User:Physics2020/Parity-Time Symmetry in Integrated Optics

Parity-Time Symmetry (PT-symmetry) in integrated optics is an optical analog of non-Hermitian quantum mechanics with PT-symmetry. Although the physics is different, the Schrodinger equation of quantum mechanics and the paraxial wave equation for guided light beams in waveguides are mathematically equivalent. This similarity can be put to use to explore symmetry ideas in the context of practical optical devices.

Let x be the position operator, let p be the momentum operator and let i be the square root of minus one. Under a parity operation, $$ p \rightarrow -p, x \rightarrow  -x. $$  Under time reversal, $$ p \rightarrow -p, i \rightarrow  -i. $$.   We write the Hamiltonian as  $$ H = p^2/2m + V_R(x)+V_I(x). $$, where $$V_R(x)$$ is the real part of the potential energy and $$V_I(x)$$ is the imaginary part. In standard textbooks on quantum mechanics, the potential energy is always real. However, if the Hamiltonian is invariant under PT-symmetry, the eigenvalues can be real even if the potential is complex. PT symmetry implies that $$V_R(-x)=V_R(x)$$ and $$V_I(-x)=-V_I(x)$$.

The Schrodinger equation for a particle in an external potential is formally equivalent to the paraxial equation for scalar optical waves in a dielectric waveguide. In place of PT symmetry constraining the form of the potential energy, it now constrains the variation in space of the dielectric. In more practical terms, it constrains the spatial variation of the optical properties of waveguide that guides the beam. If n(x) is the complex index of refraction, then $$n_R(-x)=n_R(x)$$ and $$n_I(-x)=-n_I(x)$$. How can one implement this in an actual experiment? One way would be to make two straight, parallel waveguides, one with gain and the other with loss, as in the figures below.

Let us denote the amplifying waveguide by the number 1, and let us denote the lossy waveguide by the number 2. The coupled mode equations are
 * $$ i {dE_1\over dz}-i{\gamma_{Geff} \over 2}E_1+\kappa E_2=0,\ \ i {dE_2\over dz}-i{\gamma_{L} \over 2}E_2+\kappa E_1=0.$$

PT-invariance implies $$\gamma_{Geff}=\gamma_L\equiv\gamma$$. For $$\gamma<2\kappa$$, the eigenvalues are real, and there are two linearly independent eigenmodes (although they are not orthogonal for $$\gamma \ne 0$$). At the “exceptional point”, $$\gamma = 2\kappa$$, the eigenvalues are zero and there is only one eigenvector. For $$\gamma>2\kappa$$, the eigenvalues are imaginary.

Below threshold ($$\gamma<2\kappa$$), even though the eigenvalues are real, the 2x2 coupler is a non-reciprocal two-port network. For example, if light is inserted into the lossy port, it may emerge from both ports at the other end. But if light is inserted into the amplifying port, the light may emerge from only one of the ports at the other end. This may lead to novel optical devices.