We study a Mach-Zehnder interferometer fed by a coherent state in one input port and vacuum in the other. We explore a Bayesian phase estimation strategy to demonstrate that it is possible to achieve the standard quantum limit independently from the true value of the phase shift and specific assumptions on the noise of the interferometer. We have been able to implement the protocol using parallel operation of two photon-number-resolving detectors and multiphoton coincidence logic electronics at the output ports of a weakly-illuminated Mach-Zehnder interferometer. This protocol is unbiased and saturates the Cramer-Rao phase uncertainty bound and, therefore, is an optimal phase estimation strategy.
We show that the phase sensitivity $Δθ$ of a Mach-Zehnder interferometer fed by a coherent state in one input port and squeezed-vacuum in the other one is i) independent from the true value of the phase shift and ii) can reach the Heisenberg limit $Δθ\sim 1/N_T$, where $N_T$ is the average number of particles of the input states. We also show that the Cramer-Rao lower bound, $Δθ\propto 1/ \sqrt{|α|^2 e^{2r} + \sinh^2r}$, can be saturated for arbitrary values of the squeezing parameter $r$ and the amplitude of the coherent mode $|α|$ by a Bayesian phase inference protocol.
We propose a scheme to achieve a Mach-Zehnder interferometry using a quantized Bose-Josephson junction with negative charging energy. The quantum adiabatic evolution through a dynamical bifurcation is used to accomplish the beam splitting and recombination. The negative charging energy ensures the existence of a path-entangled state which enhances the phase measurement precision to the Heisenberg limit. A feasible detection procedure is also presented. The scheme should be realizable with current technology.
We present a method for intrusion detection which is based on the Mach-Zehnder interference effect. This device provides monitored surveillance by continuously measuring the intensity of light collected by a pair of photodetectors. We find that our protocol allows for the detection of intrusion attempts which employ path- redirection and/or intercept-resend techniques. Expectation values for the registered output flux are provided for normal and interrupted operation.
We theoretically analyze a Mach-Zehnder interferometer with trapped condensates, and find that it is surprisingly stable against the nonlinearity induced by inter-particle interactions. The phase sensitivity, which we study for number squeezed input states, can overcome the shot noise limit and be increased up to the Heisenberg limit provided that a Bayesian or Maximum-Likelihood phase estimation strategy is used. We finally demonstrate robustness of the Mach-Zehnder interferometer in presence of interactions against condensate oscillations and a realistic atom counting error.
A new type of electron sources has emerged which permits to inject particles in a controllable manner, one at a time, into an electronic circuit. Such single electron sources make it possible to fully exploit the particles' quantum nature. We determine the single-particle coherence length from the decay of the Aharonov-Bohm oscillations as a function of the imbalance of a Mach-Zehnder interferometer connected to a single electron source. The single-particle coherence length is of particular importance as it is an intrinsic property of the source in contrast to the dephasing length.
We explore the possibility of implementing a Heisenberg-limited Mach-Zehnder interferometry via an array of trapped ions, which obey a quantum Ising model within a transverse field. Based upon adiabatic processes of increasing the Ising interaction and then decreasing the transverse field, we demonstrate a perfect transition from paramagnetism to ferromagnetic states, which can be used as the beam splitter for the multi-ion Mach-Zehnder interferometry. The achieved NOON state of the ions enables the Heisenberg-limited interferometry. Using currently available techniques for ultracold ions, we …
We use a novel scanning electron Mach-Zehnder interferometer constructed in a conventional transmission electron microscope to perform inelastic interferometric imaging with free electrons. An electron wave function is prepared in two paths that pass on opposite sides of a gold nanoparticle, where plasmons are excited before the paths are recombined to produce electron interference. We show that the measured spectra are consistent with theoretical predictions, specifically that the interference signal formed by inelastically scattered electrons is pi out of phase with respect to that formed by…
A coherent account of the connections and contrasts between the principles of com- plementarity and uncertainty is developed starting from a survey of the various formalizations of these principles. The conceptual analysis is illustrated by means of a set of experimental schemes based on Mach-Zehnder interferometry. In particu- lar, path detection via entanglement with a probe system and (quantitative) quan- tum erasure are exhibited to constitute instances of joint unsharp measurements of complementary pairs of physical quantities, path and interference observables. The analysis uses the repr…
Atomic three-grating Mach-Zehnder interferometry constitutes an important tool to probe fundamental aspects of the quantum theory. There is, however, a remarkable gap in the literature between the oversimplified models and robust numerical simulations considered to describe the corresponding experiments. Consequently, the former usually lead to paradoxical scenarios, such as the wave-particle dual behavior of atoms, while the latter make difficult the data analysis in simple terms. Here these issues are tackled by means of a simple grating working model consisting of evenly-spaced Gaussian sli…
Heisenberg-limited measurement protocols can be used to gain an increase in measurement precision over classical protocols. Such measurements can be implemented using, e.g., optical Mach-Zehnder interferometers and Ramsey spectroscopes. We address the formal equivalence between the Mach-Zehnder interferometer, the Ramsey spectroscope, and the discrete Fourier transform. Based on this equivalence we introduce the ``quantum Rosetta stone'', and we describe a projective-measurement scheme for generating the desired correlations between the interferometric input states in order to achieve Heisenbe…
We propose a new single-step scheme for the generation of a GHZ entangled state of three single-electron excitations (flying qubits). We also present a method to get a generalized GHZ-state. Our idea relies upon the most recent progress in the field of on-demand electron sources and mesoscopic Mach-Zehnder interferometry. We also provide the recipe for the unambiguous detection of this GHZ state via correlations measurements at the output, which imply the violation of a Bell-type inequality which is generalized to the case of three particles. We explain how such measurements can be achieved in…
Critical Microwave Mach-Zehnder-Type Interferometry with Dual-LO Rydberg Atoms
School of physics, Harbin Institute of Technology（哈尔滨工业大学物理学院）；College of Information Science and Electronic Engineering, Zhejiang University（浙江大学信息与电子工程学院）；Beijing Institute of Radio Measurement（北京无线电测量研究所）；School of Physics and Opotoelectric Engineering, Hangzhou Institute for Advanced Study, University of Chinese（中国科学院大学杭州高等研究院物理与光电工程学院）
We show how to implement a Mach-Zehnder interferometry based upon a string of trapped ions with artificial nonlinear interactions. By adiabatically sweeping down/up the coupling strength between two involved internal states of the ions, we could achieve the beam splitting/recombination. Based on current techniques for manipulating trapped ions, we discuss the experimental feasibility of our scheme and analyze some undesired uncertainty under realistic experimental environment.
A neutron interferometric test of interaction-free detection of the presence of an absorbing object in one arm of a neutron interferometer has been performed. Despite deviations from the ideal performance characteristics of a Mach-Zehnder interferometer it could be shown that information is obtained without interaction.
Entanglement distillation is the process of concentrating entanglement from a given quantum state. We present a technique for distillation of bi-partite polarization entanglement using interferometry. This technique can be optimized to extract maximal entanglement from any pure or mixed entangled state. A model for this method is presented and in particular we present experimental results for pure states when using polarizing Mach-Zehnder interferometers. These experimentally distilled states always demonstrate an increased violation of Bell's inequality.
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.
A polarized photon with well-defined orbital angular momentum that emerges from a Mach-Zehnder interferometer (MZI) is shown to circumvent wave-particle duality. Its polarization-resolved detection probability forms a non-sinusoidal interferometric pattern. For certain phase differences between the MZI arms, this pattern yields both reliable which-path information and high phase-sensitivity.
Mach-Zehnder interferometer, a powerful tool for a wide variety of measurements, has been realized with Bose-Einstein condensates in recent experiments. In this report, we propose and analyze a realizable scheme for performing a Heisenberg-limited Mach-Zehnder interferometry with dipolar spin-1 condensate. Based upon adiabatic processes of sweeping the transverse magnetic field, we demonstrate a perfect phase transition, which accomplishes the beam splitter, phase shifter and recombiner as for a Mach-Zehnder interferometer. The attractive dipolar interaction ensures the existence of a path-ent…
A novel single-photon Mach-Zehnder interferometer terminated at two different frequencies realizes the nonlinear frequency conversion of optical quantum superposition states. The information-preserving character of the relevant unitary transformation has been experimentally demonstrated for input qubits and ebits. Besides its own intrinsic fundamental interest, the new scheme will find important applications in modern quantum information technology.
We propose a new architecture for implementing electronic interferometry in quantum Hall bars. It exploits scattering among parallel edge channels. In contrast to previous developments, this one employs a simply-connected mesa admitting serial concatenation of building elements closer to optical analogues. Implementations of Mach-Zehnder and Hambury-Brown-Twiss interferometers are discussed together with new structures yet unexplored in quantum electronics.
We show that no local, hidden variable model can be given for two-channel states exhibiting both a sufficiently high interference visibility and a sufficient degree of anticorrelation in a Mach-Zehnder interferometer.
When a photon interferes with itself while traversing a Mach-Zehnder inteferometer, the output port where it emerges is influenced by the phase difference between the interferometer arms. This allows for highly precise estimation of the path length difference (delay) but is extremely sensitive to phase noise. By contrast, a delay between the arms of the two-photon Hong-Ou-Mandel interferometer directly affects the relative indistinguishability of the photon pair, affecting the rate of recorded coincidences. This likewise allows for delay estimation; notably less precise but with the advantage …
We use a parametric down-conversion source pumped by a short coherence-length continuous-wave (CW) diode laser to perform two-photon interferometry in an intermediate regime between the more familiar Franson-type experiments with a long coherence-length pump laser, and the short pulsed pump "time-bin" experiments pioneered by Gisin's group. The use of a time-bin-like Mach-Zehnder interferometer in the CW pumping beam induces coherence between certain two-photon amplitudes, while the CW nature of the experiment prevents the elimination of remaining incoherent ones. The experimental results high…
We suggest a self-testing teleportation configuration for photon q-bits based on a Mach-Zehnder interferometer. That is, Bob can tell how well the input state has been teleported without knowing what that input state was. One could imagine building a "locked" teleporter based on this configuration. The analysis is performed for continuous variable teleportation but the arrangement could equally be applied to discrete manipulations.
We demonstrate that networks of locally connected processing units with a primitive learning capability exhibit behavior that is usually only attributed to quantum systems. We describe networks that simulate single-photon beam-splitter and Mach-Zehnder interferometer experiments on a causal, event-by-event basis and demonstrate that the simulation results are in excellent agreement with quantum theory.
The employment of path entangled multiphoton states enables measurement of phase with enhanced precision. It is common practice to demonstrate the unique properties of such quantum states by measuring super-resolving oscillations in the coincidence rate of a Mach-Zehnder interferometer. Similar oscillations, however, have also been demonstrated in various configurations using classical light only; making it unclear what, if any, are the classical limits of this phenomenon. Here we derive a classical bound for the visibility of super-resolving oscillations in a Mach-Zehnder. This provides an ea…
A new scheme for generating amplitude squeezed light by means of soliton self-phase modulation is experimentally demonstrated. By injecting 180-fs pulses into an equivalent Mach-Zehnder fiber interferometer, a maximum noise reduction of $4.4 \pm 0.3$ dB is obtained ($6.3 \pm 0.6$ dB when corrected for losses). The dependence of noise reduction on the interferometer splitting ratio and fiber length is studied in detail.
In this paper, we have estimated the temperature dependent path predictability for an electronic Mach-Zehnder interferometer. The increment of path predictability can directly be associated with stronger decoherence process. We have also theoretically predicted that placing two detectors in both the paths, which are at the same equilibrium temperature with the system, erases all the memory of path information and hence acts like a quantum eraser.
We study the wave-particle duality in a general Mach-Zehnder interferometer with an asymmetric beam splitter from the viewpoint of quantum information theory. The correlations(including the classical correlation and the quantum correlation) between the particle and the which-path detector are derived when they are in pure state or mixed state at the output of Mach-Zehnder interferometer. It is found that the fringe visibility and the correlations are effected by the asymmetric beam splitter and the input state of the particle. The complementary relations between the fringe visibility and the c…
Sensitive detection of magnetic field is one of the open problem in metrology. Here, we propose an Mach-Zehnder atomic interferometry to sensitively detect the very weak local magnetic field, which is expelled by the superconductor (as the "testing magnet") due to the Meissner effect. The induced magnetic field gradient near the superconductor provides a centripetal acceleration of the atomic motion in the interferometry and thus can be detected by using the atomic interferences. Given gravity acceleration of the atoms have been measured at the accuracy of $10^{-12}$ g, the measured sensitivit…
We study the non-relativistic limit of a paradigmatic model in Relativistic Quantum Mechanics, the two-dimensional Dirac oscillator. Remarkably, we find a novel kind of Zitterbewegung which persists in this non-relativistic regime, and leads to an observable deformation of the particle orbit. This effect can be interpreted in terms of a Ramsey Interferometric phenomenon, allowing an insightful connection between Relativistic Quantum Mechanics and Quantum Optics. Furthermore, subsequent corrections to the non-relativistic limit, which account for the usual spin-orbit Zitterbewegung, can be neat…
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 give an analytical result for the quantum Fisher information of entangled coherent States in a lossy Mach-Zehnder Interferometer recently proposed by J. Joo et al. [Phys. Rev. Lett. 107, 083601(2011)]. For small loss of photons, we find that the entangled coherent state can surpass the Heisenberg limit. Furthermore, The formalism developed here is applicable to the study of phase sensitivity of multipartite entangled coherent states.
Using spontaneous parametric down conversion and a 50:50 beam splitter, we generate coaxial polarization-entangled photon pairs, of which the two photons are far separated from each other. The photons are then sent one by one through one port of a modified Mach-Zehnder interferometer. We observe interference fringes with a periodicity half of the single-photon wavelength, independent of the distance between the photons. This feature can find applications in quantum-enhanced measurement.
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.
We revisit, in the framework of Mach-Zehnder interferometry, the connection between the complementarity and uncertainty principles of quantum mechanics. Specifically, we show that, for a pair of suitably chosen observables, the trade-off relation between the complementary path information and fringe visibility is equivalent to the uncertainty relation given by Schr\"odinger and Robertson, and to the one provided by Landau and Pollak as well. We also employ entropic uncertainty relations (based on Rényi entropic measures) and study their meaning for different values of the entropic parameter. …
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.
We study the effect of quantum motion in a Mach-Zehnder interferometer where ultracold, two-level atoms cross a $π/2 $-$π$-$π/2$ configuration of separated, laser illuminated regions. Explicit and exact expressions are obtained for transmission amplitudes of monochromatic, incident atomic waves using recurrence relations which take into account all possible paths: the direct ones usually considered in the simple semiclassical treatment, but including quantum motion corrections, and the paths in which the atoms are repeatedly reflected at the fields.
We investigate the dependence of the fidelity of a Mach-Zehnder quantum interferometer on the prior information about the phase, for Fock state input and for maximally entangled (N00N) state input. For no prior information, the fidelity for Fock state input is greater than for N00N state input. In the limit of a narrow distribution describing the prior information, we find that both Fock and N00N state inputs lead to nearly equal fidelity.
The scheme of Clauser and Dowling (Phys. Rev. A 53, 4587 (1996)) for factoring $N$ by means of an N-slit interference experiment is translated into an experiment with a single Mach-Zehnder interferometer. With dispersive phase shifters the ratio of the coherence length to wavelength limits the numbers that can be factored. A conservative estimate permits $N \approx 10^7$. It is furthermore shown, that sine and cosine Fourier coefficients of a real periodic function can be obtained with such an interferometer.
Phase measurement using a lossless Mach-Zehnder interferometer with certain entangled $N$-photon states can lead to a phase sensitivity of the order of 1/N, the Heisenberg limit. However, previously considered output measurement schemes are different for different input states to achieve this limit. We show that it is possible to achieve this limit just by the parity measurement for all the commonly proposed entangled states. Based on the parity measurement scheme, the reductions of the phase sensitivity in the presence of photon loss are examined for the various input states.
A method for mitigating the readout delay characteristic of Bragg-based atom interferometry is presented, utilizing an asymmetric Mach-Zehnder interferometer sequence to generate spatial fringes that are read out while still overlapped. The time of flight after the final beamsplitter is engineered to facilitate constructive overlap of the fringes on the output states. A noise analysis performed using a precision atom interferometer is presented, comparing the traditional separated symmetric scheme with that of the separated and overlapped asymmetric scheme showing no significant increase in sh…
We experimentally demonstrate that a long distance actively stabilized Mach-Zehnder fibre optical interferometer can be used to reliably implement the GV95 quantum cryptography protocol employing orthogonal states. A proof-of-principle experiment using an interferometer composed of 1 km of spooled optical fibres connecting Alice and Bob is performed. The active stabilisation against phase drifts is done with a classical channel wavelength-multiplexed with the quantum channel and a high stable visibility implying a QBER of 2.2 % is shown.
The Mach-Zehnder interferometric setup quantitatively characterizing the wave-particle duality implements in fact a joint measurement of two unsharp observables. We present a necessary and sufficient condition for such a pair of unsharp observables to be jointly measurable. The condition is shown to be equivalent to a duality inequality, which for the optimal strategy of extracting the which-path information is more stringent than the Jaeger-Shimony-Vaidman-Englert inequality.
We have constructed an atom interferometer of the Mach-Zehnder type, operating with a supersonic beam of lithium. Atom diffraction uses Bragg diffraction on laser standing waves. With first order diffraction, our apparatus has given a large signal and a very good fringe contrast (74%), which we believe to be the highest ever observed with atom interferometers. This apparatus will be applied to high sensitivity measurements
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]
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…
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 discuss complementarity and uncertainty in a gedanken Which-Way (Welcher-Weg) experiment in a Mach-Zehnder interferometer. Although a Welcher-Weg measurement can be performed with only a negligible amount of momentum change in the detector itself, the change in the interference pattern must correspond to a change in the momentum distribution of the particle. Entanglement between the Mach-Zehnder interferometer and the Welcher-Weg measurement introduces orthogonality that disables the interference of the particle, but simultaneously, it also generates a momentum transfer between the particle…
