Nonexponential decay of a collective excitation in an atomic ensemble coupled to a one-dimensional waveguide

We study the dynamics of a single excitation coherently shared among an ensemble of atoms and coupled to a one-dimensional wave guide. The coupling between the matter and the light field gives rise to collective phenomena such as superradiant states with an enhanced initial decay rate, but also to the coherent exchange of the excitation between the atoms. We find that the competition between the two phenomena provides a characteristic dynamics for the decay of the excitations, and remarkably exhibits an algebraic behavior, instead of the expected standard exponential one, for a large number of atoms. The analysis is first performed for a chiral waveguide, where the problem can be solved analytically. Remarkably, we demonstrate that a bidirectional waveguide exhibits the same behavior for large number of atoms and, therefore, it is possible to experimentally access characteristic properties of a chiral waveguide also within a bidirectional waveguide.

Figure 1

(a) Two-level atoms coupled to a one-dimensional waveguide. The waveguide supports (in general) left- and right-propagating modes and the atoms can emit (absorb) photons into (from) both modes. (b) Effective system after the elimination of the waveguide photons. The atoms interact via an (infinite-ranged) exchange interaction $J_{jl}$ and have a correlated decay ${\mathrm{\Gamma }}_{jl}$.

Figure 2

Setup for two atoms coupled by a one-dimensional waveguide. (a) In a chiral setup (left) the atom can only emit into the forward-propagating mode with rate $\gamma$, while in the bidirectional setup (right) the atoms can emit into forward- and backward-propagating modes with rate $\gamma$ for each mode. (b) For a chiral waveguide, the subradiant and superradiant states correspond to the bright and dark states, respectively. The bright and dark states are coupled and the bright state decays with a collectively enhanced decay rate ${\mathrm{\Gamma }}_{+}=2\mathrm{\Gamma }$ and the single-atom emission rate $\mathrm{\Gamma }=\gamma$. Initially, the system is prepared in the bright state. (c) In the bidirectional case, when the atoms are on average very close to each other compared to the wavelength, there is no coupling and the bright (dark) state corresponds to the superradiant (subradiant) state. A system that is initially prepared in the bright state decays with the collectively enhanced decay rate $2\mathrm{\Gamma }$, where the single-atom decay rate is $\mathrm{\Gamma }=2\gamma$. (d) In the case where the interatomic distance is on average much greater than the wavelength, the superradiant and subradiant states are shifted with respect to each other (depending on the distance between the atoms) and emit with rates ${\mathrm{\Gamma }}_{+}$ and ${\mathrm{\Gamma }}_{-}$, respectively. Since the bright state is now a superposition of superradiant and subradiant states, these two states are coupled by the initial condition.

Figure 3

Time evolution of the population of the bright state (blue solid line), the dark state (orange dashed line), and the total population of excited states (green dashed-dotted line) for $N=2$ atoms coupled to a bidirectional waveguide in the limit $k\sigma \gg 1$. The gray dotted line shows an exponential decay with a collectively enhanced decay rate $N\mathrm{\Gamma }=2N\gamma$ expected in single-photon superradiance which appears for $k\sigma \ll 1$. (Inset) The inset shows the time evolution on a logarithmic scale. For small times $N\mathrm{\Gamma }t\ll 1$, the decay can be approximated as $1-\frac{3}{2}N\mathrm{\Gamma }t\approx e^{-\frac{3}{2}N\mathrm{\Gamma }t}$. For long times $N\mathrm{\Gamma }t\gg 1$, the populations decay much slower compared to an exponential decay with collectively enhanced decay rate $N\mathrm{\Gamma }$. The numerical calculations were performed for $k\sigma =1000$ and the positions of the atoms varied according to a Gaussian density distribution with mean 0 and variance ${\sigma }^2$. The plot shows the average over $M=1000$ realizations and convergence with respect to $M$ was checked.

Figure 4

Time evolution of the population of the bright state (blue solid line), the dark state (orange dashed line), and the total population of the excited states (green dashed-dotted line) for $N=2$ atoms coupled to a chiral waveguide. The gray dotted line shows an exponential decay with collectively enhanced decay rate $N\mathrm{\Gamma }$. (Inset) The inset shows a logarithmic plot of the time evolution of the populations. For $N\mathrm{\Gamma }t\gg 1$, the decay of the bright state population is slowed down due to the coupling to the dark state.

Figure 5

Decay dynamics of a single collective excitation of a system of $N$ atoms coupled to a chiral waveguide in the limit $N\to \infty$. The collective excitation initially decays exponentially with decay rate $\kappa$ while for long times the decay is algebraic with ${\left(\kappa t\right)}^{-3/2}$ which is shown in the inset. The dashed line shows the long-time behavior. Note that the dynamics looks qualitatively the same for finite $N$ and $N\gg 1$.

Figure 6

The blue (dark gray) line shows the time evolution of the bright state in the case of a bidirectional coupling and a normal distribution of the atoms with zero mean and variance ${\sigma }^2$ with $k\sigma =1000$ for $N=100$ atoms and averaged over $M=100$ realizations. The gray dashed line shows the corresponding time evolution for the chiral case for the same number of particles. The light gray curves in the background show trajectories for single realizations.