Transcript What Quantity is Measured in an Excess Quantum Noise

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
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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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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)
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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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
:
• Temp = 0:
S ()  G , ( < 0 only)
• Temp >> ħ: S ()  G ·Temp.
expression for S ()
(Nyquist!)
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Quantum Shot-Noise (Khlus, Lesovik)
For Fermi–Sea Conductors, different for
BEAMS in Vacuum, for same current.
μ
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.
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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.
Current Noise Measurment in Beams in
Vacuum.
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
23gate), this difference can be manipulated
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and the radiation spectrum can be
controlled.
U = gate potential
on RHS lead.
U>eV
SeV _____
Su _____ _____
U larger
eV
U

eV+U
U<eV
SeV _____
Su _____ _____ _____
U larger
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U
eV eV+U

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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Full Noise Measurement Chain
Typical experimental setup:
DC Voltage
Sample
LC Filter
Amplifier
Spectrum
analyzer
VDC

External voltage
sources (pump, idler,
FET bias,…)

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Display
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,GxsPa,  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,GxsPa,  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 N X, 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 N X, a PGx
a s Gp
choose [ X N , PN ]  - (G 2  1)i X N , PN act on the amplifier state
then
 [ X a , Pa ]  [ X N , PN ]  [ xs , ps ]  G 2i - (G 2  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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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
Ia()= I0()+G Is()
B
Is()
input siganl
Cs Ls
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A molecular or a mesoscopic
amplifier
?
Is I0() enough to supply the
necessary noise?
Ia()= I0() +G Is() + IN()
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
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(unlike most other quantum amplifiers)
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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S M ,excess (V ,  )  G  Sexcess (V ,  )
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!
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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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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
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i)
↑
(c) = Soft cutoff at p
ii)
E’0(d) = E0(d) - E0()
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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 !
1
cutoff
Increasing V
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Allowed k’s
Effect of dielectric on one side
“Macroscopic Casimir Effect”
With ():
P0(1)
P0()
=1
F
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()
() > 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!):
- ~ħp /d3
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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,
ω/ωp
light-line
ω=ck
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 (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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