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Journal of the Korean Physical Society

Classicality in imaginary weak amplification

Tao Wang, Gang Li, Chang-Bao Fu, Rui Zhang, Xuemei Su, Qian-Qian Bao

Published Sep 6, 2017• 1 citation
Publisher

Abstract

Weak amplification is an interesting phenomenon. In this paper we show an example that the imaginary weak amplification effect can be only related to the classical probability distribution of the measured quantity. A general result for any two orthogonal postselections and any meter state is also given.
DOI
10.3938/jkps.71.871

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Nearly thirty years ago the possibility of anomalous weak amplfication (AWA) was revealed by Aharonov, Albert and Vaidman [1]. Recently two papers presents two AWA schemes which are beyond the traditional proposal given by them [14, 15]. At the first glance the two papers seems very different. Ref. [14] discusses the thermal light cross-Kerr effect and finds only postselection can give the amplification effect without interference of the postselected meter states, and Ref. [15] shows that only weak interaction itself can give the amplification effect without postselection. Here the relationships between the two papers are pointed out and a generalized framework for AWA via postselecting a pair of orthogonal final states is shown.
Weak measurements with imaginary weak values are reexamined in light of recent experimental results. The shift of the meter, due to the imaginary part of the weak value, is derived via the probability of postselection, which allows considering the meter as a distribution of a classical variable. The derivation results in a simple relation between the change in the distribution and its variance. By applying this relation to several experimental results, in which the meter involved the time and frequency domains, it is shown to be especially suitable for scenarios of that kind. The practical and conceptual implications of a measurement method, which is based on this relation, are discussed.
Postselected weak measurement has been widely used in experiments to observe weak effects in various physical systems. However, it is still unclear how large the amplification ability of a weak measurement can be and what determines the limit of this ability, which is fundamental to understanding and applying weak measurements. The limitation of the conventional weak-value formalism for this problem is the divergence of weak values when the pre- and postselections are nearly orthogonal. In this paper, we study this problem by a variational approach for a general Hamiltonian ${H}_{\mathrm{int}}=gA\ensuremath{\bigotimes}\ensuremath{\Omega}\ensuremath{\delta}(t\ensuremath{-}{t}_{0}),\phantom{\rule{0.16em}{0ex}}g\ensuremath{\ll}1$. We derive a general asymptotic solution and show that the amplification limit is essentially independent of ggg and is determined by only the initial state of the detector and the number of distinct eigenvalues of AAA or $\ensuremath{\Omega}$. An example of spin-12\frac{1}{2}21​ particles with a pair of Stern-Gerlach devices is given to illustrate the results. The limiting case of continuous-variable systems is also investigated to demonstrate the influence of system dimension on the amplification limit.
We extend the idea of weak measurements to the general case, provide a complete treatment, and obtain results for both the regime when the preselected and postselected states (PPS) are almost orthogonal and the regime when they are exactly orthogonal. We surprisingly find that for a fixed interaction strength, there may exist a maximum signal amplification and a corresponding optimum overlap of PPS to achieve it. For weak measurements in the orthogonal regime, we find interesting quantities that play the same role that weak values play in the nonorthogonal regime.
Weak measurement of a quantum system followed by postselection based on a subsequent strong measurement gives rise to a quantity called the weak value : a complex number for which the interpretation has long been debated. We analyse the procedure of weak measurement and postselection, and the interpretation of the associated weak value, using a theory of classical mechanics supplemented by an epistemic restriction that is known to be operationally equivalent to a subtheory of quantum mechanics. Both the real and imaginary components of the weak value appear as phase space displacements in the postselected expectation values of the measurement device's position and momentum distributions, and we recover the same displacements as in the quantum case by studying the corresponding evolution in our theory of classical mechanics with an epistemic restriction. By using this epistemically restricted theory, we gain insight into the appearance of the weak value as a result of the statistical effects of post selection, and this provides us with an operational interpretation of the weak value, both its real and imaginary parts. We find that the imaginary part of the weak value is a measure of how much postselection biases the mean phase space distribution for a given amount of measurement disturbance. All such biases proportional to the imaginary part of the weak value vanish in the limit where disturbance due to measurement goes to zero. Our analysis also offers intuitive insight into how measurement disturbance can be minimized and the limits of weak measurement.
We extend the idea of weak measurements to the general case, provide a complete treatment and obtain results for both the regime when the PPS are almost orthogonal and the regime when they are exactly orthogonal. We surprisingly find that for a fixed interaction strength, there may exist a maximum signal amplification and a corresponding optimum overlap of PPS to achieve it. We also find interesting quantities, the orthogonal weak values, which play the role of weak values for the case when the PPS are exactly orthogonal.