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Weak values from path integrals

A. Matzkin

Published Jan 1, 2020• 1 citation
OpenAlex

Abstract

We connect the weak measurements framework to the path integral formulation of quantum mechanics. We show how Feynman propagators can in principle be experimentally inferred from weak value measurements. We also obtain expressions for weak values parsing unambiguously the quantum and the classical aspects of weak couplings between a system and a probe. These expressions are shown to be useful in quantum chaos related studies (an illustration involving quantum scars is given), and also in solving current weak-value related controversies (we discuss the existence of discontinuous trajectories in interferometers and the issue of anomalous weak values in the classical limit).

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We connect the weak measurements framework to the path integral formulation of quantum mechanics. We show how Feynman propagators can in principle be experimentally inferred from weak value measurements. We also obtain expressions for weak values parsing unambiguously the quantum and the classical aspects of weak couplings between a system and a probe. These expressions are shown to be useful in quantum chaos related studies (an illustration involving quantum scars is given), and also in solving current weak-value related controversies (we discuss the existence of discontinuous trajectories in interferometers and the issue of anomalous weak values in the classical limit).
We consider weak values in the Feynman propagator framework, to gain a broader understanding of their interpretation in terms of path integrals. In particular, we examine the phenomenon of seemingly discontinuous paths that particles take in nested Mach-Zender interferometer experiments. We extend on existing path integral approaches for weak values by deriving expressions to model a sequence of weak measurements, and study the probe shifts across the different branches of a weak value interferometer. We apply this to scrutinise two scenarios of interest, one which treats photons as measurement apparatus via their spatial projection operators, and the second treating mirrors as probes.
Weak values, obtained from weak measurements, attempt to describe the properties of a quantum system as it evolves from an initial to a final state, without practically altering this evolution. Trajectories can be defined from weak measurements of the position, or inferred from weak values of the momentum operator. The former can be connected to the asymptotic form of the Feynman propagator and the latter to Bohmian trajectories. Employing a time-dependent oscillator as a model, this work analyzes to what extent weak measurements can shed light on the underlying dynamics of a quantum system expressed in terms of trajectories, in particular by comparing the two approaches.
We investigate four key issues with using a nonzero weak value of the spatial projection operator to infer the past path of an individual quantum particle. First, we note that weak measurements disturb a system, so any approach relying on such a perturbation to determine the location of a quantum particle describes the state of a disturbed system, not that of a hypothetical undisturbed system. Second, even assuming no disturbance, there is no reason to associate the nonzero weak value of an operator containing the spatial projection operator with the classical idea of “particle presence.” Third, weak values are only measurable over ensembles, and so to infer properties of individual particles from values of them is problematic. Finally, weak value approaches to the path of a particle do not provide information beyond standard quantum mechanics (and the classical modes supporting the experiment). We know of no experiment with testable consequences that demonstrates a connection between particle presence and weak values.
We investigate four key issues with using a nonzero weak value of the spatial projection operator to infer the past path of an individual quantum particle. First, we note that weak measurements disturb a system, so any approach relying on such a perturbation to determine the location of a quantum particle describes the state of a disturbed system, not that of a hypothetical undisturbed system. Second, even assuming no disturbance, there is no reason to associate the nonzero weak value of an operator containing the spatial projection operator with the classical idea of ``particle presence.'' Third, weak values are only measurable over ensembles, and so to infer properties of individual particles from values of them is problematic. Finally, weak value approaches to the path of a particle do not provide information beyond standard quantum mechanics (and the classical modes supporting the experiment). We know of no experiment with testable consequences that demonstrates a connection between particle presence and weak values.
The two-state vector formalism provides a complete description of pre- and post-selected quantum systems and has uncovered a host of new quantum phenomena which were previously hidden. The most important feature is that any weak coupling to a pre- and post-selected system is effectively a coupling to a "weak value" which is given by a simple expression depending on the two-state vector. In particular, weak values, are the outcomes of so called "weak measurements" which have recently become a very powerful tool for ultra-sensitive measurements. Using weak values, I will show that the classical limit of quantum mechanics is a far more complicated issue; it is in fact dramatically more involved and it requires a complete revision of all our intuitions. The revised intuitions can then serve as a guide to finding novel quantum effects.