arXiv:cond-mat/0202044v1 [cond-mat.mes-hall] 4 Feb 2002 nMssoi hsc 4,i snwflyaduniversally and fully purely research now that of is years it accepted twenty [4], Physics almost Mesoscopic exactly After on are descriptions 3]. two environ- [2, or These fluctuating equivalent particle. the states the by environment’s on exerted ment the fields over random as the tracing described to either be The then to partial can due case) states. interference simplest the orthogonal the of in (in diminishing environment two the the interfer- leave if waves the only and eliminate More if will ence experiment. interaction interference this the whichspecifically, in measured etc.) vibrations, particles, not other lattice are with are interactions en- fields, which ”the E-M for (dubbed ”the freedom examples as of to vironment”, degrees referred interaction the (henceforth with an ”deco- entity particle”) to interfering or due the ”dephasing” occurs of Decoherence called being is [1]. ques- still process herence” fundamental surprisingly, This interesting are, which an debated. of is aspects below) some 1 tion, Eq. (see nwrt h ffimtv a ensgetdtheoreti- suggested An been has literature. modern affirmative in the not current can to still which answer is particle [4], test energy a lose [1] dephase can vironment decoherence. proper this of not again, distribution but, is wavelengths) enough or broad energies pro- a particle incident averaging via various example, by (For interference cesses one the classical Likewise, diminish the decoherence. of not can is half this a inter- But The will contribution). blocked. (as is vanishes paths obviously occurs interference ference two case the extreme of An one decoherence. if terms cause interference not the does modify but can solids, in defects static htdmnse h nefrneo,sy w waves two say, of, interference the diminishes What h su fwehrtezr on oe fa en- an of modes point zero the whether of issue The lmnayepaaino h nxsec fdchrnea decoherence of inexistence the of explanation Elementary ASnmes 32.k 32.x,21.m 73.21.La ,72.15.Qm, 73.20.Dx 73.23.Hk, numbers: PACS equilibrium. for in formulation properties scattering interference the to within transpo also valid the is for example corrections M¨ossbau quantum last the as well scattering, a as Bragg temperature, for are: impossible examples Prime is d energy. limit demon true zero-temperature that is the op mechanics, It approaching his quantum for elementary convenience. author and for conservation the numerou manner, approached the have pedagogical to those available hopefully of be Several to but issue. future, foreseeable the in hsnt a onwrslsadi hrfr o neddt b to intended not therefore is and results new no has note This .INTRODUCTION I. elastic cteig o xml by example for scattering, ytm ihprl lsi scattering elastic purely with systems et fCnesdMte Physics, Condensed-Matter of Dept. h ezanIsiueo science, of Institute Weizmann the Dtd coe 1 2018) 31, October (Dated: eoo 60,Israel 76100, Rehovot oehImry Yoseph nitga vrpout fcrancreainfunctions, of correlation terms certain of in ex- products formulated over no integral be and an actually that test-particle can argument the environment between above the exchanged The be degeneracy can state. large citation a ground have the the not usually of is does as environment that, environ- assumed the the is case, It nor energy. particle lose the can excitation neither ment No if exchanged 11]. [10, be an zero) can has to tends electron which linear conduction energy the excess the in (where and temperature regime and zero transport environment at the are particle both test when the exchange occur excitation not can Such the between processes environment. [2] the process) and inelastic an particle (i.e. excitation path” exchange an the ”which of by a effected called ”environment”, the be gen- by may detection very what is as decoherence understood above, erally mentioned As culation. the be- at results. contradiction vanish experiment apparent to and An theory found tween limit. and temperature [9] to has zero before due rate as calculated dephasing described been This be interactions. also electrons, can electron-electron conduction decoherence other this the due by hence mainly produced are fields coupling the the fluctuations the to these in to conductors, fluctuations For due electromagnetic solid. as zero-point interpreted the then with is which limit ehsn aeo odcineeto osntvanish not does the temperature electron experiments, the conduction when careful a rather of rate these dephasing to conduc- According of rate magnetoconductance [8]. dephasing weak-localization by the electrons determine Mohanty tion which by [7] experiments this al to in et due Interest resurfaced [5] ago [6]. has years problem vigorously 14 refuted (phonons) conduction-electron immediately vibrations a lattice and of to coupling solid the a of in case the in cally hscnrdcini o utwt pcfi oe cal- model specific a with just not is contradiction This h rnpr.Smlrsaeet apply statements Similar transport. the to odcineetosi ois The solids. in electrons conduction of rt reetadrltdpeoeaa zero at phenomena related and effect er etpril hc a o ieo lose or give not can which particle test niiul h r neetdi this in interested are who individuals s tae,uigesnilyol energy only essentially using strated, no,wihi umrzdhr na in here summarized is which inion, chrneb omlenvironment normal a by ecoherence umte oa”eerh journal ”research” a to submitted e T eotmeauefor temperature zero t → ,btrte ost finite a to goes rather but 0, 2 which can rather generally be proven to vanish as T → 0 tum interference, even truly at equilibrium, at T → 0, by [4, 12, 13, 14]. The only exceptions being an environ- a coupling to the environment [24]. It has been claimed ment having a pathologically increasing density of states that ”Vacuum fluctuations are a source of irreversibility at low energies and the above-mentioned case where the and can decohere an otherwise coherent process”. That environment has a large ground state degeneracy. A vi- statement is of course trivially valid (if certain matrix- able model for the latter are uncompensated and isolated elements do not vanish) for a system that can lose energy magnetic impurities at zero magnetic field. to the vacuum fluctuations. However, that does not in- However, the experimenters [7] have taken great pains clude an electron just on the Fermi-level. The persistent to eliminate the effect of magnetic impurities and have current in a mesoscopic ring was found [24] to be weak- presented serious arguments as to why that was not an ened at T → 0 by coupling to harmonic oscillators in important effect in their samples. An assumed high their ground state. Models invoking such oscillators with enough density of other low-energy modes, such as two- a specific coupling and density of states (DOS), chosen level systems (TLS) was also shown [14] to be able to to mimic the dissipation [3, 25, 26, 27] can be used to account, in principle, for the observed anomalies. Such describe a viscous resistance to the electron’s motion (al- low-energy modes might depend on the metallurgy of a though, at best, important modifications, such as a spa- given sample. Other experiments [15] have reported a tial distribution of these oscillators might be necessary vanishing dephasing rate as T → 0 in a different material. [12, 28]). Later experiments [16] have in fact demonstrated that Thus, the theoretical situation needs clarification, in- the T → 0 anomaly seems to depend on sample prepara- dependently of the final experimental verdict. We shall tion. Thus some lattice defects such as the TLS might be attempt in this very informal note to clarify the possi- relevant. Uncompensated magnetic impurities were also ble misunderstandings that can lead to the belief in the implicated in Ref. [16] but that possibility was argued to T → 0 decoherence. We shall show that some of the con- have been negated for the experiments of Ref.[7]. Here fusion may be due to interpreting a reduction of inter- we certainly do not attempt to pass judgement on the ference as due to dephasing, when it really is something experimental controversy. Furthermore, as indicated by else. To explain that, we shall first treat in section II an some of the early experiments [8] one must use extremely elementary model, really a rehash of the arguments of small measurement currents to be in the linear transport Ref. [6], which illustrates the whole issue very simply. regime. Ovadyahu [17] studied this issue very carefully. His findings can be summarized as showing that it may be possible to eliminate the anomalous decoherence by II. A SIMPLE MODEL: TWO-WAVE INTERFERENCE WITH VIBRATING using extra-small probing currents. However, there is a SCATTERERS large range of currents that do not heat the electrons, as found in Ref. [7], but appear nevertheless not to be small enough for the transport to be in the truly linear Let us first imagine two elastic, rigid, point scatterers placed at points ~x1 and ~x2, separated from each other by regime. Finding the mechanism leading to this, which ~ probably also involves some special low-frequency modes a vector d along the x axis. A particle wave with wave vector ~k0 impinges upon these scatterers. We look at the [17], still presents an important unsolved problem in low- ~ temperature metal Physics. Therefore, we again refrain (elastic) scattering into the state with a wavevector k. ~ ~ ~ from offering a verdict on what the final answer provided We denote the momentum-transfer vector by K = k0 −k. by all the experiments will be and we trust that that will If the scattering amplitude from each of the scatterers is be cleared up soon. From now on we confine ourselves to AK , the (lowest order) scattering probability from the the, still hotly debated, theoretical question only, which system will be proportional to ought to be decidable by applying known principles cor- rectly. 2 2 ~ ~ cl qu SK =2|AK | + 2Re|AK | exp(iK · d) ≡ SK + S , (1) That question has become controversial as well, due to K reports of calculations [18] which claimed to have pro- where the first and second terms are, in obvious notation, duced a finite dephasing rate as T → 0, in contradiction the classical and the interference contributions, bearing a with refs. [2, 9] and with the general arguments summa- strong similarity to the case of diffraction by a double slit. rized above. Refs. [19, 20] have disagreed strongly with K~ ·d~ is the phase shift due to the difference in the optical these calculations. To which criticism the authors [18] paths between the waves scattered by the two scatterers. have rebutted. The theoretical controversy is still going Somewhat similarly to the Bohr discussion of diffraction on ( e.g. [21], [22] and [23]). Unlike the experimental from two fluctuating slits, we now let each scatterer be situation, experience has shown that questions such as bound by a parabolic potential, so that the frequency of ”who has calculated the correct diagrams correctly” tend the motion in the x-direction of is ω0. The case of interest to linger and produce unnecessary controversy. This is to us here is when the scatterers are at zero temperature, compounded by further reports of the weakening of quan- but the generalization is obvious. We writex ˆi = Xi +ˆui, 3 i =1, 2. Eachx ˆi performs zero-point fluctuations (whose nature of the theory, Fermi in 1936 [35] defined a pseu- coordinate operator isu ˆi) around its equilibrium position dopotential V (r), so that the lowest-order Born approx- Xi = hxii. To obtain the new scattering pattern we must imation scattering amplitude due to it (which is propor- average the scattered intensity over the wavefunction of tional to VK ) equals the actual, correct, AK . Therefore, the scatterers. That wavefunction is a product of two using the Born approximation with the pseudopotential, gaussians, of u1 and u2. The standard deviation of each already implies taking into account all the multiple scat- 2 h¯ is σ, given by σ = 2Mω0 , where M is the mass of the tering by the same [36] scatterer. Within this picture, scatterer. Using the fact that for a gaussian distribution van Hove considered in 1954 the (in general) inelastic of v with an average hvi and a variance h∆v2i, Born approximation scattering, which for our system is proportional to

2 hexp(iv)i = exp(ihvi − h∆v i/2) (2) 2 S(K,ω)= |AK | |hf|[exp(iKxxˆ1)+ (a similar result is valid also in the quantum case [30, 31]), Xf 2 we obtain exp(iKxxˆ2)]|gi| δ(¯hω + Ef − Eg) (6)

2 2 2 where ¯hω is the energy, (¯h /2m)(k0 − k ), lost by the cl 2 ~ ~ scattered particle and Ef and Eg are the energies of the SK = hSK i + 2Re|AK | exp(iK · d) exp(−2W ), (3) final state (|fi) and initial one (|gi) of the oscillators. where 2W is the well-known Debye-Waller factor The δ function simply gives total energy conservation in the process. Since the scatterers are at T → 0, the only possible initial state is the ground state, |gi, of the two- 2 2 oscillator system. 2W = Kxσ , (4) A slightly more advanced treatment of Eq. 6 will be 2 2 ′ obtained by writing h(u1 − u2) i = 2σ , since the u s given in the next section. Right now, we simply enforce are independent random variables with zero mean value the condition of no energy exchange in the scattering. each. We remind the reader that for electrons in metals, this Thus, the quantum interference term is indeed re- most important condition follows from the fact that nei- duced by the zero-point motion. One may (as several ther the oscillators (which are at their ground states) nor researchers have done) now jump to the wrong conclu- the conduction electron at the Fermi level have any states sion that the zero-point motion indeed reduces the quan- to go down to. This is valid for an arbitrary, interact- tum interference term without apparently affecting the ing, system because there are no states below the ground classical terms. In other words, one would write: state. In the noninteracting fermion picture, this would follow formally, from the Fermi-Pauli factors (ng(1−nf )) that should then be included in Eq. 6. This enforces cl 2 the choice of only the term with f = g in the sum. As hS i =2|AK | . (5) K a result, the δ(ω) part of S(K,ω), which is the elastic This would imply decoherence by the zero-point motion! scattering cross section and the only surviving part of The (superficially valid only) reason for the above mis- S(K,ω), is given here by taken conclusion is that in the classical terms one has iKxu −iKxu he e i = 1. el 2 S ≡ dωS (K,ω)= |A | |hg|[exp(iK xˆ1)+ The fallacy in the above naive thinking is in the ap- K Z K x plication of the last seemingly innocuous equality, Eq. 5. 2 exp(iKxxˆ2)]|gi| (7) Because it automatically includes all the inelastic scat- tering [32], which should not appear. This fact has been We again have a classical term which is a sum of known for at least 45 years [29, 33] and has been re- a term due to x1 and one due to x2, and the in- inforced when the M¨ossbauer effect was discovered and terference term which is the ground-state average of 2 understood (see, for example, [30, 34]). Next, we will 2Re|AK | exp(iKxxˆ1) exp(−iKxxˆ2), yielding the Debye- explain this issue. Waller factor as before. The whole point is now, that each 2 Let us review the derivation of the inelastic nature [29] classical term is given by |AK hg| exp(iKxxi)|gi| , i = of part of the above expressions. The exact elastic scat- 1, 2. Therefore, the classical terms too are reduced tering amplitude AK of the isolated static scatterer is by the same Debye-Waller factors as the interference first parameterized in terms of a suitably defined pseu- term. Thus, again, it is not that the classical terms stay dopotential V (r), whose Fourier transform is VK . As unchanged and the interference is decreased, as would if anticipating the current sophisticated red-herring type be the case for a true decoherence, but both the non- discussions of the perturbative vs the ”nonperturbative” interfering classical part and the interference, ”quantum 4 correction” part are renormalized in the same fashion. 7), which is related to the elastic scattering. We are now Therefore, the reduction of the interference part by the ready to discuss the question of what happens when the coupling to the environment has nothing to do with deco- t → ∞ limit of the above correlators vanishes. The an- herence. Looking at the average of the classical terms, it swer depends on the physical situation considered. In the is instructive to point out that the reduction of the scat- scattering problem, the vanishing of the Debye-Waller tering in a distributed scatterer is due to the addition factor means that there is no strict Bragg-type scatter- of the scattering amplitudes from the various positions ing or M¨ossbauer effect (but interesting things can still occupied by that scatterer. These amplitudes add coher- happen, see for example Refs. [38, 39]). If the inelastic ently because the scattering is elastic. This amplitude scattering is blocked, no scattering remains and the parti- addition is familiar from diffraction theory in optics or cle can only go forward ”unscathed” through the system from antenna theory. (remaining unscattered). However, other situations can exist. One can have a vibrating double-slit system that will stop transmitting altogether. An electron in a solid III. A MORE ADVANCED TREATMENT with phonons can also localize. A sizable increase of the electron’s effective mass was found by Holstein [40] in the For a fuller treatment, van Hove [29] started by writ- small-polaron problem (which is also related to the issues ing Eq. 6 for N scatterers. The term in the square discussed here). Later, Schmid [41] found that for the ~ ˆ phonon problem similar to the one discussed here, with brackets is just i=1,N exp(iK · ~xi), which is just the Fourier transform,P nK of the particle density operator a suitable ”ohmic” spectrum, this mass renormalization ˆ is so severe that the mass diverges for a strong enough nˆ(~x) ≡ i=1,N δ(~x − ~xi). He then used a series of simple butP ingenious manipulations (Fourier-representing coupling, and the particle becomes localized. This effect the energy δ function, inserting a complete set of states appeared later, in the context of what has been called and using the Heisenberg representation for the nK ) to ”Macroscopic Quantum Coherence” [42], for mesoscopic write S(K,ω) as the Fourier transform of the temporal SQUIDS. It should perhaps be reemphasized that this correlation function, hn−K (0)nK(t)i. For phonon sys- mass renormalization and possible ensuing localization tems, where ~ui (the fluctuation of ~xi from its average po- are due to an elastic ”orthogonalization catastrophe” and sition in the lattice) is a linear superposition of harmonic not to decoherence [43]. We shall discuss this further in oscillators, the correlators relevant for inelastic neutron the next section. scattering and treated extensively in that literature in It has become customary to describe the finite resis- the late 50’s (of the last century) [30]), are therefore tance of an electron in a real metal as due to coupling with the Nyquist–Feynmann-Vernon–Zwanzig–Caldeira- Leggett oscillators [44]. We put aside, for the time be-

exp(−Jij (t)) = hexp(iK~ · ~xˆi(t)) exp(−iK~ · ~xˆj (0))i (8) ing, the serious question of how well can a resistance given physically when T → 0 by (mainly) elastic impurity The elastic part of the scattering is given by the t → ∞ scattering, be described by inelastic (and therefore phase- limit (or time-independent) part of this, since the Fourier breaking) processes. Within this model the phase decay transform of a constant is a δ function. In the modern for an electron coupled to a dissipative bath is described so-called P (E) [37] theory of phase fluctuations, similar by the temporal correlator hexp(iφˆ(t)) exp(−iφˆ(0))i. It calculations have been done and similar functions appear is governed [45] when T → 0 by with harmonic oscillators taken to represent the lossy ∞ environment. Looking at the interference between two 1 − cos ωt paths, one again sees immediately that the two-path in- exp[−const dωη(ω) ( )], (9) Z0 ω terference term is affected, but that at the same time the single-path terms are affected in the same fashion. This where η(ω) is the dissipation constant, assumed to have completes our demonstration that zero-point fluctuations a finite ω → 0 limit. We emphasize that this is do not decohere. Some details will still be presented be- the naive result, obtained by neglecting the fact that low. inelastic processes are absent. To set up an anal- It is perhaps useful to point out that the mistake in ogy with a real lattice dynamics problem we note neglecting the Debye-Waller factors in the first terms of that the ”self” correlator for the single-particle density Eqs. 1 and 3 occurs because there one looks at the equal- hexp(iKxˆi(t)) exp(−iKxˆi(0))i = exp(−Jii(t)) at T = 0 time correlators. These are related to the integrals of for a 1D Debye lattice is also (see e.g. Ref. [39]) de- ωD 1−cos ωt S(K,ω) over all frequencies [32]. Now, the inelastic parts caying (see [45]) via exp[(−const 0 dω( ω )], [45]. of S(K,ω) have to be discarded, as discussed above. The Thus, these two problems are mathematicallyR similar. t = 0 correlator (leading to Eq. 5), obtained by neglect- Both have, when T → 0, the same logarithmic long- ing this important constraint, is very different from the time behavior of the exponent, which becomes a power- relevant t → ∞ limit of the correlator (leading to Eq. law decay of the correlator. However, in the latter case, 5 these zero point-effects are very physical – for example, One may now consider a case where, in a sense, there an atom emitting radiation can produce phonons (in the are no classical terms. The sensitivity of the energy lev- non-M¨ossbauer channels) in its recoil. However, as we els of an electron on an Aharonov-Bohm (AB) ring to the explained, the electron on the Fermi level can not de- AB flux is a good example. There is no such sensitivity cay and produce an excitation. Its inelastic channels are in the classical case. This sensitivity (which determines blocked. The amplitude for the particle to go between for example the persistent current through the ring, see, two points is the sum of the amplitudes of a huge num- for example, [4]) follows from the phase shift experienced ber, N , of paths. The probability has two types of terms: by the electron going around the ring enclosing the flux. N classical ones (the absolute value squared of the am- This AB phase shift along such paths determines the flux plitude of a single path) and N 2/2 interference terms sensitivity of, say, the energy levels and thence the persis- (twice the real part of the product of a path amplitude tent current. Obviously, if decoherence is strong enough, with the complex conjugate of the amplitude of a dif- in the sense that the ring’s circumference L is much larger ferent path). Once the constraint of no energy transfer than the electron’s coherence length Lφ, these purely is imposed, there is again no major difference between quantum effects are exponentially reduced. the (Debye-Waller type) reduction factors for the classi- In the Holstein polaron model [40], the effective mass of cal and the interference terms. Hence there is no intrin- the electron increases by the coupling with the zero-point sic decoherence in the Physics of a particle coupled to a environment. Imagine putting such an electron on an AB normal bath at T → 0. Unusual low-energy properties ring. The flux-sensitivity of the energy levels is propor- of the bath might of course still give anomalous decoher- tional to the inverse mass. It will be reduced by the ence at low temperatures. Such anomalies do not exist in increase of the latter. In certain cases [41, 42] the effec- the theoretical models on which the present controversy tive mass diverges (see the previous section) and the flux rages [18, 19, 20, 21, 22, 23]. sensitivity is altogether eliminated. However, describing this electron with the measured, renormalized, effective mass, will eliminate any need to even think about deco- IV. QUANTUM EFFECTS IN CLOSED RINGS, herence. ”ANTIDECOHERENCE”??? Likewise, the coupling of a Fermi-surface electron to a T → 0 oscillator bath will reduce the amplitude of Besides real decoherence, there are other ways to re- its flux-encircling paths by the Debye-Waller type factor, duce quantum interference effects. Averaging over the Eq. 10. Other paths, not encircling the ring, will expe- energy of the incoming particle is often confused with de- rience the corresponding reductions as well, which will coherence, but it should not [4]. Using a variable energy just renormalize the physics of the ring. The magnitude filter one may recover the many interference patterns be- of the AB oscillations will be suppressed by such factors longing to different energies in the beam. This is impos- for the relevant paths. This is the effect found in Ref. sible for decoherence via ”which path” identification. A [24]. Quantum interference is indeed reduced, but this is rather aggressive way to kill quantum interference is, for not decoherence. example, to simply block (at least) one of the two inter- We now prove the incorrectness of identifying the fering paths. Alternatively, if the Fermi-surface electron Debye-Waller reduction of interference as decoherence by interacts with the T → 0 bath only when it goes via one reductio ad absurdum. It is easy to devise couplings to an of the interfering paths , the amplitude of that path will external bath which enhance the effects of quantum inter- be reduced by a Debye-Waller type factor similar to Eq. ference. In a diffusive real mesoscopic ring, the persistent 9 (see [45]), current is well known [4] to increase with the elastic mean free path ℓ. The physics of the conduction electrons is governed by two Debye-Waller renormalizations, that of ∞ 1 − cos ωt the scattering by the impurities (4), due to lattice vibra- exp[−const dωλ(ω) ( )], (10) Z0 ω tions and that of the electrons (9) due basically to the electron-electron interactions. Coming from different in- where λ(ω) contains the coupling constants and the DOS teractions, these two renormalizations are not strongly of the oscillators at frequency ω. The limit of effectively dependent (see below) on each other. Suppose, for sim- ”cutting” the path is achieved when this factor becomes plicity, that all relevant atomic masses are light enough very small. A similar reduction will be achieved if only so that the former is more important for the renormal- one of the scatterers in the simple model of section II, is ized conductivity than the latter. As found in section II, allowed to fluctuate. If the two paths are cut or severely the elastic impurity scattering rate is proportional to the weakened, a decrease of both the interference and the Debye-Waller factor (for further discussions of this, in- classical terms is obtained. It is hoped that the reader cluding the more complicated case of finite temperatures, already understands very well that this reduction is dis- see for example Refs. [49, 50]). Thus, the magnitude of tinct from decoherence. the quantum interference for T → 0 will increase with 6 the decreasing vibrational Debye-Waller factor (4) of the low-temperature decoherence for linear response. impurities. This increase at T = 0 is real, can be huge Of course, should there be special harmonic (or anhar- and may be observable, in principle. Since the effects of monic) modes with an anomalously large DOS at low en- the electron-electron interactions decrease with the elas- ergies, the dephasing rate due to them would be anoma- tic scattering strength [9], the increasing electron mean lously large at the corresponding low temperatures. It, free path will actually decrease the effect of the electronic as well as that by the TLS, should however vanish at Debye-Waller factor (9) and help the vibrational one (4) the strict T → 0 limit, unless a massive ground-state de- dominate. The analogy with the identification of the de- generacy or a pathologically increasing low energy DOS crease of quantum effects via the Debye-Waller factor as would exist. We used here a scattering picture, as in ”decoherence” would now suggest to define this increase the Landauer description: an electron comes from the as ”antidecoherence” by a suitable coupling to zero-point outside and its transmission by the system determines modes of a bath. It is hoped that by now the reader does the conductance of the latter [51]. The Fermi-surface not have to resist the urge to rush to the word processor electron comes and exits at the same energy. Its trans- in order to produce an announcement of this increase of mission amplitude across the system is given by a sum of quantum interference by the coupling to the bath, as a amplitudes, each of which multiplied by a Debye-Waller revolutionary new physical phenomenon, with a range of type factor similar to Eqs. 9,10. Since no excitations important applications. can be left by this electron at the T → 0 metal whose ground state is non-degenerate, no decoherence (in the sense of the interference terms becoming smaller than V. CONCLUSIONS, REAL METALS the ”classical” terms) can follow. This also applies when the conventional low-temperature Fermi liquid picture is Obviously, the residual (T → 0) resistivity of the metal used for the electrons in the conductor. We found that is determined by the elastic scattering off the defects, unless further special ingredients are introduced [52], this renormalized by a Debye-Waller type factor similar to picture is theoretically stable at low enough temperatures the one discussed above, including both the electron- and energies. It should be emphasized that such a scat- phonon and electron-electron interactions. All the low- tering or a quasiparticle picture is convenient in order to temperature physics of the metal is given in terms of have a clear definition of interference and decoherence. this renormalized mean-free-path (and mass). Strictly We did not mention the subtleties associated with mod- speaking, we have basically repeated and explained the els in which the interaction with the same environment arguments of refs [6]. These were devised to refute the is always on. How to define the interfering particle be- ”zero temperature decoherence” by the phonons. While comes a nontrivial issue in that case. Such subtleties are we presented the detailed discussion for a single ”Einstein theoretically interesting but are hardly of much physical model” oscillator (which is an excellent test-case for the relevance. When the electron moves in a real solid the theory), it is clear that the results are valid for an arbi- interactions are of finite range and hence it interacts with trary set of harmonic oscillators to which the conduction different degrees of freedom when it is at different posi- electron is coupled, and for which the theorem about the tions [28]. Therefore the Physics is similar to the scat- gaussian distribution we used is of course valid as well. tering situation [12] and not to models where the same Nothing of principle will change in this argument if the interactions are on all the time. phonon harmonic oscillators are replaced by another set We repeat that we did not rule out, and have of harmonic oscillators, such as those taken by Nyquist not discussed, special situations where some addi- [25], Feynmann and Vernon [3], Zwanzig [26] and Leggett tional new ingredient [52] will cause anomalous decoher- and Caldeira [27] to represent the dissipation in the solid. ence/relaxation for the low-energy electrons; nor have we Moreover, there is no need to use just an ”ohmic” model. discussed the nontrivial experimental situation in depth. An almost arbitrary set of oscillators can be used, includ- Regardless of all that, the statement that decoherence is ing ”subohmic”, ”superohmic” or whatever one pleases due to ”inelastic” processes (change of state of the envi- (see below why we used the word ”almost”). These mod- ronment) is the valid guiding principle for this problem. els are formally solvable as long as the coupling to the No ingenious model or advanced technique can go around oscillators is linear, and the solution is elementary. In that. Delving deeper into models that are argued to vio- the same fashion as with the well-known fact that the late that principle may only be useful in order to expose observed physics of an electron in vacuum automatically their limitations or as diagnostics for the calculation. We includes the effect of the zero-point motion of the lat- finally add that during the preparation of this note, the ter, the observed physics of the electron in a real piece present interpretation was conveyed to F. Guinea and of material automatically has all the renormalizations by found to be completely consistent with his model cal- whatever harmonic modes of that metal that are cou- culations [55]. These are therefore consistent with the pled to the electron. Special low energy excitations, such notion that there is no decoherence in the T → 0 limit. as TLS, magnetic modes, etc., can yield nontrivial [14] The deep question of how good is the description of 7 a real metal via coupling to harmonic oscillators was low. Decoherence is defined via the ”contrast” of the ob- avoided here. Besides not distinguishing between resis- served picture, i.e. by the relative decrease of the latter tance due to elastic or inelastic scattering, that picture compared to the former. is for sure not exact and corrections may well exist and [2] A. Stern, Y. Aharonov, and Y. Imry, Phys. Rev. A41, 3436 (1990) ; and in G. Kramer, ed. Quantum Coher- be relevant [28, 55]. For example, in the real system, ence in Mesoscopic Systems, NATO ASI Series no. 254, there will be (hopefully small) corrections to the gaus- Plenum., p. 99 (1991). sian approximation for the fluctuations [56], which have [3] R. P. Feynmann and F. L. Vernon Jr., Ann. Phys. (N.Y.) not been seriously treated either here or elsewhere. Inde- 24, 118 (1963). pendently of that, the underlying statement that without [4] Y. Imry, Introduction to Mesoscopic Physics, Oxford Un- transfer of excitation there will be no dephasing, should versity Press (1997, 2nd edition 2002). still be valid. [5] N. Kumar, D. V. Baxter, R. Richter and J. O. Strom- olsen Phys. Rev. Lett. 59, 1853 (1987). We briefly summarize the theoretical statements of this [6] J. Rammer, A. L. Shelankov and A. Schmid, Phys. Rev. note. The coupling of a particle that can not lose en- Lett. 60, 1985 (1988); G. Bergmann, Phys. Rev. Lett. ergy to environmental degrees of freedom at T → 0, can 60, 1986 (1988). modify, sometimes very seriously, the quantum behavior. [7] P. Mohanty, E.M. Jariwala and R.A. Webb, Phys. Rev. This can go in either direction and is not decoherence. Lett. 78, 3366 (1997); P. Mohanty and R.A. Webb, Phys. Rev. B55, 13452 (1997); R.A. Webb, P. Mohanty and On the other hand, the experiments still point out to E.M. Jariwala in Quantum Coherence and Decoherence, some nontrivial effects that are not understood as yet, in proceedings of ISQM, Tokyo (1998), K. Fujikawa and Y. the low-temperature Physics of disordered metals. A. Ono, eds. North Holland, Amsterdam (2000). [8] G. Bergmann, Phys. Reports 107, 1 (1984), an excellent review of weak-localization. [9] B.L. Altshuler, A.G. Aronov, and D.E. Khmelnitskii, J. Acknowledgements Phys. C15, 7367 (1982) . [10] Besides using Fermions, there are other ways to block the inelastic scattering from T → 0 systems. A simple The author was fortunate to learn about the Debye- and particularly transparent one was suggested to the Waller factor from Harry (Zvi) J. Lipkin and from the author by H. J. Lipkin, private communication. The low- late Solly G. Cohen and Israel Pelah and to collabo- est excited state of the scatterer can be taken to have rate on decoherence with Yakir Aharonov and Ady Stern. an excitation energy larger than the initial energy of the Past discussions with Walter Kley and (later) with Ted scattered particle. Schultz and collaboration with Leon Gunther and Benny [11] Actually, in more sophisticated descriptions, the lin- Gavish on these issues at low dimension have been very ear transport coefficient (e.g conductance) is described via equilibrium properties (i.e. dynamic correlation func- helpful as well. Gerd Sch¨on is thanked for useful corre- tions) of the system. Hence, there is absolutely no need spondence which helped clarify the issue. More recent to invoke nonequilibrium Physics. discussions with Michael Tinkham, with Richard Webb, [12] D. Cohen and Y. Imry, Phys. Rev. B59, 11143 (1999). with Doron Cohen and with Zvi Ovadyahu, whose exper- [13] Y. Imry, in Quantum Coherence and Decoherence, pro- iments have highlighted a very relevant issue which has ceedings of ISQM, Tokyo (1998), K. Fujikawa and Y. A. been posed already in Ref. [7]), as well as with Uri Gav- Ono, eds. North Holland, Amsterdam (2000). ish and Yehoshua Levinson, are gratefully acknowledged. [14] Y. Imry, H. Fukuyama and P. Schwab, Europhys. Lett, 47, 608 (1999). The writing of this note began when the author was in [15] Yu. B. Khavin, M. E. Gershenson and A. L. Bogdanov, the Nanoscience program at the Institute for Theoretical Phys. Rev. Lett., 81, 1066 (1998), Phys. Rev. B58, 8009 Physics, Santa Barbara and was supported there by the (1998); M. E. Gershenson et al., Sov. Phys. Uspekhi 41 NSF Grant PHY99-07949. Work at the Weizmann Insti- (2), 186 (1998). tute was supported by a Center of Excellence of the Israel [16] A. B. Gougam, F. Pierre, H. Pothier, D. Esteve and N. Science Foundation, Jerusalem and by the German Fed- O. Birge, J. Low Temp. Phys. 118, 447 (2000); see also: eral Ministry of Education and Research (BMBF) within F. Pierre, H. Pothier, D. Esteve and M. H. Devoret, ibid.; F. Pierre, Ph.D. thesis, Universit´eParis VI (2000), un- the Framework of the German-Israeli Project Coopera- published. tion (DIP). [17] Z. Ovadyahu, Phys. Rev. B63, 235403 (2001). [18] D.S. Golubev and A.D. Zaikin, Phys. Rev. Lett., 81, 1074 (1998); Phys. Rev. B59, 9195 (1999); Phys. Rev. Lett., 82, 3191 (1999); cond-mat/0103166. [19] I. L. Aleiner, B. L. Altshuler, M. E. Gershenson, Waves [1] We shall use here the terms ”dephasing” and ”decoher- in Random Media 9, 201 (1999); Phys. Rev. Lett., 82, ence” interchangeably and will not even attempt to de- 3190 (1999). cide which of them is more appropriate. The clearest [20] B. L. Altshuler, M. E. Gershenson, I. L. Aleiner, Physica definition of such a process is provided by a two-wave A3, 58 (1998). interference situation where the intensity is a sum of a [21] V. Ambegaokar and J. von Delft, private com- classical term and an interference term, as in Eq. 1 be- munication, see: http://www.theorie.physik.uni- 8

muenchen.de/∼vondelft/dephasing. particle whose velocity operator isv ˆ, by coupling it lin- [22] D.S. Golubev, A.D. Zaikin, Gerd Sch¨on On Low- early to a continuum of harmonic oscillators. The cou- Temperature Dephasing by Electron-Electron Interaction, pling to the oscillator having a very low frequency ωλ cond-mat/0110495, to be published in J. Low Temp. and position operatorx ˆλ, is taken to be cλvˆxˆλ . This

Phys. looks like a coupling to a vector potential λ cλxˆλ. For [23] T.R. Kirkpatrick and D. Belitz, Absence of electron de- a quick and dirty derivation one may followP the Onsager- phasing at zero temperature, cond-mat/0111398; Dmitri type idea of using the macroscopic laws to describe the S. Golubev, Andrei D. Zaikin, Gerd Sch¨on Comment behavior of the fluctuations. We thus write the fluctu- on ”Absence of electron dephasing at zero temperature”, ating v as the linear response to the fluctuating instan- cond-mat/0111527; T.R. Kirkpatrick and D. Belitz, Re- taneous value of the driving field, taken, in the semi- ply to comment on ”Absence of electron dephasing at zero classical case, to be a c-number. Expressing the elec- temperature”, cond-mat/0112063. tric field as the time-derivative of the vector potential, [24] P. Cedraschi, V. V. Ponomarenko and M. B¨uttiker, Phys. it follows that the power spectrum ofv ˆ is proportional 2 2 2 2 2 Rev. Lett. 84, 346 (2000) and Ann. Phys. (NY), in press. to µ λ ωλ|cωλ | (xλ)ω, where (xλ)ω is the power spec- [25] H. Nyquist, Phys. Rev. 32, 110 (1928). trumP of xλ. We remember that the latter is given by h¯ [26] R. Zwanzig, J. Stat. Phys. 9, 215 (1973). Mω [δ(ω+ωλ)(ν+1/2)+δ(ω−ωλ)ν]. Here ν is the average −1 [27] A. O. Caldeira and A. O. Leggett, Ann. Phys. (N.Y.) 149 number of excited oscillator quanta [exp(¯hω/kBT )−1] . 374; 153, 445(E) (1984). The fluctuation-dissipation theorem [47] relates [48] µ [28] D. Cohen J. Phys. A31, 8199 (1998); Phys. Rev. E55, with the (negative frequency at T → 0, and in general 1422 (1997); Phys. Rev. Lett. 78, 2878 (1997). cond- the difference betwen the negative and positive frequency mat/0103279 , to be published in Phys. Rev. E. branches of the) power-spectrum ofv ˆ, divided by ω. It [29] L. van Hove, Phys. Rev 95, 249 (1954). In this classic follows that paper, the dynamic correlation function for the density 2 M hnˆq(0)ˆn−q (t)i, was introduced, it was shown that its time |cλ| δ(ω − ωλ) ∝ . Fourier transform, S(q,ω), yields the inelastic Born scat- X ¯hµ tering cross section from the system, the properties of the If the coupling is written as: λ Cλxˆxˆλ, then the coeffi- latter were analyzed and the connection to the dissipative 2 2 2 response indicated. cients are related by |Cλ| = |Pcλ| ω . The usual explana- [30] C. Kittel, Quantum Theory of Solids Wiley (1963). tion of this model goes by noting that the excitation of [31] A. Messiah, Quantum Mechanics, North Holland, Ams- these low-frequency oscillators provides the viscous force terdam, 1961. on the slow particle. The (related) understanding employ- [32] This is, stricly speaking, valid under the proviso that ing the fluctuation-dissipation theorem, pictures the par- adding to the scattered particle a few quanta of the har- ticle as coupled to a set of oscillators designed to enable monic oscillator does not change its wavevector signif- the system to have a low-frequency absorption mimicking icantly, this has been termed by van Hove ”the static that of a diffusing particle. approximation”, very well valid for x-ray scattering, but [45] Actually, in the quantum regime the correlator in the not, for example, for slow neutron scattering in solids. time domain develops an imaginary part due to the non- This is however a technical detail which is irrelevant to commutativity of the exponent in the operator exp[iφ(t)] our discussion with itself at two different times. This results in an asym- [33] A. C. Zemach and R. J. Glauber Phys. Rev. 101, 118,129 metry in the frequency domain [46]. As a result, a term (1956). of the type i sin ωt is added [30, 37] to the cos ωt in Eq. [34] H.J. Lipkin, Quantum Mechanics, North-Holland, Ams- 9. At T = 0 this converts Eq. 9 to: terdam, 3rd printing (1992). ∞ ′ 1 − exp iωt [35] E. Fermi, Ricerca Scientifica 7, 13 (1936). exp[−const dωη(ω)( )], (11) [36] One might do better and include, in principle, the inter- Z−∞ ω scatterer multiple scattering, by using a pseudopotential which can be seen to eliminate the ω < 0 portion of specific to the given structure. This is however totally the power spectrum of exp[iφ(t)]. This portion yields irrelevant to our discussion here. We only mention it be- the energy transfer from the oscillators to the particle. cause of the misuse of the term ”nonperturbative” by Such a transfer is ruled out when the oscillators are at some of the participants in the current controversy. T = 0. At finite temperatures this imaginary part is what [37] See e.g. G.-L. Ingold and Yu. V. Nazarov, in Single brings about the correct (detailed-balance type) ratio of Charge Tunneling, eds. H. Grabert and M. Devoret, the energy-gain and energy-loss branches of the power Plenum 1992, Chapter 2. spectrum. It is not crucial, however, for the Debye-waller [38] Y. Imry and L. Gunther, Phys. Rev. B3, 3939 (1971). factor, which is the focus of our discussion here. [39] Y. Imry and B. Gavish, J. Chem. Phys. 61, 1554 (1974), [46] U. Gavish, Y. Levinson, and Y. Imry, Detection of quan- considered a combination of 1D phonons and dissipation, tum noise, Phys. Rev. B RC, R10637 (2000). in the classical limit. [47] R. Kubo, J. Phys. Soc. Japan 12, 570 (1957); 17, 975 [40] T. Holstein, Ann. Phys. (N.Y.) 8, 343 (1959). (1962). [41] A. Schmid, Phys. Rev. Lett. 51, 15061509 (1983) [48] U. Gavish, Y. Levinson and Y. Imry Quantum noise, de- [42] A. J. Leggett et al., Rev. Mod. Phys. 59, 1 (1987). tailed balance and Kubo formula in non-equilibrium quan- [43] A. Stern, Ph.D. Thesis, Tel-Aviv University (1990), un- tum systems, to be published in the Proceedings of the published. 2001 Rencontres de Moriond: Electronic Correlations: [44] The Feynmann-Vernon–Caldeira-Leggett picture models from Meso- to Nanophysics, T. Martin, G. Montambaux the finite low-frequency mobility, µ ≡ µ(ω → 0), of a and J. Trˆan Thanh Vˆan eds. EDPSciences 2001. 9

[49] Yu. Kagan and A. P. Zhernov, Zh. Eksp. Teor. Fiz. 50, there may exist factors that can destabilize these devi- 1107 (1966) [Sov. Phys. JETP 23, 737 (1966)]. ations from the FLT at low enough temperatures (such [50] P. M. Echternach, M. E. Gershenson and H. M. Bozler, as as a small interchannel coupling or a finite separation Phys. Rev. B 47, 13659 (1993). between the two states [53] in the two-channel Kondo [51] It should perhaps be emphasized that in spite of claims model, a finite length of the Luttinger liquid). Likewise, otherwise, the Landauer (two-terminal) formulation is disordered systems that have theoretically a ground-state equivalent to the linear response theory for the same ge- degeneracy (for example, the ice model [54] or glasses) ometry, at least at T → 0. We have in mind a finite sam- may have a small real-life splitting of that degeneracy or ple, with arbitrary interactions, connected by ideal and never experience it within any finite measurement time. noninteracting leads to particle reservoirs. The Landauer On the other hand, situations where truly robust devi- conductance is defined by the transmission of electrons ations from the FLT as T → 0 exist are obviously of across this sample. The linear response conductance can genouine interest. be evaluated for the same model. Their equivalence as [53] A. Zawadovski, Jan von Delft and D. C. Ralph, Phys. T → 0 for an interacting system was proven by Y. Meir Rev. Lett. 83, 2632 (1999); see also: I. L. Aleiner, B. L. and N. Wingreen, Phys. Rev. lett. 68, 2512 (1992) [See Altshuler, Y. M. Galperin and T. A. Shutenko 86, 2629 also: T. K. Ng and P. A. Lee, Phys. Rev. Lett. 61, 1768 (2001), cond-mat/0007430. (1988)]. The case of finite temperatures where inelastic [54] L. Pauling, The Nature of the Chemical Bond, Cornell scattering may occur is more complicated, but it is not University Press, Ithaca, NY (1960), 3rd edn.; G.C. Pi- relevant to the present discussion, which concerns the mental and A.L. McClellan, The Hydrogen Bond, Free- T → 0 (no inelastic scattering) limit only. This finite man, San Francisco (1960). temperature case will hopefully be treated in a future [55] F. Guinea, Aharonov-Bohm oscillations of a particle publication. coupled to dissipative environment. cond-mat/0112099, [52] In situations with such anomalous low-energy behavior, (2001). the Fermi liquid theory will break down at low temper- [56] It has been pointed out to the author by D. E. atures. Two canonical examples for that are the two- Khmel’nitskii that the treatments of Refs. [9] and [2] are channel Kondo model and the Luttinger liquid. It is per- valid only within the gaussian approximation for the fluc- haps helpful to point out, however, that in the real world, tuations.