Backreaction and the Unruh effect: New insights from exact solutions of uniformly accelerated detectors

Using nonperturbative results obtained recently for a uniformly accelerated Unruh-DeWitt detector, we discover new features in the dynamical evolution of the detector’s internal degree of freedom, and identified the Unruh effect derived originally from time-dependent perturbation theory as operative in the ultraweak coupling and ultrahigh acceleration limits. The mutual interaction between the detector and the field engenders entanglement between them, and tracing out the field leads to a mixed state of the detector even for a detector at rest in Minkowski vacuum. Our findings based on this exact solution show clearly the differences from the ordinary result where the quantum field’s backreaction is ignored in that the detector no longer behaves like a perfect thermometer. From a calculation of the evolution of the reduced density matrix of the detector, we find that the transition probability from the initial ground state over an infinitely long duration of interaction derived from time-dependent perturbation theory is existent in the exact solution only in transient under special limiting conditions corresponding to the Markovian regime. Furthermore, the detector at late times never sees an exact Boltzmann distribution over the energy eigenstates of the free detector, thus in the non-Markovian regime covering a wider range of parameters the Unruh temperature cannot be identified inside the detector.

Figure 1

Plot of $(\Delta P\Delta Q)/(\hslash /2)$ at late times with ${\Lambda }_1=0$. One can see that there exists a range of $a$ (less than about 4) where $\Delta P\Delta Q\equiv \sqrt{⟨P^2⟩⟨Q^2⟩}$ is smaller than $\hslash /2$, violating the uncertainty principle. So ${\Lambda }_1$ cannot be naively subtracted. Other parameters in this plot are taken to be $\gamma =0.1$, $\Omega =2.3$, $m_0=1$, and $\hslash =c=1$.

Figure 2

The evolution of the entropy of entanglement $\mathcal{S}(\eta )$ in (31) for a detector initially in the ground state. At late times, the entropy converges to a large number if ${\Lambda }_1$ is large. Here the parameters are taken to be $a=2$, $\gamma =0.1$, $\Omega =2.3$, $m_0=1$, ${\Lambda }_1={\Lambda }_0=10000$, and $\hslash =c=1$.