The Quantum Electrodynamics of Singular and Nonreciprocal ...

The Quantum Electrodynamics ofSingular and NonreciprocalSuperconducting Circuits

by

Martin Rymarz

Master’s Thesis in Physics

presented to

The Faculty of Mathematics, Computer Scienceand Natural Sciences

at

RWTH Aachen University

JARA-Institute for Quantum Information

September 2018

supervised by

Prof. Dr. David P. DiVincenzoProf. Dr. Fabian Hassler

Contents

1 Introduction 1

2 Qubit-resonator system realizing tunable coupling types 32.1 The different coupling types between a qubit and a resonator . . . . . . . . . . . 4

2.1.1 Transverse coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.2 Longitudinal coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.3 Proposed superconducing implementation by Richer et al. . . . . . . . . 5

2.2 Single coupling branch with one Josephson junction . . . . . . . . . . . . . . . . 62.2.1 Without intrinsic capacitances of the Josephson junction or the inductor 62.2.2 Including the intrinsic capacitance of the Josephson junction . . . . . . . 102.2.3 Including the intrinsic capacitances of the Josephson junction and the

inductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3 Qubit-resonator circuit with intrinsic capacitances . . . . . . . . . . . . . . . . . 24

2.3.1 Derivation of the Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 242.3.2 Derivation of an effective qubit-resonator Hamiltonian . . . . . . . . . . . 272.3.3 Derivation of the parameters defining the qubit-resonator model Hamil-

tonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3.4 Numerical results for CLi = 0 . . . . . . . . . . . . . . . . . . . . . . . . 332.3.5 Validity of the Born-Oppenheimer approximation . . . . . . . . . . . . . 382.3.6 Effect of non-zero CLi . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3 Quantization of nonreciprocal superconducting circuits involving gyrators 413.1 The classical ideal gyrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2 The gyrator in the Lagrangian and Hamiltonian formalism . . . . . . . . . . . . 43

3.2.1 Lagrangian formalism of the gyrator . . . . . . . . . . . . . . . . . . . . 433.2.2 Hamiltonian formalism of the gyrator . . . . . . . . . . . . . . . . . . . . 453.2.3 Construction of an appropriate basis for the Legendre transformation . . 48

3.3 Capacitances coupled by a gyrator, the C-G-C circuit . . . . . . . . . . . . . . . 513.4 Inductances coupled by a gyrator, the L-G-L circuit . . . . . . . . . . . . . . . . 533.5 LC-Resonators coupled by a gyrator, the LC-G-LC circuit . . . . . . . . . . . . 553.6 Josephson junctions coupled by a gyrator, the JJ-G-JJ circuit . . . . . . . . . . 593.7 Cooper pair boxes coupled by a gyrator, the CJJ-G-CJJ circuit . . . . . . . . . 61

3.7.1 Magnetic translation operators . . . . . . . . . . . . . . . . . . . . . . . 633.7.2 Preserved Φ0-periodicity, compact variables . . . . . . . . . . . . . . . . 653.7.3 Non-preserved Φ0-periodicity, non-compact variables . . . . . . . . . . . 663.7.4 Numerical implementation and results . . . . . . . . . . . . . . . . . . . 673.7.5 Comparison with the JJ-G-JJ circuit . . . . . . . . . . . . . . . . . . . . 693.7.6 Comparison with the C-G-C circuit . . . . . . . . . . . . . . . . . . . . . 70

3.8 Fluxoniums coupled by a gyrator, the LCJJ-G-LCJJ circuit . . . . . . . . . . . 71

i

Contents

4 Conclusion and Outlook 73

5 Acknowledgements 75

6 Appendix 776.1 The square-root capacitance matrix transformation preserves the commutation

relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776.2 The Cholesky decomposition transformation preserves the commutation relation 786.3 Relation between the square-root capacitance matrix decomposition and the

Cholesky decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.4 Tellegen’s construction of a transformer . . . . . . . . . . . . . . . . . . . . . . . 806.5 Commutation realations of the QHE variables . . . . . . . . . . . . . . . . . . . 82

ii

1 Introduction

Already in 1982 Feynman [1] introduced the idea of a working quantum computer, but cur-rently there are still huge efforts made to build one. They promise huge advantages in variouscomputational problems like e.g. prime factorization [2], cryptography [3] or simulations ofquantum systems [1].

In principle, quantum computers can be realized by several different approaches. In particular,every quantum system satisfying the DiVincenzo criteria [4] could be used to build a quantumcomputer. Currently, the probably most promising architectures for the realization of an uni-versal quantum computer are superconducting electrical circuits exhibiting quantum behavioralthough being objects of macroscopic size. Therefore, these superconducting electrical circuitshave to be cooled down to very low temperatures in order to avoid thermal radiative noise.Furthermore, it has to be ensured that such a realization in terms of superconducting electricalcircuits is well protected against any miscellaneous environmental influence.Currently, superconducting circuits can realize quantum processors with up to 72 qubits [5,6].However, the overall goal is to fabricate quantum processors with a larger number of qubits andhigh fidelity in order to achieve quantum supremacy. For this reason, the research on scalabilityof quantum devices is in great demand.

The systematic analysis and quantization of such superconducting electrical circuits has beenlargely explored [7–10] and will be considered as known for this thesis. Since superconductingarchitectures designed to meet desired properties can easily become large, also their descriptionbecomes more complex. For this reason, it is evident to look for simpler effective descriptionsor assumptions which approximate the system well. This can yield to a singular description ofthe circuit. However, common instructions of circuit quantization do not incorporate singularor nonreciprocal circuits and can be considered as not completed in that regard.

This thesis is mainly separated into two big parts.In the first part, chapter 2, we will discuss in detail an extension of a proposed singular supercon-ducting circuit architecture that realizes a qubit-resonator system with tunable coupling [11,12].In this system, not just the strength of the coupling between the qubit and the resonator istunable but also its type. Our extension will consist of the cancellation of the circuit’s singu-larity. The two descriptions of the circuit (singular and non-singular) will be compared.In the second part, chapter 3, we will introduce the gyrator in circuit quantum electrodynamicsas a lumped element. The gyrator is a fundamental nonreciprocal building block in electricalcircuit engineering which can easily yield to singular circuits as well. Therefore, we will proposea way for its quantization and afterwards quantized nonreciprocal circuits including a gyratorwill be analyzed exemplarily.

1

2 Qubit-resonator system realizingtunable coupling types

In this chapter we will analyze an extension of a superconducting circuit architecture proposedby Richer et al. [11,12] that realizes a qubit-resonator system with tunable coupling. Since thisproposed qubit-resonator system realizes not just tunable coupling in its strength but also inits types, we will briefly introduce the two possible fundamentally different coupling types andsummarize their characteristics.

Our extension of the superconducting circuit will consist of the inclusion of the Josephson ar-rays’ and inductors’ intrinsic capacitances in the description. In this context, we will introducethe concept of singular circuits which result into singular Lagrangians. Before we study thewhole circuit realizing the qubit-resonator system, we will introductorily analyze theses intrin-sic capacitances’ effects in one single building-block of the total circuit.

Finally, these results will be used to investigate the intrinsic capacitances’ effects in the totalcircuit. We will illustrate how to beneficially use the Cholesky decomposition in order to bringthe circuit’s capacitance matrix into isotropic form and to derive an effective description ofthe qubit and resonator variables. In the mean time, the exigency of the Born-Oppenheimerapproximation for systems with many variables (which are not analytically solvable or whichcannot be further simplified) will become clear.

3

2. Qubit-resonator system realizing tunable coupling types

2.1 The different coupling types between a qubit and aresonator

As already mentioned, the to be discussed superconducting circuit will effectively describe aqubit-resonator system with tunable coupling also in its types. Therefore, we will introducethe two different coupling types considered and summarize their properties in the following.Afterwards, the superconducting architecture proposed by Richer et al. will be introduced.

2.1.1 Transverse coupling

In superconducting circuit architectures the most commonly resulting coupling between a qubitand a resonator is the transverse coupling [11] in which the qubit’s σx degree of freedom couplesto the resonator. The Hamiltonian describing the qubit, the resonator and the transversecoupling between them is the Rabi Hamiltonian

H =~∆

2σz + ~ωra†a+ ~gxxσx(a† + a), (2.1)

in which ∆ and ωr denote the qubit and resonator frequency, respectively. Although the RabiHamiltonian is integrable and exactly solvable [13], one usually operates transversely coupledsystems in the dispersive regime [14], in which the coupling is small compared to the detuning,i.e. gxx/(∆−ωr) 1. In this regime one can approximately diagonalize the Rabi Hamiltonianwith a unitary Schrieffer-Wolff transformation [14,15]. Moreover, well-known readout schemesare established and applicable in the dispersive regime which profit from the dispersive shiftcausing the resonator’s frequency to depend on the qubit’s state. However, this dependencyresults in an inevitable entanglement of the qubit and the resonator which contributes to thequbit’s imperfectness (Purcell effect [16]) as well as to the imperfectness of two-qubit gates.This hinders from an easy scaling up of the system to a quantum computer with a large numberof qubits.

2.1.2 Longitudinal coupling

Another type of qubit-resonator coupling is given by the coupling of the qubit’s σz degree offreedom to the resonator and is called longitudinal coupling [11, 17]. Such a system consistingof a qubit and a resonator which are coupled longitudinally is described by the Hamiltonian

H =~∆

2σz + ~ωra†a+ ~gzxσz(a† + a). (2.2)

Again, ∆ and ωr denote the qubit and resonator frequency, respectively. Besides the factthat it is hard to achieve pure longitudinal coupling with high strength in superconductingcircuit architectures, longitudinal coupling offers various promising advantages like e.g. fastquantum nondemolition readout [18] or high scalability [11,17], which is based on the possibleexact diagonalization of the Hamiltonian Eq. (2.2) with a unitary Lang-Firsov transformation[11, 17, 19]. In a grid of such longitudinally coupled qubit-resonator systems, this results instrictly local interactions such that a single unique qubit frequency suffices for the scalabilityof such a scheme [11].

4

2.1. The different coupling types between a qubit and a resonator

2.1.3 Proposed superconducing implementation by Richer et al.

One possible implementation of a superconducting circuit architecture which realizes an effectivesystem with tunable coupling between a qubit and a resonator was presented in the work byRicher et al. [11, 12]. The most advanced circuit they proposed is shown in Fig. 2.1 togetherwith our extension.

C1

CJ1

CL1 CL2

EJq

Cq

k1EJ

1

k2EJ

2LJ1

L1 L2

LJ2

C2

CJ2

φa φe

φc

φb φd

Φext1 Φext

2

ΦextB

Figure 2.1: Superconducting circuit architecture proposed by Richer et al. [12]. It realizes aqubit-resonator system with tunable coupling. The Josephson arrays’ and linearinductors’ intrinsic capacitances are highlighted in red and are the extensions wemake. We will additionally consider these capacitances in our analysis.

It consists of a transmon qubit [20] (Josephson junction with energy EJq and capacitance Cq inparallel), which is inductively shunted by a resonator. This resonator consists of two branches,whose lumped circuit elements are denoted with an index 1 and 2, respectively (see Fig. 2.1).Each coupling branch itself consists of an inductively shunted Josephson array and a linearinductance with a capacitance in parallel to both of them. In contrast to Richer et al., we willalso include the intrinsic capacitances of these Josephson arrays and inductors in our analysis.In Fig. 2.1 they are highlighted in red. In particular, we are interested in the case in whichthese intrinsic capacitances are small and we want to compare this with the results obtainedwithin the treatment in which they are completely neglected. As we will see, the analysis forarbitrarily small but finite intrinsic capacitances differs fundamentally from the case of zerocapacitances.

Note that the circuit encloses three loops, which can be penetrated by external magnetic fluxes.The fluxes Φext

1 ,Φext2 within the single coupling branches will enable the qubit-resonator system

to operate with tunable coupling while the flux ΦextB through the big loop enhances the coupling

amplitudes as well as the anharmonicity of the qubit [12].

5


2.2 Single coupling branch with one Josephson junction

In order to investigate the effects of the additional intrinsic capacitances in the full qubit-resonator circuit (Fig. 2.1), one single coupling branch of the resonator will be analyzed first.For this purpose, we successively add the intrinsic capacitances in the description of the singlecoupling branch. For simplicity, it is appropriate to replace the Josephson array by one sin-gle Josephson junction. This replacement captures the properties which are relevant for ouranalysis. We start with the analysis in which no intrinsic capacitances are taken into account.

2.2.1 Without intrinsic capacitances of the Josephson junction or theinductor

First and for the sake of comparison, the coupling branch will be analyzed without taking theintrinsic capacitances of the Josephson junction or the inductor into account. In this case,the circuit of the single coupling branch consists of an inductance and an inductively shuntedJosephson junction with a capacitance in parallel to both of them and is depicted in Fig. 2.2.The Josephson junction and its inductive shunt form a loop which can be penetrated by anexternal magnetic flux Φext. Once the circuit has been fabricated, this external magnetic fluxcan be changed while all other circuit parameters are fixed. Therefore, it can be used as atunable parameter of the circuit.

C

Φext

EJ

LJ

Lφc φaφb

Figure 2.2: Single coupling branch without intrinsic capacitances of the Josephson junction orthe inductor. The Josephson junction is inductively shunted such that a loop isformed which can be penetrated by an external magnetic flux Φext.

Taking the node fluxes to describe the electrical circuit, we define the two independent variables

φ = φc − φa, φJ = φc − φb, (2.3)

with which the Lagrangian of the circuit reads

L =Cφ2

2− (φ− φJ)2

2L− φ2

J

2LJ+ EJ cos

(2π

Φ0

(φJ − Φext)

)=

1

2φTCφ− U(φ, φJ),

(2.4)

In the second line of Eq. (2.4) we have introduced the matrix notation

φ =

(φφJ

), C =

(C 00 0

), (2.5)

6

2.2. Single coupling branch with one Josephson junction

as well as the potential

U(φ, φJ) =1

2φTMφ− EJ cos

(2π

Φ0

(φJ − Φext)

), M =

(1L

− 1L

− 1L

1L

+ 1LJ

). (2.6)

Thus, the Lagrangian is separated into a kinetic term involving only φ (first term) and apotential term which depends on φ only (second term). The conjugate charges of φ andφJ are given by the derivative of the Lagrangian with respect to the time derivative of thecorresponding flux variable and read

Q =∂L∂φ

= Cφ, (2.7a)

QJ =∂L∂φJ

= 0. (2.7b)

Solving these for φ and φJ as functions of fluxes and conjugate charges would be equivalentto the inversion of the capacitance matrix C defined in Eq. (2.5), which is not possible sincedet(C) = 0. In particular, the capacitance matrix is positive semi-definite rather than positivedefinite and one cannot solve Eqs. (2.7) for φJ(φ, φJ , Q,QJ). For this reasons, we call thecircuit shown in Fig. 2.2 to be singular.

In fact, according to Dirac [21], any Lagrangian which does not allow for solving all the gen-eralized velocities as functions of the generalized positions and conjugate momenta is calledto be a singular Lagrangian. Singular Lagrangians imply that the physical system which isdescribed underlies some constraints [21]. This means that the variables within the Lagrangiandescription are not independent. Moreover, given a singular Lagrangian one cannot performthe Legendre transformation without further ado in order to obtain the Hamiltonian descrip-tion of the system. However, Richer et al. [11] presented a strategy which uses the classicalequations of motion for the fluxes to determine the underlying constraint and effectively reducethe number of variables in the Lagrangian. This method is equivalent to Dirac’s descriptions ofquantizing a singular Lagrangian for which one generalized conjugate momentum vanishes [21].

Calculating the Euler-Lagrange equations

d

dt

(∂L∂φ

)− ∂L∂φ

= 0,d

dt

(∂L∂φJ

)− ∂L∂φJ

= 0, (2.8)

to the Lagrangian Eq. (2.4), the classical equations of motion read

0 = Cφ+φ− φJL

, (2.9a)

0 =φJ − φL

+φJLJ

+ EJ2π

Φ0

sin

(2π

Φ0

(φJ − Φext)

), (2.9b)

and we note that Eq. (2.9b) is no differential equation but can rather be used to eliminate φJ .Although it is not analytically possible to solve for φJ , one can determine φ as a function ofφJ and invert this relation for φJ as a function of φ numerically. With the rescaling of the fluxvariables to reduced fluxes

ϕ =2π

Φ0

φ, ϕJ =2π

Φ0

φJ , ϕext =2π

Φ0

Φext, (2.10)

7


the manipulation of Eq. (2.9b) gives the relation

ϕ(ϕJ) = (1 + γ)ϕJ + β sin(ϕJ − ϕext

), (2.11)

where β = LEJ(2π/Φ0)2 denotes the screening parameter known from squid terminology [22]and γ = L/LJ is the ratio of the linear inductances. Note that β > 0 and the absence of theinductive shunt of the Josephson junction can be formally treated by setting γ = ϕext = 0. Atthis point one has to distinguish two cases:

• For β ≤ 1 + γ, the function ϕ(ϕJ) is monotonically increasing for all ϕJ and for thisreason the inverse relation ϕJ(ϕ) is well-defined.

• For β > 1 + γ, the function ϕ(ϕJ) is monotone only on the intervals

In =

[arccos

(−1 + γ

β

)+ 2πn+ ϕext,− arccos

(−1 + γ

β

)+ 2π(n+ 1) + ϕext

],

In =

[− arccos

(−1 + γ

β

)+ 2π(n+ 1) + ϕext, arccos

(−1 + γ

β

)+ 2π(n+ 1) + ϕext

],

(2.12)

with n ∈ Z. However, a piecewise inversion of ϕ(ϕJ) gives a relation ϕJ(ϕ) which ismultivalued in certain intervals and therefore not a proper function.

Examples for both cases are shown in Fig. 2.3 for γ = ϕext = 0.

-π -π -π π π π-π

-π

-π

π

π

π

(a) Function ϕ(ϕJ)

-π -π -π π π π-π

-π

-π

π

π

π

(b) Inverse ϕJ(ϕ)

Figure 2.3: (a) The function ϕ(ϕJ) and (b) the inverse relation for γ = ϕext = 0, both shownfor β = 0.7 and β = 2.0. While for β ≤ 1 + γ the function ϕ(ϕJ) is monotone andtherefore it yields to a well-defined inverse function, for β > 1+γ the function ϕ(ϕJ)possesses extrema and hence the (piecewise obtained) inverse relation is multivalued(in some intervals) and therefore not a proper function.

8


Note that the Lagrangian Eq. (2.4) does not depend on ϕJ and that ϕJ only enters in thepotential term. Therefore, after the numerically determining ϕJ as a (multivalued) function ofϕ, one can reduce the number of degrees of freedom of the Lagrangian by eliminating ϕJ in thepotential term Eq. (2.6) and introducing the effective classical potential

Ueff,cl(ϕ)

EJ=U(ϕ, ϕJ(ϕ))

EJ=

(ϕ− ϕJ(ϕ))2

2β+ γ

ϕJ(ϕ)2

2β− cos

(ϕJ(ϕ)− ϕext

)(2.13)

for the remaining variable ϕ. This effective potential will be called classical because it isobtained invoking the classical equations of motion. It is shown for the cases of β ≷ 1 + γ andγ = 0 as well as γ > 0 in Fig. 2.4.

-π -π -π π π π-

-

(a) γ/β = 0


-

(b) γ/β = 0.1

Figure 2.4: Effective classical potential Ueff,cl(ϕ) of ϕ for (a) γ/β = 0 and (b) γ/β = 0.1 atzero external flux, ϕext = 0. While for β = 0.7 the effective potential is a well-defined function of ϕ, for β = 2.0 it is multivalued in the same intervals in whichthe inverse relation ϕJ(ϕ) is multivalued. (a) Without an inductive shunt of theJosephson junction the effective potential is periodic in ϕ with periodicity 2π. (b)The effect of an inductive shunt is an additional dependence on ϕ such that theeffective potential is not periodic anymore.

While for β ≤ 1 + γ the effective classical potential is a proper function of ϕ, for β > 1 + γit is multivalued in the same intervals in which the inverse relation ϕJ(ϕ) is multivalued andtherefore not a proper function of ϕ. Without inductively shunting the Josephson junction(γ = 0) the effective classical potential is periodic in ϕ with periodicity 2π for both cases of β.This periodicity is removed by the inductive shunt of the Josephson junction (γ > 0).

Given this effective classical potential for ϕ, the Lagrangian is described by one degree offreedom only and especially not singular anymore. In particular, we are now able to performthe Legendre transformation which results in the one dimensional Hamiltonian

H = qϕ− L =(2π)2

2CΦ20

q2 + Ueff,cl(ϕ), (2.14)

describing the singular coupling branch (Fig. 2.2). It is quantized by imposing the canonicalcommutation relation [ϕ, q] = i~.

9


At this point we want to mention that multivalued physical quantities are not uncommon phe-nomena in classical as well as quantum-mechanical physics. For example, due to hysteresiseffects in paramagnets the magnetization depends on the applied magnetic field and also itshistory. However, since it is not obvious how to deal with this multivalued effective classicalpotential Richer et al. restricted themselves to the case β 1 + γ.In the following subsection we will lift this restriction by taking the Josephson junction’s in-trinsic capacitances into account, followed by the derivation of an effective quantum potential.This extended description of the coupling branch removes the singularity of the Lagrangian andexplains the appearance as well as the meaning of the multivaluedness of Ueff,cl(ϕ) for β > 1+γ.

2.2.2 Including the intrinsic capacitance of the Josephson junction

In order to interpret the multivaluedness of the effective classical potential in case of β > 1 + γa more detailed description of the system will be considered in which the intrinsic capacitanceof the Josephson junction is included. Since a Josephson junction is realized by two super-conductors separated by a thin insulating layer there is always a finite capacitance in parallelpresent due to the spatial extension of the components. Taking this intrinsic capacitance intoaccount, the resulting circuit is shown in Fig. 2.5.

C

Φext

EJ

LJ

L

CJ

φc φaφb

Figure 2.5: Single coupling branch with intrinsic capacitance of the Josephson junction. TheJosephson junction is inductively shunted such that a loop is formed which can bepenetrated by an external magnetic flux Φext.

Recalling the flux variables defined in Eq. (2.3), the Lagrangian description of the circuit reads

L =Cφ2

2+CJ φ

2J

2− (φ− φJ)2

2L− φ2

J

2LJ+ EJ cos

(2π

Φ0

(φJ − φext)

)=

1

2φTCφ− U(φ, φJ),

(2.15)

with the matrix notation

φ =

(φφJ

), C =

(C 00 CJ

), (2.16)

and the same potential U(φ, φJ) as previously defined in Eq. (2.6) (note that capacitancesdo not influence the potential). Therefore, the Lagrangian can be separated into a kineticand potential term. Note that the only but important difference compared to the analysis insubsection 2.2.1 is that the intrinsic capacitance of the Josephson junction ensures that thecapacitance matrix C is invertible, since now detC 6= 0.

10


For this reason, the Lagrangian is not singular and one can find φ as well as φJ as functions ofthe fluxes φ, φJ and the conjugate charges

Q =∂L∂φ

= Cφ, QJ =∂L∂φJ

= CJ φJ . (2.17)

This allows for straightforward application of the Legendre transformation yielding the Hamil-tonian

H = Qφ+QJ φJ − L =1

2QTC−1Q+ U(φ, φJ), (2.18)

with the conjugate charged arranged in a vector and the inverse capacitance matrix, i.e.

Q =

(QQJ

), C−1 =

(1/C 0

0 1/CJ

). (2.19)

As usual, the quantization of the Hamiltonian is obtained by imposing the canonical commu-tation relations [φ,Q] = [φJ , QJ ] = i~, whereas all other commutators vanish. Introducing therescaled, dimensionless variables

ϕ =2π

Φ0

φ, ϕJ =2π

Φ0

φJ , ϕext =2π

Φ0

Φext, n =Φ0

2π~Q, nJ =

Φ0

2π~QJ , (2.20)

the Hamiltonian reads

H =4e2

2nTC−1n+

1

2

(Φ0

2π

)2

ϕTMϕ− EJ cos(ϕJ − ϕext

). (2.21)

The rescaling transforms the commutation relation of the variables to [ϕ, n] = [ϕJ , nJ ] = iwhile, again, all other commutators vanish. Note that Φ0/2π~ = 1/2e such that n and nJcorrespond to the number of Cooper pairs on the respective capacitance.

Before we go on we make the following observation: In the limit of CJ/C → 0 the Lagrangian ofthe circuit including the Josephson junction’s intrinsic capacitance approaches the singular La-grangian of the circuit without any intrinsic capacitances (compare Eq. (2.4) with Eq. (2.15))but still remains non-singular. This correspondence does not hold for the circuits’ Hamilto-nians. In particular, the Hamiltonian Eq. (2.21) depends on two degrees of freedom and notjust a single effective one and contains a divergent term in the kinetic energy (making theterminology of singular Lagrangians clear).

However, for the comparison with the singular circuit neglecting the intrinsic capacitances weare interested in the case CJ/C 1. Although the inverse capacitance matrix is alreadydiagonal, it is appropriate to transform it such that it becomes diagonal with equal diagonalelements. This can be done by a variable transformation involving the square-root of thecapacitance matrix [23]

η = c1/2C−1/2n, f = c−1/2C1/2ϕ, (2.22)

where we introduced an arbitrary standard capacitance c such that the new variables η andf stay dimensionless. Note that the existence of the (inverse) square-root capacitance matrixC±1/2 is ensured since C is invertible. Furthermore, due to the symmetry of C the variable

11


transformation Eq. (2.22) preserves the commutation relations for the new variables, [fi, ηj] =iδij (see appendix 6.1), and brings the kinetic term of the Hamiltonian into isotropic form,

H = 4EcηTη + U(f), (2.23)

with the charging energy Ec = e2/2c and the transformed potential

U(f) =c

2

(Φ0

2π

)2

fT(C−1/2

)TMC−1/2f − EJ cos

(√c(C−1/2f)2 − ϕext

). (2.24)

Note that in our convention the actual functional form of the potential U(f) implicitly dependson its arguments. Taking the explicit form of the capacitance matrix Eq. (2.16) into account,the inverse square-root capacitance matrix is diagonal and reads

C−1/2 =

(1/√C 0

0 1/√CJ

), (2.25)

which reduces the variable transformation Eq. (2.22) to a simple rescaling and transforms thepotential Eq. (2.24) to

U(f)

EJ=

1

2EJ

(Φ0

2π

)2

fTMf − cos

(√c

CJf2 − ϕext

), (2.26)

with the transformed linear inductance matrix

M = c(C−1/2

)TMC−1/2 =

1

L

(cC

− c√CCJ

− c√CCJ

cCJ

(1 + γ)

). (2.27)

With the choice of c = C we can further rewrite to potential to read

U(f)

EJ=

1

2β

(f1 −

√C

CJf2

)2

+γ

2β

C

CJf 2

2 − cos

(√C

CJf2 − ϕext

). (2.28)

Note that within the potential the coordinate f2 only appears with a prefactor√C/CJ 1,

since we are interested in the regime where CJ/C 1. For this reason, we can assume thepotential to vary much faster in the f2 variable than in the f1 variable. Therefore, f1 and f2 willbe called the slow and the fast variable [23], respectively. To be precise, the most well-definedfast and slow directions of the potential are determined by the eigenvectors of the transformedlinear inductance matrix. In principle, one could even take the second order expansion of thepotential’s cosine into account. However, in the regime of interest, i.e. for CJ/C 1, this isnegligible.For example, for γ = 0 the transformed linear inductance matrix M has eigenvalues 0 and(1 + C/CJ)/L ≈ C/CJL and is diagonalized by the orthogonal matrix

R =1√

1 + CJ/C

1 −√

CJ

C√CJ

C1

. (2.29)

12


The corresponding variable transformation to the eigenbasis of the transformed linear induc-

tance matrix is a simple rotation by the angle θ = arctan(√

CJ/C)≈√CJ/C 1, preserving

the isotropic kinetic term of the Hamiltonian Eq. (2.23) since RT = R−1. Due to the smallrotation angle θ, the orthogonal matrix R can be approximated in zero’th order by the identitymatrix. As a consequence, the variable f1 approximately matches the direction of the eigenvec-tor with vanishing eigenvalue and therefore it is reasonable to call f1 and f2 the slow and fastvariable, respectively, even without rotating the coordinate system. Similar statements can bemade for γ > 0.Furthermore, given a vanishing eigenvalue of the transformed linear inductance matrix, it fol-lows directly that there exists a direction in which the potential U(f1, f2) is periodic.

As already mentioned, in the simple case of a diagonal capacitance matrix the variable trans-formation Eq. (2.22) is just a rescaling of the initial variables without any mixing among them.More specific, with the choice of c = C we have ensured that f1 = ϕ, i.e. the effective classicalvariable of the singular circuit (see subsection 2.2.1) is now (in good approximation) the slowvariable of the two-dimensional potential Eq. (2.28).

The isotropic kinetic part of the Hamiltonian Eq. (2.23) together with the identification of f1

and f2 as slow and fast variables allow the application of the Born-Oppenheimer approxima-tion [23]. This will reduce the Hamiltonian’s number of degrees of freedom by one, resulting inan effective one dimensional Hamiltonian for the slow coordinate. Note that the potential Eq.(2.28) is not separable, i.e. U(f1, f2) 6= U1(f1) + U2(f2), and therefore the Born-Oppenheimerapproximation is indeed an approximation and not an exact approach.

First, we perform a modified Born-Oppenheimer approximation which we call classical. There-fore, we fix the slow variable f1 and (numerically) find the minimum of the potential U(f1, f2)in the fast variable f2 instead of solving the time-independent Schrodinger equation as in thecommon Born-Oppenheimer approximation. This minimal value of the potential in f2-directionas a function of f1 will be taken as effective potential for f1. The condition for an extremumin the f2-direction is the vanishing first order derivative of the potential with respect to thisvariable,

∂

∂f2

U(f1, f2)

EJ= −

√C

CJ

1

β

(f1 −

√C

CJf2

)+γ

β

C

CJf2 +

√C

CJsin

(√C

CJf2 − ϕext

)= 0. (2.30)

This can be solved for f1 as a function of f2 and numerically inverted afterwards in order toobtain f extr

2 (f1), i.e. the location of the potential’s extrema in f2-direction as function of f1. Asa result we find that there could possibly exist multiple extrema if β > 1 + γ, which is exactlythe same condition as was found for the appearing multivaluedness of the effective classicalpotential obtained within the singular treatment of the circuit (see subsection 2.2.1).1 In thiscase, also f extr

2 (f1) is multivalued. This accordance is not accidentally but follows directly fromthe Euler-Lagrange equations Eq. (2.8).

1Note that by diagonalizing the transformed linear inductance matrix M one would get a new condition forthe possible multivaluedness which again approaches the previously discussed situation for CJ/C → 0.

13


The potential U(f1, f2) as well as the (multivalued) function f extr2 (f1) are shown in Fig. 2.6 for

the different cases of β ≶ 1 + γ, each for γ = 0 and γ > 0.

(a) β = 0.7, γ = 0 (b) β = 2.0, γ = 0

(c) β = 0.7, γ = 0.1β (d) β = 2.0, γ = 0.1β

Figure 2.6: Contour plot of the potential U(f1, f2) for CJ/C = 0.05, ϕext = 0 and differentvalues of β and γ together with the (multivalued) function f extr

2 (f1) indicating theminima (red) and maxima (yellow) of the potential in f2 direction for fixed f1.

As one can see, for β < 1 + γ there is only one minimum in f2 direction for a fixed value of f1

such that f extr2 (f1) is a proper function. In contrast to that, for β > 1 + γ there are regimes in

which f extr2 (f1) becomes multivalued, i.e. for fixed f1 coordinate there exist multiple minima

in f2 direction, separated by a maximum. Furthermore, we note that already for CJ/C = 0.05the variables f1 and f2 are indeed proper slow and fast variables, respectively. Moreover, forγ = 0 the potential has a direction in which it is periodic due to the vanishing eigenvalue of

14


the transformed linear inductance matrix M .

The actual classical Born-Oppenheimer approximation is now performed by inserting f extr2 (f1)

into the potential U(f1, f2) and neglecting the kinetic term in the f2-direction. This results inan effective Hamiltonian for the slow variable,

HclBO = 4ECη

21 + U cl

BO(f1), (2.31)

with the effective classical potential U clBO(f1) = U(f1, f

extr2 (f1)) and the charging energy EC =

e2/2C. Note that the construction of U clBO(f1) is equivalent to the construction of the effective

classical potential Ueff,cl(ϕ) within the singular treatment of the circuit neglecting any intrinsiccapacitances (see subsection 2.2.1). For this reason, recalling that f1 = ϕ, these two potentialscoincide. In particular, U cl

BO(f1) does not depend on the capacitance ratio CJ/C.

Nevertheless, there are some subtleties involving the construction of U clBO(f1). On the one hand,

the Born-Oppenheimer approximation is based on the distinction of a fast and a slow variable,which is valid for small values of CJ/C. On the other hand, taking just the minimal value of thepotential instead of solving the time-independent Schrodinger equation completely neglects thekinetic term in the Hamiltonian, giving rise to large quantum fluctuations for small values ofCJ/C. These are obviously conflicting assumptions or in other words have inconsistent regimesof validity. However, the approach of the classical Born-Oppenheimer approximation gives anintuitive understanding of the origin of the effective classical potential’s multivaluedness withinthe singular treatment.

As already mentioned, the classical Born-Oppenheimer approximation of course neglects anyquantum effects. In the following they will be taken into account within the common Born-Oppenheimer approximation. It consists of fixing the slow f1 variable and solving the time-indepent Schrodinger equation (numerically) for the fast f2 variable. The f1-dependent groundstate energy will be taken as effective potential UBO(f1) for f1 afterwards. For a better compar-ison with the effective classical potential obtained within the singular treatment of the circuitwe will subtract a constant offset energy such that UBO(f1 = 0) = Ueff,cl(ϕ = 0) = −1. Fixingthe ratio α = EJ/EC = 500, the results are shown in Fig. 2.7 for the different cases of β ≶ 1+γ,each for γ = 0 and γ > 0.

On the one hand, we observe that for both cases of γ = 0 and γ > 0 the effective potentialobtained by the Born-Oppenheimer approximation approaches the effective classical potentialfor large values of CJ/C (for which, however, the Born-Oppenheimer approximation becomesinvalid). In particular, for β > 1 + γ and intervals in which the effective classical potentialvalue is multivalued UBO(f1) approximates the lowest classical potential value. This can beexplained by a ground state wavefunction within the Born-Oppenheimer approximation whichis well localized in the global minimum (and not just a local minimum) of U(f1, f2) in thef2-direction.On the other hand, for small values of CJ/C it is plausible that UBO(f1) does not approach theeffective classical potential due to large quantum fluctuations which cause a delocalization ofthe wavefunctions within the Born-Oppenheimer approximation. For this reason, the groundstate energy is not just the minimal value of U(f1, f2) in the f2-direction but strongly dependson the potential’s overall curvature. For the case γ = 0, this explains that BBO(f1) approaches

15



-

(a) β = 0.7, γ = 0


-

(b) β = 2.0, γ = 0


-

(c) β = 0.7, γ = 0.1β


-

(d) β = 2.0, γ = 0.1β

Figure 2.7: Shifted effective potential UBO(f1) obtained by the Born-Oppenheimer approxima-tion for α = EJ/EC = 500, ϕext = 0 and exemplary values of β and γ together withthe effective classical potential Ueff,cl(ϕ) (dashed).

a cosine shape with vanishing amplitude in the limit of CJ/C → 1. In contrast, for the caseγ > 0, the effective potential BBO(f1) approaches a parabola in this limit. For both cases,compare with the potential U(f1, f2) given in Eq. (2.28).It is worth to mention that for γ = 0 the effective potential UBO(f1) is 2π periodic, sincewithin U(f1, f2) a shift of f1 by 2π can be compensated by a shift of f2 by 2π(CJ/C)1/2, i.e.U(f1 + 2π, f2 + 2π(CJ/C)1/2) = U(f1, f2).

For a further analysis of the effective classical potential’s multivaluedness for β > 1 + γ, weanalyze excited Born-Oppenheimer surfaces, i.e. instead of taking just the ground state energyof the time-independent Schrodinger equation to construct an effective potential, we will alsoconsider higher excitations with energy En (n = 0, 1, 2, . . .) which will be used to construct theeffective potentials Un

BO(f1). Fixing the ratio of capacitance ratio CJ/C = 0.1, an exemplaryresult for γ = 0 is shown in Fig. 2.8.

16


π

π π

π

-

(a) β = 0.7

π

π π

π

-

(b) β = 2.0

Figure 2.8: Born-Oppenheimer surfaces UnBO(f1) for a fixed value of CJ/C = 0.1, α = 500

and γ = ϕext = 0 with the effective classical potential Ueff,cl(ϕ) (dashed). Notethat all Born-Oppenheimer surfaces are shifted by the same offset energy such thatU0

BO(f1 = 0) = −1.

For β < 1 + γ, the Born-Oppenheimer surfaces are well-separated and the effective classicalpotential is well approximated by U0

BO(f1). Contrary, for β > 1 + γ, the Born-Oppenheimersurfaces bunch for values of f1 in which Ueff,cl(ϕ) is multivalued, especially at f1 = π. Note thatthe Born-Oppenheimer surfaces do not touch or cross like the effective classical potential does.At f1 = π, the potential U(π, f2) is a symmetric double-well potential in the f2-direction and theavoided crossing of the Born-Oppenheimer surfaces can be explained quantum-mechanically.Instead of having a degenerate ground state an energy gap between the two lowest Born-Oppenheimer surfaces emerges due to tunneling processes through the barrier separating thetwo wells. This emerging energy gap can be well described by the WKB approximation [24,25]and is shown in Fig. 2.9 for β = 2.0 and γ = ϕext = 0 as a function of CJ/C.

- - - -

-

-

-

Figure 2.9: Energy gap ∆E1,0BO(f1 = π) = U1

BO(f1 = π) − U0BO(f1 = π) between the two lowest

Born-Oppenheimer surfaces at f1 = π for α = 500, β = 2.0, γ = ϕext = 0 asfunction of the capacitance ratio CJ/C. The exact numerical value (blue) is showntogether with the value obtained by the WKB approximation (red).

17


Again, for large capacitance ratios CJ/C the energy gap becomes small, indicating the conver-gence of U0

BO(f1) to the lowest value of the effective classical potential. However, recall thatCJ/C cannot become arbitrarily large since the validity of the Born-Oppenheimer approxima-tion is justified based on this capacitance ratio to be small. We will elaborate on that later insubsection 2.3.5. Furthermore, note that the WKB approximation has a break-down for smallratios of CJ/C since it is not applicable as soon as the ground state energy considering one wellexceeds the barrier separating both the wells.

So far, this concludes the analysis of the singular treatment of the coupling branch and theresulting multivaluedness of the effective classical potential. We can conclude that the singulartreatment assumes the fast subspace to be in its classical ground state such that quantum ef-fects are completely neglected. This is plausible, since the construction of the effective classicalpotential invokes the classical equations of motion.

In the next subsection, we will additionally include the linear inductors’s intrinsic capacitanceand briefly analyze its effects.

2.2.3 Including the intrinsic capacitances of the Josephson junction andthe inductor

Similar to a Josephson junction, also a real inductor always contains an intrinsic capacitancein parallel. In this subsection we additionally take this capacitance into account and studyits effects. The resulting circuit of the coupling branch taking all intrinsic capacitances intoaccount is shown in Fig. 2.10.

C

CL

Φext

EJ

LJ

L

CJ

φc φaφb

Figure 2.10: Single coupling branch with intrinsic capacitances of the Josephson junction andthe linear inductor. The Josephson junction is inductively shunted such that aloop is formed which can be penetrated by an external magnetic flux Φext.

As in the previous subsection, the inclusion of the intrinsic capacitances results in a circuit withnon-singular Lagrangian which can be directly used for a Legendre transformation in order toobtain the Hamiltonian description of the circuit which reads

H =4e2

2nTC−1n+ U(ϕ). (2.32)

The potential U(ϕ) entering the Hamiltonian is the same potential as previously,

U(ϕ) =1

2L

(Φ0

2π

)2

(ϕ− ϕJ)2 +1

2LJ

(Φ0

2π

)2

ϕ2J − EJ cos

(ϕJ − ϕext

), (2.33)

18


however, now written as a function of the reduced flux variables. Again, the Hamiltonian isquantized by imposing the canonical commutation relations [ϕ, n] = [ϕJ , nJ ] = i (all othercommutators vanish). In comparison to the previous subsection, the inclusion of the inductor’sintrinsic capacitance changes the capacitance matrix as well as its inverse, which are given by

C =

(C + CL −CL−CL CJ + CL

), C−1 =

1

C2Σ

(CJ + CL CLCL C + CL

), (2.34)

with C2Σ = CCJ +CCL +CJCL. Note that in contrast to the previous subesction the existence

of CL > 0 allows to set CJ = 0 such that the capacitance matrix stays invertible, i.e. thecorresponding Lagrangian stays non-singular. In principle, we could proceed as before with thecalculation of the capacitance matricis square-root and the variable transformation introducedin Eq. (2.22) in order to bring the inverse capacitance matrix into an isotropic form. Eventhough this is analytically possible, the matrix elements of C±1/2 are unwieldy expressions andmake further calculations more confusing. Furthermore, C±1/2 are not diagonal which wouldimply a mixing of the variables ϕ and ϕJ such that we can not identify any transformed variablewith ϕ anymore.

With the aim to derive an effective potential for ϕ, we introduce the Cholesky decompositionas an alternative approach. It states that every positive definite matrix F can be writtenas F = ATA with an unique upper-triangular matrix A with positive diagonal entries [26].Moreover, for real F also the Cholesky decomposition A is real and its matrix elements can becalculated recursively [11], resulting in simple expressions, especially for the first rows. For theinverse capacitance matrix given in Eq. (2.34) it is straightforward to check that its Choleskydecomposition is given by

A =1

CΣ

√CJ + CL

(CJ + CL CL

0 CΣ

), (2.35)

satisfying C−1 = ATA. Note that in the special case of CL = 0 and CJ > 0 the Choleskydecomposition reduces to A = C−1/2 such that this approach reproduces the results of theprevious subsection. However, for the general case CL, CJ > 0 we perform the coordinatetransformation

η = c1/2An, f = c−1/2(AT )−1ϕ, (2.36)

which preserves the canonical commutation relations of the variables (see appendix 6.2) andtransforms the Hamiltonian to

H =4e2

2nTC−1n+ U(ϕ) =

4e2

2nTATAn+ U(ϕ)

=4e2

2cηTη + U(f),

(2.37)

which possesses an isotropic kinetic part. Again, we introduced a standard capacitance c inorder to keep the variables dimensionless. Furthermore, we have chosen to use the Choleskydecomposition of the inverse capacitance matrix appearing in the initial Hamiltonian Eq. (2.32)and not of the capacitance matrix within the Lagrangian description. This is motivated by thefact that the coordinate transformation Eq. (2.36) ensures that f1 = ϕ for c = C2

Σ/(CJ +CL).2

2Otherwise one would have to rearrange the basis of ϕ.

19


Taking this choice of c, the transformed potential reads

U(f)

EJ=

1

2EJ

(Φ0

2π

)2

fTMf − cos

(CΣ

CJ + CL

[CLCΣ

f1 + f2

]− ϕext

), (2.38)

with the transformed linear inductance matrix

M = cAMAT =1

L

1

(CJ + CL)2

(C2Lγ + C2

J CΣ(CLγ − CJ)CΣ(CLγ − CJ) C2

Σ(γ + 1)

). (2.39)

Recall the definition of γ = L/LJ , which is the ratio of the linear inductances.

As in the previous subsection, one can show that for small intrinsic capacitances CL, CJ Cthe transformed linear inductance matrix has a large and a small eigenvalue. In particular,both introduced variable transformations, i.e. the transformation based on the square-rootcapacitance matrix decomposition (Eq. (2.22)) and the transformation based on the Choleskydecomposition (Eq. (2.36)), result in transformed linear inductance matrices with equal eigen-values (see appendix 6.3).

Again, the eigenvectors of the transformed linear inductance matrix are approximately alignedalong the f1 and f2 direction, which will be again called the slow and the fast variable, re-spectively. The potential U(f) = U(f1, f2) defined in Eq. (2.38) is exemplarily shown inFig. 2.11 for the different cases of β ≶ 1 + γ, each for γ = 0 and γ > 0 for fixed values ofCJ/C = CL/C = 0.05 and ϕext = 0.

The overall shape of the potential looks similar to the potential analyzed in the previous sub-section in which CL = 0 (compare with Fig. 2.6). As already mentioned, this is due to theCholesky decomposition which approaches the inverse square-root capacitance matrix in thelimit of small CL. In particular, the variables f1 and f2 can be indeed identified as small andfast variable, respectively.

Again, we can perform the Born-Oppenheimer approximation in order to derive an effectivepotential for the variable f1 = ϕ. Before we do so, we want to discuss the kinetic part of theHamiltonian defined in Eq. (2.37). It reads

Hkin =4e2

2cηTη = 4EC

C

cηTη, (2.40)

in which EC = e2/2C denotes the charging energy with respect to the capacitance in parallelto both the Josephson junction and the linear inductor. Recall the choice of c = C2

Σ/(CJ +CL)with C2

Σ = CCJ + CCL + CJCL from which follows that C/c < 1 if CL > 0. As a result,we expect the inclusion of CL > 0 to further localize the wavefunction in the fast f2-directionwithin the Born-Oppenheimer approximation (keeping in mind that also the potential is af-fected by CL > 0 due to the variable transformation Eq. (2.36)). This expectation is confirmedby the results of the Born-Oppenheimer approximation shown in Fig. 2.12 for a fixed value ofCL/C = 0.01 and varying values of CJ/C.

20


(a) β = 0.7, γ = 0 (b) β = 2.0, γ = 0

(c) β = 0.7, γ = 0.1β (d) β = 2.0, γ = 0.1β

Figure 2.11: Contour plot of the potential U(f1, f2) for CJ/C = CL/C = 0.05, ϕext = 0 anddifferent values of β and γ together with the (multivalued) function f extr

2 (f1) indi-cating the minima (red) and maxima (yellow) of the potential in f2 direction forfixed f1.

The major difference compared to the previously analyzed subsection in which CL has notbeen taken into account is that for CJ/C → 0 the effective potential UBO(f1) does not vanish(γ = 0) or approach a parabola (γ > 0) but rather remains close to the (lowest value) of the(multivalued) effective classical potential. This can be explained by the previously mentionedenhanced localization due to CL. In fact, for CJ = 0 and CL > 0 the circuit shown in Fig. 2.10stays non-singular. However, the quantum fluctuations in the fast direction cause UBO(f1) notto coincide with the effective classical potential.

21



-

(a) β = 0.7, γ = 0


-

(b) β = 2.0, γ = 0


-

(c) β = 0.7, γ = 0.1β


-

(d) β = 2.0, γ = 0.1β

Figure 2.12: Shifted effective potential UBO(f1) obtained by the Born-Oppenheimer approxima-tion for CL/C = 0.01, α = EJ/EC = 500, ϕext = 0 and exemplary values of β andγ together with the effective classical potential Ueff,cl(ϕ) (dashed).

The enhanced localization of the wavefunction in the fast f2-direction within the Born-Oppenheimerapproximation can be also seen in the analysis of the Born-Oppenheimer surfaces, whose spac-ings become smaller compared to the case of CL = 0. This is exemplarily shown in Fig. 2.13for the Born-Oppenheimer surfaces for γ = 0 and CJ/C = 0.1, CL/C = 0.01.

22


π

π π

π

-

(a) γ = 0, β = 0.7

π

π π

π

-

(b) γ = 0, β = 2.0

Figure 2.13: Born-Oppenheimer surfaces for fixed values of CJ/C = 0.1, CL/C = 0.01, α = 500and γ = ϕext = 0 with the effective classical potential Ueff,cl(ϕ) (dashed). Notethat all Born-Oppenheimer surfaces are shifted by the same offset energy such thatU0

BO(f1 = 0) = −1.

This concludes the analysis of the single coupling branch. In the next section we will considerthe full qubit-resonator circuit.

23


2.3 Qubit-resonator circuit with intrinsic capacitances

After the analysis of the Josephson junction’s and the inductor’s intrinsic capacitances’ effectsin one single coupling branch we will now investigate their effects in the full qubit-resonatorcircuit. The circuit is shown in Fig. 2.1 and a description can be found in subsection 2.1.3.

Similar to Richer et al. [12], we will work within the classical ground state approximation ofthe Josephson array, which will be briefly summarized in the following. If all ki Josephsonjunctions of such an array are equal, its potential term can be described as

−k∑j=1

EJ cos(ϕj) = −kEJ cos

(ϕJ + 2πm

k

), (2.41)

which depends on the biasing phase variable ϕJ =∑k

j=1 ϕj only [27]. The integer numberm denotes the number of so called phase-slips [27, 28] and must be constant in time for theapproximation Eq. (2.41) to hold. This is ensured in the regime in which the Josephson en-ergy of every single junction within the array is much larger than its charging energy, sincephase-slip events, i.e. integer changes in m, are exponentially suppressed [27–29]. Without lossof generality we can choose m = 0. Note that the classical ground state approximation of theJosephson arrays completely neglects the dynamics of the Josephson arrays’ internal degrees offreedom, which is justified if the plasma frequency of each Josephson junction within the chainis much larger than the other circuit’s frequencies [30,31], i.e. the qubit and resonator frequency.

Note that the intrinsic capacitance of a Josephson array, which consists of k identical Josephsonjunctions, corresponds to the Josephson junctions’ intrinsic capacitance divided by k. Further-more, considering a Josephson array (in the classical ground state approximation) within thesingle coupling branch analysis instead of a single Josephson junction (see section 2.2) changesthe condition of the effective classical potential’s multivaluedness to β > k(1 + γ).

2.3.1 Derivation of the Hamiltonian

As a first step towards a quantum description of the full qubit-resonator circuit shown in Fig.2.1 we determine the corresponding Lagrangian. Since the circuit has five nodes it is fullydescribed by four independent variables. Adopting the convention of the previous analysis ofone single coupling branch (see section 2.2), we choose the four independent variables to be

φ1 = φc − φa, φ2 = φc − φe, φJ1 = φc − φb, φJ2 = φc − φd (2.42)

such that the Lagrangian of this full circuit reads

L =2∑i=1

[Ciφ

2i

2+CJiφ

2Ji

2+CLi(φ

2i − φ2

Ji)

2− (φ− φJi)2

2Li− φ2

Ji

2LJi

+ kiEJi cos

(2π

Φ0

(φJi − Φext

i

ki

))]+Cq(φ2 − φ1)2

2+ EJq cos

(2π

Φ0

(φ1 − φ2 + ΦextB )

).

(2.43)

24

2.3. Qubit-resonator circuit with intrinsic capacitances

It consists of the sum of the Lagrangians of the single branches and two coupling terms due tothe connection to the transmon qubit, one capacitive coupling term and one inductive couplingterm, originating from the capacitance and the Josephson junction, respectively. Since we areinterested in the effects of non-zero capacitances CJi and CLi and want to compare with thecircuit’s singular treatment, we adopt the qubit and resonator variables proposed by Richer etal. [12] which are defined as

φ1 =φr + φq

2, φ2 =

φr − φq2

, (2.44)

Taking the four independent variables describing the circuit to be ϕq, ϕr, ϕJ1 , ϕ21 , i.e. thereduced fluxes, yields to the Lagrangian in matrix notation

L =1

2

(Φ0

2π

)2

ϕTCϕ− U(ϕ) (2.45)

with the potential

U(ϕ) =1

2

(Φ0

2π

)2

ϕTMϕ− k1EJ1 cos

(ϕJ1 − ϕext

1

k1

)− k2EJ2 cos

(ϕJ2 − ϕext

2

k2

)− EJq cos

(ϕq + ϕext

B

),

(2.46)

in which we also take the external fluxes to be in their reduced form. Note that the arrangementsof the capacitance matricis and the linear inductance matricis elements of course depend onthe arrangement of the four variables within the vector ϕ, which will become crucial in thefollowing. By choosing the arrangement ϕ = (ϕJ1, ϕJ2, ϕq, ϕr)

T , the capacitance matrix andlinear inductance matrix read

C =

CJ1 + CL1 0 −1

2CL1 −1

2CL1

0 CJ2 + CL212CL2 −1

2CL2

−12CL1

12CL2

14(C1 + C2 + CL1 + CL2 + 4Cq)

14(C1 − C2 + CL1 − CL2)

−12CL1 −1

2CL2

14(C1 − C2 + CL1 − CL2) 1

4(C1 + C2 + CL1 + CL2)

,

(2.47a)

M =

1L1

+ 1LJ1

0 − 12L1

− 12L1

0 1L2

+ 1LJ2

12L2

− 12L2

− 12L1

12L2

14

(1L1

+ 1L2

)14

(1L1− 1

L2

)− 1

2L1− 1

2L2

14

(1L1− 1

L2

)14

(1L1

+ 1L2

) . (2.47b)

Recall, that in contrast to the singular treatment of the full qubit-resonator circuit the consid-eration of the Josephson arrays’ and linear inductors’ intrinsic capacitances removes the singu-larity of the Lagrangian since the capacitance matrix C is invertible. Therefore, the Legendretransformation is applicable and the derivation of the Hamiltonian is straightforward. However,before performing the Legendre transformation and progressing to the Hamiltonian descriptionof the qubit-resonator circuit, we determine the Cholesky decomposition of the capacitancematrix C and make a variable transformation. This proceeding bypasses the calculation of the

25


actual inverse capacitance matrix. The Cholesky decomposition A of the capacitance matrixC satisfying C = ATA is an upper triangular 4× 4 matrix of the form

A =

a11 a12 a13 a14

0 a22 a23 a24

0 0 a33 a34

0 0 0 a44

, (2.48)

with real-valued matrix elements. As already noted, due to the construction of the Choleskydecomposition the matrix elements of the first rows are simple expressions. Exploiting thisfact, we introduce the partial Cholesky decomposition

B =

a11 a12 a13 a14

0 a22 a23 a24

0 0√c 0

0 0 0√c

=

√CJ1 + CL1 0 −CL1

2√CJ1+CL1

−CL1

2√CJ1+CL1

0√CJ1 + CL1

CL2

2√CJ2+CL2

−CL2

2√CJ2+CL2

0 0√c 0

0 0 0√c

, (2.49)

which coincides in the first two rows with A and is proportional to the identity matrix 14×4

in the last two rows. Again, c is an arbitrary standard capacitance. This partial Choleskydecomposition will now be used to define the variable transformation

f = c−1/2Bϕ, (2.50)

with f = (f1, f2, f3, f4)T as well as ϕ = (ϕJ1, ϕJ2, ϕq, ϕr)T . In this new set of variables the

Lagrangian reads

L =1

2

(Φ0

2π

)2

fTζf − U(f), (2.51)

with the transformed capacitance matrix ζ = c(B−1)TCB−1 and the potential

U(f) =c

2

(Φ0

2π

)2

fT (B−1)TMB−1f − EJq cos(f3 + ϕext

B

)− EJ1 cos

(√c(B−1f)1 − ϕext

1

)− EJ2 cos

(√c(B−1f)2 − ϕext

2

).

(2.52)

Due to the construction of the partical Cholesky decomposition, the variable transformation Eq.(2.50) leaves the qubit and resonator variables unchanged, i.e. f3 = ϕq and f4 = ϕr, which iscrucial for an appropriate comparison with the singular treatment of the circuit. Furthermore,the capacitance matrix is transformed into a block diagonal form, decoupling the first twovariables from the last two variables in the kinetic term of the transformed Lagrangian Eq.(2.51). This can be easily seen by introducing the block matrix notation for the (partial)Cholesky decomposition and the capacitance matrix

A =

(A1 A2

02×2 A4

), B =

(A1 A2

02×2

√c12×2

), C =

(C1 C2

C3 C4

), (2.53)

in which the inverse partial Cholesky decomposition reads

B−1 =1√c

( √cA−1

1 −A−11 A2

02×2 12×2

). (2.54)

26


It follows directly that the transformed capacitance matrix

ζ = c(B−1)TCB−1 = c(AB−1)T (AB−1)

=

(c12×2 | 02×2

02×2 | AT4A4

)=

(c12×2 | 02×2

02×2 | C4 −AT2A2

)(2.55)

is indeed block-diagonal, in which the upper-left block matrix acting on the f1, f2 subspace isisotropic. Note that we do not need to evaluate A4 at any point. Considering the actual matrixelements of the block-matrix A2, one finds

AT2A2 =

1

4

(C2

L1

CJ1+CL1+

C2L2

CJ2+CL2

C2L1

CJ1+CL1− C2

L2

CJ2+CL2C2

L1

CJ1+CL1− C2

L2

CJ2+CL2

C2L1

CJ1+CL1+

C2L2

CJ2+CL2

), (2.56)

such that we can conclude that the transformed capacitance matrix ζ is diagonal if the circuit’scapacitances are symmetric, i.e. CJ1 = CJ2, CL1 = CL2, C1 = C2. After this variable transfor-mation, we derive the Hamiltonian description of the qubit-resonator circuit by means of theLegendre transformation which results in

H =4e2

2ηTζ−1η + U(f). (2.57)

Again, the Hamiltonian is quantized by imposing the commutation relations [ηi, fj] = iδij. Dueto the block-diagonal form the transformed capacitance matrix, ζ−1 is given by the inversionof the 2 × 2 block-matrices of ζ. In particular, also ζ−1 is block-diagonal. This ensures thedecoupling of the qubit and resonator variables from the remaining subspace in the kineticterm of the Hamiltonian and allows a derivation of an effective Hamiltonian for the qubit andresonator variables, which will be discussed in the following subsection.

2.3.2 Derivation of an effective qubit-resonator Hamiltonian

Given the four-dimensional Hamiltonian in Eq. (2.57) we will derive an effective two-dimensionalHamiltonian for the qubit and resonator variables f3 and f4, respectively, by performing theBorn-Oppenheimer approximation. In the parameter regime of interest, i.e. for small intrinsiccapacitances CJ1, CJ2, CL1, CL2 C1, C2, Cq, one can show that f1, f2 and f3, f4 can be con-sidered to fast and slow variables, respectively.The implementation of the multi-dimensional Born-Oppenheimer approximation is compara-ble to the one-dimensional Born-Oppenheimer approximation performed in the analysis of onesingle coupling branch (see section 2.2). This means, that we (numerically) solve the time-independent Schrodinger equation in the fast f1, f2 subspace for its ground state energy whilefixing the slow f3, f4 variables to be constant. The dependence of this ground state energyon the slow variables will be taken to be the new, effective potential U eff

BO(f1, f2) for the qubitand resonator variables. Since the numerics of the multi-dimensional Born-Oppenheimer ap-proximation is much more expensive than in case of an one-dimensional Born-Oppenheimerapproximation, we explored the following approaches for the solution of the time-independentSchrodinger equation in the fast subspace (valid in the parameter regime in which the potentialin the fast subspace has one local, well-pronounced minimum only):3

3Note that we are forced to perform a full numerical treatment or have to find different approaches for parameterregimes in which the potential has multiple local minima in the fast subspace.

27


• Expand the potential in the fast variables up to second order around f3 = f4 = 0 andcalculate this two-dimensional harmonic oscillator’s ground state energy, which will betaken as effective potential for the slow variables [23]. This approximation seems to be theonly one allowing any analytic approach. However, already for the special case CLi = 0,by construction this approach does not incorporate important effects like any longitudinalcoupling or the dependence of the qubit and resonator frequencies on CJi at all and willbe dropped for this reason.

• Find the minimal value of the potential in the fast directions numerically and makean expansion of the potential around this minimum. Expanding this minimum up tosecond order in f3, f4, the ground state energy of this resulting two-dimensional harmonicoscillator is taken to be the effective potential for the slow variables. This assumption isimproved by taking higher orders of the expansion with first order perturbation theoryinto account. We want to mention that the perturbative incorporation of higher orders inf3, f4 is particular easy since they only appear unmixed and that this approach becomesmore precise the larger the intrinsic capacitances are.

• Solve the two-dimensional Schrodinger equation in the fast subspace fully numerically.

In the following we will chose the second introduced approach up to fourth order, since itis much faster than the full numerical treatment while being sufficiently precise. Given theeffective potential derived in this way, the effective Hamiltonian for the slow qubit and resonatorvariables reads

HBO =4e2

2

(η3 η4

)ζ−1

4

(η3

η4

)+ U eff

BO(f3, f4), (2.58)

with the inverse transformed sub-capacitance matrix ζ−14 = (C4 −AT

2A2)−1 and the previouscommutation relations [ηi, fj] = iδij. This Hamiltonian is the starting point for the derivationof the qubit and resonator frequencies as well as the coupling parameters of their interaction.

Before we go on, we want to mention the following observation: For CLi = 0, the minimal valueof the four-dimensional potential U(f) within the fast subspace does not depend on CJi, sincethe variable transformation Eq. (2.50) corresponds to a simple rescaling of the variables in thiscase. Especially, similar to section 2.2, for fixed slow variables this potential’s minimal value inthe fast variables corresponds to the effective classical potential obtained within the singulartreatment of the circuit. This enables a direct comparison of the non-singular circuit with thesingular one, which, as already stated, differs in the included quantum fluctuations of the fastvariables.

2.3.3 Derivation of the parameters defining the qubit-resonator modelHamiltonian

Given the effective Hamiltonian Eq. (2.58) for the qubit-resonator system, we want to determinethe parameters of the model Hamiltonian

Hmodel =~∆

2σz + ~ωra†a+ ~gxxσx(a† + a) + ~gzxσz(a† + a), (2.59)

28


describing an ideal qubit and resonator (first and second term, respectively), which are coupledtransversely and longitudinally (third and fourth term, respectively). For this reason, Richeret al. [12] already proposed a method which is based on several crucial assumptions and willbe summarized in the following. Afterwards, an improvement in precision and generality withthe drawback of being numerically more expensive will be suggested .

Identification of the model Hamiltonian’s parameters

The method for the calculation of the model Hamiltonian’s parameters introduced by Richeret al. in [12] is only applicable if the kinetic part of the effective Hamiltonian, i.e. the inversetransformed sub-capacitance matrix ζ−1

4 , is diagonal. In that case Eq. (2.58) reduces to

H =4e2η2

3

2ζ33

+4e2η2

4

2ζ44

+ U effBO(f3, f4), (2.60)

where ζ = (ζij). As already mentioned, according to Eqs. (2.47a) and (2.56), the diagonalstructure of ζ−1

4 is given if the circuit is symmetric in its capacitances, i.e. CJ1 = CJ2 , CL1 =CL2 , C1 = C2. Furthermore, we have to demand the effective potential U eff

BO(f3, f4) to possessone local, well-pronounced minimum only, i.e. the qubit must behave like a transmon [20].Given these preliminary restrictions, we shift the origin of the slow f3, f4 subspace into theminimum of the effective potential, which will be expanded in a Taylor series

U effBO(f3, f4) =

∞∑i,j=0

κijfi3f

j4 , (2.61)

in which κ10 = κ01 = 0 due to the shift of the variables. The initial location of the minimum aswell as the expansion coefficients κij have to be found numerically. The coefficient κ00 representsthe minimal value of the potential in the fast subspace and is therefore just an offset energywhich can be discarded by setting κ00 = 0. For the definition of a proper basis set we considerthe quadratic Hamiltonian

H′ = 4e2η23

2ζ33

+4e2η2

4

2ζ44

+ κ20f23 + κ02f

24 , (2.62)

which only takes the first two, uncoupled terms of the effective potential’s series expansion Eq.(2.61) into account and describes two uncoupled harmonic oscillators with frequencies

ωq =1

~

√8e2κ20

ζ33

, ωr =1

~

√8e2κ02

ζ44

. (2.63)

The subscripts q and r already indicate that these frequencies will correspond to the qubitand resonator frequency, respectively. The verification, that these variables indeed represent aqubit and a resonator, respectively, will be given later within the analysis of the correspondinganharmonicities. In order to proceed, we introduce the ladder operators

f3 = 4

√e2

2ζ33κ20

(c† + c), η3 = i4

√ζ33κ20

8e2(c† − c), (2.64a)

f4 = 4

√e2

2ζ44κ02

(a† + a), η4 = i4

√ζ44κ02

8e2(a† − a), (2.64b)

29


in which a(†) and c(†) are the bosonic annihilation (creation) operators satisfying the bosoniccommutation relations [a, a†] = [c, c†] = 1, while all other commutators vanish. The Hamilto-nian H′ expressed in this creation and annihilation operators reads

H′ = ~ωq(c†c+

1

2

)+ ~ωr

(a†a+

1

2

), (2.65)

and its eigenstates are product states of the harmonic oscillators’ Fock states. Based on theassumption that the qubit behaves like a transmon, we define the qubit’s and resonator’sanharmonicities via the quartic deviation from the corresponding harmonic system [20], i.e. weconsider the terms κ40f

43 and κ04f

44 of the potential’s expansion. In first order perturbation

theory, these terms cause a shift of the qubit’s j’th and the resonator’s n’th eigenenergy, whichare given by

Ejq = ~ωq

(j +

1

2

)+

κ40e2

2ζ33κ20

(6j2 + 6j + 3), Enr = ~ωr

(n+

1

2

)+

κ04e2

2ζ44κ02

(6n2 + 6n+ 3),

(2.66)respectively. Finally, the relative anharmonicities are defined as [12,20]

α(r)q =

δEq21 − δE

q10

δEq10

, α(r)r =

δEr21 − δEr

10

δEr10

(2.67)

with the excitation energies

δEqji = Ej

q − Eiq, δEr

nm = Enr − Em

r . (2.68)

For a well defined resonator, the relative anharmonicity α(r)r has to be reasonable small and will

be neglected for this reason. In contrast, the qubit’s relative anharmonicity α(r)q is not desired

to be small.

Following the idea of Richer et al. [12], we truncate the potential’s series expansion Eq. (2.61)after the second order in both f3 and f4 (i, j ≤ 2). Taking the ladder operators introduced

in Eq. (2.64) into account and assuming α(r)q to be sufficient large4, we make a two-level

approximation of the qubit’s degree of freedom such that we can identify5

c†c+1

2→ σz

2+ 1, f3 ∝ (c† + c)→ σx, f 2

3 ∝ (c† + c)2 → σz + 2. (2.69)

Note that we consider the qubit’s two eigenstates with lowest eigenenergies to define the two-level system. The validity of this approximation has to be confirmed with the actual calculationof α

(r)q .

Dropping constant energy shifts and taking the shift in the qubit’s eigenenergies (Eq. (2.66))into account, we arrive at

H =~∆

2σz+~ωra†a+~gxxσx(a†+a)+~gzxσz(a†+a)+~gxzσx(a†+a)2 +~gzzσz(a†+a)2, (2.70)

4Koch et al. give an estimate of |α(r)q | ≥ 1/200π ≈ 0.0016 [20].

5Note that in our convention σz |1〉 = |1〉 , σz |0〉 = − |0〉.

30


with the corrected qubit frequency

∆ =δEq

10

~. (2.71)

The coupling parameters appearing in Eq. (2.70) are defined as

gxx =κ11

~4

√e2

2ζ33κ20

4

√e2

2ζ44κ02

, gzx =κ21

~

√e2

2ζ33κ20

4

√e2

2ζ44κ02

, (2.72a)

gxz =κ12

~4

√e2

2ζ33κ20

√e2

2ζ44κ02

, gzz =κ22

~

√e2

2ζ33κ20

√e2

2ζ44κ02

. (2.72b)

Note that they are directly related to the expansion coefficients κij. Furthermore, while gxxand gzx denote the desired transverse and longitudinal coupling, respectively, the first undesiredcouplings are proportional to the coupling coefficients gxz, gzz. We demand these undesired cou-pling coefficients to be small.

Note that the terms which have been truncated in the series expansion of the potential couldalso contribute the coupling coefficients. However, their effect is considered to be small.

Improved identification of the model Hamiltonian’s parameters

As already mentioned, the previously discussed method for the calculation of the model Hamil-tonian’s parameters is based on several assumptions: The effective potential U eff

BO(f3, f4) musthave only one distinct local minimum whose higher expansion coefficients are assumed to besufficiently small and the kinetic term of the effective Hamiltonian has to be diagonal. Tocircumvent these assumptions of applicability and provide a method which can be applied withless restrictions we propose an improved method for the calculation of the model Hamiltonian’sparameters, starting with an effective Hamiltonian in the form of Eq. (2.58), which is

H =4e2η2

3

2c33

+4e2η2

4

2c44

+4e2η3η4

c34

+ U effBO(f3, f4), (2.73)

with the capacitances cij = (ζ−14 )ij. First, we shift the origin of the f3, f4 subspace to a point

(f 03 , f

04 ), which is unspecified at this moment. Afterwards, we decompose the effective potential

in a separable part and a remaining term, i.e.

U effBO(f3, f4) = U eff

BO(f3, 0) + U effBO(0, f4) + δU(f3, f4)− U eff

BO(0, 0). (2.74)

On the right hand side of Eq. (2.74) we have subtracted U effBO(0, 0) in order to enforce the

remaining term δU(f3, f4) to vanish on the f3 and f4 axes, i.e. δU(f3, 0) = δU(0, f4) = 0. Wenow define the uncoupled effective qubit’s and resonator’s Hamiltonians

Hq =4e2η2

3

2c33

+ U effBO(f3, 0), Hr =

4e2η24

2c44

+ U effBO(0, f4), (2.75)

and denote their eigenstates (which have to be determined numerically) with |p〉 and |n〉,respectively. These eigenstates will be taken to define product states |p, n〉 = |p〉 ⊗ |n〉, which

31


will be taken to expand the initial effective Hamiltonian Eq. (2.73). For this purpose, weidentify the initial Hamiltonian to read

H = Hq +Hr +4e2η3η4

c34

+ δU(f3, f4)− U effBO(0, 0), (2.76)

as well as its separate matrix elements

〈p′, n′|Hq |p, n〉 = Epq δp′pδn′n, (2.77a)

〈p′, n′|Hr |p, n〉 = Enr δp′pδn′n, (2.77b)

〈p′, n′| 4e2η3η4

c34

|p, n〉 =−4e2

c34

∫ ∞−∞

∫ ∞−∞

df3df4χ∗p′(f3)ψ∗n′(f4)

∂χp(f3)

∂f3

∂ψn(f4)

∂f4

, (2.77c)

〈p′, n′| δU(f3, f4) |p, n〉 =

∫ ∞−∞

∫ ∞−∞

df3df4χ∗p′(f3)ψ∗n′(f4)δU(f3, f4)χp(f3)ψn(f4), (2.77d)

in which we introduced the uncoupled Hamiltonians’ wavefunctions χp(f3) = 〈f3|p〉 , ψn(f4) =〈f4|n〉 and the eigenenergies Ep

q , Enr . The matrix elements 〈p′, n′|H |p, n〉 will be utilized to

determine the model Hamiltonian’s parameters. Considering that, we restrict ourselves to thetwo-level approximation for the qubit (q = 0, 1) and compare these matrix elements with thematrix elements of the model Hamiltonian Eq. (2.59), which are given by

〈0, n′|H |0, n〉 =

−~∆

2+ ~ωrn for n = n′

−~gzx√

max(n, n′) for |n− n′| = 1

0 else

, (2.78a)

〈1, n′|H |1, n〉 =

+~∆

2+ ~ωrn for n = n′

+~gzx√

max(n, n′) for |n− n′| = 1

0 else

, (2.78b)

〈0, n′|H |1, n〉 =

+~gxx

√max(n, n′) for |n− n′| = 1

0 else. (2.78c)

The idea is now to (numerically) determine (f 03 , f

04 ) such that the matrix elements 〈p′, n′|H |p, n〉

fit the model Hamiltonian’s matrix elements defined in Eq. (2.78) best. For this optimization,one should consider matrix elements with small n, n′ only, since e.g. in the proposed applicationof longitudinal coupling [17], the resonator does not experience a high occupation. However,if the optimal choice of (f 0

3 , f04 ) is still not reasonable, one has to extent the model to a more

general one.

As already mentioned, this improved method for calculating the model Hamiltonian’s parame-ters is much more expensive, numerically. However, in principle it is applicable to more generaleffective qubit-resonator Hamiltonians than the method proposed by Richer et al. Especially,the qubit does not have to be transmon-like.Nevertheless, in the following subsection we will perform numerical calculations in a parameterregime in which the method proposed by Richer et al. works reasonable. Therefore, the couplingparameters will be calculated with the method introduced first.

32


2.3.4 Numerical results for CLi = 0

In this subsection we want to analyze the effects of the Josephson arrays’ intrinsic capacitanceswithout taking the intrinsic capacitances of the linear inductors into account. For simplicity, werestrict our considerations to a qubit-resonator circuit which is symmetric in its capacitances.In this case, the capacitance matrix C defined in Eq. (2.47a) is diagonal and its Choleskydecomposition coincides with its square-root matrix. Therefore, the variable transformation Eq.(2.50) involving the partial Cholesky decomposition results in a simple rescaling of the variables.For the following numerical analysis we consider the realistic values of the circuit’s elements(besides CLi) shown in Table 2.1 [12]. Note that the Josephson energies of the Josephson arraysare chosen to be asymmetric in order to obtain transverse and longitudinal coupling of the samemagnitude [12].

Capacitances value Inductances value Josephson junctions value

Cq 50 fF L1, L2 3.0 nH Eq h5 GHzC1, C2 65 fF LJ1, LJ2 4.5 nH EJ1 h80 GHzCL1, CL2 0 fF EJ2 h76 GHzCJ1, CJ2 varied k1, k2 5

Table 2.1: Considered values of the circuit’s elements for the numerical analysis. The capaci-tances CJ1 = CJ2 will be varied in a range CJi/Ci = 10−6...1 and the external fluxesϕext

1 , ϕext2 , ϕext

B are tunable parameters and therefore not fixed.

Varying the intrinsic capacitances of the Josephson arrays

First, we consider the exemplary case in which no external flux is present, i.e. ϕexti = ϕext

B = 0,and determine the model Hamiltonian’s parameters as functions of CJi/C. The results arepresented in Figs. 2.14-2.16 together with the results obtained within the circuit’s singulartreatment for comparison.

The frequencies of qubit and resonator as functions of the capacitance ratio CJi/Ci are shownin Fig. 2.14 together with the frequencies obtained by the singular treatment of the circuit(CJi/Ci = 0). The frequencies approach the singular results for large values of CJi/Ci. Thiscan be explained based on the single branch analysis: For large values of CJi/Ci, the effec-tive potential U eff

BO(f3, f4) obtained with the Born-Oppenheimer approximation approaches theminimal potential value of U(f) in the fast subspace for fixed f3, f4, which coincides with theeffective classical potential. However, in this limit the validity of the Born-Oppenheimer ap-proximation has to be confirmed (see subsection 2.3.5). In the opposite limit of small CJi/Cithe qubit and resonator frequencies diverge from the singular results. This also can be under-stood within the Born-Oppenheimer approximation, in which now the quantum fluctuationsin the fast f1, f2 subspace dominate the minimal potential value. Therefore, also U eff

BO(f3, f4)differs from the classical result, yielding to different frequencies.

33


- - - - - -

Figure 2.14: Qubit (blue) and resonator (red) frequency as functions of the capacitance ratioCJi/Ci (solid lines) for the values of the circuit’s elements shown in Tab. 2.1without any external flux, ϕext

i = ϕextB = 0. For comparison, also the results of the

circuit’s singular treatment are shown (dashed lines).

Similar statements can be made for the desired and undesired coupling coefficients of the qubit-resonator system, which are shown in Fig. 2.15 as functions of the capacitance ratio CJi/Citogether with the results of the circuit’s singular treatment for comparison. Note that while thecoupling parameters gxx and gzz match the results obtained within the singular treatment forlarge CJi/Ci and differ for small values of CJi/Ci, the coupling coefficients gzx and gxz vanishfor any ratio of CJi/Ci and therefore match the results of the singular treatment. This is dueto the special choice of vanishing external fluxes.

- - - - - - -

(a) Desired qubit-resonator couplings.Red: longitudinal coupling coefficientBlue: transverse coupling coefficient

- - - - - - -

-

(b) Undesired qubit-resonator couplings

Figure 2.15: (a) Desired and (b) undesired coupling coefficients of the qubit-resonator system asfunctions of the capacitance ratio CJi/Ci (solid lines) for the values of the circuit’selements shown in Tab. 2.1 without any external flux, ϕext

i = ϕextB = 0. For

comparison, also the results of the circuit’s singular treatment are shown (dashedlines).

34


Finally, also the dependence of the qubit’s and the resonator’s relative anharmonicities onthe capacitance ratio CJi/Ci, which are shown in Fig. 2.16 together with the results of thecircuit’s singular treatment for comparison, can be explained similarly. For large values ofCJi/Ci they mainly stay unaffected and resemble the singular treatment’s results. Only forsmall values of CJi/Ci they start to deviate. Furthermore, the anharmonicity of the resonatoris reasonable small, while the anharmonicity of the qubit is sufficiently large to justify thetwo-level approximation of the qubit’s degree of freedom [20].

- - - - - - -

-

-

(a) Relative qubit anharmonicity.

- - - - - - -

-

-

(b) Relative resonator anharmonicity.

Figure 2.16: Relative anharmonicities of (a) the qubit and (b) the resonator as functions of thecapacitance ratio CJi/Ci (solid lines) for the values of the circuit’s elements shownin Tab. 2.1 without any external flux, ϕext

i = ϕextB = 0. For comparison, also the

results of the circuit’s singular treatment are shown (dashed lines).

In summary, we can conclude that for our choice of the circuit elements’ values (Tab. 2.1) theeffects due to the intrinsic capacitances of the Josephson arrays are significant for very smallratios CJi/C only. In order to investigate the effect of the intrinsic capacitances for differentexternal fluxes through the small loops, we fix the capacitance ratio to CJ/C = 10−5 anddetermine all the relevant parameters as functions of these external fluxes. Note that the smallratio of CJ/C = 10−5 is experimentally already hard to achieve but will be considered sincethe effects of small intrinsic capacitances are more pronounced.

Varying the external fluxes through the small loops

As already stated by Richer et al., especially the dependencies of the coupling coefficients onthe external fluxes through the small loops play an essential role in the usability of the circuit asan effective qubit-resonator system for various tasks [12]. For example, the different couplingsobtained by tuning the external fluxes through the small loops allow for an easy readout schemeand precise two-qubit gates within one superconducting device. Therefore, we analyze if theseflux-dependencies of the coupling coefficients are still present if one consider small intrinsiccapacitances.Fixing the external flux through the big loop to vanish (ϕext

B = 0), the numerical results arepresented in Figs. 2.17-2.19 for symmetric fluxes through the small loops (ϕext

1 = ϕext2 ) together

with the results obtained within the circuit’s singular treatment for comparison. Due to theperiodicity of the potential Eq. (2.46) in the external fluxes, it is sufficient to consider the

35


regime 0 ≤ ϕexti ≤ 10π (for ki = 5).

The dependencies of the frequencies of qubit and resonator on the external flux through thesmall loops are shown in Fig. 2.17 together with the results obtained within the singular treat-ment of the circuit. Including the intrinsic capacitances CJi in the analysis of the circuit, theireffect on the qubit and resonator frequency is a flattening of the flux-dependence while theiroverall shapes stay unchanged. In particular, both the qubit and the resonator frequency, aremaximal at vanishing fluxes through the small loops and minimal at ϕext

i = 5π (both up tomultiples of 10π due to the periodicity).

π π π π π

Figure 2.17: Qubit (blue) and resonator (red) frequency as functions of equal external fluxesthrough the small loops ϕext

1 = ϕext2 (solid lines) for the values of the circuit’s

elements shown in Tab. 2.1 and fixed capacitance ratio CJi/Ci = 10−5 without anexternal flux through the big loop, ϕext

B = 0. For comparison, also the results ofthe circuit’s singular treatment are shown (dashed lines).

The coupling coefficients as functions of the external flux through the small loops are shownin Fig. 2.18 together with the results obtained within the singular treatment of the circuit.Quantitatively, the dependence of the coupling coefficients on the external flux through thesmall loops does not change drastically. Whereas the longitudinal coupling coefficient gzx hasroots at ϕext

i = 0 and ϕexti = 5π (and multiples of 10π) and therefore coincides with the results of

the singular treatment, its locations of its maximal amplitudes are slightly shifted compared tothe results of the singular treatment. The effect of the intrinsic capacitances on the transversecoupling coefficient gxx behaves vice versa: While its amplitude is locally maximal at ϕext

i = 0and ϕext

i = 5π (and multiples of 10π) and therefore at the same locations as within the singulartreatment, its roots are slightly shifted compared to the results of the singular circuit. Thisresults in the fact that the roots of gzx are still given at the location of the amplitude maximaof gxx and also approximately vice versa. Therefore, we can conclude that the presence ofthe intrinsic capacitances has no drastic effects on the desired coupling coefficients, besides asmall decrease of the maximal amplitudes and a change of the functional dependence. For theunwanted coupling coefficients gxz, gzz we make the observation that they stay reasonable small.

36


π π π π π-

-

-

(a) Desired qubit-resonator couplings.Red: longitudinal coupling coefficientBlue: transverse coupling coefficient

π π π π π-

-

(b) Undesired qubit-resonator couplings.

Figure 2.18: (a) Desired and (b) undesired coupling coefficients of the qubit-resonator couplingas functions of equal external fluxes through the small loops ϕext

1 = ϕext2 (solid lines)

for the values of the circuit’s elements shown in Tab. 2.1 and fixed capacitanceratio CJi/Ci = 10−5 without an external flux through the big loop, ϕext

B = 0. Forcomparison, also the results of the circuit’s singular treatment are shown (dashedlines).

Similar to the qubit and resonator frequencies, also the qubit’s and resonator’s relative anhar-monicities are flattened out due to the presence of the Josephson array’s intrinsic capacitances.These relative anharmonicities are shown as functions of the external flux through the smallloops in Fig. 2.19 together with the results obtained within the singular treatment of thecircuit. While the qualitative dependence of the relative qubit anharmonicity on the externalflux through the small loops and the locations of its maximum and minimum are preserved,it becomes less sensitive to ϕext

i , i.e. the difference between the maximal and minimal valuebecomes smaller. However, the absolute value of the relative qubit anharmonicity stays largeenough for the two-level approximation of the qubit to hold. Furthermore, the intrinsic ca-pacitances of the Josephson arrays cause a large decrease of the maximal resonator’s relativeanharmonicity at ϕext

i = 5π (and multiples of 10π), which is beneficial for the resonator’s quality.

We can summarize that for relatively large ratios of CJi/Ci in the order of O(CJ/C) = 10−2...1for the circuit elements’ values shown in Tab. 2.1 the effects of the Josephson arrays intrinsiccapacitances is not significant and the results for the parameters of the effective Hamiltonianmainly resemble the results obtained within the singular treatment of the circuit in which theintrinsic capacitances are not considered.

Naively, one would guess that the singular case in which CJi/Ci = 0 is resembled in the limitof CJi/Ci → 0. However, this idea is wrong due to large quantum fluctuations in the fastf1, f2-subspace, which cause a difference between the effective potential obtained by the Born-Oppenheimer approximation and the effective classical potential obtained within the singularanalysis of the circuit. Although, the results of the singular treatment are approximately re-produced for large values of CJi/Ci, this large capacitance ratio stands in contradiction to the

37


π π π π π-

-

-

-

(a) Relative qubit anharmonicity.

π π π π π-

(b) Relative resonator anharmonicity.

Figure 2.19: Relative anharmonicities of (a) the qubit and (b) the resonator as functions ofequal external fluxes through the small loops ϕext

1 = ϕext2 (solid) for the circuit

elements’ values shown in Tab. 2.1 and fixed capacitance ratio CJi/Ci = 10−5

without an external flux through the big loop, ϕextB = 0. For comparison, also the

results of the singular treatment of the circuit (CJi = 0) are shown (dashed lines).

validity of the Born-Oppenheimer approximation.

Therefore, in the following subsection we will roughly estimate the regime of CJi/Ci in whichthe Born-Oppenheimer approximation is a reasonable approach. For all capacitances ratioslarger than this threshold value, the Born-Oppenheimer approximation breaks down and is notapplicable anymore.

2.3.5 Validity of the Born-Oppenheimer approximation

In this subsection we will roughly determine the regime in which the Born-Oppenheimer ap-proximation breaks down. Therefore, the minimal excitation energy ∆Efast

min in the fast-variablesof all possible fixed pairs of slow variables will be calculated as a function of the capacitanceratio CJi/C. For the Born-Oppenheimer approximation to be valid, we demand this minimalexcitation energy to be far above any one-level excitation energy of the resulting qubit-resonatorsystem such that it is ensured that the fast subspace stays in its ground state. In principle,this analysis has to be done for every possible external flux. However, in Fig. 2.20 we showrepresentative results for zero external fluxes, i.e. ϕext

i = ϕextB = 0.

The minimal fast excitation energy is monotonically decreasing as a function of CJ/C and thecorresponding frequency is equal to the resonator frequency for CJ/C ≈ 4. This value canbe considered to be the capacitance ratio for which the Born-Oppenheimer approximation isdefinitely not applicable anymore. Of course, this is just a rough and not very precise estima-tion. However, for considerably smaller capacitance ratios, e.g. for CJ/C 10−1, the minimalfast excitation energy is already far above any one-level excitation energy of the qubit-resonatorsystem such that the validity of the Born-Oppenheimer approximation is justified. The analysisof different external fluxes gives similar results.

38


-

Figure 2.20: Minimal excitation frequency in the fast subspace (green) together with the fre-quency of qubit (blue) and resonator (red) as functions of the capacitance ratioCJ/C for zero external fluxes, ϕext

i = ϕextB = 0. The minimal excitation frequency

in the fast subspace is smaller than the resonator frequency for approximatelyCJ/C > 4, indicating the definite break-down of the Born-Oppenheimer approxi-mation in this regime.

2.3.6 Effect of non-zero CLi

We want to complete this chapter on the qubit-resonator system with a small remark consid-ering the intrinsic capacitances of the linear inductors. Their additional consideration in thedescription of the circuit can be efficiently incorporated with the partial Cholesky decompo-sition introduced in subsection 2.3.1. For a symmetric choice, i.e. CL1 = CL2, the resultinginverse capacitance matrix in the kinetic term of the effective two dimensional Hamiltonian forthe qubit and resonator variable (defined in Eq. 2.58) is diagonal. For this reason, the firstapproach presented in subesction 2.3.3 for identifying the model Hamiltonian’s parameters isapplicable.

A similar numerical analysis as presented in subsection 2.3.4 has been performed for finite fixedvalues of CLi/C > 0. The most fundamental difference compared to the case CLi = 0 is thateven for CJi = 0 the circuit stays non-singular. As a consequence, the analysis of varyingthe intrinsic capacitances of the Josephson arrays shows that for CJ/C → 0 the consideredparameter (e.g. the qubit frequency) converges a constant value. The actual value dependson the choice of CLi/C and approaches the singular result for larger values of this ratio. Thisresult matches the explanations given in subsection 2.2.3: For CJ/C → 0 the kinetic termdoes not diverge due to CLi > 0, which ensures a localization of the wavefunctions in thefast directions within the Born-Oppenheimer approximation. However, these wavefunctions’eigenenergies are not just the minimal value of the potential in the fast directions but also takeremaining quantum fluctuations into account.

39

3 Quantization of nonreciprocalsuperconducting circuits involvinggyrators

For the practical realization of a solid state quantum computer (independent of the chosenimplementation) nonreciprocal circuit elements play an essential role, especially for integratedcircuits or signal processing. For that matter, nonreciprocal circuit elements, such as e.g. cir-culators [32, 33], often constitute the interface between the quantum and classical descriptionof an electrical network. They enable the read-out of the qubits’ states while simultaneouslyprotecting the quantum system from environmental influences of the circuit’s classical partwhich could affect the quantum system drastically. This is essential for successful quantumcomputation.

In electrical network theory, the gyrator proposed by Tellegen [34] in 1948 is considered to be amore fundamental nonreciprocal circuit element than the circulator. Together with reciprocalcircuit elements only, it allows for the construction of any arbitrary nonreciprocal electricalnetwork [35]. In particular, the Carlin’s construction [36] states that a circulator can be buildout of a single gyrator and additional electric wires. For these reasons, we will focus on thegyrator in the following chapter.

Historically, first gyrators were realized by Hogan in 1952 by means of the Faraday Effect [37,38],whereas nowadays also realizations by parametric modulation [39, 40, and references therein]or by exploiting the Hall effect [36, 41, 42] are possible. The latter realization allows for greatminiaturization. As soon as the actual device size of the gyrator becomes much smaller thanthe wavelengths of interest it can be treated using a lumped-device point of view [36, 43]. Wewill thus provide a treatment using the admittance matrix rather than the scattering matrix.

Treating the gyrator as a lumped element at low energies suggests its possible incorporation inthe theory of circuit quantum electrodynamics. This will be done in this thesis chapter. Afterthe introduction of the classical ideal gyrator we will propose its possible quantization. Thereby,we will see that circuits involving gyrators can easily result in singular Lagrangians. After-wards, several fundamental, exemplary circuits involving a gyrator will be analyzed quantum-mechanically.

41

3. Quantization of nonreciprocal superconducting circuits involving gyrators

3.1 The classical ideal gyrator

The classical ideal gyrator is a four-terminal (two-port) device introduced by Tellegen to com-plete the system of electrical network elements and to capture the description of nonreciprocalphenomena [34]. Its lumped element symbol is shown in Fig. 3.1.

G

V1 V2

I1 I2

Figure 3.1: Lumped element symbol for the classical ideal gyrator proposed by Tellegen. Thevoltages and currents are related via Eq. (3.1).

It relates the currents on one port to the voltages on the other one linearly according to thedefinition [35]1 (

I1

I2

)=

(0 −GG 0

)︸︷︷︸

Y

(V1

V2

), (3.1)

with the constant, real valued gyration conductance G. Per definition, it is ensured thatI1V1 + I2V2 = 0 such that energy is neither stored nor dissipated by the gyrator. Furthermore,the anti-symmetry of the admittance matrix Y violates the reciprocity relation [35] such thatthe gyrator is indeed nonreciprocal.Already Tellegen made the observations that

• a classical ideal gyrator terminated purely inductively with inductance L on the rightport is equivalent to a capacitance C ′ = LG2,

• a classical ideal gyrator terminated purely capacitively with capacitance C on the rightport is equivalent to a an inductance L′ = C/G2,

• two cascaded classical ideal gyrators with gyration conductances G1 and G2 are equivalentto an ideal transformer with turns-ratio N = G2/G1,

which can be easily proven considering the gyrator’s admittance matrix. In the following wewill analyze the ideal gyrator within the Lagrangian as well as the Hamiltonian formalism,which immediate leads to a quantized description.

1Note that the definitions of the ideal Gyrator given in [34] and [35] differ by a simple replacement of G 7→ −G.

42

3.2. The gyrator in the Lagrangian and Hamiltonian formalism

3.2 The gyrator in the Lagrangian and Hamiltonianformalism

After the introduction to the ideal gyrator as a classical circuit element, we want to include gy-rators in electrical circuits which will be quantized. Given the fact that the gyrator is lossless, itis plausible that there should be a Lagrangian as well as Hamiltonian description of the gyrator.In fact, although the Lagrangian description of the gyrator is known since at least 1959 [44,45],up to our knowledge circuits including gyrators have not been treated quantum-mechanically,yet. Therefore, in the following we will introduce the Lagrangian description of the gyrator aswell as present a generalized Legendre transformation based on Dirac’s approach of quantizingsingular Lagrangians [21]. This is done in order to obtain the corresponding Hamiltonian de-scription of any circuit including arbitrary many gyrators. This Hamiltonian will be used forthe quantization of the circuit.At this moment, we just mention that a generalization of the Legendre transformation is neces-sary since the Lagrangians of circuits including gyrators can easily become singular. Exampleswill follow in the subsequent sections.

3.2.1 Lagrangian formalism of the gyrator

Considering the ideal gyrator as an element of a lumped circuit, within the Lagrangian formal-ism it is described by

Lgyr =G

2(φ1φ2 − φ1φ2), (3.2)

in which the fluxes φ1, φ2 denote the time-integrated voltages on the left and right port of thegyrator, respectively, see Fig. 3.2. Note that the gyrator’s nonreciprocity is manifested in theanti-symmetry by interchanging the indizes 1 and 2. It can be proven that Eq. (3.2) indeeddescribes an ideal gyrator by considering two general electrical networks with Lagrangians L1

and L2 which are coupled by a gyrator, see Fig. 3.2.

G

L1 L2

φ1 φ2

Figure 3.2: Two general electrical networks with Lagrangians L1,L2 which are coupled by agyrator. The lower terminals of the gyrator are commonly grounded.

In this configuration, the total Lagrangian reads

L = L1 + L2 + Lgyr. (3.3)

In particular, the networks’ Lagrangians Li are not just functions of φi and φi but, in general,also of some additional internal degrees of freedom, which are, however, of no further importancefor the proof. First of all, we note that the current passing through the electrical network withLagrangian Li is given by

I ′i =d

dt

(∂Li∂φi

)− ∂Li∂φi

= −Ii, (3.4)

43


and that its biasing voltage reads

Vi = φi. (3.5)

From that, we can conclude that the Euler-Lagrange equation of the total Lagrangian Eq. (3.3)with respect to the φ1-variable yields

d

dt

(∂L∂φ1

)− ∂L∂φ1

= 0,

⇔ d

dt

(∂L1

∂φ1

)− ∂L1

∂φ1

= − d

dt

(∂Lgyr

∂φ1

)+∂Lgyr

∂φ1

,

⇔ I1 = −GV2.

(3.6)

A similar evaluation of the Euler-Lagrange equation for φ2 gives

I2 = GV1, (3.7)

such that both Euler-Lagrange equation together directly recover the admittance matrix char-acterizing the classical ideal gyrator, see Eq. (3.1). Therefore, it is proven that the Lagrangianstated in Eq. (3.2) indeed describes an ideal gyrator.

After the introduction of the gyrator’s Lagrangian description we want to point out that theLagrangian formalism allows to add a total time derivative to the Lagrangian without affectingthe classical equations of motion [46]. Therefore, it is instructive to consider a more generalLagrangian describing the ideal gyrator obtained by the transformation

Lgyr → Lgyr + ad

dt

(G

2φ1φ2

)=G

2

[(1 + a)φ1φ2 − (1− a)φ1φ2

], (3.8)

with an arbitrary constant a ∈ R. Since all terms of Eq. (3.8) are linear in one single φi,the Lagrangian Lgyr can be written and interpreted as Lagrangian contribution of a ’magnetic’vector potential

Lgyr = A(φ) · φ, A(φ) =G

2

((a− 1)φ2

(a+ 1)φ1

). (3.9)

The two-dimensional vector potential A(φ) gives rise to a resulting magnetic field analog whichis orthogonal to the φ1, φ2 -plane and given by the curl of the vector potential

(∇×A(φ))3 =∂A2(φ)

∂φ1

− ∂A1(φ)

∂φ2

= G, (3.10)

which is the gyration conductance. Given this analogy between the gyration conductance andan uniform magnetic field, the transformation considered in Eq. (3.8) is equivalent to a gaugetransformation of the vector potential and the special cases of a = ±1 and a = 0 will bedenoted as Landau gauge and symmetric gauge, respectively. In fact, adding the total timederivative of any function χ(φ) to Lgyr is equivalent to adding ∇χ(φ) to the vector potential,which does not affect its curl, since ∇ ×∇χ(φ) = 0. Furthermore, the nonreciprocity of thepassive gyrator is consistent with the interpretation of the gyrator to act like an magnetic field,which breaks the time-reversal symmetry and does not store any energy.

44


3.2.2 Hamiltonian formalism of the gyrator

The Lagrangian contribution of the gyrator is the first step towards a quantized theoreticaldescription of any electric circuit containing arbitrary many gyrators. Since we want to imposethe quantization in the Hamiltonian formalism, we need to derive the corresponding Hamilto-nian, first. The Lagrangian of a general electric network with m + 1 nodes which is built outof capacitances, any sort of inductances (linear, non-linear and mutual) and gyrators can bewritten in matrix notation as

L =1

2φTCφ+ φTAφ− U(φ). (3.11)

Due to Kirchhoff’s voltage law, the electric network is fully described by at most m independentvariables. In particular, we allow the Lagrangian to be singular. In subsection 3.2.3 we willshow in general how to construct a basis in which the capacitance matrix C is block diagonaland reads

C =

(C′ 0n×k

0k×n 0k×k

), C′ = C′T , C′ > 0, (3.12)

i.e. its trivial action on the k-dimensional kernel is separated from the remaining n-dimensionalsubspace satisfying the rank-nullity theorem n = m− k. Note that the upper left block capaci-tance matrix C′ is symmetric and positive definite and therefore invertible. Simultaneously, inthis basis the vector potential matrix A is block upper triangular with block matrices of samesize and arrangement as in C and reads

A =

(A′ A

0k×n Λ′′

), A′ = −A′T . (3.13a)

While the upper left diagonal block vector potential matrixA′ is determined only up to a gaugeand can be chosen to be anti-symmetric, the upper right block matrix A does not have such agauge freedom because our basis is chosen such that the lower right diagonal block matrix Λ′′

is of the form

Λ′′ =

0l×l diag(λ1, λ2, . . . , λl) 0l×j0l×l 0l×l 0l×j0j×l 0j×l 0j×j

, λ1, λ2, . . . , λl > 0, (3.13b)

with 0 ≤ l ≤ k/2 for even k and 0 ≤ l ≤ (k − 1)/2 for odd k as well as j = k − 2l. Note thatour choice of the basis uniquely determines l and that we can conclude that j ≥ 0 for even kand j ≥ 1 for odd k. We will elaborate on that later.

The Lagrangian Eq. (3.11), written in the basis in which its capacitance matrix and vectorpotential matrix are of the form Eq. (3.12) and Eq. (3.13), respectively, is the starting pointfor the derivation of the corresponding Hamiltonian applying a straightforwardly generalizedLegendre transformation.

As usual, the conjugate charges are defined as Qi = ∂L/∂φi. However, due to the k-dimensionalkernel of the capacitance matrix C, we will see that we have to treat the first n = m− k andthe last k fluxes and conjugate charged separately. Therefore we introduce the notation

φ =

(φ′

φ′′

), (3.14)

45


in which the single and double prime compactly denote the separation φ′ = (φ1, φ2, . . . , φn)T

and φ′′ = (φn+1, φn+2, . . . , φn+k)T . Likewise, we proceed with the separation of Q. Evaluating

the first n conjugate charges, we find

Q′ = C ′φ′ +A′φ′ + Aφ′′

⇒ φ′ = C ′−1(Q′ −A′φ′ − Aφ′′),(3.15)

due to the symmetry of C′ and its invertibility. Therefore, the time derivatives of the first nfluxes can be expressed as functions of fluxes and momenta only, such that an ordinary Legendretransformation is practicable for these variables. In contrast to this result, the last k conjugatecharges evaluate inherently differently. We find, that the last k conjugate charges evaluate to

Q′′ = Λ′′φ′′, (3.16)

which does not contain a time derivative of any flux at all. In order to proceed, it will be conve-nient to further separate φ′′ into φ′′a = (φn+1, φn+2, . . . , φn+l)

T , φ′′b = (φn+l+1, φn+l+2, . . . , φn+2l)T

and φ′′c = (φn+2l+1, φn+2l+2, . . . , φn+2l+j)T , such that we can write

φ′′ =

φ′′aφ′′bφ′′c

. (3.17)

Again, similarly for Q′′. Considering the explicit form of Λ′′ (see Eq. (3.13b)), using thissubdivision yields

Q′′a = diag(λ1, λ2, . . . , λl)φ′′b , (3.18a)

Q′′b = 0l×1, (3.18b)

Q′′c = 0j×1, (3.18c)

which will be successively discussed in the following.Starting with Eq. (3.18b), we determine the conjugate charges Q′′b to vanish. Next, Eq. (3.18a)can be solved for the corresponding fluxes

φ′′b = diag(λ−11 , λ−1

2 , . . . , λ−1l )Q′′a (3.19)

as functions of the Q′′a only (recall that λi > 0 for all i = 1, 2, . . . , l). Finally, Eq. (3.18c)states that also the conjugate charges Q′′c vanish. However, at this moment we do not possessexpressions for the corresponding fluxes φ′′c as functions of all the other fluxes and conjugatecharges. For this reason, we have to solve the simplified Euler-Lagrange equations

0 =∂L

∂φn+2l+i

=∂U(φ)

∂φn+2l+i

, i = 1, 2, . . . , j, (3.20)

in order to find the constraintsφ′′c ≡ φ′′c (φ′,φ′′a,φ′′b ), (3.21)

which reduce the number of degrees of freedom in the Lagrangian by j.

46


If it is not possible to derive constrains in the form of Eq. (3.21) from Eq. (3.20), one hasto look for constraints of different form or additional constraints such that φ′′c vanishes in theLagrangian, otherwise the underlying system is physically not well described.2 Note that con-straints of different form than Eq. (3.21) can lead to a further reduction of the number ofdegrees of freedom, especially when parts of φ′′c do not enter the potential. This will be shownexemplarily in the appendix 6.4 for Tellegen’s construction of a transformer. In the followingwe assume that we can solve for constrains in the form of Eq. (3.21).

According to Dirac [21, p.36ff], a well-defined quantum theory can now be established byintroducing the Hamiltonian

H = φTQ− L

= φTQ− 1

2φTCφ− φTAφ+ U(φ)

= φ′TQ′ + φ′′TQ′′ − 1

2φ′TC ′φ′ − φ′TA′φ′ − φ′T Aφ′′ − φ′′TΛ′′φ′′ + U(φ)

= φ′T(Q′ −A′φ′ − Aφ′′ − 1

2C ′φ′

)+ φ′′TQ′′ − φ′′TΛ′′φ′′ + U(φ)

=1

2φ′TC ′φ′ + U(φ)

=1

2(Q′ −A′φ′ − Aφ′′)TC ′−1(Q′ −A′φ′ − Aφ′′) + U

((φ′T ,φ′′T

)T),

(3.22)

where for φ′′b we have to substitute the expression derived in Eq. (3.19). Therefore, we introduce

the decomposition A = (Aa,Ab,Ac) and the abbreviation d = diag(λ−11 , λ−1

2 , . . . , λ−1l ), such

that the final Hamiltonian reads

H =1

2(Q′ −A′φ′ −Aaφ

′′a −AbdQ

′′a −Acφ

′′c )TC ′−1(Q′ −A′φ′ −Aaφ

′′a −AbdQ

′′a −Acφ

′′c )

+ U((φ′T ,φ′′a

T, (dQ′′a)

T,φ′′c

T )T).

(3.23)

Note that, due to the substitution, the number of degrees of freedom is further reduced byl and recall that the φ′′c are not independent variables with their own dynamics but ratherfunctions of φ′,φ′′a and Q′′a. The Hamiltonian can now be quantized by imposing the canonicalcommutation relations [φµ, Qν ] = i~δµν , for µ, ν = 1, 2, . . . , n+ l. By considering Eq. (3.23), weobserve that the fluxes and conjugate charges appearing in the single terms of the Hamiltonianare not easily separable anymore. In particular, the potential depends on the conjugate chargesQ′′a, such that they can appear in the Hamiltonian with higher powers than two.

In the following subsection, we will show how to construct the basis, such that C and A are ofthe same form as in Eq. (3.12) and Eq. (3.13), respectively. We will see that C can be chosento be diagonal and that A′ still possesses a gauge freedom.

2In Dirac’s words: ’If we cannot do it, then we are out of luck and we cannot make an accurate quantumtheory.’ [21, p.35]

47


3.2.3 Construction of an appropriate basis for the Legendretransformation

As already stated in subsection 3.2.2, the Lagrangian of a general electric network build outof capacitances, any sort of inductances (linear, non-linear and mutual) and gyrators can bewritten in the form

L =1

2φTCφ+ φTAφ− U(φ). (3.24)

Since every single capacitor’s contribution to the Lagrangian is a positive quadratic form, thecapacitance matrix C is positive semi-definite (C ≥ 0) and can be chosen to be symmetric(C = CT ) in any initial basis. Similarly, every single gyrator’s contribution to the Lagrangiancan be written in symmetric gauge (see subsection 3.2.1), such that the resulting vector potentialmatrix A is anti-symmetric (A = −AT ). The potential U(φ) takes all the inductances of thenetwork into account.Starting with those real-valued matrices C and A, we first want to bring C into the form ofEq. (3.12), i.e. we want to decompose it into an invertible part and a trivial part.

Transformation of the capacitance matrix

Since the capacitance matrix is symmetric, due to the spectral theorem its eigenvectors canbe constructed to form a complete orthonormal basis. Therefore, there exists an orthogonalmatrix B (BT = B−1) such that

BTCB =

(C ′ 0n×k

0k×n 0k×k

), (3.25)

with a symmetric, positive definite matrix C ′ ∈ Rn×n (C ′ = C ′T ,C ′ > 0). Therefore thedimensions of the block matrices satisfy the rank-nullity theorem m = n+ k with n = rank(C)and k = dim(ker(C)), i.e. k is the total number of zero-eigenvalues of C. Note that Bcan be constructed by taking its first n columns to be normalized linear independent linearcombinations of the n eigenvectors of C with eigenvalues greater than zero, while its last kcolumns are normalized linear independent linear combinations of the k eigenvectors of C withzero-eigenvalue. The most convenient choice ofB would be to take the bare n eigenvectors of Cwith eigenvalues greater than zero to be the first n columns ofB, such thatC ′ becomes diagonalwith strictly positive diagonal-elements only. However, we will proceed with the general casein which C ′ is not diagonal. Nevertheless, C ′ is positive definite and therefore invertible. Wetake B to define the variable transformation

φ = BTφ, C = BTCB, A = BTAB, U(φ) = U(Bφ), (3.26)

and note that C is in the desired form of Eq. (3.25) and that A is anti-symmetric. In thefollowing we will omit the hats for reasons of clarity. By doing so, the transformed Lagrangianstays in form of Eq. (3.24).

We proceed with a transformation of A, such that it can be easily brought into the form of Eq.(3.13) afterwards.

48


Transformation of the vector potential matrix

Since the vector potential matrix is anti-symmetric, it can be written as

A =

(A′ 1

2A12

−12AT

12 A′′

), (3.27)

in which also the diagonal submatrices A′ ∈ Rn×n,A′′ ∈ Rk×k are anti-symmetric while theoff-diagonal block matrix A12 ∈ Rn×k is not further specified. Since A′′ is anti-symmetricand real, there exists an orthogonal matrix D′′ (D′′T = D′′−1) which transforms A′′ into theYoula-like form [47]

D′′TA′′D′′ =1

2

(Λ′′ −Λ′′T

), (3.28)

with Λ′′ defined in Eq. (3.13b). Note that D′′ can be constructed by taking its columns tobe the orthonormal eigenvectors of A′′2. The right hand side of Eq. (3.28) differs from theconventional Youla normal form by a permutation of the columns and rows (preserving theorthogonality of the transformation matrix D′′). This permutation has been performed forreasons of convenience in the subsequent derivation of the Hamiltonian, see subsection 3.2.2.Note that the non-zero eigenvalues of A′′ are given by ± i

2λ1,± i

2λ2, . . . ,± i

2λl. Therefore, we

can conclude that k = 2l + j, where j denotes the total number of zero-eigenvalues of A′′.Especially for odd k, it follows directly that j ≥ 1. We take D′′ to be part of the orthogonalmatrix

D =

(1n×n 0n×k0k×n D′′

), (3.29)

defining the variable transformation

φ = DTφ, C = DTCD, A = DTAD, U(φ) = U(Dφ). (3.30)

It is important, that this transformation leaves the capacitance matrix unaffected, i.e. C = C.Furthermore, by defining A = A12D

′′ we evaluate the transformed vector potential matrix to

A =

(A′ 1

2A12D

′′

−12D′′TAT

12 D′′TA′′D′′

)=

(A′ 1

2A

−12AT 1

2

(Λ′′ −Λ′′T

) ) , (3.31)

i.e. its diagonal submatrices are unchanged and of the desired Youla-like form, respectively.Again, we will omit the hats in the further calculations such that the transformed Lagrangianstays in form of Eq. (3.24).

In a last step, we perform a gauge transformation such that A finally obtains the desired formof Eq. (3.13).

49


Gauge transformation

Making use of the vector potential matricis gauge freedom, we can add a total time derivativeto the Lagrangian without affecting the classical equations of motion [46],

L = L+d

dt

(φTFφ

)= L+

d

dt

m∑i,j=1

φiFijφj

= L+m∑

i,j=1

(φiFijφj + φiFijφj

)= L+

m∑i,j=1

(φiFijφj + φjF

Tjiφi

)= L+ φT (F + F T )φ,

(3.32)

with a time-independent gauge matrix F ∈ Rm×m. Its appearance in the Lagrangian can beeliminated by the gauge transformation

A = A+(F + F T

). (3.33)

Specifying the gauge matrix F to read

F =

(χ′ 1

2A

0k×n12Λ′′

), (3.34)

with any arbitrary time-independent χ′ ∈ Rn×n, the gauge transformation Eq. (3.33) trans-forms the vector potential matrix to

A =

(A′ A

0k×n Λ′′

), A′ = A′ + χ′ + χ′T , (3.35)

which is exactly the desired form of Eq. (3.13). Furthermore, we observe that the upper leftdiagonal block matrix of the vector potential matrix is indeed just defined up to a gauge, whichstill can been arbitrarily chosen. Finally, omitting the hats again, the Lagrangian can still bewritten in form of Eq. (3.24), in which the capacitance matrix as well as the vector potentialmatrix are in the concrete form supposed in subsection 3.2.2.

After the derivation of the Hamilton of any general electric network including arbitrary manygyrators, we will specifically analyze simple, exemplary circuits with one gyrator only. Withinthis analysis, we will investigate the possible boundary conditions associated with a givenelectrical circuit and explore the singular description of a circuit.

50

3.3. Capacitances coupled by a gyrator, the C-G-C circuit

3.3 Capacitances coupled by a gyrator, the C-G-C circuit

Starting with a simple electric circuit containing a gyrator, we consider a gyrator where bothports are terminated by equal capacitances, see Fig 3.3. This circuit will be called C-G-Ccircuit.

Gφ1 φ2

C C

Figure 3.3: The C-G-C circuit. It consists of two equal capacitances coupled by a gyrator.

The classical treatment of the gyrator using the admittance matrix suggests that the C-G-Ccircuit should be equivalent to a LC-resonator with an effective inductance L′ = C/G2 (seesection 3.1), therefore featuring the eigenfrequency ωL′C = 1/

√L′C = G/C. The Lagrangian

of the C-G-C circuit (symmetric gauge)

L =Cφ2

1

2+Cφ2

2

2+G

2(φ1φ2 − φ1φ2), (3.36)

is equivalent to the Lagrangian of a charged particle confined to two dimensions while beingexposed to a perpendicular, homogeneous magnetic field. The analog to the resulting cyclotronfrequency is given by ωG = G/C and will be called gyration frequency. At this point we al-ready notice the equality of the two frequencies, ωL′C = ωG, confirming the agreement of theapproach invoking the admittance matrix with the Lagrangian description. From that analogyto a charged particle moving in a magnetic field it is clear that the classical trajectories fulfillingthe classical equations of motions, i.e. the Euler-Lagrange equations, are circles in φ1-φ2-space,which are passed through with the frequency ωG.

In order to obtain a quantized theory of the C-G-C circuit, we perform the Legendre transfor-mation to derive the Hamiltonian description of the circuit. The C-G-C circuit’s LagrangianEq. (3.36) is non-singular, so the conjugate charges Qi = ∂L/∂φi evaluate to

Q1 = Cφ1 −G

2φ2, Q2 = Cφ2 +

G

2φ1, (3.37)

and can be solved for the time derivatives of the fluxes

φ1 =Q1 +Gφ2/2

C, φ2 =

Q2 −Gφ1/2

C. (3.38a)

The straightforward, ordinary Legendre transformation H = Q1φ1 +Q2φ2−L gives rise to theHamiltonian

H =(Q1 + 1

2Gφ2)2

2C+

(Q2 − 12Gφ1)2

2C, (3.39)

which is quantized by imposing the canonical commutation relations [φi, Qj] = i~δij. Its energyspectrum is that of a quantum harmonic oscillator with frequency ωG, i.e. its eigenenergies are

En = ~ωG(n+

1

2

), n ∈ N0, (3.40)

51


in which the quantum number n labels the Landau levels [24, 48, 49]. The derivation of thespectrum will be delayed to section 3.5. However, we want to note that the choice of possibleboundary condition is a subtle topic and will be discussed in section 3.7.

52

3.4. Inductances coupled by a gyrator, the L-G-L circuit

3.4 Inductances coupled by a gyrator, the L-G-L circuit

After the analysis of the C-G-C circuit, we consider the related circuit in which a gyrator isshunted equally inductively on both ports, see Fig. 3.4. We call this circuit to be the L-G-Lcircuit.

Gφ1 φ2

LL

Figure 3.4: The L-G-L circuit. It consists of two equal inductances coupled by a gyrator.

Similar to the C-G-C circuit, the classical treatment of the gyrator involving the admittancematrix suggests that also the L-G-L circuit should be equivalent to a LC-resonator, in this casewith an effective capacitance C ′ = LG2 (see section 3.1), therefore featuring the eigenfrequencyωLC′ = 1/

√LC ′ = 1/GL. However, the Lagrangian of the L-G-L circuit (symmetric gauge)

L =G

2(φ1φ2 − φ1φ2)− φ2

1

2L− φ2

2

2L, (3.41)

is conceptually different from that of the C-G-C circuit, since it does not have a descriptiveanalogy with the Lagrangian of a charged particle exposed to a magnetic field. This is due tothe missing capacitances in the L-G-L circuit, from which the Lagrangian results to be singular.Nevertheless, the Euler-Lagrange equations applied to Eq. (3.41) yield the classical equationsof motion

0 = −Gφ2 +φ1

L, 0 = Gφ1 +

φ2

L, (3.42)

which can be used to derive the differential equation of a harmonic oscillator with frequencyωLC′ for φ1,

0 = G2L2φ1 + φ1. (3.43)

Its solution reads φ1(t) = A sin(ωLC′t) +B cos(ωLC′), with some parameters A,B to match theinitial conditions of φ1. Due to Eq. (3.42), the time evolution of φ2 is directly determined oncewe fixed the solution for φ1. Obviously the system underlies some constraints such that φ2 doesnot have its own dynamics and is therefore not an independent variable as presupposed for theLegendre transformation.

The failure of the ordinary Legendre transformation becomes clear if we try to derive theHamiltonian description of the L-G-L circuit in a naive way. By evaluating the conjugatemomenta Qi = ∂L/∂φi, we immediately note that it is not possible to solve

Q1 = −G2φ2, Q2 = +

G

2φ1, (3.44)

for the φi, because they do not appear. Therefore, the Lagrangian Eq. (3.41) of the L-G-Lcircuit is clearly singular. Although - neglecting the singularity of the Lagrangian for this

53


moment - the naively introduced, wrong Hamiltonian

Hwrong = Q1φ1 +Q2φ2 − L =φ2

1

2L+

Q21

G2L/2(3.45)

indeed describes a LC-resonator, the resulting eigenfrequency ωwrong = 2/GL differs from thepreviously expected eigenfrequency ωLC′ by a factor 2. This discrepancy could be removed bymodifying the classical Poisson brackets to reflect the constraints of the singular Lagrangian [21].

Instead, we will apply the results of subsection 3.2.2 to the L-G-L circuit and correctly derivethe corresponding Hamiltonian in the following. As it turns out, the universally described pro-cedure simplifies considerably as soon as a concrete electrical circuit is given.

In the Landau gauge for the gyrator’s vector potential, the Lagrangian of the L-C-L circuit

L = Gφ1φ2 −φ2

1

2L− φ2

2

2L(3.46)

is already in the desired form of Eqs. (3.12), (3.13). Note that the capacitance matrix vanishescompletely, such that m = k = 2. Furthermore, we do not have to look for further constraintssince j = 0. The Lagrangian Eq. (3.46) gives rise to the conjugate charges

Q1 = 0, Q2 = Gφ1, (3.47)

i.e. Q1 vanishes and the corresponding flux φ1 can be written as a function ofQ2 only. Therefore,we can write down the Hamiltonian

H = Q1φ1 +Q2φ2 − L =φ2

1

2L+φ2

2

2L=

Q22

2G2L+φ2

2

2L(3.48)

in which we have to discard the first degree of freedom by making the replacement φ1 =Q2/G. The Hamiltonian Eq. (3.48) is quantized by imposing the canonical commutationrelation [φ2, Q2] = i~, such that it indeed describes a harmonic oscillator with the expectedeigenfrequency ωLC′ = 1/GL.

54

3.5. LC-Resonators coupled by a gyrator, the LC-G-LC circuit

3.5 LC-Resonators coupled by a gyrator, the LC-G-LC circuit

With the aim of analyzing the physical background of the singularity of the L-G-L circuit’s La-grangian we expand the description of the system by considering a gyrator which is terminatedcapacitively and inductively on both ports. This setup results in two identical LC-resonatorswhich are coupled by a gyrator and will be called LC-G-LC circuit. It is shown in Fig. 3.5. Sim-ilar to chapter 2, the capacitances can be considered to be the inductor’s intrinsic capacitancesand are therefore small.

Gφ1 φ2

LLC C

Figure 3.5: The LC-G-CL circuit. It consists of two identical LC-resonators coupled by a gy-rator.

The LC-G-LC circuit’s Lagrangian in arbitrary gauge,

L =Cφ2

1

2+Cφ2

2

2+ A1(φ1, φ2)φ1 + A2(φ1, φ2)φ2 −

φ21

2L− φ2

2

2L, (3.49)

is non-singular and therefore the corresponding Hamiltonian

H =

(Q1 − A1(φ1, φ2)

)2

2C+

(Q2 − A2(φ1, φ2)

)2

2C+φ2

1

2L+φ2

2

2L, (3.50)

is obtained straightforwardly by an ordinary Legendre transformation. Recall that the vectorpotential has to satisfy Eq. (3.10) to describe the gyrator. Furthermore, due to the capacitancesthe Hamiltonian of the LC-G-LC circuit is equivalent to the Hamiltonian of a paraboloidallyconfined charged particle in a uniform magnetic field. As usual, the Hamiltonian Eq. (3.50) isquantized by imposing the canonical commutation relations [φi, Qj] = i~δij.

In the following we choose the symmetric gauge and derive the Hamiltonian’s spectrum. Insymmetric gauge, the LC-G-LC circuit’s Hamiltonian (3.50) can be written as a sum of twoharmonic oscillators with equal eigenfrequencies and a coupling term which mixes the fluxesand conjugate charges of the two individual harmonic oscillators,

H =2∑i=1

(Q2i

2C+

1

2CΩ2φ2

i

)+

1

2ωG (Q2φ1 −Q1φ2) (3.51)

with the frequencies

ωG =G

C, ωLC =

1√LC

, Ω =

√ω2LC +

ω2G

4. (3.52)

We observe that the eigenfrequency Ω of the harmonic oscillators does not match the bareeigenfrequency ωLC of the LC circuit but is shifted due to the coupling via the gyrator. We

55


introduce the ladder operators

a =1√2~

(√CΩφ1 +

i√CΩ

Q1

), b =

1√2~

(√CΩφ2 +

i√CΩ

Q2

), (3.53)

satisfying the bosonic commutation relations [a, a†] = [b, b†] = 1 while all other commutatorsvanish. We can write the Hamiltonian Eq. (3.51) as

H = ~Ωa†a+ ~Ωb†b− i~ωG2

(a†b− b†a

)+ ~Ω. (3.54)

The first two terms describe the individual harmonic oscillators while the third term takes theircoupling into account. The zero-point energies of the harmonic oscillators are collected into thelast term and represent a simple shift of the spectrum. Hamiltonians in form of Eq. (3.54) areso-called normal Hamiltonians [50] and can be rewritten in the matrix form

H =(a† b†

)( ~Ω − i~ωG

2i~ωG

2~Ω

)︸︷︷︸

α

(ab

)+ ~Ω, (3.55)

with the Hermitian coefficient matrix α. This normal Hamiltonian can be diagonalized by theBogoliubov-Valatin transformation matrix, which is given by the unitary matrix U diagonal-izing the coefficient matrix α [50]. Due to the structure of the coefficient matrix, it is clearthat the columns of U are the normalized eigenvectors of the Pauli σy-matrix, such that wedetermine

U =1√2

(1 1i −i

), U †αU =

(~ω+ 0

0 ~ω−

), ~ω± = ~Ω± ~ωG

2. (3.56)

Introducing a new set of annihilation operators (and creation operators, accordingly) with theBogoliubov-Valatin transformation (

cd

)= U †

(ab

)(3.57)

ensures the preservation of the bosonic commutation relations [50] and directly diagonalizes theHamiltonian Eq. (3.55) into

H =(c† d†

)(~ω+ 00 ~ω−

)(cd

)+ ~Ω

= ~ω+c†c+ ~ω−d†d+ ~Ω.

(3.58)

This diagonalized Hamiltonian describes two uncoupled harmonic oscillators with eigenfrequen-cies ω+ and ω−, respectively. Their mean is given by Ω while their difference is given by thegyration frequency ωG. The diagonalized Hamiltonian’s eigenstates |n,m〉 = |n〉c ⊗ |m〉d areproduct states of the c(†) and d(†) operators’ Fock states, respectively, and determine the LC-G-LC circuit’s spectrum to read

En,m = ~ω+n+ ~ω−m+ ~Ω, m, n ∈ N0. (3.59)

56

3.5. LC-Resonators coupled by a gyrator, the LC-G-LC circuit

Before we pursue a comparison of the CL-G-CL circuit with the circuits previously analyzed insections 3.3 and 3.4, we will determine the normalmodes of the CL-G-CL circuit as functionsof the initial fluxes and conjugate charges. Therefore, we define

M1 =

√~2

(c† + c

), N1 = i

√~2

(c† − c

), (3.60a)

M2 =

√~2

(d† + d

), N2 = i

√~2

(d† − d

), (3.60b)

where the Mi and Ni are generalized positions and momenta, respectively, satisfying the canon-ical commutation relations [Mi, Nj] = i~δij, such that the Hamiltonian Eq. (3.58) evaluatesto

H =ω+

2(M2

1 +N21 ) +

ω−2

(M22 +N2

2 ). (3.61)

Next, we use the Bogoliubov-Valatin transformation Eq. (3.57) as well as the ladder operatorsgiven in Eq. (3.53) to reverse engineer the normalmodes as linear combinations of the initialflux and conjugate charge variables,

M1

M2

N1

N2

=1√2

√CΩ 0 0 1/

√CΩ√

CΩ 0 0 −1/√CΩ

0 −√CΩ 1/

√CΩ 0

0√CΩ 1/

√CΩ 0

︸︷︷︸

J

φ1

φ2

Q1

Q2

, (3.62)

in which the Jacobian J relates the new variable set with the old one. Recall that the first(last) two entries of the vectors appearing in Eq. (3.62) correspond to generalized position(momentum) variables. At this point, we want to emphasize that the variable transformationEq. (3.62) is canonical, i.e. the Hamiltonian equations of motion are preserved, since theJacobian is symplectic,

J

(02×2 12×2

−12×2 02×2

)JT =

(02×2 12×2

−12×2 02×2

). (3.63)

Furthermore, we have determined the Jacobian J relating the initial generalized positions andmomenta with the Hamiltonian’s normalmodes by making a detour to quantum mechanics andperforming the Bogoliubov-Valatin transformation on the ladder operators associated with thevariables. However, we want to emphasize that the Hamiltonian Eq. (3.51) is quadratic andis therefore diagonalizable within a purely classical treatment [51]. The Jacobian obtained inthis way coincides with our results.

By inverting the Jacobian J , Eq. (3.62) yields to an expression of the initial fluxes and conjugatecharges in terms of linear combinations of the normal modes,

φ1

φ2

Q1

Q2

=1√2

1/√CΩ 1/

√CΩ 0 0

0 0 −1/√CΩ 1/

√CΩ

0 0√CΩ

√CΩ√

CΩ −√CΩ 0 0

M1

M2

N1

N2

, (3.64)

57


which shows that the classical solutions φi(t) are superpositions of two harmonic oscillationswith frequencies ω+ and ω−, respectively.

Finally, we want to compare the LC-G-LC circuit with the L-G-L circuit, which should berecovered in the limit of vanishing capacitances. For this reason, we identify the ratio of thebare LC-resonator’s eigenfrequency ωLC and the gyration frequency ωG, or, equivalently, theratio of the LC-resonator’s specific conductance GLC =

√C/L and the gyration conductance

G to be a small, dimensionless expansion parameter

ε =ωLCωG

=

√C

L

1

G=GLC

G 1. (3.65)

Furthermore, this parameter ε is also equivalent to the square-root of the ratio of the capacitanceC and the effective capacitance C ′ = LG2 obtained by shunting the gyrator purely inductively,see section 3.1. In the limit of Eq. (3.65), the eigenfrequencies of the LC-G-LC resonator (Eq.(3.56)) can be expanded in ε up to leading order, respectively, and evaluate to

ω+ = ωG[1 +O(ε2)

]≈ G

C, (3.66a)

ω− = ωG[ε2 +O(ε4)

]≈ 1

LG, (3.66b)

such that they exactly coincide with the eigenfrequencies of the C-G-C circuit and the L-G-Lcircuit, respectively. For this reason, the C-G-C circuit and the L-G-L circuit can be interpretedas effective high- and low-frequency projections, respectively, of the more elaborate LC-G-LCcircuit. The interpretation of the singular Lagrangian to be the low-energy projection of aphysically more detailed system in the limit of vanishing capacitances is in agreement with theresults of section 2.2.

We want to complete the section on the LC-G-LC circuit with the previously announced deriva-tion of the C-G-C circuit’s spectrum. Therefore, we formally set L =∞ which results in ωLC = 0and notice that the both circuits’ Hamiltonians coincide. This results in ω+ = ωG, ω− = 0 andΩ = ωG/2 such that the LC-G-LC circuit’s spectrum Eq. (3.59) reduces to the C-G-C circuit’sspectrum Eq. (3.40). A change of the quantum number m in Eq. (3.59) does not affect theenergy, indicating the degeneracy of the Landau levels.

58

3.6. Josephson junctions coupled by a gyrator, the JJ-G-JJ circuit

3.6 Josephson junctions coupled by a gyrator, the JJ-G-JJcircuit

In this section we will replace the linear inductances considered in section 3.4 by equal Josephsonjunctions without any intrinsic capacitances. This replacement results in the circuit shown inFig. 3.6 and will be called JJ-G-JJ circuit, accordingly.

Gφ1 φ2

EJ EJ

Figure 3.6: The JJ-G-JJ circuit. It consists of two equal Josephson junctions coupled by agyrator.

Similar to the L-G-L circuit, the JJ-G-JJ circuit’s Lagrangian is singular with a vanishingcapacitance matrix. Therefore, we directly determine the Lagrangian in Landau gauge,

L = Gφ1φ2 + EJ cos

(2π

Φ0

φ1

)+ EJ cos

(2π

Φ0

φ2

), (3.67)

such that the conjugate charges Qi = ∂L/∂φi evaluate to

Q1 = 0, Q2 = Gφ1. (3.68)

Again, since the conjugate charge Q1 vanishes and φ1 can be written as a function of φ2 andQ2 only, we can derive the JJ-G-JJ circuit’s Hamiltonian

H = −EJ cos

(2π

Φ0

Q2

G

)− EJ cos

(2π

Φ0

φ2

)(3.69)

in which we had to eliminate the number one degree of freedom, resulting in an one-dimensionalHamiltonian. Note that the Hamiltonian’s is not just periodic in the generalized position butalso in the generalized conjugate momentum. For this reason, the Hamiltonian does not possesa quadratic kinetic term and has no analogy to the description of a particle in a potential.However, Hamiltonians in the form of Eq. (3.69) are known in quantum error correction, e.g.the GKP code [52], or can result as effective Hamiltonians, e.g. for a single Bloch electron inan external magnetic field [53].

Appart from the Hamiltonian’s uncommon form, it is easy to show that it indeed leads tothe correct classical quations of motion. As before, for the quantization of the Hamiltonianwe impose the canonical commutation relation [φ2, Q2] = i~. Rewriting the first cosine of theHamiltonian Eq. (3.69) as sum of two exponentials and identifying these as the translationoperators of functions in phase representation

T (a) = eiaQ2/~, a ∈ R, (3.70)

59


we can write for the Hamiltonian

H = −EJ2

[T

(2π~Φ0G

)+ T

(− 2π~

Φ0G

)]− EJ cos

(2π

Φ0

φ2

). (3.71)

In the following we will look for the Hamiltonian’s eigenfunctions ψ(φ2) = 〈φ2|ψ〉 with eigenen-ergy E. For this purpose, we introduce the abbreviations

ε =−2E

EJ, a =

2π~Φ0G

, α =a

Φ0

, φ2 = ma, ψ(ma) = g(m), (3.72)

such that the time-independent Schrodinger equation reduces to

εg(m) = g(m+ 1) + g(m− 1) + 2 cos(2παm)g(m) (3.73)

Note that this is not a differential equation but an one-dimensional difference equation knownas Harper’s equation [53, 54]. It is a special case of the almost Mathieu operator [55, 56]3,which has been examined intensely in mathematical physics. The full spectrum of the Harper’sequation leads to the well known Hofstadter’s butterfly [53], which is shown in Fig. 3.7.

-

Figure 3.7: Hofstadter’s butterfly [53] for rational values of α with denominator less than 50.

However, recalling the Hamiltonian to describe the JJ-G-JJ circuit, it is not obvious whichboundary conditions have to be imposed. For this reason, we will lift the singularity of thecircuit by considering additional capacitances and discuss the boundary conditions with regardto this modified circuit. This will be done in the following section.

3In our case ’special’ means equal Josephson energies on both ports of the Gyrator.

60

3.7. Cooper pair boxes coupled by a gyrator, the CJJ-G-CJJ circuit

3.7 Cooper pair boxes coupled by a gyrator, the CJJ-G-CJJcircuit

To impose the boundary conditions to the Hamiltonian in the previous section, the singularity ofthe Lagrangian will be avoided by considering an invertible capacitance matrix in the circuit’sLagrangian. For this reason, we include capacitances in parallel to the Josephson junctionswhich also capture the intrinsic capacitances of the Josephson junctions. The resulting systemmodels two Cooper pair boxes coupled by a gyrator and it will be denoted from now on as theCJJ-G-CJJ circuit. It is shown in Fig. 3.8. For simplicity, we consider identical Cooper pairboxes, i.e. the capacitances and the Josephson energies on both ports of the gyrator are equal.

Gφ1 φ2

C CEJ EJ

Figure 3.8: The CJJ-G-CJJ circuit. It consists of two identical Cooper pair boxes coupled bya gyrator.

The CJJ-G-CJJ circuit’s Hamiltonian can be derived directly and it reads

H =

(Q1 − A1(φ1, φ2)

)2

2C+

(Q2 − A2(φ1, φ2)

)2

2C−EJ cos

(2π

Φ0

φ1

)− EJ cos

(2π

Φ0

φ2

)︸︷︷︸

U(φ1,φ2)

. (3.74)

We did not specify a gauge for the vector potential A(φ1, φ2) which satisfies Eq. (3.10). Weimpose the usual canonical commutation relations [φi, Qj] = i~δij to quantize this Hamiltonian.Note that this Hamiltonian is analogous the Hamiltonian of a massive, charged particle in atwo-dimensional, periodic cosine potential under the effect of a perpendicular, uniform mag-netic field.

Although the potential U(φ1, φ2) and the effective homogeneous magnetic field due to the gyra-tor are periodic with periodicity Φ0 in both the φ1 and φ2 variable, in general the Hamiltonian’seigenfunction ψ(φ1, φ2) cannot have this periodicity. This is due to the vector potential of thegyrator which breaks translation invariance. Thus, the Hamiltonian does not commute si-multaneously with both the conventional unitary translation operators Tφ1(Φ0) and Tφ2(Φ0),with

Tφi(a) = eiaQi/~, a ∈ R. (3.75)

This can be seen easily in e.g. Landau gauge. Therefore, one cannot impose the boundaryconditions of two uncoupled Cooper pair boxes

ψ(φ1 − Φ0, φ2) = Tφ1(Φ0)ψ(φ1, φ2) = eiθ1ψ(φ1, φ2), (3.76a)

ψ(φ1, φ2 − Φ0) = Tφ2(Φ0)ψ(φ1, φ2) = eiθ2ψ(φ1, φ2), (3.76b)

61


to the eigenfunctions of H. Here, we included the phases θ1, θ2 that take into account thepossible offset charges on the superconducting islands [20]. For this reason, we have to look fordifferent, appropriate boundary conditions. Therefore, we introduce a new set of variables:

Πx = Q1 − A1(φ1, φ2), Rx = φ1 +Φ2G

~Πy, (3.77a)

Πy = Q2 − A2(φ1, φ2), Ry = φ2 −Φ2G

~Πx, (3.77b)

in which ΦG =√

~/G is the analog to the magnetic length and will be denoted as gyration flux.4

These variables are analogous to the dynamical momenta (Πi) and center of mass positions (Ri)commonly used in the context of the quantum Hall effect [48, 49]. They satisfy the followingcommutation relations (see appendix 6.5):

[Πx,Πy] = i~2

Φ2G

, [Rx, Ry] = −iΦ2G, [Ri,Πj] = 0. (3.78)

In this new set of variables the CJJ-G-CJJ circuit’s Hamiltonian Eq. (3.74) yields

H =Π2x

2C+

Π2y

2C− EJ

[cos

(2π

Φ0

(Rx −

Φ2G

~Πy

))+ cos

(2π

Φ0

(Ry +

Φ2G

~Πx

))]=

Π2x

2C+

Π2y

2C− EJ

2

[exp

(−i2π

Φ0

Φ2G

~Πy

)exp

(i2π

Φ0

Rx

)+ exp

(i2π

Φ0

Φ2G

~Πx

)exp

(i2π

Φ0

Ry

)+ h.c.

],

(3.79)

where we wrote the cosines as sums of exponentials and we used the Baker-Campbell-Hausdorffformula for commuting operators. We now introduce the ladder operators associated to thedynamical momenta

Πx =~

ΦG

a† + a√2, Πy =

~ΦG

i(a† − a)√2

, (3.80)

which satisfy the common bosonic commutation relation [a, a†] = 1. They allow to define theunitary displacement operator [57]

D(α) = eαa†−α∗a, α ∈ C. (3.81)

In terms of these operators, the CJJ-G-CJJ circuit’s Hamiltonian Eq. (3.79) can be furtherexpressed as

H = ~ωG(a†a+

1

2

)− EJ

2

[D

(√2π

Φ0

ΦG

)TMy

(2πΦ2

G

Φ0

)

+D

(i

√2π

Φ0

ΦG

)TMx

(−2πΦ2

G

Φ0

)+ h.c.

].

(3.82)

4Note that we restrict ourselves to the case in which the gyration conductance is positive. The adjustments forthe treatment of negative gyration conductances are straightforward. For a vanishing gyration conductancethe CJJ-G-CJJ circuit trivially decouples into non-interacting Cooper pair boxes.

62


Note that we defined the unitary magnetic translation operators

TMx (a) = e−iaRy/Φ2G , TMy (b) = eibRx/Φ2

G , a, b ∈ R, (3.83)

and recall the gyration frequency ωG = G/C.

The CJJ-G-CJJ circuit’s Hamiltonian expressed in the form of Eq. (3.82) will be the startingpoint for determining the CJJ-G-CJJ circuit’s energy spectrum. Therefore, we will exploitthe properties of the magnetic translation operators, which will be summarized in the nextsubsection. As we will see, they give rise to well-defined boundary conditions for the eigenstatesof H.

3.7.1 Magnetic translation operators

Before analyzing the spectrum of the CJJ-G-CJJ circuit’s Hamiltonian we will briefly investi-gate the main properties of the magnetic translation operators and of their eigenstates in thissubsection. The results of this subsection are based on the ideas of [58,59].

The magnetic translation operators defined in Eq. (3.83) differ from the conventional translationoperators Eq. (3.75) by an additional term in the exponent, depending on the gauge of thevector potential. This can be seen by inserting Eq. (3.77) into Eq. (3.83), yielding to

TMx (a) = eiaQ1/~−ia[Gφ2+A1(φ1,φ2)]/~, TMy (a) = eiaQ2/~+ia[Gφ1−A2(φ1,φ2)]/~. (3.84)

The additional phase in the exponent is a function of the flux variables φ1, φ2. Because of thisphase, magnetic translations in different directions do not commute in general but generate anadditional complex phase prefactor,

TMx (a)TMy (b) = eiab/Φ2GTMy (b)TMx (a), (3.85)

which can be easily shown by applying the the trivial Baker-Campbell-Hausdorff formula. Fromthat we can conclude that a state picks up an Aharonov-Bohm like complex phase after themagnetic translation around a rectangular loop,

TMx (a)TMy (b)TMx (−a)TMy (−b) = eiab/Φ2GTMy (b)TMx (a)TMx

†(a)TMy

†(b) = eiab/Φ

2G , (3.86)

in which the complex phase ab/Φ2G = abG/~ corresponds to the gyration action of this mag-

netic translation around the rectangular loop per reduced Planck constant.5 It follows that themagnetic translation operators do not form a group like the conventional translation operatorsdo.

We have seen that a magnetic translation differs from a conventional translation by an addi-tional complex phase. Furthermore, by construction, the magnetic translation operators always

5In terminology of a massive, charged particle being exposed to a homogeneous magnetic field, the complexphase would correspond to the magnetic flux penetrating the rectangular loop per (non-superconducting) fluxquantum. However, besides the wrong units, we avoid to call abG a magnetic flux (although ab correspondsto the ’area’ of the loop and G can be interpreted as the magnetic field) since this could easily lead tomisapprehensions concerning physical fluxes like e.g. φi,Φ0,ΦG, a, b. Therefore, abG will be called gyrationaction.

63


commute with the kinetic term of the CJJ-G-CJJ circuit’s Hamiltonian Eq. (3.82), which con-tains the vector potential A(φ1, φ2). For this reasons, we require the eigenstates |ψ〉 of theCJJ-G-CJJ circuit’s Hamiltonian to satisfy the toroidal boundary conditions

TMx (Φx) |ψ〉 = eiθx |ψ〉 , TMy (Φy) |ψ〉 = eiθy |ψ〉 , (3.87)

with appropriately chosen periodicities Φx,Φy > 0 such that

[TMx (Φx),H] = [TMy (Φy),H] = 0. (3.88)

Again, we have included additional phases θx, θy ∈ [0, 2π) taking possible offset charges on thesuperconducting islands into account. These phases can be interpreted as Aharanov-Bohm likephases, which are picked up by circling the torus once in one of the two possible directions.With this interpretation, the fictive magnetic fluxes causing these phases respectively penetratethe hole of the torus and lie inside the torus.

Since both boundary conditions stated in Eq. (3.87) have to be satisfied simultaneously, thecommutator [TMx (Φx), T

My (Φy)] must vanish. According to Eq. (3.85), this is the case if and

only ifΦxΦy

Φ2G

= 2πp, p ∈ N, (3.89)

and it follows directly that the gyration action of a translation once around a unit cell, i.e. arectangle of the torus size Φx × Φy, has to be quantized in multiples of the Planck constant,

SG = ΦxΦyG = 2π~p, p ∈ N. (3.90)

Furthermore, for the toroidal boundary conditions introduced in Eq. (3.87), the condition inEq. (3.88) reduces to

Φx = nΦ0, Φy = mΦ0, m, n ∈ N, (3.91)

by taking the Hamiltonian Eq. (3.82) and applying Eq. (3.85).

At this point, we can ask what kind of magnetic translations are compatible with the toroidalboundary conditions in Eq. (3.87). Since magnetic translations along the same directionsalways commute, it is sufficient to consider orthogonal directions only, i.e. we are looking forthe operators TMx (a), TMy (b) such that

[TMx (a), TMy (Φy)] = 0, [TMy (b), TMx (Φx)] = 0. (3.92)

Considering Eqs. (3.85) and (3.89), the previous requirements reduce to the possible transla-tions by

a = Φxnxp, b = Φy

nyp, nx, xy ∈ Z. (3.93)

These considerations will be used for the definition of a basis in which the eigenstates |ψ〉 willbe expanded. We remark that in general TMx (a) and TMy (b) do not commute between eachother and thus we can use only one of them to define the basis, for example here we chooseTMy (b). Also, the smallest possible translation is obtained for ny = 1 and thus b = Φy/p.

64


Recalling the unitarity of the magnetic translation operators, we define the simultaneous eigen-states of TMx (Φx) and TMy (Φy/p) as

TMx (Φx) |kx, ky〉 = eikxΦx |kx, ky〉 , TMy (Φy/p) |kx, ky〉 = eikyΦy

p |kx, ky〉 . (3.94)

with

0 ≤ kx <2π

Φx

, 0 ≤ ky <2πp

Φy

. (3.95)

Taking Eqs. (3.85), (3.89) and (3.94) one can determine the action of TMx (Φx/p) on |kx, ky〉,namely

TMy (Φy/p)TMx (Φx/p) |kx, ky〉 = eiΦxΦy/p2Φ2

GTMx (Φx/p)TMy (Φy/p) |kx, ky〉

= eiΦyp (ky+Φx/pΦ2

G)TMx (Φx/p) |kx, ky〉

= eiΦyp

(ky+2π/Φy)TMx (Φx/p) |kx, ky〉 ,

(3.96)

from which we can deduce that

TMx (Φx/p) |kx, ky〉 = eikxΦx

p |kx, ky + 2π/Φy mod 2πp/Φy〉 . (3.97)

Note that the state maps into itself after applying TMx (Φx/p) p times. This is accounted for bythe modulo operation. Furthermore, the prefactor appearing in Eq. (3.97) is chosen such that

TMx (Φx) |kx, ky〉 =[TMx (Φx/p)

]p |kx, ky〉 = eikxΦx |kx, ky〉 . (3.98)

Finally, kx and ky have to be chosen such that |kx, ky〉 satisfies the toroidal boundary conditionsstated in Eq. (3.87). It follows that kx = θx/Φx and ky = (θy+2πn)/Φy with n = 0, 1, . . . , p−1.

We now apply this results to compute the spectrum of the CJJ-G-CJJ cirucuit’s HamiltonianEq. (3.82), which will be done in the following subsections.

3.7.2 Preserved Φ0-periodicity, compact variables

Assuming the gyrator to transfer the Φ0-periodicity of the potential in both directions to theperiodicity of the eigenstates, we have to impose the toroidal boundary conditions (Φx = Φy =Φ0)

TMx (Φ0) |ψ〉 = eiθx |ψ〉 , TMy (Φ0) |ψ〉 = eiθy |ψ〉 , (3.99)

to the Hamiltonian’s eigenstates |ψ〉. This corresponds to have compact variables φx, φy. Ac-cording to Eq. (3.90), the imposed boundary conditions immediately lead to the quantizationof the gyration conductance

G =2π~Φ2

0

p =(2e)2

hp = G0p, p ∈ N, (3.100)

in terms of integer multiples of the superconducting conductance quantum G0 = (2e)2/h, whichtakes into account that the charge is carried by Cooper pairs with total charge −2e (no spin

65

Martin Rymarz


degeneracy). Making use of this gyration conductance quantization, we find 2πΦ2G/Φ0 = Φ0/p

and the Hamiltonian Eq. (3.82) reduces to

H = ~ωG(a†a+

1

2

)− EJ

2

[D

(√2πΦG

Φ0

)TMy

(Φ0

p

)+D

(i

√2πΦG

Φ0

)TMx

(−Φ0

p

)+ h.c.

].

(3.101)

The Hamiltonian then depends on magnetic translation operators as discussed in the previoussubsection.

3.7.3 Non-preserved Φ0-periodicity, non-compact variables

If one drops the assumption that the gyrator preserves the Φ0-periodicity of the potential alsoin the eigenstates, in general all values of the gyration conductances G are allowed. In thefollowing, we will restrict ourselves to the case in which the gyration conductance is a rationalmultiple of the superconducting conductance quantum, i.e.

G = G0p

q, p, q ∈ N and co-prime. (3.102)

In this case, the gyration action of translating once around the unit cell of size Φ0 × Φ0 is arational multiple of the Planck constant 2π~, which will make the following calculations easierand is a reasonable restriction since the rational numbers are dense in R. Furthermore, allphysical parameters (so the gyration conductance) are subject to certain experimental fluctu-ations resulting in an imprecision of the actual value such that one should rather consider aninterval of possible smeared values [53].Given Eq. (3.102), it is still possible to impose the toroidal boundary conditions by enlargingthe initial φ0×φ0 unit cell to form a bigger, magnetic unit cell which contains an integer numberof flux quanta. In principle, the shape of the magnetic unit cell is arbitrary but we define itto be a rectangle of size qΦ0 × Φ0 with its long side aligned along the x-direction. If followsthat this chosen magnetic unit cell contains q initial unit cells such that the gyration action oftranslating once around it is SG = 2π~p. For this reason, we can impose the toroidal boundaryconditions (Φx = qΦ0,Φy = Φ0)

TMx (qΦ0) |ψ〉 = eiθx |ψ〉 , TMy (Φ0) |ψ〉 = eiθy |ψ〉 , (3.103)

for the Hamiltonian’s eigenstates |ψ〉 and rewrite the CJJ-G-CJJ circuit’s Hamiltonian into

H = ~ωG(a†a+

1

2

)− EJ

2

[D

(√qπ

p

)TMy

(qΦy

p

)+D

(i

√qπ

p

)TMx

(−Φx

p

)+ h.c.

].

(3.104)

Of course, this Hamiltonian coincides with Eq. (3.101) for q = 1. Also, note that the effect ofthe magnetic translation operators appearing in Eq. (3.104) on the states |kx, ky〉 defined inEq. (3.94) is known from Eqs. (3.94)-(3.98).

66


3.7.4 Numerical implementation and results

In the following, we will numerically diagonalize the Hamiltonian derived in Eq. (3.104).Therefore, we expand the Hamiltonian’s eigenstates in the product states basis of the a(†)-operator’s Fock states (which are the Landau levels) and the states |kx, ky〉 previously definedin Eq. (3.94), i.e. |m〉Π ⊗ |kx, ky〉R. Trivially, the first term of the Hamiltonian Eq. (3.104)is diagonal in the chosen product state basis. For the second term, which involves products ofoperators acting on the Π and R subspaces, the matrix elements of the displacement operatorin the Fock basis can be found analytically and they read [60]

〈m|D(α) |n〉 =

√n!

m!αm−ne−

12|α|2Lm−nn (|α|2), for m ≥ n, (3.105)

where Lm−nn denotes the associated Laguerre polynomial. From this and the unitarity of thedisplacement operator one can easily conclude that

〈m|D(α) |n〉 = 〈n|D†(α) |m〉∗ = 〈n|D(−α) |m〉∗

=

√m!

n!(−α∗)n−me−

12|α|2Ln−mm (|α|2), for m < n.

(3.106)

Given all the relevant matrix elements of the Hamiltonian in this basis, the eigenstates will beexpanded as

|ψ〉 =N−1∑m=0

p−1∑ny=0

cmny |m〉Π ⊗∣∣∣∣ θxΦx

,θy + 2πny

Φy

⟩R

(3.107)

such that they fulfill the boundary conditions Eq. (3.103) and become exact for N →∞. Thisreduces the time-independent Schrodinger equation to an eigenvalue problem of a Np × Npmatrix which determines the coefficients cmn as well as the Hamiltonian’s eigenenergies.

Note that we already can determine the energy bands to be 2π/q-periodic in the phase θy sincethe matrix representations of the Hamiltonian Eq. (3.104) coincide for states |ψ〉 defined in Eq.(3.107) and states

|ψ′〉 =N−1∑m=0

p−1∑ny=0

c′mny|m〉Π ⊗

∣∣∣∣ θxΦx

,θ′y + 2πny

Φy

⟩R

(3.108)

with θ′y = θy+2π/q. Furthermore, considering the Hamiltonian of Eq. (3.104) we expect the dif-ferent Landau Levels |m〉Π to couple with a coupling strength given by the energy ratio EJ/~ωG.

The lowest energy bands of the CJJ-G-CJJ circuit, i.e. the lowest eigenenergies as functions ofthe phases θx, θy, are shown in Fig. 3.9 for different choices of parameters.

First of all, for small ratios of EJ/~ωG one can conclude that the Landau levels indeed do notcouple strongly. However, the potential of the Josephson junctions removes the degeneracy ofthe Landau levels. Hence, every Landau level splits into p sub-bands with small bandwidthswhich are relatively well located at the energies of the Landau levels without any coupling dueto a potential, i.e. ELL

m = ~ωG(m+ 1/2) with m ∈ N0. This is shown in Fig. 3.9 (a).

67


For example, the sub-bands of the lowest Landau level with their small but finite bandwidthsare shown in Fig. 3.9 (b). These sub-bands are not just flat but possess a dependence on thephases θx, θy.

Furthermore, the expected periodicity of the energy bands in θy for rational values of G/G0 isexemplary shown in Fig. 3.9 (c) for the choice of G/G0 = 2/3. Aside from an energy rescalingdue to a different gyration conductance than in the integer case (G/G0 = 2), the dependenciesof the energy bands on θx, θy resemble a sequence of q rescaled energy bands of the integer caseif the Landau level coupling is small, compare Fig. 3.9 (c) with Fig. 3.9 (b).Note that this q-fold degeneracy can be used, in principle, to construct a wavefunction whoseabsolute value is periodic in the φ1- and the φ2-direction with the periodicity Φ0 of the potentialU(φ1, φ2). For this purpose, one has to evaluate the q degenerate eigenstates defined in Eq.(3.107) in the φ1, φ2 representation. Note that the wavefunctions

χθx,θym,ny(φ1, φ2) =

⟨φ1, φ2

∣∣∣∣m, θxΦx

,θy + 2πny

Φy

⟩, ny = 0, 1, . . . , p− 1 (3.109)

are known analytically and given by the Haldane-Rezayi wavefunctions [61,62].

Finally, a larger coupling of the Landau levels, i.e. for larger ratios EJ/~ωc, the dependence ofthe energy bands on φx, φy becomes difficult to predict analytically. Such a case of large Landaulevel coupling is shown in Fig. 3.9 (d). Note that the coupling strength can be easily increasedby increasing the capacitive shunts of the CJJ-G-CJJ circuit, since EJ/~ωc = EJC/~G.

(a) G/G0 = 2, EJ/~ωG = 0.2 (b) G/G0 = 2, EJ/~ωG = 0.2

(c) G/G0 = 2/3, EJ/~ωG = 0.2 (d) G/G0 = 2, EJ/~ωG = 20

Figure 3.9: Lowest energy bands (scaled by ~ωG) of the CJJ-G-CJJ circuit as functions of θx, θyfor different parameters.

68


We want to make the remark that the analysis of the bandwidths (scaled by ~ωG) as functions ofthe ratio q/p = G0/G for fixed constant K = −pEJ/4q~ωG coincides with the results obtainedby Springsguth et al. [63], who analyzed the Hall conductance of Bloch electrons in a magneticfield by specifying to Landau gauge.6 These results are shown in Fig. 3.10. However, ourapproach does not depend on the chosen gauge of the vector potential.

(a) K = 1 (b) K = 12

Figure 3.10: Bandwidths of the lowest energy bands (scaled by ~ωG) as functions of the ratioq/p for fixed values of K = −pEJ/4q~ωG. The plots were generated by takingdenominators up to p = 20 into account and recover the results by Springsguth etal. [63].

3.7.5 Comparison with the JJ-G-JJ circuit

In the previous subsection we have determined the energy ratio EJ/~ωc = EJC/~G to denotethe coupling strength of the different Landau levels. In theory, this energy ratio can be madearbitrarily small by considering the capacitance C to be an arbitrary small but finite intrinsiccapacitance. By completely neglecting the Landau level coupling, the starting Hamiltoniangiven in Eq. (3.104) can be projected onto the m’th Landau level such that

〈m|H |m〉 = ~ωG(m+

1

2

)− EJ

2e−|α|

2/2L0m(|α|2)

[TMx

(Φx

p

)+ TMy

(qΦy

p

)+ h.c.

]. (3.110)

6Note that Springsguth et al. use a different convention in which the roles of p and q are interchanged andthat the constant K which they fix does not really measure the strength of the Landau level coupling sinceK does not depend on the gyration conductance, i.e. the analog to the magnetic field.

69


This yields an effective Hamiltonian for the center of mass position variables Ri. The projectedeigenfunction |ψ〉m of this Hamiltonian can be written as

|ψ〉m = |m〉Π ⊗p−1∑ny=0

cmny

∣∣∣∣ θxΦx

,θy + 2πny

Φy

⟩R

, (3.111)

such that (after a rescaling of energy) the time-independent Schrodinger equation results intoa one-dimensional difference equation which is equivalent to Harper’s equation [58, 63, 64], seeEq. (3.73).

Recalling, that the CJJ-G-CJJ circuit reduces to the singular JJ-G-JJ circuit for C = 0, thisshows that the JJ-G-JJ circuit (see section 3.6) can be interpreted as an effective descriptionof the low-energy projection of the non-singular CJJ-G-CJJ circuit, which takes the small butfinite intrinsic capacitances of the Josephson junctions into account.The interpretation of the singular circuit to be a low-energy projection of the correspondingnon-singular circuit is analogous to the comparison of the L-G-L circuit with the LC-G-LCcircuit (see section 3.5).

3.7.6 Comparison with the C-G-C circuit

After the identification of the JJ-G-JJ circuit to be the effective description of the low-energyprojection of the CJJ-G-CJJ circuit in the regime of negligible Landau level coupling, we want toindicate that weak Landau level coupling can be also achieved by considering the Josephson en-ergy to be vanishing small. In case of EJ = 0, the Landau levels of the CJJ-G-CJJ circuit do notsplit into p sub-bands but remain flat and q-fold degenerate with energies ELL

m = ~ωG(m+1/2)with m ∈ N0.

Recalling that the CJJ-G-CJJ circuit reduces to the C-G-C circuit for EJ = 0, this shows thatthe energy C-G-C circuit can be interpreted as high-energy projection of the CJJ-G-CJJ circuit,which is, again, analogous to the comparison of the C-G-C circuit with the LC-G-LC circuit.However, as already mentioned in section 3.3, by imposing the toroidal boundary conditions ofEq. (3.87) to the C-G-C circuit the spectrum coincides with Eq. (3.40), but every Landau levelhas a q-fold degeneracy. In particular, assuming φ1, φ2 to be compact variables of the C-G-Ccircuit, the gyration conductance has to be quantized in integer multiples of G0.

70

3.8. Fluxoniums coupled by a gyrator, the LCJJ-G-LCJJ circuit

3.8 Fluxoniums coupled by a gyrator, the LCJJ-G-LCJJcircuit

In this section we add linear inductances on both ports of the gyrator in the CJJ-G-CJJ circuit.This circuit results in two identical fluxoniums [65] which are coupled by a gyrator and is shownin Fig. 3.11. In the following we will denote it as LCJJ-G-LCJJ circuit.

Gφ1 φ2

C EJ EJL LC

Φext1 Φext

2

Figure 3.11: The LCJJ-G-LCJJ circuit. It consists of two identical fluxoniums coupled by agyrator. The Josephson junction and the linear inductance on each port of thegyrator form a loop which can be penetrated by an external magnetic flux Φext

i .

Due to the included capacitances the LCJJ-G-LCJJ circuit is non-singular. Therefore, byapplying the Legendre transformation to the Lagrangian of this circuit we get immediately thecircuit’s Hamiltonian

H =

(Q1 − A1(φ1, φ2)

)2

2C+

(Q2 − A2(φ1, φ2)

)2

2C

+φ2

1

2L+φ2

2

2L− EJ cos

(2π

Φ0

(φ1 − Φext1 )

)− EJ cos

(2π

Φ0

(φ2 − Φext1 )

)︸︷︷︸

U(φ1,φ2)

.(3.112)

Note that we did not specify a gauge of the vector potential satisfying Eq. (3.10) yet. Fur-thermore, in our description we have included external, physical fluxes Φext

1 ,Φext2 penetrating

the loops formed by the Josephson junction and the inductor on each port of the gyrator. Thequantum mechanical description of the Hamiltonian is obtained by imposing φj, Qj to satisfyusual canonical commutation relations [φi, Qj] = i~δij. Because of the linear inductances thepotential U(φ1, φ2) is not periodic anymore. For this reason, we can demand the Hamiltonian’seigenfunctions to vanish at large absolute values of φ1, φ2, making the search of reasonableboundary conditions short.

In the following we specify to symmetric gauge and work in the basis of the LC-G-LC circuit’seigenstates, see section 3.5. Defining the flux ΦT = (~/CΩ)1/2, in this basis the Hamiltonianreads

H =~ω+

(c†c+

1

2

)+ ~ω−

(d†d+

1

2

)− EJ

[cos

(πΦT

Φ0

(c† + c+ d† + d)− ϕext1

)+ cos

(iπΦT

Φ0

(c− c† + d† − d)− ϕext2

)],

(3.113)

in which we have taken the external fluxes to be in their reduced form. By rewriting thecosines as sums of exponentials and using the Baker-Campbell-Hausdorff formula for commuting

71


operators the Hamiltonian can be written as

H =~ω+

(c†c+

1

2

)+ ~ω−

(d†d+

1

2

)− EJ

2

[A

(iπΦT

Φ0

)B

(iπΦT

Φ0

)e−iϕ

ext1 + A

(πΦT

Φ0

)B

(−πΦT

Φ0

)e−iϕ

ext2 + h.c.

],

(3.114)

in which we have identified the unitary displacement operators

A(µ) = eµc†−µ∗c, B(λ) = eλd

†−λ∗d. (3.115)

As already mentioned, for the numerical implementation one can expand the eigenstate in theproduct states of the LC-G-LC circuit, i.e.

|ψ〉 =N∑

n,m=0

αnm |n,m〉 (3.116)

with the truncation to the N ’th Fock state of the c(†)- and the d(†)-degree of freedom. Thischoice of basis is convenient because the first two terms of the Hamiltonian are already diago-nal in this basis and the matrix elements of products of the displacement operators A(µ) andB(λ) are known analytically, see Eq. (3.105). Furthermore, the off-diagonal matrix elementsconnecting different Fock states are generally smaller than the diagonal ones and they decayfor Fock states with a large difference in the occupation. This justifies the truncation of theHilbert space.

Note that by reverse engineering the Bogoliubov-Valatin transformation presented in Eq. (3.57)it is possible to evaluate the matrix elements 〈φ1, φ2|n,m〉, such that, together with the (numer-ically) determined coefficients αnm of the eigenstate |ψ〉, one can determine the wavefunctionin flux space, i.e. ψ(φ1, φ2) = 〈φ1, φ2|ψ〉.

We want to conclude this section with the presentation of an outlook for a possible continuationof the analysis of the LCJJ-G-LCJJ circuit.

The LCJJ-G-LCJJ circuit is of great interest in the parameter regime in which the inductiveenergy EL = (2π/Φ0)2/L is small compared to all other energies of the circuit. In this regime,the previously proposed numerical implementation fails if one is interested in a wide rangeof the energy spectrum, because ~ω− becomes small and one has to take a large number ofFock states into account. However, in this parameter regime the dual approach to circuitquantization using loop charges [66] is highly convenient. Given an appropriate method forderiving the Hamiltonian’s eigenstates, a promising analysis comparable to the work of Koch etal. [67] could reveal new insights into compactness within the CJJ-G-CJJ circuit. This analysisconsists of the comparison of the high-frequency responses of the LCJJ-G-LCJJ circuit and theCJJ-G-CJJ circuit for large inductances.

72

4 Conclusion and Outlook

In this thesis we have analyzed various types of singular and nonreciprocal superconductingcircuits within the scope of quantum electrodynamics.

In chapter two of this thesis, we have extended one possible realization of a superconductingcircuit architecture which realizes a qubit-resonator system with tunable coupling. The exten-sion consisted of the inclusion of intrinsic capacitances which canceled the singularity of thecircuit. We have shown that, in a wide range of actual values, these intrinsic capacitances donot have a significant effect on the resulting qubit-resonator circuit. Therefore, the singulartreatment is a reasonable effective description of the proposed circuit and we have seen thatthe fast variables are assumed to be in their classical ground state within this treatment.

In chapter three of this thesis, we have introduced the gyrator into circuit quantum electrody-namics as a fundamental nonreciprocal lumped element. We have proposed a general systematicscheme to quantize any singular, nonreciprocal circuit including arbitrary many gyrators. Af-terwards, we have analyzed basic superconducting circuits which include just one gyrator. Inthis context, we investigated possible boundary conditions of the systems. Again, we havedetermined the singular treatment of a circuit to be the effective low-energy description of thecorresponding non-singular circuit.

Although we have investigated many diverse and interesting problems within this thesis, thereare plenty possible topics for future research.

Concerning the superconducting circuit architecture realizing an effective qubit-resonator sys-tem, one could explore different values of the circuit’s elements, for example, the parame-ter regime in which the potential in the fast variables does not possess just one local, well-pronounced minimum only or values for intrinsic capacitances, which delocalize the wavefunc-tion in the fast subspace within the Born-Oppenheimer approximation. Furthermore, alsoparameter regimes in which the Born-Oppenheimer approximation is not valid are of interest.Beyond that, with the proposed improved procedure of identifying the model Hamiltonian’sparameters one could also aim for effective qubits, which are not transmon-like, with the ob-jective to realize stronger longitudinal or transverse coupling, respectively. Finally, also theclassical ground state approximation of the Josephson arrays could be lifted and the effectsof the internal degrees of freedom on the resulting qubit-resonator system could be analyzed.The overall goal would be to have an extensive theoretical knowledge of the system such thatnothing stands in the way of the actual fabrication.

With regard to non-reciprocal circuits, the compactness of the flux variables in case of non-inductively terminated ports of a gyrator is still an open question worth analyzing. This couldilluminate the possible quantization of the gyration conductance and would clarify the search of

73

4. Conclusion and Outlook

proper boundary conditions. Moreover, small-sized gyrators offer the opportunity of realizingeffective inductances with small spatial extensions and allow for a reasonable scaling up. This isfavorable in integrated circuits and the possible implementation in superconducting devices is anexciting idea. Furthermore, we only have considered symmetric circuits involving one gyrator,i.e. the ports of the gyrator were terminated identically. The generalized consideration ofasymmetric circuits could be of interest as well as the extension to circuits with many gyrators.

74

5 Acknowledgements

First of all, I like to thank David DiVincenzo for the supervision of my master’s thesis. Youhave been patient and encouraging at any time and I appreciate all the discussions we hadand the time you took for answering all my various questions or giving helpful advice. I amreally looking forward to continue working with you, also on our side projects or ’little hobbies’.There is no doubt that this will result in a promising future.

Next, I would like to thank Fabian Hassler for being my second supervisor. You did not justaccept this task formally, quite the contrary, I also profited a lot from our discussions. Thesediscussions often started during our group meetings and ended in the canteen, sometimes in acompletely different topic of philosophical nature.

Thank you Susanne for setting the foundation of this thesis and for the introduction into yourwork. The future will show which role longitudinal coupling will play for quantum computing.

Grazie to Stefano and Alessandro for several sessions of spirited discussions. You both alsohelped me a lot and you made it clear to me that gyrators can yield to pretty cool effects. Ourcultural exchange was always quite amusing.

Thank you Uta, Alexander, Benedikt, Stefano and Kerstin for sharing the work of proofreadingparts of my thesis. Hopefully, the number of typos is now minimized.

Thank you Manuel for the continuous IT-support and Lisa for sharing your knowledge aboutall possible formalities.

Thanks to all the IQI members for the incomparable and familiar working atmosphere. Thedaily coffee breaks and weekly soccer tournaments were welcome distractions - not to mentionthe cakes for special occasions.

Last but not least, I want to thank my parents and sister. You also have contributed to thecompletion of not just my master’s thesis, but the whole study. You have always supported mein everything I had in mind and I am really grateful for that. Ich danke Euch fur alles!

75

6 Appendix

6.1 The square-root capacitance matrix transformationpreserves the commutation relation

In the following we will proof that the square-root capacitance matrix transformation pre-serves the commutation relations. For this reason we rename ϕ = ϕ1, ϕJ = ϕ2. Then, thecommutation relations of the new variables Eq. (2.22) read

[fi, ηj] =[(c−1/2C1/2ϕ)i, (c

1/2C−1/2n)j

]=

[ 2∑k=1

C1/2ik ϕk,

2∑l=1

C−1/2jl nl

]

=2∑

k,l=1

C1/2ik C

−1/2jl [ϕk, nl]

= i2∑

k,l=1

C1/2ik C

−1/2jl δkl

= i∑k

C1/2ik C

−1/2jk

= i~2∑

k=1

C1/2ik C

−1/2kj

= i(C1/2C−1/2

)ij

= i (1)ij

= iδij,

(6.1)

where we used the symmetry of C−1/2. All other commutators vanish trivially.

77

6. Appendix

6.2 The Cholesky decomposition transformation preservesthe commutation relation

In the following we will proof that the Cholesky decomposition transformation preserves thecommutation relations. For this reason we rename ϕ = ϕ1, ϕJ = ϕ2. Then, the commutationrelations of the new variables read

[fi, ηj] =[(c−1/2(AT )−1ϕ)i, (c

1/2An)j

]=

[ 2∑k=1

((A−1)T

)ikϕk,

2∑l=1

Ajlnl

]

=2∑

k,l=1

(A−1)kiAj,l[ϕk, nl]

= i2∑

k,l=1

(A−1)kiAjlδkl

= i2∑

k=1

Ajk(A−1)ki

= i(AA−1)ji

= i1ji

= iδij.

(6.2)

All other commutators vanish trivially.

78

6.3. Relation between the square-root capacitance matrix decomposition and the Choleskydecomposition

6.3 Relation between the square-root capacitance matrixdecomposition and the Cholesky decomposition

In order bring any positive definite, symmetric capacitance matrix into isotropic form we in-troduced two possible transformations:The square-root capacitance matrix decomposition,

C = C1/2C1/2, φ′ = c−1/2C1/2φ, (6.3)

as well as the Cholesky decomposition,

C = ATA, φ′′ = c−1/2Aφ. (6.4)

These variable transformations also transform the linear inductance matrix according to

M ′ = c(C−1/2

)TMC−1/2, (6.5)

for the square-root capacitance matrix decomposition and

M ′′ = c(A−1

)TMA−1, (6.6)

for the Cholesky decomposition.

In the following, we will show that the transformed linear inductance matrices M ′ and M ′′

possess the same spectrum, i.e. the eigenvalues of the transformed linear inductance matrix donot depend on the chosen decomposition.

Since the capacitance matrix is real, also the Cholesky decomposition A is real [26]. Hence,its polar decomposition [26, 68] reduces to A = OX, in which O is an orthogonal matrixand X = (ATA)1/2 is positive definite and symmetric. Taking the Cholesky decompositionEq. (6.4) into account, it follows directly that X = C1/2 and therefore A = OC1/2, i.e.the Cholesky decomposition matrix and the square-root capacitance matrix are related via anorthogonal matrix. For this reason, we find

M ′′ = c(A−1

)TMA−1 = Oc

(C−1/2

)TMC−1/2OT = OM ′′OT , (6.7)

i.e. M ′ and M ′′ are related via the same orthogonal matrix, having therefore identical eigen-values.

79

6. Appendix

6.4 Tellegen’s construction of a transformer

In the following we want to present a more intuitive way of treating cascaded gyrators thanpresented in the general approach of subsection 3.2.2. This is an example in which we have todeal with different constraints than Eq. (3.21).

As mentioned in section 3.1, two cascaded ideal gyrators with gyration conductances G1 andG2 are equivalent to an ideal transformer with turns-ratio N = G2/G1. We want to show thiswithin the Lagrange description of the Gyrator. Therefore, we consider three general networkswith Lagrangians L1,L2 and L3. The first two networks are connected to the gyrators’ outerports while the third network connects the upper nodes of the gyrator’s outer ports, see Fig.6.1.

G2φG1φ1 φ2

L2L1

L3

Figure 6.1: Tellegen’s construction of a transformer. Two cascaded ideal gyrator with gyrationconductances G1 and G2 are equivalent to an ideal transformer with turns-ratioN = G2/G1. Note that the internal node corresponding to the flux φ is not allowedto be connected to any further lumped element than the two gyrators.

In Landau gauge (different Landau gauges for the two gyrators), the total Lagrangian of thissystem reads

Ltot = L1(φ1, φ1) + L2(φ2, φ2) + L3(φ1, φ1, φ2, φ2)−G1φ1φ+G2φφ2. (6.8)

In particular, all the networks’ Lagrangians can, in general, also depend on some additionaldegrees of freedom, which are, however, of no further importance. We note that the totalLagrangian Eq. (6.8) does not depend on φ. For this reason, the corresponding conjugatecharge vanishes,

Q =∂Ltot∂φ

= 0. (6.9)

This reduces the Euler-Lagrange equation for the flux φ to

0 =∂Ltot∂φ

= G2φ2 −G1φ1, (6.10)

which directly yields to the constraint of an ideal transformer,

φ1

φ2

=V1

V2

=G2

G1

. (6.11)

80

6.4. Tellegen’s construction of a transformer

Note that it is not possible to solve for φ as a function of the other fluxes and conjugate charges.However, taking the constraint Eq. (6.11) into account, the Lagrangian reads

Ltot = L1

(G2

G2

φ2 + const.,G2

G1

φ2

)+ L2(φ2, φ2) + L3

(G2

G1

φ2 + const.,G2

G1

φ2, φ2, φ2

), (6.12)

such that its number of degrees of freedom is reduced by two and not just by one.An additional integration constant might be possible.

81

6. Appendix

6.5 Commutation realations of the QHE variables

In the following we will proof that the variables Πi and Ri, which are commonly used in thecontext of the quantum Hall effect and defined in Eq. (3.77), fulfill the commutation relationsgiven in Eq. (3.78):

[Πx,Πy] = [Q1 − A1(φ1, φ2), Q2 − A2(φ1, φ2)]

= −[Q1, A2(φ1, φ2)]− [A1(φ1, φ2), Q2]

= [Q2, A1(φ1, φ2)]− [Q1, A2(φ1, φ2)]

= [Q2, φ2]∂φ2A1(φ1, φ2)− [Q1, φ1]∂φ1A2(φ1, φ2)

= −i~[∂φ2A1(φ1, φ2)− ∂φ1A(φ2, φ2)

]= i~G

= i~2G

~

= i~2

Φ2G

,

(6.13)

[Rx, Ry] =

[φ1 +

Φ2G

~Πy, φ2 −

Φ2G

~Πx

]= −

[φ1,

Φ2G

~Πx

]+

[Φ2G

~Πy, φ2

]−[

Φ2G

~Πy,

Φ2G

~Πx

]= −Φ2

G

~[φ1, Q1] +

Φ2G

~[Q2, φ2]− Φ4

G

~2[Πy,Πx]

= −2iΦ2G + i

Φ4G

~2

~2

Φ2G

= −iΦ2G.

(6.14)

The remaining, vanishing commutators can be derived in an analogous way.

82

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