ARTICLESPUBLISHED ONLINE: 20 JUNE 2010 | DOI: 10.1038/NPHYS1690

Universal resistances of the quantumresistance–capacitance circuitChristophe Mora1 and Karyn Le Hur2*Mesoscopic circuits cooled down to low temperatures witnessmarked non-local effects in their transport properties because ofelectron coherence—electron wavefunctions spread over the sample and correlate its different parts. One of the consequencesis that, in one dimension, the maximum d.c. conductance is quantized in steps of e2/h. Here we extend the concept of ‘universalquantized resistance’ to the a.c. regime. We analyse the coherent quantum resistance–capacitance circuit comprising a cavitycapacitively coupled to a gate and connected bymeans of a single spin-polarized channel to a reservoir lead.We show that, as aresult of the Coulomb interaction and global phase coherence, the charge relaxation resistance Rq is identical for weak and largetransmissions and that it smoothly changes from h/2e2 to h/e2 when the frequency exceeds the level spacing of the cavity. Forlarge cavities, we relate the resistance h/e2 to the Korringa–Shiba relation of the Kondo model. Finally, we introduce a largerclass of models with a universal charge relaxation resistance.

The Landauer–Büttiker formula for coherent d.c. transport liesat the heart of modern electronics1–3 and embodies one ofthe most marked predictions of modern condensed-matter

physics: the perfect quantization, in steps of e2/h, of the maximumelectrical conductance in one-dimensional metallic channels. It isuniversal insofar as onemay validly neglect the disruptive influencesof inelastic scattering processes within the transport process. Anelementary explanation of the quantization views the constrictionas an electron waveguide that has a non-zero resistance even thoughthere are no impurities, because of the reflections occurring whena small number of propagating modes in the waveguide is matchedto a large number of modes in the reservoirs4,5. This conductancequantization has been observed in various systems such as quantumHall states6, quantum point contacts7,8, carbon nanotubes9,10and the helical edge liquid of topological insulators11. Here, weinvestigate the a.c. regime, or more specifically the quantumresistance–capacitance (RC) circuit of Fig. 1 for spin-polarizedelectrons, and show that the charge relaxation resistance remainsquantized regardless of the mode transmission, whereas thequantized resistance in the d.c. case requires a perfectly transmittedchannel12,13. Theoretically, the study of a.c. coherent transport waspioneered in a scattering approach by Büttiker et al.14 where auniversal charge relaxation resistance of Rq=h/2e2 was predicted15for a single-mode resistor; the factor 1/2 is purely of quantumorigin and must be distinguished from spin effects. Coulombblockade effects16,17 were ignored and later they have been partiallyincluded in a Hartree–Fock theory18,19. The quantum mesoscopicRC circuit has been successfully implemented in a two-dimensionalelectron gas and the charge relaxation resistance Rq = h/2e2was measured20,21. The present work completes the proof of theuniversal quantized resistance Rq= h/2e2 by including interactionsin the cavity non-perturbatively (exactly). Moreover, we evidence amesoscopic crossover at finite frequency ω, where the charge relax-ation resistance changes from h/2e2 to h/e2 regardless of the modetransmission. In practice, the description of interactions in thedot, introduced by Matveev22,23, is extended to the investigation ofcharge dynamics and to the case of a mesoscopic cavity with a finitelevel spacing. Our findings are obtained in the two complementary

1Laboratoire Pierre Aigrain, École Normale Supérieure, Université Denis Diderot, CNRS, 24 rue Lhomond, 75005 Paris, France, 2Departments of Physicsand Applied Physics, Yale University, New Haven, Connecticut 06520, USA. *e-mail: karyn.lehur@yale.edu.

limits of weak and large transparencies at the dot–lead interface, andclose to the absolute zero to preserve the quantum coherence.

The crossover takes place when the level spacing of the cavity∆ becomes equal to hω; hereafter, we set h= 1 and h= 2π. Forsmall cavities and small frequencies, the interactingmodel results inRq=h/2e2. The metallic regime of large cavities24,25 is characterizedby a continuous spectrum. We use a mapping to the charge-Kondoeffect22,23,26–29 to justify the other universal value Rq = h/e2.Interestingly, this charge relaxation resistance is equivalent to twoSharvin–Imry contact resistances h/2e2 in series; in the metallicregime, an electron entering the cavity is disentangled from anelectron escaping the cavity. The crossover described in this articleis distinct from the crossover to the incoherent classical regime thatwas addressed for small cavities in ref. 30. The result Rq = h/e2also differs from the a.c. response of a wire coupled to two leadswhere the contact resistances are added in parallel31–34. Recently,the quantum RC circuit has also gained a growing interest in otherparametric regimes, both theoretically35,36 and experimentally37.

The potential of the reservoir is taken as a reference (V = 0) andwe vary the a.c. potential of the gate Vg. The formula that gives thecharge in the capacitor at low frequency,

Q(ω)Vg(ω)

=C0(1+ iωC0Rq)+O(ω2) (1)

for a classical circuit extends to the quantum regime with modifiedvalues of the capacitance C0 and the charge relaxation resistance Rq.In particular phase-coherent transport implies that the capacitiveand tunnelling effects cannot be disentangled. C0 describes thestatic charging of the dot22,23,38. It is generally different fromthe geometrical capacitance Cg and depends strongly on thelead–dot transparency D as shown in Fig. 2. The average of C0over oscillations as a function of Vg equals the electrochemicalcapacitance15 Cµ, given by the geometrical capacitance Cg in serieswith the quantum capacitance e2/∆. Similarly,Rq does not coincidewith the d.c. resistance h/(De2). The reason for this discrepancyis that carriers injected into the cavity may not equilibrate in a.c.transport. The productRqC0 sets the timescale for the charge to relaxand Rq controls energy dissipation during a.c. driving. Below, we

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ARTICLES NATURE PHYSICS DOI: 10.1038/NPHYS1690

QPC

2D gas

Cavity

Lead

CgV = 0

Rq

C0

V = 0

V

Vg

Cg

Vg(t)

Vg(t)

Figure 1 | The principle of the quantum RC circuit built with a mesoscopic capacitor. The quantum RC circuit realized in a two-dimensional electron gas(2D gas)20 and its equivalent circuit.

Q e

C Vg /e

0

0.5

1.0

1.5

2.0

2.5

0 0.5 1.0 1.5 2.0

µ

Figure 2 | Charge quantization in a quantum dot depending on itsopening. Schematic representation of the static charge 〈N〉= 〈Q〉/e on thecavity as function of the gate voltage. Different transparencies D=0 (plainred), intermediate D (dotted green) and D= 1 (dashed blue) arerepresented. As D increases, charge quantization is reduced. The stepsare replaced by oscillations, finally yielding a linear dependence. Thederivatives of these curves give the capacitance C0= ∂〈Q〉/∂Vg. It is flat forD= 1 (=Cµ, see main text) and builds peaks at the charge degeneracypoints as the charge becomes quantized.

provide an exact derivation of Rq and C0 from the weak and largetransmission limits.

Weak transparency. As a first example, we consider the case wherethe cavity is weakly coupled (by means of tunnel contact) tothe reservoir lead. Matveev has introduced a rigorous formalismallowing computation of the static charge fluctuations on thecavity22. We extend the analysis by computing the dynamics of thecharge fluctuations at low frequency. The Hamiltonian splits asH =H0+Hc+HT with

H0=∑p

εpcp†cp+∑k

εkdk †dk (2a)

Hc= Ec(N −N0)2 (2b)

HT= t∑k,p

(dk †cp+ cp†dk

)(2c)

The single-particle energies εk,p, respectively in the dot and inthe lead, are measured from the Fermi level. dk and cp arethe corresponding electron annihilation operators. The lead is ametallic reservoir. The level spacing in the dot is finite, ∆= πvF/Lfor a one-dimensional dot of length L. The metallic dot (largecavity) means ∆→ 0. Here, Ec = e2/(2Cg) is the charging energydetermined by the geometrical capacitive coupling to the gate;

N is the (integer) operator that gives the number of electronson the dot and N0 = CgVg/e is imposed by the gate voltage. Asobservables are periodic in Vg(N0), we restrict N0 to the window[−Cµ/(2Cg),Cµ/(2Cg)] and charge degeneracy is reached at theboundaries. The Hamiltonian part HT describes the tunnelling ofelectrons between the lead and the dot. Each tunnelling eventchanges the number of electrons on the dot by±1.

In the presence of a small time-dependent perturbation of thegate voltage, the charge on the dotQ= e〈N 〉 obeys,

Q(ω)= e2K (ω)Vg(ω) (3)

where the retarded response function, following standard linear re-sponse theoryK (t−t ′)= iθ(t−t ′)〈[N (t ),N (t ′)]〉, describes chargefluctuations at equilibrium. In the absence of electron tunnelling,the cavity charge in the ground state is 〈N 〉 = 0 and does notfluctuate, henceK =0. For weak tunnelling, the charge fluctuationson the cavity are determined using perturbation theory in HT.The second order in the static case (ω = 0) gives the capacitanceC0= e2K (0). In themetallic case, one finds22,39

C0=DCg

1/4−N 20

(4)

which diverges at the charge degeneracy points N0 = ±1/2. Thetransparency D= t 2ν0ν1 depends on the density of states ν0 (ν1)in the lead (cavity). For a finite level spacing ∆, a discretesummation over single-particle states remains in the expression ofC0 (Supplementary Section SI). We now turn to the imaginary partofK (ω) at low frequency describing dissipation. In the presence of aslowly time-varying gate voltageVg(t ), a redistribution in the virtualoccupation of excited states allows the charge Q to follow Vg andcreates dissipation40. The only way to dissipate at small frequencyω is to find a continuum of excitations with energies ∼ω closeto the ground state. States with a charge on the cavity differentfrom zero, such as those reached by second-order perturbation,exhibit gaps∼Ec�ω. Dissipation is thus carried out by states witha charge e〈N 〉 = 0 but with an extra electron–hole excitation (inthe lead or in the dot). This state is reached by applying at leasttwo times HT on the ground state. Hence, ImK (ω) is obtainedfrom the fourth order in perturbation theory. For a large cavitywhere dissipation can occur both in the lead or in the cavity, wefind ImK (ω)= (πω/2)(D/Ec)2(1/4−N 2

0 )−2. Comparing this with

equation (4) results in the illuminating formula,

ImK (ω)= 2πω[ReK (0)]2 (5)

at small frequency ω. The same calculation is carried out for a smallcavity and in this case we find:

ImK (ω)=πω[ReK (0)]2 (6)

698 NATURE PHYSICS | VOL 6 | SEPTEMBER 2010 | www.nature.com/naturephysics

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NATURE PHYSICS DOI: 10.1038/NPHYS1690 ARTICLESPhysically, states in a small cavity do not form a continuum (because∆ > ω) and therefore they do not contribute to dissipation atsmall frequencies. This explains the reduction by a factor 1/2between equations (6) and (5). The charge response to a gate-voltage oscillation in equation (3) is matched with the RC circuitformula (1). Mostly, equation (5) gives Rq= 2π/e2= h/e2 for largecavities and equation (6) gives Rq=π/e2= h/2e2 for small cavities.From the above discussion, it seems that the crossover between thetwo regimes of cavity takes place whenω∼∆where the states in thecavitymerge into a continuum able to dissipate energy.

Charge relaxation resistance and Korringa–Shiba formula. Infact, the result of equation (5) for large cavities can be inferredfrom the charge degeneracy point N0 = 1/2 where perturbationtheory breaks down and equation (4) for C0 is seen to diverge. Inthe vicinity of N0 = 1/2, the states with a charge 0 and e on thecavity give a resonant and dominant contribution to the chargefluctuations. It has been realized by Matveev22 that, by removingother charge states, a mapping to the anisotropic Kondo model canbe formulated. The fictitious spin S of the Kondo model acts on thespace formed by the charge state 〈N 〉=0 and 1 with

N =12−Sz (7)

The spin-up and spin-down conduction electrons in the Kondomodel originate from cavity and lead electrons respectively, and theassociated Hamiltonian is given by equation (2a). The tunnellingterm of equation (2c) changes the charge state corresponding to aspin-flip event. Hence, 2t plays the role of the antiferromagnetic(transverse) Kondo coupling. Finally, the vicinity to the chargedegeneracy Ec(1−2N0) gives a local magnetic field coupled to Sz .The correspondence (7) between charge and spin indicates thatthe charge fluctuations K (ω) are related to the longitudinal spinsusceptibility in the effective Kondomodel,

K (t− t ′)=χzz(t− t ′)= iθ(t− t ′)⟨[Sz(t ),Sz(t ′)

]⟩(8)

The anisotropic and isotropic Kondo models converge to the sameFermi-liquid fixed point41. Thus, at zero temperature, one may usethe Korringa–Shiba relation42 at low frequencyω

Imχzz(ω)= 2πω[Reχzz(0)]2 (9)

to recover equation (5) and therefore the universal charge relax-ation resistance Rq = h/e2. The demonstration of equation (9) inref. 40 is quite instructive. It shows that a single term accountsfor both the static susceptibility and dissipation explaining thestructure of equation (9). Moreover, this demonstration indicatesthat, similarly to the perturbative region (far from N0 = 1/2),dissipation is caused by electron–hole excitations, yielding thelinear low-frequency dependence for Imχzz(ω). The conclusionis that the Korringa–Shiba relation equation (9) generally appliesat a Fermi-liquid fixed point and in particular in the contextof the anisotropic Kondo model43 at low temperatures and inthe presence of any magnetic field40. We recall that a magneticfield in the spin model corresponds to the distance Ec(1− 2N0)to the charge degeneracy point in the original charge model.Equation (5) and the charge relaxation result Rq = h/e2 thereforeapply for all values of N0, from the vicinity of N0 = 1/2 to theperturbative region where these two formulae have been checkedexplicitly. An intimate connection between the Korringa–Shibarelation in its generalized form40 and the universal resistanceRq = h/e2 is revealed here, where the Fermi-liquid nature oflow-energy particle–hole excitations plays a striking role. Thegenerality of these arguments in fact suggests that the resultRq = h/e2 (and Rq = h/2e2) is indeed universal in the sense

that it does not depend either on the gate voltage N0 or onthe transparency D.

Large transparency. We pursue this idea and investigate theopposite limit at and close to perfect transmission. In whatfollows, we model the complete system by electrons moving alonga one-dimensional line; the lead is between −∞ and −L andthe cavity between −L and 0. The level spacing on the isolatedcavity is still ∆ = πvF/L. The interaction term (2b), and inparticular the Coulomb blockade phenomenon in the cavity, canbe treated exactly using the bosonization approach44,45. Integratingall irrelevant modes in an action formalism, at perfect transmissionD= 1, one finds the action

S0=1π

∑n

φ0(ωn)φ0(−ωn)[

|ωn|

1−e−2|ωn|L/vF+

Ec

π

](10)

where ωn = 2πTn denote bosonic Matsubara frequencies; theBoltzmann constant kB is set to unity. Here, the field φ0 is related tothe charge on the cavity, N =CµVg/e+φ0/π. From this quadraticaction, the response function equation (3) is straightforwardlycalculated. We find:

Q(ω)Vg(ω)

=Cg

1− iωπ/Ec1−e2iπω/∆

(11)

Interestingly, the response vanishes33 each time the frequency ωhits a multiple of ∆ corresponding to an eigenstate of the isolatedcavity. At low frequency ω�∆, we extract C0=Cµ—meaning thatthe Coulomb blockade effect vanishes23 for perfect transparencyD=1—andRq=h/2e2 from the comparison of equation (11) to theclassical RC circuit formula (1). We now discuss the transition tolargemetallic cavities. Equation (11) shows an oscillatory behaviourfor ω>∆. We thus average over a finite bandwidth δω, such thatω� δω�∆, and finally we find:

Q(ω)Vg(ω)

=Cg

1− iωπ/Ec(12)

This result is also obtained if one takes L→+∞ in equation (10). Itcan be checked that this correspondence extends to all correlationfunctions. Hence, equation (10) with L→+∞ defines the actionfor the large-cavity regime, as shown in the Methods section. Acomparison of the result (12) with equation (1) gives C0=Cµ=Cg,indeed corresponding to a vanishing level spacing, and Rq = h/e2.We thus recover the universal resistances h/2e2 and h/e2 for thesmall and large cavities, and the fact that the crossover takes placewhen the frequencyω becomes larger than∆.

Backscattering at the interface between the cavity and the lead (atx=−L) may be incorporated in themodel as,

SBS=−vFrπa

∫ β

0dτ cos

[2φ0(τ )+2π(CµVg/e)

](13)

and the total action is now given by S= S0+SBS with equation (10)and equation (13). Here, a is a standard short-distance cutoffand 1/a defines the region around kF where the electron bandspectrum can be linearized. r is the dimensionless strength ofbackscattering and themodel involves only a single phase φ0; see theMethods section. For large r , or small transparencyD,φ0 gets frozenaround the values that minimize equation (13). Translated into thecharge of the cavity, this gives 〈N 〉 = n ∈N and we recover chargequantization as shown in Fig. 2. The nonlinearity of equation (13)does not allow a complete analytical approach. We thus considerthe case of weak backscattering at the interface, where r � 1 andthe transparency is given byD= 1−r2.

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ARTICLES NATURE PHYSICS DOI: 10.1038/NPHYS1690

Let us first discuss the calculation of the charge relaxationresistance Rq. The fluctuations of the phase φ0 are calculatedperturbatively to second order in r by expanding the backscatteringterm SBS. They give access to the number (or charge) fluctuationsK (ω), defined by equation (3). The identification with equation (1)leads to (Supplementary Section SII)

Rq=he2

BA2

(14)

where the r and r2 corrections exactly cancel out. The dimensionlesscoefficients A and B characterize the low-frequency expansionof S0 to first order depending on whether a small or largecavity is considered: A−1 = 1 + ∆/2Ec, B−1 = 2(1 + ∆/2Ec)2and, A = 1,B = 1, respectively. The universal charge relaxationresistances h/2e2 and h/e2 for the small and large cavities are finallyrecovered from equation (14).

The capacitance C0 is determined from the static chargefluctuations (ω = 0). In contrast with the charge relaxationresistance, the result is non-universal and a continuous function ofthe ratio ∆/Ec describing a crossover between a weakly (Ec�∆)and a strongly (∆�Ec) interacting regime. It takes the form

C0

Cµ= 1−

reCg1(∆/Ec)π

2πcos(2πN0)

+

(reCg1(∆/Ec)

π

)2

g2(∆/Ec)4πcos(4πN0) (15)

where C = 0.577 ... is the Euler constant and N0 = CµN0/Cg.The function g1 smoothly connects g1(0) = 1 to g1(∞) = e−C ,and g2 connects g2(0)' 1.86 to g2(∞)= π/2. Our results for thecapacitance C0 in equation (15) are in agreement with calculationsin the metallic limit23,46 for ∆� Ec, and with calculations basedon the non-interacting a.c. scattering approach14,20 for Ec � ∆.The crossover between the finite cavity to the infinite cavitylimit is discussed thoroughly in the Methods section throughEquations (16)–(19).

Discussionandoutlook. It is important to stress that equation (14)is not a trivial result. Instead it is the result of a remarkablecancellation of all r2 corrections, which protects the value of Rq.This cancellation occurs for all (positive) values of A and B andtakes roots in the structure of the backscattering term (13). It istempting to conjecture, on the basis of our previous results in theweak-tunnelling regime, that this cancellation is present to all ordersin r , that is, possibly for all transparencies, as recent quantumMonteCarlo calculations47 also suggest. We also anticipate a family ofmodels, parameterized by the values of A and B, such that the valueof the charge relaxation resistance is universal and simply given byequation (14). For example, our analysis extends to the fractionalquantum Hall regime where the edge state, formed by the lead andcavity, embodies a chiral Luttinger liquid48 described through thesame bosonization framework45. The resulting charge relaxationresistances are renormalized by interactions in the lead, and wefind Rq = h/(2νe2) and Rq = h/(νe2) for small and large cavities(Supplementary Section SII), where ν is the bulk filling factor. Inessence, the previous arguments remain nevertheless perturbativein r , and a quantum phase transition to an incoherent regime ispredicted47 for ν < 1/2.

An interesting and non-trivial result is the quantization of Rq ata full von Klitzing resistance quantum for cavities with a dense levelspectrum. Matveev has argued23 that the large transmission regimedescribes the strong-coupling fixed point of the fictitious Kondomodel that we already discussed for small transparencies throughequations (7) and (8). The Korringa–Shiba relation equation (9)(or equation (5)) should therefore also apply for all intermediate

transparencies up to D ' 1, leading to Rq = h/e2. On the basisof these renormalization group arguments, we speculate that theKorringa–Shiba relation influences the whole phase diagram andemerges as the fundamental reason for this universal Rq. TheFermi-liquid nature of the fixed point is fundamental and it wouldbe interesting to investigate the case of non-Fermi-liquid states. Thecharge relaxation resistance appears as a fundamental observableto characterize the dynamics of strongly correlated dots. Finally,it should be noted that our results are not limited to the closevicinity of zero temperature; for large transparencies, they extend toenergies up to Ec(∆) for a large (small) cavity.

MethodsOpen-boundary bosonization. The interacting model at and close to perfecttransmission can be solved exactly by applying the open-boundary bosonizationframework44. Here, the whole system occupies a half-infinite one-dimensionalline that stops at x = 0, with only negative values of x . The dot corresponds tothe region between x =−L and 0; backscattering occurs at the entrance of thedot at x =−L. One-dimensional fermionic fields are usually decomposed interms of left- and right-moving fields, ψL(x) and ψR(x) respectively. The ideabehind open-boundary bosonization is that the semi-infinite line is unfoldedsuch that left-moving electrons become right-moving electrons on the positivex axis, namely, ψL(x)=−ψR(−x). We are left with a chiral infinite line offermions. Then, the right-moving field can be bosonized using a single boson field,ψR(x)= eiφ(x)/

√2πa, where a is a short-distance cutoff. Within these notations, the

total Hamiltonian takes the formH =H0+HBS, where

H0=vF4π

∫+∞

−∞

dx [∂xφ(x)]2+Ec

π2

(φ(L)−φ(−L)

2−πN (t )

)2

(16a)

HBS=−vFrπa

cos[φ(L)−φ(−L)] (16b)

The first term in H0 embodies the kinetic term and vF is the Fermi velocity.The second term containing Ec = e2/2Cg represents the charging energy (theinteraction on the cavity) and eN (t )= CgVg(t ). The charge on the cavitytakes the form e

∫ L−Ldx ψR

†(x)ψR(x)= e(φ(L)−φ(−L))/(2π). In addition,HBS =−vFr(ψR

†(−L)ψR(L)+h.c.) describes the backscattering of electronsat the entrance of the cavity at x =−L and r depicts the dimensionlessbackscattering strength. The strategy then is to integrate out the irrelevant modesdifferent from φ0 = (φ(L)−φ(−L))/2 in the kinetic term. Following a standardprocedure45, this results in equation (10). We consider small oscillations of thegate voltage around some constant mean value such that N (t )=N0+N1(t ) with|N1(t )| �N0. Therefore, it is convenient to shift the field φ0 by the constantπN0, leading to equation (13). Linear response theory relates the charge onthe dot to the gate voltage, Q(ω)= e2K (ω)Vg(ω). The correlation functionK (τ )= (1/π2)〈Tτφ0(τ )φ0(0)〉 is Fourier transformed and analytically continued,iωn→ω+ i0+, to produce K (ω).

Infinite metallic cavity. In the limit of a very large cavity23,26 L→+∞, one canassume that electrons entering and electrons escaping the cavity are uncorrelated.For convenience, we slightly change conventions; now the cavity lies betweenx = 0 and L→+∞ and the reservoir lead occupies the negative x values. In thiscase, we rather obtain:

H0=vF2π

∫+∞

−∞

dx π2[5(x)]2+[∂xφ(x)]2+e2

2Cg

[φ(0)π−N (t )

]2(17a)

HBS=−vFrπa

cos[2φ(0)] (17b)

Here, 5(x)= (1/π)∂xθ(x) is the momentum conjugate to the charge fieldφ(x),[φ(x),5(y)] = iδ(x− y). In the limit of very large cavities, electrons atx = L→+∞ completely decouple from the backscattering events at x = 0 andtherefore we may set φ(+∞)= 0. The charge operator on the cavity then readseφ(0)/π. It is certainly appropriate to notice the analogy between the mode φ0introduced above and φ(0). More precisely, in the case of an infinite cavity, wefind the local action:

S0=1π

∑n

φ(0,ωn)φ(0,−ωn)[|ωn|+

Ec

π

](18)

which reproduces equation (10) when L→+∞, allowing us to already justifyequation (12) at perfect transmission (r = 0).

Mesoscopic crossover. In fact, it is possible to recover the infinite metallic cavityregime from the finite dot situation by averaging the frequency ω over a finitebandwidth δω such that Ec�ω� δω�∆. Let us discuss the response functionK (ω) for r = 0. The exponential term in equation (11) gives rise to oscillations

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NATURE PHYSICS DOI: 10.1038/NPHYS1690 ARTICLESwhen the frequency ω varies on the scale of the level spacing ∆. The averagingprocedure smears out these oscillations and

K (ω)=12Ec

∫ 2π

0

dϕ2π

11− iωπ/Ec

1−eiϕ(19)

The integral is computed by changing z = eiϕ to a circle of radius one in thecomplex plane. The only pole at z = 0 then reproduces the infinite cavity resultK (ω)= (1/2Ec)(1− iωπ/Ec)−1. In fact, this smearing procedure transforms eachtermof the perturbative expansion in r to its equivalent in themetallic case.

Received 26 January 2010; accepted 4 May 2010; published online20 June 2010

References1. Landauer, R. Spatial variation of currents and fields due to localized scatterers

in metallic conduction. IBM J. Res. Dev. 1, 223–231 (1957).2. Landauer, R. Electrical resistance of disordered one-dimensional lattices.

Phil. Mag. 21, 863–867 (1970).3. Büttiker, M. Four-terminal phase-coherent conductance. Phys. Rev. Lett. 57,

1761–1764 (1986).4. Imry, Y. & Landauer, R. Conductance viewed as transmission. Rev. Mod. Phys.

71, S306–S312 (1999).5. van Houten, H. & Beenakker, C. Quantum point contacts. Phys. Today 49,

22–27 (July, 1996).6. von Klitzing, K., Dorda, G. & Pepper, M. New method for high-accuracy

determination of the fine-structure constant based on quantizedHall resistance.Phys. Rev. Lett. 45, 494–497 (1980).

7. van Wees, B. J. et al. Quantized conductance of point contacts in atwo-dimensional electron gas. Phys. Rev. Lett. 60, 848–850 (1988).

8. Wharam, D. A. et al. One-dimensional transport and the quantisation of theballistic resistance. J. Phys. C 21, L209–L214 (1988).

9. Frank, S. et al. Carbon nanotube quantum resistors. Science 280,1744–1746 (1998).

10. Kong, J. et al. Quantum interference and ballistic transmission in nanotubeelectron waveguides. Phys. Rev. Lett. 87, 106801 (2001).

11. König, M. et al. Quantum spin Hall insulator state in HgTe quantum wells.Science 318, 766–770 (2007).

12. Yacoby, A. et al. Nonuniversal conductance quantization in quantum wires.Phys. Rev. Lett. 77, 4612–4615 (1996).

13. Bachtold, A. et al. Scanned probe microscopy of electronic transport in carbonnanotubes. Phys. Rev. Lett. 84, 6082–6085 (2000).

14. Büttiker, M., Prêtre, A. & Thomas, H. Dynamic conductance and the scatteringmatrix of small conductors. Phys. Rev. Lett. 70, 4114–4117 (1993).

15. Büttiker, M., Thomas, H. & Prêtre, A. Mesoscopic capacitors. Phys. Lett. A 180,364–369 (1993).

16. Averin, D. A. & Likharev, K. K. inMesoscopic Phenomena in Solids(eds Altshuler, B., Lee, P. A. & Webb, R. A.) (Elsevier, 1991).

17. Grabert, H. & Devoret, M. H. (eds) Single Charge Tunneling (Proceedings of aNATO Advanced Study Institute, Plenum, 1992).

18. Nigg, S. E, López, R. & Büttiker, M. Mesoscopic charge relaxation.Phys. Rev. Lett. 97, 206804 (2006).

19. Ringel, Z., Imry, Y. & Entin-Wohlman, O. Delayed currents and interactioneffects in mesoscopic capacitors. Phys. Rev. B 78, 165304 (2008).

20. Gabelli, J. et al. Violation of Kirchhoff’s laws for a coherent RC circuit. Science313, 499–502 (2006).

21. Fève, G. et al. An on-demand coherent single electron source. Science 316,1169–1172 (2007).

22. Matveev, K. A. Quantum fluctuations of the charge of a metal particleunder the Coulomb blockade conditions. Zh. Eksp. Teor. Fiz. 99,1598–1611 (1990); Sov. Phys. JETP 72, 892–899 (1991).

23. Matveev, K. A. Coulomb blockade at almost perfect transmission. Phys. Rev. B51, 1743–1751 (1995).

24. Berman, D. et al. Observation of quantum fluctuations of charge on a quantumdot. Phys. Rev. Lett. 82, 161–164 (1999).

25. Lehnert, K. W. et al. Quantum charge fluctuations and the polarizability of thesingle-electron box. Phys. Rev. Lett. 91, 106801 (2003).

26. Aleiner, I. L., Brouwer, P. W. & Glazman, L. I. Quantum effects in coulombblockade. Phys. Rep. 358, 309–440 (2002).

27. Le Hur, K. & Seelig, G. Capacitance of a quantum dot from thechannel-anisotropic two-channel Kondomodel.Phys. Rev. B 65, 165338 (2002).

28. Le Hur, K. Coulomb blockade of a noisy metallic box: A realization ofBose–Fermi Kondo models. Phys. Rev. Lett. 92, 196804 (2004).

29. Li, M-R. & Le Hur, K. Double-dot charge qubit and transport via dissipativecotunneling. Phys. Rev. Lett. 93, 176802 (2004).

30. Nigg, S. E. & Büttiker, M. Quantum to classical transition of the chargerelaxation resistance of a mesoscopic capacitor. Phys. Rev. B 77, 085312 (2008).

31. Safi, I. A dynamic scattering approach for a gated interacting wire. Eur. Phys. J. B12, 451–455 (1999).

32. Burke, P. J. Luttinger liquid theory as a model of the gigahertz electricalproperties of carbon nanotubes. IEEE Trans. Nanotechnol. 1, 129–144 (2002).

33. Blanter, Ya. M., Hekking, F. W. J. & Büttiker, M. Interaction constants anddynamics conductance of a gated wire. Phys. Rev. Lett. 81, 1925–1928 (1998).

34. Pham, K. V. Interface resistances and a.c. transport in a Luttinger liquid.Eur. Phys. J. B 36, 607–618 (2003).

35. Wang, J., Wang, B. & Guo, H. Quantum inductance and negativeelectrochemical capacitance at finite frequency in a two-plate quantumcapacitor. Phys. Rev. B 75, 155336 (2007).

36. Rodinov, Ya. I, Burmistrov, I. S. & Ioselevich, A. S. Charge relaxation resistancein the Coulomb blockade problem. Phys. Rev. B 80, 035332 (2009).

37. Persson, F., Wilson, C. M., Sandberg, M., Johansson, G. & Delsing, P.Excess dissipation in a single-electron box: The Sisyphus resistance. Nano Lett.10, 953–957 (2010).

38. Grabert, H. Charge fluctuations in the single-electron box: Perturbationexpansion in the tunnelling conductance. Phys. Rev. B 50, 17364–17377 (1994).

39. Glazman, L. & Matveev, K. A. Lifting of the Coulomb blockade ofone-electron tunnelling by quantum fluctuations. Zh. Eksp. Teor. Fiz.98, 1834–1846 (1990); Sov. Phys. JETP 71, 1031–1037 (1990).

40. Garst, M., Wölfle, P., Borda, L., von Delft, J. & Glazman, L. Energy-resolvedinelastic electron scattering off a magnetic impurity. Phys. Rev. B 72,205125 (2005).

41. Nozières, Ph. A Fermi-liquid description of the Kondo problem at lowtemperatures. J. Low Temp. Phys. 17, 31–42 (1974).

42. Shiba, H. The Korringa relation for the impurity nuclear spin-lattice relaxationin dilute Kondo alloys. Prog. Theor. Phys. 54, 967–981 (1975).

43. Slezak, C., Kehrein, S., Pruschke, Th. & Jarrell, M. Semianalytical solution ofthe Kondo model in a magnetic field. Phys. Rev. B 67, 184408 (2003).

44. Fabrizio, M. & Gogolin, A. O. Interacting one-dimensional electron gas withopen boundaries. Phys. Rev. B 51, 17827–17841 (1995).

45. Giamarchi, T. Quantum Physics in One Dimension (Oxford Univ. Press, 2003).46. Clerk, A., Brouwer, P. & Ambegaokar, V. Interaction-induced restoration of

phase coherence. Phys. Rev. Lett. 87, 186801 (2001).47. Hamamoto, Y., Jonckheere, T., Kato, T. & Martin, T. Dynamic response

of a mesoscopic capacitor in the presence of strong electron interactions.Phys. Rev. B 81, 153305 (2010).

48. Wen, X. G. Chiral Luttinger liquid and the edge excitations in the fractionalquantum Hall states. Phys. Rev. B 41, 12838–12844 (1990).

AcknowledgementsWe thankM. Büttiker, G. Fève, C. Glattli, T. Kontos, T. Martin, B. Plaçais and L. Glazmanfor stimulating discussions. K.L.H. was supported by the Department of Energy, underthe grant DE-FG02-08ER46541. This work has applications for the manipulation ofquantum systems and therefore K.L.H. also thanks the Yale Center for QuantumInformation Physics (NSF DMR-0653377).

Author contributionsC.M. and K.L.H. have equally contributed to this work.

Additional informationThe authors declare no competing financial interests. Supplementary informationaccompanies this paper on www.nature.com/naturephysics. Reprints and permissionsinformation is available online at http://npg.nature.com/reprintsandpermissions.Correspondence and requests formaterials should be addressed to K.L.H.

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