Loopless multiterminal quantum circuits at odd parity

TL;DR

The work addresses loopless multiterminal superconducting devices at odd parity with time-reversal symmetry, introducing a minimal three-terminal dot model that yields a double-well SPER $U = U_0 + U_{SO}$, where $U_0 = \sum_{ij} E_0^{ij} \cos \varphi_{ij}$ and SOC introduces a full $SU(2)$ spin structure via $U_{SO} = \vec{V}_{SO} \cdot \vec{\sigma}$. By capacitively shunting the circuit, phase degrees of freedom become quantum, enabling numerical diagonalization of the full $H = T + U$ and analysis of both spinless and spinful cases. In the spinless case, the lowest-state splitting is exponentially suppressed as $E_C/E_0 \to 0$, with wavefunctions localized in two phase wells; in the spinful case, SOC splits the low-energy subspace into four spin–chirality states, and circular, electrically driven protocols can realize universal SU(4) control over this four-level subspace. The approach yields flux-noise–resilient, electrically tunable qubits and suggests pathways to large-scale multiterminal arrays for quantum simulation and improved error protection, all without relying on magnetic flux bias.

Abstract

We theoretically investigate loopless multiterminal hybrid superconducting devices at odd fermion parity with time-reversal symmetry. We find that the energy-phase relationship has a double minimum corresponding to opposite windings of the superconducting phases. Spin-orbit coupling adds multi-axial spin splittings, which contrasts with two-terminal devices where spin dependence is uniaxial. Capacitive shunting localizes quantum circuit states in the wells and exponentially suppresses their splitting. For weak spin-orbit strength, the system has a four-dimensional spin-chirality low-energy subspace which can be universally controlled with electric fields only.

Figures (4)

• Figure 1: Schematics for proximitized single-level dots with two and three terminals in the odd parity sector and its analogous circuits. (a) Two-terminal devices without SOC are typically $\pi$-junctions. (b) Three-terminal devices without SOC are analogous to a triangle of $\pi$-junctions, which hosts a double-well, spin-independent energy-phase relation. (c) Three-terminal devices with SOC include spin-dependent contributions spanning the full SU(2) structure of the spin to the spin-phase energy relation, alongside the spinless triangle of $\pi$-junctions.
• Figure 2: (a) Schematic diagram of the microscopic toy model for obtaining the Hamiltonian of the trijunction at odd parity. (b) Schematic diagram of the structure of the dot levels and the tunneling terms that are included in the model. (c) Spin-phase energy relation, $U = U_0 + U_{SO}$, for $t=0.1\Delta$, $\theta=\pi / 30$, and $\mu_1=5\Delta$. We set $\varphi_3 =0$ and so only refer to the phases of leads 1 and 2 on the axis. (d-f) Spin-dependent part of the spin-phase energy relation, $U_{SO}$, for the terms proportional to (d) $\sigma_x$, (e) $\sigma_y$, and (f) $\sigma_z$. The black and white dots in panels (c-f) mark the location of the minimum of $U_0$.
• Figure 3: (a) Circuit diagram for a capacitively-shunted Andreev trijunction. (b) The circuit spectrum of the spinless case as a function of the circuit charging energy $E_C$ for $t=0.75\Delta$. The energy splitting between the lowest pair of states is exponentially suppressed as $E_C/E_0 \rightarrow 0$. (c) Probability density of the circuit wave functions in phase space for $E_C/E_0=5\times 10^{-4}$--indicated by the blue and dot in panels (b) and (d). (d) The spectrum (black lines) and matrix elements due to charge drive at the capacitors (dashed lines) as a function of tuning one tunneling rate $t_1/t$, for $E_C/E_0 = 5\times 10^{-4}$--indicated by the blue dot in panel (b). (e) Probability density of the circuit wave functions in phase space for $t_1 / t = 1.3$ and $E_C/E_0=5\times 10^{-4}$--indicated by the orange and dot in panel (d).
• Figure 4: (a) The lowest eight energy levels for the heavy spinful circuit with weak SOC located in their respective energy wells, for $t=0.1\Delta$, $\theta=\pi/30$, $\mu_1=5\Delta$, $E_C/E_0 = 10^{-3}$. (b) Spectrum as a function of $E_C/E_0$. Unlike the spinless case, the lowest available transition frequency (from the ground state doublet to the first excited doublet) saturates for $E_C < 2V_{SO}\cdot \hat{z}\vert_{\min U}$. (c) Chirality-preserving, spin-flipping matrix elements as a function of $t_1/t$ plotted alongside the spectrum. Both drive orientations have a strong matrix element when the states are localized within the wells. These two drive orientations correspond to different Pauli matrices. (d) Spin-preserving, chirality-flipping matrix elements, which find that the adiabatic coupling (red) and one off-diagonal matrix element (yellow) are nonzero only with appreciable detunings of $t_1/t$. We used $E_C/E_0 = 10^{-3}$ in panels (c) and (d).