1. Dynamics of a Resonator Coupled to a Superconducting Single-Electron Transistor Andrew Armour University of Nottingham

2. Outline Introduction –Superconducting SET (SSET) –SSET + resonator SSET as an effective thermal bath –Fokker-Planck equation –Experimental results (mechanical resonator) Unstable regime –Numerical solution –Quantum optical analogy: micromaser –Semi-classical description

3. Superconducting SET Gate Voltage Quasiparticle tunnelling Josephson Quasiparticle Resonance [JQP] Double Josephson Quasiparticle Resonance [DJQP] Hadley et al., PRB 58 15317 Drain Source Voltage Superconducting island coupled by tunnel junctions to superconducting leads +V g

4. JQP resonance Drain source/Gate voltages tuned to: 1. Bring Cooper pair transfer across one jn resonant 2. Allow quasiparticle decays across other jn Current flows via coherent Cooper pair tunnelling+ Incoherent quasiparticle tunnelling QP CP I0I0 EE

5. LaHaye et al, Science 304, 74 Naik et al., Nature 443, 193 Nanomechanical resonator & SSET Motion of resonator affects SSET current SET suggested as ultra-sensitive displacement detector –White Jap. J. Appl. Phys. Pt2 32, L1571 –Blencowe and Wybourne APL 77, 3845 Devices fabricated so far have frequencies ~20MHz fluctuations in island charge acts back on resonator: alters dynamics

6. Superconducting resonator Can also fabricate superconducting strip-line resonators: Coupling to a Cooper-pair box achieved Resonators can be very high frequency >GHz A. Wallraff et al. Nature 431 162

7. SSET-Resonator System Three charge states involved in JQP cycle: |0>, |1> and |2> Resonator, frequency , couples to charge on SET island with strength  Charge states |0> and |2> differ in energy by E (zero at centre of resonance) Coherent Josephson tunnelling parameterised by E J links states |0> and |2>

8. Effect of resonator’s thermalized surroundings: Characterized through a damping rate,  ext and an average number of resonator quanta n Bath Quantum master equation Quasi-particle tunnelling from island to leads: 2 processes occur, |2>|1> and |1>|0> but we assume the rate is the same,  Include dissipation:

9. Effective description of resonator Can obtain effective description of resonator dynamics by taking Wigner transform of the master equation and tracing out electrical degrees of freedom Obtain a Fokker-Planck equation: Assumes resonator does not strongly affect SSET: requires weak-coupling and small resonator motion For now, will also assume the resonator is slow: << Blencowe, Imbers and AA, New J. Phys. 7 236 Clerk and Bennett New J. Phys. 7 238

10. Resonator Damping Effective damping due to SET: Negative damping tells us that resonator motion will not be captured by Fokker-Planck equation for long times Negative damping Positive damping EE

11. Effective SET temperature Quasiparticle tunnelling rate Detuning from centre of JQP resonance ‘Negative Temperature’ Positive Temperature Temperature changes sign at resonance Can obtain simple analytic expression: Minimum in T SET set by quasiparticle decay rate cf: Doppler cooling EE

12. Experimental Results Naik, Buu, LaHaye, and Schwab (Cornell) Nanomechanical Beam JQP bias point SSET gate Infer resonator properties from SSET charge noise power around mechanical frequency: known to provide good thermometry for resonator [ LaHaye et al.,Science 304 74 ] SSET island

13. Back-action: Cooling & Heating Cooling Coupling: Naik et al., Nature 443 193 Theory: T SET ~220mK But damping does not match theory so well

14. What happens to the resonator steady-state in the ‘unstable’ regime:  Bath +  SET <0 For ‘slow’ resonator can also include feedback effects in Fokker-Planck equation Can evaluate steady-state of the system by numerical evaluation of the master equation eigenvector with zero eigenvalue Instabilities turn out to be result of largely classical resonances: semi-classical description also useful Dynamic Instability Clerk and Bennett New J. Phys. 7 238; PRB 74 201301 Rodrigues, Imbers and AA PRL 98 067204 Rodrigues, Imbers, Harvey and AA cond-mat/0703150

15. Steady-state Wigner functions “Bistable”Limit-CycleFixed point + 0 EE Resonator pumped by energy transferred from Cooper pairs: E>0: CP can take energy from resonator E<0: CP can give energy to resonator Far from resonance: little current, so little pumping and external damping stabilizes resonator

16. Resonator moments I. Slow resonator limit: /<<1 Non-equilibrium/Kinetic phase transitions: Order-parameter: n mp Fixed point -> Limit cycle: Continuous Bistability: Discontinuous F=( - 2 )/ 2

17. Resonator moments II. EEEE F As  increases, resonance lines emerge: E=nh Most interesting behaviour for /~1: ~Mutual interaction strongest ~Non-classical states emerge even at low coupling -2 -1 0 +1 F<1 region

18. Analogue: Micromaser Pump parameter= (N ex ) 1/2 x coupling strength x interaction time N ex =no. atoms passing through cavity during field lifetime n/n max Stream of two-level atoms pass through a cavity resonator: can identify non- equilibrium phase transitions resonator state can be number-squeezed (F<1) Filipowicz et al PRA 43 3077; Wellens et al Chem. Phys. 268 131 N ex

19. SSET-resonator system Only 1 st transition is sharp: sharpness of transitions depends on current which decreases with  Traces of further transitions seen in n mp Well-defined region where F<1 /=1; n Bath =0

20. Semi-classical dynamics Equations of motion for 1 st moments of system –Semi-classical approx.:  Weak , Bath  resonator amplitude changes slowly: –Periodic electronic motion calculated for fixed resonator amplitude –leads to amplitude-dependent effective damping: –Good match with full quantum numerics for weak-coupling –Analytical expression available in low-E J limit

21. Limit cycles satisfy condition: Maxima in  SSET due to commensurability of electrical & mechanical oscillations Electrical oscillations: frequency  1/2 A Increasing   compresses  SSET oscillations  leads to bifurcations Origin of instabilities

22. Conclusions Despite linear-coupling SSET-resonator system shows a rich non-linear dynamics Cooling behaviour seen on ‘red detuned’ side of resonance ‘Blue detuned’ region shows rich variety of behaviours similar to micromaser Semi-classical description works (surprisingly) well Investigate dynamics further through current noise, quantum trajectories

23. Acknowledgements Collaborators –Jara Imbers, Denzil Rodrigues Tom Harvey (Nottingham) –Miles Blencowe (Dartmouth) –Akshay Naik, Olivier Buu, Matt LaHaye, Keith Schwab (Cornell) –Aashish Clerk (McGill) Funding