Non-Markovianity in a dressed qubit with local dephasing

TL;DR

The paper studies a dressed qubit realized as a spinless fermion hopping between two sites, each locally coupled to phonon baths, and analyzes non-Markovian decoherence. It uses the Lang-Firsov transformation to move to the polaron frame and a second-order time-convolutionless master equation to obtain the dynamics in the singlet-triplet basis, revealing a delocalization-to-localization transition at strong coupling and long-lived coherence. Bath spectral properties, modeled by J1(ω) = α ω^s e^{-ω^2/Ω^2} and J2(ω) = β ω^{s'} e^{-ω^2/Ω^2}, determine memory effects: sub-Ohmic baths yield pronounced non-Markovianity at moderate couplings, while Ohmic and super-Ohmic baths require larger couplings for memory, with super-Ohmic cases sometimes showing memory without coherence revivals. Non-Markovianity is quantified via the l1-norm coherence measure, highlighting that memory can persist even without visible revivals, due to information backflow and structural features of the dynamical map.

Abstract

We study the dynamics of a dressed qubit implemented by a spinless fermion hopping between two lattice sites with each site strongly coupled to a bath of phonons. We employ Lang-Firsov transformation to make the problem tractable perturbatively. Applying time-convolutionless master equation within the polaron frame, we investigate decoherence dynamics of the dressed qubit within the singlet-triplet basis of the system for a wide range of bath spectral densities. It is shown that the coherence persists for longer time scales for large coupling values and shows non-monotonic behaviour reflecting the presence of non-Markovianity in the dynamics. Non-Markovianity, characterized by coherence revivals and non-monotonic decay patterns, emerges distinctly depending on the bath spectrum and coupling strengths. Systems coupled to sub-Ohmic baths, whether both or in combination with another type, display pronounced memory effects at relatively small values of couplings. In contrast, combinations involving Ohmic and super-Ohmic baths exhibit noticeable non-Markovianity only at higher couplings.

Figures (11)

• Figure 1: Variation of the renormalized hopping amplitude, $\frac{\tilde{\mathcal{J}}}{\mathcal{J}}$, as a function of relative coupling strengths $\alpha$ and $\beta$ for the two sites, each coupled to independent phonon baths with spectral densities $J_1(\omega) = \alpha \, \omega^s e^{-\omega^2/\Omega^2}$ and $J_2(\omega) = \beta \, \omega^{s'} e^{-\omega^2/\Omega^2}$, (see section \ref{['PP']} for details). The bath exponents are chosen to represent different regimes: sub-Ohmic ($0<s,s'<1$), Ohmic ($s=s'=1$), and super-Ohmic ($s,s'>1$). We see that in all possible combinations, the effective coupling $\tilde{\mathcal{J}}$ is reduced in polaron frame.
• Figure 2: Time evolution of the population difference $P_D(t)= \langle S|\rho(t)|S\rangle - \langle T|\rho(t)|T\rangle$, populations of the $\lvert \mathcal{S} \rangle$ and $\lvert \mathcal{T} \rangle$ states, and coherence for different bath coupling strengths $\alpha$ and $\beta$, with initial state $\lvert \psi_0 \rangle = \sqrt{\frac{2}{3}}\,\lvert \mathcal{S} \rangle + \sqrt{\frac{1}{3}}\,\lvert \mathcal{T} \rangle$, for the super-Ohmic case ($s = s' = 2$). We observe a delocalization-to-localization transition: at strong couplings ($\alpha=\beta=5$), populations remain largely stationary with a persistent population difference and long-lived coherence, while at weak couplings ($\alpha=\beta=0.1$), populations equilibrate and coherence decays monotonically.
• Figure 3: Dynamics for the combination of Ohmic ($s = 1$) and super-Ohmic ($s' = 2$) spectral environments.
• Figure 4: Dynamics for Sub-Ohmic ($s = 0.5$) spectral environments.
• Figure 5: Non-Markovianity measure $\mathcal{N}$ as a function of coupling parameters $\alpha$ and $\beta$, for spectral exponent pairs $\{s, s'\} = \{0.5,0.5\}, \{0.5,1\}, \{0.5,2\}, \{1,1\}, \{1,2\}, \{2,2\}$. Non-Markovianity is prominent at intermediate couplings for sub-Ohmic baths, appears only at strong couplings for Ohmic–Ohmic cases, and remains negligible for mixed combinations. For super-Ohmic baths, non-Markovianity persists without coherence revivals, indicating subtle memory effects.