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Chinese Physics B

Scheme to measure the expectation value of a physical quantity in weak coupling regime*

Jie Zhang, Chun-Wang Wu, Yi Xie, Wei Wu, Ping‐Xing Chen

Published Dec 30, 2020• 3 citations
Publisher

Abstract

In quantum mechanics, the expectation value of an operator can be measured by using the projective measurement, if the coupling between the measured system and pointer is strong enough. However in the weak coupling regime, the pointer can not show all the eigenvalue of the physical quantity directly due to the overlapping among the pointer states, which makes the measurement of the expectation value difficult. In this paper, we propose an expectation value measurement method in the weak coupling regime inspired by the weak measurement scheme. Compared to the projective measurement, our scheme has two obvious advantages. Experimentally we use the internal state and motional state of a single trapped 40Ca+ to establish the measurement scheme and realize the proof of principle demonstration of the scheme.

DOI
10.1088/1674-1056/abd772

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In quantum mechanics, the expectation value of an operator can be measured by using the projective measurement, if the coupling between the measured system and pointer is strong enough. However in the weak coupling regime, the pointer can not show all the eigenvalue of the physical quantity directly due to the overlapping among the pointer states, which makes the measurement of the expectation value difficult. In this paper, we propose an expectation value measurement method in the weak coupling regime inspired by the weak measurement scheme. Compared to the projective measurement, our scheme has two obvious advantages. Experimentally we use the internal state and motional state of a single trapped 40Ca+ to establish the measurement scheme and realize the proof of principle demonstration of the scheme.
Abstract Weak measurements have many applications in the field of quantum information due to the peculiar properties of weak values. Quantum weak measurement can be performed by two particles or two degree of freedom of single particle. The former needs complex operations, the latter require the reconstruction of the motion state wavepacket, which consumes substantial quantum resources. This paper presents a novel weak measurement scheme that encodes multiple internal energy levels of a single particle into the measured system and the measuring pointer, while defining a finite-coupling postselected pointer ratio (FCPR). The FCPR obtained from this scheme enables both weak value amplification and rapid experimental results. To verify these advantages, we measured the FCPR using a trapped 40 Ca + experimental setup and monitored fluctuations of the laser power within the experimental system. The results show that our scheme can detect weaker laser power fluctuations, thus providing higher precision for laser power locking.
The expectation value of the product of two noncommuting observables (POTNO) can be used to test the commutation relation and dig out the correlation between two observables. However, since a POTNO is generally a non-Hermitian operator, its average cannot be measured directly according to standard quantum mechanics. Recently, a theoretical scheme has been proposed to obtain POTNOs, but the observables are limited to projection operators. In this paper, we present a scheme that employs weak measurement to directly measure the expectation value of the POTNO, where the observables are arbitrary operators. Our scheme transforms the measurement of this expectation value into the measurements of several single observable weak values. Experimentally we demonstrate our scheme by measuring the expectation value of the product of two Pauli operators and verify their commutation and anticommutation relations in a trapped-ion system. Moreover, our scheme is applicable to the cases with more than two observables.
Abstract Weak values are traditionally obtained using a weak interaction between the measured system and a pointer state. It has, however, been pointed out that weak coupling can be replaced by a carefully tailored strong interaction. This paper provides a direct comparison of two strong interaction-based approaches (strong interaction accompanied by either a suitably prepared pointer state or quantum erasure) and the traditional weak interaction-based method. The presented theoretical derivations explicitly prove analytical equivalence of these approaches, which were subsequently certified by an experiment implemented on the platform of linear optics. We find that strong-interaction-based measurements are experimentally less demanding on this platform.
The physical meaning of the quantum weak value still remains a matter of debate. Originally introduced by Aharonov et al. [Phys. Rev. Lett. 60, 1351 (1988).] as another counterintuitive feature of quantum mechanics, it eventually evolved into a practical tool that is widely used. The theoretical framework in which weak values were introduced was given by von Neumann's model of quantum measurements. In this model, a system is submitted to measurement by coupling it to a ``pointer,'' or meter. In the weak-coupling regime, one may perform a low-order Taylor expansion for the evolution operator of the system and meter. This standard approach ties weak values with weak couplings and weak measurements. We report closed-form expressions that can be used to untie weak values from weak measurements. The regime of strong measurements thereby becomes accessible to weak values, without leaving the framework in which the latter were originally introduced. The reported results should also help us to better understand the physical meaning of weak values.
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.
While the novel applications of weak values have recently attracted wide attention, weak measurement, the usual way to extract weak values, suffers from risky approximations and severe quantum noises. In this paper, we show that the weak-value information can be obtained exactly in strong measurement with postselections, via measuring the coupling-deformed pointer observables, i.e., the observables selected according to the coupling strength. With this approach, we keep all the advantages claimed by weak-measurement schemes and at the same time solve some widely criticized problems thereof, such as the questionable universality, systematical bias, and drastic inefficiency.