Finite-time Unruh effect: Waiting for the transient effects to fade off

Abstract

We investigate the transition probability rate of a Unruh-DeWitt (UD) detector interacting with massless scalar field for a finite duration of proper time, $T$, of the detector. For a UD detector moving at a uniform acceleration, $a$, we explicitly show that the finite-time transition probability rate can be written as a sum of purely thermal terms, and non-thermal transient terms. While the thermal terms are independent of time, $T$, the non-thermal transient terms depend on $(\Delta ET)$, $(aT)$, and $(\Delta E/a)$, where $\Delta E$ is the energy gap of the detector. Particularly, the non-thermal terms are oscillatory with respect to the variable $(\Delta ET)$, so that they may be averaged out to be insignificant in the limit $\Delta ET\gg 1$, irrespective of the values of $(aT)$ and $(\Delta E/a)$. To quantify the contribution of non-thermal transient terms to the transition probability rate of a uniformly accelerating detector, we introduce a parameter, ${\epsilon }_{\mathrm{nt}}$, called non-thermal parameter. Demanding the contribution of non-thermal terms in the finite-time transition probability rate to be negligibly small, \ie, ${\epsilon }_{\mathrm{nt}}=\delta \ll 1$, we calculate the thermalization time -- the time required for the detector to interact with the field to arrive at the required non-thermality, ${\epsilon }_{\mathrm{nt}}=\delta$, and the detector to be (almost) thermalized with the Unruh bath in its comoving frame. Specifically, for small accelerations, $a\ll \Delta E$, we find the thermalization time, ${\tau }_{\mathrm{th}}$, to be ${\tau }_{\mathrm{th}}\sim (\Delta E{)}^{-1}\times {\mathrm{e}}^{2\pi |\Delta E|/a}/\delta$; and for large accelerations, $a\gg \Delta E$, we find the thermalization time to be ${\tau }_{\mathrm{th}}\sim (\Delta E{)}^{-1}/\delta$. We comment on the possibilities of bringing down the exponentially large thermalization time at small accelerations, $a\ll \Delta E$.

v1: 12 pages; 2 figures

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