Download Quantum Noise and Measurement

Quantum Noise and Measurement
Rob Schoelkopf
Applied Physics
Yale 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 Measurement
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And God said:
[ a, a ] = 1
†
“Go forth, be fruitful, and multiply (but don’t commute)”
And there was light, and quantum noise…
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Manifestations of Quantum Noise
Well-known:
Spontaneous emission
Casimir effect
Lamb shift
g-2 of electron
Mesoscopic and solid-state examples (less usual?):
Shot noise
Minimum noise temperature of an amplifier
Measurement induced dephasing of qubit
Environmental destruction of Coulomb blockade
Quasiparticle renormalization of SET’s capacitance
…
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Overview of Lectures
Lecture 1: Equilibrium and Non-equilibrium Quantum Noise
in Circuits
Reference: “Quantum Fluctuations in Electrical Circuits,”
M. Devoret Les Houches notes.
Lecture 2: Quantum Spectrometers of Electrical Noise
Reference: “Qubits as Spectrometers of Quantum Noise,”
R. Schoelkopf et al., cond-mat/0210247
Lecture 3: Quantum Limits on Measurement
References: “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.
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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)
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Quantum Circuit Toolbox
L-C Resonator
Cooper-Pair Box
Vg
Single Electron
Transistor
Cg
Cge
Vds
Vge
Harmonic oscillator
Two-level
system
(qubit)
Voltage/Charge
amplifier
Superconductors: quality factor 106 or greater – levels sharp
ω > kT
1 GHz = 50 mK, very few levels populated
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The Electrical Harmonic Oscillator
Z HO
1
iω L
=
=
1/ iω L + iωC 1 − (ω / ω0 )2
t
φ (t ) = LI (t ) = ∫ V (τ )dτ
−∞
ω0 = 1
1 2 1 2
= Cφ − φ
2
2L
LC
L
Z0 =
C
C ⇔ mass
1/ L ⇔ spring constant
Q = Cφ ⇔ momentum
Thermal equilibrium:
Q 2 ~ kTC
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The Quantum Electrical Oscillator
Q2 φ 2
1⎞
⎛ †
H=
+
= ω0 ⎜ a a + ⎟
2C 2 L
2⎠
⎝
“p”
“x”
1
Q=
a − a† )
(
i 2Z 0
φ=
Z0
a + a† )
(
2
†
⎡
⎤⎦ = −i
Q
,
φ
=
−
i
a
,
a
[ ]
⎣
[ Q, H ] ≠ 0
[φ , H ] ≠ 0
Q and φ are not constants of motion!
A(t ) = eiHt / A(0)e − iHt /
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[Q(t ), Q(0)] ≠ 0
[φ (t ),φ (0)] ≠ 0
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Noise of Quantum Oscillator
What about correlation functions of φ and Q ?
e.g. for thermal equilibrium
Z0
φ (t )φ (0) =
2
⎡
⎛ ω0
⎢coth ⎜ 2kT
⎝
⎣
!?
⎤
⎞
⎟ cos (ω0t ) − i sin (ω0t ) ⎥
⎠
⎦
1) Correlator not real, how to define/interpret a spectral density?
2) Non-zero variances even at T=0
Z0
⎛ ω0 ⎞
φ (0)φ (0) =
coth ⎜
⎟
2
⎝ 2kT ⎠
⎛ ω0 ⎞
Q(0)Q(0) =
coth ⎜
⎟
2Z 0
⎝ 2kT ⎠
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Quantum Fluctuations of Charge
Q
2
⎛ ω0
=
coth ⎜
2Z 0
⎝ 2kT
hω0
⎞
⎛ ω0 ⎞
coth ⎜
⎟
⎟ = kTC
2
2
kT
kT
⎠
⎝
⎠
x
⎛ x⎞
coth ⎜ ⎟
2
⎝2⎠
Quantum:
Q
Q 2 / kTC
Thermal:
2
~
ω0
2C
Q 2 ~ kTC
ω0
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kT
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Noise and Spectral Densities Classically
Random variable
V (t )
V (t )
Auto-correlation function
Fourier transform
Spectral density
Since
V (t )
CVV (t − t ′) = V (t )V (t ′)
1
V (ω ) =
T
∫
T /2
−T / 2
t
dt eiωt V (t )
∞
SVV (ω ) = ∫ dt eiωt V (t )V (t ′) = V (ω )V ( −ω )
−∞
is classical and real,
And so:
V (t )V (t ′) = V (t ′)V (t )
SVV (ω ) = SVV ( −ω )
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Spectral Density of Classical Oscillator
for mechanical harmonic oscillator in thermal equil.:
1
mass, m
x ( t ) = x ( 0 ) cos (ω0 t ) + p ( 0 )
sin (ω0 t )
mω0
resonant
freq, ω0
p ( t ) = p ( 0 ) cos (ω0 t ) − x ( 0 ) mω0 2 sin (ω0 t )
0 in equil.
1
sin (ω0 t )
C xx ( t ) = x ( t ) x ( 0 ) = x ( 0 ) x ( 0 ) cos (ω0 t ) + p ( 0 ) x ( 0 )
mω0
1
1
1
2
2
2
k
x
m
x
=
ω0
= kT
equipartition thm:
2
2
2
kT
C xx ( t ) =
cos (ω0 t )
F.T.
2
mω0
kT
⎡δ ω − ω0 ) + δ (ω + ω0 ) ⎤⎦
S xx (ω ) = π
symmetric in ω!
2 ⎣ (
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mω0
position correlation function:
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Spectral Density of Quantum Oscillator - I
xˆ ( t ) = xˆ ( 0 ) cos (ω0 t ) + pˆ ( 0 )
1
sin (ω0 t )
mω0
pˆ ( t ) = pˆ ( 0 ) cos (ω0 t ) − xˆ ( 0 ) mω0 2 sin (ω0 t )
C xx ( t ) = xˆ ( 0 ) xˆ ( 0 ) cos (ω0 t ) + pˆ ( 0 ) xˆ ( 0 )
but because
1
sin (ω0 t )
mω0
⎡⎣ x ( 0 ) , p ( 0 ) ⎤⎦ = i
x (0) p (0) − p (0) x (0) = i
and
p ( 0 ) x ( 0 ) = −i / 2 ≠ 0
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Spectral Density of Quantum Oscillator - II
xˆ = xRMS ( a † + a )
using
with
2
xRMS
= 0 xˆ 2 0 =
2mω0
a † (t ) = eiHt / a † (0)e − iHt / = eiω0t a † (0)
2
C xx ( t ) = xˆ ( t ) xˆ ( 0 ) = xRMS
eiω0t a † (0)a(0) + e − iω0t a(0) a † (0)
(
)
2
C xx ( t ) = xRMS
nBE ( ω0 ) e + iω0t + ⎡⎣ nBE ( ω0 ) + 1⎤⎦ e − iω0t
1
where nBE ( ω0 ) = ω0 kT
e
− 1 is the Bose-Einstein occupation
2
⎡ nBE ( ω0 ) δ (ω + ω0 ) + ⎡⎣ nBE ( ω0 ) + 1⎤⎦ δ (ω − ω0 ) ⎤
S xx (ω ) = 2π xRMS
⎣
⎦
asymmetric in frequency!!
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How to Make a Resistor - 1
Caldeira-Legget prescription:
“Sum infinite number of oscillators to make continuum”
Caldeira and Leggett, Ann. Phys. 149, 374 (1983).
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How to Make a Resistor - 2
Admittance = parallel sum of series resonances
L’s and C’s chosen to give dense comb of frequencies
and the correct value of impedance/admittance
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How to Make a Resistor - 3
Transmission 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
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Quantum Noise of an Impedance
2 ω Re [ Z ]
SV (ω )
ω = kT
T ≠0
T =0
2kT Re [ Z ]
The quantum fluctuationdissipation relation:
2 ω Re [ Z ]
SV (ω ) =
− ω / kT
1− e
ω
0
Three limiting cases:
ω
kT
SV = 2kT Re [ Z ]
ω
kT
SV = 2 ω Re [ Z ]
ω
− kT
SV = 0
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Symmetrized and Antisymmetrized Noise
2 ωR
SV (ω ) =
1 − e − ω / kT
n =
1
e
ω / kT
−1
stim. emission
ω > 0:
SV ( +ω ) = 2 ω R ( n + 1)
spont. emission
ω < 0:
SV ( −ω ) = 2 ω R n
absorption
Symmetrized noise spectrum:
SVS ( ω ) = SV ( +ω ) + SV ( −ω ) ∼ 2 ω ( 2 n + 1) R
SVS ( ω ) = 2 ω R coth ⎡⎣ ω / 2kT ⎤⎦
Anti-Symmetrized noise spectrum:
A
V
S
( ω ) = S ( +ω ) − S ( −ω ) ∼ 2
V
V
ωR
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Callen and Welton,
Phys. Rev.
83, 34 (1951)
dissipation
(T indep.!)19
Symmetrized (One-Sided) Johnson Noise
SV = 2 ω R
SVS / 4kTR
2 ω R coth ⎡⎣ ω / 2kT ⎤⎦
SV = 4kTR
ω0
kT
SVS (ω → 0 )
TRJ =
=T
4kR
SVS (ω → ∞ )
ω
TQ =
=
4kR
2k
“energy per mode = ½ photon”
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Experiments On Quantum Johnson Noise
Method: measure low-freq.
noise of resistively-shunted JJ
10-21
A2/Hz
I = I C sin φ
10-22
Rectified noise from ω = 2eVDC /
1010
w/out zero-point
Frequency
1012
JJ “mixes down” noise from THz frequencies to audio
Koch, van Harlingen, and Clarke, PRL 47, 1216 (1981)
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Inferred Johnson noise
With zero-point
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Experiments On Quantum Johnson Noise
Work by Bernie Yurke et al. at Bell Labs
Josephson parametric amplifier:
19 GHz and 30 mK
observed zero-point part
of waveguide’s noise
(and then squeezed it!)
Noise power
4.5
Tamp = 0.45K = hν/2k
quantum limited amplifier!
fit to coth!
2
0
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Temperature
Movshovich et al.,
PRL 65, 1419 (1990)
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1K
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Shot Noise – “Classically”
what’s up
here?
D
n
Incident “current”
of particles
I ∼ qDn
Poisson-distributed
fluctuations
Barrier w/ finite
trans. probability
“white” noise with
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S I = 2qI
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Shot Noise is Quantum Noise
Einstein, 1909: Energy fluctuations of thermal radiation
“Zur gegenwartigen Stand des Strahlungsproblems,” Phys. Zs. 10 185 (1909)
⎡
π 2c3 2 ⎤
( ∆E ) = ⎢ ωρ (ω ) + 2 ρ (ω ) ⎥ Vdω
ω
⎣
⎦
particle term = shot noise!
wave term
first appearance of wave-particle complementarity?
2
Can show that “particle term” is a consequence of ⎡⎣ a, a † ⎤⎦ = 1
(see Milloni, “The Quantum Vacuum,” Academic Press, 1994)
∆n
2
= a aa a − a a
†
†
†
n = n = (e
2
= a (a a + 1)a − a a
†
†
†
∆n 2 = n 2 + n
†
Pn = n n / ( n + 1)
2
= a a aa + a a − a a
† †
†
ω / kT
2
− 1)
−1
n +1
a † a † aa = ∑ n(n − 1) Pn = 2n 2
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Conduction in Tunnel Junctions
I
V
I L→ R
I R→L
G
= ∫ f L (1 − f R )dE
e
G
= ∫ f R (1 − f L )dE
e
Difference gives current:
I = I L → R − I R → L = GV
Fermi functions
Assume: Tunneling amplitudes and
D.O.S. independent of energy Conductance (G)
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is constant 25
Fermi distribution
of
electrons
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Non-Equilibrium Noise of a Tunnel Junction
(Zero-frequency limit)
Sum gives noise:
S I ( f ) = 2e ( I L → R + I R → L )
⎛ eV ⎞
S I ( f ) = 2eI coth ⎜
⎟
⎝ 2 k BT ⎠
I =V / R
*D. Rogovin and D.J. Scalpino, Ann Phys. 86,1 (1974)
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Non-Equilibrium Fluctuation Dissipation Theorem
2eI
Shot Noise
Transition Region
eV~kBT
4kBTJohnson Noise
R
⎛ eV ⎞
S I ( f ) = 2eV / R coth ⎜
⎟
⎝ 2 k BT ⎠
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Noise Measurement of a Tunnel Junction
P
5µ
SEM
Al-Al2O3-Al Junction
Measure symmetrized noise spectrum at
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ω < kT
28
Seeing is Believing
δP= 1
P
Bτ
High bandwidth measurements of noise
8
B ~ 10 Hz , τ = 1 second
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δ P =10−4
P
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Test of Nonequilibrium FDT
Agreement over four decades in temperature
To 4 digits of precision
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Spietz et al., Science
R.
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300, 1929 (2003)
Comparison to Secondary Thermometers
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Two-sided Shot Noise Spectrum
(Quantum, non-equilibrium FDT)
ω + eV ) / R ( ω − eV ) / R
(
S I (ω ) =
+
−( ω + eV ) / kT
−( ω − eV ) / kT
1− e
1− e
S I (ω )
V
eV /
2 ω/R
eI
T =0
−eV /
ω =0
ω
Aguado & Kouwenhoven, PRL 84, 1986 (2000).
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Finite Frequency Shot Noise
Symmetrized Noise:
S sym = S ( +ω ) + S ( −ω )
don’t add
powers!
Shot noise
Quantum noise
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Measurement of Shot Noise Spectrum
Theory
Expt.
Schoelkopf et al., PRL 78, 3370 (1998)
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Shot Noise at 10 mK and 450 MHz
hν / kT = 2
L. Spietz, in prep.
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With An Ideal Amplifier and T=0
SI
2eI
S I = 2hν G
Quantum noise
from source
S I = 2hν G
Quantum noise
added by amplifier
eV=-hν
eV=hν
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Summary – Lecture 1
• Quantizing an oscillator leads to quantum fluctuations
present 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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