Effects of interatomic interaction on cooperative relaxation of two-level atoms

We study effects of direct interatomic interaction on cooperative processes in atom-photon dynamics. Using a model of two-level atoms with Ising-type interaction as an example, it is demonstrated that interparticle interaction combined with atom-field coupling can introduce additional interatomic correlations acting as a phase synchronizing factor. For the case of weakly interacting atoms with $J<\hslash {\omega }_0$, where $J$ is the interparticle coupling constant and ${\omega }_0$ is the atomic frequency, dynamical regimes of cooperative relaxation of atoms are analyzed in the Born-Markov approximation both numerically and using the mean field approximation. We show that interparticle correlations induced by the direct interaction result in inhibition of incoherent spontaneous decay leading to the regime of collective pulse relaxation which differs from superradiance in nature. For superradiant transition, the synchronizing effect of interatomic interaction is found to manifest itself in enhancement of superradiance. When the interaction is strong and $J>\hslash {\omega }_0$, one-particle one-photon transitions are excluded and transition to the regime of multiphoton relaxation occurs. Using a simple model of two atoms in a high-$Q$ single mode cavity we show that such transition is accompanied by Rabi oscillations involving many-atom multiphoton states. Dephasing effects of dipole-dipole interaction and solitonic mechanism of relaxation are discussed.

Figure 1

Energy spectrum of $H_A$ for the chain of six atoms. Two cases are shown: (a) weak interatomic interaction with $J∕\hslash {\omega }_0=0.1$ and (b) strong interatomic interaction with $J∕\hslash {\omega }_0=10$. The state $\mid \uparrow ⟩=\mid \uparrow ,\uparrow ,\dots ,\uparrow ⟩$ $(\mid \downarrow ⟩=\mid \downarrow ,\downarrow ,\dots ,\downarrow ⟩)$ indicates that the atoms are all in the excited (ground) state.

Figure 2

Relaxation rate $\gamma \left(t\right)$ (19) for two interacting two-level atoms at various values of the ratio $\beta \equiv J∕\hslash {\omega }_0$: (a) $\beta =0$, (b) $\beta =0.1$, and (c) $\beta =0.2$.

Figure 3

Order parameter $\tilde{C}$ defined in Eq. (35) as a function of the number of atoms $N$. The transition to the superradiant regime is shown for three different values of the ratio $\beta =J∕\hslash {\omega }_0$: (a) $\beta =0$, (b) $\beta =0.5$, and (c) $\beta =0.9$.

Figure 4

Relaxation rate $\gamma \left(t\right)$ defined in Eq. (21) as a function of time for ten interacting atoms at different values of the interaction parameter $\beta =J∕\hslash {\omega }_0$. Solid lines represent the numerical results computed by solving the system (18). The dashed line is computed from the solution of Eq. (39).

Figure 5

Relaxation rate $\gamma \left(t\right)$ defined in Eq. (21) as a function of time for 100 interacting atoms at different values of the interaction parameter $\beta =J∕\hslash {\omega }_0$ ($2{\gamma }_0$ is the spontaneous decay rate for an isolated atom). The curves are computed by solving the system (18) numerically.

Figure 6

Energy spectrum of two interacting atoms with $H_A=\hslash {\omega }_0(S_1^z+S_2^z)-J{S}_1^z{S}_2^z$.

Figure 7

(a) Relaxation rate $\gamma \left(t\right)$ as a function of time at different values of the interaction parameter $\beta$. (b) Averaged population of $i$th atom $⟨S_j^z⟩$ as a function of time and atomic number $i$ at $\beta =0.99$. The curves are computed by numerically solving Eq. (39) for a ring of 20 atoms. One atom is initially in the ground state representing a point defect in the atomic ring. At sufficiently strong interaction, relaxation of the atomic ring clearly demonstrates the solitonic mechanism when atoms undergo transition to the ground state consecutively one after another.

Figure 8

(a) Interacting particles embedded into potential lattice in the case of half-filling. Ground state is double degenerate corresponding to the particles localized in either odd or even wells($\mid \uparrow ⟩=\mid \uparrow \uparrow \cdots \uparrow ⟩$ and $\mid \downarrow ⟩=\mid \downarrow \downarrow \cdots \downarrow ⟩$). Tunneling transitions lift degeneracy so that the two lowest levels correspond to the many-particle entangled states $\mid {\psi }_{0,1}⟩\approx \frac{1}{\sqrt{2}}(\mid \uparrow ⟩\pm \mid \downarrow ⟩)$. Excited states are separated by the gap of the width $\propto J$. (b) For energetically nonequivalent wells, tunneling transitions are inelastic. For $J⪢\hslash {\omega }_0$, they can be accompanied by multiphoton processes.