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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.

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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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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)

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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!!!

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

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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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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).

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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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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 ,

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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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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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