Quantum time dilation in a gravitational field

Abstract

According to relativity, the reading of anideal clock is interpreted as the elapsedproper time along its classical trajectorythrough spacetime. In contrast, quantumtheory allows the association of many simultaneous trajectories with a single quantum clock, each weighted appropriately.Here, we investigate how the superposition principle affects the gravitational timedilation observed by a simple clock — adecaying two-level atom. Placing suchan atom in a superposition of positionsenables us to analyze a quantum contribution to a classical time dilation manifest in spontaneous emission. In particular, we show that the emission rate ofan atom prepared in a coherent superposition of separated wave packets in a gravitational field is different from the emission rate of an atom in a classical mixture of these packets, which gives rise toa quantum gravitational time dilation effect. We demonstrate that this nonclassical effect also manifests in a fractionalfrequency shift of the internal energy ofthe atom that is within the resolution ofcurrent atomic clocks. In addition, weshow the effect of spatial coherence on theatom's emission spectrum.