This paper is devoted to the analysis of an abstract formula describing quantum adiabatic charge pumping in a general context. We consider closed systems characterized by a slowly varying time-dependent Hamiltonian depending on an external parameter α. The current operator, defined as the derivative of the Hamiltonian with respect to α, once integrated over some time interval, gives rise to a charge pumped through the system over that time span. We determine the first two leading terms in the adiabatic parameter of this pumped charge under the usual gap hypothesis. In particular, in case the Hamiltonian is time periodic and has discrete non-degenerate spectrum, the charge pumped over a period is given to leading order by the derivative with respect to α of the corresponding dynamical and geometric phases.
@article{CML_2010__2_2_159_0,
author = {Alain Joye and Valentina Brosco and Frank Hekking},
title = {Abstract adiabatic charge pumping},
journal = {Confluentes Mathematici},
pages = {159--180},
publisher = {World Scientific Publishing Co Pte Ltd},
volume = {2},
number = {2},
year = {2010},
doi = {10.1142/S1793744210000156},
language = {en},
url = {https://cml.centre-mersenne.org/articles/10.1142/S1793744210000156/}
}
TY - JOUR AU - Alain Joye AU - Valentina Brosco AU - Frank Hekking TI - Abstract adiabatic charge pumping JO - Confluentes Mathematici PY - 2010 SP - 159 EP - 180 VL - 2 IS - 2 PB - World Scientific Publishing Co Pte Ltd UR - https://cml.centre-mersenne.org/articles/10.1142/S1793744210000156/ DO - 10.1142/S1793744210000156 LA - en ID - CML_2010__2_2_159_0 ER -
%0 Journal Article %A Alain Joye %A Valentina Brosco %A Frank Hekking %T Abstract adiabatic charge pumping %J Confluentes Mathematici %D 2010 %P 159-180 %V 2 %N 2 %I World Scientific Publishing Co Pte Ltd %U https://cml.centre-mersenne.org/articles/10.1142/S1793744210000156/ %R 10.1142/S1793744210000156 %G en %F CML_2010__2_2_159_0
Alain Joye; Valentina Brosco; Frank Hekking. Abstract adiabatic charge pumping. Confluentes Mathematici, Tome 2 (2010) no. 2, pp. 159-180. doi : 10.1142/S1793744210000156. https://cml.centre-mersenne.org/articles/10.1142/S1793744210000156/
[1] M. Aunola and J. J. Toppari, Phys. Rev. B 68, 020502 (2003), DOI: 10.1103/PhysRevB.68.020502.
[2] J. E. Avron and A. Elgart, Commun. Math. Phys. 203, 445 (1999), DOI: 10.1007/s002200050620.
[3] J. E. Avronet al., Comm. Pure Appl. Math. 57, 528 (2004), DOI: 10.1002/cpa.3051.
[4] J. E. Avron, R. Seiler and L. G. Yaffe, Commun. Math. Phys. 110, 33 (1987), DOI: 10.1007/BF01209015.
[5] J. E. Avron and R. Seiler, Phys. Rev. Lett. 54, 259 (1985), DOI: 10.1103/PhysRevLett.54.259.
[6] M. V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984).
[7] A. Bohm et al. (eds.) , The Geometric Phase in Quantum Systems ( Springer , 2003 ) .
[8] M. Born and V. Fock, Z. Phys. 51, 165 (1928).
[9] F. Bornemann , Homogeneization in Time of Singularly Perturbed Mechanical Systems , Lecture Notes in Mathematics 1687 ( Springer , 1998 ) .
[10] V. Broscoet al., Phys. Rev. Lett. 100, 027002 (2008), DOI: 10.1103/PhysRevLett.100.027002.
[11] P. W. Brouwer, Phys. Rev. B 58, R10135 (1998), DOI: 10.1103/PhysRevB.58.R10135.
[12] M. Büttiker, H. Thomas and A. Prêtre, Z. Phys. B 94, 133137 (1994).
[13] R. Fazio, F. W. J. Hekking and J. P. Pekola, Phys. Rev. B 68, 0545410 (2003), DOI: 10.1103/PhysRevB.68.054510.
[14] M. Governaleet al., Phys. Rev. Lett. 95, 256801 (2005), DOI: 10.1103/PhysRevLett.95.256801.
[15] G. Hagedorn and A. Joye, Recent Advances in Differential Equations and Mathematical Physics, AMS Contemporary Mathematics Series 412, eds. N. Chernovet al. (Amer. Math. Soc., 2006) pp. 183–198.
[16] A. Joye, “Geometrical and mathematical aspects of the adiabatic theorem in quantum mechanics”, EPFL thesis No 1022, 1992. http://biblion.epfl.ch/EPFL/theses/1992/1022/EPFL_TH1022.pdf.
[17] A. Joye, Commun. Math. Phys. 275, 139 (2007), DOI: 10.1007/s00220-007-0299-y.
[18] A. Joye and C.-E. Pfister, J. Math. Phys. 34, 454 (1993), DOI: 10.1063/1.530255.
[19] T. Kato, J. Phys. Soc. Jpn. 5, 435 (1950), DOI: 10.1143/JPSJ.5.435.
[20] T. Kato , Perturbation Theory for Linear Operators ( Springer , 1980 ) .
[21] S. Lang , Real Analysis ( Addison-Wesley , 1973 ) .
[22] R. Leone, L. Levy and P. Lafarge, Phys. Rev. Lett. 100, 117001 (2008), DOI: 10.1103/PhysRevLett.100.117001.
[23] A. Messiah , Quantum Mechanics ( Dover , 2000 ) .
[24] M. Möttönenet al., Phys. Rev. B 73, 214523 (2006).
[25] M. Möttönen, J. J. Vartiainen and J. P. Pekola, Phys. Rev. Lett. 100, 177201 (2008).
[26] G. Nenciu, J. Phys. A 13, L15 (1980), DOI: 10.1088/0305-4470/13/2/002.
[27] G. Nenciu, Commun. Math. Phys. 82, 121 (1981), DOI: 10.1007/BF01206948.
[28] G. Nenciu, Recent Developments in Quantum Mechanics (Poiana Braşov, 1989), Math. Phys. Stud. 12 (Kluwer, 1991) pp. 133–149.
[29] Q. Niu and D. J. Thouless, J. Phys. A 17, 30 (1984), DOI: 10.1088/0305-4470/17/12/016.
[30] J. P. Pekolaet al., Phys. Rev. B 60, R9931 (1999), DOI: 10.1103/PhysRevB.60.R9931.
[31] M. S. Sarandy and D. A. Lidar, Phys. Rev. A 71, 012331 (2005), DOI: 10.1103/PhysRevA.71.012331.
[32] B. Simon, Phys. Rev. Lett. 51, 2167 (1983), DOI: 10.1103/PhysRevLett.51.2167.
[33] S. Teufel , Adiabatic Perturbation Theory in Quantum Dynamics , Lecture Notes in Mathematics 1821 ( Springer , 2003 ) .
[34] F. Wilczek and A. Zee, Phys. Rev. Lett. 52, 2111 (1984), DOI: 10.1103/PhysRevLett.52.2111.
Cité par Sources :