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Physical Review A

Revivals and superstructures in the Jaynes-Cummings model with a small number of photons

Paweł Góra, Czesław Jędrzejek

Published Apr 1, 1994• 30 citations
Publisher

Abstract

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}${)}^{2}$\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.

DOI
10.1103/physreva.49.3046

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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 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.
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.
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.
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.
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.