An analytical solution for quantum scattering through a $\mathcal{P}\mathcal{T}$-symmetric delta potential

Abstract

We employ the Lippmann–Schwinger formalism to derive the analytical solutions of the transmission and reflection coefficients through a one-dimensional open quantum system, in which particle loss or gain on one lattice site located at x = 0, or particle loss and gain on the lattice sites located at $x=\pm \frac{L}{2}$ are considered respectively. The gain and loss on the lattice site are modeled by the delta potential with positive and negative imaginary values. The analytical solution reveals the underlying physics that the sum of the transmission and reflection coefficients through an open quantum system (even a $\mathcal{P}\mathcal{T}$-symmetric open system) may not be 1, i.e., qualitatively explains that the number of particles is not conserved in an open quantum system. Furthermore, we find that the resonance states can be formed in the $\mathcal{P}\mathcal{T}$-symmetric delta potential, which is similar to the case of real delta potential. The results of our analysis can be treated as the starting point of studying quantum transport problems through a non-Hermitian system using Green’s function method, and more general cases for high-dimensional systems may be deduced by the same procedure.

Keywords

transmission / non-Hermitian / $\mathcal{P}\mathcal{T}$-symmetry / Green function

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