Theory of single-photon transport in a single-mode waveguide. II. Coupling to a whispering-gallery resonator containing a two-level atom

We analyze the single-photon transport in a single-mode waveguide coupled to a whispering-gallery-type resonator interacting with a two-level atom. The single-photon transport properties such as the transmission and reflection amplitudes, as well as the resonator and the atom responses, are solved exactly via a real-space approach. The treatment includes the intermode backscattering between the two degenerate whispering-gallery modes of the resonator, and the dissipations of the resonator and the atom. We also show that a generalized critical coupling condition, that the single-photon transmission at the output of the waveguide goes to zero on resonance for a matched system, holds for the full coupled waveguide-ring resonator-atom system.

Figure 1

(Color online) Schematics of the system. The single-mode waveguide is denoted by the blue channel. The ring resonator is denoted by the green ring. The black dot denotes the two-level atom, of which the two energy levels are plotted on the right. The waveguiding modes are described by $c_R^{†}\left(x\right)$ and $c_L^{†}\left(x\right)$. The two degenerate whispering-gallery modes are described by $a^{†}$ (counterclockwise) and $b^{†}$ (clockwise).

Figure 2

(Color online) Fitting to experimental data of system of waveguide-microtoroidal resonator. $\Gamma \equiv {\mid V\mid }^2∕2v_g$. (a) Lower curve: $\mid h\mid ∕2\pi =5.50092\mathrm{MHz}$, $1∕{\tau }_c∕2\pi =7.49983\mathrm{MHz}$. $\Gamma ∕2\pi =6.89887\mathrm{MHz}$. ${\Gamma }^2=1.348{18}^2\sqrt{{\mid h\mid }^2+{(1∕{\tau }_c)}^2}$, which is somewhat close to the critical coupling condition. Upper curve: undercoupling. Blue curve: $\mid h\mid ∕2\pi =7.57069\mathrm{MHz}$, $1∕{\tau }_c∕2\pi =8.4642\mathrm{MHz}$. $\Gamma ∕2\pi =1.13735\mathrm{MHz}$. ${\Gamma }^2⪡{\mid h\mid }^2+{(1∕{\tau }_c)}^2$. The overall decay rate $\kappa$ in Ref. 5 is equal to $\Gamma +1∕{\tau }_c$ in this paper. (b) $\mid h\mid ∕2\pi =7.64947\mathrm{MHz}$, $1∕{\tau }_c∕2\pi =0.250879\mathrm{MHz}$, $\Gamma ∕2\pi =0.0194301\mathrm{MHz}$. ${\Gamma }^2⪡{\mid h\mid }^2+{(1∕{\tau }_c)}^2$ and $\mid h\mid ⪢1∕{\tau }_c⪢\Gamma$. The numerical values are obtained by the least squares method using the amplitude of Eq. (13).

Figure 3

(Color online) The transmission spectrum $T$ (blue curve), reflection spectrum $R$ (red curve), and the atom excitation spectrum $A\equiv {\mid e_q\mid }^2$ (green curve) of the coupled waveguide-microdisk-quantum dot system. The normalized spectrum $\frac{\Gamma }{2v_g}{\mid e_q\mid }^2$ is plotted. Upper panel, (a1)–(c1): dephasing ${\gamma }_p=0$. Lower panel, (a2)–(c2): ${\gamma }_p\ne 0$. (a1), (a2): $\Omega ={\omega }_c$. (b1), (b2): $\Omega -{\omega }_c=\mid h\mid$. (c1), (c2): $\Omega -{\omega }_c=-\mid h\mid$. Parameters used: $g∕2\pi =6$, $h∕2\pi =-9.6$, $1∕{\tau }_q∕2\pi =0.16$, $1∕{\tau }_c∕2\pi =0.76$, $\Gamma ∕2\pi =0.44$, and ${\gamma }_p∕2\pi =2.4$.

Figure 4

Map of the transmission spectrum $T$ as $g$ and $h$ are varied. $h$ is real for this map. Also, $g_a=g_b$ and $g\equiv \mid g_a\mid$. The location of $\omega ={\omega }_c$ is indicated by the gray tick in each figure.

Figure 5

Map of the group delay $d\varphi ∕d\omega$ as $g$ and $h$ are varied. $h$ is real for this map. Also, $g_a=g_b$ and $g\equiv \mid g_a\mid$. The location of $\omega ={\omega }_c$ is indicated by the gray tick in each figure.

Figure 6

Map of the atomic excitation spectrum ${\mid e_q\mid }^2$ as $g$ and $h$ are varied. $h$ is real for this map. Also, $g_a=g_b$ and $g\equiv \mid g_a\mid$. The normalized spectrum $\frac{\Gamma }{2v_g}{\mid e_q\mid }^2$ is plotted. The location of $\omega ={\omega }_c$ is indicated by the gray tick in each figure.

Figure 7

Map of the phase-matched WGM excitation ${\mid e_a\mid }^2$ as $g$ and $h$ are varied. $h$ is real for this map. Also, $g_a=g_b$ and $g\equiv \mid g_a\mid$. The normalized spectrum $\frac{\Gamma }{2v_g}{\mid e_a\mid }^2$ is plotted. The location of $\omega ={\omega }_c$ is indicated by the gray tick in each figure.

Figure 8

Anticrossing between the atomic and cavity resonances as $h$ is continuously varied. $g=2.5\Gamma$. This depicts the evolution of the resonances in the column of $g=2.5\Gamma$ in Fig. 4.

Figure 9

(Color online) Map of the transmission spectrum with intrinsic losses as $g$ and $h$ are varied. Red curves: only atom dissipation is nonzero ($1∕{\tau }_q=\Gamma$ and $1∕{\tau }_c=0$). Blue curves: only resonator dissipation is nonzero ($1∕{\tau }_c=\Gamma$ and $1∕{\tau }_q=0$). The intrinsic atom dissipation only affects resonances with atomic nature, and the intrinsic resonator dissipations only affects resonances with cavity nature. $h$ is real for this map. Also, $g_a=g_b$ and $g\equiv \mid g_a\mid$. The location of $\omega ={\omega }_c$ is indicated by the gray tick in each figure.

Figure 10

(Color online) The transmission spectrum for fixed $g$ and $h$ as the intrinsic atom and resonator losses are increased, respectively. From left to right: 0, $0.5\Gamma$, $\Gamma$, $5\Gamma$, $10\Gamma$, and $20\Gamma$. Resonances of large cavity nature are less affected by the atom dissipation; similarly, resonances of large atomic nature are less affected by the cavity dissipation. The location of $\omega ={\omega }_c$ is indicated by the gray tick in each figure.

Figure 11

The transmission spectrum when $h$ is complex. $g=2.5\Gamma$, $h=2.5e^{i{\theta }_h}\Gamma$. From left to right: $\theta =0$, $\pi ∕4$, $\pi ∕2$, $3\pi ∕4$, and $\pi$. The spectrum of $\theta =0$ is the mirror image to that of $\theta =\pi$, and $\theta =\pi ∕4$ to $3\pi ∕4$. The spectrum of $\theta =\pi ∕2$ is symmetric with respect to $\omega ={\omega }_c$. The location of $\omega ={\omega }_c$ is indicated by the gray tick in each figure.

Figure 12

The general symmetry property of the transmission spectrum when $\Delta \ne 0$. Left: $\Omega -{\omega }_c=2\Gamma$, $h=5e^{i\pi ∕4}\Gamma$. Right: $\Omega -{\omega }_c=-2\Gamma$, $h=5e^{i3\pi ∕4}\Gamma$. $1∕{\tau }_q=2\Gamma$, $1∕{\tau }_c=\Gamma$, and $g=3\Gamma$ for both plots.

Figure 13

Transmission spectrum for the lossless detuned case. $h=0$, $g_a=g_b$, and $g\equiv \mid g_a\mid$. The location of $\omega ={\omega }_c$ is indicated by the gray tick in each figure.

Figure 14

(Color online) Transmission spectrum for the detuned case with losses. Red curves: only atom dissipation is nonzero ($1∕{\tau }_q=\Gamma$ and $1∕{\tau }_c=0$). Blue curves: only resonator dissipation is nonzero ($1∕{\tau }_c=\Gamma$ and $1∕{\tau }_q=0$). $h=0$, $g_a=g_b$, and $g\equiv \mid g_a\mid$. The location of $\omega ={\omega }_c$ is indicated by the gray tick in each figure.

Figure 15

(Color online) The orientation between the $x^{\prime }\text{−}{y}^{\prime }$ and $x\text{−}y$ coordinate systems. The angle between the $x^{\prime }$ axis and $x$ axis is ${\varphi }_0$. The angle between the $x^{\prime }$ axis and the mirror plane ($y\text{−}z$ plane) is thus $\overline{\varphi }\equiv \pi ∕2-{\varphi }_0$.

Figure 16

Map of the reflection spectrum $R$ as $g$ and $h$ are varied. $h$ is real for this map. Also, $g_a=g_b$ and $g\equiv \mid g_a\mid$. $R+T=1$. The location of $\omega ={\omega }_c$ is indicated by the gray tick in each figure.

Figure 17

Map of the counterpropagating WGM excitation ${\mid e_b\mid }^2$ as $g$ and $h$ are varied. $h$ is real for this map. Also, $g_a=g_b$ and $g\equiv \mid g_a\mid$. The normalized spectrum $\frac{\Gamma }{2v_g}{\mid e_b\mid }^2$ is plotted. The location of $\omega ={\omega }_c$ is indicated by the gray tick in each figure.