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Physical review. D/Physical review. D.

Collective transitions of two entangled atoms and the Fulling-Davies-Unruh effect

Wenting Zhou, Hongwei Yu

Published Apr 20, 2020• 11 citations
Publisher

Abstract

We calculate respectively, from the viewpoints of an instantaneously/locally inertial observer and a coaccelerated observer, the transition rates of the two-atom system in the symmetric/antisymmetric entangled state [$|{\ensuremath{\psi}}_{\ifmmode\pm\else\textpm\fi{}}⟩$]. The scalar fields to which the atoms are coupled are respectively assumed to be in the vacuum state and a thermal state in the instantaneously inertial frame and in the coaccelerated frame. We show that both the upward transition $|{\ensuremath{\psi}}_{\ifmmode\pm\else\textpm\fi{}}⟩\ensuremath{\rightarrow}|{e}_{A}{e}_{B}⟩$ and the downward transition $|{\ensuremath{\psi}}_{\ifmmode\pm\else\textpm\fi{}}⟩\ensuremath{\rightarrow}|{g}_{A}{g}_{B}⟩$ take place, where $|{g}_{A}{g}_{B}⟩$ and $|{e}_{A}{e}_{B}⟩$ represent two collective eigenstates of the two-atom system with both atoms in their ground state and excited state respectively, and the transition rates observed in the two frames are identical only if there exists a thermal bath at the Fulling-Davies-Unruh temperature in the coaccelerated frame. Though both the downward and upward transition rates are characterized by the Plankian factor, this factor does not show up in the average variation rate of energy of the two-atom system, as a result of the perfect cancellation of the terms in the transition rates in the instantaneously inertial frame multiplied by the Plankian factor which corresponds to the Unruh thermal bath, and the cancellation of the temperature-dependent terms in the transition rates observed in the coaccelerated frame. A comparison of the transition rates of a static two-atom system in interaction with the massless scalar field in a thermal state at an arbitrary temperature with those of a uniformly accelerated two-atom system in interaction with the vacuum massless scalar field shows that the equivalence between the thermal bath and the uniform acceleration, which is well known to be valid for the single atom case, no longer holds for the two-atom system in the symmetric/antisymmetric entangled state. The effect of atomic uniform acceleration in the transition rates of the two-atom system, and the roles of both the downward and the upward transitions in the disentanglement of the two-atom system are also discussed.

DOI
10.1103/physrevd.101.085009

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We calculate respectively, from the viewpoints of an instantaneously/locally inertial observer and a coaccelerated observer, the transition rates of the two-atom system in the symmetric/antisymmetric entangled state [∣\ensuremathψ\ifmmode±\else\textpm\fi⟩|{\ensuremath{\psi}}_{\ifmmode\pm\else\textpm\fi{}}⟩∣\ensuremathψ\ifmmode±\else\textpm\fi​⟩]. The scalar fields to which the atoms are coupled are respectively assumed to be in the vacuum state and a thermal state in the instantaneously inertial frame and in the coaccelerated frame. We show that both the upward transition ∣\ensuremathψ\ifmmode±\else\textpm\fi⟩\ensuremath→∣eAeB⟩|{\ensuremath{\psi}}_{\ifmmode\pm\else\textpm\fi{}}⟩\ensuremath{\rightarrow}|{e}_{A}{e}_{B}⟩∣\ensuremathψ\ifmmode±\else\textpm\fi​⟩\ensuremath→∣eA​eB​⟩ and the downward transition ∣\ensuremathψ\ifmmode±\else\textpm\fi⟩\ensuremath→∣gAgB⟩|{\ensuremath{\psi}}_{\ifmmode\pm\else\textpm\fi{}}⟩\ensuremath{\rightarrow}|{g}_{A}{g}_{B}⟩∣\ensuremathψ\ifmmode±\else\textpm\fi​⟩\ensuremath→∣gA​gB​⟩ take place, where ∣gAgB⟩|{g}_{A}{g}_{B}⟩∣gA​gB​⟩ and ∣eAeB⟩|{e}_{A}{e}_{B}⟩∣eA​eB​⟩ represent two collective eigenstates of the two-atom system with both atoms in their ground state and excited state respectively, and the transition rates observed in the two frames are identical only if there exists a thermal bath at the Fulling-Davies-Unruh temperature in the coaccelerated frame. Though both the downward and upward transition rates are characterized by the Plankian factor, this factor does not show up in the average variation rate of energy of the two-atom system, as a result of the perfect cancellation of the terms in the transition rates in the instantaneously inertial frame multiplied by the Plankian factor which corresponds to the Unruh thermal bath, and the cancellation of the temperature-dependent terms in the transition rates observed in the coaccelerated frame. A comparison of the transition rates of a static two-atom system in interaction with the massless scalar field in a thermal state at an arbitrary temperature with those of a uniformly accelerated two-atom system in interaction with the vacuum massless scalar field shows that the equivalence between the thermal bath and the uniform acceleration, which is well known to be valid for the single atom case, no longer holds for the two-atom system in the symmetric/antisymmetric entangled state. The effect of atomic uniform acceleration in the transition rates of the two-atom system, and the roles of both the downward and the upward transitions in the disentanglement of the two-atom system are also discussed.
We explore, respectively, from the viewpoints of a locally inertial observer and a coaccelerated observer, the transition rates of a two-atom system in the symmetric or antisymmetric entangled state, [Formula: see text] or [Formula: see text], and coupled to the electromagnetic fields. The interatomic separation is assumed to be constant and perpendicular to the acceleration; the fields to which the atoms are coupled are assumed to be in the vacuum state in the locally inertial frame and in a thermal state in the coaccelerated frame. We find that, in the viewpoints of the two observers, both the downward transition [Formula: see text] and the upward transition [Formula: see text] can take place, where [Formula: see text] and [Formula: see text] denote two eigenstates of the two-atom system with both atoms in their ground states and excited states, respectively. With an appropriate choice of the orientations of the atomic dipole moments, coherent radiation can happen for a system in acceleration but not for one in inertial motion. By comparing the transition rates viewed by the two observers, we show that the transition rates of the accelerated two-atom system viewed by a locally inertial observer can only be recovered in the coaccelerated frame by assuming a thermal bath at the Fulling–Davies–Unruh temperature. The effects of these collective transitions on the evolution of energy of the two-atom system are also analyzed.
We investigate the transition rates of uniformly accelerated two-level single and entangled atomic systems in empty space as well as inside a cavity. We take into account the interaction between the systems and a massless scalar field from the viewpoint of an instantaneously inertial observer and a coaccelerated observer, respectively. The upward transition occurs only due to the acceleration of the atom. For the two-atom system, we consider that the system is initially prepared in a generic pure entangled state. In the presence of a cavity, we observe that for both the single and the two-atom cases, the upward and downward transitions are occurred due to the acceleration of the atomic systems. The transition rate manifests subtle features depending upon the cavity and system parameters, as well as the initial entanglement. It is shown that no transition occurs for a maximally entangled superradiant initial state, signifying that such entanglement in the accelerated two-atom system can be preserved for quantum information processing applications. Our analysis comprehensively validates the equivalence between the effect of uniform acceleration for an inertial observer and the effect of a thermal bath for a coaccelerated observer, in free space as well as inside a cavity, if the temperature of the thermal bath is equal to the Unruh temperature.
By considering the interaction between a two-atom system and the vacuum massless scalar field in the viewpoint of an instantaneously inertial observer, we study the rates of transition of a uniformly accelerated two-atom system in the symmetric/antisymmetric entangled state near a reflecting boundary and in a cavity, respectively. We find that both the downward transition | ψ ± ⟩ → | g A g B ⟩ and the upward transition | ψ ± ⟩ → | e A e B ⟩ occur for the accelerated two-atom system, as in sharp contrast with the case of a static two-atom system, in which the upward transition can never happen. Similar to the rates of transition of atoms immersed in a thermal bath with the FDU temperature, both the downward transition rate and the upward transition rate are characterized by the Plank factor ( e 2 π ω 0 / a − 1 ) − 1 . This character of the transition rates is very different from the other radiative properties of the accelerated two-atom system, such as the resonance interatomic energy, for which the revisions of the effects of uniform acceleration are never characterized by such a factor. We show with analytical and numerical results that both the downward transition and the upward transition processes can be effectively manipulated by the atomic non-inertial motion and by the presence of boundaries. By comparing the upward transition rate with the downward transition rate, we discover that, when ω 0 ≫ a , with ω 0 and a being the energy space and the proper acceleration of the two-atom system, the disentanglement caused by the upward transition is negligible, while, if ω 0 ≪ a , the disentanglement caused by the upward transition becomes as important as that caused by the downward transition.
Abstract Transition rates of a uniformly accelerated atom coupled to a vacuum scalar field are identical with those of one being static in a thermal bath at a temperature proportional to the proper acceleration a, i.e., TU=a2πT_U=\frac{a}{2\pi }TU​=2πa​ T U = a 2 π . We discuss in this paper whether there exists such an equivalence in the electromagnetic field case if more atoms are involved. To be specific, we explore the similarities and distinctions between effects of a thermal bath and those of the uniformly accelerated motion by comparing the transition rates of a two-atom system initially in the symmetric or antisymmetric entangled state [ ∣ψ±⟩|\psi _{\pm }\rangle ∣ψ±​⟩ | ψ ± ⟩ ] in two cases, i.e., two static atoms in a thermal bath [the thermal case] and two atoms uniformly accelerated in vacuum [the acceleration case]. We discover that, for some particular orientations of atomic dipole moments, coherent radiation which happens in the acceleration case never occurs in the thermal case. The rates in the acceleration case with a low acceleration which means a low TUT_UTU​ T U can be smaller or larger and even equal to their counterparts in the thermal case; while the rates in the acceleration case with a high acceleration which means a high TUT_UTU​ T U are always much larger than their counterparts in the thermal case. Our results suggest that effects of a thermal bath and those of the uniformly accelerated motion on transition properties of a two-atom system in ∣ψ±⟩|\psi _{\pm }\rangle ∣ψ±​⟩ | ψ ± ⟩ are not necessarily distinctive, but can be equivalent if the acceleration is relatively low, depending on the value of the interatomic separation.
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-like 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-like 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.
We apply the DDC formalism [proposed by Dalibard, Dupont-Roc and Cohen-Tannoudji] to study the average rate of change of energy of two identical two-level atoms interacting with the vacuum massless scalar field in synchronized motion along stationary trajectories. By separating the contributions of vacuum fluctuations and atomic radiation reaction, we first show that for the two-atom system initially prepared in the factorizable eigenstates ∣gAgB⟩|{g}_{A}{g}_{B}⟩∣gA​gB​⟩ and ∣eAeB⟩|{e}_{A}{e}_{B}⟩∣eA​eB​⟩, where ggg and eee represent the ground state and the excited state of a single atom, respectively, both vacuum fluctuations and atomic radiation reaction contribute to the average rate of change of energy of the two-atom system, and the contribution of vacuum fluctuations is independent of the interatomic separation while that of atomic radiation reaction is dependent on it. This is contrary to the existing results in the literature where vacuum fluctuations are interatomic-separation dependent. However, if the two-atom system is initially prepared in the unfactorizable symmetric/antisymmetric entangled state, the average rate of change of energy of the two-atom system is never perturbed by the vacuum fluctuations, but is totally a result of the atomic radiation reaction. We then consider two special cases of motion of the two-atom system, which is initially prepared in the symmetric/antisymmetric entangled state, i.e., synchronized inertial motion and synchronized uniform acceleration. In contrast to the average rate of change of energy of a single uniformly accelerated atom, the average rate of change of energy of the uniformly accelerated two-atom system is nonthermal-like. The effects of noninertial motion on the transitions of states of the two correlated atoms are also discussed.