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Tunable photonic cavity coupled to a voltage-biased double quantum dot system: Diagrammatic NEGF approach

We investigate gain in microwave photonic cavities coupled to voltage-biased double quantum dot systems with an arbitrary strong dot-lead coupling and with a Holstein-like light-matter interaction, by adapting the diagrammatic Keldysh nonequilibrium Green's function approach. We compute out-of-equilibrium properties of the cavity : its transmission, phase response, mean photon number, power spectrum, and spectral function. We show that by the careful engineering of these hybrid light-matter systems, one can achieve a significant amplification of the optical signal with the voltage-biased electronic system serving as a gain medium. We also study the steady state current across the device, identifying elastic and inelastic tunnelling processes which involve the cavity mode. Our results show how recent advances in quantum electronics can be exploited to build hybrid light-matter systems that behave as single-atom amplifiers and photon source devices. The diagrammatic Keldysh approach is primarily discussed for a cavity-coupled double quantum dot architecture, but it is generalizable to other hybrid light-matter systems.

Such quantum-dot circuit QED systems (QD-cQED) offer a rich platform for studying nonequilibrium open quantum systems at the interface of quantum optics and mesoscopic solidstate physics. Experiments are versatile, with a highly tunable window of parameters.
Recent breakthroughs in such devices include the observation of photon emission proceeding via the DC transport of electrons [16], and the realization of microwave lasers (masers) [24,25].
Despite ongoing theoretical advances in describing QD-cQED systems [2,17,[26][27][28], there is a compelling need for adapting well-established techniques of non-equilibrium and condensed matter physics, e.g, the diagrammatic non-equilibrium Green's function (NEGF) method, to explore the rich physics of these highly tunable and versatile hybrid light-matter systems. Existing literature typically treats electron-lead coupling in a perturbative manner (such as the Born-Markov approximation) [17,[29][30][31][32][33], further enforcing the source-drain voltage to be very high, thereby incorporating only sequential, uni-directional electron transfer across the dots. Using such approaches one potentially misses important features in the optical and electronic signals, the result of finite bias voltage and strong dot-lead couplings.
Moreover, approximate methods such as the Markovian-secular quantum master equation or mean field calculations are often uncontrolled and non-transparent. It is therefore of a great importance to introduce a systematic approach that allows for an arbitrary dotlead coupling, (especially since experiments allow tunability from weak to strong dot-lead coupling), handles finite source-drain bias voltage, and treats light-matter coupling in a systematic (even if perturbative) manner. The diagrammatic NEGF approach [34][35][36][37] is perfectly suited for this purpose. It allows us to simulate present cutting-edge experimental realizations of quantum-dot circuit-QED systems, and furthermore foresee new effects.
Our model includes two electronic levels corresponding to two quantum dots (DQD), each coupled to a primary microwave photon mode (cavity photons). This primary mode is coupled to left and right transmission lines, mimicking the openness of the cavity. A sourcedrain bias is applied across the DQD system, inducing DC electric current. This system can serve as a testbed for understanding the intricate interplay between light (cavity) and matter (voltage-biased DQD) degrees of freedom, specifically, in a non-equilibrium situation.
For a schematic representation, see Fig. 1.
The paper in organized as follows. In Sec. II we introduce our model. In Sec. III we study the optical properties of the cavity, namely, the mean photon number, power spectrum and the spectral function, transmission coefficient, and phase response. We use the Keldysh NEGF method while relying on the random-phase-approximation (RPA), which is crucial for respecting symmetry conditions. Numerical simulations demonstrate the potential to use the system and develop novel quantum devices such as microwave amplifiers, photon  (TLR). The coupling between the microwave photons of the TLR and the electronic levels of each quantum dot (QD), embedded in the nanowire, is described by a Holstein interaction of strength (g 1 , g 2 ). The DQD is driven out of equilibrium by the application of a finite sourcedrain bias ∆µ = µ L − µ R . Tunnelling rates between the dots and the electron leads (Γ L , Γ R ) and in between the dots (t) can be tuned via gate-controlled tunnel-barriers. Typical experimentally relevant values are given in the Table. (Right) A schematic representation of the model. The mesoscopic system is effectively housed in the microwave cavity. Cavity photons are coupled to the input and output ports with decay rates κ L(R) . The arrows inside the cavity represent tunnelling processes. Wavy lines indicate the light-matter coupling. sources and diodes/rectifiers. In Sec. IV we focus on the electronic part of the model, and demonstrate the influence of the cavity mode on the electric charge current. We summarize our work and provide an outlook in Sec. V.

II. MODEL
The double quantum dot setup is placed between two metal leads composed of noninteracting electrons. Electron transfer between the dots takes place via direct tunnelling.
Each dot is further coupled to a microwave cavity mode, designated as the "primary photon". This mode is coupled to two transmission lines, namely input and output ports. The total Hamiltonian is (we set = k B = e = 1 throughout the paper), where ǫ 1 , ǫ 2 are the site energies of the DQD, coupled to the left L and right R metal leads by real-valued hopping elements v l and v r , respectively.ĉ † andĉ are fermionic creation and annihilation operators for the respective dots.Ĥ ph is the Hamiltonian for the photonic degrees of freedom. It consists of the primary photon of frequency ω 0 and the secondary photon baths as two long transmission lines (K = L, R) with a symmetric coupling ν ĵ Here,â(â † ) andâ jK (â † jK ) are bosonic annihilation (creation) operators for the cavity mode and the two transmission lines.
The interaction between electrons in the junction and the primary optical mode is given byĤ withn α =ĉ † αĉ α as the level number operator and g α the coupling strength, α = 1, 2. For g 1 = −g 2 ,Ĥ el−ph describes the interaction between the microwave photon and the dipole moment of excess electrons in the DQD. Experimentally relevant parameters are given in the Table. κ = κ L = κ R is the decay rate of the cavity mode per port. In the wide-band limit we define it as κ = 2πρ|ν| 2 , where ρ is the bath density of states and ν being the average coupling between the cavity and the bath modes. In this section, we compute various experimentally-measurable properties of the cavity, namely its average photon occupation, emission (power) spectrum, spectral response function, as well as the transmission amplitude and phase response of the cavity-emitted microwave photons. It is important to mention that, while performing the phase spectroscopy (transmission amplitude and phase) we relate incoming bosonic modes of the left (L) transmission lines with the input microwave signal. While outgoing bosonic modes of the right (R) transmission lines construct the output signal [2]. For other types of photonic and electronic measurements the ports act as a source for dissipation.

A. Average photon number
In recent years, the mean photon number became an experimentally accessible quantity [21,25,38] for QD-cQED setups. We compute the mean photon number n c ≡ â †â in the cavity using the Keldysh NEGF technique. This method allows us to perform a perturbative expansion (second-order) in the electron-photon and cavity-photon bath coupling Hamiltonian, while capturing dot-lead interaction effects to all orders (non-perturbative).
We consider the contour-ordered photon Green's function, It hands over all components required to calculate various optical signals. Here, T c is the contour-ordered operator (see D 0 (τ, τ ′ ) is the Green's function of the primary photon which also includes the effect of the secondary photon modes (transmission lines). F el (τ 1 , τ 2 ) corresponds to the bubble diagrams involving the left and right dots' Green's function, see Fig. 3. It describes elastic and inelastic (energy exchange) processes, where electrons in the dots interact with the cavity mode. We will later identify the bubble diagrams, in other words, the photon selfenergy (connected part), as the density-density correlation function of electrons. In terms of the contour variables, this Green's function can be written as This function is symmetric under the exchange of the contour time parameters τ 1 and τ 2 .
G 0 ij (i, j = 1, 2) are the non-interacting electronic Green's functions (dressed by the arbitrarily strong electron-lead tunnelling Hamiltonian), defined as G 0 . The average is performed over the current-carrying steady state, determined by the inverse temperatures β L,R = 1/T L,R and chemical potentials µ L,R of the electronic leads. Components of the non-interacting electron Green's functions are given in Appendix A. In the third line of Eq. (7) we organize F el (τ 1 , τ 2 ) in a matrix form, with g and G 0 as 2 × 2 matrices with g = diag(g 1 , g 2 ). Expressions for different components of F el (τ 1 , τ 2 ) and various relations among them such as the Korringa-Shiba relation are explained in Appendix B. It is important to mention that, if the DQDs are further coupled to a phononic environment [24], G 0 (τ 1 , τ 2 ) in the bubble diagrams should be replaced by the interacting G(τ 1 , τ 2 ), dressed by the phononic interaction (assuming Wick's theorem).
In the steady state limit, different real-time components of D(τ, τ ′ ) can be obtained. The convolution in time domain results in a multiplicative form in the frequency domain. This where t,t, <, > are the time ordered, anti-time ordered, lesser and greater components of the Green's function. The primary photon retarded Green's function is given by with Σ r,a,<,> ph (ω) as different components of the self-energy, materializing due to the coupling of the cavity photon to the input and output ports. In the wide-band limit we approximate Σ r ph (ω) = −iκ (recall that κ = 2πρ|ν| 2 is the decay rate of the cavity mode per port). We also receive Σ < ph (ω) = −2 i κ n ph (ω) and Σ > ph (ω) = −2 i κ [1 + n ph (ω)] where n ph (ω) stands for the Bose-Einstein distribution function, evaluated at temperature T ph .
We now compute various components of the photon Green's function by inverting the 2 × 2 matrix in Eq. (8). We receive the greater and lesser components, as well as the retarded and advanced Green's functions Here, In what follows we show that F ′ , ′′ el play a central role in enhancing gain in the cavity mode. For later use we also define the total self-energy Π t,t,<,> (ω) which is additive in the electronic and transmission lines induced self-energies, i.e., With this at hand, we identify the mean photon number in the steady-state limit as The integration can be performed to include terms to the second order in the electron-phonon coupling (order g 2 i ) and in κ. We employ the residue theorem to perform the integration. Upto the second order the poles are located at The mean photon number is obtained as where we alternatively express it in terms of the total self-energy Π </> (ω 0 ). At equilibrium, the metallic leads are maintained at the same chemical potential (µ L = µ R ) and at the same temperature, equal to the temperature of the photonic environment (β L = β R = β ph = β).
The detailed balance condition is then satisfied for F This ensures the onset of the Bose-Einstein distribution for the cavity photon mode at equilibrium. From Eq. (15) we can further obtain the mean-square displacement of the cavity mode as In Fig. 4 we show the average cavity occupation number n c as a function of applied bias, energy detuning and dot-lead coupling. Unless otherwise stated we define here and below the detuning ǫ = ǫ 1 −ǫ 2 and enforce ǫ 1 = −ǫ 2 = ǫ/2, with matter-light coupling g = g 1 = −g 2 .
For the spectral functions of the metallic leads we make the wide-band approximation, and  Table. fix Γ L = Γ R = Γ. We also set the equilibrium Fermi energy of the metal leads at zero and change the bias symmetrically with µ L = −µ R = ∆µ/2. The temperature of the two metals and the transmission lines are chosen to be identical.
In Fig. 4(a) we study the average photon occupation as a function of bias voltage. We find that n c increases in two steps. At the first step, ∆µ > ω 0 , tunnelling electrons acquire sufficient energy to interact with the cavity mode and generate photons. The second step arises due to the additional resonance situation at ∆µ ∼ √ ǫ 2 + 4t 2 + 2 ω 0 . At this bias electrons arriving from the left metal at ∆µ/2 deposit energy (ω 0 ) to the cavity mode, allowing them to resonantly cross the junction. In the positive detuning case (ǫ > 0), examined here, n c saturates at lower values for reverse (negative) bias, in comparison to that in the forward (positive) bias. This cavity-number asymmetry with respect to bias reflects the structural asymmetry in the DQD system. With increasing tunnelling strength t, fast interdot charge transfer results in effectively weak interaction between electrons and the cavity mode, therefore showing low values for the average photon number.
In Fig. 4 The nonlinear and non-monotonic behavior of n c as a function of dot-lead coupling Γ is demonstrated in Fig. 4(d). At weak coupling and finite detuning, n c increases with Γ, as the photon number in the cavity is amplified by charge transfer through the DQD system.
In contrast, when the dot-leads coupling is strong, the renormalization and broadening of the dot energy levels allow electrons to tunnel through the DQD on a short timescale, only briefly interacting with the cavity photons, thus resulting in limited photon generation.
B. Power spectrum and spectral function for the cavity mode Next we look at the power spectrum and the spectral function of the cavity mode. The power spectrum has recently been measured for a similar setup [25]. In the stationary limit, we can immediately obtain the emission (power) spectrum in terms of the lesser Green's function It is obvious from this definition that n c = dω 2π S(ω). At the cavity frequency ω 0 , S(ω) is given by Similarly, we obtain an expression for the spectral response function of the resonator, defined as the difference between the retarded and advanced photon Green's function with the normalization condition (sum rule) dω 2π A ph (ω) = [â,â † ] = 1. This can be proved as well from the above equation invoking the residue theorem as explained before. In the absence of the light-matter interaction the cavity spectral function A cav (ω) = 2κ (ω−ω 0 ) 2 +κ 2 trivially satisfies the sum rule. The amplitude of the spectral function at ω 0 is related to the emission spectrum as A ph (ω 0 ) = S(ω 0 )/ n c . In Fig. 5(a)  The brodening, which is of the order of several MHz (1-3 MHz), results from the interplay between F ′′ el and κ, the two different sources of dissipation.

C. Phase spectroscopy: Transmission and Phase
We calculate the transmission amplitude and the phase response of the emitted microwave photons by following the input-output relations in Refs. 26, 27, and 42. Recall that for phase spectroscopy measurements (performed via heterodyne detection [16])the bosonic modes of the left (L) and the right (R) transmission lines are related with the input and output microwave signal, respectively [2]. The transmission function t(ω) reads as [26,27,42] where χ r el (ω) can be shown to be proportional to the electronic charge susceptibility as [26,27]. Here stands for the electron density response function, α, β = 1, 2 are indices for the dots, · · · el refers to the average over the electronic degrees of freedom in the nonequilibrium steady state. In our formulation, following Eq. (11) we identify the transmission to be proportional to the retarded Green's function D r (ω) of the cavity photon mode, which in the time-domain is precisely the photon response function, given by Therefore, we note that χ r el (ω) stands for the retarded component of the bubble diagram F r el (ω). Here · · · T represents average over the combined photonic-electronic steady state density operator.
We further write t(ω) = |t(ω)|e iφ(ω) and identify the real and imaginary parts of the transmission function, with the sum rule Since we are mainly interested in the absolute value of the transmission function and the value of the phase response at the frequency of the cavity mode (ω = ω 0 ), we evaluate Note that both the real and imaginary parts of F el (ω) show nontrivial dependence on bias voltage through the Keldysh component of the electronic Green's function G k 0 (ω), see Appendix B.
In Fig. 6 we plot the absolute value of the transmission and the phase response as a function of the incoming photon frequency ω under different bias voltages ∆µ. When the cavity is decoupled from the DQD, g = 0 (dashed-dotted line), the transmission reaches unity at ω = ω 0 , and the broadening is determined by κ. As well, the phase response is zero at the resonant frequency, and it approaches ±π/2 in the off-resonant regime. For finite g-yet at zero bias-charge fluctuations in the dots introduce shift in the transmission peak, further reducing the maximum amplitude. The frequency shift depends on the real part of the charge susceptibility F ′ el (ω), whereas the broadening reflects the difference between F ′′ el (ω) and κ. The phase response is zero when the transmission is at maximum. Most interestingly, we find that the absolute value of the transmission coefficient can be greatly enhanced-beyond unity-at finite bias, once F ′′ el > 0. This situation is elaborated below. Fig. 7 displays one of the central results of our work: The transmitted photon signal can be significantly enhanced at finite bias voltage, once the electronic system is fine-tuned to counteract dissipation from the (photonic) transmission lines. We study the behavior of the transmission coefficient and phase response at the bare cavity resonance frequency as a function of bias, for different cavity decay rates κ. We find that the transmission, or photon gain, increases for |∆µ| > ω 0 , and it saturates at high (positive and negative) biases. This behavior agrees with our observations for n c in Fig. 4(a). However, at a certain finite voltage (here around 30µeV ), the transmission jumps above the asymptotic value (panel a), while the phase response shows a sudden dip (dashed line in panel b). This sudden jump takes place precisely as the electronic part of the system acts to cancel out relaxation effects due to the tranmission lines, F ′′ el (ω 0 ) = κ, see panel (d). We turn to the high bias limit and study in Fig. 8 Table. amplitude. Plots of the real and imaginary components of F el , displayed as a function of ǫ, are included in Appendix B (Fig. 13). Maximum gain is achieved when F ′′ el = κ. Since F ′′ el ∝ g 2 , and typically F ′′ el < κ, increasing g shows a significant enhancement in gain and similarly loss.
The dependence of |t(ω 0 )| and φ(ω 0 ) on the system-lead coupling strength is examined in  Table. small Γ returns large values for F ′ , ′′ el , thus a significant enhancement in |t(ω 0 )| as a whole. Upon increasing the coupling strength Γ, the dots energies become broadened, thus electrons flow across the device without interacting with the cavity mode. This scenario shows small gain and loss. At very weak coupling, F ′′ el (ω 0 ) ≪ κ, the electronic medium introduces only dissipation, responsible for the sharp drop in transmission, see Fig. 5

(b)
Plots of |t| and φ, at the bare cavity frequency, as a function of ǫ 1 and ǫ 2 , reveal that degenerate quantum dots do not influence the cavity, with |t(ω 0 )| = 1 and φ(ω 0 ) = 0, see  Other parameters are the same as in the Table. - In Fig. 10 Fig. 8, where high bias was employed, we note here an additional dip at ǫ ∼ ∆µ, reflecting photon-assisted charge transfer processes from the right dot to the left dot. Correspondingly, a jump in phase is detected at the same value of detuning.

D. Special limits, scaling and universality
We discuss here scaling relations for measures related to the cavity, in different parameter regimes. We begin by considering the high bias and zero temperature limit. We further assume that the dot-lead coupling is strong, larger than detuning, Γ > ǫ. This situation is experimentally relevant and thereby potentially testable. We use Eq. (B1) along with the expressions for the non-interacting electronic Green's functions (see Appendix A). In this limit we found that withΓ = Γ/2. Further assuming that κ ≪ F </> el , the average photon number in Eq. (15) reduces to with an effective temperature T ef f = ∆µ/2. It is remarkable to note that this linear scaling, n ∝ ∆µ, is universal to the Holstein-like class of models [43][44][45][46]. In the opposite limit κ ≫ F </> el (though keeping Γ > ǫ), the electronic part effectively decouples from the cavity.
As a result, the cavity equilibrates with the secondary photon bath (ports), n c = n ph (ω 0 ), and the transmission amplitude goes to unity.
Another interesting limit is the large detuning ǫ ≫ Γ and high bias case, where we obtain In this case, F </> el (ω 0 ) ∝ 1/ǫ 6 , the electronic current is negligible, and electron-photon coupling is effectively small. Again we find that when κ ≪ F </> el , n c = 1 2 ∆µ ω 0 −1 , and in the opposite limit, κ ≫ F </> el , the cavity occupation number is thermal.

IV. ELECTRONIC PROPERTIES: STEADY STATE CHARGE CURRENT
We study the electronic properties of the DQD system, focusing on the steady state charge current at the left contact. It is given by the powerful Meir-Wingreen formula [47,48], valid for an arbitrary large light-matter interaction and dot-lead coupling Here G </> (ω) are the lesser and greater components of the electronic Green's function, fully dressed by the leads and the electron-photon interaction. As before, we obtain these components by relying on the Dyson equation, performing second order perturbation expansion in the electron-photon interaction [49,50]. We write with Σ ph (τ 1 , τ 2 ) as the nonlinear electronic self-energy arising due to the photonic degrees of freedom. Calculating it up to O(g 2 ) we receive the Hartree (H) and Fock (F) terms [51,52] (see Fig. 11), with Σ ph (τ 1 , τ 2 ) = Σ H ph (τ 1 , τ 2 ) + Σ F ph (τ 1 , τ 2 ). Following the Keldysh equation, we gather the lesser and greater components in the frequency domain as G </> (ω) = G r (ω)Σ of elastic and inelastic contributions i.e., I L = I el + I inel , with and Note that in the above equations the retarded and advanced components of the Green's functions G r,a are renormalized by the electron-photon interaction. We expand these functions following the Dyson equation, G r,a (ω) = G r,a 0 (ω) + G r,a 0 (ω)Σ r,a ph (ω)G r,a (ω), and organize the lowest order expression for the charge current in terms of the non-interacting G 0 . The elastic components of the current becomes I el = I (0) el + I (2) el , with and The inelastic component is given by In Fig. 12 Table).
By looking at different observables, we had demonstrated that the DQD electronic system can serve as a gain medium for the cavity photons when a finite source-drain bias was applied across the dots. This effect, key for realizing photon-source quantum devices, was elucidated by studying (analytically and numerically) the behavior of the mean photon number, power spectrum and the spectral function of microwave photons, as well as the transmission function and the phase response. Specifically, we had interrogated the behavior of the photon number by sweeping the bias voltage, dot-lead hybridization and detuning (bare energy difference between quantum dots), demonstrating a significant enhancement of n c upon its coupling to the voltage-biased DQD system.
The response of the cavity to signals has been investigated through its transmission coefficient and phase. We showed that the transmission could be greatly enhanced by tuning the dots' site-energies and the inter-dot tunneling energy, to satisfy a resonance condition with the frequency of the cavity. A highly nontrivial observation has been that the transmission coefficient reaches a maximum at a certain finite bias (Fig. 7), precisely once the imaginary part of the (electronic) charge susceptibility cancels out damping effects due to the (photonic) transmission lines, F ′′ el = κ. Thus, the electronic system can be manipulated to minimize dissipation effects in the cavity, to improve its coherent properties. Our method and conclusions are therefore a major step forward towards Quantum Hamiltonian and Bath Engineering of hydrid quantum systems in general and QD-cQED systems in particular. These findings are principal for the realization of microwave amplifiers and masers, and more generally, quantum devices based on electron-induced photon gain and emission.
Besides properties of the cavity, we examined the behavior of the charge current across the DQD system as a function of level detuning. Using the Meir-Wingreen formula to the lowest order in g, O(g 2 ), we identified photon-induced contributions to the current at both positive and negative detuning.
We emphasize that, for simplicity, in our treatment here we had assumed non-interacting electrons, by ignoring Coulombic repulsion energy between the dots' electrons. Once included, this interaction would further affect gain/loss in the cavity, yet general features are expected to survive. For example, the transmission coefficient should maintain its form [Eq. (20)] when electron-electron (e-e) interactions are accounted for; the electronic charge sus-ceptibility F r el will be modified, to shift features and adjust the amplitude of transmission. Our approach-valid for any value of Γ-should be contrasted with other common descriptions: In the Coulomb blockade regime (small Γ and an infinitely strong e-e repulsion) the DQD can be conveniently described in the charge-state basis, eliminating the double occupancy (one electron on each dot) state from the basis [16,17]. It would be further interesting to incorporate, within our NEGF formalism, the presense of nanowire and bulk phonons [53], influential in some quantum dot setups [17]. The existance of phonons can contain further interesting physics, for e.g, enhanced gain in signal [24], nonlinear functions such as diodes and transistors [54].
In future work we will apply the diagrammatic approach and study more compound systems, for e.g., understanding the masing behavior that has been recently observed in a double double-quantum dot system [25]. On a more general front, we plan to extend the present formalism to spin systems coupled to light degrees of freedom [55,56], further driven out of equilibrium by a source-drain voltage realized in experiments on hybrid quantum systems [25,57,58].

APPENDIX A: ELECTRON GREEN'S FUNCTIONS
In this Appendix we give expressions for different components of the non-interacting electronic Green's functions (dressed by the lead interaction) in the frequency domain. We where H S the Hamiltonian of the central DQD system and Σ r,a,</> (ω) are different components of the total self-energy which is additive in the metallic leads L and R, i.e., Σ r,a,</> (ω) = Σ r,a,</> L In writing the components Σ r,a L (ω) we ignore the real part, responsible for the renormalization of the DQD's energies. Similar expressions hold for the right lead self-energy, with Γ L → Γ R and f L (ω) → f R (ω).

KORRINGA-SHIBA RELATION
The gain in the transmission is determined by the real and imagniary components of F el (ω). Therefore understanding the relation between these components is important. Here we give compact expressions for F ′ , ′′ el (ω) at the lowest order of ω for zero bias (µ L = µ R = µ).
In the simpler setup of the Anderson impurity model, it has been shown that in this limit the F ′ el (ω) and F ′′ el (ω) components are not independent but rather related via the Korringa-Shiba relation [27,59,60]. An experimental test of this relation based on a microwave cavity setup has also been proposed in Ref. [61]. Below we discuss this realation for our DQD setup.
From Eq. (B3) in the zero temperature limit we obtain Tr g G r 0 (ω ′ ) g G r 0 (ω ′ ) − g G a 0 (ω ′ ) g G a 0 (ω ′ ) , Tr g A el (µ) g A el (µ) . (B4) where A el (ω) = i G r 0 (ω) − G r 0 (ω) is the spectral function for the DQD system. In the wide band limit (Γ(ω) = Γ), using the relation ∂G r,a 0 (ω) ∂ω = − G r,a 0 (ω) G r,a 0 (ω) (B5) one can further simplify the real and imaginary components of F el (ω). As an example, for g 1 = g 2 = g, g is proportional to an identity matrix. The above expressions then reduce to Note that the differential conductance dI/d(∆µ) is often noted to be proportional to the spectral function. This indicates that F ′ el and dI/d(∆µ) may show similar features, as confirmed in a recent experiment [14].