Quantum fluctuations and the Casimir
Effect 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
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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.
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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).
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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.
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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)
C(t’-t)=<x(t) x(t’)>
t
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Classical measurement of a time-dependent
quantity, x(t), in a stationary state.
x(t)
C(t’-t)=<x(t) x(t’)>
t
Quantum measurement of the expectation value, <xop(t)>,
in a stationary state.
<x(t)>
C(t)=?
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
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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
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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)!
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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).
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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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A Linear Amplifier Must Add Noise
(E.g., C.M. Caves, 1979)
Input
(“signal”)
Amplifier
x s , ps
X a XGx
Pa 
Gs  1
a s,GxsP,a  Gp
s Gp
[ X a , Pa ]  i  [ xs , ps ] 
i




x

p


s
s
G2
2G 2 2
Heisenberg principle is violated.
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Detector
Xa , Pa
Linear Amplifier:
But then
Output
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A Linear Amplifier Must Add Noise
(E.g., C.M. Caves)
Input
(“signal”)
Amplifier
x s , ps
X a XGx
Pa 
Gs  1
a s,GxsP,a  Gp
s Gp
[ X a , Pa ]  i  [ xs , ps ] 
i




x

p


s
s
G2
2G 2 2
Heisenberg principle is violated.

A Linear Amplifier does not exist !
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Detector
Xa , Pa
Linear Amplifier:
But then
Output
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A Linear Amplifier Must Add Noise
(E.g., C.M. Caves, 1979)
Input
(“signal”)
x s , ps
Amplifier
Output
Detector
Xa , Pa
In order to keep the linear input-output relation, with a large gain, the
amplifier must add noise
 Xs N ,PN Pa  Gps  PN
X a  Gxs  X NX, a PGx
a s Gp
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A Linear Amplifier Must Add Noise
(E.g., C.M. Caves, 1979)
Input
(“signal”)
x s , ps
Amplifier
Output
Detector
Xa , Pa
In order to keep the linear input-output relation, with a large gain, the
amplifier must add noise
 Xs N ,PN Pa  Gps  PN
X a  Gxs  X NX, a PGx
a s Gp
choose [ X N , PN ]  - (G2 1)i X N , PN act on theamplifierstate
then
 [ X a , Pa ]  [ X N , PN ]  [ xs , ps ]  G2i - (G2 1)i  i
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Cosine and sine components of any current
filtered 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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For Current Comm-s we Used Our Generalized Kubo:
S ()  S ()  2  g
where g is the differential conductance,
leads to:
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,
Average noise-power delivered to the load
(one-half in one direction)
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A molecular or a mesoscopic
amplifier
Resonant barrier coupled capacitively to an input signal
Ia()= I0()+G Is()
+back-action noise, In
B
Is()
input siganl
Cs Ls
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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
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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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The Casimir
effect in meso- and macro- systems
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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.
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The Casimir Effect
The 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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Buks and Roukes, Nature 2002
(Effect relavant to micromechanical devices)
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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.
• “Stiction” of nanomechanical devices…
• Artificial phases, soft C-M Physics.
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Casimir’s attractive force between conducting plates
i) (c)= Soft cutoff at p
ii) E 0' (d )  E 0 (d ) - E 0 ()
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Subtracted quantity  E 0 ()/ d
is radiation
pressure of the vacuum outside (Casimir, Debye,
Gonzalez, Milonni et al, Hushwater),
What is it (for volume V)?
Milonni et al (kinetic theory): momentum
delivered to the wall/unit time.
For every photon, momentum/unit time =
 E 0 ()/ d , same for many photons.
Pressure || z in k state:
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Total pressure:
Replacing sum by
integral, integrating
over angles and

changing from k to
ω,
.
with
ω=:
Defining:

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A “thermodynamic” calculation:
D(ω) is photon DOS
D(ω) extensive and >0
P0 is same order of
magnitude, but
NEGATIVE???
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Why kinetics and thermodynamics don’t agree?
M. Milgrom: ‘Thermodynamic’
calculation
valid for closed system. But states are
added (below cutoff) with increasing V !
1
cutoff

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Increasing
V
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Allowed k’s
Result for P0 is non-universal
Depends on:
 Cutoff p and details of cutoff function),
 Nature of slab,
 Dielectric function,  ( ) , of medium.
Can be used!
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Effect of dielectric on one side
“Macroscopic Casimir Effect”
With  ( ):
P0 
P0 (1)
P0 ( )
(c )
6 2c 3
3
3/ 2
d


(

)
,

 ( )  1
P0 ( ) Larger than for   1 !
 1
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F
 ( )
 Further possibilities
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Net Force on slab between different
dielectrics
 P0 ~ 0.1 N/cm2 , for  =1, p =10eV.
 Typical differences ~ 10 -5 of that.
 Force balanced by elasticity/surface
tension of materials.
 Force and slab's position depend on 's
and on slab material, differences
measurable, in principle.
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Effect of dielectric inside on the
“Mesoscopic Casimir Effect”
With  ():
Will change the sign of the
Casimir Force at large
 ( )
With  ():
enough separations,
Depending on  ( )
Interesting in static limit:
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d << c/ p
Quasistationary (E. Lifshitz, 56) regime
Length scale d << c/ p no retardation
Can use electrostatics (van Kampen et al, 68)
Casimir force becomes (no c!):
p / d 3
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Vacuum pressure on thin metal film
Quasistationary: d<<c/ωp
Surface plasmons on the
two edges
Even-odd combinations:
d
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Dispersion of thin-film plasmons
For d<<c/ωp,
light-line ω=ck
ω/ωp
1
is very steep-full
0.8
0.6
EM effects don’t
0.4
0.2
1
2
3
4
kd
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.
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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 mesoregimes. 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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