With the help of quantum entanglement, quantum dense metrology (QDM) is a technique that can perform the joint estimates of two conjugate quantities such as phase and amplitude modulations of an optical field with an accuracy beating the standard quantum limit simultaneously. SU(1,1) interferometers (SUI) can realize QDM with detection loss tolerance but is limited in absolute sensitivity. Here we present a QDM scheme with a linear interferometer (SU(2)) nested inside an SU(1,1) interferometer. By using a degenerate SUI and controlling the phase angle of the phase-sensitive amplifers in SUI, we can achieve the optimum quantum enhancement in the measurement precision of arbitrary mixture of phase and amplitude modulation.
We examine quantum detection via a Michelson interferometer embedded in a gas with Kerr nonlinearity. This nonlinear interferometer is illuminated by pulses of classical light. This strategy combines the robustness against practical imperfections of classical light with the improvement provided by nonlinear processes. Regarding ultimate quantum limits, we stress that, as a difference with linear schemes, the nonlinearity introduces pulse duration as a new variable into play along with the energy resources.
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…
We present the results of a theoretical work discussing the propagation of an electromagnetic wave through nested Mach-Zehnder interferometers using classical optics and standard quantum theory. We show that some seemingly surprising effects at first sight, which are often explained in the literature using the two-state vector formalism (TSVF), are a direct consequence of destructive or constructive interference and thus there is no need for the unconventional TSVF formulation. We show that the probability of a photon detection derived from the weak value used in TSVF can be interpreted as the…
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 …
The quantum Fisher information (QFI) in SU(2) and SU(1,1) interferometers was considered, and the QFI-only calculation was overestimated. In general, the phase estimation as a two-parameter estimation problem, and the quantum Fisher information matrix (QFIM) is necessary. In this paper, we theoretically generalize the model developed by Escher et al [Nature Physics 7, 406 (2011)] to the QFIM case with noise and study the ultimate precision limits of SU(2) and SU(1,1) interferometers with photon losses because photon losses as a very usual noise may happen to the phase measurement process. Usin…
An ideal experiment is designed to determine the past of a particle in the nested Mach-Zehnder interferometer (MZI) by using standard quantum mechanics with quantum non-demolition measurements. We find that when the photon reaches the detector, it only follows one arm of the outer interferometer and leaves no trace in the inner MZI; while when it goes through the inner MZI, it cannot reach the detector. Our result obtained from the standard quantum mechanics is contradict to the statement based on two state vector formulism, "the photon did not enter the (inner) interferometer, the photon neve…
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 …
Griffiths [Phys. Rev. A 94, 032115 (2016)] analyzed, in the framework of consistent histories interpretation, the controversy regarding the approach to the past of a quantum particle introduced by Vaidman [Phys. Rev. A 87, 052104 (2013)]. I argue that Griffith's criticism of my approach using analysis of experiments with weak probes is unfounded.
The correctness of the consistent histories analysis of weakly interacting probes, related to the path of a particle, is maintained against the criticisms in the Comment, and against the alternative approach described there, which receives no support from standard (textbook) quantum mechanics.
Potocek and Ferenczi [Phys. Rev. A 92, 023829 (2015)] provided an analysis of the experimental evidence obtained by Danan et al. [Phys. Rev. Lett. 111, 240402 (2013)] for the surprising behaviour of photons passing through an interferometer, in particular, motion on disconnected paths. Potocek and Ferenczi reproduced the results of the experiment, but when analyzing its modification, they claimed that the reasoning of Danan et al., which led to disconnected paths, is erroneous. It is argued here that the criticism of Potocek and Ferenczi is unfounded.
Bartkiewicz et al. [Phys. Rev. A 91, 012103 (2015)] provided an alternative analysis of experiment performed by Danan et al. [Phys. Rev. Lett. 111, 240402 (2013)] which presented surprising evidence regarding the past of photons passing through an interferometer. They argued that the quantity used by Danan et al. is not a suitable which-path witness, and proposed an alternative. It is argued that the quantum and classical analyses of Bartkiewicz et al. are inconsistent and both are inappropriate for describing the past of photons in a properly working interferometer.
While much of the technical analysis in the preceding Comment [1] is correct, in the end it confirms the conclusion reached in my previous work [2]: a consistent histories analysis provides no support for the claim of counterfactual quantum communication put forward in [3]
Possible paths of a photon passing through a nested Mach-Zehnder interferometer on its way to a detector are analyzed using the consistent histories formulation of quantum mechanics, and confirmed using a set of weak measurements (but not weak values). The results disagree with an analysis by Vaidman [ Phys.\ Rev.\ A 87 (2013) 052104 ], and agree with a conclusion reached by Li et al.\ [ Phys.\ Rev.\ A 88 (2013) 046102 ]. However, the analysis casts serious doubt on the claim of Salih et al.\ (whose authorship includes Li et al.) [ Phys.\ Rev.\ Lett.\ 110 (2013) 170502 ] to have constructed a …
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 develop general tools to characterise and efficiently compute relevant observables of multimode $N$-photon states generated in non-linear decays in one-dimensional waveguides. We then consider optical interferometry in a Mach-Zender interferometer where a $d$-mode photonic state enters in each arm of the interferometer. We derive a simple expression for the Quantum Fisher Information in terms of the average photon number in each mode, and show that it can be saturated by number-resolved photon measurements that do not distinguish between the different $d$ modes.
Quantum measurements have been intensively researched over decades due to quantum advantage of Heisenberg limit beating the standard quantum limit toward potential applications of quantum metrology. The kernel of quantum measurements is in the quantum correlation between bipartite photon pairs or squeezed light quenched by one parameter over corresponding noncommuting variable satisfying Heisenberg uncertainty principle. As a result, quantum measurements bring a quantum gain of the square root law in measurement sensitivity. Photonic de Broglie waves (PBW) have been the key feature of such a g…
Recently, a scheme based on the method of weak measurements to register the trajectories of photons passing through a nested Mach-Zehnder interferometer was proposed [L. Vaidman, Phys. Rev. A \textbf{87}, 052104 (2013)] and then realized [A. Danan, D. Farfurnik, S. Bar-Ad, et al., Phys. Rev. Lett. \textbf{111}, 240402 (2013)]. Interpreting the results of the experiment, the authors concluded that "the photons do not always follow continuous trajectories". It is shown in this work that these results can be easily and clearly explained in terms of traditional classical electrodynamics or quantum…
In a recent paper, arXiv:1604.04596, Griffiths questioned - based on an informative consistent-histories (CH) argument - the counterfactuality, for one of the bit choices, of Salih et al.'s protocol for communicating without sending physical particles, Phys. Rev. Lett. 110, 170502 (2013). Here, we first show that for the Mach-Zehnder version used to explain our protocol, no family of consistent histories exists where any history has the photon travelling through the communication channel, thus rendering the question of whether the photon was in the communication channel meaningless from a CH v…
We use a small atomic Bose-Einstein condensate as an interferometric scanning probe to map out a microwave field near a chip surface with a few micrometers resolution. Using entanglement between the atoms we overcome the standard quantum limit of interferometry by 4 dB and maintain enhanced performance for interrogation times up to 20 ms. This demonstrates the usefulness of quantum metrology with entangled states when the particle number is limited due to the small probe size. Extending atom interferometry to micrometer spatial resolution enables new applications in electromagnetic field sensi…
An experiment with nested Mach-Zehnder interferometer [Phys. Rev. Lett. 111, 240402 (2013)] has been recently implemented with neutrons [Phys. Rev. A 97, 052111 (2018)]. Which-path information has been extracted from faint traces the neutrons left, providing operational meaning to "the particle's path". The authors of the neutron interference experiment criticised the conclusions obtained by the authors of the optical experiment. I refute the criticism and argue that the results of the neutron interference experiment actually support the surprising picture of the past of the particle in the ne…
Two-mode interferometers, such as Michelson interferometer based on two spatial optical modes, lay the foundations for quantum metrology. Instead of exploring quantum entanglement in the two-mode interferometers, a single bosonic mode also promises a measurement precision beyond the shot-noise limit (SNL) by taking advantage of the infinite-dimensional Hilbert space of Fock states. However, the experimental demonstration still remains elusive. Here, we demonstrate a single-mode phase estimation that approaches the Heisenberg limit (HL) unconditionally. Due to the strong dispersive nonlinearity…
We argue that the modification proposed by Li et al. [Chin. Phys. Lett. 32, 050303 (2015)] to the experiment of Danan et al. [Phys. Rev. Lett. 111, 240402 (2013)] does not test the past of the photon as characterised by local weak traces. Instead of answering the questions: (i) Were the photons in A? (ii) Were the photons in B? (iii) Were the photons in C? the proposed experiment measures a degenerate operator answering the questions: (i) Were the photons in A? (ii) Were the photons in B and C together? A negative answer to the last question does not tell us if photons were present in B or C. …
By the means of the standard quantum mechanics formalism I present an explicit derivation of the structure of power spectra in Danan {\em et al.} and Zhou {\em et al.} experiments with nested dynamically changing Mach-Zehnder interferometers. The analysis confirms that we observe prominent, first-order peaks on frequencies related to some of the elements of the interferometer, but not on others. However, as I shall demonstrate, there are also other, weaker effects related to all relevant elements of the setup. In case of the Danan {\em et al.} setup, there are even peaks at all frequencies of …
In this paper, we present a coherent state-vector method which can explain the results of a nested linear Mach-Zehnder Interferometric experiment. Such interferometers are used widely in Quantum Information and Quantum Optics experiments and also in designing quantum circuits. We have specifically considered the case of an experiment by Danan \emph{et al.} (Phys. Rev. Lett. \textbf{111}, 240402 (2013)) where the outcome of the experiment was spooky by our intuitive guesses. However we have been able to show by our method that the results of this experiment is indeed expected within the standar…
Quantum number-path entanglement is a resource for super-sensitive quantum metrology and in particular provides for sub-shotnoise or even Heisenberg-limited sensitivity. However, such number-path entanglement has thought to have been resource intensive to create in the first place --- typically requiring either very strong nonlinearities, or nondeterministic preparation schemes with feed-forward, which are difficult to implement. Very recently, arising from the study of quantum random walks with multi-photon walkers, as well as the study of the computational complexity of passive linear optica…
Scattershot photon sources are known to have useful properties for optical quantum computing and boson sampling purposes, in particular for scaling to large numbers of photons. This paper investigates the application of these scattershot sources towards the metrological task of estimating an unknown phase shift. In this regard, we introduce three different scalable multimode interferometers, and quantify their quantum Fisher information performance using scattershot sources with arbitrary system sizes. We show that two of the interferometers need the probing photons to be in certain input conf…
Bragg interferometers, operating using pseudospin-1/2 systems composed of two momentum states, have become a mature technology for precision measurements. State-of-the-art Bragg interferometers are rapidly surpassing technical limitations and are soon expected to operate near the projection noise limit set by uncorrelated atoms. Despite the use of large numbers of atoms, their operation is governed by single-atom physics. Motivated by recent proposals and demonstrations of Raman gravimeters in cavities, we propose a scheme to squeeze directly on momentum states for surpassing the projection no…
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 examine the precision limits of Hong-Ou-Mandel (HOM) timing measurements, as well as precision limits applying to generalized two-photon measurements. As a special case, we consider the use of two-photon measurements using photons with variable bandwidths and frequency correlations. When the photon bandwidths are not equal, maximizing the measurement precision involves a trade-off between high interference visibility and strong frequency anticorrelations, with the optimal precision occuring when the photons share non-maximal frequency anticorrelations. We show that a generalized measurement…
We identify precision limits for the simultaneous estimation of multiple parameters in multimode interferometers. Quantum strategies to enhance the multiparameter sensitivity are based on entanglement among particles, modes or combining both. The maximum attainable sensitivity of particle-separable states defines the multiparameter shot-noise limit, which can be surpassed without mode entanglement. Further enhancements up to the multiparameter Heisenberg limit are possible by adding mode entanglement. Optimal strategies which saturate the precision bounds are provided.
In an recent work with the title "Asking Photons Where They Have Been", Danan et al. experimentally demonstrate an intriguing behavior of photons in an interferometer [Phys. Rev. Lett. 111, 240402 (2013)]. In their words: "The photons tell us that they have been in the parts of the interferometer through which they could not pass." They interpret the results using the two-state vector formalism of quantum theory and say that, although an explanation of the experimental results in terms of classical electromagnetic waves in the interferometer is possible (and they provide a partial description)…
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 an interferometric scheme for the estimation of a linear combination with non-negative weights of an arbitrary number $M>1$ of unknown phase delays, distributed across an $M$-channel linear optical network, with Heisenberg-limited sensitivity. This is achieved without the need of any sources of photon-number or entangled states, photon-number resolving detectors or auxiliary interferometric channels. Indeed, the proposed protocol remarkably relys upon a single squeezed state source, an antisqueezing operation at the interferometer output, and on-off photodetectors.
Geometric bounds on multiparameter Heisenberg scaling in optical metrology with limited squeezed resources
University of Portsmouth（朴茨茅斯大学）；Quantum Science and Technology Hub, University of Portsmouth（朴茨茅斯大学量子科学与技术中心）；Università di Bari（巴里大学）；INFN, Sezione di Bari（意大利国家核物理研究所巴里分部）；Institute of Cosmology and Gravitation, University of Portsmouth（朴茨茅斯大学宇宙学与引力研究所）
Information recycling has been shown to improve the sensitivity of atom interferometers by exploiting atom-light entanglement. In this paper, we apply information recycling to an interferometer where the input quantum state has been partially transferred from some donor system. We demonstrate that when the quantum state of this donor system is from a particular class of number-correlated Heisenberg-limited states, information recycling yields a Heisenberg-limited phase measurement. Crucially, this result holds irrespective of the fraction of the quantum state transferred to the interferometer …
Quantum metrology utilizes entanglement for improving the sensitivity of measurements. Up to now the focus has been on the measurement of just one out of two non-commuting observables. Here we demonstrate a laser interferometer that provides information about two non-commuting observables, with uncertainties below that of the meter's quantum ground state. Our experiment is a proof-of-principle of quantum dense metrology, and uses the additional information to distinguish between the actual phase signal and a parasitic signal due to scattered and frequency shifted photons. Our approach can be r…
We find a phase matching condition for enhancement of sensitivity in a Mach-Zehnder interferometer illuminated by an arbitrary state in one input port and an odd(even) state in the other port. Under this condition, the Fisher information becomes maximal with respect to the relative phase of two modes and the phase sensitivity is enhanced. For the case with photon losses, we further find that the phase matching condition keeps unchanged with a coherent state and a coherent superposition state as the input states.
In [JETP Lett. 105(3), 152 (2017)], a clear and comprehensive analysis of the paradoxical results of experiment [Phys. Rev. Lett. 111, 240402 (2013)] was carried out on the basis of the classical wave theory of light, which presupposes the continuity of possible of light paths. It was shown that the paradoxical results of the experiment are due not to the discontinuity of the trajectories of light, as claimed in [Phys. Rev. Lett. 111, 240402 (2013)], but to the used way of detecting the path of photons. The experiment modification proposed in [JETP Lett. 105(3), 152 (2017)] allows us to elimin…
Entangled photons can be used to make measurements with an accuracy beyond that possible with classical light. While most implementations of quantum metrology have used states made up of a single colour of photons, we show that entangled states of two colours can show supersensitivity to optical phase and path-length by using a photonic crystal fibre source of photon pairs inside an interferometer. This setup is relatively simple and robust to experimental imperfections. We demonstrate sensitivity beyond the standard quantum limit and show super-resolved interference fringes using entangled st…
To acquire the best path-entangled photon Fock states for robust quantum optical metrology with parity detection, we calculate phase information from a lossy interferometer by using twin entangled Fock states. We show that (a) when loss is less than 50% twin entangled Fock states with large photon number difference give higher visibility while when loss is higher than 50% the ones with less photon number difference give higher visibility; (b) twin entangled Fock states with large photon number difference give sub-shot-noise limit sensitivity for phase detection in a lossy environment. This res…
Quantum metrology allows us to attain a measurement precision that surpasses the classically achievable limit by using quantum characters. The metrology precision is raised from the standard quantum limit (SQL) to the Heisenberg limit (HL) by using entanglement. However, it was reported that the HL returns to the SQL in the presence of local dephasing environments under the long encoding-time condition. We evaluate here the exact impacts of local dissipative environments on quantum metrology, based on the Ramsey interferometer. It is found that the HL is asymptotically recovered under the long…
We propose a class of path-entangled photon Fock states for robust quantum optical metrology, imaging, and sensing in the presence of loss. We model propagation loss with beam-splitters and derive a reduced density matrix formalism from which we examine how photon loss affects coherence. It is shown that particular entangled number states, which contain a special superposition of photons in both arms of a Mach-Zehnder interferometer, are resilient to environmental decoherence. We demonstrate an order of magnitude greater visibility with loss, than possible with N00N states. We also show that t…
We propose a generalized form of entangled coherent states (ECS) and apply them in a multi-arm optical interferometer to estimate multiple phase shifts. We obtain the quantum Cramer-Rao bounds for both the linear and nonlinear parameterization protocols. Through the analysis, we find that, utilizing the simultaneous estimation, this generalized form of ECS gives a better precision than the generalized NOON states [Phys. Rev. Lett. \textbf{111}, 070403 (2013)]. Moreover, comparing with the independent estimation, both the linear and nonlinear protocols have the same advantage in the relation to…
By considering matter wave bright solitons from weakly coupled Bose-Einstein condensates trapped in a double-well potential, we study the formation of macroscopic non-classical states, including Schrödinger-cat superposition states and maximally path entangled $N00N$-states. With these macroscopic states, we examine Mach-Zehnder interferometer in the context of parity measurements, in order to obtain Heisenberg limit accuracy for linear phase shift measurement. We reveal that the ratio between two-body scattering length and intra-well hopping parameter can be measured with the scaling beyond t…
Quantum metrology overcomes standard precision limits and plays a central role in science and technology. Practically it is vulnerable to imperfections such as decoherence. Here, we demonstrate quantum metrology for noisy channels such that entanglement with ancillary qubits enhances the quantum Fisher information for phase estimation but not otherwise. Our photonic experiment covers a range of noise for various types of channels, including for two randomly alternating channels such that assisted entanglement fails for each noisy channel individually. We have simulated noisy channels by implem…
We report the experimental realization of squeezed quantum states of light, tailored for new applications in quantum communication and metrology. Squeezed states in a broad Fourier frequency band down to 1 Hz has been observed for the first time. Nonclassical properties of light in such a low frequency band is required for high efficiency quantum information storage in electromagnetically induced transparency (EIT) media. The states observed also cover the frequency band of ultra-high precision laser interferometers for gravitational wave detection and can be used to reach the regime of quantu…
We introduce the concept of entanglement enhanced interferometry from the viewpoint of the detected photons. The standard quantum limit is achieved when sequentially detected photons are assumed to be in an uncorrelated product state. However when we access the correlations between the detected photons that existed before the interferometer it becomes clear that entanglement enhanced measurement beyond the quantum limit could be achieved independent of loss. We describe possible realisations of this post-measurement entanglement detection using a small array of spin photon entangling gates. We…
We consider a nanomechanical analogue of a nonlinear interferometer, consisting of two parallel, flexural nanomechanical resonators, each with an intrinsic Duffing nonlinearity and with a switchable beamsplitter-like coupling between them. We calculate the precision with which the strength of the nonlinearity can be estimated and show that it scales as $1/n^{3/2}$, where $n$ is the mean phonon number of the initial state. This result holds even in the presence of dissipation, but assumes the ability to make measurements of the quadrature components of the nanoresonators.
