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Physical Review A

Time-averaged properties of the Jaynes-Cummings model: Effects of detuning

Mark S. Hillery, Robert John Schwartz

Published Feb 1, 1991• 12 citations
Publisher

Abstract

We consider the atomic inversion in the Jaynes-Cummings model when the field mode is detuned from the atomic transition frequency. We find that, unlike in the resonant case, the time average of the inversion depends on the initial state of the atom. This time average is compared to that obtained from a semiclassical model in which the quantized field mode is replaced by a classical driving field. The averages are found to be identical. This similarity disappears when one examines the deviation of the inversion from its mean. The inversion is found to remain much closer to its mean, on average, with a quantum field rather than with a classical one. We also compare the trajectories described by inversions at different detunings. This is done by defining a distance between different trajectories. The distance between the trajectory at zero detuning and that at a detuning of \ensuremath{\Delta} is found to take sudden, discontinuous dips at certain values of \ensuremath{\Delta}.

DOI
10.1103/physreva.43.1506

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We consider the atomic inversion in the Jaynes-Cummings model when the field mode is detuned from the atomic transition frequency. We find that, unlike in the resonant case, the time average of the inversion depends on the initial state of the atom. This time average is compared to that obtained from a semiclassical model in which the quantized field mode is replaced by a classical driving field. The averages are found to be identical. This similarity disappears when one examines the deviation of the inversion from its mean. The inversion is found to remain much closer to its mean, on average, with a quantum field rather than with a classical one. We also compare the trajectories described by inversions at different detunings. This is done by defining a distance between different trajectories. The distance between the trajectory at zero detuning and that at a detuning of \ensuremath{\Delta} is found to take sudden, discontinuous dips at certain values of \ensuremath{\Delta}.
The Jaynes-Cummings model (JCM) of a two-level atom interacting with a quantized radiation field has been investigated theoretically. Of particular interest is the transition of a quantized electromagnetic field of the model to a classical one in the limit of very large average photon number. In this thesis, two aspects of the JCM are examined. Firstly, it is shown that, in contrast to the general case, for specic initial atom-field states, there is exact correspondence between the atomic inversion predicted by the JCM and that predicted by the semiclassical model in which the field is treated classically from the outset. Secondly, the degree of quantum entanglement of the JCM is investigated quantitatively and it is shown that, at any nite absolute temperature greater than zero, the atom and field are never separable and the degree of quantum entanglement approaches zero in the high and low temperature limits
We extend the standard Jaynes-Cummings model to include transient effects due to a modulation of the atom-field coupling. The exact solution for such a generalized model is found in terms of Bessel functions for the atom-cavity mode interaction modulated by the exponential function. This function is incorporated into the basic Hamiltonian phenomenologically for modelling the processes of adiabatic switching on or switching off the interaction of a two-level atom (at rest) with a quantized mode. The dynamical behaviour of the atomic inversion for such processes is plotted and compared with the atomic dynamics for the standard model. If we take into account that, in micromaser experiments, Rydberg atoms move through a cavity with a spatially inhomogenous field, we deal with the Jaynes-Cummings model with a spatial modulation the form of which is defined by the mode-shape function depending, in turn, on the geometrical form of resonator. In the cases of a rectangular and a spherical cavity the analytical solutions for atomic inversion are found in the explicit form in the weak-coupling limit. We consider in detail the time evolution of the atomic inversion while the atom passes through a rectangular cavity. Contrary to the standard model the initial inversion can be exactly restored during the atomic transit depending on the relation between the detuning, the atomic velocity and the number of half-wavelengths of the chosen mode. The numerical solutions of the basic equations are also obtained and plotted for rectangular and spherical resonators.
Collapse and revivals in the Jaynes-Cummings model are studied within the context of the Mollow transformation for a single-mode coherent-state cavity field. The Mollow-transformed Jaynes-Cummings Hamiltonian has two atom-field terms, one corresponding to a classical field and the other to a quantized field driving the atomic transition. As such it maps onto the complementary problem of an atom in an optical cavity, whose field is initially in the vacuum state, driven by an external classical field. Both problems have the same atomic-state dynamics. It is shown that the revivals can be associated with two distinct properties of the quantized field of the transformed Hamiltonian. Revivals occurring at even (odd) multiples of the revival time are correlated with field states that are close to the initial field state and for which the average energy in the field is a minimum (maximum). Using semiclassical dressed states, we are able to map the problem for the field states onto one involving two uncoupled oscillators driven by off-resonant fields, provided effects related to dispersion are neglected. Analytic expressions for the photon number distribution of the transformed cavity field, including dispersion, are obtained for the even revival times.
Exact solutions in the form of infinite series are well known for the Jaynes-Cummings model describing the interaction of a two-level atom with a single-mode field prepared in a coherent state. While these solutions have led to the discovery of many interesting properties of the Jaynes-Cummings model as a function of time, the nature of the solutions is such that it is not a simple matter to make statements of a general nature about the properties of the system. In this paper it is shown that by taking infinite time averages of the series expressions for the atomic inversion and inversion correlation, squeezing, and antibunching, simple closed-form expressions are obtained from which general properties of the Jaynes-Cummings model can be readily inferred. In particular, new conditions are obtained under which the field is significantly squeezed on average, and further information is obtained on the long-time behavior of the revivals in the atomic inversion.
The Jaynes-Cummings model is one of the most basic systems in quantum optics. It is exactly soluble but the quantities of interest are given by infinite oscillating series which cannot be summed. This leads to problems if one wants to learn about the long-time behavior of the system. On the other hand, the time averages of those quantities can be computed exactly. Infinite time averages give information about behavior which persists for all time. We examine the time averages of the squeezing and the variance of the photon number of the field. We also use time averages to gain information about the inversion. For example, one finds that the fraction of time which the inversion spends far from its mean value decreases as the amplitude of the initial field increases. The importance of periodic behavior in the inversion also decreases as the initial amplitude increases. Finally we examine how long the system must be observed for time averages of quantities to approach their infinite time limits.
If the quantized field has initially a small mean number of photons (n\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\sim}2), quantum inversion displays distinct revivals, provided the detuning between the field and the atom is large as compared to the coupling constant [(g/\ensuremath{\Delta})2{)}^{2})2\ensuremath{\ll}1]. However, the amplitudes of these revivals are very small. For long times, individual revivals partially overlap to form fractional revivals, and for even longer times superrevivals (revivals of the revivals) appear. These phenomena result from the beating of modes of the quantized field that are not nearest neighbors. The envelope of the electric field associated with the oscillating atomic dipole exhibits similar superstructures, but with a period twice that for the atomic inversion.