Local boson-nonlocal boson coupling in a four-level system: Adiabatic, non-adiabatic, and non-Hermitian effects

Abstract

We investigates the dynamics of an open quantum system comprising a two-level electronic system coupled to local boson mode and a bosonic bath. The system is described by four distinct states, including the ground and excited electronic states, each with its corresponding zero- and one-boson vibrational levels. The dissipative dynamics arising from interactions with an external environment are modeled using two distinct theoretical frameworks: the standard Lindblad master equation and a non-Hermitian effective Hamiltonian approach. We derive the full Liouvillian superoperator for both formalisms, revealing a crucial distinction: while the Lindblad equation accounts for both state decay and repopulation via quantum jumps, the non-Hermitian formalism only captures the decay, leading to non-conservation of the total system probability.

Figures (12)

• Figure 1: The horizontal lines in each electronic potential well represent the quantized vibrational energy levels. The upper well is the excited electronic surface, and the lower well is the ground electronic surface. The two potential energy surfaces have different curvatures: The lower well is wider than the upper well (and $\omega_{0}>\omega_{1}$) since the bond becomes weaker during the transition from ground electronic state to excited electronic state. In ground electronic state, we consider the boson wavepacket locates in its lowest vibrational level at equilibrium position. In excited electronic state, we consider the boson wavepacket locates above the lowest vibrational level at nonequilibrium position.
• Figure 2: ${\rm Tr}[\rho(0)\rho(t)]$, expectation of $H$, and rate function $G(t)$ according to the lindblad master equation Eq.(\ref{['21']}) in Hermitian case. We set $\epsilon=1.5$, $\omega_{1}=0.8$, $\omega_{k}=1.2$, $\gamma=1$. The color changing from yellow to blue correspond to the increase of $\omega_{0}$ from $\omega_{0}=\omega_{1}$ to $\omega_{0}=\omega_{1}+1$. The left panels correspond to real $W=1.5$, and the right panels correspond to complex $W=1.5+10i$. A transition from exponential decay to power law decay can be observed for the survival probability decay rate $G(t)$ in early time.
• Figure 3: ${\rm Tr}[\rho(0)\rho(t)]$, expectation of $H$, and survival probability decay rate $G(t)$ according to the lindblad master equation about the transitions between the four levels (Eq.(\ref{['25']})) in Hermitian case. A transition from exponential decay to power law decay can be observed for the rate function $G(t)$ in early time. Despite the presence of Liouvilian gap causing the damping effect, the sustained oscillations for those specific modes give rise to deviation from the non-exponential decay. Here we set the temperature as $T=\omega_0$.
• Figure 4: The same with Fig.\ref{['transiH']} but at higher temperature $T=10\omega_0$.
• Figure 5: Schematic of Liouvalian in Hermitian case for transitions between boson operators (a) and between the four levels (b).