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.