Phase Transition And Landau-Zener Tunneling In PT-symmetric Waveguides With Periodic Potential

In the early 20th century,the basic theoretical system of quantum mechanics was established.Although it has been verified and developed in experiments and can be applied to many fields,in some fields there is still controversial.With the continuous development of quantum mechanics,people have gradually paid attention to non-Hermitian systems.In general,when the Hamiltonian has parity-and time-reversal?PT?symmetry,the eigenvalue is real in some region.When the parameter is larger than a critical value the eigenvalues become mutually complex conjugates,and the PT symmetry is broken.The critical value is also called the exception point.This theory has been demonstrated in experiments in optics,acoustics,and so on.In this paper,we study the dynamic evolution in PT symmetric optical waveguides,namely Laudau-Zener tunneling.First,we introduce the definition and properties of the P operator,T operator and PT combination operator,and describe the application of PT symmetry in optics.When the light propagates in the waveguide,the optical propagation equation of the Maxwell's equation under the paraxial approximation is similar to the time-dependent Schr?dinger equation.Using an optical waveguide with complex refractive index can introduce PT symmetry into the system.The Bloch theorem is used to transform the Hamiltonian from the real space to the momentum space according to the periodicity of the potential function in a one-dimensional array of curved waveguides satisfying PT symmetry.After numerically diagonalizing the Hamiltonian,it was found that when the periods of real and imaginary part of the potential function are the same and the intensity of the real part₁Vis larger than of the imaginary partV₂,the system has real energy spectrum.There exists an avoided level crossing between the two lowest bands in the PT symmetric region,and the waveguides bending degree provides the necessary driving external field.Therefore,we can approximate the analytical expression of the tunneling probability of Landau-Zenertunneling through the effective two-level Hamiltonian.Based on this,we study the effect of phase transition in PT symmetric waveguides with different periodic potentials of imaginary part.When the periods of real and imaginary parts of the potential function are different,we analyzed the eigenspectrum and the range of parameter values with the real spectrum.There still exists an avoided level crossing between the two lowest bands in the PT symmetric region,we can find that the Landau-Zenertunneling probability decreases with the increasing of₁Vand does not vary withV₂through numerical method.