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 $(ΔET)$, $(aT)$, and $(ΔE/a)$, where $ΔE$ is the energy gap of the detector. Particularly, the non-thermal terms are oscillatory with respect to the variable $(ΔET)$, so that they may be averaged out to be insignificant in the limit ΔET 1, irrespective of the values of $(aT)$ and $(ΔE/a)$. To quantify the contribution of non-thermal transient terms to the transition probability rate of a uniformly accelerating detector, we introduce a parameter, εₙₜ, called non-thermal parameter. Demanding the contribution of non-thermal terms in the finite-time transition probability rate to be negligibly small, , εₙₜ=δ1, we calculate the thermalization time -- the time required for the detector to interact with the field to arrive at the required non-thermality, εₙₜ=δ, and the detector to be (almost) thermalized with the Unruh bath in its comoving frame. Specifically, for small accelerations, aΔE, we find the thermalization time, τₜₕ, to be τₜₕ ~ (ΔE)⁻¹ × e2π|ΔE|/a/δ; and for large accelerations, a ΔE, we find the thermalization time to be τₜₕ ~ (ΔE)⁻¹/δ. We comment on the possibilities of bringing down the exponentially large thermalization time at small accelerations, aΔE.