*2.3. Calculation of Current Correlations*

Here, we consider a single-particle interference effect which takes place in a Mach–Zehnder or Michelson interferometer, but not in a Hanbury Brown and Twiss situation with a two-particle interference effect. The current fluctuations are described by the operator Δ ˆ *Iij*(*t*) − <sup>ˆ</sup>*Iij*(*t*), and the current–current correlation function is defined as [32]

$$S\_{\vec{i}\vec{j},nm}(t, t') \equiv \frac{1}{2} \left[ \langle \mathbb{I}\_{\vec{i}\vec{j}}(t) \mathbb{I}\_{nm}(t') + \mathbb{I}\_{nm}(t') \mathbb{I}\_{\vec{i}\vec{j}}(t) \rangle - 2 \langle \mathbb{I}\_{\vec{i}\vec{j}}(t) \langle \mathbb{I}\_{nm}(t') \rangle \right]. \tag{21}$$

We consider the steady currents, for which the correlation functions can be represented, in the frequency domain, by their spectral density

$$S\_{ij,nm}(\omega) \equiv 2 \int\_{-\infty}^{\infty} d\tau e^{\mu\omega\tau} S\_{ij,nm}(\tau). \tag{22}$$

In this work, we shall restrict ourselves to studying the current correlations at the zero-frequency limit *ω* = 0. As the net transport current is ˆ *Itr* = ˆ*I*12 + ˆ*I*13, its current correlation function can be expressed as a composition of the correlation functions for the bond currents

$$S\_{tr,tr} = S\_{12,12} + S\_{13,13} + 2S\_{12,13}.\tag{23}$$

The correlation functions *Sij*,*in* can be derived by means of Wick's theorem [31] and are expressed as

$$S\_{ij; \text{in}} = \frac{\varepsilon^2}{\pi \hbar} \frac{1}{2} \{ t\_{ij} t\_{\text{in}} \left( \langle c\_i^\dagger c\_j \rangle \langle c\_i c\_n^\dagger \rangle + \langle c\_i^\dagger c\_n \rangle \langle c\_i c\_j^\dagger \rangle \right) \} - t\_{ij} t\_{\text{in}} (\langle c\_i^\dagger c\_i \rangle \langle c\_n c\_j^\dagger \rangle + \langle c\_n^\dagger c\_j \rangle \langle c\_i c\_i^\dagger \rangle) \}$$

$$+ t\_{ji} t\_{\text{in}} (\langle c\_j^\dagger c\_i \rangle \langle c\_n c\_i^\dagger \rangle + \langle c\_n^\dagger c\_i \rangle \langle c\_j c\_i^\dagger \rangle)) - t\_{ji} t\_{\text{in}} (\langle c\_i^\dagger c\_i \rangle \langle c\_j c\_n^\dagger \rangle + \langle c\_j^\dagger c\_n \rangle \langle c\_i c\_i^\dagger \rangle)) \}) \, \qquad \text{(24)}$$

Once again, we use the NEGF method. As the lesser Green functions, *G*<*ji* ≡ *<sup>ı</sup>c*†*i cj*, and the greater Green functions, *G*>*ij* ≡ <sup>−</sup>*<sup>ı</sup>cic*†*j* , have the same structure, one should only exchange the Green functions in the electrodes: *g*<sup>&</sup>lt;*α* = <sup>2</sup>*π*(*g<sup>r</sup>α* − *g<sup>a</sup>α*)*f<sup>α</sup>* ↔ *g*<sup>&</sup>gt;*α* = <sup>−</sup>2*π*(*g<sup>r</sup>α* − *g<sup>a</sup>α*)(<sup>1</sup> − *fα*). Separating coefficients in front of *fL*(1 − *fL*), *fR*(1 − *fR*), and *fL*(1 − *fR*) + *fR*(1 − *fL*), and after some algebra, one can derive a compact formula for any current–current function. The auto-correlation function for the net transport current is given by the well-known Lesovik formula [30,33,34] (see also [32,35] for a multi-terminal and multi-channel case)

$$S\_{\rm tr,tr} = \frac{e^2}{\pi\hbar} \int\_{-\infty}^{\infty} dE \left\{ \mathcal{T}^2 \left[ f\_\mathcal{L} (1 - f\_\mathcal{L}) + f\_\mathcal{R} (1 - f\_\mathcal{R}) \right] + \mathcal{T} (1 - \mathcal{T}) \left[ f\_\mathcal{L} (1 - f\_\mathcal{R}) + f\_\mathcal{R} (1 - f\_\mathcal{L}) \right] \right\}, \tag{25}$$

where T is the transmission through the 3QD system. For a given temperature *TL* = *TR* = *T*, one has *fL*(1 − *fR*) + *fR*(1 − *fL*) = coth[(*μL* − *μR*)/2*kBT*](*fL* − *fR*) and, thus,

$$S\_{tr,tr} = 2I^{tr} \coth\left(\frac{\varepsilon V}{2k\_B T}\right) - \frac{\varepsilon^2}{\pi \hbar} \int\_{-\infty}^{\infty} dE \ \mathcal{T}^2 \left(f\_L - f\_R\right)^2. \tag{26}$$

When the scale of the energy dependence Δ*E* of the transmission T is much larger than both the temperature and applied voltage (i.e., Δ*E eV kBT*), one can obtain the well known explicit relation (see Blanter and Buttiker [32])

$$S\_{tr,tr} = \frac{\varepsilon^2}{\tau \hbar} \left[ 2k\_B T \, \mathcal{T}^2 (E\_F) + eV \coth \left( \frac{\varepsilon V}{2k\_B T} \right) \mathcal{S}\_{tr,tr}^{\text{sl}} \right]. \tag{27}$$

The first term is the Nyquist-Johnson noise at equilibrium and the second term, with <sup>S</sup>*shtr*,*tr* = T (*EF*)(1 − T (*EF*)), corresponds to the shot noise [30,32,34].

The correlation functions for the bond currents are calculated from Equation (24) and are expressed as

$$S\_{\vec{i}\vec{j},\vec{k}} = \frac{e^2}{\pi\hbar} \int\_{-\infty}^{\infty} dE \left\{ \mathcal{G}\_{\vec{i}\vec{j},\vec{k}}^{\text{sl}} \left[ f\_{\text{L}}(1 - f\_{\text{R}}) + f\_{\text{R}}(1 - f\_{\text{L}}) \right] + \left[ \mathcal{G}\_{\vec{i}\vec{j}}^{\text{L}} \mathcal{G}\_{\vec{i}\vec{k}}^{\text{L}} f\_{\text{L}}(1 - f\_{\text{L}}) + \mathcal{G}\_{\vec{i}\vec{j}}^{\text{R}} \mathcal{G}\_{\vec{i}\vec{k}}^{\text{R}} f\_{\text{R}}(1 - f\_{\text{R}}) \right] \right\},\tag{28}$$

where G*αij* are given by (7)–(10), and the dimensionless spectral functions of the shot noise components are

$$S\_{12,12}^{\text{sh}} = \Gamma\_L \Gamma\_R t\_{12}^2 |d\_{23,31}^2 - d\_{23,23} d\_{31,31}^\*|^2 / A^2,\tag{29}$$

$$S\_{13,13}^{\text{sh}} = \Gamma\_L \Gamma\_R t\_{31}^2 |d\_{12,23}^\* d\_{23,31}^\* + d\_{12,31} d\_{23,23}^\*|^2 / A^2,\tag{30}$$

$$\mathcal{S}\_{12,13}^{\rm cyl} = \Gamma \sqcup \Gamma \wr \mathfrak{t} \sqcup \mathfrak{z} \wr \mathfrak{k} \left[ (d\_{23,31}^2 - d\_{23,23} d\_{31,31}^\*) (d\_{12,23}^\* d\_{23,31}^\* + d\_{12,31} d\_{23,23}^\*) \right] / A^2, \text{ and} \tag{31}$$

$$\mathcal{S}\_{\rm Ir,tr}^{\rm sh} \equiv \mathcal{S}\_{12,12}^{\rm sh} + \mathcal{S}\_{13,13}^{\rm sh} + 2\mathcal{S}\_{12,13}^{\rm sh} = \Gamma\_L \Gamma\_R |t\_{12}(d\_{23,31}^2 - d\_{23,23}d\_{31,31}^\*) + t\_{31}(d\_{12,23}d\_{23,31} + d\_{12,31}^\*d\_{23,23})|^2 / A^2. \tag{32}$$

### **3. Bond Currents and Their Correlations: Driven Circular Current in the Case of Φ = 0**

Let us analyse the bond currents in detail; first in the absence of the magnetic flux, Φ = 0, and for a linear response limit *V* → 0. Using the derivations from the previous section, one can easily calculate the bond conductances and current correlation functions. The results are presented in Figure 2 for an equilateral triangle 3QD system (with all inter-dot hopping parameters *t*12 = *t*23 = *t*31 = −1, which is taken as unity in our further calculations) and for various values of the energy level *ε*3 at the 3rd QD. The central column corresponds to the case *ε*1 = *ε*2 = *ε*3 = 0, when the eigenenergies are given by *Ek* = 2*t* cos *k*, for the wave-vector *k* = 0 and the degenerated state for *k* = ±2*π*/3. It can be seen in the transmission (black curve), which is equal to T = 1 at *E* = −2 and T = 0 at *E* = 1, where the Fano resonance takes place, with destructive interference of two electron waves. At low *E* < 0, the incoming wave from the left electrode is split into two branches and the bond conductances are positive, 0 ≤ G12, G13 ≤ 1 (see the blue and green curves in the top panel of Figure 2). The cross-correlation function <sup>S</sup>*sh*12,13 (for the currents in both branches) is positive (see the red curve in the bottom panel in Figure 2). Note that, at the lowest resonant level, all correlation functions <sup>S</sup>*sh*12,12 = <sup>S</sup>*sh*13,13 = <sup>S</sup>*sh*12,13 = 0, which means that the currents in both branches are uncorrelated.

For *E* > 0, the conductances G12 and G13 can be negative and exceed unity (with their maximal absolute values inversely proportional to the coupling <sup>Γ</sup>*α*). This manifests a circular current driven by injected electronic waves to the 3QD system, which can not reach the drain electrode; therefore, they are reflected backwards to the other branch of the ring. The circular current can be characterized by the conductance (see also [16])

$$\mathcal{G}^{dr} \equiv \begin{cases} \mathcal{G}\_{12} & \text{for } \mathcal{G}\_{12} < 0, \\\ -\mathcal{G}\_{13} & \text{for } \mathcal{G}\_{13} < 0, \end{cases} \tag{33}$$

where the superscript "*dr*" marks the contribution to the circular current driven by the bias voltage, in order to distinguish it from the persistent current induced by the flux (which will be analysed later). There is some ambiguity in definition of the circular current. Our definition (33) is similar to the one given by the condition sign[G12] = −sign[G13] for the vortex flow, used by Jayannavar and Deo [36] and Stefanucci et al. [16] (see [37]–which refers to [7] ).

**Figure 2.** (Top) Transmission and dimensionless bond conductances: T —black, G12—blue, and G13—green. (Bottom) Dimensionless spectral function of the shot noise: <sup>S</sup>*shtr*,*tr*—black, <sup>S</sup>*sh*12,12—blue, <sup>S</sup>*sh*13,13—green, and −S*sh*12,13—red; calculated as a function of the electron energy *E* for the equilateral triangle system of 3QDs (with the inter-dot hopping *t*12 = *t*23 = *t*31 = −1, which is taken as unity in this paper) in the linear response limit *V* → 0. The dot levels are *ε*1 = *ε*2 = 0 and *ε*3 = −2, 0, 2, for left, center, and right columns, respectively. The coupling with the electrodes is taken to be Γ*L* = Γ*R* = 0.25. Note that the cross-correlation function *Ssh*12,13 (red) is plotted negatively to show the zero crossing more clearly.

For the considered case in Figure 2b, with *ε*3 = 0, the circular current is driven counter-clockwise for 0 < *E* < 1 and changes its direction to clockwise at the degeneracy point, *E* = 1 (i.e., when G13 becomes negative). All correlation functions are large in the presence of the circular current; their maximum is inversely proportional to <sup>Γ</sup>2*α*. The cross-correlation <sup>S</sup>*sh*12,13 is large but negative and, therefore, this component reduces the transport shot noise, <sup>S</sup>*shtr*,*tr* = <sup>S</sup>*sh*12,12 + <sup>S</sup>*sh*13,13 + <sup>2</sup>S*sh*12,13 to the Lesovik formula T (1 − T ), which reaches zero at the degeneracy point *E* = 1 (see the black curve in Figure 2e). This situation is similar to multi-channel current correlations in transport through a quantum dot connected to magnetic electrodes [38], where cross-correlations for currents of different spins usually reduce the total shot noise to a sub-Poissonian noise with Fano factor *F* < 1 (however, in the presence of Coulomb interactions, the cross-correlations can be positive and lead to a super-Poissonian shot noise with *F* > 1).

The plots on the left and right hand sides of Figure 2 give more insight into the circular current effect. They are calculated for the dot level *ε*3 = ∓2 shifted by a gate potential, which breaks the symmetry of the system and removes the degeneracy of the states. Three resonant levels can be observed with T = 1, where two of them are shifted to the left/right for *ε*3 = ∓2; however, the state at *E* = 1 is unaffected. There is still mirror symmetry, for which one gets three eigenstates, where two of them are linear compositions of all local states, but the one at *E* = 1 has the eigenvector 1/ √2(*c*† 1 − *c*† <sup>2</sup>)|0, which is separated for the 3rd QD. Therefore, the bond currents are composed of the currents through all three eigenstates, and their contribution depends on *E*. From these plots, one can see that the circular current is driven, for *E* > *ε*3, when the cross-correlation <sup>S</sup>*sh*12,13 becomes negative. The direction of the current depends on the position of the eigenlevels and their current contributions. For *ε*3 = −2, the current circulates clockwise, whereas its direction is counter-clockwise for *ε*3 = 2.

Here, we assumed a flat band approximation (FBA) for the electronic structure in the electrodes (i.e., the Green functions *g<sup>r</sup>*,*<sup>a</sup> α* = ∓*ıπρ*, where *ρ* denotes the density of states). Appendix A presents analytical results for the currents and shot noise in the fully-symmetric 3QD system coupled to a semi-infinite chain of atoms. The results are qualitatively similar. However, the FBA is more convenient for the analysis than the system coupled to atomic chains; in particular, for the cases with *ε*3 = ∓2, when localized states appear at −2.99 and 2.56 (i.e., below/above the energy band of the atomic chain).

The above analysis was performed under the assumption of a smooth energy dependence of the conductance in the small voltage limit *V* → 0 and at *T* = 0. However, the conductances exhibit sharp resonant characteristics in the energy scale Δ*E* ∝ Γ*α* and, therefore, one can expect that these features will be smoothed out with an increase of voltage bias and temperature. Figure 3 presents the Fano factor *F* = *Str*,*tr*/2*eItr*, which is the ratio of the current correlation function to the net transport current, which was calculated numerically from Equations (17) and (26). At *E* = −2, one can observe the evolution from the coherent regime, from *F* = 0 to *F* = 1/2 in the sequential regime, for *eV* Γ*α* or *kBT* Γ*<sup>α</sup>*. Quantum interference plays a crucial role at *E* = 1, leading to the Fano resonance for which the transmission T = 0 and *F* = 1 in the low voltage/temperature regime. An increase of the voltage/temperature results only in a small reduction of the Fano factor.

**Figure 3.** Fano factor as a function of the Fermi energy *EF* for the equilateral triangle 3QDs system (*<sup>t</sup>*12 = *t*23 = *t*31 = −1 and *ε*1 = *ε*2 = *ε*3 = 0) (**a**) for various bias voltages *eV* = 0.01, 0.5, 1.0, 1.5, and 2.0, at *T* = 0; and (**b**) for various temperatures *kBT* = 0.001, 0.1, 0.2, 0.3, and 0.4, for *V* → 0. The coupling to the electrodes is taken as Γ*L* = Γ*R* = 0.25, and the chemical potentials in the electrodes are *μL* = *EF* − *eV*/2 and *μR* = *EF* + *eV*/2.

### **4. Persistent Current and Its Noise: The Case** *V* **= 0**

The persistent current and its noise has been studied in many papers (e.g., by Büttiker et al. [39–42], Semenov and Zaikin [43–46], Moskalates [47], and, more recently, by Komnik and Langhanke [48]) using full counting statistics (FCS), as well as in 1D Hubbard rings by exact diagonalization by Saha and Maiti [49] (see, also, the book by Imry [50]).

Here, we briefly present the results for the persistent current and shot noise in the triangle of 3QDs. Notice that, in the considered case, the phase coherence length of electrons is assumed to be larger than the ring circumference, *Lφ L* [51]. The circular current is given by Equation (18), which shows that all electrons, up to the chemical potential in the electrodes, are driven by the magnetic flux Φ. Figure 4 exhibits the plots of *<sup>I</sup>φ*, derived from Equation (18), for different couplings with the electrodes. In the weak coupling limit, where Γ → 0 and the perfect ring is embedded in the reservoir, the persistent current can be simply expressed as

$$I^{\phi} = \varepsilon \sum\_{k} \upsilon\_{k} f\_{k} = -\frac{\varepsilon}{\hbar} \sum\_{k} 2t \sin(k + \phi/3) f\_{k} \,\,\,\,\tag{34}$$

where *fk* = 1/(exp[(*EF* − *Ek*)/*kBT*] + 1) is the Fermi distribution for the electron with wave-vector *k*, energy *Ek* = 2*t* cos(*k* + *φ*/3), and velocity *vk* = (1/¯*h*)*∂Ek*/*∂k* = (−2*t*/¯*h*) sin(*k* + *φ*/3), and where *φ* = <sup>2</sup>*π*Φ/(*hc*/*e*) is the phase shift due to the magnetic flux Φ. The sum runs over *k* = <sup>2</sup>*πn*/(*Na*) for *n* = 0, ±1, where *N* = 3 and *a* = 1 is the distance between the sites in the triangle. The current correlator is derived from Equation (24)

$$S\_{\Phi,\Phi} = \frac{e^2}{\hbar} \sum\_{k} 4t^2 \sin^2(k + \Phi/3) f\_k(1 - f\_k) \,. \tag{35}$$

This result says that fluctuations of the persistent current could occur when the number of electrons in the ring fluctuates (i.e., an electron state moves through the Fermi level and *Iφ* jumps). We show, below, that the coupling with the electrodes (as a dissipative environment) results in current fluctuations [40,41], as well.

**Figure 4.** Persistent current *Iφ* versus the flux *φ* threading the equilateral triangle system of 3QDs (*<sup>t</sup>*12 = *t*23 = *t*31 = −1 and *ε*1 = *ε*2 = *ε*3 = 0). The coupling is taken as Γ*L* = Γ*R* = Γ = 0.01 and 0.25; the Fermi energies are *EF*= −1.5 (**black**), −0.75 (**blue**), 1.5 (**red**); and *T* = 0.

At the limit, *V* → 0, the integrand function of the noise *Sij*,*in*, Equation (28), is proportional to *f*(*E*)(1 − *f*(*E*)), which becomes the Dirac delta for *T* → 0 and, therefore, one can analyze the spectral function <sup>S</sup>*φ*,*<sup>φ</sup>* = S12,12 + S13,13 − <sup>2</sup>S12,13, where the components are <sup>S</sup>*ij*,*in* = <sup>S</sup>*shij*,*in* + G*Lij*G*Lin* + G*Rij* G*Rin* (see Equations (7)–(10) and (29)–(31)). Figure 5 presents the correlation function <sup>S</sup>*φ*,*<sup>φ</sup>* and its various components for the Fermi energy *EF* = −1.5 and the strong coupling Γ*L* = Γ*R* = 1 when fluctuations are large. Notice that the fluctuations of the bond currents S12,12 and S13,13 (the blue and green curves, respectively) are different, although the average currents are equal. The cross-correlation function S12,13 is positive at *φ* = 0, but it becomes negative for larger *φ*, due to the quantum interference between electron waves passing through different states (as described in the previous section).

Figure 5 also shows (G*<sup>L</sup>*12)<sup>2</sup> (blue-dashed curve) and (G*<sup>L</sup>*12)<sup>2</sup> (blue-dotted curve), which correspond to the local fluctuations of the injected/ejected currents to/from the upper branch on the left and right junctions, respectively (see Equation (28)). The magnetic flux breaks the symmetry, inducing the persistent current and, therefore, the local conductances <sup>G</sup>*<sup>L</sup>*12 and <sup>G</sup>*<sup>R</sup>*12 are asymmetric.

**Figure 5.** Flux dependence of spectral function of the persistent current correlator <sup>S</sup>*φ*,*<sup>φ</sup>* (**black**) and its components: S12,12 (**blue**), S13,13 (**green**), -S12,13 (**red**), and S*LL* 12,12 = (G*<sup>L</sup>*12)<sup>2</sup> (**blue-dashed**), S*RR* 12,12 = (G*<sup>R</sup>*12)2, (**blue-dotted**), respectively. We assume strong coupling: Γ*L* = Γ*R* = 1.0, *EF* = −1.5, and *T* = 0.

### **5. Correlation of Persistent and Transport Currents, Φ = 0 and** *V* **= 0**

In this section, we analyze the currents and their correlations in the general case, derived from Equations (6), (14), (18), and (28), in the presence of voltage bias and magnetic flux. The results for the conductances and the spectral functions of the shot noise are presented in Figure 6. The magnetic flux splits the degenerated levels at *E* = 1 and destroys the Fano resonance. Figure 6a shows that there is no destructive interference for a small flux *φ* = 2*π*/16, and the transmission is T = 1 for all resonances. One can observe the driven circular current for *E* > 0, with negative G12 and G13, but their amplitudes are much lower than in the absence of the flux (compare with Figure 2b for *φ* = 0). For a larger flux, *φ* = 2*π*/4, there is no driven component of the circular current (see Figure 6b, where G12, G13 ≥ 0). It can also be seen that, for the state at *E* = 0, the electronic waves pass only through the lower branch of the ring, and the upper branch is blocked (with G13 = 1 and G12 = 0, respectively).

**Figure 6.** (**Top**) Energy dependence of driven conductance G12 (blue), G13 (**green**) and transmission T (**black**). (**Bottom**) Shot noise <sup>S</sup>*shtr*,*tr* (**black**) with the components: <sup>S</sup>*sh*12,12 (**blue**), <sup>S</sup>*sh*13,13 (**green**), and −S*sh*12,13 (**red**) for the considered triangular 3QD system threaded by the flux *φ* = 2*π*/16 (**left**) or *φ* = 2*π*/4 (**right**); the coupling is Γ*L* = Γ*R* = 0.25, and *T* = 0. Note that we plot <sup>−</sup>*Ssh*12,13.

The lower panel of Figure 6 presents the spectral functions of the shot noise. According the Lesovik formula, <sup>S</sup>*shtr*,*tr* = 0 at the resonant states (as T = 1). This seems to be similar to the case *φ* = 0 presented in the lower panel in Figure 2. However, there is a grea<sup>t</sup> difference in the components of the shot noise <sup>S</sup>*shij*,*in*, indicating the different nature of transport through these states and the role of quantum interference. Let us focus on the lowest resonant state, at *E* = −2, in Figure 6c, and compare with that in Figure 2e, in the absence of the flux. In the former case, the currents in both branches were uncorrelated, and <sup>S</sup>*sh*12,12 = <sup>S</sup>*sh*13,13 = <sup>S</sup>*sh*12,13 = 0. In the presence of the flux, quantum interference becomes relevant, which is seen in the shot noise (Figure 6c). Now, the currents in both branches are correlated; <sup>S</sup>*sh*12,13 is negative close to resonance and fully compensates for the positive contributions <sup>S</sup>*sh*12,12 and <sup>S</sup>*sh*13,13 at resonance. For *φ* = 2*π*/4 (see Figure 6d), all shot noise components are large, which indicates a strong quantum interference effect.

Figure 7 shows the Fano factor in the presence of the flux *φ* = 2*π*/16 and for various bias voltages. Compared with the results in Figure 3 for *φ* = 0, one can see how a small flux can destroy quantum interference and change electron transport. It is particularly seen close to *E* = −1, where the states with opposite chirality are located. In the case *φ* = 0, one can observe the Fano resonance with a perfect destructive interference, T = 0 and *F* = 1. With an increase of the flux *φ*, the Fano dip disappears, the two states are split, and transmission reaches its maximum value T = 1; the Fano factor *F* = 0 when the splitting Δ*E* > Γ*<sup>α</sup>*. A similar effect was seen in the case of Figure 2, where a change of the position of the local level *ε*3 removed the state degeneracy and destroyed the Fano resonance.

**Figure 7.** Fano factor as a function of *EF* for the considered 3QD system threaded by the flux *φ* = 2*π*/16 and for various bias voltages *eV* = 0.01, 0.5, 1.0, 1.5, and 2.0. The coupling is Γ*L* = Γ*R* = 0.25, the chemical potentials are *μL* = *EF* − *eV*/2 and *μR* = *EF* + *eV*/2, and *T* = 0.

For the strong coupling Γ*L* = Γ*R* = 1, the intensity of the transport current is comparable to the persistent current and, therefore, one can expect a significant driven circular current. Figure 8 presents the flux dependence of the total circular current *Ic* and its driven component *<sup>I</sup>dr*, as well as the transport current *<sup>I</sup>tr*, for various voltages. For the considered case *EF* = 0.9, the driven current circulates counter-clockwise and deforms the flux dependence of the circular currents, which become asymmetric.

**Figure 8.** Circular current *Ic* = *Idr* + *Iφ* (solid curves), its driven component *Idr* (dashed curves), and the net transport current *Itr* (dotted curves) versus *φ* for various bias voltages: *eV* = 0.01, 0.5, 1.0, and 1.5. We assume a strong coupling Γ*L* = Γ*R* = 1, the chemical potentials are *μL* = *EF* − *eV*/2, *μR* = *EF* + *eV*/2, *EF* = 0.9, and *T* = 0.
