Feedback of the electromagnetic environment on current and voltage fluctuations
out of equilibrium
M. Kindermann,1Yu. V. Nazarov,2and C. W. J. Beenakker1
1Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands
2Department of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands 共Received 16 June 2003; revised manuscript received 23 September 2003; published 30 January 2004兲 We present a theoretical framework for the statistics of low-frequency current and voltage fluctuations of a quantum conductor embedded in a linear electromagnetic environment. It takes the form of a Keldysh field theory with a generic low-frequency limit that allows for a phenomenological understanding and efficient evaluation of the statistics in the saddle-point approximation. This provides an adequate theoretical justification of our earlier calculation that made use of the so-called ‘‘cascaded Langevin approach.’’ We show how a feedback from the environment mixes correlators of different orders. This explains the unexpected temperature dependence of the third moment of tunneling noise observed in a recent experiment. At finite temperature, current and voltage correlators of order 3 and higher are no longer linearly related. We show that a Hall bar measures voltage correlators in the longitudinal voltage and current correlators in the Hall voltage. Next, we demonstrate that the quantum high-frequency corrections to the low-frequency limit correspond to the envi-ronmental Coulomb blockade. We prove that the leading order Coulomb blockade correction to the nth cumulant of the current fluctuations is proportional to the voltage derivative of the (n⫹1)-th cumulant. This generalizes to any n earlier results obtained for n⫽1,2.
DOI: 10.1103/PhysRevB.69.035336 PACS number共s兲: 73.50.Td, 05.40.⫺a, 73.23.⫺b
I. INTRODUCTION
A mesoscopic conductor is always embedded in a macro-scopic electrical circuit that influences its transport proper-ties. This electromagnetic environment is a source of deco-herence and plays a central role for single-electron effects.1–5 It has been noticed that the quantum mechanics of the circuit can be most generally and adequately expressed in terms of a Keldysh action where the voltage drop, or corresponding phase, across the conductor is the only variable. For super-conducting tunnel junctions this theory has been reviewed in Ref. 6. All information about electronic properties of the mesoscopic conductor is incorporated into the Keldysh ac-tion, which makes it non-Gaussian and nonlocal in time. Any conductor, not necessarily a mesoscopic one, can be de-scribed with a Keldysh action of similar structure.
Most transport studies transport address the time-averaged current. It is clear that time-dependent fluctuations of the electric current are also affected by the environment, which reduces the low-frequency fluctuations by a feedback loop: A current fluctuation ␦I induces a counteracting voltage fluc-tuation␦V⫽⫺Z␦I over the conductor, which in turn reduces the current by an amount⫺G␦V. 共Here G and Z are, respec-tively, the conductance of the mesoscopic system and the equivalent series impedance of the macroscopic voltage-biased circuit.兲 At zero temperature the macroscopic circuit does not generate any noise itself, and the feedback loop is the only way it affects the current fluctuations in the mesos-copic conductor, which persist at zero temperature because of the shot noise effect.7–9 In the second cumulant C(2), or shot-noise power, the feedback loop may be accounted for by a rescaling of the current fluctuations: ␦I→(1⫹ZG)⫺1␦I. For example, the Poisson noise C(2)⫽eI¯(1⫹ZG)⫺2 of a tunnel junction is simply reduced by a factor (1⫹ZG)⫺2
due to the negative feedback of the series impedance. We have recently discovered that this textbook result breaks down beyond the second cumulant.10 Terms appear which depend in a nonlinear way on lower cumulants, and which cannot be incorporated by any rescaling with powers of 1⫹ZG. In the example of a tunnel junction the third cumulant at zero temperature takes the form C(3)⫽e2¯(1I ⫺2ZG)(1⫹ZG)⫺4. This implies that the linear environ-ment provides an important and nontrivial effect on the sta-tistics of current and voltage fluctuations of any conductor in the low-frequency regime. In a sense, this effect is more fundamental and important than the Coulomb blockade: We will show that this environmental effect is of a classical na-ture and persists at ZⰆh/e2, whereas the Coulomb blockade is the quantum correction that dissapears in the limit of small impedance.
The concrete results given in Ref. 10 were restricted to zero temperature. In Ref. 11 we removed this restriction and showed that the feedback of the electromagnetic environ-ment on the mesoscopic conductor drastically modifies the temperature dependence of C(3). Earlier theory12–14assumed an isolated mesoscopic conductor and predicted a temperature-independent C(3) for a tunnel junction. We showed in Ref. 11 that the coupling to the environment in-troduces a temperature dependence, which can even change the sign of C(3) as the temperature is raised. No such effect exists for the second cumulant. The temperature dependence predicted has been measured in a recent experiment.15 We demonstrated in Ref. 11 that the results can be obtained in a heuristic way: under a cascade assumption one can insert by hand nonlinear terms into a Langevin equation.14 This so-called ‘‘cascaded Langevin approach’’ is not justified a pri-ori. Therefore, the purpose of the present paper is to provide an adequate theoretical framework for the statistics of
low-frequency fluctuations—a nonlinear Keldysh action that is local in time, and to support the framework with a fully quantum mechanical derivation.
The outline of this paper is as follows. In Secs. II and III we present the general framework within which we describe a broad class of electrical circuits that consist of conductors with a non-Gaussian action embedded in a macroscopic elec-tromagnetic environment. The basis is a path integral formu-lation of the Keldysh approach to charge counting statistics.16,17It allows us to evaluate correlators and cross-correlators of currents and voltages at arbitrary contacts of the circuit. We provide an instructive interpretation of the results in terms of ‘‘pseudoprobabilities.’’ Within this frame-work, we study in Secs. IV and VI series circuits of two conductors.
Further, we concentrate on the low-frequency regime and show that the path integrals over fluctuating quantum fields in this case can be readily performed in saddle-point approxi-mation. The conditions of validity for this approximation are discussed in Sec. V. We obtain general relations between third-order correlators in a series circuit and correlators of the individual isolated conductors. We concentrate on the experimentally relevant case of a single mesoscopic conduc-tor in series with a linear electromagnetic environment. Most experiments measure voltage correlators. In Sec. VII we pro-pose an experimental method to obtain current correlators, using the Hall voltage in a weak magnetic field. The funda-mental difference between current and voltage correlators rests on whether the variable measured is odd or even under time reversal. In Sec. VIII we show that Coulomb blockade effects due to the environment are accounted for by quantum fluctuations in our path integral. They renormalize of the low-frequency action.18 –20We conclude in Sec. IX.
II. DESCRIPTION OF THE CIRCUIT
We consider a circuit consisting of electrical conductors Gi, a macroscopic electromagnetic environment 关with
im-pedance matrix Z()], plus ideal current and voltage meters Mi. The current meter共zero internal impedance兲 is in series
with a voltage source, while the voltage meter共infinite inter-nal impedance兲 is in parallel to a current source. Any finite impedance of meters and sources is incorporated in the elec-tromagnetic environment. In Fig. 1 we show examples of such circuits.
The electromagnetic environment is assumed to produce only thermal noise. To characterize this noise we consider the circuit without the mesoscopic conductors, see Fig. 2. Each pair of contacts to the environment is now attached to a current source and a voltage meter. The impedance matrix is defined by partial derivatives of voltages with respect to cur-rents Z⫽
冉
ZGG ZG M ZM G ZM M冊
⫽冉
VG IG冏
I M VG IM冏
I G VM IG冏
I M VM IM冏
I G冊
. 共2.1兲共All quantities are taken at the same frequency.兲 If there is more than one pair of contacts of type G or M, then the four blocks of Z are matrices themselves. Positive and negative frequencies are related by Z␣(⫺)⫽Z␣* (). We also note FIG. 1. Electrical circuits studied in this article. The black boxes represent conductors embedded in an electromagnetic environment 共dashed rectangle兲. A voltage source is present at the contacts for a current measurement共right circuit兲 and a current source at the con-tacts for a voltage measurement 共left circuit兲. The two circuits can also be combined into one larger circuit containing two conductors and both a current and a voltage meter.
FIG. 2. Circuit used to characterize the impedance matrix of the electromagnetic environment. All contacts are now connected to a voltage meter plus a current source.
the Onsager-Casimir21 symmetry Z␣(B,)⫽Z␣(⫺B,), in an external magnetic field B. The thermal noise at each pair of contacts is Gaussian. The covariance matrix of the voltage fluctuations ␦V␣ is determined by the fluctuation-dissipation theorem
具
␦V␣共兲␦V共⬘
兲典
⫽␦共⫹⬘
兲បcotanh冉
ប 2kT冊
⫻关Z␣共兲⫹Z␣* 共兲兴, 共2.2兲 with T the temperature of the environment.We seek finite frequency cumulant correlators of the vari-ables measured at the current and voltage meters
具具
X1共1兲•••Xn共n兲典典
⫽2␦冉
兺
k⫽1 n k冊
CX (n)共1 , . . . ,n兲. 共2.3兲 Here Xi stands for either VM or IM. Fourier transforms aredefined by Xi()⫽兰dt exp(it)Xi(t). Our aim is to relate the correlators at the measurement contacts to the correlators one would measure at the conductors if they were isolated from the environment.
III. PATH INTEGRAL FORMULATION
Correlators of currents IM and voltages VM at the mea-surement contacts are obtained from the generating func-tional
ZX关 jᠬ兴⫽
具
T⫺ei兰dt[H⫹j ⫺(t)X]T⫹e⫺i兰dt[H⫹j⫹(t)X]
典
. 共3.1兲 They contain moments of outcomes of measurements of the variable X 共equal to IM or VM) at different instants of time.The symbols T⫹(T⫺) denote共inverse兲 time ordering, differ-ent on the forward and backward parts of the Keldysh con-tour. The exponents contain source terms j⫾ and a Hamil-tonian H, which we discuss separately.
The source term j⫾(t) is a charge QM⫽兰tdt
⬘
IM(t⬘
) ifX⫽VM, whereas it is a phase ⌽M⫽兰tdt
⬘
VM(t⬘
) if X⫽IM. 共We have set ប to unity.兲 The superscript ⫾
deter-mines on which part of the Keldysh contour the source is effective. The vector jᠬ⫽(jcl,jq) indicates the linear combina-tions jcl⫽1 2 t共j ⫹⫹j⫺兲, jq⫽j⫹⫺j⫺. 共3.2兲
We denote vectors in this two-dimensional ‘‘Keldysh space’’ by a vector arrow. The ‘‘classical’’ source fields jcl ⫽( j1
cl
, j2cl, . . . ) account for current or voltage sources at the measurement contacts. Cumulant correlators of the measured variables are generated by differentiation of lnZX with
re-spect to the ‘‘quantum’’ fields jq⫽( j1q, j2q, . . . ):
冓冓
k兿
⫽1 n Xk共tk兲冔冔
⫽兿
k⫽1 n ␦ ⫺i␦jkq共tk兲 lnZX兩jq⫽0. 共3.3兲The Hamiltonian consists of three parts,
H⫽He⫹
兺
i
HG
i⫺⌽GIG. 共3.4兲
The term He⫽兺j⍀jaj†aj represents the electromagnetic en-vironment, which we model by a collection of harmonic os-cillators at frequencies⍀j. The conductors connected to the
environment have Hamiltonians HGi. The interaction term
couples the phases ⌽G 共defined by i关He,⌽G兴⫽VG) to the
currents IGthrough the conductors. The phases⌽G, as well as the measured quantities X, are linear combinations of the bosonic operators aj of the electromagnetic environment
⌽G⫽
兺
j 共cj G aj⫹cj G* aj†兲, 共3.5兲 X⫽兺
j 共cj Xa j⫹cj X*a j †兲. 共3.6兲The coefficients cjGand cjXdepend on the impedance matrix of the environment and also on which contacts are connected to a current source and which to a voltage source.
To calculate the generating functional we use a Keldysh path integral formalism.17,22共See Appendix A for a brief in-troduction to this technique.兲 We first present the calculation for the case of a voltage measurement at all measurement contacts 共so Xk⫽VMk and jk⫽QMk for all k). We will then
show how the result for a current measurement can be ob-tained from this calculation. The path integral involves inte-grations over the environmental degrees of freedom aj
weighted with an influence functionalZIGdue to the
conduc-tors. Because the conductors are assumed to be uncoupled in the absence of the environment, this influence functional fac-torizes:
ZIG关⌽ᠬG兴⫽
兿
i ZIGi关⌽
ជGi兴. 共3.7兲
An individual conductor has influence functional ZIG
i⫽
具
T⫺ei兰dt[HGi⫹⌽G⫺i(t)IGi]T
⫹e⫺i兰dt[HGi⫹⌽G⫹i(t)IGi]
典
.共3.8兲 Comparing Eq.共3.8兲 with Eq. 共3.1兲 for X⫽IM, we note that
the influence functional of a conductor Gi is just the
gener-ating functional of current fluctuations in Giwhen connected
to an ideal voltage source without electromagnetic environ-ment. That is why we use the same symbol Z for influence functional and generating functional.
The integrals over all environmental fields except⌽ᠬGare
Gaussian and can be done exactly. The resulting path integral expression for the generating functional ZV
M takes the form
ZVM关QᠬM兴⫽
冕
D关⌽ᠬG兴exp兵⫺iSe关QᠬM,⌽ᠬG兴其ZIG关⌽ᠬG兴,共3.9兲 up to a normalization constant.23We use for the integration fields ⌽ᠬG the same vector notation as for the source fields
⌽ᠬG⫽(⌽G cl,⌽ G q) with ⌽ G cl⫽1 2(/t)(⌽G⫹⫹⌽G⫺) and ⌽G q
⫽⌽G⫹⫺⌽G⫺. The Gaussian environmental action Se is
cal-culated in Appendix B. The result is given in terms of the impedance matrix Z of the environment
Se关QᠬM,⌽ᠬG兴⫽ 1 2
冕
d 2关QᠬM*ZˇMMQᠬM⫹共⌽ᠬG*⫺QᠬM*ZˇMG兲Yˇ ⫻共⌽ᠬG⫺ZˇGMQᠬM兲兴, 共3.10兲 Yˇ共兲⫽冉
0 ZGG †⫺1共兲 ZGG⫺1共兲 ⫺ i 2关2N共兲⫹1兴关ZGG ⫺1共兲⫹Z GG †⫺1共兲兴冊
, 共3.11兲 ZˇMM共兲⫽冉
0 ZMM† 共兲 ZMM共兲 ⫺i 2关2N共兲⫹1兴关ZMM共兲⫹ZMM † 共 兲兴冊
, 共3.12兲 ZˇMG共兲⫽冉
⫺ZGM † 共兲 0 i 2关2N共兲⫹1兴关ZMG共兲⫹ZGM † 共兲兴 Z MG共兲冊
⫽ZˇT GM共⫺兲, 共3.13兲with the Bose-Einstein distribution N()⫽关exp(/kT) ⫺1兴⫺1. We have marked matrices in the Keldysh space by a check, for instance, Yˇ .
When one substitutes Eq.共3.10兲 into Eq. 共3.9兲 and calcu-lates correlators with the help of Eq. 共3.3兲, one can identify two sources of noise. The first source of noise is current fluctuations in the conductors that induce fluctuations of the measured voltage. These contributions are generated by dif-ferentiating the terms ofSethat are linear in QᠬM. The second
source of noise is the environment itself, accounted for by the contributions quadratic in QᠬM.
Generating functionals ZIM for circuits where currents
rather than voltages are measured at some of the contacts can be obtained along the same lines with modified response functions. It is also possible to obtain them from ZVM
through the functional Fourier transform derived in Appen-dix C,
ZIM关⌽ជM兴⫽
冕
D关QជM兴e⫺iQជM⫻⌽ជMZVM关QជM兴. 共3.14兲We have defined the cross product
Qជ⫻⌽ជ⬅
冕
dt共Qcl⌽q⫺⌽clQq兲. 共3.15兲 This transformation may be applied to any pair of measure-ment contacts to obtain current correlators from voltage cor-relators.Equation共3.14兲 ensures that the two functionals
P关V,I兴⫽
冕
D关q兴ei兰dtqVZV关Qជ⫽共I,q兲兴, 共3.16兲
P
⬘
关V,I兴⫽冕
D关兴ei兰dtIZI关⌽ជ⫽共V,兲兴 共3.17兲
are identical: P关V,I兴⫽P
⬘
关V,I兴. This functional P has an intuitive probabilistic interpretation. With the help of Eq. 共3.3兲 we obtain from P the correlators具
V共t1兲•••V共tn兲典
I⫽冕
D关V兴V共t1兲•••V共tn兲P关V,I兴冕
D关V兴P关V,I兴 , 共3.18兲具
I共t1兲•••I共tn兲典
V⫽冕
D关I兴I共t1兲•••I共tn兲P关V,I兴
冕
D关I兴P关V,I兴. 共3.19兲 This suggests the interpretation ofP关V,I兴 as a joint probabil-ity distribution functional of current and voltage fluctuations. Yet, P cannot properly be called a probability since it need not be positive. In the low frequency approximation intro-duced in the next section it is positive for normal metal con-ductors. However, for superconductors, it has been found to take negative values.24It is therefore more properly called a ‘‘pseudoprobability.’’
We conclude this section with some remarks on the actual measurement process. The time-averaged correlators 共2.3兲 may be measured in two different ways. In the first way the variable X is measured repeatedly and results at different times are correlated afterwards. In the second way共and this is how it is usually done25兲 one uses a detector that measures directly time integrals of X 共for example, by means of a
spectral filter兲. The correlators measured in the first way are obtained from the generating functional according to Eq. 共3.3兲, 2␦
冉
兺
k⫽1 n k冊
CX (n)共 1, . . . ,n兲 ⫽兿
k冋
冕
⫺⬁ ⬁ dteikt ␦ ⫺i␦jkq共t兲册
lnZX兩jq⫽0. 共3.20兲 The second way of measurement is modeled by choosing cross-impedances that ensure that an instantaneous measure-ment at one pair of contacts yields a time average at another pair, for example ZM G()⬀␦(⫺0). The resultingfre-quency dependent correlators do not depend on which way of measurement one uses.
IV. TWO CONDUCTORS IN SERIES
We specialize the general theory to the series circuit of two conductors G1 and G2 shown in Fig. 3 共lower panel兲. We derive the generating functional ZV,I for correlators of
the voltage drop V⬅VM1over conductor G1 and the current I⬅IM2 through both conductors.共The voltage drop over
con-ductor G2 equals VM2⫺VM1⬅Vbias⫺V, with Vbiasthe non-fluctuating bias voltage of the voltage source.兲 To apply the general relations of the previous section we embed the two
conductors in an electromagnetic environment, as shown in the top panel of Fig. 3. In the limit of infinite resistances R1, R2, and R3 this eight-terminal circuit becomes equivalent to a simple series circuit of G1 and G2. We take the infinite resistance limit of Eq. 共3.9兲 in Appendix D. The result
ZV,I关Qជ,⌽ជ兴⫽
冕
D关⌽ជ⬘
兴e⫺i⌽ជ⬘⫻QជZ1关⌽ជ⬘
兴Z2关⌽ជ⫺⌽ជ⬘
兴 共4.1兲 shows that the generating functional of current and voltage correlators in the series circuit is a functional integral con-volution of the generating functionals Z1⬅ZIG1 and Z2
⬅ZIG
2
of the two conductors G1and G2defined in Eq.共3.8兲. Equation 共4.1兲 implies a simple relation between the pseudo-probabilities PG1⫹G2 of the series circuit 关obtained
by means of Eq.共3.17兲 from ZV,I兩Qជ ⫽0] and the
pseudoprob-abilitiesPG
kof the individual conductors关obtained by means
of Eq.共3.17兲 from Zk]. We find
PG1⫹G2关V,I兴⫽
冕
DV⬘
PG1关V⫺V⬘
,I兴PG2关V⬘
,I兴.共4.2兲 This relation is obvious if one interprets it in terms of clas-sical probabilities: The voltage drop over G1⫹G2is the sum of the independent voltage drops over G1 and G2, so the probability PG
1⫹G2 is the convolution ofPG1 andPG2. Yet
the relation 共4.2兲 is for quantum-mechanical pseudoprob-abilities.
We evaluate the convolution 共4.1兲 in the low-frequency regime, when the functionals Z1 and Z2 become local in time,
lnZk关⌽ជ兴⬅⫺iSk关⌽ជ兴⫽⫺i
冕
dtSk„⌽ជ共t兲…. 共4.3兲We then do the path integration in saddle-point approxima-tion, with the result
lnZV,I关Qជ,⌽ជ兴⫽⫺iextr[⌽ជ⬘]
再
⌽ជ⬘
⫻Qជ⫹冕
dt关S1„⌽ជ⬘
共t兲… ⫹S2„⌽ជ共t兲⫺⌽ជ⬘
共t兲…兴冎
. 共4.4兲 The notation ‘‘extr’’ indicates the extremal value of the ex-pression between curly brackets with respect to variations of ⌽ជ⬘
(t). The validity of the low-frequency and saddle-point approximations is addressed in the next section.We will consider separately the case that both conductors G1 and G2 are mesoscopic conductors and the case that G1 is mesoscopic while G2 is a macroscopic conductor. The action of a macroscopic conductor with impedance Z is qua-dratic, Smacro关⌽ជ兴⫽ 1 2
冕
d 2⌽ជ †Yˇ⌽ជ, 共4.5兲 FIG. 3. Top panel: Circuit of two conductors G1, G2 in anelectromagnetic environment modeled by three resistances R1, R2,
R3. In the limit R1,R2,R3→⬁ the circuit becomes equivalent to the
corresponding to Gaussian current fluctuations. The matrix Yˇ is given by Eq. 共3.11兲, with a scalar ZGG⫽Z. The
corre-sponding pseudoprobabilityPmacrois positive, Pmacro关V,I兴⫽exp
再
⫺冕
d 4 兩V⫺ZI兩2 Re Z tanh冉
2kT冊冎
. 共4.6兲 Substitution ofPmacrofor PG2 in Eq.共4.2兲 gives a simpleresult forPG1⫹G2 at zero temperature,
PG1⫹G2关V,I兴⫽PG1关V⫺ZI,I兴, if T⫽0. 共4.7兲
The feedback of the macroscopic conductor on the mesos-copic conductor amounts to a negative voltage ⫺ZI pro-duced in response to a current I.
The action of a mesoscopic conductor in the low-frequency limit is given by the Levitov-Lesovik formula26,27
Smeso共⌽ជ兲⫽ 1 2 n
兺
⫽1 N冕
d ln关1⫹⌫n共eie⫺1兲nR共1⫺nL兲 ⫹⌫n共e⫺ie⫺1兲nL共1⫺nR兲兴, 共4.8兲with⌽ជ⫽(V,). The⌫n’s (n⫽1,2, . . . ,N) are the
transmis-sion eigenvalues of the conductor. The two functions nL(,T)⫽关exp(/kT)⫹1兴⫺1and nR(,T)⫽nL(⫹eV,T) are the filling factors of electron states at the left and right con-tacts, with V the voltage drop over the conductor and T its temperature.
V. VALIDITY OF THE SADDLE-POINT APPROXIMATION
The criterion for the applicability of the low-frequency and saddle-point approximations to the action of a mesos-copic conductor depends on two time scales共see Fig. 4兲. The first time scale 1⫽min(1/eV,1/kT) is the mean width of current pulses due to individual transferred electrons. 共This time scale is known as the coherence time in optics, but in mesoscopic systems that term is used in a different context.兲 The second time scale2⫽e/I⯝(e2/G)1 is the mean time between current pulses.
At frequencies below 1/1 the action of the conductor
becomes local in time, as expressed by Eq. 共4.3兲. This fol-lows from an analysis of the dependence of the actionSkon
tidependent arguments. An explicit expression for a me-soscopic conductor can be found in Ref. 20. Below the sec-ond time scale 1/2 the action of the conductor is large for values of⌽ជ where the nonlinearities become important. This justifies the saddle-point approximation. The nonlinearities in Smeso become relevant for ⯝1/e, so for time scales Ⰷ2 we indeed haveSmeso⯝Smeso⯝I⯝I/e⯝/2Ⰷ1.
These two approximations together are therefore justified if fluctuations in the path integral 共3.9兲 with frequencies above⌳⯝min(1/1,1/2) are suppressed. This is the case if the effective impedance of the circuit is small at high fre-quency: Z()Ⰶh/e2forⲏ⌳. A small impedance acts as a heavy mass term in Eq. 共4.1兲, suppressing fluctuations. This is seen from Eq.共4.5兲 for a macroscopic conductor 关note that Yˇ ()⬀Z⫺1()] and it carries over to other conductors. Physically, a small high-frequency impedance ensures that voltage fluctuations in the circuit are much slower than the electron dynamics in the conductors. Under this condition, it is sufficient to know the dynamics of the individual conduc-tors when biased with a constant voltage, as described by Eq. 共4.8兲. Effects of time-dependent voltage fluctuations in the circuit may then be neglected.
The same separation of time scales has been exploited in Refs. 28,29 to justify a cascaded average in the Langevin approach. We will see in Sec. VI that the results of both approaches are in fact identical in the saddle-point approxi-mation. The two approaches differ if one goes beyond this approximation, to include the effects of a finite high-frequency impedance. Since the path integral 共3.9兲 is micro-scopically justified at all time scales, it also allows us to calculate the corrections to the saddle-point solution 共4.4兲. These corrections are usually called the ‘‘environmental Coulomb blockade.’’ In Sec. VIII we examine the Coulomb blockade effects to lowest order in Z().
VI. THIRD CUMULANTS A. Two arbitrary conductors in series
We use the general formula 共4.4兲 to calculate the third-order cumulant correlator of current and voltage fluctuations in a series circuit of two conductors G1 and G2 at finite temperature. We focus on correlators at zero frequency共finite frequency generalizations are given later兲.
The zero-frequency correlators CX(n)(V¯ ) depend on the av-erage voltage V¯ over G1, which is related to the voltage Vbias of the voltage source by V¯⫽Vbias(1⫹G1/G2)⫺1. The aver-age voltaver-age over G2 is Vbias⫺V¯⫽Vbias(1⫹G2/G1)⫺1. Our goal is to express CX(n)(V¯ ) in terms of the current correlators C1
(n)(V) and C 2
(n)(V) that the conductors G
1 and G2 would have if they were isolated and biased with a nonfluctuating voltage V. These are defined by
具具
Ii共1兲•••Ii共n兲典典
V⫽2␦冉
兺
k⫽1n
k
冊
Ci(n)共V兲, 共6.1兲 FIG. 4. Time scales of current fluctuations in a mesoscopic
con-ductor. The time1is the duration of current pulses, whereas2is
where Iiis the current through conductor i at fixed voltage V.
To evaluate Eq. 共4.4兲 it is convenient to discretize fre-quencies n⫽2n/. The Fourier coefficients are fn
⫽⫺1兰 0
dteintf (t). The detection time is sent to infinity
at the end of the calculation. For zero-frequency correlators the sources at nonzero frequencies vanish and there is a saddle-point configuration such that all fields at nonzero fre-quencies vanish as well. We may then write Eq. 共4.4兲 in terms of only the zero-frequency fields ⌽ជ0⫽(V0,0), ⌽ជ0
⬘
⫽(V0⬘
,0⬘
), and Qជ0⫽(I0,q0), with actions⫺1S k共⌽ជ0
⬘
兲⫽Gk0⬘
V0⬘
⫹i兺
n⫽2 ⬁ 共⫺i0⬘
兲n n! Ck (n)共V 0⬘
兲. 共6.2兲 共We assume that the conductors have a linear current-voltage characteristic.兲 For ⌽ជ0⫽(Vbias,0) and Qជ0⫽(0,0) the saddle point is at ⌽ជ0⬘
⫽(V¯,0). For the third-order correlators we need the extremum in Eq.共4.4兲 to third order in0 and q0. We have to expand Sk to third order in the deviation ␦⌽ជ0⬘
⫽⌽ជ0⬘
⫺(V¯,0) from the saddle point at vanishing sources. We have to this order⫺1S 1共⌽ជ0
⬘
兲⫽G10⬘
共V¯⫹␦V0⬘
兲⫺ i 2C1 (2)共V¯兲 0⬘
2 ⫺1 6C1 (3)共V¯兲 0⬘
3⫺ i 2 d dV¯C1 (2)共V¯兲␦V 0⬘
0⬘
2 ⫹O共␦⌽ជ0⬘
4兲, 共6.3兲 ⫺1S 2共⌽ជ0⫺⌽ជ0⬘
兲 ⫽G20⬘
共Vbias⫺V¯⫺␦V0⬘
兲⫺ i 2C2 (2)共V bias⫺V¯兲0⬘
2 ⫺16C2 (3) 共Vbias⫺V¯兲0⬘
3 ⫹i 2 d dV¯C2 (2)共V bias⫺V¯兲␦V0⬘
0⬘
2⫹O共␦⌽ជ 0⬘
4兲. 共6.4兲 Minimizing the sumS1(⌽ជ0⬘
)⫹S2(⌽ជ0⫺⌽ជ0⬘
) to third order in q0 and0 we then find the required relation between the correlators of the series circuit and the correlators of the isolated conductors. For the second-order correlators we findCII(2)共V¯兲⫽共R1⫹R2兲⫺2关R1 2C 1 (2)共V¯兲⫹R 2 2C 2 (2)共V bias⫺V¯兲兴, 共6.5a兲 CVV(2)共V¯兲⫽共R1⫹R2兲⫺2共R1R2兲2关C1 (2)共V¯兲⫹C 2 (2)共V bias⫺V¯兲兴, 共6.5b兲 CIV(2)共V¯兲⫽共R1⫹R2兲⫺2R1R2关R2C2 (2)共V bias⫺V¯兲 ⫺R1C1 (2)共V¯兲兴, 共6.5c兲
with Rk⫽1/Gk. The third-order correlators contain extra
terms that depend on the second-order correlators
CIII(3)共V¯兲⫽共R1⫹R2兲⫺3关R1 3C 1 (3)共V¯兲⫹R 2 3C 2 (3)共V bias⫺V¯兲兴 ⫹3CIV (2) d dV¯ CII (2), 共6.6a兲 CVVV(3) 共V¯兲⫽共R1⫹R2兲⫺3共R1R2兲3关C2 (3) 共Vbias⫺V¯兲⫺C1 (3) 共V¯兲兴 ⫹3CVV (2) d dV¯ CVV (2) , 共6.6b兲 CVVI(3)共V¯兲⫽共R1⫹R2兲⫺3共R1R2兲2关R1C1 (3)共V¯兲⫹R 2C2 (3)共V bias ⫺V¯兲兴⫹2CVV (2)d dV¯CIV (2)⫹C IV (2) d dV¯CVV (2) , 共6.6c兲 CIIV(3)共V¯兲⫽共R1⫹R2兲⫺3R1R2关R22C2(3)共Vbias⫺V¯兲⫺R12C1(3)共V¯兲兴 ⫹2CIV (2) d dV¯ CIV (2)⫹C VV (2) d dV¯ CII (2). 共6.6d兲
These results agree with those obtained by the cascaded Langevin approach.11
B. Mesoscopic and macroscopic conductor in series
An important application is a single mesoscopic conduc-tor G1 embedded in an electromagnetic environment, repre-sented by a macroscopic conductor G2. A macroscopic con-ductor has no shot noise but only thermal noise. The third cumulantC2(3) is therefore equal to zero. The second cumu-lant C2(2) is voltage independent, given by8
C2
(2)共兲⫽cotanh
冉
2kT2冊
Re G2共兲, 共6.7兲 at temperature T2. We still assume low frequencies Ⰶmax(eV¯,kT1), so the frequency dependence of S1 can be neglected. We have retained the frequency dependence ofS2, because the characteristic frequency of a macroscopic ductor is typically much smaller than of a mesoscopic con-ductor.
From Eq. 共6.6兲 共and a straightforward generalization to frequency-dependent correlators兲 we can obtain the third cu-mulant correlators by setting C2(3)⫽0 and substituting Eq. 共6.7兲. We only give the two correlators CIII
(3)and C
VVV
(3) , since these are the most significant for experiments. To abbreviate the formula we denote G⫽G1and Z()⫽1/G2(). We find
CIII (3)共1 ,2,3兲⫽ C1 (3)共V¯兲⫺共dC 1 (2) /dV¯兲
兺
j⫽1 3 Z共⫺j兲关C1 (2)共V¯兲⫺GZ共 j兲C2 (2)共 j兲兴关1⫹Z共⫺j兲G兴⫺1 关1⫹Z共1兲G兴关1⫹Z共2兲G兴关1⫹Z共3兲G兴 , 共6.8兲 ⫺ CVVV (3) 共 1,2,3兲 Z共1兲Z共2兲Z共3兲 ⫽ C1 (3)共V¯兲⫺共dC 1 (2)/dV¯兲兺
j⫽1 3 Z共⫺j兲关C1 (2)共V¯兲⫹C 2 (2)共 j兲兴关1⫹Z共⫺j兲G兴⫺1 关1⫹Z共1兲G兴关1⫹Z共2兲G兴关1⫹Z共3兲G兴 . 共6.9兲We show plots for two types of mesoscopic conductors: a tunnel junction and a diffusive metal. In both cases it is assumed that there is no inelastic scattering, which is what makes the conductor mesoscopic. The plots correspond to global thermal equilibrium (T1⫽T2⫽T) and to a real and frequency-independent impedance Z()⬅Z. We compare CI(3)⬅CIII(3) with CV(3)⬅⫺CVVV(3) /Z3. 共The minus sign is cho-sen so that CI(3)⫽CV(3)at T⫽0.兲
For a tunnel junction one has
C1
(2)共V兲⫽GeV cotanh eV 2kT, C1
(3)共V兲⫽Ge2V. 共6.10兲
The third cumulant of current fluctuations in an isolated tun-nel junction is temperature independent,12but this is changed
drastically by the electromagnetic environment.11 Substitu-tion of Eq. 共6.10兲 into Eqs. 共6.8兲 and 共6.9兲 gives the curves plotted in Fig. 5 for ZG⫽0 and ZG⫽1. The slope dCV(3)(V¯ )/dV¯ becomes strongly temperature dependent and may even change sign when kT becomes larger than eV¯ . This is in qualitative agreement with the experiment of Reu-let, Senzier, and Prober.15In Ref. 15 it is shown that Eq.共6.9兲 provides a quantitative description of the experimental data. For a diffusive metal we substitute the known formulas for the second and third cumulants without electromagnetic environment13,14 C1 (2) 共V兲⫽13GeV共cotanh p⫹2/p兲, 共6.11兲 C1 (3)共V兲⫽e2GVp共1⫺26e 2 p⫹e4 p兲⫺6共e4 p⫺1兲 15p共e2 p⫺1兲2 . 共6.12兲 We have abbreviated p⫽eV/2kT. Plots for ZG⫽0 and ZG ⫽1 are shown in Fig. 6. The diffusive metal is a bit less striking than a tunnel junction, since the third cumulant is already temperature dependent even in the absence of the electromagnetic environment. In the limit ZG→⬁ we re-cover the result for CV(3) obtained by Nagaev from the cas-caded Langevin approach.30
VII. HOW TO MEASURE CURRENT FLUCTUATIONS
In Fig. 5 we have plotted both current and voltage corr-elators, but only the voltage correlator has been measured.15 At zero temperature of the macroscopic conductor there is no difference between the two, as follows from Eqs. 共6.8兲 and 共6.9兲: CIII
(3)⫽⫺C
VVV
(3)
/Z3 if C2
(2)⫽0, which is the case for a macroscopic conductor G2at T2⫽0. For T2⫽0 a difference appears that persists in the limit of a noninvasive measure-ment Z→0.11Since V and I in the series circuit with a mac-roscopic G2 are linearly related and linear systems are known to be completely determined by their response func-tions and their temperature, one could ask what it is that distinguishes the two measurements, or more practically: How would one measure CIII(3)instead of CVVV(3) ?
To answer this question we slightly generalize the macro-scopic conductor to a four-terminal, rather than two-terminal configuration, see Figs. 7 and 8. The voltage VM over the
extra pair of contacts is related to the current IGthrough the FIG. 5. Third cumulant of voltage and current fluctuations of a
tunnel junction共conductance G) in an electromagnetic environment 共impedance Z, assumed frequency independent兲. Both CI
(3)
and CV
(3)
are multiplied by the scaling factor A⫽(1⫹ZG)3/eGkT. The two
curves correspond to different values of ZG 共solid curve: ZG⫽1; dashed curve: ZG⫽0). The temperatures of the tunnel junction and its environment are chosen the same, T1⫽T2⫽T.
series circuit by a cross impedance VM/IG⫽ZM G. The full impedance matrix Z is defined as in Eq. 共2.1兲. For sim-plicity we take the zero-frequency limit. For this configura-tion the third cumulant CV
MVMVM (3) of V M is given by CV MVMVM (3) ZM G3 ⫽CIGIGIG (3) ⫹ZG M⫹ZM G 2ZG M
冉
CV GVGVG (3) ZGG3 ⫺CIGIGIG (3)冊
. 共7.1兲 It contains the correlator具具
␦VM()␦VG(⬘
)典典
⫽2␦(⫹
⬘
)CG M of the voltage fluctuations over the two pairs ofterminals of the macroscopic conductor, which according to the fluctuation-dissipation theorem共2.2兲 is given in the zero-frequency limit by
CG M⫽kT2共ZG M⫹ZM G兲. 共7.2兲
The correlator CG M enters since CVMVMVM
(3)
depends on how thermal fluctuations in the measured variable VM correlate
with the thermal fluctuations of VG which induce extra
cur-rent noise in G1.
We conclude from Eq. 共7.1兲 that the voltage correlator CV
MVMVM
(3)
becomes proportional to the current correlator CI
GIGIG
(3)
if ZG M⫹ZM G⫽0. This can be realized if VMis the
Hall voltage VH in a weak magnetic field B. Then ZM G
⫽⫺ZG M⫽RH, with RH⬀兩B兩 the Hall resistance. The
mag-netic field need only be present in the macroscopic conductor G2, so it need not disturb the transport properties of the mesoscopic conductor G1. If, on the other hand, VM is the
longitudinal voltage VL, then ZM G⫽ZG M⫽RL, with RLthe
longitudinal resistance. The two-terminal impedance ZGG is
the sum of Hall and longitudinal resistances, ZGG⫽RL
⫹RH. So one has CVLVLVL (3) ⫽
冉
RL RL⫹RH冊
3 CVGVGVG (3) , 共7.3兲 CV HVHVH (3) ⫽R H 3C IGIGIG (3) . 共7.4兲One can generalize all this to an arbitrary measurement variable X that is linearly related to the current IG through
G1. In a linear circuit the off-diagonal elements of the re-sponse tensor Z relating (X,VG) to the conjugated sources
are linked by Onsager-Casimir relations.21If X is even under timereversal, then ZXG⫽ZGX, while if X is odd, then ZXG
⫽⫺ZGX. In the first case CXXX
(3) ⬀C
VGVGVG
(3) , while in the sec-ond case CXXX(3) ⬀CI
GIGIG
(3) .
VIII. ENVIRONMENTAL COULOMB BLOCKADE
The saddle-point approximation to the path integral 共4.1兲 for a mesoscopic conductor G1 in series with a macroscopic conductor G2 共impedance Z) breaks down when the imped-ance at the characteristic frequency scale ⌳⫽1/max(1,2) discussed in Sec. IV is not small compared to the resistance quantum h/e2. It can then react fast enough to affect the dynamics of the transfer of a single electron. These single-electron effects amount to a Coulomb blockade induced by the electromagnetic environment.4In our formalism they are accounted for by fluctuations around the saddle point of Eq. 共4.1兲.
In Ref. 18 it has been found that the Coulomb blockade correction to the mean current calculated to leading order in Z is proportional to the second cumulant of current fluctua-tions in the isolated mesoscopic conductor (Z⫽0). More recently, the Coulomb blockade correction to the second cu-mulant of current fluctuations has been found to be propor-tional to the third cumulant.19 It was conjectured in Ref. 19 FIG. 7. Four-terminal voltage measurement.
FIG. 8. Hall bar that allows one to measure the voltage cor-relator CV(3)⬀具具VL3典典 as well as the current correlator CI(3) ⬀具具VH
3典典
.
that this relation also holds for higher cumulants. Here we give proof of this conjecture.
We show that at zero temperature and zero frequency the leading order Coulomb blockade correction to the nth cumu-lant of current fluctuations is proportional to the voltage de-rivative of the (n⫹1)-th cumulant. To extract the environ-mental Coulomb blockade from the other effects of the environment we assume that Z vanishes at zero frequency, Z(0)⫽0. The derivation is easiest in terms of the pseudoprobabilities discussed in Sec. III.
According to Eq. 共3.19兲, cumulant correlators of current have the generating functional
FG1⫹G2关⌽ជ⫽共V,兲兴⫽ln
冕
DIe⫺i兰dtIPG1⫹G2关V,I兴.共8.1兲 Zero frequency current correlators are obtained from
具具
I共0兲n典典
G1⫹G2⫽in␦n
␦关共0兲兴nFG1⫹G2关⌽ជ兴兩⫽0. 共8.2兲
We employ now Eq. 共4.7兲 and expand FG1⫹G2关⌽ជ兴 to first
order in Z, FG1⫹G2关⌽ជ兴⫽FG1关⌽ជ兴⫺
冕
DIe⫺i兰dtI冕
d 2Z共兲I共兲 ␦ ␦V共兲PG1关V,I兴冕
DIe⫺i兰dtIP G1关V,I兴 ⫽FG1关⌽ជ兴⫺i冕
d 2Z共兲 ␦2 ␦V共兲␦共兲FG1关⌽ជ兴. 共8.3兲The last equality holds since single derivatives of FG1关⌽ជ兴
with respect to a variable at finite frequency vanish because of time-translation symmetry. Substitution into Eq. 共8.2兲 gives
具具
I共0兲n典典
G1⫹G2⫽具具
I共0兲n典典
G1 ⫺冕
d 2Z共兲 ␦ ␦V共兲具具
I共兲I共0兲 n典典
G1, 共8.4兲 which is what we had set out to prove.IX. CONCLUSION
In conclusion, we have presented a fully quantum-mechanical derivation of the effect of an electromagnetic en-vironment on current and voltage fluctuations in a mesos-copic conductor, going beyond an earlier study at zero temperature.10The results agree with those obtained from the cascaded Langevin approach,11 thereby providing the re-quired microscopic justification.
From an experimental point of view, the nonlinear feed-back from the environment is an obstacle that stands in the way of a measurement of the transport properties of the me-soscopic system. To remove the feedback it is not sufficient to reduce the impedance of the environment. One also needs to eliminate the mixing in of environmental thermal fluctua-tions. This can be done by ensuring that the environment is at a lower temperature than the conductor, but this might not be a viable approach for low-temperature measurements. We have proposed here an alternative method, which is to ensure that the measured variable changes sign under time reversal. In practice this could be realized by measuring the Hall
volt-age over a macroscopic conductor in series with the mesos-copic system.
The field theory developed here also provides for a sys-tematic way to incorporate the effects of the Coulomb block-ade which arise if the high-frequency impedance of the en-vironment is not small compared to the resistance quantum. We have demonstrated this by generalizing to moments of arbitrary order a relation in the literature18,19for the leading-order Coulomb blockade correction to the first and second moments of the current. We refer the reader to Ref. 20 for a renormalization-group analysis of Coulomb blockade correc-tions of higher order.
ACKNOWLEDGMENTS
We thank D. Prober and B. Reulet for discussions of their experiment. This research was supported by the ‘‘Neder-landse organisatie voor Wetenschappelijk Onderzoek’’ 共NWO兲 and by the ‘‘Stichting voor Fundamenteel Onderzoek der Materie’’共FOM兲.
APPENDIX A: KELDYSH PATH INTEGRAL
In this appendix we give a brief introduction to the Keldysh path integral technique that we use in the text. For more details see Refs. 17,22. We restrict ourselves to a cir-cuit with just one conductor. The Hamiltonian共3.4兲 reduces to
H⫽He⫹HG⫺⌽GIG. 共A1兲
We will explain how to calculate the generating functional of the phase ⌽G. This requires the minimum amount of
vari-ables in our model, since⌽Gis needed anyway for the
cou-pling of the environment circuit to the conductor. The gen-erating function 共3.1兲 in this case takes the form
Z⌽G关 jជ兴⫽
具
T⫺e i兰⫺dt[H⫹ j⫺(t)⌽G]T ⫹e⫺i兰⫺ dt[H⫹ j⫹(t)⌽ G]典
. 共A2兲 共In the end we will take the limit →⬁.兲 Additionally, we restrict the analysis to an environment circuit that can be modeled by a single mode with HamiltonianHe⫽⍀a†a, ⌽
G⫽ca⫹c*a†. 共A3兲
Here a is the annihilation operator of a bosonic environmen-tal mode and c is a complex coefficient.
We first neglect the coupling of ⌽G to the conductor,
taking H⫽He. Equation共A2兲 can then be rewritten as a path
integral by inserting sets of coherent states共eigenstates of a), as explained, for example, in Ref. 31. In this way we intro-duce one time-dependent integration field a⫹(t) for the T⫹-ordered time evolution operator in Eq. 共A2兲 and a field a⫺(t) for the T⫺- ordered operator. These fields propagate the system forward and backward in time, respectively. The fact that we have an integration field for forward propagation as well as one for backward propagation is characteristic for the Keldysh technique.32 Equivalently, one may formulate the theory in terms of just one field that is then defined on the so-called ‘‘Keldysh-time contour’’共see Fig. 9兲. The contour runs from t⫽⫺ to t⫽ forward in time and backwards from t⫽ to t⫽⫺. The resulting path integral is共up to a normalization constant兲 Z⌽G关 jជ兴⫽
冕
Da ⫾Da⫾*关a⫾共⫺兲,a⫾*共⫺兲兴 ⫻exp再
⫺i冕
⫺ dt关a⫹*共⫺it⫹⍀兲a⫹
⫹a⫺*共⫺i
t⫺⍀兲a⫺⫹共ca⫹⫹c*a⫹*兲j⫹
⫺共ca⫺⫹c*a⫺*兲j⫺兴
冎
, 共A4兲 with关a⫹,a⫺兴 the initial density matrix of the mode a in the coherent state basis and a⫹()⫽a⫺().Following Feynman and Vernon,33one can show that the coupling to the conductor in Eq. 共A1兲 introduces an addi-tional factorZIG into the path integral, called the ‘‘influence
functional.’’ Instead of Eq. 共A4兲 we then have
Z⌽G关 jជ兴⫽
冕
Da⫾Da⫾*关a⫾共⫺兲,a⫾*共⫺兲兴 ⫻ZIG关ca⫹⫹c*a⫹*,ca⫺⫹c*a⫺*兴⫻exp
再
⫺i冕
⫺
dt关a⫹*共⫺it⫹⍀兲a⫹
⫹a⫺*共⫺i
t⫺⍀兲a⫺⫹共ca⫹⫹c*a⫹*兲j⫹
⫺共ca⫺⫹c*a⫺*兲j⫺兴
冎
. 共A5兲 The influence functional in our case is given byZIG关⌽G⫹,⌽G⫺兴 ⫽
具
T⫺ei兰⫺dt[HG⫹⌽G⫺(t)IG]T ⫹e⫺i兰⫺ dt[H G⫹⌽G⫹(t)IG]典
. 共A6兲 The density matrix of a thermal state of the environmental mode a is the exponential of a quadratic form. Therefore the integrals over the linear combinations ca⫾⫺c*a⫾* are Gaussian and can be done exactly. With the substitution ⌽G⫾⫽ca⫾⫹c*a⫾* and with the vector notation introducedin Sec. III we rewrite Eq. 共A5兲 as
Z⌽G关 jជ兴⫽
冕
D⌽ជGexp兵⫺iSe关 jជ,⌽ជG兴其ZIG关⌽ជG兴, 共A7兲with a quadratic formSe. The more general circuits of Sec.
III can be treated along the same lines, but with a multimode environmental Hamiltonian He⫽兺j⍀jaj
†
aj and sources that
couple to variables other than ⌽G. In the limit →⬁ one
arrives at Eq.共3.9兲.
APPENDIX B: DERIVATION OF THE ENVIRONMENTAL ACTION
To derive Eq.共3.10兲 we define a generating functional for the voltages V⫽(VM,VG) in the environmental circuit of Fig. 2,
Ze关Qᠬ兴⫽
具
T⫺ei兰dt[H⫹Q ⫺(t)V]T⫹e⫺i兰dt[H⫹Q⫹(t)V]
典
. 共B1兲 We have introduced sources Q⫽(QM,QG). Since theenvi-ronmental Hamiltonian is quadratic, the generating func-tional is the exponential of a quadratic form in Qᠬ,
Ze关Qᠬ兴⫽exp
冉
⫺ i 2冕
d 2Qᠬ †共兲Gˇ共兲Qᠬ共兲冊
. 共B2兲 The off-diagonal elements of the matrix Gˇ are determined by the impedance of the circuitFIG. 9. Keldysh time contour with the fields a⫹for forward and
i ␦ 2 ␦Qcl共
⬘
兲␦Qq␣*共兲lnZe兩Qᠬ ⫽0 ⫽ ␦ ␦I共⬘
兲具
V␣共兲典
⫽2␦共⫺⬘
兲Z␣共兲. 共B3兲 The upper-diagonal (cl,cl) elements in the Keldysh space vanish for symmetry reasons (Ze兩Qq⫽0⫽0, see Ref. 22兲. Thelower-diagonal (q,q) elements are determined by the fluctuation-dissipation theorem共2.2兲, ⫺ ␦ 2 ␦Q␣q*共兲␦Qq*共
⬘
兲lnZe兩Qᠬ ⫽0⫽具
␦V␣共兲␦V共⬘
兲典
⫽␦共⫹⬘
兲cotanh冉
2kT冊
关Z␣共兲⫹Z␣* 共兲兴. 共B4兲 Consequently we have Gˇ共兲⫽冉
0 Z†共兲 Z共兲 ⫺i 2cotanh冉
2kT冊
关Z共兲⫹Z †共兲兴冊
. 共B5兲 The environmental action Seis defined byZe关Qᠬ兴⫽
冕
D关⌽ᠬG兴exp共⫺iSe关QᠬM,⌽ᠬG兴⫺i⌽ᠬG⫻QᠬG兲.共B6兲 One can check that substitution of Eq. 共3.10兲 into Eq. 共B6兲 yields the sameZeas given by Eqs.共B2兲 and 共B5兲.
APPENDIX C: DERIVATION OF EQ.„3.14…
In the limit R→⬁ a voltage measurement in the circuit of Fig. 10 corresponds to a voltage measurement at contacts M and M
⬘
of the circuit C. We obtain the generating func-tional ZV of this voltage measurement from Eq. 共3.9兲. Theinfluence functional is now due to C and it equals the gen-erating functional ZI of a current measurement at contacts
M and M
⬘
of C. From Eq.共3.10兲 with ZM M⫽ZGG⫽⫺ZM G⫽⫺ZG M⫽R we find in the limit R→⬁ that the
environmen-tal action takes the simple form Se关QជM,⌽ជG兴⫽⌽ជ⫻Qជ, with
the cross-product defined in Eq. 共3.15兲. Consequently, we have
ZV关Qជ兴⫽
冕
D关⌽ជ兴e⫺i⌽ជ⫻QជZI关⌽ជ兴. 共C1兲This equation relates the generating functionals of current and voltage measurements at any pair of contacts of a circuit.
APPENDIX D: DERIVATION OF EQ.„4.1…
To derive Eq. 共4.1兲 from Eq. 共3.9兲 we need the environ-mental action Se of the circuit shown in Fig. 3. The imped-ance matrix is Z⫽ 1 R1⫹R2⫹R3
冉
R1共R2⫹R3兲 ⫺R1R2 ⫺R1共R2⫹R3兲 ⫺R1R3 ⫺R1R2 R2共R1⫹R3兲 ⫺R1R2 ⫺R2R3 ⫺R1共R2⫹R3兲 ⫺R1R2 R1共R2⫹R3兲 ⫺R1R3 ⫺R1R3 ⫺R2R3 ⫺R1R3 R3共R1⫹R2兲冊
. 共D1兲We seek the limit R1,R2,R3→⬁. The environmental action 共3.10兲 takes the form
Se关QᠬM,⌽ᠬG兴⫽⌽ជG1⫻QជM1⫹⌽ជG1⫻QជM2⫹⌽ជG2⫻QជM2. 共D2兲
Substitution into Eq.共3.9兲 gives ZVV. Employing Eq.共3.14兲 to obtain ZVIfrom ZVV we arrive at Eq.共4.1兲.
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