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arXiv:1603.00535 · Physical Review A

Maxwell boundary conditions imply non-Lindblad master equation

Motoaki Bamba, Nobuyuki Imoto

Published Sep 1, 2016• 5 citations
arXivHTMLPDF

Abstract

From the Hamiltonian connecting the inside and outside of a Fabry-P\'erot cavity, which is derived from the Maxwell boundary conditions at a mirror of the cavity, a master equation of a non-Lindblad form is derived when the cavity embeds matters, although we can transform it to the Lindblad form by performing the rotating-wave approximation to the connecting Hamiltonian. We calculate absorption spectra by these Lindblad and non-Lindblad master equations and also by the Maxwell boundary conditions in the framework of the classical electrodynamics, which we consider the most reliable approach. We found that, compared to the Lindblad master equation, the absorption spectra by the non-Lindblad one agree better with those by the Maxwell boundary conditions. Although the discrepancy is highlighted only in the ultrastrong light-matter interaction regime with a relatively large broadening, the master equation of the non-Lindblad form is preferable rather than of the Lindblad one for pursuing the consistency with the classical electrodynamics.

DOI
10.1103/physreva.94.033802

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As mentioned by De Liberato [Phys. Rev. A 89, 017801 (2014)], when we suppose the metallic thin mirror and perform the renormalization to the approach starting from the frequently used system-environment-coupling Hamiltonian, we can certainly resolve the discrepancy of its result from that obtained by the reliable approach in the main discussion of our paper [Phys. Rev. A 88, 013814 (2013)]. Although the suggested approach is currently applicable to the specific situation after checking its validity by our reliable but cumbersome approach, we instead propose to start from the system-environment-coupling Hamiltonian determined properly by the mechanism of the confinement and loss of the cavity fields. This approach is applicable to any cavity structure, in principle, and we do not face the renormalization problem appearing in the Comment.

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From the Hamiltonian connecting the inside and outside of a Fabry-P'erot cavity, which is derived from the Maxwell boundary conditions at a mirror of the cavity, a master equation of a non-Lindblad form is derived when the cavity embeds matters, although we can transform it to the Lindblad form by performing the rotating-wave approximation to the connecting Hamiltonian. We calculate absorption spectra by these Lindblad and non-Lindblad master equations and also by the Maxwell boundary conditions in the framework of the classical electrodynamics, which we consider the most reliable approach. We found that, compared to the Lindblad master equation, the absorption spectra by the non-Lindblad one agree better with those by the Maxwell boundary conditions. Although the discrepancy is highlighted only in the ultrastrong light-matter interaction regime with a relatively large broadening, the master equation of the non-Lindblad form is preferable rather than of the Lindblad one for pursuing the consistency with the classical electrodynamics.
In a recent paper, Bamba and Ogawa [Phys. Rev. A 88, 013814 (2013)] developed a microscopic model describing a photonic cavity coupled to an excitonlike resonance. One of the results they obtained studying such a model is that, in the ultrastrong coupling regime, dissipation rates calculated applying the rotating-wave approximation on the system-environment coupling qualitatively differ from the ones calculated using a microscopic theory based on the quantum electrodynamics for dielectric media. Here I show that this result is an artifact, caused by an inconsistent application of the rotating-wave approximation and by an unphysical parameter choice.
As mentioned by De Liberato [Phys. Rev. A 89, 017801 (2014)], when we suppose the metallic thin mirror and perform the renormalization to the approach starting from the frequently used system-environment-coupling Hamiltonian, we can certainly resolve the discrepancy of its result from that obtained by the reliable approach in the main discussion of our paper [Phys. Rev. A 88, 013814 (2013)]. Although the suggested approach is currently applicable to the specific situation after checking its validity by our reliable but cumbersome approach, we instead propose to start from the system-environment-coupling Hamiltonian determined properly by the mechanism of the confinement and loss of the cavity fields. This approach is applicable to any cavity structure, in principle, and we do not face the renormalization problem appearing in the Comment.
Quantum master equations are commonly used to model the dynamics of open quantum systems, but their accuracy is rarely compared with the analytical solution of exactly solvable models. In this work, we perform such a comparison for the damped Jaynes-Cummings model of a qubit in a leaky cavity, for which an analytical solution is available in the one-excitation subspace. We consider the non-Markovian time-convolutionless master equation up to the second (Redfield) and fourth orders as well as three types of Markovian master equations: the coarse-grained, cumulant, and standard rotating-wave approximation (RWA) Lindblad equations. We compare the exact solution to these master equations for three different spectral densities: impulse, Ohmic, and triangular. We demonstrate that the coarse-grained master equation outperforms the standard RWA-based Lindblad master equation for weak coupling or high qubit frequency (relative to the spectral density high-frequency cutoff ${\ensuremath{\omega}}_{c}$), where the Markovian approximation is valid. In the presence of non-Markovian effects characterized by oscillatory, nondecaying behavior, the time-convolutionless approximation closely matches the exact solution for short evolution times (in units of ${\ensuremath{\omega}}_{c}^{\ensuremath{-}1}$) even outside the regime of validity of the Markovian approximations. For long evolution times, all master equations perform poorly, as quantified in terms of the trace-norm distance from the exact solution. The fourth-order time-convolutionless master equation achieves the top performance in all cases. Our results highlight the need for reliable approximation methods to describe open-system quantum dynamics beyond the short-time limit.
Abstract The Lindblad form of the master equation has proven to be one of the most convenient ways to describe the impact of an environment interacting with a quantum system of interest. For single systems the jump operators characterizing these interactions usually take simple forms with a clear interpretation. However, for coupled systems these operators take significantly different forms and the full dynamics cannot be described by jump operators acting on the individual subsystems only. In this work, we investigate the differences between a common phenomenological model for the master equation and the more rigorous dressed-state master equation for optomechanical systems. We provide an analytical method to obtain the absorption spectrum of the system for both models and show the breakdown of the phenomenological model in both the bad cavity and the ultra-strong coupling limit. We present a careful discussion of the indirect dephasing of the optical cavity in both models and its role in the differences of their predicted absorption spectra. Our work provides a simple experimental test to determine whether the simpler phenomenological model can be used to describe the system and is a step forward toward a better understanding of the role of the coupling between subsystems for open-quantum-system dynamics.
In the standard theory of cavity quantum electrodynamics (QED), coupling between photons inside and outside a cavity (cavity system and environment) is given conserving the total number of photons. However, when the cavity photons (ultrastrongly) interact with atoms or excitations in matters, the system-environment coupling must be determined from a more fundamental viewpoint. Based on the Maxwell's boundary conditions in the QED theory for dielectric media, we derive the quantum Langevin equation and input-output relation, in which the total number of polaritons (not photons) inside the cavity and photons outside is conserved.
In Chapters 8–13 we always assumed that the nonlinear medium is contained in a unidirectional ring cavity. The case of a Fabry–Perot cavity (Fig. 8.1) is substantially more complex because the interference of the two counterpropagating fields gives rise to a standing-wave pattern that modulates the population difference and the atomic polarization to form gratings with a spatial frequency equal to the inverse of the half-wavelength and its harmonics. Therefore the atomic variables exhibit, in addition to the slow spatial variation which characterizes the envelopes, also a fast spatial variation on the wavelength scale, and the coexistence of these two spatial scales poses a highly nontrivial problem that must be treated with care. In Chapters 9 and 11 we solved exactly the Maxwell–Bloch equations with the ring-cavity boundary conditions for both the active and the passive configuration. Exact solutions of this kind have been obtained also for the equations and boundary conditions of a Fabry–Perot cavity [117–119]. However, for the sake of simplicity we will pass directly to models that are valid in the low-transmission approximation. Again for the sake of simplicity, in this chapter we assume that n B = 1, so that the light velocity in the empty part of the cavity is the same as that in the atomic sample. By following the same line of reasoning as that which we pursued in Chapter 12, in Section 14.1 we introduce a spatial transformation of the electric-field envelopes, which leads to boundary conditions without parameters, while the parameters present in the boundary conditions of the original variables appear in the envelope equations. As in the ring-cavity case, this approach allows one to introduce the cavity modes in a natural way, and to derive a set of modal equations coupled to the equations for the atomic variables. Sets of equations of this kind are common in the laser literature, but are not derived from the true boundary conditions for the cavity.