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Modifying the Time-Convolutionless Master Equation via the Moore-Penrose Pseudoinverse

Caleb Blumenfeld

TL;DR

This work addresses TCL-ME breakdown by replacing the standard inverse with the Moore-Penrose pseudoinverse and deriving a perturbative expansion via Israel and Charnes. It tests the TCL+-ME on the Jaynes-Cummings model and an Ising-bath model, finding no improvement over TCL and even worse performance as the bath size grows. The convergence analysis shows the MP-series has a stricter condition, with convergence threatened as $\|\Sigma\|$ nears $\sqrt{2}-1 \approx 0.414$. A concatenation approach between TCL and TCL+-ME is proposed but deemed impractical in general due to the unpredictable moment of breakdown. Overall, the results reveal fundamental limits of pseudoinverse-based corrections to the TCL in open quantum systems.

Abstract

We attempt to modify the time-convolutionless master equation (TCL-ME) to be more resistant to breakdown. We remove the standard assumption that a portion of the generator is invertible by instead taking the Moore-Penrose inverse. We rederive the perturbative expansion using Israel and Charnes' result, and test the equation up to sixth and fifth orders on the Jaynes-Cummings and Ising models, respectively. We find that in both cases, the modified equation fails to capture the dynamics of the exact solution compared to the standard TCL due to the terms of the modified equation scaling exponentially with the dimension of the bath, and connect this failure to a loss of convergence of the perturbative expansion.

Modifying the Time-Convolutionless Master Equation via the Moore-Penrose Pseudoinverse

TL;DR

This work addresses TCL-ME breakdown by replacing the standard inverse with the Moore-Penrose pseudoinverse and deriving a perturbative expansion via Israel and Charnes. It tests the TCL+-ME on the Jaynes-Cummings model and an Ising-bath model, finding no improvement over TCL and even worse performance as the bath size grows. The convergence analysis shows the MP-series has a stricter condition, with convergence threatened as nears . A concatenation approach between TCL and TCL+-ME is proposed but deemed impractical in general due to the unpredictable moment of breakdown. Overall, the results reveal fundamental limits of pseudoinverse-based corrections to the TCL in open quantum systems.

Abstract

We attempt to modify the time-convolutionless master equation (TCL-ME) to be more resistant to breakdown. We remove the standard assumption that a portion of the generator is invertible by instead taking the Moore-Penrose inverse. We rederive the perturbative expansion using Israel and Charnes' result, and test the equation up to sixth and fifth orders on the Jaynes-Cummings and Ising models, respectively. We find that in both cases, the modified equation fails to capture the dynamics of the exact solution compared to the standard TCL due to the terms of the modified equation scaling exponentially with the dimension of the bath, and connect this failure to a loss of convergence of the perturbative expansion.

Paper Structure

This paper contains 17 sections, 108 equations, 3 figures.

Table of Contents

  1. INTRODUCTION
  2. REVIEW OF TCL-ME DERIVATION
  3. Nakajima-Zwanzig Projection Operator Technique
  4. Derivation
  5. POWER SERIES EXPANSION FOR THE MOORE-PENROSE INVERSE
  6. SUBSTITUTION OF MOORE-PENROSE INVERSE INTO TCL DERIVATION
  7. Sorting by Powers of $\lambda$
  8. Adjoint Operators
  9. $\mathcal{P}\mathcal{L}\mathcal{P} = 0$ Relations for Adjoints
  10. TESTING
  11. Jaynes-Cummings Model
  12. Ising Model
  13. SERIES CONVERGENCE ANALYSIS
  14. CONCLUSION
  15. PROOF FOR MOORE-PENROSE INVERSE SERIES EXPANSION
  16. ...and 2 more sections

Figures (3)

  • Figure 1: Comparison of the analytical solution, TCL-2, TCL-6, and TCL+-6 with a truncated bath dimension in the Jaynes-Cummings model. Note the divergent behavior of the TCL+ as the bath dimension increases.
  • Figure 2: Simulation of the analytical solution, TCL-2,4,5, and TCL+-5 for all combinations of $N = (4,15)$ and $\beta = (1,10)$ in the Ising Model. Because the only extra term is at 5th order, we only see divergence of the TCL+ in frequency, not amplitude.
  • Figure 3: Depth constant and singular matrix convergence analyses to compare Neumann and pseudoinverse series.