Dynamics of a qubit in a high-impedance transmission line from a bath perspective

Abstract

We investigate the quantum dynamics of a generic model of light-matter interaction in the context of high impedance waveguides, focusing on the behavior of the photonic states generated in the waveguide. The model treated consists simply of a two-level system coupled to a bosonic bath (the ohmic spin-boson model). Quantum quenches as well as scattering of an incident coherent pulse are studied using two complementary methods. First, we develop an approximate ansatz for the electromagnetic waves based on a single multimode coherent state wavefunction; formally, this approach combines in a single framework ideas from adiabatic renormalization, the Born-Markov approximation, and input-output theory. Second, we present numerically exact results for scattering of a weak intensity pulse by using Numerical Renormalization Group (NRG) calculations. NRG provides a benchmark for any linear response property throughout the ultra-strong coupling regime. We find that in a sudden quantum quench, the coherent state approach produces physical artifacts, such as improper relaxation to the steady state. These previously unnoticed problems are related to the simplified form of the ansatz that generates spurious correlations within the bath. In the scattering problem, NRG is used to find the transmission and reflection of a single photon, as well as the inelastic scattering of that single photon. Simple analytical formulas are established and tested against the NRG data that predict quantitatively the transport coefficients for up to moderate environmental impedance. These formulas resolve pending issues regarding the presence of inelastic losses in the spin-boson model near absorption resonances, and could be used for comparison to experiments in Josephson waveguide quantum electrodynamics (QED). Finally, the scattering results using the coherent state wavefunction approach are compared favorably to the NRG results for very weak incident intensity. We end our study by presenting results at higher power where the response of the system is nonlinear.

Spontaneous emission of Schrödinger cats in a waveguide at ultrastrong coupling

“…, which can be solved by numerical integration [35][36][37][38] using a specially devised algorithm (see appendix for details).…”

“…In contrast to the dynamics with a single coherent state [36,38], one encounters here a computational difficulty [35], because time-derivatives ¢ f m k of all possible coherent state amplitudes enter the dynamical equation ruling a given field f j k in equation (8) through the parameter k mj in equation (9). Indeed, for stability reasons it is crucial to formulate the dynamical equations in an explicit form…”

Particle production in ultrastrong-coupling waveguide QED

“…by applying the time-dependent variational principle [56], δ dtL = 0, upon arbitrary variations of the state vector (7) with respect to its set of variational parameters. for the set of variables v = {p m , q m , f k,m , h k,m }, which can be solved by numerical integration [34,39,[57][58][59].…”

“…This problem is particularly challenging because of the combination of non-perturbative ultra-strong coupling with non-equilibrium effects that arise at finite input power. Ultra-strong coupling scattering at non-vanishing power has been addressed previously with approximate techniques [28,34,35,49] and with more advanced numerical methods [29,33]. However, systematic extraction of many-body scattering matrices has not been performed to our knowledge.…”

“…In ultra-strong wQED, many-body phenomena are expected to occur that have no counterpart in standard quantum optics [2,3] or in low-coupling superconduct-ing transmission lines [20][21][22][23][24][25][26]. A non-exhaustive list of theoretical predictions includes giant Lamb shifts [27][28][29][30][31], single-photon down-conversion [32,33], non-RWA transmission lineshapes [28,34,35], multi-mode entanglement [36][37][38], and non-classical emission [39]. The key element in all of the novel many-body phenomena in ultra-strong wQED is that the number of excitations is no longer conserved because the rotating-wave approximation is not legitimate anymore.…”