Quantum Dynamics

Goal

In this tutorial you will compute the quantum dynamics for an open quantum system using the Lindblad master equation and non-Hermitian Schrödinger equation (NHSE) solvers implemented in MQED. The Lindblad solver is built on top of the QuTiP package, while the NHSE solver uses a custom implementation. All input parameters are specified in a YAML configuration file, and the output is written to an HDF5 file.

By the end you will know how to:

• run the mqed_lindblad and mqed_nhse commands with default and custom parameters,

• switch between the Lindblad and non-Hermitian solvers,

• locate and interpret the HDF5 output file.

See also

See the companion theory documentation for the derivation of the Lindblad and non-Hermitian equations of motion (citation forthcoming).

Prerequisites

Make sure you have installed the package and activated the environment as described in Installation.

This tutorial uses example data bundled under data/example/GF_data/. To generate your own Green’s function cache, see Dyadic Green’s Function via Sommerfeld Integrals.

Tip

The example data ships with the repository under data/example/. No prior simulation run is required to follow this tutorial.

Quick start

Run the Lindblad solver from the repository root with all defaults:

mqed_lindblad

Or use the non-Hermitian approach (recommended for large systems):

mqed_nhse

Customising the run

Override individual parameters on the command line using Hydra syntax:

mqed_nhse simulation.Nmol=50 simulation.d_nm=4.0

Use a different YAML in the same config directory (configs/Lindblad/):

mqed_nhse --config-name=my_nhse_config

Use a YAML from an arbitrary directory:

mqed_nhse --config-dir=/path/to/my/configs --config-name=my_nhse_config

The same flags work for mqed_lindblad:

mqed_lindblad --config-name=my_lindblad_config simulation.Nmol=20

Tip

You can combine --config-name (or --config-dir) with individual parameter overrides. This is useful for running the same physical setup with different solver methods or lattice sizes.

Lindblad vs NHSE

Both solvers propagate the excitonic dynamics of \(N\) emitters coupled through the dyadic Green’s function.

• Lindblad — full density-matrix evolution via the Lindblad master equation, implemented with QuTiP’s mesolve. Scales as \(O(N^2)\) in memory.

• NHSE — state-vector evolution under a non-Hermitian effective Hamiltonian. Much lighter for large \(N\) and yields identical expectation values for single-excitation problems.

Tip

For transport studies with \(N \gtrsim 50\), the NHSE solver is typically an order of magnitude faster.

Configuration reference

The NHSE configuration file (configs/Lindblad/quantum_dynamics_nhse.yaml) is reproduced below. The Lindblad configuration follows the same structure and is located at configs/Lindblad/quantum_dynamics.yaml.

# ── Green's function input ─────────────────────────────
greens:
  h5_path: ${oc.env:MQED_ROOT,${hydra:runtime.cwd}}/data/example/GF_data/BEM_GF_planar_Ag_665nm_height_2nm.hdf5

height: 2                    # nm (for naming only; actual height is read from the GF file)

# ── Material information ───────────────────────────────
material:
  name: silver
  geometry: planar            # or 'sphere', 'nanorod', etc.

# ── Observables ────────────────────────────────────────
observables:
  - name: root_MSD
    kind: derived
    enabled: true

  - name: X_shift
    kind: operator

  - name: X_shift2
    kind: operator

  # Conditional versions (recommended for NHSE transport comparison)
  - name: X_shift_cond
    type: callable

  - name: X_shift2_cond
    type: callable

  - name: IPR_site
    kind: callable
    params:
      Nmol: ${simulation.Nmol}

  # Optionally: site populations
  # - name: pop_site
  #   kind: operator
  #   params:
  #     site: 1                   # 1-based site index

# ── Simulation controls ────────────────────────────────
simulation:
  # Time grid (picoseconds)
  t_ps:
    start: 0.0
    stop: 150.0
    output_step: 5e-3

  coupling_limit:
    enable: false
    V_hop_radius: 1
    keep_V_on_site: false
    Gamma_rule: leave           # "leave" | "same_as_V" | "diagonal_only" | "limit_by_hops"
    Gamma_hop_radius: null
    keep_Gamma_on_site: true

  Nmol: 100                     # number of emitter sites
  d_nm: 3                       # lattice spacing (nm)

  lambda_nm: 665                # emitter wavelength (nm)
  gf_method: BEM                # 'BEM' or 'Fresnel'

  # Dipole orientations
  mu_D_debye: 3.8
  mu_A_debye: 3.8
  theta_deg: 90.0
  phi_deg: 'magic'              # or a numeric value
  mode: stationary              # or 'disorder'
  # disorder_sigma_phi_deg: 15.0  # uncomment for disordered orientations

# ── Initial condition ──────────────────────────────────
initial_state:
  site_index: 1                 # start the exciton at site |1⟩

# ── Solver ─────────────────────────────────────────────
solver:
  method: NonHermitian           # 'Lindblad' or 'NonHermitian'

# ── Output ─────────────────────────────────────────────
output:
  filename: ${simulation.gf_method}_${material.name}_${material.geometry}_${simulation.lambda_nm}nm_N${simulation.Nmol}_height_${height}nm_inter_${simulation.d_nm}nm.hdf5

Key parameters at a glance

Parameter	Description	Default
greens.h5_path	Path to the cached dyadic Green’s function HDF5 file.	See YAML above
simulation.Nmol	Number of emitter sites in the chain.	100
simulation.d_nm	Intermolecular (lattice) spacing in nm.	3
simulation.mu_D_debye	Donor transition dipole moment (Debye). mu_A_debye is the same for the acceptor.	3.8
simulation.t_ps	Time grid: start, stop, and output_step in picoseconds.	0 – 150 ps, step 5e-3
coupling_limit.enable	Truncate dipole-dipole interaction to a finite hopping radius.	false
coupling_limit.V_hop_radius	Number of nearest-neighbour hops retained (when enable: true).	1
solver.method	Solver backend: Lindblad (QuTiP) or NonHermitian.	NonHermitian

Expected output

After the simulation finishes, a success message is printed to the terminal (and to the Hydra log file outputs/NonHermitian/.../NonHermitian.log):

2025-10-24 11:40:02.818 | SUCCESS | mqed.Lindblad.run_quantum_dynamics:app_run:229
- Simulation complete. Output saved to:
  /.../MacroscopicQED/outputs/NonHermitian/.../NAME.hdf5

The HDF5 file contains:

• time grid — the array of time points in picoseconds,

• state vectors / density matrices — the time-evolved quantum state,

• observables — MSD, IPR, site populations, and any other quantities listed in the observables section of the configuration.

What’s next?

• Plotting & Analysis — visualise MSD, participation ratio, and other transport observables from the dynamics output.