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Exceptional points in Gaussian channels: diffusion gauging and drift-governed spectrum

Frank Ernesto Quintela Rodríguez

Abstract

McDonald and Clerk [Phys.\ Rev.\ Research 5, 033107 (2023)] showed that for linear open quantum systems the Liouvillian spectrum is independent of the noise strength. We first make this noise-independence principle precise in continuous time for multimode bosonic Gaussian Markov semigroups: for Hurwitz drift, a time-independent Gaussian similarity fixed by the Lyapunov equation gauges away diffusion for all times, so eigenvalues and non-diagonalizability are controlled entirely by the drift, while diffusion determines steady states and the structure of eigenoperators. We then extend the same separation to discrete time for general stable multimode bosonic Gaussian channels: for any stable Gaussian channel, we construct an explicit Gaussian similarity transformation that gauges away diffusion at the level of the channel parametrization. We illustrate the method with a single-mode squeezed-reservoir Lindbladian and with a non-Markovian family of single-mode Gaussian channels, where the exceptional-point manifolds and the associated gauging covariances can be obtained analytically.

Exceptional points in Gaussian channels: diffusion gauging and drift-governed spectrum

Abstract

McDonald and Clerk [Phys.\ Rev.\ Research 5, 033107 (2023)] showed that for linear open quantum systems the Liouvillian spectrum is independent of the noise strength. We first make this noise-independence principle precise in continuous time for multimode bosonic Gaussian Markov semigroups: for Hurwitz drift, a time-independent Gaussian similarity fixed by the Lyapunov equation gauges away diffusion for all times, so eigenvalues and non-diagonalizability are controlled entirely by the drift, while diffusion determines steady states and the structure of eigenoperators. We then extend the same separation to discrete time for general stable multimode bosonic Gaussian channels: for any stable Gaussian channel, we construct an explicit Gaussian similarity transformation that gauges away diffusion at the level of the channel parametrization. We illustrate the method with a single-mode squeezed-reservoir Lindbladian and with a non-Markovian family of single-mode Gaussian channels, where the exceptional-point manifolds and the associated gauging covariances can be obtained analytically.
Paper Structure (39 sections, 4 theorems, 187 equations, 4 figures)

This paper contains 39 sections, 4 theorems, 187 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Bosonic Gaussian channels
  3. Gaussian channels and $(X,Y,\delta)$ parametrization
  4. Linear Lindbladians and Gaussian diffusion generators
  5. Ornstein--Uhlenbeck form and continuous-time Lyapunov gauging
  6. Linear evolution in white-noise Gaussian environments
  7. From generators to channels
  8. Noise gauging for a Gaussian channel
  9. Summary: implications for exceptional points
  10. Applications to single-mode Gaussian channels
  11. A Markovian example: Single-mode squeezed-reservoir Lindbladian
  12. Plots: EP locations and gauge eigenvalues vs. $\kappa$, $r$, $\phi$
  13. Non-Markovian single-mode Gaussian channels: EPs and analytic noise gauging
  14. A concrete non-semigroup family: $X_t=\kappa(t)e^{tB(\lambda,\omega)}$
  15. Diffusion models and analytic gauge covariance on the EP lines
  16. ...and 24 more sections

Key Result

Theorem 1

Assume eq:Hurwitz_abs and let $S$ solve eq:Lyapunov_abs. Define Then eq:OU_generator_general is similar to the diffusion-free drift generator Equivalently, at the channel level the semigroup $\{\Psi_t\}_{t\ge0}$ is similar, for every fixed $t\ge0$, to a Gaussian channel with the same drift $X_t=\mathrm{e}^{At}$ and zero diffusion.

Figures (4)

  • Figure 1: Real parts of the drift eigenvalues $\lambda_{\pm}(A)$ vs. detuning $\Delta$ at fixed $(\kappa,\epsilon)$. The EPs occur at $\Delta=\pm\epsilon$ where the square-root discriminant vanishes.
  • Figure 2: Eigenvalues of the Lyapunov gauge covariance $S$ on the EP branches $\Delta=\pm\epsilon$ for a squeezed-reservoir Lindbladian, shown as functions of (a) $\kappa$, (b) $r$, and (c) $\phi$ (with the remaining parameters held fixed as indicated).
  • Figure 3: Eigenvalues of the gauging covariance $S_t$ for a non-Markovian single-mode Gaussian channel. At fixed time $t$, the map is specified by $(X_t,Y_t)$ and $S_t$ is the unique real symmetric solution of the Stein equation $S_t=X_t S_t X_t^{\mathsf{T}}+Y_t$ in the stable regime $\operatorname{spr}(X_t)<1$. The plotted surfaces are obtained from the analytic solution of this Stein problem (closed-form expressions in \ref{['app:Stein_one_mode_vec']}). In each panel, the left (right) subplot shows $\lambda_{\min}(S_t)$ ($\lambda_{\max}(S_t)$) over the drift plane $(\lambda,\omega)$, with the EP branches $\lambda=\pm\omega$ overlaid (solid: $\lambda=\omega$, dashed: $\lambda=-\omega$). On the EP branches $X_t$ is defective and the Jordan closed form \ref{['eq:S_closed_on_Jordan']} applies. Panels: (a) isotropic diffusion, (b) anisotropic diffusion, (c) drift-aligned structured diffusion.
  • Figure 4: Non-Markovian single-mode Gaussian channel at fixed time $t$: eigenvalues $\lambda_{1,2}(S_t)$ of the gauging covariance $S_t$ along the exceptional-point branches $\lambda=\pm\omega$. The curves are obtained from the analytic solution of the Stein equation on a Jordan (defective) drift, with the corresponding closed-form expressions summarized in \ref{['app:Stein_one_mode_vec']}. Panels: (a) isotropic diffusion $Y_t=y(t)I_2$; (b) anisotropic diffusion $Y_t=\mathrm{diag}(y_q(t),y_p(t))$; (c) drift-aligned structured diffusion \ref{['eq:drift_alignedY_reorg']}. The branch-to-branch splitting in (c) reflects the sign change of the off-diagonal correlations built into $Y_t$ between the two EP branches.

Theorems & Definitions (11)

  • Theorem 1: Continuous-time (Lyapunov) diffusion gauging
  • proof
  • Theorem 2: Discrete-time (Stein) noise gauging
  • proof
  • Remark : Gauging away $Y$ does not force $X$ to be symplectic
  • Remark : Where the identities are meant to hold
  • Proposition 3: Removing diffusion without changing eigenvalues or Jordan blocks
  • proof
  • Proposition 4: Composition of multimode Gaussian channels
  • proof
  • ...and 1 more