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

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, $\varepsilon_{\rm nt}$, called non-thermal parameter. Demanding the contribution of non-thermal terms in the finite-time transition probability rate to be negligibly small, \ie, $\varepsilon_{\rm nt}=\delta\ll1$, we calculate the thermalization time -- the time required for the detector to interact with the field to arrive at the required non-thermality, $\varepsilon_{\rm nt}=\delta$, 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, $\tau_{\rm th}$, to be $\tau_{\rm th} \sim (\Delta E)^{-1} \times {\rm e}^{2\pi|\Delta E|/a}/\delta$; and for large accelerations, $a\gg \Delta E$, we find the thermalization time to be $\tau_{\rm th} \sim (\Delta E)^{-1}/\delta$. We comment on the possibilities of bringing down the exponentially large thermalization time at small accelerations, $a\ll\Delta E$.