Two-dimensional photon echoes reveal non-Markovian energy transfer in an excitonic dimer

We show that strong non-Markovian effects can be revealed by the steady-state two-dimensional (2D) photon echo spectra at asymptotic waiting times. For this, we use a simple dimer toy model that is strongly coupled to a harmonic bath with parameters typical for photoactive biomolecules. We calculate the 2D photon echo spectra employing both the numerically exact hierarchy equation of motion and the quasiadiabatic path integral approach and compare these results with approximate results from a time-nonlocal quantum master equation approach. While the latter correctly reproduces the exact population dynamics at long times, it fails at the same time to correctly describe the 2D photon echo spectra at long waiting times. The differences show that non-Markovian effects are much more important for the steady-state 2D photon echoes than for the equilibrium populations. Thus, accurate theoretical descriptions of the energy transfer dynamics in biomolecular complexes have to be based on numerically exact simulations of the environmental fluctuations when nonlinear response functions are analyzed.

Figure 1

Time-dependent probabilities of an excitation to be in monomer 1 or 2, i.e., ${\rho }_{11}\left(t\right)$ and ${\rho }_{22}\left(t\right)$, for the parameters given in the text and calculated by the TNL quantum master equation and by the HEOM technique.

Figure 2

Real part of the two-dimensional photon echo spectra (rephrasing and nonrephasing) for a dimer calculated for three different waiting times $T=0$ fs, $T=100$ fs, and $T=200$ fs (positive, solid lines; negative, dashed lines). The left column shows results obtained by numerically exact HEOM approach, while the right column shows the results obtained by the approximate TNL technique (for the parameters, see text). The transfer rate between the two exciton states was analyzed by the global fitting approach, which yields the time scale of 150 fs for both methods. We also show the frequency spectrum of the excitation laser (dashed red lines) and the linear photoabsorption spectra (solid black lines).

Figure 3

Magnitude of the peaks A and B in the photon echo spectra versus waiting time $T$. The solid and dashed lines mark the HEOM results, while the markers (square and circle) indicate the TNL results.

Figure 4

Same as in Fig. 3, but for peaks C and D.

Figure 5

Heights of the peaks A, B, and C in the photon echo spectra for an exponentially cutoff bath spectral function versus increasing waiting time. The symbols indicate the numerically exact QUAPI results and the dashed lines mark the approximate TNL results.