Canonical Circuit Quantization with Linear Nonreciprocal Devices

Canonical circuit quantization with linear nonreciprocal devices

1, 2 3, 4 1, 5, 6 A. Parra-Rodriguez, ∗ I. L. Egusquiza, D. P. DiVincenzo, and E. Solano 1Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, E-48080 Bilbao, Spain 2Department of Theoretical Physics and History of Science, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain 3Peter Grunberg¨ Institut: Theoretical Nanoelectronics, Research Center Julich,¨ D-52425 Julich,¨ Germany 4Institute for Quantum Information, RWTH Aachen University, D-52056 Aachen, Germany 5IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, 48013 Bilbao, Spain 6Department of Physics, Shanghai University, 200444 Shanghai, China Nonreciprocal devices effectively mimic the breaking of time-reversal symmetry for the subspace of dynamical variables that they couple, and can be used to create chiral information processing networks. We study the systematic inclusion of ideal gyrators and circulators into Lagrangian and Hamiltonian descriptions of lumped- element electrical networks. The proposed theory is of wide applicability in general nonreciprocal networks on the quantum regime. We apply it to pedagogical and pathological examples of circuits containing Josephson junctions and ideal nonreciprocal elements described by admittance matrices, and compare it with the more involved treatment of circuits based on nonreciprocal devices characterized by impedance or scattering matrices. Finally, we discuss the dual quantization of circuits containing phase-slip junctions and nonreciprocal devices.

I. INTRODUCTION nonreciprocal effects is imperative. We emphasize here that, even though they do not exist as such in nature, ideal gyrators Low-temperature superconducting technology [1] is on the and circulators can be useful elements to introduce in complex verge of building quantum processors [2] and simulators [3– descriptions of networks. We focus on and extend the anal- 7]; machines predicted to surpass exponentially the compu- yses of lumped-element networks of Devoret [23], Burkard- tational power of classical computers [8–11]. In electromag- Koch-DiVincenzo (BKD) [25], Burkard [26] and Solgun and netism, reciprocity is equivalent to the invariance of a system’s DiVincenzo [30]. Our extension involves, first, adding ideal linear response under interchange of sources and detectors. gyrators and circulators described by an admittance (Y) matrix Nonreciprocal (NR) elements such as gyrators and circulators to obtain quantum Hamiltonians with a countable number of [12] have been mainly used in superconducting quantum tech- flux degrees of freedom. As we will see, a bias towards a spe- nology as noise isolators and classical information routers, cific matrix description of NR devices (NRDs) appears useful i.e., out of the quantum regime, due to the size of currently when we want the Euler–Lagrange equations of motion to be available devices. Lately, there have been several proposals current Kirchhoff equations in terms of flux variables. We for building scalable on-chip NR devices based on Josephson next show how adding ideal NRDs described by impedance junction-networks [13–15], parametric permittivity modula- (Z) or scattering (S) matrices requires a more involved treat- tion [16], the quantum Hall effect [17, 18] and mechanical ment, in that the system of equations must be first properly resonators [19]. This nonreciprocal behavior presents quan- reduced. Finally, we also address canonical quantization with tum coherence properties [18] and will allow novel applica- charge variables to treat dual circuits with Z-circulators; see tions in the nontrivial routing of quantum information [20– Ref. [38] for a detailed description on circuit quantization 22]. Accordingly, there is great interest in building a general with loop charges. We apply our theory to useful, pedagog- framework to describe networks working fully on the quantum ical and pathological circuit examples that involve the main regime [23–35]. technical issues that more complex networks could eventually In this article, we use network graph theory to derive present. Hamiltonians of superconducting networks that contain both Our emphasis is on quantization of an electrical network, nonlinear Josephson junctions and ideal lineal NR devices that is to say, on quantum network analysis, and we set aside the dual problem of network synthesis. Even so, the intro-

arXiv:1810.08471v2 [quant-ph] 29 Jan 2019 with frequency-independent response [36]. The correct treatment of such ideal devices will provide us with building duction of the techniques presented here implies that more so- blocks to describe more complex nonreciprocal linear devices phisticated synthesis methods can be used for the description [37] that can be treated as linear black boxes [24, 28–30]. This of quantum devices, since our analysis can be applied to a theory lays the ground for the correct description of circuits in wider class of circuits than those previously considered. the regime where the nonreciprocal devices can be well char- Regarding the need for a more involved treatment of NRDs acterized by a linear response [13–19], even if the fundamen- with impedance or scattering matrix presentations, bear in tal nonreciprocal behavior is achieved by nonlinear elements mind that, in microwave engineering, a multiport linear [13–15]. Outside of this regime of validity, a black-box ap- (black-box) device can be always described by its scattering proach is not longer useful and a microscopic description of matrix parameters S(ω) [39], that relate voltages and currents p at its ports b = Sa, with bk = (Vk Z∗Ik)/ Re Zk and p − k { } ak = (Vk + ZkIk)/)/ Re Zk . The reference impedances can be chosen, for simplicity,{ homogeneous} and real, e.g., ∗ [email protected] Z = R Re. Simple properties of the scattering matrix re- k ∈ 2 veal fundamental characteristics of the device. For instance, a like the circulator are represented by N branches connect- network is reciprocal (lossless) when S is symmetric (unitary). ing 2N nodes pairwise [37]; see Fig.1. A spanning tree of See Fig.1 for an example of basic NR devices and their con- the graph is a set of branches that connects all nodes without ventional symbols in electrical engineering. When ports are creating loops. The set of branches in the tree are called tree impedance-matched to output transmission lines (a) a 2-port branches and all others chord branches. Making a choice for (4-terminal) ideal gyrator behaves as a perfect π-phase direc- tree and chord branches in an electrical network, we separate T T T T T T tional shifter, i. e. b2 = a1 and b1 = a2, and (b) a 3-port (6- the currents I = (Itr, Ich) and voltages V = (Vtr, Vch) terminal) ideal circulator achieves perfect− signal circulation, to write Kirchhoff’s equations as e.g. bk = ak 1 [37]. Other useful descriptions of multiport − 1 devices are the impedance Z(ω) = R(1 S(ω))− (1 + S(ω)) F Ich = Itr, (1) 1 − T − and admittance Y(ω) = Z− (ω) matrices that relate port volt- F Vtr = Vch +Φ˙ ex, (2) ages and currents as V = Z I and I = Y V respectively [39]. Although sometimes more useful, immittance descriptions of where F is the reduced fundamental loop(/cutset) matrix de- linear devices do not always exist, and working with S can scribing the topology of the graph. It contains only 0, 1, 1 { − } be unavoidable [37, 39]. This comes about whenever the S entries; see [25, 26] for details on graph theory applied to su- matrix has +1 and 1 eigenvalues. perconducting circuits. Hence we make reference to F as the − loop matrix. The vector of external fluxes Φex corresponds (a) (b) b to the set of external fluxes threading each of the loops of the 1 system. a1 I1 I2 b2 b a3 The branch charge (Q) and flux (Φ) variables are de- R 1 I fined from the flow variables I and difference variables V as V 1 V V1 V2 1 S I3 3 IX (t) = Q˙ X (t) and VX (t) = Φ˙ X (t), where the subscript X = C, L, J, G, T , R, Z, V , B denotes capacitors, inductors, Josephson junctions, nonreciprocal element branches, b a2 a1 I2 b3 1 transformer branches, resistors, two-terminal impedances, π V2 voltage sources, and current sources, respectively. For the b2 a 2 sake of simplicity we focus here on networks with passive and lossless elements, i.e., capacitors, inductors, Josephson FIG. 1. (a) A 2-port gyrator: Input ak and output bk signals are junctions, nonreciprocal element branches, and transformer related to each other through the scattering matrix b = Sa; with branches. We forward the reader to Refs. [25, 26, 29, 30] b2 = a1 and b1 = −a2, the element behaves as a perfect π-phase for the inclusion of two-terminal impedances and voltage and directional shifter (bottom) when impedance matched to transmis- current sources. sion lines at ports. (b) A 3-port circulator: Input signals transform The constitutive equations of capacitors, inductors, and into output signals cyclically, e.g. bk = ak−1. Port voltages Vk and currents Ik can be generally related to ak and bk through Eq. (8). Josephson junctions are

QC = CVC , (3) In Sec.II we present some basic aspects of network graph 1 theory, as applicable to electrical circuits, with reference to the IL = L− ΦL, (4) current literature on its use in quantization. We include non- Φ0 ΦJ = ϕ , (5) reciprocal multiport elements in the consideration. We next 2π J address, in Sec. III, the construction of the Lagrangian of cir- IJ = Icsin(ϕJ ). (6) cuits with admittance described nonreciprocal devices and the subsequent quantization. We provide specific examples of this where Icsin(ϕJ ) is the column vector with Ici sin(ϕJi) en- process. In Sec.IV we look into the issue of nonreciprocal de- tries, Ic the critical current of a junction and Φ0 the flux quan- vices with no admittance description. To this point we have tum. General multiport transformers (Belevitch transformers studied circuits with flux variables. In Sec.V the dual, charge [40]) have been previously added to the Burkard analysis in variables are investigated for their use in nonreciprocal cir- Ref. [30]. They add voltage and current constraints on the cuits. We finish with conclusions and a perspective on future right ports in terms of its left ports and vice versa, work. IR = N IL, VL = NT VR, (7) T − T T T where N is the turns ratios matrix and both left and right cur- II. NETWORK GRAPH THEORY rent directions are pointing inwards. Dual transformers exist where the left-right equations (7) are inverted [30, 40]. Pass- A lumped-element electrical network is an oriented multi- ing now to the focus of our study, the general constitutive graph [23, 25]. Each branch of the graph connects two nodes equation for the ideal (frequency-independent) nonreciprocal and has a direction chosen to be that of the current passing element branches can be retrieved from the scattering matrix through it. A one-port element will be assigned a branch. The definition choice of direction for the corresponding branch is arbitrary for symmetric elements. More generally, N-port elements (1 S) V (t) = (1 + S)R I (t), (8) − G G 3

eff eff with R a constant in resistance units. with FCL = FCL + FCT ch NFT trL and FCG = In order to carry out canonical quantization in circuits, our FCG + FCT ch NFT trG. We insert the constitutive equa- task will be to simplify Kirchhoff’s laws together with the tions (4) and the admittance version of (8), IG = YG VG, constitutive relations into a set of classical Euler-Lagrange into the reduced current equation to obtain a second-order (E-L) equations, from which Hamiltonian equations can be equation in flux variables, derived through a Legendre transformation, and canonically ¨ ˙ conjugate variables can be identified. In trivial cases, this re- CΦC = Icsin(ϕCJ ) + M0ΦC + GΦC , (11) duces to having a kinetic matrix that is non-singular. − eff 1 eff T eff eff T where M0 = FCL L− (FCL) , G = FCG YG(FCG) , and

ϕCJ = ϕJ is the vector of capacitor branch phases [related to III. NETWORKS WITH Y NRD the fluxes by (5)] in parallel with the junctions. YG is a skew- symmetric matrix (because it is the Cayley transform of an or- Given that Josephson junctions are nonlinear devices, E-L thogonal matrix S), and by construction so is G (see Appendix equations have been systematically derived in flux variables A). The antisymmetry associated with the first-order deriva- so as to have a purely quadratic kinetic sector, e.g. Refs. tives, together with the fact that these second-order equations [23, 25, 26, 29, 30]. In particular, BKD and Burkard quanti- have a non-singular kinetic matrix, allows us to derive them zation methods are constrained, with respect to Devoret’s ap- from the Lagrangian proach, to specific topological classes of circuits to make the Hamiltonian derivation even more systematic. For instance, 1 ˙ T ˙ ˙ T T L = ΦC CΦC + ΦC GΦC ΦC M0ΦC U(ϕC ). in BKD all the capacitors must be included in the tree, while 2 − − J (12) there are no capacitor-only loops; i.e., all capacitors are tree The conjugate charge variables are Q = ∂L/∂Φ˙ = branches, and no external impedance can appear in the tree, C C CΦ˙ + 1 GΦ . Notice that conjugate charge variables are while Burkard quantization has dual conditions. These as- C 2 C not necessarily identical to capacitor branch charge variables, sumptions about the assignment to tree and chord branches which are those that appear in Eq. (3). Promoting the provide us with a description of the loop matrix in block ma- variables to operators with canonical commutation relations trix form, in such a way that some of the blocks are trivial. ˆ ˆ , we derive the quantum Hamiltonian This triviality, in turn, will allow us to construct effective loop [ΦCn , QCm ] = i~δnm matrices by elimination of variables. 1 1 T 1 1 Hˆ = (Qˆ GΦˆ ) C− (Qˆ GΦˆ ) As we shall now see, those approaches can easily incorpo- 2 C − 2 C C − 2 C rate ideal NR elements described by the admittance matrix (Y 1 T + Φˆ M0Φˆ C + U(ϕˆ ). (13) devices) with the realistic assumption that all of their branches 2 C CJ are independently shunted by (parasitic) capacitors. The non linear potential is defined as U(ϕˆ C ) = For instance, the BKD formalism can be extended by as- P J EJi cos(ϕ ˆJi) and the Josephson energy of each junc- suming that all ideal NR ( ) branches are chord branches. As − i G tion is EJi = IciΦ0/(2π). Given the velocity-position cou- stated, in BKD all capacitors of the mesh have to be in the pling term arising from the G matrix, a form first devised in tree branches, whereas Josephson junctions, which are always Ref. [36], a diagonalization of the harmonic sector requires a in parallel to at least one capacitor, are chosen to be chord symplectic transformation, that can be carried out either in the branches. Inductors can be both in the tree (K) or in the chord classical variables or after the canonical quantization proce- (L) set. In the following, we sketch the derivation where all dure; see AppendixC. Notice the similarity of the G terms to a inductors are chord inductors. For pedagogical purposes, we magnetic field, and their breaking time-reversal invariance. In derive a Burkard circuit class extension in AppendixA. Fol- the same manner as a magnetic field, these gyroscopic terms lowing Ref. [30], we also include Belevitch transformers in are energy conserving. this analysis. The fundamental loop matrix of a simplified BKD circuit can be written in block matrix form as Examples F F F F ch F = CJ CL CG CT . (9) FT trJ FT trL FT trG FT trT ch These extended BKD and Burkard analyses can be directly applied to a huge family of circuits to derive Hamiltonians in Real Josephson junctions are always in parallel to capacitors, position-flux variables with Y NRDs. Up till now, most of the so that FT trJ = 0. On the other hand, if all transformer left interest in quantization of circuits has been connected with the branches can be included in the tree, while transformer right presence of Josephson junctions. In the present analysis we branches are in the chord, then FT LT R = FT trT ch = 0. We combine that presence of Josephson junctions with nonrecip- can integrate out the voltages and currents in the transformer rocal devices. We are thus motivated to keep the flux variables branches [30] inserting (7) into Kirchhoff’s equations (1,2) as the only position coordinates of a Lagrangian/Hamiltonian and write an effective loop matrix mechanical system. Here we demonstrate the quantization of two circuits consisting of two Josephson junctions cou- eff eff eff F = FCJ FCL FCG , (10) pled to (i) a general 2-port nonreciprocal black box and (ii) 4 the specific nonreciprocal impedance response of the Viola- R0 B DiVincenzo Hall effect gyrator [17]. The first circuit is a ped- ∞ Z(s) agogical and useful example where the black box, in its N- port configuration, would represent the response of any of the Cg1 R1 = 1/Ω1C1 L L given proposals in Refs. [13–19] within their valid frequency n11 n12 1 range containing two gyrators. In the second circuit, we ex- EJ1 CJ1 C1 ploit a specific 2-port impedance response, which includes a A1, B1 gyrator, to get an easy lumped-element approximation that R R n11 n12 1 can be directly quantized. Extensions of these circuits with C1 N-port Y circulators would also be readily treated by this for- Cg2 malism. We study corner cases where the circulators cannot be described by Y matrices below in Sec.IV. n21 n22 1 EJ2 CJ2 C2 L2 A2

1. NR Black-box coupled to Josephson junctions FIG. 2. Two junctions capacitively coupled to a nonreciprocal lossless impedance. Gyrator R0 implements an antisymmetric pole at infinity B∞. The network connected by gyrator R1 yields the term 2 2 The first circuit consists of a 2-port nonreciprocal loss- (sA1 +B1)/(s +Ω1). There is a pure reciprocal stage A2. Effective less impedance [41] capacitively coupled to two charge qubits tree capacitor branches are marked in red, and current directions for at its ports; see Fig.2. This is a generalization of the each branch are represented with arrows. Foster reactance-function synthesis for the 1-port reciprocal impedance Z(s), with s = iω, and a simplified version of the Brune multiport impedance expansion in Refs. [30, 42].   A lossless multiport impedance matrix can be fraction- CJ1 expanded as  CJ2     Cg1    C =  Cg2  . (15) ∞ sA + B   1 X k k  C1R  Z(s) = B + s− A0 + sA + 2 . (14) ∞ ∞ s2 + Ω  C1L  k=1 k C2

It is easy to synthesize a lumped-element circuit that has this Inductive M0 and gyration G matrices are computed using the impedance to the desired level of accuracy; see [41]. In a turn ratios matrix lossless linear system, the S matrix is unitary, and therefore nL 0 0 nL 0 0  Z(s) = Z (s) must be anti-Hermitian. If, additionally, the 11 12 † N =  0 nR 0 0 nR 0  (16) system is− reciprocal, it must be symmetric. The only com- 11 12 0 0 n 0 0 n plex parameter being s, a lossless reciprocal impedance ma- 21 22 trix must be odd in the variable s, Z( s) = Z(s). Therefore, to calculate the effective loop submatrices Feff , Feff in (10); in the fraction expansion above, the− s-odd− parts correspond to CL CG see AppendixB for an explicit form of the matrices. We recall reciprocal elements, while the s-even parts come from non- that this analysis can be completed because the constitutive reciprocity. Thus, all A matrices are symmetric and are im- equation of the nonreciprocal elements (8) simplifies to IG = plemented by reciprocal elements while B matrices are anti- T Y V , where I = (I L , I R , I L , I R ) and symmetric, and can be decomposed into networks with gyra- G G G G0 G0 G1 G1 tors. and terms correspond to the limits A0 A L2 Y 0 and , respectively,∞ in a reciprocal stage (see Fig.→2). ∞ Y = G0 , (17) C2 0 G 0 Y A requires→ special treatment, but would generally be absent G1 ∞ because of parasitic capacitors. with Y the admittance matrix for each gyrator i 0, 1 . Gi ∈ { } The general circuit implementing Z(s) contains Belevitch transformer branches [40] that can be eliminated as explained above [30] to derive a canonical Hamiltonian. An analysis 2. Hall Effect NR device of the lossless reciprocal multiport network can be found in Ref. [35]. The tree and chord branch sets are divided in The Hall effect has been proposed as instrumental in the T T T T T T T Itr = (IC , IT L ) and Ich = (IJ , IL, IT R ), with left (right) implementation of nonreciprocal devices. In Ref. [17], capac- transformer branches being tree (chord) branches. The capacitively coupled Hall effect devices were studied by Viola and itance matrix is by construction full rank and hence invertible, DiVincenzo in order to break time-reversal symmetry while 5

T keeping losses negligible. This 2-port capacitively coupled are related by ΦG = FCG ΦC , where Hall bar has an impedance matrix description [17]  1 0  0 1 FCG =   , (19) 1N 0  1 i cot(ωCL/2σ) 1 0 1N Z2P (ω) = − − , σ 1 i cot(ωCL/2σ) − (18) with 1N an N-component column vector of ones. Explicitly, where σ and CL are conductance and capacitance character- the three matrices describing the harmonic sector are the sym- istic parameters of the device, which is equivalent to that of metric an ideal gyrator with R = 1/σ connected in series to two CJL  λ/2-transmission line resonators of Z0 = 1/σ and vp/L = C C =  JR  and (20) 2σ/CL; see Fig.3( a). Lumped-element Foster expansions  CN  of the resonators can approximate the behavior of such a CN device when coupled to other lumped-element networks at 0  its ports. This connection is achieved with lumped capaci-  0  tance and inductance parameters determined by the distributed M0 =  1  (21) 2  LN−  ones as C0 = CL/2, Ck = CL/4 and Lk = CL/(σkπ) , 1 for k 1, ..., N ; see Fig.3( b). We can systemati- LN− ∈ { } matrices, and the skew-symmetric nonreciprocal

 T  0 1 0 1N (a) 1  1 0 1T 0  G =  − − N  , (22) R 0 1 0 1 1T Z2P (ω)  N N N  1 0 1 1T 0 − N − N N Z0, vp/L where we have deﬁned the capacitance submatrix CN = 1 C0diag(1, 1/2, ..., 1/2) and the inductance submatrix LN− = 1 2 1 2 L0− diag(0, 1, 4, ..., N ), N being the number of oscillators R to which we truncate the response of the resonators. Blank el-

EJ1 CJ1 CJ2 EJ2 ements of the matrices correspond to zeros. The Hamiltonian (13) can be readily computed and the canonical variables pro- 1 2 moted to quantum operators. The diagonalization of the har- HEG monic sector can be implemented through a symplectic trans- (b) formation both before or after the quantization of variables following AppendixC below. CN CN LN IV. NETWORKS WITH Z AND S NRD

C0 C0 1 2 The rules described above are useful to derive Hamiltoni- R ans of circuits containing ideal nonreciprocal devices charac-

EJ1 CJ1 CJ2 EJ2 terized by a constant skew-symmetric Y matrix. However, linear systems cannot be described by admittance matrices when 1 2 their S matrix has an eigenvalue 1. For example, ideal circulators with even (odd) number of− ports, even (odd) number of “ 1” entries and even (even) number of “1” entries in their FIG. 3. The VD Hall effect gyrator capacitively coupled to scattering− matrix admit only S-constitutive equations as in Eq. Josephson junctions. (a) An effective circuit of the device proposed (8) (both S and Z equations) [37]. by Viola and DiVincenzo matching the impedance response (18) con- We illustrate the problems arising when including circu- sists of an ideal gyrator coupled to λ/2-transmission line stubs. (b) The discrete approximate circuit based on a lumped element expan- lators without Y-descriptions with simple circuits containing sion of the transmission lines [39] that is canonically quantized. Tree 3- and 4-port circulators shunted by Josephson junctions; see (capacitor) branches are marked in red. Fig.4( a). Let us assume for concreteness that the N-port circulator is described by the scattering matrix  1 cally apply BKD theory and write a Lagrangian in terms of 1 the ﬂux branch variables of the capacitors ΦT = ΦT = N   tr C SN = ( 1)  .  , (23) . The ﬂux vari- −  ..  (ΦCJ1 ,ΦCJ2 ,Φ0L, ..., ΦNL,Φ0R, ..., ΦNR) ables at the ports of the gyrators and at the tree capacitors 1 6 blank elements being zero. This family of circulators cannot with Q = ∂L/∂f˙ the conjugated charge variables, and the be assigned a Y-matrix, nor do they have a Z-description for skew-symmetric matrix reads even N. We depart from BKD and Burkard rules and choose   as tree branches the circulator branches, Itr = IG, and ca- 1 0 0 0 pacitors and Josephson junction branches as chord branches GQ = 0 0 1 . IT = (IT , IT ). Kirchhoff’s laws can be simply written as R 0 1 0 ch J C − IG = IC + IJ and VG = VC = VJ = Φ˙ , choosing F− = F = 1. Without loss of generality and in the Had 1 not been an eigenvalue of S, all initial variables would GC GJ − interest of clarity let us assume that all Josephson junctions have been dynamical. Generally, there is a coordinate trans- have homogeneous capacitances Ci = C. Introducing Kirch- formation for any ideal circulator such that G is block diagonal, with 2 2 blocks, and, possibly, one zero in the diagonal × (a) (b) associated with eigenvalue +1 (see AppendixD).

Φ1 L1 Q1 ES SN V. DUAL QUANTIZATION IN CHARGE VARIABLES Φ3 ZG The procedures explained above are useful to derive La- Q Q2 3 grangians with flux variables as positions in a mechanical sys- ≡ tem. Equivalent descriptions of linear systems are possible Φ2 L2 L3 with charge-position variables, with E-L voltage equations, or with a mixed combination of both flux and charge vari- FIG. 4. (a) N-port S-circulator shunted by Josephson junctions. The family of S matrices of Eq. (23) does not have Y-description, ables. Indeed, fluxes have been used as position variables in nor does it have Z for even N.(b) Dual circuit with a 3-port Z- the context of superconducting qubits because the Josephson circulator shunted by phase-slip junctions in series with inductors. junction has a nonlinear current-voltage constitutive equation (6). Thus, the Lagrangian of a circuit with these elements and hoff’s and constitutive equations for capacitors (I = CΦ¨ ) Z circulators in charge variables results in nonlinear kinetic C terms. Although possible, dealing with such terms is usually and junctions (I = ΦU (Φ)) into (8) results in J ∇ more cumbersome. phase-slip R (1 + S)(CΦ¨ + ΦU(Φ)) = (1 S) Φ˙ , (24) In recent years, the (PS) junction [43, 44], a non- − ∇ − linear low-dissipative element in charge variable, has been im- T T plemented in superconducting technology [45–47]. This ele- with ΦU (Φ) = (U10 (Φ1), U20 (Φ2), ...) . Let P = v 1v 1 be the∇ projector onto the eigenspace of S such that PS −= −P, ment has a constitutive equation dual to that of the Josephson − as it is the case for the family of matrices (23). Equation (24) junction; i.e., its voltage drop is VP (t) = Vc sin(πQP /e), and implies PΦ˙ = 0; i.e., there is a frozen combination of fluxes, it is usually represented as in Fig.4( b) in green. Quantiza- which corresponds to a degenerate kinetic matrix that makes tion of circuits with PS junctions and ideal Z-NR elements in the Legendre transformation impossible to perform. A sim- charge variables can be implemented directly, using the con- ple solution is to change coordinates to single out the frozen stitutive equation VG = ZG IG. For example, the circuit in variable from the dynamical ones, and remove it through a Fig.4( b) with a ZG-circulator, the dual circuit of Fig.4( a), ˙ projection of Eqs. (24) into Q = 1 P. Integrating the frozen has the dual Lagrangian interaction term LG = (1/2)QZGQ − PN 1 and the quantum Hamiltonian variable, we can express Φ = αv 1 + n=1− wnfn, where w is a real basis expanding the− projector Q, α an initial- n Hˆ = 1 (Φˆ 1 Z Qˆ )T L 1(Φˆ 1 Z Qˆ ) + U(Qˆ ), value{ } flux constant, and f the reduced set of degrees of 2 2 G − 2 G { n} − − freedom. For the four-port case we have the following sys- ˆ T ˙ where L is the diagonal inductance matrix and U(Q) = tems of equations, v 1Φ = 0 and − P E cos(πQˆ /e). We forward the reader to Ref. [38] for − i Si i Cf¨ = ∂U˜ (f)/∂f (25) a systematic quantization method of circuits with loop charges 1 − α 1 1 [48]. Cf¨ = R− f˙ ∂U˜ (f)/∂f (26) 2 3 − α 2 1 Cf¨3 = R− f˙2 ∂U˜α(f)/∂f3 (27) − − VI. CONCLUSIONS with the definition U˜α(f) = U(Φ(α, f)) and f = . A similar system of equations can be derived (f1, f2, f3) We have presented a general framework to quantize canon- for the three-port case except for (25), associated with eigen- ically superconducting circuits with Josephson junctions and value and only appearing in the four-port case; see Ap- λ = 1 ideal linear nonreciprocal devices. We have introduced sys- pendixD for the general -port solution. Finally, the quan- N tematic rules for quantizing classes of circuits with ideal tized Hamiltonian with fully dynamical variables is admittance-described nonreciprocal devices in flux variables. T In such a scheme we have derived the Hamiltonian of Joseph- 1 1 1 Hˆ = Qˆ GQfˆ Qˆ GQfˆ + U˜ (fˆ) 2C − 2 − 2 α son junctions capacitively coupled to both a general linear 7

T T T nonreciprocal 2-port black box and the Viola-DiVincenzo gy- We eliminate ideal transformer branches IT = (IT tr , IT ch ) rator at its ports. These two examples show the crucial ele- [30], which do not store energy and are not degrees of freedom ments that we address in the general construction, and will be of the system, by making use of Kirchhoff’s current law for of interest in their own right in forthcoming experimental de- tree transformer branches and the current constraint equation vices. We have given an explicit method to quantize N-port of the transformer (7), ideal Z and S circulators shunted by Josephson junctions in I tr = (F tr I + F tr ch I ch ), (A8) ﬂux variables, by careful elimination of frozen variables. Fi- T − T C C T G G nally, we discussed an extension of these procedures to quan- IT ch = N IT tr , (A9) − tize circuits in terms of charge variables, a dual method of spe- with N the turns ratios matrix. Here we have assumed that cial importance when dealing with circuits containing nonre- transformer tree (left) branches are not shunted by transformer ciprocal elements and phase-slip junctions. Further work will chord (right) branches, i.e., F tr ch = 0 [30, 40]. We can thus be required to add distributed elements, e.g., inﬁnite transmis- T T express the current in the right branches of Belevitch trans- sion lines, to the analysis. former as

IT ch = N(FT trC IC + FT trGch IGch ). (A10) ACKNOWLEDGMENTS We write tree Josephson, inductor, and NR tree branch cur- A. P.-R. thanks Giovanni Viola, Firat Solgun and Laura Or- rents as a function of only capacitor and NR chord branch tiz for fruitful and encouraging discussions. We acknowledge currents, funding from Spanish MINECO/FEDER FIS2015-69983-P, I = I + F I + F ch I ch + F ch I ch Basque Government IT986-16 and Ph.D. Grant No. PRE- − J CJ JC C JG G JT T eff eff 2016-1-0284. We acknowledge support from the projects = ICJ + FJC IC + FJGch IGch (A11) eff eff QMiCS (820505) and OpenSuperQ (820363) of the EU Flag- I = F I + F ch I ch , (A12) − L LC C LG G ship on Quantum Technologies. eff eff I tr = F tr I + F tr ch I ch . (A13) − G G C C G G G Here, we have defined effective loop submatrices [30] APPENDIX A: EXTENDED BURKARD ANALYSIS eff FXC = FXC + FXT ch NFT trC , (A14) eff We extend Burkard [26] and Solgun-DiVincenzo [30] anal- FXGch = FXGch + FXT ch NFT trGch , (A15) yses to include ideal multiport NR Y-devices under the as- tr sumption that each branch of a NRD is shunted by a capaci- with X = J, L, G , that have real entries instead of the usual ternary{ set 1,} 1, 0 for branches with currents in the tor in the circuit independently. Relaxing the requirements of {− } the BKD analysis, we allow nonreciprocal branches to appear same or opposite direction, or out of the loop, respectively. both in the tree and in the chord set. We divide the tree and Using Kirchhoff’s current law and the capacitor constitutive chord currents and voltages for the different components of equation, we write the inductors in terms of the junction and the circuit in the following way: inductor voltages, ˙ ˙ T T T T T ICJ = QCJ = CJ VJ , (A16) Itr = IJ , IL, IGtr , IT tr , (A1) eff T eff T eff T IC = C (F ) V˙ J + (F ) V˙ L + (F tr ) V˙ Gtr . T T T T T JC LC G C I = I , I , I ch , I ch , (A2) ch CJ C G T (A17) T T T T T Vtr = VJ , VL, VGtr , VT tr , (A3) We rewrite again current-voltage constitutive relations for in- T T T T T ductors and junctions, Eqs. (4-6) in the main text (MT), for V = V , V , V ch , V ch , (A4) ch CJ C G T the symmetric elements, where we have added gyrator branches to both branch sets. IJ = Ic sin(2πΦJ /Φ0) = ΦJ U(ΦJ ), (A18) We can write Kirchhoff’s current laws without external fluxes −∇ I = L 1Φ , (A19) for simplicity, L − L while that for the Y-NR branches, Eq. (8) in the MT, can be F I = I , (A5) ch − tr decomposed into FT V = V , (A6) tr ch I tr Y tr tr Y tr ch V tr making use of the fundamental loop matrix F; see Refs. G = G G G G G . (A20) I ch Y ch tr Y ch ch V ch [25, 26] for a detailed analysis of graph theory applied to su- G G G G G G perconducting circuits, that can be partitioned as Introducing Kirchhoff’s voltage law in the current-voltage re- 1 lation for chord NR branches we derive  FJC FJGch FJT ch  eff T 0 F F ch F ch tr F =  LC LG LT  . (A7) IGch = (YGchGtr + YGchGch (FGtrGch ) ) VG (A21) 0 F tr F tr ch F tr ch  G C G G G T h eff T eff T i +Y ch ch (F ch ) VL +(F ch ) VJ . 0 FT trC FT trGch FT trT ch G G LG JG 8

Substituting Eqs. (A18-A21) in (A11, A12 and A13) we have APPENDIX B: NR 2-PORT IMPEDANCE the equations of motion of the circuit that can be derived from COUPLED TO JOSEPHSON JUNCTIONS the Lagrangian,

1 T 1 1 T We explicitly compute matrices of Hamiltonian (13) for cir- L = Φ˙ Φ˙ ΦT M Φ + Φ˙ GΦ U(Φ ), 2 C − 2 0 2 − J cuit in Fig.2, both in the MT, of a nonreciprocal 2-port loss- (A22) less impedance [41] capacitively coupled to Josephson junctions. T T T T with Φ = (ΦJ , ΦL, ΦGtr ). The symmetric capacitive and T The tree and chord branch sets are divided in Itr = inductive matrices read T T T T T T (I , I L ) and I = (I , I , I R ), with left (right) trans-   C T ch J L T CJ 0 0 former branches being tree (chord) branches. A general turns eff eff T ratios matrix for the Belevitch transformer is =  0 0 0 + C C( C ) , (A23) C 0 0 0 F F 1 T  L L  M0 = LL− L , (A24) n11 0 0 n12 0 0 I I R R N =  0 n11 0 0 n12 0  . (B1) where we defined 0 0 n21 0 0 n22  eff    FJX 0 eff eff 1 X =  FLX  , L =  L , (A25) We will calculate with it the effective loop matrix (10) and get F eff I 0 FGtrX Hamiltonian (13) in main text. The capacitance matrix is full rank and X = C, Gch . The skew-symmetric matrix is { } T eff eff T   G = tr Y tr tr tr + ch Y ch ch ( ch ) (A26) CJ1 IG G G IG FG G G FG eff T eff T  CJ2  + Gch YGchGtr Gtr + Gtr YGtrGch ( Gch ) ,   F I I F  Cg1    with the identity vector C =  Cg2  , (B2)    C1R   0   C1L  C2 Gtr =  0  . (A27) I 1Gtr where the blank elements of the matrix are zero. The inductive Explicitly, M0 matrix can be computed with the loop submatrix   GJJ GJL GJGtr T G =  GJL GLL GLGtr  , (A28) eff 0M 0 − T T FCL = FCL = (B3) G tr G tr G tr tr 0 1 − JG − LG G G where all the submatrices are defined as where M = J + g + G1 + L. J, g, G1, L are, respectively, eff eff T { } GJJ = FJGch YGchGch (FJGch ) , the number of (i) Josephson junctions (2), (ii) coupling capaci- eff eff T tors (2), (iii) gyrator-shunted capacitors (2), and (iv) inductors GJL = FJGch YGchGch (FLGch ) , eff eff T (L). 0M represents a zero square matrix of M dimension. The GLL = FLGch YGchGch (FLGch ) , skew-symmetric gyration matrix G can be computed using the eff eff T GJGtr = FJGch (YGchGtr + YGchGch (FGtrGch ) ), effective loop submatrix, eff eff T GLGtr = FLGch (YGchGtr + YGchGch (FGtrGch ) ), eff eff T eff G tr tr = F tr ch Y ch ch (F tr ch ) + Y tr tr G G G G G G G G G G FCG = FCG + FCT R NFT LG eff eff T   + FGtrGch YGchGtr + YGtrGch (FGtrGch ) . 1 0 0 0  0 1 0 0 The Hamiltonian of this system is    1 0 0 0 =  0 1 0 0 (B4) 1 1 T 1 1   H = Q GΦ − Q GΦ  L L  2 − 2 C − 2 n11 n12 1 0 1 T nR nR 0 1 + 2 Φ M0Φ + U(ΦJ ), (A29) 11 12 n21 n22 0 0 where Q = ∂L/∂Φ are the conjugate charges to the flux variables. Canonical quantization follows promoting the variables to operators with commutation relations [Φi, Qj] = i~. which is calculated through the turn ratios matrix N and the 9 submatrices or equal to those without it. In order to carry out canoni-   cal quantization of this Hamiltonian it is convenient to pro- 1 0 0 0 ceed with the symplectic diagonalization of the harmonic part. 0 1 0 0 Consider thus the matrix   1 0 0 0 F = 0 1 0 0 , (B5) CG   1 Γ 0 0 1 0 H = . (C2)   h ΓT Ω2 0 0 0 1 0 0 0 0   Since this matrix is symmetric and definite positive, the cor- 0 0 0 responding theorem of Williamson [49] holds that it can be 0 0 0   brought to the canonical form D = diag (Λ, Λ), with Λ a def- 0 0 0 inite positive diagonal matrix, by a symplectic transformation   FCT R = 0 0 0 , (B6) S. That is, ST H S = D with symplectic matrix S. The deter-   h 1 0 0 mination of the symplectic eigenvalues and of the canonical 0 1 0 symplectic transformation can be achieved by considering the 0 0 1 matrix HhJ, with 1 0 0 0 1 0 0 0 0 1 1 0 0 0 J = 1 . (C3) L   (B7) 0 FT G =   . − 0 1 0 0 0 1 0 0 Its eigenvalues form conjugate pure imaginary pairs, iλ , 0 1 0 0 ± j where the positive numbers λj are the diagonal elements This analysis can be performed because the constitu- of Λ. Choose an eigenvector vj corresponding to iλj. Its tive equation of the nonreciprocal elements (8) in the MT complex conjugate, vj∗, is an eigenvector with iλj eigenvalue. Organize the column eigenvectors in a− matrix F = could be simplified to IG = YG VG, where IG = T v1 v2 vN v∗ v∗ . Normalize the vectors by (IG0L , IG0R , IG1L , IG1R ) and ··· 1 ··· N the condition FF† = Hh. Define a matrix function SV = 1 1/2 YG0 0 F† − VD acting on unitaries V. It is clearly the case that, YG = , (B8) 0 YG1 for all unitaries V and phase choices for the eigenvectors vj, 1 1 − 1 SV† HhSV = D, since F− Hh F† = . The unitary V is with determined by the requirement that it provide us with a symplectic matrix, ST JS = J. Inter alia, this means that S is real. 1 0 1 Y = , (B9) In fact, the choice Gi R 1 0 i − 1 1 i the admittance matrix for each gyrator i 0, 1 . The final V = (C4) ∈ { } √ 1 i gyration matrix is 2 −

eﬀ eﬀ T G = FCG YG(FCG) . (B10) achieves this objective. This can be readily checked by noticing that

APPENDIX C: SYMPLECTIC DIAGONALIZATION  T  v1† + v1  .   .  We discuss now the procedure to diagonalize the quadratic 1  T  V F =  v† + v  (C5) sector of Hamiltonian (13). We can perform a canonical † †  N N  1/2 T 1/2 T √2  T  change of variables Q = C O q, ΦC = C− O f such  i v1† v1  C − −  that we diagonalize the pure capacitive and inductive sectors . of the Hamiltonian, . 1 1 T T Γ q 1 1/2 q f f (C1) is explicitly real in this case, so S− = D− V†F† is seen to H = ( , ) T 2 f + U( ), 2 Γ Ω be real. Furthermore, this choice also determines S as symplectic. 1 1/2 1/2 T 2 with the deﬁnitions Γ = OC− GC− O and Ω = 2 T T T T T 1/2 1 1/2 T 2− In the new variables, ξ π = S q f , the OC− L− C− O Γ a diagonal matrix. The conjugate variables (q, f) are canonical− in that q , f = δ . The pres- quadratic part of the Hamiltonian is diagonal. They can now { i j} ij ence of the antisymmetric matrix Γ in the harmonic part of the be canonically quantized, in the form ξn = (an + an† )/√2, Hamiltonian leads to new normal frequencies that are greater π = i(a a† )/√2. n − n − n 10

APPENDIX D: REDUCTION OF VARIABLES IN CIRCUITS where we defined Ψ = QΦ, and α is an initial-value constant WITHOUT Y IDEAL NRD in flux units. Inserting the above expression in the equation of motion and applying Q on the left, we have We formalize and generalize the problem of the quantiza- 1 Q (1 + S) Q(CQΨ¨ + Q ΨU˜ (Ψ)) = R− Q (1 S) QΨ˙ , tion of circuits in flux variables with linear NR devices that are ∇ α − − only described by a constitutive equation through S. Further (D8) below, we apply this method to the derivation of the circuits with CQ = QCQ a new symmetric reduced capacitance main Fig.4( a) in the main text. trix, and U˜α(Ψ) = U(QΦ + αv 1) the new potential. The We start from the equation of motion (24) of the main text, differential nabla operator on the− original flux variables be- that we rewrite as comes Φ = Q Ψ + v 1∂α. In this new N 1 dimensional 1 ∇ ∇ −1 − (1 + S)(CΦ¨ + ΦU(Φ)) = R− (1 S) Φ˙ , (D1) space, the remnant of Q ( + S) Q is invertible. Formally, we ∇ − − T derive in this reduced space the Euler-Lagrange equation with ΦU(Φ) = (U 0 (Φ1), U 0 (Φ2), ...) , and U 0(Φi) = ∇ 1 2 i ¨ ˜ ˙ E sin(Φ ). C is a non-degenerate capacitance matrix. An CQΨ + Q ΨUα(Ψ) = GQΨ, (D9) Ji i ∇ − ideal N-port circulator can always be described by a scatter- 1 1 with GQ = R− (Q (1 + S) Q)− (Q (1 S) Q), again under- ing matrix − stood in the reduced space. There, GQ is the Cayley transform  sN  of an orthogonal matrix, and thus a skew-symmetric matrix. Let us illustrate the procedure with the choice of a specific s1  S =  .  , (D2) decomposition of the real projector Q. Consider vk and its  ..  complex conjugate vk∗ to be orthogonal vectors in the sub- sN 1 − space complementary to P. It is then easy to prove that real Re vk = (vk + v∗)/2 and imaginary parts Im vk = where each non-zero element can only be sk = 1. By a { } k { } ± i(v v∗)/2 are orthogonal vectors, again orthogonal to correct choice of terminals, it can be proven that there are only − k − k two canonical types of ideal N-port circulators: those with the P eigenspace. This assumption will hold if the vector vk is an eigenvector of S with complex eigenvalue. If the eigenvalues (sk = 1) in all their entries, and others with all (sk = value λ = 1 is present, its associated eigenvector is also real; 1), except for one (sj = 1); see Ref. [37] for further details. The eigenvalue equation− of the scattering matrix can be re- the proof is completely analogous to the above for the eigen- N Q 1 vector v 1. Normalizing all vectors, we can write trieved noticing that S = k sk , − T P T T N Y Q = v1v1 + k xkxk + ykyk , λ = sk = 1. (D3) ± PN 1 T k = n=1− wnwn , (D10)

The eigenvalues of the scattering matrix lie on the unit circle, with xk = Re vk / Re vk and yk = iπ/N 2iπn/N { } || { }|| e e with n 0, N 1 , and either 0 or 1. The Im vk / Im vk , k running through all the vectors eigenvalue λ = 1 appears∈ { with− multiplicity} one for N even coming{ } in|| complex{ }|| conjugate pairs. In general, let us Q− Q (N odd) with sk = 1 ( sk = 1). On the other hand, the denote by wn those real orthonormal vectors spanning the eigenvalue λ = 1 is present also with− multiplicity one for N orthogonal space. Q both even and odd when sk = +1. All other eigenvalues Using this nomenclature and Eq. (D7) we write come in pairs of complex conjugate values (λ and λ ). k k∗ v P w Let us assume that S presents eigenvalue 1. We deﬁne the Φ = α 1 + n fn n, − projector P = v vT such that SP = P −= PS, where v α 1 1 1 = M , (D11) is the normalized− eigenvector− corresponding− to the eigenvalue− f 1. We complete the identity with the projector Q = 1 P, − − T which also commutes with S; [S, Q] = [S, P] = 0. It is trivial with fn = wn Φ, and M = [v 1, w1, w2, ...] an orthogonal T 1 − to prove that P is real and that thus so it is Q. If 1 is an matrix, i.e., MM = . The nabla operator can be rewritten eigenvalue, it always has multiplicity 1. Then, given− that S = as S∗,  ∂  ∂α ∂ ∂ v ∗ v v 1 T ∂α (S 1) = ∗ 1 = S ∗ 1, (D4) Φ = (M− )  ∂f1  = M . (D12) − − − − ∇  .  f Sv 1 = v 1. (D5) . ∇ − − − . The above two equations can only be true if v 1 = v∗ 1. − Finally, inserting the above decompositions (D11,D12,D10) Then, applying P to Eq. (D1), we have − in Eq. (D9), we rewrite the equation of motion PΦ˙ = 0. (D6) h i X 1 ¨ ˜ wn( + S)nm (C)mlfl + ∂fm Uα(f) This equation can be integrated, so that the ﬂux variable vector n,m,l is expressed as 1 X = R− w (1 S) f˙ , (D13) − n − nl l Φ = PΦ + QΦ = αv 1 + Ψ, (D7) n,l − 11

T with (A)rt = wr Awt, together with α˙ = 0. Multiplying The reduced capacitance matrix is T from the left with the real row vectors wn , and inverting { } 1 C2 C1 ! the ﬁrst matrix on the left-hand side, we arrive at an explicit (C + C + 4C ) − 1 3 1 2 3 √3 CQ = C2 C1 , (D17) form of Eq. (D9) − (C + C ) 2 √3 1 2 (C) f¨ + ∂ U˜ (f) = (GQ) f˙ , ml l fm α − ml l while the gyration matrix is 1 1 1 1 where we have deﬁned (GQ)ml = R− ( + S)mn− ( S)nl and we have used Einstein’s notation of summation over− re- 1 0 1 GQ = . (D18) peated indices. Here, we can identify Ψ (0, f) in Eq. (D9). R√3 1 0 ≡ − Furthermore, the matrix CQ has as matrix elements in this ba- ˜ T T sis precisely (C)ml. The Lagrangian without constraints and Finally, the potential function Uα = U(M(α, f ) ), with full-rank kinetic matrix with such equations of motion U(Φ) = P3 E cos(Φ ) and − i=1 Ji i T T 1 1 1 1 1 L = f˙ CQf˙ + f˙ GQf U˜α(f),   2 2 − √3 √3 √3 1 1 q 2 with f = (f1, f2, ...). Finally, the quantized Hamiltonian is M =   . (D19)  √6 √6 3  −1 − 1 T 0 1 1 1 1 √2 √2 Hˆ = Qˆ GQfˆ C− Qˆ GQfˆ + U˜ (fˆ), − 2 − 2 Q − 2 α ˙ again with Q = ∂L/∂f the conjugated charge variables, later b. 4-port case to be promoted to operators.

The eigenvalues and eigenvectors for N = 4 are λ4 = T Examples ( 1, 1, λ4, λ4∗) and V4 = [v 1, v1, v4, v4∗] , respectively, with− λ = i and v = ( i−, 1, i, 1)/2. The eigenvalue 4 4 − − λ = 1 of SN , present in this family of matrices, is associated Let us now use this general theory to quantize the speciﬁc − T ˙ with the constraint v 1Φ = 0 where v 1 = ( 1, 1, 1, 1)/2 cases illustrated in the main text. The scattering matrix of Eq. is the normalized eigenvector.− − − − (15) introduced in the circuits in Fig. 3(a) in the MT, The inhomogeneous capacitance matrix is  1  C1+C2+C3+C4 C4 C2 C3 C1  − − 1 4 2√2 2√2 N   C4 C2 C2+C4 SN = ( 1)  .  , (D14) C =  − 0  , (D20) −  ..  Q  2√2 2  C3 C1 C1+C3 − 0 1 2√2 2 1 has 1 eigenvalues for all N and +1 eigenvalues for even-N that reduces to CQ = C for Ci = C. On the other hand, the numbers− of ports. Notice that in the analysis of the equations gyration matrix has now a zero column and row corresponding of motion above we have not made use of the canonical form to the eigenvalue λ = +1, of S matrices mentioned earlier, and indeed this example does   not and needs not conform to that canonical presentation. 1 0 0 0 GQ = 0 0 1 . (D21) R 0 1 0 − a. 3-port case Given the complex-conjugate pairwise nature of the eigenvalues and eigenvectors, the gyration matrix can always be writ- The eigenvalues and eigenvectors for N = 3 are λ3 = T ten in a basis with 2 2 blocks, except for the row and the ( 1, λ3, λ3∗) and V3 = [v 1, v3, v3∗] , respectively, with × − 2πi/3 2πi/− 3 2πi/3 column of zeros corresponding to the +1 eigenvalue. Finally, λ3 = e and v3 = (e , e− , 1)/√3. The eigen- T T we have the potential function U˜α = U(M(α, f ) ), with value λ = 1 of SN , present in this family of matrices, P3 − T ˙ U(Φ) = i=1 EJi cos(Φi) and is associated with the constraint v 1Φ = 0 where v 1 = − − − (1, 1, 1)/√3 is the normalized eigenvector.  1 1 1 1  2 2 2 2 We can apply the theory described above to compute the −1 1 −1 1 projectors  2 2 2 2  M =  0 1 0 1  . (D22)    − √2 √2  1 1 1 1 0 1 0 T 1 − √2 √2 P = v 1v 1 = 1 1 1 , (D15) − − 3 1 1 1  2 1 1 1 − − Q = 1 P =  1 2 1 . (D16) − 3 −1 1− 2 − − 12

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