Hierarchy of Stochastic Pure States for Open Quantum System Dynamics

We derive a hierarchy of stochastic evolution equations for pure states (quantum trajectories) for open quantum system dynamics with non-Markovian structured environments. This hierarchy of pure states (HOPS) is generally applicable and provides the exact reduced density operator as an ensemble average over normalized states. The corresponding nonlinear equations are presented. We demonstrate that HOPS provides an efficient theoretical tool and apply it to the spin-boson model, the calculation of absorption spectra of molecular aggregates, and energy transfer in a photosynthetic pigment-protein complex.

Figure 1

Dynamics of the spin-boson model. (a) Nonlinear equation, (b) linear equation. In both cases, $\mathrm{\Delta }=1$, $\epsilon =0$, and the parameters of the spectral density are given by $g=2$ and $w=0.5+2i$. The blue, green, and red lines represent 100, 1000, and 10 000 realizations, respectively. The order of the hierarchy is $\mathcal{K}=8$. The inset in (a) shows the convergence (for 10 000 realizations) with respect to the order of the hierarchy. The dotted, dashed, and solid line are orders one, two, and four, respectively.

Figure 2

Transfer of electronic excitation energy within the FMO complex. The parameters are taken from Refs. [39, 40] and can be found in Supplemental Material [32]. Solid line, result of Ref. [39]; dotted line, HOPS first order; dashed line, HOPS second order.

Figure 3

Absorption of a linear chain of $N=7$ monomers with parallel transition dipoles for different values of the interaction $V$. The inset shows the spectral density used. The reorganization energy $E_r={\int }_0^{\infty }dwJ(w)/w$ is chosen as the unit of energy. The temperature is $T=0.2$. Colors indicate different orders of the hierarchy (blue, $\mathcal{K}=5$; green, $\mathcal{K}=6$; red, $\mathcal{K}=7$).