Theory of single-photon transport in a single-mode waveguide. I. Coupling to a cavity containing a two-level atom

The single-photon transport in a single-mode waveguide, coupled to a cavity embedded with a two-level atom, is analyzed. The single-photon transmission and reflection amplitudes, as well as the cavity and the atom excitation amplitudes, are solved exactly via a real-space approach. It is shown that the dissipations of the cavity and of the atom, respectively, affect distinctively on the transport properties of the photons and on the relative phase between the excitation amplitudes of the cavity mode and the atom.

Figure 1

(Color online) Schematics of the systems. A cavity interacting with a two-level atom is coupled to a single-mode waveguide in which single photons propagate in each direction. (a) Side-coupled single-mode cavity. (b) Side-coupled ring resonator. (c) Direct-coupled cases. $l$ and $r$ denote the left and right branches, respectively. The cavity is denoted by light green color, and the waveguide is denoted by the channel in light blue color.

Figure 2

(Color online) Linearization of the photonic dispersion of the waveguided mode. The dispersion relation, ${\omega }_k$, is denoted by the blue line. $\pm k_0$ are the wave vectors corresponding to an arbitrarily ${\omega }_0$. At the right branch around $k=k_0$, the dispersion is approximated by ${\omega }_{k\simeq k_0}\simeq {\omega }_0+v_g\left(k-k_0\right)$, while at the left branch around $k=-k_0$, the dispersion is approximated by ${\omega }_{k\simeq -k_0}\simeq {\omega }_0-v_g\left(k+k_0\right)$. The linearized dispersions are represented by the two red straight lines.

Figure 3

(Color online) Single-photon transmission spectrum for nondissipative case. (a) In-tune non-dissipative case ($\Omega ={\omega }_c$; $1/{\tau }_c=1/{\tau }_a=0$). At $\omega =\Omega$, the transmission is 1 and at $\omega =\Omega \pm g$, the transmission is 0. $g=0.5\Omega$ and $V^2/v_g=0.09\Omega$ are used for plotting the spectra. The conclusions however are independent of the choice of the numerical values. (b) When $g$ is small, the transmission peak becomes narrow, analogous to EIT. $g=0.03\Omega$, $V^2/v_g=0.09\Omega$. (c) Atom-cavity detuned non-dissipative case ($\Omega \ne {\omega }_c$; $1/{\tau }_c=1/{\tau }_a=0$). Left: ${\omega }_c=1.1\Omega$, $V^2/v_g=0.09\Omega$. Right: ${\omega }_c=0.9\Omega$, $V^2/v_g=0.09\Omega$. The transmission is always 1 at $\omega =\Omega$. (d) Single-photon switching. The red curve is the transmission spectrum when the atom is far detuned $\left(\left|\Omega -{\omega }_c\right|V^2/v_g⪢g^2\right)$ or is not present. The transmission of an on-resonance photon with $\omega ={\omega }_c$ thus changes from 1 to essentially 0 by tuning the atomic transition frequency.

Figure 4

(Color online) Single-photon transmission spectrum for dissipative cases. (a) In-tune dissipative cavity case ($\Omega ={\omega }_c$; $1/{\tau }_c=0.05\Omega$). The transmission is 1 at $\omega =\Omega$ and is $\simeq v_g^2/\left(V^4{\tau }_c^2\right)$ at $\omega =\Omega \pm g$. (b) Detuned dissipative atom case ($\Omega ={\omega }_c$; $1/{\tau }_a=0.05\Omega$). The transmission is $\simeq 2\left(V^2/v_g\right)/\left(g^2{\tau }_a\right)$ at $\omega =\Omega$ and is $\simeq v_g^2/\left(V^4{\tau }_a^2\right)$ at $\omega =\omega \pm g$. Also shown are the (cavity $+$ atom) component of the eigenstate and the relative phase denoted by the green and red arrows. (c) Detuned dissipative cavity and atom case ($\Omega \ne {\omega }_c$; $1/{\tau }_c=1/{\tau }_a=0.05\Omega$). The spectrum becomes asymmetric when $\Omega \ne {\omega }_c$, and the local maximum is no longer located at $\omega =\Omega$. For small dissipations, however, the transmission at $\omega =\Omega$ is insensitive to the detuning, $\delta$, as shown in the inset. $g=0.5\Omega$ and $V^2/v_g=0.09\Omega$ are used for plotting the spectra. The conclusions however are independent of the choice of the numerical values, ${\overline{V}}^2\equiv V^2/v_g$.

Figure 5

(Color online) Folding of the waveguiding paths for photons for the side-coupled case and the direct-coupled case. The $l$-branch mode of the direct-coupled case is folded from the $R$ mode of the side-coupled case, and the $r$-branch mode is folded from the $L$ mode. The orientations of $l$ and $r$ branches are such that the incident waves come from $x=-\infty$ and the outgoing waves run toward $x=+\infty$. The black dots indicate a phase shift in the reflected paths.

Figure 6

(Color online) The fitting of the transmission spectrum in Ref. [2]. The extracted experimental data are denoted by the red dots, while the blue curve indicates the fitting using Eq. (17c), with the following parameters: ${\omega }_c/2\pi =6.0446\text{GHz}$, $\Omega /2\pi =6.0444\text{GHz}$, $V^2/v_g/2\pi =0.361\text{MHz}$, ${\gamma }_c=0$, ${\gamma }_a/2\pi =0.86\text{MHz}$, and $g/2\pi =5.73\text{MHz}$ (i.e., the Rabi frequency is $2g/2\pi =11.46\text{MHz}$). These values are very close to those of experimental fitting in Ref. [2], except where an in-tune condition $\left({\omega }_c=\Omega \right)$ is assumed.