Theory of two atoms in a chiral waveguide

A theory is presented that describes the atom and field dynamics for two atoms in a chiral waveguide. A source-field approach is used that enables one to identify the various physical processes contributing to these dynamics. Each atom is prepared in an arbitrary state at $t=0$ and the field intensity and correlation functions are calculated, fully accounting for retardation. When the atoms are prepared in identical superposition states, the effects of constructive and destructive interference play a significant role on both the field intensity and second-order correlation function. It is also shown that the results can be taken over to provide a solution for the related problem of a single-photon pulse incident on an atom prepared in an arbitrary initial state.

Figure 1

Dimensionless intensity $I_{N1}$ as a function of ${\gamma }_{2}t$ for the symmetric factorized initial state with $d={\gamma }_2{X}_2/c\ll 1$, $\alpha =\beta =1/\sqrt{2}$, and $\theta =k_0{X}_2=\left(2n+1\right)\pi$, $2n\pi$.

Figure 2

Dimensionless intensity $I_{N2}$ as a function of ${\gamma }_{2}t$ for a maximally entangled initial state with $d={\gamma }_2{X}_2/c\ll 1$, $\alpha =\beta =1/\sqrt{2}$, and $\theta =k_0{X}_2=(2n+1)\pi$, $2n\pi$.

Figure 3

Dimensionless intensity $I_{N1}$ as a function of ${\gamma }_{2}t$ for the symmetric factorized initial state with $d={\gamma }_2{X}_2/c\ll 1$ and $\beta =1$ (both atoms inverted). The dashed blue curve corresponds to the spatially integrated intensity from two atoms in free space, that is, to two-atom superradiance.

Figure 4

Normalized intensity $I_{N1}$ as a function of ${\gamma }_{2}t$ for the symmetric factorized initial state with $d={\gamma }_2{X}_2/c=6,{\left|\beta \right|}^2=1/2,$ and $\theta =2n\pi$. For $d\gg 1$, the intensity depends only on the initial state populations of the atoms. The black dotted curve represents the intensity at the detector neglecting any modification of the field from atom 1 produced by atom 2.

Figure 5

Dimensionless rate of delayed coincidences ${\gamma }_2{R}_c(X_2,\tau )$ as a function of ${\gamma }_2\tau$ for $T\left(0\right)=1$ (both atoms inverted) and $d={\gamma }_2{X}_2/c=0,0.5.$

Figure 6

Dimensionless rate of delayed coincidences ${\gamma }_2{R}_c(X_2,\tau )$ as a function of ${\gamma }_2\tau$ for $T\left(0\right)=1$ (both atoms inverted) and $d={\gamma }_2{X}_2/c=2,5$.

Figure 7

Second order correlation function as a function of ${\gamma }_2\tau$ for a symmetric factorized initial state and ${\left|\beta \right|}^2=0.4$, $d\ll 1$, and $\theta =k_0{X}_2=\left(2n+1\right)\pi$. The dotted black lines are the theoretical asymptotes.

Figure 8

Second order correlation function as a function of ${\gamma }_2\tau$ for a symmetric factorized initial state and ${\left|\beta \right|}^2=0.4$, $d\ll 1$, and $\theta =k_0{X}_2=2n\pi$. The dotted black lines are the theoretical asymptotes.

Figure 9

Second order correlation function as a function of ${\gamma }_2\tau$ for a symmetric factorized initial state with $\beta =1$ and $d\ll 1$. The dotted black lines are the theoretical asymptotes.

Figure 10

Second order correlation function as a function of ${\gamma }_2\tau$ for a symmetric factorized initial state and ${\left|\beta \right|}^2=0.1$, $d=7$, and $\theta =k_0{X}_2=\left(2n+1\right)\pi$.

Figure 11

Second order correlation function as a function of ${\gamma }_2\tau$ for a symmetric factorized initial state and ${\left|\beta \right|}^2=0.1$, $d=7$, and $\theta =k_0{X}_2=2n\pi$.

Figure 12

Second order correlation function as a function of ${\gamma }_2\tau$ for a symmetric factorized initial state and ${\left|\beta \right|}^2=0.1$, $d=7$, and $\theta =k_0{X}_2=\left(n+1/2\right)\pi$.

Figure 13

Second order correlation function $g_1^{\left(2\right)}\left(0,\tau \right)$ as a function of $z=\left({\gamma }_2\tau -1\right)e^{d/2}/{\left|\alpha \right|}^2$ for a symmetric factorized initial state with ${\left|\alpha \right|}^2=0.9$ and $d=12$. Dotted black lines give theoretical values for the maxima.

Figure 14

Second order correlation function as a function of ${\gamma }_2\tau$ for a symmetric factorized initial state and ${\left|\beta \right|}^2=0.1$, $d=4$, and $\theta =k_0{X}_2=\pi ,$ showing the change as $t$ changes sign.