We study the dynamics of classical and quantum systems undergoing a continuous measurement of position by schematizing the measurement apparatus with an infinite set of harmonic oscillators at finite temperature linearly coupled to the measured system. Selective and nonselective measurement processes are then introduced according to a selection of or an average over all possible initial configurations of the measurement apparatus. At quantum level, the selective processes are described by a nonlinear stochastic Schrödinger equation whose solutions evolve into properly defined coherent states in the case of linear systems. For arbitrary measured systems, classical behavior is always recovered in the macroscopic limit.
Bell inequalities for position measurements are derived using the bits of the binary expansion of position-measurement results. Violations of these inequalities are obtained from the output state of the Non-degenerate Optical Parametric Amplifier.
We prove an uncertainty relation, which imposes a bound on any joint measurement of position and momentum. It is of the form $(ΔP)(ΔQ)\geq C\hbar$, where the `uncertainties' quantify the difference between the marginals of the joint measurement and the corresponding ideal observable. Applied to an approximate position measurement followed by a momentum measurement, the uncertainties become the precision $ΔQ$ of the position measurement, and the perturbation $ΔP$ of the conjugate variable introduced by such a measurement. We also determine the best constant $C$, which is attained for a unique p…
We present a hitherto unknown fundamental limitation to a basic measurement: that of the position of a quantum object when the total momentum of the object and apparatus is conserved. This result extends the famous Wigner-Araki-Yanase (WAY) theorem, and shows that accurate position measurements are only practically feasible if there is a large momentum uncertainty in the apparatus.
``Weak'', ``protective'', and ``delayed observation'' measurements are analyzed in the framework of the Bohm interpretation of quantum theory. It is argued that the above varieties of measurements manifest some difficulties of the Bohm interpretation since they show that Bohmian trajectories behave not as we would expect from a classical type model.
Multi-parameter estimation is necessary for force sensing due to simultaneous and nontrivial small changes of position and momentum. Designing quantum probes that allow simultaneous estimation of all parameters is therefore an important task. The optimal methods for estimation of the conjugate changes of position and momentum of quantum harmonic oscillator employ probes in entangled or quantum non-Gaussian states. We show that the same results can be obtained in a significantly more feasible fashion by employing independent sets of differently squeezed Gaussian states classically correlated to…
In this paper, we consider an interpretation of the Husimi function as the probability distribution of a successive measurement, which is clearly separated into measurements of the position and the momentum. We also show this successive measurement can be easily realized in the situation close to the experiment, and this measurement corresponds to one of Aharanov's weak measurement.
Position measurements are examined under the assumption that object position x_t and probe position X_t just after the measurement are expressed by a linear combination of positions x_0 and X_0 just before the measurement. The Heisenberg uncertainty relation between the position uncertainty and momentum disturbance holds when the measurement error ε(x_t) for the object position x_t is adopted as the position uncertainty. However, the uncertainty in the measurement result obtained for x_0 is the standard deviation of the measurement result, and not the measurement error ε(x_0). This difference …
The quantum interference of entangled photons forms a key phenomenon underlying various quantum-optical technologies. It is known that the quantum interference patterns of entangled photon pairs can be reconstructed classically by the time-reversal method; however, the time-reversal method has been applied only to time-frequency-entangled two-photon systems in previous experiments. Here, for the first time, we apply the time-reversal method to the position-wavevector-entangled two-photon systems: the two-photon Young interferometer and the two-photon beam focusing system. We experimentally dem…
The von Neumann interaction between a particle and an apparatus has been considered in the measurement of the position of a particle when the interaction lasts for a finite amount of time. When the measurement has finite duration, both the motion of the pointer and the particle influence the result of the measurement. Provided that the particle is in an eigenstate of its position at the start of the measurement, the pointer will indicate the arithmetic average between the initial and final position of the particle. Furthermore, the probability that the pointer will indicate a given average val…
We consider a particle in harmonic oscillator potential, whose position is periodically measured with an instrument of finite precision. We show that the distribution of the measured positions tends to a limiting distribution when the number of measurements tends to infinity. We derive the expression for the limiting position distribution and validate it with numerical simulation.
We study localization of atomic position when a three-level atom interacts with a quantized standing-wave field in the Ramsey interferometer setup. Both the field quadrature amplitude and the atomic internal state are measured to obtain the atomic position information. It is found that this dual measurement scheme produces an interference pattern superimposed on a diffraction-like pattern in the atomic position distribution, where the former pattern originates from the state-selective measurement and the latter from the field measurement. The present scheme results in a better resolution in th…
We study the quantum Brownian motion of a harmonic oscillator undergoing a sequence of generalized position measurements. Our exact analytical results capture the interplay of the measurement backaction and dissipation. Here we demonstrate that no freeze-in Zeno effect occurs upon increasing the monitoring frequency. A similar behavior is also found in the presence of generalized momentum measurements.
Any account of the emergence of classicality from quantum theory must address the fact that the quantum operators representing positions and momenta do not commute, whereas their classical counterparts suffer no such restrictions. To address this, we revive an old idea of von Neumann, and seek a pair of commuting operators $X,P$ which are, in a specific sense, "close" to the canonical non-commuting position and momentum operators, $x,p$. The construction of such operators is related to the problem of finding complete sets of orthonormal phase space localized states, a problem severely limited …
We describe the use of optically levitated microspheres as test masses in experiments aimed at reaching and potentially exceeding the standard quantum limit for position measurements. Optically levitated microspheres have low mass and are essentially free of suspension thermal noise, making them well suited for reaching the quantum regime. Table-top experiments using microspheres can bridge the gap between quantum-limited position measurements of single atoms and measurements with multi-kg test masses like those being used in interferometric gravitational wave detectors.
We recycle Cruz et al.'s (Phys. Lett. A 369 (2007) 400) work on the classical and quantum position-dependent mass (PDM) oscillators. To elaborate on the ordering ambiguity, we properly amend some of the results reported in their work and discuss the classical and quantum mechanical correspondence for the PDM harmonic oscillators. We use a point canonical transformation and show that one unique quantum PDM oscillator Hamiltonian (consequently, one unique ordering-ambiguity parametric set j=l=-1/4 and k=-1/2) is obtained. To show that such a parametric set is not just a manifestation of the quan…
Heisenberg formulated a noise-disturbance principle stating that there is a tradeoff between noise and disturbance when a measurement of position and a measurement of momentum are performed sequentially, and another principle imposing a limitation on the product of the uncertainties in a joint measurement of position and momentum. We prove that the former, the Heisenberg sequential noise-disturbance principle, holds when the detectors are assumed to be initially uncorrelated from each other, but that it can be violated for some properly correlated initial preparations of the detectors.
The von Neumann interaction between a particle and an apparatus, both of arbitrary mass, has been considered in the measurement of the position of a simple harmonic oscillator acted on by an external force. When the measurement has finite duration, both the motion of the pointer and the oscillator influence the result of the measurement. Provided that the oscillator is in an eigenstate of its position at the start of the measurement, the pointer will indicate the arithmetic average between the initial and final position of the particle with an added term which depends on the duration of the me…
We argue that the classical and quantum mechanical correspondence may play a basic role in the fixation of the ordering-ambiguity parameters. We use quasi-free position-dependent masses in the classical and quantum frameworks. The effective Pöschl-Teller model is used as a manifested reference potential to elaborate on the reliability of the ordering-ambiguity parameters available in the literature.
The measurement of the position-momentum EPR correlations of a two-photon state is important for many quantum information applications ranging from quantum key distribution to coincidence imaging. However, all the existing techniques for measuring the position-momentum EPR correlations involve coincidence detection and thus suffer from issues that result in less accurate measurements. In this letter, we propose and demonstrate an experimental scheme that does not require coincidence detection for measuring the EPR correlations. Our technique works for two-photon states that are pure, irrespect…
Extended abstract of "Algebraic approach to position-dependent mass systems in both classical and quantum pictures", a series of three lectures delivered by the author in the VIII School on Geometry and Physics, 24 June-8 June 2019, organized by the Department of Mathematical Physics of the University of Bialystok, in Bialowieza, Poland (http://wgmp.uwb.edu.pl/wgmp38/part_s.html)
Wigner function is a quasi-distribution that provides a representation of the state of a quantum mechanical system in the phase space of position and momentum. In this paper we find a relation between Wigner function and appropriate measurements involving the system position and momentum which generalize the von Neumann model of measurement. We introduce two probes coupled successively in time to projectors associated with the system position and momentum. We show that one can relate Wigner function to Kirkwood joint quasi-distribution of position and momentum, the latter, in turn, being a par…
In quantum theory, measuring process is an important physical process; it is a quantum description of the interaction between the system of interest and the measuring device. Error and disturbance are used to quantitatively check the performance of the measurement, and are defined by using measuring process. Uncertainty relations are a general term for relations that provide constraints on them, and actively studied. However, the true error-disturbance bound for position measurements is not known yet. Here we concretely construct linear position measurements with minimum error-disturbance in e…
We have recently presented a novel protocol to teleport an unknown atomic state via cavity QED and position measurements. Here, after a brief review of our scheme, we provide a quantitative study of its efficiency. This is accomplished by an explicit description of the measurement process that allows us to derive the fidelity with respect to the atomic internal state to be teleported.
It is proposed in this paper that without a measurement, the wave function of a system periodically transits to a bound energy eigenfunction or the complementary wave function that is orthogonal to all the bound energy eigenfunctions. Applying this assumption to analyze results of position measurement shows that when the wave function of the entangled incident object and a position detector is a linear combination of the bound energy eigenfunctions of the entangled system prior to collapses, the detection rate essentially equals the probability flux of the incident object entering the detector…
We produce polarization entangled states with variable degree of entanglement for twin photons. Entanglement in polarization is coupled to entanglement in position that produces transverse coincidence interference fringes. We show both theoretically and experimentally that, due to this coupling, we can use the interference pattern to measure the polarization degree of entanglement.
The distribution of measured values for maximally accurate, unbiased simultaneous measurements of position and momentum is investigated. It is shown, that if the measurement is retrodictively optimal, then the distribution of results is given by the initial state Husimi function (or Q-representation). If the measurement is predictively optimal, then the distribution of results is related to the final state anti-Husimi function (or P-representation). The significance of this universal property for the interpretation of the Husimi function is discussed.
Chandran et al. (SIAM J. Comput.'14) formally introduced the cryptographic task of position verification, where they also showed that it cannot be achieved by classical protocols. In this work, we initiate the study of position verification protocols with classical verifiers. We identify that proofs of quantumness (and thus computational assumptions) are necessary for such position verification protocols. For the other direction, we adapt the proof of quantumness protocol by Brakerski et al. (FOCS'18) to instantiate such a position verification protocol. As a result, we achieve classically ver…
In an article published in 2010, Kiukas and Werner claim to have shown that Bohmian Mechanics does not make the same empirical predictions as ordinary Quantum Mechanics. We explain that such claim is wrong.
Quantum theory for measurements of energy is introduced and its consequences for the average position of monitored dynamical systems are analyzed. It turns out that energy measurements lead to a localization of the expectation values of other observables. This is manifested, in the case of position, as a damping of the motion without classical analogue. Quantum damping of position for an atom bouncing on a reflecting surface in presence of a homogeneous gravitational field is dealt in detail and the connection with an experiment already performed in the classical regime is studied. We show tha…
The Dirac delta function can be defined by the limitation of the rectangular function covering a unit area with decrease of the width of the rectangle to zero, and in quantum mechanics the eigenvectors of the position operator take the form of the delta function. When discussing the position measurement in quantum mechanics, one is prompted by the mathematical convention that uses the rectangular wave function of sufficiently narrow width to approximate the delta function in order to making the state of the position physical. We argue that such an approximation is improper in physics, because …
We propose a protocol for quantum state tomography of nonclassical states in optomechanical systems. Using a parametric drive, the procedure overcomes the challenges of weak optomechanical coupling, poor detection efficiency, and thermal noise to enable high efficiency homodyne measurement. Our analysis is based on the analytic description of the generalized measurement that is performed when optomechanical position measurement competes with thermal noise and a parametric drive. The proposed experimental procedure is numerically simulated in realistic parameter regimes, which allows us to show…
We show that it is possible to find maximal violations of the CHSH-Bell inequality using only position measurements on a pair of entangled non-relativistic free particles. The device settings required in the CHSH inequality are done by choosing one of two times at which position is measured. For different assignments of the "+" outcome to positions, namely to an interval, to a half line, or to a periodic set, we determine violations of the inequalities, and states where they are attained. These results have consequences for the hidden variable theories of Bohm and Nelson, in which the two-time…
It has been suggested that the uncertainty in the measurement of a particle's momentum could be made arbitrarily small by observing the particle at two ends of an arbitrarily long flight path. However, consideration of the nature of the detection process shows that the usual limits of the uncertainty principle hold independent of the length of the flight path.
We study the minimum volume ellipsoid estimator associates to a cloud of points in phase space. Using as a natural measure of uncertainty the symplectic capacity of the covariance ellipsoid we find that classical uncertainties obey relations similar to those found in non-standard quantum mechanics.
We present a scheme for conditionally teleporting an unknown atomic state in cavity QED, which requires two atoms and one cavity mode. The translational degrees of freedom of the atoms are taken into account using the optical Stern-Gerlach model. We show that successful teleportation with probability 1/2 can be achieved through local measurements of the cavity photon number and atomic positions. Neither direct projection onto highly entangled states nor holonomous interaction-time constraints are required.
We present the quantum and classical mechanics formalisms for a particle with position-dependent mass in the context of a deformed algebraic structure (named $κ$-algebra), motivated by the Kappa-statistics. From this structure we obtain deformed versions of the position and momentum operators, which allow to define a point canonical transformation that maps a particle with constant mass in a deformed space into a particle with position-dependent mass in the standard space. We illustrate the formalism with a particle confined in an infinite potential well and the Mathews-Lakshmanan oscillator, …
A scheme for sub-wavelength position measurements of quantum particles is discussed, which operates with running-wave laser fields as opposed to standing wave fields proposed in previous setups. The position is encoded in the phase of the applied fields rather than in the spatially modulated intensity of a standing wave. Therefore, disadvantages of standing wave schemes such as cases where the atom remains undetected since it is at a node of the standing wave field are avoided. Reversing the directions of parts of the driving laser fields allows to switch between different magnification levels…
So-called quantum limits and their achievement are important themes in physics. Heisenberg's uncertainty relations are the most famous of them but are not universally valid and violated in general. In recent years, the reformulation of uncertainty relations is actively studied, and several universally valid uncertainty relations are derived. On the other hand, several measuring models, in particular, spin-1/2 measurements, are constructed and quantitatively examined. However, there are not so many studies on simultaneous measurements of position and momentum despite their importance. Here we s…
The Standard Quantum Limit (SQL) for the measurement of a free mass position is illustrated, along with two necessary conditions for breaching it. A measurement scheme that overcomes the SQL is engineered. It can be achieved in three-steps: i) a pre-squeezing stage; ii) a standard von Neumann measurement with momentum-position object-probe interaction and iii) a feedback. Advantages and limitations of this scheme are discussed. It is shown that all of the three steps are needed in order to overcome the SQL. In particular, the von Neumann interaction is crucial in getting the right state reduct…
We study complexified Harmonic Oscillator with a position-dependent mass, termed as Complex Exotic Oscillator (CEO). The complexification induces a gauge invariance [19,11]. The role of PT -symmetry is discussed from the perspective of classical trajectories of CEO for real energy. Some trajectories of CEO are similar to those for the particle in a quartic potential in the complex domain [10, 32].
We perform a quantum information analysis for multi-mode Gaussian approximate position measurements, underlying noisy homodyning in quantum optics. The "Gaussian maximizer" property is established for the entropy reduction of these measurements which provides explicit formulas for computations including their entanglement-assisted capacity. The case of one mode is discussed in detail.
The de Broglie - Bohm Interpretation of Quantum Mechanics assigns positions and trajectories to particles. We analyze the validity of a formula for the velocities of Bohmian particles which makes the analysis of these trajectories particularly simple. We apply it to four different types of particle detectors and show that three types of the detectors lead to "surrealistic trajectories", i.e., leave a trace where the Bohmian particle was not present.
The paper addresses the debate about the empirical status of particles versus wave functions in Bohmian quantum mechanics. It thereby clarifies questions and misconceptions about the role of the particles in the measurement process, the (un)reliability of position measurements ("surrealistic trajectories"), and the limited empirical access to particle positions ("absolute uncertainty"). Taking the ontological commitment of Bohmian mechanics seriously, all relevant empirical results follow from an analysis of the theory in terms of particle motions. Finally, we address the question, why particl…
In Bohmian mechanics particles follow continuous trajectories, hence 2-time position correlations are well defined. Nevertheless, Bohmian mechanics predicts the violation of Bell inequalities. Motivated by this fact we investigate position measurements in Bohmian mechanics by coupling the particles to macroscopic pointers. This explains the violation of Bell inequalities despite 2-time position correlations. We relate this fact to so-called surrealistic trajectories that, in our model, correspond to slowly moving pointers. Next, we emphasize that the nice feature of Bohmian mechanics, which do…
Measurements of the position of a relativistic particle is considered in the framework of the Restricted-Path-Integral (RPI) approach. The amplitude describing such a measurement is shown to be exponentially small outside the light cone of the space-time point corresponding to the measurement output, in a qualitative agreement with the Hellwig and Kraus' postulate of relativistic state reduction. Theory of the measurement including the probability distribution for different measurement outputs is suggested. It is shown that correct theory does not exist (for arbitrary initial states) if the er…
I utilize the Caves-Milburn model for continuous position measurements to formulate a broadband version of the Standard Quantum Limit (SQL) for monitoring the position of a free mass, and illustrate the use of Kalman filtering to recover the SQL for estimating a weak classical force that acts on a quantum-mechanical test particle under continuous observation. These derivations are intended to clarify the interpretation of SQL's in the context of continuous quantum measurement.
In recent years, novel quantifications of measurement error in quantum mechanics have for the first time enabled precise formulations of Heisenberg's famous but often challenged measurement uncertainty relation. These relations take the form of a trade-off for the necessary errors in joint approximate measurements of position and momentum and other incompatible pairs of observables. Here we review some of these error measures, examine their properties and suitability, and compare their relative strengths as criteria for "good" approximations.
We generalise a non-classicality test described by Kot et al. [Phys. Rev. Lett. 108, 233601 (2010)], which can be used to rule out any classical description of a physical system. The test is based on measurements of quadrature operators and works by proving a contradiction with the classical description in terms of a probability distribution in phase space. As opposed to the previous work, we generalise the test to include states without rotational symmetry in phase space. Furthermore, we compare the performance of the non-classicality test with classical tomography methods based on the invers…
We propose an approach to position measurements based on the hypothesis that the action of a position detector on a quantum system can be effectively described by a dissipative disordered potential. We show that such kind of potential is able, via the dissipation-induced Anderson localization, to contemporary localize the wavefunction of the system and to dissipate information to modes bounded to the detector. By imposing a diabaticity condition we demonstrate that the dissipative dynamics between the modes of the system leads to a localized energy exchange between the detector and the rest of…
We show that the position of a dark matter-wave soliton can be determined with a precision that scales with the atomic density as $n^{-3/4}$. This surpasses the standard shot-noise detection limit for independent particles, without use of squeezing and entanglement, and it suggests that interactions among particles may present new advantages in high-precision metrology. We also take into account quantum density fluctuations due to phonon and Goldstone modes and we show that they, somewhat unexpectedly, actually improve the resolution. This happens because the fluctuations depend on the soliton…
Current methods for measuring magnetic flux are based on performing many measurements over a large ensemble of electrons. We propose a novel method based on wavefunction "revival" for measuring the flux modulo hc/2e using only a single electron. A preliminary analysis of the feasibility of the experiment is provided.
We describe a technique to measure the position of a dipolar scatterer based on self-homodyne detection of the scattered light. The method can theoretically reach the Heisenberg limit, at which information gained about the position is constrained only by the back-action of the scattered light. The technique has applications in the fields of levitated optomechanics and trapped ions and is generally applicable to the position determination of confined light scatterers.
