1. Quantum Coherent Nanoelectromechanics Robert Shekhter Leonid Gorelik and Mats Jonson University of Gothenburg / Heriot-Watt University / Chalmers Univ. of Technology In collaboration with: Mechanically assisted superconductivity NEM-induced electronic Aharonov-Bohm effect Supercurrent-driven nanomechanics

2. H. Park et al., Nature 407, 57 (2000) Quantum ”bell”Single-C 60 transistor A. Erbe et al., PRL 87, 96106 (2001); Nanoelectromechanical Devices V. Sazonova et al., Nature 431, 284 (2004) B. J. LeRoy et al., Nature 432, 371 (2004) CNT-based nanoelectromechanical devices

3. Nanomechanical Shuttling of Electrons bias voltage dissipation current Gorelik et al, Phys Rev Lett 1998 Shekhter et al., J Comp Th Nanosc 2007 (review) H.S.Kim, H.Qin, R.Blick, arXiv:0708.1646 (experiment)

4. How does mechanics contribute to tunneling of Cooper pairs? Is it possible to maintain a mechanically-assisted supercurrent? Gorelik et al. Nature 2001; Isacsson et al. PRL 89, 277002 (2002)

5. To preserve phase coherence only few degrees of freedom must be involved. This can be achieved provided: No quasiparticles are produced Large fluctuations of the charge are suppressed by the Coulomb blockade:

6. Single-Cooper-Pair Box Coherent superposition of two “nearby” charge states [2n and 2(n+1)] can be created by choosing a proper gate voltage which lifts the Coulomb Blockade, Nakamura et al., Nature 1999

7. Movable Single-Cooper-Pair Box Josephson hybridization is produced at the trajectory turning points since near these points the Coulomb blockade is lifted by the gates.

8. Shuttling of Superconducting Cooper Pairs

9. Possible setup configurations A supercurrent flows between two leads kept at a fixed phase difference Coherence between isolated remote leads created by “shuttling” of Cooper pairs

10. I: Shuttling between coupled superconductors Relaxation suppresses the memory of initial conditions.

11. How does it work?

12. Resulting Expression for the Current

13. Black regions – no current. The current direction is indicated by signs

14. Mechanically Assisted Superconductive Coupling

15. Distribution of phase differences as a function of number of rotations. Suppression of quantum fluctuations of phase difference

16. Electronic Transport through Vibrating CNT Shekhter R.I. et al. PRL 97(15): Art.No.156801 (2006).

17. Quantum Nanomechanical Interferometer Classical interferometer (two “classical” holes in a screen) Quantum nanomechanical Interferometer (“quantum” holes determined by a wavefunction) Interference determines the intensity (Analogy applies for the elastic transport channel; need to add effects of inelastic scattering)

18. Model

19. Renormalization of Electronic Tunneling

20. Coupling to the Fundamental Bending Mode Only one vibration mode is taken into account CNT is considered as a complex scatterer for electrons tunneling from one metallic lead to the other

21. Theoretical Model Strong longitudinal quantization of electrons on the CNT Perturbative approach to resonant tunneling though the quantized levels (only virtual localization of electrons on the CNT is possible) Effective Hamiltonian Amplitude of quantum oscillations [about 0.01 nm] Magnetic-flux dependent tunneling

22. Linear Conductance (The vibrational subsystem is assumed to be in equilibrium) For L=1  m,  = 10 8 Hz, T = 30 mK and H = 20-40 T we estimate  G/G 0 = 1-3% The most striking feature is the temperature dependence. It comes from the dynamics of the entire nanotube, not from the electron dynamics R.I. Shekhter et al., PRL 97 (2006)

23. Backscattering of Electrons due to the Presence of Fullerene. The probability of backscattering sums up all backscattering channels. The result yields classical formula for non-movable target. However the sum rule does not apply as Pauli principle puts restrictions on allowed transitions.

24. × The applied bias voltage selects the allowed inelastic transitions through vibrating nanowire as fermionic nature of electrons has to be considered. Pauli Restrictions on Allowed Transitions Through Vibrating Nanowire

25. Magnetic Field Dependent Offset Current

26. Different Types of NEM Coupling Capacitive coupling Tunneling coupling Shuttle coupling Inductive coupling C(x)C(x) R(x)R(x) C(x)C(x)R(x)R(x) Lorentz force for given j Electromotive force at I = 0 for given v j FLFL E v H.

27. Electronically Assisted Nanomechanics From the ”shuttle instability” we know that electronic and mechanical degrees of freedom couple strongly at the nanometre scale. So we may ask.... Can a coherent flow of electrons drive nanomechanics? Does a Superconducting Nanoelectromechanical Single-Electron Transistor (NEM-SSET) have a shuttle instability? - This is an open question Electronic Aharonov-Bohm effect induced by quantum vibrations: Can resonantly tunneling electrons in a B-field drive nanomechanics? - This is an open question Can a supercurrent drive nanomechanics? - Yes! Topic for the rest of this talk

28. Supercurrent-Driven Nanomechanics Model: Driven, damped nonlinear oscillator G. Sonne et al. arXiv:0806.4680 Driving Lorentz force Induced el.motive force Energy balance in stationary regime determines time-averaged dc supercurrent Compare: NEM resonator as part of a SQUID Buks, Blencowe PRB 2006 Zhou, Mizel PRL 2006 Blencowe, Buks PRB 2007 Buks et al. EPL 2008

29. Giant Magnetoresistance V Alternating Josephson current Mechanical resonances Alternating Lorentz force, F L Force (I) leads to resonance at Force (II) leads to parametric resonance at (I) (II) Accumulation and dissipation of a finite amount of energy during one each nanowire oscillation period means that and Therefore a nonzero average (dc) supercurrent on resonance For small amplitudes (u):

30. Giant Magnetoresistance V Alternating Josephson current Mechanical resonances Alternating Lorentz force, F L Force (I) leads to resonance at Force (II) leads to parametric resonance at (I) (II) Accumulation and dissipation of a finite amount of energy during each nanowire oscillation period means that and therefore a nonzero average (dc) supercurrent on resonance

31. Giant Magnetoresistance The onset of the parametric resonance depends on magnetic field H. By increasing H the resistance jumps from to a finite value. dc bias voltage Amplitude of wire oscillations Parametric resonance Resonance ”small” H ”larger” H

32. Superconductive Pumping of Nanovibrations Mathematical formulation Introduce dimensionless variables: Equation of motion for the nanowire: (Forced, damped, nonlinear oscillator) Realistic numbers for a SWNT wire makes both parameters small:

33. Superconductive Pumping of Nanovibrations Mathematical formulation Introduce dimensionless variables: Equation of motion for the nanowire: (Forced, damped, nonlinear oscillator) Realistic numbers for a SWNT wire makes both parameters small:

34. Superconductive Pumping of Nanovibrations Resonance approximation Assuming: the equation of motion: by the Ansatz: Inserting the Ansatz in the equation of motion and integrating over the fast oscillations one gets for the slowly varying variables: Next: n=2, drop indices

35. Superconductive Pumping of Nanovibrations Resonance approximation Assuming: the equation of motion: by the Ansatz: Inserting the Ansatz in the equation of motion and integrating over the fast oscillations one gets for the slowly varying variables: Next: n=2, drop indices

36. PumpingDumping Multistability of the S-NEM Weak Link Dynamics

37. Onset of the dc Supercurrent on Resonance

38. Dynamical Bistability

39. Current-Voltage Characteristics If  ~1 GHz: V 0 ~ 5  V, 2  c ~ 50 nV If j dc ~ 100 nA  I 1,2 ~ 5 nA

40. Phase coherence between remote superconductors can be supported by shuttling of Cooper pairs. Quantum nanovibrations cause Aharonov-Bohm interference determining finite magneto-resistance of suspended 1-D wire. Resonant pumping of nanovibrations modifies the dynamics of a NEM superconducting weak link and leads to a giant magnetoresistance effect (finite dc supercurrent at a dc driving voltage). Multistable nanovibration dynamics allow for a hysteretic I-V curve, sensitivity to initial conditions, and switching between different stable vibration regimes. NEM-Assisted Quantum Coherence - Conclusions