qspec.simulate. Interaction.mc_master  (  t ,  delta = None ,  m = 0 ,  v = None ,  y0 = None ,  dynamics = False ,  ntraj = 500 ,  as_density_matrix = True  )[source]

Solver for the Monte-Carlo master equation, which is the Schrödinger equation with a non-hermitian Hamiltonian $$\begin{aligned} \frac{\partial\vec{\psi}}{\partial t} &= -\mathrm{i}(H + H_\text{leaky})\vec{\psi},\qquad \rho_{ij} = \lim\limits_{n\rightarrow\infty}\frac{1}{n}\sum\limits_{s=1}^n \langle i|\psi_s\rangle\langle\psi_s|j\rangle\\[2ex] (H_\text{leaky})_{jj} &= -\frac{\mathrm{i}}{2}\sum\limits_i\Gamma_{ij},\qquad (H_\text{leaky})_{ij}\big|_{i\neq j} = 0\\[2ex] \Gamma_{\!ij} &= \sum\limits_{Xk_{ij}} (a_{ij,k_{ij}}^{m_j - m_i})^2\,A_{ji}^{Xk_{ij}}\\[2ex] a_{ij,k_{ij}}^\lambda &= (-1)^{F_i + I + k_{ij} + J_j}\sqrt{(2F_i + 1)(2J_j + 1)} \langle F_im_ik_{ij}\lambda|F_jm_j\rangle\begin{Bmatrix}J_j & J_i & k_{ij} \\F_i & F_j & I\end{Bmatrix}\\[2ex] X_{\!ij} &= \begin{cases}+1, & \text{if electric } (\mathrm{E}k_{ij}) \\ \ \,0, & \text{if magnetic } (\mathrm{M}k_{ij})\end{cases}, \end{aligned}$$ where the Hamiltonian $H$ (2π MHz) is time-independent whenever possible, see qspec.simulate.Interaction.hamiltonian, $H_\text{leaky}$ is an imaginary diagonal operator, and $\rho$ is the density matrix in the limit of an infinite number of samples. Solutions for n samples can be calculated in parallel for nt times. The complexity of the Monte-Carlo approach only scales linearly with Atom.size, as compared with that of the exact master equation, scaling with Atom.size ** 2.

Parameters:

tarray_like

The times $t$ when to compute the solution. Any array is cast to the shape (nt, ), where nt is the size of the array t (μs).

deltaarray_like

An array of laser frequency shifts $\vec{\Delta}$. delta must be a scalar, a 1d- or 2d-array with shapes (n, ) or (n, nl), respectively, where nl is the number of lasers of the Interaction (MHz).

mOptional[int]

The index of the shifted laser. If delta is a 2d-array, m ist omitted.

varray_like

Atom velocities $\vec{v}$. Must be a scalar or have shape (n, ) or (n, 3). In the first two cases, the velocity vector(s) are assumed to be aligned with the $x$-axis (m/s).

y0array_like

The initial state of the Atom. This must be None or have shape (Atom.size, ) or (n, Atom.size). If None, all states with the same label as the first State in atom.states are populated equally.

dynamicsbool

Whether to compute the momentum dynamics of the photon-atom interactions.

ntrajint

The number of samples n to compute if no samples were given with delta, v, or y0.

as_density_matrixbool

Whether the result is returned as density matrices or as state vectors.

Returns:

rho_t(ndarray, ndarray)

The integrated Monte-Carlo master equation as a complex-valued array of shape (n, Atom.size, Atom.size, nt) or (n, Atom.size, nt).