Time-Energy Uncertainty Relation for a Quantum Clock as a Control Device

A quantum clock working as a control device is examined. The quality of the control process is characterized by the magnitude of deviation of perturbed state from unperturbed state of the controlled system. Uncertainty relations that relate the time duration of the process and energy of the clock to the quality of the control are presented.

We read a clock to know what time it is. While time is a classical external parameter in the equation of motion, we need to observe a physical object called a clock to know the time of occurrence of a certain event. In quantum theory, dynamics of any physical object, including a clock, is governed by the Schr¨odinger equation. Following the pioneering work by Salecker and Wigner [1], several versions of the quantum clock have been proposed and analyzed [2, 3]. In particular, properties of quantum clocks have been investigated in the context of the time-energy uncertainty relation [4–8]. In addition to basic roles, Peres [2] argued that a clock can be employed to control the duration of perturbation on a system. Such control dynamics is often described by introducing a time-dependent perturbation….
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