# Radiation-Reaction: Self-consistent Light-Matter interaction with real-time TDDFT¶

In this tutorial, we calculate the superradiant absorption-spectrum of a sodium-dimer chain.

The equations underlying the implementation are described in Ref. [1]. The here used parameters are coarser than the published work. This implementation is so far only available for the LCAO mode.

We use real-time TDDFT LCAO mode
and add the radiation-reaction potential for emission into a waveguide.
The available implementation adds a local potential that accounts for the recoil
that emerges as a consequence of the emission of light. This approach effectively
embeds the electromagnetic environment into the KS equation and ensures a
self-consistent interaction between light and matter.
The keyword `RRemission`

demands the cross-sectional area of the waveguide
and the polarization of the propagating electric field.
With decreasing cross-sectional area, the emission will be stronger.
Thus far, 1d-waveguide emission is the only available electromagnetic environment
but further additions are scheduled.

We will start by calculating the emission spectrum in z-direction for a single sodium dimer:

```
from ase import Atoms
from gpaw import GPAW
from gpaw.lcaotddft import LCAOTDDFT
from gpaw.lcaotddft.qed import RRemission
from gpaw.lcaotddft.dipolemomentwriter import DipoleMomentWriter
from gpaw.tddft.spectrum import photoabsorption_spectrum
# Sodium dimer chain
d = 1.104 # N2 bondlength
L = 8.0 # N2-N2 distance
atoms = Atoms('Na2',
positions=[[L, 0, +d / 2], [L, 0, -d / 2]],
cell=[L, 8.0, 8.0])
atoms.center()
# Ground-state calculation
calc = GPAW(mode='lcao', h=0.3, basis='dzp', xc='LDA',
setups={'Na': '1'},
convergence={'density': 1e-12},
txt='gs_nad.out',
symmetry={'point_group': False})
atoms.calc = calc
energy = atoms.get_potential_energy()
calc.write('gs.gpw', mode='all')
# Time-propagation calculation
# Read converged ground-state file
td_calc = LCAOTDDFT('gs.gpw', rremission=RRemission(35.05, [0, 0, 1]))
# Attach any data recording or analysis tools
DipoleMomentWriter(td_calc, 'dm_nad.dat')
# Kick
td_calc.absorption_kick([0.0, 0.0, 1e-5])
# Propagate
td_calc.propagate(20, 5000)
# Please note that those parameter values are quite course
# and should be properly converged for subsequent applications.
# Calculate spectrum with small artificial broadening
photoabsorption_spectrum('dm_nad.dat', 'spec_nad.dat',
folding='Gauss', width=0.005,
e_min=0.0, e_max=10.0, delta_e=0.005)
```

The polarization of the electric field in the waveguide is aligned with the z-axis, exerting a recoil-force on the electronic coordinates that oscillate in this direction. As a result, the electronic motion is damped as energy is emitted into the waveguide, in line with Newtons third law.

Let us next add a second dimer parallel to the first one and orthorgonal to the chain-axis (H-aggregate configuration). For reasons that will become clear in the following, we can get away with half the propagation time.

```
from ase import Atoms
from gpaw import GPAW
from gpaw.lcaotddft import LCAOTDDFT
from gpaw.lcaotddft.qed import RRemission
from gpaw.lcaotddft.dipolemomentwriter import DipoleMomentWriter
from gpaw.tddft.spectrum import photoabsorption_spectrum
# Sodium dimer chain
d = 1.104 # N2 bondlength
L = 8.0 # N2-N2 distance
atoms = Atoms('Na4',
positions=([[L, 0, +d / 2], [L, 0, -d / 2], [2 * L, 0, +d / 2],
[2 * L, 0, -d / 2]]), cell=[L + 2 * L, 8.0, 8.0])
atoms.center()
# Ground-state calculation
calc = GPAW(mode='lcao', h=0.3, basis='dzp', xc='LDA',
setups={'Na': '1'},
convergence={'density': 1e-12},
txt='gs_nad2.out',
symmetry={'point_group': False})
atoms.calc = calc
energy = atoms.get_potential_energy()
calc.write('gs.gpw', mode='all')
# Time-propagation calculation
# Read converged ground-state file
td_calc = LCAOTDDFT('gs.gpw', rremission=RRemission(35.05, [0, 0, 1]))
# Attach any data recording or analysis tools
DipoleMomentWriter(td_calc, 'dm_nad2.dat')
# Kick
td_calc.absorption_kick([0.0, 0.0, 1e-5])
# Propagate
td_calc.propagate(20, 2500)
# Please note that those parameter values are quite course
# and should be properly converged for subsequent applications.
# Calculate spectrum with small artificial broadening
photoabsorption_spectrum('dm_nad2.dat', 'spec_nad2.dat',
folding='Gauss', width=0.005,
e_min=0.0, e_max=10.0, delta_e=0.005)
```

We smoothed the spectra in both cases with a sharp Lorentzian for visual appearance but the observed width corresponds to the correct lifetimes. In the perturbative limit, this corresponds to Wigner-Weisskop theory. Two features can be observed.

1) The width of two sodium dimers is double the width of a single one. For a set of non-interacting matter-subsystems, their emission probability of a single photon scales linearly with the number of subsystem, i.e., the rate increases here linearly with the number of dimers. Intuitively, more charge oscillating provides a larger dipole which leads to a stronger emission. Clearly, the dimers are not entirely non-interacting for distance of 0.8 nm, which leads to the following observation.

2) The peak is shifted slightly to higher frequencies. The parallel configuration between the dimers results in a coupling between the dipolar excitations that shift the excitation energy to slightly higher values. This configuration is also known as H-aggregate. A head to end configuration on the other hand is known as J-aggregate, providing a red-shift.

It is important to realize here that the propagation time has to be long enough to allow the recoil-forces to substantially damp the oscillations, i.e., enough time is needed to capture the decay.

Often this might be unfeasible or unsuitable such that an extrapolation from small cross-sectional areas and quick decay to larger areas and slower decay might be beneficial. A generalized implementation, including strong coupling and complex electromagnetic environments is currently in develop.