Nondegenerate Parametric Oscillator Research Articles

Persistent dynamics in coupled non-degenerate parametric oscillators: pump saturation prevents mode competition.

The coherent dynamics in networks of coupled oscillators is of great interest in wave-physics since the coupling produces various dynamical effects, such as coherent energy exchange (beats) between the oscillators. However, it is common wisdom that these coherent dynamics are transients that quickly decay in active oscillators (e.g. lasers) since pump saturation causes mode competition that results, for homogeneous gain, in the prevalence of the single winning mode. We observe that pump saturation in coupled parametric oscillators counter-intuitively encourages the multi-mode dynamics of beating and indefinitely preserves it, despite the existence of mode competition. We explore in detail the coherent dynamics of a pair of coupled parametric oscillators with a shared pump and arbitrary coupling in a radio frequency (RF) experiment, as well as in simulation. Specifically, we realize two parametric oscillators as different frequency-modes of a single RF cavity and couple them arbitrarily using a digital high-bandwidth FPGA. We observe persistent coherent beats that are maintained at any pump level, even high above the threshold. The simulation highlights how the interplay of pump depletion between the two oscillators prevents them from synchronizing, even when the oscillation is deeply saturated.

Low-frequency fluorescence spectrum of a laser driven polar quantum emitter damped by squeezed vacuum with finite bandwidth

A two-level quantum emitter with broken inversion symmetry driven by external semiclassical monochromatic high-frequency electromagnetic (e.g., laser) field and damped by squeezed vacuum reservoir with finite bandwidth is presented. The squeezed vacuum source is assumed to be either degenerate parametric oscillator (DPO) or a non-degenerate parametric oscillator (NDPO). It is shown that the shape of low-frequency fluorescence spectrum of the emitter can be effectively alternated by controlling the degree of the squeezed vacuum source degeneration and phase of the squeezing.

Entanglement and Photon Anti-Bunching in Coupled Non-Degenerate Parametric Oscillators.

We analytically and numerically show that the Hillery-Zubairy’s entanglement criterion is satisfied both below and above the threshold of coupled non-degenerate optical parametric oscillators (NOPOs) with strong nonlinear gain saturation and dissipative linear coupling. We investigated two cases: for large pump mode dissipation, below-threshold entanglement is possible only when the parametric interaction has an enough detuning among the signal, idler, and pump photon modes. On the other hand, for a large dissipative coupling, below-threshold entanglement is possible even when there is no detuning in the parametric interaction. In both cases, a non-Gaussian state entanglement criterion is satisfied even at the threshold. Recent progress in nano-photonic devices might make it possible to experimentally demonstrate this phase transition in a coherent XY machine with quantum correlations.

Antibunching of photons in a coherent radiation field coupled to a non-degenerate parametric oscillator beyond rotating wave approximation

Under the classical (strong) pump condition, the Hamiltonian involving the signal and the idler modes of a non-degenerate parametric oscillator is exhibited. Without using the usual rotating wave approximation (RWA), the analytical solutions of the field operators are used to investigate the antibunching of photons of the input radiation field coupled to the non-degenerate parametric oscillator. By using the symbolic calculation, the antibunching of photons for both the signal and idler modes are investigated. In particular, the effects of the inclusion of rotating wave approximated terms on the antibunching of photons are clearly indicated. To substantiate the analytical results, the temporal evolution of signal photon and idler photons, and the antibunching effects of the signal and idler modes are investigated numerically by using the QuTip 3.1.0. The exact numerical results obtained by QuTip 3.1.0 matches extremely well with those of the analytical results. The present article and hence the analytical method might be of use for investigating the situations of having ultra-strongly and deep-strongly coupled systems where the possibilities of using RWA is completely ruled out.

Critical quantum fluctuations and photon antibunching in optomechanical systems with large single-photon cooperativity

A pertinent question in cavity optomechanics is whether reaching the regime of large single-photon cooperativity, where the single-photon coupling rate exceeds the geometric mean of the cavity and mechanical decay rates, can enable any new phenomena. We show that in some multimode optomechanical systems, the single-photon cooperativity can indeed be a figure of merit. We first study a system with one cavity mode and two mechanical oscillators which combines the concepts of levitated optomechanics and coherent scattering with standard dispersive optomechanics. Later, we study a more complicated setup comprising three cavity modes which does not rely on levitated optomechanics and only features dispersive optomechanical interactions with direct cavity driving. These systems can effectively realize the degenerate or the nondegenerate parametric oscillator models known from quantum optics, but in the unusual finite-size regime for the fundamental mode(s) when the single-photon cooperativity is large. We show that the response of these systems to a coherent optical probe can be highly nonlinear in probe power even for average photon occupation numbers below unity. The nonlinear optomechanical interaction has the peculiar consequence that the probe drive will effectively amplitude-squeeze itself. For large single-photon cooperativity, this occurs for small occupation numbers, which enables observation of nonclassical antibunching of the transmitted probe photons due to a destructive interference effect. Finally, we show that as the probe power is increased even further, the system enters a critical regime characterized by intrinsically nonlinear dynamics and non-Gaussian states.

Theory of quantum entanglement of intra-cavity photons produced by a non-degenerate parametric oscillator (NDPO); Semi-classical approach

The von-Neumann entropy (VNE) which is usually expressed in terms of the density operator is the mathematical tool to measure the degree of entanglement of a bi-partite system. Applying the solutions of the quantum Langevin equations, the anti-normally ordered characteristic function of the intra-cavity photons produced by a non-degenerate parametric oscillator is calculated. With the help of the resulting characteristic function, the Q function which is then used to calculate the entanglement of the intra-cavity photons using VNE is determined. Moreover, the photon number distribution of the intra-cavity photons produced by a non-degenerate parametric oscillator coupled to a two-mode squeezed vacuum reservoir is determined.

Statistics of radiation due to nondegenerate Josephson parametric down-conversion

In a process called parametric down-conversion, a dc-biased Josephson junction coupled to two microwave resonators emits photon pairs when the Josephson frequency matches the sum of the two resonance frequencies. Recent experiments have shown that such a setup permits analyzing the correlation of the radiation. Motivated by these results, we study theoretically the full counting statistics of a non-degenerate parametric oscillator below the threshold of instability. Furthermore, we analyze the second-order coherences of the radiation and discuss thermal effects on the radiation statistics. We provide results for the driving strength at which the Cauchy-Schwarz inequality is most strongly violated. Additionally, we study the impact of asymmetry in the linewidth of the modes---a distinctive property of the non-degenerate resonance effect. In particular, we find that the radiation from the mode with the larger linewidth preferably takes the total detuning, while the other mode emits photons at its resonance frequency.

The Effect of Signal-signal and Signal-idler Modes on Squeezing and Entanglement Properties of Two-mode Light

Squeezed and entangled two-mode light can be generated by the combination of degenerate and nondegenerate parametric oscillators. The correlated signal-idler modes are not only enhance the degree of quadrature squeezing and entanglement properties of the two-mode light but they are also the cause of these properties. But the effect of signal-signal modes are to enhance the degree of quadrature squeezing and to reduce the entanglement property of the two-mode light. Keywords: signal-signal modes, signal-idler modes, quadrature squeezing, entanglement DOI : 10.7176/APTA/78-02 Publication date :June 30 th 2019

Enhancement of second-harmonic generation based on the cascaded second- and third-order nonlinear processes in a multimode optical microcavity

Optical microcavities are often used to realize enhanced nonlinear optical interactions for highly efficient second-harmonic generation. With increased pump power, the efficiency of nonlinear frequency conversion can be increased further, while some other unwanted nonlinear effects will also emerge, leading to complicated dynamics or instability. Here, we study the interplay between cascaded second- and third-order nonlinear processes and investigate their impact on the second-harmonic generation in microcavities. It is found that the non-degenerate optical parametric oscillation (OPO) appears and the presence of $\chi^{(3)}$ process can modify the OPO threshold significantly when the multimode cavity is strongly pumped at the fundamental optical mode. One can even break the efficiency limitation of the second-harmonic mode restricted by the OPO by utilizing the interference between the OPO and the four-wave mixing. The present coherent interplay between nonlinear optical processes in microcavities is conducive to exploring new physics in the cavity nonlinear photonics.

Exact solution of degenerate and nondegenerate optical parametric oscillator coupled with a squeezed thermal bath

We solve exactly the master equation of the degenerate and nondegenerate parametric oscillator in the presence of a squeezed reservoir. We show that the formal solutions of both systems can be obtained in terms of the density operator by applying squeeze transformations and using the superoperators formalism.

Squeezing, mixed mode squeezing, amplitude squared squeezing and principal squeezing in a non-degenerate parametric oscillator

The Hamiltonian and hence the equation of motion of the field operators of a non-degenerate parametric oscillator (NDPO) under the influence of classical pump are formulated. The coupled operator differential equations involving the signal and idler modes are decoupled at the expense of fourth order differential equations involving the c-numbers. Without using the rotating wave approximation, the analytical solutions of the field operators are obtained. These solutions are approximated up to the second orders in dimensionless coupling constant. We investigate the squeezing, mixed mode squeezing, amplitude-squared squeezing, and the principal squeezing of the thermal and coherent light coupled to the NDPO. By using the input composite number state, we establish that the percentage and the range (interaction time) of squeezing is considerably increased with the increase of the signal photon number. For initial composite number state, the amplitude squared squeezing for Ys quadrature is obtained at the cost of canonically conjugate Zs quadrature. The percentage of amplitude squared squeezing increases significantly with the increase of signal excitation (photon). The so-called normal squeezing and the principal squeezing are also indicated for the NDPO coupled with the initially prepared composite coherent states not in the composite number states. In spite of the shortcomings of the analytical solutions, we obtain squeezing, amplitude squared squeezing and other nonclassical effects which are unavailable under the rotating wave approximation. In order to give the feelings about the analytical results (expressions), we give some symbolic calculations relevant to the possible experimental situations.

Controlled Asymmetry of Einstein-Podolsky-Rosen Steering with an Injected Nondegenerate Optical Parametric Oscillator.

We propose and analyze a nonlinear optical apparatus in which the direction of asymmetric steering is controllable within the apparatus, rather than by adding noise to measurements. Using a nondegenerate parametric oscillator with an injected signal field, we show how the directionality and extent of the steering can be readily controlled for output modes that can be up to one octave apart. The two down-converted modes, which exhibit the greater violations of the steering inequalities, can also be controlled to exhibit asymmetric steering in some regimes.

Quantum phase fluctuations of coherent and thermal light coupled to a non-degenerate parametric oscillator beyond rotating wave approximation

The essence of the rotating wave approximation (RWA) is to eliminate the non-conserving energy terms from the interaction Hamiltonian. The cost of using RWA is heavy if the frequency of the input radiation field is low (e.g. below optical region). The well known Bloch-Siegert effect is the out come of the inclusion of the terms which are normally neglected under RWA. We investigate the fluctuations of the quantum phase of the coherent light and the thermal light coupled to a nondegenerate parametric oscillator (NDPO). The Hamiltonian and hence the equations of motion involving the signal and idler modes are framed by using the strong (classical) pump condition. These differential equations are nonlinear in nature and are found coupled to each other. Without using the RWA, we obtain the analytical solutions for the signal and idler fields. These solutions are obtained up to the second orders in dimensionless coupling constants. The analytical expressions for the quantum phase fluctuation parameters due to Carruther's and Nieto are obtained in terms of the coupling constants and the initial photon numbers of the input radiation field. Moreover, we keep ourselves confined to the Pegg-Barnett formalism for measured phase operators. With and without using the RWA, we compare the quantum phase fluctuations for coherent and thermal light coupled to the NDPO. In spite of the significant departures (quantitative), the qualitative features of the phase fluctuation parameters for the input thermal light are identical for NDPO with and without RWA. On the other hand, we report some interesting results of input coherent light coupled to the NDPO which are substantially different from their RWA counterpart. In spite of the various quantum optical phenomena in a NDPO, we claim that it is the first effort where the complete analytical approach towards the solutions and hence the quantum phase fluctuations of input radiation fields coupled to it are obtained beyond rotating wave approximation. To have the feelings of the analytical solutions, we give few numerical estimates of the quantum phase fluctuation parameters relevant to a real experimental situation.

Response of a mechanical oscillator in an optomechanical cavity driven by a finite-bandwidth squeezed vacuum excitation

In this paper, we theoretically investigate the displacement and momentum fluctuations spectra of the movable mirror in a standard optomechanical system driven by a finite bandwidth squeezed vacuum light accompanying a coherent laser field. Two cases in which the squeezed vacuum is generated by degenerated and non-degenerate parametric oscillators (DPO and NDPO) are considered. We find that for the case of finite bandwidth squeezed vacuum injection, the two spectra exhibit unique features, which strongly differ from those of broadband squeezing excitation. In particular, the spectra exhibit a three-peaked and a four-peaked structure, respectively, for the squeezing injection from DPO and NDPO. Besides, some anomalous characteristics of the spectra such as squeezing-induced pimple, hole burning, and dispersive profile are found to be highly sensitive to the squeezing parameters and the temperature of the mirror. We also evaluate the mean-square fluctuations in position and momentum quadratures of the movable mirror and analyze the influence of the squeezing parameters of the input field on the mechanical squeezing. It will be shown that the parameters of driven squeezed vacuum affects the squeezing. We find the optimal mechanical squeezing is achievable via finite bandwidth squeezed vacuum injection which is affected by the intensity of squeezed vacuum. We also show that the phase of incident squeezed vacuum determines whether position or momentum squeezing occurs. Our proposed scheme not only provides a feasible experimental method to detect and characterize squeezed light by optomechanical systems, but also suggests a way for controllable transfer of squeezing from an optical field to a mechanical oscillator.

Non-Markovian dynamics and steady-state entanglement of cavity arrays in finite-bandwidth squeezed reservoirs

When two chains of quantum systems are driven at their ends by a two-mode squeezed reservoir, they approach a steady state characterized by the formation of many entangled pairs. Each pair is made of one element of the first and one of the second chain. This effect has been already predicted under the assumption of broadband squeezing. Here we investigate the situation of finite-bandwidth reservoirs. This is done by modeling the driving bath as the output field of a non-degenerate parametric oscillator. The resulting non-Markovian dynamics is studied within the theoretical framework of cascade open quantum systems. It is shown that the formation of pair-entangled structures occurs as long as the normal-mode splitting of the arrays does not overcome the squeezing bandwidth of the reservoir.

Bright entanglement generated in a four-level laser

We discuss the generation and evolution of entanglement in a four-level laser with a subthreshold nondegenerate parametric oscillator. The entanglement properties of the two-mode light generated by this scheme is studied. We show that the light produced by the present system is strongly entangled with time evolution. Especially, with the help of the parametric oscillator, the high intensity of the entangled light between the two-mode cavity can be achieved.

Enhanced entanglement in a non-degenerate four-level atom with a parametric oscillator

We discuss the generation and evolution of a macroscopic entanglement light with a subthreshold non-degenerate parametric oscillator, coupled to a vacuum reservoir. The four-level atoms driven by two classical fields interact with the parametric oscillator. The master equation for the cavity modes of the scheme is derived and analyzed, the entanglement properties of the two-mode light generated by this scheme inside and outside the cavity is studied. We show that the light produced by this system is strongly entangled with time evolution inside and outside a cavity.

The ideal three-mode nondegenerate parametric oscillator, unitary operators, nonlinear differential equations and Painlevé property

For the unitary evolution of the ideal three-mode parametric oscillator, we derive an autonomous system of first order nonlinear differential equations. We show that this autonomous system passes the Painlevé test. Then we show that the system is integrable in the sense of Liouville.

Enhanced entanglement and squeezing of two-mode field from a double-ladder four-level atom

The entanglement and squeezing properties of a two-mode field are investigated in the system of a double-ladder four-level atom embedded in a two-mode cavity. It is found that the system can reduce to the nondegenerate parametric oscillator which is responsible for the entanglement and squeezing of a cavity field under suitable conditions. Moreover, compared to the field from the three-level cascade atomic system, the degree of entanglement in the present system can be improved significantly.

Twin stimulated amplification of light and matter waves in an atom-photon pair laser

We consider ultracold atoms in a far detuned optical lattice across a high-$Q$ optical resonator. Applying an external driving laser to the atoms red detuned from the cavity mode by one vibrational quantum induces cavity-enhanced sideband cooling. For a dense and cold enough atomic ensemble we predict an oscillation threshold for optical Raman sideband lasing concurrent with coherent matter-wave amplification. Above threshold, photons and atoms in the lowest band are dominantly created pairwise via stimulated emission with a strong suppression of competing spontaneous processes. In close analogy to a nondegenerate parametric oscillator, we find sub-Poissonian photon statistics and almost perfect nonclassical atom-photon number correlations. Injecting atoms in higher vibrational bands via tunneling or incoherent scattering then leads to continuous, simultaneous generation of a coherent atom beam and laser light with nonclassical atom-field correlations.