1. Capri spring school, April 2009 With collaborators: P. Mehta - Princeton C. Bolech - Rice A. Jerez - NJIT, Rutgers G. Palacios - Rutgers N. Andrei - Rutgers Special Thanks: C. Yee -Rutgers Sung Po Chao Nonequilibrium transport in the Anderson model of a biased Quantum Dot

2. Goldhaber-Gordon et al, Cronenwet et al. van der Wiel et al. Conductance vs gate voltage - Kondo enhancement in odd valleys Differential conductance vs bias - Nonequilibrium dynamics T varies in the range 15-800mK Quantum Dot in and out of Equilibrium Equilibrium Nonequilibrium (Lee & Ng,Glazman & Raikh)

3. Hamiltonian to describe the system: Single Impurity Anderson model Scattering Bethe Ansatz approach: (P.Mehta, N.Andrei 06) 1. Diagonalize the Hamiltonian out of equilibrium: 2. Compute expectation value:

4. Current and Dot Occupation  Current and dot-occupation operators:  Expectation value expressed in terms of Bethe Ansatz density : with and given by

5. Conductance in equilibrium: Numerical results Conductance vs. gate voltage - Direct calculation from current. - Verifies Friedel SR - TBA vs SBA Von Delft, notes

6. Conductance out of equilibrium: Numerical results..2 cases Case 1: Case 2: From Kondo to mixed valence (0 refers to equilibrium Fermi surface of the leads)

7. Conductance out of equilibrium: Numerical results case 1 Obtain Tk from quadratic fit for Kondo peak nearby V=0. Observed universality for small voltage region. Conductance vs. bias voltage V : Conductance nearby zero voltage: Conductance for voltage larger than Kondo temp:

8. Conductance out of equilibrium: Numerical results case 2 Conductance vs. bias voltage : Contour plot of conductance vs source-drain voltage and gate voltage: The Kondo effect forms as is decreased, destroyed as the bias voltage is increased (M Grobis et al,PRL 100,246601)

9. Summary and Outlook Computed Nonequilibrium transport properties of Anderson impurity model. Outlook: Finite temperature/finite magnetic field. Muti-terminal quantum dot/quantum wire