Asymptotic states for stationary Unruh-DeWitt detectors

Abstract

We study the late-time asymptotic state of a stationary Unruh-DeWitt detector interacting with a field in a thermal state. We work in an open system framework, where the field plays the role of an environment for the detector. The long-time interaction between the detector and the field is modelled with the aid of a one-parameter family of switching functions that turn on and off the interaction Hamiltonian between the two subsystems, such that the long-time interaction limit is reached as the family parameter goes to infinity. In such limit, we show that if the field is in a Kubo-Martin-Schwinger (KMS) state and the detector is stationary with respect to the notion of positive frequency of the field, in the Born-Markov approximation, the asymptotic state of the detector is a Gibbs state at the KMS temperature. We then relax the KMS condition for the field state, and require only that a frequencydependent version of the detailed balance condition for the Wightman function pulled back to the detector worldline hold, in the sense that the inverse temperature appearing in the detailed balance relation need not be constant. In this setting, we show that the late-time asymptotic state of the detector has the form of a thermal density matrix, but with a frequency-dependent temperature. We present examples of these results, which include the classical Unruh effect and idealised Hawking radiation (for fields in the HHI state), and also the study of the late-time behaviour of detectors following stationary "cusped" and circular motions in Minkowski space interacting with a massless Klein-Gordon field in the Minkowski vacuum. In the cusped motion case, a frequency-dependent, effective temperature for the asymptotic late-time detector state is obtained analytically. In the circular motion case, such effective temperature is obtained numerically. * Electronic address: benito.juarez@iimas.unam.mx † Electronic address: dmoustos@upatras.gr

Unruh-like effects: effective temperatures along stationary worldlines

“…A few articles where stationary trajectories other than linear uniform acceleration (Nulltor) have been previously studied in the context of quantum field theory, detector re-JHEP06(2020)059 sponses, asymptotic states and effective recorded temperatures. (Detector responses, their asymptotic states and effective temperatures are intimately related to the temperature measured by detectors via the generalized, frequency-dependent detailed balance condition [7]). These works necessarily have some overlap with the results that we present in this paper, especially in the Parator (cusped) and Ultrator (circular) cases.…”

“…In addition to studying the Unruh effect, Louko and Satz have given a closed formula for the transition rate of a detector following the Parator (cusped) motion, from which an effective temperature can be inferred. Such effective temperature has been studied by two of us in [7], and was essentially already given in [11]. The detector responses or effective temperatures for the remaining trajectories -Ultrator, Infrator or Hypertor -have not been studied in detail in [28].…”

“…The paper is organized as follows. In section 2, we briefly describe the Unruh-DeWitt detector model that we study in this paper, and summarize the main result obtained by two of us in [7]. Namely, that detectors following stationary trajectories in Minkowski spacetime reach an equilibrium state at late times at an effective temperature that can be obtained from a generalized detailed balance condition.…”

“…For such stationary trajectories, if the generalized detailed balanced condition for the pullback of the Wightman function of the field along the detector worldline, W + (−ω) = e β(ω)ω W + (ω), is satisfied, when the switching function, χ, is stretched adiabatically for a long time period, it has been discussed in [7] that the detector will reach an equilibrium state (at leading order in the coupling c)…”

“…The sense in which such frequency-dependent function deserves to be called a temperature is discussed in detail in [7] by two of us (see also section 2 below), making use of a generalization of the detailed balance condition for Unruh-DeWitt detectors (see [8]) with a frequency-dependent function -the frequency-dependent temperature -instead of a constant temperature. The point is that detectors that interact for a long time with a quantum field will reach a Gibbs state at late times at such frequency-dependent, effective temperature, if such generalized detailed balance condition holds.…”