Multimode effects in cavity QED based on a one-dimensional cavity array

We present a microscopic model of cavity quantum electrodynamics based on a one-dimensional (1D) coupled cavity array (CCA), where a supercavity (SC) is composed by a segment of the 1D CCA with relatively smaller couplings with the outsides. The single-photon scattering problem for the SC empty or with a two level atom in it is investigated. We obtain the exact theoretical result on the transmission rate for our system, which predicts that the transmission peaks shall appear near the eigenenergies of the SC. Our numerical results further prove that the SC is a well-defined multimode cavity. When a two level atom resonant with the SC locates at the antinode of the resonant mode, the transmission spectrum shows a clear sign of vacuum Rabi splitting as expected. However, when the atom locates at the node of the resonant mode, we observe a deep valley in the transmission peak, which can be explained by the destructive interference of two transmission channels: one is the resonant mode, while the other is arising from the atom coupling with the nonresonance modes. The effect of nonresonance modes on vacuum Rabi splitting is also analyzed.

Figure 1

Schematic configuration of the single-photon scattering problem for the 1D CCA model. A single photon (filled red circle) with the wave vector $k$ injects from the left side of the supercavity composed of $N$ cavities, which is formed by a relatively small coupling strength $\eta$ with the outside cavities. A two level atom (filled blue circle) is in the $n\text{th}$ cavity of the SC. The transmission spectrum is measured by the detector on the right side of the SC. Here we take $N=5$ and $n=3$.

Figure 2

(a) The transmission rate vs the incident wave vector $k$ for the empty SC. Only the first five transmission peaks are shown. (b) Zoom in of the fourth transmission peak. Here we choose the parameters $N=31,\xi =1$, and $\eta =0.01$.

Figure 3

(a) Transmission rates vs the incident wave vector (just around the resonant mode). Blue dashed and red solid lines each stand for the atom at the antinode $\left(n=12\right)$ or node $\left(n=8\right)$ of the mode. (b) Zoom in of the red peak. Here the atom is near resonant with the fourth mode of the SC with ${\omega }_a=1.847760755\xi$, and $g=0.1$.

Figure 4

Transmission spectrum for the atom at the node of the resonant mode with $g=0$ (blue dotted line), $g=0.05$ (green dashed line), and $g=0.1$ (red solid line). (a) ${\omega }_a={\nu }_4$. (b) ${\omega }_a=1.847760755\xi$.

Figure 5

Transmission spectrum for different detunings when the atom is located in the antinode. Here we give the results for $\Delta ={\omega }_a-{\nu }_4=0$ (blue solid line), $\Delta =0.01\xi$ (red dashed line), and $\Delta =0.02\xi$ (black dotted line).

Figure 6

(a) The blue (red) solid line and triangle each represent the excitation amplitude of the SC and the atom as in state $|{\psi }_m\rangle$ $\left(|{\phi }_m\rangle \right)$. (b) The green dashed (red solid) line and triangle (circle) each represent the excitation amplitude of the SC and the atom as in state $|{\Psi }_m{\rangle }_{\text{SC}}$ in the two level approximation model (the exact model). Here, $g=0.05$ and $m=4$ while all the other parameters are the same as before.

Figure 7

Transmission rates vs the incident wave vector for different models. (a) The results under the full Hamiltonian (red solid line) and two level approximation (green dashed line). (b) The results when we only consider one channel which is supported by the $m\text{th}$ eigenstate of the SC (blue dashed line) and the atomic state (green solid line).

Figure 8

Transmission spectrum for various channels. The transition frequency of the atom ${\omega }_a=1.847760755\xi$. (a) Blue dashed line: the cavity state $|{\psi }_m\rangle$. Green solid line: the atomic state $|{\phi }_m\rangle$. (b) Green dashed line: the two level approximation. Red solid line: the exact model.