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.
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.
A model of a coupled atom---quantum-field system is presented, consisting of a time-independent Hamiltonian and a realistic dynamical initial condition. The Hamiltonian describes a fixed bound atom coupled to the complete electromagnetic field. The initial condition is a consistent quantum-mechanical description of an atom in the exact ground state of the coupled atom-field system, on which is superimposed an approaching quasiclassical excitation of the field which will cause the interesting dynamical processes later on. Tools are presented for the manipulation of this model, with emphasis towards series expansions of all relevant quantities suitable for small strength $\ensuremath{\lambda}$ of the atom-field coupling and (potentially) large magnitude $\ensuremath{\alpha}$ of the coherent field excitation, such that the "influence of the coherent excitation on the atom" (i.e., the product $\ensuremath{\lambda}\ensuremath{\alpha}$) remains constant as $\ensuremath{\lambda}$ tends to zero. The following results are obtained, at the lowest nontrivial order in $\ensuremath{\lambda}$, in an approximation suitable for times much shorter than any radiative lifetime of the atom: (a) The change in free-field energy due to the interaction with the atom is exactly compensated at all times by the change in bare-atom energy plus the atom-field coupling energy. This checks the consistency of the proposed scheme. (b) The frequency distribution of this change in free-field energy has a (somewhat unexpected) component, with a smooth spectrum, spreading over the whole width of the spectrum of the exciting pulse, in addition to $\ensuremath{\delta}$ and dispersive singularities centered on the atomic transition frequencies. This frequency distribution is discussed in detail and its features are illustrated by a number of figures, in a simple case, numerically tractable. (c) The heterodyne detection of the radiation emitted by the atom is investigated, with emphasis on the issue of causality and propagation of light. Strict causality is obtained, without further approximation, in a model with an atom-field coupling involving the field at a single point and a detector similarly sensitive to properties of the field at a single point. This result breaks down if one uses nonlocal approximations such as the rotating-wave approximation or a detection observable ${\mathbf{E}}^{\ensuremath{-}}(\mathbf{r})\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{E}}^{+}(\mathbf{r})$. Finally, an appendix is devoted to the use, in quantum mechanics, of bases moving with respect to each other.
We examine the dynamics of bipartite quantum entanglement between a two-level atom and the electromagnetic field. We begin with the Jaynes-Cummings model with a single field mode and examine in detail the exact time evolution of entanglement, including cases where the atomic state is initially mixed and the atomic transition is detuned from resonance. We then explore the effects of other nearby modes by calculating the exact time evolution of entanglement in more complex systems with two, three, and five field modes. For these cases we can obtain exact solutions which include the strong-coupling regime. Finally, we consider the entanglement of a two-level atom with the infinite collection of modes present in the intracavity field of a Fabre-Perot cavity. In contrast to the usual treatment of atom-field interactions with a continuum of modes using the Born-Markov approximation, our treatment in all cases describes the full non-Markovian dynamics of the atomic subsystem under general couplings. Only when an analytic expression for the infinite mode case is desired do we need to make a weak coupling assumption which at late times approximates Markovian dynamics.
We present a non-perturbative framework for the quantum treatment of vibrational excitation in molecular systems driven by electric fields with arbitrary time dependence. For harmonic potentials with linear dipole coupling, the field-induced dynamics is exactly described by a displaced coherent state, whose evolution is completely determined by a complex phase-space coordinate. Closed-form expressions for this coordinate reveal a clear distinction between resonant and non-resonant regimes. The resulting vibrational populations follow a Poisson distribution determined by a dimensionless time-dependent phase-space displacement, for both resonant and non-resonant driving. Resonant driving results in an enhanced phase-space displacement, whereas finite detuning between the driving and vibrational frequencies leads to an exponential suppression of the phase-space displacement and vibrational excitation. While the approach reduces to the standard time-dependent perturbation theory in the weak-field limit, it remains valid in the strong-field regime, where perturbative treatments fail. Our results provide a transparent phase-space picture of vibrational control and enable non-perturbative state preparation in ultrafast spectroscopy.
Within the framework of master equation, we study decay dynamics of an atom-molecule system strongly coupled by two photoassociation lasers. Summing over the infinite number of electromagnetic vacuum modes that are coupled to the laser-dressed atom-molecule system, we obtain an integro-differential master equation for the the system's reduced density matrix. The equation is numerically solved to describe system dynamics in the presence of decay. In particular, we discuss correlated spontaneous emission from a pair of electronically excited diatomic ro-vibrational states due to their laser induced coupling to the ground continuum of atomic scattering states. This allows us to calculate time-dependence of emitted radiation intensity. It exhibits quantum beats due to coherent dynamics. The phase difference between the two driving fields is found to significantly affect the decay dynamics and the beats. Our results demonstrate the possibility to control decay from the molecular excited states and the decoherence between them by changing the relative intensity and the phase between the lasers. We further show that, if the ground-state continuum has a shape resonance at a low energy, then the quantum beats show two distinctive time scales of oscillations in the strong coupling regime. One of the time scales originates from the energy gap between the two excited states while the other time scale corresponds to the collision energy at which free-bound Franck-Condon overlap is resonantly peaked due to the shape resonance.
Abstract This chapter presents some of the models describing the time-dependent interaction between a simple atom and a field, which may be described either classically or fully quantum mechanically. A number of equivalent interaction pictures and the important rotating-wave approximation are introduced. The two-state dynamics of an atom coupled to a classical field is solved, starting with perturbation theory. Non-perturbative behaviour is analysed, leading to Rabi oscillations of the atomic probabilities. The Jaynes–Cummings model of the dynamics of a single two-state atom coupled to a single quantized mode of the electromagnetic field is solved leading to collapses and revivals of the excited state probability. An exactly soluble model of the coupling of a single state to a Bixon–Jortner quasi-continuum of discrete levels is analysed. Finally, the coherent interactions between two field modes, of the type common in nonlinear optics, are discussed.
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.