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Physical Review A

Atoms in moderately strong electromagnetic fields: General method and its application to the two-level atom

C. Alton Coulter

Published Dec 1, 1974• 25 citations
Publisher

Abstract

The desirability of considering "dressed" rather than "bare" states of atoms in strong electromagnetic fields has been noted by others. It is suggested here that such a treatment is conveniently carried out by performing a unitary transformation of the Hamiltonian so that the "dressed" states in the new basis are precisely the "bare" states of the usual basis. A perturbative method for constructing such a transformation, closely related to a procedure described by Heitler, is proposed. The method is applied through second order to a two-level atom model, and it is verified that the transformation makes the dressed ground atomic state stable, and the excited state unstable with the proper decay probability; furthermore, the energies of the two states are renormalized by the correct amounts. The method is then compared with the "nonperturbative" momentum-translation approximation, and shown to contain already in first order the entire operator content of that approximation, but to yield different transition probabilities. Finally, the transformation method is applied through second order to the standard scalar-field model and shown to produce the usual dressing transformation, mass renormalization, and induced nucleon-nucleon potential of that model, a result which tends to substantiate the hypothesis that the transformation produces the correct dressed states of a physical system.

DOI
10.1103/physreva.10.1946

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The desirability of considering "dressed" rather than "bare" states of atoms in strong electromagnetic fields has been noted by others. It is suggested here that such a treatment is conveniently carried out by performing a unitary transformation of the Hamiltonian so that the "dressed" states in the new basis are precisely the "bare" states of the usual basis. A perturbative method for constructing such a transformation, closely related to a procedure described by Heitler, is proposed. The method is applied through second order to a two-level atom model, and it is verified that the transformation makes the dressed ground atomic state stable, and the excited state unstable with the proper decay probability; furthermore, the energies of the two states are renormalized by the correct amounts. The method is then compared with the "nonperturbative" momentum-translation approximation, and shown to contain already in first order the entire operator content of that approximation, but to yield different transition probabilities. Finally, the transformation method is applied through second order to the standard scalar-field model and shown to produce the usual dressing transformation, mass renormalization, and induced nucleon-nucleon potential of that model, a result which tends to substantiate the hypothesis that the transformation produces the correct dressed states of a physical system.
We consider a two-level atom in a strong laser field, described by the Jaynes-Cummings Hamiltonian in the rotating-wave approximation, yielding the well known dressed states. Diagonalization of the total Hamiltonian is attained by introducing dressed-atomic and photon operators. The quantum theory of measurement of finite duration is used to explore the nature of the physical observables corresponding to the dressed operators. A measuring apparatus is contrived in order to observe the bare-atomic population and the bare-photon number in a dressed state of the system. It is shown that, if the duration of the measurement is longer than the Rabi period of the system, the apparatus leads to observation of the dressed-atomic population and of the dressed-photon number rather than of the corresponding bare quantities for which it was designed. Conceptual and practical consequences of this result are discussed.
A pair of nonidentical two-level atoms, separated by a fixed distance R, interact through photon exchange. The system is described by a state vector which is assumed to be a superposition of four "essential states": (1) the first atom is excited, the second one is in the ground state, and no photon is present, (2) the first atom is in its ground state, the second one is excited, and no photon is present, (3) both atoms are in their ground states and a photon is present, and (4) both atoms are excited and a photon is also present. The system is initially in state (1). The probabilities of each atom being excited are calculated for both the minimally-coupled interaction and the multipolar interaction in the electric dipole approximation. For the minimally-coupled interaction Hamiltonian, the second atom has a probability of being instantaneously excited, so the interaction is not retarded. For the multipolar interaction Hamiltonian, the second atom is not excited before the retardation time, which agrees with special relativity. For the minimally-coupled interaction the nonphysical result occurs because the unperturbed Hamiltonian is not the energy operator in the Coulomb gauge. For the multipolar Hamiltonian in the electric dipole approximation the unperturbed Hamiltonian is the energy operator. An active view of unitary transformations in nonrelativistic quantum electrodynamics is used to derive transformation laws for the potentials of the electromagnetic field and the static Coulomb potential. For a specific choice of unitary transformation the transformation laws for the potentials are used in the minimally-coupled second-quantized Hamiltonian to obtain the multipolar Hamiltonian, which is expressed in terms of the quantized electric and magnetic fields.
The dual Dyson series [M. Frasca, Phys. Rev. A 585858 3439 (1998)] is used to develop a general perturbative method for the study of atom-field interaction in quantum optics. In fact, both the Dyson series and its dual, by renormalization-group methods to remove secular terms from the perturbation series, give the opportunity for a full study of the solution of the Schr\"odinger equation in different ranges of the parameters of the given Hamiltonian. In view of recent experiments with strong laser fields, this approach seems well-suited to give a clarification and an improvement of the applications of the dressed states as currently done through the eigenstates of the atom-field interaction, showing that these are just the leading order of the dual Dyson series when the Hamiltonian is expressed in the interaction picture. In order to best exploit the method, a study is accomplished of the well-known Jaynes-Cummings model in the rotating-wave approximation, whose exact solution is known, comparing the perturbative solutions obtained by the Dyson series and its dual with the same approximations obtained by Taylor expanding the exact solution. Finally, a full perturbative study of high-order harmonic generation is given, obtaining, through analytical expressions, a clear account of the power spectrum using a two-level model, even if the method can be successfully applied to a more general model that can account for ionization too. The analysis shows that to account for the power spectrum it is necessary to go to first order in the perturbative analysis. The spectrum obtained gives a way to measure experimentally the shift of the energy levels of the atom interacting with the laser field by looking at the shifting of hyper-Raman lines.
In recent papers we have discussed the meaning of dressed excited quantum states. We have shown that these states can be formulated in terms of distribution functions outside the Hilbert space. Our approach applies to ``nonintegrable'' systems in the sense of Poincar\'e. It involves the analytic continuation of the unitary operator U describing the transformation from bare to dressed stable states. This leads to a ``star-unitary'' operator $\ensuremath{\Lambda}.$ Interacting fields are nonintegrable systems obtained as a result of resonances. It is therefore natural to expect that our previous results remain valid for interacting fields. We consider a simple example, which corresponds to an extension of the usual Friedrichs model. This involves a local field in interaction with a bilocal scalar field. This model has been previously studied by one of the co-authors (I.P.). It is a simplified version of the model describing $\stackrel{\ensuremath{\rightarrow}}{A}B+C$ transition with quadratic interaction. The usual Bogoliubov transformation eliminates the field corresponding to A, while our method leads to strictly exponential decay of unstable field. The dressed field A corresponds to a singular distribution function outside the Liouville-Hilbert space. As in the Friedrichs model there exists two time scales---one for the preparation of a dressed state and depending on initial conditions called ``Zeno period'' and the other universal for the decay.
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.
This paper presents a nonperturbative treatment of strong-coupling induced effects in atom-field systems which cannot be seen in traditional perturbative treatments invoking compromising assumptions such as the Born-Markov, rotating wave, or Fermi Golden rule. We consider an atom whose internal degrees of freedom are modeled by a harmonic oscillator which is bilinearly coupled to a scalar quantum field, representing one of the two polarizations of an electromagnetic field. Because the whole system is Gaussian we can solve this problem exactly. Using the open quantum system conceptual framework and the influence functional formalism we derive the dynamics of the reduced density matrix for the atom which enables the calculation of atomic transition probability and other relevant physical quantities. Finding an exact solution to this problem has the distinct advantage of enabling one to capture fully the strong coupling regime and discover interesting effects such as spontaneous ground state excitation [R. Passante, T. Petrosky, and I. Prigogine, Long-time behaviour of self-dressing and indirect spectroscopy, Physica (Amsterdam) 218A, 437 (1995).] which is unfathomable in perturbative treatments. The conventional description of atomic-optical activities is predicated on the assumption that the state of the total atom-field system is a product state of the atomic excitations and the photon number states, an assumption which is valid only for vanishingly weak coupling. The correct energy eigenfunctions to use should be that of the Hamiltonian of the combined atom-field system. Other features associated with finite to strong coupling effects such as resonance peak broadening and transition from a gapped to a gapless spectrum can all be understood from this perspective. Finally, to put the issues in a proper perspective we take the perturbative limit of the exact results and compare them with those from conventional time-dependent perturbation theory (TDPT). This enables one to pin-point where the deficiencies of TDPT lie as one removes the ultraweak coupling assumption.