Quantum tunneling is one of fundamental problems in physics. In the area of atomic and molecular physics tunneling is usually associated with ionization of atoms and molecules in external electric fields. We have developed two approaches to the theoretical description of this process. One is based on the semiclassical complex-time method. In this approach the wave function in the classically forbidden region is calculated from the solution of classical equations of motion in the complex time plane. This method has been developed and applied to tunneling ionization of several atoms and molecules. The second approach is purely quantum-mechanical, and is based on the Green's function for the electron motion in the superposition of the Coulomb and laser fields. This method works well in a broad range of electric fields and allows us to get a smooth transition from the tunneling regime into the region where the ionization is allowed classically.
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