Protected Josephson Rhombus Chains

We have studied the low-energy excitations in a minimalistic protected Josephson circuit which contains two basic elements (rhombi) characterized by the $\pi$ periodicity of the Josephson energy. Novel design of these elements, which reduces their sensitivity to the offset charge fluctuations, has been employed. We have observed that the lifetime $T_1$ of the first excited state of this quantum circuit in the protected regime is increased up to $70\mu \mathrm{s}$, a factor of $\sim 100$ longer than that in the unprotected state. The quality factor ${\omega }_{01}{T}_1$ of this qubit exceeds $10^6$. Our results are in agreement with theoretical expectations; they demonstrate the feasibility of symmetry protection in the rhombus-based qubits fabricated with existing technology.

Figure 1

Top panel: The chain of two $\cos (2\varphi )$ elements; the charge of the central (common to both rhombi) island is controlled by the gate voltage. Bottom panel: Two lowest-energy wave functions in the discrete harmonic potential shown for $n_g=0$.

Figure 2

Panel (a): The $\cos (2\varphi )$ Josephson element (the Josephson rhombus) and its improved version which is less sensitive to the random offset charges on the upper and lower islands [12]. The chain contains two rhombi; the charge of the central (common to both rhombi) island is controlled by the gate voltage applied between the central conductor of the microstrip line and the ground. Panel (b): The on-chip circuit layout of the device inductively coupled to the microstrip transmission line. Panel (c): The micrograph of the two-rhombus chain coupled to the readout $LC$ resonator via the kinetic inductance $L_C$ of a narrow superconducting wire. The magnetic flux through each rhombus is ${\mathrm{\Phi }}_R$; the flux in the device loop is $\mathrm{\Phi }$.

Figure 3

The resonance frequency $f_{01}$ (blue dots) and energies $E_1$ (black triangles) and $E_2$ (red squares) versus the phase $\phi$ across the chain near full frustration (${\mathrm{\Phi }}_R\approx 0.5{\mathrm{\Phi }}_0$) at $n_g=0$. The theoretical dependences $f_{01}(\phi )$ (solid blue curve) and $E_2(\phi )$ (solid red curve) were computed for $E_2=8.5\mathrm{GHz}$ and $E_C=15\mathrm{GHz}$. The inset: The dependence $f_{01}(n_g)$ at $\phi =0$ (${\mathrm{\Phi }}_R=0.5{\mathrm{\Phi }}_0$). The energies $E_1$ and $E_2$ were extracted from fitting the dependences $f_{01}(n_g)$ measured at different $\phi$ with the calculations based on Hamiltonian diagonalization (solid black curve). The expected dependence $f_{01}(n_g)$ for the case of perfectly symmetric rhombi is shown by the red dashed curve.

Figure 4

The time-domain measurements at $\phi =0$ and ${\mathrm{\Phi }}_R=0.5{\mathrm{\Phi }}_0$. Panel (a): Energy relaxation of the first excited state after application of a $\pi$ pulse: $n_g=0.5$ (blue circles), $n_g=0$ (red squares). Inset: Pulse sequence used in the measurement. Panel (b): Ramsey ($T_2$) and spin echo ($T_{\text{echo}}$) measurements. Inset: Pulse sequence used in the measurements. For the spin echo measurement, a refocusing $\pi$ pulse (red dashed line) was applied at $t=\mathrm{\Delta }t/2$.