Two-port quantum model of finite-length transmission lines coupled to lumped circuits

We show that, in the framework of quantum circuit electrodynamics in the Heisenberg picture, a finite-length transmission line can be described as a two-port lumped element of Thévenin type. Each port consists of a resistor connected in series to a controlled voltage source. The resistance of the resistors is equal to the characteristic impedance of the line. The controlled voltage sources are governed by linear equations with delay that take into account the reflections at the line ends. We apply this model to a transmission line capacitively coupled to two lumped circuits and obtain the reduced system of Heisenberg equations that governs them. Then, we show these equations can be reformulated as a pair of quantum Langevin-like equations that are coupled through the controlled voltage sources. Finally, we apply our approach to an analytically solvable network. This approach may be useful for the modeling of quantum links between superconducting circuits.

Figure 1

A semi-infinite transmission line coupled to a lumped circuit $\mathcal{B}$ through the linear capacitor $C_c$.

Figure 2

A network composed of a finite length transmission line capacitively coupled to the lumped circuits ${\mathcal{B}}^{\prime }$ and $\mathcal{B}{}^{″}$ through the linear capacitors $C_c^{\prime }$ and $C_c^{″}$. The reference directions for the voltage $v(t;x)$ and the current intensity $i(t;x)$ are the same as those indicated in Fig. 1 for the semi-infinite line.

Figure 3

Equivalent two-port of Thévenin type of finite-length transmission lines in the Heisenberg picture. The equations ${\widehat{v}}_{\to }(t+T)={\widehat{V}}_0^{\prime }\left(t\right)-{\widehat{v}}_{\leftarrow }\left(t\right)$ and ${\widehat{v}}_{\leftarrow }(t+T)={\widehat{V}}_0^{″}\left(t\right)-{\widehat{v}}_{\to }\left(t\right)$ control the two voltage sources for $0<t<\infty$.

Figure 4

Equivalent lumped network with delay for the finite-length transmission line capacitively coupled to the lumped circuits ${\mathcal{B}}^{\prime }$ and $\mathcal{B}{}^{″}$ shown in Fig. 2.

Figure 5

A finite-length transmission line capacitively coupled to two linear LC circuits.

Figure 6

Selected entries of the impulse response matrix $\mathbf{h}\left(t\right)$ of a finite-length transmission line capacitively coupled to two identical LC circuits with parameters $g^{\prime }=0.3{\alpha }^{\prime }=2$, and ${\gamma }^{\prime }=2\pi \times 0.1$. Each impulse response is normalized to its maximum absolute value.

Figure 7

Selected entries of the impulse response matrix $\mathbf{h}\left(t\right)$ of a finite-length transmission line capacitively coupled to two identical LC circuits with parameters $g^{\prime }=0.3{\alpha }^{\prime }=2$, and ${\gamma }^{\prime }=2\pi \times 1$. Each impulse response is normalized to its maximum absolute value.

Figure 8

Selected entries of the impulse response matrix $\mathbf{h}\left(t\right)$ of a finite-length transmission line capacitively coupled to two identical LC circuits with parameters $g^{\prime }=0.3{\alpha }^{\prime }=2$, and ${\gamma }^{\prime }=2\pi \times 10$. Each impulse response is normalized to its maximum absolute value.

Figure 9

The intersections between the blue and the orange curves give the normalized natural frequencies of the modes of a finite-length transmission line connected to two equal LC circuits ($L_r^{\prime }=L_r^{″}$ and $C_r^{\prime }=C_r^{″}$) through two equal capacitors ($C_c^{\prime }=C_c^{″}$), for $g^{\prime }=0.3$, ${\alpha }^{\prime }=2$ and (a) ${\gamma }^{\prime }=0.1\times 2\pi$, (b) ${\gamma }^{\prime }=1\times 2\pi$, (c) ${\gamma }^{\prime }=10\times 2\pi$.