Blackbox quantization of superconducting circuits using exact impedance synthesis

We propose a new quantization method for superconducting electronic circuits involving a Josephson-junction device coupled to a linear microwave environment. The method is based on an exact impedance synthesis of the microwave environment considered as a blackbox with impedance function $Z\left(s\right)$. The synthesized circuit captures dissipative dynamics of the system with resistors coupled to the reactive part of the circuit in a nontrivial way. We quantize the circuit and compute relaxation rates following previous formalisms for lumped element circuit quantization. Up to the errors in the fit our method gives an exact description of the system and its losses.

Figure 1

Brune circuit (in the dotted box) shunted by a Josephson junction. The analysis of this circuit is extensively discussed in Appendix pp1.

Figure 2

Brune circuit extraction step.

Figure 3

Equivalence of a T-shaped inductive circuit in Fig. 2 to a coupled inductor.

Figure 4

A degenerate stage in a Brune circuit.

Figure 5

Geometry of the 3D transmon qubit simulated in HFSS. The light blue color indicates the perfect conductor, and the dark blue color indicates the vacuum. The qubit port terminals are defined on a dielectric substrate located at the position of the red line. Two coaxial ports are positioned symmetrically on each side of the substrate. The cavity dimensions are $\left(\mathrm{height},\mathrm{length},\mathrm{width}\right)=(4.2,24.5,42\mathrm{mm})$.

Figure 6

Fundamental mode (the TE101 mode) of the cavity with frequency $f_{\mathrm{TE}101}=6.875\mathrm{GHz}$. The green color indicates electric-field regions of higher magnitude compared to blue regions.

Figure 7

Real part of the open circuit response. The dotted green line indicates the open circuit response for the Brune circuit which we identify with the open circuit fit. The solid magenta line indicates the simulated response. The red line indicates the response of the lossy Foster circuit. TE101 and TE103 are the resonances associated with classical rectangular cavity modes [21].

Figure 8

Real part of impedance in a small range of frequencies around the qubit pole ($f_{\mathrm{qb}}=6.7052$ GHz, where $f_{\mathrm{qb}}$ is the qubit resonance for the exact fit) for the system shunted (with impedance $Z_s$) by a linear inductance $L_J=4.5\mathrm{nH}$ representing the Josephson junction for three different cases. The TE101 mode is not strongly affected by the presence of $L_J$.

Figure 9

Magnitude of the real part of qubit pole $s_{\mathrm{qb}}$ as a function of linear inductance representing the Josephson junction shunting the system for three different cases: the exact fit for the system shunted by the linear inductance, the Brune circuit shunted by the linear inductance, and the lossy Foster circuit shunted by the linear inductance. The overall decreasing trend of this rate is simply due to the movement of the qubit pole to decreasing frequency as $L_J$ is increased. The $T_1$ relaxation rate of the qubit is given by $T_1^{-1}={\omega }_{\mathrm{qb}}/Q_{\mathrm{qb}}$, where the quality factor $Q_{\mathrm{qb}}={\omega }_{\mathrm{qb}}/\left|{\xi }_{\mathrm{qb}}\right|$ with ${\xi }_{\mathrm{qb}}=\mathrm{Re}\left[s_{\mathrm{qb}}\right]$ and ${\omega }_{\mathrm{qb}}=\mathrm{Im}\left[s_{\mathrm{qb}}\right]$ is the frequency of the qubit mode.

Figure 10

Modified Brune circuit. Tree branches are shown in black, and chord branches are shown in green. Current directions are chosen to have the matrix ${\mathcal{F}}_C$ in Eq. (A1) with all positive entries. Formal capacitance $C_{M+1}$ is introduced for a technical reason: with the substitution $C_{M+1}=\frac{1}{i\omega R_{M+1}}$ we are able to compute the dissipation rate due to $R_{M+1}$ in the formalism of [9]. After the coordinate transformations we take the $C_J\to 0$ limit. To get ${\mathcal{F}}_C$ in Eq. (A1) with all positive entries we reversed the direction of currents through and inverted the polarity of voltages across right coupled inductor branches, which requires the update $M_j\to -M_j$ for mutual inductances. See Fig. (13) for the definition of the coupled inductor.

Figure 11

Lossless part of the Brune circuit. It is this circuit that is described by the “system” Hamiltonian Eqs. (3.1) and (3.2). As discussed above, we take the limit $C_J\to 0$ so that this element is removed. The lossless circuit is obtained from Fig. 10 by taking $R_1,R_2,...R_M\to 0$ and $R_{M+1}\to \infty$. It is these different limiting treatments that require that the descriptions of $R_1-R_M$ follow the low-impedance treatment as in [10], while the description of $R_{M+1}$ needs the high-impedance treatment as in [9].

Figure 12

Circuit identity showing that tightly coupled inductor pairs simplify in the case of a turns ratio equal to 1; in this case Fig. 11 becomes identical to one of the classic lossless Foster canonical forms [3, 5].

Figure 13

Generic two-port coupled inductor with the convention chosen for current directions and voltage polarities.

Figure 14

Generic shunt resonant stage in a lossy Foster circuit.

Figure 15

Lossy Foster circuit.