View
2
Download
0
Category
Preview:
Citation preview
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…
Noise and Quantum MeasurementR. Schoelkopf
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…
Noise and Quantum MeasurementR. Schoelkopf
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)
Noise and Quantum MeasurementR. Schoelkopf
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
Noise and Quantum MeasurementR. Schoelkopf
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:
Noise and Quantum MeasurementR. Schoelkopf
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−=
Noise and Quantum MeasurementR. Schoelkopf
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⎛ ⎞= ⎜ ⎟⎝ ⎠
Noise and Quantum MeasurementR. Schoelkopf
9
Quantum Fluctuations of Charge2 0 0 0
0
coth coth2 2 2 2
hQ kTCZ kT kT kT
ω ω ω⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
Noise and Quantum MeasurementR. Schoelkopf
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:
Noise and Quantum MeasurementR. Schoelkopf
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
Noise and Quantum MeasurementR. Schoelkopf
13
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⎡ ⎤⎡ ⎤= + + + −⎣ ⎦⎣ ⎦
Noise and Quantum MeasurementR. Schoelkopf
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).
Noise and Quantum MeasurementR. Schoelkopf
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
Noise and Quantum MeasurementR. Schoelkopf
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
Noise and Quantum MeasurementR. Schoelkopf
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= + + − +∼
Noise and Quantum MeasurementR. Schoelkopf
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⎡ ⎤⎣ ⎦
Noise and Quantum MeasurementR. Schoelkopf
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)
Noise and Quantum MeasurementR. Schoelkopf
21
Experiments On Quantum Johnson NoiseWork by Bernie Yurke et al. at Bell Labs
Josephson parametric amplifier:19 GHz and 30 mK
Noise and Quantum MeasurementR. Schoelkopf
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”
Noise and Quantum MeasurementR. Schoelkopf
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
Noise and Quantum MeasurementR. Schoelkopf
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
Noise and Quantum MeasurementR. Schoelkopf
27
Noise Measurement of a Tunnel Junction
SEM5µ
P
Al-Al2O3-Al Junction
Measure symmetrized noise spectrum at kTω <
Noise and Quantum MeasurementR. Schoelkopf
28
Seeing is Believing
1PP Bδ
τ=
High bandwidth measurements of noise
Noise and Quantum MeasurementR. Schoelkopf
29
410PPδ −=8
,~ 10 zB H τ = 1 second
Test of Nonequilibrium FDT
Noise and Quantum MeasurementR. Schoelkopf
30
Agreement over four decades in temperature
To 4 digits of precision
L. Spietz et al., Science 300, 1929 (2003)
Comparison to Secondary Thermometers
Noise and Quantum MeasurementR. Schoelkopf
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
Noise and Quantum MeasurementR. Schoelkopf
33
Symmetrized Noise: ( ) ( )symS S Sω ω= + + −
Shot noise
Quantum noise
don’t addpowers!
Measurement of Shot Noise Spectrum
Noise and Quantum MeasurementR. Schoelkopf
34
Theory Expt.
Schoelkopf et al., PRL 78, 3370 (1998)
Noise and Quantum MeasurementR. Schoelkopf
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
Noise and Quantum MeasurementR. Schoelkopf
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!
Noise and Quantum MeasurementR. Schoelkopf
37
Recommended