and 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 TALK (Gavish, Levinson) DEPHASING (Stern, Aharonov) • Quantum noise, Physics of Power • Inelasticity - change of state of Spectrum environment. • Fluctuation-Dissipation Theorem, in • Disordered conductors -- especially steady state low dimensions. • Shot-Noise, Excess noise, • Nonequilibrium dephasing by dependence on full state of system “quantum detector”. • What is detected in a quantum noise • Low-temperature limit ??? and in an excess noise measurement? • Recovery of Interference. • Heisenberg Constraints on Quantum Amps’ (Yurke) 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”exist due to flux. • They do not decay by impurity scattering ( BIL ).

Two-slit interference--a quintessential QM example:

““Two slit formula ”” When is it valid??? Closed system! scatterer

scatterer h/e osc. –mesoscopic fluctuation. Compare: h/2e osc. – impurity-ensemble average,

Altshuler, Aronov, Spivak, Sharvin 2 Aharonov Bohm Oscillation in mesoscopic interferometer (Heiblum)

B = Φ0 / A 0.110 (a.u.) C I

0.105

Collector Collector Current, 0.100

-10 -5 0 5 10 Magnetic Field, B [mT]

δ

Visibility = δ/ “NANO”:

Electron Coral (Eigler & Co) Matter- (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 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 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. Second Motivation

Breakdown of FLT in glassy,

“aging”, systems:

Can we salvage the proper FLT?

(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 (I, ρ, ρq…), in T-invariant system

Necessary for 2 nd 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 N ω 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 This is the Kubo formula (cf AA ’82)!

Fluctuation-Dissipation Theorem (FDT)

Valid in a nonequilibrium stationary state!!

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

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

• Need just a STEADY STATE SYSTEM:

Density-matrix diagonal in the energy representation.

“States |i> with probabilities P i , 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 .

eV ≡≡≡ 1- 2 Equilibrium Noise in the Landauer Picture

2 2 2 2 2 2 | jll | = | jll | =(evT ) ; | jlr | = | jrl | =(ev T(1-T) )

Since T(1-T) + T 2 = T, from van Hove-type expression for S (ωωω) : • Temp = 0: S (ωωω) ∝∝∝ G ωωω, ( ωωω < 0 only) • Temp >> ħωωω: S (ωωω) ∝∝∝ G ·Temp. (Nyquist!) Quantum Shot-Noise (Khlus, Lesovik) For Fermi–Sea Conductors, different for BEAMS in Vacuum, for same current. Left-coming Scattering state

2 2 || = v F TR, for (k- k’ << 1/L)

→ S( ω) = 2e( e2V/ πħ) T(1-T), ω <

= 0 , ω >V . This is Excess Noise.

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 conductor From 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 loc öloc and loc ö continuum matrix elements.

Now : transition between two resonances. Interference among continuum ö continuum 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 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)!) 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 differential conductance, 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 • What spoils the 2Re( Ψ1 Ψ * ) interference? Ψ=Ψ1+Ψ2 2 Leaving a ("which

|env1> path") trace in the 1 environment : Φ = 0 2 |env2> Inducing uncertainty in the relative phase,

intensity 2 |Ψ| arg(Ψ1 Ψ2* ) 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, φ 2=O(1)  =0 t U = T exp[-(1/ ħ)∫∫∫ HI(t’) dt’]  U induces changes in the environment state FLUCTUATION-DISSIPATION  and creates an uncertainty in the phase,

 Φ=arg(Ψ1 Ψ2*) THEOREM (FDT)

 determined by the dynamic correlators of HI(t). Physical Remarks

 No dephasing if identical  Reabsorbing the excitation excitation is produced by 2 restores the phase. paths.  But: After interaction is Switched off, environment  How much energy transferred becomes irrelevant. Is irrelevant!  Special effects: Retrieval of interference by  Excitation should resolve measurements on env. the 2 paths: (epr: Stern, Hackenbroich, Rosenow &Weidenmuller, k•(x1 – x2) ~ π back later!) 1/ τφ ~ rate for particle to excite environment (and lose phase!) Probability to excite the 2 1/τϕ = dq dω |V q| environment till time t, ∫ ∫ for a particle moving in Sp(-q, - ω) · Ss(q, ω). medium, can be calculated  via the particle env. S (q, ω)=dynamic structure factor = Fermi Golden Rule F.T [density-density corr. Fcn] Measures the corr. of space-time Results produce all known density fluctuations  much cases (dirty metals, any d) physical info. Known for models. (see later…) Agreement (of AAK results ) with experiments:

Narrow wire (“quasi 1D”):

2/3 1/ τφ ~ T

Very nontrivial (FLT???)

What does exp say? LOWLOW----TEMPSATURATIONOFTEMPSATURATIONOF ??? τττφφφ Mohanty, Jariwala and Webb (1997) and many others.

• Must rule out: EXTERNAL NOISE, MAGNETIC IMPURITIES... •DISAGREESWITHUSUAL 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 2 nd law, makes S( ω>0) = 0, at T=0. No Dephasing as T ö 0 !

Starting from our expression: S environment particle 2 1/ τφ = ∫ dq∫ dω |V q| SP(-q, -ω) · Ss(q, ω), we see that supports of two S’s

DO NOT OVERLAP 1/ τφ = 0. ω Unless having g.s. degeneracy (spins…). Within FGR dephasing: Tô0 deph ruled out Neither environment nor by laws of thermodynamics! particle can transfer anything to the other! Experiment: T ô 0 deph is an interesting

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

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

Nonmagnetic (in InO 3-x).

E • 1mm ≈≈≈ kBT !!! What can cause apparent saturation of true 1/ 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, B Halperin and Varma for the low-T properties of glasses. Should exist due to disorder!

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

Proper distribution of B and o can explain apparent saturation We used equilibrium correlators to prove the vanishing of 1/ τφ when 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!!! •For the “no Au” sample, an unusually minute driving field is necessary for No Au Linear transport (note apparent constancy at higher fields!). •But, electrons are not heated (confirmed from AA corections to (T), as Mohanty et al did). 2% Au •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:

eE Λ << kBT What is Λ?

Experimental result: Λ = Ler , (for Ler << Lsample )

The length to transfer the field-supplied energy away.

Based on a thorough study, unexpected theoretically .

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 with more Au, leads to quasi- saturation of 1/ τφ , but followed by a rapid decrease at lower T! (as 0% Au in the IFS model)

3% Au • Au goes into a O (or O 2) vacancy– a large rattling cage – may have a few minima structure. ' -1 -1 RW/R ( s(%ec). ) - - - I - 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 5 ...... x 6 4 2 2 4 6 8 0 0 2 4 6 8 0 2 - 10 10 10 2 0 2

Exp. results, disordered InO:Au,. Zvi Ovadyahu . 5 0 . 11 10 11 5 x - 10 2 L L . S I I 0 = O - 0 = 2400 1 1490 . 5 - 1 Å . 5 Å - 1 1 . 0 . 0 T T = = - 0 2 0 10 . . K 1 5 3 H . 0 5 K 0 T (  .  0 T (  ) 2 K . 0 0 ) . 5

f f E i i t t x " pe w 1 i r t 2 . . hou 0 . 5 t K 1 ondo . 5 3 . " 0 2 . 0 10 3 2 . . 5 5 1    A Double-minimum TLS model (IFS)

Ω0 is the tunneling matrix- element between the two wells. A Born-appr calculation for n impurities of x-section σ in s 0 a unit volume, for electrons

with Fermi velocity vF at temp B T, gives (4 α2β2 is the well asymmetry parameter Ω0/2 (=1 in the symmetric case): Averaging the result over the TLS distribution:

We use a conventional TLS distribution, as in the theory of 1/f noise: B and ln  are uniform

between 0 and Bmax and min and max . It was suggested by IFS to be relevant for the low-temp Dephasing problem. Decoherence, CONCLUSIONS

• Mesoscopic Physics helps us • Interesting Physics understand fully the issue of for low T! decoherence (limiting the quantum behavior), which

happens around τΦ, the (de)coherence time. • Decoherence rate vanishes, when T ô0 !!! 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 nonequilibrium behavior. • A larger body decoheres faster. How can we avoid that? Correlated states? LRO???