We apply the momentous quantum mechanics formalism to the non-perturbative study of superconducting circuits in the transmon regime, deriving effective equations of motion for the bare Josephson junction (JJ), the cavity--JJ system, and a dissipative resonator. A Gaussian closure on the hierarchy of quantum moments resums the full cosine nonlinearity into the closed-form effective Hamiltonian $\tfrac{1}{2}[V(φ+φ_s)+V(φ-φ_s)]$, which is non-perturbative and preserves the periodicity and boundedness of the Josephson potential at all phase amplitudes; the standard Kerr (Duffing) approximation is recovered as a special case. We derive the quantum-dressed frequency $ω_{\rm eff}=Ω_p\sqrt{\cos(φ_{\rm zpf}/φ_0)}$, and benchmark the effective dynamics against exact Mathieu-function diagonalization, with the quantum width $G^{2,0}(t)$ providing the most sensitive diagnostic of the closure's validity and marks the boundary of the Gaussian approximation more sharply than $\langle\hatφ\rangle(t)$ does. We compare the Caldirola--Kanai, Bateman, and Lindblad descriptions: the Bateman dynamics, quantized with a switched symplectic structure, preserves the Heisenberg bound, admits exact closed-form moment solutions, and reproduces the Lindblad benchmark for weak damping, i.e., within the analytic error bound $(λ/ω_1)^2\,φ_{\rm zpf}^2$.
We apply the momentous quantum mechanics formalism to the non-perturbative study of superconducting circuits in the transmon regime, deriving effective equations of motion for the bare Josephson junction (JJ), the cavity--JJ system, and a dissipative resonator. A Gaussian closure on the hierarchy of quantum moments resums the full cosine nonlinearity into the closed-form effective Hamiltonian 21[V(φ+φs)+V(φ−φs)], which is non-perturbative and preserves the periodicity and boundedness of the Josephson potential at all phase amplitudes; the standard Kerr (Duffing) approximation is recovered as a special case. We derive the quantum-dressed frequency ωeff=Ωpcos(φzpf/φ0), and benchmark the effective dynamics against exact Mathieu-function diagonalization, with the quantum width G2,0(t) providing the most sensitive diagnostic of the closure's validity and marks the boundary of the Gaussian approximation more sharply than ⟨φ^⟩(t) does. We compare the Caldirola--Kanai, Bateman, and Lindblad descriptions: the Bateman dynamics, quantized with a switched symplectic structure, preserves the Heisenberg bound, admits exact closed-form moment solutions, and reproduces the Lindblad benchmark for weak damping, i.e., within the analytic error bound (λ/ω1)2φzpf2.
Superconducting circuits consisting of a few low-anharmonic transmons coupled to readout and bus resonators can perform basic quantum computations. Since the number of qubits in such circuits is limited to not more than a few tens, the qubits can be designed to operate within the dispersive regime where frequency detunings are much stronger than coupling strengths. However, scaling up the number of qubits will bring the circuit out of the regime, and this invalidates current theories. We develop a formalism that allows to consistently diagonalize the superconducting circuit Hamiltonian beyond the dispersive regime. This will allow to study qubit-qubit interaction unperturbatively, therefore, our formalism remains valid and accurate at small or even negligible frequency detuning; thus, our formalism serves as a theoretical ground for designing qubit characteristics for scaling up the number of qubits in superconducting circuits. We study the most important circuits with single- and two-qubit gates, i.e., a single transmon coupled to a resonator and two transmons sharing a bus resonator. Surprisingly, our formalism allows to determine the circuit characteristics, such as dressed frequencies and Kerr couplings, in closed-form formulas that not only reproduce perturbative results, but also extrapolate beyond the dispersive regime and can ultimately reproduce (and even modify) the Jaynes-Cumming results at resonant frequencies.
We present a Heisenberg-Langevin formalism to study the effective dynamics of a superconducting qubit coupled to an open multimode resonator, without resorting to the rotating wave, two level, Born or Markov approximations. Our effective equations are derived by eliminating resonator degrees of freedom while encoding their effect in the Green's function of the electromagnetic background. We account for the openness of the resonator exactly by employing a spectral representation for the Green's function in terms of a set of non-Hermitian modes. A well-behaved time domain perturbation theory is derived to systematically account for the nonlinearity of weakly nonlinear qubits like transmon. We apply this method to the problem of spontaneous emission, capturing accurately the non-Markovian features of the qubit dynamics, valid for any qubit-resonator coupling strength. Any discrete-level quantum system coupled to the electromagnetic continuum is subject to radiative decay and renormalization of its energy levels. When inside a cavity, these quantities can be strongly modified with respect to vacuum. Generally, this modification can be captured by including only the closest resonant cavity mode. In circuit-QED architecture, with substantial coupling strengths, it is however found that such rates are strongly influenced by far off-resonant modes. A multimode calculation over the infinite set of cavity modes leads to divergences unless an artificial cutoff is imposed. Previous studies have not pointed out what the source of this divergence is. We show that unless the effect of A2 is accounted for up to all orders exactly, any multimode calculations of circuit-QED quantities is bound to diverge. Subsequently, we present the calculation of finite radiative corrections to qubit properties that is free of an artificially introduced high frequency cut-off.
A Josephson junction embedded in a dissipative circuit can be externally driven to induce nonlinear dynamics of its phase. Classically, under sufficiently strong driving and weak damping, dynamic multistability emerges, associated with dynamical bifurcations so that the often-used modeling as a Duffing oscillator, which can exhibit bistability at most, is insufficient. The present work analyzes in this regime corresponding quantum properties by mapping the problem onto a highly nonlinear quasienergy operator in a rotating frame. This allows us to identify in detail parameter regions where simplifications such as the Duffing approximation are valid, to explore classical-quantum correspondences, and to study how quantum fluctuations impact the effective junction parameters as well as the dynamics around higher-amplitude classical fixed points.
Josephson junctions are the central nonlinear element of superconducting qubits, yet their standard description is reached through a chain of approximations. We propose a purely quantum-mechanical model of the junction, formulated directly in Hilbert space: a charged particle moves on a one-dimensional loop, and the junction is a point at which the wave function obeys self-adjoint boundary conditions, labelled by a coupling strength λ, which controls the transparency of the junction, and a phase jump δ, which plays the role of the superconducting phase difference. For a loop with a single junction we solve the spectral problem and show that the ground-state momentum expectation, identified with the supercurrent, factorises exactly as ⟨p^0⟩=Ic(δ,λ)sinδ. For opaque junctions Ic becomes independent of δ and the first Josephson relation I=Icsinδ is recovered; for strongly attractive coupling, where the ground state is bound to the junction, the sinδ law is also recovered, but with an exponentially suppressed amplitude. For more transparent junctions, however, the model \emph{predicts} deviations from the pure sinδ law: Ic acquires a dependence on δ already at order λ−2, compatible with the higher Josephson harmonics recently found necessary to describe superconducting qubit spectra, and a fully transparent contact yields a current linear in δ. When the phase varies in time, a gauge transformation shows that the voltage drop across the junction is proportional to δ˙, so that the second Josephson relation emerges without being imposed. The boundary-condition approach thus reproduces both Josephson relations from first principles and predicts corrections to them, offering a route towards higher-fidelity models of superconducting qubits.
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.
Building on the first-order circuit quantization method [arXiv:2304.12252, arXiv:2401.09120], we provide simple formulas to construct exact Hamiltonians for Josephson-junction-based superconducting qudits capacitively, inductively, or galvanically coupled to passive linear environments. These environments may be multiport, multimode, discrete or continuous, reciprocal or nonreciprocal, and are characterized directly by their impedance or admittance matrices. In the weak-coupling regime, we further derive \emph{divergence-free} dispersive Hamiltonians for mode-resolved environments and transition-resolved weak-coupling master equations for dissipative continua. Mode structure, frequency renormalizations, environment-mediated interactions, decay rates, and directional cross couplings then follow from the same causal immittance response, while spurious Lamb-shift divergences arising from uncontrolled approximations in previous treatments are made explicit and avoided. We apply the theory to a set of illustrative circuits comprising a discrete resonator filter, finite-band metamaterial environments, nonreciprocal waveguide-QED systems, and superconducting giant atoms, for which analytical response matrices can be obtained, although the method is particularly well suited to numerical responses from electromagnetic solvers or experimental characterization. We thereby extend the black-box quantization framework to multiport, dissipative, and nonreciprocal settings, establishing a simple and scalable route toward optimized and automated electromagnetic design of large-scale superconducting quantum hardware.