Superconducting circuits offer the opportunity to study quantum mechanics on mesoscopic scales unimpeded by dissipation. This fact and the nonlinearity of the Josephson inductance make it possible to use superconducting circuits as artificial atoms whose long-lived states can be selectively addressed and studied. A pronounced nonlinearity of the energy spectrum, however, requires quantum fluctuations of the flux across the Josephson junction which are large on the scale of the superconducting flux quantum $\Phi_Q = h/2e$. This implies charge fluctuations below the single Cooper-pair limit via flux-charge duality. The localization of charge leads to a strong susceptibility to interactions with charges in the environment which has motivated the search for schemes to decouple charges from their environment. This thesis is concerned with theoretical challenges arising from two complementary approaches to this problem: the realization of large impedances and the fractionalization of electrons by means of Majorana bound states.In recent years, the decoupling of charges from the environment through reactive large impedances, so-called ``superinductances'' $L$, has attracted much interest. These inductances feature small parasitic capacitance $C$ such that the characteristic impedance $\sqrt{L/C}$ is much larger than the superconducting resistance quantum $R_Q = h/4e^2$. Superinductances have various applications ranging from qubit designs such as the $0$-$\pi$ qubit or the fluxonium to impedance matching, Bloch oscillations and the stabilization of phase slips in superconducting nanowires. Although there exists a well-established formalism for the quantization of superconducting circuits in terms of node fluxes, this formalism is ill-suited for the description of fast flux transport with localized charges in large-impedance environments. In particular, the nonlinear capacitive behavior of phase slip junctions cannot be modeled in a straightforward way using node fluxes. In view of the ever growing interest in superinductances, in the first part of the thesis, we present a recipe for quantization of planar circuits in terms of loop charges. As we will show, the loop charge approach is dual to the usual node flux formalism and well-adapted to a large impedance setting.In the second part of the thesis, we turn to a complementary approach of charge decoupling by means of Majorana bound states (MBS). MBS solve the decoupling problem by encoding a fermionic mode nonlocally into two bound states with large spatial separation such that a local coupling to the stored charge is no longer possible. It has been shown that despite the apparent nonlocality of the fermionic mode, transport through the mode remains local unless the MBS are coupled by a global perturbation like a finite charging energy. Here we show that, even in absence of charging energy, decoupling the superconductor from the ground plane achieves subtle coupling of the MBS that leads to nonlocal transport.Finally, in the last part of the thesis, we turn to two mesoscopic applications related to supersymmetric quantum mechanics. The simultaneous presence of a fermionic mode due to the MBS and a bosonic mode due to the Cooper-pair condensate makes systems involving MBS appealing candidates for the realization of supersymmetric quantum mechanics. For a Majorana Cooper-pair box, we discuss an unusual ``bosonic'' supersymmetry and its experimental signatures. Since MBS remain challenging to realize experimentally, we show that a similar supersymmetry can even be realized in a setup using standard superconducting circuitry without MBS.
Superconducting circuits offer the opportunity to study quantum mechanics on mesoscopic scales unimpeded by dissipation. This fact and the nonlinearity of the Josephson inductance make it possible to use superconducting circuits as artificial atoms whose long-lived states can be selectively addressed and studied. A pronounced nonlinearity of the energy spectrum, however, requires quantum fluctuations of the flux across the Josephson junction which are large on the scale of the superconducting flux quantum ΦQ=h/2e. This implies charge fluctuations below the single Cooper-pair limit via flux-charge duality. The localization of charge leads to a strong susceptibility to interactions with charges in the environment which has motivated the search for schemes to decouple charges from their environment. This thesis is concerned with theoretical challenges arising from two complementary approaches to this problem: the realization of large impedances and the fractionalization of electrons by means of Majorana bound states.In recent years, the decoupling of charges from the environment through reactive large impedances, so-called superinductances'' $L$, has attracted much interest. These inductances feature small parasitic capacitance $C$ such that the characteristic impedance $\sqrt{L/C}$ is much larger than the superconducting resistance quantum $R_Q = h/4e^2$. Superinductances have various applications ranging from qubit designs such as the $0$-$\pi$ qubit or the fluxonium to impedance matching, Bloch oscillations and the stabilization of phase slips in superconducting nanowires. Although there exists a well-established formalism for the quantization of superconducting circuits in terms of node fluxes, this formalism is ill-suited for the description of fast flux transport with localized charges in large-impedance environments. In particular, the nonlinear capacitive behavior of phase slip junctions cannot be modeled in a straightforward way using node fluxes. In view of the ever growing interest in superinductances, in the first part of the thesis, we present a recipe for quantization of planar circuits in terms of loop charges. As we will show, the loop charge approach is dual to the usual node flux formalism and well-adapted to a large impedance setting.In the second part of the thesis, we turn to a complementary approach of charge decoupling by means of Majorana bound states (MBS). MBS solve the decoupling problem by encoding a fermionic mode nonlocally into two bound states with large spatial separation such that a local coupling to the stored charge is no longer possible. It has been shown that despite the apparent nonlocality of the fermionic mode, transport through the mode remains local unless the MBS are coupled by a global perturbation like a finite charging energy. Here we show that, even in absence of charging energy, decoupling the superconductor from the ground plane achieves subtle coupling of the MBS that leads to nonlocal transport.Finally, in the last part of the thesis, we turn to two mesoscopic applications related to supersymmetric quantum mechanics. The simultaneous presence of a fermionic mode due to the MBS and a bosonic mode due to the Cooper-pair condensate makes systems involving MBS appealing candidates for the realization of supersymmetric quantum mechanics. For a Majorana Cooper-pair box, we discuss an unusual bosonic'' supersymmetry and its experimental signatures. Since MBS remain challenging to realize experimentally, we show that a similar supersymmetry can even be realized in a setup using standard superconducting circuitry without MBS.
This thesis describes a series of theoretical proposals of novel circuits that embed ultrathin superconducting nanowires with coherent quantum phase-slips (QPS). The motivation for our proposals is twofold: firstly, to facilitate unambiguous experimental verification of coherent phase-slips. Secondly, to suggest superconducting devices with new features and functionalities. The first circuit is a driven LC oscillator with the superconducting inductance subject to QPS. We have studied the effect of QPS in two limits: for small and for large phase-slip amplitudes. We expect that weakQPS (whereweakmeans that the QPS amplitude is small compared to the resonance frequency) have ameasurable effect when the oscillator is resonantly driven. The charge induced by the gate affects the quantum interference of QPS with opposite shifts and thus the effect of QPS is periodic in gate voltage. The experimental observation of such dependence would unambiguously identify the quantum coherence of phase-slips. The other crucial effect of QPS is that they produce uncommon non-linearities in the proposed device. The distinctive feature of these non-linearities is the oscillatory dependence on number of photons n with a local period of the order of pn. We analyzed the regime where the QPS correction becomes large by employing a semiclassical approximation as well as by solving the full quantum mechanical equations. At semiclassical level, the QPS-induced non-linearities persist until very high photon number and they result in amultitude ofmetastable states. This was illustrated in the responses of the oscillator: the QPS-induced non-linearities cause deviations fromthe otherwise Lorentzian curve that represents the number of photons N versus detuning. At sufficiently large phase-slip amplitudes this results in an impressive characteristic, a "corkscrew" shape. At any given detuning one finds a multitude of states that differ in the number of photons. About half of these states are stable. At quantum level, we have demonstrated that there is indeed a single quantum state corresponding to the semiclassical metastable states. We have found that these states are robust, their switching time is exponentially long, although they encompass only a few photons. The QPS oscillator is highly tunable: small changes of the driving force, detuning, or charge induced change the number of stable states, thereby enabling easy manipulation of the number of photons. All these features of the phase-slip oscillator make it useful for a wide range of applications, such as ultra-sensitive measurements, quantum manipula- tion and naturally, an unambiguous experimental verification of coherent QPS. In Chapter 3 we propose a series of devices that illustrate the emergence of Coulomb blockade from coherent QPS in thin superconducting wires. The basic ideas is that QPS create an isolation in a wire which is similar to the isolation required by Coulomb blockade. We exploit this idea and suggest setups that are derived from Cooper-pair box and Cooper-pair transistor, so we refer to them as QPS-box and QPS-transistor, respectively. Our main goal was to demonstrate that the devices exhibit sensitivity to a charge induced by a gate electrode, this being the main signature of Coulomb blockade. We analyze the emergence of discrete charging in the limit of strong phaseslips. We investigated six distinct regimes that are realized depending on the relation between the three characteristic energy scales: inductive and charging energy, and QPS amplitude. In both cases the charge sensitivity appears already for small QPS amplitudes as a perturbative correction to the ground-state energy. In both cases, the charge-sensitive part of the perturbative correction is exponentially suppressed in the limit of low impedance. In contrast to the QPS-box, the QPStransistor exhibits both flux and charge sensitivity that makes it potentially useful for measurements. However, if the QPS amplitude becomes of the order of either charging energy (large impedance regime) or inductive energy (small impedance regime), both devices showdiscrete charge states that followthe commonCoulomb blockade pattern of "crossing parabolas". The crossover to Coulomb blockade occurs differently in the limits of large and small impedance. For QPS-transistor, we have analyzed the flux sensitivity (superconducting current) as well as combined flux-charge sensitivity. For a symmetric QPS-transistor we have found a variable separation at specific values of induced charge that leads to double degeneracy of the states at q/e = 1 and half-integer external flux. We have calculated the superconducting current through QPS-transistor to show the non-triviality of the device. Experimental realization of these devices (achievable with the state-or-theart technology), will unambiguously prove the Coulomb blockade as an effect of coherent phase-slip processes. For completeness, we discuss the Josephson-based devices that are dual to QPS-box and QPS-transistor. In Chapter 4 we propose a way to realize quantum synchronization of Josephson and Bloch oscillations in a superconducting device. Essentially, this implies the synchronization of quantumconjugated variables: phase and charge. A circuit comprising of a QPS junction, series resistor and biased by a d.c. voltage exhibits Bloch physics of the charge variable. To ensure well-defined classical oscillations, the resistor needs to be large compared to the resistance quantum. We couple this Bloch subcircuit with a Josephson subcircuit in which the phase variable performs thewell-known Josephson oscillations. Themain effect of the coupling is the transfer of oscillating voltage/current from Josephson/Bloch to Bloch/Josephson part, whereby the voltage/current is multiplied with an amplification coefficient. This amplification coefficient is required to be high for stable synchronization. This is achieved by using an LC-resonator with high quality factorQ. The synchronization takes place in a rather broad interval of frequencies near the LC oscillator’s resonant value. Owing to this synchronization, the transresistance of our device exhibits a typical devil’s staircase curve, very similar to (fractional) Quantum Hall samples. In principle, fluctuations of phase and charge could could destroy the synchronization. Using the quantum description in the framework of Keldysh action formalism, we have investigated the effect of fluctuations on our circuit. The minimum synchronization error rate is shown to be exponential in Q. There is also significant practical interest in the field on metrology. The suggested device can be used to close the famous "metrological triangle" for threemetrological standards: those of resistance, voltage and current—within a single device.
Abstract The non-dissipative nonlinearity of Josephson junctions1 converts macroscopic superconducting circuits into artificial atoms2, enabling some of the best-controlled qubits today3, 4. Three fundamental types of superconducting qubit are known5, each reflecting a distinct behaviour of quantum fluctuations in a Cooper pair condensate: single-charge tunnelling (charge qubit6, 7), single-flux tunnelling (flux qubit8) and phase oscillations (phase qubit9 or transmon10). Yet, the dual nature of charge and flux suggests that circuit atoms must come in pairs. Here we introduce the missing superconducting qubit, ‘blochnium’, which exploits a coherent insulating response of a single Josephson junction that emerges from the extension of phase fluctuations beyond 2π (refs. 11–14). Evidence for such an effect has been found in out-of-equilibrium direct-current transport through junctions connected to high-impedance leads15–19, although a full consensus on the existence of extended phase fluctuations is so far absent20–22. We shunt a weak junction with an extremely high inductance—the key technological innovation in our experiment—and measure the radiofrequency excitation spectrum as a function of external magnetic flux through the resulting loop. The insulating character of the junction is manifested by the vanishing flux sensitivity of the qubit transition between the ground state and the first excited state, which recovers rapidly for transitions to higher-energy states. The spectrum agrees with a duality mapping of blochnium onto a transmon, which replaces the external flux by the offset charge and introduces a new collective quasicharge variable instead of the superconducting phase23, 24. Our findings may motivate the exploration of macroscopic quantum dynamics in ultrahigh-impedance circuits, with potential applications in quantum computing and metrology.
The conventional approach to circuit quantization is based on node fluxes and traces the motion of node charges on the islands of the circuit. However, for some devices, the relevant physics can be best described by the motion of polarization charges over the branches of the circuit that are in general related to the node charges in a highly nonlocal way. Here, we present a method, dual to the conventional approach, for quantizing planar circuits in terms of loop charges. In this way, the polarization charges are directly obtained as the differences of the two loop charges on the neighboring loops. The loop charges trace the motion of fluxes through the circuit loops. We show that loop charges yield a simple description of the flux transport across phase-slip junctions. We outline a concrete construction of circuits based on phase-slip junctions that are electromagnetically dual to arbitrary planar Josephson junction circuits. We argue that loop charges also yield a simple description of the flux transport in conventional Josephson junctions shunted by large impedances. We show that a mixed circuit description in terms of node fluxes and loop charges yields an insight into the flux decompactification of a Josephson junction shunted by an inductor. As an application, we show that the fluxonium qubit is well approximated as a phase-slip junction for the experimentally relevant parameters. Moreover, we argue that the $0\ensuremath{-}\ensuremath{\pi}$ qubit is effectively the dual of a Majorana Josephson junction.
Quantum mechanics reveals a world that defies classical determinism, where uncertainty, superposition, and fluctuations are fundamental aspects. Engineering devices that harness these quantum features requires not only precision, but also a deep understanding of how they interact with their surrounding environment. Superconducting circuits, which exploit macroscopic quantum coherence in low-loss superconducting materials, provide a scalable platform for implementing such systems. Among the critical elements in these circuits, superinductors—high-impedance, dissipation-free inductive components—play a central role by suppressing charge fluctuations. They allow quantum states to be delocalized in phase space, protect qubits from environmental noise, and facilitate access to phenomena such as dual Josephson physics and ultra-strong coupling regimes. This thesis explores two complementary implementations of high-impedance circuits: geometric superinductors, demonstrating that high impedance can be achieved beyond kinetic inductance, and Josephson junction chains, used to investigate both microwave mode properties and DC transport across the superconductor-to-insulator transition. Part I addresses geometric superinductors. Contrary to the common belief that high-impedance superconducting circuits require kinetic inductance, we demonstrate that purely geometric designs can achieve characteristic impedance exceeding the resistance quantum. By exploiting mutual coupling between adjacent turns, coil-based inductors achieve enhanced self-inductance, creating a reliable platform for qubits and resonators. Modeling, simulation, fabrication, and characterization confirm that these elements behave as superinductor. With low loss, high linearity, and minimal stray capacitance, these elements are reproducible, free of uncontrolled tunneling events, and capable of strong magnetic coupling. This establishes geometric superinductors as robust, single-wave-function superconducting devices suitable for hardware protected qubits and hybrid systems. Part II presents classical numerical simulations of a Quantum Phase Slip circuit to study dual Shapiro steps. The circuit consists of an ideal Quantum Phase Slip element embedded in a resistive-inductive environment with a parasitic capacitance. Part III extends the investigation of high characteristic-impedance circuit elements to one-dimensional Josephson junction chains, which act as a quantum simulator for many-body physics and the superconductor–insulator transition. Different devices are realized on both sides of the DC phase transition, showing either a supercurrent branch or Coulomb blockade at zero bias. The effect of the crossover on microwave modes, however, remains insufficiently investigated. Studying these modes provides insight into the interplay between disorder and phase-slip events. Small differences in circuit component sizes determine which side of the transition a device falls on, making these results relevant not only for fundamental understanding but also for the design of quantum devices, emphasizing the crucial role of the electromagnetic environment in stabilizing and controlling fragile quantum states. Together, these results illustrate how carefully engineered high characteristic-impedance elements provide a link between macroscopic circuits and the inherently uncertain quantum world, enabling experiments that probe, control, and ultimately exploit quantum fluctuations for applications in quantum information, metrology, solid state physics and beyond.
This work is concerned with two fascinating circuit quantum electrodynamics components, the Josephson junction and the geometric superinductor, and the interesting experiments that can be done by combining the two. The Josephson junction has revolutionized the field of superconducting circuits as a non-linear dissipation-less circuit element and is used in almost all superconducting qubit implementations since the 90s. On the other hand, the superinductor is a relatively new circuit element introduced as a key component of the fluxonium qubit in 2009. This is an inductor with characteristic impedance larger than the resistance quantum and self-resonance frequency in the GHz regime. The combination of these two elements can occur in two fundamental ways: in parallel and in series. When connected in parallel the two create the fluxonium qubit, a loop with large inductance and a rich energy spectrum reliant on quantum tunneling. On the other hand placing the two elements in series aids with the measurement of the IV curve of a single Josephson junction in a high impedance environment. In this limit theory predicts that the junction will behave as its dual element: the phase-slip junction. While the Josephson junction acts as a non-linear inductor the phase-slip junction has the behavior of a non-linear capacitance and can be used to measure new Josephson junction phenomena, namely Coulomb blockade of Cooper pairs and phase-locked Bloch oscillations. The latter experiment allows for a direct link between frequency and current which is an elusive connection in quantum metrology. This work introduces the geometric superinductor, a superconducting circuit element where the high inductance is due to the geometry rather than the material properties of the superconductor, realized from a highly miniaturized superconducting planar coil. These structures will be described and characterized as resonators and qubit inductors and progress towards the measurement of phase-locked Bloch oscillations will be presented.
A practical quantum computer demands a physical quantum bit (qubit) that is scalable, has a well-defined initial state, and brings together coherence with reliable control and read-out. A compelling idea is to put together a locally protected Majorana qubit with a superconducting qubit. The superconducting qubit is used to read out the Majorana states while the Majorana qubit may act as a quantum memory. This thesis presents a series of studies on quantum transport in semiconducting nanowires coupled to superconductors to investigate various platforms for realizing Majorana. These studies are followed by microwave measurements on superconducting circuits compatible with semiconducting-superconducting heterostructures. We first evaluate the potential of Ge/Si core/shell nanowires by achieving induced superconductivity as well as estimating spin-orbit coupling. Next we explore the transport mediated by Andreev bound states formed in InSb nanowire quantum. A subgap negative differential conductance is investigated together with the coalescing Andreev resonances at zero bias relevant for the correct interpretation of Majorana experiments done on the same structures. We conclude our studies of semiconducting nanowires by exploring tunnel junctions in Sn-InSb nanowires that are prepared by in-situ shadowing using nearby nanowires and flakes. Tin shells are found to induce a hard superconducting gap persisting up to high magnetic fields. We observe the two-electron charging effect from a small superconducting island of Sn-InSb. This effect is attributed to charge parity stability, which makes this nanowire system an intriguing candidate for superconducting and topological quantum circuits. These Sn-InSb junctions are then used as the nonlinear element in a transmon qubit design where we observe a dispersive coupling between this nanowire Josephson junction and a superconducting resonator. We also present our progress towards building a magnetic field resilient superconducting circuit that allows integration with semiconducting and topological structures. In the last chapter, InSb semiconducting nanowires are utilized as shadow masks prior to superconductor deposition on an InAs quantum well. We study Josephson current properties in Josephson junctions made from these nanowire shadows. Our results point to highly transparent junctions that can be developed further for hybrid superconductor-semiconductor qubit systems.