# 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.

(1)\[H_{\rm eff}=H_{\rm sys}+ \frac{i\hbar}{2}\left( -\sum_{n}C^{+}_{n}C_{n}+ |C_{n} \psi |^2\right).\]

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

(2)\[\delta p_{n} = \left<\psi(t)|C_{n}^{+}C_{n}|\psi(t)\right> \delta t\]

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,

(3)\[\delta \psi = \left( \frac{C_n \psi} {\left| C_n \psi \right|} - \psi \right)\]

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

(4)\[\delta \psi(t) = - i H_{\rm sys} \psi(t) \delta t + \sum_n \left( -\frac{C_n^{+} C_n}{2} \psi(t) \delta t + \frac{ \left| C_n \psi \right| ^2}{2} \delta t + \delta N_n \left( \frac{C_n \psi} {\left| C_n \psi \right|} - \psi \right)\right),\]

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

(5)\[L_{\rm eff}(\rho(t)) = L_{\rm sys}(\rho(t)) + \sum_{n}\left( \rm{tr} \left(C_{n}^{+}C_{n} \rho C_{n}^{+}C_{n} \right) - C_{n}^{+}C_{n} \rho C_{n}^{+}C_{n} \right),\]

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

(6)\[\delta p = \rm{tr} \left( C \rho C^{+} \right) \delta t.\]

After a jump, the density matrix become

\[\rho' = \frac{C \rho C^{+}}{\rm{tr} \left( C \rho C^{+} \right)}.\]

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

(7)\[\rho(t + \delta t) = \rho(t) + L_{\rm eff}(\rho) \delta t + \delta N \left(\frac{C \rho C^{+}}{\rm{tr} \left( C \rho C^{+} \right)} - \rho \right).\]