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Physical Review A

Weak measurements and quantum weak values for NOON states

L. Rosales-Zárate, Bogdan Opanchuk, M. D. Reid

Published Mar 26, 2018• 14 citations
Publisher

Abstract

Quantum weak values arise when the mean outcome of a weak measurement made on certain preselected and postselected quantum systems goes beyond the eigenvalue range for a quantum observable. Here, we propose how to determine quantum weak values for superpositions of states with a macroscopically or mesoscopically distinct mode number, that might be realized as two-mode Bose-Einstein condensate or photonic NOON states. Specifically, we give a model for a weak measurement of the Schwinger spin of a two-mode NOON state, for arbitrary $N$. The weak measurement arises from a nondestructive measurement of the two-mode occupation number difference, which for atomic NOON states might be realized via phase contrast imaging and the ac Stark effect using an optical meter prepared in a coherent state. The meter-system coupling results in an entangled cat-state. By subsequently evolving the system under the action of a nonlinear Josephson Hamiltonian, we show how postselection leads to quantum weak values, for arbitrary $N$. Since the weak measurement can be shown to be minimally invasive, the weak values provide a useful strategy for a Leggett-Garg test of $N$-scopic realism.

DOI
10.1103/physreva.97.032123

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Quantum weak values arise when the mean outcome of a weak measurement made on certain preselected and postselected quantum systems goes beyond the eigenvalue range for a quantum observable. Here, we propose how to determine quantum weak values for superpositions of states with a macroscopically or mesoscopically distinct mode number, that might be realized as two-mode Bose-Einstein condensate or photonic NOON states. Specifically, we give a model for a weak measurement of the Schwinger spin of a two-mode NOON state, for arbitrary NNN. The weak measurement arises from a nondestructive measurement of the two-mode occupation number difference, which for atomic NOON states might be realized via phase contrast imaging and the ac Stark effect using an optical meter prepared in a coherent state. The meter-system coupling results in an entangled cat-state. By subsequently evolving the system under the action of a nonlinear Josephson Hamiltonian, we show how postselection leads to quantum weak values, for arbitrary NNN. Since the weak measurement can be shown to be minimally invasive, the weak values provide a useful strategy for a Leggett-Garg test of NNN-scopic realism.
In standard quantum weak measurements, preselection and postselection of quantum states are implemented in the same photon. Here we go beyond this restrictive setting and demonstrate that the preselection and postselection can be performed in two different photons, if the two photons are polarization entangled. The Pancharatnam-Berry phase metasurface is incorporated in the weak measurement system to perform weak coupling between probe wave function and spin observable. By introducing nonlocal weak measurement into the microscopy imaging system, it allows us to remotely switch different microscopy imaging modes of pure-phase objects, including bright-field, differential, and phase reconstruction. Furthermore, we demonstrate that the nonlocal weak-measurement scheme can prevent almost all environmental noise photons from detection and thus achieves a higher image contrast than the standard scheme at a low photon level. Our results provide the possibility to develop a quantum nonlocal weak-measurement microscope for label-free imaging of transparent biological samples.
The weak value of an operator is the average result of measuring that operator on a quantum system given specified initial and final states. The overlap between these states appears in the denominator of the weak value, and thus if that overlap is small, the weak value can be arbitrarily large. In 1988, Albert, Aharonov, and Vaidman proposed a weak Stern-Gerlach experiment which would be able to measure a so-called anomalous weak value, a measurement result which lies out- side of the eigenvalue spectrum of the operator being measured. Though anomalous weak values have been measured in numerous optical systems, here we present the first such realization of this experiment as originally proposed. By placing a Bose-Einstein condensate of 87Rb in a magnetic gradient and performing an unlikely postselection, we measure an effective magnification of a weak magnetic momentum kick (~ 8 μμμm/s) by a factor of 14. We also demonstrate that increasing the momentum kick decreases the "weakness" of the measurement and reduces the amplification.
We consider tests of Leggett-Garg's macrorealism and of macroscopic local realism, where for spacelike separated measurements the assumption of macroscopic noninvasive measurability is justified by that of macroscopic Bell locality. We give a mapping between the Bell and Leggett-Garg experiments for microscopic qubits based on spin-1/21/21/2 eigenstates and gedanken experiments for macroscopic qubits based on two macroscopically distinct coherent states $|\ensuremath{\alpha}\ensuremath{\rangle}$ and $|\ensuremath{-}\ensuremath{\alpha}\ensuremath{\rangle}$ (as $\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\infty}$). In this mapping, the unitary rotation ${U}_{\ensuremath{\theta}}$ of the Stern-Gerlach analyzer or polarizing beam splitter is realized by a local interaction $H=\mathrm{\ensuremath{\Omega}}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}^{4}$ where $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}$ is the number of quanta. By adjusting the time of interaction, one alters the measurement setting $\ensuremath{\theta}$. We thus predict violations of Leggett-Garg and Bell inequalities in a macroscopic regime where coarse-grained measurements $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}$ need only discriminate between two macroscopically distinct coherent states. To interpret the violations, we distinguish between different definitions of macroscopic realism. Deterministic macroscopic local realism (dMR) assumes the system is in a state with a definite outcome ${\ensuremath{\lambda}}_{\ensuremath{\theta}}$ for the measurement $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}$ prior to the unitary rotation ${U}_{\ensuremath{\theta}}$, and is negated by the violations. Weak macroscopic realism (wMR) assumes a definite outcome for systems prepared in a superposition ${\ensuremath{\psi}}_{\text{pointer}}$ of two macroscopically distinct eigenstates of $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}$, after the unitary rotation ${U}_{\ensuremath{\theta}}$. We find that wMR can be viewed consistently with the violations and with the predictions of quantum mechanics for two tests of wMR. A model is presented in which wMR holds and the macroscopic violations emerge over the course of the unitary dynamics occurring at both sites. Implications for the realism of the micro-macro state generated during a quantum measurement are discussed. Finally, we point out an EPR-type paradox, which demonstrates inconsistency between wMR and the completeness of quantum mechanics at a microscopic level.
We investigated precision measurements in a two-level system coupled to a single-photon-added coherent state (SPACS) under postselected weak measurement. We analyzed strategies for improving measurement precision, including parameter estimation and the signal-to-noise ratio (SNR) in postselected weak measurements, using the photon statistics of SPACS as the meter. Our results demonstrate that SPACS-based postselected weak measurements can outperform conventional measurement schemes in terms of precision. Additionally, we explicitly introduced an alternative weak measurement method, which is commonly applied in dispersive light-atom interactions. Our work presents a novel approach to addressing fundamental issues in quantum precision measurement based on photon statistics, and it provides a method for extracting the photon and phase shifts of radiation fields through the weak values of system observables.
Entanglement and nonlocality are studied in the framework of pre-/postselected ensembles with the aid of weak measurements and the two-state-vector formalism. In addition to the EPR–Bohm experiment, we revisit the Hardy and Cheshire Cat experiments, whose entangled pre- or postselected states give rise to curious phenomena. We then turn to even more peculiar phenomenon suggesting “emerging correlations” between independent preand postselected ensembles of particles. This can be viewed as a quantum violation of the classical “pigeonhole principle.” Introduction It is seldom acknowledged that seven years before the celebrated Bell paper [1], Bohm and Aharonov [2] published an analysis of the EPR paradox [3]. They suggested an experimental setup, based on Compton scattering, for testing nonlocal correlations between the polarizations of two annihilation photons. In 1964, Bell proposed his general inequality, thereby excluding local realism. During the same time, Aharonov et al. constructed the foundations of a time-symmetric formalism of quantum mechanics [4]. While Bell's proof utilizes entanglement to demonstrate nonlocal correlations,wewill describe inwhat follows the emergence of nonlocal correlations between product states. For this purpose, however, we shall invoke weak measurements of pre- and postselected ensembles. In classical mechanics, initial conditions of position and velocity for every particle fully determine the time evolution of the system. Therefore, trying to impose an additional final condition would lead either to redundancy or to inconsistency with the initial conditions. This is radically different in quantum mechanics. Because of the uncertainty principle, an initial state vector does not fully determine, in general, the outcome of a future measurement. However, adding a final (backward-evolving) state vector results in a more complete description of the quantum system in between these two boundary conditions, which has a bearing on the determination of measurement outcomes. The basis for this time-symmetric formulation of quantum mechanics was laid down by Aharonov, Bergman, and Lebowitz (ABL), who derived a symmetric probability rule concerning measurements performed on systems, while taking into account the final state of the system, in addition to the usual initial state [4].
It is argued that a weak value of an observable is a robust property of a single pre- and postselected quantum system rather than a statistical property. During an infinitesimal time a system with a given weak value affected other systems as if it had been in an eigenstate with eigenvalue equal to the weak value. This differs significantly from the action of a system preselected only and possessing a numerically equal expectation value. The weak value has a physical meaning beyond a conditional average of a pointer in the weak measurement procedure. The difference between the weak value and the expectation value has been demonstrated on the example of photon polarization. In addition, the weak values for systems pre- and postselected in mixed states are considered.