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Physical Review A

General framework for interatomic interaction energy of two ground-state atoms in a thermal bath

Shijing Cheng, Wenting Zhou, Hongwei Yu

Published Jan 19, 2023• 5 citations
Publisher

Abstract

We give a general derivation of the fourth-order DDC formalism (a formalism proposed by Dalibard, Dupont-Roc, and Cohen-Tannoudji) for calculating the interaction energy between two ground-state multilevel atoms which are coupled to electromagnetic fields in a thermal bath at temperature $T$. Both the contributions of the thermal field fluctuations and the radiation reaction of atoms are separately identified. As an application of the formalism, we revisit the interaction energy of two static ground-state two-level atoms in a free space and discover new behaviors such as the $\ensuremath{\sim}T{L}^{\ensuremath{-}2}$ behavior of the van der Waals interaction energy in the region where ${\ensuremath{\lambda}}^{3/4}{T}^{\ensuremath{-}1/4}\ensuremath{\ll}L\ensuremath{\ll}\ensuremath{\lambda}$, with $\ensuremath{\lambda}$ the transition wavelength of the atoms and $L$ the interatomic separation, and an oscillatory behavior of the Casimir-Polder interaction energy as $L$ varies which is superimposed on the monotonic $\ensuremath{\sim}T{L}^{\ensuremath{-}6}$ dependence when $L\ensuremath{\gg}\ensuremath{\lambda}\ensuremath{\gg}{T}^{\ensuremath{-}1}$.

DOI
10.1103/physreva.107.012815

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We give a general derivation of the fourth-order DDC formalism (a formalism proposed by Dalibard, Dupont-Roc, and Cohen-Tannoudji) for calculating the interaction energy between two ground-state multilevel atoms which are coupled to electromagnetic fields in a thermal bath at temperature TTT. Both the contributions of the thermal field fluctuations and the radiation reaction of atoms are separately identified. As an application of the formalism, we revisit the interaction energy of two static ground-state two-level atoms in a free space and discover new behaviors such as the \ensuremath∼TL\ensuremath−2\ensuremath{\sim}T{L}^{\ensuremath{-}2}\ensuremath∼TL\ensuremath−2 behavior of the van der Waals interaction energy in the region where \ensuremathλ3/4T\ensuremath−1/4\ensuremath≪L\ensuremath≪\ensuremathλ{\ensuremath{\lambda}}^{3/4}{T}^{\ensuremath{-}1/4}\ensuremath{\ll}L\ensuremath{\ll}\ensuremath{\lambda}\ensuremathλ3/4T\ensuremath−1/4\ensuremath≪L\ensuremath≪\ensuremathλ, with \ensuremathλ\ensuremath{\lambda}\ensuremathλ the transition wavelength of the atoms and LLL the interatomic separation, and an oscillatory behavior of the Casimir-Polder interaction energy as LLL varies which is superimposed on the monotonic \ensuremath∼TL\ensuremath−6\ensuremath{\sim}T{L}^{\ensuremath{-}6}\ensuremath∼TL\ensuremath−6 dependence when L\ensuremath≫\ensuremathλ\ensuremath≫T\ensuremath−1L\ensuremath{\gg}\ensuremath{\lambda}\ensuremath{\gg}{T}^{\ensuremath{-}1}L\ensuremath≫\ensuremathλ\ensuremath≫T\ensuremath−1.
Abstract We generalize the formalism proposed by Dalibard, Dupont-Roc, and Cohen-Tannoudji (the DDC formalism) in the fourth order for two atoms in interaction with scalar fields in vacuum to a thermal bath at finite temperature T, and then calculate the interatomic interaction energy of two ground-state atoms separately in terms of the contributions of thermal fluctuations and the radiation reaction of the atoms and analyze in detail the thermal corrections to the van der Waals and Casimir–Polder interactions. We discover a particular region, i.e. λ 3 β 4 ≪ L ≪ λ with L, β and λ denoting the interatomic separation, the wavelength of thermal photons and the transition wavelength of the atoms respectively, where the thermal corrections remarkably render the van der Waals force, which is usually attractive, repulsive, leading to an interesting crossover phenomenon of the interatomic interaction from attractive to repulsive as the temperature increases. We also find that the thermal corrections cause significant changes to the Casimir–Polder force when the temperature is sufficiently high, resulting in an attractive force proportional to TL −3 in the λ ≪ β ≪ L region, and a force that can be either attractive or repulsive and even vanishing in the β ≪ λ ≪ L region depending on the interatomic separation.
We generalize the formalism proposed by Dalibard, Dupont-Roc, and Cohen-Tannoudji (the DDC formalism) to the fourth order of the coupling constant, which can be used to study the interatomic interaction of two ground-state atoms coupled with the vacuum scalar fields. We show that the interatomic potential can be attributed to the joint effect of both vacuum fluctuations and the radiation reaction of atoms. Remarkably, the formulas we derived for the contributions of vacuum fluctuations and the radiation reaction to the interatomic potential upon which future research on fourth-order effects in particular circumstances can be based differ from those in the existing literature [Phys. Rev. D 95, 085014 (2017)].
We study the modification by the presence of a plane wall of energy-level shifts of two-level atoms which are in multipolar coupling with quantized electromagnetic fields in a thermal bath in a formalism which separates the contributions of thermal fluctuations and radiation reaction and allows a distinct treatment to atoms in the ground and excited states. The position-dependent energy shifts give rise to an induced force acting on the atoms. We are able to identify three different regimes where the force shows distinct features and examine, in all regimes, the behaviors of this force in both the low-temperature limit and the high-temperature limit for both the ground-state and excited-state atoms, thus providing some physical insights into the atom-wall interaction at finite temperature. In particular, we show that both the magnitude and the direction of the force acting on an atom may have a clear dependence on the atomic polarization directions. In certain cases, a change in relative ratio of polarizations in different directions may result in a change in direction of the force.
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.
Abstract We study the interaction potential of two nonidentical ground-state atoms coupled to a scalar field in a vacuum by separately calculating the contributions of vacuum fluctuations and those of the radiation reaction of the atoms. Both cases of atoms in a free space and in parallel or vertical alignment to a reflecting boundary are considered. For the former case, we find that the leading-order interaction potential in the region λ A ≪ L ≪ λ B exhibits the same separation-dependence as that in the region L ≪ λ A ≪ λ B , where L, λ A and λ B are respectively the interatomic separation and the transition wavelengths of two atoms with λ A ≪ λ B . For the latter case, we find that boundary-induced modifications are very remarkable when L ≫ z, with z characterizing the separation between the two-atom system and the boundary. Particularly, when L further satisfies L ≫ λ A and L ≫ λ B , the interaction potential in the parallel- and the vertical-alignment cases respectively scales as z 4 L −7 and z 2 L −5, the L-dependence of which is one order higher than those of two atoms in regions where L ≫ z and meanwhile L ≪ λ A or/and L ≪ λ B . Our results suggest that retardation for the interaction potential of two nonidentical atoms with remarkably distinctive transition frequencies happens only when the interatomic separation is much greater than the transition wavelengths of both atoms.
We investigate the radiative processes of a quantum system composed by two identical two-level atoms in the de Sitter spacetime, interacting with a conformally coupled massless scalar field prepared in the de Sitter-invariant vacuum. We discuss the structure of the rate of variations of the atomic energy for two static atoms. Following a procedure developed by Dalibard, Dupont-Roc, and Cohen-Tannoudji, our intention is to identify in a quantitative way the contributions of vacuum fluctuations and the radiation reaction to the generation of quantum entanglement and to the degradation of entangled states. We find that when the distance between two atoms larger than the characteristic length scale, the rate of variation of atomic energy in the de Sitter-invariant vacuum behaves differently compared with that in the thermal Minkowski spacetime. In particular, the generation and degradation of quantum entanglement can be enhanced or inhibited, which are dependent not only on the specific entangled state but also on the distance between the atoms.