The Phase-Coupled Caldeira-Leggett Model: Non-Markovian Open Quantum Dynamics beyond Linear Dissipation

TL;DR

This work introduces the Phase-Coupled Caldeira-Leggett model with an exponential system–bath coupling $H_{ ext{SB}} \\propto e^{i\\lambda \\hat{F}}$ and develops an exact, nonperturbative framework using the dissipaton formalism and generalized normal ordering to obtain a closed equation of motion for the reduced density operator that fully preserves non-Markovian effects. The approach reveals a phase-mediated renormalization of the system Hamiltonian, nontrivial steady states that do not commute with $H_S$, and coherent dynamics distinct from the conventional Caldeira–Leggett model, including an effective $H_S^{\\text{eff}} \\approx H_S + 2g\\,S$ with $g = e^{-\\lambda^2\\langle \\hat{F}^2\\rangle_B/2}$. These findings demonstrate that nonlinear, phase-type environment couplings can qualitatively alter decoherence and relaxation, with implications for quantum transport, strong light–matter coupling, and molecular junctions.

Abstract

We introduce the \textit{Phase-Coupled Caldeira-Leggett} (PCL) model of quantum dissipation and develop an exact framework for its dynamics. Unlike the conventional Caldeira-Leggett model with linear system-bath coupling $H_{\mathrm{SB}}\propto\hat F$, the PCL model features an exponential interaction $H_{\mathrm{SB}}\propto e^{iλ\hat F}$, where $\hat F$ denotes the collective bath coordinate. This model unifies concepts from quantum Brownian motion and polaron physics, providing a general platform to study phase-mediated dissipation and decoherence beyond the linear-response regime. Despite its nonlinear system-bath coupling, the Gaussian nature of the environment allows a nonperturbative and non-Markovian treatment of PCL model within the algebra of dissipative quasiparticles. We obtain an exact closed-form equation of motion for the reduced density operator, and numerical simulations reveal distinctive dynamical behaviors that deviate markedly from those predicted by the conventional Caldeira-Leggett model.

Figures (4)

• Figure 1: (a) Schematic illustration of the PCL model [cf. Eq. (\ref{['HSB']})] and DD of bath [cf. Eq. (\ref{['map1']})]. A quantum system interacts with a harmonic bath through an exponential, phase-type coupling. This interaction mediates dissipation through phase modulation rather than linear displacement, distinguishing the PCL model from the conventional CL framework. The bath is further decomposed into dissipatons via the DD, preserving complete non-Markvian informations. (b) Illustration of the equations of motion Eq. (\ref{['eom']}) in a hierarchical structure, showing the coupings across different levels of the dynamical varibales, including coupling of the nearest neighbor layer, next-nearest neighbor layer, next-next-nearest neighbor layer, and so on.
• Figure 2: Numerical results for the two-level system dynamics under the PCL and CL system–bath interactions. The system Hamiltonian is $H_{\hbox{\tiny S}} = \epsilon_{\hbox{\tiny S}}\hat{\sigma}_z$ with the coupling operator $\hat{S} = \alpha\hat{\sigma}_x$. The bath is modeled by the Drude spectral density, $J(\omega) = \xi \omega / (\omega^2 + \gamma^2)$. The parameters are chosen as $\alpha = \epsilon_{\hbox{\tiny S}}$, $\gamma = \epsilon_{\hbox{\tiny S}}$, $k_BT = 2\epsilon_{\hbox{\tiny S}}$, $\xi = 1$, and $\lambda = 0.5$. Here, we evaluate the transient expectations of Pauli matrices, $\bar{\sigma}_i(t) \equiv {\rm tr}_{\hbox{\tiny S}}[\hat{\sigma}_i\rho_{\hbox{\tiny S}}(t)]$. For both the PCL and CL Hamiltonians, panels (a) and (b) show the population dynamics characterized by $\bar{\sigma}_z(t)$ and $|\bar{\bm\sigma}(t)| \equiv \sqrt{\bar{\sigma}_x^2(t) + \bar{\sigma}_y^2(t) + \bar{\sigma}_z^2(t)}$, respectively. Panels (c) and (d) depict the corresponding trajectories within the Bloch sphere for the PCL and CL cases. Remarkably, under the PCL interaction, the eigenvectors of the steady state $\lim_{t\to+\infty}\rho_{\hbox{\tiny S}}(t)$ deviate significantly from the eigenstates of $H_{\hbox{\tiny S}}$.
• Figure 3: Population of system density operator in the instantaneous eigenbasis and von Neumann entropy calculated with $\lambda = 0.5, 1$, and $2$. Other parameters are given by $\alpha = 2\epsilon_{\hbox{\tiny S}}$, $\gamma = \epsilon_{\hbox{\tiny S}}$, $k_BT = 2\epsilon_{\hbox{\tiny S}}$, and $\xi = 1$. The steady state under the PCL model shows a nonmonotonic dependence on $\lambda$, remaining low entropy in both the weak and strong coupling limits.
• Figure 4: Population of system density operator in the instantaneous eigenbasis and von Neumann entropy calculated with $\alpha = 0.5, 1, 1.5$, and $2\epsilon_{\hbox{\tiny S}}$. Other parameters are given by $\lambda = 0.5$, $\gamma = \epsilon_{\hbox{\tiny S}}$, $k_BT = 2\epsilon_{\hbox{\tiny S}}$, and $\xi = 1$. The equilibrium entropy decreases monotonically when increasing $\alpha$.