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arXiv:2008.09726 · Physical Review A

Variational approach to time-dependent fluorescence of a driven qubit

Yiying Yan, Lipeng Chen, JunYan Luo, Yang Zhao

Published Aug 21, 2020• 15 citations
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Abstract

We employ the Dirac-Frenkel variational principle and the multiple Davydov ansatz to study time-dependent fluorescence spectra of a driven qubit in the weak to strong qubit-reservoir coupling regimes, where both the Rabi frequency and the spontaneous decay rate are comparable to the transition frequency of the qubit. Our method agrees well with the time-local master-equation approach in the weak coupling regime, and offers a flexible way to compute the spectra from the bosonic dynamics instead of two-time correlation functions. While the perturbative master equation breaks down in the strong coupling regime, our method actually becomes more accurate due to the use of bosonic coherent states under certain conditions. We show that the counter-rotating coupling between the qubit and the reservoir has considerable contributions to the photon number dynamics and the spectra under strong driving conditions even when the coupling is moderately weak. The time-dependent spectra are found to be generally asymmetric, a feature that is derived from photon number dynamics. In addition, it is shown that the spectral profiles can be dramatically different from the Mollow triplet due to strong dissipation and/or multiphoton processes associated with the strong driving. Our formalism provides a unique perspective to interpret time-dependent spectra.

DOI
10.1103/physreva.102.023714

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We employ the Dirac-Frenkel variational principle and the multiple Davydov ansatz to study time-dependent fluorescence spectra of a driven qubit in the weak to strong qubit-reservoir coupling regimes, where both the Rabi frequency and the spontaneous decay rate are comparable to the transition frequency of the qubit. Our method agrees well with the time-local master-equation approach in the weak coupling regime, and offers a flexible way to compute the spectra from the bosonic dynamics instead of two-time correlation functions. While the perturbative master equation breaks down in the strong coupling regime, our method actually becomes more accurate due to the use of bosonic coherent states under certain conditions. We show that the counter-rotating coupling between the qubit and the reservoir has considerable contributions to the photon number dynamics and the spectra under strong driving conditions even when the coupling is moderately weak. The time-dependent spectra are found to be generally asymmetric, a feature that is derived from photon number dynamics. In addition, it is shown that the spectral profiles can be dramatically different from the Mollow triplet due to strong dissipation and/or multiphoton processes associated with the strong driving. Our formalism provides a unique perspective to interpret time-dependent spectra.
The time-resolved fluorescence spectrum of a driven two-level system is investigated using a time-dependent variational method. We first establish the transient build-up of the Mollow triplet via the Lindblad master equation under the rotating-wave approximation, providing a complete visualization of the spectral evolution from initial turn-on to steady state. To go beyond the perturbative regime, we employ the multiple Davydov D2 ansatz (multi-D2), which uses the σz eigenstates as the basis and naturally accommodates arbitrary system–bath coupling types and spectral densities. The multi-D2 method converges with M = 4 multiplicities in studied cases, outperforming the multi-D1 ansatz (M = 8) for the σx coupling benchmark. For pure σz (dephasing) coupling under resonant driving, we find that the time-resolved spectrum reveals a distinct fluorescence peak at the Rabi frequency—a signature of dressed-state transitions induced by the dephasing channel that remains hidden in population dynamics. Under mixed σx–σz coupling, the spectrum exhibits combined features of both Mollow triplet and σz-mediated emission. The effects of sub-Ohmic, Ohmic, and super-Ohmic spectral densities are systematically compared. While the resonant spectral weight J(ω0) governs the overall dissipation rate, a controlled comparison at fixed J(ω0) reveals that the super-Ohmic regime exhibits an intrinsic shape-dependent suppression of sideband emission under resonant driving, highlighting an asymmetric role of the spectral density exponent in engineering transient fluorescence. Our work establishes the multi-D2 ansatz as a versatile tool for simulating time-resolved fluorescence in complex bosonic environments.
We study the emission spectrum of two distant qubits strongly coupled to a waveguide by using the numerical and analytical approaches, which are beyond the Markovian approximation and the rotating-wave approximation. The numerical approach combines the Dirac-Frenkel time-dependent variational principle with the multiple Davydov D1{D}_{1}D1​ ansatz. A transformed rotating-wave approximation (TRWA) treatment and a standard perturbation (SP) are used to analytically calculate the emission spectrum. It is found that the variational approach and the TRWA treatment yield accurate emission spectra of the two distant qubits in certain strong-coupling regimes while the SP breaks down. The emission spectrum is found to be asymmetric irrespective of the two-qubit distance and exhibits a single peak, double peaks, and multiple peaks depending on the two-qubit distance as well as the initial states. In sharp contrast with the single-qubit case, the excited-state populations of the two qubits can ultraslowly decay due to the subradiance even in the presence of a strong qubit-waveguide coupling, which in turn yields an ultranarrow emission line. Our results provide insights into the emission spectral features of the two distant qubits in the strong light-matter coupling regime.
We study the emission spectra of the dissipative Rabi and Jaynes-Cummings models in the non-Markovian and ultrastrong coupling regimes. We have derived a polaron-transformed Nakajima-Zwanzig master equation (PTNZE) to calculate the emission spectra, which eliminates the well-known limitations of the Markovian approximation and the standard second-order perturbation. Using the time-dependent variational approach with the multiple Davydov ansatz as a benchmark, the PTNZE is found to yield accurate emission spectra in certain ultrastrong coupling regimes where the standard second-order Nakajima-Zwanzig master equation breaks down. It is shown that the emission spectra of the dissipative Rabi and Jaynes-Cummings models are, in general, asymmetric under various initial conditions. Direct comparisons of spectra for the two models illustrate the essential role of the qubit-cavity counter-rotating term and the spectral features under different qubit-cavity coupling strengths and system initial conditions.
We study the fluorescence spectrum of a strongly driven two-level system (TLS) with modulated transition frequency, which is a bichromatically driven TLS and has multiple resonance frequencies. We are aiming to provide a reliable description of the fluorescence in a regime that is difficult to tackle with perturbation theory and the rotating-wave approximation (RWA), and illustrate the spectral features of the fluorescence under off- and multiphoton-resonance conditions. To go beyond the RWA, we use a semianalytical counter-rotating-hybridized rotating-wave method that combines a unitary transformation and Floquet theory to calculate the two-mode Floquet states and quasienergies for the bichromatically driven TLS. We then solve the master equation accounting for the spontaneous decay in the bases of the two-mode Floquet states, and derive a physically transparent fluorescence spectrum. In comparison with the numerically exact spectrum from the generalized Floquet-Liouville approach, the present spectrum is found to be applicable in a wide range of the parameters where the RWA and the secular approximation may break down. We find that the counter-rotating (CR) terms of the transverse field omitted in the RWA have non-negligible contributions to the spectrum under certain conditions. Particularly, at the multiphoton resonance the width of which is comparable with the Bloch-Siegert shift, the RWA and non-RWA spectra markedly differ from each other because of the CR-induced shift. We also analyze the symmetry of the spectrum in terms of the transition matrix elements between the two-mode Floquet states. We show that the strict symmetry of the spectrum cannot be expected without the RWA but the almost symmetric spectrum can be obtained at the single-photon resonance that takes the Bloch-Siegert shift into account if the driving is moderately strong and at the multiphoton resonance with a sufficiently weak transverse field.
Lindblad master equations describing driven quantum systems coupled to multiple bosonic modes are central to modeling cavity quantum electrodynamics and pseudomode models. In this work, we employ the density-operator-based Dirac-Frenkel time-dependent variational principle combined with the multiple Davydov D2Ansatz to provide optimal solutions for Lindblad master equations in both multimode and driven scenarios. Our approach is benchmarked against numerically "exact" methods in two representative models. The first describes a driven qubit subject to spontaneous emission and coupled to a lossy cavity, while the second corresponds to the pseudomode Lindblad master equation derived from the driven spin-boson model, incorporating seven discrete pseudomodes. The proposed variational approach achieves excellent agreement with numerically "exact" results. Furthermore, we analyze the accuracy of the variational solutions using an error metric based on the Frobenius norm, confirming their reliability. Overall, the present method offers an accurate and computationally efficient framework for simulating Lindblad master equations in complex open quantum systems.
The pseudomode framework provides an efficient description of open quantum systems in non-Markovian and nonperturbative coupling regimes by utilizing simple Markovian Lindblad master equations that incorporate a set of damped bosonic pseudomodes. Developing an efficient numerical scheme to solve these equations accurately is highly desirable. In this work, we propose a modified multiple Davydov D1 density operator ansatz and combine it with the Dirac-Frenkel time-dependent variational principle to simulate the dynamics governed by these master equations. We apply this variational method to the spin-boson model, both with and without a periodic external driving field, and compute both single- and two-time expectation values in non-Markovian, strong-coupling regimes. The accuracy of our variational approach is validated by comparing its predictions with results obtained from the small matrix path integral method. We demonstrate that the variational and path-integral approaches are in excellent agreement for both single- and two-time expectations across nondriven and driven cases, spanning both low- and high-temperature regimes. Our results establish the reliability and broad applicability of the pseudomode-based variational approach for calculating two-time correlation functions in open quantum systems.