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Open harmonic chain without secular approximation

Melika Babakan, Fabio Benatti, Laleh Memarzadeh

Abstract

We study particle and energy transport in an open quantum system consisting of a three harmonic oscillator chain coupled to thermal baths at different temperatures placed at the ends of the chain. We consider the exact dynamics of the open chain and its so-called local and global Markovian approximations. By comparing them, we show that, while all three yield a divergence-like continuity equation for the probability flow, the energy flow exhibits instead a distinct behavior. The exact dynamics and the local one preserve a standard divergence form for the energy transport, whereas the global open dynamics, due to the rotating wave approximation (RWA), introduces non-divergence sink/source terms. These terms also affect the continuity equation in the case of a master equation obtained through a time-coarse-graining method whereby RWA is avoided through a time-zoom parameter $Δt$. In such a scenario, sink and source contributions are always present for each $Δt>0$. While in the limit $Δt\to+\infty$ one recovers the global dissipative dynamics, sink and source terms instead vanish when $Δt\to 0$, restoring the divergence structure of the exact dynamics. Our results underscore how the choice of the dissipative Markovian approximation to an open system dynamics critically influences the energy transport descriptions, with implications for discriminating among them and thus, ultimately, for the correct modeling of the time-evolution of open quantum many-body systems.

Open harmonic chain without secular approximation

Abstract

We study particle and energy transport in an open quantum system consisting of a three harmonic oscillator chain coupled to thermal baths at different temperatures placed at the ends of the chain. We consider the exact dynamics of the open chain and its so-called local and global Markovian approximations. By comparing them, we show that, while all three yield a divergence-like continuity equation for the probability flow, the energy flow exhibits instead a distinct behavior. The exact dynamics and the local one preserve a standard divergence form for the energy transport, whereas the global open dynamics, due to the rotating wave approximation (RWA), introduces non-divergence sink/source terms. These terms also affect the continuity equation in the case of a master equation obtained through a time-coarse-graining method whereby RWA is avoided through a time-zoom parameter . In such a scenario, sink and source contributions are always present for each . While in the limit one recovers the global dissipative dynamics, sink and source terms instead vanish when , restoring the divergence structure of the exact dynamics. Our results underscore how the choice of the dissipative Markovian approximation to an open system dynamics critically influences the energy transport descriptions, with implications for discriminating among them and thus, ultimately, for the correct modeling of the time-evolution of open quantum many-body systems.
Paper Structure (22 sections, 155 equations, 2 figures)

This paper contains 22 sections, 155 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Markovian global and local master equations
  4. Global master equation
  5. Local master equation
  6. Time coarse-graining master equation
  7. Probability density continuity equation
  8. Exact reduced dynamics
  9. Local approach
  10. Global approach
  11. Time-coarse-graining approach
  12. Energy continuity equation
  13. Exact reduced dynamics
  14. Local approach
  15. Global approach
  16. ...and 7 more sections

Figures (2)

  • Figure 1: The schematic representation of a three interacting harmonic oscillators which also interact with multi-mode external thermal baths at the two ends with temperatures $T_\ell$ and $T_r$.
  • Figure 2: Here, we plot $\expval{Q_\alpha}_\infty^{(\beta)}$ for $\alpha=L,R$ and $\beta=\rm{tcg},\rm{glb}$ with the steady state $\rho_\infty^{(\rm tcg)}$ and $\rho_\infty^{(\rm glb)}$, respectively. The blue and orange lines are $\expval{Q_L}_\infty^{(\rm{tcg})}$ and $\expval{Q_R}_\infty^{(\rm{tcg})}$, while the green dashed line and red dotted-dashed lines are $\expval{Q_L}_\infty^{(\rm{glb})}$ and $\expval{Q_R}_\infty^{(\rm{glb})}$, respectively. We set $\lambda=0.1$, $\omega_0=1$, $\omega_c=3$, $T_L=10$ and $T_R=1$.

Theorems & Definitions (2)

  • Remark 1
  • Remark 2