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