Protocol for detection of nonsecular conversion through coherent nanooptical spectroscopy

The theoretical description of dynamics in open quantum systems becomes very demanding upon inclusion of non-Markovian effects. To simplify the computational implementation for density matrix equations of motion, the secular approximation is often applied. An experimental verification of its validity though remains difficult due to uncertainties in the system parameters and the absence of qualitatively distinct features. In this paper, we present the proposal for an experimental detection protocol sensitive to nonsecular processes neglected in the secular approximation. The protocol uses a combination of multidimensional coherent spectroscopy and nanoplasmonics. It allows for studies of nonsecular processes in various systems and provides a tool to experimentally verify the validity of the secular approximation. We apply this protocol to a system of CdSe/ZnS nanostructures and discuss the particular features originating from nonsecular processes on the resulting two-dimensional spectra.

Figure 1

Principle of the detection protocol. Through a particular pulse sequence, the system is prepared in a coherence $\left|A\right.〉\left.〈B\right|$ between two excited-state manifolds $A$ and $B$. Only nonsecular processes are able to convert the coherences into populations $\left|A\right.〉\left.〈A\right|$ and $\left|B\right.〉\left.〈B\right|$ in either manifold. By only measuring the resulting populations, secular processes are filtered out. Conversion into other coherences $\left|A^{\prime }\right.〉\left.〈B^{\prime }\right|$ are in principle also possible, but cannot be detected.

Figure 2

Schematic representation of the incoming laser pulses for the detection protocol. The pulses are centered around the times $t_k$, with time differences between the pulses ${\tau }_k=t_{k+1}-t_k$. For detection via phase cycling, each pulse has to have a defined phase shift ${\varphi }_k$ with respect to all other pulses. The first and fourth pulse are exciting bright transitions (light red), whereas the second and third pulse are exciting dark transitions (dark blue).

Figure 3

Level scheme of the lowest excitonic and biexcitonic states. Solid arrows indicate dipole-allowed transitions and dashed arrows dipole-forbidden transitions. The relevant optical transition elements for each excitation are labeled accordingly. The small dashed arrows indicate possible conversion processes.

Figure 4

Sketch of the plasmonic dolmen structure (a) in three dimensions with the placement and direction of the QD/NR indicated in light red at ${\mathit{r}}_0$. (b) Top view with all relevant dimensions. The coordinate system is centered in the middle of the structure. The dimensions of the dolmen structure and material parameters (corresponding to gold [84]) used in the simulations are shown in (c).

Figure 5

Left side: Comparison between (a) $|{\mathbf{\mu }}_{e_{b}g}·{\mathit{E}}_{\mathit{r}}^x|$ and (b) $|{\mathbf{\mu }}_{e_{b}g}·{\mathit{E}}_{\mathit{r}}^y|$ in the $y-z$ plane of the dolmen structure. Right side: Comparison between (c) $|Q_{e_{d}g}:{\mathbf{\nabla }}^S{\mathit{E}}_{\mathit{r}}^x|$ and (d) $|Q_{e_{d}g}:{\mathbf{\nabla }}^S{\mathit{E}}_{\mathit{r}}^y|$ in the $y-z$ plane. The microscopic dipole matrix element ${\mathit{d}}_{cv}^{+}$ corresponds to spin-up excitation. An enhancement of dipole transitions for $x$-polarized light (a) can be seen around ${\mathit{r}}_0=(0,-38\mathrm{nm},0)$, which gets suppressed upon excitation with $y$-polarized light (b). Whereas the quadrupole excitation is negligible for $x$-polarized light (c), an enhancement of quadrupole excitations is visible around ${\mathit{r}}_0$ in $y$ polarization (d). The position and size of the QD at ${\mathit{r}}_0$ is marked as a gray dot.

Figure 6

Double-sided Feynman diagrams for the remaining Liouville pathways in the PEEM measurement with the phase-cycling condition ${\varphi }_s=-{\varphi }_1+{\varphi }_2+{\varphi }_3-{\varphi }_4$. The photon-ionization process is not depicted. The diagrams correspond to (I) GSB, (II) ESE, (III) and (IV) ESA. Coloring of the light pulses corresponds to bright (light red) and dark (dark blue) state excitation.

Figure 7

Feynman diagram for pathway III (a) in secular approximation and (b) including nonsecular dynamics. The labeling of the states corresponds to the formation of peak A in the 2D spectra (Figs. 8 and 9). Conversion between states in the secular approximation (a) is restricted to population relaxation, prohibiting the stimulated emission at $t_4$ and leading to a vanishing signal in PEEM. In (b), nonsecular conversion from ${\rho }_{f_{dd}{e}_b}\to {\rho }_{f_{bb}{e}_b}$ during ${\tau }_3$ produces a measurable signal (indicated by a color gradient from dark blue to light red during ${\tau }_3$).

Figure 8

2D spectra $|n_e(0,{\mathrm{\Omega }}_3,{\tau }_2,{\mathrm{\Omega }}_1)|$ including nonsecular terms in the ideal case of $Q_{mn}:{\mathbf{\nabla }}^S{\mathit{E}}_{\mathit{r}}^x={\mathbf{\mu }}_{mn}·{\mathit{E}}_{\mathit{r}}^y=0$ for (a) ${\tau }_2=72\mathrm{fs}$ and (b) ${\tau }_2=94\mathrm{fs}$. Two distinct features A and B are visible. Destructive interference with the excitonic states leads to a quantum beating behavior. In secular approximation, both spectra vanish.

Figure 9

2D spectrum $|n_e(0,{\mathrm{\Omega }}_3,{\tau }_2=72\mathrm{fs},{\mathrm{\Omega }}_1)|$ for realistic switching quality ${\eta }_{{\nu }_k}$, Eq. (15) (a) including nonsecular processes and (b) in secular approximation. The realistic switching in (a) introduces only a minor error compared to Fig. 8. This error is directly represented through (b).