Analysis of a tunable coupler for superconducting phase qubits

This paper presents a theoretical analysis of the recently realized tunable coupler for superconducting phase qubits [R. C. Bialczak et al., arXiv:1007.2219 (unpublished)]. The coupling can be turned off by compensating a negative mutual inductance with a tunable Josephson inductance. The main coupling in this system is of the $XX$ type and can be zeroed exactly, while there is also a small undesired contribution of the $ZZ$ type. We calculate both couplings as functions of the tuning parameter (bias current) and focus on the residual coupling in the OFF regime. In particular, we show that for typical experimental parameters the coupling OFF/ON ratio is few times ${10}^{-3}$, and it may be zeroed by proper choice of parameters. The remaining errors due to physical presence of the coupler are on the order of ${10}^{-6}$.

Figure 1

The analyzed scheme of two-qubit coupling, which is controlled by the bias current $I_B$ of the coupling Josephson junction. Indices 1 and 2 refer to the two qubits and index 3 refers to the coupling junction. Coupling inductors $L_4$ and $L_5$ have a negative mutual inductance $-M$. The current $I_B$ controls the Josephson inductance $L_3$ of the coupling junction, which effectively adds to $-M$. An additional small coupling is via capacitance $C_a$.

Figure 2

(Color online) The coupling ${\Omega }_{XX}/2\pi$ and ${\Omega }_{ZZ}/2\pi$ (multiplied by 5 for clarity) as functions of the bias current $I_B$. Solid lines show numerical results while dashed lines show analytics using Eqs. (25, 29). The circuit parameters are: $I_{1,\text{cr}}=I_{2,\text{cr}}=1.5\mu \text{A}$, $I_{3,\text{cr}}=3\mu \text{A}$, $C_1=C_2=1\text{pF}$, $C_3=0.1\text{pF}$, $C_a=0$, $L_1=L_2=0.7\text{nH}$, $L_4=L_5=3\text{nH}$, $M=0.2\text{nH}$; ${\varphi }_{e,4}={\varphi }_{e,5}=0$, $N_1=N_2=5$ $\left({\omega }_1/2\pi ={\omega }_2/2\pi =6.59\text{GHz}\right)$. The panel (b) is a blow-up of the panel (a) near the crossing points at positive $I_B$. We see that ${\Omega }_{XX}=0$ at $I_B/I_{3,\text{cr}}=0.759$ and the residual coupling (square symbol) is ${\Omega }_{ZZ}^{\text{res}}/2\pi =-172\text{kHz}$.

Figure 3

(Color online) (a) Same as in Fig. 2b, but for a circuit with $C_3=0.3\text{pF}$. This makes positive ${\Omega }_{ZZ}^{\text{res}}/2\pi =49\text{kHz}$ (square symbol). (b) Same as in (a), but with added coupling capacitance $C_a=0.155\text{pF}$, that produces ${\Omega }_{ZZ}^{\text{res}}=0$, i.e., the couplings ${\Omega }_{XX}$ and ${\Omega }_{ZZ}$ are zeroed simultaneously.

Figure 4

(Color online) Solid lines: numerical results for ${\Omega }_{XX}^{+}$, ${\Omega }_{XX}^{-}$, and ${\Omega }_{ZZ}$ (multiplied by 10) as functions of the qubit detuning ${\omega }_1-{\omega }_2$ for the parameters of Fig. 3b at the point where ${\Omega }_{XX}={\Omega }_{ZZ}=0$. Dashed lines show the analytics for ${\Omega }_{XX}^{\pm }\approx \pm \Delta {\Omega }_{XX}/2$ from Eq. (34). Error state occupation due to nonzero ${\Omega }_{XX}^{\pm }$ is $1.5\times {10}^{-6}$.