Documentation

Theoretical Framework

Mathematical and physical foundations for nonlinear gain/loss dynamics in topological photonic lattices.

Lattice Models

Non-Reciprocal SSH Model

The NRSSH model uses intra-cell hoppings \(v\) and \(u\), inter-cell hopping \(r\), and onsite potential \(\epsilon\). Setting \(v = u\) recovers the SSH model, while \(v = u = r\) gives the trivial tight-binding limit.

NRSSH lattice structure
NRSSH lattice structure.
\[ H = \sum_i \left( v|2i\rangle\langle 2i+1| + u|2i+1\rangle\langle 2i| + r(|2i+1\rangle\langle 2i+2| + |2i+2\rangle\langle 2i+1|) \right) + \epsilon I \]

Diamond Model

The Diamond model uses hoppings \(t_1\), \(t_2\), \(t_3\), and \(t_4\). Facing dimerization satisfies \(t_1 = t_4\) and \(t_2 = t_3\); neighbouring dimerization satisfies \(t_1 = t_3\) and \(t_2 = t_4\); and the intra/inter case satisfies \(t_1 = t_2\) and \(t_3 = t_4\).

Diamond lattice structure
Diamond lattice structure.
\[ H = \sum_i \left( t_1(|3i\rangle\langle 3i+1| + |3i+1\rangle\langle 3i|) + t_2(|3i\rangle\langle 3i+2| + |3i+2\rangle\langle 3i|) + t_3(|3i+1\rangle\langle 3i+3| + |3i+3\rangle\langle 3i+1|) + t_4(|3i+2\rangle\langle 3i+3| + |3i+3\rangle\langle 3i+2|) \right) + \epsilon I \]

Gain and Loss

Nonlinear saturable gain is controlled by gain \(\gamma_1 \in (0, 1]\), loss \(\gamma_2 \in (0, 1]\), and saturation \(S \geq 0\). The onsite nonlinear gain term is:

\[ \frac{i\gamma_1}{1 + S|\varphi_m|^2} \]

NRSSH applies gain and loss on all sites. Diamond applies nonlinear saturable gain on A sites and constant loss \(\gamma_2\) on B and C sites.

Time Evolution

Natural units set \(\hbar = 1\) and often \(c = 1\), so \(p = \hbar k\) lets \(k\)-space represent momentum-space. The Schrodinger equation is:

\[ i\frac{d\varphi}{dt} = H\varphi \]

The first-order operator is:

\[ U(t) = I - i\,dt\,H(t) \]

The simulations use a second-order operator based on the \(\pm dt / 2\) update:

\[ \varphi(t + dt/2) = (I - i\,dt\,H / 2)\varphi(t) \]
\[ \varphi(t + dt/2) = (I + i\,dt\,H / 2)\varphi(t + dt) \]
\[ U(t) = (I - i\,dt\,H / 2)(I + i\,dt\,H / 2)^{-1} \]

Phase Classification