Two-Qubit Spin-Boson Model in the Strong Coupling Regime: Coherence, Non-Markovianity, and Quantum Thermodynamics

TL;DR

This work analyzes a two-qubit spin-boson model in the strong-coupling, non-Markovian, and non-equilibrium regime using two nonperturbative methods, HEOM and reaction-coordinate mapping (RCM). It characterizes coherence via the $l_1$-norm and non-Markovianity via the BLP trace-distance measure, and examines entropy production and non-equilibrium steady-state transport (spin and heat currents) between two baths at different temperatures. Key findings include robust coherence generation that increases with tunneling amplitudes $oldsymbol{elta}$ and is best sustained for weak system-bath coupling and strong inter-qubit coupling, as well as positive entropy production across all regimes; the spin and heat currents reveal a breakdown of simple current relations when $oldsymbol{elta_1} eq 0$, illustrating intricate thermodynamic behavior beyond the weak-coupling limit. Overall, the results provide guidance for engineering quantum thermal devices, such as diodes and transistors, that operate in the strong-coupling and non-Markovian regime with controlled coherence and transport properties.

Abstract

We investigate the dynamics of a two-qubit open quantum system, in particular the two-qubit spin-boson model in the strong coupling regime, coupled to two thermal bosonic baths under non-Markovian and non-equilibrium conditions. Two complementary approaches, the Hierarchical Equations of Motion (HEOM) and Reaction Coordinate Mapping (RCM), are employed to examine various coupling regimes between the qubits and their respective baths. The dynamical features of the model and the impact of the tunneling amplitude on quantum coherence of the system are probed using the $l_1$-norm of coherence. The model is further shown to have non-Markovian evolution. The nontrivial task of calculating entropy production in the strong-coupling regime is performed using auxiliary density operators in HEOM. Motivated by the realization of a quantum thermal device in the strong-coupling regime, the non-equilibrium steady-state behavior of the system is investigated. Furthermore, the relationship between the heat and spin currents and the tunneling amplitude is probed.

Figures (7)

• Figure 1: A schematic diagram of the two-qubit spin-boson model, depicting an interaction of the qubits with their respective bosonic baths at different temperatures. $K_1, K_{12}$, and $K_2$ characterize the interaction strengths between qubit 1 and the cold bath, between qubits 1 and 2, and between qubit 2 and the hot bath, respectively.
• Figure 2: Variation of the expectation value $\left\langle\sigma^{(1)}_z\right\rangle$ with time using both RCM and HEOM techniques. The parameters are taken to be: $\Delta_1 = 1.0, \epsilon_1 = 0.5\Delta_1, \Delta_2 = 1.0,\epsilon_2 = 0.4\Delta_2, J = 0.1/\pi, \alpha_1 = 0.2/\pi, \alpha_2 = 0.2/\pi, \omega_{c_1} = 0.05\Delta_1, \omega_{c_2} = 0.10\Delta_2, T_1 = 1.04, T_2 = 1.39$. The excited state is taken to be the initial state of the system.
• Figure 3: Variation of $l_1$-norm of coherence ($C_{l_1}$) with time for different values of tunneling amplitudes $\Delta_1 = \Delta_2 = \Delta$ and in (a) SSS, (b) SWS, (c) WSW, and (d) WWW coupling regimes. $\ket{00}_{12}$, where $\ket{0} = 10$, is taken to be the initial state of the two-qubit system. The parameters are discussed in Sec. \ref{['setup']}.
• Figure 4: Variation of trace distance $T\left[\rho_{S, 1}(t), \rho_{S, 2}(t)\right]$ with time for different coupling regimes. Here, $\ket{00}_{12}$, where $\ket{0} = 10$, and $\ket{++}_{12}$, where $\ket{+} = \frac{1}{\sqrt{2}}11$, are taken as the initial states of the two-qubit system. The parameters for different coupling regimes are discussed in Sec. \ref{['setup']}.
• Figure 5: Variation of von Neumann entropy $S[\rho_S(t)]$ with time for the two-qubit spib-boson model in SSS, SWS, WSW, WWW coupling regimes. $\ket{00}_{12}$ is taken as the initial state of the system.