Non-Hermitian Hamiltonians and Quantum Transport history
Overview

Traditional treatment of quantum transport is based on the scattering theory [1]. A correspondence between the scattering matrix (S-matrix) and Hamiltonian approaches is established within the framework of Fano–Feshbach formalism [2][3][4]. In this formalism, an effective non-Hermitian Hamiltonian is introduced, whose complex eigenvalues coincide with scattering matrix poles. Non-Hermitian Hamiltonians are of great interest in modern quantum physics, as they can describe various phenomena beyond the traditional paradigm of Hermitian operators in a very robust and illustrative way [5]. Non-Hermitian Hamiltonians typically appear in the study of open quantum systems (OQS), where the total Hermitian Hamiltonian of the whole system is projected on the states of its subsystem of interest [2] resulting in a non-Hermitian effective Hamiltonian. OQS being a part of a bigger system, does not have stationary eigenstates. Eigenstates of the projected effective Hamiltonian are called resonant states, and corresponding eigenvalues are complex, with the real part indicating the energy and the imaginary part showing the decay rate (outgoing momentum flux [6]). However, incoming and outgoing (scattered) waves are characterized by real energies. Hence, the connection between complex eigenvalues of an effective Hamiltonian (poles of S-matrix) with real energies of transmission peaks/dips is of high importance. Usually, one associates energies of tunneling transmission resonances with real parts of the S-matrix poles. This interpretation is adequate only in the case of well-separated and narrow resonances. If perfect (unity-valued) resonances become wider and closer to each other, they can coalesce, resulting in a single transmission peak with amplitude smaller than unity [7]. This phenomenon cannot be detected from the analysis of the S-matrix poles alone [8]. In complex systems, where destructive quantum interference (DQI) is possible, much more complicated interference phenomena are expected, so the traditional S-matrix (or effective Hamiltonian) point of view cannot handle all the variety of possible interference effects in quantum transport.

This video is adapted from 10.3390/e22040459

References

  1. Datta, S. Electronic Transport in Mesoscopic Systems; Cambridge Studies in Semiconductor Physics; Cambridge University Press: Cambridge, UK, 1997.
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  3. Fano, U. Effects of Configuration Interaction on Intensities and Phase Shifts. Phys. Rev. 1961, 124, 1866–1878.
  4. Miroshnichenko, A.E.; Flach, S.; Kivshar, Y.S. Fano resonances in nanoscale structures. Rev. Mod. Phys. 2010, 82, 2257–2298.
  5. Moiseyev, N. Non-Hermitian Quantum Mechanics; Cambridge University Press: Cambridge, UK, 2011.
  6. Hatano, N.; Sasada, K.; Nakamura, H.; Petrosky, T. Some Properties of the Resonant State in Quantum Mechanics and Its Computation. Prog. Theor. Phys. 2008, 119, 187–222.
  7. Gorbatsevich, A.; Zhuravlev, M.; Kapaev, V. Collapse of resonances in semiconductor heterostructures as a transition with symmetry breaking in an open quantum system. J. Exp. Theor. Phys. 2008, 107, 288–301.
  8. Gorbatsevich, A.; Shubin, N. Coalescence of resonances in dissipationless resonant tunneling structures and PT-symmetry breaking. Ann. Phys. 2017, 376, 353–371.
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