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Open quantum systems beyond equilibrium: Lindblad equation and path integral molecular dynamics

Benedikt M. Reible, Somayeh Ahmadkhani, Luigi Delle Site

Abstract

The Lindblad equation determines the time evolution of the density operator of open quantum systems. While valid for any system size, its use is, in practice, restricted to prototype/surrogate models with the aim of tackling specific aspects of the overall quantum complexity of a multi-atomic system. Path integral molecular dynamics (PIMD) instead provides static and dynamical quantum statistical averages of physical observables for systems in equilibrium composed of up to thousands of atoms over timescales up to nanoseconds, under the condition that short-time quantum coherence is not relevant for the properties of interest. PIMD relies on the well-established technique of molecular dynamics (MD) with its associated classical trajectories. However, it cannot describe a direct time evolution of a system and its convergence to a stationary state in situations out of equilibrium. In this work, we analyze the link between the Lindblad equation and PIMD; specifically, we will discuss how PIMD can actually be used to calculate the time evolution of ensemble-averaged physical observables and their convergence to a stationary state for situations out of equilibrium, bypassing the need of explicitly solving the Lindblad equation. Yet, at the same time, the Lindblad equation and PIMD are linked to one another through a formal relation of equivalence, which provides an argument for the consistency of PIMD results, namely the positivity of the density operator at any time. A numerical study of a prototype system, which is of interest in chemical physics, will be used to showcase the method.

Open quantum systems beyond equilibrium: Lindblad equation and path integral molecular dynamics

Abstract

The Lindblad equation determines the time evolution of the density operator of open quantum systems. While valid for any system size, its use is, in practice, restricted to prototype/surrogate models with the aim of tackling specific aspects of the overall quantum complexity of a multi-atomic system. Path integral molecular dynamics (PIMD) instead provides static and dynamical quantum statistical averages of physical observables for systems in equilibrium composed of up to thousands of atoms over timescales up to nanoseconds, under the condition that short-time quantum coherence is not relevant for the properties of interest. PIMD relies on the well-established technique of molecular dynamics (MD) with its associated classical trajectories. However, it cannot describe a direct time evolution of a system and its convergence to a stationary state in situations out of equilibrium. In this work, we analyze the link between the Lindblad equation and PIMD; specifically, we will discuss how PIMD can actually be used to calculate the time evolution of ensemble-averaged physical observables and their convergence to a stationary state for situations out of equilibrium, bypassing the need of explicitly solving the Lindblad equation. Yet, at the same time, the Lindblad equation and PIMD are linked to one another through a formal relation of equivalence, which provides an argument for the consistency of PIMD results, namely the positivity of the density operator at any time. A numerical study of a prototype system, which is of interest in chemical physics, will be used to showcase the method.
Paper Structure (14 sections, 1 theorem, 39 equations, 5 figures)

This paper contains 14 sections, 1 theorem, 39 equations, 5 figures.

Table of Contents

  1. Introduction
  2. From the Lindblad equation to PIMD
  3. Path integral representation of the density matrix
  4. Path Integral Molecular Dynamics
  5. Going beyond equilibrium
  6. Classical D-NEMD method
  7. Extension to quantum systems
  8. Positivity of rho(t)
  9. Accuracy
  10. Application of the method
  11. Conclusions
  12. Derivation of the Redfield and Lindblad equation
  13. Limit to equilibrium in the NPI method
  14. Technical details of the simulation

Key Result

Theorem 1

Suppose that $\dim \mathcal{H}_S < + \infty$. Then the time-evolution $\hat{\rho}_S(t) = \Phi(t) \hat{\rho}_S(0)$ is completely positive, meaning that $\hat{\rho}_S(t)$ is a positive operator for all $t > 0$, if and only if the generator $\mathcal{L}$ of $(\Phi(t))_{t \ge 0}$, defined via $\frac{\ma with $\lambda_j \ge 0$, $\hat{L}_j$ bounded operators on $\mathcal{H}_S$, and $\hat{H}$ a self-adjo

Figures (5)

  • Figure 1: Panel (a) illustrates the mapping from the classical representation of an atom as a spherical object to the quantum representation as a polymer ring whose beads are connected by harmonic oscillators, here for $P = 6$. Panel (b) shows how interactions are modeled in the polymer ring representation, here for $N = 2$ particles and $P = 6$: only beads with the same index on different polymer rings interact with each other.
  • Figure 2: Illustration of the D-NEMD protocol. The horizontal curve at the top represents the equilibrium trajectory. The vertical curves, originating from temporally equidistant points along the former, represent trajectories where the external perturbation acts on the particles. The non-equilibrium statistical average of a physical observable is calculated using the values of the observable on points of the branched trajectory on the horizontal dotted lines.
  • Figure 3: Temperature profile of a one-dimensional chain of 87 water molecules, each one modeled as a polymer ring (pictorially visualized above the $x$-axis), in a thermal gradient for different values of the number of beads $P$. On the left-hand side of the chain is a "hot" temperature reservoir at $330 \, \text{K}$, and on its right-hand side is a "cold" reservoir at $280 \, \text{K}$. As expected, the chain reaches a steady state with a uniform temperature at around $305 \, \text{K}$, and the higher the value of $P$, the smaller the deviation of the temperature along the chain from this steady-state value.
  • Figure 4: Heat flux along the one-dimensional chain of water molecules for different values of the bead number $P$. The data illustrates that for $P = 64$, the heat flux has essentially converged.
  • Figure 5: Illustration of the limit to equilibrium of the D-NEMD method. The same explanations as given in \ref{['fig:d-nemd']} apply, but now the vertical lines correspond to trajectories that are equivalent to the equilibrium trajectory starting from the point taken from the horizontal curve, which is the main equilibrium trajectory.

Theorems & Definitions (2)

  • Remark
  • Theorem