We propose two schemes to generate entanglement between a pair of mechanical oscillators using parametric amplification. In contrast to existing parametric drive-based protocols, both schemes operate in the steady-state. Using a detuned parametric drive to maintain equilibrium and to couple orthogonal quadratures, our approach can be viewed as a two-mode extension of previous proposals for parametric squeezing. We find that robust steady-state entanglement is possible for matched oscillators with well-controlled coupling. In addition, one of the proposed schemes is robust to differences in the damping rates of the two oscillators.
Recently it has been demonstrated that the combination of weak-continuous position detection with detuned parametric driving can lead to significant steady-state mechanical squeezing, far beyond the 3 dB limit normally associated with parametric driving. In this work, we show the close connection between this detuned scheme and quantum non-demolition (QND) measurement of a single mechanical quadrature. In particular, we show that applying an experimentally realistic detuned parametric drive to a cavity optomechanical system allows one to effectively realize a QND measurement despite being in t…
High-performance two-qubit gates have been reported with superconducting qubits coupled via a single-transmon coupler (STC). Most of them are implemented for qubits with a small detuning since reducing residual $ZZ$ coupling for highly detuned qubits by an STC is challenging. In terms of the frequency crowding and crosstalk, however, highly detuned qubits are desirable. Here, we numerically demonstrate a high-performance parametric gate for highly detuned fixed-frequency qubits using a recently proposed tunable coupler called a double-transmon coupler (DTC). Applying an ac flux pulse, we can p…
We consider ultracold atoms in a far detuned optical lattice orientated across a high-Q optical resonator. Applying an external driving laser to the atoms, which is 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 this oscillation threshold photons and atoms in the lowest band are dominantly created pairwise via stimulated emission with a strong suppression of competing s…
Cavity optomechanical phenomena, such as cooling, amplification or optomechanically induced transparency, emerge due to a strong imbalance in the dissipation rates of the parametrically coupled electromagnetic and mechanical resonators. Here we analyze the reversed dissipation regime where the mechanical energy relaxation rate exceeds the energy decay rate of the electromagnetic cavity. We demonstrate that this regime allows for mechanically-induced amplification (or cooling) of the electromagnetic mode. Gain, bandwidth, and added noise of this electromagnetic amplifier are derived and compare…
We propose a scheme for quantum estimation by means of parametric amplification in circuit Quantum Electrodynamics. The modulation of a SQUID interrupting a superconducting waveguide transforms an initial thermal two-mode squeezed state in such a way that the new state is sensitive to the features of the parametric amplifier. We find the optimal initial parameters which maximize the Quantum Fisher Information. In order to achieve a large number of independent measurements we propose to use an array of non-interacting resonators. We show that the combination of both large QFI and large number o…
Linear parametric amplification is a key operation in information processing. Our interest here is quantum-limited parametric amplification, $i.e.$, amplification of quantum signals while adding the minimum amount of noise allowed by quantum mechanics, which is essential for any viable implementation of quantum information processing. We describe parametric amplifiers based on the dispersive nonlinearity of Josephson junctions driven with appropriate tones playing the role of pumps. We discuss two defining characteristics in the architecture of these amplifiers: the number of modes occupied by…
A Kerr-nonlinear parametric oscillator (KPO) is one of the promising devices to realize qubits for universal quantum computing. The KPO can stabilize two coherent states with opposite phases, yielding a quantum superposition called a Schrödinger cat state. Universal quantum computing with KPOs requires three kinds of quantum gates: $R_z, R_x$, and $R_{zz}$ gates. We theoretically propose a two-qubit gate $R_{zz}$ for highly detuned KPOs. In the proposed scheme, we add another two-photon drive for the first KPO. This leads to the $R_{zz}$ gate based on the driving of the second KPO depending on…
Josephson meta-materials have recently emerged as very promising platform for superconducting quantum science and technologies. Their distinguishing potential resides in ability to engineer them at sub-wavelength scales, which allows complete control over wave dispersion and nonlinear interaction. In this article we report a versatile Josephson transmission line with strong third order nonlinearity which can be tuned from positive to negative values, and suppressed second order non linearity. As an initial implementation of this multipurpose meta-material, we operate it to demonstrate reversed…
This short and opinionated review starts with a concept of quantum signals at microwave frequencies and focuses on the principle of linear parametric amplification. The amplification process arises from the dispersive nonlinearity of Josephson junctions driven with appropriate tones. We discuss two defining characteristics of these amplifiers: the number of modes receiving the signal, idler and pump waves and the number of independent ports through which these waves enter into the circuit.
We carry out analysis of optomechanical system formed by moveable mirror of Fabry-Perot cavity pumped by detuned laser. Optical spring arising from detuned pump creates in the system several eigen modes which could be treated as high-Q oscillators. Modulation of laser power results in parametric modulation of oscillators spring constants thus allowing to squeeze noise in quadratures of the modes. Evidence of the squeezing could be found in the light reflected from the cavity.
Nonlinear forces allow motion of a mechanical oscillator to be squeezed below the zero-point motion. Of existing methods, mechanical parametric amplification is relatively accessible, but previously thought to be limited to 3dB of squeezing in the steady state. We consider the effect of applying continuous weak measurement and feedback to this system. If the parametric drive is optimally detuned from resonance, correlations between the quadratures of motion allow unlimited steady-state squeezing. Compared to back-action evasion, we demonstrate that the measurement strength, temperature and eff…
We derive analytic solutions for Heisenberg evolution under the trilinear parametric Hamiltonian which are correct to second order in the interaction strength but are valid for all pump amplitudes. The solutions allow pump depletion effects to be incorporated in the description of parametric amplification in experimentally relevant scenarios and the resulting new phenomena to be rigorously described.
Breaking the symmetry in a coupled wave system can result in unusual amplification behavior. In the case of difference parametric amplification the resonant pump frequency is equal to the difference, instead of the sum, frequency of the normal modes. We show that sign reversal in the symmetry relation of parametric coupling give rise to difference parametric amplification as a dual of optical parametric amplification. For optical systems, our result can potentially be used for efficient XUV amplification.
We present an impedance engineered Josephson parametric amplifier capable of providing bandwidth beyond the traditional gain-bandwidth product. We achieve this by introducing a positive linear slope in the imaginary component of the input impedance seen by the Josephson oscillator using a $λ/2$ transformer. Our theoretical model predicts an extremely flat gain profile with a bandwidth enhancement proportional to the square root of amplitude gain. We experimentally demonstrate a nearly flat 20 dB gain over a 640 MHz band, along with a mean 1-dB compression point of -110 dBm and near quantum-lim…
The low-noise amplification of weak microwave signals is crucial for countless protocols in quantum information processing. Quantum mechanics sets an ultimate lower limit of half a photon to the added input noise for phase-preserving amplification of narrowband signals, also known as the standard quantum limit (SQL). This limit, which is equivalent to a maximum quantum efficiency of $0.5$, can be overcome by employing nondegenerate parametric amplification of broadband signals. We show that, in principle, a maximum quantum efficiency of 1 can be reached. Experimentally, we find a quantum effic…
We report the parametric amplification of a microwave signal in a Kerr medium formed from superconducting qubits. Two mutually coupled flux qubits, embedded in the current antinode of a superconducting coplanar waveguide resonator, are used as a nonlinear element. Shared Josephson junctions provide the qubit-resonator coupling, resulting in a device with a measured gain of about 20 dB. We argue, that this arrangement represents a unit cell which can be straightforwardly extended to a quasi one-dimensional quantum metamaterial with a large tunable Kerr nonlinearity.
We present a full operator approach to treatment of the cross-Kerr interaction combined with parametric amplification. It is shown that this problem can be exactly integrated using the method of higher-order operators. While the initial basis is infinite-dimensional, an orthogonal transformation can reduce the problem exactly into a six-dimensional basis which can be integrated conveniently.
Spin-spin interactions generated by a detuned cavity are a standard mechanism for generating highly entangled spin squeezed states. We show here how introducing a weak detuned parametric (two-photon) drive on the cavity provides a powerful means for controlling the form of the induced interactions. Without a drive, the induced interactions cannot generate Heisenberg-limited spin squeezing, but a weak optimized drive gives rise to an ideal two-axis twist interaction and Heisenberg-limited squeezing. Parametric driving is also advantageous in regimes limited by dissipation, and enables an altern…
It is demonstrated that the nature of optical parametric amplification is a quantum phenomenon. The system Lagrangian can be constructed by the path integral of coherent state. The equations of motion for photon operators are indeed the Euler-Lagrange equations of a Lagrangian. The quantum state evolution equation can also be obtained without resorting to quantum Hamiltonian or Lagrangian. Starting with classical Newton equation, quantum transition amplitude of the system can be educed by surface integral.
There is a serious flaw in the proposal [arXiv:1603.06857] for the achievement of unity efficiency in SPDC. This is a replacement due to mistakes in the table of probabilities. Numbers have been corrected.
We investigate the multiparticle quantum superposition and the persistence of multipartite entanglement of the quantum superposition generated by the quantum injected high-gain optical parametric amplification of a single photon. The physical configuration based on the optimal universal quantum cloning has been adopted to investigate how the entanglement and the quantum coherence of the system persists for large values of the nonlinear parametric gain g.
We show the possibility of noiseless amplification of an optical image in cavities containing a parametric oscillator. We consider a confocal ring cavity with plane mirrors and compare with the case of planar cavity. In the latter case the system operates with severe spatial limitations, while in the confocal case, there is the possibility of preserving the signal-to-noise ratio while amplifying uniformely the entire image.
We theoretically and experimentally demonstrate coherence phenomena in optical parametric amplification inside a cavity. The mode splitting in transmission spectra of phase-sensitive optical parametric amplifier is observed. Especially, we show a very narrow dip and peak, which are the shape of $δ$ function, appear in the transmission profile. The origin of the coherence phenomenon in this system is the interference between the harmonic pump field and the subharmonic seed field in cooperation with dissipation of the cavity.
We study experimentally parametric amplification in the continuous regime using a transverse-degenerate type-II Optical Parametric Oscillator operated below threshold. We demonstrate that this device is able to amplify either in the phase insensitive or phase sensitive way first a single mode beam, then a multimode image. Furthermore the total intensities of the amplified image projected on the signal and idler polarizations are shown to be correlated at the quantum level.
Phase-sensitive optical parametric amplification of squeezed states helps to overcome detection loss and noise and thus increase the robustness of sub-shot-noise sensing. Because such techniques, e.g., imaging and spectroscopy, operate with multimode light, multimode amplification is required. Here we find the optimal methods for multimode phase-sensitive amplification and verify them in an experiment where a pumped second-order nonlinear crystal is seeded with a Gaussian coherent beam. Phase-sensitive amplification is obtained by tightly focusing the seed into the crystal, rather than seeding…
Stimulated emission of two photons is observed experimentally in the parametric amplification process and is compared to a three-photon interference scheme. We find that the underlying physics of stimulated emission is simply the constructive interference due to photon indistinguishability. So the observed signal enhancement upon the input of photons is a result of multi-photon interference of the input photons and the otherwise spontaneously emitted photon from the amplifier.
In the Schroedinger picture, we find explicit solutions for two models of degenerate parametric oscillators in the case of multiparameter 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 is briefly discussed too.
Polarization squeezing is shown to occur in non-degenerate parametric amplification of coherent light and the degree of squeezing at interaction time $T$ can be as large as $1-e^{-2T}$. This gives $86.4\%$ polarization squeezing for $T=1$ and $98.2\%$ for $T=2$. One simple case when this occurs is on taking initially plane polarized light having equal amplitudes in the two modes that finally has equal intensities of two circular polarizations. This suggest the experimental settings of parameters to achieve this extent of polarization squeezing in coherent light.
In the framework of the paraxial and of the slowly varying envelope approximations, with reference to a normally dispersive medium or to vacuum, the electromagnetic field is given as a continuous quantum superposition of non-dispersive and non-diffracting wave-packets (namely X-waves). Entangled states as pairs of elementary excitations traveling at (approximately) the same velocity are found in optical parametric amplification.
We propose how to achieve strong phonon blockade (PB) in a hybrid spin-mechanical system in the weak-coupling regime. We demonstrate the implementation of magnetically-induced two-phonon interactions between a mechanical cantilever resonator and an embedded nitrogen-vacancy (NV) center, which, combined with parametric amplification of the mechanical motion, produces significantly enlarged anharmonicity in the eigenenergy spectrum. In the weak-driving regime, we show that strong PB appears in the hybrid system along with a large mean phonon number, even in the presence of strong mechanical diss…
The Hanbury Brown Twiss (HBT) interferometer was proposed to observe intensity correlations of starlight to measure a star's angular diameter. As the intensity of light that reaches the detector from a star is very weak, one cannot usually get a workable signal-to-noise ratio. We propose an improved HBT interferometric scheme introducing optical parametric amplifiers into the system, to amplify the correlation signal, which is used to calculate the angular diameter. With the use of optical parametric amplifiers, the signal-to-noise ratio can be increased up to 400 percent.
We propose a method for the fast generation of nonclassical ground states of the Rabi model in the ultrastrong and deep-strong coupling regimes via the shortcuts-to-adiabatic (STA) dynamics. The time-dependent quantum Rabi model is simulated by applying parametric amplification to the Jaynes-Cummings model. Using experimentally feasible parametric drive, this STA protocol can generate large-size Schrödinger cat states, through a process that is 10 times faster compared to adiabatic protocols. Such fast evolution increases the robustness of our protocol against dissipation. Our method enables o…
High-gain optical parametric amplification is an important nonlinear process used both as a source of coherent infrared light and as a source of nonclassical light. In this work, we experimentally demonstrate an approach to optical parametric amplification that enables extremely large parametric gains with low energy requirements. In conventional nonlinear media driven by femtosecond pulses, multiple dispersion orders limit the effective interaction length available for parametric amplification. Here, we use the dispersion engineering available in periodically poled thin-film lithium niobate n…
Second harmonic generation and optical parametric amplification in negative-index metamaterials (NIMs) are studied. The opposite directions of the wave vector and the Poynting vector in NIMs results in a "backward" phase-matching condition, causing significant changes in the Manley-Rowe relations and spatial distributions of the coupled field intensities. It is shown that absorption in NIMs can be compensated by backward optical parametric amplification. The possibility of distributed-feedback parametric oscillation with no cavity has been demonstrated. The feasibility of the generation of ent…
We develop a model for parametric amplification, based on a density matrix approach, which naturally accounts for the peculiarities arising for matter waves: significant depletion and explicit time-dependence of the source state population, long interaction times, and spatial dynamics of the amplified modes. We apply our model to explain the details in an experimental study on twin-atom beam emission from a one-dimensional degenerate Bose gas.
We propose an optical scheme, employing optical parametric down-converters interlaced with nonlinear sign gates (NSGs), that completely converts an $n$-photon Fock-state pump to $n$ signal-idler photon pairs when the down-converters' crystal lengths are chosen appropriately. The proof of this assertion relies on amplitude amplification, analogous to that employed in Grover search, applied to the full quantum dynamics of single-mode parametric down-conversion. When we require that all Grover iterations use the same crystal, and account for potential experimental limitations on crystal-length pr…
We present novel experimental results demonstrating the coherence properties of the nonlinear emission from semiconductor microcavities in the strong coupling regime, recently interpreted by parametric polariton four-wave mixing. We use a geometry corresponding to degenerate four-wave mixing. In addition to the predicted threshold dependence of the emission on the pump power and spectral blue shift, we observe a phase dependence of the amplification which is a signature of a coherent polariton wave mixing process.
Parametric Amplification of Spin-Motion Coupling in Three-Dimensional Trapped-Ion Crystals
Institute of Science and Technology Austria（奥地利科学技术研究所）；Department of Physics, Boston University（波士顿大学物理系）；Perimeter Institute for Theoretical Physics（理论物理 perimeter 研究所）；JILA, National Institute of Standards and Technology（美国国家标准与技术研究院 JILA）；Department of Physics, University of Colorado（科罗拉多大学物理系）；Center for Theory of Quantum Matter, University of Colorado（科罗拉多大学量子物质理论中心）；National Institute of Standards and Technology（美国国家标准与技术研究院）；Department of Physics, Indian Institute of Technology Madras（印度马德拉斯理工学院物理系）；Center for Quantum Information, Communication and Computing, Indian Institute of Technology Madras（印度马德拉斯理工学院量子信息、通信与计算中心）
In the last few years, several groups have proposed and developed their own platforms demonstrating quantum-limited linear parametric amplification, with evident applications in quantum information and computation, electrical and optical metrology, radio astronomy and basic physics concerning axion detection. Here we propose a short review on the physics behind parametric amplification via metamaterials composed by coplanar wave-guides embedding several Josephson junctions. We present and compare different schemes that exploit the nonlinearity of the Josephson current-phase relation to mix the…
Quantum tomography is the standard method of reconstructing the Wigner function of quantum states of light by means of balanced homodyne detection. The reconstruction quality strongly depends on the photodetectors quantum efficiency and other losses in the measurement setup. In this article we analyse in detail a protocol of enhanced quantum tomography, proposed by Leonhardt and Paul in 1994, which allows one to reduce the degrading effect of detection losses. It is based on phase sensitive parametric amplification, with the phase of the amplified quadrature being scanned synchronously with th…
We consider the possibility of parametric amplification of a mechanical vibration mode of a nanowire due to its interaction with a Bose-Einstein condensate (BEC) of ultracold atoms. The magneto-mechanical coupling is mediated by the vibrationally modulated magnetic field around the current-carrying nanowire, which can induce atomic transitions between different hyperfine sublevels. We theoretically analyze the limitations arising from the fact that the spin inverted atomic medium which feeds the mechanical oscillation has a finite bandwidth in the range of the chemical potential of the condens…
We theoretically investigate the creation of squeezed states of a Bose-Einstein Condensate (BEC) trapped in a magnetic double well potential. The number or phase squeezed states are created by modulating the tunnel coupling between the two wells periodically with twice the Josephson frequency, i.e., through parametric amplification. Simulations are performed with the multi configurational Hartree method for bosons (MCTDHB). We employ optimal control theory to bring the condensate to a complete halt at a final time, thus creating a highly squeezed state (squeezing factor of 0.12, $ξ_S^2=-18$ dB…
Noise usually has an unwelcome influence on system performance. For instance, noise inevitably affects the low-frequency mechanical freedom in optomechanical experiments. However, we investigate here the beneficial effects of thermal noise on a basic optomechanical system with parametric instability. In a regime near parametric instability, it is found that thermal noise in the mechanical freedom can sustain long-term quasi-coherent oscillations when the system would otherwise remain in the equilibrium state. In an overlapping regime of parametric instability and bistability, intermittent swit…
We report on the use of parametric excitation to coherently manipulate the collective spin state of an atomic vapour at room temperature. Signatures of the parametric excitation are detected in the ground-state spin evolution. These include the excitation spectrum of the atomic coherences, which contains resonances at frequencies characteristic of the parametric process. The amplitudes of the signal quadratures show amplification and attenuation, and their noise distribution is characterized by a strong asymmetry, similarly to those observed in mechanical oscillators. The parametric excitation…
We study polarization squeezing of a light beam initially in the coherent state using the nonlinear interaction hamiltonian $ H=k\big(\hat a_{x}^{\dagger2}+{\hat a_{x}}^2\big)$. For the degree of polarization squeezing, we use a definition written in the final form by the authors and also used in earlier papers. We find that the polarization squeezing can be very high and the degree of polarization squeezing can be less than unity by a very small amount. We achieve the polarization squeezing even for very low beam intensity under some conditions involving phase angles in the polarization modes…
In this paper, the scheme of a force sensor is proposed which has been composed of a hybrid optomechanical cavity containing an interacting cigar-shaped Bose-Einstein condensate (BEC) where the \textit{s}-wave scattering frequency of the BEC atoms as well as the spring coefficient of the cavity moving end-mirror (the mechanical oscillator) are parametrically modulated. It is shown that in the red-detuned regime and under the so-called impedance-matching condition, the mechanical response of the system to the input signal is enhanced substantially, which leads to the amplification of the weak i…
The 'optical spring' results from dynamical back-action and can be used to improve the sensitivity of cavity-enhanced gravitational-wave detectors. The effect occurs if an oscillation of the cavity length results in an oscillation of the intra-cavity light power having such a phase relation that the light's radiation pressure force amplifies the oscillation of the cavity length. Here, we analyse a Michelson interferometer whose optical-spring cavity includes an additional optical-parametric amplifier with adjustable phase. We find that the phase of the parametric pump field is a versatile para…
We propose a scheme to enhance the single- and two-photon blockade effect significantly in a standard optomechanical system (OMS) via optical parametric amplification (OPA). The scheme does not rely on the strong single-photon optomechanical coupling and can eliminate the disadvantages of suppressing multi-photon excitation incompletely. Through analyzing the single-photon blockade (1PB) mechanism and optimizing the system parameters, we obtain a perfect 1PB with a high occupancy probability of single-photon excitation, which means that a high quality and efficient single-photon source can be …
Quantum battery optimized by parametric amplification
College of Physics and Electronic Engineering, Northwest Normal University（西北师范大学物理与电子工程学院）；Gansu Provincial Research Center for Basic Disciplines of Quantum Physics（甘肃省量子物理基础学科研究中心）；Key Laboratory of Quantum Theory and Applications of MoE, Lanzhou Center for Theoretical Physics, and Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University（兰州大学理论物理省部共建教育部重点实验室、兰州理论物理中心）；Lanzhou University（兰州大学）
