Microwave Spectroscopy Reveals the Quantum Geometric Tensor of Topological Josephson Matter

Multiplet Supercurrents in a Josephson Circuit

Multiterminal Josephson circuits have been proposed as a promising platform to host synthetic topological phases of matter, Floquet states, and multiplet supercurrents that are mediated by pairs of Cooper pairs. Here, we explore a Josephson circuit in which three superconducting electrodes are connected through Josephson junctions to a common superconducting island. We demonstrate the dynamic generation of the multiplet supercurrents, which are found to be robust to elevated temperatures and are confirmed by exhibiting the expected Shapiro step quantization under a microwave drive. We also find an unexpected novel supercurrent, which couples a pair of contacts that are both voltage-biased with respect to the common superconducting island. We show that this supercurrent also results from synchronization of the phase dynamics of the device, opening pathways to study the interplay of nonlinear, quantum, and topological physics.

Experimental demonstration of topological bounds in quantum metrology

Quantum metrology is deeply connected to quantum geometry, through the fundamental notion of quantum Fisher information. Inspired by advances in topological matter, it was recently suggested that the Berry curvature and Chern numbers of band structures can dictate strict lower bounds on metrological properties, hence establishing a strong connection between topology and quantum metrology. In this work, we provide a first experimental verification of such topological bounds, by performing optimal quantum multi-parameter estimation and achieving the best possible measurement precision. By emulating the band structure of a Chern insulator, we experimentally determine the metrological potential across a topological phase transition, and demonstrate strong enhancement in the topologically non-trivial regime. Our work opens the door to metrological applications empowered by topology, with potential implications for quantum many-body systems.

Nonreciprocal Josephson Linear Response

We consider the finite-frequency response of multiterminal Josephson junctions and show how nonreciprocity in them can show up at linear response, in contrast to the static Josephson diodes featuring nonlinear nonreciprocity. At finite frequencies, the response contains dynamic contributions to the Josephson admittance, featuring the effects of Andreev bound state transitions along with Berry phase effects, and reflecting the breaking of the same symmetries as in Josephson diodes. We show that outside exact Andreev resonances, the junctions feature nonreciprocal reactive response. As a result, the microwave transmission through those systems is nondissipative, and the electromagnetic scattering can approach complete nonreciprocity. Besides providing information about the nature of the weak link energy levels, the nonreciprocity can be utilized to create nondissipative and small-scale on-chip circulators whose operation requires only rather small magnetic fields.

Phase-engineering the Andreev band structure of a three-terminal Josephson junction

In hybrid Josephson junctions with three or more superconducting terminals coupled to a semiconducting region, Andreev bound states may form unconventional energy band structures, or Andreev matter, which are engineered by controlling superconducting phase differences. Here we report tunnelling spectroscopy measurements of three-terminal Josephson junctions realised in an InAs/Al heterostructure. The three terminals are connected to form two loops, enabling independent control over two phase differences and access to a synthetic Andreev band structure in the two-dimensional phase space. Our results demonstrate a phase-controlled Andreev molecule, originating from two discrete Andreev levels that spatially overlap and hybridise. Signatures of hybridisation are observed in the form of avoided crossings in the spectrum and band structure anisotropies in the phase space, all explained by a numerical model. Future extensions of this work could focus on addressing spin-resolved energy levels, ground state fermion parity transitions and Weyl bands in multiterminal geometries.

Quantum Fluctuation of the Quantum Geometric Tensor and Its Manifestation as Intrinsic Hall Signatures in Time-Reversal Invariant Systems

In time-reversal invariant systems, all charge Hall effects predicted so far are extrinsic effects due to the dependence on the relaxation time. We explore intrinsic Hall signatures by studying the quantum noise spectrum of the Hall current in time-reversal invariant systems, and discover intrinsic thermal Hall noises in both linear and nonlinear regimes. As the band geometric characteristics, quantum geometric tensor and Berry curvature play critical roles in various Hall effects; so do their quantum fluctuations. It is found that the thermal Hall noise in linear order of the electric field is purely intrinsic, and the second-order thermal Hall noise has both intrinsic and extrinsic contributions. In particular, the intrinsic part of the second-order thermal Hall noise is a manifestation of the quantum fluctuation of the quantum geometric tensor, which widely exists as long as Berry curvature is nonzero. These intrinsic thermal Hall noises provide direct measurable means to band geometric information, including Berry curvature related quantities and quantum fluctuation of quantum geometric tensor.

Transport Theory for Topological Josephson Junctions with a Majorana Qubit

We construct a semiclassical theory for the transport of topological Josephson junctions starting from a microscopic Hamiltonian that comprehensively includes the interplay among the Majorana qubit, the Josephson phase, and the dissipation process. With the path integral approach, we derive a set of semiclassical equations of motion that can be used to calculate the time evolution of the Josephson phase and the Majorana qubit. In the equations we reveal rich dynamical phenomena such as the qubit-induced charge pumping, the effective spin-orbit torque, and the Gilbert damping. We demonstrate the influence of these dynamical phenomena on the transport signatures of the junction. We apply the theory to study the Shapiro steps of the junction, and find the suppression of the first Shapiro step due to the dynamical feedback of the Majorana qubit.

Dynamical Stabilization of Multiplet Supercurrents in Multiterminal Josephson Junctions

The dynamical properties of multiterminal Josephson junctions (MT-JJs) have attracted interest, driven by the promise of new insights into synthetic topological phases of matter and Floquet states. This effort has culminated in the discovery of Cooper multiplets in which the splitting of a Cooper pair is enabled via a series of Andreev reflections that entangle four (or more) electrons. Here, we show that multiplet resonances can also emerge as a consequence of the three-terminal circuit model. The supercurrent appears due to correlated phase dynamics at values that correspond to the multiplet condition nV₁ = -mV₂ of applied bias. Multiplet resonances are seen in nanofabricated three-terminal graphene JJs, analog three-terminal JJ circuits, and circuit simulations. The stabilization of the supercurrent is purely dynamical, and a close analog to Kapitza's inverted pendulum problem. We describe parameter considerations that optimize the detection of the multiplet lines both for design of future devices.

Multiterminal Inverse AC Josephson Effect

When a Josephson junction is exposed to microwave radiation, it undergoes the inverse AC Josephson effect─the phase of the junction locks to the drive frequency. As a result, the I-V curves of the junction acquire "Shapiro steps" of quantized voltage. If the junction has three or more superconducting contacts, coupling between different pairs of terminals must be taken into account and the state of the junction evolves in a phase space of higher dimensionality. Here, we study the multiterminal inverse AC Josephson effect in a graphene sample with three superconducting terminals. We observe robust fractional Shapiro steps and correlated switching events, which can only be explained by considering the device as a completely connected Josephson network. We successfully simulate the observed behaviors using a modified two-dimensional RCSJ model. Our results suggest that multiterminal Josephson junctions are a playground to study highly connected nonlinear networks with novel topologies.

Zero Crossing Steps and Anomalous Shapiro Maps in Graphene Josephson Junctions

The AC Josephson effect manifests itself in the form of "Shapiro steps" of quantized voltage in Josephson junctions subject to radiofrequency (RF) radiation. This effect presents an early example of a driven-dissipative quantum phenomenon and is presently utilized in primary voltage standards. Shapiro steps have also become one of the standard tools to probe junctions made in a variety of novel materials. Here we study Shapiro steps in a widely tunable graphene-based Josephson junction in which the high-frequency dynamics is determined by the on-chip environment. We investigate the variety of patterns that can be obtained in this well-understood system depending on the carrier density, temperature, RF frequency, and magnetic field. Although the patterns of Shapiro steps can change drastically when just one parameter is varied, the overall trends can be understood and the behaviors straightforwardly simulated, showing some key differences from the conventional RCSJ model. The resulting understanding may help interpret similar measurements in more complex materials.