Superconducting circuit quantum electrodynamical (cQED) platforms present a persistent modeling challenge: the intrinsic nonlinearity of the Josephson potential couples to a dissipative electromagnetic environment in ways that resist both perturbative treatment and naive Markovian reduction. Standard approaches either scale poorly with system size or absorb undeclared approximations about the noise structure into their master equations. In this work, we generalize Garraway's pseudomode construction to accommodate nonlinearly intercoupled auxiliary modes, providing a nonperturbative and systematically reducible framework for open-system cQED dynamics. The key observation is that pseudomode elimination is not fundamentally tied to linearity but to representability: any eliminated sector whose influence on the retained subsystem admits a rational self-energy can be replaced by a finite set of damped auxiliary modes, independent of the internal nonlinear structure of the retained Hamiltonian. We develop the general theory in the Heisenberg picture via a Dyson equation for the retained-mode Green's function, then demonstrate closed-form elimination for two-, three-, and four-mode Kerr-coupled systems with bilinear exchange and three-wave mixing interactions. The resulting framework substantially reduces the computational overhead of open-system cQED modeling while remaining faithful to the underlying physics, provided the spectral description of the eliminated sector is chosen to match the experimentally measured response functions of the hardware.
Superconducting circuit quantum electrodynamical (cQED) platforms present a persistent modeling challenge: the intrinsic nonlinearity of the Josephson potential couples to a dissipative electromagnetic environment in ways that resist both perturbative treatment and naive Markovian reduction. Standard approaches either scale poorly with system size or absorb undeclared approximations about the noise structure into their master equations. In this work, we generalize Garraway's pseudomode construction to accommodate nonlinearly intercoupled auxiliary modes, providing a nonperturbative and systematically reducible framework for open-system cQED dynamics. The key observation is that pseudomode elimination is not fundamentally tied to linearity but to representability: any eliminated sector whose influence on the retained subsystem admits a rational self-energy can be replaced by a finite set of damped auxiliary modes, independent of the internal nonlinear structure of the retained Hamiltonian. We develop the general theory in the Heisenberg picture via a Dyson equation for the retained-mode Green's function, then demonstrate closed-form elimination for two-, three-, and four-mode Kerr-coupled systems with bilinear exchange and three-wave mixing interactions. The resulting framework substantially reduces the computational overhead of open-system cQED modeling while remaining faithful to the underlying physics, provided the spectral description of the eliminated sector is chosen to match the experimentally measured response functions of the hardware.
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.
We introduce a numerically exact and computationally feasible nonlinear-response theory developed for lossy superconducting quantum circuits based on a framework of quantum dissipation in a minimally extended state space. Starting from the Feynman-Vernon path-integral formalism for open quantum systems with the system degrees of freedom being the nonlinear elements of the circuit, we eliminate the temporally nonlocal influence functional of all linear elements by introducing auxiliary harmonic modes with complex-valued frequencies coupled to the nonlinear elements. In our work, we propose a concept of time-averaged observables, inspired by experiment, and provide an explicit formula for producing their quasiprobability distribution. We illustrate the consistency of our formalism with the well-established Markovian input-output theory by applying them the dispersive readout of a superconducting transmon qubit. For an important demonstration of our approach beyond weak coupling, we analyze the low-frequency linear response of a capacitively and resistively shunted Josephson junction and observe signatures of a much-debated quantum phase transition at a finite temperature. The developed framework enables a comprehensive fully quantum-mechanical treatment of nonlinear quantum circuits coupled to their environment, without the limitations of typical approaches to weak dissipation, high temperature, and weak drive. This versatile tool paves the way for accurate models of quantum devices and increased fundamental understanding of quanutm mechanics such as that of the quantum measurement.
This thesis develops model reduction methods for open quantum system dynamics, yielding efficient models for bipartite systems, with direct application to circuit QED and broader applicability beyond this platform. First, we consider dissipative dynamics characterized by timescale separation, which translates to a gap on the real axis of the spectrum of the Liouvillian generating it. This allows to compute the reduced evolution for a target subsystem by adiabatically eliminating fast-decaying degrees of freedom. A Sylvester equation integral representation yields closed forms for the reduced generators and explicit reconstruction maps to arbitrary order without a preliminary interaction-picture averaging step, providing a linear center manifold description that separates long-lived from fast-decaying modes by the real parts of the spectrum. The reduced dynamics takes place on the center manifold, that is, the spectral subspace associated with eigenvalues whose real parts are small (long-lived modes), while placing no restriction on imaginary parts, thus allowing for fast unitary dynamics on the target subsystem. We further demonstrate the equivalence of such a geometric approach to adiabatic elimination with the time-convolutionless master equation framework, by showing that the maps defining the reduced model satisfy the same invariance equation in both frameworks. The time-convolutionless formulation of adiabatic elimination extends to cases where the target system undergoes fast unitary dynamics and retains explicit time dependence in the reduced model, hence capturing initial transients that are neglected in time-independent geometric reductions. Second, we apply a unitary chain mapping of microscopic system–environment Hamiltonians to a nearest-neighbor form, enabling efficient time evolution exploiting tensor network time evolution algorithms based on the time dependent variational principle. Accuracy of the tensor network representation, and thus the size of the reduced model, is controlled by entanglement entropy growth, which we quantify numerically. While the methods are general and apply beyond the specific platform considered, we focus on dispersive readout in circuit QED (a qubit dipolarly coupled to a driven, lossy cavity). We quantify the dependence of the relaxation time T1 of the qubit on intracavity photon number nˉ (and thus on the drive amplitude) and on the spectral structure of the environment, and compute the emission spectrum directly from environmental observables. Simulations including a Purcell notch filter predict a decrease of T1 with increasing drive power, a qualitatively different behavior with respect to Born–Markov–secular GKSL predictions but consistent with experimental trends, indicating that explicit filter modeling is required in this case. Together, the results specify when reduced Liouvillian models under a dissipative gap are reliable and when microscopic, non-perturbative simulations are required.
We introduce a hybrid qubit-boson beam-splitter gate in which a microwave cavity mode couples to an exchange-dressed two-level subsystem of an interacting two-qubit system in the presence of Kerr nonlinearity. Starting from a general circuit quantum electrodynamics (cQED) model, we derive the corresponding hybrid qubit-cavity interaction, develop its open-system description including photon- and qubit-sector-bath-induced dissipation and obtain in the weak-dissipation regime an analytical expression for the average gate fidelity. We further identify carbon-nanotube circuit QED as a concrete platform for implementing and controlling the gate, provide a representative operating regime and perform noiseless and noisy numerical simulations to study the gate dynamics and benchmark the analytical results. Beyond this implementation route, the proposed hybrid primitive provides a natural building block for many-body dynamics, including quantum-cellular-automaton (QCA) and lattice-gauge-inspired architectures and, through its collision-model reformulation, also suggests connections to noisy QCA, non-Markovian extensions and reservoir-style quantum information processing.
We introduce a new framework to study the dynamics of open quantum systems with linearly coupled Gaussian baths. Our approach replaces the continuous bath with an auxiliary discrete set of pseudomodes with dissipative dynamics, but we further relax the complete positivity requirement in the Lindblad master equation and formulate a quasi-Lindblad pseudomode theory. We show that this quasi-Lindblad pseudomode formulation directly leads to a representation of the bath correlation function in terms of a complex weighted sum of complex exponentials, an expansion that is known to be rapidly convergent in practice and thus leads to a compact set of pseudomodes. The pseudomode representation is not unique and can differ by a gauge choice. When the global dynamics can be simulated exactly, the system dynamics is unique and independent of the specific pseudomode representation. However, the gauge choice may affect the stability of the global dynamics, and we provide an analysis of why and when the global dynamics can retain stability despite losing positivity. We showcase the performance of this formulation across various spectral densities in both bosonic and fermionic problems, finding significant improvements over conventional pseudomode formulations.
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.