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arXiv:1811.03085 · Physical review. B./Physical review. B

Equation of motion approach to black-box quantization: Taming the multimode Jaynes-Cummings model

Fabian Hassler, Jakob Stubenrauch, Alessandro Ciani

Published Jan 28, 2019• 9 citations
arXivHTMLPDF

Abstract

An accurate modeling of a Josephson junction that is embedded in an arbitrary environment is of crucial importance for qubit design. We present a formalism to obtain a Lindblad master equation that describes the evolution of the system. As the qubit degrees of freedom oscillate with a well-defined frequency ${\ensuremath{\omega}}_{q}$, the environment only has to be modeled close to this frequency. We show that this goal, differently from alternative approaches, can be achieved by modeling the environment with only a few degrees of freedom. We treat the example of a transmon qubit coupled to a stripline resonator. We derive the parameters of a dissipative single-mode Jaynes-Cummings model starting from first principles. We show that the leading contribution of the off-resonant modes is a correlated decay process involving both the qubit and the resonator mode. In particular, our results show that the effect of the off-resonant modes in the multimode Jaynes-Cummings model is perturbative in $1/{\ensuremath{\omega}}_{q}$.

DOI
10.1103/physrevb.99.014515

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An accurate modeling of a Josephson junction that is embedded in an arbitrary environment is of crucial importance for qubit design. We present a formalism to obtain a Lindblad master equation that describes the evolution of the system. As the qubit degrees of freedom oscillate with a well-defined frequency \ensuremathωq{\ensuremath{\omega}}_{q}\ensuremathωq​, the environment only has to be modeled close to this frequency. We show that this goal, differently from alternative approaches, can be achieved by modeling the environment with only a few degrees of freedom. We treat the example of a transmon qubit coupled to a stripline resonator. We derive the parameters of a dissipative single-mode Jaynes-Cummings model starting from first principles. We show that the leading contribution of the off-resonant modes is a correlated decay process involving both the qubit and the resonator mode. In particular, our results show that the effect of the off-resonant modes in the multimode Jaynes-Cummings model is perturbative in 1/\ensuremathωq1/{\ensuremath{\omega}}_{q}1/\ensuremathωq​.
The Jaynes-Cummings and quantum Rabi models are fundamental to cavity and circuit quantum electrodynamics, as they describe the simplest form of light-matter interaction, where a single qubit is coupled to a single bosonic mode. A scenario that is commonly encountered in the experimental practice arises when the bosonic mode interacts with an external dissipative thermal bath, making the qubit-boson system open. In this work, we present a new analytical solution of the Lindblad master equations for the open dispersive Jaynes-Cummings model and a perturbative analytical solution of the open quantum Rabi model in the limit of weak qubit-boson coupling ggg, using the holomorphic formalism in Bargmann space. Specifically, we derive the most general solution of the local Lindblad master equation for the open dispersive Jaynes-Cummings model coupled to a thermal bath, with the only assumptions that the initial state of the qubit-boson system is separable. Additionally, we obtain a perturbative analytical solution for the open quantum Rabi model up to second order in ggg. Notably, our findings include a new formula for the qubit's steady state at zeroth order, showing that the stationary populations depend on both qubit and boson frequencies in the quantum Rabi model, but not in the Jaynes-Cummings model, regardless of the value of ggg. Our results are of general interest to the study of open quantum systems in the context of light-matter interaction.
We consider a particularly simple exactly solvable model for a qubit coupled to sequentially nested environments. The purpose is to exemplify the coherence conserving effect of a central system, that has been reported as a result of increasing the coupling between near and far environment. The paradigmatic example is the Jaynes–Cummings Hamiltonian, which we introduce into a Kossakowski–Lindblad master equation using alternatively the lowering operator of the oscillator or its number operator as Lindblad operators. The harmonic oscillator is regarded as the near environment of the qubit, while effects of a far environment are accounted for by the two options for the dissipative part of the master equation. The exact solution allows us to cover the entire range of coupling strength from the perturbative regime to strong coupling analytically. The coherence conserving effect of the coupling to the far environment is confirmed throughout.
We study the dynamics of a transmon qubit that is capacitively coupled to an open multimode superconducting resonator. 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 dissipation of the resonator exactly by employing a spectral representation for the Green's function in terms of a set of non-Hermitian modes and show that it is possible to derive effective Heisenberg-Langevin equations without resorting to the rotating-wave, two-level, Born, or Markov approximations. A well-behaved time-domain perturbation theory is derived to systematically account for the nonlinearity of the 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.
Environmental influences on the dynamics of a coupled qubit-oscillator system are studied analytically in a regime where deviations from the exactly solvable Jaynes–Cummings model are important. We investigate the case of a quasi-degenerate qubit within the ultra-strong coupling regime for which the qubit frequency is much smaller than the frequency of the oscillator, and the coupling between the qubit and the oscillator is large, both of which invalidate the usually employed rotating wave approximation. In contrast to the standard quantum optics master equation, we explicitly take the qubit-oscillator coupling into account while microscopically deriving a dressed state master equation. Using the derived master equation, we discuss a spectroscopic technique which can be used to probe the dressed energy-level structure of the qubit-oscillator system.
The study of phase transitions in dissipative quantum systems based on the Liouvillian is often hindered by the difficulty of constructing a time-local master equation when the system-environment coupling is strong. To address this issue, the complex discretization approximation for the environment is proposed to study the quantum phase transition in the extended Jaynes-Cumming model with an infinite number of boson modes. This approach yields a non-Hermitian effective Hamiltonian that can be used to simulate the dynamics of the spin. It is found that the ground state of this effective Hamiltonian determines the spin dynamics in the single-excitation subspace. Depending on the opening of the energy gap and the maximum population of excitations on the spin degree of freedom, three distinct phases can be identified: fast decaying, localized, and stretched dynamics of the spin. This approach can be extended to multiple excitations, and similar dynamics were found in the double-excitation subspace, indicating the robustness of the single-excitation phase.