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arXiv:1406.4016 · Journal of Mathematical Physics

SU(4) based classification of four-level systems and their semiclassical solution

Surajit Sen, Helal Ahmed

Published Dec 1, 2014• 2 citations
arXivHTMLPDF

Abstract

We present a systematic method to classify the four-level system using SU(4) symmetry as the basis group. It is shown that this symmetry allows three dipole transitions which eventually leads to six possible configurations of the four-level system. Using a dressed atom approach, the semi-classical version of each configuration is exactly solved under rotating wave approximation and the symmetry among the Rabi oscillation among various models is studied.

DOI
10.1063/1.4903000

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We present a systematic method to classify the four-level system using SU(4) symmetry as the basis group. It is shown that this symmetry allows three dipole transitions which eventually leads to six possible configurations of the four-level system. Using a dressed atom approach, the semi-classical version of each configuration is exactly solved under rotating wave approximation and the symmetry among the Rabi oscillation among various models is studied.
A system of three-level atoms interacting with a one-mode electromagnetic field in the dipolar and rotating wave approximations is studied. The order of the quantum phase transitions is determined explicitly for each of the configurations Ξ, Λ and V , with and without detuning. A semi-classical approximation to the exact quantum solution is built by projecting coherent field and matter states to a specific value of the total number of excitations. The states in parameter-space and the expectation values of relevant observables are obtained explicitly, and it is shown that those for the projected states offer an excellent approximation to the exact quantum solution. The exact form of the ground state near the triple point in the Ξ configuration is given.
We considered the semiclassical Rabi model, that is, a two-level system interacting with a single-mode, classical field. In resonance and for an arbitrary initial state of the system, we obtained a simple, approximate, and analytic solution that takes into account the counterrotating terms and that is accurate as long as the Rabi frequency is smaller than or equal to 1/2 (1/4) of the angular frequency of the field and 1 (10) Rabi oscillations are considered. In addition, the approximate solution has the same level of complexity as that obtained with the rotating-wave-approximation (RWA) and allows one to describe the evolution of the Bloch vector of the system in terms of a \textit{slow-precessional} and a \textit{fast-nutational} motion similar to that of a symmetric top with one point fixed. Finally, the approximate solution also leads to a simple criterion that allows one to determine the accuracy of the solution in the RWA.
We consider N a three-level atoms (or systems) interacting with a one-mode electromagnetic field in the dipolar and rotating wave approximations. The order of the quantum phase transitions is determined explicitly for each of the configurations Ξ, Λ and V , with and without detuning. The semi-classical and exact quantum calculations for both the expectation values of the total number of excitations and photon number have excellent correspondence as functions of the control parameters. We prove that the ground state of the collective regime obeys sub-Poissonian statistics for the and n distribution functions. Therefore, their corresponding fluctuations are not well described by the semi-classical approximation. We show that this can be corrected by projecting the variational state to a definite value of .
We considered the semiclassical Rabi model: a two-level atom fixed at a position and interacting with a classical, monochromatic electromagnetic field. In resonance and for an arbitrary initial state of the system, we obtained an approximate, analytic solution that takes into account the counterrotating terms and that is accurate as long as the Rabi frequency is smaller than or equal to 1/21/21/2 (1/41/41/4) of the angular frequency of the field and 111 (101010) Rabi oscillations are considered. In addition, the approximate solution has the same level of complexity as that obtained with the rotating-wave-approximation (RWA) and the approximate solution also leads to a simple criterion that allows one to determine the accuracy of the solution in the RWA. We found that the approximate solution is composed of the the solution in the RWA plus terms that oscillate at the angular frequency of the field and at the Rabi frequency. Moreover, these corrections are proportional to the Rabi frequency divided by two times the angular frequency of the field.
One of the simplest nontrivial problems involving the atom–field interaction is the coupling of a two-level atom with a single mode of the electromagnetic field. A two-level atom description is valid if the two atomic levels involved are resonant or nearly resonant with the driving field, while all other levels are highly detuned. Under certain realistic approximations, it is possible to reduce this problem to a form which can be solved exactly; allowing essential features of the atom-field interaction to be extracted. In this chapter we present a semiclassical theory of the interaction of a single two-level atom with a single mode of the field in which the atom is treated as a quantum two-level system and the field is treated classically. A fully quantum mechanical theory will be presented in Chapter 6. A two-level atom is formally analogous to a spin-1/2 system with two possible states. In the dipole approximation, when the field wavelength is larger than the atomic size, the atom–field interaction problem is mathematically equivalent to a spin-1/2 particle interacting with a time-dependent magnetic field. Just as the spin-1/2 system undergoes the so-called Rabi oscillations between the spin-up and spin-down states under the action of an oscillating magnetic field, the two-level atom also undergoes optical Rabi oscillations under the action of the driving electromagnetic field. These oscillations are damped if the atomic levels decay. An understanding of this simple model of the atom–field interaction enables us to consider more complicated problems involving an ensemble of atoms interacting with the field.
Abstract We study two models describing the interaction of a two-level system with two quantum field modes in a parallel or orthogonal setup. We show that both models present a partial two-mode SU (2) symmetry and that they can be solved in the exceptional case of fields with the same frequency. We study their ground state configurations; that is, we find the quantum precursors of the corresponding semi-classical phase transitions, as well as their whole spectra to infer their integrability. We show that the first model is isomorphic with the quantum Rabi model and shows the standard crossover from a vacuum to a non-vacuum ground state configuration. The second model shows a crossover involving four ground state configurations: one vacuum, two non-vacuum single modes and one non-vacuum dual mode. We give analytic and numerical pointers that may suggest its integrability in certain regimes. We also show that, in the single excitation subspace, an excited two-level system may entangle two initial vacuum fields even in the ultra-strong coupling regime.