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Markovianity and non-Markovianity of Particle Bath with Dirac Dispersion Relation

Takano Taira, Naomichi Hatano, Akinori Nishino

Abstract

The decay rate of quantum particles in open quantum systems has traditionally been known as exponential, based on empirical predictions from experiments and theoretical predictions from the Markovian dynamics of the corresponding quantum states. However, both theoretical predictions and experimental observations suggest deviations from this exponential decay, particularly in the short and long time regimes. In this study, we explore the spontaneous emission of a single Dirac particle within an environment characterized by an energy spectrum with a gap $m$ and an energy cutoff $L$. Our results reveal that high-energy structures, such as the spectral cutoff $L$, play a critical role in driving the short-time non-exponential decay. In contrast, the long-time decay is predominantly influenced by low-energy structures, such as the Dirac gap $m$. Surprisingly, we find that in the limits where the energy cutoff $L$ is infinite and the energy gap $m$ is zero, the decay dynamics of massless Dirac particles exhibit Markovian behavior without the need for conventional approximations like the Born-Markov approximation. This work underscores the complex interplay between particle energy properties and decay dynamics, providing new insights into quantum decay processes.

Markovianity and non-Markovianity of Particle Bath with Dirac Dispersion Relation

Abstract

The decay rate of quantum particles in open quantum systems has traditionally been known as exponential, based on empirical predictions from experiments and theoretical predictions from the Markovian dynamics of the corresponding quantum states. However, both theoretical predictions and experimental observations suggest deviations from this exponential decay, particularly in the short and long time regimes. In this study, we explore the spontaneous emission of a single Dirac particle within an environment characterized by an energy spectrum with a gap and an energy cutoff . Our results reveal that high-energy structures, such as the spectral cutoff , play a critical role in driving the short-time non-exponential decay. In contrast, the long-time decay is predominantly influenced by low-energy structures, such as the Dirac gap . Surprisingly, we find that in the limits where the energy cutoff is infinite and the energy gap is zero, the decay dynamics of massless Dirac particles exhibit Markovian behavior without the need for conventional approximations like the Born-Markov approximation. This work underscores the complex interplay between particle energy properties and decay dynamics, providing new insights into quantum decay processes.

Paper Structure

This paper contains 14 sections, 52 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Survivial probability of two-level system with half-filled Dirac environment
  3. Model
  4. Master equation
  5. Spectural structure and Markovianity
  6. Low energy structure and Markovianity: Lg limit
  7. High energy structure and Markovianity: Lg case
  8. Pole contributions
  9. Branch cut contribution
  10. Spectrual structure and Markovianity: Numerics for Lg
  11. Conclusion
  12. Quantum Zeno Effect
  13. Contour deformation
  14. Laplace transform

Figures (9)

  • Figure 1: Panel (a) shows the contour, branch cuts, and poles of the Bromwich integral (\ref{['Eq:Simpler_solution']}), which is equivalent to the contour shown in panel (b).
  • Figure 2: Phase diagram of solutions to the fourth-order polynomial $\mathcal{P}(z) = (z+i \omega_0 )^2(m^2 + z^2 ) - g^4 \pi^2 z^2$, where we put $\hbar=\omega_0=1$.
  • Figure 3: All three panels are showing the plot for $\hbar=1,\omega_0 =0.1, m=1, g= 0.6$ in different time regimes. (a) the plot of the survival probability $P(t)$ given by Eq. (\ref{['Eq:L_infity_exact_solution']}). The horizontal red dashed line indicates the absolute value squared of the pole contribution (\ref{['Eq:L_infty_bound']}) (b) the log-log plot of $1-P(t)$ for the exact solution (\ref{['Eq:L_infity_exact_solution']}), plotted along $t$ and $t^2$. (c) the log plot of the numerical solution and approximations up to $t^{-3/2}$and upto $t^{-5/2}$ in Eq. (\ref{['Eq:L_infty_approx']}).
  • Figure 4: Panel (a) and (b) show the plot of poles on the principle Riemann surface for $L=2, g=1, \omega_0 =0$ and $L=2, g=1, \omega_0 =8$ respectively.
  • Figure 5: (a) Contour plot of $\text{exp}(-|x_1 - x_2|)$, showing the periodic regions where $x_1 \sim x_2$ and bound region where $x_1$ and $x_2$ are different. Two points indicate the parameter choices for the plot of the survival probability in Fig.\ref{['fig: L_finite_Prob']}. (b) Contour plot of $-\log (|\text{Pole}(0)|^2)$, with overlap of the contour lines of $\text{exp}(-|x_1 - x_2|)$.
  • ...and 4 more figures