Weak localization, Aharonov–Bohm oscillations and decoherence in arrays of quantum dots

Combining scattering matrix theory with non-linear σ-model and Keldysh technique we develop a unified theoretical approach enabling one to non-perturbatively study the effect of electron–electron interactions on weak localization and Aharonov–Bohm oscillations in arbitrary arrays of quantum dots. Ou...

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Date:	2010
Main Authors:	Golubev, D.S., Semenov, A.G., Zaikin, A.D.
Format:	Article
Language:	English
Published:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2010
Series:	Физика низких температур
Subjects:	Quantum coherent effects in superconductors and normal metals
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/117519
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Cite this:	Weak localization, Aharonov–Bohm oscillations and decoherence in arrays of quantum dots / D.S. Golubev, A.G. Semenov, A.D. Zaikin // Физика низких температур. — 2010. — Т. 36, № 10-11. — С. 1163–1183. — Бібліогр.: 64 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-1175192025-02-09T16:34:23Z Weak localization, Aharonov–Bohm oscillations and decoherence in arrays of quantum dots Golubev, D.S. Semenov, A.G. Zaikin, A.D. Quantum coherent effects in superconductors and normal metals Combining scattering matrix theory with non-linear σ-model and Keldysh technique we develop a unified theoretical approach enabling one to non-perturbatively study the effect of electron–electron interactions on weak localization and Aharonov–Bohm oscillations in arbitrary arrays of quantum dots. Our model embraces weakly disordered conductors, strongly disordered conductors and (iii) metallic quantum dots. In all these cases at T→0 the electron decoherence time is found to saturate to a finite value determined by the universal formula which agrees quantitatively with numerous experimental results. Our analysis provides overwhelming evidence in favor of electron–electron interactions as a universal mechanism for zero temperature electron decoherence in disordered conductors. This work was supported in part by RFBR grant 09-02-00886. 2010 Article Weak localization, Aharonov–Bohm oscillations and decoherence in arrays of quantum dots / D.S. Golubev, A.G. Semenov, A.D. Zaikin // Физика низких температур. — 2010. — Т. 36, № 10-11. — С. 1163–1183. — Бібліогр.: 64 назв. — англ. 0132-6414 PACS: 73.23.–b, 73.21.La, 73.20.Fz https://nasplib.isofts.kiev.ua/handle/123456789/117519 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Quantum coherent effects in superconductors and normal metals Quantum coherent effects in superconductors and normal metals
spellingShingle	Quantum coherent effects in superconductors and normal metals Quantum coherent effects in superconductors and normal metals Golubev, D.S. Semenov, A.G. Zaikin, A.D. Weak localization, Aharonov–Bohm oscillations and decoherence in arrays of quantum dots Физика низких температур
description	Combining scattering matrix theory with non-linear σ-model and Keldysh technique we develop a unified theoretical approach enabling one to non-perturbatively study the effect of electron–electron interactions on weak localization and Aharonov–Bohm oscillations in arbitrary arrays of quantum dots. Our model embraces weakly disordered conductors, strongly disordered conductors and (iii) metallic quantum dots. In all these cases at T→0 the electron decoherence time is found to saturate to a finite value determined by the universal formula which agrees quantitatively with numerous experimental results. Our analysis provides overwhelming evidence in favor of electron–electron interactions as a universal mechanism for zero temperature electron decoherence in disordered conductors.
format	Article
author	Golubev, D.S. Semenov, A.G. Zaikin, A.D.
author_facet	Golubev, D.S. Semenov, A.G. Zaikin, A.D.
author_sort	Golubev, D.S.
title	Weak localization, Aharonov–Bohm oscillations and decoherence in arrays of quantum dots
title_short	Weak localization, Aharonov–Bohm oscillations and decoherence in arrays of quantum dots
title_full	Weak localization, Aharonov–Bohm oscillations and decoherence in arrays of quantum dots
title_fullStr	Weak localization, Aharonov–Bohm oscillations and decoherence in arrays of quantum dots
title_full_unstemmed	Weak localization, Aharonov–Bohm oscillations and decoherence in arrays of quantum dots
title_sort	weak localization, aharonov–bohm oscillations and decoherence in arrays of quantum dots
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2010
topic_facet	Quantum coherent effects in superconductors and normal metals
url	https://nasplib.isofts.kiev.ua/handle/123456789/117519
citation_txt	Weak localization, Aharonov–Bohm oscillations and decoherence in arrays of quantum dots / D.S. Golubev, A.G. Semenov, A.D. Zaikin // Физика низких температур. — 2010. — Т. 36, № 10-11. — С. 1163–1183. — Бібліогр.: 64 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT golubevds weaklocalizationaharonovbohmoscillationsanddecoherenceinarraysofquantumdots AT semenovag weaklocalizationaharonovbohmoscillationsanddecoherenceinarraysofquantumdots AT zaikinad weaklocalizationaharonovbohmoscillationsanddecoherenceinarraysofquantumdots
first_indexed	2025-11-27T23:54:48Z
last_indexed	2025-11-27T23:54:48Z
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fulltext	© D.S. Golubev, A.G. Semenov, and A.D. Zaikin, 2010 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11, p. 1163–1183 Weak localization, Aharonov–Bohm oscillations and decoherence in arrays of quantum dots D.S. Golubev1, A.G. Semenov2, and A.D. Zaikin1,2 1Institute for Nanotechnology, Karlsruhe Institute of Technology (KIT), Karlsruhe 76021, Germany E-mail: andrei.zaikin@kit.edu 2I.E. Tamm Department of Theoretical Physics, P.N. Lebedev Physics Institute, Russian Academy of Sciences 53 Leninskii Pr., Moscow 119991, Russia Received May 7, 2010 Combining scattering matrix theory with non-linear σ-model and Keldysh technique we develop a unified theoretical approach enabling one to non-perturbatively study the effect of electron–electron interactions on weak localization and Aharonov–Bohm oscillations in arbitrary arrays of quantum dots. Our model embraces weakly disordered conductors, strongly disordered conductors and (iii) metallic quantum dots. In all these cases at T → 0 the electron decoherence time is found to saturate to a finite value determined by the universal formula which agrees quantitatively with numerous experimental results. Our analysis provides overwhelming evidence in favor of electron–electron interactions as a universal mechanism for zero temperature electron decoherence in disordered conductors. PACS: 73.23.–b Electronic transport in mesoscopic systems; 73.21.La Quantum dots; 73.20.Fz Weak or Anderson localization. Keywords: quantum interference of electrons, Aharonov–Bohm oseillations, quantum dots. 1. Introduction Quantum interference of electrons is a fundamentally important phenomenon which can strongly influence on the electron transport in disordered conductors [1–3]. Quantum coherent effects are mostly pronounced at low temperatures in which case certain interaction mechanisms are «frozen out» and, hence, do not anymore limit the ability of electrons to interfere. However, there exists at least one mechanism, electron–electron interactions, which remains important down to lowest temperatures and may destroy quantum interference of electrons down to = 0T . In a series of papers [4] two of the present authors formu- lated a general theoretical formalism which allows to de- scribe electron interference effects in the presence of dis- order and electron–electron interactions at any temperature, including the limit 0T → . This approach extends Chakravarty–Schmid description [2] of weak loca- lization (WL) and generalizes Feynman–Vernon path integral influence functional technique [5] to fermionic systems with disorder and interactions. With the aid of this technique it turned out to be possible to quantitatively ex- plain low temperature saturation of WL correction to con- ductance ( )WLG Tδ commonly observed in diffusive me- tallic wires [6,7]. It was demonstrated [4] that this satura- tion effect is caused by electron–electron interactions. It is worth pointing out that low temperature saturation of WL correction and of the electron decoherence time ϕτ (extracted from ( )WLG Tδ or by other means) has been repeatedly observed not only in metallic wires but also in virtually any type of disordered conductors ranging from individual quantum dots [8] to very strongly disordered 3D structures and granular metals [9]. Hence, it is plausi- ble that in all these systems we are dealing with the same fundamental effect of electron–electron interactions. In order to test this conjecture it is necessary to develop a unified theoretical description which would cover essen- tially all types of disordered conductors. Although the approach [4] is formally an exact procedure treating elec- tron dynamics in the presence of disorder and interactions, in some cases, e.g., for quantum dots and granular metals, it can be rather difficult to directly evaluate ( )WLG Tδ within this technique. One of the problems in those cases is that the descrip- tion in terms of quasiclassical electron trajectories may become insufficient, and electron scattering on disorder D.S. Golubev, A.G. Semenov, and A.D. Zaikin 1164 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 should be treated on more general footing. In addition, within the approach [4] disorder averaging is (can be) postponed until the last stage of the calculation which is convenient in certain physical situations. In other cases — like ones studied below — it might be, in contrast, more appropriate to perform disorder averaging already in the beginning of the whole analysis. Finally, it is desirable to deal with the model which would embrace various types of conductors with well defined properties both in the long and short wavelength limits. Below we will elaborate an alternative approach which combines the scattering matrix and Keldysh techniques with the description of electron–electron interactions in terms of quantum Hubbard–Stratonovich fields. Note that previously a similar type of approach was employed in order to de- scribe Coulomb effects in tunnel junctions, see, e.g., [10,11]. Here we will describe a disordered conductor by means of an array of (metallic) quantum dots connected via junctions (scatterers) with an arbitrary distribution of transmissions of their conducting channels. This model will allow to easily crossover between the limits of granular metals and those with point-like impurities and to treat spatially restricted and spatially extended conductors within the same theoretical framework. Electron scattering on each such scatterer will be treated within the most general scattering matrix formal- ism [12,13] adopted to include electron–electron interaction effects [14–21]. Averaging over disorder will be performed within the non-linear σ-model technique in Keldysh formu- lation. This method has certain advantages over the imagi- nary time approach since it allows to treat both equilibrium and non-equilibrium problems and also enables one to in- clude Coulomb interaction between electrons in a straight- forward manner [22]. In this paper we will review and extend our analysis of weak localization effects and Aharonov–Bohm oscillations in systems composed of metallic quantum dots [23–27]. In Sec. 2 we will construct a theory for essentially non- interacting electrons including interaction effects only phenomenologically by introducing an effective electron dephasing time ϕτ as an independent parameter. In Sec. 3 we will develop a systematic unified analysis of the effect of electron–electron interactions on weak localization and Aharonov–Bohm oscillations in both quantum dots and extended diffusive conductors. Section 4 is devoted to a comparison of our results with experimental observations. 2. Weak localization in quantum dot arrays 2.1. The model and basic formalism Let us consider a 1d array of connected in series chaotic quantum dots (Fig. 1). Each quantum dot is characterized by its own mean level spacing nδ . Adjacent quantum dots are connected via barriers which can scatter electrons. Each such scatterer is described by a set of transmissions of its conducting channels ( )n kT (here k labels the chan- nels and n labels the scatterers). Below we will ignore spin-orbit scattering and focus our attention on the case of 1D arrays. If needed, generalization of our analysis to sys- tems of higher dimensions can be employed in a straight- forward manner [23]. An effective action [ ]S Q of an array depicted in Fig. 1 depends on the fluctuating 4 4× matrix fields [19,23] 1 2( , )nQ t t defined for each of the dots ( = 1,..., 1n N − ). Each of these fields is a function of two times 1t and 2t and obeys the normalization condition 2 = 1.nQ (1) The action of an array can be represented as a sum of two terms [ ] = [ ] [ ].d tiS Q iS Q iS Q+ (2) The first term, [ ],diS Q describes the contribution of bulk parts of the dots. This term reads 1 2 2 =1 [ ] = Tr ([ , ]) . N d n n n nn iS Q Q H A Q t − π ∂⎡ ⎤− α⎢ ⎥δ ∂⎣ ⎦ ∑ (3) Here H is an external magnetic filed, =nα 2 2 2 2( / ) min{ , },n F n e nb e c d l d= v nb is a geometry de- pendent numerical prefactor [13], nd is the size of nth dot, el is the elastic mean free path in the dot, and A is 4 4× matrix: 1 0 0 0 0 1 0 0 = . 0 0 1 0 0 0 0 1 A ⎛ ⎞ ⎜ ⎟−⎜ ⎟ ⎜ ⎟ ⎜ ⎟ −⎝ ⎠ (4) The second term in Eq. (2), [ ],tiS Q describes electron transfer between quantum dots. It has the form [28] ( ) 1 =1 1[ ] = Tr ln 1 ({ , } 2) . 2 4 nN k t n n n k T iS Q Q Q− ⎡ ⎤ ⎢ ⎥+ − ⎢ ⎥⎣ ⎦ ∑∑ (5) Note that here the magnetic field H is included only in the term (3) describing the quantum dots while it is ignored in the term (5). Usually this approximation remains appli- cable at not too low magnetic fields. An equilibrium saddle point configuration 1 2( )t tΛ − of the matrix field 1 2( , )Q t t depends only on the time differ- Fig. 1. 1D array of N – 1 quantum dots coupled by N barriers. Each quantum dot is characterized by mean level spacing nδ . Each barrier is characterized by a set of transmissions of its con- ducting channels ( ).n kT ... d1 d2 dN–2 dN–1 T k (1) T k (2) T k (3) T k (N – 2) T k (N – 1) T k (N) Weak localization, Aharonov–Bohm oscillations and decoherence in arrays of quantum dots Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 1165 ence and has the form 1 0 0 0 0 1 0 0 ( ) = e , ( ) 0 1 02 0 ( ) 0 1 iEt K K dEt g E g E − −⎛ ⎞ ⎜ ⎟ ⎜ ⎟Λ ⎜ ⎟π ⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ∫ (6) where ( ) = 2[1 2 ( )] = 2 tanh ( / 2 ).K Fg E f E E T− This choice of the saddle point corresponds to the following structure of the 4 4× matrix Green function :G * * * 0 0 0 0 0 = . 0 0 0 0 A A K R K R G G G G G G G ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟− ⎜ ⎟⎜ ⎟ ⎝ ⎠ T T T T T T (7) Here we defined the time inversion operator :T ( ) = ( ),ff t f t t−T (8) where ft will be specified later. Note that the function G in Eq. (7), defined for a given disorder configuration, should be contrasted from the Green function 12 = 2 2Q e iG i Q t m − ⎡ ⎤∂ ∇ + +⎢ ⎥ ∂ τ⎢ ⎥⎣ ⎦ (9) defined for a given realization of the matrix field Q . In Eq. (9) we also introduced the electron elastic mean free time .eτ 2.2. Gaussian approximation In order to evaluate the WL correction to conductance we will account for quadratic (Gaussian) fluctuations of the matrix field nQ . This approximation is always suffi- cient provided the conductance of the whole sample ex- ceeds 2 / ,e h in certain situations somewhat softer applica- bility conditions can be formulated. Expanding in powers of such fluctuations we introduce the following paramete- rization = e eiW iWn nnQ −Λ = 2 31= [ , ] { , } ( ). 2n n n ni W W W W O WΛ + Λ + Λ − Λ + (10) It follows from the normalization condition (1) that only 8 out of 16 matrix elements of W are independent parame- ters. This observation provides certain freedom to choose an explicit form of this matrix. A convenient parameteriza- tion to be used below is 1 1 2 2 1 1 1 2 2 2 0 0 0 0 = . 0 0 0 0 n n n n n n n n n n n u b u b W a b a b ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟+ ⎜ ⎟ +⎝ ⎠ v v (11) With this choice the quadratic part of the action takes the form (2)(2) (2)= [ , ] [ , ],uabiS iS a b iS u+ v v (12) where (2)[ , ]abiS a b does not depend on H and describes diffuson modes, while (2)[ , ]uiS uv v is sensitive to the mag- netic field and is responsible for the Cooperons. The diffu- son part of the action (2)[ , ]abiS a b was already analyzed be- fore [19] and will be omitted here. Below we will focus our attention on the Cooperon contribution which reads 1 (2) 2 1 2 1 2 =1 2[ , ] = Tr[ [ , ] 16 ] N u n n n nn iS u u u H u u t − π ∂ − α + δ ∂∑v v 1 2 2 1 1 2 =1 2 Tr[ [ , ] 16 ] N n n n nn H t − π ∂ + − α − δ ∂∑ v v v v 1 1, 1 2 2, 1 =1 Tr [( )( ) 2 N n n n n n n g u u u u− −− − − +∑ 1 1, 1 2 2, 1( )( )],n n n n− −+ − −v v v v (13) where ( ) 2= 2 = 2 /n n nk k g T e Rπ∑ is the dimensionless conductance of nth barrier. With the aid of the action (13) we can derive the pair correlators of the fields 1,2u and 1,2 :v 1 1 2 2 1 2 1 2( , ) ( , ) = ( , ) ( , )n m n mu t t u t t t t t t′ ′′ ′ ′′〈 〉 〈 〉 =v v 1 2 1= ( ) ( ) , 2 m nmt t t t C t t δ ′ ′′ ′′δ − + − − π (14) where we defined a discrete version of the Cooperon ( )nmC t obeying the equation 1 1 1 [( ) 4 n nm n n nm Hn n C g g C t + ϕ ⎛ ⎞ δ∂ + + + + −⎜ ⎟⎜ ⎟∂ τ τ π⎝ ⎠ 1, 1 1, ] = ( ).n n m n n m nmg C g C t− + +− − δ δ (15) This equation should be supplemented by the boundary condition ( ) = 0nmC t which applies whenever one of the indices n or m belongs to the lead electrode. Here 2= 1/16Hn n Hτ α is the electron dephasing time due to the magnetic field. In Eq. (15) we also introduced an addition- al electron decoherence time in nth quantum dot nϕτ which can remain finite in the presence of interactions. In this section we account for electron decoherence only phe- nomenologically by keeping the parameter nϕτ in the equ- ation for the Cooperon. Rigorous description of quantum decoherence by electron–electron interactions will be car- ried out in Sec. 3. 2.3. Weak localization corrections to conductance Let us now derive an expression for WL correction to the conductance in terms of the fluctuating fields u and v. In what follows we will explicitly account for the discrete D.S. Golubev, A.G. Semenov, and A.D. Zaikin 1166 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 nature of our model and specify the WL correction for a single barrier in-between two adjacent quantum dots in the array. We start, however, from the bulk limit, in which case the Kubo formula for the conductivity tensor αβσ reads ( , ') = ( ) t i dt t tαβ −∞ ′ ′σ − − ×∫r r ( , ') ( , ) ( , ) ( , ') .j t j t j t j tβ α α β′ ′× 〈 − 〉r r r r (16) Following the standard procedure [1,2], approximating the Fermi function as ( ) / ( )Ff E E E−∂ ∂ ≈ δ (which effectively implies taking the low temperature limit) and using a phe- nomenological description of interactions as mediated by external (classical) fluctuating fields [29], from Eq. (16) one can derive the WL correction in the form: 2 2( , ') = 4 t WL e dt dt m αβ −∞ ′ ′′δσ − × π ∫ ∫r r = = ' = ' = '' '1 2 1 21 2 1 2 ( ) ( )β βα α× ∇ −∇ ∇ −∇ ×r r r r r rr r r r 1 2 1 2 dis, max, cross ( , ; , ' ) ( , ' ; , ) ,R AG t t G t t′′ ′× r r r r (17) which implies summation over all maximally crossed dia- grams, as indicated in the subscript. At the same time, av- eraging over fluctuations of Q within Gaussian approxi- mation is equivalent to summing over all ladder diagrams. Since we are not going to go beyond the above approxima- tion, we need to convert maximally crossed diagrams in Eq. (17) into the ladder ones. Technically this conversion can be accomplished by an effective time reversal proce- dure for the advanced Green function which can be illu- strated as follows. Consider, e.g., the second order correction to AG in the disorder potential dis ( )U x 2 (2) 3 3 1 2 2 1 2 1( , ' ; , ) = t A t t G t t i d d d d τ ′ ′ ′δ − τ τ ×∫ ∫ ∫r r x x 1 1 1 dis 1 1 1 2 2( , ' ; , ) ( ) ( , ; , )A AG t U G′× τ τ τ ×r x x x x dis 2 2 2 2( ) ( , ; , ).AU G t× τx x r (18) Making use of the property * 1 2 2 1( , ) = ( , ),A RG X X G X X we get 2 (2) 3 3 1 2 2 1 2 1( , ' ; , ) = t A t t G t t i d d d d τ ′ ′ ′δ − τ τ ×∫ ∫ ∫r r x x * * 2 2 2 dis 2 2 2 1 1( , ; , ) ( ) ( , ; , )R RG t U G× τ τ τ ×r x x x x * dis 1 1 1 1( ) ( , ; , ' ) .RU G t ′× τx x r (19) Setting = ,ft t t′+ we rewrite this expression as fol- lows 2 (2) 1 2 2 1( , ' ; , ) = t tf A t t t tf f G t t i d d ′− τ − − ′δ − τ τ ×∫ ∫r r 3 3 * 2 1 2 2 2( , ; , )R fd d G t t′× − τ ×∫ x x r x * dis 2 2 2 1 1( ) ( , ; , )RU G× τ τ ×x x x * dis 1 1 1 1( ) ( , ; , ' ).R fU G t t× τ −x x r (20) Close inspection of the right hand side of Eq. (20) allows to establish the following relation (2) (2) * 1 2 2 1( , ' ; , ) = ( , ; , ' ) ,A RG t t G t t′ ′δ δr r r rT T (21) which turns out to hold in all orders of the perturbation theory in dis .U As before, the time inversion operator T is defined in Eq. (8) with = .ft t t′+ As a result, the expression for WL αβδσ takes the form: 2 2( , ') = 4 t WL e dt dt m αβ −∞ ′ ′′δσ − × π ∫ ∫r r = = ' = ' = '' '1 2 1 21 2 1 2 ( ) ( )β βα α× ∇ −∇ ∇ −∇ ×r r r r r rr r r r * 1 2 2 1 dis, ladder ( , ; , ' ) ( , ; , ' ) .R RG t t G t t′′ ′× r r r rT T (22) Rewriting Eq. (22) in terms of the matrix elements of the Green function (7), we obtain 2 2( , ') = 4 t WL e dt dt m αβ −∞ ′ ′′δσ − × π ∫ ∫r r = = ' = ' = '' '1 2 1 21 2 1 2 ( ) ( )β βα α× ∇ −∇ ∇ −∇ ×r r r r r rr r r r 33 1 2 44 2 1 dis, ladder( , ; , ' ) ( , ; , ' ) .G t t G t t′′ ′× r r r r (23) Our next step amounts to expressing WL correction via the Green function QG (9). For that purpose we will use the following rule of averaging 33 1 2 44 2 1 dis( , ; , ' ) ( , ; , ' )G t t G t t′′ ′ =r r r r 33; 1 2 44; 2 1= ( , ; , ' ) ( , ; , ' )Q Q Q G t t G t t′′ ′ −r r r r 34; 1 1 43; 2 2( , ; , ' ) ( , ; , ' ) .Q Q Q G t t G t t′ ′′− r r r r (24) One can check that within our Gaussian approximation in u and v the first term in the right hand side of Eq. (24) does not give any contribution. Hence, we find Weak localization, Aharonov–Bohm oscillations and decoherence in arrays of quantum dots Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 1167 2 2( , ') = 4 t WL e dt dt m αβ −∞ ′ ′′δσ × π ∫ ∫r r = = ' = ' = '' '1 2 1 21 2 1 2 ( ) ( )β βα α× ∇ −∇ ∇ −∇ ×r r r r r rr r r r 34; 1 1 43; 2 2( , ; , ' ) ( , ; , ' ) .Q Q Q G t t G t t′ ′′× r r r r (25) Let us now turn to our model of Fig. 1 in which case the voltage drops occur only across barriers. In this case Eq. (25), which only applies to bulk metals, should be ge- neralized accordingly. Consider the conductance of an in- dividual barrier determined by the following Kubo formula = ( ) ( , ) ( , ) ( , ) ( , ) . t G i dt t t I t x I t x I t x I t x −∞ ′ ′ ′ ′ ′ ′− − 〈 − 〉∫ (26) Here ( , )I t x is the operator of the total current flowing in the lead (or dot) and x is a longitudinal coordinate chosen to be in a close vicinity of the barrier. Due to the current conserva- tion the conductance G should not explicitly depend on x and .x′ Comparing Eqs. (26) and (16), and making use of Eq. (25) and the relation 2( , ) = ( , , ),xI t x d j t x∫ z z where xj is the current density in the x-direction and z is the vector in the transversal direction, we conclude that WL correction to the conductance of a barrier between the left and right dots should read 2 2 2 2= 4 t WL LR eG dt dt d d m −∞ ′ ′′ ′δ × π ∫ ∫ ∫ z z = = = =1 2 1 2 1 2 1 2 ( ) ( )x x x x x x x x x x′ ′ ′ ′ ′× ∇ −∇ ∇ −∇ × 34; 1 1 43; 2 2( , , ; , , ) ( , , ; , , ) .Q Q Q G t x t x G t x t x′ ′ ′ ′′ ′ ′× z z z z (27) In what follows we will assume that both coordinates x and x′ are on the left side from and very close to the cor- responding barrier. Let us express the Green function in the vicinity of the barrier in the form 1( , , ; , , ') = {exp( ) ( , , , )Q n m mn nm G t x t x ip x ip x t t x x++′ ′ ′ ′ ′− +∑z z G exp ( ) ( , , , )n m mnip x ip x t t x x−−′ ′ ′+ − + +G exp ( ) ( , , , )n m mnip x ip x t t x x+−′ ′ ′+ + +G exp ( ) ( , , , )} ( ) ( ') ,n m mn n mip x ip x t t x x−+′ ′ ′+ − − Φ Φz zG (28) where ( )nΦ z are the transverse quantization modes which define conducting channels, np is projection of the Fermi momentum perpendicular to the surface of the barrier, and the semiclassical Green function mn αβG slowly varies in space. Eq. (27) then becomes 2 2= 4 t WL LR eG dt dt m −∞ ′ ′′δ × π ∫ ∫ = 1 ( )( )n k m l mnkl p p p p αβγδ ± × α − γ β − δ ×∑ ∑ ;34 ;43( , , , ) ( , , , )mn kl Q t t x x t t x xαβ γδ′ ′ ′′ ′× ×G G 1 1 2 2 = = ; = =1 2 1 2 exp( ) \| .n m k l x x x x x xi p x i p x i p x i p x ′ ′ ′′ ′× α − β + γ − δ (29) Next we require WL LRGδ to be independent on x and ,x′ i.e., in Eq. (29) we omit those terms, which contain quickly oscillating functions of these coordinates. This requirement implies that = 0n kp pα + γ and = 0.m lp pβ + δ These con- straints in turn yield = ,γ −α = ,δ −β =k n and = .l m Thus, we get 2 2 = 1 = t WL LR n m mn eG dt dt p p m αβ ± −∞ ′ ′′δ αβ × π ∑ ∑ ∫ ∫ , ;34 ;43( , , , ) ( , , , ) .mn nm Q t t x x t t x xαβ −α −β′ ′ ′′ ′× G G (30) Let us choose the basis in which transmission and ref- lection matrices t̂ and r̂ are diagonal. In this basis the semiclassical Green function is diagonal as well, ,mn nn nm∝ δG G and Eq. (30) takes the form 22 2= t WL n LR n peG dt dt m −∞ ′ ′′δ × π ∑ ∫ ∫ , ;34 , ;43 , ;34 , ;43( , ) ( , ) ( , ) ( , )L nn L nn L nn L nnt t t t t t t t++ −− −− ++′ ′′ ′ ′′× 〈 + −G G G G , ;34 , ;43 , ;34 , ;43( , ) ( , ) ( , ) ( , ) .L nn L nn L nn L nn Qt t t t t t t t+− −+ −+ +−′ ′′ ′ ′′− − 〉G G G G (31) What remains is to express the WL correction in terms of the field Q only. This goal is achieved with the aid of the following general relation [23] 2 = t WL LR n eG dt dt −∞ ′ ′′δ − × π ∑ ∫ ∫ 1 2 1 2[ ( , ) ( , ) ( , ) ( , )]n L R R LT t t t t t t t t′ ′′ ′ ′′×〈 + +v v v v 2 1 1 2 2[ ( , ) ( , )][ ( , ) ( , )] .n L R L RT t t t t t t t t′ ′′ ′ ′′+ − − 〉v v v v (32) Note that the contribution linear in ,nT which contains the product of the fluctuating fields on two different sides of the barrier, vanishes identically provided fluctuations on one side tend to zero, e.g., if the barrier is directly attached to a large metallic lead. In contrast, the contribution 2 nT∝ in Eq. (32) survives even in this case. Finally, applying the contraction rule (14) we get 2 2 0 = { [ ( ) ( )] 4 WL LR R LR L RL e gG dt C t C t ∞ δ − β δ + δ + π ∫ (1 )[ ( ) ( )]}.R RR L LLC t C t+ − β δ + δ (33) D.S. Golubev, A.G. Semenov, and A.D. Zaikin 1168 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 Here ,L Rδ is the mean level spacing in the left/right quan- tum dot, = 2 k k g T∑ (34) is the dimensionless conductance of the barrier and = (1 ) /k k k k k T T Tβ −∑ ∑ (35) is the corresponding Fano factor. Likewise, the WL correction to the nth barrier conduc- tance in 1D array of 1N − quantum dots with mean level spacings nδ connected by N barriers with dimensionless conductances ng and Fano factors nβ reads 2 1, 1 , 12 0 = { [ ( ) ( )] 4 WL n n n n n n n n n e g G dt C t C t ∞ − − −δ − β δ + δ + π ∫ 1 1, 1(1 )[ ( ) ( )]}.n n nn n n nC t C t− − −+ −β δ + δ (36) So far we discussed the local properties, namely WL corrections to the conductivity tensor, , ( , '),WL α βδσ r r and to the conductance of a single barrier, .WL LRGδ Our main goal is, however, to evaluate the WL correction to the conduc- tance of the whole system. For bulk metals one finds that at large scales the WL correction (17) is local, , ( , ') ( ').WL α βδσ ∝ δ −r r r r In general though, there can exist other, non-local, contributions to the conductivity tensor [30]. Without going into details here, we only point out that, even if these non-local terms are present, one can still apply the standard Ohm's law arguments in order to obtain the conductance of the whole sample. Specifically, in the case of 1D arrays one finds [23] (see also [31]) 1 1 =1 =1 1 1= ( ) WL N N WL n n n n n G G G G− − δ − = + δ∑ ∑ 2 =1 2 =1 / = higher order terms . 1/ N WL n n n N n n G g g δ + ⎛ ⎞ ⎜ ⎟⎜ ⎟ ⎝ ⎠ ∑ ∑ (37) Equations (33), (36), and (37) will be used to evaluate WL corrections for different configurations of quantum dots considered below. 2.4. Examples 2.4.1. Single quantum dot. We start from the simplest case of a single quantum dot depicted in Fig. 2. In this case the solution of Eq. (15) reads 11( ) = exp , D H t t tC t ϕ ⎡ ⎤ − − −⎢ ⎥ τ τ τ⎢ ⎥⎣ ⎦ (38) where 1 2= 4 / ( )D dg gτ π + δ is the dwell time, and dδ is the mean level spacing in the quantum dot. All other com- ponents of the Cooperon are equal to zero. From Eq. (33) we get 2 1 1 1 2 (1 ) 1= , 1/ 1/ 1/4 WL d D H e g G ϕ −β δ δ − τ + τ + τπ 2 2 2 2 2 (1 ) 1= . 1/ 1/ 1/4 WL d D H e g G ϕ −β δ δ − τ + τ + τπ (39) According to Eq. (37) the total WL correction becomes ( ) 2 22 1 2 1 1 2 2 2 2 1 2 (1 ) (1 ) = . 4 ( ) 1/ 1/ 1/ WL D H g g g geG g g ϕ −β + −βδ δ − π + τ + τ + τ (40) Since 21 / ,H Hτ ∝ the magnetoconductance has the Lo- rentzian shape [13]. In the limit = 0H and in the absence of interactions ( ϕτ →∞ ) Eq. (40) reduces to [32] 2 22 1 2 1 1 2 2 3 1 2 (1 ) (1 ) = . ( ) WL g g g geG g g −β + −β δ − π + (41) As one can see for the case of low transmissions (for ex- ample in case of tunneling barriers) the WL corrections equals to zero. 2.4.2. Two quantum dots. Next we consider the most general setup composed of two quantum dots with the cor- responding conductances and Fano factors defined as in Fig. 3. The Cooperon is represented as a 2 2× matrix which zero frequency component satisfies the following equation 11 12 1 11 12 21 2221 22 2 y y y y g g g g C C C Cg g g g + + + γ −⎛ ⎞⎛ ⎞ =⎜ ⎟⎜ ⎟⎜ ⎟− + + + γ ⎝ ⎠⎝ ⎠ 1 2 4 / 0 = , 0 4 / π δ⎛ ⎞ ⎜ ⎟π δ⎝ ⎠ (42) where 1,2 1,2 1,2 1,2 4 1 1= . H ϕ ⎛ ⎞π γ +⎜ ⎟⎜ ⎟δ τ τ⎝ ⎠ (43) Defining Fig. 2. Single quantum dot connected to the leads via two barriers. �d T k (1) T k (2) Weak localization, Aharonov–Bohm oscillations and decoherence in arrays of quantum dots Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 1169 2 11 12 1 21 22 2= ( )( ) ,y y yg g g g g g gΔ + + + γ + + + γ − we get 21 22 2 1 211 12 21 22 1 11 12 1 2 ( ) / /4= . / ( ) / y y y y g g g gC C C C g g g g + + + γ δ δ⎛ ⎞⎛ ⎞ π ⎜ ⎟⎜ ⎟ ⎜ ⎟δ + + + γ δΔ⎝ ⎠ ⎝ ⎠ With the aid of Eq. (33) we can derive WL corrections for all five barriers in our setup which we do not specify here for the sake of brevity (see [23] for further details). WL correction to the conductance of the whole struc- ture WLGδ is obtained from the general expression for the conductance determined by Ohm's law: 11 12 21 22 21 22 11 12= [ ( ) ( )G G G G G G G G G+ + + + 12 22 11 21 11 12 21 22( )( )] /[( )( )yG G G G G G G G G+ + + + + + 11 12 21 22( )].yG G G G G+ + + + (44) Substituting WL ij ij ijG G G→ +δ into this formula and ex- panding the result to the first order in ,WL ijGδ we get , =1,2 = .WL WL WL ij y ij yi j G GG G G G G ∂ ∂ δ δ + δ ∂ ∂∑ (45) This general result for the WL correction to the conduc- tance is illustrated in Fig. 4 for a particular choice of the system parameters. Of particular importance for us here is the system of two quantum dots connected in series, as shown in Fig. 5, i.e., in the general structure of Fig. 3 we set 12 21= = 0,G G 11 1= ,G G 2= ,yG G 22 3= ,G G 11 1= ,β β 2=yβ β and 22 3= .β β We also assume = 0H and = .ϕτ ∞ WL correc- tions to the barrier conductances then take the form 2 1 2 3 1 1 1 2 2 3 1 3 ( )(1 ) = ,WL g g geG g g g g g g + −β δ − π + + 22 2 1 3 2 2 2 1 2 2 3 1 3 ( )(1 ) 2 = ,WL g g g geG g g g g g g + −β + δ − π + + 2 3 1 2 3 3 1 2 2 3 1 3 ( )(1 ) = ,WL g g geG g g g g g g + −β δ − π + + (46) while Eq. (44) reduces to 1 2 3 1 2 1 3 2 3 = . G G G G G G G G G G+ + (47) WL correction for the whole system then reads Fig. 3. Most general system with two quantum dots. g ,11 �11 �� g , �12 12 g ,21 �21 g , �22 22 �� g , by y –10 –5 0 5 10 –0.50 –0.40 –0.30 –0.20 –0.10 0 H/H1 g /g = 100y 0 g /g = 5y 0 g /g = 1y 0 g /g = 0.5y 0 g = 0y Fig. 4. The magnetoconductance of two dots of Fig. 3 for 1 2, ,ed d l 1 2/ = 5,d d 0= ,ijg g = 0,ijβ = 0,yβ 1 =ϕτ 2 = .ϕ= τ ∞ Here 1 1 1= 1/ 4 DH α τ is the field at which weak localization is effectively suppressed in the first dot. For = 0yg the magnetoconductance is given by superposition of two Lorent- zians with different widths (decoupled dots), while for large yg only one Lorentzian survives corresponding to the contribution of a one «composite dot». Fig. 5. Two quantum dots in series. �1 �2 T k (1) T k (2) T k (3) D.S. Golubev, A.G. Semenov, and A.D. Zaikin 1170 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 2 22 1 2 3 2 3 1 3 1 2 2 3 1 3 ( )(1 ) = ( ) WL g g g g geG g g g g g g + −β δ − − π + + 2 22 1 2 3 1 3 2 3 1 2 2 3 1 3 ( )(1 ) ( ) g g g g ge g g g g g g + −β − − π + + 2 22 1 2 3 1 2 3 3 1 2 2 3 1 3 ( )(1 ) ( ) g g g g ge g g g g g g + −β − − π + + 2 2 22 1 2 3 3 1 2 2 3 1 3 2 . ( ) g g ge g g g g g g − π + + (48) In the limit of open quantum dots, i.e., 1,2,3 = 0,β we re- produce the result [31]. It is easy to see that provided the conductance of one of the barriers strongly exceeds two others, Eq. (48) reduces to Eq. (41). If all three barriers are tunnel junctions, 1,2,3 1,β → the first three contributions in Eq. (48) vanish, and only the last contribution — indepen- dent of the Fano factors — survives in this limit. If, on top of that, one of the tunnel junctions, e.g., the central one, is less transparent than two others, 2 1 3, ,g g g the result acquires a particularly simple (non-Lorentzian) form ( ) ( ) 22 2 1 1 3 2 2= ,WL geG g g δ − π + γ + γ (49) with 1,2γ defined in Eq. (43). Note that 2 2 ,WLG gδ ∝ i.e., this result is dominated by the second order tunneling processes across the second barrier. 2.4.3. 1D array of identical quantum dots. Let us now turn to 1D arrays of quantum dots depicted in Fig. 1. For simplicity, we will assume that our array consists of 1N − identical quantum dots with the same level spacing n dδ ≡ δ and of N identical barriers with the same dimen- sionless conductance ng g≡ and the same Fano factor .nβ ≡ β We will also assume that the quantum dots have the same shape and size so that Hn Hτ ≡ τ and .nϕ ϕτ ≡ τ For this system the Cooperon can also be found exactly. The result reads 1 =1 sin sin2( ) = . 1 cos1 1 N nm q H D qn qm N NC qN Ni − ϕ π π ω π − − ω+ + + τ τ τ ∑ (50) Here = 2 /D dgτ π δ and 2= 1 /16 .H Hτ α The WL correc- tion then takes the form 2 1 2 2 =1 cos 1 = . 2 1 cos1 1 N WL d q H D q e g NG qN N − ϕ π β + −βδ δ − ππ − + + τ τ τ ∑ (51) The sum over q can be handled exactly and yields 2 2 2 2 2 2 1 1= 1 1 N WL N e u uG N N u u ⎡⎛ ⎞+ + δ − − ×⎢⎜ ⎟⎜ ⎟π − −⎢⎝ ⎠⎣ 2 2 (1 ) 2(1 ) ( 1) ] , 1 u u N u β + + −β × − − β − (52) where 2 = 1 1 1.D D D D H H u ϕ ϕ ⎛ ⎞τ τ τ τ + + − + + −⎜ ⎟⎜ ⎟τ τ τ τ⎝ ⎠ (53) In the tunneling limit = 1β and for ϕτ →∞ our result defined in Eqs. (52) and (53) becomes similar — though not exactly identical — to the corresponding result [33]. If ϕτ is long enough, namely Th1/ ,Eϕτ where 2 2 Th = / 2 DE Nπ τ is the Thouless energy of the whole array, in Eqs. (51) and (52) it is sufficient to set = .ϕτ ∞ In this case the magnetic field H significantly suppresses WL correction provided Th1 / H Eτ or, equivalently, if 1, = . 8 d N N g H H H N π δ α (54) In the opposite limit Th1/ Eϕτ we find 2 2 1 1 = . 1 1 D D HWL D D H eG N ϕ ϕ ⎡ ⎤ ⎛ ⎞τ τ⎢ ⎥β + + + −β⎜ ⎟⎢ ⎥⎜ ⎟τ τ⎝ ⎠⎢ ⎥δ − −β⎢ ⎥π ⎛ ⎞⎢ ⎥τ τ + + −⎜ ⎟⎢ ⎥⎜ ⎟τ τ⎢ ⎥⎝ ⎠⎣ ⎦ (55) In particular, in the diffusive limit ,H Dϕτ τ τ we get 2 = ,HWL H DeG Nd ϕ ϕ τ τ δ − π τ + τ (56) where we introduced the diffusion coefficient 2= / 2 .DD d τ (57) Equation (56) coincides with the standard result for quasi- 1D diffusive metallic wire. Note, however, that the values of Hτ within our model may differ from those for a metal- lic wire. The ratio of the former to the latter is met fl/ / ,qd H DHτ τ τ τ∼ where fl / Fdτ v∼ is the flight time through the quantum dot. Since typically fl < Dτ τ we con- clude that for the same value of D the magnetic field de- phases electrons stronger in the case of an array of quan- tum dots. For a single quantum dot ( = 2N ) Eq. (52) reduces to 2 (1 ) 1= 4 1 WL D D H eG ϕ −β δ − π ⎛ ⎞τ τ + +⎜ ⎟⎜ ⎟τ τ⎝ ⎠ (58) in agreement with Eq. (40). Weak localization, Aharonov–Bohm oscillations and decoherence in arrays of quantum dots Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 1171 For two identical quantum dots in series we obtain 2 2 2 / 3= , 2 2 2 29 1 1 3 3 WL D D D D H H eG ϕ ϕ ⎡ ⎤ ⎢ ⎥−β −β⎢ ⎥δ − + τ τ τ τ⎢ ⎥π + + + +⎢ ⎥τ τ τ τ⎣ ⎦ (59) i.e., the magnetoconductance is just the sum of two Lo- rentzians in this case. Finally, in the absence of any interactions ( =ϕτ ∞ ) and at = 0H we obtain 2 2 1 1 1= . 3 3 WL eG N N ⎡ β ⎤⎛ ⎞δ − − + β −⎜ ⎟⎢ ⎥π ⎝ ⎠⎣ ⎦ (60) In the limit N →∞ this result reduces to the standard one for a long quasi-1D diffusive wire [34] while for any finite N we reproduce the results for tunnel barriers [33] ( 1)β → and open quantum dots [35] ( 0).β→ The magnetoconductance of a 1D array of 1N − iden- tical open quantum dots in the absence of interactions is also illustrated in Fig. 6. 3. Quantum decoherence by electron–electron interactions 3.1. Qualitative arguments Let us now include electron–electron interactions and analyze their impact on loss of phase coherence of elec- trons' wave functions. Before turning to a detailed calcula- tion it is instructive to discuss a simple qualitative picture demonstrating under which conditions decoherence by electron–electron interactions is expected to occur. Consider first the simplest system of two scatterers se- parated by a cavity (quantum dot, Fig. 7) The WL correc- tion to conductance of a disordered system WLG is known to arise from interference of pairs of time-reversed electron paths [2]. In the absence of interactions for a single quan- tum dot of Fig. 7 this correction was evaluated in the pre- vious sections (see Eq. (41)). The effect of electron– electron interactions can be described in terms of fluctuat- ing voltages. Let us assume that the voltage can drop only across the barriers and consider two time-reversed electron paths which cross the left barrier (with fluctuating voltage ( )LV t ) twice at times it and ,ft as it is shown in Fig. 7. It is easy to see that the voltage-dependent random phase factor exp ( ) ft L ti i V t dt ⎛ ⎞ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠ ∫ acquired by the electron wave function Ψ along any path turns out to be exactly the same as that for its time-reversed counterpart. Hence, in the product ΨΨ these random phases cancel each other and quantum coherence of electrons remains fully pre- served. This implies that for the system of Fig. 9 fluctuat- ing voltages (which can mediate electron–electron interac- tions) do not cause any dephasing. This qualitative conclusion can be verified by means of more rigorous considerations. For instance, it was demon- strated [18] that the scattering matrix of the system remains unitary in the presence of electron–electron interactions, which implies that the only effect of such interactions is transmission renormalization but not electron decoherence. A similar conclusion was reached [36] by directly evaluat- ing the WL correction to the system conductance. Thus, for the system of two scatterers of Fig. 7 electron–electron interactions can only yield energy dependent (logarithmic at sufficiently low energies) renormalization of the dot channel transmissions [18,20] but not electron dephasing. Let us now add one more scatterer and consider the sys- tem of two quantum dots depicted in Fig. 8. We again as- sume that fluctuating voltages are concentrated at the bar- riers and not inside the cavities. The phase factor accumulated along the path (see Fig. 8) which crosses the central barrier twice (at times it and > it t ) and returns to the initial point (at a time ft ) is [ ( ) ( )]e ,i t tiϕ −ϕ where / = ( )e V tϕ is the fluctuating voltage across the central barrier. Similarly, the phase factor picked up along the –6 –4 –2 0 2 4 6 –0.35 –0.30 –0.25 –0.20 –0.15 –0.10 –0.05 0 H/HN N = 200 N = 10 N = 4 N = 2 Fig. 6. Magnetoconductance of a 1D array of 1N − identical open ( = 0)β quantum dots in the absence of interactions ( ).ϕτ → ∞ The field NH is defined in Eq. (54). Fig. 7. Single quantum dot and a pair of time-reversed electron paths. Fluctuating voltages LV and RV are assumed to drop only across the barriers and not inside the dot. gL gR VL VR D.S. Golubev, A.G. Semenov, and A.D. Zaikin 1172 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 time-reversed path reads [ ( ) ( )] e . i t t t tf i fϕ + − −ϕ Hence, the overall phase factor acquired by the product ΨΨ for a pair of time-reversed paths is totexp( ),iΦ where tot ( , , ) = ( ) ( ) ( ) ( ).i f i f i ft t t t t t t t tΦ ϕ −ϕ −ϕ + − +ϕ Averaging over phase fluctuations, which for simplicity are assumed Gaussian, we obtain 2 tot tot 1exp [ ( , , )] = exp [ ( , , ) ] 2i f i fi t t t t t t〈 Φ 〉 − 〈Φ 〉 = = exp [ 2 ( ) 2 ( ) ( ) ( 2 )],i f f i f iF t t F t t F t t F t t t− − − − + − + + − (61) where we defined the phase correlation function 2( ) = ( ( ) (0)) / 2.F t t〈 ϕ − ϕ 〉 (62) Should this function grow with time the electron phase coherence decays and, hence, WLG has to be suppressed below its non-interacting value due to interaction-induced electron decoherence. The above arguments are, of course, not specific to sys- tems with three barriers only. They can also be applied to any system with larger number of scatterers, i.e., virtually to any disordered conductor where — exactly for the same reasons — one also expects non-vanishing interaction- induced electron decoherence at any temperature including = 0.T Below we will develop a quantitative theory which will confirm and extend our qualitative physical picture. We are going to give a complete quantum mechanical analysis of the problem which fully accounts for Fermi statistics of electrons and treats electron–electron interac- tions in terms of quantum fields produced internally by fluctuating electrons. 3.2. Nanorings with two quantum dots 3.2.1. The model and basic formalism. Consider the system depicted in Fig. 9. The structure consists of two chaotic quantum dots (L and R) characterized by mean level spacing Lδ and Rδ which are the lowest energy pa- rameters in our problem. These (metallic) dots are inter- connected via two tunnel junctions J1 and J 2 with con- ductances 1tG and 2tG forming a ring-shaped con- figuration as shown in Fig. 9. The left and right dots are also connected to the leads (LL and RL) respectively via the barriers JL and JR with conductances LG and .RG We also define the corresponding dimensionless conductances of all four barriers as 1,2 1,2=t t qg G R and , 1,2= ,L R t qg G R where 2= 2 /qR eπ is the quantum resistance unit. The whole structure is pierced by the magnetic flux Φ through the hole between two central barriers in such way that electrons passing from left to right through different junctions acquire different geometric phases. Applying a voltage across the system one induces the current which shows AB oscillations with changing the external flux Φ . Note that in the absence of the magnetic flux the system just reduces to that of two connected in series quantum dots (cf. Fig. 5) which is also subject to weak localization effects. Thus, the model considered here allows to analyze WL and AB effects within the same formalism to be de- veloped below. The system depicted in Fig. 9 is described by the effective Hamiltonian: , = , ˆ ˆ ˆ ˆ ˆ= 2 ij i j LL RL i j L R C H + + +∑ V V H H = , ˆ ˆ ˆ ˆ ,j L R j L R + + + +∑ H T T T (63) where ijC is the capacitance matrix, ( ) ˆ L RV is the electric potential operator on the left (right) quantum dot, †3 ,, = , ˆ ˆ ˆ ˆ= ( )( ) ( ),LL LL LL LLLL LL d H eV αα α ↑ ↓ Ψ − Ψ∑ ∫H r r r †3 ,, = , ˆ ˆ ˆ ˆ= ( )( ) ( )RL RL RL RLRL RL d H eV αα α ↑ ↓ Ψ − Ψ∑ ∫H r r r are the Hamiltonians of the left and right leads, ,LL RLV are the electric potentials of the leads fixed by the external voltage source, †3 ,, = , ˆ ˆ ˆ ˆ ˆ= ( )( ) ( )j j j jj j d H e αα α ↑ ↓ Ψ − Ψ∑ ∫H r r V r defines the Hamiltonians of the left ( = )j L and right ( = )j R quantum dots and gL gR VL V VR gt Fig. 8. Two quantum dots and a typical electron path. Fluctuating voltages LV , V, and RV are again assumed to drop only across the barriers. Fig. 9. Two quantum dots with magnetic flux. LL L R RL JR J2 J1 JL Weak localization, Aharonov–Bohm oscillations and decoherence in arrays of quantum dots Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 1173 2ˆ[ ( )] ˆ = ( ) 2j j ep A r c U r m μ μ− −μ +H is the one-particle Hamiltonian of electron in jth quantum dot with disorder potential ( ).jU r Electron transfer be- tween the left and the right quantum dots will be described by the Hamiltonian †2 ,, = , 1 2 ˆ ˆ ˆ= [ ( ) ( ) ( ) c.c.].RL J J d t αα α ↑ ↓ + Ψ Ψ +∑ ∫T r r r r The Hamiltonian ( ) ˆ L RT describing electron transfer be- tween the left dot and the left lead (the right dot and the right lead) is defined analogously. The real time evolution of the density matrix of our sys- tem is described by means of the standard equation ˆ ˆ 0ˆ ˆ( ) = e e ,iHt iHtt −ρ ρ (64) where Ĥ is given by Eq. (63). Let us express the opera- tors ˆ e iHt− and ˆ eiHt via path integrals over the fluctuating electric potentials ,F B jV defined respectively on the for- ward and backward parts of the Keldysh contour: ˆ 0 ˆe = exp ( ) , t iHt F F j jDV T i dt H V t− ⎧ ⎫⎪ ⎪⎡ ⎤′ ′−⎨ ⎬⎣ ⎦⎪ ⎪⎩ ⎭ ∫ ∫ ˆ 0 ˆe = exp ( ) . t iHt B B j jDV T i dt H V t ⎧ ⎫⎪ ⎪⎡ ⎤′ ′⎨ ⎬⎣ ⎦⎪ ⎪⎩ ⎭ ∫ ∫ (65) Here expT ( expT ) stands for the time ordered (anti- ordered) exponent. Let us define the effective action of our system 0 ˆ[ , ] = ln tr exp ( ) t F B F jiS V V T i dt H V t ⎛ ⎡ ⎧ ⎫⎪ ⎪⎜ ⎡ ⎤⎢ ′ ′− ×⎨ ⎬⎣ ⎦⎜ ⎢ ⎪ ⎪⎩ ⎭⎣⎝ ∫ 0 0 ˆˆ exp ( ) . t B jT i dt H V t ⎞⎤⎧ ⎫⎪ ⎪ ⎟⎡ ⎤ ⎥′ ′× ρ ⎨ ⎬⎣ ⎦ ⎟⎥⎪ ⎪⎩ ⎭⎦ ⎠ ∫ (66) Integrating out the fermionic variables we rewrite the ac- tion in the form 1 ext= 2Tr ln .CiS iS iS −⎡ ⎤+ + ⎣ ⎦G (67) Here CS is the standard term describing charging effects, extS accounts for an external circuit and 1 † 1 1 † 1 † 1 ˆ ˆ 0 0 ˆˆ ˆ 0 = ˆˆ ˆ0 ˆˆ0 0 LL L LL R R RLR G T T G T G T G T T G − − − − − ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ (68) is the inverse Green–Keldysh function of electrons propa- gating in the fluctuating fields. Here each quantum dot as well as two leads is represented by the 2 2× matrix in the Keldysh space: 1 ˆ 0ˆ = ˆ0 F t i i i B t i i i H eV G i H eV − ⎛ ⎞∂ − + ⎜ ⎟ ⎜ ⎟− ∂ + −⎝ ⎠ . (69) 3.2.2. Effective action. Let us expand the exact action iS (67) in powers of ˆ.T Keeping the terms up to the fourth order in the tunneling amplitude, we obtain † ext ˆ ˆˆ ˆ2trC L R L RiS iS iS iS iS G TG T⎡ ⎤≈ + + + − −⎣ ⎦ † †ˆ ˆ ˆ ˆˆ ˆ ˆ ˆtr .L R L RG TG T G TG T⎡ ⎤− ⎣ ⎦ (70) Here ,L RiS are the contributions of isolated dots, the terms 2t∝ yield the Ambegaokar–Eckern–Schön (AES) action [10] AESiS described by the diagram in Fig. 10,a, and the fourth order terms 4t∝ (diagrams in Fig. 10,b,c) account for the weak localization correction to the system conduc- tance [24,25]. It is easy to demonstrate [26] that after disorder averag- ing AESiS becomes independent of Φ and, hence, it does not account for the AB effect investigated here. After aver- aging the last term in Eq. (70) over realizations of trans- mission amplitudes and over disorder only the contribution generated by the diagram (c) keeps depending on the mag- netic flux and yields [26] ( ) ( )2 ( )1 2 2 , =1,2 = e 4 n miWL g gt t m nL R ig g iS N N ϕ −ϕ Φ − × π ∑ 1 2 1 4 1 2... ( ) ( )L Rd d dt dt C C× τ τ τ τ ×∫ ∫ ( ( ) ( ) ( ) ( )) 12 3 4 1 ( ) e sin 2 i t t t t t−+ + + +ϕ −ϕ +ϕ −ϕ ϕ × × ( ) ( )2 2 2 21 2 1 1 2 1( )e ( )e t t i i h t t f t t − −ϕ ϕ − ⎡ ⎤ ⎢ ⎥ × − − τ + − − τ ×⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ( )3 22 3 2 3 4 1( )e ( ) t i h t t f t t −ϕ − ⎡ ⎢ × − − τ − + τ −⎢ ⎢ ⎣ ( )3 22 3 2 3 4 1( )e ( ) t i f t t h t t −ϕ ⎤ ⎥ − − − τ − + τ ×⎥ ⎥ ⎦ ( ) ( )4 4 2 24 1 2 4 1 2e ( ) e ( ) t t i i f t t h t t − −ϕ ϕ − ⎡ ⎤ ⎢ ⎥ × − + τ + − + τ +⎢ ⎥ ⎢ ⎥ ⎣ ⎦ { , },L R ± ±+ ↔ ϕ → −ϕ (71) D.S. Golubev, A.G. Semenov, and A.D. Zaikin 1174 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 where , ( )L RC t the Cooperons in the left and right dots, ( ) = ( ) / 2Ff t f E dE π∫ is the Fourier transform of the Fermi function ( )Ff E and ( ) = ( ) ( ).h t t f tδ − Here we also introduced the geometric phases (1,2) = ( ) , R g L e dx A x c μ μϕ ∫ (72) where the integration contour starts in the left dot, crosses the first (1)( )gϕ or the second (2)( )gϕ junction and ends in the right dot. The difference between these two geometric phases is (1) (2) 0= 2 / .g gϕ −ϕ πΦ Φ In addition, we defined the «clas- sical» and the «quantum» components of the fluctuating phase ( ) = ( ( ) ( )) / 2F Bt t t+ϕ ϕ + ϕ and ( ) = ( ) ( ),F Bt t t−ϕ ϕ − ϕ where the phases , , , 0 ( ) = [ ( ) ( )] t F B F B F B R Lt e d V Vϕ τ τ − τ∫ are defined on the forward and backward parts of the Keldysh contour. The above expression for the action WLSΦ (71) fully ac- counts for coherent oscillations of the system conductance in the lowest non-vanishing order in tunneling. The WL contribution to action of two quantum dots is recovered in exactly the same way [24]. The result is the similar except geometric phases should be omitted and the combination 1 2t tg g should be substituted by 2 1tg or 2 2 .tg 3.2.3. Aharonov–Bohm conductance and WL correction. Let us now evaluate the current I through our system. This current can be split into two parts, 0= ,I I I+ δ where 0I is the flux-independent contribution and Iδ is the quantum correction to the current sensitive to the magnetic flux Φ . This correction is determined by the action ,WLiSΦ i.e., 2 [ , ][ , ] = e . ( ) WL iSS I e t + − + −± ϕ ϕΦ − δ ϕ ϕ δ − ϕ δϕ∫D (73) In order to evaluate the path integral over the phases ±ϕ in (73) we restrict our consideration to the most inter- esting for us metallic limit assuming that dimensionless conductances ,L Rg are much larger than unity, while the conductances 1tg and 2tg are small as compared to those of the outer barriers, i.e., 1 2, 1, , .L R t tg g g g (74) In the limit (74) phase fluctuations can be considered small down to exponentially low energies [14,37] in which case it suffices to expand both contributions up to the second order .±ϕ Moreover, this Gaussian approximation be- comes exact [15,18,20,21] in the limit of fully open left and right barriers with , 1.L Rg Thus, in the metallic limit (74) the integral (73) remains Gaussian at all relevant energies and can easily be performed. This task can be accomplished with the aid of the fol- lowing correlation functions ( ) = , ( ) = 0,t eVt t+ −〈ϕ 〉 〈ϕ 〉 (75) ( ( ) (0)) (0) = ( ),t F t+ + +〈 ϕ − ϕ ϕ 〉 − (76) ( ) (0) ( ) (0) = 2 (\| \|),t t iK t+ − − +〈ϕ ϕ + ϕ ϕ 〉 (77) ( ) (0) ( ) (0) = 2 ( ) ,t t iK t+ − − +〈ϕ ϕ − ϕ ϕ 〉 (78) ( ) (0) = 0,t− −〈ϕ ϕ 〉 (79) where the last relation follows directly from the causality principle [4]. Here and below we define = RL LLV V V− to be the transport voltage across our system. Note that the above correlation functions are well famili- ar from the so-called P(E)-theory [10,38] describing electron tunneling in the presence of an external environment which can also mimic electron–electron interactions in metallic conductors. They are expressed in terms of an effective im- pedance ( )Z ω «seen» by the central barriers J1 and J2 2 1 cos( )( ) = coth [ ( )] , 2 2 d tF t e Z T ω ω − ω ℜ ω π ω∫ (80) 2 sin( )( ) = [ ( )] . 2 d tK t e Zω ω ℜ ω π ω∫ (81) Further evaluation of these correlation functions for our system is straightforward and yields 4 sinh( )( ) ln , RC TtF t g T ⎛ ⎞π + γ⎜ ⎟⎜ ⎟π τ⎝ ⎠ (82) 2( ) sign ( ),K t t g π (83) where we defined 2= 4 / (0)g e Zπ and 0.577γ is the Euler constant. Neglecting the contribution of external leads and making use of the inequality (74) we obtain 2 / ( ).L R L Rg g g g g+ We observe that while ( )F t grows with time at any temperature including = 0,T the function ( )K t always remains small and it can be safely ignored in the leading order in 1 / 1.g After that the Fig. 10. Diagrammatic representation of different contributions originating from expansion of the effective action in powers of the central barrier transmissions: second order (AES) terms (a) and different fourth order terms (b) and (c). a b c Weak localization, Aharonov–Bohm oscillations and decoherence in arrays of quantum dots Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 1175 Fermi function ( )Ff E drops out from the final expression for the quantum correction to the current [24–26]. Hence, the amplitude of AB oscillations is affected by the elec- tron–electron interaction only via the correlation functions for the «classical» component of the Hubbard–Strato- novich phase .+ϕ The expression for the current takes the form 0 1 2( ) = cos(4 / ) ,AB WL WLI I I Iδ Φ − πΦ Φ − − (84) where the first — flux dependent — term in the right-hand side explicitly accounts for AB oscillations, while the terms 1,2WLI represent the remaining part of the quantum correc- tion to the current [24] which does not depend on .Φ Let us restrict our attention to the case of two identic- al quantum dots with volume ,V dwell time Dτ and di- mensionless conductances = = 4 / ,L R Dg g g≡ π δτ whe- re = 1/δ νV is the dot mean level spacing and ν is the electron density of states. In this case the Cooperons take the form /( ; , ) = ( ; , ) = ( ( ) / )e .t DL RC t C t t − τθx y x y V We obtain [26] 1 2 ( , )2 2 1 21 2 1 23 0 = e , 4 t t DAB e g g V I d d τ +τ ∞ − − τ τ τδ τ τ π ∫ F (85) 1 22 2 2 ( , )1 21,2 1,2 1 23 0 = e . 8 t DWL e g V I d d τ +τ ∞ − − τ τ τδ τ τ π ∫ F (86) where 1 2 1 2 1 2= 2 ( ) 2 ( ) ( ) ( ).F F F Fτ + τ − τ − τ − τ + τF In the absence of electron–electron interactions this formula yields (0) 2 2 1 2= 4 / ( ).t tABI e g g V gπ In order to ac- count for the effect of interactions we substitute Eq. (82) into Eq. (85). Performing time integrations at high enough temperatures we obtain 8 8/ 1 1 (0) 1/2 1 (2 ) e , , 1 4 /= 1 , , 2 g g RC D RC AB D AB RC RC D T T I T g I g T T γ − − − − ⎧ π τ⎪ τ τ⎪ + π τ⎪ ⎨ ⎪ τ⎛ ⎞ τ⎪ ⎜ ⎟τ⎪ ⎝ ⎠⎩ (87) while in the low temperature limit we find 8 8/ 1 (0) 2 = e , . g g RCAB D DAB I T I γ − −⎛ ⎞τ τ⎜ ⎟τ⎝ ⎠ (88) Essentially the same results follow for 1,2.WLI These results demonstrate that interaction-induced suppression of both AB oscillations and WL corrections in metallic dots with RC Dτ τ persists down to = 0.T The fundamental rea- son behind this suppression is that the interaction of an elec- tron with an effective environment (produced by other elec- trons) effectively breaks down the time-reversal symmetry and, hence, causes both dissipation and dephasing for inte- racting electrons down to = 0T [4]. In this respect it is also important to point out a deep relation between interaction- induced electron decoherence and the P(E)-theory [10,38] which was already emphasized elsewhere [24–26]. 3.3. Arrays of quantum dots and diffusive conductors One of the main conclusions reached above is that the electron decoherence time is fully determined by fluctua- tions of the phase fields +ϕ (and the correlation function ( )),F t whereas the phases −ϕ (and the response function ( ))K t are irrelevant for ϕτ causing only a weak Coulomb correction to .WLG This conclusion is general being inde- pendent of a number of scatterers in our system. Note that exactly the same conclusion was already reached in the case of diffusive metals by means of a different approach [4]. Thus, in order to evaluate the decoherence time for interacting electrons in arrays of quantum dots it is suffi- cient to account for the fluctuating fields V + totally ignor- ing the fields .V − The corresponding calculation is pre- sented below. 3.3.1. 1D structures. Let us consider a 1D array of 1N − quantum dots by N identical barriers as shown in Fig. 1. For simplicity, we will stick to the case of identical barriers (with dimensionless conductance 1g and Fano factor )β and identical quantum dots (with mean level spacing δ and dwell time = 2 /D gτ π δ ). The WL correc- tion to the system conductance has the form (see Eq. (36)): 2 2 2 =1 0 = 4 N WL n e gG dt N ∞δ − × π ∑ ∫ 1, , 1 1, 1{ [ ( ) ( )] (1 )[ ( ) ( )]}.n n n n nn n nC t C t C t C t− − − −× β + + −β + (89) The Cooperon ( )nmC t is determined from a discrete ver- sion of the diffusion equation. For non-interacting elec- trons and in the absence of the magnetic field this equation reads 1, 1,2 = ( ). 2 nm n m n mnm nm D C C CC t t − +− −∂ + δ δ ∂ τ (90) The boundary conditions for this equation are = 0nmC as long as the index n or m belongs to one of the bulk elec- trode. The solution of Eq. (90) with these boundary condi- tions can easily be obtained. We have 1 (0) =1 sin sin2( ) = e . 2 1 cos N i t nm q D qn qm d N NC t qN Ni − − ω π π ω ππ − − ω+ τ ∑ ∫ (91) This solution can be represented in the form (0) ( ) =nmC t bulk bulk( ) ( ),n m n mC t C t− += − where D.S. Golubev, A.G. Semenov, and A.D. Zaikin 1176 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 1 bulk =1 ( )cos1( ) = e . 2 1 cos N i t n m q D q n m d NC t qN Ni − − ω − π − ω ππ − − ω+ τ ∑ ∫ (92) In the limit of large N the term bulk ( )n mC t+ can be safely ignored and we obtain bulk( ) ( ).nm n mC t C t−≈ Let us express the contribution bulk ( )n mC t− as a sum over the integer valued paths ( )ν τ , which start in the mth dot and end in the nth one (i.e., (0) = ,mν ( ) =t nν ) jumping from one dot to another at times .jt This expression can be recovered if one expands Eq. (92) in powers of 1 cos[ / ]D q N−τ π with subsequent summation over q in every order of this ex- pansion. Including additional phase factors acquired by electrons in the presence of the fluctuating fields ,V + ν we obtain =\| \| ( ) ( ) = 1( ) = (0) = (2 ) nm k k n m D t n C t m ∞ − ν τ ν × ν τ ∑ ∑ 3 2 1 2 1 0 0 0 0 t t tt k k kdt dt dt dt−× ×∫ ∫ ∫ ∫… 1 2 2 2 1 1 e e e e e t t t t t t t t tk k k D D D D D − − − −−− − − − − τ τ τ τ τ× ×… ( ) ( ) 0 exp [ ( ) ( )] . t ti d eV eV+ + ν τ ν −τ ⎧ ⎫⎪ ⎪× τ τ − τ⎨ ⎬ ⎪ ⎪⎩ ⎭ ∫ (93) Averaging over Gaussian fluctuations of voltages V + and utilizing the symmetry of the voltage correlator 1 2 1 21 2 2 1 ( ) ( ) = ( ) ( ) ,V V V V+ + + + ν ν ν ν〈 τ τ 〉 〈 τ τ 〉 we get / =\| \| ( ) ( ) = e( ) = (0) = (2 ) t D nm k k n m D t n C t m − τ∞ − ν τ ν × ν τ ∑ ∑ 3 2 1 2 1 0 0 0 0 t t tt k k kdt dt dt dt−× ×∫ ∫ ∫ ∫… 2 1 2 1 2( ) ( )1 2 0 0 exp [ ( ) ( ) t t e d d V V+ + ν τ ν τ ⎧⎪× − τ τ 〈 τ τ 〉 −⎨ ⎪⎩ ∫ ∫ 1 2( ) ( )1 2 ( ) ( ) ] .tV V+ + ν τ ν −τ ⎫⎪− 〈 τ τ 〉 ⎬ ⎪⎭ (94) The correlator of voltages can be derived with the aid of the σ-model approach developed in Sec. 2 of this paper. Integrating over Gaussian fluctuations of the Q-fields one arrives at the quadratic action for the fluctuating fields V + which has the form =1 2= [4 1 cos 2 N g q i d qiS C C N N ω π⎛ ⎞− + +⎜ ⎟π ⎝ ⎠ ∑∫ 2 1 cos ] ( ) ( ) 1 cos D q q D q g e N V V qi N + − π −τ + ω −ω − ππ − ωτ + − 2 2 2 =1 2 2 1 cos coth 2 2 2 1 cos N D q D q g ed N T N q N π ω⎛ ⎞− ω⎜ ⎟τω ⎝ ⎠− × π π π⎛ ⎞ω τ + −⎜ ⎟ ⎝ ⎠ ∑∫ ( ) ( ).q qV V− −× ω −ω (95) Here we defined 1 =1 ( ) = sin e ( ). N i t q n n qV dt V t N − ± ω ±π ω ∑ ∫ (96) The action (95) determines the expressions for both corre- lators V V+ +〈 〉 (F-function) and V V+ −〈 〉 (K-function) responsible respectively for decoherence and Coulomb blockade correction to WL. Since our aim is to describe electron decoherence, only the first out of these two corre- lation functions is of importance for us here. It reads 1 ( )1 21 2 =1 2( ) ( ) = e 2 N i t t n m q dV t V t N − − ω −+ + ω 〈 〉 × π∑ ∫ 2 2 2 1 cos sin sin 1 cos 4 1 cos 1 cos D g D ge q qn qm N N N q g eq NC C qN i N π π π⎛ ⎞−⎜ ⎟π ⎝ ⎠× × π −τπ⎛ ⎞− + +⎜ ⎟ ππ⎝ ⎠ − ωτ + − 2 2 2 2 coth 2 . 1 cos D D T q N ω τ ω × π⎛ ⎞ω τ + −⎜ ⎟ ⎝ ⎠ (97) In the continuous limit 1N and for sufficiently low frequencies 1 / Dω τ both correlators V V+ +〈 〉 and V V+ −〈 〉 defined by Eq. (95) reduce to those of a diffusive metal [4]. To proceed let us consider diffusive paths ( ),ν τ in which case one has 1 1 2 1 2( ) ( )1 2 , =1 1( ) ( ) ( ) ( ) 1 N n m n m V V V V N − + + + + ν τ ν τ〈 τ τ 〉 ≈ 〈 τ τ 〉 × − ∑ 1 2(\| \|),nmD× τ − τ (98) where ( )nmD τ is the diffuson. For 0H → it exactly coin- cides with the Cooperon for non-interacting electrons (91), (0) ,( ) = ( ),nm n mD t C t i.e., Weak localization, Aharonov–Bohm oscillations and decoherence in arrays of quantum dots Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 1177 1 =1 sin sin2( ) = e . 2 1 cos N i t nm q D qn qm d N ND t qN Ni − − ω π π ω ππ − − ω+ τ ∑ ∫ (99) Substituting Eq. (98) into (94), we obtain (0) ( )( ) ( ) e ,t nm nmC t C t −≈ F (100) where 2 1 1 2 1 2 , =1 0 ( ) = ( ) ( ) 1 tN n m n m et dt dt V t V t N − + +〈 〉 × − ∑ ∫F 1 2 1 2[ (\| \|) (\| \|)]nm nmD t t D t t t× − − − − (101) is the function which controls the Cooperon decay in time, i.e., describes electron decoherence for our 1D array of quantum dots. The WL correction WLG in the presence of electron–electron interactions is recovered by substituting the result (100) into Eq. (89). Since the behavior of the latter formula was already analyzed in details earlier there is no need to repeat this analysis here. The dephasing time ϕτ can be extracted from the equation ( ) = 1.ϕτF From Eq. (101) with a good accuracy we obtain 2 1 , =1 1 = ( ) (0) ( ). 1 N n m nm n m e d V V D N − + + ϕ τ〈 τ 〉 τ τ − ∑ ∫ (102) Combining this formula with Eqs. (97) and (99), in the most interesting limit 0T → and for (4 )D gR C Cτ + we find 21 0 =1 1 1 2= ln , 2 ( 1) 4 1 cos N D q g e g N qC C N − ϕτ τ − ⎛ π ⎞⎛ ⎞δ − +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ∑ which yields 0 2 4= = , ln(4 / ) ln(4 / ) D C C g E Eϕ τ π τ δ δ δ (103) where 2= / 2C gE e C for gC C and 2= / 4CE e C in the opposite case .gC C In order to determine the dephasing length =L Dϕ ϕτ let us define the diffusion coefficient 2 2 = = , 2 4D d d gD δ τ π (104) where 1/3d ≡ V is the average dot size. Combining Eqs. (103) and (104), at = 0T we obtain 0 0= = / ln(4 / ).CL D d g Eϕ ϕτ δ (105) At non-zero T thermal fluctuations provide an addi- tional contribution to the dephasing rate 1/ .ϕτ Again substituting Eqs. (97) and (99) into (102), we get 0 1 1 min{ , }, ( ) 3 T N N T g ϕ ϕ ϕ π + τ τ (106) where = / / DN L dϕ ϕ ϕτ τ∼ is the number of quantum dots within the length Lϕ . We observe that for sufficiently small <N Nϕ (but still 1N ) the dephasing rate in- creases linearly both with temperature and with the number N . At larger > / ln[4 / ]CN g E δ and/or at high enough temperatures Nϕ becomes smaller than N and Eq. (106) for ϕτ should be resolved self-consistently. In this case we obtain 2/3(3 / )Dg Tϕτ τ π (107) thus reproducing the well known result [29]. Equation (106) also allows to estimate the temperature 02 / [ min{ , }]T g N Nϕ ϕπ τ at which the crossover to the temperature-independent regime (103) occurs. We find * 3ln[4 / ] , , 2 ln[4 / ] C D C E gT N N E δ π τ δ 3/2 * 3 [4 / ]ln , . ln[4 / ]2 C CD E gT N Eg δ δπτ (108) 3.3.2. Good metals and granular conductors. The above analysis and conclusions can be generalized further to the case 2D and 3D structures. This generalization is absolute- ly straightforward (see, e.g., [23]) and therefore is not ela- borated here. At 0T → one again arrives at the same re- sult for 0ϕτ (103). Now we discuss the relation between our present results and those derived earlier for weakly disordered metals by means of a different approach [4]. Let us express the dot mean level spacing via the average dot size d as 3 0= 1 / N dδ (where 2 0 = / 2FN mp π is the electron densi- ty of states at the Fermi level). Then we obtain 0 = . 4 gD N dπ (109) Below we consider two different physical limits of ( )a good metals and ( )b strongly disordered (granular) con- ductors. For the model ( )a we assume that quantum dots are in a good contact with each other. In this case g scales linearly with the contact area 2= ,dγA where γ is a nu- merical factor of order (typically smaller than) one which particular value depends on geometry. For weakly disor- dered metals most conducting channels in such contacts can be considered open. Hence, 2= / 2Fg p πA and = / 4 ,FD dγv (110) D.S. Golubev, A.G. Semenov, and A.D. Zaikin 1178 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 i.e., .D d∝ If most channels are not fully transparent, then the factor γ in (110) also accounts for their transmissions. Comparing Eq. (110) with the standard definition of D for a bulk diffusive conductor, = / 3,FD lv we immediately observe that within our model the average dot size is com- parable to the elastic mean free path, ,l dγ∼ as it should be for weakly disordered metals. Expressing 0ϕτ (103) via D, in this limit we get 2 3 0 3 2 1 64= , ln( / )cF m D D Dϕτ πγ v (111) where m is the electron mass and 1cD is constant which depends on .CE Estimating, e.g., 2 / 2 ,CE e d≈ one ob- tains 1 1 0= 4 2 / .c FD e N− γv Note that apart from an unimportant numerical pre- factor and the logarithm in the denominator of Eq. (111) the latter result for 0ϕτ coincides with that derived for a bulk diffusive metal within the framework of a completely different approach [4]. Within that approach local proper- ties of the model remain somewhat ambiguous and, hence, in the corresponding integrals in [4] we could not avoid using an effective high frequency cutoff procedure. This cutoff yields the correct leading dependence 3 0 Dϕτ ∝ and it only does not allow to recover an additional loga- rithmic dependence on D in (111). Our present approach is divergence-free and, hence, it does not require any cu- toffs. We can also add that Eq. (103) also agrees with our ear- lier results [4] derived for quasi-1D and quasi-2D metallic conductors. Provided the transversal size a of our array is smaller than d one should set da∼A for 2D and 2a∼A for 1D conductors. Then Eq. (103) yields 2 0 / lnD Dϕτ ∝ and 0 / lnD Dϕτ ∝ respectively in 2D and 1D cases. Up to the factor ln D these dependencies coincide with ones derived previously [4]. Now let us turn to the model ( )b of strongly disordered and/or granular conductors. In contrast to the situation (a), we will assume that the contact between dots (grains) is rather poor, and inter-grain electron transport may occur only via limited number of conducting channels. In this case the average dimensionless conductance g can be approximated by some A -independent constant = .cg g Substituting cg instead of g into Eq. (109) we observe that in the case of strongly disordered structures one can expect 1/ .D d∝ Accordingly, for 0ϕτ (103) one finds 3 0 2 2 3 0 2 = , 32 ln( / ) c c g N D D D ϕτ π (112) where 2cD again depends on CE . For 2 / 2CE e d≈ we have 1 2 0= 2 2 / .c cD N eg− π Hence, the dependence of 0ϕτ on D for strongly disordered or granular conductors (112) is it qualitatively different from that for sufficiently clean metals (111). One can also roughly estimate the crossover between the regimes ( )a and ( )b by requiring the values of = / 4FD dγv (110) and 0= / 4cD g N dπ to be of the same order. This condition yields 2( ) 2 / ,F cp d gπ γ∼ and we arrive at the estimate for D at the crossover 0.6 .cg D m ≈ γ (113) Here we restored the Planck constant set equal to unity elsewhere in our paper. 3.3.3. Ring composed of quantum dots. Now let us turn to a ring-shaped nanostructure as shown in Fig. 10. For simplicity we will consider the case of identical quantum dots (with mean level spacing δ and dwell time = 2 / ( )D gτ π δ ) coupled by junctions with conductances tg and the Fano-factor .tβ Leads are coupled to the ring at the dots with numbers 1 and 1L + by junctions with con- ductance .g The interference correction to the conduc- tance of nth junction nGδ was already derived in Sec. 2 by means of the non-linear sigma-model approach. We obtain 2 , 1 02 0 = ( ) exp[(4 ) / ] 4 t n t n n e g G dt C t i N ∞ + ⎧δ ⎪δ − β π Φ Φ +⎨ ⎪π ⎩ ∫ , 1, 1(1 )( ( ) ( ))t n n n nC t C t+ ++ −β + + 1, 0( ) exp[– (4 ) / ( )] ,t n nC t i N+ ⎫⎪+ β π Φ Φ ⎬ ⎪⎭ (114) where , ( )m nC t is the cooperon. The quantum correction to conductance of the whole system can be obtained with the aid of the Kirchhoff's law. For the case tNg g consi- dered here one finds 2 2 2 2 ( ) ( )= . (2 ( ) ) 4 n t t NL N L g L N L gG g G Ng L N L g Ng − − δ δ ≈ δ + − (115) Further procedure is analogous to that implemented above for 1D arrays. The main difference of the present ring-shaped geometry just concerns the form of diffusons ( ),mnD t cooperons (0) ( )mnC t and the fluctuating voltage correlators ( ) = ( ) (0) .mn m n V F t V t V+ + +〈 〉 We obtain 2 ( ) =1 e( ) = , 2 ( ) iqi t m nN ND mn Dq dD t N i q π − ω + − τ ω π − ωτ + ε∑∫ (116) LL L RLL + 11 L + 2 2 gtgt gt gt g g N Fig. 11. Ring composed of N quantum dots. Weak localization, Aharonov–Bohm oscillations and decoherence in arrays of quantum dots Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 1179 2 ( ) (0) 0=1 e( ) = , 2 ( 2 / ) iqi t m nN ND mn Dq dC t N i q π − ω + − τ ω π − ωτ + ε − Φ Φ∑∫ (117) and 2 ( ) 2 2 2 =1 ( )e( ) = e coth , 2 2 ( ) iq m nN Ni tD mn q D d f qF t N T q π − − ωτ ω ω ω π ω τ + ε ∑∫ (118) where 2 2 ( )( ) = , (4 ( ) ) t D g g e qf q C q C τ ε π ε + (119) 2 ( )( ) = ( ) 4 ( ) t D g g e qq q C q C τ ε ε ε + π ε + (120) and ( ) = 1 cos[2 / ].q q Nε − π As above, here C and gC denote respectively the junction and the dot capacitances. The above equations are sufficient to evaluate the func- tion ( )tF in a general form. Here we are primarily inter- ested in AB oscillations and, hence, we only need to ac- count for the flux-dependent contributions determined by the electron trajectories which fully encircle the ring at least once. Obviously, one such traverse around the ring takes time 2 .Dt N≥ τ Hence, the behavior of the function ( )tF only at such time scales needs to be studied for our present purposes. In this long time limit ( )tF is a linear function of time with the corresponding slope 2( )Dt N′ ≥ τ ≈F 2 2 1 2 2 2 2 2 2 =1 ( ) ( ) coth2 2 . 2 ( ( ))( ( )) N D q D D f q qe d T N q q − ω ε ωτ ω ≈ π ω τ + ε ω τ + ε ∑ ∫ (121) This observation implies that at such time scales electron– electron interactions yield exponential decay of the coope- ron in time (0)( ) ( )exp( / )mn mnC t C t t φ≈ − τ (122) where 21 = ( )Dt N φ ′ ≥ τ τ F (123) is the effective dephasing time for our problem. In the case gC C and 22 / ( )D RC g tC e gτ τ ≡ π from Eq. (124) we obtain 4 ln , 1/ , 1 = , 1/ , 3 C D D t E T N NT T N g φ δ⎧ τ⎪π δ⎪ ⎨ πτ ⎪ τ ⎪⎩ (124) where 2= / (2 ).C gE e C These expressions are, of course, fully consistent with the results derived above in the case of 1D chains of quantum dots and weakly disordered diffu- sive conductors, cf. also [4]. Let us emphasize again that the above results for ( )tF apply at sufficiently long times which is appropriate in the case of AB conductance oscillations. At the same time, other physical quantities, such as, e.g., weak localization correction to conductance can be determined by the func- tion ( )tF at shorter time scales. Our general results allow to easily recover the corresponding behavior as well. For instance, at DT τ and 2 Dt N τ we get 1/2 3/24 2( ) ... 3 t D Tt t g ⎛ ⎞π ≈ +⎜ ⎟τ⎝ ⎠ F (125) in agreement with the results [25]. This expression yields the well known dependence 2/3T− φτ ∝ which — in con- trast to Eq. (124) — does not depend on N and remains applicable in the high temperature limit. To proceed further let us integrate the expression for the Cooperon over time. We obtain 0 ( ) =mnC t dt ∞ ∫ 0=1 2exp ( ) = , ( 2 / ) / / ( ) N D D tq iq m n N N q g g Nφ π⎡ ⎤−⎢ ⎥τ ⎣ ⎦ ε − Φ Φ + τ τ +∑ (126) where the term / ( )tg g N in the denominator accounts for the effect of external leads and remains applicable as long as .tNg g Combining Eqs. (114), (115), and (126) after summation over q we arrive at the final result 2 2 2 ( )= 2 AB t e L N L gG Ng − δ × π 0 2 0 ( 1 )( cos(4 / )) , 1( 2cos(4 / )) N t t N N z z z − − β α + −β − πΦ Φ × α − + − πΦ Φ (127) where = 1 / / ( )D tg g Nφα + τ τ + and 2= 1.z α + α − This equation with Eq. (124) fully determines AB oscilla- tions of conductance in nanorings composed of metallic quantum dots in the presence of electron–electron interac- tions. Expanding Eq. (127) in Fourier series we obtain ( )( ) 0 =1 = cos 4 /AB k k G G k ∞ δ δ π Φ Φ∑ (128) where 2 2 ( ) \| \| 2 2 ( ) ( 1 ) = . 2 1 k N kt t t e L N L g G z Ng −− β α + −β δ − π α − (129) D.S. Golubev, A.G. Semenov, and A.D. Zaikin 1180 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 In the limit Dφτ τ we have 1 2 / ...Dz φ≈ + τ τ + , hence ( )kGδ behaves as \| \| (2 / )( ) e , N k DkG − τ τφδ ∝ (130) i.e., at hight temperatures log \| \|Gδ scales with N as 3/2N while at low temperatures it scales as .N The tem- perature dependence of the first three harmonics of AB conductance in the presence of electron–electron interac- tions is depicted in Fig. 12. The results obtained here allow to formulate quantita- tive predictions regading the effect of electron–electron interactions on Aharonov–Bohm oscillations of conduc- tance for a wide class of disordered nanorings embraced by our model. Of particular interest is the situation of large number of dots 1N which essentially mimics the beha- vior of diffusive nanostructures. In order to establish a di- rect relation to this important case it is instructive to intro- duce the diffusion coefficient 2= / (2 )DD d τ and define the electron density of states 3= 1/ ( ),dν δ where d is a linear dot size. Then we obtain with exponential accuracy: ( ) 3/2 exp ( \| \| ( / )) , / ( ), exp ( \| \| ( / ) ) , / ( ). k k T D d G k T D d φ φ −⎧⎪δ ⎨ −⎪⎩ L L L L L L ∼ Here we introduced the ring perimeter = NdL and the effective decoherence length ( ) 1/2 3 1/32 2 4 / ln , / ( ), = 12 / , / ( ). CE d D T D d d D T T D d φ ⎧⎛ ⎞⎪ πν⎜ ⎟⎪ δ⎝ ⎠⎨ ⎪ ν⎪⎩ L L L Note in the high temperature limit / ( )T D dL the above results match with those derived earlier for metallic nanorings with the aid of different approaches [39,40]. On the other hand, at lower T our results are different. This difference is due to low temperature saturation of φτ which was not accounted for in [39,40]. A non-trivial fea- ture predicted here is that — in contrast to weak localiza- tion [4] — the crossover from thermal to quantum dephas- ing is controlled by the ring perimeter L . This is because only sufficiently long electron paths fully encircling the ring are sensitive to the magnetic flux and may contribute to AB oscillations of conductance. We believe that the quantum dot rings considered here can be directly used for further experimental investigations of quantum coherence of interacting electrons in nanoscale conductors at low temperatures. 4. Comparison with experiments and concluding remarks Turning to the experimental situation in the field, it is im- portant to emphasize again that low temperature saturation of the electron decoherence time has been repeatedly observed in numerous experiments and is presently considered as firm- ly established and indisputably existing phenomenon. Al- though in some cases this phenomenon can be attributed to various extrinsic mechanisms, like magnetic impurities, overheating etc., in the vast majority of cases none of such extrinsic mechanisms can reasonably account for experimen- tal observations. On the other hand, it was demonstrated above that electron–electron interactions universally provide non-vanishing electron dephasing down to = 0T in all types of disordered conductors. Therefore, it would be interesting to perform quantitative comparison between our universal formula for 0 ,ϕτ Eq. (103), and experimental values of the electron decoherence time measured in different structures. Note that in some of our earlier publications [4,41,42] we have already demonstrated a good quantitative agree- ment between our theoretical predictions [4] and experi- mental data for 0ϕτ obtained for numerous metallic wires and quasi-1D semiconductors. Here we address the expe- riments on quantum dot structures as well as on both weak- ly and highly disordered metals. First turning to quantum dots, we recall that in all 14 sam- ples reported in experiments with open quantum dots per- formed by different groups [8,43–46] the values 0ϕτ were found to rather closely follow a simple dependence [46] 0 .Dϕτ ≈ τ (131) This approximate scaling was observed within the interval of dwell times Dτ of about 3 decades, see Fig. 5 in [46]. Our Eq. (103) essentially reproduces this scaling, especial- ly having in mind that the dimensionless conductance g was of order one (or slighlty larger) in almost all samples [8,43–46]. To the best of our knowledge no alternative explanation for the scaling (131) has been offered until now. Thus, we conclude that our theory is clearly consis- tent with the available experimental data on zero tempera- ture electron dephasing in open quantum dots. Fig. 12. Temperature dependence of the first three harmonics of AB conductance for = 500tg , = 30g , = 10,N = 1tβ and / = 120D RCτ τ . � G (k ) / � G 0(k ) 1 10 –1 10 –2 10 –3 10 –4 10 –5 T�D 10 –2 10 –1 1 10 k = 1 k = 2 k = 3 Weak localization, Aharonov–Bohm oscillations and decoherence in arrays of quantum dots Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 1181 Let us now consider spatially extended disordered con- ductors. As our theory of dephasing by electron–electron interactions predicts a rather steep increase of 0ϕτ with the system diffusion coefficient D, for most weakly disordered metals as 3 0 ,Dϕτ ∝ we can conclude that for a large number of disordered conductors 0ϕτ strongly increases with increasing D. This trend is indeed quite obvious for relatively weakly disordered conductors. On the other hand, Lin and coworkers [9,47–49] analyzed numerous experimental data for 0ϕτ obtained by various groups in rather strongly disordered conductors with 10D cm 2 /s and observed systematic decrease of 0ϕτ with increasing D. The data could be fitted by the dependence 0 D−α ϕτ ∝ with the power 1.α This trend is clearly just the oppo- site to one observed in less disordered conductors with 10D cm2/s. In Fig. 13 we collected experimental data for 0ϕτ ob- tained in about 130 metallic samples with similar Fermi velocities and diffusion coefficients varying by 3∼ dec- ades, from 0.3D ≈ cm2/s to 350D ≈ cm2/s. The data were taken from about 30 different publications listed in the figure caption. We see that the measured values of 0ϕτ strongly depend on D. Furthermore, this dependence turns out to be non-monotonous. For relatively weakly disor- dered structures with 10D cm2/s 0ϕτ increases with increasing D, while for strongly disordered conductors with 10D cm2/s the opposite trend takes place. In addi- tion to the data points in Fig. 13 we indicate the dependen- cies 0 ( )Dϕτ (111) and (112) for two models ( )a and ( )b discussed above. We observe that for 10D cm2/s the data points clearly follow the scaling (111). Practically all data points remain within the strip between the two lines corresponding to Eq. (111) with = 1γ (dashed line) and = 0.2γ (solid line). On the other hand, for more disordered conductors with 10D cm 2 /s the data are consistent with the scaling (112) obtained within the model ( )b . We would like to emphasize that theoretical curves (111) and (112) are presented in Fig. 13 with no fit parameters except for a geometry factor γ for the first dependence and the value 150cg ≈ for the second one. This value of cg was estimated from the crossover con- dition (113) with 10D∼ cm2/s and 1γ ∼ . Now let us consider the data for strongly disordered conductors with < 10D cm2/s. As we already pointed out, the agreement between the data and the dependence (112) predicted within our simple model ( )b is reasonable, in particular for samples with < 3D cm2/s. At higher diffu- sion coefficients most of the data points indicate a weaker dependence of 0ϕτ on D which appears natural in the vicinity of the crossover to the dependence (111). The best fit for the whole range 0.3 cm2/s < < 10D cm2/s is achieved with the function 0 D−α ϕτ ∝ with the power α ≈ 1.5–2. Thus, we conclude that our theory allows to qualitative- ly understand and explain seemingly contradicting depen- dencies of 0ϕτ on D observed in weakly and strongly disordered conductors. While the trend «less disorder–less decoherence» (111) for sufficiently clean conductors is quite obvious, the opposite trend «more disorder–less de- coherence» in strongly disordered structures requires a comment. The latter dependence may indicate that with increasing disorder electrons spend more time in the areas with fluctuating in time but spatially uniform potentials. As we already discussed in the beginning of Sec. 3, such fluc- tuating potentials do not dephase and thus 0ϕτ gets effec- tively increased. In other words, in this case the corres- ponding dwell time Dτ in Eq. (103) becomes longer with increasing disorder and, hence, the electron decoherence time 0ϕτ does so too. Note that since local conductance fluctuations increase with increasing disorder, several grains can form a cluster with internal inter-grain conductances strongly exceeding those at its edges. In this case fluctuating potentials remain almost uniform inside the whole cluster which will then play a role of an effective (bigger) grain/dot. Accordingly, the average volume of such «composite dots» 1/∝ δV may grow with increasing disorder, electrons will spend more time in these bigger dots and, hence, the electron decoherence time (103) will increase. The above comparison with experiments confirms that our previous quasiclassical results [4] for 0ϕτ are applicable to relatively weakly disordered structures with Fig. 13. The low temperature dephasing times observed in various experiments for the following samples: Au-1 to Au-6 [6], Au-7 [50], Au-8 and Au-10 [51] (■ ); 44 samples (AuPd and AgPd) [47] ( ); 18 samples [48]: Au2Al ( ), Sb ( ), Sc85Ag15 ( ), V3Al ( ); 9 samples (CuGeAu) [49] ( ); 15 samples (Au, Ag and Cu) [52] and AgMI6N [53] ( ◊ ); CF-1 and CF-2 [54] (Δ ); A, B (Au) [55], Au1 [7], Ag1 [56] and Ag2 [57] ( ); S, M and L (Pt) [58] ( • ); D and F [59] ( ); Ag, AgFe1 and AgFe2 [60] (∇ ); 10 samples [61] within the box ( ); 2 (Au) [62] and 1 (Au) [63] (▲ ); Al-1 [64] (♦). Our Eq. (111) for = 0.2γ and 1 is indicated respectively by solid and dashed lines, while Eq. (112) for = 150cg is depicted by dashed- dotted line. 1 10 10 2 10 –1 10 –2 1 10 10 2 10 –3 D, cm /s 2 � � 0 , n s D.S. Golubev, A.G. Semenov, and A.D. Zaikin 1182 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 10D cm2/s, while for conductors with stronger disorder different expressions for 0ϕτ (e.g., Eq. (112)) should be used. Our analysis also allows to rule out scattering on magnetic impurities as a cause of low temperature satura- tion of ϕτ . The latter mechanism can explain neither strong and non-trivial dependence of the electron decohe- rence time on D nor even the level of dephasing observed in numerous experiments, e.g., in order to be able to attribute dephasing times as short as 12 0 10 s− ϕτ to magnetic impurities one needs to assume huge concentra- tion of such impurities ranging from few hundreds to few thousands ppm which appears highly unrealistic, in partic- ular for systems like carbon nanotubes, 2DEGs or quantum dots. Similar arguments were independently put forward by Lin and coworkers [47,49]. Thus, although electron dephasing due to scattering on magnetic impurities is by itself an interesting issue, its role in low temperature saturation of ϕτ in disordered conduc- tors is sometimes strongly overemphasized. 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Weak localization, Aharonov–Bohm oscillations and decoherence in arrays of quantum dots

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