Superconducting qubit readout via low-backaction
electro-optomechanical transduction
byRobert Delaney
BSc, University of British Columbia, 2014
MSc., University of British Columbia, 2016

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
2022
Committee Members:
Konrad Lehnert, Chair
Cindy Regal
John Teufel
Graeme Smith
András Gyenis

iiDelaney, Robert (Ph.D., Physics)
Superconducting qubit readout via low-backaction electro-optomechanical transduction
Thesis directed by Prof. Konrad Lehnert

Entangling superconducting quantum processors via light would enable new means of securecommunication and distributed quantum computing. However, transducing quantum signals between these disparate regimes of the electromagnetic spectrum remains an outstanding goal andinterfacing superconducting qubits with electro-optic transducers presents significant challenges dueto the deleterious effects of optical photons on superconductors. An ideal transducer should leave
the state of the qubit unchanged: more precisely, the backaction from the transducer on the qubit
should be minimal.
In this work, I demonstrate readout of a superconducting transmon qubit via a low-backaction
electro-optomechanical transducer. Requirements for integrating technology from circuit quantum
electrodynamics are discussed, and the results of superconducting qubit readout via an electro-optic
transducer are presented. The modular nature of the transducer and circuit QED system used inthis work enable complete isolation of the superconducting qubit from optical photons, and the
backaction on the qubit from the transducer is less than that imparted by thermal radiation fromthe environment. I show that only moderate improvements in transducer bandwidth and added
noise should enable the transduction of non-classical signals from a superconducting qubit to theoptical domain.

Acknowledgements

There are many people I’d like to thank, who have helped significantly with the work thatwent into this thesis, and who helped me with the completion of my PhD. First off, my experience
working at JILA over the past 5 plus years has been really incredible. There are too many people toinclude here, but really everyone that works at JILA contributes to the goal of completing amazing
research, and it really has been a pleasure to work here with so many amazing people.
Before I even arrived at CU Boulder, I learned a huge amount about low temperature physics
from my previous supervisors Douglas Bonn and Sarah Burke, as well as James Day and Pinder
Dosanjh. Much of the work in this thesis would have been far more difficult to complete without
the things I learned from them about the assembly of fragile low temperature physics experiments.
When I first started working in the lab Adam Reed was incredibly helpful, and taught me
how to keep a rickety old dilution refrigerator going. He also taught me the inner workings of
our electromechanical experiments. Lucas Sletten, Eric Rosenthal, Brad Moores, Dan Palken, Ben
Chapman and Xizheng Ma worked alongside me in the lab through much of my PhD. I learned ahuge amount from all of them, and really enjoyed my time working in the lab with them.
During the later part of my PhD I joined the microwave-to-optical transduction team, which
has been an amazing experience, where I got to work with many great people. Jon Kindem, Max
Urmey, Benjamin Brubaker, Peter Burns and Sarang Mittal have all been incredible teammates,
and the work presented in this thesis would not have been in any way possible without each of
their contributions. It has also been a pleasure to get to know and work with the newer membersof the microwave-to-optical transduction team Luca Talamo, Kazemi Adachi and Sheng-Xiang Lin.

iv
Joining the microwave-to-optical transduction team has also given me the privilege of getting towork with Cindy Regal. Her incredible insight into any and all aspects of experimental quantum
optics has been extremely helpful, and she has really helped broaden my knowledge of the field.
I’d like to thank the other members of my thesis committee for taking the time out of
their busy schedules to be on my committee: John Teufel, Graeme Smith and András Gyenis. In
particular, we have often leaned on John Teufel for his expertise in electromechanics, and he has
given incredibly useful feedback on both of the manuscripts that are described in this thesis.
Finally, I feel very lucky that I ended up in Konrad’s group for my PhD, and working
for him has really made my PhD an enjoyable experience. While he always guided me towards
certain experimental goals, he has also provided the freedom to explore new ideas. Through ourmany meetings I’ve learned a huge amount from him about physics in general, and I’m continually
impressed by his knowledge of experimental physics.

v

Contents

Chapter
1 Introduction
1.1

1.2

1

Communication with electromagnetic waves . . . . . . . . . . . . . . . . . . . . . . .

1

1.1.1

Superconducting quantum computers . . . . . . . . . . . . . . . . . . . . . .

2

1.1.2

The quantum internet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.1.3

Electro-optic transducers for the quantum internet . . . . . . . . . . . . . . .

5Thesis overview and contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62 Electro-optic transduction theory

8

2.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.2

A simple beamsplitter interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.3

Transduction through an intermediary mode . . . . . . . . . . . . . . . . . . . . . . . 12

2.4

2.3.1

Electro-optomechanical transduction . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.2

Beamsplitter interaction from real electro-optomechanical coupling . . . . . . 13

State-space model for the electro-optomechanical transducer . . . . . . . . . . . . . . 14
2.4.1

Approximate frequency domain solution . . . . . . . . . . . . . . . . . . . . . 16

2.4.2

Time domain calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Integrating Circuit QED technology with electro-optomechanical transducers: experimental
requirements
3.1

20

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

vi
3.2

Efficiently transducing signals from a circuit QED system . . . . . . . . . . . . . . . 21
3.2.1

Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.2

Tunable frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3

Input referred added noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.4

Interaction between the added noise and the efficiency . . . . . . . . . . . . . . . . . 28

3.5

Repetition rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4 Optically mediated superconducting qubit readout

32

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2

Dispersive readout of a superconducting qubit . . . . . . . . . . . . . . . . . . . . . . 32

4.3

4.2.1

Dispersive Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2.2

Optically mediated dispersive readout . . . . . . . . . . . . . . . . . . . . . . 33

Demonstrating readout of a superconducting qubit via the electro-optomechanicaltransducer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.4

Two equivalent descriptions of linear measurement . . . . . . . . . . . . . . . . . . . 37
4.4.1

4.5

Modeling the transducer’s added noise . . . . . . . . . . . . . . . . . . . . . . 39

Characterizing the quantum efficiency: theory . . . . . . . . . . . . . . . . . . . . . . 40
4.5.1

Superconducting qubit readout signal-to-noise ratio. . . . . . . . . . . . . . 40

4.5.2

Signal-to-noise ratio for a general pulsed measurement . . . . . . . . . . . . . 41

4.5.3

Measurement induced dephasing . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.6

Measurement-induced dephasing: experimental results . . . . . . . . . . . . . . . . . 42

4.7

Characterizing the quantum efficiency of the optical readout apparatus . . . . . . . . 43
4.7.1

4.8

Technical details of the quantum efficiency calibration . . . . . . . . . . . . . 45

Contributions to the quantum efficiency . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.8.1

Bandwidth limitations: ηbw . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.8.2

Transducer efficiency: ηt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.8.3

Transducer gain: ηG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

vii

4.9

4.8.4

Microwave transmission loss: ηmic

. . . . . . . . . . . . . . . . . . . . . . . . 48

4.8.5

Optical detection efficiency: ηopt . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.8.6

Circuit QED system cavity loss: ηcav . . . . . . . . . . . . . . . . . . . . . . . 50

The transmission loss and comparison to other results . . . . . . . . . . . . . . . . . 51

5 Characterization of backaction from the electro-optic transducer

55

5.1

Motivation

5.2

Backaction in circuit QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.2.1

5.3

5.5

Implications for the remote entanglement of qubits . . . . . . . . . . . . . . . 57

Isolation and filtering to reduce backaction
5.3.1

5.4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

. . . . . . . . . . . . . . . . . . . . . . . 58

Reducing backaction through modularity . . . . . . . . . . . . . . . . . . . . 59

Sources of excess backaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.4.1

Coherent backaction from the electro-optic transducer . . . . . . . . . . . . . 60

5.4.2

Thermal backaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Future improvements and considerations . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.5.1

Filtering and interferometric cancellation . . . . . . . . . . . . . . . . . . . . 65

5.5.2

Excess backaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.5.3

Residual thermal occupancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6 Transient electromechanical amplification

69

6.1

Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.3

Electromechanical device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.4

Pulsed measurement and squeezing of motion . . . . . . . . . . . . . . . . . . . . . . 72

6.5

Added noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.6

Quantum state tomography of motion . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.7

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

viii
7 Conclusion

80

Bibliography

83

Appendix
A Technical specifications of the circuit QED system and electro-optic transducer

97

A.1 Circuit QED system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
A.2 Electro-optic transducer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
A.3 Experimental layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
B Qubit Fabrication procedure

104

C Transient electromechanical amplification experimental details

111

C.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
C.1.1 Two-tone electromechanical equations of motion . . . . . . . . . . . . . . . . 111
C.1.2 Additional single mode squeezing due to detuning of microwave pumps . . . . 113
C.2 Pump dependent cavity frequency shifts . . . . . . . . . . . . . . . . . . . . . . . . . 114
C.3 Variance normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
C.4 Mechanical occupancy, added noise and gain calibrations . . . . . . . . . . . . . . . . 116
C.4.1 Calibration of sideband cooling and two-quadrature measurement noise . . . 117
C.4.2 Inference of mechanical squeezing . . . . . . . . . . . . . . . . . . . . . . . . . 118
C.4.3 Inference of the measurement noise . . . . . . . . . . . . . . . . . . . . . . . . 119
C.4.4 Direct measurement of mechanical variance . . . . . . . . . . . . . . . . . . . 119
C.4.5 Estimating thermal occupancy of the mechanical oscillator . . . . . . . . . . 120
C.4.6 Gain calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
C.5 Quadrature extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
C.6 Tracking measurement and squeezing axes . . . . . . . . . . . . . . . . . . . . . . . . 121

ix
C.7 Quantum state tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
C.8 Experimental apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
C.8.1 Electromechanical device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
C.8.2 Arbitrary microwave signal generation . . . . . . . . . . . . . . . . . . . . . . 123

xTablesTable
4.1

Contributions to the quantum efficiency . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2

Recent electro-optic transduction results . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.1

Circuit QED system parameters important for backaction estimation . . . . . . . . . 68

A.1 Circuit QED system and electro-optic transducer parameters. . . . . . . . . . . . . . 102
C.1 Parameters of the electromechanical device. . . . . . . . . . . . . . . . . . . . . . . . 124
C.2 Tomographically reconstructed squeezed state parameters . . . . . . . . . . . . . . . 124

xiFiguresFigure2.1

Pulse transduction efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1

Apparatus for readout of a superconducting qubit via the electro-optomechanicaltransducer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2

Tunable microwave cavity coupling rates . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3

Lifetime and coherence time of the qubit contained within Cavity A. . . . . . . . . . 25

3.4

Cavity A dispersive shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.5

Microwave LC circuit noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.6

Distributed beamsplitter model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.7

Frequency and time domain operation of the transducer . . . . . . . . . . . . . . . . 31

4.1

Dispersive readout of a superconducting qubit . . . . . . . . . . . . . . . . . . . . . . 33

4.2

Dispersive readout of a superconducting qubit with loss and noise . . . . . . . . . . . 34

4.3

Single-shot optical readout of a transmon qubit . . . . . . . . . . . . . . . . . . . . . 36

4.4

Effective beamsplitter model

4.5

Measurement induced dephasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.6

Measurement induced dephasing Ramsey fringe data . . . . . . . . . . . . . . . . . . 44

4.7

Quantum efficiency of the optically mediated qubit readout . . . . . . . . . . . . . . 45

4.8

Characterization of the quantum efficiency of the microwave readout apparatus . . . 51

4.9

Differences in efficiency characterization . . . . . . . . . . . . . . . . . . . . . . . . . 53

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

xii
4.10 Transmission loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.1

Isolation and pump cancellation scheme . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.2

Excess backaction from pump photon leakage . . . . . . . . . . . . . . . . . . . . . . 62

5.3

Qubit coherence when operating the electro-optic transducer . . . . . . . . . . . . . 63

5.4

Qubit decay times versus temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.5Transducer backaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.1

Transient electromechanical amplification schematic . . . . . . . . . . . . . . . . . . 71

6.2

Transient electromechanical amplification pulse sequence . . . . . . . . . . . . . . . . 74

6.3

Transient electromechanical amplification: added noise and squeezing . . . . . . . . . 75

6.4

Quantum state tomography of motion . . . . . . . . . . . . . . . . . . . . . . . . . . 77

A.1 Image of the optical access dilution refrigerator . . . . . . . . . . . . . . . . . . . . . 99
A.2 CAD model of the circuit QED system . . . . . . . . . . . . . . . . . . . . . . . . . . 100
A.3 Experimental schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
C.1 Typical measured histograms when inferring squeezed state variance . . . . . . . . . 124
C.2 Transient electromechanical amplification Y-factor measurement. . . . . . . . . . . . 125
C.3 Electromechanical temperature sweep . . . . . . . . . . . . . . . . . . . . . . . . . . 126
C.4 Microwave and mechanical noise comparison . . . . . . . . . . . . . . . . . . . . . . . 127
C.5 Transient electromechanical amplification gain

. . . . . . . . . . . . . . . . . . . . . 127

C.6 Transient electromechanical amplification experimental schematic . . . . . . . . . . . 128

Chapter 1

Introduction

1.1

Communication with electromagnetic waves
Analog and digital communication through electro-magnetic waves has existed for centuries

in various different forms. Optical telegraphs, or semaphores, used lights modulated by manual
shutters to distribute digitally encoded information over long distances through a series of stations[1]. This basic demonstration of digital communication via light was followed by more advanced
technologies such as the telegraph [2], where radio frequency information was digitally encoded via
Morse code [3], and sent through radio frequency transmission lines. Today a huge portion of theelectromagnetic spectrum is used for classical communication, with frequencies from a few hertz inunder water or geological applications all the way up to terahertz scale optical frequencies used by
the modern classical internet.
Although the bandwidth, speed and ubiquity make the modern day internet seem completely
different from these early technologies, it is nonetheless all based upon encoding digital informationin electromagnetic waves. Relatively recent technological advances over the past few decades haveenabled the fast distribution of classical information through the internet, and much of this progresshas largely been enabled by the use of fiber optic cable. Computers operate at radio frequencies,
where the classical bits are operated at microwave clock frequencies, but this information then getsconverted to infrared light for transmission through optical fibers near a wavelength of 1550 nm. The
bandwidth over which signals can be encoded is much larger at optical frequencies than at microwave
frequencies, meaning that a much larger number of bits can be sent in parallel. Furthermore, when

2
compared to coaxial transmission lines, optical fiber offers extremely low loss transmission, withstate of the art fibers demonstrating only 0.14 dB/km of loss [4].
An equivalent dichotomy between microwave frequency computation and optical communication is arising in the field of quantum information, where it would be extremely useful if ananalogous quantum internet [5, 6, 7, 8] could be developed. As in the classical case, it is best
to encode quantum information in optical light for transmission, but due to additional and more
stringent constraints on noise when transmitting quantum states, if quantum computations are
performed at microwave frequencies, the quantum states must be transmitted optically in order toprevent the quantum state from being corrupted by thermal noise at ambient temperature.

1.1.1

Superconducting quantum computers

In classical computing, one of the first transistors [9] was held together by paper clips and gold
foil, but transistors quickly progressed over more than 70 years to precisely fabricated, nanometerscale devices, with modern transistors spanning less than 14 nm, where now millions of transistors
can be placed on a single chip. Today, quantum computing has progressed far past this stage ofhastily prepared one-off devices [9], and we are now seeing the demonstration of modest quantum
processors [10, 11, 12, 13] with capabilities that may soon be able to exceed those of classicalcomputers [14] for certain computational tasks [15]. As a result, much research effort is being
invested in rapidly scaling these capabilities.
Although there exist a wide range of platforms that are being investigated for scalable quantum computation, including trapped ions [16], neutral atoms [17] and optical qubits [18], one
commonly pursued technique involves superconducting qubits [19, 20], in which the Josephson effect [21] can be used to create an artificial atom. The transition between the lowest two energy
levels of these anharmonic inductor-capacitor (LC) circuits [22] is at microwave frequencies, typically in the range of 4−12 GHz. Analogous to the modern transistor it is possible to fabricate thesesuperconducting qubits using standard lithographic techniques [23, 24, 25], meaning that a large
number of precisely engineered devices can be fabricated all on the same chip [12]. Furthermore,

3
the microwave transition frequencies enable the use of standard microwave technology similar tothat used in cellphones for the control and readout of qubits. This contrasts systems involving
atoms or trapped ions, which rely upon an atomic transition fixed by the laws of nature to form
the physical qubit. Although this can have advantages in terms of the lifetime and reproducibility of the effective qubit, the ability to engineer the device that is needed, rather than trying toaccommodate a physical system that already exists, is invaluable, and has spurred much researchand investment in superconducting quantum computation.
However, the major drawback of superconducting quantum computers is their reliance upon
low temperature for proper operation. There exists two main reasons for this constraint. First,
a microwave resonator will not be in its quantum ground state (the thermal occupancy of theresonator is nth  1) until its temperature is below 100 mK, meaning that some sort of cooling isrequired. Note that this can be true in atomic systems as well, where the motional state of the atom
is typically far from the quantum ground state at ambient temperature, however this is typicallyaddressed through techniques such as laser cooling [26, 27]. Second, the reliance these qubits haveon superconductivity, means that the material used must be in the superconducting state, and infact, must be deep in the superconducting state to avoid the deleterious effects of quasiparticles inthe superconductor [28, 29, 30, 31]. Since aluminum is by far the best material for the fabricationof Josephson junctions, due to the clean aluminum oxide layers that can be easily formed throughoxidizing the material, this necessitates operating superconducting qubits at a temperature far
below that of the critical temperature of aluminum (Tc = 1.2 K), or essentially again also at atemperature below 100 mK. Thus far, this has not impeded progress in developing superconducting
quantum computers, as standard tools from low temperature physics such as dilution refrigerators
exist, and can easily cool devices down below 50 mK. There do exist questions about just how far
this technology can be scaled, as being able to address millions of qubits (in order to form far lesslogical qubits via error correction [32, 33, 34, 35]) presents difficulties for the thermalization of thecontrol lines connected to the processor [36]. However, this is an active field of research [37, 38, 39],
and has not yet proven to be a limiting factor.

4
1.1.2

The quantum internet

The speed of the classical internet is limited by the physical laws of nature, as a single bitof information cannot be sent faster than the speed of light. Furthermore, the total bandwidth of
the transmission medium determines the number of bits that can be sent in parallel. The quantum
internet is constrained as above, but also must meet a much more stringent set of requirements.
For example classical bits can be made robust to errors and noise by simply making the pulse oflight encoding the state 1 bright enough such that it is not possible to mistake it for the state 0. As
long as there aren’t constraints on the power of the classical signal being transmitted, errors can bemade arbitrarily small during classical communication. Indeed, the invention of the transistor was
motivated by the need to amplify analog classical information being transmitted through phonelines [9].
However, when transmitting quantum states, that are encoded in quantum excitations, orquanta, of the electromagnetic field, loss or added noise will irreversibly corrupt the quantum
state of interest. Although loss can typically be detected and then subsequently compensated forthrough heralding techniques, where photon counting can be used to discard events in which no
photon was detected [40, 41, 42, 43], even small amounts of noise (at even single photon levels) willquickly prevent the transmission of quantum information [44]. This is the why superconducting
quantum computers must be operated at such low temperatures, as even if superconductivity was
not a limiting factor (though it also is!), thermal noise would corrupt the states encoding thequantum information. A cryogenic link has recently been demonstrated between two separate
dilution refrigerators [38] in the same room, and this will certainly be useful for expanding the local
capabilities of superconducting quantum computers. However, this likely will not be scalable overlong distances since cryogenic interconnects at millikelvin temperatures require extensive radiation
shielding and vacuum systems that will be challenging to implement over kilometer scale distances.
In contrast to microwave frequency quantum devices, optical resonators are in their quantum
ground state at ambient temperature, and optical quantum networks are an already proven tech-5
nology [45, 46, 47], where qubits encoded in propagating optical photons can be sent through either
optical fibers [48, 49, 50] or through free space [51]. Thus a natural way to network superconducting quantum computers would involve converting the quantum information encoded at microwave
frequencies to optical frequencies for transmission through fiber optic cable, exactly in the samemanner as in the classical internet. Thus, a necessary tool for the networking of superconducting
quantum computers will be an electro-optic transducer that is capable of transducing quantum
states between the microwave and optical domains.
Such a transducer must add close to zero noise, so as not to corrupt the fragile quantum states
it is converting. Ideally, it should be efficient, such that it transduces nearly all of the incident
signal and finally it should be compatible with superconducting qubits. More precisely, it should
impart little to no backaction on the qubit, or more simply put, operating the transducer shouldhave no effect upon the state of the superconducting qubit in any basis. In more direct analogy
with the classical internet, ideally such a transducer should have a large enough bandwidth suchthat information can be encoded over a relatively large bandwidth.
It is interesting to note that one of the most promising candidates for building a successful
quantum computer uses fabricated devices that operate at microwave frequencies, and the only
reasonable technique for networking these quantum computers over any appreciable distance isthrough optical links—exactly what classical computing has converged upon. Thus for the quantum
internet we desire all of the capabilities of today’s classical internet such as large bandwidth, fast
data transmission and reliability, but there also exist new requirements in order to preserve thequantum information being transmitted.

1.1.3

Electro-optic transducers for the quantum internet

Although much progress has been made in the realm of superconducting quantum computers,
as of yet qubits and superconducting processors have only been connected in networks distributed
within the same dilution refrigerator [52] or the same room [38]. The pursuit of a truly distributed(over kilometer distance scales) quantum network of superconducting qubits has thus given rise

6
to a rich research field [53] searching for efficient, low-noise electro-optic transduction techniques[54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65]. Due to the orders of magnitude difference in energy scale
between microwave and optical signals, these electro-optic transducers must use intermediary systems that simultaneously interact with both microwave and optical photons, where the interaction
strength is enhanced parametrically by microwave and optical pumps. Examples of electro-optic
transduction platforms include non-linear solid state materials [66, 65], Rydberg atom ensembles[55], magnonic systems [54], rare earth ions [64], or mechanical resonators [58, 67, 56, 61, 62]. Whensuch quantum enabled transducers become available, the engineerability of microwave frequency
superconducting qubits can then be combined with the advantages of certain optical elements
such as state of the art photon counting and low loss optical transmission over optical fiber [68].
These different technologies can then be combined to perform new and exciting quantum optics
experiments, and to create networks of superconducting quantum processors.

1.2

Thesis overview and contents
Networking superconducting quantum computers is very much in the early stages of devel-

opment. This is mainly due to the fact that, although electro-optic transducers are now quantum
enabled [65], or nearly quantum enabled [62, 69], they are not yet able to distribute quantum signals
from superconducting qubits to optical communication channels.
This thesis work will describe experiments in which a superconducting qubit is integrated
with an electro-optic transducer based upon an electro-optomechanical device. More broadly I use
the framework of linear quantum optics [70] to describe the process of reading out a superconducting qubit, converting that propagating microwave signal encoding the state of the qubit to an
optical pulse and then measuring the resulting optical output. Although the microwave-to-opticaltransducers being developed at JILA [67, 56, 69] are not yet quantum enabled, it is still possible
to learn valuable information about the integration of superconducting qubits with electro-optic
transducers through the process of reading out the qubit optically.
In Chapter 2 I outline the theoretical requirements for transducing signals from a circuit

7
QED system to the optical domain. In Chapter 3 I discuss experimental requirements and figuresof merit for the transduction of signals from the circuit QED system. In Chapter 4 the results of
optically mediated single-shot readout of a superconducting qubit are discussed and by using thesuperconducting qubit as a non-Gaussian resource, the quantum efficiency of the optical measurement apparatus is characterized. In Chapter 5 the backaction from the electro-optic transducer on
the qubit is measured and discussed. Chapter 6 describes a purely electromechanical experimentfrom earlier in my PhD, where a new technique for pulsed measurement of the motion of a mechanical oscillator was developed. The thesis is concluded in Chapter 7, where the outlook for quantum
enabled electro-optic transduction of photons from a circuit QED system is discussed.

Chapter 2

Electro-optic transduction theory

2.1

Overview
In this chapter I describe the necessary theoretical requirements to achieve transduction be-

tween the microwave and optical domain, as well as the additional constraints that apply when
integrating superconducting qubits with electro-optic transducers. Ideally, an electro-optic transducer performs unitary operations on the microwave and optical signals, with no signal lost, or
noise added by the transduction process, but of course in a real transducer this is not the case, andin this Chapter the non-idealities of the transduction process are discussed.
The needed interactions for electro-optic transduction can be achieved in several different
ways. As is common in devices that use the direct electro-optic effect [65, 66, 71], a direct interaction
between microwave and optical fields can be mediated, which enables swapping between microwave
and optical photons. Such devices typically rely upon the Pockels effect, where an electric field can
produce birefringence in an optical medium [72]. Alternatively, one can introduce an intermediateresonant mode, frequently a mechanical oscillator [69, 62, 60], which interacts simultaneously withmicrowave and optical fields to mediate the transduction process.

2.2

A simple beamsplitter interaction
To achieve ideal electro-optic transduction, a unitary transformation between an input field at

microwave (optical) frequencies and an output field at optical (microwave) frequencies is required.
The most basic version of this unitary operation is described by a beam-splitter interaction in terms

9of bosonic operators â and b̂:∗ †Hbs /~ = gbs âb̂† + gbsâ b̂,(2.1)

where optical excitations (â) are swapped with microwave excitations (b̂) with coupling strength
gbs .
With the Hamiltonian describing the system, one can use the Heisenberg-Langevin equations[73] to describe the equations of motion of the system. Where for an arbitrary operator Ô,
N

X√
i
κtot
˙
Ô = [Ĥ, Ô] −Ô +κj Ôin,j .
~
2

(2.2)

j

Here, there are N total ports coupled to the system with individual coupling rates κj , and the(bosonic) mode has a total linewidth of:κtot =

N
X

κj .

(2.3)

j

For ports that the experimentalist has physical access to, the input-output relations can also be
usefully described by:Ôin,j + Ôout,j =√κj Ô.

(2.4)

Thus, for the simple beamsplitter interaction described in Equation 2.1, the Heisenberg-Langevinequations yield:√
κo∗b̂ + κo âinâ˙ = − â + igbs
2√κe˙b̂ = − b̂ + igbs â + κe b̂in ,
2

(2.5)
(2.6)

and
âin + âout =b̂in + b̂out =√√κo â,(2.7)

κe b̂,(2.8)

where κo (κe ) is the coupling of the optical (microwave) resonator to its external port.
Assuming steady state input fields, one can look at the steady state response of the optical˙and microwave fields such that â˙ = b̂ = 0. This gives:
â =

∗
2igbs
2
b̂ + √ âin
κo
κo

(2.9)

10
b̂ =

2igbs
2
â + √ b̂in .
κeκe(2.10)

Assuming that gbs is real (which amounts to a choice of arbitrary phase), and I can also define thecooperativity as:
C=

2
4gbs
.
κe κo

(2.11)

Using Equations. 2.7, 2.8, 2.9 and 2.10, I can solve for the optical and microwave output fields âout
and b̂out in terms of the optical input âin and the microwave input b̂in , which gives:
1  √2i C b̂in + (1 − C) âin ,
1+C

1  √
2i Câin + (1 − C) b̂in ,b̂out =
1+C

âout =

(2.12)(2.13)and provides the desired linear relation between microwave input fields and the optical output field.
It is clear that this can be written in terms of a unitary matrix such that: âout  âin  = Û  b̂outb̂in ,(2.14)

where the Û is given by:√
1 1 − C 2i C 
Û =
.
 √
1+C
2i C 1 − C(2.15)

It is critical that the matrix Û be unitary as it must preserve the bosonic commutation relationsof the output fields [74].
Furthermore, it is clear that at unit cooperativity Equations 2.12 and 2.13 simplify drasticallyto:âout = ib̂in ,(2.16)b̂out = iâin ,(2.17)

which is of course an ideal bidirectional transduction operation.

11
Of equal importance is the noise properties of such a transducer, which can be found throughthe auto-correlation functions for the optical and microwave fields:hâin (t)â†in (t0 )i = δ(t − t0 )(2.18)hâ†in (t)âin (t0 )i = 0(2.19)hb̂in (t)b̂†in (t0 )i = (1 + nmic )δ(t − t0 )(2.20)hb̂†in (t)b̂in (t0 )i = nmic δ(t − t0 ),(2.21)

where nmic describes the thermal occupancy of the microwave field and I have assumed that the
optical field is in its ground state. For microwave-to-optical transduction, the variance of the optical
output field is:(1 − C)2
4C
†
0
(t
)i
+
h
b̂
(t)
b̂
hâin (t)â†in (t0 )iinin(1 + C)2(1 + C)2
4C(1 − C)2†0hâout (t)âout (t )i =(1 + nmic ) +δ(t − t0 ),(1 + C)2(1 + C)2hâout (t)â†out (t0 )i =(2.22)(2.23)

4C
Note, that there is gain G = (1+C)
2 (or if one prefers, attenuation, since G ≤ 1), which is only

unity for C = 1. Dividing by this gain gives noise referred to the input microwave port of:hâout (t)â†out (t0 )i/G =

(1 − C)2
1 + nmic +4Cδ(t − t0 ).

(2.24)

Note that the factor of unity is simply the contribution from microwave vacuum fluctuations, andis an inherent part of the signal, thus the added noise from transduction is:
nadd = nmic +(1 − C)2,4C(2.25)

which can be made arbitrarily small by ensuring low microwave thermal occupancy and the C ≈ 1limit.
An example of this ’direct’ beamsplitter between microwaves and optics involves the useof whispering gallery mode resonators, where the microwave field can directly modulate opticallight via an inherent electro-optic non-linearity of the material such as lithium niobate [66]. The
free spectral range of an optical whispering gallery mode resonator matches that of a microwave

12resonator, allowing the electro-optic effect to be resonantly enhanced by pumping one of the modes,
yielding the direct beamsplitter interaction between microwaves and optics described above. While
this technology is promising, and recently demonstrated quantum enabled microwave to opticaltransduction for the first time [65], it has yet to be integrated with superconducting qubits, which,as will be shown in this work, provide significant additional constraints.

2.3

Transduction through an intermediary mode
An alternative technique for microwave-to-optical transduction, is to instead swap excitations

to an intermediary system that interacts with both microwave and optical photons, achieving
the same result as a direct beamsplitter. In the electro-optomechanical system used here, thisintermediary coupling is mediated by a high quality factor mechanical mode, and gets describedby the interaction Hamiltonian:
Hint /~ = G∗o â† ĉ + Go âĉ† + G∗e b̂† ĉ + Ge b̂ĉ† .

(2.26)

where Go and Ge are now the strength of the optomechanical and electromechanical beamsplitterinteractions respectively.

2.3.1

Electro-optomechanical transduction

Another avenue towards quantum enabled electro-optic transduction involves the use of mechanical elements simultaneously coupled to optical and microwave light [56, 69, 62]. At JILA,
membrane in the middle style devices have been pursued [75, 67, 56, 69, 76], where the motion ofa partially transparent silicon nitride membrane inside of a Fabry-Pérot cavity modulates its resonance frequency. To achieve electromechanical coupling, a part of the silicon nitride membrane ismetalized to form the capacitor in a microwave frequency LC circuit. Motion of the silicon nitride
membrane changes the parallel plate capacitance and hence the resonance frequency of the LC
circuit. For a more detailed description of the electro-optomechanical transducer see [67, 56, 69].

13
2.3.2

Beamsplitter interaction from real electro-optomechanical coupling

The interaction described above is a pure beamsplitter interaction between a mechanicalmode and optical/microwave electromagnetic modes. However, in practice electro-optomechanics
achieves a Hamiltonian that instead approximates the beamsplitter interaction. The coupling
between motion and light comes in the form of a position dependent frequency shift:
Heo = ~ωo â† â + ~ωe b̂† b + ~ωm b̂† b + ~go â† â(ĉ + ĉ† ) + ~ge b̂† b̂(ĉ + ĉ† ).

(2.27)

Where the Hamiltonian above of course does not provide the desired beamsplitter interaction.
Furthermore, the electromechanical (optomechanical) coupling ge (go ) is very small relative to κoand κe . For example in [77, 69] the device had electromechanical and optomechanical coupling
rates of ge = 2π × 1.6 Hz and go = 2π × 60 Hz respectively. Thus, both to enhance this interaction
strength and to yield the desired swapping interaction, strong microwave and optical pumps can beapplied at frequencies ωp,e and ωp,o respectively, causing a displacement of the fields in the opticaland microwave resonators such that â → ā + â and b̂ → b̄ + b̂. Linearizing the Hamiltonian aroundthese strong pumps yields:
Heo = ~∆o â† â + ~∆e b̂† b + ~go ā(â + â† )(ĉ + ĉ† ) + ~ge b̄(b̂ + b̂† )(ĉ + ĉ† )(2.28)

where the detunings are given by ∆o = ωo − ωp,o and ∆e = ωe − ωp,e . In this linearized Hamiltonian
I now get the desired beamsplitter interaction, along with a two mode squeezing interaction where:int
Heo= Hbs + Htms

(2.29)

Hbs = ~go ā(â† ĉ + âĉ† ) + ~ge b̄(b̂† ĉ + b̂ĉ† )(2.30)

Htms = ~go ā(â† ĉ† + âĉ) + ~ge b̄(b̂† ĉ† + b̂ĉ).

(2.31)

By choosing the detuning of the laser and microwave pumps, such that they are red-detuned with∆o = ∆e = −ωm , the beamsplitter interaction can be resonantly enhanced, while suppressing thetwo mode squeezing interaction.

14

2.4

State-space model for the electro-optomechanical transducer
To systematically solve the equations of motion describing the electro-optomechanical trans-

ducer and a coupled circuit QED system, techniques from linear control theory can be used [78].
From the Heisenberg-Langevin equations and the state-space model formalism the equations of
motion can be written as:ẋ(t) = Ax(t) + Bxin (t)(2.32)

xout (t) = Cx(t) + Dxin (t),(2.33)

where x = (X̂1 , Ŷ1 , Ẑ1 , X̂2 , Ŷ2 , Ẑ2 )T is a vector of quadrature amplitudes, with X̂l , Ŷl and Ẑl(l = 1, 2) corresponding to the dimensionless optical, microwave and mechanical mode quadratures
respectively. The optical quadratures can be defined in terms of creation/annihilation operatorsas X̂1 = 21 (â† + â) and X̂2 = 2i (â† − â), with the microwave and mechanical quadratures defined
analogously. The input and output fields may also be written in vector form:
xin = (X̂1,in,ext , X̂1,in,int , Ŷ1,in,ext , Ŷ1,in,int , Ẑ1,in,int , X̂2,in,ext , X̂2,in,int , Ŷ2,in,ext , Ŷ2,in,int , Ẑ2,in,int )T(2.34)
xout = (X̂1,out,ext , Ŷ1,out,ext , X̂2,out,ext , Ŷ2,out,ext )T ,(2.35)The transducer can be viewed as a phase-preserving amplifier with near unity gain [74], so
the equations of motion are independent of the phases of the optical and microwave pumps. Thus
under an appropriate choice of these arbitrary phases, the elements of the matrices A, B, C, D in
the state-space model can all be made real. Transforming to a rotating frame removes the freeevolution of the quadratures, enabling the state-space model to be described with the following

15
matrices:



κo
− 2



−go ā


− κ2e
0
0
0
−ge b̄ γm0− 2 −ge b̄ −go ā
0 


κo
0
go ā − 2
0
0 



0
ge b̄
0
− κ2e
0 


γm
ge b̄
0
0
0
− 2


Q U A = ARWA + 

U −Q
0

0

0

0



 0



 0
ARWA = 

 0



 0


go ā



(2.37)
−go ā sin(2ωm t)

00−ge b̄ sin(2ωm t) −go ā sin(2ωm t) −ge b̄ sin(2ωm t)
0


0




Q=



(2.36)

0

0




U =



go ā cos(2ωm t)

00
ge b̄ cos(2ωm t) go ā cos(2ωm t) ge b̄ cos(2ωm t)0√ κo,ext
M = 00

√(2.38)

0





M
B=
0

0

M

√

(2.39)(2.40)
κo,int00√00 κo,ext√κe,ext
 0
C=

 0
0


0
0

0

κe,ext√0

κe,int
0

0 


0 

√ 
γm

(2.41)


0

0

0

0

0

0

0
0

√

κo,ext00√κe,ext

0


0


0


0

(2.42)

16


−1


0
D=

0


0


0 0


0 −1 0 0 0 0 0 0 0.

0 0 0 0 −1 0 0 0 0

0 0 0 0 0 0 −1 0 0

0

0

0 0

0

0

0

(2.43)

Where, κo (κe ) is the total optical (microwave) cavity linewidth, the subscript ‘ext’ indicates
coupling to an external port, the subscript ‘int’ indicates coupling to an internal loss port andthe mechanical linewidth is given by γm .

2.4.1

Approximate frequency domain solution

As has been shown previously [67], the electro-optomechanical transducer has an approximately Lorentzian response in the frequency domain, where I can describe the transmission(|Soe (ω)|2 ) between the microwave and optical domain as a function of frequency as|Soe (ω)|2 = κe,ext κo,ext
Γe Γo
ηg ,
κe
κo ω 2 + Γ2T /4(2.44)

where ω = 0 corresponds to a signal on resonance with the microwave LC circuit, and the Lorentzian
has a characteristic linewidth of ΓT = Γe + Γo + γm . ηG is a small transduction gain factor due tofinite sideband resolution [79, 67]:

ηG =

1+

κe
4ωm

2 !


1+

κo
4ωm

2 !
.

(2.45)The strength of the optical (microwave) beamsplitter interaction go ā (ge b̄) sets the total bandwidthof the transduction process, and determines the optomechanical and electromechanical damping
rates respectively [67]:Γo ≈4go2 ā2κo(2.46)

Γe ≈4ge2 b̄2.
κe(2.47)

17
2.4.2

Time domain calculations

While solving the state-space model in the frequency domain provides insight into the bandwidth of the electro-optomechanical transducer, and can be done so exactly [67], for pulsed qubit
readout experiments the signals are more easily represented in the time domain. Thus I elect tosolve the state-space model in the time domain. To do so, I must also find the equations of motion
for the circuit QED system’s microwave cavity described by:
κ

√
√
˙
c
dˆ = −+ iχ dˆ + κc,ext dˆin + κc,w dˆin,w
2(2.48)√
dˆout,w + dˆin,w = κc,w dˆ

(2.49)√
ˆdˆout + dˆin = κc,ext d,(2.50)

where the subscript ‘w’ indicates the weakly coupled port to which the readout pulse is applied.
The cavity is strongly overcoupled to its external coupling port (labeled ’ext’), which is routed
towards the electro-optomechanical transducer. Note, that the sign in front of χ will depend upon
whether the qubit is in the ground or the excited state, but because I am only interested in theenergy transmitted, the phase of the readout pulse is irrelevant to the total amount of transmitted
energy. The output field of the circuit QED system can then be split into quadratures such that:1Ŵ1,out = (dˆout + dˆ†out )2(2.51)iŴ2,out = (dˆ†out − dˆout )2(2.52)The output quadrature fields of the qubit cavity can then be routed as inputs to the state-spacemodel describing the electro-optomechanical transducer. I can then define the pulse transduction
efficiency as:Rηp =  R

X̂1,out,ext (t)2 dt +RX̂2,out,ext (t)2 dt

Ŵ1,out,ext (t)2 dt +RŴ2,out,ext (t)2 dt,(2.53)

where I have included the optical cavity mode matching factor  in ηp .
The pulse transduction efficiency depends strongly upon the temporal envelope of the signal
emitted by the circuit QED system, and the properties of the qubit dictate the maximum length

18of a useful qubit readout pulse Tp , and hence its minimum bandwidth ∆ωp . There exist two main
factors limiting the length of Tp . First, in order to efficiently read out the state of the qubit,
the signal-to-noise ratio (SNR) of the measurement does not increase significantly for Tp > T1 ,where T1 is the lifetime of the qubit [80]. Second, I wish to characterize the quantum efficiencyof the optically mediated qubit readout scheme (detailed in Chapter 4), which requires injecting aweak readout pulse of length Tp into a Ramsey sequence, and measuring the resulting amplitudeof Ramsey oscillations. Thus the SNR of this measurement decreases exponentially as a functionof the T2 time of the qubit, and for Tp ∼ T2 , the integration times become prohibitively long. It
is the combination of these two factors that sets a minimum bandwidth of the qubit readout pulse∆ωp ∼ T12 .
Both the circuit QED system and electro-optomechanical transducer’s state-space models
can be solved numerically to then calculate ηp . Where I have included the optical cavity mode
matching  as an additional factor included in the pulse efficiency. It can also be useful to express
ηp in terms of approximate expressions from the various different efficiency contributions:ηp ≈ ηt ηbw ηG
!



 !
 !

1 − e−ΓT Tp /2
κo,ext κe,ext 4Γe Γo
κe 2
κo 2
ηp ≈ 
1−2
1+
1+
κo
κe
ΓT Tp
4ωm
4ωm
Γ2T
|
{z
}|
{z
}|
{z
}
ηt

ηbw

(2.54)(2.55)

ηG

The first term is the transduction efficiency ηt , which characterizes the efficiency when transducing narrowband signals [67, 56, 69]. The second term ηbw , is the efficiency due to bandwidth
limitations. To retrieve a tractable expression for this I approximate the output of the circuit QED
system as a square pulse, and assume that the transducer has an entirely Lorentzian response withlinewidth ΓT (see Equation 2.44). The final term constitutes the gain of the transducer ηG due
to finite sideband resolution [79]. These theoretical contributions to the quantum efficiency of the
optically mediated readout, along with additional experimental contributions, will be describedfurther in Chapter 4.
Figure 2.1 shows a theoretical calculation of ηp = ηt ηbw ηG (see Equation 2.55) as Γe is19varied. The curves are calculated through the state-space model described in Section 2.4 using
experimentally measured parameters as inputs. Γo > Γe for all data points except the two rightmost
points of the Γo = 640 Hz curve, which limits ηt . Additionally, there is insufficient total bandwidthΓT since the bandwidth of the qubit readout pulse (the readout pulse has length Tp = 15 µs) greatly
exceeds the total bandwidth of the transducer. The combination of these effects causes ηp to peakat approximately 0.04.



ηp

Γo = 2π × 0.6N+]Γo = 2π × 1.4N+]Γo = 2π × 2.5N+]Γo = 2π × 5.0N+]









Γe/2πN+]





Figure 2.1: Pulse transduction efficiency. Numerically calculated pulse efficiency including
contributions from bandwidth limitations, mismatch between Γe and Γo , optical mode matching,Γe -dependent LC circuit loss and transducer gain. Since the pulse used to readout the state of thequbit can have a maximum (useful) length of approximately the T1 ∼ T2 time of the qubit, thebandwidth of the qubit pulse significantly exceeds that of the electro-optomechanical transducer,
the primary limit to the ηp .

Chapter 3

Integrating Circuit QED technology with electro-optomechanical transducers:
experimental requirements

3.1

Overview
Having demonstrated that electro-optomechanical transduction is theoretically feasible withclassical Gaussian fields I now move onto the experimental requirements for coupling circuit QEDtechnology to the microwave-to-optical transducer. The end goal is to efficiently transduce quantum states from a microwave frequency superconducting qubit through the electro-optomechanicaltransducer to the optical domain. Once upconverted to an optical signal these states can be detectedwith a linear optical detector, or alternatively via photon counting [62, 81].
Since quantum enabled electro-optic transduction is still in the early stages of development[82], and has yet to be achieved outside of the context of Gaussian fields [65], it is not yet possible
to demonstrate non-classical optical photon statistics that are correlated with the state of a superconducting qubit. Nonetheless, by sending classical states from a circuit QED system throughthe electro-optomechanical transducer, one can still learn both about the properties of the transducer itself, and the effect of the transducer on the qubit. It is in this framework of classical qubit
readout that I operate in order to learn about the integration of superconducting qubits with theelectro-optomechanical transducer.
In this section I outline the necessary properties of a circuit QED system for coupling to
the electro-optomechanical transducer, as well as the relevant performance metrics for the electrooptomechanical transducer itself. Figure 3.1 provides an experimental schematic demonstrating

21
the important elements of the optically mediated superconducting qubit readout experiment. The
experiment can be broken into four main components, with the first being the circuit QED system(see Figure 3.1(a)), containing a transmon qubit dispersively coupled (see Section 4.2 for an exact
definition of dispersive coupling) to a resonant microwave cavity. The isolation and coupling elements (see Figure 3.1(b)) enable proper routing of signals between the circuit QED system and theelectro-optomechanical transducer, and isolate the circuit QED system from the strong microwave
pumps required for transduction. The electro-optomechanical transducer itself is formed by amechanically compliant silicon nitride membrane, which simultaneously modulates the resonancefrequency of the optical Fabry-Perot cavity and a microwave frequency inductor-capacitor (LC)circuit (see Figure 3.1(c) and Section A.2). Finally, signals emitted by the electro-optomechanicaltransducer are detected via optical heterodyne detection (see Figure 3.1(d)).

3.2

Efficiently transducing signals from a circuit QED system

3.2.1

Bandwidth

The transducer has finite bandwidth, which limits the frequency span, and hence the minimum time duration of signals that it can efficiently transduce. The bandwidth of the transducer isgiven by ΓT = Γe + Γo + γm , and ideally signals to be transduced through the converter will havebandwidth Γsig  ΓT . Additionally, the transducer’s efficiency depends strongly on what amounts
to an impedance matching condition:ηt = 

κo,ext κe,ext 4Γe Γo
,
κo
κe
Γ2T

(3.1)

where maximal transduction efficiency ηt occurs for Γo = Γe . This requirement of Γe = Γo for
maximal ηt results from the simultaneous beamsplitter interactions, where for efficient transduction,
optical and microwave photons must be swapped into the mechanical resonator at equal rates. Thus,
if only one of Γe or Γo is limited for various technical reasons, the total bandwidth of the transducer
will be limited when trying to maximize ηt . This was exactly the case in [77, 69], where ge = 1.6 Hz
was significantly smaller than in previous devices, where for example in [56] ge = 3.8 Hz. Since

22
a

b

c
Microwave
measurement

Sapphirerod
Transmonqubit

Microwave
pump
Cancel

ωd ωc

d

LO
Opticalpump

Opticalheterodyne
detection

Input
line

Figure 3.1: Apparatus for readout of a superconducting qubit via the electrooptomechanical transducer. (a) Circuit QED system consisting of a transmon qubit dispersively coupled to a 3D coaxial quarter-wave cavity resonator. A translatable sapphire rod tunes
the frequency of the cavity in situ. Qubit preparation pulses with frequency ωd (gold), followedby readout pulses with frequency ωc (green), are injected through the circuit QED system’s input
line. (b) Isolation between the circuit QED system and the transducer is achieved using circulators, along with a directional coupler for interferometric cancellation. The directional coupler isalso used to apply the microwave pump to the transducer, while the circulator enables microwave
heterodyne measurement of reflected signals. (c) The electro-optomechanical transducer consists ofan optical cavity and a flip-chip microwave LC circuit resonator simultaneously coupled to a single
mode of a high-quality factor silicon nitride membrane with frequency ωm /2π = 1.45 MHz. (d)Optical pump and heterodyne detection scheme.

Γe ≈ 4ge2 |b̄|2 /κe , this reduced the achievable damping rate by a factor of 5.6 for a given pump power|b̄|2 , reducing the maximum achievable bandwidth of the transducer.

3.2.2

Tunable frequency

Due to the narrow bandwidth of the electro-optomechanical transducer detailed above it iscritical that signals sent from a circuit QED system be on resonance with the microwave resonanceof the transducer. Currently the electro-optomechanical transducers made in JILA are rather
bespoke devices, as the fabrication process involves many steps and separate components, followedby extensive screening of devices to find a device with ideal parameters. Furthermore, the microwave
presonant frequency of the transducer is formed by an LC circuit where ωe = 1/ LC(x) andC(x) =

o A
C(0)
=
,
d−x
1 − xd(3.2)

23
where the resonance frequency shifts significantly with the parallel plate capacitor separation d.
In fact, it is desirable for the resonance frequency to vary significantly as a function of d, so as tomaximize electromechanical coupling which scales as [83]:ge = xzp
where xzp =dωeωe= xzp ,
dx x=02d(3.3)

p~/2ωm m is the zero-point motion of the mechanical oscillator, determined by theexact geometry, mass and tension of the silicon nitride membrane [83]. Thus it is ideal for d to be
as small as possible to maximize this coupling. However, as the parallel plate capacitor geometry isformed by two separate chips that are epoxied together, it has been found that when cooling thesedevices to temperatures below 100 mK, the resonant frequency is subject to at a minimum 100 MHz
variation between thermal cycles of the same device. Furthermore, when screening devices it ispossible for the resonance frequency of the device to end up anywhere in the 4 − 8 GHz microwave
measurement band of the optical access dilution refrigerator. Thus a circuit QED system witha tunable resonance frequency is extremely important for efficiently coupling between the qubit
and the transducer. Ideally, multiple circuit QED systems are prepared that can cover the whole
microwave measurement band, and can be extensively tuned at the base temperature of the dilution
refrigerator to bring the circuit QED system into resonance with the transducer.
The necessary tunability is achieved by attaching a sapphire rod to a piezoelectric stepping
module with a maximum travel of approximately 5 mm at the base temperature of the dilution
refrigerator. By advancing this sapphire rod so that it participates more in the circuit QED cavity’s
microwave resonance, it lowers the resonant frequency of the cavity. Using this technique I was able
to fabricate two cavities: cavity A, which is tunable between 5.7 − 7.2 GHz and cavity B, tunable
between 6.8 − 8.5 GHz, covering a large fraction of the microwave measurement band. Figure 3.2
demonstrates the internal loss rate κc,int and total linewidth κc of cavity A as a function of its
resonant frequency ωc . It is clear that the internal loss remains low and roughly constant over theentire tuning range of the cavity, with κc.int /2π ≈ 4 − 10 kHz, corresponding to internal quality
factors of Qint ∼ 106 . Typically devices are pre-screened, and then one of these cavities can be24FRXSOLQJUDWHN+]



κc, int/2πκc/2π

















ωc/2π*+]
Figure 3.2: Tunable microwave cavity coupling rates. Internal loss rate κc,int and total
linewidth κc of cavity A. The external coupling rate changes significantly as a function of thesapphire rod position as it changes the distribution of the fields relative to the coupling pin—the
non-monotonic dependence on frequency is expected from finite element analysis simulations. The
blue points represent the internal loss (including the weakly coupled port, which may constitute alarge fraction of this loss) of the resonator, which remains nearly constant over the entire tuning
range of the cavity.

chosen and cooled down alongside the selected device. The non-monotonic behavior of κc as afunction of ωc (see Figure 3.2), which is dominated by changes in κc,ext , is due to the changing
field distribution in the microwave cavity as the sapphire rod is translated, and is expected fromfinite element analysis simulations. Once the frequency of the electro-optomechanical transducer isknown at low temperature, the sapphire rod can be tuned with the piezoelectric stepping module(attocube ANPz101 linear z-nanopositioner) such that ωc = ωe . See Section A.1 for technicaldetails of the circuit QED system and Figure A.2 for a cross sectional view of a CAD model of thedevice.
While the cavity frequency is being tuned, it is also important that the properties of the25superconducting qubit remain relatively unchanged over the set of possible cavity frequencies. Bychanging the resonance frequency of the cavity, the detuning between the cavity and the qubit(∆ = ωc − ωq ) changes, which alters both the dispersive shift χ, and affects the Purcell limit for
the qubit [84]: g 2
1
=
κc .
T1
∆

(3.4)

Thus it is important to check that both the lifetime and the coherence time of the qubit are notsignificantly affected by tuning the cavity. Such a measurement is shown in Figure 3.3 for cavity
A, where he T1 time of the qubit is largely unchanged over nearly the entire tuning range of thecavity.

μ1μ2'HFD\WLPH

μs

















ωc*+]





Figure 3.3: Lifetime and coherence time of the qubit contained within Cavity A. The T1
and T2 times are shown as a function of the cavity’s resonant frequency.

Interestingly, while the coherence time remains relatively high for all frequencies of the cavity,
there is a non-monotonic dependence of the T2 time of the qubit on cavity frequency ωc . While thedispersive shift between the fundamental mode of the cavity (see Figure 3.4) does have the expected
monotonic dependence, the linewidth of the cavity κc (as seen in Figure 3.2) is not monotonic, andmay explain the behavior of the T2 time of the qubit. Additionally, there could be significant
contributions to the dephasing of the qubit from higher modes of the cavity as the sapphire rod is26
moved. Unfortunately, because these modes are outside of the 4 − 8 GHz microwave measurement
band it is not possible to characterize the effects of these modes quantitatively.

Figure 3.4: Cavity A dispersive shift. Dispersive shift χ of cavity A as a function of its frequency.

3.3

Input referred added noise
Ideally the transduction process should be entirely noiseless so that only signal photons arepresent at the output of the electro-optomechanical transducer. In practice, noise needs to be
mitigated to a level where it does not significantly harm the fidelity of the state transfer, with theexact amount of added noise that can be tolerated depending upon the exact protocol being used[82]. In the case of linear detection, if more than 50% of the signal is noise such that the added noisein two quadratures nadd > 1/2 is larger than that of vacuum fluctuations, then the non-classicalproperties of a quantum state will be lost. For example, there will no longer be any negativity in
the Wigner function [85, 86], a frequently used measure of non-classicality.
In current devices [87, 69] the primary factor limiting quantum enabled transduction is noise
generated in the LC circuit from the microwave pumps required for transduction. In Figure 3.5 thenoise emitted by the microwave LC circuit Np is shown both as a function of the number of pump
photons circulating in the LC circuit nint (top axis) and as a function of Γe (bottom axis). Note,27
at Γe = 50 Hz, Np ≈ 1 photons/s/Hz, which precludes quantum transduction [82] even over a very
narrow bandwidth.
Figure 3.5 also demonstrates how increases in ge lead to direct reductions in Np . SinceΓe ∝ ge2 nint , for the same level of added noise Np , a higher Γe can be achieved when ge is increased.
In the device described in [77] ge = 2π × 1.6 Hz was significantly suppressed relative to evenpreviously fabricated devices, where ge = 2π × 3.8 Hz [56]. Ideally ge can be made large enoughsuch that it is possible to operate with Γe ≥ 1 kHz, but with the noise significantly below Np < 1.
This would require an increase of ge by approximately a factor of 10, assuming that all else remains
constant. The anomalously low ge = xzp ωe /2d = 2π × 1.6 Hz in this work is likely due to ananomalously large capacitor plate separation, equating to d ≈ 1 µm, meaning that ideally one
would like d ≈ 100 nm to reduce the effects of this noise emitted by the microwave circuit. Wafer























Np

SKRWRQVV+]



nint









Γe/2π+]









Figure 3.5: Microwave LC circuit noise. Parameter noise emitted by the microwave circuit Np
as a function of the number of photons circulating in the LC circuit nint (top axis) and Γe (bottomaxis).

bonding techniques are currently being explored to decrease the separation of the plates of thecapacitor, but thermal contraction of the many different layers used in the flip chip geometry makes
achieving this a challenge [69]. Another avenue for reducing Np is to reduce the noise emitted by

28
the microwave circuit for a given value of nint , in what amounts to essentially improving the power
handling of the LC circuit. Because the source of Np is currently not well understood this remains
an outstanding challenge.

3.4

Interaction between the added noise and the efficiency
Loss is going to be a nearly fundamental component of any physically realizable quantum

network since fiber optic cables suffer photon loss of at least 0.2 dB/km. However, in principle,
heralding protocols [40, 41, 42, 88] can tolerate arbitrarily high levels of signal loss, so long as darkcounts on the photon counter heralding the entanglement are far more infrequent than the arrivalof real signal photons. However, in practice, these dark counts impose significant constraints on
the amount of loss that can be tolerated in a remote entanglement experiment [89, 90, 91, 92].
This is because the noise at the output of the circuit QED system is the relevant metric forquantum transduction [82], but depends explicitly upon the transmission efficiency ηloss such that
NcQED = Ndet /ηloss , where here Ndet is the noise at the input of the optical detector.
Dark counts at the detector can be generated intrinsically by the photon counting module,
and in state of the art devices occur at mHz to Hz rates [93, 94, 95]. Alternatively, the electrooptomechanical transducer can itself generate noise photons, which behave equivalently to darkcounts on the photon counting module. In the JILA transducers [67, 56, 69], the source of thisnoise is primarily due to the strong microwave pump generating noise in the microwave LC circuit(see Section 3.3), which is then transduced over to the optical domain. However, there are additional
contributions from a small amount of laser induced noise [69] and finite sideband resolution causing
the transducer to act as an amplifier, which means noise must be added during transduction [74].
To illustrate how noise and loss enters the system and influences the noise at the output of thecircuit QED system NcQED , a distributed beamsplitter model can be used. As an example, suppose
there are two dominant sources of loss η1 and η2 , which can be modelled as a beamsplitters withtransmissivity ηk respectively. This model is illustrated in Figure 3.6, and at each beamsplitter,noise Ni can also enter through the dark port of the beamsplitter. I can then write the noise referred29
to the output of the circuit QED system as:
NcQED = N1 +

N2
.
η1

(3.5)

Hence the noise added later on in the transmission chain has a larger effect on the input referred
added noise than the noise added at the signal input. This occurs because the noise N1 gets
attenuated the same amount as the signal before reaching the detector. This is a rather trivial
result and is just the inverse effect of cascaded amplifiers, where due to the gain of the amplifier,input referred noise added after the initial amplifier is suppressed by the gain of earlier stages ofthe measurement chain [96].
This makes dark counts inherent to photon counters particularly damaging to the input referred added noise when transducing states from a circuit QED system to the optical domain. If thetotal transmission process has efficiency ηloss , then any dark noise at the input of the photodetector
Nd will be enhanced such that NcQED = Nd /ηloss . This also makes it highly important to reduce
the loss between the circuit QED system and the transducer, as any loss between these two devices
will also increase NcQED . This is one advantage of systems in which the superconducting qubit isintegrated onto the same chip as the electro-optic transducer [62], as it can be engineered so thatthe majority of the signal makes it to the electro-optic transducer for frequency conversion.
1-η1
signal input

1-η2
η1

η2
photon counter

N1

N2

Figure 3.6: Distributed beamsplitter model demonstrating how multiple sources of loss/added
noise contribute to the noise seen at the photon counting module/optical detector and at the signal
input.

30

3.5

Repetition rate
The repetition rate of the experiment is a critical performance metric when coupling circuit

QED systems to the electro-optomechanical transducer, as Γrep determines integration times for any
experiment that requires averaging of many trials of the signal. Furthermore, in photon counting
protocols, it also imposes a strict upper bound on the possible entanglement rates between remote
qubits. In a real heralding experiment, any achieved entanglement rates will likely be several
orders of magnitude lower than Γrep due to various sources of inefficiency reducing the probabilityof success.
Small repetition rates are actually a major limiting factor in several electro-optic transduction experiments. For example, in impressive recent results [62, 65] pulsed protocols were required
to mitigate optical heating, since optical photons can easily generate quasiparticles in superconductors, which act as a source of noise in the microwave circuit. Additionally, optical dissipationin the dielectrics of the mechanical oscillator cause heating and can elevate the mechanical modetemperature [97]. To mitigate this, the laser field can be pulsed on for only a short period of timeto avoid heating the sample. Assuming sufficiently large bandwidth, the short transduction pulse
may be quite efficient, but the effects of quasiparticle generation take up to several milliseconds todissipate, so Γrep is significantly reduced. Techniques to increase Γrep by switching to materials inwhich the superconductor has a much shorter quasiparticle lifetime are a current topic of researchin the field.
The experiments in this work are also limited to a fixed experimental repetition rate, butinstead this is due to the finite bandwidth of the electro-optomechanical transducer. The rather
unique etalon design for the Fabry–Pérot cavity prevents light from the optical cavity from scattering onto the superconductor, and combined with the use of NbTiN as the superconductor in
the microwave circuit, the transducer can be run continuously with little effect from the laser [69].
The energy decay rate of the silicon nitride membrane is determined by the total damping rateΓT , which means that one must wait for a time of at least Twait > 5/ΓT so that the energy from(a)277 THz

Γe

Γo

ωe
(b)
circuit QED
system input
transducer
pumps

31

ωm

ωm

τ

Tr

Tp

ωo

electromechanical pumpfrequency

time

optomechanical pump
optical field
amplitude
time

Figure 3.7: Frequency and time domain operation of the transducer. (a) Transducer
pumps and experimental readout signals represented in the frequency domain. The electro(opto)mechanical damping rates Γe (Γo ) are controlled by the strength of the respective pumps, both
red-detuned by ωm to transduce microwave signals to the optical domain. (b) A pulse timing
diagram illustrates the qubit (gold) and readout (green) pulses, with the experiment conductedwith a repetition time of Tr = 0.4 − 2 ms and steady-state pumps. The upconverted readout pulse
is filtered by the transducer’s frequency response (bottom row), setting the minimum repetition
time of the experiment Tr .

the pulse has decayed to less than 1% of its initial value. This is demonstrated schematically in
Figure 3.7, which shows that one must wait for the amplitude of the filtered readout pulse to
decay to a sufficiently low level before the protocol can be repeated. In the current device, sinceΓT ≤ 2π × 6.1 kHz for all measurements, this meant that the electro-optic transducer was indeed
the limiting factor in the duty cycle as the decay rate of the qubit (Γ1 = T11 = 2π × 9.3 kHz) was
larger. However in future devices, where we desire a qubit with T1 > 100 µs [98, 99, 100], thequbit will actually be the limiting factor when it comes to duty cycle. However this can be easily
addressed through active qubit reset protocols [101, 102, 103, 104, 88].

Chapter 4

Optically mediated superconducting qubit readout

4.1

Introduction
In this chapter I demonstrate optically mediated readout of a superconducting qubit, and thenuse the qubit as a non-Gaussian resource [77] to characterize the quantum efficiency of the electrooptomechanical transducer and optical measurement apparatus. This technique was originallydeveloped to characterize the efficiency of parametric amplifiers used in superconducting qubit
experiments [105, 106, 107, 108, 109], where it exploits the direct relationship between the dephasing
induced by a weak measurement of the qubit and the signal-to-noise ratio of dispersive readout.

4.2

Dispersive readout of a superconducting qubit

4.2.1

Dispersive Hamiltonian

The Jaynes-Cummings model describes the interaction of a two level system with a resonantcavity [110]. In circuit QED, the two level system is the formed by the lowest two energy levelsof an anharmonic LC circuit [111, 112, 113, 22], where the inductor consists of a single Josephson
junction and the capacitor is a standard linear capacitor. In the rotating wave approximation the
Jaynes-Cummings Hamiltonian is given by:
HJC = ~ωc â† â + ~ωqσ̂z+ ~g(âσ̂+ + â† σ̂− ).
2

(4.1)

It is frequently beneficial to operate with the atom and the resonator relatively far detuned fromeach other (ie. |∆| = |ωc − ωq |  g, κc , κq ) such that instead of resonantly swapping atomic states

33

2|α|sin(θ)
Y2

|α|

θθ

|α|
Y1

Figure 4.1: Dispersive readout of a superconducting qubit. The states |ei and |gi get mapped
onto two phase shifted coherent states in the microwave cavity.

for excitations in the cavity, the Jaynes-Cummings interaction simply dresses the states of the cavity
and the atom. To see this, the dispersive transformation can be performed such that H → U † HU .
The transformation is chosen to ’complete the square’, and diagonalize the Hamiltonian to firstgorder in ∆[114]:

U = expg∆(σ̂− â† − âσ̂ + ) ,(4.2)

which gives the dispersive Hamiltonian:


 2
g2
σ̂z
g
†
Hd = ~ ωc + σ̂z â â + ~ωq + O
.
∆
2
∆2

(4.3)

Where now the frequency of the resonator depends upon the state of the qubit, or equivalently,
the average frequency of the qubit is dependent upon the average number of photons in the cavityhâ† âi. Thus, as is shown in Figure 4.1, if a coherent state α is injected into the microwave resonatorwith frequency near ωc , the phase of the resulting output field will vary based upon the state of
the qubit.

4.2.2

Optically mediated dispersive readout

This dispersive Hamiltonian forms the basis for much of modern circuit QED [20, 115], andenables readout of the superconducting qubit through a coupled microwave resonator. Inside of
the microwave cavity, this readout process can be described by two coherent states α|gi and α|ei of34
equal amplitude but differing phase (see Figure 4.1) corresponding to the qubit being in either the
ground or the excited state. The angle θ shown in Figure 4.1 is determined by the dispersive shiftχ and the linewidth of the circuit QED system’s cavity κc :
θ = arctan

2χ
κc


(4.4)

However, by the time the coherent state propagates through the electro-optomechanical transducer and makes it to the optical detector (See Figure 3.1 for a schematic of the experimental
apparatus), the optical heterodyne voltages are significantly modified by the act of transduction.
First, the amplitude of the coherent state is reduced by loss through the measurement apparatus
ηloss . Second, added noise converts the coherent states into mixed thermal states with variancelarger than vacuum (see Figure 4.2). This noise can result from the optical detector itself, wherefor example heterodyne detection requires that noise equivalent to the variance of vacuum fluctuations be added [74], or result from the electro-optomechanical transducer imprinting noise onto thetransduced readout signal—see Section 3.3.

(a)

(b)
2ηloss|α|sin(θ)

2|α|sin(θ)
Y2
|α|

Q

|α|
θθ
Y1

ηloss|α| θ θ ηloss|α|I

Figure 4.2: Dispersive readout of a superconducting qubit with loss and noise. (a)The states |ei and |gi get mapped onto two coherent states of opposite phase in the microwave
domain with quadratures Y2 and Y1 . (b) The microwave state is converted to the optical domain
and detected on an optical heterodyne detector with voltage quadratures I and Q. Signal loss ηloss
reduces the amplitude of the coherent state, while added noise increases the variance of the detectedstate. Both effects serve to reduce the SNR of the measurement.

35

4.3

Demonstrating readout of a superconducting qubit via the electrooptomechanical transducer
I demonstrate readout of the superconducting qubit by using the electro-optomechanicaltransducer to perform optically measured superconducting qubit experiments. To initialize thequbit state, a short qubit drive pulse with frequency ωd and duration τ (gold pulse in Figure 4.3(a))
can be applied to the input line of the circuit QED system to induce Rabi oscillations in the qubit
at a rate Ωr . For τ = π/Ωr , the qubit population is inverted, and prepared mainly in the excitedstate |ei. When no pulse is applied (τ = 0), the qubit is prepared mainly in the ground state|gi. State preparation is followed by a square readout pulse of length Tp = 15 µs (green pulse in
Figure 4.3(a)). The maximum useful length of the readout pulse, and hence the minimum pulse
bandwidth, is determined by the lifetime of the qubit (T1 = 17 µs) [80]. The pulse then travels
through the cavity and is upconverted as described above, and the demodulated optical signal isdigitized and integrated to extract a single voltage encoding the state of the qubit.
I first use this protocol to demonstrate single-shot optical readout of the superconducting
qubit. By recording multiple voltage traces to form histograms of the qubit state-dependent optical
heterodyne voltage, I can estimate the single-shot probability P (e), and choose a voltage threshold
Vthresh (see dashed line in Figure 4.3(b)) to maximize the fidelity Fopt = 1 − P (e|g) − P (g|e),where P (e|g) (P (g|e)) is the probability of measuring |ei (|gi) given that |gi (|ei) was prepared.
In Figure 4.3(b) I show histograms of the optical heterodyne voltage when preparing the qubit
in either |ei or |gi. The amplitude of the microwave readout pulse incident on the circuit QED√
system’s microwave cavity is n̄r = 19 photons1/2 , and the electro-optomechanical transducer isoperated continuously with (Γe , Γo )/2π = (0.5, 2.4) kHz to transduce the emitted microwave field
to the optical domain for detection. A bimodal distribution is clearly visible in each histogram,with the two modes corresponding to the ground and excited states of the qubit.
p
The maximum optical readout fidelity of F = Fo erf( 2ηq nr ) ≈ 0.4 is consistent with themeasured quantum efficiency and a residual excited state population of 10 − 15% in the qubit.

36aCircuit QED
system input

Ωr

ωd ωc
Time

Opticalheterodyne
voltageGroundExcitedVthreshCountsb

TimeSingle-shot optical probability

Optical heterodyne voltage (V)cFigure 4.3: Single-shot optical readout of a transmon qubit. (a) A microwave drive pulse
with frequency ωd near the qubit frequency and length τ is applied to drive Rabi oscillations in
the qubit at rate Ωr . A microwave readout pulse centered around ωc is sent through the cavity
and subsequently transduced to the optical domain for optical heterodyne detection. The phase of
the optical pulse depends on the state of the qubit. Pulse widths are not to scale for clarity. (b)Histogram of optical heterodyne voltage when reading out the superconducting qubit through the√electro-optomechanical transducer. A 15 µs microwave pulse ( n̄r = 19 photons1/2 ) is applied tothe microwave cavity to optically read out the state of the superconducting qubit when preparing itin the ground state (teal curve) or excited state (purple curve), with (Γe , Γo )/2π = (0.5, 2.4) kHz.
The dashed line represents the voltage threshold Vthresh for single-shot readout. (c) Opticallymeasured Rabi oscillations with the same readout pulse and transducer parameters above.

This residual occupancy is likely due to the relatively low attenuation (51 dB) between ambient
temperature and the dilution refrigerator base plate on the microwave pump and cancellation lines[116], which is required to deliver the high-power microwave pump to the electro-optomechanicaltransducer. However, further work is needed to determine whether other aspects of the opticalaccess cryostat contribute to this thermal population in the qubit.
I can then use single-shot readout to measure Rabi oscillations of the superconducting qubit
and demonstrate the stability of the optical measurement apparatus. To measure Rabi oscillations

37
I vary the qubit drive pulse length τ and the pulse frequency ωd (Figure 4.3(c), with the same pulse
amplitude and transducer damping rates as in Figure 4.3(b)). This measurement was taken overapproximately 1.5 hours using a single threshold value chosen at the beginning of the experiment,
indicating stability of the optical cavity lock and electro-optomechanical transducer over this periodof time.

4.4

Two equivalent descriptions of linear measurement
When performing a linear measurement, the relevant figure of merit is typically the signal-

to-noise ratio. Instead of characterizing both the noise and loss of the measurement apparatus
separately, it can be useful to describe the measurement apparatus in terms of a single quantity
known as the the quantum efficiency ηq , which contains two separate factors: the transmission loss
ηloss and the contribution from the noise ηnoise . Here I show how the noise measured at the inputof the optical heterodyne detector, or equivalently at the output of an effective amplifier with gain
ηloss (see Figure 4.4), can be expressed as a contribution to the quantum efficiency ηq = ηloss ηnoise ,in which the optical readout apparatus can be modeled as an effective beamsplitter (see Figure 4.4).

1-ηq
ηq

signal input

photo detector

signal input

ηloss

signal output
Ndet

Figure 4.4: Effective beamsplitter model. A linear measurement can be characterized as aneffective beamsplitter with efficiency ηq (top), or can equivalently be mapped onto to an effective
amplifier model characterized by transmission efficiency due to loss ηloss and noise added at theoutput of the effective amplifier Ndet (bottom). Both models are equivalent descriptions of linear
measurement and transduction.

The voltage records obtained from the optical heterodyne detection of the upconverted qubit

38
readout pulse can be written in the form:

where

I|ki (t) =


p √
Go
κc ηloss Re(α|ki (t)) + ζ̂I (t)(4.5)

Q|ki (t) =


p √
Go
κc ηloss Im(α|ki (t)) + ζ̂Q (t) ,

(4.6)

√
Go is an overall gain factor and the index k = {g, e} labels whether the qubit was preparedin the ground state or the excited state. ηloss includes all sources of loss between the circuit QED
system and the ideal optical detector, while ζ̂I and ζ̂Q are noise operators whose autocorrelation
functions include contributions from vacuum fluctuations, the added noise of an ideal heterodyne
detector, and transducer added noise Nt :1hζ̂I (t)ζ̂I (t0 )i = hζ̂Q (t)ζ̂Q (t0 )i = (1 + Nt )δ(t − t0 ).
2

(4.7)

Eqs. (4.5), (4.6), and (4.7) define the amplifier model described in Figure 4.4 in which thetransduction process is characterized by an efficiency ηloss and total two-quadrature noise Ndet =1 + Nt at the output of the ideal amplifier. However, as described above, for qubit readout it isoften more convenient to combine these metrics into a single figure of merit. To this end, I can
rescale Eqs. (4.5), (4.6) to obtainκc ηloss
Re(α|ki (t)) + ζ̃I (t)1 + Ntrpκc ηloss0
Im(α|ki (t)) + ζ̃Q (t) ,Q(t) = Go
1 + Ntp
I(t) = G0o

where

p

G0o =

p

r

(4.8)

(4.9)

Go (1 + Nt ) is a modified overall gain factor, and ζ˜I and ζ˜Q are noise oper-

ators containing only contributions from vacuum fluctuations and ideal heterodyne detection:hζ̃I (t)ζ̃I (t0 )i = hζ̃Q (t)ζ̃Q (t0 )i = 12 δ(t − t0 ). From Eqs. (4.8) and (4.9) it is clear that ηq can be
defined asηq = ηloss ηnoise = ηloss ×

1.
1 + Nt(4.10)

Thus through a simple rescaling, the amplifier model can be directly mapped onto an effective
beamsplitter describing the measurement apparatus.

39
4.4.1

Modeling the transducer’s added noise

To obtain an expression for Nt , one can begin with the spectral density of the noise at the
input of the optical heterodyne detector, which is measured directly at each pair of Γe , Γo values,
corresponding to a total electro-optomechanical linewidth of ΓT = Γo + Γe + γm . When normalized
to the measured shot noise of the LO beam, the single-quadrature power spectral density in unitsof photons/s/Hz is given byΓ2 St (0)1Sout (ω) = (1 + Sb ) + T Γ2,2
4 T + ω2

(4.11)

4

where St (0) is the amplitude of the Lorentzian frequency response on resonance. The first term isthe sum of equal contributions from vacuum fluctuations and the added noise of an ideal heterodyne
detector, Sb encodes a small contribution from phase noise on the optical pump, and the Lorentzianterm is due to fluctuations in the motion of the membrane imprinted on the reflected optical pump
[69, 117]. The noise operator autocorrelation functions are obtained from the inverse Fouriertransform of this spectrum:hζ̂I (t)ζ̂I (t0 )i =1
ΓT0
(1 + Sb ) δ(t − t0 ) +St (0)e−ΓT |t−t |/2 ,2
4(4.12)and hζ̂Q (t)ζ̂Q (t0 )i = hζ̂I (t)ζ̂I (t0 )i.
To obtain a single value encoding the state of the qubit, I perform a weighted integral of thetwo quadrature voltage records of the form [106]Z TintV|ki (Tint ) =
WI (t)I|ki (t) + WQ (t)Q|ki (t) dt,(4.13)

0where I integrate the signal for a time Tint long enough such that all of the energy in the qubit
readout pulse has decayed. The weights WI (t) and WQ (t) are chosen to optimize the SNR of thequbit readout [80] and to simplify the calculation of the quantum efficiency [106]:WI (t) = hI|ei (t)i − hI|gi (t)i

(4.14)WQ (t) = hQ|ei (t)i − hQ|gi (t)i.

(4.15)

40
The variance of this integrated voltage is then given by2h∆V|ki(Tint )i =Z Tint Z Tint0
WI (t)WI (t0 )hζ̂I (t)ζ̂I (t0 )i + WQ (t)WQ (t0 )hζ̂Q (t)ζ̂Q (t0 )i dt dt0(4.16)

0Thus, I can define the total noise (in two quadratures) Ndet at the input of the ideal optical
heterodyne detector (where any inefficiency in the real optical detector has been included in ηopt )in units of photons/s/Hz as
Ndet = 1 + Nt ,(4.17)

where Nt is the noise added by the transducer,
ΓT St (0)Nt =4G(Tint )Z Tint Z Tint00
WI (t)WI (t0 ) + WQ (t)WQ (t0 ) e−ΓT |t−t |/2 dt dt0 + Sb ,(4.18)0and
1
G(Tint ) =2

Z TintWI (t)2 + WQ (t)2 dt.

(4.19)

0

4.5

Characterizing the quantum efficiency: theory

4.5.1

Superconducting qubit readout signal-to-noise ratio

In a dispersive measurement, a coherent state |αi is entangled with the state of the qubit. If
initially the qubit is prepared in a superposition state |ψq i = √12 (|ei + |gi), its interaction with thedriven cavity mode causes it to evolve into the state1|ψm i = √ (|ei|αe i + |gi|αg i) ,2

(4.20)

where |αe i and |αg i are coherent states with equal magnitude but phases shifted by θ± = ± arctan 2χ/κc ,√with the + (−) sign corresponding to the qubit in |gi (|ei). The readout amplitude n̄r = |α| sin θdetermines the separation in phase space between these two coherent states. The degree to which
the two Gaussian distributions can be resolved in phase space is quantified by the signal-to-noise
ratio (SNR): loss reduces the SNR by narrowing the phase space separation, while noise reduces
the SNR by increasing the variance of each distribution. The SNR is defined as
SNR =|µe − µg |,σge(4.21)

41
where σge = σe = σg is the standard deviation of the Gaussian-distributed optical heterodyne
voltage corresponding to the qubit in either the ground or the excited state, and µg and µe are
the mean values of the two distributions. Noise Nt gets added by the transducer, and some of thesignal is lost such that |α| → ηloss |α|; thus the SNR for a steady state readout tone can be writtenas√
2 2ηloss |α| sin θ
√
SNR =
.
1 + Nt4.5.2(4.22)

Signal-to-noise ratio for a general pulsed measurement

For pulsed measurements the signal-to-noise ratio of qubit readout becomes slightly more
complicated, where now using Equations 4.13, 4.14 and 4.15 and one can find that the readout
contrast is given by:
S = |hV|ei (Tint )i − hV|gi (Tint )i|.
Z TintS = κc ηq G0|αe (t0 ) − αg (t0 )|2 dt0 .

(4.23)(4.24)

0

The noise was calculated in Equation 4.16, but can be simplified by using the vacuum noise autocorrelation functions hζ̃I (t)ζ̃I (t0 )i = hζ̃Q (t)ζ̃Q (t0 )i = 12 δ(t − t0 ) along with the weights in Equations 4.14
and 4.15, giving the noise:s
N = G01
κc ηq
2

Z Tint|αe (t0 ) − αg (t0 )|2 dt0 .

(4.25)

0Thus the signal-to-noise ratio for any arbitrary pulsed measurement is given by:s
Z Tint
SNR = 2κc ηq|αe (t0 ) − αg (t0 )|2 dt0 .

(4.26)

0

4.5.3

Measurement induced dephasing

Reading out the state of the qubit through the cavity necessarily causes fluctuations in thefrequency of the qubit due to the dispersive interaction between the qubit and the cavity—see42Equation 4.1. This leads to measurement-induced dephasing [118], where a projective (strong)measurement completely destroys the coherence of any initial superposition of the qubit. Usinga master equation approach [118, 119] it can be shown that the effect of the measurement pulse
dephases the qubit such that:
Z TintΓφ = 2χIm(αe∗ (t0 )αg (t0 ))dt0 .

(4.27)

0

As was shown in [106], one can use the equations of motion for the qubit cavity:α̇e (t) = −i(∆ − χ)αe (t) −√κcαe (t) + κc αin
2(4.28)α̇g (t) = −i(∆ + χ)αg (t) −√κcαg (t) + κc αin ,2(4.29)

where ∆ is the detuning from the microwave resonator’s center frequency. Along with the additional
requirement that all energy from the readout pulse decays on a timescale shorter than Tint , one canthen directly relate Γφ to the signal-to-noise ratio of the qubit measurement such that:
Z TintΓφ = 2χκc
Im(αe∗ (t0 )αg (t0 ))dt0 =20

Z Tint|αe (t0 ) − αg (t0 )|2 dt0 .

(4.30)

0

Using the above equation to eliminate the integral term in Equation 4.26 gives a very simple
expression for the quantum efficiency [106]:
ηq =

4.6

4SNR2
Γφ

(4.31)

Measurement-induced dephasing: experimental results
The off diagonal elements of a density matrix are the so-called coherence elements, which

describe the (basis dependent) level of superposition between different quantum states. A measurement of length Tp has the effect of reducing the amplitude of these coherence elements [106, 109]by an amount:ρ̂ge (Tp ) = ρge (0)e−Γφ (Tp ) .

(4.32)

To estimate |ρge /ρge (0)| one can perform a Ramsey sequence, where the coherence elements are
given by |ρge | = h|σz |i/4. If one then injects a weak measurement pulse of length Tp into the Ramsey

43sequence (see Figure 4.5(a)), and then measures the resulting amplitude of the Ramsey oscillations(see Figure 4.5(b) and Figure 4.6), this will give an estimate of the decay of ρge . Figure 4.5(b) shows
the results of these measurements, using the microwave readout apparatus and varying the strengthof the weak 15 µs readout pulse. I observe a clear Gaussian decay of ρge as a function of readout
amplitude√

nr . This measurement-induced dephasing quantifies the amplitude of the readout pulse

inside of the circuit QED system’s microwave cavity, while the SNR of the qubit readout quantifies
the distinguishability of the two (thermal) coherent states at the optical detector.
(a)proj. meas.
weak measurement
τ

time

(b)

Figure 4.5: Measurement induced dephasing. (a) Protocol for measurement-induced dephasing
calibration. A weak measurement pulse is injected into a Ramsey sequence. The Ramsey sequenceis then followed by a strong projective measurement of the qubit. (b) The coherence of the qubit√
ρge decays as a Gaussian function of the weak measurement amplitude n̄r . The points are data,
while the line is a Gaussian fit.

4.7

Characterizing the quantum efficiency of the optical readout apparatus
Having established the quantum efficiency as an unforgiving performance metric describing

the optical measurement apparatus I plot ηq as a function of the total electro-optomechanicaldamping Γe + Γo in Figure 4.7, where Γe is varied at several different fixed values of Γo . Using the44











√ nr


ρge



í

í

í

í











(μs)





5DPVH\WLPH

Figure 4.6: Measurement induced dephasing Ramsey fringe data. As the measurement
strength is increased the amplitude of the Ramsey oscillations decreases as a Gaussian function√of measurement amplitude nr . The Ramsey time (the separation between π/2 pulses) starts at
20 µs since the measurement pulse must fit within the Ramsey sequence.
.

state-space model (see Section 2.4) I perform a fit to ηq as Γe + Γo is varied, with ηmic as the only
free parameter. The fit results are plotted as dashed lines in Figure 4.7(c). This yields an apparent
app
microwave loss of ηmic
= 0.17, in slight tension with ηmic = 0.34 inferred from the microwave

measurement described in Section 4.8.4. This discrepancy may be due to slight drifts in the opticalcavity mode matching or microwave apparatus gain over time since the experiment was performed
over the course of approximately 3 months. Two notable features are present in the quantum
efficiency of the optically mediated superconducting qubit readout. First, the efficiency drops off
significantly for Γe  Γo , which is due to both mismatched transduction (see Equation 4.37), as
well as small transduction bandwidth ΓT = Γe + Γo + γm . The efficiency also plateaus for large
Γe due to Γe -dependent LC circuit loss and noise in the LC circuit that increases with Γe (see

45
Figure 3.5).

Figure 4.7: Quantum efficiency of the optically mediated qubit readout. The quantum
efficiency is calibrated by comparing the signal-to-noise ratio of an optical qubit measurement to
the total measurement-induced dephasing. For 20 Hz ≤ Γe /2π ≤ 1100 Hz, Γe is varied for several
fixed values of Γo and perform qubit readout measurements. The points are data from thesemeasurements, while the dashed lines are a model using independently measured parameters of
the electro-optomechanical transducer. The model includes all sources of inefficiency such as loss,
added noise, and finite transduction bandwidth. All error bars represent one standard deviation.

4.7.1

Technical details of the quantum efficiency calibration

To control the readout amplitude√
nr a drive voltage V is varied via an IQ mixer. The SNR

and Gaussian decay of ρge can be recast in terms of this drive voltage as:
SNR = aV

(4.33)
V2

ρge = ρge (0)e− 2σ2(4.34)

46
2

V
which implies that Γφ = 2σ
2 . Using Equations 4.33 and 4.34 in Equation 4.31 then gives a simple

expression for the quantum efficiency:ηq =σ 2 a2,2(4.35)

where in this convention an ideal heterodyne detector would have ηq = 1. Thus to measure thedata in Figure 4.7, I first measure the effects of a measurement pulse on the coherence of the qubit
as in Figure 4.5. This can be completed using the microwave measurement apparatus, which issignificantly more efficient than the optical measurement apparatus, as this measurement is only
concerned with the effects of the pulse on the qubit. Second, for each Γe and Γo point, I sweep themeasurement voltage V , and record histograms when preparing the qubit in both the ground andthen the excited state (see Figure 4.3(b)). Finding the SNR as a function of V then enables theextraction of a, and subsequently ηq .

4.8

Contributions to the quantum efficiency
Since the quantum efficiency describes all sources of loss and added noise resulting from theoptically mediated readout, there exist many different contributions to the final value, all of which
are described in this Section. The quantum efficiency of the optically mediated qubit readout
apparatus can be split into seven main components, ηq = ηbw ηt ηG ηmic ηopt ηcav ηnoise . I describe eachcontribution below, and the contributions are also summarized in Table 4.1. Note the first threecontributions to the quantum efficiency ηbw , ηt and ηG are theoretically calculated from the known
transducer operating parameters and are discussed in Section 2.4. The rest of the contributions
are technical sources of loss and added noise.

4.8.1

Bandwidth limitations: ηbw

A significant contribution to ηq is due to the mismatch in bandwidth between the transducer and the readout pulse emitted by the qubit cavity. In the rotating wave approximation,

47
Table 4.1: Contributions to the quantum efficiency ηq = ηbw ηt ηG ηmic ηopt ηcav ηnoise . Numericalvalues are not reported for contributions that depend on Γe and Γo .
Parameter
Finite bandwidth
Transducer efficiency
Transducer gain
Microwave transmission loss
Optical detection efficiency
Circuit QED system cavity loss
Added noise

Symbol
ηbw
ηt
ηG
ηmic
ηopt
ηcav
ηnoiseValue
see Equation (4.36)
See Equation (4.37)
See Equation (4.38)
ηmic = 0.34
ηopt = 0.28
ηcav > 0.96
−1
ηnoise = (1 + Nt )−1 = Ndet

approximating the circuit QED system’s output as a square pulse with width Tp , obtaining
1 − e−ΓT Tp /2
ηbw ≈ 1 − 2.
(4.36)
ΓT Tp
Using the state space model outlined in Section 2.4, I solve for the full dynamics without the above
approximations to give a contribution to the efficiency in the range 0.02 < ηbw < 0.19 over the rangeof ΓT values plotted in Figure 4.7. It is possible to increase ηbw significantly by either increasing
the length of the signal emitted by the circuit QED system Tp , or by increasing the total bandwidthof the electro-optomechanical transducer ΓT .

4.8.2

Transducer efficiency: ηt

The relatively low electromechanical coupling in this device (ge = 2π × 1.6 Hz) limits the
maximum achievable electromechanical damping rate to Γe = 2π×1.1 kHz. To maximize bandwidth
the transducer is often operated in a mismatched mode with Γo  Γe , but this comes at a cost to
the narrowband signal efficiency of the transducer [69]:ηt = 

κo,ext κe,ext 4Γe Γo
,
κo
κe
Γ2T

(4.37)

which sets an upper bound on the efficiency with which broadband signals can be transduced.
Equation (4.37) also depends on Γe implicitly due to the pump power-dependent LC circuit loss.
This contribution is responsible for the sharp drop in efficiency for Γe  Γo on each curve in
Figure 4.7, and is largely responsible for the plateau at high Γe for each fixed value of Γo .

48
4.8.3

Transducer gain: ηG

Due to the finite sideband resolution of the electro-optomechanical system [79], the transducer
has gain [67]

ηG =

1+

κe
4ωm

2 !


1+

κo
4ωm

2 !
.

(4.38)

This two-quadrature gain is undesirable for transduction, as it is necessarily accompanied by unwanted added noise [74], but it is a small effect, varying in the range 1.3 < ηG < 1.5 with microwave
pump power. This gain is intrinsically linked to the added noise, as the transducer can be be described as a phase preserving amplifier with gain ηG [74].

4.8.4

Microwave transmission loss: ηmic

The experimental system also permits qubit readout through a microwave readout apparatus
by reflecting the readout signal off of the electro-optomechanical transducer (shown in Figure 3.1).
Its quantum efficiency ηqmic is defined to include all sources of loss and noise that affect the readout
pulse as it emerges from the circuit QED system’s cavity, reflects off the transducer’s LC circuit,and is detected using the microwave heterodyne measurement apparatus. The quantum efficiencyof the microwave apparatus can be split into four separate components such that:ηqmic = ηmic ηHEMT ηnoise ηref .

(4.39)

The first two parameters characterize transmission losses, where ηmic represents the loss between
the circuit QED system and the electro-optomechanical transducer, and is the parameter I wish
to extract from these quantum efficiency measurements. ηHEMT accounts for the loss between theelectro-optomechanical transducer and the high-electron-mobility transistor (HEMT) amplifier, andis characterized by an independent electro-mechanical temperature sweep to be ηHEMT = 0.26 [69].
The contribution from added noise ηnoise takes into account the added noise of the HEMT andnoise emitted by the LC circuit. The circuit reflection loss ηref is a function of the power dependent

49
linewidth of the LC circuit since in the steady state:ηref = |S11 |2 =∆2 + (κe − 2κe,ext )2 /4,∆2 + κ2e /4(4.40)

where ∆ = ω − ωe is the detuning of the probe field. Since the circuit is close to critically coupledat the maximum incident pump power of -45 dBm (Γe = 2π × 1.1 kHz), with κe /2κe,ext = 0.97, thisLC circuit reflection loss is a large contribution to ηqmic at high pump powers and is responsible forthe shape of the curve in Figure 4.8.
The signal emitted by the qubit cavity has finite bandwidth, and as a result a significant
fraction of the energy of the qubit readout pulse will be detuned from the resonator. To account forthis I estimate the energy of the circuit QED system’s output field by solving the set of equations
for the output of the qubit cavity:

κc 
√
√
α̂˙ = −i(∆ ± χ) −α̂ + κc,w α̂in,w (t) + κc,ext α̂in (t)2κc,w α̂(4.42)

κc,ext α̂,(4.43)

α̂out,w + α̂in,w =α̂out + α̂in =√√(4.41)

where the qubit readout pulse is applied to the weak port with coupling rate κc,w and the otherport of the qubit cavity directs energy towards the electro-optomechanical transducer with coupling
rate κc,ext . The parameter ∆ is the detuning of the pulse from cavity resonance, while the + (−)
sign corresponds to the case where the qubit is prepared in the ground (excited) state.
α̂in,w (t) = α̂in,o (Θ(t) − Θ(t − Tp ))(4.44)

is described by a square pulse of length Tp , where Θ(t) is the Heaviside step function. Equivalently
the equations of motion can be solved in reflection off the electro-optomechanical transducer where
the fields are described by:

κe 
√
˙
b̂ = − i∆ +b̂ + κe,ext b̂in2√
b̂out + b̂in = κe,ext b̂,(4.45)(4.46)

50
and the efficiency ηref for a broadband pulse can be calculated as:R|b̂out (t)|2 dt
Rηref =.
|α̂out (t)|2 dt(4.47)

Measuring ηqmic allows for the retrieval of the microwave transmission loss ηmic between thecircuit QED system and the transducer. As shown in Figure 4.8, ηqmic decreases significantly withincreasing Γe . This data is fit to a model (purple curve in Figure 4.8) that includes the effects of Γe dependent LC circuit reflection loss, noise emitted from the microwave port of the transducer, andindependently calibrated microwave measurement apparatus added noise referred to the transducer
output [69], with ηmic as the only free parameter, yielding ηmic = 0.34 (4.7 dB). This relatively highlevel of loss can be mitigated by reducing the total number of microwave connectors in the signal
path, switching to superconducting cables, and removing filters from the signal path and placing
them instead on the pump and cancel lines.

4.8.5

Optical detection efficiency: ηopt

The optical detection efficiency is given by the product of factors encoding optical transmission losses between the transducer and the balanced heterodyne detector, the inefficiency ofthe heterodyne detector itself, and the detector’s dark noise: together these three factors yieldηopt = 0.28. This figure excludes optical cavity losses and imperfect mode matching, which areincluded in Equation (4.37) as they are an inherent part of the electro-optomechanical transducer.

4.8.6

Circuit QED system cavity loss: ηcav

Through the measured attenuation on the lines, the sum of the of the weak port coupling andthe internal loss to κc,int + κc,w < 2π × 15 kHz can be bounded. Thus ηcav = 1 −κc,int +κc,w> 0.96κcdoes not significantly affect ηq . Although there are no detailed measurements as a function of theresonant frequency of cavity B, from measurements on cavity A, the loss is well below 15 kHz overthe entire tuning range of the cavity—see Figure 3.2.

51
PRGHO

TXELWPHDVXUHPHQW


ηqmic










í



í



Γe/2πN+]

í







Figure 4.8: Characterization of the quantum efficiency of the microwave readout apparatus. The microwave readout efficiency ηqmic is measured as a function of the electromechanicaldamping rate Γe . The points are data, while the line is a model including partial absorption of
the readout pulse by the pump power-dependent reflection loss of the LC circuit, power-dependent
added noise, fixed transmission losses and the independently measured added noise of the microwave
heterodyne measurement apparatus. The shape of the curve is dominated by power-dependent LC
circuit reflection loss, with the quantum efficiency of the microwave readout apparatus approachingzero at high power because the LC circuit is nearly critically coupled. From this, I estimate 4.7 dBof loss (ηmic = 0.34) between transducer and the circuit QED system.

4.9

The transmission loss and comparison to other results
For comparison to other results, I can independently measure the output noise of the trans-

ducer Nt , and divide out its contribution to the quantum efficiency (see Section 4.4) to retrieve thetransmission loss:
ηloss = ηq /ηnoise = ηq (1 + Nt )(4.48)

This transmission loss is an important figure of merit as it ultimately will contribute to an upperbound on possible entanglement rates between remote qubits (see Section 3.5), and it represents the52practical amount of signal that one has transduced all the way to the end of the optical measurement
apparatus. I plot the transmission efficiency as a function of Γe + Γo , where a maximum value ofapproximately 2 × 10−3 is achieved.
While the above result for ηloss may seem low, especially when compared to previous resultsfrom JILA where the transducer efficiency was ηt = 47 % [56], ηloss takes into account all sourcesof inefficiency from the transduction process. Thus ηt is only a small contribution to ηloss =ηbw ηt ηG ηmic ηopt ηcav , since it contains many different contributions, nearly all of which are far fromunity. This technique of characterizing the total transmission loss contrasts previous results fromJILA [67, 56, 69], as well as other groups [65] where the performance of only the electro-optic
transducer itself was characterized. It is perfectly valid to characterize just the performance of thetransducer itself (characterized by ηt ), but it does not capture all of the sources of loss that will be
relevant for networking remote superconducting qubits.
In Table 4.2 I compare the results shown in this work with other recent results in the field.
The important figures of merit are the transmission efficiency ηloss , the input referred added noiseof the transducer, the repetition rate of the experiment, and whether qubits were integrated. Notethat the integration of qubits entails extra experimental constraints, and can significantly reduce theefficiency and increase the added noise. For example [69] had qubits integrated with the transducer,
but the qubit was unused during the characterization of the transducer, which moves the reference
plane for the efficiency and added noise up to the input of the converter—see Figure 4.9. That fact,
combined with the use of narrowband signals, lead to orders of magnitude higher efficiency andorders of magnitude lower added noise than in [77], which used the same experimental apparatus.
This is why the results that do no integrate qubits [69, 65] may seemingly have significantly better
performance than the two lone results that have integrated qubits into the experiment [77, 62].

53

Table 4.2: Recent electro-optic transduction results
Referenceηloss

Delaney et al. (2021), [77]Mirhosseini et al. (2021), [62]Sahu et al. (2021), [65]Brubaker et al. (2021), [69]Forsch et al. (2020), [60]

2 × 10−3
9 × 10−5
0.09-0.15
0.47
1.6×10−5

Added
noise
at input (photons/s/Hz)
740
0.6
0.16
3.2
> 100000

Repetition Integrates
rate (kHz) qubits
2.5
0.1
0.01-0.5
continuous
continuous

yes
yes
no
no
no

ηloss when characterizing the transducer

ηloss when characterizing the qubit measurement apparatus
Figure 4.9: Differences in efficiency characterization. The large orange box highlights allof the components that are included when calibrating ηloss of the qubit measurement apparatus[77]. While the small purple box highlights the efficiency number that is calibrated when simply
characterizing the electro-optic transducer [69, 56, 67]. The different reference planes are usedbecause all sources of loss are important when sending signals from a superconducting qubit to an
optical detector, but when characterizing the performance of only an electro-optic transducer it isimportant to characterize the performance of the device itself.

54

ηloss × 103


















(Γe + Γo)/2πN+]

Figure 4.10: Transmission loss. Transmission loss ηloss (scaled by a factor of 103 ) as a functionof the total electro-optomechanical damping rate ΓT = (Γe + Γo )/2π. ηloss includes all sources ofloss between the circuit QED system and the optical heterodyne detector at the end of the opticalmeasurement apparatus. The points are data from qubit and optical noise measurements. Thelines are the independently measured and calculated efficiency using only linear measurements.

Chapter 5

Characterization of backaction from the electro-optic transducer

5.1

Motivation
Backaction is fundamental to quantum mechanics, as the act of measurement necessarily

involves interaction between the system of interest and the detector, and according to the Heisenberg
uncertainty principle, as you learn more about an observable (for example the position of a particlex̂) there becomes more uncertainty about the value of the conjugate observable (in this case, themomentum p̂). No matter the system, the backaction depends fundamentally upon the strength ofthe measurement, and the system upon which the backaction is imparted is agnostic as to whether
the measurement is being performed by an experimentalist carefully monitoring the signal, or by theenvironment, through degrees of freedom that the experimentalist has no access to. It is this later
form of ’measurement’ backaction by the environment that is most problematic in circuit QED,
as thermally generated photons from noise propagating down signal and control lines [120, 121]
can make it into the circuit QED system, and ’measure’ the state of the qubit. It is also thisunintentional injection of photons into the circuit QED system by the electro-optic transducer,
either coherently from the microwave pumps, or through heating, that this chapter is concerned
with.
In the context of electro-optic transduction, backaction is an extremely important figure of
merit. An electro-optic transducer in principle can impart no backaction, as transduction is apurely unitary operation (see Section 2.2 and [122]). However, due to experimental non-idealities,
including accidental measurement, or heating by laser and/or microwave pumps, backaction can be56rather significant and is an important performance metric for a real electro-optic transducer.
As an example, the microwave pump used to mediate the transduction process in the experiments shown in this thesis has a maximum incident power of Pinc = −45 dBm, which if it wereincident on the circuit QED system would result in ∼ 2×108 photons in the circuit QED system’s microwave cavity. This level of power in the circuit QED system vastly exceeds ncrit = ∆2 /4g 2 = 302,
the photon number at which the dispersive approximation breaks down [123, 22]. Thus, not only isthere significant potential for backaction on the qubit from the pumps, it is possible to drive the thequbit out of the dispersive regime, driving the qubit out of its computational basis and effectively
rendering it useless. Additionally, if optical photons are incident on the superconducting qubit,or
significantly heat the substrate, as was the case in [62], then quasiparticles will be generated, also
effectively rendering the qubit unusable for a several millisecond time scale until the quasiparticles
have decayed—preventing continuous operation of the transducer. In both of these cases this ismuch worse than backaction—which simply erases phase information—and instead is destructive
to the state (and even the existence) of the qubit during operation of the electro-optic transducer.

5.2

Backaction in circuit QED
The coherence time (T2 time) of a qubit describes the characteristic time that phase infor-

mation can be stored in a qubit, and is determined both by its lifetime and any additional sourcesof dephasing Tφ , such that:
1
1
1
=
+
.
T2
2T1 Tφ(5.1)

In circuit QED, shot noise, which arises from the discrete nature of photons, randomly shifts theresonance frequency of the qubit coupled to the microwave cavity through the dispersive interaction.
This shot noise can result from measurement induced dephasing, where a coherent state dephases
the qubit, and will be discussed in Section 5.4.1, and has been studied in both circuit QED [118]and cavity QED [124, 125, 126]. Additionally, as will be discussed in Section 5.4.2, dephasing canoccur due to thermal photons incident on the circuit QED system [127], with this thermal dephasing

57
being one of the primary factors limiting coherence times in circuit QED systems today [120, 121].

5.2.1

Implications for the remote entanglement of qubits

Preserving the qubit in its prepared state is particularly important for the implementation ofheralding protocols when encoding qubits in bosonic fields [128, 88, 129], or for heralding remoteentanglement between two spatially separated qubits [43]. For example, using diamond NV centers[130, 131], a memory qubit is required to store the quantum state for entanglement distillation
between remote qubits. However, if the operation of the transducer disrupts the state of thememory qubit, then this will also certainly disrupt the entanglement between the remote qubits.
A simple example of this, more centered around transmon qubits, involves driving the blue
sideband transition between the microwave cavity at frequency ωc and qubit at ωq to generate
single photons [132, 133, 134, 88, 135]. This driving field induces joint Rabi oscillations between
the states |e, 1i and |g, 0i. Thus the joint system can be prepared in|ψi = sin θ|e, 1i + cos θ|g, 0i.

(5.2)

Operating in the regime where θ is small [43], and assuming the standard single cavity circuit QED
regime, where the qubit has a much longer lifetime than the cavity such that κc  1/T1 , |e, 1iwill decay to the state |e, 0i at a rate κc  1/T1 . Thus, if a photon counter detects the emitted
photon—after having traveled through the electro-optic transducer—this heralds the state |ei inthe qubit. If this is completed simultaneously with two identical remotely located systems, and thephotons are injected into a 50-50 beamsplitter before detection, then this is a basic protocol [43]for generating heralded Bell states
1|ψBell i = (|e, gi + |g, ei),
2

(5.3)

where the fidelity of the bell state can be exchanged for the success rate, and the protocol isinsensitive (in terms of fidelity) to photon loss. However, if the act of transducing the photons
destroys the state of the qubit, then entanglement cannot be heralded between the two remote
qubits.

58

5.3

Isolation and filtering to reduce backaction
The cryogenic portion of the experiment consists of two interconnected modular systems,

the first being the electro-optic transducer, in which microwave light can be coupled through an
SMA port, and optical light can be coupled in through free space. The other device is a circuit
QED system, where the microwave cavity is dispersively coupled to the qubit, and energy fromthe microwave cavity leaks out a port strongly coupled to a transmission line directed towards theelectro-optic transducer. These two modular systems are connected via a superconducting coaxial
transmission line through a single-junction circulator connected in series with a triple-junction
circulator (see Figure 5.1). This provides a total of 63 dB of isolation to shield the circuit QED
system from the strong microwave pump (up to Pinc = −45 dBm incident on the LC circuit), which
is routed to the transducer through a directional coupler. The 63 dB of isolation is significantly
less than the nominally specified value of 77 dB, and is likely due to unintended standing wave
resonances between the circulators that reduce total amount of nonreciprocity.
The circulators are quite important for proper operation of the experiment. First, the isolation of the strong microwave pumps is critical to prevent dephasing of the qubit by the electro-optic
transducer. Secondly, the circulators are needed for signal routing, so that signals reflected off ofthe electro-optic transducer can be routed towards a microwave amplifier for detection.
However, the circulators alone do not provide sufficient isolation between the circuit QED
system and the transducer, and thus interferometric techniques must be used to reduce the amplitude of the microwave pump before it reaches the circulators. To achieve this additional isolation
I send a cancellation tone through the second port of the directional coupler (labeled “cancel” inFigure 5.1(a)) to interferometrically cancel [136] the reflected microwave pump. By sending a signalof equal amplitude, but opposite phase to that of the reflected microwave pump, the signals will
coherently cancel each other to null the field propagating towards the circuit QED system. The
total amount of cancellation is determined by the degree to which the arms of the interferometercan be balanced.

59

microwave
measurement(a)cancelincident
microwave
pump

Pref

reflected microwave pump(b)

cancellation tone
reflected microwave pump

Pref

Figure 5.1: Isolation and pump cancellation scheme. (a) Circulators (total isolation of 63dB) and interferometric cancellation are used to prevent the strong microwave pump from reaching
the circuit QED system and dephasing the qubit. Cancellation is achieved by sending a microwave
cancellation tone with amplitude equal to that of the reflected microwave pump but opposite phase
into the second arm of the directional coupler. The phasor sum of this cancellation tone and thereflected microwave pump determines the pump power Pref propagating towards the circuit QED
system. (b) The directional coupler acts as an interferometer, enabling interference between thereflected microwave pump and the cancellation tone. During normal operation of the transducer
the cancellation tone is tuned to minimize Pref , but to estimate the effect of pump photons on
the qubit the cancellation tone can be tuned to only partially interfere and tune Pref over several
decades. The phasor sum shown here is for the case of constructive interference.

5.3.1

Reducing backaction through modularity

A major advantage of the modular design used in this work is the ability to nearly completely
isolate the qubit from the electro-optic transducer, while still being able to send signals from thecircuit QED system through the electro-optic transducer with relatively high efficiency. Though

60
the modularity and isolation comes at a small cost to signal transfer efficiency between the circuit
QED system and the electro-optic transducer, where ηmic = 0.34 (see Section 4.8.4) is a significant
source of loss, this still compares relatively favorably to [62]—the only other example where signals
from a circuit QED system have been detected optically—where the equivalent efficiency for sending
signals from a qubit to the transducer was 0.75. However, a major difference, is that in our modular
system the qubit is nearly unaffected (the degree to which there is backaction will be discussed in
Sections 5.4.1 and 5.4.2) by the operation of the transducer. In contrast, in [62], the act of turningon the transducer heats the qubit and completely destroys any stored information, making further
error correction and heralding operations challenging.

5.4

Sources of excess backaction
I now discuss the possibility for different sources of backaction from the electro-optic trans-

ducer to influence the circuit QED system. This can come from three separate sources: the opticalpumps heating the circuit QED system, coherent backaction from the microwave pump or heating
induced by the microwave pump.

5.4.1

Coherent backaction from the electro-optic transducer

Due to the power of the microwave pumps used to drive the electro-optic transducer (Pinc =−45 dBm), there is a distinct possibility of coherent pump photons traveling through the isolation(described in Section 5.3), entering the circuit QED system’s microwave cavity, and dephashing thequbit. In this case coherent photons will dephase the qubit via the coherent AC stark effect, andinduce a frequency shift of the form [118]:∆ω = 2χn̄pχ
1− 2
κc /4 + χ2 + ∆2≈ 2χn̄p ,(5.4)

where the fractional term can be neglected as the pump is detuned by |∆| = ωm = 2π × 1.45 MHz,such that |∆|  κc , χ. The pump photons in the circuit QED cavity are given by n̄p = (n̄+ + n̄− )/2,

61
where
n± =

κc,ext |αin |2.
κ2c /4 + (∆ ± χ)2(5.5)

Note that n̄+ ≈ n̄− insofar as χ/|∆|  1. See Table 5.1 for the relevant circuit QED system parameters. In order to separate out thermal backaction (which will also shift the qubit’s frequency) fromcoherent backaction, I tune the electromechanical pump to a power such that Pinc = −77.5 dBmand heating of the dilution refrigerator’s base plate by the pump is negligible. Then, withoutchanging the incident power (which could possibly change the thermal backaction) I tune the interferometric cancellation to tune the reflected microwave pump power Pref over several decades(see Figure 5.1(a),(b)). Thus, under these operating conditions, any frequency shift of the qubit isentirely due to coherent backaction from the microwave pump.
Figure 5.2 shows the expected linear dependence of coherent photon number n̄p (as measured
via a Ramsey experiment) on Pref . Note that during normal operation of the transducer I tune
the cancellation so that Pref < −95 dBm. There is no data at these low values of Pref , sinceresolving such small frequency shifts is not easily achieved, but I can extrapolate the linear fit to−105 dBm ≤ Pref ≤ −95 dBm to infer n̄p at the typical operating point of the interferometer.
From this data, it is clear that np < 1 × 10−3 , indicating minimal coherent backaction from themicrowave pump during normal operation of the electro-optic transducer.

5.4.2

Thermal backaction

The main source of backaction present in more standard circuit QED experiments is backaction from thermally generated photons [120]. In the experiments shown here, since the basetemperature of the dilution refrigerator is raised by both the microwave and optical pumps used
to mediate transduction, it is certainly possible that the operation of the electro-optic transducer
imparts thermal backaction on the qubit. These thermal photons cause dephasing of the qubit of
the form [137, 138]
s


2
κc
2iχ
8iχneff
Γφ = Re 
1++− 1 .
2
κc
κc

(5.6)

62

typicaloperatingpoint

Figure 5.2: Excess backaction from pump photon leakage. The number of pump photons
in the circuit QED system n̄p is measured as Pref is varied. During transducer operation thecancellation is typically tuned to achieve −105 dBm < Pref < −95 dBm or equivalently n̄p <1 × 10−3 . The cancellation does not stay completely fixed over the course of the experiment due tosmall thermal drifts slightly changing the amplitude of Pref .

Where neff = n̄th + ∆n is the effective occupancy of the circuit QED system’s microwave cavity
mode. This effective occupancy is the sum of two different contributions: the thermal occupancyof the resonator due to its environment nth , and excess backaction from the converter ∆n. In thelimit where neff  1 Equation 5.6 can be Taylor expanded such that:1
κc χ2= Γφ = κ2
neff .
c
Tφ
+ χ2

(5.7)

4

To measure the effects of thermal dephasing, I first measure the effect of the optical pumpon the circuit QED system, with the microwave pump off. In Figure 5.3 a microwave-domain
Ramsey experiment is performed to measure the coherence time of the qubit when the laser is off(red circles) and when it is on with Γo /2π = 5.0 kHz (cyan diamonds), achieved with 11 mW ofcirculating power in the optical cavity. I find T2,on = 20.7±0.2 µs and T2,off = 20.4±0.2 µs, showing
no measurable change in the qubit coherence time due to laser illumination. This is the hoped for

63

Figure 5.3: Qubit coherence when operating the electro-optic transducer. Ramsey fringes
under different operating conditions of the electro-optic transducer. Red circles are with the transducer completely off. Cyan diamonds are with the laser on (Γo = 2π × 5.0 kHz). The blue squares
are with the transducer on with (Γe , Γo )/2π = (1.1, 5.0) kHz

result as the circuit QED system is sealed in a light tight box and spatially separated from theoptical port. The blue points are taken with the transducer on, such that Γo /2π = 5.0 kHz andΓe /2π = 1.1 kHz. In this case, the coherence time is slightly reduced to T2,tr = 19 ± 1 µs. The
single-shot readout contrast is reduced significantly due to microwave pump power-dependent loss
and noise in the LC circuit (see Section 4.8.4), but this is not an effect of the pump on the qubit.
The scope of this measurement can be expanded by additionally measuring the lifetimeof the qubit to determine its dephasing rate Γφ = Tφ−1 = T2−1 − 21 T1−1 , and infer the effective
occupancy neff of the circuit QED system’s microwave cavity using Equation (5.7). In Figure 5.4
I show the results of sweeping the dilution refrigerator’s base temperature, where the T1 , T2 andTφ time of the qubit are shown under various different operating conditions of the transducer.
There is a strong dependence of the T2 time and Tφ time on the base temperature of the dilution

64
refrigerator as expected. I can then convert the decay time measurements to effective occupancy

Figure 5.4: Qubit decay times versus temperature. Qubit Lifetime (T1 , black circles), qubit
Ramsey time (T2 , green circles), dephasing time with the laser and microwave pumps off (Tφ , redcircles), dephasing time with the laser on (Γo = 2π × 5.0 kHz, cyan diamond) and dephasing timewith the Transducer on (Γe = 2π × 1.1 kHz, Γe = 2π × 5.0 kHz).

of the circuit QED system’s microwave resonator through Equations 5.1 and 5.7. In Figure 5.5
I plot the effective occupancy neff as the temperature of the dilution refrigerator’s base plate Tbp
is varied. At low temperature, the inferred thermal occupancy plateaus at nth ≈ 0.019 photons(Teff = 95 mK), though this plateau may be caused by intrinsic sources of qubit dephasing otherthan an elevated temperature [20]. The cyan diamond in Figure 5.5 demonstrates that there isno measurable excess backaction from the laser (Γo /2π = 5.0 kHz) on the qubit. I then turn on
the microwave pump (Γe /2π = 1.1 kHz, leaving Γo unchanged) and the blue square in Figure 5.5

65
indicates excess backaction of ∆n = (3 ± 1) × 10−3 photons. This excess backaction must comelargely from local heating of microwave components by the strong microwave pump with incident
power Pinc = −45 dBm, since in Section 5.4.1 it was established that backaction from coherent
pump photons is negligible. It is worth emphasizing that these backaction measurements are taken
at the largest possible values of Γo and Γe used in this work, and backaction will likely be evenlower when operating at smaller Γe .

7UDQVGXFHURII
7UDQVGXFHURQ



/DVHURQ

neff














TbpP.







Figure 5.5: Transducer backaction. Effective thermal occupancy of the circuit QED system’s
microwave cavity as a function of the dilution refrigerator base plate temperature Tbp . Red circles
are data obtained with the transducer off, while the red line is theory. The cyan diamond indicatesthe occupancy of the microwave cavity with Γo /2π = 5.0 kHz, indicating no additional dephasingof the qubit from the optical field. The blue square is the effective occupancy while operating thetransducer at (Γe , Γo )/2π = (1.1, 5.0) kHz. There is a small amount of excess backaction equivalent
to ∆n = (3 ± 1) × 10−3 photons in the microwave cavity. All error bars represent one standard
deviation.

5.5

Future improvements and considerations

5.5.1

Filtering and interferometric cancellation

Filters are placed between the circuit QED system and the transducer (see Figure A.3) toeliminate the propagation of high-frequency thermal radiation along the coaxial cable connecting

66
the two systems. These filters consist of a custom built eccosorb filter, which becomes lossy atfrequencies above 10 GHz, and a low pass filter from K & L with a cutoff frequency at 10 GHz.
See Figure A.3 for details on the location and type of filtering. It may be prudent to remove thesefilters from the signal path in future experiments as they are likely responsible for a large portionof the microwave loss (ηmic = 0.34), which is discussed in 4.8.4. The filters could instead be placedon all of the input lines. Assuming that the dominant source of noise is coming from the pumpand control lines, they would function equally well in such a configuration, and ηmic would likely
improve. However, if the electro-optic transducer is a source of high frequency radiation itself,
which is a possibility due to heating from microwave and optical pumps, then the coherence timeof the superconducting qubit may be reduced in this configuration.
Interferometric cancellation (see Figure 5.1) turned out to be critical for the operation of theexperiment, and yet it also could be improved further. Currently it is tuned by hand by adjusting
the phase and amplitude of the cancellation signal until the pump signal measured via the microwave
measurement chain is minimized. This enables upwards of a 40 dB reduction in Pref , but tends
to slowly drift over time as the helium level in the cryostat varies. Furthermore, each time theelectromechanical damping is changed, this changes the settings required for cancellation, requiring
the cancellation to be adjusted, which depending on the set of measurements being completed, canbe quite tedious and inefficient. Thus a software controlled cancellation scheme might be quite
important in future iterations of this experiment. Not only would it allow for higher levels ofcancellation and isolation between the transducer and the circuit QED system through the use ofactive feedback, but it would also streamline certain experiments where Γe must be varied.

5.5.2

Excess backaction

The excess backaction of ∆n = (3 ± 1) × 10−3 from the transducer is already very low, but
further improvements to the experimental apparatus can improve upon these results. Improvements
in ge should directly lower ∆n since Γe ∝ ge2 Pinc , meaning that lower Pinc will be required to achieve
the same value of Γe . This may be relatively important since the circuit QED system used in this67
work had quite a narrow linewidth, with κc = 2π×380 kHz, which also provided significant isolationfrom the pump detuned by 1.45 MHz. If in the future, more modern circuit QED technology is used,
where multiple cavities are coupled to the qubit, the resonator that is coupled to the transmission
line may have a linewidth closer to 2 MHz. If such a system were used, then the backaction fromcoherent pump photons would be enhanced by roughly a factor of 60. Thus, the isolation provided
by the resonant cavity of the circuit QED system must be considered when designing devices for
integration with the electro-optic transducer.

5.5.3

Residual thermal occupancy

If one calculates the thermal occupancy of a 7.938 GHz microwave resonator (the cavity
frequency used in this work) at 40 mK, then one finds that n̄th ≈ 8 × 10−5 photons. However,under real experimental conditions it is difficult to thermalize microwave components to the basetemperature of the dilution refrigerator, and thermal occupancies well above this level in the rangeof 10-3 < nth < 10-1 are frequently seen [127, 92, 139]. Indeed in this experiment, the thermal
occupancy of the microwave cavity plateaus at an effective occupancy of neff = 0.019 photons,
more than two orders of magnitude higher than the expected value. This is particularly concerning,
because with the same qubit and circuit QED system cooled down in a dilution refrigerator without
optical access, the T2 time of the qubit was significantly higher at T2 = 34 µs, while T1 = 21 µs,
which yields a dephasing time of 179 µs—significantly larger than what is seen in the optical accessdilution refrigerator (see Figure 5.4). This yields an effective occupancy of neff = 5 × 10−3 , which
is significantly lower.
The reason for this disparity in performance between the optical access dilution refrigerator
and a more ’normal’ dilution refrigerator likely has nothing to do with the optical access port.
Instead, the rather large level of pump power required for the microwave pumps necessitates lessattenuation than normal on the microwave pump and cancellation lines. In this work that attenuation was measured to be 51 dB. Whereas in a normal qubit experiment, one would ideally place
70 − 80 dB of total attenuation [116, 120] (typically divided between the 4 K, still and < 50 mK

68stages) between 300 K and the base temperature of the dilution refrigerator. Currently, the attenuation at the base temperature of the dilution refrigerator is supplied by a 20 dB directional
coupler. Switching this out to a 30 dB or 40 dB directional coupler may have significant benefits
for the coherence time of the qubit in the optical access dilution refrigerator, though this will haveto be carefully balanced with the capabilities of the microwave generator at ambient temperature,
where ∼ 27 dBm of power is already being sourced to achieve the highest incident microwave powerof Pinc = −45 dBm.
Table 5.1: Circuit QED system parameters important for backaction estimation
Parameter
Qubit frequency
Qubit-cavity coupling
Qubit anharmonicity
Dispersive shift
Cavity frequency
Cavity linewidth
Weak port coupling
Cavity internal loss
Maximum microwave pump power

Symbol
ωq
gqc
ν
χ
ωc
κc
κc,w
κc,intPinc

Valueωq /2π = 5.632 GHz
gqc /2π = 66.4 MHz
ν/2π = 228 MHz
χ/2π = 172 kHz
ωc /2π = 7.938 GHz
κc /2π = 380 kHz
κc,w /2π < 5 kHz
κc,int /2π < 10 kHz
Pinc = −45 dBm

Chapter 6

Transient electromechanical amplification

6.1

PrefaceThis work was completed during the early stages of my doctoral work, predating my workon integrating superconducting qubits with electro-optic transducers. Although it is not directly
related to quantum networking and electro-optic transduction, this work describes a technique forpulsed readout of an electromechanical element at the noise limits imposed by quantum mechanics.
Pulsed protocols may be needed to mitigate noise in future iterations of the microwave to opticaltransduction project, or for experiments involving continuous variable entanglement between microwave and optical modes [140, 141, 142, 143]. Furthermore, an ideal transducer can be viewed
as the limit of a measurement without gain [74], and thus many of the ideas in this work can be
related closely to what one would like to achieve with an electro-optic transducer.

6.2

Introduction
The past ten years has seen a dramatic improvement in the ability to measure and control thequantum state of macroscopic mechanical oscillators. Much of this progress results from advancesin the parametric coupling of these oscillators to optical cavities or resonant electrical circuits.
These related fields of optomechanics and electromechanics have demonstrated the ability to coolmechanical oscillators to near their motional ground state [117, 144, 97, 69], entangle mechanical oscillators with each other [145, 146, 143] or with other degrees of freedom [141], and create
squeezed states of motion [147, 148, 149]. To verify the successful creation of these non-classical70
states, electromechanical and optomechanical methods have also enabled measurements of mechanical motion with near 50% quantum efficiency [150, 151], or equivalently an added noise equal to
the zero-point motion of the oscillator, the quantum limit for simultaneous measurement of bothmechanical quadratures [74].
These advances have encouraged notions of using non-classical states of motion to test quantum mechanics at larger scales, sensing forces with quantum enhanced precision, and enabling
quantum transduction between disparate physical systems [56]. But as mechanical oscillators areprepared in more profoundly quantum states [152, 153], with finer features in oscillator phase-space,
the measurement efficiency must further improve to resolve these fine features and to use them torealize a quantum advantage.
Reaching higher levels of efficiency with existing methods is hindered by fundamental andtechnical limitations, which seem difficult to overcome. In electromechanical and optomechanicaldevices, the state of motion can be converted without gain or added noise into a propagating electricfield, and one quadrature component of the field can be measured nearly noiselessly [141, 150].
However, the loss experienced by the field traveling between the device and the amplifier has
prevented quantum efficiency much greater than 50%. To improve measurement efficiency, thedevice can be used as its own parametric amplifier, emitting an electric field that encodes anamplified copy of the mechanical oscillator’s state, thereby overcoming any subsequent loss andinefficiency of the following measurement chain. Using this strategy, both quadratures can bemeasured simultaneously with added noise very close to the quantum limit [151]. For steadystate monitoring of a single quadrature, backaction evading schemes are in principle, noiseless[154, 155, 156, 148]. However, unwanted parametric effects, both parasitic [157, 158] and intrinsic
to the electromechanical Hamiltonian [159, 160], have prevented measurements with noise far below
the quantum-limited value.
In this Chapter, I demonstrate an efficient measurement of a single mechanical quadrature,monitoring mechanical motion with an added noise of −8.5 ± 2.0 dB relative to zero-point motion,
and a quantum efficiency of ηq = 88 ± 5 %. By generating mechanical dynamics equivalent to the71time-reverse of dissipative squeezing [161], I intentionally induce mechanical instability throughthe electromechanical interaction, allowing for a pulsed measurement of the initial state of themechanical oscillator. I term this protocol transient electromechanical amplification (TEA), anddemonstrate the resolution of fine features in phase space by using TEA to perform quantum state
tomography [162] on a dissipatively squeezed state of the mechanical oscillator, from which thedensity matrix of the mechanical oscillator can be reconstructed.

6.3

Electromechanical device
(a)

pumps

detector

L
C(x)

A(t)
(c)

Γem>0

squeeze
time

A(t)
(e)

(b)

A(ω)
Γem<0

amplifytime

A(ω) ω - ω
c
m
(d)

(f)

frequency
ωc+ ωmfrequency

Figure 6.1: Transient electromechanical amplification schematic. (a) Schematic of experiment consisting of the electromechanical circuit (green) inductively coupled to a transmission line.
Pump tones are applied through a directional coupler, while outgoing microwave signals are directed to a chain of conventional microwave amplifiers and mixer circuits, forming a microwave
receiver, which adds noise much larger than zero-point fluctuations. (b) False-color micrographof aluminum drum. The white bar corresponds to a distance of approximately 10 µm. (c) Timeand (d) frequency domain representation of temporally overlapping dissipative squeezing pumptone amplitudes (A(t) and A(ω)). (e) Time and (f) frequency domain representation of transient
electromechanical amplification (TEA) pump tone amplitudes.

The device (shown schematically in Figure 6.1a is an aluminum inductor-capacitor (LC)circuit composed of a spiral inductor and a compliant vacuum gap capacitor, which couples electricalenergy to motion. The LC circuit has a resonant frequency of ωc ≈ 2π × 7.4 GHz, and is coupled
to a transmission line at a rate κext ≈ 2π × 3.1 MHz. The compliant top-plate of the capacitor(shown in Figure 6.1(b)) is free to vibrate with a fundamental mechanical resonant frequency of72ωm ≈ 2π × 9.4 MHz and mechanical linewidth of γm ≈ 2π × 21 Hz. The electromechanical system is
attached to the base plate of a dilution refrigerator, resulting in a mechanical occupancy of nm ≤ 40
in thermal equilibrium.

6.4

Pulsed measurement and squeezing of motion
The electromechanical circuit is in the resolved sideband regime [79], enabling coherent control

of motion with microwave tones. Applying a red detuned microwave pump to the LC circuit(∆ ≡ ωc − ωp = −ωm ) allows for sideband cooling [117], and state transfer between mechanical andmicrowave fields [163, 164], where ωp is the frequency of the pump tone. A blue detuned microwave
pump (∆ = +ωm ) creates entanglement between mechanical and microwave fields [141], and realizes
a quantum limited phase-insensitive amplifier of mechanical motion [151]. Combining these twointeractions, with simultaneous application of red and blue detuned pump tones, addresses two
orthogonal mechanical quadratures X̂+ = √i2 (ĉ† − ĉ) and X̂− = √12 (ĉ† + ĉ) independently, andenables backaction evading measurement, dissipative squeezing and TEA.
The type of interaction is determined by the sign of Γem (t) = Γ− (t) − Γ+ (t), where Γ± (t)are the electromechanical growth and decay rates caused by the blue (+) and red (-) detuned
microwave tones respectively [141]. Dissipative squeezing occurs when Γem (t) > 0, which cools
the mechanical oscillator towards a squeeezed vacuum state [161]. The microwave control tones
that enable dissipative squeezing are shown schematically in the time and frequency domain in
Figures 6.1(c) and 6.1(d). Ideal backaction evasion occurs when Γem = 0, where perfect constructive interference between sidebands decouples one mechanical quadrature from microwave vacuum
fluctuations, producing an amplified noiseless representation of a single mechanical quadrature in asingle microwave quadrature [155]. Finally, TEA occurs when Γem (t) < 0, amplifying motion withenergy gain G ≈ e|Γem |t . Figures 6.1(e) and 6.1(f) show the microwave pump tones used in the timeand frequency domain for TEA.
For both TEA and backaction evading measurement, the motion of a single mechanicalquadrature X̂ is encoded in a single microwave quadrature Û . The variance of Û can then be

73
written as the sum of the noise contributions from the signal and added noise:2h∆Û 2 i = Gtot h∆X̂ 2 i + h∆X̂add
i ,

(6.1)

where Gtot is the total gain of the microwave receiver chain in units of V 2 /quanta. If the total
2 i is known, then the variance of the mechanical state h∆X 2 i canadded measurement noise h∆X̂add

be inferred. Equivalently, by preparing a mechanical state with known variance the added mea2 i = 0
surement noise can be characterized. For an ideal single quadrature measurement h∆X̂add

and U faithfully records one quadrature of of the mechanical state. Approaching this ideal behavior is highly desirable for characterizing quantum states of motion, as the number of repeatedmeasurements required to reconstruct a quantum state grows rapidly with added noise. Furthermore, assigning meaningful uncertainties to the extracted density matrix after any inference ordeconvolution procedure is complicated and subtle, diminishing confidence in the inferred state.
For the two special quadratures X̂± , the noise properties of TEA are determined by the
relative strength of Γ+ and Γ− . Assuming optimal detuning of the microwave tones by exactly
2
i referred to±ωm and Γ±  κ/2 (avoiding the strong coupling regime), the added noise h∆X̂add,±

the input of TEA is given by(2i≈h∆X̂add,±

√

Γ+ ±

√

Γ− )2 + γm (2nm + 1).
2|Γem + γm |

(6.2)

2
i willIn analogy with the high cooperativity limit, if γm (2nm + 1)/|Γem + γm |  1, then h∆X̂add,−

be less than the zero-point motion. In the case where Γ− = 0, equal noise will be added to both
quadratures, enabling nearly quantum limited phase-insensitive amplification [151]. However, if thepump frequencies deviate from optimal detuning, either through an initial detuning, or throughpump-power induced shifts in the circuit’s resonance frequency, Equation 6.2 is not valid, andtheory including pump induced mechanical and cavity frequency shifts is required (Appendix C.2).
Similarly, the variance of the squeezed and anti-squeezed quadratures after dissipative squeezing2h∆X̂sq,±i takes the same form as Equation 6.2, but with Γ− > Γ+ . In Figure 6.2(a) I demonstrate,

in a three step protocol, the control of the mechanical oscillator needed to study TEA. An initial pair

voltage (V)

74

(a)

250
prepare0
0.5

amplifytransfer(b)0Γ-(t) (kHz)Γ+(t) (kHz)250

extract
quadratures

-0.5
-80

-40

0
time (μs)

4080Figure 6.2: Transient electromechanical amplification pulse sequence. (a) General pulse
protocol for characterization of TEA and squeezing. The first overlapping red/blue-detuned pulses
determine whether the mechanical oscillator is squeezed, cooled or allowed to thermalize with its
environment. The second set of pulses tune the gain and added noise of the measurement. The
final red-detuned pulse transfers the amplified state of the mechanical oscillator to the microwave
field for quadrature extraction. The pulse lengths are chosen so that e|Γem |t is large, and provides
sufficient amplification gain or dissipative squeezing and cooling to overwhelm the thermal noiseof the mechanical environment. The pulse envelopes drawn are slower than in the experimentfor visual clarity. (b) A single experimental voltage trace of the down-converted microwave field(ωhet = 2π × 1.8 MHz) showing the resulting exponential growth and decay of the microwave field
due to the amplification and transfer pulses respectively.

of pulses prepares the mechanical oscillator in a desired state, by either sideband cooling (Γ− > 0and Γ+ = 0), dissipatively squeezing (Γ− > Γ+ > 0) or letting the mechanical oscillator reachequilibrium with its thermal environment (Γ+ = Γ− = 0). Following state preparation, the motionof the mechanical oscillator and the amplitude of the microwave field, are amplified by applying
red and blue pumps such that Γ+ > Γ− . After a short delay, the red-detuned pump is pulsed on
to transfer the previously amplified state of the mechanical oscillator to the microwave field [164].
After further amplification by a high-electron-mobility transistor (HEMT) amplifier, and a room
temperature measurement chain, the signal is mixed down to ωhet = 2π × 1.8 MHz, allowing thetwo mechanical quadratures to be extracted from the exponentially decaying microwave field shownin Figure 6.2(b).

(a)

75

〈 ΔX 2〉 (dB)
add,±

30

cool

Γ±

20
10

zero-point motion

0〈 ΔX 2〉
add,1.1

1

1.3
Γ+ /Γ-

1.2

20
Γ±

15
10
〈 ∆X 2sq ,± 〉 (dB)transfer

time
2
〈 ΔX
〉
add,+

-10

(b)

amplifysqueeze

amplify

1.5

1.6transfertime〈 ∆X 2sq ,+ 〉5

1.4

zero-point motion

0-5〈 ∆X 2sq ,- 〉

-10
0

0.2

0.4

Γ+ /Γ-

0.6

0.8

1

Figure 6.3: Transient electromechanical amplification: added noise and squeezing. (a)Total added noise referred to the input of TEA relative to zero-point motion. Γ+ is varied, whileΓ− = 2π ×181 kHz is held constant. The circles are data, while theory from Equation 6.2 (including
HEMT noise contributions) is shown without any free parameters as the dashed lines, and deviates
significantly because the pump power is large enough to induce additional parametric processes.
The solid lines are theory including parametric effects (with free parameters). The inset illustrates
2
i. Here, I obtain a minimum added noise
the pulse sequence used for the inference of h∆X̂add,±22of h∆X̂add,− i = −8.5 ± 2.0 dB. (b) Inferred variance of the squeezed h∆Xsq,−
i and anti-squeezed2h∆Xsq,+ i quadratures after dissipatively squeezing. Γ+ is varied, while Γ− = 2π × 154 kHz is held
2
constant. The minimum squeezed variance is h∆Xsq,−
i = −7.9 ± 1.4 dB. The circles are the data,
while theory is shown without any free parameters as the dashed lines, with the expected agreement
at low pump powers. The solid lines are theory including parametric effects (with free parameters).
The inset illustrates the pulse sequence for the inference of squeezing.

6.5

Added noise2
I can determine experimentally the total noise h∆X̂add,±
i added during TEA by separately

preparing the mechanical oscillator in both a thermal state and through sideband cooling. By
comparing the variance of these two states in a ratio, the added noise can be inferred (see Ap-

76pendix C.4). Figure 6.3(a) shows the total added noise as a function of the ratio of red and blue
pump power. With the optimal ratio of the red and blue-detuned pumps I find total added noise2
relative to zero-point motion of h∆X̂add,−i = −8.5 ± 2.0 dB. I compare these results to the pre-diction of Equation 6.2 with no adjustable parameters, illustrating poor quantitative agreement. I
attribute this discrepancy to additional squeezing of the mechanical oscillator caused by non-linear
mixing of the microwave pumps. I find good agreement in a fit to a more general theory that
includes such processes (and see Appendix C.2). The two theories deviate significantly from eachother, but TEA nevertheless achieves a minimum added noise equivalent to that predicted by the2
i is the total noise added by the entire meaideal case in Equation 6.2. I emphasize that h∆Xadd,±

surement chain, and for Γ+ /Γ− > 1.3 TEA has large enough gain to overwhelm the noise addedby the HEMT amplifier (see Appendix C.4.6).
Avoiding the noise associated with the simultaneous measurement of non-commuting observables is of particular importance when measuring mechanical states with a width in phase spaceless than the zero-point motion of the oscillator [165], and is desirable for many quantum state
tomography protocols [166]. Thus, to test the effectiveness of TEA on states with variance below
zero-point fluctuations, I prepare squeezed states of motion using the dissipative procedure illustrated in the inset of Figure 6.3(b). To infer the total amount of squeezing, the motion is firstsqueezed for 90 µs, then a 30 µs blue-detuned microwave pulse (Γ+ = 2π × 73 kHz and Γ− = 0)
is applied to amplify both motional quadratures. The variance associated with zero-point motion,
which must be added by the phase-insensitive amplifier, is subtracted to infer the variance of thesqueezed and anti-squeezed quadratures, which is shown in Figure 6.3(b). I obtain a maximum2
inferred vacuum squeezing of h∆X̂sq,−i = 7.9 ± 1.4 dB below the zero-point motion of the me-

chanical oscillator. I am able to far surpass the so-called steady state 3 dB squeezing limit bothbecause of the use of pulsed operations, and more than a single mode is involved during dissipative
squeezing [167]. Theory without any free parameters is plotted as the dashed lines in Figure 6.3(b),which agrees well at low pump powers. The solid lines show predicted squeezing when including
additional parametric effects induced by nonlinear mixing of the two microwave pumps (with free77
parameters).
counts80

X(φ ) (quanta)1/2

4 (a)
0

40

-4

0
0.4
(b)

X(φ ) (quanta)1/2

4

ρnn

0

〈 ∆X(φ )2 〉 (dB)

-4
10

(d)

ρnn
0.5

0

-5

0
1

Xsq,- X(φ )

φ

Xsq,+

(e)

0.2

squeezedsqueezed

5(c)squeezed(f)cooledcooledcooled0π2πφ (rad)3π
200 2 4 6 8 10
phonon number

Figure 6.4: Quantum state tomography of motion. Measurement of mechanical squeezed andsideband cooled states with single quadrature TEA. (a) A density plot of the marginal distributionsof a sideband cooled state as a function of tomography angle φ. (b) A density plot of the marginal
distributions of a squeezed state using single quadrature TEA as a function of φ. (c) The schematicshows the rotation of the single quadrature measurement axis X(φ) relative to the prepared squeezed
state. The squeezed variance is represented as the blue ellipse. (d) The total measured varianceof a sideband cooled state with nsb ≈ 0.02 (green) and squeezed vacuum (blue), which exhibits
squeezing 2.8 ± 0.3 dB below zero-point motion. The data points are the circles, while theory (withno free parameters) is the solid line. (e) Squeezed and (f) sideband cooled diagonal density matrix
elements are inferred from tomographic reconstruction of the covariance matrix. The error bars
represent 90% confidence intervals estimated with an empirical bootstrap of the tomography data.

6.6

Quantum state tomography of motion
Having demonstrated that I can prepare a squeezed state with variance below zero-point

motion, the ability of TEA to resolve fine phase space features can be tested by performing quantum
state tomography on the squeezed mechanical state. By rotating a noiseless single quadrature
measurement through all possible measurement axes, a set of phase space marginals can be recorded,
and the density matrix can be reconstructed via quantum state tomography [168, 169, 170, 136, 171].
Figures 6.4(a) and 6.4(b) show histograms of a sideband cooled and a dissipatively squeezed state

78of the mechanical oscillator as a function of the tomography angle φ. Figure 6.4(c) demonstrates
the rotation of the single quadrature measurement axis relative to the prepared squeezed state byφ.
The minimum width that can be resolved in the tomography data h∆X̂min (φ)2 i is an important figure of merit for single quadrature measurements in the quantum regime. In Figure 6.4(d)the total variance as a function of tomography angle is computed with theory (using independently
measured parameters) shown as the solid blue line. The squeezed quadrature has a total variance22of h∆X̂min (φ)2 i = h∆X̂sq,−i + h∆X̂add,−i = 2.8 ± 0.3 dB below the zero-point motion of the me-

chanical oscillator. I emphasize that this represents the total reduction in noise that is present at
the end of the conventional microwave receiver and no noise is subtracted to find this result.
The marginal distributions (1.4 × 105 points in total) can be used to reconstruct the density
matrix of the quantum state in the number basis. For a general quantum state, iterative methods of tomographic reconstruction [162]–based upon maximum likelihood–are a reliable method of
estimating quantum states [166], and are guaranteed to produce a physical density matrix. However, tomographic reconstruction of squeezed states in the Fock basis requires estimating density
matrix elements up to very high phonon number [172, 136]. To avoid calculating large density
matrices I assume Gaussian Wigner quasiprobability distributions [70], and estimate the density
matrix through reconstruction of the covariance matrix [173]. The covariance matrix is then used
to infer the Fock basis density matrix of the mechanical oscillator. In Figures 6.4(e) and 6.4(f) I
plot the inferred diagonal density matrix elements for the squeezed vacuum and sideband cooled
states, with the error bars on the measurements representing 90 % confidence intervals from anempirical bootstrap procedure (see Appendix C.7). From the density matrix I also infer the purityof the squeezed state to be µ = 0.53 ± 0.03 (see Appendix C.7), where nsq is the equivalent thermal occupation of the squeezed state. This demonstrates the direct resolution of features in phasespace with a width approximately half that of zero-point fluctuations and the ability to resolve thesqueezed character in the number basis.

79

6.7

Outlook
Mechanical devices are increasingly being integrated into circuit QED systems as resource

efficient elements, transducers and quantum memories, which offer access to new regimes of circuit
QED [153, 152]. By directly using mechanical instability as a probe, TEA can efficiently measure motion in the presence of additional nonlinear effects. Combining TEA with already demonstrated [151] quantum state transfer techniques provides a path towards efficient tomography of
non-Gaussian states in macroscopic mechanical oscillators.

Chapter 7

Conclusion

In this work I coupled technology from cavity quantum electrodynamics with a modular
electro-optomechanical transducer and demonstrated readout of a superconducting qubit in the
optical domain. Although the electro-optic transducer still adds slightly too much noise to enable
the transduction of quantum states, this work firmly establishes that the integration of superconducting qubits with optical technologies is possible. This thesis provides a clear path forward toquantum enabled transduction of photons from a circuit QED system. In Chapter 2 I establishedthe theoretical requirements for interfacing electro-optic transducers with superconducting qubits.
In Chapter 3 I outlined the relevant performance metrics and experimental elements that are required when interfacing superconducting qubits with the electro-optic transducer. In Chapter 4 I
demonstrated optical readout of a superconducting qubit, and found that the quantum efficiency isvery close to what is expected when using the qubit as a non-Gaussian resource for the calibrationof the optical readout apparatus. In Chapter 5 I verified that the backaction from the electro-optic
transducer on the qubit is negligible.
In Chapter 6 I presented work from the beginning of my PhD on transient electromechanicalamplification. This focused on efficient pulsed single quadrature measurement of the motion ofa mechanical oscillator, and forms a useful technique for performing tomography on mechanicaloscillators. Given the use of electromechanical elements in both Chapter 6 and by the electro-optic
transducer, this work could have applications in the microwave-to-optical transduction project for
both the distribution and measurement of continuous variable entanglement and squeezing.

81
This work also establishes experimental requirements for the transduction of quantum statesfrom the microwave to the optical domain. As described in Chapter 4, a major limiting factor was
the relatively narrow bandwidth of the electro-optic transducer relative to the bandwidth of thepulse emitted by the circuit QED system. Additionally, this inefficiency had a significant effect on
the input-referred added noise (noise referred to the output of the circuit QED system) NcQED =Nt /ηloss , which when maximizing for the efficiency of the transduction process, yielded NcQED =740 photons. This is of course far from the value required for quantum-enabled transduction [82]and is limited by insufficient bandwidth, LC circuit loss, transducer efficiency, and pump powerdependent noise generated by the LC circuit [69]. However, there is significant room for optimism,
as even moderate improvements in the vacuum electromechanical coupling ge will greatly improve
both the efficiency of the transducer and its added noise NcQED , and may soon enable quantum
transduction.
An increase in ge would provide many simultaneous improvements to the operation of thetransducer by reducing the number of microwave pump photons required to achieve a given transduction bandwidth. For example, ge /2π = 8 Hz would enable Γe = Γo = 2π × 5 kHz and reduce
pump power-dependent LC circuit loss, yielding ηt ≈ 0.6. If accompanied by an increase in thequbit decay times to T2 = T1 = 100 µs, I expect ηbw = 0.9, and the transducer to be quite close
to quantum-enabled with NcQED < 10. Further reductions in added noise, which will be needed
to approach the regime such that NcQED < 1, are currently being explored and include the useof wafer bonding to achieve more reliable and even larger electromechanical coupling rates. Given
that the microwave-to-optical transduction team has fabricated electromechanical devices with ge
at these levels already [76]—just not in a full working electro-optic transducer—these required improvements seem quite achievable, and quantum enabled transduction of photons from a microwave
frequency superconducting qubit to the optical domain should be well within reach.
Perhaps the most exciting finding of this work is that with proper isolation (described in
Chapter 5) superconducting qubits are completely compatible with this type of electro-optic transducer, as indicated by the minimal backaction of the transducer on the qubit. Due to the vastly

82
different energy scales between microwave and optical photons, coherent pump fields are required
universally by all current techniques for electro-optic transduction. Given that superconducting
qubits are extremely sensitive to microwave photons, and completely incompatible with opticalphotons, it is reasonable to be concerned about the proper operation of the circuit QED system
when running the electro-optic transducer. However, I found that with sufficient non-reciprocal
isolation between the two systems, there exists only a small amount of heating and backaction fromthe microwave pump, and there is no measurable influence of the optical pump on the superconducting qubit. Thus, with the aforementioned improvements to the transduction efficiency andadded noise, continuous transduction of quantum signals from a circuit QED system to the opticaldomain should be possible.

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Appendix A

Technical specifications of the circuit QED system and electro-optic transducer

A.1

Circuit QED system
The circuit QED system consists of a planar superconducting transmon qubit coupled to thefundamental mode of a seamless 3D aluminum cavity. The qubit was fabricated on a 14 × 2 mm2 cplane sapphire chip. Using a single e-beam lithography step, the Dolan bridge process ([174] and see
Appendix B) was used to fabricate an Al AlOx Al Josephson junction with area of 150 × 250 nm2 .
The qubit capacitance comes from two rectangular 500 × 700 µm pads separated by 200 µm.
The cavity, a 40 mm long circular waveguide with radius 4.7 mm, was machined out of asingle piece of 99.999% purity aluminum. The waveguide is open at one end, with a 7.25 mmlong post with 1.5 mm radius at the other end; this geometry defines the fundamental mode as aquarter-wave resonance with evanescent coupling to the waveguide above [98]. A cross section of
the CAD model for the circuit QED system is shown in Figure A.2 alongside of a photo of one of
the cavities, while Figure A.2 shows a picture of the optical access dilution refrigerator containing
both the circuit QED module and the electro-optic transducer. To reduce surface loss, the cavity
was etched in Transene aluminum etchant Type A for 24 hours at ambient temperature. A sapphire
rod with radius 2 mm is epoxied to a copper plate and fastened to a piezoelectric stepping module
above the waveguide with a maximum travel of 5 mm. This assembly allows the sapphire rod to betranslated along the center of the waveguide to tune the fundamental mode’s resonant frequency ωcin situ by up to 1.7 GHz. The cavity mode has a total linewidth of κc = κc,w + κc,int + κc,ext and isstrongly overcoupled to the transmission line connected to the electro-optic transducer. To achieve

98
coupling to the weak and strong ports at rate κc,w and κc,ext respectively, the outer conductor
and dielectric are removed from the end of a superconducting coaxial cable such that the center
conductor can be inserted into the cavity through a hole in the bottom of the quarter wave resonator(see Figure A.2). Since the coupling pin is being injected at a node of the resonance it is criticalthat the pin be superconducting, as the pin extends several millimeters into the cavity to achieve
the appropriate amount of coupling. This technique for coupling to the resonant mode was chosenover more standard evanescent coupling techniques at the anti-node of the microwave resonance[175] due to spatial constraints in the optical access dilution refrigerator.
The circuit QED system can be described by the Jaynes-Cummings model with interaction
JC = ~g (dˆ† σ̂ − + dσ̂ˆ + ). When the qubit is far detuned from the resonator (gqc Hamiltonian Ĥint
qc|∆qc |, where ∆qc = ωq − ωc ), the Jaynes-Cummings model can be described under the dispersive
JC = −~χσ̂ dˆ† d.
ˆ The dispersive shift χ is given by χ = g 2 ν/ (∆qc (∆qc − ν))approximation Ĥintzqc[22], where the qubit anharmonicity ν encodes the contribution from higher energy levels of thetransmon qubit. See Extended Data Table A.1 for a list of parameters describing the circuit QED
system.
The qubit lifetime (T1 = 17 µs) is far from Purcell-limited [84], likely due to participationof lossy dielectrics that remain after e-beam lithography and aluminum deposition. The coherencetime of the qubit is T2R ≈ T2echo ≈ 20 µs, and no difference is seen when measured using a Hahnecho versus a standard Ramsey sequence.
The circuit QED system is contained in a magnetic shield comprising an outer shield ofhigh-permeability Amumetal 4K and an inner shield made of pure aluminum in order to minimize
stray magnetic fields near the circuit QED system during cooling of the device through its criticaltemperature.

A.2

Electro-optic transducer
The electro-optic transducer consists of a microwave resonator and an optical resonator si-

multaneously coupled to a single mode of a micromechanical oscillator. It is described by the

99

Figure A.1: Image of the optical access dilution refrigerator. The circuit QED module iscovered in copper tape and on the left. The copper box in the center of the base plate contains theelectro-optic transducer.

HamiltonianĤ= ωo â† â + ωe b̂† b̂ + ωm ĉ† ĉ + (go â† â + ge b̂† b̂)(ĉ + ĉ† ),
~

(A.1)

where â, b̂ and ĉ are the annihilation operators for the optical, microwave and mechanical modes,
respectively, and go (ge ) is the vacuum optomechanical (electromechanical) coupling rate.
The mechanical oscillator is a 100 nm thick, 500 µm wide square silicon nitride membranesuspended from a silicon chip. Phononic shielding is patterned into the silicon chip, isolating
the membrane mode used for transduction, with resonant frequency ωm = 2π × 1.45 MHz, fromvibrational modes of the chip. This shielding results in a quality factor Qm = 1.3 × 107 for themode of interest, or equivalently an energy dissipation rate γm = 2π × 0.11 Hz.
The microwave resonator is a superconducting flip-chip LC circuit whose capacitance is modulated by the motion of a superconducting pad deposited on the membrane. The strength of theelectromechanical coupling is determined by the capacitor gap spacing between this pad and asecond nearby chip hosting the rest of the circuit, which is ∼ 300 nm for most devices. In the

100

Figure A.2: CAD model of the circuit QED system. (a) The circuit QED system is formed
by a superconducting qubit coupled to a 3d quarter wave post resonator evanescently coupled to
the wave guide above. A sapphire rod can be translated towards the end of the post resonator inorder to tune its frequency. The sapphire rod is epoxied to a clamp with Stycast 2850 and thenthe clamp is attached to the Attocube translation stage. A lid with a narrow hole for the sapphire
rod to pass through is attached to the top of the waveguide to prevent coupling of modes in the
Attocube mount to modes within the waveguide and the quarter wave resonator. (b) Photo of thecircuit QED system and Attocube mount. A copper braid is attached to the gold plated copper
qubit clamp in order to better thermalize the superconducting qubit to the base temperature of
the dilution refrigerator.

transducer used in this work the capacitor gap was unusually large, resulting in a low vacuum
electromechanical coupling rate ge = 2π × 1.6 Hz. The LC circuit is coupled to a microwave transmission line at rate κe,ext = 2π × 1.42 MHz, and its total linewidth κe varies between 1.6 MHz and
2.7 MHz due to dependence of the internal loss κe,int on the power of the strong microwave pumpused to mediate transduction.
The optical resonator is a Fabry-Pérot cavity defined by high-reflectivity ion beam sputteredmirror coatings, which are chosen to be very asymmetric such that the cavity mode couples preferentially out the front mirror with external coupling κo,ext = 2π × 2.1 MHz. The total cavity
linewidth is κo = κo,ext + κo,int = 2π × 2.7 MHz, where κo,int includes scattering and absorption
losses as well as transmission through the back mirror. The membrane is positioned at a maximum
in the intensity gradient of the intracavity light, yielding a vacuum coupling rate go = 2π × 60 Hz.

101
The heterodyne mode matching factor quantifying the overlap of the propagating optical cavity
mode and the local oscillator beam (LO) is  = 0.8.
During operation, strong microwave and optical pumps are applied near the corresponding
electromagnetic resonances to enhance the vacuum coupling rates go and ge [67]. Transforming to arotating frame to remove the free evolution of the fields, â → âe−iωo t , b̂ → b̂e−iωe t and ĉ → ĉe−iωm t ,the Hamiltonian can then be linearized around the strong pumps to yield
lin
Ĥint
= go ā(â† + â)(ĉ + ĉ† ) + ge b̄(b̂† + b̂)(ĉ + ĉ† ),
~

(A.2)

where ā (b̄) is the optical (microwave) mode coherent state amplitude due to the incident pump.
Both pumps are red-detuned from the respective resonances by ωm , resonantly enhancing theoptomechanical and electromechanical beamsplitter terms in the Hamiltonian [79]. In the resolved
sideband limit at this optimal detuning, the electromechanical (optomechanical) damping rate
is then given by Γo = 4go2 ā2 /κo Γe = 4ge2 b̄2 /κe . The transducer bandwidth is given by ΓT =
Γe + Γo + γm .
The electro-optic transducer parameters are summarized in Tab. A.1. See Ref. [69] for amore detailed description of the fabrication, assembly, and characterization of the electro-optic
transducer, as well as the theory of transducer operation.

A.3

Experimental layout
A schematic of the experimental setup is shown in Supplementary Figure A.3. Here I briefly

summarize the details of the optical setup, discussed further in Ref. [69].
A beam sourced by a low-noise external cavity diode laser (Toptica Photonics CTL) operated
at a wavelength λ = 1084 nm is passed through an optical filter cavity (not shown) to further reduce
laser phase noise, and then split three ways to obtain pump (red), lock (yellow), and local oscillator(LO, maroon) beams. The pump and lock beams are frequency shifted relative to the LO beam by
acousto-optic modulators, such that the lock beam is resonant with the optical cavity mode, thepump beam is red-detuned by ωm , and the LO beam is detuned from the pump beam by 12.8 MHz

102
Table A.1: Circuit QED system and electro-optic transducer parameters.
Parameter
Qubit frequency
Qubit-cavity coupling
Qubit anharmonicity
Dispersive shift
Cavity frequency
Cavity linewidth
Weak port coupling
Cavity internal loss
Qubit lifetime
Ramsey time
Optical cavity frequency
Optical cavity external coupling
Optical cavity linewidth
LC circuit frequency
LC circuit external coupling
LC circuit linewidth (low pump power)
LC circuit linewidth (high pump power)
Mechanical mode frequency
Intrinsic mechanical dissipation rate
Vacuum optomechanical coupling
Vacuum electromechanical coupling
Optical cavity mode matching

Symbol
ωq
gqc
ν
χ
ωc
κc
κc,w
κc,intT1
T2
ωoκo,extκoωeκe,ext
κe
κe
ωm
γm
go
ge


Value
ωq /2π = 5.632 GHz
gqc /2π = 66.4 MHz
ν/2π = 228 MHz
χ/2π = 172 kHz
ωc /2π = 7.938 GHz
κc /2π = 380 kHz
κc,w /2π < 5 kHz
κc,int /2π < 10 kHz
T1 = 17 µs
T2 = 20 µs
ωo /2π = 277 THz
κo,ext /2π = 2.12 MHz
κo /2π = 2.68 MHz
ωe /2π = 7.938 GHz
κe,ext /2π = 1.42 MHz
κe /2π ≈ 1.6 MHz
κe /2π ≈ 2.7 MHz
ωm /2π = 1.45 MHz
γm /2π = 0.11 Hz
go /2π = 60 Hz
ge /2π = 1.6 Hz
 = 0.80

to enable heterodyne measurement. The lock and pump beams are orthogonally polarized and thus
can be combined and routed to separate detectors using a polarizing beamsplitter.
The Pound-Drever-Hall technique [176] is employed to simultaneously lock the laser to boththe optical filter cavity and the transducer’s optical cavity. The error signal is used to feed back to
the laser wavelength, the filter cavity length, and the rf drive to an acousto-optic modulator. The
absence of tuning elements on the transducer’s optical cavity leads to improved stability relative
to cavities in previous devices [56].
As shown in Supplementary Figure A.3, the qubit control and readout pulses are generated
via individual sources and controlled and gated by mixers and switches [20].

103
Readout(Agilent E8257D)
Qubit drive(Keysight N5172B)

Q

L

I

Q

R

I

50 Ω termination

Variable attenuator

10 GHz LPF

Variable phase shifterRF switch

180° hybrid

Transducer pump
(R & S SMA100B)

L

R

b

Directional couplerIF

Ecco-sorb filter

R
Photodetector

Polarizing Beamsplitter

Digitizer

Digitizer(AlazarTech(AlazarTech

ATS9462)

ATS9462)+30 dB

1.9 MHz+30 dB

1.9 MHz

1.9 MHz

20 dB

10 dB

d
Local
oscillator(Agilent E8257D)

I

Q
L

in

HEMT

20 dB

20 dB

4K
20 dB

c

Q

Lock-in Amplifier(Zurich Instruments
HF2LI)IR

1.9 MHz

L

AWG(Zurich Instruments
HDAWG)

Ecco

a

20 dB

800 mK-

40 mK

Faraday
rotatorλ/2

pump

SapphirerodcancelTransmon
qubit

Ecco

30 dB

e

Faraday
rotatorλ/2

Opticalpump

LO

To optical cavity
lock feedback

Ecco
x4

20 dB
Resonant
lock

Figure A.3: Experimental schematic. (a) Microwave layout demonstrating the exact configuration of the qubit readout/control pulses and the pumps for the electro-optic transducer. (b) Legendfor various different microwave and optical components. (c) Cryogenic portion of the experiment.
(d) Demodulation and detection scheme. The two digitizers allow for simultaneous measurement of
the microwave and optical signals emitted from the transducer. (e) Simplified schematic of opticalbeam layout and balanced heterodyne detector.

Appendix B

Qubit Fabrication procedure

Here I describe the procedure used to fabricate qubits in the JILA cleanroom using wafer scaletechniques and the standard Dolan bridge process [174, 177]. Similar procedures for fabricating
basic transmon qubits through electron-beam lithography can be found here [178]. The processdescribed here can easily be generalized to fabricating individual qubit chips.
(1) Sonicate the sapphire wafer at high power in acetone and then IPA for two minutes in eachsolvent. Depending on the wafer supplier additional cleaning may be needed.
(2) Blow the wafer dry after the IPA clean to ensure all solvent is removed.
(3) Place wafer on 3 glass slides (for a 3” wafer) on top of a hot plate at 180 ◦ C to pre-bakethe wafer for 3 minutes. The exact baking time is not particularly important as this is justto ensure that the wafer is entirely dry before spinning resist.
(4) Remove the wafer from the hot plate, let it cool for at least one minute.
(5) Program the spin coater to run at 4000 RPM and accelerate at 1000 RPM/s, with a 45 s
spin time. Place the wafer onto the vacuum chuck, turn on the vacuum to hold the wafer.
(6) Pour the co-polymer (MMA(8.5) EL 10) directly onto the wafer, making sure that the vast
majority of the wafer is coated in resist to ensure uniformity of the resist across the entire
wafer after spinning.

105(7) Close the spin coater lid, and run the spin cycle. Once completed, turn off the vacuum
chuck, and place the wafer on the hot plate.
(8) Bake for 6 minutes at 180 ◦ C.
(9) Remove the wafer, and let it cool for one minute.
(10) Repeat the previous four steps but with PMMA A6 instead of the co-polymer.
(11) The wafer can now be loaded into the vacuum chamber for deposition of a 15 nm aluminum dissipation layer. This layer prevents charging of the wafer during electron beamlithography.
(12) After aluminum deposition, the wafer can be stored in a black wafer carrier (PMMA isnominally not sensitive to UV light, but there is no need to expose it unnecessarily) for as
long as is needed.
(13) Using a diamond scribe or metal tweezers, put 6 scratches onto the outer radius of thewafer for future focusing purposes.
(14) Load the wafer into the SEM, focus and adjust stigmators on the auxiliary gold sample.
(15) Move over to the wafer and align to the flat of the wafer so that the junctions will beparallel with the wafer flat.
(16) Move to the Faraday cup. Use NPGS to measure the beam current at a spot size of 1.1
and 5.0, though these may vary based upon the SEM emission current. Essentially, it isbest to write junctions and the fingers connecting them to the pads with less than 100 pAof current, and it is best to write the transmon pads with approximately 2.5 nA of current.
Using more current than this to write the pads typically ruins the junctions (likely fromstray charge incident on the junction region), while less current than this makes writing
take a prohibitively long time.

106(17) Return to the gold sample, check if the stigmators need to be adjusted further, or if the
focus has changed. This can occasionally occur when switching the spot size of the beam.
(18) Once focused, move to the wafer’s edge and find one of the scratches. Avoid traversing thecenter of the wafer so as not to expose the resist.
(19) By adjusting the Z-position of the SEM stage (not the focus knob!) bring the SEM into
focus on a loose piece of aluminum near the scratch. Do not try to focus on the inside of
the scratch as this is not conducting and will charge up and obscure the image.
(20) In the SEM software, record this position as focused using the direct stage control functionality.
(21) Move to the next scratch by traveling around the outer edge of the wafer. The sample
should in all likelihood not be in focus due to tilt of the wafer. Find another flake ofaluminum at the new scratch, but this time use the fine focus knob to focus on the sample,
again record on the direct stage control feature that the sample is in focus.
(22) Repeat this procedure above (without changing the Z-position of the SEM stage) on all
6 scratches so that the SEM can fit a plane to find the tilt and automatically adjust thefocus for a given X-Y coordinate.
(23) Switch the control of the software to NPGS mode, and ensure that the beam blanker is set
to external (NPGS) control mode.
(24) Go to the center of the wafer, and run the desired NPGS write program.
(25) Once the electron beam writing is completed prepare four crystallizing dishes with MF-26A(dilute TMAH based photoresist developer), water, MIBK (1-3 parts MIBK to IPA) and
IPA. Turn a hot plate on to 100 ◦ C.
(26) Etch the aluminum in MF-26A. Once the aluminum is no longer visible, etch for 10 additional seconds. Remove the wafer and place in a dish of water for 30 s.

107(27) Remove the wafer from the water and blow dry with nitrogen. Be certain that all TMAH
has been diluted/rinsed off with water before doing so as you do not want to spray thistoxic chemical into the air or onto anything else.
(28) Once the wafer is dry, submerge the wafer in MIBK for 45 s. The precise amount ofdevelopment time will determine the size of the junctions, so it is important to keep thistime consistent.
(29) After development, immediately submerge the wafer in IPA for 30s.
(30) Remove the wafer and blow dry with nitrogen gas.
(31) Bake the wafer on the hot plate at 100 ◦ C for one minute.
(32) Inspect the resist under a microscope, looking to see that the features appear as expected.
You will not be able to resolve the junctions directly, but you should be able to see that
there is a wire connecting both pads to each other through the junction.
(33) Once visual inspection is passed, bring the wafer to the reactive ion etcher for O2 plasma
cleaning.
(34) Using 50 w and 50 SCCM of O2 for 10 seconds, run the O2 plasma cleaning cycle.
(35) Immediately the place wafer in the deposition chamber, being careful to align the wafer
flat with the vertical grooves on the wafer holder, and then place the wafer in the chamber
and pump out to ideally better than 2 × 10−7 Torr.
(36) Evaporate 30 nm of aluminum with sample at 20◦ .
(37) Quickly stop the evaporation by going down to 30 mA of electron-beam current on thealuminum crucible—note, this current value is for the JILA evaporator. This will keep the
crucible hot, but stop the evaporation process. Close the gate valve between the evaporator
and the sample chamber. Let in approximately 7 Torr of oxygen to the sample chamber to108oxidize the aluminum for three minutes. The time and pressure can be varied to control
the junction resistance. Alternatively the junction size can also be varied to adjust thejunction resistance.
(38) Right before the oxidation time is up turn the evaporator off completely.
(39) Pump the oxygen out of the chamber, and when the chamber is below 8 × 10−2 Torr,the valve to the roughing pump can be closed, and the gate valve can be opened. The
evaporator should be turned on and ramped to its maximum current immediately after
the vacuum interlock allows one to do so, and evaporation should resume relatively quickly
because the aluminum crucible was kept hot.
(40) Once a sufficiently large evaporation rate is achieved (typically in the range of 0.3 nm/s withthe shutter closed), the sample can be rotated to 340◦ to evaporate 60 nm of aluminum.
(41) With the second layer of aluminum complete, the evaporator can be ramped down to
30 mA. Once the deposition rate has returned to zero, close the gate valve between thesample chamber and the evaporation chamber and oxidize the sample for 10 minutes in
7 Torr of oxygen.
(42) Pump out the oxygen with the rough pump and then vent the sample chamber to remove
the sample.
(43) Place wafer upright in a teflon stand in large beaker of PG remover at 80◦ C for lift-off.
It’s best to leave this overnight, though several hours typically works.
(44) Perform liftoff using a transfer pipette to gently ’blow’ off the remaining aluminum on thewafer inside of the solvent.
(45) Clean the wafer in IPA and methanol at low power in the sonicator, and then blow dry.
(46) Spin a protective layer of thick positive photoresist onto the wafer. It is important to use
positive photo resist as the wafer is going to be exposed to UV light when dicing, which

109
could make a negative photoresist difficult to remove due to cross-linking of the resist.
(47) Using heat sensitive double sided tape, fix the sapphire wafer to a thick silicon wafer. Affix
this to the dicing saw with the vacuum chuck.
(48) Dice the chips.
(49) Blow off all of the water using dry nitrogen, and place the silicon wafer onto a hot plate at
100 ◦ C.
(50) Once the heat tape releases, remove the silicon wafer from the hot plate, and transfer thechips over to the machined aluminum cleaning jig. The wire shorting the pads together
can be used to determine which side of the chip the aluminum was deposited on if needed.
(51) Soak and sonicate the chips in acetone. Replacing the acetone 2-3 times to ensure that the
dissolved photoresist is largely removed.
(52) Transfer the chip cleaning jig to PG remover at 80◦ C for at least several hours, preferably
overnight.
(53) Transfer the chip cleaning jig to IPA (acetone does not dissolve PG remover). Sonicate at
low power for several minutes.
(54) Sonicate the chips in acetone. It can also be worthwhile to let the chips soak in acetone forseveral more hours or overnight.
(55) Sonicate the chips in IPA and/or Methanol.
(56) Blow each chip dry individually, and carefully place face up on a microscope slide.
(57) Place the slides with the chips in the reactive ion etcher. Using an oxygen atmosphere at
100 W and 100 SCCM of O2 for 3 minutes, run the reactive-ion etcher. Repeat this cycle
2-3 times. This step is critical for the lifetime of the qubit as during our dicing process110some sort of surface contamination that remains after the above cleaning steps seems to be
removed by this plasma cleaning process.
(58) Place chips in gelpack for storage in dry box or for transport to the probing station.
(59) Probe chips to select for appropriate junction resistance. [179].

Appendix C

Transient electromechanical amplification experimental details

C.1

TheoryC.1.1

Two-tone electromechanical equations of motion

Starting with the linearized Heisenberg-Langevin equations of motion [79] in a frame rotating
at a center frequency ωr I find that
 κ

√
√
˙
dˆ = − + i∆ dˆ + ig0 α(t)(ĉ + ĉ† ) + κex dˆin + κ0 fˆin2p˙ĉ = −iωm − Γm ĉ + ig0 (α∗ (t)dˆ + α(t)dˆ† ) + Γm ĉin ,
2

(C.1)
(C.2)

while in a two tone driving scheme:α(t) = α+ e−iω+ t + α− e−iω− t ,(C.3)

where ∆ = ωr − ωc and ω± = ωr ± (ωm + δm ). Going to a rotating frame for the mechanical modesuch that ĉ → ĉe−i(ωm +δm )t and ignoring all fast counter rotating terms:
 κ

√
√
˙
dˆ = − + i∆ dˆ + ig0 (α− ĉ + α+ ĉ† ) + κex dˆin + κ0 fˆin2p˙ĉ = iδm − Γm ĉ + ig0 (α∗ dˆ + α+ dˆ† ) + Γm ĉin .
−
2

(C.4)
(C.5)

I define the mechanical and cavity quadratures:
1X̂− = X̂1 = √ (ĉ† + ĉ)
2

(C.6)

i
X̂+ = X̂2 = √ (ĉ† − ĉ)
2

(C.7)

112
1
ˆ
Û1 = √ (dˆ† + d)2iÛ2 = √ (dˆ† − d).
2

(C.8)
(C.9)

Specializing to the case δm = 0 and ω± = ωc ± ωm (as in [161]) the linearized Heisenberg-Langevinequations can be calculated in the rotating wave approximation:
√ 
p
p 
κ p
Γm
Γ+ − Γ− Û2 −X̂1 + Γm X̂1,in
2
2
√ 
p
p 
κ p
Γm
˙
Γ+ + Γ− Û1 −X̂2 + Γm X̂2,in
X̂2 =
2
2
√ 

p
p
√
√
κ
κ
˙
Û1 =
Γ+ − Γ− X̂2 − Û1 + κext Û1,in + κ0 fˆ1,in
2
2
√ 

p
p
√
√
κ
κ
˙
Û2 =
Γ+ + Γ− X̂1 − Û2 + κext Û2,in + κ0 fˆ2,in .
2
2
˙
X̂1 =

(C.10)(C.11)(C.12)(C.13)

Where Γ± = 4g02 n± /κ depends upon the average number of photons n± circulating in the LC
circuit from the red (-) and blue (+) detuned pumps respectively. This experiment stays out of thestrong coupling regime such that κ/2 >> Γ± . The cavity decay rate is much larger than all otherdecays rates in the system, thus the cavity follows the state of the mechanical oscillator, allowing
˙
for adiabatic elimination of the cavity amplitude such that Û1 = 0 and U̇2 = 0. This allows for
simplification of the system of equations above into two independent equations for the mechanicalquadratures:pp  √√
Γ+ − Γ− − Γm
1 p
˙
X̂1 =
X̂1 + Γm X̂1,in + √Γ+ − Γ−κext Û2,in + κ0 fˆ2,in2κpp  √√1 p
Γ+ − Γ− − Γm˙X̂2 + Γm X̂2,in + √Γ+ + Γ−κext Û1,in + κ0 fˆ1,in .
X̂2 =
2
κ

(C.14)(C.15)

To calculate the expected variance of the mechanical quadratures the differential equations
can be solved and the noise correlation functions of the cavity and mechanical quadratures can be
used:


1
nc +
δ(t − t0 )
2


1
†
†
0
0
ˆ
ˆ
hf2,in (t)f2,in (t )i = hÛ2,in (t)Û2,in (t )i = nc +δ(t − t0 )
2


1
†
0
hX̂1,in (t)X̂1,in (t )i = nm +δ(t − t0 )2††
hfˆ1,in(t)fˆ1,in (t0 )i = hÛ1,in(t)Û1,in (t0 )i =(C.16)(C.17)(C.18)

113†
hX̂2,in(t)X̂2,in (t0 )i =


nm +

1
2



δ(t − t0 ).

(C.19)

Computing the variance gives:h∆X̂12 i =h∆X̂1 (0)2 ie(Γ+ −Γ− −Γm )t +(C.20)pp1( Γ+ − Γ− )2 (2nc + 1) + Γm (2nm + 1) (e(Γ+ −Γ− −Γm )t − 1)2(Γ+ − Γ− − Γm )h∆X̂22 i =h∆X̂2 (0)2 ie(Γ+ −Γ− −Γm )t +(C.21)pp1( Γ+ + Γ− )2 (2nc + 1) + Γm (2nm + 1) (e(Γ+ −Γ− −Γm )t − 1).
2(Γ+ − Γ− − Γm )In the large gain limit, such that the energy gain G = e(Γ+ −Γ− −Γm )t  1 the noise added at
the input of transient electromechanical amplification (TEA) for the two preferred quadratures
becomes:2i=h∆X̂add,±p
p1( Γ+ ± Γ− )2 (2nc + 1) + Γm (2nm + 1) .
2|Γ+ − Γ− − Γm |(C.22)

If instead, I squeeze the motion of the mechanical oscillator with e(Γ+ −Γ− −Γm )t ≈ 0, then thevariance of each preferred quadrature is given by:2h∆X̂sq,±
i=

C.1.2

p

p
1
( Γ+ ± Γ− )2 (2nc + 1) + Γm (2nm + 1) .
2|Γ+ − Γ− − Γm |(C.23)

Additional single mode squeezing due to detuning of microwave pumps

The equations of motion in Section C.1 are valid for κ >> Γ± , but even if adiabatic elimination of the cavity mode remains valid, as the pump power is increased, both the cavity resonant
frequency and the mechanical resonant frequency depend on the number of photons circulating in
the LC circuit. The data can be described by adding in the effects of these pump induced frequency
shifts. The following analysis is similar to that in [159] where single mode squeezing of the mechanical oscillator was demonstrated in the large detuning limit ∆ >> κ where adiabatic elimination of
the cavity field is also valid. I operate in the limit such that κ >> ∆ to demonstrate an equivalent

114
result. Assuming that κ2 >> ∆, Γ± , Γm , I can still adiabatically eliminate the cavity fluctuations[180, 181]:†

ig0 (α− ĉ + α+ ĉ ) +dˆ ≈κ/2 − i∆

√ ˆ
κdin

,

(C.24)

where I have ignored the small internal loss of the cavity. This gives an independent equation of
motion for the mechanical field as:p˙ĉ = Γ+ − Γ− − Γm ĉ + i δm − ∆ (Γ+ + Γ− ) ĉ − 2i∆ Γ+ Γ− ĉ† +
2
κ
κ
s
s




Γ−
2i∆ ˆ
Γ+
2i∆ ˆ†
i
1
+
1
−
d
+
i
din ,
in
2
2
κ
κ
1 + 4∆2
1 + 4∆2
κ(C.25)(C.26)

κ

where I choose a global measurement phase and assume that α± are both real. In this form I canidentify two additional terms in the equation for the cavity field that are caused by detuning both
microwave tones from cavity resonance. These two terms correspond to a mechanical frequency
shift δm,tot and a single-mode squeezing term χ given by:∆(Γ+ + Γ− )
κ
2∆ p
χ=
Γ+ Γ− .
κδm,tot = δm −(C.27)(C.28)

I combine these effects into a coupled set of equations of motion to describe the two cavity quadratures.
  
 
√
˙
Γ+ −Γ− −Γm∆ √2
δm + κ ( Γ+ − Γ− )  X̂1 
2
X̂1    = +√√˙
Γ+ −Γ− −Γm2
X̂2−δm − ∆
(
Γ
+
Γ
)
X̂2
+
−
κ
2
√


√ 

Γ+ − Γ−
√
2∆
q
Γ
X̂
+
Û
+
Û
2,in
 m 1,in2
κ 1,in 
1+ 4∆2


κ
√

√
.


Γ+ + Γ− √2∆Γm X̂2,in + q 4∆2
Û1,in − κ Û2,in
1+(C.29)(C.30)

κ2

The additional single mode squeezing term and frequency shifts cause significant deviation fromthe equations of motion in section C.1.1 as the power of the two tones is increased.

C.2

Pump dependent cavity frequency shifts
I estimate the cavity frequency shifts induced by the two microwave tones from optomechan-ical theory [79] by noting that there is a nonlinear dependence of the detuning from the cavity on

115
microwave pump power, where α and β are the coherent displacements of the cavity and mechanicaloscillators respectively:√
κex αinα=
κ/2 − i∆eff(C.31)−ig0 |α|2
Γm /2 − iωm(C.32)β=∆eff = ∆0 + g0 (β + β ∗ ).

(C.33)

Solving these equations (neglecting Γm ) the effective detuning is given by∆eff ≈ ∆0 +2g02 2κ|α| = ∆0 +(Γ+ + Γ− ),ωm
2ωm(C.34)

demonstrating the implicit dependence of the cavity resonant frequency on the pump power sent
to the circuit. This also demonstrates that a detuned back-action evading measurement [156] willcause degenerate parametric amplification of the mechanical mode. I use the result from Equation
C.34 to solve the full set of equations in Equation C.30. With free parameters δm = 300 Hz and∆0 = −74 kHz I get good agreement between theory and experiment. Given that there are two
adjustable parameters to make the theory agree, other parametric effects such as the ones describedin [158] may also contribute. Thermal effects (such as [157]) are less likely because only a small
1 dωmshift in mechanical resonant frequency with temperature is seen, with 2π
dT ≈ 4 Hz/mK.

C.3

Variance normalization
When preparing or measuring states at or near the limits imposed by quantum mechanics

the relevant scale for the variance is the zero-point motion of the oscillator. This is the appropriate
scale for squeezing because vacuum squeezing reduces the variance in a single quadrature below
the zero-point motion. It is also the relevant scale for measurement noise, because a high gain
phase preserving amplifier (simultaneous measurement of both quadratures) must add at least theequivalent of zero-point fluctuations at the input of the measurement [74]. Thus I normalize all
2 i = 1/2. Explicitly, this gives the measured
variance measurements to zero-point motion h∆X̂zp

116
values in decibels as:
2h∆X̂add,db
i = 10 log102h∆X̂sq,db
i = 10 log10

2 ih∆X̂add!

2 ih∆X̂zp!
2 ih∆X̂sq2 ih∆X̂zp(C.35).

(C.36)In Figure 6.4(d) I characterize the variance of the squeezed state hX(φ)2 i (measured by single
quadrature TEA) as a function of phase angle φ. To see that I can directly measure a squeezed
state without any inference I include measurement noise, but still normalize to zero-point motion:!
2h∆X(φ)2 i + h∆X̂add,−i2hX(φ)db i = 10 log10.
(C.37)
2 ih∆Xzp
2 i provides a direct measure of the squeezed state without any
The quantity h∆X(φ)2 i + h∆X̂add

inference or noise subtraction.

C.4

Mechanical occupancy, added noise and gain calibrations
To characterize the noise added by TEA and the total amount of mechanical squeezing,states with known variance are required to calibrate these unknown values. To accurately infer
the variance of a prepared mechanical state h∆X(φ)2 i the added measurement noise h∆X(φ)2add i
must be known. Conversely to characterize the noise added by measurement, the variance of the
prepared mechanical state must be known. I achieve this by comparing the variance of a calibrated
mechanical thermal state, to the state I wish to characterize. In microwave engineering this is
known as a Y-factor measurement [96]:r(φ) =

2h∆X̂thermi + h∆X(φ)2add i.
h∆X(φ)2 i + h∆X(φ)2add i(C.38)The variance of the mechanical thermal state is:2h∆X̂therm
i = nm + 1/2,

(C.39)

where nm is the independently calibrated thermal occupancy (described in Section C.4.5) of themechanical oscillator.

117
C.4.1

Calibration of sideband cooling and two-quadrature measurement noise

In order to fully characterize the added noise of TEA I prepare the mechanical oscillator instates with two different temperatures: a thermal state and a sideband cooled state. A thermal state
can be prepared and calibrated by allowing the mechanical oscillator to reach thermal equilibrium
2 i, I measure both(see Section C.4.5). To calibrate the variance of a sideband cooled state h∆Xsb

quadratures of motion simultaneously with the blue detuned pump (Γ+ > 0, Γ− = 0). This nearly
quantum limited amplifier adds a total noise to one quadrature of2h∆X̂addi = nadd + 1/2,

(C.40)

where nadd is any noise the amplifier adds above zero-point fluctuations. While the variance of the
sideband cooled state is2
i = nsb + 1/2,h∆X̂sb

(C.41)

where nsb is the remaining thermal occupancy after sideband cooling. The ratio of the measured
thermal state variance to the sideband cooled variance is:
r=

2 i
2
i + h∆X̂addh∆X̂therm
2 i + h∆X̂ 2 ih∆X̂sbadd=

nm + nadd + 1.
nsb + nadd + 1(C.42)

From the measurements it is clear that r  1, indicating that nm  nadd , which allows nadd to beneglected in the numerator, and I find that nsb + nadd is given by:1
nsb + nadd ≈ (nm + 1) − 1.
r

(C.43)

Measurements of this number yield a value:
nsb + nadd = 0.05 ± .03 quanta(C.44)

It isn’t possible to differentiate between nadd and nsb , but by comparing it to the minimum
theoretically possible number it is possible to demonstrate that the mechanical oscillator is beingnearly perfectly sideband cooled. To find this lower bound I use the theory of electromechanical

118
sideband cooling [79], and find (assuming negligible cavity occupancy) that in the resolved sideband
regime the minimum mechanical occupancy after sideband cooling is:
nsb,min ≈

Γm
κ2
nm +
= .012 ± .002,2
Γ− + Γ m
16ωm(C.45)

while equivalently the effective added occupancy at the input of the phase-preserving amplification
pulse is given by:
nadd,min ≈

Γm
κ2
nm +
= .018 ± .002.
2
Γ+ − Γm
16ωm(C.46)

This gives a total minimum remaining occupancy of:nsb,min + neff,min = .030 ± .004.

(C.47)

Thus the total remaining occupancy after sideband cooling and measurement in Equation C.44 isnearly equivalent to the minimum possible value nsb + neff ≈ nsb,min + neff,min , suggesting that themechanical oscillator is being cooled to very close to the limit of resolved sideband cooling.

C.4.2

Inference of mechanical squeezing

To infer mechanical squeezing and anti-squeezing I use TEA to measure both quadratures
simultaneously (Γ+ > 0, Γ− = 0) and obtain histograms for three separately prepared mechanicalstates. These histograms can then be used to calibrate the variance and added noise. First, I
2prepare a mechanical thermal state with variance hXtherm
i = nm + 1/2 = 36.5 by allowing themechanical oscillator to thermalize with its environment for 250 ms. This is repeated 2048 times
to obtain a histogram and estimate the variance of the thermal state in units of V 2 /quanta. Witha 2 ms repetition time the same procedure is performed with a dissipatively squeezed state. The
increased speed of this measurement is enabled by the increased mechanical damping rate fromsqueezing. The variances of these two histograms are compared as a ratio:
r± =

2
2 i
h∆X̂thermi + h∆X̂add
22 ih∆X̂sq,±i + h∆X̂add=

nm + nadd + 1
2h∆X̂sq,±i + nadd + 1/2.

(C.48)

I assume that nadd is equal to its lowest possible value in Equation C.46. This means that any
deviation from the ideal perfect two-quadrature measurement will decrease the inferred squeezing.

119
2
For each measurement of h∆Xsq,±
i I obtain a histogram of a sideband cooled state, so I can monitor

that the added measurement noise remains consistent with the ideal case as in Section C.4.1.

C.4.3

Inference of the measurement noise

I use a similar procedure to Section C.4.2 to infer the noise added by TEA. I obtain histogramsof 2048 points for a mechanical thermal state and a sideband cooled state. This results in a ratioof variances of:r± =nm + 1/2 + h∆X̂add,± i
nsb + 1/2 + h∆X̂add,± i,(C.49)

where X̂± are the two preferred quadratures of TEA. At low gains the added noise of the HEMT issignificant, and is included in the theory for single quadrature measurement noise. For the inference2
i I assume perfect sideband cooling as in Equation C.47 to provide a conservativeof h∆X̂add,±

estimate of the added noise. For each variance data point I also make a separate 2048 point twoquadrature measurement of a thermal state and sideband cooled state to infer that nsb , nadd andnm remain constant over the course of the entire measurement.

C.4.4

Direct measurement of mechanical variance

For a direct measurement of a mechanical state without any inference I use single quadratureamplification (Γ+ > 0 > Γ− > 0) to acquire histograms of a thermal, sideband cooled and squeezed
states of motion as in Section C.4.2 and Section C.4.3. I also separately use a two-quadrature
measurement to obtain histograms of a mechanical thermal state and sideband cooled state tomonitor that nm , nsb and nadd remain constant over the course of the entire measurement. Again,
I can then compute the ratio of the variance of a mechanical thermal and the squeezed state:r(φ) =

2h∆X̂thermi + h∆X̂add,− ih∆X(φ)2sq i + h∆X̂add,− i≈2h∆X̂thermi2h∆X(φ)2sq i + h∆X̂add,−i.

(C.50)

2
2
The approximation is valid because h∆X̂add,−
i is negligible compared to the h∆X̂thermi. I can thensolve for the total variance of the squeezed state (as measured by TEA):
2h∆X(φ)2 i = h∆X(φ)2sq i + h∆X̂add,−i,(C.51)

120
which is the plotted result in Figure 6.4(d). The blue points in Figure C.2 show r(φ) as a functionof tomography angle, while the red points are an equivalent measurement of a sideband cooled state(nearly mechanical vacuum) using single quadrature amplification. The black points are the ratio of
variance of a thermal state and a sideband cooled state obtained from a simultaneous measurementof both quadratures (Γ+ > 0 and Γ− = 0).

C.4.5

Estimating thermal occupancy of the mechanical oscillator

I independently measure the thermal occupancy of the mechanical oscillator in two separate
ways. First, I use the same procedure as in [151, 163], where a very weak continuous red detuned
pump is applied such that Γm >> Γ− . The ratio of the mechanical sideband power Pm and theresulting power circulating in the cavity Pc is measured. The temperature sweeps in Figure C.3a
demonstrate that the mechanical oscillator is thermalized to 18 mK (the temperature of the dilution
refrigerator) and allows extraction of the electromechanical coupling rate of g0 = 2π × (287 ± 11)
Hz.
Alternatively, I verify the temperature of the mechanical oscillator by using the blue detuned
pump as a quantum limited amplifier2
h∆X̂1,measi = eΓ+ t (nm + nadd + 1) .

(C.52)The measurement of a mechanical thermal state nm is repeated 2048 times at each different dilution
refrigerator base plate temperature to compute the variance. Drift in the gain is corrected at eachtemperature by measuring the exponential envelope of the amplification pulse to retrieve Γ+ . Figure
C.3b shows the result of a temperature sweeps performed in this way. The linear dependence ofthe variance on temperature demonstrates that the mechanical device is thermalized to the basetemperature of the dilution refrigerator (16 mK).

121
C.4.6

Gain calibration

From the calibration of the total noise, the gain of TEA can be estimated, and can thenbe compared with the expected results from theory. In Figure C.5 I compare this independently
measured gain to the gain expected from G− = e(Γ+ −Γ− )t and find reasonable agreement. The
deviation for large Γ+ /Γ− may be due to additional parametric effects becoming relevant at highpump powers [157, 158].

C.5

Quadrature extraction
I perform heterodyne detection (ωhet = 2π · 1.8 MHz) on the microwave frequency signal

emerging from the electromechanical circuit. There are two temporally separated components ofthe down converted microwave field. First, is a pulse with an exponentially growing envelope that
results from the amplification pulse. The second exponentially decreasing microwave pulse encodes
the amplified state of the mechanical oscillator that was transferred to the microwave field [164].
Both of these pulses encode the motion of the mechanical oscillator, but due to the additionalparametric effects described in C.1.2 the added noise is not exactly equivalent. Figure C.4 shows
a comparison of the added noise of TEA when measuring the amplified microwave field and thetransferred mechanical field. For all measurements in the main text the transferred mechanicalfield was used as it exhibited slightly better performance.

C.6

Tracking measurement and squeezing axes
The measurement and squeezing axes are controlled by the average phase φavg = (φ+ +φ− )/2of the red and blue detuned pumps. This phase is monitored by mixing both pumps down with thelocal oscillator (ωLO = ωr + ωhet ) and then measuring the phase of the resulting down converted
tones with frequencies Ω± = ωm ± ωhet on a separate channel of the data acquisition card (seeFigure C.6 for experimental schematic). I use this independent measurement of the phase of the
microwave tones to track the squeezing and measurement axes as a function of time. Due to a122small frequency difference between the local oscillator (Holzworth HS9000) and the red and blue
detuned pump microwave generators (Agilent E8257D) the measurement and squeezing axes drift
linearly with time. I measure this phase drift and correct all data points by rotating them back to
the initial measurement axis.

C.7

Quantum state tomography
Squeezed and thermal states are inherently Gaussian states, thus to infer the density matrixof these states I assume a Gaussian quasiprobability distribution and perform tomography as in[173]. Using the set of tomography data points xk (φj ), I define the two matrix quantities:D=

S T (φj )|wk ihwk |S(φj )hwk |S(φj )GRT (φj )|wk i + δη2(C.53)RT (φj )|wk ihwk |S(φj )xk (φj )2
hwk |S(φj )GRT (φj )|wk i + δη2(C.54)

X
k,j

R=

X
k,j

where S(φ) is the 2d rotation matrix, δη2q = (1 − ηm )/(2ηm ) is a function of the measurement
efficiency ηm and G is the covariance matrix to be iterated over. To find the covariance matrix
describing the data I start with a vacuum covariance matrix and then iterate such that:
G(i+1) = (Di )−1 Ri Gi Ri (Di )−1 ,(C.55)

from which the values of D, R and G are iteratively calculated. As was shown in [173] this procedure
will compute the covariance matrix that underlies the data. The covariance matrix for a squeezed
thermal state is given by:− sinh(2r) sin(φ)
1 cosh(2r) + sinh(2r) cos(φ)
G = nsq +,2− sinh(2r) sin(φ)
cosh(2r) − sinh(2r) cos(φ)(C.56)

with nsq , r and φ inferred from the tomographic reconstruction. The matrix elements of a squeezed
thermal state can be calculated as in [172]. The squeezing parameters inferred from tomography aredisplayed in Table C.2. The 90% confidence intervals are found from a case re-sampling bootstrap
algorithm [182].

123

C.8

Experimental apparatus

C.8.1

Electromechanical device

The electromechanical device is mounted to the base plate of a dilution refrigerator and held at
a temperature of T ≤ 18 mK. The design and operation of the electromechanical device is describedin [163, 151]. Table C.1 shows the relevant parameters of the electromechanical device.

C.8.2

Arbitrary microwave signal generation

I generate the red and blue detuned microwave tones ω± = ωc ± ωm with two separate
microwave generators (Agilent E8267D). The microwave generators are pulsed on and off with anarbitrary waveform generator (Tektonix AWG 5014C). To control the temporal envelope of thesignals the microwave generators act as local oscillators on two separate double-balanced mixers(Marki-0626H), and the IF port of each mixer is controlled with a baseband signal from the AWG.
I use square pulses with Gaussian edges (σ = 200 ns) to avoid driving the mechanical oscillator into
a coherent state. The pulses are then sent through two cavity filters to reduce pump fluctuationsat ωc . After filtering, the pulses are split into three separate lines. The first two lines are sent
into the fridge to pump the electromechanical circuit and to cancel the reflected microwave pumps.
The third line is mixed back down against the local oscillator for independent measurement of
the phase of the red and blue detuned microwave tones. See Figure C.6 for a full diagram of theexperimental schematic. The AWG controls the timing of all pulses, and runs at a 4 Hz repetition
rate when measuring thermal states (to allow the mechanical oscillator to reach equilibrium with
its environment) and a 500 Hz repetition rate when measuring squeezed and sideband cooled states.
To maintain phase coherence between all signals, the frequencies of all generators are set to integer
multiples of 500 Hz.

124
Table C.1: Parameters of the electromechanical device.
Symbolωc /2πκ/2πκext /2π
ωm /2πΓm /2πg0 /2πDescription
Circuit resonant frequency
Circuit decay rate
Circuit decay rate into the transmission line
Mechanical resonant frequency
Mechanical decay rate
Electromechanical coupling

9.3608 MHz
21 Hz
287 Hzthermal state
sideband cooledsqueezed

10

X2 (quanta) 1/2

Value and units
7.376 841 GHz
3.4 MHz
3.1 MHz

0

-10

-10

0
X1 (quanta) 1/2

10

Figure C.1: Typical measured histograms when inferring squeezed state variance. To
estimate squeezing a thermal state (magenta points, nm = 36) is compared with a sideband cooled
state (black points, nsb > .02) and a dissipatively squeezed state (blue). I estimate and diagonalize
the covariance matrix to retrieve the variances of the three data sets for comparison in the yfactor measurement. This measurement was performed using only the blue detuned pump tosimultaneously amplify both quadratures, thus noise equivalent to the mechanical zero-point motion
is added to all of these measurements.

Table C.2: Tomographically reconstructed squeezed state parameters
Symbolr
nsq
φ

Description
Squeezing parameter
Thermal occupancy
Squeezing angle

Value and units
0.661 ± 0.008
0.44 ± 0.05 quanta
1.481 ± 0.005

125

sideband cooled/both quadratures
sideband cooled/single quadraturesqueezed/single quadrature
150

r(φ)

100

50

0

0

100

φ (deg)200

300

Figure C.2: Transient electromechanical amplification Y-factor measurement. The blue
points show r(φ) (see Equation C.48) for a squeezed state measured with single quadrature TEA.
The red points are r(φ) for a sideband cooled state. The black points are r(φ) (see Equation C.42)
for a sideband cooled state measured using two-quadrature TEA.

126

(a)

6

×10-7

(b)
140

sweep 15

sweep 2

120

×10

Pm/Pc

4

nm+nadd+1 (quanta)

-7

3
2

100
80
60
40

1
0

20

0

20

40
Temperature (mK)

60

80

0

0

20
40
Temperature (mK)60

Figure C.3: Electromechanical temperature sweep. (a) Ratio of power in the red sideband
Pm to power circulating in cavity Pc due to red-detuned pump. This measurement is used to
extract the electromechanical coupling rate g0 = 2π × (287 ± 11) Hz and verify the steady state
thermal occupancy of the mechanical oscillator. The blue and green points represent two separate
temperature sweeps. The error bars on the green points are smaller than the data points. These
measurements also demonstrate that the mechanical oscillator is thermalized to the base temperature of the fridge. (b) Temperature sweep using blue-detuned pump as a quantum limited amplifier,demonstrating that the mechanical oscillator remains thermalized to the base temperature of thefridge. The linear fit extrapolated to nm = 0 yields the amplifier added noise of nadd = 1.4 ± 2.5,
which is consistent with the quantum limited amplification that was inferred in Section C.4.

127

〈 ΔX 2〉add,-

5added noise (dB)voltage (V)

(a)

10

U

0.5

X

-0.5
0

-80

40
time (μs)

80zero-point motion/vacuum fluctuations

0

〈 ΔU2
〉
add,-

-5

-10
1

1.1

1.2

1.3
Γ+ /Γ-

1.4

1.6

1.5

Figure C.4: Microwave and mechanical noise comparison. Electromechanical temperature sweep. Comparison of the added noise of TEA when directly measuring the microwave fieldU or the transferred mechanical field X. The blue diamonds are the added noise of TEA whenusing the microwave field to measure motion. The teal circles correspond to the case where thequadratures are extracted from the transferred mechanical field. The inset demonstrates the two
separate microwave pulses containing X and U. Comparable levels of added noise are achieved, but
are not equivalent due to the additional parametric effects (described in Section C.1.2) acting on
the mechanical oscillator.

10log10 (e(Γ+ −Γ− ) t )80

gain (dB)measured gain
60
40

20
0

1.0

1.2

Γ+/Γ-

1.4

1.6

Figure C.5: Transient electromechanical amplification gain. Characterization of TEA gain.
The blue points are inferred from the the added noise measurements. The red line is theory for theenergy gain using only independently measured parameters.

128

Tektronix AWG 5014c

Cancel3+27 dB-6 dB
L+20 dB

Local
oscillator

RI
L-20 dB

I

R

1 MHz

Q

Coherent
tone
20 dB
Ecco

Notch filter

50 Ω termination

Variable attenuator

DC block

Variable phase shifter

Filter cavity

SwitchEcco

10 GHz LPF

180° hybrid
Ecco-sorb filter

10 dB

Bandpass filter<25 mK

Ecco

Directional coupler

Ecco

Anritsu K250
bias tee

6 dB

800 mK

HEMT
20 dB

20 dB
Ecco

4K

200MHz2

180 kHz 80 MHz

1

20 MHzRed-detunedpump

20 MHz-6 dB

15 MHz+27 dBVdc+30 dBPump
3

1.9 MHz2b+32 dB1-10 dB

Blue-detuned
pump

Digitizer(AlazarTech
ATS 9462)c
Vrf

180 kHz

Analog channels
1 2 3 418.7 MHzd...

-20 dB

JPA

18.7 MHzaElectromechanical device

Figure C.6: Transient electromechanical amplification experimental schematic. (a) Electromechanical device (b) dc actuation line that allows for the tuning of the mechanical and microwave resonant frequencies, which is not used in this work. (c) rf actuation line allowing forrf modulation of the mechanical resonant frequency, which is not used in this work. (d) Roomtemperature microwave control and measurement electronics.