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Dynamics of a Resonator Coupled to a Superconducting
Single-Electron Transistor
Andrew ArmourUniversity of Nottingham
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
Superconducting SET
Gate Voltage
Quasiparticle tunnelling
JosephsonQuasiparticleResonance [JQP]
Double JosephsonQuasiparticleResonance [DJQP]
Hadley et al., PRB 58 15317Drain SourceVoltage
Superconducting island coupledby tunnel junctions tosuperconducting leads
+Vg
JQP resonance
Drain source/Gate voltages tuned to:1. Bring Cooper pair transfer across one jn resonant2. Allow quasiparticle decays across other jn
Current flows via coherent Cooper pair tunnelling+Incoherent quasiparticle tunnelling
QP
QP
CP I0
E
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
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
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>
Effect of resonator’s thermalized surroundings:Characterized through a damping rate, ext and an average number of resonator quanta nBath
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:
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
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
Effective SET temperature
Quasiparticle tunnelling rate Detuning from centre of JQP resonance
‘NegativeTemperature’
PositiveTemperature
• Temperature changes sign at resonance
• Can obtain simple analytic expression:
• Minimum in TSET set by quasiparticle decay rate• cf: Doppler cooling
E
Experimental Results
Naik, Buu, LaHaye, and Schwab (Cornell)
NanomechanicalBeam
-5 -4 -3 -2 -1 0 1 2 3 4 50.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
Vg (mV)
Vds
(m
V)
0
0.5
1
1.5
2
2.5
3
3.5
4x 10
-9 A
JQP bias pointSSET 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
Back-action: Cooling & Heating
SETbath
SETbathNR
SETbathTT
T
Cooling
Coupling:
Naik et al., Nature 443 193
• Theory: TSET~220mK• But damping does not match theory so well
• 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
Steady-state Wigner functions
“Bistable” Limit-Cycle Fixed pointFixed point
+0E
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
Resonator moments I.
• Slow resonator limit: /<<1• Non-equilibrium/Kinetic phase transitions:
Order-parameter: nmp
Fixed point -> Limit cycle: ContinuousBistability: Discontinuous
F=(<n2>-<n>2)/<n>2
Resonator moments II.
E E
<n> 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
Analogue: Micromaser
Pump parameter= (Nex)1/2x coupling strength x interaction timeNex=no. atoms passing through cavity during field lifetime
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)
Filipowicz et al PRA 43 3077; Wellens et al Chem. Phys. 268 131
Nex
SSET-resonator system
• 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
/=1; nBath=0
Semi-classical dynamics• Equations of motion for 1st moments of system
– Semi-classical approx.: <x02> <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
• Limit cycles satisfy condition:
• Maxima in SSET due to commensurability of electrical & mechanical oscillations
• Electrical oscillations: frequency 1/2A• Increasing compresses SSET oscillations leads to bifurcations
Origin of instabilities
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
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