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

September 2020

By : Shahab Ramezanpour

Department of Physics and Engineering ITMO University

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

Shahab Ramezanpour Tuning 2ND AND 3RD Order Exceptional Point by Kerr-Nonlinearity

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➢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

• bserved in optics and photonics containing resonators with gain

and loss.

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2 1 1 1 2 2 2 2

2 | | | | i g i t g              − +      =       − +      

2 1 1 1 1 1 2 2 2 2 2 2 2 1

( | | ) ( | | ) d i i g i dt d i i g i dt             = − − − − = − − − −

2 1 1 1 2 2 2 2

2 | | | | i g g               − +     =      − +      

2 1 1 1 1 1 2 2 2 2 2 2

| | | | i g i g                − +     =      − +      

d i dt  → −

Coupled-Mode Theory with Kerr-nonlinearity Assuming monochromatic excitation nonlinear Schrodinger equation (NLSE)

• r Gross–Pitaevskii equation

Applying on a two level system Time-Independent Form Analogous between Nonlinear Shrodinger Equation in Quantum Mechanics and Nonlinear Coupled Mode Approach in Optics

Shahab Ramezanpour Tuning 2ND AND 3RD Order Exceptional Point by Kerr-Nonlinearity

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Matrix Form of Nonlinear Coupled Mode Theory The matrix equation, describes a nonlinear non-Hermition system

2 2

| | 2 | | a a g a i b b g b         + −     =       − +      

( )

2 2

| a | | | 2 g b − +

shift the energy levels of the Hamiltonian by

2 c i a a c b b         + −       =       − −      

/ 2 c g =

2 2

| | | | a b  = −

After some algebraic manipulation (Stokes parameters)

2 2 4 3 2 2 2 2 2

( ) 2 (1 ) 2 g gh h g gh h       + + + + − − − − = , , c g h       = = = There can be two up to four real roots and each of them is connected to a complex eigenvalue by

(1 ) c i      = + − +

Shahab Ramezanpour Tuning 2ND AND 3RD Order Exceptional Point by Kerr-Nonlinearity

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v = 1.01 > γ v = 1.00 = γ The coupling and dissipation factors are unequal The coupling and dissipation factors are equal

Real part of eigenvalues Imaginery part of eigenvalues Real part of eigenvalues Imaginery part of eigenvalues

EP

Graefe, Czechoslovak Journal

• f Physics, 2006

EP

For large value of |ε|, we consider g1=g2=0 Using the eigenfunction in nonlinear problem Finding eigenvalues and eigenfunctions Convergence to a specific eigenvalue and eigenfunction

decreasing |ε|

2 1 2 2

| | 2 | | a a g a i b b g b         + −     =       − +      

1   = =

1 2

1.8 g g = =

Shahab Ramezanpour Tuning 2ND AND 3RD Order Exceptional Point by Kerr-Nonlinearity

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1 2 3 4

( , )

c c

a i b i +  +  +  + 

At the discontinuities, we consider the eigenfunction 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

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1 2

2.3, 1.8 g g = =

EP EP

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2 1 2 2 2 3

| | | | | | i g a a a g b b b i g c c c           + +           =             − +      

1 2 2

0.1, g g g = = =

2   =

1  =

Shahab Ramezanpour Tuning 2ND AND 3RD 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.

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Shahab Ramezanpour Tuning 2ND AND 3RD Order Exceptional Point by Kerr-Nonlinearity

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