Dynamics of four- versus two-terminal transport through chaotic quantum cavities M. L. Polianski 1 , 2 and M. B¨ uttiker 3 1 Niels Bohr Institute, NBIA, Blegdamsvej 17, DK-2100 Copenhagen Denmark 2 Laboratoire de Physique des Solides, Universit´ e Paris-Sud, 91405 Orsay, France 3 D´ epartement de Physique Th´ eorique, Universit´ e de Gen` eve, CH-1211 Gen` eve 4, Switzerland (Dated: November 23, 2018) We consider multi-terminal mesoscopic transport through a well-conducting chaotic quantum cavity using random matrix theory. Four-probe resistance vanishes on the average and is not affected by weak localization. Its fluctuations are given by a single expression valid for arbitrary temperature, ac frequency, non-ideal coupling of the contacts, and in the presence of floating probes; surprisingly, they are governed by the dwell time only. In contrast, the two-probe transport additionally depends on the RC-time, which is interpreted as a property of the measurement scheme. We also predict a universal mesoscopic distribution of the phase of transmitted voltage in an ac experiment. PACS numbers: 73.23.-b,05.60.Gg,73.63.Kv Introduction. Mesoscopic transport and its sample- to-sample fluctuations are fundamentally important for understanding quantum effects in electronic transfer through nano-structures. However, when a voltage is measured between the source and drain in a two-terminal geometry, a big classical contribution to transport shad- ows quantum effects of order e $^{2}$/h [1, 2]. The visibility of interference effects can be enhanced by diminishing the width of the contacts to reservoirs, but this also leads to Coulomb blockade of electrons [2]. Another approach is to keep the contacts well-conducting and apply a multi- terminal scheme with a voltage through the sample mea- sured by two additional contacts, see Fig. 1(a). The thus defined four-terminal resistance R$_{4t}$ , well-known to ex- perimentalists, has some benefits over the two-terminal one, R$_{2t}$ , which is more popular among theoreticians. PSfrag replacements f r V 0 ˜ V η dc two-terminal conductance shows a weak-localization correction (WL) ∼ e $^{2}$/h due to an enhanced return prob- ability for an electron, and universal conductance fluctu- ations (UCF) ∼ ( e $^{2}$/h ) $^{2}$; a magnetic field destroys WL and reduces UCF by a factor 2 [16]. These results for ballistic quantum point contacts (QPCs) can be gener- alized to include either (i) a dephasing (floating) probe coupling [19], or (ii) non-ideal QPCs [20], or (iii) a low- frequency ac setup [21]. In (iii) the averaged conduc- tance [21] and shot-noise [22] in their leading order de- pend only on the charge relaxation time (RC-time) τ$_{RC}$ , sensitive to the Coulomb interaction energy e $^{2}$/ 2 C of the capacitor. Its appearance is natural: at high frequen- cies ω ≳ 1 /τ$_{RC}$ the capacitor conducts better than the contacts. However, WL and UCF, as well as the third current cumulant, additionally depend on the dwell time τ$_{d}$ an electron typically spends in the dot [21, 23, 24]. As a result, even in linear transport there is no unique time-scale for dispersion of various quantities. At first one expects R$_{4t}$ to share the features of two- terminal WL and UCF. However, these expectations are The central advantage of R$_{4t}$ is to exclude the contact properties. Indeed, classical resistances are local, and a four-terminal scheme removes the effect of the contacts and therefore probes the sample’s properties only [3]. Quantum mechanically resistance is not local, and this scheme does reduce the role of contacts but they still af- fect the measurement. Nevertheless even for a coherent sample there are of course profound differences between R$_{2t}$ and R$_{4t}$ . The former determines Joule heating and is therefore always positive and symmetric to the magnetic field inversion. In contrast, the latter need not be pos- itive [4, 5] and experiments demonstrate its fluctuations around zero [6–8] and magnetic field asymmetry [9]. Non-universal fluctuations in R$_{4t}$ were widely investi- gated in metallic diffusive samples, where small quantum effects on top of a large classical average depend on the probe locations [10–15]. On the other hand, in a chaotic cavity [16, 17] the exact positions of probes are irrele- vant, the classic voltage drop vanishes, and an experi- ment would measure quantum effects directly. Remark- ably, R$_{4t}$ in this generic geometry remained unexplored except for a recent initial experiment [18]. What could one expect from R$_{4t}$ in a quantum dot? Its FIG. 1: (a) A multi-terminal chaotic cavity with a gate (ca- pacitance C ). A current through source-drain contacts 1,2 and a voltage drop between 3,4 (or 1,2) determine the four- probe R 4t = V 34 /I (or two-probe R 2t = V 12 /I ) resistances. Additional floating probes draw no current. (b) The meso- scopic distribution P ( φ ) (rescaled by × ωτ$_{d}$ ) of phase φ of the transmitted ac voltage as a function of frequency, for ωτ$_{d}$ = 0 . 1 , 0 . 3 , 1, full, long- and short-dashed, respectively. arXiv:1001.2492v1 [cond-mat.mes-hall] 14 Jan 2010 1 3 (b) (a) 4 2 0.5 1 1.5 P( f ) f p /3 p /2 -p /2 -p /3 -p /6 V$_{12}$ V$_{34}$ V$_{0}$ I C p /6