Superconducting circuits exhibit quantum properties on a macroscopic scale, and are natural candidates for solid state quantum computing. Their low-energy physics can be described in terms of the phase of the order parameter, a single collective degree of freedom associated with billions of coherently paired electrons. In practice, however, on top of the superfluid condensate there are single-particle excitations (quasiparticles) with a continuous energy spectrum; the quasiparticles are coupled to the phase degree of freedom. The presence of quasiparticles in the system sets serious constraints on the performance of superconducting charge qubits. In this thesis, we study the kinetics of superconducting quantum circuits, and discuss the fundamental limitations on the energy and phase relaxation times in the presence of quasiparticles.
Superconducting circuits exhibit quantum properties on a macroscopic scale, and are natural candidates for solid state quantum computing. Their low-energy physics can be described in terms of the phase of the order parameter, a single collective degree of freedom associated with billions of coherently paired electrons. In practice, however, on top of the superfluid condensate there are single-particle excitations (quasiparticles) with a continuous energy spectrum; the quasiparticles are coupled to the phase degree of freedom. The presence of quasiparticles in the system sets serious constraints on the performance of superconducting charge qubits. In this thesis, we study the kinetics of superconducting quantum circuits, and discuss the fundamental limitations on the energy and phase relaxation times in the presence of quasiparticles.
We investigate the energy and phase relaxation of a superconducting qubit caused by a single quasiparticle. In our model, the qubit is an isolated system consisting of a small island (Cooper-pair box) and a larger superconductor (reservoir) connected with each other by a tunable Josephson junction. If such a system contains an odd number of electrons, then even at lowest temperatures a single quasiparticle is present in the qubit. Tunneling of a quasiparticle between the reservoir and the Cooper-pair box results in the relaxation of the qubit. We derive master equations governing the evolution of the qubit coherences and populations. We find that the kinetics of the qubit can be characterized by two time scales---quasiparticle escape time from the reservoir to the box ${\ensuremath{\Gamma}}_{\text{in}}^{\ensuremath{-}1}$ and quasiparticle relaxation time $\ensuremath{\tau}$. The former is determined by the dimensionless normal-state conductance gT of the Josephson junction and one-electron level spacing ${\ensuremath{\delta}}_{r}$ in the reservoir $({\ensuremath{\Gamma}}_{\text{in}}\ensuremath{\sim}{g}_{T}{\ensuremath{\delta}}_{r})$, and the latter is due to the electron-phonon interaction. We find that phase coherence is damped on the time scale of ${\ensuremath{\Gamma}}_{\text{in}}^{\ensuremath{-}1}$. The qubit energy relaxation depends on the ratio of the two characteristic times $\ensuremath{\tau}$ and ${\ensuremath{\Gamma}}_{\text{in}}^{\ensuremath{-}1}$ and also on the ratio of temperature T to the Josephson energy EJ.
Superconducting circuits are one of the leading candidates for scalable quantum computation, owing to their long coherence time and compatibility with fabrication techniques. Despite their experimental success, the conventional theoretical framework for superconducting circuit quantisation relies on a semiclassical procedure that requantises the phenomenological Ginzburg-Landau order parameter, introducing the postulated canonical relations and leaving the microscopic origins of circuit parameters hidden. In the present thesis, we develop a rigorous and first-principles approach for the quantisation of superconducting circuits, directly grounded in Bardeen-Cooper-Schrieffer (BCS) theory for superconductivity, and extend this framework to investigate the dynamics of the superconducting Higgs amplitude mode. The central methodological contribution of this work is a projective approach, in which the full many-body electronic Hamiltonian of a superconducting system is projected onto a carefully constructed low-energy BCS Hilbert space, HBCS, spanned by phase parameterised BCS ground states. We systematically characterise the kinematic properties of this subspace, quantify the orthogonality and completeness of the phase basis via the BCS overlap and the projection error, and show that the effective dimensionality of the space is scaled as the square root of the number of single electronic modes. This approach does not impose a spontaneously broken symmetry so that it consistently describes quantised circuits that support superpositions of phases, and the canonical commutation relations between phase and number are derived from the underlying fermionic commutation properties, rather than imposed. This establishes the fully quantum mechanical foundation upon which the standard circuit quantisation approach implicitly rests. The projective formalism is subsequently applied to derive the effective Hamiltonian for superconducting circuit elements. For point-like elements, \eg, capacitive islands and Josephson junctions, projection of the fermionic Hamiltonian onto HBCS yields analytical expressions for the charging energy and the Josephson energy in terms of microscopic material parameters. The resulting Hamiltonian reproduces the standard Cooper-pair-box and transmon Hamiltonians, but now on an unambiguous microscopic footing. We further extend the approach to spatially distributed elements. Inductance is derived from the constitutive response to spatially varying phase torsion in a superconducting wire, yielding the lumped-element inductive energy from first principles. This analysis also identifies the emergent low-energy Hilbert space of the inductor. Combining the results, a complete microscopic derivation of the standard superconducting inductance-capacitance-junction (LCJ) circuit Hamiltonian is recovered. The final part of this thesis extends the projective methodology to the superconducting Higgs mode associated to the amplitude dynamics of the superconducting order parameter. A general BCS space parameterised by phase and gap amplitude is constructed, onto which we project the microscopic Hamiltonian. This yields a generalised eigenvalue problem governing the dynamics of the gap amplitude. The Higgs mode of a superconducting island is thereby shown to behave as a quantum harmonic oscillator. We derive the analytical expressions for the superconducting Higgs mass, spring constant, and oscillation frequency of the gap dynamics. While our results reveal a logarithmic correction to the Higgs frequency reported in previous literature, we numerically validate these results, and compute quantum anharmonic corrections to the Higgs frequency for higher excitations. This thesis proposes a novel, formal and quantitative circuit quantisation theory for superconducting circuits, based on which we explore the Higgs amplitude mode dynamics. The approach is capable of resolving the fundamental question about the compactness of superconducting phase and is extendable to cover quasiparticle contributions. This microscopic circuit theory equips us with a new tool to understand superconducting devices at a level of microscopic detail inaccessible to the standard semiclassical framework, paving the way for a principled treatment of quasiparticle losses and the exploration of exotic superconducting circuit elements.
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 theory of superconducting phase qubits---also known as current-biased Josephson junctions---is presented. In the first part of this thesis, I introduce quantum computation, quantum simulation, and their deep connection with symplectic integration. I then consider the fundamental many-body theory of superconductivity and Josephson junctions and show how the quantum dynamics of a single macroscopic degree of freedom, the gauge invariant phase difference, emerges. A complete study of the Hilbert space structure of such a variable is performed for the current-biased junction. The resulting resonance structure is studied in detail, using various formalisms including the WKB approximation, instanton methods, the complex scaling transformation, basis set stabilization, numerical integration, and dynamical simulation using Lie algebraic wave-packet propagation. The second part of this thesis explores how the current-biased junction can be used as an element of a quantum computer---a quantum bit (qubit). Single qubit operations are studied, followed by the presentation of the theory of coupled qubit devices. My key result is the design and optimization of quantum logic gates with high fidelity (F ~ 0.9999) for capacitively coupled phase qubits with short gate times (~ 10 ns). Finally, I examine an advanced qubit-coupling scheme, a resonant coupling method utilizing a harmonic oscillator as the auxiliary degree of freedom. The models and methods presented here have been developed in direct collaboration with an experimental program. These experiments are the first to show spectroscopic evidence for entanglement between two and three macroscopic degrees of freedom in a superconducting circuit.
Micro-fabricated superconducting circuits have emerged as a leading candidate for quantum computers. However, typical error rates of 10-3 for quantum gates remain a challenge for performing error correction and need to be further decreased for achieving practical quantum computation. Improving qubit coherence times for superconducting qubits is a continuous endeavor in order to decrease error rates. As part of this effort, a topic addressed in this dissertation is the study of the generation mechanism and dynamics of non-equilibrium quasiparticles (QPs), which are electron-like excitations of a superconductor. This source of decoherence was studied by measuring the rate of single-electron changes in the parity state of multiple charge-sensitive transmon qubits under different conditions. The presence of non-equilibrium QPs in superconducting resonators and qubits operating at millikelvin temperature has been known for decades. Understanding the sources and dynamics of these QPs is crucial to evaluate and improve the performance of the devices. Specifically, for superconducting qubits, a single QP tunneling across the Josephson junction changes the total charge parity from even to odd or vice versa, and can induce energy relaxation and dephasing, limiting the coherence of the device. To study the dynamics of non-equilibrium QPs, I utilized a Ramsey-like pulse sequence to measure the rate of single-electron parity changes in transmon qubits, which had a weak charge dispersion, with different shunting capacitor geometries and materials. I started out by measuring a charge parity rate of ~ 33 s-1 for a grounded "x-mon" qubit and ~ 17 s-1 for a galvanically isolated "two-pads" qubit with our standard setup. These rates decreased to ~ 5 s-1 and ~ 2 s-1 for the x-mon and two-pads respectively when IR-filters and extra layers of shielding were installed. From my measurements, I found that the charge parity rates are sensitive to the geometry of the shunting capacitor, weakly depend on the amount of shielding surrounding the device, and independent of using Al or Ta for the superconducting shunting capacitor. To distinguish between two possible mechanisms for the parity switching, I measured the parity rates after cooling one of the qubit devices in an external magnetic field to induce vortices in the superconducting electrodes. After increasing the magnetic field above 23 μT, I measured a faster QP trapping rate and a much smaller T1, suggesting that at least one vortex was added to the superconducting electrode near the Josephson junction. On the other hand, the measured charge parity rates exhibited no clear dependence on the background QP trapping rates. These data suggest that a large portion of parity switching events occurs from QPs being produced at the Josephson junction, where pair-breaking radiation is rectified into QPs by the junction, as opposed to tunneling of QPs residing in the electrodes. To understand the difference in the measured parity rates, I simulated the absorption efficiency of pair-breaking radiation for my two designs and observed a similar difference in absorption rates when a single-mode blackbody radiation with an effective temperature T* ~ 300 mK was assumed to be illuminating the devices. This large T*, which is significantly higher than the device's nominal temperature of 20 mK, is consistent with values reported in the literature. However, its physical origin remains not fully understood. For the other part of my dissertation, I developed a new simplified process for making superconducting air bridges. Air bridges are used to suppress unwanted extraneous high frequency chip modes parasitic to superconducting quantum circuits that can couple to qubits and cause decoherence. Traditional ways of making air bridges utilize thermal reflow to create arched photoresist scaffolds that are limited in heights. This results in insufficient mechanical strength of the bridges which tend to collapse when spanning to larger lengths. My simplified fabrication method that utilizes two-photon lithography, a technique that enables creation of three-dimensional micron-scale structures, relaxes this limitation and is capable of making mechanically stable bridges with lengths up to 100 μm. The resulting Al air bridges, having a superconducting transition temperature of Tc = 1.08 mK and a dc residual resistivity ratio of 3.85, were confirmed to form a good electrical contact with the base metal layers and exhibit a small measurable amount of rf loss when placed over coplanar waveguide resonators, on par with other air bridge fabrication methods.
In a quantum computer, the information carriers, which are bits in ordinary computers, are implemented as devices that exhibit coherent superpositions of physical states and entanglement. Such components, known as quantum bits or qubits, can be realized with various different types of two-state quantum systems. Quantum computers will be built for computational speed, with hoped for applications especially in cryptography and in other tasks where classical computers remain inefficient. Circuit quantum electrodynamics (cQED) is a quantum-computer architecture which employs superconducting electronic components and microwave photon fields as building blocks. Compared to cavity quantum electrodynamics (CQED), where atoms are trapped in physical cavities, cQED is more attractive in that its qubits are tunable and conveniently integrable with the electronics already in use. This architecture has shown some of the most promising qubit designs, despite their coherence times reaching tens of microseconds, are still below the state of the art with spin qubits, which reach milliseconds. Coherence times are historically the most relevant parameters describing the fitness of a qubit, although these days not necessarily the limiting factor. This thesis presents a comprehensive set of theoretical and experimental methods for measuring the characteristic parameters of superconducting qubits. We especially study transmission-line-shunted plasma oscillation qubits, or transmons, and presents experimental results for a single sample. Transmon capacitively couples a superconducting quantum interference device (SQUID) with a coplanar waveguide (CPW) resonator, often with added frequency tunability utilizing an external magnet. The number of superconducting charge carriers tunnelled through a junction in the SQUID are used as qubit degrees of freedom. Readout of the qubit state is carried out by measuring transmission through the CPW. A cryogenic setup is employed with measurement and driving pulses delivered from microwave sources. Steady-state spectroscopy is employed to determine the resonance frequencies of the qubit and the resonator, qubit-resonator coupling constants, and the energy parameters of the qubit. Pulse-modulated measurements are employed to determine the coherence times of the qubit. The related analysis- and simulation programs and scripts are collected togithub.com/patomaki.