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.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

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

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

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

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

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