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Quadrature and polarization squeezing in a vectorial Kerr cavity

Ferran V. García-Ferrer, Isabel Pérez–Arjona, Germán J. de Valcárcel, Eugenio Roldán

Published Jun 17, 2007• 0 citations
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Abstract

Polarization squeezing [1] is attracting much attention in the last few years because of its potential applications in quantum information. We reconsider here the analysis of quantum fluctuations in dispersive optical bistability by generalizing the standard model in order to take into account fluctuations in the two orthogonal polarization components of the intracavity field, and not only in the pumped one as it is customary. In other words, we consider a relatively simple model for a vectorial Kerr cavity: The Kerr nonlinearity is the responsible for dispersive bistability, and two extra nonlinear terms, describing cross-phase modulation and four-wave mixing, couple the intense (pumped) intracavity field (with polarization parallel to that of the field injected in the cavity) with the orthogonally polarized mode. The weight of the new nonlinearities is controlled by the Maker-Terhune coefficients, which verily A + B!2 = 1 for an isotropic material, which we consider. This system exhibits a very rich variety of bifurcations [2], In this communication we describe the simplest results.

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•Jan 1, 1984•0 citations
Polarization squeezing [1] is attracting much attention in the last few years because of its potential applications in quantum information. We reconsider here the analysis of quantum fluctuations in dispersive optical bistability by generalizing the standard model in order to take into account fluctuations in the two orthogonal polarization components of the intracavity field, and not only in the pumped one as it is customary. In other words, we consider a relatively simple model for a vectorial Kerr cavity: The Kerr nonlinearity is the responsible for dispersive bistability, and two extra nonlinear terms, describing cross-phase modulation and four-wave mixing, couple the intense (pumped) intracavity field (with polarization parallel to that of the field injected in the cavity) with the orthogonally polarized mode. The weight of the new nonlinearities is controlled by the Maker-Terhune coefficients, which verily A + B!2 = 1 for an isotropic material, which we consider. This system exhibits a very rich variety of bifurcations [2], In this communication we describe the simplest results.
We theoretically study quadrature and polarization squeezing in dispersive optical bistability through a vectorial Kerr cavity model describing a nonlinear cavity filled with an isotropic ${\ensuremath{\chi}}^{(3)}$ medium in which self-phase- and cross-phase-modulation, as well as four-wave mixing, occur. We derive expressions for the quantum fluctuations of the output field quadratures as a function of which we express the spectrum of fluctuations of the output-field Stokes parameters. We pay particular attention to study how the bifurcations affecting the non-null linearly polarized output mode squeezes the orthogonally polarized vacuum mode, and show how this produces polarization squeezing.
Summary form only given. In a recent paper [1] it has been shown how a bichromatic fast driving of optomechanical (optical domain) and superconducting circuit systems (microwave domain), operating in a limit where they present a non-linear Kerr-type interaction, can give rise to very strong vacuum squeezing. The driving with two close frequencies of a Kerr cavity changes the usual bistability bifurcation behaviour that takes place under monochromatic driving, into a degenerate four-wave mixing bifurcation, where a phase-bistable component starts oscillating spontaneously at a frequency that lies halfway between the two driving frequencies [2]. This resembles the physics of the optical parametric oscillator threshold, devices that has produced experimentally the strongest squeezing up to now. Associated to this bifurcation there is a strong reduction of the quantum fluctuations for a given quadrature of the electromagnetic field, resulting in vacuum squeezing, the reduction takes place at the central non-injected frequency.Here we show the application of this bichromatic driving protocol to a semiconductor micro-cavity system [3,4], where the nonlinear interaction between photons is caused through the excitons induced at the semiconductor embedded in the optical cavity.
We investigate the formation of the polarization-squeezed state of light in different schemes of perfect spatial overlap of quadrature-squeezed ultrashort light pulses produced by self-phase modulation effect in electronic Kerr media. The light polarization is treated in terms of four Stokes operators whose spectra of quantum fluctuations are investigated for each configuration of interest. We establish that the frequency at which the suppression of quantum fluctuations of Stokes operators is the greatest can be controlled by adjusting the linear phase difference between initial coherent pulses. The increase of the intensity of one pulse can squeeze efficiently the quantum fluctuations of Stokes operators. We study the spatial overlap of quadrature-squeezed pulses produced in an anisotropic electronic Kerr medium and show that the cross-phase modulation which occurs in addition to the self-phase one can be employed to control the fluctuation spectra of Stokes operators. The change of the nonlinear phase shift per photon for one pulse controls effectively the polarization squeezing. The spatial overlap of a coherent pulse with an interference one, produced by mixing two quadrature-squeezed pulses on a beam splitter, is analysed. It is found that squeezing is produced in three of the four Stokes operators, with the squeezing in the first two being simultaneous.
We analyze the classical behavior and the quantum fluctuations for the system of degenerate four-wave mixing inside a doubly resonant optical cavity. Two frequency-degenerate but orthogonally polarized modes are coupled together in a ${\mathrm{\ensuremath{\chi}}}^{(3)}$ medium. We include all significant phase-matched interactions that occur in a third-order polarization (self-phase modulation, cross-phase modulation, and four-wave mixing). Two critical points are predicted to occur in the system. The first occurs for low pumping, associated with the excitation of the single mode. The second critical point occurs for higher pumping, associated with the onset of self-pulsing. Quantum fluctuations are analyzed for the various regions of pumping. At the first critical point, perfect squeezing is possible in both the pump and the signal modes when the damping of the signal mode is greater than the damping of the pump (${\ensuremath{\gamma}}_{\mathit{a}}$>${\ensuremath{\gamma}}_{\mathit{b}}$). In the case where the damping of the signal mode is much greater than that of the pump (${\ensuremath{\gamma}}_{\mathit{a}}$\ensuremath{\gg}${\ensuremath{\gamma}}_{\mathit{b}}$), nearly perfectly squeezed states of the signal mode are possible in the entire region between the two critical points.
The authors present a quantum analysis of four-wave mixing in a non-resonant nonlinear medium inside a cavity which is driven by two driving fields. In this process a nonlinear medium couples two pump modes with a signal mode of the frequency equal to the half-sum of the two pump field frequencies. Three types of stable steady-state solutions of the deterministic problem, describing three regimes of oscillation, are found: one below threshold and two different above-threshold regimes. The spectra of fluctuations of the quadrature-phase amplitudes for all three modes are calculated. It is found that in the above-threshold regime each of three modes may display a squeezed noise reduction. Below threshold a squeezing is realized for the central mode.
In Chapter 1 we consider the question of squeezing in a two-photon laser. A two-photon laser is modelled as N two-level atoms interacting via a two-photon transition with a single resonant cavity field mode. A Fokker-Planck equation containing all leading noise terms is derived. A linearised fluctuation analysis reveals that there is no squeezing in a two-photon laser with injected signal. A small amount of squeezing is present in two-photon optical bistability. In Chapter 2 a microscopic model of degenerate four wave mixing including the quantisation of the medium is presented. Time-dependent solutions show ideal squeezing to be obtained in the limits of large detuning, low saturation and zero loss. The effect of nonideal atomic parameters is analysed. Equations describing the spatial propagation of the field through the medium are also derived. In Chapters 3 and 4 we are concerned with quantum effects, in particular violation of Bell's Inequalities, present in optical systems. An optical field produced by an intracavity four wave mixing is shown to exhibit the nonclassical features of photon antibunching, squeezing and violation of Cauchy-Schwarz and Bell's Inequalities. These intrinsic quantum mechanical effects are shown to be associated with the nonexistence of a positive normalizable Glauber-Sudarshan P-function.