We calculate the energy eigenvalues and eigenstates corresponding to coherent single and multiple excitations of a number of different arrays of N identical two-level atoms (TLA's) or qubits, including polygons, ``diamond'' structures, polygon multilayers, icosahedra, and dodecahedra. We assume only that the coupling occurs via an exchange interaction which depends on the separation between the atoms. We include the interactions between all pairs of atoms, and our results are valid for arbitrary separations relative to the radiation wavelength.
Using the formalism developed earlier, we treat spontaneous emission from a pair of identical two-level atoms
A1,
A2, whose separation
r21 can be comparable to the wavelength $\ensuremath{\lambda}$. We obtain expressions for time-dependent intensities and damping rates with the initial conditions (a) both atoms inverted, (b) prior excitation by a short $\frac{1}{2}\ensuremath{\pi}$ pulse, and (c) only
A1 inverted. The results in (a) are compared with those obtained for a model consisting of two initially excited harmonic oscillators
O1,
O2. The atoms exhibit superradiant behavior, whereas
O1,
O2 tend to trap radiation. In (a), the intensity pattern $\mathcal{R}(\ensuremath{\theta},t)$ develops lobes in different directions at different times, so that the spatial distribution of photons at time $t\ensuremath{\rightarrow}\ensuremath{\infty}$ is the same as in the independent-atom case ${r}_{21}\ensuremath{\gg}\ensuremath{\lambda}$. For the oscillators, the lobes of $\mathcal{R}(\ensuremath{\theta},t)$ do not change direction, but only become more pronounced as time increases. In (b) and (c), the lobes oscillate back and forth at frequency $2{\ensuremath{\Omega}}_{12}$ corresponding to the shifts $\ifmmode\pm\else\textpm\fi{}{\ensuremath{\Omega}}_{12}$ of the triplet and singlet states due to the ${A}_{1}\ensuremath{-}{A}_{2}$ interaction. The intensity can therefore have a sinusoidal component. Field correlation functions calculated for (a) and (c) show that
A2 and
A2 radiate simultaneously around the frequencies $\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}{\ensuremath{\Omega}}_{12}$, where $\ensuremath{\epsilon}$ is the single-atom resonant frequency. The spectrum is calculated for case (c), and shows the effects of coherent linewidth enhancement in addition to the frequency shifts.
A method utilising the coherent state variables for the electromagnetic field and appropriate spin variables for two-level atomsis developed for obtaining the dynamical evolution of the field and the atoms separately. The case of a single radiation mode interacting with two two-level atoms is studied. The time evolution of the photon number, starting from n photons and both atoms at their higher levels, is given. Furthermore, when the initial state consists of n photons and one of the atoms in its excited state with the other in its ground state, the temporal development of the photon number, as well as the motion of each atom, are obtained. In the latter case it becomes apparent that energy exchange takes place among the atomic systems, while the field acts as a transfer agent.
In an earlier paper, we considered a Hamiltonian trilinear in boson operators, which described several physical systems of interest such as interaction of
N two-level atoms with a single-mode resonant radiation field, parametric amplification, frequency conversion, Raman and Brillouin scattering, etc. We found that the dynamics of the system can almost exactly be determined if either one of the three modes is strong for all times or any two modes are separately strong in two complementary time domains. In the present paper, we determine the statistical behavior of the system in each case. We give explicit results for a number of initial states (e.g., coherent, chaotic, a mixture of coherent and chaotic, number state, etc.) of the modes 1, 2, and 3.
We study an aggregate of N identical two-level atoms (TLA's) coupled by the retarded interatomic interaction, using the Lehmberg-Agarwal master equation. First, we calculate the entangled eigenstates of the system; then, we use these eigenstates as a basis set for the projection of the master equation. We demonstrate that in this basis the equations of motion for the level populations, as well as the expressions for the emission and absorption spectra, assume a simple mathematical structure and allow for a transparent physical interpretation. To illustrate the use of the general theory in emission processes, we study an isosceles triangle of atoms, and present in the long wavelength limit the (cascade) emission spectrum for a hexagon of atoms fully excited at
t=0. To illustrate its use for absorption processes, we tabulate (in the same limit) the biexciton absorption frequencies, linewidths, and relative intensities for polygons consisting of
N=2,…,9 TLA's.
The interaction is considered between
N stationary two-level atoms and a radiation field described by a single cavity mode. The state vector for the complete system of atoms plus radiation is expressed as a linear superposition of states constructed from a product of photon states in the
n representation and products of Pauli spin eigenstates describing all combinations of atoms in the lower and upper energy levels. Equations for the corresponding probability amplitudes are derived by substituting this superposition into the Schr\"odinger equation. The resulting equations are combined into bilinear form and phenomenological damping contributions are added. After the neglect of certain of the bilinear quantities, a master equation is derived which governs the probability
Pmn of having
m atoms in the lower level and
n photons in the mode. This master equation takes account of multiple single-quantum absorption and emission processes but not of simultaneous multiple processes involving two or more atoms at a time. The equation which governs the expected number of photons
〈n〉 derived from the master equation bears a close resemblance to a rate equation. The effect of radiation loss from the cavity is incorporated into the master equation. Numerical calculations for a
Q-spoiled laser show that the statistics of the number of photons in the mode bear a qualitative resemblance to Poisson statistics.