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Physical Review A

Loss of state purity and regularity in the Jaynes-Cummings model

Julio R. Gea-Banacloche

Published Dec 1, 1992• 15 citations
Publisher

Abstract

It is established here that a recently derived asymptotic solution for the Jaynes-Cummings model, for large quasiclassical fields, holds over a length of time which is proportional to n\ifmmode\bar\else\textasciimacron\fi{}, the average number of photons in the field. When the initial state of the atoms is one of the semiclassical eigenstates, the purity of the atomic state is found to decay with time in a form approximately Gaussian, with a characteristic time proportional to n\ifmmode\bar\else\textasciimacron\fi{}. When the initial state is not one of the semiclassical eigenstates, the collapses and revivals of the population inversion, for which the model is best known, become irregular also over the same time scale; an explanation for this fact using the asymptotic solution is presented here.

DOI
10.1103/physreva.46.7307

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It is established here that a recently derived asymptotic solution for the Jaynes-Cummings model, for large quasiclassical fields, holds over a length of time which is proportional to n\ifmmode\bar\else\textasciimacron\fi{}, the average number of photons in the field. When the initial state of the atoms is one of the semiclassical eigenstates, the purity of the atomic state is found to decay with time in a form approximately Gaussian, with a characteristic time proportional to n\ifmmode\bar\else\textasciimacron\fi{}. When the initial state is not one of the semiclassical eigenstates, the collapses and revivals of the population inversion, for which the model is best known, become irregular also over the same time scale; an explanation for this fact using the asymptotic solution is presented here.
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.
An approximate solution is given for the Jaynes-Cummings model with cavity losses, i.e., the problem of a two-level atom interacting with a single mode of the quantized radiation field, in the rotating-wave approximation, when the field is damped by a reservoir at zero temperature. The approximate solution is derived for initial coherent field states with moderately large numbers of photons. It is simpler in form than earlier results derived by other authors and, over the appropriate parameter range, substantially more accurate than some of them, as shown by direct numerical integration of the master equation. In particular, it is found that an earlier treatment of this problem based on a secular approximation is seriously flawed, in that the conditions for its validity are much more restrictive than was previously believed. Among the results derived it is shown that, just as for the lossless case, when the atom is initially prepared in one of the semiclassical eigenstates the evolution is very simple, with the field and the atomic dipole drifting together in phase. For moderate losses this leads, as in the lossless case, to a ``state preparation''; i.e., to a good approximation, the state of the atom at a specific time can be made independent of its initial state. The effect of losses on the recently discovered ``Schr\"odinger cat'' state of the field is also analyzed. It is found that, although the dissipation destroys the coherence of the macroscopic superposition very rapidly, preparation and observation of the ``cat'' should be possible with the cavity quality factors reported in recent micromaser experiments.
For large initial fields, the evolution of the Jaynes-Cummings model is shown to simplify considerably. For two special initial atomic states (the eigenstates of the semiclassical problem) one finds a disentangled, quasipure-state evolution for both the field and atom. For an arbitrary initial atomic state the evolution is a linear super-imposition of these two privileged alternative paths. New results include the fact that the state of the atom at half-revival time is pure and independent of the initial state (nonunitary evolution), and that the state of the field at that time is a coherent super-imposition of macroscopically distinct states (Schrodinger cat).
We use the recently-established asymptotic solution (AS) for large fields to study squeezing in the Jaynes-Cummings model. Arbitrary initial atomic states (including atomic coherence) are considered. We find that most features of squeezing can be understood from the properties of the asymptotic field states, and from the observation, which we establish numerically, that for large fields the interference between the two branches of the wavefunction does not contribute significantly to the squeezing (not even near the revival times, when the two field states being superposed have very nearly the same phase). We obtain a limit on the time over which the field may show any squeezing and find that it scales as the 3/4 power of the number of photons. We also obtain a bound on the maximum achievable squeezing for a given number of photons, which is an improvement over earlier estimates. Finally, we discuss some of the shortcomings of the AS, and in particular the existence of a small ghost peak in the field Q function for initial conditions where the AS predicts a single peak, which is enough to destroy the squeezing for certain times.
Many field states in quantum optics may be regarded as quasiclassical in that they possess a fairly well-defined phase and amplitude in the limit in which the average number of photons is large. Examples include coherent states and squeezed states that have a coherent component substantially larger than the squeezed noise contribution. For such states, calculations involving sums over the coefficients in the Fock-state basis can be considerably simplified in the limit of large photon number. The Jaynes-Cummings model and its extensions (to, e.g., three-level, two-mode configurations) may be advantageously studied from this perspective. New results on the three-level, two-mode problems, on the influence of cavity losses, and on the various time scales of interest will be presented here. The method typically identifies a set of privileged, long-lived atomic states, which lead to a quasiclassical evolution that agrees with the prediction of semiclassical treatments over a characteristic time scale; linear combinations of the long-lived states, on the other hand, become superpositions of macroscopically distinct states in time and result in an evolution that exhibits many uniquely quantum effects.