Correlated photon emission by two excited atoms in a waveguide

Systems of atoms coupled to a single or a few waveguide modes provide a test-bed for physically and practically interesting interference effects. We consider the dynamics of a pair of atoms, approximated as two-level quantum emitters, coupled to a linear guided mode. In particular, we analyze the evolution of an initial state in which both atoms are excited, which is expected to decay into an asymptotic two-photon state. We investigate the lifetime of the initial configuration and the properties of the asymptotic photon correlations, and analyze the probability that the two photons are emitted in the same or in opposite directions. We find that the ratio $R$ between parallel and antiparallel emission probabilities is maximal when the interatomic distance is an odd multiple of a quarter wavelength of the emitted light. In such a case, $R=3$ in the small-coupling regime.

Figure 1

Pictorial representation of two atoms $A$ and $B$, modeled as two-level fixed quantum emitters, embedded in a linear waveguide at a distance $d$ from each other. The atoms have identical excitation energy ${\omega }_0$, and we assume that the atomic transition between the excited state $|e\rangle$ and the ground state $|g\rangle$ is coupled only to one mode of the waveguide. In the article, we will consider the evolution from the initial double-excitation state $|e_A,e_B\rangle$.

Figure 2

Contributions of order $V^2$ [diagram (a)] and $V^4$ [diagrams (b), (c), and (d)] to the double-excitation propagator $G_2\left(z\right)$ [Eq. (17)]. The diagrams (a), (b), and (c), without external legs, represent the $O\left(V^4\right)$ contributions to the self-energy ${\mathrm{\Sigma }}_2\left(z\right)$. The wavy lines represent photons; the double horizontal line represents the free propagator $G_2^{\left(0\right)}\left(z\right)$ in (11); the single horizontal line with a wavy line represents the free propagator (12) of $|{\mathrm{\Psi }}^s;k\rangle$, namely, the state with one photon and a single atomic excitation with sign $s$; and the dashed line with two wavy lines represents the free propagator $G_1^{\left(0\right)}(z,k)$ in (13). The crosses represent one of the vertices $v_s\left(k\right)$ in (14, 15, 16). Processes (b) and (c) preserve the sign $s$ (i.e., $s^{\prime }=s$), while in diagram (c) the relative sign of $s$ and $s^{\prime }$ is arbitrary.

Figure 3

Renormalization scheme for the two-excitation state propagator $G_2\left(z\right)$.

Figure 4

Diagrammatic representation of the two-photon amplitude $A(k_1,k_2,z)=\langle k_1,k_2|{(z-H)}^{-1}|e_A,e_B\rangle$. The dressed propagators and the dressed vertex are defined in Eqs. (19, 20, 21) and represented in Fig. 3.

Figure 5

Representation of the processes that contribute to the probability of parallel ($P_{\iff }$) and antiparallel ($P_{\rightleftharpoons }$) photon pair emission.

Figure 6

The upper panel displays the density plot of the two-photon distribution $P(k_1,k_2)$, normalized to its maximal value, for ${\omega }_0=1.1M$, $\lambda ={10}^{-2}M$, and $k_{0}d=\pi /2$. Colors range from blue (normalized density equal to 0) to white (normalized density equal to 1). It is evident that the probability is concentrated in very small areas of linear size $O\left({\lambda }^2\right)$ around the on-shell points. The lower panels (a) and (b) represent magnifications of the on-shell peaks highlighted in the upper plot, around $(-k_0,k_0)$ and $(k_0,k_0)$, respectively. The different shape of these peaks is due to the fact that emission with intermediate symmetric and antisymmetric atomic states interfere constructively in the parallel case (b) and destructively in the antiparallel case (a). For such choice of parameters, the ratio $R\left(\lambda \right)$ between the total parallel and antiparallel emission is close to 3.

Figure 7

Plots of the ratio $R=P_{\iff }/P_{\rightleftharpoons }$ in different conditions. Panel (a): Dependence of $R$ on $\lambda$ at fixed ${\omega }_0=1.1M$, for $d=\pi /\left(2k_0\right)$ (solid orange line), $d=3\pi /\left(2k_0\right)$ (dashed blue line), and $d=5\pi /\left(2k_0\right)$ (dot-dashed red line). Panel (b): Dependence of $R$ on $\lambda$ in the case $d=5\pi /\left(2k_0\right)$ for ${\omega }_0=1.1M$ (solid orange line), ${\omega }_0=1.2M$ (dashed blue line), ${\omega }_0=1.3M$ (red dot-dashed line), and ${\omega }_0=1.4M$ (dotted black line). Panel (c): Dependence of $R$ on $k_{0}d$, at fixed ${\omega }_0=1.1M$, for $\lambda ={10}^{-3}M$ (solid orange line), $\lambda ={10}^{-2}M$ (dashed blue line), and $\lambda =2\times {10}^{-2}M$ (red dot-dashed line).