Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Exact diagonalization of a non-quadratic bosonic Liouvillian with two-body loss

Masaaki Tokieda

Abstract

We present the full diagonalization of a non-quadratic bosonic Liouvillian with a two-body loss term. The Liouvillian is shown to be exactly diagonalizable in terms of left and right confluent hypergeometric functions, whose distinction arises from the noncommutative nature of superoperators. The resulting spectral decomposition yields the general solution of the master equation, extending previous results. We further investigate the construction of a non-Gaussian open system model through the lens of nonlinear pseudomodes.

Exact diagonalization of a non-quadratic bosonic Liouvillian with two-body loss

Abstract

We present the full diagonalization of a non-quadratic bosonic Liouvillian with a two-body loss term. The Liouvillian is shown to be exactly diagonalizable in terms of left and right confluent hypergeometric functions, whose distinction arises from the noncommutative nature of superoperators. The resulting spectral decomposition yields the general solution of the master equation, extending previous results. We further investigate the construction of a non-Gaussian open system model through the lens of nonlinear pseudomodes.

Paper Structure

This paper contains 13 sections, 95 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Exact diagonalization
  3. Setups
  4. Weak symmetry and spectrum
  5. Eigenvectors
  6. Exact solution of the master equation
  7. Characterization of Non-Gaussian open system dyanmics
  8. Conclusion
  9. Similarity transformations with $e^{\mathcal{A}}$
  10. Eigenvectors with confluent hypergeometric function
  11. Proof of Eq. (\ref{['eq:diag_F-1']})
  12. Explicit forms of eigenvectors with $k = 0$ and $1$
  13. Exact time evolution

Figures (1)

  • Figure 1: Comparison of the probability density $P(x)$ for the linear ($\kappa_2 = 0$, red) and nonlinear ($\kappa_2 = 10$, blue) cases. Here $\omega = 1$ and $U = 0$ in the unit $\kappa_1 = 1$, and the system is initialized in the vacuum state.