arXiv:1001.2664v2[cond-mat.mes-hall]30 Apr 2010 Charge and spin dynamics in interacting quantum dots Janine Splettstoesser 1 , $^{2}$, Michele Governale$^{3}$, J¨ urgen K¨ onig $^{4}$, and Markus B¨ uttiker 5 1 Institut f¨ ur Theoretische Physik A, RWTH Aachen University, D-52056 Aachen, Germany 2 JARA - Fundamentals of Future Information Technology, Germany 3 School of Physical and Chemical Sciences and MacDiarmid Institute for Advanced Materials and Nanotechnology, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand 4 Theoretische Physik, Universit¨ at Duisburg-Essen and CeNIDE, D-47048 Duisburg, Germany $^{5}$D´epartement de Physique Th´eorique, Universit´e de Gen`eve, CH-1211 Gen`eve 4, Switzerland (Dated: October 30, 2018) The transient response of a quantum dot with strong Coulomb interaction to a fast change in the gate potential, as well as the stationary ac-response to a slow harmonic variation in the gate potential are computed by means of a real-time diagrammatic expansion in the tunnel-coupling strength. We find that after a fast switching, the exponential relaxation behavior of charge and spin are governed by a single time constant each, which differ from each other due to Coulomb repulsion. We compare the response to a step potential with the RC time extracted from the ac response. PACS numbers: 73.23.-b,73.23.Hk Single electron charging of nanoscale devices driven by a time-dependent signal provides novel electron sources for the investigation of mesoscopic transport. 1,2 Under- standing their charging behavior is of crucial importance to assess maximum operation speed. From the theoreti- cal perspective such an investigation can shed light on the influence of quantum-mechanical processes on the time-dependent behavior of nanoscale systems. In this work, we focus on the charging of a quantum dot, which acts as a mesoscopic capacitor plate, the other plate is a macroscopic gate. 3,4 This system, when subject to dy- namical modulations, is the key ingredient to realize a co- herent single-electron source, as shown both experimen- tally 1 and theoretically. 5 The previous studies of meso- scopic capacitors showed that the linear low-frequency response regime is characterized by a quantum electro- chemical capacitance and a charge relaxation resistance, cf. the circuit in Fig. 1. When the capacitor is at- tached to a single-channel lead, the charge relaxation resistance in the fully coherent regime is found to be quantized 3,4,6,7 demonstrating quantum mechanical non- locality of impedances. Theoretical discussions have ad- dressed the regime of weak Coulomb interaction taken into account on the Hartree-Fock level 6 or more recently in the Luttinger-liquid regime, which can be treated by bosonization methods. 8 In this paper, we study the time response of a single- level quantum dot driven out of equilibrium by either the fast switching (step pulse) of the gate potential or by a slow periodic modulation. The charge relaxation time after a fast switching has been experimentally extracted in, e.g., Ref. 1. Also the time scale of spin relaxation in a single quantum dot was measured, 9 where the relaxation of the spin is due to spin-orbit or hyperfine interaction. For theoretical in- vestigations of these subjects, see, e.g., Refs. 10. Spin relaxation due to tunneling through a double dot in the While Coulomb interaction is weak in some of the ex- periments, 1,4 side-gate coupled semiconductor quantum dots 2 often exhibit strong local Coulomb interaction ef- fects and, therefore, a theory applicable to this regime is highly desirable. We consider a quantum dot with arbitrarily strong onsite Coulomb interaction, coupled weakly to a reservoir by a tunneling contact. We use a diagrammatic real-time approach to study the relax- ation behavior of such a mesoscopic capacitor beyond the Markov limit, taking into account tunneling processes up to next-to-leading order in the tunnel coupling (i.e. fourth order in the tunnel matrix elements). This ex- pansion permits arbitrary interaction strength U . Both U and temperature are assumed to be large compared spin blockade regime was analyzed in Ref. 11. 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