Time derivatives of weak values

Area of Science:

• Quantum Mechanics
• Quantum Information Theory
• Theoretical Physics

Background:

• Weak values offer insights beyond expectation values.
• The time derivative of physical properties can yield new meaningful properties.
• Gauge invariance is crucial in electromagnetic interactions.

Purpose of the Study:

• To investigate physical properties obtainable from the time derivative of weak values.
• To define and analyze gauge-invariant time derivatives of weak values.
• To establish a connection between weak values and local dynamics.

Main Methods:

• Definition of left- and right-hand time derivatives for weak values.
• Derivation of conditions for gauge-invariant weak values.
• Development of finite-difference approximations for time derivatives.
• Formulation of a local Ehrenfest-like theorem.

Main Results:

• The time derivative of a gauge-invariant weak value is generally not a weak value or gauge-invariant.
• Conditions were found to ensure gauge-invariant weak values for time derivatives.
• A local Ehrenfest-like theorem was derived, interpreting time derivatives of weak values.
• Measuring position weak values provides access to local velocity and acceleration weak values.

Conclusions:

• The time derivative of weak values offers a new route to understanding quantum properties.
• This work provides a theoretical framework and practical guidelines for experimentalists.
• Findings support the development of novel quantum technologies by linking weak value measurements to local dynamics.