Stopping single photons in one-dimensional circuit quantum electrodynamics systems

We propose a mechanism to stop and time reverse single photons in one-dimensional circuit quantum electrodynamics systems. As a concrete example, we exploit the large tunability of the superconducting charge quantum bit (charge qubit) to predict one-photon transport properties in multiple-qubit systems with dynamically controlled transition frequencies. In particular, two qubits coupled to a waveguide give rise to a single-photon transmission line shape that is analogous to electromagnetically induced transparency in atomic systems. Furthermore, by cascading double-qubit structures to form an array and dynamically controlling the qubit transition frequencies, a single photon can be stopped, stored, and time reversed. With a properly designed array, two photons can be stopped and stored in the system at the same time. Moreover, the unit cell of the array can be designed to be of deep subwavelength scale, miniaturizing the circuit.

Figure 1

Transmission spectrum of two closely spaced $(d⪡\lambda )$ qubits coupled to a waveguide. A schematic of the two qubits in the waveguide is shown. The qubits have transition frequencies ${\Omega }_1$ and ${\Omega }_2$, respectively. ${\Omega }_0\equiv ({\Omega }_1+{\Omega }_2)∕2$. The normalized detuning $\Delta \equiv \mid {\Omega }_1-{\Omega }_2\mid ∕(V^2∕v_g)$. (a) Small detuning regime. Solid curve: $\Delta =0.2$; dashed curve: $\Delta =0.6$. (b) Large detuning regime. $\Delta =100$. The transmission spectrum is obtained from Eq. (1). $\varphi \left(d\right)$ is taken as ${10}^{-2}$ in both figures. The two minima in each transmission spectrum are at ${\Omega }_1$ and ${\Omega }_2$, respectively.

Figure 2

(Color online) Schematic of a tunable one-dimensional qubit-waveguide system used to stop and store a single photon. The system consists of a periodic array of unit cells of length $D$. Each unit cell consists of two qubits, $A$ and $B$, separated by a distance $d_1$. The three horizontal bars represent the one-dimensional transmission line waveguide.

Figure 3

The photonic bands of the system related to tuning scheme (i). The qubits frequencies are given by ${\Omega }_{A,B}={\omega }_c\pm (1∕2)\Delta (V^2∕v_g)$. $V^2∕v_g=2\times {10}^{-4}{\omega }_0$. ${\omega }_0\equiv 2\pi v_g∕D$. $d_1=0.3D$ in all cases. (a) Large detuning regime. $\Delta =100$. ${\omega }_c\equiv 0.1{\omega }_0$. (b) Small detuning regime. $\Delta =0.6$. ${\omega }_c\equiv 0.1{\omega }_0$.

Figure 4

The photonic bands of the system related to tuning scheme (ii). The circles denote the photon states in phase space. $k_c$ for the open circle is chosen as $0.4[KD∕\left(2\pi \right)]$ and is indicated by the vertical line. $\Delta \equiv \mid {\Omega }_1-{\Omega }_2\mid ∕(V^2∕v_g)=100$. ${\omega }_0\equiv 2\pi v_g∕D$. $d_1=0.3D$. $V^2∕v_g=2\times {10}^{-4}{\omega }_0$. (a) ${\omega }_c=0.6{\omega }_0$. (b) ${\omega }_c=0.51{\omega }_0$. (c) ${\omega }_c=0.42{\omega }_0$. (d) ${\omega }_c=0.38{\omega }_0$.