Superconducting ‘twin’ qubit

We study a flux qubit consisting of a symmetrical pair of superconducting loops, with two Josephson junctions in each, joined by a common Josephson junction—a ‘twin’ flux qubit. The qubit is capacitively coupled to a transmission line, which allows us to characterize the spectrum of the device by measuring the scattering of propagated electromagnetic waves. We perform a detailed analytical analysis of the double-loop system, revealing its properties, and compare experimental results with numerical simulations. At half-flux quantum bias of both loops, the qubit is protected against global and local magnetic field fluctuations with much less sensitivity to the global field in the second order. The system selection rules allow even-odd transitions and prohibit transitions between even-even or odd-odd levels due to the symmetry of the device.

Figure 1

Geometry of a ‘twin’ qubit: (a) Scanning electron microscope image of the ‘twin’ qubit. The $\mathrm{Al}-{\mathrm{AlO}}_x$-Al JJs are highlighted in red and yellow. ${\phi }_l$ and ${\phi }_r$ are phases induced in the left and right loops by an external magnetic field. (b) Each of the qubits is coupled to the transmission line with a T-shaped capacitor (light blue). (c) The ‘twin’ qubit is a symmetrical arrangement of two individual flux qubits [8] sharing the central JJ. The qubit has four islands, which are labeled with Cooper pair (CP) occupation $\vec{n}=\left|n_1,n_2,n_3\right.〉$, and phases across its junctions: ${\phi }_{ij}$, where $i$ and $j$ are the island indices 0, 1, 2, or 3. The central junction is geometrically a factor $\alpha$ larger than the outer ones, resulting in energies $\alpha E_C$ and $\alpha E_J$.

Figure 2

Spectrum of the quantum system: (a) The resonance frequencies (${\omega }_{10}$) in the vicinity of ${\mathrm{\Phi }}_0/2$ (blue points). An inset exemplifies the power transmission coefficient ${\left|t\right|}^2$ for the $\left|0\right.〉 \leftrightarrow \left|1\right.〉$ transition, taken in the weak driving limit. The transition frequencies ${\omega }_{21}$ (red) are obtained in a two-tone measurement. (b) The spectrum measured in a wide flux bias range. Experimental data (circles) are compared with simulations (solid lines) for ${\omega }_{10}$ (blue) and ${\omega }_{21}$ (red). Asymmetry in the flux penetrating the left and right loops results in the gradual change of transition frequencies with every ${\mathrm{\Phi }}_0$ period: ${\omega }_{10}$ creeps up, while ${\omega }_{21}$ creeps down, breaking the usual periodicity seen in flux qubits.

Figure 3

Stability diagrams of the system: (a) The minimal value of Josephson potential is plotted as a function of externally induced phase in the left (${\phi }_l$) and right (${\phi }_r$) loops. The stable flux configurations are denoted as $|00\rangle , |01\rangle , |10\rangle$, and $|11\rangle$. When the loops are symmetrically biased, in the case of a uniform global magnetic field and identical loop areas, the potential takes values along the red arrowed line. An inset shows the energies along the dashed yellow line. The blue curve is the minimized potential and the red curves are parabolic approximations for $\left|01\right.〉$ and $\left|10\right.〉$ flux configurations. Red diamonds mark the ends of the $\left|01\right.〉-\left|10\right.〉$ degeneracy line. (b)–(d) show phases ${\phi }_{02}, {\phi }_{01}+{\phi }_{12}$, and ${\phi }_{03}+{\phi }_{32}$. The abrupt phase change by $\sim 2\pi$ corresponds to a flux quantum jumping through the junctions.

Figure 4

Potential minimum at the $\left({\phi }_l=\pi ,{\phi }_r=\pi \right)$ degeneracy point: The plots show 2D Josephson potential in ${\phi }_{02}-{\phi }_{01}$ coordinates and 1D potentials along the minimum as a function of ${\phi }_{02}$ (light blue line) for: (a) $\alpha =0.8$; (b) $\alpha =1.0$; (c) $\alpha =1.2$. A double well emerges for $\alpha >1.0$ (light blue points).

Figure 5

Energies of ground-to-excited state transitions: (a),(b) Transition energies as functions of external phase biases (${\phi }_l, {\phi }_r,{\phi }_{+}=({\phi }_l+{\phi }_r)/2,{\phi }_{-}=({\phi }_l-{\phi }_r)/2$). (c) Transition energies along lines denoted I, II, III, and IV. Curve I shows the transition energy as a function of the global magnetic flux, which increases with ${\phi }_{+}$. Curves II, III, and IV show the transition energy as a function of the difference in magnetic fluxes, denoting local flux fluctuations. The qubit is much better protected against the global field fluctuations as curve I has a much smaller curvature.

Figure 6

Energies of the system and the corresponding transition matrix elements: (a) As a function of the local magnetic flux deviations with ${\phi }_{+}=\pi$. (b) As a function of the global magnetic flux bias with ${\phi }_{-}=0$.

Figure 7

Rabi oscillations: taken at the degeneracy point by driving the qubit with resonant microwaves pulses for fixed time periods $\mathrm{\Delta }t$. The decoherence time of ${\tau }_{\text{dec}}=42\text{ns}$ is extracted from the decay envelope $e^{-\mathrm{\Delta }t/{\tau }_{\text{dec}}}$ of the oscillations.