Development of a Josephson Parametric Amplifier
for the Preparation and Detection of NonclassicalStates of Microwave Fields
byManuel Angel Castellanos Beltran
B.S., Tecnológico de Monterrey, 2002

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
2010This thesis entitled:
Development of a Josephson Parametric Amplifier for the Preparation and Detectionof Nonclassical States of Microwave Fields
written by Manuel Angel Castellanos Beltran
has been approved for the Department of Physics

Prof. Konrad W. Lehnert

Prof. James K. Thompson

Date

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

iii
Castellanos Beltran, Manuel Angel (Ph.D., Physics)
Development of a Josephson Parametric Amplifier for the Preparation and Detection of
Nonclassical States of Microwave Fields
Thesis directed by Prof. Konrad W. Lehnert

Recent innovations in the technology of superconducting circuits have made it
possible to create nonclassical states of microwave light fields. These states are usuallyassociated with the subject of quantum optics. The ability to manipulate quantum
states of microwave light fields holds out the promise of building dense integrable circuits to process quantum information electrically. However, a major difficulty lies in thefact that there is no general purpose method of efficiently measuring the quantum stateof a microwave field. In this thesis, I describe the development of an amplifier, basedon a Josephson parametric amplifier. This new parametric amplifier enables this kindof efficient measurement. Although parametric amplifiers are narrowband (only amplifying signals in a narrow frequency range around a central frequency), I have developed
an amplifier whose central frequency is widely tunable (a full octave, 4–8 GHz), greatly
improving its usefulness. I have studied the gain, bandwidth, dynamic range, and added
noise of the amplifier. I have shown that when operated in a particular manner, known
as degenerate mode, the parametric amplifier adds less noise than the noise associatedwith the quantum fluctuations of the electromagnetic vacuum. In addition, its gain islarge enough to ensure that the amplified vacuum noise overwhelms the noise added byconventional amplifiers that follow the parametric amplifier. Together, these two features make it possible to measure either the phase or amplitude of a microwave signal
where the dominant uncertainty comes from the quantum noise of the signal itself. In
addition, a degenerate parametric amplifier also prepares particular nonclassical statescalled squeezed states. Our parametric amplifier prepares squeezed states, which haveiv
(in one of its quadratures) noise with a variance less than one tenth that of the vacuum
noise. I have employed the parametric amplifier in two experiments that require its ability to make efficient measurements. First, I used the parametric amplifier to determine
the full density matrix of a squeezed state generated by a second parametric amplifier.
Second, I used the amplifier to enable a measurement of position with precision better than the value at the standard quantum limit. These demonstrations are powerfulevidence that we have indeed realized a general-purpose quantum-efficient method of
measuring microwave fields.

Dedication

Para mi esposa Kassi y mis papás que siempre me apoyaron en todas mis empresas

vi

Acknowledgements

In these last seven years, I have had the opportunity of meeting and working
with great people. First I would like to thank my advisor Konrad Lehnert who taught
me all I know about low-temperature and microwave measurements, and some more I
already forgot. His passion and dedication to science always surprised and encouraged
me. Also, I would like to thank Dan Dessau and Jim Shepard for encouraging me tocome to Boulder for grad school. I also had the pleasure of working with a few greatpost-docs such as Dan Schmidt, Cindy Regal, John Teufel, François Mallet and Tobias
Donner. I want to specially thank John Teufel for all his help; it didn’t matter what he
was doing, any time I had a question he would take the time to answer it (as long as I
asked him in the right way: “John, you are an expert in INSERT ANYTHING HERE,...”). It was also a pleasure to work with the other graduate students in the lab, Nathan
Flowers-Jacobs Jennifer Harlow and Hsiang-Sheng Ku, although some times we foughtfor cables and equipment we had to share. I also would like to thank Hsiang-Sheng Ku
for checking some of the early chapters of this thesis. Our collaborators from NIST, Kent
Irwin, Gene Hilton and Leila Vale, made and helped us design the second generationof devices. I also had helpful conversations with (and borrowed multiple microwave
components from) Joe Aumentado and Lafe Spietz, from NIST as well. To all thispeople (and others I am forgetting) thank you for your help.
Finally, these past years would not have been the same without the support ofall my friends and family. I first want to thank my wife Kassi whose support helped

vii
me through all the most difficult parts of my graduate career. Thank you for all yourpatience and understanding especially the times I was running the experiments, when I
would spend (a lot) more time with Konrad than with you. I also want to thank my bestfriend Luis Miaja Ávila with whom I came to Boulder, first as an undergrad exchange
student, and then as a graduate student. I want to thank my parents and siblings for
all their help and support. It was great to have my brother Nacho live in Boulder withme in the last couple of years. To all my family and friends, thank you as I would not
be where I am, or who I am, without you.

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Contents

Chapter
1 Introduction

1

2 Quantization of electrical circuits and amplifiers

8

2.1

2.2

2.3

2.4

Electrical simple harmonic oscillator . . . . . . . . . . . . . . . . . . . .

10

2.1.1

Classical normal mode description of a LC-resonator . . . . . . .

10

2.1.2

Description of the problem in quantum mechanics language . . .

12Transmission line theory . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.2.1

Classical analysis of transmission lines . . . . . . . . . . . . . . .

14

2.2.2

Quantization of a lossless transmission line . . . . . . . . . . . .

15

Input-Output theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.3.1

Harmonic oscillator coupled to a transmission line . . . . . . . .

20

2.3.2

Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.3.3

Alternative approach . . . . . . . . . . . . . . . . . . . . . . . . .

27

Quantum-limited amplification: Haus-Caves theorem . . . . . . . . . . .

30

2.4.1

Phase insensitive amplifier . . . . . . . . . . . . . . . . . . . . . .

31

2.4.2

Phase sensitive amplifier . . . . . . . . . . . . . . . . . . . . . . .

35

2.4.3

Quadrature squeezing . . . . . . . . . . . . . . . . . . . . . . . .

37

3 Theory of Josephson parametric amplifiers
3.1

Non-linear driven oscillators (Duffing oscillator) . . . . . . . . . . . . . .

39
39

ix
3.2

3.3

General theory of Josephson junctions . . . . . . . . . . . . . . . . . . .

45

3.2.1

Simple junction theory . . . . . . . . . . . . . . . . . . . . . . . .

45

3.2.2

dc-SQUID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

Josephson junction oscillator . . . . . . . . . . . . . . . . . . . . . . . .

51

3.3.1

Circuit theory of a driven Josephson junction in a RLC-resonator

52

3.3.2

Parametric processes in the Josephson junction oscillator . . . .

55

3.3.3

Kerr-medium equivalence of a Josephson junction oscillator . . .

56

3.3.4

Necessary conditions on the amount of non-linearity to have agood parametric amplifier . . . . . . . . . . . . . . . . . . . . . .

4 Previous and other current work on Josephson parametric amplifiers

72
77

4.1

Noise rise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

4.2

Previous successful implementations . . . . . . . . . . . . . . . . . . . .

80

4.2.1

Squeezing of thermal and vacuum noise . . . . . . . . . . . . . .

80

4.2.2

Parametric amplifier with Josephson junction arrays . . . . . . .

81

Current work on Josephson parametric amplifiers . . . . . . . . . . . . .

83

4.3.1

Josephson ring modulator . . . . . . . . . . . . . . . . . . . . . .

83

4.3.2

Flux driven Josephson parametric amplifier . . . . . . . . . . . .

85

4.3.3

dc-SQUID stripline resonator . . . . . . . . . . . . . . . . . . . .

86

4.3.4

Josephson bifurcation amplifier . . . . . . . . . . . . . . . . . . .

87

4.3.5

Parametric amplification with superconducting resonators and weak-

4.3

links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Design of JPA
5.1

89
91

Microwave environment design . . . . . . . . . . . . . . . . . . . . . . .

91

5.1.1

Transmission line resonators . . . . . . . . . . . . . . . . . . . . .

92

5.1.2

Coplanar waveguide transmission lines . . . . . . . . . . . . . . .

95

5.1.3

Coupling a cavity to a transmission line . . . . . . . . . . . . . .

99

x
5.2

Cavity JPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.2.1

Widely tunable resonator . . . . . . . . . . . . . . . . . . . . . . 106

5.2.2

SQUID arrays: a metamaterial with a Kerr-nonlinearity . . . . . 111

5.2.3

Loss and other ports . . . . . . . . . . . . . . . . . . . . . . . . . 115

6 Experimental results of JPAs
6.1

6.2

6.3

6.4

6.5

123

Instrumentation and measurement techniques . . . . . . . . . . . . . . . 124
6.1.1

Microwave measurement at ultra-low temperatures . . . . . . . . 124

6.1.2

Microwave measurements at room temperature . . . . . . . . . . 135

Characterization of the resonator . . . . . . . . . . . . . . . . . . . . . . 139
6.2.1

dc-Characterization of Josephson junctions . . . . . . . . . . . . 139

6.2.2

Linear measurements of the resonators . . . . . . . . . . . . . . . 143

6.2.3

Non-linear characterization . . . . . . . . . . . . . . . . . . . . . 145

Characterizing the amplifier . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.3.1

Signal and image gain . . . . . . . . . . . . . . . . . . . . . . . . 146

6.3.2

Phase dependence gain . . . . . . . . . . . . . . . . . . . . . . . . 148

6.3.3

Noise measurements . . . . . . . . . . . . . . . . . . . . . . . . . 150

6.3.4

Squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

6.3.5

Amplifier saturation . . . . . . . . . . . . . . . . . . . . . . . . . 156

Aluminum resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.4.1

dc-measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

6.4.2

Linear measurements of the resonator . . . . . . . . . . . . . . . 159

6.4.3

Non-linear characterization . . . . . . . . . . . . . . . . . . . . . 162

6.4.4

Gain measurements . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.4.5

Estimate of the added noise of the amplifier . . . . . . . . . . . . 166

6.4.6

Final comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Niobium resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

xi
6.5.1

Description of the device . . . . . . . . . . . . . . . . . . . . . . . 168

6.5.2

dc-Characterization of Josephson Junctions . . . . . . . . . . . . 168

6.5.3

Linear measurements of the resonators . . . . . . . . . . . . . . . 170

6.5.4

Nonlinear characterization . . . . . . . . . . . . . . . . . . . . . . 174

6.5.5

Gain measurements . . . . . . . . . . . . . . . . . . . . . . . . . 176

6.5.6

Measurement of the added noise of the amplifier . . . . . . . . . 178

6.5.7

Inferred squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . 182

6.5.8

Amplifier saturation . . . . . . . . . . . . . . . . . . . . . . . . . 183

6.5.9

Final comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

7 Applications I. Microwave homodyne tomography of squeezed states
7.1

7.2

185

Quasiprobability distributions . . . . . . . . . . . . . . . . . . . . . . . . 186
7.1.1

Wigner function . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

7.1.2

Other distribution functions . . . . . . . . . . . . . . . . . . . . . 192

Quantum tomography: using a homodyne detector to measure the Wigner
function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

7.3

7.2.1

Histograms and radon transformation . . . . . . . . . . . . . . . 195

7.2.2

Inverse radon transformation . . . . . . . . . . . . . . . . . . . . 198

7.2.3

Inefficiencies and measurement issues . . . . . . . . . . . . . . . . 199

7.2.4

Implementing an ideal homodyne tomography detector with a JPA204

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
7.3.1

Details of the experiment . . . . . . . . . . . . . . . . . . . . . . 207

7.3.2

Histograms of the traces . . . . . . . . . . . . . . . . . . . . . . . 212

7.3.3

Quasiprobability distribution reconstruction . . . . . . . . . . . . 216

8 Applications II. Increasing the sensitivity in the detection of mechanical motionof a nanomechanical wire.
8.1

222

Optomechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

xii
8.2

Nanomechanical resonators inside microwave resonators . . . . . . . . . 229

8.3

Experimental results. Increasing the sensitivity and getting closer to a
shot-noise limited detection. . . . . . . . . . . . . . . . . . . . . . . . . . 234

9 Conclusions and future directions.

240

9.1

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

9.2

Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

Bibliography

248

Appendix
A Multimode fields

254

A.1 Multimode signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
A.2 Squeezed states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
A.3 Quantum limits for multimode linear amplifiers . . . . . . . . . . . . . . 258
A.4 Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
B Thermal sweep and loss calibration

261

B.1 HEMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
B.1.1 Hot load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
B.1.2 Cold load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
B.2 JPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
B.2.1 Cold load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
B.2.2 Hot load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
B.2.3 Low-frequency noise . . . . . . . . . . . . . . . . . . . . . . . . . 268
B.3 Chapter 7 calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
B.3.1 HEMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

xiii
B.3.2 JPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
B.3.3 Squeezer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
B.4 Chapter 8 calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
C Sample Fabrication

276

C.1 Aluminum resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
C.2 Niobium resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
D List of symbols and abbreviations

280

xiv

FiguresFigure1.1

Amplification process . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2

Josephson tunnel junction . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.1

Simple harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.2Transmission line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.3

Generalized flux

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2.4

Noise model of a resistor . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.5

LC resonator coupled to a transmission line . . . . . . . . . . . . . . . .

20

2.6

Input-output theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

2.7

Quantum limits in amplification. . . . . . . . . . . . . . . . . . . . . .

31

2.8

Squeezed states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

3.1

Nonlinear response of the Duffing oscillator . . . . . . . . . . . . . . . .

43

3.2

Bifurcation of the Duffing oscillator . . . . . . . . . . . . . . . . . . . . .

44

3.3

Josephson junction as a nonlinear oscillator . . . . . . . . . . . . . . . .

46

3.4

SQUID

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

3.5

Flux dependance of the critical current of a SQUID . . . . . . . . . . . .

50

3.6

Flux dependance of the critical current of a SQUID with finite loop in-

3.7

ductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

Nonlinear electrical resonator with a Josephson junction . . . . . . . . .

53

xv
3.8

Nonlinear response of the Josephson oscillator . . . . . . . . . . . . . . .

54

3.9

Microwave cavity with a Kerr nonlinearity . . . . . . . . . . . . . . . . .

57

3.10 Phase of the nonlinear reflection coefficient . . . . . . . . . . . . . . . .

59

3.11 Signal and image gain . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

3.12 Squeezing of the internal mode . . . . . . . . . . . . . . . . . . . . . . .

67

3.13 Squeezing of the external field . . . . . . . . . . . . . . . . . . . . . . . .

69

3.14 Spectrum of the two kind of signals applied to the amplifier. . . . . . . .

70

3.15 Parallel Josephson junction oscillator . . . . . . . . . . . . . . . . . . . .

73

3.16 Other electrical environments for a Josephson junction resonator . . . .

75

4.1

Circuit schematic of previous implementations of a JPA . . . . . . . . .

80

4.2

Parametric amplifier with Josephson junction arrays . . . . . . . . . . .

81

4.3

Josephson ring modulator . . . . . . . . . . . . . . . . . . . . . . . . . .

84

4.4

Flux driven Josephson parametric amplifier . . . . . . . . . . . . . . . .

85

4.5

dc-SQUID stripline resonator . . . . . . . . . . . . . . . . . . . . . . . .

86

4.6

Josephson bifurcation amplifier . . . . . . . . . . . . . . . . . . . . . . .

88

4.7

Parametric amplification with superconducting resonators and weak-links 89

5.1

Transmission line quarter-wave resonator. . . . . . . . . . . . . . . . .

93

5.2

Transmission line half-wavelength resonator . . . . . . . . . . . . . . . .

94

5.3

Coplanar waveguide geometry . . . . . . . . . . . . . . . . . . . . . . . .

96

5.4

Skinny ground planes used to avoid flux trapping . . . . . . . . . . . . .

98

5.5

Coupling between the transmission line resonator and the driving microwave line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.6

Reflection coefficient for different coupling coefficients . . . . . . . . . . 102

5.7

Lumped element model of the series array of SQUIDs used to create anonlinear medium

5.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Unit cell of the microwave network used for the ABCD matrix analysis . 110

xvi
5.9

Resonance frequency fres as function of flux for there different networks

111

5.10 Nonlinear cavity coupled to two ports . . . . . . . . . . . . . . . . . . . 115
5.11 Loss modeled as a beam- splitter at the input of JPA . . . . . . . . . . . 118
5.12 Squeezing and added noise of a JPA when losses are included . . . . . . 120
5.13 Effect of the nonlinear loss in the reflection coefficient Γ and resonancefrequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.1

Dilution cryostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.2

Sample box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.3

Low pass powder filters . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.4

Directional coupler and circulator . . . . . . . . . . . . . . . . . . . . . . 131

6.5

Noise temperature of an amplifier chain . . . . . . . . . . . . . . . . . . 132

6.6

Heatsinking of the semi-rigid coaxial cables . . . . . . . . . . . . . . . . 135

6.7

Mixer up-conversion process . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.8

Mixer down-conversion process . . . . . . . . . . . . . . . . . . . . . . . 140

6.9

dc-measurements setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6.10 Simplified schematic used to measure the S-parameters . . . . . . . . . . 143
6.11 Schematic of the setup used to measure the signal and image gain . . . 147
6.12 Schematic used to measure the phase dependant gain, the added noise of
the amplifier and squeezing . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.13 Transformation performed by an ideal JPA . . . . . . . . . . . . . . . . 152
6.14 Image of a test aluminum sample . . . . . . . . . . . . . . . . . . . . . . 157
6.15 dc-measurements of the aluminum device . . . . . . . . . . . . . . . . . 158
6.16 Linear measurements for the aluminum resonator . . . . . . . . . . . . . 160
6.17 Resonance frequency versus flux for the aluminum resonator . . . . . . . 161
6.18 Linear measurements for the aluminum resonator made on January, 2007(second cooldown) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

xvii
6.19 Power dependance of the resonance circuit . . . . . . . . . . . . . . . . . 163
6.20 Gain performance of the amplifier. . . . . . . . . . . . . . . . . . . . . 164

6.21 Gain as a function of the phase and added noise estimate . . . . . . . . 166
6.22 Images of the niobium device . . . . . . . . . . . . . . . . . . . . . . . . 168
6.23 I-V curves of test niobium devices . . . . . . . . . . . . . . . . . . . . . 169
6.24 S-parameters for the half-wavelength niobium resonator . . . . . . . . . 171
6.25 Model to include standing waves in the fits for the S-parameters . . . . 172
6.26 Fits to the reflection coefficient of a quarter-wave resonator using themodel of figure 6.25

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

6.27 Unexpected nonlinear behavior of the resonators at low resonance frequency175
6.28 Power dependance of the resonator . . . . . . . . . . . . . . . . . . . . . 176
6.29 Gain measurements of the niobium JPA . . . . . . . . . . . . . . . . . . 177
6.30 Signal gain for different pump powers for the quarter-wave resonator . . 178
6.31 Simplified diagram of the setup used for the Y -factor measurement . . . 178
6.32 Y -factor measurement done at 7 GHz . . . . . . . . . . . . . . . . . . . 180
6.33 System added noise and added noise at different resonance frequencies . 181
6.34 Demonstration of squeezing . . . . . . . . . . . . . . . . . . . . . . . . . 182
6.35 Saturation of the amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7.1

Wigner functions of different states . . . . . . . . . . . . . . . . . . . . . 190

7.2

Tomographic projection of the Wigner function of a coherent state . . . 196

7.3

Schematic diagram for Wigner function reconstruction . . . . . . . . . . 197

7.4

Effect of a beam splitter on a Wigner function

7.5

Effective model for losses in the detector . . . . . . . . . . . . . . . . . . 202

7.6

Schematic diagram with a JPA used as a preamplifier . . . . . . . . . . 206

7.7

Schematic of the experimental setup for our implementation of quantum-. . . . . . . . . . . . . . 200

state tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

xviii
7.8

Resonance frequency for the JPA and the squeezer as a function of thecurrent though a coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

7.9

Schematic of our implementation for a high efficiency microwave homodyne detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

7.10 Noise at the output of our microwave homodyne detector as a functionof the JPA-pump and LO phase . . . . . . . . . . . . . . . . . . . . . . . 213
7.11 Measured real squeezing ζ . . . . . . . . . . . . . . . . . . . . . . . . . . 215
7.12 Probability distribution for an ideal squeezed state . . . . . . . . . . . . 217
7.13 Normalized variance of the quadrature variable . . . . . . . . . . . . . . 219
7.14 Wigner function reconstructed from the data . . . . . . . . . . . . . . . 220
8.1

Schematic of a typical optomechanical setup . . . . . . . . . . . . . . . . 223

8.2

Shot-noise limited phase detection . . . . . . . . . . . . . . . . . . . . . 225

8.3

Spectral density at the output of an interferometer used to measure mechanical motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

8.4

Microwave cavity optomechanics device . . . . . . . . . . . . . . . . . . 230

8.5

Transmission coefficient for the feedline geometry . . . . . . . . . . . . . 232

8.6

Simplified diagram of measurement setup . . . . . . . . . . . . . . . . . 233

8.7

Detailed diagram of the measurement setup . . . . . . . . . . . . . . . . 234

8.8

Fluctuations at the output of the mixer in arbitrary units (V2 /Hz) andthermal sweep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

8.9

Displacement sensitivity below that at the Standard Quantum Limit . . 238

9.1

Schematic of the new devices to be tested . . . . . . . . . . . . . . . . . 244

9.2

Gradiometer SQUIDs

9.3

Passive microwave components developed by our group . . . . . . . . . . 247. . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

B.1 Simplified diagram of the measurement when only the HEMT is used . . 262

xix
B.2 Simplified diagram of the measurement when both the HEMT and a
Josephson parametric amplifier . . . . . . . . . . . . . . . . . . . . . . . 265
B.3 Spectral density of the output of the mixer when the JPA is used . . . . 268
B.4 Thermal sweep using only the HEMT amplifier . . . . . . . . . . . . . . 269
B.5 Thermal sweep with the JPA . . . . . . . . . . . . . . . . . . . . . . . . 270
B.6 Thermal sweep with the squeezer . . . . . . . . . . . . . . . . . . . . . . 272
B.7 Model for the setup when all the losses and amplifiers are included . . . 272
B.8 Squeezing as a function of the squeezer gain . . . . . . . . . . . . . . . . 273
B.9 Thermal sweep using only the HEMT amplifier . . . . . . . . . . . . . . 274
B.10 Thermal sweep with the JPA . . . . . . . . . . . . . . . . . . . . . . . . 275
C.1 Dolan bridge technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
C.2 SEM and evaporator used to fabricate the aluminum devices . . . . . . 278
C.3 Niobium trilayer process . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

Chapter 1
Introduction

The theory of quantum mechanics emerged to explain primarily macroscopic degrees of freedom, such as atoms and molecules. In the past thirty years there have been
increasingly successful attempts to build macroscopic electrical circuits that nonethelessbehave quantum mechanically. Recently, this trend has only accelerated with the emergence of the field of quantum information. This field seeks to exceed classical bounds
on communication, cryptography, and other information processing tasks.
In contrast to microscopic degrees of freedom, these quantum electrical circuits are
excited by much smaller energies. The transition frequencies between the eigenstates lie
in the microwave rather than optical part of the electromagnetic spectrum. Because theenergy of a microwave photon is so small, detecting fields at the level of single photons
is one of the major problems that need to be solved to store and process quantum
information with electrical circuits. In this thesis I address this measurement problem.
I developed a method of measuring microwave fields where the quantum fluctuationsin the microwave field are the dominant source of noise: in this sense, a measurement
operating at the single photon level. The key element in our measurement scheme
is an amplifier called the Josephson parametric amplifier (JPA), which is capable ofamplifying a signal, in principle, without adding any noise.
At the microwave frequencies we will discuss in this thesis (4–8 GHz), there
is not a quantum-limited photodetector such as in optics as the energy per photon

2of a microwave signal is too small to be individually detected in the same way as
an optical photon. Therefore, the detection of weak microwave signals needs to bepreceded by amplification. The process of amplification usually consists of the task of
weakly coupling many microscopic degrees of freedom (for example, the electrons of atransistor) to an input signal, and channeling energy from those microscopic degrees
to the input signal. However, amplification is usually accompanied by the unfortunate
addition of noise to the signal (Fig. 1.1a). Most of the time, this added noise will comefrom the lack of control of each individual microscopic degree of freedom. However,even in the case when the control is ideal, quantum mechanics may impose limits on
the minimum added noise of an amplifier depending on the kind of amplification thatis performed.
In this thesis, I will be only interested in linear amplifiers. Linear amplifiers arethose whose output is linearly related to its input signal. I will be mostly interested
in phase-sensitive amplifiers, whose response depends on the phase of the input signal
and which can circumvent the quantum limit on the noise added by phase insensitive
amplifiers. One simple way to understand the concept of a phase sensitive amplifier is
to first define the quadratures of a signal. In general, we can decompose a voltage signal
with frequency components close to a frequency ωo as
Vs (t) = 2V0 [X1 (t) cos(ωo t) + X2 (t) sin(ωo t)](1.1)

where X1 and X2 are the quadratures of the signal. A phase sensitive amplifier amplifies
the quadrature components of the signal by different amounts. I will discuss only a specific type of phase-sensitive amplifier, one that will only amplify one of the quadraturesand de-amplify the other one.
The quantum noise associated with amplification has been discussed extensively
by several authors [1–3]. The quantum fluctuations limit the ultimate performanceof any amplifier and hence these fluctuations play an important role in high-precision

3(a)

G (Signal+Noise)
Signal
G
Noise

(b)

(c)

X2

X2

bs
as

as

bs
X1

X1

Figure 1.1: (a) In general, the amplification process will degrade the signal to noise
ratio by adding certain amount of noise to the signal before amplifying it. (b) Quantum
mechanics places a restriction on the minimum amount of this added noise when theamplifier amplifies both quadratures of the signal. When an amplifier achieves this limit,
it is said to be quantum limited. (c) On other hand, if the amplifier is a phase-sensitive
amplifier, and only amplifies one of the quadratures, then it can do that ideally without
adding any noise.

measurements. The exact manifestation of this quantum limit depends on the actual
amplification process. Quantum mechanics does not impose any restriction on theultimate precision of a single measured quantity (for example, position or momentumof a particle), but only in the combined precision of two conjugate variables [4]. The
typical case that interests us is a sinusoidal signal with the information encoded in theamplitude or phase. If we care about the information contained in both quadratures,
Eq. (1.1), then we would need to use a linear phase-insensitive amplifier (Fig. 1.1b),
and the minimum amount of noise (in energy units) that an amplifier must add to asignal at frequency ωs is
EN =

~ωs
2

4
where EN is the noise that the amplifier adds to the signal before being amplified (Fig.
1.1a). This is often called the standard quantum limit of amplifiers and applies only
to phase-insensitive amplifiers. However, if one is only interested in the information
contained in one of the quadratures, the situation is very different (Fig. 1.1c). In
this case, there is no restriction on the added noise of the amplifier. Essentially, as
long as the information in one of the components of the signal is lost, the information
contained in the other quadrature can be amplified with no degradation of the signal tonoise ratio. In the ideal case, these amplifiers can be built from a system with a single
well-controlled degree of freedom (the resonant-mode of a microwave circuit). Anotherinteresting consequence of the behavior of phase sensitive amplifiers is that while oneof the quadratures is noiselessly amplified, the other one can be squeezed. That is, thefluctuations at the output of the amplifier would be smaller than those fluctuations at
the input, even if at the input there were only vacuum fluctuations. Such squeezed
states are nonclassical states of light.

S
I
S
Nonlinear
Inductor

Figure 1.2: A Josephson tunnel junction consists of two superconductors separated bya small insulating barrier. For the most part, we will treat the Josephson junction as anonlinear inductor.

In this thesis, I will describe how we built a microwave amplifier with the bestreported noise-performance ever achieved. Its noise performance is so good and it canbe operated at high enough gains that it is now possible to measure microwave signal
where the dominant noise source is the quantum fluctuations of that microwave signal

5
itself. In addition, we have shown that it is a practical, useable amplifier which can be
integrated with other low temperature microwave experiments.
The way we approached this problem of fabricating an ultra-low noise microwave
amplifier was by creating a parametric amplifier using Josephson junctions (Fig. 1.2).
Josephson junctions play a key role in the fabrication of low-noise microwave amplifiers, being the only nonlinear dissipationless element usable at microwave frequencies.
The name parametric amplifier is associated with a class of amplifying and frequencyconverting devices that use the properties of nonlinear or time-varying reactances. If areactive parameter oscillates at twice the resonance frequency, energy can be pumped
into (or out of) the input mode, realizing an amplifier. In practice, this time dependence
is often generated through a nonlinear reactance, either inductance or capacitance. If
the nonlinear reactance is proportional to the intensity rather than the amplitude of themode, then an intense pump tone applied at the resonance frequency ω automaticallycreates the necessary 2ω parametric oscillation. In analogy with optics, we describe thiseffect as a Kerr nonlinearity. The nonlinear current-dependent inductance of a Josephson junction provides such a Kerr nonlinearity. By detecting only one quadrature ofa signal, degenerate parametric amplifiers can add even less noise than the minimum
required by quantum mechanics when detecting both quadratures. [5].
We are not the first group to study these kind of amplifiers. Josephson parametric amplifiers have already been operated with near quantum-limited sensitivity [6]and have been used to squeeze both thermal and vacuum noise [6–8]. However, inspite of these promising results they were not widely adopted since they suffer fromtwo main disadvantages: a small dynamic range and narrow band gain. Furthermore,compelling applications that would benefit from a lower-noise microwave amplifier have
only recently been developed: with the advent of quantum information processing using
superconducting circuits [9, 10], or microwave-cavity optomechanics [11], there is now aneed for practical amplifiers that operate at the limits imposed by quantum mechanics.

6
Also in contrast with the previous work where the JPA was only tested but never used
to measure something else, we operated our JPA in two low-temperature experimentsthat require its unprecedented sensitivity. First, I efficiently reconstruct the quantum
state of a nonclassical microwave signal. Second, I collaborate with other members of
my lab to measure the position of an object with precision better than the quantum
zero-point motion of that object.
While the devices that I create and study are practical amplifiers, we have also
tested their performance in creating nonclassical states of the microwave field: squeezed
states. In fact, the nonclassical microwave fields whose quantum state I reconstructedusing my amplifier were themselves created by another JPA. These nonclassical states
are useful resources for quantum information processing. The continuation of the workthat I started during my Ph.D. will use the nonclassical state to process quantum
information.

Thesis overview
In the succeeding chapters I will explore the operating principles of these devices,present a few design criteria for their construction and discuss two particular areasof application. This thesis is organized as follows. In chapter 2, I will describe thetheoretical aspects related to the quantum description of microwave circuits, including
quantum network theory and input-output theory. I will also discuss the origin of thelimits on amplification as well as the generation of squeezed states using a degenerateparametric amplifier, a type of phase-sensitive amplifier. In chapter 3, I will treat very
generally a type on nonlinear oscillators called Duffing oscillators. I will describe, in
general terms, what a Josephson junction is and discuss how an electrical resonator witha Josephson junction embedded in it behaves, within some limit, like a Duffing oscillator.
I will then describe how the theory discussed in chapter 2 can be used to analyze thebehavior of the Josephson resonator as a degenerate parametric amplifier. In chapter 4,

7
I discuss some of the previous and present work regarding the use of Josephson junctions
for parametric amplifiers. In chapter 5, I will describe the design and implementationof the amplifier, including a brief discussion of microwave resonators using distributed
circuits. In previous chapters, the analysis of the JPA has been done assuming an
ideal behavior, that is without including the possibility of loss in the system. In thischapter I also discuss the effects of loss on the performance of a JPA. In chapter 6, I
will describe the experiments done to characterize the samples that we have measured.
This includes the linear and nonlinear behavior of the cavities, and the performance ofthe nonlinear resonators as amplifiers, including gain, bandwidth, noise properties andsaturation behavior. I will also describe the measurement of quadrature squeezing, butusing a commercial microwave amplifier. Therefore, even though the inferred squeezing
is about 10 dB, the real squeezing at the microwave detector at room temperature
is only 2%. In chapter 7, I will discuss how we try to increase the sensitivity of our
microwave homodyne measurement by using another JPA as a pre-amplifier for the chainof commercial amplifiers used in the experiment. This allows us to detect real (rather
than inferred) squeezing of 37% and to perform what is called in the optics community
Wigner-function reconstruction. In chapter 8, I will also use a JPA to increase thesensitivity of a microwave interferometer used to detect the motion of a nanowire. By
using the JPA, our interferometer is less than a factor of 4 away from being shot-noise
limited. Finally, I provide the conclusions of this thesis in chapter 9 as well as somedirections for the future of these experiments.

Chapter 2
Quantization of electrical circuits and amplifiers

The theory of parametric amplification is intimately related to the theory of coupled modes of resonant systems. So our study begins with the study of the simplest
resonant system: the simple harmonic oscillator (SHO). In our case, our SHO consistof circuits made out of microfabricated structures, such as for example a parallel LCoscillator as shown in figure 2.1a. Quantum mechanics has been generally associatedwith the description of microscopic particles such as electrons and atoms. Nevertheless,recently the concept of macroscopic quantum (mesoscopic) systems has been used todescribe systems that have an intermediate status between the realms of microscopic
particles and classical macroscopic objets [12]. By macroscopic I mean that these systems contain a large number of microscopic particles and have been fabricated according
to certain specifications. However, their behavior is described in terms of variables thatconsist of collective degrees of freedom that behave quantum-mechanically. For typicalvalues and dimensions, the components that make up these systems can have resonancesof a few gigahertz and dimensions of 1-100 microns. Because the centimeter-long wavelength of the probing microwave excitation is much larger than the circuits, we are wellinto the lumped element limit.1 The state of the circuit in figure 2.1a can be specified
by only two collective degrees of freedom: the charge Q in the capacitor and flux Φ1

We can also fabricate SHO out of microwave cavities. In that case the dimension of the circuit is
too big to be in the lumped element limit. However, we still characterize the behavior in terms of single
collective excitations.

9in the inductor. In order to probe the quantum mechanical behavior of such a simple
circuit, we need to cool it down to milliKelvin temperatures such that kB T ¿ ~ω0 .
However, this last condition is not enough to guarantee that the collective degrees of
freedom Φ and Q need to be treated as a quantum variables; the separation between
the different energy levels ∆E/~ must be bigger than their width γ (Fig. 2.1b). Thiscondition means that the quality factor Q of the LC oscillator needs to satisfy Q À 1, aconstraint on the damping (and the coupling to the external world) of the oscillator [12].

(a)

(b)

I

Ψ (Φ)

U(Φ)

8

L

C
V

Φ

Q

ω0 =

V=

6

1
LC

4

2

dΦ
dt

2γ < ∆E
-4

-2

2

4

Φ

Figure 2.1: (a) Circuit representation of our simple harmonic oscillator. The systemcoordinates Φ and Q are used to specify the state of the system. (b) One requiredcondition to observe the quantum nature of a system is that the width of the different
energy states need to be smaller than their separation Q = ∆E/~2γ À 1.

In this chapter I will develop the classical and quantum description of electricalcircuits, including a quantum description of dissipation. I will begin by studying a dissipationless classical and quantum lumped element circuit. Then I will give a descriptionof transmission lines in the classical and quantum mechanical regime. This description
will allow me to model dissipation as a semi-infinite transmission line –a model thatis compatible with quantum mechanics. I will then discuss a method that describes
how to treat quantum-mechanically the coupling of an arbitrary system (in this case,a SHO) to the external world called Input-Output theory (IOT). Finally, in the thirdpart of the chapter, I will discuss the quantum limits for both phase-insensitive and10phase-sensitive amplifiers. These limits emerge from the simple constraint that boththe input and output modes must satisfy the appropriate commutation relations. I will
finish this part with a small description of the squeezed states generated by a degenerateparametric amplifier, a kind of phase-sensitive amplifier.

2.1

Electrical simple harmonic oscillator

2.1.1

Classical normal mode description of a LC-resonator

We will begin by deriving classical and quantum equations of motion for thesimple circuit in figure 2.1. More complex circuits can described as combinations of thissimple circuit. The Lagrangian of the circuit shown in figure 2.1a is
L =
=

1
1
CV 2 − LI 2 ,
2
2
µ ¶2
1
dΦ
1 Φ2
C
,
−
2
dt
2 L

(2.1)

where in the second line the Lagrangian is expressed in terms of the generalized coordinate Φ. In analogy to the mechanical case of a SHO, the electrostatic energy plays
the role of the kinetic energy, while the magnetic energy plays the role of the potential
energy. From Eq. (2.1), we can derive the canonical conjugate momentum of Φ given
by
dΦ
∂L
=C
= Q.
dt
∂ Φ̇
Using the canonical conjugate momentum Q, we can obtain the HamiltonianH=

1 Φ2 1 Q 2
+
2 L
2 C

(2.2)

and the Hamiltonian form of the equations of motion for this simple oscillator:
dQ
dt
dΦ
dt

∂H
Φ
=− ,
∂Φ
L
∂H
Q
= .
∂Q
C

= −
=

(2.3)

11
Since the signs of the coupling coefficients in Eqs. (2.3) are opposite, it follows thatthe system will oscillate with energy periodically exchanged between the electric andmagnetic fields. Solving the equations of motion (EOM) for the LC-resonator is very
simple however one thing to notice is that Eqs. (2.3) are coupled. The simple oscillator
can be described by two decoupled first-order differential equations. This is the normal
mode form of EOM [13]. One way to derive this form is to find the linear combinationsof the Hamiltonian equations that will decouple them. First, we define the complex
variables a and a∗ :1
a = √ (Φ + iZr Q),
2 L
1a∗ = √ (Φ − iZr Q),
2 L
qwhere Zr =

(2.4)

L
C . Replacing (2.4) in (2.3) yields(d+ iω0 )a = 0
dt

(2.5)

and its complex conjugate (c.c.). The choice of normalization constants for the normal
modes will become apparent shortly.
Eqs. (2.5) and its complex conjugate are the two first-order decoupled differential
equations that I was looking for. They are said to be the normal mode form and describe
the resonator equally as well as Eqs. (2.3). The quantities a(t) and a∗ (t) are the normal
mode amplitudes and are basically the classical version of the the operators â and â†often used to study the quantum mechanical version of the simple harmonic oscillator(SHO). They are linear combinations of the generalized coordinate and momentum(in this case flux of the inductor and charge in the capacitor) and can be pictured as
two counterrotating phasors of constant length. The magnitudes of these phasors arenormalized so that the sum of their squares gives the total energy stored in the system.

12
That is,E =1 Φ(t)2 1 Q(t)2
+
,
2 L
2 C

= |a(t)|2 + |a∗ (t)|2 .

(2.6)
(2.7)

Furthermore, |a|2 and |a∗ |2 represent the energy stored in the a and a∗ modes respectively. Since both V and I are real variables, |a|2 = |a∗ |2 , and the energy stored in eachmode is equal. The solutions of the normal modes are justa(t) = a(0)e−iωt ,(2.8)

a∗ (t) = a∗ (0)eiωt ,

(2.9)

where
1
a(0) = √ (Φ(0) + iZr Q(0)) =
2 L

r

·
µ
¶¸
E
Zr Q(0)exp i arctan2
Φ(0)with E being the total energy.
Although it may seem artificial to call a and a∗ different normal modes, thisseparation of modes offers the advantage of visualizing Φ(t) + iZr Q(t) and its c.c. as
two different counterrotating phasors; this will be useful when we analyze parametric
processes in coupled resonators. As we will see, modes that rotate in the same sense in
non-degenerate parametric amplifiers will couple more effectively than modes that rotate
in opposite senses; for degenerate parametric amplifiers, we only have one resonant
circuit, and in that case the coupling of modes will be between the a and a∗ mode [13].
Therefore dividing modes up in this fashion will aid in understanding and simplifying
more involved coupled systems.

2.1.2

Description of the problem in quantum mechanics language

We can quantize the system of figure 2.1a following the usual procedure for canonical quantization [12, 14, 15]; Q and Φ become operators satisfying the commutation

13
relations:[Φ̂, Q̂] = i~.

(2.10)In analogy to the classical we define the quantum operator for the normal modes interm of the charge and flux operatorsr

1(Φ̂ + iZr Q̂),
2~Zr
r
1
†
(Φ̂ − iZr Q̂).
â =
2~Zr
â =

(2.11)The normalization factor in this case is slightly different than in Eq. (2.4) to be consistent with the usual quantum mechanical definitions for these operators. They satisfy
the bosonic commutation relations[â, â† ] = 1.
Replacing this expression in the Hamiltonian for the system (2.2) we obtain the familiar
expression for the Hamiltonian of the quantum SHO:H=1~ω(ââ† + â† â) = ~ω(a† a + ).
2
2

(2.12)In order to simplify the notation, I will drop the hat on the operators, unless thedistinction between classical c-numbers and quantum mechanical operator is not obvious
from the context.

2.2

Transmission line theory
The main difference between circuit theory and transmission line theory is elec-

trical size. For circuit theory, the main assumption is that the physical dimension of thecircuit network is much smaller than the wavelength of the electrical signal used to drive
the circuit. In a transmission line, the length of the network is large enough that thevoltages and currents can vary in magnitude and phase over its length. In the systems
we will be studying, there will be a combination of sizes in the building blocks that will

14
require us to make use of both circuit and transmission line theory. In the first part forthis section I will first derive the equations of motion of a classical transmission line andthen introduce quantum mechanics following the procedure for canonical quantization.

2.2.1

Classical analysis of transmission lines(a)I(z,t)

(b)
Ll dz

+
V(z,t)_z=0
I(z+dz,t)

VR

+
Cl dz

v, Zc

V(z+dz,t)_I(z,t)
V(z,t) +
_

Zload

VL
dz

Figure 2.2: (a) Lumped element equivalent circuit of a lossless transmission line; theinductance and capacitance of the circuit shown are given by Ll dz and Cl dz respectively,
where Ll is the inductance per unit length, Cl the capacitance per unit length and dz theinfinitesimal length the section of transmission line. (b) A transmission line terminatedat z = 0 by a load of impedance Zload . Current I and voltage V are indicated as definedin the main text in terms of the right and left traveling voltage waves VR and VL .

Fields inside a transmission line satisfy the 1-D wave equation. We can derive
this result by modeling the transmission line as a series of lumped element circuits oflength dz, as shown in figure 2.2a, where Cl and Ll is the capacitance and inductance
per unit length. Applying Kirchhoff’s voltage and current law on one of the cells results
into the derivations of the so called telegrapher equations [16]:∂V (z, t)∂z∂I(z, t)∂z

∂I(z, t),∂t
∂V (z, t)
= −Cl
.
∂t
= −Ll

(2.13)

Combining these equations, we can derive the 1-D wave equation for both voltages andpcurrents with a propagating velocity of v = 1/Ll Cl :∂ 2 V (z, t)
1 ∂ 2 V (z, t)
=
.
∂z 2
v2
∂t2

(2.14)

15
We know that the solution for the wave equation can be decomposed into a left-travelingpart and a right-traveling part:zz
V (z, t) = VR (t − ) + VL (t + ),
v
v
1
z
z
I(z, t) ={VR (t − ) − VL (t + )},Zcvv
where Zc is the characteristic impedance of the transmission line given by Zc =(2.15)p
Ll /Cl ,and relates the voltage and current on the line as:
Zc =

VR
VL
=− .
IR
IL

VL and VR are arbitrary functions of their argument and they represent signals being
launched in the transmission line.
For an infinite transmission line, the right and left-traveling waves are independent; however, when at some coordinate z (that for our convenience, we will define thispoint as z = 0) the transmission line is terminated by some circuit then these solutions
are not independent (Fig. 2.2b). They both have to fulfill certain boundary condition
imposed by the circuit with a frequency dependant impedance Zload (ω):
Zload =V (z = 0, ω).
I(z = 0, ω)(2.16)The instantaneous net power flow transported by the transmission line is given by
P =1(VR (z, t)2 − VL (z, t)2 )Zc(2.17)

where our convention is that positive carried power goes to the right. So far our description of a transmission line has been completely classical. In the next section wewill treat it as a quantum system.

2.2.2

Quantization of a lossless transmission line

Before quantizing the system consisting of a transmission line, we will write its
Hamiltonian and Lagrangian. We will see that an infinite transmission line can be16+ ∆V _I(z,t)+
V(z,t)

I(z+dz,t)−∆V = V ( z + dz ) − V ( z )= ∂ t φ( z + dz ) − ∂ t φ( z )+L=dzLlV(z+dz,t)= dz∂ z ∂ t φ( z )__dz∂ t ∂ z φ( z ) = −dzLl ∂ t I ( z, t )∆V = ( dzLl )∂ t I ( z, t )Figure 2.3: The relation between the current I(z, t) and the generalized flux variable
φ(z, t) is described.

studied as a collection of an infinite number of independent harmonic oscillators andhence it can be easily quantized.2 Using the generalized flux variable defined as [3, 12]
Z t

dt0 V (z, t0 )φ(z, t) ≡(2.18)−∞

where V (z, t0 ) = ∂φ(z,t)
is the local voltage of a transmission line at position z and at∂ttime t0 . The relation between the current I(z, t) and φ(z, t) is derived in figure 2.3 andis given by
I(z, t) = −1 ∂φ(z, t).
Ll ∂z

Then the Lagrangian density for a transmission line is
Cl
L=
2

µ

∂φ(z, t)
∂t

¶2

1
−
2Ll

µ

∂φ(z, t)∂z

¶2.

(2.19)

Using Eq. (2.19), it can be shown using the Euler-Lagrange equation that φ satisfies a
1-D wave equation:1 ∂ 2 φ(z, t)∂ 2 φ(z, t)=.
∂z 2v 2 ∂t2(2.20)The momentum conjugate variable of the flux φ(z, t) is the charge density q(z, t):
∂φ
δL
= Cl
= Cl V (z, t) = q(z, t)
δ∂t φ
∂t
2

The only difference between the quantization of electromagnetic modes of vacuum in free space
and the ones studied here that in our case photons do not carry a polarization label and travel in one
dimension yielding a different density of states as a function of frequency [3].

17from which we can then derive the Hamiltonian density:
1
H=
2Ll

µ

∂φ(z, t)
∂z

¶2
+

1
q(z, t)2 .
2Cl(2.21)We have already stated that since the fields in the transmission line satisfy thewave equation, then their solutions consist of left and right-traveling waves (2.15). We
can introduce quantum mechanics into the analysis of the transmission line again following the usual procedure of canonical quantization:[φ(z 0 , t), q(z, t)] = i~δ(z − z 0 ),[q(z, t), q(z 0 , t)] = [φ(z, t), φ(z 0 , t)] = 0.
We can fourier expand these fields following the same procedure described in Refs.
[3, 15, 17]. The only distinction is that I have chosen to use the voltage V (z, t) =q(z, t)/Cl as my quantized field variable so that our analysis follows closely the language
used in microwave theory
r

Z ∞

dω √
ω(aR (ω)e−iω(t−z/v) + H.c.),
2π
0
r
Z ∞
~Zc
dω √
VL (z, t) =ω(aL (ω)e−iω(t+z/v) + H.c.),2 0 2πVR (z, t) =~Zc
2

(2.22)

where H.c. stands for Hermitian conjugate, and the operators aL (ω) and aR (ω) are theannihilation operators for the signal quanta propagating to the left and to the right,
respectively. They satisfy the standard commutation relations for a 1-D scalar fieldgiven below, where we have used the convention from Ref. [3, 18]:[aα (ω), aα0 (ω 0 )] = 0,(2.23)[aα (ω), a†α0 (ω 0 )] = 2πδ(ω − ω 0 )δα,α0 ,(2.24)ha†α (ω 0 )aα0 (ω)i =1
exp( k~ω)−1BTh{aα (ω), a†α0 (ω 0 )}i = coth(2πδ(ω − ω 0 )δα,α0 ,~ω)2πδ(ω − ω 0 )δα,α0 ,2kB T(2.25)(2.26)

18
where α and α0 stand for either R or L. Using expressions (2.22)-(2.24) and replacing
the field operators into the expression of the Hamiltonian density (2.21), we can derive
our expression for the Hamiltonian of an infinite transmission line:
Z ∞
H=

Z ∞
dzH =

−∞

0

dω
~ω[a†L (ω)aL (ω) + a†R (ω)aR (ω)]
2π(2.27)

where I have dropped a constant term that arises from the commutation relations. Eq.
(2.27) shows what I mentioned above: a transmission line can be modeled as a systemcomposed of an infinite number of harmonic oscillators.
Once we have quantized the fields inside a transmission line, we can calculate
statistical values for some important quantities. For a given thermal state, VR and VR2
have the following expectation values:
hVR i = 0,
Z
Z
dω ~ω
~ω
dω
2
SVR VR = Zc
coth(),
hVR i =
2kB T
B 2π 2B 2π(2.28)(2.29)

where SVR VR is the symmetrized spectral density of right traveling voltage signal, andthe integration has been limited to the bandwidth Bω of the measuring instrument. We
observe then that a thermal state with temperature T has a noise power per unit band~ω
width of ~ω
2 coth( 2kB T ) propagating to the right and to the left along the transmission

line. We can use the previous results to estimate the voltage fluctuations generated by aresistor [3]. In order to do that, let’s consider the example of a transmission line terminated with a resistor at one end. The terminating resistor acts as a black body emitting
thermal noise down the transmission line. If the resistor is held at temperature T it willbring the transmission line modes into equilibrium at the same temperature. Consider
figure 2.2b where the load Zload is replaced by a resistor like the one shown in figure
2.4. We can derive the voltage noise of a resistor by considering the voltage noise at the
open termination of a transmission line with Zc = R. For the open termination of thetransmission line, the left and right moving signals are not independent, but VR = VL ,

19
so that the voltage at the end is given by
V = VR + VL = 2VR
and thus, using Eqs. (2.29), we have
SV V (ω) = 4SVR ,VR (ω) = 4Zc

~ω
~ω
coth(
).
2
2kB T

(2.30)

It is these fluctuations from a resistor that will allow us to calibrate the noise added by
our amplifier in chapter 6.
Another important point is that transmission lines can act as a simple resistor
in which instead of dissipating energy, it carries it away as propagating waves. Thiseffective dissipation is a consequence of its semi-infinite length and since it has a conservative Hamiltonian, it can be used to treat dissipation within a quantum theory
framework [3, 12, 15]. This model for resistors (called Nyquist model [19]) is completely
equivalent to modeling dissipation as coupling to a thermal bath or reservoir with an
infinite number of harmonic oscillators [20].

R
VR
+
V _

SVV

Zc

VN

VL
Figure 2.4: A noisy resistor R is coupled to a transmission line with a characteristic
impedance of Zc . The noise of the resistor is modeled as a voltage source with a spectral
density given by SV V .

2.3

Input-Output theory
In this section I will consider the problem of coupling a system, e.g. a simple

harmonic oscillator, to the fields of a transmission line using Input-Output theory (IOT).

20
IOT is an approximate method that solves an important problem: how to calculate thequantum statistical properties of both, the field inside a cavity and the field exiting thecavity given the input field. In this section I will first treat the useful example of couplinga simple harmonic oscillator, in the form of an LC resonator, to external signals comingfrom a transmission line. Due to its simplicity, I can solve this problem exactly using
quantum network theory [15].3

Then I will explain a series of approximations madeto the EOM that allows us to cast them into a form identical to the EOM derived inquantum optics books for a cavity coupled a reservoir modeled as a collection of infinite
harmonic oscillators [22]. This serves as a prelude for the next part of this section where
I describe another more direct approach of quantizing systems coupled to thermal baths
in a language more akin to that of quantum optics.

2.3.1

Harmonic oscillator coupled to a transmission line

z=0
Vin

IC

IL

I(z=0,t)

L

Φ

V

C

Vout
Figure 2.5: A SHO in the form of an LC resonator is coupled to a transmission line.
The resonator is described using as generalized coordinates the flux Φ of the inductor
and the charge Q of the capacitor.

The system analyzed is the one shown in figure 2.5. It consist of an LC resonator
coupled to a transmission line. The equations of motion can be derived using current
3

Quantum network theory was developed as quantum extension of the classical theory of electrical
networks and as a quantum extension of linear response theory in the context of electrical circuits. It
allows us to study fluctuations entering a system as input signal fields from a transmission line [15, 21]

21conservation law4 :−I(z = 0, t) + IL (t) + IC (t) = 0,
Φ d(CV )
+
L
dt
Φ
d2 Φ
−I(z = 0, t) + + C 2
L
dt−I(z = 0, t) +(2.31)= 0,(2.32)= 0.

(2.33)

Now, using expressions (2.15), we can solve for I(z = 0, t) = I(t) in terms of theinternal V (z = 0, t) and incoming Vin (z = 0, t) fields (where for convenience I am using
the subscripts “in” and “out” instead of “R” and “L” to make the distinction between
the input and output fields clearer). Solving for I(t), the result isI(t) =

2Vin (t) − V (t).
Zc

Using the known relation between voltage and flux V = dΦdt , our EOM for the internal
field Φ is then
C

d2 Φ
2
1 dΦ Φ+ =Vin (t)
+
2
dt
Zc dt
L
Zc

(2.34)The term Z2c Vin (t) is the driving force that the incoming fields exert on the harmonic
oscillator. For the case when the transmission line represents a thermal bath, Vin is
the fluctuation or Langevin force and Eq. (2.34) can be consider just an example ofa Langevin equation. Solving the EOM of the LC-resonator allows us to predict theoutgoing field. In this rather simple example, the response of the oscillator is linear andcan be characterized by a complex frequency dependant impedance Z(ω) (Eq. 2.16)
Z(ω) =V (z = 0, ω).
I(z = 0, ω)

From basic microwave theory using expressions (2.15) and (2.16), it can be shown thatthe relation between the incoming and outgoing signal is
Vout (z = 0, ω) = Γ(ω)Vin (z = 0, ω)
4(2.35)A Quantum Network theory approach to a problem usually starts by writing the Lagrangian of
the system, and using the Euler-Lagrange equation, the EOM are then derived [12, 15]. However, thisexample is simple enough that we can skip this step.

22
where Γ(ω) is the called the reflection coefficient and is defined asΓ(ω) =

Z(ω) − Zc.
Z(ω) + Zc(2.36)For our example, from circuit theory we know that the impedance of a parallel LC
resonator is given byZ(ω) =jLω
1 − (ω/ω0 )2(2.37)√
where I am using the electrical engineering sign convention of j = −i and ω0 = 1/ LC.
In the limit of low damping (Q = ω0 Zc C À 1), we expect the resonator to respond to
inputs only near the frequency ω0 where the impedance is large compared to Zc , andwe may approximate [3, 16]Z(ω) =jLω02
1
jLωω02≈=(ω0 − ω)(ω0 + ω)2(ω0 − ω)
2Cj(ω − ω0 )which then yields for the reflection coefficient in the high Q limit
Γ(ω) =

Z(ω) − Zc
γ − j(ω − ω0 )≈Z(ω) + Zc
γ + j(ω − ω0 )(2.38)

1
defines the amplitude decay rate, or half width at half maximum
where γ = 2CZc(HWHM).
So far, the analysis has been completely classical. We start the quantum mechanical analysis by making the input and output fields quantum mechanical signals thatare represented in the fourier domain as shown in expression (2.22). In this case, theincoming field consist of the right-propagating signals at z = 0r
Vin (t) =

~Zc
2

Z ∞
0

dω √
ω(ain (ω)e−iωt) + H.c.).
2π(2.39)

Solving for internal fields in Eq. (2.34) in the frequency domain, the result isΦ(ω) =

1
C

†

1
C

Φ (ω) =rrain (ω)2~ω
,2
Zc −ω − i2γω + ω02(2.40)a†in (ω)
2~ω
.
Zc −ω 2 + i2γω + ω02(2.41)

23
From these expressions we can derive the time-dependant solution for Φ as shown belowr Zµ¶
1 2~ ∞ dω √ain (ω)e−iωtΦ(t) =
ω+ H.c. .
(2.42)C Zc 0 2π−ω 2 − i2γω + ω02
Now consider the flux’s conjugate momentum given by the charge of the capacitor
Q = CV = C(dΦ/dt). Using the commutation relations (2.23 and 2.24) we can showthat Q and Φ have the expected commutation relation:
4γ~[Φ(t), Q(t)] = i
π

Z ∞
dω
0

ω2
.
(ω 2 − ω02 )2 + (2γω)2(2.43)= i~.
This is the electrical equivalent to [x, p] = i~. Eqs. (2.34-2.42) constitute the solution
for the quantum problem of an LC circuit coupled to a transmission line. We cancalculate expectation values for the internal fields, and together with Eq. (2.15) we cancalculate expectation values for the output fields (VL → Vout ).
2.3.2

Approximations

So far, I have not made any approximations in the analysis, and we have beenable to solve this problem exactly due to its simplicity. However for more complicatedsystems (e.g. a non-linear harmonic oscillator), the exact solution is not possible unlesssome approximations are made. I introduce these approximations in this section in
the same context of a SHO coupled to a transmission line from which I will derive
expressions similar to those found in the quantum optics literature describing a cavity
coupled a thermal bath.
Consider again the EOM for the LC-harmonic oscillator coupled to a transmission
line.
C

d2 Φ
1 dΦ Φ
2++ =Vin (t).
dt2
Zc dt
L
Zc

We now write the expression for Φ(t) asrhω0 LΦ(t) =[c(t)e−iω0 t + c† (t)eiω0 t ]2(2.44)

24
where the normalization constant has been chosen according to Eq. (2.11), so that theenergy in the internal field will be E = ~ω0 c† c. In this expression, the amplitude c(t) isa slowly varying function of time.
Let’s consider the incoming field Vin (t) as defined in Eq. (2.39). When the Q of
the resonator is high, one expects it to respond to inputs only near the frequency ω0 .
√√With this in mind, we replace ω with ω0 :r
Z
~Zc ∞ dω √(2.45)Vin (t) =ω(ain (ω)e−iωt + a†in (ω)eiωt ),2 0 2πr
Z~Zc ω0 −iω0 t ∞ dω 00≈{ebin (ω 0 )e−iω t2−ω0 2πZ ∞0dω † 0 iω0 t+eiω0 tbin (ω )e },(2.46)−ω0 2πwhere we have made a change of variables (ω → ω 0 +ω0 ) and defined the new annihilation
operatorbin (ω) = ain (ω + ω0 )which has the same commutation relations as ain (Eqs. 2.23, 2.24). Further, because theoscillator will only respond to a driving signal whose frequency is close to ω0 , that is atfrequencies ω 0 = 0 for the new frequency variable, we can extend the integral lower limit
to −∞ without introducing a large error. Thus, we arrive to the following expression
for the incoming fieldrVin (t) =~Zc ω0 −iω0 t[ebin (t) + eiω0 t b†in (t)]
2

where
1
bin (t) =
2π

Z ∞
−∞

dωbin (ω)e−iωt .

(2.47)(2.48)

We can picture bin (t) as the slowly varying part compare to ω0 of the input field [17].
Since bin (ω) satisfy the commutation relations[bin (ω), bin (ω 0 )] = 0,(2.49)[bin (ω), b†in (ω 0 )] = 2πδ(ω − ω 0 ),(2.50)

25
the analysis of the temporal modes (2.48) shows that their quantum commutator obeys[bin (t), bin (t0 )] = 0,(2.51)[bin (t), b†in (t0 )] = δ(t − t0 ).

(2.52)

It can be shown that the normalization of bin (t) results in a incoming power for a
monochromatic signal at frequency ω of [3]Pin = ~ωhb†in (t)bin (t)i.

(2.53)

The approximations used to derive Eqs. (2.46), (2.47), (2.48) and (2.50) arereferred to as the Markov approximation [3, 17, 22, 23]. A similar line of reasoning,with the appropriate changes can be applied to obtain the expression for the outgoing
field and we get exactly the same equations as Eqs. (2.47-2.53) with the subscript “in”
replaced by “out”.
Our next set of approximations will let us write the EOMs for these coupledsystems in a form similar to those found in quantum optics literature [3, 17, 22–24]. If
we replace Eqs. (2.44) and (2.47) into (2.34) and separating the parts oscillating ate−iω0 t and eiω0 t , we obtain:C

1 dce−iω0 t2d2−iω0 t−iω0 t
(ce
)
+
(ce
)
+
=√
bin (t)e−iω0 t .
2
dt
Zc dt
L
LZc

(2.54)

The following approximations are made:2d−iω0 t ) ≈ −ω 2 ce−iω0 t − i2ω dc e−iω0 t
• dt
0 dt
2 (ce
0

• | 1c dc
dt | ¿ ω0
These approximations constitute the “slowly varying envelope approximation” [13, 17].
It basically means that the amplitude c(t) is going to be a slow varying function of timecompared to the time scale of 1/ω0 . After substituting these equations in (2.54), we
obtainpdc+ γc = i 2γbin (t).
dt

(2.55)

26
The equivalent expression for the boundary conditions (2.15) in terms of the operatorsc, bin and bout ispbout (t) + bin (t) = −i 2γc.

(2.56)

We can derive an expression for Q(t) equivalent to (2.44) within the same set of approximations. Substituting both expressions for Q and Φ in Eq. (2.43), it can be deduced
that [c, c† ] = 1, as expected.
The previous expressions have been derived in the ω0 -rotating frame. One cantransform back to the non-rotating frame through the following transformationsa(t) = e−iω0 t c(t),ain (t) = e−iω0 t bin (t),aout (t) = e−iω0 t bout (t),with the same commutation relations as their counterparts in the rotating frame. The
equations of motion (2.55) and (2.56) in the non-rotating frame becomepda+ iω0 a + γa = i 2γain (t)dtpaout (t) + ain (t) = −i 2γa.

(2.57)(2.58)

We can go on to solve exactly Eqs. (2.57) and (2.58); however, I will not dothat now, but in the following section. The point I wanted to make in this section
was to explain the approximations that allowed us to reduce expressions (2.34) and(2.15) derived using quantum network theory to a form that is commonly encounteredin quantum optics when analyzing a cavity mode (in our case the cavity’s role playedby the LC-resonator) coupled to an external field through a partly transparent mirror.
What I will show in the next section is that Eqs. (2.57) and (2.58) can also be derived
in a much more direct way when one is willing to make the approximations explained inthis section since the very beginning. This method is what is more commonly described
as Input-Output theory.

27
2.3.3

Alternative approachaout(t)test port

cavityγ

H = h ω a† aain(t)

Figure 2.6: The typical example in quantum optics used to introduce Input-Output
theory consists of a cavity whose Hamiltonian is that of harmonic oscillator coupled toa thermal bath via a test port. The coupling between the reservoir and the cavity is
described by frequency-independent term γ.

In this section I describe a more direct way to derive the equivalent equations
to (2.57) and (2.58) but for a generic system (not necessarily a SHO). The present
discussion follows closely standard references on the subject [3, 17, 22–24]. In order todrive a cavity like the one shown in figure 2.6, we need to open a port which exposes thecavity to both an external drive and the noise of the thermal bath it is now coupled to.
Input-Output theory allow us to formally separate the degrees of freedom into internal
cavity modes and external bath modes. The Hamiltonian of a generic high-Q cavity
coupled to a thermal bath can written in the following form
H = Hsys + Hbath + Hint
where Hsys is the Hamiltonian for the system under study. In our case, Hsys is given by
Eq. (2.12),
HB =X Z ∞ dωn

~ωa†n (ω)an (ω)(2.59)[κ∗n a†n (ω)g + κn g † an (ω)](2.60)−∞ 2π

is the sum over the n thermal baths andHint = ~X Z ∞ dωn−∞ 2πwhere g = g(a, a† ) and g † are the system operators the bath couples to and κn is thecoupling constant between the n thermal bath and the system. I have assumed that all

28
the baths couple to the same operator in the system under study. This Hamiltonian
gives rise to linear coupling between the system variables and the bath modes. The anoperators are the boson annihilation operators for the n-different baths the system iscoupled to and they obey the usual commutation relations:[an (ω), a†n0 (ω 0 )] = 2πδn,n0 δ(ω − ω 0 ),(2.61)[an (ω), an0 (ω 0 )] = [a†n (ω), a†n0 (ω 0 )] = 0.
Now there are two important points about the interaction and thermal bath Hamiltonian:(1) The range of integration is from −∞ to ∞ as opposed to from 0 to ∞: the cavity
resonance only responds to frequencies within certain range. Thus, the lower
integration limit has been extended from 0 to −∞ as we did in the previous
section under the “slowly varying envelope approximation.”(2) The coupling constants κn are frequency independent: we have assumed theMarkov approximation by assuming κn is a constant over the range of frequencies that are relevant to the system.
Since the creation and annihilation operators have no explicit time dependance,
we can derive their Heisenberg equations of motion fromi~dO
= [O, H]dt

from which one obtains the following EOM for the bath mode operatorsdan (ω)= −iωan (ω) − iκ∗n g
dt(2.62)

and for the system operator f (not necessarily g), one obtains
X Z ∞ dωdf (ω)i= − [f, Hsys ] − i{κ∗n a†n (ω)[f, g] + κn [f, g † ]an (ω)}.
dt
~
2π
−∞
n

(2.63)

29
The bath mode fields have yet to be separated into incoming and outgoing bath fields.
The procedure through which this is done can be reviewed in the extensive literature[3,17,22,23]. We do not review it here since the derivation is not so intuitively appealing
as in the case explained above for the transmission line coupled to a LC-resonator; wejust quote the resulting EOM for the internal field in terms of the input fields:Xdfi= − [f, Hsys ] +
dt
~
n

½·

¸
·
¸¾
|κn |2 †|κn |2∗ in††in
g + iκn an (t) [f, g] − [f, g ]g − iκn an (t)22(2.64)and the boundary condition that relates the input, output and internal fields
out
∗
ain
n (t) + an (t) = −iκn g(t)(2.65)

with[aαn (t), aαn0 (t0 )] = 0(2.66)

000[aαn (t), aα†n0 (t )] = δn,n δ(t − t )(2.67)

for α → in, out
So far, Eqs. (2.64-2.65) have been expressed for a very general case. To touchground, let’s consider the equivalent case of a harmonic oscillator coupled to a single
thermal bath (Fig. 2.6). In that case we drop the index n for multiple thermal baths,
Hsys = ~ω0 a† aand replace g for a in Eq. (2.60). Expressions (2.64) and (2.65) for the case of f = a
reduces toda|κ|2= −iω0 a −a + iκain (t)dt
2(2.68)ain (t) + aout (t) = −iκa(t).

(2.69)

andEqs. (2.68) and (2.69) are exactly the same equations we derived in the previous√
section (2.57) and (2.58) with the coupling constants κ being defined as κ = 2γ. They

30
can be solved by Fourier transformation for both the internal field and the output field
yieldinga(ω) =
aout (ω) =√2γiain (ω)i(ω0 − ω) + γi(ω − ω0 ) + γ
ain (ω).
−i(ω − ω0 ) + γ(2.70)(2.71)

From Eq. (2.71), the ratio of the input and output field isaouti(ω − ω0 ) + γ=
ain−i(ω − ω0 ) + γ
which is exactly the reflection coefficient Γ(ω) that was derived for a parallel LC resonator coupled to a transmission line (2.38)
The reason the alternative approach is useful, is that it gives us a recipe for how
to add ports, without worrying about the details of the coupling and it lets us treat
more complicated systems than a SHO in a much simpler way. For these reasons, wewill be doing extensive use of expressions (2.64) and (2.65) in the following chapters.

2.4

Quantum-limited amplification: Haus-Caves theorem
A comprehensive review on quantum limits of amplification can be found in Ref.

[2]. This section will review the fundamental properties of amplification. The generalquantum limit (Haus-Caves theorem) is first discussed for phase insensitive amplifiers,
where the origin of the half a quantum of added noise is clarified. Then we will discuss
the concept of a noiseless amplifier in the form of a phase sensitive amplifier. Finally
we will discuss the generic example of a degenerate parametric amplifier as a phasesensitive amplifier, and its properties as a generator of quadrature-squeezed states. In
this section, I will only discuss the simple case of a single-mode system. However, real
signals and real amplifiers have a nonzero bandwidth, and a multimode analysis of theamplifiers is necessary. The generalization of the results derived here to the multimodecase is considered in appendix A.

31

(a)

(b)
X2

X2

bs
as

as
X1

bs
X1

Figure 2.7: (a) Input and output signals for a linear phase insensitive amplifier. The
input noise (red) is amplified (pink). However the output contains an amount on noise
larger than just the amplified noise due to the added noise of the amplifier (blue). (b)For an ideal phase sensitive amplifier (such as a degenerate parametric amplifier), theoutput does not contain any amplifier noise. However, this comes with the restrictionthat only one of the quadratures (X1 ) is amplified, while the other is deamplified.

2.4.1

Phase insensitive amplifier

A phase insensitive amplifier is one that for a given input, the output will not show
any phase preference; that is, for a given phase shift on the input signal, the only effectis an equivalent phase shift of the output [2]. Another requirement for an amplifier tobe phase insensitive is that it must add equal amount of noise to both quadratures [25].
In general, we can define a voltage signal with the time dependance explicitly includedas³´
Vs = V0 as e−iωo t + a†s eiωo twhere as (0) ≡ as and a†s (0) ≡ a†s . We can decompose this voltage signal into two phaseor quadrature operators:
Vs = 2V0 (X1 cos(ωo t) + X2 sin(ωo t)) .

(2.72)In the previous expressions, the signal Vs has been normalized so that a†s as = X12 +X22 − 1/2 corresponds to the photon number operator in the signal mode [26]. The

32
quadrature operators are hermitian (and therefore observables) defined by
1
X1 = (as + a†s ),2
1
X2 = (as − a†s ).
2i(2.73)(2.74)

From Eq. (2.72), it is evident that the quadrature operators are associated with field
amplitudes oscillating 90◦ out of phase. Let’s consider a linear amplifier with gain G1
for the cosine quadrature component and gain G2 for the sine quadrature component.
The two quadrature components Y1 and Y2 of the output amplified signal are given by:
Y1 =
Y2 =

p

G1 X1 + F1 ,(2.75)p

G2 X2 + F2 ,where F1 and F2 are the fluctuation operators of the amplifier internal modes. The
input and output mode operators must satisfy the boson commutation relationsi[X1 , X2 ] = [Y1 , Y2 ] = .
2(2.76)In order for Y1 and Y2 to satisfy the commutation relations, F1 and F2 cannot be zero,but they need to satisfy
p pi[F1 , F2 ] = [1 − G1 G2 ]2(2.77)

where I have assumed that the input mode is independent of the amplifier internal mode([Xi , Fj ] = 0).5 Eq. (2.77) implies that the product of the fluctuations in F1 and F2
must satisfy the following inequalityh∆F12 ih∆F22 i ≥G1 G2
1(1 − √ √ )2 .
16
G1 G2

(2.78)

When quoting a noise number for an amplifier, this is usually referred to the
input of the amplifier; that is, the output of the amplifier is divided by its gain, and5Carlton Caves mentions the possibility of systems amplifier-input field where it would be possible
to have less noise added than the limit we will derive. However, this would only be possible if the input
noise and the internal-mode noise of the amplifier were correlated [2].

33that way we can do a fair comparison in the noise performance of different amplifiers,
independent of their gains. Following the same reasoning, we can define the normalized
output signals as
X1,ef f=X2,ef f=YF
√ 1 = X1 + √ 1 ,
G1
G1
Y2
F2
√
= X2 + √ .
G2
G2(2.79)

Again, the uncertainty product for the new defined normalized output modes is givenbyh∆F22 ih∆F12 i)(h∆X22 i +
),
G1
G2
1
1
(2 − √ √ )2 .
16
G1 G2

2
2
2
h∆X1,eff ih∆X2,ef f i = (h∆X1 i +≥(2.80)

Now we can observe, in the limit of high gain, what Eq. (2.80) implies: for G1 = G2 À 1
the uncertainty product (2.80) will be greater or equal to 1/4 which is to be comparedwith the original uncertainty product in the quadrature components of the incoming
field h∆X12 ih∆X22 i ≥ 1/16. The equality is true when two requirements are fulfilled:
the input is a minimum uncertainty state andh∆Fi2 i= h∆Xi2 iGiwhere i = 1, 2. In such a case, it can be shown that the noise added by the amplifier toboth quadratures of the signal is exactly half a quantum of noise, AN = 12 where AN is
the added noise number (figure 2.7a). Some authors prefer to specify the added noise
as a noise temperature. In most cases, the noise temperature is defined as
TN =

AN ~ω
.
kB

Now the general case for a finite gain depends on whether the amplifier is phase√
√
√
√
√
√
preserving ( G1 = G2 = G) or a phase conjugating ( G1 = G = − G2 ) amplifier. The fundamental theorem is
AN =

1
1
h∆F12 i h∆F22 i+≥ |1 ∓ |Gi
Gi
2
G(2.81)

34
where the upper (lower) sign holds for phase-preserving (phase-conjugating) amplifiers[2].
Haus-Caves theorem is just a consequence of a more general quantum-mechanicaluncertainty relationship for the simultaneous measurement of two conjugate variables:
the uncertainty product of two simultaneously measured conjugate observables is afactor of 2 larger than that dictated by the Heisenberg uncertainty principle [4, 27].
One explanation for this is that the additional noise must be introduced so that these2 observables measured simultaneously commute; what I mean by this statement canbe better described if we calculate the commutation relation of the normalized output
fields in Eq. (2.79):i1√ √2 G1 G2√ √where we can observe that in the limit of | G1 G2| À 1, the two normalized output[X1,ef f , X2,ef f ] =signal operators will commute: after large enough amplification, the quantum signals
X1 and X2 get to a “classical power level”, where simultaneous measurements can be
performed with no additional introduction of uncertainty. The quantum bounds on
the simultaneous measurement of the input modes are not violated due to the noise
apparently added at the amplifiers input [27].
The simplest realization of a linear phase insensitive amplifier is a nondegenerateparametric amplifier [28]. An example of such consists of three resonant modes coupled
together by nonlinear elements. The three modes are called the pump, the signal andthe idler. I will not discuss all the details of nondegenerate parametric amplifier but
only the transformation it does on the input idler and signal modes, ignoring the factthat the pump mode is the one that does the necessary work to do the amplification.
A nondegenerate parametric amplifier performs the following transformation of

35
the input modes [2]:










 bs   cosh r sinh r   as 

=


†
†
bi
sinh r cosh rai(2.82)

where s stands for signal mode, i for idler mode, and r is a real constant that determines√√√√
the gain of the amplifier.6 The gain G1 = G2 = G is given by G = cosh r. In
this case, the noise operator Fi from Eq. (2.75) is given bysinh r(ai + a†i ) = X1,i sinh r,
2
i sinh r= −(ai − a†i ) = X2,i sinh r.
2

F1 =
F2

In this set of transformations, it is obvious that the added noise is introduced throughthe idler mode, and the minimum added noise will be given when this idler mode is in
the vacuum state ha†i ai i = 0. The added noise number is then given byAN==

1 sinh2 r,
2 cosh2 r
1
1(1 − ),
2
G

which is the Haus-Caves theorem, Eq. (2.81), for a phase preserving amplifier. In
chapter 7 we will use the nondegenerate parametric amplifier to model the inefficiencies
associated with any linear amplifier, including those whose added noise is larger thanvacuum noise.

2.4.2

Phase sensitive amplifier

A phase sensitive amplifier is one in which the output depends in an essential
way on the phase of the signal at its input. Consider Eqs. (2.75), in the special caseof a phase sensitive amplifier with G1 G2 = 1. The output modes Y1 and Y2 will satisfy
the commutation relation without the need of the operators F1 and F2 responsible
6

The most generic transformation done by a nondegenerate parametric amplifier would include thepossibility of complex amplitude gains in the transformation matrix of Eq. 2.82. However, this would
not change any of our conclusions.

36of the added amplifier noise. In this case, X1 is amplified with no added noise andcan be measured as a classical signal after the amplification process obtaining all theinformation stored in this quadrature. However, X2 is deamplified, therefore losing all
the information in that other quadrature. This is necessary in order not to violate thequantum limit on the simultaneous measurement of two conjugate variables [4, 27].
One example of such a noiseless amplifier is a degenerate parametric amplifier(DPA), an example that will occupy us during most of the rest of this thesis. For now I
just give a taste of the transformation that a DPA imposes on the input mode, regardlessof the way we are implementing it (that will be described in the next chapter). The
generic scattering matrix for a lossless degenerate parametric amplifier is given by [2, 8]

 


 b s   G M   as 
(2.83)

=


†
†
∗
∗
bs
M G
as
where bs is the signal output mode and as is the signal input mode. In order to maintainthe commutation relations [as , a†s ] = [bs , b†s ] = 1, and [as , as ] = [bs , bs ] = 0, we requirethat|G|2 − |M|2 = 1.
For simplicity, and without loss of generality, call G =√√G, and therefore M = G − 1.

The resulting gains for both quadratures is then given byp√√
Y1 = ( G + G − 1)X1 = G1 X1 ,(2.84)

p√√
Y2 = ( G − G − 1)X2 = G2 X2 .
The two gains satisfy the condition G1 G2 = 1 and therefore there is no additional noise
term in the output quadratures: the quadrature X1 is amplified but X2 is deamplified,
as shown in figure 2.7b. In addition, for a gain larger than unity the fluctuations of theoutput quadrature Y2 will have less fluctuations than vacuum fluctuationsh∆Y22 i <14

37
Quantum states for which this condition is fulfilled are called (quadrature) squeezed
states.

2.4.3

Quadrature squeezing

For the two quadrature operator X1 and X2 of a system, i.e. SHO, that satisfy
the commutation relation Eq. (2.76), it follows thath∆X12 ih∆X22 i ≥1.
16(2.85)

A state of that system is said to be squeezed if for either of the quadratures operators1h∆Xi2 i ≤ .
4Obviously, because of Eq. (2.85), we cannot have both variances being squeezed. Thosestates for which one of the quadratures is squeezed and the equality in Eq. (2.85) holds
are called ideal squeezed states. States for which one of the conditions holds will haveless “noise” in one of the quadratures than for a coherent state or a vacuum state, withthe requirement that the fluctuations in the other quadrature must be enhanced in order
not to violate the uncertainty relation.
It is sometimes convenient to define a generic quadrature operator
1
X(θ) = (ae−iθ + a† eiθ )2(2.86)

where X1 = X(0) and X2 = X(π/2). We characterize the squeezing by introducing theparameter ξ(θ)ξ(θ) =h∆X(θ)2 i.
1/4(2.87)

Squeezing exists for some θ whenever 0 ≤ ξ < 1.
Most schemes for generation of quadratures squeezed light are based on some sortof parametric process using various types of nonlinear devices. A degenerate parametricamplifier can be used for this process of squeezing. Without loss of generality, consider

38

Y2

X2
Y1

φ

X1

Figure 2.8: Schematic of the noise in the quadrature variable space for a vacuum state(circle) and a squeezed state (ellipse) for r = 1. The squeezed state is rotated by anangle φ according to Eqs. (2.88-2.89). The quadrature axes Yi are defined according to
Eq. (2.86), Y1 = X(φ) and Y2 = X(φ + π/2).

the DPA described in the previous section with G = cosh r and M = −eiφ sinh r, where
φ defines the axis where the squeezing happens. In this case, b = a cosh r − a† ei2φ cosh r
and we obtainh∆X12 i =h∆X12 i|φ=0 =h∆X22 i =h∆X22 i|φ=0 =1(cosh2 r + sinh2 r − 2 sinh r cosh r cos 2φ),
4
1 −2r
e ,
4
1
(cosh2 r + sinh2 r + 2 sinh r cosh r cos 2φ),
4
1 2r
e ,
4

(2.88)(2.89)

where for φ = 0 evidently squeezing exists in the X1 quadrature. For φ = π/2, thesqueezing appears in the X2 quadrature. In general, for an arbitrary angle φ, thesqueezing will appear along the x-axis of a frame rotated by an angle φ. A representationof the squeezed vacuum state where the fluctuations in one of the quadratures is reduced
is shown in figure 2.8. The states generated by the parametric process of a lossless DPA
are ideal squeezed states. However, in general, squeezed states do not need to equalize
the uncertainty relation (2.85).

Chapter 3
Theory of Josephson parametric amplifiers

In this chapter I introduce a generic form of the Josephson parametric amplifier. I
start by discussing the driven Duffing oscillator. I will then give a brief description of thenonlinear circuit element we will use to create our parametric amplifier: the Josephson
junction. Finally, I will discuss the Josephson non-linear oscillator in an electric circuits
language, and its behavior as an amplifier, including the necessary conditions that it
must fulfill in order to be a useful parametric amplifier. In this last section I will
describe how the analysis of the Josephson parametric amplifier can be discussed using
IOT and discuss the equivalence of both approaches, IOT and circuit theory. I will thenderive expressions for the gains of the JPA, showing how it satisfies the condition for adegenerate parametric amplifier, Eq. (2.83). I will discuss some technical aspects of theamplifier, such as bandwidth and added noise, as well as squeezing of the internal andoutput modes.

3.1

Non-linear driven oscillators (Duffing oscillator)
The nonlinear system we will be studying is the Duffing oscillator:ẍ + 2γ ẋ + ω02 x + βx3 = F cos(ωt).

(3.1)

In mechanical terms, the left hand side (lhs) of the Duffing equation can be thoughtof as a damped nonlinear spring; a hard spring for β > 0 and a soft spring for β < 0.

40
In the next section, I will show that the Josephson non-linear oscillator is equivalent
to a soft spring Duffing oscillator, consequently I will assume β < 0 in the rest of thechapter. In a typical problem of forced oscillations in a linear RLC circuit like the one
studied in chapter 2 (Duffing equation with β = 0), the standard approach used to findthe solution of the EOM is to assume that enough time has passed so that we can ignore
the transient solution and seek the steady state periodic solution vibrating at the same
frequency ω as that of the driving force. A nonlinear oscillator can respond in many
additional ways that are not possible for a linear system. Nevertheless, we begin ourstudy of forced duffing oscillator with a similar approach to that of the linear oscillator
by seeking the harmonic solution which has the same frequency as the driving term.
In order to obtain the approximate analytic form for the harmonic solution, we
will used the “iteration method” [29]. We write the Duffing equation in the followingform:ẍ = −2γ ẋ − ω02 x − βx3 + F cos(ωt)(3.2)

and assume that to lowest (zeroth) order, the solution is given by x0 = A cos(ωt + φ)and substitute it in the rhs of equation 3.2. Then we integrate the resulting equation
generating a first order solution x1 (t) which should represent a better approximationof x(t) provided β satisfies certain conditions.1 The iteration procedure can be thenrepeated until a sufficiently accurate solution for x has been achieved (after n of suchiterations, the solution x(t) ' xn (t)).
Now to keep the analysis simple, I will temporarily set the damping coefficient γ
equal to zero.
ẍ = −ω02 x − βx3 + F cos(ωt).

(3.3)

Since there is no loss, the response is expected to be either in phase or 180◦ , so I can
1

The integration constants that result from the integration of the rhs are set equal to zero in order
to avoid the appearance of secular terms which would spoil the periodicity of our desired solution [29].

41
safely assume that x0 = A cos(ωt). From Eq. (3.3), we obtain
µ
¶
3 3
1
2
x¨1 = − ω0 A + βA − F cos(ωt) − βA3 cos(3ωt).
44This equation is easily integrated twice from which we obtain
x1 (t) = A1 cos(ωt) +1 βA3
cos(3ωt)36 ω 2

(3.4)

where
A1 ≡

3
1 2
(ω0 A + βA3 − F ).
2
ω
4

Now, x2 (t) can be derived from x1 (t) and so on. I won’t derive this long series, butrather argue that for small nonzero β, if x0 (t) = A cos(ωt) is a reasonable zeroth orderapproximation, then the coefficient A1 in x1 (t) should differ very little from A, so weset A = A1 , and A now is solved from the following expression,¶µ
3β 2
A A = F.
ω02 − ω 2 +4Then Eq. 3.4 reduces tox1 (t) ≈ A cos(ωt) +1
βA3
cos(3ωt).
36 ω02 + 43 βA2 − FA

The condition that the zeroth order solution is close to the full solution is equivalent tosaying that the harmonic term in Eq. 3.4 is much larger than the third harmonic term.
This can be expressed as a restriction on the size of β,
βA2
¿ 1.
36ω 2

(3.5)

I will return to this restriction when discussing about the requirements on the size ofthe non-linearity required for a good parametric amplifier.2
I will now relax the assumption that γ = 0. Just like in the driven damped simple
harmonic oscillator, when damping is included the displacement and the driving force
2

The actual expression for the third-harmonic correction is slightly different [30]. In order to obtainit, we just need to do the next iteration keeping both the harmonic ω term and the next 3ω term.
However, for our purposes βA2 /36ω 2 is close enough and gives us a good order of magnitude restriction
on β.

42
will be out of phase, with an angle between 0 and 180◦ . The approximate series solution
is similar to the lossless case, only adding the possibility of a phase shift between the
response and the drive. However, in order to make the derivation simpler we fix thephase of the response and leave the phase of the driving force to be determined:x¨1 = −2γ x˙0 − ω02 x0 − βx30 + F cos(ωt + φ),= −2γ x˙0 − ω02 x0 − βx30 + H cos(ωt) − G sin(ωt),
with x0 (t) = A cos(ωt). Again integrating the rhs yields
x1 (t) = A1 cos(ωt) + B1 sin(ωt) +1 βA3
cos(3ωt)36 ω 2with
1 2 3 3
(ω + βA − H),
ω2 0 41≡ − 2 (2γωA − G).
ω

A1 ≡
B1

If the zeroth order solution is a reasonable starting point, then A1 ' A and B1 = 0.
We can solve for H and G, and remembering that F 2 = G2 + H 2 , to this order, we canderive an expression that relates the amplitude A to the driving force F and the phase
φ:
G
2γω
= 2
,
H
(ω0 − ω 2 ) + 43 βA2
µ
¶
3 3 2
2
2
=
(ω0 − ω )A + βA
+ (2γωA)2 .
4tan(φ) =

(3.6)

F2

(3.7)

We plot the solution for the amplitude and phases given by Eq. (3.7) and (3.6)
in figure 3.1. The nonlinear resonance curves look similar to those corresponding tolinear resonance curves. For a small drive, the response curve is the familiar Lorentzian
response of the simple harmonic oscillator. For large drives, the resonance curves tilt
to the left; this is equivalent to saying that the nonlinearity has made the resonanceof the system depend on the amplitude of the internal field A. In the high-Q limit

43

A

8

4

0
0.95

1

1.05

ΩΩ0
Figure 3.1: First order nonlinear response curves of a duffing oscillator with finite
damping for 6 different driving forces. The nonlinear resonator variables are Q = 100,
β = −0.001, and the driving forces are equally spaced between 0.25Fc and 1.5Fc . The
resonance curves bend over as the drive amplitude is increased and become multi-valued
for certain drive amplitudes.

(Q = ω0 /2γ),3 the following approximations can be made: ω0 + ω ≈ 2ω0 and ω/ω0 ≈1. These are equivalent to the “slowly varying envelope approximation” explained in
section 2.3.2. Eqs. (3.6) and (3.7) reduce to2γω0,2ω0 (ω0 − ω) + 34 βA2
µ
¶
3 22
2
=
[2ω0 (ω0 − ω) + βA ] + (2γω0 ) A2 .
4tan(φ) =

(3.8)

F2

(3.9)

I can verify that the analysis of the duffing oscillator is reasonable by setting the nonlinearity parameter to zero in Eq. (3.9). With β = 0, the expression obtained for A2 isthe familiar Lorentzian response of a high-Q SHO with a resonance frequency ω0 and alinewidth (HFHM) of γ. In the case of β 6= 0, we can define the resonance frequency as
3

In the previous chapter, the quality factor was represented by Q, in order to avoid confusion withthe charge variable Q. However, since that will not be a problem here, I will go back to the usual
convention of representing the quality factor with the variable Q.

44
the maximum of the response curve (dA/dω = 0). This condition yields
ω0 − ω +

3β A2
=0
4ω0 2

(3.10)

that is, the peak of the resonance curve is shifted by an amount given by 3βA2 /8ω.
Note also that for a drive above a critical force value Fc , curves shown in figure 3.1 have
two points at which the slope dA/dω is infinite. It can also been shown that the region
lying between these two points is unstable while the regions outside these two points
correspond to stable solution (Fig. 3.2). This behavior leads to bistability. When we
operate our parametric amplifier, we will operate below the critical point4 so I will nottreat the region of bistability. References [18, 29, 31] and references therein treat thebistable region.

8

A

æ

4

æ

0
0.95

1

1.05

ΩΩ0
Figure 3.2: Nonlinear response for a duffing oscillator and a driving force F = 1.5Fc
such that for certain pump frequencies the response is multi-valued. The red-dashedsection represents the unstable solution of the resonator. The dots represent the points
where the slope dA/dω is infinite.

4
Large parametric gain is achieved close to the points where the slope dA/dω = ∞, but in orderto remain stable we need to operate below the critical point with the parameters drive frequency andpower chosen so the Duffing curve does not have a bistable region.

45
We can estimate the critical drive by looking for the points that satisfydω/dA = 0.

(3.11)

This would give me the two points of instability shown in figure 3.2. In addition, we needd2 ω/dA2 = 0 for the two points of instability to coincide. When these two conditions are
satisfied, it is straightforward to derive the following expressions for the critical drive,
critical amplitude and critical frequency detuning:ωc = ω0 −
Fc2 =
A2c =

√
3γ,

256γ 3 ω03√,
9 3 |β|16γ ω0√.
3 3 |β|(3.12)(3.13)(3.14)

So far I have discussed the response of a nonlinear duffing oscillator without
discussing the origin of the nonlinearity. In the next section I will describe the way wecreate our nonlinear resonators.

3.2

General theory of Josephson junctions
As described in the previous section, in order to create a parametric amplifier weneed a non-linear element embedded in an electrical resonator. This nonlinear element
will provide the coupling between the two normal modes of the resonator. In this section
I explain the nonlinear circuit element we will use: the Josephson junction [32].

3.2.1Simple junction theory

The Josephson junction is made out of two superconducting electrodes separated
by a thin insulating barrier (Fig. 3.3a). In a superconducting metal, the electrons bind
together forming a bosonic condensate of Cooper pairs whose behavior is described bya single wave function Ψ(r) = |Ψ(r)|eiϕ(r) . Due to the small thickness of the insulating
barrier, both wave functions of the superconducting electrodes can overlap, allowing

46

(a)

(b)

S
I

2e

I0

CJ

S

Figure 3.3: (a)A Josephson junction consists of two superconducting leads separated
by an insulator layer. Cooper pairs can tunnel through the barrier, generating a dissipationless supercurrent. (b) Circuit element that represents a Josephson junction. In
general, a Josephson junction is pictured as a tunneling element and a capacitor inparallel. This means that the Josephson junction has an intrinsic resonance frequency,
called the plasma frequency. This plasma frequency is usually in the tens of Gigahertz,
so for our purposes, this extra resonance will be neglected in the analysis.

the tunneling of Cooper pairs. The constitutive relations that describe the current of
Cooper pairs across the barrier are
I(t) = I0 sin(δ(t)),
Z t1
dt0 V (t0 ),δ(t) =
ϕ0 −∞(3.15)(3.16)

where I is the current, V the voltage and δ = ϕ2 − ϕ1 the gauge-invariant phasedifference5 across the junction. In these expressions ϕ0 = ~/2e is the reduced flux
quantum, and I0 is the junction critical current which gives the maximum current thatthe junction can support in its superconducting state. At zero magnetic field, the criticalcurrent is given by the Ambegaokar-Baratoff relation
I0 =

∆(T )
π∆(T )
tanh(
)
2eRn
2kB T

(3.17)

5

The concept of generalized flux quanta that was introduced in the previous chapter in Eq. (2.18)
is closely related to the condensate phase ϕ at a given node of a superconducting circuit. The twoquantities satisfy the relation
ϕ = φ mod 2π.
For more details on the definition of the generalized flux and its relation to the superconducting phase
ϕ for superconduction circuits, see Ref. [12].

47
where ∆(T ) is the superconducting gap of the material used for the electrodes and Rn
is the normal state resistance of the junction. We operate at temperatures much smaller
than the superconducting critical temperature, so we use the T = 0 limit of Eq. (3.17),
I0 = π∆/2eRN .
Now let’s consider the Josephson junction effective impedance. There is someambiguity on the definition of the Josephson junction inductance LJ . We start with
V = ϕ0 dδ/dt and I(t) = I0 sin(δ). Solving for I in terms of the gauge-invariant phasedifference δ, the voltage can be expressed asV =d(ϕ0 arcsin(I(t)/I0 )) .
dt

(3.18)

Here is where the two different definitions in the usual literature appear. If we define
the total Josephson inductance LJ as the ratio between the flux ΦJ (t) = ϕ0 δ and thecurrent I(t), LJ ≡ ΦJ (t)/I(t), as done in Refs. [8, 33, 34], we obtain
LJ = LJ0

arcsin(I/I0 )I/I0(3.19)

with
LJ0 = ϕ0 /I0 .

(3.20)On the other hand, we could perform the differentiation in Eq. (3.18), which leads us
toV=

ϕ0 /I0pdI

1 − (I/I0 )2 dt,

LJ0 dI,cos(δ) dt
dI= L0J ,
dt=

where the other definition for the Josephson inductance
L0J =

LJ0
cos(δ)(3.21)

is then the ratio of the V and dI/dt. Either definition can be used as long as the meaning
is understood. In a sense, the definition of L0J can be considered the effective differential

48
inductance. Depending on the context one definition or the other can be more useful.
For example if I wanted to know the effective impedance of the Josephson junction whenprobed with a small rf tone, L0J would be more useful. However, to calculate the flux Φacross a junction, LJ is more convenient.
I will now make a few remarks about the energy associated with the Josephson
junction [35]. Consider a Josephson junction connected to a current source. The energychange due to the work done by the current generator is given by dE = I(t)V (t)dt,
where I is the current imparted by the source and V is the voltage across the junction.
Using Eq. (3.18), the total energy associated with a biased Josephson junction is given
by:
Z
E =

I(t)

d
(ϕ0 arcsin(I(t)/I0 )) dt,
dt= −EJ cos(δ(t)) + constant(3.22)

where in the second line I have used Eqs. (3.15) and (3.16), and
EJ = ϕ0 I0 .

(3.23)

The constant of integration is usually ignored as it does not influence the dynamic of
the junction.

3.2.2

dc-SQUID

A dc-SQUID (Superconducting Quantum Interference Device)) is created by
combining two Josephson junctions in parallel, as shown in figure 3.4. When a magnetic
field is applied on the SQUID, the maximum supercurrent through the SQUID showsa periodic behavior. We now explain that behavior. The total current I though the
SQUID is the sum of the current though both branches, and using Eq. (3.15) this isequal toI = I01 sin(δ1 ) + I02 sin(δ2 ).

(3.24)

49

I1 = I 01 sin( δ1 )I = I1 + I 2
L1IΦ = Φ ext + Lloop I circΦ

Lloop = L1 + L2
L2

Φ ext = BA
I 2 = I 02 sin( δ2 )Figure 3.4: A SQUID is formed by two Josephson junctions embedded in a superconducting wire loop. This loop has an area of A and an intrinsic inductance given by
Lloop = L1 + L2 . A magnetic field perpendicular to the loop of magnitude B is applied.
The total flux Φ in the SQUID loop consist of the self-induced and applied flux.

Due to the fact that the total magnetic flux though a superconducting loop must be
quantized, the relation between the gauge-invariant phase differences of each junctionisδ2 − δ1 = 2πn + 2πΦ.
Φ0(3.25)

Eqs. (3.24) and (3.25) can be simplified if some assumptions are made regarding thecritical currents of the Josephson junctions and the loop inductance.
Let’s consider first a SQUID that has asymmetric junctions
I = I01 sin(δ1 ) + I02 sin(δ1 − 2π

Φ
).
Φ0

(3.26)

The analysis is greatly simplified in the case when the loop inductance is negligible. Inthat case, the flux threading the SQUID loop only comes from the field Φext applied(Φ = Φext ). At a given Φext , the maximum current, and therefore the equivalent criticalcurrent of the SQUID I0S is found by maximizing Eq. (3.26) with respect to δ1 . This
yieldss
I0S =2 + I 2 + 2I I cos
I01
01 02
02

µ

¶
2πΦext.
Φ0(3.27)

50
If the two junctions have identical critical currents, I01 = I02 = I0 , Eq. 3.27 reduces to
¯
µ
¶¯
¯
πΦ ¯¯
S
¯
I0 = 2I0 ¯cos.
(3.28)
Φ0 ¯The dependance of the SQUID critical current with magnetic field is shown in figure 3.5.
From expressions (3.27) and (3.28), we see that the SQUID will behave as a junction
with tunable current. In the case of a SQUID with asymmetric junctions, the only
difference is that the modulation of the current is not as deep as in the symmetricjunctions case. The effective SQUID inductance is just given by Eq. (3.19) with I0
replaced by I0S .

0

I S/I

01

2

1

0
−2

−1

0
flux (Φext/Φ0)

1

2

Figure 3.5: Flux dependance of the critical current of a SQUID for the case of negligible
loop inductance. The two traces shown for the case of I01 = I02 (red) and I01 = 2I02
(blue).

When we include the loop inductance, it turns out that solving exactly for the flux
dependance of the maximum current I0S is not possible. It must be found numerically.
For simplicity, let’s assume that I01 = I02 = I0 , and the loop inductance Lloop is
symmetrically divided in the loop (L1 = L2 = Lloop /2). If the current divides equallyin the loop, there is no circulating current and the fluxes created by both current willcancel. In the general case when this does not happen, I1 6= I2 , we can define a

51circulating current given
1
Icirc = (I2 − I1 ).
2
This current produces a flux given by ΦL = Lloop Icirc . The total flux in the loop is thesum of the applied flux and the self-induced flux:1Φ = Φext + Lloop I0 {sin(δ2 ) − sin(δ1 )}2
The phase differences δ1 and δ2 are related by Eq. (3.25):
δ2 = δ1 −1
2π[Φext + Lloop I0 {sin(δ2 ) − sin(δ1 )}]Φ0
2(3.29)In order to obtain I0S , we need to solve Eq. (3.29) and (3.26) self-consistently, andmaximizing with respect to δ1 . The flux dependance is shown in figure 3.6a. Again,
the modulation of I0S is not complete. For small Lloop , the minimum critical current isSapproximately given by I0,min= πLloop I0 /Φ0 , as can be observed in figure 3.6b. When

we include the loop inductance in the problem, it turns out that the nonlinear Josephson
inductance is not exactly given by just Eq. 3.19 with I0 just replaced by I0S , but for theparameters of our SQUIDs, it is a good approximation.

3.3

Josephson junction oscillator
Now that I have explained how a Josephson junction can be considered a nonlinear

inductor, we can start analyzing the RLC-nonlinear resonator that includes a Josephson
junction as the inductor. In this section I will first write the EOM that describes thesystem. I will approximate the non-linearity of the Josephson inductance to include only
the first non-linear order; this approximation reduces the EOM of the RLC-nonlinear
resonator to the duffing equation (3.1). Then I will analyze the system in a circuit
theory language similar to the one used in section 2.3.1 following closely Refs. [18]and [31]. Before I calculate the gain performance of the amplifier, I will analyze theJosephson Duffing oscillator problem applying IOT (2.3.3) on a completely equivalent

52

(a)

(b)
2

I /I

I0S

S
0 01

2

1

0
−1

1

−0.5
0
0.5
flux (Φ /Φ )
ext

0

1

0
0

πL

5

10
I /Φ

Loop 00

Figure 3.6: (a) Flux dependance of the critical current of a SQUID for the case offinite loop inductance (πLloop I0 /Φ0 = 0.5) and junctions with the same critical current(blue). For comparison, the negligible loop inductance case is also plotted (red). (b)The minimum (cyan) and maximum (blue) SQUID critical currents as a function ofSloop inductance. The case shown in figure (a) are represented by a circle (I0,max) and aS
cross (I0,min ). For small loop inductance, the minimum critical current is approximately
a linear function of Lloop (green).

system of a cavity with a Kerr nonlinearity showing that both approaches are completely
equivalent. I will then derive expressions for the gain of the amplifier, bandwidth andnoise temperature. Finally I will discuss the requirements the resonator needs to fulfill
in order to be a usable JPA.

3.3.1

Circuit theory of a driven Josephson junction in a RLC-resonator

The nonlinear Josephson oscillator is shown in figure 3.7a. The equation that
describes its behavior is similar to Eq. (2.33) except that the current through the linear
inductor is replaced by IJ = sin(Φ/2π) = sin(δ), where I will be using δ = Φ/ϕ0 , thesuperconducting gauge-invariant phase across the junction as my generalized coordinate:Cϕ0

d2 δ(t)+ I0 sin(δ(t)) = I(z = 0, t)
dt2

53
z=0

(a)

(b)
IJ

I(z=0,t)
RS=Zc
VS (t)

IJ
Zc

V δ

C

IC

IC

IS (t)

I0

Zc δ

I0

C

Figure 3.7: (a) Josephson junction with a critical current I0 shunted by a capacitor C
connected to a microwave source through a transmission line with an impedance Zc .
The microwave source has a source resistance RS = Zc . (b) Norton equivalent of thecircuit shown in (a).

We can use the Norton equivalent of the circuit (Fig. 3.7b), which yields the following
equation for a current source driving a non-linear RLC resonator:
Cϕ0

d2 δ(t) ϕ0 dδ(t)+ I0 sin(δ(t)) = IS (t).
+
dt2
Zc dt

(3.30)

where Zc is the characteristic impedance of the transmission line, I0 is the criticalcurrent of the junction and C is the total capacitance shunting the junction and IS (t)is the microwave power going into the resonator. It will consist of two components: alarge pump and a small signal IS (t) = Ip (t) + is (t). We now approximate the sin(δ)
term by its first two component of the Taylor expansion.
µ
¶
Ip
dδ(t)
δ 3 (t)2d2 δ(t)2+ 2γ+ ω0 δ(t) −− ω02 cos(ωp t) =v in (t)
dt2
dt
6
I0
Cϕ0 Zc s(3.31)

where I have written the signal component in terms of the voltage waves coming into
the resonator from the transmission line. From the point of view of microwave circuits,
it is more convenient to use the incoming and outgoing modes, especially when I analyze
the gain of the amplifier and quantize the noise coming in from the input port. The firsttask is to analyze the effect of the large pump on the circuit, so I will set vs = 0. Eq.
(3.31) is just the Duffing equation we studied in section 3.1, with x → δ and β → −ω02 /6.
Using the same procedure as shown in that section, we assume a zeroth order solution
δp (t) = δa cos(ωp t − φ) and solve of the amplitude δa for a given drive F = ω02 Ip /I0 .

54This is shown in figure 3.8. As in the Duffing oscillator, the resonant response shifts tolower frequency as the pump amplitude is increase. Using expressions 3.12-3.14, I canwrite the equivalent critical expressions for the case of the Josephson resonator:
ωc = ω0 −
2
Ip,c
=
2
δa,c
=

√
3γ,

(3.32)

512γ 3 264√ 3 I0 = √
I02 ,
3
3 3ω0
3 3Q
32γ
16
√
=√ .
3ω0
3Q

(3.33)(3.34)∆a 2Π

0.1

0.05

0
0.95

1

1.05

ΩΩ0
Figure 3.8: Frequency response of a driven Josephson oscillator as a function of pump
detuning ωp /ω0 for multiple drive powers. The driving amplitudes of the incoming
microwave pump are equally spaced between 0.25Ip,c and 1.5Ip,c . The quality factor is
Q = 100.

We incorporate the signal by treating it as a small perturbation around the solution of the pump (this is equivalent to making an undepleted pump approximation);that is, δ = δp + δs , where δp is the internal field solution associated with the pump andis given by the zeroth order solution of the Duffing equation, and δs is the internal field
associated with the small signal. Substituting the pump solution into Eq. (3.31) andretaining terms only to first order in δs , the following equation for the dynamics of δs

55
is obtained [18]:2d2 δs (t)dδs (t)+ ω12 (δa , t)δs (t) =+ 2γv in (t)dt2dt
Cϕ0 Zc swhereδ2ω12 (δa , t) ≡ ω02 (1 − a )
4

µ
1+

¶
δa2
cos(2ωp − 2φ)4(1 − δa2 /4)(3.35)(3.36)and φ is given by Eq. (3.6). Eq. (3.35) describes the dynamic of our small perturbation
δs and consist of a parametrically driven oscillator [13,18,30,31,36], where the resonancefrequency of the system ω(t) = ω02 (1 + h cos(νt)) is a function of time (h ¿ 1 ). It
explains the origin of the parametric gain we will see in the signals coming in. For aparametric resonance to be the strongest, the frequency ν needs to be close to twice
ω0 [36]. This of course occurs naturally in our case if our pump is close to the resonancefrequency of the oscillator ωp ' ω0 + δω where the detuning δω is of the order of theresonator linewidth γ.

3.3.2

Parametric processes in the Josephson junction oscillator

We are now ready to start the derivation of the expressions that describe theamplifier performance of the JPA. In the following paragraphs, we will see that whena signal is applied at a frequency ωs , an additional tone will be created at a frequency
ωi = 2ωp − ωs , called the image tone. The creation of this extra tone is a consequenceof the coupling between normal modes in a parametrically driven oscillator. Because
our pump is at a frequency close to our signal tone, the mode of operation of our JPA
is sometimes called four-wave or four-photon mixing [8] since four photons are involved:
one at ωs , one at ωi and two photons of the pump at ωp . Another mode called the threewave or three-photon mixing also exists where ωs + ωi = ωp . This requires the energy ofthe non-linear element to have at least some odd symmetry in the pump current. This
could be achieved if we were to apply a dc-current to the Josephson junction [8, 37]. As
we did not operate our amplifiers in this mode, I will not discuss this case.

56
The solution for Eq. (3.35) is easily found in the frequency domain. We define
for our internal fieldZ ∞δs (t) =0dωδs (ω)e−iωt + H.c.
2π

Taking the fourier transform of Eq. (3.35) and using the convolution theorem yields:√ r
ω02 ² 2iφ †4γ Zc ~ω22
(ω0 (1 − ²) − ω − i2γω)δs (ω) −e δs (2ωp − ω) =ain (ω)(3.37)2
ϕ02
where I have used Eq. (2.39) for the input signal vs (t) and defined ² = δa2 /4. In
deriving equation (3.37), I have used the rotating wave approximation in order to ignore
terms whose frequency is close to 3ω0 . Before we move on, I will make the samehigh-Q approximation, which essentially is equivalent to the “slowly varying envelope”
approximation described in section 2.3.2 (ωp + ω ≈ 2ωp , ω/ω0 ≈ 1). I will also define
the internal c(ω 0 ) and incoming cin (ω 0 ) mode operators in the rotation frame of the√pump δ(ωp + ω 0 ) = ~ω0 2EJ c(ω 0 ) and ain (ωp + ω 0 ) = cin (ω 0 ). In order to make notation
simpler, I will drop the prime symbol from the ω variable, but note that from now on,ω represents a fourier frequency offset from the pump frequency ωp . Eq. (3.37) reducesto³
ω0 − ωp − ω −´pω0 ² −2iφ †ω0 ²− iγ c(ω) −ec (−ω) = 2γcin (ω).
2
4

(3.38)

Eq. (3.38) has a very important feature mentioned in the previous chapter: the pumpof this parametric driven system will actively couple the two normal modes c(ω) andc† (−ω) [13]. We can now solve for the internal field c(ω) in terms of the input fields cin (ω)
and c†in (−ω). However, before I do that I will describe how to derive this expression
within the Input-Output theory framework.

3.3.3

Kerr-medium equivalence of a Josephson junction oscillator

So far we have analyzed the RLC-resonator in a simple circuit language. In
section 2.3.3, I described the method of Input-Output theory that allows to analyze
the same system in an equivalent way. The language used to describe such nonlinear

57
systems is slightly different and I discuss it here following closely Refs. [17,38,39]. I put
special attention in showing the equivalence between the two methods.

nonlinear cavity

aout(t)test portγ

H = hωA† A +ain(t)hK † †A A AA
2

Figure 3.9: A nonlinear cavity is driven by an open port characterized by a linewidthγ = ω/(2Q). The incoming power is given by Eq. (2.53), Pin = ~ωhb†in (t)bin (t)i, with asimilar expression for the outgoing power.

Consider Eq. (3.1). Ignoring dissipation, the system can be described by the
HamiltonianH=mω02 2 mβx4
p2
+
x +
.
2m
2
4

(3.39)

Using the expressions for the normal modes, Eq. (2.11), this Hamiltonian may be
written in the form ofK(A + A† )4 ,12K
' ~ωA† A + ~ A† A† AA,
2

H = ~ω0 A† A + ~(3.40)

where I have dropped the zero-point motion term since it does not affect the dynamics
of the system and made the rotating wave approximation to ignore rapidly oscillating
terms in the quartic term. The constant K is referred to as the Kerr constant. In thiscase, it is given by
K=

3 ~β
.
4 mω02

Now lets consider the case when we couple this nonlinear resonator to a port (Fig. 3.9).
Using the IOT theory described in section 2.3.3, we can write the EOM for the cavity
mode A aspdA= −iω0 A − iKA† AA − γA + i 2γain (t).
dt

(3.41)

58
In order to solve this nonlinear problem, we follow an analogue of the procedure described in the previous section. We first solve the problem of a large classical pump,and then linearize the solution for the small signal:
ain = ain,p + ain,s .
Let’s first discuss the classical response to the pump in the absence of any signal. I
write for the incoming and internal fields:
ain,p = bin e−i(ωp t+ψ) ,A = Be−i(ωp t+φ) .
Since I am only interested in the steady state solutions, bin and B are (real) constants.
Substituting these expressions into Eq. (3.41) yields(i(ω0 − ωp ) + γ)B + iKB 3 = −ip2γbin ei(φ−ψ) .

(3.42)

Multiplying Eq. (3.42) by its complex conjugate, one obtains the following equation
where I have defined N = A∗ A = B 2
K 2 N 3 + 2(ω0 − ωp )KN 2 + [(ω0 − ωp )2 + γ 2 ]N = 2γb2in(3.43)

and for the phase between the pump and the internal field:tan(φ − ψ) = −γ.
ω0 − ω + KN

(3.44)

Expressions (3.43) and (3.44) are exactly equivalent to 3.8 and 3.9, with F → 2γb2in ,A2 /2 → N and K → 3β/4ω0 .6 Therefore we expect the same behavior for the internal
field of the resonator as shown in figures 3.1 and 3.8. Defining our output mode field asaout = bout e−i(ωt+ψ)6
Note that these expression have already built in the high-Q approximations, since these approximations are already assumed in IOT.

59
where bout can be complex, we can estimate our reflection coefficient Γ = bout /bin using
the boundary condition Eq. (2.69). Since we have only one port and no intrinsic
loss in the resonator (|Γ(ωp )| = 1), all the information about the resonator is in thephase of Γ(ωp ). This is plotted in figure 3.10 for three different drives where we defineδωp = ωp /ω0 − 1. Equivalent expressions for the critical points are given by:
√ K
3γ
,
|K|
4 γ2
√
,
3 3 |K|
√
2 3γ
.
3|K|

ωp,c = ω0 +(3.45)

b2in,c =(3.46)

Nc =(3.47)

Using Eq. (2.53), we can then calculate the critical power that we need to apply to ourresonator as Pin,c = ~ωp b2in,cphase (degrees)

360

180

0
−0.02

−0.01

δωp

0

0.01

Figure 3.10: Phase of the reflection coefficient Γ as a function of the pump detuningδωp = (ωp −ω0 )/ω0 for a very low power, 0.5bin,c , 0.9bin,c and 1.25bin,c . For bin = 1.25bin,c ,the unstable solution is shown as a dashed line. The quality factor is Q = 100.

Now, we linearize the equation for a small incoming signal δain . We write the60
incoming and internal fields as
ain = bin e−i(ωp t+ψ) + cin (t)e−iωp t ,aout = bout e−i(ωp t+ψ) + cout (t)e−iωp t ,A = Be−i(ωp t+φ) + c(t)e−iωp t ,and keep only terms linear in cin and c. B and bout constitute the solution to the classicalpump in the absence of any signal. In this case, c(t) and cin (t) are the internal mode andincoming mode operators in the pump rotating frame and are not constants. Replacingthese expressions in (3.41), we obtain the following linearized equation for c [38]:pdc= [−i(ω0 − ωp ) + γ]c − 2iKN c − iKN e−i2φ c† − i 2γcin .
dt(3.48)

Eq. (3.48) describes the dynamics of the internal field and it is equivalent to Eq. (3.35),
in the Markovian and “slowly varying envelope” approximation. The equivalence ofboth expressions can be more easily seen if we transform Eq. (3.48) into the frequency
domain:[(ω0 − ωp ) − ω − iγ + 2KN ]c(ω) + KN e−i2φB c† (−ω) =p2γcin (ω).

(3.49)

Eqs. (3.38) and (3.49) can be made identical by replacing KN with its circuit equivalent−ω0 ²/4. So far I have redone the analysis for a Duffing oscillator circuit using quantum
network theory and shown that within the high-Q approximation it is exactly equivalent
to treating a system with a Kerr nonlinearity using IOT. Generalizing the system toinclude other nonlinearities as well as loss is much easier using IOT; thus, I will follow
this method. Before continuing, let’s defineW= i(ω0 − ωp ) + γ + 2iKN

(3.50)V= iKN e−2iφ(3.51)

Then Eq. (3.49) reducesp−iωc(ω) + W c(ω) + V c† (−ω) = i 2γcin (ω).

61
We can solve for the internal field in terms of the input field:´√ ³
i 2γ cin (ω)(W ∗ − iω) − V c†in (−ω)c(ω) =,(−iω + λ0 )(−iω + λ1 )= ν(ω)cin (ω) + µ(ω)c†in (−ω),
with(3.52)(3.53)q
λ0/1 = γ ∓K 2 N 2 − (ω0 − ωp + 2KN )2 .

(3.54)

It is important to notice that λ0 tends to zero as one approach a point of infinite
susceptibility, Eq. (3.11). This trend represents the slowing dynamics as the parametricamplifier approaches the critical power. I will come back to this point when I discuss
the bandwidth of the amplifier.
Parametric gain

Using the boundary condition Eq. (2.56), I can solve for the output field cout .
µcout (ω) =−1 +¶2γ(−iω + W ∗ )cin (ω)(−iω + λ0 )(−iω + λ1 )2γV+c† (−ω)(−iω + λ0 )(−iω + λ1 ) in

= G(ω)cin (ω) + M(ω)c†in (−ω)(3.55)(3.56)

where G and M are accordingly defined. It is straightforward to prove that G(ω) andM(ω) satisfy the expected relations for a degenerate parametric amplifier [2, 8]:|G(ω)|2 − |M(ω)|2 = 1,

(3.57)

G(ω)M(−ω) = G(−ω)M(ω).

(3.58)As explained at the beginning of this section, in a degenerate parametric amplifier, fora signal at a frequency ω0 + ω another tone at the image frequency ωi = ω0 − ω is also
created. Signal and image gains can be defined by taking the operators cin and cout torepresent classical c-numbers for the input and the output. Let’s define for the signal

62
power gain:
cout (ω)= |G(ω)|2cin (ω)(3.59)

cout (ω)= |M(ω)|2 .
cin (−ω)(3.60)Gs (ω) =and for the image power gain:
Gi (ω) =We plot both signal and image gains for different pump detuning and pumpmagnitude in figure 3.11a and 3.11b. and signal frequency ω = 0. From expressions(3.57) and (3.58), it can be shown that the the signal and image gains are symmetric interms of the fourier frequency ω.
Gs (ω) = Gs (−ω),
Gi (ω) = Gi (−ω).
This is shown in figure 3.11c and 3.11d for both the signal and image gain, respectively.
When ω = 0, both gains will have in the denominator λ20 λ21 . As we approach the criticaldetuning and critical drive, both gains will diverge. It is close to the critical point wheredB/dω diverges that the amplifier achieves the highest gains. Of course, in practice thegain would never diverge; rather, amplified quantum noise would begin to saturate theamplifier.
Bandwidth

As we can observe in figure 3.11c and 3.11d, the bandwidth of the amplifier getssmaller and smaller as we increase its gain. As I mentioned at the beginning of thissection, when we approach a critical point, i.e. for large gains, λ0 sets the bandwidth of
the amplifier. Using Eq. (3.57) in the large gain limit, the expression for Gs ≈ Gi = G
and are approximately given byG(ω) =≈

4γV 2,(ω 2 + λ20 )(ω 2 + λ21 )G(0).
1 + ( B2ωω )2(3.61)

63
(a)

(b)

40

20

Gi (dB)Gs (dB)

40

20

0
0

−20
−2

−1
∆ωp/γ

0

1

(c)

−2

0

1

(d)
40

Gi (dB)40

Gs (dB)

−1
∆ωp/γ

20

0

−2

20

0

−1

0
(ω−ω )/γ

1

2

−2

p

−1

0
(ω−ω )/γ12pFigure 3.11: (a) and (b) Signal and image gain as a function of pump detuning ∆ωp =ωp − ω0 for a signal frequency of ω = 0 for and incoming power b2in equal to 0.5bin,c ,
0.8bin,c , 0.9bin,c and 0.99b2in,c . (c) and (d) Signal and image gain as function of signal
frequency ω. The pump detuning was chose by finding the maximum gain from plots(a) and (b). For b2in ' b2in,c , the optimal pump detuning is given by Eq. (3.45).

The 3-dB bandwidth is Bω = 2λ0 . Substituting in Eq. (3.54) expressions that are
correct close to a critical point, Eqs. (3.47-3.45), yields
4γBω = √ p.
3 G(0)(3.62)

This expression shows that the bandwidth of the amplifier is inversely proportional to the√amplitude gain G. This gain-bandwidth compromise is a common property of many
amplifier, including parametric amplifiers both degenerate and non-degenerate. Becauseparametric amplifiers usually operate only in a narrow band around a center frequency,

64
we decided to develop parametric amplifiers with widely tunable center frequencies.
Squeezing

Squeezed state generation involves the creation of correlations between the signal
ωp +ω and image ωp −ω frequency. When squeezed states are created their spectrum cango below the vacuum/thermal noise spectrum for a properly chosen phase quadrature.
When describing the generation of squeezing in a cavity parametric amplifier like theone I am studying, I need to make a distinction between the internal field and theoutput field. Confusion regarding this distinction was the origin of the erroneous belief
that cavity parametric amplifiers were bad for squeezing, as early work in the fieldclaimed that the maximum amount of squeezing was 50% [40]. However, this limit
only applies for the internal field. It was shown that for the output field the squeezing
could be arbitrarily large. This large squeezing in the output field is a consequenceof an interference effect between the reflected field and the squeezed field exiting thecavity [17, 24, 40, 41].
The kind of squeezing the JPA performs is quadrature squeezing, as explained inchapter 2. From Eq. (2.73), the intracavity quadrature operators are given by1
X1 (t) = (a(t) + a† (t))
2
1
X2 (t) = (a(t) − a† (t))
2i
In the following calculations, I will assume that the incoming noise from the test port
shown in figure 3.9 consist of only thermal or vacuum noise. This implies that in a
linear cavity, the variances of both quadratures would beh∆X12 i = h∆X22 i =

1
~ω
coth(
)
4
kB T

(3.63)

Now let’s consider our degenerate parametric amplifier. The amplified quadrature is
not necessarily neither X1 or X2 but maybe a combination of them. For that reason, it

65
is more convenient to define a rotated quadrature given by
1Xθ (t) = (e−iθ a(t) + eiθ a† (t))
2(3.64)

Transforming Eq. (3.64) into the frequency domain and into a rotated frame aroundthe pump frequency the quadrature operator is given by
1Xθ (ω) = (e−iθ c(ω) + eiθ c† (−ω))
2(3.65)

where ω represents a detuning with respect to the pump frequency ωp (see appendix
A). We can then calculate Xθ (ω) in terms of the input noise using Eq. (3.53). In orderto calculate the spectral density of the quadrature operators, we need to calculate theanticommutator of X(ω) such as in the case of the spectral density of ain (ω) in Eq.
(2.26) [2, 8]. However, within the high-Q approximation, Eq. (2.26) is approximated byh{ain (ωp + ω), a†in (ωp + ω 0 )}i ' 2π coth(~ωp)δ(ω − ω 0 ),2kB T

= 2πSa,a δ(ω − ω 0 ).

(3.66)(3.67)

In making this approximation, we are assuming that we are dealing with a narrow bandof frequencies ω in the vicinity of ωp . Using Eqs. (3.66) and (3.67), we can then computethe spectral density of the quadrature operator
SXθ ,Xθ (ω)2πδ(ω − ω 0 ) = h{Xθ (ω), Xθ† (ω 0 )}i(3.68)


2
2
2
2
1
 |ν(ω)| + |ν(−ω)| + |µ(ω)| + |µ(−ω)| , 
Sa,a 
= .(3.69)
4−2θ
+2Re[e (ν(ω)µ(−ω) + ν(−ω)µ(ω))]
For the intracavity field, in order to estimate the amount of squeezing, we need tocompute h∆Xθ2 i. This can be computed by integrating SXθ ,Xθ (ω) over all frequencies.
I now redefine X1 as Xθ for θ = θ1 that maximizes the internal quadrature varianceand X2 as Xθ for θ = θ2 that minimizes the internal quadrature variance. Defined
this way, X1 is the amplified intracavity quadrature and X2 is the squeezed intracavity

66
quadrature. We then can derive the expected variances for the both quadratures which
are given by:h∆X12 iZ ∞=h∆X22 i =

~ωp
1
|W |
coth(
)
,
4
2kB T |W | − |V |
0
~ωp
1
|W |
coth(
)
,
4
2kB T |W | + |V |dωSX1 ,X1 =(3.70)(3.71)

where W and V are given by Eqs. (3.50) and (3.51). The limits of integration are basedon the fact that our definition of the quadrature operators in the fourier domain only
include positive frequencies as shown in Eqs. (A.5-A.6). We observe that the maximum
amount of squeezing is given at a critical point given by Eqs. (3.45-3.47), at which|W | = |V |. At this configuration, h∆X12 i diverges and h∆X22 i = 1/8Sa,a . This implies
that the maximum squeezing for the intracavity field is of only 50%(Fig. 3.12).
Now, let’s consider the output itinerant-wave field. In this case our definition
for the quadrature variables are slightly different as we are considering a traveling-wave
which is intrinsically multimode (see appendix A). We define the multimode quadrature
variable using the same convention as in Refs. [8] and [38] given by:
1
Xout,θ (ω) = (cout (ω)eiθ + c†out (−ω)e−iθ ).
2(3.72)

Again, we calculate the spectral density of the output modes by calculating the anticommutator. The result is
2 + |G(−ω)|2 + |M(ω)|2 + |M(−ω)|2|G(ω)|1
SXout,θ ,Xout,θ (ω) = Sa,a .
4+2Re[e−2θ (G(ω)M(−ω) + G(−ω)M(ω))](3.73)

With Eqs. (3.57) and (3.58), we can simplify Eq. (3.73):
1
SXout,θ ,Xout,θ (ω) = Gθ (ω) Sa,a2

(3.74)

where Gθ (ω) is an effective quadrature gain defined as:pGθ (ω) = 2Gs (ω) − 1 + 2 Gs (ω)(Gs (ω) − 1) cos(2θ + 2φ)(3.75)

67
(a)

(b)

2

2

SX , X

1

10

5

10

3

10

2

1

SX ,X

1

10

1

10
0

10

−10

−5

0
ω/γ

5

10

−10

−5

0
ω/γ

5

10

2

(c) 10

1

1

4∆ X2

10

0

10

−1

10

0

0.5
bin/bcin

1

Figure 3.12: Squeezing of the internal field when the incoming thermal noise is that ofa vacuum state (Sa,a = 1). Spectrum for the (a) squeezed and (b) amplified internal
field quadrature for pump amplitudes bin equal to 0, 0.5bin,c and bin,c . In the case ofno pump power, two cases are shown, one for which the quadrature is defined arounda frequency√ωp = ω0 (blue) and for the quadrature defined around the critical detuningωp = ω0 − 3γ (red). For the other two traces, the quadrature is defined around thecritical detuning. (c) Internal field quadrature as a function of incoming pump power.
The pump detuning is defined as the critical detuning. Even at the critical pump
amplitude, the maximum squeezing is only 50%.

where φ here is defined as the angle between the two complex gains G(ω) and M(−ω):G(ω)M(−ω) =p p
Gs Gs − 1ei2φ .

The angle for which the maximum of the gain Gθ (ω) occurs defines the amplified quadrature, while the minimum defines the squeezed quadrature. Let’s define again X1,out as
the amplified quadrature and X2,out as the squeezed one. Squeezing as a function of the68pump power is shown in figure 3.13a. Using the expressions for the output field, Eq.
(3.56), it can be shown that close to a critical point where the maximum of the gain for
X1,out occurs (SX1,out ,X1,out (ω = 0) = ∞), the spectral density of the squeezed output
field is given by
SX2,out ,X2,out (ω) =ω2.
4γ 2 + ω 2(3.76)

Note that this suggest that the output field can be arbitrarily squeezed for frequencies
close to the pump frequency (Fig. 3.13b). Nonetheless, in practice, other issues would
prevent us from achieving arbitrarily high squeezing such as pump depletion or gainstability issues, since while we are squeezing one quadrature, we are amplifying theother one. Noise from the losses and gain saturation will limit the maximum degree ofsqueezing to be below that theoretically calculated here.
For Gs À 1, Eq. 3.75 reduces to two very simple expressions for X1 and X2 :G1 = 4Gs ,(3.77)

1.
4Gs(3.78)

G2 =This shows that the quadrature gain is expected to be 6 dB higher than the signal andimage gains and therefore its gain-bandwidth product will be 3 dB higher as the gain
is a factor of two larger but the effective bandwidth reduces to half.
Added noise

It is characteristic of parametric amplifiers that both a signal and an image mode
exist. The energy in both of these modes or channels grows at the expense of the pumpenergy. This means that either channel can be used for amplification. The effective
noise performance of the amplifier will be different depending on what kind of linear
combinations of input fourier modes (modes labeled by fourier frequency ω) is inputed
as a signal. Only when the input signal is a linear combinations of fourier frequencies
symmetric around the pump and with the right phase the amplifier is noiseless (Fig.

69
(a)

(b)

1

1
2

,X

out

(ω)

out

2

0.5

2

2SX

X ,X

2

0.5

0

10

S

2

X ,X

(ω=0)

10

S

−2

10

0

10

0

0

0.5
bin/bcin

1

0

0

5
ω/γ

1

10
G1

2

10

10

Figure 3.13: Squeezing of the external field. (a) Squeezing of the ω = 0 frequency component of the noise as a function of pump amplitude bin for a constant pump detuningequal to the critical pump detuning (blue), and for a variable pump detuning chosen
to maximize the gain Gs as shown in figure 3.11a. By letting the pump detuning vary
the squeezing is optimized for a pump amplitude lower than the critical amplitude.
(b) Spectrum of the squeezed output field. Inset: Squeezed (blue) and amplified (red)quadrature spectrum for ω = 0 as a function of the quadrature gain GX1 .

3.14a). On the other hand, when the applied signal contains only 1 single frequency
component (Fig. 3.14b), the amplifier behaves as a nondegenerate parametric amplifier,adding at least half a quantum of noise. In order to prove these statements in the caseof an amplifier with a finite bandwidth we need to consider the fact that the quadratureof an incoming mode at a frequency ω is given by an expression similar to that of Eq.
(3.72):
1
Xin,θ (ω) = (cin (ω)eiθ + c†in (−ω)e−iθ )2(3.79)

where the angle θ is defined with respect to the phase of the pump. When the signal isapplied to both channels (Fig. 3.14b), we are exciting a mode like the one defined by(Eq. 3.79). In this case, the gain will depend on the phase θ and the added noise of theamplifier is zero. In the case when the incoming signal is applied to only one channel(Fig. 3.14a), then the gain will not depend on phase, and in this case the amplifier does
add noise. Next I will explained these claims in more detail.

70
(a)

(c)

S(ω)
ωi = −ωs

X2

ω
ωi
pump
(b)

ωs

S(ω)

ω
pump

X1

ωs

Figure 3.14: Spectrum of the two kind of signals applied to the amplifier. (a) For a linear
combination of two fourier-frequency signals symmetric around the pump frequency, theamplifier will act as a degenerate parametric amplifier. (b) For a single tone detuned
a frequency ω from the pump, the amplifier will add at least half a quantum of noise.
This noise enters the amplifier at the image frequency ωi = −ω. (c) When the signal
is a linear combination of two fourier-frequency tones, depending on the phase between
the two tones, the signal can be pictured as the slow amplitude-fluctuations (blue) orphase-fluctuations (green) of a small carrier (red). The carrier can be ignored, since all
the information is in its amplitude or phase fluctuations; however, I draw it for clarity.
If the JPA only amplifies the quadrature labeled as X1 , then only the projection of thesignal (dashed lines) onto the quadrature X1 is amplified. By setting the phase of thecarrier properly, then the added noise of an ideal JPA can be made arbitrarily small.

In the case of a single-fourier-frequency signal, the JPA acts as a non-degenerateparametric amplifier, amplifying the signal regardless of the its phase with respect tothe pump7 and adding the half a quantum of noise Haus-Caves theorem states, Eq.
(2.81). Consider the case when one is only interested in the signal that is at a frequency
ω detuned from the pump. In order to compute the noise performance of the amplifier,
we can calculate the output noise of the amplifier at a frequency ω for an incoming
thermal field with a spectral density Sain ,ain given by Eq. (3.66). From Eq. (3.56), we
7

In fact, in this case the phase between the signal and the pump is not really defined, since they are
not at the same frequency, and this phase evolves constantly with time.

71
can calculate the noise at the output at the frequency ω, Saout ,aout (ω) given by
~ωp
~ωp
~ωp
Saout ,aout (ω) = Gs (ω)
Sain ,ain (ω) + Gi (ω)
Sain ,ain (−ω).
222(3.80)In this expression, it becomes obvious from where the extra noise is coming. Whenlooking at the output spectral density at a frequency ω, the parametric amplifier adds
some extra noise that enters the JPA at the image frequency (−ω). Any noise at afrequency ω is part of the signal so it is not added noise. Using Eqs. (3.57) and (3.58),
we can rewrite Eq. (3.80),
~ωp
~ωp
~ωp
Saout ,aout (ω) = Gs (ω)
Sain ,ain (ω) + (Gs (ω) − 1)
Sain ,ain (−ω),
2
2
µ2
¶
~ωp
~ωp
1
= Gs (ω)
Sain ,ain (ω) + (1 −)Sain ,ain (−ω) (3.81).
2
Gs (ω) 2
Taking~ωp
2 Sain ,ain (ω) as the input noise, then Eq. (3.81) shows that the added noise in

number of quanta is given by·
AN = 1 −¸
1
1
Sa ,a (−ω).
G(ω) 2 in in(3.82)

We make the same approximation as in the previous section, Sain ,ain (ω) = Sain ,ain (−ω) =coth(~ωp /2kB T ). Eq. (3.82) becomes·
AN = 1 −

¸
~ωp
1
1
coth(
)
G(ω) 2kB T

which for the case of T = 0, reduces to the quantum limit of a phase preserving amplifier,Eq. (2.81). On the other hand, note that if we decide to consider the input to be
Sain ,ain (ω), but measure at the image frequency −ω, the effective added noise would be
¸
~ωp
1
1
coth(
)
AN = 1 +G(ω) 2kB T·

which is again the quantum limit, but for a phase conjugating amplifier.
In the case where our signal consist of two tones as shown in figure 3.14a, thenthe gain will depend on the effective phase between the pump and the signal. In orderto quantify the added noise, we can again consider the case when the incoming signal

72
is thermal noise. This is exactly the case treated when analyzing squeezing, so we canuse those same expressions. The only difference is that in order to study the amplified
quadrature, we need to consider the case where the angle is rotated 90◦ compared to thesqueezed quadrature. Again for the incoming signal quadrature spectral density being
given by
SXin ,Xin =

~ωp
1
coth(
)
2
2kB T

the output amplified quadrature noise is given by
SXout,θ ,Xout,θ (ω) = Gθ (ω)SXin ,Xin
where the quadrature gain Gθ (ω) is given by 3.75, where the phase θ is chosen to
maximize the gain. From this expression one thing is obvious: the lack of a term
representing any added noise; thus, the amplifier will not add any noise! This last claim
assumes that the desired information is contained entirely in the amplified quadrature.
For example, if the desired information is encoded as amplitude or phase modulationof a tone (Fig. 3.14b), all of that information is contained in a single quadrature as
the combination of two fourier frequencies, Eq. (3.79), and can therefore be amplified
noislessly. Otherwise, the noise will be amplified more than the signal and then thiswould appear as extra added noise.

3.3.4

Necessary conditions on the amount of non-linearity to have agood parametric amplifier

After solving the equations of the parametric amplifier, it is worth discussing somedetails about the necessary condition to have a device that can perform appropriately.
First, we need to set a limit on how small the nonlinearity can be. Following closely
the line of thought in Ref. [31], consider a slightly different circuit, shown in figure
3.15. The main difference is that I have included an extra inductor in parallel to theJosephson junction to include the possibility of diluting the nonlinearity by adding extra

73

Is(t)

Re

Ce

Le

Lp =

Le LJ 0
Le + LJ 0

LJ0

Figure 3.15: Parallel Josephson junction oscillator. An extra inductance has been added
to model extra reactive elements in the circuit. By added this extra inductance, thenonlinearity is diluted. This is quantified by the participation ratio p = LJ0 /Lp . For
the parallel resonator, the nonlinear Josephson junction is separated in a linear term
LJ0 in parallel with a nonlinear term, following the same convention is an in Ref. [31].

linear inductance (effectively β goes from being −ω02 /6 to being −ω02 /6 × (Lp /LJ0 )).
We use the quantity called participation ratio as defined in [31], p = Lp /LJ0 . Using
equation 3.34, we can define, for a small critical amplitude δc , a critical current given
by Ic = I0 sin(δc ) ' I0 δc . If we want the approximation of the nonlinear inductance tobe accurately modeled by the Taylor expansion up to third order, we need Ic /I0 ¿ 1.
This translates into a restriction on the quality factor and the nonlinearity given by:
4
31/4

1
√
¿1
Qp

(3.83)A second condition we need to consider is related to Eq. 3.35. In section 3.1,
I mentioned that the zeroth order solution for the duffing oscillator would be a good
approximation if Eq. (3.5) was satisfied. However this only sets a limit on how accurate
the zeroth order solution is. There is another concern on this non-linear systems whichis the stability of the solution and the possibility for chaotic behavior. The following
results are from Refs. [30, 31, 36] and are not derived in this thesis.
In solving Eq. (3.1) and (3.31) we have assumed that they have a solution with
only odd multiples of the driving frequency:y(t) =∞
X

A2k−1 cos((2k − 1)ωt + φ2k−1 )k=1with the coefficients A2k−1 decreasing in magnitude with k. The first harmonic is our
zeroth order solution, and its amplitude is given by Eq. (3.7). It has been observed that

74
in the Duffing oscillator, the transition to chaotic behavior can occur via consecutive
period doubling, and period doubling is always preceded by the transition of the solution
from having only odd-harmonics to one containing both odd and even harmonics. In
a sense, the appearance of even-harmonics in the solution is a precursor of chaotic
behavior in the case of the Duffing oscillator [30, 31]. It is usually claimed that we arein the weak nonlinear regime if our solution only has odd harmonics. The procedure
is similar to the procedure of linearizing the Duffing oscillator for a small perturbation
δs around the large pump solution δp . It implies analyzing the stability regions of theMathieu equation 3.35, with vin = 0. I will not embark myself in such an endeavor since
it will takes us far from the goal of this thesis. I only quote the results from Ref. [31].
The restriction in our pump amplitude δa and pump detuning to avoid the instability
regions that would lead us to chaos are
3 2
x < O(1)2w2
ωwhere w = ωp0 and x =

1
<w<1
2

(3.84)pp/6δa . These restrictions define the boundary between thesolution having only odd-harmonics and the generation of even harmonics.
The high gain region of a parametric amplifier happens close to the bifurcation
point. We want to be able to access it assuming that the amplitude of the internal
fields is small enough that the third order approximation of the nonlinearity is accurateenough, and that at the same time we are far from the instabilities that can cause
chaotic behavior. In Ref. [31] it is shown that we can avoid these issues if we fulfill thefollowing restrictions:(1) Quality factor much greater than one.
(2) Large product of participation ratio and quality factor pQ À 1.
(3) Oscillation amplitude δa much smaller than 1.
(4) The relative detuning of the pump is of the order of the linewidth.

75
Other circuits

(a)

(b)
Z(ω)
Vs(t)

Y(ω)

Is(t)
(c)

Is(t)

(d)

Re

Ce

Le

Vs(t)

Re

Ce

Le

Figure 3.16: (a) and (b) Norton and Thevenin equivalents of a Josephson junction
embedded in an arbitrary microwave network represented by the admittance Y (ω) or
impedance Z(ω). We assume that the microwave networks have a resonance. Close tothat resonance, they can be modeled by parallel or series resonators as shown in figure(c) and (d).

So far in our analysis, we have just studied the case when a junction is shunted
by a capacitor, and damped via frequency independent resistance R = Zc . This canbe extended to the general case when the Josephson inductance is embedded in a moregeneral electromagnetic environment with a frequency dependant impedance Z(ω) or
admittance Y (ω) as shown in figure 3.16a and 3.16b. This reactive impedance allows
us to tune the resonance frequency and quality factor of our resonator. For example,
Z(ω) could represent a transmission line resonator in series with a coupling capacitor.
Now, if Z(ω) or Y (ω) have a resonance, close to that resonance frequency they can bemodeled by an effective series or parallel resonator (Fig. 3.16c and Fig. 3.16d), so a lotof insight can be gained by just analyzing these simpler cases.
The parallel case has already being analyzed in this chapter. The series resonator

76
can be studied using the same tools. The EOM of the circuit shown in figure 3.16d is"#
LJ0
d2 q(t)dq(t) q(t)V (t) = Le + p+ Re+(3.85)
2
2
2
dt
dt
C
1 − (dq(t)/dt) /I0"µ¶ #LJ0 dq(t) 2 d2 q(t)dq(t) q(t)= Le + LJ0 + 2+ Re+(3.86)
2
dt
dt
dt
C
I0
where q is the charge on the capacitor and in the second line I have made an approximation to second order in dq/dt.
Making an analogy with the mechanical harmonic oscillator, in Eq. 3.86 would
be equivalent to one where the nonlinearity does not affect the restoring force but themass of the particle. Even though it is not a Duffing oscillator, it can be shown that at
low drives, large Q and the single harmonic approximation in the solution, we recover
the same dynamics as in the Duffing oscillator [31]. Similar performance as parametricamplifiers is expected and equivalent restrictions can be derived regarding the conditions
that the electromagnetic environments must satisfy. For more details see Ref. [31].

Chapter 4
Previous and other current work on Josephson parametric amplifiers

4.1

Noise rise
In the 1970’s there was a strong effort to develop Josephson parametric amplifiers.

Because of their theoretically predicted low added noise, they were expected to havepromising performance for practical applications such as microwave/millimeter-wave detection in microwave astronomy and analog signal processing [42]. JPAs achieved goodsignal gain in 1-100 GHz band, but unfortunately these amplifiers proved impracticalbecause of a poorly understood difficulty with their added noise [43, 44]. It was foundthat for moderate to high gains, the amplifiers’ noise temperatures were significantly
larger than the environment temperature. This noise problem is referred to in the literature as “noise rise.” As the bias point was varied, the output noise of the device
had a number of sharp peaks whose origin was unknown [45]. This behavior was foundin most implementations. However the literature does contain examples of JPAs operated with low noise temperatures between 10 and 30 K and signal gains exceeding
20 dB [34,46–48]. Nonetheless, good performance was only attainable if the parametersthat controlled the amplifier gain such as current through the Josephson junctions ormagnetic field applied to the SQUID loops were tuned to a narrow range. It was notunderstood why the amplifier only obtained good performance on a narrow range of

78
parameters.1
The phenomenon referred to as “noise rise” was also observed to have the featurethat the noise temperature Tn was directly proportional to gain [44]. There were different theories that tried to explain the “noise rise” problem and they can be roughly
divided in three different groups:2(1) Amplified noise, possibly including amplified thermal radiation that saturated
the amplifier.
(2) Instability caused by noise in the parameters that determine the operating point,e.g. fluctuations in the pump power.
(3) Intrinsic instabilities and chaotic behavior associated with the nonlinear dynamics of pumped Josephson junctions.
The first case seems self-explanatory, and except for some papers published in the late1970’s (see Ref. [44] and references therein), this line of thought does not seem to havebeen given much continuity. Regarding the second group of theories, the intuitive notionis that when an amplifier is biased for high gain, the gain itself must be very sensitive
to external influences that vary the operating point. Therefore, the amplifier is more“unstable”, meaning small changes in the pump power can cause large fluctuations in
the operating point of the amplifier. Feldman and coworkers predicted that these large
fluctuations would cause a noise temperature proportional to the gain [45].
The group of theories that seemed to have more support from the community were
the ones that considered the “noise rise” problem a consequence of intrinsic nonlinear
dynamics of these systems. These explanations included phase instabilities, hopping1

Also, some attention was given to the effect of saturation due to thermal noise from room temperature [47] which raises the question of maybe inappropriate filtering as a possible root of stability
problems specially in the 3-photon mode of operation where a stable dc-bias needs to be applied.
2
However, it is not entirely clear that all the excess noise found in these first experiments had thisfeature and more than one theory may be needed to explain all the issues regarding the poor noise
performance on JPAs [44].

79
between coexisting attractors, deterministic chaos, and universal features of systems
operated near certain types of bifurcations. Next is a brief description of all thesehypothesis.
Regarding deterministic chaos, Huberman and coworkers suggested that the large
broad-band noise-rise phenomenon could be understood in terms of the existence of
chaotic solutions to the full nonlinear junction dynamics, associated with the appearanceof a strange attractor in phase space [49]. We have seen in section 3.3.1 that in thelimit of small amplitudes of phase oscillations δ, it is possible to study the stability ofequation (3.30) by converting it into a Mathieu type equation, 3.35. They show thatas δ increases, the first two terms of the Taylor expansion of the sin δ term lead to acascade of bifurcations into a chaotic regime which cannot be obtained via perturbation
theory [49]. Pederson and coworkers also suggested similar arguments agreeing that theso called noise-rise observed in connection with high gains in almost all JPAs had its
origin in chaos [50]. On the other hand, other people working in the field asserted thatchaotic behavior could not account for noise temperature proportional to gain. Instead
they argue that noise rise should occurs with no gain. In fact noise rise without any
gain has been experimentally observed (see for example Ref. [51]).
The phase instability theories [44, 51], noise-induced hopping between coexisting attractors [52] and universal behavior near instabilities or bifurcations [53, 54] aresomewhat similar. The essential ingredients of these theories is that the amplifier dynamics can have multiple solutions (coexisting attractors) and external noise can kickthe system back and forth between stable solutions. Bryant and Yurke in Ref. [54] show
qualitative agreement between their theory and experiment, and a correlation at highenough gains between the increase of noise and the increase of signal gain close to thebifurcation point. The details of each theory are outside the scope of this thesis andthe reader is referred to the literature.

80

4.2

Previous successful implementations
In spite of the “noise-rise issue” attempts to fabricate low-noise JPAs continued

with some of those attempts being somewhat successful [55]. Smith and coworkers
created the lowest noise temperature microwave amplifier, other than maser based amplifiers, and published their results in 1985. They used an rf-SQUID and operated at 8
GHz reaching a noise temperature of 6 K similar to the temperature of operation (4.2
K) [42]. This opened the possibility to generate squeezed states of microwave light andoperate amplifiers with quantum-limited performance.

4.2.1

Squeezing of thermal and vacuum noise
Iφ

Zs

Zp
-Ic

+Ic
Ls

L

C

J

Ls
J

C

L

Figure 4.1: Circuit schematic of the device used in Refs. [6] and [7]. Two Josephson
junctions form a dc-SQUID. Magnetic flux is coupled into this SQUID via the inductors
Ls through the +Ic -Ic current bias line, adjusting the critical current of the SQUID
formed by the parallel combination of the 2 junctions. The phase δ of the effective
junction is controlled via the current bias line Iφ . Two transmission line with different
impedances are used for the pump and for the signals. The JPA is operated as areflection amplifier.

Yurke and coworkers were one of the few groups that continued their researchon JPAs. The device studied consisted on a RLC-circuit as the one shown in figure4.1. Their main goal was the generation of squeezed states of microwave fields. The
JPA was operated in both modes three-photon and four-photon mode [8]. In the case81
of squeezing, it was operated in the three-photon mode where the applied pump hada frequency of 38.8 GHz. In the first experiment that showed squeezing of microwave
fields, 42 % squeezing of 4.2 K thermal noise was demonstrated and a noise temperatureof Tn = 0.28 K [7]. This noise temperature implied that the JPA exhibited less internallygenerated noise than a linear phase insensitive amplifier. However, in this case, theyreported that at 4.2 K, the highest gain achieved before the generation of excess noisedue to the onset of chaos was 12 dB [8].
They also demonstrated squeezing of quantum noise at 19.16 GHz [6]. Operatingin the degenerate mode (the pump frequency being twice the signal carrier frequency),
they demonstrated squeezing of 47 % below the vacuum-noise floor. A noise temperatureof 0.446 K is accounted by the vacuum noise emitted by the losses. In this same paper,
they reported that they have operated the amplifier with gains as high as 37 dB andwith a noise temperature of TN = 0.5 K [6, 54]. The device used had, according to theauthors, the same design as the one used earlier to do classical squeezing of thermal
noise. However, they do not provide details on why this new sample shows a better
performance.

4.2.2

Parametric amplifier with Josephson junction arrays
Zc

ZJ

_14 λ
l~
Figure 4.2: A Josephson transmission line made out of 1000 Junctions is directly coupled
to a 50 Ω transmission line at one end and shorted at the other end. Due to theJosephson non-linearity, the Josephson transmission line will act as a “distributed”
parametric amplifier operated in reflection.

Most of the JPA implementations consisted of resonant circuits as the one shown82
in 4.1. One drawback of such implementations is their narrow bandwidth. In orderto obtain large bandwidths, it is desirable to avoid the use of resonant structures toimpedance match the signals from a 50 Ohm transmission line to the low impedance exhibited by a Josephson junction. It had been realized that impedance matching problems
could be circumvented through the use of a series array Josephson junctions. Arrays ofJosephson junctions had been used previously for JPAs, but with sizes small enough tostill treat them as lumped elements [34, 46, 47]. In this new configuration suggested by
Sweeny and Mahler [56], the array was significantly larger than the wavelength of the
microwave field, and constituted a distributed parametric amplifier.
Yurke and coworkers implemented this idea, fabricating an array of 1000 Josephson junctions and a length of 9 mm, which greatly exceeded the wavelength of 0.65 mm
for the 19 GHz signals. The amplifier was pumped at the signal carrier frequency, thatis, it was operated in the four-photon mode [57]. With a critical current of I0 = 12 µA
per junction, the zero-current inductance per unit length was Ll = 34.5 µH/m. The
capacitance per unit length was Cl = 187 pF/m which resulted in a characteristic transmission line impedance of 430 ohms and a phase velocity of v = 1.25 × 107 m/s. 3
They then terminated one end with a short (Fig. 4.2).4

The Josephson transmission

line was operated at zero bias as a reflection parametric amplifier, where a single port
was used for both pump and signal.
In this implementation, they demonstrated very low noise performance of Tn =0.5 K consistent with internally generated noise from losses. Their implementation also
showed a bandwidth of 125 MHz with a gain of 12 dB, or an equivalent gain-bandwidth
product of 500 MHz. However, some peculiar behavior regarding the noise and gain
3

The phase velocity and impedance given here are the same as the ones given in Ref. [57]. However, they do not agree with the critical current of the junctions the authors claim, I0 = 12 µA. The
inconsistency is fixed if a critical current of I0 = 1.2 µA is assumed instead.
4
I believe it is debatable calling this implementation a non-resonant parametric amplifier, as the
impedance mismatch of both transmission lines is large enough to be able to create well defined resonantmodes inside the Josephson transmission line. However the amplifier would be operated at an unusuallylarge mode of the resonator.

83
performance were also observed. As the pump power was increased from low values,
the gain of the amplifier first increased and then, after passing through a maximum, it
decreased. They called this point at which the maximum gain is achieved the thresholdvalue [57]. Good noise performance is reported only past the threshold point. Below
threshold the amplified showed excess noise as large as an extra 6 K.

4.3

Current work on Josephson parametric amplifiers
After the promising results of vacuum noise squeezing and large gain with low

noise performance shown in Ref. [7], research on Josephson parametric amplifiers dwindled. However, the interest in them has been growing the past few years with several
groups interested in developing low-noise microwave parametric amplifiers for quantum
information applications and quantum limited detection of microwave signals [9, 11].
Next I describe some of the most successful recent implementations.

4.3.1

Josephson ring modulator

The Josephson ring modulator has been developed by N. Bergeal and collaborators for quantum computing applications [58]. The four-photon mode operation of
the JPA implies the presence of a large pump tone in the middle of the amplifier band
which for some applications can be a problem. For some applications, it is necessary
to separate the modes associated with the signal, idler5 and pump so that all three
tones are at a very different frequency. One approach is to fabricate a nondegenerateparametric amplifier where each of the modes correspond to a different resonator. The
Josephson ring modulator (JRM) archives this by separating all 3 modes as observed infigure 4.3a. The JRM consists of four nominally identical Josephson junctions designed
into a ring threaded by a magnetic flux Φ. The ring has only three orthogonal electricalmodes that couple to the junctions: two differential ones, X and Y , and a common one,
5

In non-degenerate parametric amplifiers the image tone is usually referred as the idler tone.

84
(a)

(X)

(Y)
C1

R1

1

2
L1

V1

C1

4

Φ

2

Ip

(b)

+

3

C2

0

0

δ X = δ1 − δ2

External
drive

+

+

X

V2

(Z)

0

0

R2

2
2

2

External
drive

-

C2

+

Y

δY = δ4 − δ3Z

δ Z = δ1 + δ2 − δ3 − δ4

Figure 4.3: (a) Lumped element circuit schematic of the Josephson ring modulator. The
ring modulator is coupled to two LC resonators and a pump drive. (b) There are four
orthogonal modes, but only three of them couple to the Josephson junctions: the twodifferential modes X and Y used to couple to the two LC resonators, and the common
Z mode is used for the pump drive.

Z (Fig. 4.3b). In the circuit schematic shown in figure 4.3a, the two resonator circuits
X and Y are used as the signal and the idler modes, while the input Z for the pumpdrives the common mode.
The JRM amplifier is operated usually at a flux of Φ = Φ0 /2. At this flux bias,
and for mode intensities δX , δY and δZ much smaller than 1, the ring hamiltonian has
approximately the following form:·

δ 2 + δY2 + δZ2
Hring ∝ δX δY δZ − X2¸

where terms of order higher than three have been neglected. The second term corresponds to a “contamination term” whose only effect is to renormalize the resonancefrequency of the resonators. The first part is a pure nonlinearity that couples all threemodes, and in the coupling each mode appears only linearly. The bridge configuration

85
proposed by Bergel, et. al.

has the advantage of eliminating unwanted terms like2 δ , δ 2 δ , δ 2 δ and δ 2 δ
δX
Z
Y Z
X Y
Y X which would introduce other, unwanted types of mixing.

One interesting consequence is that the resonance frequency of the circuit used for thesignal mode will not have the observed change in frequency as a function of amplitudeof internal fields, as the one observed in the Duffing oscillator. This implementation cantheoretically achieve the minimum added noise limit set by Haus-Caves theorem (2.81).

4.3.2

Flux driven Josephson parametric amplifier(b)(a)
signal in

Cc

pump
Zc

signal out
λ
l~
4

2f0

f0
Φ

dc flux bias0

Φ / Φ00.5

Figure 4.4: (a) The flux-driven parametric amplifier consists of a transmission line
resonator terminated by dc-SQUID. By applying a dc flux bias, the boundary conditionof the cavity resonator changes, modifying the resonance frequency. A microwave pumptone is inductively applied to the SQUID which modifies the resonance frequency at
fpump = 2f0 parametrically amplifiying a signal applied at the input port through thecoupling capacitor Cc . (b) In order to operate this amplifier in a three-wave mixing
mode, a dc flux bias must be applied to break the even symmetry of f0 versus flux.

This implementation of the Josephson parametric amplifier has been studied by
T. Yamamoto and collaborators [59]. It consist of a superconducting transmission-line
resonator terminated by a dc-SQUID (Fig. 4.4). Unlike our implementation, their pumpdoes not modulate the current through the SQUID, but it is applied as a flux modulation.
A dc-flux bias breaks the even-symmetry of the resonance frequency-versus-flux relation.
Applying a microwave pump at twice the resonance frequency fpump = 2f0 ∼ 20 GHz
to the inductively coupled pump line makes the cavity’s resonance to oscillate at 2f0 .

86This modulation does parametric work on a signal applied at the input port of thecavity. As the pump is applied at twice the resonance frequency of the JPA, this typeof operation is a three-wave mixing or degenerate mode, as opposed to the four-wave
mixing or doubly degenerate mode.
We can distinguish an important difference between our implementation and thisone. As the signal and the pump are very different in frequency (fpump = 2fsignal ), andapplied at different ports, it is simpler to separate the amplified signal from the pump.
Also, the cavity resonator is a quarter-wave resonator and the next resonance mode is ata frequency f1 ' 3f0 , so the leakage of the pump microwave power into the signal port issmall due to the absence of a resonance close to the pump frequency. The performanceof the amplifier has been described in Ref. [59]. They have observed a maximum gainof 17 dB. For the noise performance they have estimated an upper bound in the noise
temperature of TN < 0.87 or in terms of added quanta, AN < 1.7 quanta.
4.3.3

dc-SQUID stripline resonatorsignal +pump1 inlc ~ 3 λsignal outΦdc flux bias +pump2

Figure 4.5: The device consist of circular microstrip resonator with a dc-SQUID locatedin a current antinode of the third resonance mode. The parametric amplifier is operated
in two different ways. First in the double degenerate mode with the pump applied close
the signal frequency and at the same port as the signal (pump1 ). The second mode is
the three wave mixing, where the pump is applied in the flux line (pump2 ).

87This implementation is in principle very similar to the previous one. The JPA
consists of a stripline microwave superconducting resonator integrated with a dc-SQUID[60] (Fig. 4.5). The dc-SQUID sits at a current antinode of the third mode resonance.
Abdo, et. al. studied the signal and image gain as a function of pump power observing
gains as high as 24 dB.
They studied the noise behavior as function of pump power showing instabilities
close to the critical pump power. They suggest that the reason is due to the factthat at high critical powers, the amplitude of the current inside the cavity is large
compared to the critical current of the SQUID (I0S ) [60]. This would suggest that this
implementation does not satisfy all the requirements given by Ref. [31] and described
previously in section 3.3.4 .
In addition, they also operate their amplifier in the three-photon mode, where thepump is applied as a microwave flux bias, similar to Ref. [59]. They show a signal gainof 38 dB but with a lot of other tones at other harmonics. They also show a signal tonoise improvement of around 23 dB. Yet, without a noise temperature of the amplifier
chain it is not possible to estimate the added noise of their JPA.

4.3.4

Josephson bifurcation amplifier

Developed by Siddiqi and coworkers [61, 62], the Josephson bifurcation amplifierconsists of a nonlinear oscillator formed by a capacitively shunted Josephson junction(Fig. 4.6a). This is exactly the same circuit we have been analyzing so far and we know
it behaves as a parametric amplifier. The name Josephson bifurcation amplifier (JBA)
distinguishes the regime in which it is operated. The resonant circuit is biased with astrong microwave current to access the nonlinear regime as shown in figure 4.6b. The
pump power and pump detuning are set in the parameter space to be in the bistableregime. In this regime, two possible dynamical states are accessible. A weak signal isthen coupled to the amplifier junction critical current, where figure 4.6a shows just one

88
(a)

Zc

signal in

0.1

δa /2πZc

outputJBA
Rs=Zc

phase (degrees)

(b)
OH

360
180

0.05

0
0.96

0.98

1

ω0 / ωp

OL

Vp(t)
0
0.95

1

1.05

ω0 / ωp

Figure 4.6: (a) Circuit schematic of a possible implementation of the Josephson bifurcation amplifier. The input line is inductively coupled to critical current of the Josephson
element in the JBA. A large pump rf-bias the resonator in the nonlinear regime where
two dynamical states are possible for the JBA. (b) Nonlinear resonant curve of the
JBA showing a possible bias point for the JBA (solid line). A small signal coupled tothe critical current will make the resonance frequency, and consequently its response,change (dashed line). If properly biased, the oscillator will change from a low amplitudestate (OL ) to a high-amplitude state (OH ). Inset: Phase of the reflected pump tone forthe given pump power. The two possible dynamical states are indicated (red and greendots), showing the large and detectable change in phase that allows the JBA to measurea weak signal due to small changes in the effective critical current of the junction.

example of a possible implementation of the signal-I0 coupling. This signal coupling
results in a dispersive shift of the circuit resonance frequency and consequently in theoscillator evolving from the state OL to OH , thus realizing a sensitive threshold detector
for I0 . The dynamical state of the JBA can be measured by the large difference in phaseof the reflected pump as can be observed on the inset of figure 4.6b.
However, the JBA can be operated as a Josephson parametric amplifier: instead of
coupling an input signal to the critical current of the Josephson junction, we can apply
a weak signal at the same port as the pump and amplify that signal as a reflection
amplifier. Vijayaraghavan has operated the JBA in the parametric amplifier regime.
His results have not been published, but are available in Ref. [18]. He reported a gain ofover 25 dB and a gain-bandwidth product of 280 MHz. The estimated noise temperature

89
in the nondegenerate mode was Tn = 0.08−0.15 K, which compared to quantum-limited
value of 0.072 K, it implies a performance close to the theoretical limit. They have alsomeasured squeezing of vacuum noise.

4.3.5

Parametric amplification with superconducting resonators andweak-links

Figure 4.7: Picture of the sample used by Tholén, et. al. to study parametric amplification. The resonators are made of Niobium and 10 weak-links are fabricated at thecenter of the resonator where there is a current antinode. Reproduced with permission
from Ref. [63].

It is a known phenomenon that thin superconducting films present nonlinear effects at high enough currents [64]. These nonlinear effects arise from current induced
magnetic field penetration into the superconducting thin films and give rise to a nonlinear inductance. This current dependent kinetic inductance can be modeled as aneffective kerr nonlinearity [38]. Tholén and collaborators have used this Kerr effect insuperconducting films to fabricate nonlinear cavities that can be used for parametricamplification [65]. Using the intrinsic nonlinearity in superconducting transmission line
resonators in their first implementation they achieved 22.4 dB of maximum gain (signal
and image gain). Due to a lack of calibrated noise source, they were unable to estimate

90
the added noise of the amplification process.
In order to increase the nonlinearity of their resonators, in their second implementation they fabricated a parametric amplifier based on the nonlinear kinetic inductanceof a weak link (Fig. 4.7). A weak link consists of a Josephson element where the weakelectrical contact between superconducting electrodes is provided by some small metallic
constriction with dimensions of the order of the superconducting coherence-length [66].
The name is used to distinguish non-tunnel from tunnel junctions Josephson elements.
This type of Josephson element also behaves as a non-dissipative nonlinear inductance.
However the strength of the nonlinearity is a strong function of the geometry and thecurrent-phase relation is not necessarily a sinusoidal function as in Eq. (3.15) for a
simple tunneling junction. In this second implementation of parametric amplifiers,
Tholén reports an improved performance with a maximum signal and image gain of
25 dB [63].

Chapter 5
Design of JPA

In this chapter I describe the process of designing our Josephson parametric amplifier. I will begin by discussing microwave cavity resonators including the processof coupling power to them. I will also discuss some non-ideal features present in our
devices such as loss. In the second part of the chapter, I discuss the integration of
the Josephson junction into the microwave cavities. I will then generalize the resultsof chapter 3 to include both loss and the presence of additional ports. Finally, I will
discuss the effect of these extra ports in the performance of the amplifier.

5.1

Microwave environment design
In chapter 3, I discussed the case of a lumped-element parallel Josephson resonatorand its behavior as an amplifier. However, at microwave frequencies the wavelengths
are not necessarily much bigger than the size of the circuits elements that are used. At
these frequencies, the electrical response of a structure becomes highly dependent on its
geometry. Thus, in order to fabricate microwave resonators, we use distributed elementsin the form of transmission lines, instead of using lumped elements such as capacitor
and inductors. I next discuss the two types of transmission line resonators that we use
in the experiments.

92
5.1.1

Transmission line resonators

Consider figure 5.1a. Assume that there exists a right traveling wave with aphasor representation given by VR = V0R e−jkz . We have seen that the ratio of voltage
to current for such a transmission line is the characteristic impedance Zc . But when theline is terminated in an arbitrary load, ZL 6= Zc , the ratio of voltage to current must be
ZL . Thus a reflected wave needs to be created with the appropriate amplitude and phaseto satisfy this condition. The incident and reflected waves will interfere destructively
and constructively generating a standing wave pattern through the transmission line.
This pattern can be found by calculating the reflection coefficient Γ for a given load
(Eq. 2.37).
V (z) = V0R e−jkz + V0L ejkz ,= V0R (e−jkz + Γejkz ),I(z) ==

V0R −jkz V0L jkz
e
−
e ,
Zc
Zc
V0R −jkz(e− Γejkz ).
Zc

(5.1)

(5.2)

From the definition of impedance, Z = I/V , we can see that the observed impedancelooking into the transmission line must vary with position. At a distance z = −l fromthe load, the input impedance is given by
1 + Γe−2jkl
Zc ,1 − Γe−2jkl
ZL + jZc tan(kl)= Zc.
Zc + jZL tan(kl)

Zin =

(5.3)
(5.4)

The boundary condition imposed at both z = −l and at z = 0 define the impedance
seen and the type of resonator we have created.
Short-circuited quarter-wavelength resonator

Figure 5.1 shows a transmission line terminated at one end in a load impedance
ZL = 0, which is usually referred as a short circuit. The other end is left open. This

93

(a)

VR

(b)

V ( z ), I ( z )
Zc

Z in

Z in

R

Z in ≈

R
1 + 2 j ( ω − ω0 ) RC

L

C

VL
z = -l

l=

z=0

λ
4

ω0 = 1/ LC

Figure 5.1: (a) A short-circuited transmission line of length l = λ/4. (b) A parallel
RLC-resonator. In the high-Q limit, the impedance Zin of the resonator will resemble
the impedance Zin of the short-circuited section of transmission line shown in (a).

configuration creates a standing wave resonance for a signal whose wavelength is λ =4l/n for n an odd integer. We will discuss only the case for n = 1, since it is the mode
we will use. Therefore, l = λ/4 and this resonator is called a quarter-wavelength (λ/4)
resonator. To make the analysis more accurate, I will consider a lossy transmission line.
The only change in the previous expression is to replace jkl with (α + jk)l, where α isthe attenuation constant of the transmission line and whose origin can be either losses
in the conductor or dielectric of the line. For a shorted end, the input impedance seenat z = −l is given, from Eq. (5.4) by:
Zin = Zc tanh(αl + jkl),= Zc

1 − j tanh(αl) cot(kl).
tanh(αl) − j cot(kl)

(5.5)
(5.6)

We can then assume that l = λ/4 at ω = ω0 , and ω = ω0 + ∆ω. With these assumptions
kl =

π π∆ω
+
.
2
2ω0

For a small amount of loss, tanh(αl) ≈ αl, and expanding to first order in ∆ω/ω0
1 + jαlπ∆ω/2ω0,
αl + jπ∆ω/2ω0Zc.
αl + jπ∆ω/2ω0

Zin = Zc
≈

(5.7)

94
Eq. (5.7) has the same form as the impedance of a parallel RLC resonator circuit in
the high-Q limit (Fig. 5.1b)
RLC
Zin
=

1
.
1/R + 2j∆ωC

Note that in this expression, we have included internal losses of the circuit as a resistor R.
This resistor R is not related with the characteristic impedance Zc of the transmission
line coupled to our resonator. We can identify the resistance, capacitance, inductance,and quality factor of the equivalent circuit as:
R = Zc /αl.
C =
L =

π
.
4ω0 Zc1.
2
ω0 C

Ql = ω0 RC =

(5.8)
(5.9)
(5.10)π,4αl(5.11)

showing that the quality factor Ql decreases as the attenuation of the line α increases.
Here, Ql is the intrinsic quality factor associated with internal losses of the resonator,and not the coupling quality factor Q we used in Chapter 2. In the next section, I will
describe how we couple power to these resonators.
Open-circuited half-wavelength resonatorVRV ( z ), I ( z )
Zc

Z in
VL
z = -l

l=

λ
2

z=0

Figure 5.2: An open-circuited length of transmission line. For lengths of l = nλ/2, thistransmission line will be a resonator.

95
Another kind of resonator we will be using is the open-circuited half-wavelength(λ/2) resonator, as shown in figure 5.2. Such a resonator will behave as well as a parallel
resonant circuit, when the length is l = λ/2 or multiples of λ/2. The input impedanceis given by
Zin = Zc coth(αljkl).
In this case,kl = π +π∆ω.
ω0Doing similar approximations to the ones done for the λ/4 resonator, the input impedance
seen at z = −l is
Zin =

Zc
.
αl + j∆ωπ/ω0(5.12)

Again, comparing it with the impedance of a parallel RLC resonator, the equivalentvalues for the resistance, capacitance, inductance and quality factor are
R = Zc /αl,
C =
L =

π
,
2ω0 Zc1,2
ω0 C

Ql = ω0 RC =5.1.2

(5.13)(5.14)(5.15)π.
2αl(5.16)

Coplanar waveguide transmission lines

So far the description of the transmission line resonator has been rather general,
without describing what kind of structure we are using. In any physical implementationof a transmission line, the geometry must be chosen to best fulfill our needs. Dueto several reasons, our choice was to use coplanar waveguide transmission lines (Fig.
5.3). A first advantage arises from the fact that the signal ground and conductors lie
in the same plane, which makes it simple to create an electrical short on a chip. In
addition the dimensions of the coplanar waveguide can be scaled while preserving the

96
(a)

(b)

c
b
a

b
a

εr

h

εr

Figure 5.3: Coplanar waveguide geometry. (a) Ideal case, where the ground planes andthe dielectric are assumed to extend to infinity. (b) A more realistic case of a coplanar
waveguide transmission line where both the ground planes and the dielectric are finite.
The dielectric is backed by a ground plane.

wave impedance. As we will see, the impedance of the transmission line depends on theratio of the width of the center conductor to the gaps. As long as that ratio is kept
the same, the geometry of the transmission line can be scaled from large structures towhich we can attach macroscopic wires down to very small structures compatible withthe micron size of the Josephson junctions.
The ideal coplanar waveguide (CPW) consists of two slots each of width W =(b − a)/2 printed on a dielectric substrate (Fig. 5.3a) . The spacing between theslots is denoted by a. The ground plates extend to infinity as well as the dielectric(infinitely thick substrate). While any physical implementation can be treated using
the equations that describe a transmission line, the exact expressions for the phasevelocity and impedance depend on the geometry. Approximate expressions for both the
impedance and the phase velocity can be found using conformal transformations [67].
Assuming non-magnetic substrates for which µ = µ0 , for the ideal coplanar waveguide97
shown in figure 5.3a the expressions are
²re =
v =
Zc =

²r + 1
,
2
c
√ ,
²re
30π K 0 (κ1 ),√²re K(κ1 )(5.17)(5.18)(5.19)

with K and K 0 being the complete elliptic integral of the first kind and its complement,
and
κ1 =

a
.
b

If we introduce the necessary modifications to include the fact that our dielectric is not
infinite and neither are the ground planes (Fig. 5.3b), the expressions are sligthly more
complicated [67]. Defining κ2 and κ3 ass
a 1 − b2 /c2
κ2 =,b 1 − a2 /c2ssinh(πa/2h) 1 − sinh2 (πb/2h)/ sinh2 (πc/2h).
κ3 =sinh(πb/2h) 1 − sinh2 (πa/2h)/ sinh2 (πc/2h)
The new expressions for the phase velocity and characteristic impedance are²re = 1 +v =Zc =²r − 1 K(κ3 ) K 0 (κ2 ),2 K 0 (κ3 ) K(κ2 )

c
√ ,
²re
30π K 0 (κ2 ).
√²re K(κ2 )(5.20)(5.21)(5.22)For our resonators, we designed our coplanar waveguide with ground planes of
reduced width, as can be seen in figure 5.4. The main reason that we used such reducedwidth ground planes is to reduce the potential influence of trapped vortices of magnetic
flux. To elaborate, superconductors will exhibit a characteristic referred to as Meissner
effect which consists of the expulsion of a magnetic field from the superconductor [33].
In general, however, superconductors rarely exhibit a complete field expulsion; grainboundaries, normal inclusions, and other defects serve to trap magnetic flux within

98

(b)

(a)

10 µm
10 µm

Figure 5.4: Pictures of the two kind of devices discussed in chapter 6. In both cases
5 µm-wide ground planes were used in order to avoid flux trapping close to the SQUIDs.

the superconductor. Flux expulsion can be further decreased by the geometry of thesample. Thus, for actual experiments, the Meissner effect is typically incomplete andstrongly dependent on sample preparation and geometry. In our case, this concern isvery important as flux trapped in the ground planes close to the SQUIDs could introduce
flux noise in the SQUID loops. Although the resonators will in general be kept in a
low-magnetic field environment, the situation is always critical as the superconductor
is cooled though the critical temperature Tc . In Ref. [68], the authors studied flux
expulsion in the case of a thin-film superconducting strip of width W when a strip iscooled through Tc . In that Letter, it is shown that such a strip exhibits a complete
Meissner expulsion of vortices below a critical field Bm , independent of the details ofmaterial parameters. Furthermore, this expulsion has a universal characteristic, withBm

= Φ0 /W 2 . For a width of 5 µm, this maximum field for vortex expulsion is

0.8 gauss, slightly bigger than the 0.3-0.6 gauss of Earth’s magnetic field. We designed
the CPW so that we could cooldown through Tc with a magnetic field equal to thatof the Earth’s magnetic field without having trapped vortices close to the SQUIDs. In
order to further protect the sample we have two levels of magnetic shielding: a cryoperm
shield around the sample that sits at 15 mK and a µ-metal shield at room temperature.

99
5.1.3

Coupling a cavity to a transmission line

As yet I have not described the way we will excite the resonators. We could directly connect a transmission line to the resonator, as it was done in chapter 2. However,this would allow the energy stored in the cavity to leak out too quickly; that is, theresonator would be heavily loaded, and the Q would be too low. To reduce the couplingstrength, we use a small capacitor to connect the resonator to the transmission line weuse to launch signals as shown in figure 5.5. The impedance mismatch that the capacitor represents is analogous to a dielectric mirror that is partially transparent. Since anideal capacitor is lossless, but transmits a small amount of power, we can inject power
to the cavity or allow energy to leak out of the cavity through this “microwave mirror.”
Another advantage over directly connecting the resonator and the transmission line isthat the coupling capacitor serves as a dc-filter for the Josephson junctions that will be
inside the cavity.
In order to derive an expression for the quality factor Q, there are a variety of
approaches, of which I will describe two for being the most useful. First I will consider
the case of a lossy λ/4 resonator and then a lossless λ/2 resonator. The reason for thisis that by comparing the final expression of the two resonators, it will become obviousthat the intrinsic loss of the resonator can be modeled as just an extra port. This
fictitious port allows energy to leak out of the cavity (through dissipation) as well as
admits energy in the form of thermal noise.
Single port

Next we consider a λ/4 short-circuited resonator coupled to a transmission line
through a capacitor C1 (Fig. 5.5a). The normalized input impedance seen by thefeedline with impedance Zc isz(ω) =jZcµ¶
1
Zr tan(kl) −bc(5.23)

100(a)port 1Zc

Zr
z = -l

(b)

l=

λ
4

port 2port 1Zc

z=0

Zrz = -l

l=

λ
2

Zc
z=0

Figure 5.5: (a) λ/4-resonator coupled to a transmission line of impedance Zc by acapacitor. (b) λ/2-resonator coupled to two transmission lines of impedance Zc by two
capacitor. Port 1 is defined as the strongly coupled port and port 2 is the weakly coupled
port.

where bc = Zc ωC1 is the normalized susceptance of the coupling capacitor. Resonanceoccurs whencot(kl) − bc = 0.
Solutions to this transcendental equation can only be found numerically. However, inpractice bc ¿ 1. This is equivalent to saying that the coupling quality factor from port 1√
Q1 À 1. This requirement allows us to assume that the new resonance frequency will
be close to the frequency for which kl = π/2 (the first resonance frequency of theunloaded resonator ω0 ). By doing a Taylor expansion around the new loaded resonancefrequency ωr , it can be shown that the normalized impedance is given byz(ω) = jπ(ω − ωr ).
2ωr Zc Zr (ωr C1 )2

So far we have not included losses. For high-Q cavities, loss can be easily included
following the common recipe of replacing the resonant frequency ωr with the complex
resonance frequency ωr (1 + j/2Ql ) [13, 16]. In this substitution, Ql is the quality factor

101
associated with the intrinsic loss of the resonator, Eq. (5.11). Then, the normalized
impedance of the lossy λ/4 resonator is
Q1z(ω) =Qlµ¶(ω − ωr )1 + 2jQl
ωr(5.24)

where the quality factor associated with the coupling port Q1 is given by
Q1 =

π
1
4 Zc Zr (ωr C1 )2(5.25)

I can then calculate the reflection coefficient using Eq. (5.24),
Γ=Γ0 + 2jQδω
1 + 2jQδω(5.26)

where δω = (ωr − ω)/ωr , Q is the total quality factor
1
1
1
=
+
Q
Q1 Ql

(5.27)and Γ0 is the reflection coefficient at the resonance frequency, and it is given byΓ0 =Q1 − Ql.
Q1 + Ql(5.28)

Note also that in this case, there are two types of quality factors we are interested
in: the internal quality factor, Qi and the coupling quality factor, Q1 . Defining thecoupling coefficient g as
g=

Ql
Q1

(5.29)

we find there are three different cases.
(1) g < 1 The resonator is undercoupled to the feedline.
(2) g = 1 The resonator is critically coupled to the feedline.
(3) g > 1 The resonator is overcoupled to the feedline.
The reflection coefficient will be very different depending on the value of g as can beseen in figure 5.6.

102(b) 360
phase (degrees)

Γi

(a) 1

0

180

0
−1
−1

0
Γ

r

1

−90
−3

−1.5

0
δω

1.5

3
−3

x 10

Figure 5.6: (a) Parametric plot of the real and imaginary part of the reflection coefficientΓ as a function of frequency for different values of coupling coefficient: g = ∞ (blue),g = 3 (red), g = 1 (green) and g = 1/3 (black). (b) Phase of Γ as of function of thedetuning δω = (ω − ωr )/(ωr ) for the same coupling coefficients as in (a). Regardless ofthe exact value, for g > 1 the phase evolves a full 360◦ as we sweep the frequency. This
is not the case for g ≤ 1, in which the phase only evolves 180◦ or less as we sweep thefrequency through resonance.

Double port

We could use a similar method to the one described above for the single port
to calculate the coupling quality factors associated with the two ports connected via
coupling capacitors to a λ/2-resonator as shown in figure 5.5b. However, I will use
energy consideration so that the meaning of quality factor becomes obvious [69]. For
now, let’s assume a lossless transmission line. This loading (or coupling) calculation
yields the coupling quality factors, Qj , where j = 1, 2 denotes the port.
Let’s assume that the resonator has been initially excited, and some energy isstored. Due to the coupling, this energy is going to leak out through the capacitors. In
a λ/2-resonator we have seen that the voltage antinodes are located at the end of theresonator, where the capacitors are located, and using the shape of the voltage standing

103
wave we can estimate the electric energy stored inside.
1
Ec = Cl2

Z 0

dzhV 2 (z, t)i−lwhere the averaging hV 2 (z, t)i is done over one cycle of oscillation. Assuming thatthe coupling is weak enough that the normal modes of the resonator are close to theunperturbed one, the shape is just one half of a sinusoidal wave, so we can perform theintegration. Furthermore, on resonance, the electric and magnetic energy stored in thetransmission line are the same, so the total energy is given by
1
E = Cl lhV 2 (t)i2where the extra factor of 2 is from the space integration. In order to calculate thepower dissipated through the coupled ports, we calculate the current flowing in thetransmission line, P = h(I12 + I22 )iZc , where Ij indicates the current flowing out of port
j. Assuming that the coupling capacitors are the dominant impedance (basically thatZc ¿ 1/ωCj ), Ij ≈ V ωCj , and the power dissipated is thenP = h(C1 V (t)2 + C2 V (t)2 )iωZc .
The total quality factor Q of the resonator is defined as [16]average energy stored
energy loss/second
1/2Cl lhV 2 (t)i= ω0h(C1 V (t)2 + C2 V (t)2 )iωZc

Q = ωThis expression reduces to
1
1
1
=
+
Q
Q1 Q2

(5.30)

with the coupling Qj given by
Qj

=
=

Cl l
2Zc Cj2 ω0π.
2Zr Zc (Cj ω0 )2(5.31)(5.32)

104
Comparing Eqs. (5.30) and (5.27), we can see that in terms of the total quality factor,
port 2 just looks like extra loss. This is even more obvious when we calculate thereflection coefficient S11 .1

S11 can be calculated by doing a similar procedure toestimate the impedance of the λ/2 resonator close to the new loaded resonance frequency
ωr as it was discussed for the λ/4 resonator. The result isS11 =Γ0 + 2jQδω
1 + 2jQδω(5.33)

where Γ0 is given byΓ0 =Q1 − Q2.
Q1 + Q2

The fact that loss can be modeled as an extra port will simplify the analysis of thenonlinear resonator in the next section.
Intrinsic Resonator Losses

The measured Q of a resonator is expected to depend on the total energy loss
per cycle of the resonator. We expect energy to leak out through several paths. Somepaths, like one through the coupling capacitor, are designed by us, while others, likethe loss of energy to electromagnetic radiation, are governed by geometry and material
parameters. The quality factors loss terms add as follows:2
1
1
=
+
Q
Q1

X
all other paths

1.
Qj

Ideally the quality factor of a resonator would be something that we would completely
control by adjusting the coupling capacitors. However, this is not necessarily the case.
There are three primary mechanisms of loss in superconducting microwave resonators:
1
For the λ/2-resonator, I will use S11 to refer to the reflection coefficient as opposed to Γ as inthe λ/4-resonator. This is done to distinguish the fact that in the λ/2-resonator, there is actually two
possible scattering parameters we will measure, S11 and S12 .
2
The equivalent expression for the linewidths is simpler as they just add:Xγ = γ1 +γj .
all other paths

105
resistive, dielectric and radiative losses. In principle, resistive losses should be negligible
as we are dealing with superconducting devices. It has been shown that radiative losses
would limit the quality factor of a resonator like the ones studied in this thesis to106 [69], so we do not consider them as limiting factors as they are several orders ofmagnitude larger than our designed coupling Qs.
Dielectric loss can also affect the Q of the resonator. Microwave materials areusually characterized by specifying the real permittivity and the loss tangent all as acomplex number² = ²0 − j²00 = ²0 (1 − j tan δ)
where tan δ is referred to as the loss tangent. In order to reduce these losses, we use highresistivity silicon for our resonators, which have been shown to achieve Qs as high as
100,000 [70]. However, due to the trilayer process used to fabricate the second generationof devices (see appendix C), we need a layer of deposited SiO2 as an insulating layer.
This layer is known to be extremely lossy (see for example, Ref. [59]) and could be alimiting factor for our internal quality factor.

5.2

Cavity JPA
If we were to put a Josephson junction inside a microwave cavity resonator, theanalysis of the JPA could be done in terms of the quantum network theory describedin sections 2.3.1 and 3.3.1. Close to the resonance frequency the full circuit could bemapped into one of the two circuits shown in figure 3.16. Such an implementation has
been used in the bifurcation limit as the JBA explained in section 4.3.4 [71]. However,
one of the problems with such amplifiers is that they will have a very narrow bandwidth. In order to partially circumvent this problem, a SQUID (or maybe even a few
SQUIDS) instead of a Josephson junction can be used. This modification provides sometunability to the resonance frequency, and thus in the amplified band of the JPA [72,73].

106
Nevertheless, in such implementations, the Josephson inductance is heavily diluted by
the geometrical inductance of the microwave cavity (small participation ratio p). Thus,
for low magnetic flux (Φ close to zero), the resonance frequency is rather insensitive toflux and it is only when the applied flux is close to Φ0 /2 that large changes of resonancefrequency can be achieved. Yet, for such fluxes, the critical current of the SQUID isvery small, and therefore the critical power of the resonator is very small as well. Verysmall critical power has the effect of making the amplifier saturation power too small
to be useful as an amplifier. In addition, in those implementations, it has also been observed that close to Φ0 /2, the intrinsic losses of the resonator increase. This additionalloss would hurt the performance of the amplifier [72, 73]. When we were designing ourparametric amplifier, we were trying to fabricate a JPA that could be used over a wide
range of frequencies without a large degradation of the amplifier performance. I next
describe our approach.

5.2.1

Widely tunable resonator

In order to create an amplifier with a tunable band, we decided to create atransmission line whose phase velocity across the whole resonator could be tuned by
an external knob that we had control over. Then for a fixed length of a transmission
line resonator, by adjusting the phase velocity we could adjust the resonance frequency.
Therefore, instead of embedding 1 or a few SQUIDs inside a cavity, our approach was totake a CPW transmission line and replace the center conductor with a long series arrayof SQUIDs (Fig. 5.7a). We have seen in section 3.2 that the small signal inductance ofa Josephson junction is given by LJ0 = φ0 /I0 . I also showed that the critical current ofa SQUID I0S can be tuned by applying a magnetic field. Therefore, a resonator madeout of a transmission line like the one shown in figure 5.7a would allow us to adjust thephase velocity and the resonator resonance frequency [74]. By adjusting the resonancefrequency, we adjust the amplified band of the JPA. This feature will allow us to use

107
the amplifier over the whole tunable band.
(a)

LSQUID
Lg ∆z

Cl ∆z

∆z
(b)

I1
+
V
_1

A1 B1
C1 D1

I2
+
V
_2

A2 B2

An Bn

AT BT

C2 D2

Cn Dn

CT DT

Figure 5.7: (a) Lumped element model of the series array of SQUIDs used to fabricate
a JPA. In each cell, the capacitance to ground of the center conductor is modeled by
the geometrical capacitance Cl ∆z of the CPW. The inductance of the cell is modeledby the Josephson inductance of the SQUID and the small geometrical inductance Ll ∆zof the CPW. (b) The behavior of such periodic structure can be easily described by
calculating the ABCD-matrix of each cell and calculating the total ABCD-matrix ofthe whole array.

At microwave frequencies the SQUIDs are spaced much less than a wavelengthapart, so we treat the array as a continuous medium with flux tunable and intensity
dependent phase velocity (c.f. an optical Kerr medium). We call this nonlinear effective
medium a Josephson metamaterial. The resonance frequency (fres ) of this cavity is
proportional to the phase velocity. As the inductance per unit length Ll and therefore
the phase velocity are nonlinear functions of the currents due to the SQUIDs in thecenter conductor, fres is a function of the intensity of the microwave field stored in thecavity. When a large tone (the pump) is applied through the coupled port close to theresonance frequency of the cavity, it makes the resonance frequency oscillate at twice
the pump frequency just as in the case of a lumped-element resonator (Eq. 3.35). This

108
process creates parametric gain [13].
Josephson parametric amplifiers using series arrays have been suggested in theliterature [56] and even implemented [48, 57]. However, in Ref. [48], the array wastreated as a lumped element rather than as a medium. In Refs. [56, 57], the array istreated as a nonlinear medium, as in our case. Nonetheless, in their implementation
Josephson junctions instead of SQUIDs were used. In Ref. [57], it is argued that byusing a distributed parametric amplifier with a nonlinear medium, the problem of asmall bandwidth is eliminated. However, distributed parametric amplifiers still have alimited gain-bandwidth product [57] and other concerns such as matching the Josephson metamaterial to the feedline transmission line arise.3

In addition a distributed

parametric amplifier requires at least one junction per wavelength, and length of several
wavelengths. From a fabrication point of view, it is hard to do at microwave frequencies
on a chip.
In order to estimate the magnetic field dependance of the resonator and justify
treating the array as a continuous medium, we model the cells as shown in figure 5.7a,
where we have included the small geometrical inductance that the cells have (for thisanalysis only the linear part of the SQUID inductance is used). The arrays are madeout of hundreds of such cells, with the last of these cells loaded by a boundary condition
that will depend on the type of resonator we are building. As an example we will modela shorted λ/4 resonator. Such periodic structures as the one shown in figure 5.7a, canbe easily analyzed using the method of ABCD matrix [16]. (Fig. 5.7b). The ABCD
matrix consist of a 2 × 2 transmission matrix defined only for a two-port network. For a
microwave network that consists of a cascade connection of multiple two-port networks
as the one shown in figure 5.7, the analysis is simplified by using this approach. The
3

If the characteristic impedance of the transmission line used to send and recover the signals fromthe distributed JPA and the characteristic impedance of the Josephson metamaterial are too different,
well defined standing waves modes appear and the Josephson metamaterial would just be a very strongly
coupled microwave cavity JPA.

109
ABCD matrix of the whole network is found by multiplying the ABCD matrices of its
components. The components of the ABCD matrix are defined using the convention
shown in figure 5.7bV1 = A1 V2 + B1 I2 ,I1 = C1 V2 + D1 I2 .
The full network is then given by
 


 
 AT BT   A1 B1   A2 B2   AN
 ... 

=

C1 D1
C2 D2
CN
CT DT


BN 
.
DN

(5.34)For a simple cell as shown in figure 5.8a, the ABDC matrix can be easily calculated:  A B   1 + Z1 /Z2 Z1 
(5.35)

=

C D
1/Z2
1
where Z1 consists of the total inductance of the cell, the small geometrical inductanceLl ∆z and the linear part of the SQUID inductance LSJ0 ; and Z2 is the small capacitanceto ground of the transmission line Cl ∆z . For the simple case where all the cells areexactly the same, it is possible to show that a large enough array can be treated as acontinuous medium with an effective phase velocity approximately given by∆z

vp = q(5.36)(LSJ0 + Ll ∆z )(Cl ∆z )where I have assumed that the wavelength of microwave modes we are interested in ismuch bigger than the cell spacing ∆z.4 In this case, a simple expression can be found
for the resonance frequencys0fres (Φ) = fress0
= fresLT,0,
Ll ∆z + LSJ0 (Φ)
L0T
Ll ∆z + LS,0J0 /| cos(πΦ/Φ0 )|(5.37),(5.38)

4
We can make an analogy with a similar system such as a one-dimensional array of masses connectedwith springs as the one studied in the context of phonons. In that case, for very large wavelength phonons
or frequencies much smaller than the Debye frequency of the system, it can be shown that the dispersion
relation reduces to a linear expression between the frequency and the wavelength. Eq. 5.36 is completely
equivalent to this limit in the phonon case.

110S
where LT,0 = Ll ∆z + LSJ0 (Φ = 0), LS,0J0 = LJ0 (Φ = 0) and in the second line I haveassumed the most ideal case, which implies identical junctions and negligible loop inductance. In order to take into account loop inductance or different junctions in the
SQUID, we only need to substitute the appropriate flux-dependant value LSJ0 (Φ) in Eq.
(5.37). Examples for the ideal case, Eq. (5.38), and the case of finite loop inductanceare plotted in figure 5.9.
(a)

LJ
Ll ∆z
Z1
Cl ∆z

Z2

∆z
(b)

LJ1

LJ2
Ll ∆z1Cl ∆z1∆z1

Ll ∆z2Cl ∆z2∆z2

Figure 5.8: (a) The unit cell of the microwave network used for the ABCD matrix
analysis in the ideal case of completely uniform cells. (b) A more complicated caseconsist of a network with two different kind of cells (both the length of the cell and the
SQUIDs are different). The primitive cell consists of two of such smaller cells grouped
together.

Asymmetries in the junctions, or finite loop inductance only prevent a complete
suppression of the critical current of the SQUIDs (and therefore, put a limit on thelowest achievable resonance frequency). However, other more serious problems canarise in such large arrays. One in particular that we have observed is the case of non-

111
identical SQUID areas. For example, one of the devices discussed in chapter 6 exhibited
this problem. The analysis with ABCD matrices can be done similarly as in the previous
case, as long as in the matrix-multiplication we consider the fact that there are going
to be two different ABCD matrices, one for each cell. It is also possible to consider alarger cell of size ∆zT = ∆z1 + ∆z2 that encompasses both cells (Fig. 5.8b). Again, inthe case where the wavelength is much larger than the cell size, the array will behave
as a continuous medium with a phase velocity approximately given by∆zT

vp = p(LJ2 + LJ1 + Ll ∆zT )(Cl ∆zT ),from which the resonance frequency can be easily obtained.

f res (GHz)

8

4

0
−1

−0.5

0
Φ/Φ0

0.5

1

Figure 5.9: Resonance frequency fres as function of flux for there different networks: an
uniform array with identical junctions and negligible loop inductance (blue), an uniform
array with identical junctions and Lloop I0 /Φ0 = 0.1 (red), and an array that presents
two size of cells, ∆z1 = 1.67∆z2 (green).

5.2.2

SQUID arrays: a metamaterial with a Kerr-nonlinearity

So far we have only considered the linear limit of the Josephson junction arrayresonator. In order to consider the nonlinear limit, microwave circuit theory becomes

112
extremely cumbersome if we want to analyze the 400 degrees of freedom that describe
the 400 SQUIDs. Instead, we use the formalism of IOT. In order to do that, we model
the Josephson metamaterial as a Kerr-medium. We will estimate the effective Kerr
constant following closely Ref. [38] where they use the methods of quantum network
theory (Ref. [15]) to analyze a nonlinear transmission line.
The Lagrangian of a lossless transmission line with a length l is given by Eq.
(2.19)
Z 0
L =

dz
−l

=

1
2

Z 0
−l

Cl
2

µ

∂φ(z, t)
∂t

¶2
−

1
2L0l

µ

∂φ(z, t)∂z

¶2,dzCl V 2 − L0l I 2 ,(5.39)

where Cl and L0l are the capacitance and inductance per unit length. However, in thiscase L0l is a nonlinear function of the current in the transmission line given byµ ¶2I0Ll = Ll0 + ∆Ll
I0(5.40)

where I is the current in the transmission line, and I0 is the critical current of themedium.5

The quantization of Eq. (5.39) is described in [38] and is not reproduced

here, but only briefly described. As in the linear resonators, the imposed boundary
conditions set the resonance frequency as well as the basis used for expanding I(z, t):I(z, t) =X

vn (t)un (z).

n

The normal modes in the z-coordinate are the solutions of
µ
¶
1 dun
d
= −ωn2 Ll0 un
dz Cl dz

(5.41)

with the following orthonormalization condition:
Z 0
Ll0 un um = δn,m .
−l5Reference [38] was meant to study the parametric gain observed in resonators made out of thin-film
superconductors which exhibit a nonlinearity described by Eq. (5.40). In that case I0 is just a parameterthat describes the nonlinearity of the film and it is a function of geometry and current distribution in
the superconducting film [64]. In our case, the nonlinearity of the medium comes primarily from thenonlinear Josephson inductance of the SQUIDs and therefore, I0 will be given by the critical current of
the SQUIDs I0S .

113
In Eq. (5.41), Cl and Ll0 can be space-dependant. Nonuniform transmission lines can beused to to avoid strong intermode coupling when the resonators are driven non-linear,as the different n-modes would not be equally spaced. However, the resonators that
we studied had uniform transmission lines. Ignoring the intermode coupling and all theother modes of the resonator except for the first mode, the Hamiltonian is approximately
given by an expression similar to Eq. (3.40)~
H = ~ω0 A† A + KA† A† AA
2
where K is the effective Kerr constant. K is given by [38]~ω 2
K = − 20I0

Z l0

dzu0 (z)4 ∆Ll .

(5.42)For the the half-wavelength resonator, the boundary conditions used are
u0 (0) = u0 (−l) = 0,
and for the quarter-wavelength resonator
u0 (−l) =du0 (0)= 0.
dz

However, the final result for the Kerr constant K is equal for both cases, assuming an
uniform transmission line, and it is given byK=−

3 ~ω02 ∆Ll
2 I02 L2l0 l(5.43)In order to determine the Kerr constant for a series array of SQUIDs (Fig. 5.7a),let’s first consider the Lagrangian per cell of an array of Josephson junctions:
Cl ∆z

V2
I2
− Ll ∆z + EJ cos(δ).
22

In the previous expression, I used Eq. (3.22) for the energy of the Josephson junction
and Ll is the geometrical linear inductance per unit length of the transmission line.
Again, doing a Taylor expansion and ignoring the constant terms that do not affect

114
the dynamics of the system, we obtain that the Lagrangian for a cell of size ∆z with aJosephson junction is"µ ¶ #V 2 I2
LJ0 I 2Cl ∆z−
Ll ∆z + LJ0 +
.
2
2
4
I0

(5.44)

Comparing Eq. (5.44) with Eq. (5.40), we deduce that in our case in the continuummedium limit
LJ0
,
∆z
LJ0
∆Ll ≈
.
4∆z

Ll0 ≈ Ll +(5.45)(5.46)In the case of a SQUID, the Kerr constant will be a function of the magnetic flux
through its dependance of the SQUID critical current (Eq. 3.23). In the case that theself inductance of the SQUID loop is negligible compared to the Josephson inductancesof the SQUID junctions, the SQUID can be described as a single junction with effectiveflux-dependent Josephson energy [75]:s
EJS =2 + E 2 + 2E E
EJ,1J,1 J,2 cos
J,2

µ

¶
2πΦext,Φ0where EJ,i is the Josephson energy of the i = 1, 2 junction. We then would obtain
similar expressions to Eqs. (5.44-5.46), with I0 → I0S (Φ) and LJ0 → LSJ0 (Φ).
Once we have estimated the effective nonlinearity of our Josephson metamaterial,
we can derive equivalent conditions to those explained in section 3.3.4. The main restriction comes form the need of having the current through the SQUIDs be much smaller
than their critical currents when the amplifier is operated close to the bifurcation point.
Assuming a current standing wave with an amplitude Ia , the total energy inside theresonator can be calculated by integrating over the standing wave shape. This yields:
1
E = (lLl )Ia2
4
where I have assumed I(z) = Ia cos(πz/2l) and averaged over a cycle of oscillation. This
energy is also equal to the number of photons inside the resonator times the energy per115photon, E = ~ω0 N . Using Eq. (3.47) for the critical number of photons, Nc , as well as
Eq. (5.43) for the Kerr constant, I can then derive a condition similar to Eq. (3.83) sothat Ia /I0 ¿ 1. This restriction is:
s
Ia
=
I0

16 γ Ll0√¿1
3 3 ω0 ∆Ll(5.47)

Defining an effective participation ratio6
p=

LJ0
,
Ll ∆z + LJ0

then the restriction on the product of participation ratio and quality factor of theresonator is

s
32 1
√
¿ 1.
3 3 QpNote that this expression is, except for a factor of the order of unity, equal to Eq. (3.83).
The extra numerical factor is related to the averaging over the mode shape.

5.2.3

Loss and other ports

anl,in(t)a1,out(t)test port 1 γ1nonlinear cavity

a1,in(t)H = hωA† A +K
h A† A† AA
2a2,out(t)test port 2 γ2a2,in(t)γnl nonlinear dissipation
anl,out(t)al,in(t)γllinear dissipational,out(t)Figure 5.10: Nonlinear cavity coupled to two ports plus two additional loss ports: alinear loss port characterized by a coupling γl and a nonlinear loss port characterized
by a coupling γnl .

The inclusion of loss in the theory is easily done with the methods of IOT. As itwas shown in section 5.1.1 when we compared the results of the λ/4 and λ/2 resonator,
6

The participation ratio defined here is similar to the one expected for a series RLC-resonator as
the one shown in figure 3.16d (see Ref. [31]).

116
the loss behaves as an extra port (one whose only input is noise). Therefore, we modela resonator with loss by adding an extra port as shown in figure 5.10. The analysis isvery similar to the one port case, with the only difference being that the sum in Eqs.
(2.59) and (2.60) is done over all the n-ports (including the loss-ports). The analysis of
the critical points gives similar results to Eqs. (3.45-3.47)
√ K
3γ
,
|K|
4
γ3
√
,
3 3 γi |K|
√
2 3γ
,
3|K|

ωp,c = ω0 +(bci,in )2 =Nc =(5.48)(5.49)(5.50)

where i denotes the index port (i = 1, 2), and γ = γ1 + γ2 + γl .
For simplicity, I will consider the case where there is one driving port and one
loss-port. In that case, it can be shown that the equivalent equation to Eq. (3.56) isgiven byµc1,out (ω) =−1 +¶2γ1 (−iω + W ∗ )c1,in (ω) +(−iω + λ0 )(−iω + λ1 )µ √¶2 γ1 γl (−iω + W ∗ )2γ1 V†c (−ω) +cl,in (ω) +(−iω + λ0 )(−iω + λ1 ) 1,in(−iω + λ0 )(−iω + λ1 )√2 γ1 γl Vc† (−ω)(5.51)(−iω + λ0 )(−iω + λ1 ) l,in

= G1 c1,in (ω) + M1 c†1,in (−ω) + Gl cl,in (ω) + Ml c†l,in (−ω).

(5.52)The signal and image gain are still defined the same way as in Eqs. (3.59) and(3.60). From the amplifier’s point of view, the effect of linear loss (or even extra ports
such as in a lossless λ/2 resonator) is to generate extra-added noise as well as to reduce
the amount of squeezing. Calculating the spectral density of the output quadrature
operator (Eq. 3.72), I obtain³´
1
SXout,θ ,Xout,θ (ω) = Sa1,in ,a1,in |eiθ G1 (−ω) + e−iθ M∗1 (ω)|2 + |eiθ G1 (ω) + e−iθ M∗1 (−ω)|2
4³´
1Sal,in ,al,in |eiθ Gl (−ω) + e−iθ M∗l (ω)|2 + |eiθ Gl (ω) + e−iθ M∗l (−ω)|2
4(5.53)

117
where Sa1,in ,a1,in is the spectrum of the incoming field of port 1 and Sal,in ,al,in is the
spectrum for the incoming field of the port representing the intrinsic resonator loss.
These spectrums are given by Eq. (3.67). Eq. (5.53) allow us to compute the noisepower spectrum at any quadrature Xθ for any set of device parameters. It is importantto point out that due to the inclusion of loss, the signal gain G1 and image gain M1 will
not fulfill anymore the relations (3.57) and (3.58). Eq. (5.53) can be further simplified
in the case of ω = 0:
1
Sa ,a |eiθ G1 + e−iθ M∗1 |2
2 1,in 1,in
1
+ Sal,in ,al,in |eiθ Gl + e−iθ M∗l |2 ,2
1
1=
Sa ,a Gθ + Sal,in ,al,in Gl,θ .
2 1,in 1,in
2

SXout,θ ,Xout,θ (0) =(5.54)(5.55)

Eq. (5.55) will allow us to calculate the added noise of the amplifier as well as the
maximum amount of squeezing. In general, the added noise and squeezing will not only
depend on the gain of the maximum amplified quadrature G1 = Gθ=θ1 as in the caseof the lossless JPA, but it will also depend on the pump-detuning and pump powers;that is, for the same gain G1 , there is more than one possible value of squeezing andadded noise. This is shown in figure 5.12. However, the maximum squeezing and added
noise become single-valued in the limit of high gain. For the amplified quadrature, inthe high-gain limit, the added noise is given by:7
1
AN =
2

µ

¶
~ωp
γl
coth(
) .
γ1
2kB TlFor the case of multiple ports, Eq. (5.56) is easy generalized to


X
~ωp
~ωp 
1 
AN =
γj coth() + γl coth() .
2γ1
2kB Tj
2kB Tl(5.56)(5.57)

j6=1

From Eq. (5.57) we can conclude that increasing the number of ports will just hurt thenoise-performance of the amplifier. It can be tempting to add a second port so that we
7

Note that we are using a single-sided spectral density and AN is really the added noise per unitbandwidth as defined in appendix A. In this convention, the added noise per quadrature per unitbandwidth of a quantum-limited phase-insensitive amplifier is 1/2.

118
have a port dedicated for the pump (port 2) and another for only the signals (port 1).
However, if such a thing is necessary or desired, care must be taken to make the ports
highly asymmetric so that γ2 /γ1 ¿ 1.
In the case of squeezing, again for large gains and ω = 0, Eq. 5.55 simplifies toγl
SXout,2 ,Xout,2 (0) =
γ

µ

¶
~ωp
1
coth(
)
2
2kB Tl(5.58)

where the output is specified in number of quanta. Eq. (5.58) shows the effect of loss inthe squeezing. In the case of Sa1,in ,a1,in = Sal,in ,al,in , the maximum squeezing is given byξ = γl /γ (Eq. 2.87). The generalization to multiple ports is easily done by adding to
Eq. (5.58) the noise at the input of each extra ports multiplied by γi /γ. An interesting
thing, however, is that regardless of the number of ports, the maximum squeezing of
the internal field is always 50%, just as in the case of a single-port lossless parametricamplifier.
Sloss

η
Sin

Lossless
JPA

Sout

Sloss

Figure 5.11: Loss model for a JPA where the loss is included as a beam splitter at the
input of a lossless amplifier (see Ref. [8]).

The inclusion of loss has complicated the expressions for the squeezing and added
noise, specially since they do not have a simple closed form unless we are in the largegain limit. There is another way to include loss that is somewhat simpler. Following
the model used by Yurke, et. al., in Ref. [8], the losses of an amplifier could be modeledby placing a beam splitter at the input of the amplifier. In chapter 7, I will discuss

119
more details of how inefficiencies of an amplifier can be modeled with a beam splitter.
For now, let’s just assume it is possible to model it this way. The effect of a beam
splitter is described by the relation between the input and output ports of the beam
splitter [26, 28, 76, 77].


 
√
− 1 − η  a 
 


= √
 
√
b0
1−η
η
b




a0

√
η

(5.59)

where a represents the operator of the modes we want to amplify, and b the operatorof the noise that always accompanies dissipation. The effect of an amplifier with losses
on an input signal is then modeled as an ideal lossless JPA with a beam splitter at its
input, as shown in figure 5.11. One port is the signal port and the other port is just anoise-port (this can also be considered a model for an attenuator). Assuming that bothnoise-ports have the same spectrum (same temperature Tl ), the output power at thequadrature Xθ (ω) is given by
1
1
SXout,θ ,Xout,θ (ω) = η 2 Gθ (ω) Sa1,in ,a1,in + (1 − η)(ηGθ (ω) + 1) Sal,in ,al,in22(5.60)

where Gθ is given by the lossless case of a JPA, Eq. (3.75). For the amplified quadrature,
X1 , and large gains, the added noise is given by
AN='

SXout,θ ,Xout,θη 2 G11−ηSal,in ,al,inη(5.61)

where the second line has been expanded to the leading order in G1 . For the squeezed
quadrature X2 , again in the case of large gain, the output quadrature noise is
SXout,θ ,Xout,θ ' (1 − η)Sal,in ,al,in .

(5.62)

Both equations, Eqs. (5.61) and (5.62) can model Eqs. (5.56) and (5.58) by setting thevalue of the beam splitting transmission to
η=

γ1
.
γ

120
Note, however, that this is only valid at high gains. The functional form of the output
power spectrum evaluated from Eqs. (5.54) and (5.60) will in general not be identical.
The exact form depends on where the losses are embedded in the microwave network [8].
However, the beam splitter model has the advantage of having closed forms that can beused to understand semi-quantitatively the effect of losses in the detection of a signal
and the squeezing of noise.
(a)

(b)

1

AN

ξ

0.2

0.5

0.1

0

0

10
G (dB)
1

20

0

0

10
20
G (dB)

30

1Figure 5.12: (a) Squeezing ξ as a function quadrature gain G1 for the case of cavity
with loss γ1 = 13.6γl . The blue-shaded area represents all the possible values that ξ cantake for a given gain. The dashed-black line represent the theoretical high-gain limit
given by Eq. (5.58) and the red line represents the squeezing ξ according to the loss
model of a beam-splitter. η is given by γ1 /γ. (b) Added noise AN as a function of G1
for the same case as in (b). The dashed-black line represents the theoretical high-gain
limit according to Eq. (5.56). The dashed-cyan line represents the effective added noisea linear cavity would have just due to its attenuation on resonance |Γ0 |2 , where Γ0 isgiven by Eq. (5.28). The value of γ1 = 13.6γl was chosen based on the fact that mostof the data shown in chapter 6 for the λ/2-resonator was for a frequency at which theresonator linewidths had these particular values.

Finally, one phenomenon that has been observed in these microwave cavities is
power-dependant loss. Power-dependant loss has been observed in superconducting resonators [69,70,78] where the loss decreases as the power of the probing tone is increased.
However, the effect observed in our resonators has the opposite power-dependance: as
we increase the probing microwave tone power, the loss of the resonator seems to in-121crease. We do not have an explanation of this phenomenon. Although we do not know
the specific process that causes this nonlinear loss, we can still model it as it is done
in Ref. [38]. The most obvious effect of this nonlinear loss is a deviation from the expected behavior of the resonance frequency (Fig. 5.13). In a nonlinear resonator with
only linear loss, as we increase the pump power, the Kerr nonlinearity makes the resonance frequency decrease according to the equation ω0 − ωm + KN = 0, where ωmis the frequency at which Γ is minimum, K is the Kerr constant, and N is the energy
stored in the resonator in number of quanta [38]. In the case of zero non-linear loss, thepump-amplitude dependance of the resonance frequency would be purely quadratic:
!2
Ã
b1,in
8γ
ωm − ω0 = − √3 3 bc1,in
where bc1,in is the critical amplitude of the pump given by (5.49). Nonlinear loss ismodeled by defining a nonlinear loss coefficient γnl such that the total loss is given byγ = γ1 + γl + γnl N.
In this case the simple quadratic dependance of the resonance frequency ωm is no longer
valid, and it then becomes a slightly more complicated expression:
ωm − ω0 = −γ[g(x) − 2]3ε(5.63)

where ε = γnl /|K| (See Ref. [38]). The function g(x) is given by:√
1 + (|x| − x2 − 1)2/3√
g(x) =(|x| − x2 − 1)1/3
and x is given by36
1 + ε2√x = −√ ε
3 (1 − 3ε)3

Ã

b1,in
bc1,in!2− 1.

The observed effect due to nonlinear loss in the first experiments using aluminum resonator was not small, thus a nonlinear term was included in their analysis. However,for the second generation of devices (niobium resonators) the nonlinear loss was small
so I will not include it in their analysis.

Γi

(a) 1

0

(b)

0

∆ωres/γ

122

−1

−2

−1
−1

0
Γ

r

1

−3

0

0.5
b1,in /bc1,in1

Figure 5.13: (a) Parametric plot of the real and imaginary part of the reflection coefficient Γ as a function of frequency for 4 different values of pump amplitude b1,in :b1,in /bc1,in ' 0 (blue), b1,in /bc1,in = 0.5 (red), b1,in /bc1,in = 0.75 (green), and b1,in /bc1,in = 1(black). The resonator does not have any linear loss, γl = 0, and the nonlinear loss is
ε = 0.2. As the the pump power is increased, the circle that represents the reflection
coefficient shows an increase of the losses represented by the reduction in size of the circle. (b) Change of resonance frequency ωm as function of normalized pump amplitude
b1,in /bc1,in for ε = 0 (red) and ε = 0.2 (blue).

Chapter 6
Experimental results of JPAs

In previous chapters I have laid the necessary background to understand the experiments that I will now describe. But before I describe the experiment, it is necessary
to discuss the different experimental techniques used to characterize the amplifier.
The experiments I now discuss are, in a sense, very similar to quantum optic
experiments, with the very important difference of the frequency of light used. Maxwell’sequations do not show a preference for a particular frequency, but practical issues domatter when it comes to doing the experiments. The frequency band where we operate
is between 4 to 8 GHz. Higher microwave frequencies imply a more challenging design,
as parasitic reactances are harder to prevent and parasitic resonance modes can affectthe measurement of the Josephson parametric amplifiers. Lower microwave frequencies
could be used. However at much lower frequencies, even at the temperatures of a dilution
refrigerator (Tfridge = 10 mK) it would not be possible to achieve a true vacuum stateof the electromagnetic environment (kB Tfridge /~ ' 2π × 200 MHz).
In this chapter I will first discuss the requirements to make our measurements
at cryogenic temperatures and the apparatus used to analyze the measurements atroom temperature. Then I will discuss, in general terms, all the measurements used
to characterize the Josephson parametric amplifier, and then discuss individually thedifferent samples that we have tested.

124

6.1

Instrumentation and measurement techniques
Performing quantum optics experiments at GHz frequency demands careful fil-tering and attenuation of microwave signals generated a room temperature. First thethermal energy at room temperatures kB Troom is much greater than photon energy ~ω
at microwave frequencies (300 GHz> ω >300 MHz). In addition, Josephson junctions,which provide the crucial dissipationless nonlinearity at microwave frequencies, only
work at low enough temperatures for the metal used in the fabrication to become a superconductor. These facts imply that the experiments discussed here are all performed
at cryogenic temperatures, with temperatures as low as 10 mK. Thus, there are strict
filtering requirements in order to ensure that the electromagnetic environment that the
sample sees is mostly devoid of thermal photons.

6.1.1

(a)

Microwave measurement at ultra-low temperatures(b)
4K plate

Still (800 mK)
100mK plate

mixing chamber
plate (15 mK)
Figure 6.1: The dilution cryostat is a Cryoconcept system. (a) The full cryostat isshown with a typical measurement setup. The metallic cylinder around the dewar is aµ-metal magnetic shield. (b) The main part of the fridge sits inside a vacuum can (notshown). The sample is located at the coldest stage, the mixing-chamber plate.

125
All the experiments I describe in this thesis were performed in a Cryoconcept
200 µW dilution cryostat with a base temperature of 10 mK (Fig. 6.1). The cryostat
contained a Nb-Ti magnet capable of generating magnetic fields as high as 9 T. But
it was only used to apply small fields (a few gauss) to tune the critical current of the
SQUIDs inside the resonator. However, for the most recent experiments, the 9-teslamagnet was removed and replaced by a much smaller magnet manually built using
superconducting wire with a copper core (Fig. 7.8). Due to certain problems withthe cryostat, its performance is somewhat degraded and the base temperature is only
15 mK.
Launching and sample box

The sample was placed at the mixing chamber stage (Fig. 6.1b). The chip whichcontained the device was located inside a light-tight sample box where a printed circuitboard (PCB) with the appropriate traces was located (see Fig. 6.2a). The sample box
is a gold-plated OFHC (oxygen-free-high-conductivity) copper enclosure that provides
a shielded environment to the chip as well as to the circuit board to which the chip iswirebonded (Fig. 6.2b).
The main purpose of the PCB is to implement the transition of the signals fromthe semirigid coaxial cables that transport them, via connectors, to the circuit board.
The transition from the cables to the circuit board is accomplished using a right angle
surface mount SMP connector from Rosenberger. These connectors have good performance up to 20 GHz. In order to suppress parasitic modes between the ground planes,
the top ground plane is shunted periodically to the bottom plate using vias as shownin figure 6.2b. However, this does not prevent possible microwave sample-box modes.
Initially, we had not been concerned about this possibility, but measurements performedby other members of our group [79] found that the sample box initially used for themeasurements did contain 3 resonant modes in the 4-9 GHz range (Fig. 6.2c). Another

126

(a)

(b)

(c)
-40

S21 (dB)

-60
-80
-100
-120
-140
4

5

6
7
Frequency (GHz)89Figure 6.2: The sample box is a light-tight enclosure. It was built out of highconductivity-oxygen-free copper and gold-plated for maximal thermal conductivity. The
circuit board has 4 coplanar waveguide lines, 3 lines for coaxial dc-lines, and 8 lines for
dc-wires. This last lines have not been used yet, so a copper plate is screwed on the lidof the sample box to keep it light-tight. (b) A chip is glued down to the sample box with
GE-varnish or rubber cement. It is electrically connected to the PCB with aluminum
wirebonds. The chip is placed in a cut-out in the circuit board to align the traces on
the PCB and on the chip as closely as possible. (c) The sample box was cooled down to77 K with no PCB or chip inside. Using ports (1) and (2) shown in (a) the transmission
parameter S21 was measured showing three well defined resonant modes of the sample
box.

127
concern is the impedance mismatches when coupling to the chip using wirebonds. The
inductance of these wirebonds is approximately proportional to their length. Their inductance per unit length is of the order of the free space value, µ0 = 4π × 10−7 H/m.
In order to minimize the length of the bonds as much as possible as well as match themodes of the chip and circuit board as closely as possible, the CPW traces of the chip
are aligned to the CPW traces in the PCB (Fig. 6.2b). Also, the chip is located in ahole cut out in the circuit board that closely matches the size of the chip (Fig. 6.2a).
The circuit board is connected to the chip using as many wirebonds as we can fit aroundin the center connector (2-3 wirebonds depending on the graduate student wirebonding
skills) and the skinny ground-planes (5-8 wirebonds in each skinny-ground-plane), tolower the inductance as much as possible.
Filtering and attenuation

As I mentioned at the beginning of the chapter, filtering is an integral part of any
cryogenic experiment. In our experiments, all the signals (both dc and rf signals) are
carried using semi-rigid coaxial cable. Even though the sample box provides a shieldedenvironment, the cables provide a path for the noise to couple to the JPA. In general,
we only have rf-lines. However, for some diagnostic measurements, we use dc-lines.
The rf-lines are divided into input and output lines. The filtering and attenuation of input rf-lines is related to the removal thermal noise from room temperature.
At frequencies of interest (4-8 GHz), the power has an exponential dependance withtemperature. Hence the filtering requirements become very important. At a frequencyof 8 GHz and temperatures above 400 mK, thermal Johnson noise will be significant.
Since the signal and noise are in the same band, in the case of the rf-lines rather thanfiltering we attenuate the noise coming from room temperature. The amount of attenuation implies a compromise, as the more attenuation one puts, the more power one
has to be able to generate at room temperature to send down the heavily attenuated128
input lines. Also, if too much attenuation is used, that means there will be a significant
amount of power dissipated at the different cold stages of the cryostat. This dissipatedpower may be so large, that the temperature of the cryostat may increase substantially.
Ideally, the necessary attenuation is the ratio of the black body spectra at the twotemperatures [80] of interest, 10 mK and 290 KÃ!
ehf /kB T1 − 1
A(f ) = −10 log.
ehf /kB T2 − 1
A frequency of f = 8 Ghz implies a total of 140 dB of attenuation, which is from a practical point of view, impossible. The attenuation that we use is much less than that. Nevertheless, the estimated quanta of noise per unit bandwidth.SX,X = 1/2 coth(hf /kB T )is still close to the 1/2 of vacuum noise1 In order to go from room temperature (290 K)
to liquid helium temperature (4 K), a 20 dB attenuator is placed at the 4 K plate.
To go from 4 K to 15 mK another 20-40 dB of attenuation was used, depending on
the experiment. The attenuators were placed at the mixing-chamber plate. The cables
provide an additional 10 dB of attenuation, for a total of 50-70 dB. In spite of the large
amount of attenuation, the power of the tones sent to probe the JPA was low enoughthat heating of the mixing chamber stage was not observed. When only 40 dB of attenuation was used, the estimated incident noise in the device was Sin = 0.65 − 0.7 quanta.
When 60 dB was used, the thermal noise due to the attenuated noise coming from room
temperature was insignificant, with Sin = 0.502 quanta.2
The attenuation of the rf-input lines is required to remove classical noise frommicrowave tones. In our case, we use an Agilent E8257D microwave analog signal
generator. However, the signals at the output of these microwave generators are notcoherent states, having large amount of phase and amplitude noise, as can be observed in1

In the rest of chapter 6, 7 and appendix B, I will change the convention to name the single-sided
spectral densities of a quadrature SX,X . The main reason is that I will use the subscript index to referto the different possible locations in the experiment setup.
2
As we will see in this chapter, depending on the attenuation used for the noise-calibration line, themain source of thermal noise can be the losses of the microwave switch used to estimate the added noiseof the amplifier.

129
their data sheet [81]. In order to obtain a coherent state at the sample inside the dilution
refrigerator, we must attenuate the output of the microwave generator to remove thisextra phase and amplitude noise. The amplitude and phase noise are defined relative
to the carrier, in units of dBc/Hz. Thus, they are approximately independent of thecarrier power. Nonetheless, the voltage fluctuations associated with the AM or PM
noise do depend on carrier power. It is the critical power of the parametric amplifierthat determines the amount of attenuation required to reduce the amplitude and phasenoise of the generator below the level of vacuum fluctuations. In the case of the Niobium
resonators, the critical power using the strongly coupled port in the case of the halfwavelength resonator, or using the only port in the quarter-wavelength resonator, isabout -90 dBm or less. From the data sheet of the signal generator, for an 8 GHz
carrier tone the manufacturer’s estimate of phase noise is -120 dBc/Hz at 100 kHz
offset from the carrier. The phase noise decreases at larger offset frequencies; i.e. it is-140 dBc/Hz at a 500 kHz offset. The amplitude noise is always smaller than the phasenoise. The question is how does the amplitude and phase noise of a -90 dBm carrier
tone compare to the noise of vacuum flucuations at 8 GHz, ~ω/2 = 2.65 × 10−24 W/Hz?
At 100 kHz, -210 dBm/Hz is 1 × 10−24 W/Hz, or slightly more than 1/3 of vacuum
noise. However, at 500 kHz, the extra phase noise is 2 orders of magnitude smaller,
which for our purpose is rather insignificant.3
Filtering of the dc-lines is very different. The only kind of dc-measurement we
performed was current versus voltage (I-V) curves. These measurements do not require
high bandwidth, but they employ much higher voltages. We use low pass filters withvery low attenuation at dc, but very large attenuation at microwave frequencies wheremost of the thermal noise power is located. There are many different types of low pass3

Note however that when performing a homodyne measurement with a mixer, if both the LO andpump of the JPA come from the same signal generator, we become immune to the phase noise in thedetection, but when two different signal generators are used, we will filter the low frequency phase noiseat the output of the detector.

130
(a)

(b)

-50

21

|S | (dB)

0

-100
6

10

8

10
frequency (Hz)10

10Figure 6.3: (a) Picture of one of the low pass powder filters (PF) used for the dcmeasurements in our cryostat. (b) |S21 | as a function of frequency for the filter shownin (a). Transmission data from 100 MHz to 40 GHz was taken with the Anritsu 37269B
network analyzer. Data below 100 MHz was taken with a Hewlett Packard 8712B network analyzer. While this data was taken at room temperature, there is only a smallchange in when used at cryogenic temperatures.

filters available but for this experiment we use copper powder filters (Fig. 6.3a) [82].
These filters consist of a long (1 m) copper wire that is coiled and immersed in epoxy(Emmons and Cummins 2150) and copper mesh powder (spherical APS 10 microns)
mixture. This structure is housed in a copper box with SMA connectors at both ends.
The high frequency loss comes from the powder itself, and the filters achieve attenuation
greater than 80 dB at frequencies greater than 1 GHz (Fig. 6.3b). These filters have acutoff at around 5-10MHz.
Separation of input and output signals

The Josephson parametric amplifier is operated as a reflection amplifier, and inorder to characterize it, we need to be able to separate the incoming and outgoing signals in the same cable. This is accomplished using directional couplers and microwave
circulator. Directional couplers are passive microwave components used for power division or power combining (Fig. 6.4a). A directional coupler is a four port network with

131
(a)

(b)
2
CoupledP3
3

Isolated
P4
4
1

1
Input
P1

C
1-C

2
Through
P2

3

Figure 6.4: For the separation of the input and output signals, two microwave devices
are used. (a) The directional coupler is characterized by the coupling C, the directivity
D and the isolation Is. The coupling indicates the fraction of the input power P1 thatis coupled to port 3 (P3 ). The isolation is a measure of the directional coupler’s ability
to avoid power-leakage from port 1 into port 4 (P1 /P4 ) as is the directivity (P3 /P4 ).
The ideal directional coupler would have infinite directivity and isolation, so it would
be completely characterized by the coupling factor C. (b) A circulator is a passive
non-reciprocal three-port device. Power entering any port is ideally only transmitted to
the next port in rotation. In reality, some power leaks to the other port. The amountof leakage power is determined by the isolation Is of the circulator.

one of the ports usually terminated with a 50 Ω resistor; it is the microwave analog ofa beam splitter. The basic operation of an ideal directional coupler is the following.
Power supplied to port 1 is coupled to port 3 with the coupling factor of C, while theremainder (1 − C) is delivered to port 2. In an ideal directional coupler, no power is delivered to port 4. The directional coupler used in our experiment was a MAC C2056-20,
with C = 0.01 = −20 dB of coupling.
A circulator is a three-port non-reciprocal device (Fig. 6.4b). In an ideal circulator, all the power entering port 1 will exit port 2, powering entering port 2 will exit port
3 and powering entering port 3 will exit port 1. And assuming all ports are matched, nopower flows in the reverse direction. The nonreciprocal nature of the circulator requiresa magnetized material to break time reversal symmetry; therefore, they emit large straymagnetic fields.4 For that reason, we place them outside the magnetic shield far from4

Shielded circulators can be purchased, but the shielding is big and bulky. This bulk can be a

132
the sample. The circulator used at low temperatures is a Pamtech CTH-1394K. It is
specified to work at low temperatures. It provides 18-30 dB (a specified minimum of
18 dB) of isolation and has a bandwidth of one octave (4-8 GHz).
Low temperature amplification(a)=
AN ,1 , G1(b)

AN,2 , G2

AN,n , Gn

Asys , Gsys(c)

Figure 6.5: (a) The effective noise temperature of the whole chain of amplifiers is calculated by referring each amplifier’s noise to the input of the entire chain. Ideally, thefirst amplifier has enough gain that it sets the noise-floor. (b) Picture of the HEMT
amplifier. It was purchased from Sandy Weinreb’s group in Caltech. (c) Gain and noise
temperature measured by the manufacturer at 12 K of both HEMTs our group owns.
The used amplifier performance is shown as the red-trace.

As I mentioned in chapter 2, any phase-preserving amplification must necessarily
result in an increase in the noise of the measurement. If there are several amplifiers inseries, the first stage of amplification is the most important in determining the noisenuisance in the small space inside a dilution refrigerator.

133of the measurement. Ignoring possible losses between different amplifiers, the added
noise of the amplifier chain with n amplifiers can be calculated by referring the total
output noise at the output of the chain to the input of the first amplifier (Fig. 6.5a). By
dividing the total output noise by the total gain, then it is easy to see that the effective
added noise Asys is given by
Asys = AN,1 +

AN,2
AN,n
+ ... +.
G1
G1 G2 ...Gn−1

Ideally, the first amplifier is operated with high enough gain such that it sets the noise
floor of the detector, Asys ' AN,1 .
Our first amplifier (ignoring the JPA) is a high-electron-mobility-transistor (HEMT)
amplifier with a noise temperature of TN = 5 − 10 K, depending on the frequency (Fig.
6.5b). We did the appropriate tests to ensure that less than three percent of the system
noise was contributed by the amplifiers and components that follow the HEMT amplifier. It is thermally anchored to the 4 K plate, the lowest temperature at which it ispossible to dissipate the 20 mW of amplifier bias power without raising the temperature.
In order to prevent the noise emitted by the input of the HEMT amplifier from exciting
the JPA, a circulator with the third port terminated with a 50 Ω resistor is placed atbase temperature (in this configuration, the circulators are called isolators). Thus, thenoise seen by the device would ideally be the Johnson noise of the 50 Ω resistor.
Thermal anchoring

One final situation that deserves some discussion is the thermal anchoring of
the cables. At low temperatures, thermal conductivity, especially due to phonons, issubstantially reduced. This means that care must be taken to ensure that cables andcomponents are properly heat sunk. The outside of the cables is easily heat-sunk to
the different stages5 by using brackets that are attached to the plates of the cryostat
5

The outer conductor is anchored to the all the plates 4 K, still, 100mK as well as the mixing
chamber in order to minimize the total heat load.

134
as shown in figure 6.6a. However, a constant problem is how to heat-sink the center
conductor of the cables. The attenuators provide a resistive connection through whichheat can be dissipated from the center pin to the cable shield (and cryostat). Both theoutside shield and center pin of the circulators are heat-sunk as well. The outside isheat-sunk using copper braid and copper brackets, and the center pin is heat-sunk when
they are used as isolators as the third port is terminated with a 50 Ω resistor to ground.
This termination is anchored to the base temperature. In the case of the dc-lines, theheat-sinking is accomplished via the filtering elements. The copper powder filters allow
the center pin to come to thermal equilibrium.
In some instances, we need to thermalize the center conductor of a coaxial cablewithout using an attenuator. In those cases, we use a microwave component called bias
tee. Bias tees consist of an inductor and a capacitor connected as shown in figure 6.6b
and are usually used to supply dc currents or voltages to rf-devices. However, we use
it to heat-sink the center conductors by shorting the dc-port to the fridge ground. The
rf-signals are injected though the rf-port and come out of the rf+dc port with very low
attenuation. The insertion loss of the the bias tee used for this purpose (Anritsu K250)
is about 0.5 dB at room temperature at the frequencies we used.
In order to thermally isolate the different stages of the fridge the choice of semirigid cable is also important. For this purpose, we can divide the cryostat in 4 sections.
For the cables from room temperature to the liquid-helium bath, we use stainless steel
outer, silver-plated-copper-weld (SPCW) inner conductor. In the helium bath, we used
cable with copper outer and copper, or SPCW, inner conductor. The cables chosen to
go between the 4 K plate to the mixing chamber must be of a low thermal conductancematerial for both the center pin and the outer conductor. Stainless steel inner andouter conductor (ss-ss) cables were used for the input lines. Unfortunately, its highthermal resistance comes with a large electrical resistance making them quite lossy, and135
attenuating a signal around 2 dB/ft at 8 GHz.6 This extra attenuation is not a concern
for the input lines, but in the output lines it would increase the noise temperature of
the amplifying chain. For the output lines, superconductor cables, niobium, or niobiumtitanium (NbTi) alloy, were used to avoid the attenuation of the stainless steel cables.
(a)

(b)

HEMT
rf+dc
Copper heat-sinksdc

Stainless steel
coaxial cable
NbTi coaxial cablerf

Figure 6.6: (b) Heatsinking of the semi-rigid cables is done using copper-brackets machined from OHFC copper. The HEMT is heatsunk with a copper bracket for the outershell. The center pin is thermalized to 4-K using a bias-tee with the dc-lines shortcircuited. (a) Picture and schematic of a bias tee used for thermal sinking. The model
used is an Anritsu K250.

6.1.2

Microwave measurements at room temperature

The generation of microwave tones, in order to probe the behavior of the JPAs,
is accomplished either using the microwave signal generators I described in the previous section, or using a vector network analyzer (Anrtisu 37269B). The vector network
analyzer is a two-channel microwave receiver, which is used to measure the scatteringparameters of a microwave network. It is designed to provide phase and magnitude
6

This value was estimated based on a ss-ss cable cooled down to 77 K.

136
information by generating a signal at one of its ports and analyzing the power transmitted to a second port and the power reflected back to the first port. However, for mostof our measurements, the microwave tones are created by signal generators. To create multiple copies of the same tone, we split the generators output using 3-dB power
splitters and we attenuate and phase-shift the tones for their different purposes: thepump, the mixer LO, the small gain-calibration tone and the carrier-cancelation tone.
For the carrier cancelation tones used in experiments described in chapter 7, variableattenuators Arra 5814-20A-1A and variable phase shifter Arra 9425A are used to set
the appropriate amplitude and phase.
(a)

(b)
S(ω)

ωIF

X2

Low-frequency
microwave toneIF
RF
LO
High-frequency
microwave tone

S(ω)

S(ω)

ωIF
ωLO

amplitude-modulated(am) test tone

φ
X1

ωLO

Figure 6.7: (a) Up-conversion using a mixer. A low frequency tone enters the IF-port
and is multiplied by a the local oscillator (LO). The output of the mixer is an amplitude
modulated signal with power in its spectrum at frequencies ωLO ± ωIF . (b) In principle,
two different kind of modulated signals can be used to probe the phase-dependent gainof the JPA: amplitude-modulated (blue) and phase-modulated (red) tones. For our
experiments, we only used AM-tones. However, in chapter 8, the signal I will amplify
using a JPA consists of phase-modulated information on top of a large carrier (red).
The only difference is a 90◦ phase shift in the phase dependance of the JPA gain.

For the detection of the signals at room temperature, we use commercial spectrum
and network analyzers. A spectrum analyzer gives a frequency-domain representationof an input signal and displays the average power spectral density as a function of

137
frequency. The spectrum analyzer we used was Hewlett Packard E4407B spectrum
analyzer. We also implement our own analyzer using a mixer and digitalization cards.
A mixer is a device used to provide frequency conversion between an input signal andan output signal. We use mixers for both up-conversion and down-conversion. Upconversion refers to the process of taking a low-frequency signal and transforming it intoa higher frequency one (Fig. 6.7a). There are two options to do this, using amplitude
modulated or phase modulated signals (Fig. 6.7b). We chose amplitude modulated
signals for the test-tone to measure the phase-dependant gain (the only difference inusing a phase modulated signal is that the maximum gain would occur with a 90◦ phaseshift in the LO). The output of the mixer is an AM microwave tone with the carrier
strongly suppressed. It is basically the same as the AM-tone shown in figure 6.7b, withthe exception that the modulation amplitude is large compared to the average value ofthe carrier (indicated by the red phasor). The up-conversion in the case of an AM-tone
is equivalent to multiplying the LO tone by the input of the IF-port. The output port in
the up-conversion is the RF-port. The microwave tone at the RF-port A(t) cos(ωLO +φ)is given by
A(t) cos(ωLO + φ) = cos(ωIF ) × cos(ωLO + φ)=1[cos(ωLO − ωIF + φ) + cos(ωLO + ωIF + φ)] .
2

(6.1)
(6.2)

In Eq. (6.1), it is clear that even though the signal consists of two tones at different
frequencies ωLO ± ωIF , the phase of the signal is given by the LO phase φ, and theamplitude of the signal is A(t) = cos(ωIF ), a slow function of time (ωIF ¿ ωLO ). The
AM-tone generated allows us to apply a signal with the appropriate linear combinationof input fourier modes (Eq. 3.79) in order to measure the phase-dependance of the JPA
gain Gθ . The mixer used for the up-conversion is a Mini-Circuits ZMX-10G.
For the detection of microwave fields, the mixer is used for down-conversion. In
the down-conversion process, the input is the RF port and the output is the IF port.

138
Next a brief and qualitative description of the down-conversion operation of a mixer isdiscussed; for a full explanation of the operation of a mixer, the reader is referred toRef. [16]. For a given RF and LO microwave tone frequencies, two nominally equalamplitude output signals are produced at the sum and at the difference of the RF and
LO frequencies. This is effectively the same as saying that the mixer output consists of
the result of multiplying the input signal A cos(ωs t + φ) by the LO
1
A cos(ωs t + φ) × cos(ωLO t) = A [cos([ωs − ωLO ]t + φ) + cos([ωs + ωLO ]t + φ)] .
2
In the down conversion process, the second term is removed by a low pass filter. In
most of our cases ωLO = ωs . This case is usually referred in the literature as homodyne
detection, and the case where ωLO 6= ωs is called heterodyne detection. We use a pair ofmixers configured in a particular way to form a so-called IQ-mixer. In this configuratoin,
the LO is split in two tones and one of them is phase-shifted by 90◦ . Then, the input
signal at the RF-port is also split in two and each multiplied by the two LO-tones. The
two output ports are given byI(t) = B(A(t) cos(φ))(6.3)

Q(t) = B(A(t) sin(φ))(6.4)

where B is just the mixer gain.7

From Eqs. (6.3) and (6.4), we can then deduce that

the I-port just gives the X1 quadrature and the Q-port the X2 quadrature relative to
the LO (Fig. 6.8). With this down-conversion process, the amplitude and phase of theoriginal signal have been mapped onto two lower frequency signals. Both, the amplitude
7
In reality, the IQ-mixers are not ideal and contain both phase and amplitude unbalance. That isthat the real outputs of the I and Q port are given byI=

I0 + BI A(t) cos(φ)

Q=

Q0 + BQ A(t) sin(φ + ϑ)

where I0 and Q0 are dc-offsets, BI 6= BQ and ϑ 6= 0. In the cases that it is necessary, such as when
implementing a network analyzer with the mixer, this imperfections are corrected in the computer by
calibrating the mixer before taking the data. The calibration is similar to the one explained in appendix
F of Ref. [83].

139
A (AM-tone) and the phase φ (PM-tone) of the signal can be a slow (compared to ωs )function of time (Fig. 6.8b). In this case, the spectral content associated with the slow
variations in A(t) and φ(t) appear with a frequency offset ω as the spectral content at
the I and Q ports of the mixer (Fig. 6.8b). The mixers used for the down-conversion
are Marki Microwave IQ0307LXP and IQ4509LXP.
The output of the I and Q ports are first amplified (to compensate for the 6 dB
conversion-loss of the mixer), low-passed filtered by an anti-alising filter that sets the
bandwidth of the measurement, and then digitized using a digitizer card (DAC). For
all the experiments in this chapter, a NI-6122 and an Acquiris DP-240 card were used.
The NI-6122 had a maximum sampling rate of 500 kHz and a 16-bit resolution. This
card was used for the aluminium resonators, which had much smaller bandwidth. The
Acquiris DP-240 was used for the niobium resonators since it had a maximum sampling
rate of 1 GHz, but it was an 8-bit resolution card. For the experiments of chapter 7, an
Alazar ATS460 card with 14-bit resolution was used in order to improve the linearity
and accuracy of the measurements.

6.2

Characterization of the resonator
In order to characterize the resonators, we need to perform multiple kinds ofmeasurements. The first ones I will describe are dc-measurements of test junctions. I
will then describe the measurements done on the JPA that test only its linear and non
linear resonator behavior. Finally I will describe all the tests made to characterize theamplifier performance.

6.2.1

dc-Characterization of Josephson junctions

In order to estimate the resonance frequency of the resonator, I need to know thetotal inductance of the Josephson junctions. For that, we placed a few test junctions in
the chip. Once fabrication is completed, whether it is done by me or our colleagues at

140

(a)

(b)

Low-frequency
quadrature-phase (Q)output
Q

Q
S(ω)

ω
ωLO

RF
LO

φ
I

amplitude or phasemodulated
input tone

I Low-frequency
in-phase (I)outputS(ω)ω

High-frequency
Local-oscillator tone

S(ω)

ωLO

Figure 6.8: (a) Signal entering the RF-port of the mixer. Besides the carrier (redphasor), the signal contains information in the form of amplitude modulation (blue)
and phase modulation (red). The projections of those modulations onto the I-axis areshown. (b) Down-conversion process. A tone is applied to the RF-port of the mixer.
Regardless of whether it is phase or amplitude modulated, its spectrum has two tones
equally spaced from the main carrier. After going through the mixer the information
contained in those tones are shifted down in frequency. The main carrier (red) willappear at dc (ω = 0), while the side-tones will appear at a frequency ω. The magnitudeof the power at ω will be proportional to the projection of the modulation onto the I
or Q channel.

141
NIST, these test junctions are probed at room temperature to ensure that the resistances
are as designed. This is done with a probe station connected to the same equipmentused for the dc-measurements in the cryogenic experiments (Fig. 6.9a). Two kind of test
structures were probed: single SQUIDs and series arrays of SQUIDs. Particular caremust be taken not to damage the devices with electrostatic discharge during handlingand dc measurement. Single SQUID test structures are the most susceptible to damage.
The arrays are more robust.
The dc measurements setup is shown in figure 6.9a. All measurements were
single-ended using coaxial cables (either BNC when probing the junctions at room
temperature, or semi-rigid cable with SMA or SMP connectors when tested in thecryostat). The devices are wirebonded to the cryostat ground through the PBC ground.
The I-V curves are taken by measuring the voltage drop across a series bias resistor atroom temperature. In order to have a very stable dc-bias, a Yokogawa 7651 dc-voltage
source is used. Using low noise instrumentation amplifiers, the voltage across the bias
resistor and across the device are measured with Agilent 34401A digital multimeters.
When their I-V curves are measured at 15 mK, we can observe the characteristic featuresof Superconducting-Insulator-Superconducting junctions (Fig. 6.9b).
There are essentially three distinct regimes. The first region is the dissipationlesssupercurrent at V = 0. In the second region, the subgap region 0 < |V | < 2∆/e, thecurrent drops many orders of magnitude. As the voltage bias reaches 2∆/e, there isa sudden increase of quasiparticle tunneling current. At higher voltages, the dynamic
resistance asymptotically approaches the normal state resistance Rn of the junction
above the transition temperature Tc [33]. Our dc-measurement measures all three regions, providing a direct measurement of the two main parameters for characterizing
a junction, Rn and I0 . Yet, there is a few caveats I should mention. The two sharp
features of the I-V curves, the gap-rise current at V = 2∆/e and the supercurrent at
V = 0 are very susceptible to noise. Specifically, the supercurrent itself is an extremely

142
(a)

DMM
Voltagedivider

Vbias

Rbias

+
_
DMM

DUT

(b)

I

Superconducting
current

V2∆/e

Figure 6.9: (a) Setup for the dc-measurements of the test SQUIDs. The output of afloating voltage source (Yokogawa 7651 dc-voltage source) is sent to a “bias-box”, shownas the blue dashed-line box in the figure. In the bias-box, the voltage is attenuated witha voltage resistive-divider. The attenuated voltage is then connected to the device
under test (DUT) via a bias resistor Rbias . The voltages across the bias resistor andthe device are amplified for simultaneous measurement of the current and voltage andmeasured with two digital multimeters (DMM) Agilent 34401A. The output of the biasbox is also splitted and sent to two unity-gain-isolation amplifiers, referred to as theiso-box (and shown as the red dashed-line box in the figure). The iso-box is used toprevent a ground-loop between the oscilloscope and the cryostat. (b) Characteristic I-V
curve of Superconducting-Insulator-Superconducting junction with a normal resistanceof Rn , (blue) and the expected linear response of a resistor with the same value. The
measurement of the I-V curve using the setup just described would use a load-line(green) with a slope of 1/Rbias .

sensitive threshold detector and spurious noise can make the junction to switch from thesupercurrent state to a dissipative state prematurely, that is before the current reaches
the theoretical maximum of I0 . Likewise, the gap-rise also represents a sharp featurethat can easily be rounded. In some cases, due to self-heating, the gap-rise shows a“backbend” caused by an excess of quasiparticles, locally reducing the gap.88This is particularly true in the SQUID arrays.

There-

143
fore, it is important to distinguish the measured maximum supercurrent and the onethat we would theoretically expect from the junction resistance and Eq. (3.17). Also,
there will be features that I will miss in the measured I-V curve, as the load line of
the measurement will determine exactly how close to the supercurrent one can stably
bias (Fig. 6.9b). In order to estimate the Josephson inductance, the resistance of thejunction and Eq. (3.20) is used.

6.2.2

Linear measurements of the resonators
Anritsu
NWA

20 dB

4.1 K

HEMT

20 dB

15 mK
circulator
20 dB
in

20 dB
Φ

JPA

in

directional
coupleroutport 1

port 25 mm

Figure 6.10: Simplified schematic used to measure the S-parameters and nonlinear behavior of the resonator with a Network analyzer. The JPA is measured in a cryostat
held at 15 mK. We can send the testing signal to port 1 or 2 of the resonator to measure
S11 or S12 . In the case of a quarter-wave resonator, it only has one port. Hence only
S11 is measured.

The linear parameters of the resonators are measured using a vector network analyzer using a probing signal with very low power (compared to the critical power of theresonator). By measuring the reflection and transmission coefficient of the resonators,
we can estimate the linear coupling parameters γi and the loss γl . Depending on the144kind of resonator (λ/4 or λ/2), only the reflection coefficient S11 , or both the reflection and transmission S12 coefficient can be measured. A simplified schematic of themeasuring setup is shown in figure 6.10.
Extraction of linewidths, or equivalently the quality factors Q = ω0 /2γ, based on
the S-parameters is done by fitting the S-parameters to the expected transmission andreflection expression. For transmission, we expect a Lorentzian response
S12 =

T0
1 + 2jQ(ω − ω0 )/ω0

(6.5)

where Q is the total quality factor given by
1
1
1
1
+
+
=
Q
Q1 Q2 Qland T0 is given by
2Qc Ql
.
T0 = √Q1 Q2 (Qc + Ql )Qi , i = 1, 2, is the quality factor associated with the ports 1 and 2, Ql represents theinternal losses of the resonator, and Qc is the coupling quality factor associated withboth ports
Qc =

Q1 Q2
.
Q1 + Q2

As it was describe in chapter 5, the expected reflection response, is given by
S11 =Γ0 + 2jQ(ω − ω0 )/ω0
1 + 2jQ(ω − ω0 )/ω0

Γ0 =1/Q2 + 1/Ql − 1/Q1.
1/Q2 + 1/Ql + 1/Q1

(6.6)

where Γ0 is given by

In the case of a single ended cavity, there is not transmission coefficient S21 to measure.
But S11 has the same response, with Q2 → ∞. In order to more accurately estimate
the reflection coefficient S11 and remove delay effects due to the finite length of thesignal path, we first calibrate the latter by tuning the resonator out of the measurement
frequency range. This is done by applying a current to the magnetic coil that tunes

145
the resonance frequency of the resonator. By applying a large enough current, we movethe resonance frequency several linewidths from the frequency range of interest. Then,the resonator behaves as a mirror, reflecting all the signal and allowing us to calibrate
the signal path. Then the resonance frequency is tuned back to the expected frequency.
Finally the two traces are divided.

6.2.3

Non-linear characterization

The final feature of our resonators to probe are its nonlinear characteristics: Kerr
constant and the nonlinear loss [38]. Both of these characteristics are estimated basedon the power dependence of the resonance frequency. The measuring setup is exactly
the same as the one to measure the S-parameters, the only difference being that we vary
the power of the probing signal. As we increase the pump power, the Kerr nonlinearity
makes the resonance frequency decrease according to the equation ω0 − ωm + KN = 0,
where ωm is the frequency at which Γ is minimum, K is the Kerr constant, and N isthe energy stored in the resonator in number of quanta [38]. Defining, K 0 = K/~ω0 ,then the change in resonance frequency is given byω0 − ωm + K 0 E = 0,
where E is the energy of the resonator in energy units and ~K 0 is a unit-less number.
Above the critical power Pc , S11 and S12 are discontinuous, and the resonators response
is bistable. From the onset of this bi-stability we can estimate the critical power, andhence the Kerr constant using Eq. (5.49).
cPi,in= ~ω(bci,in )2 ,==4~ω γ 3√,3 3 γi |K|4γ3√.
3 3 γi |K 0 |(6.7)

Then nonlinear loss is estimated based on Eq. (5.63): when the loss is only linear,

146
the pump-amplitude dependance of the resonance frequency would be purely quadratic:
Ã
!2
b1,in
8γ
ωm − ω0 = − √3 3 bc1,in
where bc1,in is the critical amplitude of the pump given by (5.49). However, when there
exist non-linear loss, this simple quadratic dependance is no longer valid. From thatdeviation, we can estimate ε = γnl /|K| (See Ref. [38]).

6.3

Characterizing the amplifier
The analysis of the parametric amplifier closely follows the theory developed by

Yurke and Buks for parametric amplification in superconducting resonators [38]. In
their model, the Kerr nonlinearity is provided by the intrinsic kinetic inductance of asuperconducting film, while in our case it arises from the nonlinear Josephson inductanceof the SQUIDs. The full characterization of the amplifier implies measuring the signal
Gs and image Gi gain, measuring the correlation between these two gains by observing
the phase-dependance of the quadrature gain Gθ , measuring the added noise of theamplifier, the squeezing of noise on one of the quadratures and finally the saturation of
the amplifier. Next is a brief description of each measurement.

6.3.1

Signal and image gain

Once the critical power has been found by finding the bifurcation point, then we
can set the pump power as close to it as we want in order to generate the appropriate
gain. Then, it is a matter of sending a small test signal to measure the amplifier gain.
The setup used is shown in figure 6.11. The schematic is drawn assuming a λ/2-resonatorand pumping the resonator through the weakly coupled port. However, the pump could
also be applied though the signal port. This is the way the λ/4-resonator is pumped.
The only change in the setup is the requirement to use a power-splitter to add the twotones, the test signal and the pump, outside the cryostat. Two signal generators create

147
Spectrum
Analyzer
NWA
pump

290 K

20 dB

4.1 K

20 dB HEMT

15 mK

20 dB

20 dB
Φ

test signal
JPA

in

out
port 1

port 25 mm

Figure 6.11: Schematic of the setup used to measure the signal and image gain. Two
microwave generators are used to study the JPA. One generator creates the pump,while the second creates a gain-test tone. The pump tone is injected through port 2of the JPA, while the calibration tone is incident on port 1 of the JPA through a 20
dB direction coupler. Signals emerging from port 1 pass through a circulator and arethen amplified by a HEMT amplifier and a room temperature amplifier before entering
input of a spectrum analyzer. Another setup to measure the gain only uses one signal
generator and a network analyzer. The generation of the probing tone and its detection
is done by the network analyzer.

two tones, a pump at frequency fp and a signal at fs . The two tones are injected into
the dilution refrigerator. The pump is attenuated by 20 dB at 4 K and at 15 mK.
The signal is attenuated by 20 dB at 4 K and a directional coupler at 15 mK provides
an additional 20 dB of attenuation and separates the incident tones from the reflected
tones.
The image gain Gi and signal gain Gs can be predicted from the resonator’s
parameters measured using the S-parameters. We define Gs as the ratio between thereflected signal power with the pump on and the incident signal power; Gi is the ratio
between the image tone and the incident signal. Initially, I used a spectrum analyzer todo the measurement. This way I could measure both the direct and image gain. Yet,

148
since both give us the same information, we found it was simpler and faster to use the
network analyzer to measure Gs only. Our network analyzer is not capable of measuring
the tone at the image frequency 2ωs − ωp , making it unable to measure Gi . In orderto calibrate the gain, we do a similar procedure as the one to measure the reflection
coefficient. Once the amplified signal is measured, the resonance frequency is tuned outof the frequency range of interest by applying a different current to the coil, therefore,changing the flux in SQUID loops. Then, we measure the signal reflected from theresonator to calibrate the signal path.
We generally do two kind of signals and image gain measurements: a bandwidthvs gain where we observe the expected behavior of limited gain-bandwidth product(Eq. 3.62); and a measurement of gain versus pump-detuning, which it is only used tocompare it with the theory from Ref. [38] described in the previous chapters.

6.3.2

Phase dependence gain

The phase dependance gain is a consequence of the correlation between the signal
and image gain. This correlation is measured using the setup shown in figure 6.12a.
The correlation between Gi and Gs can be measured by applying a single tone slightly
detuned from the pump and measuring the output in either the I-channel or Q-channel.
The power gain in the I-channel is then given byp p
Gmixer ∝ (Gs + Gi + 2 Gs Gi cos(2θ + 2ψ))

(6.8)

where θ is the LO-phase [8]. The proportionality constant consist of the the gain of
the whole amplifier chain after the JPA and the term between parenthesis is the phasedependant quadrature gain of the JPA (Gθ ). ψ is a phase offset that depends on thesignal-path length and the difference in phase between the signal and image amplitudegain. Without loss of generality and for simplicity, in the rest of the chapter I will assume
ψ = 0. Another way to measure these correlations is to apply the appropriate linear

149
I

(a)
LO

pump

θ

Low-frequency
generator

(b)

Q
rf

IF
LO
290 K

20 dB

20 dB HEMT

4.1 K

High-frequency
generator

sw

15 mK

α

S 4K

(c)
α'

circulator
20 dB

20 dB
test tone
Φ

JPA

in

directional
coupler

Switchρ

outport 1

port 2

amplitude-modulated(am) test tone

S'in

JPA

AJ , GJβ

S outθ
AH , GH

4K
15 mK

5 mm

Figure 6.12: (a) The schematic used to measure the phase dependant gain, the added
noise of the amplifier and squeezing is very similar to the one used to measure thesignal and image gain. The main change is the incorporation of a mixer to perform
homodyne detection of the output of the JPA. The pump tone is injected through port
2 of the JPA, while the calibration tone is incident on port 1 of the JPA through a
20 dB direction coupler. Signals emerging from port 1 pass through a circulator andare then amplified by a cryogenic high-electron-mobility transistor amplifier (HEMT)and a room temperature amplifier before entering the radio-frequency (rf) port of amixer. The noise incident on port 1 of the JPA which comes primarily from the isolated
port of the circulator can be switched (sw) to come from two different resistors held at
different temperatures. The mixer LO comes from either the same signal generator as
the pump, in which case the phase is tuned by a manual phase-shifter or from anotherphase-locked signal generator. In the latter case, the phase is changed by using thephase-knob of the apparatus. (b) The calibration tone can be a single frequency tone ora linear combination of two fourier frequencies around the carrier frequency fpump . The
latter is realized by using a mixer as an up-converter. By using a two-fourier-frequency
signal, we can measure the phase-dependant gain of the JPA. (c) The noise model forthe input noise-power of the JPA consist of two calibrated noise generators in the formof two resistor connected to a switch. The switch located at 4 K has a loss ρ that adds
(1 − ρ)1/2 coth(~ω/kB T4K ) of thermal noise. The output of the switch is attenuatedby an attenuator and the loss of the circulator and goes into the port 1 of a directional
coupler. At the output of the directional coupler and the input of the JPA, the noise isformed by 99% of the input noise plus 1% of the noise coming from the 4 K plate. This
noise, plus the added noise of the JPA is amplified by the JPA and the amplifier chain,down-converted by the mixer and appears at the output I and Q channels. Considering
θ is chosen by
only one of the channels (say I), the noise spectral density measured Sout
the angle θ of the LO. To measure the amplified quadrature, θ = 0, while to measure
the squeezed quadrature, θ = π/2.

150
combination of input fourier modes (Eq. 3.79). As I mentioned in section 6.1.2, there
are two options to do this: using amplitude modulated or phase modulated signals. We
chose amplitude modulated signals with suppressed carrier to test the phase-dependance
gain. The setup to generate this AM-tone is shown in figure 6.12b. The LO of the mixer
used to generate the test-am-tone is assumed to have the same frequency as the JPA
pump; in fact, both tones come from the same signal generator. In Eq. (6.1), it was
shown that even though the signal consists of two tones at different frequencies ωp ±ωIF ,the phase of the signal is given by the LO phase φ, and the effective amplitude of thesignal, A(t) is just a slow function of time (in our case ωIF = 10 − 200 kHz). Note that,even though the two ways of measuring the correlations of the signal and image gain arecompletely equivalent, it is only for signal modes that correspond to particular linear
combinations of frequencies above and below the pump (Eq. 6.2), that the JPA will not
be quantum limited; that is, it will be able to amplify those signal with less added noise
than a quantum-limited amplifier.

6.3.3

Noise measurements

The correlations measured in the phase-dependant gain can also manifest themselves in the form of noise squeezing and amplification, depending of what quadratureof the JPA we measured. So far the analysis of the mixer as done in section 6.1.2 wasrather classical. Now, I use a quantum-language (where the physics variables are now
considered operators) to explain the effect of an ideal homodyne detector. Imagine we
could have an ideal mixer right at the output of a JPA operated in the homodyne modein which ωLO = ωp are equal and in which the input at both the signal frequency ωp + ωand image frequency signal frequency ωp − ω are both regarded as the signal. The
output of the I port in the frequency domain is given byI(ω) = B[b(ω)eiθ + b† (−ω)e−iθ ]151where B is just a normalization factor [8, 84, 85]. If we chose the normalization factor
so that the expected value of the anticommutator of I of a vacuum state is (assuming
a single-sided spectral density convention)h0|{I † (ω), I(ω)}|0i =~ω,2then I gives the quadrature Xθ of the field at the input of the mixer (Fig. 6.13).
I(ω) = X1 (ω) cos θ − X2 (ω) sin θ.
where Xj (ω) are defined as in appendix A. However, at microwave frequencies, there
is not such a thing as an ideal mixer, and homodyne detections are far from quantumlimited. Besides, the mixer is not at all located right at the output of the JPA. Instead,at the output of the I and Q-port of the mixer, what we really have isI(ω) =p

GT (X̂1 (ω) cos θ − X̂2 (ω) sin θ + χI )andQ(ω) =p

GT (X̂1 (ω) sin θ + X̂2 (ω) cos θ + χQ ),where GT is the total power gain of the commercial amplifiers and mixer, θ is the phasebetween the pump and the LO, and χI and χQ are random variables whose variance are
the spectral density of the noise of the amplifier chain. Both will have a variance withan average value of AT , where AT is the noise number [2] of the commercial amplifier
chain expressed as noise added at the output of the JPA.
Let’s consider first the amplified quadrature, and study the JPA’s added noise
and the total noise of the amplifier chain as a function of the JPA gain. In thesemeasurements, the LO and pump come from the same signal generator, and a manualphase shifter is used to send all the amplified quadrature into the I channel of the mixer.
By switching between two calibrated noise sources (Fig. 6.12a and c), we can change theincident noise on the JPA. This method to estimate the added noise of an amplifier is

152

Q
X2

G2 =1 G1

out

in

X1

G1

θ
I

Figure 6.13: (a) This diagram represents the operation of an ideal degenerate parametricamplifier which transforms an input state
ellipse)√ (red circle) into an output state (green √by√amplifying the X1 component by G1 and squeezing the X2 component by G2 =1/ G1 . The mixer projects the output state into axes rotated by θ.

called Y -factor measurement. The microwave switch used for our Y -factor measurement
is a Radiall R573423600. The change in noise power at the output of the mixer allows
us to estimate the noise AJ just added by the JPA, the JPA gain9 GJ ≡ G1 , the total
added noise of both the JPA and the commercial amplifier chain Asys (i.e., the totalnoise referred to the input of the JPA), and the total gain GJ × GH × β = GJ × GT ,where GT = βGH is the effective gain taking into account the loss between the JPA andthe HEMT. The calibrated noise sources are derived from two resistors held at different
temperatures. The resistances are matched to the wave impedance of the cables thatcarry signals to the JPA input. The noise emitted by both resistors is attenuated by anattenuator held at a temperature Tα = 15 mK and then passed to the JPA through a
microwave circulator which separates the incident and reflected signals from the JPA.
Ideally, the power spectral density in units of noise quanta incident on the JPA through9

During the rest of this chapter, I define GJ as the maximum quadrature gain of the JPA G1 .

153
the circulator isSin (Tload ) =
=

1
~ω
~ω
(α coth+ (1 − α) coth)2
2kB Tα2kB Tload111+ (1 − α),+ α ~ω
~ω2e kB Tα − 1
e kB Tload − 1(6.9)

where Tload = Tc = 15 mK or Tload = Th = 4.1 K depending on the position of theswitch shown in Fig. 6.12a and c, and α is the attenuation that we have intentionallyplaced in the mixing chamber to prevent saturation of the JPA with thermal noise when
the Y -factor measurement is done at high gains.
The attenuation α also reduces the thermal noise originated in the loss of theswitch. This loss can actually be a problem since even in the case of the switch being
positioned in the cold load, the finite (unknown) loss of the switch can be large enoughto generate significant thermal noise. The first term in equation (6.9) is the quantum
noise, while the second and third terms are the thermal noise which is small comparedto the quantum noise at Tload = Tc . Yet, due to loss of the circulator, the attenuation
α of Eq. (6.9) needs to be replaced by an α0 > α that takes into account this extra
sources of loss. Also, a small amount of noise that comes from the coupled port of thedirectional coupler as shown in figure 6.12c needs to be added.10 Hence, the complete
expression for the noise incident on the JPA when the non-ideal behavior of the all the
microwave components is taken into account includes contributions from: thermal noise
introduced with the calibration tone via the directional coupled port, the insertion loss
from the circulator, directional coupler and the switch. Expression (6.9) then becomes
µ
¶
1
~ω
~ω
~ω
0
Sin
(Tload ) =[ (ρ coth+ (1 − ρ) coth
)α0 + (1 − α0 ) coth×2
2kB Tload
2kB 4.1
2kB Tcry~ω(1 − C) + C coth](6.10)
2kB T4K
The unknown insertion loss at cryogenic temperatures of the circulator and of the microwave switch are the dominant uncertainties in the added noise and squeezing mea10

When an extra 20 dB are placed between the attenuator at the 4 K plate and the directional
coupler, the noise coming through the 20 dB-coupled port is negligible, as it is possible to model it as
extra attenuation at 15 mK whose added noise is just vacuum noise.

154
surements. In the data shown in the following sections, the extreme values represented
by the error bars are derived by assuming: first, that there in no insertion loss, and then,that the insertion loss has not changed between room temperature and the operating
temperature. In the case of the circulator, the data-sheet specifies the maximum loss
at 77 K.
By selecting θ = 0, the amplified quadrature appears at the I port of the mixer(Fig. 6.12a). We measure the ratio Y of the noise power at the output I with the hotload incident on the JPA divided by the noise power with the cold load incident on the
JPA. This Y -factor measurement allows us to extract00Asys = (Sin
(Th ) − Y Sin(Tc ))/(Y − 1)

(6.11)and the total gain GJ GT . We then adjust the cavity’s resonance frequency about 10
linewidths away from the LO frequency and turn off the pump. Because the JPA now
acts as a passive mirror, we can perform a Y -factor measurement on our commercial
amplifier chain, finding AT and GT . From both measurements, we extract the JPA
gain GJ and AJ (Fig. 6.12c). To avoid unknown loss and noise in the cables carrying
the calibrate noise signals, superconducting Niobium coaxial cables were used to carrythese signals between the 15 mK region of the refrigerator and 4 K region. A calibrated
ruthenium oxide thermometer measured the refrigerators base temperature and thetemperature of the helium bath (4.1 K), to which the 4 K resistor and microwave switch
were thermally anchored. A calibration tone, (Fig. 6.12a) is used to verify the gain
inferred from the Y -factor measurements.

6.3.4

Squeezing

We now turn our attention to the squeezed quadrature X2 when the incident
noise is mostly quantum noise (Tload = Tc ). When θ = 0, X2 appears at the Q port
(Fig. 6.13). The noise at the Q port referred to the input of the HEMT is composed

155of the effective added noise of the commercial amplifiers (AH /β = AT ≈ 26) and2
noise in the rotated quadrature Xθ+π/2 coming from the JPA, ∆Xθ+π/2= ∆X12 sin2 θ +∆X22 cos2 θ (Fig. 6.13). In order to illustrate the JPA’s unequal effect on the noise of both2
2
quadratures, we re-define the quantity ξ(θ) = (∆Xθ+π/2)pump on /(∆Xθ+π/2)pump off . In
2order to measure (∆Xθ+π/2)pump off , I turn the JPA pump off and adjust its resonancefrequency so that the resonator behaves as a mirror. When θ = 0, we will observethat ξ < 1 which means that the noise at the output of the JPA has been squeezed. If
instead θ = π/2 radians, we will find ξ ≈ GJ > 1. For an ideal JPA ξ(π/2) = GJ [8].
(Because a small fraction of the incident noise on port 2 is either lost through port 1
or absorbed in the cavity, for our device there is a small difference between ξ(π/2) and
GJ .) We also measured the squeezing ξ as a function of gain GJ and compare it withtheory (Ref. [38]).
In this measurement, we tested both setups shown in figure 6.12a, using the same
signal generator for the pump and the LO, and a manual phase shifter to change thephase in the LO . We also used a different signal generators (with their internal clocks
locked) and change the phase with the built-in phase option of the generators. In bothcases we got the same answer, but in the second option the measurement could be
automated so we chose the second option. Another advantage is that all manual phaseshifters have some phase-dependance insertion loss as high as 0.2 dB, which could affectthe “observed squeezing” unless care was taken in adding another signal to monitored
the conversion-loss of the mixer as the phase (and hence the LO power) was changed.
We do not have this problem if we use two different signal generators as the output
power does not depend on its output phase. One caveat, though, is that when two
signal generators are used, the phase-noise of the signal generators is observed at theoutput of the mixer. This phase noise looks like a phase-dependance noise on top of thenoise at the input of the mixer. This would not look like squeezing in the sense that
it would not make the noise to go below the vacuum noise level. However, if it is large

156
enough, it could prevent us from measuring squeezing. In our case, for frequencies larger
than 100 kHz, the phase noise of the signal generators is not an issue. A calibration
tone was added to the measurement chain to measure the gain of the squeezer and toensure that the noise reduction arises from genuine squeezing and not saturation of theamplifiers.

6.3.5

Amplifier saturation

The amplifier saturation was measured using the same setup used to measure theamplifier’s gain with a network analyzer as shown in figure 6.11. We measure the direct
gain as a function of the signal power as we increase it. We cannot compare it with theory
as the theory used to analyze the amplifier assumes an undepleted-pump approximation.
The effect of saturation on parametric amplifiers has not been extensively treated in
the literature, with only a few exceptions for very particular implementations [47, 86].

6.4

Aluminum resonators
This sample was fabricated in March, 2006, with the original idea of testing phasetransition physics in arrays of Josephson junctions. We cooled it down first on April,2006 and then again in January 2007. This device was fabricated out of aluminum, andthe SQUIDs were made from aluminum-aluminum oxide-aluminum junctions. They
were fabricated using traditional electron beam lithography techniques using a doubleangle evaporation (Dolan bridge technique [87]). A picture of a similar device to theone measured is shown in figure 6.14a. The details of the fabrication are describedin appendix C. Next I discuss all the measurements done on this particular sample.
It consists of a quarter-wave CPW resonator whose center conductor is an array of
400 SQUIDs in series (Fig. 6.14). Typically, the resistance of the device tends toincrease over time. This was definitely observed in this sample as the critical current
per SQUID dropped from 1.9 to 1.5 µA. In the first cooldown, only its linear properties

(a)

(c)

(b)

100 µm157

1.5 µm
8.0 µm18.0 µm

Figure 6.14: (a) Image of a test sample. An array is also shown on the left of thedevice. However, the I-V curves at 15 mK were taken from an array exactly as theone used for the rf-measurements (including the ground planes) with the only difference
being that the capacitance was replaced by a short circuit. (b) Zoom-in of the array,
where the two periodicity of the SQUID loop size is shown. This imperfection in the
fabrication is responsible for the “beating” manifested in the magnetic-field dependanceof the resonance frequency. (c) The tested device is a CPW transmission line resonator
where the center conductor is a series array of 400 SQUIDs. The associated dimensionsof the CPW are shown in the figure. The resonator’s ground plane is made out ofaluminum, and the SQUIDs are made from Al/AlOx/Al junctions. They are fabricatedusing e-beam lithography and double angle evaporation on an oxidized silicon substrate.

158
were tested while in the second cooldown, we measured its performance as a parametricamplifier. All the data shown is from the second cooldown where the device was used
as an amplifier, unless it is specifically stated that it is from the first cooldown. The
resonance frequency can be adjusted over an octave in frequency, between 4 and 7.8
GHz by applying a magnetic field (4.5 and 8.8 GHz before the junctions aged).

6.4.1

dc-measurements(a)(b) 104
R=89 kΩ5
I (nA)I (µ A)

2

0

0

−5
−2
−0.2

R=71 kΩ
−0.1

0
V

0.1

0.2

−10
−3

−1.5

0
V (mV)

1.5

3

Figure 6.15: (a) IV characteristic of an array of 400 SQUIDs measured in the superconducting state at 15 mK. Data from both cooldowns is plotted in order to show theobserved aging of the junctions. Three I-V curves for three different applied magnetic
fields is shown for the case of R = 71 kΩ to show the modulation of the critical currentof the SQUIDs. An offset has been added to the I-V curve of the R = 89 kΩ case forclarity. (b) IV curve of a co-processed SET (2 junctions in series) at 15 mK. From thisIV curve, the size of the superconducting gap was estimated to be ∆ = 215 µeV.

We performed dc-measurements as described in section 6.2.1. We measured arraysof SQUIDs to find the average resistance of the junctions. There were two type of arrays,
some small ones like the ones shown on the left of the resonator in figure 6.14a, andother arrays that were identical to the resonators, but with the capacitor replaced with a159
short-circuit. Only the latter were measured at 15 mK. The IV-curves for both cooldown
are shown in figure 6.15. The estimated total resistance per SQUID is that of the array
divided by the 400 SQUIDs per array. The size of the superconducting gap is measured
using the I-V curves of a co-fabricated single-electron-transistor (SET), in a different
chip (Fig. 6.15b), from which we estimate ∆ = 215 µeV. Using Eq. (3.17), we estimate
the critical current Ic0 for one SQUID to be 1.5 µA (1.9 µA for the SQUIDs in thefirst cooldown). When one measures a single SQUID or an series array of SQUIDs, themaximum measured current at V = 0, that is, the current at which the junctions switchfrom the supercurrent branch to a resisitive state is not a way to estimate Ic since it
dependes on how well isolated the junction is. Instead, I will use the resistance of the
SQUID and the Ambegaokar-Baratoff expression for the critical current, Eq. (3.17).
From the dc measurements, we estimate that the resulting metamaterial has azero-flux inductance per unit length of Ll = 0.9 mH/m ≈ 700µ0 . The CPW wasdesigned to have a capacitance per unit length of Cl = 0.11 nF/m, yielding a zero-flux√
phase velocity of vph = 1/ Ll Cl = 0.01c and a zero-flux characteristic impedance of
Zc = 2.9 kΩ. The reason for the rather large characteristic impedance of this sample isthat it was originally designed to study phase transitions in Josephson junction arrays[72, 74].

6.4.2

Linear measurements of the resonator

Since this device was a quarter-wave resonator, we only measure its reflection
coefficient Γ using the setup shown in figure 6.10. We apply a low enough incident
power P , in order to assure that the resonator response is linear (by low enough power I
mean that I only use P < Pc /1000). From the imaginary and real part of Γ, the quality
factors associated with losses Ql and capacitively coupled port Q1 are calculated. In thisdevice, both turn out to be very similar (the resonator is slightly overcoupled). Hence,
the resonance frequency fres is easily identified by a dip in |Γ|. Figure 6.16(a) shows how

160
(a)

(b)
1

|Γ|

f res (GHz)

8

6

0.5

4
0

50

100
B (G)

150

0

6.57

6.59
6.61
frequency (GHz)

Figure 6.16: Linear measurements for the aluminum resonator made on April, 2006. (a)Resonance frequency as a function of flux. (b) Magnitude of the reflection coefficient
showing the expected Lorentzian behavior. Data shown in blue and fitting in red. From
the fitting, the quality factors are calculated: Q1 = 1530 and Ql = 2170.

the resonance frequency behaved as a function of the applied magnetic field B in the first
cooldown of this device. A few things are worth mentioning. In figure 6.14b is easy to
see that the SQUIDs loops are not uniform but have two different sizes, consequence ofa small difference in the two angles of evaporation during the fabrication. This creates abimodal distribution in the SQUID-loop area that manifests as a two-periodicity in themagnetic field dependance of the critical currents of the SQUID array and the resonancefrequency of the resonator. This is observed in the figure 6.16a. In this plot, it is also
obvious that there are certain frequency ranges (7.3-8.2 GHz) where the plot does notshow measured resonance frequencies. The reason for that is that we were unable tosee a resonance frequency in that region,11

which we refer to as the “desert.” For

frequencies close to them, the resonators losses rise significantly. We do not understandthis behavior. However, away from these regions, the resonator behaved normally. The
11
There were other regions where a resonance frequency was not observed only at certain higher
magnetic fields. For example, in the second (small) lobe around B = 40 G, no resonance frequency wasobserved below 5.8 Ghz, even though they could be observed in the first and third large lobes. This,
however, is thought to be related with a large increase in internal losses when the magnetic field wasclose to half a flux quantum for some of the SQUIDs.

161
reflection coefficient Γ for a particular field is plotted in figure 6.16b. Based on themeasured reflection coefficient and Eq. (6.6), we extract the quality factors Q1 and Ql .
For this case, the values of quality factor are Q1 = 1530 and Ql = 2170. In general,
the internal quality factors were over 2000, except for regions close to the desert andat magnetic fields close to half a flux quantum for both types of SQUIDs (basically for
fres < 5.2 Ghz). The origin of the desert is not yet understood.
(a)

(b)

8

fres (GHz)fres (GHz)

7.4
6

4

−30

7.2
7
6.8

−15

0
B (G)

15

30

9

11
B (G)

13

Figure 6.17: (a) Resonance frequency as a function of flux for the second cooldown of
the device. Due to aging, the zero-flux resonance frequency decreased from 8.8 to 7.8
GHz. At first sight, the “desert” seemed to have disappeared. However, if we looked
more carefully in the regions enclosed by the square, we find that at B = 11 G a strange
feature appears (b). We believe this feature is the same as the one that we called the“desert” in the previous cooldown, but that it has changed due to the aging of thejunctions.

As mentioned at the beginning of this section, we cooled down this same device
10 months later and it showed signs of aging. However, the increase of the resistance
was not the only change. The desert itself changed also as it is shown in figure 6.17,
assuming the feature called previously as the desert and this one are the same. Again,
we do not understand it, but it seems to be something intrinsic to the resonator, as itchanges with the aging of the junctions. We did not look into this feature as we weremore interested in studying it as a parametric amplifier. Therefore, we stayed away

162
from this feature at B = 11 G. We estimated the quality factors and the linewidths(HWHM) γ = ω0 /2Q. For example, for a flux of Φ = 0.2Φ0 , we find the resonator’slinear parameters fres = 6.952 GHz, Q1 = 1830, and Ql = 3160 (Fig. 6.18a). To
demonstrate the tunability of the JPA, we also test the performance of the amplifier atlower frequencies. For example, for Φ = 0.35Φ0 , the resonance frequency is ω0 /2π =5.203 GHz. A similar analysis as the one described for Φ = 0.2Φ0 gives the following
parameters: Q1 = 2740, Ql = 3060 (Fig. 6.18b). The quality factors did not change
dramatically from the previous cooldown, except in the regions close to the desert, where
now the high-loss is only reduced to a small range of only ∼ 100 MHz (Fig. 6.17b).
Nonetheless, we still do not have an explanation for the losses in this resonator.
(a)

(b)
1

|Γ|

|Γ|

1

0.5

0.5

0

6.935

6.95
6.965
frequency (GHz)

0

5.197

5.202
5.207
frequency (GHz)

Figure 6.18: Linear measurements for the aluminum resonator made on January,
2007 (second cooldown). Magnitude of the reflection coefficient showing the expected
Lorentzian behavior for 2 different resonance frequencies. Data shown in blue and fitting
in red. From the fitting, the quality factors are calculated. (a) At fres = 6.951 GHz,
Q1 = 1650 and Ql = 3170. (b) At fres = 5.202 GHz, Q1 = 3160 and Ql = 3200.

6.4.3

Non-linear characterization

From the frequency and power dependence of Γ, we estimate the critical powerand (modified) Kerr constant K 0 as defined by Eq. (6.7). We performed this test at

163
(b)

(a) 1
fres (GHz)

|Γ|

6.95
0.1 Pc0.5

0.5 Pc6.9475

0.7 Pc
1.0 P

6.945

c

0
6.94

6.95
Frequency (GHz)

6.96

0

0.5
b

1
/bc
in
in

Figure 6.19: Power dependance of the resonance circuit. (a) Reflection coefficient magnitude as a function of frequency at different pump powers for B = 14.4 G. (b) By doinga fitting to the expected power dependance of the resonance frequency, Eq. (5.63), theratio of nonlinear loss and kerr constant γnl /|K| is estimated. For this case, from thefitting we estimate that γnl /|K| = 0.03

three different frequencies. At Φ = 0.2Φ0 and fres = 6.952 GHz, the critical power andKerr constant were Pc = 3±1.3 fW and ~K 0 = −1.6±0.7×10−5 , respectively. The large
uncertainty comes from the 4 dB uncertainty of the incident power on the resonator.
From appendix A in Ref. [38], we can calculate our estimate for the Kerr constantusing Eqs. (5.44-5.46). The expected value for the Kerr constant is ~K 0 = −1 × 10−5 ,in agreement with our measurement. To model more completely the behavior of theresonator, we also include a nonlinear dissipation term γnl = 0.03 |K|.
A similar analysis as the one described for Φ = 0.2Φ0 gives the following parameters for Φ = 0.05Φ0 : fres = 7.756 GHz, the critical power was Pc = 3.5 ± 1.5 fW, theKerr constant was ~K 0 = −1.9 ± 0.8 × 10−5 and the nonlinear loss was γnl = 0.02 |K|.
For Φ = 0.35Φ0 , the resonance frequency is ω0 /2π = 5.203 GHz and the nonlinear
parameters are: Pc = 0.5 ± 0.2 fW, ~K 0 = −9 ± 4 × 10−5 and γ3 = 0.145 |K|. For
frequencies of 5.5 GHz and below, we have observed an increase of the nonlinear loss,
which starts affecting the performance of the amplifier.

164
Since this device has very small critical powers, the phase noise of the generator
is not an issue, as can be easily proved by seeing that at 10 kHz, the phase noise is thesame as at 100 kHz [81], which compared to the vacuum noise is of the order of 0.1%.

6.4.4

Gain measurements
GGs

Gain (dB)

20

i

(a)

(b)

10

0
−2

0
Signal detuning (MHz)

2

(c)

−2

0
Signal detuning (MHz)

2

−6
−4
Pump detuning (MHz)

−2

(d)

Gain (dB)

20

0

−20
−8

−6
−4
Pump detuning (MHz)−2−8Figure 6.20: Performance of the amplifier. (a) and (b) Gs and Gi as functions of signal
detuning (points) and predictions of Ref. [38] (lines) for two different pump powers. Gsand Gi for P = 0.9 Pc (black) and for P = 0.75 Pc (red). (c) and (d) Gs and Gi as
functions of pump detuning (points) for P = 0.95 Pc (blue) and P = 0.5 Pc (red) andprediction (lines) of Ref. [38].

The parametric amplifier is operated in reflection mode, as shown in Fig. 6.11(a).
We operate it first by applying a calibrating tone slightly detuned from the pump andmeasure the frequency dependence of both gains in two different ways. In the first case,the signal and the pump frequencies are different. In the first test, we apply the pumpat a frequency close to ωm and analyze Gs and Gi as we detune the signal frequency

165
from the pump by an amount ∆f . In Figs. 6.20(a) and 6.20(b), we plot both Gs andGi as a function ∆f = fsignal − fpump for two different pump powers. We also plot thepredictions from the theory in Ref. [38] where the parameters in the theory are extracted
from the measurements of Γ. From this plot, we estimate the 3 dB bandwidth to beabout 300 kHz when Gs and Gi are 18 dB. Next, we measure Gi and Gs as a functionof pump detuning, i.e., the difference in frequency between the applied pump and fres .
In this test, the signal and the pump frequency differ by a fixed amount, ∆f = 10 kHz,(Figs. 6.20(c) and 6.20(d)). The agreement seen in Fig. 6.20 between the data andtheory explained in chapter 3 and 5, suggests that Ref. [38] provides an appropriate
model for our device.
In the second test, for fres = 6.95 GHz we measured the phase dependance ofthe gain of the degenerate parametric amplifier. In this case the the pump and thesignal frequencies are the same (Fig. 6.12a and b). The gain of the parametric amplifier
is sensitive to the phase between the pump and the signal. To measure this phasedependence, we amplitude modulate the signal at 20 kHz and adjust the phase of thepump relative to the signal. We define the gain as the ratio of the am-modulation
sideband power with the pump on and pump off. From Eq. (6.8), for Gs ∼ Gi À 1,
the quadrature gain GJ is 6 dB larger GJ ≡ GX1 ≈ 4Gs , (see Eq. 3.77), if the phasebetween the pump and signal is tuned for maximum gain. The phase dependence of thegain for a pump power close to Pc is shown in Fig. 6.21(a); there, it is evident that wesee deamplification, a hallmark of degenerate parametric amplifiers.
To demonstrate the tunability of the JPA, we also test the performance of theamplifier at lower frequencies. However, as mentioned above, the performance is greatly
affected for resonance frequencies fres < 5.2 GHz due to the onset of large nonlinear
loss. This increase of nonlinear loss made the measured gains smaller than the ones at
6.95 GHz. The highest Gi and Gs observed at this frequency are both 12 dB.

166
6.4.5

Estimate of the added noise of the amplifier

(a)

(b)
Signal10 -7

Gain (dB)

2

PSD (V /Hz)

20

Pump onSignal0

10

Pump off-9

HEMT Noise0
Phase (degrees)

100

0

50
frequency (kHz)

Figure 6.21: (a) Gain as a function of the phase between the pump and the signal(points) and prediction (line) from Ref. [38] showing the expected behavior for degenerate parametric amplifiers (P = 0.9Pc ). (b) Power spectral density of the demodulator
output for the cases when the pump is on (P = 0.95Pc ) and off. The gain in this caseis 630 (28 dB). The applied signal power is 1.6 ± 0.7 × 10−20 watts.

In Fig. 6.21(b), we plot the power spectral density (PSD) of the demodulated
signal with the pump off and pump on for P = 0.95Pc , where the signal-pump phasehas been adjusted for maximum gain (28 dB). At this gain, the HEMT amplifier’s input
noise is overwhelmed by the noise at the output of the parametric amplifier, effectively
improving the signal-to-noise ratio (S/N) by 16 dB. A definitive measurement of thenoise added by our parametric amplifier requires a calibrated noise source. In thiscooldown, we did not have such calibrated noise source. However, by measuring the
S/N with the pump off, we find that the noise referred to the input of the HEMTis TN = 12 ± 5 K. From the S/N improvement with the pump on, we estimate thetotal noise referred to the input of the JPA as 300 ± 130 mK. This value suggeststhat the parametric amplifier adds an amount of noise comparable to the vacuum noise(hfres /2kB = 166 mK), which must be present at the input of the JPA. The error bars
were estimated based on the losses of the semi-rigid cables. We measure their loss at167
room temperature and then at 77 K.

6.4.6

Final comments

With this aluminum sample, we demonstrated a widely tunable parametric amplifier based on a coplanar waveguide resonator whose inner conductor is made from a
SQUID array. We observed good gains and performance that was understood within
the theory frame of Ref. [38]. Although some features were not understood, such as theorigin of the loss in this superconducting devices, or the region called the “desert,” itstill performed well enough to act as a low noise preamplifier for the quietest commercial
amplifier available. For the next generation of devices, we increased the critical powerof the resonators by increasing the critical current of the junctions and we increased the
bandwidth by using a much larger coupling capacitor. Also, the devices were made outof niobium, which has a larger critical temperature than aluminum. We do not have thecapability of making niobium Josephson junctions at JILA so for this new generationof resonators we collaborated with Kent Irwin’s group from NIST.

6.5

Niobium resonators
In this case, two different realization of a JPA were measured, a quarter-wavelengthand a half-wavelength cavity. The inductance per unit length of this transmission line
comes mostly from the nonlinear Josephson inductance of the SQUIDs, but unlike thealuminum device, at zero-flux the geometrical inductance does provide a non-negligible
amount of inductance (30% at zero flux). Unless, explicitly said, the data shown is fromthe half-wavelength resonator. We create the half-wavelength microwave cavity fromthis metamaterial by interrupting the center conductor with two capacitors, one muchbigger than the other, thus creating a strongly coupled port and a weakly coupled port
(Fig. 6.22).

168
6.5.1

Description of the device

(a)

(b)

30 µm

Figure 6.22: Images of the device. (a) The images are a picture of the full device, anda magnified image of the weakly coupled port, a few SQUIDS and the strongly coupled
port. (b) SEM picture of a pair of SQUIDs (picture courtesy of Leila Vale).

Devices were fabricated at NIST Boulder on a high purity uncompensated siliconwafer with resistivity > 17 kΩcm. Standard Nb/AlOx/Nb trilayer process [88] was used,
modified by eliminating the shunt resistor layer and minimizing deposited oxides [89](see appendix C). The device studied here was coprocessed with 10 other devices used
for dc characterization. Initially, only a µ-metal shield was used. Recently, a cryoperm
metal shield from the company Amuneal has been added at the mixing chamber in order
to increase the supression of external magnetic noise. The center conductor is a series
array of 480 SQUIDs (in the case of the quarter-wave resonator, the array contains 250
SQUIDs). Next is a summary of the multiple tests performed on these devices.

6.5.2

dc-Characterization of Josephson Junctions

A total of 92 chips were fabricated with three different values of overlap (junction)
size. The reason for the three “flavors” of SQUIDs was that the fabrication process forthis junctions was somewhat different from the usual process used at NIST. Since wewere trying to get Ic = 30 µA per SQUID, the three different junction sizes gave us a

169
(b)

(a)

100
50

50

I (µ A)I (µ A)

100

0
−50

0
−50

−100
−100
−10

−5

0
5
V (mV)

10

−1

−0.5

0
0.5
V (Volt)

1

Figure 6.23: I-V curves of test niobium devices. (a) I-V curve of a single SQUID for B =0. From this measurement we estimate the resistance per SQUID to be Rn ' 73 Ω. The
measured superconducting gap was 2∆ = 2.9 mV. (b) I-V curve of an array. The blue
curve was sweep increasingly in voltage and the red curve the opposite direction, showing
hysteresis. The measured resistance and superconducting gap were not consistent withthe expected number of SQUIDs (250) but with a significantly smaller number. We
suspect some of the Josephson junctions turned into short-circuits. The reason whysome of the SQUIDs are dying to short circuits is not understood.

larger target to shoot at. The junction with the closest value had a size of 2 × 2 µm2 .
The dc-measurements on single SQUIDs indicate an average resistance 73 Ω and asuperconducting gap of 2∆ = 2.9 mV. This implies an average critical current Ic per
SQUID in the JPA of 31 µA (Fig. 6.23a). The transmission lines from which the cavities
were built were designed to have a capacitance per unit length of Cl = 0.15 nF/m anda geometrical inductance per unit length of Ll = 0.49 µH/m. For Ic = 31 µA, theJosephson inductance per SQUID is LJ = 10.6 pH and each cell had a length of 7
µm. These values imply a zero-flux impedance of Zc = 117 Ω. In the case of thehalf-wavelength resonator, these values predicts a first-mode resonance frequency of
8.23 GHz when the loading the capacitors is included, and when no flux is applied.
This value is within 3% of the observed value. A slightly smaller resonance frequency is
expected for the quarter-wavelength as it is more strongly loaded by the large capacitor.

170
While this particular device worked well, the device yield from the wafer waspoor. Of the 92 chips, one half had the right value of junction size. However, only 3
chips were found to have working resonators. We have also observed problems with thetest devices fabricated for dc-measurement. We measured arrays of 250 SQUIDs, as
shown in figure 6.23b. These measurements show that if we assume that there are 250
SQUIDs, the resistance per SQUID is less than the expected value. In the case of thearray shown in figure 6.23b with a total resistance of 10.3 kΩ, a resistance per SQUID of
41 Ω would be then estimated if we assumed that the array had 250 working SQUIDs.
But if we measure the superconducting gap, then this would also suggest a much smallergap, 2∆ = 1.6 mV. This reduction of the superconducting gap seems much bigger thanwhat could be explained by an increase in the temperature on the central junctions bybias heating. A possible explanation for such a small measured superconducting gap,
which would also explain the fact that only 3 out of 46 chips had working devices, could
be that some of the junctions are dying and turning into short-circuits. Whether thishappens during fabrication or handling of the chips is still under investigation.12 Ourcolleagues at NIST that do the fabrication are studying this problem.

6.5.3

Linear measurements of the resonators

Before operating the device as a parametric amplifier, we first characterize its
linear behavior using the same setup as in figure 6.10. In figures 6.24a and b, we measure
the cavity’s response both by measuring the reflectance of a signal on port 2 (S22 ) andthe transmittance from port 1 to port 2 (S21 ) as functions of frequency, for two different
magnetic fields. The powers used are low enough so that the cavity’s response is still
linear. From this measurement we can extract the cavity’s resonance frequency fres , thequality factors Q1 and Q2 that describe the cavity power loses through both ports as
12

Although the devices also die during handling due to electrostatic discharge, in previous devices
we have found that arrays are extremely hard to kill. Also, resonators are isolated from us, speciallythe half-wavelength resonators, since they do not have a dc-connection to the outside world.

S parameters

(a)

1

0.5

0

6.95

7
7.05
frequency (GHz)(b)

1

S parameters

171

0.5

0
6.14

6.18
6.22
frequency (GHz)

(c)

f res

8

6

4
−10

−5

0
flux (Φ/Φ )

5

10

0

Figure 6.24: We plot the fraction of microwave amplitude transmitted from port 2 toport 1 |S12 | (blue) and the fraction reflected from port 2 |S11 | (black) as functions offrequency at two particular values of applied magnetic flux: (a) Φ = 0.29Φ0 and (b)Φ = 0.37Φ0 . (a) By fitting this data to a model of a two port cavity (red lines), we
extract fres = 7.0038 GHz, Q2 = 8010, Q1 = 330 and Ql = 10, 300. (b) From the fittings,
we extract fres = 6.184 GHz, Q2 = 9570, Q1 = 396, and Ql = 2420. (c) Measured valueof fres as a function of applied magnetic flux Φ in units of the magnetic flux quantum
Φ0 . We use the obvious periodicity (in the first few periods) in this data to infer themagnetic flux enclosed by the SQUIDs. The inevitable imperfections in fabrication start
to manifest after 6 flux-quanta, where interference between the different SQUID sizes
start to affect the magnetic field dependance.

172
Resonator

Standing waves

ZΓ

Figure 6.25: As the experiment has become more involved (see for example, theschematic of the next two chapters), imperfections in the microwave components startadding up causing standing waves in the measured reflection coefficient. In order to doa better job in the fitting, the slightly more complicated model shown here is used.

well as through internal loss processes Ql . When operating as a parametric amplifier,the center of the amplified band will be close to fres , and the unity gain signal bandwidth
will be limited by the total linewidth γ1 + γ2 + γl = ωres (1/2Q1 + 1/2Q2 + 1/2Ql ). By
applying magnetic flux we can move the center of the amplified band over a large rangeof frequencies (Fig. 6.24c).
The flux dependence shown in figure 6.24c is also a demonstration of the uniformity in the junctions and in the SQUIDs areas. Unlike the aluminum device where
the two different sizes of SQUIDs areas were so different that a “beating” of the two
periods was obvious at very small magnetic fluxes, in this devices it is necessary to go
to fluxes larger than 5Φ0 to observe an obvious “beating” between the different periodsin the flux dependance of the multiple SQUIDs in the array. The lowest resonancefrequency in the resonators is consistent with the critical current of the SQUIDs being
limited by the finite loop inductance of the SQUID loop LLoop = 13 pH. This valueof loop inductance is consistent with simulations using Fast Henry, a software capableof computing the frequency-dependant self and mutual inductances of superconducting
structures. Finally, another important difference is that we avoided having large characteristic impedances in the coplanar waveguide that formed the resonator even at thelowest resonance frequency (Zc < 0.3 kΩ). Finally, in these devices we do not observe
any “desert” region, rather a clear cavity resonance is visible at all values of applied
flux.

173
(a)

(b)
1

Γr,i

Γr,i

1

0

−1
7.15

7.525
frequency (GHz)

−1
7.15

7.9

(c)

0

7.525
frequency (GHz)

7.9

7.525
frequency (GHz)

7.9

(d)
0.5

Residuals

Residuals

0.5

0

−0.5
7.15

7.525
frequency (GHz)

7.9

0

−0.5
7.15

Figure 6.26: Fits to the resonance frequency of a quarter-wave resonator using the modelof figure 6.25. (a) Fits to the imaginary Γi and real part Γr of Γ using only Eq. (6.6).
Data in red for Γr and blue for Γi . Fits are plotted in black. The standing waves on
top of the plots are easily observed especially in the residual of the plots shown in (c).
(b) Fits to the data using the model of figure 6.25 and (d) residuals showing that the
new model used for the fits now does a much better job.

The coupling capacitances were estimated to be 4.6 fF and 22.6 fF using microwave simulations in Microwave Office. Based on measurements of the S-parameters
for low powers these values are very close to the observed ones. For recent experiments
where the setup has increasingly become more complex and more microwave connectors
are used, the measured reflection coefficients are not as clean as those shown in figure
6.24, presenting a standing wave structure on top of the expected Lorentzian response.
This can interfere with our attempts to do an appropriate fitting to obtain the liner

174parameters of the resonator. In order to solve this problem we use a somewhat more
complicated model shown in figure 6.25. In this model we assume that an impedancemismatch a length ` away from the input port of the resonator shunts the center conductor of the transmission line connected to the resonator, generating the standing waves
observed in the data. This model allows us to do a better fitting as can be observed infigure 6.26.
In general, the quality factors associated with the intrinsic losses of the resonatorare over Ql > 2000 and at some frequencies we have measured Ql > 10000 as shownin figure 6.24. A possible reason for the loss observed is the lossy dielectric used in
the fabrication process (see appendix C). However, we do not completely understandthe large variation of loss over the whole 4-GHz band of the resonator. On possible
explanation is the existence of other resonant modes, for example those observed in thesample-box (Fig. 6.2c).

6.5.4

Nonlinear characterization

We measured the nonlinear behavior of the resonator as well. Based on the criticalpowers measured and the linear S-parameters, we estimated the values of the Kerr
constant for different frequencies. Next, I show examples of the measured Kerr constantfor six different frequencies using Eqs. (5.44-5.46). The results of this measurement are
shown in the following table.
fres
8.3 GHz
8.0 GHz
7.0 GHz
6.2 GHz
5.1 GHz
4.1 GHz

Measured ~K 0
−2.8 ± 0.3 × 10−7
−3.4 ± 0.4 × 10−7
−4.4 ± 0.5 × 10−7
−5.8 ± 0.8 × 10−7
−12.5 ± 2 × 10−7
−57 ± 7 × 10−7

Expected ~K 0
−2.4 × 10−7
−2.7 × 10−7
−3.7 × 10−7
−4.9 × 10−7
−6.8 × 10−7
−10 × 10−7

We observe that there is some quantitative agreement with the expected Kerr constantat low magnetic fluxes. However, as we increase the flux through the SQUID loops to

175
−4

|S12|2 (a.u.)

1

x 10

0.5

0

3.87

3.89
frequency (GHz)

3.91

2 |for the λ/2 resonator for Φ =Figure 6.27: Magnitude squared of the transmission |S21
0.48Φ0 . At low powers (P ≈ −125 dBm) the response has the expected lorentzian
shape (blue). For 27 dB higher power (red), the response has the expected Duffinglike response of a resonator about to reach the bifurcation state. However, before thebifurcation happens, the response becomes unpredictable.

suppress the critical current I0S of the SQUIDs to tune down the resonance frequency,
the disagreement grows. For example, at 4.1 GHz the difference between the estimatedKerr constant K 0 and the expected Kerr constant is a factor of close to 6. This is not the
only unexpected behavior at fluxes close to Φ = Φ0 /2. Close to this values of flux, thenonlinear behavior of the resonator becomes somewhat unpredictable as can be observedin figure 6.27. However, this behavior is only observed at frequencies fres < 4 GHz.
Unlike the aluminum devices, in this device we do not observe a large increaseof nonlinear loss for the lower resonance frequencies (between 5 and 4 GHz). This isobserved in figure 6.28, where even at fres = 4.5 GHz, close to the lowest frequencies
observed, the nonlinear loss is only γnl /|K| = 0.007. However, it is hard to measure thenonlinear dependance of the resonance at frequencies below 4 GHz due to the behavior
observed at those frequencies. Thus, we cannot estimate whether the nonlinear loss
increases at those frequencies.

176
0

(a)

(b)

−50

4.502
fres (GHz)

Arg(Γ)

−100
−150
−200
−250

4.5
4.498
4.496

−300
−350

4.494
4.45

4.5
frequency (GHz)

4.55

0

0.5
b /bc
in

1

in

Figure 6.28: Power dependance of the resonator. (a) Phase of the reflection coefficient
as a function of frequency at different pump powers for Φ/Φ0 = 0.46. (b) By fitting
to the expected power dependance of the resonance frequency, Eq. (5.63), the ratio of
nonlinear loss and kerr constant γnl /|K| is estimated. From the fit, we estimate that
γnl /|K| < 0.01

6.5.5

Gain measurements

We study the amplifier’s properties using the apparatus shown in Fig. 6.11 using
the spectrum analyzer [90]. As described in Ref. [91], we determine the critical pump
power at which the JPA gain diverges. We then characterize the image gain (Gi ) andsignal gain (Gs ) for pump powers less than this critical power. To characterize thebandwidth of the parametric amplifier, we measure the frequency dependance of bothgains. For sufficiently large gains, these are Lorentzian functions of the signal-pump
detuning ∆f = fsignal − fpump between the signal and pump frequencies. For each pump
power we determine the 3 dB bandwidth ∆f1/2 and gain at ∆f = 0. We observed thatwith increasing gain, ∆f1/2 is reduced. We find that the expressionp

Gs ∆f1/2 = αγ(6.12)

is correct for a factor α of the order of two [38]. The exact expression is not given by√√Gs ∆f1/2 = 4/ 3γ, Eq. (3.62), since we derived that expression for the case of no losses

177

30

(a)

(b) 30

30

30
15

15

6.17

6.19

15

6.17

6.19
fpump

i

fpump

G (dB)15s

G (dB)

0
0

0
0
−15

0
∆f (MHz)

15

−15

0
∆f (MHz)

15

Figure 6.29: Gain measurements. (a) and (b) Gs and Gi as functions of signal detuning(points) and predictions of [38] (lines) for three different pump powers. Gs and Gi for
P = 0.95 Pc (black), P = 0.9 Pc (red) and for P = 0.6 Pc (green). For each pump power,
we find an optimum pump frequency which decreases as a function of pump power due
to the Duffing-like behavior of the resonator [38]. Insets: Gs and Gi as a function ofpump detuning for the same pump powers as in the main plot. The device shows the
expected behavior described by [38]. The test signal was detuned from the pump by
100 kHz in all the measurements. The data shown in this figure was taken using the
setup shown in figure 6.11 with the spectrum analyzer.

inside the resonator. Eq. (6.12) is also correct for Gi if√
Gi > 2. These results are shownin Figs. 6.29(a) and 6.29(b), where we plot both Gi and Gs as a function ∆f for threedifferent pump powers for fres = 6.184 GHz (Φ = 0.37Φ0 ). Qualitatively, this behavior is
expected for parametric amplifiers, and we find quantitative agreement with the theoryof [38]. From this plot, we estimate the 3 dB bandwidth to be about 1.38 MHz when Gs√
and Gi are 22 dB, or a showing a gain-bandwidth product of Gs Bω = 17.5 MHz [90].
The setup of Fig. 6.11 with the network analyzer only allow us to measure thedirect gain of the amplifier. However, the image gain does not provide any extra information. Additionally, this setup is much faster. Therefore, for the last experiments thisis the only way we have characterized the bandwidth of the amplifier. An example forthe quarter-wave resonator of this measurement is shown in figure 6.30. This resonatoris expected to have a bandwidth at least a factor of two larger, which is observed in the

178
(b)

(a)

Gs Bw

40

20

p

Gs

40

20

0
7.82

7.84
f signal

0

7.86

10

20
30
Gs (dB)40

Figure 6.30: (a) Signal gain for different pump powers for the quarter-wave resonator.
The data shown in this figure was taken using the setup shown in figure 6.11 with the
network analyzer. (b) Signal amplitude-gain-bandwidth product as a function of power
gain. The dashed line represents the theoretical limit of the gain-bandwidth product fora lossless parametric amplifier (Eq. 3.62). For this resonator, the internal loss quality
factor of the resonator is significantly larger than the coupling quality factor, so we
expect to get close to this limit, as observed in the√graph. The √total quality factor wasmeasured to be Q = 210, so the theoretical limit is Gs Bω = 4/ 3fres /(2Q) = 43 MHz.

measurements, showing a gain-bandwidth product of

6.5.6

√
Gs Bω = 42 MHz.

Measurement of the added noise of the amplifier

S'in

JPA

Amplifier Chain
and mixer

SoutI

S 'in (Tload = 0.015) = 0.67 ± 0.03
S 'in (Tload = 4.1) = 1.75 ± 0.05

AJ , GJ

AT , GT

Figure 6.31: Simplified diagram of the setup shown in figure 6.12a and c. The input
noise is modeled as the noise coming from a resistor with an effective temperature Tr
0 at its
whose noise is found using equation 6.10. The JPA adds AJ to the noise power Sin
input. Both noises are multiplied by the JPA power gain GJ before entering the inputof the commercial amplifier chain. The cascaded commercial amplifiers and mixer canbe modeled as a single amplifier with added noise AT and gain GT

Using the setup shown in figure 6.12, and switching between two calibrated noise

179
sources we can change the incident noise on the JPA and measure the added of theamplifier [92]. The change in noise power at the output of the mixer allows us to√estimate the noise AJ just added by the JPA, the JPA gain GJ , the total added noiseof both the JPA and the commercial amplifier chain Asys (i.e., the total noise referred√to the input of the JPA), and the total gain GJ × GT . The effective temperature of
the incident noise is calculated using Eq. (6.10). In this case, the attenuation α place inthe calibration line was 10 dB. The circulator maximum insertion loss was -0.4 dB andthe switch’s room-temperature maximum insertion loss is -0.3 dB. Using these values,
0 (Twe can estimate the incident noise quanta Sinload ), where Tload = Tc = 15 mK or

Tload = Th = 4.1 K depending on the position of the switch shown in Fig. 6.22a.
By selecting θ = 0, the amplified quadrature appears at the I port of the mixer(Fig. 6.31). The spectral densities of the noise at the output if the I-channel of the mixer
is shown in figure 6.32. We measure the ratio Y as defined by Eq. (6.11) to extract Asys√and the total gain GJ × GT . We then then perform a Y -factor measurement on ourcommercial amplifier chain, finding AT and GT (Fig. 6.32b). From both measurements,√
we extract the JPA gain GJ (Fig. 6.32c) and AJ (Fig. 6.32d). As observed infigure 6.32d, our JPA adds less than 1/2 a quantum of noise for fourier frequencies
above 100 kHz. We did this measurement for various gains, finding only a very weakgain dependance. However, the added noise is significantly higher than the theoreticalminimum, AJ = 0.04 quanta.
In addition, the amplifier can can be operated at large enough gain to amplify
the quantum noise above the commercial amplifier’s noise (Fig. 6.33a). We performed
similar Y -factor measurements over the whole 4 − 8 GHz band where the amplifier
operates. The results of this measurements are shown in figure 6.33b. Over the wholeband, the estimated added noise was less that half a quantum of noise [92]. However,
the added noise was always bigger than expected from the theory of Ref. [38] (see Eq.
5.57). The origin of this noise is not yet understood and is under investigation. Another

180
(a)

(b)

−10

−11

10

−12

10

−13

10

4

10
(c)

5

(d)

AJ (Quanta)

15

0
4
10

0

6

5

6

10
10
frequency (Hz))

x 10

2

10
10
frequency (Hz))

30

G1 (dB)

4

PSD (V2/Hz)

PSD (V2/Hz)

10

−13

0

1

1
2
frequency (MHz))

3

0.5
0.25
0
0
10

0.5

0

0

G

1

1
2
frequency (MHz))

3

10

3

Figure 6.32: Y -factor measurement done at 7 GHz. Power spectral density of thedemodulator output for the cases when the pump is on (a) and off (b), and the switch
is turned to the cold resistor (blue) and the hot resistor (red). For the case of the pumpoff, the resonanance has been moved to avoid the reduction of the hot-load noise due tolosses in the resonator. (c) Using the difference in noise between the pump off and pumpon spectral density, the gain of the amplifier can be measured. In this case the gain is
28 dB. (d) Using Eq. (6.11), we estimate the added noise of the JPA as a function of
fourier frequency (x-axis). Below 100 kHz 1/f n noise is observed, whose origin is notknown. But for fourier-frequencies larger than 100 kHz, the estimated added noise is
AJ ' 0.23. This added noise is larger than the expected added noise according to Eq.
(5.57), (green-dashed line). Inset: added noise for different JPA gains.

point of concern is the low frequency noise that is easily observed in figure 6.32d below
100kHz. A thermal sweep was used to see its temperature dependance. This is shown infigure 6.33c where it is shown that this noise presents a linear temperature dependance.
One hypothesis is that it consist of current noise present in any closed-loop of metal

181(like for example the circular walls of the sample box) which inductively couples noise
to the SQUID loop. But this still needs to be studied.
(a)

Asys

10

1

0.1

1

2

10

3

10

10

G

J

(c)

(b)

10
AJPA

P.S. (a.u.)

0.5

0.25

0

4

6
fres (GHz)

8

5

0

0

0.2

0.4

0.6

T (K)

Figure 6.33: (a) We plot Asys when fres = 7.00 GHz versus GJ (points). At thisfrequency AT = 26 and is dominated by the HEMT. With increasing G, this noise
becomes negligible compared to AJ = 0.23, where we expect Asys = AJ + AT /GJ (blueline). At the largest gains Asys < 0.5 (green line) and AT /G = 0.04. (b) We measure AJ
at nine different values of fres between 4 and 8 GHz and observe that the added noise isless than half a quantum (green line). The error bar applies to all points; it represents
the systematic uncertainty introduced primarily by the unknown loss of the microwave
components when operated cryogenically. (c) We performed a thermal sweep in orderto study the low frequency noise that appears in figure 6.34a and d . If I integrate thespectral density noise below 100 kHz, we observe a linear dependance with respect totemperature.

182
(a)

(b) 1

ξ(0)

AT+2∆X22

30

25

20

0.5

0.75
frequency (MHz)

1

0.5

0

0

10
G (dB)

20

(c)

ξ(θ)

10

1

0.1
0

90
θ (degrees)

180

Figure 6.34: Demonstration of squeezing. (a) Power spectral density versus frequency
at the Q output of the mixer referred to the input of the commercial amplifier chain inunits of quanta, measured with the pump on (red) and the pump off (blue) for θ = 0,fres = 7 GHz and GJ = 15.3 dB. (b) The squeezing ξ as a function of GJ (points) andthe maximum predicted squeezing from Ref. [38] (line). The error bars represent thesame systematic error as in Fig. 6.31b. This uncertainty varies smoothly between theerror bars shown in selected data points. (c) ξ(θ) as a function of θ for three different
gains (points) and GJ = 34 (black), GJ = 4 (green), GJ = 1.4 (cyan) and the expectedθ dependance (lines) [8, 38].

6.5.7

Inferred squeezing

We now turn our attention to the squeezed quadrature X2 . The squeezing datais shown in figure 6.34. The spectral density at the output of the Q port when the LOphase is set to θ = 0 is shown in figure 6.34a. There it is obvious a small reduction ofclose to 2% of the noise. Based on the Y -factor measurements done on the HEMT (Fig.
6.32b), we convert it into a squeezed-noised quanta ξ(θ = 90◦ ) at the output of the183
JPA. As we increase the JPA gain, we observe an increase in the noise squeezing up to
G = 16 dB (Fig. 6.34b) where we observe an squeezing of about 90% [92]. For G < 16 dB
the squeezing we observe is consistent with the expected behaviour from a model ofparametric amplification that includes loss in the JPA but no other imperfections [38].
However, for G > 16 dB we observe less than the predicted squeezing probably due toinstability in the phase acquired by the pump as it passes through the cryostat in thetime required to complete one measurement. In Fig. 6.34c, we show ξ(θ) as a functionof the LO phase θ demonstrating that the fluctuations at the output of the JPA are
indeed squeezed along the axis defined by the pump (Fig. 6.13).

6.5.8

Amplifier saturation

We have also estimated the dynamic range and saturation power of this amplifier
at 2 different frequencies: 6.184 and 8.012 GHz. We determine the saturation power as
the input power for which the gain is reduced by 1 dB (6.35b). In Fig. 6.35b we plot
Gs as function of input signal power for two different cavity’s resonance frequency. At
8.012 GHz and Gs = 27 dB, the 1 dB compression point is −130 dBm. From the noise
measurements previously described, the minimum detectable signal for this amplifier in
1 Hz band is 0.77~ω. From these two measurements, we estimate the dynamic range tobe 74 dB/Hz. At 6.18 GHz and Gs = 26 dB, the 1 dB compression point was −137 dBm
and the dynamic range is 67 dB/Hz. The reduction in dynamic range is expected: when
we tune down the resonance frequency by applying a magnetic field, we are reducing
the critical currents of the SQUIDs and the critical power of the resonator.
As a rule of thumb, we have observed that the saturation powers for large gains
can be found as Psat ≈ (γ2 /γ1 )Pc /(20GD ), where Pc is the critical pump power. The
factor γc1 /γc2 arises because we applied the pump to the weakly coupled port, but
applied the signal to the strongly coupled port. In the case of applying the pump in
the strongly coupled port (as it is necessary in the λ/4-resonator), this rule of thumb

184
simplifies to Pc /(20Gs ), where Pc is the critical pump power associated to the signal(strongly coupled) port. This expression is reasonable as we expect to observe saturation
behavior when the amplified signal power becomes a significant fraction of the pump
power. For the case of the quarter-wave resonator, since its critical power is 3 dB higher
and have similar noise performance, its dynamic range is about 3 dB higher.
(a)

(b)

30

Gs (dB)Gs (dB)

25
25

20

20
8.0106

8.012
fs (GHz)8.0136

15
−170

−150
−130
Input power (dBm)

Figure 6.35: (a) Direct gain as a function of signal-pump detuning for Φ = 0.15Φ0 for
three different signal powers, showing the expected saturation behavior at large enoughsignal powers. Signal powers shown are -145 dBm (dots), -130 dBm (circles) and -125
dBm (crosses).(b) Direct gain as a function of input signal powers for fres = 6.184 GHz(circles) and fres = 8.040 GHz (dots).

6.5.9

Final comments

Once we have characterized the behavior of the JPA and found that it has agood noise performance we can now use it for doing other interesting experiments.
For example, the measurement of the squeezed state would have proceeded much more
rapidly if we had used a subquantum limited amplifier, such as a second JPA, insteadof a HEMT amplifier. This experiment will be the main subject of our next chapter. In
addition to characterizing states of microwave fields as I will show in chapter 7, the JPA
is also well suited to amplifying the signal generated by a nanomechanical beam moving
in a microwave cavity [11], enabling a quantum-limited measurement of position. Thisapplication will be the subject of chapter 8.

Chapter 7
Applications I. Microwave homodyne tomography of squeezed states

In the previous chapter I showed how it is possible to generate microwave squeezed
states with Josephson parametric amplifiers. However, we have only characterized thesqueezed states by measuring the variance or spectral density of the itinerant squeezed
mode. These measurements do not fully characterize the states; to fully characterize
the state of a light field, we need to determine the density matrix which will account fornoise that is both quantum and classical in origin. In this chapter, I will describe how weapproach this problem of fully characterizing a squeezed state. Rather than measuring
the density matrix in a photon number basis, we measure the Wigner function using
our own implementation of “microwave homodyne tomography.” The Wigner function
is a particular representation of density matrix
In the first part of this chapter I describe the Wigner function as well as someof its more important properties. I will also discuss other possible representations of
the density matrix in the form of the P , Q and s-parameterized distribution functions.
The experimental estimation of Wigner functions for quantum states of light basedon real physical measurements is called quantum tomography and will be the subjectof the second part of this chapter. It consists of measuring complementary features(position and momentum, for example) in identically prepared quantum objects in aseries of distinct experiments. I will explain how quantum tomography allows us to use acomplete set of observations to reconstruct the Wigner function. I will also describe how186
imperfections in the measurement can affect our reconstruction of the Wigner function.
Finally, I will explain our experimental implementation of quantum tomography usinga Josephson parametric amplifier as the one described in the previous chapter.

7.1

Quasiprobability distributions
In the standard formulation of quantum mechanics, (for a pure state) the prob-

ability density ρ(x) in position space x is given by the square of the magnitude of the
wave function ρ(x) = |ψ(x)|2 . Thus, knowing ψ(x) it is easy to visualize the distribution
ρ(x). We can easily obtain the distribution in the momentum p space by performing
the Fourier transform on ψ(x) → ϕ(p). Although straightforward, knowing ψ(x) does
not make it easy to visualize ϕ(p). It would be desirable to have a function that displays the probability distribution simultaneously of x and p. The Wigner function
allows us to do that. It could be described as an attempt to describe the quantum
mechanical state of a system in almost a classical language. In classical optics as wellas in classical mechanics we can characterize the statistics of a system by introducing
a phase-space distribution W (q, p). The distribution W (q, p) quantifies the probabilityof finding a particular value of q and p in a simultaneous measurement. This meansthat the phase-space distribution describes the state in classical physics. However, thisis more subtle in quantum mechanics. Due to the Heisenberg’s uncertainty principle,
two conjugate variables values cannot be simultaneously known precisely. This meansthat defining a quantum phase-space distribution is more difficult and, as we will see,it may be ill-defined in the sense that W (p, q) could become negative. For this reason,
we will call it a quasiprobability distribution. We can still try to use W (q, p) to makestatistical predictions as if it were a classical distribution; however, these predictions
will only coincide with the classical prediction in special cases. Furthermore, there areinfinitely many ways of defining quasiprobability distributions (simply because there is
no way to define them uniquely) [76]. However, we will mainly discuss one option, the187
Wigner function and mention very briefly other possible definitions of quasiprobability
distributions.

7.1.1

Wigner function

The Wigner function was introduced by Wigner in 1932 with the original goalof finding quantum corrections to classical statistical mechanics where the Boltzmann
factors contain energies which are expressed as functions of both x and p [93]. As it iswell known from the Heisenberg uncertainty principle, there must be constraints on this
distribution and thus on the Wigner function. One way to define a quantum quasiprobability distribution W (p, q) is by postulating its properties. For the Wigner function,
just one postulate turns out to be enough [76, 94]: let’s assume that W (q, p) behaves
like a joint probability distribution of both quadratures. In that case, the marginalR
R
distributions defined as W (x1 , x2 )dx1 or W (x1 , x2 )dx2 must yield the probability
distributions for x2 and x1 respectively. In addition, we can perform a phase shift
θ for all complex amplitudes and accordingly the phase-space distribution will be expected to rotate in the two-dimensional phase space of x1 and x2 . Hence, the way todefine the Wigner function is by postulating the existence of the following probability
distribution [76]:pr(x1 , θ) ≡ hx1 |Û (θ)ρ̂Û † (θ)|x1 i

(7.1)
(7.2)

Z ∞
=
−∞

W (x1 cos θ − x2 sin θ, x1 sin θ + x2 cos θ)dx2 .

(7.3)

where ρ̂ is the density operator defining the quantum state of the system and U (θ) =exp(−iθâ† â) is the phase-shifting operator [76]. These equations also define how the
Wigner function is related to a quantum mechanics description of a state. I will notgive a detailed explanation of how Eqs.(7.1) and (7.3) are sufficient to postulate theexistence of the Wigner function or that it has a one-to-one correspondence with the188density operator. For that, I refer the reader to Ref. [76]. I will only mention some of its
more important properties. Eqs.(7.1) and (7.3) tie W (x1 , x2 ) to physical observations.
For example, consider the special cases of θ = 0
Z ∞−∞and θ = π/2

W (x1 , x2 )dx2 = hx1 |ρ̂|x1 i

Z ∞−∞W (−x2 , x1 )dx2 = hx2 |ρ̂|x2 i.

Both integrals show how to derive the probability distribution for both quadratures. The
previous integral transformations are called Radon transformations and they are studied
in the context of tomographic imaging. The inversion of the Radon transformation iswhat we need to reconstruct the Wigner function from physical observations. This will
be explained in the next section.
It can be shown that the relation between the Wigner function and the density
operator of a system is given by1
W (x1 , x2 ) =
π

Z ∞

y
y
dye2iyx2 hx1 − |ρ̂|x1 + i
2
2
−∞

(7.4)

where this expression is sometimes referred to as the Weyl-transform of the density
operator ρ̂ [95]. First, we note that the Wigner function is real for hermitian operatorsρ̂ and normalized

Z ∞Z ∞−∞−∞

W (x1 , x2 )dx1 dx2 = 1

because the density operator ρ̂ is by definition normalized: tr{ρ̂} = 1. Another interesting property is the overlap formula for two arbitrary operators F̂1 and F̂2 :Z ∞Z ∞
tr{F̂1 F̂2 } = π−∞−∞

WF1 (x1 , x2 )WF2 (x1 , x2 )dx1 dx2for the Wigner function of the operators F̂1 and F̂2 .1(7.5)

The usefulness of the overlap

1
The Wigner function of an arbitrary operator Ô is defined using equation 7.4 with ρ̂ replaced by Ô.
However, unless we make the distinction, we use the term Wigner function only to refer to the special
case when the operator Ô is the density operator.

189
formula is that it can be used to calculate expectation values
Z ∞Z ∞
tr{ρ̂F̂ } = π−∞−∞

W (x1 , x2 )WF (x1 , x2 )dx1 dx2 .

(7.6)

This equation shows that the Wigner function plays the role of a classical phase-spacedensity with WF (x1 , x2 ) being the physical quantity that is averaged with respect toW (x1 , x2 ) in a “classical-like” fashion. A special case happens when the operator F̂ is
Weyl, or symmetrically, ordered in terms of the operators x̂1 and x̂2 . For example, forthe operator F̂ = x̂1 x̂2 + x̂2 x̂1 , Eq. (7.6) becomes
Z ∞Z ∞
hx̂1 x̂2 + x̂2 x̂1 i =−∞−∞2x1 x2 W (x1 , x2 )dx1 dx2 .

which then resembles closely the averaging of a physical quantity with a classical distribution. Another simple consequence of Eq. (7.5) is the expression
Z ∞Z ∞2|hψ1 |ψ2 i| = π−∞−∞

Wψ1 (x1 , x2 )Wψ2 (x1 , x2 )dx1 dx2

However, this quantity must be zero when both states |ψ1 i and |ψ2 i are orthogonal.
Because the overlap Eq. (7.5) cannot be zero for two positive functions Wψ1 and Wψ2 ,this means that the Wigner function cannot be in general always positive for all valuesof x1 and x2 .2This case shows why the Wigner function cannot be regarded as aprobability distribution function since we would expect it to have only positive values
in all its domain in x1 and x2 . Negative regions in the Wigner function of a given state
can be seen as signatures of nonclassical behavior.
The overlap formula can also be used to estimate the purity of state2

Z ∞Z ∞

tr{ρ̂ } = π−∞−∞

W (x1 , x2 )2 dx1 dx2 .

The purity tr{ρ̂2 } ranges between zero and one, and equals unity only if the state ispure (ρ̂ = |ψihψ|). One more characteristic of the Wigner function is that their values
2

In fact, for pure states, only those having Gaussian wave functions have nonnegative Wigner
functions [76].

190
are constrained, that is|W (x1 , x2 )| ≤2πThis constraint shows that the Wigner functions cannot be highly peaked meaning it
cannot approach delta functions or other problematic functions.

Figure 7.1: Wigner functions of a vacuum state (a), a coherent state with α = 1 + i (b),thermal state with ~ω/kB T = 1 (c), and an ideal squeezed state with ∆x21 =1/10(d).
All four states have Gaussian Wigner functions.

Now that I have defined what the Wigner function is, I will describe the Wigner
function of a few important examples: vacuum and thermal states, coherent statesand squeezed states. One interesting thing about all these states is that their Wigner
function consist of two-dimensional Gaussian functions. For a vacuum state, the Wigner

191
function is given by
W0 (x1 , x2 ) =2exp(−2x21 − 2x22 ).
π

(7.7)

In this case, the variance of each variable x1 and x2 is given by ∆x2j = 1/4 as expected(Fig. 7.1a). For a coherent state |αi, the Wigner function is given by
Wc (x1 , x2 ) =£¤2exp −2(x1 − Re(α))2 − 2(x2 − Im(α))2 .
π

(7.8)

As expected, the Wigner function of the coherent state consists of the Wigner functionof vacuum displaced by an amount α from the origin (Fig. 7.1b). For a thermal statewith a density operator in the number basis given by
¶X
µ
∞
~ω
−n~ω
)
)
ρ̂ = 1 − exp(−|nihn| exp(
kB T
kB T
n=0

the Wigner function is given by"#
2
2
Wthermal (x1 , x2 ) =exp −(x21 + x22 )~ω
π coth( 2k~ω
)
coth(
)
2kB T
BT

(7.9)

with both quadrature variances being larger than vacuum’s, ∆x2j = 1/4 coth(~ω/2kB T ).
Finally, for the case of an ideal squeezed state (one for which the equality holds in Eq.
(2.85)), the Wigner function is given by
Ws (x1 , x2 ) =2exp(−2e2r x21 − 2e−2r x22 ).
π(7.10)The Wigner function is a Gaussian distribution, but with unequal variances, ∆x21 =1/4e−2r and ∆x22 = 1/4e2r , indicating the effect of quadrature squeezing. Using Eq.
(7.3) it is easy to estimate the marginal distribution for the rotated quadrature xθ :
x2
1
exp(−
)
pr(x, θ) = q2∆x2θ
2π∆x2θ
where the phase-dependent variance of the Gaussian probability distribution is∆x2θ ==

1 2r 2(e sin θ + e−2r cos2 θ)41[cosh(r) + sinh(r) cos(2θ)]
4(7.11)(7.12)

192
7.1.2

Other distribution functions

From a practical point of view, the Wigner function can be considered as thebest compromise between a classical phase-space density and the correct quantummechanical behavior: it generates the right marginal distributions and it can be easily
used to estimate expectation values using the overlap formula Eq. (7.5). However, there
are other definitions of quasiprobability distributions, the more important being the P ,Q and s-parametrized distribution. I will describe them very briefly, mainly regarding
their relation with the Wigner function.
Q function

The Q function can be obtained from the Wigner function by convolving it witha Gaussian distribution with the same width as vacuum2
Q(x1 , x2 ) =π

Z ∞Z ∞−∞−∞

W (z1 , z2 ) exp(−2(z1 − x1 )2 − 2(z1 − x1 )2 )dz1 dz2 .

(7.13)

Comparing the second part of Eq. (7.13) with Eq. (7.8), we see that the Q function
gives simply the overlap of the density operator ρ and the projection operator for thecoherent state |αi, where α = x1 + ix2 :Q ==1
tr{ρ|αihα|}π1hα|ρ|αiπ

Clearly the Q function will not be negative and since the coherent states |αi forman over-complete basis [26], it can be shown that the Q function is normalized. Thissmoothing with the Wigner function of a vacuum state effectively averages the values ofW (z1 , z2 ) in a circular area around (x1 , x2 ) with a radius given by the vacuum fluctuations. Because any negativity disappears after this procedure, the negative values of a
Wigner function must be concentrated in an area smaller than vacuum fluctuations.3
3Note that this point puts a restriction in the kind of function that can be considered to represent
the Wigner function of a state. It means that the negative values of a valid Wigner function cannot be

193
Looking from an experimentalist point of view, this means that resolving these negativevalues of the Wigner function requires an accuracy on the order of vacuum fluctuations.
The smoothing of the Wigner function is also clearly seen in the Fourier domain.
Performing the Fourier transform on the Q function
Z ∞Z ∞Q̃(u, v) =Q(x1 , x2 ) exp(−iux1 − ivx2 )dx1 dx2
−∞ −∞
·
¸
1 2
2
= W̃ (u, v) exp − (u + v ) .
8This shows that the details in the Wigner function that correspond to high “frequencies”
of u and v are suppressed in the Q representation.
P function

The P function has a very important role in the context of calculating expectation
values of normally ordered operators in quantum optics [26, 76]. I will not discuss its
properties other than its relation to the Wigner function. In the Fourier domain, it canbe shown that·¸
1W̃ (u, v) = P̃ (u, v) exp − (u2 + v 2 )8(7.14)

One feature of the P function is that it can be very ill-behaved. For example, from Eq.
(7.14), we see that the Wigner function is a smoothed version of the P function. But
if the Wigner function can be negative for nonclassical states, the P function may not
only be negative, but it may be more singular than a delta function [26, 76]; that is, itwill not be possible to express it in terms of ordinary functions. For this reason, there
is no practical way to reconstruct it from experimental data.
s-parameterized quasiprobability distributions

I have shown how the relation between the P , the Wigner, and the Q functions
is just a smoothing process with a Gaussian that represents the Wigner function ofextended over areas significantly wider than 1/4 measured in quadrature units; otherwise the Q function
could be negative.

194
vacuum. However, I could also convolve the Wigner function with a Gaussian having
a different width. I then obtain a family of distributions called the s-parameterized
distributions W (x1 , x2 , s). In the Fourier domain this family of distributions is definedas:W̃ (u, v, s) ≡ W̃ (u, v) exp

hs
8i(u2 + v 2 )(7.15)

where s < 0 for smoothing of the Wigner function. All previous examples correspondto the parameters(1) s=1: P function,(2) s=0: Wigner function,(3) s=-1: Q function.
In this new family of distributions the s parameter is defined over the whole range ofthe real numbers. However, for our purposes in this thesis I will only consider the casesof s ≤ 0. In this case, consider the definition for the s-parameterized distribution2
W (x1 , x2 , s) = −πsµZ ∞Z ∞−∞−∞

W (z1 , z2 ) exp

2(z1 − x1 )2 + 2(z1 − x1 )2s¶
dz1 dz2(7.16)

provided s ≤ 0 so that the integral converges. We can then define the marginal distribution of the smoothed Wigner function asZ ∞pr(x1 , θ; s) ≡−∞W (x1 cos θ − x2 sin θ, x1 sin θ + x2 cos θ; s)dx2

s

=

1
π

2
|s|

¸
2
2
pr(x, θ) exp − (x1 − x) dx.
|s|
−∞

Z ∞

·

(7.17)The main reason we are interested in the s-parameterized distributions is that, as I willshow in the next section, inefficiencies in the process of measuring the Wigner function
can be mapped into an effective smoothing. The smoothing parameters s is a functionof such inefficiencies. These inefficiencies will have two main culprits: losses at the input

195
of the detector and added noise of the amplifiers used to measure the (microwave) lightfield.

7.2

Quantum tomography: using a homodyne detector to measure
the Wigner function
As a fundamental feature of quantum mechanics, we cannot know all the in-formation about a system simultaneously. If we observe its position, then we cannot
gain momentum information at the same time, or at least not with arbitrary precision.
However, there is no principle that prevents us from seing all complementary aspectsof a system in a series of distinct experiments on identically prepared objects. The
question is then, can we infer the quantum state from such a set of observation? The
answer is yes. In the same way medical tomographic imaging can reconstruct the shapeof a hidden object inside a patient’s body by gathering projection data from multiple
directions, in quantum optics, tomography has been experimentally applied to reconstruct the quantum state of light from a complete set of measured quantities. Quantum
tomography explains how quantum states can be tomographically reconstructed fromexperimental data. The first theoretical suggestion about how to reconstruct the Wigner
function using homodyne detection was done by Vogel and Risken [96] and the experimental demonstration that proved this method was published in 1993 by Smithey,et.al. [97, 98]. Yet, these experiments have always been done in the optical frequencies,
where highly efficient detectors exist. In the next section I describe the basics of quantum tomography and demonstrate that it is now possible to perform quantum state
tomography at microwave frequencies.

7.2.1

Histograms and radon transformation

In order to understand the concept of tomography in this context, we need toremember Eq. (7.1) that explains how to obtain the distribution function for the values

196

Figure 7.2: Tomographic projection of the Wigner function of a coherent state forθ = 0. Tomography is a method that allows us to infer the “shape” of an object bytaking multiple cross-sectional images or projections of that object. In quantum-state
tomography, the Wigner function of the state plays the role of the “shape” of the object,and its projections are the quadrature probability distributions.

of a quadrature in terms of the Wigner function. In this case, we imagine the “shape”
of the quantum object to be the Wigner function and the transmission profiles are themarginal distributions
Z ∞pr(x1 , θ) =−∞W (x1 cos θ − x2 sin θ, x1 sin θ + x2 cos θ)dx2

which in a sense are the projections of W (x1 , x2 ) onto a line of the quantum space (Fig.
7.2) [76]. As mentioned previously, integrals as the one described in this equation are
called Radon transformations and are the basis for the mathematical theory of tomographic imaging. Although we cannot observe the Wigner function directly, we canmeasure the marginal distributions as quadrature histograms by performing homodyne

197(quadrature) measurements (Fig. 7.3). By varying the phase θ, we can observe thequantum object under different angles and measure the quadrature histograms at allthese different angles. Given the distributions pr(x1 , θ), the mathematics of computerized tomography can be used to estimate the Wigner function (basically, by performing
the inverse radon transformation on the multiple histograms).

Optical homodyne detector

Voltagemarginal
distribution

θ
LO

time

Figure 7.3: Schematic diagram for Wigner function reconstruction. A light signal fromthe prepared state enters a balanced homodyne detector where the projection of the
Wigner function is set by the phase of the local oscillator (LO). The voltage output of
the homodyne detector is measured and its histogram is created. This histogram is an
estimate of the quadrature distributions for that particular phase.

In optics experiments, the quadrature of a light field can be measured using thebalanced homodyne detection [99]. In the microwave world, this is done by using the
microwave mixer that we described in section 6.1.24 However, a microwave mixer has
a very poor efficiency, so in order to measure the nonclassicality of quantum states in
microwave fields, we need to add a few elements to the measurement scheme as we
will describe in the next section. Just like a microwave mixer (only better), an opticalhomodyne detector measures a quadrature component xθ of the incoming field. The
reference phase θ is provided by the local oscillator (LO) and it can be adjusted bychanging its phase. To measure the quadrature distribution, we can either fix the angle
4
It is interesting to notice that even though it can be said that balanced homodyne detection is
intrinsically more efficient in the quantum optics world, the idea of balanced homodyne detection was
actually born in the context of microwave technology development during World War II [100].

198of the LO or sweep it as a function of time, and perform a series of homodyne measurements at all the required different phases. From these measurements, we then build
up a histogram that in the limit of a large number of measurements will approximate
pr(x1 , θ) (Fig. 7.3). The homodyne measurement must be repeated many times to gain
sufficient statistical information about the quadrature values and at sufficient numberof phases in order to accurately reconstruct the Wigner function. These last details will
be discussed in a later section.
7.2.2

Inverse radon transformation

The general idea of quantum tomography has already been explained by defining
the probability distribution for a quadrature measurement as the Radon transform of
the Wigner function, Eq. (7.1). Fourier transforming the marginal distribution results
in
Z ∞
pr(ξ,
˜
θ) =

pr(x, θ)e−iξx dx−∞= W̃ (ξ cos θ, ξ sin θ).

(7.18)In order to obtain the Wigner function we need to perform the inverse Fourier transformin polar coordinates of the measured marginal distribution
Z ∞ Z 2π1
W (x1 , x2 ) =W̃ (ξ cos θ, ξ sin θ)eiξ(x1 cos θ+x2 sin θ) ξdξdθ(2π)2 0Z ∞ Z0 π
1W̃ (ξ cos θ, ξ sin θ)eiξ(x1 cos θ+x2 sin θ) |ξ|dξdθ=(2π)2 −∞ 0
¸
Z π·
Z ∞
1
iξ(x1 cos θ+x2 sin θ)
=
pr(ξ,
˜
θ)e
|ξ|dξ dθ(7.19)(2π)2 −∞
0where in the second line we have used the property that W̃ (ξ, θ + 180◦ ) = W̃ (−ξ, θ).
The integral of Eq. 7.19 may be expressed as
Z π
W (x1 , x2 ) =Hθ (x1 cos θ + x2 sin θ)dθ

(7.20)0

where
1Hθ (t) =
(2π)2

Z ∞
−∞

pr(ξ,
˜
θ)eiξt |ξ|dξ(7.21)

199
This estimate of the Wigner function W (x1 , x2 ) based on the Fourier transform of
the measured histograms pr(x, θ) has a simple interpretation. It represents a filtering
operation where the frequency response of the filter is given by |ξ|. The term Hθ isusually referred as the “filtered projection.” The phase integration is approximated bya sum over the measured phases θ to generate the estimate of W (x1 , x2 ).
Expression (7.21) presents certain mathematical difficulties. From the convolution
theorem it can be expressed asZ ∞Hθ (t) =pr(x, θ)K(t − x)dx−∞

where K(t) is nominally the inverse Fourier transform of |ξ|. However, this kernel K(t)exists only in the sense of a generalized function. I will not get into this detail orthe regularization of the generalized function K(t). I refer the interested reader to theappropriate literature [76,101]. Numerical recipes for the proper implementation of Eq.
7.20 are easy to find in the literature and will not be reviewed here. In our case, we did
not write our own implementation, but used the filter back-projection algorithm alreadyprogrammed as the function iradon in MATLAB 7.7.0. This particular implementation
is explained in Ref. [101].

7.2.3

Inefficiencies and measurement issues

In our experiment where we try to detect microwave squeezed states there are twomain sources of inefficiencies in detection: losses between the amplifiers and the JPA
that generates the squeezed states, and added noise of our amplifiers. Losses between
the squeezer and amplifiers can be easily modeled in the same way losses are modeled in
optical systems: by incorporating a fictitious beam splitter (the microwave equivalent
being a directional coupler) in front of the detector. This beam splitter accounts forthe losses by letting only a fraction η of the incoming intensity pass and diverting thefraction (1 − η) of lost intensity to an unobserved port of a beam splitter (Fig. 7.4).

200
From the fluctuation-dissipation theorem, the loss must be accompanied by fluctuations,which are modeled as noise introduced at the fourth port of the beam splitter. Thesefluctuations are described as the vacuum noise that enters the detector via the otherport of the beam splitter or directional coupler.5

W ( x1 , x2 )η

+ W

output( x1′, x2′ , y1′, y2′ ) = W ( x1′, x2′ )W0 ( y1′, y2′ )W0 ( y1 , y2 )Figure 7.4: Effect of a beam splitter on a Wigner function W (x1 , x2 ). The second port
is assumed unused and therefore only vacuum fluctuations enter this port represented
by the Wigner function of vacuum W0 (y1 , y2 ). The effect of the beam-splitter losses (η)
manifests in the transformation {x1 , x2 , y1 , y2 } → {x01 , x02 , y10 , y20 }, Eq. (7.22)

The effect of a beam splitter on the Wigner function is easily found by using therelation between the input and output ports of the beam splitter [26, 28, 76, 77].

 
 
√
√
0
η
− 1 − η  a 
 a  (7.22)

= √
 
√
b0
1−η
η
b
We are only interested in one of the outputs of the beam splitter a0 =√√
ηa − 1 − ηb,the other output representing the absorbed power. The Wigner function for the joint
system of the input modes is given by Winput (x1 , x2 , y1 , y2 ) = W (x1 , x2 )W0 (y1 , y2 ),where W0 (y1 , y2 ) is the vacuum Wigner function given by Eq. (7.7). We express the
Wigner function in terms of the total Wigner function for both output fields of the beam
5

The loss we are modeling sits inside the dilution refrigerator at 15 mK. Therefore, it is accurate
to model the noise introduced by the loss as vacuum noise given that ~ω À kB T for the frequencies weare interested.

201
splitter. The output Wigner function consists ofpp√√
Woutput (x01 , x02 , y10 , y20 ) = W ( ηx01 − 1 − ηy10 , ηx02 − 1 − ηy20 )pp√√
×W0 ( 1 − ηx01 + ηy10 , 1 − ηx02 + ηy20 ).
Because we only measure one of the outputs of the fictitious beam splitter, the measured
distribution Wmeas (x1 , x2 ) is
Z ∞Z ∞
Wmeas (x1 , x2 ) =

−∞

−∞

p
p
√
√
W ( ηx1 − 1 − ηy1 , ηx2 − 1 − ηy2 )pp√√
×W0 ( 1 − ηx1 + ηy1 , 1 − ηx2 + ηy2 )dy1 dy2 .

(7.23)We see that the measured Wigner function is really a smoothed version of the ideal
distribution W (x1 , x2 ). By making a change of variables and comparing Eq. (7.23)
with the definition of the s-parameterized distribution, Eq. (7.16), it is possible to showthat after including losses, the measured distribution is not the Wigner function but as-parameterized distribution given by
Wmeas (x1 , x2 ) =

x1 x2
1
1
W ( √ , √ , s = 1 − ).
ηηηη

(7.24)How losses affect the Wigner function has a simple interpretation: the rescaling corresponds to the linear loss in the intensity, whereas the s-parameter smoothing (s < 0)corresponds to the introduction of fluctuations that averages the details in the Wigner
function. Losses and inefficiencies smooth the Wigner function of the considered quantum state and lead ultimately to a supression of nonclassical behavior. It is interesting
to note that for 50% detection efficiency, the distribution obtained is a scaled Q function. This means that for η = 0.5, the amount of noise introduced in the measurement
is enough to wash out all the nonclassicality of the Wigner function (recall that the
Q-function is everywhere positive).
So far I have only discussed efficiency problems due to losses. In order to do
our measurements outside the dilution fridge, we amplify our signals before going into

202

W ( x1 , x2 )θηIdeal homodyne
detector+

W0 ( y1 , y2 )Wmeas ( x1 , x2 , y1 , y2 ) = ∫ W ( x1′, x2′ )W0 ( y1′, y2′ )dy1dy2
Figure 7.5: Effective model for losses in the detector. The signal is attenuated andcontaminated by vacuum fluctuations entering the unused port (y1 , y2 ) of the fictitious
beam splitter. The transmittance of the beam splitter represents the homodyne detector
efficiency.

the mixer that will allow us to measure the quadrature of the microwave fields. What
sets the noise floor of our measurement is the first amplifier in the measuring chain(assuming it has enough gain to overcome the noise of the next amplifies). This noise
floor represents the added noise of the amplifier AN and unless AN = 0, this added
noise can be modeled as an effective loss. In order to explain this claim, we will model
the noisy amplifier (in this case our HEMT amplifier) as a nondegenerate parametricamplifier.6

In chapter 2, I described the transformation that a parametric amplifier

does to the incoming fields (Eq. 2.82):

 


0
a
cosh(r)sinh(r)
a

 



=


b0†
sinh(r) cosh(r)b†(7.25)

where a represents the mode we are interested in. Although the origin of the noise may
be a very large amount of degrees of freedom, we can model it as a single fictitious
bosonic mode b, the idler mode of this nondegenerate parametric amplifier. This linear
6

Nondegenerate parametric amplifiers can be used to modeled any linear amplification processcaused by Gaussian reservoirs [28, 102].

203
amplification transformation is very similar to the transformation performed by a beam
splitter, Eq. (7.22). Thus, following a similar approach to the one for the beam splitter,
the Wigner function of the amplified signal field is given by integrating over the idler
mode variables [28]Wampl (x01 , x02 ) =1πAN

Z ∞Z ∞−∞−∞

W (x1 , x2 ) exp(−(y12 + y22 ))dy10 dy20
AN(7.26)

where x0i and yi0 are the modes transformed by Eq. (7.25), and the added noise is givenby the noise in the idler mode11~ωAN = h{b̂, b̂† }i = coth().
22
kB Tidler
This integral can be mapped into a scaled s-parameterized distribution,
Wampl (x1 , x2 ) =1
x1
x2
sinh2 (r) AN
W
(
,
,
−
)
cosh(r) cosh(r) cosh2 (r) 1/2
cosh2 (r)(7.27)

with the smoothing s-parameter given bys=−sinh2 (r) AN.
cosh2 (r) 1/2

For large gain, cosh(r) ' sinh(r), so the effective efficiency for the amplification process
becomesη=1.
1 + 2AN(7.28)For a linear phase preserving amplifier, the smallest possible added noise (standard
quantum limit) is AN = 1/2, consequently η = 0.5. We then see that even for aquantum-limited amplifier, amplifying a signal removes all the nonclassicality of the
Wigner function, because the resulting quasiprobability distribution is the everywherepositive Q function. In our case, when we use the HEMT amplifier to measure squeezed
states our efficiency is less than 2%, assuming an added noise of AN = 26 quanta. In
order to estimate the total efficiency of a measurement, we only need to estimate theefficiency of the separate components, ηloss and ηamp noise . The total efficiency is then

204
given by
η = ηloss × ηamp noise .

(7.29)

Next, I will explain how we improve this efficiency in our measurements.

7.2.4

Implementing an ideal homodyne tomography detector with a JPA

So far it has not been possible to do much with the microwave squeezed states that
we have generated since the detectors at microwave frequencies are highly inefficient.
Unlike optics, where the large energy per photon can be used to create highly efficient
detectors (efficiencies as high as η = 0.97 [103]), the energy of one microwave photon is sosmall that a plausible scheme for a single-photon detector has yet to be developed. The
way small signals at microwave frequencies are measured is by measuring voltages after
the signals have been amplified. Yet, the amplification process is rather noisy; stateof-the-art commercial microwave amplifiers have an estimated added noise number of
10-20 quanta at 5-10 GHz. To provide a sense of the problem, 20 quanta of added noise
is equivalent to only 2.5% efficiency, Eq. (7.28). However, the same type of JPA thatwas used to generates squeezed states can also be used to improve the efficiency of thedetection.
In homodyne detections, only one quadrature is measured. We have seen that,in principle, it is possible to noiselessly amplify one of the quadratures of a signal
and squeeze the other one. Therefore, a way to circumvent issues of efficiency in thehomodyne detection is to noiselessly amplify the measured quadrature by a large enoughfactor so that the smoothing due to inefficiencies in the detector is negligible. The
transformation on the signal mode a and a† described for an ideal generic parametricamplifier was discussed in chapter 2 (Eq. 2.83) and it can be re-expressed as in section

205
2.4.3 in the following way

 


0
a
cosh(r)sinh(r)
a

 



=


a0†
sinh(r) cosh(r)a†(7.30)

where r is just a parameter that determines the gain of the amplifier. Based on thistransformation, the quadrature modes are then transformed in the following way
x01 = er x1 , x02 = e−r x2 ,showing that x1 is amplified and x2 is squeezed. Defining(7.31)√
G1 = er as the quadrature

gain, the transformed Wigner function at the output of the squeezer is given byx0 p
W 0 (x01 , x02 ) = W ( √ 1 , G1 x02 ).
G1(7.32)The proposed set-up shown in figure 7.6 would use a JPA as pre-amplifier for the inefficient homodyne detector (here modeled as a black box that represents the amplification
chain and mixer combined and a beam splitter that models the inefficiencies of the detector). The phase of both the JPA pump and the LO of the homodyne detector must
be properly adjusted so that the measured quadrature corresponds to the amplified
quadrature and not to the squeezed one. The effect of both the noiseless amplifica√tion G1 and absorbtion of the loss η on the Wigner function W (x1 , x2 ) resulting ina squeezed-smoothed Wigner function Wmeas can be easily computed. However, we are
more interested on the final histogram prmeas (x01 , θ) measured:Z ∞prmeas (x, θ) =Wmeas (x, y)dy
−∞
s
µ
¶
Z ∞
2
2ηG1 0x 2
0=
pr(x , θ) exp −[x − √] dx0 (7.33)π(1 − η) −∞1−ηηG1where pr(x0 , θ) is the original marginal distribution of W (x1 , x2 ). From Eq. (7.33), it
can be observed that the smoothing is reduced by increasing the gain G1 . In the limit ofhigh gain, the exponential approaches a delta function resulting in measuring a scaled
ideal distribution
x
1
pr( √
, θ).
prmeas (x, θ) = √
ηG1
ηG1

206
This means that using an ideal degenerate parametric amplifier as a pre-amplifier would
allow us to measured the marginal distributions of the original Wigner function, withthe appropriate rescaling.

−1/ 2

1/ 2

W ( x1 , x2 )W (G1 x1 , G1

Squeezer

Losses and otherx2 ) inefficiencies
ηJPA
gain=G1

θ

Ideal homodyne detector

prmeas ( x, θ) =Vacuum

1
x
pr(
, θ)
η G1 η G1

θ
LO

pump

Figure 7.6: Schematic diagram with a JPA used as a preamplifier for an inefficienthomodyne detector. For large enough JPA gain G1 , the measured histograms approachthe quadrature probability distribution of the ideal Wigner function W (x1 , x2 ), hence
allowing for its tomographical reconstruction.

Comparing Eq. (7.33) and Eq. (7.17) we can see that if we were to use thehistograms from Eq. (7.33) measured with this set-up, we would be able to reconstruct
the following s-parameterized distribution
W 0 (x1 , x02 ) =

x1
x2
1−η
1
W(√ , √ ; −
).
ηG
ηG
ηG ηG

(7.34)

We observe that in the limit of high gain, the smoothing factor s approaches zero and thereconstructed Wigner function is the original, but scaled, Wigner function W (x1 , x2 ).7

7.3

Results
Using a parametric amplifier as a preamplifier to increase the efficiency of thedetectors has already been suggested in quantum optics [28], but it has never been
implemented at microwave frequencies. Next, I will show our implementation of thisidea using Josephson parametric amplifiers to increase efficiency of the measurement of
microwave squeezed states as well as the reconstruction of their Wigner function.
7

Note, however, that this tomographically reconstructed Wigner function is not the Wigner function
Wmeas at the input of the homodyne detector.

207
7.3.1

Details of the experimentφcirculator HEMT
JPA
6 dB

directional
coupler
φ

θ LO

rf

I

room
Q
temperature
4 K amplifer

Dilution fridge

Squeezer

15 mK

Figure 7.7: Schematic of the experimental setup for our implementation of quantumstate tomography of squeezed states with a high-efficiency microwave homodyne detector. Two signal generators are used in the experiment: one as the pump of thesqueezer and the other one as the pump of the JPA (anti-squeezer) and the LO of the
microwave mixer. A pair of noise sources (resistors) are used to estimate the added
noise of the different amplifiers that determine our detector efficiency. All four ports ofthe directional couplers are used since we need to perform carrier-cancelation of both
JPA and squeezer pumps. This is accomplished by splitting the pumps and sending a
copy of the pumps to the fourth port of the directional coupler. The appropriate phaseand magnitude of this cancelation tones are adjusted with manual variable attenuatorsand phase shifters. The incoming and outgoing tones are separated in the same way it
was explained in chapter 6 using microwave circulators. Due to losses in the different
microwave components and cables and added noise of the JPA, our detector inefficiency
cannot be made arbitrarily small even at the highest JPA gain (see appendix B)

The full schematic of the measurement is shown in figure 7.7. It is similar to theone shown in chapter 6 (fig. 6.12). However, we have added an extra JPA in order toamplify the squeezed state generated by the first JPA (from now on referred only as
squeezer). The resonance frequencies of both the squeezer and the JPA meet at two
points for low currents: fres ' 7.45 GHz and fres ' 8.15 GHz, as shown in figure 7.8.
Since the performance of the circulators was better at 7.45 GHz, we operated at this208
frequency. For this reason the performance of the HEMT, JPA and squeezer was tested
close to this particular frequency as explained in appendix B. We are operating the
JPA and squeezer in the four-photon mode (ωp ' ωresonance ), thus in order to prevent
saturation of the JPA with the squeezer pump we need to suppress it by sending aninterfering or cancelation tone using a directional coupler (Fig. 7.7). However, themicrowave circulators are not ideal, having only 20 to 30 dB of isolation. This means
there is some leakage of the JPA pump into the squeezer. Although small, it could
affect the magnitude of the squeezer pump as we switch the phase of one with respect
to another due to interference modifying the amount of squeezing. Hence we also send acarrier cancelation tone for the JPA pump. Nonetheless, the most important cancelationis the squeezer pump cancelation.

(a)

(b)

f res (GHz)864Squeezer
Amplifier-10
I (mA)

Amplifier
Squeezer

1

Figure 7.8: (a) Resonance frequency for the JPA and the squeezer as a function of thecurrent though a coil installed in the mixing chamber. Both resonance meet at multiple
currents, but only for two values if we restrict the current to be smaller than the needed
to create a flux quanta at the resonators. These two values are shown by arrows. (b)The difference in the current dependance for both resonators was achieved by placing
the samples at different locations relative to the magnetic field coil.

We use two signal generators for the measurement setup. In principle the JPA
pump, the squeezer pump, and the LO of the mixer could all come from the same

209
microwave generator. Yet, the carrier cancelations and calibration measurements weresignificantly simplified by using two signal generators. One provides both the LO andthe pump of the JPA. The other provides the squeezer pump. The cancelation tone
for each pump came from the same signal generator as the respective pump (Fig. 7.7).
The appropriate amplitude and phase to suppress the pumps are chosen with variableattenuators and variable phase shifters. The cancelation could be as good as 60 dB,
but it was only stable for a few seconds (long enough to do our measurements). Still, astable cancelation of 40 dB was achieved for several minutes.
Based on the discussion of the previous section, it can be concluded that knowing
the losses as well as other inefficiencies such as amplifier noise in the path of yoursignal is very important. We performed some tests including thermal sweeps and Y factor measurements as the ones described in the previous chapter. These tests are
summarized in appendix B. From that analysis, I will model our setup as shown infigure 7.9. For the analysis I will assume the following values of attenuation (obtained
from the thermal sweep data): α + β ' 3 ± 0.2 dB, α ' 1.7 ± 0.1 dB, β = 1.3 ± 0.3 dB.
Note that due to imperfections in the JPA (AJ 6= 0) and losses between the squeezer
and JPA, the equivalent model (Fig. 7.6) of the microwave homodyne detector will havean additional beam-splitter at the input the JPA. This means that even with the largest
gain possible of the JPA, it will not be possible to reconstruct the ideal Wigner function
coming from the squeezer.
Another important point that needs to be addressed is to avoid saturation of the
JPA with the squeezer amplified noise. Even though one of the quadratures of a squeezed
state may have fluctuations below those of a vacuum state, the photon distribution ofa squeezed state can be rather large (Fig. 7.12). In order to calculate the power at the
input of the JPA, we can calculate the power in the amplified quadrature noise at theoutput of the squeezer. At the input of the squeezer there is a total of around 1 quantaof noise (the 0.5-0.8 quanta of incoming noise plus the 0.2 quanta of added noise). If

210
JPA pump and LO

S in
Teff

ξ

Squeezer

Tf

Tf

α

β
JPA
AJ , GJ

θ

rf

LO
Sout

Amplifiers + mixer
AH , GH

microwave homodyne detector

Figure 7.9: Schematic of our implementation for a high efficiency microwave homodyne
detector. The microwave homodyne detector consist of the Josephson parametric amplifier operating as the JPA plus the commercial microwave amplifiers and the mixer. The
phase of the mixer needs to appropriately chosen to measure the amplified quadratureof the JPA. The noise at the input of the squeezer is modeled as noise coming froma resistor with an effective temperature Teff . The main reason Teff 6= Tc is due to theunknown insertion losses at 4.1 K.

the squeezer has a gain GSQ , the noise power per unit bandwidth at the input of the
JPA is then G~ω (for simplicity, I am ignoring the loss α between the squeezer and
JPA and the Lorentzian frequency dependance of the squeezer gain, as the squeezer isoperated at a small gain, so its bandwidth is bigger than the JPA amplifier bandwidth).
Assuming this noise at the input of the JPA, this means that if the total (fullwidth)
amplifier bandwidth of the JPA is γGain , the total amplified power at the output of
the JPA is GJ (GSQ ~ω)πγGain /2. Here we have again used the convention of defining
the quadrature gain G1 of the JPA as GJ as in chapter 6. The JPA has a saturation
power at the output of about -95 dBm. For a JPA quadrature gain of around 23 dB(direct gain of 17 dB) and a half-width of around 3.5 MHz, this means that we may
be saturating the amplifier with a squeezer gain of 15 dB or a squeezer power gain of
GSQ = 30. However, from appendix B, I know that the a maximum squeezing of about
80% is achieved at a gain of about 10 dB (see appendix B), so we should be able tooperate at this gain with only small saturation problems.
We can now estimate the expected observed squeezing. For a given frequency
we did a Y -factor measurement (sect. 6.3.3) and estimate the HEMTs noise. We did

211
as well a Y -factor on the JPA.8 From figure 7.9, the output power of our microwave
homodyne detector isSout = GH {(Sin ξα + (1 − α)Pf + AJ )GJ β + (1 − β)Pf + AH }= GH GJ αβ{Sin ξ +(1 − α)AJ(1 − β)
AH
Pf +
+
Pf +
}
α
α
GJ βα
GJ βα

= Geff {Sin ξ + Aeff }(7.35)(7.36)(7.37)

squeezer
where ξ = Sout
/Sin , Sin = 12 coth( kBhfTeff ), and Sc = 21 coth( kBhfTc ) ' 21 with Tc

being the temperature of the cryostat (15 mK). Aeff is the effective added noise of thehomodyne detector. In the limit of high JPA gain, Aeff can be approximated as:
Aeff '(1 − α)
AJ
AH
Sc +
+
.
α
α
GJ βα

(7.38)The effective added noise of the detector has three components: the HEMT noise divided
by the effective gain of the JPA (GJ αβ), the added noise of the JPA scaled by the loss α
and the noise coming from this loss Sc (1−α)/α. The effective efficiency of the homodyne
detector is then given, according to Eq. (7.28), byη ==1
1 + 2Aeffα.
1 + 2(AJ + AH /GJ β)

Note that the effective efficiency has the form of Eq. (7.29) for two main sources ofnoise: an amplifier with an added noise of AJ + AH /GJ β and loss α at its input.9
Using values for all the attenuation and noises of amplifiers, with a JPA gain of 180,
α = 1.7 dB, β = 1.3 dB, AJ= 0.25 quanta and AHEM T = 17.3 quanta, the expected
efficiency is 38%. Operating at higher gains would have increased the efficiency up to8
We also compared these values with the ones obtained from the thermal sweeps making sure theyagreed within the error of the measurements.
9
In Eq. (7.28), it is obvious that the origin of the numerical factor of 2 is the fact that there is halfa quantum of noise in the vacuum state. Thus, there can be some ambiguity about whether to use 2Aeff
or 4Aeff in Eq. (7.28), as that equation was obtained for a phase-insensitive amplifier, which adds AN /2
to each quadrature (compared to the 1/4 quanta of noise in the quadrature of a vacuum state), andour quoted noise is the noise added to a single quadrature. However, since we are using a single-sided
spectrum convention, the total quanta per unit bandwidth in one quadrature for a vacuum state is 1/2(see appendix A) so η = 1/(1 + 2Aeff ) is the right expression to use in our case.

212
45% but at a much reduced bandwidth and the difficulty of achieving perfect cancelationof the JPA pump, so the highest gain we used was GJ = 180.
To measure the quadrature histograms there are two possible approaches. We
can fix the phase between the squeezer and the homodyne detector and perform aseries of homodyne measurements at this particular phase to build up a quadrature
histogram pr(x, θ). Then, the LO and JPA pump phase should be changed and repeat
this procedure as many times as the required numbers of phases. Another possibility
is to sweep the phase in a known way while measuring the output of the homodyne
detector, each sample representing a different phase. We used both approaches, but allof the data shown below comes from the second option where the phase was swept at
a known rate.10 The squeezer and JPA pumps were detuned by 100 kHz. This meansthat when we take a time trace, each point corresponds to a different phase between
the squeezer and JPA pumps (basically a different phase between the squeezed noise
and the homodyne detector). The time traces were 0.2 seconds long and digitized at a
10 MHz sample rate with a data acquisition Alazar card (ATS460) with 14 bit resolution
and a 200 mV voltage range. The bandwidth was set by our 5 MHz anti-alias filter.
With a detuning of 100 kHz, we have a full 360 degrees of rotation between the pumps
every 100 samples, which means that in 1 time trace there are 20000 realizations foreach phase.11

7.3.2

Histograms of the traces

The analysis of the time traces include some digital filtering in order to remove
the low frequency noise observed in the amplification and squeezing process explainedin the previous chapter. Besides, it helps remove the remaining squeezer pump thatwas not possible to suppress, which appears as small tone at ∆f = 100 kHz. We also10

We chose the second option since the cancelation of both pumps was better accomplished.
Actually, since the histograms pr(x, θ) and pr(x, θ + π) are not independent the only phases thatmatter for the Wigner function reconstruction are 0 − 180◦ and we then have 50 independent phaseswith 40000 realizations
11

213
(a)

(b)

Vθ

0.08
2000

0.04
0

N

−0.04

−4

x 10

(c)

1000

σ2V

θ

2
1
0

0

2

θ (rad)

4

6

0

−0.05

0
V

0.05

Figure 7.10: Noise at the output of our microwave homodyne detector as a function of
the JPA-pump and LO phase. Blue points are the measured noise when the squeezer
pump is applied and the red points are measured when the squeezer pump is turned off.
The detection bandwidth is 5 MHz. (b) Histograms of the voltage fluctuations. Circles(o) are data and lines represent Gaussian fittings. The red circles are taken whenmeasuring the amplified quadrature of the squeezer and the blue circles represent thesqueezed quadrature. The black circles represent the calibration noise measured when
the squeezer pump is turned off. (c) Variance of the Gaussian fittings as a function ofphase. Both figures (a) and (c) share the same x-axis.

need to do a calibration of the noise level. Traditionally, in the optical experiments
this is done by blocking the input of the homodyne detector and measure the “shotnoise” level, which represents the vacuum noise level. A similar approach is done in
our case. However, we can not go and block the signal path since it sits at 15 mK
inside the dilution refrigerator. What we do is we turn off the squeezer pump andmeasure the noise level at the output of the microwave homodyne detector with the
JPA pump on. Unfortunately, due to issues12 in the calibration (Y -factor) line, weestimate that the input noise is not a vacuum state, but a low occupancy thermal state
12

These issues are mainly the losses at 4.1 K due to the microwave switch and the bias tee used toheatsink the center conductor of the coaxial lines.

214(hni + 1/2 = 0.65 − ±0.15) quanta of noise. For the rest of this chapter I will assume
that hni + 1/2 = 0.65. Assuming an efficiency of 38 %, that means that the noiselevel we measure of our calibration represents 0.56 quanta. The data was taken with 2
different JPA quadrature power gains, GJ = 50 and GJ = 180. Most of the shown data,
however, is from the highest gain since it has a better effective efficiency. For each JPA
gain, 6 different squeezer power gains were taken between GSQ = 1.1 and GSQ = 14.
The gain of the squeezer was modified by adjusting the pump power, but keeping thesame frequency of the pump.
A characteristic time trace for both the squeezer pump on and off is shown infigure 7.10a. For the case when the squeezer pump is turned on, the time trace reveals
the characteristic phase-dependance noise of a squeezed state. A Gaussian fitting to
the points representing a phase is used to estimate the variance of the histograms (Fig.
7.10b). The variance of the Gaussian fitting as a function of phase is shown in figure
7.10c where it is clear the squeezing of noise below our calibration noise.
We can now estimate the amount of squeezing we expect to see. We first startwith the low JPA gain (G = 50). In this case, the efficiency of the detector is η = 0.28.
The input power is Sin = 0.65 quanta. The ratio between the variance of the signal atR
the output of the homodyne detector (σV ∝ Bω Sout (ω)dω) with the squeezer on and
off is defined using Eq. 7.37 asζ=σV (ξ, θ).
σV (ξ = 1)

(7.39)In this case, the maximum observed squeezing should be 26 ± 4%. The observed maximum squeezing as a function of gain is shown in figure 7.11a for the low JPA gain,
and the phase dependance of ζ is shown for two different squeezer gains in figure 7.11b.
In the case of large JPA gain (G = 180), we already estimated the efficiency, which isη = 0.38. The expected squeezing ζ is 36 ± 5%. The observed squeezing ζ as a functionof squeezer gain is shown in figure 7.11c, and its phase dependance for two different

215
(b)

1

ζ(θ)

ζ(GSQ )

(a)

0.5

0

5

10

10

1

0.1
0

15

2

G

SQ

(d)

1

ζ(θ)

ζ(GSQ )

(c)

0.5

0

5

10

15

θ (rad)

4

6

4

6

10

1

0.1
0

GSQ

2

θ (rad)Figure 7.11: (a) Measured real squeezing ζ (circles) as defined by Eq. (7.39) andprediction (lines) using Ref. [38] as a function of squeezer gain. The solid line is assuming
an incident noise of 0.65 quanta, while the dashed lines represent the uncertainty on
this value. (b) Squeezing ζ as a function of phase for two different gains. The gains
shown in (b) correspond to the same colors as the circles shown in (a). (c) and (d) aresame as (a) and (b) but for a JPA gain of 180.

gains is shown in figure 7.11d. In general, we observe the expected behavior and thedisagreement with theory observed is of the order of a 1 % for high gains and up to
4% at lower gains. The maximum observed real squeezing is ζ = 37% as observed infigure 7.11c, which is close to the estimated squeezing. We were unable to go further insqueezer gain for two main reasons. The observed squeezing stops growing for higher
squeezer gains, as was shown in the previous chapter.13

And in the case of high JPA

gain we observed a small saturation (reduction of the JPA gain by 0.1 dB) that we
believe was not related to a poor cancelation of the squeezer pump but caused due to
13

In this particular cooldown, the maximum squeezing happens at a gain of about GSQ = 10 asshown in appendix B, not at GSQ = 30 as observed in the previous cooldown described in chapter 6.

216
saturation from the large amplified noise coming from the squeezer.14
So far, our analysis of the time traces has not been very different from the previous
chapter as comparing the variance of the calibration and the squeezed state histograms
is effectively the same as comparing its power spectral density. The only difference
has been that by using a JPA as a preamplifier for our HEMT amplifier, our effective
efficiency has been increased from 2% to 38%, which allow us to have an observed
real squeezing of ζ = 37%. However, this increment in efficiency can allow us to start
reconstructing quasiprobability distribution functions. Note, though, that since ourefficiency is only η ' 0.4, the reconstructed distribution is really an s-parameterizedfunction, with s = 1 − 1/η ' −1.5. But for simplicity I will call this reconstruction,
Wigner function reconstruction.

7.3.3

Quasiprobability distribution reconstruction

For the reconstruction of the Wigner function based on the histograms, we needto first consider if the number of phases and the number of points are enough for areliable reconstruction. In this case, since we are analyzing squeezed state, which has aGaussian Wigner function, the reconstruction is very forgiving.
What is the effect of only using d phases for the Wigner function reconstruction? One first consequence is that using a phase resolution of π/d would prevent us
from properly discerning details with phase oscillations in the quadrature distribution
pr(x, θ) with frequencies higher than d as they would be aliased. In our case, the phasedependant variance of the Gaussian distribution of a quadrature squeezed state only
oscillates once in a π period, so this possible error of aliasing certain details of the
Wigner function is not a concern. Leonhardt in Ref. [76] derives a rule-of-thumb for theminimum amount of phase values d in order to reconstruct the density state based on
14

In the previous section I estimated that a squeezer gain of 15 dB or GSQ = 30 could start saturating
the JPA. This was assuming the saturation point as the 1 dB compression point. This estimation is not
far from the observed gain that started showing saturation behavior.

217
quantum tomography. This rule-of-thumb reads “the number of phase must be equal
to the minimum dimension to accurately approximate density matrix of the measured
state.” However, although this a sufficient condition, it is not a requirement [76], especially for Gaussian states such as squeezed states. In any case, even with this constraint,
figure 7.12 shows that the 50 phases taken over a range of π are more than enough.

0.75

P10=0.01

Pn

0.5

0.25

0

0

2

4

6
n

8

10

Figure 7.12: Probability distribution for and ideal squeezed state with ∆x21 = 1/4e−2 ,or squeezing of 84 %. Setting an accuracy for the tomography detection of Pn > 0.01,
suggest that a density matrix of dimension n = 10 would be sufficient for an accurate
Wigner function reconstruction.

Regarding the statistical error due to finite samples of length N for a given phaseθ, how much confidence do we have in the reconstruction of the quasiprobability distribution. Dividing the measurements in bins of size δx, the rate at which the quadrature
values fall into that bin are governed by the expectation value pr(x, θ). This process isa Poisson counting process, with the measured prmeas (x, θ) being a fluctuation quantity.
As a Poisson process, I expect the value of the histogram representing the bin δxj tohave fluctuations σj2 of the order of Nj . For the measured distribution prmeas (x, θ) which
is formed by the histogram normalized by the total number of measured samples and218
the uniform bin size δx, I expect then its variance to be given by2
σpr=meas (xj ,θ)pr(xj , θ)2,N δx(7.40)

showing the expected result that the fluctuations in the measured marginal distributions prmeas (x, θ) will have a 1/N dependance. The question is, though, how are thesefluctuations going to affect the reconstruction of a Wigner function. Again Ref. [76]gives a rule-of-thumb based on Eq. (7.40) but using the photon number distribution
ρn,m . There it is shown that the variance of ρn,n is bounded by
2
σnn
≤

4
N

and since for an experiment we want the statistical error to be smaller than the measured
quantity,
σnn < ρnn .
This estimate implies that the required number samples per phase interval scales with
the inverse of the photon-number probability squared
Nmin ∼

4
ρ2n,n.

If I want to resolve states with an accuracy of Pn > 0.01 (Fig. 7.12), this implies a
total number of samples per phase interval of N = 40000 over a phase range of π. Eachof our time traces have this amount of samples per phase, and in order to make thereconstruction better, we took 10 time traces making a total of 400000 samples perphase interval.
Once we have decided that we have enough phase resolution and samples for areliable construction of the quasiprobability distribution we can use the inverse radonalgorithm to reconstruct it. We first need to scale the measured quadratures accordingly.
Assuming again that our incoming noise is a thermal state with hni = 0.15, then thevariance of the Gaussian histogram measured with η efficiency is given by1∆x2meas = (1 − η) + η∆x2therm
4(7.41)

219

0

θ

∆ x2

10

−1

10

0

1

2

3

θ (rad)4

5

6

Figure 7.13: Assuming the Sin = 0.65 quanta and an efficiency of η = 0.38, we use thethe noise calibration to estimate the real quadrature values of the time traces measured
with our microwave homodyne detector. The solid line represents our calibration noise(∆x2meas = 0.28) the dashed line represent our estimate of the shot noise limit. Fromthe squeezed traces, we observe a minimum real squeezing of ζ = 37% which represents
30% below the vacuum level.

where∆x2therm

=
=

µ
¶
1
~ω
coth
,
4
2kB T
1
(1 + 2hni),4= 0.33.
In our case, because our calibration noise has a variance of ∆x2therm = 0.33, ourmeasured variance is equivalent to ∆x2meas = 0.28. Note that if we had been able touse a vacuum state, Eq. (7.41) would have given us the “shot-noise” limit ∆x2meas =1/4. The calibrated squeezing then looks like shown in figure 7.13. Based on thisnormalization, we estimate that the maximum real squeezing is 30% below the vacuum
level. Compared with the 2% of real squeezing that was observed using only the HEMT
amplifier (Fig. 6.34b), we then can conclude that using the JPA as a preamplifier has
increase significantly our detection efficiency.

220

Figure 7.14: Wigner function reconstructed from the data for the squeezed state (a)and the calibration noise (b). The projection of the Wigner functions are shown in (c)and (d) including contour plots.

With calibrated histograms now, I use the MATLAB function iradon to obtain the
Wigner functions of the prepared states. This is shown in figure 7.14 for the “calibrationnoise” and squeezer gain of GSQ = 14 and a JPA gain of GJ = 180. Both distributions
were calculated by averaging the distributions reconstructed from each of the 10 time
traces in the case of the squeezed state, and the 4 time traces in the case of the calibrationnoise. In the case of the squeezed states, the phase offset for each time trace was different
so we first did a fitting to the variance as a function of phase using an expression similar
to 7.12 but without assuming intelligent states (the variances h∆x2i i do not fulfill the221equality h∆x21 ih∆x22 i = 1/16). Once the phase offsets are calculated for each trace, thenthis phase offset is included in the reconstruction of the Wigner function by providing
the appropriate phase vector to the MATLAB iradon algorithm.

Chapter 8
Applications II. Increasing the sensitivity in the detection of
mechanical motion of a nanomechanical wire.

In this chapter I will discuss the use of a Josephson parametric amplifier in order
to increase the sensitivity of the detection of motion of a nanomechanical wire. The
coupling of light and mechanical degrees of freedom is the underlying principle of manytechniques to measure mechanical displacement and sits at the core of the research field
called Optomechanics. Explaining all the interesting possibilities that emerge from these
coupling of optical and mechanical degrees of freedom would take far from the main goalof this thesis. Instead, I will first give a very brief description of the concepts that will
helps us understand the importance of using a JPA in the detection of motion of ananomechanical harmonic oscillator. This will be done in the first part of this chapter.
Sometimes I will be using a language that seems closer to that used in optics. The
reason for this is that, as it is obvious from the name, optomechanics is a field that has
been dominated by implementations that use light at optical frequencies to probe themotion of mechanical oscillators. However, in the case of our group, the implementationis all done with microwave circuit elements. In the second part of this chapter I will
describe some features of this implementation, specifically the use of microwave cavities
and the way we implement the coupling of the mechanical oscillator to the microwavelight degrees of freedom. In the last section I will discuss the results, specifically how theimprecision of our measurement was below the imprecision at the Standard Quantum

223
Limit (SQL).
One point that I want to emphasize is that I am not part of the group that
performs the microwave-cavity optomechanics experiments. In the experiments that I
will describe in this chapter, I only helped in the implementation of the JPA in themeasurement scheme and some of the experimental work. For this reason, I will not
discuss details such as the fabrication of the devices, or the specific decisions regarding
the design of the sample. For those details, I refer the reader to the paper published byour group regarding this experiment [104].

8.1

Optomechanics

CavityMechanicalOscillator

Laser Input

Phase detector

Figure 8.1: Schematic of a typical optomechanical setup. An optical cavity is formedby two parallel mirrors, one of which is attached to a mechanical oscillator (red). Light
entering the cavity (cyan) induces radiation pressure that affects the motion of themechanical oscillator, which in turn alters the mode frequency relative to the frequencyof the incident light. The effect on the reflected light is measured with a phase detector,
for example a Mach-Zehnder interferometer.

Optomechanics is concerned with the effects that emerge from having a coupling between mechanical and electromagnetic degrees of freedom, such as the effectsof radiation pressure on a mechanical object. We are mostly concerned with those inconnection with micro- and nanomechanical structures coupled to a resonant cavities

224(cavity-optomechanics). A standard setup of cavity-optomechanics used to measuredisplacement is a Fabry-Perot interferometer such as the one shown in figure 8.1. It
consists of on optical cavity driven by a laser impinging on the cavity through a fixedmirror. The other mirror of the cavity is movable (for example, it may be attached toa cantilever). In such a setup, the mechanical effects of light are enhanced, as the lightfield inside the cavity is resonantly increased and each photon interacts with the movablemirror in each of the multiple (of the order of Q) reflections it undergoes before leaving
the cavity. The coupled nanomechanical-cavity system is described by a Hamiltonianof the form
1
1
H = ~ω0 (a† a + ) + ~ωm (b† b + ) − ~gc a† a(b† + b)δxzp ,
2
2

(8.1)

where the cavity and mechanical modes are described respectively by a and b, ω0 is
the bare cavity resonance, ωm is the mechanical resonance, and gc = ∂ωc /∂x is thecoupling between the mechanical harmonic oscillator displacement x = (b† + b)δxzp andthe perturbed cavity’s resonance frequency ωc . δxzp is the oscillator zero point motionpδxzp = ~/2mωm . In Eq. (8.1) I have left out the additional terms that describe thedriving of the cavity by an outside light source (see section 2.3.3).
In terms of mirror displacement measurement, it is possible to show that a singleport cavity described by Eq. (8.1) will have a linear dependance between the the phaseshift φ of the reflected signal from the cavity and the position x of the oscillator:φ ≈ 4Q

gc
x.
ω0

(8.2)

The previous expression can be derived using the tools of IOT on the Hamiltonian given
by Eq. (8.1) [3]. Two fundamental sources of imprecision exist for the measurementof φ and hence x. First, there is the detector noise that, for an ideal coherent lightsource (emitting a coherent state) and an ideal detector, is given by the random arrivalof photons at the detector; i.e., shot noise (Fig. 8.2). The phase imprecision expressed225
in terms of single-sided power spectral densities is given by [3]:
imp
Sφ,φ
=

1
2Ṅ

(8.3)

where Ṅ is the photon flux at the incident on the detector. An interferometer with aphase uncertainty given by Eq. (8.3) is said to be shot-noise limited. We can find an
expression for the imprecision of our measurement of the position1 x, by multiplying Eq.
(8.3) by a transducing factor ∂φ/∂x that relates the displacement x in the mechanicaloscillator and the phase φ of the signal at the output of the cavity (Eq. (8.2), forexample). Hence the imprecision of our measurement for x is given by [104]
imp
Sx,x
=

=

1
1
2Ṅ (∂φ/∂x)2
1
1
2Pin /~ωc (∂φ/∂x)2

(8.4)
(8.5)

where in the second line, I have expressed the phase imprecision in terms of the incident
power in the detector rather than the photon flux.

X2

∆φ2=

1/ 4
α2

α
Vaccum noise

X1
Figure 8.2: A coherent state contains a Poisson distribution of the number of p
photons.
Thus the fluctuations and the mean number of photon are related as ∆N = hN i =|α|. This meansp that there is an uncertainty in any measurement of the phase givenby ∆φ = 1/2 hN i. The equivalent of this statement in terms of spectral densities
associated with a real experiment is given by Eq. (8.3), where (∆φ)2 → Sφ,φ and(∆N )2 → SN = 2Ṅ [3]. Note, however, that we use a single-sided spectral density.

In order to land some of these ideas, let’s consider what it is actually measured.
1

It is important to point out that the measurement discussed here is what it is known as a weakcontinuous measurement which means that the position x of the mechanical oscillator undergoes multiple
cycles before we can acquire significant information [3].

226
According to the fluctuation-dissipation theorem, a mechanical oscillator with a mechanical damping rate of γm will experiment a random force with a power spectral
density of
th
SF,F

1
= 4~ωm (nT + )
2

(8.6)

≈ 4kB T mγm ,

(8.7)

~ωm

where nT = 1/(e kB T − 1). In the second line, I have done the classical approximation,
consistent with the equipartition theorem, since our mechanical resonator will behaveclassically due to a high occupancy nT (kB T À ~ωm ). The response of the resonator isgiven by
Sx,x = SF,F |H(ω)|2
¯
¯2
¯
¯
1
¯
¯
= SF,F ¯
2 − iωγ ) ¯m(ω 2 − ωm
m

(8.8)
(8.9)

where H(ω) is called the mechanical susceptibility. We use a single-sided convention
such that

Z ∞
0

kB T
dω
Sx,x =
22πmωm(8.10)

In our experiments, when the spectral density of the phase of the reflected light Eq. (8.2)
is measured by an interferometer, what will be measured is the Lorentzian response ofth on top of a background imprecision
the mechanical oscillator to the thermal force SF,F

noise floor (Fig. 8.3). Such measurement will allows us to calibrate the coupling of themechanical oscillator to the cavity and assuming we know the temperature, we will beable to calibrate the imprecision of our measurement (the background noise floor).
As can be deduced from Eq. (8.2), the detector imprecision decreases with larger
incident power. Increasing power, however, comes at the expense of increased intracavity power, causing a back-action onto the mirror: every time a photon reflects off
the movable mirror, it will transfer momentum to it. This leads to a second source

227

Sφ,φ (rad2/Hz)

area
proportional
to T

0Measurement
imprecision1
frequency (ω/ω )2mFigure 8.3: Example of the phase spectral density at the output of an interferometerused to measure a systems such as the one shown in figure 8.1. The Lorentzian peakcorresponds to the thermal motion of the mechanical oscillator, while the backgroundis our imprecision of the measurement, ideally just given by photon shot-noise.

of imprecision: the resulting momentum kicks of reflected photons create a mirror displacement noise, causing the mechanical oscillator to be driven and thus effectively
heated. Although this noise can contain classical contributions due to uninterestingsources of noise (excess phase or amplitude noise, in the driving light, for example), it isultimately limited by the quantum nature of light. Hence, it is termed quantum backaction [105]. Since the back-action does affect the quantity being measured (position),
there is a limit on the accuracy on the measurement. This limit is described by the
Heisenberg uncertainty limit, which ensures that an increasingly precise measurementof a harmonic oscillator’s displacement is accompanied by an increasingly large force
acting back on that oscillator [106]. This principle limits the total noise energy addedin a measurement of displacement to be at least half a quantum ~ωm /2. The limit
between the imprecision of the measurement and its back-action is better described, in
imp b.a.
imp
spectral density units, as Sx,xSF,F ≥ ~2 , where Sx,xis the displacement imprecisionb.a. the back-action force spectral density, both specified as single-sided spectral
and SF,F

densities. A measurement reaches this limit, called the standard quantum limit (SQL)228
when the imprecision and the driven motion from the back-action force each add precisely one quarter of a quantum of energy ~ωm /4. Therefore, the imprecision at the
SQL provides a natural scale (which we will be using) to which to compare the noiseof the measurement. On resonance, and in units of displacement spectral density, anoise energy ~ωm /4 corresponds to a displacement noise of Sx,x = ~/(mωm γm ). It is
worth noting that although the total added noise of the measurement has a minimum
T = S imp + S b.a. = 2~/mω γ , the imprecision can be made arbitrarily small comSx,x
x,x
m m
x,x
SQL
pared to that at the SQL, Sx,x
= ~/mωm γm . A system described by the Hamiltonian(8.1) can in principle reach the SQL. Physically, at the SQL, our knowledge of the mirror
position is limited equally by shot-noise in the measured signal and by the motion of ofthe mirror due to the random kicks the mirror receives from the photons. On the other
hand, if the strength of the drive used to probe the cavity is too small, the imprecisionof the measurement is dominated by the photon shot-noise Ṅ . If the drive is too big,
the back-action noise perturbs the oscillator so strongly that the position uncertainty
increases beyond the minimum.
Although the concept of quantum back-action is very interesting and its observation is a goal pursued in our lab, I will not discuss it here. For the effects of quantum
back-action to become most apparent, we would need two things: first we would need toenter the regime where quantum fluctuations (zero-point motion) become the dominant
fluctuations felt by the mechanical oscillator (kB T ¿ ~ωm ), or equivalently a thermal occupation less than unity (something our lab is working on to achieve [107, 108]).
And second, we would need a quantum-limited light detector, such as for example a
Josephson parametric amplifier.
Most experimental implementations of such optomechanical systems are madeusing light at optical frequencies. Photons at optical frequencies are very well suited
to measure mechanical displacement for several reasons. For example, because of thehigh energy of optical photons (∼ 1 eV), thermal occupation is negligible at room229
temperature, present-day laser sources are available that offer noise performance that islimited only by quantum noise, and detectors which a high efficiency allow to measure
at the quantum-level, that is being only limited by the vacuum noise of the light used.
However, devices in the microwave domain such as a nanomechanical resonator coupled
to a superconducting microwave resonator have an advantage in some regards. For
example, for such microwave implementations, the measurement setup is compatible
with dilution-refrigerator temperatures, automatically cooling the mechanical motion
by four orders of magnitude. Next I will discuss our implementation of the Hamiltonian
given by Eq. (8.1).

8.2

Nanomechanical resonators inside microwave resonators
The microwave analysis of the resonator is similar to the ones done in chapter 5.

The implementation of the experiment is shown in figure 8.4a. For simplicity we model
the microwave resonator as a lumped element resonator (Fig. 8.4b). The geometry
used to excite the resonator is also somewhat different from the ones we have described.
Instead of measuring the resonator in reflection, or in transmission by using a two-port
resonator, we place the microwave resonator as a shunt across a transmission line. Thisgeometry (referred to as the feedline geometry), allows us to measure more than one
cavity by putting several of them in the feedline. Each resonator is designed to havea slightly different resonant frequency. As long as the resonance frequencies of each of
the microwave cavities are separated in the frequency axis by several linewidths, thenit is possible to individually address them [69, 70].
We measured the transmission coefficient from port 1 to port 2, S21 which is givenby [16]:S21 =2
2 + Zc /Zr(8.11)

where Zr is the impedance of the circuit shunting the feedline. For now let’s assume a

230
(a)

(b)

Port 2

Port 1Zc
feedline

Zc

Cb ( x )

Lr

Cc

Cr

Rr

(c)

x

Figure 8.4: (a) Optical image of the inductor-capacitor (LC) resonant circuit, which
behaves as a microwave cavity with ωc = 2π × 7.49 GHz. The cavity linewidth is givenby γc = 2π ×2.88 MHz. The microwave-cavity was fabricated from superconducting aluminum on top of a silicon substrate. The nanomechanical oscillator consists of nanowire
embedded in the microwave resonator near the CPW feedline used to excite the resonator. The wire has dimensions of 150µm × 170nm × 160 nm, yielding a total mass
m = 11 ± 2 pg and a mechanical resonance frequency ωm = 2π1.04M Hz. (b) Electrical
diagram of the device. The resonator is coupled to the feedline by a capacitor Cc . The
effect of the nanowire on the circuit is equivalent to that of a small capacitor Cb inparallel to the LC resonator. (c) The mechanical degree of freedom is the fundamental
mode of the freely suspended aluminum nanowire.

lossless cavity (Rr → ∞). The bare cavity resonance is given by1.
ω0 = √
Lr Cr
However, by finding the location of the minimum of |S21 |, we can find the new resonancefrequency of the cavity loaded by both capacitors:
ωc = ω0 (1 −

Cb
Cc
−
).
2Cr
2Cr

(8.12)

If we now consider finite loss Rr , but still working in the high-Q limit, we find that the

231
quality factors associated with internal losses and the coupling capacitor:
Qc =

2Cr
ω0 Cc2 Zc

and
Ql = ω0 Rr Cr .
Based on this expression, Eq. (8.11) reduces toS21 =S0 + 2iQ(ω − ωc )/ωc
1 + 2iQ(ω − ωc )/ωc

(8.13)

where Q is the total quality factor given by Eq. (5.27)
Q=

Qc Ql
Qc + Ql

S0 =Qc.
Qc + Ql

and

From the measurements of the scattering parameter S21 , the internal and coupling
quality factors of the cavity used for the experiment were Qc = 3.0 × 103 and Ql =2.6×104 . Yet, in the case of Ql , it varies by as much as 20% for the range of temperaturesand powers used in the experiment.
Eqs. (8.12) and (8.13) and figure 8.4c allows us to understand how it is that
we can infer the motion of the beam from the information encoded in the measured
S21 . For that, let’s plot the phase and amplitude we expect for some usual parameters(Fig. 8.5). As it can be seen in those plots, the amplitude is rather insensitive to smallchanges of resonance frequency. However, the phase depends linearly to changes in theresonance frequency (to first order). Therefore, the motion of the beam (Fig. 8.4c) will
generate changes in the resonance frequency, Eq. (8.12) due to changes in the capacitor
Cb (x). When measuring a cavity of the kind shown in figure 8.4a, usually the frequencyof the microwave probing tone remains constant. Hence, the motion of the beam will
be encoded in the phase of the transmitted signal (Fig. 8.5c).

232
(a)

(b)

2

|S12|

1

φ (radians)

1

0.5

−1
0

φ (radians)

(c)

−5

0
∆ω/γ

5

−5

0
∆ω/γ

5

1
0.5
0
∆ φ
−0.5
−1
−1

−0.5

0
∆ω/γ

0.5

1

Figure 8.5: (a) Magnitude |S21 | and (b) phase φ of the transmission coefficient as afunction of frequency for the feedline geometry used to probe the microwave resonators.
The values used for the quality factors were the same as the ones found from themeasurement of S21 , Qc = 3000 and Ql = 26000. (c) The phase is a very sensitive
function of the resonance frequency. Small changes in ωc , correspond to large changes
in phase ∆φ.

The simplified diagram of the microwave setup used in shown in figure 8.6. Itis essentially the microwave analog of a Mach-Zehnder interferometer (MZI) typicallyused to measure phase-shifts. However, at microwave frequencies, from the same reasons
there is not a commercial mixer with quantum-limited sensitivity (sect. 6.3.3), there
is not such a thing as a shot-noise limited phase detector. As shown in figure 8.6,
the signal at the output of the cavity is actually amplified by a chain of commercial
amplifiers (same as explained in chapter 6) before entering the RF-port of the mixer.
Calling AN the noise floor of the amplifier chain (including the mixer) , the equivalentof Eq. (8.2) for a microwave interferometer that is not shot-noise limited because of this

233
φ

AN

Dilution fridge

Figure 8.6: Simplified diagram of measurement setup. We detect the motion of theoscillator with a microwave analog of a Mach-Zehnder interferometer. A microwave toneis split and injected into the two arms of the interferometer. One arm forms the phasereference (LO of an IQ-mixer), and the other arm contains the cavity optomechanicalsystem shown as an LC-microwave resonant circuit with an embedded nanomechanicaloscillator. Displacement of the oscillator changes the capacitance of the electrical circuitand therefore its resonance frequency, Eq. (8.12). The microwave tone excites thecircuit and acquires a phase shift proportional to the mechanical motion. That phasemodulated microwave signal is first amplified by a chain of microwave amplifiers andthen mixed with the phase reference.

extra added noise AN is
imp
Sx,x
=

1
AN + 1/2
Pin /(~ωc ) (∂φ/∂x)2(8.14)

Although the sensors for the applications mentioned above are governed by classical physics, the imprint of quantum phenomena upon them can be readily seen in thelaboratory if care is taken in preparing the mechanical oscillator in the ground state.
As I said before, there is an ongoing effort in the field (and in our lab) to accomplish
this endeavor. But, even if we were able to do it, in order to reach this quantum limit,
the detector must be quantum limited. Here is where the use of a quantum-limited
amplifier becomes necessary in experiments used at microwave frequencies. The signalthat comes from the cavity is just a phase-modulated (PM) signal, and all we want to
do is to measure it with the best sensitivity possible. Since, the signal consists of alinear combination of fourier frequencies around a carrier tone (Fig. 3.14), I have shownthat the most efficient way to measure such a tone is by using a degenerate parametric234
amplifier like the JPA. As long as the phase between the pump of the JPA and carrier of
the PM-modulated signal is chosen appropriately, we could ideally amplify that signal
with no added noise, and could recover the ideal case for the imprecision in Eq. (8.2).

8.3

Experimental results. Increasing the sensitivity and getting
closer to a shot-noise limited detection.

θInterferometer
input

JPA pump
andcancellation
tone

room
θ LO
temperatureamplifer
rf
φ

HEMT
20 dB
4K

3 dB

20 dB
Isolators

15 mK

JPA
CavityDilution fridge

Figure 8.7: Detailed diagram of the measurement setup. A microwave generator is used
to generate a tone that gets split into three main arms plus one carrier-cancelation arm.
The first is used to excite a microwave cavity with a coupled nanomechanical oscillator.
The second arm is used to pump the JPA and the third arm provides a phase reference
for the mixer at room temperature. The carrier-cancelation arm is used to suppress thetone from the first arm in order to prevent saturation of the JPA. The setup for the Y factor measurement (switch plus two terminations held at two different temperatures)is connected to the main port of the first directional coupler.

The detailed diagram of the setup used is shown in figure 8.7. A large microwave
tone is used to probe the cavity, and only a fraction (1 − S02 ) is transmitted. The

235
transmitted power has the information regarding the motion of the nanomechanicaloscillator in the form of phase modulation. Before reaching the JPA (a λ/4 resonator),a carrier-cancelation similar to that described in chapter 7 needs to be done in order toprevent saturation of the JPA. Once the carrier-cancelation is performed, the JPA pumpphase is chosen appropriately so that the phase-modulation information is amplified.
The gain of the JPA was chosen close to Gs (0) = 14 dB, or equivalently, a quadrature
gain 6 dB higher, GX1 = 20 dB (higher gains were used for some measurements but
a proper carrier cancelation was too hard to achieve for the highest incident power on
the cavity-optomechanics system). After the JPA, the measurement setup is similar tothe one described in section chapter 6. The amplified signal emerging form the cryostat
is down-converted using an IQ-mixer (as a homodyne detector, ωc = ωp = ωLO ). We
can convert voltage spectral density at the output of the mixer into spectral density ofcavity fluctuations from the following expression,Sωc (ωm ) =2 /γ 2
1 + 4ωmc
SV,V (ωm )(∂V /∂ωc )2(8.15)

2 /γ 2 is included to account for the fact that since ω /γ ∼where the factor 1 + 4ωmm ccO(1), the cavity field cannot respond instantaneously to the changes in ωc due to themechanical-motion of the nanowire. Hence, the frequency fluctuations Sωc (ωm ) are
2 /γ 2 )filtered by the Lorentzian response of the cavity, the filtering factor being 1/(1+4ωmc[109]. The voltage V at the output of the mixer will be given by V = V0 Im(S21 ),where V0 is a function of the incident power on the cavity as well as the attenuation
and amplification through the whole signal path. It can be found by measuring thetransmission |S21 | far from resonance.
Once we know how to convert the voltage fluctuations at the output of the mixer
into cavity frequency fluctuations, we can use the thermal motion of the nanowire (Eq.
8.10) to estimate the optomechanical coupling gc = ∂ωc /∂x used in Eq. (8.1). Thisis shown in figure 8.8. The thermal sweep shown in figure 8.8b yields a value of gc =

236
−3

× 10

−2

(b)
10
2

(δωc /2π) (kHz )

−3

2

10

−4

10

1.0435

5

2

SV,V (V2/Hz)(a) 101.0437frequency (MHz) )1.0439

0

0

0.15
0.3
Temperature (K) )Figure 8.8: Fluctuations at the output of the mixer in arbitrary units (V2 /Hz) for a
fridge temperature of T = 0.150 K. The red data was taken using the JPA and the blue
only using the HEMT. By using the JPA, we observe an improvement in the sensitivityof over 13 dB with respect to the HEMT for similar incident powers. With Eqs. (8.10)
and (8.15), we can calibrate the interferometer and convert the voltage fluctuations at
the output of the mixer into fluctuations in the cavity’s resonance frequency due tothe thermal motion of the mechanical oscillator as shown in (b). From a linear fit tothese fluctuations as they vary with temperature, we can determine the optomechanicalcoupling constant gc . The lowest three temperature points are excluded from the fit as
the mechanical mode is no longer in thermal equilibrium with the cryostat.

2π × 32 ± 3 kHz/nm. One thing to notice is that at the lowest temperature of the fridge,
the beam is no longer in thermal equilibrium with the cryostat. Knowing the valueof gc will let us directly measure the temperature on the mode from the area under
the Lorentzian curve in the displacement spectral density Sx,x . Also, it will help us
determine the imprecision of the measurement. This can can be shown by using the
expected dependance of φ from Eq (8.13). Then, Eq. (8.14) becomes [109]
µ
¶2
¡
¢
S0 ωc
AN + 1/2
imp
2
Sx,x =
1 + 4ωm/γc2
Pin /(~ωc ) 2Q(1 − S0 )gc(8.16)

Eq. (8.16) demonstrates that the absolute imprecision is reduced by measuring with alarger power, a stronger coupling gc and minimum added noise AN .
However, even with the JPA, the microwave interferometer is not shot-noise limited. We estimate the noise-floor of the amplifier chain as explained in section 6.3.3

237
using a cryogenic switch that can toggle between two 50 Ω terminations held at two
different temperatures. However, in order to accurately refer the noise to the outputof the cavity, we need to measure the microwave loss. This is done by doing a thermal
sweep and measuring the output noise of the mixer as a function of Tfridge , when theswitch is connected to hot-load. The thermal sweep calibration is similar to the oneused to estimate the microwave losses in the experiment explained in chapter 7, and it isexplained in appendix B. We estimate that there is 6 dB of total loss between the switch
and the JPA (that includes the 3 dB-attenuator placed at the output of the switch) of
which 2 dB is between the cavity and the JPA, and 1.5 dB from the JPA to the HEMT.
From a Y -factor measurement, we infer that the JPA adds AJ = 0.3 microwave quanta
and that the HEMT adds AH = 24.5 microwave quanta of noise. The effective added
noise of the HEMT to our measurement is obtained by referring its noise number to the
output of the cavity, that is by dividing its noise number AH by GX1 and multiplying
by the loss between the the HEMT and the cavity output. Including all the losses, weestimate that the interferometer adds 1.16 microwave quanta of noise. Yet, in addition
to the noise added by the amplifier chain, there is a small amount of thermal noisefrom the input signal. The signal line is attenuated by about 50 dB, reducing the larger
room-temperature thermal noise to about 0.17 quanta on top of the vacuum noise. In
total, we estimate the number of noise quanta added to be AN = 1.3.
Once all the optomechanical-system and interferometer parameters have been
determined, we can estimate the displacement fluctuations of the nanomechanical wirefrom the phase fluctuations at the output of the mixer. I display the main result of thischapter in figure 8.9a. There we demonstrate that this setup is capable of achieving
imp
SQL
SQL
SX
< SX
. The minimum measured imprecision is (0.8 ± 0.03) × SX
, or 0.2

mechanical quanta. From the Lorentzian peak, we can infer the temperature of thewire’s thermal motion, which corresponds to T = 130 mK, or thermal occupancy of
nth = 2600 mechanical quanta.

238
(a)

(b) 10

4

4

10

3

SQL

2

10

Sx/Sx

Sx/SSQL
x

10

2

10

1

10
0

10

0

10

SQL

SQL

−1

1.038

1.041
frequency (MHz) )

1.044

10

0

10

1

2

10
10
P (pW) )3

10

Figure 8.9: (a) Displacement sensitivity below that at the Standard Quantum Limit.
The data shown is the spectral density of the displacement fluctuations Sx,x in units
SQL
Sx,xversus fourier frequency. In black, we show a fit to the Lorentzian thermal-response
plus a frequency-independent background. This background represents the imprecisionSQL
imp
of our measurement, and lies below Sx,x
(black dashed line). (b) Imprecision Sx,x
SQL
(blue circles) and total noise Sx,x (red circles) in units of Sx,x
as a function of incident
microwave power at the resonator. The data points at the highest power shown as asquare were extracted from the data shown in (a). The statistical error is smaller than
the size of the symbols. Two theoretical predictions are shown, one for a shot-noise
limited interferometer (solid-green) and for an interferometer with AN = 1.3. Thisvalue was independently measured as explained in appendix B.

The quoted imprecision is estimated based on the measurement of the thermal
motion of the wire. However, we can compare it with the predictions from Eq. 8.16
assuming an interferometer with AN = 1.3. This is done in figure 8.9b, where we plotted
our measured imprecision and the expected imprecision based on Eq. 8.16, for different
incident powers. We also display the predictions for shot-noise-limited interferometer
AN = 0. Figure 8.9b shows that our model accurately predicts the observed imprecision.
The predictions shown are not straight lines since they take into consideration changes
in both the cavity and the mechanical oscillator parameters (γc and γm ) as a higher
incident power is applied. Also, the temperature of thermal motion of the wire increases
with incident power. Therefore, a quantum efficient measurement which minimizes thepower needed to reach a given imprecision is always desirable. To quantify how close we239
are to a shot-noise limited interferometer, we can estimate the effective efficiency of thedetector using Eq. (7.28) for AN = 1.3. This gives an efficiency of η = 0.28, comparable
to the one observed in chapter 7, for a similar JPA gain.
In conclusions, in this chapter I have shown the JPA can be used to boost thesensitivity in the detection of motion of a nanomechanical oscillator embedded in a
microwave cavity. It has allowed us to get to the SQL by using a much lower power
than if were only using the HEMT amplifier. The use of the JPA has also made our
microwave interferometer only a factor of ∼ 4 from being shot-noise limited.

Chapter 9
Conclusions and future directions.

In this thesis, I described our approach to develop an ultra-low noise microwave
amplifier. Even though our results show great experimental progress, they also revealed
areas that require further research. We achieved our overall goal of characterizing theperformance of a new design of Josephson parametric amplifiers for a microwave signal.
After the characterization, we used it for two specific applications: microwave homodyne tomography for quantum state reconstruction and improving the sensitivity of a
microwave interferometer used in the detection of mechanical motion of a nanomechanical oscillator. In this chapter I will review some of the more interesting results presented
in previous chapters and discuss the possible future directions for JPA in our lab.

9.1

Conclusions
Although the focus of the initial measurements of a microwave cavity with aseries SQUID array as the center conductor of the transmission line was different, theexperiments done with the first device were very encouraging. With our first JPA we
demonstrated that a Josephson parametric amplifier based on a coplanar waveguideresonator whose inner conductor is made from a SQUID array could be operated withgood performance and that it was tunable over a full octave of bandwidth. We observed
gains as high as 28 dB and although the resonator is composed of discrete elements,
its behavior is well described by a continuum theory of parametric amplification [38].

241
In this first experiment, we did not have a calibrated noise source that would allow us
to estimate the added noise of the JPA. Nonetheless, compared to the measurementof a weak microwave signal using only a low-noise HEMT, we observed that the JPA
was 16 dB more sensitive than a HEMT amplifier. From this improvement of thesignal-to-noise ratio we estimated a total noise at the output of the JPA referred to the
input of slightly less than 1 quanta of noise, suggesting that the JPA added an amountof noise comparable to the vacuum noise. One very important difference with previous
implementations was that in our case we were able to operate the JPA with high enoughgain that the very small amount of noise at its input, after amplification, overwhelmed
the noise of the following amplifier chain. Therefore, the JPA set the noise-floor of themeasurement.
This first device showed very promising performance, but also some problems,
specifically some strange behavior in the magnetic-field dependance of the resonancefrequency. It also had very low power-handling capacity, which was good enough forsome applications (for example for amplifying the signals generated by superconducting
qubits), but not for some of the applications we were intended to use it for. Also, its gainbandwidth product was rather small. Hence, we decided to modify the design to increase
both the power-handling capacity and the bandwidth. In the second implementation ofthis amplifier we used niobium instead of aluminum for the superconducting resonant
structure as well as a larger SQUID current. The coupling quality factor Q was also
increased by a factor of 10. All these changes increased the critical powers, and hence
the power-handling capacity, by about 20 dB. For this new design, we collaborated
with Kent Irwin’s group at NIST. The fabrication of these devices was done using
a more industrial-like procedure, which would give us a more homogeneous array of
SQUIDs, hoping to avoid issues such as the non-ideal magnetic-field dependance of thefirst resonator.
We tested two different kinds of niobium JPAs. A quarter-wavelength and an242
asymmetrically-coupled half-wavelength resonator. With these new devices, we also
implemented a setup to measure the added noise of the JPA by using a microwave
switch that could toggle between two noise sources (50 Ω resistors) held at two different
temperatures. From these noise measurements, we estimated that the added noise of
the amplifier was as low as 0.23 quanta. This is a very important result as it makesthese JPA the lowest-noise microwave amplifiers ever reported. Again, we were able tooperate the amplifier with high gains (Gs as high as 40 dB) and set the noise floor of themeasurement. The larger critical powers also gave us a better power-handling capacity.
The amplifier showed at 8 GHz a 1-dB compression point of -130 dBm with a gainof 27 dB. We also tested their squeezing performance: being a degenerate parametricamplifiers, the JPA can amplify one quadrature with less added noise than that of aquantum-limited amplifier, while it squeezes the fluctuations at the other quadrature.
We measured squeezing of noise of as much as 10 dB.
Once we characterized the performance of the JPA we decided it was possible
to use it for two particular applications. For most measurements, the low added noiseof the JPA is helpful, but for some measurements it is crucial. For example, to fully
characterize the quantum state of non-Gaussian states, such as the Fock states generated
by superconducting qubits [110], a subquantum limited amplifier is a necessity [111].
Hence, we decided to use the JPA to implement what, in optics, is called homodynetomography for the Wigner function reconstruction of quantum-states of light. In ourcase, those quantum states were squeezed states generated by a second JPA. Using the
JPA as an integral part of microwave homodyne tomography lead to a dramatic increaseof the performance of the Quantum State Tomography. It has enabled us to reconstruct
the Wigner function of a squeezed state of the microwave field and observe real squeezingof the input noise of 37%, compared to the 2% real squeezing observed using only the
HEMT amplifier. In this case, the quantum efficiency of the state tomography detector
was 38%.

243
In addition to characterizing non-Gaussian states of microwave fields, the JPA is
also well suited to amplifying the signal generated by a nanomechanical beam moving
in a microwave cavity [11], enabling a quantum-limited measurement of position, whichlead us to the second application I discussed in this thesis. Here, I presented displacement measurements of a nanomechanical oscillator with an imprecision below that at
the standard quantum limit. We infer the motion from the phase modulation imprinted
on a microwave signal by that motion. The modulation is enhanced by embedding theoscillator in a high-Q microwave cavity. We achieve the low imprecision by reading out
the modulation using a JPA, realizing a microwave interferometer that operates near
the shot-noise limit. The apparent motion of the mechanical oscillator, due to the interferometer’s noise, is now less than its zero-point motion, making future detection ofquantum states feasible. In addition, the phase sensitivity of the demonstrated interferometer is 30 times higher than previous microwave interferometers, providing a criticalpiece of technology for many experiments investigating quantum information encodedin microwave fields.

9.2

Future directions
Although the good performance of the JPAs described here have let us improvethe detection of squeezed states and the detection of mechanical motion, we still needto improve certain things about the setup in general. First, the amplifiers added noise
is higher from what we would expect from a model that only includes loss as the contributor of added noise. The 0.23 quanta of added noise already sets the maximum
efficiency of a homodyne detector using these amplifiers at about 70%. As we do notknow where this extra added noise is coming from, it is hard to decide what the next
step is. However, since we have had some yielding problems during fabrication, one
obvious step would be to reduce the number of SQUIDs in the resonator as shown infigure 9.1. This would mostly affect the range of tuning for the resonance frequency.

244

(b)

(a)
signal in Cc

pump
Zc

signal out

l~

2f0

f0

λ
4
dc flux bias0

Φ / Φ00.5

Figure 9.1: (a) In our new devices, we reduce the number of SQUIDs to increase theyield of devices per wafer. (b) In these new devices, we will study the possibility of
implementing a 3-photon mode parametric amplifier, applying the pump as a 2-ω toneon the magnetic flux line, in order to eliminate the necessity of doing carrier-cancelation
when multiple JPAs are used.

However, for the applications we are planning on doing, we do not need a full octaveof tuning. Treating the small SQUID-array as a lumped element [48], in this case thechange in resonance frequency comes from a change in the boundary-condition of theresonator rather than from a change in the phase velocity of the metamaterial [72, 73].
Another point our group will study is the possibility of implementing a three-photon
mode Josephson parametric amplifier by using the magnetic-coupling line to apply amicrowave pump close to twice the resonance frequency of the resonator [59]. The main
reason to attempt this is that although the carrier cancelation works, it would be rather
hard to implement it when more than two JPAs are used. This will be the case in the
future experiments our group is planing on doing for certain quantum communication
protocols.
Another issue we want to eliminate is the low-frequency noise observed in theseamplifiers (Fig. 6.33c). One hypothesis is that it consists of current noise present inany closed-loop of metal (like for example the circular walls of the sample box), which
inductively couples noise to the SQUID loop. One way to test this hypothesis is to use
dc-SQUID gradiometers (Fig. 9.2). In the gradiometer configurations shown in figure

245
(a)

(b)

I

I

dc flux bias
Φ

Φ
Bz x

Figure 9.2: One strategy we will implement to reduce the low-frequency noise observedin the JPA is to use dc-SQUID gradiometers. (a) Parallel and (b) series gradiometer.
Due to their geometry they will not couple to space-homogeneous magnetic fields but
rather to the first derivative ∂Bz /∂x of the field.

9.2, the dc-SQUIDs would not couple to space-homogeneous magnetic fields, but rather
to the first derivative ∂Bz /∂x. If our hypothesis is right, we should see a significant
reduction of the low-frequency noise. The only disadvantage of using gradiometersis that, due to the more complicated fabrications, they tend to be larger and hence
have larger loop-inductance, limiting the suppression of the supercurrent and hence thetuning of the effective Josephson inductance. However, as I said before, the amount ofresonance frequency tuning we are trying to achieve is much less than the one obtainedin the devices described in this thesis.
Another modification in the setup is to use a better calibration. Due to theunknown losses in the microwave components at 4.1 K that are used to toggle between
the two calibrated noise sources, the uncertainty in the added noise is too large. If
we want to really nail down the origin of the extra added noise observed we need toreduce this uncertainty. For that we are planing on implementing a varying temperature
resistor which is thermally weakly coupled to the fridge using a superconducting coaxial
cable. By removing the need of going back and forth between the 4K plate and mixing
chamber plate, we will be able to reduce the uncertainty in the losses between the noise
source and the JPA.

246
Thinking ahead in terms of applications for degenerate parametric amplifiers, ourgroup is planing on using the squeezed states at the output of the JPA as building blocks
for quantum information experiments. These states, which are highly non-classical, form
EinsteinPodolskyRosen (EPR)-states when combined together and thus are the basisof a complete quantum information processing strategy, known as Continuous Variables Quantum Information [112]. Quantum information processing aims to exploit
entanglement, the unique feature of quantum mechanics, to enhance our abilities forcommunication and computation. In the strategy of continuous-variables quantum information processing, an entangled state of the electromagnetic field, can be generated
by combining two squeezed states on a beam splitter. In this thesis, I showed that wehave the ability to generate squeezed states at microwave frequencies using a Josephson
Parametric Amplifier. However, generating EPR states is more demanding. Passive
elements, like directional couplers and 90-degree-hybrid couplers using superconducting
elements have to be developed such that they can be integrated on the same chip as the
JPA, as the loss in the coaxial cables and commercial microwave components can reallydegrade the entanglement. Our group is also starting to test these passive components’performance (Fig. 9.3).

247





Figure 9.3: Passive microwave components being developed by our group for applications in continuous-variables quantum information processing. (a) 90-degree-hybrid
coupler and (b) directional coupler.

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Appendix A
Multimode fields

In this appendix we discuss the broadband analysis of signals and the necessary
corrections on the single-mode analysis done in chapter 2. These correction will set new
limits on amplification and squeezing of vacuum noise. For a more in-depth explanationof the results quoted here, I refer the reader to Refs. [2, 84, 112, 113].

A.1

Multimode signals
The idea is to find the corrections to the single-mode analysis which results fromthe nonzero bandwidth that any real signal and real amplifier have. In general, the input
and output mode frequencies do not have to be the same. However, in our experiments
this is usually the case so I will not make a distinction between the input and output
mode frequencies. Consider a broadband signal whose voltage, according to Eq. (2.39)
can be expressed asZ
V (t) = V0dω 0 √ 000ω [a(ω 0 )e−iω t + a† (ω 0 )eiω t ]
2π
Bω

(A.1)

where I assume ω 0 > 0 only. The limits of the integral are labeled by Bω which represents
the bandwidth of integration of the measuring instrument used. Here the annihilationand creation operators (now, no longer dimensionless but each in units of root of time,√t) satisfy the commutation relation given by Eqs. (2.23) and (2.24):[a(ω 0 ), a† (ω 00 )] = 2πδ(ω 0 − ω 00 )

255
and
[a(ω 0 ), a(ω 00 )] = 0.
I assume that associated with the signal, there is a carrier with frequency ω0 . The
carrier frequency ω0 is essentially some preferred frequency, say the pump of a degenerate
Josephson parametric amplifier, or the local oscillator of a homodyne detector (a mixer).
The quadrature phases are to be defined relative to this frequency. The quadrature
operators X1 (t) and X2 (t) of the field amplitude components can be introduced bywriting our expression for the input signal V (t) in the follwing form√
V (t) = 2 ω0 V0 [X1 (t) cos(ω0 t) + X2 (t) sin(ω0 t)] .

(A.2)

Using Eqs. (A.1) and (A.2), we obtain explicit expression for the quadrature operators X1 (t) and X2 (t) deriving what essentially is a continuum generalization of thequadrature operatorsrdω 1 ω0 + ω
X1 (t) =[a(ω0 + ω)eiωt + H.c.]
2π
2
ω
0
Bω
r
Z
dω 1 ω0 + ω[−ia(ω0 + ω)eiωt + H.c.]
X2 (t) =
2π
2
ω
0
Bω
Z

(A.3)
(A.4)

where now the fields are described as functions of the modulation frequency ω which
consists of an offset frequency respect to ω0 , ω 0 = ω0 ± ω. The previous expressions canbe simplified significantly in the case where the bandwidth of the signal is symmetric
about ω0 , and if we also define the operators in the rotating frame about the central
frequency ω0 as it was done in section 2.3.2:
b(t) = a(t)eiω0 t .
We use the following definition of the fourier transformZ1b(ω) =
dtb(t)eiωt .
2π
In this case, the rotating frame operators follow the commutation relation[b(ω), b† (ω 0 )] = 2πδ(ω − ω 0 )256
Based on the steps described above, we obtain from Eqs. (A.3) and (A.4)
Z Bω /2
X1 (t) =

(A.5)

0

dω
[X1 (ω)e−iωt + H.c.]
2π

(A.6)

0

dω
[X2 (ω)e−iωt + H.c.]
2π

Z Bω /2
X2 (t) =where
X1 (ω) =X2 (ω) =r·r¸
1ω0 + ω
ω0 − ω †b(ω) +b (−ω)
2
ω0
ω0
r
¸
·r
ω0 + ω
ω0 − ω †1b(ω) −b (−ω)
2i
ω0
ω0

(A.7)
(A.8)

Eqs. (A.5-A.8) are the multimode generalization of the single-mode quadrature operators, Eq. (2.73). The generalization to a rotated-quadrature operator Xθ (ω) is similar
to the single-mode quadrature, Eq. (2.86)
1Xθ (ω) =2·rω0 + ω
b(ω)eiθ +ω0r

ω0 − ω †b (−ω)e−iθ
ω0

¸
(A.9)

where it is easy to show that X1 = Xθ=0 and X2 = Xθ=−π/2 .
Although the multimode generalization X1 (t) and X2 (t) can have reduced fluctuations (squeezing) just like in the single-mode case, it can be shown [2] that thosefluctuations cannot be reduced to zero as in the single-mode case. There are also corrections to the minimum amount of of the noise that a linear amplifier adds, which
are bandwidth dependant and emerge naturally from the multimode analysis. These
new limits and corrections can be inferred by calculating the spectral density for thefluctuations of the quadratures SXi ,Xi (ω) (i = 1, 2) of the mean-square fluctuations of
Xi (t) fromh{Xi (ω), Xi† (ω 0 )}i = 2πSXi ,Xi (ω)δ(ω − ω 0 )(A.10)

where we are using a single-sided spectral density convention. Then, it can be shown
that
SXi ,Xi (ω) ≥

1 ω
2 ω0

(A.11)

257
Eq. (A.11) is the limit I mentioned, and it will be the origin of the new limits on added
noise for a phase-sensitive amplifier and the minimum squeezing it can generate.

A.2

Squeezed states
The transformation of a single mode degenerate parametric amplifier, Eq. (2.83),

to create squeezed states can be generalized to the broadband case. Consider the vacuum
state for the operator b(ω)b(ω)|0i = 0.
Squeezed states can be generated from this vacuum state via a mode transformation
similar to the single-mode caseb0 (ω) = G(ω)b(ω) + M(ω)b† (−ω)(A.12)

Under this transformation, |0i, which is a vacuum state for the b(ω) operator, becomesa squeezed state for the b0 (ω) operator. Since both operators b(ω) and b0 (ω) must satisfy
commutation relations of the form (2.61), the quantities G(ω) and M(ω) need to fulfill
the following relations|G(ω)|2 − |M(ω)|2 = 1
and
G(ω)M(−ω) = G(−ω)M(ω).
Substituting Eq. (A.12) into Eq. (A.9), the transformed quadrature operator is given
by
r
µr
¶
1
ω0 + ω
ω0 − ω ∗iθ
G(ω)e +M (−ω)e−iθ b(ω)
Xθ (ω) =
2
ω0
ω0
r
µr
¶
1
ω0 + ω
ω0 − ω ∗iθ−iθ
M(ω)e +G (−ω)eb† (−ω)2ω0ω0Hence, from Eq. (A.13), we obtain that


s
µ ¶2
1
ω
SX,X (ω, θ) = |G(ω)|2 + |M(ω)|2 + 2 1 −Re(G(ω)M(−ω)e2iθ )
2
ω0(A.13)(A.14)

258
The termp
1 − (ω/ω02 )2 consist of the correction due to the multimode analysis, andit sets a fundamental upper bound on how small SX,X (ω, θ) can be made [2, 84]. It is
possible to show that
SX,X (ω, θ) ≥

1 ω
.
2 ω0

as it was mentioned above, Eq. (A.11).

A.3

Quantum limits for multimode linear amplifiers
The correction from the multimode analysis in the case of amplifiers is very dif-ferent whether is phase-sensitive or phase-insensitive. In the case of phase-insensitive
amplifiers, the only real change is that rather than specifying an added number ofquanta, what it’s specified is an added number of quanta per unit bandwidth. The
fundamental theorem for phase-insensitive linear amplifiers is then given by [2]
¯
¯
¯
¯
1
¯
SA (ω) ≥ ¯¯1 ∓G(ω) ¯(A.15)

where the upper (lower) sign holds for phase-preserving (phase-conjugating) amplifiers.
SA (ω) is the added noise single-sided spectral density (in units of number-quanta perunit-bandwidth). If the input signal has phase-independent noise, then the output
spectral density is given by
SO (ω) = G(ω)[SI (ω) + SA (ω)]
where the superscripts I and O are used to designate the spectral densities of the input
and output signals.
In the case of a phase-sensitive amplifier such as a degenerate parametric amplifiers, the situation is different. In this case, it can be shown that the multimodecorrection to the single mode result is¯¯1 ¯¯1 ω ¯¯1−
SXi ,A =2 ω0 ¯Gi (ω) ¯

259
where the subscript i is used to designate the quadrature, and Gi is the gain for thequadrature Xi . Note that unlike the single-mode case, there is a fundamental limit on
amount of noise for modulation-frequencies ω > 0. The purpose of this limit is to ensurethat the output mode still follows Eq. (A.11) [2].
Some literature uses a different convention for the spectrum of an operator Ŷ ,where the symmetrized noise SY,Y is actually defined as
1h{Y (ω), Y † (ω 0 )}i = 2πSY,Y (ω)δ(ω − ω 0 ).
2With this definition, the limit for phase insensitive amplifiers SA becomes 1/2 quanta
per unit bandwidth (just like the single mode case). Also the numerical factor in Eq.
(A.11) becomes 1/4 as opposed to 1/2. We do not use this convention, except for in
Eqs. (3.80-3.82) where the extra factor of 1/2 was added to compare the added noise of
the JPA with and ideal single-mode phase-insensitive amplifier. However, for the rest of
the thesis I have used the convention of Eq. (A.10). In this convention, then the added
noise per unit bandwidth in each quadrature of a quantum-limited phase-insensitive
amplifier is 1/2 quantum, which is the value we compare the added noise of our JPA.

A.4

Approximation
Although this fundamental limit is important from a theoretical point of view,

from an experimental point of view ω ¿ ω0 . In the rotating frame, using a(ω0 ± ω) =b(±ω), the frequency resolved quadrature amplitudes may be written in a form morereminiscent of the discrete quadratures:
1
X1 (ω) ' (b(ω) + b† (−ω))
21X2 (ω) ' (b(ω) − b† (−ω))
2i(A.16)(A.17)

These are the definitions I used for the quadrature operators in the frequency domain
in this thesis. They are also the definitions usually used to analyze the experimentally

260
measured quadrature noise [8, 38]. In our case, for all the noise spectral densities thatwere measured ω/ω0 < 1/1000, and it is justified to use the approximated quadrap
ture operators. Within this approximation, the term 1 − (ω/ω02 )2 from Eq. (A.14)
disappears and we obtain Eqs. (3.74) and (3.75).

Appendix B
Thermal sweep and loss calibration

As described in section 6.3.3, we use a cryogenic switch to toggle between two
terminations held at two different temperatures to estimate the added noise of theHEMT and the JPA. However, in order to accurately refer this noise to the input ofthe different amplifiers, one must also measure the microwave loss of the components
between the calibrated noise sources and the input of the amplifiers. Knowing theamount of loss between the different sections of the experiment is also very importantin order to accurately predict the amount of squeezing observed. The determinationof the amount of loss is accomplished by measuring the change in the measured noiseat the output of the amplifier chain as a function of the cryostat temperature Tc when
the switch is connected to the hot load at Th = 4.1 K. In the presence of loss the noisefrom the hot termination at Th will be partially absorbed. In addition, this loss will
emit noise with a spectrum given by the temperature of the loss (Tc ). In this appendix,
I will describe a set of thermal-sweep measurements that help us estimate the loss
between the switch and the amplifiers. From the thermal sweeps, we can also estimate
the added noise of amplifiers and compare it with what we measure using the Y -factor
measurement described in section 6.3.3 to make sure both methods give similar answers.
In the following calculations, I will assume that the HEMT sets the noise floorof the measurement. For this reason and for simplicity I label with a subindex H thegain and added noise of the commercial amplifier chain, starting with the HEMT and262
ending at the digitalization card used to measure the output of the mixer.

B.1

HEMT
T
c

η

S out
HEMT +
other commercial
amplifiers

Switchρ

4K

Tc

Figure B.1: Simplified diagram of the measurement when only the HEMT is used. Bymeasuring the noise at the output of the amplifier chain for both loads, as we change
the temperature of the fridge Tc , we can estimate the loss η.

During the thermal sweeps we do a Y -factor measurement on the amplifiers, inthis case the HEMT. This means that we have access to two possibilities for Sout (Fig.
B.1). One is when we are observing at the “cold load” which sits at a temperatureof Tc and the second when we are observing at the hot load with a temperature of
Th = 4.1 K. When we measure the cold load at a temperature Tc , we can estimate the
HEMT’s noise (assuming a few approximations I will describe below). Then, when wemeasure the hot load. If there was no loss between the switch and the HEMT, ideally
we would not see any change of Sout when we change the temperature of the fridge.
Thus, when measuring the the hot load, it is the change we measure in Sout that allows
us to estimate the loss η. The ouput of the HEMT is given by Sout ,Sout = GH [AH + (ρSload + (1 − ρ)Sρ )η + (1 − η)Sη ](B.1)

where AH is the added noise of the HEMT in noise quanta, GH is the gain of theHEMT (more accurately is the gain of all the amplifier chain, but for simplicity let’s263
call it the HEMT gain), Sload is the noise coming from the load (it can be at the fridge
temperature or at 4.1 K), ρ is the loss in the switch, η is the loss from the switch to
the HEMT. Sρ is the noise attributed to the switch loss, and Sη is the noise attributed
to the loss between the switch and the HEMT. Sload , Sη and Sρ are in units of noise
quanta and are given by:

Si =

1
hf
)
coth(
2
2kB Ti(B.2)

Unfortunately, with the following analysis, we will not be able to estimate theloss of the microwave components that sit at 4.1 K. This consists of the loss of theswitch as well as the loss from a bias-tee used to heat sink the center-conductor of thesuperconducting cables and the switch. Thus, I will do fits assuming worst and bestcase scenario based on the factory specifications of the microwave components: a total
insertion loss of −0.8 dB for the worst case, or ρ ' 0.83, and for the ideal case ρ = 1.

B.1.1

Hot loadIn this case when we are measuring the hot load, then Sload = Sρ , and Eq B.1
yieldsSout = GH [AH + Sρ η + (1 − η)Sη ](B.3)) is the term that will change with the fridge temperature.
where Sη = 12 coth( 2khf
B Tc
Now, this expression could be used to do a fits to obtain the HEMT’s added noise andthe loss. However, since we do not know exactly the gain of the amplifier chain, thevariable GH in the fitting expression is also a fitting parameter. This means that there
is one too many fitting parameters which give us too much freedom in the fits. So, I do
first a fitting to the cold load measurement, which sets (approximately) GH and AH .

264
B.1.2

Cold load

In this case, Sload = Sη , and then Eq B.1 reduces toSout = GH [AH + (1 − η + ηρ)Sη + (1 − ρ)ηSρ ](B.4)

If the switch and bias-tee at 4.1 K did not have any loss (ρ = 1), then this expression
would be much simpler Sout = GH {AH + Sη } and the fitting to a simple expression
with only 2 fitting parameters (GH and AH ) would work fine. However, since I cannot
assume that there is no loss at 4.1 K, things are slightly more complicated. In order tosimplify the expression I will regroup some of the terms in Eq. (B.4):
Sout = GH [AH + (1 − η + ηρ)Sη + (1 − ρ)ηSρ ]·¸
AH + (1 − ρ)ηSρ
= GH (1 − η + ηρ)+ Sη
(1 − η + ηρ)= Gfit {Afit + Sη }

(B.5)
(B.6)
(B.7)

with
Gfit = GH (1 − η + ηρ)(B.8)

AH + (1 − ρ)ηSρ(1 − η + ηρ)(B.9)

Afit =

Now I have a simple expression with only 2 fitting parameters. The procedure toestimate the loss is the follwing:(1) I do a fit to the cold load, from which I get Gfit and Afit ;(2) Using Gfit and Afit , I do a fit to the hot load, where I replace GH and AH using
Eqs. (B.8) and (B.9) assuming ρ = 1 from which I get a value of the attenuation
η;(3) Then I do a fit to the hot load assuming ρ = 0.83 from which I get anothervalue of the attenuation η.
Since ρ is close to one and η < 0.1, (1 − η + ηρ) ' 1 this will make Gfit ' GH and
Afit ' AH .

265

B.2

JPA
We can try to do a similar analysis when we operate the JPA to estimate the added

noise of the JPA and the attenuation between the different locations of the circuit shownin figure B.1. The output of the amplifier chain (again labeled by the subindex H) andincluding the JPA is given by the following expression:Sout = GH [AH + (1 − η2 )Sc + η2 GJ (AJ + (1 − η1 )Sc + η1 {(1 − ρ)Sρ + ρSload })](B.10)
where η1 is the attenuation between the switch and the JPA, η2 is the attenuation
between the JPA and the HEMT, and Si = 12 coth( 2khf). For simplicity, since the noise
B Ti
from both losses η1 and η2 are the same and are function of the fridge temperature, I
have used Sc to label their noise. Sload is the load thermal noise, which can be at the
fridge temperature Tc or at 4.1 K, and Sρ is the thermal noise (with T=4.1 K) from theloss at 4.1 K (switch and bias-tee).
T

T

J

c

η1

η2
Josephson
Parametric
Amplifier(JPA or Squeezer)

Switchρ

Tc

AJ , G
c

A ,G
H
H

Sout
HEMT +
other commercial
amplifiers

4K

Figure B.2: Simplified diagram of the measurement when both the HEMT and a Josephson parametric amplifier (either the one labeled the squeezer or the JPA) are used. Bymeasuring the noise at the output of the amplifier chain for both loads, as we change
the temperature of the fridge Tc , we can estimate the loss η1 . From the measurement
done only with the HEMT where we estimate η = η1 + η2 we can then know η2 .

266
B.2.1

Cold load

In this case Sload = Sc and equation B.10 reduces to:Sout = GH {AH + (1 − η2 )Sc + η2 GJ (AJ + (1 − η1 )Sc + η1 [(1 − ρ)SS + ρSc ])}= GH [g(1 − η1 (1 − ρ)) + (1 − η2 )]{AH + gAJ + gη1 (1 − ρ)Sρ+ Sc }g(1 − η1 (1 − ρ)) + (1 − η2 )= G0 (A0 + Sc )where g = GJ η2 . Now, for reasonable values for η1 , ρ and η2 , then A0 can be simplified:A0 ='AH + gAJ + gη1 (1 − ρ)Sρg(1 − η1 (1 − ρ)) + (1 − η2 )AH
+ (1 − ρ)η1 Sρ + AJ
g

(B.11)(B.12)

Finally, when taking the data, we are modifying not only the temperature of the fridge
and the cold-resistor but also the temperature of the JPA. We do not know the temperature dependance of the added noise of the JPA. Although our model includes sometemperature dependance of the added noise, Eq. (5.56), the measured added noise islarger than what theory predicts. I know that the added noise of the amplifier is (idehf
γloss 1ally) of the order of AJ = γcoupling
2 coth( 2kB Tc ). So, for these fits, I will assume that atleast this much of the added noise of the JPA must depend with temperature. Then I
can use the following expression to fit the data:·
AJ = A0J + κ

¸
1
hf
coth(
)
2
2kB Tcryo(B.13)

γloss
is calculated from fits to the reflection coefficient and A0J is assumed
where κ = γcoupling

temperature independent. Hence, my fitting function for the cold-load noise measurement is:
cold
Sout
= Gfit [Afit + (κ + 1)Sc ]where Afit = AgH + (1 − ρ)η1 Sρ + A0J .

(B.14)

267
B.2.2

Hot loadIn this case Sload = Sρ , so equation B.10 reduces toSout = GH {AH + (1 − η2 )Sc + η2 GJ (AJ + (1 − η1 )Sc + η1 [(1 − ρ)Sρ + ρSρ ])}= GH [g + (1 − η2 )]{AH + gAJ
gη1
gη1+
Sρ + (1 −)Sc }g + (1 − η2 ) g + (1 − η2 )g + (1 − η2 )= G00 (A00 + η10 Sload + (1 − η10 )Sc )With some approximations that are valid for high gain g and reasonable values of η1 , ρ
and η2 , this expression can be written as follows:
hot
Sout
=

Gfit
{Afit (1 − η1 (1 − ρ)) − η1 (1 − ρ)Sρ + η10 Sρ + (1 − η10 )Sc }1 − η1 (1 − ρ)

(B.15)

where Gfit and Afit will not be fitting parameters in this case but are obtained fromthe fit to the cold-load data. Finally, the last thing to do is to notice that η10 can beapproximated by:
η10 =

gη1
g + (1 − η2 )' η1Therefore, expression B.15 reduces to
hot
Sout
=

Gfit
{Afit (1 − η1 (1 − ρ)) + η1 ρSρ + (1 − η1 + ρ(1 − η1 (1 − ρ)))Sc } (B.16)
1 − η1 (1 − ρ)which is the expression I use to fit to the hot load with one single fitting parameter,
η1 .1
The procedure for the fits of the data in the case of the JPA is similar to the
HEMT amplifier case.
(1) I do a fit to the cold load, from which I get Gfit and Afit ;(2) Using Gfit and Afit , I do a fit to the hot load assuming ρ = 1 from which I get
a value of the attenuation η1 ;1

The term (1 − η1 (1 − ρ))) is a number very close to one, so this extra terms does not affect theresults of the fits.

268(3) Then I do a fit to the hot load assuming ρ = 0.83 from which I get anothervalue of the attenuation η1 ;(4) η2 can be obtained from the value of η = η1 + η2 given by the fits to the HEMT
amplifier thermal sweep data.

B.2.3

Low-frequency noise

(a)

(b)
10

−10

P.S. (a.u.)

P.S.D ( V2 )

10

−12

10

−14

10

4

10

5

6

10
10
frequency (Hz) )

5

0

0

0.2

0.4

0.6

T (K)

Figure B.3: (a)Spectral density of the output of the mixer when the JPA is used. Below
100 kHz there is an extra contribution from an unknown source. (b) This extra noise
has a linear dependance with temperature which, unless avoided, would interfere with
our fits to Eqs. (B.14) and (B.16).

Before we move into fitting the data to Eqs. (B.14) and (B.16), there is something
I need to discuss which is the low frequency noise we observe in the spectral density at
the output of our amplifier chain (fig. B.3). If we integrate over only the low frequency
part (below 100 kHz), then we obtain a linear T -dependance as shown in figure B.3b.
Thus, if I want to fit to the expressions (B.14) and (B.16) then I can not integrate over
the whole band measured but at least above 200 kHz. What I actually do is I fit to thelow frequency part which gives a dependance of ∼ 1/f 1.8 . Then I compare how muchof the total noise at, for example, 500 kHz comes from the 1/f 1.8 part. For this case, it269
is less that 5%, and since I integrate over a band bigger than 1 MHz, at the end, after
integrating above 500 kHz, the part of the noise that comes from the ∼ 1/f 1.8 part isnegligible.

B.3

Chapter 7 calibration
In all the data that I will show, we used a calibration tone that allowed us toremove drifting in the gain of the whole amplifier chain (as the measurements took
several hours).

B.3.1

HEMT
−5

x 10

P.S. (a.u.)

4.4

4.2

4

3.8

0

0.25

0.5
Tc (K) )0.75

1Figure B.4: Thermal sweep using only the HEMT amplifier. The JPA and the squeezer
resonances have been tuned far from the frequency of interest by applying a current tothe coil. Thus, the resonators behave as mirrors reflecting all the incident power. The
data shown consist of the power spectrum over a 5 MHz bandwidth when the switch isconnected to the cold load (dots) and the hot load (circles) as well as their fits to Eqs.
(B.16) and (B.14) (red lines). The error bars are smaller than the data points.

270
The fits to the data taken with the HEMT are shown in figure B.4. In order
to avoid the losses of the resonators, we applied a current to the coil that move theresonance frequency of both the JPA and squeezer far from the frequency at which wewere measuring (f = 7.44 GHz). Thus, both resonators behaved as mirrors, reflecting all
the noise incident on them. By doing a fit to Eqs. (B.3) and (B.4), we estimate that theadded noise of the HEMT is AH = 17.3 ± 0.2, where the uncertainty is predominantly
from the unknown loss of the microwave components at 4.1 K. Regarding the loss ηfrom figure B.1, our estimate is η = −11.7 ± 0.3 dB.

B.3.2

JPA
−4

−4

x 10

x 10

(b)

(a)

5
P.S. (a.u.)

P.S. (a.u.)

1

0.5

0

0

0.3

0.6
Tc (K) )

0.9

2.5

0

0

0.3

0.6
Tc (K) )0.9

Figure B.5: Thermal sweep with a JPA gain of (a) GJ = 60 and (b) GJ = 250. These
values of the JPA gain consist of the average gain over the integrated band. The data
shown consist of the power spectrum over over a 2.5 MHz bandwidth (from 500 kHz to
3 MHz), when the switch is connected to the cold load (dots) and the hot load (circles)
as well as their fits to Eqs. (B.16) and (B.14) (red lines).

In this experiment we have two different parametric amplifiers one as a squeezer(and hence will be only referred as the squeezer) and one as an amplifier (JPA). We

271
first analyze the one that is used as an amplifier. When we took this data we took
thermal sweeps with 2 different gains. All through the thermal sweep we kept the gains
the same. From the fits to the data (Fig. B.5) we estimate the following parameters:
AJ = 0.24 ± 0.03, where again the largest contribution to the uncertainty comes fromthe unknown losses at 4.1 K. Regarding the losses, we obtain, η1 = 10.4 ± 0.2 dB andη2 = 1.3 ± 0.3 dB. In order to obtain AJ , η1 , η2 , we used the results obtained from theHEMT-thermal-sweep analysis.
One important point that is still under discussion is that I have assumed thatthe noise of the JPA is formed by one part that is constant with temperature andone part that depends in temperature as shown in Eq. (B.13). This is a somewhat
arbitrary model. The second part κ( 21 coth( 2khf)) is what I expect from theory, andB Tc
since we see extra noise I have added a constant, but this “constant” A0J could very well
be a function of temperature, which would then change the results previously shown.
However, the obtained attenuation is compatible with measurements of the attenuationof the microwave components when measured at 77 K. Also, assuming the estimatedattenuation from the fits, we can do Y -factor measurements on the JPA and the obtained
added noise of AJ ' 0.25 ± 0.05 is compatible with the estimated from the fits. So I
will use these values to analyze the data in chapter 7.

B.3.3

Squeezer

In the case of the squeezer, the analysis from the fits shown in figure B.6 yields
AJ = 0.26 ± 0.03, η1 = −8.7 ± 0.3, and η2 = 3.0 ± 0.2 dB (note that η1 and η2 inthis case will be different from the ones obtained for the JPA). Just as in the previous
case, by performing a Y -factor measurement, we have observed consistency between themeasured loss and the measured added noise from the thermal sweeps, and the measured
added noise from the Y -factor measurement.

272
−4

−3

x 10

1
P.S. (a.u.)

P.S. (a.u.)

2

1

0

x 10

(b)

(a)

0

0.3

0.6
Tc (K) )

0.5

0

0.9

0

0.3

0.6
Tc (K) )0.9

Figure B.6: Thermal sweep with a squeezer gain of (a) GJ = 70 and (b) GJ = 350.
These values of the JPA gain consist of the average gain over the integrated band.
The data shown consist of the power spectrum over over a 2.5 MHz bandwidth (from500 kHz to 3 MHz), when the switch is connected to the cold load (dots) and the hotload (circles) as well as their fits to Eqs. (B.16) and (B.14) (red lines).

The estimated values of η for the HEMT, and η1 and η2 for both the JPA andthe squeezer can be used to estimate the losses between all the three amplifiers, as
shown in figure B.7. For the analysis done in chapter 7, we assumed α = 1.7 ± 0.1,
β = η2JPA = 1.3 ± 0.3 and Sin = 0.65 ± 0.15.
JPA pump and LO

S in
Teff

ξ

Squeezer

Tc

Tc

α

β
JPA
AJ , GJ

θ

rf

LO
Sout

Amplifiers + mixer
AH , GH

microwave homodyne detector

Figure B.7: From the thermal sweeps then we can estimate the loss between the squeezer
and the JPA (α), and the loss between the JPA and the HEMT (β).

273Squeezing

We also measured the squeezing just using the HEMT to observe whether thesqueezer behave as in the previous experiments. The observed squeezing as a function ofgain is shown in figure B.8 and compared with theory. We observe a similar behavior as
in figure 6.34b. The squeezing increases as a function of gain up to gains of GJ ' 10 dB,
and the it decreases for higher gains. The maximum estimated observed squeezing is
80%.

ξ

1

0.5

0

0

10

1

10
GJ

2

10

Figure B.8: Squeezing as a function of the squeezer gain. The maximum squeezing
observed is 80%.

274

B.4

Chapter 8 calibration
−5

x 10

P.S. (a.u.)

7.1
6.9
6.7
6.5
6.3

0

0.3
Tc (K) )0.6

Figure B.9: Thermal sweep using only the HEMT amplifier. Only one JPA is used inthis experiment and its resonance has been tuned far from the frequency of interest by
applying a current to the coil. The data shown consist of the power spectrum over a
5 MHz bandwidth when the switch is connected to the cold load (dots) and the hot load
(circles) as well as their fits to Eqs. (B.16) and (B.14) (red lines).

We also performed a thermal sweep for the experiment described in chapter 8.
Following are the plots with the results from the fits using the procedure previously
described. Based on the thermal-sweep data for the HEMT and other commercial amplifiers (Fig. B.9), we get the following results for the effective added noise of the commercial amplifier chain and attenuation η between the switch and the HEMT amplifier.
AH = 24.5 ± 0.5, and η = 7.6 ± 0.2.
In the case of the JPA, we took the thermal-sweep data for two different gains.
From the fits shown in figure B.10, we estimate the following values: AJ = 0.33 ± 0.04
and η1 = 6 ± 0.3 dB. The values of the estimated added noise of both the JPA andthe HEMT are consistent with the values from Y -factor measurements assuming the275
given values of attenuation of η1 ' −6 dB and η = η1 + η2 = −7.6 dB. These are
the values assumed for the estimate of the effective added noise AN in the microwave
interferometer for figure 8.9.
−3

−4

(a) 3

x 10

x 10

(b)

2

P.S. (a.u.)

P.S. (a.u.)

1.5

1

0

1

0.5

0

0.3
Tc (K) )

0.6

0

0

0.3
Tc (K) )0.6

Figure B.10: Thermal sweep with a JPA gain of (a) GJ ≈ 115 and (b) GJ ≈ 400.

Appendix C
Sample Fabrication

In this appendix I discuss some details regarding the fabrication of the samples.
First I will describe in some detail the fabrication of the aluminum samples which I did
in the JILA cleanroom. Then I will only briefly describe the process involved in the
fabrication of the niobium samples done by our collaborators at NIST.

C.1

Aluminum resonators
This sample was fabricated with electron beam (e-beam) lithography. The firststep is choosing a substrate. We used a 300 µm thick silicon substrate with a thermallygrown oxide. The thickness of the oxide was 150 nm. On the wafer, we deposit a protecting layer resist. At this point, the wafer was diced into individual 1/4 in2 chips. Then
individual chips are prepared for the fabrication of the devices. Each chip is thoroughly
cleaned with acetone and isopropanol before depositing the layers of photoresist. First,
a thick (∼ 5000 Å) layer of polymethylglutarimide (PMGI) SF-9 from Micro-Chem isspun at 3000 rpm for 40 seconds and baked for 10 minutes on the substrate using a
hot plate at 190◦ C; then a second thinner (∼ 3000 Å) layer of polymethyl methacrylate(PMMA-A5.5) is spun at 4000 rpm and baked at 180◦ C for 30 minutes. These twolayers form a resist bilayer necessary to create suspended structures (Fig. C.1). Thenthe chip is exposed to an e-beam using a scanning electron microscope (SEM) as shownin figure C.2a. The PMMA is then developed for 1:20 minutes using PMMA-developer

277
Areas exposed by the e-beam

PMMA
PMGI
Silicon Oxide
SiliconAluminum
Aluminum Oxide

Al-AlOx-Al junctions

Figure C.1: Dolan bridge technique. A bilayer of PMGI/PMMA resist is spun andbaked on a silicon chip with a thermally grown oxide. The pattern is written with ascanning electron microscope, exposed and developed to create the suspended patternshown in cross section. Then a layer of aluminum is evaporated at an angle to create
the bottom electrode of the tunnel junction. The next step is to create an oxide layer byintroducing oxygen into the evaporation chamber. Next, a second layer of aluminum isevaporated to form the counter electrode. Finally, the resist pattern is lifted to leaving
the finished tunnel junction and other structures of the device.

(MIBK). The areas of PMMA exposed are then removed and the PMGI is developed for
3:00 minutes using PMGI developer. This creates a the suspended pattern as shown incross section in figure C.1. After that the sample is placed in an evaporator (Fig. C.2b).
I evaporate aluminum at two different angles and allow an oxide to form in between.

278This oxidation is accomplished by letting oxygen into the evaporating chamber causing
the aluminum to oxidize. The thickness of each of the layers of evaporated aluminumwas 100 nm. The region of overlap results in a tunnel junction. This technique iscalled Dolan-bridge technique [87]. Finally, the resist pattern is lifted using acetone and
PMGI-remover leaving the tunnel junctions and the resonator structure.

(a)

(b)

Figure C.2: (a) Scanning electron microscope, JEOL JSM-6400, used in the fabricationof the aluminum devices. (b) Oxford evaporator. It is set up to do both thermal ande-beam evaporation. The device described in chapter 6 was thermally evaporated.

C.2

Niobium resonators
Niobium devices were fabricated at NIST Boulder using optical-lithography witha standard NbAlOxNb trilayer process [88], modified by eliminating the shunt resistor
layer and minimizing deposited oxides [89]. Trilayers are deposited in a dedicated loadlocked vacuum system. The first base-electrode layer consist of a 200 nm thick Nbfilm; then a 6-10 nm thick Al film is deposited and oxidized. Finally a 50-100 nm
thick Nb counter-electrode deposition completes the trilayer process. The pattern of
the resonator is done by performing an etch process in a load-locked reactive-ion-etchsystem (Fig. C.3).

279
Our role in the fabrication of the Niobium resonator consist of designing thestructures. For that, we used commercial software named Xic Graphical Input Editor,
from Whiteley Research Inc. In there we plot all the structure including the layers usedin the fabrication. Beside the trilayer described in the previous paragraph, which only
creates the junctions, we also used two extra layers, a dialectric insulating layer (SiO2 )and a niobium wiring layer used to connect the multiple junctions. The dialectric SiO2
layer is deposited using plasma-enhanced-chemical-vapor-deposition (PECVD) with anelectron-cyclotron-resonance (ECR) reactor. Before depositing the wiring-layer, vias are
etched in the SiO2 layer. For more details in the fabrication process of these devices,
contact Leila Vale at NIST.

Thermally-grown
Silicon Oxide
SiliconAluminum
Aluminum Oxide
Niobium
ECR-based PECVD
Silicon Oxide

vias

Figure C.3: Niobium trilayer process. First a layer of niobium is deposited followedby an aluminum layer which is oxidized and finally another niobium layer completingthe trilayer process. However, this creates the junctions as well as the other resonatorstructures, but leaves the junctions disconnected from the rest of the device. Hence, alayer of SiO2 is deposited followed by an etch to creates vias and the deposition of aniobium-wiring layer that connects the junctions to form the SQUIDs and to connect
them to the rest of the resonator circuitry.

Appendix D
List of symbols and abbreviations

∆
γ
ζ
η
η
ξ
Φ
Bω
C
Cl
CP W
EOM
DPA
gc
G1
G2
Gθ
Gi
GJ
Gs
HEMT
I
I0
I0S
IOT
JPA
L
Ll

Superconducting gap
linewidth defined as the amplitude decay rate,or half width at half maximum (HWHM). γ = ω0 /2Q
Real squeezing observed at the output of the homodyne detector.
Transmission coefficient of a beam splitter.
Efficiency of the homodyne measurement modeled by a beamsplitter at the input of an ideal homodyne detector.
Squeezing at the output of the JPA
Flux
Bandwidth
Capacitance
Capacitance per unit length
Coplanar waveguide
Equations of motion
Degenerate parametric amplifier
Coupling between the mechanical harmonic oscillator displacement
and the cavity’s resonance frequency, gc = ∂ωc /∂x.
Maximum quadrature power gain
Maximum de-amplification. In an ideal degenerate parametricamplifier G2 = 1/G1
Phase-dependant quadrature power gain
Image power gain
JPA gain, defined as the maximum quadrature gain. GJ = G1
Signal power gain
High electron mobility transistor amplifier
Current
Critical current of a Josephson junction
Critical current of a SQUID
Input Output Theory
Josephson parametric amplifier
InductanceInductance per unit length

LJ
LJ0
Q
Q ,Q
SHO
S/N
SQL
Sy,y
Sz

T
vph
W (x1 , x2 )
Xi
xi
Xθ
Zc

281
Nonlinear Josephson inductanceLinear part of the Josephson inductance
ChargeQuality factor
Simple Harmonic Oscillator
Signal-to-noise ratio
Standard Quantum Limit
Single-sided spectral density for an operator ŷ, Sŷ,ŷ = h{ŷ, ŷ}iQuadrature single-sided spectral density, where z defines alocation in the experiment setup. This variable-naming scheme is
only used in chapter 6, 7 and appendix B.
TemperaturePhase velocity of a transmission line
Wigner function
Quadrature operator, i = 1, 2
Quadrature variable, i = 1, 2. The Wigner function is a functionof xi .
Phase-dependant quadrature
Characteristic impedance of a transmission line