Conductance and Kondo Interference beyond Proportional Coupling

TL;DR

This paper derives a conductance formula for interacting nanostructures with complex geometries, demonstrating Kondo effects and conductance modulations in a quantum dot coupled to electronic cavity modes, beyond traditional proportional coupling assumptions.

Contribution

It introduces a conductance expression applicable to systems with non-proportional coupling geometries, expanding understanding of Kondo physics in complex nanostructures.

Findings

Kondo effect persists across a wide range of coupling strengths.

Conductance exhibits strong modulations and asymmetries.

Robust Kondo singlet correlations are maintained despite modulations.

Abstract

The transport properties of nanostructured systems are deeply affected by the geometry of the effective connections to metallic leads. In this work we derive a conductance expression for interacting systems whose connectivity geometries do not meet the Meir-Wingreen proportional coupling condition. As an interesting application, we consider a quantum dot connected coherently to tunable electronic cavity modes. The structure is shown to exhibit a well-defined Kondo effect over a wide range of coupling strengths between the two subsystems. In agreement with recent experimental results, the calculated conductance curves exhibit strong modulations and asymmetric behavior as different cavity modes are swept through the Fermi level. These conductance modulations occur, however, while maintaining robust Kondo singlet correlations of the dot with the electronic reservoir, a direct consequence of the lopsided nature of the device.

Equations

$$G=\frac{2e^2}{\hslash }\frac{{\tilde{\Gamma }}_L({\epsilon }_F){\tilde{\Gamma }}_R({\epsilon }_F)}{{\tilde{\Gamma }}_L({\epsilon }_F)+{\tilde{\Gamma }}_R({\epsilon }_F)}\int d\omega \left(-\frac{\partial f_0}{\partial \omega }\right)A_d(\omega ),$$

$$J_{L(R)}=\frac{ie}{h}\int d\omega$$

$$\displaystyle\left.+f_{L(R)}(\omega)\left[\mathbf{G}^{r}(\omega)-\mathbf{G}^{a}(\omega)\right]\Big{\}}\right)\,,$$

$${\mathbf{G}}^{<}(\omega )\approx {\mathbf{G}}_{\mathrm{eq}}^{<}(\omega )-\frac{\partial f_0}{\partial \omega }\mathbf{\Pi }(\omega )\Delta \mu +O(\Delta {\mu }^2),$$

$$H_{\mathrm{dot}}$$

$$H_{\mathrm{cavity}}$$

$$H_{\mathrm{leads}}$$

$$H_{\mathrm{coupling}}=$$

$$+\Omega \sum _{j,\sigma }{c}_{d\sigma }^{†}{a}_{j\sigma }^{}+H.c..$$

$${\Sigma }_d^{(0)}(\omega )=-\frac{i}{2}\left({\Gamma }_{dL}+{\Gamma }_{dR}\right)+{\left(\Omega -\frac{i}{2}\sqrt{{\Gamma }_{dR}{\Gamma }_{cR}}\right)}^2\tilde{S}(\omega ),$$

$$\begin{gathered} {\tilde{\Gamma }}_R(\omega )={\Gamma }_{dR}+{\Gamma }_{cR}{\left|\tilde{S}(\omega )\right|}^2\left({\Omega }^2+\frac{{\Gamma }_{cR}{\Gamma }_{dR}}{4}\right)+ \\ \sqrt{{\Gamma }_{dR}{\Gamma }_{cR}}\tilde{S}(\omega )\left(\Omega -\frac{i}{2}\sqrt{{\Gamma }_{cR}{\Gamma }_{dR}}\right)+\text{H.c.}. \end{gathered}$$

$${\psi }_{n,j}(r,\theta )$$

$${\epsilon }_{n,j}$$

$$z_{n,j}=\frac{3\pi }{4}+\frac{\pi }{2}(n+2j)=\frac{3\pi }{4}+\frac{\pi }{2}l\equiv z_l.$$

$${\epsilon }_l=\frac{{\hslash }^2}{2m{\ell }^2}{\left(\frac{3\pi }{4}+\frac{\pi }{2}l\right)}^2.$$

$${\delta }_{cav}={\epsilon }_{l+2}-{\epsilon }_l=\frac{{\hslash }^2}{2m{\ell }^2}\frac{{\pi }^2}{2}(5+2l).$$

$$G_{A,B}(\omega )\equiv ⟨⟨A;B⟩{⟩}_{\omega },$$

$$\omega ⟨⟨A;B⟩⟩=$$

$$=$$

$$(\omega -{\epsilon }_i)G_{id}(\omega )=$$

$$(\omega -{\epsilon }_{\alpha \mathbf{k}})G_{\alpha \mathbf{k},d}(\omega )=$$

$$(\omega -{\epsilon }_i)G_{i,j}(\omega )=$$

$$(\omega -{\epsilon }_{\alpha \mathbf{k}})G_{\alpha \mathbf{k},j}(\omega )=$$

$$G_d^{(0),r}(\omega )=\frac{1}{\omega -{\epsilon }_d-{\Sigma }_d^{(0),r}(\omega )},$$

$$J_{\alpha }=-e⟨\frac{d}{dt}{N}_{\alpha }⟩$$

$$J_{R}(t)=-e\sum_{{\bf k}\sigma}\Big{[}\sum_{i}V_{iR}G^{<}_{R{\bf k}\sigma,i\sigma}(t,t)+V_{dR}G^{<}_{R{\bf k}\sigma,d\sigma}(t,t)-{\rm H.c.}\Big{]},$$

$$G_{R\mathbf{k}\sigma ,i\sigma }^{<}(t,t^{\prime })=$$

$$G_{R\mathbf{k}\sigma ,d\sigma }^{<}(t,t^{\prime })=$$

$$J_{R}=2e\,{\rm Re}\Bigg{\{}\sum_{{\bf k}\sigma}\int\!\frac{d\omega}{2\pi\hbar}\Bigg{[}\sum_{i}V_{iR}G^{<}_{R{\bf k}\sigma,i\sigma}(\omega)+\\ V_{dR}G^{<}_{R{\bf k}\sigma,d\sigma}(\omega)\Bigg{]}\Bigg{\}}.$$

$$J_L=2e\mathrm{Re}\left\{\sum _{\mathbf{k}\sigma }\int \frac{d\omega }{2\pi \hslash }\left[V_{dL}{G}_{L\mathbf{k}\sigma ,d\sigma }^{<}(\omega )\right]\right\}.$$

$$G_{R\mathbf{k},d}(\omega )=\frac{V_{dR}^{\ast }}{\omega -{\epsilon }_{R\mathbf{k}}}{G}_{dd}(\omega )+\frac{1}{\omega -{\epsilon }_{R\mathbf{k}}}\sum _i{V}_{iR}^{\ast }{G}_{id}(\omega )=g_{R\mathbf{k}}(\omega )V_{dR}^{\ast }{G}_{dd}(\omega )+g_{R\mathbf{k}}(\omega )\sum _i{V}_{iR}^{\ast }{G}_{id}(\omega )$$

$$G_{R\mathbf{k},j}(\omega )=\frac{V_{dR}^{\ast }}{\omega -{\epsilon }_{R\mathbf{k}}}{G}_{dj}(\omega )+\frac{1}{\omega -{\epsilon }_{R\mathbf{k}}}\sum _i{V}_{iR}^{\ast }{G}_{ij}(\omega )=g_{R\mathbf{k}}(\omega )V_{dR}^{\ast }{G}_{dj}(\omega )+g_{R\mathbf{k}}(\omega )\sum _i{V}_{iR}^{\ast }{G}_{ij}(\omega ),$$

$$(\omega -{\epsilon }_j)G_{jd}(\omega )=V_{jd}^{\ast }{G}_{dd}(\omega )+\sum _{\mathbf{k}}\frac{V_{jR}{V}_{dR}^{\ast }}{\omega -{\epsilon }_{R\mathbf{k}}}{G}_{dd}(\omega )+\sum _{\mathbf{k},i}{V}_{jR}\frac{1}{\omega -{\epsilon }_{R\mathbf{k}}}{V}_{iR}^{\ast }{G}_{id}(\omega ).$$

$$\begin{gathered} \sum _i\left[(\omega -{\epsilon }_i){\delta }_{ij}-\sum _{\mathbf{k}}{V}_{jR}\frac{1}{\omega -{\epsilon }_{R\mathbf{k}}}{V}_{iR}^{\ast }\right]G_{id}(\omega )= \\ \left[V_{jd}+\sum _{\mathbf{k}}{V}_{jR}\frac{1}{\omega -{\epsilon }_{R\mathbf{k}}}{V}_{dR}^{\ast }\right]G_{dd}(\omega ) \end{gathered}$$

$$(\omega -{\epsilon }_i)G_{ij}(\omega )={\delta }_{ij}+\sum _{\mathbf{k}}{V}_{iR}\frac{1}{\omega -{\epsilon }_{R\mathbf{k}}}{V}_{dR}^{\ast }{G}_{dj}(\omega )+\sum _{\mathbf{k},l}{V}_{iR}\frac{1}{\omega -{\epsilon }_{R\mathbf{k}}}{V}_{lR}^{\ast }{G}_{lj}(\omega )+V_{id}^{\ast }{G}_{dj}(\omega ).$$

Full text

The quantum coupling of spatially localized discrete levels to cavity modes has emerged as a key tool for quantum information processing in different contexts, from cavity systems in atoms Raimond et al. (2001) and semiconductor quantum dots Hennessy et al. (2007) to exciton-polariton condensates in optical systems Kasprzak et al. (2006). Similarly, coherent coupling of electronic modes to discrete quantum systems has been explored in quantum corrals created on metallic surfaces Heller et al. (1994), allowing the manipulation and control of quantum information over regions a few nanometers across Manoharan et al. (2000). Recent experiments have extended this fascinating line of inquiry to systems implemented on two-dimensional electronic structures in semiconductors Rössler et al. (2015); Brun et al. (2016). These new systems have paved the way for quantum engineering in integrated, scalable nanoscale systems with great flexibility on geometries and interesting physical behavior.

The control of quantum dot (QD) characteristics in these systems, such as the tunnel coupling to external current leads, have also allowed the experimental study of the Kondo regime, an emblematic many-body effect Goldhaber-Gordon et al. (1998); Cronenwett et al. (1998). In this regime, the net magnetic moment of an unpaired spin in the QD becomes effectively screened by the conduction electrons in the leads, forming a delocalized quantum singlet that involves correlations with the electronic spins in the lead reservoirs Hewson (1997). Moreover, the coupling of a QD to reservoirs with non-trivial energy dependence gives rise to a variety of interesting effects on the ensuing Kondo state, including the appearance of zero-field splittings of the Kondo resonance Thimm et al. (1999); Dias da Silva et al. (2006, 2008). As QD systems are designed to interact with increasingly complex structures, one is led to ask how such many-body correlations would evolve.

The standard theoretical tool for the description of the two-terminal conductance through interacting regions is the Meir-Wingreen (MW) generalization of the Landauer formula for correlated systems Meir and Wingreen (1992). The MW expression is particularly useful in cases where the coupling matrix elements between the leads and the system are related to each other by a multiplicative factor. This condition was later dubbed “proportional coupling” (PC) Meir et al. (1993) and it is essential in writing the conductance in terms of the system’s retarded Green’s function. In many cases, however, the PC description is inadequate Komijani et al. (2013) and the evaluation of the conductance requires an alternative treatment.

A remarkable example of a nanoscale device with non-PC geometry was recently investigated in Ref. Rössler et al. (2015). They demonstrated coherent coupling between a QD in the Coulomb blockade regime and a larger, cavity-like region inscribed electrostatically onto the same two-dimensional electron gas (2DEG). The QD is coupled to two metallic leads while the cavity itself is coupled to only one of them, clearly breaking the PC condition. The size of the cavity and its coupling to the QD can be controlled by gate voltages on the device, allowing for fine control over the spacing between cavity resonances, the tunnel rate of electrons between cavity and QD, and the dot-cavity coupling over a wide range, while studying the conductance of the entire structure.

In this paper we extend the applicability of the MW expression to a large class of non-PC cases, providing theoretical tools to analyze the transport properties and temperature dependence of systems with a single interacting level (such as a QD) embedded in complex structures, as some studied recently Rössler et al. (2015); Brun et al. (2016). We find it is possible to write the linear conductance of such systems as

[TABLE]

where $f_{0}$ is the equilibrium Fermi function, the couplings ${\tilde{\Gamma}}_{L,R}(\varepsilon_{F})$ are effective hybridization functions to left ($L$) and right ($R$) leads, $A_{d}(\omega)=(-1/\pi)\mbox{Im}\,G_{d}^{r}(\omega)$ the spectral function, and $G^{r}_{d}$ is the retarded Green’s function at the QD. The latter two functions can be accurately calculated through a variety of techniques, such as Wilson’s numerical renormalization group (NRG) Bulla et al. (2008).

Although deceptively similar to the MW conductance formula for a single-level QD Meir et al. (1993), this expression incorporates the connection of the entire complex system to each lead through the effective hybridization functions ${\tilde{\Gamma}}_{L,R}(\varepsilon_{F})$. A crucial difference is that, in the original formula Meir and Wingreen (1992), the hybridization is represented by matrices of functions $\mathbf{\Gamma}^{L,R}$ involving the couplings and the density of states in the leads. Here, such complexities are encoded in the intricate energy structure of ${\tilde{\Gamma}}_{L,R}(\omega)$. As we will see below, these functions can be obtained after careful consideration of the effective connectivity of the system.

Next, we use this approach to successfully describe and provide further insight on conductance measurements of a QD coupled to a cavity Rössler et al. (2015). We implement a realistic model of the curved electrostatic reflector used to define the cavity in experiments, utilizing both analytical and numerical approaches. We further calculate the QD spectral density required by Eq. (1) by applying NRG to an effective Anderson model that incorporates the cavity. Our results show contrasting transport properties in the weak- and strong-coupling regimes, in excellent agreement with experiments. As the coupling to the cavity sets in, the conductance is strongly modulated, especially as different cavity resonances are swept through the Fermi level in the leads by applied gates Rössler et al. (2015). Moreover, the NRG calculations allow us to relate the conductance behavior to other intrinsic characteristics, such as the Kondo temperature $T_{K}$. We find that even as the conductance peaks are strongly distorted due to the interaction with the cavity modes, the Kondo screening remains robust, with larger $T_{K}$ values for stronger cavity coupling.

MW formula beyond proportional coupling.

Proportional coupled systems are those in which the coupling matrices of the interacting system to $L$ and $R$ leads are proportional to each other, namely, $\mathbf{\Gamma}^{R}(\omega)=\lambda\mathbf{\Gamma}^{L}(\omega)$ where $\lambda$ is a constant factor Meir and Wingreen (1992). This condition is clearly violated in the case of a QD connected to a cavity on only one lead, such as in Fig. S1. An electron in the dot is transmitted from $L$ by a direct tunneling process regulated by the coupling matrix element $V_{dL}$ and the density of states in that lead. In contrast, the transmission to the right involves the coherent interference between multiple paths that include the cavity resonances and states in $R$. Figure 1(b) indicates the different dot-lead ($V_{dR}$), and cavity-lead ($V_{cR}$) couplings that enter as non-zero elements in $\mathbf{\Gamma}^{R}$, while the cavity-lead couplings are zero in $\mathbf{\Gamma}^{L}$, thereby making the system evidently non-proportional Sup .

The main technical difficulty in obtaining a transport formula is the calculation of the lesser Green’s functions matrix $\mathbf{G}^{<}$ for the interacting region, which appears in the general expression for the current Meir and Wingreen (1992). The latter gives the current through the $L$ ($R$) lead as

[TABLE]

where $\mathbf{G}^{r(a)}$ is the retarded (advanced) Green’s function matrix Sup and $f_{L(R)}$ is the Fermi distribution at the $L(R)$ lead with chemical potential $\mu_{L(R)}$. Proportional coupling and current conservation make possible to simplify the calculation by ingeniously writing $J_{L(R)}$ in terms of $\mathbf{G}^{r(a)}(\omega)$. In contrast, for interacting non-PC systems away from equilibrium, the elimination of $\mathbf{G}^{<}$ is in general not possible. However, in the linear response regime it can be achieved by recalling that Komijani et al. (2013)

[TABLE]

where $\Delta\mu=\mu_{L}-\mu_{R}$ and $\mathbf{\Pi}(\omega)$ has a slow $\omega$ dependence within energy windows of $k_{B}T$ corresponding to the experiments of interest. These conditions eventually lead to Eq. (1); the detailed derivation is provided in the supplement Sup . Notice that the structure of the system may result in a cumbersome derivation of the ${\tilde{\Gamma}}_{L,R}(\omega)$ entering Eq. (1). We now specify the QD-cavity model that exemplifies this treatment.

Resonant cavity modes.

The key experimental element is a “mirror” that focuses resonant modes onto the QD, both elements electrostatically defined on a 2DEG. The cavity has a length \ell\sim 2$$\mum and angular aperture $\theta_{C}\sim 45^{\circ}$, as indicated in Fig. S1(a). Assuming circular symmetry, the normal modes are given by Bessel functions, $\psi_{n,j}(r,\theta)\!\simeq\!J_{n}(k_{n,j}r)\sin(n\theta)$. The dot-cavity coupling is maximal for modes with largest amplitude in the vicinity of $r\approx 0$, and dominated by resonances with $n=1$, given that $J_{n}(kr)\!\propto\!(kr)^{n}$ for $kr\ll 1$. These modes have a characteristic energy spacing $\delta_{cav}\approx 200~{}\mu$eV for a cavity with these dimensions, in agreement with the resonance separations in the experiment Rössler et al. (2015) and confirmed by Kwant calculations Sup ; Groth et al. (2014).

It is remarkable that although the cavity is immersed in the $R$-lead, it can be tuned to produce sharply peaked resonances that strongly modify $\tilde{\Gamma}_{R}(\omega)$, providing different electronic paths for the current. In the experiment, a gate voltage shifts the cavity resonance levels and the coupling to the QD. This tunability can be incorporated in the interacting QD model as follows.

Interacting quantum impurity model.

The Hamiltonian for this system can be written as $H=H_{\rm dot}+H_{\rm cavity}+H_{\rm leads}+H_{\rm coupling}$, where

[TABLE]

Here $c^{\dagger}_{d\sigma}$, $a^{\dagger}_{j\sigma}$, and $c^{\dagger}_{\alpha\mathbf{k}\sigma}$ create a spin-$\sigma$ electron in the dot, the $j$th mode of the cavity, and each of the leads $\alpha=L,R$. The resonances are assumed equally spaced, $\varepsilon_{j}=\epsilon_{c}+(j-1)\delta_{cav}$, where $\epsilon_{c}$ is shifted by a gate voltage; leads have a flat density of states $\rho(\omega)=\rho_{0}\Theta(D-|\omega|)$, symmetric about the Fermi energy ($\omega=0$). For simplicity all couplings are assumed local, real and independent of either momentum in the leads or cavity-mode index $j$. The coupling Hamiltonian is then, see Fig.  S1(b),

[TABLE]

QD effective decay widths.

As the Coulomb interactions are localized in the QD, one can find its effective couplings to $L$ and $R$ leads and the cavity, by calculating the dot retarded Green’s function for the system with $U\!=\!0$, $G^{(0),r}_{d}(\omega^{+})\!\equiv\!\langle\!\langle c^{\phantom{\dagger}}_{d\sigma};c^{\dagger}_{d\sigma}\rangle\!\rangle_{\omega}$. In the wide-band limit for the leads, $\sum_{\mathbf{k}}(\omega^{+}-\varepsilon_{\mathbf{k}})^{-1}\rightarrow-i\pi\rho_{0}$, we obtain $G^{(0),r}_{d}(\omega)=({\omega-\varepsilon_{d}-\Sigma^{(0)}_{d}(\omega)})^{-1}$, where

[TABLE]

is the non-interacting self-energy. Here, $\Gamma_{(c,d)\alpha}\equiv 2\pi\rho_{0}|V_{(c,d)\alpha}|^{2}$, for $\alpha=L,R$, with the cavity structure contained in $S(\omega)\equiv\sum_{j}({\omega-\varepsilon_{j}})^{-1}$ and $\tilde{S}(\omega)=S(\omega)\left(1+iS(\omega)\Gamma_{cR}/2\right)^{-1}$. The hybridization function of the (non-interacting) dot with the effective fermionic system is given by $\Delta(\omega)=-\mbox{Im}\Sigma^{(0)}_{d}(\omega)$. This approach can be extended to the interacting Green’s function Dias da Silva et al. (2006, 2008), as long as the interactions are restricted to the QD.

The interference of cavity modes and states in the leads is contained in the structure of $\Delta(\omega)$, which yields a highly structured density of states of the “effective” Fermi reservoir in which the QD is embedded Sup . Most importantly, the structure in $\Delta(\omega)$ affects strongly the Kondo state in the system once interactions set in. $\Delta(\omega)$ reliably describes the experimental system once cavity parameters are extracted either from a microscopic model, and/or determined from experiments 111 The structure studied in Ref. Rössler et al. (2015) has charging energy $U\!=\!700\,\mu$eV; dot-source (left) and dot-drain (right) couplings $\Gamma_{dL}\!\approx\!\Gamma_{dR}\!\approx\!87\,\mu$eV; cavity broadening $\Gamma_{cR}\!\approx\!40\,\mu$eV; and cavity mode spacing $\delta_{cav}\!\approx\!\,220\mu$eV. Using $U\!=\!0.5D\approx\!700\,\mu$eV, the NRG scales are then set as $\Gamma_{dL}\!=\!\Gamma_{dR}=0.125U$, $\Gamma_{cR}\!=\!0.06U$ and $\delta_{cav}\!=\!0.32U$..

Conductance for the interacting system.

Eq. (1) determines the conductance through the system under different cavity+QD coupling regimes. The QD coupling to the left (source) reservoir is simply $\tilde{\Gamma}_{L}=\Gamma_{dL}$. In contrast, the coupling to the right (drain) reservoir requires the full Green’s function and results in Sup

[TABLE]

This expression encodes information about all non-trivial interference processes taking place during transport. The energy dependence of $\tilde{\Gamma}_{R}(\omega)$ prevents the use of the PC simplification, demanding the more general approach we put forward here. The spectral function needed in Eq. (1) is obtained by an NRG approach that uses the full intricate structure of the effective hybridization function $\Delta(\omega)$ coupling the interacting QD to the environment.

Before discussing the conductance, we analyze the QD spectral function. In general, $A_{d}(\omega)$ shows a sequence of asymmetric features whenever $\epsilon_{c}$ shifts cavity modes near the Fermi level ($\omega\!=\!0$), with characteristic shape and width that changes strongly with coupling $\Omega$. Figure 2 illustrates this behavior for weak ($\Omega<\Gamma_{cR}/2$) and strong ($\Omega>\Gamma_{cR}/2$) dot-cavity coupling regimes. For weak coupling [Fig. 2(a)&(c)], the modulation is marked by diagonal “valleys” whenever a cavity mode contributes to $\Delta(0)$, separated by bright peaks in $A_{d}$. The large $\Omega$ regime [Fig. 2(b)&(d)] is drastically different: $\Delta(0)$ exhibits Fano asymmetric lineshapes as a function of $\epsilon_{c}$, leading to sharp asymmetric peaks in $A_{d}(\omega\!<\!T_{K})$ Sup .

This behavior can be qualitatively understood in terms of the Friedel sum rule (FSR) Dias da Silva et al. (2006); Vaugier et al. (2007); Dias da Silva et al. (2013), as $A_{d}(0)$ is inversely proportional to $\Delta(0)$. Accordingly, when a resonant peak of $\Delta(\omega)$ lies close to the Fermi energy, it causes a downturn in the spectral function, and a consequent splitting of the Kondo peak may appear in $A_{d}$ in the $\omega<T_{K}$ range Dias da Silva et al. (2006). Such splittings do appear for some $\epsilon_{c}$ values, where $A_{d}$ shows two local maxima away from the $\omega\!=\!0$ mark in Fig. 2(b) (see details in Sup ). Nonetheless, even at these points, $A_{d}(\omega)$ shows fully-developed Kondo resonances of width $\sim T_{K}$ in between Hubbard peaks (insets in Figs. 2(a) and (b)).

The resulting conductance $G$ (in units of $G_{0}=2e^{2}/h$) is shown in Fig. 3 vs cavity voltage $\varepsilon_{c}$, for $\Omega$ values from $0.01D$ (weak) to $0.2D$ (strong coupling) and for $T=0$ & 250mK. At low temperatures and small $\Omega$, the conductance exhibits a quantized peak whenever a cavity resonance is near the Fermi level, in agreement with the experimental result Rössler et al. (2015). The conductance drops away with $\epsilon_{c}$ as destructive interference sets in and results in a non-zero scattering shift associated with the strongly asymmetric $A_{d}(\omega)$, as expected from the FSR. Conversely, when a cavity resonance is aligned with the Fermi level in the strong coupling regime, a Fano-like dip appears in the conductance, with a width much smaller than the cavity level spacing. This feature is also consistent with the experimental data of Ref. Rössler et al. (2015). Finite temperatures do not result in qualitative changes of this picture, but suppress the magnitude of $G$, as one would expect, with a larger effect for $T_{K}$ values below the temperature of the reservoir (here 250mK).

Notice that the spinful QD remains in the Kondo regime over this range of coupling to the cavity. In fact, the Kondo screening is stronger for larger $\Omega$, as monitored by the value of $T_{K}$. To quantify this, we calculate $T_{K}$ from the magnetic susceptibility curves obtained from NRG, a procedure that focuses on how the Kondo fixed point is reached at lower energies, and does not rely on the behavior of the spectral density Bulla et al. (2008). The inset in Fig. 3 shows $T_{K}$ increasing rapidly with larger QD-cavity coupling $\Omega$. For $\Omega=0.15D-0.2D$, we obtain $T_{K}\sim 0.0048U-0.03U$; with the experimental $U=0.7$meV, this translates into $T_{K}\sim 40-240$mK, which is consistent with the observed value of $\sim 100$mK, obtained from the conductance peak width (see supplement in Rössler et al. (2015)). Our calculations also show $T_{K}$ to depend weakly on $\epsilon_{c}$. This might appear counterintuitive, as $\Delta(0)$ is strongly modulated by changes in $\epsilon_{c}$, but the explanation is simple: The effective coupling defining the Kondo temperature (e.g., $\Gamma$ in Haldane’s expression Haldane (1978)) is given not by $\Delta(0)$, but rather by an integral over the full bandwidth, $\Gamma\!\propto\!{\int\Delta(\omega)d(\omega/D)}$ Gonzalez-Buxton and Ingersent (1998). This “$\Gamma$” depends strongly on the dot-cavity coupling $\Omega$ (thereby giving the strong variation of $T_{K}$ with $\Omega$) while only weakly with $\epsilon_{c}$, whose main effect is to shift the peaks in $\Delta(\omega)$.

The increasing $T_{K}$ indicates that the screening of the QD spin by the composite cavity-lead environment is in fact more robust for larger $\Omega$, which is confirmed by an NRG analysis of the thermal properties of the QD. This is remarkable behavior, as the strong variation in $A_{d}(\omega)$ and resulting conductance are drastically different from the simply-connected QD in the Kondo regime.

Discussion.

We have presented an approach that allows one to calculate the linear conductance through interacting systems beyond the proportional coupling approximation. This opens the possibility of studying interesting systems with complex geometries where quantum interference introduces non-trivial energy dependence on the effective decay widths ${\tilde{\Gamma}}_{\alpha}$. We have illustrated the power of the method by analyzing a recent experiment with very interesting geometry Rössler et al. (2015). Despite the observed splitting and strong modulation of conductance peaks for growing cavity coupling, we find that the Kondo screening is in fact strengthened, as characterized by a larger $T_{K}$. This interpretation is supported by calculations of the conductance in excellent agreement with experiment. It would be interesting to be able to measure the expected phase shifts introduced by the interaction with the cavity to provide further insights into the coherent interference that these many-body coupled systems experience.

We acknowledge useful discussions with C. Rössler, T. Ihn, K. Ensslin, and N. Sandler. LDS acknowledges support from CNPq grants 307107/2013-2 and 449148/2014-9, PRP-USP NAP-QNano and FAPESP grant 2016/18495-4. SEU received support from NSF grant DMR 1508325, and the Aspen Center for Physics, NSF grant PHY-1066293. CHL is supported by CNPq grant 308801/2015-6 and FAPERJ grant E-26/202.917/2015. GJF and EV acknowledge financial support from CNPq and FAPEMIG.

I System geometry and model parameters

The model parameters we use in this paper have been inferred from the experimental data from Ref. Rössler et al., 2015 combined with analytical estimates and numerical calculations. Here, we provide more details on the numerical simulations.

The experimental setup of Ref. Rössler et al., 2015 consists of a cavity focusing resonant modes into a quantum dot, both set on a GaAs two-dimensional electron gas (2DEG). The cavity has a radius $\ell\sim 2~{}\mu\text{m}$ and an angular aperture $\theta_{C}\sim 45^{\circ}$, as indicated in Fig. S1(a). To obtain a simple, yet accurate description of the non-interacting modes of the cavity, we consider its eigenstates to be approximately given by Bessel functions. The Bessel approximation becomes exact for a large aperture $\theta_{C}\rightarrow 180^{\circ}$, as the cavity approaches a semi-circle shape [Fig. S1(b)]. In the following we show that this approximation leads to a level spacing that agrees remarkably well with the experimental Rössler et al. (2015) peak energy splitting $\delta_{cav}\approx 220~{}\mu\text{eV}$.

Assuming hard-wall boundary conditions, the solution for the Schrödinger equation in cylindrical coordinates results in eigenstates $\psi_{n,j}(r,\theta)$ given by Bessel functions $J_{n}(z)$, and eigenenergies $\varepsilon_{n,j}$ set by the $j^{\text{th}}$ zero $z_{n,j}$ of $J_{n}(z)$ at $r=\ell$, which reads

[TABLE]

where $k_{n,j}=\sqrt{2m\varepsilon_{n,j}/\hbar^{2}}$, and $C_{n,j}$ is a normalization constant. To satisfy the boundary condition at the linear wall of the semi-circle ($x=0$), the index $n=1,2,3,4,\dots$ must be a non-zero integer.

Near the Fermi level $k_{n,j}\approx k_{F}=2\pi/\lambda_{F}$, where $\lambda_{F}$ is the Fermi wavelength of the 2DEG under the resonant cavity. For $\ell\approx 2~{}\mu\text{m}$ one gets $k_{n,j}\ell\approx 2\pi\ell/\lambda_{F}\gg 1$, which allow us to use the asymptotic limit of the Bessel functionsAbramowitz and Stegun (1964) to find analytical expression for the zeros $z_{n,j}$. Since $J_{n}(z)\approx\sqrt{2/\pi z}\cos(z-n\pi/2-\pi/4)$ we find

[TABLE]

The $n$ and $j$ quantum labels become degenerate, and the Bessel zeros become simply $z_{l}$, with $l=(n+2j)$. The integer $l$ is odd (even) whenever $n$ is odd (even). Consequently $\varepsilon_{n,j}\rightarrow\varepsilon_{l}$ near the Fermi level,

[TABLE]

The coupling of the dot with the resonant modes of the cavity occurs via the split-gate set by the linear electrodes in Fig. S1(a). Therefore the relevant quantity is the LDOS $\propto|\psi_{n,j}(r,\theta)|^{2}$ of the cavity modes in the vicinity of this region, i.e. $\bm{r}\sim 0$. Figure S2 shows $|\psi_{n,j}(r,\theta)|^{2}$ for different $n$ and $j$. Since $J_{n}(kr)\propto(kr)^{n}$ for $kr\ll 1$, near $r=0$ the dominant coupling must be given by $n=1$, yielding odd $l$.

We conclude that the energy spacing between cavity resonant modes that are effectively coupled to the dot is

[TABLE]

Considering the experimental data of Ref. Rössler et al., 2015, $\lambda_{F}=53$ nm and $\ell=2~{}\mu\text{m}$, we obtain $\varepsilon_{F}\approx 8$ meV and $l\sim 150$ for $\varepsilon_{l}\sim\varepsilon_{F}$, corresponding to 75 even $l$ and 75 odd $l$ occupied resonant modes. From these we find $\delta_{cav}\approx 200~{}\mu\text{eV}$, which matches the experimental energy splitting between resonances reported in Ref. Rössler et al., 2015.

We compare the Bessel function approximation with a numerically calculated LDOS implemented using the Kwant code Groth et al. (2014). Figure S3 shows a remarkably good agreement for both low energies and energies close the $\varepsilon_{F}$, corresponding to panels (a) and (b). Note that the even $l$ states (blue dots) do not contribute to the LDOS near $r=0$ as expected from the discussion based on Bessel eigenmodes. Panels (c)-(f) show the full LDOS map on the cavity for small energies, also in good agreement with the Bessel solutions shown in Fig. S2.

II Green’s Functions and Equations of Motion

Our approach combines the equations-of-motion (EOM) with the numerical renormalization group (NRG) method to find the linear response current in strongly interacting systems. The EOM method allows us to assess the “single-particle” interference processes for arbitrarily complicated geometries and cast them in terms of effective energy dependent hybridization functions. The NRG, on the other hand, provides an robust approach to tread strongly correlated many-body systems and is amenable for including non-trivial geometric effects beyond the wide band limit.

Before presenting the details of the calculation of the current, let us address the hybridization function of the experimental system of interest and discuss some of the implications of our findings.

Let us begin by writing the Green’s functions in the Zubarev notation, namely,

[TABLE]

with the corresponding equations of motion (EOMs)

[TABLE]

that have the same form for the retarded, advanced, and time-ordered Green’s functions (GFs). These GFs are computed for all combinations of creation and annihilation operators in our model system. (The later correspond to $d_{\sigma},c_{\alpha{\bf k}\sigma}$, and $a_{j\sigma}$ that are defined in the main text.) In what follows, we shall omit the spin label $\sigma$, and indicate the type of Green’s function only when necessary.

Using these results, one readily obtains a set of coupled Green’s functions for our model Hamiltonian, defined in paper. These read

[TABLE]

We use the indices $i$ and $j$ to label cavity modes and $d$ to denote the quantum dot level. In the main text, we use the standard shorthand notation $G_{d}\equiv G_{dd}$ for the quantum dot Green’s function.

Using the expressions above, we can “close” the EOMs (for $U=0$) and write the retarded quantum dot Green’s function for the fully connected system in the absence of electron-electron interactions as

[TABLE]

where the expression for $\Sigma^{(0),r}_{d}(\omega)$ is given in the main text. We define the energy-dependent effective hybridization function $\Delta(\omega)\!\equiv\!-\mbox{Im }\Sigma^{(0),r}_{d}(\omega)$.

III Numerical calculations

The exact $U\!=\!0$ analytical expression for $\Delta(\omega)\!\equiv\!-\mbox{Im }\Sigma^{(0),r}_{d}(\omega)$ is used as input in the $U\!\neq\!0$ NRG calculations to capture the Kondo regime. To this end, we make a slight simplification in the model and consider equal couplings between all cavity levels $i=1,N$ and the right reservoir. This amounts into setting $V_{iR}=V_{cR}$ in Eqs. S9–S11. We will refer this approximation as the “simplified model” hereafter.

A key ingredient influencing the interacting do spectral function if the value of the hybridization function at the Fermi energy $\Delta(\omega=0)$ ($\omega\!=\!0$ is the Fermi level). Illustrative examples of $\Delta(0)$ vs $\epsilon_{c}$, are shown in Fig. S4 for both weak ($\Omega=0.01D$) and strong-coupling ($\Omega=0.15D$) regimes. The other parameters used are those mentioned in the main paper, namely $\Gamma_{dL}\!=\!\Gamma_{dR}\!=\!0.0625D$, $\Gamma_{cR}\!=\!0.6D$ and $\delta_{cav}\!=\!0.16D$.

The drastically different dependence on $\epsilon_{c}$ in both cases is also reflected in contrasting $\omega$ dependence at fixed cavity parameters (not shown), which strongly affects the effective spin fluctuations that set in once interactions are considered. As we will show below, this behavior has important consequences for the zero-bias conductance of the system, among other observables.

From the geometry of the device and the size of the cavity, it is natural to expect the cavity-reservoir coupling to be much larger than the dot-reservoir coupling, such that $\Gamma_{cR}\gg\Gamma_{dR}$. Surprisingly, as a result of interference effects in the structure of $\Delta(\omega)$, such relative large cavity-reservoir couplings translate into small widths in the peaks of $\Delta(0)$ in the weak cavity-dot coupling regime . In fact, the calculated widths of the peaks in Fig. S4-a ($\Omega=0.01D$) are $\Delta\epsilon_{c}\sim 0.022D\ll\Gamma_{cR}$.

One can show that, in the non-interacting expression, the widths of the peaks in $\Delta(0)$ roughly translate into the width of the conductance peaks through the device in the weakly cavity-dot coupling regime. These were dubbed “$\Gamma_{\rm cav}$” in Ref. Rössler et al., 2015. Using the experimental estimate of $U\sim 700\mu$eV and taking $D=2U$, the widths in Fig. S4-a are $\approx 31\mu$ eV, which is comparable to the experimental value for the conductance peak broadening $``\Gamma_{\rm cav}"\!\sim\!40\mu$eV in Ref. Rössler et al., 2015. As we show in the text, the widths of the interacting conductance peaks in the weak dot-cavity coupling regime are of the same order $\sim 56\mu$eV.

III.1 Details of the NRG calculations

The NRG calculations were carried out using an effective single-site Anderson model for a symmetric impurity ($\varepsilon_{d}=-U/2$) with an hybridization function given by $\Delta(\omega)$. The discretization of the effective band was carried out as discussed in Refs.Gonzalez-Buxton and Ingersent, 1998; Dias da Silva et al., 2008, 2006 using a discretization parameter $\Lambda=2.5$ and z-trick averaging ($N_{z}=5$). In the calculations, we explored charge and $SU(2)$ spin symmetries and up to 1000 $Q,S$ states were retained at each NRG iteration.

The spectral density data shown in the paper were obtained using the DM-NRG method.Hofstetter (2000) Additional runs using the CFS approachPeters et al. (2006); Weichselbaum and von Delft (2007) were also performed to check convergence of the results.

Examples for the results (data in Fig. 2 of the main paper) are presented in Fig. S5. Notice the formation of the Kondo resonance in the insets, with a broader peak for for $\Omega\!=\!0.15D$ indicating a larger $T_{K}$, as discussed in the main text.

IV Calculation of the current through the system

IV.1 Extension of the Meir-Wingreen formalism

In general, the current flowing from the $\alpha$ contact can be written as

[TABLE]

where $N_{\alpha}=\sum_{{\bf k}\sigma}c_{\alpha{\bf k}\sigma}^{\dagger}c_{\alpha{\bf k}\sigma}$ counts the number of electrons at lead $\alpha$. Let us start with the right lead ($\alpha=R$). Using the Heisenberg picture, where $i\hbar\dot{c}_{R{\bf k}\sigma}=[c_{R{\bf k}\sigma},H]$, one obtains

[TABLE]

where the following Green’s functions were introduced,

[TABLE]

The current $J_{R}$ is real, since Haug and Jauho (1996) $G_{ab}^{<}(t,t)=-[G_{ba}^{<}(t,t)]^{*}$.

We are interested in the stationary regime, where $J_{R}$ does not depend on time. Thus, it is convenient to write Eq. (S14) in the frequency representation,

[TABLE]

The above equation is the generalization of the two-terminal Meir-Wingreen formula Meir and Wingreen (1992) for our model system, where the right lead ($\alpha=R$) is coupled to both the dot and the cavity; see Fig. 1 of the paper.

In contrast, the left lead ($\alpha=L$) is only coupled to the dot. Consequently, the current $J_{L}$ is given by the standard expression

[TABLE]

Next, we use the method of equations of motion (EOM) and the Langreth rules Lan ; Langreth (1976) to express the Green’s function $G^{<}_{i\sigma,R{\bf k}\sigma}(\omega)$ in a convenient form.

Using the results of Section II, the contact Green’s functions $G_{R{\bf k},d}$ and $G_{R{\bf k},j}$ that appear in Eq. (S17) can be expressed as

[TABLE]

and

[TABLE]

where $g_{R{\bf k}}(\omega)$ is the free Green’s function at the terminal $R$.

Recall that in the simple two-terminal case one has to deal only with $G_{R{\bf k},d}=g_{R{\bf k}}V^{*}_{dR}G_{dd}$. This means that the problem is reduced to the calculation of $G_{dd}$, see Section VI. Our goal here is similar: we want to eliminate all hybrid (or contact) Green’s function and express the current in terms of $G_{dd}$ only. This is always possible, as long as interactions are local and restricted to the QD.

Let us now solve for $G_{jd}$. By inserting Eq. (S19) into (S8) we write

[TABLE]

Hence

[TABLE]

Next, let us solve for $G_{ij}$. By inserting Eq. (S20) into (S10) we get

[TABLE]

Hence,

[TABLE]

Before we proceed, let us simplify the notation by introducing the resonance self-energies

[TABLE]

where $\nu=i,d$. Let us also define

[TABLE]

Collecting the results, we obtain

[TABLE]

and

[TABLE]

Note that the integrand in Eq. (S17) contains the Green’s functions $G^{<}_{i\sigma,R{\bf k}\sigma}(\omega)$ and $G^{<}_{d\sigma,R{\bf k}\sigma}(\omega)$. Using the Langreth rules Lan and the Eqs. (S19) and (S20) we write

[TABLE]

and

[TABLE]

where the free Green’s functions are given by

[TABLE]

and $g_{\alpha{\bf k}}^{<}(\omega)=2\pi i\delta(\omega-\varepsilon_{\alpha{\bf k}})f_{\alpha}(\omega)$.

In the wide band limit, we can evaluate the self-energies as

[TABLE]

where $\rho_{\alpha}$ is density of states of the reservoir $\alpha=R,L$. From this expression we can define $\Sigma_{\nu\nu}^{\alpha,r}(\omega)=-i\pi V_{\nu\alpha}\rho_{\alpha}V^{*}_{\nu\alpha}\equiv-i\Gamma_{\nu\alpha}/2$, and assuming the couplings real, $\Sigma_{id}^{\alpha,r}(\omega)=-i\pi V_{i\alpha}\rho_{\alpha}V^{*}_{d\alpha}=-i\sqrt{\Gamma_{iR}\Gamma_{dR}}/2$. Notice that this definition of $\Gamma_{\nu\alpha}$ (more frequently used in transport works) carries an extra factor of 2 as compared to the definition commonly used by the strongly-correlated systems community (“$\Gamma=\pi\rho|V|^{2}$” ).

We now introduce the new self-energies

[TABLE]

and

[TABLE]

Using the Langreth rulesLan we are able to express $G_{id}$ and $G_{ij}$, given by Eqs. (S28) and (S27), in terms of free Green’s functions (that we know analytically) and of $G_{dd}$. Combining Eqs. (S27) and (S28) with (S33)- (S35) we can write

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

We are now ready to return to Eq. (S17) and calculate the current $J_{R}\equiv J_{R}^{(1)}+J_{R}^{(2)}$, with

[TABLE]

and

[TABLE]

where have used the wide flat band approximation to get rid of the Cauchy principal value contribution.

We now convert the summations over ${\bf k}$ into energy integrations, namely

[TABLE]

For notational simplicity, let us assume that all coupling matrix elements $V$ are real to write

[TABLE]

and

[TABLE]

We recall that $[G_{a,b}^{<}(\omega)]^{*}=-G_{b,a}^{<}(\omega)$. Therefore $G_{dd}^{<}(\omega)$ is pure imaginary. For the non-diagonal terms we use Re$(ic)=i(c-c^{*})/2$ to write

[TABLE]

Therefore,

[TABLE]

and

[TABLE]

and finally

[TABLE]

This lengthy expression reduces to the standard expression for the current found in Meir-Wingreen when one considers the simple case without cavity, that is, $\Gamma_{iR}=0$.

Using $G^{a}(\omega)=[G^{r}(\omega)]^{\dagger}$ one could simplify somewhat the second line of Eq. (IV.1) to obtain

[TABLE]

The current from the left lead is simple because $\Gamma_{iL}=0$, so we have for $J_{L}$,

[TABLE]

Notice that Eqs. (S49) and (S50) can be written in the matrix form used in Ref. Meir and Wingreen, 1992:

[TABLE]

where matrices $\mathbf{\Gamma}^{R,L}$ and the interacting Green’s functions $\mathbf{G}^{\alpha=r,a,<}$ are given by:

[TABLE]

In this notation, it is clear that the system cannot be proportionally coupled since $\mathbf{\Gamma}^{L}\neq\lambda\mathbf{\Gamma}^{R}$ always.

Our task now is to rewrite Eq. (S49) in terms of dot Green’s functions only. Inserting the expressions for the lesser Green’s function in Eq. (S49) we obtain

[TABLE]

In the above we need to know that

[TABLE]

correspond to the cavity Green’s functions in the absence of the dot.

IV.2 Expressions for the current

We now write the results of Eq. (S53) for the simplified model. First note that by assuming that the coupling of all the cavity levels with the right reservoir are equal, i.e., $V_{iR}=V_{cR}$ for all $i$ in the cavity. Then $\Sigma_{ij}^{R}(\omega)=-i\Gamma_{cR}/2$, $\Sigma_{id}^{R}(\omega)=-i\sqrt{\Gamma_{dR}\Gamma_{cR}}/2$. The lesser GF can be written as $\Sigma_{ij}^{R,<}(\omega)=if_{R}(\omega)\Gamma_{cR}$ using the fact that $\sum_{{\bf k}}g^{<}_{{\bf k},R}(\omega)=2i\pi\rho_{R}f_{R}(\omega)$. Assuming also $V_{di}=\Omega$ for all dot-level coupling matrix elements, the self-energies defined in Eqs.  (S33) to (S35) can also be simplified,

[TABLE]

We can use the method of equations of motion (EOM) to write $J_{R}$ and $J_{R}$ in terms of the Green’s function for the dot. Using the Eqs. (S57) into Eq. (S26) we write

[TABLE]

with $\tilde{S}(\omega)=S(\omega)(1+iS(\omega)\Gamma_{cR}/2)^{-1}$, where $S=\sum_{i}(\omega-\varepsilon_{i})^{-1}$. The lesser GF becomes

[TABLE]

From there, we can write the other Green’s functions we are going to need in terms of dot’s GFs:

[TABLE]

[TABLE]

We are now in a position of re-writing Eq. (S53) for this simplified model. Let us start with by writing it in terms of the GFs defined above (sums included). Using Eqs. (S32), (S57), and (S58), the Eq. (S53) becomes

[TABLE]

We now substitute Eqs. (S59)–(S62) into Eq. (IV.2) and collect the terms in $G^{r}_{dd}$, $G^{r}_{dd}$ and $G^{<}_{dd}$. Using the limit $\lim_{\eta\rightarrow 0}S(\omega+i\eta)$ (i.e., taking the analytic continuation of $S(\omega)$ to the real axis), and after some long but straightforward algebra, we obtain $J_{R}$ in a nice, compact form:

[TABLE]

In the equation above, $J^{(0)}_{R}$ is a background contribution coming from the terms in Eq. (IV.2) that do not involve dot’s Green’s functions:

[TABLE]

In fact, as we will show below, $J^{(0)}_{R}$ vanish, explicitly, for $\eta\rightarrow 0$. The effective coupling ${\tilde{\Gamma}}_{R}(\omega)$ is a real algebraic function of the parameters, given by:

[TABLE]

From Eq. (S50) it is straightforward to show that the effective coupling to the left lead is simply ${\tilde{\Gamma}}_{L}(\omega)\!=\!\Gamma_{dL}$.

These expressions are all exact as long as there are no interactions in the cavity. At this point, a fair question is “What are the gains by performing such transformations”? The advantage here is that now $J_{R}$, given by Eq. (S64), is written in terms of dot’s Green’s functions only, in the same structure as $J_{L}$ (for which $\Gamma_{iL}=0$) given by Eq. S50. As shown below, this is a crucial step in the elimination of $G_{dd}^{<}(\omega)$ in the current expression.

V Fluctuation-dissipation theorem

An important consistency check for the expressions given in the previous section is the applicability of the fluctuation-dissipation theorem (FDT). For instance, the expression for $J_{R}$ in Eq. (S64) vanishes in equilibrium, when the fluctuation-dissipation theorem (FDT) applies.

Just a reminder: the FDT states that, for a system in thermal equilibrium with a reservoir described by a Fermi distribution $f_{R}(\omega)$, the lesser Green’s function is proportional to the spectral density

[TABLE]

where $A_{\nu}(\omega)=(-1/\pi)$Im $G^{r}_{\nu}(\omega)$.

We can put the FDT in terms of retarded and advanced Green’s functions. Using $G^{a}_{\nu}(\omega)=(G^{r}_{\nu}(\omega))^{*}$, the FDT implies

[TABLE]

This is important as a consistency check for the current calculations. Applying Eq. (S68), the current to/from a single lead should vanish (which is the correct result in equilibrium). This can be readily verified, for instance, for $J^{(0)}_{R}$ defined in Eq. (S65) and for $J_{L}$ [Eq. (S50)].

In fact, this consistency check can be applied to each of the three terms in Eq. (S49) by verifying that the FDT is satisfied for each of the Green’s functions involved. Note that the first term in in Eq. (S49) involves diagonal (dot) GFs and is clearly consistent with the FDT: it vanishes if $G^{<}_{dd}(\omega)=-f_{R}(\omega)[G^{r}_{dd}(\omega)-G^{a}_{dd}(\omega)]$.

The second term involves non-diagonal Green s functions. We can then explicitly show that

[TABLE]

The right-hand side of the above expression can be easily calculated using Eq. (S61). Using the short-hand notations

[TABLE]

we have

[TABLE]

Thus, the left-hand side can be calculated with the help of Eqs. (S36) and (S38) and from Eqs. (S57) to (S59), giving

[TABLE]

Now, using $G^{<}_{dd}(\omega)=-f_{R}(\omega)[G^{r}_{dd}(\omega)-G^{a}_{dd}(\omega)]$, Eq. (S73) reduces to Eq. (S72). In order to show that, we take the limit $S(\omega)=\lim_{\eta\rightarrow 0}S(\omega+i\eta)$ and then use the following properties:

[TABLE]

A similar calculation can be done to show that the third term in Eq. (S49) also satisfies the FDT.

VI Meir-Wingreen-like elimination of $G^{<}$

Let us consider the current formula for a single-resonance QD Meir and Wingreen (1992)

[TABLE]

[TABLE]

In the steady state, charge conservation implies that $J_{L}=-J_{R}$, hence

[TABLE]

or, in general $J_{L}=xJ_{L}-(1-x)J_{R}$, where $x$ is arbitrary.

We stress that $J_{L}=-J_{R}$ is the same as

[TABLE]

which does not mean that $I_{L}(\omega)=-I_{R}(\omega)$ for a given energy $\omega$.

Let us restrict ourselves to the linear response regime and write

[TABLE]

We recall that the fluctuation-dissipation theorem gives

[TABLE]

allowing us to write the current $J_{L(R)}$, Eq. (S76), as

[TABLE]

where $\mp$ refer to the sign of chemical potential offset of $L$ and $R$ terminals with respect to the Fermi energy. Affleck and collaborators Komijani et al. (2013) claim that $\partial G^{<}/\partial\mu$ is expected to have the form $(-\partial f_{0}/\partial\omega)\Pi(\omega)$, where (in general) $\Pi(\omega)$ has a smooth energy dependence on the scale of $kT$. For now, we assume this is true.

Let us assume that $\tilde{\Gamma}_{L(R)}(\omega)$ varies slowly with $\varepsilon$ over energies scales of the order of $kT$, which is a condition met in almost all situations of interest. In this scenario it is safe to approximate

[TABLE]

We now use the general relation $J_{L}=xJ_{L}-(1-x)J_{R}$ to write

[TABLE]

To eliminate the $G^{<}$ term one needs $x\tilde{\Gamma}_{L}-(1-x)\tilde{\Gamma}_{R}=0$, yielding $x=\tilde{\Gamma}_{R}/(\tilde{\Gamma}_{L}+\tilde{\Gamma}_{L})$. Hence

[TABLE]

This expression is the same as the one obtained by Meir and Wingreen Meir and Wingreen (1992) using the proportional coupling trick, namely, by assuming that $\tilde{\Gamma}_{L}(\omega)=\lambda\tilde{\Gamma}_{R}(\omega)$, where $\lambda$ does not depend on energy.

From the expression for the current [Eq. (S85)] we can readily derive the corresponding expression for the conductance through the system:

[TABLE]

where $f_{0}$ written in terms of the dot spectral density $A_{d}(\varepsilon,T)=(-1/\pi)\mbox{Im }G_{dd}^{r}(\omega)$ at temperature T that can be calculated with NRG.

For the cavity system studied in this work, we have ${\tilde{\Gamma}}_{L}(\omega)\!=\!\Gamma_{dL}$ and ${\tilde{\Gamma}}_{R}(\omega)$ is given by Eq. (S66). We note, however, that this approach is very generic and can be applied to a large class of systems with arbitrarily complex geometries and for which interactions are restricted to a single level.