Spectral Density \(J(\omega)\)

Goal

In this tutorial you will learn how to:

1. Compute the spectral density \(J_{\alpha\beta}(\omega)\) from pre-computed dyadic Green’s function (GF) data stored in HDF5 files.

2. Plot the resulting spectral density curves with customisable layout, labelling, and styling.

The spectral density links the electromagnetic environment (encoded in the imaginary part of the dyadic Green’s function) to the energy-transfer coupling between quantum emitters. It is defined as

\[J_{\alpha\beta}(\omega) = \frac{\omega^{2}}{\pi\,\hbar\,\varepsilon_{0}\,c^{2}}\; \boldsymbol{\mu}_{\alpha} \cdot \operatorname{Im}\!\bigl[\mathbf{G}(\mathbf{r}_{\alpha}, \mathbf{r}_{\beta};\omega)\bigr] \cdot \boldsymbol{\mu}_{\beta}\]

where \(\boldsymbol{\mu}_{\alpha,\beta}\) are the transition dipole moments of the donor and acceptor, and the result is expressed in units of eV.

Prerequisites

• A working MQED installation (see /getting_started).

• A dyadic Green’s function HDF5 file produced by one of the GF tutorials (e.g. Dyadic Green’s Function via Sommerfeld Integrals). The file must contain the imaginary part of the GF together with the energy and position grids.

Quick start

Tip

If you have already run the Dyadic Green’s Function via Sommerfeld Integrals tutorial, the default configuration expects the output at ./data/gf_data.h5. Adjust the input_file path if your file is elsewhere.

Step 1 — Compute the spectral density

python -m mqed.analysis.spectral_density

This reads the default configuration from configs/analysis/spectral_density.yaml, loads the GF data, evaluates the spectral density, and writes the result to an HDF5 file.

Step 2 — Plot the spectral density

python -m mqed.plotting.plot_spectral_density

This reads the default configuration from configs/plots/spectral_density.yaml, loads the spectral density HDF5 file produced in Step 1, and saves a publication-ready PNG figure.

Configuration reference — analysis

The analysis step is configured by configs/analysis/spectral_density.yaml. A minimal version looks like:

# configs/analysis/spectral_density.yaml

input_file:  '${oc.env:MQED_ROOT,./data}/gf_data.h5'
output_file: 'spectral_density.h5'

orientations:
  donor:
    theta_deg: 90.0
    phi_deg:   0.0
  acceptor:
    theta_deg: 90.0
    phi_deg:   0.0

Analysis parameters

Parameter	Default	Description
input_file	./data/gf_data.h5	Path to the GF HDF5 file (supports ${oc.env:MQED_ROOT} interpolation).
output_file	spectral_density.h5	Output HDF5 file for the computed spectral density.
orientations.donor.theta_deg	90.0	Polar angle \(\theta\) (degrees) of the donor transition dipole moment.
orientations.donor.phi_deg	0.0	Azimuthal angle \(\phi\) (degrees) of the donor transition dipole moment.
orientations.acceptor.theta_deg	90.0	Polar angle \(\theta\) (degrees) of the acceptor transition dipole moment.
orientations.acceptor.phi_deg	0.0	Azimuthal angle \(\phi\) (degrees) of the acceptor transition dipole moment.

Tip

You can specify the magic angle orientation (\(\theta \approx 54.74°\)) by using the keyword magic in place of a numeric value in any theta_deg or phi_deg field.

Specifying dipole orientations

The orientation of each transition dipole is given in spherical coordinates \((\theta, \phi)\). Common choices:

• x-oriented dipole — theta_deg: 90, phi_deg: 0

• y-oriented dipole — theta_deg: 90, phi_deg: 90

• z-oriented dipole — theta_deg: 0, phi_deg: 0

• Magic angle — theta_deg: magic (≈ 54.74°, isotropic average)

To override the orientations from the command line:

python -m mqed.analysis.spectral_density \
    orientations.donor.theta_deg=0 \
    orientations.acceptor.theta_deg=0

This sets both donor and acceptor dipoles to the z-direction.

Using a custom configuration file

You can point Hydra at a different configuration directory or override the config name:

python -m mqed.analysis.spectral_density \
    --config-path /absolute/path/to/my_configs \
    --config-name my_spectral_density

Or override individual values via the command line:

python -m mqed.analysis.spectral_density \
    input_file=/path/to/my_gf_data.h5 \
    output_file=my_spectral_density.h5

Configuration reference — plotting

The plotting step is configured by configs/plots/spectral_density.yaml. A minimal version looks like:

# configs/plots/spectral_density.yaml

input_file: '${oc.env:MQED_ROOT,./outputs}/spectral_density/spectral_density.h5'

font:
  family:       Arial
  labelsize:    18
  ticksize:     16
  legendsize:   14
  titlesize:    18
  labelweight:  bold
  titleweight:  bold

plot_settings:
  separation_indices: [0]
  pair_indices:       [[0, 0]]
  label_template:     'Rx = {Rx:.1f} nm'
  xlabel:             'Energy (eV)'
  ylabel:             'J (eV)'
  title:              'Spectral Density'
  xscale:             linear
  yscale:             linear
  figsize:            [8, 5]
  lw:                 1.5
  dpi:                300
  grid:               true
  save_plot:          true
  filename:           'spectral_density.png'

Plotting parameters

Parameter	Default	Description
input_file	(see YAML)	Path to the spectral density HDF5 file.
font.family	Arial	Matplotlib font family.
font.labelsize	18	Font size for axis labels.
font.ticksize	16	Font size for tick labels.
font.legendsize	14	Font size for the legend.
font.titlesize	18	Font size for the plot title.
font.labelweight	bold	Font weight for axis labels.
font.titleweight	bold	Font weight for the plot title.
plot_settings.separation_indices	[0]	List of separation (Rx) indices to plot when using the separation-indexed GF layout.
plot_settings.pair_indices	[[0, 0]]	List of emitter-pair index pairs [i, j] to plot when using the pair-indexed GF layout.
plot_settings.label_template	Rx = {Rx:.1f} nm	Python format string for the legend label. {Rx} is replaced by the Rx value in nanometres.
plot_settings.xlabel	Energy (eV)	Label for the x-axis.
plot_settings.ylabel	J (eV)	Label for the y-axis.
plot_settings.title	Spectral Density	Plot title.
plot_settings.xscale	linear	Scale for the x-axis (linear or log).
plot_settings.yscale	linear	Scale for the y-axis (linear or log).
plot_settings.figsize	[8, 5]	Figure size in inches [width, height].
plot_settings.lw	1.5	Line width for the plotted curves.
plot_settings.dpi	300	Resolution of the saved figure.
plot_settings.grid	true	Whether to display a grid on the plot.
plot_settings.save_plot	true	Whether to save the figure to disk.
plot_settings.filename	spectral_density.png	Filename for the saved figure.

Customising the plot from the command line

Override any plotting parameter via Hydra:

python -m mqed.plotting.plot_spectral_density \
    plot_settings.xscale=log \
    plot_settings.yscale=log \
    plot_settings.title='Log-scale Spectral Density' \
    plot_settings.filename=spectral_density_log.png

Expected output

After running Step 1 (analysis), the output HDF5 file (spectral_density.h5) contains:

• J_eV — the spectral density array in eV. The shape depends on the GF layout:

  • Separation layout [K, M]: K separation indices, M energy points.

  • Pair layout [N, N, M]: all emitter pairs for N emitters over M energy points.

• energy_eV — the energy grid in eV.

• Rx_nm (separation layout only) — the donor–acceptor separations in nanometres.

After running Step 2 (plotting), you will find a PNG file (default: spectral_density.png) showing \(J_{\alpha\beta}(\omega)\) versus energy.

Using the output in downstream analyses

The spectral density is a key input for quantum-dynamics simulations. You can load it in your own scripts:

import h5py
import numpy as np

with h5py.File('spectral_density.h5', 'r') as f:
    J_eV     = np.array(f['J_eV'])
    energy_eV = np.array(f['energy_eV'])

# Use J_eV and energy_eV in your quantum-dynamics workflow
# e.g., as input to the Lindblad master equation solver

See also

• Dyadic Green’s Function via Sommerfeld Integrals — computing the dyadic Green’s function that serves as input.

• Quantum Dynamics — using the spectral density in a quantum-dynamics simulation.

• Plotting & Analysis — general plotting utilities in MQED.