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.. _example17-label:


Optical Bloch equations: Lindblad dynamics using the `matvec` function
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This example uses the `tools.misc.matvec()` function to define a Lindblad equation and solve the ODE using the `tools.evolution.evolve()` function. 

Consider the the two-level system: 

.. math::
	H &= \delta\sigma^z + \Omega_0\sigma^x, \\
	L &= \sigma^+.

where :math:`H` is the Hamiltonian of the two-level system, and :math:`L` is the Lindblad (or jump) operator. The Lindblad equation is a non-unitary extension of the Liouville-von Neumann equation for the density matrix :math:`\rho(t)`:

.. math::
	\partial_t\rho(t) = -i[H,\rho(t)] + 2\gamma\left(L\rho(t)L^\dagger - \frac{1}{2}\{L^\dagger L,\rho(t) \} \right),

where :math:`[\cdot,\cdot]` is the commutator, and :math:`\{\cdot,\cdot\}` is the anti-commutator. The system of equations for this specific problem is also known as the optical Bloch equations.

Below, we provide three ways to define the function for the Lindblad ODE:
	* the first version is very intuitive as it follows the definition of the Lindblad equation above; but it is rather slow; 
	* the second version uses the `hamiltonian.dot()` and `hamiltonian.rdot()` functions and is a bit more sophisticated and a bit faster; 
	* the third version uses the `matvec()` function and is faster than the previous two (but may not be memory efficient for large systems). Notice that there are different kinds of `matvec` functions for sparse matrices, depending on the format (csc, csr, etc.). In using them, one should keep in mind the format of the matrix, which can be determined using the `get_matvec_function`. 

Note that this way of simulating the Lindblad equation has severe limitations for many-body systems. An alternative, parallelizable way to effectively simulate a subset of Lindblad dynamics using unitary evolution is described in 
`arXiv:1608.01317 <https://arxiv.org/pdf/1608.01317.pdf>`_.

Script
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:download:`download script <../../../examples/scripts/example17.py>`

.. literalinclude:: ../../../examples/scripts/example17.py
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