Black-Box Superconducting Circuit Quantization

We present a semiclassical method for determining the effective low-energy quantum Hamiltonian of weakly anharmonic superconducting circuits containing mesoscopic Josephson junctions coupled to electromagnetic environments made of an arbitrary combination of distributed and lumped elements. A convenient basis, capturing the multimode physics, is given by the quantized eigenmodes of the linearized circuit and is fully determined by a classical linear response function. The method is used to calculate numerically the low-energy spectrum of a 3D transmon system, and quantitative agreement with measurements is found.

Figure 1

Cartoon of a JJ at the center of a broadband dipole antenna inside a 3D microwave cavity. The presence of the antenna alters the geometry of all cavity modes. This is illustrated with the lowest energy dressed mode [full (red) curve]. Capturing this effect in the usual method would require the inclusion of many bare modes of the empty cavity (dashed curves). This resummation is done automatically in the method presented here.

Figure 2

(a) Schematics of a JJ ((red) boxed cross) coupled to an arbitrary linear circuit (striped disk). (b) The Josephson element is replaced by a parallel combination of: a linear inductance $L_J$, a linear capacitance $C_J$ and a purely nonlinear element with energy $E_J[1-\cos (\phi )]-(E_J/2){\phi }^2$, represented by the spider symbol. (c) The linear part of the circuit shown in (b) is lumped into an impedance $Z(\omega )$ seen by the nonlinear element. (d) Foster-equivalent circuit (pole-decomposition) of the impedance $Z(\omega )$.