Metastability and self-oscillations in superconducting microwave

Eran SegevQuantum Engineering Laboratory, Technion, Israel

resonators integrated with a dc-SQUID

Quantum Measurements of Solid-State Devices

• Indirect measurements approach:– Resonance Readout - The quantum device is

coupled to a superconducting resonator.

• Direct measurements of solid-state quantum devices has many drawbacks.

V

0 Input ProbeOutput SignalFreq

Resonance Curve

S12

1ZInput Probe

Output Signal

Quantum Device

0Z 0Z

– The state of the device modifies the resonance frequencies.– Readout is done by probing these resonance frequencies.

Resonance readout and Thermal Instability

0Z0Z

1µmFeed lineResonator

Weak link : Micro-Bridge

• Nonlinear thermal instability is expected under dc current bias.

A. VI. Gurevich and R. G. Mints, Rev. Mod. Phys. 59, 941 (1987)

2Q T T I• Heat production:

• Heat balance condition:

T con TW st • Heat transfer to a coolant

j

hot spot

Q T W THeat production Cooling power

T

WQ

unstable

CT

Q

• Test bed for resonance readout – Superconducting micro-bridge as artificial weak link.

Self-Oscillations

p

SC Threshold

NC Threshold

S.C PhaseN.C Phase

Pres

T

Oscillation Cycle1. Energy Buildup + Temperature increase2. Switching the NC phase at T >= Tc3. Energy relaxation + Temperature cool down4. Switching back to the SC phase at T <= Tc

0Z 0ZPower

• When embedded in a resonator, the resonator applies negative feedback to the thermal instability mechanism, leading to self-oscillations.

Measurement Setup

SpectrumAnalyzer

Synthesizer

~ 4.2 K300 K

0Z 0ZOscilloscope

pumpP

• E. Segev et al., Euro. Phys. Lett. 78, (2007)• E. Segev et al. , J. Phys.: Condense. Matter 19, (2007)

Feed Line

Self-Modulation - Time Domain I

-50 0 50

-80

-60

-40

Frequency [MHz]

Pow

er [d

Bm

] Frequency domain

0 2 4 6 8 10

-20

0

20

40

Time [Sec]

|B|2

Time Domain @ -28.01[dBm] Pump Power

SpectrumAnalyzer

~Oscillo-scope

Time Domain

Frequency Domain

pumpP

Self-Modulation - Time Domain II

Time Domain

Frequency Domain

-50 0 50

-80

-60

-40

Frequency [MHz]

Pow

er [d

Bm

] Frequency domain

0 2 4 6 8 10

-20

0

20

40

Time [ Sec]

|B|2

Time Domain @ -27.85[dBm] Pump Power

pumpP

SpectrumAnalyzer

~Oscillo-scope

Self-Modulation - Time Domain III

Time Domain

Frequency Domain

-50 0 50-70-60-50-40-30

Frequency [MHz]

Pow

er [d

Bm

] Frequency domain

0 200 400 600 800 1000

-20

0

20

40

Time [nSec]

|B|2

Time Domain @ -27.72[dBm] Pump Power

pumpP

SpectrumAnalyzer

~Oscillo-scope

1thP

Self-Modulation - Time Domain IV

Time Domain

Frequency Domain

-50 0 50

-80

-60

-40

Frequency [MHz]

Pow

er [d

Bm

] Frequency domain

0 100 200 300 400 500

-20

0

20

40

Time [ nSec]

|B|2

Time Domain @ -21.81[dBm] Pump Power

pumpP

SpectrumAnalyzer

~Oscillo-scope

1thP

Self-Modulation - Time Domain V

Time Domain

Frequency Domain

-50 0 50

-80

-60

-40

Frequency [MHz]

Pow

er [d

Bm

] Frequency domain

0 100 200 300 400 500

-20

0

20

40

Time [nSec]

|B|2

Time Domain @ -19.35[dBm] Pump Power

pumpP

SpectrumAnalyzer

~Oscillo-scope

1thP

2thP

Self-Modulation – Power Dependence

SpectrumAnalyzer

~Oscillo-scope

System Model

B B in pb

outb

12T

inb p

T

Control parameters

Input signal amplitudeInput signal frequency

Internal variablesB Mode Amplitude

Micro-Bridge TemperatureParameters

1 Coupling rate to environment

2 Coupling rate to losses

0 Resonance frequency

C Heat capacity of micro-bridge

H Heat Transfer rate

•Resonance mode amplitude EOM

0 1 2 12dd np ii T

tiB B bT

ForceStored amplitude (energy)

•Thermal balance EOM

20 02

dd

2 T T BTCt

H T T

heating power cooling power

Equations of motion

Stability diagram

mono-stable (S)

mono-stable (N)

bi-stable

unstable

2inb

p

bi-stable

0

B Binboutb

1 2T

MB is superconducting

MB is normal-conducting

MB is either super or normal-conducting.

MB oscillates between super and normal-conducting states.

•The shape of the stability diagram may vary depending on the tunability strength of the resonance frequency.

Self-Modulation Frequency

2inb

pms (S)

ms (N)bsbs

us

E15

E16

E16

E15

m-s (S)

m-s (N) bsbs

us

2inb

p

Self-Oscillation Frequency1. Resonance frequency is

negligibly tuned.

2. Resonance frequency is substantially tuned.

Theory vs. Experiment – Time Domain

2inb

p

mono-stable

(S)

mono-stable (N) bistablebistable

Un-s

working point

Theoretical Results

Experimental Results

t [Sec]0 0.2 0.4 0.6 0.8 1 0

0.20.4 (ii)

0 0.1 0.2 0.3 0.4 0.5-0.2

00.2 (iii)

0 0.1 0.2 0.3 0.4 0.5-0.2

00.2

P ref [

a.u.

]

(vii)0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1 0

0.20.4 (vi)

0 5 10 150

0.20.4

t [Sec]

(i)

0 5 10 150

0.20.4 (v)

0 0.2 0.4 0.6 0.8 1

00.20.4 (ii)

0 0.1 0.2 0.3 0.4 0.5-0.2

00.2 (iii)

0 0.1 0.2 0.3 0.4 0.5-0.2

0

0.2

P ref [n

.u.]

(vii)0 0.2 0.4 0.6 0.8 1

-0.4-0.2

0 (iv)

0 0.2 0.4 0.6 0.8 1 -0.4-0.2

0 (viii)

0 0.2 0.4 0.6 0.8 1 0

0.20.4 (vi)

Theory vs. Experiment – Threshold phenomenon

2inb

p

mono-stable

(S)

mono-stable (N) bistablebistable

un-stable working

point

Theoretical Results

Experimental Results

Noise is added to simulation

Noise is added to simulation

Thermal instability as sensitive detection mechanism

SpectrumAnalyzer

~4.2 K

0Z 0ZOscilloscope

pumpP Weak AM modulation

• The AM creates small oscillations around the working point.

2inb

p

mono-stable

(S)

mono-stable (N) bistablebistable

un-stable

x

x

The thermal non-linearity in our device has two advantages in terms of detection.1. The response of the system to a detectable stimulation is fast and strong.2. The system has a natural feedback mechanism that drives it back to its original

state once the response to the stimulation is ended.

Amplification mechanism

P ref

[a.u

.]

xun-s

ms

pump THP P

P ref

[dB

m]

• E. Segev et al., Phys Rev B. 77 (2008)

ExperimentsSimulation

Strongest amplification at the threshold of self-oscillations

Non-Linear Optical detection

SpectrumAnalyzer

Synthesizer

~ 4.2 K

0Z 0ZpumpP

Modulated IR Illumination

0.5 1 1.5 2 2.5 3 3.5 4

10-1

100

101

102

Optical Modulation Frequency [GHz]

NEP

[pW

Hz-0

.5]

38 fWNEPHz

• E. Segev et al., IEEE Trans. Appl. Supercond., 16 (2006).

• E. Segev et al., IEEE Trans. Appl. Supercond., 17 (2007).

The amplitude modulation is replaced by modulated IR laser illumination

Threshold of Self-

Oscillations

Fresnel Zone Plate

Optical fiber

1550 nm

NbN meander

Superconducting detectors must be kept small. Therefore:1. Signal degraded due to light beam expansion between fiber tip and

detector.2. Cryogenic alignment between fiber and detector is needed.Problem is reduced by an order of magnitude using Fresnel zone

plate.• Alignment between FZP and detector is done in lithography.

NbNNbN

Optical fiber

1550 nm

Fresnel zone plate

NbN meander

Additional thermal driven non-linear phenomena

Noise Squeezing Mode coupling

-2000 -1500 -1000 -500 0 500 1000 1500 2000

-100

-90

-80

-70

-60

-50

fc [Hz]

P ref [d

Bm

]

Period doubling and Stochastic resonanceSub-Harmonics

Unusual escape rate

Low noise non-linearity

Input Probe

Output Signal0Z 0Z

• Strong nonlinearity.• Self-Oscillations.• Strong non-linear amplification and detection

• But – Thermal noise creates a major drawback.Solution – Inductive nonlinearityInput Probe

Output Signal0Z 0Z

• Thermal (resistive) driven nonlinearity.

( , )f I T

• SQUIDs - Superconducting Quantum Interference Devices may behave as ideal non-dissipative inductors.• In practice – SQUID dynamics might be hysteretic and dissipative.

x( , , )f I T

SQUID: The ideal nonlinear Inductor

2

1xIx

Im

xI

JJ C JJ

0 JJJJ

sin Josephson Current

Josephson Voltage2

I IdVdt

JJ CharacterDC SQUID Model

JJ JJ( )

0 JJJJ

C JJ

12 cos

L

IVI t

SEM Image of a DC-SQUID

Nano-Bridge based JJx External Flux

x External currentI

80nm

60nm

Resonance Frequency Tuning

• E. Segev et al., Appl. Phys. Lett. 95, (2009)

S11 vs. magnetic field

Input Probe

Output Signal 0Z 0Z

SEM Image of a DC-SQUID

Mag

netic

Flu

x [a

.u.]

11Output PowInput P r

erowe

S

Self-Oscillations in superconducting resonator integrated with a DC-SQUID

Flux dependant self-oscillations

Flux triggering of self-oscillations

Self-Oscillation without magnetic flux

Simulation of flux dependant self-oscillations

50

SpectrumAnalyzer

~

•Resonance mode amplitude EOM

0 x 1 2

1

xd ,d

2

,p

in

TB i Bti b

T

0 x 2 x

2

0

d 2 ,d

,TTC Bt

H T T

T

•Thermal balance EOM

Experiment Simulation

Physical model of DC-SQUID

1 2

Control parametersBias currentxI Magnetic fluxx

Internal variable

20 L 00

cos cos sin sin / /x x CU I IE

Sine Term Quadric Term Source

2

1xIx

Im

xI

Squid Potential:

x xU @ 0.05 , 0, 80, 0.1C LI I

SQUID C Critical currentI

L Hysteresis Parameters_______________________

x xU @ 0.05 , 0, 80, 0.1C LI I

DC-SQUID Potential – Roll of Hysteresis Parameter

-5 0 50

200

400

600

800

L=3

L=10

L=20

L=80

+

U

X2D Potential @ 0, 0

L

200

L 0cos cos sin sin / /x x CU I IE

Sine Term Quadric Term Source

Hysteretic parameter that control the degree of metastability.

DC-SQUID Potential – Roll of control parameters

Static Zonex cI I

x xU@ 0.6 , 0CI I

x xU @ 1.1 , 0CI I

Free Running zonex cI I x x 0U @ 0, 0.9I

Tilt by Current

Tilt by magnetic Flux

Control parametersBias currentxI External magnetic fluxx

2

1xIx

Im

xI

x xU @ 0.05 , 0CI I

DC-SQUID Equations of Motions

JR JC

1I

JC2CI

2I

JR

1CI

xI

2L

2L

x

Im

xI

DC-SQUID EOM

x1 D 1 1 1 2 x 0

L

11 sin 2 /C

I noiseI

x2 D 2 2 1 2 x 0

L

11 sin 2 /C

I noiseI

DC-SQUID Circuit Model

JJ Current Coupling

Circuit model includes:• RJ – Shunting resistor.• CJ – JJ capacitance.• L – Self-inductance.

Control parametersBias currentxI

External fluxx

th JJ phasek k Internal variable

DampingD 1 2c cI I

C Critical currentI L Hysteresis

Parameters

Kirchhoff Equations

-2 -1 0 1

-2

-1

0

1

2

1

-1

2

0

-2

+/

-/

Stability boundaries – Phase space

1 2det , 0,

tr 0

H f

H

2 2

21 1 2

2 2

21 2 2

d u d ud d d

Hd u d u

d d d

Hessian Local Stable Zones

1 2 x x, , , 0dU f Id

Local Extremum Points

0dUd

Stability Diagram in the plane ofStability Diagram in the plane of x x,I ,

-1 0 1-6

-4

-2

0

2

4

61

-1

2

0

-2

Ix/Ic

x/

0

Local stability zones

-1 0 1-6

-4

-2

0

2

4

61

-1

2

0

-2

Ix /Ic

x/

0Stability boundaries – Alternating excitation

Stability Diagram in the plane of x x,I

-2 -1 0 1

-2

-1

0

1

2

1

-1

2

0

-2

+/

-/$

$

$

$

$

$

$

$

$

$$

$$

$

$$

$

Stability Diagram in the plane of ,

Periodic dissipative zone – Static stability zones were dissipation of energy occurs under periodic excitation.

Numerical results

0.7 0.8 0.9 1-1.5

-1

-0.5

0

0.5

1

1.5

|Ix|/I

c

x/

0

-1 0 1-6

-4

-2

0

2

4

61

-1

2

0

-2

|Ix|/Ic

x/

0 Periodic dissipative static zones

-2 -1 0 1

-2

-1

0

1

2

1

-1

2

0

-2

+/

-/

0.96 0.97 0.98 0.99 1

-2

-1

0

1

2

Ix/IC

x/

0Periodic dissipative static zone

Periodic non-dissipative static zone

Periodic dissipative static zone

E38 Parameters:

L 722 0.025

Free running

zonePeriodic dissipative

static zone

Periodic dissipative static zoneExperimental data Vs. Simulation

SimulationExperiment

• E. Segev et al., arxiv:1007.5225v1 (2010)

4.2K

Lockin Amplifier

Oscillo-scope

1k

1M

Silicon Wafer

LPF

300K

1M

xLPF

XI

Spectrume analyzer

0.7 0.8 0.9 1-1.5

-1

-0.5

0

0.5

1

1.5

|Ix|/Ic

x/

0

Double Threshold to Oscillatory Zone Periodic non-Dissipative

Static zone

Periodic Dissipative Static zone

Oscillatory zoneOscillatory zone

only for negative excitation values

Oscillatory zone only for positive excitation values

0 1 2 3-1

-0.50

t/Tx

V SQD [a

.u.]

0 1 2 3

00.5

1

t/Tx

V SQD [a

.u.]

0 1 2 3-3-2-10

t/TxV SQ

D [a

.u.]

0 1 2 3

-1012

t/Tx

V SQD [a

.u.]

Double Threshold to Oscillatory Zone

0.7 0.8 0.9 1-1.5

-1

-0.5

0

0.5

1

1.5

|Ix|/I

c

x/

0

Experimental Results

Simulation Results

Split Threshold

Hybrid zonesExperimental Results

SQUID Voltage Noise Level TD Statistics

Parametric Excitation Of Superconducting Resonator

• The reflected tone is measured with a spectrum analyzer.• The reflected power has many sidebands originated by the nonlinear

mixing

dc acx x x px

px 0

0

cos ,

2

Resonance frequency ~ 3GHz

t

• Magnetic flux can be used to create parametric excitation of superconducting resonators

SpectrumAnalyzer

Synthesizer

~ 0Z 0ZpumpP

SEM Image of a DC-SQUID

Current Sources ~

X

Stability Diagram for Parametric Excitation

• Only flux excitation: dc acx x x pxcos t

-1 0 1-6

-4

-2

0

2

4

6

|Ix|/Ic

x/

0

dcx ac

x

6 6.5 7 7.5 8 8.5

-2

-1

0

1

2

3

|xac|/0

xdc

/0

0

Stability diagram in the plane of________dc acx x,

• No Free-Running Zone

• The current through the SQUID is negligible.

Periodic non-dissipative static zone

Periodic dissipative static zone

6 6.5 7 7.5 8 8.5

-2

-1

0

1

2

3

|xac|/0

xdc

/0

Parametric excitation – Numerical results

Simulation results

Stability diagram in the plane of dc acx x,

• The effect of SQUID inductivity emerges at high frequencies.• Boundaries between local stable zones are observed in the

periodic non-dissipative zone.• The variance of the SQUID inductance within a local stable

state is observed.PNDSZ PDSZ

PNDSZ PDSZ

Parametric excitation – Experimental results

Location and shape of threshold is different

Simulation results Exc. Heat Production

Experimental results

• Many features agree between simulation and experimental results, but: • Location and shape of PDSZ threshold is different.• Different βL fits the PNDSZ and the PDSZ.

Different βL fits the PNDSZ and the PDSZ.L 45

L 35

Threshold point to PDSZ

Heat relaxation rates are comparable to the excitation

frequency!

Only heat degree of Freedom Can explain this change

DC-SQUID Model inc. heat balance equationEOM for the Josephson junction phases

1L0

x1 D 1 1 1 2 x 0

0

11 sin 2 /C

yII noise

2L0

x2 D 2 2 1 2 x 0

0

11 sin 2 /C

yII noise

JJ Current Coupling

3/2 1/22 2

0 0

; 1 1kCkk k k k

C k

yIy yI y

represents the dependence of the kth JJ critical current of the temperature. ky

Heat balance EOMs

20 , 1,2H

Ck

Dkk k

Heat Production Heat transfer to coolant

Heat capacitanceC

Heat transfer rateH

0 Base temperature

Parameters

0 1 2 3 4 5 6-0.4-0.2

00.20.4

t/Tx

V s

6 6.5 7 7.5 8 8.5

-2

-1

0

1

2

3

|xac|/

0

xdc

/0

Numerical results inc. Heat production

PNDSZ PDSZ

+

-2 -1 0 1

-1

0

1

2

1

2

-1

0

+

-

-2 -1 0 1

-1

0

1

2

1

2

-1

0

+

-

Stability diagram in the plane of dc acx x,

Time domain simulation

Stability Diagram in the plane of ,

0 1 2 3 4 5 6-0.4-0.2

00.20.4

t/Tx

V s

First Cycle

Additional Cycles

Legend

Heat dependant Hysteresis

0 1 2 3 4 5 6-0.4-0.2

00.20.4

t/Tx

V s

3840

42

44

e L

ac Lx 02

6.5 7 7.5 8 8.5|x

ac|/0

6

-2

-1

0

1

2

3

xdc

/0

+

•The hysteresis parameter depends on temperature.

L C0

L I

•When βL decreases the stability diagram shifts to the left.

•The effective working point corresponds to enhanced number of transitions between LSZs.

Stability diagram in the plane of dc acx x,

•Transitions between local stable states produces heat.

•The heat induces transient and average changes in the local temperature of the SQUID.

Future Research – Quantum Nano-Mechanics• Quantum Nano-Mechanics – emerging research field in which

quantum phenomena are measured in nano-mechanical beams.• Question – Does stress or strain in Nano-beams affects material

coherency ?• Method – Study the effect of a mechanical degree of freedom on the

Aharonov-Bohm effect.

100nm

1um

30nm-thick Aluminum

IV

V I

Side electrode

Side

el

ectr

ode

1um2 AB rings, 30x90nm2 cross section.

Future Research – Quantum Nano-Mechanics• Question – Can nano-mechanical beam behave like two level system,

showing superposition of states?• Method – Suspend one side of a DC-SQUID embedded in a resonator.

Summary• Thermal (resistive) nonlinearity.

• Metastable and Hysteretic SQUID

Self-Oscillations Detection and amplification

Periodic dissipative stability zone

Tunable resonators and self-oscillations

Parametric Excitation

Publication List

1. E. Segev, B. Abdo, O. Shtempluck, and E. Buks, 'Fast Resonance Frequency Modulation in Superconducting Stripline Resonator', IEEE Trans. Appl. Sup., 16 (3), P. 1943 (2006).

2. E. Segev, B. Abdo, O. Shtempluck, and E. Buks 'Novel Self-Sustained Modulation in Superconducting Stripline Resonators', Europhys. Lett. 78, 57002 (2007).

3. E. Segev, B. Abdo, O. Shtempluck, and E. Buks 'Thermal Instability and Self-Sustained Modulation in Superconducting NbN Stripline Resonators', J. Phys. Cond. Matt. 19, 096206 (2007).

4. E. Segev, B. Abdo, O. Shtempluck, and E. Buks 'Extreme Nonlinear Phenomena in NbN Superconducting Stripline Resonators', Phys. Lett. A 366, pp. 160-164 (2007).

5. E. Segev, B. Abdo, O. Shtempluck, E. Buks, and B. Yurke 'Prospects of Employing Superconducting Stripline Resonators for Studying the Dynamical Casimir Effect Experimentally', Phys. Lett. A 370, pp. 202-206 (2007).

6. E. Segev, B. Abdo, O. Shtempluck, and E. Buks 'Utilizing Nonlinearity in a Superconducting NbN Stripline Resonator for Radiation Detection' , IEEE Trans. Appl. Sup., 17, pp. 271-274 (2007).

7. E. Segev, B. Abdo, O. Shtempluck, and E. Buks 'Stochastic Resonance with a Single Metastable State: Thermal instability in NbN superconducting stripline resonators', Phys. Rev. B 77, 012501 (2008).

8. E. Segev, O. Suchoi, O. Shtempluck, and E. Buks ‘Self-oscillations in a superconducting stripline resonator integrated with a dc superconducting quantum interference device', Appl. Phys. Lett. 95, 152509  (2009).

9. E. Segev, O. Suchoi, O. Shtempluck, Fei Xue, and E. Buks ‘Metastability in a nano-bridge based hysteretic DC-SQUID embedded in superconducting microwave resonator, arXiv:1007.5225v1 (2010).

Publication List10. E. Buks, S. Zaitsev, E. Segev, B. Abdo, and M. P. Blencowe, ‘Displacement Detection with a

Vibrating RF SQUID: Beating the Standard Linear Limit’, Phys. Rev. E 76, 026217 (2007).11. E. Buks,  E. Segev, S. Zaitsev, B. Abdo, and M. P. Blencowe, ‘Quantum Nondemolition

Measurement of Discrete Fock States of a Nanomechanical Resonator’, EuroPhys. Lett., 81 10001 (2008).

12. B. Abdo, E. Segev, O. Shtempluck, and E. Buks, ‘Observation of Bifurcations and Hysteresis in Nonlinear NbN Superconducting Microwave Resonators’, IEEE Trans. Appl. Sup., 16 (4), p. 1976, (2006).

13. B. Abdo, E. Segev, O. Shtempluck, and E. Buks, ‘Nonlinear dynamics in the resonance line-shape of NbN superconducting resonators’,  Phys. Rev. B 73, 134513 (2006).

14. B. Abdo, E. Segev, O. Shtempluck, and E. Buks, ‘Intermodulation gain in nonlinear NbN superconducting microwave resonators’, App. Phys. Lett. 88 , 022508 (2006).

15. B. Abdo, E. Segev, O. Shtempluck, and E. Buks, ‘Escape rate of metastable states in a driven NbN superconducting microwave resonator’, J. App. Phys., 101, 083909 (2007).

16. B. Abdo, E. Segev, O. Shtempluck, and E. Buks, ‘Signal Amplification in NbN superconducting resonators via Stochastic Resonance’, Phys. Lett. A 370, p. 449 (2007).

17. B. Abdo, O. Suchoi, E. Segev, O. Shtempluck, M. Blencowe and E. Buks, ‘Intermodulation and parametric amplification in a superconducting stripline resonator integrated with a dc-SQUID’, Europhys. Lett. 85, 68001 (2009).

18. G. Bachar, E. Segev, O. Shtempluck, S. W. Shaw and E. Buks, ‘Noise Induced Intermittency in a Superconducting Microwave Resonator’, Europhys. Lett. 89, 17003 (2009).

19. Oren Suchoi, Baleegh Abdo, Eran Segev, Oleg Shtempluck, Miles Blencowe and Eyal Buks, ‘Intermode Dephasing in a Superconducting Stripline Resonator’, Phys. Rev. B 81, 174525 (2010).