Dephasing in quantum chaotic transport: A semiclassical approach

Dephasing in quantum chaotic transport: A semiclassical approach

Robert S. Whitney,¹ Philippe Jacquod,²and Cyril Petitjean^3,4

1Institut Laue-Langevin, 6 rue Jules Horowitz, Boîte Postale 156, 38042 Grenoble, France

2Physics Department, University of Arizona, 1118 East 4th Street, Tucson, Arizona 85721, USA

3Département de Physique Théorique, Université de Genève, CH-1211 Genève 4, Switzerland

4Institut I-Theoretische Physik, Universität Regensburg, Universitätsstrasse 31, D-93040 Regensburg, Germany 共Received 7 September 2007; published 15 January 2008兲

We investigate the effect of dephasing共decoherence兲 on quantum transport through open chaotic ballistic conductors in the semiclassical limit of small Fermi wavelength to system size ratio,␭F/LⰆ1. We use the trajectory-based semiclassical theory to study a two-terminal chaotic dot with decoherence originating from共i兲 an external closed quantum chaotic environment,共ii兲a classical source of noise, and共iii兲a voltage probe, i.e., an additional current-conserving terminal. We focus on the pure dephasing regime, where the coupling to the external source of dephasing is so weak that it does not induce energy relaxation. In addition to the universal algebraic suppression of weak localization, we find an exponential suppression of weak localization ⬀exp关

−˜␶/␶␾兴, with the dephasing rate ␶_␾⁻¹. The parameter˜␶depends strongly on the source of dephasing. For a voltage probe,˜␶is of order the Ehrenfest time⬀ln关L/␭F兴. In contrast, for a chaotic environment or a classical source of noise, it has the correlation length␰of the coupling or noise potential replacing the Fermi wavelength

␭F. We explicitly show that the Fano factor for shot noise is unaffected by decoherence. We connect these results to earlier works on dephasing due to electron-electron interactions and numerically confirm our findings.

DOI:10.1103/PhysRevB.77.045315 PACS number共s兲: 05.45.Mt, 03.65.Yz, 73.23.⫺b, 74.40.⫹k

I. INTRODUCTION A. Dephasing in the universal regime

Electronic systems in the mesoscopic regime are ideal testing grounds for investigating the quantum-to-classical transition from a microscopic coherent world共where quan- tum interference effects prevail兲 to a macroscopic classical world.¹On one hand, their size is intermediate between mac- roscopic and microscopic 共atomic兲 systems; on the other hand, today’s experimental control over their design and pre- cision of measurement allows one to investigate them in re- gimes ranging from almost fully coherent to purely classical.^2–4The extent to which quantum coherence is pre- served in these systems is usually determined by the ratio

␶_␾/␶cl of the dephasing time ␶_␾ to some relevant classical time scale ␶cl. For instance, ␶cl can be the traversal time through one arm of a two-path interferometer^5–7or the aver- age dwell time spent inside a quantum dot.^8–12 In a given experimental setup, ␶␾ can often be tuned from ␶␾⬎␶cl

共quantum coherent regime兲 to ␶_␾Ⰶ␶cl 共purely classical re- gime兲by varying externally applied voltages or the tempera- ture of the sample.

Coherent effects abound in mesoscopic physics, the most important of them being the weak localization, universal conductance fluctuations, and Aharonov-Bohm interferences in transport, as well as persistent currents.^2–4The disappear- ance of these effects as dephasing processes are turned on has raised a lot of theoretical^8–25and experimental^26–33inter- est. Focusing on transport through ballistic systems, dephas- ing is usually investigated using mostly phenomenological models of dephasing,^16–21 the most successful of which are the voltage-probe and dephasing-lead models.^16,17 In these models, a cavity is connected to two external, left 共L兲 and right共R兲, transport leads, carryingN_RandN_Ltransport chan-

nels, respectively. Dephasing is modeled by connecting a third “fictitious” lead to the system, with a voltage set such that no current flows through it on average. Electrons leaving the system through this third lead are thus reinjected at some later time, with a randomized phase 共and randomized mo- mentum兲. These models of dephasing present the significant advantage that the standard scattering approach to transport can be applied as in fully coherent systems, once it is prop- erly extended to account for the presence of the third lead.

Using random matrix theory 共RMT兲, the voltage- and dephasing-probe models³⁴ predict an algebraic suppression of the weak-localization contribution to the conductance共in units of 2e²/h兲,³⁵

g_RMT^wl = − N_RN_L

关N_R+N_L兴²关1 +␶D/␶_␾兴⁻¹, 共1兲 where −NRNL/关NR+NL兴² is the weak-localization correction in the absence of dephasing. Similarly, universal conduc- tance fluctuations become¹¹

␦^gRMT2 = 2

␤

N_R²N_L²

关NR+NL兴⁴关1 +␶D/␶␾兴⁻² 共2兲 and are thus damped below their value 共2/␤兲NR2N_L²/关NR

+NL兴² 共in units of 4e⁴/h²兲 for fully coherent systems with 共␤= 1兲or without共␤= 2兲 time-reversal symmetry. For a fic- titious lead connected to a two-dimensional cavity共a lateral quantum dot兲 via a point contact of transparency␳ ^{and car-} ryingN₃channels, one has␶_␾=mA/ប␳^N3, with the electron massmand the areaAof the cavity. Similarly, the dwell time through the cavity is given by␶D=mA/ប共NL+N_R兲.

The dephasing- and voltage-probe models account for dephasing at the phenomenological level only, without refer- ence to the microscopic processes leading to dephasing. At

1098-0121/2008/77共4兲/045315共22兲 045315-1 ©2008 The American Physical Society

sufficiently low temperature, it is accepted that the dephasing arises dominantly from electronic interactions, which, in dif- fusive systems, can be well modeled by a classical-noise potential.^8,9 Remarkably enough, this approach reproduces the RMT results of Eqs.共1兲and共2兲with␶␾set by the noise power. These results are moreover quite robust in diffusive systems. They are essentially insensitive to most noise- spectrum details and hold for various sources of noise such as electron-electron and electron-phonon interactions, or ex- ternal microwave fields. For this reason, it is often assumed that dephasing is system independent and exhibits a charac- ter of universality well described by the RMT of transport applied to the dephasing- and voltage-probe models.³⁵

B. Departure from random matrix theory universality According to the Bohigas-Giannoni-Schmit surmise,³⁶ closed chaotic systems exhibit statistical properties of Her- mitian RMT³⁷in the short wavelength limit. Opening up the system, transport properties derive from the corresponding scattering matrix, which is determined by both the Hamil- tonian of the closed system and its coupling to external leads.³⁸ It has been shown that for not too strong coupling, and when the Hamiltonian matrix of the closed system be- longs to one of the Gaussian ensembles of random Hermitian matrices, the corresponding scattering matrix is an element of one of the circular ensembles of unitary random matrices.³⁹ One thus expects that, in the semiclassical limit of large ratioL/␭F of the system size to Fermi wavelength, transport properties of quantum chaotic ballistic systems are well described by the RMT of transport. This surmise has recently been verified semiclassically.⁴⁰

The regime of validity of RMT is generally bounded by the existence of finite time scales, however, and it was no- ticed by Aleiner and Larkin that, while the dephasing time␶_␾ gives the long time cutoff for quantum interferences, an Ehrenfest timescale appears in the quantum chaotic system in the deep semiclassical limit, which determines the short- time onset of these interferences.¹⁴ The Ehrenfest time ␶E

corresponds to the time it takes for the underlying chaotic classical dynamics to stretch an initially localized wave packet to a macroscopic, classical length scale. In open cavi- ties, the latter can be either the system sizeLor the widthW of the opening to the leads. Accordingly, one can define the closed cavity,␶E

cl=␭⁻¹ln关L/␭F兴, and the open cavity Ehren- fest time,␶Eop=␭⁻¹ln关W²/␭FL兴.^41,42The emergence of a finite

␶E strongly affects quantum effects in transport, and recent analytical and numerical investigations of quantum chaotic systems have shown that weak localization^14,43–46 and shot noise^47–50are exponentially suppressed⬀exp关−␶E/␶D兴in the absence of dephasing 共␶_␾→⬁兲. Interestingly enough, the deep semiclassical limit of finite␶Esees the emergence of a quantitatively dissimilar behavior of weak localization and quantum parametric conductance fluctuations, the latter ex- hibiting no␶Edependence in the absence of dephasing.^51–54 These results are not captured by RMT; instead, one has to rely on quasiclassical approaches^14,44 or semiclassical methods43,45,46,55,56to derive them.

C. Dephasing in the deep semiclassical limit

The behavior of quantum corrections to transport at finite

␶Ein the presence of dephasing was briefly investigated ana-

lytically in Ref.14, for a model of classical noise with large angle scattering, and numerically in Ref. 23, for the dephasing-lead model with a tunnel barrier. Intriguingly enough, the two approaches delivered the same result that quantum effects are exponentially suppressed

⬀exp关−␶E/␶␾兴. This suggested that dephasing retains a char- acter of universality even in the deep semiclassical limit.

More recent investigations have, however, showed that at finite Ehrenfest time, Eq.共1兲becomes²⁵ 共see also Ref.24兲

g^wl= − NRNL

关NR+NL兴²

exp关−共␶E

cl/␶D+^˜␶/␶_␾兲兴

1 +␶D/␶␾ , 共3兲 with a strongly system-dependent time scale˜. Reference␶ 25 showed that, for the dephasing-lead model, ^˜␶⁼␶Ecl+共1

−␳兲␶E

op in terms of the transparency␳ of the contacts to the leads, which provides theoretical understanding for the nu- merical findings of Ref. 23. If, however, one considers a system-environment model, where the environment is mim- icked by electrons in a nearby closed quantum chaotic dot, one has^˜␶=␶_␰, where

␶␰=␭⁻¹ln关共L/␰兲²兴, 共4兲 in terms of the correlation length␰of the interdot interaction potential.

On the experimental front, an exponential suppression

⬀exp关−T/Tc兴of weak localization with temperature has been reported in Ref. 29. Taking ␶_␾⬃T⁻¹ as for dephasing by electronic interactions in two-dimensional diffusive systems, this result was interpreted as the first experimental confirma- tion of Eq.共3兲. There is no other theory for such an expo- nential behavior of weak localization; however, the tempera- ture range over which this experiment has been performed makes it unclear whether the ballistic,^13,15 ␶_␾⬃T⁻², or the diffusive dephasing time determines the Ehrenfest time de- pendence of dephasing共see the discussion in Ref.24兲.

D. Outline of this paper

In the present paper, we amplify on Ref. 25and extend the analytical derivation of Eq. 共3兲 briefly presented there.

We investigate three different models of dephasing and show that the suppression of weak-localization corrections to the conductance is strongly model dependent. First, we consider an external environment modeled by a capacitively coupled, closed quantum dot. We restrict ourselves to the regime of pure dephasing, where the environment does not alter the classical dynamics of the system. Second, we discuss dephasing by a classical-noise field. Third, following Ref.

56, we provide a semiclassical treatment of transport in the dephasing-lead model. For these three models, we reproduce Eq.共3兲and derive the exact dependence of^˜␶on microscopic details of the models considered. All our results are summa- rized in TableI.

The outline of this paper is as follows. In Sec. II, we present the treatment of the system-environment model, fo- cusing, in particular, on the construction of a scattering ap- proach to transport that incorporates the coupling to external degrees of freedom. We apply this formalism to a model of

an open quantum dot coupled to a second, closed quantum dot. We present a detailed calculation of the Drude conduc- tance and the weak-localization correction, including coher- ent backscattering, which explicitly preserves the unitarity of the S-matrix, and hence current conservation. This calcula- tion is completed by a derivation of the Fano factor, showing that, in the pure dephasing limit, shot noise is insensitive to dephasing to leading order. In Sec. III, we present a model of dephasing via a classical-noise field 共such as microwave noise兲. We consider classical Johnson-Nyquist noise models of dephasing due to electron-electron interactions within the system and dephasing due to charge fluctuations on nearby gates. In Sec. IV, we present a trajectory-based semiclassical calculation of conductance in the dephasing-lead model, both for fully transparent barriers and tunnel barriers. We also comment on dephasing in multiprobe configurations. Finally, Sec. V is devoted to numerical simulations confirming our analytical results. Summary and conclusions are presented in Sec. VI, while technical details are presented in the Appen- dix.

II. TRANSPORT THEORY FOR A SYSTEM WITH ENVIRONMENT

In the scattering approach to transport, the system is as- sumed fully coherent and all dissipative processes occur in the leads.⁵⁸Apart from its coupling to the leads, the system is isolated. Here, we extend this formalism to include coupling to external degrees of freedom in the spirit of the standard theory of decoherence. The coupling to an environment can induce dephasing and relaxation. Here, we restrict ourselves to pure dephasing, where the system-environment coupling does not induce energy nor momentum relaxation in the sys- tem. In semiclassical language, we assume that classical tra- jectories supporting the electron dynamics are not modified by this coupling.

The starting point of the standard theory of decoherence is the total density matrix ␩tot that includes both system and

environmental degrees of freedom.¹The observed properties of the system alone are contained in the reduced density matrix␩sys, obtained from ␩tot by tracing over the environ- mental degrees of freedom. This procedure is probability conserving, Tr关␩sys兴= 1, but it renders the time evolution of

␩sys nonunitary and, in particular, the off-diagonal elements of␩sysdecay with time. This can be quantified by the basis independent purity,⁵⁹ 0艋Tr关␩sys2 兴艋1, which remains equal to 1 only in the absence of environment. We generalize this standard approach to the transport problem.

A. Scattering formalism in the presence of an environment We consider two capacitively coupled chaotic cavities, as sketched in Fig.1. The first one is the system共sys兲, an open, two-dimensional quantum dot, ideally connected to two ex- ternal leads. The second one is a closed quantum dot, which TABLE I. Summary of the known results to date on the nature of the exponential term exp关−˜␶/␶␾兴in the

dephasing关cf. Eq.共3兲兴. Results that are not referenced are obtained in the present paper and in Ref.25. Here, we list the value of˜␶for different transport quantities and different sources of dephasing, all in the pure dephasing regime共the phase-breaking regime of Ref.17兲. The parameter␰differs slightly from system to system共see text for details兲; however, it is always related to the correlation length of the interaction with the environment. The results of Ref.24neglect␶_␰contributions共so “0” could indicate a˜␶of order␶_␰兲; they are also only valid for␶E

clⰇ␶␾.

Weak localization Conductance fluctuations Shot noise

System with environment ˜=␶ ␶_␰ — No dephasing

Classical noise共microwave, etc.兲 ˜=␶ ␶_␰ ˜␶⬃0, Ref.24 No dephasing e-e interactions within system ˜␶=␶Ecl+₂¹␶_␰ ˜␶⬃¹2␶Ecl, Ref.24 No dephasing System-gate e-e interactions ˜=␶ ␶_␰ ˜␶⬃2¹␶_␰, follows

from Ref.24

No dephasing

Dephasing lead:

No tunnel barrier ˜␶=␶E

cl ˜= 0␶ No dephasing, Ref.57

Low transparency barrier ˜␶=␶Eop+␶Ecl⬃2␶Ecl ˜␶⬃2␶Ecl共numerics兲, Ref.23 No dephasing, Ref.57

FIG. 1. 共Color online兲 Schematic of the system-environment model. The system is an open quantum dot that is coupled to an environment in the shape of a second, closed dot.

plays the role of an environment 共env兲. The two dots are capacitively coupled and, in particular, they do not exchange particles. Thus, current through the system is conserved. We require that the size of the contacts between the open system and the leads is much smaller than the perimeter of the sys- tem cavity but is still semiclassically large, so that the num- ber of transport channels satisfies 1ⰆN_L,RⰆL/␭F. This en- sures that the chaotic dynamics inside the dot has enough time to develop,␭␶DⰇ1, with the classical Lyapunov expo- nent␭. Electrons in the leads do not interact with the second dot. Few-electron double-dot systems have recently been the focus of intense experimental efforts.⁶⁰ Parallel geometries, of direct relevance to the present work, have been investi- gated in Refs.61and62.

The total system is described by the following Hamil- tonian:

H=H_sys+H_env+U. 共5兲 Inside each cavity, the chaotic dynamics is generated by the corresponding one-particle Hamiltonian H_sys,env. We only specify that the capacitive coupling potentialU is a smooth function of the distance between the particles. It is character- ized by its magnitudeUand its correlation length␰^{such that} its typical gradient isU/␰. Physically,␰is determined by the electrostatic environment of the system, such as electric charges on the gates defining the dots and the amount of depletion of the electrostatic confinement potential between the gates and the inversion layer in semiconductor hetero- structures. Generally speaking,Uand␰are independent pa- rameters and can have different values in different systems and might even be tuned by applying external backgate volt- ages on a given system.

In the standard scattering approach, the transport proper- ties of the system derive from its共N_L+N_R兲⫻共N_L+N_R兲 scat- tering matrix¹⁶

Sˆ =

冉

冊

which we write in terms of transmission共t=s_LR兲and reflec- tion共r=s_␣,␣,␣苸兵L,R其兲 matrices. From Sˆ , the system’s di- mensionless conductance共conductance in units of 2e²/h兲is given by

g= Tr共t^†t兲. 共7兲 To include coupling to an environment in the scattering ap- proach, we need to define an extended scattering matrix S that includes the external degrees of freedom. This is for- mally done in the Appendix, and our starting point is Eq.

共A11兲 for the case of an initial product density matrix ␩tot

=␩_sys^共n兲丢␩env, with ␩_sys^共n兲=兩n典具n兩, n苸兵1 , . . . ,NL其, and ␩env the initial density matrix of the environment. We define the con- ductance matrix

g_nm^共r兲=具m兩Trenv共S关␩sys共n兲丢␩env兴S^†兲兩m典, 共8兲 where Tr_envstands for the trace over the environmental de- grees of freedom. From this matrix, the dimensionless con- ductance is then given by

g=

兺

n=0 N_R

m=0

兺

N_L

g_nm^共r兲, 共9兲

and Eq.共A11兲reads

g=_n苸

兺

R;m苸L

冕

共10兲

Equation共10兲is the generalization of the Laudauer-Büttiker formula in the presence of an external environment. It con- stitutes the backbone of our trajectory-based semiclassical theory of dephasing.

B. Drude conductance

The semiclassical derivation of the one-particle scattering matrix has become standard.^63–65 Once we introduce the en- vironment, we deal with a bipartite problem; here, we use the two-particle semiclassical propagator developed in the framework of entanglement and decoherence.^66,67 The ex- tended scattering matrix elements can be written as

Smn共q,q0兲= −i

冕

⬁

dt

冕

dy₀

冕

dy 具m兩y典具y₀兩n典 共2␲ប兲^{共dsys−1兲/2}

⫻

兺

⌫

A_␥A_⌫

共2␲ប兲^Nd/2e^{i共S␥+S⌫+S␥,⌫兲/ប}, 共11兲

where we take a d_sys-dimensional, one-particle system 共throughout what follows, we take d_sys= 2兲, and a d-dimensional,N-particle environment. At this point, S de- pends on the coordinates of the environment and is given by a sum over pairs of classical trajectories, labeled ␥ ^{for the} system and⌫for the environment. The classical paths␥^and

⌫connecty₀共on a cross section of leadL兲andq₀共anywhere in the volume occupied by the environment兲toy共on a cross section of leadR兲 andq 共anywhere in the volume occupied by the environment兲in the timet=t_␥=t_⌫. For an environment ofNparticles inddimensions,qis anNdcomponent vector.

In the regime of pure dephasing, these paths are solely de- termined byH_sysandH_env. Each pair of paths gives a contri- bution weighted by the square rootA_␥A_⌫ of the inverse de- terminant of the stability matrix^68,69and oscillating with one- particle共S_␥andS_⌫, which include Maslov indices兲and two- particle 共S_␥,⌫=兰₀^td␶U关y_␥共␶兲,q_⌫共␶兲兴兲 action integrals accumulated along␥^and⌫.

We insert Eq.共11兲into Eq.共10兲, sum over channel indices with the semiclassical approximation^45,56 兺n

N_L具y0兩n典具n兩y₀⬘典

⬇␦共y₀⬘^−y0兲. For the environment, we make the random ma- trix ansatz that 具q₀兩␩env兩q₀⬘典⬇共2␲ប兲^Nd⌶_env⁻¹␦共q₀⬘^−q0兲, where

⌶_env is the environment phase-space volume. The dimen- sionless conductance then reads

g=共2␲ប兲⁻¹

⌶env

冕

⬁

dtdt⬘

冕

dq₀dq

冕

dy₀

冕

dy

⫻

兺

␥,⌫;␥⬘^,⌫⬘

A_␥A_⌫A_␥⬘A_⌫⬘e^{i共⌽sys+⌽env+⌽U兲}. 共12兲 This is a quadruple sum over classical paths of the system 共␥^and␥⬘, going fromy₀to y兲and the environment共⌫ and

⌫⬘, going fromq₀ toq兲with action phases

⌽sys=关S_␥共y,y₀;t兲−S_␥⬘共y,y0;t⬘^{兲兴/ប, 共13a兲}

⌽env=关S_⌫共q,q₀;t兲−S_⌫⬘共q,q₀;t⬘兲兴/ប, 共13b兲

⌽_U=关S_␥,⌫共y,y0;q,q₀;t兲−S_{␥⬘,⌫⬘}共y,y0;q,q₀;t⬘兲兴/ប. 共13c兲 We are interested in quantities averaged over variations in the energy or cavity shapes. For most sets of paths, the phase of a given contribution will oscillate wildly with these varia- tions, so the contribution averages to zero. In the semiclas- sical limit, Eq.共12兲is thus dominated by terms which satisfy a stationary phase condition共SPC兲, i.e., where the variation of ⌽sys+⌽env+⌽_U has to be minimized. In the regime of pure dephasing, individual variations of⌽sys,⌽env, and⌽_U are uncorrelated. They are moreover dominated by variations of ⌽sys and⌽env, on which we therefore enforce two inde- pendent SPC’s.

The dominant contributions that survive averaging are the diagonal ones. They give the Drude conductance. Indeed, setting ␥⁼␥⬘ ^and ⌫=⌫⬘ straightforwardly satisfies SPC’s over⌽sysand⌽env. These two SPC’s requiret=t⬘and lead to an exact cancellation of all the phases ⌽sys=⌽env=⌽_U= 0.

The dimensionless Drude conductance is given by g^D=

冕

⬁

dt

冕

dq₀dq

冕

dy₀

冕

dy

兺

␥,⌫

A_␥²A_⌫² 共2␲ប兲⌶env

. 共14兲 From here on, the calculation proceeds along the lines of Ref.45. The main idea is to relate semiclassical amplitudes with classical probabilities. This is done by the introduction of two sum rules that express the ergodic properties of open cavities, Eq.共15a兲, and of closed ones, Eq.共15b兲,

兺

冕

␲/2

d␪0d␪P_sys共Y,Y0;t兲关¯兴Y₀, 共15a兲

兺

冕

Here, P_sys共Y,Y₀;t兲=pFcos␪0⫻P˜

sys共Y;Y₀;t兲, and P˜

sys共Y,Y₀;t兲and˜P

env共Q,Q₀;t兲 are the classical probability densities. For the system, we need to take into account the fact that particles are injected, which is why the classical probability density must be multiplied with the initial system momentum pFcos␪0 along the injection lead.⁶⁴ The phase pointsY₀=共y0,␪0兲andY=共y,␪兲are at the boundary between the system and the leads. In contrast, Q₀=共q0,p₀兲 and Q

=共q,p兲 are inside the closed environment cavity. The mo-

menta are integrated over the entire environment phase space, while P˜

env共Q,Q₀;t兲 will always contain a␦ ^function which restricts the final energy to equal the initial one共i.e., 兩p兩=兩p0兩if allNenvironment particles have the same mass兲.

The average ofP_sys over an ensemble of systems or over energy gives a smooth function. For a chaotic system, we write

具P˜

sys共Y;Y0;t兲典= cos␪ 2共W_L+W_R兲␶D

e^−t/␶D. 共16a兲 Likewise, the average ofP˜

envgives 具P˜

env共Q;Q0;t⬘兲典=␦共兩p兩−兩p₀兩兲

⌶_env^兩p0兩 , 共16b兲 where ⌶_env^兩p0兩 is the size of the hypersurface in the environ- ment’s phase space defined by 兩p兩=兩p₀兩 关for d= 2, ⌶_env^兩p0兩

=共2␲^pFenv⍀env兲^Nwhere⍀env is the area of the environment兴.

Inserting Eqs.共15a兲,共15b兲,共16a兲, and共16b兲into Eq.共14兲, we can perform all integrals using 兰envdQ₀⬅兰envdq₀dp₀=⌶env

and 兰envdQ␦共兩p兩−兩p0兩兲=⌶_env^兩p0兩. Then, since N_L,R=kFW_L,R/␲^, we recover the classical Drude conductance,

g^D= NLNR

NL+NR

. 共17兲

C. Overview of the effect of environment on weak localization

The leading-order weak-localization corrections to the conductance were identified in Refs.14,55, and70as those arising from trajectories that are paired almost everywhere except in the vicinity of an encounter. An example of such a trajectory is shown in Fig.2. At the encounter, one of the trajectories 共␥⬘^兲 intersects itself, while the other one 共␥兲 avoids the crossing. Thus, they travel along the loop they form in opposite directions. For chaotic ballistic systems in the semiclassical limit, only pairs of trajectories with small crossing angle⑀contribute significantly to weak localization.

In this case, trajectories remain correlated for some time on both sides of the encounter, the correlated region indicated in pink in Fig.2. In other words, the smallness of⑀requires two minimal times,TL共⑀兲 to form a loop andTW共⑀兲 in order for the legs to separate before escaping into different leads. In the case of a hyperbolic dynamics, one estimates⁷¹

TL共⑀兲 ⯝␭⁻¹ln关⑀⁻²兴, 共18a兲 TW共⑀兲 ⯝␭⁻¹ln关⑀⁻²共W/L兲²兴. 共18b兲 As long as the system-environment coupling does not generate energy and/or momentum relaxation, the presence of an environment does not significantly change this picture.

However, it does lead to dephasing via the accumulation of uncorrelated action phases, mostly along the loop, when ␥ and␥⬘are more than a distance␰apart. This is illustrated in Fig.2for the case when ␰is less thanW. We define a new time scaleT_␰as twice the time between the encounter and the start of the dephasing,

T_␰共⑀兲 ⬇␭⁻¹ln关⑀⁻²共␰/L兲²兴. 共19兲

Dephasing occurs mostly in the loop part. However, if ␰

⬍⑀LandT_␰⬍0, dephasing starts before the paths reach the encounter. We discuss this point in more detail below in Sec.

II G.

D. Calculating the effect of the environment on weak localization

In the absence of dephasing, each weak-localization con- tribution accumulates a phase difference

␦⌽sys=EF⑀²/共␭ប兲.^55,70In the presence of an environment, an additional action phase difference ␦⌽_U is accumulated. In- corporating this additional phase into the calculation of weak localization does not require significant departure from the theory at U= 0. We extend the theory of Ref.45to account for this additional phase.

We follow the same route as for the Drude conductance, but now consider the pairs of paths described in Sec. II C above and shown in Fig. 2, while the environment is still treated within the diagonal approximation, ⌫⬘^{=⌫. The sum} rule of Eq.共15b兲still applies. Though the sum over system paths is restricted to paths with an encounter, we can still write this sum in the form given in Eq.共15a兲, provided that the probability P_sys共Y,Y₀;t兲 is restricted to paths which cross themselves. To ensure this, we write

P_sys共Y,Y0;t兲=pFcos␪0

冕

dR₂dR₁P˜

sys共Y,R2;t−t₂兲

⫻P˜

sys共R2,R₁;t₂−t₁兲P˜

sys共R1,Y₀;t₁兲.

共20兲 Here, we useR=共r,␾兲,␾苸关−␲^,␲兴, for phase-space points inside the cavity, whileYlies on the lead as before. We then restrict the probabilities inside the integral to trajectories which cross themselves at phase-space positions R_1,2 with the first 共second兲 visit to the crossing occurring at time t₁ 共t2兲. We can write dR₂=v_F²sin⑀dt₁dt₂d⑀ and set R₂

=共r1,␾1±⑀兲. Then, the weak-localization correction to the dimensionless conductance in the presence of an environ- ment is given by

g^wl=共␲ប兲⁻¹

冕

dY₀

冕

with

F共Y0,⑀兲= 2v_F²sin⑀

冕

⬁

dt

冕

dt₂

冕

dt₁

⫻pFcos␪0

冕

dY

冕

dR₁P˜

sys共Y,R2;t−t₂兲

⫻P˜

sys共R₂,R₁;t₂−t₁兲P˜

sys共R₁,Y₀;t₁兲

⫻

冕

˜P

env共Q,Q0;t兲exp关i␦⌽_U兴. 共21b兲 Comparison with Eq.共34兲of Ref.45shows that the effect of the environment is entirely contained in the last line of Eq.

共21b兲. At the level of the diagonal approximation for the environment,⌫⬘^{=⌫, one has}

␦⌽_U= 1 ប

冕

t

d␶兵U关r_␥共␶兲,q_⌫共␶兲兴−U关r_␥⬘共␶兲,q_⌫共␶兲兴其, 共21c兲 wherer_␥共␶兲andq_⌫共␶兲parametrize the trajectories of the sys- tem and of the environment, respectively. We note that in the absence of coupling,␦⌽_U= 0, the integral over the environ- ment is 1, and we recover the weak-localization correction of an isolated system关cf. Eq.共35兲of Ref.45兴.

To evaluate Eqs.共21a兲–共21c兲, we need the average effect of the environment on the system, after one has traced out the environment. For a single measurement, this average is an integral over all classical paths followed by the environ- ment, starting from its initial state at the beginning of the measurement. We therefore also average over an ensemble of initial environment states or an ensemble of environment Hamiltonians which corresponds to performing many mea- surements. For compactness, we define具¯典env as this inte- gral over environment paths and the ensemble averaging over the environment,

FIG. 2.共Color online兲A semiclassical contribution to weak lo- calization for the system-environment model. The paths are paired everywhere except at the encounter, where one path crosses itself at angle ⑀, while the other one does not 共going the opposite way around the loop兲. Here, we show␰⬎⑀L, so the dephasing 共dotted path segment兲starts in the loop共T_␰⬎0兲.

具¯典env=

冕

具P˜

env共Q;Q0;t兲关¯兴Q₀典. 共22兲 Without loss of generality, we assume that for allr, the in- teractionU共r,q兲is zero upon averaging over allq. We can ensure that an arbitrary interaction fulfills this condition by moving any constant term inU into the system Hamiltonian 共these terms do not lead to dephasing兲. Since the environ- ment is ergodic, we have

具U关r_␥共t兲,q_⌫共t兲兴典env= 0. 共23兲 Now, we use the chaotic nature of the environment to give the properties of the correlation function 具U关r_␥共t兲,q_⌫共t兲兴U关r_␥⬘共t⬘^兲,q⌫共t⬘^兲兴典env.

共i兲 Correlation functions typically decay exponentially fast with time in chaotic systems, with a typical decay time related to the Lyapunov exponent.⁷² The precise functional formJ共␭env兩t⬘^−t兩兲of the temporal decay of the coupling cor- relator depends on details of H_env and U; however, for all practical purposes, it is sufficient to know that it decays fast, and we approximate it by a ␦ ^{function, J}共␭env兩t⬘^−t兩兲

⯝␭_env⁻¹␦共t兲.

共ii兲 We argue that the spatial correlations of U for two different system paths at the samet also decay because the averaging over many paths and many initial environment states act like an average overq. We define K共兩r⬘^−r兩/␰兲as the functional form of this decay of spatial correlations, with K共0兲= 1. The precise form ofK共x兲depends on details ofH_sys, H_env, andU. In particular, the typical length␰of this decay is of order the scale on which U共r,q兲 changes between its maximum and minimum values.

Given these arguments, we have

具U关r_␥共t兲,q_⌫共t兲兴U关r_␥⬘共t⬘兲,q_⌫共t⬘兲兴典env

=具U²典

␭env

K共兩r_␥⬘共t兲−r_␥共t兲兩/␰兲␦共t−t⬘兲. 共24兲 Then,

具␦⌽_U²典env= 1 ប²

冕

t

d␶2d␶1

具

⫻兵U关r_␥共␶1兲,q_⌫共␶1兲兴−U关r_␥⬘共␶1兲,q_⌫共␶1兲兴其

典

= 2具U²典

␭env

冕

d␶关1 −K共兩r_␥⬘共␶兲−r_␥共␶兲兩/␰兲兴. 共25兲 We further make the following step-function approxima- tion forK:

K共x兲=⌰共1 −x兲, 共26兲 where the Euler⌰function is 1共zero兲for positive共negative兲 arguments. In principle, this is unjustifiable forx⬃1; how- ever, since the paths diverge exponentially from each other, the time during whichx⬃1 is of order␭⁻¹, while dephasing happens on a time scale ␶_␾ which is typically of order the dwell time,␶D. Thus the step-function approximation ofK共x兲

will have corrections of order 共␭␶D兲⁻¹Ⰶ1, which we there- fore neglect. Once we have made the approximation in Eq.

共26兲, we see that nonzero contributions to具␦⌽_U²典come from regions where the distance between␥^and␥⬘is larger than␰^. We are now ready to calculate dephasing for those system paths shown in Fig.2. As defined above,t₁andt₂are the two times at which the path␥⬘crosses itself. Dephasing acts on the loop formed by ␥⬘ and, as just argued, it acts once the distance between␥ ^and␥⬘is greater than␰, i.e., in the time window from共t₁+T_␰/2兲 to共t₂−T_␰/2兲, where T_␰共⑀兲 is given in Eq.共19兲. We average Eq.共21b兲over the environment and use the central limit theorem to evaluate the action phase due to the coupling between system and environment,

具exp关i␦⌽_U兴典env= exp

关

兴

␶_␾⁻¹⬃ប⁻²␭_env⁻¹具U²典. 共28兲 Given that具¯典envis defined in Eq.共22兲, we can substitute Eq. 共27兲 directly into Eqs. 共21a兲–共21c兲. We thereby reduce the problem to an integral over system paths, which is almost identical to the equivalent integral for U= 0. Assuming phase-space ergodicity for the system, we get the probabili- ties

具P˜

sys共R1,Y₀;t₁兲典= e^−t1/␶D 2␲⍀sys

, 共29a兲

具P˜

sys共R2,R₁;t₂−t₁兲典=e^{−共t2−t1−TW/2兲/␶D} 2␲⍀sys

, 共29b兲

具P˜

sys共Y,R2;t−t₂兲典=cos␪^{e−共t−t2−TW/2兲/␶D} 2共W_L+W_R兲␶D

, 共29c兲

with⍀sysbeing the real space volume occupied by the sys- tem共the area of the cavity兲. At this point, the integral is the same as without dephasing, except that during the time 共t2

−t₁−T_␰兲, the inverse dwell time is replaced by 共␶D−1+␶_␾⁻¹兲.

Thus, when evaluating the共t2−t₁兲integral, we get the extra prefactor exp关−TL共⑀兲/␶␾兴/共1 +␶D/␶␾兲 compared with the equivalent integral result without dephasing. Thus, we have

具F共Y0,⑀兲典⬀sin⑀^{e−TL共⑀兲/␶D−关}

T_L共⑀兲−T_␰共⑀兲兴/␶_␾

1 +␶D/␶␾ . 共30兲 Since 关TL共⑀兲−T_␰共⑀兲兴=␶_␰ with ␶_␰ defined in Eq. 共4兲, the ⑀ dependence of the ␶_␾⁻¹ term drops out. This means that 具F共Y₀,⑀兲典simply differs from its value without dephasing by a constant factor,e^{−␶␰/␶␾}/共1 +␶D/␶␾兲. Thus, the integral over

⑀ ^{in Eq.} 共21a兲 is identical to the one in the absence of the environment and takes the form⁴⁵ Re兰₀^⬁d⑀⑀^{1+2/共␭␶D兲}exp关iEF⑀²/共␭ប兲兴, where we have assumed

⑀Ⰶ1. The substitution z=EF⑀²/共␭ប兲 immediately yields a dimensionless integral and an exponential term,e^{−␶Ecl/␶D} 共ne- glecting as usualO关1兴 terms in the logarithm in ␶Ecl兲. From this analysis, we find that the weak-localization correction is given by

g^wl= g₀^wl

1 +␶D/␶_␾^exp关−␶_␰/␶_␾兴, 共31兲 whereg₀^wl is the weak-localization correction at finite␶E

clin the absence of dephasing,

g₀^wl= − NLNR

共NL+NR兲²exp关−␶E

cl/␶D兴. 共32兲

We see that the dephasing of weak localization is not expo- nential with the Ehrenfest time; instead, it is exponential with the␭F-independent scale␶_␰given in Eq.共4兲. In all cases where␰is a classical scale共i.e., of similar magnitude toW,L rather than␭F兲, we see that␶_␰is much less than the Ehrenfest time,␶_␰Ⰶ␶_␾. In such cases, the exponential term in Eq.共31兲 is much less noticeable than the universal power-law sup- pression of weak localization.

E. Weak localization for reflection and coherent backscattering

We show explicitly that our semiclassical method is prob- ability conserving and thus current conserving, also in the presence of dephasing. We do this by calculating the leading- order quantum corrections to reflection, showing that they enhance reflection by exactly the same amount that transmis- sion is reduced. There are two leading-order off-diagonal corrections to reflection. The first one reduces the probability of reflection to arbitrary momenta共weak localization for re- flection兲, while the second one enhances the probability of reflection to the time reversed of the injection path共coherent backscattering兲. The distinction between these two contribu- tions is related to the correlation between the path segments when they hit the leads. For coherent-backscattering contri- butions, these segments are correlated 共see Fig. 3兲, but for

weak localization contributions, they are not.

The derivation of the weak localization for reflectionr^wl is straightforward and proceeds in the same way as the deri- vation for g^wl given above, replacing the factor NR/共NR

+NL兲byNL/共NR+NL兲. We thus get r^wl= r₀^wl

1 +␶D/␶_␾^exp关−␶␰/␶␾兴, 共33兲 where r₀^wl= −exp关−␶E

cl/␶D兴NL

2/共NL+NR兲² is the finite-␶E cl cor- rection in the absence of dephasing.

We next calculate the contributions to coherent back- scattering, extending the treatment of Ref.45to account for the presence of dephasing. As before, the environment is treated in the diagonal approximation. The coherent- backscattering contributions correspond to trajectories where legs escape together within TW/2 of the encounter. Such a contribution is shown in Fig.3. The correlation between the system paths at injection and exit induces an action differ- ence␦⌽sys=␦^Scbsnot given by the Richter-Sieber expression.

It is convenient to write this action difference in terms of relative coordinates at the lead共rather than at the encounter兲.

The system action difference is then ␦^Scbs= −共p_0⬜

+m␭r_0⬜兲r_0⬜, where the perpendicular difference in position and momentum are r_0⬜=共y0−y兲cos␪0 and p_0⬜=pF共␪−␪0兲.

As with weak localization, we can identify three time scales, T_L⬘^,TW⬘^,T_␰⬘, associated with the time for paths to spread to each of three length scales, L,W,␰. However, unlike for weak localization, we define these time scales as a time mea- sured from the lead rather than from the encounter. Thus, we have

T_ᐉ⬘共r_0⬜,p_0⬜兲 ⯝␭⁻¹ln关共m␭ᐉ兲²/兩p_0⬜+m␭r_0⬜兩²兴, 共34兲 withᐉ=兵L,W,␰其. Writing the integral overY₀as an integral over 共r_0⬜,p_0⬜兲 and using pFsin␪0dY₀=dp_0⬜dr_0⬜, the coherent-backscattering contribution is

r^cbs=

冕

dp_0⬜dr_0⬜

pFsin␪0

Re关e^i␦Scbs兴具F^cbs共Y₀兲典. 共35兲 After integrating out the environment in the same manner as for weak localization, we get

F^cbs共Y0兲=

冕

dY

冕

⬁

dt具Psys共Y,Y0,t兲典exp关−共t−T_␰⬘兲/␶_␾兴

=N_Lp_Fsin␪0

␲共NL+NR兲

exp关−共TL⬘^−TW⬘/2兲/␶D−共TL⬘^−T_␰⬘兲/␶␾兴 1 +␶D/␶␾ .

共36兲 Now, we can proceed as forU= 0, pushing the momentum integral’s limits to infinity and evaluating the r_0⬜ integral over the rangeW, with the help of a Euler ⌫ function. We finally obtain

r^cbs= r₀^cbs

1 +␶D/␶␾exp关−␶_␰/␶␾兴, 共37兲 in terms of r₀^cbs= exp关−␶Ecl/␶D兴 NL/共NL+NR兲, the finite-␶Ecl

coherent-backscattering contribution in the absence of FIG. 3. 共Color online兲A semiclassical contribution to coherent

backscattering for the system-environment model. It involves paths which return to close, but antiparallel to themselves at leadL. The two solid paths are paired共within W of each other兲 in the cross- hatched region. Here, we show ␰⬎⑀L, so the dephasing 共dotted path segment兲starts in the loop共T_␰⬎0兲. In the basis parallel and perpendicular to␥at injection, the initial position and momentum of path ␥ at exit are r_0⬜=共y₀−y兲cos␪0, r_0储=共y₀−y兲sin␪0, and p_0⬜=p_F共␪−␪0兲.