Valley-dependent time evolution of coherent electron states in tilted anisotropic Dirac materials

The effect of the Dirac cone tilt of anisotropic two-dimensional materials on\nthe time evolution of coherent electron states in the presence of electric and\nmagnetic fields is studied. We propose a canonical transformation that maps the\nanisotropic Dirac-Weyl Hamiltonian with tilted Dirac cones to an effective and\nisotropic Dirac Hamiltonian under these fields. In this way, the well-known\nLandau-level spectra and wave functions allow calculating the Wigner matrix\nrepresentation of Landau and coherent states. We found a valley dependency in\nthe behavior of the Wigner function for both Landau and coherent electron\nstates. The time evolution shows that the interplay of the Dirac cone tilt and\nthe electric field keeps the uncertainties of both position and momentum in one\nvalley significantly lower than in the other valley. The increment of quantum\nnoise correlates with the emergence of negative values in the Wigner function.\nThese results may help us to understand the generation of coherent electron\nstates under the interaction with electromagnetic fields. The reported\nvalley-dependent signatures in the Wigner function of materials with tilted\nDirac cones may be revealed by quantum tomography experiments, even in the\nabsence of electric fields.\n