1. Theoretical study of the phase evolution in a quantum dot in the presence of Kondo correlations Mireille LAVAGNA Work done in collaboration with A. JEREZ (ILL Grenoble and Rutgers University) and P. VITUSHINSKY (CEA-Grenoble)

2. Determining the phase of a QD by using a two path Aharonov-Bohm interferometer  QD Aharonov-Bohm oscillations of the conductance as a function of the magnetic flux  ref source drain the phase  introduced by the QD is deduced from the shift of the oscillations with magnetic field Experimental context: quantum dots studied by Aharonov-Bohm interferometry

3. allows to determine the phase and visibility of the QD Quantum interferometry

4. Unitary limit Kondo regimeCoulomb blockade Ji, Heiblum et Shtrikman PRL 88, 076601 (2002) Evolution of the phase when reducing coupling strength Uncomplete phase lapse plateau

5. for the Kondo effect in QD NRG and Bethe-Ansatz calculations Theoretical context Langreth PR 150, 516 (66) and Nozières JLTP 17, 31 (74) for the Kondo effect in bulk metals Gerland, von Delft, Costi, Oreg PRL 84, 3710 (2000)

6. 2-reservoir Anderson model Glazman and Raikh JETP Lett. 47, 452 (88) Ng and Lee PRL 61, 1768 (88) where 1-reservoir Anderson model Theoretical interpretation

7. In the case when there is no magnetic moment in the dot (for instance in the Kondo regime at T=0), spin-flip scattering cannot occur incoming outgoing Asymptotic solutions Scattering theory in 1D

8. (Friedel sum rule see Langreth Phys.Rev.’66) Scattering theory Using exact results on Fermi liquid at T=0, one can show that For the symmetric QD, following Ng and Lee PRL ’88 Scattering theory in 1D ^ ^ ^ Denoting the phase of by, one gets

9. Using trigonometric arguments Using again exact results on Fermi liquid at T=0, one can show Putting altogether, one gets

10. Scattering off a composite system Generalized Levinson’s theorem where is the number of bound states is the number of states excluded by the Pauli principle 1s " +1s # 1s " +1s " H + e Example: scattering of an electron by an atom of hydrogen Phase shift “Singlet” scattering: S z tot =0 “Triplet” scattering : S z tot =1 1s 1s 2 0 0 1s " is the ground state of a hydrogen atom Levinson’49 Swan ’55 Rosenberg and Spruch PRA’96

11. Quantum dot = Artificial atom Generalized Levinson’s theorem Scattering theory in 1D The single level Anderson model (SLAM) is not sufficient to capture the whole physics contained in the experimental device which can be viewed as an artificial atom. One may try to start with a many level Anderson model (MLAM) description of the system. We have chosen another route and introduced the missing ingredients through an additional multiplicative factor in front of the S-matrix of the SLAM.

12. with is chosen in order that satisfies the generalized Levinson theorem. It is easy to show that

13. Aharonov-Bohm interferometry Landauer formula Consequences (at T=0, H=0) Phase shift measured Conductance measured Scattering theory in 1D

14. P. Vitushinky, A.Jerez, M.Lavagna Quantum Information and Decoherence in Nanosystems, p.309 (2004) Experimental check of the prediction Scattering theory in 1D

15. We have numerically solved the Bethe ansatz equations to derive n 0 and hence  /  as a function of the parameters of the model (Wiegmann et al. JETP Lett. ’82 and Kawakami and Okiji, JPSJ ’82) Particle-hole symmetry Bethe-Ansatz solution at T=0 symmetric limit A.Jerez, P.Vitushinsky, M.Lavagna PRL 95, 127203 (2005)

16. Asymptotic behavior in the limit n 0 0 Universal behavior occurs when The existence of both those universal and asymptotic behavior is of valuable help in fitting the experimental data In the asymmetric regime,, n 0 shows a universal behavior as a function of the renormalized energy Bethe-Ansatz solution at T=0

17. Fit in the unitary limit and Kondo regimes All the experimental curves are shifted in order to get  =  at the symmetric limit Bethe-Ansatz solution at T=0

18. (a) Unitary limit (b) Kondo regime Fit in the unitary limit and Kondo regimes Very good agreement in presence of a single fitting parameter  /U (we consider linear correspondence between  0 and V G ) A.Jerez, P.Vitushinsky, M.Lavagna, PRL’05 Bethe-Ansatz solution at T=0

19. Conclusions 1.We have shown that there is a factor of 2 difference between the phase of the S-matrix responsible for the shift in the AB oscillations and the phase controlling the conductance. 2.This result is beyond the simple single-level Anderson model (SLAM) description and supposes to consider the generalisation to the multi-level Anderson model (MLAM). Done here in a minimal way by introducing a multiplicative factor in front of the S-matrix in order to guarantee the generalized Levinson theorem. 3.Then the phase measured by A.B. experiments is related to the total occupation n 0 of the dot which is exactly determined by Bethe- Ansatz calculations. We have obtained a quantitative agreement with the experimental data for the phase in two regimes. 4.We have also checked the prediction with experimental data on G(V G ) and  (V G ) and also found a very good agreement.