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Dynamics of a Resonator Coupled to a Superconducting Single-Electron Transistor

Dynamics of a Resonator Coupled to a Superconducting Single-Electron Transistor. Andrew Armour University of Nottingham. Outline. Introduction Superconducting SET (SSET) SSET + resonator SSET as an effective thermal bath Fokker-Planck equation

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Dynamics of a Resonator Coupled to a Superconducting Single-Electron Transistor

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

  4. JQP resonance I0 QP CP QP E • Drain source/Gate voltages tuned to: • Bring Cooper pair transfer across one jn resonant • Allow quasiparticle decays across other jn • Current flows via coherent Cooper pair tunnelling+ • Incoherent quasiparticle tunnelling

  5. 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 • LaHaye et al, Science 304, 74 Naik et al., Nature 443, 193

  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 EJ links states |0> and |2>

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

  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 E Positive damping Negative damping

  11. Effective SET temperature • Temperature changes sign at resonance • Can obtain simple analytic expression: ‘Negative Temperature’ Positive Temperature E Quasiparticle tunnelling rate Detuning from centre of JQP resonance • Minimum in TSET set by quasiparticle decay rate • cf: Doppler cooling

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

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

  14. Dynamic Instability • 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 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. Fixed point “Bistable” Limit-Cycle Fixed point + E 0 Steady-state Wigner functions • 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. F=(<n2>-<n>2)/<n>2 • Slow resonator limit: /<<1 • Non-equilibrium/Kinetic phase transitions: • Order-parameter: nmp • Fixed point -> Limit cycle: Continuous • Bistability: Discontinuous

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

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

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

  20. Semi-classical dynamics • Equations of motion for 1st moments of system • Semi-classical approx.: <x02> <x><02> • 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-EJ limit

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

  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

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