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Quantum fluctuations and the Casimir Effect in meso- and macro- systems Yoseph Imry. - PowerPoint PPT Presentation
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QUANTUM-NOISE-051
Quantum fluctuations and the Casimir Effect in meso- and macro- systems
Yoseph Imry
QUANTUM-NOISE-052
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
-----------------------------
QUANTUM-NOISE-053
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
Understanding The Physics of
Noise-Correlators, and relationship
to DISSIPATION:
QUANTUM-NOISE-057
Classical measurement of time-dependent quantity, x(t), in a stationary state.
x(t)
t
C(t’-t)=<x(t) x(t’)>
QUANTUM-NOISE-058
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)=?
QUANTUM-NOISE-0510
The crux of the matter:
From Landau and Lifshitz,Statistical Physics, ’59(translated by Peierls and Peierls).
------
QUANTUM-NOISE-0511
Van Hove (1954), EXACT:
QUANTUM-NOISE-0512
QUANTUM-NOISE-0513
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-0514
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-0515
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-0516
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-0517
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)!)
QUANTUM-NOISE-0518
Important Topic:
Fundamental Limitations
Imposed by the Heisenberg Principle on Noise and Back-Action in Nanoscopic
Transistors.
Will use our generalized FDT for this!
QUANTUM-NOISE-0519
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”)
QUANTUM-NOISE-0520
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-0521
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
QUANTUM-NOISE-0522
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
QUANTUM-NOISE-0523
Cosine and sine components of any currentfiltered with window-width
QUANTUM-NOISE-0524
For phase insensitive linear amp:
gL and gS are load and signal conductances (matched to those of the amplifier). G2 = power gain.
QUANTUM-NOISE-0525
For Current Comm-s we Used Our Generalized Kubo:
where g is the differential conductance, leads to:
,( ) ( ) 2S S g
QUANTUM-NOISE-0526
Average noise-power delivered to the load
(one-half in one direction)
QUANTUM-NOISE-0529
A molecular or a mesoscopic amplifier
Resonant barrier coupled capacitively to an input signal
Is()
Ia()= I0()+G Is()
Cs Ls
B
input siganl
+back-action noise, In
QUANTUM-NOISE-0530
A new constraint on transistor-type amplifiers
Coupling to signal = γ
Noise is sum of original shot-noise I0~ γ0 and
“amplified back-action noise” In~ γ2
QUANTUM-NOISE-0531
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.
QUANTUM-NOISE-0532
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.
QUANTUM-NOISE-0534
The Casimir effect in meso- and macro- systems
QUANTUM-NOISE-0535
Even at T=0, we are sorrounded by huge g.s. energy of various fields.
No energy is given to us (& no dephasing!). But: various renormalizations, Lamb-shift…Casimir: If g.s. energy of sorrounding fields
depends on system parameters (e.g. distances…) – a real force follows!
This force was measured, It is interesting and important.
Will explain & discuss some new features.
QUANTUM-NOISE-0536
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)
QUANTUM-NOISE-0539
Buks and Roukes, Nature 2002(Effect relavant to micromechanical devices)
From:
QUANTUM-NOISE-0540
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.• “Stiction” of nanomechanical devices…• Artificial phases, soft C-M Physics.
QUANTUM-NOISE-0541
Casimir’s attractive force between conducting plates
'0 0 0
i) (c)= Sof t cutoff at
ii) ( ) ( ) - ( )
p
E d E d E
QUANTUM-NOISE-0542
QUANTUM-NOISE-0543
What is it (for volume V)?
Pressure || z in k state:
Milonni et al (kinetic theory): momentum delivered to the wall/unit time.
0Subtracted quantity ( )/ is radiation
pressure of the vacuum (Casimir, Debye,
Gonzalez, Milonni et al,
outsi
Hushwater),
de
E d
0
For every photon, momentum/ unit time =
( )/ , same f or many photons.E d
QUANTUM-NOISE-0544
Total pressure:
Replacing sum by integral, integratingover angles and changing from k to ω,with ω=:
.
Defining:
QUANTUM-NOISE-0545
A “thermodynamic” calculation:
D(ω) is photon DOS
D(ω) extensive and >0
P0 is same order of magnitude, but NEGATIVE???
QUANTUM-NOISE-0546
Why kinetics and thermodynamics don’t agree? M. Milgrom: ‘Thermodynamic’
calculation valid for closed system. But states are added (below cutoff) with increasing V!
Allowed k’s
Increasing V
cutoff
1
QUANTUM-NOISE-0547
Result for P0 is non-universal
p
Depends on:
Cutoff and details of cutoff f unction),
Nature of slab,
Dielectric f unction, , of medium.
C
( )
an
be used!
QUANTUM-NOISE-0548
Effect of dielectric on one side“Macroscopic Casimir Effect”
( )3 3/ 2
0 2 3
0
With ( ):
( ) ,6
( ) 1
( ) Larger than f or 1 !
Further possibilities
c
P dc
P
0( )P 0(1)P
1 ( )
F
QUANTUM-NOISE-0549
Net Force on slab between different dielectrics
20 p
-5
~ 0.1 N/ cm , f or =1, =10eV.
Typical diff erences ~ 10 of that.
Force balanced by elasticity/ surf ace
tension of materials.
Force and slab's position depend on 's
and on
P
slab material,
diff erences
measurable, in principle.
QUANTUM-NOISE-0550
Effect of dielectric inside on the“Mesoscopic Casimir Effect”
With ( ):
p
Will change the sign of the
Casimir Force at large
enough separations
I nteresting in static limi
,
Depen
ding
t:
on ( )
d << c/
( )
With ( ):
QUANTUM-NOISE-0551
Quasistationary (E. Lifshitz, 56) regime
p
3p
(van Kampen et al,
Length scale no retardation
Can use electrostatics
Casimir f orce becomes (no c!):
d <<
c/
68)
/ d
QUANTUM-NOISE-0552
Vacuum pressure on thin metal film
d
Quasistationary: d<<c/ωp
Surface plasmons on the two
edges
Even-odd combinations:
QUANTUM-NOISE-0553
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
QUANTUM-NOISE-0554
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.
QUANTUM-NOISE-0555
Conclusions• EM Vacuum pressure is positive, like kinetic calculation
result. It is the Physical subtraction in Casimir’s calculation. Depends on properties of surface!
• 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.
QUANTUM-NOISE-0556
END, Thanks for attention!