Stochastic Solver: Mixing stochastic and deterministic equations¶

Copyright (C) 2011 and later, Paul D. Nation & Robert J. Johansson

Direct photo-detection¶

Here we follow an example from Wiseman and Milburn, Quantum measurement and control, section. 4.8.1.

Consider cavity that leaks photons with a rate $\kappa$. The dissipated photons are detected with an inefficient photon detector, with photon-detection efficiency $\eta$. The master equation describing this scenario, where a separate dissipation channel has been added for detections and missed detections, is

$\dot\rho = -i[H, \rho] + \mathcal{D}[\sqrt{1-\eta} \sqrt{\kappa} a] + \mathcal{D}[\sqrt{\eta} \sqrt{\kappa}a]$

To describe the photon measurement stochastically, we can unravelling only the dissipation term that corresponds to detections, and leaving the missed detections as a deterministic dissipation term, we obtain [Eq. (4.235) in W&M]

$d\rho = \mathcal{H}[-iH -\eta\frac{1}{2}a^\dagger a] \rho dt + \mathcal{D}[\sqrt{1-\eta} a] \rho dt + \mathcal{G}[\sqrt{\eta}a] \rho dN(t)$

or

$d\rho = -i[H, \rho] dt + \mathcal{D}[\sqrt{1-\eta} a] \rho dt -\mathcal{H}[\eta\frac{1}{2}a^\dagger a] \rho dt + \mathcal{G}[\sqrt{\eta}a] \rho dN(t)$

where

$\displaystyle \mathcal{G}[A] \rho = \frac{A\rho A^\dagger}{\mathrm{Tr}[A\rho A^\dagger]} - \rho$

$\displaystyle \mathcal{H}[A] \rho = A\rho + \rho A^\dagger - \mathrm{Tr}[A\rho + \rho A^\dagger] \rho $

and $dN(t)$ is a Poisson distributed increment with $E[dN(t)] = \eta \langle a^\dagger a\rangle (t)$.

Formulation in QuTiP¶

The photocurrent stochastic master equation is written in the form:

$\displaystyle d\rho(t) = -i[H, \rho] dt + \mathcal{D}[B] \rho dt

• \frac{1}{2}\mathcal{H}[A^\dagger A] \rho(t) dt

• \mathcal{G}[A]\rho(t) d\xi$

where the first two term gives the deterministic master equation (Lindblad form with collapse operator $B$ (c_ops)) and $A$ the stochastic collapse operator (sc_ops).

Here $A = \sqrt{\eta\gamma} a$ and $B = \sqrt{(1-\eta)\gamma} $a.

In QuTiP, the monte carlo solver can solve this equation when the deterministic part is passed as a liouvillian.

Highly efficient detection¶

Highly inefficient photon detection¶

Efficient homodyne detection¶

The stochastic master equation for inefficient homodyne detection, when unravaling the detection part of the master equation

$\dot\rho = -i[H, \rho] + \mathcal{D}[\sqrt{1-\eta} \sqrt{\kappa} a] + \mathcal{D}[\sqrt{\eta} \sqrt{\kappa}a]$,

is given in W&M as

$d\rho = -i[H, \rho]dt + \mathcal{D}[\sqrt{1-\eta} \sqrt{\kappa} a] \rho dt

\mathcal{D}[\sqrt{\eta} \sqrt{\kappa}a] \rho dt

\mathcal{H}[\sqrt{\eta} \sqrt{\kappa}a] \rho d\xi$

where $d\xi$ is the Wiener increment. This can be described as a standard homodyne detection with efficiency $\eta$ together with a stochastic dissipation process with collapse operator $\sqrt{(1-\eta)\kappa} a$. Alternatively we can combine the two deterministic terms on standard Lindblad for and obtain the stochastic equation (which is the form given in W&M)

$d\rho = -i[H, \rho]dt + \mathcal{D}[\sqrt{\kappa} a]\rho dt + \sqrt{\eta}\mathcal{H}[\sqrt{\kappa}a] \rho d\xi$

Standard homodyne with deterministic dissipation on Lindblad form¶

Versions¶