### Time Integration Methods for Quantum Systems Out of Equilibrium

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Thomas Blommel, University of Michigan
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Information about quantum systems out of equilibrium is contained within the non-equilibrium Green's functions. From the one particle non-equilibrium Green's function we can extract observables such as electron density, current, and the time-dependent spectrum. The one-particle Green's function is a two-time quantity which sits on the three-legged Keldysh contour and its equations of motion are a coupled set of integro-differential equations known as the Kadanoff-Baym equations. We have implemented a time integration method which allows us to numerically solve the Kadanoff-Baym equations given an input Hamiltonian. We apply our timestepping scheme to molecular systems and couple the system to external electric fields using a classical dipole coupling. This coupling allows us to calculate spectra corresponding to time-resolved photoabsorption spectra. We do our calculations within the second order self-consistent interaction expansion and use orbital basis sets commonly used in quantum chemistry.

**Abstract Author(s): **Thomas Blommel