Julien Laurat, Thomas Coudreau, Claude Fabre
Type-II Optical Parametric Oscillators are efficient sources of quadrature squeezed or polarization squeezed light, intensity correlated beams, and entangled light. We review here the different levels of quantum correlations and entanglement that are reached in this device, and present some applications.
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The spectral-temporal correlation and the correlation time of a biphoton wavepacket generated in the process of parametric down-conversion (PDC), is of great importance for a broad range of quantum experiments. We utilise an integrated PDC source to generate biphotons with different types of spectral-temporal correlations and probe their respective correlation times. The outcomes confirms that the correlation time is independent of the coherence time of the pump light, and it is only determined by the waveguide length and its dispersion properties. Furthermore, we investigate the properties of…
The transversely confined propagating light modes of a nano-photonic optical waveguide or nanofiber can mediate effectively infinite-range forces. We show that for a linear chain of particles trapped within the waveguide's evanescent field, transverse illumination with a suitable set of laser frequencies should allow the implementation of a coupled-oscillator quantum simulator with time-dependent and widely controllable all-to-all interactions. At the example of the energy spectrum of oscillators with simulated Coulomb interactions we show that different effective coupling geometries can be em…
We study the quantum correlations induced by spontaneous parametric down-conversion (SPDC) of a frequency comb. We derive a theoretical method to find the output state corresponding to a pump with an arbitrary spectral profile. After applying it to the relevant example of a spectrally chirped pump, we run an optimization algorithm to numerically find the pump profiles maximizing some target functions. These include the number of independently squeezed modes and the variances of nullifiers defining cluster states used in many continuous-variable quantum information protocols. To assess the adva…
We study microwave response of a Josephson parametric oscillator consisting of a superconducting transmission-line resonator with an embedded dc-SQUID. The dc-SQUID allows to control the magnitude of a Kerr nonlinearity over the ranges where it is smaller or larger than the photon loss rate. Spectroscopy measurements reveal the change of the microwave response from a classical Duffing oscillator to a Kerr parametric oscillator in a single device. In the single-photon Kerr regime, we observe parametric oscillations with a well-defined phase of either $0$ or $π$, whose probability can be control…
In this article we explore the quantum properties of a degenerate optical parametric oscillator when it is tuned to the first family of transverse modes at the down converted frequency.
Time--dependent integrals of motion which are linear forms in position and momentum are discussed for Husimi parametric forced oscillator. Generalization of these integrals of motion for q--oscillator is presented. Squeezing and quadrature correlation phenomena are discussed on the base of Schrödinger uncertainty relation. The properties of the generalized correlated states, squeezed states, even and odd coherent states (the Schrödinger cat states) are reviewed. The relation of the constructed nonclassical states to representations of the symplectic symmetry group and finite symmetry groups is…
Glauber coherent states of quantum systems are reviewed. We construct the tomographic probability distributions of the oscillator states. The possibility to describe quantum states by tomographic probability distributions (tomograms) is presented on an example of coherent states of parametric oscillator. The integrals of motion linear in the position and momentum are used to explicitly obtain the tomogram evolution expressed in terms of trajectories of classical parametric oscillator
This work analyses the quantum properties of counter-propagating twin beams generated by a Mirrorless Optical Parametric Oscillator in the continuous variable regime. Despite the lack of the filtering effect of a cavity, we show that in the vicinity of its threshold it may generate high levels of narrowband squeezing and Einstein-Podolsky-Rosen (EPR) correlation, completely comparable to what can be obtained in standard optical parametric oscillators.
We consider preparation of nonclassical oscillatory states in a degenerate parametric oscillator combined with phase modulation. In this scheme intracavity oscillatory mode is excited by train of Gaussian laser pulses through degenerate down-conversion process and phase modulation element inserted in a cavity leads to anharmonicity of oscillatory mode. We demonstrate production of nonclassical oscillatory states with two-fold symmetry in phase-space including the superposition of Fock states and quantum localized states on the level of few excitation numbers and in over-transient dissipative r…
We study the dissipative dynamics of a single quantum harmonic oscillator subjected to a parametric driving with in an effective Hamiltonian approach. Using Liouville von Neumann approach, we show that the time evolution of a parametrically driven dissipative quantum oscillator has a strong connection with the classical damped Mathieu equation. Based on the numerical analysis of the Monodromy matrix, we demonstrate that the dynamical instability generated by the parametric driving are reduced by the effect of dissipation. Further, we obtain a closed relationship between the localization of the…
New families of time-dependent potentials related to the parametric oscillator are introduced. This is achieved by introducing some general time-dependent operators that factorize the appropriate constant of motion (quantum invariant) of the parametric oscillator, leading to new families of quantum invariants that are almost-isospectral to the initial one. Then, the respective time-dependent Hamiltonians are constructed, and the solutions of the Schrödinger equation are determined from the intertwining relationships and by finding the appropriate time-dependent complex-phases of the Lewis-Ries…
The effects of an intracavity photonic crystal in a multimode optical parametric oscillator are studied, with a special focus on quantum fluctuations. The capability to either stimulate or inhibit the spatial instability, lowering or increasing the parametric threshold, allows to control the intensity fluctuations and correlations. A significative quantum noise reduction and an increase of the range of squeezed quadratures are found above threshold where spatial Einstein-Podolsky-Rosen entanglement and inseparability are found.
We use a versatile model to evaluate the multipartite entanglement and the nonclassical light generation in optical parametric oscillators, exploring the differences between doubly and triply resonant cavity configurations. We demonstrate the entanglement of the pump mode with converted fields in both situations, and the fundamental differences of oscillators using parametric down conversion and four wave mixing processes as the intracavity amplification technique. The strong correlations involving the sidebands of the pump and converted fields gives the signatures of a rich dynamic of multipa…
We present a multimode treatment of the optical parametric oscillator, which is valid for both pulsed and continuous-wave pump fields. The two-time correlation functions of the output field are derived, and we apply the theory to analyze a scheme for heralded production of non-classical field states that may be subsequently stored in an atomic quantum memory.
We generate squeezed state of light at 860 nm with a monolithic optical parametric oscillator. The optical parametric oscillator consists of a periodically poled KTiOPO_4 crystal, both ends of which are spherically polished and mirror-coated. We achieve both phase matching and cavity resonance by controlling only the temperature of the crystal. We observe up to -8.0 dB of squeezing with the bandwidth of 142 MHz. Our technique makes it possible to drive many monolithic cavities simultaneously by a single laser. Hence our monolithic optical parametric oscillator is quite suitable to continuous-v…
We analyze the above-threshold behavior of a mirrorless parametric oscillator based on resonantly enhanced four wave mixing in a coherently driven dense atomic vapor. It is shown that, in the ideal limit, an arbitrary small flux of pump photons is sufficient to reach the oscillator threshold. We demonstrate that due to the large group-velocity delays associated with coherent media, an extremely narrow oscillator linewidth is possible, making a narrow-band source of non-classical radiation feasible.
The output field from a continuously driven linear parametric oscillator may exhibit considerably more squeezing than the intracavity field. Inspired by this fact, we explore the nonclassical features of the steady-state output field of a driven nonlinear Kerr parametric oscillator using a temporal wave packet mode description. Utilizing a new numerical method, we have access to the density matrix of arbitrary wave packet modes. Remarkably, we find that even though the steady-state cavity field is always characterized by a positive Wigner function, the output may exhibit Wigner negativity, dep…
Theory of Stationary Photon Emission from a Steadily Driven Parametric Oscillator Based on the Complex Spectral Analysis of the Heisenberg Equation
本文利用 Heisenberg 方程的复谱分析，揭示稳态驱动参量振荡器的复本征频率跨 Riemann 叶移动如何抑制指数增长并产生恒定光子通量。
A degenerate optical parametric oscillator network is proposed to solve the NP-hard problem of finding a ground state of the Ising model. The underlying operating mechanism originates from the bistable output phase of each oscillator and the inherent preference of the network in selecting oscillation modes with the minimum photon decay rate. Computational experiments are performed on all instances reducible to the NP-hard MAX-CUT problems on cubic graphs of order up to 20. The numerical results reasonably suggest the effectiveness of the proposed network.
We suggest a variant of the recently proposed experiment for the generation of a new kind of Schroedinger-cat states, using two coupled parametric down-converter nonlinear crystals [F. De Martini, Phys. Rev. Lett. 81, 2842 (1998)]. We study the parametric oscillator case and find that an entangled Schroedinger-cat type state of two cavities, whose mirrors are placed along the output beams of the nonlinear crystals, can be realized under suitable conditions.
For $N$-coupled generalized time-dependent oscillators, primary invariants and a generalized invariant are found in terms of classical solutions. Exact quantum motions satisfying the Heisenberg equation of motion are also found. For number states and coherent states of the generalized invariant, the uncertainties in positions and momenta are obtained.
The parametric ladder climbing (successive Landau-Zener-type transitions) and the quantum saturation of the threshold for the classical parametric autoresonance due to the zero point fluctuations at low temperatures are discussed. The probability for capture into the chirped parametric resonance is found by solving the Schrodinger equation in the energy basis and the associated resonant phase space dynamics is illustrated via the Wigner distribution. The numerical threshold for the efficient capture into the resonance is compared with the classical and quantum theories in different parameter r…
In this paper we study the nondegenerate optical parametric oscillator with injected signal, both analytically and numerically. We develop a perturbation approach which allows us to find approximate analytical solutions, starting from the full equations of motion in the positive P-representation. We demonstrate the regimes of validity of our approximations via comparison with the full stochastic results. We and that, with reasonably low levels of injected signal, the system allows for demonstrations of quantum entanglement and the Einstein-Podolsky-Rosen paradox. In contrast to the normal opti…
We report the first experimental observation of bright EPR beams produced by a type-II optical parametric oscillator operating above threshold at frequency degeneracy. The degenerate operation is obtained by introducing a birefringent plate inside the cavity resulting in phase locking. After filtering the pump noise, which plays a critical role, continuous-variable EPR correlations between the orthogonally polarized signal and idler beams are demonstrated.
We solve analytically master equations that describe a cavity filled with a kerr medium, taking into account the dissipation induced by the environment, and parametric down conversion processes. We use superoperator techniques.
Using a continuous-wave type-II optical parametric oscillator below threshold, we have demonstrated a novel source of heralded single-photons with high-fidelity. The generated state is characterized by homodyne detection and exhibits a 79% fidelity with a single-photon Fock state (91% after correction of detection loss). The low admixture of vacuum and the well-defined spatiotemporal mode are critical requirements for their subsequent use in quantum information processing.
We propose novel types of parametric oscillators generating both three-photon and four-photon bright light which are accessible for an experiment. The devices are based on the cascaded down-conversion processes and consist of second-order media inserted in two-resonant mode cavity. Discussion of dissipation and quantum features of the system are performed by the quantum-jump simulation method and concerns to the Wigner functions. The phase-space multistabilities and critical threshold behavior of three- and four-photon subharmonics are obtained.
We study how an optical parametric amplifier inside the cavity can affect the normal mode splitting behavior of the coupled movable mirror and the cavity field. We work in the resolved sideband regime. The spectra exhibit a double-peak structure as the parametric gain is increased. Moreover, for a fixed parametric gain, the double-peak structure of the spectrum is more pronounced with increasing the input laser power. We give results for mode splitting. The widths of the split lines are sensitive to parametric gain.
We derive the steady state solution of the Fokker-Planck equation that describes the dynamics of the nondegenerate optical parametric oscillator in the truncated Wigner representation of the density operator. The adiabatic limit of strong pump damping is assumed. This phase space image provides a clear view of the intracavity two-mode entangled state valid in all operating regimes of the OPO. A nongaussian distribution is obtained for the above threshold solution.
We present squeezed light that is robust against spatial mode mismatch (beam displacement, tilt, and beam-size difference), which is generated from a self-imaging optical parametric oscillator below the threshold. We investigate the quantum properties of the generated light when the oscillator is detuned from the ideal self-imaging condition for stable operation. We find that the generated light is more robust to mode mismatch than single-mode squeezed light having the same squeezing level, and it even outperforms the single-mode infinitely squeezed light as the strength of mode mismatch incre…
We study the squeezing as well as the statistical properties of the output radiation from a degenerate parametric oscillator coupled to a squeezed vacuum reservoir employing the stochastic differential equations associated with the normal ordering. It is found that the degree of squeezing of the output radiation is less than the corresponding cavity radiation. However, for output radiation the correlation of the quadrature operators evaluated at different times also exhibits squeezing, which is the reason for quenching of the overall noise in one of the quadrature components of the squeezing s…
Quantum dynamics of a driven parametric oscillator in a Kerr medium
本文研究Kerr介质中受时变力驱动的参量振荡器量子动力学，通过时间变换关联一般模型，并在小χ与小k两种区域结合解析与数值方法分析态的统计特性。
We study experimentally the spatial distribution of quantum noise in the twin beams produced by a type II Optical Parametric Oscillator operating in a confocal cavity above threshold. The measured intensity correlations are at the same time below the standard quantum limit and not uniformly distributed inside the beams. We show that this feature is an unambiguous evidence for the multimode and nonclassical character of the quantum state generated by the device.
We propose a scheme to modulate the entanglement between two oscillators separated in space via the squeezing cavity field generated by the optical parametric amplifier instead of injecting the squeezing field directly with the assistance of Coulomb interaction. We show that the Coulomb interaction between the oscillators is the essential reason for the existence of entanglement. Due to the gain of the optical parametric amplifier and the phase of the pump driving the optical parametric amplifier can simultaneously modulate the squeezing cavity field, the radiation pressure interaction between…
An experimental scheme is proposed for building massively multipartite entangled states using both the spatial and the frequency modes of an optical parametric oscillator. We provide analytical forms of the entangled states using the squeezed eigenmodes of Heisenberg equations, a.k.a. the nullifiers of the corresponding graph state. This scheme can generate, in parallel, several cluster states described by sparsely connected, bicolorable graph states, usable for one-way quantum computing. We indicate the experimentally accessible quantum graphs, depending on the squeezing parameter.
We present a theoretical treatment of conditional preparation of one-photon states from a continuous-wave non-degenerate optical parametric oscillator. We obtain an analytical expression for the output state Wigner function, and we maximize the one-photon state fidelity by varying the temporal mode function of the output state. We show that a higher production rate of high fidelity Fock states is obtained if we condition the outcome on dark intervals around trigger photo detection events.
Quantum-correlated twin beams were generated from a triply resonant optical parametric oscillator with an a-cut KTP crystal pumped by a frequency-doubled diode laser. A total output of 5.1 mW was obtained in the classical-to-nonclassical-light conversion system driven by a 50-mW diode laser at 1080 nm. Quantum noise reduction of 4.3 dB (63%) in the intensity difference between the twin beams was successfully observed at the detection frequency of 3 MHz.
We study a quantum system of coupled oscillators subject to a periodic excitation of its parameters. Using Floquet-Lyapunov theory we derive the linear integrals of motion of the system and relate their covariance matrix to that for the canonical observables. The operator integrals allows us to construct the intelligent (minimum uncertainty) states of the system and the corresponding photon-added states. We found explicit expressions for the wavefunction, Wigner function and covariance matrix of these states.
Mode-selective excitation in parametrically driven coupled quantum oscillators
通过调制两个耦合量子谐振子之间的耦合强度，实现频率选择性地激发任意简正模，同时保持其他模接近基态。
We perform the noise spectroscopy of a Josephson parametric oscillator (JPO) by implementing a microwave homodyne interferometric measurement scheme. We observe the fluctuations in the self-oscillating output field of the JPO for a long 10 s time interval in a single shot measurement and characterize the phase and amplitude noise. Furthermore, we investigate the effects of the pump strength on the output noise power spectra of the JPO. We found strong fluctuations in the phase with a $1/f^2$ characteristics in the phase noise power spectrum, which is suppressed by increasing the pump strength.
The evolution of a quantum oscillator, with periodically varying frequency and damping, is studied in the two cases of parametric resonance (PR) producing a limited, or unlimited stretching of the wave function. The different asymptotic behaviors of the energy distribution, show the non trivial interplay between PR and the initial quantum state. In the first case, the oscillator's mean energy tends asymptotically to a fully classical value, independent of the initial state, with vanishing relative quantum fluctuations. In the second case, the memory of the initial state persists over arbitrary…
Optomechanical simulation of a parametric oscillator
研究支持多模式的光机械器件，通过独立建模光学模式的相干驱动获得有效哈密顿量，提出在有损腔和平均场近似下的双色驱动方案，实现参数可光学控制的参量振荡器量子模拟。
We observed continuous-variable entanglement between the bright beams emitted above threshold by an ultrastable optical parametric oscillator, classically phase-locked at a frequency difference of 161.8273240(5) MHz. The amplitude-difference squeezing is -3 dB and the phase-sum one is -1.35 dB. Besides proving entanglement in a new physical system, the phase-locked OPO, such unprecedented frequency-difference stability paves the way for transferring entanglement between different optical frequencies and densely implementing continuous-variable quantum information in the frequency domain.
We measure the complete quantum state for six modes of the electromagnetic field produced by an optical parametric oscillator. The investigation involves the sideband of the intense pump, signal, and idler fields generated by stimulated parametric downconversion inside a triply resonant optical resonator. We develop a theoretical model to successfully interpret the experimental results. The model takes into account the coupling of the field modes to the phonon bath of the nonlinear crystal, clearly showing the roles of different physical effects in shaping the structure of the quantum correlat…
Quantum models for synchronously pumped type I optical parametric oscillators (SPOPO) are presented. The study of the dynamics of SPOPOs, which typically involves millions of coupled signal longitudinal modes, is significantly simplified when one considers the ?supermodes?, which are independent linear superpositions of all the signal modes diagonalizing the parametric interaction. In terms of these supermodes the SPOPO dynamics becomes that of about a hundred of independent, single mode degenerate OPOs, each of them being a squeezer. One derives a general expression for the squeezing spectrum…
We address the exploitation of an optical parametric oscillator (OPO) in the task of mitigating, at least partially, phase noise produced by phase diffusion. In particular, we analyze two scenarios where phase diffusion is typically present. The first one is the measurement of the phase of a noisy optical field, while the second involves a quantum estimation scheme of a phase shift imposed on a noisy probe. In both cases, we prove that an OPO may lead to a partial or full compensation of the noise.
Non-Gaussian continuous variable states play a central role both in the foundations of quantum theory and for emergent quantum technologies. In particular, "cat states", i.e., two-component macroscopic quantum superpositions, embody quantum coherence in an accessible way and can be harnessed for fundamental tests and quantum information tasks alike. Degenerate optical parametric oscillators can naturally produce single-mode cat states and thus represent a promising platform for their realization and harnessing. We show that a dissipative coupling between degenerate optical parametric oscillato…
I consider several interesting aspects of a new light source, a two-level atom, or N two-level atoms inside an Optical Parametric Oscillator. We find that in the weak driving limit, detection of a transmitted or fluorescent photon generates a highly entangled state of the atom and the cavity. This entanglement can be used with beamsplitters to create more complex quantum states and implement teleportation protocols. Also, one can store a single photon in the atoms, along the lines of recent slow and stopped light proposals and experiments.
Spectroscopic estimation of the photon number for superconducting Kerr parametric oscillators
本文提出通过将辅助量子比特耦合到超导Kerr参量振荡器并进行光谱测量，从而实现对KPO光子数的可行估计，为基于KPO的量子退火提供精确控制手段。
We propose and analyse 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 which can be up to one octave apart. The two downconverted modes, which exhibit the greater violations of the steering inequalities, can also be controlled to exhibit asymmetric steering in some regimes.
We calculate the quantum correlations existing among the three output fields (pump, signal, and idler) of a triply resonant non-degenerate 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 info…
Quantum computers with Kerr-nonlinear parametric oscillators (KPOs) have recently been proposed by the author and others. Quantum computation using KPOs is based on quantum adiabatic bifurcations of the KPOs, which lead to quantum superpositions of coherent states, such as Schrodinger cat states. Therefore, these quantum computers are referred to as "quantum bifurcation machines (QbMs)." QbMs can be used for qauntum adiabatic optimization and universal quantum computation. Superconducting circuits with Josephson junctions, Josephson parametric oscillators (JPOs) in particular, are promising fo…
We analyze the three-mode correlation properties of the electromagnetic field in a optical parametric oscillator below threshold. We employ a perturbative expansion of the Itô equations derived from the positive-P representation of the density matrix. Using the generalized Cauchy-Schwarz inequality, we investigate the genuine quantum nature of the triple correlations between the interacting fields, since in this case continuous variable entanglement is not detected by the van Loock-Furusawa criterion [Phys. Rev. A {\bf 67}, 052315 (2003)]. Although not being a necessary condition, these triple…
Quantum repeaters - fundamental building blocks for long-distance quantum communication - are based on the interaction between photons and quantum memories. The photons must fulfil stringent requirements on central frequency, spectral bandwidth and purity in order for this interaction to be efficient. We present a design scheme for monolithically integrated resonant photon-pair sources based on parametric down-conversion in nonlinear waveguides, which facilitate the generation of such photons. We investigate the impact of different design parameters on the performance of our source. The genera…
Intracavity and external third order correlations in the damped nondegenerate parametric oscillator are calculated for quantum mechanics and stochastic electrodynamics (SED), a semiclassical theory. The two theories yield greatly different results, with the correlations of quantum mechanics being cubic in the system's nonlinear coupling constant and those of SED being linear in the same constant. In particular, differences between the two theories are present in at least a mesoscopic regime. They also exist when realistic damping is included. Such differences illustrate distinctions between qu…
We study the temporal evolution of a coherent state under the action of a parametric oscillator and a nonlinear Kerr-like medium. We make use of the interaction picture representation and use an exact time evolution operator for the time independent part of the Hamiltonian. We approximate the interaction picture Hamiltonian in such a way as to make it a member of a Lie algebra. The corresponding time evolution operator behaves like a squeezing operator due to the temporal dependence of the oscillator's frequency. We analyze the probability amplitude and the auto correlation function for differ…
Modulating the frequency of a harmonic oscillator at nearly twice its natural frequency leads to amplification and self-oscillation. Above the oscillation threshold, the field settles into a coherent oscillating state with a well-defined phase of either $0$ or $π$. We demonstrate a quantum parametric oscillator operating at microwave frequencies and drive it into oscillating states containing only a few photons. The small number of photons present in the system and the coherent nature of the nonlinearity prevents the environment from learning the randomly chosen phase of the oscillator. This a…
We propose a protocol for quantum state tomography of nonclassical states in optomechanical systems. Using a parametric drive, the procedure overcomes the challenges of weak optomechanical coupling, poor detection efficiency, and thermal noise to enable high efficiency homodyne measurement. Our analysis is based on the analytic description of the generalized measurement that is performed when optomechanical position measurement competes with thermal noise and a parametric drive. The proposed experimental procedure is numerically simulated in realistic parameter regimes, which allows us to show…
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.
The influence of the phase fluctuation of the pump laser on the phase-correlation between the signal and idler modes of the output fields from as non-degenerate optical parametric oscillator operating above oscillation threshold was experimentally investigated. The noise spectra of the intensity-difference and the phase-sum of the entangled optical beams were measured with a pair of unbalanced fiber Match-Zehnder interferometers specifically designed. The experimental results proved the previously theoretical prediction and are in reasonable agreement with the calculation based on semiclassica…
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