Microtoroidal cavity QED with fiber overcoupling and strong atom-field coupling: A single-atom quantum switch for coherent light fields

We propose a scheme for single-atom, quantum control of the direction of propagation of a coherent field incident, via a tapered fiber, upon a microtoroidal whispering-gallery-mode (WGM) resonator. The scheme involves overcoupling of the fiber taper to the resonator and strong coupling of an atom to the evanescent field of the WGM, i.e., an atom-field coupling that exceeds the total WGM linewidth. In contrast to previous, related schemes that operate in the bad-cavity regime, the proposed scheme can operate effectively with much stronger incident fields, while also preserving their coherent nature. It can also serve to prepare an entangled state of the atom and coherent optical pulses propagating in opposite directions along the fiber. We evaluate the fidelity of preparation of such a state taking into account absorption and atomic spontaneous emission and demonstrate that high fidelities should be possible with realistic parameters.

Figure 1

Schematic of the microtoroid, atom, and tapered-fiber coupler. Whispering gallery modes, $a$ and $b$, couple with strength $g$ to an atom through their evanescent fields. The modes are coupled via intrinsic scattering at rate $h$ and suffer intrinsic loss at rate ${\kappa }_{\mathrm{i}}$ into the output fields $a_{\mathrm{out},\mathrm{i}}$ and $b_{\mathrm{out},\mathrm{i}}$. Evanescent coupling of the modes to the tapered fiber corresponds to an extrinsic loss at rate ${\kappa }_{\mathrm{ex}}$, while atomic spontaneous emission into free space (output field $c_{\mathrm{out}}$) occurs at rate $\gamma$. Fields propagating along the fiber towards and away from the microtoroid are denoted by $\{a_{\mathrm{in},\mathrm{ex}},b_{\mathrm{in},\mathrm{ex}}\}$ and $\{a_{\mathrm{out},\mathrm{ex}},b_{\mathrm{out},\mathrm{ex}}\}$, respectively.

Figure 2

Normalized power transmission $T_{\mathrm{F}}$ (blue) and reflection $T_{\mathrm{B}}$ (red) as a function of probe detuning in the weak-driving limit (using the linearized theory) for critical coupling (${\kappa }_{\mathrm{ex}}=\sqrt{{\kappa }_{\mathrm{i}}^2+h^2}$) and ${\omega }_{\mathrm{A}}={\omega }_{\mathrm{C}}$. Parameters are $\{{\kappa }_{\mathrm{ex}},{\kappa }_{\mathrm{i}},h,\gamma \}/2\pi =\{10,10,0,5.2\}$ MHz. Solid lines are for $g_{\mathrm{tw}}/2\pi =100$ MHz [$sin\left(kx\right)=0$] and dashed lines are for $g_{\mathrm{tw}}=0$. For large $g_{\mathrm{tw}}$, the maximum increase in $T_{\mathrm{F}}({\Delta }_{\mathrm{C}}=0)$ is 0.25.

Figure 3

Normalized power transmission $T_{\mathrm{F}}$ (blue) and reflection $T_{\mathrm{B}}$ (red) as a function of probe detuning in the weak-driving limit (using the linearized theory) for strong overcoupling (${\kappa }_{\mathrm{ex}}\gg {\kappa }_{\mathrm{i}},h$) and ${\omega }_{\mathrm{A}}={\omega }_{\mathrm{C}}$. Parameters are $\{{\kappa }_{\mathrm{ex}},{\kappa }_{\mathrm{i}},h,\gamma \}/2\pi =\{20,0.2,0,5.2\}$ MHz. Solid lines are for $g_{\mathrm{tw}}/2\pi =100$ MHz [$sin\left(kx\right)=0$] and dashed lines are for $g_{\mathrm{tw}}=0$. For large $g_{\mathrm{tw}}$, the maximum change in $T_{\mathrm{F}}({\Delta }_{\mathrm{C}}=0)$ is close to 1.

Figure 4

(a) Normalized power transmission $T_{\mathrm{F}}$ (blue) and reflection $T_{\mathrm{B}}$ (red), (b) output photon correlation functions $g_{\mathrm{FF}}^{\left(2\right)}\left(0\right)$ (blue) and $g_{\mathrm{BB}}^{\left(2\right)}\left(0\right)$ (red), (c) intracavity field amplitudes $\langle a\rangle$ (blue) and $\langle b\rangle$ (red), (d) normal-mode amplitudes $\langle A\rangle$ (blue) and $\langle B\rangle$ (red), and (e) atomic excited-state population as a function of atom-field coupling strength $g_{\mathrm{tw}}$ for strong overcoupling (${\kappa }_{\mathrm{ex}}\gg {\kappa }_{\mathrm{i}},h$) and ${\omega }_{\mathrm{A}}={\omega }_{\mathrm{C}}={\omega }_{\mathrm{p}}$ (${\Delta }_{\mathrm{C}}={\Delta }_{\mathrm{A}}=0$). Parameters are $\{{\kappa }_{\mathrm{ex}},{\kappa }_{\mathrm{i}},h,\gamma ,E_{\mathrm{p}}\}/2\pi =\{30,0.5,0,5.2,10\}$ MHz. The horizontal dashed lines in (c) show the approximations to $\langle a\rangle$ and $\langle b\rangle$ from Table 1 for the limit of large $g_{\mathrm{tw}}$. The vertical dashed line in each plot indicates the value of ${\kappa }_{\mathrm{ex}}$.

Figure 5

(a) Normalized power transmission $T_{\mathrm{F}}$ (blue) and reflection $T_{\mathrm{B}}$ (red), (b) output photon correlation functions $g_{\mathrm{FF}}^{\left(2\right)}\left(0\right)$ (blue) and $g_{\mathrm{BB}}^{\left(2\right)}\left(0\right)$ (red), (c) intracavity field amplitudes $\langle a\rangle$ (blue) and $\langle b\rangle$ (red), (d) normal-mode amplitudes $\langle A\rangle$ (blue) and $\langle B\rangle$ (red), and (e) atomic excited-state population as a function of probe strength $E_{\mathrm{p}}$ for strong overcoupling (${\kappa }_{\mathrm{ex}}\gg {\kappa }_{\mathrm{i}},h$) and ${\omega }_{\mathrm{A}}={\omega }_{\mathrm{C}}={\omega }_{\mathrm{p}}$ (${\Delta }_{\mathrm{C}}={\Delta }_{\mathrm{A}}=0$), with $g_{\mathrm{tw}}/2\pi =50$ MHz (solid line), 100 MHz (dashed line), and 150 MHz (dot-dashed line). Other parameters are $\{{\kappa }_{\mathrm{ex}},{\kappa }_{\mathrm{i}},h,\gamma \}/2\pi =\{30,0.5,0,5.2\}$ MHz.

Figure 6

Optical bistability curves for $g_{\mathrm{tw}}/2\pi =50$ MHz (solid line), 100 MHz (dashed line), and 150 MHz (dot-dashed line) with strong overcoupling (${\kappa }_{\mathrm{ex}}\gg {\kappa }_{\mathrm{i}},h$) and ${\omega }_{\mathrm{A}}={\omega }_{\mathrm{C}}={\omega }_{\mathrm{p}}$ (${\Delta }_{\mathrm{C}}={\Delta }_{\mathrm{A}}=0$). Inset: The case $g_{\mathrm{tw}}/2\pi =100$ MHz on a larger vertical scale. Other parameters are $\{{\kappa }_{\mathrm{ex}},{\kappa }_{\mathrm{i}},h,\gamma \}/2\pi =\{30,0.5,0,5.2\}$ MHz.

Figure 7

Normalized power (a) transmission $T_{\mathrm{F}}$ and (b) reflection $T_{\mathrm{B}}$, and output photon correlation functions (c) $g_{\mathrm{FF}}^{\left(2\right)}\left(0\right)$ and (d) $g_{\mathrm{BB}}^{\left(2\right)}\left(0\right)$, for $g_{\mathrm{tw}}/2\pi =150$ MHz with strong overcoupling (${\kappa }_{\mathrm{ex}}\gg {\kappa }_{\mathrm{i}},h$), ${\omega }_{\mathrm{A}}={\omega }_{\mathrm{C}}$, and $E_{\mathrm{p}}/2\pi =10$ MHz (solid line), 50 MHz (dashed line), and 100 MHz (dot-dashed line). Other parameters are $\{{\kappa }_{\mathrm{ex}},{\kappa }_{\mathrm{i}},h,\gamma \}/2\pi =\{30,0.5,0,5.2\}$ MHz.

Figure 8

Fidelity $F$ of entangled-path coherent-state preparation as a function of the mean photon number, ${\left|\alpha \right|}^2$, in the incident pulse for $g_{\mathrm{tw}}/2\pi =\{50,100,150\}$ MHz (curves are labeled by these values in units of MHz). Dashed lines are the approximate expression (67). Other parameters are $\{{\kappa }_{\mathrm{ex}},{\kappa }_{\mathrm{i}},h,\gamma \}/2\pi =\{30,0.5,0,5.2\}$ MHz, $t_{\mathrm{p}}=318$ ns ($\kappa t_{\mathrm{p}}\simeq 60$), and ${\omega }_{\mathrm{A}}={\omega }_{\mathrm{C}}$.

Figure 9

Fidelity $F$ of entangled-path coherent-state preparation as a function of the mean photon number, ${\left|\alpha \right|}^2$, in the incident pulse for $g_{\mathrm{tw}}/2\pi =100$ MHz, with ${\kappa }_{\mathrm{i}}/2\pi =\{0.25,0.5,1.0,2.0\}$ MHz (curves are labeled by these values in units of MHz). Dashed lines are the approximate expression (67). Other parameters are $\{{\kappa }_{\mathrm{ex}},h,\gamma \}/2\pi =\{50,0,5.2\}$ MHz, $t_{\mathrm{p}}=159$ ns ($\kappa t_{\mathrm{p}}\simeq 50$), and ${\omega }_{\mathrm{A}}={\omega }_{\mathrm{C}}$.

Figure 10

Input photon flux $|\langle a_{\mathrm{in},\mathrm{ex}}\left(t\right)\rangle {|}^2$ (blue, short-dashed curve) and output photon fluxes $|\langle a_{\mathrm{out},\mathrm{ex}}^{\left(0\right)}\left(t\right)\rangle {|}^2$ (blue, long-dashed curve) and $|\langle b_{\mathrm{out},\mathrm{ex}}\left(t\right)\rangle {|}^2$ (red solid curve), as a function of time for parameters $\{g_{\mathrm{tw}},{\kappa }_{\mathrm{ex}},{\kappa }_{\mathrm{i}},h,\gamma \}/2\pi =\{100,50,0.5,0,5.2\}$ MHz, $E_{\mathrm{p}}/2\pi =28$ MHz, and $t_{\mathrm{p}}=159$ ns ($\kappa t_{\mathrm{p}}\simeq 50$), with ${\omega }_{\mathrm{A}}={\omega }_{\mathrm{C}}$. The mean photon number in the input pulse (i.e., the area under the dashed curve) is ${\left|\alpha \right|}^2=20$. The output pulses have mean photon numbers $|{\alpha }_{\mathrm{ex}}^{\left(0\right)}{|}^2=19.2$ and $|{\beta }_{\mathrm{ex}}{|}^2=19.3$, respectively.