Radiative process of two entangled uniformly accelerated atoms in a thermal bath: a possible case of anti-Unruh event

Abstract

We study the radiative process of two entangled two-level atoms uniformly accelerated in a thermal bath, coupled to a massless scalar field. First, by using the positive frequency Wightman function from the Minkowski modes with a Rindler transformation we provide the transition probabilities for the transitions from maximally entangled symmetric and anti-symmetric Bell states to the collective excited or ground state in (1 + 1) and (1 + 3) dimensions. We observe a possible case of anti-Unruh event in these transition probabilities, though the (1 + 1) and (1 + 3) dimensional results are not completely equivalent. We infer that thermal bath plays a major role in the occurrence of the anti-Unruh effect, as it is also present in the transition probabilities corresponding to a single detector in this case. Second, we have considered the Green's functions in terms of the Rindler modes with the vacuum of Unruh modes for estimating the same. Here the anti-Unruh effect appears only for the transition from the anti-symmetric state to the collective excited or ground state. It is noticed that here the (1 + 1) and (1 + 3) dimensional results are equivalent, and for a single detector, we do not observe any anti-Unruh effect. This suggests that the entanglement between the states of the atoms is the main cause for the observed anti-Unruh effect in this case. In going through the investigation, we find that the transition probability for a single detector case is symmetric under the interchange between the thermal bath's temperature and the Unruh temperature for Rindler mode analysis; whereas this is not the case for Minkowski mode. We further comment on whether this observation may shed light on the analogy between an accelerated observer and a real thermal bath. An elaborate investigation for the classifications of our observed anti-Unruh effects, i.e., either weak or strong anti-Unruh effect, is also thoroughly demonstrated.

Role of thermal field in entanglement harvesting between two accelerated Unruh-DeWitt detectors

“…( 15). However, the Green's function obtained in this way, for thermal field, is not time translational invariant in terms of proper time and a prescription to obtain unit time detector response using them is not possible [49]. On the other hand, one can also express the scalar field Φ(x) in terms of the Rindler modes and operators for which the corresponding vacuum is the Rindler vacuum.…”

“…This way of construction provides the proper time translation invariance in a natural way and analysis becomes analytically more tractable (e.g. see [49]).…”

“…This will serve the purpose of approaching towards more realistic situation and thereby helping to know the exact features of our surroundings. In this regard, it may be mentioned that the thermal nature of fields has already been included in various investigations related to Unruh-De Witt detectors; like calculation of response functions in case of single detector [45][46][47][48] and two entangled detectors [49]. In both [43] and [44] it is predicted that the entanglement extraction gets depleted with increasing temperature of the thermal field.…”

“…However, if we consider similar description of the Feynman propagators and the Wightman functions for thermal fields, the resulting Green's functions do not remain time translational invariant with respect to the detectors' proper times. This issue is also discussed in [49] where the authors have considered Rindler modes with the vacuum for the Unruh modes to provide a description of the Wightman functions correspond-ing to accelerated observers in thermal bath, which are time translational invariant. This method circumvents the previously mentioned issue by expressing the Green's functions in terms of modes and their momentum space integrals rather than a position space representation, in line with the chain of thoughts also presented in [41].…”

“…In particular, we have considered the interaction between the two-level point-like detectors and the scalar field to be of monopole type, but it is observed that the specific form of this monopole moment operator is not needed to understand the role of the spacetime trajectories and the thermal bath in entanglement extraction. We use the prescription as provided in [49] for the construction of the Green's functions and follow the formulation of articles [40,42] for entanglement harvesting. We arrive at the same assertions that entanglement extraction is possible only for the anti-parallelly accelerated detectors and not for the parallelly accelerated ones, and also encounter the phenomena of degrading entanglement extraction with increasing temperature of the thermal bath [43,44].…”