Generalized Schrieffer-Wolff formalism for dissipative systems

We present a formalized perturbation theory for Markovian master equations in the language of a generalized Schrieffer-Wolff (SW) transformation. A nonunitary rotation decouples the unperturbed steady states from all fast degrees of freedom, in order to obtain an effective Liouvillian that reproduces the exact low excitation spectrum of the system. The transformation is derived in a constructive way, yielding a perturbative expansion of the effective Liouville operator. The presented formalism realizes an adiabatic elimination of fast degrees of freedom to arbitrary order. We exemplarily employ the SW formalism to two generic open systems and discuss general properties of the different orders of the perturbation.

Figure 1

Comparison of the exact evolution according to Eq. (45) (dashed) with the approximate solution up to second (dotted) and third (solid) order [Eqs. (55) and (57)]. The figure shows the radiation intensity emitted from the electron spin ($\propto d/dt\langle I_z\rangle$) for a system of $N=100$ nuclear spins, initially fully polarized in the $z$ direction. The emission intensity shows clearly the characteristic superradiant burst [31]. Parameters are $\omega =\gamma /5$ and $\sqrt{N}g=0.2\gamma$. The second-order effective evolution ($L_2^{\mathrm{eff}}$) shows significant deviation from the exact dynamics. Addition of the third-order $L_3^{\mathrm{eff}}$, improves the approximation substantially.