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QUANTUM-NOISE-051
Quantum fluctuations in meso- and macro- systems Yoseph Imry
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I. Noise in the Quantum and Nonequilibrium Realm, What is Measured? Quantum Amplifier Noise. work with: Uri Gavish, Weizmann (ENS)
Yehoshua Levinson, Weizmann B. Yurke, Lucent Thanks: E. Conforti, C. Glattli, M. Heiblum, R. de Picciotto, M. Reznikov, U. Sivan
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II. Sensitivity of Quantum Fluctuations to the volume:
Casimir Effect
Y. Imry, Weizmann Inst.
Thanks: M. Aizenman, A. Aharony, O. Entin, U. Gavish Y. Levinson, M. Milgrom, S. Rubin, A. Schwimmer, A. Stern, Z. Vager, W. Kohn.
QUANTUM-NOISE-054
Quantum, zero-point fluctuations
Nothing comes out of a ground state system, but:
Renormalization, Lamb shift,
Casimir force, etc.
No dephasing by zero-point fluctuations!
How to observe the quantum-noise?
(Must “tickle” the system).
QUANTUM-NOISE-055
Outline:• Quantum noise, Physics of Power
Spectrum, dependence on full state of system
• Fluctuation-Dissipation Theorem, in steady state
• Application: Heisenberg Constraints on Quantum Amps’
•Casimir Forces.
QUANTUM-NOISE-056
Direct observation of a fractional charge (also in
Saclay).R. de-Picciotto, M. Reznikov, M.
Heiblum, V. Umansky, G. Bunin & D. Mahalu
Nature 1997 (and 1999 for 1/5)
QUANTUM-NOISE-057
A recent motivation How can we observe fractional charge (FQHE,
superconductors) if current is collected in normal leads?
Do we really measure current fluctuations in normal leads?
ANSWER: NO!!!
SOMETHING ELSE IS MEASURED.
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Second Motivation
Breakdown of FLT in glassy,
“aging”, systems:
Can we salvage the proper FLT?
(not a stationary system)
Needs Work, but…
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Understanding The Physics of
Noise-Correlators, and relationship
to DISSIPATION:
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Classical measurement of time-dependent quantity, x(t), in a stationary state.
x(t)
t
C(t’-t)=<x(t) x(t’)>
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Classical measurement of a time-dependent quantity, x(t), in a stationary state.
x(t)
t
C(t’-t)=<x(t) x(t’)>
Quantum measurement of the expectation value, <xop(t)>, in a stationary state.
<x(t)>
t
C(t)=?
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The crux of the matter:
From Landau and Lifshitz,Statistical Physics, ’59(translated by Peierls and Peierls).
------
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Van Hove (1954), EXACT:
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Emission = S(ω) ≠ S(-ω) = Absorption,(in general)
From field with Nω photons, net absorption
(Lesovik-Loosen, Gavish et al):
Nω S(-ω) - (Nω + 1) S(ω)
For classical field (Nω >>> 1):
CONDUCTANCE [ S(-ω) - S(ω)] / ω
QUANTUM-NOISE-0517
This is the Kubo formula (cf AA ’82)!
Fluctuation-Dissipation Theorem (FDT)
Valid in a nonequilibrium steady state!!
Dynamical conductance - response to “tickling”ac field, (on top of whatever nonequilibrium state).
Given by S(-ω) - S(ω) = F.T. of the commutator of the temporal current correlator
QUANTUM-NOISE-0518
Nonequilibrium FDT
• Need just a STEADY STATE SYSTEM: Density-matrix diagonal in the energy representation.
“States |i> with probabilities Pi , no coherencies”
• Pi -- not necessarily thermal, T does not appear in this
version of the FDT (only ω)!
QUANTUM-NOISE-0519
Landauer: 2-terminal conductance = transmission
G I/V = (e2/πħ) |t|2 , with spin. eV μ1- μ2
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Equilibrium Noise in the Landauer Picture
| jll |2 = | jll |2 =(evT )2 ; | jlr |2 = | jrl |2 =(ev T(1-T) )2
Since T(1-T) + T 2 = T, from van Hove-type
expression for S () :
• Temp = 0: S () G , ( < 0 only)
• Temp >> ħ: S () G ·Temp.
(Nyquist!)
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Quantum Shot-Noise (Khlus, Lesovik)
For Fermi–Sea Conductors, different for BEAMS in Vacuum, for same current.
Left-coming Scattering state
|<lk| j |rk’>| 2 = vF2 TR, for (k- k’ << 1/L)
→ S(ω) = 2e(e2V/πħ) T(1-T), ω <<V
= 0, ω >V . This is Excess Noise.
μ
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Exp confirmation, of T(1-T) Reznikov et al, WIS, 1997
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Current Noise Measurment in Quantum Point Contact
Shot noise measurment.
Pauli blocking effect between all particles.
Charge 1/3 detection (Weizmann – Saclay).
Interaction effects.
Shot noise measurment.
Pauli blocking effect between the particles in the current only.
Charge 1/3 detection: impossible.
Interaction effects.
Is the current noise identical to a beam in vacuum?
Answer: NO.The Pauli principle blocks more transitions in the point-contact, so a different noise is emitted. By changing the occupancy at the sink (with a gate), this difference can be manipulated and the radiation spectrum can be
controlled.
Current Noise Measurment in Beams in Vacuum.
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U eV eV+U
U<eVSeV _____ Su _____ _____ _____
U>eV
eV U eV+U
U larger
SeV _____ Su _____ _____
U larger
U = gate potentialon RHS lead.
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Partial Conclusions
• The noise power is the ability of the system to emit/absorb (depending on sign of ω).
FDT: NET absorption from classical field. (Valid also in steady nonequilibrium States)• Nothing is emitted from a T = 0 sample, but it may absorb…• Noise power depends on final state filling.• Exp confirmation: deBlock et al, Science 2003, (TLS with SIS detector).
QUANTUM-NOISE-0526
A recent motivation How can we observe fractional charge (FQHE, superconductors) if current is collected in normal
leads?
Do we really measure current fluctuations in normal leads?
ANSWER: NO!!!
THE EM FIELDS ARE MEASURED.
(i.e. the radiation produced by I(t)!)
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Important Topic:
Fundamental Limitations
Imposed by the Heisenberg Principle on Noise and Back-Action in Nanoscopic
Transistors.
Will use our generalized FDT for this!
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Typical experimental setup:
VDC
Sample LC Filter
Amplifier
External voltage sources (pump, idler, FET bias,…)
Spectrum analyzer
DC Voltage
Display
Full Noise Measurement Chain
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Linear Amplifier:
But then
Heisenberg principle is violated.
A Linear Amplifier Must Add Noise (E.g., C.M. Caves, 1979)
Amplifier Output
Xa , Pa
Detector
xs , ps
Input (“signal”)
22 ],[ ],[
22
Gpx
G
ipxiPX ssssaa
sasa GpPGxX , 1 , GGpPGxX sasa
Amplifier Output
Xa , Pa
Detector
xs , ps
Input (“signal”)
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1 , GGpPGxX sasa
Linear Amplifier:
But then
Heisenberg principle is violated.
A Linear Amplifier does not exist !
A Linear Amplifier Must Add Noise (E.g., C.M. Caves)
Amplifier Output
Xa , Pa
Detector
xs , ps
Input (“signal”)
22 ],[ ],[
22
Gpx
G
ipxiPX ssssaa
sasa GpPGxX ,
QUANTUM-NOISE-0531
In order to keep the linear input-output relation, with a large gain, the amplifier must add noise
A Linear Amplifier Must Add Noise (E.g., C.M. Caves, 1979)
Amplifier Output
Xa , Pa
Detector
xs , ps
Input (“signal”)
, NsaNsa PGpPXGxX , NsaNsa PGpPXGxX
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In order to keep the linear input-output relation, with a large gain, the amplifier must add noise
choose
then
A Linear Amplifier Must Add Noise (E.g., C.M. Caves, 1979)
Amplifier Output
Xa , Pa
Detector
xs , ps
Input (“signal”)
, NsaNsa PGpPXGxX
stateamplifier on theact , )1(- ],[ 2NNNN PXiGPX
)1(-],[],[],[ 22 iiGiGpxPXPX ssNNaa
, NsaNsa PGpPXGxX
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Cosine and sine components of any currentFiltered with window-width
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For phase insensitive linear amp:
gL and gS are load and signal conductances (matched to those of the amplifier). G2 = power gain.
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Average noise-power delivered to the load
(one-half in one direction)
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Our Generalized Kubo:
where g is the differential conductance, leads to:
,
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From our Kubo-based commutation rules:
Hence:
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This generalizes results on photonic amps, where the current
commutators are c-numbers.
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A molecular or a mesoscopic amplifier
Resonant barrier coupled capacitively to an input signal
Is()
Ia()= I0()+G Is()
Cs Ls
B
input siganl
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A molecular or a mesoscopic amplifier
Ia()= I0() +G Is() + IN() ?
Is I0() enough to supply the necessary noise?
G Is()
IN() 2
This question is important for molecular or a mesoscopic amplifier because of two specific characteristics:
1. There is a current flowing even without coupling to the signal.
2. The amplified signal is proportional to the coupling (unlike most other quantum amplifiers)
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Constraint on this amplifier:
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General Conclusion: one should try and keep the ratio between old shot-noise and the amplified signal constant, and not much smaller than unity.
In this way the new shot-noise, the one that appears due to the coupling with the signal, will be of the same order of the old shot-noise and the amplified signal and not much larger.
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Amp noise summary
• Mesoscopic or molecular linear amplifiers must add noise to the signal to comply with Heisenberg principle.
• This noise is due to the original shot-noise, that is, before coupling to the signal, and the new one arising due to this coupling.
• Full analysis shows how to optimize these noises.
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Noise Conclusions
• Understand meaning of correlators, power-spectrum.• Derived generalization of FDT to steady-states.• Generalized FDT used to get constraints on amps’.• New constraints on mesoscopic transistor-type amp.
• Amplification process gives inherent noise• Since power, not accumulated charge, is measured →
can get fractional charges in spite of leads!
),( ),( , VSGVS excessexcessM
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The Casimir EffectThe attractive force between two surfaces in a vacuum - first predicted by Hendrik Casimir over 50 years ago - could affect everything from micromachines to unified theories of nature.(from Lambrecht, Physics Web, 2002)
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Accoustic Casimir Force:• Modes of drum
membrane depend on where 2 weights are placed (also for 1 weight).
• From dependence of g.s. energy on weights’ positions force between two weights.
• Same with weights on a tight string…
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Buks and Roukes, Nature 2002(Effect relavant to micromechanical devices)
From:
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Why interesting?
• (Changes of) HUGE vacuum energy—relevant
• Intermolecular forces, electrolytes.• Changes of Newtonian gravitation at
submicron scales? Due to high dimensions.• Cosmological constant.• “Vacuum friction”; Dynamic effect (Unruh).• “Stiction” of nanomechanical devices…• Artificial phases, soft C-M Physics.
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Casimir’s attractive force between conducting plates
↑(c) = Soft cutoff at p
E’0(d) = E0(d) - E0()
i)
ii)
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Subtracted quantity is radiation pressure of the vacuum outside, What is it?
D(ω) is photon DOS
D(ω) extensive and >0
↓
P0 is NEGATIVE!!!
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Kinetic theory, momentum delivered to the wall/unit time:
For every photon, momentum/unit time =
-E/V, same for many photons.
Milonni et al PRA (88): same order of magnitude, but P0 > 0 !
Why kinetics and thermodynamics don’t agree (for the whole system)?
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Kinetic calculation misses added states (below cutoff) with
increasing V!
Increasing V
cutoff1
Allowed k’s
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Effect of dielectric on one side“Macroscopic Casimir Effect”
()=1
F
P0()P0(1)
With ():
() > 1
|P0()| Larger than for =1!
Further possibilities
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Effect of dielectric outside on the“Mesoscopic Casimir Effect”
With ():
Will change the sign of the Casimir Force at large enough separations,Depending on ()!Interesting in static limit: d << c/p
()()
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Quasistationary (E. Lifshitz, 56) regime
Length scale d << c / p – no retardation
Can use electrostatics (van Kampen et al, 68)
Casimir force becomes (Lifshitz (56), no c!):
- ~ħp /d3
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Vacuum pressure on thin metal film
d
Quasistationary: d<<c/ωp
Surface plasmons on the two
edges
Even-odd combinations:
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Dispersion of thin-film plasmons ω/ωp
1 2 3 4
0.2
0.4
0.6
0.8
1
kd
For d<<c/ωp, light-line
ω=ck
is very steep-full
EM effects don’t
Matter-- quasi
stationary appr.Note: opposite dependence of 2 branches on d
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Casimir pressure on the film, from derivative of total zero-pt plasmon energy:
Large positive pressures on very thin metallic films, approaching eV/A3 scales for atomic thicknesses (Thin-film tech).
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Conclusions, Casimir part
• EM Vacuum pressure is negative, unlike kinetic calculation result. It is the Physical subtraction in Casimir’s calculation. Depends on properties of surface! MACROSCOPIC CASIMIR FORCE.
• Effects due to dielectrics in both macro- and meso- regimes. Some sign control.
• Large positive vacuum pressure due to surface plasmons, on thin metallic films.
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END, Thanks for attention!
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