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Physical Review A

Quantum transitions and nonlocality

Tomio Yamakoshi Petrosky, Gonzalo E. Ordonez, Ilya Prigogine

Published Sep 15, 2000• 26 citations
Publisher

Abstract

As shown by Hegerfeldt, positivity of energy in quantum mechanics leads to nonlocal effects, where a state that is initially localized immediately develops infinite tails. In a previous paper we have described the mechanism of such effects for relativistic free fields. In this paper we discuss the nonlocal effects for a more complex system that consists of a two-level atom interacting with a radiation field. We study the dressing process of the bare ground state in terms of the photon density. In this process there appears a nonlocal cloud that remains around the atom and ``Zeno'' photons that move away from the atom with the speed of light. Nonlocality is seen to be to due to a ``curtain'' effect where the cloud is initially covered by the Zeno photons, and immediately afterwards is uncovered as the photons move away from the atom. No superluminal speeds are involved. We also consider nonlocal effects associated with a partially dressed state where the atom is excited starting from the true ground state. The excitation of the atom produces a nonlocal rearrangement of the photon cloud, which leads to precursors to the usual retarded field.

DOI
10.1103/physreva.62.042106

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As shown by Hegerfeldt, positivity of energy in quantum mechanics leads to nonlocal effects, where a state that is initially localized immediately develops infinite tails. In a previous paper we have described the mechanism of such effects for relativistic free fields. In this paper we discuss the nonlocal effects for a more complex system that consists of a two-level atom interacting with a radiation field. We study the dressing process of the bare ground state in terms of the photon density. In this process there appears a nonlocal cloud that remains around the atom and Zeno'' photons that move away from the atom with the speed of light. Nonlocality is seen to be to due to a curtain'' effect where the cloud is initially covered by the Zeno photons, and immediately afterwards is uncovered as the photons move away from the atom. No superluminal speeds are involved. We also consider nonlocal effects associated with a partially dressed state where the atom is excited starting from the true ground state. The excitation of the atom produces a nonlocal rearrangement of the photon cloud, which leads to precursors to the usual retarded field.
We study the radiative emission due to the self-dressing of a two-level atom, initially in its bare ground state, interacting with the zero-point electromagnetic field. Evolution in time leads to the formation of a dressed ground state of lower energy. This energy difference between bare and dressed ground state is taken into account by the emission of real photons. In order to describe this aspect of the self-dressing process we study the transition probability amplitude from the initial bare state to an asymptotic state consisting of the atom in its dressed ground state plus some real photons. Adopting nonperturbative techniques based on the resolvent method we find that the bare-dressed transition occurs at the lowest order with the emission of two photons. The spectral distribution obtained shows that there is a probability for emission of photons at frequency of the order of the atom's transition frequency.
We consider a system formed by a two-level atom initially in its bare ground state and the electromagnetic field in the absence of radiation. The interaction of the atom with the radiation field leads to the formation of a dressed ground state. To describe the dynamics of self-dressing, we study the time evolution of the permanence amplitude in the initial state. We adopt non-relativistic QED and take the atom-field interaction Hamiltonian in the minimal coupling form, within the electric dipole approximation. In order to follow the evolution of the permanence amplitude for long enough time periods we use the van Hove resolvent technique with appropriate partial summation of Feynman graphs. We show that the self-dressing process presents the typical behaviour of a decay process and occurs with a timescale a *-1 , which is long compared with both the atom's inverse transition frequency 0 -1 and the excited state decay time -1 . This time is comparable with the lifetime of the excited state due to ordinary two-photon decay. We consider the reasons for such a long time period of self-dressing and make comparison with previous results.
We consider influence of nonlocal (retardation) effects caused by the finite value of the speed of light on collective emission of a single photon by atomic ensembles. Using a fully quantum mechanical description of light and atoms we obtain an evolution equation for the atomic system that takes into account retardation. We found an exact analytical solution of this equation for the atomic slab geometry that yields insight on how the crossover between local and nonlocal dynamics occurs. In particular, it shows that initially nonlocal evolution, accompanied by collective oscillations of atomic population, becomes local at large time. In addition, atomic excitation in some parts of the sample rises above its initial value. We propose an experiment in which the transition between local and nonlocal regimes can be observed by increasing the size of the atomic sample or atomic density.
In a previous paper we have studied dressed excited states in the Friedrichs model, which describes a two-level atom interacting with radiation. In our approach, excited states are distributions (or generalized functions) in the Liouville space. These states decay in a strictly exponential way. In contrast, the states one may construct in the Hilbert space of wave functions always present deviations from exponential decay. We have considered the momentum representation, which is applicable to global quantities (trace, energy transfer). Here we study the space-time description of local quantities associated with dressed unstable states, such as, the intensity of the photon field. In this situation the excited states become factorized in Gamow states. To go from local quantities to global quantities, we have to proceed to an integration over space, which is far from trivial. There are various elements that appear in the space-time evolution of the system: the unstable cloud that surrounds the bare atom, the emitted real photons and the ``Zeno photons,'' which are associated with deviations from exponential decay. We consider a Hilbert space approximation to our dressed excited state. This approximation leads already to decay close to exponential in the field surrounding the atom, and to a line shape different from the Lorentzian line shape. Our results are compared with numerical simulations. We show that the time evolution of an unstable state satisfies a Boltzmann-like H\mathcal{H}H theorem. This is applied to emission and absorption as well as scattering. The existence of a microscopic H\mathcal{H}H theorem is not astonishing. The excited states are ``nonequilibrium'' states and their time evolution leads to the emission of photons, which distributes the energy of the unstable state among the field modes.
An atom or a molecule is constituted by a set of bound electric charges with dynamics governed by the laws of quantum mechanics. These charges are sources of the quantized electromagnetic field which also binds them together, and thus the effects of their interaction with the field cannot be disregarded in principle. Consequently even overall neutral atoms, on which we focus our attention, are driven by a dynamics which is inextricably related to the dynamics of the quantized electromagnetic field. One of the most prominent aspects of the atom-held interaction is the existence of a cloud of virtual photons which dresses the atom even in the lowest possible energy state of the system. In this paper we review the static as well as the dynamic aspects of the theory of the virtual cloud around the neutral atoms. We begin by reviewing various forms of the atom-field coupling as well as various models of simplified atoms which will be used in the rest of the paper. The question then arises as how to characterize quantitatively the shape of the virtual cloud, and we show that the energy density of the electromagnetic field is a physical quantity suitable for this purpose. First a perturbative approach to calculating this shape is developed and applied to several physical models of a ground-state dressed atom. The next step is to consider virtual clouds which are out of equilibrium and examine their time development. This leads to the concept of half-dressed sources, which are discussed in a different physical context both in the absence and in the presence of real photons. In particular the role of the virtual cloud in ensuring causality of the field propagating in dressing and undressing processes is emphasized. Finally, the nature of the virtual cloud is further discussed in the light of theories concerning the dynamics of an atomic pair, the quantum theory of measurement and the effects of a driving electromagnetic field.
We consider a multilevel atom such as an hydrogen atom, interacting with the quantum electromagnetic field, in the dressed ground state of the interacting system. We evaluate by perturbation theory the dressed ground state of the system, within dipole approximation; we investigate and review the effect of the self-dressing of the atom on several field and atomic observables. Specifically, we obtain general expressions of the renormalized electric and magnetic field fluctuations and energy densities around the atom, and analyze their scaling with the distance from the atom, obtaining approximated expressions in the so-called near and far zone. We also investigate nonlocal spatial field correlations around the atom. We stress how the quantities we evaluate can be probed through two- and three-body nonadditive Casimir-Polder dispersion interactions. We also investigate the effect of the self-dressing, i.e. of the virtual transitions occurring in the dressed ground state, on atomic observables, for example the average potential energy of the electron in the nuclear field; this also allows us to obtain a more fundamental quantum basis for the Welton interpretation of the Lamb shift of a ground-state hydrogen atom, in terms of the atomic self-dressing processes.