Subtleties in the pseudomodes formalism

Abstract

The pseudomode method for open quantum systems, also known as the mesoscopic leads approach, consists in replacing a structured environment by a set of auxiliary "pseudomodes" subject to local damping that approximate the environment's spectral density. Determining what parameters and geometry to use for the auxiliary modes, however, is non-trivial and involves many subtleties. In this paper we revisit this problem of pseudomode design and investigate some of these subtleties. In particular, we examine the scenario in which pseudomodes couple to each other, resulting in an effective spectral density that is no longer a sum of Lorentzians. We show that non-diagonalizability of the pseudomodes' effective single-particle non-Hermitian Hamiltonian can lead to terms in the effective spectral density which cannot be obtained by diagonalizable non-Hermitian Hamiltonians. We also present a method for constructing the pseudomode parameters to exactly match a fit to a spectral density, and in doing so illuminate the enormous freedom in this process. The case of many uncoupled pseudomodes evenly distributed in energy is explored, and we show how, contrary to conventional assumption, the effective spectral density does not necessarily converge in the limit of an infinite number of pseudomodes distributed this way. Finally, we discuss how the notion of effective spectral densities can also emerge in the context of scattering theory for non-interacting systems.

Figures (6)

• Figure 1: (a) Standard setup of a quantum system coupled to two reservoirs. (b) The pseudomodes model with uncoupled pseudomodes (the Diagonal case), where each bath has been replaced by a finite set of damped pseudomodes. (c) The general pseudomode model, where the pseudomodes are allowed to couple to each other arbitrarily in addition to the residual baths.
• Figure 2: (a) Comparison of fits to an effective spectral density from a diagonal configuration (sum of Lorentzians, with uncoupled pseudomodes) and from a diagonalizable one (sum of Lorentzians and Anti-Lorentzians, with pseudomodes coupled to one another). The true spectral density is $J(\omega)=\sqrt{1-\omega^2}$, which both fits model using 6 pseudomodes. The diagonal-case fit was obtained by brute-force optimization for the parameters $|\zeta_k|^2,\varepsilon_k,\gamma_k$, while for the diagonalizable case, Prony's method (described in section \ref{['sec:fitting']} and appendix \ref{['apx:prony']}) was used to obtain the best-fit $\kappa_k,\varepsilon_k,\gamma_k$. (b) Contributions to the effective spectral density in the diagonal case from each of the 6 pseudomodes; each mode contributes a Lorentzian term. (c) Contributions to the effective spectral density in the diagonalizable case from each of the 6 pseudomodes; each mode contributes the sum of a Lorentzian term and an Anti-Lorentzian term. The parameters from optimization used in these fits are given in Appendix \ref{['apx:data']}.
• Figure 3: (a) The most general non-diagonalizable pseudomode configuration, consisting of two pseudomodes with $W$ given by \ref{['eqn:ND2W']}. (b) The simplest non-diagonalizable pseudomode configuration, with $\zeta_2=\eta=0$. One mode is coupled to the system and the other is coupled to the residual bath. If the energies and couplings are chosen so that $W$ is non-diagonalizable, then the system feels an effective spectral density which is a squared Lorentzian, as in \ref{['eqn:squared-lorentzian']}.
• Figure 4: Fits to (a) a memory kernel of the form \ref{['eqn:numex-X']} and (b) the corresponding spectral density \ref{['eqn:numex-j']}, with $\Delta=0.04$ and $\delta=6.0$. The blue curves are obtained by fitting \ref{['eqn:numex-X']} with the ESPRIT algorithm (see Appendix \ref{['apx:prony']}). After following the inversion procedure of section \ref{['ssec:inversion']}, optimizing over the vector $u$\ref{['eqn:vfromu']} and matrix $B$\ref{['eqn:STS_e']} to minimize any negative elements of $\Gamma$, one obtains the pseudomode parameters $\Lambda,\zeta,\Gamma$ corresponding to that fit. Finally, setting any negative elements of the matrix $\Gamma$ (which would cause negative rates in the corresponding master equation) gives rise to the modified fit shown in red. These fits are compared with the best fit using uncoupled modes, which was obtained using brute-force optimization. The pseudomode parameters appearing in these fits are given in Appendix \ref{['apx:data']}.
• Figure 5: Effective spectral density obtained by applying the many-mode approximation procedure described above to a semi-elliptical spectral density, for 100 pseudomodes with $\omega_{m/M}=\pm 1.4$. The effective spectral density oscillates rapidly about the true spectral density.