Superconducting circuit quantization conventionally starts from classical Euler-Lagrange circuit equations of motion. Invoking the correspondence principle yields a canonically quantized description of circuit dynamics over a bosonic Hilbert space. This approach has been very successful for describing experiments, but implicitly starts from the classical Ginsberg-Landau mean-field theory for the circuit. Here, we employ a different approach that starts from a microscopic fermionic Hamiltonian for interacting electrons, whose ground space is described by the Bardeen-Cooper-Schrieffer (BCS) many-body wave function that underpins conventional superconductivity. We introduce the BCS ground space as a subspace of the full fermionic Hilbert space, and show that projecting the electronic Hamiltonian onto this subspace yields the standard Hamiltonian terms for Josephson junctions, capacitors, and inductors, from which standard quantized circuit models follow. This approach does not impose a spontaneously broken symmetry so that it consistently describes quantized circuits that support superpositions of phases, and the canonical commutation relations between phase and charge are derived from the underlying fermionic commutation properties, rather than imposed. By expanding the projective subspace, this approach can be extended to describe phenomena outside the BCS ground space, including quasiparticle excitations.
Superconducting circuit quantization conventionally starts from classical Euler-Lagrange circuit equations of motion. Invoking the correspondence principle yields a canonically quantized description of circuit dynamics over a bosonic Hilbert space. This approach has been very successful for describing experiments, but implicitly starts from the classical Ginsberg-Landau mean-field theory for the circuit. Here, we employ a different approach that starts from a microscopic fermionic Hamiltonian for interacting electrons, whose ground space is described by the Bardeen-Cooper-Schrieffer (BCS) many-body wave function that underpins conventional superconductivity. We introduce the BCS ground space as a subspace of the full fermionic Hilbert space, and show that projecting the electronic Hamiltonian onto this subspace yields the standard Hamiltonian terms for Josephson junctions, capacitors, and inductors, from which standard quantized circuit models follow. This approach does not impose a spontaneously broken symmetry so that it consistently describes quantized circuits that support superpositions of phases, and the canonical commutation relations between phase and charge are derived from the underlying fermionic commutation properties, rather than imposed. By expanding the projective subspace, this approach can be extended to describe phenomena outside the BCS ground space, including quasiparticle excitations.
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.
The quantization of superconducting transmission-line resonators is usually introduced phenomenologically by modeling the resonator as an effective LC circuit and imposing canonical commutation relations on macroscopic variables such as charge and flux. Although this approach is highly successful, it leaves open why these macroscopic variables should obey quantum commutation relations and how this behavior emerges from the superconducting state. In this work, starting from the microscopic pairing Hamiltonian underlying BCS superconductivity, we derive the low-energy effective Hamiltonian of a circuit-QED architecture containing a superconducting transmission line with distributed capacitive and inductive elements. We establish quantitative relations between macroscopic observables, including current and voltage, and the spatially local superconducting phase, as well as the microscopic parameters of the electron-phonon system. We then extend the third quantization of the superconducting order parameter, introduced in Paper (I) for the global phase, to the spatially local case. This gives a macroscopic field quantization of the superconducting phase. We show that, after restriction to the low-energy excitation subspace, the local superconducting phase becomes a genuine quantum dynamical variable. Thus, the quantum behavior of transmission-line resonators need not be postulated at the macroscopic level, but follows from the third quantization of the superconducting order parameter. These results suggest that capacitive and inductive superconducting circuit elements share the same microscopic origin, providing a unified framework for superconducting circuit quantization.
The quantum mechanics of superconducting circuits is derived by starting from a classical Hamiltonian dynamical system describing a dissipationless circuit, usually made of capacitive and inductive elements. However, standard approaches to circuit quantization treat fluxes and charges, which end up as the canonically conjugate degrees of freedom on phase space, asymmetrically. By combining intuition from topological graph theory with a recent symplectic geometry approach to circuit quantization, we present a theory of circuit quantization that treats charges and fluxes on a manifestly symmetric footing. For planar circuits, known circuit dualities are a natural canonical transformation on the classical phase space. We discuss the extent to which such circuit dualities generalize to nonplanar circuits.
We extend a recently developed projective circuit quantisation approach to incorporate superconducting Higgs modes associated to gap dynamics. This approach starts from a microscopic fermionic Hamiltonian for mesoscopic superconductors, and projects the system onto its low-energy "BCS" Hilbert space. We derive analytical results for the superconducting Higgs mass, "spring" constant, and oscillation frequency of the gap dynamics, which we validate numerically. We compute anharmonic corrections to the Higgs frequency for higher excitations of small superconducting islands, and compare our results to previous long-wavelength calculations.
The quantized Josephson junction equation that underpins the behaviour of charge qubits and other tunnel devices is usually derived through canonical quantization of the classical macroscopic Josephson relations. However, this approach may neglect effects due to the fact that the charge qubit consists of a superconducting island of finite size connected to a large superconductor. We show that the well-known quantized Josephson equation can be derived directly and simply from a microscopic many-body Hamiltonian. By choosing the appropriate strong-coupling limit we produce a highly simplified Hamiltonian that nevertheless allows us to go beyond the mean-field limit and predict further finite-size terms in addition to the basic equation.
In the quest to produce quantum technology, superconducting networks, working at temperatures just above absolute zero, have arisen as one of the most promising physical implementations. The precise analysis and synthesis of such circuits have required merging the fields of physics, engineering, and mathematics. In this dissertation, we develop mathematically consistent and precise Hamiltonian models to describe ideal superconducting networks made of an arbitrary number of lumped elements, such as capacitors, inductors, Josephson and phase-slip junctions, gyrators, etc., and distributed ones like transmission lines. We give formal proofs for the decoupling at high and low frequencies of lumped degrees of freedom from infinite-dimensional systems in different coupling configurations in models based on the effective Kirchhoff's laws. We extend the standard theory to quantize circuits that include ideal nonreciprocal elements all the way to their Hamiltonian descriptions in a systematic way. Finally, we pave the way on how to quantize general frequency-dependent gyrators and circulators coupled to both transmission lines and other lumped-element networks. We have explicitly shown, that these models, albeit ideal, are finite and present no divergence issues. We explain and dispel misunderstandings from the previous literature. Furthermore, we have demonstrated the usefulness of a redundant basis for performing separation of variables of the transmission line (1D) fields in the presence of point-like (lumped-element) couplings by time-reversal symmetry-breaking terms, i.e. nonreciprocal elements.