-cubed squeezing in stimulated Raman and in spontaneous Raman scattering

Coherent light coupled to a third-order nonlinear medium is responsible for spontaneous and for stimulated Raman processes. The analytical solutions up to second order in coupling constants of various field modes for spontaneous Raman and for stimulated Raman processes are utilized to obtain the amplitude-squared and amplitude-cubed squeezing of the input coherent light. The amplitude-squared and amplitude-cubed squeezing are completely ruled out for all modes for spontaneous Raman processes. For stimulated Raman processes, we observe that the squeezing for amplitude-squared and amplitude-cubed are possible for pump and for vibrational phonon modes. It is found that the percentage of higher ordered amplitude squeezing decreases with the increase of the orders. The squeezing in Stokes and in anti-Stokes modes are ruled out even for stimulated Raman processes.     

Quantum Theory of Hyper-Raman Scattering

Abstract

We develop the quantum theory of stimulated hyper-Raman scattering (HRS) and discuss the quantum statistics of pump and Stokes fields. We show that the initially coherent pump field acquires a strong squeezing that exceeds the similar squeezing in other nonlinear processes also involving two-photon absorption. Stokes field statistics are analysed in detail as a function of the initial statistics of both modes. Bunching effects are shown to be amplified in the short-time region, while in asymptotics the antibunching of Stokes photons becomes dominant, which in many cases results in sub-Poissonian statistics of Stokes mode. We also find a number of correlation effects between the photons of the two modes.     

Quantum Statistical Properties of Mixed Degenerate and Non-degenerate Parametric Processes

Abstract

Quadrature and integrated intensity fluctuations are found for mixed degenerate and non-degenerate optical parametric processes in single and compound modes. Additionally to quantum statistical effects for single processes new phenomena can occur arising from the coupling of both the processes, which can support squeezing of vacuum fluctuations and sub-Poisson photon statistics in single and compound modes.     

Squeezing from Raman Scattering in Optical Parametric Oscillators

Abstract

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.     

Non-classical behaviour and switching in Kerr nonlinear couplers

Abstract

Directional nonlinear couplers composed of two Kerr nonlinear waveguides are studied. Quantum consistent solutions of the equations of motion are obtained showing the collapses and revivals of the field mode amplitudes. The approximate method is also adopted for the contradirectional geometry and the switching accompanied by collapses of the modes is revealed. Kerr couplers with varying linear coefficients are examined and possible controls of the switching of modes, for instance, digital switching and fast switching are demonstrated. The two-mode squeezing of vacuum fluctuations and the generation of pure states are shown.     

Quantum Statistical Properties of Photon and Phonon Fields in Brillouin Scattering with Intense Pumping

The quantum statistical properties of Brillouin scattering of intense laser light are derived including the coupling of Stokes, anti-Stokes and phonon modes, if the anti-Stokes interaction prevails. Making use of the coherent-state technique, the Heisenberg equations of this process are solved neglecting the loss mechanism, and the normal quantum characteristic function and the normal generating function are derived. The time dependences of the photon distribution and its factorial moments are demonstrated if the phonon, Stokes and anti-Stokes modes are initially in a coherent state and periodical anti-bunching of the field is found when the phases of the incident fields fulfil certain phase conditions; the field can also return to the coherent state again.     

Quantum phase measurements and non-classical polarization states of light

Abstract

In our paper we consider the non-classical behaviour of both the Hermitian (observable) Stokes parameters of light and the phase difference of two modes that describe the quantum polarization states of optical field. To characterize the degree of polarization of light we introduce a new quantity taking into account the quantum properties of different quantum states of two orthogonally polarized modes. The problem of determination of the phase difference in two modes of optical field for the quantum polarization states of light is discussed. To describe in general such a quantum field we introduce two pairs of the phase operators: the phase angles for the Stokes parameters of light in a three-dimensional picture of the Poincaré sphere. We also consider a special type of the eight-port polarization interferometer (polarimeter) for simultaneous homodyne detection of both the Stokes parameters of light and the polarization phase operators and their fluctuations as well. Using an anisotropic (spatioperiodic) Kerr-like nonlinear medium associated with the polarization interferometer we could generate and also observe the polarization-squeezed phase states of light. The fluctuations in the phase difference between two orthogonally polarized modes for these non-classical states are less than the fluctuations for light in coherent state.     

SU(1, 1) and SU(2) Squeezing in Trilinear Optical Processes

Abstract

The squeezing properties in terms of SU(1, 1) and SU(2) operators for the case of trilinear processes are studied. The initial state of the system is supposed to be a coherent state in one of the modes and number states in the remaining modes. It is pointed out that in several cases a considerable amount of squeezing can be achieved. Due to the common mathematical structure the case of a two-mode coupler with intensity dependent coupling is also analysed.     

Non-classical Properties of Two-mode SU(1, 1) Coherent States

Abstract

We examine the non-classical properties of two-mode coherent states based on different unitary irreducible representations of SU(1, 1). Such states are generated by the action of the two-mode squeezing operator on initial states of the field containing arbitrary numbers of photons in each of the two modes. If the initial state of the field is a two-mode vacuum state, the final state is of course the two-mode squeezed vacuum. An initial occupation generalizes the idea of a squeezed vacuum to the SU(1, 1) coherent states. We show that fields in such states have remarkable quantum properties such as sub-Poissonian statistics, violations of the Cauchy-Schwarz inequality, strong correlations in the photon number fluctuations and squeezing. Using information theory formalism, we show that these coherent states are less correlated than the usual two-mode squeezed vacuum. Moreover, we show that an initial coherent amplitude contribution, in a large amplitude limit, can result in the reduction of correlations between modes.     

On the Squeezing of Coherent States by the Multiphoton Jaynes-Cummings Hamiltonian with an Intensity-dependent Coupling

Abstract

The squeezing properties of the multiphoton Hamiltonian with intensity-dependent coupling are evaluated for the [xcirc] and [pcirc] _x quadratures, for the initial state of a coherent electromagnetic field and an atom in the ground state. Two measures of squeezing: the percentage of total squeezing and the squeezing time-period percentage, are introduced. Interesting squeezing properties with respect to [xcirc] are observed for real coherent states when the time evolution of the above measures and of the time-averaged squeezing are analysed. The multiphoton intensity-dependent coupling Hamiltonian is found to be almost independent of the specific powers of the annihilation and creation operators, as long as the sum of the powers is kept constant.     

Scheme to measure quantum stokes parameters and their fluctuations and correlations

Abstract

We propose a scheme to measure quantum Stokes parameters, their fluctuations and correlations. The proposal involves measurements of intensities and intensity-intensity correlations for suitably defined modes, which can be produced by a combination of half-wave and quarter-wave plates.     

Parametric oscillation with squeezed vacuum reservoirs

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.     

Higher-order squeezing of the electromagnetic field in spontaneous and stimulated Raman processes

The occurrence of fourth and fifth-order squeezing in the fundamental mode of the electromagnetic field in spontaneous and stimulated Raman processes under the short-time approximation is investigated on the basis of a fully quantum-mechanical approach. In an idealized model, the Raman process is looked upon as a two-photon on a one phonon interaction process. The coupled Heisenberg equations of motion involving real and imaginary parts of the quadrature operators are established. The dependence of fourth- and fifth-order squeezing in terms of the signal-to-noise ratio on the number of photons is investigated. We have also established the amplitude squeezing effects in the Stokes and vibrational modes, which is found to be dependent on higher-order difference squeezing showing a difference squeezing that can be converted to normal squeezing.     

Forward and Backward Four-wave Mixing of Non-classical Light with Pump Depletion

Abstract

Photon statistics and non-classical behaviour of forward and backward four-wave mixing are examined including the coupling of modes in the approximation of small fluctuations around a stationary point, which is analytically determined and conditions for its stability are obtained. The depletion of pump modes is involved and effects of nonlinear dynamics, non-classical behaviour of input radiation, external noise and losses are included.     

Order-parameter collective modes in 3He-A and their effect on nuclear magnetic resonance (NMR) and ultrasonic attenuation

A generalization of our previous microscopic theory is used to derive the spin and density correlation functions for the Anderson-Brinkman-Morel (ABM) state. The coupling of the 18 fluctuation components of the order parameter due to the nuclear dipole interaction is taken into account rigorously. The results are valid for arbitrary temperature, frequency, wave number (with q parallel to the anisotropy axis), and magnetic field. Damping by pair-breaking processes is calculated explicitly; however, quasiparticle collisions are neglected. The frequencies of the clapping modes that couple to spin, and the frequencies of the flapping modes that couple to density, are found to split in a magnetic field. Near T _c the linewidth of the clapping mode and, at low T, the linewidth of the normal-flapping mode become small while the linewidth of the super-flapping mode is broad at all temperatures. The linewidth of the transverse NMR and the ultrasound attenuation coefficient, considered as functions of temperature, exhibit pronounced peaks at low temperatures. These arise from a coupling, induced by the dipole interaction, between spin or density fluctuations and the normal-flapping mode. Our results for the linewidth and line shift of the longitudinal NMR at low temperatures due to orbital effects are in agreement with results of Combescot.     

Back-action-induced Squeezed Light in a Detuned Quantum Non-demolition Scheme

Abstract

We describe the experimental observation and theoretical interpretation of a squeezing effect which occurs through the coupling of two light beams in a three-level atomic system. The origin of this effect can be attributed to the transfer of the intensity fluctuations of one beam to the phase fluctuations of the other one, followed by an optical feedback onto the intensity of the initial beam. The first step of the information transfer is similar to the one which occurs in a quantum non-demolition (QND) measurement of the intensity. The feedback effect is obtained through mixing of the phase and intensity quadratures, due to the detuning of the optical cavity which contains the nonlinear medium. Therefore the information obtained by the QND measurement is used to correct the intensity fluctuations of the signal beam by a build-in mechanism, which does not require any use of external electronics.     

Phase Fluctuations and Squeezing

Abstract

We investigate the relationship between squeezing and reduced phase fluctuations for various states of the single-mode electromagnetic field, including the strongly-squeezed vacuum and phase states. We find that, although squeezing the fluctuations of the electric field that arise from the vacuum guarantees a more well-defined phase, reducing phase fluctuations does not guarantee a squeezed electric field. We also investigate the evolution of the electric field and its fluctuations for a phase state. Our results show that even though the electric field fluctuations never vanish for a phase state, the times when the electric field changes sign are precisely defined. We also discuss why it is not always possible to attribute physical properties to certain states, such as simple superpositions of phase states.     

Random Multimode Optical Media: I. Mode Coupling Process in Slab Waveguide with Stochastic Wall Perturbations

Abstract

The scope of the paper is to present in explicit form the mode coupling coefficients in slab waveguide, in the case when both boundaries of guiding core or gradient refractive index distribution are perturbed by random deviations from an ideal model. The central idea is to present the relationships between the processes, depicting the waveguide deformations and the estimators of the mode coupling processes. The correlation functions and the spectral density functions of both groups of processes are linearly interdependent; we examine the complex coefficients of these relations. In particular, the TE modes are analysed in detail in step-index slab. For the case of parabolic profile (n(x) = A(1 ? Bx²), within the core) the method which derives the statistic integral form of coupling coefficients is presented (also for guided TE modes). All modes are assumed to be monochromatic, i.e. time-dependence is exp (?iωt).     

Non-classical Features in the Three-level Jaynes-Cummings Model

Abstract

The eigenfunctions and eigenenergies of the system composed of a cascade three-level atom and bichromatic or monochromatic field with arbitrary detunings are presented. The development of the state vector of the system with arbitrary initial conditions is obtained. The development of the quantum fluctuations and the second-order coherence degree for the field is calculated. The dependence of the squeezing and antibunching on the detunings, initial intensities and initial squeezing is studied. The contribution of single-photon and two-photon transitions to, and the influence of, the a.c. Stark effect on the squeezing are discussed.