Stochastic Master equation

# Stochastic Master equation

A stochastic master equation has the general form

$\dot{\rho} = -\frac{i}{\hbar} \big[H,\rho\big] + \mathcal{L}[\rho] + \mathcal{H}[\rho]$

where

$\mathcal{L}[\rho]=\sum_i \left( J_i \rho J_i^\dagger - \frac{1}{2} J_i^\dagger J_i \rho - \frac{1}{2} \rho J_i^\dagger J_i \right)$

and

$\mathcal{H}[\rho] = \sum_n\left[ \left(C_n\rho + \rho C_n^\dagger\right) - \langle C_n + C_n^\dagger\rangle\rho\right]\xi_n(t)$

Here, $J_i$ are the Lindblad damping operators, while $C_n$ are operators proportional to the white noise terms $\xi_n(t)$. The last term in $\mathcal{H}[\rho]$ (the expectation value) ensures trace conservation.

In order to describe a measurement, one has to set the operators $C_n$ proportional to collapse operators . The superoperator $\mathcal{H}$ then describes the information gain due to a measurement. For example, to describe unit-efficiency homodyne detection in a single cavity mode, one has to set $C = \sqrt{\kappa}a e^{-i\theta}$, where $\kappa$ is the damping rate, $a$ the photon annihilation operator and $\theta$ is the phase difference between the signal field and the local oscillator.

Note, that (if desired) stochastic terms $H_n^s$ in the Hamiltonian can be included by setting $C_n = -i H_n^s$. If the operators $H_n^s$ are Hermitian, then the $\mathcal{H}[\rho]$ simply becomes a commutator and the expectation value vanishes (since the commutator preserves the trace).

The function that implements this equation is very similar to timeevolution.master.

stochastic.master(tspan, ρ0, H, J, C; dt=dt)

The only additional argument here is C, which is a vector containing the operators $C_n$ which constitute the superoperator defined in the stochastic master equation above.

For time-dependent problems, one can make use of the dynamic version. You simply need to define two functions for the deterministic and the stochastic part of the master equation, respectively.

function fdeterm(t, rho)
# Calculate time-dependent stuff
H, J, Jdagger
end

function fstoch(t, rho)
# Calculate time-dependent stuff
C, Cdagger
end
stochastic.master_dynamic(tspan, ρ0, fdeterm, fstoch; dt=dt)

Note, that C and Cdagger have to be vectors of equal length containing the operators $C_n$. Optionally, one can include rates in the function output, fdeterm(t, rho) = H, J, Jdagger, rates for the deterministic part.

Like in the stochastic.schroedinger_dynamic function, if you want to avoid initial calculation of the given functions in order to find the total number of noise processes, you can do so by passing the noise_processes keyword argument. Note, that noise_processes has to be equal to length(C).

## References

 Jacobs, K. and Steck, D. A. A straighforward introduction to continuous quantum measurements, Contemporary Physics, 47:5, 279-303, (2006). URL: https://arxiv.org/abs/quant-ph/0611067