Quantum signal processing with mechanical oscillators
byReed W. Andrews
B.A., Macalester College, 2008
M.S., University of Colorado, 2012

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
2015This thesis entitled:
Quantum signal processing with mechanical oscillators
written by Reed W. Andrews
has been approved for the Department of Physics

Konrad Lehnert

Cindy Regal

Date

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

iii
Andrews, Reed W. (Ph.D., Physics)
Quantum signal processing with mechanical oscillators
Thesis directed by Prof. Konrad Lehnert

This thesis experimentally demonstrates how macroscopic mechanical oscillators can be used
to process and manipulate electromagnetic signals, i.e. light. Light shined upon an object cancause it to move or, as in the case of a mechanical oscillator, vibrate. This interaction allows light
to be converted between vastly different frequencies, and provides an opportunity to manipulate
the time and frequency content of electromagnetic signals. The following chapters describe theconstruction and measurement of systems that use electromagnetic resonators and highly tensioned
membranes as mechanical oscillators to manipulate electromagnetic signals with high efficiency
and extremely low added noise. These results show that mechanical oscillators can bring needed
signal processing capabilities to quantum information networks, where they can be used to connect
separate, physically distant, and otherwise mismatched cQED systems.

iv

Dedication

I would like to dedicate this dissertation to the car that I will hopefully be able to purchasewhen I am no longer limited to the salary of a graduate student.
Also, to my wife Anna. We make a good team, even if she is way cooler than me.

v

Acknowledgements

I’d like to thank Konrad Lehnert for hiring me into his lab back in 2009. Although I spent
my first year in graduate school far more concerned with coursework than with lab work, I like tothink I’ve made up for this in the years since. Konrad has an incredibly keen sense of how currents
and voltages propagate through circuits. I’ve tried to acquire some of this knowledge, but instead
I seem to have acquired a penchant for using over-the-top analogies to explain my feelings on a
subject.
I’d also like to thank Cindy Regal for a fruitful collaboration that combined microwaves,lasers, and membranes. I may still carry a small grudge from when she banned everyone in her lab
from working on (or speaking to me about) the project for a small portion of 2012, but once she
has read and signed this thesis all transgressions real and imagined will be forgotten.
I’d also like to thank all members of the JILA shop. I am a skeptical person and sometimes
ignored their advice. In most cases, however, if I was smarter I would have followed it.
I’d also like to thank all the labmates I’ve worked with throughout my time at graduate
school. Jen Harlow and Tobias Donner had the patience to let me make mistakes, even when it
slowed their progress. Joe Kerckhoff always asked a variety of questions, and in trying to answer
them I had to thoroughly consider many different experimental options. These questions helped
counter my eternal temptation to solve problems with the most familiar method rather than withthe best/cheapest/most reliable method. Adam Reed has displayed a great deal of patience withmy sometimes unhelpful direction and advice. Tom Purdy and Robert Peterson gave me a crashcourse in optics and building optical cavities, and didn’t object too strongly when I tried to map

vieverything onto notions of microwave network analysis. Kat Cicak taught me much of what I know
about fabrication techniques.
I’d also like to thank Stephen Pordes and everyone at the PAB for introducing me to experimental physics.
I’ve had the good fortune to meet and work with people who have been willing to share
their knowledge and expertise. Perhaps we’ll have future opportunities to collaborate. But, in themeantime, thank you for helping me earn a doctorate.

Contents

Chapter
1 Introduction1

1.1

Cavity optomechanics in quantum networks . . . . . . . . . . . . . . . . . . . . . . .

4

1.2

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2 Mechanical oscillators and light confined together
2.1

Tensioned membranes as mechanical oscillators . . . . . . . . . . . . . . . . . . . . .
2.1.1

2.2

8
9

Tension increases the quality factor of membrane vibrations . . . . . . . . . . 11

Electromagnetic resonators: cavities and circuits . . . . . . . . . . . . . . . . . . . . 14
2.2.1

Coupled-mode description of electromagnetic resonators . . . . . . . . . . . . 15

2.3

Mechanical oscillator displacement readout using electromagnetic fields . . . . . . . . 17

2.4

Dynamical backaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5

2.4.1

Weak-coupling regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4.2

Resolved-sideband regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Heisenberg-Langevin equations of motion . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5.1

Transfer function of optomechanical system . . . . . . . . . . . . . . . . . . . 25

2.5.2

Time-dependent system dynamics . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Microwave to optical frequency conversion with a mechanical oscillator

29

3.1

The need for a microwave-to-optical quantum link . . . . . . . . . . . . . . . . . . . 29

3.2

A frequency converter forms a two-port network . . . . . . . . . . . . . . . . . . . . 31

viii
3.3

3.4

Construction of electro-opto-mechanical frequency converter . . . . . . . . . . . . . . 33
3.3.1

Construction of microwave circuit. . . . . . . . . . . . . . . . . . . . . . . . 34

3.3.2

Construction of optical cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3.3

Initial characterization of frequency converter . . . . . . . . . . . . . . . . . . 39

3.3.4

Future improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Radiation pressure enables frequency conversion. . . . . . . . . . . . . . . . . . . . 43

3.4.1

Frequency conversion in the resolved-sideband and weak-coupling regime. . 433.4.2

Frequency conversion in the weak-coupling regime . . . . . . . . . . . . . . . 47

3.4.3

Noise added during frequency conversion . . . . . . . . . . . . . . . . . . . . . 49

3.5

Measurement network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.6

Microwave to optical frequency conversion of electromagnetic signals . . . . . . . . . 52

3.7

3.6.1

Bidirectional frequency conversion . . . . . . . . . . . . . . . . . . . . . . . . 52

3.6.2

Bandwidth of frequency conversion . . . . . . . . . . . . . . . . . . . . . . . . 54

3.6.3

Noise added during frequency conversion . . . . . . . . . . . . . . . . . . . . . 56

Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4 Temporal and spectral mode conversion of microwave signals

60

4.1

Analog quantum signal processing is needed in quantum networks

4.2

A tunable microwave-frequency optomechanical system . . . . . . . . . . . . . . . . . 62

4.3

4.2.1

Limitations of electrostatically actuated capacitors . . . . . . . . . . . . . . . 63

4.2.2

Construction of temporal and spectral mode converter . . . . . . . . . . . . . 65

4.2.3

Initial characterization of mode converter . . . . . . . . . . . . . . . . . . . . 67

Radiation pressure interaction is naturally tunable . . . . . . . . . . . . . . . . . . . 70
4.3.1

4.4

Relation between optomechanical coupling and mechanical resonant frequency 72

Measurement network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.4.1

4.5

. . . . . . . . . . 60

Calibration of microwave receiver . . . . . . . . . . . . . . . . . . . . . . . . . 74

Temporal mode conversion of microwave signals . . . . . . . . . . . . . . . . . . . . . 76

ix
4.6

4.7

Temporal and spectral mode conversion of microwave signals . . . . . . . . . . . . . 78
4.6.1

Wigner-Ville distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.6.2

Temporal and spectral mode conversion . . . . . . . . . . . . . . . . . . . . . 81

4.6.3

Operation at the quantum level . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5 Prosepects for quantum networking with mechanical oscillators
5.1

Conclusion

87. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Bibliography

90

Appendix
A Construction details of circuits and membranes

107

B Predicted frequency tuning of the temporal and spectral mode converter

110

C Measurement network for microwave-to-optical frequency converter112

C.1 Optical network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
C.2 Microwave network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
C.3 Signal generation and detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
D Measurement network for temporal and spectral mode conversion

117

D.1 Microwave receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
D.2 Microwave generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
D.3 Mode converter and resistive load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
D.4 Actuation line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

TablesTable
3.1

Parameters of electro-opto-mechanical frequency converter . . . . . . . . . . . . . . . 44

4.1

Parameters of temporal and spectral mode converter . . . . . . . . . . . . . . . . . . 70

A.1 Deposition and etch processes for frequency converter . . . . . . . . . . . . . . . . . 108
A.2 Deposition and etch processes for temporal and spectral mode converter . . . . . . . 109

Figures

Figure1.1

Laser interferometer for gravitational wave detection . . . . . . . . . . . . . . . . . .

2

1.2

A cavity optomechanical system. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3

Schematic of quantum information network . . . . . . . . . . . . . . . . . . . . . . .

5

2.1

A coordinate system for a square membrane . . . . . . . . . . . . . . . . . . . . . . . 11

2.2

Examples of tensioned membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3

Ringdown of a silicon nitride membrane . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4

Diagrams of electromagnetic resonators . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5

Displacement readout with electromagnetic fields . . . . . . . . . . . . . . . . . . . . 19

2.6

Radiation pressure-induced damping of a mechanical oscillator . . . . . . . . . . . . 22

2.7

Diagram of an optomechanical system . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1

Scattering parameters of a frequency converter . . . . . . . . . . . . . . . . . . . . . 32

3.2

Schematic of a microwave-to-optical frequency converter . . . . . . . . . . . . . . . . 34

3.3

Assembly of a ‘flip-chip’ microwave circuit . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4

Assembled flip-chip circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.5

Packaged flip-chip circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.6

Electro-opto-mechanical frequency converter . . . . . . . . . . . . . . . . . . . . . . . 40

3.7

Compatibility of microwave circuitry with optical frequency light. . . . . . . . . . . . 41

3.8

Vibrational modes of membrane used in frequency converter. . . . . . . . . . . . . . 42

xii
3.9

Three coupled harmonic oscillators form the frequency converter . . . . . . . . . . . 44

3.10 Frequency domain schematic of frequency conversion . . . . . . . . . . . . . . . . . . 46
3.11 Expected frequency conversion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.12 Frequency domain schematic of frequency conversion with gain . . . . . . . . . . . . 49
3.13 Minimum gain during frequency conversion . . . . . . . . . . . . . . . . . . . . . . . 50
3.14 Measurement network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.15 Measured scattering parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.16 Bidirectional and efficient frequency conversion . . . . . . . . . . . . . . . . . . . . . 55
3.17 Bandwidth of microwave-to-optical frequency conversion . . . . . . . . . . . . . . . . 57
3.18 Signal to noise ratio and added vibrational noise . . . . . . . . . . . . . . . . . . . . 58
4.1

Schematic of temporal and spectral mode converter . . . . . . . . . . . . . . . . . . . 62

4.2

Instability of electrostatically actuated capacitor . . . . . . . . . . . . . . . . . . . . 64

4.3

Completed temporal and spectral mode converter . . . . . . . . . . . . . . . . . . . . 66

4.4

Mode converter geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.5

Frequency tuning of mode converter . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.6

Time-dependent coupling between propagating electromagnetic fields and a mechanical oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.7

Protocol for temporal mode conversion . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.8

Measurement network for mode converter . . . . . . . . . . . . . . . . . . . . . . . . 74

4.9

Calibration of microwave receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.10 Test of microwave receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.11 Storage of microwave signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.12 Sub-optimal temporal mode conversion of a microwave signal . . . . . . . . . . . . . 79
4.13 Temporal mode conversion of a microwave signal . . . . . . . . . . . . . . . . . . . . 79
4.14 Protocol for temporal and spectral mode conversion . . . . . . . . . . . . . . . . . . 80
4.15 Temporal and spectral mode conversion of a microwave signal . . . . . . . . . . . . . 82

xiii
4.16 Noise added during mode conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.1

Electromagnetic signal router based on a mechanical oscillator . . . . . . . . . . . . 88

A.1 Construction details of microwave circuit for frequency conversion . . . . . . . . . . 108
A.2 Construction details of temporal and spectral mode converter . . . . . . . . . . . . . 109
B.1 Simplified mode converter geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
C.1 Measurement network for microwave-to-optical frequency conversion . . . . . . . . . 114
D.1 Measurement network for temporal and spectral mode conversion . . . . . . . . . . . 118
D.2 Circuit diagram of mode converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Chapter 1

Introduction

Light carries momentum. It was—and perhaps still is—a strange concept: light shined
upon an object can cause it to move, change course. This peculiar ability of light was perhaps firstobserved by Johannes Kepler when he noted that the tail of a comet is seemingly “an effluvium fromthe head, expelled through the rays of the sun...” [1, 2]. As early as 1754, laboratory experiments
attempted to confirm that light can move an object [3, 4]. But these initial experiments could
not disentangle the effect of light from that of residual air currents, in part because there was noquantitative prediction of how much momentum light should carry.
In 1873, Maxwell’s theory of electromagnetism predicted that electromagnetic waves–that
is, light—should indeed carry a very specific amount of momentum [5]. A few decades later, theexistence and predicted magnitude of the ‘radiation pressure’ of light were experimentally confirmed[6, 7, 8, 9]. Soon after the invention of the laser, radiation pressure was used to manipulate andcontrol small, micron-scale objects [10], and this spawned proposals that used radiation pressure tocool the motion of atoms [11, 12]. Radiation pressure cooling of trapped atoms was demonstrated
in the 1970s [13], and has since become widely used [14].
Far from manipulating single atoms, radiation pressure can control the motion of macroscopic
objects. The mathematical understanding of their control and manipulation developed in thecontext of highly sensitive interferometers designed to detect small strains induced by gravitational
waves [15, 16]. An early gravitational wave detector is diagrammed in Fig. 1.1. The position ofsuspended mirrors used in the interferometer is influenced by radiation pressure: Reflected light2

Figure 1.1: Laser interferometer for gravitational wave detection, from [16]. The radiation pressureof the light reflected from the mirrors affects their position. This effect limits the theoreticalsensitivity of such interferometers, and can also be used to control the motion of the mirrors.

transfers momentum to the mirrors and alters their position. At the quantum level, fluctuationsof the electromagnetic field create a fluctuating force on the mirrors that limits the sensitivity of
the interferometer as expected from the Heisenberg uncertainty principle [17, 18]. In addition to
disturbing the position of the mirror, radiation pressure changes the dynamical behavior of themirror’s vibrations and can provide damping or anti-damping of vibrations [19]. This phenomenon,
termed ‘dynamical backaction’, was first demonstrated in 1970 [20]. Initial work has given rise to
the field of cavity optomechanics, a field that focuses on manipulating macroscopic objects withradiation pressure.
The prototypical cavity optomechanical system contains an optical cavity with one mirror
harmonically bound, as diagrammed in Fig. 1.2. The mirror forms a macroscopic mechanicaloscillator whose motion couples to the electromagnetic field confined in the cavity. The radiationpressure coupling is characterized by a quantity G that expresses the resonant frequency shift of thecavity per unit motion of the mechanical oscillator. Originally, the goals of cavity optomechanics
were to study this interaction in order to make better detectors of force [21, 22], acceleration,
mass, and anything that could be somehow coupled to the motion of a macroscopic object [23, 24].
Other more exotic goals include using radiation pressure to generate nonclassical states of motion
in a macroscopic mechanical oscillator [25, 26] and using these nonclassical states to probe the3

MirrorSpringFigure 1.2: A cavity optomechanical system. The prototypical cavity optomechanical system contains a cavity formed from two high-reflectivity mirrors. One mirror is fixed, and the other isharmonically bound.

intersection of gravity and quantum mechanics [27, 28]. A first major goal of cavity optomechanics
was to use radiation pressure to cool a macroscopic mechanical oscillator to near its ground stateof motion.
To cool a mechanical oscillator to its ground state, the damping provided by the electromagnetic field must exceed the decoherence rate of the mechanical oscillator. This requirement is often
written as Γ > nm κm , where κm is the coupling of the mechanical oscillator to its thermal environment and nm ≡ 1/(e~ωm /kB Tenv − 1) is the phonon occupation number of the mechanical oscillator
for environment temperature Tenv and mechanical oscillator frequency ωm . The optomechanicaldamping Γ depends on the strength of a monochromatic electromagnetic field, often referred toas a pump: Γ ∝ n, where n is the number of photons induced in the cavity by the pump. When
Γ > nm κm , the system is ‘quantum-enabled’ as electromagnetic signals couple to the mechanicaloscillator faster than the mechanical oscillator acquires one phonon from its environment. Reaching
this regime took a concerted experimental and theoretical effort. Early work probed the feasibility of quantum-enabled devices and established methods for characterizing optomechanical systems[29, 30, 31, 32] and understanding their limitations [33, 34, 35]. In 2011 two systems achieved groundstate cooling of a macroscopic mechanical oscillator. One experiment used a microwave-frequency
superconducting circuit coupled to a vibrating aluminum film [36], and the other a photonic andphononic crystal cavity [37]. These achievements have opened up the way for more tantalizing and4exotic proposals to become reality.
However, the leap between using radiation pressure for ground state cooling and using it
to create nonclassical states of motion has proved challenging. Fundamentally, cooling exploits alinearized radiation pressure interaction that scales with pump power. The nonlinear interaction
necessary to create nonclassical states of motion requires Gxzp > κ, where xzp are the zero-point
fluctuations of the mechanical oscillator and κ is the energy decay rate of the electromagnetic
resonator [38, 39]. Even for state-of-the-art systems, reaching this new regime of optomechanicalcoupling seems a daunting task. New methods, new ways of thinking, and new creative designs[40] are beginning to emerge, and hopefully some will begin to probe quantum phenomena on amacroscopic scale.
If new methods do not work as hoped, there is still an incredible opportunity to use the linear,pump-enhanced radiation pressure interaction to manipulate (or process) electromagnetic signals.
For a quantum-enabled optomechanical system, the manipulation can be clean and extraordinarily
low noise as it only involves two degrees of freedom: the electromagnetic field and single mechanicaloscillator. A growing portion of the cavity optomechanics community explores electromagnetic
signal processing with quantum-enabled optomechanical systems. The signal processing capabilities
have many potential near-term applications [24], but in particular they can be used to extend theabilities of quantum networks.

1.1

Cavity optomechanics in quantum networks
Quantum information networks are physically extended networks that use quantum nodes

for processing information and quantum channels to link nodes [41], as diagrammed in Fig. 1.3.
Quantum networks allow communication between nodes with security guaranteed by physical laws
[42, 43], and can increase the capacity of communication channels beyond the classical Shannon
bound [44]. Technologies for the nodes of a quantum information network are usually quantum
two-level systems (qubits) such as nitrogen vacancy centers, Josephson junctions, or suitable atoms[45]. The communication channels that connect the nodes are almost invariably photonic, as5ChannelNode

Figure 1.3: Schematic of a quantum information network. The network consists of nodes that process and store quantum information, and quantum channels that are used to transmit information
between nodes [49, 41].

electromagnetic fields provide fast, low-loss transmission and they can be easily routed between
nodes with cables or optical fibers [46]. A quantum network constructed using these elements
requires control over both the temporal and spectral content of the quantum electromagnetic signals
exchanged between nodes. Temporal control allows entanglement to be deterministically distributed
throughout the network [47], and spectral control allows different types of qubits to be used in
the same quantum network. For example, spectral control is needed to connect qubits based on
Rubidium atoms with those based on Cesium, or with those based on quantum dots, as each typeof qubit interacts with electromagnetic fields at a different frequency [48]. Network componentsthat can arbitrarily shape the temporal and spectral content of quantum electromagnetic fields area necessary component of complex quantum information networks.
Quantum-enabled cavity optomechanical systems can control both the temporal and spectral
content of electromagnetic fields. Initial interest in using mechanical oscillators for this task was
motivated by the broad number of systems that couple to motion of macroscopic objects. It wasenvisioned that mechanical oscillators could mediate qubit-light interactions and thus enable long
distance quantum communication [50, 51, 52]. This led to the idea of ‘optomechanical transducers’and ‘photon-phonon translators’ that could be used in a quantum network. Experiments in quantum signal processing with mechanical oscillators have demonstrated that optomechanical systems
can be used to delay electromagnetic signals at both optical [53, 54] and microwave frequencies

6[55, 56]. Other work has demonstrated time domain protocols that store electromagnetic signals
as vibrations of a mechanical oscillator [57], and this capability has recently been extended toquantum-enabled systems [58, 59]. In addition, cavity optomechanical systems have been used toconvert electromagnetic signals between different optical frequencies [60, 61, 62]. There has been
an experimental push to extend the signal processing capabilities of cavity optomechanical systems
and to integrate them into quantum information networks.
This thesis advances two goals in quantum signal processing. First, it extends the frequency
conversion capabilities of cavity optomechanics to over four orders of magnitude in frequency,
and in particular demonstrates bidirectional frequency conversion with record-breaking efficiency
between signals at microwave and optical frequencies. As discussed in more detail in Chapter 3,
this capability can provide the backbone of a hybrid quantum network that can exploit the benefitsof all types of qubits, whether they be made from artificial atoms or real ones. Second, thisthesis extends the time-domain processing of electromagnetic signals to show that beyond storage
and delay, cavity optomechanics can be used to arbitrarily manipulate the temporal and spectral
content of microwave-frequency electromagnetic signals. This new capability provides a way tocreate microwave-frequency quantum information networks.

1.2

Contents
Chapter 2 introduces the mechanical oscillators used in this work and how they can be incor-

porated into electromagnetic resonators. It also establishes a mathematical basis for the radiationpressure interaction in cavity optomechanics. This introductory chapter is not meant to be a rigorous explanation of all the phenomena of cavity optomechanics, but instead lays a framework forunderstanding the experiments of later chapters. Chapter 3 describes a demonstration of bidirectional and efficient conversion of electromagnetic signals between microwave and optical frequencies,
and its relevance for quantum networks. Chapter 4 demonstrates that a mechanical oscillator in a
microwave circuit can be used to control both the temporal and spectral content of electromagnetic
signals. Finally, Chapter 5 looks to future experiments and how optomechanical systems can be7integrated into quantum information networks.

Chapter 2

Mechanical oscillators and light confined together

A mechanical oscillator is a prototypical physical system: An object of mass m, constrained
to move in one dimension, is bound by a Hooke’s law restoring force. If the mass interacts withlight at all, light shined upon the mass can give it momentum. For a single photon, the maximum
momentum transfer is ∆p = 2~ω/c, where ω is the frequency of the photon, and c is the speedof light. This additional momentum causes the oscillator to vibrate with amplitude 2~ω/(mωm c),
where ωm is the oscillation frequency of the mechanical oscillator. The mechanical oscillator ‘remembers’ the interaction until the vibrations of the oscillator decay away. This basic example
highlights the scales that affect the radiation pressure interaction. Mechanical oscillators with asmall mass are more easily driven by the momentum of light, and oscillators with low dissipation
serve as better memories for photons.
Yet even low mass, high quality mechanical oscillators still only feebly interact with light. A
single photon at 282 THz (1064 nm) only gives a vibrational amplitude of ∼10−22 m to a nanogram
mechanical oscillator with ωm = 2π × 1 MHz. The photon-induced vibrations are completelypmasked by zero-point fluctuations of the mechanical oscillator, xzp = ~/2mωm ≈ 10−16 m. The
momentum kick imparted by a single photon can be amplified by incorporating the mechanicaloscillator into an electromagnetic resonator. The resonator recirculates or recycles photons, giving each photon many interactions with the mechanical oscillator and amplifying the momentum
transfer. A suitable mechanical oscillator and a high quality electromagnetic resonator form thebasic cavity optomechanical system [23, 24].

9
Constructing low mass, high quality mechanical oscillators and integrating them with electromagnetic resonators is a key endeavor for creating quantum-enabled cavity optomechanical systems. Advances in microfabrication techniques have produced a variety of optomechanical systems[23]. For this thesis, we work exclusively with mechanical oscillators made from tensioned membranes. Section 2.1 describes some favorable properties of tensioned membranes, and section 2.2
describes how they can be integrated into electromagnetic resonators. Sections 2.3 through 2.5 givea mathematical framework for the radiation pressure interaction between electromagnetic fields andvibrations of a mechanical oscillator.

2.1

Tensioned membranes as mechanical oscillators
A membrane is a thin film with motion that is constrained by boundary conditions. Common

examples of membranes include drumheads and trampolines. The motion of a membrane can bemodeled as a collection of mechanical oscillators. For a membrane in two dimensions, its transverse
vibrations are described by a wave equation [63]:
−

2
D
∂2
∇2 W + T ∇2 W = ρ 2 W
h
∂t

(2.1)

where W describes the transverse displacement of the film at a particular location and time, h is
the thickness of the film and ρ its mass per unit area, and ∇2 is the two-dimensional Laplacian.
The parameters D and T control how much energy is required to deflect the film by a given
amount. The flexural rigidity, D, depends on the membrane’s mechanical properties and thickness:
D = Eh3 /(12(1 − ν 2 )), where E is the Young’s modulus of the membrane and ν its Poisson ratio.
The flexural rigidity scales as h3 , indicating that a thin membrane deforms more readily than athick one. The tension T of the membrane also resists deformations, and for a thin membrane orlarge tension, the tension dominates the dynamics of the transverse vibrations.
Consider a square membrane in the y-z plane where each side of the membrane has length L,
and is oriented as shown in Fig. 2.1. If we look for solutions W (y, z, t) that are harmonic in time

10
so that W (y, z, t) = W (y, z) × cos(ωt), then the equation for the transverse displacement becomes−

2D∇2 W + T ∇2 W = −ρω 2 W.
h

(2.2)

Eigenmodes of Eqn. 2.2 depend on the boundary conditions. For the membrane in Fig. 2.1 with all
four edges clamped, displacement W (y, z) must vanish and ∂W/∂y = 0 or ∂W/∂z = 0 at all fouredges. Under these clamped boundary conditions, the approximate eigenmodes can be expressedusing hyperbolic functions and sinusoids [64]. However, for a membrane with dynamics dominatedby tension with T  D/hL2 , the eigenmodes are approximately sinusoidal [65, 66]:
 nπz lπy
Wl,n (y, z) ∝ sin
× sin
L
L

(2.3)

where l and n are positive integers. In this high-tension approximation, the eigenvalues are given
by
2
ωl,n
≈


T π2 2l + n2 .
2
ρL

(2.4)

Each eigenmode of the transverse vibrations can be considered a mechanical oscillator with resonant
frequency ωl,n .
To completely map the vibrations of the membrane onto a collection of one-dimensional
mechanical oscillators, we must assign each eigenmode an effective mass, meff , and a single displacement coordinate, x. With these assignments, the potential energy U of an eigenmode should
2 x2 /2, the potential energy of a one-dimensional mechanical oscillator.
be given by U = meff ωn,m

Although the coordinate x can be assigned in any way, it should be physically motivated. For
RL
example, x can be the average displacement of the membrane: x(t) = L−2 0 Wl,n (y, z, t)dydz.
The convention used here is that x(t) = max(Wl,n (y, z, t)). However x is assigned, it is always truethat the potential energy of the membrane is well defined and can be expressed as [67, 23]
Z L
1 2
1
2
U = ωl,nx(t)2 .
ρWl,n (y, z, t)dydz = meff ωl,n2
2
0
For the chosen convention of x(t) = max(Wl,n (y, z, t)), this gives an effective mass of
2
Z L 
Wl,n
meff =ρdydz.
max(Wl,n )
0

(2.5)

(2.6)

11

x

0

L

z

L

y

Membrane

Figure 2.1: A coordinate system for a square membrane. The membrane is oriented in the y-z
plane and each side has length L.

For the eigenmodes of Eqn. 2.3, meff = m/4 where m is the physical mass of the membrane,RL
m = 0 ρdydz. With this mapping, each eigenfunction Wn,m of the membrane can be considereda mechanical oscillator with mass m/4 and frequency ωl,n . In a cavity optomechanical system,
we usually study the interaction between the electromagnetic field and one particular vibrational
eigenmode (or mode, for short) of the membrane.

2.1.1

Tension increases the quality factor of membrane vibrations

The quality factor of a membrane is crucial in determining whether a cavity optomechanicalsystem is quantum enabled. For a quantum-enabled system, Γ > nm κm , as introduced in Chapter 1. For temperatures kB Tenv > ~ωm , nm ≈ kB Tenv /~ωm . In this case, a quantum-enabled system
requires Γ > kB Tenv /~Q, where Q ≈ ωm /κm is the quality factor of a particular vibrational mode of
the membrane. Mechanical oscillators with higher quality factors reduce the optomechanical damping Γ necessary to make the system quantum-enabled. As outlined below, tensioned membranes
generally have higher quality factors than untensioned structures.
Consider a membrane as in Fig. 2.1 that is simply supported so that at the boundaries
W = 0. In this case, the eigenfunctions of Eqn. 2.1 are simply sinusoids as in Eqn. 2.3 and haveeigenfrequencies [63, 68]
2
ωl,n
≈

 Dπ 4 2
2
T π2 22l+n+
l + n2
2
4
ρL
hρL

(2.7)

12
where l and n are positive integers. Material properties affect the resonant frequency ωl,n . In particular, materials that absorb vibrational energy can be modeled with a complex Young’s modulus
where the imaginary part represents loss [69, 65]. Any imaginary part of the Young’s modulus
causes the resonant frequecy of the membrane ωl,n to also be complex, and this causes vibrations

of the membrane to damp over time as Wl,n (y, z, t) = Wl,n (y, z) × Re eıωl,n t . This loss directly
affects the quality factor of the vibrations [70, 71]:Im(ωl,n )1≡2.
Q
Re(ωl,n )(2.8)

Although the material used in the membrane is lossy, tension appears to have little or no imaginary
part. Thus, greater tension increases the real part of the resonant frequency and improves thequality factor of the vibrations as can be seen from Eqns 2.7 and 2.8. In particular, in the high√tension limit, Q ∝ T .
√Even though the result that Q ∝ T was found using simple boundary conditions, it isapproximately true for a variety of conditions. The same relationship between quality factor andtension has been predicted using clamped boundary conditions [65]. Moreover, tension has been
observed to increase the quality factor of vibrations [72, 66]. In real systems, however, the ultimate
quality factor can be limited by factors beyond membrane material loss. For example, imperfectboundary conditions allow membrane vibrations to radiate out into the environment [73]. This
limits the quality factor achievable through tension alone. Nonetheless, the high-Q vibrationalmodes achieved with tensioned membranes [65, 74] help create quantum-enabled optomechanicalsystems.
Two examples of membranes used in this thesis are shown in Fig. 2.2. The vibrational modesof the silicon nitride membrane are approximately described by Eqn. 2.3. Although the aluminum
membrane has a much different support structure and boundary conditions, its vibrational modes
still form mechanical oscillators. For both these types of membranes, tension develops from differential thermal contraction between the thin film and the substrate that supports it [68, 36], andthis tension allows high-Q vibrational modes. A ringdown measurement of a vibrational mode of13

Figure 2.2: Two examples of tensioned membranes. a, Photograph of a silicon nitride (Si3 N4 )membrane supported by a silicon frame. The lowest frequency vibrational mode for this membrane
is ωm ≈ 2π × 400 kHz. b, Scanning electron micrograph of an aluminum membrane on a sapphire
substrate. The lowest frequency vibrational mode for this membrane is ωm ≈ 20 MHz. A second
aluminum film can be seen under the membrane. This structure forms a parallel-plate capacitor
where one plate of the capacitor is free to vibrate. Image courtesy of John Teufel and Aric Sanders.

Position (V)0.1ωm=2� × 660 kHz
Q =1.5 × 107

0

-0.1

0

20
Time (s)40

Figure 2.3: Ringdown of a silicon nitride (Si3 N4 ) membrane. The membrane is cooled to <25
millikelvin in a dilution fridge and measured. The mechanical oscillator has a resonant frequencyof ωm = 2π × 660 kHz, and an energy decay rate of κm = 2π × 0.04 Hz, yielding a quality factor of
approximately 15 million.

a silicon nitride membrane is shown in Fig. 2.3. The fabrication of silicon nitride membranes isdiscussed in more detail in section 3.3, and the fabrication of aluminum membranes in section 4.2.2.

14

a

b
Membrane
L

L
MirrorCdMembrane

Figure 2.4: Diagrams of electromagnetic resonators. a, A Fabry-Perot cavity formed from two
mirrors. One of the mirrors is fixed; the other is part of a membrane that is free to vibrate.
Electromagnetic waves are coupled into and out of the cavity via a partially transmissive input
mirror. The length L of the cavity depends on the position of the membrane. b, An inductorcapacitor circuit. The capacitor is a parallel plate capacitor where one of the plates is a membranethat is free to vibrate. The separation d between the capacitor plates depends on the position of
the membrane. Electromagnetic waves are coupled into and out of the resonator through a nearby
transmission line, which is coupled to the inductor via a mutual inductance.

2.2

Electromagnetic resonators: cavities and circuits
In order to enhance the radiation pressure interaction, we must couple the membrane to anelectromagnetic resonator. The membrane can be coupled to a Fabry-Perot cavity as diagrammedin Fig. 2.4a. This resonator geometry is often used to confine optical frequency electromagnetic
fields [75], but in principle it can confine electromagnetic fields at any frequency [70]. Fabry-Perotcavities have been used to create high-finesse cavities for frequencies of several tens of gigahertz[76], but at lower frequencies the cavities become large and unwieldy as the wavelength of theelectromagnetic fields increases. For these lower frequencies, other types of cavities (such as empty
metal cylinders) more effectively confine electromagnetic fields [77]. Taken to the extreme, electricalcomponents such as inductors and capacitors can be used to confine electromagnetic fields at theinductor-capacitor resonant frequency. This last type of resonator, which is commonly used in thisthesis, is less of a ‘cavity’ and more of a ‘circuit’. A membrane can be integrated into a circuit bymaking it one plate of a parallel plate capacitor, as diagrammed in Fig. 2.4b.
Electromagnetic resonators based on circuit components can be extremely compact with

15
dimensions much less than a wavelength. For example, the length L of the Fabry-Perot cavity
diagrammed in Fig. 2.4a is usually many wavelengths but at a very minimum L = λ/2 where λis the wavelength of the electromagnetic field at the cavity’s resonant frequency of ω0 . However,there is no such restriction on the separation between the capacitor plates used in the microwave
circuit and generally d  λ. The length scales of electromagnetic resonators are critical design
parameters as they set the scale of the optomechanical interaction. For the Fabry-Perot cavity,
when the membrane moves and alters the cavity length by a distance ∆L, the resonant frequency
is changed by ∆ω0 /∆L = ω0 /L. For the microwave circuit, an equivalent amount of motion ∆dalters the resonant frequency by ∆ω0 /∆d = ω0 /2d. For both resonators, smaller length scales yieldlarger optomechanical coupling.
Electromagnetic resonators based on circuit components are suitable for frequencies . 100
GHz, as low-loss circuit components for these frequencies can be microfabricated using superconducting materials. Such resonators are often formed by patterning metal films on dielectric
substrates [78, 79]. Electromagnetic waves are coupled into and out of the resonator via a nearby
transmission line, as is discussed in sections 3.3.1 and 4.2.2. The coupling between the resonatorand the transmission line serves the same purpose as a partially transmissive mirror in a FabryPerot cavity. When cooled to cryogenic temperatures so that the metal film superconducts, thesecircuits form high quality microwave-frequency electromagnetic resonators.
Although many different forms of electromagnetic resonators exist, they all can be mathematically handled the same way. A confined electromagnetic mode couples to freely propagating
electromagnetic fields. In the following description of electromagnetic resonators, based on coupledmode theory [80], I’ll refer to a microwave resonator like the one diagrammed in Fig. 2.4b, but themathematics applies equally well to other electromagnetic resonators.

2.2.1

Coupled-mode description of electromagnetic resonators

Energy in an electromagnetic resonator can be described with a complex field amplitude a(t),where the normalization of a(t) is chosen so that |a(t)|2 gives the number of photons contained in

16
the resonator at time t. As the field amplitude evolves, energy in the resonator decays away. Someof this energy exits as electromagnetic fields that propagate down the transmission line. This loss
can be chosen during resonator design, and I’ll refer to it as the external loss rate. However,energy in the resonator can also be absorbed by, for example, two-level systems that form on the
microwave circuit’s surface [81, 82]. This type of loss is intrinsic to the exact construction of theelectromagnetic resonator, and I’ll refer to it as internal loss. The total energy loss rate is the sumof internal and external loss rates: κ = κint + κext . Incorporating this loss rate into an equation of
motion for field amplitude a(t),κȧ(t) = −ıω0 −a(t)2√aout (t) = κext a(t)


(2.9)

where aout is the amplitude of the field in the transmission line, and ω0 is the resonant frequency of
the resonator. The field amplitude aout is normalized so that |aout |2 has units of photons/second.
If κext = κ, then all the energy initially in the resonator will come out of the transmission line:R∞|a(0)|2 = 0 |aout (t)|2 dt.
Importantly, all loss channels for the resonator field a(t) also drive it. Including driving terms
into Eqns. 2.9 allows for a fully quantum description of a resonator that can be formally derived
with input-output theory [83] (see section 2.5). Neglecting these driving fluctuations, we can still
coherently drive the resonator via its coupling to a transmisssion line. This driving field can beincorporated as√κȧ(t) = −ıω0 −a(t) + κext ain (t)2√aout (t) = κext a(t) − ain (t)(2.10)

where ain is the incoming field amplitude, normalized to have units of photon flux like aout . The
output field now consists of the reflected input field and fields emitted by the resonator. Eqns. 2.10
provide a description of electromagnetic resonators such as those shown in Fig 2.4. However,Eqns. 2.10 can be modified to incorporate different configurations of resonators and transmission

17lines [23].

2.3

Mechanical oscillator displacement readout using electromagnetic fields
When a mechanical oscillator is embedded in an electromagnetic resonator as diagrammed inFig. 2.4, the position of the oscillator affects the resonator’s frequency. For example, an oscillator
with a position x(t) = x0 sin(ωm t) will modulate the resonant frequency and the equation of motion
for the resonator field is now√κȧ(t) = −ı (ω0 + Gx0 sin(ωm t)) −a(t) + κext ain (t)2(2.11)

where G = ∂ω0 /∂x is a coupling constant with units of resonant frequency shift per unit motion.
√
For the circuit diagrammed in Fig. 2.4b, with resonant frequency ω0 ≈ 1/ LC, G ≈ ω0 /2d whered is the separation between the capacitor plates. Similarly, G ≈ ω0 /L for the Fabry-Perot cavity
diagrammed in Fig. 2.4a.
Motion of the mechanical oscillator modulates the amplitude and phase of the energy inthe electromagnetic resonator, and this motion appears in the electromagnetic field that leaves
the resonator via the transmission line. This provides a way to observe motion of the mechanicaloscillator via electromagnetic fields.
To observe mechanical motion, we pump the resonator with a non-zero ain (t). Information
about the mechanical oscillator will be contained in aout (t), and this can be found by solving
Eqn. 2.11. The homogeneous part of equation 2.11 has solutions



κ
ıGx0
ah (t) ∝ exp − − ıω0 t +cos(ωm t) .
2
ωm

(2.12)

Using the ansatz a(t) = ã(t)ah (t), we have
√
˙
ã(t)
= κext āin exp

κ2+ ıω0 − ıωin




ıGx0
t−
cos(ωm t) ,ωm

(2.13)

where I’ve assumed a monochromatic pump, ain (t) = āin e−ıωin t . Although Eqn. 2.13 can be directly
integrated using a cylindrical wave expansion [84], if the motion of the mechanical oscillator shifts

18
the electromagnetic resonator’s frequency by much less than ωm (Gx0  ωm ), then
˙ã(t) ≈ √κext āin e( κ2 −ı∆)t 1 − ıGx0 eıωm t + e−ıωm t
2ωm(2.14)

where ∆ = ωin −ω0 . Eqn. 2.14 can be directly integrated and used to find the output field amplitude.
Once transients have decayed away, the output field amplitude isaout (t) =ı∆ + κext − κ/2āin e−ıωin t−ı∆ + κ/2
Gx0
κext
āin
+
2 −ı∆ + κ/2e−ı(ωin +ωm )te−ı(ωin −ωm )t−−ı(∆ + ωm ) + κ/2 −ı(∆ − ωm ) + κ/2!
.

(2.15)The output field consists of the reflected input field at frequency ωin , and two modulation‘sidebands’ at frequencies ωin + ωm and ωin − ωm , as diagrammed in Fig. 2.5. These sidebands areweighted by the cavity Lorentzian of (−ıω + κ/2)−1 where ω is the sideband frequency. The powerin the upper sideband at frequency ωin + ωm is, in photons per second,
Pusb =

Gx0
2

2

κ2ext |āin |2∆2 + κ2 /4



1(∆ + ωm )2 + κ2 /4


(2.16)

And similarly, the power in the lower sideband at frequency ωin − ωm is, in photons per second,
Plsb =

Gx0
2

2

κ2ext |āin |2∆2 + κ2 /4

1(∆ − ωm )2 + κ2 /4


(2.17)The power in the sidebands provides information about the amplitude of the oscillator’svibrations, and the phase of the sidebands relative to the pump provides information about thephase of the mechanical oscillator’s vibrations. A larger pump field, āin increases the power in the
sidebands, giving more and more precise information about the motion of the mechanical oscillator.
Eventually, however, any measurement of the mechanical oscillator, if done stongly enough, must
provide a momentum kick to the mechanical oscillator so that the position-momentum uncertainty
relation is preserved [85, 72, 86]. This necessary dynamic is not included in this simple modelof displacement readout. To be accurate, the radiation pressure force must be included in themathematical description.

19

Figure 2.5: Displacement readout with electromagnetic fields. A monochromatic pump at frequency
ωin is injected into the electromagnetic resonator. The field emitted from the resonator is modulated
by motion of the mechanical oscillator, producing ‘sidebands’ at frequencies ωin + ωm and ωin − ωm .
The amplitude of the sidebands is filtered by the resonator’s frequency response, shown as the black
curve.

2.4

Dynamical backaction
The photons in the electromagnetic resonator exert a force on the mechanical oscillator

proportional to the average number of photons |a|2 in the electromagnetic resonator. For anelectromagnetic resonator, the force takes on the general form Frp = −~G|a|2 (the negative sign isa convention that a greater radiation pressure force decreases the resonant frequency of the cavity
for positive G). Another way to view this force is that as the mechanical oscillator vibrates, it does
work on the electromagnetic field [68]. Incorporating the radiation pressure force into an equationof motion for a damped mechanical oscillator, the equations of motion for the cavity opomechanicalsystem are√κȧ(t) = −ı (ω0 + Gx(t)) −a(t) + κext ain (t)22ẍ(t) = −κm ẋ(t) − ωmx(t) − ~G|a(t)|2 /meff(2.18)

where meff is the effective mass of the mechanical oscillator [23, 67], km its damping rate, ωm its
resonant frequency.
The equations of motion are nonlinear—the postion x(t) depends on the square of the field
amplitude a(t). The radiation pressure interaction can be considered a type of Kerr nonlinearity,

20
as the electromagnetic resonant frequency is a function of the intensity of light in the resonator. In
most experiments to date, the radiation pressure force is enhanced by driving the electromagnetic
resonator with a strong, monochromatic pump. This pump effectively consumes the nonlinearity,
thus rendering the equations of motion for the electromagnetic and mechanical modes linear, but
it also enhances the radiation pressure interaction.
We mathematically pump the resonator with ain (t) = āin eıωin t , as in section 2.3. This pump
populates the electromagnetic resonator with a steady state field amplitude āe−ıωin t , and creates a
net force on the mechanical oscillator. The equilibrium field amplitude ā and oscillator position x̄
can be found by setting the time derivatives Eqns. 2.18 to zero.
√ā =κext āin,−ı∆ + κ/2x̄ =−~G|ā|2,2
meff ωm(2.19)

where ∆ = ωin − ω0 − Gx̄. We can examine the dynamics of the fluctuations of the field amplitude
and oscillator postion about their steady state values, provided the steady-state solution is stable[87, 88]. Assuming a steady state solution exists, let a → e−ıωp t (ā + a(t)) and x → x̄ + x(t) so thata(t) and x(t) represent fluctuations about equilibrium. Using these replacements in Eqns. 2.18, thedynamics of the fluctuations are described byȧ(t) = (ı∆ − κ/2)a(t) − ıGāx(t)2ẍ(t) = −ωmx(t) − κm ẋ(t) −~G(āa∗ (t) + ā∗ a(t))
meff(2.20)

where I’ve linearized the equations of motion by assuming the fluctuations |a(t)|  |ā|. Although
the pump simplifies the dynamics of the cavity optomechanical system, it also greatly increases thecoupling between the electromagnetic and vibrational modes. Every occurrence of the coupling G
in Eqns. 2.20 is now accompanied by ā or its conjugate, meaning that a stronger pump effectively
increases the magnitude of the radiation pressure interaction.
The dynamics of the fluctuations a(t) and x(t) are easily observed in the frequency domain.

21
Transforming Eqns. 2.20 using f (ω) =R∞−∞ f (t)eıωt dt,a(ω)(−ı(∆ + ω) + κ/2) = −ıGāx(ω)(2.21)a∗ (ω)(ı(∆ − ω) + κ/2) = ıGā∗ x(ω)(2.22)

2x(ω)(−ω 2 + ωm
− ıωκm ) = Frp (ω)/meff(2.23)

where Frp (ω) is the force due to radiation pressure.
Frp (ω) = −~G(āa∗ (ω) + ā∗ a(ω))


∆+ω
∆−ω
2
2
x(ω)
+
= −~G |ā|(∆ + ω)2 + κ2 /4 (∆ − ω)2 + κ2 /4


κ/2
κ/2
+ı~G2 |ā|2−x(ω)(∆ + ω)2 + κ2 /4 (∆ − ω)2 + κ2 /4(2.24)The linearized radiation pressure force, Frp is only non-zero when the mechanical oscillator moves
away from its equilibrium position: Frp (ω) ∝ x(ω). As such, the radiation pressure force consistsof a real part that alters the mechanical resonant frequency and an imaginary part that alters
mechanical damping.
Consider the susceptibility of the mechanical oscillator to an external force F (ω). The positionof the oscillator can be described using x(ω) = χ(ω)F (ω), where χ(ω) is the mechanical oscillator’ssusceptibility. In the presence of Frp (ω),χ(ω)−1 = −ω 2 + (ω + Ω(ω))2 − ıω(κm + Γ(ω))(2.25)

where Ω(ω) = −Re[Frp (ω)]/(2meff ωm x(ω)) and Γ(ω) = Im[Frp (ω)]/(meff ωx(ω)). This radiationpressure-induced change in mechanical frequency and damping is the essence of dynamical backaction.

2.4.1

Weak-coupling regime

If the electromagnetic resonator has a broader frequency response than the mechanical oscillator, we can approximate Ω(ω) and Γ(ω) in Eqn. 2.25 as frequency-independent parametersnear the mechanical oscillator’s resonant frequency. This approximation requires weak radiation

22

Figure 2.6: Radiation pressure-induced damping of a mechanical oscillator. The oscillator couples
to its environment through its intrinsic loss at rate κm , but also to propagating microwave fields at
rate Γ.

pressure coupling: κm + Γ  κ. In this weak-coupling regime, the mechanical oscillator experiencesa radiation pressure-induced frequency shift and damping of∆ + ωm∆ − ωmΩ = |g|+(∆ + ωm )2 + (κ/2)2 (∆ − ωm )2 + (κ/2)2


κ
κ
2
Γ = |g|−(∆ + ωm )2 + (κ/2)2 (∆ − ωm )2 + (κ/2)22

where g =p

(2.26)~/2meff ωm Gā.

The additional damping Γ can be either positive or negative. For positive damping, which isused throughout this thesis, energy stored as vibrations of the mechanical oscillator leaves throughtwo separate channels as diagrammed in Fig. 2.6. First, energy is lost to the mechanical oscillator’s
environment at a rate κm . Second, energy is converted into electromagnetic fields at a rate Γ. The
additional damping provided by radiation pressure can be used to cool the mechanical oscillator:
Motion induced by the oscillator’s thermal environment is rapidly converted into electromagnetic
fields, leaving the oscillator with less net motion as if it has been cooled [29, 30, 31].

2.4.2

Resolved-sideband regime

The additional damping Γ has both positive and negative contributions as seen in Eqn. 2.26.
Γ = Γ+ − Γ− whereΓ+ = |g|2

κ,(∆ + ωm )2 + (κ/2)2

Γ− = |g|2

κ.
(∆ − ωm )2 + (κ/2)2(2.27)

23
The negative contribution to damping Γ− amplifies the mechanical oscillator’s motion and thepositive contribution Γ+ damps and cools the oscillator. The ratio of the amplification rate to thecooling rate determines how much entropy can be removed from the mechanical oscillator [33, 34].
When 4ωm  κ, the system is in the resolved sideband regime and to a good approximation
either Γ+ or Γ− can be heavily suppressed. In the resolved-sideband regime with ∆ = −ωm , Γ− isnegligible and Γ = 4|g|2 /κ. Cavity optomechanical systems in this regime have been used to coola mechanical oscillator to near its ground state of motion [35, 36, 37].

2.5

Heisenberg-Langevin equations of motion
The electromagnetic resonator and the mechanical oscillator have similar dynamics: Bothcan be modeled as harmonic oscillators. However, sections 2.3 and 2.4 emphasize the position of
the mechanical oscillator. To highlight the similarity between the electromagnetic resonator andthe mechanical oscillator mode, we can reformulate Eqns. 2.20 using a field amplitude c(t) rather
than displacement x(t). The two representations are related through x(t) = xzpf (c(t) + c∗ (t)),
pwhere xzpf = ~/(2meff ωm ) are the zero-point fluctuations of the mechanical oscillator. Using thisrelation,ȧ(t) = (ı∆ − κ/2)a(t) − ıg(c(t) + c∗ (t)) +√κext ain (t) +ċ(t) = (−ıωm − κm /2)c(t) − ı (ga∗ (t) + g ∗ a(t)) +√√κint a0 (t)κm cin (t)(2.28)

Eqn. 2.28 includes all the possible couplings that can drive the electromagnetic resonator or themechanical oscillator, as diagrammed in Fig. 2.7. The terms ain (t) and a0 (t) drive the electromagnetic resonator through the coupled transmission line and internal resonator loss, respectively.
These terms are in the same rotating frame as the fluctuations a(t). Additionally, cin (t) drives
the mechanical oscillator through its internal loss mechanism. Eqns. 2.28, along with the relation
aout (t) =√κext a(t) − ain (t), form a complete description of linear optomechanics.

The resonator field amplitude a(t) and a∗ (t) and the mechanical oscillator amplitude c(t) andc∗ (t) can be directly promoted to creation and annihilation operators. The resulting Heisenberg-

24

Figure 2.7: An optomechanical system consists of two coupled harmonic oscillators: an electromagnetic resonator, described by field amplitude a(t), and a mechanical oscillator described byfield amplitude c(t). Both the electromagnetic resonator and the mechanical oscillator are coupled through some intrinsic loss rate to their environment, and the electromagnetic resonator isadditionally coupled to a transmission line.

Langevin equations of motion can be formally derived starting from a Hamiltonian formulation andinput-output theory [89, 83, 90].
Eqns. 2.28 are first order, linear differential equations. As such, much of linear control theory[91, 92] can be applied. The framework of control theory provides an extremely general and powerfultool for predicting and modeling the behavior of optomechanical systems. System dynamics aredescribed with a state space model that has the general structureẋ(t) = A(t)x(t) + B(t)u(t)y(t) = C(t)x(t) + D(t)u(t)(2.29)

Here, u(t) is a vector of inputs that drive the system, y(t) is a vector of outputs, and x(t) is thevector that describes the state of the system. The matrix A describes how the state of the systemevolves in time, and B controls how the inputs to the system affect the state of the system. Matrices
C and D control how the state of the system and the inputs map onto the outputs of the system.

25
The dynamics of a cavity optomechanical system can be described usingain (t)a0 (t)  a(t) 












a
(t)
c
(t)
 c(t) 


out
in






x=
, u = , y =  ††a† (t)
a (t)
aout (t) in 




 † 
†


c (t)
 a0 (t) c†in (t)(2.30)−ıg(t)
0−ıg(t) ı∆ − κ/2




√


∗
κext 000−ıωm − κm /2
−ıg
0
 −ıg

, C = 
A=




√
∗
∗


0
0
κext 00ıg−ı∆ − κ/2ıg (t) 



ıg ∗
0
ıg
ıωm − κm /2



√
√
κ
κ
0
0
0
0
int
 ext





√


0
κm
0
0
0 
−1
0
0
0
0
0
 0

, D = 
B=

.


√
√
 0

0
0
κ
κ
0
0
0
0
−1
0
0
ext
int




√
0
0
0
0
0
κm

(2.31)

These vectors and matrices when used in Eqns. 2.29 reproduce Eqns. 2.28 and the input-output
relation aout (t) =

2.5.1

√

κext a(t) − ain (t).

Transfer function of optomechanical systemIf matrices A, B, C, and D are time invariant, a system transfer function Ξ(ω) compactly
describes how inputs u are taken to outputs y: y(ω) = Ξ(ω)u(ω). 1

1

Ξ(ω) = C (ıωI − A)−1 B + D(2.32)

R∞f (t)e−st dt. This is equivalent to Fourier

The transfer function is computed in Laplace space, where f (s) =space so long as we take s → −ıω.

−∞

26
In the context of cavity optomechanics, Ξ(ω) describes how a signal at frequency ω above thepump will propagate from an input to the outputs. This method is used in chapter 3 to describe
the dynamics of microwave to optical conversion.

2.5.1.1

Transfer function and scattering parameters

For time-invariant systems, the transfer function can be measured in the lab by injecting
a monochromatic signal and measuring the emitted signal. Comparing the emitted signal to theinjected signal yields the amplitude and phase shift of the signal which can be recast as the realand imaginary parts of one component of the transfer function matrix Ξ(ω). This measurement iscommonly performed in microwave-frequency laboratories with a network analyzer [70]. The close
tie between mathematical model and experimental characterization make the transfer function an
extremely useful tool. In microwave network analysis, the transfer function Ξ(ω) is sometimes
referred to as the scattering matrix, and each element of the matrix as a scattering parameter.

2.5.1.2

Noise analysis using the transfer function

Noise properties of an output electromagnetic field can also be predicted using the transfer
function formalism. The spectral density S(ω) of the output fields is [93, 94]
DE
DE
2πS(ω)δ(ω − ω 0 ) = [y(ω 0 )]† yT (ω) = Ξ∗ (ω 0 ) [u(ω 0 )]† uT (ω) ΞT (ω)(2.33)

More compactly, S(ω) = Ξ∗ (ω)ΣΞT (ω) where Σ = (2πδ(ω − ω 0 ))−1 [u(ω 0 )]† uT (ω) .
The correlation matrix Σ describes the noise properties of the inputs. If the system is driven
by noise from its thermal environment—that is, blackbody radiation or Johnson noise [95]—theelements of Σ have a simple form [93]. For example, [âin (ω 0 )]† âin (ω) = 2πδ(ω − ω 0 )nin , where nin
is the Bose-Einstein occupation factor for the transmission line modes. Thus, the (1, 1) componentof Σ will have the value nin = 1/(e~ω/kB T − 1) where T is the temperature of the modes in thetransmission line. If the electromagnetic resonator has a narrow freqeuency response (κ  ω0 ), we
can approximate ω ≈ ω0 in the expression for nin . For kB T  ~ω0 , nin ≈ kB T /~ω0 , which is the27same as Johnson noise expressed in units of photons/sec·Hz. If all the ports of the optomechanicalsystem are driven by thermal noise, then




0
0
0
0 
nin 0
 0 n0 0
0
0
0 






0 nm
0
0
0 
0
.
Σ=


0
0
0 nin + 1
0
0 






0
0
0
0
n0 + 1
0 




0
0
0
0
0
nm + 1(2.34)In order to perform low-noise measurement and manipulation of a mechanical oscillator using an electromagnetic resonator, it is crucial to minimize nin and n0 . For a microwave circuitwith ω0 = 2π × 7 GHz, n0 < 1 requires a temperature of T < 300 mK. For this reason, manipulation of mechanical oscillators using a microwave frequency electromagnetic resonator is often
performed at low temperatures of <50 mK attained with a dilution refrigerator [96]. Furthermore,
the transmission line connecting to the electromagnetic resonator must be driven by Johnson noisefrom a cold (ideally <50 mK) source resistance so that nin  1. Achieving this in practice means
utilizing components like cryogenic attenuators, directional couplers, and circulators as shown inAppendix D.
Even though the electromagnetic resonator can be driven by vacuum fluctuations (nin = n0 =0), the mechanical oscillator can have a significant driving term nm  1. Even at a temperature
T = 50 mK, a mechanical oscillator with frequency ωm = 2π ×10 MHz will have an average thermal
occupancy of nm ≈ 100. This additional noise can be avoided by using mechanical oscillators withresonant frequencies of several GHz [97, 98, 99]. For mechanical oscillators with resonant frequencies
from kHz to many MHz, the thermal fluctuations driving the mechanical oscillator are almost always
the largest noise source for the optomechanical system.
Finally, I want to point out that the ‘quantum’ aspect of this input-output formalism is theasymmetry between the operators for the input fields. For example, [âin (ω 0 )]† âin (ω) = 2πδ(ω −ω 0 )nin and âin (ω 0 )[âin (ω)]† = 2πδ(ω − ω 0 )(nin + 1). This asymmetry exists, of course, because the28
creation and annihilation operators do not commute [94]: ain (ω 0 ), [ain (ω)]† = 2πδ(ω −ω 0 ). Without
this asymmetry in the correlation matrix Σ, all the dynamics and noise properties of the cavity
optomechanical system are entirely classical.

2.5.2

Time-dependent system dynamics

Although in our experiments the matrices B, C, and D in Eqn. 2.29 are approximately time
independent, A can be time dependent. The mean field amplitude in the electromagnetic resonator,ā, is determined by the strength of the pump, as seen in Eqns. 2.19. By varying the amplitude of
the pump, g = Gāxzpf can be made time dependent so that g → g(t) and A → A(t). Additionally,
the pump has a phase and this determines the phase of ā. Altering the phase of the pump by πchanges the sign of ā and this changes the sign of the pump-enhanced radiation pressure coupling.
These dynamics provide an extremely convenient way to control and tune the radiation pressureinteraction.
For time domain signal processing demonstrated in chapter 4, it is necessary to modulate
the coupling g(t) between the electromagnetic resonator and the mechanical oscillator. Although
the transfer function approach of section 2.5.1 does not fail with a time-dependent A matrix, it is
no longer as useful in understanding the system dynamics. Instead, equations 2.29 can be directly
integrated or numerically solved to predict the system dynamics. For example, equations 2.29
are discretized and numerically solved to predict the effect of the time-dependent optomechanicalinteraction used in chapter 4.

Chapter 3

Microwave to optical frequency conversion with a mechanical oscillator

The radiation pressure interaction between light and a mechanical oscillator is inherently
broadband: electromagnetic signals at most frequencies can interact with a mechanical oscillator, as initially indicated in chapter 2. In this chapter, we utilize the broadband nature of theradiation pressure interaction to convert electromagnetic signals between microwave and opticalfrequencies. We couple two electromagnetic resonators—one at a microwave frequency, one at an
optical frequency—to the same mechanical oscillator. The dynamical backaction of the resonatorfields on the mechanical oscillator, as introduced in section 2.4, allows a microwave signal sent into
the electro-opto-mechanical frequency converter to stream through the mechanical oscillator andemerge as an optical signal, and vice-versa. This process can be remarkably efficient, and we demonstrate photon number conversion efficiencies of ∼10%. Although the demonstrated conversion adds
significant noise due to thermally driven vibrations of the mechanical oscillator, conversion can inprinciple be noiseless and we anticipate nearly noiseless frequency conversion is achievable withmodest modifications.

3.1

The need for a microwave-to-optical quantum link
Modern communication networks manipulate information at several gigahertz with micropro-

cessors and distribute information at hundreds of terahertz via optical fibers. The division of labor
between these different frequencies did not occur by chance, but was in part directed by nature.
Silica has a natural minimum of absorption and scattering of light at wavelength λ = 1550 nm. This30
has led to low-loss optical fibers suitable for distributing signals at this wavelength [100]. Similarly,
the rise of silicon-based integrated circuits has enabled low-cost manipulation of electrical signals[101]. Combining the processing power available to electrical signals with the efficient distribution
available to optical signals has ushered in an era of high-speed internet connectivity.
A similar frequency dichotomy is developing in quantum information processing. Superconducting qubits operating at several gigahertz have recently emerged as promising high-fidelity andintrinsically scalable quantum processors [102, 103, 104]. Conversely, optical frequencies provide
access to low-loss transmission [46, 100] and long-lived quantum-compatible storage [45, 105, 106].
Converting information between gigahertz-frequency ‘microwave light’ that can be deftly manipulated and terahertz-frequency ‘optical light’ that can be efficiently distributed will potentiallyenable small-scale quantum systems [107, 108, 109] to be combined into larger, fully-functional
quantum networks [41, 110]. But no current technology can transform information between thesevastly different frequencies while preserving the fragile quantum state of the information. For thisdemanding application, a frequency converter must provide a near-unitary transformation between
microwave light and optical light; that is, the ideal transformation is coherent, lossless and noiseless.
Traditionally, nonlinear materials such as lithium niobate are used to create electro-optic
modulators (EOMs) that connect microwave and optical light. Although EOMs are in principle capable of quantum frequency conversion [111, 112], such conversion has not yet been demonstrated,and current EOMs [113, 114, 115] have predicted photon number efficiencies of only a few 10−4
[114, 111, 112]. Although these technical challenges might be overcome in the future, there is a technological vacuum that has been filled with proposed routes to a quantum link between microwave
and optical light. Proposed methods of frequency conversion utilize clouds of ultracold atoms[116, 117, 118], nitrogen-vacancy centers in diamond [119], rare-earth doped crystals [120, 121],and mechanical oscillators [52, 122, 99]. All these proposed frequency converters face the challenge
of integrating optical light with the cryogenic temperatures needed for low-noise microwave signals
and superconducting circuitry.
Experimental interest in using mechanical oscillators for microwave-to-optical frequency con-31
version began around 2011 when both microwave light and optical light were used to cool mechanical oscillators to near the ground state of motion [55, 37]. Since this time, experiments havedemonstrated coupling between MHz-frequency signals and optical light using mechanical oscillators [123, 124, 125], and show promise for microwave-to-optical frequency conversion. Anotherapproach uses a combination of piezoelectricity and radiation pressure to couple microwave signals to optical frequencies [99]. This chapter describes the construction and measurement of a
microwave-to-optical converter based on a mechanical oscillator [122]. The converter operates at
cryogenic temperatures and the frequency conversion is shown to be both bidirectional and highly
efficient.

3.2

A frequency converter forms a two-port network
From an experimental point of view, a frequency converter can be viewed as a ‘black box’ thathas a microwave frequency port and an optical frequency port, as diagrammed in Fig. 3.1a. The
converter must be characterized using these ports alone. We can characterize the coherent responseof the converter by, for example, injecting a microwave field at frequency ω into the converter andcomparing it to the outgoing optical field at frequency ωc + ω, where ωc is the frequency change
added by the converter. This measurement yields one element of the transfer function matrix Ξ(ω)
as defined in section 2.5.1. In the language of microwave engineering, each element of the transfer
function corresponds to a single scattering parameter of the network. For a two port network suchas the frequency converter, four S-parameters describe the network [70] as detailed in Fig. 3.1b.
Without knowledge of the internal dynamics of the converter, we can still generate criteria
for an ideal frequency converter. First, we would like the converter to preserve the number ofphotons in a signal. The fields bin (ω) and aout (ω) in Fig. 3.1b have units of (number×sec)1/2 asin section 2.2.1, and so the apparent photon number efficiency for upconversion can be expressed
as |S21 (ω)|2 = |aout (ωc + ω)/bin (ω)|2 . Ideally, |S21 (ω)|2 = 1 and the converter has no loss and no
photon number gain. Although gain might be beneficial for some applications [126], gain necessarily
adds noise [127] that will corrupt quantum signals. If conversion has no gain and has unit efficiency,

32

aConverterb
Microwave Portbin(ω)
bout(ω)Converter

S11

S21

S22

S12

Optical Port
aout(ωc+ω)ain(ωc+ω)Figure 3.1: Scattering parameters of a frequency converter. a, All frequency converters have a
microwave port and an optical port, here represented by an SMA connector and a fiber connector,
respectively. b, Transmission and reflection of a frequency converter can be characterized byscattering parameters. For example, the transmission for upconversion, S21 (ω), can be found bymeasuring the quantity aout (ωc +ω)/bin (ω) with ain (ω) = 0, where ωc is the frequency change addedby the converter.

there can be no reflected microwave signal: The ideal converter is impedance matched and S11 (ω) =bout (ω)/bin (ω) = 0. Finally, if frequency conversion is lossless and adds no noise to the converted
signal, it can be reversed and undone. As an initial indication of reversibility, the conversion process
should be reciprocal or bidirectional: S21 (ω) = S12 (ω). A perfectly efficient, impedance-matched,and bidirectional converter is a significant step toward quantum frequency conversion.
These three criteria can be satisfied with a frequency converter based on a mechanical oscillator. Section 3.3 describes the construction of an electro-opto-mechanical frequency converter.
Section 3.4 predicts the performance of this frequency converter with respect to the three criteria,
and section 3.6 demonstrates that our converter begins to meet all criteria. However, frequency
conversion with a mechanical oscillator can add noise to the converted signal, as discussed in
section 3.4.3. Reducing added noise is a crucial challenge, and in section 3.6.3 we demonstrate areduction of added noise that can in principle be extended to create a quantum frequency converter.
Although low added noise and high conversion efficiency imply the converter is suitable for33
conversion of quantum states, an unambiguous demonstration of quantum frequency conversion isnecessary. The experiments presented here are progress toward this goal, but quantum frequency
conversion is a task for future graduate students.

3.3

Construction of electro-opto-mechanical frequency converter
Coupling electromagnetic fields and mechanical oscillators was pioneered in the 1970s and1980s [20, 128, 88], but not until recently has the strength of the radiation pressure coupling beenenhanced to the point that technological goals, such as quantum frequency conversion [129, 130, 131,
132, 90], seem within reach. A key development was the discovery that silicon nitride membranes
have outstanding mechanical and optical properties [133]. When placed between two mirrors thatform a high-finesse Fabry-Perot cavity, the vibrations of the silicon nitride membrane couple to theelectromagnetic field confined in the cavity [134, 135]. At microwave frequencies, state-of-the-art
experiments use metallic membranes to create low-loss capacitors that naturally couple membrane
vibrations to electrical signals [136, 55]. These experiments require cryogenic temperatures bothto eliminate Johnson noise at microwave frequencies and to create low-loss superconducting circuits. Our frequency converter leverages advances in both types of optomechanical systems. It
includes a Fabry-Perot cavity that forms an optical-frequency electromagnetic resonator and an
inductor-capacitor circuit that forms a microwave-frequency electromagnetic resonator at cryogenic temperatures. A silicon nitride membrane spans these two electromagnetic resonators, as
shown in Fig. 3.2.
The challenge of constructing an electro-opto-mechanical system is integration of the twoelectromagnetic resonators. The optical cavity must operate at cryogenic temperatures needed by
the microwave circuit, and the microwave circuit must operate in close proximity to the FabryPerot cavity, where light in the cavity can potentially interact with the microwave circuit. In
particular, high-frequency photons absorbed by a superconducting microwave circuit adversely
affect its performance [137]. Photons with energy greater than the superconducting gap (which forniobium corresponds to a photon frequency of ≈ 1 THz) can break cooper pairs apart into their

34
MirrorC
Circuit

7 GHz

MembraneL

282 THz

Figure 3.2: Schematic of a microwave-to-optical frequency converter. The converter consists of asilicon nitride membrane coupled to a microwave-frequency electromagnetic resonator formed froman inductor-capacitor circuit and an optical-frequency electromagnetic resonator formed from two
high-reflectivity mirrors. Propagating electromagnetic fields are coupled to the microwave resonatorwith an inductive coupler, and to the optical resonator with a slightly transmissive input mirror.

individual electrons. Unpaired electrons cause additional loss in the circuit in a manner similar to
an increased temperature of the entire superconductor [138]. Although changes in the number of
unpaired electrons can be used to detect photons [78, 79], for a frequency converter we need themicrowave circuit to be unaffected by the amount of light in the Fabry-Perot cavity.
We begin constuction of the frequency converter with a physically large silicon nitride membrane (dimensions of 1 mm by 1 mm by 90 nm thick) so that the superconducting microwave
circuit and the Fabry-Perot cavity can couple to physically distant portions of the silicon nitride
membrane. The membrane is fabricated using standard photolithographic techniques [139, 140]
that create a silicon nitride membrane held in place by a silicon frame. Additionally, we cover a
portion of the membrane with niobium for use in the superconducting circuit. A picture of themetallized silicon nitride membrane is shown in figure 3.3a.

3.3.1

Construction of microwave circuit

To couple the silicon nitride membrane to a microwave circuit, we pattern a thin film of
niobium on a separate silicon substrate as shown in Fig. 3.3b. This substrate contains the majority

35

Figure 3.3: Construction of a ‘flip-chip’ microwave circuit. a, A silicon frame supports the silicon
nitride membrane. The silicon nitride appears clear, but the niobium metallization appears as agray-colored bowtie shape on the silicon nitride. b, Partially complete microwave circuit createdby patterning a niobium film on a silicon substrate. The microwave circuit is completed when themembrane is flipped over onto this circuit and attached. c, Schematic depiction of the completed
microwave circuit. d, Expanded view of microwave circuitry.

36of the microwave circuitry, including a transmission line that allows microwave signals to be coupled
into and out of the circuit, and is designed to mate with the metallized silicon nitride membrane
to form a complete inductor-capacitor circuit. The circuitry and the membrane are brought into
close proximity to form what we call a ‘flip-chip’ microwave circuit. An electrical diagram of theflip-chip is shown in Fig. 3.3c.
We require a large electromechanical coupling Ge = ∂ωe /∂x where ωe is the resonant frequency of the microwave circuit and x describes the position of the silicon nitride membrane. For
the inductor and parallel-plate capacitor circuit shown in Fig. 3.3c, the electromechanical couplingis Ge ≈ ωe /(2d) where d is the separation between the two plates of the parallel plate capacitor,
indicating that small plate separations are necessary for large Ge . A key engineering aspect of
creating small separations is removal of any unnecessary material. Less material at the interface
between the circuitry and membrane reduces the chance that particulate will prevent the capacitor plates from reaching the desired separation distance. Using material removal as a design rule,
the flip-chip microwave circuits consistently have parallel-plate capacitors with plate separationsof d ≈ 400 nm. This separation yields an electromechanical coupling of Ge ≈ 2π × 4 MHz/nm.
An assembled flip-chip and its profile are shown in figure 3.4, and fabrication details are given in
Appendix A.
Vibrational excitations of the membrane can radiate out into an acoustically lossy environment, which can limit the the quality factor of vibrational modes of the membrane [65, 73]. In
an attempt to decouple vibrational modes of the membrane and its supporting silicon frame fromother lossy vibrational modes, we assemble the flip chip with a small amount of Stycast 2850FT
epoxy. We have found this assembly process to be compatible with high quality factor vibrationalmodes of the membrane, as shown in Fig. 2.3.
To integrate the flip-chip with an optical cavity, we must mount and connectorize it in an
appropriate way. We attach the flip-chip with epoxy to a copper-plated Invar bracket designed for
direct integration with a Fabry-Perot cavity. Invar is our material of choice for use in the FabryPerot cavity (see section 3.3.2), but copper is necessary to create a controlled electromagnetic

37

Figure 3.4: a, Picture of an assembled flip-chip. Note the large, approximately 1mm by 1mm
central area that will allow optical frequency light to interact with the silicon nitride membrane. b,
Schematic profile of the assembled flip-chip. Importantly, only the posts that hold the chips apartand the capacitor plates of the microwave circuit are within 400 nm of each other (d ≈ 400 nm).
All other surfaces are > 25 µm apart.

Figure 3.5: Packaged flip-chip microwave circuit. a, Flip-chip is mounted on a copper-plated Invar
substrate, which simultaneously provides a low-loss electromagnetic environment for the microwave
circuit and low thermal contraction required for the cryogenically compatible Fabry-Perot cavity.
A flexible microstrip transmission line connects to the microwave circuit, and also connects to a
coaxial cable via a commercially available connector (not shown). b, A niobium shim, once attachedto the top of the copper-plated Invar, blocks most stray light from reaching the microwave circuit.
The niobium shim is necessary to protect the superconductivity of the microwave circuit when itis placed in a Fabry-Perot cavity (see section 3.3.3).

38
environment for the microwave circuit. Without a high-conductivity ground plane, magnetic fields
in the microwave circuit will induce eddy currents in the Invar that will dissipate energy in the
microwave circuit. We couple microwave signals into and out of the circuit via a flexible circuitboard made from copper rolled onto a polyimide substrate [141]. The copper is patterned to forma microstrip transmission line. Although the microstrip transmisison line forms a robust electricalconnection to the microwave circuit, it creates only a weak mechanical connection. This ensuresthat forces applied to the circuit board will not influence the exact position of the silicon nitride
membrane, whose placement is critical for successful operation of the optical cavity. A packaged
flip-chip is shown in figure 3.5. Importantly, the entire structure can be placed in a high-finesse
Fabry-Perot cavity: The flip-chip and package contain a large central area that is completely freeof metal and silicon as can be seen in Fig. 3.4a and Fig. 3.5a. This open area will allow opticallight to interact with the silicon nitride membrane without additionally interacting with silicon,
niobium, or other materials.

3.3.2

Construction of optical cavity

To create a robust, cryogenically compatible Fabry-Perot cavity, we construct a cavity usingmaterials with low thermal expansion, especially Invar [142, 96]. An Invar cavity spacer rigidly
holds two high-reflectivity mirrors, one of which is attached to a linear translation piezoelectricelement that adjusts the cavity length. The packaged flip-chip slides between the cavity mirrors
and is attached to the Invar spacer via Invar plates and a second linear translation piezoelectricelement that moves the silicon nitride membrane along the cavity length. Including only two
adjustable degrees of freedom allows for construction of a rigid Fabry-Perot cavity that is resistantto environmental vibrations. However, once the components of the cavity are aligned and fixed inplace with Stycast 2850FT epoxy, any subsequent misalignment of the silicon nitride membrane
relative to the cavity mirrors cannot be corrected [74]. Such misalignment is a common resultof cooling to cryogenic temperatures, when differential thermal expansion of cavity components
alters their relative positions. Constructing the cavity using materials with low thermal expansion

39
minimizes the potential for misalignment. With careful construction, a cavity can be cycled fromcryogenic temperatures to room temperature multiple times without degredation in performance.
A diagram of the cavity is shown in Fig. 3.6a.
The optomechanical coupling Go = ∂ωo /∂x, where ωo is the resonant frequency of the FabryPerot cavity, is predominantly determined by the length of the cavity and the thickness of the silicon
nitride membrane [143, 68]. We use a Fabry-Perot cavity ≈ 5 mm long and a membrane ≈ 90 nm
thick, which together yield a coupling of Go ≈ 2π × 40 MHz/nm.

3.3.3

Initial characterization of frequency converter

As noted in section 3.3, the superconducting microwave circuit can fail when exposed to toomuch optical frequency light. To test compatibility of the microwave circuit with light confinedin the Fabry-Perot cavity, we cool the frequency converter to 4 kelvin using a liquid helium bathcryostat so that the microwave circuit superconducts (niobium superconducts at temperatures below
about 9 kelvin). We monitor the response of the microwave circuit with a network analyzer as anincreasing amount of 1064 nm light passes through the unmetallized portion of the silicon nitride
membrane. From the circuit’s response, we determine the energy decay rate of the microwave
circuit due to internal loss. Results of this measurement are shown in figure 3.7.
During initial tests we found that the superconducting circuit was inadequately isolated fromthe light in the optical cavity. Specifically, only a few milliwatts of circulating power in the cavity
drastically increased the internal loss rate of the microwave circuit. Internal loss of the microwave
circuit only became sufficiently decoupled from the amount of light in the Fabry-Perot cavity when
we incorporated an additional niobium shim designed to prevent stray optical light from reaching
the microwave circuit. The niobium shim (shown in Fig. 3.5b) did not affect light confined in the
Fabry-Perot cavity. These data support previous observations that stoichiometric silicon nitride
has low absorption at optical frequencies and the membrane does not experience significant bulk
heating when irradiated [135, 74].
Additionally, we characterize the vibrational modes of the silicon nitride membrane. We

40

Figure 3.6: Electro-opto-mechanical frequency converter. a, Exploded view of a CAD renderingof the converter. Silver corresponds to Invar, yellow to piezoelectric material, and light gray to
fused silica (the substrate for the high-reflectivity mirrors). The flip-chip microwave circuit andmounting are also shown. b, Picture of completed converter, which during operation is enclosed ina copper box. A coaxial cable carries microwave signals to and from the converter, and a hole inthe copper box allows optical signals to enter and exit converter.

41

Microwave resonatorinternal loss rate (MHz)2
InitialIntermediate
Current device(added niobium shim)

1

0.2

0

20

40
60
80
Power through Si3N4 membrane (mW)

100

Figure 3.7: Compatibility of microwave circuitry with optical frequency light. Optical power confined in the cavity had dramatically less effect on the microwave circuit when we incorporated aniobium shim that prevented any stray light from reaching the microwave circuit (see Fig. 3.5).

drive the linear translation piezoelectric element connected to the silicon nitride membrane at afrequency ω and simultaneously drive the optical cavity with a monochromatic pump. Motion of
the silicon nitride membrane appears as ‘sidebands’ on the pump as discussed in section 2.3. A
peak in sideband power occurs when the signal driving the piezoelectric element has a frequency
near ω = ωm . With this method, we measure the resonant frequency of several vibrational modesof the silicon nitride membrane. By comparing the measured resonant frequencies with a finite
element simulation of the membrane, we identify several vibrational modes suitable for frequency
conversion. These modes are shown in Fig. 3.8.

3.3.4

Future improvements

We have constructed a microwave-to-optical frequency converter by simultaneously couplinga silicon nitride membrane to both a superconducting microwave circuit and a high-finesse optical cavity. Our frequency converter provides a potential route to quantum-compatible frequency
conversion; however, a few aspects need improvement. The flip-chip microwave circuit does not provide a reproducible acoustic environment for the silicon nitride membrane. The silicon frame that

42

Figure 3.8: Simulated vibrational modes of membrane used in electro-opto-mechanical frequency
converter. a—d, Contour plots of simulated membrane displacement, normalized to unity, forfour different vibrational modes of the partially metallized silicon nitride membrane used in thefrequency converter. Outline of the niobium metallization is indicated with red bowtie shapes, andthe outline of the optical beam waist is indicated with red circles.

supports the membrane can potentially be modified to include a phononic crystal [144] that will
acoustically isolate the membrane from the epoxy connections [145, 146]. This modification is likely
compatible with a flip-chip microwave circuit, and could result in reproducibly high quality factors
for vibrational modes of the silicon nitride membrane. Second, although the open design of the
Fabry-Perot cavity makes incorporating the microwave circuit simple, it sacrifices some mechanicalrigidity and makes the converter susceptible to failure during cooling to cryogenic temperatures.
We think that building a more integrated device, rather than attaching a packaged microwave
circuit as shown in Fig. 3.5, will improve converter performance and simultaneously simplify its

43assembly.

3.4

Radiation pressure enables frequency conversion
The electro-opto-mechanical converter described in section 3.3 consists of a mechanical oscilla-

tor coupled to two electromagnetic resonators. We can model the converter using the mathematicalformalism of cavity optomechanics, introduced in section 2.2. In particular, equations 2.28 can be
modified to include a second electromagnetic resonator. This results inȧ(t) = (ı∆o − κo /2)a(t) − ıgo (c(t) + c∗ (t)) +ḃ(t) = (ı∆e − κe /2)b(t) − ıge (c(t) + c∗ (t)) +√√κo,ext ain (t) +κe,ext bin (t) +√√κo,int a0 (t)κe,int b0 (t)ċ(t) = (−ıωm − κm /2)c(t) − ı(go a∗ (t) + go∗ a(t)) − ı(ge b∗ (t) + ge∗ b(t)) +√κm cin (t),(3.1)

where the subscript ‘e’ indicates parameters associated with the electrical circuit and the subscript‘o’ to indicate parameters associated with the optical cavity. The field amplitudes b(t) and a(t) are
the fluctuations about the microwave and optical pumps, respectively. The pumps populate themicrowave circuit and optical cavity with mean field amplitudes b̄ and ā, as described in section 2.4.
The mean field amplitude controls the strength of the optomechanical coupling: ge = Ge xzpf b̄ andgo = Go xzpf ā. Fig. 3.9 visually depicts Eqns. 3.1, and table 3.1 contains parameters of the electroopto-mechanical frequency converter described in section 3.3.

3.4.1

Frequency conversion in the resolved-sideband and weak-coupling regime

The dynamics of frequency conversion can be more easily understood with a few experimentally justified approximations to Eqn. 3.1. First, I’ll set the frequency of the microwave andoptical pumps so that ∆e = ∆o = −ωm as diagrammed in Fig. 3.10. In the resolved-sideband
regime (4ωm  κe , κo ), these detunings maximally damp the mechanical oscillator via dynamical
backaction as discussed in section 2.4. To simplify the dynamics, I’ll remove the free evolutionof the field amplitudes by moving to a new rotating frame: a(t) → a(t)e−ıωm t , b(t) → b(t)e−ıωm t ,c(t) → c(t)e−ıωm t , and so forth for the other field amplitudes. With these adjustments Eqns. 3.1

44

Figure 3.9: Three coupled harmonic oscillators form the frequency converter. The electromagnetic
field in the microwave circuit is described by field amplitude b(t), the position of the mechanicaloscillator is described by field amplitude c(t), and the electromagnetic field in the optical cavity
is described by field amplitude a(t). The microwave circuit can be driven via its coupling to atransmission line and the optical cavity can be driven via a partially transmissive input mirror,
as diagrammed in Fig. 3.2. The electromagnetic resonators and the mechanical oscillator are also
coupled to the their environment.

Table 3.1: Parameters of electro-opto-mechanical frequency converter. Sets of values apply to threedifferent vibrational modes of the mechanical oscillator we used for frequency conversion.
Parameter

Value

Units

ge
ωe
κe,extκe,intκegoωoκo,extκo,intκoωmκm

2π×80*
2π×7.06
2π×12202π×370κe,ext + κe,int
2π×80†
2π×2822π×3802π×1270κo,ext + κo,int
2π×{380, 560, 1240}
2π×{0.42, 4.0, 4.2}

kHz
GHz
kHz
kHz
kHz
THz
kHz
kHz
kHz
Hz

*Adjustable up to this value via the microwave pump.
†

Adjustable up to this value via the optical pump.

45
becomeȧ(t) = −(κo /2)a(t) − ıgo (c(t) + c∗ (t)e2ıωm t ) +√ḃ(t) = −(κe /2)b(t) − ıge (c(t) + c∗ (t)e2ıωm t ) +√κo,ext ain (t)κe,ext bin (t)ċ(t) = −(κm /2)c(t) − ı(go a∗ (t)e2ıωm t + go∗ a(t)) − ı(ge b∗ (t)e2ıωm t + ge∗ b(t)) +√κm cin (t).

(3.2)

Eqns. 3.2 can be simplified with a rotating wave approximation that neglects the terms oscillating
at frequency 2ωm . For this approximation to be valid, these terms must oscillate faster thanother rates in the problem, leading to 4ωm  κo , κe . This approximation places the model in the
resolved-sideband regime. Although the frequency converter we have constructed does not operate
deep in the resolved-sideband regime (see Table 3.1), many aspects of frequency conversion in the
resolved-sideband regime appear in the unresolved-sideband regime. If the optomechanical coupling
rates are much slower than the energy decay rates of the electromagnetic resonators, κo  go andκe  ge , then the dynamics of the electromagnetic resonators can be adiabatically eliminated by
setting ȧ(t) = ḃ(t) = 0 [147]. In this weak-coupling regime, we have−2ıgo2√
κo,ext ain (t)c(t) +
κo
κo
−2ıge
2√
b(t) =
c(t) +
κe,ext bin (t)
κe
κe


√
√
4|go |2 4|ge |2
12ıge∗ κe,ext
2ıgo∗ κo,ext
κm +ċ(t) = −+ain (t) − √bin (t)c(t) − √
√
√
2
κo
κe
κo
κo
κe
κe

a(t) =

Eqns. 3.3 are simply solved in the frequency domain.
aout (ω) =

√

(3.3)

Using the input-output relation

κo,ext a(ω) − ain (ω) with ain (ω) = 0, we can find an expression for upconversion.
√aout (ω)Γo Γe√ıφS21 (ω) ==e ×× ηo ηebin (ω)−ı(ω − ωm ) + (Γo + Γe + κm ) /2

(3.4)

where φ = Arg(−go ge∗ ), Γo = 4|go |2 /κo and Γe = 4|ge |2 /κe , and ηe = κe,ext /κe and ηo = κo,ext /κo .1
Γe and Γo are exactly the additional mechanical damping rates predicted in section 2.26 with
∆ = −ωm and 4ωm  κ. Eqn. 3.4 is in the frame of the pumps so that bin (ω) and aout (ω) are a
1

The optical cavity additionally emits light into a particular spatial mode. If this mode does not perfectly matchthe spatial mode of the incident light field, ηo = κo,ext /κo where  is the optical mode matching.

46

Figure 3.10: Frequency domain schematic of frequency conversion. A strong microwave pump (bluearrow) is applied below the microwave resonance (response shown as black curve) with detuning∆e . Similarly, a strong optical pump (red arrow) is applied below the optical resonance (response
shown as a black curve) with detuning ∆o . This enables an incoming microwave frequency signalbin (ω) to be upconverted to an outgoing optical frequency signal aout (ω). The expression forfrequency upconversion (or downconversion) in the weak-coupling (κo  go and κe  ge ) and
resolved-sideband (4ωm  κe , κo ) regimes with ∆e = ∆o = −ωm is given in equation 3.4

frequency ω above the microwave and optical pumps, respectively. In this analysis, upconversion ismathematically identical to downconversion, S21 (ω) = S12 (ω), meaning that frequency conversion
is bidirectional. A frequency domain schematic of frequency conversion is shown in Fig. 3.10.
Using Eqns. 3.3 with bout (ω) =√κe,ext b(ω)−bin (ω), we can find an expression for the reflected

microwave signal.
S11 (ω) =bout (ω)−Γe ηe=+ 2ηe − 1.
bin (ω)−ı(ω − ωm ) + (Γe + Γo + κm ) /2

(3.5)

The scattering parameters S11 (ω) and S21 (ω) (= S12 (ω)) are shown in Fig. 3.11 for matched(Γe = Γo ) optomechanical coupling rates.
Photon number conversion efficiency |S21 (ω)|2 is greatest when the signal is a mechanicalresonant frequency above the microwave and optical pumps (ω = ωm ). Additionally, the rates
Γe and Γo must be matched to each other to maximize conversion efficiency. This ‘cooperativity
matching’ [60, 130] can be accomplished by adjusting the strength of the microwave or opticalpump. Unit conversion efficiency is only reached when the microwave circuit, optical cavity, andmechanical resonator have negligible loss: ηe = ηo ≈ 1 and Γo , Γe  κm . When unit transfer

47

Figure 3.11: a, Expected scatterering parameters for electro-opto-mechanical frequency conversion
with matched damping rates (Γe = Γo ). a, Frequency conversion occurs over a bandwidth Γo +
Γe + κm and with unit photon number efficiency when ηo = ηe = 1 and Γ0 , Γe  κm . b, Reflected
microwave power diminishes as transfer efficiency increases.

efficiency is reached, there is no reflected microwave power, as can be seen in Fig. 3.11. This
indicates nearly ideal conversion, as the entire microwave signal is transformed into an opticalsignal. However, the converter is only active over a limited bandwidth set by the total dampingof the mechanical oscillator, Γo + Γe + κm . This bandwidth can be increased by using stronger
microwave and optical pumps; however, eventually it is limited by the bandwidths of the microwave
circuit and optical cavity.

3.4.2

Frequency conversion in the weak-coupling regime

A full calculation of the transfer function using equations 3.1 without any additional simplifying assumptions can be performed using the methods presented in section 2.5.1 [148]. If we assume
the weak coupling regime, a valid approximation given parameters of our frequency converter (seetable 3.1), we can write down a relatively succinct expression for S21 (ω).
√
paout (ω)Γo ΓeıφS21 (ω) ==e ×× Aηo ηe ,0bin (ω)−ı(ω − ωm ) + (Γo + Γe + κm ) /2

(3.6)

0 = ω + Ω + Ω is the radiation pressure-modified mechanical resonant frequency andwhere ωmmeo

A = Ae Ao is conversion gain. Frequency shifts Ωe and Ωo , and coupling rates Γo and Γe , are48
determined by the dynamical backaction of the electromagnetic fields as given in equation 2.26.
∆ e + ωm∆e − ωm+(∆e + ωm )2 + (κe /2)2 (∆e − ωm )2 + (κe /2)2
∆o + ωm∆o − ωmΩo = go2+(∆o + ωm )2 + (κo /2)2 (∆o − ωm )2 + (κo /2)2Ωe = ge2




κe
κe
−
(∆e + ωm )2 + (κe /2)2 (∆e − ωm )2 + (κe /2)2


κo
κo
2
Γo = go−.
(∆o + ωm )2 + (κo /2)2 (∆o − ωm )2 + (κo /2)2Γe = ge2

(3.7)



(3.8)

Additionally, the gains are given by(∆e − ωm )2 + (κe /2)2−4∆e ωm(∆o − ωm )2 + (κo /2)2Ao =.
−4∆o ωm
Ae =

(3.9)

Although there is a small shift in the center frequency of the conversion window of Eqn. 3.6
relative to Eqn. 3.4, the most important difference is that frequency conversion in the unresolvedsideband regime has photon number gain as indicated by the factor A. Gain becomes appreciable
when 4ωm < κo , κe or when pump detunings are small and ωm > 4∆o , 4∆e . From an experimental
point of view, gain occurs when a converted signal appears at a frequency ω both above and below
the microwave or optical pump, as diagrammed in Fig. 3.12. The gains Ae and Ao in Eqns. 3.9
can be formulated as a ratio of power in the upper sideband at frequency ω above the pump, andpower in the lower sideband at frequency ω below the pump.
As introduced in section 2.3, if a mechanical oscillator in a cavity optomechanical systemvibrates with a fixed amplitude at frequency ωm , it scatters energy from the pump into frequencies
ωm above and below the pump. The amount of power scattered is given in Eqns. 2.16 and 2.17
for a generic electromagnetic resonator with linewidth κ and pump detuning ∆. If we identify theresonator as the optical cavity of the electro-opto-mechanical frequency converter, then
Ao =

Pusb
Pusb − Plsb

(3.10)

49

Figure 3.12: Frequency domain schematic of frequency conversion with gain. A strong microwave
pump (blue arrow) is applied below the microwave resonance (response shown as black curve) withdetuning ∆e . Similarly, a strong optical pump (red arrow) is applied below the optical resonance(responce shown as a black curve) with detuning ∆o . This enables an incoming microwave frequency
signal bin (ω) to be upconverted to an outgoing optical frequency signal at ωo + ω. Conversion also
produces a lower frequency signal at ωo −ω when the optical resonator response does not completely
suppress it, i.e., when ωm . κo . This additional sideband indicates the conversion has gain, as seenin Eqn. 3.6.

and similarly for the microwave circuit. Gain A is only unity when there is no power present
in the lower sideband, and suppressing the lower sideband requires the resolved sideband regime
4ωm  κe , κo . The minimum expected conversion gain A as a function of the resolved-sideband
parameter ωm /κe is shown in Fig. 3.13 for κe = κo .
In section 3.6.1, we demonstrate frequency conversion using three different vibrational modesof the silicon nitride membrane. One vibrational mode with ωm = 1240 kHz approaches theresolved-sideband regime with A < 2. We use the other two vibrational modes to explore theminimum bound on conversion gain.

3.4.3

Noise added during frequency conversion

Phase-insensitive gain during frequency conversion means that noise must be added to thesignal during conversion [127]. This noise manifests as extraneous vibrations of the mechanicaloscillator. We quantify the vibrational noise added to a signal during upconversion by calculating
the spectral density of the output optical field with no input fields, [aout (ω)]† aout (ω) with ain (ω) =50Figure 3.13: Minimum gain during frequency conversion, assuming κe = κo . In the resolved
sideband limit of 4ωm  κe , κo , the minimum gain approaches unity.

bin (ω) = 0, and referring it to the input of the converter [60]. In units of quanta of added noise (aquantum corresponding to one photon per second in a 1 Hz bandwidth [93]), upconversion adds
[aout (ω)]† aout (ω)
1nvib ==|S21 (ω)|2
Ae ηe




κm nenv+ (Ae − 1) + (Ao − 1)Γe(3.11)

for matched coupling rates, where nenv ≡ 1/(e~ωm /kB Tenv − 1) and Tenv is the temperature of themechanical oscillator’s environment. Any conversion gain adds noise given by the last two terms in
Eqn. 3.11. Environment-driven vibrations of the mechanical oscillator, indicated by the mechanicaldecoherence rate κm nenv , also add noise. Only with unit gain (ηe  Ae − 1 and ηe  Ao − 1)and negligible mechanical decoherence (ηe Γe  κm nenv ) do vibrations of the mechanical oscillator‘freeze out’.
Frequency conversion with low gain and low vibrational noise is closely related to resolvedsideband cooling of a mechanical oscillator. A gain of A < 2 implies that the oscillator can, in
principle, be cooled to its ground state of motion using radiation pressure [23, 33, 34]. Low noise
frequency conversion can only occur when the mechanical oscillator is simultaneously cooled to nearits ground state of motion by both the microwave circuit and the optical cavity. In addition to low
vibrational noise, noiseless frequency conversion requires unit conversion efficiency (ηe = ηo = 1).

51

3.5

Measurement network
In order to measure the scattering parameters of the electro-opto-mechanical frequency con-verter, we must integrate it into a larger measurement network. To properly predict converter
performance, we must specify which components are part of the converter, and which are part ofthe measurement network. For the calculations in sections 3.4, the converter includes the microwave
circuit, the mechanical oscillator, and the optical cavity. Converter performance thus includes internal losses in the microwave, optical, and mechanical resonators and imperfect optical mode
matching, but excludes losses and gains in other components that are used in our measurement.
This is a natural definition for the converter, and one that makes it a stand-alone component thatcan be readily integrated into other networks.
A microwave signal generator and a Nd:YAG laser are used to pump the microwave circuitand Fabry-Perot cavity, respectively. To generate a coherent signal at a frequency ω ≈ ωm above thepumps, we generate a signal at ω ≈ ωm and use it to modulate the microwave or optical pump via
single-sideband modulation (accomplished with standard microwave and optical components [148]).
After interacting with the converter, the outgoing signals are detected using homodyne detection(for microwave frequencies) and direct detection with a photodiode (for optical frequencies). The
detected signal is referenced to the initial signal to obtain a magnitude and phase shift. A diagramof the measurement network is shown in Fig. 3.14, and a detailed diagram is included in appendix C.
Components in the measurement network add additional gains and losses to a measured
S-parameter, as diagrammed in Fig. 3.14b, and this makes it difficult to independently and accurately calibrate the quantities S21 (ω) and S12 (ω). However, we can exploit the symmetry of
the network to calibrate the product S12 (ω)S21 (ω): To isolate the converter performance fromthe gains and losses of other components used in our measurement, we measure the forward andreverse transmission, αS21 δ and γS12 β, and the off-resonant reflections, αβ and γδ. This gives
S21 S12 = (αS21 δ)(γS12 β)/(αβ)(γδ), a quantity insensitive to imperfect measurement of the separate gains and losses α, β, γ, δ. The off-resonant reflections are measured by tuning the injected

aMicrowave Port
ω

52

b

Optical Port

Microwave PortPD

1064 nmloss αbin(ω)Outω

Refgain βω
7 GHzConverterR

S11

bout(ω)Optical Port

S21

S22

S12

aout(ω)ain(ω)

gain δ

loss γI
L

Figure 3.14: Measurement network. a, A signal at frequency ω ≈ ωm modulates a microwave or
optical pump. After interacting with the converter, the output signal is referenced to the initialsignal (Ref) to obtain a magnitude and phase shift. b, Schematic of the measurement network.
To make a calibrated measurement of the S-parameters of the converter, we must remove theadditional gains and losses added by the measurement network. This can be done by exploiting
the symmetry of the network.

signal away from the resonator’s central frequency, where the converter acts as a near-perfect mirror. With this method, we can determine S21 (ω)S12 (ω) to about 6% [148]. We find this calibration
procedure appealing because it makes equivalent use of both the forward and reverse transmissionof the converter, demonstrating and emphasizing its bidirectional nature.

3.6

Microwave to optical frequency conversion of electromagnetic signals

3.6.1

Bidirectional frequency conversion

By injecting a microwave or optical field and monitoring the outgoing fields, we measure
the scattering parameters that give the number efficiency of the converter. Initially, we use theωm = 2π × 560 kHz vibrational mode, shown in Fig. 3.15d, for frequency conversion. By sweeping
the frequency of the injected microwave signal near ω ≈ ωm , we confirm that forward transmis0 . Reverse
sion S21 (ω) has the expected Lorentzian lineshape of Eqn. 3.6, with a peak at ω = ωmtransmission S12 (ω) is the same to within our measurement error, giving initial indication thatthe conversion process is bidirectional. We also monitor microwave reflection S11 (ω) during up-

53

Figure 3.15: Measured scattering parameters. a − c, Upconversion S21 (ω), downconversion S12 (ω),
and microwave reflection S11 (ω) for the ωm = 2π×560 kHz vibrational mode with Γe = 2π×1.2 kHz
and Γo = 2π × 2.8 kHz. d, Contour plot of membrane displacement, normalized to unity, for theωm = 2π × 560 kHz vibrational mode. The outlines of the niobium metallization (red bowtie shape)and optical beam waist (red circle) are included.

0 , there is a corresponding dip in reflected
conversion. When conversion efficiency peaks at ω = ωm

microwave power, as seen in Fig. 3.15c, further demonstrating that some of the injected microwave
signal is indeed being converted. In addition to the magnitudes of the converted and reflectedsignal, the measurements also yield phase information. We find that, up to a frequency-dependent
phase shift, the phase is preserved during conversion, indicating the conversion between the two
frequency regimes is coherent.
Frequency conversion in Fig. 3.15 is performed with specific microwave and optical pump
powers that yield optomechanical damping rates of Γe = 2π × 1.2 kHz and Γo = 2π × 2.8 kHz.
However, as indicated by Eqn. 3.6, the maximum conversion efficiency is expected to occur nearΓe = Γo . By changing the microwave pump power while holding the optical pump power fixed, we54vary the ratio Γe /Γo and observe that maximum conversion efficiency occurs near Γe = Γo , as seenin Fig. 3.16b, confirming the cooperativity matching condition predicted by Eqn. 3.6.
To observe the presence of conversion gain, we perform frequency conversion using different
vibrational modes of the silicon nitride membrane. For the ωm = 2π × 380 kHz vibrational mode,ωm /κe ≈ ωm /κo ≈ 0.25 and a photon number gain of A = 3 is expected as indicated in Fig. 3.13.
This results in an apparent efficiency of 0.25, as seen in the peak value of |S21 S12 | for this vibrationalmode (Fig. 3.16c). For conversion with the higher frequency ωm = 2π ×1.24 MHz vibrational mode,ωm /κe ≈ ωm /κo ≈ 0.8 and the converter approaches the resolved-sideband limit. This reduces thephoton number gain to A = 1.4. With this lower gain comes a lower apparent efficiency, but
the absorption of microwave power occurs nearer the peak in conversion efficiency, as shown in
Fig. 3.16a, indicating more ideal conversion.
The converter reaches nearly unit internal efficiency as Γe , Γo  κm . For the ωm = 2π × 560
kHz vibrational mode, the converter reaches a matched (Γe = Γo ) bandwidth of approximately
30 kHz, overwhelming the intrinsic mechanical decay rate of κm = 2π × 4 Hz by nearly fourorders of magnitude. For this vibrational mode, we expect the maximum apparent conversion
efficiency to be |S21 S12 | = Aηe ηo = 0.169 ± 0.012, which corresponds well with the measured
maximum of 0.178 ± 0.011. The true efficiency ηe ηo , which does not include conversion gain A, islimited by absorption and scattering of light in the electromagnetic resonators (κe,ext /κe = 0.76 andκo,ext /κo = 0.23), and imperfect optical mode-matching ( = 0.47). These factors combine to givean expected efficiency ηe ηo = 0.082 ± 0.006. This corresponds well with the measured maximum
apparent efficiency divided by the expected gain of 0.086 ± 0.007.

3.6.2

Bandwidth of frequency conversion

The converter achieves a maximum conversion bandwidth of Γe + Γo = 2π × 32 kHz, as shownin Fig. 3.17. Beyond these damping rates, microwave pump power begins to break superconductivity
in the microwave circuit. Although this places a limit on the bandwidth of frequency conversion witha single vibrational mode of the mechanical oscillator, other vibrational modes of the mechanical55

Figure 3.16: Bidirectional and efficient conversion. (a) Contour plot of simulated membrane displacement, normalized to unity, for the three vibrational modes used for frequency conversion.
Outlines of the optical mode (dashed circle) and niobium metallization (dashed bowtie) are shown.
0 )S (ω 0 )|, and microwave reflection coefficient,(b) Maximum apparent transfer efficiency, |S21 (ωm
21 m
0
2|S11 (ωm )| , as a function of damping ratio for the three vibrational modes. Measurement error issmaller than the plotted points. The black lines are expected values given independently measured
system parameters. Gray regions express the uncertainties of our system parameters, which areapproximately 10% [148]. Pumps are red-detuned with ∆e ≈ −ωm and ∆o /2π = −730 kHz.

56
oscillator also contribute to frequency conversion, as can be seen in Fig. 3.17b. This suggests thepossibility of improving conversion bandwidth by using multiple vibrational modes of the silicon
nitride membrane for frequency conversion.

3.6.3

Noise added during frequency conversion

Noise added during frequency conversion will appear in the spectral density of the output
optical or microwave fields. We can specify the noise added to a signal during conversion witha signal to noise ratio (SNR) defined as SNR=Sin /nvib where Sin is the spectral density of the
input signal and nvib is the vibrational noise added during conversion as given by Eqn. 3.11. To
characterize noise added by vibrations of the mechanical oscillator, we inject a monochromatic
microwave field into the microwave port of the converter and measure the spectral density of theoutput optical field. We measure vibrational noise added during upconversion (as opposed todownconversion) because near quantum-limited photodetection allows us to quickly characterize
0 above the microwave
noise at optical frequencies. The microwave field is injected at a frequency ωmpump with matched coupling rates (Γe ≈ Γo ). As shown in the inset of Fig. 3.18, the converted
signal power appears on a noise background that is dominated by thermal fluctuations filtered by
the mechanical oscillator’s susceptibility. As we increase the coupling rates Γe and Γo (while keeping
the injected microwave signal power constant), we observe an increased SNR (Fig. 3.18). Fitting
the observed SNR with nvib as given by Eqn. 3.11 allows us to estimate the added vibrational noise.
Vibrational noise is non-zero during 4 K operation because the mechanical resonator is still‘hot’ and driven by thermal fluctuations. While this effect can be substantially reduced by increasing
Γe and Γo , colder temperatures are needed to reduce added vibrational noise to the single quantum
level. Thermalizing the vibrational modes of the mechanical resonator to a dilution refrigerator
temperature of 40 mK with the achieved Γe = Γo = 2π × 15 kHz would ideally result in nvib ≈ 1.
We view colder temperatures as a necessity for quantum frequency conversion since temperaturesof Tenv < (2π × 7 GHz)~/kB ≈ 300 mK are required to eliminate Johnson noise at microwave
frequencies that corrupts weak (quantum) signals.

57a

S21(ω)

�

0
1

0
1

0.1

0.1

Magnitude

Phase

�

450

b
Magnitude

S12(ω)

500

550
ω/2� (kHz)600

650 450500550
ω/2� (kHz)

600

650

|S21(ω)|

1

1

0.1

0.01
250

0.1

350

450

550
ω/2� (kHz)

650

750

850

0.01

Figure 3.17: (a) Forward transmission S21 (ω) and reverse transmission S12 (ω) for the ωm = 2π×560
kHz vibrational mode with 32 kHz of transfer bandwidth. A Lorentzian fit (blue) is included. (b)Upconversion over a wide range of frequencies; conversion is visible for three vibrational modes ofthe mechanical oscillator at ωm /2π = 380 kHz, 560 kHz, and 760 kHz.

90
PSD (dBm/Hz)

SNR (Linear Units)

60

-105

45

560

Signal-115

Vibrational
noise

-125

1500
1800
2500

561
ω/2� (kHz)

30

3700

15

7400

1
0

200

Гe/2� (Hz)400

Added vibrational noise(photons/sec∙Hz)

75

58

1200

1.1×105
600

Figure 3.18: Signal to noise ratio (SNR). SNR for an upconverted microwave signal as a functionof coupling rate Γe ; injected signal power is kept constant and Γo ≈ Γe . A fit with nvib as given by
Eqn. 3.11 is shown (line), and the extracted added vibrational noise (nvib ) is shown on the rightvertical axis. Inset: power spectral density (PSD) of an upconverted microwave signal.

3.7

Future work
We have demonstrated a converter that provides a bidirectional, efficient, and impedance-

matched link between microwave and optical frequencies. Additionally, the bidirectionality of
the conversion, its phase-preserving nature, and the absorption of injected signal power during
conversion provide firm evidence that state transfer is occurring, and that we have accessed thebeam-splitter Hamiltonian [147] that is fundamentally capable of quantum frequency conversion.
The ability to efficiently and coherently exchange information between microwave and optical light
opens new possibilities for quantum information, in particular entanglement between microwave
and optical photons [132, 149, 150, 151].
Although the converter functions well, some modest changes can significantly improve performance. Removing internal loss in the electromagnetic resonators will improve ηe and ηo , whichcurrently limit conversion efficiency. Lower temperatures will improve the microwave resonator,
which at 4 K suffers from thermally excited quasiparticles that limit ηe . More careful placementof the flip-chip circuit inside the optical cavity can decrease scattered light and improve opticalmode matching [86]. Added vibrational noise can be reduced by using mechanical oscillators with

59
higher quality factors. In particular, the type of silicon nitride membrane we use is capable ofextremely high quality factors of Q > 107 [133, 74, 65], about two orders of magnitude greaterthan the quality factors used in this work. With our modular converter design, improvements toany single converter component can be carried out independently. This inherent flexibility will
hopefully allow future work to explore new materials and methods and to demonstrate quantum
frequency conversion between microwave and optical light.

Chapter 4

Temporal and spectral mode conversion of microwave signals

The radiation pressure interaction between light and a mechanical oscillator can be variedin time, as initially pointed out in section 2.5. Additionally, a mechanical oscillator can be used
to create a tunable capacitor. In this chapter, we combine these two capabilities and create a‘temporal and spectral mode converter’ that can manipulate the temporal and spectral content of
microwave-frequency electromagnetic signals. As a demonstration of our mode converter, we use it
to implement a particular protocol that converts the temporal envelope of a 7 GHz microwave signal
from a decaying exponential to a Gaussian, and also shifts the signal frequency by up to 250 MHz.
We characterize the noise acquired by low-amplitude classical signals during mode conversion, andwe find the total added noise is less than one quantum for frequency conversions up to ∼100 MHz,suggesting that our mode converter is suitable for use with fragile quantum signals.

4.1

Analog quantum signal processing is needed in quantum networks
Cavity quantum electrodynamics (cQED) and its low-frequency on-chip counterpart circuit

quantum electrodynamics (also cQED) form prototypical quantum systems that have enabled many
tests of quantum theory and advanced goals of quantum computation [152, 103]. A single cQED
system links a quantum two-level system (a qubit) to the fields in an electromagnetic resonator.
Because they naturally interface with electromagnetic fields, multiple cQED systems can be used
to form the nodes of a quantum information network linked by photonic channels [41, 107].
To deterministically distribute information over a network made from cQED systems, elec-

61
tromagnetic signals emitted by one node must be completely absorbed by another [47]. For electromagnetic resonators used in cQED, absorption is the time reverse of emission. For example, theelectromagnetic signal in a resonator naturally exits with a temporal envelope of e−κt for resonator
energy decay rate κ, but the same electromagnetic resonator only completely absorbs a signal withthe time reverse envelope, eκt , as can be seen in Eqns. 2.10. Unless somehow time-reversed, a signal
emitted by one resonator will not be completely absorbed by another, even if they share the same
resonant frequency and energy decay rate. Moreover, because different types of qubits interact
with electromagnetic fields at different frequencies [48], both temporal and spectral manipulationof electromagnetic signals are necessary to construct a quantum information network.
Tools that tailor quantum electromagnetic signals have been developed since the late 1990s.
At optical frequencies, the most general tools typically use nonlinear materials such as lithium niobate [153, 154], although three-level atomic systems can be used to control the temporal envelopeof single photon signals [155, 156]. However, there is a lack of technology that controls the temporal and spectral content of quantum microwave signals. Recent experiments control the temporalcontent of signals using tunable inductors made from Josephson junctions [157, 158, 159]; othermethods exploit the flexibility of superconducting qubits to control the release of single photon
signals [160, 161]. In separate experiments, Josephson junctions have been used to convert microwave signals between different frequencies [162, 163, 164]. However, frequency conversion has
only been demonstrated between two fixed frequencies, and such conversion is not necessarily compatible with state-of-the-art circuit QED systems which currently use electromagnetic resonatorswith frequencies that are imprecisely fixed during fabrication [165, 166].
This chapter describes the construction and measurement of a mode converter that canarbitrarily manipulate the temporal and spectral content of electromagnetic signals. The mode
converter utilizes the radiation pressure interaction and an electrostatically tunable capacitor tomanipulate electromagnetic signals both efficiently and with low added noise, qualities that suggest
it can be used with quantum electromagnetic signals. Section 4.2 describes how a membranemade from a tensioned metallic film can couple to a microwave-frequency electromagnetic resonator

62
7 GHz

Membrane

L

C
Actuation
Vdc

Figure 4.1: Schematic of temporal and spectral mode converter. An aluminum membrane is used
to form the capacitor of an inductor (L) and capacitor (C) electromagnetic resonator (shown inblue). Additionally, an actuation circuit (shown in black) can alter the position of the aluminum
membrane and thus change the resonant frequency of the microwave circuit.

and simultaneously form a tunable capacitor. Section 4.3 describes how the radiation pressureinteraction can be used to manipulate the temporal content of electromagnetic signals. Sections 4.5
and 4.6 demonstrate efficient and low noise manipulation of electromagnetic signals.

4.2

A tunable microwave-frequency optomechanical system
Thin aluminum membranes have been successfully integrated into microwave frequency elec-

tromagnetic resonators [136, 167], and our mode converter builds on these advances. We use analuminum membrane to form two separate parallel-plate capacitors, as diagrammed in Fig. 4.1.
One capacitor is used in the microwave-frequency electromagnetic resonator, and as the membrane
vibrates it alters this capacitance and thus the resonant frequency, yielding the usual cavity optomechanical coupling. The second capacitor consists of the membrane and an actuation electrode.
When a voltage is applied to the actuation electrode, it creates a force on the membrane that pulls
the membrane toward the actuation electrode. The net displacement of the membrane increases
the capacitance of the microwave circuit and lowers its resonant frequency. Tuning the resonant
frequency in this way will allow us to manipulate the spectral content of electromagnetic signals.

63
4.2.1

Limitations of electrostatically actuated capacitors

Electrostatic actuation of the membrane using a voltage is convenient and relatively easy
to implement, but it suffers from practical limitiations. In particular, electrostatic actuation cancause the membrane to collapse and stick to the actuation electrode [168, 169]. Additionally, if thevoltage across the collapsed capacitor exceeds the breakdown voltage of any dielectric separating
the capacitor [170, 171], the resulting electric discharge typically destroys the entire circuit. Finally,
voltage fluctuations on the actuation electrode can incoherently drive vibrations of the aluminum
membrane. If not controlled, these fluctuations corrupt electromagnetic signals processed with themembrane.
To understand and design around these problems, consider the simplified system diagrammedin Fig. 4.2a, which shows a parallel plate capacitor formed by an actuation electrode and a membrane. In the absence of any voltage on the actuation electrode, the membrane experiences anapproximately harmonic restoring force described by a potential energy U (x) = k(d − x)2 /2, where
k is the spring constant of the mechanical oscillator formed by the membrane and d is the separation
between the membrane’s equilibrium position and the actuation electrode. For the membranes weuse, the harmonic potential is predominantly provided by tension in the aluminum. When a voltage
2 /2, where C
is applied, U (x) = k(d − x)2 /2 − Cdc (x)Vdc
dc is the capacitance between the membrane

and the actuation electrode1 . For a parallel plate capacitor, Cdc ∝ 1/x. For any Vdc 6= 0, theabsolute minimum energy configuration of the system is at x = 0. However, a meta-stable equilibrium position of the mechanical oscillator near x ≈ d can still exist, as shown in Fig. 4.2b, but the
dynamics of the membrane are modified. The membrane has a new equilibrium position at x < d,and the resonant frequency of the membrane is lowered. Both these effects can be visually seen in
Fig. 4.2b, where the harmonic potential is effectively widened with a shifted local minimum.
Beyond a certain voltage, known as the ‘pull-in’ voltage, Vpi , the attractive force between
the membrane and the actuation electrode overwhelms the restoring force provided by tension in
1
2
The energy stored in the capacitor is of course C(x)Vdc/2, but a voltage source is also connected to the system.
2
This results in an effective potential of −C(x)Vdc/2 [172].

a

64

b
Membranex
Vdc

Vdc<Vpi

U(x)
d

0

0

Vdc=Vpi

U(x)
0

d
2d
Membrane position, x0d2dMembrane position, xMembrane potential energy, U(x)Contribution to U(x) from tension in membrane
Contribution to U(x) from attraction between electrodes

Figure 4.2: Instability of electrostatically actuated capacitor. a, A schematic of a tunable capacitor. The aluminum membrane is harmonically bound with frequency ωm , and a nearby actuationelectrode forms a capacitance with the membrane. b, Potential energy of membrane, with specificcontributions indicated as dashed lines, for two different values of the actuation voltage Vdc . For
Vdc < Vpi , there is a stable equilibrium for the membrane at x 6= 0. For Vdc > Vpi , there is no stableequilibrium other than x = 0.

the membrane. The membrane is ‘pulled in’ to the actuation electrode because there is no stableequilibrium position other than x = 0, as indicated in Fig. 4.2c. For the simple schematic in
Fig. 4.2a, when Vdc = Vpi , the equilibrium position of the membrane is x = 2d/3 [173].
The pull-in effect limits the dynamic range of an electrostatically actuated capacitor. If sucha capacitor is used in a microwave circuit, under ideal circumstances the resonant frequency ofthe circuit can only be continuously tuned between ω0 and approximately 0.8ω0 , where ω0 is theresonator’s frequency with Vdc = 0. We would like a capacitor that can ideally tune the resonant
frequency between ω0 and 0. Such tunability can potentially be achieved with the circuit shownin Fig. 4.1, provided the separation between the membrane and the actuation electrode is at least
three times greater than the separation between the membrane and the microwave electrode. In thiscase, before the electrostatic instability ‘pulls in’ the membrane, the capacitance in the microwave
circuit has been tuned over its entire working range.
Additionally, the pull-in voltage Vpi determines the sensitivity of the membrane to fluctuations

65
2 /2. For a noisy
in Vdc . The force on the membrane is given by F = −∂U (x)/∂x = ∂Cdc /∂x × Vdc

actuation voltage, Vdc → Vdc + Vn where Vn is a noise voltage that is assumed small relative tothe actuation voltage Vdc . The noise Vn creates a force, Fn , that drives the membrane. For theschematic shown in Fig. 4.2,
Fn ≈

∂Cdc
Vdc VnVdc Vn ∝ kd
.
∂x
Vpi2

(4.1)

For a given separation d and noise voltage Vn , a larger pull-in voltage results in a lower Fn . Ideally,
Fn should be less than the force fluctuations provided by the membrane’s coupling to its thermal
environment. With a sufficiently large Vpi and careful filtering of noise on Vdc , this can be achieved(see Appendix D for filtering details).

4.2.2

Construction of temporal and spectral mode converter

We create our mode converter by patterning thin films of aluminum on a sapphire substratefollowing a technique originally developed at NIST Boulder [136]. Multiple layers of aluminum are
initially separated by a layer of silicon nitride. Once the silicon nitride is removed, a suspended
aluminum membrane remains [174], as shown in Fig. 4.3. (A suspended aluminum membranewithout microwave circuitry is shown in Fig. 2.2b.) Two electrodes reside under the aluminum
membrane. One completes the capacitor of the microwave-frequency electromagnetic resonator,
and a separate annular electrode actuates the membrane. Additional circuitry couples microwave
signals to the resonator, and an inductor-capacitor low-pass filter prevents signals in the resonatorfrom coupling to the actuation line (further discussed in Appendix D).
The exact placement of the actuation electrode relative to the aluminum membrane is criticalfor circuit tunability. As shown in Fig. 4.4a, the actuation electrode is formed by a thin layer ofaluminum below the edge of the aluminum membrane. At room temperature both the actuationelectrode and the electrode of the microwave resonator are separated from the membrane by approximately 200 nm, as diagrammed in the lower panel of Fig. 4.4a. However, upon cooling tocryogenic temperatures needed to both operate the circuit and remove microwave frequency John-

66

Figure 4.3: Completed temporal and spectral mode converter. a, A schematic of the circuit,showing a separate annular actuation electrode and a central microwave capacitor. b, Scanningelectron micrograph of aluminum membrane, oriented similarly to panel a. Actuation electrodeand microwave capacitor plate are below the membrane and not directly visible. Only their outline
appears in the membrane in the micrograph. c, d, Progressively zoomed-out view of the aluminum
membrane. Signals are coupled to the inductor-capacitor electromagnetic resonator via a transmission line that passes near the inductor (visible as a thin line in c), and an inductor-capacitor
low-pass filter prevents signals in the resonator from coupling to the actuation line (the capacitorof the filter is visible in the upper portion of c). Multiple circuits couple to the same microwave
feedline, as visible in d. e, Photograph of completed sapphire chip that contains the temporal andspectral mode converter.

67
son noise, the differential thermal contraction between sapphire and aluminum greatly alters thegeometry. A finite element simulation of this thermal contraction is shown in Fig. 4.4b. Due to thealtered geometry, the separation between the membrane and the actuation electrode is greater than
the separation between the membrane and the capacitor plate. Ideally, this geometry circumventsthe limitation of electrostatically tunable capacitors discussed in section 4.2.1.
Although this design avoids the instability introduced by electrostatic tuning of the capacitor,
any attractive force that increases as two metal films are brought into proximity can affect thecircuit’s properties. For example, the Casimir force also provides an instablity [175, 176, 177], as
do patch potentials [178, 179] that can develop between the aluminum membrane and the electrodes
beneath it.
The finite element simulations of the mode converter geometry guide other aspects of contruction. In particular, alumium films with different thickness have different bending moments andreact differently to the strain created during cooling to cryogenic temperatures. For certain film
thicknesses, the capacitor is extremely likely to collapse during cooling to cryogenic temperatures(that is, the displacement of the membrane in Fig. 4.3b is greater than the separation between the
two aluminum films). For example, aluminum films with thickness 200 nm separated by 200 nm
are predicted to collapse during cooling. The film thicknesses used in this device represent a design
choice that creates an aluminum membrane that is free to vibrate at crogenic temperatures, but
this choice may not be optimal in some respects.

4.2.3

Initial characterization of mode converter

To verify tunability of the mode converter, we cool it to <25 mK in a dilution refrigerator
and measure properties of the mode converter as a voltage is applied to the actuation electrode.
As expected, the mode converter responds similarly to both positive and negative voltages (the2 ; see section 4.2.1), and greater magnitude voltages tune the frequency
electrostatic force is ∝ Vdc

of the electromagnetic resonator, ωe , lower as shown in Fig. 4.5. Unfortunately, the resonator only
tunes to about 75% of its maximum resonant frequency before an instability ‘pulls in’ the membrane

68

Figure 4.4: Mode converter geometry. a, Diagram of the mode converter, and cross section of
the parallel plate capacitor that is formed by depositing aluminum films (gray) on a sapphire
substrate (light blue). At room temperature, there is approximately 200 nm of separation between
the aluminum films. b, Finite element simulation of device geometry after cooling from room
temperature to cryogenic temperatures. The parallel plate capacitor in the microwave circuit has
a separation of approimately 40 nm, while the separation between the actuation electrode and thealuminum membrane is approximately 120 nm.

and collapses the capacitor. The instability also affects the membrane’s vibrational modes. At
Vdc = 0, the lowest frequency vibrational mode of the aluminum membrane has a frequency ofωm ≈ 2π × 9.6 MHz, a frequency set by the 200 MPa of tension created by differential thermal
contraction between the sapphire substrate and the aluminum films. As attractive forces between
the capacitor plates overwhelm the tension in the membrane, the membrane’s vibrational frequency
decreases. Just before the capacitor ‘pulls in’, the membrane’s vibrational frequency approaches
zero as can be seen in Fig. 4.2b and the data in Fig. 4.5. The pull-in voltage of approximately 17 V
represents a rough compromise between low sensitivity to noise and safe, easily generated voltages.
Given the geometry of the aluminum membrane and the underlying electrodes shown in
Fig. 4.4, we can predict the frequency tuning of the electromagnetic resonator and the vibrationalmodes of the membrane. A diagram of the geometry used for finite element simulation is includedin Appendix B. If we include only the electrostatic force between the metal films, the performance
predicted with a finite element simulation (shown as dashed lines in Fig. 4.5) deviates from the

69

7

ωm/2� (MHz)ωe/2� (GHz)

10

6
5
-20

-10

0
Vdc (V)

10

20

8
6
4
-20

-10

0
Vdc (V)

10

20

Figure 4.5: Frequency tuning of mode converter. Measured microwave resonant frequency (ωe ) andmechanical resonant frequency (ωm ) of the mode converter as a function of Vdc . Solid lines show
the expected peformance as calculated with a finite element simulation that includes an attractive
force that scales like the Casimir force and is within a factor of two of an estimated Casimir force
magnitude. Neglecting this additional attractive force results in the much different behavior shownwith the dashed lines.

observed frequency tuning: The predicted pull-in voltage and tuning of the microwave resonator are
significantly greater than observed. Only when we include an additional attractive force that scales
like the Casimir force and is within a factor of two of an estimated Casimir force magnitude [180]does the performance predicted with a finite element simulation (shown as solid lines in Fig. 4.5)
agree with the observed frequency tuning.
Although incorporation of the Casimir force accounts for our data reasonably well, we can not
rule out some contribution from patch potentials [178]. For this reason we view the data in Fig. 4.5
not as a measurement of the Casimir force, but as strong evidence that the Casimir force plays acrucial role in determining performance of the mode converter. Given its significance, the Casimirforce can potentially be used to counter the ‘pull-in’ effect from electrostatic forces: it is perhaps
possible to engineer repulsive Casimir forces that prevent the capacitor from collapsing [181, 182].
Such a repulsive force could be used to increase the frequency tunability of the electromagnetic
resonator. Nevertheless, the current device can be tuned over approximately 2 GHz, an amountsufficient to demonstrate spectral control of electromagnetic fields.

70
Table 4.1: Parameters of temporal and spectral mode converter.
Parameter

Value

Units

g
ωe
κext
κint
κ
ωm
κm
nm

2π×1*
2π×7.34†2π×2.32π×200κext + κint
2π×9.6†
2π×25
36

MHz
GHz
MHz
kHz
MHz
Hz

*Adjustable up to this value via the microwave pump.
†

4.3

Quoted values are with Vdc = 0 V

Radiation pressure interaction is naturally tunable
The mode converter described in section 4.2 consists of a mechanical oscillator coupled to atunable electromagnetic resonator. Although the tunable resonant frequency will allow for spectral
control of electromagnetic signals, the radiation pressure coupling provides temporal control. As
noted in section 2.4, a time dependent radiation pressure coupling arises when the nonlinear interaction between light and a mechanical oscillator is linearized in the presence of a large pump. To
see how this tunability can be used to manipulate electromagnetic signals, consider the equationsof motion for a mechanical oscillator and an electromagnetic resonator in the presence of a strong
pump. Using Eqns. 3.3 with only one electromagnetic resonator, we have−2ıg2√c(t) +
κext ain (t)
κ
κ


√
1
4|g|2
2ıg ∗ κextċ(t) = −κm +c(t) − √√ ain (t)2κκ
κa(t) =

Using the relation aout =

√

(4.2)

κext a(t) − ain (t), the dynamics of the system are described byp1ċ(t) = − (κm + Γ(t)) c(t) − e−ıψ ηΓ(t)ain (t)2paout (t) = eıψ ηΓ(t)c(t) + (2η − 1)ain (t)(4.3)

where Γ(t) = 4|g|2 /κ, η = κext /κ, and ψ = Arg(−ıg). A schematic of Eqns. 4.3 is shown in Fig. 4.6.

71

Figure 4.6: Time-dependent coupling between propagating electromagnetic fields and a mechanicaloscillator. a A microwave pump (blue arrow) is applied at a frequency ωm below the microwave
resonator (response shown as a black curve). b The microwave pump determines Γ(t), which
couples incoming and outgoing microwave fields to vibrations of the mechanical oscillator.

Eqns. 4.3 describe how a freely propagating electromagnetic field, ain (t), couples to vibrations of a mechanical oscillator, c(t). There is a strong formal similarity between Eqns. 4.3 andthose of an electromagnetic resonator coupled to a transmission line, Eqns. 2.10: Both describe
a freely propagating electromagnetic field coupled to a resonator. For the formal mapping to benear-exact, we require Γ(t)  κm and η ≈ 1. Parameters of our mode converter (see table 4.1)
approximately satisfy these requirements. This formal simlarity implies that both a simple electromagnetic resonator and our mode converter have similar signal processing capabilities. However,for the mode converter, both the strength of the coupling Γ(t) and its phase ψ can be modulated
by changing the amplitude and phase of a microwave pump. This tunability is generally not available with electromagnetic resonators, where the coupling to propagating fields is often fixed during
construction.
A tunable coupling is key to manipulating the temporal content of electromagnetic signals[183, 184]. Consider an incoming signal ain (t) with a temporal envelope as shown in Fig. 4.7. We
can use the mode converter to change the temporal content of this signal by implementing thefollowing protocol, which is diagrammed in Fig. 4.7. First, Γ(t) is modulated so that the incoming
signal, ain (t), is optimally captured as a vibration of the mechanical oscillator. If η ≈ 1, optimally

72
capturing the signal is equivalent to minimizing the reflected signal, aout (t), for times 0 < t < t1 .
Rt
Minimizing the energy in the reflected signal, 0 1 |aout (t)|2 dt, with η = 1 requires a coupling Γ(t)of [185]2
Γ(t) = a (0)2
in

Γ(0)

eκm t |ain (t)|2Rt+ 0 eκm t0 |ain (t0 )|2 dt0(4.4)

where Γ(0) is the coupling at t = 0. Once the incoming signal is captured, Γ(t) = 0 for t1 < t < t2 .
During this time, the captured signal is stored as a vibration of the mechanical oscillator. Beginningat time t = t2 , Γ(t) is modulated to recover the signal with the desired waveform. An experimental
implementation of this temporal mode conversion protocol is given in section 4.5.
For the temporal mode conversion protocol to be successful, it must manipulate electromagnetic signals before they are corrupted or lost. For an optomechanical mode converter, the entire
protocol should be completed before the mechanical oscillator acquires one phonon from its environment, which occurs at the mechanical decoherence rate nm κm . However, signals can only be
manipulated with bandwidth Γ. Arbitrary control over the temporal content of signals thus requiresΓ  nm κm . A similar figure of merit is used in optical signal processing, where a large bandwidthdelay product (BDP) indicates large signal processing power [186]. For an optomechanical system,
BDP ≈ Γ/nm κm . Our mode converter approaches BDP  1 with a maximum Γ ≈ 2π × 1 MHz
and a decoherence rate nm κm ≈ 2π × 1 kHz, or a BDP of 103 .

4.3.1

Relation between optomechanical coupling and mechanical resonant frequency

Eqns. 4.3 allow temporal control of electromagnetic fields; however, these equations of motion
are an approximation of the radiation pressure interaction assuming the resolved-sideband regime,
4ωm  κ. In this regime, with the specified pump detuning ∆ = −ωm , the mechanical oscillator
experiences an additional damping Γ(t) = 4|g(t)|2 /κ and a frequency shift Ω = 0, as given by
Eqns. 2.26. Even though our mode converter is in the resolved-sideband regime (see table 4.1), the2This
R t coupling can be found by using the Euler-Lagrange equation to find a stationary point of the the reflectedenergy, 0 1 |aout (t)|2 dt, with aout (t) as given by Eqns. 4.3.

73

ωm
ωe

|ain(t)|

Γ(t)

ωm

Γ(t)

ωe

Capture Store Shape &release|aout(t)|

√Γ(t)
0

t1

Time

t2

t3

Figure 4.7: Protocol for temporal mode conversion. An incoming microwave signal, ain (t), at
frequency ωe is first captured as a vibration of the mechanical oscillator by modulating Γ(t) (bluewaveform). After an adjustable storage time, the signal is released from the mechanical oscillator
into outgoing microwave signal, aout (t), by modulating Γ(t).

microwave pump still slightly alters the frequency of the mechanical oscillator. For a pump withdetuning ∆ = −ωm , Ω(t)/Γ(t) ≈ −κ/8ωm , a ratio that is independent of pump amplitude. As
Γ(t) is modulated to shape an outgoing microwave signal, the corresponding change in Ω(t) ‘chirps’
the frequency of the signal. This effect can be corrected by altering the frequency of the pump by−Ω(t) = κΓ(t)/8ωm . We use this correction when operating our mode converter.

4.4

Measurement network
We create a measurement network that allows us to inject microwave signals into the mode

converter and carefully measure signals reflected or emitted from the mode converter. The main
components of the network are diagrammed in Fig. 4.8. A combination of arbitrary waveform

74

Figure 4.8: Measurement network. A signal and a pump are created using standard microwave
techniques and injected into the mode converter. A microwave receiver measures signals reflectedor emitted from the mode converter. To calibrate the efficiency ζ and gain G, a resistive load with avariable temperature injects Johnson noise into the microwave receiver. A detailed schematic that
includes components used in microwave generation is included in Appendix D

generators and microwave mixers generates a microwave pump and signal with programmableamplitude, frequency, and phase. A microwave receiver measures signals after they have interactedwith the converter. The performance of the receiver is characterized by an efficiency ζ, and a gain,
G. The efficiency can in principle range between 0 (no measurement) and 1 (perfect measurement),
and the gain expresses how a spectral density at the input of the receiver is converted to a spectral
density at the output. To calibrate the microwave receiver, the measurement network includes a
resistive load whose temperature TL can be varied. Once a signal is amplified, it is downconvertedwith a frequency mixer and digitized. Data collected thus consist of time-stamped voltage values.
A detailed schematic of the measurement network is included in Appendix D.

4.4.1

Calibration of microwave receiver

The microwave receiver employs a Josephson parametric amplifier (JPA) [187]. The JPA
amplifies one quadrature of a microwave signal and deamplifies the other, where a quadrature iscomposed of linear combinations of signals at frequencies above and below the JPA’s pump fre-

75

Figure 4.9: Calibration of microwave receiver. The output spectral density of the microwave receivervaries with the temperature of the load. A fit to Eqn. 4.6 (solid line) shows the receiver has anefficiency of ζ 0 = 0.6 and a gain G = 7(µV)2 /(Hz · quanta).

quency [188, 189]. Before amplification or deamplification, the spectral density of each quadratureemitted by the resistive load is given by, in units of quanta (a quanta corresponding to a photon
per second per unit bandwidth),1
SL = coth2~ω2kB TL(4.5)

where TL is the temperature of the resistive load [189]. At temperatures TL  ~ω/kB , SL approaches
1/2 quantum. At these low temperatures, the total spectral density of both quadratures is thus
1 quantum, consistent with the convention that each unit of bandwidth has a minimum spectral
density of 1/2 quantum [93]. Similarly, at temperatures TL  ~ω/kB , the total power emitted inboth quadratures by the resistive load is 2kB TL /~ω, the classical Johnson noise formula for twounits of bandwidth.
Once measured by the microwave receiver, we expect to observe a spectral density given by[190]
Sout = G
1 − ζ00+ ζ SL .
2(4.6)

Fitting this model to the measured receiver output at ω = 2π × 7.34 GHz yields ζ 0 = 0.6 and
G = 7.0 (µV)2 /(Hz · quanta), as shown in Fig. 4.9.

76
We use the microwave receiver to measure signals reflected or emitted from the mode converter. These signals are entirely contained in frequencies above the JPA pump frequency. As such,the measured signal is contained neither in the amplified or deamplified quadratures, but in both.
The signal in the deamplified quadrature is not measured and this loss of information reduces theefficiency of our microwave receiver by a factor of two, so ζ = ζ 0 /2 = 0.30. The maximum possible
efficiency for this mode of operation is ζ = 0.5 [127].

4.5

Temporal mode conversion of microwave signals
As an initial test of our measurement network, we generate a microwave signal and measure

it with the microwave receiver. The signal is centered at the mode converter’s resonant frequencyof ωe = 2π × 7.34 GHz (with Vdc = 0) and has an exponentially rising temporal envelope: |ain (t)| ∝(1 − Θ(t − t1 ))e(t−t1 )γ/2 , where Θ(t) is the Heaviside step function and γ = 2π × 24 kHz. This
waveform is shown in Fig. 4.10a. Initially, we inject no microwave pump. With no pump, Γ(t) = 0
and aout (t) ≈ ain (t) as indicated by Eqns. 4.3. The measured signal, shown in Fig. 4.10b, isconsistent with this prediction and confirms the signal generation and microwave receiver function
as expected.
In order to manipulate the temporal content of the incoming microwave signal, we mustmodulate Γ(t). To capture an incoming signal with a rising exponential envelope as a vibrationof the mechanical oscillator, we inject a constant microwave pump so that Γ(t) = γ during signal
capture. By pulsing Γ(t) between Γ(t) = γ and Γ(t) = 0, we capture, store, and retrieve theincoming microwave signal, as shown in Fig. 4.11. This protocol stores the incoming microwave
signal and reverses its temporal envelope [58].
In order to arbitrarily manipulate the temporal content of an incoming signal, Γ(t) must be
carefully chosen. Consider, for example, an incoming microwave signal with a decaying exponential
temporal envelope: |ain (t)| ∝ Θ(t)e−γt/2 . This temporal envelope is ubiquitous, and in particular
highly coherent circuit QED systems produce quantum electromagnetic signals with similar envelopes [166, 165]. A microwave pump pulsed so that Γ(t) = γ or Γ(t) = 0 does not efficiently

a

77

b

Input

Measured output

0.04|ain(t)|

Volts

0

0
0

50
Time (μs)

√Γ(t)

100

0
0

50
Time (μs)

100Figure 4.10: Test of microwave receiver. a, A signal with a rising exponential temporal envelope(green waveform) is injected into the mode converter. Because no pump is injected, Γ(t) = 0
and the incoming signal should reflect off the mode converter. a, Outgoing microwave signal asmeasured with our microwave receiver. As expected, the outgoing signal has the same temporalenvelope as the injected signal.

a

b

Input

Measured output

0.04|ain(t)|

Volts

0

0

0
√Γ(t)

50
Time (μs)

100

0
0

50
Time (μs)

100Figure 4.11: Storage of microwave signals. a, A microwave signal (green waveform) and a pump are
injected into the mode converter. b, By pulsing Γ(t), the signal is temporarily stored as a vibrationof the mechanical oscillator and later recovered with a decaying exponential temporal envelope.

capture an incoming signal with a decaying exponential temporal envelope. As demonstrated inFig. 4.12, much of the incoming signal is reflected and only a fraction is stored as a vibration of themechanical oscillator and later retrieved. To efficiently capture this signal, Γ(t) must be modulated
as prescribed by Eqn. 4.4:
Γ(t) = Θ(t)γe−γt1 − e−γt + γ/Γ(0)

(4.7)

78
where I’ve assumed γ  κm and Γ(0) is an input; Γ(0)  γ yields a high capture efficiency. We
use γ = 2π × 24 kHz and Γ(0) ≈ 2π × 500 kHz. As demonstrated in Fig. 4.13a, this choice of Γ(t)efficiently captures the incoming signal as a vibration of the mechanical oscillator. We choose torelease the signal with a Gaussian temporal envelope, as shown in Fig. 4.13b, because Gaussianenvelopes are time-reverse symmetric and are relatively compact in both time and frequency. The
coupling necessary to produce a Gaussian envelope centered at time t = to can be found using
Eqn. 4.4:22γe−γ (t−to )Γ(t) =
1 − erf (γ(t − to )) + δ

(4.8)
2 2

where erf(t) is the error function, γ = 2π × 24 kHz, and δ ≈ γ/Γ(0)e−γ to [185].

4.6

Temporal and spectral mode conversion of microwave signals
To incorporate spectral mode conversion into our protocol for temporal mode conversion, we

utilize the signal storage capability of the mode converter. As can be seen in Figs. 4.11 and 4.13,
our temporal mode conversion protocol stores microwave signals for approximately 30 µs. We canuse this time to alter the center frequency of the electromagnetic resonator by adjusting Vdc . This
allows a microwave signal captured at one frequency to be released from the mode converter at
another. A protocol for temporal and spectral mode conversion is diagrammed in Fig. 4.14.

4.6.1

Wigner-Ville distribution

Signals emitted from the mode converter have different temporal and spectral content depending on the details of the mode conversion protocol. To clearly represent the voltage waveform
measured by our microwave receiver, we construct the Wigner-Ville distribution using the disR∞
cretized version of W (t, f ) = −∞ v(t + τ /2)v ∗ (t − τ /2)e2πıf τ dτ , where W (t, f ) is the Wigner-Ville
distribution and v(t) is the analytic voltage measured at time t. The analytic voltage is constructedusing a Hilbert transform on the measured voltage [191]: v(t) = V (t) + ıH(V (t)), where V (t) isthe voltage measured at time t and H is the Hilbert transform. The resulting time-frequency

a

79

b

Input

Measured output

0.04|ain(t)|

Volts

0

0

0
√Γ(t)

50
Time (μs)

100

0
0

50
Time (μs)

100Figure 4.12: Sub-optimal temporal mode conversion of a microwave signal. a, A signal with adecaying exponential temporal envelope (green waveform) and a pulsed pump (blue waveform) areinjected into the converter. b, Much of the incoming signal is reflected from the mode converter;
only a fraction is captured as a vibration of the mechanical oscillator and later released.

a

b

Input

Measured output

0.04|ain(t)|

Volts

0

0

0
√Γ(t)

50
Time (μs)

100

0
0

50
Time (μs)

100Figure 4.13: Temporal mode conversion of a microwave signal. a, A signal with a decaying exponential envelope (green waveform) and an optimally shaped microwave pump (blue waveform)
are injected into the mode converter. b, The vast majority of the incoming microwave signal iscaptured as a vibration of the mechanical oscillator, and emitted as an outgoing microwave signal
with a Gaussian temporal envelope.

distribution is similar to the position-momentum Wigner quasiprobability distribution.
Although there are many ways to represent the temporal and spectral content of signals [192],
the Wigner-Ville distribution has several convenient properties. The marginal distributions of the
Wigner-Ville distribution give the temporal envelope and energy spectral density of the signal:
R∞R∞|v(t)|2 = −∞ W (t, f )df and |v(f )|2 = −∞ W (t, f )dt. Additionally, the Wigner-Ville distribution

80

a
ωm

Vdc

Capture Store &
shift
ωe,1

ωm

Γ(t)

ωe,2

Shape &
releaseωe,2

Pump
Amplitude

Resonant
frequency

ωe,1

Γ(t)

30
60
Time (μs)

0

b

ωm

90
ωm

ωe,1

ωe,2

Figure 4.14: Protocol for temporal and spectral mode conversion. a, An incoming signal at frequency ωe,1 (green sinusoid) is first converted into a vibration of the aluminum drumhead (blacksinusoid). Γ(t) is adjusted by varying the power in a pump at frequency ωe,1 − ωm (red waveform).
Then, we alter the frequency of the microwave circuit by changing Vdc . During this time, the signal
is stored as a vibration of the mechanical oscillator; the change in Vdc adiabatically changes ωmby approximately 2π × 300 kHz. Finally, the signal is released from the mechanical oscillator into
an outgoing microwave signal at frequency ωe,2 by adjusting the power in a pump at frequency
ωe,2 − ωm (blue waveform). b, Frequency domain picture of mode conversion. Microwave pumps(red and blue arrows) applied at a frequency ωm below the microwave circuit (response shown as
the red and blue curves) allow a signal at frequency ωe,1 to be converted to a signal at frequency
ωe,2 (green arrows).

provides high temporal and spectral resolution [193]. However, the marginal properties and resolution of the Wigner-Ville distribution necessarily imply W (t, f ) can be negative [193]. Unlike theposition-momentum Wigner quasiprobability distribution, nothing ‘quantum’ is associated withnegative values of W (t, f ).

81
4.6.2

Temporal and spectral mode conversion

We generate a signal near 7.08 GHz that has a decaying exponential envelope (power decayrate of 2π × 24 kHz) and inject it into the mode converter. Initially, we set Vdc so that the
microwave circuit has a resonant frequency substantially different from the injected signal, and thesignal reflects off the mode converter and is measured by our microwave receiver. The measured
Wigner-Ville distribution of the injected signal is shown in Fig. 4.15a, with the data normalizedR∞
by Esig = −∞ |vsig (t)|2 dt.
To perform mode conversion, we set Vdc ≈ 10 V so the microwave circuit is centered near
7.08 GHz, and send in the decaying exponential envelope signal. We apply a microwave pump witha time-dependent amplitude so that Γ(t) is given by Eqn. 4.7. Only a small amount (about 4%) of
the signal energy is reflected off the converter, as demonstrated in Fig. 4.15b. We then set Vdc = 0
so that the microwave circuit is centered near 7.34 GHz, and again apply a microwave pump witha time dependent so that Γ(t) is given by Eqn. 4.8. The Wigner-Ville distribution of the signal
emitted by the mode converter is plotted in Fig. 4.15b. From the relative energies in the reflected
and converted signals, we find the photon number efficiency of the conversion process to be 0.81, in
good agreement with an expected efficiency of (κext /κ)2 × 0.96 = 0.81 where 0.96 is the efficiency
with which we capture the injected signal [185].
The mode converter can be used to change signals between arbitrary temporal and spectralmodes. We demonstrated frequency conversion over approximately 250 MHz; conversion over alarger window is feasible but the mode converter becomes more sensitive to fluctuations in Vdc at
higher bias. And although we altered the temporal envelope of a microwave signal from a decaying
exponential to a Gaussian, other envelopes are possible, only limited by the bandwidth of theconverter. The maximum bandwidth is set by the rate at which information can be exchanged
between the transmission line and the drumhead, which for this device is limited by the microwave
circuit’s bandwidth of Γ < κ/2 ≈ 2π × 1 MHz. The minimum bandwidth is set by the mechanicaldecoherence rate. Even though energy in the drumhead is lost to the environment at the modest

82

a

b
εt (μs-1)εt (μs-1)

Without mode conversion
0.1

07.34185

With mode conversion

0.1
0

0.5

Frequency (GHz)7.34160W(t, f )/Esig 0-0.5

7.34135

0 10

7.078400 10

7.07815

7.07790050Time (μs)

100

0 10
εf (MHz-1)

050Time (μs)

100

0 10
εf (MHz-1)

Figure 4.15: Temporal and spectral mode conversion of a microwave signal. a, Wigner-Ville distribution of a signal that reflects off the mode converter (so no mode conversion is performed) and ismeasured with a microwave receiver. The marginal distributions of the Wigner-Ville distribution
show the signal is a decaying exponential in time, and a Lorentzian in frequency; t = |v(t)|2 /Esig
and f = |v(f )|2 /Esig . The signal is centered near 7.07815 GHz and has a power decay rate of
2π × 24 kHz. There is no signal content near 7.34160 GHz. b, Wigner-Ville distribution of a modeconverted signal. When the mode converter is used, the decaying exponential microwave signal
is captured as a vibration of the mechanical oscillator, and the signal content at 7.07815 GHz isalmost entirely absent. Later in time, we recover the signal that is now centered near 7.34160 GHz
and has a Gaussian temporal envelope and Gaussian spectral content with width 2π × 24 kHz.

83
rate of κm = 2π × 25 Hz, fluctuations of the environment incoherently drive vibrations of thedrumhead. To avoid these environment-driven vibrations, signals must be processed faster thanthe mechanical decoherence rate of nm κm = 2π × 900 Hz. Avoiding excess vibrations is essential
when manipulating fragile quantum signals.

4.6.3
4.6.3.1

Operation at the quantum level
Quadrature amplitudes of microwave signals

In order to analyze signals at the quantum level, we would like to express a measured signal
in units of photon flux per unit bandwidth, or quanta, referred to the input of the receiver. For
a continuous wave signal, we can express the signal in quanta using a Fourier transform of themeasured voltage waveform. In this case, we define quadrature amplitudes at a frequency ω:
r
X1 =
r
X2 =

2
·
ζG

r

2
·
ζG

r

N

δt X
·
y(k) · cos(ω · k · δt)
N
δt
·
N

k=1
N
X

y(k) · sin(ω · k · δt)

(4.9)

k=1

where y(k) is the measured voltage waveform of length N , and 1/δt is the sample rate. The factorp√2/(ζG) converts the voltage waveform into units of quanta · Hz at the input of the receiver (thefactor of two is present because G was calculated using a single-sided spectral density). X1 andX2 are proportional to the real and imaginary parts of the complex Fourier amplitude, and can beconsidered to represent the magnitude and phase of the signal: X12 + X22 yields an estimate of thenumber of quanta contained in the signal, and arctan(X2 /X1 ) the phase of the signal.
Eqns. 4.9 are appropriate when the signal is continuous and at a known frequency. When asignal has a particular temporal envelope, this definition no longer provides an accurate descriptionof the signal. To compensate for the broader bandwidth of the signal, we define quadrature amplitudes using a Fourier transform with a window function where the window function matches the
temporal envelope of the signal. This window function prevents the quadrature amplitudes from

84
being affected by noise that occurs either before or after the signal.
r
X1 =
r
X2 =

2
·
ζG

r

2
·
ζG

r

N

δt X
·
f (k) · y(k) · cos(ω · k · δt)
C
δt
·
C

k=1
N
X

f (k) · y(k) · sin(ω · k · δt)(4.10)

k=1

where f (k) is the temporal envelope of the signal and C =NP|f (k)|2 is a factor that compensates

k=1

for the noise equivalent bandwidth of f (k).
Using the window function f (k) with a Fourier transform is equivalent to using least squares
parameter estimation, where the parameters to be estimated are the amplitudes of f (k)·cos(w·k·δt)
and f (k) · sin(w · k · δt). This method of parameter estimation is appropriate when noise in thevoltage waveform is Gaussian, as is the case for our measurements [194].

4.6.3.2

Noise added during mode conversion

To verify that our converter operates quickly and quietly enough to avoid corrupting signals,
we collect statistics on single-shot measurements of mode conversion using low amplitude (∼ 10
quanta of energy) signals. We inject a low-amplitude signal with a decaying exponential envelope
into the converter, shift its frequency by 90 MHz, and recover a signal with a Gaussian envelope.
The central frequency and temporal envelope of the converted signal are already known, and we
use this information and Eqns. 4.10 to estimate the quadrature amplitudes X1 and X2 . The resultof each single shot measurement and quadrature amplitude estimation appears as a single point in
Fig. 4.16a. The scatter of the quadrature amplitudes indicates the total amount of noise added
to the signal during preparation, mode conversion, and measurement. By comparing the total
variance, Var(X1 ) + Var(X2 ), of the converted signal with that of measured vacuum noise, we infer
the mode converter adds 0.92 ± 0.07 quanta of noise during conversion with a 90 MHz frequency
shift, indicating compatibility with fragile quantum states [195].

a

85

b

X1 (quanta1/2)

10

0

-10
-10
Added Noise(quanta)c

0
10
X2 (quanta1/2)

0 130
counts

1
0.5
0

0

50
Frequency shift (MHz)100

Figure 4.16: Noise added during mode conversion. a, Quadrature amplitudes for 500 independent repetitions of the mode conversion protocol. Results of conversion with a 90 MHz frequency
shift with a signal containing ∼10 quanta (blue points) and ∼100 quanta (green points) of energy.
Low and high amplitude signals are intentionally generated with different phases. For reference, avacuum input into our microwave receiver yields the gray points. The total variance of the quadrature amplitudes, Var(X1 )+Var(X2 ), subtracted from the variance of measured vacuum indicatesthe amount of noise added during signal generation and mode conversion, which we find to be
0.92 ± 0.07 quanta. b, X1 marginal of the quadrature amplitudes with Gaussian fits (black curves).
c, For frequency shifts less than 90 MHz, we observe less added noise. Error bars, s.e.m. Dashedgray line indicates expected added noise from a mechanical decoherence rate of nth κm = 2π × 900
Hz.

4.7

Future work
We have demonstrated a temporal and spectral mode converter whose operating frequency,

bandwidth, and low noise properties make it compatible with many circuit QED systems. As anext step, our mode converter could be integrated with a single circuit QED system consistingof a superconducting qubit embedded in a high-Q microwave resonator. This could enable thepreparation of non-classical states of motion of a macroscopic mechanical oscillator. For example,
circuit QED has been used to generate many-photon Schrodinger cat states of the electromagnetic

86
field [196]. By releasing these quantum signals into a transmission line and then capturing them asvibrations of the mechanical oscillator, we could in principle prepare the mechanical oscillator in adisplacement superposition. This superposition could be used to probe the intersection of gravity
and quantum mechanics [27], one of the original motivations of cavity optomechanics.
From an experimental point of view, even if non-classical states of motion can be prepared,
they must also be detected. Measuring the position of a mechanical oscillator with sensitivity
great enough to detect non-classical motion is an outstanding challenge for the field of cavity
optomechanics [197, 198, 199, 167]. A related approach to measuring the motion of a mechanicaloscillator amplifies motion via parametrically pumping the oscillator’s resonant frequency [200,201]. This latter approach can be implemented with our mode converter by utilizing the actuationelectrode. This offers a potential route to nearly noiseless, phase-sensitive detection that is capableof detecting nonclassical motion.
Although the temporal and spectral mode converter functions well and adds less than one
quanta of noise to the converted signal, the mechanical decoherence rate limits the added noise
to &0.4 quanta, as shown in Fig. 4.16b. A mode converter that utilizes membranes with higher
quality factor vibrational modes could thus reduce added noise. In particular, vibrational modesof silicon nitride membranes have quality factors well in excess of those used here. Silicon nitride
membranes have already been integrated with superconducting microwave circuits [148, 202]. These
construction techniques, combined with advances in both understanding and fabricating silicon
nitride membranes [65, 146, 145], could enable mode converters with greatly improved performance.

Chapter 5

Prosepects for quantum networking with mechanical oscillators

The microwave-to-optical frequency conversion demonstrated in Chapter 3 and the arbitrary
temporal and spectral mode conversion presented in Chapter 4 together offer a route to overcome
many of the signal processing challenges inherent in quantum information networks. However,the signal manipulation presented here utilizes low-amplitude classical signals. A next task is tocombine macroscopic mechanical oscillators into networks that manipulate quantum signals. As
suggested in section 4.7, a simple quantum network could be formed from a circuit QED system andour temporal and spectral mode converter. Such a network could be used to generate nonclassicalstates of macroscopic motion, but also confirm that mechanical oscillators can process quantum
electromagnetic signals.
For a cavity optomechanical system to manipulate quantum signals with high fidelity, it must
add much less than 1/2 quantum of noise to the signals [195]. Although the temporal and spectralmode converter can add approximately 0.4 quanta of noise, for high-fidelity manipulation boththe mode converter and the microwave-to-optical frequency converter must operate deep in thequantum-enabled regime with Γ  nm κm . Key to reaching deeper into this limit is the use ofhigh-Q mechanical oscillators, as well as boosting Γ. However, for the type of signal processing
presented here, there is a strict frequency hierarchy: 2Γ < κ  4ωm , as set by the weak-coupling
and resolved-sideband requirements introduced in section 2.4. High bandwidth manipulation of
propagating electromagnetic signals thus requires a large ωm . By utilizing recent advances in
microfabrication and understanding of membrane mechanical oscillators, we hope to combine elec-

88Port 1

Port 3
L1

Port 2

L3
C1

C3

C2

C4

L2

Port 4L4

membrane
Figure 5.1: Electromagnetic signal router based on a mechanical oscillator. The radiation pressureinteraction can be used to route quantum signals. For example, a signal entering through port 1
could be captured as a vibration of the mechanical oscillator and then released out any of the four
ports.

tromagnetic resonators and mechanical oscillators that have Q > 107 and ωm > 2π×20 MHz. Thesesystems could bring truly quantum and useful capabilities to electromagnetic signal processing.
If mechanical oscillators can be used successfully in small quantum networks, their signal
processing capabilities can be expanded to be relevant to larger and more complex networks. For
example, large networks need devices that can route quantum signals between many different
locations and components [203, 204]. An optomechanical router can potentially be created by
combining a mechanical oscillator with multiple electromagnetic resonators. A sketch of such arouting circuit is shown in Fig. 5.1. The flexibility of microfabricated circuits and the radiationpressure interaction may allow optomechanical systems to play a key role in the development of
quantum information networks.

5.1

Conclusion
The radiation pressure interaction between light and a mechanical oscillator provides a general

method for manipulating quantum electromagnetic signals. The microwave-to-optical frequency

89
conversion of Chapter 3 demonstrates the broadband nature of the interaction, and the temporaland spectral mode conversion of Chapter 4 highlights the time-domain control available with theradiation pressure interaction. Next steps include integrating cavity optomechanical systems into
small quantum networks, and extending their signal processing capabilities to include amplification
and routing. Given recent progress in cavity optomechanics, I think mechanical oscillators will havea place in future quantum information networks.

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Appendix A

Construction details of circuits and membranes

Figs. A.1 and A.2 overview the microfabrication steps used to create the microwave circuit
used in the microwave-to-optical frequency converter and the temporal and spectral mode converter,
respectively. The figures show profile views starting from the initial wafer, which is made fromeither silicon or sapphire. Layers of material are added and removed using standard deposition[205, 206] and etching [207] processes. Tables A.1 amd A.2 give the deposition processes andetch/removal processes for different materials used during construction of the frequency converter
and temporal and spectral mode converter, respectively. All fabrication was performed at the
NIST Boulder Microfabrication Facility (BMF). Additional information about construction of the
microwave circuit is provided in section 3.3.1 (for microwave-to-optical frequency conversion) andsection 4.2.2 (for temporal and spectral mode conversion).

108

Deposit 100 nm Si3N4

Deposit ≈400 nm Al and pattern

Deposit ≈30 nm Nb and pattern

Deposit ≈200 nm Nb and pattern

Etch Si ≈25 μm

Etch Si3N4 and ≈25 μm Si

Etch through wafer

Etch Si with KOH

Assembly
Si3N4
SiliconNiobium
AluminumEpoxyd300 μm25 μm

Assembly

Figure A.1: Construction details of microwave circuit for frequency conversion. Both structures
begin with high-resistivity silicon wafers. With an aluminum layer of ≈ 400 nm, the separation
distance d ≈ 400 nm.

Table A.1: Deposition and etch processes used during construction of the microwave circuit used
for frequency conversion.
Material

Deposition process

Removal process

Silicon nitride (Si3 N4 )Niobium

LPCVD
DC magnetron sputtering

Aluminum
SiliconElectron beam deposition

RIE with CF4
RIE with 30% O2 and 60% CF4 or
ICP+RIE with SF6
Wet etch with Transene type A
DRIE with SF6 and C4 F8 or
Wet etch with KOH

109

Deposit ≈100 nm Al and pattern

Selectively deposit ≈25 nm Al

Deposit ≈100 nm SiN and pattern

Deposit ≈100 nm Al and pattern

Etch SiN
SiN
Sapphire
Niobium
Aluminum

Figure A.2: Construction details of temporal and spectral mode converter. Construction begins
with a sapphire wafer. The two initial layers of aluminum form the microwave electrode and theactuation electrode, visible in Fig. 4.3. The final layer of aluminum forms the membrane once thesilicon nitride is removed.

Table A.2: Deposition and etch processes used during construction of the temporal and spectralmode converter.
Material

Deposition process

Removal Process

Aluminum, 100 nm layers
Aluminum, 25 nm layer
Silicon nitride (SiNx )DC magnetron sputtering
electron beam deposition
PECVD

RIE with 25% Cl2 and 75% BCl3
RIE with 60% O2 and 40% CF4 or
RIE with SF6

Appendix B

Predicted frequency tuning of the temporal and spectral mode converter

The approximate geometry of the temporal and spectral mode converter is outlined inFig. 4.4a. With this geometry, we use a finite element simulation to predict the effect of thermal
contraction between the aluminum and the sapphire substrate. In simulation, the total thermal
contraction is adjusted so that the central portion of the drumhead deflects by ≈160 nm, leavinga ≈40 nm separation between the plates of the microwave capacitor, a value set by the measured
Gxzp = 2π × 270 Hz. The total thermal contraction needed to create this amount of deflection
is seven parts per thousand, a value approximately consistent with that of bulk aluminum. The
predicted effect of thermal contraction on device geometry is shown in Fig. 4.4b.
Using relevant parameters from the thermally-altered geometry of Fig. 4.4, we create a simplified model to predict the frequency tuning of the mode converter. This model, detailed inFig. B.1, is used in a finite element simulation to estimate the membrane’s vibrational frequency
and deflection. As the membrane deflects with increasing voltage, it alters the capacitance in the
microwave circuit and thus its resonant frequency. This effect is modeled by assuming a parallel
plate capacitance (Cm ) that is modulated by changes in the position of the aluminum membrane,
and an additional parasitic capacitance (Cp ) provided by, for example, the coils of the inductor.
With no applied voltage, Cm / (Cp + Cm ) = 0.65, estimated from electromagnetic simulations of the
microwave circuit. Using this information, we predict the frequency tuning shown as dashed lines
in Fig. 4.5.
In order to correctly predict the frequency tuning of the device, we must include the effects of

111

Figure B.1: Simplified model of the aluminum membrane and actuation electrode used in the mode
converter. The membrane is under tension with fixed edges so that with Vdc = 0, ωm = 2π × 9.56
MHz. The membrane is periodically perforated so that it resembles Fig. 4.3. A voltage Vdc canbe applied between the membrane and the actuation electrode. Additional attractive forces areincorporated by distributing the force over the central area of the membrane that overlaps withthe capacitor plate (the capacitor plate is shown for clarity, but is not used during simulation).
For example, to model a Casimir force, P → −0.7 × ~cπ 2 /240(d − x)4 , where d = 40 nm is theinitial separation distance with Vdc = 0 and x is the local deflection of the membrane away fromits equilibrium position.

the Casimir force. We include an attractive pressure of approximately 0.7 × Pcas in a finite element
simulation of the membrane in the presence of a DC voltage, where Pcas = −~cπ 2 /240a4 is the
Casimir pressure for two parallel, perfectly conducting plates separated by a distance a of vacuum.
(For imperfectly conducting plates, a reduction in the Casimir force for small separations d . λp is
expected, where λp is the plasma wavelength of the conducting plates [180].) With this additionalpressure, we predict the frequency tuning shown as solid lines in Fig. 4.5.

Appendix C

Measurement network for microwave-to-optical frequency converter

Fig. C.1 shows a detailed diagram of the network used to measure the microwave-to-opticalfrequency converter presented in Chapter 3. The network contains components that function at
microwave frequencies (≈7 GHz), optical frequencies (≈282 THz), and mechanical frequencies (≈1MHz). The necessary optical components are discussed in section C.1, and the necessary microwave
components in section C.2. Mechanical frequency signals are written onto microwave or opticalsignals and then injected into the converter before being measured, as discussed in section C.3.

C.1

Optical network
The optical network utilizes two laser beams: A pump (Fig. C.1, red line) and a locking signal(Fig. C.1, green line), both of which are derived from a single 1064 nm Nd:YAG laser (Coherent,
Inc. Mephisto). The laser beam is double-passed through a 200 MHz AOM that locks the laser
frequency to the cavity, and then split into the pump and locking signal using two AOMs. The
AOMs are modulated at approximatey 80 MHz with a relative detuning of 1 MHz, so the pump
has a frequency 1 MHz below the locking signal. Polarizing beamsplitters (PBSs), and a half-wave
plate are used to inject the pump and locking signal into orthogonal polarization modes of the
optical cavity. Light reflected or emitted from the cavity is separated from incoming light usinga Faraday rotator and detected with photodetectors. Transmitted light is also detected and used
to measure optomechanical parameters. The measured locking signal provides feedback to the 200
MHz AOM and locks the frequency of the locking signal to the cavity’s resonance using the Pound-113
Drever-Hall technique. The cavity has a birefringent splitting of 270 kHz, and so the optical pump
has a detuning of ∆o = −2π × 730 kHz. The locking signal is low power and near the opticalcavity’s resonant frequency so the dynamical backaction from this signal can be neglected.

C.2

Microwave network
The microwave network utilizes an Agilent PSG signal generator to create a microwave pumpat ≈ 7 GHz. This pump drives the microwave port of the converter, and additionally provides a local
oscillator for two quadrature-IF mixers (Marki IQ-4509LXP) that are used to both upconvert anddownconvert the signals that interact with the converter. Microwave signals reflected or emitted
from the cavity are amplified using a room-temperature low noise amplifier (Miteq, Inc.). The
microwave cavity is intrinsically stable (environmental fluctuations change the microwave resonant
frequency by much less than its linewidth), and no stabilization technique is needed.

C.3

Signal generation and detection
A network analyzer generates a swept signal near the mechanical oscillator’s resonant fre-quency. This signal is upconverted to either microwave or optical frequencies via single-sideband(SSB) modulation of the optical or microwave pump, a process where a signal (at frequency ω1 )modulates a pump (at frequency ω2 ) to create an ‘upper sideband’ on the pump (at frequency
ω1 + ω2 ). The signal directly modulates the microwave pump using a microwave-frequency mixer.
However, to modulate the optical pump, the signal first modulates an ≈80 MHz pump that drives
an AOM. Any stray lower sideband modulation (at frequency ω2 − ω1 ) is suppressed at least 20 dB(10 dB) below the upper sideband for SSB modulation at microwave (optical) frequencies. Afterinteracting with the converter, reflected and transmitted signals are demodulated and detectedusing homodyne (direct) detection for microwave (optical) signals. The spectral density of outgoing optical signals can also be measured with a spectrum analyzer. Three switches allow themeasurement network to change between measuring microwave or optical frequency signals.
For a measurement of S21 (ω) = aout (ω)/bin (ω), we generate an upper sideband on the mi-

114
Network Analyzer
in Q

c

SW2

in I

out

SW3
I

Spectrum
Analyzer

Q
L

R

PD

b
7 GHz

20 dB
-20 dB

SW1

-90°
0°

L

-90°
0°

L

Q
I
Q
I

R

R
Faraday rotator

λ/2 PBSPD

87 MHz
PBS

a

PD

EOM

-86 MHzAOM
Nd:YAG
1064 nm

200 MHzAOM

to lock

50:50 BS-87 MHz
AOM

λ/2
4K
77K
Cryostat

Converter

λ/4

PD Photodetector

Amplifier

Mixer

SplitterDirectional
coupler

Circulator

Attenuator

50 Ω termination

RF (1 MHz)
Microwave (7 GHz)
Optical (282 THz)
Fiber coupler

20 dB

Figure C.1: Measurement network for the microwave-to-optical frequency converter. The network isdivided into three general sections: a, Optical network. b, Microwave network. c, Signal generationand detection.

crowave pump (with amplitude and phase given by bin (ω)) and measure the upper sideband on
the outgoing optical pump (which yields the amplitude and phase of aout (ω)). In general, theconversion process also produces a non-zero lower sideband; that is, the transfer function elementthat connects bin (ω) to aout (−ω) is nonzero (see Fig. 3.12). During downconversion (S12 ) andmicrowave reflection (S11 ), we collect both in-phase and quadrature data using homodyne detec-

115
tion. No information is lost during measurement, and we can reconstruct the amount of power
present in the upper- and lower- sidebands (bout (ω) and bout (−ω), respectively) of the reflected microwave pump. During upconversion (S21 ), the reflected optical pump and sidebands are focused
onto a photodetector, and we only detect amplitude modulation of the pump. In order to assign
a measured amplitude modulation an equivalent upper-sideband amplitude, we need to know the
relationship between the upper- and lower- sidebands.
The photocurrent we measure is proportional to the incident photon flux (in units of number/sec). The photon flux isDEf (t) ≡ d†out (t)dout (t)(C.1)

where dout (t) is the annihilation operator for the light field incident on the detector and has unitspof number/sec, as introduced in Chapter 2. During an upconversion measurement, an opticalpump ain (t) = āin e−ıωin t is injected into the optical cavity. If the mechanical oscillator is vibrating
0 t + Φ), the outgoing field can be described bywith displacement x0 sin(ωmdout (t) =ı∆o + κo,ext − κo /2 √āin e−iωin t−i∆o + κo /2√
κo,extGo x 0+āin
2 −ı∆o + κo /200e−ı(ωin −ωm )t+ıΦ)
e−ı(ωin +ωm )t−ıΦ−0 ) + κ /20 ) + κ /2−ı(∆o + ωm−ı(∆o − ωm
o
o

!
,

(C.2)

in close analogy to Eqn. 2.15 where I’ve additionally included the effect of modematching as given√by . The non-modematched light in the optical pump does not beat with the sidebands on thephotodetector, and appears as a zero-frequency photocurrent. Defining θ = Arg[(ı∆o + κo,ext −κo /2)/(−ı∆o + κo /2)], ψ = Arg[κo,ext /(−ı∆o + κo /2)], and neglecting the zero-frequency portionof the photocurrent,
f (t) = Be

0 t+Φ)−i(ωm2

B =  |āin |




0
2i(θ−ψ) −i(∆o + ωm ) + κo /2+ c.c.
1−e0 ) + κ /2i(∆o − ωmoı∆o + κo,ext − κo /2−i∆o + κo /2

κo,extGo x0−i∆o + κo /22e−i(θ−ψ)0 ) + κ /2−i(∆o + ωm
o

(C.3)
!

0 , we would detect
In a measurement where we do not detect the lower sideband at frequency ωp −ωma photocurrent proportional to fssb (t). The ratio of these two photon fluxes is (using complex

116
notation to express a phase difference)
fssb (t)
1

.
=
0
−i(∆
o +ωm )+κo /2f (t)
1 − e2i(θ−ψ)(C.4)

0 )+κ /2i(∆o −ωmoThis ratio allows us to convert between the measured photon flux f (t) and an equivalent uppersideband photon flux fssb (t).

Appendix D

Measurement network for temporal and spectral mode conversion

Fig. D.1 shows a detailed diagram of the network used to measure the temporal and spectralmode converter presented in Chapter 4. The network consists of four main components: Signals
reflected or emitted from the mode converter are measured with a sensitive microwave receiver(subsection D.1). Microwave signals that interact with the mode converter are created using a
combination of arbitrary waveform generation and upconversion (subsection D.2). The mode converter and a resistive load are mounted to the base stage of a dilution refrigerator (subsectionD.3). A voltage bias on the actuation electrode controls the center frequency of the mode converter(subsection D.4).

D.1

Microwave receiver
Microwave signals reflected or emitted from the mode converter are measured via a sensitive

microwave receiver. The first component of the receiver is a Josephson parametric amplifier (JPA),
followed by additional amplifiers, a downconverting mixer, and a digitizer. A microwave pump
detuned 1 MHz below the converter’s resonant frequency of ωe,2 = 2π × 7.34 GHz drives the JPA.
This pump also serves as the phase reference for the downconverting mixer. We adjust the phaseof the pump entering the local oscillator (LO) port of the downconverting mixer so that the signal
amplified by the JPA exits via the in-phase channel of the mixer. This channel is sampled by ahigh speed digitizer (AlazarTech ATS 9462).
The JPA has a limited dynamic range [189], and the pumps used with the mode converter

118

c

Arbitrary waveform generator(Tektronix 5014c)
Analog channels
1

1

2

2

3

3

4

4

d

Digitaltriggersb
Digitizer(AlazarTech
ATS 9462)

70 MHzBandpass(Yokogawa
7651)

Analog switches

DAQ burst
Switch trigger
To I/Q mixers

Pulse on/off

Local oscillator

JPA pump

Pump 120 dB

200MHz80 MHz

20 dB
20 dB
HEMT

<25 mK
Calibration signal
20 dB
20 dB

3 dB

800 mK

10 dB
6 dB

Pump 2

JPA

10 dB

I
R
Q

20 dB

4K

I
R
Q
CavityfilterL

180 kHzJPA cancellation

Cavityfilter

L

Q

...

I
L R
Q

Input signal

20 dB

a

180 kHz

I
R
Q

6 dB

L

I
R

1 MHz

L

50 Ω
50 Ω termination

Variable attenuation

DC block

Variable phase

Mode converter

Figure D.1: Measurement network for temporal and spectral mode conversion. a, Microwave
receiver. b, Microwave generation. c, Mode converter at <25 mK and resistive load at temperature
TL . d, Actuation line.

119
carry enough power to saturate the JPA’s gain. We use variable attenuators and phase shifters tocreate cancellation tones that reduce the pump power incident on the JPA by 50 to 60 dB, which
is sufficient for proper JPA operation. We monitor the gain of the JPA (for both gain drift andsaturation) with a weak calibration signal.

D.2

Microwave generation
Waveforms with programmable amplitude, phase, and frequency are generated by a Tektronix

5014c arbitrary waveform generator (AWG). However, the AWG can only produce waveforms withcenter frequencies up to approximately 300 MHz. We must upconvert these waveforms so they interact with the mode converter, which has a center frequency of approximately 7 GHz (see Fig. 4.5).
Upconversion is accomplished via single-sideband modulation, a process where the original waveform (centered at frequency ω1 ) modulates a constant microwave tone (at frequency ω2 ) to produce
a final waveform (centered at frequency ω1 + ω2 ). Agilent PSG signal generators provide theconstant microwave tones, and microwave mixers (Marki Microwave IQ4509-MXP) combine themicrowave tones and the waveforms from the AWG. The relative phase, amplitude, and DC offsetsof the AWG waveforms are adjusted to compensate for non-idealities in the mixer.
Channels 1 and 2 of the AWG produce four waveforms centered at 70 MHz that create themicrowave pumps used during our mode conversion protocol (see section 4.6). Bandpass filters(KR Electronics 2657) at the outputs of the AWG attenuate phase and amplitude noise by >40 dB
for frequencies outside their passband (≈20 MHz centered at 70 MHz). Although the filters limit
the bandwidth of waveforms produced by the AWG, this bandwidth is sufficient for our mode
conversion protocol. Once the waveforms are upconverted using single-sideband modulation, they
are amplified (Mini-Circuits ZVA-183V) and additionally filtered [77]. Ideally, this procedure results
in a microwave-frequency pump at frequency ωe,1 − ωm or frequency ωe,2 − ωm that is >170 dB
more powerful than at frequency ωe,1 or ωe,2 . This large dynamic range is necessary to produce a
powerful microwave pump whose phase and amplitude noise do not incoherently drive vibrationsof the mechanical oscillator. When no pump is necessary, the Agilent PSGs are pulsed off so that

120
they emit no microwave tone.
Channels 3 and 4 of the AWG produce waveforms centered at 30 MHz that create the signals
used during our mode conversion protocol. Once upconverted, the waveforms produced by the
AWG are at frequency ωe,1 or ωe,2 . These signals are neither amplified nor filtered because thesignals can be attenuated to reduce phase and amplitude noise. The signal at frequency ωe,1 ismanipulated with the mode converter. The signal at frequency ωe,2 is used to monitor the gainof the JPA and additionally used to correct small (< 10 ps) timing errors that occur between the
AWG and the Agilent PSGs during each repetition of our mode conversion protocol.

D.3

Mode converter and resistive load
The mode converter and a resistive load are mounted to the base stage of a dilution re-frigerator and cooled to <25 millikelvin. A low insertion loss (<0.30 dB) 2-way switch (Radiall
R577433000) connects the microwave receiver to either the mode converter or the resistive load.
The temperature of the resistive load, TL , can be varied without significantly affecting the temperature of the mode converter or fridge, as it has a weak thermal link to base stage of the dilution
refrigerator and has an attached heating element. The resistive load injects a known amount ofnoise into the microwave receiver, and this noise allows us to calibrate the efficiency and gain of
the microwave receiver (see section 4.4.1).

D.4

Actuation line
The voltage on the actuation line controls the microwave circuit’s resonant frequency. We use

a stable voltage source (Yokogawa 7651) to provide a constant Vdc . Solid state switches (Maxim
Integrated 313) change the voltage on the actuation line between Vdc and 0 V. The state of theswitch network is controlled by a digital output of the AWG.
The actuation line is filtered to ensure voltage fluctuations in Vdc do not incoherently drive
the mechanical oscillator. In particular, the mechanical oscillator is sensitive to voltage fluctuations
near its resonant frequency of ωm ≈ 2π × 9 MHz. Room temperature filters attenuate fluctuation in

121
7 GHz

Membrane

L

C
Cp

Cf Lf

< 200 kHz

Figure D.2: Circuit diagram of mode converter, including a low-pass filter on the actuation line.
Parasitic capacitance, Cp , between the actuation electrode and the microwave electrode causes
microwave energy in the inductor-capacitor electromagnetic resonator (shown in blue) to exit via
the actuation line (shown in black). For the design shown in Figs. 4.3 and 4.4, Cp ≈ 3 fF. The
low-pass filter has Lf ≈ 20 nH and Cf ≈ 1 pF. When the filter is terminated with 50 Ohm load,
as done in our experiments with a bias tee, the energy loss rate is relatively small (estimated to be≈10 kHz)Vdc by more than 100 dB at frequencies near ωm , and cryogenic low-pass filters provide additional
attenuation near ωm . These filters are designed so the voltage noise seen by the mode converter (atfrequencies near ωm ) is ideally given by Johnson noise from a few Ohms at <25 mK. Additionally,
the filters provide enough bandwidth to quickly (<30 µs) change the voltage on the actuation line.
The external filters sufficiently suppress voltage noise; however, the actuation line must also
not dissipate microwave energy stored in the electromagnetic resonator. To limit the loss of microwave energy, we include a superconducting inductor-capacitor low-pass filter on the actuationline during construction of the mode converter. A circuit diagram of the mode converter and filter
is shown in Fig. D.2. With this filter, we estimate a microwave energy loss rate through the DC
line of ∼10 kHz, much less than the total energy loss rate of κ = 2π × 2.5 MHz (see table 4.1).