Quantum clocks observe classical and quantum time dilation

Alexander R H Smith,Mehdi Ahmadi

doi:10.1038/s41467-020-18264-4

Alexander R H Smith, Mehdi Ahmadi

Open Access

https://doi.org/10.1038/s41467-020-18264-4

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Abstract

At the intersection of quantum theory and relativity lies the possibility of a clock experiencing a superposition of proper times. We consider quantum clocks constructed from the internal degrees of relativistic particles that move through curved spacetime. The probability that one clock reads a given proper time conditioned on another clock reading a different proper time is derived. From this conditional probability distribution, it is shown that when the center-of-mass of these clocks move in localized momentum wave packets they observe classical time dilation. We then illustrate a quantum correction to the time dilation observed by a clock moving in a superposition of localized momentum wave packets that has the potential to be observed in experiment. The Helstrom-Holevo lower bound is used to derive a proper time-energy/mass uncertainty relation.

Highlights

At the intersection of quantum theory and relativity lies the possibility of a clock experiencing a superposition of proper times
Time observables are identified with positive-operator valued measures (POVMs) that transform covariantly with respect to the group of time translations acting on the employed clock system[8,9]
We introduce a proper time observable defined as a covariant POVM on the internal degrees of freedom of a relativistic particle moving through curved spacetime

Summary

Introduction

At the intersection of quantum theory and relativity lies the possibility of a clock experiencing a superposition of proper times. Extending the operational view to quantum theory, one is led to define time through measurements of quantum systems serving as clocks[3] Such descriptions of quantum clocks have been developed in the context of quantum metrology[4,5,6,7]. In this regard, time observables are identified with positive-operator valued measures (POVMs) that transform covariantly with respect to the group of time translations acting on the employed clock system[8,9]. Covariant time observables allow for a rigorous formulation of the time-energy uncertainty relation[4,5,6,7], circumvent Pauli’s infamous objection to the construction of a time operator[11,12], and play an important role in relational quantum dynamics[13,14,15,16,17]

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