Engineering and manipulating exciton wave packets

When a semiconductor absorbs light, the resulting electron-hole superposition amounts to a uncontrolled quantum ripple that eventually degenerates into diffusion. If the conformation of these excitonic superpositions could be engineered, though, they would constitute a new means of transporting information and energy. We show that properly designed laser pulses can be used to create such excitonic wave packets. They can be formed with a prescribed speed, direction, and spectral make-up that allows them to be selectively passed, rejected, or even dissociated using superlattices. Their coherence also provides a handle for manipulation using active, external controls. Energy and information can be conveniently processed and subsequently removed at a distant site by reversing the original procedure to produce a stimulated emission. The ability to create, manage, and remove structured excitons comprises the foundation for optoexcitonic circuits with application to a wide range of quantum information, energy, and light-flow technologies. The paradigm is demonstrated using both tight-binding and time-domain density functional theory simulations.

Figure 1

Material settings for quasi-one-dimensional exciton dynamics. (a) Possible implementations include inorganic quantum-well superlattices (top), Rydberg atoms in optical lattices (second), atoms in strongly coupled optical cavities (third), and organic molecular chains (bottom). (b) Finite chain lattice of interest with photon absorption only at leftmost site. (c) Ring lattice construct used to design laser pulses. Sites are shown in red with structured excitons in blue.

Figure 2

Dispersion relations on a finite lattice. The dispersion relation of Eq. (20) is plotted for three values of hopping parameter $\chi$. When $2\chi >1$ (red), there exist positive temporal frequencies with negative counterparts leading to the generation of multiple excitons. When $2\chi \le 1$ (blue, green), single exciton packets will be generated.

Figure 3

Quantum interference evanescence. The finite lattice of Fig. 1 is excited with a (nonphysical) laser pulse composed of only the second term $-E^{*}$ of Eq. (21). The resulting disturbance is dominated by an exponential decay of the maximum exciton occupancy that each site attains for the first few sites nearest to the excitation source. Here $N=200$, $\mathrm{\Delta }=2\chi$, and $\sigma =10.0$. The central wave number of the packet is labeled for each simulation set, and the straight lines are exponential fits.

Figure 4

Quantum interference evanescence. The finite lattice of Fig. 1 is excited with each of the terms of Eq. (21) separately. (a) Laser pulse consists of only the first term in Eq. (21). The central wave number of the ring exciton is $k_0=2.19$ which is also the phase difference between adjacent sites since the characteristic length is the lattice spacing $a$. (b) Laser pulse consists of only the second term in Eq. (21). The amplitude of adjacent sites are out of phase by $\pi$, and this is the source of the exponential decay shown in Fig. 3. For both plots, $N=200$, $\mathrm{\Delta }=2\chi$, and $\sigma =10.0$.

Figure 5

Fourier transform of dipole moment for a single methane molecule. A discrete Fourier transform (FT) of the dipole moment versus time, after a delta kick polarized in $x$ direction, is used to determine the lowest excitation energy, 10.05 eV. Discrete points are the actual Fourier data while the red lines are a guide to the eye.

Figure 6

Rabi oscillation in methane dimer. A set of Gaussian-shaped laser pulses [Eq. (36)] were applied to the left site of the methane dimer to determine the energy best able to create a clean exciton oscillation. The laser was polarized along $x$ direction with $F=0.5\mathrm{V}/\AA$, $\tau =1.97$ fs, and $t_0=9.87$ fs. The optimum value of $\hslash \omega$ was found to be 9.6 eV and the associated oscillations are shown. The red (green) line is the exciton occupation on the right (left) site. The isosurfaces of electron (green) and hole (red) density at $0.008/{\mathrm{bohr}}^3$ are shown underneath these curves.

Figure 7

Oscillation in an $N$-site chain. The initial condition of Eq. (39) was applied to a 15-site chain to generate the oscillation pattern shown.

Figure 8

Oscillation in trimer. A Gaussian laser pulse polarized along $x$ direction, $E_x\left(t\right)=Fcos\left(\omega t\right)e^{\frac{-{(t-t_0)}^2}{2{\tau }^2}}$ with strength $F=0.5$ V/Å, $\hslash \omega =9.6$ eV, $\tau =1.97$ fs, and $t_0=9.87$ fs, is applied on the middle site of the trimer. Oscillation between even and odd sites results. Red indicates the exciton number on the middle site, orange is the exciton number on either side, green denotes the exciton number on the left site, and blue line is the exciton number on the right site.

Figure 9

Evolution of a laser-induced exciton wave packet. The finite lattice of Fig. 1 is excited with a laser pulse at left. The result is an exciton (green) that is plotted for several time slices along with the original exciton (black) of Fig. 1. Here $N=200$, site energy $\mathrm{\Delta }=2\chi$, and the exciton footprint has a standard deviation $\sigma =10.0a$. The central wave number of the ring exciton is $k_0=1.58/a$. The initial exciton occupation fraction of the ring is 0.5, although this can be set to any value.

Figure 10

Tunable exciton speed. The finite lattice of Fig. 1 is excited with twice the real part of the laser pulse of Eq. (17) for a range of central wave numbers. The numerically measured exciton speeds (green) are compared with the speed predicted from the continuum dispersion relation (black). The speed was changed by over a factor of 5 in the simulations carried out. Here $N=200$, $\mathrm{\Delta }=2\chi$, and $\sigma =10.0a$.

Figure 11

Exciton annihilation. A exciton wave packet (green) travels to the right on the finite lattice of Fig. 1. The same exciton (black) is considered on the superimposed ring geometry of Fig. 1. This is used with Eq. (43) to design a laser pulse that will extract the excitonic energy from the rightmost site. Several time slices show that the resulting stimulated emission removes the excitonic wave packet. Here $N=200$, $\mathrm{\Delta }=2\chi$, and $\sigma =10.0a$. The central wave number of the ring exciton is $k_{0}a=2.19$.

Figure 12

Control over wave packet character. The finite lattice of Fig. 1 is excited with a laser pulse at left defined by Eq. (15). This creates a triplet of overlapping Gaussian excitations (green) plotted for several time slices along with the original ring wave packet (black) of Fig. 1. The central packet was intentionally tuned so that it travels slightly faster than its neighbors. Here $N=200$, $\mathrm{\Delta }=2\chi$, and $\sigma =10.0a$. The central wave numbers of the ring exciton are, from left to right, $k_{0}a=2.03$, $k_{0}a=1.59$, and $k_{0}a=2.03$. Units of time are $\hslash /\chi$.

Figure 13

Five-site benzene chain. Uncontrolled laser-generated exciton wave packet moves down a chain of five co-facial benzene molecules. The green and red are the isosurfaces with value 0.002 electrons per ${\mathrm{bohr}}^3$ for electron and hole densities, respectively.

Figure 14

20-site methane chain. Laser pulse shape and composition is modified so as to vary exciton speed on a 20-site chain of methane molecules. Solid curve is from dispersion relation for an infinite chain [Eq. (20)], hollow blue squares are from 20-site TB model, and filled magenta triangles are from RT-TDDFT simulations.

Figure 15

Excitonic stop and pass bands. (a) A region of homogeneous sites (white, left) is joined to a superlattice composed of alternating crystalline layers (light/dark gray, right). (b) The eigenvalues of the stand-alone superlattice (black lines) are plotted along with those of the entire system (green lines). An exciton is constructed that is composed of eigenvalues lying in a stop band (projection magnitudes in magenta) or a pass band (projection magnitudes in brown) of the superlattice. (c) The stop band exciton (real part in magenta, absolute value in black) is plotted three times to show that it is reflected at the phase boundary. (d) The pass band exciton (real part in brown, absolute value in black) is plotted three times to show that it is largely transmitted at the phase boundary. Here $N=400$, ${\mathrm{\Delta }}_1=\chi$, ${\mathrm{\Delta }}_{2a}=0.5\chi$, ${\mathrm{\Delta }}_{2b}=1.5\chi$, and $\sigma =30.0a$. There are 5 and 15 sites in the superlattice, and the central wave numbers of the exciton are $k_0=2.36/a$ (c) $k_0=2.15/a$ (d).

Figure 16

Exciton dissociation using quantum interference. Left panels show snapshots of a density plot of the probability distribution of the exciton along the chain. The green trajectory line is a guide to the eye. Right panels depict the same dynamics representing the electron (hole) probability projections in green (red) and the corresponding wave packet velocity for times shown at left (top to bottom). Here $N=100$, ${\mathrm{\Delta }}_1^0=2\chi$, ${\mathrm{\Delta }}_1^1=2\chi$, ${\mathrm{\Delta }}_{2a}^0=0.0$, ${\mathrm{\Delta }}_{2b}^0=2\chi$, ${\mathrm{\Delta }}_{2a}^1=1.6\chi$, ${\mathrm{\Delta }}_{2b}^1=2.4\chi$, $\chi$ is the same for both bands, $\sigma =10.4a$, and the Coulomb interactions have been turned off—the simplest setting for dissociation. There are five sites in each layer of phase 2, and the central wave number of the ring exciton is $k_0=1.63/a$.