We study a nonlinear interferometer consisting of two consecutive parametric amplifiers, where all three optical fields (pump, signal and idler) are treated quantum mechanically, allowing for pump depletion and other quantum phenomena. The interaction of all three fields in the final amplifier leads to an interference pattern from which we extract the phase uncertainty. We find that the phase uncertainty oscillates around a saturation level that decreases as the mean number $N$ of input pump photons increases. For optimal interaction strengths, we also find a phase uncertainty below the shot-noise level and obtain a Heisenberg scaling $1/N$. This is in contrast to the conventional treatment within the parametric approximation, where the Heisenberg scaling is observed as a function of the number of down-converted photons inside the interferometer.
Absolute sensitivity is measured for the phase measurement in an SU(1,1) type interferometer and the results are compared to that of a Mach-Zehnder interferometer operated under the condition of the same intra-interferometer intensity. The interferometer is phase locked to a point with the largest quantum noise cancellation, and a simulated phase modulation is added in one arm of SU(1,1) interferometer. Both the signal and noise level are estimated at the same frequency range, and we obtain 3dB improvement in sensitivity for the SU(1,1) interferometer over the Mach-Zehnder interferometer. Our …
The sensitivity properties of an SU(1,1) interferometer made of two cascaded parametric amplifiers, as well as of an ordinary SU(2) interferometer preceded by a squeezer and followed by an anti-squeezer, are theoretically investigated. Several possible experimental configurations are considered, such as the absence or presence of a seed beam, direct or homodyne detection scheme. In all cases we formulate the optimal conditions to achieve phase super-sensitivity, meaning a sensitivity overcoming the shot-noise limit. We show that for a given gain of the first parametric amplifier, unbalancing t…
We propose a possible approach to achieve an 1/N sensitivity of Michelson interferometer by using a properly designed random phase modulation. Different from other approaches, the sensitivity improvement does not depend on increasing optical powers or utilizing the quantum properties of light. Moreover the requirements for optical losses and the quantum efficiencies of photodetection systems might be lower than the quantum approaches and the sensitivity improvement is frequency independent in all detection band.
SU(1,1) interferometers, based on the usage of nonlinear elements, are superior to passive interferometers in phase sensitivity. However, the SU(1,1) interferometer cannot make full use of photons carrying phase information as the second nonlinear element annihilates some of the photons inside. Here, we focus on improving phase sensitivity and propose a new protocol based on a modified SU(1,1) interferometer, where the second nonlinear element is replaced by a beam splitter. We utilize two coherent states as inputs and implement balanced homodyne measurement at the output. Our analysis suggest…
For a squeezing-enhanced SU(2) interferometer, we theoretically investigate the possibility to broaden the phase range of sub-shot-noise sensitivity. We show that this goal can be achieved by implementing detection in both output ports, with the optimal combination of the detectors outputs, leading to a phase sensitivity independent of the interferometer operation point. Provided that each detector is preceded by a phase-sensitive amplifier, this sensitivity could be also tolerant to the detection loss.
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.
We consider various configuration of quantum-enhanced interferometers, both linear (SU(2)) and non-linear (SU(1,1)) ones, as well as hybrid SU(2)/SU(1,1) schemes. Using the unified modular approach, based on the Quantum Cramer-Rao bound, we show that in all practical cases, their sensitivity is limited by the same equations (95) or (97) which first appeared in the pioneering work by C.Caves [Phys.Rev.D 23, 1693 (1981)].
We provide general bounds of phase estimation sensitivity in linear two-mode interferometers. We consider probe states with a fluctuating total number of particles. With incoherent mixtures of state with different total number of particles, particle entanglement is necessary but not sufficient to overcome the shot noise limit. The highest possible phase estimation sensitivity, the Heisenberg limit, is established under general unbiased properties of the estimator. When coherences can be created, manipulated and detected, a phase sensitivity bound can only be set in the central limit, with a su…
A Michelson-type interferometer with two-mode squeezed coherent state input is considered. Such an interferometer has a better phase sensitivity over the shot-noise limit by a factor of $e^{2r}$, where $r$ is the squeezing parameter [Phys. Rev. A 102,022614 (2020)]. We show that when photon loss and noise in the two arms is asymmetric an optimal choice of the squeezing angle can allow improvement in phase sensitivity without any increase in input or pump power. In particular, when loss occurs only in one arm of the interferometer, we can have improvement in phase sensitivity for photon loss up…
We theoretically study the phase sensitivity of the SU(1,1) interferometer with a coherent light together with a squeezed vacuum input case using the method of homodyne. We find that the homodyne detection has better sensitivity than the intensity detection under this input case. For a certain intensity of coherent light (squeezed light) input, the relative phase sensitivity is not always better with increasing the squeezed strength (coherent light strength). The phase sensitivity can reach the Heisenberg limit only under a certain moderate parameter interval, which can be realized with curren…
We study the dynamics of Bose-Einstein condensates in time-dependent microtraps for the purpose of understanding the influence of the mean field interaction on the performance of interferometers. We identify conditions where the non-linearity due to atom interactions increases the sensitivity of interferometers to a phase shift. This feature is connected with the adiabatic generation of a dark soliton. We analyze the robustness of this phenomenon with respect to thermal fluctuations, due to excited near fields in an electromagnetic surface trap.
We propose a theoretical scheme to enhance the phase sensitivity by introducing a Kerr nonlinear phase shift into the traditional SU(1,1) interferometer with a coherent state input and homodyne detection. We investigate the realistic effects of photon losses on phase sensitivity and quantum Fisher information. The results show that compared with the linear phase shift in SU(1,1) interferometer, the Kerr nonlinear case can not only enhance the phase sensitivity and quantum Fisher information, but also significantly suppress the photon losses. We also observe that at the same accessible paramete…
With photon-number resolving detectors, we show compression of interference fringes with increasing photon numbers for a Fabry-Perot interferometer. This feature provides a higher precision in determining the position of the interference maxima compared to a classical detection strategy. We also theoretically show supersensitivity if N-photon states are sent into the interferometer and a photon-number resolving measurement is performed.
The light--matter-wave Sagnac interferometer based on ultra-slow light proposed recently in (Phys. Rev. Lett. 92, 253201 (2004)) is analyzed in detail. In particular the effect of confining potentials is examined and it is shown that the ultra-slow light attains a rotational phase shift equivalent to that of a matter wave, if and only if the coherence transfer from light to atoms associated with slow light is associated with a momentum transfer and if an ultra-cold gas in a ring trap is used. The quantum sensitivity limit of the Sagnac interferometer is determined and the minimum detectable ro…
We theoretically study the effects of loss on the phase sensitivity of an SU(1,1) interferometer with parity detection with various input states. We show that although the sensitivity of phase estimation decreases in the presence of loss, it can still beat the shot-noise limit with small loss. To examine the performance of parity detection, the comparison is performed among homodyne detection, intensity detection, and parity detection. Compared with homodyne detection and intensity detection, parity detection has a slight better optimal phase sensitivity in the absence of loss, but has a worse…
Recently, biased weak measurement(BWM) has shown higher precision than both conventional measurement and standard weak measurement(SWM) in optical metrology. In this paper, we propose a scheme of detecting rotation velocity in Sagnac's interferometer with BWM. In particular, BWM employs an additional reduction of photons in the post-selection by introducing a pre-coupling, and the remaining photons have been shown to be extremely sensitive to the estimated parameter. In addition, our numerical results show that the scheme with BWM can obtain a higher sensitivity than the scheme with SWM.
We study the sensitivity of a Mach-Zehnder interferometer that contains in addition to the phase shifter a non-linear element. By including both elements in a cavity or a loop that the light transverses many times, a non-linear kicked version of the interferometer arises. We study its sensitivity as function of the phase shift, the kicking strength, the maximally reached average number of photons, and damping due to photon loss for an initial coherent state. We find that for vanishing damping Heisenberg-limited scaling of the sensitivity arises if squeezing dominates the total photon number. F…
Many works have stated that nonlinear interactions can improve phase sensitivity beyond the Heisenberg limit scaling of $1/N$ with $N$ being the mean photon number. This raises some open questions---among them the conclusive sensitivity limits with respect to single-mode inputs. Namely, when one of two inputs is vacuum, is there a shot-noise-style sensitivity bound on a nonlinear Mach-Zehnder interferometer? Within the reach of second-order nonlinear phase shifts, we make an attempt to provide an answer to this question. Based upon phase-averaging approach, this puzzle is partially resolved wi…
We present experimental and theoretical results on a new interferometer topology that nests a SU(2) interferometer, e.g., a Mach-Zehnder or Michelson interferometer, inside a SU(1,1) interferometer, i.e., a Mach-Zehnder interferometer with parametric amplifiers in place of beam splitters. This SU(2)-in-SU(1,1) nested interferometer (SISNI) simultaneously achieves high signal-to-noise ratio (SNR), sensitivity beyond the standard quantum limit (SQL) and tolerance to photon losses external to the interferometer, e.g., in detectors. We implement a SISNI using parametric amplification by four-wave …
We theoretically studied the quantum Cramér-Rao bound of an actively correlated Mach-Zehnder interferometer (ACMZI), where the quantum Fisher information obtained by the phase-averaging method can give the proper phase-sensing limit without any external phase reference. We numerically calculate the phase sensitivities with the method of homodyne detection and intensity detection in the presence of losses. Under lossless and very low loss conditions, the ACMZI is operated in a balanced case to beat the standard quantum limit (SQL). As the loss increases, the reduction in sensitivity increases. …
We theoretically investigate an interferometer composed of four four-wave-mixers by Lie group method. Lie group SU(1,2) characterizes the mode transformations of this kind of interferometer. With vacuum state inputs, the phase sensitivity of SU(1,2) interferometer achieves the Heisenberg limit, and the absolute accuracy beats SU(1,1) interferometer because of higher intensity of light inside the interferometer. For different input cases, the optimal combination of output photon number for detection to obtain the best phase sensitivity is calculated. Our research on SU(1,2) interferometer sheds…
Sensitivity Evaluation of SU(1,1) Interferometers with Arbitrary Input Probe State and Homodyne Detections
Indian Institute of Technology Kharagpur（印度理工学院卡拉格普尔分校）；Université Paris-Saclay（巴黎萨克雷大学）；CNRS（法国国家科学研究中心）；CentraleSupélec（中央理工-高等电力学院）；Centre for Nanosciences and Nanotechnologies, CNRS, Université Paris-Saclay（法国国家科学研究中心-巴黎萨克雷大学纳米科学与工程中心）
Quantum-enhanced phase sensitivity in an all-fiber Mach-Zehnder interferometer
Université Côte d’Azur, CNRS, Institut de physique de Nice（尼斯大学，法国国家科学研究中心，尼斯物理研究所）；Télécom Paris-LTCI, Institut Polytechnique de Paris（巴黎电信学院-信息与通信研究所，巴黎理工学院）
An ${\rm SU(1,1)}$ interferometer uses a sequence of two optical parametric amplifiers for achieving sub-shot-noise sensitivity to a phase shift introduced in between. We present the first realization of a wide-field ${\rm SU(1,1)}$ interferometer, where the use of a focusing element enables spatially multimode operation within a broad angle. Over this angle, the interference phase is found to be flat. This property is important for the high sensitivity to the phase front disturbance. Further, $-4.3\pm0.7$ dB quadrature squeezing, an essential requirement to the high sensitivity, is experiment…
We demonstrate the applicability of the EPR entanglement squeezing scheme for enhancing the shot-noise-limited sensitivity of a detuned dual-recycled Michelson interferometers. In particular, this scheme is applied to the GEO\,600 interferometer. The effect of losses throughout the interferometer, arm length asymmetries, and imperfect separation of the signal and idler beams are considered.
We investigate quantum sensing of rotation with a multi-atom Sagnac interferometer and present multi-partite entangled states to enhance the sensitivity of rotation frequency. For studying the sensitivity, we first present a Hermitian generator with respect to the rotation frequency. The generator, which contains the Sagnac phase, is a linear superposition of a z component of the collective spin and a quadrature operator of collective bosons depicting the trapping modes, which enables us to conveniently study the quantum Fisher information (QFI) for any initial states. With the generator, we d…
We theoretically study the phase sensitivity of an SU(1,1) interferometer with a thermal state and squeezed vacuum state as inputs and parity detection as measurement. We find that phase sensitivity can beat the shot-noise limit and approaches the Heisenberg limit with increasing input photon number.
We study the phase sensitivity in the conventional $SU(2)$ and nonconventional $SU(1,1)$ interferometers with the coherent and squeezed vacuum input state via the quantum Cramer-Rao bound. We explicitly construct the detection scheme that gives the optimal phase sensitivity. For practical purposes, we show that in the presence of photon loss, both interferometers with proper homodyne detections, are nearly optimal. We also find that unlike the coherent state and squeezed vacuum state, the effects of the imperfect detector on the phase sensitivity cannot be asymptotically removed for a generic …
Quantum entanglement has the potential to revolutionize the entire field of interferometric sensing by providing many orders of magnitude improvement in interferometer sensitivity. The quantum-entangled particle interferometer approach is very general and applies to many types of interferometers. In particular, without nonlocal entanglement, a generic classical interferometer has a statistical-sampling shot-noise limited sensitivity that scales like $1/\sqrt{N}$, where $N$ is the number of particles passing through the interferometer per unit time. However, if carefully prepared quantum correl…
Quantum metrology promises greater sensitivity for optical phase measurements than could ever be achieved classically. Here we present a theory of the phase sensitivity for the general case where the detection probability is given by an $N$ photon interference fringe. We find that the phase sensitivity has a complex dependence on both the intrinsic efficiency of detection $η$ and the interference fringe visibility $V$. Most importantly, the phase that gives maximum phase sensitivity is in general not the same as the phase at which the slope of the interference fringe is a maximum, as has previ…
The high inertial sensitivity of atom interferometers has been used to build accelerometers and gyrometers but this sensitivity makes these interferometers very sensitive to the laboratory seismic noise. This seismic noise induces a phase noise which is large enough to reduce the fringe visibility in many cases. We develop here a model calculation of this phase noise in the case of Mach-Zehnder atom interferometers and we apply this model to our thermal lithium interferometer. We are thus able to explain the observed dependence of the fringe visibility with the diffraction order. The dynamical…
Interferometers are crucial for precision measurements, including gravitational waves, laser ranging, radar, and imaging. The phase sensitivity, the core parameter, can be quantum-enhanced to break the standard quantum limit (SQL) using quantum states. However, quantum states are highly fragile and quickly degrade with losses. We design and demonstrate a quantum interferometer utilizing a beam splitter with a variable splitting ratio to protect the quantum resource against environmental impacts. The optimal phase sensitivity can reach the quantum Cramér-Rao bound of the system. This quantum in…
In a conventional atomic interferometer employing $N$ atoms, the phase sensitivity is at the standard quantum limit: $1/\sqrt{N}$. Using spin-squeezing, the sensitivity can be increased, either by lowering the quantum noise or via phase amplification, or a combination thereof. Here, we show how to increase the sensitivity, to the Heisenberg limit of $1/N$, while increasing the quantum noise by $\sqrt{N}$, thereby suppressing by the same factor the effect of excess noise. The protocol uses a Schrödinger Cat state representing a superposition of two collective states of $N$ atoms, behaving as a …
In this article we show that, in a two-arm interferometer, pure quantum states of perfect path distinguishability (particles) are geometrically equidistant from all states with constant path distinguishability D. This property is not shared by other states, such as perfect fringe-visibility (waves) or maximally entangled quantum states (entanglon). Indeed, the Bures distance between a particle and any other state depends only the distinguishability of the latter. On the contrary, the Bures distance between a wave or an entanglon, and any other single photon state depends on other set of parame…
For a generic interferometer, the conditional probability density distribution, $p(ϕ|m)$, for the phase $ϕ$ given measurement outcome $m$, will generally have multiple peaks. Therefore, the phase sensitivity of an interferometer cannot be adequately characterized by the standard deviation, such as $Δϕ\sim 1/\sqrt{N}$ (the standard limit), or $Δϕ\sim 1/N$ (the Heisenberg limit). We propose an alternative measure of phase sensitivity--the fidelity of an interferometer--defined as the Shannon mutual information between the phase shift $ϕ$\ and the measurement outcomes $m$. As an example applicati…
We consider a solvable model of a laser-interferometer measuring a weak classical force. The model takes into account dissipation of the energy by transfer to the environment at zero temperature. The sensitivity (the signal-to-noise ratio) of the device is defined as the corresponding ratio between the mean value and the variance of a certain observable. We analyze the dependence of the sensitivity upon the duration of the measurement and the photon number. For parameters typical for the LIGO project, we discuss numerical estimates.
A fundamental limit to the sensitivity of optical interferometry is thermal noise that drives fluctuations in the positions of the surfaces of the interferometer's mirrors, and thereby in the phase of the intracavity field. Schemes for reducing this thermally driven phase noise are presented in which phase shifts from concomitant strains at the surface and in the bulk of the substrate compensate the phase shift due to the displacement of the surface. Although the position of the physical surface fluctuates, the optical phase upon reflection can have reduced sensitivity to this motion.
Quantum entanglement is a resource in quantum metrology that can be distributed to two orthogonal physical quantities for the enhancement of their joint measurement sensitivity, as demonstrated in quantum dense metrology. On the other hand, we can also devote all the quantum resource to phase measurement only for optimum measurement sensitivity. Here, we experimentally implement a dual-beam scheme in an SU(1,1) interferometer for the optimum phase measurement sensitivity. We demonstrate a 3.9-dB improvement in signal-to-noise ratio over the optimum classical method and this is 3-dB better than…
We propose an enhanced optical interferometer based on tailored non-classical light generated by nonlinear dynamics and projective measurements in a three-level atom cavity QED system. A coherent state in the cavity becomes dynamically entangled with two ground states of the atom and is transformed to a macroscopic superposition state via a projective measurement on the atom. We show that the resulting highly non-classical state can improve interferometric precision measurements well beyond the shot-noise limit once combined with a classical laser pulse at the input of a Mach-Zehnder interfero…
The quantum correlation of light and atomic collective excitation can be used to compose an SU(1,1)-type hybrid light-atom interferometer, where one arm in optical SU(1,1) interferometer is replaced by the atomic collective excitation. The phase-sensing probes include not only the photon field but also the atomic collective excitation inside the interferometer. For a coherent squeezed state as the phase-sensing field, the phase sensitivity can approach the Heisenberg limit under the optimal conditions. We also study the effects of the loss of light field and the dephasing of atomic excitation …
High precision interferometers are the building blocks of precision metrology and the ultimate interferometric sensitivity is limited by the quantum noise. Here we propose and experimentally demonstrate a compact quantum interferometer involving two optical parametric amplifiers and the squeezed states generated within the interferometer are directly used for the phase-sensing quantum state. By both squeezing shot noise and amplifying phase-sensing intensity the sensitivity improvement of $4.86\pm 0.24$ dB beyond the standard quantum limit is deterministically realized and a minimum detectable…
The SU(1,1) interferometer can be thought of as a Mach-Zehnder interferometer with its linear beamsplitters replaced with parametric nonlinear optical processes. We consider the cases of bright and vacuum-seeded SU(1,1) interferometers using intensity or homodyne detectors. A simplified, truncated scheme with only one nonlinear interaction is introduced, which not only beats conventional intensity detection with a bright seed, but can saturate the phase sensitivity bound set by the quantum Fisher information. We also show that the truncated scheme achieves a sub-shot-noise phase sensitivity in…
We present a new quantum control strategy for increasing the shot-noise-limited sensitivity of optical interferometers. The strategy utilizes active phase-insensitive quantum filtering of the signal inside the interferometer and does not rely on optical squeezing. On the example of the coupled-cavity resonators, employed in the gravitational-wave detectors, we show that fully causal and stable phase-insensitive filters can improve the interferometer sensitivity by more than an order of magnitude. The role of the phase-insensitive component in such systems is to provide frequency-dependent comp…
