Degenerate Parametric Oscillator Research Articles

Radiation statistics of a degenerate parametric oscillator at threshold

As a function of the driving strength, a degenerate parametric oscillator exhibits an instability at which spontaneous oscillations occur. Close to threshold, both the nonlinearity as well as fluctuations are vital to the accurate description of the dynamics. We study the statistics of the adiation that is emitted by the degenerate parametric oscillator at threshold. For a weak nonlinearity, we can employ a quasiclassical description. We identify a universal Liouvillian that captures the relevant long-time dynamics for large photon-numbers. We find that the cumulants obey a universal power-law scaling as a function of the nonlinearity. The Fano factor shows a maximum close, but not coinciding, with the threshold. Moreover, we predict a certain ratio of the first three cumulants to be independent of the microscopic details of the system and connect the results to experimental platforms.

Turing instability in quantum activator–inhibitor systems

Turing instability is a fundamental mechanism of nonequilibrium self-organization. However, despite the universality of its essential mechanism, Turing instability has thus far been investigated mostly in classical systems. In this study, we show that Turing instability can occur in a quantum dissipative system and analyze its quantum features such as entanglement and the effect of measurement. We propose a degenerate parametric oscillator with nonlinear damping in quantum optics as a quantum activator–inhibitor unit and demonstrate that a system of two such units can undergo Turing instability when diffusively coupled with each other. The Turing instability induces nonuniformity and entanglement between the two units and gives rise to a pair of nonuniform states that are mixed due to quantum noise. Further performing continuous measurement on the coupled system reveals the nonuniformity caused by the Turing instability. Our results extend the universality of the Turing mechanism to the quantum realm and may provide a novel perspective on the possibility of quantum nonequilibrium self-organization and its application in quantum technologies.

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.

Squeezing properties of nondegenerate three-level laser with degenerate parametric amplifier

We study the squeezing properties of the light generated by nondegenerate three-level laser in which the three-level atoms are in cascade configuration with degenerate parametric oscillator. We obtain the c-number Langevin equations associated with the normal ordering using the pertinent master equation. Making use of the solutions of the c-number Langevin equations, we calculate the quadrature variances. The results show that the two-mode light produced by the system under consideration is in squeezed state and the squeezing occurs in the minus quadrature. The degree of squeezing increase with the linear gain coefficient and almost perfect squeezing can be obtained for large values of the linear gain coefficient and the atoms initially prepared in lower level slightly greater than in the upper level. Moreover, the presence of nonlinear crystal increases the degree of squeezing for single mode and two-mode lights.  Â

Deterministic generation of parametrically driven dissipative Kerr soliton

Abstract We theoretically study the nature of parametrically driven dissipative Kerr soliton (PD-DKS) in a doubly resonant degenerate micro-optical parametric oscillator (DR-DμOPO) with the cooperation of χ (2) and χ (3) nonlinearities. Lifting the assumption of close-to-zero group velocity mismatch (GVM) that requires extensive dispersion engineering, we show that there is a threshold GVM above which single PD-DKS in DR-DμOPO can be generated deterministically. We find that the exact PD-DKS generation dynamics can be divided into two distinctive regimes depending on the phase matching condition. In both regimes, the perturbative effective third-order nonlinearity resulting from the cascaded quadratic process is responsible for the soliton annihilation and the deterministic single PD-DKS generation. We also develop the experimental design guidelines for accessing such deterministic single PD-DKS state. The working principle can be applied to different material platforms as a competitive ultrashort pulse and broadband frequency comb source architecture at the mid-infrared spectral range.

Floquet theory for temporal correlations and spectra in time-periodic open quantum systems: Application to squeezed parametric oscillation beyond the rotating-wave approximation

Open quantum systems can display periodic dynamics at the classical level either due to external periodic modulations or to self-pulsing phenomena typically following a Hopf bifurcation. In both cases, the quantum fluctuations around classical solutions do not reach a quantum-statistical stationary state, which prevents adopting the simple and reliable methods used for stationary quantum systems. Here we put forward a general and efficient method to compute two-time correlations and corresponding spectral densities of time-periodic open quantum systems within the usual linearized (Gaussian) approximation for their dynamics. Using Floquet theory we show how the quantum Langevin equations for the fluctuations can be efficiently integrated by partitioning the time domain into one-period duration intervals, and relating the properties of each period to the first one. Spectral densities, like squeezing spectra, are computed similarly, now in a two-dimensional temporal domain that is treated as a chessboard with one-period x one-period cells. This technique avoids cumulative numerical errors as well as efficiently saves computational time. As an illustration of the method, we analyze the quantum fluctuations of a damped parametrically-driven oscillator (degenerate parametric oscillator) below threshold and far away from rotating-wave approximation conditions, which is a relevant scenario for modern low-frequency quantum oscillators. Our method reveals that the squeezing properties of such devices are quite robust against the amplitude of the modulation or the low quality of the oscillator, although optimal squeezing can appear for parameters that are far from the ones predicted within the rotating-wave approximation.

Spontaneous parametric down-conversion induced by optomechanical gradient forces in nanophotonic waveguides

Optomechanical gradient forces arise from evanescent fields of guided waves in parallel photonic waveguides. When designed to be of attractive nature, they increase exponentially as the gap between the waveguides decreases. Moreover, the amplitude of the gradient force can be well controlled due to its linear dependence on the input laser power. Here, we propose to exploit the intrinsic nonlinear nature of the optomechanical gradient force to induce a tunable 3-wave coupling between the fundamental modes of two doubly clamped nanophotonic beams. For one of the beams having half the width of the other beam, the 1:2 internal resonance between the fundamental modes supports degenerate spontaneous parametric down-conversion (SPDC). We theoretically explore the main feature of the dissipative phase diagram of the underlying degenerate parametric oscillator model to show that the critical point of the SPDC occurs at parameters which are well in reach of state-of-the-art experiments.

Overcoming decoherence of Schrödinger cat states formed in a cavity using squeezed-state inputs

A cat state is a superposition of two coherent states with amplitudes α0 and −α0. Recent experiments create cat states in a microwave cavity field using superconducting circuits. As with degenerate parametric oscillation (DPO) in an adiabatic and highly nonlinear limit, the states are formed in a signal cavity mode via a two-photon dissipative process induced by the down conversion of a pump field to generate pairs of signal photons. The damping of the signal and the presence of thermal fluctuations rapidly decoheres the state, and the effect on the dynamics is to either destroy the possibility of a cat state or else to sharply reduce the lifetime and size of the cat states that can be formed. In this paper, we study the effect on both the DPO and microwave systems of a squeezed reservoir coupled to the cavity. While the threshold nonlinearity is not altered, we show that the use of squeezed states significantly lengthens the lifetime of the cat states. This improves the feasibility of generating cat states of large amplitudes and with a greater degree of quantum macroscopic coherence, which is necessary for many quantum technology applications. Using current experimental parameters for the microwave setup, which requires a modified Hamiltonian, we further demonstrate how squeezed states enhance the quality of the cat states that could be formed in this regime. Squeezing also combats the significant decoherence due to thermal noise, which is relevant for microwave fields at finite temperature. By modeling a thermal squeezed reservoir, we show that the thermal decoherence of the dynamical cat states can be inhibited by a careful control of the squeezing of the reservoir. To signify the quality of the cat state, we consider different signatures including fringes and negativity, and the Cl1 measure of quantum coherence.14 MoreReceived 31 May 2020Accepted 20 October 2020DOI:https://doi.org/10.1103/PhysRevResearch.2.043387Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasNonlinear opticsOpen quantum systems & decoherenceQuantum channelsQuantum coherence & coherence measuresQuantum foundationsQuantum information processing with continuous variablesQuantum-to-classical transitionQuantum InformationGeneral PhysicsAtomic, Molecular & Optical

Dynamics of transient cat states in degenerate parametric oscillation with and without nonlinear Kerr interactions

A cat-state is formed as the steady-state solution for the signal mode of an ideal, degenerate parametric oscillator, in the limit of negligible single-photon signal loss. In the presence of the signal loss, this is no longer true over timescales much longer than the damping time. However, for sufficient parametric nonlinearity, a cat-state can exist as a transient state. In this paper, we study the dynamics of the creation and decoherence of cat-states in degenerate parametric oscillation, both with and without the effect of a Kerr nonlinearity that applies to recent superconducting-circuit experiments generating cat-states in microwave cavities. We determine the time of formation and the lifetime of a cat-state in terms of three dimensionless parameters $\lambda$, $g$ and $\chi$. These relate to the driving strength, the parametric nonlinearity, and the Kerr nonlinearity, respectively. We find that the Kerr nonlinearity has little effect on the threshold parametric nonlinearity ($g>1$) required for the formation of cat-states, and does not significantly alter the decoherence time of the cat-state, but can reduce the time of formation. The quality of the cat-state increases with the value $g$, and can also improved by the Kerr nonlinearity. To verify the existence and quality of the cat-state, we consider several signatures, including interference fringes and negativity, and show how they can be computed. We simulate a superconducting-circuit experiment using published experimental parameters and found good agreement with experimental results, indicating that a nonclassical cat-like state with a small Wigner negativity is generated in the experiment. A stronger nonlinearity would lead to a cat-state with convincing cat-state signatures. Finally, we explore the feasibility of creating large cat-states with a coherent amplitude of 20, corresponding to 400 photons.

On persistence of superoscillations for the Schrödinger equation with time‐dependent quadratic Hamiltonians

In this work, we prove the persistence in time of superoscillations for the Schrödinger equation with time‐dependent coefficients. In order to prove the persistence of superoscillations, we have conditioned the coefficients to satisfy a Riccati system, and we have expressed the solution as a convolution operator in terms of solutions of this Riccati system. Further, we have solved explicitly the Cauchy initial value problem with three different kinds of superoscillatory initial data. The operator is defined on a space of entire functions. Particular examples include Caldirola‐Kanai and degenerate parametric harmonic oscillator Hamiltonians, and other examples could include Hamiltonians not self‐adjoint. For these examples, we have illustrated numerically the convergence on real and imaginary parts.

Squeezed vacuum states from a whispering gallery mode resonator

Squeezed vacuum states enable optical measurements below the quantum limit and hence are a valuable resource for applications in quantum metrology and also quantum communication. However, most available sources require high pump powers in the milliwatt range and large setups, which hinders real world applications. Furthermore, degenerate operation of such systems presents a challenge. Here, we use a compact crystalline whispering gallery mode resonator made of lithium niobate as a degenerate parametric oscillator. We demonstrate about 1.4 dB noise reduction below the shot noise level for only 300 $\mu\text{W}$ of pump power in degenerate single mode operation. Furthermore, we report a record pump threshold as low as 1.35 $\mu\text{W}$. Our results show that the whispering gallery based approach presents a promising platform for a compact and efficient source for nonclassical light.

Discrete time symmetry breaking in quantum circuits: exact solutions and tunneling

We discuss general properties of discrete time quantum symmetry breaking in degenerate parametric oscillators. Recent experiments in superconducting quantum circuit with Josephson junction nonlinearities give rise to new properties of strong parametric coupling and nonlinearities. Exact analytic solutions are obtained for the steady-state of this single-mode case of subharmonic generation. We also obtain analytic solutions for the tunneling time over which the time symmetry-breaking is lost above threshold. We find that additional anharmonic terms found in the superconducting case increase the tunneling rate, and can also lead to new regimes of tristability as well as bistability. Our analytic results are confirmed by number state calculations.

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

Hybrid Systems for the Generation of Nonclassical Mechanical States via Quadratic Interactions.

We present a method to implement two-phonon interactions between mechanical resonators and spin qubits in hybrid setups, and show that these systems can be applied for the generation of nonclassical mechanical states even in the presence of dissipation. In particular, we demonstrate that the implementation of a two-phonon Jaynes-Cummings Hamiltonian under coherent driving of the qubit yields a dissipative phase transition with similarities to the one predicted in the model of the degenerate parametric oscillator: beyond a certain threshold in the driving amplitude, the driven-dissipative system sustains a mixed steady state consisting of a "jumping cat," i.e., a cat state undergoing random jumps between two phases. We consider realistic setups and show that, in samples within reach of current technology, the system features nonclassical transient states, characterized by a negative Wigner function, that persist during timescales of fractions of a second.

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.

Time-delayed quantum coherent Pyragas feedback control of photon squeezing in a degenerate parametric oscillator

Quantum coherent feedback control is a measurement-free control method fully preserving quantum coherence. In this paper we show how time-delayed quantum coherent feedback can be used to control the degree of squeezing in the output field of a cavity containing a degenerate parametric oscillator. We focus on the specific situation of Pyragas-type feedback control where time-delayed signals are fed back directly into the quantum system. Our results show how time-delayed feedback can enhance or decrease the degree of squeezing as a function of time delay and feedback strength.

Degenerate parametric oscillation in quantum membrane optomechanics

The promise of innovative applications has triggered the development of many modern technologies capable of exploiting quantum effects. But in addition to future applications, such quantum technologies have already provided us with the possibility of accessing quantum-mechanical scenarios that seemed unreachable just a few decades ago. With this spirit, in this work we show that modern optomechanical setups are mature enough to implement one of the most elusive models in the field of open system dynamics: degenerate parametric oscillation. The possibility of implementing it in nonlinear optical resonators was the main motivation for introducing such model in the eighties, which rapidly became a paradigm for the study of dissipative phase transitions whose corresponding spontaneously broken symmetry is discrete. However, it was found that the intrinsic multimode nature of optical cavities makes it impossible to experimentally study the model all the way through its phase transition. In contrast, here we show that this long-awaited model can be implemented in the motion of a mechanical object dispersively coupled to the light contained in a cavity, when the latter is properly driven with multi-chromatic laser light. We focus on membranes as the mechanical element, showing that the main signatures of the degenerate parametric oscillation model can be studied in state-of-the-art setups, thus opening the possibility of studying spontaneous symmetry breaking and enhanced metrology in one of the cleanest dissipative phase transitions.

Dual-pumped degenerate Kerr oscillator in a silicon nitride microresonator.

We demonstrate a degenerate parametric oscillator in a silicon nitride microresonator. We use two frequency-detuned pump waves to perform parametric four-wave mixing and operate in the normal group-velocity dispersion regime to produce signal and idler fields that are frequency degenerate. Our theoretical modeling shows that this regime enables generation of bimodal phase states, analogous to the χ(2)-based degenerate OPO. Our system offers potential for realization of CMOS-chip-based coherent optical computing and an all-optical quantum random number generator.

Degenerate parametric amplification of squeezed photons: Explicit solutions, statistics, means and variances

In the Schrödinger picture, we find explicit solutions for two models of degenerate parametric oscillators in the case of multi-parameter squeezed input photons. The corresponding photon statistics and Wigner's function are also derived in coordinate representation. Their time evolution is investigated in detail. The unitary transformation and an extension of the squeeze/evolution operator are briefly discussed.

Non-Classical Formulation of Photon Energy for the Degenerate Parametric Oscillator

We have analyzed photon statistics and quadrature squeezing of the signal mode produced by one-mode subharmonic generator. It is found that the mean photon energy of the signal mode is twice of a twin signal light beam. And the energy fluctuations of the two signal light beams are four times that of one of the signal light beams. We have also shown that the photon energy quadrature squeezing of the output signal mode is exactly equal to that of the cavity signal mode. In light of this, the global quadrature squeezing of the signal mode is independent of the mean photon energy. On the basis of the above results, the quantum analysis of a one-mode sub-harmonic generating system should somehow be modified.