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Physical Review A

Superstructures, fractional revivals, and optical Schrödinger-cat states in the Jaynes-Cummings model

Paweł Góra, Czesław Jędrzejek

Published Oct 1, 1993• 37 citations
Publisher

Abstract

It is shown that if the initial photon distribution is narrow (sub-Poissonian), long-time behavior of the atomic inversion is dominated by very spectacular and almost-periodical superstructures. These superstructures generalize phenomena described previously (fractional revivals). They result from beating of not-nearest-neighbor eigenstates of the system, and we show which beats lead to particular types of revivals. In addition, the cavity field develops macroscopically distinguishable states (optical Schr\"odinger-cat states). These stages emerge when the fractional revivals with a frequency of 2/${\mathit{t}}_{\mathit{R}}$, 3${\mathit{t}}_{\mathit{R}}$, and 4/${\mathit{t}}_{\mathit{R}}$, where ${\mathit{t}}_{\mathit{R}}$ is the single-revival period, appear for the second, fourth, sixth, etc., time in the superstructure pattern.

DOI
10.1103/physreva.48.3291

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Physical Review A•Mar 1, 1994•71 citations
It is shown that if the initial photon distribution is narrow (sub-Poissonian), long-time behavior of the atomic inversion is dominated by very spectacular and almost-periodical superstructures. These superstructures generalize phenomena described previously (fractional revivals). They result from beating of not-nearest-neighbor eigenstates of the system, and we show which beats lead to particular types of revivals. In addition, the cavity field develops macroscopically distinguishable states (optical Schr"odinger-cat states). These stages emerge when the fractional revivals with a frequency of 2/tR{\mathit{t}}_{\mathit{R}}tR​, 3tR{\mathit{t}}_{\mathit{R}}tR​, and 4/tR{\mathit{t}}_{\mathit{R}}tR​, where tR{\mathit{t}}_{\mathit{R}}tR​ is the single-revival period, appear for the second, fourth, sixth, etc., time in the superstructure pattern.
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.
It is shown that the phenomenon of fractional revivals, which was observed in recent laser experiments on atomic wave packets, determines the long-time behavior of the Jaynes-Cummings model for a two-level atom interacting with a quantized cavity field. This provides an alternative mechanism for the generation of coherent superpositions of macroscopically distinguishable states of the field (so-called ``optical Schr\"odinger cats'') via resonant processes. The emergence of these states is associated with the appearance of atomic inversion revivals following each other 2, 3, 4, . . . times faster than the usual ones.
We analyze the quantum dynamics of the fractional-time Jaynes-Cummings model using a recent unitary framework for the fractional-time Schrödinger equation. We examine how the fractional derivative order ααα influences non-classical features under different initial conditions. For an initial Fock state, fractional evolution introduces transient dynamics and heightened sensitivity to coupling strength. Through an inverse problem approach, we interpret these effects as arising from an effective time-dependent coupling with a strong initial pulse. For an initial coherent state, the fractional order tunes the system between dynamical regimes, with a transition at $α= 0.50 $ where standard collapse-and-revival is replaced by stable, periodic evolution. This regime enhances non-classical field properties, including stronger sub-Poissonian statistics, periodic quadrature squeezing, and the formation of Schrödinger cat states.
The Jaynes-Cummings model of optical resonance describes the simplest fully quantized interaction between two quantum systems of different nature: a two-level atom (fermionic system) and a quantized field mode (bosonic system). This interaction leads to extreme quantum entanglement of the atom and field. However, the model also predicts that, at precisely half of the revival time, the atom and field become asymptotically disentangled. This disentanglement becomes more exact as the coherent-state amplitude increases. In this paper we investigate the nature of the pure-field-state superposition generated at such times. We show that this superposition is of distinguishable states of the field with the same amplitude but opposite phase. Interference between these components leads to nonclassical oscillations in photon-number distributions and squeezing in quadratures of the field. The Schr\"odinger-cat states of the field are highly transient, and depend very sensitively on the interaction time, the initial intensity of the field, and the atom-field detuning.
A variant Jaynes-Cummings model (JCM) where the two-state atom coupled to the cavity mode is driven by a strong external field reveals many interesting behaviors. On resonance, a large scale oscillatory behavior of the mean cavity photon number is predicted. It varies sinusoidally with the amplitude of 2(/spl epsi//g)/sup 2/, centered at the same value, where /spl epsi/ is the (real) driving field amplitude and g the vacuum Rabi frequency. The period of the oscillation was found to be exactly twice the "revival time" of the atomic inversion, i.e., T/sub 0/=(4/spl pi//g)(/spl epsi//g). Furthermore, the oscillation then shows another kind of "collapse-and-revival" type behavior as time proceeds. It is very similar to that of the atomic inversion, but the size of oscillation and the time scales are fundamentally different. The "revival" time of the oscillation, for instance, is found to be exactly one half of the so-called "fractional revival" time of the atomic inversion. Our current interest is directed to the radiative behavior of the driven JCM in a more general situation where dissipation as well as detuning is introduced.
We study the dynamics of the driven Jaynes-Cummings model where an atom interacts with a single-mode cavity field but is also directly driven by an external radiation field. We show that the dynamic variables can be described in terms of ordinary Jaynes-Cummings dynamic variables with a different initial condition. We investigate the special case of an initial coherent state. We find collapses and revivals in the average number of photons at a much larger time scale than their Jaynes-Cummings model counterparts. We show that these ``super-revivals'' are related to the large-time-scale revivals that occur in the average electric field in the ordinary Jaynes-Cummings model. We also examine the photon statistics in the driven Jaynes-Cummings model and show that the cavity field is sub-Poissonian for certain interaction times.