Coherence and Quantum Noise in Mesoscopic Physics

Yoseph Imry, WIS, 01/08

• Most of Mesoscopics has to do with interference--coherence necessary.

• Decoherence due to “inelastic” scattering -- interaction with the dynamic noise of the environment.

OUTLINE OF TALKQUANTUM NOISE (Gavish, Levinson)• Quantum noise, Physics of Power

Spectrum• Fluctuation-Dissipation Theorem, in

steady state• Shot-Noise, Excess noise,

dependence on full state of system • What is detected in a quantum noise

and in an excess noise measurement?• Heisenberg Constraints on Quantum

Amps’ (Yurke)

DEPHASING (Stern, Aharonov)• Inelasticity - change of state of

environment.• Disordered conductors -- especially

low dimensions.• Nonequilibrium dephasing by

“quantum detector”.• Low-temperature limit ???• Recovery of Interference.

A-B Flux in an isolated ring

• A-B flux equivalent to boundary condition.

• Physics periodic in flux, period h/e (Byers-Yang).

• “Persistent currents”existdue to flux.

• They do not decay by impurity scattering (BIL).

Two-slit interference--a quintessential QM example:

““ Two slit formula ””When is it valid???

h/e osc. –mesoscopic fluctuation. Compare:

h/2e osc. – impurity-ensemble average,

Altshuler, Aronov, Spivak, Sharvin2

scatterer

scatterer

Closed system!

-10 -5 0 5 10

0.100

0.105

0.110

Magnetic Field, B [mT]

Col

lect

or C

urre

nt,

I C

(a.u

.)

Aharonov Bohm Oscillation in mesoscopic interferometer(Heiblum)

∆ ΦB A= 0 /

δ∆

Visibility = δ/ ∆

“NANO”:

Electron Coral(Eigler & Co)

Matter-wave interference (Ketterle’s group)

Interference of two expanding, overlapping BEC’s, which started as independent

A. Tonomura: Electron phase microscopy

Each electron produces a seemingly random spot, but:Single electron events build up to from an interference pattern in the double-slit experiments.

Quantum, zero-point fluctuations(with Gavish and Levinson)

Nothing comes out of a ground state system, but:

Renormalization, Lamb shift,

Casimir force, etc.

No dephasing by zero-point fluctuations (LATER!)

How to observe the quantum-noise?

(Must “tickle” the system, or amplify its output).

Noise Part Outline:• Quantum noise, Physics of Power Spectrum, dependence on full state of system

• New Results on noise in 2-level systems

• Fluctuation-Dissipation Theorem, in steady state

• Application: Heisenberg Constraints on Quantum Amps’

Direct observation of a fractional charge

R. de-Picciotto, M. Reznikov, M. Heiblum, V. Umansky, G. Bunin & D. Mahalu

Nature 1997 (and 1999 for 1/5)

A recent motivationHow can we observe fractional charge (FQHE,

superconductors) if current is collected in normal leads?

Do we really measure current fluctuationsin normal leads?

ANSWER: NO!!!

SOMETHING ELSE IS MEASURED.

Second Motivation

Breakdown of FLT in glassy,

“aging”, systems:

Can we salvage the properFLT?

(not a stationary system)

Needs Work, but…

Understanding The Physics of

Noise-Correlators, and relationship

to DISSIPATION:

The crux of the matter:

From Landau and Lifshitz,Statistical Physics, ‘59

------

Van Hove (1954), EXACT:

Detailed-balance condition

S(ω) = S(-ω) exp(-ħω/kT)

Valid in equilibrium for power spectrum of any operator (I, ρ, ρq…), in T-invariant system

Necessary for 2nd law—will come back to it later.

Emission = S(ω) ≠ S(-ω) = Absorption,(in general)

Therefore, symmetrizing the power spectrum, can lose physical info.

NOT RECOMMENDED, although a common practice!

Cf, Lesovik-Loosen, Aguado-Kouwenhoven.

Exp confirmation: deBlock et al-03 , Billangeon et al-06

Emission = S(ω) ≠ S(-ω) = Absorption,(in general)

From field with Nωphotons, net absorption(Lesovik-Loosen, Gavish et al):

NωS(-ω) - (Nω + 1) S(ω)

For classical field (Nω >>> 1):

CONDUCTANCE ∝∝∝∝ [ S(-ω) - S(ω)] / ω

This is the Kubo formula (cf AA ’82)!

Fluctuation-Dissipation Theorem (FDT)

Valid in a nonequilibrium stationary state!!

Dynamical conductance - response to “tickling”acfield, (on top of a class of nonequilibrium states).

Given by S(-ω) - S(ω) = F.T. of the commutator ofthe temporal current correlator

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)!

Landauer: 2-terminal conductance = transmission

G ≡≡≡≡ I/V = (e2/πħ) |t|2 , with spin.eV ≡≡≡≡ µ1- µ2

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) + T2 = T, from van Hove-type

expression for S(ωωωω) :

• Temp = 0: S(ωωωω) ∝∝∝∝ G ωωωω, (ωωωω < 0 only)• Temp >> ħωωωω: S(ωωωω) ∝∝∝∝ G ·Temp.

(Nyquist!)

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 isExcess Noise.

µ

Exp confirmation, of T(1-T)Reznikov et al, WIS, 1997

New Results: Noise in two-level systems

(with O Entin-Wohlman, A Aharony, S

gurvitz). MOTIVATION:

Durkan and Welland, 2001

In a single-channel conductorFrom current matrix elements and unitarity:

New Interference Effect in Transition Amplitudes (generalization of Fano).

In systems having two levels

A variation on the Fano Effect

Fano: transition between local state

and a resonance (state coupled to continuum). Interference of locöloc

and loc ö continuum matrix elements.

Now: transition between two resonances. Interference among continuum öcontinuum matrix elements.

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); Billangeon et al-06

A recent motivationHow can we observe fractional charge (FQHE,

superconductors) if current is collected in normal leads?

Do we really measure current fluctuationsin normal leads?

ANSWER: NO!!!

THE EM FIELDS ARE MEASURED.

(i.e. the radiation produced by I(t)!)

Important Topic:

Fundamental Limitations

Imposed by the Heisenberg Principle on Noise and Back-Action in Nanoscopic

Transistors.

Can use our generalized FDT for this!

Our Generalized Kubo:

where g is the differentialconductance, leads to:

,

From our Kubo-based commutation rules:

Hence:

Main idea of the derivation

Following that of Heisenberg principle except for the fact that instead of using a commutator which is an imaginary number, we use a commutator of which the expectation value in a stationary state is an imaginary number.

This generalizes results on photonic amps, where the current commutators are c-

numbers,and establishes the link between the FDT and amplifier noise

theory.

“Decoherence”, by environment

(via cplg to all other degrees of freedom)

Two-wave interference

intensity

Φ

1

2

|Ψ| 2

|env1>

|env2>

Ψ = Ψ1 + Ψ2

• What spoils the 2Re(Ψ1Ψ2* ) interference?

Leaving a ("which path") trace in the environment : <env1|env2> = 0

Inducing uncertainty in the relative phase,

arg(Ψ1Ψ2* )

Electromagnetic

Coupling to other

degrees of freedom

This is what

charged Particles

always do!!!

These two statements are exactly equivalent (SAI, 89)

Proof: by considering the time evolution operator,

U = T exp[-(1/ħ)∫∫∫∫t H I(t’) dt’]

U induces changes in the environment state

and creates an uncertaintyin the phase,

Φ=arg(Ψ1Ψ2*)

determined by the dynamic correlators of H I(t).

2=O(1) <env1|env2>=0

FLUCTUATION-DISSIPATION

THEOREM (FDT)

φ∆

Physical Remarks

Reabsorbing the excitation restores the phase.

But: After interaction isSwitched off, environmentbecomes irrelevant.

Special effects: Retrievalof interference by measurements on env.(epr: Stern, Hackenbroich, Rosenow &Weidenmuller, back later!)

No dephasing if identicalexcitation is produced by 2 paths.

How much energy transferredIs irrelevant!

Excitation should resolve

the 2 paths:

k• (x1 – x2) ~ π

1/τφ ~ rate for particle to excite environment (and lose phase!)

Probability to excite the environment till time t, for a particle moving in medium, can be calculated via the

Fermi Golden Rule

Results produce all knowncases (dirty metals, any d)

1/ = dq dω |Vq|2

Sp(-q, -ω) · Ss(q, ω).

particle env.S (q, ω)=dynamic structure factor =

F.T [density-density corr. Fcn]Measures the corr. of space-time density fluctuations much physical info. Known for models.

(see later…)

ϕτ ∫ ∫

Agreement (of AAK results ) with experiments:

Narrow wire (“quasi 1D”):

1/ τφ ~ T2/3

Very nontrivial (FLT???)

What does exp say?

LOWLOWLOWLOW----TEMP SATURATION OF TEMP SATURATION OF TEMP SATURATION OF TEMP SATURATION OF ττττφφφφ ????Mohanty, Jariwala and Webb (1997) and many others.

• Must rule out: EXTERNAL NOISE, MAGNETIC IMPURITIES...

• DISAGREES WITH USUAL DISAGREES WITH USUAL DISAGREES WITH USUAL DISAGREES WITH USUAL

THEORY!THEORY!THEORY!THEORY!

• Debye-Waller-type phenomenon?

• Unexpected low-energy excitations?

Detailed-balance condition

S(ω) = S(-ω) exp(-ħω/kT)

Valid in equilibrium for power spectrum of any operator (I, ρ, ρq…), in T-invariant system

Necessary for 2nd law, makes S(ω>0) = 0, at T=0.

No Dephasing as Tö 0 !

Starting from our expression:

1/ τφ= dq dω |Vq|2

SP(-q, -ω) · Ss(q, ω),

we see that supports of two S’s

DO NOT OVERLAP1/ τφ = 0.Unless having g.s. degeneracy

(spins…). Tô0 deph ruled outby laws of thermodynamics!

ω

Sparticleenvironment

∫ ∫

Within FGR dephasing:Neither environment nor particle can transfer anything to the other!

Experiment: Tô 0 deph is an interesting

artifact

E • 1mm ≈≈≈≈ kBT !!!

Pierre et al, 2003,Magnetic impurities in the metal. (ppm level)

Ovadyahu, 2001 (T ≈ .3K):Nonequilibrium effect.i.e. out of linear transport!Nonmagnetic (in InO3-x).

What can cause apparent saturation of true1/ τφ ?Need abundance of “soft” low-energy modes

• Can be magnetic impurities,• Or (YI, Fukuyama, Schwab, 99)

Two-level systems (TLS),as suggested by Anderson, Halperin and Varma for the

low-T properties of glasses.

Should exist due to disorder!

• In both cases,1/ τφwill vanish when T ö 0.

B

∆

Proper distribution of B and Ωo can explain apparent saturation

We used equilibrium correlators to prove the vanishing of 1/ τφwhen Tô 0.

Such correlators determine the linear response conductance (and magnetoconductance).

It becomes crucial that the exps probe the linear response (I, V ô 0) regime. Finite V opens more inelastic (hence, dephasing) channels!

Ovadyahu’s results show that the conditions for that are more strict than usually expected!!!

No Au

2% Au

•For the “no Au” sample, an unusually minute driving field is necessary forLinear transport (note apparent constancy at higher fields!).•But, electrons are not heated(confirmed from AA corections to (T), as Mohanty et al did).•Adding gold, facilitates getting linear!•Systematic studies produced the very nontrivial condition for linearity of the transport, in terms of the electric field used for the measurement.

Experimental condition for linearity of the transport (NOT HEATING):

Can always be written in terms of a (surprisingly long) length:

eEΛ << kBTWhat is Λ?Experimental result: Λ = Ler , (for Ler << Lsample)

The length to transfer the field-supplied energy away.

Based on a thorough study, unexpectedtheoretically.

The Real Question:

• How can the electric field cause dephasing without heating?

• Possible in principle! Precise answer here seems to depend on TLS’s

Qualitative explanation

• e’s and TLS are well coupled.

• TLS weakly coupled to bath, via τi,TLS (but better than e’s!), their cv >> that of the e’s.

• e-TLS-bath channel gives dominant energy-relaxation.

• E’s dump all field energy into TLS, whose temp changes little, rate of relaxation to bath: τi,TLS.

excitation (dephasing) with no heating!

Another Intriguing Exp Result:

• Doping the samples withmore Au, leads to quasi-saturation of 1/ τφ , but followed by a rapid decrease at lower T! (as in the IFS model)

• Au goes into a O (or O2) vacancy– a large rattling cage – may have a few minima structure.

0% Au

3% Au

Exp

. res

ults

, dis

orde

red

InO

:Au,

Zvi

Ova

dyah

u

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

T=0.3K

'R/R

(%

)

H (T)

-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

T=2K

LII=2400Å L

SO=1490Å

Exper. fit fit "without Kondo"

2.5x10-1

100

101

1010

1011

5x1011

W-1 I

(se

c.-1)

T (K)

A Double-minimum TLS model (IFS)

∆

B

Ω0 is the tunneling matrix-element between the two wells.A Born-appr calculation forns impurities of x-section σ0 ina unit volume, for electrons with Fermi velocity vF at temp T, gives (4α2β2 is the well asymmetry parameter Ω0/2∆(=1 in the symmetric case):

Averaging the result over the TLS distribution:

We use a conventionalTLS distribution, as in the theory of 1/f noise:B and ln are uniform between 0 and Bmaxandmin and max. It was suggested by IFS to be relevant for the low-tempDephasing problem.

Decoherence, CONCLUSIONS

• Mesoscopic Physics helps us understand fully the issue of decoherence (limiting the quantum behavior), which happens around τΦ, the (de)coherence time.

• Decoherence rate vanishes, when Tô0 !!!

• Interesting Physics for low T!

Idea by Hans Weidenmüller (following work on photons):

Note: detector has finite number of states

Recovery of interference from the correlation of detector-interferometer signals (Neder et al. 06)

AB interference dephased by path-detection (by another edge-channel)

And recovered by correlation with the detector signal

Questions for the future:

• What are the physically relevant “soft”impurity potentials?

• Fuller understanding of nonequilibriumbehavior.

• A larger body decoheres faster. How can we avoid that? Correlated states? LRO???