Widely tunable on-chip microwave circulator for
superconducting quantum circuitsbyBenjamin J. Chapman
B.A., Dartmouth College, 2009
M.S., University of Colorado, 2015

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2017This thesis entitled:
Widely tunable on-chip microwave circulator for superconducting quantum circuits
written by Benjamin J. Chapman
has been approved for the Department of Physics

Konrad W. Lehnert

John Price

Date

The final copy of this thesis has been examined by the signatories, and we find that both thecontent and the form meet acceptable presentation standards of scholarly work in the above
mentioned discipline.

iii
Chapman, Benjamin J. (Ph.D., Physics)
Widely tunable on-chip microwave circulator for superconducting quantum circuits
Thesis directed by Prof. Konrad W. Lehnert

This thesis develops theory for and experimentally demonstrates a new way to break Lorentz
reciprocity—the symmetry, in an electrical network, under exchange of source and detector. The
approach is based on the sequential application of frequency conversion and delay; as frequency
and time are Fourier duals, these operations do not generally commute. We apply this method in
the construction of an on-chip superconducting microwave circulator, a critical component for theunidirectional routing of quantum information in superconducting networks. The device requires
neither permanent magnets nor microwave control tones, allowing on-chip integration with othersuperconducting circuits without expensive control hardware. Isolation in the device exceeds 20
dB over a bandwidth of tens of MHz, and its insertion loss is small, reaching as low as 0.9 dB atselect operation frequencies. Furthermore, the device is linear with respect to input power for signal
powers up to many hundreds of fW (≈ 103 circulating photons), and the direction of circulation
can be dynamically reconfigured. We demonstrate its tunability with operation at a selectionof frequencies between 4 and 6 GHz. Given the current status of quantum error-correction andarchitectures for quantum information processing with superconducting circuits, such scalable nonreciprocal devices will almost certainly be necessary for construction of a superconducting quantum
computer intended to be more than a proof-of-principle.

iv

Dedication

For my sister, Ellie, and my parents, Frankie and Bill.

v

Acknowledgements

I benefited from the help of a great many people during the course of this thesis. It would be
impractical to recognize now all of those sources of support; I am blessed with thoughtful colleagues
and friends, and singularly blessed in family. Here there is space only to acknowledge those thathave been central to this project.
First, I would like to thank my undergraduate advisors, Alex Barnett and Miles Blencowe,as well as the NMR group at Schlumberger-Doll Research, in particular Pabitra Sen and MartinHürlimann, for their encouragement to attend graduate school and their patience with a fledgling
physicist.
I am also obliged to two of our group’s former graduate students, Will Kindel and Adam Reed,for speaking with me about joining Konrad’s lab, and helping me learn the basics of microwave
engineering. When I arrived, I was fortunate to begin work immediately on the circulator projectwith post-doc Joe Kerckhoff. I am indebted to Joe not only for the rich intellectual ground inwhich the work is planted, but also for his mentorship.
Collaboration with the Quantum Devices group at NIST gave a tangible form to our many
hours of design. Leila Vale fabricated the circuits and Gene Hilton and Ben Mates shared theirexpertise in superconducting circuit layout. Kent Irwin and Dale Li also shared with us theirconsiderable experience in this field.
Much of the device design and testing was done in the last two years, with help from my lab
mate Eric Rosenthal. Eric’s creativity and energy have amplified our progress on the project. More
broadly, I would like to acknowledge all of my lab mates. The open culture of ideas, the readinessvi
of others to offer help or advice, and the general sense of camaraderie make it a pleasure to work
here.
Finally, I would like to thank my advisor Konrad. It’s difficult for me to express how grateful
I am for the chance to work with him. Rather than attempt in this public setting, I will just say“Konrad: thank you.”

Contents

Chapter
1 Introduction

1

2 Quantum information processing

6

2.1

The road to quantum computation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.2

Quantum error correction and hardware architectures . . . . . . . . . . . . . . . . .

8

2.3

Circuit quantum electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Lorentz reciprocity

15

3.1

The scattering matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2

The Lorentz reciprocity theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3

Methods for breaking Lorentz reciprocity . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4

3.3.1

Magneto-optic devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3.2

Nonlinear devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3.3

Active devices

3.3.4

Connection with time-reversal symmetry . . . . . . . . . . . . . . . . . . . . . 27

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Common non-reciprocal devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4.1

Gyrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4.2

Isolators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4.3

Circulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

viii
4 Theory of operation

31

4.1

Model system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2

Superconducting implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3

4.2.1

Multiplying elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2.2

Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2.3

Assembly of the full circulator . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Circuit analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5 Circulator Layout

45

5.1

Design philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.2

The Nb/AlOx /Nb trilayer process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.3

5.4

5.2.1

Josephson junctions in the trilayer process . . . . . . . . . . . . . . . . . . . . 48

5.2.2

SQUID arrays in the trilayer process . . . . . . . . . . . . . . . . . . . . . . . 49

5.2.3

Additional wiring layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.2.4

Best practices for trilayer process layout . . . . . . . . . . . . . . . . . . . . . 52

Design considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.3.1

Capacitor design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.3.2

Use of normal metal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.3.3

Bias line design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Design iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.4.1

Generation I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.4.2

Generation II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.4.3

Generation III. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.4.4

Generation IV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6 Experimental results

77

6.1

Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.2

Tune-up procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

ix
6.3

Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.4

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.4.1

Device limitations and deviation from theoretical models. . . . . . . . . . . 86

6.4.2

Comparison with the fourth-generation, version b device . . . . . . . . . . . . 89

6.4.3

Filtering, attenuation, and power-consumption considerations, in the context
of scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

7 Conclusion and Outlook

94

Bibliography

98

Appendix
A Graph theory eigenfrequency analysis for a half-circulator

110

B Design rules of the NIST Nb trilayer process

115

C Measurement details

119

D Calibration of network parameter measurements

121

D.1 Transmission calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
D.2 Reflection calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
D.3 Calibration of group delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

TablesTable
3.1

Symbols and scattering matrices for three common non-reciprocal devices . . . . . . 28

5.1

Layer stackup for the NIST niobium trilayer process . . . . . . . . . . . . . . . . . . 47

5.2

Design features and performance of the first-generation circulator . . . . . . . . . . . 63

5.3

Design features and performance of the second-generation circulator . . . . . . . . . 65

5.4

Design features and performance of the third-generation circulator . . . . . . . . . . 71

5.5

Design features and performance of the fourth-generation circulator

5.6

Design features and performance of the alternate design of the fourth-generation. . . . . . . . . 75

circulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.1

Power budget for an active circulator . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

B.1 Design rules of the NIST niobium trilayer process . . . . . . . . . . . . . . . . . . . . 115

Figures

Figure1.1

A single-mode Fabry-Perot cavity exhibits a fundamental challenge of precision measurement: isolating the experimental system from the environment, while maintaining the ability to rapidly control or measure it . . . . . . . . . . . . . . . . . . . . . .

2

2.1

The path to fault-tolerant quantum computation . . . . . . . . . . . . . . . . . . . .

7

2.2

Signal-routing of a circulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3

Schematic illustrating a simplified measurement setup for readout of a superconducting qubit in the circuit quantum electrodynamics architecture . . . . . . . . . . . . . 13

3.1

Diagram illustrating a bounded network which electromagnetic signals can enter orexit through guided modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2

Faraday rotation and optically active media . . . . . . . . . . . . . . . . . . . . . . . 21

3.3

Requirements for non-reciprocity in cascades of linear systems . . . . . . . . . . . . . 23

4.1

Model system which demonstrates gyration with frequency-conversion and delay . . 32

4.2

Superconducting lumped-element implementation of multipliers and delays . . . . . 35

4.3

Model system realized as a lumped-element network of capacitors and dynamicallytunable inductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.4

Schematic for a circuit analysis of the circulator . . . . . . . . . . . . . . . . . . . . . 41

5.1

Josephson junctions in the Niobium trilayer process. . . . . . . . . . . . . . . . . . 48

xii
5.2

Layer stack-up of SQUIDs in the Niobium trilayer process . . . . . . . . . . . . . . . 50

5.3

Layout of SQUIDs in the Niobium trilayer process . . . . . . . . . . . . . . . . . . . 51

5.4

Capacitor design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.5

Effect of normal metal on the stability of a circulator’s resonant modes . . . . . . . . 57

5.6

Bridge circuit design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.7

Optical micrograph of the first-generation circulator . . . . . . . . . . . . . . . . . . 62

5.8

Hysteresis in the first-generation circulator with respect to sweep-direction of bias
currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.9

AC modulation of the first-generation circulator . . . . . . . . . . . . . . . . . . . . . 66

5.10 Optical micrograph of the second-generation circulator . . . . . . . . . . . . . . . . . 67
5.11 Performance of the second-generation circulator . . . . . . . . . . . . . . . . . . . . . 68
5.12 Angular dependence of the “edge” feature in the second-generation circulator . . . . 70
5.13 Optical micrograph of the third-generation circulator . . . . . . . . . . . . . . . . . . 72
5.14 Performance of the third generation circulator . . . . . . . . . . . . . . . . . . . . . . 73
5.15 Optical micrograph of the fourth-generation circulator . . . . . . . . . . . . . . . . . 74
5.16 Optical micrograph of the alternate design of the fourth-generation circulator . . . . 76
6.1

Experimental schematic for circulator measurements . . . . . . . . . . . . . . . . . . 78

6.2

Measurements of the circulator’s tunable resonant-delay . . . . . . . . . . . . . . . . 79

6.3

A dynamically reconfigurable circulator . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.4

Scattering parameter measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.5

Circulator tunability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.6

Circulator theory and experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.7

Scattering parameters of the fourth-generation circulator, version b . . . . . . . . . . 89

6.8

Performance of the fourth-generation circulator, version b . . . . . . . . . . . . . . . 90

7.1

Schematic for a lossless broadband superconducting circulator . . . . . . . . . . . . . 96

xiii
A.1 Analytical circuit analysis of a half circulator . . . . . . . . . . . . . . . . . . . . . . 111
C.1 Detailed experimental schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

List of Symbols

The tables below catalogue the symbols used throughout this thesis. They are organizedby the chapters in which the symbols are introduced (and not the chapters in which the symbols
appear). For example, kB is introduced as Boltzmann’s constant in Chapter 2 and appears only inthe table of symbols for Chapter 2, despite being used in later chapters as well. Although efforts
were made to keep notation unambiguous, some symbols are used to represent different quantities
in different chapters.
Chapter 1
Tltransmission of the left mirror in Fig. 1.1

Trtransmission of the right mirror in Fig. 1.1

aannihilation operator for the cavity mode in Fig. 1.1

ainincoming field in Fig. 1.1

aoutoutgoing field in Fig. 1.1

κexternal coupling rate for the cavity mode in Fig. 1.1

ωcresonant frequency of the the cavity mode in Fig. 1.1

Chapter 2
kB

Boltzmann’s constantTtemperature~

Planck’s constant divide by 2πωfrequency scale for a weakly anharmonic microwave resonatorxvωqtransmon qubit frequency (frequency separation between the ground andfirst excited states of the transmon)

ωrfrequency of the microwave resonator coupled to the transmon qubitaannihilation operator for the microwave resonator coupled to the transmon
qubit

σz

Pauli z operator for the transmon qubitχfrequency scale for the dispersive coupling between the transmon qubit
and the microwave resonator

Chapter 3Ωssurface that bounds the network in Fig. 3.1µ

mode index for the guided modes in Fig. 3.1

ET,µcomponent of the electric field tangential to the waveguide carrying theµth guided modeHT,µcomponent of the magnetic field tangential to the waveguide carrying theµth guided modeeT,µcomponent of the electric field tangential to the waveguide carrying theµth guided mode with transverse dependencehT,µcomponent of the magnetic field tangential to the waveguide carrying theµth guided mode with transverse dependenceaµcomplex amplitude of the wave travelling into the network in the µth waveguidebµcomplex amplitude of the wave travelling out of the network in the µth
waveguideβµpropagation constant for the µth guided modeδµν

the Kronecker deltaPthe net power entering the network in Fig. 3.1

xvi
S

the scattering matrix

a0

excitation state of a first hypothetical scattering processa00

excitation state of a second hypothetical scattering processb0

output state of a first hypothetical scattering processb00

output state of a second hypothetical scattering process

E0electric field of a first hypothetical scattering process

E00

electric field of a second hypothetical scattering process

H0

magnetic field of a first hypothetical scattering process

H00

magnetic field of a second hypothetical scattering processµ(H)magnetic permeability tensor

(E)electric permittivity tensor

ωfrequency of the harmonic fieldsAa general linear systemBa second general linear systemhA

impulse response of A

hB

impulse response of Buinitial state of a linear systemyintermediate state of a linear systemz

final state of a linear systemt, 0 t0 , η, η 0 , ξvarious variables for timehBA

impulse response for the composite-system formed by the cascade of systems A and B

hAB

impulse response for the composite-system formed by the cascade of systems B and A

ωiresonant frequency of the ith mode in a system of parametrically-coupledresonant modesxvii
linewidth of the ith mode in a system of parametrically-coupled resonant

κi

modes
HI

interaction Hamiltonian for a system of parametrically coupled modes in
the resolved sideband limitθphase of the beamsplitter interaction between modes i and jaiannihilation operator for the ith mode in a system of parametricallycoupled resonant modes

Chapter 4
ωpfrequency of the input microwave fieldΩfrequency of bias signal for the multiplying elementsθphase of a multiplying element’s bias signal

τduration of the delay provided by the delay elementl+inductance of one pair of inductors in a tunable inductive bridgel−inductance of a second pair of inductors in a tunable inductive bridgeδ

imbalance parameter for a tunable inductive bridgel0base inductance parameter for a tunable inductive bridgeccapacitance of the capacitor in Fig. 4.2b

Φuuniform magnetic flux created by an off-chip magnetic coil, used to tune
an inductive bridge realized with SQUIDs

Φggradiometric magnetic flux created by an on-chip bias line, used to tune
an inductive bridge realized with SQUIDs

Z0characteristic impedance of a transmission lineωiangular frequency of an input microwave tone√−1linductance of an N SQUID array threaded by magnetic flux ΦNnumber of SQUIDs in an array with inductance lxviiiΦ

flux threading the SQUIDs in an array with inductance l

ϕ0

reduced flux quantum ~/(2e)I0Josephson junction critical currentφphase of the gradiometric bias flux

δ0

amplitude of the sinusoidally varying imbalance parameter δαπΦu /Φ0βπΦg /Φ0

Φ0

the flux quantum 2πϕ0 = h/(2e)Jnthe nth Bessel function of the first kind∠phase (of a complex number)

ω0resonant frequency of the resonant delay

Ii

current through port i

φibranch flux across port iIl,eeven current through the left port

Il,oodd current through the left port

Ir,eeven current through the right port

Ir,oodd current through the right port

φl,eeven branch flux through the left port

φl,oodd branch flux through the left port

φr,eeven branch flux through the right port

φr,oodd branch flux through the right portI+

co-rotating current through ports q and pI−

counter-rotating current through ports q and p

φ+co-rotating branch flux through ports q and pφ−counter-rotating branch flux through ports q and p

A, B, C, D

block matrices

Yadmittance matrix

xix
So

odd scattering matrixSeeven scattering matrix

U

unitary (change-of-basis) matrix that transforms between numbered ports
and left/right, even/odd ports

Chapter 5rrelative permittivity of the silicon wafer in the NIST Nb trilayer process

Tctransition temperature of a superconductor

L

inductance of a superconducting loop

B0

magnitude of a magnetic fieldw

width of a microstripeelectron chargeQ

quality factorllength of a resistor parallel to the direction of current flow

ξc“clean” coherence lengthlnmean free pathvF

Fermi velocity

ξd“dirty” coherence lengthIn

critical current of an SNS tunnel junction

Rnroom-temperature resistance of an SNS junctionlφinelastic scattering length

EJ

Josephson energy of an SNS junctionR

low-temperature resistance of an SNS junctionΩfrequency of the on-chip gradiometric flux lines

Iindcurrent induced in the microwave lines by the RF bias signals

MAmutual inductance between a bias line and a large circuit loop

Ma

mutual inductance between a bias line a SQUID loop

xxA

area of a large circuit loopaarea of a SQUID loop

E

EMF around a large circuit loopIg

current amplitude in the on-chip bias linepparticipation ratio for the large circuit loop

ZA

impedance of the large circuit loop

Is

critical current of a SQUID

Bg

magnetic field created by the shielded, on-chip bias liner

distance from the shielded bias line

separation between the inner and outer lines in the shielded configuration

Φedge
g

amplitude of gradiometic flux where the “edge” feature appears

Chapter 6Φ̃ueffective uniform magnetic flux used for the predictions in Fig. 6.2

Rreflection coefficientTtransmission coefficientgfrequency splitting of the circulator’s resonant modes

SIAN

sideband noise power caused by amplitude fluctuations of the gradiometric
flux

SIPN

sideband noise power caused by phase fluctuations of the gradiometric flux

I1dB

signal current in the device at its 1 dB compression pointSI

Johnson noise current spectral density

nphotons of added noise

Chapter 7
Zm

impedance transformer in Fig. 7.1

T

modulation period 2π/Ωxxi
Appendix A
Ii

current in the ith element of Fig. A.1

Vivoltage in the ith element of Fig. A.1

Vb

vector of branch voltages

Vc

vector of chord voltages

Ib

vector of branch currents

Ic

vector of chord currentsrresistance of the resistors in Fig. A.1

l1

inductance of the first kind of inductors in Fig. A.1

l2

inductance of the second kind of inductors in Fig. A.1

ccapacitance of the capacitors in Fig. A.1

F

matrix relating chord currents to branch currents through Kirchoff’s current law

Zbmatrix of branch impedancesZcmatrix of chord impedancess

Laplace variablesi

ith eigenfrequency of the circuit

Appendix C
Tn

Noise temperature of the input lines

Appendix D
Rbalmeasured reflection of the circulator when all bridges are balancedGgain of the reflection measurement chainΓbalreflection coefficient of the circulator when all bridges are balanced

Sbal

scattering matrix of the circulator when all bridges are balanced

xxii
l

effective inductance of one arm of an inductive bridge in the circulator’sbalanced (unbiased) stateRopmeasured reflection during operation

Γopreflection coefficient during operation

τ

duration of the circulator’s resonant delay

τdelectrical delay of the measurement chain

Chapter 1

Introduction

The idea to use quantum systems for computation was first proposed in the 1980s by notablescientists such as Feynman [1, 2] and Deutsch [3, 4]. The notion gained major traction in 1994
when Peter Shor showed how such a machine could efficiently factor composite integers [5]. (TheRSA encryption scheme [6]—a widely utilized encryption method for commerce and other forms
of sensitive communication—is founded on the difficulty of this task.) Factoring large composite
integers is but one example from a list of difficult problems that could be solved by these efforts:
others include efficient search algorithms [7], quantum chemistry and catalyst design [8, 9, 10], andcommunication whose security is guaranteed by physical laws [11]. For these reasons, efforts tocreate analog and digital quantum computers, quantum simulators, and quantum annealers span
many realms of physics.
Across all of the proposed platforms for constructing a quantum computer, one of the key
challenges is engineering a quantum system which is both well isolated from its environment (toprotect it from dissipation and decoherence) and rapidly measurable (and controllable). This
challenge is not unique to problems specific to quantum computing; rather, its roots extend into
the theory of quantum measurements. According to Heisenberg, measurements of a system are
perturbing, as illustrated famously (and controversially) by his microscope thought experiment [12].
The problem is also present in purely classical systems. Heuristically, oscillators that interact
weakly with their environment respond slowly to control fields, and oscillators that respond quickly
to control fields rapidly dissipate energy into their environment. This is no conspiracy: from the2perspective of the oscillator, the experimentalist is a part of the environment.
To illustrate this issue more quantitatively, consider a system formed by a single electromagnetic mode in a Fabry-Perot cavity, where the left mirror is perfectly reflective (vanishing transmission) and the right mirror has a transmission Tr (Fig. 1.1). The quantum Langevin equations

a

ain

Tr

Tl=0

aout

Figure 1.1: A single electromagnetic mode in a Fabry-Perot cavity, characterized by the annihilation
operator a. The left mirror is perfectly reflective, and the right mirror has transmission Tr , which
couples the cavity mode to an incoming field ain and an outgoing field aout .

that describe the time-evolution of the cavity field a and the outgoing field aout are [13]ȧ = iωc a −aout =√√
κa + κain2κa − ain .

(1.1)
(1.2)

Here, ωc is the frequency of the cavity mode, and κ is a rate determined by Tr which describes thecoupling between the cavity and the incoming and outgoing fields ain and aout .
In this example, the cavity mode is the system of interest, which an experimentalist may
hope to study via measurements of the outgoing field aout . The prospects for that aspiration canbe understood by examination of two limiting cases, in which the transmission of the right mirror
is small or large.
When Tr approaches 0, the coupling rate κ also vanishes. This is beneficial from the perspective of isolating the system of interest from its environment: the dynamics of the cavity mode, given
in Eq. (1.1), are now uncoupled from the external field ain . Unfortunately, the experimentalist has
also forfeited the ability to infer the state of the cavity from measurements of the outgoing field
aout (Eq. (1.2)).

3
In the opposite limit where Tr is large, κ may become non-negligible with respect to thecavity frequency ωc . The outgoing field now contains considerable information about the state of
the cavity mode. The cost of this encoding, however, is that the time-evolution of the cavity mode
is now subject to the “slings and arrows” of the environment, in the form of the incoming field ain .
The link between the environment’s effect on the cavity mode (through ain ) and the cavity
mode’s radiation into the environment (through aout ) is an instance of the fluctuation-dissipation
theorem [14, 15]. The right mirror’s transmission creates dissipation in the cavity mode (which
may be measured and recorded by an experimentalist) but also causes fluctuations in the mode,
driven by the incoming field.
As an aside, we note that in this example, measurement of the cavity state requires that it
decays into a propagating mode. This is not a general requirement for measurement of a quantum system. For example, quantum non-demolition measurements [16, 17] allow measurement
without dissipation [18]. Measurements are not necessarily accompanied by dissipation; they areaccompanied by disturbance.
Returning to the measurement of the cavity mode, the examined limiting cases of large
and small κ suggest that our measurement dilemma cannot be resolved by optimization of thecavity’s parameters. The solution is external, and involves directing the outgoing field aout tothe measurement apparatus, while ensuring that the incoming field ain is supplied by a very coldbath. In a classical analysis, this strategy can protect the cavity mode from its environment, whilestill allowing an experimentalist to study the state of the mode. Quantum mechanically, vacuum
fluctuations persist even at zero temperature. But the influence of these vacuum fluctuations on thecavity’s dynamics are small compared to, for example, the black-body radiation of the measurement
apparatus.
In the vacuum of free-space, however, Maxwell’s equations make this kind of directionalsignal routing impossible. Practically, an interaction between electromagnetic waves and matteris required to separate signals based on their direction of propagation. Devices which do so arecalled non-reciprocal, in reference to their violation of Lorentz reciprocity [19]. In the context of

4quantum networks, non-reciprocal devices serve a variety of purposes beyond the fundamental one
described above. For example, they enable protocols for the generation of remote entanglement [20,21, 22] and proposals for the construction of rich quantum many-body systems [23] that rely on
directional signal routing, and allow for the use of quantum-limited reflection amplifiers [24, 25].
Such applications impose a strict set of design constraints: for quantum measurements, the ideal
non-reciprocal device has no loss and no added noise. These requirements make familiar nonreciprocal devices like diodes and op-amps unsuitable for use with quantum systems.
More broadly, enforcing the unidirectional flow of energy and information is a critical signal processing primitive in a variety of networks. A common use in telecommunication networks,
for instance, is antennae duplexing, in which the incoming and outgoing signals from an antenna
are separated, allowing for simultaneous transmission and reception. Given their utility, it shouldcome as no surprise that commercial non-reciprocal devices have been developed across the electromagnetic spectrum, and are widely used in a variety of experimental contexts. For example,
in experiments on quantum superconducting circuits—one of the most promising platforms forthe development of a quantum computer—non-reciprocal devices are ubiquitous. State-of-the-art
commercial non-reciprocal devices, however, are constructed with large permanent magnets andgyrotropic media, precluding their miniaturization for on-chip applications, as well as their construction with superconducting materials.
The optimal architecture for a quantum computer built with superconducting circuits is verymuch still an open research question. The surface code, though, is one of the leading contenders,
and would require over a hundred million physical qubits to accomplish the task [26].1

As eachof these qubits must be simultaneously shielded from its environment and controlled in hardware,
the need for a scalable non-reciprocal technology is pressing.
To address that need we have developed an alternative method for routing propagating electromagnetic fields with minimal loss or added noise, which uses no permanent magnets and can be
1

This estimate is for a computer that could execute a practical quantum computation—in this case, Shor’salgorithm on an integer with 600 digits—in approximately one day.

5
integrated on-chip with superconducting circuits in a scalable way. In this thesis we describe thedesign, construction, and measurement of a non-reciprocal device which is based on this method.
Design goals for the circuit were set with a superconducting qubit application in mind, andin the context of modular quantum networks. But the device will also find use in other lowtemperature microwave networks, especially ones which require high measurement efficiency. Examples include rapid single flux quantum logic [27]; microwave kinetic inductance detectors [28],transition edge sensors [29] and other astronomical detector arrays; dark-matter searches [30]; andtests of Lorentz invariance with high-quality microwave resonators [31].
The structure of the thesis is as follows: Ch. 2 gives a brief overview of quantum information
processing, and efforts to process quantum information with superconducting circuits, broadly
referred to as circuit quantum electrodynamics. The assumptions and consequences of the Lorentz
reciprocity theorem are discussed in Ch. 3, along with other proposals and demonstrations forcreating non-reciprocal devices. Ch. 4 describes the theory of operation for our device, and Ch. 5
describes its realization as a superconducting circuit, as well as relevant layout considerations.
Experimental results are summarized in Ch. 6. Finally, conclusions and an outlook for the work
are presented in Ch. 7.

Chapter 2

Quantum information processing

The enormous computational power of a quantum processor has attracted interest from avariety of subfields within physics and other disciplines. In the atomic domain, work is underway
to store quantum bits of information (qubits) in the electronic states of trapped ions and Rydberg atoms. Notable solid-state platforms include nitrogen (and other) vacancy centers, in which
information is encoded in the composite spin of the vacancy; semiconductor quantum dots, where
the qubit is formed by the singlet state and the projection-less triplet state of two confined electron spins; and superconducting circuits, where information is stored in the ground or first excitedstate of an anharmonic microwave circuit. Many other platforms are also being investigated (optical, NMR, etc); Refs. [32, 33] provide a high-level overview of these approaches. Even within
the experimental systems noted above, there exist a variety of ways to encode a bit of quantum
information. 1

More exotic proposals also exist, in which the braiding of topological excitations

with non-Abelian statistics provides the basis for computation [37]. While the listed platforms andphysical realizations of a qubit are intended to be representative of the field’s breadth, they are far
from exhaustive.
1

For example, some researchers in the field of superconducting circuits use microwave resonators as the memory
elements, and construct the computational basis {|0i = |αi + |−αi , |1i = |iαi + |−iαi} with even superpositions ofcoherent states |αi—so-called cat codes [34], named for the cat states [35] that comprise them. Other resonator-based
schemes also exist; Ref. [36] provides an overview and comparison of several prominent codes.

7

2.1

The road to quantum computation
Across all of these platforms, however, the ultimate goal of fault-tolerant quantum computa-tion is shared. Fig. 2.1, from Ref. [38], illustrates one conception of the path to this goal.

Fault-tolerant quantum computation

Complexity

Algorithms on multiple logical qubits
Operations on single logical qubits
Logical memory w/ longer lifetime than physical qubits
QND measurements for error correction and control
Algorithms on multiple physical qubits
Operations on single physical qubits

Time
Figure 2.1: One conception for the path to fault-tolerant quantum computation, divided into sevensteps. The y-axis of the graphic illustrates the relative complexity of each stage—advancing along
the path requires mastery of the previous steps. The x-axis, time, indicates that active efforts toimprove at each stage are ongoing. After [38].

The first step involves operations (also known as gates) on single physical qubits. This isfollowed by algorithms (long sequences of gates) on multiple physical qubits. Completing these
initial two steps requires satisfaction of the first five DiVincenzo criteria [39] (a list of five necessary
conditions for constructing a quantum computer, plus two additional conditions for quantum communication). Following this is a more advanced stage, requiring quantum non-demolition (QND)
measurements of qubits and advanced control. This stage enables, for example, QND measurementsof error syndromes like parity, and the stabilization of arbitrary quantum states. Demonstrationsof this third stage have been made with trapped-ions [40], Rydberg atoms [41], Nitrogen vacancy
centers [42], and superconducting qubits [43, 44, 45].
The fourth step is the creation of a logical memory with a lifetime that exceeds the physicalqubits which comprise it. This step, which entails encoding information redundantly among mul-

8
tiple physical qubits and actively correcting for errors in that encoding, is critical, as it allows thepreservation of quantum information for timescales long enough to enable meaningful computation.
Two groups have demonstrated this stage with superconducting circuits [46, 47].
The final two stages—operations on single logical qubits and algorithms on multiple logicalqubits—mirror the initial two stages of the progression, but for logical qubits in place of physicalqubits. One caveat for these steps: they must be completed without degrading the lifetime of thelogical qubits. The first demonstrations of operations on single logical qubits [48, 49] were madethis year with superconducting circuits. As of yet, no experimental platforms have demonstrated
the sixth stage.
Naturally, the different experimental platforms have their specific affordances and drawbacks.
The purpose of Fig. 2.1 is not to compare progress among subfields, but rather to emphasize thecommonalities among the different approaches, highlight the progress that the field as a whole
has made, and motivate superconducting circuits as a viable platform for quantum information
processing. Indeed its promise as a subfield has attracted interest outside the academic realm: avariety of private enterprises (IBM, Google, Rigetti Computing, DWAVE Systems, etc.) are now
actively engaged in the construction of a superconducting quantum computer.

2.2

Quantum error correction and hardware architectures
As efforts in industry and academic settings look forward and plan their ascent to the upperechelon’s of Fig. (2.1), their hardware requirements will depend strongly on the architectures they
employ, especially in the context of error correction. A detailed review of these proposals is beyond
the scope of this thesis, but a brief overview is helpful in estimating the hardware resources required
for a fault-tolerant quantum computer.
It is now widely accepted that error correction will occupy a majority of the resources ofa fault-tolerant quantum computer [38]. One approach to error-correction involves the use ofstabilizer codes [50, 51, 52], in which a logical qubit is formed by redundantly encoding information
in a register of entangled physical qubits. The Steane code, for example, can be implemented with

9seven physical qubits in its register [51]. Under the assumption that error rates are sufficiently low
such that at most one error occurs during each pass of the code, errors are detected by measuringcollective properties of the register (the eponymous “stabilizer operations”). Additional gates canthen be applied to correct the error, if needed.
The assumption that errors occur singly is, however, quite stringent. For the range of currently conceivable error rates, its satisfaction requires that the protocol be nested, such that eachelement in the error correcting register is itself a logical qubit, stabilized by its own register, which
could itself be composed of logical qubits, and so on. As the stabilizer codes are also fairly operation intensive—that is, they drastically increase the number of gates required to implement
an algorithm—stabilizer codes are recognized as a feasible but challenging way to perform error
correction.
A more recent but related approach is known as the surface code, in which identical physicalqubits are arranged in a rectangular array, which forms a surface [53, 26]. Connecting groupsof four nearby qubits in a particular way allows rapid measurements of their parity to detect
errors. The advantages of this approach are its flat hierarchy, which can be easily scaled once
demonstrations of a unit cell are made, and its relatively relaxed requirements for single-qubit
error rates (with respect to the stabilizer codes, for example). The drawbacks, however, are thata very large number of qubits (current estimates are on the order of 108 or 109 physical qubits fora modest quantum computation [26]) are required to reach a fault-tolerant level with the protocol,and that these benefits do not become substantial in systems with less than hundreds or thousandsof physical qubits [38]. Nonetheless, efforts to demonstrate proofs-of-principle are already underway
at Google [46] and IBM [54]. Ref. [55] gives a good overview and summary of the technical challenges
relevant to this architecture.
A third strategy, which may be called the modular approach, is based on networks of nested
modules, which rely heavily on hierarchy. In this scheme, information is stored in memory qubits
which interact with the greater network via intermediate communication qubits. Computation is
performed via manipulations of the communication qubits, which distributes entanglement across10
the network. In some sense, this strategy represents the antithesis of the surface code approach: its
affordances are the capability to test and iterate individual modules and processes, and possibly a
lower requirement of physical qubits, at the cost of increased architectural and process complexity.
Ultimately, as with the choice of experimental platforms, selection of the optimal architecture
for fault-tolerant quantum computing is an open question. But while the precise structure is yet
to be determined, it seems clear that the necessary hardware resources will be immense.

2.3

Circuit quantum electrodynamics
To provide a more concrete notion of these hardware needs, we specialize now to a discussionof quantum information processing with superconducting circuits. Detailed reviews on this subject
may be found elsewhere [56, 57]. Here we provide only an informal introduction, intended to give
context for the design of a scalable circulator for superconducting quantum circuits.
The qubits in this platform are usually formed by slightly anharmonic electromagnetic oscillators. At sufficiently low temperatures, kB T < ~ω, the quantized energy levels in these circuits areresolvable. Here, kB is Boltzmann’s constant, T is the temperature of the environment, ~ is Plank’s
constant over 2π, and ω is a frequency scale for the anharmonic oscillator. Modern dilution refrigerators allow for experiments at temperatures in the tens of mK, which makes the energy available
from the thermal bath about ten times less than a photon at 5 GHz. Hence, the oscillators aredesigned to have resonant frequencies in the microwave range, typically between 1 and 20 GHz.
Like atoms, superconducting qubits come in a variety of forms. Most, though, can be modelled as a network of capacitors, inductors, and Josephson junctions. The Josephson junction [58]provides a dissipationless nonlinearity to the circuit, which breaks the harmonic spacing of its energy levels and allows individual transitions to be addressed with coherent microwave tones. A
qubit can then be created by restricting attention to a pair of energy eigenstates (typically, the twolowest energy levels).
Readout of such a qubit, in the circuit quantum electrodynamics (cQED) scheme [59], is11accomplished by dispersively coupling the qubit to a microwave cavity.2

In the dispersive limit,

the cavity’s resonant frequency is dressed by the state of the qubit, and a transmission measurementof the cavity can therefore indicate the state of the qubit in a QND manner [60, 61].
For a transmon3 in the dispersive limit, the effective Hamiltonian that governs the qubitcavity system is given by [62]H = ~ωq σz + ~ (ωr + χσz ) â† â,= ~ ωq + χâ† â σz + ~ωr â† â.

(2.1)

Here ωq is the qubit frequency (the difference between the ground and first excited states of thetransmon), ωr is the resonator’s frequency, â is the annihilation operator for the cavity, σz is thePauli z operator for the qubit, and χ is the dispersive shift—a parameter that characterizes thecoupling between the qubit and the cavity. In the first line of Eq. (2.1), the Hamiltonian is factored
to make explicit the qubit-state-dependent dressing of the cavity’s resonant frequency.
The same effect, however, also dresses the qubit frequency by an amount proportional to thephotonic occupation in the cavity, as shown in the second line of Eq. (2.1). This allows the qubit
to act as a photon counter/detector [22], and reveals the importance of controlling the photonic
occupation in the cavity: a “hot” cavity, occupied by a high-temperature thermal state, will containa broad distribution of Fock states, effectively broadening the qubit transition. This effect is so
pronounced that superconducting qubits have become the most sensitive thermometers in theseexperiments. Measurable reductions in qubit coherence are observed with average cavity-photon
occupations of 10−3 [63].4
In this light, one can see that the need to enforce the directional propagation of signals
in cQED measurements arises directly from the Hamiltonian in Eq. (2.1). To probe the state of
the qubit, electromagnetic waves departing the resonant cavity must be directed to a microwave
2

Typically a distributed resonator formed, for example, by a quarter-wave transmission line, or a literal (threedimensional) metal cavity.
3
One example, chosen from a selection of different superconducting qubits (see, for example, Sec. 2 in Ref. [57]).
4
As an aside, it may be noted that in some implementations this scheme is inverted, and a microwave resonatorserves as the memory element, while a qubit such as the transmon provides a dispersive nonlinearity that allows
individual transitions in the resonator to be addressed [64]. This is advantageous as coherence times in microwave
cavities currently exceed those in superconducting qubits [65].

12
receiver. At the same time, however, electromagnetic waves propagating toward the cavity should
originate from a cold bath—in practice, a 50 Ohm resistor to ground, well-thermalized at the baseof a dilution refrigerator—to avoid dephasing the qubit.
Experimentally, this uni-directional signal flow is enforced by non-reciprocal circuit elements
known as circulators and isolators. Isolators are impedance-matched two port devices which transmit a signal incident on one of their ports, but absorb signals incident on their other port. Circulators are impedance-matched n-port devices with three or more ports (n ≥ 3), which transmit
signals incident on port m ≤ n to port 1 + (m modulo n): they “circulate” an incident signal,
directing it out of one of the two adjacent ports.
Fig. 2.2 depicts this process for an n = 3 port circulator. In Fig. 2.2a, a signal incident on
the circulator’s first port is transmitted to its second port. In Fig. 2.2b, a signal incident on port
2 is routed to port 3. Fig. 2.2c shows a signal incident on port 3 directed to port 1. A circulator
may be configured as an isolator by terminating all but two adjacent ports in 50 Ohms.

...

(c)

...

port 3

(b)

...

(a)

port 3

...

...

...

port 1

port 2port 1

port 3...

port 2...

port 1...

port 2

Figure 2.2: Schematic displaying the way in which an n = 3 port counter-clockwise circulator (circle
with blue counter-clockwise arrow) routes incident fields. (a) A field incident on the circulator’s
first port is transmitted out the second port. (b) A field incident on the second port is routed out
the third port. (c) A field incident on the third port is routed out the first port.

To illustrate how circulators and isolators are utilized in the cQED architecture, Fig. 2.3
shows a simplified experimental schematic for readout of a superconducting qubit coupled to aresonant cavity. To reduce the thermal population of states in the qubit-cavity system, the devices
are mounted at the base of a dilution refrigerator. In the readout procedure, a signal generator atroom temperature first creates a microwave tone which propagates to the base of the refrigerator

13
and impinges on the weakly-coupled port of the resonant cavity (depicted as a Fabry-Perot cavity
in Fig. 2.3). The tone then traverses the cavity, and a portion of its power exits out the second,
strongly-coupled cavity port. Here the tone is transmitted through an isolator and directed by acirculator to a “quantum” reflection amplifier, where the signal is reflected with gain of approximately 20 dB. Here, “quantum” is used to mean an amplifier with added noise approximately equal
to half a photon [66]—for example, a Josephson parametric amplifier [24] or a Josephson parametricconverter [25]. Next, the circulator routes the tone through a second isolator, is amplified a second
time by a high-electron mobility transistor (HEMT) amplifier, and finally propagates up the fridgefor further room-temperature amplification and digitization. In various schemes, the phase or amplitude of the transmitted tone can be made to encode the state of the qubit, via the interaction
described in Eq. (2.1).

ADC

300 K

4K

50

50

10 mK

HEMT
amplifierquantumreflection
amplifierFigure 2.3: Simplified experimental schematic illustrating readout of a superconducting qubit in adilution refrigerator. Signal filtering and attenuation, and control lines of the reflection amplifier,are omitted for clarity. The superconducting qubit (green cross with capacitive paddles) is situatedinside a microwave resonant cavity (illustrated graphically by the green Fabry-Perot resonator)
with two ports. A measurement of the transmission through the cavity can encode the state of
the qubit. Circulators and isolators are used to direct the signal that exits the cavity out to a
detector, while ensuring that the reverse-propagating signals originate from a cold-bath (and arenot, for example, carrying the Johnson noise of the HEMT amplifier, or the strong pump tones in
the reflection amplifier).

14
In this measurement, the signal routing capabilities of circulators are used in several ways,
as evidenced by the multiple circulators employed in the readout of a single qubit. The first
circulator, configured as an isolator, directs reflected signals and the strong pump tones powering
a quantum reflection amplifier into a cold bath (serving as an entropy dump), while supplying thestrongly coupled port of the qubit/cavity system with Johnson noise from this cold bath. The
second circulator separates the incoming and outgoing signals for a quantum reflection amplifier,transforming that one-port device into a usable amplifier. A third circulator, again configured as
an isolator, directs the Johnson noise emitted by the HEMT amplifier into an entropy dump, andreplaces it with the Johnson noise from a cold bath.
It should be emphasized that the schematic in Fig. 2.3 illustrates the need for three circulators in a relatively simple single-qubit experiment. Experiments with multiple qubits, or additional requirements for directional signal routing, will employ more circulators. For example,
a recent demonstration of the generation of remote entanglement between two superconducting
qubit/cavities, using a third qubit/cavity as a detector, employed seven circulators [22]. Even withmultiplexed readout and broadband (several GHz) circulators, a surface code quantum computerwith modest capabilities [26] could still easily require millions of circulators.

Chapter 3

Lorentz reciprocity

First observed in optical systems, where it is referred to as Helmholtz reciprocity, StokesHelmholtz reciprocity, or the principle of reversibility, Lorentz reciprocity can be summarized colloquially as “if I can see you, you can see me” [67, 68]. In a more general sense, Lorentz reciprocity is
the electromagnetic version of a symmetry common to a variety of physical systems, which implies
that the response at point a due to an impulse at b is identical to the response at b due to an
impulse at a. The prevalence of reciprocity theorems stems from the invariance of many systems
under time-reversal [69].
As in the electromagnetic case, breaking reciprocity can be of fundamental scientific interest
and great technological use. Efforts to generate non-reciprocity are therefore also underway in, forexample, acoustical and mechanical systems [70, 71]. Connections between non-reciprocity and theone-way propagation of edge-states in topological systems have also been observed, and are driving
research in topological acoustics [72, 73], metamaterials [74], and non-reciprocal devices based on
the quantum Hall effect [75, 76, 77].
In the electromagnetic context, Lorentz reciprocity is essentially a statement about the timereversibility of Maxwell’s equations [19], which occurs under certain assumptions: in a framework
where electromagnetic fields propagate in guided modes into and out of a bounded network at ports,
Lorentz reciprocity implies that the scattering between a pair of ports is invariant upon exchangeof the source port and the detection port [78, 79]. This statement can be proven and expressedsuccinctly in a scattering matrix formalism, which we now introduce, following Ref. [80].

16

3.1

The scattering matrix
Consider a bounded region of space, enclosed by the surface Ωs , which contains a linear

electrical network with time-independent components (Fig. 3.1). Energy can be exchanged between
the network and the environment (the region outside Ωs ) only through lossless and reciprocal
waveguides, which we refer to as ports. (We assume the surface is sufficiently large such thatradiative modes may be neglected, and shaped such that each waveguide is normal to Ωs ).
aM
Ωs

...

bM
port 3b2

port Ma3
port 1a1...

port 2b1 a 2

...

...

b2

Figure 3.1: Model of an electrical network as a region of space bounded by the surface Ωs . Electromagnetic signals can propagate into and out of the network in guided modes, referred to as ports.
After [80].

For each waveguide, we imagine a coordinate system with the z-axis oriented along the
waveguide, and directed into Ωs . These waveguides carry guided modes which are eigensolutionsof Maxwell’s equations, propagating in the +ẑ and −ẑ directions. The components of the electromagnetic fields tangential to Ωs have the formaµ e−iβµ z + bµ eiβµ eT,µ (x, y),HT,µ (x, y, z) =aµ e−iβµ z − bµ eiβµ hT,µ (x, y).
ET,µ (x, y, z) =



(3.1)

Here µ is the mode index, aµ and bµ are the complex amplitudes of the waves travelling into andout of the network, and βµ is the propagation constant. The modes are normalized such that

17Z(eµ × h∗ν ) dA = 2δµν ,

(3.2)

Ωs

where ∗ indicates complex conjugation and δµν is the Kronecker delta.
Eq. (3.2) has two interesting consequences. First, at the surface Ωs the tangential componentsof the fields have the simple formET =X

(aµ + bµ ) eT,µ ,

(3.3)

µ

HT =

X

(aµ − bµ ) hT,µ ,µwhich is useful in deriving one of the principal consequences of the Lorentz reciprocity theorem for
networks.
Second, the power P carried into the network is just the difference of the wave power entering
the surface and the wave power exiting the surface:XP =|aµ |2 − |bµ |2
µ

(3.4)

= a∗t a − b∗t b.
Here in the second line we’ve introduced a vector notation a and b for the complex amplitudes aµand bµ , and superscript t indicates vector transposition.
Our assumption that Ωs encloses a passive and linear network ensures that the vector b iscompletely determined by a. If this were not the case, the network could scatter an excitation a into
distinct outputs b and b0 . The linearity of the network implies that a null excitation (0 = a − a)could therefore scatter into b − b0 . Eq. (3.4) shows that the power carried out of the network inthis event is non-zero, in violation of the assumption that Ωs encloses a passive network.
As the network is assumed to be linear, there is thus a unique matrix S which maps a to b:
Sa = b.

(3.5)

The matrix element Sµν is the ratio of the outgoing field at port µ to the incident field at port ν.
The matrix S is known as the scattering matrix, for its description of how the network bounded byΩs scatters incident fields.

18

3.2

The Lorentz reciprocity theorem
Consider two scattering processes, with excitation states a0 , a00 , output states b0 , b00 , andtheir corresponding fields E0 , E00 and H0 , H00 . With no electromagnetic sources, the time-harmonic
form of Maxwell’s equations for the fields of the first scattering process is∇ × E0 = −iωµ(H0 )H0 ,(3.6)∇ × H0 = iω(E0 )E0 .

(3.7)

Here µ(H) and (E) are the magnetic permeability and electrical permittivity tensors of the mediain Ωs . Taking the inner product [·] of Eq. (3.6) with H00 and Eq. (3.7) with E00 and summing themyields
H00 · ∇ × E0 + E00 · ∇ × H0 = iω E00 · (E0 )E0 − H00 · µ(H0 )H0 .

(3.8)

Repeating this procedure with the second scattering process yields the same result, with interchanged primes. When the two equations are subtracted, one obtains
∇ · E0 × H00 − E00 × H0 = iω E00 · (E0 )E0 − E0 · (E00 )E00 − H00 · µ(H0 )H0 + H0 · µ(H00 )H00 .
(3.9)
A vector identity for the divergence of the cross-product of two vectors was used to simplify theleft-hand side of this expression. It may be further simplified to have a vanishing right-hand side,
if the two following conditions are met:(1)  and µ are symmetric tensors:  = t and µ = µt .
(2)  and µ describe linear media.

19
To see this, consider one of the terms on the right-hand side:
E00 · (E0 )E0 =E00 · E0= E00 · E0= E00t E0tt= E0t t E00

= E0 · E00


=E0 ·  E00 E00 .

(3.10)

Here we’ve used the linearity of  in the first line and the the fact that scalars are invariant undertransposition in the second line. The second to last line follows from the symmetry of . A magnetic
term can be similarly commuted, to obtain the Lorentz reciprocity theorem:∇ · E0 × H00 − E00 × H0 = 0.

(3.11)

Eq. (3.11) gains a clear physical interpretation when it is expressed in the scattering matrix
formalism. Integrating it over the volume enclosed by Ωs and applying the divergence theorem
yieldsZE0 × H00 − E00 × H0 · dA = 0.

(3.12)Ωs

When the modal expansions in Eq. (3.4) are substituted into Eq. (3.12), the minus sign ensuresthat the cross-terms add constructively, while the other terms add destructively:
0 = 2

X

b0ν a00µ − a0µ b00νZeµ × hν dA,(3.13)

Ωs

µ,ν
0t 00

= b a − a0t b00 ,= a0t St − S a00
Here we’ve used the orthogonality condition in Eq. (3.2) and the definition of the scattering matrix
in Eq. (3.5). As a0 and a00 are arbitrary, the scattering matrix must be symmetric:
S = St .

(3.14)

20
This symmetry ensures that the scattering between a pair of ports is invariant under exchange ofthe source port and the detection port: Sµν = Sνµ .

3.3

Methods for breaking Lorentz reciprocity
Concepts for non-reciprocal devices may be broadly divided into categories, based upon whichassumption of the Lorentz reciprocity theorem they violate.

3.3.1

Magneto-optic devices

Virtually all the circulators and isolators currently used in cQED experiments are constructedwith magneto-optic materials. A common example of this is the Faraday-effect [81], in which left
and right circularly polarized fields propagating in a gyrotropic material (which is biased by amagnetic field parallel to the propagation direction) have different group velocities. Careful choiceof the propagation length and or magnetic field strength therefore allows the polarization of the
field to be rotated by the desired angle.
Critically, the polarization’s sense of rotation depends only on the orientation of the magnetic
field, and not on the direction of propagation. Fig. 3.2a and Fig. 3.2b illustrate this process. The
effect, from the perspective of an observer receiving the propagating field, is therefore a propagationdirection-dependent rotation of the polarization. The observer’s perspective is relevant here as thisrepresents the perspective of a detector receiving the propagating field, and Lorentz reciprocity is
an invariance upon exchange of source and detector.
In optical applications, this phenomenon is combined with a pair of polarizers to create anisolator. In the microwave domain, it is commonly used to couple signals (non-reciprocally) between
orthogonally-polarized waveguides, realizing a circulator. Ref. [83] provides a detailed descriptionof ferrite-junction circulators that operate via the Faraday effect.
The Faraday effect may be contrasted with a similar (and reciprocal) phenomenon in optically
active materials [81], in which the chirality of the crystal structure rotates the polarization of the
propagating field. This causes the polarization vector’s sense of rotation to reverse when the21aobserver/detector𝐁bsource𝐁𝛽𝛽source2𝛽observer/detectordcobserver/detector𝛽source𝛽sourceobserver/detector

Figure 3.2: Faraday rotation and a seemingly similar but distinct phenomenon observed in optically
active media. (a) An electromagnetic wave propagating from left to right through a Faraday-active
medium in the presence of an applied on-axis magnetic field B. From the perspective of an observer(or a detector) receiving the field, the polarization of the wave rotates clockwise by an angle β. (b)If the signal is reflected and propagates back through the medium (from right to left), the observer
perceives a counter-clockwise rotation of the wave’s polarization by the same angle β. Transmission
is non-reciprocal. (c) A wave propagating left to right through an optically active medium, where amaterial chirality (depicted as a helix) determines the sense of rotation. An observer receiving thewave perceives a clockwise rotation of the polarization. (d) Reflection of the wave and right to left
propagation results in the same clockwise rotation of the polarization, and reciprocal transmission.
After [82].

propagation direction is inverted, in the same way that pushing or pulling on a right-handed screw
will induce clockwise or counter-clockwise rotation. From the perspective of an observer receiving
the propagating field, however, the rotation-direction is independent of the propagation direction,
as depicted in Fig. 3.2c and Fig. 3.2d.
Magneto-optic devices are unconstrained by reciprocity because the permeability tensor µ is
not symmetric in the presence of a static magnetic field. This leads to a non-vanishing right-handside of Eq. (3.9), and a breakdown of the reciprocity theorem.
With respect to the design needs of superconducting qubit applications, magneto-optic non-

22
reciprocal devices are attractive in that they may have broad operation bandwidths, low (butnot vanishing) insertion loss on the order of several tenths of a dB, and they are passive devices.
Their drawbacks include their roughly 10 cubic cm volumes, and their reliance on large permanent
magnets, which makes them difficult to miniaturize (despite many efforts [84, 85, 86, 87, 88, 89])and integrate with superconducting circuits, and precludes the elimination of their insertion loss
by replacing their metallic components with superconductors.

3.3.2

Nonlinear devices

Nonlinear devices can break reciprocity because the intensity-dependence of their material
properties prevents the cancellation of terms like
E00 · (E0 )E0 − E0 · (E00 )E00 .

(3.15)As in the case of magneto-optic devices, this leads to a non-vanishing right-hand side of Eq. (3.9).
In practice, this is leveraged by introducing some kind of spatial asymmetry into the designof the device. A simple example of this is a finite-length rectangular waveguide which is filled by
distinct dielectrics on its two ends [90]. The device is reciprocal, but careful choice of the dielectrics
and the relative fractions of the waveguide that they fill makes the spatial profile of the electricfield dependent on the propagation direction of an incident signal. By positioning a nonlinear
resonator at a location within the waveguide where the intensity of the electric field is strong whena signal is incident on one port of the waveguide but weak when a signal is incident on its otherport, a propagation-direction-dependent loss-channel can be introduced to the system—usually
by arranging the frequency-pulling of the nonlinear resonator to move the resonator in or out of
resonance with the incident signal. Different implementations of this general scheme have resulted
in a variety of realizations for optical diodes constructed with silicon [91, 92, 93, 90].
The advantages of this approach include ease of miniaturization, and compatibility withsuperconducting circuits and fabrication methods. The disadvantages are an inherent restriction
on linearity (such schemes only work when higher-power incident fields or pump tones activate a23material nonlinearity), and typically narrow operation bandwidths.

3.3.3

Active devices

Given the intimate connection between reciprocity and time-reversal, it seems intuitive that
active devices—that is, devices with parameters that vary in time—could break reciprocity. Thiscan be seen explicitly with a simple example of two general cascaded linear systems A and B,
which we describe following Ref. [94], and depict in Fig. 3.3. The impulse responses of A and B
are denoted hA and hB , and u, z, and y describe the input, intermediate, and output states of thesystem.

u

z

A

y

B

Figure 3.3: A cascade of two linear systems. The impulse response of the composite system is nonreciprocal when the impulse responses of the individual systems are time-dependent. After [94].

The output y can be written in terms of the input u and the impulse responses hA and hB :Z ∞hB (t, η)z(η)dη(3.16)
y(t) =−∞

Z ∞z(t) =hA (t, ξ)u(ξ)dξ.

(3.17)−∞

Eliminating z in Eq. (3.16) yields
Z ∞ Z ∞

y(t) =hB (t, η)hA (η, ξ)dη u(ξ)dξ,−∞(3.18)−∞and the bracketed term in that expression may be identified as the impulse response hBA of thecomposite system BA.
When A and B are time-invariant systems, their impulse responses depend only on the
difference of their arguments,
Z ∞hB (t − η)hA (η − ξ)dη.

hBA =
−∞

(3.19)

24
Changing integration variables to t0 = t − ξ and η 0 = η − ξ, Eq. (3.19) becomes
Z ∞
hBA =

hB (t0 − η 0 )hA (η 0 )dη 0 .

(3.20)−∞

which is a convolution. In this form (or from the convolution theorem), it is clear that the system
is reciprocal: hBA = hAB . For the time-dependent case, though, hBA is not in general equal tohAB .
The dependence of reciprocity on time-invariance is reflected in the arguments of Sec. 3.1
and Sec. 3.2, which respectively assume passive networks and harmonic fields. Efforts in the 1960s
extended these arguments to time-dependent systems, and showed that reciprocity also places
symmetry and time-invariance constraints on the scattering matrices of time-variable networks [95].
Since then, a variety of schemes have been developed for generating non-reciprocity with actively
modulated circuits. We do not attempt a complete catalogue of them here, and instead limit
discussion to two particular approaches.

3.3.3.1

Parametric coupling of resonant modes in the resolved sideband limit

The first of these involves the parametric coupling of resonant modes in the resolved sideband
limit, which is a widely used strategy in radio frequency [96, 97], superconducting microwave [82, 98,
99, 100, 101], electroacoustic [102], electromechanical [103, 104, 105], and optomechanical [106, 107]
devices. Excepting the circulator discussed in this thesis, most state-of-the-art non-reciprocal
circuits for superconducting circuits are implemented with this approach.
An excellent conceptual description of this strategy is given in Ref. [108], which provides agraph-theoretical description for the Langevin equations of parametrically coupled resonant systems. Rather than reproduce that treatment here, we attempt a succinct and general summary of
the concept.
Consider a system of two or more resonant modes (frequencies {ωi } and linewidths {κi }) inthe resolved sideband limit. By this, we mean the frequencies in the set {ωi } are separated by manytimes their respective linewidths, and that the linewidths of each resonator are narrower than all

25
the resonant frequencies. Now imagine a parametric interaction, which allows variation of a systemparameter to shift the frequencies of the resonant modes {ωi } by approximately their linewidths∆ωi ≈ κi . Varying this parameter at the difference frequency ωi − ωj between a pair of modes
induces a beam-splitter interaction, allowing excitations to be exchanged between the two resonant
systems:
HI ∝ eiθ ai a†j + e−iθ a†i aj .

(3.21)

Here, ai is the annihilation operator of the ith resonant mode, and θ is the phase of the parametricinteraction. In practice, this phase can be adjusted by tuning the phase of the electrical signal
which is controlling the parametric interaction.
Examination of Eq. (3.21) shows that the phase of the beamsplitter interaction depends onthe direction in which excitations are exchanged. For example, if the phase θ = π/2, excitations
which originate in mode i and are transferred to mode j will be π out of phase with excitations
which originate in mode j and are transferred to mode i. If desired, a frequency diplexer can thenbe used to route the different resonant modes out separate physical ports of the device.
There is a complication, though, in generating non-reciprocity in this way: the phase referenceof the parametric interaction is arbitrary. Said another way, a gauge may always be chosen in which
the phase θ is zero, making the beamsplitter interaction reciprocal. This difficulty may be resolved
with the introduction of another parametric interaction between modes i and j (or to a third modek). Experimentally, this can be accomplished by cascading a duplicate of the original device in series
with the original [101]. The second parametric interaction also has a phase, and while its phasereference is also arbitrary, the difference between the phases of the two parametric interactions isa gauge-invariant.
A description of this approach can also be couched in the language of geometric phases createdby synthetic magnetic fields [109, 110, 111, 102, 99, 107]. In that framework, the coupled resonantmodes are viewed as nodes of a lattice, with a hopping interaction determined by the system’s
parametric modulation. The phase of the parametric modulation sets a Peierls’s phase [112], and26a Berry curvature (synthetic vector potential) is quantified by the total Peierls’s phase acquired in
the circumnavigation of one plaquette of the lattice.
The affordances of using parametrically coupled modes to generate non-reciprocity are therelative simplicity of the circuit designs, and a wide versatility in function: in addition to realizing
elementary non-reciprocal devices such as isolators [103] and circulators [99, 100], the parametricinteraction can also be used to drive parametric down-conversion, enabling processes like directional
amplification [98, 99, 100, 113]. This flexibility, and the ability to dynamically switch between thevarious modes of operation, has led some groups to refer to these devices as “field-programmable,”in reference to multi-function field-programmable gate arrays [100]. The major drawback inherent
to this method is its reliance on resonant modes in the resolved sideband limit, which constrain
the device’s bandwidth. A second drawback, in some situations, is the frequency-conversion whichaccompanies some implementations of this approach [99, 100].

3.3.3.2

Frequency conversion and delay

A second approach for the generation of non-reciprocity with active devices is enabled via
successive translations in frequency and time, or in other words, a combination of frequency conversion and delay. As frequency and time are Fourier duals, successive translations in these quantities
do not, in general, commute. Signals traversing a network in which a series of these translations
are made may encounter them in a different time-ordering, depending on the port of the network
at which they are arrive. Such a dependence on propagation direction violates Lorentz reciprocity.
The promise of generating non-reciprocity in this way was first observed in the 1960s, incascades of symmetric lattices with dual arms of time-varying inductors and capacitors [114]. In
these networks, a tank circuit resonance provides the delay, and the active modulation of thecircuit elements create frequency sidebands on the input signal. In more recent years, the idea
has been leveraged in a general-purpose circulator proposal which is amenable for implementationin superconducting circuits [115]. The realization of that proposal is the subject of this thesis.
Related proposals in the microwave [116], millimeter [?], and optical [117, 118, 119, 120] domains

27
have utilized the same concept with non-resonant delays.
The affordances of the frequency-conversion and delay approach are a general absence of
bandwidth constraints, stemming from the potential to implement the approach without any resonant physics. Related to this, resonant-based implementations are in general not subject to theconstraints of the resolved sideband-limit, allowing low-frequency modulation of the active components. This is an especially attractive feature for superconducting circuits, where high-bandwidth
dilution-refrigerator lines are a precious resource.
Its drawbacks are the care required in coherently erasing (or otherwise suppressing) sidebands
created by the frequency converting elements, and the general circuit complexity which is sometimes needed for this task. An additional drawback, which is also common to circuits based on theparametric coupling of resonant modes, or in general any active device, is the power consumptionof the control signals. This is not a problem for some applications, but in a cryogenic setting additional heat loads on the mixing chamber of a dilution refrigerator are an important consideration,
especially in light of the large number of circulators which may be required in a superconducting
fault-tolerant quantum computer.

3.3.4

Connection with time-reversal symmetry

As reciprocity arises in physical systems which are invariant under time-reversal, another way
to classify non-reciprocal systems is via the mechanism by which they break this invariance. This
perspective is utilized in recent demonstrations [107] and proposals [121, 122, 123], which break
Lorentz reciprocity with judicious use of dissipation, referred to as reservoir engineering.
Generating non-reciprocity, however, does not in general require dissipation, or a complete
absence of time-reversal symmetry. For example, many approaches break Lorentz reciprocity bycoupling electromagnetic fields to matter with an interaction that is odd under time-reversal. Examples of this are magnetic fields, as in circulators built with gyrotropic media and quantum Hall
effect circulators [75, 76, 77]; linear momentum [124, 125, 126, 127], for example through dispersion engineering [125, 126, 127] or radiation pressure in an optomechanical interaction [124]; and28
angular momentum [96, 70, 115], for example, by coupling electromagnetic fields to non-degenerateclockwise and counter-clockwise propagating modes in a ring resonator. The degeneracy of thesemodes can then be lifted with physical rotation, as in a Sagnac interferometer [128] (or in acoustic
demonstrations, with the motion of the acoustic medium [70]). It may also be lifted in other ways,
such as active modulation of circuit parameters, which can simulate a sense of rotation [115] or asynthetic gauge field [96].

3.4

Common non-reciprocal devices
As both Lorentz reciprocity and scattering matrices have now been discussed, we introduce

in this section the scattering matrix description of several common non-reciprocal devices. Tab. 3.1
shows the microwave circuit symbols and scattering matrices for a gyrator, an isolator, and acirculator.
gyratorisolator

circulator

50

3

𝜋

microwave circuit symbol

1

2
1


ideal scattering matrix

0 1
−1 0



2



0 0
1 0



1

2



0 0 1
1 0 0
0 1 0Table 3.1: Symbols and scattering matrices for three common non-reciprocal devices

3.4.1

Gyrators

Gyrators are impedance-matched two-port devices that transmit signals in the forward direction without changing their phase, but impart a π phase shift to signals transmitted in the reverse
direction [78]. This action is visible in their scattering matrix: for a gyrator, S21 and S12 are π outof phase. Gyrators were first proposed in 1948 by Tellegen as a hypothetical fifth element of basic
circuit theory, supplementing the resistor, inductor, capacitor, and transformer [129]. Addition of

29
the gyrator to this set allows for the realization of passive non-reciprocal networks, and also reduces
the number of basic circuit elements to three: the gyrator can replace the transformer and either
the capacitor or the inductor.
In practice, gyrators are used frequently in the design of active filters, and in the constructionof other non-reciprocal circuits, such as isolators and circulators. This can be done directly, by tieing
together the grounds of the gyrator’s two ports and defining a third port as the differential voltage
between them, resulting in a three-port circulator. Alternatively, the gyrator can be inserted
between two beamsplitters (or microwave hybrids) to create a four-port circulator. This procedure
is known as the Hogan construction of a circulator [130].

3.4.2

Isolators

Isolators are impedance-matched two-port devices that act as one-way valves for classicalelectromagnetic signals. Signals propagating in the forward direction through an isolator are transmitted unchanged, while signals propagating in the reverse direction are completely absorbed. Thisaction is visible in the off-diagonal elements of the scattering matrix: S21 = 1, while S12 = 0.
Isolators are used in a variety of ways across the electromagnetic spectrum: in optical experiments, they are commonly employed as a way to prevent reflections in an optical network fromdisturbing the source laser. In telecommunication networks, where signals are frequently amplified
and attenuated by many orders of magnitude, isolators serve as a way of protecting sensitive (andlower signal-amplitude) components in the network from amplifier reflections. More generally, as
discussed in Ch. 1, isolators are of fundamental use in precision measurement.

3.4.3

Circulators

Circulators are impedance matched devices with three or more ports, that provide the directional signal routing needed for a variety of electromagnetic networks. As the scattering matrix in
Tab. 3.1 shows, signals incident on one of the circulator’s ports will be routed counter-clockwisearound the device and transmitted out the subsequent port.

30
Circulators can be used to make isolators (by depositing the signal at all but two of thecirculator ports into an entropy dump) or gyrators (by combining them with 180-degree microwave
hybrids/beamsplitters). The need for a high-quality and scalable circulator which could supplantthe ferrite-based commercial circulators currently used in superconducting qubit experiments is the
motivation for this thesis.

Chapter 4

Theory of operation

The circulator presented in this thesis may be understood in terms of “synthetic rotation”
created by the active modulation of the circuit, and analyzed rigorously with a lumped-element
circuit theory or an input-output formalism [115]. Before performing a formal analysis we provide
a complementary explanation for its operation based upon the frequency-domain dynamics of ananalogous model system.

4.1

Model system
The model is a lumped-element network of multipliers and delays (Fig. 4.1a) which createsa gyrator (see Sec. 3.4.1). Gyration in the model system arises from the non-commutation of
successive translations in frequency and time [116]: the multiplying circuits operate as frequency
converters, translating an input signal up and down in frequency, and the delays translate fields
forward in time. As frequency and time are Fourier duals, the time-ordering of these translations
matters (the two operations do not generally commute). Transmission through the network thus
depends on the propagation direction of the incident signal, breaking Lorentz reciprocity.
To see that non-reciprocity explicitly, frequency-phase diagrams are used to calculate themodel’s scattering parameters. The diagrams follow an incident signal at frequency ωp as it propagates through the device, tracking its amplitude, frequency, and phase in a frame rotating at
ωp .
The insets in Fig. 4.1a depict the way that the model system’s two constituent elements—

32

Re
input

1

(a)

(b)
−sin(Ω𝑡)𝜔Im𝜔p
output

Ω

input
𝜔

i.

𝜔p
multiplycos(Ω𝑡)Ω𝜏 = 𝜋/2

iii.
delay

iii.
cos(Ω𝑡)sin(Ω𝑡)multiply

2Ω

𝜔

i.

iv.
output2Ωii.
multiply

output2Ω

multiply

delay

input

output

iv.

Ω

ii.
𝜏

𝜏

(c)

𝜔p
input

Figure 4.1: Conceptual diagram of non-reciprocity generated with frequency conversion and delay.
(a) Lumped-element network that forms a gyrator. The insets show how fields are transformed by
the network’s two components: multiplying elements and delays. In the upper inset, an input field
at frequency ωp (top panel) is multiplied by cos(Ωt) to create a field with spectral components at
ωp ± Ω (bottom panel). The real and imaginary axes of the plot shows the phase of these spectral
components in a frame rotating at ωp . In the lower inset, a delay of length τ = π/ (2Ω) advances(retards) the phase of spectral components at ωp + Ω (ωp − Ω) by π/2. (b) Calculation of theforward-scattering parameter for the network in (a), by following an incident field at frequency
ωp as it propagates through the device. Purple (green) arrows indicate fields propagating in theleft (right) arm of the network. Fields are forward transmitted with amplitude and frequency
unchanged, but phase shifted by π. (c) Backward transmission through the network in (a). Fields
are transmitted with amplitude, frequency, and phase unchanged.

multipliers and delays—transform input fields to output fields. In the multiplying elements, thattransformation occurs via multiplication by a bias signal—in this case, cos(Ωt). The trigonometricproduct-to-sum identity statescos(ωp t) cos(Ωt + θ) =1cos ([ωp + Ω] t + θ) + cos ([ωp − Ω] t − θ) ,
2

(4.1)

which has a simple interpretation in the frequency domain: multiplication creates two sidebands,
each detuned from ωp by the bias frequency Ω. Importantly, the phases of these sidebands dependon the phase θ of the multiplier’s bias signal. We choose a phase reference such that multiplicationby cos(Ωt) creates two sidebands with the same phase.
In the delay elements, inputs are transformed to outputs by way of a phase shift. In the

33rotating frame, a delay of length τ = π/(2Ω) leaves the phase of spectral components at ωp unchanged, while advancing the phase of spectral components in the upper sideband ωp + Ω by π/2,
and retarding the phase of components in the lower sideband ωp − Ω by π/2.
With the action of the multiplying and delay elements defined, calculation of the scattering parameters is straightforward. Forward transmission through the model system is shown in
Fig. 4.1b. A signal incident on port 1 with frequency ωp (Fig. 4.1b, i.) is first divided equally
into the network’s two arms. Fields in both arms encounter a first multiplying element, a delay, asecond multiplying element, and are then recombined.
Critically, the modulation sidebands at ωp ±2Ω created in the network’s two arms are π out ofphase and interfere destructively at the device’s output (Fig. 4.1b, iv). Conversely, the components
at the frequency ωp interfere constructively. Comparison of Fig. 4.1b, iv. with Fig. 4.1b, i. showsthat the incident signal has been transmitted through the device with its frequency and amplitude
unchanged, but its phase shifted by π. The scattering parameter S21 for the network is therefore−1.
The reverse path is traced out in Fig. 4.1c, for a signal incident on the network’s second port.
As with forward transmission, destructive interference occurs at ωp ± 2Ω (Fig. 4.1c, iv.). Likewise,
this is accompanied by constructive interference at the frequency ωp . Now, however, comparison ofFig. 4.1c, iv. with Fig. 4.1c, i. shows that the frequency, amplitude, and phase of the incident signal
were unchanged by the network. Therefore, in contrast to the forward transmission, the backwards
transmission is characterized by a scattering parameter S12 = 1. The network in Fig. 4.1a is thus
described by the gyrator scattering matrix in Tab. 3.1, and forms an ideal gyrator.
In the model system, the convert-delay-convert process happens simultaneously in both armsof the network. Consequently, each arm is individually non-reciprocal. Alone, though, a single arm
creates unwanted modulation sidebands. To create an ideal gyrator, two arms, with the bias signalsof their multiplying elements separated in phase by π/2, are connected in parallel. This balanced
architecture engineers destructive interference of the spectral components at ωp ± 2Ω.
Such a strategy for suppressing the creation of spurious sidebands, which we refer to as “co-34
herent cancellation,” may be contrasted with that used in non-reciprocal devices that operate withthe parametric coupling of resonant modes in the resolved-sideband limit, discussed in Sec. 3.3.3.1.
In that scheme, parametric modulation of a resonant system creates sidebands at the parametricdrive frequency, and a second resonant mode is used to enhance the density of states at the desired frequency, while simultaneously diminishing it at the undesired frequency. To work in the
resolved sideband limit, however, the parametric modulation must be many times the resonant system’s linewidth. In microwave frequency implementations, this typically requires GHz modulation
tones. In contrast, the coherent cancellation approach lifts the resolved-sideband constraint, andcan therefore be used with lower-frequency control tones.

4.2

Superconducting implementation
We make use of the unique properties of superconducting circuitry to realize compact on-

chip multiplier and delay elements. Specifically, Josephson junctions form widely tunable inductors,
while vanishing conductor loss permits high-quality on-chip microwave resonators. Fig. 4.2 shows
how a single arm of the model system (Fig. 4.2a) is made with a network of capacitors and dynamically tunable inductors (Fig. 4.2b).

4.2.1

Multiplying elements

The multiplying elements in the circuit representation are created with reactive bridge circuits(also known as symmetrical lattices [114]), built with two tunable pairs of nominally identicalinductors l+ and l− arranged opposite one-another (gray box in Fig. 4.2b). Two differential ports
are defined by the left-and-right and top-and-bottom bridge nodes. Importantly, the inductors
tune in a coordinated fashion: when one pair of inductors increases, the other pair decreases. We
parametrize this tuning with a base inductance l0 and an imbalance variable δ, by writingl± = l0 /(1 ± δ).

(4.2)

35
(a)

1

2

(b)
𝑙+

𝑙—

𝑙—

𝑙+

𝑐

1
off-chip coil

...

...

...

...

(d)
delay [ns]bias line

Φu − ΦgΦu + Φg

(c)

2
3

2

1

0

6
7
8
frequency [GHz]

Figure 4.2: Multiplying elements and delays realized in a superconducting lumped-element circuit.
(a) The model system (Fig. 4.1a) is constructed from two parallel instances of this network. (b)A lumped-element version of the network in (a), created with capacitors and tunable inductors
arranged in a bridge geometry. (c) To create an inductive bridge circuit in a superconducting
microwave environment, four series-arrays of SQUIDs are arranged in a figure-eight geometry, andtuned with an off-chip magnetic coil producing a uniform flux Φu and an on-chip bias line creating a
gradiometric flux ±Φg . (d) Simulated group delay for the circuit in (b) when its ports are connected
to 50 Ohm transmission lines. The bridge inductors are parametrized according to Eq. (4.2), withc = 1 pF, l0 = 1 nH, and δ = 0.2.

When a bridge is coupled to transmission lines of characteristic impedance Z0 , its forward scatteringparameter at angular frequency ω isS21 =2iδωl0 Z02−Z0 (1 − δ 2 ) − 2iωl0 Z0 + ω 2 l02=−δ

2iωl0 Z0
+ O(δ)2 .
(Z0 + iωl0 )2(4.3)

As the imbalance in the bridge determines its transmission, changing δ allows the circuit to act asa switch or a multiplying element [131, 132].
The bridge circuit’s tunable inductors are realized with series-arrays of superconducting quan-

36
tum interference devices (SQUIDs), formed by the parallel arrangement of two Josephson junctions.
Arrays are used in place of individual SQUIDs to increase the linearity of the inductors [115]. The
inductance l of an N SQUID array depends on the magnetic flux Φ that threads through each
SQUID [58]:
ϕ0
l=N
sec
2I0



Φ
2ϕ0



+ O(I/I0 )2 .

(4.4)

Here ϕ0 = ~/2e is the reduced flux quantum, and the Josephson junctions in each SQUID areassumed to be identical with critical current I0 and negligible geometric inductance. The SQUIDs
are also assumed to be identical.
To realize the coordinated tuning of inductors described in Eq. (4.2), we arrange the SQUID
arrays in a figure-eight geometry (Fig. 4.2c). Two flux controls determine the imbalance in thebridge. First, an off-chip coil threads a uniform magnetic flux Φu through all the SQUIDs. Second,an on-chip bias line, which bisects the figure-eight, threads a gradiometric flux Φg through the
SQUIDs. SQUIDs on the left side of the bias line therefore experience an overall magnetic flux
which is the sum of the uniform and gradiometric contributions, whereas SQUIDs to the right of
the line are threaded by the difference of the uniform and gradiometric fluxes.
When the gradiometric bias line is driven with a sinusoidal current source at frequency Ω, theflux through the SQUIDs varies in time as Φ = Φu ± Φg cos(Ωt + φ). This process creates a bridgeof inductors which tune according to Eq. (4.2), with a simple sinusoidal variation in the imbalanceδ = δ0 cos(Ωt + φ) and a rescaling of the base inductance l0 . Such a periodic variation in δ makes
SQUID bridges biased in this way into dissipationless multiplying elements. The precise mapping
between the flux controls Φu & Φg and the circuit parameters l0 & δ0 is [133]J1 (β)+ O(β 2 ),J0 (β)ϕ01= N+ O(β 2 ),2I0 cos(α)J0 (β)δ0 = −2 tan(α)
l0

(4.5)

37
with
Φu
,
Φ0
Φg
β ≡ π ,
Φ0

α ≡ π

(4.6)

and Jn the nth Bessel function of the first kind.

4.2.2

Delays

The second primitive needed for the model system is a delay, realized in our circuit witha resonant mode. Conveniently, the SQUIDs in the bridge circuits are inductive, so the additionof a single capacitor is enough to create a resonance. This resonance delays fields near its center
frequency by a timescale τ characterized by the inverse of its linewidth. More quantitatively, whena harmonic field incident on port ν of a resonant network is scattered to port µ, it acquires agroup delay τ = d∠Sµν /dω [78]. Here ω is the frequency of the harmonic field, and ∠Sµν is thephase of Sµν . Fig. 4.2d shows delay as a function of frequency, simulated for the resonant circuit inFig. 4.2b. Fields near the circuit’s resonant frequency experience a delay of several nanoseconds.
As a confirmation of these simulations, an analytical study of the same circuit was also made, usinga graph theoretical approach [134, 135] to solve for the circuit’s eigenfrequencies with Kirchoff’s
current and voltage laws. This is described in App. A.
Delays realized with resonant modes allow for a deeply sub-wavelength implementation, whichis critical for the “coherent cancellation” approach. While these lumped-element delays are necessarily narrower in bandwidth than those created with, for example, a length of transmission line,
their finite bandwidth is mitigated by the tunable inductance of the bridge circuits, which allows
the frequency ω0 of the resonant delay to be tuned (over several GHz) with the uniform magnetic
flux Φu . As the multiplying elements are broadband [132], this tunability of the delay is inheritedby the full circulator. Likewise, the duration τ of the delay depends on the imbalance in the bridges,
and may be tuned with the gradiometric flux Φg , facilitating satisfaction of the requirement thatτ = π/ (2Ω).

38
Tuning of the resonant delay takes a simple form when expressed in terms of the circuit
parameters l0 and δ0 . When two of the arms in Fig. 4.2b are combined in parallel to create thefully assembled circuit, the resonant delay occurs at the frequency [115]s
4 − δ02
ω0 =
,
2l0 c

(4.7)

and its duration τ is approximately the inverse of the resonant mode’s linewidth,
τ≈

8Z0 c
.
δ02

(4.8)

Here Z0 is the characteristic impedance of the surrounding transmission lines.

4.2.3

Assembly of the full circulator

Construction of a superconducting version of the model system requires the parallel combination of two of the circuits shown in Fig. 4.2. This process is depicted Fig. 4.3, which shows
the model system (Fig. 4.3a) alongside a lumped-element schematic of an equivalent network madewith variable inductors and capacitors (Fig. 4.3b).
In principle, a circulator could be created from the gyrator shown in Fig. 4.3b using anyof the methods detailed in Sec. 3.4.1. Alternatively, the same circuit could be designed without
differential ports, where instead each of the four nodes define their own port via comparison to
a common ground, as shown in Fig. 4.3c. Such a design allows direct realization of a four-port
circulator, without any additional embedding network. The ideal scattering matrix for this fourport clockwise circulator is

 0


 0
S = 

 0


1


1 0 0 
0 1 0 ,

0 0 1 


0 0 0

(4.9)

and reversing the direction of gyration also allows the same network to act as a counterclockwise
circulator with scattering matrix ST . This device is the four-port version of the circulator describedin Tab. 3.1.

39𝑐sin(Ω𝑡)cos(Ω𝑡)δ0sin(Ωt)δ0cos(Ωt)

(b)

(a)

𝜏
1

1

2

2

𝑐

𝜏

cos(Ω𝑡)−sin(Ω𝑡)−δ0sin(Ωt)(c)δ0sin(Ωt)δ0cos(Ωt)δ0cos(Ωt)

(d)

𝑐

1

a

∆

3

b

Σ

𝜋

∆

a

2

Σ

b

4

4

2

3

1

𝑐

−δ0sin(Ωt)δ0cos(Ωt)

Figure 4.3: (a) The model system of Sec. 4.1 forms a gyrator, which breaks Lorentz reciprocity
with a combination of frequency conversion and delay. (b) A realization of the network shown in(a), made with tunable, reactive circuit elements. (c) When four ports are formed by comparing
the nodes of the circuit in (b) to a common ground, the network forms a four port circulator.
Circulation is accomplished through the interference of the common and differential modes of the“left” (ports 1 & 3) and “right” (ports 2 & 4) circuit ports. This is analogous to the circulation
generated by a Hogan circulator (d), which comes from the interference between two arms of aninterferometer, one of which contains a gyrator, and the other of which is a direct short between
ports.

The transformation between gyrator and circulator can be understood in the following way:
driving any one of the four ports in Fig. 4.3c involves simultaneously exciting the common anddifferential modes of the circuit. The gyrating differential mode is non-reciprocal, whereas theprompt scattering of the non-resonant common mode is reciprocal. The interference of these two
scattering processes results in circulation [115].

40
In a sense, this method for realizing a four-port circulator is a virtual form of the Hogan
construction [130] depicted in Fig. 4.3d. In (a physical implementation of) the Hogan construction,
an incident signal is routed through a 180◦ hybrid and directed out its differential (∆) and common(Σ) ports. The differential port leads to a gyrator and then to the differential port of a second
hybrid, whereas the common port is directly connected to the second hybrid’s common port. The
result is an interferometer with two arms: the ∆ arm contains a gyrator (and is therefore nonreciprocal), and the Σ arm is a direct connection between the two microwave beamsplitters, whichis reciprocal. The circulator in Fig. 4.3c relies on the same interference effect, but in this casethe two “arms” of the interferometer are not spatially separated. Instead, they are encoded in
the even and odd excitations of ports 1 & 3 and 2 & 4. In both cases, however, the circulation isaccomplished through the interference of a non-reciprocal (gyrating) signal pathway and a reciprocal
signal pathway.

4.3

Circuit analysis
We have attempted in the preceding sections to establish intuition for how the circuit inFig. 4.3c realizes a four-port circulator. To support that argument we now directly calculate theelements of its scattering matrix using circuit theory, following Ref. [115].
Consider the bridge circuit shown in Fig. 4.4a. The constitutive relations for this circuit are


 

1  1 δ(t)   φ1 (t)   I1 (t) (4.10)


=
.
l
δ(t) 1φq (t)Iq (t)The matrix on the left-hand side of Eq. (4.10) is the reluctance of the circuit. The reluctance matrix
gives the current response Ii to an applied flux φi , defined as the time integral of the voltage at
Rt
port i: φi ≡ −∞ Vi (τ )dτ . It is a sort of inverse inductance, what was historically called “magnetic
resistance” before the term “reluctance” was coined by Heaviside [136].
When four of these circuits are connected together, in the manner shown in Fig. 4.4b, the

(a)

41

(b)

+

-1

l0[1+δ0cos(Ωt)]

+

q

q

l0[1–δ0cos(Ωt)]-1

–
–

1

–

+

3

4

2

l0[1+δ0sin(Ωt)]-1l0[1–δ0sin(Ωt)]-11+l0[1+δ(t)]-1pl0[1–δ(t)]-1–

Figure 4.4:

constitutive equation for the network is










2/δ0
0
cos(Ωt) sin(Ωt) − cos(Ωt) − sin(Ωt)   φp (t)   Ip (t) 

 



 









0
2/δ
sin(Ωt)−cos(Ωt)−sin(Ωt)cos(Ωt)
φ
(t)
I
(t)
0
  q


 q
 



 



cos(Ωt)sin(Ωt)
3/δ
−1/δ
−1/δ
−1/δ
φ
(t)
I
(t)






0
0
0
0
1
1
δ0 
=
.

 


l 

 sin(Ωt) − cos(Ωt) −1/δ0



3/δ0
−1/δ0
−1/δ0   φ2 (t)   I2 (t) 


 



 





 − cos(Ωt) − sin(Ωt) −1/δ0



−1/δ
3/δ
−1/δ
φ
(t)
I
(t)
0
0
0
  3


 3
 



− sin(Ωt) cos(Ωt)
−1/δ0
−1/δ0
−1/δ0
3/δ0
φ4 (t)I4 (t)(4.11)
Analysis is simplified by changing to a left-right, even-odd basis for ports 1-4, and a rotating,circular basis for ports p and q: Il,e (t) 
 1






 Ir,e (t)  01
= √ 2
 I (t) 
 1
l,o






Ir,o (t)
0










0   I1 (t) 




1 0
1   I2 (t) 
,





0 −1 0   I3 (t) I4 (t)
1 0 −10eiΩt

1  I+ (t) 
= √ 
2
I− (t)e−iΩt1−ieiΩt(4.12)

  Iq (t) 

,
−iΩt
ie
Ip (t)(4.13)and similarly for the branch fluxes. In these new bases, Eq. (4.11) separates into two uncoupledsystems of equations:

42




 

1
−1
φ
(t)
I
(t)
2
  l,e  l,e



=
,
l
−1 1
φr,e (t)Ir,e (t)




1

l




2

0

δ0

0

2

δ0

δ0

δ0

4

−iδ0 iδ0

0


 

iδ0   φ+ (t)   I+ (t) 

 


 

−iδ0   φ− (t)   I− (t) 

=
.

 





0   φl,o (t)   Il,o (t) 


 

4
φr,o (t)Ir,o (t)(4.14)(4.15)

To complete the circulator, we shunt ports p and q with capacitances c. This fixes the relation
between the currents and the branch fluxes at these ports:−cd2
φq,p = Iq,p ,
dt2(4.16)

or equivalently in the circular, rotating basis,− c(d∓ iΩ)2 φ± = I± .
dt

(4.17)

With this constraint, Eq. (4.15) has a simple representation in the frequency domain, which we
indicate with square brackets [·]:


 

2
0
δ0 iδ0   φ+ [ω]  0 2 − lc(ω − Ω)



 







2 δ
0
2
−
lc(ω
+
Ω)
−iδ
φ
[ω]
0






0
0
−
1

=
.






l





δ
δ
4
0
φ
[ω]
I
[ω]
0
0
l,o
l,o


 



 

−iδ0
iδ0
0
4
φr,o [ω]Ir,o [ω](4.18)At this point we pause and comment on the constitutive relations for the even and odd
excitations of the left and right ports of the circuit, given in Eq. (4.14) and Eq. (4.18), respectively.
Even excitations of the left and right ports are uncoupled from nodes p and q, and are therefore
non-resonant. Contrastingly, odd excitations of the left and right ports depend on the branch fluxes
across nodes p and q, and on the value of the capacitors c. These dynamics are resonant. To showthat they are also non-reciprocal, we write Eq. (4.18) as a 2×2 system of block matrices


 

1  A B   φ±   0 


=
,
l
C D
φl,r
Il,r

(4.19)

43
where each matrix element in the reluctance matrix is itself a 2×2 matrix (bold variables denotematrices), and the internal degrees of freedom (φ+ and φ− ) and the external degrees of freedom(φl,o and φr,o ) are contained in the 2×1 vectors φ± and φl,r .
Written in this way, it is clear that the internal degrees of freedom φ± can be eliminated,yielding a 2×2 matrix equation
D − CA−1 B φl,r = Il,r .

(4.20)

Expanding this to first order in Ω/ω, and evaluating the output on resonance when ω equalsr
4 − δ02
ω0 ≡,(4.21)
2lc
we obtain the reluctance relation

iω0









16cΩ  0 1   φl,o [ω]   Il,o [ω] 


=
.
δ02
−1 0
φr,o [ω]Ir,o [ω](4.22)

This is the reluctance relation of a gyrator, with gyration conductance 16cΩ/δ02 [129]. For harmonic
fields, the reluctance matrix is iωY, where Y is the system’s admittance matrix.
If ports 1 through 4 are now connected to transmission lines of characteristic impedance Z0 ,the network’s scattering matrix S can be calculated:
S = (1 + Z0 Y)−1 (1 − Z0 Y) .

(4.23)

Here, 1 is the 2×2 identity matrix. Using this expression, the scattering matrix So (here thesubscript indicates the odd or differential two-port network) for the reluctance relation in Eq. (4.22)
is





 0 −1 
So = 

1 0(4.24)

provided that the gyration conductance is set to Z0−1 . Repeating this process for the even dynamics
in Eq. (4.14), the even scattering matrix isSe = 

ωlωl−4iZ0

4Z0
4Z0 +iωl

4Z0
4Z0 +iωlωlωl−4iZ0,(4.25)

44
which to zeroth order in ωl/Z0 is the scattering matrix of a direct (matched) connection between
the left and right ports:

 0 1 Se = 
+O
1 0



ωl
Z0


.

(4.26)

Finally, the full 4×4 scattering matrix in the numbered port basis can be recovered by reversing
the change-of-basis procedure:
S = U−1 Se,o U,(4.27)

where U is the unitary matrix in Eq. (4.12) that transforms between the even-odd and numbered
port bases, and Se,o is a 4×4 block-diagonal scattering matrix formed from the 2×2 scatteringmatrices of the even and odd dynamics: Se 0 
Se,o = .
0 So(4.28)

When Se and So are given by Eq. (4.26) and Eq. (4.24), Eq. (4.27) reduces to the scattering matrix
for an ideal four port counterclockwise circulator, the transpose of the expression in Eq. (4.9). The
same result can be obtained via a time-domain analysis in an input-output formalism [115], but we
do not reproduce that treatment here.

Chapter 5

Circulator Layout

In this chapter we discuss the layout of the superconducting microwave circulator describedin Ch. 4. Many of the design considerations are common to a variety of other monolithic microwave
integrated circuits. Others are rooted in the device’s use of superconducting materials, and superconducting tunnel junctions. Before we enumerate these considerations, and the design measures
undertaken to address them, we give a brief summary of the overarching design philosophy adopted
in this project, and the trilayer process used to fabricate the devices. We then describe the design’s
evolution over the course of four design and measurement cycles.

5.1

Design philosophy
The construction of a superconducting circulator is part of a broader effort within our group to

improve the measurement of superconducting qubits with a hierarchical, network-based approach.
In the past three years, this has led to proposals for superconducting [115] and broadband [116]circulator designs; and an autonomous feedback scheme for qubit state preparation and preservation [137]; as well as superconducting demonstrations of single-pull, single-throw switches forin-fridge signal processing [131]; cross-over switches for code-domain multiplexing of qubit readout [131]; single-sideband modulators for frequency-domain multiplexing of qubit readout [132];and cryogenic circulators [138].
These experimental efforts are greatly enhanced by a collaboration with the Quantum Devices
group at the National Institute of Standards (NIST) in Boulder. The superconducting circuits

46
for the above demonstrations were fabricated at NIST in a standard niobium trilayer process (seeSec. 5.2). The design rules for this process (listed in App. B) impose some constraints on our layouts,
but the process is wafer-scale, high-yield, and allows for multiple wiring layers and thousands ofJosephson junctions on a single chip [139].
The complexity that the process affords is therefore quite powerful. But undisciplined application of that complexity makes it difficult to maintain microwave hygiene and retain physicalintuition about a circuit. To leverage the advantages of the trilayer process while avoiding thesepitfalls, we construct integrated circuits from repeated instances of the same basic circuits, withhierarchy utilized whenever possible. For example, the bridge-circuit multiplying element discussed
in Sec. 4.2.1 functions as a single microwave switch [131], but two instances of the bridge circuit
can be embedded between a pair of microwave hybrids to create a single-sideband modulator [132],and four instances can be used to create a tunable [115, 138] or broadband [116] circulator. Other
primitives developed in the lab, such as the nonlinear resonator formed from an array of Josephson junctions in series with a capacitance (dubbed a tunable Kerr circuit), have similarly been
instanced in the construction of parametric amplifiers [140, 24], tunable couplers for mechanicaloscillators [141], and autonomous feedback controllers [137]. The bridge circuits and tunable Kerr
circuits are themselves composed of repeated instances of SQUIDs, made up two Josephson junctions in parallel.

5.2

The Nb/AlOx /Nb trilayer process
The circuits and structures discussed in this thesis were designed with open source layout

software (KLayout), and fabricated by Leila Vale at the NIST Boulder cleanroom using opticallithography and a standard Nb/AlOx /Nb trilayer process [142, 143]. Circuit boards for the chips
were also designed in KLayout, and printed by Sierra Circuits on a Rogers 3010 substrate.1
Tab. 5.1 overviews the design layers available within the trilayer process.
1

KLayout produces gds2 files which can be converted to the gerber format preferred by boardhouses with standard
software PCB software, such as LinkCad.

47
Table 5.1: Names, abbreviations, materials, and functions of the layers used in the NIST niobium
trilayer process. Layers are in process-order. a-Si stands for amorphous silicon.
layer namecounter electrodeabbrev.
CEmaterialNbfunction
The counter (upper) electrode for Josephson junctions.

counter electrode overflash

CEO

Nb

Overflash mask
counter electrode.

aluminumAL

Al

Aluminum layer that supports a surface oxide, whichforms the insulating barrier
for the Josephson junctions.

aluminum overflash

ALO

Al

Overflashminum.

base electrodeBENb

The base (bottom) electrodefor Josephson junctions. Also
serves as the primary wiring
layer.

normal metal

R1

Au, Cu, or AuPd

Normal metal layer helpful
for breaking supercurrent
loops or engineering dissipation.

insulator 1

I1

SiO2 , Si3 N4 ,or a-Si

Etch of first PECVD dielectric layer for creation of vias
between BE and W1.

wiring 1

W1

Nbinsulator 1 excess

I1X

SiO2 , Si3 N4 ,or a-Si

Additional etch to remove
excess dielectric in the first
PECVD layer.

insulator 2

I2

SiO2 , Si3 N4 ,or a-Si

Dielectric etch for creation ofvias between W1 and W2.

wiring 2

W2

Nbinsulator 2 excess

I2X

SiO2 , Si3 N4 ,or a-Si

Secondarylayer.

Tertiarylayer.

mask

for

for

niobium

niobiumthealu-wiringwiringAdditional etch to remove
excess dielectric in the second
PECVD layer.

48
5.2.1

Josephson junctions in the trilayer process

As the name suggests, the process allows for the creation of superconducting circuits withsuperconducting-insulating-superconducting Josephson tunnel junctions [58], formed by sandwiching niobium layers around an insulating aluminum oxide. Fig. 5.1 shows the layer stack-up for thisprocess, which begins with thermal growth of an oxide layer on a circular silicon chip (diameter3”, relative electric permittivity r = 11.45).2

The trilayers are then deposited in a load-locked

vacuum system, with an oxidation step after the aluminum deposition to create the aluminum oxide
layer. The result is a “vertically” oriented Josephson junction which spans the entire wafer, oriented such that at sufficiently cold temperatures, cooper pairs may tunnel back and forth between
the stacked layers of niobium. The lower niobium layer, known as the base electrode, also serves as
the primary wiring layer for the circuit. The upper niobium layer is called the counter electrode.
(a) Nb (CE)
AlO2
Al
Nb (BE)
SiO2
Si
(c)

120 nm(b)

7 nm
200 nm20 nm
380 𝜇m(d)Figure 5.1: Josephson junctions in the Niobium trilayer process. (a) Layer stack-up of the trilayer.
(b) The counter-electrode etch. (c) The aluminum/aluminum oxide etch. (d) The base electrodeetch. The junction, indicated by the black ‘x’, is formed at the aluminum oxide interface. Graphics
courtesy of Maxime Malnou.

The circuit is then defined via a subtractive process. First, a stepper flashes the counterelectrode mask. To reduce the amount of counter electrode which unintentionally survives thesubsequent etch, a second mask (the counter electrode overflash) is also flashed. This step is less2

This oxide growth is sometimes performed at NIST, and sometimes done by an external vendor—recently, Rogue
Valley Microdevices.

49
about overcoming inconsistencies in the photoresist’s reaction with light, and more about making
the process robust to imperfections in the masks. It helps ensure that no “extra” unwanted junctions
remain on the wafer.
The same process is then repeated with the aluminum layer, including an additional mask
for an aluminum overflash. Finally, the base electrode mask is also flashed, and the niobium isetched, leaving the desired pattern of Josephson junctions, as well as any circuit wiring done on thebase electrode layer. If the circuit includes any normal metal (the R1 layer), this is now deposited,masked, and etched, leaving only the pattern defined by the R1 mask.

5.2.2

SQUID arrays in the trilayer process

To forms SQUIDs from these Josephson junctions, four junctions are arranged at the cornersof a rectangle, as illustrated in Fig. 5.2a. A dielectric is then deposited via an electron cyclotron
resonance plasma enhanced chemical vapor deposition (PECVD) process (Fig. 5.2b). Vias tothe junction counter electrodes are etched through this layer (defined with the I1 mask), and asecondary niobium layer, called W1, is deposited on top of the dielectric and etched according to
the W1 mask (Fig. 5.2c). This niobium layer galvanically connects the four counter electrodes.
Finally, an additional etch removes the excess dielectric in portions of the circuit that containneither SQUIDs nor the W1 layer, reducing dielectric loss.
The result of this process is a series-array of two SQUIDs, each formed from a pair of Josephson junctions (Fig. 5.2e). Current in the first SQUID flows from the base electrode up to the firstwiring layer, and then flows back to the base electrode through the second SQUID. Longer arraysof SQUIDs may be created with multiple instances of this pattern. Secondary wiring can also be
accomplished on the W1 layer. Fig. 5.3 shows a layout view for a cell containing a series-array of
two SQUIDs, as depicted schematically in Fig. 5.2.
This cell was used to create the SQUID arrays in the fourth-generation circulator. The
Josephson junctions are 2 by 2 microns with nominal critical current density 0.48 µA/µm2 , yielding
1.92 µA junctions and SQUIDs with maximum critical current of 3.84 µA.

50
(a)

(b)
SiO2

(c)

Nb (W1)

(d)

{

{

{

(e)

BE

W1

BE

Figure 5.2: SQUIDs in the Niobium trilayer process are fabricated in pairs. (a) Four Josephson
junctions are arranged on the corners of a rectangle. (b) A 300 nm-thick layer of dielectric (siliconoxide, in this case) is deposited on the wafer. This is the layer of insulation etched by I1 and I1X. (c)Vias (defined by I1) are etched through the dielectric layer to the junction counter electrodes. These
vias are obscured in the figure by the W1 layer, and are not visible. A third layer of niobium—theW1 layer—is then deposited on top of the silicon oxide, and etched with the mask defined by W1.
(d) A second etch on the silicon dioxide (defined by I1X) removes excess dielectric. (e) The resultof this process is a series-array of two SQUIDs. Current in the first SQUID flows from the baseelectrode up to the first wiring layer, and then flows back to the base electrode through the second
SQUID. Trilayer graphics courtesy of Maxime Malnou.

51
CEO ALO

AL

I1
CE

BE

2 𝜇m
W1

I1X

Figure 5.3: Layout of the SQUID cell used in the fourth-generation circulator, depicted schematically in Fig. 5.2. The CEO and ALO layers are directly on top of one-another, and demarcated by
the largest rectangle.

Typical variation in junction critical currents with the NIST trilayer process is below 5% forjunctions less than 1 mm apart. Furthermore, the frequency of high critical current outliers is also
low: less than 1% of junctions have critical currents that exceed the target value by more than30%. As the tunable inductors needed to realize the bridge circuit multipliers are formed by arraysof SQUIDs, the variation in array inductance is further reduced by averaging: if variations in thejunction critical current are assumed to be uncorrelated across the array, an array of N SQUIDs will√
have a fractional uncertainty in its total inductance which is 1/ 2N times the fractional uncertaintyof the inductance of a single junction.

5.2.3

Additional wiring layers

If a tertiary wiring layer is desired, a second PECVD dielectric layer is then deposited. Vias(defined by the I2 mask) are etched through this layer, and a fourth niobium layer, W2, is thendeposited and etched, leaving only the wiring pattern defined by the W2 mask. A final dielectricetch (defined by I2X) removes the excess dielectric everywhere except below W2.

52
5.2.4

Best practices for trilayer process layout

We conclude discussion of the trilayer process with a list of “best-practices” for trilayer layout.
Many of these are specific to the equipment and practices at NIST Boulder, but some are more
broadly applicable to general layout of superconducting circuits.
• Although BE, W1, and W2 are all composed of niobium, they are deposited on different
substrates, and this gives the BE layer qualitatively different material properties from W1and W2. (BE is deposited on thermally grown silicon oxide, whereas W1 and W2 resideon dielectrics deposited with a plasma-enhanced, chemical vapor deposition technique.) In
particular, BE has lower dielectric losses than W1 and W2. For this reason,∗ Circuit connectivity should be done whenever possible on BE.
∗ If a crossover is needed between the bias lines and microwave lines, make sure that itis the bias lines that make the cross over, since loss is typically more tolerable in thatchannel.
∗ Crossovers should be as short as possible. A good rule of thumb is to place vias 2 µm
away from the line which is being crossed.
• When making resistors with normal metal, one must consider the superconducting-normal
metal-superconducting junction physics caused by the induced superconductivity of theproximity effect. A detailed discussion of this is deferred to Sec. 5.3.2. But the induced
quasiparticle transport in the superconductors must also be considered. When possible,
widen niobium traces that have normal metal overlap to mitigate this effect.
• Different layers should not share edges.
• The wafer dicing saw has a width of 40 µm. When designing chips, move the edge of thedesign 20 µm in from the die edge. This facilitates dicing and prevents problems such as
ground plane peeling, and can be accomplished by including alignment marks of width 20µm on the corners of the chip design.

53• In general, geometric splits (e.g. variations in number of SQUIDs or capacitor dimensions)are easy to fabricate, whereas splits with different materials, oxidation time, or numbers oflayers are more time consuming. When possible, design splits with geometric variations.
For example, to design a spread of resonance frequencies in a wafer of our circulators, vary
the capacitor dimension, not the critical current density of the Josephson junctions. Limit
geometric splits to 6 or less, when possible. Splits should be labelled with names that are
explanatory, but also easily distinguishable for collaborators not familiar with the work.
For example, in a wafer with four splits, the labels
A Cap95Res10, B Cap95Res15, C Cap115Res10, and D Cap115Res15
are preferable to the labels
Cap95Res10, Cap95Res15, Cap115Res10, and Cap115Res15.
• To avoid lift-off problems, do not layout W1 or W2 layers above the R1 layer. This can bedone by removing the normal metal directly beneath the wiring layer.
• Normal metal should be more than 4 µm away from vias.
• Vias from W2 to BE should be done in two steps, and not a single tall via.
• Avoid designs with different chip sizes on a single wafer.
• Utilize hierarchy; design with cells.
• Consider the effect of trapped flux upon cooling through the superconductor’s criticaltemperature Tc . This can be done rapidly by comparing the equipartition energy kB Tc /2
to the energy stored in the magnetic degree of freedom for one loop in the circuit, Φ20 /2L,with L the inductance of the loop. If kB Tc /2 > Φ20 /2L the available thermal energy will
excite higher energy flux states as the circuit cools through Tc , and each time the device
cools down it will be initialized in a different state.
• Consider the effect of vortex formation as superconducting films are cooled through theirtransition temperature Tc . Ref. [144] provides a rule of thumb for this process in designs

54with niobium: in a magnetic field B0 , vortex formation is suppressed in a niobium microstripof width w when B0 w2 < Φ0 = h/(2e).

5.3

Design considerations
Broadly speaking, the three major challenges in the layout of a superconducting circulator

based on the approach discussed in Ch. 4 are(1) retaining circuit symmetry,(2) maintaining the stability of the flux environment, and(3) suppressing the coupling between the microwave signal lines and the radio-frequency bias
lines.
In this section, we detail the design measures undertaken to overcome these difficulties.

5.3.1

Capacitor design

To realize capacitors of capacitance c for the circulator’s lumped element representation(Fig. 4.3c), we layout parallel-plate capacitors in a metal-insulator-metal geometry with niobium
plates sandwiched around the dielectric SiO2 (Fig. 5.4a). In the frequency range of 4 to 8 GHz,
roughly pF capacitances are required to create capacitor-impedances near 50 Ohms. Making a pF
capacitor with SiO2 in the Nb trilayer process requires capacitor plates that are roughly 100 µm ona side—large enough to trap magnetic flux vortices when cooled through niobium’s superconducting
transition temperature Tc in earth’s magnetic field [144].
To avoid trapping flux vortices, we pattern slots in the capacitor electrodes, such that the
Nb strips that form the electrodes never exceed a width w  100 µm. This ability to suppress
vortices in non-zero magnetic fields is important for our layout, as the circulator is actuated withflux controls which can be spoiled by a static and unremovable flux gradient. Choosing w = 5 µm
ensures that the capacitor electrodes trap no magnetic flux vortices when the capacitor is cooled
through Tc in our experiment’s modestly shielded magnetic environment.

55
(a)

Nb

(not to scale)

Nb

SiO2

Si

(b)

50 µm

Figure 5.4: Capacitor design. (a) Schematic illustrating the symmetrization of the parallel plate
capacitors used in the device. To create a capacitor c, two capacitors with half the desired capacitance are connected in parallel, such that the upper (lower) plate of the first capacitor is galvanicallyconnected to the lower (upper) plate of the second. This procedure gives each side of the capacitor
the same parasitic capacitance to the ground plane. (b) False-color scanning electron microscope
image showing one of the c/2 capacitors used in the fourth-generation device (blue), and the joint
connecting it to another c/2 capacitor (green), as described in (a). The plates in both capacitors
are formed from narrow niobium strips of width w = 5 µm to prevent trapping flux vortices [144].

To further symmetrize the circuit, each parallel-plate capacitor is then divided into two
capacitors of capacitance c/2, and connected in parallel, such that the upper plate of the first(second) capacitor is galvanically linked to the lower plate of the second (first) capacitor (Fig. 5.4b).
This procedure gives each side of the composite capacitor the same parasitic capacitance to ground,and is essential for preserving the symmetry on which the concept of the device relies.
A scanning electron microscope image of a capacitor is shown in Fig. 5.4b, which shows the
Nb strips that form the top plate of one of the c/2 parallel plate capacitors. In the right side of
the image, the capacitor is connected to a second parallel plate capacitor (mostly out of view) in
the manner described above.

56
5.3.2

Use of normal metal

Superconducting loops in the circulator can trap magnetic flux and lead to an unstable
flux environment, interfering with the flux biasing used to control the device. To avoid trapping
unwanted flux, we layout the circulator using small amounts of a normal metal—in earlier versionof the circulator, gold-palladium, and in later versions, gold. Fig. 5.5 illustrates the effect ofusing normal metal in this way: it shows measurements of the phase of forward transmission as
a function of a static gradiometric flux Φg , in two circulators which are identical except for theabsence (Fig. 5.5a) or presence (Fig. 5.5b) of normal metal. The phase wrap of the resonant delay
is visible in both devices, but in Fig. 5.5a, the circuit’s resonant frequency shifts from scan to scan,
resulting in jagged edges visible in the transmission’s phase. In contrast, the phase in Fig. 5.5b is
stable from scan to scan, which we interpret as evidence of a a stable flux-environment.
The thickness (height) of the gold layer is d = 225 nm, giving it a sheet resistance of 60
mOhms/square. To reduce resistive losses in this layer, 11 Au squares are placed in parallel,
yielding a total film resistance of about 10 mOhms. Four of these resistors are placed in each
Wheatstone bridge (resistor symbols in Fig. 5.6b) to break supercurrent loops and maintain thesymmetry required by the circuit. Estimates with time-domain numerical simulations (Simulink)indicate that the addition of these resistors causes the dissipation in the circuit to increase by 0.1
dB, limiting the internal Q of the circuit to be less than 2000.
To design the normal metal resistors in a way that prevents proximitization by the nearby
niobium, the resistor lengths l—defined as its dimension parallel to the flow of current—is constrained to be l  ξd . Here ξd is the coherence length of Au calculated in a dirty limit, where themetal film’s mean free path ln is less than the clean-limit coherence length [145]
ξc =

~vF
.
2πkB T(5.1)

In the above, vF is the Fermi-velocity of the metal, kB is Boltzmann’s constant, and T is the metal’s
temperature. We justify this treatment with the observation that the Au film’s mean free pathln ≈ 600 nm  6 µm ≈ ξc . This estimate for ξc is made with the assumption that vF = 1.4 × 106

(a)

2𝜋

57

0.30

Φg/Φ0

0.15
𝜋

0
-0.15
-0.30

0

(b)

2𝜋

0.30

Φg/Φ0

0.15
0

𝜋

-0.15
-0.30
0
4

6

5
frequency [GHz]

Figure 5.5: The effect of normal metal on the stability of a circulator’s resonant modes. (a) The
phase of forward transmission ∠S21 as a function of probe frequency and a static gradiometric flux
Φg , with Φu fixed at 0.2Φ0 . The device under test is a fourth-generation circulator design, identicalto that discussed in Ch. 6, with the exception that it contains no normal metal. (b) The samemeasurement as (a) on a fourth-generation circulator with normal metal inserted into the bridge
circuits, as described in the text.

m/s in Au [146], and the temperature T set to 300 mK. The mean free path is calculated with theDrude model [146] and the film’s resistivity.
In the dirty-limit, the coherence length ξd is essentially a geometric mean of the clean-limit
coherence length and the metal’s mean free path [145]:
r
ξd =

ln ξc
,
3

(5.2)

which comes out to ξd ≈ 1 µm for the above values of ξc and ln . This value is comparable with

58
measurements of the dirty-limit coherence length in thin films of a similar elemental metal, copper,
when one corrects for sample thickness [147, 148].
The condition l  ξd can be made quantitative with consideration of the superconductingnormal-superconducting (SNS) junction physics which govern the Nb-Au-Nb interface. Unlike asuperconducting-insulator-superconducting junction, which is governed by an energy scale set by
the superconducting gap, the natural energy scale for the proximity effect in an SNS junction is
the Thouless energy [148]. For junctions of reasonable size (l > ξd ), the critical current of the SNSjunction is
2πkb T
In =
Rn q



ξd
l

2

e−l/ξd e−l/lφ .

(5.3)

Here q is the electron charge, Rn is the room temperature resistance of the junction, lφ ≈ 2 µm
is the inelastic scattering length of gold at 300 mK [149], and the Thouless energy is expressedin terms of the coherence length ξd and the junction length l. The critical current In sets the
Josephson energy EJ = ϕ0 In of the SNS junction, and the resistance R of the junction scales inrelation to the Josephson energy and the energy in the thermal environment:
R = Rn e−EJ /kB T .

(5.4)

Along with the Josephson inductance of a SQUID array (Eq. (4.4)), the resistance in Eq. (5.4)
sets an L/R time which characterizes the time required for trapped-flux to dissipate out of thecircuit. Choosing l > 5.5ξd ensures that L/R time is less than 1 second. In preliminary designs,
we therefore set l = 10 µm. In later designs we found experimentally that l = 5 µm also prevents
flux-trapping, likely due to a dirty-limit coherence length which is less than our 1 µm estimate.
The device presented in the following chapter has l = 5 µm.

5.3.3

Bias line design

The circulator’s active components are actuated with flux controls created by a pair of on-chipbias lines. Design of these bias lines involves two important layout considerations: namely, isolating

59
the microwave fields from the bias lines, and preventing the RF bias signals from interfering withthe operation of the microwave circuit.
Isolating the circulator’s microwave fields from the bias lines is important because from theperspective of the microwave circuit, coupling to the bias lines acts as an additional loss channel.
To reduce losses of this kind low-pass filters (20 nH spiral inductors) are inserted into the bias lines
as they enter and exit the chip (pink inductor symbols in Fig. 5.6a). These simple filters present
an impedance of approximately 15 Ohms to the bias signals at Ω = 2π × 120 MHz, whereas at
microwave frequencies in the 4 to 8 GHz band their impedance exceeds 500 Ohms. Simulations
using commercial planar method-of-moments solvers (AWR Microwave Office) indicate that these

(b)

...

...

...

(a)

...

filters limit microwave transmission out the bias lines to less than −20 dB.

(c)

MA
Ma
A

a

5 µm
bias line

Figure 5.6: (a) Schematic showing the position of normal metal resistors in the inductive bridges,
and the routing of a quadrupole-source bias line which couples strongly to the SQUIDs it encloses
and weakly to surrounding loops of the circuit. (b) False color scanning electron microscope image
showing two adjacent SQUID arrays (green) and the microstrip lines (pink) that flux-bias them.
(c) Schematic illustrating the challenge of coupling a bias line strongly to a small SQUID loop while
coupling it weakly to a larger circuit loop.

The challenge of the second consideration—preventing bias signals from interfering with thecircuit’s microwave operation—is illustrated in Fig. 5.6c. The lumped-element representation of the

60
circulator (Fig. 4.3c) contains tunable inductors, realized with flux-modulated SQUIDs, as well aslarger circuit loops which are (partly) comprised of SQUIDs. For simplicity, we consider the effectof a bias line on one such loop of area A which includes a SQUID with area a inside it (Fig. 5.6c).
To operate the circulator, the bias line must dynamically thread a flux through the SQUID,
on the order of a tenth of a flux quantum Φg ≈ Φ0 /10, at a rate Ω. If the mutual inductance
between the bias line and the SQUID loop is denoted as Ma , this requires an AC bias current withamplitude Ig ≈ Φ0 / (10Ma ).
The time-dependent gradiometric flux, however, also threads through the larger circuit loopof area A. Faraday’s law describes the electromotive force induced around this loop, which fora cosinusoidal bias current is E = ΩMA Ig sin(Ωt). We assume the impedance of the loop ZA is
entirely inductive in origin. The loop inductance LA is the sum of its geometric Lg and Josephson
inductance LJ , which we write in terms of the participation ratio p ≡ LA /LJ as LA = pLJ , yielding
ZA = iΩpLJ . Ohm’s law then allows a calculation of the AC current induced around the loop ofarea A, which with the appropriate substitutions has an amplitude of
Iind ≈

2π MA
Is ,10p Ma(5.5)

with Is the critical current of the SQUID.
When the induced currents approach the SQUID critical currents in magnitude, the higher
order corrections in Eq. (4.4) become significant, and when it exceeds the critical currents, theSQUIDs become dissipative elements. From Eq. (5.5) one can see that the bias signals will couple
to the microwave circuit and interfere with the circulator’s operation unless the prefactor on theequation’s right-hand side is much less than one. The circulator’s performance (in particular, theability to impedance match the device) requires that the participation ratio p not be much greaterthan one. The only way to satisfy the requirement, then, is to engineer the mutual inductancesA
such that M
Ma  1. This is challenging, as the size of the parallel plate capacitors and the SQUID

arrays mandates that A/a ≈ 103 .
To overcome the disparity in loop areas and satisfy the coupling condition MA /Ma  1, we61layout the bias lines in a symmetric way, such that their currents create magnetic quadrupoles.
The layout of the shielded bias lines is shown schematically in Fig. 5.6a, and is also visible in the
SEM image in Fig. 5.6b. The central bias line, bisecting the bridge, carries the full bias current Ig
across the chip, and then splits into two parallel arms, each carrying a current Ig /2 on the outsideedges of the bridge. As the currents in these lines flow in opposite directions, the magnetic fieldBg from this shielded configuration scales asBg =  4µ0 Ig   2
+O
,
2πr r
r

(5.6)

where  is the separation between the inner and outer bias lines, and µ0 is the vacuum permeability.
We make  as small as possible in our layouts, given the requirement that the SQUID arrays must
reside between the inner and outer bias lines. These constraints result in the choice  = 17.5 µm.

5.4

Design iterations
In the following sections we catalogue the previous design iterations. The intent is to provide

some record of the design process chronology, and the insights which led to the design considerations
discussed in the preceding section. In the interest of brevity, extensive details about measurementsof past devices are not provided. Likewise, details about the experimental set-up are deferred toCh. 6.

5.4.1

Generation I.

Fig. 5.7 shows a false-color optical micrograph of the first-generation circulator. The micrograph is taken after a protective photoresist is removed from the chip, via a standard sequence ofsolvent rinses (30 seconds in acetone, 30 seconds in acetone, 30 seconds in isopropanol, dry withnitrogen gas). The main design features and observed performance are summarized in Tab. 5.2.
When the resonant delay was tuned with a static gradiometric flux, its frequency dependence
agreed qualitatively with a circuit model. These measurements, however, exhibited a clear hysteresis
with respect to the sweep-direction of the bias current, as illustrated in Fig. 5.8.

21

bridgecircuit

62

ports

4

20 SQUID
array
cap.

100 µmcosinebias
line

normal metal

sine
bias
line

3
sine bias line

cosine bias line

...

...

...

...

Figure 5.7: False-color optical micrograph of the first-generation circulator layout. The pop-out
displays how the cosine bias line couples to a sinusoidally modulated bridge circuit in an undesirable
way.

When the device was configured as a gyrator and biased near the operation point expectedfrom theoretical considerations (e.g. with φ = π/2, Ωτ = π/2, as described in Ch. 4), the resonant
delays were visible. Fig. 5.9a shows the phase of transmission (color), as a function of frequency andthe magnitude of the AC gradiometric flux Φg . At gradiometric fluxes above 0.08 Φ0 , two resonantmodes separated by 2Ω/(2π) are visible in the plot. At lower gradiometric fluxes, though, the flux
environment of the resonator appears to be unstable. The largest non-reciprocity is observed inthis unstable region, as shown in Fig. 5.9b, which plots |S21 − S12 | [dB] over the same region of

63
Table 5.2: Design features and performance of the first-generation circulator.
Design features• Capacitors symmetrized tobalance parasitic capacitanceto ground, as described inSec. 5.3.1

Observed performance• Resonant modes tune withstatic flux, in qualitative agreement with circuit model, but
exhibit hysteresis• Normal metal (AuPd) inserted
to break all supercurrent loops
except those inside the bridge
circuits, which are suppressedby gradiometry of figure-eightlayout (see Sec. 4.2.1)• Transmission strongly attenuated by dynamical flux modulation

2𝜋

(a)
0.04

Φg/Φ0

0.02
𝜋

0
-0.02
-0.04

0
(b)

2𝜋
0.04

Φg/Φ0

0.02
𝜋

0
-0.02
-0.04

0
5.7

6

6.3

frequency [GHz]

Figure 5.8: Hysteresis in the first-generation circulator with respect to sweep-direction of bias
currents. (a) Phase of transmission (color) through the device, as a function of frequency and an
upwardly swept static gradiometric flux Φg . (b) The same measurement, for a downwardly swept
static gradiometric flux Φg .

64parameter space. Even there, though, its magnitude is still far from the maximum theoretical valueof 3 dB. This is not due to an equality in the phases of forward and reverse transmission, as canbe seen in Fig. 5.9c, where ∠S21 − ∠S12 is displayed for the same measurement.
Hysteresis in the DC flux sweeps and instability and weak transmission in the AC flux sweeps
likely resulted from a flux-stability issue. Inspecting the layout in Fig. 5.7, one can see that thecosine (sine) bias line couples strongly to the sinusoidally (cosinusoidally) modulated bridges. Thisis illustrated in the pop-out below the micrograph. There, the sinusoidal bias line modulates theflux through the SQUIDs in the desired fashion, but does not thread flux through the figureeight because of the circuit’s gradiometric design (the twist at the center of the figure-eight). The
cosinusoidal bias line, however, threads a large flux through the bridge-circuit’s figure eight. As
these bridges have no internal normal metal to break super-current loops, and as that couplingthreads many flux-quantum through those loops,3

the result is an unstable (and dynamically

evolving) flux-environment.

5.4.2

Generation II.

To address this issue, the second generation device was a large-scale redesign. The figureeight-geometry of the bridge circuits was retained, as was the symmetrization of the capacitors,
but otherwise the circuit layout was completely rearranged. Fig. 5.10 shows a false-color optical
micrograph of the second-generation circulator. The main design features and observed performanceare summarized in Tab. 5.3.
In this device, again configured as a gyrator, the observed performance improved dramatically. Transmission was still strongly attenuated (-26 dB of insertion loss), but the resonant delays
were visible during AC modulation of the bias currents, and the device exhibited the desired π
difference between forward and backward transmission, characteristic of gyrators. Despite these
improvements, we still observed instabilities in the flux environment, evident in sub-linewidth-scale3

To see this, recall that the bias lines are designed to thread a sizable fraction of a flux-quantum through the
SQUIDs that they address, and these SQUIDs are orders of magnitude smaller in area than the bridge circuit.

65

Table 5.3: Design features and performance of the second-generation circulator.
(New) design features• Smaller footprint (tens of nH
of geometric inductance eliminated)

Observed performance• Resonant modes tune withstatic flux, in quantitative
agreement with circuit model• New layout allows bias lines
to run in straight, parallellines and drastically reduces
mutual inductance between cosine (sine) bias line and sinusoidally (cosinusoidally) modulated bridges• Resonant modes tune with dynamic flux modulation, but
shift in frequency from scan toscan by fractions of a linewidth.

• Straight bias lines enabled byinversion of one pair of bridge
circuit electrodes, which introduces an effective π phase shift
into the modulation (realizingthe − sin(Ωt) modulation witha sinusoidal bias current)• Slotted parallel-plate capacitors, to suppress the creationof flux vortices, as described inSec. 5.3.1• Gyration (π phase shift between forward and backward
transmission) with -26 dB insertion loss

2𝜋

(a)

66

Φg/Φ0

0.2

𝜋
0.1

0

0
(b)

0
0.2

Φg/Φ0

-20

0.1

-40

0

-60

(c)

2𝜋

Φg/Φ0

0.2

𝜋
0.1

0
5.7

6

6.3

0

frequency [GHz]

Figure 5.9: AC modulation of the first-generation circulator. (a) Phase of transmission (color)through the device, as a function of frequency and AC gradiometric flux of amplitude Φg . (b)Non-reciprocity |S21 − S12 | [dB] as a function of frequency and AC gradiometric flux of amplitude
Φg . (c) Angular non-reciprocity ∠S21 − ∠S12 over the same parameter space.

shifts in the resonant frequency of the circuit between scans.

2

1

67

ports
20 SQUID arraybridge circuitcosine
bias
line

cap.

normal metal

sine
bias
line

‘twist’
100 µm4

3

Figure 5.10: False-color optical micrograph of the second-generation circulator layout.

In the first and second generation designs, normal metal had been inserted to break all of thesuperconducting loops in the circuit, except those within the bridges themselves. The idea behind
this had been that as these structures were folded into a symmetrical figure-eight, their gradiometry
would prevent the flow of screening currents.
Motivated by the observed flux-instabilities, we reconsidered this conclusion. As the uniformbackground magnetic field must be strong enough to thread a sizable fraction of a flux quantum
through each SQUID, and as the SQUIDs are more than three orders of magnitude smaller inarea than the bridge circuits, a gradiometric approach requires gradiometry better than one partin 103 . For bridge circuits which are hundreds of microns long and tens of microns wide, thatentails fabrication with sub-micron resolution, unachievable with the optical lithography used in
the trilayer process. We therefore resolved to move the normal metal into the bridge circuits, andbreak all of the device’s supercurrent loops in the subsequent design cycle.
When considering what limited the insertion loss of the device, we observed that the measured
operation point was far from the operation point expected from the theoretical model: namely,Ωτ  π/2. Attempts to increase bias frequency Ω or the duration of the resonant delay τ (which

68

(a)

2𝜋

Φg/Φ0

0.08

𝜋

0.04

0

4.4

4.8

5.2

0

frequency [GHz](b)
Het [𝜇V]

150
50
-50
0

200
time [ns]400

Figure 5.11: Performance of the second-generation circulator (configured as a gyrator). (a) Measurements of ∠S21 (color) as a function of frequency and the magnitude of AC gradiometric flux
Φg . The phase wrap from the two resonant delays is visible around 4.8 and 5.2 GHz, respectively.
An “edge” feature is also visible in the data when the gradiometric flux is around Φedge= 0.065Φ0 .
g(b) Heterodyne measurement of forward (red circles, dashed red line) and backward (yellow circles,
solid yellow line) transmission through the device. The two signals are π out of phase, as desired
for a gyrator. The maxima on the y-axis, however, indicate the received power has been attenuatedby 26 dB relative to the input.

may be tuned by increasing the degree to which the bridges are imbalanced), led to a breakdown inthe device, visible in the sudden disappearance of the resonant delay and any kind of non-reciprocaltransmission (e.g. the “edge” feature in Fig. 5.11a). In dedicated studies of this breakdown, we
observe that the location of the edge (in parameter space) depended on Ω and τ , but also on thephase φ between the gradiometric flux drives.
Fig. 5.12 illustrates how such an angular dependence could arise, if the edge resulted fromthe total bias flux (i.e. the interference between the two gradiometric flux lines) through a large

69
loop in the circuit exceeding some critical value—for example, a value set by the SQUID criticalcurrents, in the manner discussed in Sec. 5.3.3. To see this, note that the flux Φbias through thelarge loop in Fig. 5.12a can be expressed with a sum-to-product identity asΦbias = MA Ig [cos (Ωt) − cos (Ωt + φ)]= −2MA Ig sin (φ/2) sin (Ωt + φ/2).

(5.7)

Eq. (5.7) predicts a half-angle scaling of the magnitude of the total bias flux with φ. To test thatprediction, we perform sweeps like that shown in Fig. 5.11a, with a variety of different phases φand bias frequencies Ω. For each sweep, we record the gradiometric flux Φedge
at which the “edge”
gfeature appears.
If, as discussed in Sec. 5.3.3, the edge results from a changing flux through one of these largeasloops inducing a current that exceeds some threshold value, then a plot of the inverse of Φedgega function of φ should exhibit the same half-angle dependence predicted in Eq. (5.7). Fig. 5.12bshows that this is indeed the case. Each circle is a separate measurement of Φedge, and the differentgtraces correspond to bias frequencies between 2π × 20 MHz and 2π × 120 MHz. The solid lines
without circles are guides to the eye. They show the expected half-angle dependence sin(φ/2) for
three different magnitudes, and are included to facilitate comparison with the measured data. Thisobservation supports the conjecture that the edge was caused by induced currents in the microwave
circuit which exceeded the critical current of the SQUIDs.

5.4.3

Generation III.

To combat this failure-mode, in the third design cycle we implemented the quadrupoleshielded bias lines discussed in Sec. 5.3.3. We also moved the normal metal inside the bridge
circuits, and switched from AuPd to Au, to decrease resistance of the normal metal inserts from
400 mOhm to 10 mOhm. Other than these changes, and the addition of low-pass filters in thebias lines, the overall architecture was largely unchanged from generation two. Fig. 5.13 shows afalse-color optical micrograph of the third-generation circulator. Its principal design features and

70

(a)
Ig cos(Ωt)

Ig cos(Ωt +ϕ)

MA

MA

A

“cosine” bias line

“sine” bias line(b)

Φ0/Φgedge

80

40

0

0

𝜋

2𝜋

ϕ

Figure 5.12: Angular dependence of the “edge” feature in the second-generation circulator. (a)Schematic illustrating how flux driven through a large circuit loop (containing a SQUID) by thetwo bias lines can interfere, to produce a total flux that scales as sin (φ/2). (b) Measurementsof the “edge” feature (like the one shown in Fig. 5.11), as a function φ. Circles are individual
measurements of Φedge, and solid lines are theory curves that scale as sin (φ/2). The differentgtraces correspond to repeated measurements with different bias frequencies Ω.

observed performance are summarized in Tab. 5.4.
Generation three also improved substantially on its predecessor. When configured as a gyrator, the circuit’s resonant frequencies were visible and stable under active modulation of thegradiometric flux Φg (Fig. 5.14a), and the device exhibited the expected non-reciprocal transmission with -4 dB of insertion loss. Heartened by this performance, we also tested the device as a

71

Table 5.4: Design features and performance of the third-generation circulator.
(New) design features• Used Au instead of AuPd tobreak supercurrent loops• Moved normal metal inside thebridge circuit, to break all off
the circuit’s superconducting
loops

Observed performance• Resonant modes tune withstatic flux, in quantitative
agreement with circuit model• Resonant modes tune with dynamic flux modulation; resonant frequencies are stable• Implemented the shielded bias
lines discussed in Sec. 5.3.3 byrunning bias currents on thesecondary wiring layer (W1)

• Gyration (π phase shift between forward and backward
transmission) with -4 dB insertion loss

• Inserted simple low-pass filters(20 nH spiral inductors) on thebias lines to reduce the approximately 3 dB of estimated microwave loss through this channel• 13 dB of isolation, 3 dB insertion loss, when operated as an
isolator• Linear in microwave power for
signals up to -86 dBm

72
bridge circuit

1

2

ports
20 SQUID arraycosine
bias
line

cap.

normal metal

100 µm
4

3

sine
bias
line

Figure 5.13: False-color optical micrograph of the third-generation circulator layout.

circulator. Fig. 5.14b shows transmission through the device from port 1 to 2 (dashed line) and port
2 to 1 (solid line), as a function of the bias phase φ. The approximately 10 dB of contrast between
the two traces reveals a violation of Lorentz reciprocity. The device’s dynamical reconfigurability
is also evident—operation points for both clockwise and counter-clockwise circulation are visible inthe sweeps. At these points, isolation exceeds -13 dB and insertion loss is less than 3 dB.
Nevertheless, the operation point where optimal performance was observed deviated from theexpected parameters for ideal operation—though the degree of deviation was far less than in thesecond generation device. Hypothesizing that the performance breakdown discussed previously was
still a limiting factor, we attempted to further reduce the parasitic coupling between the bias lines
and larger circuit loops, by reducing the separation between the primary and current returning
microstrips of the bias lines from  = 23 µm to  = 17.5 µm (reducing the quadrupole moment of
the bias line). We also reduced the number of SQUIDs in each array to N = 12, to reduce the areaof the loops which couple to the bias lines. To maintain the inductance of the arrays at 1 nH, wemade a concomitant change in the junction critical current, reducing it from 3 µA to 1.92 µA.

73

(a)

2𝜋

Φg/Φ0

0.12

𝜋

0.08

0.04

6.5

7

7.5

8

0

frequency [GHz]
(b)
-2
𝑆21

𝑆𝜇𝜈 [dB]

𝑆12

0
-8

-14
0

𝜋/2

𝜋

3𝜋/2

2𝜋

ϕ

Figure 5.14: Performance of the third generation circulator. (a) The phase of transmission ∠S21
as a function of frequency and the magnitude of the AC gradiometric flux Φg . Unlike previous
generations, the resonant response is completely stable from scan to scan. (b) Transmission as afunction of phase φ, from the first port to the second port (dashed line) and the second port to thefirst port (solid line). Two clear operation points are visible, where clockwise and counter-clockwise
circulation is realized.

5.4.4

Generation IV.

These changes are reflected in the layout of the fourth-generation device, depicted in Fig. 5.15
alongside the circulator’s lumped-element representation. They are also recorded in Tab. 5.5, with74
the device’s observed performance. As that performance is the subject of the following chapter, wemomentarily defer its discussion.
(a)δ0cos(Ωt)δ0sin(Ωt)(b)δ0cos(Ωt)

14

δ0cos(Ωt)

𝑐

1

4

3

𝑐

2

𝑐

𝑐

−δ0sin(Ωt)(c)−δ0sin(Ωt)δ0cos(Ωt)
bridge circuit

1

4

2

3

δ0sin(Ωt)

12 SQUID arraycosine
bias
line

cap.
normal metal

sine
bias
line

2

3

ports

100 µmFigure 5.15: Layout of the fourth generation circulator. (a) Lumped-element representation of thecirculator. (b) The lumped-element circulator representation in (a), rearranged to better matchthe device’s layout. (c) False-color optical micrograph of the fourth-generation circulator layout.

An additional split was also designed for this wafer, which makes a more radical departurefrom the layout in generation three. We call this device version b of the fourth-generation circulator.
An optical micrograph of a version b chip is shown in Fig. 5.16. Its design features and observed
performance are summarized in Tab. 5.6, and are discussed in more detail in the following chapter.

75

Table 5.5: Design features and performance of the fourth-generation circulator.
(New) design features• Reduced junction critical current from 3 µA to 1.92 µA andnumber of SQUIDs in each array from N = 20 to N = 12,
to keep arrays at 1 nH of inductance while reducing circuit
footprint

Observed performance• Resonant modes tune withstatic flux, in quantitative
agreement with circuit model• Moved bias lines to tertiary
wiring layer (W2) to tighten thequadrupole layout (reduce , inthe language of Sec. 5.3.3)• >40 dB of isolation, less than 1
dB insertion loss, at select frequencies• Resonant modes tune with dynamic flux modulation; resonant frequencies are stable• Tunable operation frequency
between 4 and 5.6 GHz• Linear in microwave power for
signals up to -90 dBm

Table 5.6: Design features and performance of the fourth-generation circulator, version b.
(New) design features• Interdigitated capacitors• Bias lines split off chip allow for
50% reduction in geometric inductance• Bias line “quadrupole” created
without any current division

Observed performance• Optimal operation phases separated by π
• Resonant modes tune with dynamic flux modulation; resonant frequencies are stable• >40 dB of isolation, less than 1
dB insertion loss, at select frequencies• Tunable operation frequency
between 4 and 5.6 GHz• Linear in microwave power for
signals up to -90 dBm

76

1

ports

2

bridge circuit

12 SQUID array

sine
bias
line

cosine
bias
line

cap.

cosinebias
line

normal metal

100 µm

3

sine
bias
line

4

Figure 5.16: False-color optical micrograph of the fourth-generation circulator (version b) layout.

Chapter 6

Experimental results

In this chapter we describe the performance of the fourth-generation circulators. We begin
by describing the basic experimental setup. Next the tune-up procedure (that is, the process bywhich the device’s control parameters are set) is discussed. Experimental results follow this, including measurements of the circulator’s scattering matrix elements and a characterization of its
transmission spectrum and linearity. These measurements are performed over a range of different
operation frequencies and with the circulator configured for clockwise and counterclockwise circulation, highlighting the device’s tunability and the capability to dynamically reconfigure its senseof circulation in-situ. We conclude with a discussion of the measurements, including the device’s
current limitations, and scaling considerations.

6.1

Experimental setup
To measure the circulator, two of its four ports are terminated in 50 Ohm loads and thecircuit is mounted at the base of a 3 He cryostat. A simplified schematic of the experimental setup
is shown in Fig. 6.1 (additional details are provided in App. C). Two switches and a directional
coupler allow for measurement of the four accessible scattering parameters with a network analyzer.

6.2

Tune-up procedureThree straightforward steps are required to prepare the circulator for operation. First, thefrequency of the resonant delay ω0 is tuned to the desired operation frequency. Second, the duration

78
Φgcos(Ω𝑡+𝜙)Φgcos(Ω𝑡)
50

VNA

300 K

Φu

4K

300 mK

Figure 6.1: Simplified experimental schematic for circulator measurements in a 3 He cryostat (attenuation, filtering, and isolation omitted). Two switches and a directional coupler allow measurementof four of the circulator’s scattering parameters.

of the resonant delay τ is set to π/2Ω. Finally, the phase difference φ between the gradiometricflux control drives is set to ±π/2.
We illustrate the first two of these steps in Fig. 6.2, which shows in color the group delay τ
acquired during transmission through the device at different probe frequencies ωp , and for different
values of a static gradiometric flux Φg applied to all four of the inductive bridges. The measurement
is shown for three different values of the uniform flux Φu .
Two features are immediately evident in the data. First, the resonant nature of the delay isclear: for fixed values of Φu and Φg , fields at most probe frequencies are off-resonance and theirgroup delays are less than 5 ns, as visible in the black background of the color plot. Against
this background, three arches are visible, which show the resonant delay tuning with Φg for thethree measurements at distinct Φu . The shapes of these arches are qualitatively captured by thetheoretical predictions in dashed gray lines, which are made with Eq. (4.7) and the relations thatmap Φg & Φu to δ0 & l0 in Eq. (4.5) and Eq. (4.6).
Second, when |Φg | approaches 0, the group delay vanishes. A gradiometric bias with magnitude much less than Φ0 = 2πϕ0 results in approximately balanced bridges (δ  1). As theexternal-coupling of the resonant mode depends on δ 2 (Eq. (4.8)), balanced bridges result in undercoupled resonant modes, which strongly attenuates transmission through the resonant differential𝜏 (ns)0.25

Φu/Φ0= 0.20
Φu/Φ0= 0.33

Φg/Φ0

Φu/Φ0= 0.42

79

100
50

0

10
-0.25
4

5
frequency [GHz]6

5

Figure 6.2: Measurements of the circulator’s group delay τ (color, log scale) as a function of
the probe frequency and a static gradiometric flux Φu applied to all four of the bridge circuits.
The duration and center frequency of the resonant delay depend on the uniform and gradiometricflux, allowing the circulator’s operation frequency to be tuned between 4 and 6 GHz. Dashed graylines are predictions of Eq. (4.7) which use the mapping in Eq. (4.5) and Eq. (4.6). To accountfor geometric inductance in the circuit (which is not present in the model), the dashed lines arecalculated with effective uniform fluxes Φ̃u /Φ0 = 0.38, 0.33, and 0.28, chosen to match the frequencyof the measured and predicted delays when Φg = 0.

modes. (The internal quality factor of the circuit is estimated to be 400 when the resonant delay istuned to 5 GHz.) Power is still transmitted through the non-resonant common mode, but withoutacquiring resonant delay.
As circulation bandwidth scales with the linewidth of the resonant delay, for the measurements in this paper we operate the device with a relatively brief delay on the order of several
nanoseconds, with Ω = 2π × 120 MHz. This choice has the additional benefit of reducing theinfluence of internal losses by keeping the circuit strongly over-coupled.
The final step of the tune-up procedure involves selection of the relative phase φ between thegradiometric flux controls. Fig. 6.3 shows a sweep of φ when the resonant delay ω0 is set near 2π ×4GHz and the duration of the delay τ is fixed at several ns with Ω = 2π × 120 MHz. The color scalein Fig. 6.3a shows the magnitude of S21 as a function of this phase and the probe frequency: S12 isshown in Fig. 6.3b. (Calibration of network parameter measurements is discussed in Appendix D.)

80

𝑆𝜇𝜈 [dB]

-30

0
𝑆12

frequency [GHz](a)-15 dB

4.1

-3 dB
-1 dB

4 -20 dB
-30 dBfrequency [GHz]

(b)

𝑆21

4.1

4

0 (c)

𝑆𝜇𝜈 [dB]

-15

𝑆21

-30

𝑆12
-45
0

𝜋/2

𝜋
ϕ

3𝜋/2

2𝜋

Figure 6.3: Measurements of a dynamically reconfigurable circulator. The phase φ between thegradiometric flux lines determines the direction of circulation. (a)-(b) S12 and S21 a function of
the probe frequency and the phase φ. Horizontal and vertical lines indicate the location of linecuts
plotted in (c) and Fig. 6.4(a). (c) Linecuts of S21 (dashed) and S12 (solid) at probe frequency
ωp = 2π × 4.044 GHz.

The color plots in Fig. 6.3 reveal two regions of parameter space in which operating points
can be chosen. At these phases, the insertion loss is less than 1 dB and the isolation exceeds 30

81
dB. They can therefore be interpreted as the phases which realize a clockwise or counterclockwise
circulator. To illustrate this, Fig. 6.3c shows frequency linecuts at 4.044 GHz from both transmission measurements. Importantly, the linecuts show that high transmission in the counterclockwise(clockwise) direction is accompanied by strong isolation in the clockwise (counterclockwise) direction. They also illustrate how toggling the phase φ allows dynamical reconfiguration of the device’s
sense of circulation. Interestingly, one can see that the strongest non-reciprocity is observed atphases near but distinct from the expected operating points at π/2 and 3π/2. Sec. 6.4 describes
how geometric inductance in the circuit causes this discrepancy.

6.3

Performance
The frequency-dependence of the non-reciprocity is visible in linecuts of Fig. 6.3a, b, taken

at the two optimal operating phases. These linecuts are shown alongside reflection measurements
in Fig. 6.4a, and characterize four of the device’s sixteen scattering parameters. Different ports
were probed in a separate cooldown, with similar results (presented momentarily).
In the transmission measurements (top right and bottom left plots), high transmission (> −1dB) and robust isolation (> 20 dB) are observed in a 50 MHz window around 4.044 GHz. Thesefeatures are approximately coincident with −11 dB dips in the reflection measurements (top left
and bottom right plots). Together, power collected in the transmission and reflection measurements
account for 90% of the injected signal power.
To determine if the remaining power is dissipated or scattered to other frequencies, a spectrum
analyzer is used to measure the transmission of the circulator at frequencies ωp ± mΩ which aredetuned from the input frequency by integer multiples of the modulation rate Ω. Fig. 6.4b shows
the power of these spectral components, relative to the transmitted spectral component at ωp . The
device suppresses spurious sidebands by more than 20 dB. The spectral purity of the output—inparticular, the suppression of spectral components at ωp ± 2Ω—is a testament to the high-degreeof symmetry in the circuit. From this measurement, we conclude that the remaining 10% of input
power is dissipated into heat or other radiation modes.

82

0
(a)

4
-20

1

3

-40

2

𝑆12 [dB]

𝑆11 [dB]

0

4
-20

1
𝑆21 [dB]-40

P(𝜔)/P(𝜔p) [dB]44.1
frequency [GHz]

0

3

3

2

𝑆22 [dB]4
4.1
frequency [GHz]

(b)

-20

-40

-4Ω -3Ω -2Ω -Ω 0
Ω 2Ω 3Ω 4Ω
detuning from 𝜔p = 2𝜋 x 4.038 GHz

𝑆𝜇𝜈 [dB]

0

(b)
(c)

-20

-40 2
10

𝑆21 cw
𝑆21 ccw
𝑆12 cw
𝑆12 ccw
103
on-chip power [fW]104

Figure 6.4: Performance of an on-chip superconducting circulator tuned to operate near 4 GHz.
(a) Frequency dependence of four of the circulator’s 16 scattering parameters, when configured as acounterclockwise circulator (blue traces) and a clockwise circulator (orange traces). (b) Transmission spectrum of the circulator at ωp = 2π × 4.044 GHz, measured at frequencies ωp ± mΩ with ma positive integer. Spectral components are normalized by the power transmitted at ωp . Spurious
sidebands are suppressed by approximately 20 dB. (c) Transmission as a function of probe power,with probe frequency fixed at ωp = 2π × 4.044 GHz. 1 dB compression occurs around 1 pW.

Finally, Fig. 6.4c displays the dependence of clockwise and counterclockwise transmission on
the power of the probe signal. Fixing the probe frequency at ωp = 2π×4.044 GHz, the measurement

83
is repeated for both clockwise and counterclockwise operation. In both cases, 1 dB compression of
the transmitted signal occurs at input powers around 1 pW. As the input power approaches thisvalue, we also observe a degradation in the circulator’s isolation, which drops below 20 dB at a
power again roughly equal to 1 pW. In analogy with the 1 dB compression point, we refer to thispower as the 20 dB expansion point of the circulator. Expressed in terms of photon number, thislinearity allows the circulator to process over 103 photons per inverse of its bandwidth.
For reference, the typical power in a microwave tone used for dispersive readout [59] of asuperconducting qubit is between 100 and 1000 aW (few photon level) [150]. The three orders ofmagnitude that separate this power scale from the 1 dB compression and 20 dB expansion pointsof the device are critical for one attractive application of a monolithic superconducting circulator:
on-chip integration with a quantum-limited reflection amplifier, such as a Josephson parametricamplifier. The high power handling of the circulator allows it to route qubit readout tones evenafter they reflect off a Josephson parametric amplifier and are amplified by 20 dB.
To demonstrate the circulator’s tunability, we operate the device at a variety of frequencies
between 4 and 6 GHz and repeat the measurements shown in Fig. 6.4. Fig. 6.5 summarizes theperformance of the device across this band. Insertion loss is shown in Fig. 6.5a. Transmission
is greatest at the lowest frequency, and decreases with frequency until it approaches -3 dB. We
attribute this trend to the geometric inductance present in the circuit, which limits the degree towhich the bridges can be imbalanced. This inhibits impedance matching and reduces the degree
to which the resonant differential modes are over-coupled. At lower operation frequencies, theJosephson inductance comprises a greater fraction of the bridge’s total inductance, mitigating thiseffect.
This interpretation is supported by the power dissipation that we estimate at each operation
frequency (Fig. 6.5b), computed as the sum of the reflection R and transmission T coefficients−10 log10 R2 + T 2 . (Power transmitted to sidebands of the modulation frequency is suppressedby over 20 dB, and is therefore neglected in this accounting). Reflections, visible in the discrepancy
between insertion loss and transmission, are larger at higher frequencies, where the role of geometric

(d)

(e)

dissipation [dB]max. iso. [dB]

(c)

50

2

1

0
4

50

2
2
50

0
70

1

50

35

2
0
60

iso. BW [MHz](b)sidebandsupp. [dBc](a)ins. loss [dB]

84
4

4

30

1

0
30

50

50

20
10

4

0
power [fW]

(f)

1

103

50

50
102

1 dB compression
44.5

20 dB expansion

5frequency [GHz]5.5

Figure 6.5: Performance of a widely tunable on-chip circulator. The minimum insertion loss (a),dissipation (b), maximum isolation (c), isolation bandwidth (d), sideband suppression (e), andpower handling (f) are plotted at frequencies between 4 and 6 GHz, for clockwise (orange andpurple rectangles) and counterclockwise (blue and green circles) operation. Isolation bandwidthrefers to the bandwidth over which isolation exceeds 20 dB. Sideband suppression refers to the contrast between transmitted spectral components at the input frequency ωp and the largest spurious
sideband of the form ωp ± nΩ. In (e), dotted (solid) lines indicate the device’s 1 dB compression(20 dB expansion) point.

85
inductance is more pronounced. Dissipation is also greater at higher operation frequencies, where
the external coupling of the resonant mode is lower. We believe the dominant source of dissipationin the circuit is the dielectric loss of SiO2 .
The circulator’s maximum isolation is plotted in Fig. 6.5c. Below 5.5 GHz, isolation exceeds
35 dB for both device configurations. Critically, isolation is achieved over a bandwidth of several
tens of MHz, much greater than the bandwidths typical for strongly-coupled cavity ports in dispersive qubit readout, which range up to several MHz [46, 47, 151]. Fig. 6.5d shows the frequency
interval over which the isolation exceeds 20 dB.
It should be noted this isolation is achieved concurrent with the performance shown in the restof Fig. 6.5: all specifications are measured at two fixed operation phases, which realize clockwise andcounterclockwise circulation. To select these operation phases in a quantitative manner, we write acost function to simultaneously balance the benefits of low insertion loss, high isolation, and broad
bandwidth, for both clockwise and counterclockwise operation. Ultimately, different applications
will prioritize the relative importance of these specifications in different ways, allowing trade-offs
in performance specifications, for example, between insertion loss and isolation. Similarly, if thedevice’s reconfigurability is not needed, performance will generally exceed that shown in Fig. 6.5.
Fig. 6.5e characterizes the spectral purity of transmitted fields at each operation frequency.
It shows the size of the largest spurious sideband, (relative to the power transmitted at the probe
frequency), which we call the sideband suppression. Harmonics of the modulation frequency Ω arestrongly suppressed across the operation band, typically by about 20 dB.
Lastly, Fig. 6.5f shows how the power-handling of the circulator depends on the operation
frequency. Frequencies between 4 and 5 GHz have 1 dB compression points and 20 dB expansion
points around 1 pW, roughly three orders of magnitude above the power level used for dispersive
readout of a superconducting qubit.

86

6.4

Discussion
Applications and prospectives for an on-chip superconducting circulator are discussed in the

final chapter of this thesis. Here we comment on issues currently limiting device performance, andon the attenuation and filtering considerations relevant to scalability.

6.4.1

Device limitations and deviation from theoretical models

In this section we discuss non-idealities observed in the circulator, in which the network parameter measurements depart from the theoretical predictions of the scattering matrix, obtainedwith the analytical model in Ref. [115]. The reference predicts the dependence of S on the parameters l0 , δ, and Ω. Using the relations in Eq. (4.5) and Eq. (4.6), l0 and δ can be mapped to
the flux controls Φu and Φg . To facilitate this comparison, Fig. 6.6 shows measured and predictedtransmission parameters, as a function of the probe frequency and the gradiometric flux Φg .
Qualitatively, the experiment and model agree fairly well: all four plots show a pair of resonantmodes split by twice the modulation rate Ω/2π, in analogy with a Sagnac interferometer [128]. As
Φg increases, these modes shift down in frequency and broaden. Furthermore, the device’s nonreciprocity is evident in both the model and in experiments: at the lower frequency mode, S21 islarge in magnitude at the same frequency and gradiometric flux that S12 is small.
One can also see aspects of the experimental data which are not captured in the model. For
example, as Φg approaches 0 the resonant modes become increasingly difficult to perceive in theexperimental data. In the model, though, the modes become narrower as Φg decreases, but remain
well distinguished from the off-resonant transmission. This discrepancy is a result of the fact thatinternal losses are not included in the theoretical model. In the measurement, the presence of lossmeans that for small enough Φg , the modes become under-coupled and are difficult to detect.
Another discrepancy is the slight splitting (2g ≈ 2 × 2π × 17 MHz) of each resonant mode,
visible in the experimental plots (Fig. 6.6a and b). We attribute this splitting to a hybridizationof the circuit’s two degenerate resonant modes, which is not included in the model.

(a) 0.15

87

1

(b)

0.8
0.1
Φg/Φ0

0.6
0.4

0.05

(c)

0.2
0
1

(d)

0.15

0.8
0.1
Φg/Φ0

0.6
0.4

0.05
0.2

3.7

4
4.3
frequency [GHz]4.6 3.7

4

4.3
frequency [GHz]4.6

0

Figure 6.6: Measurements (a,b) and theoretical predictions (c,d) of |S21 | and |S12 |, as a function ofprobe frequency and amplitude of the oscillatory gradiometric flux, when the device is configured asa counterclockwise circulator. Theoretical predictions are made with the expressions in Ref. [115]and the mapping in Eq. (4.5) and Eq. (4.6). Circuit parameters are fixed at their design targets(l0 = 1 nH, c = 1 pF) and the flux controls are set to match the measurements in (a) and (b):
Ω = 2π × 120 MHz, Φu = 0.38Φ0 .

The sharp “edge” visible at large Φg /Φ0 ≈ 0.15 in the measurements is an additional difference between the model and experiments. As discussed in Ch. 5, this edge feature is likely caused
by induced currents in the microwave circuit which exceed the critical current of the SQUIDs. Refinements in the layout can reduce these induced currents, though device operation would still belimited in the amplitude of the applied gradiometric flux; the application of a total external flux
Φu ±Φg with magnitude greater than Φ0 /2 causes a deviation from the simple flux tuning described

88
in Fig. 4.2. When the total flux exceeds this threshold, further increase in Φg serves to balance
the inductive bridges, rather than imbalance them, and a departure from the model is expected inthis regime.
A final difference between the model and experiments is visible in the scaling of the resonantmodes with the gradiometric flux. The modes in the theory plots are more sensitive to Φg , bending
down to lower frequencies than the measured modes. They also broaden and merge, to a degree
which is not apparent in the measurements. We attribute this discrepancy to geometric inductancein the circuit which reduces the tunability of the resonant delay and restricts the modal linewidth.
This interpretation is supported by our observation of optimal circulator performance atdrive phases φ distinct from the theoretically expected values at π/2 and 3π/2. When geometricinductance restricts the linewidths of the circulator’s resonant modes, it prevents the creation of
the brief (2 ns) resonant delay needed to satisfy the convert-delay operation condition: Ωτ = π/2.
The condition can be met with reduction of Ω, but this is undesirable for two reasons: first,
the circulator’s bandwidth is proportional to Ω; second, device performance requires that themodulation rate Ω exceed the internal splitting g of the hybridized resonant modes: Ω  g.
A simple extension of the theory discussed in Sec. 4.1 shows how the circulator’s transmission
depends on φ and Ωτ in the general case when Ωτ takes values other than π/2:
S21 ∝
S12 ∝


1
1 − cos (Ωτ + φ)2
1
1 − cos(Ωτ − φ) .
2

(6.1)

From these expressions, it is clear that if Ωτ is forced to take values greater than π/2, improved
counterclockwise (clockwise) circulation can be obtained with phases greater (less) than π/2 (3π/2).
Our observation of optimal control phases at ≈ 2π/3 and 4π/3 corresponds to a minimum achievabledelay of about 3 ns.

89
6.4.2

Comparison with the fourth-generation, version b device

The performance of the fourth generation version b chip is also consistent with this argument.
In that device, geometric inductance was reduced by approximately 50% (see Fig. 5.16 for an optical
micrograph) by dividing the sine and cosine bias lines off-chip, allowing the bridge circuits to beimbalanced to a greater degree.
Fig. 6.7 shows its scattering matrix when the device was configured as an isolator. Over
40 dB of contrast may be observed between the forward transmission and reverse isolation at thecenter of the device’s bandwidth. Importantly, the difference between the optimal clockwise andcounterclockwise operation phases for this device is 0.9972π ≈ π. In sum, we expect shorter delays
to be possible with this device, and observe no deviation in the operation phases from theoretical
expectations.
0
4

-20
1

3

-40
2

-60

𝑆11 [dB]

𝑆12 [dB]

0

4
-20

1

3

3

-40
-60
5.7

𝑆21 [dB]
5.8
5.9
frequency [GHz]

2
6 5.7

𝑆22 [dB]

5.8
5.9
frequency [GHz]6

Figure 6.7: Scattering parameters of the fourth-generation circulator, version b, tuned to operatre
near 5.9 GHz. Frequency dependence of four of the circulator’s 16 scattering parameters, when
configured as a counterclockwise circulator (blue traces) and a clockwise circulator (orange traces).

Fig. 6.8 shows the performance of this circulator over a wide operation band, between 4 and
7 GHz. Insertion loss and dissipation are plotted in Fig. 6.8a, the maximum isolation is plotted inFig. 6.8b, and the bandwidth over which isolation exceeds 20 dB is plotted in Fig. 6.8c.

90loss [dB](a)insertion loss [dB]dissipation [dB]

6
4

50

2
0

1

50

max. iso. [dB]

70
(b)

2
35
50
0

(c)

iso. BW [MHz]

1
200

50

2

100
0
4

5

6

7

frequency [GHz]

Figure 6.8: Performance of the fourth-generation circulator, version b. Insertion loss (a, solidlines) and dissipation (a, dashed lines), maximum isolation (b), and isolation bandwidth (c), as afunction of operation frequency. Isolation bandwidth is defined as the frequency interval over which
the isolation exceeds 20 dB.

In many ways, its performance is comparable to the other generation-four design (discussed
in Sec. 6.3). The predominant difference is a slight increase in the reflection coefficient of the version
b device, which results in greater insertion loss. (Version b exhibits insertion losses between 2 and6 dB, compared to 0.8 to 3 dB insertion losses in the other device).
The discrepancy is especially apparent at low operation frequencies. One explanation for thisobservation is the difference in capacitance values for the two designs (0.96 pF in the first device
and 0.6 pF in the version b design). The smaller capacitance in the version b device allows it tooperate up to 7 GHz, and necessitates the application of a large uniform flux to shift the circuit’s
resonant frequency down to 4 GHz. At this frequency, the SQUID arrays have a reactance of about
100 Ohms, which leads to a total breakdown of the expansion in Ch. 4 in the parameter ωl/Z0 .

91
In other words, as the two devices have different maximum operation frequencies, a comparison
at 4 GHz may not be a direct comparison: the device with the greater operation frequency mustbe tuned to a greater degree to reach 4 GHz, which can cause a breakdown of a needed high-Q
approximation. Nevertheless, even near its maximum operation frequency the version b device is
not as well matched to 50 Ohms.
One surprising similarity in the performance of the two devices is that their measured dissipation is quite similar. Measurements with test resonators (SQUID array bridge circuits shunted
by parallel-plate capacitors) indicate an internal Q between 300 and 400, which was assumed to belimited by dielectric loss in the silicon oxide. Given the external coupling rate of the circulator’s
resonant modes, this would lead to approximately 0.5 dB of insertion loss. Unexpectedly, the version b device (which uses interdigitated capacitors) shows no reduction in dissipation. A detailed
explanation for the insertion loss in the two devices remains an open question.

6.4.3

Filtering, attenuation, and power-consumption considerations, in the context
of scaling

One of the costs associated with replacing passive ferrite circulators with active on-chip
circulators is the power consumption of the control tones, and the heat loads this creates in adilution refrigerator. Estimating that power consumption requires a discussion of the attenuationand filtering of the control lines.
To determine the attenuation required to keep the added noise below half a photon, the added
noise is estimated as a function of the temperature T to which the control lines are thermalized.
Scaling and filtering considerations are then discussed, in light of this result.
For simplicity, consider the noise added by the circulator during transmission from its firstport to its second port. Fluctuations of the bias current amplitude and relative phase between thetwo bias signals will modulate a transmitted tone, thus creating noisy modulation sidebands of thetone. The sideband noise powers caused by amplitude fluctuations SIAN and phase fluctuations SIPN

92
are (at most)
SIAN
SIPN


=

=

∂S21
I1dB
∂Ig

2

∂S21 I1dB
∂φ Ig

SI ,
2
SI .

(6.2)

Here, I1dB is the signal current in the device at its 1 dB compression point and SI = 4kB T /Z0 isthe current spectral density of the Johnson noise (in the bias lines) at a temperature T [152, 14].
We say the sideband noise powers are bounded above by the relations in Eq. (6.2) because thesenoise powers are multiplicative, and it is reasonable to assume the microwave signal power incident
on the device (during normal operation) is at most I1dB .
Because we operate near a maximum in |S21 |, the dominant effect of noise in both theamplitude and phase of the bias currents is the phase modulation of the transmitted tone; i.e.,
both SIAN and SIPN are predominantly phase noise in the transmitted tone.
The partial derivatives in Eq. (6.2) can be calculated directly from measurements of thescattering parameters, made as a function of the bias current amplitude Ig and the phase between
the bias lines φ (shown, for example, in Fig. 6.6 and Fig. 6.3). After these numerical derivatives
are calculated, the sideband noise powers may be divided by 2~ωp /Z0 to convert them to photon
numbers. In our measurements, where the bias lines are thermalized to T ≈ 300 K, this results in
n = 14 photons of added noise, with SIAN accounting for 95% of the noise.
Positioning 40 dB of attenuation at room temperature and 20 dB at the four Kelvin stageof a dilution refrigerator would result in a noise temperature of T ≈ 7 K, or in units of photons,
n = 0.3. This level of attenuation is reasonable for modern dilution refrigerators, as the circulator
operates with gradiometric currents Ig on the scale of 100 µA: the heat load caused by a 20 dB
attenuator at the four-K stage is 50 µW, which is much less than the Watt-scale cooling power
available at that stage.
With superconducting twisted-pairs to carry the bias currents from the four-K stage to themixing chamber plate, and a contact resistance of 10 mOhm at the chip interface, the heat loadon the mixing chamber plate is 100 pW. This load is also much less than the roughly 50 µW of

93available cooling power on a 50 mK mixing chamber plate. These considerations are summarizedin Tab. (6.1), which presents a power budget for an active circulator with control lines thermalized
as described above.
Table 6.1: Power budget for an active circulator with control lines thermalized to T ≈ 7 K as
described in the text. In this configuration, the circulator adds 0.3 photons of noise. A contact
resistance of 10 mOhm is assumed at the chip interface. Cited cooling powers are for an Oxford
cryogen-free dilution refrigerator, and are approximate.
T [K]
300
4
0.05

Ig [A]
10−1
10−3
10−4

heat load [W]
5 × 10−1
5 × 10−510−10cooling power [W]n/a7.5 × 10−1
5 × 10−5This analysis indicates the feasibility of operating 103 on-chip circulators in a single dilution
refrigerator, each with less than half a photon of added noise. We emphasize that this is one ofmany possible design choices and it is possible to reduce the added noise and dissipated power inseveral different ways. For example, the bias lines could be filtered to reject the noise below 50
MHz, which adds noise in the circulator’s band, while still passing 100 MHz bias tones.

Chapter 7

Conclusion and Outlook

In this thesis we realize the on-chip superconducting circulator proposed in Ref. [115]. Lorentz
reciprocity is broken in the circuit with sequential translations in frequency and time, which we showwith a standard circuit analysis and with a simple model system composed of just two components:
multiplying elements and delays. We describe how both of these elements can be created in acryogenic microwave environment, and then characterize the performance of a circulator built fromthese components. We observe low insertion loss and over 20 dB of isolation over a bandwidth ofapproximately 50 MHz. The device is linear with respect to input power for fields up to 1 pW
in power, and its transmission spectrum is spectrally pure, in the sense that spurious harmonics
created by the device’s RF control tones are suppressed by more than 20 dB. Finally, we demonstratethat all of these performance specifications can be achieved over a frequency range in excess of 1
GHz, and in clockwise or counterclockwise configurations. This ability to dynamically reconfigure
the sense of circulation is a feature unavailable in commercial ferrite circulators which are typicallyconstructed with permanent magnets, and enables new possibilities for the connectivity of advanced
networks.
As the device requires no microwave frequency control tones, and as it is orders of magnitude
more compact than commercial ferrite circulators, this superconducting circulator is a scalable alternative to signal routing with ferrite junction circulators. We estimate that with superconducting
twisted pairs carrying the low-frequency control tones, 103 of these circulators could be operated
in a single dilution refrigerator.

95
From a technological perspective, the superconducting circulator constructed for this thesis
represents a new level of complexity for analog integrated circuits in the niobium trilayer process. It
also provides a demonstration of a general-purpose prescription for circulation [115], which need not
be implemented with superconducting materials. For example, in room temperature applications
where tunable capacitors are a widely available component, the dual of this circuit can be used
to realize a scalable, on-chip circulator. Transmission and reception of signals in the antennas ofcellular phones and cellular base stations is currently multiplexed in frequency, due to the needfor high channel isolation and the dearth of compact, integrated, non-reciprocal components at
microwave frequencies. Duplexing with a circulator would obviate the need for this multiplexing,
doubling the effective bandwidth of the antenna. Such an adaptation of the circulator technology
could be lucrative, as 5G telecommunication technologies are likely to gain mass-market coverageby 2020, with a projected $250 billion USD market size [153].
Looking forward, the work suggests several immediate extensions. In a future design, layoutchanges could improve device performance: dielectric loss can be reduced with the use of low-loss
dielectrics like amorphous silicon [100], or with interdigitated capacitors. Similarly, dividing thepower in the gradiometric flux lines off-chip and delivering the control tones with four dedicatedon-chip bias lines removes layout constraints, and enables the design of a circuit with approximately
half the geometric inductance.
Even with the device’s existing performance, another obvious extension is on-chip integration of the circulator with a quantum-limited amplifier. This further miniaturizes the amplification
chain, and removes unwanted sources of loss from normal-metal connections between the superconducting reflection amplifier and circulator.
Finally, the essential concept of frequency conversion and delay can be adapted to a losslessand broadband design, using non-resonant delays, as suggested in Ref. [116]. Fig. 7.1a illustrates
one possible design for such a broadband, lossless circuit. The device uses the same bridge-circuit
multipliers as the circulator discussed in the previous chapter, but requires only two of the bridges.
The element Zm is an impedance included to impedance match the bridge circuits, needed because

96of the finite-tunability of the SQUID-array inductors. As the Bode-Fano criterion [154, 155] isrelatively lenient for low quality-factor circuits, we estimate that appropriate choice of Zm would
allow reflections to be limited to less than -20 dB over a bandwidth of 8 GHz (see supplementary
information in Ref. [132]).
The delays, rather than realized with a resonant mode, are transmission lines connecting thetwo bridges. A final difference is that the bias signals are square wave, rather than sinusoidal.
Biased in this way, the impedance-matched bridge circuits function as cross-over switches.

3

Zm
Zm
Zm
𝜏
2

(b)

(d)

(f)

(c)

(e)

(g)

1

-1
𝜏 = 𝜋/(2Ω)= T/4𝜏modulation

1

Zm

modulation

δ0sign(cos[Ωt])

(a)

Zm
Zm

Zm
Zm

δ0sign(sin[Ωt])

4

1

-1
0

T/2
time

T0

T/2
time

T0

T/2
time

T

Figure 7.1: Lossless, broadband circulation. (a) Schematic for a lossless, broadband, and superconducting circulator, inspired by ideas in Refs. [116, 156, 120]. Circulation arises from interference
between the reciprocal even-mode and the gyrating odd-mode of the circuit. (b)-(g) Describe how
non-reciprocity arises in the odd-mode. (b) Modulation profiles of the two square-wave biased
bridge circuits when the delay separating the bridges is τ = 0. (c) Product of the modulation
profiles in (b). Transmission is reciprocal, and power is scattered into modulation sidebands. (d)Modulation profile for a signal exciting the differential mode of the upper port (odd excitation ofports 1 & 3) when the delay separating the bridges is a quarter of the modulation period, τ = T /4.
(Graphically, this shifts the orange trace back by a quarter-period.) (e) The product of the modulation profiles in (d). As the product is 1 for the entire modulation period, the signal is effectively
unmodulated. (f) Modulation profile for a signal exciting the differential mode of the lower port
(odd excitation of ports 2 & 4) when the delay separating the bridges is a quarter of the modulation
period, τ = T /4. (Graphically, this shifts the blue trace back by a quarter-period.) (e) The productof the modulation profiles in (d). The product is constant in time, as in (e), but the minus sign
imparts a π phase shift on the incident signal. The odd-mode of the circuit therefore realizes agyrator.

97
As in the analysis of Ch. 4, the circuit’s operation is greatly simplified by transforming to abasis of even and odd excitations of the top (ports 1 & 3) and bottom (ports 2 & 4) ports. Viewed
in that light, circulation again arises from a “virtual” Hogan construction, e.g. the interference ofa reciprocal common mode and a gyrating differential mode. To show how non-reciprocity arisesin the differential mode, the square-wave modulation of transmission realized by the two bridge
circuits is plotted in Fig. 7.1b. If no delay is included between them, a differential signal on theupper (odd excitation of ports 1 & 3) or lower (odd excitation of ports 2 & 4) is modulated intime according to Fig. 7.1c, and the network is reciprocal (and scatters the incident signal into
sidebands spaced by odd multiples of the modulation rate Ω). If, however, a delay τ of one-quarterthe modulation period T /4 = π/(2Ω) is inserted between the two bridge circuits, reciprocity isbroken, and the incident signal’s frequency is unchanged. A signal incident on the upper port ismodulated by (Fig. 7.1d and Fig. 7.1e)sign(cos[Ωt]) × sign(sin[Ωt + π/4]) = 1,

(7.1)

whereas a signal incident on the lower port is modulated by (Fig. 7.1f and Fig. 7.1g)
sign(sin[Ωt]) × sign(cos[Ωt + π/4]) = −1.

(7.2)

The circuit’s odd-mode therefore realizes a gyrator.
Prospects for such a device are extremely attractive given the high power-handling of these
SQUID-array based devices, as their integration with a broadband low-noise amplifier [157] could
enable scalable frequency-domain multiplexing of many-qubit systems with near-unit measurement
efficiency. Given the current state of quantum error-correction and architectures for quantum
information processing with superconducting circuits, such signal-processing innovations will almost
certainly be necessary for construction of any superconducting quantum computer intended to be
more than a proof-of-principle.

Bibliography[1] R. P. Feynman, “Simulating physics with computers,” International Journal of
Theoretical Physics, vol. 21, no. 6, pp. 467–488, Jun 1982. [Online]. Available:
https://doi.org/10.1007/BF02650179[2] ——, “Quantum mechanical computers,” Foundations of Physics, vol. 16, no. 6, pp. 507–531,
Jun 1986. [Online]. Available: https://doi.org/10.1007/BF01886518[3] D. Deutsch, “Quantum theory, the church-turing principle and the universal quantum
computer,” Proceedings of the Royal Society of London A: Mathematical, Physical andEngineering Sciences, vol. 400, no. 1818, pp. 97–117, 1985.
[4] ——, “Quantum computational networks,” Proceedings of the Royal Society of London
A: Mathematical, Physical and Engineering Sciences, vol. 425, no. 1868, pp. 73–90, 1989.
[Online]. Available: http://rspa.royalsocietypublishing.org/content/425/1868/73[5] P. Shor, “Lecture notes in computer science,” in Algorithmic Number Theory: FirstInternational Symposium, L. M. Adleman and M.-D. A. Huang, Eds. New York: Springer,
1994, p. 289.
[6] R. L. Rivest, A. Shamir, and L. Adleman, “A method for obtaining digital signatures andpublic-key cryptosystems,” Commun. ACM, vol. 21, no. 2, pp. 120–126, Feb. 1978. [Online].
Available: http://doi.acm.org/10.1145/359340.359342[7] L. K. Grover, “Quantum mechanics helps in searching for a needle in a haystack,”
Phys. Rev. Lett., vol. 79, pp. 325–328, Jul 1997. [Online]. Available:
https://link.aps.org/doi/10.1103/PhysRevLett.79.325[8] M. H. Yung, , J. D. Whitfield, S. Boixo, D. G. Tempel, and A. Aspuru-Guzik, Advancesin Chemical Physics, Volume 154: Quantum Information and Computation for Chemistry,
K. B. Whaley, A. R. Dinner, and S. A. Rice, Eds. John Wiley & Sons, 2014.
[9] M. B. Hastings, D. Wecker, B. Bauer, and M. Troyer, “Improving quantum algorithms forquantum chemistry,” Quantum information and computation, vol. 1, no. 1-2, p. 00010021,
2015.
[10] D. Poulin, M. B. Hastings, D. Wecker, N. Wiebe, A. C. Doherty, and M. Troyer, “Thetrotter step size required for accurate quantum simulation of quantum chemistry,” Quantum
information and computation, vol. 15, no. 5-6, p. 03610384, 2015.

99[11] A. K. Ekert, “Quantum cryptography based on bell’s theorem,” Phys. Rev. Lett., vol. 67,
pp. 661–663, Aug 1991. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevLett.
67.661[12] W. Heisenberg, The physical principles of the quantum theory. Courier Corporation, 1949.
[13] C. Gardiner and P. Zoller, Quantum noise: a handbook of Markovian and non-Markovian
quantum stochastic methods with applications to quantum optics. Springer Science & Business Media, 2004, vol. 56.
[14] H. Nyquist, “Thermal agitation of electric charge in conductors,” Phys. Rev., vol. 32, pp.
110–113, Jul 1928. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRev.32.110[15] H. B. Callen and T. A. Welton, “Irreversibility and generalized noise,” Phys. Rev., vol. 83,
pp. 34–40, Jul 1951. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRev.83.34[16] Braginskiı̆, “Quantum-mechanical limitations in macroscopic experiments and modern experimental technique.”[17] V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne, “Quantum nondemolition
measurements,” Science, vol. 209, no. 4456, pp. 547–557, 1980. [Online]. Available:
http://science.sciencemag.org/content/209/4456/547[18] G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. Raimond, and S. Haroche,“Seeing a single photon without destroying it,” Nature, vol. 400, no. 6741, p. 239, 1999.
[19] H. A. Lorentz, “The theorem of poynting concerning the energy in the electromagnetic field
and two general propositions concerning the propagation of light,” Amsterdammer Akademie
der Wetenschappen, vol. 4, p. 176, 1896.
[20] J. Kerckhoff, L. Bouten, A. Silberfarb, and H. Mabuchi, “Physical model of continuous
two-qubit parity measurement in a cavity-qed network,” Phys. Rev. A, vol. 79, p. 024305,
Feb 2009. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevA.79.024305[21] N. Roch, M. E. Schwartz, F. Motzoi, C. Macklin, R. Vijay, A. W. Eddins, A. N. Korotkov,
K. B. Whaley, M. Sarovar, and I. Siddiqi, “Observation of measurement-induced entanglement
and quantum trajectories of remote superconducting qubits,” Physical review letters, vol. 112,
no. 17, p. 170501, 2014.
[22] A. Narla, S. Shankar, M. Hatridge, Z. Leghtas, K. M. Sliwa, E. Zalys-Geller, S. O. Mundhada,
W. Pfaff, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret, “Robust concurrent remoteentanglement between two superconducting qubits,” Physical Review X, vol. 6, no. 3, p.
031036, 2016.
[23] H. Pichler, T. Ramos, A. J. Daley, and P. Zoller, “Quantum optics of chiralspin networks,” Phys. Rev. A, vol. 91, p. 042116, Apr 2015. [Online]. Available:
https://link.aps.org/doi/10.1103/PhysRevA.91.042116[24] M. A. Castellanos-Beltran, K. D. Irwin, G. C. Hilton, L. R. Vale, and K. W. Lehnert, “Amplification and squeezing of quantum noise with a tunable josephson metamaterial,” Nature
Physics, vol. 4, no. 12, pp. 929–931, 2008.

100[25] N. Bergeal, R. Vijay, V. E. Manucharyan, I. Siddiqi, R. J. Schoelkopf, S. M. Girvin, andM. H. Devoret, “Analog information processing at the quantum limit with a josephson ring
modulator,” Nature Physics, vol. 6, no. 4, pp. 296–302, 2010.
[26] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: Towards
practical large-scale quantum computation,” Phys. Rev. A, vol. 86, p. 032324, Sep 2012.
[Online]. Available: https://link.aps.org/doi/10.1103/PhysRevA.86.032324[27] K. K. Likharev and V. K. Semenov, “Rsfq logic/memory family: a new josephson-junction
technology for sub-terahertz-clock-frequency digital systems,” IEEE Transactions on Applied
Superconductivity, vol. 1, no. 1, pp. 3–28, March 1991.
[28] J. Zmuidzinas, “Superconducting microresonators: Physics and applications,” Annu. Rev.
Condens. Matter Phys., vol. 3, no. 1, pp. 169–214, 2012.
[29] W. Holland, D. Bintley, E. Chapin, A. Chrysostomou, G. Davis, J. Dempsey, W. Duncan,
M. Fich, P. Friberg, M. Halpern et al., “Scuba-2: the 10 000 pixel bolometer camera on thejames clerk maxwell telescope,” Monthly Notices of the Royal Astronomical Society, vol. 430,
no. 4, pp. 2513–2533, 2013.
[30] B. M. Brubaker, L. Zhong, Y. V. Gurevich, S. B. Cahn, S. K. Lamoreaux, M. Simanovskaia,
J. R. Root, S. M. Lewis, S. Al Kenany, K. M. Backes, I. Urdinaran, N. M. Rapidis,
T. M. Shokair, K. A. van Bibber, D. A. Palken, M. Malnou, W. F. Kindel, M. A.
Anil, K. W. Lehnert, and G. Carosi, “First results from a microwave cavity axion
search at 24 µeV,” Phys. Rev. Lett., vol. 118, p. 061302, Feb 2017. [Online]. Available:
https://link.aps.org/doi/10.1103/PhysRevLett.118.061302[31] P. Wolf, S. Bize, A. Clairon, A. N. Luiten, G. Santarelli, and M. E. Tobar, “Tests of lorentz
invariance using a microwave resonator,” Phys. Rev. Lett., vol. 90, p. 060402, Feb 2003.
[Online]. Available: https://link.aps.org/doi/10.1103/PhysRevLett.90.060402[32] R. Hughes, “A quantum information science and technology roadmap,” Los Alamos National
Laboratory, Tech. Rep., 2004.
[33] P. Zoller, T. Beth, D. Binosi, R. Blatt, H. Briegel, D. Bruss, T. Calarco, J. I. Cirac,
D. Deutsch, J. Eisert, A. Ekert, C. Fabre, N. Gisin, P. Grangiere, M. Grassl, S. Haroche,
A. Imamoglu, A. Karlson, J. Kempe, L. Kouwenhoven, S. Kröll, G. Leuchs, M. Lewenstein,
D. Loss, N. Lütkenhaus, S. Massar, J. E. Mooij, M. B. Plenio, E. Polzik, S. Popescu,
G. Rempe, A. Sergienko, D. Suter, J. Twamley, G. Wendin, R. Werner, A. Winter,
J. Wrachtrup, and A. Zeilinger, “Quantum information processing and communication,” The
European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics, vol. 36, no. 2,
pp. 203–228, Nov 2005. [Online]. Available: https://doi.org/10.1140/epjd/e2005-00251-1[34] Z. Leghtas, G. Kirchmair, B. Vlastakis, R. J. Schoelkopf, M. H. Devoret, and M. Mirrahimi,“Hardware-efficient autonomous quantum memory protection,” Phys. Rev. Lett., vol. 111, p.
120501, Sep 2013. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevLett.111.
120501[35] S. Haroche and J.-M. Raimond, Exploring the quantum: atoms, cavities, and photons. Oxford university press, 2006.

101[36] V. V. Albert, K. Noh, K. Duivenvoorden, R. T. Brierley, P. Reinhold, C. Vuillot, L. Li,
C. Shen, S. M. Girvin, B. M. Terhal, and L. Jiang, “Performance and structure of bosoniccodes,” ArXiv e-prints, Aug. 2017.
[37] A. Kitaev, “Fault-tolerant quantum computation by anyons,” Annals of Physics, vol. 303,
no. 1, pp. 2 – 30, 2003. [Online]. Available: http://www.sciencedirect.com/science/article/
pii/S0003491602000180[38] M. H. Devoret and R. J. Schoelkopf, “Superconducting circuits for quantum information: an
outlook,” Science, vol. 339, no. 6124, pp. 1169–1174, 2013.
[39] D. P. DiVincenzo et al., “The physical implementation of quantum computation,” arXiv
preprint quant-ph/0002077, 2000.
[40] D. Nigg, M. Müller, E. A. Martinez, P. Schindler, M. Hennrich, T. Monz,
M. A. Martin-Delgado, and R. Blatt, “Quantum computations on a topologicallyencoded qubit,” Science, vol. 345, no. 6194, pp. 302–305, 2014. [Online]. Available:
http://science.sciencemag.org/content/345/6194/302[41] C. Sayrin, I. Dotsenko, X. Zhou, B. Peaudecerf, T. Rybarczyk, S. Gleyzes, P. Rouchon,
M. Mirrahimi, H. Amini, M. Brune et al., “Real-time quantum feedback prepares and stabilizes photon number states,” Nature, vol. 477, no. 7362, pp. 73–77, 2011.
[42] G. Waldherr, Y. Wang, S. Zaiser, M. Jamali, T. Schulte-Herbrüggen, H. Abe, T. Ohshima,
J. Isoya, J. Du, P. Neumann et al., “Quantum error correction in a solid-state hybrid spin
register,” Nature, vol. 506, no. 7487, pp. 204–207, 2014.
[43] M. D. Reed, L. DiCarlo, S. E. Nigg, L. Sun, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf,“Realization of three-qubit quantum error correction with superconducting circuits,” Nature,
vol. 482, p. 382, 2012.
[44] D. Ristè, C. C. Bultink, K. W. Lehnert, and L. DiCarlo, “Feedback control of a solid-state
qubit using high-fidelity projective measurement,” Phys. Rev. Lett., vol. 109, p. 240502, Dec
2012. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevLett.109.240502[45] P. Campagne-Ibarcq, E. Flurin, N. Roch, D. Darson, P. Morfin, M. Mirrahimi, M. H.
Devoret, F. Mallet, and B. Huard, “Persistent control of a superconducting qubit bystroboscopic measurement feedback,” Phys. Rev. X, vol. 3, p. 021008, May 2013. [Online].
Available: https://link.aps.org/doi/10.1103/PhysRevX.3.021008[46] J. Kelly, R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y.
Mutus, B. Campbell, Y. Chen et al., “State preservation by repetitive error detection in asuperconducting quantum circuit,” Nature, vol. 519, no. 7541, pp. 66–69, 2015.
[47] N. Ofek, A. Petrenko, R. Heeres, P. Reinhold, Z. Leghtas, B. Vlastakis, Y. Liu, L. Frunzio,
S. Girvin, L. Jiang et al., “Extending the lifetime of a quantum bit with error correction insuperconducting circuits,” Nature, 2016.
[48] R. W. Heeres, P. Reinhold, N. Ofek, L. Frunzio, L. Jiang, M. H. Devoret, and R. J. Schoelkopf,“Implementing a universal gate set on a logical qubit encoded in an oscillator,” Nature
Communications, vol. 8, 2017.

102[49] S. Touzard, A. Grimm, Z. Leghtas, S. Mundhada, P. Reinhold, R. Heeres, C. Axline,
M. Reagor, K. Chou, J. Blumoff et al., “Coherent oscillations in a quantum manifold stabilized
by dissipation,” arXiv preprint arXiv:1705.02401, 2017.
[50] P. W. Shor, “Scheme for reducing decoherence in quantum computer memory,”
Phys. Rev. A, vol. 52, pp. R2493–R2496, Oct 1995. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevA.52.R2493[51] A. Steane, “Multiple-particle interference and quantum error correction,” Proceedings of theRoyal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 452, no.
1954, pp. 2551–2577, 1996.
[52] E. Knill and R. Laflamme, “Theory of quantum error-correcting codes,” Phys. Rev. A,
vol. 55, pp. 900–911, Feb 1997. [Online]. Available: https://link.aps.org/doi/10.1103/
PhysRevA.55.900[53] M. H. Freedman and D. A. Meyer, “Projective plane and planar quantum codes,”Foundations of Computational Mathematics, vol. 1, no. 3, pp. 325–332, Jul 2001. [Online].
Available: https://doi.org/10.1007/s102080010013[54] M. Takita, A. D. Córcoles, E. Magesan, B. Abdo, M. Brink, A. Cross, J. M. Chow, andJ. M. Gambetta, “Demonstration of weight-four parity measurements in the surface codearchitecture,” Physical review letters, vol. 117, no. 21, p. 210505, 2016.
[55] J. M. Gambetta, J. M. Chow, and M. Steffen, “Building logical qubits in a superconducting
quantum computing system,” NPJ Quantum Information, vol. 3, no. 2, 2017.
[56] U. Vool and M. Devoret, “Introduction to quantum electromagnetic circuits,” International
Journal of Circuit Theory and Applications, vol. 45, no. 7, pp. 897–934, 2017, cta.2359.
[Online]. Available: http://dx.doi.org/10.1002/cta.2359[57] X. Gu, A. F. Kockum, A. Miranowicz, Y.-x. Liu, and F. Nori, “Microwave photonics withsuperconducting quantum circuits,” arXiv preprint arXiv:1707.02046, 2017.
[58] K. K. Likharev, Dynamics of Josephson junctions and circuits. Gordon and Breach sciencepublishers, 1986.
[59] A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, “Cavity quantum
electrodynamics for superconducting electrical circuits: An architecture for quantum computation,” Physical Review A, vol. 69, no. 6, p. 062320, 2004.
[60] R. Vijay, D. H. Slichter, and I. Siddiqi, “Observation of quantum jumps in a superconducting
artificial atom,” Phys. Rev. Lett., vol. 106, p. 110502, Mar 2011. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevLett.106.110502[61] D. Ristè, J. G. van Leeuwen, H.-S. Ku, K. W. Lehnert, and L. DiCarlo, “Initialization bymeasurement of a superconducting quantum bit circuit,” Phys. Rev. Lett., vol. 109, p. 050507,
Aug 2012. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevLett.109.050507[62] J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H.
Devoret, S. M. Girvin, and R. J. Schoelkopf, “Charge-insensitive qubit design derived fromthe cooper pair box,” Phys. Rev. A, vol. 76, p. 042319, Oct 2007. [Online]. Available:
https://link.aps.org/doi/10.1103/PhysRevA.76.042319

103[63] A. P. Sears, A. Petrenko, G. Catelani, L. Sun, H. Paik, G. Kirchmair, L. Frunzio, L. I.
Glazman, S. M. Girvin, and R. J. Schoelkopf, “Photon shot noise dephasing in thestrong-dispersive limit of circuit qed,” Phys. Rev. B, vol. 86, p. 180504, Nov 2012. [Online].
Available: https://link.aps.org/doi/10.1103/PhysRevB.86.180504[64] C. Wang, Y. Y. Gao, P. Reinhold, R. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor,
J. Blumoff, K. Sliwa et al., “A schrödinger cat living in two boxes,” Science, vol. 352, no.
6289, pp. 1087–1091, 2016.
[65] M. Reagor, W. Pfaff, C. Axline, R. W. Heeres, N. Ofek, K. Sliwa, E. Holland,
C. Wang, J. Blumoff, K. Chou, M. J. Hatridge, L. Frunzio, M. H. Devoret,
L. Jiang, and R. J. Schoelkopf, “Quantum memory with millisecond coherence incircuit qed,” Phys. Rev. B, vol. 94, p. 014506, Jul 2016. [Online]. Available:
https://link.aps.org/doi/10.1103/PhysRevB.94.014506[66] C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D, vol. 26, pp. 1817–
1839, Oct 1982. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevD.26.1817[67] G. G. Stokes, “On the perfect blackness of the central spot in newton’s rings, and on theverification of fresnel’s formulae for the intensities of reflected and refracted rays,” Cambridgeand Dublin Mathematical Journal, vol. 4, pp. 1–14, 1849.
[68] H. Von Helmholtz, Handbuch der physiologischen Optik.

Voss, 1867, vol. 9.

[69] H. B. G. Casimir, “On onsager’s principle of microscopic reversibility,” Rev. Mod. Phys.,vol. 17, pp. 343–350, Apr 1945. [Online]. Available: https://link.aps.org/doi/10.1103/
RevModPhys.17.343[70] R. Fleury, D. Sounas, C. F. Sieck, M. R. Haberman, and A. Alù, “Sound isolation and giant
linear nonreciprocity in a compact acoustic circulator,” Science, vol. 343, no. 6170, 2013.
[71] C. Coulais, D. Sounas, and A. Alù, “Static non-reciprocity in mechanical metamaterials,”Nature, vol. 542, no. 461, 2017.
[72] Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topologicalacoustics,” Phys. Rev. Lett., vol. 114, p. 114301, Mar 2015. [Online]. Available:
https://link.aps.org/doi/10.1103/PhysRevLett.114.114301[73] P. Wang, L. Lu, and K. Bertoldi, “Topological phononic crystals with one-way elasticedge waves,” Phys. Rev. Lett., vol. 115, p. 104302, Sep 2015. [Online]. Available:
https://link.aps.org/doi/10.1103/PhysRevLett.115.104302[74] S. A. Cummer, J. Christensen, and A. Alù, “Controlling sound with acoustic metamaterials,”Nature Reviews Materials, vol. 1, p. 16001, 2016.
[75] G. Viola and D. P. DiVincenzo, “Hall effect gyrators and circulators,” Physical Review X,
vol. 4, no. 2, p. 021019, 2014.
[76] A. C. Mahoney, J. I. Colless, S. J. Pauka, J. M. Hornibrook, J. D. Watson, G. C. Gardner,
M. J. Manfra, A. C. Doherty, and D. J. Reilly, “On-chip microwave quantum hall circulator,”Phys. Rev. X, vol. 7, p. 011007, Jan 2017.

104[77] A. C. Mahoney, J. I. Colless, L. Peeters, S. J. Pauka, E. J. Fox, X. Kou, L. Pan, D. G.-G.
K. L. Wang, and D. J. Reilly, “Zero-field edge magnetoplasmons in a magnetic topologicalinsulator,” arXiv preprint arXiv:1703.03122, 2017.
[78] D. M. Pozar, “Microwave engineering. 4th,” 2011.
[79] D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni,
J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is and what is
not an optical isolator,” Nature Photonics, vol. 7, p. 579582, 2013.
[80] C. Vassallo, Optical Waveguide Concepts, ser. 5. Sara Burgerhartstraat 25 P.O. Box 211,
1000 AE Amsterdam, The Netherlands: Elsevier, 1991, vol. 1, ch. 1, pp. 15–23.
[81] F. L. Pedrotti, L. S. Pedrotti, and L. M. Pedrotti, Introduction to optics, 3rd ed.
Hall, 2007.

Prentice[82] A. Kamal, J. Clarke, and M. H. Devoret, “Noiseless non-reciprocity in a parametric active
device,” Nature Physics, vol. 7, no. 4, pp. 311–315, 2011.
[83] C. E. Fay and R. L. Comstock, “Operation of the ferrite junction circulator,” Microwave
Theory and Techniques, IEEE Transactions on, vol. 13, no. 1, pp. 15–27, 1965.
[84] J. Fujita, M. Levy, R. M. O. Jr., L. Wilkens, and H. Dtsch, “Waveguide optical isolator
based on machzehnder interferometer,” Applied Physics Letters, vol. 76, no. 16, pp.
2158–2160, 2000. [Online]. Available: http://dx.doi.org/10.1063/1.126284[85] L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. Ross, “Onchip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nature
Photonics, vol. 5, no. 12, pp. 758–762, 2011.
[86] Y. Shoji, T. Mizumoto, H. Yokoi, I.-W. Hsieh, and R. M. O. Jr., “Magneto-optical isolator
with silicon waveguides fabricated by direct bonding,” Applied Physics Letters, vol. 92,
no. 7, p. 071117, 2008. [Online]. Available: http://dx.doi.org/10.1063/1.2884855[87] W. Śmigaj, J. Romero-Vivas, B. Gralak, L. Magdenko, B. Dagens, and M. Vanwolleghem,“Magneto-optical circulator designed for operation in a uniform external magnetic
field,” Opt. Lett., vol. 35, no. 4, pp. 568–570, Feb 2010. [Online]. Available:
http://ol.osa.org/abstract.cfm?URI=ol-35-4-568[88] Z. Wang and S. Fan, “Optical circulators in two-dimensional magneto-optical photonic
crystals,” Opt. Lett., vol. 30, no. 15, pp. 1989–1991, Aug 2005. [Online]. Available:
http://ol.osa.org/abstract.cfm?URI=ol-30-15-1989[89] K. Fang, Z. Yu, V. Liu, and S. Fan, “Ultracompact nonreciprocal optical isolator
based on guided resonance in a magneto-optical photonic crystal slab,” Opt.
Lett., vol. 36, no. 21, pp. 4254–4256, Nov 2011. [Online]. Available:
http://ol.osa.org/abstract.cfm?URI=ol-36-21-4254[90] A. M. Mahmoud, A. R. Davoyan, and N. Engheta, “All-passive nonreciprocal metastructure,”
Nature Communications, vol. 6, no. 8359, 2015.

105[91] K. Gallo, G. Assanto, K. R. Parameswaran, and M. M. Fejer, “All-optical diode in aperiodically poled lithium niobate waveguide,” Applied Physics Letters, vol. 79, no. 3, pp.
314–316, 2001. [Online]. Available: http://dx.doi.org/10.1063/1.1386407[92] L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An
all-silicon passive optical diode,” Science, vol. 335, no. 6067, pp. 447–450, 2012. [Online].
Available: http://science.sciencemag.org/content/335/6067/447[93] Y. Zhang, D. Li, C. Zeng, Z. Huang, Y. Wang, Q. Huang, Y. Wu, J. Yu, and J. Xia, “Silicon
optical diode based on cascaded photonic crystal cavities,” Opt. Lett., vol. 39, no. 6, pp.
1370–1373, Mar 2014. [Online]. Available: http://ol.osa.org/abstract.cfm?URI=ol-39-6-1370[94] L. A. Zadeh and C. A. Deoser, Linear system theory.

McGraw Hill, 1963.

[95] B. D. O. Anderson and R. W. Newcomb, “On reciprocity and time-variable networks,”
Proceedings of the IEEE, vol. 53, no. 10, pp. 1674–1674, 1965.
[96] D. L. Sounas, C. Caloz, and A. Alù, “Giant non-reciprocity at the subwavelength scale using
angular momentum-biased metamaterials,” Nature communications, vol. 4, p. 2407, 2013.
[97] K. Fang, Z. Yu, and S. Fan, “Experimental demonstration of a photonic aharonov-bohm
effect at radio frequencies,” Phys. Rev. B, vol. 87, p. 060301, Feb 2013.
[98] B. Abdo, K. Sliwa, S. Shankar, M. Hatridge, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret, “Josephson directional amplifier for quantum measurement of superconducting circuits,” Physical review letters, vol. 112, no. 16, p. 167701, 2014.
[99] K. M. Sliwa, M. Hatridge, A. Narla, S. Shankar, L. Frunzio, R. J. Schoelkopf, and M. H.
Devoret, “Reconfigurable josephson circulator/directional amplifier,” Phys. Rev. X, vol. 5, p.
041020, 2015.
[100] F. Lecocq, L. Ranzani, G. A. Peterson, K. Cicak, R. W. Simmonds, J. D. Teufel,
and J. Aumentado, “Nonreciprocal microwave signal processing with a field-programmablejosephson amplifier,” Phys. Rev. Applied, vol. 7, p. 024028, Feb 2017. [Online]. Available:
https://link.aps.org/doi/10.1103/PhysRevApplied.7.024028[101] B. Abdo, M. Brink, and J. M. Chow, “Gyrator operation using josephson mixers,”
Phys. Rev. Applied, vol. 8, p. 034009, Sep 2017. [Online]. Available:
https://link.aps.org/doi/10.1103/PhysRevApplied.8.034009[102] E. Li, B. J. Eggleton, K. Fang, and S. Fan, “Photonic aharonov–bohm effect in photon–
phonon interactions,” Nature communications, vol. 5, 2014.
[103] G. A. Peterson, F. Lecocq, K. Cicak, R. W. Simmonds, J. Aumentado, andJ. D. Teufel, “Demonstration of efficient nonreciprocity in a microwave optomechanicalcircuit,” Phys. Rev. X, vol. 7, p. 031001, Jul 2017. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevX.7.031001[104] N. R. Bernier, L. D. Toth, A. Koottandavida, M. A. Ioannou, D. Malz, A. Nunnenkamp,
A. Feofanov, and T. Kippenberg, “Nonreciprocal reconfigurable microwave optomechanicalcircuit,” arXiv, 2017.

106[105] S. Barzanjeh, M. Wulf, M. Peruzzo, M. Kalaee, P. B. Dieterle, O. Painter, and J. M. Fink,“Mechanical on-chip microwave circulator,” arXiv preprint arXiv:1706.00376, 2017.
[106] F. Ruesink, M.-A. Miri, A. Al, and E. Verhagen, “Nonreciprocity and magnetic-free isolation
based on optomechanical interactions,” Nature communications, vol. 7, p. 13662, 2016.
[107] K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter,“Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nature Physics, vol. 13, no. 5, pp. 465–471, 2017.
[108] L. Ranzani and J. Aumentado, “Graph-based analysis of nonreciprocity in coupled-mode
systems,” New Journal of Physics, vol. 17, no. 2, p. 023024, 2015.
[109] K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling thephase of dynamic modulation,” Nature Physics, vol. 6, no. 11, pp. 782–787, 2012.
[110] ——, “Photonic aharonov-bohm effect based on dynamic modulation,” Phys. Rev. Lett., vol.
108, p. 153901, Apr 2012.
[111] L. D. Tzuang, K. Fang, P. Nussenzveig, S. Fan, and M. Lipson, “Non-reciprocal phase shift
induced by an effective magnetic flux for light,” Nature photonics, vol. 8, no. 9, 2014.
[112] R. Peierls, “Zur theorie des diamagnetismus von leitungselektronen,” Zeitschrift für Physik
A Hadrons and Nuclei, vol. 80, no. 11, pp. 763–791, 1933.
[113] S. Khorasani, “Diamond configuration for non-reciprocal transmission,” IEEE Journal of
Quantum Electronics, 2017.
[114] B. Anderson and R. Newcomb, “The time-variable lattice and nonreciprocal rlc networks,”
Circuit Theory, IEEE Transactions on, vol. 13, no. 2, pp. 233–235, 1966.
[115] J. Kerckhoff, K. Lalumière, B. J. Chapman, A. Blais, and K. W. Lehnert, “On-chipsuperconducting microwave circulator from synthetic rotation,” Phys. Rev. Applied, vol. 4,
p. 034002, Sep 2015. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevApplied.
4.034002[116] E. I. Rosenthal, B. J. Chapman, A. P. Higginbotham, J. Kerckhoff, and K. W. Lehnert,“Breaking lorentz reciprocity with frequency conversion and delay,” Phys. Rev. Lett.,
vol. 119, p. 147703, Oct 2017. [Online]. Available: https://link.aps.org/doi/10.1103/
PhysRevLett.119.147703[117] S. K. Ibrahim, S. Bhandare, D. Sandel, H. Zhang, and R. Noe, “Non-magnetic 30 db integrated optical isolator in iii/v material,” Electronics Letters, vol. 40, no. 20, pp. 1293–1294,
2004.
[118] S. Bhandare, S. K. Ibrahim, D. Sandel, H. Zhang, F. Wust, and R. Noé, “Novel nonmagnetic 30-db traveling-wave single-sideband optical isolator integrated in iii/v material,” IEEE
Journal of selected topics in quantum electronics, vol. 11, no. 2, pp. 417–421, 2005.
[119] C. R. Doerr, N. Dupuis, and L. Zhang, “Optical isolator using two tandem phasemodulators,” Opt. Lett., vol. 36, no. 21, pp. 4293–4295, Nov 2011. [Online]. Available:
http://ol.osa.org/abstract.cfm?URI=ol-36-21-4293

107[120] Y. Yang, C. Galland, Y. Liu, K. Tan, R. Ding, Q. Li, K. Bergman, T. Baehr-Jones, andM. Hochberg, “Experimental demonstration of broadband lorentz non-reciprocity in an integrable photonic architecture based on mach-zehnder modulators,” Optics express, vol. 22,
no. 14, pp. 17 409–17 422, 2014.
[121] A. Metelmann and A. Clerk, “Nonreciprocal photon transmission and amplification via reservoir engineering,” Physical Review X, vol. 5, no. 2, p. 021025, 2015.
[122] A. Metelmann and H. E. Türeci, “Nonreciprocal signal routing in an active quantum network,” arXiv preprint arXiv:1703.04052, 2017.
[123] A. Metelmann and A. A. Clerk, “Nonreciprocal quantum interactions and devices via
autonomous feedforward,” Phys. Rev. A, vol. 95, p. 013837, Jan 2017. [Online]. Available:
https://link.aps.org/doi/10.1103/PhysRevA.95.013837[124] S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanicalstructures,” Phys. Rev. Lett., vol. 102, p. 213903, May 2009. [Online]. Available:
https://link.aps.org/doi/10.1103/PhysRevLett.102.213903[125] Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nature Photonics, vol. 3, 2009.
[126] ——, “Optical isolation based on nonreciprocal phase shift induced by interband photonic
transitions,” Applied Physics Letters, vol. 94, no. 17, p. 171116, 2009.
[127] H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced byinterband photonic transition on a silicon chip,” Phys. Rev. Lett., vol. 109, p. 033901, Jul
2012. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevLett.109.033901[128] M. O. Scully and M. S. Zubairy, Quantum optics.

Cambridge University Press, 1997.

[129] B. D. H. Tellegen, “The gyrator, a new electric network element,” Philips Res. Rep, vol. 3,
no. 2, pp. 81–101, 1948.
[130] C. L. Hogan, “The ferromagnetic faraday effect at micro- wave frequencies and its applications,” Rev. Mod. Phys., vol. 25, 1953.
[131] B. J. Chapman, B. A. Moores, E. I. Rosenthal, J. Kerckhoff, and K. W. Lehnert, “General
purpose multiplexing device for cryogenic microwave systems,” Applied Physics Letters, vol.
108, no. 22, p. 222602, 2016.
[132] B. J. Chapman, E. I. Rosenthal, J. Kerckhoff, L. R. Vale, G. C. Hilton, and K. W. Lehnert, “Single-sideband modulator for frequency domain multiplexing of superconducting qubit
readout,” Applied Physics Letters, vol. 110, p. 162601, 2017.
[133] K. Lalumière, “Électrodynamique quantique en guide d’onde,” Ph.D. dissertation, Université
de Sherbrooke, 2015.
[134] G. Burkard, R. H. Koch, and D. P. DiVincenzo, “Multilevel quantum description of
decoherence in superconducting qubits,” Phys. Rev. B, vol. 69, p. 064503, Feb 2004. [Online].
Available: https://link.aps.org/doi/10.1103/PhysRevB.69.064503

108[135] G. Burkard, “Circuit theory for decoherence in superconducting charge qubits,” Phys. Rev.
B, vol. 71, p. 144511, Apr 2005. [Online]. Available: https://link.aps.org/doi/10.1103/
PhysRevB.71.144511[136] O. Heaviside, Electrical Papers.

MacMillan and Co, 1894, vol. 2.

[137] C. K. Andersen, J. Kerckhoff, K. W. Lehnert, B. J. Chapman, and K. Mølmer, “Closing
a quantum feedback loop inside a cryostat: Autonomous state preparation and long-time
memory of a superconducting qubit,” Phys. Rev. A, vol. 93, p. 012346, Jan 2016. [Online].
Available: https://link.aps.org/doi/10.1103/PhysRevA.93.012346[138] B. J. Chapman, E. I. Rosenthal, J. Kerckhoff, B. A. Moores, L. R. Vale, G. C. Hilton,
K. Lalumière, A. Blais, and K. W. Lehnert, “Widely tunable on-chip microwave circulator
for superconducting quantum circuits,” arXiv preprint arXiv:1707.04565, 2017.
[139] W. Holland, M. MacIntosh, A. Fairley, D. Kelly, D. Montgomery, D. Gostick, E. AtadEttedgui, M. Ellis, I. Robson, M. Hollister, A. Woodcraft, P. Ade, I. Walker, K. Irwin,
G. Hilton, W. Duncan, C. Reintsema, A. Walton, W. Parkes, C. Dunare, M. Fich, J. Kycia,
M. Halpern, D. Scott, A. Gibb, J. Molnar, E. Chapin, D. Bintley, S. Craig, T. Chylek,
T. Jenness, F. Economou, and G. Davis, “Scuba-2: a 10,000-pixel submillimeter camera
for the james clerk maxwell telescope,” pp. 62 751E–62 751E–15, 2006. [Online]. Available:
http://dx.doi.org/10.1117/12.671186[140] M. A. Castellanos-Beltran and K. W. Lehnert, “Widely tunable parametric amplifier basedon a superconducting quantum interference device array resonator,” Applied Physics Letters,
vol. 91, no. 8, p. 083509, 2007. [Online]. Available: http://dx.doi.org/10.1063/1.2773988[141] J. Kerckhoff, R. W. Andrews, H. S. Ku, W. F. Kindel, K. Cicak, R. W. Simmonds,
and K. W. Lehnert, “Tunable coupling to a mechanical oscillator circuit using a coherent
feedback network,” Phys. Rev. X, vol. 3, p. 021013, Jun 2013. [Online]. Available:
https://link.aps.org/doi/10.1103/PhysRevX.3.021013[142] J. E. Sauvageau, C. J. Burroughs, P. A. A. Booi, M. W. Cromar, R. P. Benz, and J. A.
Koch, “Superconducting integrated circuit fabrication with low temperature ecr-based pecvd
sio/sub 2/dielectric films,” Applied Superconductivity, IEEE Transactions on, vol. 5, no. 2,
pp. 2303–2309, 1995.
[143] J. A. B. Mates, G. C. Hilton, K. D. Irwin, L. R. Vale, and K. W. Lehnert, “Demonstrationof a multiplexer of dissipationless superconducting quantum interference devices,” Applied
Physics Letters, vol. 92, no. 2, p. 023514, 2008.
[144] G. Stan, S. B. Field, and J. M. Martinis, “Critical field for complete vortex expulsion fromnarrow superconducting strips,” Phys. Rev. Lett., vol. 92, p. 097003, Mar 2004. [Online].
Available: https://link.aps.org/doi/10.1103/PhysRevLett.92.097003[145] T. Van Duzer and C. W. Turner, Principles of superconductive devices and circuits, 2nd ed.
Prentice Hall, 1981.
[146] N. W. Ashcroft and N. D. Mermin, Solid state physics. Holt, Rinehart and Winston, 1976.

109[147] H. Pothier, S. Guéron, D. Esteve, and M. H. Devoret, “Flux-modulated andreev current
caused by electronic interference,” Phys. Rev. Lett., vol. 73, pp. 2488–2491, Oct 1994.
[Online]. Available: https://link.aps.org/doi/10.1103/PhysRevLett.73.2488[148] P. Dubos, H. Courtois, B. Pannetier, F. K. Wilhelm, A. D. Zaikin, and G. Schön, “Josephson
critical current in a long mesoscopic s-n-s junction,” Phys. Rev. B, vol. 63, p. 064502, Jan
2001. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevB.63.064502[149] P. Mohanty and R. A. Webb, “High-field measurements of electron decoherence time in
metallic nanowires: Switching off magnetic impurity spins,” Phys. Rev. Lett., vol. 91, p.
066604, Aug 2003. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevLett.91.
066604[150] D. Riste, M. Dukalski, C. A. Watson, G. de Lange, M. J. Tiggelman, Y. M. Blanter, K. W.
Lehnert, R. N. Schouten, and L. DiCarlo, “Deterministic entanglement of superconducting
qubits by parity measurement and feedback,” Nature, vol. 502, no. 7471, pp. 350–354, 2013.
[151] S. Hacohen-Gourgy, L. S. Martin, E. Flurin, V. V. Ramasesh, K. B. Whaley, and I. Siddiqi,“Quantum dynamics of simultaneously measured non-commuting observables,” Nature, vol.
538, no. 7626, pp. 491–494, 2016.
[152] J. B. Johnson, “Thermal agitation of electricity in conductors,” Phys. Rev., vol. 32, pp.
97–109, Jul 1928. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRev.32.97[153] Z. E. Research,“Strong growth potential for u.s. telecom industry,”
September 2016. [Online]. Available:
https://www.zacks.com/commentary/89921/
strong-growth-potential-for-us-telecom-industry[154] H. W. Bode et al., Network analysis and feedback amplifier design.

van Nostrand, 1945.

[155] R. M. Fano, “Theoretical limitations on the broadband matching of arbitrary impedances,”
Journal of the Franklin Institute, vol. 249, no. 1, pp. 57–83,139–154, 1950.
[156] T. Dinc, M. Tymchenko, A. Nagulu, D. Sounas, H. Krishnaswamy, and A. Alù, “Synchronized
conductivity modulation to realize broadband lossless magnetic-free nonreciprocity,” Nature
Communications, vol. 8, no. 275.
[157] C. Macklin, K. O‘Brien, D. Hover, M. E. Schwartz, V. Bolkhovsky, X. Zhang, W. D. Oliver,and I. Siddiqi, “A near–quantum-limited josephson traveling-wave parametric amplifier,”
Science, vol. 350, no. 6258, pp. 307–310, 2015.

Appendix A

Graph theory eigenfrequency analysis for a half-circulator

Consider the network in Fig. A.1a. Its resonant frequencies may be calculated with a graphtheoretical approach [134, 135], in which each element in the network is designated as a (tree)
branch or a chord. Together, the branches form a tree—an object which encloses no circuit loops,
but which creates a loop with the addition of any chord. Note that for networks of non-trivial size,
there are multiple ways to define a tree.
We choose a tree with five branches (black circuit elements in Fig. A.1b) and four chords (graycircuit elements in Fig. A.1b). Each circuit element is labeled with an integer, to allow indexingof the voltage across (or current through) the element. Arrows indicate the defined direction of
positive current flow.
Each chord completes a circuit loop, and Kirchoff’s voltage law thus associates an equation
with each of the chords. Take, for example, the chord associated with current I6 . Addition of thischord to the tree creates a loop, with associated equation
V3 + V6 − V1 − V2 = 0.

(A.1)

It is helpful to collect the equations for each chord in a matrix notation, writing
FT V b = V c ,(A.2)

where the vectors Vb and Vc are the voltages of the branches and chords. For the network in

111

(a)
𝑙1

𝑙2

𝑙1

𝑙2

𝑙1

I3

I4

I5

I8

I9

𝑙1

𝑟 𝑙2

𝑙2

𝑙1

𝑙1

𝑙2

𝑙2

I1

𝑐

(b)
I7

(c)

I2

I6

𝑙1

𝑟 𝑙2

𝑙2

𝑙1

𝑐

Figure A.1: (a) General form of the circuit depicted in Fig. 4.2b, isolated from any transmission
lines (or equivalently, terminated in transmission lines of very large characteristic impedance). (b)The circuit in (a), with currents at each chord and branch labelled. Arrows indicate the (arbitrary)choice of direction for which current flow is positive. (c) The circuit in (a), connected to transmission
lines of characteristic impedance r, as indicated by resistor symbols.

Fig. A.1b,
 
V1 
 
 
V2 
 
 

Vb = 
V3  ,
 
 
V4 
 
 
V5
 
V6 
 
 
V7 

Vc = 
 ,
V8 
 
 
V9

(A.3)

(A.4)

112
and

1


1
t
F =

1


1

1 −1
1

0

1

0

1

0


0


0
0
.

−1 0 


0 −1
0

(A.5)

This is convenient, as the deliberate manner in which the equations in Eq. (A.2) were collected
ensures that the currents in the branches and chords satisfy a complementary equation,
FIc = −Ib .

(A.6)

The next step is to relate the currents and voltages of the chords (branches) in the Laplacedomain:Vb = Zb Ib ,(A.7)

Vc = Zc Ic .
In this example,



l
0
0
0
0

1


0
0
0


sl1
 0 l2 0 0 0 




0


 0 1/cs 0
Zb = s  0 0 l2 0 0  , Zc = 



0
0
sl2 0



 0 0 0 l1 0 





0
0
0 sl1
0 0 0 0 l2


0


0
.

0


0

(A.8)

Eqs. A.2, A.6, and A.8 may now be combined into a single matrix equation:
FIc = −Ib ,FZc −1 Vc = −Zb −1 Vb ,FZc −1 FT Vb = −Zb −1 Vb ,FZc −1 FT Vb + Zb −1 Vb = 0,FZc −1 FT Vb Zb + 1 Vb = 0.

(A.9)

113
Here, 1 is the identity matrix and the superscripts denote matrix inversion. By the invertible
matrix theorem, the determinant of the matrix on the left-hand side must vanish for a non-trivial
solution. The requirement that,
Det[FZc −1 FT Vb Zb + 1] = 0.

(A.10)

yields a characteristic equation in s, the roots of which are the eigenfrequencies of the network. In
this example, those frequencies arer
s1 = i2 (1 − δ 2 ),lc(A.11)

s2 = s1 ∗ .
when the inductances l1 and l2 are parametrized according to Eq. (4.2). Here the ∗ indicatescomplex conjugation.
If the same process is repeated for the network in Fig. A.1c, and an expansion is performed
to second order in δ and first order in 1/r, the resulting eigenfrequencies arer
s1 = − ,lr 2δ 2s2 = − +
,
l
rcr


2
δ2
δ2
s3 = − + i
1−
,
rc
lc
2

(A.12)

s4 = s3 ∗ .
The real and imaginary components of these eigenfrequencies indicate the dissipation and oscillation
rates of the eigenmodes. Their quality factors are therefore the ratio of their imaginary and realparts. While the modes associated with s1 and s2 are purely dissipative, modes 3 and 4 have alinewidth that scales with the square of the imbalance in the bridges. Comparison with Eq. (A.11)
shows that the imaginary components of the eigenfrequencies are Taylor expansions of the resistorless network’s eigenfrequencies, to second order in δ.

114
Comparison with the group delay plotted in Fig. 4.2d can now be made. With c = 1 pF,
l0 = 1 pH, and δ = 0.2, mode s3 is resonant at 6.97 GHz and has an inverse linewidth of 1.25 ns,
in agreement with the simulation.

Appendix B

Design rules of the NIST Nb trilayer process

Further details on the design space and design rules for the trilayer process are cataloguedin Tab. B.1.
Table B.1: Layer names, abbreviations and associated design
rules for the NIST niobium trilayer process.

layer

notes

CE

design rules
minimum width is 2.5 µm
minimum space (from other CE features) is 2 µm
must be overlapped by BE by 1.25 µm
must be overlapped by AL by 1 µm
must be overlapped by CEO by 0.5 µm

CEO

AL

minimum width is 3 µm

Thickness ranges from 6 to 10 nm.

minimum width is 4 µm
must be overlapped by BE by 0.25 µm
must be overlapped by ALO by 0.5 µm

116

ALO

BE

minimum width is 4.5 µm

200 nm thick.

minimum width is 2 µm
minimum space (from other BE features) is 1 µm

R1Thickness ranges between 135 and 225

minimum width is 2 µm

nm, allowing sheet resistances between

minimum space (from other R1 fea-

2 and 1.2 Ohms/square (AuPd) andtures) is 4 µm

100 and 60 mOhms/square (Au). Au

minimum overlap with BE is 1 µm

has stronger temperature dependence
than AuPd—if this is an issue, andresistor footprint is not a constraint,interdigitation of AuPd resistors allows for fabrication of low-resistance,
temperature-independent resistors.

117

I1

Thickness ranges from 300 to 500 nm.

must overlap with BE, CE, or W1

Relative dielectric constant of SiO2 isminimum width is 0.8 µm in junctions

r = 3.9. Loss tangent is temperature

minimum width is 1 µm otherwise

and power dependent [144].

must be overlapped by CE by 0.75 µm

Withhigh-power measurements of test res-must be overlapped by BE by 1 µm

onators at 300 mK we estimate it tomust be overlapped by W1 by 1 µm

be 3 × 10−3 . SiN [144] and amorphoussilicon [100] are reported to have lower
loss tangents, though amorphous silicon
has poor etch selectivity with Nb. Thiswas only recently overcome [100].

W1When possible use BE for wiring, as theminimum width is 2 µm in junctions

Nb in that layer is higher quality.

minimum space (from W1 features) is
1.5 µmminimum space (from BE edges) is 0.3µm

I2 or I1X

400 nm thick. In practice, this is thelayer that covers W1.

W2

When possible use BE for wiring, as

minimum width is 2 µm in junctions

the Nb in that layer is higher quality.

minimum space (from W2 features) is

350 nm thick.

1.5 µmminimum space (from BE edges) is 0.3
µm

118

I2X

This is the layer that covers W2.

Appendix C

Measurement details

A detailed schematic of the experimental setup is shown in Fig. C.1. A single-pull doublethrow switch, a cross-over switch (Pasternack PE7152), and a directional coupler (MAC C3205-20)
allow for measurement of the four accessible scattering parameters with an Agilent ENA5071C network analyzer. For measurements of the transmission spectrum, a Tektronix RSA6100A spectrum
analyzer serves as the microwave detector.
Φgcos(Ω𝑡+𝜙)ΦuΦgcos(Ω𝑡)
VNA

50

50

RSA
300 mK
4K

300 K

Figure C.1: Detailed experimental schematic for the measurements discussed in Ch. 5 and Ch. 6.

A Yokogawa 7651 current source provides the current for an off-chip, home-wound electromagnet, which creates the uniform flux Φu . A two-channel arbitrary waveform generator (Agilent
33500A), creates a pair of radio-frequency bias signals at frequency Ω, with phase difference φ,providing the gradiometric flux Φg .

120
Both (transmission and reflection) input lines are attenuated by 20 dB at the 4K plate, andan additional 20 dB at the 300 mK base plate of the 3 He cryostat, allowing for a noise temperature
Tn as low as
Tn > 300 K × 10−4 + 4 K × (1 − 10−2 ) × 10−2 + 300 mK × (1 − 10−2 )(C.1)≈ 370 mK
The probe signals are then routed to the microwave receiver, where they propagate through an isolator (Raditek RADC-4-8-Cryo-(30mK-4K)-S23-1WR-MS-b) and are amplified by a Caltech HEMT
amplifier (CRYO 4-12) at 4 K and a room temperature amplifier (Miteq AMF-3F-04000800-0710P). Bias lines are filtered with 350 MHz low-pass powder filters, and have a noise temperatureof roughly 300 K.

Appendix D

Calibration of network parameter measurements

In this appendix we describe the calibration procedures used for analysis of the network
parameter measurements described in Ch. 6.

D.1

Transmission calibration
To remove the gain of the measurement chain in transmission measurements, a bypass switch(formed from two Radiall 570443000 single-pull double-throw switches) is mounted at the base of
the cryostat, which routes fields through a 5 cm SMA cable instead of the circulator. We alsouse dedicated through measurements, (made in a separate cooldown) in which the circulator chipis exchanged for a like-sized circuit board traversed by a single 50 Ohm transmission line. Using
these techniques, the reference plane for transmission measurements is moved (approximately) tothe edge of the chip.

D.2

Reflection calibration
To remove the gain G from reflection measurements, we measure the reflection Rbal off thecirculator when no bias current is applied to the on-chip bias lines. In this unbiased state, all four
inductor bridges are balanced, and the reflection coefficient Γbal is the diagonal entry in each rowof the balanced scattering matrix Sbal . The matrix Sbal may be calculated by setting δ0 = 0 in
Eq. (4.11) and substituting the resulting admittance matrix into Eq. (4.23). This procedure yields

122Γbal =iωl + 2Z0.
iωl − 4Z0

(D.1)

As
Rbal = GΓbal ,(D.2)

and the gain G of the reflection measurement chain is assumed to be independent of the circulator’s
state, the reflection coefficient Γop at arbitrary operation points is related to the measured reflection
Rop by

Γop =

Rop
Rop
= Γbal
.
G
Rbal

(D.3)

To account for geometric inductance in the bridges, the inductance l in Eq. (D.1) is estimatedusing measurements of the unbalanced circuit’s resonant frequency, the capacitance design value of
1 pF, and Eq. (4.7).

D.3

Calibration of group delay
Preparing the circulator for operation requires correctly setting the duration τ of the resonant

delay. Measurements of the circulator’s group delay are used for this purpose. To separate the nonresonant delays of the finite-length measurement chain from the resonant delay τ , we multiply themeasured transmission data by eiωτd , where τd = 62 ns is the time required for an off-resonant
microwave field to propagate through the measurement chain. In the absence of circuit resonances,
this multiplication makes the phase of the transmission flat as a function of frequency, zeroing thegroup delay.