Post on 06-Apr-2020
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,…
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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)
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Fundamental Noise SourcesJohnson-Nyquist Noise
• Frequency-independent• Temperature-dependent• Used for thermometry
4( ) BI
k TS fR
=2A
H z⎡ ⎤⎢ ⎥⎣ ⎦
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• Frequency-independent • Temperature independent
( ) 2IS f eI=Shot Noise
2AH z
⎡ ⎤⎢ ⎥⎣ ⎦
Shot Noise – “Classically”
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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
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Conduction in Tunnel Junctions
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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
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Non-Equilibrium Fluctuation Dissipation Theorem
( ) 2 / coth2I
B
eVS f eV Rk T
⎛ ⎞= ⎜ ⎟
⎝ ⎠
Johnson Noise
2eIShot Noise
4kBTR
Transition RegioneV~kBT
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Noise Measurement of a Tunnel Junction
SEM5µ
P
Al-Al2O3-Al Junction
Measure symmetrized noise spectrum at kTω <
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Seeing is Believing
1PP Bδ
τ=
High bandwidth measurements of noise
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410PPδ −=8
,~ 10 zB H τ = 1 second
Test of Nonequilibrium FDT
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Agreement over four decades in temperature
To 4 digits of precision
L. Spietz et al., Science 300, 1929 (2003)
Self-Calibration Technique
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P(V) = Gain( SIAmp+SI(V,T) )
2 /Bk T e
4 AmplifierI
BG Sk TR
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
+
P(V)
2eI
V
Comparison to Secondary Thermometers
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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
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Finite Frequency Shot Noise
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Symmetrized Noise: ( ) ( )symS S Sω ω= + + −
Shot noise
Quantum noise
don’t addpowers!
Measurement of Shot Noise Spectrum
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Theory Expt.
Schoelkopf et al., PRL 78, 3370 (1998)
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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
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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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Additional material on Johnson noise follows
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Nyquist’s Derivation of Johnson’s Noise
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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
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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
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∼ 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
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