Following the analytical work of Ref. [1], a numerical analysis of squeezing and quantum entanglement in a continuous wave singly-resonant optical parametric oscillator approaching threshold is provided. The singly resonant case is mainly relevant to largely non-degenerate signal and idler modes (two-colour output). As the threshold of oscillation is approached the numerical spectra of the intensity difference confirm squeezing of quantum fluctuations and a progressive line-narrowing in the linear case. In the nonlinear case entanglement is confirmed although progressively reduced when approaching threshold with the squeezing spectra still displaying a narrowing of the spectral line. Modification of quantum entanglement approaching threshold is also evaluated via the condition of state inseparability.
Following the analytical work of Ref. [1], a numerical analysis of squeezing and quantum entanglement in a continuous wave singly-resonant optical parametric oscillator approaching threshold is provided. The singly resonant case is mainly relevant to largely non-degenerate signal and idler modes (two-colour output). As the threshold of oscillation is approached the numerical spectra of the intensity difference confirm squeezing of quantum fluctuations and a progressive line-narrowing in the linear case. In the nonlinear case entanglement is confirmed although progressively reduced when approaching threshold with the squeezing spectra still displaying a narrowing of the spectral line. Modification of quantum entanglement approaching threshold is also evaluated via the condition of state inseparability.
We apply the input-output theory of optical cavities to formulate a quantum treatment of a continuous-wave singly resonant optical parametric oscillator. This case is mainly relevant to highly nondegenerate signal and idler modes. We show that both intensity and quadrature squeezing are present and that the maximum noise reduction below the standard quantum limit is the same at the signal and idler frequencies as in the doubly resonant case. As the threshold of oscillation is approached, however, the intensity-difference and quadrature spectra display a progressive line narrowing which is absent in the balanced doubly resonant case. By use of the separability criterion for continuous variables, the signal-idler state is found to be entangled over wide ranges of the parameters. We show that attainable levels of squeezing and entanglement make singly resonant configurations ideal candidates for two-color quantum information processes, because of their ease of tuning in experimental realizations.
Many theoretical and experimental approaches to quantum entanglement in Optical Parametric Oscillators (OPO) concern the degenerate or the slightly non-degenerate cases where a single local oscillator can be used to measure quadrature correlations for the characterization of fluctuations below the Standard Quantum Limit (SQL) in the frequency domain. In the strongly non-degenerate case, alternative characterizations of quantum entanglement are needed. We provide evidence that correlation functions corresponding to coincidence measurements of the intensity difference between signal and idler fields should be able to assess quantum entanglement and at the same time be experimentally viable in the case of strongly non-degenerate OPO.
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.
We study theoretically and experimentally the quantum properties of a type-II frequency-degenerate optical parametric oscillator below threshold with a quarter-wave plate inserted inside the cavity which induces a linear coupling between the orthogonally polarized signal and idler fields. This original device provides a good insight into general properties of two-mode Gaussian states, illustrated in terms of the covariance matrix. We report on the experimental generation of a two-mode squeezed vacuum on nonorthogonal quadratures depending on the plate angle. After a simple operation, the entanglement is maximized and put into standard form, i.e., quantum correlations and anticorrelations on orthogonal quadratures. A half sum of squeezed variances as low as $0.33\ifmmode\pm\else\textpm\fi{}0.02$, well below the unit limit for inseparability, is obtained and the entanglement measured by the entropy of formation.
The dynamical behavior of Kerr optical frequency combs is very well understood today from the perspective of the semiclassical approximation. These combs are obtained by pumping an ultrahigh-Q whispering-gallery mode resonator with a continuous-wave laser. The long-lifetime photons are trapped within the toruslike eigenmodes of the resonator, where they interact nonlinearly via the Kerr effect. In this article, we use quantum Langevin equations to provide a theoretical understanding of the nonclassical behavior of these combs when pumped below and above threshold. In the configuration where the system is under threshold, the pump field is the unique oscillating mode inside the resonator, and it triggers the phenomenon of spontaneous four-wave mixing, where two photons from the pump are symmetrically up- and down-converted in the Fourier domain. This phenomenon, also referred to as parametric fluorescence, can only be understood and analyzed from a fully quantum perspective as a consequence of the coupling between the field of the central (pumped) mode and the vacuum fluctuations of the various side modes. We analytically calculate the power spectra of the spontaneous emission noise, and we show that these spectra can be either single- or double-peaked depending on the value of the laser frequency, chromatic dispersion, pump power, and spectral distance between the central mode and the side mode of interest. We also calculate as well the overall spontaneous noise power per side mode and propose simplified analytical expressions for some particular cases. In the configuration where the system is pumped above threshold, we investigate the phenomena of quantum correlations and multimode squeezed states of light that can occur in the Kerr frequency combs originating from stimulated four-wave mixing. We show that for all stationary spatiotemporal patterns, the side modes that are symmetrical relative to the pumped mode in the frequency domain display quantum correlations that can lead to squeezed states of light under some optimal conditions that are analytically determined. These quantum correlations can persist regardless the dynamical state of the system (rolls or solitons), regardless of the spectral extension of the comb (number side modes) and regardless of the dispersion regime (normal or anomalous). We also explicitly determine the phase quadratures leading to photon entanglement and analytically calculate their quantum-noise spectra. For both the below- and above-threshold cases, we study with particular emphasis the two principal architectures for Kerr comb generation, namely the add-through and add-drop configurations. It is found that regardless of the configuration, an essential parameter is the ratio between out-coupling and total losses, which plays a key role as it directly determines the efficiency of the detected fluorescence or squeezing spectra. We finally discuss the relevance of Kerr combs for quantum information systems at optical telecommunication wavelengths below and above threshold.
We calculate the quantum correlations existing among the three output fields (pump, signal, and idler) of a triply resonant nondegenerate optical parametric oscillator operating above threshold. By applying the standard criteria [P. van Loock and A. Furusawa, Phys. Rev. A 67, 052315 (2003)], we show that strong tripartite continuous-variable entanglement is present in this well-known and simple system. Furthermore, since the entanglement is generated directly from a nonlinear process, the three entangled fields can have very different frequencies, opening the way for multicolored quantum information networks.