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Physical Review A

Exchange effects on cooperative radiation interaction in a two-atom model

Thomas Östreich

Published Sep 1, 1994• 1 citation
Publisher

Abstract

We study cooperative effects on the radiation interaction of a pair of two-level atoms with a spatial separation r\ensuremath{\ll}${\ensuremath{\lambda}}_{0}$ much closer than the optical wavelength of the dipole transition. In this regime we allow for charge exchange between the atoms (hybridization), which leads to qualitatively different behavior in contrast to an isolated pair of atoms because of the presence of the monopole Coulomb interaction. The interaction of the atoms with the radiation field and an external classical driving field is treated within a master equation approach. The numerical as well as the analytical results indicate the existence of quantum beats in the radiative decay of the system. In the case of an external driving field the system shows a dark resonance for a fixed detuning independent of the strength of the external field.

DOI
10.1103/physreva.50.2621

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We study cooperative effects on the radiation interaction of a pair of two-level atoms with a spatial separation r\ensuremath{\ll}\ensuremathλ0{\ensuremath{\lambda}}_{0}\ensuremathλ0​ much closer than the optical wavelength of the dipole transition. In this regime we allow for charge exchange between the atoms (hybridization), which leads to qualitatively different behavior in contrast to an isolated pair of atoms because of the presence of the monopole Coulomb interaction. The interaction of the atoms with the radiation field and an external classical driving field is treated within a master equation approach. The numerical as well as the analytical results indicate the existence of quantum beats in the radiative decay of the system. In the case of an external driving field the system shows a dark resonance for a fixed detuning independent of the strength of the external field.
Microscopic models based on multilevel atoms are central to optimizing nonlinear optical responses and the coherent control of light. These models are traditionally based on single-atom effects that are parametrically extrapolated to include collective effects, such as an enhanced response or propagation within atomic media. In this work, we present a systematic analysis of the cooperative effects arising in driven systems composed of multilevel atoms coupled via a common electromagnetic environment. The analysis is based on an interplay between dressed states induced by the driving field and photon exchanges, and collective decay channels. This theory is applied to the case of four-wave mixing induced by a pair of lasers acting on an atomic pair with internal levels in the diamond configuration. The effect of interatomic correlations and collective decay over the photons created in this nonlinear process is then explored. We identify three regions of operation: (i) laser-dominated, (ii) intermediate, and (iii) dipole-dominated. The dependence of single- and two-photon correlations is studied in detail for each region by varying atomic orientations and laser parameters consistent with current experiments involving atomic gases. Photonic correlation functions are shown to exhibit a transition from a Lorentz-like dependence on the two-photon detuning—with general features that can be obtained in an isolated atom scheme—to a two-peaked distribution when the dipole-dipole interactions become relevant. For weak Rabi frequencies whose value is smaller than the highest collective decay rate, the atoms are trapped inside their ground state as they approach each other. It is found that the anisotropy of the dipole-dipole interaction and its wave nature are essential to understand the behavior of the photon correlations. Signatures of these processes are identified for existing experimental realizations. The intuition obtained from this connection helps to uncover relevant parameters that could be exploited for quantum control protocols based on dispersive and dissipative cooperative effects in multilevel systems.
We follow in detail the spontaneous decay of fully excited samples of two and three resonant two-level atoms separated by distances much smaller than a wavelength. We include both the static dipole-dipole interaction and an in rinsic detuning (inhomogeneous broadening) of the atoms. We find that the Coulomb interaction may have both a decohering and a recohering effect. The decohering effect is seen in three atoms not equivalently placed. Here the Coulomb interaction diminishes super-radiance by subjecting the different atoms to different frequency shifts. The recohering effect is seen in all cases and consists in a protection of super-radiance against an inhomogeneous broadening far in excess of the inverse radiation time. Both effects have analogs in small many-atom samples studied in previous literature.
The interaction between electromagnetic radiation and two three-level atoms is investigated. The atom-atom separation is assumed to be smaller than the corresponding mean resonance wavelength. The total radiance rate of the atomic system is calculated as a function of time. Explicit results are given for several different initial states of the atomic system. Some of them exhibit superradiance, and some act as photon-trapping states. Instead of exponential decay, the oscillations appear in the time evolution of the radiation rate due to the correlation between atoms. The decay constants and coupling decay constants show strong effects on the emission rates. Completely differing from two identical double-level atoms, the cooperative effects always exist in the three-level atomic system except for the case in which the coupling decay constants ${\ensuremath{\gamma}}_{12}$ and ${\ensuremath{\gamma}}_{21}$ are zero.
Spontaneous emission from two non-identical two-level atoms is examined from the viewpoint of quantum beats and super-radiant effects. Our analytical solution, illustrated graphically, shows that the total radiation rate exhibits temporal sinusoidal modulation (beats). The beats result from the fact that the atoms have different transition frequencies and different natural linewidths. For small interatomic separations, the total radiation rate is greater than its initial value. This ‘super-radiant’ property is absent in the case of noninteracting atoms, as well as for two identical atoms. In the case when the atoms are separated by distances much smaller than the resonant wavelength, the population trapping effect can take place in such a system.
The collective decay effects between the dipole-active three-level subsystems in the nonlinear interaction with dipole-forbidden transitions, like 2 S − 1 S of hydrogen-like radiators, are proposed, taking into consideration the cooperative exchanges between two species of atoms through the vacuum field in the scattering and the two-photon resonance processes. One of them corresponds to the situation when the total energy of the emitted two photons by the three-level radiator in the cascade configuration enters into the two-photon resonance with another type of dipole-forbidden transitions of hydrogen-like (or helium-like) atoms. The similar situation appears in the cooperative scattering between two species of quantum emitters when the difference of the excited energies of the two dipole-active transitions of the three-level radiators is in the resonance with the dipole-forbidden transitions of the Hydrogen-like radiators. These effects are accompanied by the interference between single- and two-quantum collective transitions of the inverted radiators from the ensemble. The two-particle collective decay rate is defined in the description of the atomic correlation functions taking into consideration the phase retardation between them. The kinetic equations which describe the cooperative processes as the function of time and correlation are obtained. The behavior of the system of radiators at short and long time intervals in comparison with the retardation time between them is studied.
When atoms are coupled to a common electromagnetic environment, the exchange of photons through dipole-dipole interactions leads to the emergence of cooperative effects. As a particular example, superradiance arises from spontaneous emission when this exchange leads to constructive interference of the emitted photons. Here, we introduce an integrated method for studying cooperative radiation in many-body systems. This method, which allows to study extended systems with arbitrarily large number of particles, can be formulated by an effective, nonlinear, two-atom master equation that describes the dynamics using a closed form which treats single- and many-body terms on an equal footing. We apply this method to a homogeneous gas of initially inverted two-level atoms, and demonstrate the appearance of both superradiance and subradiance, identifying a many-body coherence term as the source of these cooperative effects. We describe the many-body induced broadening -- which is analytically found to scale with the optical depth of the system -- and light shifts, and distinguish spontaneous effects from induced ones. In addition, we theoretically predict the time-dependence of subradiance, and the phase change of the radiated field during the cooperative decay.