Analysis and control of electron dynamics
K. Krieger, M. Hellgren, M. Odashima, D. Nitsche, A. Castañeda MedinaIn the mid eighties, Runge and Gross developed the mathematical foundation of
timedependent density functional theory ^{[1]}.
Traditional density functional theory exclusively deals with groundstate properties of atoms,
molecules and solids. Our timedependent generalization allows the abinitio calculation of
excitedstate properties and thereby makes a whole new arena of phenomena accessible to
a firstprinciples treatment. In the nineties, simultaneously with Marc Casida, we devised an
explicit scheme of calculating molecular excitation energies ^{
[2],
[3],
[4]} and oscillator strengths which, by now, has become a standard technique
in quantum chemistry. Hundreds of groups worldwide use this method to calculate molecular
excitation spectra.
Timedependent density functional theory (TDDFT), however, is not restricted to weak probes
of molecules or solids (which yield the excitation spectra). Nonlinear optical properties as well
as interactions of strong laser pulses with matter can be treated as well. In the past years, we
have successfully described strongfield phenomena, such as the generation of high harmonics.
We have developed a theoretical tool, the socalled timedependent electron localization
function (TDELF), which allows the timeresolved observation of the laserinduced formation
or breaking of chemical bonds, thus providing a visual understanding of the dynamics of excited
electrons
^{
[5],
[6],
[7]}.
Questions like: "How, and how fast, does an electron travel from one state to another?"
have been successfully answered with this approach. As an example, the figure below shows
snapshots of a laserinduced ππ^{∗} transition in acetylene.
The toruslike structure between the carbon atoms in the first picture shows the triple bond,
while the two tori in the last picture clearly represent an antibonding state. It is clear from the
plots that the transition is completed after about 3.9 fs. Information of this type may soon
become experimentally available with attosecond pumpprobe spectroscopy.
We also have applied the concept of the ELF to 2D systems. The figure below on the
righthand side shows the ELF of a 12electron quantum dot with four minima.
One can clearly identify a region of enhanced electron localization in the center,
i.e., a feature similar to a chemical bond. Like for chemical bonds, the localization is
visible only in the ELF, not in the plot of the density on lefthand side.
Most recently our work on timedependent effects has focused on the study of timedependent
transport phenomena. Using our TDDFT approach to quantum transport
^{[8]},
we calculate the electronic current through a single molecule attached to (semiinfinite) metallic
leads. We have studied a variety of timedependent effects (see also the section on quantum
transport), such as electron pumping
^{[9]}.
An electron pump is a device which, without any static bias, generates a (macroscopic)
net current upon application of a timedependent field. An example is given in the figure
below which shows (a) a schematic picture of the experiment and (b) the corresponding AFM
picture of a carbon nanotube on a piezoelectric surface. When a sound wave is generated on
the surface, a running electric wave along the nanotube is produced by the piezoelectric effect.
This leads to a measurable net current through the nanotube.
 Figure from: P.J. Leek, M.R. Buitelaar, V.I. Talyanskii, C.G. Smith, D. Anderson, G.A.C. Jones, J. Wei and D.H. Cobden, Phys. Rev. Lett. 95, 256802 (2005)
A very interesting aspect, which can be explained by our abinitio theory of quantum transport,
is the fact that the current can have a direction opposite to the running wave. The figure
below shows a snapshot of the excess density (in red), i.e. the timedependent density
minus the initial density, as it moves into the left lead whereas the timedependent electric
wave superimposed to the lattice potential of the carbon nanotube (in green) moves to the right.
Depending on the frequency of the electric wave and on the Fermi energy of the leads,
the current may go to the left (negative current) or to the right (positive current).
The figure below shows the calculated timeaveraged current
^{[9]}
as a function of the
Fermi energy (corresponding to different values of the gate voltage in the experiment).
The above calculation is a simple example of controlling the electronic dynamics with
external parameters, i.e. the direction of the current is controlled by the value of the
applied gate voltage.
This type of control can be taken to a much more sophisticated level by combining
TDDFT with quantum optimalcontrol theory
^{[10],
[11]
}.
Here the goal is not to predict and/or analyze timedependent processes resulting from
a given driving field, but rather to determine the "best" laser pulse to achieve a given
control target. Questions like: "Starting from a given initial state, by which laser pulse can
one achieve maximal population of a given final state at the end of the pulse?" can be
addressed in this way. The snapshots below show how the chirality of the current in a
2D quantum ring of 44 nm diameter can be switched by a suitably shaped laser pulse,
calculated with our optimal control algorithm
^{
[12]
}.
Arrows represent the current density,
the color indicates the magnitude of the density, and the time scale is in effective atomic units.
Interestingly, after about half of the pulse length (t=28 a.u.), one finds the current density
going in opposite directions on opposite sides of the ring, leading to a zero net current.
Using an analogous algorithm, we also have controlled the electron localization in a
double quantum dot
^{
[13]}.
In a generalization of the control algorithms to quantumclassical
systems we recently succeeded to selectively break molecular bonds by ultrashort laser pulses
^{ [14]},
formulating the control target in terms of classical nuclear degrees of freedom.
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References

[1]
 Density functional theory for timedependent systems, E. Runge and E.K.U. Gross, Phys. Rev. Lett. 52,
997 (1984).

[2]
 Excitation energies from timedependent densityfunctional theory, M. Petersilka, U.J. Gossmann
and E.K.U. Gross, Phys. Rev. Lett. 76, 1212 (1996).

[3]
 Molecular excitation energies from timedependent densityfunctional theory, T. Grabo, M. Petersilka and
E.K.U. Gross, J. Mol. Struc. (Theochem) 501, 353 (2000).

[4]
 Excitations in TimeDependent DensityFunctional Theory, H. Appel, E.K.U. Gross and K. Burke,
Phys. Rev. Lett. 90, 043005 (2003).

[5]
 Timedependent electron localization function, T. Burnus, M.A.L. Marques and E.K.U. Gross,
Phys. Rev. A (Rapid Comm.) 71, 010501 (2005).

[6]
 Timedependent electron localization functions for coupled nuclearelectronic motion, M. Erdmann,
E.K.U. Gross and V. Engel, J. Chem. Phys. 121, 9666 (2004).

[7]
 Timedependent electron localization function: A tool to visualize and analyze ultrafast processes, A. Castro,
T. Burnus, M.A.L. Marques and E.K.U. Gross, in: Analysis and Control of Ultrafast Photoinduced Reactions, O. Kühn
and L. Wöste, ed(s), (Springer Series in Chemical Physics, vol. 87, 2007) p. 553574.

[8]
 Timedependent quantum transport: A practical scheme using density functional theory, S. Kurth, G. Stefanucci,
C.O. Almbladh, A. Rubio and E.K.U. Gross, Phys. Rev. B 72, 035308 (2005).

[9]
 A timedependent approach to electron pumping in open quantum systems,
G. Stefanucci, S. Kurth, A. Rubio and E.K.U. Gross, Phys. Rev. B 77, 075339 (2008).

[10]
 Quantum Optimal Control, J. Werschnik and E.K.U. Gross, J. Phys. B 40, R175 (2007).

[11]
 Optimal control of timedependent targets, I. Serban, J. Werschnik and E.K.U. Gross,
Phys. Rev. A 71, 053810 (2005).

[12]
 Optimal control of quantum rings by terahertz laser pulses, E. Räsänen, A. Castro, J. Werschnik, A. Rubio and E.K.U. Gross,
Phys. Rev. Lett. 98, 157404 (2007).

[13]
 Optimal LaserControl of Double Quantum Dots E. Räsänen, A. Castro, J. Werschnik, A. Rubio and E.K.U. Gross, Phys. Rev. B 77, 085324 (2008).

[14]
 Optimization schemes for selective molecular cleavage with tailored ultrashort laser pulses, K. Krieger, A. Castro, E.K.U. Gross,
Chem. Phys. 391, 50 (2011).
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