K. Kowalski, J. Rembielinski
The uncertainty relations for the position and momentum of a quantum particle on a circle are identified minimized by the corresponding coherent states. The sqeezed states in the case of the circular motion are introduced and discussed in the context of the uncertainty relations.
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It is shown by examples that the position uncertainty on a circle, proposed recently by Kowalski and Rembieliński [J. Phys. A 35 (2002) 1405] is not consistent with the state localization. We argue that the relevant uncertainties and uncertainty relations (UR's) on a circle are that based on the Gram-Robertson matrix. Several of these generalized UR's are displayed and related criterions for squeezed states are discussed.
In the preceding Comment (quant-ph/0209032) Trifonov disputes our recently proposed uncertainty relations for a quantum particle on a circle. He states that (i) the quantity $Δ^2(\hatϕ)$ introduced by us representing the uncertainty of the angle is not a proper measure of the position uncertainty and therefore the proposed inequalities can hardly be qualified as a relevant uncertainty relations on a circle; and that (ii) the most suitable uncertainty relations on a circle are those based on the Gram-Robertson matrix. It is shown by examples that both points are erroneous.
A discrete completeness relation and a continuous one with a positive measure are found for the photon-added squeezed vacuum states. Extension to the photon-added squeezed one-photon states is considered. Photon-added coherent states on a circle are introduced. Their normalization and unity resolution relation are given.
For a charged particle in a homogeneous magnetic field, we construct stationary squeezed states which are eigenfunctions of the Hamiltonian and the non-Hermitian operator $\hat{X}_Φ = \hat{X} \cos Φ+ \hat{Y} \sin Φ$, $\hat{X}$ and $\hat{Y}$ being the coordinates of the Larmor circle center and $Φ$ is a complex parameter. In the family of the squeezed states, the quantum uncertainty in the Larmor circle position is minimal. The wave functions of the squeezed states in the coordinate representation are found and their properties are discussed. Also, for arbitrary gauge of the vector potential we…
Mechanical Squeezed-Fock Qubit: Towards Quantum Weak-Force Sensing
本文提出利用参数驱动非线性机械振子中的 squeezed Fock 状态来增强机械量子比特的非线性与非谐振性，从而实现更灵敏的弱力传感。
We experimentally investigate a mechanical squeezed state realized in a parametrically-modulated membrane resonator embedded in an optical cavity. We demonstrate that a quantum characteristic of the squeezed dynamics can be revealed and quantified even in a moderately warm oscillator, through the analysis of motional sidebands. We provide a theoretical framework for quantitatively interpreting the observations and present an extended comparison with the experiment. A notable result is that the spectral shape of each motional sideband provides a clear signature of a quantum mechanical squeezed …
Mechanical Squeezed-Fock Gravimeter
提出利用杜芬振荡器驱动的机械压缩福克态量子比特重力梯度仪，通过压缩增强重力诱导跃迁率并保持质量标度，分析阻尼下信号放大与退相干速率的权衡。
A challenge of modern physics is to investigate the quantum behavior of a bulk material object, for instance a mechanical oscillator. We have earlier demonstrated that by coupling a mechanical oscillator to the energy levels of embedded rare-earth ion dopants, it is possible to prepare such a resonator in a low phonon number state. Here, we describe how to extend this protocol in order to prepare momentum- and position squeezed states, and we analyze how the obtainable degree of squeezing depends on the initial conditions and on the coupling of the oscillator to its thermal environment.
We study an optomechanical system for the purpose of generating a nonclassical mechanical state when a mechanical oscillator is quadratically coupled to a single-mode cavity field driven by a squeezed optical field. The system corresponds to a regime where the optical dissipation dominates both the mechanical damping and the optomechanical coupling. We identify that multi-phonon processes emerge in the optomechanical system and show that a mechanical oscillator prepared in the ground state will evolve into an amplitude-squared squeezed vacuum state. The Wigner distribution of the steady state …
We use a reservoir engineering technique based on two-tone driving to generate and stabilize a quantum squeezed state of a micron-scale mechanical oscillator in a microwave optomechanical system. Using an independent backaction evading measurement to directly quantify the squeezing, we observe $4.7\pm0.9$ dB of squeezing below the zero-point level, surpassing the 3 dB limit of standard parametric squeezing techniques. Our measurements also reveal evidence for an additional mechanical parametric effect. The interplay between this effect and the optomechanical interaction enhances the amount of …
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 …
Squeezed Coherent States in Supersymmetric Quantum Mechanics with Position-Dependent Mass
本文在位置依赖质量的超对称量子力学框架下，利用变形代数结构推广产生与湮灭算子构建了压缩相干态，揭示了受变形参数和质量函数影响的非经典特征，为扩展相干态理论至复杂量子系统提供了途径。
Some predictions of quantum mechanics are in contrast with the macroscopic realm of everyday experience, in particular those originated by the Heisenberg uncertainty principle, encoded in the non-commutativity of some measurable operators. Nonetheless, in the last decade opto-mechanical experiments have actualized macroscopic mechanical oscillators exhibiting such non-classical properties. A key indicator is the asymmetry in the strength of the motional sidebands generated in an electromagnetic field that measures interferometrically the oscillator position. This asymmetry is a footprint of th…
Cavity optomechanics provides a unique platform for controlling micromechanical systems by means of optical fields that crosses the classical-quantum boundary to achieve solid foundations for quantum technologies. Currently, optomechanical resonators have become promising candidates for the development of precisely controlled nano-motors, ultrasensitive sensors and robust quantum information processors. For all these applications, a crucial requirement is to cool the mechanical resonators down to their quantum ground states. In this paper, we present a novel cooling scheme to further cool a mi…
Multimode Gaussian steady state engineering in optomechanical systems with a squeezed reservoir
本文研究了在多模式光学机械系统中利用单个 squeezed 库实现机械量子态耗散稳定化的理论协议，通过分析该协议在生成矩形图定义的机械簇态中的性能。
Quantum enhanced sensing is a powerful technique in which nonclassical states are used to improve the sensitivity of a measurement. For enhanced mechanical displacement sensing, squeezed states of light have been shown to reduce the photon counting noise that limits the measurement noise floor. It has long been predicted, however, that suppressing the noise floor with squeezed light should produce an unavoidable increase in radiation pressure noise that drives the mechanical system. Such nonclassical radiation pressure forces have thus far been hidden by insufficient measurement strengths and …
We propose an efficient scheme for generating spin-squeezed states at steady state in a spin-mechanical hybrid system, where an ensemble of SiV centers are coupled to a strongly damped nanomechanical resonator. We show that,there exists a collective steady state in the system, which is exactly formed by the collective spin states plus the zero excitation state of the mechanical mode. The generation of the steady spin-squeezed state is based on a dissipative quantum dynamical process in which the mechanical dissipation plays a positive role but without destroying the target state. We demonstrat…
Mechanical Squeezed Kerr Oscillator based on Tapered Ion Trap
本文提出在锥形离子阱中用单个离子实现机械压缩Kerr振子，通过阱几何导致的模式耦合产生可控Kerr非线性，实现非高斯量子门，并利用非线性实现运动压缩，且频率间隙增大有利于快速量子门。
Quantum fluctuations of the electromagnetic vacuum impose an observable quantum limit to the lowest temperatures that can be reached with conventional laser cooling techniques. As laser cooling experiments continue to bring massive mechanical systems to unprecedented temperatures, this quantum limit takes on increasingly greater practical importance in the laboratory. Fortunately, vacuum fluctuations are not immutable, and can be "squeezed" through the generation of entangled photon pairs. Here we propose and experimentally demonstrate that squeezed light can be used to sideband cool the motio…
One examines putative corrections to the Bell operator due to the noncommutativity in the phase-space. Starting from a Gaussian squeezed envelop whose time evolution is driven by commutative (standard quantum mechanics) and noncommutative dynamics respectively, one concludes that, although the time evolving covariance matrix in the noncommutative case is different from the standard case, the squeezing parameter dominates and there are no noticeable noncommutative corrections to the Bell operator. This indicates that, at least for squeezed states, the privileged states to test Bell correlations…
Quantum mechanics on a circle with a finite number of α-uniformly distributed points
本文研究了具有有限个α-均匀分布点的圆上的量子力学，定义了角度和平移算符，利用离散角度表示构造了两类离散角动量算符和厄米哈密顿量，并计算了自由粒子的能级。
The interference pattern in electron double-slit diffraction is a hallmark of quantum mechanics. A long standing question for stochastic electrodynamics (SED) is whether or not it is capable of reproducing such effects, as interference is a manifestation of quantum coherence. In this study, we use excited harmonic oscillators to directly test this quantum feature in SED. We use two counter-propagating dichromatic laser pulses to promote a ground-state harmonic oscillator to a squeezed Schrödinger cat state. Upon recombination of the two well-separated wavepackets, an interference pattern emerg…
What Bohmian mechanic says about arrival times of 1D vacuum squeezed states
本文基于玻姆力学形式体系，利用薛定谔表示下的一维真空压缩态波函数与辛群幺正表示，计算了量子粒子的到达时间概率分布，并得出区别于标准量子力学的闭合表达式。
We propose and analyse a feasible experimental scheme for a quantum force sensor based on the elimination of back-action noise through coherent quantum noise cancellation (CQNC) in a hybrid atom-cavity optomechanical setup assisted with squeezed vacuum injection. The force detector, which allows for a continuous, broad-band detection of weak forces well below the standard quantum limit (SQL), is formed by a single optical cavity simultaneously coupled to a mechanical oscillator and to an ensemble of ultracold atoms. The latter acts as a negative-mass oscillator so that atomic noise cancels exa…
In these notes, we discuss squeezed states using the elementary quantum language based on one-dimensional Schrödinger equation. No operators are used. The language of quantum optics is mentioned only for a hint to solve a differential equation. Sections II and III (squeezed vacuum and pure squeezed states) can be used in a standard undergraduate course on quantum mechanics after discussion of a harmonic oscillator. Section IV discusses density matrix of mixed squeezed states in $x$-representation. These notes present explicit (therefore somewhat lengthy) step-by-step derivations. These notes a…
We report on a hitherto unexplored application of squeezed light: for quantum-enhancement of mechanical transduction sensitivity in microcavity optomechanics. Using a toroidal silica microcavity, we experimentally demonstrate measurement of the transduced phase modulation signal with a sensitivity $-0.72(\pm 0.01)$\,dB below the shot noise level. This is achieved for resonant probing in the highly under-coupled regime, by preparing the probe in a weak coherent state with phase squeezed vacuum states at sideband frequencies.
Creation of Two-Mode Squeezed States in Atomic Mechanical Oscillators
本研究以二维光晶格中的原子为量子寄存器，通过频率突变产生单模压缩态、再经相对相位受控投影制备双模压缩态，经纠缠导引判据和Fock态分析验证，可应用于量子传感等领域。
Laser cooling is a fundamental technique used in primary atomic frequency standards, quantum computers, quantum condensed matter physics and tests of fundamental physics, among other areas. It has been known since the early 1990s that laser cooling can, in principle, be improved by using squeezed light as an electromagnetic reservoir; while quantum feedback control using a squeezed light probe is also predicted to allow improved cooling. Here, we implement quantum feedback control of a micro-mechanical oscillator for the first time with a squeezed probe field. This allows quantum-enhanced feed…
We perform a reconstruction of the polarization sector of the density matrix of an intense polarization squeezed beam starting from a complete set of Stokes measurements. By using an appropriate quasidistribution, we map this onto the Poincare space providing a full quantum mechanical characterization of the measured polarization state.
Conditional Motional Squeezing of an Optomechanical Oscillator Approaching the Quantum Regime
本研究通过优化光机械参数与滤波技术，将50纳克悬臂制备至条件压缩态，最小方差较前低3个数量级，证明测量协议可有效制备宏观振荡器量子压缩态。
We report electromagnetically induced transparency using quantized fields in optomechanical systems. The weak probe field is a narrow band squeezed field. We present a homodyne detection of EIT in the output quantum field. We find that the EIT dip exists even though the photon number in the squeezed vacuum is at the single photon level. The EIT with quantized fields can be seen even at temperatures of the order of 100 mK paving the way for using optomechanical systems as memory elements.
Broadband squeezed light field by magnetostriction in an opto-magnomechanical
本文提出一种在光-磁-力学系统中利用磁致伸缩效应产生宽带压缩光场的新机制，该压缩光场对温度鲁棒，可用于量子精密测量等应用。
We construct families of squeezed quantum states on an interval (depending on boundary conditions, we interpret the interval as a circle or as the infinite square potential well) and obtain estimates of position and momentum dispersions for these states. A particular attention is paid to the possibility of proper localization of a particle in nanoscale space domains. One of the constructed family of squeezed states is based on the theta function. It is a generalization of the known coherent and squeezed states on the circle. Also we construct a family of squeezed states based on truncated Gaus…
We discuss some basic tools for an analysis of one-dimensionalquantum systems defined on a cyclic coordinate space. The basic features of the generalized coherent states, the complexifier coherent states are reviewed. These states are then used to define the corresponding (quasi)densities in phase space. The properties of these generalized Husimi distributions are discussed, in particular their zeros.Furthermore, the use of the complexifier coherent states for a semiclassical analysis is demonstrated by deriving a semiclassical coherent state propagator in phase space.
To gain an advantage, quantum technologies utilize phenomena particular to quantum mechanics. Two such phenomena are squeezing and entanglement. Having generated states that exhibit these features, verification of their generation with local measurements can be a difficult process. Here we consider the states that are generated using the two-qudit single-axis squeezing Hamiltonian, that not only produces entangled two-qudit squeezed states but also results in various forms of interesting entangled states. We show how one can use local measurements to both efficiently verify and directly estima…
Fast quantum squeezing of a nanomechanical oscillator with an inverted potential
本文提出利用光学倒置势阱对室温真空光悬浮二氧化硅纳米颗粒的摆动模式进行指数加速压缩，在250纳秒内实现低于真空涨落11 dB的量子压缩，为机械运动量子压缩和量子增强传感提供新范式。
We investigate quantum-squeezing-enhanced weak-force sensing via a nonlinear optomechanical resonator containing a movable mechanical mirror and an optical parametric amplifier (OPA). Herein, we determined that tuning the OPA parameters can considerably suppress quantum noise and substantially enhance force sensitivity, enabling the device to extensively surpass the standard quantum limit. This indicates that under realistic experimental conditions, we can achieve ultrahigh-precision quantum force sensing by harnessing nonlinear optomechanical devices.
We rederive uncertainty relations for the angular position and momentum of a particle on a circle by employing the exponential of the angle instead of the angle itself, which leads to circular variance as a natural measure of resolution. Intelligent states minimizing the uncertainty product under the constraint of a given uncertainty in angle or in angular momentum turn out to be given by Mathieu wave functions. We also discuss a number of physically feasible approximations to these optimal states. The theory is applied to the orbital angular momentum of a beam of photons and verified in an ex…
We develop a theory of optomechanical cooling with a squeezed input light field. We show that Stokes heating transitions can be \emph{fully} suppressed when the driving field is squeezed below the vacuum noise level at an appropriately selected squeezing phase and for a finite amount of squeezing. The quantum backaction limit to laser cooling can be therefore moved down to zero and the resulting final temperature is then solely determined by the ratio between the thermal phonon number and the optomechanical cooperativity parameter, independently of the actual values of the cavity linewidth and…
Optimized mechanical quadrature squeezing beyond the 3-dB limit via a gradient-descent algorithm
本文提出利用梯度下降算法在腔光力学系统中寻找最优腔场驱动脉冲，实现了超越3-dB稳态限制的机械正交压缩，并在热声子占据数为100时于一个振荡周期内超快产生压缩，推动了最优量子控制的应用。
Enhancing the quantum entanglement and EPR steering of a coupled optomechanical system with a squeezed vacuum field
本文研究了在考虑机械模式引起的失谐位移时，利用弱压缩真空场增强耦合光机械系统的量子纠缠与EPR导引，发现参考相位比压缩参数起更关键的作用。
The q-deformed coherent states for a quantum particle on a circle are introduced and their properties investigated.
Preparation of conditionally-squeezed states in qubit-oscillator systems
本文提出在量子谐振子系统中制备条件性压缩态的协议，并探讨其在量子纠错码中的应用及退相干下的鲁棒性。
Many non-classicality indicators are used to measure quantum effects of different systems. Kenfack's and Sadeghi's non-classicality indicators are introduced in terms of the amount of Wigner function's negativities and interferences in phase space quantum mechanics, respectively. They are, effectively, applied for some real distribution functions. In this paper, we compared these non-classicality indicators for the entangled state of photonic number states in the Wigner, Husimi and Rivier representations. It is shown that for a two-level entangled state, Sadeghi's indicator has more benefits w…
A squeezed mechanical oscillator with milli-second quantum decoherence
本文提出低退相干、大耦合的超导电路光机械平台，实现毫秒级量子态寿命、93%保真度基态制备和−2.7 dB机械压缩。
We address local quantum estimation of bilinear Hamiltonians probed by Gaussian states. We evaluate the relevant quantum Fisher information (QFI) and derive the ultimate bound on precision. Upon maximizing the QFI we found that single- and two-mode squeezed vacuum represent an optimal and universal class of probe states, achieving the so-called Heisenberg limit to precision in terms of the overall energy of the probe. We explicitly obtain the optimal observable based on the symmetric logarithmic derivative and also found that homodyne detection assisted by Bayesian analysis may achieve estimat…
Optomechanical cooling with simultaneous intracavity and extracavity squeezed light
本文提出利用腔内简并光学参量放大与腔外压缩光驱动的联合压缩方案，通过量子干涉抑制斯托克斯加热并增强反斯托克斯冷却，在深度未分辨边带条件下实现机械振子高效基态冷却，显著提升冷却速率并降低耦合强度要求。
This study is aimed at investigating the feasibility of generating quantum entanglement between macroscopic mechanical mirrors in optomechanical systems while under continuous measurement and feedback control. We carefully derive a covariance matrix for mechanical mirrors in a steady state, employing the Kalman filtering problem with an assumed dominant cavity photon dissipation, such that the common and differential modes of the mirrors are squeezed by the action of measuring the output light beams. We demonstrate that entanglement between the mechanical mirrors is generated when the states o…
Quantum ground-state cooling of macroscopic mechanical resonators is of essential importance to both fundamental physics and applied science. Conventional method of laser cooling is limited by the quantum backaction, which requires mechanical sideband resolved in order to cool to ground state. This work presents an idea to break the quantum backaction limit by engineering intracavity optical squeezing. It gives rise to quantum interference for all the dissipation channels, and under certain circumstances can totally remove the influence of the cavity dissipation and the resultant quantum backa…
In this paper, we show the relation between $sp(4,\mathbb{R})$, the Lie algebra of the symplectic group, and the elements of $Sp(4,\mathbb{R})$. We use this result to obtain some special cases of symplectic matrices relevant to the study of squeezed states. In this regard, we provide some applications in quantum mechanics and analyze the squeezed polymer states obtained from the polymer representation of the symplectic group. Remarkably, the polymer's dispersions are the same as those obtained for the squeezed states in the usual representation.
The disputed question of uncertainty relations (UR) on a circle is regarded as a particular element of a more general problem which refers to the quantum description of angular observables $L_z$ and $ϕ$. The improvised $L_z-ϕ$ UR are found to be affected by unsourmontable shortcomings. Also in contradiction with a largely accepted belief it is proved that the usual procedures of quantum mechanics are accurately applicable for the $L_z-ϕ$ pair. The applicability regards both the known circular motions and the less known non-circular rotational motions. The presented facts contribute as argument…
We introduce a high-dimensional quantum encoding based on coherent mode-dependent single-photon subtraction from multimode squeezed states. This encoding can be seen as a generalization to the case of non-zero squeezing of the standard single-photon multi-rail encoding. The advantage is that the presence of squeezing enables the use of common tools in continuous-variable quantum processing, which in turn allows to show that arbitrary $d$-level quantum states can be generated and detected via simply tuning the classical fields that gates the photon-subtraction scheme. Therefore, the scheme is s…
Impulse measurements enhanced with squeezed readout light
研究了压缩光如何降低机械探测器冲量测量中的量子噪声至标准量子极限以下，发现频率相关压缩优于频率无关压缩，并揭示了压缩效益受系统机械特性限制的基本极限。
Understanding Squeezed States of Light Through Wigner's Phase-Space
本文利用维格纳相空间分布函数，从经典到量子力学的转变出发，探讨了相空间中的相干态、单模和双模压缩态及压缩真空，并指出双模压缩态自然产生纠缠，同时涉及耦合谐振子和对称群的应用。
We describe a six-parameter family of the minimum-uncertainty squeezed states for the harmonic oscillator in nonrelativistic quantum mechanics. They are derived by the action of corresponding maximal kinematical invariance group on the standard ground state solution. We show that the product of the variances attains the required minimum value 1/4 only at the instances that one variance is a minimum and the other is a maximum, when the squeezing of one of the variances occurs. The generalized coherent states are explicitly constructed and their Wigner function is studied. The overlap coefficien…
Hybrid Optomechanical Cooling with Kerr Magnons and Squeezed Vacuum
该研究提出基于含YIG球的混合光机械系统的冷却策略，利用克尔磁振子和压缩真空实现深度未分辨边带区的高效冷却，性能优于现有方案。
In this note we establish a relation between two exactly-solvable problems on circle, namely singular Coulomb and singular oscillator systems.
Transfer of quantum states and stationary quantum correlations in a hybrid optomechanical network
本研究探讨了通过公共机械振子耦合的混合光机械网络中量子态传输与稳态同步效应，发现可实现高保真度量子态传输，且同步与纠缠相互关联，结果可应用于量子信息与计量学。
Single quantum emitters like atoms are well-known as non-classical light sources which can produce photons one by one at given times, with reduced intensity noise. However, the light field emitted by a single atom can exhibit much richer dynamics. A prominent example is the predicted ability for a single atom to produce quadrature-squeezed light, with sub-shot-noise amplitude or phase fluctuations. It has long been foreseen, though, that such squeezing would be "at least an order of magnitude more difficult" to observe than the emission of single photons. Squeezed beams have been generated usi…
Einstein-Podolski-Rosen steering is a form of quantum correlation exhibiting an intrinsic asymmetry between two entangled systems. In this paper, we propose a scheme for examining dynamical Gaussian quantum steering of two mixed mechanical modes. For this, we use two spatially separated optomechanical cavities fed by squeezed light. We work in the resolved sideband regime. Limiting to the adiabatic regime, we show that it is possible to generate dynamical Gaussian steering via a quantum fluctuations transfer from squeezed light to the mechanical modes. By an appropriate choice of the environme…
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