Superradiant decay and dipole-dipole interaction of distant atoms in a two-way cascaded cavity QED system

We investigate a two-way cascaded cavity QED system consisting of microtoroidal resonators coupled through an optical fiber. Each microtoroidal cavity supports two counterpropagating whispering-gallery modes coupled to single atoms through their evanescent fields. We focus on a pair of atom-microtoroid systems and compute the spectrum of spontaneous emission into the fiber with one atom initially excited. Explicit results are presented for strong-coupling and bad-cavity regimes, where the latter allows the effective atom-atom interaction to be controlled through the atom-cavity coupling and detuning: the atoms exhibit either collective spontaneous emission with no dipole-dipole interaction or a (coherent) dipole-dipole interaction and independent (single-atom) emission. This capacity for switching the character of the interaction is a feature of bidirectional coupling and connects our two-way cascaded system to work on one-dimensional waveguides. Building upon our bad-cavity results, we generalize to many atom-microtoroid systems coupled through an optical fiber.

Figure 1

Schematic of the cascaded atom-microtoroid system. Counterpropagating microtoroid modes couple through their evanescent fields to two two-state atoms and the input-output fields of an optical fiber.

Figure 2

Spectrum of spontaneous emission into the fiber from the numerical solution of the full set of coupled amplitude equations; for (in units of $2\pi$ MHz) ${\gamma }_{\mathrm{A}}=5,g_1=(g_2,i{g}_2)=50$ (upper, lower), ${\kappa }_{\mathrm{ex}}^{(1,2)}=5,{\kappa }_{\mathrm{in}}^{(1,2)}=0.1$, and all the other parameters zero. The blue solid (green dashed) curves plot the spectrum for the fiber output traveling from left to right (right to left).

Figure 3

Spectra of spontaneous emission into the fiber from the numerical solution of the full set of coupled amplitude equations; for (in units of $2\pi$ MHz) ${\gamma }_{\mathrm{A}}=5,g_1=(g_2,-i{g}_2)=50$ (upper, lower), ${\kappa }_{\mathrm{ex}}^{(1,2)}=500,{\kappa }_{\mathrm{in}}^{(1,2)}=0.5$, and all other parameters zero. The blue solid (green dashed) curves plot the spectrum for the fiber output traveling from left to right (right to left). The red solid and dashed lines are plotted from Eq. (52) (upper) and Eqs. (55a) and (55b) (lower).

Figure 4

Spectra of spontaneous emission into the fiber in the bad-cavity regime for three atoms. Curves plotted from the numerical solution of the full set of coupled amplitude equations; for (in units of $2\pi$ MHz) ${\gamma }_{\mathrm{A}}=5,g_1=g_2=g_3=50,{\kappa }_{\mathrm{ex}}^{(1,2,3)}=500,{\kappa }_{\mathrm{in}}^{(1,2,3)}=0.5$, and all other parameters zero. The blue solid (green dashed) curves plot the spectrum for the fiber output traveling from left to right (right to left). The initial state is ${\sigma }_1^{+}|G\rangle$ (upper) and ${\sigma }_2^{+}|G\rangle$ (lower).

Figure 5

As in Fig. 4 but with $g_1=-i{g}_2=g_3=50$.