Abstract
Hybrid devices combining quantum dots with superconductors are important building blocks of conventional and topological quantuminformation experiments. A requirement for the success of such experiments is to understand the various tunnelinginduced nonlocal interaction mechanisms that are present in the devices, namely crossed Andreev reflection, elastic cotunneling, and direct interdot tunneling. Here, we provide a theoretical study of a simple device that consists of two quantum dots and a superconductor tunnelcoupled to the dots, often called a Cooperpair splitter. We study the three special cases where one of the three nonlocal mechanisms dominates, and calculate measurable groundstate properties, as well as the zerobias and finitebias differential conductance characterizing electron transport through this device. We describe how each nonlocal mechanism controls the measurable quantities, and thereby find experimental fingerprints that allow one to identify and quantify the dominant nonlocal mechanism using experimental data. Finally, we study the triplet blockade effect and the associated negative differential conductance in the Cooperpair splitter, and show that they can arise regardless of the nature of the dominant nonlocal coupling mechanism. Our results should facilitate the characterization of hybrid devices, and their optimization for various quantuminformationrelated experiments and applications.
Introduction
Superconducting hybrid nanodevices provide a promising platform for quantum architectures. While superconductors (SCs) allow for a spatially extended coherent state, nanodevices provide the confinement of electrons into 1D or 0D. The interplay of these properties is a key ingredient of novel promising qubit realizations, such as Majorana qubits [1,2] and Andreev qubits [3].
The basic physical mechanism behind these applications is the Andreev reflection, when a Cooper pair from the SC is transformed to two electrons in the normal conductor. The conversion of Cooper pairs has a special form called crossed Andreev reflection (CAR), when the two electrons originating from the Cooper pair ends up in spatially separated normal parts [47].
CAR is also a potential resource for quantum hardware. On one hand, it naturally generates entangled spatially separated electron pairs [7]. On the other hand, several novel topological superconducting proposals are based on CAR processes, such as the poor man’s Majorana setup [8], the Majorana chain [9], Tritops [1014], Majorana states in graphene [1517] and devices with even more exotic nonAbelian excitations, such as parafermions [1820]. CAR was studied experimentally in metallic nanostructures [2124] and later in socalled Cooperpair splitter devices, where two quantum dots (QDs) are weakly tunnelcoupled to a superconductor in a QD–SC–QD geometry [2532]. The QD–SC–QD setup serves also as the basic building block of the poor man’s Majorana setup [8] and the Majorana chain [9].
Strong tunnel coupling between a QD and a SC leads to the formation of Andreev bound states (ABSs) [3351] via local Andreev reflection (LAR). Due to the charging energy on the QDs, the QD–SC–QD geometry prefers CAR over the LAR and leads to the expectation that CAR hybridizes the states of the two QDs, generating the socalled Andreev molecular state [5254]. The first experimental realization towards such a state is already reported [55]. However, CAR is not the only coupling mechanism between the QDs. Electrons can be transmitted from one of the QDs to the other via virtual intermediate SC quasiparticle states, via the socalled elastic cotunneling (EC) process [21,23,5668]. Furthermore, if there is direct tunnel coupling between the dots, as in certain experimental realizations [69,70], then this interdot coupling (IT) also influences the spectrum and the dynamics.
Several previous works on similar systems completely neglect the possibility of EC processes [7,5254,71,72]. In other works EC coupling is taken into account [44,61,64,65,67,68,73], but in most of the cases it is defined as a constant coupling term that is equivalent to the IT coupling used here, therefore they can be incorporated to the same term. However, as we demonstrate below, in certain models the EC term is not constant, but it can depend on the onsite energies of the QDs. To keep the generality, we treat the energydependent EC and the independent IT term separately.
In an experimental realization of a QD–SC–QD setup, any of the three nonlocal coupling mechanisms (CAR, EC or IT) could dominate. The focus of this paper is to calculate measurable quantities and explore differences between the individual fingerprints of the three nonlocal processes, which allow one to identify and quantify the dominant nonlocal term. In particular, we describe the groundstate properties (phase diagram, average electron occupation) of the system, the zerobias conductance describing electron transport through the device in the presence of tunnelcoupled normal leads, and the excitation spectrum that is accessible via finitebias transport measurements.
The paper is organized as follows: First, we introduce the microscopic model of the Cooperpair splitter, and outline how to derive a simple effective Hamiltonian, which describes the nonlocal coupling mechanisms as individual Hamiltonian terms, from the microscopic model. Then, we analyze the groundstate properties, and the phase diagram and average electron occupation of the QD–SC–QD system. Subsequently, we outline the transport model describing the setup where each QD is coupled to a normal lead. Finally, we analyze the transport signatures of the different nonlocal coupling terms via the zero bias conductance, the finitebias conductance, and the triplet blockade effect.
Model
Microscopic Hamiltonian of the proximitized double quantum dot
Throughout this work, we study a standard Cooperpair splitter device. The setup is shown in Figure 1a. It consists of two QDs, each of them tunnelcoupled to its own normal (N) lead, and a common superconducting lead (SC). The Hamiltonian of the system is:
We assume that the level spacings of the dots are large, i.e., each QD has a single spinful orbital, which can be occupied by 0, 1 or 2 electrons. The first term in the Hamiltonian reads as
where ε_{α} is the onsite energy of QD_{α} (α = L,R), d_{ασ} () annihilates (creates) an electron on QD_{α} with spin σ, and U_{α} is the onsite Coulomb repulsion energy. Note that the interdot Coulomb repulsion in neglected here, since the SC lead between the QDs screens this interaction. Throughout this work, we assume identical Coulomb repulsion energies in the two dots, and use this energy scale U = U_{L} = U_{R} as the unit of energy.
The SC lead is described by the standard meanfield Bardeen–Cooper–Schrieffer (BCS) Hamiltonian
where ε_{Sk} is the dispersion of conduction electrons in the SC, c_{Skσ} () annihilates (creates) an electron in the SC with momentum k and spin σ, and Δ is the superconducting order parameter. Only one SC lead is present in our setup, hence Δ is chosen to be real. H_{SC} can be diagonalized by a Bogoliubovtransformation, where
resulting in
where is the quasiparticle energy.
The Hamiltonian of the normal leads is
where α = L,R is the lead index, ε_{α}_{k} is the dispersion relation, and c_{α}_{k}_{σ} () annihilates (creates) an electron with momentum k and spin σ in lead N_{α}.
Tunneling between the three leads and the two dots is described by the following terms:
where H_{T,SC} describes the tunneling between the QDs and the SC lead, while H_{T,N} between the dots and the N leads, with t_{Sα} (t_{Nα}) being the tunneling amplitude between the SC (N_{α}) lead and QD_{α}. Tunneling to the SC will be treated coherently, while tunneling to the N leads is assumed to be weak, and are treated by Fermi’s Golden Rule in the transport model outlined below.
Finally,
describes interdot tunneling (IT), i.e., direct tunneling between the QDs, with an amplitude t_{LR}.
Effective Hamiltonian of the proximitized double quantum dot
The complete Hamiltonian H specified above is infinitedimensional. However, if the temperature T is low and the superconducting gap Δ is large, then one may simplify the Hamiltonian by eliminating the superconducting quasiparticles from the description. Technically, this is done by integrating out the quasiparticles using secondorder perturbation theory in the SCQD tunneling term H_{T,SC}. This procedure yields a 16dimensional lowenergy effective Hamiltonian for the double QD, which describes the superconducting proximity effect of the SC lead on the double QD. Here, we describe this effective Hamiltonian and the procedure to obtain it.
For this, we consider the Hamiltonian without the N leads, H_{QD} + H_{SC} + H_{T,SC} + H_{IT}. (We will take into account the N leads later to describe transport.) Assuming Δ U and further neglecting the QDSC tunneling H_{T,SC}, the 16dimensional quasiparticlefree lowenergy subspace is energetically wellseparated from other states containing a finite number of quasiparticles. The lowenergy subspace is spanned by the product basis, the products of particlenumber eigenstates of each QD, namely,
where the arrows denote the spin states of the electrons. We will use the notation We perform secondorder Schrieffer–Wolff perturbation theory in the tunneling term H_{T,SC} to obtain the effective Hamiltonian for the 16dimensional lowenergy subspace. See Supporting Information File 1 for the derivation and the validity conditions. As a result of this procedure, we find that the QDSC tunneling H_{T,SC} generates three coupling terms in the effective Hamiltonian: (i) a local (singledot) pairing term, called local Andreev reflection (LAR), (ii) a nonlocal (interdot) pairing term, called crossed Andreev reflection (CAR), and (iii) an effective interdot tunneling term, called elastic cotunneling (EC):
In H_{eff}, the second and third terms read as
The effective parameters Γ_{LAR,α} and Γ_{CAR} are related to each other on the level of the presented model, i.e., neglecting the spatial separation of the QDs (see, e.g., [52] and Supporting Information File 1). However, for the remainder of this work, we will consider Γ_{LAR,α} and Γ_{CAR} as independent parameters, since the CAR mechanism is expected to be suppressed, when a finite distance between the QDs is introduced [74].
The EC term describes singleelectron tunneling between the QDs via SC. This term, in contrast to the LAR and CAR coupling, has a strong dependence on the onsite energies ε_{L}, ε_{R} of the QDs. For example, the EC matrix element coupling the and states is well approximated by
whereas the matrix element coupling the and states is
where the strength of the EC mechanism is characterized by the dimensionless parameter γ_{EC} = Γ_{CAR}/Δ. See Supporting Information File 1 for the derivation and for the complete list of the matrix elements.
To illustrate the dependence of these matrix elements on the QD onsite energies, we plot the two matrix elements shown in Equation 11 and Equation 12 in Figure 1b, using Δ = 5U and Γ_{CAR} = 0.1U. The two matrix elements, determined from secondorder perturbation theory, plotted in Figure 1b vanish at ε_{L} = −ε_{R} and ε_{L} + U_{L} = −ε_{R}−U_{R}, respectively. Of course, if such a contribution vanishes, then higherorder terms neglected here may actually be important.
Importantly, fermion parity and spin are conserved in our effective model. This implies that the 16dimensional effective Hamiltonian has a block structure; more precisely, there are six orthogonal subspaces that are not mixed by the effective Hamiltonian. These invariant subspaces are shown in Figure 1c. The first invariant subspace (singlet  S, left panel of Figure 1c) contains the five spinsinglet states with even number of electrons on the QDs: the empty and the doubly occupied states and the spinsinglet combination of the (1,1) states, The second and third invariant subspaces (doublet  D, right panel of Figure 1c) contain the states with odd number of electrons. Since we do not account for a magnetic field, these eight states are decomposed into two invariant subspaces with different total spin z component
Each energy eigenvalue in one doublet subspace has an equal partner in the spectrum of the other doublet subspace. The three spintriplet combinations of the (1,1) states, i.e., and
remain uncoupled from each other and from the other invariant subspaces, and these three states have the same energy eigenvalue.
In Figure 1c, the arrows visualize the tunnelinginduced matrix elements coupling the basis states of the effective Hamiltonian. Note that in Figure 1c, we use the singlet–triplet basis instead of the product basis. The tunneling processes giving rise to the coupling matrix elements indicated by red arrows on Figure 1c, are illustrated in Figure 1d. For instance, one of the LAR matrix elements corresponds to transferring a Cooper pair from the SC to QD_{R} through a virtual intermediate state, in which one electron occupies QD_{R} and one quasiparticle is present in the SC. The analogous CAR matrix element corresponds, again, to extracting a Cooper pair from the SC, but in this case the electrons end up in different QDs. Both the EC and the IT matrix elements correspond to the transfer of an electron from one QD to the other. In the case of EC, there is an intermediate virtual state with one quasiparticle in the SC, but in the case of IT, the tunneling is direct. The difference between EC and IT processes results an important difference of their matrix elements: for EC, they depend on the onsite energies [see, e.g., Equation 11 and Equation 12], while for IT they do not (see Equation 8).
In what follows, we will rely on the numerically obtained eigenvalues E_{χ} and eigenstates of the effective Hamiltonian H_{eff} of Equation 9. Furthermore, we will use U = U_{L} = U_{R} as the energy unit, Γ_{LAR,}_{L} = Γ_{LAR,}_{R} = 0.25U, and will focus on the parameter range Γ_{CAR} [0,0.1]U, γ_{EC} [0,0.15], t_{LR} [0,0.1]U. To convert our results to physical units, we can use, e.g., U = 1 meV; then the above numbers correspond to an experimentally realistic parameter set.
Groundstate Properties
Here we analyze the groundstate properties of the effective Hamiltonian H_{eff}, namely, the groundstate degeneracy, the fermion parity and the average electron occupations of the QDs, as functions of the onsite energies ε_{L} and ε_{R}, for different values of the nonlocal coupling terms CAR, EC and IT (see Figure 2a–e). First we analyze the fingerprints of the three nonlocal coupling mechanisms, onebyone, and finally we show an example where all coupling terms are finite. We show that although CAR and EC couple different states (see Figure 1b), they produce rather similar phase diagrams, but IT can be clearly distinguished from the previous two. In an experimental situation when one of these mechanisms is dominant, our results can be used to identify that dominant mechanism. We also note that in an actual experiment, it is challenging to effectively tune the ratio of these parameters.
The left column of Figure 2 shows the phase diagram, the middle column shows the zerobias conductance of QD_{L}, and the right column shows the average electron occupation of QD_{L} for different Γ_{CAR}, γ_{EC} and t_{LR} values. Note that for all cases the local pairing term is finite, Γ_{LAR,}_{L} = Γ_{LAR,}_{R} = 0.25U. In this subsection, we discuss the phase diagram and the average electron occupation, and return to the zerobias conductance results later, after introducing the transport model.
The phase diagrams of the QD–SC–QD system display the dependence of two groundstate properties on the onsite energies ε_{L} and ε_{R}: (i) the degree of degeneracy of the ground state, and (ii) the fermion parity of the ground state. For example, Figure 2a shows the phase diagram without nonlocal couplings, i.e., Γ_{CAR} = γ_{EC} = t_{LR} = 0. Different colors correspond to different groundstate degeneracies: Yellow denotes a twofold degenerate ground state, dark blue denotes a nondegenerate ground state, and light blue denotes a fourfold degenerate ground state. The ground state has even (odd) fermion parity in the dark blue and light blue (yellow) regions. Note that such a phase diagram should be regarded as the generalization of the stability diagrams of nonsuperconducting double QDs, see, e.g., Figure 2 of [75].
In Figure 2a, where nonlocal couplings are absent, the two QDs are independent, thus the phase boundaries are vertical and horizontal lines. (Recall that interdot Coulomb repulsion is neglected.) The dark blue regions correspond to a singlet (S), unique ground state, where both QDs are in the bonding combination of the states and Note that in the absence of nonlocal couplings, the state remains uncoupled from the other four states in the singlet subspace, and degenerate with the three triplets and As a consequence, the groundstate degeneracy in the central, light blue region is fourfold. The two ground states of the yellow regions are drawn from the doublet (D) subspace of Figure 1b [35].
In general, if a nonlocal coupling is turned on, then the light blue regions (fourfold degenerate ground state) disappear, see Figure 2b–e. The reason for that is as follows. Due to the nonlocal coupling, the state couples to the other four singlet states Therefore, the energy of the lowestenergy singlet eigenstate will be lower than the energy of the triplets. This mechanism leaves two possibilities for the ground state: nondegenerate singlet, or a twofold degenerate doublet. The only exception is a line, when EC is the only finite nonlocal mechanism, where the fourfold degeneracy is preserved, which we discuss below.
The case of finite CAR coupling is presented in Figure 2b with Γ_{CAR} = 0.1U. Besides the disappearance of the fourfold degenerate light blue region, another difference compared to Figure 2a is the merging of the singletphase regions along the diagonal (i.e., the ε_{L} = ε_{R} line) and the merging of the doubletphase regions parallel to the diagonal. These features are consequences of the CAR coupling, and can be understood via simple perturbative arguments.
For example, consider the top left quadruple point in the phase diagram of Figure 2a. At this point, the most relevant doublet states are and These states are coupled directly by CAR, as shown in Figure 1b; as a consequence, the bonding combination of these will form the doublet ground state, with an energy lowered by approximately Γ_{CAR} due to the nonlocal coupling. On the other hand, the most relevant singlet states are and S(1,1), which are not coupled directly by CAR, see Figure 1b. In conclusion, the doublet ground state will have a lower energy than the singlet ground state in the top left quadruple point, explaining the merge of the doublet phase parallel to the diagonal in Figure 2b. Similar considerations apply to the other three quadruple points of Figure 2a.
Consider now the case when the only nonlocal coupling mechanism in the setup is EC. For this case, we can infer the groundstate character from a perturbative consideration similar to the one above. This consideration yields the expectation that the singlet (doublet) phases would merge along (parallel to) the skew diagonal (i.e., the line ε_{R} = U − ε_{L}) of the phase diagram, in contrast to the case of CAR. However, the numerically evaluated phase diagram for this case (γ_{EC} = 0.15), plotted in Figure 2c, shows that the phase diagram is actually very similar to Figure 2b.
To understand this surprising feature, we have to (i) go beyond the previous firstorder perturbative analysis, and (ii) go beyond the qualitative arguments based on the selection rules of Figure 1b, i.e., taking into account the onsite energy dependence of the EC coupling matrix elements, exemplified in Equation 11 and Equation 12. For example, consider the top right quadruple point in Figure 2a. Here, the lowestenergy states are dominantly and from which the doublet states and should be coupled directly by EC (see Figure 1b), but the corresponding matrix elements are proportional to ε_{L} + ε_{R} (see Equation 11), and hence are strongly suppressed. On the other hand, the two other states, and are coupled by LAR and EC in second order via intermediate states, by (cf. Figure 1b)
which includes the matrix element, which is not suppressed. Hence, at the top right quadruple point, this secondorder hybridization results in a lowered energy of the singlet ground state. Similar considerations explain the features of the phase diagram in Figure 2c at all four quadruple points.
A difference between Figure 2b and Figure 2c is the presence of a light blue skew diagonal line at the central region of Figure 2c. Along this line, the EC matrix element vanishes (see Equation S13 in Supporting Information File 1), and therefore the state is decoupled from the other singlet states, and remains degenerate with the triplets, preserving the fourfold degeneracy.
Figure 2d shows the phase diagram for the case when the only nonlocal coupling mechanism is IT, for t_{LR} = 0.1U. Here, the singlet and doublet phases merge parallel to the skew diagonal. This is explained by arguments analogous to the firstorder perturbative considerations outlined above, keeping in mind that the nonlocal coupling matrix elements of IT do not depend on the QD onsite energies.
As we discussed in the introduction the nature of the EC coupling is model dependent, in certain cases it has no dependence on the onsite energies, hence it is indistinguishable from IT coupling. In such cases the phase diagram is sufficient to distinguish between the CAR and the EC/IT couplings.
In conclusion, if we assume that only one nonlocal coupling mechanism is present, then CAR and onsite energy dependent EC produce rather similar phase diagrams, but they can be clearly distinguished from the case of IT.
All phase diagrams of Figure 2a–d are symmetric in two ways: (i) for the transformation (ε_{L},ε_{R}) (ε_{R},ε_{L}), and (ii) for the transformation (ε_{L},ε_{R}) (−U_{L}−ε_{L},−U_{R}−ε_{R}). The property (i), which we call the left–right symmetry, originates from the symmetric choice of local parameters, i.e., U_{L} = U_{R}, Γ_{LAR,}_{L} = Γ_{LAR,}_{R}. The property (ii) is the result of a particle–hole symmetry of the system, as discussed in the following.
A particle–hole transformation converts the filled electron states to empty ones and vice versa, i.e., exchange the role of creation and annihilation operators. Eight different particle–hole transformations are introduced and discussed in Supporting Information File 1. One example is the transformation (iv) in Table S1 of Supporting Information File 1, corresponding to the mapping Each transformation can be represented as a unitary transformation W on the 16dimensional Fock space. For each Hamiltonian terms H(ε_{L},ε_{R}), these transformations connect the inverted points of the phase diagram (see Supporting Information File 1 for details), namely
These transformations correspond to an inversion in the phase diagram to the central point (ε_{L},ε_{R}) = (−U_{L}/2,−U_{R}/2), usually called particle–hole symmetric point. We say that the transformation W is the particle–hole symmetry of the Hamiltonian term H, if Equation 13 is an equality. All coupling Hamiltonians and H_{IT} have a few such particle–hole symmetries, but each term has a different set of those, see Table S1 of Supporting Information File 1. If there exists a single particle–hole transformation that is a particle–hole symmetry of all coupling terms forming the Hamiltonian, then the phase diagram (along with other quantities) reflects the particle–hole symmetry. For example, the phase diagram in Figure 2d, where IT is the only nonlocal coupling mechanism, shows particle–hole symmetry, since transformation (iv) in Table S1 of Supporting Information File 1 is a particle–hole symmetry of the Hamiltonian from which and are omitted.
Finally, consider the general case, having all nonlocal couplings finite, Γ_{CAR} = t_{LR} = 0.1U and γ_{EC} = 0.02. Figure 2e shows the phase diagram for this case. First, the left–right symmetry is apparent that is still a consequence of the left–right symmetric choice of the parameter values. Second, the particle–hole symmetry is absent in Figure 2e. This is consistent with the fact that none of the eight particle–hole transformations is a particle–hole symmetry of all terms in this general Hamiltonian (see Table S1 in Supporting Information File 1).
The phase boundaries between ground states of different fermion parities can be mapped by lowenergy transport [75]. The expected zerobias conductance, with peaks following the evenodd phase boundaries, is illustrated by the middle column of Figure 2 and will be discussed in more detail below. A drawback of transport analysis is that the coupling to the electrodes lead to the broadening of conductance peaks (which is neglected from the model presented here). One may reduce broadening by decreasing the coupling at the price of decreasing the currents, too.
Charge sensing [76,77] is another method to map out the boundaries of the phase diagram, as we illustrate in the right column of Figure 2. A charge sensor is usually engineered to be mostly sensitive to the average electron occupation of one of the quantum dots, say, QD_{L}. Compared to the conductance measurement through the QD–SC–QD system, charge sensing has the advantage of conceptual simplicity, and the measurability without additional N leads attached to the QD–SC–QD system; but might have the disadvantage of a more complex device design, since the charge sensor is an additional device element. Similar methods, yielding information related to average electron occupation, are based on reflectometry with electromagnetic radiofrequency signals [78,79] or microwave resonators [80,81]; those are not discussed further in this work.
The groundstate average electron occupation in QD_{L} is expressed as
We plot as the function of the onsite energies ε_{L} and ε_{R} in the right column of Figure 2, for the parameter values providing the previously discussed phase diagrams.
In the absence of nonlocal couplings (Figure 2a), the QDs are independent, therefore does not depend on ε_{R}. We emphasize, since it is not apparent in the density plot in Figure 2a, that values of are not restricted to the integer values 0, 1 and 2. This is because the LAR mechanism provides coherent coupling within a given fermionparity sector between states with different electron numbers, see Figure 1b. In fact, as a function of ε_{L} slightly decreases in the yellow ( ≈ 2) and black ( ≈ 0) regions in the right panel of Figure 2a, its value is strictly = 1 in the green region, and jumps abruptly at the boundaries.
The nonlocal couplings are switched on in Figure 2b–e. Similarly to Figure 2a, the average electron occupation decreases as ε_{L} is increased, and the jump locations in follow the phase boundaries. The jumps are more pronounced along the vertical phase boundaries, i.e., in a chargesensing experiment the measurement of QD_{L} maps out the vertical phase boundary lines more efficiently. Due to the finite nonlocal couplings, the variation of the average electron occupation within a given fermionparity sector is smooth, as shown in Figure 2b–e.
Due to the left–right symmetry, for all cases presented here, the average electron occupation of QD_{R}, that is, , can be obtained by mirroring to the diagonal. Therefore, in the map the horizontal phase boundary lines are more pronounced. This allows for the measurement of the phase boundaries by measuring the occupation of the two QDs independently.
In this subsection, we have shown that on contrary to expectations, the CAR and the EC mechanisms produce rather similar phase diagrams as functions of ε_{L} and ε_{R}, but IT can be clearly distinguished from the previous two mechanisms. Furthermore, the measurement of the average electron occupation of the QDs allows for determining the phase boundaries, even in the absence of normal electrodes tunnelcoupled to the QD–SC–QD system.
Transport Calculation
As pointed out earlier, a chargesensing measurement is demanding, since the addition of the charge sensor complicates device fabrication. However, the groundstate phase diagrams discussed above can also be explored experimentally by electronic transport measurements, utilizing two additional N leads besides the SC lead (see Figure 1a) and low bias voltages. We will demonstrate this using the results shown in the middle column of Figure 2. Transport measurements using a sufficiently large bias voltage allow to determine energy gaps above the ground state. We will show (see below Figure 4) that such finitebias measurements can distinguish the CARdominated and ECdominated cases, i.e., the two cases that are not distinguished by the groundstate properties shown in Figure 2b,c.
The transport setup we wish to describe is shown in Figure 1a. Throughout this work, we assume that the SC lead is grounded, μ_{SC} = 0, and the two N leads are biased symmetrically, μ_{NL} = μ_{NR} = μ_{N}, with the convention that for positive (negative) μ_{N}, the electrons tend to flow into (out from) the SC lead.
Our transport model is based on the effective Hamiltonian H_{eff} of Equation 9 describing the QD–SC–QD system. In addition, here we also take into account the Nlead Hamiltonians and the leadQD tunneling Hamiltonians, that is, H_{N} + H_{T,N}. We describe the electronic transport in this device using a classical master equation, where the tunnel rates between the N leads and the QD–SC–QD system are obtained perturbatively, from Fermi’s golden rule.
The classical master equation describes the time evolution of the occupation probabilities P_{χ}(t) of the 16 energy eigenstates of H_{eff}, and reads
with the normalization condition The transition rates
are obtained from the leadingorder term in Fermi’s golden rule as
where α {L,R} is the lead index, is a characteristic tunneling rate between the normal lead N_{α} and the SC–QD–SC system, ρ_{Nα} is the density of states at the Fermi energy in the lead N_{α}, and is the Fermi function.
We calculate the stationary (dP_{χ}/dt = 0) solution of Equation 14 to obtain the steadystate occupation probabilities P_{χ}, and use the latter to evaluate the steadystate current in lead N_{α} via
Since we use a leadingorder Fermi’s golden rule, each nonzero transition rate corresponds to the tunneling of a single electron between a lead and the QD–SC–QD system, therefore connects states with different fermion parities. The differential conductance of lead N_{α} is calculated by numerically differentiating the current by the chemical potential of the normal leads, i.e.,
We plot the conductance in the units of the conductance quantum G_{0} = 2e^{2}/h.
In the following, we will use Γ_{NL} = Γ_{NR} = Γ_{N} = 0.005U and k_{B}T = 0.005U. If our energy unit is U = 1 meV, then Γ_{N} = 5 μeV. Note that in our model, the currents and differential conductances are simple linear functions of Γ_{N}, since we employ a leadingorder Fermi’s golden rule. Furthermore, the broadening of the conductance resonances in our results is caused only by the finite temperature of k_{B}T = 0.005U ≈ k_{B}·60 mK, since the presented model neglects the lifetime broadening. Experimentally, lifetime broadening of the conductance resonances might be dominant over the thermal broadening. In this case, the line width can be reduced by decreasing the tunnel coupling to the N leads, at the price of suppressing the currents.
Results of the Transport Simulation
Here, we present the results we obtained from the transport model of the previous section. First, we discuss how to experimentally map the phase boundaries using zerobias conductance measurements, as presented in Figure 2. Second, we demonstrate that finitebias differential conductance measurements provide a means to distinguish a system dominated by CAR from one dominated by EC. Third, we demonstrate and analyze the appearance of negative differential conductance in our setup, which is often attributed to the triplet blockade [52,53] and the CAR mechanism; here we show that not only CAR but any of the three nonlocal coupling mechanism can result in triplet blockade and a corresponding negative differential conductance.
Zerobias conductance
The zerobias conductance of lead N_{L}, G_{L} is shown in the middle column of Figure 2, for the five previously discussed cases, i.e., in the absence of nonlocal couplings Figure 2a, when having only one of them turned on (Figure 2b–d), and the general case of all three having finite values (Figure 2e). For all five cases, G_{L} shows a resonant enhancement along the phase boundaries. The conductance in Figure 2b–e shows further enhancement in the vicinity of the quadruple points seen in the left panel of Figure 2a.
Due to the left–right symmetric choice of the parameter values, similarly to the average electron occupation, the conductance G_{R} of QD_{R} can be obtained from G_{L} by mirroring the latter to the diagonal ε_{L} = ε_{R} line.
In conclusion, the zerobias measurement is a sufficient tool to locate the phase boundaries in the ε_{L}–ε_{R} plane, and to determine whether the interaction between the dots is due to the proximity of the superconductor (CAR or EC dominates over IT), or not (IT dominated).
Finitebias conductance
Here, first we show how the presence nonlocal couplings affect the finitebias transport for a general case, then we show that CAR and EC provide different fingerprints, and therefore these measurements can distinguish between a CARdominated setup and an ECdominated setup.
Figure 3a shows the finitebias conductances G_{L} and G_{R} of the two leads, in the absence of nonlocal couplings, for a fixed value ε_{R} = −1.2U, as the function of ε_{L} and the bias voltage μ_{N}. Figure 3b shows the same quantities, for a general case when all three nonlocal couplings are switched on.
Without nonlocal couplings (Figure 3a), Andreev bound states (ABSs) are formed on each QD due to the coupling to the superconductor. The ABSs on the two QDs are independent. In Figure 3a, the conductance G_{L} in the left panel shows an eyeshaped resonance with two crossing points at zero bias. This characteristic resonance is the usual fingerprint of an ABS in a single QD (see, e.g., [35]). The resonance maps the excitation energy between the two lowestenergy eigenstates of the QD_{L}SC subsystem: the bonding linear combination of the singletlike state, and the doublets, In the central region, the doublets are the ground states, while on the sides, the ground state is the singlet Note that the higherenergy antibonding singlet state, can be accessed from the doublet states, but the corresponding conductance resonance lies out of the energy window used here. Note also that in our model, transitions between the two singlet states are forbidden, since these two states both have even fermion parity.
As noted earlier, the positive μ_{N} sides of the finitebias conductance plots of Figure 3 correspond to transport by adding electrons from the N_{L} to QD_{L}, while the negative μ_{N} sides represent the opposite processes. Although the finitebias conductance plots are not symmetric to the zerobias axis, the conductance lines are positioned symmetrically, since (i) the spectrum does not depend on μ_{N} and (ii) transition between the eigenstates with different fermion parity is possible either by adding or by removing one electron. Generally, the tunnel rate for adding an electron is different from the tunnel rate for removing an electron, therefore the heights of the conductance resonances at positive and negative bias are different.
A complete transport cycle constitutes of the transport of two electrons, e.g., for positive bias, the first electron enters the QD with positive energy, bringing it to an excited state, while the second electron enters with negative energy, making the system relax back to its ground state. These two electrons, which reside in the leads at the beginning of the cycle, are absorbed by the SC as a Cooper pair by the end of the cycle. Analogously, for negative bias, the first (second) electron leaves the QD with negative (positive) energy.
The finitebias conductance plots in the absence of nonlocal couplings (Figure 3a) exhibit an inversion symmetry, G_{L}(ε_{L}, μ_{N}) = G_{L}(−U_{L} − ε_{L}, −μ_{N}). We attribute this to the particle–hole symmetry discussed above, with an extension of the particle–hole transformations to the lead electrons. In the presence of nonlocal couplings, if the Hamiltonian is particle–hole symmetric, then the inversion symmetry G_{L}(ε_{L}, ε_{R}, μ_{N}) = G_{L}(−U_{L} − ε_{L}, −U_{R} − ε_{R}, −μ_{N}) would be reflected in the finitebias conductance plots (not shown).
In the finitebias conductance plot of QD_{R} only two horizontal lines are present (see the right panel in Figure 3a). This is due to the independence of the QDs: ε_{L} has no effect on the transport via QD_{R}. The position and the amplitude of the lines is the same as the ones in the left panel of Figure 3a, at ε_{L} = −1.2U.
When the nonlocal couplings are switched on, they induce a crosstalk between the two QDs, resulting in a strong modification of the finitebias conductance plots. This is shown in Figure 3b, for a general case with all three nonlocal coupling being finite, Γ_{CAR} = 0.1U, γ_{EC} = 0.02 and t_{LR} = 0.06U. The previously discussed features, i.e., the eyeshaped resonances in G_{L} and the horizontal lines in G_{R}, appear in both finitebias conductance plots of Figure 3b. In addition, anticrossings between conductance resonances are openened by the nonlocal couplings wherever the excitation energies of the two uncoupled dots would coincide. The two most pronounced anticrossings are marked with arrows in the left panel of Figure 3b, open at ε_{L} = ε_{R} = −1.2U and ε_{L} = −U − ε_{R} = 0.2U.
The analysis of the conductance resonances as functions of the onsite energies allows one to distinguish between the coupling terms. This is shown in Figure 4, where the finitebias conductance along the skew diagonal ε_{R} = −U − ε_{L} of the phase diagram is shown, for only one finite nonlocal coupling. Note that along this skew diagonal, we have G_{L} = G_{R}. Figure 4a, Figure 4b and Figure 4c) show the case of CAR, EC and IT coupling, with Γ_{CAR} = 0.1U, γ_{EC} = 0.15 and t_{LR} = 0.1U, respectively. The conductance resonances in Figure 4a and Figure 4b show an appreciable splitting, which is absent in the right panel. The size of the splitting is constant for CAR, while for EC it is the largest at the particle–hole symmetric point, ε_{L} = ε_{R} = −U/2 and decreases further from it. This difference between the finitebias conductance can used to identify whether the dominant nonlocal coupling mechanism is CAR or EC. As expected from the qualitatively different nature of the phase diagrams of Figure 2b and Figure 2c compared to Figure 2d, the case of IT is also qualitatively different in terms of finitebias conductance (Figure 4a and Figure 4b compared to Figure 4c).
Further finite bias data and discussion on the features of Figure 3 and Figure 4 is provided in Supporting Information File 1.
Triplet blockade
In certain cases, the finitebias conductance plots show lineshaped regions of negative differential conductance (NDC). Examples, marked with white circles, are shown in the right panel of Figure 3b and in Figure 5. Further examples are shown on Figure S1 and S2 in Supporting Information File 1.
The NDC lines appear when a socalled blocking state becomes energetically available as μ_{N} reaches its excitation energy. The blocking state starts to be populated, but the rates of transitions out of this state are small, therefore the population is accumulating in the blocking state, reducing the current. When the blocking states are spintriplets, then this effect is called triplet blockade. The triplet blockade is often referred to as a hallmark of the CAR coupling in the literature. It is argued that the spin incompatibility of the triplet state in the QDs and the spinsinglet Cooper pairs in the SC prohibit CAR coupling of the QDs [52,53]. Here, we show that the presence of NDC lines is not exclusive for the CAR, but they can appear for all three nonlocal couplings.
Figure 5a shows an example of tripletblockaderelated NDC lines in the CAR coupled case, for Γ_{CAR} = 0.1U and ε_{R} = −U/2. Let us focus on the cut at ε_{L} = 0.1U, marked with the dashed orange vertical line. For these parameter values, we show the relevant energy levels in the left panel of Figure 5b. The ground state is a doublet,
and the first excited state is a singlet, − − − + with a dominant contribution. The second excited level is the threefold degenerate triplet level. All excited states are reachable from the ground state via electron tunneling, but tunneling transitions between two different excited states are forbidden, as both states have even fermion parity.
In the left panel of Figure 5b, we illustrate the tunneling transitions (arrows) relevant for electron transport at this parameter point. If the bias voltage is sufficiently large to induce a transition to the singlet excited state, then a wellconducting transport channel opens up, characterized by the uphill rate W_{SD} and the downhill rate W_{DS}. In this transport cycle, the uphill transition, is dominated by the process, while the downhill transition, is dominated by and processes. W_{SD} = 0.383Γ_{N} and W_{DS} = 0.297Γ_{N} are the relevant transition rates.
When the bias voltage is further increased, then the triplet states are also populated, see Figure 5b. In this case, the uphill transition from the ground state to the triplet is dominated by the process, with the rate W_{TD} = 0.465Γ_{N} rate. The downhill transitions from the triplets are possible, and their rates are dominated by the and processes, but these rates are small, W_{DT} = 0.019Γ_{N}, more than one order of magnitude smaller than the other transition rates. This small downhill transition rate results in the accumulation of the population in the triplet states, and the reduction the singlet and doublet populations that would provide efficient conduction. Therefore, the net effect of the triplet states becoming available upon increasing the bias voltage is the reduction of the current, and as a consequence, the appearance of the NDC lines in the differential conductivity.
We show the level diagram and the transitions, in the absence of CAR coupling, in the right panel of Figure 5b. In this case, due to the absence of CAR, the component is missing from the ground states, and the is missing from the excited states, which are parts of the two main downhill processes enabling the transition, as we have seen above. Therefore all four, now degenerate, (1,1) states block the transport, as illustrated on the right panel of Figure 5b.
The comparison of the two cases is shown in Figure 5c, where the current I_{L} in the left lead is plotted as the function the bias voltage μ_{N} with blue (orange) line without nonlocal coupling (with Γ_{CAR} = 0.1U) for the same ε_{L} = 0.1U and ε_{R} = −U/2 as above. In the presence of finite CAR, as the bias voltage is increased from zero, first a well conducting channel opens around μ_{N} ≈ 0.1U, and the current jumps down at μ_{N} ≈ 0.17U due to the triplet blockade. In contrast, in the absence of nonlocal couplings, above μ_{N} ≈ 0.15U the current is comparably small as in the blockaded case. Note that in the presence of CAR coupling, the triplet excitation is shifted to somewhat higher energy, since the energy of the ground state is lowered due to the nonlocal coupling.
The orange line in Figure 5c indicates a negative differential conductance. However, the current does not drop to zero but forms a finite plateau for μ_{N} > 0.17U, i.e., the triplet blockade is ‘incomplete’, and there is a finite ‘leakage current’. As discussed by Trocha and Weymann in [53], this leakage current is due to the fact that double occupancy of the QDs is allowed in this model. In contrast, in models neglecting double occupancy, e.g., by assuming infinite onsite Coulomb repulsion U→∞, this leakage current vanishes exactly [52].
As we have seen, the presence of the high current due to the wellconducting singlet–doublet channel requires that (i) the singlet energy is brought below the energy of the triplets, and (ii) the state is mixed with the empty or double occupied states. These conditions can be induced not only by CAR, but by any of the three nonlocal couplings described here. Thus, for all the three cases, tripletblockadeinduced NDC lines can appear. In Figure 5c–e we map the onsite energy regions where the triplet blockade is present for the three different couplings. We marked with black those (ε_{L},ε_{R}) values where there exists a bias voltage, at which the absolute value of the current I_{L} decreases, and the triplet occupation of the QD–SC–QD system increases by at least 0.25 simultaneously. The black regions of Figure 5d–f are not symmetric for mirroring to the diagonal ε_{L} = ε_{R} line, thus the NDC lines are present in different (ε_{L},ε_{R}) regions for QD_{L} and QD_{R}. Note that due to the leftright symmetry the NDC maps calculated from I_{R} would be the same as the presented ones mirrored to the diagonal.
In our setup, NDC can also appear in cases where it is not caused by triplet blockade. In the main text, we have been focusing on tripletblockadeinduced NDC, but an example for nontripletinduced NDC is show in Supporting Information File 1 in Figure S1b.
In conclusion, we have shown that the triplet blockade is not exclusive for CAR, but can appear also in the presence of any of the three nonlocal coupling mechanisms. In fact, a blockade can arise even in the absence of nonlocal couplings, even though the singlet state is also blocking in that case. The presence of a nonlocal coupling makes the singlet state well conducting, and makes the triplet blockade effect easily observable in a transport experiment by the appearance of NDC lines.
Conclusion
We have analyzed the spectrum of a QD–SC–QD system in the presence of different nonlocal coupling mechanisms: CAR, EC and IT. Our aim was to calculate the effects of these on measurable quantities (phase diagrams, average electron occupations, zerobias and finitebias conductance), with the goal of identifying features that are characteristic for each nonlocal coupling mechanism.
The phase diagram of the system can be mapped via chargesensing or zerobias conductance measurements. We find that CAR and EC produces very similar phase diagrams, thus measuring the phase diagram alone would not allow one to distinguish between these two nonlocal coupling mechanisms. However, IT produces a qualitatively different phase diagram, and hence it should be straightforward to identify if the nonlocal couplings are dominated by IT in a device. Furthermore, we have demonstrated that finitebias measurements could be used to distinguish between the cases when CAR or EC dominates the nonlocal couplings.
In the literature, triplet blockade is often linked to the presence of the CAR mechanism. Here we have shown that the suppression of the current due to the population of the triplet states is not specific to CAR coupling. In fact, such a current suppression can appear even without any nonlocal mechanism present in the device. In the presence of nonlocal processes, the current suppression can be interpreted as a triplet blockade, and it is observed via NDC lines, and this effect is not unique for CAR, but EC and IT can also generate it.
We expect that the results presented here will facilitate the accurate characterization of hybrid superconductor–quantum dot devices, which are likely to be used as building blocks of future conventional and topological quantuminformation schemes.
Supporting Information
Supporting Information File 1: Derivation of the effective Hamiltonian, details of the particlehole symmetry and additional transport data.  
Format: PDF  Size: 2.6 MB  Download 
Acknowledgements
We acknowledge the fruitful discussions with Takis Kontos, Pascu C. Moca and Gergely Zaránd. AP was supported by the National Research Development and Innovation Office of Hungary (NKFIH) Grants 105149 and 124723, and the ÚNKP174III New National Excellence Program of the Ministry of Human Capacities of Hungary. This work was supported by NKFIH within the Quantum Technology National Excellence Program (Project No. 20171.2.1NKP2017 00001), by the COST action NanoCoHybri CA16218, and QuantERA network ’SuperTop’ (NN 127900).
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