Transcript Dynamics of a nanomechanical resonator coupled to a single

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
– Experimental results (mechanical resonator)
• Unstable regime
– Numerical solution
– Quantum optical analogy: micromaser
– Semi-classical description
Superconducting SET
+Vg
Superconducting island coupled
by tunnel junctions to
superconducting leads
Quasiparticle
tunnelling
Josephson
Quasiparticle
Resonance [JQP]
Drain
Source
Voltage
Hadley et al., PRB 58 15317
Gate Voltage
Double
Josephson
Quasiparticle
Resonance [DJQP]
JQP resonance
I0
QP
CP
QP
E
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
Nanomechanical resonator & SSET
LaHaye et al, Science 304, 74
Naik et al., Nature 443, 193
• 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>
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
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
E
Positive damping
Negative damping
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
Experimental Results
Naik, Buu, LaHaye, and Schwab (Cornell)
SSET gate
1.2
x 10
4
JQP bias point
1.1
-9
A
3.5
1
3
0.9
2.5
Vds (mV)
0.8
0.7
2
0.6
1.5
0.5
1
0.4
0.5
0.3
SSET island
Nanomechanical
Beam
0.2
-5
-4
-3
-2
-1
0
Vg (mV)
1
2
3
4
5
0
Infer resonator properties from SSET charge noise power around
mechanical frequency: known to provide good thermometry for
resonator [LaHaye et al.,Science 304 74]
Back-action: Cooling & Heating
Cooling
Coupling:
 bathT   SET T
TNR 
 bath   SET
bath
SET
• Theory: TSET~220mK
• But damping does not match theory so well
Naik et al., Nature 443 193
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
Clerk and Bennett New J. Phys. 7 238; PRB 74 201301
• Can evaluate steady-state of the system by numerical
evaluation of the master equation eigenvector with
zero eigenvalue
Rodrigues, Imbers and AA PRL 98 067204
• Instabilities turn out to be result of largely classical
resonances: semi-classical description also useful
Rodrigues, Imbers, Harvey and AA cond-mat/0703150
Steady-state Wigner functions
Fixed point
“Bistable”
Limit-Cycle
E
Fixed point
0
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.
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
Resonator moments II.
-2
-1
0
+1
<n>
E
F
E
F<1 region
• As  increases, resonance lines emerge: E=nh
• Most interesting behaviour for /~1:
~Mutual interaction strongest
~Non-classical states emerge even at low coupling
Analogue: Micromaser
Filipowicz et al PRA 43 3077; Wellens et al Chem. Phys. 268 131
n/nmax
Nex
Stream of two-level atoms
pass through a cavity
resonator:
• can identify nonequilibrium phase transitions
• resonator state can be
number-squeezed (F<1)
Pump parameter= (Nex)1/2x coupling strength x interaction time
Nex=no. atoms passing through cavity during field lifetime
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
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
Origin of instabilities
• 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
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