Time Derivatives of Weak Values

TL;DR

This paper investigates the physical meaning of the time derivatives of weak values in quantum mechanics, establishing conditions for gauge invariance and deriving an Ehrenfest-like theorem, with implications for quantum measurement and technology.

Contribution

It introduces conditions under which the time derivative of weak values remains gauge-invariant and derives a local Ehrenfest-like theorem for weak values, linking derivatives to physical properties.

Findings

Time derivatives of weak values can reveal local velocity and acceleration.

Conditions for gauge invariance of weak value derivatives are identified.

An example shows how to determine electromagnetic fields from weak value derivatives.

Abstract

The time derivative of a physical property often gives rise to another meaningful property. Since weak values provide empirical insights that cannot be derived from expectation values, this paper explores what physical properties can be obtained from the time derivative of weak values. It demonstrates that, in general, the time derivative of a gauge-invariant weak value is neither a weak value nor a gauge-invariant quantity. Two conditions are presented to ensure that the left- or right-time derivative of a weak value is also a gauge-invariant weak value. Under these conditions, a local Ehrenfest-like theorem can be derived for weak values giving a natural interpretation for the time derivative of weak values. Notably, a single measured weak value of the system's position provides information about two additional unmeasured weak values: the system's local velocity and acceleration, through the first- and second-order time derivatives of the initial weak value, respectively. These findings also offer guidelines for experimentalists to translate the weak value theory into practical laboratory setups, paving the way for innovative quantum technologies. An example illustrates how the electromagnetic field can be determined at specific positions and times from the first- and second-order time derivatives of a weak value of position.