Theory of light emission from quantum noise in plasmonic contacts: above-threshold emission from higher-order electron-plasmon scattering
TTheory of light emission from quantum noise in plasmonic contacts: above-thresholdemission from higher-order electron-plasmon scattering
Kristen Kaasbjerg
1, 2, ∗ and Abraham Nitzan Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel School of Chemistry, The Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel (Dated: February 24, 2015)We develop a theoretical framework for the description of light emission from plasmonic contactsbased on the nonequilibrium Green function formalism. Our theory establishes a fundamentallink between the finite-frequency quantum noise and AC conductance of the contact and the lightemission. Calculating the quantum noise to higher orders in the electron-plasmon interaction, weidentify a plasmon-induced electron-electron interaction as the source of experimentally observedabove-threshold light emission from biased STM contacts. Our findings provide important insightinto the effect of interactions on the light emission from atomic-scale contacts.
PACS numbers: 73.63.-b, 72.70.+m, 73.20.Mf, 68.37.Ef
Introduction. —Atomic-size contacts [1], realized, e.g.,in scanning tunneling microscopy (STM), provide aunique platform to study fundamental quantum trans-port phenomena such as, e.g., conductance quantiza-tion [2], suppression of shot noise [3], and vibrationalinelastic effects on the conductance and shot noise [4, 5].Recently, light emitted from STM contacts due to inelas-tic electron scattering off localized plasmons was used asa probe of the shot noise at optical frequencies [6]. Apartfrom standard emission due to one-electron scatteringprocesses with photon energy (cid:126) ω < eV , the observationof above-threshold emission with (cid:126) ω > eV indicates thatinteraction effects on the noise are probed [6, 7]. Emissionfrom atomic-scale contacts is thus well-suited for study-ing the fundamental properties of high-frequency quan-tum noise.The role of quantum noise as excitation source of elec-tromagnetic fields is known from theoretical work onmesoscopic conductors [8–11]. The emission is related tothe positive frequency part, S ( ω > asymmetric quantum shot noise S ( ω ) = (cid:90) dt e − iωt (cid:104) δI (0) δI ( t ) (cid:105) , (1)where δI = I −(cid:104) I (cid:105) and I is the current operator, whereasthe absorption is connected to the negative frequencypart, S ( ω < Theory. —We consider a single radiative LSP modewith frequency ω pl (see Fig. 1(a)) represented by thequantized vector potential field A ( r ) = ξ pl ( r ) (cid:115) (cid:126) (cid:15) ω pl (cid:0) a † + a (cid:1) , (2)where Ω is a quantization volume and ξ pl is a mode vector j el γ rad γ eh pl ω Ve 2eV subtip −+ + + + + + + V − −−−− −−− (a) (b) B( )
A(r) ω FIG. 1. (Color online) (a) Plasmonic STM contact. Theinset illustrates the spectrum of the localized surface-plasmonpolarition (LSP) supported by the contact. (b) Schematicillustration of the one and two-electron scattering processesresponsible for the 1 e ( (cid:126) ω < eV ; dashed lines) and 2 e ( eV < (cid:126) ω < eV ; full lines) light emission. a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b describing the spatial distribution of the field [40]. Theinteraction between the LSP and the tunnel current isgiven by the standard coupling term H el-pl = (cid:90) d r j el ( r ) · A ( r ) , (3)where j el = j ∇ el + j A el is the electronic current density,and j ∇ el and j A el the paramagnetic and diamagnetic com-ponents, respectively.The excitation dynamics of the LSP and the emissionis encoded in the LSP GF D ( τ, τ (cid:48) ) = − i (cid:104) T c A ( τ ) A ( τ (cid:48) ) (cid:105) where A = a + a † and T c is the time-ordering operatoron the Keldysh contour. The retarded and lesser compo-nents of the GF are given by their respective Dyson andKeldysh equations, D r ( ω ) = d r ( ω ) + d r ( ω )Π r ( ω ) D r ( ω ) , (4)and D < ( ω ) = D r ( ω )Π < ( ω ) D a ( ω ) , (5)where d r ( ω ) = ω pl ( ω + i + ) − ω is the bare GF and Π =Π el + Π rad + Π eh is the irreducible self-energy whichaccounts for the interaction with the tunneling current(Π el ) as well as radiative decay (Π rad ) and decay intobulk electron-hole pair excitation (Π eh ). The latter twoare modeled with phenomenological damping parameters γ rad/eh with self-energies Π r rad/eh ( ω ) = − iγ rad/eh sgn( ω ) / >/< rad/eh ( ω ) = − iγ rad/eh | n B ( ∓ ω ) | , where n B is theBose-Einstein distribution function, and give rise to abroadened LSP, D r ( ω ) = ω pl ω − ω − iω pl γ , with width γ = γ rad + γ eh .The excitation, damping and broadening of the LSPdue to the el-pl interaction (3) are governed by the el-pl self-energy, Π el = Π ∇ + Π A . To lowest order in theinteraction, the two contributions are given by [41]Π ∇ ( τ, τ (cid:48) ) = − i (cid:104) T c δj ( τ ) δj ( τ (cid:48) ) (cid:105) , (6)and Π A ( τ, τ (cid:48) ) = δ ( τ − τ (cid:48) ) (cid:104) ρ ( τ ) (cid:105) , (7)respectively, where δj = j − (cid:104) j (cid:105) , j = (cid:115) (cid:126) (cid:15) ω pl (cid:90) d r ξ pl ( r ) · j ∇ el ( r ) (8)is the projection of the current operator onto the LSPmode vector, ρ = − e m e (cid:126) (cid:15) ω pl (cid:90) d r ξ pl ( r ) · ξ pl ( r )Ψ † ( r )Ψ( r ) (9)is the mode-weighted electron-density operator and (cid:104)·(cid:105) is the expectation value in the absence of the el-pl inter-action (3). The diamagnetic self-energy does not affect the LSP dynamics as it only depends on the static chargedensity (cid:104) ρ (cid:105) , and is here neglected.From a perturbative expansion of the LSP GF in theparamagnetic interaction, we find that the exact param-agnetic self-energy is given by [42]Π ∇ ( τ, τ (cid:48) ) = − iS irr ( τ, τ (cid:48) ) , (10)valid to all orders in the el-pl interaction. Here, S irr is the irreducible part of the current-current correlationfunction S ( τ, τ (cid:48) ) = (cid:104) T c δj ( τ ) δj ( τ (cid:48) ) (cid:105) [43] and the expec-tation value (cid:104)·(cid:105) is with respect to the Hamiltonian thatincludes the el-pl interaction (3).Focusing on the case where the el-pl coupling (8) isproportional to the current operator I , the lesser and re-tarded components of the correlation function S are givenby the quantum noise in Eq. (1) and the response func-tion K r ( t − t (cid:48) ) = − i Θ( t − t (cid:48) ) (cid:104) [ I ( t ) , I ( t (cid:48) )] (cid:105) , respectively.From the important formal result (10), it then followsthat the lesser Π ∇ ,< = − iS < irr and retarded Π ∇ ,r = − iS r irr self-energies are directly related to these quantities, thusconnecting the damping of the LSP to the dissipativereal part of the AC conductance G ( ω ) = i K r ( ω ) /ω of thecontact. These quantities are connected via [44] − ω Re G = 2Im K r ( ω ) = S ( ω ) − S ( − ω ) , (11)which may be regarded as a nonequilibrium fluctuation-dissipation relation. This demonstrates a fundamentalconnection between quantum noise and AC conductancepreviously discussed in Refs. 8 and 45.To connect the quantum noise to the light emission,we consider the radiative decay of the LSP into a reser-voir of far-field modes, H far-field = (cid:80) λ (cid:126) ω λ a † λ a λ , with theassociated exchange rate per unit frequency given byΓ rad ( ω ) = Π < rad ( ω ) D > ( ω ) − Π > rad ( ω ) D < ( ω ) . (12)Here, the two terms account for absorption and emission,respectively, in agreement with Eq. (1) [44]. The emittedlight is thus governed by D < ( ω ) = − B ( ω ) Π < ( ω )2ImΠ r ( ω ) T =0 ,ω> = − B ( ω ) Π < el ( ω )2ImΠ r ( ω ) , where B = − D r is the LSP spectralfunction (see Fig. 1(a)), showing that it resembles theLSP spectrum and is driven by Π < el . With the above, wehave established the link between plasmonic light emis-sion and the quantum noise and AC conductance. Generic model and results. —With the formal theoryestablished, we go on to study the light emission andfinite-frequency noise in a generic model for an atomic-scale STM contact consisting of a spin-degenerate elec-tronic state/conduction channel coupled to bulk leadreservoirs of the STM tip and substrate, H = ε (cid:88) σ d † σ d σ + (cid:88) α H α + H T + H el-pl + (cid:126) ω pl a † a. (13)Here, ε = 0 is the energy of the electronic leveland H α = (cid:80) k ε k,α c † kα c kα , α = { tip , sub } is the = + Π ∇ Π ∇ = − i × (cid:20) + + + . . . (cid:21) → FIG. 2. (Color online) (top) Dyson equation for the LSP GFwhere Π ∇ denotes the paramagnetic el-pl self-energy. (cen-ter) Perturbation expansion of the self-energy showing thediagrams up to 4’th order in the el-pl interaction. (bot-tom) The 6’th order diagrams responsible for the 2 e emis-sion can be obtained by replacing the broadened LSP GFwith its 2’nd order correction (a plasmon excited by elec-tron tunneling) in the 4’th order diagrams. Feynman dictio-nary: • : el-pl interaction; solid lines: electronic contact GF G ij ( τ, τ (cid:48) ) = − i (cid:104) T c c i ( τ ) c † j ( τ (cid:48) ) (cid:105) , i, j = tip , sub , d ; thin wig-gly lines: broadened LSP GF D ( τ, τ (cid:48) ) = − i (cid:104) T c A ( τ ) A ( τ (cid:48) ) (cid:105) . Hamiltonian of the reservoirs with chemical potentials µ tip/sub = ± V /
2. The coupling to the reservoirs, H T = (cid:80) α,k t αk ( c † kα d + h.c.), leads to tunnel broadenings Γ α =2 πρ α | t α | which define tunneling (Γ sub (cid:28) Γ tip ) and con-tact (Γ sub ∼ Γ tip ) regimes. For Γ = Γ tip + Γ sub (cid:29) eV, k B T , the DC conductance is given by G = G T where G = 2 e /h and T = 4Γ tip Γ sub / Γ is the transmissioncoefficient.The el-pl interaction takes the form H el-pl = (cid:80) α M α I α ( a † + a ), where I α = i (cid:80) k ( t α c † kα d − h.c.) isthe paramagnetic current operator at reservoir α and M α = e (cid:126) (cid:113) (cid:126) (cid:15) ω pl l α is a dimensionless coupling con-stant with l α a characteristic length scale for the inter-action [46]. To describe the experimental light emis-sion [6, 7], we take the el-pl coupling to be given by | M tip/sub | = M with M tip = − M sub , which implies thatthe LSP couples to the total current I tot = I tip − I sub through the contact [47]. The paramagnetic self-energycan now be written as a sum over lead-lead components,Π ∇ = (cid:80) αβ Π ∇ αβ , whereΠ ∇ αβ ( τ, τ (cid:48) ) = − iM α M β S irr αβ ( τ, τ (cid:48) ) , (14)with S αβ ( τ, τ (cid:48) ) = (cid:104) T c δI α ( τ ) δI β ( τ (cid:48) ) (cid:105) and δI α = I α − (cid:104) I α (cid:105) .With the above-mentioned assumption for the couplingconstant, we have Π ∇ ( τ, τ (cid:48) ) = − iM S irr ( τ, τ (cid:48) ) where S ( τ, τ (cid:48) ) = (cid:104) T c δI tot ( τ ) δI tot ( τ (cid:48) ) (cid:105) .In the following, we proceed with a perturbative cal-culation of the irreducible el-pl self-energy illustrated interms of Feynman diagrams in Fig. 2 (see Ref. [42] fordetails). We focus on the regime Γ (cid:29) (cid:126) ω pl , eV, k B T andtake k B T = 0, corresponding to the experimentally rele-vant situation k B T (cid:28) (cid:126) ω pl where the current is the onlyexcitation source for the LSP.Figure 3(a) shows the numerically calculated ( irre-ducible ) emission noise to different orders in the el-pl Frequency (eV) S < ( e V ) V =1.3 VV =1.5 VV =1.7 V n =0n =2n =4 -2 -1 Conductance (G ) -10 -8 -6 -4 -2 E m i ss i o n n o i s e
1e (n =0)1e (n =2)2e (n =4) ∼G∼G Transmission F a n o f a c t o r
1e (n =0)1e (n =2)2e (n =4) Conductance (G ) I n t e g r a t e d e y i e l d Exp.Num.T (1−T ) (c) (d)(a) (b) FIG. 3. (Color online) (a) Noise spectrum to different ordersin the el-pl interaction ( n = 0: solid, n = 2: dashed, and n = 4: dotted lines) for a contact with T ∼ .
2. The insetshows a zoom of the noise spectrum at ω (cid:38) eV . (b) and (c)Integrated 1 e and 2 e emission noise and Fano factors at V =1 . e emission yield vs conductance at V = 1 . e yield from Ref. [6]. In(c) and (d), the Fano factors and yields have been normalizedto unity at their maximum value (the 1 e Fano factors havebeen normalized with the n = 0 maximum). Parameters: ω pl = 1 . γ = 0 . M = 0 .
1, Γ tip = 10 eV. interaction (the order of the corresponding self-energyis n + 2) for a contact with T ∼ . n = 0) noninteracting quantum noise (full lines inFig. 3(a)) is given by the bare bubble diagram. In thelimit Γ (cid:29) eV, (cid:126) ω, k B T , we find, in agreement with previ-ous works [9, 48], that the noninteracting noise spectrumis given by [49] S < ( ω ) ≈ × π (cid:2) T (1 − T ) [ H ( ω + eV ) + H ( ω − eV )]+ 2 T H ( ω ) (cid:3) , (15)where H ( x ) = xn B ( x ). At k B T = 0, the emission partsimplifies to S < ( ω > ∼ T (1 − T )Θ( eV − ω )( eV − ω ),and is hence suppressed at perfect transmission and cutoff at ω = eV , i.e. in this order only emission with ω
3. Theintegrated Fano factors shown in Fig. 3(c) are in excel-lent agreement with our expectations, confirming the an-ticipated scaling of the 2 e emission noise. The ratio ofthe 1 e and 2 e emission noise thus scales with the cou-pling constant and transmission coefficient as S < e /S < e ∼ M T (1 − T ).Next, we discuss the emission spectrum shown in Fig. 4as a function of bias voltage and conductance. As ex-pected, the emission which resembles the LSP spectrum Frequency (eV) V o l t a g e ( V ) eV =ħω eV =2ħω -10 -8 -6 -4 -2 0.0 0.5 1.0 1.5 2.0 2.5 Frequency (eV) C o n d u c t a n c e ( G ) -10 -8 -6 -4 -2 (a) (b) FIG. 4. (Color online) (a) Emission spectrum vs applied volt-age for a contact with T ∼ .
2. (b) Emission spectrum vstransmission coefficient at V = 1 . rad ( ω ) ∝ − Im D < ( ω ) in units of γ rad on alogarithmic scale. Parameters: ω pl = 1 . γ = 0 . M = 0 .
1, Γ tip = 10 eV. has a dominant 1 e component which is driven by the non-interacting quantum noise, and a weaker 2 e componentdriven by the higher-order quantum noise. Due to thepredicted T dependence of the quantum noise, both the1 e and 2 e emission peak at G ∼ . G and are stronglyreduced at G ∼ G . This counter intuitive behavior atperfect transmission where the current is maximized andone naively would expect the same for the emission, is aunique fingerprint of the quantum noise origin.The tunneling-induced damping of the LSP associatedwith the dissipative part of the AC conductance gives riseto an additional spectral broadening γ el = − ∇ ,r . Inthe large Γ-limit and to lowest order in the el-pl inter-action, Re G ( ω ) = G T and γ el = 8 ω/πM T . Contraryto the emission noise, it does not vanish at T = 1 andis independent of the bias voltage. Due to the nondis-sipative part of the AC conductance (real part of theself-energy) the LSP resonance redshifts ( ∼ . e emis-sion yield, defined as emission per current, as a functionof conductance together with the experimental 2 e pho-ton yield from Ref. [6]. Compared to the Fano factor inFig. 3(c), the above-mentioned spectral changes result ina slight left shift of the curve for the yield. The agreementwith the experimental 2 e yield is very good, indicatingthat we have identified the mechanism responsible for 2 e emission. At G ∼ G , however, the experimental yieldsdo not show complete suppression (see Ref. [6]). This dis-crepancy can be due to experimental factors such as: (i)imperfect or additional transmission channels [6], and/or(ii) changes in the LSP mode and el-pl coupling as thetip-substrate distance is reduced [57]. Summary. —To summarize, we have presented a frame-work based on the Keldysh GF formalism for the de-scription of light emission from plasmonic contacts andestablished the connection between the quantum noiseand AC conductance of the contact and the light emis-sion. Studying a generic contact model, we have identi-fied a plasmon-induced el-el interaction associated withthe higher-order quantum noise as the mechanisms be-hind the experimentally observed above-threshold emis-sion [6, 7]. Our approach, which can be generalized tomore complex situations, paves the way for a better un-derstanding of the effect of interactions on light emissionand quantum noise in atomic-scale and molecular con-tacts.We would like to thank W. Belzig, T. Novotn´y,M. Galperin, N. A. Mortensen, J. Paaske and M. Brand-byge for fruitful discussions, R. Berndt for providing uswith the original data from Ref. 6 and M. H. Fischerfor comments on the manuscript. 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G.Borisov, “Robust subnanometric plasmon ruler by rescal-ing of the nonlocal optical response,” Phys. Rev. Lett. , 263901 (2013).[56] T. V. Teperik, P. Nordlander, J. Aizpurua, and A. G.Borisov, “Quantum effects and nonlocality in stronglycoupled plasmonic nanowire dimers,” Optics Express ,27306 (2013).[57] J. Aizpurua, G. Hoffmann, S. P. Apell, and R. Berndt,“Electromagnetic coupling on an atomic scale,” Phys.Rev. Lett. , 156803 (2002). upplemental material for “ Theory of light emission from quantum noise inplasmonic contacts: above-threshold emission from higher-order electron-plasmonscattering”
Kristen Kaasbjerg
1, 2 and Abraham Nitzan Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot, Israel 76100 School of Chemistry, The Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel (Dated: January 25, 2015)
I. LSP GREEN’S FUNCTION
The central object of interest for our description ofplasmonic light emission from biased STM contacts, isthe contour-ordered GF for the localized surface-plasmonpolariton (LSP) of the contact represented by the quan-tized vector potential A ( r ) = ξ pl ( r ) s ~ (cid:15) ω pl (cid:0) a † + a (cid:1) , (1)where Ω is a quantization volume and ξ pl is the modevector. The contour-ordered GF is defined by D ( τ, τ ) = − i h T c A ( τ ) A ( τ ) i , (2)where A = a + a † and T c is the time-ordering operatoron the Keldysh contour. In the presence of interactions,it obeys the Dyson equation D ( τ, τ ) = d ( τ, τ ) + Z dτ Z dτ × d ( τ, τ )Π( τ , τ ) D ( τ , τ ) , (3)where d is the bare GF and Π is the irreducible self-energy.
A. Electron-plasmon self-energy
In order to establish an exact expression for the el-plself-energy, we start from the perturbation expansion of the LSP GF in terms of the S -matrix on the Keldyshcontour, S c = T c exp (cid:20) − i Z c dτ V ( τ ) (cid:21) , (4)where V = Z d r j el ( r ) · A ( r ) (5)is the el-pl interaction accounting for the interaction withthe tunnel current, j el = j ∇ + j A , which is a sum of para-magnetic and diamagnetic components. As explainedin the main text, we here neglect the diamagnetic com-ponent and focus on the paramagnetic self-energy, Π ∇ ,which governs the excitation dynamics of the LSP.Introducing the current operator j = s ~ V (cid:15) ω pl Z d r ξ pl ( r ) · j ∇ ( r ) , (6)the S -matrix expansion of the LSP GF in the paramag-netic interaction can be written D ( τ, τ ) = − i h T c S c A ( τ ) A ( τ ) i = − i ∞ X n =0 ( − i ) n n ! Z dτ · · · Z dτ n h T c A ( τ ) V ( τ ) · · · V ( τ n ) A ( τ ) i con0 = − i ∞ X n =0 ( − i ) n (2 n )! Z dτ · · · Z dτ n h T c A ( τ ) A ( τ ) · · · A ( τ n ) A ( τ ) i h T c j ( τ ) · · · j ( τ n ) i , (7)where the sum is over connected diagrams and, in the last equality, we have replaced n → n (the expectation valueof an odd number of boson operators is zero, implying that only even orders contribute in the perturbation series).In order to identify the bosonic self-energy from the perturbation series, the expectation value of the time-orderedbosonic operators is rewritten as h T c A ( τ ) A ( τ ) · · · A ( τ n ) A ( τ ) i = 2 n (2 n − h T c A ( τ ) A ( τ ) i h T c A ( τ ) · · · A ( τ n − ) i h T c A ( τ n ) A ( τ ) i (8)The factor of 2 n (2 n −
1) on the right-hand side comes from number of ways two internal times can be paired upwith the two external times τ, τ in the Wick’s contraction of the bosonic expectation value. For each pairing τ, τ n and τ , τ n , the contractions of the remaining 2 n − D ( τ, τ ) = d ( τ, τ ) + Z dτ Z dτ d ( τ, τ )Π ∇ ( τ , τ ) d ( τ , τ ) (9)where the reducible self-energy is defined byΠ ∇ ( τ, τ ) = i ∞ X n =1 ( − i ) n (2 n )! 2 n (2 n − Z dτ · · · Z dτ n − h T c j ( τ ) j ( τ ) · · · j ( τ n − ) j ( τ ) i h T c A ( τ ) · · · A ( τ n − ) i = − i ∞ X n =0 ( − i ) n (2 n )! Z dτ · · · Z dτ n h T c j ( τ ) j ( τ ) · · · j ( τ n ) j ( τ ) i h T c A ( τ ) · · · A ( τ n ) i ≡ − iS ( τ, τ ) . (10)Here, we have identified the sum over connected dia-grams in the second line as the perturbation expansion ofthe correlation function S ( τ, τ ) = h T c δj ( τ ) δj ( τ ) i where δj = j − h j i . Writing out the correlation function wehave S ( τ, τ ) = h T c j ( τ ) j ( τ ) i − h j i . Here, the last termcancels the disconnected diagrams from the perturbationexpansion of the first term which do not appear in theexpansion for the LSP GF in Eq. (7).From the above, it follows trivially that the irreducible self-energy is given byΠ ∇ ( τ, τ ) ≡ − iS irr ( τ, τ ) , (11)where S irr is the irreducible part of the correlation func-tion S . B. Self-energy for the contact model
For the simple contact model considered in the maintext, the paramagnetic part of the interaction can bewritten on the form V = X α M α I α ( a † + a ) (12)where M α is the coupling constant at lead α and theparamagnetic current operator is defined by I α = i X k h t α c † αk d − h.c i , (13)where the sum is over states k in the lead. It is hereassumed that the tunnel coupling t α and the couplingconstant M α are independent on the state index k . Repeating the perturbation expansion for the plas-monic GF in the preceding section with the interac-tion in Eq. (12), we find that the paramagnetic self-energy can be written as a sum over lead components,Π ∇ = P αβ Π ∇ αβ , whereΠ ∇ αβ ( τ, τ ) = − iM α M β S irr αβ ( τ, τ ) , (14)and S αβ ( τ, τ ) = h T c δI α ( τ ) δI β ( τ ) i is the reducible cor-relation function with δI α = I α − h I α i .
1. Identities for the self-energy
From the hermitian property of the paramagneticcurrent operator I α , the following set of identities be-tween the different lead-lead components of the re-tarded/advanced and lesser/greater self-energies can bederived.For the retarded/advanced components we haveΠ rαβ ( ω ) = (cid:2) Π aβα ( ω ) (cid:3) ∗ = Π aβα ( − ω ) = (cid:2) Π rαβ ( − ω ) (cid:3) ∗ . (15)For the lesser/greater components,Π <αβ ( ω ) = Π >βα ( − ω ) . (16)In addition, the components of the lesser self-energy infrequency domain are related as[Π <αβ ( ω )] ∗ = − Π <βα ( ω ) . (17)Hence, the diagonals Π <αα are purely imaginary while theoff-diagonal elements, in general, have both a real and animaginary part. Note, however, that the real parts of theoff-diagonal components cancel each other, implying thatthe total lesser self-energy becomes purely imaginary. II. PERTURBATION SERIES FOR THESELF-ENERGY
The perturbation series and the rules for evaluating thecorresponding Feynman diagrams are most easily devel-oped by writing the el-pl interaction (12) on the generalform V = X ij M ij c † i c j ( a † + a ) (18)where i, j = α, d and α = L, R is a composite lead/stateindex, α = ( α, k ). With the interaction written on this form, the el-pl interaction (12) can be represented by thecoupling matrix M = i − t L M L − t R M R t L M L t R M R , (19)in the ( d, L, R ) basis.We can now write up the perturbation series for thelead-lead components of the self-energy and getΠ ∇ αβ ( τ, τ ) = − iM α M β ∞ X n =0 ( − i ) n n ! Z dτ · · · Z dτ n h T I α ( τ ) V ( τ ) · · · V ( τ n ) I β ( τ ) i con0 = − i X ij ∈{ α,d } X i j ∈{ β,d } M ij M i j ∞ X n =0 ( − i ) n (2 n )! Z dτ · · · Z dτ n X i j ··· i n j n M i j · · · M i n j n × h T c c † i ( τ ) c j ( τ ) c † i ( τ ) c j ( τ ) · · · c † i n ( τ n ) c j n ( τ n ) c † i ( τ ) c j ( τ ) i h T c A ( τ ) · · · A ( τ n ) i . (20)Here the first two sums over i, j and i , j are associatedwith the current operators appearing explicitly in thecorrelation function S αβ . A. The contact GF
Given the structure of the perturbation series in (20)above, the fundamental building block in the Feynmandiagrams for the self-energy is identified as the contactGreen’s function defined by G ij ( τ, τ ) = − i h T c c i ( τ ) c † j ( τ ) i , (21)where i, j = α, d .In the absence of interactions, i.e. only tunneling be-tween the leads and the level is included, the componentsof the contact GF are given by G dd ( τ, τ ) = g d ( τ, τ )+ Z dτ Z dτ g d ( τ, τ )Σ( τ , τ ) G dd ( τ , τ ) (22) G αd ( τ, τ ) = Z dτ t α g α ( τ, τ ) G dd ( τ , τ ) (23) G dα ( τ, τ ) = Z dτ G dd ( τ, τ ) t ∗ α g α ( τ , τ ) (24) G αβ ( τ, τ ) = δ αβ g α ( τ, τ )+ Z dτ Z dτ g α ( τ, τ ) t α G dd ( τ , τ ) t ∗ α g β ( τ , τ ) (25) where Σ = Σ L + Σ R , Σ α = P k | t α | g α is the self-energydue to the coupling to the leads here described withinthe wide-band limit, and g α/d is the bare lead/dot GFs. B. Feynman rules
The perturbation series for the self-energy in (20) canbe represented by the Feynman diagrams familiar fromperturbation expansions of other two-particle GFs in thepresence of an electron-boson interaction (see Fig. 2 ofthe main text) . Here, we give the Feynman rules thatapply to the present case: • At order 2 n , draw 2 n + 2 vertices. • Label each vertex with i, j indices and multiply by M ij . • Connect vertices with the components of the con-tact GFs (21) matching the in/out going vertex in-dices. • Connect internal vertices with bare boson GFs. • At each vertex, sum over the states of the involvedlead. The appearence of P k t α / P k t ∗ α at each ver-tex, implies that the component of the contact GFentering/exiting the vertex with a lead index canbe replaced according to: – G αβ ( τ, τ ) → δ αβ Σ α ( τ, τ )+ R dτ R dτ Σ α ( τ, τ ) G dd ( τ , τ )Σ β ( τ , τ ) – G αd ( τ, τ ) → R dτ Σ α ( τ, τ ) G dd ( τ , τ ) – G dα ( τ, τ ) → R dτ G dd ( τ, τ )Σ α ( τ , τ ) ,i.e., only the k -summed lead-dot/dot-lead/lead-lead GFs appear in the perturbation series. • Integrate over all internal contour times. • Factor of − • Factor of − i × i n +2 at order 2 n .The evaluation of the self-energy (retarded/advanced andlesser/greater) on the real-time axis can be accomplishedusing Keldysh GFs . III. GFS IN KELDYSH SPACE
In a perturbative calculation of the self-energy, it canbe advantageous to work with the GFs in Keldysh spacespanned by the forward ( − ) and backward (+) branchesof the Keldysh contour . The GFs are here expressed as2 × G = (cid:18) G −− G − + G + − G ++ (cid:19) , (26)with the components corresponding to the different com-binations for the positions of the two time arguments onthe contour.This formulation formally arises from the contour in-tegrals which can be recast to real-time integrals as R c dτ i → R − dt − i + R + dt + i = R dt − i − R dt + i . For an ex-pression on the contour like X ( τ, τ ) = Z c dτ dτ · · · dτ n × A ( τ, τ ) A ( τ , τ ) · · · A n ( τ n , τ ) (27)which contain n contour integrals, the different real-timecomponents in Keldysh space can hence be obtained as X σσ ( t, t ) = X σ σ ...σ n η η · · · η n Z dt dt · · · dt n × A σσ ( t, t ) A σ σ ( t , t ) · · · A σ n σ n ( t n , t ) , (28)where σ i = − / + is the contour branch index and η i =+ / − is the accompanying sign. The prefactor in front ofthe integrals hence keeps track of the overall sign aris-ing from the contour variables residing on the backwardbranch. The GF in Keldysh space obeys the Dyson equationˇ G = ˇ g + ˇ g ˇ σ ˇΣˇ σ ˇ G, (29)where ˇ σ i denotes the Pauli matrices, and can hence beobtained in the usual way asˇ G − = ˇ g − − ˇ σ ˇΣˇ σ , (30)where g is the bare GF and Σ is the self-energy account-ing for interactions and couplings to, e.g., external leads.The self-energy is defined asˇΣ = (cid:18) Σ −− Σ − + Σ + − Σ ++ (cid:19) , (31)and the matrix product with the Pauli matrices in (30)adds a minus sign to the off-diagonal elements in orderto account for the above-mentioned sign arising from thecontour integration in the Dyson equation.When the bare GF ˇ g is diagonal, the full GF is givenby ˇ G = 1 G × (cid:18) − ( g ++0 ) − + Σ ++ Σ − + Σ + − − ( g −− ) − + Σ −− (cid:19) , (32)where the denominator of the prefactor is given by G = Σ − + Σ + − − (cid:2) ( g −− ) − − Σ −− (cid:3) (cid:2) ( g ++0 ) − − Σ ++ (cid:3) (33)For cases where Σ −− = Σ ++ and g −− = − g ++0 (see,e.g., below), the denominator of the prefactor simplifiesto G = Σ r Σ a − ( g −− ) − ( g ++0 ) − .The above also holds for a bosonic GFs.In the subsections below, we give expressions for someof the Keldysh space GFs relevant for the present work. A. GF for electronic level coupled to leads
For a single electronic level the, the GF in the absenceof coupling to leads and interactions is given by ˇ g ( ε ) = (cid:18) g −− ( ε ) 00 g ++ ( ε ) (cid:19) . (34)where g −− ( ε ) = ε − ε + i sgn( ε ) and g ++ ( ε ) = − g −− ( ε ).The self-energy due to coupling to leads isˇΣ α ( ε ) = (cid:18) Λ α ( ε ) 00 − Λ α ( ε ) (cid:19) + i (cid:18) Γ α ( ε ) [ f α ( ε ) − / − Γ α ( ε ) f α ( ε )Γ α ( ε ) [1 − f α ( ε )] Γ α ( ε ) [ f α ( ε ) − / (cid:19) . (35)With this, we can calculate the GF G for the coupled level and findˇ G ( ε ) = 1 G ( ε ) × (cid:18) ε − ε − P α Λ α ( ε ) + i P α Γ α ( ε ) [ f α ( ε ) − / i P α Γ α ( ε ) f α ( ε ) − i P α Γ α ( ε ) [1 − f α ( ε )] − ε + ε + P α Λ α ( ε ) + i P α Γ α ( ε ) [ f α ( ε ) − / (cid:19) . (36)Note that the off diagonals are given by the usual lesser and greater GFs as expected. In the wide-band limit thedenominator of the prefactor reduces to G ( ε ) = ( ε − ε ) + (Γ / . B. GF for noninteracting lead
The Keldysh GF for noninteracting lead fermions in equilibrium is given by ˇ g k ( ε ) = P ε − ε k (cid:18) − (cid:19) + 2 πiδ ( ε − ε k ) (cid:18) f ( ε ) − / f ( ε ) − (1 − f ( ε )) f ( ε ) − / (cid:19) . (37)In the wideband limit, the real part is thrown away. C. Level-lead GF
The contour-ordered level-lead GF is given by G αd ( τ, τ ) = Z dτ g α ( τ, τ ) t α G ( τ , τ ) . (38)In Keldysh space it can be written on the formˇ G αd ( ε ) = t α ˇ g α ( ε )ˇ σ ˇ G ( ε ) . (39)When multiplied by t ∗ α and summed over k this can beexpressed in terms of the coupling self-energy as X k t ∗ α ˇ G αd ( ε ) = ˇΣ α ( ε )ˇ σ ˇ G ( ε ) . (40) D. GF for damped plasmon
The bare
GF for a plasmon with frequency ω is givenby ˇ d ( ω ) = (cid:18) d −− ( ω ) 00 d ++0 ( ω ) (cid:19) (41)where d −− ( ω ) = ω − ω + iδ − ω + ω − iδ = ω ω − ω + iδ and d ++0 ( ω ) = − d −− ( ω ).The self-energy due to damping mechanisms describedby a phenomenological damping rate γ is given byˇΠ damp ( ω ) = (cid:18) − iγ [ F ( ω ) + sgn( ω ) / iγF ( ω ) iγF ( − ω ) − iγ [ F ( ω ) + sgn( ω ) / (cid:19) (42)where F ( ω ) = | n B ( ω ) | and n B ( x ) = ( e βx − − is the Bose-Einstein distribution function ( β = 1 /k B T ). For thedamped plasmon GF, we then findˇ D ( ω ) = 2 ω D ( ω ) (cid:18) ω − ω − ω iγ [ F ( ω ) + sgn( ω ) / − ω iγF ( ω ) − ω iγF ( − ω ) − ω + ω − ωiγ [ F ( ω ) + sgn( ω ) / (cid:19) , (43)where D ( ω ) = ( ω − ω ) + 4 ω ( γ/ . IV. EXPRESSION FOR THENONINTERACTING QUANTUM NOISE
In the following, we derive an expression for the non-interacting finite-frequency noise for the single-channelcontact considered in the main text. We consider here the total noise S < = S Many-particle Physics (Springer, 2010), 3rded. J. Rammer and H. Smith, Rev. Mod. Phys. , 323 (1986). H. Haug and A.-P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors (Springer, Berlin, 1998). R. Aguado and L. P. Kouwenhoven, Phys. Rev. Lett. ,1986 (2000). Y. M. Blanter and M. B¨uttiker, Phys. Rep.336