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International Journal of Modern Physics B

QUASIPROBABILITY DISTRIBUTIONS FOR THE THREE-LEVEL JAYNES-CUMMINGS MODEL

Nguyen Dinh Huyen, Ho Trung Dung

Published Apr 30, 1994• 0 citations
Publisher

Abstract

The Q-function and Wigner function for a three-level system coupled to a one-mode coherent field under two-photon resonance are treated for various initial atomic states. It is shown that the dynamical behavior of the quasiprobability distributions reflects the collapses and revivals in the time histories of the Rabi oscillations in a very interesting way. The structured quasiprobabilities are reviewed in the light of the semiclassical eigenstates.

DOI
10.1142/s0217979294000609

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The Q-function and Wigner function for a three-level system coupled to a one-mode coherent field under two-photon resonance are treated for various initial atomic states. It is shown that the dynamical behavior of the quasiprobability distributions reflects the collapses and revivals in the time histories of the Rabi oscillations in a very interesting way. The structured quasiprobabilities are reviewed in the light of the semiclassical eigenstates.
In this work, we study the quantum Rabi model, a simple two-state system coupled to a harmonic oscillator. With increasing interaction strength between the subsystems, the model undergoes a quantum phase transition (QPT), a phenomenon explained by a change in the shape of the potential in the semiclassical limit. We numerically cal- culate the Wigner quasiprobability distribution of the oscillator subsystem for various eigenstates and coupling strengths, demonstrate its nontrivial structural change as the QPT is crossed, and compare it with the Husimi Q function. Finally, we employ the Wigner negativity to quantify the nonclassicality of the states and observe that the neg- ativity is enhanced near the QPT. This work can contribute to the understanding of the Wigner distribution and Wigner negativity in a system relevant to quantum technology and computing.
The Jaynes-Cummings model with cavity damping is investigated in the rotated-wave approximation. First we introduce six appropriate combinations of the matrix elements of the density operator, which are still operators with respect to the light field. With the help of the s-parametrized quasiprobability distributions of Cahill and Glauber [Phys. Rev. 177, 1882 (1969)] the equations of motion for the density operator transform to six coupled partial-differential equations. By expanding the quasiprobability distributions into two suitable sets, we obtain six tridiagonally coupled differential equations for the expansion coefficients, which are solved by a Runge-Kutta method. Starting with an initial coherent state of the cavity field and the atom in its upper state, we find that the initially one-peak quasiprobability function splits into two peaked functions counterrotating in the complex plane and, depending on the damping constant, spiraling into the origin. Revivals of the inversion oscillation are found for those times, when the two peaks collide. The time dependence of the inversion and the intensity as well as some special distributions of interest are also discussed.
We compare the semiclassical and quantum predictions for the unitary dynamics of a two-level atom interacting with a single-mode electromagnetic field under parametric modulation of the atomic parameters in the regime of multiphoton atom–field resonances. We derive approximate analytic solutions for the semiclassical Rabi model when the atomic transition frequency and the atom–field coupling strength undergo harmonic external modulations. These solutions are compared to the predictions of the quantum Rabi model, which we solve numerically for an initial coherent state with a large average photon number (on the order of 104), in the regime of three-photon resonance. We show that, for short enough times and sufficiently intense coherent states, the semiclassical dynamics agrees quite well with the quantum dynamics, although it inevitably fails at longer times due to the absence of collapse–revival behavior. Furthermore, we describe how the field state evolves throughout the interaction, presenting numerical results for the average photon number, entropies (related to atom–field entanglement), and other quantities characterizing the photon number statistics of the electromagnetic field.
The evolution of the atomic state in the resonant Jaynes-Cummings model (a two-level atom interacting with a single mode of the quantized radiation field) with the field initially in a coherent state is considered. It is shown that the atom is to a good approximation in a pure state in the middle of what has been traditionally called the ``collapse region.'' This pure state exhibits no Rabi oscillations and is reached independently of the initial state of the atom. For most initial states a total or partial ``collapse of the wave function'' takes place early during the interaction, at the conventional collapse time, following which the state vector is recreated, over a longer time scale.