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dq.lindbladian 

lindbladian(H: QArrayLike, jump_ops: list[QArrayLike], rho: QArrayLike) -> QArray

Applies the Lindbladian superoperator to a density matrix.

The Lindbladian superoperator \(\mathcal{L}\) is defined by: $$ \mathcal{L} (\rho) = -i[H,\rho] + \sum_{k=1}^N \mathcal{D}[L_k] (\rho), $$

where \(H\) is the system Hamiltonian, \(\{L_k\}\) is a set of \(N\) jump operators (arbitrary operators) and \(\mathcal{D}[L]\) is the Lindblad dissipation superoperator (see dq.dissipator()).

Note

This superoperator is also sometimes called Liouvillian.

Parameters

  • H (qarray-like of shape (..., n, n)) –

    Hamiltonian.

  • jump_ops (list of qarray-like, each of shape (, ..., n, n)) –

    List of jump operators.

  • rho (qarray-like of shape (..., n, n)) –

    Density matrix.

Returns

(qarray of shape (..., n, n)) Resulting operator (it is not a density matrix).
See also
  • dq.slindbladian(): returns the Lindbladian superoperator in matrix form (vectorized).

Examples 

>>> a = dq.destroy(4)
>>> H = a.dag() @ a
>>> L = [a, a.dag() @ a]
>>> rho = dq.fock_dm(4, 1)
>>> dq.lindbladian(H, L, rho)
QArray: shape=(4, 4), dims=(4,), dtype=complex64, layout=dense
[[ 1.+0.j  0.+0.j  0.+0.j  0.+0.j]
 [ 0.+0.j -1.+0.j  0.+0.j  0.+0.j]
 [ 0.+0.j  0.+0.j  0.+0.j  0.+0.j]
 [ 0.+0.j  0.+0.j  0.+0.j  0.+0.j]]