QUANTUM-NOISE-051

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 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).

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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)

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

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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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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 ,

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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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For Current Comm-s we Used Our Generalized Kubo:

where g is the differential conductance, leads to:

,( ) ( ) 2S S g

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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

Is()

Ia()= I0()+G Is()

Cs Ls

B

input siganl

+back-action noise, In

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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 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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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.• “Stiction” of nanomechanical devices…• Artificial phases, soft C-M Physics.

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Casimir’s attractive force between conducting plates

'0 0 0

i) (c)= Sof t cutoff at

ii) ( ) ( ) - ( )

p

E d E d E

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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

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Total pressure:

Replacing sum by integral, integratingover 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!

Allowed k’s

Increasing V

cutoff

1

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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!

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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

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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.

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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 ( ):

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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

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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.

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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 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!