quantum noise and measurement · 2019. 12. 18. · experiments on quantum johnson noise work by...

Quantum Noise and Measurement

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

And God said:

†[ , ] 1a a =

“Go forth, be fruitful, and multiply (but don’t commute)”

And there was light, and quantum noise…


2

Manifestations of Quantum Noise

Spontaneous emissionCasimir effectLamb shiftg-2 of electron

Well-known:

Mesoscopic and solid-state examples (less usual?):Shot noiseMinimum noise temperature of an amplifierMeasurement induced dephasing of qubitEnvironmental destruction of Coulomb blockadeQuasiparticle renormalization of SET’s capacitance…


3

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, & RS.Noise and Quantum Measurement

R. Schoelkopf4

Outline of Lecture 1

• Quantum circuit intro and toolbox

• Electrical quantum noise of a harmonic oscillator (L-C)

• How to make a quantum resistor (= the vacuum!)

• Noise of a resistor:the quantum Fluctuation-Dissipation Theorem (FDT)

• Experiments on the zero point noise in circuits

• Shot noise and the nonequilibrium FDT (time permitting)


5

Quantum Circuit ToolboxSingle Electron

TransistorL-C Resonator Cooper-Pair Box

Vg

Vge

Vds

Cg Cge

Two-levelsystem (qubit)

Voltage/Chargeamplifier

Harmonic oscillator

Superconductors: quality factor 106 or greater – levels sharp

kTω > 1 GHz = 50 mK, very few levels populated


6

The Electrical Harmonic Oscillator

( )20

11/ 1 /HO

i LZi L i C

ωω ω ω ω

= =+ −

( ) ( ) ( )t

t LI t V dφ τ τ−∞

= = ∫2 21 1

2 2C

Lφ φ= −

0 1 LCω =

massC ⇔ 1/ L ⇔ spring constant0

LZC

=Q Cφ= ⇔ momentum

2 ~Q kTCThermal equilibrium:


7

The Quantum Electrical Oscillator2 2

†0

12 2 2QH a aC L

φ ω ⎛ ⎞= + = +⎜ ⎟⎝ ⎠

“p” “x”

( )†

0

12

Q a ai Z

= − ( )†0

2Z a aφ = +

[ ] †, ,Q i a a iφ ⎡ ⎤= − = −⎣ ⎦

[ ][ ]

, 0

, 0

Q H

Hφ

≠

≠Q and φ are not constants of motion!

[ ][ ]

( ), (0) 0

( ), (0) 0

Q t Q

tφ φ

≠

≠/ /( ) (0)iHt iHtA t e A e−=


8

Noise of Quantum OscillatorWhat about correlation functions of φ and Q ?

e.g. for thermal equilibrium

( ) ( )0 00 0( ) (0) coth cos sin

2 2Zt t i t

kTω

φ φ ω ω⎡ ⎤⎛ ⎞= −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

!?

1) Correlator not real, how to define/interpret a spectral density?

2) Non-zero variances even at T=0

0 0(0) (0) coth2 2Z

kTωφ φ ⎛ ⎞= ⎜ ⎟

⎝ ⎠

0

0

(0) (0) coth2 2

Q QZ kT

ω⎛ ⎞= ⎜ ⎟⎝ ⎠


9

Quantum Fluctuations of Charge2 0 0 0

0

coth coth2 2 2 2

hQ kTCZ kT kT kT

ω ω ω⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠


10

2 /Q kTC

0

kTω

2 ~Q kTC

2 0~2

QCω

Thermal:

Quantum:coth

2 2x x⎛ ⎞

⎜ ⎟⎝ ⎠

Noise and Spectral Densities Classically

t

( )V t

( )V t

Auto-correlation function

Random variable

( ) ( ) ( )VVC t t V t V t′ ′− =

/ 2

/ 2

1( ) ( )T i t

TV dt e V t

Tωω

−= ∫Fourier transform

( ) ( ) ( ) ( ) ( )i tVVS dt e V t V t V Vωω ω ω

∞

−∞′= = −∫Spectral density

( )V t ( ) ( ) ( ) ( )V t V t V t V t′ ′=Since is classical and real,

( ) ( )VV VVS Sω ω= −And so:Noise and Quantum Measurement

R. Schoelkopf11

Spectral Density of Classical Oscillatorfor mechanical harmonic oscillator in thermal equil.:

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

0 00

20 0 0

10 cos 0 sin

0 cos 0 sin

x t x t p tm

p t p t x m t

ω ωω

ω ω ω

= +

= −

mass, mresonant

freq, 0ω

position correlation function:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )0 00

10 0 0 cos 0 0 sinxxC t x t x x x t p x tm

ω ωω

= = +

0 in equil.

2 2 20

1 1 12 2 2

k x m x kTω= =equipartition thm:


12

( ) ( )020

cosxxkTC t t

mω

ω=

( ) ( ) ( )0 020

xxkTS

mω π δ ω ω δ ω ω

ω⎡ ⎤= − + +⎣ ⎦

F.T.

symmetric in ω!

Spectral Density of Quantum Oscillator - I( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

0 00

20 0 0

1ˆ ˆ ˆ0 cos 0 sin

ˆ ˆ ˆ0 cos 0 sin

x t x t p tm

p t p t x m t

ω ωω

ω ω ω

= +

= −

( ) ( ) ( ) ( ) ( ) ( ) ( )0 00

1ˆ ˆ ˆ ˆ0 0 cos 0 0 sinxxC t x x t p x tm

ω ωω

= +

( ) ( )0 , 0x p i⎡ ⎤ =⎣ ⎦but because

( ) ( ) ( ) ( )0 0 0 0x p p x i− =

( ) ( )0 0 / 2 0p x i= − ≠and


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Spectral Density of Quantum Oscillator - II

( )†ˆ RMSx x a a= +using

2 2

0

ˆ0 02RMSx xmω

= =with

0† / † / †( ) (0) (0)i tiHt iHta t e a e e aω−= =

( ) ( ) ( ) 0 02 † †ˆ ˆ 0 (0) (0) (0) (0)i t i txx RMSC t x t x x e a a e a aω ω−= = +

( ) ( ) ( )( )0 020 0 1i t i t

xx RMS BE BEC t x n e n eω ωω ω+ −⎡ ⎤= + +⎣ ⎦

( )001

1BE kTne ωω =

−where is the Bose-Einstein occupation

( ) ( ) ( ) ( ) ( )20 0 0 02 1xx RMS BE BES x n nω π ω δ ω ω ω δ ω ω⎡ ⎤⎡ ⎤= + + + −⎣ ⎦⎣ ⎦


14asymmetric in frequency!!

How to Make a Resistor - 1Caldeira-Legget prescription:

“Sum infinite number of oscillators to make continuum”

Caldeira and Leggett, Ann. Phys. 149, 374 (1983).


15

How to Make a Resistor - 2Admittance = parallel sum of series resonances

L’s and C’s chosen to give dense comb of frequenciesand the correct value of impedance/admittance


16

How to Make a Resistor - 3Transmission line = infinite LC ladder

L and C are constants (all same) =to the inductance and capacitance per unit length,

calculated from electro/magneto-statics of the particular transmission line

the “vacuum”or a perfect blackbody!

Line needs to be infinite – no reflections/memory and infinite number of d.o.f. to make reservoir


17

0T ≠0T =

Quantum Noise of an Impedance

kTω =

( )VS ω

0 ω

[ ]2 RekT Z

[ ]2 Re Zω

The quantum fluctuation-dissipation relation:

Three limiting cases:

[ ]2 ReVS kT Z=kTω

( ) [ ]/

2 Re1V kT

ZS

e ω

ωω −=

−[ ]2 ReVS Zω=kTω

0VS =kTω −Noise and Quantum Measurement

R. Schoelkopf18

Symmetrized and Antisymmetrized Noise

( ) /

21V kT

RSe ω

ωω −=−

0 :ω >

/

11kTn

e ω=−

( ) ( )2 R 1VS nω ω=+ +stim. emission

spont. emission

( ) 2 RVS nω ω=−0 :ω < absorption

Symmetrized noise spectrum:

( ) ( ) ( ) ( )2 2 1 RSV V VS S S nω ω ω ω= + + − +∼


19

Callen and Welton,Phys. Rev.

83, 34 (1951)

( ) 2 R coth / 2SVS kTω ω ω= ⎡ ⎤⎣ ⎦

Anti-Symmetrized noise spectrum:

( ) ( ) ( ) 2 RAV V VS S Sω ω ω ω= + − − ∼ dissipation

(T indep.!)

Symmetrized (One-Sided) Johnson Noise

0

kTω

/ 4SVS kTR

4VS kTR=

2VS Rω=

( )04

SV

RJS

T TkRω →

= =( )4 2

SV

QS

TkR k

ω ω→∞= =

“energy per mode = ½ photon”

2 R coth / 2kTω ω⎡ ⎤⎣ ⎦


20

Experiments On Quantum Johnson Noise

Method: measure low-freq.noise of resistively-shunted JJ

sinCI I φ=

Rectified noise from 2 /DCeVω =

With zero-point

w/out zero-point

Frequency1010 1012

Infe

rred

Joh

nson

noi

se

10-22

10-21

A2 /H

z

JJ “mixes down” noise from THz frequencies to audioKoch, van Harlingen, and Clarke, PRL 47, 1216 (1981)


21

Experiments On Quantum Johnson NoiseWork by Bernie Yurke et al. at Bell Labs

Josephson parametric amplifier:19 GHz and 30 mK


22

observed zero-point part of waveguide’s noise

(and then squeezed it!)

Movshovich et al., PRL 65, 1419 (1990)

Tamp = 0.45K = hν/2kquantum limited amplifier!

Noi

se p

ower

Temperature0 1K

fit to coth!

2

4.5

Shot Noise – “Classically”


23

D

what’s up here?

n I qDn∼

Poisson-distributedfluctuations

Incident “current”of particles

Barrier w/ finitetrans. probability

“white” noise with

2IS qI=

Shot Noise is Quantum NoiseEinstein, 1909: Energy fluctuations of thermal radiation

“Zur gegenwartigen Stand des Strahlungsproblems,” Phys. Zs. 10 185 (1909)

( )2 3

2 22( ) ( )cE Vdπωρ ω ρ ω ω

ω⎡ ⎤

∆ = +⎢ ⎥⎣ ⎦

particle term = shot noise! wave termfirst appearance of wave-particle complementarity?

†, 1a a⎡ ⎤ =⎣ ⎦Can show that “particle term” is a consequence of (see Milloni, “The Quantum Vacuum,” Academic Press, 1994)

( ) 1/ 1kTn n e ω −= = −

( ) 1/ 1 nnnP n n += +

22 † † †n a aa a a a∆ = −2† † †( 1)a a a a a a= + −

2† † † †a a aa a a a a= + − † † 2( 1) 2na a aa n n P n= − =∑2 2n n n∆ = + Noise and Quantum Measurement

R. Schoelkopf24

Conduction in Tunnel Junctions


25

Assume: Tunneling amplitudes and D.O.S. independent of energyFermi distribution of electrons

V

I

(1 )

(1 )

L R L R

R L R L

GI f f dEeGI f f dEe

→

→

= −

= −

∫

∫

L R R LI I I GV→ →= − =Difference gives current:

Conductance (G)is constant

Fermi functions

Non-Equilibrium Noise of a Tunnel Junction

Sum gives noise:

( ) 2 coth2I

B

eVS f eIk T

⎛ ⎞= ⎜ ⎟

⎝ ⎠

( ) 2 ( )I L R R LS f e I I→ →= +

/I V R=

(Zero-frequency limit)

*D. Rogovin and D.J. Scalpino, Ann Phys. 86,1 (1974)Noise and Quantum Measurement

R. Schoelkopf26

Non-Equilibrium Fluctuation Dissipation Theorem

( ) 2 / coth2I

B

eVS f eV Rk T

⎛ ⎞= ⎜ ⎟

⎝ ⎠

Johnson Noise

2eIShot Noise

4kBTR

Transition RegioneV~kBT


27

Noise Measurement of a Tunnel Junction

SEM5µ

P

Al-Al2O3-Al Junction

Measure symmetrized noise spectrum at kTω <


28

Seeing is Believing

1PP Bδ

τ=

High bandwidth measurements of noise


29

410PPδ −=8

,~ 10 zB H τ = 1 second

Test of Nonequilibrium FDT


30

Agreement over four decades in temperature

To 4 digits of precision

L. Spietz et al., Science 300, 1929 (2003)

Comparison to Secondary Thermometers


31

Two-sided Shot Noise Spectrum(Quantum, non-equilibrium FDT)

( ) ( )( )

( )( )/ /

/ R / R1 1I eV kT eV kT

eV eVS

e eω ω

ω ωω − + − −=

+ −+

− −

ω0ω =

( )IS ω

/eV

/eV−

V 2 / Rω

eI0T =

Aguado & Kouwenhoven, PRL 84, 1986 (2000). Noise and Quantum Measurement

R. Schoelkopf32

Finite Frequency Shot Noise


33

Symmetrized Noise: ( ) ( )symS S Sω ω= + + −

Shot noise

Quantum noise

don’t addpowers!

Measurement of Shot Noise Spectrum


34

Theory Expt.

Schoelkopf et al., PRL 78, 3370 (1998)


35

Shot Noise at 10 mK and 450 MHz

/ 2h kTν =

L. Spietz, in prep.

With An Ideal Amplifier and T=0

VeV=hνeV=-hν

2eIIS

2IS h Gν=

2IS h Gν=

Quantum noisefrom source

Quantum noiseadded by amplifier


36

Summary – Lecture 1

• Quantizing an oscillator leads to quantum fluctuationspresent even at zero temperature.

• This noise has built in correlations that make it very different from any type of classical fluctuations, and these cannot be represented by a traditional spectral density- requires a “two-sided” spectral density.

• Quantum systems coupled to a non-classical noise source can distinguish classical and quantum noise, and allow us to measure the full density – next lecture!


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quantum noise and measurement · 2019. 12. 18. · experiments on quantum johnson noise work by...

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