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Annual Meeting Optical Society of America

Time averaged quantities in the Jaynes-Cummings model

James D. Cresser, Mark S. Hillery

Published Jan 1, 1988• 0 citations
Publisher

Abstract

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.

DOI
10.1364/oam.1988.ms4

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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.
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.
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}.
Squeezed states of the Jaynes-Cummings (JC) model are constructed and the statistical properties of the field oscillator in these states are studied. The presence of interaction between the field oscillator and the atom results in the appearance of slow temporal variations in the average values of the dynamic quantities. Two cases essentially differ: When only one of two modes of the JC model is excited, the variations in time, in particular, of the average value of the coordinate are governed by the mode anharmonicity. In the case when both modes of the JC model with close frequencies are excited, the temporal variations represent collapse and revival of oscillations with the frequency 2kn+12k\sqrt {n + 1} 2kn+1​ , where k is the constant of the atom-field oscillator interaction and n is the number of photons. The constructed squeezed vacuum states are not two-photon states, because the probability of finding an odd number of photons is not equal to zero and varies with time.
Amplitude-squared (AS) squeezing has been investigated for the m-photon Jaynes-Cummings model assuming the field to be initially in the squeezed states. The role played by intensity-dependent coupling has also been discussed. It has been shown that for the large initial average photon number [Formula: see text] with odd values of m, AS squeezing revokes permanently whereas with even values it recurs periodically. As m increases the revocation is hastened and the duration of occurrence decreases. Higher values of m for the initial field in a squeezed vacuum state can make one of the quadrature permanently squeezed. The AS squeezing behavior for two initial states of the atom, i.e., ground state versus excited state is also compared.
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.
Amplitude-squared squeezing is investigated for the two-photon Jaynes-Cummings model with the field initially in a squeezed vacuum or a more general squeezed state. For small values of the coherence parameter and large values of the squeeze parameter, the amplitude-squared variances show threefold oscillatory behavior on the long time scale. It is found that a ground-state atom (as against an excited one) leads to considerable enhancement of the input amplitude-squared squeezing. The behavior of photon number squared and its relationship to the amplitude-squared squeezing is commented upon.