Coherent Single Photon Transport in a One-Dimensional Waveguide Coupled with Superconducting Quantum Bits

A recent theoretical analysis [A. Blais et al., Phys. Rev. A 69, 062320 (2004)] and experimental results [A. Wallraff et al., Nature (London) 431, 162 (2004)] show that interesting transport properties of a single microwave photon emerge when a quantum bit in a cavity is coupled to a one-dimensional waveguide. Here we adopt a real-space model Hamiltonian to give a unified approach which accounts for the experimental results, and make new predictions on the properties of single photon transport, such as the general Fano line shape, symmetric vacuum Rabi splitting for a leaky cavity at resonance, and a one-photon switching capability.

Figure 1

Schematics of the systems. (a) A Cooper pair box is embedded in a cavity coupled to a one-dimensional transmission line waveguide. The three horizontal bars indicate the waveguide. The cavity is formed by two capacitive open gaps in the center line. (b) Without cavity.

Figure 2

The transmission spectrum (dashed line) and the reflection spectrum (solid line) for a two-level system in a one-dimensional waveguide as shown in Fig. 1b.

Figure 3

The transmission spectra through the total system corresponding to Fig. 1a. The frequency is in units of ${\omega }_0\equiv 2\pi v_g/d$, $V^2/v_g={10}^{-5}{\omega }_0$, $Z_1=50$, and $Z_2=-1500i$. (a) Gray line: the cavity mode without the Cooper pair box. The resonance peak is located at ${\omega }_r=1.01148738{\omega }_0$. Solid line: vacuum Rabi splitting. The Cooper pair is in resonance with the cavity. $X=0.2$ ($|\overline{r}|\simeq 0.9996$). (b) Off resonance. Solid line: $\Omega ={\omega }_r$. Dashed line: $\Omega =1.0125{\omega }_0$. Gray line: $\Omega =1.0146{\omega }_0$. (c) Off resonance for leaky cavity. $\Omega =1.01148738{\omega }_0$. Solid line: $X=0.5$ ($|\overline{r}|\simeq 0.996$). Dashed line: $X=0.7$ ($|\overline{r}|\simeq 0.984$). Gray line: $X=0.96$ ($|\overline{r}|\simeq 0.525$). Inset shows the details of the overlapping peaks at normalized frequency around 1.01. (d) On resonance (zero detuning) for the leaky cavity. Solid line: $\Omega =1.017631{\omega }_0$, $X=0.5$ ($|\overline{r}|\simeq 0.996$). Gray line: $\Omega =1.030651{\omega }_0$, $X=0.7$ ($|\overline{r}|\simeq 0.984$). Inset shows the case when the coupling is 5 times larger: $V^2/v_g=25\times {10}^{-5}{\omega }_0$, $\Omega =1.030651{\omega }_0$, $X=0.7$.

Figure 4

One-photon switching. (a) Energy diagram of the single Cooper pair box. The dashed and the solid curves represent before and after the change of magnetic flux, respectively. The gray curve denotes the noninteracting case. $4E_c$ is adopted as the energy unit. (b) The transmission peaks for a qubit in a cavity coupled to the waveguide. The solid and dashed curves correspond to those in (a), respectively. The gray line denotes the change of transmission.