Separation of relaxation timescales via strong system-bath coupling: Dissipative three-level system as a case study

TL;DR

This work investigates how strong system–bath coupling reshapes relaxation in a three-level impurity by employing the reaction-coordinate polaron-transform (RCPT) to map a strongly coupled bath onto an effective weakly dissipative system. It analytically derives two distinct relaxation timescales, τ_1 and τ_2, where the fast timescale speeds up with SBC while the slow timescale grows (exponentially at strong coupling) due to anisotropy between dissipative channels characterized by p and q. A metastable state with long-lived coherence emerges between the two timescales, and under sufficiently strong coupling, steady-state coherences persist; these findings are confirmed by RC-QME numerical simulations. The results offer a route for bath-engineered quantum state preparation and provide a unified microscopic framework for understanding metastable dynamics across weak- and strong-coupling limits.

Abstract

We analytically demonstrate that strong system-bath coupling separates the relaxation dynamics of a dissipative quantum system into two distinct regimes: a short-time dynamics that, as expected, accelerates with increasing coupling to the environment, and a slow dynamics that, counterintuitively, becomes increasingly prolonged at sufficiently strong coupling. Using the reaction-coordinate polaron-transform mapping, we uncover the general mechanism behind this effect and derive accurate expressions for both relaxation timescales. Numerical simulations confirm our analytical predictions. From a practical perspective, our results suggest that strong coupling to a dissipative bath can autonomously generate and sustain long-lived quantum coherences, offering a promising strategy for bath-engineered quantum state preparation.

Figures (8)

• Figure 1: (a) A schematic diagram of the three-level system in its initial representation. It consists of a single ground state $\ket{1}$ at zero energy and two excited states, $\ket{2}$ and $\ket{3}$. The splittings $v$ and $v - \Delta$ separate the levels $\ket{1} \leftrightarrow \ket{3}$ and $\ket{1} \leftrightarrow \ket{2}$, respectively. Thick red arrows indicate transitions induced by interaction with a thermal bath at temperature $T$. (b) The model after the RCPT mapping, which reorganizes the energy levels and introduces a non-diagonal term in the effective system Hamiltonian. This latter term connects $\ket{2}$ and $\ket{3}$ with amplitude $h$ (blue curly arrows). The three-level system still interacts with the bosonic environment via the same coupling operator as in the original representation. This operator induces the same transition in the system, but with a modified amplitude, indicated by dashed red arrows. (c) Effective system in a diagonal basis. In this representation, the bath induces anisotropy between transitions $\ket{E_0}\leftrightarrow\ket{E_-}$ and $\ket{E_0}\leftrightarrow\ket{E_+}$, with transition matrix elements denoted by $p$ (fading dashed arrow) and $q$ (thick dashed arrow), respectively.
• Figure 2: Eigenvalues of $\hat{H}^\text{eff}_S$ as a function of $\lambda$. Parameters are $v=1$, $\Delta=0.01$, and $\Omega=10$. Stronger coupling pushes $E_0$ and $E_-$ down to lower energies, while suppressing the splitting between the two. $E_+$ remains relatively unchanged as a function of $\lambda$.
• Figure 3: (a) $p^2$ and $q^2$ as a function of $\lambda$ for $\Delta=\{0.01,0.02,0.03,0.1,0.5\}$ in fading order indicated by the arrows. For large $\lambda$, $p^2\rightarrow 0$ while $q^2\rightarrow1$. (b) Bohr frequencies as a function of $\lambda$ at different values of $\Delta$ [identical to that of panel $(a)$]. For a wide range of coupling strength, the Bohr frequencies are of $\mathcal{O}(v)$. Here, $v=1$, $\Omega=10$, and $\Gamma=0.05$.
• Figure 4: [(a)-(d)] The relaxation timescales $\tau_1$ and $\tau_2$ presented as a function of $\lambda$ at different temperatures and at different choices of $\Delta \in \{0.01,0.02,0.03,0.1,0.5\}$ in fading order indicated by the arrow. Note, $\tau_1$ is almost independent of $\Delta$ while $\tau_2$ is pulled towards shorter timescales upon increasing $\Delta$. It also depends nonmonotonically on $\lambda$. [(e)-(h)] The dissipative timescales $\tau_1$ and $\tau_2$ presented as a function of $\Delta$ at different temperatures and for different choices of $\lambda\in\{0.1,1,3,5,10\}$ in fading order indicated by the arrows. As observed in [(a)-(d)], $\tau_1$ is almost independent of $\Delta$ while $\tau_2$ decreases as a function of $\Delta$ at large enough $\lambda$.
• Figure 5: (a) Comparison of two timescales described by the full expression Eq. \ref{['eq: two timescales']} to the approximated versions. (a) Low temperature comparison using Eq. \ref{['eq: low temperature limit']}. The arrows indicate the direction of increasing $\Delta$. At $T=0.1$, the low temperature limit accurately describes all the key features of the full expression. (b) Comparison at high $T$, $T=5$, given by Eq. \ref{['eq: high T limit final']}. For both limits [(a) and (b)], the ultrastrong coupling shows a slight deviation from the actual values. Parameters are identical to those of Fig. \ref{['fig:figure 3']}.