Ole Steuernagel
Two mode squeezed states can be used to achieve Heisenberg limit scaling in interferometry: a phase shift of $δϕ\approx 2.76 / < N >$ can be resolved. The proposed scheme relies on balanced homodyne detection and can be implemented with current technology. The most important experimental imperfections are studied and their impact quantified.
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We study the sensitivity and resolution of phase measurement in a Mach-Zehnder interferometer with two-mode squeezed vacuum (<n> photons on average). We show that super-resolution and sub-Heisenberg sensitivity is obtained with parity detection. In particular, in our setup, dependence of the signal on the phase evolves <n> times faster than in traditional schemes, and uncertainty in the phase estimation is better than 1/<n>.
We propose a hybrid quantum-classical atomic clock protocol where the interrogation of an ensemble of uncorrelated atoms in a spin-coherent state is used to feedback one (or more) spin-squeezed atomic ensembles toward their optimal phase sensitivity point. This protocol overcomes the stability of a single Ramsey clock and it reaches a Heisenberg-limited stability while avoiding non-destructive measurements. Analytical predictions are compared with numerical simulations of clocks operations including correlated 1/f local oscillator noise.
The interference between coherent and squeezed vacuum light can produce path entangled states with very high fidelities. We show that the phase sensitivity of the above interferometric scheme with parity detection saturates the quantum Cramer-Rao bound, which reaches the Heisenberg-limit when the coherent and squeezed vacuum light are mixed in roughly equal proportions. For the same interferometric scheme, we draw a detailed comparison between parity detection and a symmetric-logarithmic-derivative-based detection scheme suggested by Ono and Hofmann.
We demonstrate an angular momentum-enhanced protocol that permits an angular displacement estimation by using tunable squeezed Bell state and parity detection. We consider the resolution and the sensitivity, super-resolution is presented along with Heisenberg scaling sensitivity for arbitrary tunable factor, the tunable factor which can optimize the sensitivity is also discussed. Additionally, we analyze the advantages of using angular momentum via considering and comparing simulation results. Under the situation of the optimal tunable factor, the Heisenberg-limited sensitivity and $2\left(\el…
We report on an orbital-angular-momentum-enhanced scheme for angular displacement estimation based on two-mode squeezed vacuum and parity detection. The sub-Heisenberg-limited sensitivity for angular displacement estimation is obtained in an ideal situation. Several realistic factors are also considered, including photon loss, dark counts, response-time delay, and thermal photon noise. Our results indicate that the effects of the realistic factors on the sensitivity can be offset by raising orbital angular momentum quantum number $\ell$. This reflects that the robustness and the practicability…
Multiparameter quantum metrology at Heisenberg scaling for an arbitrary two-channel linear interferometer with squeezed light
利用 squeezed 光在任意双通道线性干涉仪中实现 Heisenberg 缩放的多参数量子计量，为高精度光学干涉计量提供实验可行的设计原则。
We propose a multi-parameter quantum metrological protocol based on a Mach-Zehnder interferometer with a squeezed vacuum input state and an anti-squeezing operation at one of its output channels. A simple and intuitive geometrical picture of the state evolution is provided by the marginal Wigner functions of the state at each interferometer output channel. The protocol allows to detect the value of the sum $β=\frac{1}{2}(φ_1+φ_2)+θ_\mathrm{in}-θ_\mathrm{out}$, of the relative phase $θ_\mathrm{in}-θ_\mathrm{out}$ between the two squeezers, and the average of the phase delays $φ_1,φ_2$ in the tw…
Two path interferometry with coherent states and squeezed vacuum can achieve phase sensitivities close to the Heisenberg limit when the average photon number of the squeezed vacuum is close to the average photon number of the coherent light. Here, we investigate the phase sensitivity of such states in the presence of photon losses. It is shown that the Cramer-Rao bound of phase sensitivity can be achieved experimentally by using a weak local oscillator and photon counting in the output. The phase sensitivity is then given by the Fisher information F of the state. In the limit of high squeezing…
Geometric bounds on multiparameter Heisenberg scaling in optical metrology with limited squeezed resources
研究多参量同时估计中，在多模高斯网络里给定压缩探测器时，有多少参数组合能继承海森堡标度。通过分析量子费希尔信息矩阵，得出海森堡标度系数秩的界限，确定饱和条件并构建干涉仪族。
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。
Full-Period Optical Phase Estimation with Heisenberg Scaling Using Displaced Squeezed States and Gaussian Measurements
提出固定能量约束下单模高斯态和高斯测量的全周期光学相位估计两阶段优化策略，经推导广义克拉美罗界，位移压缩态在第一阶段可降低优化界。
When measuring a time-varying phase, the standard quantum limit and Heisenberg limit as usually defined, for a constant phase, do not apply. If the phase has Gaussian statistics and a power-law spectrum $1/|ω|^p$ with $p>1$, then the generalized standard quantum limit and Heisenberg limit have recently been found to have scalings of $1/{\cal N}^{(p-1)/p}$ and $1/{\cal N}^{2(p-1)/(p+1)}$, respectively, where ${\cal N}$ is the mean photon flux. We show that this Heisenberg scaling can be achieved via adaptive measurements on squeezed states. We predict the experimental parameters analytically, a…
We show that the two-axis counter twisting interaction squeezes a coherent spin state into three states of interest in quantum information, namely, the twin-Fock state, the equally-weighted superposition state, and the state that achieves the Heisenberg limit of optimal sensitivity defined by the Cramer-Rao inequality in addition to the well-known Heisenberg-limited state of spin fluctuations.
It is a major goal in quantum thermometry to reach a $1/N$ scaling of thermometric precision known as Heisenberg scaling but is still in its infancy to date. The main obstacle is that the resources typically required are highly entangled states, which are very difficult to produce and extremely vulnerable to noises. Here, we propose an entanglement-free scheme of thermometry to approach Heisenberg scaling for a wide range of $N$, which has built-in robustness irrespective of the type of noise in question. Our scheme is amenable to a variety of experimental setups. Moreover, it can be used as a…
After reviewing parity measurement based interferometry with twin-Fock states, which allows for super-sensitivity (Heisenberg-limited) and super-resolution, we consider interferometry with two different superpositions of twin-Fock states, namely two-mode squeezed vacuum states and pair coherent states. This study is motivated by the experimental challenge of producing twin-Fock states on opposite sides of a beam splitter. We find that input two-mode squeezed states, while allowing for Heisenberg-limited sensitivity, do not yield super-resolutions, whereas we that find both are possible with in…
We propose an approach to quantum phase estimation that can attain precision near the Heisenberg limit without requiring single-particle-resolved state detection. We show that the "one-axis twisting" interaction, well known for generating spin squeezing in atomic ensembles, can also amplify the output signal of an entanglement-enhanced interferometer to facilitate readout. Applying this interaction-based readout to oversqueezed, non-Gaussian states yields a Heisenberg scaling in phase sensitivity, which persists in the presence of detection noise as large as the quantum projection noise of an …
We show how to use two-mode squeezed light to exponentially enhance cavity-based dispersive qubit measurement. Our scheme enables true Heisenberg-limited scaling of the measurement, and crucially, is not restricted to small dispersive couplings or unrealistically long measurement times. It involves coupling a qubit dispersively to two cavities, and making use of a symmetry in the dynamics of joint cavity quadratures (a so-called quantum-mechanics-free subsystem). We discuss the basic scaling of the scheme and its robustness against imperfections, as well as a realistic implementation in circui…
We show that the phase sensitivity $Δθ$ of a Mach-Zehnder interferometer fed by a coherent state in one input port and squeezed-vacuum in the other one is i) independent from the true value of the phase shift and ii) can reach the Heisenberg limit $Δθ\sim 1/N_T$, where $N_T$ is the average number of particles of the input states. We also show that the Cramer-Rao lower bound, $Δθ\propto 1/ \sqrt{|α|^2 e^{2r} + \sinh^2r}$, can be saturated for arbitrary values of the squeezing parameter $r$ and the amplitude of the coherent mode $|α|$ by a Bayesian phase inference protocol.
Current metrological bounds typically assume full control over all particles that are involved in the protocol. Relaxing this assumption we study metrological performance when only limited control is available. As an example, we measure a static magnetic field when a fully controlled quantum sensor is supplemented by particles over which only global control is possible. We show that even for a noisy quantum sensor, a protocol that maps the magnetic field to a precession frequency can achieve transient super-Heisenberg scaling in measurement time and Heisenberg scaling in the particle number. T…
We consider the problem of estimating the phase of squeezed vacuum states within a Bayesian framework. We derive bounds on the average Holevo variance for an arbitrary number $N$ of uncorrelated copies. We find that it scales with the mean photon number, $n$, as dictated by the Heisenberg limit, i.e., as $n^{-2}$, only for $N>4$. For $N\leq 4$ this fundamental scaling breaks down and it becomes $n^{-N/2}$. Thus, a single squeezed vacuum state performs worse than a single coherent state with the same energy. We find the optimal splitting of a fixed given energy among various copies. We also com…
Adaptive Phase Estimation with Squeezed Vacuum Approaching the Quantum Limit
针对光学相位估计中最优量子测量物理实现未知的问题，提出一种基于局部最优POVM、零差与外差测量及自适应框架的压缩真空高斯测量策略，在$[0, π)$全相位范围逼近量子极限。
Approaching the double-Heisenberg scaling sensitivity in the Tavis-Cummings model
本文证明Tavis-Cummings模型借助相干平均机制可在平均光子数和原子数上同时实现双重海森堡标度灵敏度，并通过量子压缩在实验上实现，为腔QED和超导电路提供计量应用。
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…
The use of quantum resources can provide measurement precision beyond the shot-noise limit (SNL). The task of ab initio optical phase measurement---the estimation of a completely unknown phase---has been experimentally demonstrated with precision beyond the SNL, and even scaling like the ultimate bound, the Heisenberg limit (HL), but with an overhead factor. However, existing approaches have not been able---even in principle---to achieve the best possible precision, saturating the HL exactly. Here we demonstrate a scheme to achieve true HL phase measurement, using a combination of three techni…
Minimum uncertainty states and squeezed states from the sum uncertainty relation
本文研究了求和不确定性关系下的最小不确定性态，证明其与传统乘积不确定性关系下的最小不确定性态一致，并展示了压缩态不受影响。
Within a self-consistent framework of q-deformed Heisenberg algebra and its equivalent framework of q-deformed boson commutation relations, which relate to the under-cutting phenomenon of Heisenberg's minimal uncertainty relation, special q-deformed squeezed states are constructed. Besides the similar local maximum squeezing as the one in the undeformed case, new strong squeezing appears when the amplitude of the related coherent state increases to large values. A critical phenomenon appears at a large value of the amplitude: the variance of one component of the quadrature of the light field a…
We have previously shown that quantum-enhanced atom interferometry can be achieved by mapping the quantum state of squeezed optical vacuum to one of the atomic inputs via a beamsplitter-like process [Phys.~Rev.~A \textbf{90}, 063630 (2014)]. Here we ask the question: is a better phase sensitivity possible if the quantum state transfer (QST) is described by a three-mode-mixing model, rather than a beamsplitter? The answer is yes, but only if the portion of the optical state not transferred to the atoms is incorporated via information recycling. Surprisingly, our scheme gives a better sensitivit…
Toward Heisenberg-Limited Interferometry with Dual Squeezers
本文提出双压缩干涉仪方案，克服直接光子数差检测中的极限问题，实现海森堡极限的相位灵敏度，同时具备抗噪声能力。
Enhanced Phase Estimation via Photon-Added Two-Mode Squeezed States and Kerr Nonlinearity
本文提出一种利用光子添加双模压缩态和 Kerr 非线性性提升相位估计精度的量子计量学方案，实现超越标准量子极限和超 Heisenberg 极限的高灵敏度测量。
Criterion for qubit-assisted quantum metrology approaching Heisenberg scaling
研究探针系统与辅助量子比特耦合时的计量精度，发现限制探针-量子比特耦合方向是达到海森堡极限的充分判据，并在玻色子和自旋系综系统中验证了量子Fisher信息的海森堡标度行为。
Employing the quantum Hamiltonian describing the interaction of two-mode light (signal-idler modes) generated by a nondegenerate parametric oscillator (NDPO) with two uncorrelated squeezed vacuum reservoirs (USVR), we derive the master equation. The corresponding Fokker-Planck equation for the Q-function is then solved employing a propagator method developed in Ref. \cite{1}. Making use of this Q-function, we calculate the quadrature fluctuations of the optical system. From these results we infer that the signal-idler modes are in squeezed states and the squeezing occurs in the first quadratur…
A proposed phase-estimation protocol based on measuring the parity of a two-mode squeezed-vacuum state at the output of a Mach-Zehnder interferometer shows that the Cramér-Rao sensitivity is sub-Heisenberg [Phys.\ Rev.\ Lett.\ {\bf104}, 103602 (2010)]. However, these measurements are problematic, making it unclear if this sensitivity can be obtained with a finite number of measurements. This sensitivity is only for phase near zero, and in this region there is a problem with ambiguity because measurements cannot distinguish the sign of the phase. Here, we consider a finite number of parity meas…
Two-parameter estimation with single squeezed-light interferometer via double homodyne detection
本文研究了单压缩光Mach-Zehnder干涉仪中通过双端口零差检测同时估计双参数的问题，发现压缩真空态能进一步提高相位灵敏度，在特定条件下可超越Heisenberg极限。
We consider sub-Poissonian single-mode laser with external synchronization and analyze its applicability to the problems of quantum information. Using Heisenberg-Langevin theory we calculate the quadrature variances of the field emitted by this laser. It is shown that such systems can demonstrate strong quadrature squeezing. Taking into account that the emitted field is temporally multi-mode the application of such sources to multichannel quantum teleportation and dense coding protocols is discussed.
A Mach-Zender interferometer with a gaussian number-difference squeezed input state can exhibit sub-shot-noise phase resolution over a large phase-interval. We obtain the optimal level of squeezing for a given phase-interval $Δθ_0$ and particle number $N$, with the resulting phase-estimation uncertainty smoothly approaching $3.5/N$ as $Δθ_0$ approaches 10/N, achieved with highly squeezed states near the Fock regime. We then analyze an adaptive measurement scheme which allows any phase on $(-π/2,π/2)$ to be measured with a precision of $3.5/N$ requiring only a few measurements, even for very la…
We derive a general expression of the quantum Fisher information for a Mach-Zehnder interferometer, with the port inputs of an \emph{arbitrary} pure state and a squeezed thermal state. We find that the standard quantum limit can be beaten, when even or odd states are applied to the pure-state port. In particular, when the squeezed thermal state becomes a thermal state, all the even or odd states have the same quantum Fisher information for given photon numbers. For a squeezed thermal state, optimal even or odd states are needed to approach the Heisenberg limit. As examples, we consider several…
We consider a degenerate parametric oscillator whose cavity contains a two-level atom. Applying the Heisenberg and quantum Langevin equations, we calculate in the bad-cavity limit the mean photon number, the quadrature variance, and the power spectrum for the cavity mode in general and for the signal light and fluorescent light in particular. We also obtain the normalized second-order correlation function for the fluorescent light. We find that the presence of the two-level atom leads to a decrease in the degree of squeezing of the signal light. It so turns out that the fluorescent light is in…
Precision bounds for quantum phase estimation using two-mode squeezed Gaussian states
本文研究双模压缩高斯态量子相位估计的精度界限，通过最大化量子Fisher信息发现两个单模压缩真空态为最优输入，其精度比Heisenberg极限高2倍，且在光子损耗率低于0.4时仍优于散粒噪声极限。
Heisenberg scaling under collective non-parallel directional noise via geometric state design
本文针对经典集体定向噪声下的参数估计，提出基于几何态设计的抗噪机制，可实现探针数依赖的海森堡标度估计精度，最优性能可通过测量单个集体自旋分量达成。
In this study, we explore the theoretical application of entangled multi-mode squeezed light for label-free optical super-resolution imaging. By generating massively entangled multi-mode squeezed light through an array of balanced beam splitters, using a single-mode squeezed light input, we create a multi-mode quantum light state with exceptional entanglement and noise suppression below the shot noise level. This significantly reduces imaging measurement errors compared to classical coherent state light imaging when the same number of photons are used on the imaging sample. We demonstrate how …
We theoretically investigate the phase sensitivity with parity detection on an SU(1,1) interferometer with a coherent state combined with a squeezed vacuum state. This interferometer is formed with two parametric amplifiers for beam splitting and recombination instead of beam splitters. We show that the sensitivity of estimation phase approaches Heisenberg limit and give the corresponding optimal condition. Moreover, we derive the quantum Cramér-Rao bound of the SU(1,1) interferometer.
In this paper we discuss the possibility of building a linear cluster state based on modes with a certain orbital angular momentum (OAM). We show that in the system under consideration a field with a rich mode structure is generated in the cavity. We also analyze the conditions under which an infinite system of Heisenberg-Langevin equations describing the dynamics of intracavity fields can be shortened and solved analytically. To analyze the genuine number of quantum degrees of freedom, we use the supermodes technique. This approach allows us to build the most entangled cluster state.
Atom interferometers provide exquisite measurements of the properties of non-inertial frames. While atomic interactions are typically detrimental to good sensing, efforts to harness entanglement to improve sensitivity remain tantalizing. Here we explore the role of interactions in an analogy between atomic gyroscopes and SQUIDs, motivated by recent experiments realizing ring shaped traps for ultracold atoms. We explore the one-dimensional limit of these ring systems with a moving weak barrier, such as that provided by a blue-detuned laser beam. In this limit, we employ Luttinger liquid theory …
We propose a quantum process tomography scheme that utilizes two-mode squeezed vacuum to realize the parameter estimation with Heisenberg scaling. The objective is to estimate a rotating angle of polarization and parity detection is used as the detection strategy. With the help of symplectic matrix theory, we discuss the estimation visibility and sensitivity of output signal in lossless situation, the quantum Fisher information is also given via calculation. Finally, the impacts of two realistic factors on both visibility and sensitivity are also considered, including photon loss during the in…
Laser-free method for creation of two-mode squeezed state and beam-splitter transformation with trapped ions
本研究提出一种基于时变电场和离子晶体非线性耦合的无激光方案，可制备双模压缩态、实现分束器变换，达到海森堡精度极限，还可用于构建三量子比特Fredkin门。
Heisenberg-scaling sensitivity in the estimation of two parameters in a Mach-Zehnder interferometer
本文提出一种实验可行的方案，利用压缩态和相干态光输入及零差探测，在马赫-曾德尔干涉仪中同时估计两个相位参数，实现海森堡极限灵敏度。
As an "input" state of a linear (Mach-Zehnder or Ramsey) interferometer, the phase-squeezed state proposed by Berry and Wiseman exhibits the best sensitivity approaching to the Heisenberg limit [Phys. Rev. Lett. 85, 5098 (2000)]. In this paper, we find that it can be generated dynamically with atomic Bose-Einstein condensates confined in a symmetric double well. Similar with the Berry and Wiseman's state, the prepared states show the squeezing along spin operator S_y and the anti-squeezing along S_z, leading to a sub-shot-noise of the phase sensitivity.
Gaussian Formalism: Joint Measurement for Heisenberg's Uncertainty Relation for Errors by Squeezed Coherent States
本文提出高斯波包形式可实现位置与动量的联合测量，基于POVM在投影测量间平滑插值，首次获得LT误差与精化Lee误差，发现精化Lee不确定性关系提供与海森堡值一致的常数下界。
Quaternary-Squeeze Quantum Identity Authentication: Direction-Scaling Security via Single-Mode Squeezed Light
本文针对量子身份认证提出基于压缩光场非经典特性的协议，利用正交压缩相干态特性及诱饵态保真度参数等，能抗高斯克隆攻击、检测窃听，四方向压缩可提高安全阈值，相比二元压缩协议有优势，便于实际应用。
We theoretically present the quantum Cramér-Rao bounds (QCRB) of an SU(1,1) interferometer for Gaussian states input with and without the internal photonic losses. The phase shifts in the single arm and in the double arms are studied and the corresponding analytical expressions of quantum Fisher information with Gaussian input states are presented. Different from the traditional Mach-Zehnder interferometer, the QCRB of single arm case is slightly higher or lower than that of double arms case depending on the input states. With a fixed mean photon number and for pure Gaussian state input, the o…
Squeezed states for Frenkel-like two-fermion composite bosons
研究双费米子复合玻色子的压缩态，重点在弗伦德尔-like复合玻色子，探讨Pauli阻塞和非标准交换代数对压缩态的影响，推导四阶方差表达式，并展示其对不确定性原理的约束。
Adaptive protocols for SU(11) interferometers to achieve ab initio phase estimation at the Heisenberg limit
本文提出两种 SU(11) 干涉仪自适应协议，通过光子数测量配合反馈相位序列及无需调谐的最优测量，放宽相位大小限制，实现海森堡极限下的从头相位估计。
This study significantly concentrates on cryogenic InP HEMT high-frequency circuit analysis using quantum theory to find how the transistor nonlinearity can affect the quantum correlation of the modes generated in the circuit. Firstly, the total Hamiltonian of the circuit is derived, and the dynamic equation of the motion contributed is examined using the Heisenberg-Langevin equation. Using the nonlinear Hamiltonian, some components are attached to the intrinsic internal circuit of InP HEMT to fully address the circuit characteristics. The components attached are arisen due to the nonlinearity…
We propose a metrological strategy reaching Heisenberg scaling precision in the estimation of functions of any number $l$ of arbitrary parameters encoded in a generic $M$-channel linear network. This scheme is experimentally feasible since it only employs a single-mode squeezed vacuum and homodyne detection on a single output channel. Two auxiliary linear network are required and their role is twofold: to refocus the signal into a single channel after the interaction with the interferometer, and to fix the function of the parameters to be estimated according to the linear network analysed. Alt…
Super-Heisenberg scaling in a triple point criticality
研究了三重临界点中量子临界性对测量精度的增强作用，提出绝热演化协议实现指数超海森堡标度，并指出参数调制不当会导致标度退化。
We present in this letter a scheme for optical interferometry. We utilize coherent-beam-stimulated two-mode squeezed light, which interacts with a phase shifter and is then squeezed again before detection. Our theoretical device has the potential to reach far below the shot noise limit (SNL) in phase sensitivity. This new proposal avoids the pitfalls of other setups, such as difficulty in creating the required resource. Furthermore, our scheme requires no complicated detection protocol, relying instead only on simple intensity measurement. Also, bright, coherent sources "boost" squeezed light,…
Quantum-enhanced estimation of signal field amplitudes with critical squeezed states of photonic modes
本文提出一种利用临界性增强的量子传感协议，通过qubit-光子相互作用系统估计外部信号场振幅，利用压缩真空态实现高精度测量。
We analyze the Heisenberg limit on phase estimation for Gaussian states. In the analysis, no reference to a phase operator is made. We prove that the squeezed vacuum state is the most sensitive for a given average photon number. We provide two adaptive local measurement schemes that attain the Heisenberg limit asymptotically. One of them is described by a positive operator-valued measure and its efficiency is exhaustively explored. We also study Gaussian measurement schemes based on phase quadrature measurements. We show that homodyne tomography of the appropriate quadrature attains the Heisen…
Three-Axis Spin Squeezed States Associated with Excited-State Quantum Phase Transitions
本文提出三轴自旋压缩态，结合各向异性模型，实现了量子相变与高精度计量的统一。
Heisenberg limit in phase measurements: the threshold detection approach
本文研究了单臂和双臂光学干涉仪中利用高斯态进行相位测量的精度极限，比较了常规混频测量与非高斯阈值检测方法，发现通过反称压缩输入的双臂干涉仪可同时实现海森堡极限灵敏度和宽高灵敏度范围。
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