Quickstart

This example constructs a two-mode Gaussian state, defines a dissipative system, evolves the state in time, and evaluates simple observables from the resulting trajectory.

Imports

import numpy as np

from gaussian_systems.initial_state import GaussianCVState
from gaussian_systems.systems import GaussianCVSystem

1. Construct a Gaussian state

Create a two-mode vacuum state and apply single-mode squeezing to both modes.

state = GaussianCVState.vacuum(2)

state.single_mode_squeeze((1.0, 0.0), 1)
state.single_mode_squeeze((1.0, 0.0), 2)

The library uses 1-indexed mode labels, so the two modes are labeled 1 and 2.

State transformations act in place and return self, which allows chaining if desired.

state = GaussianCVState.vacuum(2)
state.single_mode_squeeze((1.0, 0.0), 1).single_mode_squeeze((1.0, 0.0), 2)

2. Construct a Gaussian system

Create a two-mode system with free evolution and add a collective annihilation dissipator.

system = GaussianCVSystem.free_evolution(2, np.array([0.0, 0.0]))
system.multi_annihilation_dissipator((1, 2), 1.0)

More structured models can be built by adding Hamiltonian couplings and different dissipative channels.

3. Evolve the state

Choose a time grid and evolve the state.

t_eval = np.linspace(0.0, 20.0, 500)
solution = system.evolve_state(state, t_eval)

The returned GaussianSolution object stores:

  • the time grid

  • the evolved mean vectors

  • the evolved covariance matrices

4. Evaluate observables

Compute a few trajectory-level quantities.

entanglement = solution.entanglement_time_trace((1, 2))
purity = solution.purity_time_trace()
entropy = solution.entropy_time_trace()

These are evaluated directly from the evolved covariance matrices.

5. Compare to a reference state

You can also compute fidelity to a fixed Gaussian state.

target = GaussianCVState.vacuum(2)
fidelity = solution.fidelity_time_trace_fixed(target)

Notes

  • Phase-space ordering is always

\[ (x_1, \dots, x_n, p_1, \dots, p_n) \]

  • Subsystems are specified using 1-indexed mode labels.

  • Evolution checks covariance matrices for quantum physicality at each time step.

  • The library is restricted to Gaussian states and time-independent generators.

Next Steps

For the mathematical conventions used throughout the package, see the Core Concepts page.

For more complete demonstrations, including structured reservoirs and finite-memory embeddings, see the Examples page.