The protection of superconducting qubits from certain noise sources often comes at the cost of increased sensitivity to other decoherence channels. Here, we explore a route to avoid this tradeoff by encoding quantum information in quantum states of a transmon entangled with the spin of a trapped Andreev quasiparticle. We term such devices spinmons. We lift the spinmon Kramers degeneracy by introducing a Zeeman field and develop two routes for full qubit control via electrostatic gates and an AC flux drive, providing multiple directions for experimental implementations. Finally, we compute coherence times and verify the qubit robustness against flux and charge noise sources.

The superconductor-insulator-superconductor Josephson junction is the fundamental nonlinear element of superconducting circuits. Connecting two junctions in series gives rise to higher-harmonic content in the total energy-phase relation, enabling new design opportunities in multimode circuits. However, the double-junction element hosts an internal mode whose spectrum is set by the finite capacitances of the individual junctions. Using a Born-Oppenheimer approximation that treats the additional mode as fast compared to the qubit mode, we analyze the double-junction circuit element shunted by a large capacitor. Here, we derive an effective single-mode model of the qubit containing a correction term owing to the presence of the internal mode. We explore experimentally relevant parameter regimes and find that our model accurately describes the low-energy spectrum of the qubit. We further discuss how eliminating the internal degree of freedom affects the system's periodic boundary conditions and how this leads to non-uniqueness in performing the Born-Oppenheimer approximation. Finally, we analyze the harmonic content of the double-junction element and discuss its sensitivity to charge noise.

Andreev bound states are coherent electron-hole superpositions that form in a normal metal through repeated Andreev reflection at a superconducting interface. When the length of a superconducting segment is comparable to the coherence length, the bound states on opposite sides of the segment hybridize through quasiparticle tunneling. In a periodic array, these hybridized Andreev bound states form energy bands below the superconducting gap. We develop a theoretical framework for transport in such Andreev crystals. We demonstrate that, at high interface transparency, a constant phase bias between neighboring superconductors renders the bands directional: one band contains only right-moving and the other only left-moving electronic states. This property leads to a directional conductance that enables the device to operate as a flux- and bias-voltage-tunable filter that allows signal transmission in only one direction.

Demonstration of a nontrivial result of quasiparticle exchange (or braiding) is usually considered the definitive proof of a topological phase with non-Abelian excitations, such as Majorana bound states (MBSs). However, in finite systems with disorder and smooth potential variations, the MBSs are imperfect in the sense that they are not fully isolated in space and can, to a varying degree, resemble conventional fermions. The main goal of this work is to reveal the braiding properties of isolated MBSs, regular fermions, and anything in between. We start from an experimentally attractive braiding protocol and find a way to compensate for the undesired splitting of the ground-state degeneracy which occurs during the protocol for imperfect MBSs. This leads to a braiding outcome that depends on the degree of MBS isolation but remains robust and non-Abelian except in the perfect fermion limit. Our protocol could be implemented in different platforms with non-Abelian excitations, including quantum-dot-based minimal Kitaev chains.

Superconducting quantum circuits derive their nonlinearity from the Josephson energy-phase relation. Besides the fundamental cos ϕ term, this relation can also contain higher Fourier harmonics cos ( k ϕ ) corresponding to correlated tunneling of k Cooper pairs. The parity of the dominant tunneling process, i.e. whether an odd or even number of Cooper pairs tunnel, results in qualitatively different properties, and controlling this opens up a wide range of applications in superconducting technology. However, access to even-dominated regimes has remained challenging and has so far relied on complex multi-junction or all-hybrid architectures. Here, we demonstrate a simple "harmonic parity qubit"; an element that combines two aluminum-oxide tunnel junctions in parallel to a gate-tunable InAs/Al nanowire junction forming a SQUID, and use spectroscopy versus flux to reconstruct its energy-phase relation at 85 gate voltage points. At half flux quantum, the odd harmonics of the Josephson potential can be suppressed by up to two orders of magnitude relative to the even harmonics, producing a double-well potential dominated by even harmonics with minima near  ± π/2. The ability to control harmonic parity enables supercurrent carried by pairs of Cooper pairs and provides a new building block for Fourier engineering in superconducting circuits.

When studying non-Hermitian electronic systems, an obvious question is how various non-Hermitian effects affect measurable quantities like the conductance. Here, we show that uniformly dissipative circuits can exhibit nonreciprocal conductance, meaning that the two nonlocal conductances are different. We describe how this happens through a difference in transmission times between left-moving and right-moving electrons. We consider a specific case of a dissipative Rashba nanowire with a skewed magnetic field, and show how this difference in transmission times comes about through interference inside the circuit, and how this is modified as the dissipation strength changes.

Open quantum systems governed by non-Hermitian effective Hamiltonians exhibit unique phenomena, such as the non-Hermitian skin effect, where eigenstates localize at system boundaries. We investigate this effect in a Rashba nanowire coupled to a ferromagnetic lead and demonstrate that it can be detected via nonlocal transport spectroscopy: while local conductance remains symmetric, the nonlocal conductance becomes nonreciprocal. We account for this behavior using both conventional transport arguments and the framework of non-Hermitian physics. Furthermore, we explain that exceptional points shift in parameter space when transitioning from periodic to open boundary conditions, a phenomenon observed in other non-Hermitian systems but so far not explained. Our results establish transport spectroscopy as a tool to probe non-Hermitian effects in open electronic systems.

Tunable Josephson harmonics open new avenues for qubit design. We demonstrate a superconducting circuit element consisting of a tunnel junction in series with a superconducting quantum interference device (SQUID) loop, yielding a Josephson potential whose harmonic content is strongly tunable by magnetic flux. Through spectroscopy of the first four qubit transitions, together with an effective single-mode model renormalized by the internal mode, we resolve a second harmonic with an amplitude up to ∼ 10 % of the fundamental. We identify a flux sweet spot where the ground-state dispersive shift vanishes, achieved by balancing the dispersive couplings to the internal and qubit modes. This highly tunable element provides a route toward protected qubits and customizable nonlinear microwave devices.

In optical diffraction, the phase difference between sources in a grating or multi-slit mask is determined by the angle to the imaging screen, yielding the familiar multi-lobed diffraction image. Here, we realize a similar phenomenon in a superconductor-semiconductor hybrid circuit configured to allow Andreev scattering from multiple parallel scatterers. Phase differences between scatterers are set by tapping off of a remote superconducting meander. We investigate arrays with two, three, four, and ten Andreev scatterers, examining local and nonlocal diffraction patterns, finding good agreement with a theory of multiple Andreev scattering, not to be confused with multiple Andreev reflection. Adding current-carrying taps to the meander allows individual phase control.

Technology involving hybrid superconductor-semiconductor materials is a promising avenue for engineering quantum devices for information storage, manipulation, and transmission. Proximity-induced superconducting correlations are an essential part of such devices. While the proximity effect in the conduction band of common semiconductors is well understood, its manifestation in confined hole gases, realized for instance in germanium, is an active area of research. Lower-dimensional hole-based systems, particularly in germanium, are emerging as an attractive platform for a variety of solid-state quantum devices, due to their combination of efficient spin and charge control and long coherence times. The recent experimental realization of the proximity effect in germanium thus calls for a theoretical description that is tailored to hole gases. In this work, we propose a simple model to describe proximity-induced superconductivity in two-dimensional hole gases, incorporating both the heavy-hole (HH) and light-hole (LH) bands. We start from the Luttinger-Kohn model, introduce three parameters that characterize hopping across the superconductor-semiconductor interface, and derive explicit intraband and interband effective pairing terms for the HH and LH bands. Unlike previous approaches, our theory provides a quantitative relationship between induced pairings and interface properties. Restricting our general model to an experimentally relevant case where only the HH band crosses the chemical potential, we predict the coexistence of s-wave and d-wave singlet pairings, along with triplet-type pairings, and modified Zeeman and Rashba spin-orbit couplings. Our results thus present a starting point for theoretical modeling of quantum devices based on proximitized hole gases, fueling further progress in quantum technology.

We propose Josephson junction arrays as realistic platforms for observing nonequilibrium scaling laws characterizing the Kardar-Parisi-Zhang (KPZ) universality class, and space-time soliton proliferation. Focusing on a two-chain ladder geometry, we perform numerical simulations for the roughness function. Together with analytical arguments, our results predict KPZ scaling at intermediate time scales, extending over sufficiently long time scales to be observable, followed by a crossover to the asymptotic long-time regime governed by soliton proliferation.

Tunable Josephson harmonics open up for new qubit design. We demonstrate a superconducting circuit element with a tunnel junction in series with a SQUID loop, yielding a highly magnetic-flux tunable harmonic content of the Josephson potential. We analyze spectroscopy of the first four qubit transitions with a circuit model which includes the internal mode, revealing a second harmonic up to ∼10% of the fundamental harmonic. Interestingly, a sweet spot where the dispersive shift vanishes is achieved by balancing the dispersive couplings to the internal and qubit modes. The highly tunable set-up provides a route toward protected qubits, and customizable nonlinear microwave devices.

We propose a nanodevice based on multi-terminal Josephson junctions for the creation and detection of poor man's Majorana (PMM) modes. The device consists of a double three-terminal Josephson junction (3TJJ) embedded in a planar semiconductor that engineers a two-site Kitaev chain. We identify the conditions necessary for the emergence of PMM modes by precisely tuning the superconducting phases of the terminals and the inter-junction coupling between the 3TJJs. To validate our findings, we perform numerical simulations that account for disorder in a more general geometry, and test experimentally feasible protocols based on conductance measurements to reliably detect these modes. Unlike traditional approaches that rely on magnetic fields, our design eliminates the need for such fields, potentially increasing the energy-level resolution and significantly expanding the range of compatible semiconductor materials, such as Ge-based heterostructures.

Demonstration of a nontrivial result of quasiparticle exchange (or braiding) is usually considered the definitive proof of a topological phase with nonabelian excitations, such as Majorana bound states (MBSs). However, in finite systems with disorder and smooth potential variations, the MBSs are imperfect in the sense that they are not fully isolated in space and can, to a varying degree, resemble conventional fermions. Here, we study the braiding properties of isolated MBSs, regular fermions, and anything in between. We find a way to compensate for the undesired splitting of the ground-state degeneracy which occurs during the protocol for imperfect MBS. This leads to a braiding outcome that depends on the degree of MBS isolation but remains robust and nonabelian except in the perfect fermion limit. Our protocol could be implemented in different platforms with nonabelian excitations, including quantum-dot-based minimal Kitaev chains.

We study a non-Hermitian, multiterminal superconducting-normal system in order to identify experimental signatures of exceptional points. We focus on a minimal setting with a single spinful level, spin-dependent normal leads, and a noncollinear magnetic field. This system hosts both topologically protected and symmetry-protected exceptional points. Using an exact transport formalism, we show that the exceptional points leave signatures visible through spectroscopy of the Andreev states but that they have a minor effect on the Josephson current. We also argue that these findings hold with interactions.

If a quantum dot is coupled to a topological superconductor via tunneling contacts, each contact hosts a Majorana zero mode in the limit of zero transmission. Close to a resonance and at a finite contact transparency, the resonant level in the quantum dot couples the Majorana modes, but a ground state degeneracy per fermion parity subspace remains if the number of Majorana modes coupled to the dot is five or larger. Upon varying shape-defining gate voltages while remaining close to resonance, a nontrivial evolution within the degenerate ground-state manifold is achieved. We characterize the corresponding non-Abelian holonomy for a quantum dot with chaotic classical dynamics using random matrix theory and discuss measurable signatures of the non-Abelian time-evolution.

The possibility to engineer artificial Kitaev chains in arrays of quantum dots coupled via narrow superconducting regions has emerged as an attractive way to overcome the disorder issues that complicate the realization and detection of topological superconducting phases in other platforms. Although a true topological phase would require long chains, already a two-site chain realized in a double quantum dot can be tuned to points in parameter space where it hosts zero-energy states that seem identical to the Majorana bound states that characterize a topological phase. These states were named "poor man's Majorana bound states" (PMMs) because they lack formal topological protection. In this work, we propose a roadmap for next-generation experiments on PMMs. The roadmap starts with experiments to characterize a single pair of PMMs by measuring the Majorana quality, then moves on to initialization and readout of the parity of a PMM pair, which allows measuring quasiparticle poisoning times. The next step is to couple two PMM systems to form a qubit. We discuss measurements of the coherence time of such a qubit, as well as a test of Majorana fusion rules in the same setup. Finally, we propose and analyse three different types of braiding-like experiments which require more complex device geometries. Our conclusions are supported by calculations based on a realistic model with interacting and spinful quantum dots, as well as by simpler models to gain physical insight. Our calculations show that it is indeed possible to demonstrate nonabelian physics in minimal two-site Kitaev chains despite the lack of a true topological phase. But our findings also reveal that doing so requires some extra care, appropriately modified protocols and awareness of the details of this particular platform.

Fast, high fidelity control and readout of protected superconducting qubits are fundamentally challenging due to their inherent insensitivity. We propose a flux qubit variation which enjoys a tunable level of protection against relaxation to resolve this outstanding issue. Our qubit design, the double-shunted flux qubit (DSFQ), realizes a generic double-well potential through its three junction ring geometry. One of the junctions is tunable, making it possible to control the barrier height and thus the level of protection. We analyze single- and two-qubit gate operations that rely on lowering the barrier. We show that this is a viable method that results in high fidelity gates as the non-computational states are not occupied during operations. Further, we show how the effective coupling to a readout resonator can be controlled by adjusting the externally applied flux while the DSFQ is protected from decaying into the readout resonator. Finally, we also study a double-loop gradiometric version of the DSFQ which is exponentially insensitive to variations in the global magnetic field, even when the loop areas are non-identical.

Bound states in quantum dots coupled to superconductors can be in a coherent superposition of states with different electron number but with the same fermion parity. Electrostatic gating can tune this superposition to a sweet spot, where the quantum dot has the same mean electric charge independent of its electron-number parity. Here, we propose to encode quantum information in the local fermion parity of two tunnel-coupled quantum dots embedded in a Josephson junction. At the sweet spot, the qubit states have zero charge dipole moment. This protects the qubit from dephasing due to charge noise acting on the potential of each dot, as well as fluctuations of the (weak) interdot tunneling. At weak interdot tunneling, relaxation is suppressed because of disjoint qubit states. However, for strong interdot tunneling the system is protected against noise affecting each quantum dot separately (energy-level noise, dot-superconductor tunneling fluctuations, and hyperfine interactions). Finally, we describe initialization and readout as well as single-qubit and two-qubit gates by pulsing gate voltages. Published by the American Physical Society 2024

In analogy to conventional semiconductor diodes, the Josephson diode exhibits superconducting properties that are asymmetric in applied bias. The effect has been investigated in number of systems recently, and requires a combination of broken time-reversal and inversion symmetries. We demonstrate a dual of the usual Josephson diode effect, a nonreciprocal response of Andreev bound states to a superconducting phase difference across the normal region of a superconductor-normal-superconductor Josephson junction, fabricated using an epitaxial InAs/Al heterostructure. Phase asymmetry of the subgap Andreev spectrum is absent in the absence of in-plane magnetic field and reaches a maximum at 0.15 T applied in the plane of the junction transverse to the current direction. We interpret the phase diode effect in this system as resulting from finite-momentum Cooper pairing due to orbital coupling to the in-plane magnetic field, without invoking Zeeman or spin-orbit coupling.