Enhancing quantum metrology for multiple frequencies of oscillating magnetic fields by quantum control

Quantum multi-parameter estimation has recently attracted increased attention due to its wide applications, with a primary goal of designing high-precision measurement schemes for unknown parameters. While existing research has predominantly concentrated on time-independent Hamiltonians, little has been known about quantum multi-parameter estimation for time-dependent Hamiltonians due to the complexity of quantum dynamics. This work bridges the gap by investigating the precision limit of multi-parameter quantum estimation for a qubit in an oscillating magnetic field model with multiple unknown frequencies. As the well-known quantum Cramer–Rao bound is generally unattainable due to the potential incompatibility between the optimal measurements for different parameters, we use the most informative bound instead which is always attainable and equivalent to the Holevo bound in the asymptotic limit. Moreover, we apply additional Hamiltonian to the system to engineer the dynamics of the qubit. By utilizing the quasi-Newton method, we explore the optimal schemes to attain the highest precision for the unknown frequencies of the magnetic field, including the simultaneous optimization of initial state preparation, the control Hamiltonian and the final measurement. The results indicate that the optimization can yield much higher precisions for the field frequencies than those without the optimizations. Finally,we study the robustness of the optimal control scheme with respect to the fluctuation of the interested frequencies, and the optimized scheme exhibits superior robustness to the scenario without any optimization.

Quantum metrology with a non-Markovian qubit system

The dynamics of the quantum Fisher information(QFI) of phase parameter estimation in a non-Markovian dissipative qubit system is investigated within the structure of single and double Lorentzian spectra. We use the time-convolutionless method with fourth-order perturbation expansion to obtain the general forms of QFI for the qubit system in terms of a non-Markovian master equation. We find that the phase parameter estimation can be enhanced in our model within both single and double Lorentzian spectra. What is more, the detuning and spectral width are two significant factors affecting the enhancement of parameter-estimation precision.

Quantum metrology with coherent superposition of two different coded channels

We investigate the advantage of coherent superposition of two different coded channels in quantum metrology.In a continuous variable system,we show that the Heisenberg limit 1/N can be beaten by the coherent superposition without the help of indefinite causal order.And in parameter estimation,we demonstrate that the strategy with the coherent superposition can perform better than the strategy with quantum switch which can generate indefinite causal order.We analytically obtain the general form of estimation precision in terms of the quantum Fisher information and further prove that the nonlinear Hamiltonian can improve the estimation precision and make the measurement uncertainty scale as 1/N^(m) for m≥2.Our results can help to construct a high-precision measurement equipment,which can be applied to the detection of coupling strength and the test of time dilation and the modification of the canonical commutation relation.

Micron-sized fiber diamond probe for quantum precision measurement of microwave magnetic field

We present a quantitative measurement of the horizontal component of the microwave magnetic field of a coplanar waveguide using a quantum diamond probe in fiber format.The measurement results are compared in detail with simulation,showing a good consistence.Further simulation shows fiber diamond probe brings negligible disturbance to the field under measurement compared to bulk diamond.This method will find important applications ranging from electromagnetic compatibility test and failure analysis of high frequency and high complexity integrated circuits.

Experimental demonstration of nonlinear quantum metrology with optimal quantum state

Nonlinear quantum metrology may exhibit better precision scalings. For example, the uncertainty of an estimated phase may scale as △φ∝ 1/N2 under quadratic phase accumulation, which is 1/N times smal-ler than the linear counterpart, where N is probe number. Here, we experimentally demonstrate the non-linear quantum metrology by using a spin-I（I 〉 1/2） nuclear magnetic resonance （NMR） ensemble that can be mapped into a system ofN = 2I spin-1/2 particles and the quadratic interaction can be utilized for the quadratic phase accumulation. Our experimental results show that the phase uncertainty can scale as △φ∝1/（N2-1） by optimizing the input states, when N is an odd number. In addition, the interferomet-tic measurement with quadratic interaction provides a new way for estimating the quadrupolar coupling strength in an NMR system. Our system may be further extended to exotic nonlinear quantum metrology with higher order many-body interactions.

The Influence of Quantum Technology on the Development of Metrology and Measuring Science

In the new century, quantum technology has developed rapidly and has been applied in many fields. As an important aspect of the aerospace science, metrology and measuring science is a field which is influenced by the quantum technology dramatically. The new generation of the International System of Units will be redefined on the basis of the quantum theory. More and more new sensing techniques are developed taking into account quantum principles. In this paper, the influence of quantum technology on metrology and measuring science is introduced.

Holevo bound independent of weight matrices for estimating two parameters of a qubit

Holevo bound plays an important role in quantum metrology as it sets the ultimate limit for multi-parameter estimations,which can be asymptotically achieved.Except for some trivial cases,the Holevo bound is implicitly defined and formulated with the help of weight matrices.Here we report the first instance of an intrinsic Holevo bound,namely,without any reference to weight matrices,in a nontrivial case.Specifically,we prove that the Holevo bound for estimating two parameters of a qubit is equivalent to the joint constraint imposed by two quantum Cramér–Rao bounds corresponding to symmetric and right logarithmic derivatives.This weightless form of Holevo bound enables us to determine the precise range of independent entries of the mean-square error matrix,i.e.,two variances and one covariance that quantify the precisions of the estimation,as illustrated by different estimation models.Our result sheds some new light on the relations between the Holevo bound and quantum Cramer–Rao bounds.Possible generalizations are discussed.

Nonlinear time-reversal interferometry with arbitrary quadratic collective-spin interaction

Atomic nonlinear interferometry has wide applications in quantum metrology and quantum information science.Here we propose a nonlinear time-reversal interferometry scheme with high robustness and metrological gain based on the spin squeezing generated by arbitrary quadratic collective-spin interaction,which could be described by the Lipkin–Meshkov–Glick(LMG)model.We optimize the squeezing process,encoding process,and anti-squeezing process,finding that the two particular cases of the LMG model,one-axis twisting and two-axis twisting outperform in robustness and precision,respectively.Moreover,we propose a Floquet driving method to realize equivalent time reverse in the atomic system,which leads to high performance in precision,robustness,and operability.Our study sets a benchmark for achieving high precision and high robustness in atomic nonlinear interferometry.

Beating standard quantum limit via two-axis magnetic susceptibility measurement

We report a metrology scheme which measures the magnetic susceptibility of an atomic spin ensemble along the x and z directions and produces parameter estimation with precision beating the standard quantum limit.The atomic ensemble is initialized via one-axis spin squeezing with optimized squeezing time and parameterΦ(to be estimated)assumed as uniformly distributed between 0 and 2πwhile fixed in each estimation.One estimation ofΦcan be produced with every two magnetic susceptibility data measured along the two axes respectively,which has an imprecision scaling(1.43±0.02)/N^(0.687±0.003)with respect to the number N of the atomic spins.The measurement scheme is easy to implement and is robust against the measurement fluctuation caused by environment noise and measurement defects.

Towards quantum-enhanced precision measurements:Promise and challenges

Quantum metrology holds the promise of improving the measurement precision beyond the limit of classical ap- proaches. To achieve such enhancement in performance requires the development of quantum estimation theories as well as novel experimental techniques. In this article, we provide a brief review of some recent results in the field of quantum metrology. We emphasize that the unambiguous demonstration of the quantum-enhanced precision needs a careful analysis of the resources involved. In particular, the implementation of quantum metrology in practice requires us to take into ac- count the experimental imperfections included, for example, particle loss and dephasing noise. For a detailed introduction to the experimental demonstrations of quantum metrology, we refer the reader to another article ＇Quantum metrology＇ in the same issue.

Information-theoretic perspective on performance assessment and fundamental limit of quantum imaging [Invited]

Performance assessment of an imaging system is important for the optimization design with various technologies.The information-theoretic viewpoint based on communication theory or statistical inference theory can provide objective and operational measures on imaging performance. These approaches can be further developed by combining with the quantum statistical inference theory for optimizing imaging performance over measurements and analyze its quantum limits, which is demanded in order to improve an imaging system when the photon shot noise in the measurement is the dominant noise source. The aim of this review is to discuss and analyze the recent developments in this branch of quantum imaging.

Quantum vector DC magnetometry via selective phase accumulation

Precision measurement of magnetic fields is a crucial issue in both fundamental scientific research and practical sensing technology.The sensitive detection of a vector magnetic field poses a significant challenge in quantum magnetometry,particularly in estimating a vector DC magnetic field with high precision.Here,we propose a comprehensive protocol for quantum vector DC magnetometry,utilizing selective phase accumulation in both non-entangled and entangled quantum probes.Building upon the principles of Ramsey interferometry,our protocol enables the selective accumulation of phase for a specific magnetic field component by incorporating a meticulously designed pulse sequence.In the individual measurement scheme,we employ three individual quantum interferometries to independently estimate each of the three magnetic field components.Alternatively,in the simultaneous measurement scheme,the application of a pulse sequence along different directions enables the simultaneous estimation of all three magnetic field components using only one quantum interferometry.Notably,by employing an entangled state(such as the Greenberger-Horne-Zeilinger state)as the input state,the measurement precisions of all three components may reach the Heisenberg limit.This study not only establishes a general protocol for measuring vector magnetic fields using quantum probes,but also presents a viable pathway for achieving entanglement-enhanced multi-parameter estimation.

Phase sensitivity of two nonlinear interferometers with inputting entangled coherent states

We investigate the phase sensitivity of the SU（1,1） interfereometer [SU（1,1）I] and the modified Mach-Zehnder in- terferometer （MMZI） with the entangled coherent states （ECS） as inputs. We consider the ideal case and the situations in which the photon losses are taken into account. We find that, under ideal conditions, the phase sensitivity of both the MMZI and the SU（1,1）I can beat the shot-noise limit （SNL） and approach the Heisenberg limit （HL）. In the presence of photon losses, the ECS can beat the coherent and squeezed states as inputs in the SU（1,1）I, and the MMZI is more robust against internal photon losses than the SU（1,1）I.

Super-sensitive phase estimation with coherent boosted light using parity measurements

We consider a passive and active hybrid interferometer for phase estimation, which can reach the sub-shot-noise limit in phase sensitivity with only the cheapest coherent sources. This scheme is formed by adding an optical parametric amplifier before a Mach-Zehnder interferometer. It is shown that our hybrid protocol can obtain a better quantum Cramer- Rao bound than the pure active （e.g., SU（1, I）） interferometer, and this precision can be reached by implementing the parity measurements. Furthermore, we also draw a detailed comparison between our scheme and the scheme suggested by Caves [Phys. Rev. D 23 1693 （1981）], and it is found that the optimal phase sensitivity gain obtained in our scheme is always larger than that in Caves＇ scheme.

Quantum-limited resolution of partially coherent sources

Discriminating two spatially separated sources is one of the most fundamental problems in imaging.Recent research based on quantum parameter estimation theory shows that the resolution limit of two incoherent point sources given by Rayleigh can be broken.However,in realistic optical systems,there often exists coherence in the imaging light field,and there have been efforts to analyze the optical resolution in the presence of partial coherence.Nevertheless,how the degree of coherence between two point sources affects the resolution has not been fully understood.Here,we analyze the quantum-limited resolution of two partially coherent point sources by explicitly relating the state after evolution through the optical systems to the coherence of the sources.In particular,we consider the situation in which coherence varies with the separation.We propose a feasible experiment scheme to realize the nearly optimal measurement,which adaptively chooses the binary spatial-mode demultiplexing measurement and direct imaging.Our results will have wide applications in imaging involving coherence of light.

Multilevel atomic Ramsey interferometry for precise parameter estimations

Multi-path(or multi-mode)entanglement has been proved to be a useful resource for sub-shot-noise sensitivity of phase estimation,which has aroused much research interest in quantum metrology recently.Various schemes of multi-path interferometers based on optical systems have been put forward.Here,we study a multi-state interferometer with multilevel atoms by projective measurements.Specifically,we investigate its ultimate sensitivity described by quantum Fisher information theory and find that the Cramer-Rao bound can be achieved.In particular,we investigate a specific scheme to improve the sensitivity of magnetometery with a three-state interferometry delivered by a single nitrogen-vacancy(NV)center of diamond with tailor pulses.The impacts of imperfections of the atomic beam-splitter,described by the three-level quantum Fourier transform,on the sensitivity of phase estimation is also discussed.

Super-sensitivity measurement of tiny Doppler frequency shifts based on parametric amplification and squeezed vacuum state

The precision measurement of Doppler frequency shifts is of great significance for improving the precision of speed measurement.This paper proposes a precision measurement scheme of tiny Doppler shifts by a parametric amplification process and squeezed vacuum state.This scheme takes a parametric amplification process and squeezed vacuum state into a detection system,so that the measurement precision of tiny Doppler shifts can exceed the Cram′er–Rao bound of coherent light.Simultaneously,a simulation study is carried out on the theoretical basis,and the following results are obtained:for the signal light of Gaussian mode,when the amplification factor g=1 and the squeezed factor r=0.5,the measurement error of Doppler frequency shifts is 14.4%of the Cramer–Rao bound of the coherent light in our system.At the same time,when the local light mode and squeezed vacuum state mode are optimized,the measurement precision of this scheme can be further improved by√(2n+1)/(n+1)times,where n is the mode-order of the signal light.

Super-resolution and super-sensitivity of entangled squeezed vacuum state using optimal detection strategy

Interference metrology is a method for achieving high precision detection by phase estimation. The phase sensitivity of a traditional interferometer is subject to the standard quantum limit, while its resolution is constrained by the Rayleigh diffraction limit. The resolution and sensitivity of phase measurement can be enhanced by using quantum metrology. We propose a quantum interference metrology scheme using the entangled squeezed vacuum state, which is obtained using the magic beam splitter, expressed as ｜ψ〉=（｜ξ〉｜0〉＋｜0〉｜ξ〉）/√2＋2/coshr, such as the N00 N state. We derive the phase sensitivity and the resolution of the system with Z detection, project detection, and parity detection. By simulation and analysis, we determine that parity detection is an optimal detection method, which can break through the Rayleigh diffraction limit and the standard quantum limit.

Quantum multiparameter estimation with multimode photon catalysis entangled squeezed state

We propose a method to generate the multi-mode entangled catalysis squeezed vacuum states(MECSVS)by embedding the cross-Kerr nonlinear medium into the Mach–Zehnder interferometer.This method realizes the exchange of quantum states between different modes based on Fredkin gate.In addition,we study the MECSVS as the probe state of multi-arm optical interferometer to realize multi-phase simultaneous estimation.The results show that the quantum Cramer–Rao bound(QCRB)of phase estimation can be improved by increasing the number of catalytic photons or decreasing the transmissivity of the optical beam splitter using for photon catalysis.In addition,we also show that even if there is photon loss,the QCRB of our photon catalysis scheme is lower than that of the ideal entangled squeezed vacuum states(ESVS),which shows that by performing the photon catalytic operation is more robust against photon loss than that without the catalytic operation.The results here can find applications in quantum metrology for multiparatmeter estimation.

Quantum information processing and metrology with color centers in diamonds

The Nitrogen Vacancy （NV） center is becoming a promising qubit for quantum information processing. The defect has a long coherence time at room temperature and it allows spin state initialized and read out by laser and manipulated by microwave pulses. It has been utilized as a ultra sensi- tive probe for magnetic fields and remote spins as well. Here, we review the recent progresses in experimental demonstrations based on NV centers. We first introduce our work on implementation of the Deutsch- Jozsa algorithm with a single electronic spin in diamond. Then the quantum nature of the bath around the center spin is revealed and continuous wave dynamical decoupling has been demonstrated. By applying dynamical decoupling, a multi-pass quantum metrology protocol is realized to enhance phase estimation. In the final, we demonstrated NV center can be regarded as a ultra-sensitive sensor spin to implement nuclear magnetic resonance （NMR） imaging at nanoscale.