dq.slindbladian

slindbladian(H: ArrayLike, jump_ops: ArrayLike) -> Array

Returns the Lindbladian superoperator (in matrix form).

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.sdissipator()).

The vectorized form of this superoperator is: $$ -i (I_n \otimes H) + i (H^\mathrm{T} \otimes I_n) + \sum_{k=1}^N \left( L_k^* \otimes L_k - \frac{1}{2} (I_n \otimes L_k^\dag L_k) - \frac{1}{2} (L_k^\mathrm{T} L_k^* \otimes I_n) \right). $$

Note

This superoperator is also sometimes called Liouvillian.

Parameters

H (array_like of shape (..., n, n)) –
Hamiltonian.
jump_ops (array_like of shape (N, ..., n, n)) –
Sequence of jump operators.

Returns

(array of shape (..., n^2, n^2)) Lindbladian superoperator.