научная статья по теме DEPHASING IN THE SEMICLASSICAL LIMIT IS SYSTEM-DEPENDENT Физика

Текст научной статьи на тему «DEPHASING IN THE SEMICLASSICAL LIMIT IS SYSTEM-DEPENDENT»

Pis'ma v ZhETF, vol.86, iss. 10, pp.736-740

© 2007 November 25

Dephasing in the semiclassical limit is system-dependent

C. Petitjean, P. Jacquod+, R. S. Whitney* Département de Physique Théorique, Université de Genève, CH-1211 Genève 4, Switzerland + Physics Department, University of Arizona, Tucson, AZ 85721, USA *Institut Laue-Langevin, 38042 Grenoble, France Submitted 8 October 2007

We investigate dephasing in open quantum chaotic systems in the limit of large system size to Fermi wavelength ratio, L/Xf » 1. We semiclassically calculate the weak localization correction gwl to the conductance for a quantum dot coupled to (i) an external closed dot and (ii) a dephasing voltage probe. In addition to the universal algebraic suppression gwl oc (1 + td/t>)-1 with the dwell time td through the cavity and the dephasing rate r^1, we find an exponential suppression of weak localization by a factor oc exp[—f/r^], with a system-dependent f. In the dephasing probe model, r coincides with the Ehrenfest time, r oc ln[L/Ap], for both perfectly and partially transparent dot-lead couplings. In contrast, when dephasing occurs due to the coupling to an external dot, r oc ln[_L/£] depends on the correlation length £ of the coupling potential instead of Ap.

PACS: 03.65.Yz, 05.45.Mt, 73.23.^b, 74.40.+k

Introduction. Electronic transport in mesoscopic systems exhibits a range of quantum coherent effects such as weak localization, universal conductance fluctuations and Aharonov-Bohm effects [1, 2]. Being intermediate in size between micro- and macroscopic systems, these systems are ideal playgrounds to investigate the quantum-to-classical transition from a microscopic coherent world, where quantum interference effects prevail, to a macroscopic classical world [3]. Indeed, the disappearance of quantum coherence in mesoscopic systems as dephasing processes set in has been the subject of intensive theoretical [4-7] and experimental [8-10] studies. When the temperature is sufficiently low, it is accepted that the dominant processes of dephasing are electronic interactions. In disordered systems, dephasing due to electron-electron interactions is known to be well modeled by a classical noise potential [4], which gives an algebraic suppression of the weak localization correction to conductance through a diffusive quantum dot,

5wl=5owl/(l + rD/r0). (1)

Here, ^q 1 is the weak localization correction (in units of 2e2/h) without dephasing, the dephasing time r^ is given by the noise power, and td is the electronic dwell time in the dot. Eq. (1) is insensitive to most noise-spectrum details, and holds for other noise sources such as electron-phonon interactions or external microwave fields.

Other, mostly phenomenological models of dephasing have been proposed to study dephasing in ballistic systems [5-7], the most popular of which, perhaps, being the dephasing lead model [5, 6]. A cavity is connected to two external L (left) and R (right) leads of widths W"i.. W"b • A third lead of width W3 is connected to the system via a tunnel-barrier of transparency p. A voltage is applied to the third lead to ensure that no current flows through it on average. A random matrix theory (RMT) treatment of the dephasing lead model leads to Eq. (1) with td = to£/(Wl+Wr) and r^ = ToL/pW3, in term of the dot's time of flight To [6, 11]. Thus it is commonly assumed that dephasing is system-independent. The dephasing lead model is often used phenomenolog-ically in contexts where the source of dephasing is unknown.

Our purpose in this article is to revisit dephasing in open chaotic ballistic systems with a focus on whether dephasing remains system-independent in the semiclassical limit of large ratio L/Xp of the system size to Fermi wavelength. This regime sees the emergence of a finite Ehrenfest time scale, = A-1 ln[L/Ap] (A is the Lyapunov exponent), in which case dephasing can lead to an exponential suppression of weak localization, oc exp[—Tg1 /t^]/(1 + td/t^) [12]. Subsequent numerical investigations on the dephasing lead model support this prediction [13]. Here we analytically investigate two different models of dephasing, and show that the suppression of weak localization is strongly system-dependent. First, we construct a new formalism that

Dephasing in the semiclassical limit is system-dependent...

incorporates the coupling to external degrees of freedom into the scattering approach to transport. This approach is illustrated by a semiclassical calculation of weak localization in the case of an 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 provide the first semiclassical treatment of transport in the dephasing lead model. We show that in both cases, the weak localization correction to conductance is

5wl = 5owl exp[-r/T0] /(1 + td/t0), (2)

where 1 is the finite-r^1 correction in absence of de-phasing. The time scale f is system-dependent. For the dephasing lead model, f = Tg + (1 in terms of the

transparency p of the contacts to the leads, and the open system Ehrenfest time TgP = A-1 \n\W2/X-pL}. This analytic result fits the numerics of Ref. [13], and (up to logarithmic corrections) is in agreement with Ref. [12]. Yet for the system-environment model, f = A-1 ln[(L/£)2] depends on the correlation length £ of the inter-dot coupling potential. We thus conclude that dephasing in the semiclassical limit is system-dependent.

Transport theory for a system-environment model. In the standard theory of decoherence, one starts with the total density matrix rjtot including both system and environment degrees of freedom [3]. The time-evolution of rjtot is unitary. The observed properties of the system alone are given by the reduced density matrix %ys, obtained by tracing rjtot over the environment degrees of freedom. This is probability conserving, Tr jfeys = 1, but renders the time-evolution of t]sys non-unitary. The decoherence time is inferred from the decay rate of its off-diagonal matrix elements [3]. We generalize this approach to the scattering theory of transport.

To this end, we consider two coupled chaotic cavities as sketched in Fig. la. Few-electron double-dot systems similar to the one considered here have recently been the focus of intense experimental efforts [14]. One of them (the system) is an open quantum dot connected to two external leads. The other one (the environment) is a closed quantum dot, which we model using RMT. The two dots are chaotic and capacitively coupled. In particular, they do not exchange particles. We require that Af -C M'l.r -C l, so that the number of transport channels satisfies 1 -C iVi^R -C L/Xp and the chaotic dynamics inside the dot has enough time to develop, Atd 1. Electrons in the leads do not interact with the second dot. Inside each cavity the dynamics is generated by chaotic Hamiltonians Hsys and Henv. We only specify

737

Fig.l. (a) (left panel) 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 quantum dot. (b) (right panel) Schematic of the dephasing lead model

that the capacitative coupling potential, U, is smooth, and has magnitude U and correlation length

The environment coupling can be straightforwardly included in the scattering approach, by writing the scattering matrix, S, as an integral over time-evolution operators. We then use a bipartite semiclassical propagator to write the matrix elements of § for given initial and final environment positions, (q0,q), as

Smn(qo,q)=(27r)_1 / dt dy0 / dy {m|y) (y0|n) x JO J l J r

x ^(C7 CT)1/2 exp[j {S7 + ST + 57>r}]. (3)

7,r

This is a double sum over classical paths, labeled 7 for the system and F for the environment. For pure dephasing, the classical path 7 (F) connecting y0 (qo) to y (q) in the time t is solely determined by Hsys (Henv). The prefactor C7Cr is the inverse determinant of the stability matrix, and the exponent contains the non-interacting action integrals, S7, Sr, accumulated along 7 and F, and the interaction term, <S7jr = /pOiTWfy.ytTKqrtT)].

Since we assume that particles in the leads do not interact with the second cavity, we can write the initial total density matrix as rj^ = rjly} ® rjenv, with ijsys = \n){n\, n = 1,2,...JVl. We take rjenv as a random matrix, though our approach is not restricted to that particular choice. We define the conductance matrix as the following trace over the environment degrees of freedom,

g£>n = (m\Tienv[SVW §i]\m). (4)

The conductance is then given by g = ^2mnglnli-This construction is current conserving, however the environment-coupling generates decoherence and the suppression of coherent contributions to transport. To see this we now calculate the conductance to leading order in the weak localization correction.

We insert Eq. (3) into Eq. (4), perform the sum over channel indices with the semiclassical approximation EnL(yoln)(nlyo) ~ S(y'o ~ yo) [15], and use the RMT result (qo|»?env|qo) « fi^tf^ - qo), where Oe„v is the environment volume [16]. The conductance then reads

g= (47T2 fienv) 1 / dtdt1 / dq0 dq dy0 dyx Jo J Q„v J l J r

x J2 (C-r Ci' CT')1/2 e,'(*«"+*»'+*"). (5)

This is a quadruple sum over classical paths of the system (7 and 7', going from y0 to y) and the environment (F and F', going from q0 to q), with action phases $sys = S,7(y0,y;i)^57'(y0,y;i'), $env = 5r(q0,q;i) - Sr'(q0,q;i') and = = <S7jr(yo,y;qo,q;i) - <57',r'(yo,y;qo,q;i')- We are interested in the conductance averaged over energy variations, and hence look for contributions

to Eq. (5)

with stationary #sys,#env The first such contributions are the diagonal ones with 7 = 7' and F = F', for which = 0. They are ¿/-independent and give the classical, Drude conductance, g° = NlNr/(Nl + JVr). The leading order correction to this comes from weak-localization paths

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