On the statistics of quantum transfer of non-interacting fermions in multi-terminal junctions
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n On the statistics of quantum transfer of non-interacting fermions in multi-terminaljunctions
Dania Kambly and D. A. Ivanov
Institute of Theoretical Physics, Ecole Polytechnique F´ed´erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland (Dated: October 7, 2009)Similarly to the recently obtained result for two-terminal systems, we show that there are con-straints on the full counting statistics for non-interacting fermions in multi-terminal contacts. Incontrast to the two-terminal result, however, there is no factorization property in the multi-terminalcase.
Introduction.
The problem of full counting statis-tics (FCS) of electronic charge transfer has been ad-dressed since long time, and the particular model of non-interacting fermions has been studied in detail in varioussetups. The FCS for transfer of non-interacting fermionsis given by the Levitov–Lesovik determinant formula valid at arbitrary temperature and for an arbitrary timeevolution of the scatterer. Recently, some properties ofthis result have been elucidated. First, in the particu-lar case of charge transfer driven by a time-dependentbias voltage at zero temperature, the resulting FCS en-joys certain symmetries. Second, in the more generalcase of an arbitrary time-dependent scatterer and at ar-bitrary temperature, it has been shown that the FCS isfactorizable into independent single-particle events.
In the present work, we generalize the result of Refs. 8,9to a multi-terminal setup. As in those works, we ad-dress the problem of determining which multi-channelcharge transfers are possible and which are not in an ar-bitrary quantum pump, in the model of non-interactingfermions. In the two-terminal case, the constraint derivedin Refs. 8,9 is exact. In the multiterminal case, how-ever, the problem is more complicated, and we have onlypartially solved it: we have formulated a necessary con-straint (a “convexity condition”) on the charge-transferstatistics, without a proof (or a counterexample) that thisconstraint is sufficient. Also, there is no obvious physicalinterpretation of this constraint: we show that, unlikein the two-terminal case, our constraint cannot be inter-preted as a factorization property of the charge-transferstatistics.This work is partly based on the results reported inRef. 10.
Determinant formula.
We first introduce notationand review the Levitov–Lesovik determinant formula for charge transfer of non-interacting fermions in appli-cation to a multi-lead setup. The notation and argumentis fully parallel to that in Ref. 9 where the two-lead casewas considered.We consider a contact with L leads, connected by anarbitrary time-dependent scatterer (see Fig. 1). To eachlead (numbered i = 1 , . . . , L ) we associate a “countingfield” λ i and a projector operator P i acting in the single-particle Hilbert space. The leads are defined in such a λ P , λ P , λ P ,
L L
FIG. 1: A schematic figure of the multiterminal contact. Toeach of the L leads, there corresponds a single-particle pro-jector P i and a counting variable λ i . way that L X i =1 P i = . (1)Then the probabilities of the multi-lead charge transferscan be determined from the generating function χ ( λ , . . . , λ L ) = Tr (cid:16) ˆ ρ ˆ U † e iλ ˆP ˆ U e − iλ ˆP (cid:17) . Tr ˆ ρ . (2)Here the trace is taken in the multi-particle Fock space,ˆ ρ is the initial density matrix, ˆ U is the multi-particleevolution operator. We also use the shorthand notation λ ˆP = P i λ i ˆ P i , where ˆ P i is the multi-particle operator(a fermionic bilinear ) constructed from the projector P i (it counts the particles in the lead i ). As in the two-leadproblem, under the assumption that ˆ ρ commutes withˆ P i (the absence of entanglement in the initial state), theFourier components of the generating function (2) givethe charge-transfer probabilities P q ,...,q L , χ ( λ , . . . , λ L ) = ∞ X q ,...,q L = −∞ P q ,...,q L exp i L X i =1 λ i q i ! . (3)Those probabilities are only non-zero for charge-conserving transfers with P i q i = 0. This charge con-servation corresponds to the symmetry of the generatingfunction with respect to a simultaneous shift of all vari-ables, χ ( λ , . . . , λ L ) = χ ( λ + δλ, . . . , λ L + δλ ) . (4)As in the two-lead case, we define the complex variables u i = e iλ i , i = 1 , . . . , L , (5)and consider the generating function as a function of u i .As in Ref. 9, we assume, in addition to the absenceof entanglement of the initial state, that both ˆ ρ and ˆ U are exponentials of fermionic bilinears (which reflects ourassumption of non-interacting fermions). Under those as-sumptions, we repeat the calculation of Ref. 9 and arriveat the resulting determinant formula χ ( λ , . . . , λ L ) = det h n F ( U † e iλ P U e − iλ P − i , (6)which involves only operators in the single-particleHilbert space with the occupation-number operator n F = ρ ρ + 1 . (7) Convexity condition.
Similarly to the trick employedin Ref. 9, we can rewrite the determinant formula bydefining the hermitian “effective-transparency operators”˜ X ( i ) = (1 − n F ) P i + n / F U † P i U n / F . (8)After simple algebra [using the completeness relation (1)],one can re-express the generating function (6) as χ ( u , . . . , u L ) = det " e − iλ P L X i =1 u i ˜ X ( i ) . (9)The eigenvalues of the operators ˜ X ( i ) are bounded be-tween 0 and 1, which allows us to prove a certain con-straint on the zeroes (roots) of the generating function(9). An elegant form of this constraint can be formulatedin terms of the convex envelope (convex hull) H c ( X ) ofa given set of complex numbers X : a minimal convexset containing X (see Fig. 2a). The constraint may nowbe cast in the form of two conditions that need to besatisfied:1. For any root of the characteristic function χ ( u , . . . , u L ) = 0, the convex envelope H c ( { u , . . . , u L } ) contains zero.2. If χ ( u , . . . , u L ) = 0 and if zero belongs to the boundary of H c ( { u , . . . , u L } ), then those of thepoints { u , . . . , u L } that do not lie on the straightsegment of the boundary of H c ( { u , . . . , u L } ) con-taining zero, can be arbitrarily changed while stillsatisfying the equation χ ( u , . . . , u L ) = 0 (Fig. 2b).The proof of Condition 1 is easy: if | Ψ i is a zero modeof the operator in the determinant (9), then L X i =1 u i h Ψ | ˜ X ( i ) | Ψ i = 0 . (10)Since all the coefficients h Ψ | ˜ X ( i ) | Ψ i are non-negative realnumbers (whose sum equals one), zero belongs to theconvex envelope of u , . . . , u L . u u u u u u u u u uu u 0(b)(a) FIG. 2: (a):
Illustration of the definition of the convex en-velope (convex hull). The shaded region shows the convexenvelope of the points u , . . . , u in the complex plane. If thepoints u , . . . , u correspond to a root of the generating func-tion, then Condition 1 of the constraint claims that zero mustbelong to the shaded region. (b): Illustration of Condition2 of the constraint. In this figure (with the points u , . . . , u corresponding to a root of the generating function), the points u , u , and u can be changed arbitrarily, and the new set ofpoints will still give a root of the generating function. S FIG. 3: A counterexample proving non-factorizability of thefull counting statistics for multi-terminal contacts. Twofermions are sent to a time- and energy-independent L -leadscatterer (with L ≥
3) along the leads 1 and 2 in the shapeof exactly identical wave packets, synchronized in time.
To prove Condition 2, consider again a root( u , . . . , u L ) of the generating function and the corre-sponding zero mode | Ψ i . If zero lies at the boundaryof the convex envelope H c ( { u , . . . , u L } ), then the lin-ear combination (10) contains nonvanishing coefficients h Ψ | ˜ X ( i ) | Ψ i only for variables u i which belong to the samestraight segment of the boundary containing zero. All theother coefficients necessarily vanish, which, by virtue ofthe non-negativity of ˜ X ( i ) , implies ˜ X ( i ) | Ψ i = 0. There-fore all those variables u i may be changed arbitrarilywhile | Ψ i will remain a zero mode. This completes theproof of Condition 2.We can make several comments on the obtained re-sult. First, in the particular case of two leads ( L = 2),this constraint is equivalent to that found in Ref. 9 (thevariable u in that work corresponds to the ratio u /u in our present notation). Second, while our constraint isa necessary condition for realizability of a given statis-tics in a non-interacting fermionic system, we could notdetermine if it is also a sufficient one. Moreover, we donot have any algorithm which would determine if a givencharge-transfer statistics is realizable (or design a suit-able quantum evolution if it is). Those interesting ques-tions are left for future studies. Third, our criterion istechnically difficult to check in its full formulation for allroots ( u , . . . , u L ). However, for practical applications,one may test the constraint on suitably chosen familiesof roots (e.g., one-parametric families ), either analyti-cally or numerically. Non-factorizability.
In the two-terminal case, the“convexity condition” derived above implies a factoriz-ability of the charge transfer statistics: the probabilitiesof a given charge transfer are the same as in a superpo-sition of some single-electron transfer processes (whosetransfer probabilities depend in a non-trivial way on theevolution of the quantum system). One can see that it isnot the case in the multi-terminal (
L >
2) case.This can be most easily demonstrated with a counter-example involving only a finite number of electrons (inthe wave packet formalism of Ref. 11, to which our re-sult is also applicable). Consider two fermions sent intoa stationary multi-terminal contact along two terminals(labeled 1 and 2) with exactly the same shape of wavepackets (Fig. 3). Then, due to the Fermi statistics ofparticles, the probabilities to have both fermions scat-tered to the same lead vanish. The resulting generatingfunction will therefore have the form χ ( u , . . . , u L ) = 1 u u X i To summarize, we have consideredthe problem of possible full counting statistics for non-interacting fermions in coherent multi-terminal systems.We have obtained a necessary condition for a full count-ing statistics to be realizable. Like in the two-terminalcase, this condition may be used to prove impossibilityof certain sets of charge-transfer probabilities (one caneasily construct examples of such impossible statistics).At the same time, the problem of designing an ac-tual “quantum pump” for a given charge-transfer statis-tics (or even merely proving its possibility ) appears muchmore difficult in the multi-terminal case than in the two-terminal one. While in the two-terminal case, the fullcounting statistics of non-interacting fermions is conve-niently parameterized by the spectral density of “effectivetransparencies”, we are not aware of a similar parame-terization in the multi-terminal case. In the formula-tion with a finite number of particles , even the ques-tion of the dimensionality of the space of all possible fullcounting statistics remains open. All those interestingquestions deserve further study, in particular in the con-text of using quantum contacts for generating entangledstates. L. S. Levitov and G. B. Lesovik, Pis’ma v ZhETF , 225(1993) [JETP Lett. , 230 (1993)]. Charge distribution in quantum shot noise. D. A. Ivanov and L. S. Levitov, Pis’ma v ZhETF , 450(1993) [JETP Lett., , 461 (1993). The reader should becareful about an excessive number of typos in the Englishtranslation]. Statistics of charge fluctuations in quantum transport inan alternating field. L. S. Levitov, H.-W. Lee, and G. B. Lesovik, J. Math. Phys. , 4845 (1996). Electron counting statistics and coherent states of electriccurrent. D. A. Ivanov, H.-W. Lee, and L. S. Levitov, Phys. Rev. B , 6839 (1997). Coherent states of alternating current. I. Klich, in “Quantum noise in mesoscopicphysics”, ed. Yu. V. Nazarov, Springer (2003)[arXiv:cond-mat/0209642]. Full Counting Statistics: An elementary derivation ofLevitov’s formula. M. Vanevi´c, Yu. V. Nazarov, and W. Belzig, Phys. Rev. B , 245308 (2008). Elementary charge-transfer processes in mesoscopic con-ductors. M. Vanevi´c, Yu. V. Nazarov, and W. Belzig, Phys. Rev.Lett. , 076601 (2007). Elementary events of electron transfer in a voltage-drivenquantum point contact. A. G. Abanov and D. A. Ivanov, Phys. Rev. Lett. ,086602 (2008). Allowed charge transfers between coherent conductorsdriven by a time-dependent scatterer. A. G. Abanov and D. A. Ivanov, Phys. Rev. B , 205315(2009). Factorization of quantum charge transport for non-interacting fermions. D. Kambly, Master thesis, EPFL, Lausanne, (2009). F. Hassler, M. V. Suslov, G. M. Graf, M. V. Lebedev,G. B. Lesovik, and G. Blatter, Phys. Rev. B , 165330(2008). Wave-packet formalism of full counting statistics. I. Klich and L. Levitov, Phys. Rev. Lett. , 100502(2009), Quantum noise as an entanglement meter .in “Advances in Theoretical Physics: Landau Memo- rial Conference”, eds. V. Lebedev and M. V. Feigelman,AIP Conference Proceedings , 36 (2009) [e-printarXiv:0901.3391]. Many-Body entanglement: a new application of the fullcounting statistics. A good practical test of the constraint is obtained by re-stricting u i to a one-parameter family u i = a i z + b i for some fixed sets of real numbers a i and b i with the condi-tion that all a i ≥ b i ). Thenthe equation χ ( z ) = 0 must either have only real roots z orbe identically satisfied for all z . This form of the constraintis convenient for numerical tests by using a large numberof randomly chosen vectors a i and b ii