Shot Noise and the Non-Equilibrium FDT · 2019-12-18 · • Shot noise is quantum noise • Shot...

Shot Noise and the Non-Equilibrium FDT

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

Outline

• Shot noise is quantum noise

• Shot noise of a tunnel junction

• Measurements of shot noise – testing the non-eq. FDT

• “Quantum shot noise” – measuring the frequency dependence of shot noise

• Experiments on the zero point noise in circuits

• Shot noise and the nonequilibrium FDT (time permitting)


2

Fundamental Noise SourcesJohnson-Nyquist Noise

• Frequency-independent• Temperature-dependent• Used for thermometry

4( ) BI

k TS fR

=2A

H z⎡ ⎤⎢ ⎥⎣ ⎦

3

• Frequency-independent • Temperature independent

( ) 2IS f eI=Shot Noise

2AH z

⎡ ⎤⎢ ⎥⎣ ⎦

Shot Noise – “Classically”


4

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. Schoelkopf5

Conduction in Tunnel Junctions


6

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. Schoelkopf7

Non-Equilibrium Fluctuation Dissipation Theorem

( ) 2 / coth2I

B

eVS f eV Rk T

⎛ ⎞= ⎜ ⎟

⎝ ⎠

Johnson Noise

2eIShot Noise

4kBTR

Transition RegioneV~kBT


8

Noise Measurement of a Tunnel Junction

SEM5µ

P

Al-Al2O3-Al Junction

Measure symmetrized noise spectrum at kTω <


9

Seeing is Believing

1PP Bδ

τ=

High bandwidth measurements of noise


10

410PPδ −=8

,~ 10 zB H τ = 1 second

Test of Nonequilibrium FDT


11

Agreement over four decades in temperature

To 4 digits of precision

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

Self-Calibration Technique

12

P(V) = Gain( SIAmp+SI(V,T) )

2 /Bk T e

4 AmplifierI

BG Sk TR

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

+

P(V)

2eI

V

Comparison to Secondary Thermometers


13

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. Schoelkopf14

Finite Frequency Shot Noise


15

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

Shot noise

Quantum noise

don’t addpowers!

Measurement of Shot Noise Spectrum


16

Theory Expt.

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


17

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


18

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!


19

Additional material on Johnson noise follows


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Nyquist’s Derivation of Johnson’s Noise


21

Connection Between Johnson Noise and Blackbody Radiation*

*R. H. Dicke, Rev. of Sci. Instrum. 17, 268 (1946)

AntennaCoaxial-Line

Resistor Temperature TR

Enclosure Walls Black Temperature Tbb


22

Johnson Noise Power ≡ Blackbody Radiation PowerTR = Tbb

Connection Between Johnson Noise and Blackbody Radiation in Rayleigh-Jeans Limit

Antenna

Resistor Temperature TR

Coaxial-Line

νT h /kR BEnclosure Walls Black

4 /I BS k T R= Temperature TbbRayleigh-Jeans limit

∼ TP k Bbb-Rayleigh- eans B bbJNoise and Quantum Measurement

R. Schoelkopf23

∼ TP k BJohnson RB

Radiative Cooling of a Resistor?

P = kTB T=0T > 0

maxkTkTB kTh

= ×Total radiated power:

Total conductance:2

/photonkG dP dT Th

= =

One quantum of thermal conductance per electromagnetic mode

More correct:/

0 0

( )1h kT

hP h n d de ν

νν ν ν ν∞ ∞

= =−∫ ∫

Schmidt, Cleland, and Schoelkopf, PRL 93, 045901 (2004)


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Resistor as Ideal Square Law DetectorEγ

~ / Hn E kTγ γ

eC

T=0

/H eT E C=Single photon heats one resistor to

If no other thermal conductances, cools entirely by radiation!

~ / Hn E kTγ γPhoton number gain is large!:

Where’s the nonlinearity?Noise and Quantum Measurement

R. Schoelkopf25

Shot Noise and the Non-Equilibrium FDT · 2019-12-18 · • Shot noise is quantum noise • Shot...

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