qspec.simulate. Interaction.hamiltonian  (  t ,  delta = None ,  m = 0 ,  v = None  )[source]

The interaction Hamiltonian of the coherent light-matter interaction in frequency units $$\begin{align} H &= H_\text{diagonal} + H_\text{off-diagonal}\\[1ex] &= \operatorname{diag}(A\vec{\omega}_0 + B\vec{\omega}^\prime) + \frac{1}{2}\sum\limits_m\Omega_m,\\[2ex] \omega^\prime_m &= 2\pi(\nu_m + \Delta_m) \gamma(\vec{v})(1 - \hat{k}_m\cdot\frac{\vec{v}}{c}) \end{align}$$ where $A$ (Interaction.atommap) is a matrix with shape (atom.size, atom.size), mapping the atomic frequencies $\vec{\omega}_0$ onto the diagonal of the Hamiltonian, $B$ (Interaction.deltamap) is a matrix with shape (atom.size, #lasers), mapping the laser frequencies in the rest-frame of the atom $\vec{\omega}^\prime$ onto the diagonal of the Hamiltonian, $\vec{v}$ is the velocity vector of the atom, $\hat{k}_m$ is the direction of laser m, $\Delta_m$ is the detuning of lasers m, $\gamma(\vec{v})$ is the time-dilation factor qspec.physics.gamma_3d, and $\Omega_m$ is the complex Rabi-frequency matrix of laser m, see qspec.simulate.Interaction.rabi.

If the Hamiltonian is time-dependent, because two or more lasers drive the same transition or form loops within the atom, the off-diagonal Hamiltonian becomes $$ (H_\text{off-diagonal})_{ij} = \frac{1}{2}\sum\limits_m(\Omega_m)_{ij} \exp\left[\operatorname{sign}(j - i)\,\mathrm{i}t\left(B\vec{\omega}^\prime\,- (T_m)_{ij}\,\omega^\prime_m\right)\right]\quad\text{for all }i\neq j, $$ where $T_m$ is a matrix for laser m that maps the laser frequency onto the transitions $|i\rangle\rightarrow |j\rangle$, whose entries take values $0,\pm 1$, depending on the energetic order of the two involved states and if the transition is driven by laser m.

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).

Returns:

HNone

The (time-dependent) Hamiltonian(s) for n samples and nt times in the shape (n, atom.size, atom.size, nt) ($2\pi\,\mathrm{MHz}$).