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We propose a double quantum dot (DQD) in parallel as a tractable model for a mesoscopic ring with an embedded quantum dot (QD), which functions as a QD interferometer. One of the DQDs (QD 1) represents the embedded QD with energy level *ε*_{1} and Coulomb interaction *U*, whereas the other (QD 2) acts as a reference arm with an enlarged linewidth due to the tunnel coupling to external leads. The conductance at temperature *T* = 0 is formulated simply in terms of the retarded and advanced Green’s functions of the DQD, which are exactly evaluated using the Bethe ansatz solution for the Kondo effect. Using this model, we address two controversial issues concerning the QD interferometer. The first issue concerns the shape of the conductance peak as a function of *ε*_{1}. We show a crossover from an asymmetric Fano resonance (Fano–Kondo effect) to a symmetric Breit–Wigner resonance (Kondo plateau) in the absence (presence) of *U*, by decreasing the connection between the QDs through the leads. The second is on the measurement of the transmission phase shift through QD 1 by a “double-slit experiment”, which is impossible in the two-terminal geometry because of the Onsager’s reciprocity theorem. In a three-terminal geometry, we discuss a possible observation of phase locking at *π*/2 in the Kondo regime.

Transport through an Aharonov–Bohm (AB) ring with an embedded quantum dot (QD), the so-called QD interferometer, has been one of the most important subjects in mesoscopic physics.^{1}^{–}^{4}^{)} This issue involves the interference effect, resonant tunneling through a discrete energy level in the QD, and the strong Coulomb interaction in the QD. Despite long-term experimental and theoretical studies, there still remain controversial issues. One concerns the measurement of the transmission phase shift through the QD as a double-slit experiment. Another is on the shape of the Coulomb peak, i.e., conductance peak as a function of energy level in the QD.

It is well known that the transmission phase shift through the QD cannot be measured by a QD interferometer in the two-terminal geometry.^{1}^{)} This is due to the restriction by the Onsager's reciprocity theorem. The conductance *G* satisfies ^{3}^{,}^{4}^{)} The phase measurement was first reported using the interferometer in a four-terminal geometry^{2}^{)} and has recently been examined in detail using a three-terminal device.^{5}^{–}^{7}^{)} In the Kondo regime, the phase shift through the QD should be locked at ^{8}^{–}^{10}^{)} This phase locking was also investigated experimentally using four- or three-terminal QD interferometers.^{6}^{,}^{11}^{–}^{15}^{)} It is nontrivial, however, how precisely the phase shift is measured using the multiterminal interferometers.

Another issue concerns the shape of the Coulomb peak in the QD interferometer. Kobayashi et al. observed an asymmetric shape with a peak and a dip due to the Fano resonance.^{16}^{)} The Fano resonance stems from the interference between a discrete energy level in the QD and continuum energy states in the ring.^{17}^{,}^{18}^{)} Remarkably, the magnetic flux Φ penetrating the ring changes the shape of the Coulomb peak, which can be expressed using a complex Fano factor.^{16}^{,}^{19}^{)} However, the other groups observed symmetric Coulomb peaks, which can be fitted to the Lorentzian function of Breit–Wigner resonance, e.g., Refs. 6 and 7. No criteria have been elucidated regarding the Fano or Breit–Wigner resonance in the QD interferometer.

Theoretically, Bułka and Stefański^{20}^{)} and Hofstetter et al.^{21}^{)} reported pioneering works on the Kondo effect in the two-terminal QD interferometer. Hofstetter et al. found an asymmetric Fano–Kondo resonance by applying the numerical renormalization group calculation to a model in which a QD is coupled to leads *L* and *R* (by *W*).^{21}^{)} Their works were followed by many theoretical studies, e.g., to elucidate various aspects of the Kondo effect,^{22}^{–}^{29}^{)} fluctuation theorem,^{30}^{)} and dynamics of electronic states.^{31}^{)} Recently, the Fano resonance was proposed to detect the Majorana bound states.^{32}^{,}^{33}^{)} The Fano–Kondo effect, however, has not been observed experimentally.

Two of the present authors extended the model of Hofstetter et al.^{21}^{)} to include the multiple conduction channels in the leads, to address the above-mentioned issues.^{34}^{)} The model is depicted in Fig. 1(c) and is the same as the previous model except that the tunnel couplings, *W*, depend on the states in leads *L* and *R*.^{35}^{)} The state dependence of the tunnel couplings results in a parameter *α* (*U* in the QD. Thus, the different experimental results for the shape of the Coulomb peak could be explained using

Figure 1. (a) Model of DQD in parallel for an AB ring with an embedded QD, the so-called QD interferometer, in the two-terminal geometry. The DQD consists of QD 1 with energy level *L* and *R*. The tunnel couplings between QD *j* and lead *L* [*R*], *k* [^{38}^{)} (b) Model of DQD in parallel in the three-terminal geometry with leads *L*,

In this paper, we propose another useful model for the QD interferometer, a double quantum dot (DQD) in parallel, shown in Fig. 1(a). One of the DQDs (QD 1) represents the embedded QD with energy level *U* in QD 1 is taken into account exactly using the Bethe ansatz solution for the Kondo effect. The advantages of this model are as follows. (i) The conductance at temperature *α*. The Coulomb interaction *U* in QD 1 can be involved in the Green's functions ^{34}^{,}^{35}^{)} (ii) The multiple channels in lead *α* can be taken into account by the linewidth function ^{36}^{)} (iii) Although we assume a single conduction channel in the AB ring in the present paper, our model can be straightforwardly extended to more general situations:^{37}^{)} e.g., *N* channels in the AB ring could be described by *N* energy levels in QD 2.

This paper is organized as follows. The DQD model and calculation method are given in Sect. 2. We introduce the Keldysh Green's functions to formulate the electric current under a finite bias voltage *V*. Then the simple formulae are derived for the conductance in the limit of *U*, changing the parameter *U*. In Sect. 4, we discuss the measurement of the transmission phase shift through QD 1 in a three-terminal geometry. We show that the phase locking at

We examine a model of DQD for the QD interferometer, depicted in Figs. 1(a) and 1(b) for two- and three-terminal geometries, respectively. The DQD consists of QD 1 with energy level

For the two-terminal system, the Hamiltonian is given by *L* and *R*, and *σ* in QD *j*, with *α* (*k* and spin *σ*, whose energy is denoted by *U* is present only in QD 1. In *j* and lead *α* depends on state *k* in the lead. ^{38}^{)}

We define the linewidth function *α*, which is represented by a *ε*-dependence is weak around the Fermi level (*j* due to the tunnel coupling to lead *α*. We set

For the off-diagonal elements of ^{36}^{)} characterizes the connection between the two QDs through lead *α* (*α* at energy ^{39}^{)} The interference by the AB effect is maximal when *k* in ^{34}^{)} Although

For the three-terminal system, lead *R* is divided into leads *k* belongs to lead *k* in lead

For both two- and three-terminal systems, the total linewidth function is defined by

The electrochemical potential in lead *α* is given by

We use the Keldysh Green's functions to formulate the electric current in the stationary state. The retarded, advanced, lesser, and greater Green's functions of the DQD are denoted by

Let us focus on the retarded Green's function *U*, *ϕ*-dependent effective energy level

The electron–electron interaction *U* modifies *z* is a factor of wavefunction renormalization by *U*.^{40}^{–}^{42}^{)} Since the phase shift ^{43}^{,}^{44}^{)} Hence, we can precisely calculate

In both the absence and presence of *U* in QD 1, the other elements of

The electric current is formulated using the Keldysh Green's functions along the lines of Ref. 45 for both two- and three-terminal geometries. The current from lead *α* to the DQD in Figs. 1(a) or 1(b) is given by *α*. In the stationary state, *α*,

In the absence of electron–electron interaction *U*, Eq. (28) turns out to be a well-known formula. In the two-terminal geometry in Fig. 1(a),

Now, we formulate the conductance at *U* in QD 1. The corresponding Green's functions are given by Eqs. (B·3), (B·4), (B·7), and (B·8) in Appendix B, respectively, using the self-energies due to the interaction *U*, *U* or higher, the term *U* although the effect of *U* is taken into account in

Similarly, the current in the three-terminal geometry in Fig. 1(b) is given by

In the present and next sections, we examine our model of DQD to simulate the QD interferometer in which the reference arm of the AB ring is weakly coupled to the leads, in accordance with previous works.^{21}^{,}^{34}^{)} We choose

In this section, we show the calculated results for the two-terminal system depicted in Fig. 1(a). We discuss the shape of the conductance peak as a function of energy level

*U*

We begin with the case without electron–electron interaction *U* in QD 1. Figure 2 shows conductance *G* as a function of

Figure 2. Calculated results for conductance *G* in the two-terminal geometry in the absence of *U*. *G* at temperature

In Fig. 2(a), the conductance peak has an asymmetric shape with a dip and a peak for the AB phase ^{17}^{,}^{18}^{)} In our model, QD 2 plays the role of the reference arm of the interferometer, in which the continuous states are present in the energy scale of ^{16}^{)}

The conductance shape becomes less asymmetric with decreasing *ϕ*, similarly to the Lorentzian shape of the Breit–Wigner resonance. The weak connection between QD 1 and QD 2 (or reference arm of the interferometer) causes negligible asymmetric Fano resonance. This should be the case of the experimental results obtained by Takada and co-workers.^{6}^{,}^{7}^{)} Note that the conductance can exceed

In the previous paper,^{34}^{)} qualitatively the same results were given for the shape of the conductance peak, using the model in Fig. 2(c). This means that our model of DQD is suitable for the description of the QD interferometer.

*U*

In the presence of *U* in QD 1, the conductance at *G* as a function of

Figure 3. Calculated results for conductance *G* in the two-terminal geometry in the presence of *U*. *G* at temperature

In the Coulomb blockade regime with an electron in QD 1 (^{21}^{)} In Fig. 3(a), the conductance shows an asymmetric shape with a dip, a gradual slope around the center of the Coulomb valley, and a peak in the absence of magnetic field (^{21}^{)}

With decreasing *ϕ*-dependent Fano–Kondo effect to the *ϕ*-independent Kondo plateau while changing the parameters

We should comment on the Kondo temperature *ϕ*-dependent Kondo temperature,^{40}^{,}^{46}^{)} *ϕ* dependence of ^{22}^{,}^{47}^{–}^{49}^{)}

In this section, we discuss the measurement of the transmission phase shift through a QD using the QD interferometer in a three-terminal geometry, as a double-slit interference experiment. We study a specific situation that the tunnel coupling between QD 2 and lead

Figure 4. Transmission phase shift through the DQD from lead *L* to lead *U*. We set ^{50}^{)}

We define the measured transmission phase shift by the AB phase *π*. In the three-terminal geometry with

For the intrinsic transmission phase shift, we consider *U*, *U*, *ϕ*, its *ϕ* dependence is negligibly small in the situations discussed below.^{50}^{)}

*U*

In Fig. 4, we plot the measured transmission phase shift *U*. We set (a) *π* owing to the restriction by the reciprocity theorem, as shown in Fig. 4(a). With an increase in

The intrinsic transmission phase shift,

*U*

Finally, we discuss the measured phase shift *U*. The intrinsic transmission phase shift

Figure 5. Transmission phase shift through the DQD from lead *L* to lead *U* in QD 1. We set ^{50}^{)}

We find that the measured phase

We studied the transport through a DQD in parallel as a model for the QD interferometer, in which one of the DQDs (QD 1) represents the embedded QD and the other (QD 2) plays the role of a reference arm with an enlarged linewidth *U* in QD 1, we derived the formulae of the conductance *G* at temperature

In the two-terminal geometry, we showed that the parameters *L* or *R*, are relevant to the shape of the conductance peak as a function of energy level *U*, conductance *G* shows an asymmetric peak of the Fano resonance for *U*, we observed a crossover from an asymmetric shape of the Fano–Kondo effect to the Kondo plateau. Adopting the three-terminal geometry, we demonstrated the measurement of the transmission phase shift through a QD by a “double-slit experiment” and showed a possible observation of the phase locking at

These results are in good agreement with those of a previous study using the model in Fig. 1(c),^{34}^{)} which indicates the validity of the DQD model for the QD interferometer. In the DQD model, the conductance formulae in Eqs. (34) and (36) are much simpler than those for the model in Fig. 1(c).^{34}^{)} Moreover, our model can be straightforwardly extended for more general situations, e.g., in the presence of multiple conduction channels in the upper arm of the ring.

To derive the conductance formulae, we neglected the term *U* implies that

## Acknowledgments

We appreciate fruitful discussions with Dr. Akira Oguri. This work was partially supported by JSPS KAKENHI Grant Numbers JP15H05870, JP20K03807, and JP21K03415 and JST-CREST Grant Number JPMJCR1876.

*U*

We derive the Keldysh Green's functions in the absence of *U*. We omit the spin index *σ* in this section.

First, we introduce the Green's functions of isolated lead *α*, without the tunnel coupling to the DQD, *α*, given by Eq. (29). The Fourier transformation (

We also introduce the Green's functions of isolated QD *j*, e.g., *j*.

In the presence of

Note that the same technique can be applied to *U* in QD 1. This procedure leads to Eqs. (23)–(25).

In the matrix form, Eq. (A·13) is rewritten as *t* results in

To derive the lesser and greater Green's functions, we consider a complex-time contour from ^{51}^{,}^{52}^{)} We introduce the contour-ordered Green's function by ^{51}^{–}^{53}^{)} we obtain

In the same way, the greater Green's function is given by

It is worth mentioning the *ϕ*-dependent real number for ^{54}^{–}^{56}^{)}

*U*

Now we consider the electron–electron interaction *U* in QD 1. We denote the self-energies due to the interaction by *U* by adding the superscript (0), e.g., Eq. (A·24) reads

For the contour-ordered Green's function, the Dyson equation is given by ^{51}^{–}^{53}^{)} we obtain

From Eq. (B·5),

In the same way, the greater Green's function is given by

The electric current *α* to the DQD is defined by Eq. (26). It is written as

In the absence of *U*, Eqs. (A·24) and (A·18) hold for

In the presence of *U*, we use Eq. (B·7) for

In the current formulae in Eqs. (32) and (35), the term *U* or higher. Therefore, we expect that the term can be neglected when obtaining the conductance

First of all, note that the self-energies due to the interaction *U*, *U* is present in QD 1 only in our model. Hence, using Eqs. (23)–(25),

Apart from a constant *α* (*L*) when *R*).

We prepare a supplementary system in a magnetic field in the opposite direction, or with the AB phase of

We comment on *π* for any

In the stationary state, ^{54}^{,}^{57}^{)} From Eq. (D·10), therefore, we conclude that

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