# Stochastic Solver - Photocurrent¶

Photocurrent method, like monte-carlo method, allows for simulating an individual realization of the system evolution under continuous measurement.

## Closed system¶

Photocurrent evolution have the state evolve deterministically between quantum jumps. During the deterministic part, the system evolve by schrodinger equation with a non-hermitian, norm conserving effective Hamiltonian.

With \(C_{n}\), the collapse operators. This effective Hamiltonian is equivalent to the monte-carlo effective Hamiltonian with an extra term to keep the state normalized. At each time step of \(\delta t\), the wave function has a probability

of making a quantum jump. \(\delta t\) must be chosen small enough to keep
that probability small \(\delta p << 1\). *If multiple jumps happen at the
same time step, the state become unphysical.*
Each jump result in a sharp variation of the state by,

The basic photocurrent method directly integrates these equations to the first-order. Starting from a state \(\left|\psi(0)\right>\), it evolves the state according to

for each time-step. Here \(\delta N = 1\) with a probability of \(\delta \omega\) and \(\delta N_n = 0\) with a probability of \(1-\delta \omega\).

Trajectories obtained with this algorithm are equivalent to those obtained with
monte-carlo evolution (up to \(O(\delta t^2)\)).
In most cases, `qutip.mcsolve`

is more efficient than
`qutip.stochastic.photocurrent_sesolve`

.

## Open system¶

Photocurrent approach allows to obtain trajectories for a system with both measured and dissipative interaction with the bath. The system evolves according to the master equation between jumps with a modified liouvillian

with the probability of jumps in a time step \(\delta t\) given by

After a jump, the density matrix become

The evolution of the system at each time step if thus given by