Full control of qubit rotations in a voltage-biased superconducting flux qubit

We study a voltage-controlled version of the superconducting flux qubit [Chiorescu et al., Science 299, 1869 (2003)] and show that full control of qubit rotations on the entire Bloch sphere can be achieved. Circuit graph theory is used to study a setup where voltage sources are attached to the two superconducting islands formed between the three Josephson junctions in the flux qubit. Applying a voltage allows qubit rotations about the $y$ axis, in addition to pure $x$ and $z$ rotations obtained in the absence of applied voltages. The orientation and magnitude of the rotation axis on the Bloch sphere can be tuned by the gate voltages, the external magnetic flux, and the ratio $\alpha$ between the Josephson energies of the junctions via a flux-tunable junction. We compare the single-qubit control in the known regime $\alpha <1$ with the unexplored range $\alpha >1$ and estimate the decoherence due to voltage fluctuations.

Figure 1

(Color online) The voltage-biased SC flux qubit (schematic). The circuit consists of a SC ring with three Josephson junctions $J_1$, $J_2$, and $J_3$, threaded by an external magnetic flux ${\Phi }_x$. The Josephson energy of the middle junction $J_3$ differs from the other two by a factor of $\alpha$. A voltage bias $V_i$ is applied to each of the two islands formed by the three junctions via a capacitor $C_i$.

Figure 2

(a) Circuit of a voltage-biased flux qubit (Fig. 1). The main loop contains three Josephson junctions and a (chord) inductance (K). An external magnetic flux ${\Phi }_x$ threads the SC loop. The junctions $J_1$ and $J_2$ are biased by two electrostatic gates, representing the main feature of the circuit. Solid lines represent the tree of the circuit graph, while dotted lines are the chords. (b) Each thick solid line represents a Josephson junction shunted by a capacitance $C_J$.

Figure 3

(Color online) Plot of the potential $\mathcal{U}\left(\mathbf{\phi }\right)$ for ${\phi }_x=\pi$ along the line ${\phi }_1+{\phi }_2=0$ as a function of ${\phi }_{\Vert }=\frac{1}{\sqrt{2}}({\phi }_1-{\phi }_2)$ for several values of $\alpha$. In the curve for $\alpha =0.5$ the two minima are degenerate, while for $\alpha >0.5$ they split showing the double well. The inset is a density plot of the potential for $\alpha =0.8$, showing the two minima and the line ${\phi }_1+{\phi }_2=0$.

Figure 4

(Color online) Density plot of the double well potential $U({\phi }_1,{\phi }_2)$ in units of $E_J$ for $\alpha =0.8$ on a logarithmic scale. The periodicity of the potential is evident; each unit cell contains two minima (black). The primitive vectors of the Bravais lattice are denoted ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ while $t_1$ and $t_2$ are the tunneling matrix elements between the nearest-neighbor minima.

Figure 5

The ratio $t_2∕t_1$ between the tunneling matrix elements, plotted as a function of $\alpha ⩽1$ for several values of $E_J∕E_C$.

Figure 6

The ratio $t_2∕t_1$ between the tunneling matrix elements, plotted as a function of $E_J∕E_C$ for several values of $\alpha ⩽1$.

Figure 7

Plot of the real and imaginary part of $\Delta$ as a function of $Q_1∕2e=C{V}_1∕2e$ for $C{V}_2∕2e=0.5$ for (a) $\alpha =0.95$, $E_J∕E_C=35$; (b) $\alpha =0.95$, $E_J∕E_C=10$; and (c) $\alpha =1$, $E_J∕E_C=15$.

Figure 8

Plot of the real and imaginary part of $\Delta$ as a function of $\delta Q∕2e=C(V_1-V_2)∕2e$ for $V_1+V_2=0$ choosing (a) $\alpha =0.95$, $E_J∕E_C=35$, (b) $\alpha =0.95$, $E_J∕E_C=10$ and (c) $\alpha =1$, $E_J∕E_C=15$.

Figure 9

Plot of the gap versus $\delta Q∕2e=C(V_1-V_2)∕2e$ (solid line) and $C(V_1+V_2)∕2e$ (dashed line), for $\alpha =1$ and $E_J∕E_C=15$. In this case both the amplitude of oscillation and the cross region of the curves are appreciable.

Figure 10

The ratio $t_1∕t_2$ between the tunneling matrix elements, plotted as a function of $\alpha ⩾1$ for several values of $E_J∕E_C$.

Figure 11

The ratio $t_1∕t_2$ between the tunneling matrix elements, plotted as a function of $E_J∕E_C$ for several values of $\alpha ⩾1$.

Figure 12

(Color online) Density plot of the double well potential $U({\phi }_1,{\phi }_2)$ for $\alpha =1.4$, on a logarithmic scale. Two equivalent one-dimensional chains with nearest neighbor interaction are highlighted in the figure.