Controllable Coupling of Superconducting Flux Qubits

We have realized controllable coupling between two three-junction flux qubits by inserting an additional coupler loop between them, containing three Josephson junctions. Two of these are shared with the qubit loops, providing strong qubit-coupler interaction. The third junction gives the coupler a nontrivial current-flux relation; its derivative (i.e., the susceptibility) determines the coupling strength $J$, which thus is tunable in situ via the coupler’s flux bias. In the qubit regime, $J$ was varied from $\sim 45$ (antiferromagnetic) to $\sim -55\mathrm{mK}$ (ferromagnetic); in particular, $J$ vanishes for an intermediate coupler bias. Measurements on a second sample illuminate the relation between two-qubit tunable coupling and three-qubit behavior.

Figure 1

Circuit design of sample 1. Junctions 123 (456) form qubit $a$ ($b$). The large qubit junctions have areas $S_{1,3,4,6}=150\times 700{\mathrm{nm}}^2$, while ${\alpha }_{a,b}\equiv S_{2,5}/S_1=0.65$. Junctions 789 form the coupler (red), with areas $S_{7,9}=150\times 2000{\mathrm{nm}}^2$ and $S_8=150\times 400{\mathrm{nm}}^2$. The ratio between the coupler and qubit loop areas is $1:2$. The tank circuit has inductance $L=90\mathrm{nH}$ and capacitance $C=470\mathrm{pF}$, yielding a resonance at ${\omega }_{\mathrm{T}}/2\pi =20.76\mathrm{MHz}$ with quality $Q=300$. The mutual inductances, deduced from the flux periodicity, are $M_{a,\mathrm{T}}=M_{b,\mathrm{T}}=98$, $M_{\mathrm{c},\mathrm{T}}=51$, $M_{a,\mathrm{b}1}=0.85$, $M_{b,\mathrm{b}1}=0.17$, $M_{\mathrm{c},\mathrm{b}1}=0.19$, $M_{a,\mathrm{b}2}=0.23$, $M_{b,\mathrm{b}2}=1.2$, $M_{\mathrm{c},\mathrm{b}2}=0.25$, $M_{a,\mathrm{b}3}=0.42$, $M_{b,\mathrm{b}3}=0.39$, and $M_{\mathrm{c},\mathrm{b}3}=0.56$ (all in pH).

Figure 2

Black: the current-flux relation $I(f_{\mathrm{c}})$ of a coupler with ${\alpha }_{\mathrm{c}}=S_8/S_9=0.2$ and $I_{\mathrm{c}}=1\mu \mathrm{A}$. Red-dashed line: the coupling energy $J(f_{\mathrm{c}})$ obtained from Eq. (3) using this $I(f_{\mathrm{c}})$ and the loop currents $I_{\mathrm{p}a,b}$ found independently from the qubit response. Circles: experimental $J(f_{\mathrm{c}})$ obtained from Fig. 3.

Figure 3

$-\tan \theta (f_a,f_b)$ for sample 1 at coupler bias $f_{\mathrm{c}}=-0.08$, $-0.30$, and $-0.38$, with a manifest change in coupling sign. A theory fit as in Fig. 4 yields couplings $J=45$, 0, and $-55\mathrm{mK}$. The excess response in the $|\uparrow \uparrow ⟩$ quadrant for $f_{\mathrm{c}}=-0.38$ is due to the coupler.

Figure 4

Theoretical fit for Fig. 3a. The extracted parameters are $T_{\mathrm{eff}}=70\mathrm{mK}$, ${\Delta }_a=300\mathrm{mK}$, $I_{\mathrm{p}a}=75\mathrm{nA}$, ${\Delta }_b=55\mathrm{mK}$, $I_{\mathrm{p}b}=180\mathrm{nA}$, and $J(-0.08)\approx 45\mathrm{mK}$.

Figure 5

$-\tan \theta (I_{\mathrm{bT}},I_{\mathrm{b}1})$ for sample 2. (a) $I_{\mathrm{b}2}=0$. At codegeneracy, the coupler flux is $f_{\mathrm{c}}=0.237$, and the induced coupling $J\approx 85\mathrm{mK}$. (b) and (c) $I_{\mathrm{b}2}=450\mu \mathrm{A}$. (b) Close-up of the codegeneracy point [dashed box in (c)], where $f_{\mathrm{c}}=0.347$ and $J\approx 44\mathrm{mK}$. (c) Overview and stability diagram. At bias points outside the boxed area, the coupler itself can become bistable, and contribute its own anticrossing to the impedance response. The arrow sizes denote the relative magnitudes of the loop currents in qubit $a$, coupler, and qubit $b$, respectively.