1. Hisao Hayakawa (YITP, Kyoto University) based on collaboration with T. Yuge, T. Sagawa, and A. Sugita 1/24 44 Symposium on Mathematical Physics "New Developments in the Theory of Open Quantum Systems" Toruń, June 20-24, 2012 (June 24)

2. Tetsuro Yuge (YITP->Osaka Univ.) Ayumu Sugita (Osaka City Univ.) Takahiro Sagawa (YITP) & Ryosuke Yoshii (YITP) I would like to thank all these collaborators on this problem. 2/24

3.  Introduction  Geometric Pump for Fermion Transport ◦ Setup ◦ Main Results ◦ Special Cases & Example  Application to Entropy Production  Discussion  Conclusion 3/24

4.  Introduction  Geometric Pump for Fermion Transport ◦ Setup ◦ Main Results ◦ Special Cases & Example  Application to Entropy Production  DIscussion  Conclusion 4/24

5.  In mesoscopic systems, a current can exist even at zero bias. This effect is called the quantum pumping. 5/24 Nano-machine to extract work

6.  Adiabatic quantum pump ◦ Geometric effect is important (P. W. Brower, PRB58, 10135 (1998)). ◦ Control of system parameters  Can we get the pump effect by controlling reservoir parameters? 6/24

7.  Introduction  Geometric Pump for Fermion Transport ◦ Setup ◦ Main Results ◦ Special Cases & Example  Application to Entropy Production  Discussion  Conclusion 7/24

8.  Projection measurement  Counting: ◦ Number of spinless electrons transfer from L to R  Statistics & cumulant generating function 8/24

9.  We assume that the total Hamilitonian satisfies von-Neumann equation.  We calculate the modified von-Neumann equation via the counting field:  Ref. 9/24

10. 10/24 χ

11. Control parameters 11/24

12. where 12/24

13.  Based on FCS Born-Markov approximation + rotational wave approximation (RWA), we obtain 13/24

14. 14/24

15. 15/24

16. 16/24

17.  Introduction  Geometric Pump for Fermion Transport ◦ Setup ◦ Main Results ◦ Special Cases & Example  Application to Entropy Production  Discussion  Conclusion 17/24

18.  The method we adopted can be used for the calculation of any other quantities.  We can discuss the path dependence of the nonequilibrium entropy production.  Namely, the entropy is a geometric quantity under a nonequilibrium situation.  Note that the entropy production is a non- conserved quantity.  See Sagawa and HH, PRE 84, 051110 (2011). 18/24 heat

19. Path-dependence quasi-static process parameters space 19/24

20.  Introduction  Geometric Pump for Fermion Transport ◦ Setup ◦ Main Results ◦ Special Cases & Example  Application to Entropy Production  Discussion  Conclusion 20/24

21.  Effects of spins and many-body interactions ◦ We have already calculated Kondo problem (R. Yoshii and HH, in preparation). ◦ The many-body effect can be absorbed via Schrieffer-Wolff transform.  Without the potential scattering term, the result is unchanged.  If we introduce the term, the symmetry of evolution matrix is changed. So there is possibility to have the geometric effect. 21/24

22.  So far, we assume that particles are Fermions.  However, our analysis is based on RWA (quasi-classical) and the result contains only distribution function of reservoirs.  We expect that the geometric effect can appear for Bosons.  See Jie Ren et al., PRL 104, 170601 (2010). 22/24

23.  Introduction  Geometric Pump for Fermion Transport ◦ Setup ◦ Main Results ◦ Special Cases & Example  Application to Entropy Production  Discussion  Conclusion 23/24

24.  We have analyzed a quantum pump effect on Fermion transport.  We have found that spinless Fermions without interactions do not have any geometric effect if we control reservoir parameters.  We confirm that there exist geometric effects for the control of system parameters.  Such an idea can be used for entropy production. ◦ Geometric effects are important.  We are now calculating the Kondo problem. 24/24