Time-Evolving Quantum Superpositions in Open Systems and the Rich Content of Coherence Maps

Abstract

We discuss the general features of the time-evolving reduced density matrix (RDM) of multistate systems coupled to dissipative environments and show that many important aspects of the dynamics are visualized effectively and transparently through coherence maps, defined as snapshots of the real and imaginary components of the RDM on the square grid of system sites. In particular, the spread, signs, and shapes of the coherence maps collectively characterize the state of the system and the nature of the dynamics, as well as the equilibrium state. The topology of the system is readily reflected in its coherence map. Rows and columns show the composition of quantum superpositions, and their filling indicates the extent of the surviving coherence. Linear combinations of imaginary RDM elements specify instantaneous population derivatives. The main diagonal comprises the incoherent component of the dynamics, while the upper/lower triangular areas give rise to coherent contributions that increase the purity of the RDM. In open systems, the coherence map evolves to a band surrounding the principal diagonal whose width decreases with increasing temperature and dissipation strength. We illustrate these behaviors with examples of 10-site model molecular aggregates with Frenkel exciton couplings, where the electronic states of each monomer are coupled to harmonic vibrational baths.

Coherence Maps and Flow of Excitation Energy in the Bacterial Light Harvesting Complex 2

“…In this Letter, we supplement our earlier findings with an analysis of the excitation energy transfer (EET) pathways based on coherence maps . A coherence map (C-map) is a pair of snapshots that depict the real and imaginary parts of the system’s reduced density matrix (RDM) ρ ̃( t ) on the square grid of system states at a particular time t .…”

“…45 In this Letter, we supplement our earlier findings with an analysis of the excitation energy transfer (EET) pathways based on coherence maps. 49 A coherence map (C-map) is a pair of snapshots that depict the real and imaginary parts of the system's reduced density matrix (RDM) ρ(t) on the square grid of system states at a particular time t. The temporal evolution of the C-map offers a very effective visualization of a dynamical process in a multistate system because the magnitude and distribution of real and imaginary components encodes a wealth of dynamical information: 49 Further, when the coupling to the environment is diagonal in the basis of system states, as in the case of the Frenkel-type exciton−vibration Hamiltonian, 50,51 the time evolution of site populations is completely determined by the imaginary components of off-diagonal RDM elements weighted by state-to-state electronic couplings H jk e according to the relations 52…”

Impact of Spatial Inhomogeneity on Excitation Energy Transport in the Fenna-Matthews-Olson Complex

“…The usual practice is to focus the numerical studies on the evolution of the population in the excited states of each of the chromophores. Here, we additionally use the recently developed coherencebased analytic techniques [43][44][45][46] to unravel the pathways that the excitation takes and the impact of the vibrational modes. Baker and Habershon [47] have explored these pathways in FMO using the Lindblad master equation.…”

“…Wu et al [43] have built flux networks to analyze the flows along different pathways. More recently, Dani and Makri have shown that the time-derivative of the on-site populations is related to linear combinations of the off-diagonal terms of the reduced density matrix [44], and visualized the coherences in forms of maps that encode information about the dynamics of the system [45]. We extended these ideas to partition the time-dependent population change on a site in terms of transport along different state-to-state channels, which is useful for understanding the instantaneous population transfer in these systems with complex interconnects [46].…”

“…There is a limit to the amount of information that can be extracted from the direct population dynamics. We have used the recently introduced ideas of relating the coherence to the population transport [44][45][46] to explore the routes of transfer that the molecular excitation takes. Given the very complex interactions between these chromophores, it is not trivial to use an exact numerical computation to attribute the excitation transport to the different directed path-ways that exist.…”