In this work, we theoretically and numerically investigate nonclassical phonon states created in the collective vibration of a Raman medium by the generation of a dual-pumped Raman optical frequency comb in an optical cavity. This frequency comb is generated by cascaded Raman scattering driven by two phase-locked pump lasers that are separated in frequency by three times the Raman phonon frequency. We characterize the variety of conditioned phonon states that are created when the number of photons in all optical frequency modes except the pump modes are measured. Almost all of these conditioned phonon states are extremely well approximated as three-phonon-squeezed states or Schrödinger-cat states, depending on the outcomes of the photon number measurements. We show how the combinations of first-, second-, and third-order Raman scattering that correspond to each set of measured photon numbers determine the fidelity of the conditioned phonon state with model three-phonon-squeezed states and Schrödinger-cat states. All of the conditioned phonon states demonstrate preferential growth of the phonon mode along three directions in phase space. That is, there are three preferred phase values that the phonon state takes on as a result of Raman scattering. We show that the combination of Raman processes that produces a given set of measured photon numbers always produces phonons in multiples of three. In the quantum number-state representation, these multiples of three are responsible for the threefold phase-space symmetry seen in the conditioned phonon states. \n \nWith a semiclassical model, we show how this three-phase preference can also be understood in light of phase correlations that are known to spontaneously arise in single-pumped Raman frequency combs. Additionally, our semiclassical model predicts that the optical modes also grow preferentially along three phases, suggesting that the dual-pumped Raman optical frequency comb is partially phase-stabilized.
In this work, we theoretically and numerically investigate nonclassical phonon states created in the collective vibration of a Raman medium by the generation of a dual-pumped Raman optical frequency comb in an optical cavity. This frequency comb is generated by cascaded Raman scattering driven by two phase-locked pump lasers that are separated in frequency by three times the Raman phonon frequency. We characterize the variety of conditioned phonon states that are created when the number of photons in all optical frequency modes except the pump modes are measured. Almost all of these conditioned phonon states are extremely well approximated as three-phonon-squeezed states or Schrödinger-cat states, depending on the outcomes of the photon number measurements. We show how the combinations of first-, second-, and third-order Raman scattering that correspond to each set of measured photon numbers determine the fidelity of the conditioned phonon state with model three-phonon-squeezed states and Schrödinger-cat states. All of the conditioned phonon states demonstrate preferential growth of the phonon mode along three directions in phase space. That is, there are three preferred phase values that the phonon state takes on as a result of Raman scattering. We show that the combination of Raman processes that produces a given set of measured photon numbers always produces phonons in multiples of three. In the quantum number-state representation, these multiples of three are responsible for the threefold phase-space symmetry seen in the conditioned phonon states. \n \nWith a semiclassical model, we show how this three-phase preference can also be understood in light of phase correlations that are known to spontaneously arise in single-pumped Raman frequency combs. Additionally, our semiclassical model predicts that the optical modes also grow preferentially along three phases, suggesting that the dual-pumped Raman optical frequency comb is partially phase-stabilized.
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.
The dynamical behavior of Kerr optical frequency combs is very well understood today from the perspective of the semi-classical approximation. In this article, we provide a theoretical understanding of the phenomena of quantum correlations, multimode entanglement and squeezed states of light that can occur in these frequency combs when quantum noise is accounted for. We prove that for all stationary spatio-temporal patterns, the side-modes that are symmetrical relatively to the central (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. We show that these quantum correlations can persist regardless the dynamical state of the system (rolls or solitons), regardless of the spectral extension of the comb (number sidemodes), and regardless of the dispersion regime (normal or anomalous). We study with particular emphasis the two principal architectures for Kerr comb generation, namely the add-through and add-drop configurations. We show that in both cases, the ratio between out-coupling and total losses plays a key role as it directly determines the efficiency of the squeezing. We also explicitly determine the phase quadratures leading to photon entanglement, and analytically calculate their quantum noise spectra. The relevance of Kerr combs for quantum information systems at optical telecommunication wavelengths is also discussed.
Quantum statistical properties are derived for stimulated Raman scattering by squeezed phonons including squeezing of vacuum fluctuations in radiation modes and sub-Poisson photon behaviour.Initial radiation can be in squeezed thermalized and/or sub-Poisson states, phonon system can also be squeezed.In particular, effects of initial phase conditions, losses, initial nonclassical behaviour and phonon squeezing are demonstrated in terms of reduced factorial moments up to the fifth order in combined photon-photon and photon-phonon modes.Also photon number distribution, quadrature variances and principał squeeze parameters are shown.
Nonclassical light is now available in a number oflaboratories, being generated by Frank—Hertz experiment, optical parametric processes, four-wave mixing including optical fibres, by optical system with feedback, etc. Therefore it is senseful to reconsider various nonlinear optical interactions involving initially nonclassical light ,originally solved with coherent (laser) light. Adopting tools of quasidistributions related to antinormal ordering, the photocount distributions, its factorial moments and characteristics of squeezing of vacuum fluctuations can be investigated assuming initially nonclassical (squeezed) light in the framework of the Heisenberg—Langevin of Schrödinger approaches. Photon statistics of stimulated Raman scattering of squeezed light can be derived assuming phonon system in chaotic, coherent or squeezed states and including additive noise. The influence of nonlinear dynamics can be established on initially inclassical light and it can be shown that the initially nonclassical effects are encresed by the nonlinear dynamics provided that the Stokes mode is less amplified than the anti-Stokes mode is damped, whereas in the opposite case the nonclassical features are smoothed out. Also frequency detuning can be optimized to have maximum nonclassical effects, which are missing with increasing phonon number, even if the phonon system is squeezed. Small initial squeezing can by increased by the nonlinear dynamics, whereas strong squeezing can only be diminished.
With the development of quantum optics, photon correlations acquired a prominent role as a tool to test our understanding of physics, and played a key role in verifying the validity of quantum mechanics. The spatial and temporal correlations in a light field also reveal information about its origin, and allow us to probe the nature of the physical systems interacting with it. Additionally, with the advent of quantum technologies, they have acquired technological relevance, as they are expected to play an important role in quantum communication and quantum information processing. This thesis develops techniques that combine spontaneous Raman scattering with Time Correlated Single Photon Counting, and uses them to study the quantum mechanical nature of high frequency vibrations in crystals and molecules. We demonstrate photon bunching in the Stokes and anti-Stokes fields scattered from two ultrafast laser pulses, and use their cross-correlation to measure the 3.9 ps decay time of the optical phonon in diamond. We then employ this method to measure molecular vibrations in CS2, where we are able to excite the respective vibrational modes of the two isotopic species present in the sample in a coherent superposition, and observe quantum beating between the two signals. Stokes scattering, when combined with a projective measurement, leads to a well defined quantum state. We demonstrate this by measuring the second order correlation function of the anti-Stokes field conditional on detecting one or more photons in the Stokes field, which allows us to observe a phonon modeâ s transition form a thermal state into the first excited Fock state, and measure its decay over the characteristic phonon lifetime. Finally, we use this technique to prepare a highly entangled photon-phonon state, which violates a Bell-type inequality. We measure S = 2.360 ± 0.025, violating the CHSH inequality, compatible with the non-locality of the state. The techniques we developed open the door to the study of a broad range of physical systems, where spectroscopic information is obtained with the preparation of specific quantum states. They also hold potential for future technological use, and promote vibrational Raman scattering to a resource in nonlinear quantum optics -- where it used to be considered as a source of noise instead.