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CNRS-UJF team (Grenoble). CRTBT - LCMI - LP2MC. PhD : J. Claudon A. Fay A. Ratchov. permanent : W. Guichard F.W.J. Hekking L. Lévy O. Buisson. Noise and decoherence in a dc SQUID. | 4 . | 3 . | 2 . | 1 . | 0 . Introduction. dc - SQUID. artificial atom. =. 10 GHz.

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Noise and decoherence in a dc squid

CNRS-UJF team (Grenoble)

CRTBT - LCMI - LP2MC

PhD :

J. Claudon

A. Fay

A. Ratchov

permanent :

W. Guichard

F.W.J. Hekking

L. Lévy

O. Buisson

Noise and decoherencein a dc SQUID


Introduction

|4

|3

|2

|1

|0

Introduction

dc - SQUID

artificial atom

=

10 GHz

Low energy sub-space { |0 , |1 }

a phase Qubit (cf. J. Martinis et al.)

Higher energy sub-space { |0 , |1 ,|2 , ... }

more complex dynamics : eg. multilevel coherent oscillations


The josephson junction jj

S1

S2

The Josephson Junction (JJ)

A S-I-S junction

CJ model

Ib

Ib

Ox

C0

Ic

U()

U (Ib)

|3

|2

 dynamics?

mechanical analogy

|1

|0

  • fictitious particle

  • mass  C0

  • position 

  • potential U()

p(Ib)


The current biased dc squid a tunable artificial atom
The current biased dc SQUID:a tunable artificial atom

Ib

DU (Ib,Fb)

Ls

Fb

I0 , C0

I0 , C0

JJ1

p (Ib,Fb)

shape of the

anharmonic well

bias point (Ib,Fb)


The current biased dc squid a tunable artificial atom1
The current biased dc SQUID:a tunable artificial atom

Ib

DU (Ib,Fb)

Ls

F(t)

MW

Fb

I0 , C0

I0 , C0

n,W1

JJ1

p (Ib,Fb)

shape of the

anharmonic well

bias point (Ib,Fb)

MW flux F(t)

excitation


Outline
Outline

  • I. Experimental setup

  • II. Finite lifetime of the ground state

    • MQT measurement and LF noise

  • III. Quantum dynamics

    • Multilevel coherent oscillations

    • Quantum or classical oscillations?

  • IV. Incoherent processes in the 2 level limit

    • Low power spectroscopy

    • Energy relaxation

|2

MQT

|1

|0

MW

|2

|1

|0

|1

Gr

|0


Sample realization

Process

  • e-beam lithography

  • Al shadow evaporation

Sample realization

SiO2

200 m

MW antenna

bias lines

15 m

JJs

15 μm²

superconducting loop


Vs

MW line

dc - 20 GHz

heavily filtered bias lines

dc - 200 kHz

  • MW excitation

  • fast measurement of

  • the quantum state

  • SQUID bias(Ib,Fb)

  • voltage readout

Experimental setup (simplified)

~ 25 mK

Ze()

Loc

Ib

Fhf

Ib

50 Ω

Vhf

Fb

Cp

coil


Voltage states of a hysteretic dc squid

Ib-Vs hysteretic characteristic

3.0

1.5

0

-1.5

-3.0

Vs (mV)

-500

-250

0

250

500

Voltage states of a hysteretic dc-SQUID

Ic

Ib (mA)

0-voltage state

resistive state

|1

|0


Voltage states of a hysteretic dc squid1

Ib-Vs hysteretic characteristic

3.0

3.0

2.5

1.5

2.0

1.5

0

1.0

-1.5

0.5

-3.0

Vs (mV)

-500

-250

0

250

500

0.0

-0.5

0

0.5

Bias point (Ib, b) potential shape

Voltage states of a hysteretic dc-SQUID

“Phase diagram”

Ic

Ic

Ib (mA)

Ib (mA)

Fb/F0

0-voltage state

resistive state

Ic =f(b) SQUID electrical parameters

I0 = 1.242 A

C0= 560 fF

LS = 250 pH

 = 0.414

|1

|0


Escape of the trapped fictitious particle

Measurement of : principle

Ib

T=50 μs

TA

~ 4000 repetitions

(1 kHz)

|1

|0

escape

Vs

MQT

t

1

0.9

Pesc

0.5

0.1

ΔI

0

2.38

2.4

2.42

Ib (µA)

Escape of the trapped fictitious particle

Dependence on the bias current Ib

During ΔT, escape probability :

ΔI dominant escape

mechanism


Escape at low temperature

30

20

10

-0.4

-0.2

0

0.2

0.4

20

18

16

14

12

10

-0.5

0

0.5

Escape at low temperature

~ 40 mK

F. Balestro’s SQUID

MQT for all fluxes

fit : no free parameters

DI(nA)

MQT

F. Balestro et al, PRL 2003

Fb/F0

~ 25 mK

This sample

small deviation from MQT which scales as

DI(nA)

flux noise

MQT

Fb/F0


Noise and escape measurements
Noise and escape measurements

Noise on bias parameters :

or

  • equivalent effects

  • difference : flux sensitivity is strongly modulated by

  • Hypothesis :

  • adiabatic : frequency

  • gaussian fluctuations

Effect on escape curve : depends on noise frequencies

compared to Dt


Noise and escape measurements1

I

(t)

b

Dt

...

...

...

t

broadening ~

0

I

b

1

0.5

0

1.7

1.72

1.74

1.76

1.78

Noise and escape measurements

Low frequency limit (< 1/Dt)

MQT &

LF noise

Pesc

MQT

Ib (mA)


Noise and escape measurements2

I

(t)

I

(t)

b

b

Dt

Dt

...

...

...

...

t

t

broadening ~

0

0

I

I

1

b

b

1

0.5

0.5

0

0

1.7

1.72

1.74

1.76

1.7

1.72

1.74

1.76

1.78

Noise and escape measurements

Low frequency limit (< 1/Dt)

High frequency limit (>1/Dt)

MQT &

LF noise

Pesc

Pesc

MQT &

HF noise

MQT

MQT

Ib (mA)

Ib (mA)

shift ~


Interpretation of escape measurements low frequency flux noise

20

MQT & LF flux noise

18

16

DI(nA)

14

12

MQT

This sample

5.5×10-4

10

-0.5

0

0.5

F. Balestro’s SQUID

< 3×10-4

Fb/F0

Delft flux Qubit

(loop area normalization)

~ 2×10-4

Interpretation of escape measurements :low frequency flux noise

DI measurements : a probe

to LF noise [0.2Hz,20kHz]

  • no significant LF current noise

(fit uncertainty)

  • yes LF flux noise :


Outline1

|2

MQT

|1

|0

Outline

  • I. Experimental setup

  • II. Finite lifetime of the ground state

    • MQT measurement and LF noise

  • III. Quantum dynamics

    • Multilevel coherent oscillations

    • Comparison to classical oscillations

  • IV. Incoherent processes in the 2 level limit

    • Low power spectroscopy

    • Energy relaxation

MW

|2

|1

|0

|1

Gr

|0


Quantum dynamics:typical sketch of experiments

3

3

3

1

2

2

1

Fb is fixed

current

Ip(t)

Ib

t

2

(t)

Fm

Ic

flux

Fb

Fb+Fm

bias point :

shape of the potential

manipulation :

(deep well)

adiabatic deformation :

selective escape of excited states

population of excited states

MW

initial state =|0

O. Buisson etal, Phys. Rev. Lett. 2003


Example of flux sequence

1

Vhf (V)

n = 5 GHz

0.5

t (ns)

0

0

2

4

6

8

Example of flux sequence

Voltage signal(entrance of cryostat)

excitation MW pulse

measurement pulse

risetime = 1 ns

duration~ 1 to 300 ns

risetime = 1.6 ns

duration~ 1.5 ns

amplitude : ~10-3F0

amplitude : ~10-2F0


Coherent oscillations

60

40

20

0

Coherent oscillations

Bias point : Ib = 2.222 mA , Fb = -0.117 F0

n01 = 8.287 GHz

(low power spectroscopy)

E/h (GHz)

7 levels

5

2

1

n01-n12 = 160 MHz

n01

0

MW excitation

  • n = n01

  • amplitude : W1

  • duration TMW : variable


Coherent oscillations1

0.8

0.6

60

0.4

40

0.2

0.8

20

0.6

0

0.4

0.2

0.8

0.6

0.4

0.2

0

20

40

60

80

Coherent oscillations

P = -6 dBm

W1/2p = 65 MHz

Bias point : Ib = 2.222 mA , Fb = -0.117 F0

ncoh = 66 MHz

Pesc

n01 = 8.287 GHz

(low power spectroscopy)

E/h (GHz)

7 levels

P = 0 dBm

W1/2p = 130 MHz

5

Pesc

2

1

n01-n12 = 160 MHz

n01

0

ncoh = 122 MHz

MW excitation

P = +6 dBm

W1/2p = 260 MHz

  • n = n01

  • amplitude : W1

  • duration TMW : variable

Pesc

ncoh = 208 MHz

Attenuation time ~ 20 ns

Tmw (ns)


Multilevel coherent oscillations

250

200

How many levels?

150

anharmonicity

excitation

amplitude

100

coherent superposition

of excited states

50

|0

0

0

50

100

150

200

250

300

Multilevel coherent oscillations

solid line = multilevel theory

(F. Hekking et al.)

J. Claudon etal, Phys. Rev. Lett. 2004

ncoh (MHz)

n01

MW amplitude : W1/2p (MHz)


Multilevel coherent oscillations1

250

200

150

100

coherent superposition

of excited states

50

|0

0

0

50

100

150

200

250

300

Multilevel coherent oscillations

solid line = multilevel theory

(F. Hekking et al.)

2-level limit

2

J. Claudon etal, Phys. Rev. Lett. 2004

ncoh (MHz)

1

n01

MW amplitude : W1/2p (MHz)

W1«n01-n12

Rabi oscillations between |0 and |1

cf. J. Martinis et al.


Multilevel coherent oscillations2

250

200

150

100

coherent superposition

of excited states

50

1

|0

p1

0

0.5

Populations

0

50

100

150

200

250

300

p2

0

0

5

10

15

20

25

30

Tmw (ns)

Multilevel coherent oscillations

solid line = multilevel theory

(F. Hekking et al.)

2-level limit

2

3

J. Claudon etal, Phys. Rev. Lett. 2004

ncoh (MHz)

n01

MW amplitude : W1/2p (MHz)

p2 < 12 %

close to a 2-level oscillation


Multilevel coherent oscillations3

250

200

150

100

coherent superposition

of excited states

50

1

|0

0

p1

0.5

Populations

0

50

100

150

200

250

300

p2

p3

0

0

2

4

6

8

10

12

14

16

Tmw (ns)

Multilevel coherent oscillations

solid line = multilevel theory

(F. Hekking et al.)

2-level limit

2

3

J. Claudon etal, Phys. Rev. Lett. 2004

ncoh (MHz)

MW amplitude : W1/2p (MHz)

p3 < 3 %

3-level oscillation


Multilevel coherent oscillations4

250

200

150

100

coherent superposition

of excited states

50

|0

0

0

50

100

150

200

250

300

Multilevel coherent oscillations

solid line = multilevel theory

(F. Hekking et al.)

2-level limit

2

3

J. Claudon etal, Phys. Rev. Lett. 2004

ncoh (MHz)

4

MW amplitude : W1/2p (MHz)

W1»n01-n12

# involved levels  1

oscillations still exist

classical description ?


Classical rabi type oscillations
Classical Rabi-type oscillations

U(j)

n, W1

j(t)

j(t)

ncoh

  • At t=0, MW on

  • transient : anharmonicity of the well

  • modulation of the phased-locked state

  • energy modulation with frequency  ncoh

(a. u.)

  • small excitation power & n = wp/2p

Rabi-type oscillations in a

classical Josephson junction

N. Grønbech-Jensen and M. Cirillo (2005)

cond-mat/0502521

A. Ratchov and F. Faure (LP2MC-Grenoble)

unpublished


Quantum vs classical description

250

classical theory

with n=wp/2p

200

150

100

50

0

0

50

100

150

200

250

300

Quantum VS classical description

# involved levels

2

3

4

quantum theory

with n=n01

ncoh (MHz)

αΩ12/3

  • Low excitation power :

  • quantum description

  • Higher excitation power

  • qualitative agreement

  • between classical and

  • quantum descriptions

n01-n12 = 160 MHz

αΩ1

MW amplitude : W1/2p (MHz)


Outline2

|2

MQT

|1

|0

Outline

  • I. Experimental setup

  • II. Finite lifetime of the ground state

    • MQT measurement and LF noise

  • III. Quantum dynamics

    • Multilevel coherent oscillations

    • Comparison to classical oscillations

  • IV. Incoherent processes in the 2 level limit

    • Low power spectroscopy

    • Energy relaxation

MW

|2

|1

|0

|1

Gr

|0


Low power spectroscopy resonance frequency

12

3

Ib (mA)

2

11

Ic

1

10

0

8

-0.5

0

0.5

9

Fb/F0

6

8

7

4

solid lines : semi-classical theory for

X2X3 potential

0.5

1

1.5

2

2.5

2

SQUID electrical parameters

8

8.1

8.2

8.3

8.4

8.5

I0 = 1.242 A

C0= 560 fF

LS = 250 pH

 = 0.414

Low power spectroscopy : resonance frequency

n01 (GHz)

Ib (mA)

n01

~ gaussian

shape

Pesc (%)

n (GHz)


Low power spectroscopy resonance linewidth

12

3

Ib (mA)

2

11

Ic

1

10

0

8

-0.5

0

0.5

9

250

Fb/F0

200

6

8

150

7

4

100

50

2

8

8.1

8.2

8.3

8.4

8.5

0

0.5

1

1.5

2

2.5

Low power spectroscopy : resonance linewidth

n01 (GHz)

~ gaussian

shape

Dn (MHz)

Pesc (%)

Dn

Ic

Ic

n (GHz)

Ib (mA)


Energy relaxation

8

12

n01

11

6

Dm

10

120

4

9

100

8

2

80

7

60

0

0

0.1

0.2

0.3

0.4

0.5

0.6

40

0.5

1

1.5

2

2.5

Ib (mA)

Energy relaxation

Ib = 2.222 mA, Fb = -0.117 F0

1/Gr for different bias points

n01 (GHz)

Pesc (%)

1 / Gr (ns)

Dm (ms)

solid line

Ic

Ic

1 / Gr = 90 ns

Qr = 6000


current

fluctuations

flux

fluctuations

Transverse and longitudinal coupling

SQUID

coupling terms

environment

linear

eigenbasis { |0 , |1 }

transverse

longitudinal

relaxation

“pure” dephasing


Relevant fluctuations sources

LF flux

fluctuators

Relevant fluctuations sources

Rmcs

Rp(w)

Loc

Lf

dI

dF

Cp

Cmsc

chip

~ 25mK

Heavy filtering and shielding

significant fluctuations sources located close to the SQUID

quantum fluctuation

dissipation theorem

  • electrical circuit

  • LF flux fluctuators

MQT measurements

[0.2Hz,20kHz]


Energy relaxation1

12

Loc

11

Rp(w)

10

120

9

100

8

80

7

60

40

0.5

1

1.5

2

2.5

Energy relaxation

Environment at high frequencies (> 5GHz)

gold

capacitor

n01 (GHz)

hypothesis :no high frequency flux noise

1 / Gr (ns)

Ic

Ic

consistent with skin effect estimations

Ib (mA)

Dissipation is induced by

the gold filtering capacitor


Analysis of spectroscopic results

STEP 1

Analysis of spectroscopic results

reduced density matrix :

(interaction frame)

NOISE

inelastic

processes

“pure”

dephasing

hypothesis :

  • gaussian fluctuations

  • linear coupling

0.2 Hz

20 kHz

flux noise :

f

neglected

(static approximation)


Analysis of spectroscopic results1
Analysis of spectroscopic results

current noise :

STEP 2

Linear response

NOISE

NO MW

NOISE

MW (linear regime)

Fourier transform

resonance line shape

Dn


12

11

10

9

250

200

8

150

7

100

50

0

0.5

1

1.5

2

2.5

Low power spectroscopy for various bias points

n01 (GHz)

  • fit without free parameter

  • Ib Ic :

Dn (MHz)

longitudinal noise sensitivity

and

Ic

Ic

Dn

Ib (mA)


Conclusion

MQT & LF flux noise

250

20

DI(nA)

MQT

18

200

Fb/F0

16

150

2

3

4

250

14

ncoh (MHz)

αΩ12/3

100

200

12

αΩ1

150

50

10

W1/2p (MHz)

100

0

-0.5

0

0.5

50

0

50

100

150

200

250

300

Dn (MHz)

0

0.5

1

1.5

2

2.5

Ic

Ic

Ib (mA)

Conclusion

  • MQT measurements : a probe to low frequency noise

  • Multilevel coherent oscillations

  • Incoherent processes in the 2 level limit


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