Single-photon scattering on a qubit: Space-time structure of the scattered field

We study the space-time structure of the scattered field induced by the scattering of a narrow single-photon Gaussian pulse on a qubit embedded in a one-dimensional open waveguide. For weak excitation power we obtain explicit analytical expressions for the space and time dependence of reflected and transmitted fields which are, in general, different from plane traveling waves. The scattered field consists of two parts: a damping part which represents spontaneous decay of the excited qubit and a coherent, lossless part. We show that for a large distance $x$ from the qubit and at times $t$ long after the scattering event our theory provides the result which is well known from stationary photon transport. However, the approach to the stationary limit is very slow. The scattered field decreases as the inverse powers of $x$ and $t$ as both the distance from the qubit and the time after the interaction increase.

Figure 1

Schematic representation of a single-photon Gaussian pulse interacting with a two-level atom with energy levels $|g\rangle$ and $|e\rangle$. $\mathrm{\Omega }$ is the separation between the energy levels. Long horizontal lines denote the waveguide geometry.

Figure 2

Two-dimensional map of the transmittance calculated from (36) for (a) $t=1$ ns and (b) $t=5$ ns. The color bar shows the value $|T({\omega }_S,x,t){|}^2$. $\mathrm{\Gamma }/2\pi =0.01\mathrm{GHz}$, $\mathrm{\Omega }/2\pi =5\mathrm{GHz}$, and $\lambda =6$ cm.

Figure 3

Two-dimensional map of the reflectance calculated from (42) for (a) $t=1$ ns and (b) $t=5$ ns. The color bar shows the value $|R({\omega }_S,x,t){|}^2$. $\mathrm{\Gamma }/2\pi =0.01\mathrm{GHz}$, $\mathrm{\Omega }/2\pi =5\mathrm{GHz}$, and $\lambda =6$ cm.

Figure 4

Dependence of (a) transmittance $|T({\omega }_S,x_0,t\ge t_0){|}^2$, Eq. (36), and (b) reflectance $|R({\omega }_S,x_0,t\ge t_0){|}^2$, Eq. (42) ($|x_0|=1$ mm, $t_0=10$ ps), for different frequencies: (1) ${\omega }_S=\mathrm{\Omega }$, (2) ${\omega }_S=\mathrm{\Omega }+0.5\mathrm{\Gamma }$, (3) ${\omega }_S=\mathrm{\Omega }+\mathrm{\Gamma }$, and (4) ${\omega }_S=\mathrm{\Omega }+1.5\mathrm{\Gamma }$; $\mathrm{\Gamma }/2\pi =0.01\mathrm{GHz}$, $\mathrm{\Omega }/2\pi =5\mathrm{GHz}$, and $T_{\mathrm{\Omega }}=2\pi /\mathrm{\Omega }$.

Figure 5

The dependence of the transmittance (43) (solid black line) and reflectance (44) (dashed red line) on the photon frequency for different distances of the field point from the qubit: (a) $x=1$ mm, (b) $x=5$ mm, and (c) $x=10$ mm. $\mathrm{\Gamma }/2\pi =0.01\mathrm{GHz}$, $\mathrm{\Omega }/2\pi =5\mathrm{GHz}$, and $\lambda =6$ cm.

Figure 6

The influence of the interference on the reflectance (44) at 1-mm distance from the qubit. Dashed red line, labeled 1, the reflectance $|R({\omega }_S,x,t){|}^2$; solid black line, labeled 2, the reflectance $|R\left({\omega }_S\right){|}^2$ in the absence of interference; dashed blue line, labeled 3, the term ${\left|z\right|}^2$; solid green line, labeled 4, the interference term $2|R\left({\omega }_S\right){|}^2\text{Re}\left(z\right)$. $\mathrm{\Gamma }/2\pi =0.01\mathrm{GHz}$, and $\mathrm{\Omega }/2\pi =5\mathrm{GHz}$.

Figure 7

Spatial dependence of (a) reflectance (44) and (b) transmittance (43) for off-resonant conditions. Solid black line (labeled 1), ${\omega }_S=\mathrm{\Omega }$; dashed red line (labeled 2), ${\omega }_S=\mathrm{\Omega }+0.5\mathrm{\Gamma }$; dotted blue line (labeled 3), ${\omega }_S=\mathrm{\Omega }+\mathrm{\Gamma }$; solid green line (labeled 4), ${\omega }_S=\mathrm{\Omega }+1.5\mathrm{\Gamma }$. $\mathrm{\Gamma }/2\pi =0.01\mathrm{GHz}$, $\mathrm{\Omega }/2\pi =5\mathrm{GHz}$, and $\lambda =6$ cm.

Figure 8

The influence of the interference on the reflectance (44) at $x=\lambda /2$ distance from the qubit. Dashed red line (labeled 1), the reflectance $|R({\omega }_S,x,t){|}^2$; solid black line (labeled 2), the reflectance $|R\left({\omega }_S\right){|}^2$ in the absence of the interference; dashed blue line (labeled 3), the term ${\left|z\right|}^2$; solid green line (labeled 4), the interference term $2|R\left({\omega }_S\right){|}^{2}Re\left(z\right)$. $\mathrm{\Gamma }/2\pi =0.01\mathrm{GHz}$, $\mathrm{\Omega }/2\pi =5\mathrm{GHz}$, and $\lambda =6$ cm.

Figure 9

Spatial dependence of transmittance (solid black line, labeled 1) and reflectance (dashed red line, labeled 2) for the off-resonant condition ${\omega }_S=\mathrm{\Omega }+0.5\mathrm{\Gamma }$, calculated from (45) and (46), respectively. The solid magenta line (labeled 5) and solid cyan line (labeled 6) show the asymptotic behavior of transmittance and reflectance, respectively. The dashed blue line (labeled 3) and green line (labeled 4) are the corresponding envelopes. $\mathrm{\Gamma }/2\pi =0.01\mathrm{GHz}$, $\mathrm{\Omega }/2\pi =5\mathrm{GHz}$, and $\lambda =6$ cm.

Figure 10

Plane of the complex $\omega$.