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.
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.
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.
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.
The proximity effect in hybrid superconducting--normal-metal structures is shown to affect strongly the coherent oscillations of the superconducting order parameter $\mathrm{\ensuremath{\Delta}}$ known as the Higgs modes. The standard Higgs mode at frequency $2\mathrm{\ensuremath{\Delta}}$ is damped exponentially by the quasiparticle leakage from the primary superconductor. Two new Higgs modes with the frequencies depending on both the primary and induced gaps in the hybrid structure are shown to appear due to the coherent electron transfer between the superconductor and the normal metal. Altogether, these three modes determine the long-time asymptotic behavior of the superconducting order parameter disturbed either by the electromagnetic pulse or the quench of the system parameters and thus are of crucial importance for the dynamical properties and restrictions on the operating frequencies for superconducting devices based on the proximity effect used, e.g., in quantum computing, in particular, with topological low-energy excitations.
In superconductors gauge U(1) symmetry is spontaneously broken. According to Goldstone theorem this breaking of a continuous symmetry establishes the existence of the Bogoliubov phase mode while the gauge-invariant response also includes the amplitude fluctuations of the order parameter. The latter, which are also termed ’Higgs’ modes in analogy with the standard model, appear at the energy of the spectral gap 2Δ when the superconducting ground state is evaluated within the weak coupling BCS theory and are therefore damped. Previously we have shown that in the time-dependent Gutzwiller approximation (TDGA) Higgs modes appear inside the gap with a finite binding energy relative to the quasiparticle continuum. Here we show that the binding energy of the Higgs mode becomes exponentially small in the weak coupling limit converging to the BCS solution. On the other hand, well-defined undamped amplitude modes exist in strongly coupled superconductors when the interaction energy becomes of the order of the bandwidth.
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.
Collective modes in a superconductor correspond to fluctuations in the amplitude (Higgs mode) or the phase (Goldstone mode) of the order parameter. Starting from the BCS Hamiltonian, we derive a microscopic Hamiltonian for the coupled Higgs-Goldstone-electron system that contains all interactions among these degrees of freedom. The Higgs subsystem is a weakly interacting Bose gas: Higgs modes experience a mutual attraction mediated by the virtual exchange of Bogoliubov quasi-particle pairs. They also interact with external electromagnetic-fields with a coupling constant that we derive explicitly. We use this theory to examine whether an optically pumped gas of Higgs modes can undergo Bose-Einstein condensation. We identify a density window, bounded below by the equilibrium condensation criterion and above by collapse due to the attractive interaction, in which condensation can occur and we derive the expected density of Higgs-modes due to external pumping. This pumping depends crucially on non-parabolic corrections to the electron dispersion around the Fermi-energy. We show that the pumped mode density is compatible with condensation for realistic material parameters. We suggest possible experimental signatures of this non-equilibrium condensate.