Far-Field Signatures of a Two-Body Bound State in Collective Emission from Interacting Two-Level Atoms on a Lattice

The collective emission from a one-dimensional chain of interacting two-level atoms is investigated. We calculate the light scattered by dissipative few-excitation eigenstates in the far field, and, in particular, focus on signatures of a lattice two-body bound state. We present analytical results for the angle-resolved, temporal decay of the scattered light intensity. Moreover, we find that the steady-state emission spectrum that emerges when the system is probed by a weak, incoherent driving field exhibits a distinct signature for the existence of a bound state, and allows us to determine the momentum distribution of the two-body relative wave function. Intriguingly, our study does not rely on single-atom addressability and/or manipulation techniques.

Figure 1

(a) 1D lattice of two-level atoms (dipole moments aligned parallel to the $x$ axis). (b) An incoherent drive (pump rate $|\mathcal{P}{|}^2$, angle ${\beta }_{\mathrm{exc}}\ne 0$) “imprints” the wave number $k_P$ so that the relevant Hilbert space comprises only two-excitation states with $K=2k_P$. The pump’s electric field polarization vector lies in the $x\text{−}z$ plane and is perpendicular to $\mathbf{k}$.

Figure 2

(a) Complex two-excitation dispersion relation as a function of the center-of-mass momentum $K({\lambda }_{\mathrm{at}}/a=0.5)$. The different black and blue lines denote different relative momenta $p$ for the scattering states. The bound states’ energies (real parts) are detached from the scattering states. (b) Simplified level scheme for ${\omega }_0$, $U\gg {\gamma }_0$ (see text for details).

Figure 3

(a) Spontaneous emission pattern $G^{(1)}/4{\xi }^2|\mathbf{w}{|}^2$ according to Eq. (6) (plotted for $t_{\mathrm{ret}}=0$ and $K=0$) as a function of the detection angle $\beta$ and the emission wavelength ${\lambda }_{\mathrm{at}}$ (in units of $a$). As ${\lambda }_{\mathrm{at}}/a$ increases, less Bragg orders become visible but the width of the emission peaks increases. (b) Angle dependence of the emitted radiation $G^{(1)}{r}^2/4{\xi }^2|\mathbf{d}{|}^2$. The black dashed line indicates the direction of the atomic chain and the blue line denotes the alignment of the dipole moments. The pattern exhibits a toroidal-like structure (a remnant of the single-dipole pattern) with lobes resulting from the properties of the bound state’s momentum distribution.