The Energy of an Arbitrary Electrical Circuit, Classical and Quantum
TThe Energy of an Arbitrary Electrical Circuit, Classical and Quantum
Matteo Mariantoni ∗ Institute for Quantum Computing, University of Waterloo,200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada andDepartment of Physics and Astronomy, University of Waterloo,200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada (Dated: July 20, 2020)In this Note, I show an algorithmic method to find the energy and, thus, the Hamiltonian of anarbitrary electrical circuit based on the so-called incidence matrix and the circuit’s total power. Thismethod does not require to find any Lagrangian; instead, it is based on the concept of generalizedlinear momenta for the kinetic and co-kinetic energy of a circuit. The method can account forsuperconducting loops by a simple extension of Faraday-Henry-Neumann’s law. Auxiliary (i.e.,parasitic) circuit elements are required to deal with circuits with an incomplete set of generalizedvelocities resulting in an incomplete set of canonical coordinates. This method can be readilyautomatized to obtain the Hamiltonian of arbitrarily complicated circuits. I also show how toquantize the circuit associated with a resonator capacitively coupled with a qubit.
I. INTRODUCTION
The problem of finding the energy of an arbitraryelectrical circuit has been addressed by many authorsin the past century. The first article I am aware of isby D.A. Wells in 1938 [1]; this article treats linear cir-cuits. Over the next several decades, a large body ofwork culminated with the theory of nonlinear circuits byB.M. Maschke et al. in Ref. [2]. More recently, the devel-opment of quantum computers based on superconductingcircuits has renewed the interest in this topic; the worksby Burkard et al. in Ref. [3] and by U. Vool and M.H. De-voret in Ref. [4] explore similar approaches, although theformer follows a very rigorous method and the latter amore practical one. Other works include the quantumnetwork theory by Yurke and Denker [5] as well as theFoster representation method of Russer and Russer [6].The aim of this Note is to describe the simplest methodan electrical engineer would follow to obtain the energyand, then, the Hamiltonian of any circuit. This approachrequires only a basic knowledge of circuit theory as inchapter one of Ref. [7], as well as a very limited knowledgeof classical mechanics as in chapter five of Refs. [8, 9]. Ihave been using the method presented here for more thana decade in summer schools and for my own research: Iwant to share it here because I find it the most practicalapproach among all those mentioned above.In order to explain my method, I will describe threeexamples: First, a parallel circuit with an inductor L ,a capacitor C , and a Josephson tunnel junction J , or LCJ circuit { and the two limiting cases of a simple LC resonator and the circuit of a transmon quantumbit (qubit) [10] } ; second, a DC superconducting quan-tum interference device (SQUID) with DC current bias;third, two LC parallel resonators coupled by means ofa third LC resonator (i.e., the “Vool-Devoret circuit” ∗ Corresponding author: [email protected] revisited). Each of these examples allows me to intro-duce the main concepts of the method: The derivationof the incidence matrix from the circuit’s digraph, fol-lowed by finding the circuit’s total power and, by timeintegration, the instantaneous energy of the circuit; thedefinition of generalized linear momenta for the kineticand co-kinetic (to be introduced below) energy of the cir-cuit, allowing to obtain the Hamiltonian; the extensionof Faraday-Henry-Neumann’s law (simply Faraday’s lawin the English-speaking literature) to include supercon-ducting loops; the concept of auxiliary circuit elementsto complete an incomplete set of generalized velocitiesresulting in an incomplete set of canonical coordinates,i.e., to generate a complete set of independent degrees offreedom (DOF).Additionally, I show the procedure to quantize the cir-cuit associated with a resonator coupled with a qubit.This is an important example as it leads to the Jaynes-Cummings Hamiltonian, which is one of the foundationsof circuit quantum electrodynamics and superconductingquantum computing.In this Note, we indicate a potential difference (or volt-age drop, or simply voltage) as v ; a current as ı (withouta dot, not to get confused with time derivatives); a fluxas φ ; a charge as q . L C J
FIG. 1. An
LCJ circuit. a r X i v : . [ phy s i c s . c l a ss - ph ] J u l LCJ circuit.
II.
LCJ
PARALLEL CIRCUIT
Figure 1 shows an
LCJ circuit. The constitutive rela-tion for the capacitor is ı = C ddt v = C • v , (1)and for the inductor v = L ddt ı = L • ı . (2)The physics of the Josephson tunnel junction is wellexplained, e.g., in Ref. [11] and its constitutive relationis given by ı ( φ ) = I c0 sin( k J φ ) = I c0 sin( ϕ ) , (3)where I c0 is the junction’s critical current, k J = 2 π/ Φ isthe non-normalized Josephson constant, and ϕ is calledthe gauge-invariant phase difference across the junction;Φ = h/ (2 e ) is the superconducting magnetic flux quan-tum ( h is Planck’s constant and e the electron charge).From Faraday-Henry-Neumann’s law, the voltageacross the junction is given by v = • φ = • ϕk J . (4)Equation (4) is often referred to as the “second Josephsonequation,” whereas Eq. (3) is called the first Josephsonequation. I prefer to call Eq. (4) simply Faraday-Henry-Neumann’s law because it is nothing more than that [12].I call Eq. (3) simply the Josephson equation.Figure 2 illustrates the simple digraph associated withthe physical circuit in Fig. 1 [7]. The digraph is com-prised of two nodes, 1 and 2 , as well as three ori-ented branches, 1, 2, and 3; we set 2 → datum, i.e.,to a reference node experimentally realized by earthingor grounding the circuit. The sign convention for theoriented branches is that any branch entering a node is given the value − (cid:126)ı = ı ı ı (cid:126)q = q q q (cid:126)v = v v v (cid:126)φ = φ φ φ (cid:126)e = [ e ] . (5a)(5b)(5c)Following the sign convention for the digraph outlinedabove, the incidence matrix for this digraph reads A a = (cid:18) → − − − → (cid:19) . (6)The reduced incidence matrix A is obtained by strikingout the row in A a associated with the datum, resultingin A = (cid:2) − − − (cid:3) . (7)Kirchhoff’s current law (KCL) reads A (cid:126)ı = 0 , (8)from which we obtain the constraint for the currents, ı = − ı − ı . (9)Kirchhoff’s voltage law (KVL) reads (cid:126)v = A T (cid:126)e , (10)where A T is the transpose of A , from which we obtainthe constraint for the voltages, v = v = v = v (11)(parallel circuit).By means of Eq. (4), the constraints given by Eq. (11)can be written as the system ddt ( φ − φ ) = 0 ddt ( φ − φ ) = 0 . (12a)(12b)From Eqs. (12a) and (12b) it follows that φ = φ + φ ∼ φ = φ + φ ∼∼ , (13a)(13b)where φ ∼ and φ ∼∼ are constant flux offsets. These offsets mayresult in DC currents in the equations of motion for thecircuit (as shown below). In particular, from Eq. (13b)we find φ = φ − φ ∼∼ . (14)In the coordinates given by Eqs. (5a) and (5b), thebranch equations can be written as ı = • q = C • v v = • φ = L • ı = − L ( • ı + • ı ) = − L ( • q + • q ) ı = • q = I c0 sin( k J φ ) . (15a)(15b)(15c)Accounting for the constraints given by Eqs. (9), (11),and (14) as well as for the branch Eqs. (15a), (15b), and(15c), the circuit’s total power is P LCJ = v ı + v ı + v ı = v C • v + L ( • ı + • ı )( ı + ı )+ v I c0 sin[ k J ( φ − φ ∼∼ )] . (16)The instantaneous energy of the circuit is then ob-tained by time integration from the initial time τ = 0 to ageneric time τ = t . Expressing all coordinates in terms offluxes and charges and with the usual trick for a contin-uous function f that αf • f = d ( αf / /dt (with α ∈ R ), the total instantaneous energy is E LCJ ( t ) = (cid:90) τ = tτ =0 dτ ddτ (cid:20) C • φ ( τ ) (cid:21) + ddτ (cid:26) L [ • q ( τ ) + • q ( τ )] (cid:27) + (cid:90) τ = tτ =0 dτ • φ ( τ ) I c0 sin { k J [ φ ( τ ) − φ ∼∼ ] } = 12 C • φ ( t ) + 12 L (cid:2) • q ( t ) + 2 • q ( t ) • q ( t ) + • q ( t ) (cid:3) − C • φ (0) − L (cid:2) • q (0) + 2 • q (0) • q (0) + • q (0) (cid:3) − I c0 k J cos { k J [ φ ( t ) − φ ∼∼ ] } + I c0 k J cos { k J [ φ (0) − φ ∼∼ ] } = 12 C • φ ( t ) + 12 L (cid:2) • q ( t ) + 2 • q ( t ) • q ( t ) + • q ( t ) (cid:3) − E J0 cos { k J [ φ ( t ) − φ ∼∼ ] }− K LCJ , (17)where E J0 = I c0 k J = I c0 Φ π (18)is called the Josephson energy and K LCJ = 12 C • φ (0) − L (cid:2) • q (0) + 2 • q (0) • q (0) + • q (0) (cid:3) + I c0 k J cos { k J [ φ (0) − φ ∼∼ ] } (19)is a constant of integration.Setting K LCJ = 0, defining • q = • q , using Eq. (15c)with φ given by Eq. (14), and keeping all variables asimplicit time-dependent functions, Eq. (17) reads E LCJ = 12 C • φ + 12 L (cid:110) • q + 2 • qI c0 sin[ k J ( φ − φ ∼∼ )] + I sin [ k J ( φ − φ ∼∼ )] (cid:111) − E J0 cos[ k J ( φ − φ ∼∼ )]= 12 C • φ + 12 L • q + LI c0 • q sin[ k J ( φ − φ ∼∼ )] + 12 LI sin [ k J ( φ − φ ∼∼ )] − E J0 cos[ k J ( φ − φ ∼∼ )] . (20)The single DOF of this circuit is clearly associated withthe voltage v = v = • φ and the associated current ı = ı = • q . In this case, the total energy obtained by direct in-tegration of the power does not coincide with the circuit’sHamiltonian. In fact, the power integration results in afunction E LCJ ( • φ, • q, φ ), i.e., of mixed coordinates insteadof a function of either the canonical coordinates( v = • φ, ı = • q ) (21) or (cid:18) q = (cid:90) dτ ı, φ = (cid:90) dτ v (cid:19) (22)only.In order to obtain the circuit’s Hamiltonian, we needto express • φ and • q in terms of q and φ , respectively. Ifonly terms such as E C = 12 C • φ E L = 12 L • q , (23a)(23b)were present, we could simply use the definitions of ca-pacitance and inductance, C = q • φL = φ • q , (24a)(24b)respectively, and write E C = • q C E L = • φ L . (25a)(25b)However, the presence of the term LI c0 • q sin[ k J ( φ − φ ∼∼ )] (26)complicates the procedure. In the following section, weoutline a general method to find the Hamiltonian of a cir-cuit from the total energy obtained by integrating the cir-cuit’s power. Then, we specialize that method to find theHamiltonian associated with the total energy of Eq. (20). III. THE KINETIC AND CO-KINETICENERGY TRANSFORMATION
Suppose that P is the total power of a given circuit. Wedefine ξ k as the generalized coordinates of the circuit (infact, our argument would apply to any generic system, in-cluding mechanical systems) and • ξ k as the correspondinggeneralized velocities ( k ∈ N ); we additionally define p k as the conjugate momenta associated with ξ k and • p k thefirst time derivatives of the conjugate momenta [13]. Byintegrating the total power of any arbitrary circuit weobtain (cid:90) τ = tτ =0 dτ P ( τ ) = E tot [ ξ k ( t ) , • ξ k ( t ); p k ( t ) , • p k ( t ); t ] . (27)In Eq. (27), the functional dependence from, e.g., a sub-set of the generalized coordinates ξ k , { ξ (cid:96) } , or a subsetof the conjugate momenta p k , { p m } , can be an arbitraryfunction (e.g., of trigonometric type; i.e., not a simplepolynomial), E tot ( ξ k (cid:54) = (cid:96) , f ( ξ (cid:96) ) , • ξ k ; p k (cid:54) = m , f ( p m ) , • p k ; t ); theremaining variables, including the time derivatives, areinstead characterized by a polynomial functional depen-dence. Notably, in Eq. (27) the sets { ξ k , • ξ k } and { p k , • p k } represent two distinguished sets of generalized coordi-nates and their corresponding time derivatives, while, atthe same time, the variables p k are also the conjugatemomenta of the generalized coordinates ξ k . Therefore, { ξ k , p k } is a set of canonical coordinates.Due to the homogeneity of time [14], Eq. (27) does notexplicitly depend on the time t , i.e., E tot = E tot ( ξ k , • ξ k ; p k , • p k ) , (28)where the time dependence of all the variables in paren-thesis is not explicitly shown to simplify the notation.Equation (28) is not the system’s Hamiltonian, whichmust be only a function either of the set of canonicalcoordinates { ξ k ; p k } or of the set of canonical coordi-nates { • ξ k def = Ξ k ; • p k def = P k } , but not a combination ofboth sets. We additionally note that Eq. (28) could beconfused for the system’s Routhian [14], but it is not.For example, the circuit studied in Sec. II is character-ized by a total energy in the form of Eq. (28). For anytotal energy of this type, the kinetic-energy terms areall those containing time derivatives of first and secondorder such as those reported in Table I.In typical applications, mixed terms such as • ξ (cid:96) • p m areabsent from Eq. (28). This is true even for relatively com-plicated systems comprising Josephson tunnel junctions[see, e.g., Eq. (20)] or, as we will show in Sec. V, for sys-tems with multiple DOF. Qualitatively, this makes sensebecause a Josephson tunnel junction is characterized bya current that does not depend on the time derivativeof any generalized coordinate and purely inductive or ca-pacitive elements (even when coupled) always simplify sothat their energy only contains either • ξ (cid:96) - or • p (cid:96) -terms.In general, all the terms originating from a quadraticform are kinetic-energy terms. In Eq. (20), the term12 LI sin [ k J ( φ − φ ∼∼ )] , (29)which stems from a quadratic form is also a kinetic-energy term, although it does not comprise any time-derivative of the generalized coordinates or of their con-jugate momenta (this term can easily, but erroneously beconfused for a potential energy term!).Once all the kinetic-energy terms are identified, theremaining terms in any total energy expressed as inEq. (28) are potential-energy terms (see Appendix A).It is worth pointing out already at this stage that, whenanalyzing any circuit, it is not necessary to identify the TABLE I. First- and second-order time derivatives participat-ing in a general kinetic-energy term ( (cid:96), m ∈ N ). • ξ (cid:96) • p (cid:96) • ξ (cid:96) • ξ m • ξ (cid:96) • p m • p (cid:96) • p m ( • ξ (cid:96) ) ( • p (cid:96) ) potential-energy terms and distinguish them from thekinetic-energy terms. The general method we outline inthe following works regardless of any classification of theenergy terms in Eq. (28).It is rare to encounter circuits where the kinetic-energyterms are comprised of complicated functions of • ξ k or • p k , or both (at least, I never encountered such a case);typically, the kinetic-energy terms are second-order poly-nomial functions and, sometimes, even simple homoge-neous polynomials of degree two . Circuits characterizedby kinetic-energy terms with a non-polynomial functionaldependence must be linearized by means of, e.g., a Taylorseries expansion around a given operation (bias) point.In Eq. (20), for example, the set of generalized veloc-ities { • ξ k } is characterized by one element only • ξ = • φ .The kinetic energy associated with this generalized ve-locity is T ( • φ ) = 12 C • φ ; (30)the set of generalized velocities { • p k } , instead, is char-acterized by a 2-tuple ( • p = • q = • q , • p = • q = I c0 sin[ k J ( φ − φ ∼∼ )]). The corresponding kinetic energy isthus T ∗ ( • q , • q ) = 12 L ( • q + 2 • q • q + • q )= 12 L • q + LI c0 • q sin[ k J ( φ − φ ∼∼ )]+ 12 LI sin [ k J ( φ − φ ∼∼ )] , (31)where we define T ∗ as the co-kinetic energy of the system.In general, any term of the form T ( (cid:126)ξ, (cid:126) • ξ ) = N ξ (cid:88) (cid:96),m =1 A (cid:96),m ( (cid:126)ξ, (cid:126)p ) • ξ (cid:96) • ξ m + N ξ (cid:88) (cid:96) =1 B (cid:96) ( (cid:126)ξ, (cid:126)p ) • ξ (cid:96) + C ( (cid:126)ξ, (cid:126)p ) (32) or T ∗ ( (cid:126)p, (cid:126) • p ) = N ξ (cid:88) (cid:96),m =1 D (cid:96),m ( (cid:126)ξ, (cid:126)p ) • p (cid:96) • p m + N ξ (cid:88) (cid:96) =1 E (cid:96) ( (cid:126)ξ, (cid:126)p ) • p (cid:96) + F ( (cid:126)ξ, (cid:126)p ) (33)is a kinetic energy term. In this two equations, (cid:126)ξ =( ξ , ξ , . . . ξ k , . . . ξ N ξ ) and (cid:126) • ξ = ( • ξ , • ξ , . . . • ξ k , . . . • ξ N ξ ) aretwo vectors ( N ξ -tuples) the components of which are allthe N ξ generalized coordinates and generalized veloci-ties, respectively. Similarly, (cid:126)p = ( p , p , . . . p k , . . . p N ξ )and (cid:126) • p = ( • p , • p , . . . • p k , . . . • p N ξ ) are two vectors the com-ponents of which are all the N ξ conjugate momenta(here interpreted as second set of generalized coordi-nates) and their first time derivatives (here interpretedas a second set of generalized velocities). Clearly,since the components of (cid:126)p are the conjugate momentaof the components of (cid:126)ξ , these two vectors must havethe same dimension N ξ (similarly for their derivatives).The coefficients A (cid:96),m ( (cid:126)ξ, (cid:126)p ), B (cid:96) ( (cid:126)ξ, (cid:126)p ), and C ( (cid:126)ξ, (cid:126)p ) as wellas D (cid:96),m ( (cid:126)ξ, (cid:126)p ), E (cid:96) ( (cid:126)ξ, (cid:126)p ), and F ( (cid:126)ξ, (cid:126)p ) are functions of thegeneralized coordinate vectors (cid:126)ξ and (cid:126)p , but not of theirtime derivatives [all first time derivatives are supposed tobe quadratic terms, i.e., the terms multiplying A (cid:96),m ( (cid:126)ξ, (cid:126)p )and D (cid:96),m ( (cid:126)ξ, (cid:126)p ) or linear terms, i.e., the terms multiply-ing B (cid:96) ( (cid:126)ξ, (cid:126)p ) and E (cid:96) ( (cid:126)ξ, (cid:126)p ) in Eqs. (32) and (33), respec-tively].As already noted, the kinetic energy T ∗ ( (cid:126)p, (cid:126) • p ) is a func-tion of (cid:126)p , which is the conjugate momentum vector as-sociated with the generalized coordinate vector (cid:126)ξ . Forthis reason, we call T ∗ ( (cid:126)p, (cid:126) • p ) the co-kinetic energy associ-ated with the kinetic energy T ( (cid:126)ξ, (cid:126) • ξ ). In absence of anyterm ∝ • ξ (cid:96) • p m the total kinetic energy of a general systemis given by the sum of Eqs. (32) and (33), i.e., of thekinetic and co-kinetic energies, T tot ( (cid:126)ξ, (cid:126) • ξ ; (cid:126)p, (cid:126) • p ) = T ( (cid:126)ξ, (cid:126) • ξ ) + T ∗ ( (cid:126)p, (cid:126) • p ) . (34)Depending on the context, we use either thevector notation (cid:126)ξ, (cid:126) • ξ, (cid:126)p , and (cid:126) • p or the equivalentindex notation either as sets of indexed vari-ables { ξ k } , { • ξ k } , . . . { ξ k , • ξ k } , { p k , • p k } , . . . { ξ k , p k } orsimply as indexed variables ξ k , • ξ k , p k , and • p k .The coefficients A (cid:96),m ( (cid:126)ξ, (cid:126)p ), B (cid:96) ( (cid:126)ξ, (cid:126)p ), and C ( (cid:126)ξ, (cid:126)p ) asso-ciated with the kinetic energy T ( (cid:126)ξ, (cid:126) • ξ ) can be derived byanalogy with the kinetic energy of a mechanical systemdescribed by the generalized coordinate vector (cid:126)ξ [8] T ( (cid:126)ξ, (cid:126) • ξ ) = N (cid:88) n =1 M n (cid:126)v n • (cid:126)v n = N (cid:88) n =1 M n N ξ (cid:88) (cid:96) =1 ∂P n ( ξ (cid:96) ; t ) ∂ξ (cid:96) • ξ (cid:96) + ∂P n ( t ) ∂t · N ξ (cid:88) m =1 ∂P n ( ξ m ; t ) ∂ξ m • ξ m + ∂P n ( t ) ∂t = N ξ (cid:88) (cid:96),m =1 A (cid:96),m ( (cid:126)ξ, (cid:126)p ) • ξ (cid:96) • ξ m + N ξ (cid:88) (cid:96) =1 B (cid:96) ( (cid:126)ξ, (cid:126)p ) • ξ (cid:96) + C ( (cid:126)ξ, (cid:126)p ) , (35)where M n is the inertial mass of the n -th particle of asystem of N ∈ N point-like particles, each with vectorvelocity (cid:126)v n ; the position in the three-dimensional Eu-clidean space of a particle is given by the generic positionpoint P n , which is a function of the N ξ generalized co-ordinates ξ k (which, e.g., can be Cartesian coordinates)and time t , P n ( ξ k ; t ). With (cid:96), m = 1 , , . . . N ξ , the coeffi-cients A (cid:96),m ( (cid:126)ξ, (cid:126)p ), B (cid:96) ( (cid:126)ξ, (cid:126)p ), and C ( (cid:126)ξ, (cid:126)p ) in this mechanicalanalog are given by A (cid:96),m ( (cid:126)ξ, (cid:126)p ) = N (cid:88) n =1 M n ∂P n ∂ξ (cid:96) ∂P n ∂ξ m B (cid:96) ( (cid:126)ξ, (cid:126)p ) = N (cid:88) n =1 M n ∂P n ∂ξ (cid:96) ∂P n ∂tC ( (cid:126)ξ, (cid:126)p ) = 12 N (cid:88) n =1 M n ∂P n ∂t ∂P n ∂t . (36a)(36b)(36c)Following a similar mechanical analogy, a comparable setof equations can be derived for the coefficients D (cid:96),m ( (cid:126)ξ, (cid:126)p ), E (cid:96) ( (cid:126)ξ, (cid:126)p ), and F ( (cid:126)ξ, (cid:126)p ) associated with the co-kinetic en-ergy T ∗ ( (cid:126)p, (cid:126) • p ).In the kinetic-energy expression of Eq. (35), the co-efficients given by Eqs. (36a), (36b), and (36c) can de-pend on the generalized coordinates ξ k and p k [simi-larly, the coefficients D (cid:96),m ( (cid:126)ξ, (cid:126)p ), E (cid:96) ( (cid:126)ξ, (cid:126)p ), and F ( (cid:126)ξ, (cid:126)p ) canalso depend on the same two sets of generalized coordi-nates]. These coefficients can be arbitrary functions, e.g.,trigonometric functions as in the example of Subsec. II;remarkably, this fact has minimal consequences when at-tempting to obtain the Hamiltonian of a circuit from itstotal energy. In particular, no linearization methods arerequired to deal with the coefficients A (cid:96),m ( (cid:126)ξ, (cid:126)p ), B (cid:96) ( (cid:126)ξ, (cid:126)p ),and C ( (cid:126)ξ, (cid:126)p ) or D (cid:96),m ( (cid:126)ξ, (cid:126)p ), E (cid:96) ( (cid:126)ξ, (cid:126)p ), and F ( (cid:126)ξ, (cid:126)p ). As ex-pected, the kinetic-energy terms depend on the first timederivatives of the generalized coordinates, • ξ k or • p k , onlylinearly and quadratically. As already mentioned, theanalysis would require a linearization step in presence of(rare) higher-order polynomial or non-polynomial func-tions of the time derivatives of the generalized coordi-nates.A useful property of the coefficients A (cid:96),m ( (cid:126)ξ, (cid:126)p ) and D (cid:96),m ( (cid:126)ξ, (cid:126)p ) is that A (cid:96),m ( (cid:126)ξ, (cid:126)p ) = A m,(cid:96) ( (cid:126)ξ, (cid:126)p ) D (cid:96),m ( (cid:126)ξ, (cid:126)p ) = D m,(cid:96) ( (cid:126)ξ, (cid:126)p ) . (37a)(37b) This property can be readily proven from the definitionof Eq. (36a) [or a similar definition for D m,(cid:96) ( (cid:126)ξ, (cid:126)p )], andthe commutative property of the product of the partialderivatives of P n in that definition.In classical mechanics, the linear momentum magni-tude for the n -th particle of a mechanical system com-prised of N particles is ddv n (cid:18) M n v n (cid:19) = dd • r n (cid:18) M n • r n (cid:19) = M n • r n = M n v n = p n , (38)where r n is the magnitude of a position vector (cid:126)r n (froman arbitrary origin) associated with P n . Since the kineticenergy of a circuit can be cast into that of a mechanicalsystem, as shown by Eq. (35), we can extend the con-cept of linear momentum magnitude given by Eq. (38)to that of generalized linear momentum p k for T ( ξ k , • ξ k )and generalized linear momentum ξ k for T ∗ ( p k , • p k ). Asa consequence, ∂∂ • ξ k E tot ( ξ k , • ξ k ; p k , • p k ) = ∂∂ • ξ k T ( ξ k , • ξ k ) = p k ∂∂ • p k E tot ( ξ k , • ξ k ; p k , • p k ) = ∂∂ • p k T ∗ ( p k , • p k ) = ξ k . (39a)(39b)These expressions clearly show that it is not even nec-essary to actually identify the kinetic– and co-kinetic–energy terms in Eq. (28) to perform the derivatives withrespect to • ξ k or • p k ; the generalized linear momentaare simply obtained by deriving the general expressionfor E tot as found from circuit analysis.In order to change from the two sets of generalized co-ordinates and velocities { ξ k , • ξ k } and { p k , • p k } in Eq. (28)to the set of canonical coordinates { ξ k , p k } used inthe Hamiltonian, it is necessary to solve the system ofEqs. (39a) and (39b). Using the general expression forthe kinetic energy T ( (cid:126)ξ, (cid:126) • ξ ) given by Eq. (35) and an analo-gous expression for T ∗ ( (cid:126)p, (cid:126) • p ) and remembering Eqs. (37a)and (37b), it can be readily shown that ∂∂ • ξ k T ( ξ k , • ξ k ) = N ξ (cid:88) (cid:96) =1 A k,(cid:96) • ξ k + B k = p k ∂∂ • p k T ∗ ( p k , • p k ) = N ξ (cid:88) (cid:96) =1 D k,(cid:96) • p k + E k = ξ k . (40a)(40b)This system of two linear, non-homogeneous alge-braic equations can be re-written by bringing theterms B k ( (cid:126)ξ, (cid:126)p ) and E k ( (cid:126)ξ, (cid:126)p ) to the right-hand side ofEqs. (40a) and (40b), respectively, N ξ (cid:88) (cid:96) =1 A k,(cid:96) ( (cid:126)ξ, (cid:126)p ) • ξ k = p k − B k ( (cid:126)ξ, (cid:126)p ) N ξ (cid:88) (cid:96) =1 D k,(cid:96) ( (cid:126)ξ, (cid:126)p ) • p k = ξ k − E k ( (cid:126)ξ, (cid:126)p ) . (41a)(41b)As in the case of mechanical Newtonian systems, ∂ ∂ • ξ k • ξ (cid:96) T = A k,(cid:96) ( (cid:126)ξ, (cid:126)p ) ∂ ∂ • p k • p (cid:96) T ∗ = D k,(cid:96) ( (cid:126)ξ, (cid:126)p ) (42a)(42b)are the elements of two definite positive matrices A ( (cid:126)ξ, (cid:126)p )and D ( (cid:126)ξ, (cid:126)p ), respectively. Thus, both these matrices arecharacterized by a nonzero determinant, (cid:40) det A ( (cid:126)ξ, (cid:126)p ) (cid:54) = det D ( (cid:126)ξ, (cid:126)p ) (cid:54) = , (43a)(43b)which is a necessary and sufficient condition for the sys-tem of Eqs. (41a) and (41b) to have a unique non-trivial solution. IV. THE HAMILTONIAN OF THE
LCJ
CIRCUIT, FINALLY
In order to find the Hamiltonian of the
LCJ circuitstudied in Sec. II, we first solve the system of Eqs. (39a)and (39b) for the total energy of Eq. (20). This reads ∂∂ • φ E LCJ = C • φ = q∂∂ • q E LCJ = L • q + LI c0 sin[ k J ( φ − φ ∼∼ )] = φ . (44a)(44b)This system is in the form of Eqs. (41a) and (41b),with A ( φ ; q ) = C , B ( φ ; q ) = 0, D ( φ ; q ) = L , and E ( φ ; q ) = LI c0 sin[ k J ( φ − φ ∼∼ )]. By solving the systemwith respect to • φ and • q , we obtain • φ = qC • q = φ − LI c0 sin[ k J ( φ − φ ∼∼ )] L . (45a)(45b)We now substitute the results of Eqs. (45a) and (45b)into Eq. (20) and obtain the Hamiltonian H LCJ = q C + 12 L { φ − LI c0 sin[ k J ( φ − φ ∼∼ )] } L + LI c0 φ − LI c0 sin[ k J ( φ − φ ∼∼ )] L sin[ k J ( φ − φ ∼∼ )] + 12 LI sin [ k J ( φ − φ ∼∼ )] − E J0 cos[ k J ( φ − φ ∼∼ )]= q C + φ L − E J0 cos[ k J ( φ − φ ∼∼ )] . (46)Although in this case the Hamiltonian is fairly simpleand we could have guessed it without resorting to themethod introduced here, the simplicity of this examplehelps to show the working principle of the method.We can derive the equation of motion for this circuitusing the usual Hamilton equations, ∂∂q H LCJ = qC = • φ∂∂φ H LCJ = φL + I c0 sin[ k J ( φ − φ ∼∼ )] = − • q . (47a)(47b)Solving Eq. (47a) with respect to q , deriving the re-sult with respect to time one time, and substituting thisderivative into Eq. (47b), we obtain the single second- order differential equation C •• φ + φL + I c0 sin[ k J ( φ − φ ∼∼ )] = 0 . (48)A similar result can be found directly from Eq. (9), ı + ı + ı = C •• φ + φ L + I c0 sin( k J φ )= C •• φ + φ − ˜ φL + I c0 sin[ k J ( φ − φ ∼∼ )]= 0 . (49)It is worth noting that following this direct approach re-sults in the extra term − ˜ φ/L , which corresponds to a DCcurrent. Since there are no DC current sources attached L CI b J J FIG. 3. LC circuit with DC current source and DC SQUIDin parallel. to the LCJ circuit considered here, this term has to beset to zero by assuming ˜ φ = 0. This step is unnecessarywhen using our method, which gives an equation of mo-tion already without any DC current source (note thatthe φ ∼∼ in the argument of the sine does not lead to any DCcurrent, as it can be shown from a simple trigonometricidentity). A. LC Circuit with DC SQUID in Parallel
Consider the circuit of Fig. 3, first assuming the DCcurrent source with current I b to be an open circuit. Thecircuit elements highlighted in red indicate the parallelconnection of two identical Josephson tunnel junctions J and J , each with a critical current I c0 (the generaliza-tion to the case of non-identical junctions is left to thereader as an exercise). This parallel connection is called a DC SQUID. Maintaining the same notation as in thecase of the LCJ circuit, we now have one more flux as-sociated with J , φ .In this case, ddt ( φ − φ ) = 0 ddt ( φ − φ ) = 0 . (50a)(50b)However, this time we need to extend the conditions ofEqs. (13a) and (13b) to account for the flux quantizationof a superconducting loop [15]. It follows that φ − φ = φ ∼∼ φ − φ = k Φ , (51a)(51b)with k ∈ Z . The flux quantization condition of Eq. (51b)is a special case of the condition following from Faraday-Henry-Neumann’s law.From Eq. (51b), it also follows that φ + φ φ − k Φ φ − φ k Φ . (52a)(52b)Using simple trigonometric identities as well asEqs. (52a), (52b), and (51a), the energy of theDC SQUID is given by E SQUID ( t ) = (cid:90) τ = tτ =0 dτ • φ ( τ ) I c0 { sin[ k J φ ( t )] + sin[ k J φ ( t )] } = (cid:90) φ ( t ) φ (0) dφ I c0 sin (cid:18) k J φ + φ (cid:19) · cos (cid:18) k J φ − φ (cid:19) = (cid:90) φ ( t ) φ (0) dφ I c0 sin (cid:20) k J (cid:18) φ − k Φ (cid:19)(cid:21) · cos (cid:18) k J k Φ (cid:19) = (cid:90) ϕ ( t ) /k J ϕ (0) /k J dϕ I c0 k J sin ( ϕ − kπ ) · cos ( kπ )= (cid:90) ϕ ( t ) /k J ϕ (0) /k J dϕ I c0 k J sin ϕ = (cid:90) φ ( t ) φ (0) dφ I c0 sin[ k J ( φ − φ ∼∼ )]= − E J0 cos[ k J ( φ ( t ) − φ ∼∼ )] + K SQUID , (53)where K SQUID is a constant of integration. From thisresult, it is clear that the pair of junctions in the SQUIDcan be treated as a single junction with twice the Joseph-son energy of each junction.Flux tunability can be included by coupling a current source to the DC SQUID by means of the mutual induc-tance between the SQUID loop and an external inductorin series with the source. B. LCJ
Circuit with Current Source
We now consider the circuit of Fig. 3 in presence ofthe DC current source I b . Since we can represent theDC SQUID as an effective single Josephson junction, wesimply assume that the source is connected in parallelwith L , C , and a single J .The energy generated by the DC source can easily befound from E b ( t ) = (cid:90) τ = tτ =0 dτ • φ ( τ ) I b = I b (cid:90) φ ( t ) φ (0) dφ = I b [ φ ( t ) − φ (0)] (54)and can readily be added to the other terms in Eq. (17),directly leading to the tilted-washboard potential (atleast in the case L → + ∞ ) [15]. C. Limiting Cases: LC Resonator and TransmonQubit
From Eq. (46) it is straightforward to show that bychoosing I c0 = 0, we obtain the Hamiltonian of a simple(linear) resonator, or harmonic oscillator, H LC = q C + φ L . (55)The same result is found by setting the Josephson tunneljunction in Fig. 1 to an open circuit.Similarly, by setting L → + ∞ (and for a suitablechoice of the values of C and I c0 ) in Eq. (46), we ob-tain the Hamiltonian of a transmon qubit H q = q C − E J0 cos[ k J ( φ − φ ∼∼ )] . (56)The same result is found by setting the inductor in Fig. 1to an open circuit. This circuit behaves as a nonlinearresonator. V. THE “VOOL-DEVORET CIRCUIT”REVISITED
Figure 4 shows the famous example in the review arti-cle by U. Vool and M.H. Devoret of Ref. [4]. This circuitis comprised of a set of three capacitors, { C , C , C } ,and a set of three inductors, { L , L , L } .Figure 5 illustrates the digraph associated with thephysical circuit in Fig. 4. The digraph is comprised ofthree nodes, 1 and 2 , and 3 , as well as six branches,1, 2, 3, 4, 5, and 6; we set 3 → datum. The branches areindicated by the letter b = 1 , , . . . n = 1 ,
2, and 3 [not to be confused with the numberof particles in the mechanical analogy of Eq. (35)]. C L C L L C FIG. 4. The “Vool-Devoret Circuit.”
The branch currents (as well as charges), the branchvoltages (as well as fluxes), and the node-to-datum volt-ages are represented by the vectors (cid:126)ı = ı ı ı ı ı ı (cid:126)q = q q q q q q (cid:126)v = v v v v v v (cid:126)φ = φ φ φ φ φ φ (cid:126)e = (cid:20) e e (cid:21) . (57a)(57b)(57c)The incidence matrix for this digraph reads A a = → − − → − − → − − . (58)
31 51 234 62FIG. 5. Digraph of the “Vool-Devoret Circuit.” A is obtained by strikingout the row in A a associated with the datum, resultingin A = (cid:20) − − − − (cid:21) . (59)From KCL, Eq. (8), we obtain the constraint for thecurrents, (cid:26) − ı − ı + ı + ı + 0 + 0 = 00 + 0 − ı − ı + ı + ı = 0 (60a)(60b)From KVL, Eq. (10), we obtain the constraint for the voltages, v = − e v = − e v = e − e v = e − e v = e v = e . (61a)(61b)(61c)(61d)(61e)(61f)The constitutive relations for all the elements in thecircuit are ı = C • v v = L • ı ı = C • v v = L • ı ı = C • v v = L • ı . (62a)(62b)(62c)(62d)(62e)(62f)In matrix form, these relations read C C C
00 0 0 0 0 0 • v • v • v • v • v • v + − − − v v v v v v (63)+ L L L • ı • ı • ı • ı • ı • ı + − − − ı ı ı ı ı ı = ( M D + M ) (cid:126)v + ( N D + N ) (cid:126)i = (cid:126) , (64)where D is the differential operator d/dt and it isstraightforward to find M , M , N , and N from the one-to-one correspondence between Eqs. (63) and (64).We can now combine together KCL, KVL, and thebranch equations in matrix form to obtain1 − A T D + M N D + N (cid:126)e(cid:126)v(cid:126)ı = − − − − − − − − C D − − L D C D − − L D C D − − L D e e v v v v v v i i i i i i = (cid:126) P V D = (cid:88) b =1 v b ı b . (66)Since we do not have any Josephson tunnel junctionin the circuit considered at present, the circuit’s totalinstantaneous energy is derived by integrating P V D overtime and using the trick αf • f = d ( αf / /dt , E V D = 12 C v + 12 C v + 12 C v + 12 L ı + 12 L ı + 12 L ı . (67)By writing v = − v − v from Eqs. (61a), (61c), and(61e), finding ı from Eq. (60a), expressing ı using Eq. (62a), ı using Eq. (62c) and furthermore substitut-ing • v with the constraint of Eq. (61c) where e and e are written in terms of v and v from Eqs. (61a) and(61e), leaving ı as an independent coordinate, finding ı from Eq. (60b), and expressing ı using Eq. (62e), weobtain E V D = 12 C v + 12 C ( − v − v ) + 12 C v + 12 L [ − C • v + C ( − • v − • v ) + ı ] + 12 L ı + 12 L [ C ( − • v − • v ) + ı − C • v ] . (68)By performing simple algebraic calculations and substi-tuting • v = ı /C and • v = ı /C from Eqs. (62a) and(62e), respectively, we finally find E V D = 12 C v + 12 C v + C v v + 12 C v + 12 C v + 12 L ı + L C C ı + L C C ı ı − L ı ı + 12 L C C ı + L C C C ı ı + 12 L C C ı − L C C ı ı − L C C ı ı + 12 L ı + 12 L ı + 12 L C C ı + L C C C ı ı + 12 L C C ı − L C C ı ı − L C C ı ı + L C C ı ı + L C C ı + 12 L ı − L ı ı + 12 L ı , (69)2where the terms colored in blue correspond to the energystored in the capacitive subnetwork of the circuit, or ki-netic energy, and those colored in red to that stored inthe inductive subnetwork embedded within the capacitivesubnetwork, or co-kinetic energy. It is worth noting thatin a circuit such as that in Fig. 4 the inductive subnet-work typically stems from a parasitic mutual inductancebetween two capacitively coupled circuits, where the mu-tual inductance can be cast into an equivalent T -networkof three inductances (see, e.g., page 456 in Ref. [7]), whichcan finally be represented as the Π-network of induc-tances L , L , and L .By inspecting all terms in Eq. (69), it appears evidentthat we are in presence of an incomplete vector of gener-alized velocities (we remind that any v = • φ or any ı = • q is a generalized velocity). In fact, we have (cid:126) • ξ = ( • ξ = • φ = v , • ξ = absent , • ξ = • φ = v ) (cid:126) • p = ( • p = • q = ı , • p = • q = ı , • p = • q = ı ) . (70a)(70b)As a consequence, if we were to use Eq. (69) when at-tempting to solve the system of Eqs. (39a) and (39b), wewould obtain an incomplete set of canonical coordinates { ξ = φ , ξ = φ , ξ = φ ; p = q , p = absent , p = q } . (71)In order to obtain a complete set of canonical coor-dinates, we must introduce an auxiliary circuit elementin correspondence with the missing generalized veloc-ity • ξ = v . Since we are dealing with a missing voltage,the natural choice is to add an auxiliary capacitor withcapacitance (cid:101) C in series with L (similarly, if we weremissing a current, we would add an auxiliary inductor;whether an auxiliary element has to be added in seriesor parallel depends on the specific case). This may seeman artifact to make the method to work. However, as al-ready pointed out but not explicitly explored in Ref. [4],every inductor is inevitably characterized by a parasiticcapacitor and every capacitor by a parasitic inductor.For example, consider a small-coil antenna used inAM radios. The coil mostly forms an inductor, but ifwe were not to account for the parasitic capacitor asso-ciated with it, we would miss a natural resonance of theantenna. This resonance can interfere with the operationfo the antenna, which may become selective in the wrongregion of the frequency spectrum (e.g., away from thecarrier frequency of the radio station we intend to tunein with). This issue is typically solved by connectingthe small-coil antenna with an ad hoc shunting capaci-tor (much larger or smaller than the parasitic capacitor, depending whether the shunt capacitor is connected inparallel or series with the parasitic capacitor). Notably,this is what we do when we build a transmon quibit: Weconnect a Jospephson tunnel junction, which is charac-terized by an intrinsic (“parasitic”) capacitor, in parallelwith a much larger capacitor effectively behaving as thequbit capacitor!Figure 6 shows the auxiliary circuit element (cid:101) C neededin our example. The instantaneous energy associatedwith this element is E aux = 12 (cid:101) C ˜ v , (72)where ˜ v can easily be found from − v = v + ˜ v (73)(or, equivalently, by extending the digraph by includingthe branch associated with (cid:101) C and proceeding with all theusual steps, KCL, KVL, etc.). As before, by writing v = − v − v from Eqs. (61a), (61c), and (61e), we find˜ v = v − v + v (74)and, thus, E aux = 12 (cid:101) C v + 12 (cid:101) C v + 12 (cid:101) C v − (cid:101) C v v + (cid:101) C v v − (cid:101) C v v . (75)The complete expression for the total instantaneousenergy of the entire circuit, including the auxiliary el-ement, is therefore [written in terms of the generalizedvelocities of Eqs. (70a) and (70b) and with • φ = v ] C C L L C e C ˜ v L v FIG. 6. The auxiliary circuit element for the “Vool-DevoretCircuit.” E V D = 12 C • φ + 12 C • φ + C • φ • φ + 12 C • φ + 12 C • φ + 12 (cid:101) C • φ + 12 (cid:101) C • φ + 12 (cid:101) C • φ − (cid:101) C • φ • φ + (cid:101) C • φ • φ − (cid:101) C • φ • φ + 12 L • q + L C C • q + L C C • q • q − L • q • q + 12 L C C • q + L C C C • q • q + 12 L C C • q − L C C • q • q − L C C • q • q + 12 L • q + 12 L • q + 12 L C C • q + L C C C • q • q + 12 L C C • q − L C C • q • q − L C C • q + • q + L C C • q • q + L C C • q + 12 L • q − L • q • q + 12 L • q . (76)This equation contains a complete vector of generalizedvelocities, both for the kinetic and for the co-kinetic en- ergy, (cid:126) • ξ = ( • ξ = • φ , • ξ = • φ , • ξ = • φ ) = (cid:126) • φ(cid:126) • p = ( • p = • q , • p = • q , • p = • q ) = (cid:126) • q . (77a)(77b)We can now use Eq. (76) to solve the system ofEqs. (39a) and (39b). We find, (cid:16) C + C + (cid:101) C (cid:17) • φ + (cid:16) − (cid:101) C (cid:17) • φ + (cid:16) C + (cid:101) C (cid:17) • φ = q (cid:16) − (cid:101) C (cid:17) • φ + (cid:16) − (cid:101) C (cid:17) • φ + (cid:16) − (cid:101) C (cid:17) • φ = q (cid:16) C + (cid:101) C (cid:17) • φ + (cid:16) − (cid:101) C (cid:17) • φ +( C + C ) • φ = q (78a)(78b)(78c)for the kinetic energy and4 (cid:18) L + 2 L C C + L C C + L C C (cid:19) • q + (cid:18) − L − L C C − L C C (cid:19) • q + (cid:18) L C C + L C C C + L C C C + L C C (cid:19) • q = φ (cid:18) − L − L C C − L C C (cid:19) • q + (cid:18) L + L + L (cid:19) • q + (cid:18) − L C C − L C C − L (cid:19) • q = φ (cid:18) L C C + L C C C + L C C C + L C C (cid:19) • q + (cid:18) − L C C − L C C − L (cid:19) • q + (cid:18) L C C + L C C + 2 L C C + L (cid:19) • q = φ (79a)(79b)(79c)for the co-kinetic energy. The systems of Eqs. (78a) to(78c) and of Eqs. (79a) to (79c) can be written in a morecompact fashion as M • φ + M • φ + M • φ = q M • φ + M • φ + M • φ = q M • φ + M • φ + M • φ = q (80a)(80b)(80c)for the kinetic energy and N • q + N • q + N • q = φ N • q + N • q + N • q = φ N • q + N • q + N • q = φ (81a)(81b)(81c)for the co-kinetic energy. The constants M k and N k for k = 1 , , . . . M = M M M M M M M M M (82) and N = N N N N N N N N N , (83)the systems of Eqs. (80a) to (80c) and of Eqs. (81a) to(81c) can be finally written as M (cid:126) • φ T = (cid:126)q T N (cid:126) • q T = (cid:126)φ T , (84a)(84b)where (cid:126)q = ( q , q , q ) (cid:126)φ = ( φ , φ , φ ) . (85a)(85b)By solving the systems of Eqs. (84a) and (84b) with re-spect to (cid:126) • φ and (cid:126) • q by simple matrix inversion and insertingthe results into Eq. (76), we finally obtain the Hamil-tonian H V D of the circuit for the canonical coordinatevectors (cid:126)φ and (cid:126)q .5 VI. “SECOND QUANTIZATION” OF AN LC RESONATOR COUPLED TO A TRANSMONQUBIT
The circuit in Fig. 7 shows a simple L r C r resonatorcoupled by means of a capacitor with capacitance C rq toa transmon qubit with qubit capacitance C q and criticalcurrent I c0 . The digraph associated with this circuit isdisplayed in Fig. 8. We leave all the details of the prob-lem of finding the Hamiltonian of this circuit using theincidence matrix method as an exercise to the reader.Instead, we focus here on the final result and its quanti-zation.Before writing down the final result for the circuitHamiltonian and proceeding with its quantization, it isworth performing an approximation that, without loos-ing generality for the quantization procedure, it allowsus to greatly simplify the notation (the approximationcaptures well many experimental implementations of thiscircuit, an area of study sometimes called circuit quan-tum electrodynamics). After realizing all the steps of theincidence matrix method, we end up obtaining an induc-tive energy of the form T ∗ = (cid:32) L r + L r C rq C r + 12 L r C C (cid:33) ı + (cid:32) L r C rq C q + L r C C r C q (cid:33) ı ı + (cid:32) L r C C (cid:33) ı , (86)where ı and ı are the currents associated withbranches 1 and 4 in the digraph of Fig. 8. We approxi-mate this energy by assuming that C rq (cid:28) C r , C q (weakcoupling approximation); the inductive energy reduces to T ∗ = 12 L r ı . (87)With this approximation, the circuit Hamiltonian be- C rq L r C r C q J q FIG. 7. An LC resonator coupled to a transmon qubit. 1 51 233 42FIG. 8. Digraph of an LC resonator coupled to a transmonqubit. comes H rq = φ L r + 1det C (cid:20) C q + C rq q + q C rq q + C r + C rq q (cid:21) − E J0 cos[ k J ( φ )] , (88)where φ , φ , q , and q form a complete set of canonicalcoordinates (we can set any φ ∼ phase in the cosine termdue to Faraday-Henry-Neumann’s law between J q and C q to zero for simplicity; additionally, note that in thiscase there is not a simple connection between φ and φ because J q and C r are not in parallel as in the casestudied in Sec. II) and C = (cid:34) ( C r + C rq ) C rq C rq ( C q + C rq ) (cid:35) . (89)We now follow a standard quantization procedure as,e.g., in Ref. [16]. The classical canonical coordinates canbe promoted to quantum-mechanical operators as ( φ , φ ) → (cid:16) ˆ φ , ˆ φ (cid:17) ( q , q ) → (ˆ q , ˆ q ) = (cid:18) − j (cid:126) ∂∂ ˆ φ , − j (cid:126) ∂∂ ˆ φ (cid:19) , (90a)(90b)where j = − (cid:126) = h/ (2 π ).The quantum-mechanical Hamiltonian then readsˆ H rq = ˆ φ L r + 1det C (cid:20) C q + C rq q − j (cid:126) ˆ q C rq ∂∂ ˆ φ − C r + C rq (cid:126) ∂ ∂ ˆ φ (cid:35) − E J0 cos[ k J ( ˆ φ )] . (91)The Hamiltonian of Eq. (91) can be written asˆ H rq = ˆ H r + ˆ H i + ˆ H q , (92)6where the resonator, interaction, and qubit Hamiltonianare defined as ˆ H r = ˆ φ L r + C q + C rq C ˆ q ˆ H i = − j (cid:126) C rq det C ˆ q ∂∂ ˆ φ ˆ H q = − (cid:126) C r + C rq C ∂ ∂ ˆ φ − E J0 cos[ k J ( ˆ φ )] . (93a)(93b)(93c)Under the weak coupling approximation we can treatˆ H r and ˆ H q separately, and consider ˆ H i as a perturbativeinteraction. Thus, ˆ H r can be quantized using the stan-dard creation and annihilation operators ˆ a † and ˆ a of asimple quantum-mechanical harmonic oscillator,ˆ H r = hf r (cid:18) ˆ a † ˆ a + 12 (cid:19) , (94)where f r = 12 π (cid:113) L r (cid:101) C , (95)with (cid:101) C = det C / ( C q + C rq ) (as expected, (cid:101) C = C r when C q = 0).By diagonalizing ˆ H q we obtain a set of eigenstates {| (cid:96) (cid:105)} = {| g (cid:105) , | e (cid:105) , | f (cid:105) , . . . } , (96)where | g (cid:105) is the energy ground state, | e (cid:105) the energy firstexcited state, and | f (cid:105) the next excited state, etc. By defin-ing the eigenfrequencies associated with these eigenstatesas f (cid:96) , we have ˆ H q = h (cid:88) (cid:96) f (cid:96) | (cid:96) (cid:105) (cid:104) (cid:96) | . (97)By imposing twice the completeness relation for theset of eigenstates of Eq. (96) around ˆ H i ,ˆ H i = (cid:88) (cid:96),m | (cid:96) (cid:105) (cid:104) (cid:96) | (cid:18) − j (cid:126) C rq det C ˆ q ∂∂ ˆ φ (cid:19) | m (cid:105) (cid:104) m | = (cid:88) (cid:96),m | (cid:96) (cid:105) (cid:104) (cid:96) | (cid:18) − j (cid:126) C rq det C ∂∂ ˆ φ (cid:19) | m (cid:105) (cid:104) m | ⊗ ˆ q , (98)where ˆ q = q (ˆ a † + ˆ a ) and the zero-point charge is givenby q = (cid:112) C r hf r / g (cid:96),m = − j C rq q π det C (cid:104) (cid:96) | (cid:18) ∂∂ ˆ φ (cid:19) | m (cid:105) (99)as the coupling coefficients in unit hertz,ˆ H i = h (cid:88) (cid:96),m g (cid:96),m | (cid:96) (cid:105) (cid:104) m | ⊗ (ˆ a † + ˆ a ) . (100) In some cases, it is possible to simplify the totalHamiltonian of this circuit by making a two-level ap-proximation, {| (cid:96) (cid:105)} = {| g (cid:105) , | e (cid:105)} , by neglecting the coef-ficients g gg and g ee , by offsetting the energy differencebetween the energy ground and first excited state suchthat f ge = f e − f g = f q , and by assuming g ge = g eg = g rq ,ˆ H rq = hf q | e (cid:105) (cid:104) e | − | g (cid:105) (cid:104) g | )+ hg ge ( | g (cid:105) (cid:104) e | + | e (cid:105) (cid:104) g | ) ⊗ (ˆ a † + ˆ a )+ hf r (cid:18) ˆ a † ˆ a + 12 (cid:19) = hf q σ z + hg rq ˆ σ x ⊗ (ˆ a † +ˆ a )+ hf r (cid:18) ˆ a † ˆ a + 12 (cid:19) , (101)where ˆ σ x and ˆ σ z are the usual Pauli operators.By performing a rotating wave approximation we fi-nally find the Jaynes-Cummings Hamiltonianˆ H JC = hf q σ z + hg rq (ˆ σ − ˆ a † + ˆ σ + ˆ a ) + hf r (cid:18) ˆ a † ˆ a + 12 (cid:19) , (102)where ˆ σ + and ˆ σ − are the usual two-level raising and low-ering operators. VII. CONCLUSIONS
I find it rather interesting that when deriving the totalinstantaneous energy of a circuit from its power writtenusing the incidence matrix, which is nothing by a neatway to use KCL and KVL, we obtain a kinetic and aco-kinetic energy term.As for the “Vool-Devoret circuit,” I was not too sur-prised to find three DOF. In fact, that circuit is basicallythe series of three LC resonators and, thus, should com-prise three modes, i.e., three DOF. It is worth noting thatwhen the effect of C and L is small compared to that ofthe other circuit elements in the system, the circuit effec-tively becomes two weakly coupled LC resonators and,thus, with just two effective DOF.The first time I realized that auxiliary (in real life, par-asitic) circuit elements can be a key to find the Hamil-tonian of a circuit is when a theorist back in 2012 askedme: “How do you quantize a single capacitor C ?” Youcan not, unless you include the parasitic inductor L p as-sociated with it. Then the problem becomes that of asimple harmonic oscillator as you are just dealing withan L p C resonator. The next question was, “How can youunveil the quantum-mechanical nature of a single capac-itor?” Now, the resonance frequency of the capacitor-with-parasitic-inductor resonator, f = 1 / (2 π (cid:112) L p C )may be very high or very low. For example, if it is verylow you will need to cool it down to extremely low tem-peratures to unveil its quantum-mechanical nature.Sometimes, in real applications, the parasitic elementsresult in resonance frequencies (modes) very far from allthe frequencies of interest for a certain experiment and,7thus, do not effectively participate much in the system’sdynamics of interest. This suggests that, in the methoddiscussed in this Note, we could effectively “eliminate”their presence by means of a limiting process; this is atopic that I am presently investigating in detail and willeventually be added to the Note. ACKNOWLEDGEMENTS
I would like to thank J.H. B´ejanin, Y. Ayadi,M.M. Ayyash, as well as the present and past members ofmy team at the University of Waterloo for their fruitfuldiscussions. I am indebted to Prof. A. Premoli for teach-ing an excellent, and incredibly tough, course on circuittheory back in 1998 at the Politecnico di Milano. Addi-tionally, I thank A.V. Bardysheva for helping with thefigures.
Appendix A: Potential Energy
In Eq. (20), the potential energy associated with theonly element of the generalized coordinate set { ξ k } , ξ = φ is U ( φ ) = − E J0 cos[ k J ( φ − φ ∼∼ )] . (A1)More in general, any term that can be written as U ( (cid:126)ξ ) = N ξ (cid:88) k =1 f ( ξ k ) U ( (cid:126)p ) = N ξ (cid:88) k =1 f ( p k ) (A2a)(A2b)is a potential energy term. Appendix B: Lagrangian
We do not need to find any Lagrangian with thismethod. Although unnecessary, I will make a connec-tion between the energy and Hamiltonian as found hereand their corresponding Lagrangian in a future appendixto this note. [1] D. A. Wells, Journal of Applied Physics , 312 (1938),https://doi.org/10.1063/1.1710422.[2] B. M. Maschke, A. J. van der Schaft, and P. C. Breed-veld, IEEE Transactions on Circuits and Systems I: Fun-damental Theory and Applications , 73 (1995).[3] G. Burkard, R. H. Koch, and D. P. DiVincenzo, Phys.Rev. B , 064503 (2004).[4] U. Vool and M. Devoret, International Journal ofCircuit Theory and Applications , 897 (2017),https://onlinelibrary.wiley.com/doi/pdf/10.1002/cta.2359.[5] B. Yurke and J. S. Denker, Phys. Rev. A , 1419 (1984).[6] J. A. Russer and P. Russer, in (2011) pp. 1153–1156.[7] L. O. Chua, C. A. Desoer, and E. S. Kuh, Linear andNonlinear Circuits (McGraw-Hill, Inc., USA, 1987).[8] C. Cercignani,
Spazio, Tempo, Movimento. Introduzionealla Meccanica Razionale (Nicola Zanichelli S.p.A.,Bologna, Italy, 1976).[9] This excellent textbook is only in Italian, but “equationsall same.”.[10] J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I.Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin,and R. J. Schoelkopf, Phys. Rev. A , 042319 (2007).[11] A. Barone and G. Patern`o, Physics and Applications ofthe Josephson Effect (John Wiley & Sons, Inc., USA, 1982).[12] The only reason Eq. (4) may aspire to be called with itsown name is because of the k J coefficient that relates φ to ϕ .[13] Not to get confused with the charge variables, we do notuse the standard notation where the generalized coordi-nates are indicated by q k . Similarly, we use the index k instead of i not to get confused with the current variablesor the imaginary unit.[14] L. Landau and E. Lifshitz, Mechanics - Third Edi-tion; Course of Theoretical Physics, Vol. 1 (ElsevierButterworth-Heinemann, Linacre House, Jordan Hill,Oxford UK, 1976).[15] M. Tinkham,
Introduction to Superconductivity - SecondEdition (Dover Publications, Inc., Mineola, New YorkUSA, 1996).[16] R. Loudon,
The Quantum Theory of Light (OUP OxfordUK, 2000).[17] This follows from the fact that the sum of the capacitiveand inductive mean energies (cid:104)E C0 (cid:105) + (cid:104)E L0 (cid:105) = hf r / (cid:104)E C0 (cid:105) = (cid:104)E L0 (cid:105) , as well as (cid:104)E C0 (cid:105) = q / (2 C rr