Tuning 2 ND AND 3 RD Order Exceptional Points by Kerr- Nonlinearity - PowerPoint PPT Presentation

ITMO University Department of Physics and Engineering Tuning 2 ND AND 3 RD Order Exceptional Points by Kerr- Nonlinearity By : Shahab Ramezanpour September 2020

Shahab Ramezanpour Tuning 2 ND AND 3 RD Order Exceptional Point by Kerr-Nonlinearity ➢ The exceptional point (EP) is a degeneracy in non-Hermitian systems at which the eigenvectors become parallel. ➢ It is different from degeneracy in Hermitian systems where the eigenvectors are orthogonal. ➢ The abrupt phase transitions around this point in photonic systems leads to exotic functionalities such as unidirectional invisibility, laser mode selectivity and sensitivity enhancement. ➢ Although EP is introduced in Quantum Mechanics, but it can be observed in optics and photonics containing resonators with gain and loss. 1/9

Shahab Ramezanpour Tuning 2 ND AND 3 RD Order Exceptional Point by Kerr-Nonlinearity Analogous between Nonlinear Shrodinger Equation in Quantum Mechanics and Nonlinear Coupled Mode Approach in Optics nonlinear Schrodinger equation (NLSE) Coupled-Mode Theory with or Gross – Pitaevskii equation Kerr-nonlinearity  d = −  −  −   −  2 1 ( | | ) i i g i 1 1 1 1 2 dt  d = −  −  −   −  2 2 ( | | ) i i g i 2 2 2 2 1 dt Applying on a two level system      −  +        2 Assuming monochromatic excitation 2 | | i g = 1 1 1      i    − +    2      | |  t g 2 2 2 d → −  i dt Time-Independent Form          −  +        −  +       2 2 2 | | | | i g i g =  =  1 1 1 1 1 1 1    1            −  +    2  − +    | |      2 i g      | |  g 2 2 2 2 2 2 2 2 2/9

Shahab Ramezanpour Tuning 2 ND AND 3 RD Order Exceptional Point by Kerr-Nonlinearity Matrix Form of Nonlinear Coupled Mode Theory    + −       2 The matrix equation, describes a nonlinear | | 2 a a g a i =        non-Hermition system  − +  2      | |  b b g b ( ) g shift the energy levels of the Hamiltonian by − + 2 2 | a | | | b 2 = / 2  +  −   c g       2 c i a a =         − −    = − 2 2       | | | | c b b a b +   +  + + −  −  −  − = 2 2 4 3 2 2 2 2 2 ( ) 2 (1 ) 2 0 g gh h g gh h After some algebraic manipulation   c (Stokes parameters) = =  = , , g h    There can be two up to four real roots  and each of them is connected to a  = + −  +  (1 ) c i  complex eigenvalue by 3/9

Shahab Ramezanpour Tuning 2 ND AND 3 RD Order Exceptional Point by Kerr-Nonlinearity The coupling and dissipation factors are unequal v = 1.01 > γ Imaginery part of eigenvalues Real part of eigenvalues The coupling and dissipation factors are equal v = 1.00 = γ Real part of eigenvalues Imaginery part of eigenvalues Graefe , Czechoslovak Journal of Physics , 2006 EP EP 4/9

Shahab Ramezanpour Tuning 2 ND AND 3 RD Order Exceptional Point by Kerr-Nonlinearity    + −       2 | | 2 a a g a i =  1       For large value of  − +  2      | |  b b g b 2 | ε |, we consider g 1 =g 2 =0  =  = 1 = = 1.8 g g 1 2 Finding eigenvalues and eigenfunctions decreasing | ε | Using the eigenfunction in nonlinear problem Convergence to a specific eigenvalue and eigenfunction 5/9

Shahab Ramezanpour Tuning 2 ND AND 3 RD Order Exceptional Point by Kerr-Nonlinearity At the discontinuities, we consider the eigenfunction +  +  +  +  ( , ) a i b i 1 2 3 4 c c     , , , as initial value of the next step, and change the 1 2 3 4 with considering the changing behavior of eigenfunctions in the previous steps. EP EP 6/9

Shahab Ramezanpour Tuning 2 ND AND 3 RD Order Exceptional Point by Kerr-Nonlinearity = = 2.3, 1.8 g g 1 2 EP EP 7/9

Shahab Ramezanpour Tuning 2 ND AND 3 RD Order Exceptional Point by Kerr-Nonlinearity    +  +      2 | | 0 i g a a a 1         =  2 | | g b b b       2        −  + 2 0 | |     i g c c c   3  = = = =  =  1 0.1, 0 g g g 2 1 2 2 8/9

Shahab Ramezanpour Tuning 2 ND AND 3 RD Order Exceptional Point by Kerr-Nonlinearity Conclusion ➢ We proposed a numerical method based on SFC and iteration methods to solve nonlinear non-Hermitian eigenvalue problems. This method is performed in two stages. ➢ It is observed that both 2nd and 3rd order EP can be tuned by the contrast between the Kerr nonlinearities in the matrix equation. 9/9

Thank you