Merging diabolical points of a superconducting circuit

We present the first theoretical study of the merging of diabolical points in the context of superconducting circuits. We begin by studying an analytically solvable four-level model which may serve as theoretical pattern for such a phenomenon. Then, we apply it to a circuit named Cooper pairs pump, whose diabolical points are already known.


I. INTRODUCTION
Superconducting circuits 1 become more and more central in modern quantum physics. Their principal building blocks are ultra-small Josephson junctions 2 which can be assembled in a variety of different ways, each of them defining a specific quantum system. Often easier to manipulate, these solid state devices tend to supplant atomic and photonic systems, the "old paragons" of quantum mechanics. Notably, superconducting circuits are widely used to engineer qubits [3][4][5][6][7][8] : the nonlinear behavior of Josephson junctions serves to isolate couples of levels in a Hamiltonian spectrum. They are also used to perform analogs of cavity quantum electrodynamics 9-11 (a qubit plays the role of an artificial atom while a transmission line carries artificial photon modes), (non-) Abelian holonomies [12][13][14][15][16] , (non-) Abelian quantum charge pumpings [17][18][19][20][21][22][23][24] , etc. In brief, they are good candidates to implement quantum logic operations 7,25 and also promising for applications in electrical metrology 26 . The Cooper pairs pump (CPP) which we will consider in this article is an archetype of quantum circuit having a few (collective) degrees of freedom. It has been demonstrated theoretically in Ref. 22 a possible topological quantization of the pumped charge, through an invariant called first Chern number (or Chern index) 22,27 . It relies on the existence of diabolical points 28 in the three-dimensional parameter space of the system, i.e. on double degeneracies characterized by a linear dispersion in all directions of that space.
Relatively recently, G. Montambaux et al. have demonstrated the possibility of merging Dirac points in certain two-dimensional crystals, especially in hexagonal -graphene-like -lattices [29][30][31][32] (see also Refs. 33 and 34). Dirac points are nothing else but diabolical points in the reciprocal space of crystals. They are "naturally" located at points of high symmetry -e.g. at vertices of a regular hexagonal lattice. But, in accordance with the famous Wigner-von Neumann theorem 35 , they may move, driven by well-chosen additional parameters. In the graphene example, the two triangular sublattices carry non equivalent Dirac points. The merging of two neighboring non equivalent Dirac points evokes the meeting of a knot and its anti-knot: monitored by a merging parameter, they move closer together, then merge into a single degeneracy and finally disappear. At the transition, the single degeneracy is characterized by a quadratic dispersion in the merging direction. Inspired by the works of G. Montambaux et al., we present in this paper a theoretical study of a merging of diabolical points in the context of superconducting circuits. The choice of the CPP was motivated by its well-known diabolical points located in a hexagonal lattice, a property which confers to that circuit a great similarity with graphene.
This paper is organized as follows. Section II is devoted to the short study of a merging of diabolical points using a four-level model. It is interesting by itself since it provides a "universal Hamiltonian" which is susceptible to be realized in different quantum contexts as ours. It gives en passant the opportunity to briefly review some characteristics of double degeneracies in a parameter space. In Sec. III, we present the CPP and emphasize the symmetry origin of its "mobile diabolical points". Finally, via a modification of the circuit, we suggest in Sec. IV a way of merging these points. This will be done through an effective Josephson energy as merging parameter.

II. THE FOUR-LEVEL MODEL
We consider a model Hamiltonian depending on a triple of tunable parameters R = (X, Y, Z) and having the form in an orthonormal basis {|e 1 , |e 2 , |e 3 , |e 4 }. Here, F = 0 and ξ are constants, and G is an additional tunable parameter. The latter is dubbed merging parameter for a reason which will appear shortly. The set of vectors R forms the natural parameter space of the problem. In this space, the spectrum of H possess the symmetry D 2h . Indeed, H is (anti)unitary transformed under signreversing of X, Y and Z. Explicitly, we have 1. H(X, Y, −Z) = K H(X, Y, Z) K † where K is the complex conjugation operator with respect to the basis {|e 1 , |e 2 , |e 3 , |e 4 }; Plots of the two lowest energy levels as functions of X and Y , with Z = 0 and ξ = |F |, for different values of the merging parameter G. As long as G > Gc the distance between the diabolical points D + and D − decreases with G. They merge for G = Gc into a single degeneracy characterized by a quadratic dispersion δ(X) in the X-direction and disappear as In particular, under R-inversion, we observe the very simple unitary equivalence H(−R) = T H(R)T † . Since e 1 |(H − µ)(H − ν)|e 4 = 2 F 2 e 2iZ = 0, for any µ and ν, H has at least three distinct eigenvalues. Thus, λ is a (doubly) degenerate eigenvalue of H if and only if (iff) it exists a real β > 0 such that In this case, −λ − β and −λ + β are the other eigenvalues and λ is the smallest one iff β < −2λ. The matrix element of Eq. (2) between e 1 | and |e 4 imposes λ = −ξ − G cos Z. This value is a root of H's characteristic polynomial iff one of these three conditions is verified: (i) Y = G = 0 ; (ii) X = ±(2ξ + G cos Z) ; (iii) Y = 0 and Z = 0 mod π. A study of Eq. (2) within conditions (i) or (ii) shows that they can not give rise to a ground degeneracy. We now consider the last possibility. It is sufficient to restrict ourself to the case Y = Z = 0 since Z → Z + π amounts to change the sign of F and G. After a little algebra, we find that the ground level of H is degenerate iff G is greater than the critical value G c = 2F 2 + ξ 2 −ξ and X = ±X d with As long as G > G c , the two distinct points D ± , located at R ± = (∓X d , 0, 0), are isolated degenerate points in the R-space. They illustrate a classic theorem of von Neumann and Wigner 35 which states that, generically, twofold degeneracies have codimension three. Alternatively stated, in an N -dimensional parameter space, it exists generically submanifolds of dimension N − 3 over which a level is doubly degenerate. Here, N = 3 and the degenerate subspaces are points (dimension: 3 − 3 = 0). One should also think about a bigger space collecting all the parameters entering in H, as the six-dimensional space of vectors R = (X, Y, Z, ξ, F, G). The submanifold verifying simultaneously F = 0, Y = Z = 0, G G c (ξ, F ), and X = ±X d (ξ, F, G) carries a degenerate ground level and has a (co)dimension 3 in the Rspace, as expected.
The points D ± move closer together when we reduce G, while conserving the symmetry relations [H, U] = [H, K] = 0 (see Fig. 1). It corresponds to a generic situation: if they would deviate from the planes Y = 0 or Z = 0, each of them would split into 2 (or 4) distinct degenerate points. They merge at R = 0 for G = G c and finally disappear as soon as G < G c , the minimal gap between the two lowest levels being Let Π be the projector into H(R + )'s ground eigenspace and {|1 , |2 } an orthonormal basis of that subspace. Let σ = (σ x , σ y , σ z ) be the triple of operators represented by the usual Pauli matrices in the basis. Up to an unimportant component along Π, it exists a unique fixed real matrix M of order 3 such that Π ∇H(R + ) · (R − R + ) Π reads σ · M(R − R + ). Obviously, M depends on the choice of the basis, but it is a simple task to show that the signum of its determinant is intrinsic to the degeneracy. It is the signature 36 of the degenerate point D + in the R-space. One can explicitly choose where β is given by Within this choice, one finds As long as G > G c , the signature of D + is +1. In its vicinity, the two lowest levels are close together and σ · M(R − R + ) is an accurate Hamiltonian for the states belonging to them. Since D + has a nonzero signature, the level splitting around it is effective from the first order in R − R + in all directions of the parameter space: D + is a diabolical point 28 . By symmetry, so does D − , whose signature is found to be −1. For G = G c , the single degeneracy located at the origin has a vanishing signature, because the dispersion in the merging direction is quadratic. A perturbative analysis shows that a deviation (0, 0, 0) → (X, 0, 0) opens a gap Before introducing the system which will serve to realize our four-level model, let us end this section by two remarks. The first one is peculiar to the model: if Y , Z are suppressed and ξ, F tunable, the Hamiltonian may be used to construct non-Abelian holonomies 37,38 over the manifold satisfying simultaneously F = 0, G G c (ξ, F ) and X = ±X d (ξ, F, G). The model may also serve to implement non-Abelian pumpings having Z as pumping parameter 38 . The second remark is more general and concerns the signature. Consider some Hamiltonian H continuously defined over a R-space. Suppose the existence of a nonsingular transformation t : R → R ′ associated with a fixed symmetry operator T , such that H(R ′ ) = T H(R)T † . If T is unitary, it is straightforward to verify that the signatures are conserved by the transformation if t is orientation-preserving and reversed otherwise. If T is antiunitary, the contrary occurs. In our example, the signature of D − is due to the orientation-reversing map R → −R associated with the unitary operator T . Moreover, we will below use successive orientation-preserving transformations of the parameter space, without incidence on the signature.

A. Basic settings
We consider one of the simplest implementation for a CPP, represented in Fig. 2. It is a small-inductance superconducting loop (L → 0), threaded by a magnetic flux Φ x and broken by three ultra-small Josephson junctions 2 . The junctions are assumed non-dissipative. They enclose two superconducting islands, polarized by gate voltages V gk through low gate capacitances C gk (k = 1, 2). We set the corresponding gate charges in unit of 2e (> 0). The vanishing loop inductance leads to a biasing phase 2 ϕ x = 2π Φx Φ0 accross the CPP, where Φ 0 = h 2e is the quantum of flux. We suppose Φ x , V g1 and V g2 independently tunable. The system has two collective degrees of freedom -one for each island -and depends on three external parameters that we choose to be n g1 , n g2 and ϕ x rather than V g1 , V g2 and Φ x . The conjugated operators assigned to the degrees of freedom are n k and ϕ k (k = 1, 2): the number of Cooper pairs in excess (with respect to charge neutrality) and the phase of the superconducting parameter of the k th island, respectively. They verify the commutation relations [n j , ϕ k ] = i δ j,k . We will study the CPP in charge representation; |n 1 , n 2 will represent the fundamental charge states defined such that e ±iϕ1 |n 1 , n 2 = |n 1 ∓ 1, n 2 and e ±iϕ2 |n 1 , n 2 = |n 1 , n 2 ∓ 1 . Since eigenvalues of n 1 and n 2 can theoretically cover all the range of Z, the orthonormal basis B = {|n 1 , n 2 | (n 1 , n 2 ) ∈ Z 2 } spans the whole Hilbert space of the problem. Throughout this article, C Σ = C L + C + C R will be the capacitance unit and E C = (2e) 2 CΣ the energy unit. The latter is a typical charging energy of the circuit. We will study the system in the Coulomb blockade regime, characterized by Josephson energies small in comparison to E C = 1. Using the canonical quantization procedure, a Hamiltonian H = H(n g1 , n g2 , ϕ x ) may be derived for the system. It splits into two parts: a charging Hamiltonian H C = H C (n g1 , n g2 ) and a Josephson tunneling Hamiltonian H J = H J (ϕ x ). Neglecting the gate capacitances in comparison to C Σ = 1 and using the notation α = (α 1 , α 2 ), the former is where C is the capacitance matrix: The charging Hamiltonian is obviously diagonal in the basis B and verifies H C (n g + a) = e −ia·ϕ H C (n g ) e ia·ϕ for any integer vector a. Over the n g -plane the energy surface of the eigenstate |0 = |0, 0 is an elliptic paraboloid centered at n g = 0. Thus, the energy surface of |n = |n 1 , n 2 = e −in·ϕ |0 is simply the translation by n of this paraboloid. Two different states |n and |n ′ are degenerate on a straight line characterized by n, n ′ and the capacitances. Then, one easily checks that |n 1 , n 2 is the ground state of H C in a hexagon hex(n 1 , n 2 ) centered at n g = n. That defines the well-known honeycomb lattice of the CPP. It is graphically obtained by integer translations of two nonequivalent lattice points T ± whose coordinates are The lattice picture is useful if we identify each fundamental state |n 1 , n 2 with its corresponding hexagon hex(n 1 , n 2 ). With respect to H C , the common side of two neighboring hexagons is a piece of the degeneracy line between the states, while the vertices are points of triple degeneracy. Introducing the distance induced by the scalar product (x|y) = 2 −1/2 x · C −1 y in the plane, this picture allows to interpret the charging energy of |n 1 , n 2 as the squared distance between n g and the center of hex(n 1 , n 2 ). "Branching" H J , which can be brought into the form couples the neighboring states and lifts generically the degeneracies of H C . Explicitly, U (ϕ x ) = e iκ.·nϕx , with κ 1 = 1 − C R (C −1 ) 12 and κ 2 = 2 − C R (C −1 ) 22 . In the Coulomb blockade regime, H J is seen as a perturbation of H C . As a good approximation, the Hilbert space may be reduced to its subspace spanned by a few number of fundamental states in the neighborhood of n g . To this end, we only take into account the states |n 1 , n 2 at a distance of n g shorter than a certain value 39 .
The honeycomb lattice of the CPP over the ng-plane. It is generated by the triple points T ± whose coordinates are given in the text. Each state |n1, n2 is the ground eigenstate of HC in the hexagon hex(n1, n2). In the space of vectors r = (ng, ϕx), the spectrum of the Hamiltonian H possess the symmetry C 2h . It becomes D 2h = C 2h ×σv if the CPP is mirror symmetric (the vertical reflection planes are represented by thick lines).
Since H(n g + a, ϕ x ) = e −ia·ϕ H(n g , ϕ x ) e ia·ϕ , translations of lattice vectors a leaves the physics unchanged up to a displacement |n → |n + a of the fundamental charge states. Moreover, performing the gauge transformation |n 1 , n 2 → U (ϕ x )|n 1 , n 2 , the Hamiltonian is invariant under the translations ϕ x → ϕ x + 2kπ (k ∈ Z). Thus, the spectrum of H possesses the translational symmetry of a hexagonal prism lattice in the space of vectors r = (n g , ϕ x ). In the new representation, let us introduce the complex conjugation operator K, the "sign change operator" S : |n 1 , n 2 → | − n 1 , −n 2 and the "charge exchange operator" P : |n 1 , n 2 → |n 2 , n 1 . Taking the ϕ x -axis vertical, the spectrum of H possesses the point symmetry C 2h = σ h × ι: under the reflection σ h and the inversion ι, we have H(r) = K † H(σ h r)K = S † H(ιr)S. In particular, the symmetry C 2 = ι • σ h implies that the Hamiltonians at (T + , ϕ x ) and (T − , ϕ x ) are antiunitary equivalents, and even unitary equivalents iff ϕ x = 0 mod π.
If the CPP is mirror symmetric, that is to say, if the "exterior junctions" are identical (E JL = E JR and C L = C R ), the symmetry D 2h = C 2h × σ v is reached. Indeed, the reflection σ v , shown in Fig. 3, exchanges n g1 and n g2 , inducing the transformation

B. The diabolical points
The CPP is said totally symmetric if the three junctions are identical.
In this specific case, one has n g (T ± ) = ±( 1 3 , 1 3 ). As shown in Fig. 4, the orientation- The regularized honeycomb lattice of the totally symmetric CPP. The new parameters X and Y are notably chosen so that T + is located at the origin.
preserving map (n g1 , n g2 ) → (X, Y ), such that X = √ 3 2 (n g2 + n g1 − 2 3 ) and Y = 1 2 (n g2 − n g1 ), makes the hexagons regular in the (X, Y )-plane and places the origin at T + . The charging energy of a state |n 1 , n 2 becomes the usual squared distance between (X, Y ) and the center of hex(n 1 , n 2 ).
Setting Z = ϕ x , the spectrum possesses the sym- Fig. 4). The rotation C 3 induces the transfor- with R : |n 1 , n 2 → |1 − n 1 − n 2 , n 1 . The two symmetry operators P and R generate an unitary representation Γ of the group D 3 = C 3 × σ v in the Hilbert space, such that Γ(σ v ) = P and Γ(C 3 ) = R. At the high symmetry points (T + , ϕ x = 0 mod π), the Hamiltonian is invariant under D 3 . From the non-abelianity of D 3 and the faithfulness of Γ, it exists necessarily degeneracies in the spectrum of H(T + , ϕ x ), when ϕ x = 0 mod π. More precisely, D 3 admitting only irreducible representations of degree 1 or 2, the symmetry allows only double degeneracies (see the character table I).
The ground eigenspace of H C (T + ) -spanned by |0, 0 , |1, 0 , and |0, 1 -is an invariant subspace of Γ. If Γ g is the restriction of Γ to this space, we have in the basis {|0, 0 , |1, 0 , |0, 1 }. These matrices do not commute, so Γ g is a non-Abelian representation of D 3 . Hence, it contains necessarily the two-dimensional irreducible representation (irrep) E. The trace of Γ g (σ v ) shows that Γ g contains also the totally symmetric irrep A. Obviously, the state belongs to A. Then, we complete a basis of Γ g by choosing two orthonormal states belonging to E: In the basis {|A , By the invariance of H J (0) and H J (π) under D 3 , any Josephson coupling between the states |A , |E 1 and |E 2 is forbidden at the high symmetry points. Since E α |H J (π)|E α − A|H J (π)|A = − 3 2 < 0 (α = 1, 2), up to the first order in E J , the ground level of H(T + , π) belongs to E while the first excited one belongs to A. The contrary occurs for H(T + , 0). Thus, the half-fluxoid condition Φ x = Φ0 2 mod Φ 0 guarantees the double degeneracy of the ground level at the point T + . The same conclusion holds at T − from the equivalence between H(T − , ϕ x ) and H(T + , ϕ x ).
Let us analyze the signatures of the degenerate points D ± = (T ± , π). Redefining, for convenience, Z as ϕ x − π, D + is located at the origin of the new R-space. Using the same notations as in Sec. II, basis states |1 and |2 of H(D + )'s ground level are partners of the irrep E = Π Γ Π. They may -and they will -be chosen so that the matrices of E are given by (5) in the basis {|1 , |2 }. Thereby, we have |α = |E α + O(E J ), α = 1, 2. The (anti)unitary transformations of H under the action of D 3h imply the existence of two reals α and β such that is +1. Returning to the natural r-space, the points D ± are located at r ± = (n g (T ± ), π). Since r − = C 2 r + and The existence of "signed degeneracies" is fundamental for the physics of quantum pumpings. They quantize the pumped charge along classes of cycles in the parameter space 22 . They are robust in the sense that their existence is guaranteed by the Wigner-von Neumann theorem even though the D 3 symmetry is broken. In some ways, one should say that the symmetry plays the important role of producing signed degeneracies which become "accidental" as soon as the symmetry is broken. Under continuous variations of the circuit's characteristics (capacitances and Josephson energies) they move continuously in the plane ϕ x = π, conserving their signature and the relation [H, K] = 0 (but loosing the regularity of the conical intersection over the (X, Y )-plane).

IV. MERGING DIABOLICAL POINTS OF A COOPER PAIRS PUMP
As a first approximation in the close vicinity of T ± , the whole Hilbert space can be reduced to the ground eigenspace of H C (T ± ). In these three-level models, the positions of D ± are easily found. They are located in the plane ϕ x = π at .
Suppose that the CPP is mirror symmetric and E J tunable. The above formula illustrates that the displacement of D ± conserves the symmetry [H, P] = 0. Furthermore, reducing E J improves n g1 (D + ) = n g2 (D + ) as much as it reduces n g1 (D − ) = n g2 (D − ). So the two diabolical points D + and D − shown in Fig. 5 6. The new circuit. The central junction of the CPP is replaced by a dc SQUID threaded by a controllable flux Φ ′ x . It behaves essentially as an effective junction having a capacitance C and a Josephson energy tunable between 0 and 2 EJE. merge symmetrically at their midpoint I located at ( 1 2 , 1 2 ) in the n g -plane.
Let E JE = E JL = E JR be the Josephson energy of the exterior junctions. By the definition of the capacitance unit, their capacitance is 1−C 2 . Around the point I, an approximate Hamiltonian is the restriction of H to the subspace spanned by the basis {|0, 0 , |1, 0 , |0, 1 , |1, 1 }. Up to an unimportant shift of the zero of energies, the truncated Hamiltonian has the form (1) in this basis, where X and Y are redefined as follows: The other parameters are ξ =

C
(1−C)(1+3C) , F = EJE 2 and G = EJ 2 . In this context, U(Z) and T are the respective restrictions of V(ϕ x )P and e −i(ϕ1+ϕ2) U(Z)SP. Within the four-level approximation, if E J is greater than the critical value the degeneracies are located at R ± = ∓(X d , 0, 0), with The merging of the diabolical points D + and D − is possible if E J is adjustable. It is well-known that a tunable effective Josephson coupling can be realized via two junctions in a loop configuration (a dc SQUID). Such a circuit element is de facto interesting to tune couplings between superconducting qubits 40,41 . It has also demonstrated its utility for Cooper pairs pumping in the socalled Cooper pairs sluice 19 . As shown in Fig. 6, we replace the central junction by a dc SQUID and assume all the junctions of the circuit identical. To be consistent with our previous notations, we set E JE the Josephson energies and C 2 the capacitances of all the junctions. The new central element has a capacitance C and an effective Josephson energy E J = 2 E JE cos . The charging and Josephson Hamiltonians still read (3) and (4) after the replacements ϕ x → ϕ x + ϕ ′ x 2 and U (ϕ x ) → U (ϕ x , ϕ ′ x ), the exact definition of the last unitary operator being irrelevant for our purpose. We also have C Σ = 2 C and ξ = 0.4.
The merging is done by tuning the central coupling (through ϕ ′ x ) while we use ϕ x to maintain the new halffluxoid condition ϕ x + ϕ ′ x 2 = π. A numerical simulation of the process was made, using the 62 closest states of the point I to define the truncated Hilbert space. The results are in good accordance with the four-level model in the Coulomb blockade regime. For example, it is shown in Fig. 7 the plot of the critical value E Jc as a function of E J : the numerical result coincides with the expression (7) in the limit E J ≪ 1.

V. CONCLUSION
We have demonstrated the possibility of merging diabolical points of a superconducting quantum circuit. We have emphasized the role played by the symmetry for that phenomenon. In an experimental perspective, the principal difficulty to overcome is the mirror symmetry since it is impossible to fabricate two identical junctions. This problem can be partially eliminated by using balanced SQUIDs 42 . The theoretical study was accurately based on a four-level model whose eigenproblem is exactly solvable within the constraint of a degeneracy. For subsequent works, it may serve as a formal model to implement mergings of diabolical points in different contexts, such as quantum circuits or cold atoms. It may also be used to produce non-Abelian holonomies as well as non-Abelian pumpings.