Noise and Measurement Efficiency of a Partially Coherent Mesoscopic Detector A. A. Clerk and A. D. Stone Departments of Applied Physics and Physics, Yale University, New Haven CT, 06511, USA Jan. 19, 2004 4 0 We study the noise properties and efficiency of a mesoscopic resonant-level conductor which is 0 usedasaquantumdetector,intheregimewheretransportthroughthelevelisonlypartiallyphase 2 coherent. Wecontrast models in which detector incoherence arises from escape to a voltage probe, n versusthoseinwhichitarisesfromarandomtime-dependentpotential. Particularattentionispaid a to the back-action charge noise of the system. While the average detector current is similar in all J models,wefindthatitsnoisepropertiesandmeasurementefficiencyaresensitivebothtothedegree 0 of coherence and to the nature of the dephasing source. Detector incoherence prevents quantum 2 limited detection, except in the non-generic case where the source of dephasing is not associated withextraunobservedinformation. Thislattercasecanberealizedinaversionofthevoltageprobe ] model. l l a PACSnumbers: h - s I. INTRODUCTION conductor,the influence ofdephasing onthe output cur- e m rent noise has received considerable attention27; in con- trast, its influence on the back-action charge noise has . Motivated primarily by experiments involving t only been addressed in a limited number of cases15,16. a solid-state qubit systems, attention has recently NotethatarecentexperimentbySprinzaket. al17 using m turned to examining the properties of meso- a point-contact detector suggests that the back-action - scopic conductors viewed as quantum detectors or d amplifiers1,2,3,4,5,6,7,8,9,10,11,12,13,14. Of particular inter- noise is independent of dephasing. n est is the issue of quantum limited detection– does a To study the role of detector incoherence, we focus o particular detector have the minimum possible back- here on the case where the mesoscopic scattering detec- c [ action noise allowed by quantum uncertainty relations? tor is a non-interacting, single-level resonant tunneling Reachingthe quantumlimit is crucialto the success ofa structure, with the signalof interest (e.g., a qubit) mod- 2 numberofpotentialexperimentsinquantuminformation ulating the energy of the level. This model provides an v physics, including the detection of coherent qubit oscil- approximate description of transport through a quan- 3 0 lations in the noise of a detector10. The measurement tum dot near a Coulomb blockade charge degeneracy 1 efficiency of a number of specific mesoscopic detectors point, in the limit where the dot has a large level spac- 1 has been studied1,6,7,9,11, as have general conditions ing. Such a system could act as a quantum detector of, 0 needed for quantum limited detection12,13. Recent e.g., a double-dot qubit. The resonant level detector is 4 studies of a broad class of phase coherent mesoscopic also conceptually similar to detectors using the Joseph- 0 scattering detectors11,12 have helped establish a general son quasiparticle (JQP) resonance in a superconducting / t relation between back-action noise, the quantum limit singleelectrontransistor18,19,20,ashavebeenusedinsev- a m and information. The back-action charge noise of these eral recent qubit detection experiments21,22. In these detectors was found to be a measure of the total acces- systems the resonance is between two transistor charge - d sible information generated by the detector’s interaction states,oneofwhichisbroadenedbyquasiparticletunnel- n with a qubit, and the quantum limit condition to imply ing, and the signal of interest modulates the position of o the lack of any “wasted”information in the detector not the resonance. Despite the incoherenceofthe resonance- c revealed at its output. broadeninghere,ithasbeenshowntheoreticallythatone : v An important unansweredquestion regardsthe roleof can still make a near quantum-limited measurement us- i ing the JQP process20. X detector coherence– does a departure from perfectly co- herent transport in the detector necessarily imply a de- The detector properties of a fully coherent resonant r a viation from the quantum limit? One might expect that level model were studied comprehensively by Averin in dephasing will have a negative impact, as there will now Ref. 7,whofoundthatdetectioncanbequantum-limited beextraneousnoiseassociatedwiththesourceofdephas- inthe smallvoltageregime(asfollowsalsofromthe gen- ing. However, if there were unused phase information in eralanalysisinRefs. 11and12), andnear-quantumlim- thecoherentsystem,onemightexpecttheadditionofde- ited in the large voltage regime. We are interested now phasingtobringthedetectorclosertothequantumlimit, in how the addition of dephasing changes these conclu- asthis unusedphaseinformationwillbeeliminated. Ad- sions. The influenceofdephasingonthe noiseproperties dressingtheinfluenceofdephasingconcretelyrequiresan of the resonant level model also has intrinsic interest, understanding of its effects on the noise properties of a as this is one of the simplest systems with non-trivial detector. In the case where the detector is a mesoscopic energy-dependent scattering. Standard treatments28 in- 2 dicate that the effects of dephasing on the resonantlevel v(t) which produces a weak potential in the scattering cannotbeidentifiedviatheenergy-dependenceoftheav- region will lead to a change in average current given by erage current– both the coherent and incoherent models δhI(t)i=λv(t), where λ is the zero-frequency gain coef- yieldaLorentzianformfortheconductance. Incontrast, ficient of the detector. In the case where the potential the noise properties of the coherent and incoherent res- created by the signal (i.e. qubit) is smooth in the scat- onant level models are significantly different; while this tering region, and where the drain-source voltage tends is known for the current noise, we show that it can also to zero (i.e. µ = µ +eV,eV → 0) the zero frequency 1 2 hold for the back-actioncharge noise. noise correlatorsof the system are given by11,12: To study the effects ofdephasing,we will use twogen- eral models. The first corresponds to dephasing due to e3V 2 e3V λ = ∂ [s (ε)] = ∂ T (ε) (2a) ε 0 12 ε 0 unobserved escape from the level, and will be modelled h h using the voltage probe model developed by Bu¨ttiker28. 2e3V (cid:12) (cid:12) S = T(cid:12)(ε)R (ε)(cid:12) (2b) The second will correspond to dephasing arising from I h 0 0 a random time-dependent external potential. This ap- e3V~ proach is particularly appealing, as it allows a simple SQ = × π heuristic interpretation of the effects of dephasing, and allows for a clear separation between pure dephasing ef- [∂εT0(ε)]2 +T (ε)R (ε)[∂ β(ε)]2 (2c) 0 0 ε fects and inelastic scattering. 4T0(ε)R0(ε) ! In general, we find that both the magnitude of de- phasing and the nature of the dephasing source need Here, S and S are the zero-frequency output current I Q to be determined in order to evaluate the effect of in- noiseandback-actionchargenoise,R ≡1−T isthere- 0 0 coherence on the quantum limit. Dephasing prevents flectioncoefficient, and allfunctions shouldbe evaluated ideal quantum-limited detection, except in a simple, at ε = µ, where µ is the chemical potential of the leads. physically-realizable version of the voltage probe model. Note that the chargeQ here refers to the total charge in Thelattermodelisunique,asonecanshowthatthereis thescatteringregion,andisnotsimplytheintegralofthe no extra unobserved information produced by the addi- source-drain current I. Also note that throughout this tionofdephasing(i.e. theadditionofthevoltageprobe). paper, we concentrate on the case of zero temperature. We alsofindthatthe voltageprobemodelcanyieldvery We are interested in the measurement efficiency ratio similar noise properties to a sufficiently “slow” random χ, defined as potentialmodelinthesmallvoltagelimit;thiscorrespon- dence however is lost at larger voltages. ~2λ2 χ≡ . (3) S S I Q In the case of a qubit coupled to the detector, χ repre- II. COHERENT DETECTOR sentstheratioofthemeasurementratetotheback-action dephasing rate in a quantum non-demolition setup8,9,12, Webeginbybrieflyreviewingthepropertiesofacoher- and the maximum signal to noise ratio in a noise- ent mesoscopic scattering detector, as discussed in Refs. spectroscopy experiment10. χ is rigorously bounded by 11 and 12. In the simplest case, the detector is a phase unity12,13, and reaching the quantum limit corresponds coherent scattering region coupled to two reservoirs (1 to having χ = 1. If we view our detector as a linear and 2) via single channel leads, and is described by the amplifier, achieving χ = 1 is equivalent to having the scattering matrix: minimum possible detector noise energy9,12,23. Even in the V → 0 limit, the general 1D scatter- s (ε)=eiα0(ε) eiβ0(ε) 1−T0(ε) −i T0(ε) . ing detector described above fails to reach the quantum 0 (cid:18) −ipT0(ε) e−iβ0(ε)p1−T0(ε)(cid:19) limitbecauseofunusedinformationavailableinthephase (1) p p β(ε): Therearethreeparameterswhichdetermines : theover- 0 all scattering phase α0(ε), the transmission coefficient ~2λ2 1 T (ε), and the relative phase between transmission and χ≡ = (4) 0 S S 2 reflection, β0(ε). The only assumption in Eq. (1) is that I Q 1+ 2T0R0∂∂εεTβ00 time-reversal symmetry holds; as discussed in Ref. 12, (cid:16) (cid:17) the presence or absence of time-reversal symmetry is ir- As noted, β is the energy-dependent relative phase be- relevant to reaching the quantum limit. Note that if s tween reflection and transmission, and in principle is 0 has parity symmetry, the phase β is forced to be zero, accessible in an experiment sensitive to interference be- 0 which implies that there is no energy-dependent phase tweentransmittedandreflectedcurrents17. Thepresence difference between transmitted and reflected currents. of parity symmetry would force β = 0, and would thus Theconductordescribedbys issensitivetochangesin allowanarbitraryone-channelphase-coherentscattering 0 thepotentialinthescatteringregion,andmaythusserve detector to reach the quantum limit in the zero-voltage as a detector of charge. A sufficiently slow input signal limit6,11,12. Note that this discussion neglects screenings 3 effects; such effects have been included within an RPA III. DEPHASING FROM ESCAPE scheme in Ref. 11, where it was shown they did not ef- fect χ. A. General setup We now specialize to the case where our scattering re- gion is a single resonant level. Taking the tunnel matrix We first treat the effects of dephasing by using the elements tobe independent ofenergy,the scatteringma- phenomenological voltage probe model developed by trixintheabsenceofdephasingisdeterminedintheusual Buttiker28. A fictitious third lead is attached to the res- way by the retardedGreen function of the level. Letting onant level, with its reservoir chosen so that there is no 1 (2) denote the L (R) lead, we have: net current flowing through it. Nonetheless, electrons may enter and leave this third lead incoherently, lead- ing to a dephasing effect. We term this dephasing by i Γ Γ s (ε)=ˆ1−i Γ Γ ·GR(ε)=ˆ1− i j (5) escape, as it models electrons actually leaving the level. 0 i j ε−εpd+iΓ0/2 In practice, the Green function of the level is broadened p by an additional amount Γ over the elastic broadening ϕ Γ ; one also has to explicitly consider the contribution Here, Γ (Γ ) is the level broadening due to tunneling 0 L R to the current and noise from electrons which enter and to the left (right) lead; Γ = Γ + Γ represents the 0 L R leave the voltage probe incoherently. totalwidthofthelevelduetotunneling. Theparameters Moreconcretely,assumingthatthereareM propagat- appearing in Eq. (1) are given by: ing channels in the voltage probe lead, our detector is now described by a (2+M)×(2+M) scattering matrix Γ s . Weassumethroughoutthepresenceoftime-reversal 0 big α (ε) = −arctan (6a) 0 2(ε−ε ) symmetry; the results presented here are independent of (cid:18) d (cid:19) this assumption. The (non-unitary) 2×2 sub-matrix of Γ Γ L R T (ε) = (6b) s describing direct, coherent scattering between the 0 (ε−ε )2+Γ2/4 big d 0 physical leads is given by29: Γ −Γ R L β (ε) = arctan (6c) 0 (cid:18)2(ε−εd)(cid:19) s˜0(ε)≡s0(ε+iΓϕ/2) (8) where s (ε) is given by Eq. (5). This matrix continues 0 Notethatthereisonlyonenon-trivialeigenvalueofs0(ε), to have a decoupled scattering channel (i.e. eigenvalue given by e2iα0 (i.e. there is a scattering channel which 1), while the eigenvalue of the coupled channel becomes: decouples from the level). One finds for the measurement efficiency at zero volt- age: e2iα0(ε) → 1−Tϕ(ε)e2iα(ε) (9) q with (ε −µ)2 d χ= (7) 1 (εd−µ)2+(ΓL−ΓR)2/4 G˜R(ε) = (10) ε−ε +iΓ/2 d Γ Γ As per our general discussion above, the coherent reso- T (ε) = ϕ 0 (11) ϕ (ε−ε )2+(Γ/2)2 nantleveldetectorisonlyquantumlimitedifthereispar- d ity symmetry, i.e. ΓL =ΓR; if this condition is not met, α(ε) = 1 argG˜R(ε)−arctan Γ0−Γϕ (12) there is unused information available in the phase β(ε). 2 2(ε−ε ) (cid:20) (cid:18) d (cid:19)(cid:21) For the resonant level model, the effects of this unused information can be minimized by working far from reso- Here,Γ=Γ0+Γϕ representsthetotalwidthofthelevel, nance (i.e. |ε −µ|≫0), as this suppresses the informa- G˜R(ε) is the retarded Green function of the level in the d tion in the phase β(ε) faster than that in the amplitude presenceofdephasing,andT parameterizesthestrength ϕ T (ε). Itis worthnoting thatintroducinga thirdleadto of the dephasing. Transmission into the voltage probe 0 representdephasing(aswedointhenextsection)breaks from the physical leads will be described by an M ×2 parityinasimilarmannertosimplyhavingΓ 6=Γ ,and sub-matrix of s which we denote t . Using a polar L R big ϕ its effect may be understood in similar terms. Note also decomposition30, it may in general be written as: that for Γ = Γ ,χ = 1 for any value of µ. This may L R seem surprising as the gain vanishes when εd = µ (i.e. [tϕ]mj = Vmk(ε) Tϕ,k(ε)UkTj(ε) (13) the peakofthe resonancelineshape). However,the noise kX=1,2 q alsovanishesatthispointinjustthe mannerrequiredto maintain χ=1. This feature does not carry over in any Here, U and V are unitary matrices which parameterize ofthemodelsofdephasing;χwilldependontheposition the preferred modes in the leads, and T are the two ϕ,k of µ and will not be maximized for ε =µ. transmission eigenvalues characterizing the strength of d 4 transmission into the voltage probe. These transmission 1(cid:13) eigenvalues are uniquely specified by s˜ (ε): 0 0.8(cid:13) T (ε) = δ T (ε) (14) ϕ,k k1 ϕ A general result of voltage probe models is that the 0.6(cid:13) χ average current hIi and current noise S are indepen- I dent of the matrices U and V appearing in the polar 0.4(cid:13) decomposition27,32. This is convenient, because in gen- eral, these matrices are not uniquely determined by the 0.2(cid:13) form of the coherent scattering matrix s˜ (ε). However, 0 inthepresentproblemwearealsointerestedback-action charge noise S , which in general does depend on these Q 0.2(cid:13) 0.4(cid:13) (cid:13) 0.6(cid:13) 0.8(cid:13) 1(cid:13) matrices. For the dephased resonant-level,the matrix U Γ / Γ ϕ is completely specified by s˜ (ε): 0 FIG. 1: Quantumefficiency ratio χ versus Γ /Γ for theres- ϕ eiα(ε)/2cosθ −e−iα(ε)/2sinθ onant level model, in the limit eV/Γ 0, and with sym- U(ε)=eiα(ε)/2 eiα(ε)/2sinθ e−iα(ε)/2cosθ (15) metric couplings ΓL = ΓR. The solid r→ed curve corresponds (cid:18) (cid:19) to the “pure escape” voltage probe model, the short-dashed greencurvetotheinelasticvoltageprobemodel,andthelong- where the angle θ = tan−1 Γ /Γ parameterizes the L R dashedbluecurvetothedephasingvoltageprobemodel;each asymmetry in the coupling to the leads. p has εd µ = Γ/2. In addition, the dotted red curve corre- ThematrixV remainsunknown;tospecifyit,wemake − sponds to the “pure escape” model at ε µ = 10Γ. For d the additional assumption that the voltage probe is cou- strong dephasing, only the “pure escape”−model is able to pled to the resonant level via a tunneling Hamiltonian remain near quantum-limited. with energy-independent tunnel matrix elements. This yields: 0.25(cid:13) V =−i Γme−iα(ε)eiargG˜R(ε), (16) m1 sΓϕ 0.2(cid:13) with Γ = M Γ . Again, the ambiguity in V(ε) has χ 0.15(cid:13) ϕ m=1 m no affect on the average current through the system or on the currPent noise. It will however be important in 0.1(cid:13) determining the charge noise of the system; the choice 0.05(cid:13) givenin Eq. (16) represents a best case scenario,in that it minimizes the charge noise. Finally, we must specify the distribution function in thereservoirattachedtothethirdlead. We willcontrast 0.2(cid:13) 0.4(cid:13) (cid:13) 0.6(cid:13) 0.8(cid:13) 1(cid:13) Γ / Γ threedifferentchoiceswhichallyieldavanishingaverage ϕ current in the third lead, and which correspond to dif- FIG.2: QuantumefficiencyratioχversusΓ /Γforthereso- ϕ ferent physical mechanisms of dephasing27,32. The first nant levelmodel, in thelimit eV/Γ 0, and with Γ Γ . L R correspondstoaphysically-realizablesituationwherethe We have chosen ε = Γ/(2√3) to m→aximize gain. Th≫e dif- d reservoir attached to the third lead has a well defined ferentcurvescorrespondtodifferentvoltageprobemodels,as chemicalpotential; it is chosento yield a vanishing aver- labelledinFig. 1. Notethemarkednon-monotonicbehaviour age current into the probe. We term this the “pure es- of χ in the “pureescape” voltage probemodel. cape”voltageprobemodel. Thesecondmodelissimilar, except one now also enforces the vanishing of the probe current at each instant of time by allowing the voltage from the scattering matrix s describing the dephased big associatedwiththethird-leadtofluctuate31. Thismodel resonant level detector. The standard relations between is usuallytakento giveagooddescriptionofinelasticef- thesequantitiesandthescatteringmatrixaregiven,e.g., fects, and is knownas the inelastic voltage probe model. in Ref. 12. We will study the effects of dephasing by Finally,inthe thirdmodelonealsoenforcescurrentcon- keepingthetotalwidthofthelevelΓconstant,andvary- servation as a function of energy. This is achieved by ing its incoherent fraction Γ /Γ. This is equivalent to ϕ assigning a non-equilibrium distribution function to the asking how the noise and detector properties of a given reservoir32. Thismodelisthoughttowell-describequasi- Lorentzian conductance resonance of fixed width Γ de- elastic dephasing effects, and is known as the dephasing pends on the degree to which it is coherent. Of course, voltage probe model. simply increasing dephasing while keeping the coupling In what follows, we calculate the averagecurrent(and tothe leadsfixed(e.g. byincreasingtemperature)would hence the gain λ), current noise S and charge noise S also cause the overall width Γ to increase. I Q 5 B. Results from the “pure escape” voltage probe ameasureofthetotalaccessibleinformationinascatter- model ing detector. Thus, there is no “conservationofinforma- tion” as dephasing is increased in the present model. In For simplicity, we focus throughout this subsection on the strongdephasinglimit, SQ is suppressedin the same thezerovoltagelimit. Ingeneral,theaveragecurrenthas way as hIi and SI, that is by a factor Γ0/Γ. We again both a coherentcontribution(involvingonly the scatter- emphasize that this result corresponds to a physically- ing matrix s˜ ) and an incoherentcontribution, whichin- realizable setup, where a third lead is attached to the 0 volves transmission into the voltage probe lead27. These levelandassignedawell-definedchemicalpotential. The combine to yield the simple result28: effect of a similar dephasor on the charge fluctuations of a quantum-point contact was studied experimentally by e2V 2Γ Γ Sprinzak et. al in Ref. 17; in contrast to the result of hIi = L R −Im G˜R(ε=µ) h Γ +Γ Eq. (21) for the resonant level detector, they found that L R e2V Γ h Γ2/4 i theadditionofdephasingdidnotappreciablychangethe = 0 sin22θ (17) back-action noise of a quantum point-contact detector. h Γ (µ−ε )2+Γ2/4 (cid:18) (cid:19) d Ofcourse,the systemstudied in Ref. 17is verydifferent from the one studied here. Nonetheless, our result indi- The conductance continues to have a Lorentzian form cates that, at the very least, the insensitivity of charge even in the presence of dephasing, though its overall fluctuations to dephasing seen in this experiment is not weight is suppressed by a factor (Γ /Γ); the gain λ will 0 generic to all mesoscopic conductors. be suppressed by the same factor. Note that this sup- Finally, turning to the measurementefficiency ratioχ, pression is indistinguishable from simply enhancing the wenotethatsinceeachofλ,S andS aresuppressedas asymmetry between the couplings to the leads. I Q Γ /Γ, turning on dephasing does not lead to a paramet- The effect of dephasing on the current noise is more 0 ric suppression of χ (see Figs. 1 and 2). In the strong pronounced. In the present model, even though the av- dephasing limit, we find: erage current into the voltage probe vanishes, there may nonetheless exist fluctuatingcurrents into andout of the 1 (µ−ε )2 d voltageprobe. The resultisthat measuringdifferentlin- χ→ , (22) 2f (µ−ε )2+Γ2/4 earcombinations ofthe currentinthe left andrightlead d will yield different values for the current noise, though wheretheFanofactorf isgiveninEq. (20). Inthestrong they all yield the same average current. Choosing the dephasinglimitthere iswastedphaseinformationdueto measured current to be the linear combination: thestrongasymmetrybetweenthecouplingtothephysi- calleads andto the voltageprobe lead. Nonetheless,the I =(sin2α)I +(cos2α)I , (18) meas L R effectsofthiswastedphaseinformationcanbeminimized by working far from resonance. Thus, for dephasing due andwriting the currentnoise interms ofthe Fanofactor to true escape (i.e. due to a simple third lead), one can f: approach the quantum limit even in the strongly incoher- entlimit. Inthislimit,alltransportinvolvesenteringthe S =2efhIi (19) I voltageprobe,andthenleavingitincoherently. Nonethe- less, in the small voltage limit we consider, there is no onefindsthatinzerodephasingcase,f isindependentof wastedamplitude informationinthe presenceofdephas- αandis givenbythe coherentreflectionprobability(1− ing. Though for energies in the interval [µ ,µ ] there is T (ε)) (c.f. Eq. (6b)), whereas in the strong dephasing 2 3 0 a current I flowing into the voltage probe lead, one limit (Γ /Γ→0), it is given by: 3,in 0 cannotlearnanythingnew bymeasuringit, asanidenti- f →sin4α+cos4α (20) cal current exits the voltage probe in the energy interval [µ ,µ ] and contributesdirectly to the measuredcurrent 3 1 Thisformreflectsthesuppressionofcorrelationsbetween flowingbetweentheleftandrightcontacts. Itisthislack I and I ; f ranges from a minimum of 1/2 (for a sym- ofwastedamplitudeinformationthatallowsonetoreach L R metric combination of I and I ) to a maximum of 1 if the quantum limit even for strong dephasing. L R one measures either I or I . L R Turning to the charge noise, we find: C. Results from the inelastic voltage probe model 2 1− Γϕ Γ2 SQ = e32Vπ~ 2ΓΓL2ΓR [(µ(cid:20)−ε(cid:16))2Γ+(cid:17)Γ(cid:21)2/4]2 (21) is Aidsendtiisccaulssteodtehaerl“iepru,rteheesicnaeplae”stimcovdoeltlaogfetphreoblaesmt soudbe-l (cid:18) 0 (cid:19) d section,exceptthatwe nowalsorequirethatthe current Within the “pure escape” voltage probe model, S de- into the voltage probe vanishes at each instant of time. Q creases monotonically with increasing dephasing, regard- This is achieved on a semi-classical level by assigning lessofasymmetryorthepositionofthelevel. Aswasdis- the voltageprobea fluctuating-in-timevoltagechosento cussedinRef. 12,thechargenoiseS canberegardedas exactlyenforcethis additionalconstraint31. The average Q 6 currentisindependentofthisadditionalstep,andisiden- where hIi is given in Eq. (17), and where we have again tical to that in the previous section, Eq. (17). For the taken the zero voltage limit. In the strong dephasing currentnoise,aswenowhaveI =I atalltimes,S be- limit this contribution dominates, and we find: L R I comes independent of the particular choice of measured current. In essence, the effect of the fluctuating voltage S →4e~2 |hIi| (28) in the probe is to simply choose a particular value of α Q (Γ )2 0 in Eq. (18)31. In the strong dephasing limit, the Fano factor is given by: Unlike Eq. (21) for the intrinsic charge fluctuations, whicharesuppressedwithincreaseddephasing,Eq. (27) Γ2 +Γ2 describing the induced fluctuations of Q diverges in the f → L R (23) (ΓL+ΓR)2 strong dephasing limit as Γ/Γ0. As a result, the mea- surement efficiency χ tends to zero as (Γ /Γ)2, and one 0 Not surprisingly, this is the classical Fano factor corre- is far fromthe quantumlimit for strongdephasing. This sponding to two Poisson processes in series. One also result is not surprising. The fluctuating voltage in the finds this Fano factor in the large voltage regime of the probe lead represents a classical uncertainty in the sys- coherent resonant level model, where the model may be tem stemming from the source of dephasing; this uncer- treated using classical rate equations24,26. Note that de- tainty in turn leads to extraneous noise in Q, leading pendingonthepositionofε andtheratioΓ /Γ ,f can departure from the quantum limit. On a heuristic level, d L R either increase or decrease with increasing dephasing. thereis unusedinformationresidinginthe voltageprobe Finally, we turn to the calculation of the charge noise. degrees of freedom responsible for generating the fluctu- The effects ofthe fluctuating voltageonS are far more atingpotential. Ofcourse,ifthisclassicalnoiseisad-hoc Q extremethanonS 34,asitleadstoanew,classicalsource suppressed,oneisbackatthe“pureescape”modelofthe I for charge fluctuations. Note that in general we have: previous section, which can indeed reach the quantum limit even for strong dephasing. Γ f (ε)+Γ f (ε)+Γ f (ε) L L R R ϕ 3 hQi=e dερ(ε) Γ Z (cid:18) (cid:19) (24) D. Results from the dephasingvoltage probe model where ρ(ε) = −1ImG˜R(ε) . Fluctuations of the volt- π Inthedephasingvoltageprobemodel,thedistribution age in the pro(cid:16)be lead will c(cid:17)ause the probe distribution function in the voltage probe reservoir is chosen so that function f (ε) to fluctuate, and will in turn causehQi to 3 the current flowing into the voltage probe vanishes at fluctuate. Letting ∆Q(t) denote the fluctuating part of the charge Q, one has16: each energy27. One obtains a non-equilibrium distribu- tion function f (ε) defined by: 3 Γ ϕ ∆Q(t)=[∆Q(t)]bare+e Γ dερ(ε)(∆f3(ε,t)) (25) f (ε)=f¯(ε)≡ ΓLfL(ε)+ΓRfR(ε) (29) Z 3 Γ +Γ L R Thefirsttermarisesfromfluctuationsinthetotalcurrent incident on the scattering region; it is the only contribu- Similar to the inelastic voltage probe, we also enforce a tion present in the absence of a fluctuating potential, vanishing current into the voltage probe at each instant and is identical to what is found in the “pure escape” of time by having f3(ε) fluctuate in time32. This model model. The second term describes fluctuations of hQi isusuallythoughttobettermimicpuredephasingeffects than the inelastic voltage probe, as the coupling to the arising from voltage fluctuations in the voltage probe. voltageprobedoesnotleadto aredistributionofenergy. These voltage fluctuations are in turn completely deter- Fortheaveragecurrentinthesmallvoltageregime,we mined by the requirementthat the currentinto the volt- age probe vanish at all times31. Note that as a result, againobtainEq. (17),asinthe “pureescape”model. In the finite voltage regime we obtain the simple result: the two terms in Eq. (25) will be correlated. Including these effects, one finds that the charge fluc- e 2Γ Γ tuations are given by: hIi= L R dε(f (ε)−f (ε)) −Im G˜R(ε) , 1 2 hΓ +Γ L R Z (cid:16) (cid:17)(30) S =[S ] +[S ] (26) Q Q bare Q class where again G˜R(ε) is the Green function of the level in Here, [S ] describes the intrinsic charge fluctuations the presence of dephasing (c.f. Eq. (10)). Eq. (30) is Q bare (i.e. from the first term in Eq. (25)); its value is given identicaltotheformallyexactexpressionderivedbyMeir by Eq. (21). The term [S ] arises from fluctuations and Wingreen33 for the current through a single level Q class of ∆f (i.e. second term in Eq. (25)), and includes the havingarbitrary on-siteinteractions. Thus,wecanthink 3 effect of their correlation with [∆Q] . It is given by: of the dephased Green function G˜R in the voltage probe bare approach as mimicking the effects of dephasing due to [S ] =4e~2 Γϕ 1+ Γ0Γϕ/2 |hIi| (27) interactions. The simplicity of both the Meir-Wingreen Q class (Γ )2Γ ω2+(Γ/2)2 and voltage-probe approaches (i.e. hIi ∝ GR) may be 0 (cid:18) (cid:19) 7 simplyextendedtothe multi-levelcaseiftheasymmetry 0.7(cid:13) in the tunnel couplings is the same for each level. 0.6(cid:13) Turning to the noise, the main effect of the non- equilibrium distribution function f3 will be to increase 0.5(cid:13) the noise over the previous two models. On the level of χ 0.4(cid:13) the calculation, these new contributions arise from diag- onal elements of the current I and charge Q operators, 0.3(cid:13) considered in the basis of scattering states. More physi- 0.2(cid:13) cally, this additional noise is due to the fact that the de- tectorplusvoltageprobesystemisnolongerdescribedby 0.1(cid:13) a single pure state. Rather, the distribution function f 3 correspondsto a statisticalensemble ofstates,with each 0.2(cid:13) 0.4(cid:13) (cid:13) 0.6(cid:13) 0.8(cid:13) 1(cid:13) stateoftheensembleyieldinga(possibly)differentquan- Γ / Γ tum expectation of Q and I. The extra noise produced ϕ by this “classical”uncertainty correspondsto unusedin- FIG. 3: Quantumefficiency ratio χ versus Γ /Γ for theres- ϕ formation (e.g. if the non-purity of the detector density onant level model, in the large-voltage “cotunneling” limit: matrix results from entanglement with a reservoir, the µ µ =100Γ,andε =µ +Γ. Thelong-dashedbluecurve L R d L − missing information resides in the reservoir degrees of corresponds to thedephasing voltage probe model, while the freedom), and thus we anticipate a departure from the solid black curve corresponds to the slow random potential quantum limit even if we neglect the extraneous charge model (with Γϕ λϕ). Despite the large voltage, χ is still → noisearisingfromthefluctuatingvoltageprobepotential. suppressedtozerobydephasinginbothmodels;notealsothe large differencebetween thetwo models. We find that the modification of the current noise is ratherminimal– inthe strongdephasinglimit, one again obtains the classical Fano factor of Eq. (23). The effect voltage in the voltage probe (i.e. enforce current conser- on the charge noise is more pronounced. Similar to the vation in energy but not in time), and hence had only inelasticvoltageprobe,therewillbetwocontributionsto the intrinsic contribution to S , one would still have a S , one stemming fromfluctuations ofthe totalincident Q Q parametric suppresion of χ at strong dephasing; this is current on the level, the other from fluctuations of the in contrast to the inelastic voltage probe model, where distributionfunctionf (ε). Separatingthetwocontribu- 3 it is only [S ] that prevents reaching the quantum tions toS asinEq. (26), wefindforarbitrary voltages: Q class Q limitatstrongdephasing. Including alltermsinS , one Q finds that the meaurement efficiency for the dephasing [S ] = 4e2~ dεf¯(ε) 1−f¯(ε) − Im GR(ε) 2 voltage probe model is always less than that for the in- Q bare π elastic dephasing voltage probe model (see Figs. 1 and Z (cid:0) (cid:1)(cid:16) (cid:17)(31) 2). As discussed above, these result follows from the ad- e ditional classical uncertainty resulting from the voltage 4e2~ Γ [S ] = ϕ dεf¯(ε) 1−f¯(ε) probe plus detector system being in a mixed state. Q class π ΓΓ0 Z Finally, it is interesting to consider the case of a large 2(cid:0)Γ Γ (cid:1) voltage (µ − µ = eV ≫ Γ) “co-tunneling” regime, − Im GR(ε) 1+ 0 ϕ (32) L R ε2+(Γ/2)2 wherethe levelisplacedslightlyabovethehigherchemi- (cid:16) (cid:17)(cid:18) (cid:19) cal potential µ . Averin7 demonstrated that in the fully L Interestingly,wefindethatinthedephasingvoltageprobe coherent case, one can still come close to the quantum model, the intrinsic charge fluctuations [SQ]bare (Eq. limit (i.e. χ→3/4) in this regime despite the loss of in- (31)) are independent of dephasing strength; the addi- formationassociatedwith the large voltageand the con- tionalnoiseduetothenon-equilibriumdistributionfunc- sequent energy averaging. One might expect χ to be tionf¯exactlycompensatesthesuppressionofthe intrin- insensitive to dephasing in this regime, given the large sic noise found in the “pure escape” model and inelas- voltage. This is not the case; as is shown in Fig. 3, χ is tic voltage probe models (c.f. Eq. (21)). Recall that again suppressed to zero as (Γ /Γ)2. 0 S = [S ] in the absence of voltage fluctuations in Q Q bare the voltage probe lead. The extrinsic contribution [S ] (Eq. (32)) is sim- Q class IV. DEPHASING FROM A FLUCTUATING ilar in form to the corresponding contribution for the POTENTIAL inelastic voltage probe model. In the strong dephasing limit this term dominates, and at arbitrary voltage, we A. General setup againobtainthedivergentresultofEq. (28)forS . Sim- Q ilartotheinelasticvoltageprobemodel,the fluctuations of Q induced by the fluctuating probe voltage will cause We now consider an alternate model of dephasing in a suppression of χ as (Γ /Γ)2 in the strong dephasing whichthe resonantleveldetectorissubjecttoarandom, 0 limit. Note that even if one did not use a fluctuating Gaussian-distributed, time-dependent potential. This 8 modelrepresentstheclassical(high-temperature)limitof thelargestscaleintheproblem,wemaysolvetheHeisen- dephasinginducedbyabathofoscillators(e.g. phonons), berg equations. Setting ~=1, we find: andis attractiveas it allowsa simple semiclassicalinter- pretation of the influence of dephasing on noise. It also allows one to make a clear distinction between pure de- d(t)= dt′GR(t,t′) Γα dεe−iεt′c˜ (ε) (34) 2π α phasing effects and inelastic scattering. Surprisingly, we Z α r Z X find that for noise properties, dephasing froma fluctuat- ingrandompotentialis notcompletely equivalenttoany where ofthevoltageprobemodels,evenifonechoosesa“slow” potentialwhichdoesnotgiverisetoinelasticeffects; one GR(t,t′) = −iθ(t−t′)e−Γ0(t−t′)/2e−iεd(t−t′)× finds a reasonable agreement between the models only t in the small voltage regime. The upshot of the analy- exp −i dτη(τ) (35) sis is that the noise properties and detector efficiency of (cid:18) Zt′ (cid:19) the resonant level model is sensitive both to the degree and where the c˜ (ε) operators describe conduction elec- of detector coherence and the nature of the dephasing α trons in the leads in the absence of tunneling. Expecta- source. Study of the random potential model also helps tions of the c˜ operators obey Wick’s theorem, and are elucidatetheoriginofthetwocontributionstothecharge α given in terms of the lead distribution functions in the noise found in voltage probe models (c.f. Eq. (26)). usualmanner. Wewillassumethroughoutthatthenoise Notethatamodelsomewhatsimilartothatconsidered here was used by Davies et. al35 to study the effect of η(t) is stationary and Gaussian with an auto-correlation function J(τ): dephasing on current noise in a double tunnel-junction structure. Unlike the present study, they focused on the largevoltageregimeeV ≫Γ, andcouldnotconsiderthe hη(t)η(t′)i=J(t−t′)=J(t′−t) (36) effect of varying the timescale of the random potential. Note that whereas voltage probe models are essentially The effect of a fluctuating potential on current noise in characterizedbya singleenergyscaleΓϕ, herethe band- a mesoscopicinterferometer was also recently studied by widthandmagnitudeofJ(t=0)givetwodistinctscales. Marquardtet. al.36. Thesituationhereisquitedifferent, as the scattering has a marked energy dependence, and we are also interested in the back-action charge fluctua- 1. Average current tions. The time-dependent Hamiltonian for the system is: Definingthecurrentinleadαasthetimederivativeof D the particle number in lead α yields: H(t) = [ε +η(t)]d†d + dǫ ǫ·c†(ǫ)c (ǫ) + d α α α=L,RZ−D X (cid:2) (cid:3) Γ Γα dǫ d†c (ǫ)+c†(ǫ)d (33) Iα(t)/e=−Γαd†(t)d(t)+i 2πα dω eiωtc˜†α(ω)d(t)− h.c. α=L,Rr2π Z α α r Z (cid:2) (37) (cid:3) X (cid:2) (cid:3) Takingthe expectationofEq. (37)for the currentinthe Assuming that the conduction electron bandwidth D is right lead, we find: ∞ ∞ hI (t)i = −eΓ dω dt Γ f¯(ω) dt GA(t ,t;ω)GR(t,t ;ω)+2f (ω) ImGR(t,t ;ω) (38) α α 0 0 b b 0 α 0 Z Z−∞ " Z−∞ # where f¯ is defined in Eq. (29), and the additional Using this to simplify Eq. (38) and then averaging over argument ω in GR/A indicates that one should shift the random potential η yields: ε → ε − ω in Eq. (35). Given the simple form of d d GR(t,t′), there is an optical theorem which relates these hhI ii = e 2ΓRΓL dω(f (ω)−f (ω)) R η R L two contributions for arbitraryη(t): 2π Γ0 Z ∞ dτ Im hGR(τ,0;ω)i (40) dt dt GA(t ,t;ω)GR(t,t ;ω)= (39) η a b b a Z0 Z Z We have used the fact that the η-averaged value of GR −2 dt e−Γ0t0 dτ Im GR(t−t ,t−t −τ;ω) 0 0 0 is invariant under time-translation. Eq. (40) for hIi is Z Z 9 identical to the expression emerging from the dephasing the second describes a process where an electron leaves voltageprobe model (c.f. Eq. (30)), with the η-averaged thelevelattime0afteradwelltimeτ . Eq. (43)thusex- 2 Greenfunctionplayingthe roleofthe “dephased”Green presses the chargenoise as a sum over pairs of tunneling function G˜R in the latter model. On a heuristic level, events occurring at different times. Unlike the average Eq. (40) indicates that tunneling processes with differ- current (c.f. Eq. (38)), the charge noise is sensitive to ent dwell times τ on the level contribute to hIi; the ran- interference between tunnel events which have different dom phases picked up during these events will cause a exit times from the level. suppression of the current. It is useful to write the Green function factor in Eq. Averaging the η dependent parts of GR yields: (43) as: hGR(τ)iη = e−iεdτe−Γ0τ/2× (41) Im GR(t,t−τ1;ω¯) Im GR(0,−τ2;ω¯) = 1 ∞ sin(ωt/2)2 η exp − dω J(ω) D WD(t;τ ,τ )−WI(t;τE,τ ) (cid:18) π Z−∞ ω2 (cid:19) Re 1 2 2 1 2 (44) Notsurprisingly,thefactorinhGRiarisingfromtheaver- where aginghas an identicalformto whatis encounteredwhen studying the dephasing of a spin coupled to a random WD(t;τ ,τ ) = GR(t,t−τ )GA(0,−τ ) (45a) 1 2 1 2 potential or to a bosonic bath. D E WI(t;τ ,τ ) = GR(t,t−τ )GR(0,−τ ) (45b) 1 2 1 2 D E 2. Charge noise Similar to standard disorder-averaged calculations, we have both a “diffuson” (WD) and an “interference” Ingeneral,thenoiseinaquantityX inthefluctuating (WI) contribution. Averaging over the random poten- potential model will have two distinct sources: tial yields: Var X2 = hX2i−hXi2 + hXi2 −hhXii2 WD/C(t;τ1,τ2)=e−i(εd−ω¯)(τ1∓τ2)e−Γ0(τ1+τ2)/2 η η η 1 ∞ ∞ (cid:2) (cid:3) D(cid:2) (cid:3)E (cid:18)D E (42(cid:19)) exp − dt1 dt2χ∓(t1)χ∓(t2)J(t1−t2) (46) 2 The firsttermdescribesthe intrinsicnoiseforeachgiven (cid:20) Z−∞ Z−∞ (cid:21) realizationofη(t), whereastheseconddescribestheclas- where sical fluctuation of the average value of X from realiza- tion to realization. In what follows, we will focus on the χ (t) = χ (t)±χ (t) (47a) ± [t−τ1,t] [−τ2,0] first, more intrinsic effect; this corresponds to an exper- χ (t) = θ(t−t )−θ(t−t ) (47b) iment where the noise (i.e. variance) is calculated for [ti,tf] i f each realization of η(t), and is only then averaged over In what follows, it is useful to make the shift t → t+ different realizations. Note that the neglected classical (τ −τ )/2 in Eq. (43). Simplifying, and using the fact 1 2 contribution to the noise (i.e. the second term in Eq. that the η-averaged Green function is time-translation (42)) is positive definite; including it will only push the invariant, we then have: system further from the quantum limit. Noting that here, Q(t) ≡ d†(t)d(t), we find using Eq. WD(t;τ ,τ ) = C(t;τ ,τ ) hGR(τ )ihGA(τ )i(48a) 1 2 1 2 1 2 (34): WI(t;τ ,τ ) = hGR(τ1)ih(cid:2)GR(τ2)i (cid:3)(48b) ∞ 1 2 C(t;τ ,τ ) (cid:2) 1 2 (cid:3) S ≡ 2 dt [hQ(t)Q(0)i−hQ(t)ihQ(0)i] Q Z−∞ D Eη with 2e2 ∞ ∞ ∞ ∞ ∞ = dt dω¯ d(∆ω)ei∆ωt dτ dτ ∞ ∞ π2 Z−∞ Z−∞ Z−∞ Z0 1Z0 2C(t;τ1,τ2) = exp dt1 dt2 f¯(ω¯ +∆ω/2) 1−f¯(ω¯ −∆ω/2) e−i∆ω(τ1−τ2)/2 hZ−∞ Z−∞ Im GR(t,t−(cid:0)τ1;ω¯) Im GR(0,−(cid:1)τ2;ω¯) (43) χ[t−τ1+2τ2,t+τ1−2τ2](t1)χ[−τ2,0](t2)J(t1−t2) η 2 ∞ i D E = exp dωJ(ω) We have used the optical theorem of Eq. (39) to express "π Z−∞ S in terms of a product of two (as opposed to four) Q sin(ωτ /2)sin(ωτ /2) Green functions. Similar to Eq. (40) for the current, 1 2 × cosωt (49) Eq. (43) for SQ may be given a simple heuristic inter- ω2 # pretation. The first factor of GR in Eq. (43) yields the amplitude of an event where an electron leaves the level Eq. (46)indicatesthatthegeneraleffectoftherandom at time t after having spent a time τ on the level, while potential is to suppress the contribution of each pair of 1 10 tunneleventstoS . Thecorrelationbetweentherandom If, in addition, we have the reasonable result that there Q phases acquired by each of the two events plays a cen- arenophasecorrelationsinthelongtime-separationlimit tralrole,andisdescribedbythefactorC(t;τ ,τ ),where (i.e. C(t;τ ,τ ) →1 as t → ∞), the elastic contribution 1 2 1 2 τ ,τ are the dwell times of the two events, and t is the becomes: 1 2 difference in exit times. Eq. (49) gives a simple expres- sionforC(t;τ ,τ )intermsofthe spectraldensityofthe 1 2 random potential. The existence of phase correlations allows pairs of tunnel events to interfere constructively when contributing to the diffuson contribution; thus, as can be seen in Eqs. (48), phase correlations tend to en- hance the diffuson contribution relative to the interfer- ence contribution. 4e2 ∞ 2 In addition, the phase correlation factor C(t;τ1,τ2) is SQ elast = π dωf¯(ω)(1−f¯(ω)) Im GR(ω) the only t dependent factor remaining in the expression (cid:12) Z−∞ D E(51) for S after η averaging. Thus, it will completely deter- (cid:12) Q ThisexpressionfortheelasticcontributiontoS isiden- Q minethecontributionofinelasticprocessestothecharge tical to what is obtained for the intrinsic charge fluctua- noiseS (i.e. termswith∆ω 6=0inEq. (43)). Suchpro- Q tions [S ] in the dephasing voltage probe model (c.f. Q bare cessescorrespondtotheabsorptionoremissionofenergy Eq. (31)), if we associate the η-averaged Green func- by the random potential. Note the formal similarity be- tion with the dephased Green function G˜R in the latter tween the correlation kernel C(t;τ ,τ ) in Eq. (43), and 1 2 model. Recall that the in the voltage probe model, the thekernelappearingintheP(E)theorydescribingtheef- intrinsic contribution [S ] arises fromfluctuations of fectofenvironmentalnoiseonelectrontunneling37. Here, Q bare thetotalcurrentincidentonthelevel,andisindependent notsurprisingly,the probabilityofaninelastictransition offluctuationsoftheprobevoltage. Wethushavethatif dependsonthedwelltimesτ ,τ ofthetwotunnelevents. 1 2 therearenolong-timephasecorrelations,thepurelyelas- We can now straightforwardly identify the “pure de- tic effect of a random potential is captured (in form) by phasing” (i.e. elastic) contribution to S for an arbi- Q the intrinsicchargefluctuationsinthe dephasingvoltage trary J(ω) by keeping only the t-independent part of probe model. Note that in Eq. (51) both the interfer- C(t;τ ,τ ) when evaluating Eq. (43). This is equiva- 1 2 ence and diffuson terms contribute equally. The lack of lent to replacing C(t;τ ,τ ) by C(t → ∞;τ ,τ ) in Eq. 1 2 1 2 any phase correlations implies that there is no relative (43). We thus define: enhancement of WD over WI. 2e2 ∞ Finally, we may define a purely inelastic contribution S = dωf¯(ω)(1−f¯(ω)) dτ dτ (50) to S for an arbitrary J(ω) by simply subtracting off Q elast π 1 2 Q Z−∞ Z Z the elastic contribution defined in Eq. (50) from the full (cid:12)(cid:12) Re WD(t→∞;τ1,τ2)−WI(t→∞;τ1,τ2) expression for SQ. We find: (cid:2) (cid:3) 2e2 ∞ ∞ ∞ ∞ S | = dω¯ d(∆ω) dτ dτ f¯(ω¯ +∆ω/2) 1−f¯(ω¯ −∆ω/2) (52) Q inelast 1 2 π Z−∞ Z−∞ Z0 Z0 (cid:0) (cid:1) Re P (∆ω;τ ,τ ) GR(τ ;ω¯) GA(τ ;ω¯) −P (∆ω;τ ,τ ) GR(τ ;ω¯) GR(τ ;ω¯) D 1 2 1 2 I 1 2 1 2 " η η η η# D E D E D E D E where we have defined the real-valued functions: 3. Current noise P (∆ω;τ ,τ )= (53) D/I 1 2 Weagainfocusonthe“intrinsic”fluctuations(e.g. the ei∆ωt dt C(t;τ ,τ )±1−C(t→∞;τ ,τ )±1 first term in Eq. (42)) and ignore the additional contri- 1 2 1 2 2π Z h i bution to the current noise arising from variations of hIi in different realizations of the random potential η. Sim- Though they are not necessarily positive definite, the ilar to the case of the charge noise, the current noise S I functions P (∆ω;τ ,τ ) may be interpreted as the may be expressedin terms of products of GR and GA at D/I 1 2 quasi-probabilityofobtaininganinelasticcontributionof differenttimes. Also,adirectcalculationshowsthatcur- size ∆ω from (respectively) the diffuson or interference rent conservation holds for the fluctuations: the current contribution. noise is independent ofthe leadin whichit is calculated.