Optomechanical-like coupling between superconducting resonators

We propose and analyze a circuit that implements a nonlinear coupling between two superconducting microwave resonators. The resonators are coupled through a superconducting quantum interference device that terminates one of the resonators. This produces a nonlinear interaction of the standard optomechanical form, where the quadrature of one resonator couples to the photon number of the other resonator. The circuit therefore allows for all-electrical realizations of analogs to optomechanical systems, with coupling that can be both strong and tunable. We estimate the coupling strengths that should be attainable with the proposed device, and we find that the device is a promising candidate for realizing the single-photon strong-coupling regime. As a potential application, we discuss implementations of networks of nonlinearly coupled microwave resonators, which could be used in microwave-photon-based quantum simulation.

Figure 1

Schematic illustration of the system, which consists of two superconducting transmission-line resonators $A$ and $B$. The resonators are coupled to each other through the SQUID terminating resonator $A$. The coupling mechanism is the following: part of the magnetic field generated by the signal in resonator $B$ threads the SQUID loop that terminates resonator $A$, changing the phase across the SQUID. This phase determines the boundary condition for resonator $A$. The result is an interaction where the amplitude of resonator $B$ couples to the photon number of resonator $A$.

Figure 2

Circuit diagram of the coupler part of the circuit in Fig. 1, where resonator $B$ meets the SQUID embedded in resonator $A$.

Figure 3

${\Phi }_{\mathrm{ext}}$ dependence of the factor $F\left({\Phi }_{\mathrm{ext}}\right)$ that appears in the coupling strength [see Eq. (31)].

Figure 4

Two possible coupling geometries. (a) An inductive coupling design, where the magnetic field from resonator $B$ couples inductively to the SQUID loop. (b) An alternative coupling design, with potentially larger coupling strength, where the SQUID loop is galvanically connected to resonator $B$.

Figure 5

Normalized coupling strength $g_0/{\omega }_A$ as a function of the external flux bias ${\Phi }_{\mathrm{ext}}$, for the galvanic (blue solid) and inductive (green dashed) coupling designs. The parameters used to evaluate Eq. (54) were $Z_0\approx 50\Omega$, ${\omega }_A=10$ GHz, ${\omega }_B=1$ GHz, $d_A=d_B/20=3$ mm, $L_0=4.57\times {10}^{-7}$ H/m, $C_0=1.46\times {10}^{-10}$ H/m, $\Delta /d_B=10$%, and $E_J=4.11\times {10}^{-22}$ J.

Figure 6

Contours of $g_0/{\omega }_B$, the ratio of the fundamental coupling strength to the frequency of resonator $B$, as a function of the ratio between the frequencies of resonator $B$ and $A$, ${\omega }_B/{\omega }_A$, and the external flux bias ${\Phi }_{\mathrm{ext}}^0/{\Phi }_0$, for the galvanic coupling design. Here we used the same parameters as in Fig. 5, except that here ${\omega }_A$ is varied.

Figure 7

Three possible networks of nonlinearly coupled microwave resonators. These circuits represent all-electrical networks of resonators that are analogous to arrays of optomechanical systems. The electrical implementation here replaces the mechanical resonator with an electrical resonator, while keeping the nonlinear interaction. With different layouts it is possible to create networks where the “mechanical” (a), “optical” (b), or both (c) systems are coupled.