Effective Interactions
Effective interactions occur in the equations of motion of large spin systems that have certain symmetries so that the dynamics of every single spin is identical:
\[\begin{aligned} \langle\dot{\sigma^x}\rangle &= \Omega^{\mathrm{eff}}\langle\sigma^y\rangle\langle\sigma^z\rangle -\frac{1}{2} \Big( \gamma -\Gamma^{\mathrm{eff}}\langle\sigma^z\rangle \Big) \langle\sigma^x\rangle, \\ \langle\dot{\sigma^y}\rangle &= -\Omega^{\mathrm{eff}}\langle\sigma^x\rangle\langle\sigma^z\rangle -\frac{1}{2} \Big( \gamma -\Gamma^{\mathrm{eff}}\langle\sigma^z\rangle \Big) \langle\sigma^y\rangle, \\ \langle\dot{\sigma^z}\rangle &= -\gamma \big(1 + \langle\sigma^z\rangle\big) -\frac{1}{2} \Gamma^{\mathrm{eff}} \Big(\langle\sigma^x\rangle^2 + \langle\sigma^y\rangle^2\Big). \end{aligned}\]
These quantities encapsulate the influence of all spins onto one single spin:
\[\begin{aligned} \Omega^\mathrm{eff} = \sum_{j=2}^N \Omega_{1j} \\ \Gamma^\mathrm{eff} = \sum_{j=2}^N \Gamma_{1j}. \end{aligned}\]
The following functions can be used to easily calculate them for common examples:
CollectiveSpins.effective_interaction.triangle_orthogonalCollectiveSpins.effective_interaction.square_orthogonalCollectiveSpins.effective_interaction.rectangle_orthogonalCollectiveSpins.effective_interaction.cube_orthogonalCollectiveSpins.effective_interaction.box_orthogonalCollectiveSpins.effective_interaction.chainCollectiveSpins.effective_interaction.chain_orthogonalCollectiveSpins.effective_interaction.squarelattice_orthogonalCollectiveSpins.effective_interaction.hexagonallattice_orthogonalCollectiveSpins.effective_interaction.cubiclattice_orthogonalCollectiveSpins.effective_interaction.tetragonallattice_orthogonalCollectiveSpins.effective_interaction.hexagonallattice3d_orthogonal
Rotated effective interactions
If we allow for the individual atomic states to bare a spatially dependent phase of $\Delta \phi$ on the excited state, i.e. $|\psi_k{\rangle} = \frac{1}{\sqrt{2}} \left( |g{\rangle} + \exp (i \phi_k) |e{\rangle} \right)$, we can absorb this into our equations efficiently. Using the abbreviations $\Omega_{kj}^\mathrm{cos} = \Omega_{kj} \cos(\phi_k - \phi_j)$ and $\Omega_{kj}^\mathrm{sin} = \Omega_{kj} \sin(\phi_k - \phi_j)$ we obtain the following modified equations of motion
\[\begin{aligned} \frac{d}{dt}\langle\tilde{\sigma}_k^x\rangle &= \sum_{j;j \neq k} \Omega_{kj}^\mathrm{sin} \langle\tilde{\sigma}_j^x\sigma_k^z\rangle + \sum_{j;j \neq k} \Omega_{kj}^\mathrm{cos} \langle\tilde{\sigma}_j^y\sigma_k^z\rangle -\frac{1}{2} \gamma \langle\tilde{\sigma}_k^x\rangle +\frac{1}{2} \sum_{j;j \neq k} \Gamma_{kj}^\mathrm{cos} \langle\tilde{\sigma}_j^x \sigma_k^z\rangle -\frac{1}{2}\sum_{j;j \neq k} \Gamma_{kj}^\mathrm{sin} \langle\tilde{\sigma}_j^y \sigma_k^z\rangle \\ \frac{d}{dt}\langle\tilde{\sigma}_k^y\rangle &= -\sum_{j;j \neq k} \Omega_{kj}^\mathrm{cos} \langle\tilde{\sigma}_j^x\sigma_k^z\rangle + \sum_{j;j \neq k} \Omega_{kj}^\mathrm{sin} \langle\tilde{\sigma}_j^y\sigma_k^z\rangle -\frac{1}{2} \gamma \langle\tilde{\sigma}_k^y\rangle +\frac{1}{2} \sum_{j;j \neq k} \Gamma_{kj}^\mathrm{sin} \langle\tilde{\sigma}_j^x \sigma_k^z\rangle +\frac{1}{2} \sum_{j;j \neq k} \Gamma_{kj}^\mathrm{cos} \langle\tilde{\sigma}_j^y \sigma_k^z\rangle \\ \frac{d}{dt}\langle\sigma_k^z\rangle &= -\sum_{j;j \neq k} \Omega_{kj}^\mathrm{sin} ( \langle\tilde{\sigma}_j^x \tilde{\sigma}_k^x\rangle + \langle\tilde{\sigma}_j^y \tilde{\sigma}_k^y\rangle) +\sum_{j;j \neq k} \Omega_{kj}^\mathrm{cos} ( \langle\tilde{\sigma}_j^x \tilde{\sigma}_k^y\rangle - \langle\tilde{\sigma}_j^y \tilde{\sigma}_k^x\rangle) \\&\qquad -\gamma (1+ \langle\sigma_k^z\rangle) -\frac{1}{2} \sum_{j;j \neq k} \Gamma_{kj}^\mathrm{cos} ( \langle\tilde{\sigma}_j^x \tilde{\sigma}_k^x\rangle + \langle\tilde{\sigma}_j^y \tilde{\sigma}_k^y\rangle) -\frac{1}{2} \sum_{j;j \neq k} \Gamma_{kj}^\mathrm{sin} ( \langle\tilde{\sigma}_j^x \tilde{\sigma}_k^y\rangle - \langle\tilde{\sigma}_j^y \tilde{\sigma}_k^x\rangle). \end{aligned}\]
We see that the following definitions prove to be very helpful
\[\begin{aligned} \Omega_k^\mathrm{cos} &= \sum_{j;j \neq k} \Omega_{kj} \cos(\phi_k-\phi_j) \qquad \Omega_k^\mathrm{sin} = \sum_{j;j \neq k} \Omega_{kj} \sin(\phi_k-\phi_j) \\ \Gamma_k^\mathrm{cos} &= \sum_{j;j \neq k} \Gamma_{kj} \cos(\phi_k-\phi_j) \qquad \Gamma_k^\mathrm{sin} = \sum_{j;j \neq k} \Gamma_{kj} \sin(\phi_k-\phi_j) \end{aligned}\]
Again, if we consider highly symmetric configurations where $\Omega^\mathrm{f} = \Omega^\mathrm{f}_k$ and $\Gamma^\mathrm{f} = \Gamma^\mathrm{f}_k$ and the rotated states are initially identical we can define the effective rotated quantities
\[\begin{aligned} \tilde{\Omega}^\mathrm{eff} &= \Omega^\mathrm{cos} - \frac{1}{2} \Gamma^\mathrm{sin} \\ \tilde{\Gamma}^\mathrm{eff} &= \Gamma^\mathrm{cos} + 2 \Omega^\mathrm{sin} \end{aligned}\]
which lead to a closed set of simplified effective equations as well, i.e.
\[\begin{aligned} \frac{d}{dt}\langle\tilde{\sigma}^x\rangle &= \tilde{\Omega}^{\mathrm{eff}}\langle\tilde{\sigma}^y\rangle\langle\sigma^z\rangle -\frac{1}{2} \gamma \langle\tilde{\sigma}^x\rangle +\frac{1}{2} \tilde{\Gamma}^{\mathrm{eff}} \langle\tilde{\sigma}^x\rangle\langle\sigma^z\rangle \\ \frac{d}{dt}\langle\tilde{\sigma}^y\rangle &= -\tilde{\Omega}^{\mathrm{eff}}\langle\tilde{\sigma}^x\rangle\langle\sigma^z\rangle -\frac{1}{2} \gamma \langle\tilde{\sigma}^y\rangle +\frac{1}{2} \tilde{\Gamma}^{\mathrm{eff}} \langle\tilde{\sigma}^y\rangle\langle\sigma^z\rangle \\ \frac{d}{dt}\langle\sigma^z\rangle &= -\gamma \big(1 + \langle\sigma^z\rangle\big) -\frac{1}{2} \tilde{\Gamma}^{\mathrm{eff}} \Big(\langle\tilde{\sigma}^x\rangle^2 + \langle\tilde{\sigma}^y\rangle^2\Big) \end{aligned}\]
The calculation of these quantities for a few systems is implemented by: