Quantum Spectrometers of Electrical Noiseboulderschool.yale.edu/sites/default/files/files/... · 2019. 12. 18. · Outline of Lecture 2 • A spin or two-level system (TLS) as spectrometer

Quantum Spectrometers of Electrical Noise

Rob SchoelkopfApplied PhysicsYale University

Gurus: Michel Devoret, Steve Girvin, Aash Clerk

And many discussions with D. Prober, K. Lehnert, D. Esteve, L. Kouwenhoven, B. Yurke, L. Levitov, K. Likharev, …

Thanks for slides: L. Kouwenhoven, K. Schwab, K. Lehnert,…

Noise and Quantum MeasurementR. Schoelkopf

1

Overview of LecturesLecture 1: Equilibrium and Non-equilibrium Quantum Noise

in CircuitsReference: “Quantum Fluctuations in Electrical Circuits,”

M. Devoret Les Houches notes

Lecture 2: Quantum Spectrometers of Electrical NoiseReference: “Qubits as Spectrometers of Quantum Noise,”

R. Schoelkopf et al., cond-mat/0210247

Lecture 3: Quantum Limits on MeasurementReferences: “Amplifying Quantum Signals with the Single-Electron Transistor,”

M. Devoret and RS, Nature 2000.“Quantum-limited Measurement and Information in Mesoscopic Detectors,”

A.Clerk, S. Girvin, D. Stone PRB 2003.

And see also upcoming RMP by Clerk, Girvin, Devoret, & RSNoise and Quantum Measurement

R. Schoelkopf2

Outline of Lecture 2• A spin or two-level system (TLS) as spectrometer

• The meaning of a two-sided spectral density

• The Cooper-pair box (CPB): an electrical TLS

• Using the CPB to analyze quantum noise of an SETMajer and Turek, unpub.

• Other quantum analyzers:• SIS junction: a continuum edge

(DeBlock, Onac, & Kouwenhoven)• Nanomechanical system: a harmonic oscillator

(Schwab et al.; Lehnert et al.)Noise and Quantum Measurement

R. Schoelkopf3

The Electrical Engineer’s Spectrum Analyzers

FFT

( )VS ω

ω0ω =( )V t


4

Both measure only the symmetrized spectral density:

( ) ( ) (0) (0) ( )S i tVVS dt e V t V V V tωω

∞

−∞= +∫

( )VS ω

ω0ω =

tunedfilter( )V t

( ) ( )VV VVS Sω ω= + + −

The Quantum Mechanic’s Analyzer – a Spin


5

tomagnetometer

0 ˆB z

( )xB t( )V t

=

Spins only absorb at Larmor frequency01 0 /g Bω ω µ= =

Resolution = inverse of spin coherence time

Able to measure both sides of a spectral density!

Spin as Spectrum Analyzer - II0 ˆB z

( )xB t( )V t

010 2 z

H ω σ= −

1 ( ) xH AV t σ=

( )( )

g

e

tt

αψ

α⎛ ⎞

= ⎜ ⎟⎝ ⎠

10

( ) (0) ( ) (0)tit d H tψ ψ τ ψ= − ∫with initial condition:(0) gψ =

01

0 0

( ) ( ) ( ) ( )t t

ie x

iA iAt d e g V d e Vω τα τ σ τ τ τ τ= − = −∫ ∫

01 1 2

2 2( )

1 2 1 2 012 20 0

( ) ( ) ( ) ( )t t

ie V

A Ap t d d e V V t Sω τ ττ τ τ τ ω− −= = −∫ ∫

( )012

2 Vg e SA

ω→Γ −= ( )012

2 Ve g SA

ω→Γ +=


6

Interpretation of Two-Sided Spectrum

0T ≠0T =

01ω+01ω−

absorption by spin emission by spin

↑Γ ↓Γ

g

e

01ω( ) /

2 R1V kT

Se ωωω −= −

kTω =

( )VS ω

0 ω

absorption by sourceemission by source

( )01VS ω↓Γ +∝( )01VS ω↑Γ −∝


7

Polarization of Spin and Noise Spectra

↑Γ ↓Γ

g

e

01ω

eg e

ge g

dp p pdtdp

p pdt

↑ ↓

↓ ↑

= Γ − Γ

= Γ − Γ

g ep p↑ ↓Γ = ΓSteady-state:

g eP p p= −

Define polarization of spin:

If noise is truly classical,

g ep p≡ and no polarization!

01 // kTe gp p eω−=Thermal equilibrium:

01 /01 01( ) ( )

kTV VS e S

ωω ω+ = −Requires particular asymmetry!Noise and Quantum Measurement

R. Schoelkopf8

Polarization of Spin - IIDefine steady-state polarization:

( ) ( )( ) ( )

01 01

01 01SS

S SPS S

ω ωω ω

↓ ↑

↓ ↑

Γ −Γ + − −= =Γ + Γ + + −

due to relative asymmetry of noise

( ) ( ) 1( ) ( ) ( )d P t P t P t

dt ↓ ↑∆

= −∆ Γ + Γ = −∆ Γ

( ) ( )[ ]2

1 01 0121

1 A S ST

ω ωΓ = = + + −

( ) ( ) SSP t P t P∆ = −Define deviation from steady state:

So relaxation rate (Γ1) due to total noise (and coupling)Noise and Quantum Measurement

R. Schoelkopf9

Ways to Characterize a Quantum Reservoir

Fermi’s Golden Rule

Fluctuation-Dissipation RelationNMR

Harmonic Oscillator

Quantum Optics10

( )2

VA S ω↑

⎛ ⎞Γ = −⎜ ⎟⎝ ⎠

( )2

VA S ω↓

⎛ ⎞Γ = +⎜ ⎟⎝ ⎠

2( )

n ↑↓ ↑

Γ=

Γ −Γ [ ] 2( )

Re ( )ZA

ωω

↓ ↑Γ −Γ=

P ↓ ↑↓ ↑

Γ −Γ=Γ +Γ

( ) 11T−

↑ ↓= Γ +Γ

( )2( )

Eω ↑ ↓

↓ ↑

Γ + Γ=

Γ −Γ( )

Qωγ ↓ ↑= = Γ −Γ

EinsteinA ↓ ↑= Γ −ΓEinsteinB ↑= Γ

Cooper Pair Box as Two-Level System

11 /g g gn C V e=

2~ 5

2( )jc

gE e GHz

C C=

+

4 ( / )ˆc gel gE n C V eE = −

CgBox

Vg

2 ˆq en= ˆ ˆ2box xelEH σ=

Cj

Vg

1

0n =

1n =

4 CEE

nerg

y

(Buttiker ’87; Bouchiat et al., 98)

Cooper Pair Box as Two-Level Systemˆ ˆ ˆ

2 2el J

box x zHEE σ σ= −

2~ 5

2( )jc

gE e GHz

C C=

+


12 /g g gn C V e=

CgBox

Vg


Cj

2 5 GHz4 JJ e R

E π∆= ≈

Vg4 CE

1

EJ

0 1−

0 1+E

nerg

y



2 2el J


2~ 5

2( )jc

gE e GHz

C C=

+


13 /g g gn C V e=

CgBox

Vg


Cj

2 5 GHz4 JJ e R

E π∆= ≈

Vg

1

EJ 4 CE

↓

↑E

nerg

y



2 2el J


2~ 5

2( )jc

gE e GHz

C C=

+


14 /g g gn C V e=

CgBox

Vg


Cj

2 5 GHz4 JJ e R

E π∆= ≈

Vg

1

EJ 4 CE

↓

↑E

nerg

y


Cooper-pair Box Coupled to an SET

Box

SET Vg Vge

Cg Cc CgeVds

Box SET Electrometer

Superconducting tunnel junction

SET TransistorCooper-pair BoxQuantum state readout

orQubit


15

Nonequilibriumnoise source

Quantum spectrum analyzer

What Does SET Measure?

16

effB

effB

ggC Ve1

0

1

1

0 1

0E

ggC Ve

2JE

( )1 / 2xn σ= +Measure box charge

2ˆ ˆ ˆ

2x ze JlH E Eσ σ= −


Excited state

Ground state

2elE a b c

a b c

n

2JE

2JE

x̂

ẑ

a 0 2effB

2elE

b

c


17

Box Gate Charge (e)

Ene

rgy

2

0

Cha

rge

2

0

10

1 2with microwave

Spectroscopy of Box

( )VS ω

ω

Spectrum of oscillator

peaks saturateto q=1e


18

Effects of Voltage Noise on Pseudo-Spin

θ

sinelB E⊥ =δµ δ θcoselB Eδµ δ θ=

slow fluctuations of B dephasing

resonant fluctuations of B⊥ transitions

( )2

201

1 sinboxmix V

mix

e ST

ω θ⎛ ⎞= Γ = ⎜ ⎟⎝ ⎠

01 effBω µ= ( )2

21 0 cosboxV

e ST ϕϕ

ω θ⎛ ⎞= Γ = →⎜ ⎟⎝ ⎠

2el boxeE V=δ δ

effBẑ

2JE

θ

01

sin JEθω

=

2elE x̂

Spontaneous Emission of Cooper-pair Box1 xH AVσ=


19

Box 50 envR ≈ Ω0T =

( )1 2 sin

/ sin

gC g x

g g x

CH E V

ee C C V

θ δ σ

θ δ σΣ

=

=

gC

Vg

01( ) 0envVS ω− =

01 01( ) 2 (50 )envVS ω ω+ = × Ω

Excited-statelifetime, T1

( )2

201

1

21 sinenvV

eS

Tκ ω θ↓= Γ = +⎛ ⎞⎜ ⎟⎝ ⎠

1 0.1 1 sT µ≈ −/ 12%gC Cκ Σ= ∼ estimate:

Polarization = 100%

Cooper-pair Box at Finite Temperature0T >


20

Box 50 envR ≈ Ω

Vg

gC

( )01 01( ) 2 1envVS R nω ω+ = +01 01( ) 2envVS R nω ω− =

( ) ( )( ) ( )

01 01 01

01 01

1 tanh2 1 2

S S n nPS S n kT

ω ω ωω ω+ − − + − ⎡ ⎤= = = ⎢ ⎥+ + − + ⎣ ⎦

( )2

2

1

201

12 sin 2 1e

TR nκ θω↓ ↑= Γ Γ =

⎛ ⎞+ +⎜ ⎟⎝ ⎠

Excited-state Lifetime Measurement of Box

21

Pea

k he

ight

(e)

0 time 10 µs

2ggC V

e

n

0 10.50

1

0.3e

follow peak height after turning

off microwaves

with continuous measurement

K. Lehnert et al., PRL 90,

027002 (2003).

1 1.3 sP ~ 1

T

(@ 76 GHz)

µ=

901

01

( ) 5 10 pairs /

( ) ~ 0box

box

n

n

S Hz

S

ω

ω

−+ ×

−

∼

Coupling of SET Backaction to Box

SET Box


22

Cge-

Cc

2e

VgeenvR

Charge fluctuationson SET island with

( )01QS ω±Environmentrelaxes box

SET couplesbackaction to box

23

Double JQP Process in the SSET

Reflected power from RF-SET ~ Conductance of SSET

qpΓ0 2

0 2→

1 1→ −

2 /cE e

JQP

DJQP

Quantum Shot Noise of DJQP* Process*Double Josephson-quasiparticle cycle: (A. Clerk et al. PRL 89 176804 (2002))

24

Excitationof box

Relaxation of box

0ω =

ω = +∆ω = −∆

( )log VS ω⎡ ⎤⎣ ⎦

2ggC V

e

n

1

0

Predicted box chargeSET noise spectrum

on resonanceoff resonance

SET Determines Relaxation Time (T1)Calculated symmetric noise at 30 GHz

25

low highH. Majer and B. Turek, unpublished

T1 ~ 1 µs

T1 < 100 ns

Theory of quantum noise for DJQPA. Clerk et al., PRL 89 176804 (2002)

What About Asymmetry in SET Noise?Calculated 5 GHz asymmetric noise

Measured reflected power

( ) ( )Q QS Sω ω+ −−

Blue:relaxes

Red:excites

qpΓ0 2 26

Below resonance SET relaxes boxabove resonance SET excites…

Population Inversion

Calculated asymmetric noise at 5 GHz

Measured box charge

negative

positive

SE

T excitesQ

ubit S

ET relaxes

Qubit

SET Gate Charge (e)

SET creates an negative effective spin temperature27

Inversion: Theory and Experiment

28

Numerical calculation including strong coupling to SET + environmental relaxation (A. Clerk)

Other Quantum Spectrometers – Delft Group

Equivalent AC circuit

Deblock, Onac, Gurevich, and Kouwenhoven, Science 301, 203 (2003)

29

SIS detection principleSIS detection principle

High-frequency detection based on Photon Assisted Tunneling

30

Density of States

superconductor

insulator

superconductorhν

VSIS

Voltage biased SIS junction

Numerical simulations

QuasiQuasi--particle shot noiseparticle shot noise

White noise fit

31

SI as fit parameter

Only emission part: SI(ω)=eI

Resolution: 80 fA2/Hz(3mK on a 1 kΩ resistor)

Summary of Lecture 2• Positive frequency noise = reservoir absorbs• Negative frequency noise = reservoir emits

• A quantum system can act as a “quantum spectrometer,”able to measure both positive and negative frequency components of a non-classical noise source.

• Using CPB (= electrical TLS) as quantum spectrometer

• Observed quantum noise (at > GHz) of SET backaction- SET controlling relaxation time of box- SET can create population inversion in box

due to asymmetry of noise


32

Single Quantum Dot as Noise DetectorSingle Quantum Dot as Noise Detector

Device picture

Quantum Dot in CB regime: noise detector

Quantum Point Contact: noise source

Inverse picture: QPC as acharge detector

Detector back-action on the studied quantum system

34

J.M.Elzerman et.al, Nature 430, 431 (2004)

Single Quantum Dot as Noise DetectorSingle Quantum Dot as Noise Detector

35

Transport through orbital states QPC transmission dependence

eVdotδ1

δ2

κ2κ1

κ1= κ2 = 0.0167

κ∗1= κ∗2 = 0.0048

VQPC= 1.27 meVVdot= 30 µeVTemperature=200 mKδ1= 245 meV ~ 60 GHzδ2= 580 meV ~ 140 GHzΓgs = 0.575 GHz

Γ1es = 5.75 GHzΓ2es = 4.025 GHz

Set of fitting parameters

Double Quantum Dot DetectionDouble Quantum Dot Detection

• Non-coherent limit ε »TC ; Γi«ΓL, ΓR

• Iinel = e/ħ (ΓL-1+ Γi-1+ ΓR-1)-1 ≈ e/ħ Γi-1

• P(ε) probability for energy exchange with the enviroment

• Circuit transimpedanceSV(ω) = |Z(ω)|2 SI(ω)|Z(ω)|2 = κ2 RK2

36R. Aguado and L.P. Kouwenhoven, PRL 84, 1986 (2000)

Overview of LecturesOutline of Lecture 2The Electrical Engineer’s Spectrum AnalyzersThe Quantum Mechanic’s Analyzer – a SpinSpin as Spectrum Analyzer - IIInterpretation of Two-Sided SpectrumPolarization of Spin and Noise SpectraPolarization of Spin - IIWays to Characterize a Quantum ReservoirCooper Pair Box as Two-Level SystemCooper Pair Box as Two-Level SystemCooper Pair Box as Two-Level SystemCooper Pair Box as Two-Level SystemCooper-pair Box Coupled to an SETSpectroscopy of BoxDouble JQP Process in the SSETQuantum Shot Noise of DJQP* ProcessSET Determines Relaxation Time (T1)What About Asymmetry in SET Noise?Population InversionInversion: Theory and ExperimentOther Quantum Spectrometers – Delft GroupSIS detection principleQuasi-particle shot noiseSummary of Lecture 2Single Quantum Dot as Noise DetectorSingle Quantum Dot as Noise DetectorDouble Quantum Dot Detection

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Quantum Spectrometers of Electrical Noiseboulderschool.yale.edu/sites/default/files/files/... · 2019. 12. 18. · Outline of Lecture 2 • A spin or two-level system (TLS) as spectrometer