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arXiv:quant-ph/9904067 · Journal of Modern Optics

Dressed-state approach to population trapping in the Jaynes-Cummings model

Daniel Jonathan, K. Furuya, A. Vidiella-Barranco

Published Oct 1, 1999• 12 citations
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Abstract

The phenomenon of atomic population trapping in the Jaynes-Cummings model is analysed from a dressed-state point of view. A general condition for the occurrence of partial or total trapping from an arbitrary, pure initial atom-field state is obtained in the form of a bound to the variation of the atomic inversion. More generally, it is found that in the presence of initial atomic or atom-field coherence the population dynamics is governed not by the field's initial photon distribution, but by a ‘weighted dressedness’ distribution characterizing the joint atom-field state. In particular, individual revivals in the inversion can be analytically described to good approximation in terms of that distribution, even in the limit of large population trapping. This result is obtained through a generalization of the Poisson Summation Formula method for an analytical description of revivals developed by Fleischhauer and Schleich (1993, Phys. Rev A, 47, 4258).

DOI
10.1080/09500349908231366

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The phenomenon of atomic population trapping in the Jaynes-Cummings model is analysed from a dressed-state point of view. A general condition for the occurrence of partial or total trapping from an arbitrary, pure initial atom-field state is obtained in the form of a bound to the variation of the atomic inversion. More generally, it is found that in the presence of initial atomic or atom-field coherence the population dynamics is governed not by the field's initial photon distribution, but by a ‘weighted dressedness’ distribution characterizing the joint atom-field state. In particular, individual revivals in the inversion can be analytically described to good approximation in terms of that distribution, even in the limit of large population trapping. This result is obtained through a generalization of the Poisson Summation Formula method for an analytical description of revivals developed by Fleischhauer and Schleich (1993, Phys. Rev A, 47, 4258).
The existence of revivals in the atomic inversion of the Jaynes-Cummings model (JCM) has been pinpointed on the numerical display of its expectation value. However, until recently its individual identification, isolation, and analytical representation was successful if the initial photon statistics allowed the use of saddle point techniques. 1 The recent introduction of a new analytical method,2 free of such restriction, and its successful use with thermal JCM revivals allows us to study the mixed thermal coherent JCM revivals, drawing a direct relation to the saddle point technique. We compare our results with previous ones of the pure thermal case,2 and the coherent limiting cases, in particular with the systematic technique proposed by Fleischhauer and Schleich.
An expression for the population inversion in the Jaynes-Cummings model and the most general initial state of the atom-field system is presented. States that maintain the same level of population inversion when the atom and the field are initially decoupled are found. When the atom is prepared in a coherent superposition of its states and the field is in an eigenstate of the Susskind-Glogower phase operator, for a certain choice of the relative phase between the atomic dipole and the field, coherent trapping occurs. The evolution of the squeezing of the field for these states is given, and the dependence on the phase and on the mean photon number of the field is studied.
An electromagnetic field state is found which maintains the population inversion of the atom stationary during the interaction with the field through a Jaynes-Cummings model (JCM) with a Kerr type nonlinearity in the cavity. The condition of stationarity of the population inversion includes the phase coupling of atomic dipole with the field. We have shown that the Kerr nonlinearity in the cavity field significantly modifies the photon statistics of the trapped field state through an intensity dependent detuning in the field compared to the normal JCM trapping state. We have also demonstrated the novel features of sub-Poissonian character and the squeezing of the trapped field state. The dynamics of the initial trapped field is studied in terms of squeezing and the Q-function.
An asymptotic result is derived for the Jaynes-Cummings model of a two-level atom interacting with a quantized single-mode field, which is valid when the field is initially in a coherent state with a large average photon number. It is shown that for certain initial atomic states the joint atom-field wave function factors into an atomic and a field part throughout the interaction, so that each system remains separately in a pure state. The atomic part of the wave function displays a crossing of trajectories in the atom Hilbert space that leads to a unique state for the atom, independent of its initial state, at a specific time t0{\mathit{t}}_{0}t0​ (equal to half the revival time). The field part of the wave function resembles a crescent squeezed state. The well-known collapses and revivals are investigated from this perspective. The collapse appears to be associated with a ``measurement'' of the initial state of the atom with the field as the measuring apparatus. The measurement is not complete for finite average photon number: the system is instead left in a coherent superposition of macroscopically distinct states. At the half-revival time t0{\mathit{t}}_{0}t0​ this superposition of states is entirely in the field part of the state vector, so that the (pure) state of the field at that time is of the form sometimes referred to as a ``Schr\"odinger cat.'' The revivals of the population inversion are seen to be entirely due to the fact that the linear superposition of the two macroscopically distinct field states is coherent (i.e., a pure state), as opposed to an incoherent mixture.
We investigate the phenomenon of quantum revivals in the Jaynes-Cummings model for an arbitrary quantized field mode. With the help of the Poisson summation formula, we cast the infinite sum determining the atomic inversion into an infinite sum of integrals. Each integral, when evaluated using the method of stationary phase, yields under appropriate conditions one revival. We present simple approximate analytical expressions for these revivals and illustrate this general technique by the examples of a coherent and a highly squeezed state. The oscillatory photon distribution of the latter creates slightly different Rabi frequencies which give rise to a beat note; that is, echos in the revivals. We obtain the photon statistics of the quantized field by ``measuring'' the atomic collapse of a single revival--a technique which might be applicable in the realm of the one-atom maser.
The emission of light from an atom represents a fundamental process that provides valuable insights into the atom–light interaction. The Jaynes–Cummings model is one of the simplest fully quantized models to deal with these interactions, allowing for an analytical solution, while exhibiting notable nontrivial effects. We explore new, to our knowledge, features in the fluorescence emission spectrum for initial “trapping states,” which suppress the atomic population inversion. Despite the seemingly dormant activity of the atom, the resulting emission spectra exhibit rich features, and using a dressed-state coordinates formalism, we are able to quantitatively explain the different profiles in the spectrum. We generalize the trapping conditions for nonzero atom-field detuning and also unveil two types of trapping states that lead to spectra with three peaks, in contrast to previously known states: a center peak and one secondary peak on each side. These are a trapping state formed by a Schrödinger cat state with Poissonian statistics (Yurke–Stoler state) and also a different type of “perfect trapping state.”