FerBo: a noise resilient qubit hybridizing Andreev and fluxonium states

Abstract

We propose a novel superconducting quantum circuit that should be robust against both relaxation and dephasing over a wide and experimentally accessible parameter range. The circuit consists of a parallel arrangement of a large inductance, a small capacitor, and a well-transmitting Josephson weak link. Protection against relaxation arises from the hybridization between the fermionic degree of freedom associated with Andreev levels in the weak link and the bosonic electromagnetic mode of the LC circuit, hence its name: FerBo. Furthermore, as in the fluxonium qubit, delocalization of the wavefunctions in phase space provides resilience against dephasing.

Figures (8)

• Figure 1: Principle of protection. (a) Electrical circuit for the light-fluxonium qubit. (b) Potential energy $V(\varphi)$ of a "light" fluxonium ($Z\gg R_Q$) for $\varphi_{\rm ext}=0$, together with the two lowest eigenstates wavefunctions. $V(\varphi)$ is the sum of a parabolic term associated to the inductance (dashed line) and a $\cos(\varphi)$ term associated to the Josephson energy. The delocalization of the wavefunctions in phase strongly reduces the qubit's sensitivity to external flux noise. (c) Idealized FerBo circuit: light fluxonium incorporating an Andreev ballistic weak link. Andreev reflections at the normal-superconducting interfaces give rise to the formation of Andreev bound states ($\ket{-},\ket{+}$). (d) FerBo potential depends on Andreev level occupation. The Josephson cosine term is replaced by $\mp\cos(\varphi/2)$ terms associated with the two Andreev bound states. The FerBo ground-state wavefunction resides in the $\ket{-}$ Andreev manifold, whereas its first excited state resides in the $\ket{+}$ one. Their simultaneous phase delocalization and disjoint support in the Andreev sector suppress both flux-noise-induced dephasing and charge-noise-induced relaxation.
• Figure 2: Transition energy $E_{01}$ between the ground and first excited states of the FerBo circuit normalized by $\Delta_{\mathrm{eff}}$ for $\Gamma/E_C=0.75$; $\epsilon_r/E_C=0.13$, $E_C=15$ GHz and $\delta\Gamma/\Gamma=0.01$) as a function of normalized external magnetic flux shown for different values of the $LC$-mode impedance $Z/R_Q$. The dotted line corresponds to the Andreev transition energy for perfect phase bias. Inset: flux dispersion amplitude $\delta E_{01}=E_{01}(0)-E_{01}(\pi)$ as a function of reduced impedance $Z/R_Q$.
• Figure 3: Relaxation and dephasing susceptibilities. (a) Normalized charge relaxation susceptibility $|\bra{0}\hat{n}\ket{1}|^2$ as a function of the external flux for three values of $\epsilon_r/E_C=0.01(), 0.05 (), 0.25()$ at fixed LC-impedance ($Z/R_Q=5$), $\delta\Gamma/\Gamma=0.01$ and $\Gamma/E_C=0.75$. At $\varphi_{ext}=0$ the susceptibility drops sharply, and the circuit becomes protected against relaxation due to charge and flux noise (see main text). (b) Color map of $|\bra{0}\hat{n}\ket{1}|^2$ at $\varphi_{\rm ext}=0$ as a function of level position $\epsilon_r/E_C$ and of the reduced impedance $Z/R_Q$. Light-colored regions indicate negligible susceptibility, whereas dark regions are unprotected. White dashed line corresponds to $Z/R_Q=2 E_C/\pi \epsilon_r$. (c) Grayscale map of the normalized second-order flux dephasing susceptibility $(1/E_C)|\partial^2 E_{01}/\partial \varphi_{\rm ext}^2|$. The FerBo regime corresponds to the upper-left region of the color plots, where the circuit is simultaneously protected against relaxation and dephasing.
• Figure 4: Symmetry transition underlying flux-noise protection. (a) Color map of the matrix element $\left|\langle \mathrm{0}|\hat{n}|\mathrm{1}\rangle\right|^{2}$as a function of the LC-impedance and the resonant level position $\epsilon_r$ (top) (b) Eigenenergies of the three lowest states of the FerBo ($\Gamma/E_C=0.75$ and $\delta\Gamma/\Gamma=0.01$ ) at a fixed impedance, $Z/R_Q=5$, depicted as a function of $\epsilon_r$. The sharp boundary between "protected" and "unprotected" regions coincides with the avoided crossing between the first and second excited states, highlighted by the red vertical dashed line. (c) and (d): eigenstates wavefunctions below and above the (avoided) crossing, respectively. Solid lines are the real parts; dashed lines (magnified by a factor of 20) are the imaginary parts. Wavefunctions are projected onto the $\ket{-}$ and $\ket{+}$ sectors. To the left of the anticrossing (c) the ground state (blue) is mainly localized in the $\ket{-}$ sector and resembles $\ket{g_-}$, while the first excited state (red), with even symmetry, resides mostly in the $\ket{+}$ sector and resembles $\ket{g_+}$. The second excited state (green), with odd symmetry, is also predominantly located in the $\ket{+}$ sector and resembles $\ket{e_+}$. To the right of the anticrossing (d), the symmetries swap: the first excited state becomes odd like $\ket{e_+}$, and the second excited state becomes even like $\ket{g_+}$. The same qualitative behavior occurs at fixed $\epsilon_r$ varying the LC-mode impedance Z.
• Figure 5: Eigenstates and potential landscapes of the FerBo for an intermediate transmission ($E_C=20$ GHz $E_L=1$ GHz, $\Gamma=5$ GHz, $\delta\Gamma=0$, $\epsilon_r=4.5$ GHz, $\varphi_{\text{ext}}=0$, corresponding to $\tau=0.55$). (a) Full energy spectrum of $\hat{H}'$ showing the probability densities of the wavefunctions of the 3 lowest energy states $\ket{0},$$\ket{1},$$\ket{2},$ projected on the Andreev basis: $\braket{\hat{\varphi},A_- |i}$, in blue, and $\braket{\hat{\varphi},A_+|i}$ in red. The global ground state $\ket{0}$ and first excited state $\ket{1}$ have the same parity (even), protecting them from direct dipole transitions. (b) Potential $V_{A_-}(\varphi)$ for the lower Andreev sector ($\sigma_z=1$) hosting the dominant component of $\ket{0}$. (c) Potential $V_{A_+}(\varphi)$ for the upper Andreev sector ($\sigma_z=-1$) hosting the dominant component of $\ket{1}$.