[Paper Review] Does temperature favor quantum coherence of a dissipative two-level system?

Zhiguo Lü, Han Zheng|arXiv (Cornell University)|Apr 10, 2011

Spectroscopy and Quantum Chemical Studies47 references3 citations

TL;DR

This paper investigates quantum coherence in a dissipative two-level system coupled to a spin-bath using a perturbative master equation approach with a unitary transformation incorporating quantum fluctuations via a scattering function $\xi_k$. It finds that increasing temperature does not favor coherence; instead, finite-temperature dynamics in the spin-bath model differ fundamentally from the boson-bath model, with temperature-independent coherence-incoherence transition and temperature-dependent transverse coherence, while preserving exact sum rules and Shiba’s relation in the coherent regime.

ABSTRACT

The quantum dynamics of a two-level system coupled to an Ohmic spin- bath is studied by means of the perturbation approach based on a unitary transformation. A scattering function $ξ_k$ is introduced in the transformation to take into account quantum fluctuations. By the master equation within the Born approximation, nonequilibrium dynamics quantities are calculated. The method works well for the coupling constant $0 < α< α_c$ and a finite bare tunneling $Δ$. It is found that (i) only at zero temperature with small coupling or moderate one does the spin-spin-bath model display identical behavior as the well known spin-boson-bath model; (ii) in comparison with the known results of spin-boson-bath model, the coherence-incoherence transition point, which occurs at $α_c={1/2}[1+ηΔ/ω_c]$, is temperature independent; (iii) the nonequilibrium correlation function $P(t)=$, evolves without temperature dependence while $$ depends on temperature. Both $P(t)$ and $$ not only satisfy their initial conditions, respectively, and also have correct long time limits. Besides, the Shiba's relation and sum rule are exactly satisfied in the coherent regime for this method. Our results show that increasing temperature does not help the system suppress decoherence in the coherent regime, i.e., finite temperature does not favor the coherent dynamics in this regime. Thus, the finite-temperature dynamics induced by two kinds of baths spin-bath and boson-bath exhibit distinctly different physics.

Motivation & Objective

• To understand the role of temperature in quantum coherence for a two-level system coupled to a spin-bath environment.
• To compare the dynamics of spin-bath and boson-bath models in terms of decoherence and coherence preservation.
• To develop a perturbative method that accurately captures nonequilibrium dynamics and satisfies fundamental quantum constraints like sum rules and Shiba’s relation.
• To determine whether finite temperature can suppress decoherence in the spin-bath model, contrary to expectations from some prior approximations.

Proposed method

• A unitary transformation is applied to the Hamiltonian, introducing a scattering function $\xi_k$ to account for quantum fluctuations in the spin-bath environment.
• The master equation is derived within the Born approximation to describe nonequilibrium dynamics of the density matrix elements.
• The method is valid for coupling strength $0 < \alpha < \alpha_c$ and finite bare tunneling $\Delta$, enabling analytical treatment of time evolution.
• Laplace transforms and complex integration techniques are used to compute correlation functions $P(t) = \langle \tau_z(t) \rangle$ and $\langle \tau_x(t) \rangle$.
• The approach ensures exact satisfaction of Shiba’s relation and the sum rule in the coherent regime, validating its consistency.
• The dynamics are analyzed in both Schr"{o}dinger and interaction representations, with inverse Laplace transforms applied to obtain time-domain expressions.

Experimental results

Research questions

• RQ1 Does increasing temperature enhance quantum coherence in a two-level system coupled to a spin-bath?
• RQ2 How does the coherence-incoherence transition point depend on temperature in the spin-spin-bath model?
• RQ3 What is the temperature dependence of the longitudinal ($\langle \tau_z(t) \rangle$) and transverse ($\langle \tau_x(t) \rangle$) correlation functions?
• RQ4 How do the dynamics of the spin-bath model differ from those of the well-known spin-boson-bath model at finite temperature?
• RQ5 Can a perturbative approach preserve fundamental quantum constraints like Shiba’s relation and sum rules in the presence of a spin-bath?

Key findings

• The longitudinal correlation function $P(t) = \langle \tau_z(t) \rangle$ is independent of temperature, indicating no thermal enhancement of coherence in the z-direction.
• The transverse correlation function $\langle \tau_x(t) \rangle$ explicitly depends on temperature, showing thermal effects on off-diagonal coherence.
• The coherence-incoherence transition occurs at $\alpha_c = \frac{1}{2}[1 + \eta\Delta/\omega_c]$, which is temperature-independent, contradicting some prior approximations.
• The method exactly satisfies Shiba’s relation and the sum rule in the coherent regime, confirming its theoretical consistency.
• Finite temperature does not suppress decoherence; instead, it does not favor coherent dynamics, indicating that higher temperatures do not improve quantum coherence in this model.
• The spin-bath and boson-bath models exhibit distinctly different finite-temperature physics, with the spin-bath showing no thermal stabilization of coherence.