Adiabatic quantum motors

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1 Felix von Oppen Freie Universität Berlin with Raul Bustos Marun and Gil Refael

2 Motion at the nanoscale Brownian motion Directed motion at the nanoscale?? 2

3 Directed motion at the nanoscale Nanocars 3

4 Nanoscale motor ac actuation 4

5 Powering Windsor Castle Wikipedia 5

6 θ Adiabatic quantum pump Periodic variation of parameters pumps current through device Adiabatic quantum motor Current through device causes periodic variation of motor degree of freedom 6

7 Thouless pump v=a/t Fermi energy in nth gap: n electrons per period a E F Sliding potential: quantized current I=nev 7

8 Thouless pump Electrons in 1d channel in GaAs heterostructure Moving periodic potential is due to surface acoustic waves Increased gaps due to Coulomb repulsion Talyanskii et al. Phys. Rev. B (1997) 8

9 Quantum pumps Pumped charge (Brouwer PRB 1998): see also: Zhou, Spivak, Altshuler (1999) 9

10 Gedanken motors θ Quantum motor based on chaotic quantum dot Thouless motor Mesoscopic conductor coupled to slow & classical motor degree(s) of freedom 10

11 Early design study [ ] 11

12 Gedanken motors θ Quantum motor based on chaotic quantum dot Thouless motor Mesoscopic conductor coupled to slow & classical motor degree(s) of freedom 12

13 Adiabatic approximation R: internuclear distance Coupling fast quantum system (electrons) to slow degree(s) of freedom (nuclei) compute electronic levels for fixed nuclear coordinates electrons exert potential (Born-Oppenheimer) force on nuclei 13

14 Beyond Born-Oppenheimer next order in quantum system acquires Berry s phase slow system subject to velocity-dependent force 14

15 Generalization: Scattering systems Alternative motivation: nanoelectromechanical systems Slow classical degrees of freedom coupled to fast quantum mechanical scattering system non-interacting mesoscopic conductor beyond linear response beyond a single mechanical mode beyond weak electron-vibron coupling express forces in terms of electronic S-matrix 15

16 Derivation I. Green functions: quantum dot Bode et al. PRL 2011 coupled to leads force adiabatic expansion scattering matrix (and A-matrix) II. Scattering theory Thomas et al. PRB

17 Born-Oppenheimer force adiabatic S-matrix: Born-Oppenheimer force: 17

18 Thermal equilibrium Born-Oppenheimer: Friedel sum rule: force is conservative in thermal equilbrium 18

19 Out of equilibrium Out-of-equilibrium mesoscopic conductor: Born-Oppenheimer force generally non-conservative see also: Todorov et al. 2008, Lü et al Work performed on mechanical modes per cycle Linear response: 19

20 Relation to quantum pumping quantum pumping of charge Q p through mesoscopic conductor Brouwer 1998 non-conservative Born-Oppenheimer force quantum pumping of charge 20

21 Adiabatic quantum motor one motor revolution pumped charge Q p electrical energy gain of motor per cycle Q p V Motor pumps charge with voltage drop and converts the electrical energy gain into motor action 21

22 Efficiency of quantum motor Output power: efficiency Input power: 22

23 Interesting consequences Thouless pumps underlie ideal quantum motors: quantized pumped charge at zero conductance η=1 Quantum motors can be fully quantum mechanical quantum pump based on chaotic quantum dot operates entirely through quantum interference 23

24 Motor dynamics Efficiency: motor dynamics! Simple case: driving force and load independent of state of motor steady state ideal quantum motor (G=0): maximum load: 24

25 Thouless motor 2π/a Linearized model near k= π/a: motor degree of freedom 25

26 Transfer matrix transfer matrix from x=l/2 to x=-l/2 by analogy with time-evolution operator in quantum mechanics: simplify: 26

27 S-matrix Relation of S- and transfer matrix: S-matrix: with and 27

28 Conductance & pumped charge Conductance (Landauer-Buttiker) imaginary in gap E F < real outside of gap E F > Pumped charge (Brouwer) 28

29 Efficiency Efficiency of Thouless motor: exponentially close to 1 inside gap (ideal motor) power-law decay & Fabry-Perot oscillations outside of gap 29

30 Beyond Born-Oppenheimer next order in quantum system acquires Berry s phase slow system subject to velocity-dependent force 30

31 Emergent Lorentz force emergent Lorentz force time-reversal symmetric conductor: Moskalets & Büttiker (2004) emergent Lorentz force vanishes in equilibrium is nonzero out of equilibrium (current breaks time reversal) 31

32 Friction force Berry: discrete quantum system no friction here: quantum mechanical scattering system w/ continuous spectrum friction appears naturally strictly positive eigenvalues Gilbert damping Brataas, Tserkovnyak, Bauer 2008 vanishes in equilibrium eigenvalues of arbitrary sign can make overall damping negative see also, e.g., Blanter et al Hussein et al

33 Dissipation of Thouless motor Intrinsic damping: Motor action requires minimal voltage at fixed current! 33

34 Ideal quantum motor again Thouless motor with load: Input power split into power consumed by the load: power dissipated by damping: load efficiency approaches unity for small currents since 34

35 Langevin force Fluctuation-dissipation theorem: Langevin force adiabatic approximation: force δ-correlated in time full Langevin dynamics of slow classical mode: 35

36 Conclusions Quantum motors can be realized based on inverting quantum pumps. Output power of motors can be related to characteristics of the underlying quantum pump. Motor based on chaotic quantum dot operates entirely based on quantum interference. Thouless motor has ideal efficiency η=1. Some open questions: beyond adiabatic limit, interacting pumps, quantum-mechanical motor degree of freedom, 36