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Journal of Modern Optics

Squeezing in the Jaynes-Cummings model for Large Coherent Fields

Clark W. Woods, Julio R. Gea-Banacloche

Published Dec 1, 1993• 14 citations

Abstract

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.

DOI: 10.1080/09500349314552391

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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.

Using the well-known solution for a two-level atom interacting with a single mode of radiation (Jaynes-Cummings model), we reanalyze the possibility of squeezing. We found that if only the initial coherent state has a mean photon number exceeding 10 (for a resonant case) significant squeezing can be achieved. Its strength can be arbitrarily large for increasing average photon number. Finally, we solve our model in the situation when the em field is damped and analyze the impact of damping on both squeezing and revival phenomena.

A complete description of the electric field in the Jaynes-Cummings model is presented using the exact operator solutions for the field variables. Two-time field correlation functions are calculated which display the destruction and subsequent partial revival of field coherence. It is shown that squeezing of the field can be enhanced for small n\ifmmode\bar\else\textasciimacron\fi{} when the atom is initially in a coherent superposition state. Amplitude-squared squeezing of both field quadratures occurs for all values of n\ifmmode\bar\else\textasciimacron\fi{} with these initial conditions. The squeezing is not related to collapse and revival phenomena.

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

{\mathit{t}}_{0}

(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

{\mathit{t}}_{0}

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.

First, the effect of detuning on squeezing of coherent light is studied by Jaynes-Cummings model in a large N (initial mean photon number) limit. Analytical formulae for quadrature variances are obtained both in one-photon and two-photon transition cases. These formulae show that the effect is quite different in these two cases. In addition, explicit expressions for the k-th minimum of the suppressed variance are given when k is fairly large, which obviously shows it tends to zero according to 1/k2. Second, squeezing of coherent light field in a cavity by repeated interaction with atoms is studied. Numerical calculation shows the variance (delta) can be reduced by this process to a certain extent for suitably chosen interaction time t. But the squeezing becomes worse if the number of times K is too large. In the two-photon transition case, with a favorably chosen value of t, analytical formulae for (delta) (N, K) are obtained in large N approximation. These formulae show definitely that the least variance (delta) goes down when K increases up to quite a large value, and the achievable minimum value of (delta) 2 is in the order of 1/N1/2. Besides, the light field so obtained is nearly in a pure state with a minimum uncertainty product (delta) 1(delta) 2 equals 1/4.

An m-photon Jaynes-Cummings model is analyzed assuming the field to be initially in the squeezed vacuum state or in a more general squeezed state. Collapses and revivals of the Rabi oscillations in the atomic population inversion appear for a general squeezed input, but not for a squeezed vacuum input. The second-order coherence function is found to exhibit collapses and revivals for large initial average photon numbers. With a low-intensity input field and large m, one can achieve nearly steady-state squeezing.