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Driven Duffing Oscillator
|A|2
ω - ω0
Cf f>
|A|2
ω - ω0
Cf f<
•The response in the absence of noise
•Consider an oscillator with cubic nonlinearity driven by an harmonic force
•The classical response is calculated employing the slow envelope approximation
|A|2
ω - ω0
Cf f=
critical point
Nonlinear Resonators -From Nanomechanical to Superconducting Stripline
Eyal BuksDepartment of Electrical Engineering, Technion
100 µm
NbN
Collaborators:Technion Ronen Almog, Stav Zaitsev, Baleegh Abdo, Eran Segev, Oleg ShtempluckBell Labs Bernard Yurke
Intermodulation amplifiers•Gain•Dissipation (linear and nonlinear)•Noise squeezing
100 µm
Nanomechanical Resonators -Fabrication and Characterization
~dcV
Si
Si3N4
Si3N4
Au
spectrumanalyzer
ine-beam
secondary electrondetector
175000 176000 177000 1780000.00
0.02
0.04
0.06
0.08
Spec
trum
Ana
lyze
r Sig
nal [
a.u.
]
Frequency [Hz]
M.L. Roukes, 2000 Solid State Sensor and Actuator Workshop
Damping in Nanomechanical Resonators
•For intermodulation amplifiers Nonlinear Dampingmay contribute to the total noise.
•Surface properties are important in NEMS.
•Damping mechanisms: bulk and surface defects,thermoelastic damping, nonlinear coupling to other modes, phonon-electron coupling, clamping loss, etc.
•Nanomechanical systems suffer from low quality factor Q relative to their macroscopic counterparts.
Nonlinear Damping
|A|2
|A|2
|A|2
ω - ω0 ω - ω0 ω - ω0
Cf f< Cf f= Cf f>
critical point
•Bistable regime is accessible only when p<1, where
•A nonlinear damping term is added to the equation of motion
Bernard Yurke and EB, unpublishedStav Zaitsev and EB, cond-mat/0503130 (2005)
~e-beam
secondary electrondetector lockin
amp.in
dcV
Extracting the Parameter p - I
Stav Zaitsev and EB, cond-mat/0503130 (2005)
Extracting the Parameter p - II
Stav Zaitsev and EB, cond-mat/0503130 (2005)
Extracting the Parameter p - III
Stav Zaitsev and EB, cond-mat/0503130 (2005)
•Nonlinear damping plays an important role !
•What are the underlying mechanisms ?
Intermodulation Characterization - I
e-beam
secondary electrondetector
spectrumanalyzer
in
Pumppower
combiner
Signal
Signal
Pump
Idler
Frequency
Power spectrum
offset
•The frequencies of the pump, signal and idler are all within the bandwidth of the fundamental mode.
Intermodulation Characterization - II
Freq. sweep up
Freq. sweep down5.4 5.45 5.5 5.55 5.6 5.65 5.7 5.75 5.8
-100
-80
-60
-40
frequency [Hz]
Pum
p [d
Bm
]
x 105
5.4 5.45 5.5 5.55 5.6 5.65 5.7 5.75 5.8-90
-80
-70
-60
frequency [Hz]
Sign
al [d
Bm
]
5.4 5.45 5.5 5.55 5.6 5.65 5.7 5.75 5.8
x 105
-85
-80
-75
-70
frequency [Hz]
Idle
r [dB
m]
x 105
Signal
Pump
Idler
Intermodulation Characterization - III
•The signal gain and intermodulation gain are both limited by pump depletion !
-63dbm
-65dbm
-65dbm
Hysteresis and BistabilityAmplitude sweep up
Amplitude sweep down
Bistability
12
34
1
2
4
31
23
4
Superconducting Stripline Resonator
Nb / NbN
Nb / NbN
Sapphire
SapphireNb / NbN
E. Buks unpublished results
Nonlinear Response of Nb Resonator
2.63 2.635 2.64 2.645 2.65 2.655-24
-22
-20
-18
-16
-14
-12
-10
-8
-6
-4
-15 dBm
-10
-5
0
5
10
15
frequency [GHz]
S11
[dB
]
inPT=4.2K
Nb
inP inPS11
vector network analyzer
• Kerr Nonlinearity (kinetic inductance)
• Nonlinear dissipation:
• Meissner effect leads to non-uniform current profile with very high current density near the edges of the stripline.
1 2 3 4 5 6 7 8 9 10-60
-50
-40
-30
-20
-10
0
10
-10 dbm-9 dbm-8 dbm-7 dbm-6 dbm-5 dbm-4 dbm-3 dbm-2 dbm-1 dbm0 dbm1 dbm2 dbm3 dbm4 dbm5 dbm6 dbm7 dbm8 dbm9 dbm10 dbm
Frequency [GHz]
|S11
| [dB
]
+5.6304
+2.5812
+8.4188NetworkAnalyzer
Lets look closer at this resonance!
NbN Resonator - I
8.282 8.284 8.286 8.288 8.29 8.292 8.294 8.296-180
-160
-140
-120
-100
-80
-60
-40
-20
0
20
-23.5 dbm-23.25 dbm-23 dbm-22.75 dbm-22.5 dbm-22.25 dbm-22 dbm-21.75 dbm-21.5 dbm-21.25 dbm-21 dbm-20.75 dbm-20.5 dbm-20.25 dbm-20 dbm-19.75 dbm-19.5 dbm-19.25 dbm-19 dbm-18.75 dbm-18.5 dbm-18.25 dbm-18 dbm
Frequency [Ghz]
|S11
| [db
]
NbN Resonator - II
•Onset of bistability - 3 orders of magnitude lower than Nb !
•Critical coupling
1 2 3 4 5 6 7 8 9 10-120
-100
-80
-60
-40
-20
0
20
-10 dbm-9 dbm-8 dbm-7 dbm-6 dbm-5 dbm -4 dbm-3 dbm-2 dbm-1 dbm0 dbm1 dbm2 dbm3 dbm4 dbm 5 dbm 6 dbm7 dbm8 dbm9 dbm
Frequency [GHz]
|S11
| [db
]
+2.5152 +4.1963 +4.425+6.3806
+8.176 10dbm NetworkAnalyzer
Lets look closer at this resonance!
NbN Resonator - III
4.37 4.375 4.38 4.385 4.39 4.395 4.4 4.405 4.41-70
-60
-50
-40
-30
-20
-10
0
10
-9.5 dbm-9 dbm-8.5 dbm-8 dbm-7.5 dbm-7 dbm-6.5 dbm-6 dbm-5 dbm-4.5 dbm-4 dbm-3.5 dbm-3 dbm-2.5 dbm-2 dbm-1.5 dbm
Frequency [GHz]
|S11
| [dB
]
-5.5 dbm
Critical coupling –Baleegh Abdo, Eran Segev, Oleg Shtempluck, EB, cond-mat/0501236 (2005)
critical coupling
NbN Resonator - IV
4.36 4.37 4.38 4.39 4.4 4.41 4.42 4.43 4.44-160
-140
-120
-100
-80
-60
-40
-20
0
20
-8.05 dbm
Frequency [GHz]
|S11
| db
-8.04 dbm-8.03 dbm-8.02 dbm-8.01 dbm-8 dbm-7.99 dbm-7.98 dbm-7.97 dbm-7.96 dbm-7.95 dbm-7.94 dbm-7.93 dbm-7.92 dbm-7.91 dbm-7.9 dbm-7.89 dbm-7.88 dbm-7.87 dbm-7.86 dbm-7.85 dbm-7.84 dbm-7.83 dbm-7.82 dbm-7.81 dbm-7.8 dbm
CW frequency scan in both directions
CW ForwardCW Backward
Clockwise hysteresis loop
No hysteresis loop
Counter clockwisehysteresis loop
Hysteresis Loop Changes Direction
Multiple Jumps
1 .4 1 .4 5 1 .5 1 .5 5 1 .6 1 .6 5 1 .7 1 .7 5 1 .8-4 0
-3 5
-3 0
-2 5
-2 0
-1 5
-1 0
-5
0
1 .4 9 d b m
F re q u e n c y [G h z ]
|S11
| [db
]
C W s c a n fo rw a rdC W s c a n b a c k w a rd
Baleegh Abdo, Eran Segev, Oleg Shtempluck, EB, cond-mat/0501114 (2005)
Contrary to the case of Nb
1. A simple model of a one-dimensional Duffing resonator cannot account for
the experimental results of NbN resonators.
2. Besides kinetic inductance, another mechanism contributes to nonlinearity.
NbN vs. Nb
What is the underlying mechanism ?
Is it Heating ?
10*lo
g|V ou
t|Frequency [Ghz]
4.38 4.385 4.39 4.395 4.4 4.405 4.41 4.415-34
-32
-30
-28
-26
-24
-22
-20
-7.5 dbm
Dynamic state 1
Dynamic state 2
signal generator
FM
circulator
signal generator
powerdiode
LoadScope
1 2sync
RF output
Dewar 4.2K
stripline resonator
2 sµ
•Global heating effect? - unlikely …
Baleegh Abdo, Eran Segev, Oleg Shtempluck, EB, cond-mat/0504582 (2005)
•However, local heating of hot spots in the NbN film (out of equilibrium) are not rolled out
/ 5 nsCdτ α= ≅
4.32 4.33 4.34 4.35 4.36 4.37 4.38 4.39 4.4-200
-150
-100
-50
0
50
5.299k5.35k5.4k5.449k5.499k5.55k5.599k5.65k5.699k5.749k5.8k5.849k5.899k
Frequency [Ghz]
|S11
| [db
]
Constant P in=-10dbm
Temp.[k]
Strong Dependence on Temperature
Baleegh Abdo, Eran Segev, Oleg Shtempluck, EB, cond-mat/0504582 (2005)
4 .3 7 4 .3 7 5 4 .3 8 4 .3 8 5 4 .3 9 4 .3 9 5 4 .4 4 .4 0 5 4 .4 1-8 0
-7 0
-6 0
-5 0
-4 0
-3 0
-2 0
-1 0
0
1 0
0 m T0 .9 0 9 m T1 .8 2 m T2 .7 3 m T3 .6 4 m T4 .5 5 m T5 .4 5 m T6 .3 6 m T7 .2 7 m T8 .1 8 m T9 .0 9 m T1 0 m T1 0 .9 m T1 1 .8 m T1 2 .7 m T1 3 .6 m T
F r e q u e n c y [G h z ]
|S11
| [db
]
C o n s ta n t p o w e r o f -5 d b m
Strong Dependence on Magnetic Field
Jump disappears11.8 mT
Weak Link Hypothesis
•Microscopic Josephson junctions forming at the grain boundaries of the columnar structure of the NbN film are suspected to be responsible for the observed behavior.
bI
R cI( )tIN CV
bI
R cI( )tIN CV
θ
U [I
cφ0/ π
]
-8 -6 -4 -2 0 2 4 6 8-1
-0.5
0
0.5
1
1.5
2
2.5Potential EnergyQuadratic approximation
b
C MassR frictionI driving force
∼∼∼
cross section NbN film
RCSJ with AC bias current driven Duffing oscillator
IMD Measurement Setup“Pump”
“Signal”
Isolator
Isolator
Powercombiner Circulator
Spectrum analyzer
Dewar 4.2K
Superconducting resonator
signalpumpidler
Offset
Pump power [dBm] Pump power [dBm]
Pump power [dBm]
Freq
uenc
y [G
Hz]
Freq
uenc
y [G
Hz]
Freq
uenc
y [G
Hz]
[dB] [dB]
[dBm]
A A” A A”
A A”
IMD Gain - I
idler gain signal gain
reflected pump
-26 -25 -24 -23 -22 -21 -20 -19 -18-30
-20
-10
0
10
20
Inte
rmod
ulat
ion
gain
[dB
]
Pump power [dBm]
A-A" : Frequency 2.5879 GHz
-26 -24 -22 -20 -18-55
-50
-45
-40
-35
-30
Ref
lect
ed p
ump
pow
er [d
Bm
]
-26 -25 -24 -23 -22 -21 -20 -19 -18-10
0
10
20
Sign
al g
ain
[dB
]
Pump power [dBm]
A-A" : Frequency 2.5879 GHz
-26 -25 -24 -23 -22 -21 -20 -19 -18-60
-50
-40
-30
Ref
lect
ed p
ump
pow
er [d
Bm
]
14.99 dB13.91 dB
IMD Gain - IIidler gain signal gain
Pump power [dBm]Pump power [dBm]
Freq
uenc
y [G
Hz]
Freq
uenc
y [G
Hz]R
efle
cted
pum
p po
wer
[dB
m]
Ref
lect
ed p
ump
pow
er [d
Bm
]
Decreasing pump Increasing pump
Hysteresis
Bistability of a Duffing Resonator
|A|2
ω - ω0
Cf f>
•In the bistable regime the phase space contains two basins of attaraction.
•Can noise induce transitions ?
Adding White Noise …
Miteqamplifier
Uniphaseamplifier
50 ohm
Powercombiner
Circulator
Networkanalyzer
Dewar 4.2K
Superconducting resonator
Baleegh Abdo, Eran Segev, Oleg Shtempluck, EB, cond-mat/0504582 (2005)
Noise Induced Transitions
2.57 2.58 2.59 2.6 2.61-140
-120
-100
-80
-60
-40
-20
0-23.9 dbm
Frequency [Ghz]
|S21
| [db
]
-23.6 dbm
-23.3 dbm
-23 dbm
-22.7 dbm
-22.4 dbm
-22.1 dbm
-21.8 dbm
-21.5 dbm
-21.2 dbm
-20.9 dbm
-20.6 dbm
-20.3 dbm
-20 dbm
With -80 dbm white noise
cw forwardcw backward
Baleegh Abdo, Eran Segev, Oleg Shtempluck, EB, cond-mat/0504582 (2005)
2.57 2.58 2.59 2.6 2.61-140
-120
-100
-80
-60
-40
-20
0-23.9 dbm
Frequency [Ghz]
|S21
| [db
]
-23.6 dbm
-23.3 dbm
-23 dbm
-22.7 dbm
-22.4 dbm
-22.1 dbm
-21.8 dbm
-21.5 dbm
-21.2 dbm
-20.9 dbm
-20.6 dbm
-20.3 dbm
-20 dbm
With -58 dbm white noise 1410 KeffT =
frequent transitions
hysteresiseliminated
Nonlinear resonator
γ1
γ2
γ3
Test port
Linear dissipation port
1a2a
Nonlinear dissipation port
3a
H0 = ~ ω 0A†A + (~/2) KA†A†AA
Nonlinear Resonator Model
linear dissipation
Kerr Nonlinearity
nonlinear dissipation
B. Yurke and EB, unpublished
• Heisenberg equation of motion:
Classical Response in the Absence of Noise
-0.1 -0.05 0 0.05 0.10
0.05
0.1
|A|
-0.1 -0.05 0 0.05 0.10
0.5
1
-0.1 -0.05 0 0.05 0.10
0.05
0.1
|A|
-0.1 -0.05 0 0.05 0.10
0.5
1
-0.1 -0.05 0 0.05 0.10
0.05
0.1
|A|
∆ ω /ω 0-0.1 -0.05 0 0.05 0.10
0.5
1
∆ ω /ω 0
cavity mode amplitude reflection
|aou
t /ain |
11
|aou
t /ain |
11
|aout /a
in|
11
criticalpoint ∞=
∂∂
ωA
B. Yurke and EB, unpublished
Nonlinear resonator
γ1
γ2
γ3
Test port
Linear dissipation port
1a2a
Nonlinear dissipation port
3a
H0 = ~ ω 0A†A + (~/2) KA†A†AA
The Correlation Function - I
•Define
where
•The correlation function
•The low frequency limit of the power spectrum of
-0.1 0 0.10
10
20
30
40
b1in=2b1c
in
∆ ω / ω0
B
(e)
-0.1 0 0.1-5
0
5
10
15
20
∆ ω / ω0
(f)
-0.1 0 0.10
10
20
30
40
b1in=b1c
in
∆ ω / ω0
B
(c)
-0.1 0 0.1-5
0
5
10
15
20
∆ ω / ω0
(d)
-0.1 0 0.10
10
20
30
40
b1in=0.5b1c
in
∆ ω / ω0
B
(a)
-0.1 0 0.1-5
0
5
10
15
20
∆ ω / ω0
(b)
Log[
S 0(0)]
Log[
S 0(0)]
Log[
S 0(0)]
The Correlation Function - II
criticalpoint
S0(0) diverges at the bifurcation points !
Intermodulation Gain
SA
-0.1 0 0.10
10
20
30
40
-0.1 0 0.1-505
101520
-0.1 0 0.10
10
20
30
40
-0.1 0 0.1-505
101520
-0.1 0 0.10
10
20
30
40
-0.1 0 0.1-505
101520
•In the limit where the offset frequency ω → 0 the intermodulation gain close to the critical point is
•Thus GI diverges at the critical point.
•However, the assumption that the idler amplitude is small is violated, and the model thus breaks down close to the critical point.
criticalpoint
B. Yurke and EB, unpublished
Noise Squeezing
•Assume for simplicity
•The noise properties can be characterized by homodyning with coherent radiation.
•Thus Pmin/Pmax → 0 at the bifurcation point.
•In the limit where the offset frequency ω → 0 close to a bifurcation point
t
t
t
coherent state
squeezed state with reduced phase uncertainty
squeezed state with reduced amplitude uncertainty
B. Yurke and EB, unpublished
t
signal power –2.27 dBm
4.122 4.123 4.124 4.125 4.126 4.127-45
-40
-35
-30
-25
-20
-15
Freq [GHz]
S11
[dB]
pump off
2.5 3 3.5 4 4.5-6
-5
-4
-3
-2
-1
0
1
2
3
Freq. [GHz]
S11
[dB
]
Inter-mode Coupling - I
powercombiner
circulator
networkanalyzer
Dewar 4.2K
Superconducting resonator
signal
Lets look closer at the signal mode
signal mode
4.1245 GHz
pump mode
2.7528 GHz
pump
pump on
pump power 0 dBm
Inter-mode Coupling - II
bridge QD QPC
•In the rotating wave approximation the nonlinear coupling between the modes is given by
•Since V commutes with the Hamiltonian of the system, such a measurement is a quantum non-demolition one [Sanders and Milburn, PRA 39, 694 (’89)].
•The pump mode can be considered as a detector measuring the number of photons in the signal mode Ns.
•The coupling also leads to dephasing induced on the signal mode,with a rate
•The dephasing rate diverges at bifurcation points.
SummaryIntermodulation amplification is demonstrated for both nano-mechanical
resonators and superconducting stripline resonators.
Nonlinear damping in nanomechanical resonators plays an important role.
High gain is observed near the bifurcation points of both nanomechanical and superconducting stripline resonators.
Microscopic Josephson junctions forming at the grain boundaries of the columnar structure of the NbN films are suspected to be responsible for the nonlinear response.
The noise at the output of the intermodulation amplifiers is strongly squeezed.
Inter-mode coupling may induce strong dephasing when driving the pump mode close to a bifurcation.
Injected noise induces transitions between basins of attraction.