PULSUS

PULSUS: PUlse-dressed Liouvillian Spectroscopy Using Superoperators.

PULSUS is a Python package for direct frequency-domain spectroscopy of open quantum systems using Liouville-space superoperators.

PULSUS v0.1.1 focuses on linear and third-order spectroscopy with time-independent Markovian Liouvillian dynamics, including finite Gaussian pulses, impulsive limits, rotating-wave approximations, arbitrary field-sign sectors, and complementary impulsive time-domain calculations.

Core idea

For a field-free Liouvillian

[ \mathcal{L}, ]

PULSUS evaluates frequency-domain propagation using resolvent actions

[ G_\eta(\omega)

\left[ (\eta-i\omega)I-\mathcal{L} \right]^{-1}, ]

without explicitly constructing the matrix inverse.

Finite Gaussian pulses are represented by Liouvillian dressing operators. For example,

[ F_1^{(s)}

C e^{is\Phi} \exp\left[ \frac{\sigma^2}{2} \left( \mathcal{L} +i(\omega_1-s\omega_L)I \right)^2 \right]. ]

The production finite-pulse implementation uses the physical field-free Liouvillian inside the pulse dressing,

[ F_j = F_j(\mathcal{L}), ]

while the resolvents may use a small positive broadening parameter (\eta).

Standard finite-pulse assumption

The standard finite-pulse formulation assumes that the state immediately before the pulse sequence is stationary under the field-free Liouvillian,

[ \mathcal{L}|\rho_0\rangle\rangle = 0. ]

For dissipative systems this is typically the stationary state of the Liouvillian.

PULSUS can also accept explicitly supplied initial density matrices for calculations where this assumption is not desired.

Third-order spectroscopy

PULSUS currently provides four principal third-order models:

Function	Pulse treatment	Light-matter interaction
finite_pulse_spectrum	finite Gaussian pulses	full interaction
impulsive_spectrum	impulsive	full interaction
short_pulse_rwa_spectrum	Gaussian spectral envelopes	RWA
impulsive_rwa_spectrum	impulsive	RWA

The impulsive approximation and the rotating-wave approximation are treated as distinct approximations.

In particular,

[ \sigma \rightarrow 0 ]

for the full finite-pulse theory approaches the impulsive full-interaction response, not automatically the impulsive RWA response.

Linear spectroscopy

The corresponding linear-response calculations are available through:

pulsus.finite_pulse_linear_spectrum
pulsus.impulsive_linear_spectrum
pulsus.short_pulse_linear_rwa_spectrum
pulsus.impulsive_linear_rwa_spectrum

Installation

Install the released package from PyPI with:

pip install pulsus-spec

The PyPI distribution is named pulsus-spec, while the Python package is imported as:

import pulsus

For development from the repository source:

python -m venv .venv
source .venv/bin/activate
python -m pip install -e ".[dev]"

Run the test suite with:

pytest -q

Basic usage

A spectroscopy model is defined using SpectroscopySystem:

import pulsus

system = pulsus.SpectroscopySystem(
    H=H,
    dipole=mu,
    collapse_ops=collapse_ops,
)

The stationary state can be obtained from:

rho_ss = system.stationary_state()

A Gaussian pulse is specified by its carrier frequency, temporal width, amplitude, and phase:

pulse = pulsus.GaussianPulse(
    omega_L=1.10,
    sigma=1.25,
    E0=1.0,
    phase=0.0,
)

A finite-pulse nonrephasing spectrum can then be evaluated directly:

import numpy as np

omega1 = np.linspace(0.7, 1.5, 81)
omega3 = np.linspace(0.7, 1.5, 81)

S_NR = pulsus.finite_pulse_spectrum(
    system=system,
    pulse1=pulse,
    pulse2=pulse,
    pulse3=pulse,
    omega1=omega1,
    omega3=omega3,
    T=20.0,
    eta=0.02,
    pathway="NR",
)

The returned complex array has shape:

(len(omega3), len(omega1))

with convention

[ S[j,i]

S(\omega_{3,j},\omega_{1,i}). ]

Rephasing and nonrephasing convention

PULSUS uses signed first-coherence frequencies.

For the nonrephasing sector,

[ \omega_1>0, ]

with field-sign signature

[ (+,-,+). ]

For the rephasing sector,

[ \omega_1<0, ]

with field-sign signature

[ (-,+,+). ]

For example:

omega1_NR = np.linspace(0.7, 1.5, 81)
omega1_R = -omega1_NR

The rephasing spectrum may be displayed against the positive spectroscopic excitation coordinate -omega1_R, but the calculation itself uses signed negative first-coherence frequencies.

Arbitrary field-sign sectors

PULSUS is not restricted to R and NR pathways.

Any third-order signature

[ (s_1,s_2,s_3), \qquad s_j=\pm1, ]

can be supplied explicitly.

For example:

signature = (+1, +1, +1)

S = pulsus.finite_pulse_spectrum(
    system=system,
    pulse1=pulse,
    pulse2=pulse,
    pulse3=pulse,
    omega1=omega1,
    omega3=omega3,
    T=20.0,
    eta=0.02,
    signature=signature,
)

The aliases

pathway="NR"
pathway="R"

remain available for the standard nonrephasing and rephasing sectors.

Sign-reversed conjugation symmetry

For the tested coupled-dimer model, the field-sign sectors satisfy

[ S_{(-s_1,-s_2,-s_3)} (-\omega_1,-\omega_3)

S_{(s_1,s_2,s_3)} (\omega_1,\omega_3)^* ]

to floating-point precision.

This groups the eight possible signatures into four sign-reversed conjugate pairs.

Full interaction and RWA

Full-interaction calculations require only the total dipole operator

[ \mu. ]

RWA calculations additionally require a decomposition

[ \mu = \mu_+ + \mu_-. ]

For example:

system = pulsus.SpectroscopySystem(
    H=H,
    dipole=mu,
    collapse_ops=collapse_ops,
    dipole_plus=mu_plus,
    dipole_minus=mu_minus,
)

PULSUS currently requires this decomposition to be supplied explicitly rather than inferring it automatically.

Time-domain calculations

PULSUS also provides impulsive time-domain response functions.

For linear spectroscopy:

R_t = pulsus.impulsive_linear_rwa_time_signal(
    system=system,
    times=times,
    sign=+1,
)

For third-order spectroscopy:

R_t3_t1 = pulsus.impulsive_third_order_rwa_time_signal(
    system=system,
    t1=t1,
    t3=t3,
    T=20.0,
    pathway="NR",
)

These provide a complementary time-domain representation of the same impulsive Liouvillian dynamics used in the direct frequency-domain formulation.

Pathway polarization in laboratory time

A selected RWA pathway can also be followed through the sequence of three impulsive interactions:

P, components = pulsus.impulsive_pathway_polarization(
    system=system,
    times=times,
    pulse_times=(5.0, 15.0, 35.0),
    signature=(+1, -1, +1),
    return_components=True,
)

The resulting polarization is piecewise associated with the perturbative state active between pulses:

[ P_{\mathrm{path}}(t)

\begin{cases} 0, & t<\tau_1,\ P^{(1)}(t), & \tau_1\le t<\tau_2,\ P^{(2)}(t), & \tau_2\le t<\tau_3,\ P^{(3)}(t), & t\ge\tau_3. \end{cases} ]

This is the polarization of the selected pathway, not the total physical polarization from all perturbative orders and all field-sign sectors.

Complex spectra and plotting

PULSUS retains the complete complex response.

A spectrum can be plotted with:

pulsus.plot_spectrum(
    omega1=omega1,
    omega3=omega3,
    spectrum=S_NR,
    component="imag",
    normalize=True,
)

Supported displayed components are:

• "real"
• "imag"
• "abs"

Normalization is optional and disabled by default.

Numerical implementation

Two-dimensional spectrum calculations exploit the separable dependence on the excitation and detection frequencies.

For a third-order response, the grid can be written schematically as

[ S_{ji}=L_jR_i, ]

where (R_i) contains the (\omega_1)-dependent operations and (L_j) contains the (\omega_3)-dependent operations.

This avoids repeating expensive matrix exponentials and resolvent solves independently at every pair

[ (\omega_1,\omega_3). ]

The optimized implementation reproduces the original point-by-point calculations to floating-point precision.

Validation

PULSUS is tested against independently validated coupled-dimer calculations, frozen reference spectra, and explicit time-domain propagation.

For the finite-pulse development benchmark at

[ \sigma=1.25, \qquad T=20, \qquad \eta=0.02, ]

the production finite-pulse calculation gives complex spectral-shape discrepancies relative to an explicit finite-pulse time-domain ODE calculation of approximately

[ 0.61% \quad \text{(NR)} ]

and

[ 0.64% \quad \text{(R)}. ]

For comparison:

Model	NR error	R error
Finite-width PULSUS	0.61%	0.64%
Short-pulse RWA	5.51%	5.81%
Impulsive RWA	9.60%	9.83%

These values characterize this development benchmark and are not universal accuracy bounds.

Time-domain versus frequency-domain validation

For impulsive linear RWA response, numerical time propagation followed by an FFT agrees with the direct frequency-domain resolvent calculation with a relative complex error of approximately

[ 0.0035%. ]

For impulsive third-order NR response, two-dimensional time propagation followed by a 2D FFT agrees with the direct frequency-domain result with a relative complex error of approximately

[ 0.015%. ]

These comparisons independently validate the time/frequency sign and transform conventions used in PULSUS.

Test suite

The current test suite contains 63 passing unit and regression tests covering:

• Liouville-space vectorization and superoperators
• Lindblad Liouvillian construction
• stationary states
• resolvent actions
• Gaussian pulse dressing
• pulse-integral identities
• linear response
• third-order response
• R and NR sign conventions
• arbitrary field-sign signatures
• finite-pulse, impulsive, short-pulse, and RWA models
• two-dimensional spectrum construction
• optimized versus point-by-point evaluation
• plotting and array orientation
• frozen finite-pulse regression data
• impulsive time-domain propagation
• time-domain pathway polarization

Examples

The examples/ directory contains the current development and validation examples.

Main examples

01_finite_pulse_dimer.py

Builds the coupled dimer and calculates finite-pulse NR and R spectra through the PULSUS API.

03_reproduce_figure6.py

Compares finite-pulse, short-pulse RWA, and impulsive RWA spectra.

06_all_third_order_signatures.py

Calculates all eight third-order field-sign sectors and checks the sign-reversed conjugation relation.

09_linear_td_fft_vs_fd.py

Validates impulsive linear time-domain propagation and FFT against the direct frequency-domain resolvent calculation.

10_third_order_td_fft_vs_fd.py

Validates third-order NR time-domain propagation and a 2D FFT against the direct frequency-domain calculation.

11_impulsive_polarization_traces.py

Visualizes the selected impulsive pathway in laboratory time and separates first-, second-, and third-order polarization stages.

Validation and diagnostic examples

02_compare_notebook04.py

Compares the production finite-pulse implementation with frozen explicit-ODE validation data.

04_linear_response_comparison.py

Compares full-interaction and RWA linear spectra.

05_linear_rwa_quadratures.py

Examines real and imaginary quadratures of the linear full-interaction and RWA responses.

07_full_vs_rwa_signatures.py

Compares finite-pulse full-interaction and short-pulse RWA results across third-order signatures.

08_approximation_ladder.py

Separates the effects of the full-interaction, RWA, short-pulse, and impulsive approximations.

Current scope

PULSUS v0.1.1 assumes:

• finite-dimensional quantum systems
• time-independent Liouvillian dynamics
• Markovian Lindblad evolution
• electric-dipole light-matter coupling
• Gaussian finite-duration pulses for the finite-pulse frequency-domain formulation
• temporally ordered, separated pulses
• linear and third-order response
• full-interaction and RWA calculations
• arbitrary third-order field-sign signatures
• impulsive time-domain propagation

Finite-duration driven time-domain propagation is not currently part of the public production API.

Public API philosophy

The principal user-facing objects and functions are:

pulsus.SpectroscopySystem
pulsus.GaussianPulse

pulsus.finite_pulse_spectrum
pulsus.impulsive_spectrum
pulsus.short_pulse_rwa_spectrum
pulsus.impulsive_rwa_spectrum

pulsus.finite_pulse_linear_spectrum
pulsus.impulsive_linear_spectrum
pulsus.short_pulse_linear_rwa_spectrum
pulsus.impulsive_linear_rwa_spectrum

pulsus.impulsive_linear_time_signal
pulsus.impulsive_third_order_rwa_time_signal
pulsus.impulsive_pathway_polarization

pulsus.plot_spectrum

Additional lower-level response, Liouville-space, resolvent, and pulse-dressing functions remain available for advanced calculations and validation.

Repository structure

.
├── src/
│   └── pulsus/          PULSUS source code
├── tests/               unit and regression tests
├── examples/            usage and validation examples
├── notebooks/           numerical validation notebooks
├── figures/             manuscript figures
├── results/             benchmark results
├── main.tex             manuscript / technical development
├── references.bib       bibliography
└── literature_notes.md  literature notes

PULSUS is under active development.