Fig. 2. Tunable complex-phase element.

a Hermitian hopping term a ± i b which is to be implemented between two nodes 1 and 2 in an electric circuit. b Symbol for the circuit element corresponding to the hopping term with e^iϕ ∝ a + ib. The impedance representation is given by Eq. (3) with $a=\cos \left(\varphi \right)/\left(\mathrm{i}\omega L\right)$ and $b=\sin \left(\varphi \right)/\left(\mathrm{i}\omega L\right)$. c Implementation of the circuit element using four analog multipliers (represented by the circles with a cross symbol). We choose R = ωL. The voltages V_a and V_b tune the phase $\varphi =\arctan$V_b/V_a. This circuit implements the complex coupling from node 1 to 2 with phase e^iϕ as well as the back-direction from 2 to 1 with phase e^−iϕ.