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.
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.
In this paper we address the problem of optimizing an unbalanced Mach-Zehnder interferometer, for a given pure input state and considering a specific detection scheme. While the optimum transmission coefficient of the first beam splitter can be uniquely determined via the quantum Fisher information only [Phys. Rev. A 105, 012604 (2022)], the second beam splitter transmission coefficient is detection-scheme dependent, too. We systematically give analytic solutions for the optimum transmission coefficient of the second beam splitter for three types of widely used detection schemes. We provide de…
Application of frequency-dependent squeezed vacuum improves the force sensitivity of optomechanical interferometer beyond the standard quantum limit by a factor of $e^{-r}$, where $r$ is the squeezing parameter. In this work, we show that the application of squeezed light along with quantum optical restoring force can enhance the sensitivity beyond the standard quantum limit by a factor of $\sqrt{e^{-2r}ζ/4Δ}$, where $0< ζ/Δ<1$, with $ζ$ as the optomechanical cavity decay rate and $Δ$ as the detuning between cavity eigenfrequency and driving field. The technique described in this article is re…
The phase uncertainty of an unseeded nonlinear interferometer, where the output of one nonlinear crystal is transmitted to the input of a second crystal that analyzes it, is commonly said to be below the shot-noise level but highly dependent on detection and internal loss. Unbalancing the gains of the first (source) and second (analyzer) crystals leads to a configuration that is tolerant against detection loss. However, in terms of sensitivity, there is no advantage in choosing a stronger analyzer over a stronger source, and hence the comparison to a shot-noise level is not straightforward. In…
Interferometry is a widely-used technique for precision measurements in both classical and quantum contexts. One way to increase the precision of phase measurements, for example in a Mach-Zehnder interferometer (MZI), is to use high-intensity lasers. In this paper we study the phase sensitivity of a MZI in two detection setups (difference intensity detection and single-mode intensity detection) and for three input scenarios (coherent, double coherent and coherent plus squeezed vacuum). For the coherent and double coherent input, both detection setups can reach the quantum Cramer-Rao bound, alt…
We propose an alternative scheme for phase estimation in a Mach-Zehnder interferometer (MZI) with photon recycling. It is demonstrated that with the same coherent-state input and homodyne detection, our proposal possesses a phase sensitivity beyond the traditional MZI. For instance, it can achieve an enhancement factor of 9.32 in the phase sensitivity compared with the conventional scheme even with a photon loss of 10% on the photon-recycled arm. From another point of view, the quantum Cramer-Rao bound (QCRB) is also investigated. It is found that our scheme is able to achieve a lower QCRB tha…
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.
In this work, we discuss two phase-measurement methods for the Mach-Zehnder interferometer (MZI) in the presence of internal losses and give the corresponding optimum conditions. We find theoretically that when the core parameters (reflectivities, phase difference) are optimized, the phase sensitivity of the two methods can reach a generalized bound on precision: standard interferometric limit (SIL). In the experiment, we design an MZI with adjustable beam splitting ratios and losses to verify phase sensitivity optimization. The sensitivity improvements at loss rates from 0.4 to 0.998 are demo…
The Cramér-Rao bound and the quantum Fisher information have been tools used extensively for interferometric phase sensitivity. Most scenarios considering a Mach-Zehnder interferometer with two input sources focused on the phase-matched case, when the Fisher information is maximal. Under this constraint, the best sensitivity is achieved for a balanced (50/50) input beam splitter. In this paper, we take a different approach: we allow the beam splitter transmission coefficient as well as the input phase mismatch to be variable parameters. We then search for a pair of these parameters that maximi…
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…
Nonlinear SU(1,1) interferometers are fruitful and promising tools for spectral engineering and precise measurements with phase sensitivity below the classical bound. Such interferometers have been successfully realized in bulk and fiber-based configurations. However, rapidly developing integrated technologies provide higher efficiencies, smaller footprints, and pave the way to quantum-enhanced on-chip interferometry. In this work, we theoretically realised an integrated architecture of the multimode SU(1,1) interferometer which can be applied to various integrated platforms. The presented int…
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.
Nonlinear interferometers with correlated photons hold a promise to advance optical characterization and metrology techniques by improving their performance and affordability. Nonlinear interferometers offer the sub-shot noise phase sensitivity and enable measurements in detection-challenging regions using inexpensive and efficient components. The sensitivity of such interferometers, defined by the ability to measure small shifts of interference fringes, can be significantly enhanced by using multiple nonlinear elements, or crystal superlattices. However, to date, the experiments with no more …
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…
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…
The best performance of a Mach-Zehnder interferometer is achieved with the input state |N_T/2 + 1>|N_T/2-1 > + |N_T/2 - 1>|N_T/2+1>, being N_T the total number of atoms/photons. This gives: i) a phase-shift error confidence C_{68%}=2.67/N_T with ii) a single interferometric measurement. Different input quantum states can achieve the Heisenberg scaling ~ 1/N_T but with higher prefactors and at the price of a statistical analysis of two or more independent measurements.
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）
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 study the ultimate bounds on the sensitivity of a Bloch-oscillation atom interferometer where the external force is estimated from the measurement of the on-site atomic density. For external forces such that the energy difference between lattice sites is smaller than the tunneling energy, the atomic wave-function spreads over many lattice sites, increasing the separation between the occupied modes of the lattice and naturally enhancing the sensitivity of the interferometer. To investigate the applicability of this scheme we estimate the effect of uncontrolled fluctuations of the tunneling e…
A new type of quantum entangled interferometer was recently realized that employs parametric amplifiers as the wave splitting and recombination elements. The quantum entanglement stems from the parametric amplifiers, which produce quantum correlated fields for probing the phase change signal in the interferometer. This type of quantum entangled interferometer exhibits some unique properties that are different from traditional beam splitter-based interferometers such as Mach-Zehnder interferometers. Because of these properties, it is superior to the traditional interferometers in many aspects, …
Atom interferometers have been used to measure acceleration with at best a $T^2$ scaling in sensitivity as the interferometer time $T$ is increased. This limits the sensitivity to acceleration which is theoretically achievable by these configurations for a given frequency of acceleration. We predict and experimentally measure the acceleration-sensitive phase shift of a large-momentum-transfer atom interferometer based upon Bloch oscillations. Using this novel interferometric scheme we demonstrate an improved scaling of sensitivity which will scale as $T^3$. This enhanced scaling will allow an …
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 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-n…
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.
