Master equation

Master time evolution

The dynamics of open quantum systems are governed by a master equation in Lindblad form:

\[\dot{\rho} = -\frac{i}{\hbar} \big[H,\rho\big] + \sum_i \big( J_i \rho J_i^\dagger - \frac{1}{2} J_i^\dagger J_i \rho - \frac{1}{2} \rho J_i^\dagger J_i \big)\]

It is implemented by the function

timeevolution.master(tspan, psi0, H, J)

The arguments required are quite similar to the ones of timeevolution.schroedinger. tspan is a vector of times, rho0 the initial state and H the Hamiltonian. We now also need the vector J that specifies the jump operators of the system.

The additional arguments available are

The first specifies the decay rates of the system with default values one. If rates is a vector of length length(J), then the i th entry of rates is paired with the i-th entry of J, such that J_i decays with $\gamma_i$. If, on the other hand, rates is a matrix, then all entries of J are paired with one another and matched with the corresponding entrie of rates, resulting in a Lindblad term of the form $\sum_{i,j}\gamma_{ij}\left(J_i\rho J_j^\dagger - J_i^\dagger J_j\rho/2 - \rho J_i^\dagger J_j/2\right)$.

The second keyword argument can be used to pass a specific set of jump operators to be used in place of all $J^\dagger$ appearances in the Lindblad term.

We can pass an output function just like the one for a Schrödinger equation. Note, though, that now the function must be defined with the arguments fout(t, rho).

Furthermore, a time-dependent Hamiltonian can also be implemented analogously to a Schrödinger equation using timeevolution.master_dynamic(tspan, rho0, f).

For performance reasons the solver internally first creates the non-hermitian Hamiltonian $H_\mathrm{nh} = H - \frac{i\hbar}{2} \sum_i J_i^\dagger J_i$ and solves the equation

\[\dot{\rho} = -\frac{i}{\hbar} \big[H_\mathrm{nh},\rho\big] + \sum_i J_i \rho J_i^\dagger\]

If for any reason this behavior is unwanted, e.g. special operators are used that don't support addition, the function timeevolution.master_h (h for hermitian) can be used.