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Advanced Quantum Technologies

Cavity Modified Dissipative Dynamics of the Two‐Spin Spin‐Boson Model

Ningshuang Zhao, Yiying Yan, Zhiguo Lü, Yang Zhao

Published Nov 20, 2025• 2 citations
Publisher

Abstract

ABSTRACT We investigate the dynamics of two spins that are simultaneously coupled to a high‐Q cavity and an Ohmic phonon bath using the Dirac–Frenkel time‐dependent variational principle together with two types of the multiple Davydov ansatz. In our framework, the cavity mode can either be treated as part of the bath modes within the multi‐D 2 ansatz or regarded together with the spins as a dressed composite system within the multi‐D 1 ansatz. Although both approaches yield consistent results for the spin, cavity, and phonon observables, the multi‐D 1 ansatz proves to be numerically much more efficient than the multi‐D 2 ansatz. In the regime of relatively weak spin‐bath coupling, we find that the cavity induces intricate beat patterns in the spin population dynamics as well as multipeaked phonon spectra. These cavity‐related features can also be reproduced by a polaron‐like transformed Nakajima–Zwanzig master equation, where the multipeaked phonon spectra arise from transitions between dressed states. By contrast, under relatively strong spin‐bath coupling, both the beat patterns and the multipeaked phonon spectra become suppressed. Furthermore, even with strong spin‐cavity coupling, sufficiently strong spin‐bath interaction leads to spin localization, reminiscent of the single‐spin spin‐boson model. Overall, our results reveal how spin‐bath and spin‐cavity couplings compete to shape the dissipative dynamics.

DOI
10.1002/qute.202500727

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ABSTRACT We investigate the dynamics of two spins that are simultaneously coupled to a high‐Q cavity and an Ohmic phonon bath using the Dirac–Frenkel time‐dependent variational principle together with two types of the multiple Davydov ansatz. In our framework, the cavity mode can either be treated as part of the bath modes within the multi‐D 2 ansatz or regarded together with the spins as a dressed composite system within the multi‐D 1 ansatz. Although both approaches yield consistent results for the spin, cavity, and phonon observables, the multi‐D 1 ansatz proves to be numerically much more efficient than the multi‐D 2 ansatz. In the regime of relatively weak spin‐bath coupling, we find that the cavity induces intricate beat patterns in the spin population dynamics as well as multipeaked phonon spectra. These cavity‐related features can also be reproduced by a polaron‐like transformed Nakajima–Zwanzig master equation, where the multipeaked phonon spectra arise from transitions between dressed states. By contrast, under relatively strong spin‐bath coupling, both the beat patterns and the multipeaked phonon spectra become suppressed. Furthermore, even with strong spin‐cavity coupling, sufficiently strong spin‐bath interaction leads to spin localization, reminiscent of the single‐spin spin‐boson model. Overall, our results reveal how spin‐bath and spin‐cavity couplings compete to shape the dissipative dynamics.
Dynamics of the two-spin spin-boson model in the presence of Ohmic and sub-Ohmic baths is investigated by employing a multitude of the Davydov D1 trial states, also known as the multi-D1 Ansatz. Its accuracy in dynamics simulations of the two-spin SBM is improved significantly over the single D1 Ansatz, especially in the weak to moderately strong coupling regime. Validity of the multi-D1 Ansatz for various coupling strengths is also systematically examined by making use of the deviation vector which quantifies how faithfully the trial state obeys the Schrödinger equation. The time evolution of population difference and entanglement has been studied for various initial conditions and coupling strengths. Careful comparisons are carried out between our approach and three other methods, i.e., the time-dependent numerical renormalization group (TD-NRG) approach, the Bloch-Redfield theory, and a method based on a variational master equation. For strong coupling, the multi-D1 trial state yields consistent results as the TD-NRG approach in the Ohmic regime while the two disagree in the sub-Ohmic regime, where the multi-D1 trial state is shown to be more accurate. For weak coupling, the multi-D1 trial state agrees with the two master-equation methods in the presence of both Ohmic and sub-Ohmic baths, but shows considerable differences with the TD-NRG approach in the presence of a sub-Ohmic bath, calling into question the validity of the TD-NRG results at long times in the literature.
The Dirac-Frenkel time-dependent variation is employed to probe the dynamics of the zero temperature sub-Ohmic spin-boson model with strong friction utilizing the Davydov D1 ansatz. It is shown that initial conditions of the phonon bath have considerable influence on the dynamics. Counterintuitively, even in the very strong coupling regime, quantum coherence features still manage to survive under the polarized bath initial condition, while such features are absent under the factorized bath initial condition. In addition, a coherent-incoherent transition is found at a critical coupling strength α ≈ 0.1 for s = 0.25 under the factorized bath initial condition. We quantify how faithfully our ansatz follows the Schrödinger equation, finding that the time-dependent variational approach is robust for strong dissipation and deep sub-Ohmic baths (s ≪ 1).
Developed originally for the Holstein polaron, the Davydov D1 ansatz is an efficient, yet extremely accurate trial state for time-dependent variation of the spin-boson model [N. Wu, L. Duan, X. Li, and Y. Zhao, J. Chem. Phys. 138, 084111 (2013)]. In this work, the Dirac-Frenkel time-dependent variational procedure utilizing the Davydov D1 ansatz is implemented to study entanglement dynamics of two qubits under the influence of two independent baths. The Ohmic spectral density is used without the Born-Markov approximation or the rotating-wave approximation. In the strong coupling regime finite-time disentanglement is always found to exist, while at the intermediate coupling regime, the entanglement dynamics calculated by Davydov D1 ansatz displays oscillatory behavior in addition to entanglement disappearance and revival.
Abstract A Rabi dimer is used to model a recently reported circuit quantum electrodynamics system composed of two coupled transmission‐line resonators with each being coupled to one qubit. In this study, a phonon bath is adopted to mimic the multimode micromechanical resonators and is coupled to the qubits in the Rabi dimer. The dynamical behavior of the composite system is studied by the Dirac–Frenkel time‐dependent variational principle combined with the multiple Davydov D2 ansätze. Initially, all the photons are pumped into the left resonator, and the two qubits are in the down state coupled with the phonon vacuum. In the strong qubit–photon coupling regime, the photon dynamics can be engineered by tuning the qubit–bath coupling strength α and photon delocalization is achieved by increasing α. In the absence of dissipation, photons are localized in the initial resonator. Nevertheless, with moderate qubit–bath coupling, photons are delocalized with quasiequilibration of the photon population in two resonators at long times. In this case, high‐frequency bath modes are activated by interacting with depolarized qubits. For strong dissipation, photon delocalization is achieved via frequent photon‐hopping within two resonators and the qubits are suppressed in their initial down state.
ABSTRACT Markovian pseudomode models are shown to accurately describe non‐Markovian and non‐perturbative reduced dynamics and multi‐time correlation functions under certain constraints. However, it is challenging to solve the Lindblad master equation for such models when a number of pseudomodes are involved. A variational approach based on multiple Davydov states is used to solve the master equation of pseudomode models for the multi‐mode Jaynes–Cummings (JC) model, and the spin‐boson system across four distinct configurations, enabling computation of the two‐time correlation function that incorporates bath phonon occupation dynamics. In parallel, calculations of the unitary evolution of full Hamiltonians are chosen as benchmark references. Benchmark comparisons demonstrate that the variational approach for Markovian pseudomode models successfully predicts both the single‐time and two‐time expectations within the multi‐mode JC model. In contrast, in the spin‐boson model, the variational simulations for the pseudomode system and the full Hamiltonian yield inconsistent bath occupation numbers in a non‐perturbative strong coupling regime despite that these two kinds of simulations predict almost identical reduced dynamics, resulting from a fitting artefact in the pseudomode model. Our findings offer insights into the applicability of the non‐unitary variational approach on the simulation of two‐time correlation functions within the pseudomode framework.
The dynamics of the (sub-)Ohmic spin-boson model under various bath initial conditions is investigated by employing the Dirac–Frenkel time-dependent variational approach with the multiple Davydov D1 Ansatz in the interaction picture. The validity of our approach is carefully checked by comparing the results with those of the hierarchy equations of motion method. By analyzing the features of nonequilibrium dynamics, we identify the phase diagrams for different bath initial conditions. We find that for the spectral exponent s < sc, there exists a transition from coherent to quasicoherent dynamics with increasing coupling strengths. For sc < s ≤ 1, the coherent to incoherent crossover occurs at a certain coupling strength and the quasicoherent dynamics emerges at much larger couplings. The initial preparation of the bath has a considerable influence on the dynamics.