Reducing Added Noise in a Microwave-Mechanical-Optical
ConverterbyPeter Schaunaman Burns
B.A., University of Montana, 2013
M.S., University of Colorado Boulder, 2016

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2019This thesis entitled:
Reducing Added Noise in a Microwave-Mechanical-Optical Converter
written by Peter Schaunaman Burns
has been approved for the Department of Physics

Prof. Konrad Lehnert

Prof. 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
Burns, Peter Schaunaman (Ph.D., )Reducing Added Noise in a Microwave-Mechanical-Optical Converter
Thesis directed by Prof. Konrad Lehnert

Viewed as resources for quantum information processing, microwave and optical fields offer
complementary strengths. Quantum technologies operating at microwave frequencies have shown
the ability to create and manipulate quantum information, making them leading candidates forbuilding a quantum computer. Optical fields can transmit quantum information over long distances with low loss making them ideal for use as links between quantum nodes. Therefore theability to transduce quantum information between microwave and optical frequencies is an essential
technology in realizing an extended quantum network. To date there has been no demonstrationof a microwave to optical link that is capable of quantum transduction.
This thesis describes the design and characterization of a device that couples one modeof a micromechanical oscillator to a resonant microwave circuit and a high-finesse optical cavity,
thereby realizing a mechanically mediated electro-optic converter. We have operated this converter
at T < 100 mK, and demonstrated unprecedented conversion efficiency of 47% and added noise of
38 photons. I discuss how noise correlations can be exploited to develop a feedforward protocol.
I also discuss the limitations on the noise performance of the device used to achieve these results,
and describe recent improvements in the fabrication process and optical and microwave resonator
designs aimed at reducing added noise. These improvements should produce a device that is near
the threshold for quantum conversion.

iv

Acknowledgements

There is not enough space to thank the many people that contributed to the science and to
my well being, up to and during my time on this project. Even if there was, I don’t know if I
have the words to truly express how lucky I have been to be surrounded and supported by such
impressive individuals. My PhD would not have been possible without all of them, but here I makea meager attempt to thank the specific individuals who made this project possible. However, I
must also express my gratitude to my parents and brother. Without their support and belief inme, none of this would have happened.
First, I would like to thank my committee members Graeme Smith, John Teufel, and Victor
Bright for generously taking time out their busy schedules to provide feedback and guidance on thisproject and for the invaluable insights and discussions throughout my PhD. Cindy Regal’s joint
leadership of this project has been invaluable both to the productivity of the group, and to mygrowth as scientist.
Due to its fabrication-heavy nature, this project owes a debt of gratitude to all the staff at
the NIST BMF cleanroom. Their patience and willingness to help allowed this project to succeed,
and also made the cleanroom a welcoming place to work. A special thanks goes to Kat Cicak who
I shadowed for my training and taught me nearly everything I know about fabrication.
I am grateful to all of the Lehnert and Regal lab members for their willingness to have open,
engaging physics discussions at any moment’s notice and for the sincere camaraderie that has been
built both in the lab trenches and on the athletic pitch. In particular, I would like to thank themembers of the microwave-to-optics team. Bob Peterson, Tim Menke, Oliver Wipfli, and Andrew

v
Higginbotham all made this project a joy to be a part of in the first half of my time here. I owe a
special thanks to Andrew, not only for his direct efforts in lab, but more importantly for shaping
me as a scientist and for the many wonderful conversations. To the newer members of the team,
Max Urmey, Ben Brubaker, Sarang Mittal, and Jon Kindem, thanks for carrying me through tothe finish line. Your creativity and energy made everyday exciting and thanks to you I have great
confidence about the future of the project.
Finally, there is not a way to accurately recognize my advisor, Konrad Lehnert, for all he hasdone for me. Konrad’s leadership as an advisor goes well beyond the level of scientific guidance.
His excitement for all physics, both big and small, is contagious and meant there was rarely a dull
discussion.
To all of you: thank you.

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Contents

Chapter
1 Microwave-Optical Transducers as Tools for Quantum Networks

1

1.1

Quantum information networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Microwave-Optical transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3

Optomechanics for conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52 Theory of Operation

9

2.1

Electromagnetic resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

Cavities coupled to mechanical oscillators . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3

Bidirectional microwave optical conversion . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Converter Device Used for Highest Efficiency, Lowest Added Noise, and Feedforward

9

17

3.1

Electromechanical transducer chip . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2

Wireless microwave coupling to the circuit . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3

Optical cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.4

Data acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 State-of-the-Art Conversion
4.1

30

Converter parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.1.1

Microwave cavity parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.1.2

Optical cavity parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

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4.2

Measurement procedure and converter operation . . . . . . . . . . . . . . . . . . . . 33

4.3

Extracting bandwidth and efficiency from scattering parameters . . . . . . . . . . . . 35

4.4

4.5

4.3.1

Optomechanical response theory . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.3.2

Mode matching and efficiency calibration . . . . . . . . . . . . . . . . . . . . 38

Added noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.4.1

Measuring added noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.4.2

Mechanical dissipation and added noise . . . . . . . . . . . . . . . . . . . . . 43

4.4.3

Operating without matching criteria for technical noise reduction . . . . . . . 44

Measurement efficiency calibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.5.1

Microwave and optical calibrations . . . . . . . . . . . . . . . . . . . . . . . . 46

4.5.2

Checking the consistency of the calibrations . . . . . . . . . . . . . . . . . . . 48

5 Correlations and Feedforward

50

5.1

Feedforward introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.2

Electro-optic correlation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.3

Feedforward harnesses correlations to reduce added noise

5.4

Feedforward theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.4.1

Feedforward eliminates thermal noise . . . . . . . . . . . . . . . . . . . . . . . 60

5.4.2

Gaussian states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.4.3

Qubit up-conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6 Converter Limitations
6.1

6.2

. . . . . . . . . . . . . . . 56

68

Conversion pumps lead to added noise . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.1.1

Microwave circuit parameter noise . . . . . . . . . . . . . . . . . . . . . . . . 72

6.1.2

Microwave circuit noise induced by laser light . . . . . . . . . . . . . . . . . . 74

Capacitor gap spacing is unreliable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.2.1

Stycast affects capacitor gap spacing . . . . . . . . . . . . . . . . . . . . . . . 77

6.2.2

Chip frequency spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

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6.2.3
6.3

Deformation of top chip reduces Qm . . . . . . . . . . . . . . . . . . . . . . . 80

Loss in optical cavity limits efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

7 Technical Improvements to Reduce Added Noise
7.1

84

Fabrication methods to improve Qm and reduce capacitor gap separation . . . . . . 85
7.1.1

Improved Qm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

7.1.2

Increasing electromechanical coupling through smaller gaps . . . . . . . . . . 86

7.1.3

Fabrication challenge and solution: silicon frames patterned with holes . . . . 88

7.2

Parameter noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

7.3

Bonding technique for flip-chips and optical cavity assembly . . . . . . . . . . . . . . 95
7.3.1

Etalons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7.3.2

Chip bonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

7.4

Frequency-independent coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

7.5

Improvements summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

8 Conclusion and Outlook

104

Bibliography

106

Appendix
A Fabrication Details

117

A.1 Process sheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

ix

TablesTable
4.1

Converter parameters for high efficiency ηM = 0.43 device configuration. . . . . . . . 31

6.1

Added noise budget for each contributing source . . . . . . . . . . . . . . . . . . . . 69

7.1

Added noise budget including all improvements . . . . . . . . . . . . . . . . . . . . . 102

x

FiguresFigure1.1

Canonical optomechanics picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.1

Microwave optical transducer coupling diagram . . . . . . . . . . . . . . . . . . . . . 10

2.2

Optical cavity coupling schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1

Profile view of flip-chip device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2

Photo of fabricated top and bottom chip and flip-chip . . . . . . . . . . . . . . . . . 19

3.3

Circuit coupling diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4

3D cavity coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.5

Converter microwave and optical cavity designs . . . . . . . . . . . . . . . . . . . . . 23

3.6

3D cavity theoretical coupling description . . . . . . . . . . . . . . . . . . . . . . . . 24

3.7

Varied LC couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.8

Measurement Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.1

Microwave scattering parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2

Optical cavity linewidth characterization . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3

Simplified measurement network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.4

Converter scattering parameters

4.5

Converter efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.6

Optical scattering parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.7

Visibility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

xi
4.8

Optical power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.9

Laser Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.10 Temperature sweep for calibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.11 Microwave chain noise for calibration check . . . . . . . . . . . . . . . . . . . . . . . 49
5.1

Feedforward schematic and measurement network . . . . . . . . . . . . . . . . . . . . 52

5.2

Electro-optic correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.3

Time domain feedforward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.4

Feedforward variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.5

Feedforward operation of a microwave-mechanical-optical converter . . . . . . . . . . 60

5.6

Quantum feedforward observable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.1

Microwave noise squashing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6.2

Added noise from parameter noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.3

Broadband parameter noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.4

Laser Heating of LC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.5

Thermal contraction bows chip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.6

Epoxy placement determines bow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.7

Qm is correlated to LC frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.8

Cavity couplings dependence on membrane position . . . . . . . . . . . . . . . . . . 83

7.1

Initial tether designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

7.2

Relieved silicon structure accommodates shrinking epoxy

7.3

One sided etch jig

7.4

Profile schematic of wafer scale KOH etch . . . . . . . . . . . . . . . . . . . . . . . . 91

7.5

NbTiN vs Nb added noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

7.6

Nb NbTiN loss and noise comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

7.7

Etalon is naturally aligned . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

. . . . . . . . . . . . . . . 88

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

xii
7.8

Self-aligned etalon cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7.9

Etalon assembly methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

7.10 Flip-chip bonded without epoxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.11 High inductance coupler reduces frequency dependence . . . . . . . . . . . . . . . . . 101
7.12 Photo of improved transducer chip . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
A.1 Construction details for converter chips . . . . . . . . . . . . . . . . . . . . . . . . . 119
A.2 Bonding Assembly for flip-chips and etalon . . . . . . . . . . . . . . . . . . . . . . . 120

Chapter 1

Microwave-Optical Transducers as Tools for Quantum Networks

1.1

Quantum information networks
We are in the middle of the second quantum revolution [1]. The tools being developed inlabs today are quickly finding their way into the growing field of commercially-available, quantumassisted technologies. Rather than just describing and understanding the world using the frameworkof quantum mechanics, the second quantum revolution is based on harnessing quantum mechanics
to build useful technologies. This change in our technical ability to control and manipulate systems
at the quantum level has been advancing for nearly a century. The first quantum revolution was
one of knowledge, as our views of the physical world were profoundly changed by the discoveryof quantum theory at the turn of the 20th century. In its early stages, this theory was thought
to describe very small, discrete systems, but the consequences of applying it to the macroscopic
world were fiercely debated [2]. Schrödinger’s famous cat is an example of the kind of paradoxthat seems to exist when the laws of quantum mechanics are used to describe macroscopic systems.
Schrödinger himself even had this to say in 1952 about the idea of having experimental access toquantum degrees of freedom:
We never experiment with just one electron or atom or (small) molecule. In
thought-experiments we sometimes assume that we do; this invariably entails ridiculous consequences... We are not experimenting with single particles, any more thanwe can raise Ichthyosauria in the zoo [3].
Since that time, we have gained sufficient control over the quantum degrees of freedom of not just2microscopic atoms and molecules, but over semi-macroscopic objects. It has forced us to address
the “ridiculous consequences,” and more importantly it has allowed us to take advantage of them.
One of the areas that is benefiting from the consequences is the field of quantum information
science. Because of the many superposed degrees of freedom, simulating a quantum system witha classical computer quickly becomes intractable. Simulating quantum systems on classical computing machines is inherently insufficient as the machinery is fundamentally different than what isbeing represented. This was realized in the 1980s as Feynman [4, 5] and Deutsch [6, 7] proposedusing quantum systems themselves as a platform for simulation. Beyond just simulating quantum
systems to learn more about the physics, the field gained considerable traction when an algorithm
for efficiently factoring large numbers was developed [8]. The ability to efficiently factor large numbers sparked interest due to the fact that common encryption techniques such as RSA are based on
factoring being a classically hard problem [9]. In parallel to the development of quantum computing
theory, the field of cryptography also began realizing benefits from incorporating quantum information. The quantum key distribution protocol first proposed in 1984 [10], which takes advantageof the no-cloning theorem [11] to establish physically secure communication, is still the basis for
multi-city [12, 13], and even satellite-based [14] quantum key distribution. While secure communication and encryption breaking are still driving forces in developing quantum technology, quantum
science will also benefit other fields including quantum metrology [15] and force sensing [16], searchalgorithms [17], and chemical simulation [18, 19, 20].
A long standing goal for the field of quantum information is to realize an extended quantum
network. A quantum network entangling spatially separated quantum nodes would permit securecommunication and distributed quantum computing [21, 22, 23, 24]. The nodes create, store andmanipulate quantum information, while links between them transmit information and allow forentanglement between nodes [25]. A quantum internet could follow the same road map as themodern internet, using local computers, operating on electronic signals, connected around theglobe via fiber optic links. Just like classical information, quantum information has a physicalplatform. The information, however is independent of the physical implementation. We take this3fact for granted with classical information as signals in our computer are converted all the timefrom different physical states such as magnetic memory, electronic signals on chips, or fiber optic
communication networks. Likewise, quantum information theory does not presuppose a physicalsystem and as such there have been developments in quantum proving grounds across a variety ofphysical platforms. Propagating optical fields are the natural choice for quantum links in such a
network, as they enable entanglement distribution at room temperature and over kilometer-scaledistances. Promising qubits have been demonstrated across a variety of solid state implementations
including superconducting qubits [26, 27, 28], spin qubits [29, 30], and charge qubits in various
material systems [31, 32]. These solid state qubits typically have excitations energies corresponding
to a range from a few GHz to a tens of GHz, suggesting that the most powerful nodes will use
microwave-frequency excitations, and therefore will need to be operated in ultralow-temperatureenvironments (T < 100 mK), so that thermal fluctuations don’t obscure quantum signals. Making
a large network all housed at these ultralow temperatures is fairly impractical, implying that aquantum state preserving electro-optic converter is a crucial element needed for a future quantum
internet.

1.2

Microwave-Optical transducers
Given that classical networks already convert information between microwaves and optics,

it is worth pointing out why the current commercially-available technology is not compatible withquantum signals. The technology used in modern communication networks works for classicaltransduction because efficiency and noise can be overcome by amplification. A quantum signals
however, requires much more stringent conversion requirements as quantum states can not be
amplified or cloned. Error correcting quantum algorithms exist [33], but still typically require anerror rate less than 1% depending on the type of error [34]. If one is limited to Gaussian resources, aconverter will only have quantum capacity if the efficiency is greater than 50% [35]. Furthermore, aquantum transducer needs to have a large enough bandwidth to convert a signal before it is lost. Fora first-generation converter to be compatible with leading superconducting qubit lifetimes [36], a4conversion bandwidth of at least 10 kHz is required, however for long term applicability even larger
bandwidths would be necessary. These stringent requirements for a quantum microwave-opticallink make this a challenging task and has created an active and open field of research pursuing thiseffort.
Frequency conversion between the two domains is not possible without a significant nonlinearity, and the source of the nonlinearity is a useful way to categorize the different experimental
approaches. The nonlinearity can come from a material susceptibility, or through an intermediarymode. The material susceptibility for microwave-optical conversion can either be a bulk effect thattakes advantage of the electro-optic effect, or a resonant effect of a multi-level system. A parametric processes, such as coupling through an intermediary mechanical mode, can also give rise toa nonlinearity. In most schemes the single photon nonlinearity is very small, and therefore theseschemes benefit from increasing the effect with the use of a resonant cavity, which enhances thephoton interaction and increases the optical density of states.
Conversion based on materials with optical nonlinearities can further be divided into bulk
solid state effects or resonant effects in atomic clouds or atom-like defects in solid-state hosts. Fora solid state system, the susceptibility χ(ω) of a crystal describes the polarization in response to analternating electric field. The bonds in the crystal can respond in harmonic and anharmonic ways
which leads to frequency mixing. The goal of electro-optic converters is to find materials with low
loss and high χ(2) , where χ(2) is the second order susceptibility that leads to the electro-optic effect,
or Pockels effect. Experimental conversion efforts have shown success using whispering gallery mode
optical resonators made out of LiNbO3 , which has a high χ(2) , placed inside a microwave cavity [37].
The challenge of such an approach comes from matching the resonance conditions between the
microwave frequency and the spacing between optical modes, but an efficiency of η = 1 × 10−3
has still been achieved [38]. Another approach uses lithographically defined microwave resonators
patterned on AlN which allows for higher mode overlap between the microwave and optical fields.
This approach has lead to the highest efficiency in an electro-optic effect device of η = 1×10−2 [39].
Approaches in atomic like systems use a three level λ system where the spacing between the two5lowest levels is in the microwave range. Prospective conversion technologies using this technique
are ultracold atoms [40, 41], or optically active spins in solids [42, 43, 44, 45, 46]. So far the strict
resonant conditions of these systems have also limited there efficiency to η = 5 × 10−2 , albeit with
15 MHz bandwidth [47].
Another approach (and the one pursued in this thesis) relies on coupling vibrational modes
to microwaves and optical fields. The optomechanical interaction is appealing because it is mathematically symmetric between the microwaves and optical modes and the coupling is independentof intrinsic resonances and therefore can be tuned across a range of frequencies. The approachrelies on mechanical motion parametrically coupling to one or both electromagnetic modes. For
example, confining the mode to a nanomechanical element that uses piezoelectric coupling works
well for mechanical modes in the GHz range. These converters benefit from the high mechanicalfrequencies that allow for less thermal occupation of the mode and for higher bandwidths, but areusually limited in efficiency and by heating of the mechanical mode from the optical field [48, 49].
In this thesis, our approach is to simultaneously couple a mechanical mode to both a microwave and optical cavity as shown in Figure 2.1. We use a doubly parametric approach thatcan reduce mode overlap between mechanics and optics and therefore shows less signs of heating.
High quality mechanical modes combined with low-loss cavities have allowed for the highest efficiency and lowest added noise to date [50]. In a mechanical converter, low added noise requires
high electromechanical and optomechanical cooperativities [51, 52], which have yet to be achieved
together in a single device. In optomechanical systems, alternatives to high cooperativity have been
explored to cool mechanical modes [53, 54, 55, 56, 57, 58, 59]. Our feedforward protocol makes useof the two ports of our device to present an alternative to the high cooperativity requirement for
conversion.

1.3

Optomechanics for conversion
Our approach to use a mechanical element to couple between the optical and microwave

domain is built on the field of optomechanics [60, 61, 62]. This field comes from two simple ideas:

6

mirrors

springFigure 1.1: Canonical optomechanics picture. Two mirrors define a single mode optical cavity.
The mirror on the left is partially transmissive and allows light to enter and exit. The mirror on
the right behaves like a mass on a spring. Light in the cavity can push on the mirror throughradiation pressure.

motion can influence light, and therefore electromagnetic signals, and light carries momentum.
The first has been used to aid the technology revolution as mechanical elements are ubiquitousin modern devices from clocks, to sensors. The second was predicted by Maxwell’s descriptionof electromagnetism in 1873 [63], and was noticed by Johannes Kepler in the observation thatcomets’ tails point away from the sun [64]. Experimental efforts to use light to affect motion
and motion to affect light began being used to trap objects, starting with the microspheres [65]and developing into quantum control over single atoms [66, 67, 68, 69]. In parallel to using lightto control objects, the quantum limits of continuous displacement detection were largely worked
out in the context of interferometers used for gravitational wave detection. Developing theory
and experimental platforms to study gravitational waves has led to much of the language used forstudying optomechanics. For example, the standard quantum limit for measurement of the positionof mechanical oscillator was developed in terms of the precision of an interferometer for searching for
gravitational waves [70]. The advancement of the field fueled the search for gravitational waves [71,72] and led to the experimental discovery and study of them [73]. Interferometers like LIGO are
now so precise that quantum mechanical fluctuations of the light field are now the limiting factor
in some parts of their sensitivity range [74, 75].
From the initial efforts to improve measurement sensitivity, emerged the broader fields of op-

7
tomechanics and electromechanics. Efforts to improve the sensitivity of optomechanical detectors
is still a driving force, but has also led to other interesting uses for optomechanics–for example,
putting mechanical objects into profoundly non-classical states [76, 77]. The canonical optomechanics experiment is an optical cavity in which one mirror is free to vibrate as shown in Figure 1.1.
The motion of the mirror affects the light in the cavity, and therefore the light exiting the cavity
holds information about the position of the mirror. The same interactions that were developed forcooling and trapping ions in harmonic potentials can be used to affect the motion of the mirror.
In the microwave regime, the same picture applies, but with an LC circuit where one plate of thecapacitor is free to vibrate. Experiments have developed to the level of control that the mechanical modes of these systems can be cooled to their quantum ground state [78, 79]. Mechanicalresonators can also be used to entangle motion with microwave fields [80]. Preparing non-classicalstates of light has improved force sensitivity, but has also shown promise in quantum information
fields as mechanical oscillators can be used for quantum memories [81] or dynamic signal processing
including frequency conversion [82, 83, 84]. Quantum information can be transferred between microwaves and mechanical motion and between optical fields and mechanical motion, accomplishingboth these quantum coherent couplings, simultaneously, to the same mechanical oscillator would
achieve a quantum, electro-optic converter.
In this thesis, we explore electro-optic conversion in a mechanically mediated transducer. We
operate a converter device at T < 100 mK, as required for compatibility with superconducting
qubits. We demonstrate an unprecedented conversion efficiency of 47 ± 1%, and implement a feedforward protocol that exploits noise correlations between the two converter output ports to reduce
the noise added to a signal referred to the converter input to Nadd = 38 photons. These figuresof merit represent significant technical progress relative to the 8% efficiency and 1500 photons ofadded noise achieved in a prototype system operated at 4 K [85]. Using the noise correlations,
we explore an alternative feedforward framework where quantum tasks could be performed even if
thermal-mechanical noise yields Nadd > 1, provided threshold efficiencies are reached. We characterize the limitations of this converter in terms of the sources of added noise and make technical8
improvements to each part individually such that when put all together should be on the thresholdof a quantum-enabled converter.

Chapter 2Theory of Operation

This chapter lays out the theory needed to describe the operation of the converter. The
converter is a system of three linearly coupled harmonic oscillators. In order to understand thephysics of the full device, Section 2.1 starts by describing a single resonator coupled to internaland external ports. Section 2.2 introduces a mechanical oscillator and describes how it mediates
the coupling between two electro-magnetic modes. The chapter finishes with a full description of
the transducer as well as the mathematical machinery needed to describe correlations measured by
exploring the system of three coupled oscillators in Section 2.3.

2.1

Electromagnetic resonators
The coupling between each component of the converter is shown in Figure 2.1. The resonators

are shown as circles to highlight the fact that although physically quite different, mathematicallythey are equivalent. Each resonator couples to the environment at some internal loss rate κint . The
mechanical resonator couples to each electromagnetic resonator at rate Γ, and the electromagnetic
resonators couple to propagating fields at rate κext .
Intrinsically, we care about the interaction between photons and motion. This effect is enhanced by confining the harmonic oscillator to high-Q, compact resonators. Although a single
photon transfers a minuscule amount of momentum that would be obscured by the zero-point motion for all but the smallest mechanical oscillators, a cavity allows a photon to interact many times
with the oscillator, transferring momentum with each pass. In the optical domain the resonator

10

Γe

κext,eωemechanicalresonator

Γoκext,oωmLCresonator

ωoγmκint,eκint,oopticalresonatorkBT

Figure 2.1: Microwave optical transducer coupling diagram. The two electromagnetic resonators (blue and red circles) are coupled to the mechanical resonator (green circle) at rates Γ.
They each have there own loss rates coupled to the thermal environment. Propagating fields arecoupled to the LC and optical cavity at rates κ.

can be formed by two mirrors, or in the microwave domain it can be a lumped element inductor
capacitor circuit. Making the cavities more compact increases the sensitivity to displacement because a small change in position corresponds to a fractionally large change in a dimension thatsets the resonant frequency. For an optical cavity this means making small spacings between the
mirrors and for an LC circuit this means making a narrow gap capacitors.
An electromagnetic resonator and a mechanical oscillator both exhibit harmonic behaviour
and therefore can be described by the same equations of motion. In a mechanical oscillator theoscillating quantities are position and momentum, while in an electrical oscillator they are theelectric and magnetic fields. This can be described using input-output formalism and for thederivation of bare electromagnetic resonators and cavities interacting with mechanical motion wefollow Ref. [62]. The mathematical notation describes equally well optical light confined between
two mirrors in a cavity and electrical energy stored in inductor capacitor resonant circuits.
The state of the cavity, shown in Figure 2.2, can be described in a basis of coherent states|αi, which most closely approaches the classical notion of a cavity in a state of definite amplitude
and phase, and described by a complex number α. In quantum optics it is more natural to work

11

a

ain
aout

κο

Figure 2.2: Optical cavity coupling schematic. A single electromagnetic mode in a Fabry-Perotcavity, characterized by the annihilation operator a which couples at rate κo to an incoming field
ain and an outgoing field aout .

in the Heisenberg picture, where the equations of motion are written in terms of operators, inthis case the annihilation operator a(t), whose eigenstates are coherent states a |αi = α |αi. The
Heisenberg-Langevin equation of motion for a single cavity mode is:√√κȧ(t) = −iωcav −a(t) + κext ain (t) + κint fin ,
2

(2.1)

where wcav is the resonant frequency, and a† (t)a(t) = |a(t)|2 is number of photons in the cavity,
and a†in (t)ain (t) = |ain (t)|2 is the incident power in units of photon number per second. Incident
fields couple to the cavity at a rate κext that we refer to as the external coupling rate. The external
coupling rate is usually an engineered parameter and is determined by geometrical or material
properties of the particular cavity. Fields inside the cavity are coupled to various loss mechanisms
such as absorption or scattering to other modes. The loss rate κint can also be driven by bathmodes fin . The total cavity coupling rate is the sum of external couplings and internal couplingκ = κint + κext . The dynamics of the cavity field are contained in the Heisenberg-Langevin equation
along with the instantaneous relationship between the input fields, cavity fields, and output fields
aout (t) =√κext a(t) − ain (t)(2.2)

where aout is the outgoing field and is normalized such that |aout |2 has units of photons/second.
We solve for the transfer function in the frequency domain (denoted by tilde symbols)
Ξ1port (ω) =ãout (ω)
2i∆ + κint − κext=−.
ãin (ω)
2i∆ + κint + κext

(2.3)

12
Where ω is Fourier transform variable of t and it has been shifted from the cavity frequency ωcavby ∆ by working the rotating frame. By measuring the amplitude and phase response of a cavity,
we characterize the external and internal coupling rates. There are three regimes of coupling thatare commonly referred to, overcoupled, critically coupled, and undercoupled. Overcoupled is whenthe external coupling rate is the dominant loss rate for photons in the cavity and is usually desired
as it means most the information leaves the cavity into a mode that can be measured. Criticallycoupled means the two loss rates are equal, and undercoupled means that the internal loss rate of
the cavity dominates.

2.2

Cavities coupled to mechanical oscillators
The coupling of a cavity to a mechanical oscillator is examined here. An uncoupled mechan-ical oscillator is describable with the same equations of motion as the cavities. Here we use the
Hamiltonian formalism to calculate the coupling between a harmonic oscillator and a cavity. Thiscan be used to describe the dynamics of optomechanical systems in both classical and quantum
regimes. The optical and mechanical modes used in this derivation are completely generic. The
Hamiltonian for an uncoupled system is just the sum of the two harmonic oscillators, and without
constant terms is written as
H = ~ωcav a† a + ~ωm c† c.

(2.4)

Where c is the mechanical oscillator’s annihilation operator, and ωm is the mechanical oscillator’s
resonance frequency.
We now introduce the coupling of the cavity’s resonance frequency to the position of themechanical element. We find an interacting term linear in x by Taylor expanding the x dependent
cavity resonance frequency to first order in x. The cavity resonance frequency as a function of theposition is written asωcav (x) = ω0 + x∂ωcav (0)+ O x2 .
∂x

(2.5)

13
Substituting this in and only keeping terms to first order gives an interaction Hamiltonian as
Hint = −~Gxa† a,(2.6)

where G = − ∂ω∂xcav is the frequency shift per unit displacement. The coupling term can also
be understood because the radiation pressure of light is proportional to the intensity, thus theinteraction Hamiltonian is the number operator a† a multiplied by the position operator. The
position x is written in terms of creation and annihilation operators as x = xZPF (c+c† ), where xZPF
pis the mechanical oscillators zero-point fluctuation amplitude. It has the form xZPF = ~/2Meff ωmwhere Meff is the effective mass of the oscillator. We also define a single photon phonon coupling
rate g0 = GxZPF which is a useful parameter to characterize the strength of the optomechanicalinteraction. We change the description of the optical mode by switching to a frame rotating at thepump or laser frequency ωL . Now the interaction Hamiltonian takes the following form

Hint = −~g0 a† a(c + c† ).

(2.7)

The interaction is simplified by linearizing the optical field by splitting it into a classicalamplitude and a remaining fluctuating term, a = α + δa. The classical component sets the scale ofthe interaction. Terms proportional to |α|2 represent static shifts that we will absorb into a newcavity resonance frequency and equilibrium oscillator displacement. Terms proportional to δa2 are
dropped. The terms of the form δaα† and c.c. contain the interesting dynamics. In the rotating
frame around the laser frequency we introduce detuning ∆ = ωL − ωcav . With these changes, wewrite the Hamiltonian√
H lin = −~∆δa† δa + ~ωm c† c − ~g0 ncav (δa† + δa)(c† + c).

(2.8)

Where ncav = |α|2 is the intracavity photon number. Here one could return to using input-output
formalism to write down the full Heisengberg-Langevin equations of motion. Instead we look at
the interaction for different detunings to elucidate the effects of laser detuning.
The detuning of the pump tone leads to different forms of the Hamiltonian which havedifferent physical effects. First, if the pump is detuned below the cavity by the mechanical resonator14frequency ∆ = −ωm , and the rotating wave approximation is used, the dominant term is the
Hamiltonian of two oscillators at the same frequency that can exchange energy with the interaction
red
Hint= −~g(δa† c + δac† )

(2.9)

√
at rate g = g0 ncav . From the language of quantum optics this is the beamsplitter Hamiltonian.
It swaps a mechanical excitation with an electromagnetic excitation. If the fluctuating piece δais in its vacuum state, then this cools the mechanical oscillator as it swaps thermal phonons for
optical vacuum states. This is also the interaction used to transduce electromagnetic excitations
into mechanical oscillations that we take advantage of in our converter operation. This coolingprocess can be understood from a conservation of energy argument. Photons from the pump areconverted to photons at the pump frequency plus mechanical frequency. This difference in energy
is provided by one mechanical phonon..
The interaction takes a different form if the pump is detuned above the cavity by the frequencyof the mechanical oscillator ∆ = ωm . This is referred to as blue detuned and the interaction has
the form
blue
Hint= −~g(δa† c† + δac).

(2.10)

This interaction creates pairs of entangled excitations. It represents two mode squeezing andparametric amplification [86]. Again, this can be understood from energy conservation. The energy
is scattered into a lower frequency sideband and therefore gives up energy to create a mechanicalexcitation, entangling mechanical motion with light. In the language of atomic physics, the twointeractions are the stokes and anti-stokes sidebands and in the same way are used to cool or excite
mechanical motion of trapped atomic systems.

2.3

Bidirectional microwave optical conversion
For understanding the behavior of the converter we derive condensed equations of motion. We

start with Hamiltonian again and then move to a state-space model, from which the full dynamics

15
can be implemented with computation software and are used to check our results of operating a
converter.
The Hamiltonian with an additional electromagnetic resonator coupled to the same mechanical mode is written as
H = ~ωo â† â + ~ωe b̂† b̂ + ~ωm ĉ† ĉ + ~Go (â† â)x̂ + ~Ge (b̂† b̂)x̂(2.11)

where, â, b̂, and ĉ denote annihilation operators for the optical, electrical and mechanical modes,
respectively. Frequencies ωi describe the mode resonant frequencies, with subscripts i = {o, e, m}denoting optical, electrical, and mechanical. The couplings from the optical and microwave resonators to the mechanical mode are denoted Go and Ge .
The Hamiltonian used to calculate the Heisenberg-Langevin equations of motion [87] canthen be linearized about a strong, coherent pump field as in Section 2.2. The resulting system offirst-order, linear differential equations are concisely described as a state space model [88].
ȧ(t) = Aa(t) + Bain (t)(2.12)

aout (t) = Ca(t) + Dain (t)(2.13)

where a = (â, b̂, ĉ, â† , b̂† , ĉ† ) is a vector of the resonator modes that represents the internal state of
the converter, and ain (aout ) is a vector of input (output) fields. See Ref. [85] for full matrix forms.
The state-space model reduces to a transfer function
aout (ω) = Ξ(ω)ain (ω)Ξ(ω) = C(−iωI − A)−1 B + D
Where we use f (ω) =R∞−∞ f (t)e−iωt dt.

(2.14)(2.15)

This formalism reflects our measurement, where we ap-

ply fields (i.e. modify ain ) and measure fields (i.e. measure components of aout ). It provides us
with a framework to understand the performance of our converter device. For example, the efficiency, characterized by the transmission coefficient corresponds to the transfer function element
Ξaout,L ,bin,R (ω) = aout,L (ω)/bin,R (ω).

16Frequency-dependent fluctuations leaving each port are useful for describing added noise,
two mode squeezing and mechanical temperature. The spectra of the output fields are expressedin terms of a matrix S(ω), whose elements are given by1
Sij (ω)δ(ω − ω 0 ) = h{[aout,i (ω 0 )]† , aout,j (ω)}i.
2(2.16)

Chapter 3

Converter Device Used for Highest Efficiency, Lowest Added Noise, andFeedforward

This chapter describes the apparatus used to couple a harmonic oscillator to both microwaves
and light. Section 3.1 explains the transducer chip geometry that forms the LC circuit and mechanical membrane. The wireless coupling method is outlined in Section 3.2, which allowed us toreach the highest efficiency converter. Next, is a discussion of the optical cavity geometry in Section 3.3, and how the integration with the 3D microwave cavity allowed for higher optomechanicalcoupling and greater optical stability. The chapter finishes by laying out the data acquisition setup
in our measurement network in Section 3.4. Iterations and improvements made to the device will
be discussed later in Chapters 6 and 7

3.1

Electromechanical transducer chip
The basic concept and layout of the transducer chip is described in detail by Andrews [89].

Therefore we will summarize the device and highlight salient features and changes that madethe results of this thesis possible. One challenge of a mechanical microwave optical transducer isworking with mechanical modes that couple to both microwave and optical photons. Silicon nitride(SiN) membranes have performed well in pure optomechanical applications.. They have low loss at
optical wavelengths and can be made with high stress to increase their quality factors. Althoughground state cooling has been achieved for SiN membranes with optics [90], and for aluminum
drum heads with microwaves [78], a mechanical oscillator that couples to both microwaves and18siliconSiN
superconductor
stycastlaser
mirror

300 nmFigure 3.1: Profile view of flip-chip device. The transducer flip-chip is formed by two piecesof silicon. The bottom has an inductor loop and one part of a capacitor. The top chip consistsof a high stress membrane stretched across a silicon frame with one portion covered by a patch ofsuperconductor. When the top chip is brought very close, it forms a LC circuit whose resonanceis modified by the motion of the membrane. It is aligned to the mode of an optical cavity so thatthe optical resonance is modulated by the motion of the membrane. The areas of silicon that are
not critical for the posts or circuit are recessed away to reduce the chance of dust dictating theseparation.

optics has yet to be ground state cooled. Our approach is to couple a SiN membrane to a highfinesse optical cavity, with one corner coated in a superconducting thin film to form one plate of aparallel plate capacitor. In this way when the membrane moves from regions of high intensity to low
in the standing wave of the optical cavity, it changes the optical length of the cavity and therefore
modulates the resonant frequency. Likewise when the metal plate moves, it changes the capacitanceof the microwave circuit and modulates its resonance frequency. This realizes the physics describedin Section 2.2 where in both cases deflection causes a linear change in cavity resonance frequency.
We realize the electromechanical converter hardware by microfabricating a transducer chip.
The transducer chip consists of two silicon chips: one with an inductive wire loop and the bottom
plates of the capacitor, the other with a suspended slab of dielectric with a patch of metal for the top
capacitor plate. The mechanical oscillator is a 100 nm thick, 500 µm wide silicon nitride membranesuspended from a silicon chip. A 25 nm thick, niobium film which serves as one capacitor pad isfabricated on one quadrant of the membrane before it is released. The membrane chip is flipped19

LC
circuitflip-chipbottomtop
Figure 3.2: Photo of fabricated top and bottom chip and flip-chip. The top and bottom
chips are shown with zoomed in regions of the centers. The flip-chip is formed by epoxying the top
chip to the bottom chip.

over and affixed to a second silicon chip, on which a microfabricated niobium circuit comprising
an inductor and a second capacitor pad was previously patterned using standard lithographic
techniques. Figure 3.1 shows a profile schematic view of the assembled flip-chip device in an opticalcavity, while Figure 3.2 shows the top and bottom chips and assembled flip-chip. The areas of the
top and bottom chip that do not have the capacitor plate or posts are recessed away to reduce thechance of dust limiting the chip separation.
The flip-chip assembly is constructed with a West Bond manual die bonder, and the chips
are affixed using Stycast 2850. In the fully assembled flip-chip device, the two niobium pads form aparallel-plate capacitor with a plate spacing of 300 nm, and the resonant frequency of the resulting
LC circuit is modulated by vibrations of the mode of interest.

3.2

Wireless microwave coupling to the circuit
By using a 3D cavity to couple to the LC circuit on the transducer chip [91], we were able toreduce the optical cavity length, which improved the optomechanical coupling but more importantly
reduced the optical cavity loss. Because the cavity removes the need for a galvanic connection to20
the chip, we refer to it as wireless coupling. Here we motivate the need for wireless coupling anddescribe the coupling method.
The total converter efficiency is limited by our ability to make strongly overcoupled microwave
and optical cavities. The external coupling rate is set by design, but the internal loss rate isdependent on the optical cavity stability to misalignment. In the first iteration of the converter theefficiency was limited to 8.5% [85] mainly due to loss in the optical cavity. In that experiment, theoptical cavity had a length that was limited to greater than 4 mm to accommodate the components
needed for the microwave chip and the transmission lines. This relatively long cavity length andnumerous components made the cavity more sensitive to geometric misalignment. The chip wasconnected to microwave transmission lines by flexible circuit boards of gold coated copper printed
on polyimide film. This cavity design became more misaligned upon cooling down than otherpure optomechanics experiments by our group. This could have been because the flexible circuitboards stressed the chip holder, or could just be from the long cavity having so many components,
each with its own epoxy joint with the potential to misalign as the epoxy and components shrank
when cooled down. Moving to a smaller cavity geometry improves the cavity coupling efficiency
by making it less sensitive misalignment. However, even moving to 2 mm in the old geometry,a reduction by a factor of two, would not have been possible as the chip holder and components
between the mirrors were too thick. In order to reduce the optical cavity length the microwave
coupling scheme was changed to remove components needed for the wired connection.
Previously, the coupling from the LC circuit to the transmission line was via an on-chip coplanar waveguide that was terminated with the center conductor forming a loop to ground (Figure 3.3).
Mutual inductance between the loop and the inductor of the circuit sets the external coupling rateof the circuit. The chip was then wire-bonded to a flexible circuit board. Our strategy to remove
bulky components and reduce connections to the chip was to use a 3D cavity with a circulating
magnetic field to inductively couple to the chip. The cavity mode and coupling can be understoodby considering the cavity as a distributed LC circuit, where the inductive current path is throughthe cavity and the capacitance is set by the area and separation of the central post. This produces

21

port
Z0

Lc

M

L

C

Figure 3.3: Circuit coupling diagram. Port with impedance Z0 is terminated in inductor LC
with mutual inductance M to circuit formed by inductor L and mechanically compliant capacitorC

cavity modes with well separated electric and magnetic fields. The circulating magnetic field couples effectively to the magnetic antennae formed by the wired loop inductor on chip. A schematicof the coupling cavity and the chip are shown in Figure 3.4. The resonance of the cavity is set
by the path length inductance the capacitance of the center posts. Being able to make changes tothe resonance frequency by only adjusting either the capacitance or inductance makes this cavity
geometry easy to work with as these can be adjusted in machining or by making modular pieces.
The cavity also couples to external microwave ports via loop antennas. The cavity is made of gold
and copper-coated invar. The invar was used to reduce misalignment of the optical cavity fromthermal contraction, while the coatings were used for their low loss microwave properties.
Figure 3.5a presents the integration of the microwave cavity with an optical Fabry-Pérot
cavity. The exploded view in Figure 3.5b shows how the microwave and optical parts of the device
have been physically decoupled. This greatly reduces the complexity of the setup. The opticalcavity and membrane on the chip can now be aligned and assembled in the same way as a pure
optomechanics system [92]. The small number of interfaces and symmetric design of the mount are
expected to minimize misalignment that arises from thermal contraction. Parts E-F, dubbed the“hat” of the microwave cavity, contain the center posts and connect the device to the microwave
circuitry via coaxial loop couplers. We can readily change to a hat with different center post
separation without interfering with any other part of the device. The external coupling can also
be adjusted by changing the size of the loop couplers. In this way, we have precise control over the

22

b

a
z
y
x

y
z

x

d
Figure 3.4: 3D cavity coupling. a, The microwave-to-optics transducer is assembled from twosilicon chips (gray). A silicon nitride membrane (light blue) is suspended on the top chip anda niobium circuit (dark blue) is patterned on the chips. b, Cutaway drawing of the re-entrant
microwave cavity. The cavity (gray) can be modeled as an effective LC circuit (yellow line), spatiallyseparating electric field (red) and magnetic field (blue arrows).

microwave cavity’s resonant frequency and coupling to the external ports.
Figure 3.6a outlines the relevant coupling rates of the 3D microwave cavity and electromechanical chip. The cavity has two ports and couples to them with external coupling rates κcav,1 and κcav,2 ,respectively. The coupling rates are chosen such that κcav,1  κcav,2 . In this way, transmission
and reflection measurements can be performed while minimizing the signal that is lost in reflection.
In addition, there is an internal cavity loss rate κcav,loss due to absorption and radiation throughseams. In total, energy stored in the cavity decays at the rate κcav,tot = κcav,1 +κcav,2 +κcav,loss . The
inductive coupling rate between the cavity and LC is quantified by the frequency g. It depends
on the size and position of the LC circuit’s loop inside the re-entrant cavity. The external andinductive coupling rates can be estimated by finite element simulations to guide the design process
The coupling of the re-entrant microwave cavity mode to both the transmission line and the
LC circuit gives rise to an effective, wireless coupling of the external microwave ports to the LC

23

z

x

b

a
y
z

y
z

x

Figure 3.5: Converter microwave and optical cavity designs. a, Cutaway drawing of thecomplete assembly. The device mount consists of pieces labeled A-E. The re-entrant cavity volume
is formed in the middle with magnetic field (blue arrows). The cavity couples to microwave input
and output (dark blue arrows) via loop antennas made from coaxial cables. The transducer chip(dark gray) is located inside an optical Fabry-Pérot cavity. The mirrors (light blue) are brought inthrough holes in the microwave cavity walls. They are connected to piezoelectric crystals (yellow)
via aluminum and fused-silica spacers (dark gray and turquoise, respectively). The optical pathis shown as a red arrow. b, Partially exploded view of the device, showing how the microwave
and optical parts separate. The optical cavity is mounted on parts A-B. Excess piezo wiring (notshown) can be wrapped around the grooves on the sides of parts A-B, relieving stress from theconnection between piezo and wire. Part C serves as the sample holder for the transducer chip. At
the same time, parts C-F form the microwave cavity volume. The “hat” consisting of parts E-F
contains the re-entrant cavity’s center posts and has holders for the loop antennas attached. The
microwave and optical parts are in contact only via part C.

24a

Microwave cavity

Port 2

LC resonator

MechanicsbTransmission

Port 1Figure 3.6:
3D cavity theoretical coupling description. a, Schematic overview of thecomponents and coupling rates involved in the setup. The microwave cavity (gray) couples tothe transmission line (dark blue) via two ports, with external coupling rates κcav,1 and κcav,2 ,respectively. It also couples to the LC resonator (light blue) with rate g. This results in effective
coupling rates κeff,1 and κeff,2 of the two ports to the LC resonator, given by Eq. (3.1). Additionally,
the microwave cavity and LC resonator have internal decay rates κcav,loss and κLC,bare , respectively(not shown). In electromechanical measurements, the LC resonator also couples to the mechanicalmode of the membrane. b, Magnitude squared of the theoretical transfer function in absence of
mechanical coupling. Cavity linewidth κcav,tot , effective detuning ∆eff , LC linewidth κLC,tot labeled.

circuit. The cavity resonant frequency ωcav and the LC resonator frequency ωLC have been shiftedfrom their bare values by the coupling, and their effective detuning is ∆eff = ωcav − ωLC . For
sufficient detuning ∆eff & κcav,tot , g, the effective coupling of the LC circuit to the cavity ports isgiven by
κeff,i = κcav,ig2,∆2eff + (κcav,tot /2)2

(3.1)

where i ∈ {1, 2, loss}. We call the couplings κeff,1 , κeff,2 to the transmission line “wireless” because
there is no direct connection of the LC circuit to a waveguide or coaxial cable. Note that theeffective coupling leads to an LC dissipation rate κLC,loss = κLC,bare + κeff,loss that is higher thanthe bare loss rate κLC,bare of the circuit.
The design targets an effective coupling κeff,1 /2π in the 1.5-2.0 MHz range. In this way, the LC
circuit is well over-coupled with respect to its expected internal loss rate, which is typically 0.4 MHz
for our niobium circuits. The total LC decay rate κLC,tot = κeff,1 + κeff,2 + κLC,loss then also fulfills
the resolved sideband criterion: κLC,tot  4ωm , where the chip is designed to have a mechanical

a 4

25

b
100
linewidth [MHz]transmission

3
2
0.4

1

0.2
0.0
7.36

0
5.5

7.37

6.0

7.38

7.39

6.5
frequency

10

1

0.1
7.0

7.5

8.0

detuning

Figure 3.7: Varied LC couplings a, Normalized transmitted power versus frequency for four
different center post separations: 238 µm (blue), 270 µm (orange), 300 µm (green), 316 µm (red).
The broad peak corresponds to the cavity resonance, and the narrow peak corresponds to the LC
resonance. Offsets added to curves for clarity. Inset: Zoom of LC line shape for 270 µm post
separation, with FWHM, κLC,tot , labeled. Fit of the theoretical transfer function shown in black.
Averaging over fits to all four curves gives the cavity-LC coupling g/2π = 57(2) MHz, and the bare
LC loss rate κLC,bare /2π = 0.48(3) MHz. b, LC linewidth versus effective detuning. Colored squares
indicate the κLC,tot obtained for the different post separations. Lines show theoretical predictions
for the total LC linewidth (κLC,tot , solid), effective coupling (κeff,1 , dashed) and loss rate (κLC,loss ,dot-dashed) with g and κLC,bare fixed from (a). The measured range of effective couplings contains
the target regime 1.5-2.0 MHz.

resonance frequency ωm /2π ∼ 1.5 MHz. The last inequality ensures that electro-optic conversion
occurs without photon number gain, as required for quantum state preservation. From Eq. (3.1)
we understand that we need to couple the re-entrant cavity strongly to both the LC circuit and thetransmission line. In particular, the re-entrant cavity should couple to port 1 much more strongly
than it couples to its internal loss such that most of the LC decay is due to its coupling to thetransmission line: κcav,1  κcav,loss . In order to limit the dissipation added to the LC resonator,
an internal Q-factor of about 500 for the re-entrant cavity is targeted. Finite element simulations
show that an inductive coupling g theo /2π = 60 MHz and external coupling κtheocav,1 /2π = 150 MHz toport 1 can be achieved. A desired effective coupling then requires a detuning ∆theoeff /2π ∼ 600 MHz.
The device behaviour is accurately modeled by Eq. (3.1). Figure 3.7b shows the LC linewidths
κLC,tot that are extracted directly from the measured transmission spectra. These are compared to26
a plot of the theoretical total loss rate κLC,tot that is obtained from Eq. (3.1) using the fitted model
parameters. Both agree well for large detuning. The desirable and undesirable LC loss are plottedas κeff,1 and κLC,loss in the figure. Note that the wireless coupling has been varied by over an orderof magnitude by using the different hats. In the target regime of 1.5-2.0 MHz for the effective
coupling, the unwanted dissipation rate κLC,loss constitutes at most 30 % of the total LC loss rate.
The dissipation contribution is reduced to 17 % when operating the device in the millikelvin regime.

3.3

Optical cavity
The optical cavity design and assembly is described in Ref. [93]. Here we discuss it in enoughdetail to motivate its limitations and improvements. Using a 3D cavity to mediate coupling between
the chip and a transmission line allowed for the removal of all other components between the mirrors
except the chip itself. The chip is 0.7 mm thick which allowed for a reduced the cavity length bya half to 2 mm, any smaller and the cavity assembly becomes unwieldy. Figure 3.5a shows a sideview of the optical Fabry-Pérot cavity. The cavity consists of two stacks of components endingin highly reflective surfaces. The stacks have piezoelectric elements for adjustments of the opticalcavity length and membrane position, as well as invar and fused silica spacers to shorten the cavity.
The cavity components are all referenced to parts A and B. Piezoelectric elements are mounted
outside of the microwave cavity volume.
The optical cavity is comprised of two mirrors, with 29 ppm and 98 ppm power transmission
respectively, which can give a range of external couplings κext /2π = 1.0 − 2.0 MHz depending on
membrane position (Figure 6.8). The chip assembly is placed in the standing wave of the opticalcavity with the membrane 750 µm from the high-transmission mirror, such that the membrane’s
vibrations modulate the cavity’s resonant frequency [94]. The optical mode intersects the membranein the quadrant opposite the capacitor, and these two regions of the membrane move in phase forthe mechanical mode of interest. We refer to this as a three-element cavity because the mirrors andmembranes are all fixed relatively separate to each other. The main challenge here is to control
the alignment of the three surfaces while they are referenced to larger invar pieces that are 3 cm

27
further away. Each joint of the stack of components is held together with stycast and represents a
possible source of misalignment upon cooling down.
Unlike the microwave cavity which needs to be tuned to the right frequency at room temperature and then is inaccessible cold, the optical cavity parameters are partially adjustable. By
applying voltage to the piezoelectric actuators, the mirrors can be moved. Common-mode motion
moves where the membrane is in the standing wave and differential motion changes the resonant
frequency. During operations, one of these actuators is used in a Pound-Drever-Hall cavity locking
scheme to keep the optical pump at a fixed detuning from the cavity resonance [95]; the othermay be used to change the membrane position within the optical standing wave of the cavity. The
locking scheme uses an optical beam to monitor the cavity and slow drifts are fed back to the piezo,
while fast changes are fed directly to the laser. The locking beam is injected on resonance with thecavity and is always present during converter operation. This is important to highlight because itmeans that even with no conversion beam, there is a fixed level of optical power in the cavity fromthe lock beam that affects the LC circuit in the form of added noise that depends on the amountof locking power.

3.4

Data acquisition
The device is characterized by making scattering parameter measurements and by collecting

spectral data. Measurements are made by sweeping the frequency of a tone incident on a network
component. The phase and amplitude of the reflected or transmitted signal are measured andused to extract scattering parameters. For example, the converter box is characterized by a seriesof probe-tone scattering parameter measurements Sij (δ), where i = e, o is the measured port,
j = e, o is the excited port, and δ is the frequency of the probe relative to the pump [85]. Spectral
information is acquired by acquiring a signal at the output of a network port for some amount oftime and then taking the Fourier transform to acquire power spectral density information.
The full experimental setup is detailed in Figure 3.8. Data acquisition is performed witha Zurich Instruments HF2LI lock-in amplifier. Scattering parameters are measured by sweeping

28
an output tone and demodulator frequency, and spectra can be measured by taking a time trace
and computing the Fourier transform. The converter is operated at the base temperature of adilution refrigerator with optical access, and a cryogenic HEMT amplifier is used for microwave
measurements. Microwave signals are measured by demodulating with an I/Q mixer, remodulating
at a frequency of 10 MHz, then feeding into the HF2LI. The demodulate-remodulate procedureallows both microwave quadratures to be measured by a single physical channel, and makes thesignal processing functions performed by the HF2LI symmetric between the microwave and opticaldomains. Optical signals are measured by combining the beam exiting the cavity with an LO on abeamsplitter and reading out the difference port of a balanced heterodyne detector.
When computing device output spectra, a signal to be measured s(t) is demodulated at afrequency fd = fc +fm , where fc is the carrier frequency: fc = 10 MHz for microwave measurements
and fc = 12 MHz for optical measurements. The resulting quadratures, q(t) and p(t), are used tocompute,a(t) = (q(t) + ip(t))ei2πfd t ,(3.2)

which is the complex-equivalent signal of s(t),a(t) = s(t) + iH[s(t)],where H denotes the Hilbert transform. The Fourier transform of a(t), a(ω) =

(3.3)
R +∞

−∞ a(t)e−iωt dt isthen computed approximately using DFT. a(ω) is a complex-valued function,a(ω) = X(ω) + iY (ω),(3.4)

with real component X(ω) and imaginary component Y (ω), and is zero when ω < 0. In the limitof narrow-band demodulation the quadrature variances satisfy e.g., Var(qo ) = hXo (ωm )Xo (ωm )i. A
demodulation bandwidth of 2π × 50 Hz is used for all quadrature data. Note that away from mechanical resonance, X(ω) is not just the Fourier transform of q(t), but rather contains contributions
from both q(t) and p(t). Spectra are displayed shifted so that fc appears at zero frequency.

29

Cryostat
ZI lock-in

R

mw in

Q out
I

opt in

Q
L

I

I

Q
L

4K
R

HEMT

10 MHz-20 dB
800 mK

6 GHz

-

PD

L

L

Q
I
Q
I

PBS
λ/2

R

35 mK
20 dB

PD

κex,e

PBS

Microwave SSB
Optical SSB

Detection

R

Faraday
Rotatorκint,ePBS

75 MHzλ/2

PBS
Faraday
Rotator-75 MHzAOM
EOM-74 MHzAOMκex,oConverterγm
λ/2 PBSκint,oPDκB,ocameraPD

to lock

PBS

87 MHzAOM
LO beamNd:YAG
1064 nm

200 MHzAOM

λ/4

RF (MHz)Microwave (7 GHz)
Optical (282 THz)

Figure 3.8: Measurement Network. A Zurich Instruments Lock-in Amplifier (ZI) is used as aspectrum and network analyzer. Outputs are driven 90◦ out of phase to produce a single sideband
probe tone. In the microwave single sideband generator (blue box), a mixer is used to produce asingle upper sideband directly on the microwave pump. The optical single sideband modulation(red box) is first mixed with an intermediate 75 MHz tone to drive an AOM. A lock beam (green) isused to reference and lock the optical cavity. The converter (purple box) is thermalized to the baseplate of an optical access dilution refrigerator (grey dashed box), and microwave and optical tones
are routed in with appropriate filtering. A directional coupler is used at base to provide filteringand an effective cold load. Transmission through the optical cavity can be used for characterization.
Reflected and converted signals are routed by circulators to detection (yellow). A cryogenic HEMT
amplifier is used for microwave amplification. The microwave output signal is demodulated withan I/Q mixer, remodulated at a frequency of 10 MHz, and then routed to the ZI. The optical beam
is mixed with a local oscillator beam (orange) and measured in heterodyne. Spectrum analyzer
measurements can be made by turning the ZI outputs off and fixing the demodulation frequency.

Chapter 4

State-of-the-Art Conversion

This chapter characterizes the conversion performance of the device described in Chapter 3.
The metrics described are converter bandwidth, efficiency, and added noise. The device parametersand coupling terms are laid out in Section 4.1. Next the measurement network and procedure isdiscussed in Section 4.2. Experimental results follow, highlighting our highest efficiency device (4.3)
as well as added noise performance (4.4). Section 4.5 concludes with a summary of the calibrations
used to determine the added noise.

4.1

Converter parameters
The device described in Chapter 3 is housed in a cryostat with base temperature T = 35 mK.

This realizes microwave and optical cavities simultaneously coupled to a single vibrational mode ofa suspended dielectric membrane. Each cooldown represents a separate realization of the converter
device as the device parameters change from cooldown to cooldown both due to our design choices
and uncontrolled changes to physical geometries upon thermal cycling. Therefore, the table 4.1
lists the parameters from one device, in this case our highest efficiency device. The microwave andoptical cavities are characterized first in the absence of the optomechanical interaction.

4.1.1

Microwave cavity parameters

The microwave network is characterized by reflection, and scattering measurements. Offresonance of the LC circuit, all the power is reflected, allowing us to use this value to calibrate the net

31
Table 4.1: Converter parameters for high efficiency ηM = 0.43 device configuration.
κe /2πκext,e /2π
κint,e /2π∆e /2πGe xzp /2π
κo /2πκext,o /2πκB,o /2π + κint,o /2π∆o /2πGo xZPF /2π
ωm /2π = fm
γm /2π

2.5
2.3
0.2
-1.47
3.8
2.1
1.1
0.87
1.0
-1.11
6.6
1.4732
11

MHz
MHz
MHz
MHz
Hz
MHz
MHz
MHz
MHz
Hz
MHz
Hz

loss and gain of the microwave chain. By normalizing to the off-resonance value the response of the
LC cavity can be isolated and is described by the transfer function of a one-port cavity as in Eq. 2.3.
Measuring the amplitude and phase response and fitting to the transfer function (see Figure 4.1),
extracts internal and external coupling rates, κext,e /2π = 2.3 MHz and κint,e /2π = 0.2 MHz. Narrow
band measurements around the LC resonance frequency have background levels that match theresonant dip value, |See (0)|2 = 0.69.

4.1.2

Optical cavity parameters

Unlike in the microwave domain, the optical measurement does not have a well calibrated
frequency knob. Because our experiment does not have some fixed transition like an atomic energy
level, the laser and cavity are allowed to drift slightly as long as they track together. This makes
wide band scattering parameter style measurements of the optical cavity difficult. Instead theoptical cavity is characterized by sweeping a beam with sidebands of a known frequency acrossresonance as in Figure 4.2. The optical beam is modulated with an EO so that sidebands show
up at a known frequency spacing. As the frequency of the carrier is swept across resonance thereflected voltage is measured as a function of time. Three dips appear corresponding to the main

32

a

b

1.0

data
data
fit
fit

0.8

φ (S ee ) (rad)|S ee |

2

2
data
fit

0
-2

6.14

6.15

6.16
f (GHz)

6.17

6.14

6.15

6.16
f (GHz)6.17

Figure 4.1: Microwave scattering parameters. Measured scattering parameter of microwave
circuit (black), with fit (blue) to extract cavity coupling rates. (a) Power response shows Lorentzian
dip with some background ripple characteristic of wide band measurement. (b) Phase response.

Reflection (V)

0.12

16.591 MHz

0.10

0.08
3.8

3.9

-3

4.0x10Time (s)Figure 4.2: Optical cavity linewidth characterization. Reflection of the optical cavity ismeasured on a photodetector. Carrier beam is modulated using EO to produce sidebands of known
frequency. Carrier beam frequency is swept through cavity resonance. The x-axis is the realmeasured time. Fitting to the three peaks allows for calibration for the x-axis into frequency units
as the peak spacing is well known. A fit to the center peak extracts optical cavity linewidth.

33beam and the two sidebands sweeping across the cavity. The two sidebands are used to calibrate
the x-axis from time to frequency units. Then, a fit to the Lorentzian dip extracts the optical cavity
coupling rates. The optical response is further complicated compared to the microwave cavity by
the mode matching between the incident beam and cavity mode. This is discussed in more detailin Section 4.3.2.

4.2

Measurement procedure and converter operation
The device is characterized by making scattering parameter measurements or by looking at

the output spectrum from either port. In scattering parameter measurements we measure thefraction of power transmitted or reflected by the converter. Noise characteristics can be madeby looking at the output spectrum of the device. A simplified measurement setup showing signal
routing to and from the device is shown in Figure 4.3. The mechanical oscillator is shown coupled
to both the LC circuit and optical cavity. Strong pump tones are single-sideband modulated to
produce a probe tone whose frequency can be tuned. Signals are directionally routed to heterodyne
measurement chains discussed in more detail in Section 3.4. The device is studied by making
reflection or transmission measurements or by measuring the power spectra at either port.
The converter is turned on by using two strong pump tones, one optical and one microwave.
These pump tones are detuned from there respective cavities by roughly the mechanical resonatorfrequency. Along with the two pump tones, a weak probe tone, incident on either the microwave
or optical cavity, is measured in heterodyne detection, and used for characterization. The probe
frequency is swept relative to the pump tone to measure response as a function of frequency. Signals
from the device are routed to measurement chains that measure scattering parameters or spectra.
With a strong red-detuned pump beam incident on the optical cavity, the optomechanicalinteraction couples the mechanical oscillator to propagating optical fields at a rate Γo that exceeds
the intrinsic mechanical damping rate γm = 2π × 11 Hz. At pump detuning ∆o = −ωm and in the

34

6 GHz

35 mK
LO
-

282 THzmeasurement

LO

Figure 4.3: Simplified measurement network.
Microwave (optical) signal routing chain
in blue (red). Grey box represents base of dilution refrigerator. Carrier tones are single sideband
modulated to produce probe tone at frequency δ. Measurements are made in heterodyne. Reflection
and transmission measurements can be made of converter device.

resolved sideband limit (4ωm /κo  1), the optomechanical damping rate has the simple form[85]4G2o x2zp αo2
,
Γo =
κo

(4.1)

where αo is the intracavity pump amplitude and xzp is the zero-point amplitude of the membranemode. The electromechanical damping in the presence of a strong red-detuned microwave pumptone has the same form:
Γe =4G2e x2zp αe2
.
κe

(4.2)

In practice, our converter is moderately sideband-resolved (4ωm /κo ≈ 4ωm /κe ≈ 2.5), with pump
detunings ∆e /2π = −1.47 MHz and ∆o /2π = −1.11 MHz. These details change the precise formof the expressions for Γo and Γe but not the essential physics. The full details are derived fromequations in Chapter 2. Imperfect sideband resolution leads to some power gain in the converter
ih
i
h
κo 2
κe 2
which in the resolved sideband regime and ∆ = −ωm simplifies to A = 1 + ( 4ω
)
1
+
(
)
.
4ωm
m
We set the optical pump detuning ∆o to maximize the optomechanical damping rate per incident
photon. In the resolved sideband limit, this maximum damping occurs at ∆o = −ωm , but this is
not generally the case with imperfect sideband resolution.

35|See | 2|Seo | 2|Soe | 2|Soo | 2| S |2

1.0

0| S |2

1.0

0
1.469

fm

fm

1.476 1.469
1.476/2 (MHz)/2 (MHz)

Figure 4.4: Converter scattering parameters.
Measured converter scattering parameters
versus probe frequency δ: microwave reflection (See ), optical reflection (Soo ), microwave-to-opticaltransmission (Soe ), and optical-to-microwave transmission (Seo ). Colored trace is Γe ≈ Γo . Gray
trace is Γe  Γo , with same Γo . Shaded region highlights induced absorption of incident power
when conversion rates are matched.

4.3

Extracting bandwidth and efficiency from scattering parameters
A series of probe-tone scattering parameter measurements Sij (δ) characterize the converter

box as described in Section 3.4. The device performance is explored for a range of pump powers, andtherefore conversion rates, in order to extract bandwidths and efficiencies. As shown in Figure 4.4,
when Γe ≈ Γo , a dip in microwave reflection occurs near δ/2π = fm , with a corresponding peak in
microwave-to-optical transmission, indicating the absorption of signals in the microwave port andtheir emission at the optical port. A nearly identical optical-to-microwave transmission signal is also
observed. The peak transmission |t|2 = 0.55 corresponds to a conversion efficiency of η = |t|2 /A =0.41, where A is the independently measured converter gain due to imperfect sideband resolution.
The converter bandwidth (measured as the full width at half maximum of the transmission peak)is given by the total linewidth of the optomechanically and electromechanically damped membrane

36

b
0.5
M

a
0.5

0.3
0

1

T

/2 (kHz) 10

1

o

/2

(MHz) 5Figure 4.5: Converter efficiency. a, Converter efficiency η versus total damping ΓT . ΓT is swept
by tuning Γe with Γo fixed. Black line is a fit to Eq. 4.3. Purple and gray points correspondto data in Figure 4.4. b, Matched efficiency ηM versus optical cavity linewidth κo . The ratio of
external optical cavity coupling to internal loss varies with κo . Black dotted line indicates quantum
feedforward threshold at ηM = 0.5, discussed in Chapter 5. Horizontal error bars represent thestandard deviation of several repeated linewidth measurements. Vertical error bars are obtainedby propagating standard error in individual scattering parameter measurements.

mode, ΓT = Γe + Γo + γm .
For comparison, if Γe is decreased such that Γe  Γo , microwave reflection becomes almost
flat, with a value determined by the microwave cavity coupling, and nearly zero power is transmitted. At the same time, a peak is observed in optical reflection resulting from optomechanicallyinduced transparency effects. The peak height exceeds one due to converter gain from not being
sideband resolved (see Section 4.3.1). The optical reflection peak is suppressed when Γe ≈ Γo ,constituting electromechanically induced optical absorption.
The efficiency of the device is characterized by the total efficiency from the microwave port
to the optical port. Inefficiencies in the measurement chain are not counted against the efficiencyof the device. However, we do consider external coupling efficiency from both the microwave andoptical cavities rather than just the internal efficiency of resonant microwave fields converted toresonant optical field. There are several compromises that are made in building a full converter incomparison to pure optical or microwave experiments and only looking at the internal efficiency
would not accurately represent the potential use of this device.
The efficiency is extracted from peak transmission for a range of Γe with Γo = 2π × 725 Hz37
fixed (figure 4.5a). For fixed optical cavity parameters, efficiency is maximized when damping rates
are matched (Γe = Γo , ΓT = 2π × 1.45 kHz). The conversion efficiency is fit to [85]η=4Γe Γo4(ΓT − Γo − γm )Γo
ηM ,
ηM =
2
(Γe + Γo + γm )
Γ2T

(4.3)

where γm and Γo are fixed from independent measurements and ηM is the only fit parameter. For
our converter Γe , Γo  γm , and thus η = ηM when the converter is matched. Therefore we referto ηM as the matched efficiency. In this low mechanical dissipation regime, there is negligible
energy loss in the electro-optic transduction process itself, but some energy is absorbed in eachelectromagnetic resonator, and the spatial profile of the optical cavity mode does not perfectly
match that of the external modes used for measurement and signal injection. This is described in
more detail in Section 4.3.2. We thus expect ηM = (κext,o /κo )(κext,e /κe ), where κo and κext,o (κeand κext,e ) are the optical (microwave) cavity linewidth and external coupling respectively, and 
parameterizes the optical cavity mode matching. The fit result, ηM = 43 ± 1%, agrees with thetheoretical expectation, ηM = 43 ± 4% obtained from independent measurements of , κext,o /κo ,and κext,e /κe .
Tuning the membrane position in situ within the optical cavity changes both κext,o and κodue to interference effects (see Limitations chapter), and therefore alters the matched conversion
efficiency (Figure 4.5b). Matched conversion efficiency initially increases with κo , reaching a maximum at κo = 2π × 2.7 MHz, and then decreases as internal optical cavity loss begins to dominate.
The in situ tuning of the optical cavity also changes the optomechanical coupling and thus theachievable converter bandwidth ΓT . For intermediate linewidths near the highest efficiencies, theoptical cavity became unstable, possibly due to large optomechanical coupling in these regions. The
peak conversion efficiency achieved was 47 ± 1%, approaching the quantum feedforward thresholdefficiency discussed in Chapter 5. At peak conversion efficiency, the matched converter bandwidth
was 12 kHz, while a 100 kHz bandwidth was achieved at κo = 2π × 3 MHz. As a figure of merit,the peak efficiency represents significant technical progress relative to the 8% efficiency achieved ina prototype system operated at 4 K [85].

38
theorydata|S oo |

2

1.0
0.5
0
1.470

1.475
/2 (MHz)

Figure 4.6: Optical scattering parameter. Measured scattering parameter of the optical port
vs probe frequency δ (blue) showing optomechanically induced transparency. Black line shows
expected response from independent parameter measurements. Grey band expresses uncertainty ofsystem parameters.

4.3.1

Optomechanical response theory

Reflection measurements of the optical cavity can be understood using the mechanicallymediated state transfer defined in Section 2.3. The optical cavity has a Lorentzian response andshows up as dip in reflection, from which we detune a pump tone and study mechanical response.
Figure 4.6 shows narrow-band measured and expected scattering parameters of the mechanical response for the optical port, with Γe  Γo . Fixing all the cavity and pump parameters in
the equations of motion generates a line shape that reproduces salient features observed in theexperiment. The peak value exceeds one due to gain from imperfect sideband resolution, and the
Fano-like shape arises from the mechanical response not being on the flat bottom of the dip, buton the slope of the optical cavity response (due to ∆o 6= −ωm ).

4.3.2

Mode matching and efficiency calibration

Careful calibration of probe-tone scattering measurements requires correctly treating various
optical mode matching factors. Mode matching is important for calibration as it affects reflection
measurements of the optical port. Specifically, we care about mode matching between input light39
and cavity mode, and between the signal and the heterodyne LO beam. Any light that is notproperly matching to the mode of the cavity is promptly reflected by the input mirror and does notmake it into the cavity. As a result, the measured dip in reflection on resonance will be measured
to be smaller than expected We parameterize the mode matching between two optical fields E1 ,E2 of the same polarization by the overlap of the electric fields, given byR ∗E1 E2 d2 x
 = qR.
R|E1 |2 d2 x |E2 |2 d2 x

(4.4)

We can extract the mode matching through a interference measurement. Here, we direct the two
beams on to a single photon detector The beams are set to different frequencies and their power isbalanced. If the two beams are mode matched, then they will interfere and a beat note will appear
on a scope measuring the voltage output of a photo detector as seen in Figure 4.7. The visibility
is given byv=

Imax − Imin,
Imax + Imin

(4.5)

where Imax(min) is the maximum (minimum) intensity of oscillations. The mode matching is thesquare root of the fringe visibility so that maximal mode matching corresponds to perfect extinction
at the fringe minimum.
The converter is calibrated by dividing the transmission in each direction by the off-resonant
reflection off each port [85]. In this way the individual gains and losses of the measurement networkneed not be known, and yet they can still be calibrated out. With this calibration procedure, mode
matching d between the incident beam and the cavity mode is automatically included in thedownconversion efficiency, and mode matching u between the outgoing cavity mode and the LO
is included in the upconversion efficiency. In practice u ≈ d , and we define the mode matching
factor  that appears in the expression for the bidirectional conversion efficiency ηM as  =√u d .

We must also account for the imperfect mode matching optical heterodyne measurement
LO between the local oscillator and the reflection from the optical port, which only affects theoff-resonant reflection measurement used to calibrate out path losses. Since LO is not an inherent
quality of the converter, it should not be included in the measured conversion efficiency, and must

40

Figure 4.7: Visibility. As two tones interfere they create a beat note. The x-axis is the phase ofthis beat note. The phase can be tied to some physical change in time, for example the end mirrorof one arm of a Michelson interferometer moving. A beat note will also show up if the two tones
are at slightly different frequency in which case the phase oscillates at the difference frequency
between them. Our measurement uses beams of different frequency. Visibility and mode matchingare extracted from the maximum and minimum of this curve. Figure from Wikipedia.

be calibrated out separately. An independent measurement of the interference visibility between the
LO and a far-detuned beam reflected off the cavity yields LO = 0.83, which is similar to  = 0.87
as expected.

4.4

Added noise
Although this device shows exceptional efficiency, that is not the current limiting factor for

quantum operation. With efficiency of less than 50% entanglement distillation and heralding [96,97, 98, 99] could still allow for quantum enabled operation. However, for a device to work in thequantum regime, the noise added by the converter needs to be less than one photon (per photon
converted). Heralding and distillation help combat loss but do not address added noise. This strict
requirement comes from the fact that quantum signals are not as robust to added noise as classicalsignals. In classical signals, the signal to noise ratio is what matters, but in quantum signals, evenone photon of added noise is enough to destroy the information. For example if you’re encoding a41
qubit as 0 + 1 photon, then 1 photon added noise ruins the fidelity of the qubit.
Due to the high Q of our mechanical resonators, an insignificant amount of information leaks
out to the environment through the internal loss of the mechanics. This allows us to easily accessa regime where the conversion rate is much higher than the intrinsic damping Γ  γm . However,the low frequency of the mechanical element means that the thermal occupation of the device isstill nth,m >> 1 even at dilution refrigerator temperatures because kB T >> ~ωm . This meansthat although the rate for a signal excitation (γm ) lost to the environment is relatively low, thenumber of phonons leaking into the signal are scaled by the number of phonons in the bath (nth )and therefore there are many opportunities for excitations to degrade the signal. For this reason,
to achieve added noise less than one, the damping rate needs to exceed not only the mechanicaldissipation rate, but also needs to reach the requirement Γ  nth γm . If this requirement is notmet, a large portion of the mechanical oscillator’s motion will be due to fluctuations from thethermal environment instead of coherent state information from the conversion process, i.e. you
transduce these excess phonons in addition to the photons you wanted. In addition to these thermal
fluctuations, noise can be added from other technical imperfections.

4.4.1

Measuring added noise

Instead of sweeping a probe tone across resonance to characterize the efficiency, the noiseproperties are inferred from measuring the power spectral density at the output of the converter.
An example of the power spectral density of a upconverted microwave tone measured optically isshown in Figure 4.8. The narrow spike is the signal tone corresponding to converted microwave
photons Ideally the spectrum would show only the conversion tone on a flat background which
would correspond to no added noise. Instead we see the signal riding on a Lorentzian skirt at
the frequency of the mechanical oscillator. This skirt is a measure of how well we are transducing
thermo-mechanical motion into optical photons, and is also a measure of the added noise.
Characterization of the added noise is done in the high-stability configuration with ηM = 43%
described in Section 4.3. We perform heterodyne measurements of the noise exiting both converter

90

2

Xo (f) + Yo (f) (nph)

42

2

50

10
1.472
ω (MHz)1.476

Figure 4.8: Optical power spectrum, Xo2 (ω) + Yo2 (ω), for a upconverted microwave signal measured in the optical domain. Narrow spike corresponds to signal while Lorentzian skirt is a measureof the added noise. This is done with Γe ≈ Γo and ΓT = 550 Hz.

ports, and analyze the results in the frequency domain: X(ω) and Y (ω) will denote respectively
the real and imaginary parts of the Fourier transform of the noise time stream at detuning +ω fromthe pump. The total power spectral density is then S(ω) = hX 2 (ω)i + hY 2 (ω)i, and is reported inunits of photons/s/Hz, or more simply photons, referred to the converter output. For example, anideal heterodyne measurement of S(ω) gives a background noise level of 1 photon, and 50% loss inthe measurement chain would double the background level. The added noise of the converter canbe calculated by
Nadd = (S(fm ) − 2no )/(A · η)

(4.6)

Where no is the single quadrature noise of the optical measurement, and A is the independently
measured converter gain due to imperfect sideband resolution. The added noise depends on calibrations of the measurement chains, and therefore as we approach the threshold for a quantum
protocol, Nadd < 1, these calibrations need to be accurate. Our calibration methods are detailedin Section 4.5.
We record the power spectral density for a range of pump powers. The damping rates are43matched so that efficiency is always maximized. As the pump powers are turned up, the conversion
rate increases relative to the fixed thermal decoherence rate. In this way the added noise peak is
broadened and the peak height is reduced due to cooling of the mechanical oscillator. As shownin Figure 4.9, increasing the total damping, ΓT , while maintaining matching, Γe ≈ Γo , improvesnoise performance while increasing bandwidth. The peak center frequency moves down due to theoptical spring effect [62]. Ideally we would keep turning up the pumps until the high cooperativity
regime is reached and the added noise was below one photon. However, Figure 4.9 also shows how
technical noise is limiting the performance. Starting at the lowest bandwidth, as the conversion
rate is doubled the height of the peak drops by a factor of two as expected. When the bandwidth isdoubled to the highest shown bandwidth, the peak height is no longer reduced by the same factor.
This is the consequence of technical noise sources from the strong pump tones and is exploredfurther in Chapter 6.
For reporting noise performance of the converter, the relevant metric is noise referred toconverter input (right side of Figure 4.9). This reference point allows us to quote the added noiseof our device independent of measurement chain noise, but importantly it takes into account theconverter efficiency. A poor efficiency will correspond to a worse added noise for all else being
equal. The added noise can be calculated using Eqn. 4.6 from the output noise on resonance bydividing by the apparent converter efficiency, A · η. For this device, laser cooling achieved a bestvalue of Nadd = 34 photons of input-referred added noise. Again, this figure of merit represents
significant technical progress relative to the 1500 photons of added noise achieved in a prototype
system operated at 4 K [85].

4.4.2

Mechanical dissipation and added noise

In the presence of technical noise that limits damping rates, the performance of laser coolingis set by nth,m γm , where nth,m is the thermal phonon occupancy of the membrane. This means
reductions to the added noise could be achieved by cooling the mode or by decreasing the mechanicaldissipation. Ideally, the final membrane phonon occupancy nf,m scales as nth,m γm /ΓT . At ΓT /2π =Xo2 ( ) +Yo2 ( ) (n ph )44ΓT = 200 Hz
ΓT = 550 Hz
ΓT = 1.7 kHz
ΓT = 3.5 kHz

laser
cooling
50

Nadd
34

10
1.472

/ 2 (MHz)

1.476

Figure 4.9: Laser Cooling. e, Total noise power spectra, Xo2 (ω)+Yo2 (ω), for different laser coolingrates, ΓT , while maintaining Γe ≈ Γo . Best added noise in photons referred to converter input,
Nadd , indicated on right axis.

3.5 kHz, laser cooling reduced the membrane’s phonon occupancy from nth,m ≈ 1200 to a best valueof nf,m = 18.6. Laser cooling the mechanical mode is limited by technical noise, but another way to
improve the added noise metric is by reducing γm so that phonons leak in at a slower rate. Indeed,
we have studied laser cooling in a different converter with 10 times lower mechanical dissipation
but a lower conversion efficiency of 12%, and achieved an Nadd = 13 photons of added noise. This
device is not highlighted as much in this thesis because of the low efficiency, but it is important tohighlight as it is the lowest added noise yet achieved.

4.4.3

Operating without matching criteria for technical noise reduction

With the strong pump powers needed for our experiment, two distinct effects lead to excess
noise within the LC circuit bandwidth. First, we observe excess phase noise on resonance whichscales with the microwave pump power and overwhelms the intrinsic phase noise of the pump signal
generator. We also observe excess noise in the LC circuit that scales with the total laser powerincident on the membrane, to which both the lock beam and the optical pump contribute. Both ofthese effects are explored in more detail in chapters 6 and 7. Without any further mention of their

45
cause we point out a way to address technical noise in converters. If the technical noise is due to
one pump more than the other, then the added noise can be reduced by reducing power on thatpump and compensating for it with the other. If one of the microwave or optical pumps contributes
more technical noise than the other, then the added noise can be reduced by reducing power onthat pump and compensating for it by increasing the other to keep the total converter bandwidth
constant. In this way the added vibrational noise is constant, but the technical noise is reduced.
However, this reduction does not come for free. By moving away from the matched condition, theefficiency reduces which impacts the added noise referred to converter input. Because the efficiency
reduces as 1/Γe,o (see Eq. 4.3), this approach only helps if the technical noise scales higher thanmonotonically with the damping rate.

4.5

Measurement efficiency calibrations
The performance quoted is sensitive to the calibration of the microwave and optical measure-ment chain and is therefore detailed here. The scattering parameters are nearly self calibrating up
to some factors of gain and mode matching, but added noise depends on independent calibrationsof the measurement chains. The goal of the calibration is to find the efficiency after the output of
the converter. Many microwave amplifier characterizations are done with a known noise source at
the input of the amplifier by varying the temperature of a resistor [100]. Comparing the measured
noise to the expected allows for a calibration in terms of photons at the amplifier input. For our
device, we use the mechanical bath as the known noise source. As the base temperature of thefridge is swept, the occupation of the thermal bath that the mechanics couples to varies. Thissignal along with independent measurements of the cavity parameters allows us to calibrate themeasured signal peak height to the expected number of photons at the device. The calibration andconsistency checks are described in detail here.

46
4.5.1

Microwave and optical calibrations

A temperature sweep of the base plate of the dilution refrigerator is used to calibrate outputreferred noise spectra like those shown in Figure 4.10. Raw spectra are normalized to the noiselevel with the pump off, to which the only contributions are vacuum noise and measurement chain
added noise; the peak height is σ in these units. The peak exhibits a linear temperature dependence
due to the thermally driven motion of the mechanical oscillator (figure 4.10), except at the lowest
temperatures where the membrane evidently falls out of equilibrium with the dilution refrigerator(open circles, Figure 4.10c and d). The temperature below which the membrane falls out of equilibrium is higher in the optical measurement, indicating that optical heating is a significant factor
below T = 100 mK. We find that this membrane heating occurs whenever the locking beam isintroduced, but is independent of damping power, consistent with the observations in Ref. [90].
The spectra are calibrated by fitting the observed peak heights σ(T ) to an expected linear
dependence obtained from combining the full optomechanical equations of motion [85] with independently measured parameters in Table 4.1 [92]. The only fit parameter is a proportionality factor
that converts uncalibrated thermal peaks to units of photons at the converter output; the normalization of the uncalibrated spectra implies that the single-quadrature background noise (vacuum
noise plus measurement chain added noise) is simply given by the calibration factor for each measurement. This procedure yields a single-quadrature background noise ne = 29.6 photons at the
microwave output port and no = 2.7 photons at the optical output port. Inverting this calibration
procedure, we can associate the number of output-referred photons on resonance in spectra with
an effective mechanical bath temperature.
The microwave cavity coupling ratio κext,e /κe is easily measured using the HF2LI as a network
analyzer (see Section 4.1.1). The optical cavity parameters are needed to predict the optical field
output noise using optomechanical theory [62]. The mode matching between the cavity and thepump beam is obtained by directing a transmitted beam and the reflected pump onto the heterodyne
detector, and measuring the visibility of the resulting interference pattern (see Section 4.3.2). The

47

20

b
microwaves
400 mK
300 mK
220 mK
90 mK
40 mK

10

σe (norm )100

optics
400 mK
300 mK
220 mK
90 mK
40 mK

50

0

0

c

X o 2 (f) + Y o 2 (f) (norm )

30

1.472

1.473
f (MHz)

1.474

1.472

1.474
f (MHz)

d

30

100

microwaves
n e = 29.6 photons

σo(norm )X e 2 (f) + Y e 2 (f) (norm )a20

opticsn o = 2.7 photons

50

10
0

0
0

T (mK)

400

0

T (mK)

400

Figure 4.10: Temperature sweep for calibrations.(a) Total microwave noise power spectra,
Xe2 (f )+Ye2 (f ), shown for a range of base plate temperatures. Optical pump off. Noise is normalized
to unity with pumps off. (b) Total optical noise power spectra, Xo2 (f ) + Yo2 (f ), shown for a rangeof base plate temperatures, with the same normalization procedure. Microwave pump off. (c)Microwave peak height σe vs. temperature T , with a single-parameter linear fit to extract totalnoise (see text). Open data point, indicating imperfect thermalization to the cryostat at low
temperature, is excluded from fit. (d) Optical peak height σo vs. temperature T , with a similar
fit. Open data point, indicating imperfect thermalization to the cryostat at low temperature, is
excluded from fit.

48
optical cavity linewidth is measured by sweeping a probe with two sidebands at a known frequency
through resonance. The optical cavity coupling ratio κext,o /κo , which is used to determine κext,oin Table 4.1, is then fixed based on the measured converter efficiency. As a check, κext,o /κo is
independently extracted from reflection and transmission measurements of both ports of the opticalcavity [101]. This procedure yields κext,o /κo = 0.53, which can also be used to determine theexpected microwave-optical conversion efficiency, (κext,o /κo )(κext,e /κe ) = 43 ± 4%.

4.5.2

Checking the consistency of the calibrations

As a check of the optical calibration, we independently measure the optical path loss andheterodyne detector dark noise. This measurement gives an effective transmission of 16%, or
equivalently an expected single-quadrature background noise of (1/4 × 2) × 1/0.16 = 3.1 photons,
where the three factors correspond to the single-quadrature vacuum variance, inefficiency of an ideal
heterodyne detector, and effective loss, respectively. The added noise Figure 4.9 is a two quadrature
measurement and therefore has a background value 2no . This is in reasonable agreement with thethermal calibration.
As a check of the microwave calibration, the amplifier chain can be characterized by measuring
off-resonance noise, as shown in Figure 4.11. Data are fit toSout = G

1
coth
2



hf
2kB T




+ NHEMT ,(4.7)

where the two fit parameters are the gain G and two-quadrature receiver added noise NHEMT ,which we expect is dominated by the added noise of the HEMT preamplifier. We obtain NHEMT =20 photons, which implies 10 photons of added noise in a single-quadrature measurement. Comparing this result to the single-quadrature noise in Section 4.5.1, ne = 29.6 photons, naively implies
4.6 dB of loss from the LC circuit output to the effective cold load, a somewhat high value. This
loss inference should be treated with caution, as the temperature sweep in Figure 4.11 was performed with some stages of the dilution refrigerator slightly colder, which lowers total chain noisecompared to Figure 4.10.

49

2

Noise ( µV )

90

89

88

87
100

200
300
T (mK)

400

Figure 4.11: Microwave chain noise for calibration check. Noise measured in the microwave
domain off LC resonance as temperature is varied, fit to Eq. 4.7.

Chapter 5

Correlations and Feedforward

This chapter focuses on the experimental identification of correlations in the noise at eachport of the converter. A feedforward protocol harnesses the discovery that noise emitted from thetwo converter output ports is strongly correlated because both outputs record thermal motion of thesame mechanical mode as introduced in Section 5.1. The mathematical framework needed to understand the correlations is examined in Section 5.2. An experimental demonstration in Section 5.3
leverages the correlations to reduce the added noise. Section 5.4 discusses the theory of feedforward
and shows that a quantum feedforward protocol that, given high system efficiencies, would allow
quantum information to be transferred even when thermal phonons enter the mechanical element
faster than the electro-optic conversion rate.

5.1

Feedforward introduction
We consider classical and quantum feedforward protocols in a microwave-mechanical-opticalconverter in which the mechanical element is thermally occupied (figure 5.1a). When the mechanicaloscillator coupling rate to the hot bath is small compared to the opto-electromechanical damping
rates, the thermal bath acts as classical random force on the oscillator. The force noise is thenpresent in both the optical and electrical fields exiting the converter, and as such measuring them inthe electrical domain should permit you to remove them in the optical domain. During operation,
the signal to be converted is injected into the microwave port and an ancilla state is injected into
the optical port. The upconverted signal, with noise added, is emitted from the optical port.

51
The ancilla state, contaminated with the same added noise, is emitted from the microwave port.
Measuring the downconverted ancilla and feeding forward to the propagating optical mode canremove this correlated added noise.
Classical and quantum feedforward protocols are distinguished by the choice of ancilla. Simply choosing the ancilla to be optical vacuum causes zero-point fluctuations (vacuum noise) to be
fed forward along with the thermal noise. In this case, although thermal noise may be completely
removed, vacuum noise from the ancilla is necessarily written onto the upconverted signal. Choosing vacuum as the ancilla therefore prohibits upconversion of a state with negative Wigner function
or squeezing, so we refer to it as classical feedforward. Classical feedforward is a resource for recovering classical signals, whose performance — interestingly — depends on system efficiencies rather
than temperature, quality factor, or cooperativity. However, classical feedforward is unhelpful for
quantum tasks.
A different choice of ancilla would allow quantum tasks to be accomplished, even in thepresence of thermal noise. For example if the ancilla is an infinitely squeezed vacuum state, one
quadrature can be fed forward noiselessly, permitting noiseless measurement of a single upconverted
quadrature or upconversion of a squeezed quadrature without high cooperativity. This quantum
protocol does however place limits on measurement and converter efficiencies, as any loss will add
noise to the feedforward process. In the limit of perfect measurement efficiencies and squeezing,a converter efficiency η > 50% is still required to upconvert a squeezed quadrature or measure a
remotely prepared microwave quadrature with added noise less than vacuum. We therefore refer
to 50% as a quantum threshold efficiency for feedforward protocols. More sophisticated tasks canbe imagined with quantum feedforward, albeit with more stringent efficiency requirements. In
particular, quantum feedforward could be used to upconvert a qubit state given a qubit encoding
robust to noise in one quadrature. One example of such a feedforward-based qubit upconversion
scheme is explored in Section 5.4.3.
Experimentally we focus on the identification of correlations that allow feedforward, andperform classical feedforward with a vacuum ancilla. For this demonstration we measure both

52aelectrosig

anc + therm
therm

anc

optic

FF
sig + therm

sig

b
6 GHz

35 mK
LO
-

282 THz

LO

Figure 5.1: Feedforward schematic and measurement network. a, A signal incident on the
microwave port (blue sinusoid) is output from the optical port (red sinusoid) with thermal noise
coupled in from the internal port (maroon disk). Optical ancilla (black circle) is simultaneously
downconverted with correlated noise added. The ancilla is measured and a feedforward protocol(FF) is applied. Dotted line is the converter box. Blue and red arrows show microwave andoptical reflections, and purple arrows show conversion process. b, Feedforward operations areindicated by beige-shaded regions. For our classical demonstration these operations are optical
heterodyne measurement and subtraction in post-processing. The corresponding operation in aquantum feedforward protocol would be a unitary displacement of the optical field conditioned on
the microwave measurement result.

53
converter outputs in heterodyne detection and perform subtraction in post-processing to remove
the correlated noise (figure 5.1b). A quantum protocol would instead entail homodyne measurementof the ancilla at the microwave port, and conditional unitary displacement at the optical port [102].

5.2

Electro-optic correlation theory
Here the full input output formalism developed in Section 2.3 is leveraged to describe thecorrelations in noise from each port.
Spectra of the output fields are expressed in terms of a matrix S(ω), whose elements are
given by1
Sij (ω)δ(ω − ω 0 ) = h{[aout,i (ω 0 )]† , aout,j (ω)}i,
2

(5.1)

where aout = (âout,B , âout,F , b̂out , â†out,B , â†out,F , b̂†out )T is a vector of optical (âout ) and microwave(b̂out ) output field operators. For completeness optical operators are written for both the front,âout,F , and back, âout,B , optical ports, but in most of this thesis we make single port measurements
and therefore the front port is simply referred to as ‘external,’ as it is the only port where signals
are measured, and the back port is included in the loss budget. Output fields are related to input
fields by the matrix Ξ(ω), aout (ω) = Ξ(ω)ain (ω), calculated from linearized equations of motion
following Ref. [85]. HereTain = âin,B , âin,F , âin,int , b̂in,ex , b̂in,int , ĉin , â†in,B , â†in,F , â†in,int , b̂†in,ex , b̂†in,int , ĉ†in(5.2)

is a vector of input fields for optical (âin ), microwave (b̂in ), and mechanical (ĉin ) modes.
In terms of input fields, the spectral matrix is then
1
S(ω)δ(ω − ω 0 ) = Ξ∗ (ω 0 ) h[ain (ω 0 )]† aTin (ω)i + h[ain (ω)[aTin (ω 0 )]† ]T i ΞT (ω),
2

(5.3)

where transposes act only on matrices and adjoints act only on operators. The input fields arethermal states, satisfying for example h[cin (ω 0 )]† cin (ω)i = δ(ω − ω 0 )nth,m and hcin (ω)[cin (ω 0 )]† i =δ(ω − ω 0 )(nth,m + 1), where nth,m is the number of thermal phonons in the mechanical oscillator.

54
Integrating over ω 0 , Eq. 5.3 can then be rewritten
1
S(ω) = Ξ∗ (ω)ΞT (ω) + Ξ∗ (ω)ΣΞT (ω).
2

(5.4)

The first term originates from quantum noise of the input modes, whereas the second is a thermal
contribution that vanishes at zero temperature, due to the fact that


N
0


Σ=
,
0 N

(5.5)

where N = Diag[nth,o , nth,o , nth,o , nth,e , nth,e , nth,m ] is a diagonal matrix of thermal noise contributedby each input mode.
Note that the diagonal entries of S(ω) are real, whereas the off-diagonal components are in
general complex. In terms of quantities measured, for example, [âout,F (ω)]† âout,F (ω) = Xo2 + Yo2 ,whereas[âout,F (ω)]† b̂out (ω) = Xo Xe + Yo Ye + iXo Ye − iYo Xe .

(5.6)

In practice one can adjust the optical and microwave demodulator phases to make all entries real,
which corresponds to removing X-Y microwave-optical correlations, hXo Ye i = hYo Xe i = 0. In theexperiment, the demodulator phases are adjusted to approximately fulfill this condition.
While the full expressions for the matrix elements of Eq. 5.4 are cumbersome, in the limit
of optimal detuning (∆o = ∆e = −ωm ), weak damping (Γe , Γo  κe , κo ), and resolved sidebands(κo , κe  ωm ), on mechanical resonance the electro-optic correlations take the simple form
Seo (ωm ) =1
I2 + S(th) (ωm ),
2

(5.7)


(th)

S

(ωm ) =

4nth,m γm Γo(Γo + Γe + γm )2 √
Γe Γo

√


Γo Γe 
,
Γe

(5.8)

where I2 is the 2x2 identity, representing the contribution of quantum noise. The diagonal elementsof S(th) have a straightforward interpretation. Thermal noise added to the optical output mode,(th)S1,1 , is due to the resonant transmission of mechanical noise nth,m from the bath, coupled at rate
γm , to the optical mode, coupled at rate Γo . Similarly, thermal noise added to the microwave output

55
(th)

mode, S2,2 , is due to the resonant transmission of mechanical noise from the bath to the microwave
mode, coupled at rate Γe . Since the mechanical resonator couples to the optical, microwave, andbath modes, its total linewidth is Γo + Γe + γm .
The elements of S(th) satisfy the property(th)S1,2 =q(th) (th)S1,1 S2,2 .

(5.9)
(th)

As a consequence, the smallest eigenvalue of Seo is equal to 1/2 photon. If S1,2 were to exceedq(th) (th)S1,1 S2,2 in magnitude, then the smallest eigenvalue of Seo would be less than 1/2 photon,signaling the presence of two-mode squeezing. Eq. 5.9 therefore represents the condition for maximalclassical correlations, as they are as large as possible without implying the presence of squeezed
quantum noise.
While Eq. 5.8 is derived under restricted assumptions, we have verified numerically thatEq. 5.9 is satisfied for arbitrary detunings and linewidths, including for the system parameters
(th)

(th)

reported in Table 4.1. Interestingly, the property S1,1 = S2,2 can also be obtained under moregeneral conditions, provided an appropriate detuning is used. For instance, if we had instead chosen∆o = −0.38 MHz but kept all other parameters the same as Table 4.1 we would have obtained
(th)

(th)

S1,1 = S2,2 at the expense of weaker optomechanical damping per incident photon.
Importantly, imperfect sideband resolution does not weaken the strength of the classical correlations. With imperfect sideband resolution, arbitrary detunings, and cavity loss, the correlation
matrix still can still be written in the form
10
Seo (ωm ) = I2 + S (ωm ),2(5.10)

0with S satisfying
0

S1,2 =q

0

0

S1,1 S2,2 .

(5.11)

The correlations are still classically maximal, but, in contrast to Eq. (5.9), there are now contributions both from thermal and quantum noise.

56

5.3

Feedforward harnesses correlations to reduce added noise
We now turn to experimental measurements of the correlations in the noise and explore aclassical feedforward in a microwave-mechanical-optical converter in which the mechanical element
is thermally occupied (figure 5.1). In order to explore correlations in this noise, we turn off the
weak probe tone and again work in a high-stability configuration with ηM = 43%, but smaller pump
power (Γe ≈ Γo , ΓT = 2π × 200 Hz). In this configuration electro/optomechanical cooling of themechanical mode is diminished, while still maintaining Γe , Γo  γm .
The microwave real power spectral density, hXe (ω)Xe (ω)i, exhibits a peak of width ΓT /2π =(th) 2200 Hz around ω/2π = fm (figure 5.2a). The peak height relative to background, h(Xe) i =69.2 photons, is obtained from a Lorentzian fit to the data and attributed to thermally driven
mechanical motion at a bath temperature T = 87 ± 4 mK (see 4.5), which exceeds the cryostat
base temperature of 35 mK. The elevated membrane temperature is consistent with independently
measured optical heating (see 4.5). The background noise level, 31.8 photons, corresponds tone = 29.6 photons from vacuum noise and the added noise of the microwave measurement chain,with the remaining 2.2 photons due primarily to parameter noise in the LC circuit (independently
calibrated, see 6.1.1).
The optical real spectral density, hXo (ω)Xo (ω)i, shows a similar peak (figure 5.2b). The peak(th) 2height above noise, h(Xo) i = 33.1 photons, provides a second measure of the bath temperature,

T = 80 ± 4 mK, consistent with Figure 5.2a. The background noise, no = 2.7 photons, corresponds
to vacuum noise plus the effect of loss in the optical measurement chain. Note that no < ne ,indicating that the optical measurement apparatus is closer to ideal.
The real cross-spectral density, hXe (ω)Xo (ω)i, has a 47.7 photon peak at ω/2π = fm , indicating that the thermal fluctuations are common to both outputs (figure 5.2c). The difference betweenqq(th) 2(th)
the observed correlations and their maximum classical value, hXe Xo i − h(Xe ) i h(Xo )2 i =−0.2 ± 0.3 photons, is consistent with zero, indicating that thermal noise is perfectly correlated
between the two outputs, as expected from optomechanical theory. Away from mechanical res-Xe ( )Xo ( ) (nph )

100
75
50
25
1.471 fm 1.475/ 2 (MHz)Xo ( )Xo ( ) (nph )

a

b
30
15

c

40
20
0
1.471 fm 1.475/ 2 (MHz)0
1.471 fm 1.475/ 2 (MHz)dcov (nph )Xe ( )Xe ( ) (n ph )

57

80
40
0

Xe

Ye

Xo

Yo

Xo
X Ye

Yo

e

Figure 5.2: Electro-optic correlations. a, Microwave real spectral density, hXe (ω)Xe (ω)i,in units of photons referred to converter output, showing thermal-mechanical noise peak andbackground noise from microwave measurement chain. b, The optical real spectral density,
hXo (ω)Xo (ω)i, exhibits similar features. c, Real cross-spectral density, hXe (ω)Xo (ω)i, illustrating that thermal noise exhibits perfect classical correlations between outputs and background noise
is uncorrelated. d, Covariance matrix obtained from averaging the spectral and cross-spectral
densities around ωm with a 2π × 50 Hz bandwidth, indicating similar behavior for the imaginary
microwave and optical spectral densities and the imaginary cross-spectral density; no correlations
exist between real and imaginary parts.

58

feedforward off

feedforward on

t*

0

v

qo (n1/2
ph )

20

-20
0

1

t (s)

2

3

Figure 5.3: Time domain feedforward. Fed-forward optical quadrature q̌o versus time withfeedforward off (t < t∗ ) and feedforward on (t > t∗ ). A weak signal incident on the microwave port
is recovered at the optical port. Red trace shows what signal form should be to guide eye.

onance the cross-correlation vanishes, as expected for uncorrelated noise from the independentmeasurement chains. Similar behavior is observed for hYe (ω)Yo (ω)i. The full covariance matrix(figure 5.2d), obtained from averaging the spectral and cross-spectral densities around ωm with a
2π × 50 Hz bandwidth, shows that the imaginary spectral densities are also positively correlated,but that no correlations exist between real and imaginary parts due to our choice of demodulator
phase
Harnessing the observed correlations, we use classical feedforward to recover a weak upconverted signal (figure 5.1). With the weak signal incident on the microwave port and detuned fromthe pump by δ/2π = fm + 5 Hz, microwave reflection and converter transmission to the opticalport are simultaneously measured. The microwave reflection is fed forward to remove noise fromthe upconverted optical signal. In a quadrature picture where demodulated microwave and opticalfields are described in the time domain by (q(t), p(t)) position-momentum pairs, the fed-forward
optical quadrature q̌o is given byrq̌o = qo − w

no
qe ,
ne

(5.12)

where qo is the measured optical quadrature, qe is the microwave quadrature, ne (no )is the added
noise of the microwave (optical) measurement chain, and w is the feedforward weight; a similar
definition is used for p̌.

59

a

b

v

po (n 1/2
ph )

20

-20
-20

v

1/2

qo (n ph )

20 0

1/2

0.1

P (nn ph )Figure 5.4: Feedforward variance. a, Repeated measurements of both optical quadratures(q̌o , p̌o ) with and without feedforward. Signal tone turned off. b, P (p̌o ), the inferred probability
density of p̌o , with and without feedforward, with Gaussian fits. The variance decreases by 59%
with feedforward on. Error bars obtained from Poisson counting statistics.

Initially, with feedforward turned off (w = 0 for t < t∗ ), the upconverted signal, a 2π × 5 Hz
quadrature oscillation, is difficult to resolve (figure 5.3). After feedforward is turned on (w = 1.6
for t > t∗ ), the weak signal becomes clearly visible. The red line is on the figure to help guidethe eye and does not represent real data. To quantify the improvement, the optical quadratures(q̌o , p̌o ) are repeatedly measured without a signal tone, as shown in Figure 5.4a. The quadraturesare Gaussian distributed (figure 5.4b), and feedforward reduces each quadrature variance by 59%.
Feedforward performance is limited mainly by the addition of uncorrelated noise from imperfect measurement chains. This limitation can be understood by examining the power spectral
density of the feedforward signal, hX̌o2 i + hY̌o2 i, where=nohXo2 i + w2 hXe2 i − 2w
ne

r

no
hXe Xo i,
ne
r
no
no
hY̌o2 i = hYo2 i + w2 hYe2 i − 2w
hYe Yo i,nenehX̌o2 i(5.13)(5.14)

in terms of the measured real and imaginary spectral densities. As shown in Figure 5.5, as thefeedforward weight w is increased, the peak noise power near fm decreases due to the presence ofcorrelations near mechanical resonance. However, Figure 5.5 also illustrates that the noise power off
resonance increases with increasing w. At w = 1, the background noise receives equal contributions

feedforward

=0
= 0.9
= 1.3
= 1.6

50

v

v

Xo2 ( ) +Yo2 ( ) (n ph )

60

Nadd
38

10
1.471

/ 2 (MHz)

1.475

Figure 5.5: Feedforward operation of a microwave-mechanical-optical converter. Total
noise power spectra, X̌o2 (ω) + Y̌o2 (ω), for different feedforward weights, w, with ΓT = 2π × 200 Hz.
Best added noise in photons referred to converter input, Nadd , indicated on right axis.

from the optical measurement noise and fed-forward noise from the microwave measurement chain;
for w > 1 the noise introduced by feedforward will dominate off-resonance while the thermal noisearound fm continues to decrease.

5.4

Feedforward theory
The feedforward framework is developed further in order to explore the potential and limi-

tations of the technique.

5.4.1

Feedforward eliminates thermal noise

After passing through a sideband-resolved and impedance-matched converter, the signal andancilla states are described by the quadratures
qs,out =qa,out =√ηqs,in +√ηqa,in +p

1 − ηV1 + qth,out ,p

1 − ηV2 + qth,out ,(5.15)(5.16)

where qs,in and qa,in are the input signal and ancilla quadratures, V1 and V2 are vacuum noise
introduced by imperfect conversion, and qth,out is the thermal noise in the output spectrum. It

61
has been assumed that the pumps are configured to give identical thermal noise on microwave
and optical outputs. For vacuum signal and ancilla, the quadratures in Eqs. (5.15-5.16) givea correlation matrix of the appropriate form, as discussed in Section 5.2. Assuming noiselesssingle-quadrature measurement and perfect converter efficiency, one can construct the fed-forward
quadrature q̌s = qs − wqa . Choosing w = 1 yields
q̌s = qs,q − qa,q ,(5.17)

so thermal noise can be completely eliminated using feedforward. With a non-ideal measurement
or converter, some additional noise will be fed forward along with the quantum noise of the ancilla;
w = 1 will not in general be the optimal value for cancelling the thermal noise.
In the more realistic case of imperfect sideband resolution and imperfect cavities, gain andloss terms are introduced which cause an impedance mismatch. Due to the mismatch, some signal
is reflected at peak efficiency. Reflected signals are inevitably fed forward along with the correlated
noise, and feedforward is therefore degraded. Reflections can be reduced by operating away frompeak efficiency, but this also degrades feedforward. Interestingly, an adaptive control strategy
that avoids the matching requirement has recently been proposed [103]. The compatibility of thistechnique with feedforward, and other considerations for optimally handling imperfect sideband
resolution, are interesting problems for future investigation.
Because our device is well matched, signal reflections are small for the feedforward experiment,
and feedforward performance is overwhelmingly limited by measurement efficiency. Equations (5.155.16) therefore represent a useful limit for understanding the experiment and its main limitations.
A full treatment should include explicitly the effects of gain and loss, and thoroughly explore thetrade-offs for optimizing feedforward, which remains an open problem. It should be emphasizedthat the experimental characterization of feedforward performance does not require any assumptions
about sideband resolution: we explicitly measure the variance of the fed-forward optical quadratureand observe that it is reduced.

62
5.4.2

Gaussian states

For a gaussian signal and ancilla, one need only keep track of quadrature variances. For theideal case introduced above, the variance of the fed-forward quadrature is22
hq̌s2 i = hqs,qi + hqa,qi.

(5.18)

When the ancilla is simply vacuum, feedforward adds vacuum noise, while removing all thermal
noise. Our experimental efforts have only used vacuum ancilla and therefore are all referred to asclassical feedforward. If a squeezed ancilla is chosen, then in the limit of perfect squeezing where
2 i = 0, the squeezed quadrature can be fed forward noiselessly. This enables upconversion of ahqa,qsqueezed state or noiseless measurement of a remotely prepared microwave quadrature.
The squeezed protocol requires a threshold converter efficiency to be reached. For a finite
conversion efficiency, but with an otherwise perfect homodyne measurement apparatus, the fedforward variance is22
hq̌s2 i = ηhqs,in
i + ηhqa,in
i + 2(1 − η)hV 2 i,

(5.19)

where qs,in is the input signal quadrature and qa,in is the input ancilla quadrature, and hV 2 i denotes
2 i = 0, less than one vacuum of
the vacuum variance. In the limit of perfect ancilla squeezing, hqa,innoise is only added when η > 1/2. This is a threshold for measuring squeezing in an upconverted
signal, or more generally measuring a remotely prepared microwave state with added noise lessthan 1/2 photon, referred to as a quantum threshold.
If one considers separate efficiency thresholds for microwave-optical and optical-microwave
conversion and operates in a regime where thermal noise is added asymmetrically, it is possible
to relax the threshold η = 1/2. If much less noise is added to one output port than the other,performance is improved when the low-noise output is used for the signal, so a lower efficiency
can be tolerated in that direction. However, if the high-noise output is used for the signal, ahigher efficiency is required. Strictly, η = 1/2 is therefore the minimum threshold for bi-directional
conversion of squeezed states.

63
5.4.3

Qubit up-conversion

With a squeezed ancilla, the feedforward protocol can be used to remove noise in one quadrature. The task then is to encode a qubit in a single quadrature, and to find an encoding that makes
this qubit robust to noise in that same quadrature. Here we examine one encoding to demonstrate that upconversion of a quantum signal in the low-cooperativity limit is possible in principle,
without regard to technical feasibility. While our encoding has the virtue of formal simplicity, it
is unlikely to be optimal with regard to efficiency requirements. We speculate that a GottesmanKitaev-Preskill qubit encoding may improve efficiency thresholds for qubit conversion as it would
be robust to displacements in both quadratures [104, 105].
Consider using a squeezed ancilla to eliminate classical noise in the Y quadrature, leavingclassical noise in the X quadrature. The noise operator N is of the formZ
N (ρ) =Uβ ρUβ† p(β)dβ,(5.20)

where ρ is the density operator of the qubit, Uβ |xi → |x + βi, andp(β) = √122e−β /(2σX ) .
2πσX(5.21)

Our goal is to show that, with a suitable decoder, this channel can transmit a qubit. Thisis equivalent to showing the channel can transmit a half of a maximally entangled state with highfidelity. It is formally simplest to demonstrate the required fidelity for an encoding expressed interms of X eigenstates, X|xi = x|xi, but similar encodings for coherent states should work as well.
A decoder should preserve the quantum state, but should allow for information to be gained
about β so that the state can be shifted back into its original space. What we want now is a pairof logical states |0̄i and |1̄i and a decoding operation D : H → A, where H is the output space of
N and A is a single-qubit space (spanned by |0̄i and |1̄i), such that the entanglement fidelity
Fent = hφ0 | I ⊗ (D ◦ N )(|φ0 i hφ0 |) |φ0 i ≈ 1,
where |φ0 i = √12 (|00i + |11i). This fidelity can be rewritten as(5.22)

64

1

Fent =1

1 XX
hs̄| D ◦ N (|s̄i ht̄|) |t̄i ,4(5.23)

s=0 t=0

which will make it easier to calculate.
The encoding we will choose depends on an X eigenvalue b > 0, and is of the form
|0̄i = |bi(5.24)|1̄i = |−bi .

(5.25)

Noise of the form Eq. (5.20) will shift this codespace to a new space spanned by |b + βi and |−b + βi.
Choosing b  σX will enable us to figure out the displacement β and shift the codespace back.
An excellent observable to measure to determine the value of β isZ bAb =γPγ dγ.

(5.26)−b

Here, Pγ = |b + γihb + γ| + | − b + γih−b + γ| is the projector onto the codespace after being displacedby γ in the x direction. Given outcome γ (which we will show below will almost surely equal thetrue shift β), our decoding will be completed by applying a conditional displacement Uγ to map Pγ
back to the original codespace P0 .
It is useful to re-express Ab in terms of the X operator as follows:

Z b
Ab =

γPγ dγ(5.27)

γ (|b + γihb + γ| + | − b + γih−b + γ|) dγ(5.28)

−b
Z b

=
−b
Z b

=

Z b
γ| − b + γih−b + γ|dγ

(5.29)(x + b)|xihx|dx(5.30)

γ|b + γihb + γ|dγ +
−b
Z 2b

−b

Z 0
(x − b)|xihx|dx +

=

−2b

0

Z 2b
=

fb (x)|xihx|dx(5.31)fb (x)|xihx|dx(5.32)

−2b
Z ∞

=
−∞

= fb (X),(5.33)

65
where




x − b for 2b ≥ x ≥ 0





fb (x) = x + b for 0 ≥ x ≥ −2b







0 for |x| > 2b.

(5.34)

The function fb (x) is plotted in Figure 5.6.
To derive a result for the entanglement fidelity we would like to evaluate Eq. (5.23). The
following calculation will be useful: fixing |s̄i and requiring |β| < b and |γ| < b, we havehs̄| Uγ† Pγ Uβ |s̄i = hs̄| (|bi hb + γ| + |−bi h−b + γ|) Uβ |s̄i

(5.35)= h(−1)s b| (|bi hb + γ| + |−bi h−b + γ|) |β + (−1)s bi(5.36)= h(−1)s b + γ|β + (−1)s bi(5.37)= hγ|βi .

(5.38)

Our decoder D is implemented by a two-step process. The first step is the quantum nondemolition measurement of Ab = fb (X) which results in an estimate γ of the noise β that was applied.
The second step is a translation Uγ† to map back to the original codespace. We now evaluate thefidelity achieved by this decoding. To evaluate Eq.(5.23), we first fix |s̄i and |t̄i,and computeZZ

hs̄| D ◦ N (|s̄i ht̄|) |t̄i ≥
|β|<b

Z

|γ|<b

(5.39)

hγ|βi hβ|γi p(β)dγdβ(5.40)

Z=|β|<bhs̄| Uγ† Pγ Uβ |s̄i ht̄| Uβ† Pγ Uγ |t̄i p(β)dγdβ

|γ|<b

Z
=

p(β)dβ(5.41)

|β|<b



1
β2
√ exp − 2 dβ
2σX
2π
|β|<b


b
= 1 − erfc
σX



β2
= 1 − O exp − 2
.
2σX
Z

=

Averaging over s̄ and t̄ gives us the result(5.42)(5.43)(5.44)

66

γ

-b

b

x

Figure 5.6: Quantum feedforward observable. Plot of measurement outcome for ideal observable fb (x) (black) and approximated observable f˜b (x) (blue).


2Fent ≥ 1 − O(exp −b2 /(2σX) ),(5.45)

which will be close to 1, for b  σX . In other words, if the displacement b far exceeds thermal
noise, errors can be diagnosed and corrected with an appropriately chosen decoding operation.
A key step in the decoding procedure described above is the QND measurement of fb (X),
but implementing this measurement could be a challenge. We therefore consider a more physicaldecoding procedure, and argue that it will give similar fidelity to the one described above. In
particular, we let
11
f˜b (x) = 2 x3 − x.
2b2

(5.46)

This is a good approximation to fb near ±b (see Figure 5.6): for |γ|  b we have f˜b (±b + γ) = γ,
which is exactly what we require. Because of our choice b  σX , we are very likely to get ameasurement outcome in this range if we measure f˜b (X), so doing so is a good proxy for measuring
fb (X).

67
Our decoding procedure is again two-step: First, do a QND measurement of
Ãb =

1 3 1
X − X,
2b2
2(5.47)

obtaining a result γ. Second, apply Uγ† . This gives a transmitted fidelity close to 1 as long asb  σX .

Chapter 6

Converter Limitations

This chapter discusses the limitations of the transducer in order to highlight why quantum
operation has eluded us. First, Section 6.1 lays out a detailed study of the sources of added noise.
Next, Section 6.2 delineates the challenges of making a flip-chip with a reliable, reproducible gap
separation. Section 6.3 contains a brief discussion of some of the consequences of the compromises
made in building the optical cavity around a microwave enabled chip to motivate the improvements
discussed in Chapter 7.

6.1

Conversion pumps lead to added noise
Although feedforward and laser cooling achieve comparable noise performance in the current

setup, their limitations are quite different. In the presence of technical noise that limits damping
rates, the performance of laser cooling at a given pump power is set by nth,m γm , where nth,m isthe thermal phonon occupancy of the membrane. In feedforward, the main limitation is noise fromthe measurement chain, which is indistinguishable from the deviations being corrected for. For thequantum feedforward protocol, we considered the case with perfect measurement efficiency more
broadly however, the product of both measurement efficiencies and converter efficiency needs to be
above 50%. Our work has not focused on improving the measurement chains and therefore the focusof this chapter will be on the limitations of the more standard laser cooling method. Although manyof the technical limitations for laser cooling are not the limiting factor for feedforward, addressing
them would still help the feedforward method.

noise source

noise photons
high efficiency high Qm

parameter noise
generator phase noise
thermal motion
laser heating

23
2
7
2

1
0
11
1

total

34

13

69

Table 6.1: Added noise budget for each contributing source. The high efficiency device
reached η = 47% and was the focus of Chapter 4. The Qm device reached lower added noise but
had lower efficiency. Noise budgets are shown for the best laser cooled added noise configuration ineach device. That is, the point where the pumps are turned up as high as possible before technicalnoise dominates.

In the low conversion rate regime, the transducer is dominated by thermal noise from themechanical oscillator. As the conversion rate is turned up, the contribution from thermal motion
is reduced, but the pumps also introduce potential noise sources. These are referred to as technicalnoise sources because, unlike the thermal motion, they are not inherent to the device but rather arise
due to the non-ideal behaviour from the pumps needed to operate the device. The technical noise
sources include: classical phase noise from our microwave source, laser heating of the LC circuit,and parameter noise in our LC circuit due to the high microwave pump power. In the absenceof technical sources of added noise, the pump powers could be increased until the conversion rate
dominates over the fixed thermal decoherence rate to overcome the thermal added noise. The pumps
can introduce noise simply by being spectrally dirty or through more complicated processes thatinduce fluctuations in the device. Table 6.1 shows a budget of the noise figure quoted in Chapter 4.
The total noise figure is categorized in order that solutions can address the independently.
The optomechanical cooperativity compares the conversion rate to the intrinsic mechanicaldamping rate and is useful in understanding the sources of added noise as it highlights the highpump powers needed (Eq. 6.1). The criterion of low mechanical added noise is analogous to the morecommon optomechanics goal of ground state cooling. Ground state cooling is when the mechanicaloscillator is damped by the pumps until it spends the majority of its time in the ground state.
If a mechanical oscillator is in the ground state then information can be transmitted through it

70
without the environment adding photons to the signal. Therefore the limit of no added mechanicalnoise is the limit of spending 100% of the time in the ground state. This is also equivalent to thehigh cooperativity regime: the optomechanical damping rate being much higher than the intrinsic
dissipation,
Γ> γm .
nth(6.1)

4g02 ncav
κT

(6.2)

Here we expressΓ=

to highlight technical values needed, where ncav is the number of photons in the cavity. Due to thelow frequency of our mode, the thermal occupation is fairly large even at millikelvin temperatures.
The mechanical excitations are bosonic and therefore follow Bose-Einstein statistics
nth (ω = 2π × 1.5 MHz, T = 100 mK) =1
e~ω/kB T − 1≈ 1400.

(6.3)

Combined with the modest quality factors we achieve thus far (γm = 11 × 2π Hz for the highefficiency device), it takes a high damping rate of Γ ∼ 15 × 2π kHz to reach this regime. Feasibly
we can reach the cooperativity requirement, but it is achieved with very high pump powers thatcause technical added noise. The single photon-phonon coupling rate is very small for our device,(see table 4.1) which means very a high number of photons in the resonator (ncav ∼ 108 ) are needed
to reach the high cooperativity regime. Ideally this would not be a problem, and indeed for matched
conversion the highest damping rate achieved was ΓT = 100 × 2π kHz. However, experimentally
we find that the high pump powers introduce technical noise sources that are converted along withthe signal that lead to higher added noise and preclude ground state cooling.
One example of technical noise is phase noise from the microwave pump. This source ofnoise is essentially a dynamic range problem. In which which the pump power needed is weighedagainst the excitation of a single photon. Because 108 photons in the resonator are needed forhigh cooperativity and the pump is off resonance, this means the pump tone needs 1016 photons

71
incident, or Pin = −42 dBm. For a detuned pump, the number of photons in the cavity is given by
ncav =

Pin
κext
.
2
~ω ∆ /4 + κ2T(6.4)

The power from a single photon in a 1 Hz bandwidth, P/BW, can be obtained for using BoseEinstein statistics,
nM W =

1.
e~ω/(P/BW ) − 1(6.5)

Solving for the power in a 1 Hz bandwidth gives −204 dBm This corresponds to a difference of
162 dB between the power in pump tone and the PSD that can tolerated detuned by the mechanicalfrequency of 162 dB. This kind of dynamic range is difficult to achieve in microwave generators. The
phase noise measured for the generator we use (Agilent PSG E8257D) at 1.5 MHz is −144 dBc/Hz.
This means that for a given pump power, if a PSD is measured 1.5 MHz from the tone, there
will be a noise floor 144 dB below the peak pump power. The noise will be converted along withthe signal and would look like added noise in the converter. The best added noise performance inTable 6.1 was achieved with pump powers of −57 dBm incident on the cavity. This means that
there is −201 dBm of noise 1.5 MHz from the carrier, which is equivalent to 2 photons of noise at
6 GHz.
Options for reducing generator phase noise are either buying a lower-noise microwave sourceor filtering the pump. Commercially available generators cannot offer much greater dynamic range
than ours. There are optically generated microwave signals that can offer the dynamic range
required, but are limited in power [106]. If we were to use the powers needed for high cooperativity
with our current generator we would need nearly 20 dB of filtering. For narrow filter windows very
close to the pump this requires high Q filter cavities. This possible but as the phase noise is not our
main limiting factor it has only lightly been pursued. Increasing the electromechanical coupling
would also relax this constraint by reducing the microwave power needed.

2

150
100

2

Xe (f) + Ye (f) (nph )7250ΓΤ = 50 HzΓΤ = 550 HzΓΤ = 5.0 kHz

1.460

1.466

1.472
1.478
ω/2π (MHz)1.484

Figure 6.1: Microwave noise squashing. Noise measured at the microwave converter output.
As power is increased, the mechanical peak is damped and the background noise increases. At thehighest power, the mechanical peak becomes a dip in the background noise.

6.1.1

Microwave circuit parameter noise

Most of the conversion added noise plots shown so far show the noise emitted at the optical
port. For the converter used in our high efficiency device, we can also look at the noise coming out
the microwave port. The spectrum at the optical port looks distinctly different than the spectrumat the microwave port. The optical port shows the mechanical mode with a flat background as seenin Figure 4.9. In contrast, the microwave port shows the mechanical mode on a background thatis dependent on microwave power. Spectra at the microwave port for increasing pump power are
shown in Figure 6.1. In the microwave output spectrum the background increases, but looking at
the optical spectrum the noise only increases in the converter bandwidth, which is evidence thatthe noise is present in the LC circuit. This background noise is due to an unanticipated sourceof fluctuations in the LC circuit. Under the strong microwave pump required for operating in thehigh cooperativity regime, the circuit’s resonance frequency and quality factor both fluctuate with
substantial frequency components around 1 MHz. Such fluctuations are essentially indistinguishable
from fluctuations in the oscillator’s position.
Fluctuations in the LC circuit parameters convert pump photons to power at the mechanicaloscillators resonance frequency, therefore adding noise to the conversion process. At high pump

73

Nadd (photons)

100

10

1
2

10

3

10

4

10

Γe/2π (Hz)Figure 6.2: Added noise from parameter noise. The noise coming out the microwave port as
the electromechanical damping rate is increased shown in units of conversion added noise.

powers, it looks as if the parameters of the LC circuit, either the Q or the frequency, are fluctuatingin time, and we therefore refer to this noise as parameter noise. This can be understood byconsidering the pump from the frame of reference of the LC. From the fixed LC frame of reference
it looks as if the pump power and frequency is fluctuating in time, and therefore this noise hurts us
in exactly the same way as classical noise on the pump. The excess noise in the microwave circuit
is correlated to an increase in loss, which is explored in more detail in 7.2. The noise budget for
LC parameter noise was studied by monitoring the noise coming out the microwave port as thepump power was increased. Figure 6.2 shows the added noise vs damping rate. (Note, by referring
to converter input we are referencing the noise at the microwave port, which conveniently is thesame reference place made if we measure microwave noise and calibrate out the measurement chain.
Therefore, the same calibration allows us to quote the excess noise in terms of added noise photons
for a conversion process.)
Further evidence that the noise originates in the LC circuit can be seen from looking at awider spectrum showing that the noise is filtered by the susceptibility of the LC circuit. The added
noise shows up in the microwave spectrum as an excess of power around the LC frequency as shown

74

PSD (mW)

-13

4x10

2

0

0.5

1.0

1.5

2.0

ω/2π (MHz)Figure 6.3: Broadband parameter noise. Noise measured for high pump power showing noise isoriginating in LC. This measurement was made on a device with 500 kHz linewidth. The LC shapeshows up above the background noise floor. Sharp features are mechanical modes and electronicnoise.

in Figure 6.3. The noise has the characteristic width of the LC circuit indicating that the noise
is either originating in the LC, or is at least filtered by it, which eliminates sources of noise on
the pump before the LC. Another way to tell that the noise is in the LC circuit is from looking
at the spectrum around the mechanical frequency at the microwave port as a function of damping
power. Unlike in measuring the noise in the optical domain, the noise increases off mechanicalresonance with increasing pump power. As the power is increased, first the mode is broadened andthe peak is reduced as in normal operation, but at higher powers noise squashing is observed. Noisesquashing is interference between noise in the LC circuit coupling directly to the transmission line
and coupling first to the mechanical oscillator and then to the transmission line [55, 107, 108, 109].
The noise is present in the LC circuit, so in the optical spectrum, the background remains constantwith increasing power, while the transduced noise filtered by the mechanical susceptibility increases.

6.1.2

Microwave circuit noise induced by laser light

Perhaps the easiest source of noise to predict is laser heating of the LC circuit. Most superconducting experiments go to great lengths to isolate there quantum circuits from higher frequency

75
photons. They are placed inside a dilution refrigerator in light tight boxes and even the filteringon the cabling is an open area of research to reduce noise [110, 111, 112]. In our experiment wehave done the complete opposite: we have a window out to room temperature as well as a brightlaser shining less than a half millimeter away from the superconducting circuit. Therefore it is perhaps not surprising that turning on the laser adds some amount of noise in the LC circuit that isthen upconverted along with the signal. The size of this effect, however, is determined empirically.
There were concerns that this would make our approach impossible, and it was not until we cooleda converter device down in a dilution refrigerator and studied the noise that we understood theamount of noise that we would need to contend with.
The laser heating is one of the lower contributors of added noise (Table 6.1). This is reassuringin the sense that it is not orders of magnitude away from what is needed for quantum operation. The
contribution from laser heating is measured by shining light into the optical cavity and measuring
the PSD coming out the microwave port. This is similar to the conversion data except it is done
without the microwave pump on, which allows the noise figure to be extracted independent of
all other sources (thermal motion, generator phase noise, and parameter noise). Optical photons
heating the superconductor convert optical photons to microwave photons, but in a noisy incoherent
way that is detrimental to the conversion process.
Figure 6.4 shows the effect of the laser light on the microwave cavity linewidth and the noise
coming out the microwave port. This is taken with the optical cavity configured in the highstability regime used for efficiency measurements (Table 4.1). The LC linewidth increases by about
10% as the optical damping rate is increased. On the graph, both the noise and linewidth have
two points that correspond to zero damping. The first point is the LC parameters with all lasers
blocked, while the second, higher point is with the lock beam incident on cavity resonance, butwith zero damping power. The initial increase from the lock is worrisome because it is alreadyat a low lock power operating point of 500 nW. After the initial jump the added noise increases
gradually. This is not understood, but is reassuring in that if the baseline can be lowered thenthe laser heating may not be the limiting factor, even at high pump powers. In this device we

76

b

1.5

Noise (x10

-12

2

V)

2.8
2.7
2.6
0

500

1000

ΓO/2π (Hz)

94

1.0
0.5

92

Nadd

LC linewidth (MHz)

a

0.0
0

500

1000

ΓO/2π (µW)

Figure 6.4: Laser Heating of LC. a, Laser light increases loss in the microwave circuit. There
are two points at zero damp power. The lower one corresponds to no lasers, while the higher one iswith the cavity locked with 500 nW. b, Noise increase for given damping rate. The left axis shows
uncalibrated microwave noise while the right axis is in added photons.

were not able to lock at high optical damping rates but, this problem has been fixed by modifying
room temperature electronics which has also allowed us to operate at lower lock powers. Moving
to a higher optomechanical coupling rate should also lower this curve as less optical power will beneeded for the same damping rate.
The laser heating has not been extensively explored. There is a bottle neck in acquiring
more data as full devices with microwave and optical capabilities are difficult and time consuming
to make. In the experiment, we can adjust the membrane’s location within the optical standing
wave. This changes the internal and external coupling rate but when normalized to photons lostthe noise is relatively constant. Reducing the loss of the cavity should allow for a more strongly
overcoupled cavity and fewer photons lost inside, minimizing the chance of depositing energy into
the superconductor. The path a stray photon takes could be absorption in the silicon nitride
that then leads to high energy phonons hitting the superconductor, the silicon nitride acting as awaveguide for scattered photons, or maybe multiple reflections leading to the superconducting pad.
In future tests we plan to put a metal wall on the membrane meant to differentiate between energy
pathways in the membrane and scattered photons.

77

6.2

Capacitor gap spacing is unreliable
One of the largest challenges of this experiment that I spent considerable effort grapplingwith is reliably achieving small capacitor gap sizes for high mechanical coupling rates. The chips
are order cm in size and the intended gap separation is 100 nm, a five order-of-magnitude difference.
That is equivalent to holding a kilometer long stick by the ends and trying to accurately control the
separation near the middle of it down to the cm level. This analogy shows just how many thingsneed to be accounted for to get this small gap. There can’t be anything higher than a centimeterover that whole kilometer, things like the bow over that distance will matter, and upon cooling
down the different coefficients of thermal expansion cause stress or strain on the chips and candistort the separation. Our flip-chip devices are fabricated with this in mind. In order to limit the
chances of dust getting between the chips and defining a larger gap, there are 15 µm trenches cut
out of the chips everywhere except for the posts that define the separation and capacitor plate as
seen in Figure 3.1. This reduces the chance that any dust on the chip will be in a region that would
dictate the separation.
This strategy allowed us to achieve a gap separation of a few hundred nanometers at room
temperature. The first devices cooled down all had high resonant frequencies corresponding to large
gap separations. A white light interferometer revealed that the separation matched the design atroom temperature indicating that the separation was increasing upon cooling down. In order toverify that the SiN was not causing an error in the interferometer measurements, the most accurate
method was to break the membrane and measure the height difference between the shards of SiN
remaining on the membrane frame and the bottom capacitor.

6.2.1

Stycast affects capacitor gap spacing

Mismatches in the coefficients of thermal expansion cause the epoxy to strain the chips
and change the capacitor gap separation. Even though the separation is set by a thin layer ofaluminum, the posts are actually 30 um tall (because of the trenched away areas), which means

78
warm

cold
b

a

c

Figure 6.5: Thermal contraction bows chip. a, Profile view of assembled flip chip. At room
temperature the chips are well aligned and have a capacitor gap spacing set by the chips posts.
Grey is silicon top and bottom chip. Blue is posts defining separation and capacitor pads. Blackis epoxy. b, In the corner epoxy design, when the epoxy contracts relative to the posts, the chipbows and the middle is bowed up, increasing the capacitor gap. c, Epoxying on the edge of the
top chip pulls the capacitor pads closer together as the epoxy bows the chips. Epoxy on edge looks
like it is between the post when viewed in profile. See Figure 6.6 for top layout.

that the stycast drops span that same height. The thermal contraction of silicon is a few parts per
million while stycast which is an order of magnitude higher. This means that the epoxy shrinks on
the order of a few hundred nanometers more than the post. The geometry of the posts causes thestycast to pull the corners down, and the middle of the chip is levered up, increasing the capacitor
separation. Figure 6.5 shows the a profile view of the chip flex affecting the capacitor separation,
while Figure 6.6 shows a finite element simulation of the silicon chip flexing for a given amount ofstrain at the epoxy joint. Switching the epoxy from the corners to the edges changes from a lever
flexing up, to pulling the middle down (Figure 6.6) which allowed for a solution to the problem
with the existing chips. There are small posts right near the membrane and therefore the spacing
should be set by forcing those together.

79
chip displacement

chip displacement
100 nm
0 nm

glue

posts
-400 nm

Figure 6.6: Epoxy placement determines bow. Finite element simulations of ship bowing due
to stress at points of contact with epoxy. Epoxying on the corners levers the middle of the chipup, leading to increased capacitor gap separation. Epoxy on the edges pulls the middle of the chip
down.

6.2.2

Chip frequency spread

The change of epoxy placement solved the high resonance problem, but the fact that ourchips spacing was dependent on a relatively uncontrolled amount of epoxy meant that the spacing
was still different from chip to chip and from cooldown to cooldown. In fact, the force from the chip
flexing down was enough to squish the posts near the membrane in some devices, leading to lower
resonance frequencies. We confirmed that the low frequencies were due to the posts deforming bybreaking apart flip-chip devices after cooling down and measuring the height of the center post. In
some devices it had reduced by up to 80 nm.
The uniqueness of each device leads to a range of LC frequencies across the 4-8 GHz band.
The modular hats of the 3D cavity were designed to account for this spread, allowing us to chose a
hat to match the chip frequency but the change from cooldown to cooldown could not be predicted
or addressed. The 3D cavity coupling depends on the detuning between the LC and the 3D cavity
resonance, as well as on the external coupling (see equation 3.1). A range of external couplings

80
work for our converter device, but there is an optimum for efficiency and added noise. The largest
linewidths we could achieve for our 3D cavity were about 250 MHz, meaning that our device couldchange by about 50 MHz from the goal and we would still have a converter that had reasonablesideband resolution and external coupling. Our flip-chips changed by up to 500 MHz from cooldown
to cooldown. Patience and multiple tries allowed this procedure to work well enough for a few
experimental runs but was too tedious for long term viability. In addition, to help address technicalnoise sources we aim to reduce the design gap separation to increase the mechanical coupling rate,
meaning fractionally we would be even more vulnerable to gap changes. Therefore the chips needhigher reliability or the coupling method needs to be independent of the LC frequency. The 3D
cavity allowed us to move to shorter cavities, so any solution that moved away from the 3D cavity
needs to preserve the optical cavity length to avoid a step backwards in progress.

6.2.3

Deformation of top chip reduces Qm

Strain in the top chip at the location of the membrane limits Qm . There are two mechanisms
by which the radiative loss could be changing. Deforming the chip down increases the contact between the top and bottom chip and could increase radiative loss. However, we observe deformation
away from the bottom chip also lowers the Qm , indicating that strain in the chip is the cause. If
the central posts are clamped tightly down, then there is a more efficient link to the bottom chipright near the membrane and more mechanical loss is present. If this was the only effect then all
chips above the design frequency should have high Qm as well. The low Qm of chips above thedesign frequency suggests that some amount of stress in the top chip from bending and flexing leads
to more loss as well. In testing a range of devices we found a trend of lower Qm the further thedevice resonance was from the design frequency. This was true for devices with frequencies above
as well as below the design goal, as shown in Figure 6.7. Both points suggest a solution that moreaccurately hits the design frequency and reduces local strain near the membrane.

81

Qm (millions)

1.0

0.5

0
-1.5

-1.0

-0.5

0

0.5

Frequency error (GHz)

Figure 6.7: Qm is correlated to LC frequency. Qm for various devices is shown as a functionof measured LC frequency compared to the design goal. The devices with the highest Qm are theones closest to the design frequency.

6.3

Loss in optical cavity limits efficiency
The limitation of the optical cavity geometry is discussed here to motivate the switch to an

etalon cavity. The inability to make a strongly overcoupled cavity limits the over all conversion
efficiency. The conversion efficiency scales as the microwave and optical coupling efficienciesη∝κext,o κext,e
.
κo
κe

(6.6)

The limiting factor is set by the uncontrolled loss of the optical cavity. Our group has become very
good at making strongly overcoupled cavities with membranes in the middle [90, 92]. However,the challenge of integrating the microwave components leads to higher internal loss and a lower
coupling ratio. Even with microwave components, the optical cavities are overcoupled at room
temperature, but upon cooling down the membrane and mirrors misalign relative to each otherand lead to an increase in internal loss. For example, we can look at the cavity used for thehigh efficiency conversion described in Chapter 4, which had the highest optical efficiency yet fora converter device, but still shared the characteristic challenges of integrating all the components
together. The empty cavity had an external coupling rate κext = 1.1 × 2π MHz and an internal
coupling rate κint = 0.2 × 2π MHz. Adding the membrane increased the internal loss rate to82κint ≈ 0.4 × 2π MHz. The approximation symbol is used to highlight that once the membrane
is in the cavity, the external coupling and the loss rate depend on the membrane spacing (seeFigure 6.8), but for concreteness we quote values at the converter operating point. Cooling down
increased the loss further to κint ≈ 1.0 × 2π MHz, which leaves us with a marginally overcoupledcavity. The frequency of the mechanics and our desire to stay sideband resolved prohibit us from
simply moving to a more strongly overcoupled cavity by increasing the external coupling.
The optical cavity parameters are partially adjustable cold by moving the mirrors. Both optical cavity mirrors can be translated longitudinally via piezoelectric actuators. During operations,one of these actuators is used for the locking scheme, and the other is used to change the membrane
position within the optical standing wave of the cavity and then held stationary. The membraneand stationary mirror effectively constitute a low-finesse etalon whose transmission depends ontheir separation.
Figure 6.8 shows the theoretical and measured optical linewidth and the theoretical optomechanical coupling rate as a function of membrane-mirror separation. As the membrane position isvaried, the light energy is stored predominately to one side or the other of the membrane [92],and the external coupling factors associated with the high-throughput input mirror and the lowthroughput back mirror (κext,o and κB,o , respectively) vary accordingly. In order to further increaseκext,o /κo , the optical cavity mirrors have unequal transmission rates. This cavity asymmetry resultsin variation in the total optical cavity linewidth κo as the membrane position is varied. Furthermore,
our membrane is placed closer to the input mirror, resulting in enhanced optomechanical coupling
Go when κext,o /κo is maximized. Ideally, we would operate at a position where Go and κext,o are
simultaneously maximized. However, due to optical instability, a configuration with ηM = 0.43 andκext,o /κo = 0.54 was used instead for noise measurements.
Upon cooling down, misalignment leads to an increase in the internal loss across the membrane position as the light is scattered into higher order modes, but also leads to sharp deviations
from the expected cavity linewidths as we sweep the membrane through the cavity (Figure 6.8). This
is indicative of a misaligned membrane that is causing coupling to higher order lossy modes [113].

83

к /2 (MHz)3.5κoκ ex,o

2.5

1.5

0.5

|G/Gmax |

1.0
0.5
0

0.0

0.5

1.0

x/λ

Figure 6.8: Cavity couplings dependence on membrane position. Upper plot: simulated(blue line) and measured (blue points) optical cavity linewidth, and simulated external coupling
rate (purple line) as membrane distance from mirror, 750 µm + x, is varied. Difference between
measured and expected is a result of increased loss. Error bars represent the standard deviation ofseveral repeated linewidth measurements. Lower plot: absolute value of theoretical optomechanicalcoupling normalized to maximum coupling.

Although moving to the 3D microwave cavity allowed us to shorten the optical cavity and produce
our first overcoupled optical cavity in a converter experiment, it still showed some characteristics of being misaligned. All of these things combined lead us to push for a more robust opticalcavity design so that we can have higher overall efficiency, as well as work at higher mechanicalcoupling rates, and therefore need less optical power, which will help with laser heating of thesuperconductor.

Chapter 7

Technical Improvements to Reduce Added Noise

This chapter discusses the steps taken to address each limitation laid out in Chapter 6. The
parameter space is interconnected and not all problems could be solved independently. Fabrication
methods to improve Qm , the capacitor gap separation, and microwave circuit parameter noise are
discussed in sections 7.1 and 7.2. Fabrication techniques to improve the optical cavity and flip-chip
assembly are introduced in Section 7.3. These improvements allow for a more robust LC coupling
method to be used, which is described in Section 7.4. The chapter concludes in Section 7.5 with aprojected noise budget for a new converter device that will test the range of improvements that I
have worked on.
For completeness, here is a list of the problems and implemented solutions covered in thischapter• Implementing suppression of acoustic radiation to reduce mechanical loss.
• Reducing capacitor gap separation for improved electromechanical coupling.
• Relieving silicon in top chip to improve capacitor gap accuracy.
• Changing superconductor to Niobium Titanium Nitride (NbTiN) to reduce LC parameter
noise.
• Introducing etalons to improve optical cavity stability.
• Developing bonding methods for glueless etalons and flip-chips.

85

7.1

Fabrication methods to improve Qm and reduce capacitor gap separation

7.1.1

Improved Qm

We developed a method that improves the mechanical quality factor by an order of magnitude
and allows us to reliably reach Qm > 106 in our flip-chip geometry. Reducing the mechanicalcoupling to the environment improves the added noise contribution from thermally driven motion.
In our standard membrane designs Qm was limited by radiation loss due to the variability in affixing
the top and bottom chip as discussed in Section 6.2. The variability produced a range of quality
factors, where almost all chips had Qm > 105 , and occasionally some reached Qm > 106 . Whilethe quality factors achieved in our flip-chips are not too much worse than other simple membranegeometries, there is clearly room for improvement as evidenced by the multiple approaches thatnow use soft clamping [114] and phononic shielding [115] to achieve Qm ≥ 108 .
Incorporating a PNC structure into our flip-chips would be advantageous for several reasons.
A PNC opens up a band gap around the frequency of the membrane mode and therefore reduces
radiation loss as the substrate does not support those modes [116]. The elimination of thermallyoccupied substrate modes near the membrane mode of interest is also valuable in and of itself,
beyond the enhancement of mechanical Q. In order to maximize the benefit of a PNC, it isdesirable to have the only contacts from the membrane to the outside world be through the PNC.
However, in order to avoid collapsing the capacitor when working with very small gap separations,
we have always fabricated posts very near the membrane in our flip-chip devices. Whether a PNC
would still improve Qm in the presence of these posts is not clear, and the reliability of flip-chips
with the points of contact outside multiple PNC unit cells has not been demonstrated.
We design a geometry that improves Qm and tests fabrication techniques needed to incorporate phononic crystal (PNC) structures in our flip-chips. Because of these uncertainties, we decided
to start with a simpler approach to improving Qm : connecting the membrane to the top chip frame
only through four narrow tethers (Fig. 7.1). Because the tether fabrication process is quite similar
to the fabrication required for a PNC, the simple tether design also enabled us to test fabrica-

a

b

86

Figure 7.1: Initial tether designs. Different orientation of tethers to test whether Qm depended
on tether orientation along a nodal line of the mechanical mode. The two designs were tested on
two different size chips. Two of four design configurations shown here. Light grey is Si, dark regions
are holes that are etched away, light blue is capacitor pad, yellow shows areas that are not etchedaway for trenches (only shown on left design).

tion techniques needed to eventually incorporate PNC structures into our flip-chips. Tether/PNC
fabrication challenges and our solutions are discussed further in section 7.1.3.
Figure 7.1 shows two designs for a top chip with cutouts for improving mechanical isolation.
There are several distinct physical mechanisms through which these tethers might improve Qm :by approximating a single-pole PNC, by making the silicon frame a less rigid boundary condition(i.e., approximating soft clamping), or simply by reducing strain from the bowing silicon chip. The
two different designs were to test whether tethers along nodal or anti-nodal lines produced higher
isolation, but initial tests were made for both designs and each showed reliable Qm ∼ 106 . This
improvement reduces the contribution from thermal phonons by an order of magnitude.

7.1.2

Increasing electromechanical coupling through smaller gaps

We improve the electromechanical coupling rate by reducing the capacitor gap. The single
phonon-photon coupling rate for the microwave circuit isg0 =w0
Cp xzp ,
2d

(7.1)

87
where Cp < 1 is the participation ratio of the mechanically compliant capacitance to the total
capacitance of the circuit, d is the capacitor gap separation, and w0 is the LC resonant frequency.
The zero-point motion was maximized for our membrane size by shaping the metal pad to have highspatial overlap with the mechanical mode of interest. The participation ratio is set by the geometryof the LC circuit can only be improved marginally from its present value Cp = 0.8. Withoutchanging the size or thickness of the membrane or in some other way changing the microwave
circuit the most direct way to improve the electromechanical coupling is to reduce the gap size.
Increasing the frequency of the microwave circuit would also increase g0 , and tradeoffs with otherconverter parameters are being explored to determine if this would result in a net improvement.
The mechanical coupling rate g0 is doubled by reducing the capacitor gap by a factor of two
from 300 nm in previous designs to 150 nm. However, flip chips with small gap separations are more
susceptible to the capacitor collapsing. For comparison, aluminum drumhead devices collapse with
gap separations below 40 nm [89], and they are usually an order of magnitude smaller in transverse
dimension. A few test devices have been designed for 100 nm gap separation, but only one out offour tests led to a working device, suggesting that it is too ambitious a goal for reliably making a
converter. A separation of 150 nm reduces the power needed to achieve a given electromechanicaldamping by a factor of 4 and reliably produces working devices.
Moving to smaller gap separations would not have been possible without solving the reliability
issues, discussed in Section 6.2, that plagued devices designed for 300 nm gap spacings. These issues
are addressed by engineering the top chip bulk substrate and the post geometry. The position of
the posts and epoxy blobs affect how the strain due to thermal contraction is distributed overthe chip, and thus how the capacitor gap changes. These positions can be adjusted to minimize
strain near the membrane. An alternative and complementary approach to reducing strain near
the membrane is to make the top chip more compliant. One way to do this is to make the chiplarger, which provides a longer lever arm to take up the strain at the epoxy blob, and thus reduces
the local curvature. This is difficult because larger chips means that the intrinsic bow of the wafer
starts to matter and the area needed to keep clean increases. Alternatively, the silicon structure

88
chipchipdisplacementbowing

100 nm0 nm-400 nm

Figure 7.2: Relieved silicon structure accommodates shrinking epoxy. Simulation of chipbowing due to epoxy shrinking. Cutouts allow silicon to locally deform around the epoxy blobs (onthe top and bottom edges) without transferring strain to middle of the chip. The post spacing has
also been adjusted to null out motion at center of chip.

itself can be made more compliant by cutting out material, so that deformations are local and do
not extend to the middle of the chip near the membrane. In this way the epoxy is able to shrink,deforming the top chip locally, without transferring that strain to the posts or to the middle ofthe chip. We also switched to only epoxying in two places to not over constrain the ways the chipcan bend. Figure 7.2 shows a simulation of the chip bow for the final design of the top chip. The
post spacing is set to null out motion near the membrane. By keeping the epoxy on the edge butchanging the location of the posts the force on the middle of the chip can be swept from up to
down through zero, which allows us to null out deformation at the membrane.

7.1.3

Fabrication challenge and solution: silicon frames patterned with holes

The cutouts used to increase chip compliance and improve mechanical isolation rely on new
fabrication techniques in order to make them compatible with the flip-chip geometry. The new
top chip fabrication involves three etches that are deeper than a few microns: one to etch trenches
for dust, one for releasing the membrane, and the new one for the cutouts. Standard lithographic

89

Figure 7.3: One sided etch jig. Profile view. The jig uses o-rings to seal around the wafer.
When lowered into an etch solution, only one side is exposed. The equilibrium tube is for balancing pressure on the final released membranes. The concept is the same for wafers or forchips. We fabricate different jigs to accommodate different size chips. Photo from A.M.M.T
https://www.ammt.com/products/wet-etching/single-series/

techniques require relatively smooth surfaces, and deep features are usually reserved for the laststep. If that is not possible there do exist some solutions: spray-on resist and resist tape. Resist
tape did not work for our application because it could not protect the already released membranes,
and the spray-on technique was not pursued. We were able to implement two deep etches in our
initial fabrication process by using a backside etch to release the membrane so that each deep etchlithography could be done on a smooth surface.
We developed a way to protect the released membranes and allow for an additional lithographic step over the uneven surface. The solution not only allowed for the fabrication of membranes
with arbitrary cutouts in the surrounding silicon frame, but also reduced the longest single step inthe fabrication from ∼ 6 hours to ∼ 1 hr and enabled a wafer-scale process for this step. The KOH
etch that releases the membrane attacks silicon along its crystalline axis, which means that any
exposed Si will be etched and the trenched side of the wafer must be protected. In our initial fab
process, this KOH etch was the last step, and was done chip-scale with jigs to enable a one-sided

90
etch as shown in Figure 7.3. The slow etch rate of KOH (∼ 1 µm/minute through a 380 µm chip)
together with the chip-scale nature and 60% yield of this process made it painfully slow comparedto the rest of the wafer-scale fabrication. We reduced the etch time by using deep reactive ion
etching (DRIE) to etch through all but the last ∼ 10 µm of silicon. Using a one sided jig fora wafer scale etch is not possible due to the low yield. If even a single membrane breaks, KOH
can leak through and expose the entire side that is supposed to be protected by the jig. Instead,
we protect the trenched side of the wafer by bonding it to a sapphire backing wafer using wax(CrystalbondTM 509). The wax dissolves in KOH so this is not a standard practice. However, thebacking wafer confines the exposed area to a very thin surface of wax that it dissolves slowly simply
due to lack of access. Over the duration of the shorter etch enabled by the DRIE technique, only
on the outer ∼ 1 mm of the wafer is the wax dissolved and the exposed silicon etched. This is not
a problem because no chips are patterned that close to the edge of the wafer. Compared to a jig
method, a membrane breaking is not catastrophic with the wax method. If a membrane breaks the
KOH only reaches ∼ 1 mm around the membrane which is not a problem for the rest of the wafer
as seen in Figure 7.4.
After the membranes are released, the final step is to fill the membrane holes with resist sothat another lithography and etch step can dice the chips and pattern the tethers. At this pointin the fab process the wafer has deep trenches on the membrane side and holes all the way to themembrane on the other making standard lithography difficult. The released membranes are also
very fragile and need to be protected for further processing. The problem of patterning on theuneven surface is solved by unconventional lithography methods. The wafer is again bonded to asapphire backing wafer with wax, with the membrane side down for protection. The membraneholes are then filled individually with photoresist and a paintbrush. This can be a very challenging
and tedious task as a wafer can have as many as 70 chips on it and the membrane hole is 0.5 mm
on a side. This process needs to be done iteratively by applying resist, then baking, then moreresist until each membrane hole is full. If it is not done meticulously enough, bubbles build up in
the resist that burst when baking and expose the membrane to the etch. Finally, the surface is

91

pre KOH etch:post KOH etch:wax dissolved,
membrane broken

superconductor
SiN
Si
wax
saphireFigure 7.4: Profile schematic of wafer scale KOH etch. Not to scale. A sapphire wafer isaffixed with wax onto the silicon wafer of top chips. Wafer is placed in wet KOH etch. The wafer
in the diagram has three chips across for simplicity. The actual wafer has 70 chips on it. Prior tothis step the wafer has a ∼ 10 µm layer of silicon over membranes that is etched by KOH to release
them. The KOH dissolves the wax from the edges and etches the exposed silicon. Wax dissolves
slow enough that only edge of wafer is affected and chips are not compromised. If a membrane
breaks, the etch only gets through in a small region around the broken membrane and does nottouch any other chips.

smooth enough and even layer of resist can be applied using standard techniques.

7.2

Parameter noise
High-Q and low-noise superconducting microwave resonators are desirable for many appli-

cations, ranging from quantum computing [117] to particle acceleration [118] to photon detection [119]. Although the details of noise mitigation depend sensitively on geometry, reducing
noise and loss invariably involves making very clean superconductors and concentrating fields awayfrom lossy dielectrics. Therefore, fabrication of low-loss circuits with little excess noise requires
addressing each potential source by optimizing the surface cleanliness [120, 121, 122], the etchchemistry [122, 123, 124], the surface interfaces [125], and the choice of superconductor [123, 126].
For a good review of parameter noise in superconducting circuits see Gao’s thesis [127]. One com-

parameternoise_bigfont.pdf from fitting_04Jun18.pxp

Nb
NbTiN

100
Nadd (ph)

92

10
1

1

10

2

10

3

10
Γe (Hz)4

10

Figure 7.5: NbTiN vs Nb added noise. Parameter noise contribution to added noise budget
for NbTiN devices compared to Nb. There is a factor of 4 reduction for a given electromechanical damping rate. The NbTiN is not measured in a converter device but is inferred from fromnoise performance of test devices assuming constant electromechanical coupling g0 , and that the
microwave circuit performance is unaffected by integration into an optical cavity.

mon source of noise is two level system (TLS) dissipation, and therefore much of the literature isdedicated to understanding and reducing this noise source. Our circuits show some of the characteristics of TLS noise at low power, but deviated from the expectation at higher powers. The
high-power regime is where parameter noise is our limiting factor, and therefore we focus on finding
a solution for that range. Even though the origin of the parameter noise observed at high pump
powers is not understood, borrowing practices from the literature to make high-Q, low-noise resonators still helped alleviate our problem. We measured frequency noise and loss as a functionof power for a variety of devices with different circuit dimensions, etch processes, and materials,
using the number of noise photons that would be added to a conversion process as our final figureof merit. The change that had the largest positive effect was changing the superconductor from
niobium to NbTiN.
Switching our superconductor to NbTiN reduced the parameter noise for a given drive powerby 6 dB. The superconducting properties of NbTiN are very process-dependent. We follow a93
process that was developed at NIST [128] and was modified from a titanium nitride (TiN) film
deposition also developed at NIST [126]. The investigation for using nitrided super conductors was
first pursued at NIST based on work out of JPL using TiN for microresonator detectors [129]. The
deposition is done by co-sputtering pure Nb and Ti targets at 500 ◦ C in an Ar : N2 atmospherewith 4 W RF, 100 V DC self bias on the substrate. The substrate is first cleaned in HF to remove
native oxides. The deposition starts by growing a thin layer of SiN which selects the correct grain
structure of NbTiN. The Nb:Ti ratio is controlled by the RF power on each gun. This recipe has
been optimized to give a TC of 13.8 K. Because any dry etch has the potential to roughen the SiN
membrane and increase optical loss, a wet etch is used to define the NbTiN circuit. We use a SiO2
hard mask and a NH4 OH wet etch. Full fabrication details are in appendix A.
Figure 7.5 shows the projected added noise for a NbTiN device compared to Nb. For speedof testing we measured parameter noise in devices without released membranes or optical cavity
integration. Therefore the added noise for the NbTiN device is the projected added noise forconverter if the only change was switching to NbTiN.
The improvement from NbTiN can be further understood by looking at internal loss andfrequency noise in our NbTiN resonators. The first indicator that it is an improvement is seen in
the internal loss shown in Figure 7.6a. The NbTiN device has lower loss overall which is generally
good and will result in lower loss in the microwave part of the converter, but more importantly it
doesn’t exhibit the sharp increase at high powers. The sharp increase in loss is correlated to theexcess noise we observe. The low-power behaviour of the NbTiN device is characteristic of TLS
noise. As the power is increased the loss decreases at first as loss from TLSs is saturated. The loss
increases again at high power, but not as drastically as in the Nb device. The parameter noise may
also be characterized in terms of the LC circuit’s fractional frequency noise spectrum evaluatedat the mechanical resonance frequency, shown in Figure 7.6b. The fractional frequency noise isindependent of specific device parameters such as circuit linewidths and couplings and thus is amore accurate measure of the underlying material noise processes. We see that the NbTiN device
levels off to a lower value at high power. The fractional frequency noise at low powers matches whatanbtinvsnbloss.pdf
94 from fitting_04JubSδf/f at 1.5 MHz (1/Hz)

κint /2π (kHz)

-16

250

200

150
1

10

3

10

5

10

7

10

Photons in resonator

9

10

10

-17

10

-18

10

-19

10

NB
NbTiN

-20

10

2

10

4

10

6

10

8

10

Photons in resonator

Figure 7.6: Nb NbTiN loss and noise comparison. a, Internal loss as number of photons inresonator is varied. Blue trace is NbTiN. Red is Nb. b, Fractional frequency noise at 1.5 MHz asfunction of photons in resonator.

is expected for TLS noise and the deviation from this at high powers corresponds to the excessparameter noise we observe.
Prior to settling on NbTiN, we tried several different modifications to our fabrication procedure to diagnose and try to mitigate parameter noise. Here we motivate a few of the tests andhighlight results that may have not solved the parameter noise but were still helpful. Parameter
noise has been studied extensively in CPW resonators whose geometry is quite different from ours.
In the CPW resonators, the field lines mainly go through the metal-insulator interface due to thehigh dielectric constant of the substrate compared to air. In our geometry most of the electricfield lines live in the gap between the two capacitor plates and therefore pass through the metalair interface. Therefore, we worried about surface cleanliness such as dirty oxides, residues frometches, and electric field density. Most of our test devices exhibited no change in noise performance.
Some devices of interest are discussed briefly here. One was a flip chip assembled from an NbTiN
bottom chip and a Nb top chip. This had low loss at low powers but had the sharp uptick athigh powers, suggesting that the loss and parameter noise are therefore not associated only withthe bottom chip. Another device of interest used 100 nm of Nb on the top chip compared to theusual 25 nm. This chip also had lower loss than most Nb chips, but did not have improved noise

95
performance, suggesting that increasing the metal thickness on the top chip could be beneficial forreasons unrelated to parameter noise. Realizing this benefit will required careful optimization as
increasing the thickness of the top chip metal also reduces the membrane’s zero point motion and
g0 .
The contribution to the total added noise budget due to LC parameter noise has been reduced
by a factor of four by switching from Nb to NbTiN. While the factor of four improvement isobviously helpful, it will not be sufficient for quantum operation without an improvement in either
the coupling rate or the mechanical dissipation, either of which would mean it would reduce theelectromechanical pump power required to achieve the same level of thermal added noise.

7.3

Bonding technique for flip-chips and optical cavity assembly
We implemented SiO2 -Si bonding process to create etalons and flip-chips without epoxy. This

procedure allows us to produce flip-chip devices that are immune to the effects of epoxy shrinking
on the capacitor gap separation. It also produces etalons in a highly scalable way with very goodalignment between the mirror and membrane (< 1 mrad). The alignment is ensured because thebond depends on the two flat surfaces being in intimate contact, so if the bond forms the surface
will be aligned to the membrane to the degree that the wafers are flat. Removing the epoxy between
the surfaces also ensures that the etalon remains well-aligned upon cooling down.
The recipe we follow for the bonding process was developed at NIST for wafer scale bonding to
make frequency combs in suspended silicon waveguides [130]. A summary of the technique follows
and full details are in appendix A. The substrates are cleaned with an ultrasound solvent bathand an aggressive O2 plasma to remove any organics. Surface bonds are activated in a atmosphericplasma system in a N2 : H2 plasma making it hydrophilic. Substrates are rinsed in water to activateand saturate −OH groups. An initial bond is formed by Van der Waals forces between the hydroxyl
groups which is then made permanent by annealing at 150 ◦ C for an hour. The annealing processreplaces the OH-OH bonds with stronger Si-O-Si bonds. The strength of the final bond can byincreased by annealing at higher temperature for more time, but 150 ◦ C is both strong enough formirrorflip-chip

96

mirrorθ~ mradetalonFigure 7.7: Etalon is naturally aligned. Three-element cavity on left shows potential for misalignment. Two-element cavity on right consists of a mirror and a etalon. The etalon is formed
between the membrane and the flat mirror. These two pieces are structurally robust to misalignment and thus the whole thing behaves as a stable cavity.

our purposes and compatible with the metal thin films of our circuit. For example, annealing at
900 ◦ C can increase the bond strength up to the fracture strength of the wafers.

7.3.1

Etalons

Moving from a three-element cavity to a two-element cavity makes the optical resonator
more robust to misalignment and importantly facilitates making the cavity shorter. We refer to thenew design as a two-element cavity with the two elements being a mirror and an etalon as shownin Figure 7.7. The etalon is really two elements in itself, the other mirror and the membrane,
but by making them structurally the same piece, they behave like a single mirror with periodic
wavelength-dependent transmission The short cavity is important for optomechanical coupling,
while the robustness to misalignment is important for reducing cavity scattering losses. The etalon
allowed us to move back to wired microwave coupling with a shorter optical cavity than in thefirst-generation design by reducing the number of components in the physical support structure.
The etalons have shown to be much more robust to misalignment on cooling down but thiscomes at the cost of cavity tunability. Using an etalon leaves us with only one cavity tuning knob
while cold. In the three-element cavity there is a piezoelectric actuator on each mirror. One is used

/2

(MHz)

97

4
sim
meas

3
2
1
1062.51063

1063.51064

1064.5

Wavelength offset (nm)Figure 7.8: Self-aligned etalon cavity. Measured and simulated cavity linewidth as a function of
wavelength. This is equivalent to moving membrane in the standing wave of the cavity. Cavity still
has excess loss, but this loss is independent of membrane position in the optical fringe. Compare
to Figure 6.8

for locking the cavity to the laser while the other is used for moving the membrane in the standing
wave. The piezo for locking has so far been critical, and therefore adopting etalons we forfeit theability to move the membrane. An equivalent method to move the standing wave relative to themembrane is to tune the wavelength of the laser. This is the technique used to produce Figure 7.8.
The cavity linewidth as a function of laser wavelength in two-element cavities is well behaved. Thereis still the excess loss above the expected value from mirror transmissions alone, but there are nolonger narrow regions with large loss indicative of hybridization when compared to Figure 6.8. We
associate this with a more well aligned cavity.
We tried two other etalon assembly methods (shown in Figure 7.9) before settling on thebonding method discussed above. Our first attempt involved simply setting a mirror on a chipand applying epoxy to the edge. This produced etalons with angular misalignment between themembrane the flat mirror ranging from 0.1 mrad to 0.8 mrad, mainly limited by epoxy wicking
between the two surfaces. The low end of this range works for our application, but we still pushed fora more reliable and less time-consuming fabrication method. Our second approach was opticallycontacting chips to mirrors on fused silica substrates. Optical contacting relies on two surfaces

a

b

98

Figure 7.9: Etalon assembly methods. a, Epoxied etalon. No intimate contact between mirror
and chip. Epoxy wicking under leads to misalignment. b, Optically contacted etalon backed up
with epoxy. Has reliable, robust mirror-membrane alignment.

being extremely flat and clean, so that by touching them together the two surfaces stick together
by Van der Waals forces. We subsequently painted epoxy around the bonded surface to protect
the relatively weak bond during thermal cycling, and verified this epoxy did not affect the etalonalignment. This produced good alignment at reliably < 0.3 mRad, but the yield was limited dueto the strict cleanliness requirements that were difficult to achieve even in a cleanroom. Our third
approach was SiO2 bonding, which preserved the alignment reliability and works at the wafer scale,
dramatically increasing production numbers. We switched our mirror substrates from fused silica
pieces that were the size of a single chip, to 3” silicon wafers. The wafer scale bonding techniquedescribed above produced an entire set of bottom chips with mirrors attached. Flip-chips are thenproduced with the well-aligned etalon built in.

7.3.2

Chip bonding

Perhaps one of the largest contributions of this thesis has been simply reducing the numberof epoxy joints in the electro-optic converter. Epoxy has been a constant source of variability from

99

Figure 7.10: Flip-chip bonded without epoxy. Test flip-chip bonded with SiO2 posts. Noticelack or epoxy anywhere on edges or corners. No membrane in test chips. These chips were used totest microwave properties, infer gap separations, and ensure bonds survived thermal cycling.

device to device and misalignment of both the LC circuit and optical cavity, and using bonded
etalons reduced the number of epoxy joints in the optical cavity by half. The bonding method has
also allowed us to make flip-chips without epoxy. This eliminates flexing of the top and bottom
chips relative to each other on thermal cycling, improves the number of thermal cycles the chips canhandle, and reduces variability from thermal cycle to thermal cycle. The bonding technique wasdeveloped for wafer-scale bonding of pristine surfaces, but we were able to adapt it for chip-scale
fairly easily. A flip-chip device using the bond process is shown in Figure 7.10.
Bonded flip-chips have not been integrated into the full converter yet, but they have beentested for robustness and for the effect on Qm . The bond is formed over four 100 µm diameter posts.
Smaller post areas were not robust enough for handling, and larger post areas made it difficult toachieve small capacitor gap separations, probably due to dust on the posts. In order to integrate thebonding process into our chip design, we replaced the Al posts previously used to set the flip-chip
separation with SiO2 posts deposited using electron cyclotron resonance plasma enhanced chemicalvapour deposition (ECR-PECVD) deposited SiO2 . The ECR-PECVD process was chosen for its
clean, reliable-thickness depositions, but also importantly for its low-energy growth. Thermally100grown or high-energy depositions would ruin the NbTiN thin film which is constrained to be thefirst step due to its high-temperature deposition. The chips that were temperature cycled repeatedly
and showed gap separations as reliable as our best devices with the stress compensation for epoxy.
Bonded chips produced mechanical quality factors that were orders of magnitude worse thanour improved tether designs. Only one of the tested devices achieved Qm > 5 × 105 and that device
had the middle posts etched away to eliminate contact area inside the tethers. This suggests thatwhen the chips really are in intimate contact, placement of points of contact relative to the tethers
matters a lot. This was obviously not what we hoped to find, but is valuable for a few reasons.
It suggests that while bonding does limit the achievable Qm with standard chip designs, there areroutes to improve it. For example a phononic shield in the top chip inside of the joints should reduce
the radiative loss. Another more extreme example would be make the top chip much smaller andput a phononic shield outside it in the bottom chip.

7.4

Frequency-independent coupling
A method of coupling to the LC circuit that is less frequency dependent is developed. Even

with the capacitor gap improvements, the capacitance is still a relatively uncontrolled variableleading to a range of frequencies and therefore to a range of external couplings, and this variation
is exacerbated by smaller gap spacings. The coupling method determines the frequency scaling
for external coupling dependence. The wireless coupling allowed us to make smaller optical cavity
length, produced our lowest loss and most stable optical cavity. However, because the microwave
circuit is resonantly coupled to the 3D cavity, the external coupling rate to the transmission line
the external coupling rate is highly sensitive to the frequency of the LC circuit. The etalon cavity
geometry allows us to preserve the short optical cavity length while returning to wired connections
to the chip.
In contrast to the 3D cavity coupling, inductive coupling on chip can be designed to havenearly no dependence on the LC frequency. The 3D wireless cavity is resonantly coupled and101standard couplerhigh inductance coupler
0.5 mm

0.5 mm
0.2 mm
0.1 mm

0.6 mm

L = 1.5 mm

0.5 mm

L = 1.5 mm

Figure 7.11: High inductance coupler reduces frequency dependence. Designs are shown forthe bottom chip including the CPW, coupler and resonator. The left shows our standard coupler
used in wired designs. The right shows a higher inductance coupler that reduces the frequency
dependence of a variable capacitor gap separation.

therefore scales as 1/∆2 (equation 3.1). For an on-chip inductive coupler, the external coupling isκext =

1 ω 2 M 2 Z0.
L Z02 + ω 2 L2c(7.2)

Where M is the mutual inductance, Lc is the coupling inductance, Z0 is the impedance of thetransmission line, and the expression should be evaluated at ω = ωLC . Our original wired coupling
design with a small inductive loop coupler operated in the ωLc  Z0 limit, whereκext ∼

1 ω2M 2
.
L Z0

(7.3)


If the frequency changes due to a change in the capacitance then the coupling will scale as O ω 2 .
In the limit that ωLc  Z0κext ∼

Z0 M 2
.
L2c

(7.4)

In this case, if the capacitance changes then the external coupling will remain constant. In our
experiment the capacitance is an unknown quantity until we cool down, so in order to set the
coupling constant across the frequency band we should work in the regime where ωLc  Z0 . This
corresponds to making a very large coupling inductor. In practice one can’t make Lc arbitrarily
large before its self capacitance will cause it to look like a resonator and we would be back to thesame resonant coupling scheme of the 3D cavity.

noise source

noise photons
high efficiency device projected improvements

parameter noise
generator phase noise
thermal motion
laser heating

23
2
7
2

1.4
0.5
0.7
2.0

total

34

4.6

102

Table 7.1: Added noise budget including all improvements. Measured added noise budget forour high-efficiency device and projected noise budget for a device that implements all improvements.
The high efficiency device reached η = 47% and was the focus of Chapter 4. The converter efficiency
is taken to be the same for the device with improvements.

The coupling design is shown if Figure 7.11. This larger loop is not quite in the regime whereωLc  Z0 , but it still reduces the frequency dependence. The coupling is simulated and measured
to have a roughly linear dependence on ω dependence across the 4 − 8 GHz frequency band. This
allows us to make devices that all have a external coupling rate that will leave us highly overcoupled
but still sideband resolved.

7.5

Improvements summary
Although a converter experiment has not been run with these improvements, they have all

been independently tested and are ready to be implemented. A finished flip-chip with all these
improvements (except the etalon) is shown in Figure 7.12. The noise budget of the high-efficiency
device is shown in table 7.1, along with a projected noise budge for a converter incorporating allthese improvements, using a conservative estimate of the efficiency. The efficiency used to calculate
the projected noise budget is η = 47% as in our best device because even using two-element opticalcavities, we still have yet to achieve the best expected optical efficiency due to excess loss. The
expected optical efficiency corresponds to an overall efficiency of 75%, which would further reduce
the total added noise by a factor of 1.5. The parameter noise has been reduced by a factor of four
from reducing the gap size by a factor of two and another factor of four from switching from Nb to
NbTiN. Smaller gap size also reduces the contribution from generator phase noise. Improvements

103

Figure 7.12: Photo of improved transducer chip. Photo of fabricated device with NbTiN,
wired coupling, strain relief, and mechanical isolation cutouts.

in Qm reduce the contribution from thermal phonons by a factor of 10. The laser heating of the
LC circuit could improve with the switch to NbTiN or by the improvements to the optical cavity,
but since it has not been tested the value is left the same. The total of the noise budget is still
greater than one, which would preclude any quantum operation. However, the order of magnitudeimprovement is a large milestone towards quantum operation, and serves to elucidate where weneed to focus our efforts moving forward.

Chapter 8

Conclusion and Outlook

This thesis shows improvements to our microwave optical converter that allowed for thehighest conversion efficiency η = 47% and a record lowest added noise Nadd = 13 photons. The
discovery of correlations in the noise at each port led to a feedforward protocol that presents analternative path to quantum operation, even if thermal noise couples into the mechanical oscillator
faster than the conversion rate. The device was studied in depth to understand the added noise
limitations and to make improvements that address each source. The solutions found represent
significant technical progress that would reduce the added noise to nearly the quantum threshold.
With a little luck or ingenuity, this device could even operate below one photon with little to nofurther improvements than developed here.
The improvements made individually address each source of added noise and present a path
towards reducing added noise to close to a single photon. Making a microwave optical converter thatrealizes all the improvements simultaneously has not yet been achieved. The largest impedimenthas been the effect of the etalons on the microwave chips. Bonding mirrors to the bottom chip
reduces Qm and alters the capacitor gap separation. Different coefficients of thermal expansion of
the mirror substrate and coating flex the flip-chip causing the capacitor to collapse. The mirror
also reduces the mechanical quality factor by an order of magnitude (Qm ∼ 104 ) compared to theplain membranes.
There are hopeful solutions to both these problems. The collapse issue has been solved in test
devices that use Si mirror substrates with coating layers removed to match thermal contractions.

105
The mechanical dissipation can be improved again by implementing a PNC design as discussed in
Chapter 7.
The next steps involve further reducing the added noise and integrating this converter into
more complex experimental setups. Even if the added noise is still above one photon there areinteresting experiments that could be done. For instance, an ongoing effort in our group is to use
the converter to optically readout a superconducting qubit. The experimental effort would test
qubit operation in the same dilution refridgerator as an optical component, which is an importantquestion given the long term goal of linking superconducting quantum computers via optical fiber.
The converter could also be used with optical single photon counting infrastructure, which gives
access to discrete variable entanglement schemes that are more robust to photon loss and canachieve high fidelity entanglement at the cost of lower success rate without needing high efficiency
transduction [96, 97, 98, 99].

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Appendix A

Fabrication Details

Standard lithographic fabrication techniques are used to make top and bottom chips. Microfabrication techniques involve depositing or etching layers or material in selective ways. Patterning
is all done via photo lithography using a maskless contact aligner (Heidelberg MLA150). Figure A.1
shows profile schematic of fabrication process of both top and bottom chips. All fabrication was performed at the NIST Boulder Microfabrication Facility (BMF). Full etch chemistry and deposition
parameters can be found in the process sheets included in Section A.1.
The bonding technique produces etalons or chips without epoxy holding them together. Figure A.2 shows a completed flip chip with mirror bonded to form an etalon, and a flip-chip assembled
without epoxy. The bond is formed between a layer of SiO2 grown on both chips. In the etalon,
the mirror coating is capped with a thin layer of SiO2 that bonds to the silicon of the bottom chip.
The surface treatment for preparing the bond is as follows:(1) Ace/ipa sonication of both wafers, followed by SRD.
(2) Inspect/make sure no big particles.
(3) Ontos O2 clean on both wafers. Opto pr strip recipe(4) Put both wafers in Ontos H2/N2 treatment(5) SRD both wafers.
(6) Place one wafer right side up on the bonding jig.
(7) Move the flags over the wafer with the 1 toggle switch.
(8) Place the other wafer on top of the flags, upside down over the first wafer, and align the

118
flats
(9) Press together with a metal tweezers and hold for a few seconds.
(10) Retract the flags fully with the 1 toggle switch.
(11) Tap around the wafer a few times to make sure its pressed together uniformly(12) Inspect on IR cam setup.
(13) Anneal 150C for 1 hr on hotplate.
(14) Inspect again.

119

Deposit 100 nm Si3N4

Deposit ≈30 nm NbTiN and 100 nm SiO2

Pattern hard mask for NbTIN etch

Deposit ≈200nm NbTiN and pattern

Etch Si3N4 and ≈25 μm Si

Etch Si with DRIE

Deposit ≈150 nm Al and pattern

Etch Si with KOH

Etch Si ≈25 μm

Etch through wafer

Etch through wafer

Assembly
Si3N4
SiliconNbTiN
Aluminum
Epoxy
SiO2

300 μm25 μm

Assemblyd

Figure A.1: Construction details for converter chips. Profile view of fabrication process.
Layers are not to scale. Both chips start with high-resistivity silicon wafers. Layers are depositedand etched way to produce top and bottom chips with cutouts, released membrane, and LC circuit.
Final flip-chip assembled as shown with d = 150nm

120
mirror bonded to bottom chip
Si3N4
Silicon
NbTiN
Epoxy
SiO2
Mirror Coating
Mirror substratebonded flip-chip

300 μm25 μm
dFigure A.2: Bonding Assembly for flip-chips and etalon. Profile views of assembled etalon
and bonded flip-chip. Layers not to scale. Mirror is bonded to back of bottom chip to makenaturally aligned etalon. Flip-chip is bonded by SiO2 layers on posts to reduce effect of epoxy ond.

A.1

Process sheets

121

NbTiN Q Membrane Chip wfr_________________________________________________
Date/Initials:_____________
Project name: PSB020819.Membrane.wafer.NbtiN.Q.x.gds
Circle Wafer Type: Oxidized Si,

Unoxidized Si,

Sapphire, Other___________

Stepper job file:jb110815.MemLC.rwa
Layer / PM/MLA150
ONE
Image
B1
B2
TR

xic Image (1 of 6)

Process Outline
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•

Sputter NbTiN in PVD system
Deposit SiO2 in PEVCD Oxford
Pattern 1um resist
Etch SiO2 in Trion Fl
Etch NbTiN in NH4OH
Etch SiO2 in HF
Pattern 1 um resist
Cut through nitride (with axic)
Pattern 7um resist
RIE 15um Si
Glue to sapphire backing wafer (membranes down)(back side) Pattern 1 um resist
Cut through nitride (with axic)
Pattern 7um resist
Etch into silicon substrate (STS ASE) until 10 microns left
Koh, Protect with wax and backing wafer
pattern 7 micron resist
Dice/trampoline with DRIE

Thickness of:Wafer center:
Wafer edge:
LPCVD stoich. Nitride:_________________________________________
Capacitor pads (Nb):____________________________________________
Si trench depth (Si):________________________________________

122

Sputter NbTiN in big PVD System
Date/Initials________________

•
•
•
•
•

•

•
•

Stress data:____________________________________________________
Inspect wafer surface under microscope:_________________________________________________
Clean wafer in automated washer/drier
Load wafer into
Load Lock p = _____________________(< 5x10-6 ) torr
Main chamber base p = ______________(< 5x10-8 torr)
Recipe:Ti Power:
Nb Power:Time:
Pressure:
N2 Flow:
Record parameters:
RF Clean:__________(10)mTorr, _______(45)W, _______(-298)V, ________(120)sec
pre-dep: ________( )mTorr, Nb: _______( )W, _______( )V, ________( )A, ________( )sec
Ti: _______( )W, _______( )V, ________( )A,
dep: ________( )mTorr, Nb: _______( )W, _______( )V, ________( )A, ________( )sec
Ti: _______( )W, _______( )V, ________( )A,

Inspect under microscope: ____________________________________________________PECVD SiO2 in Oxford ICP
Date/Initials________________

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•
•
•
•

•

•
•

Stress data:____________________________________________________
Inspect wafer surface under microscope:_________________________________________________
Clean wafer in automated washer/drier
Load wafer into:
Load Lock p = _____________________(< 5x10-6 ) torr
Main chamber base p = ______________(< 5x10-8 torr)
Recipe:DC Power:
RF Power:Time:
Pressure:
Gas flows:
Record parameters:
RF Clean:__________(10)mTorr, _______(45)W, _______(-298)V, ________(120)sec
dep: ____________(5.7)mTorr, _______(250)W, _______(305)V, ________(0.86)A, _______(130)sec
Inspect under microscope: ____________________________________________________

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Pattern SPR 660L Resist
Date/Initials________________

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•
•

•
•

Clean masks in automated washer/drier 100s wash, 200s dry
Clean wafer in automated washer/drier
Pattern ____1____-layer resist on a wafer:a) Rinse spinner nozzle with acetone, dry with a wipe, and run program PRG-660 a few times on dummy
wafer to purge resist line.
b) Load wafer and spin P-20 primer and then SPR660L resist at 2600 rpm for 40s (run program 3A26P )c) Place wafer in hotplate at 95 C for 60 sec (load into hotplate and run).
d) Expose on stepper:i) Masks: _FCM07B4-00___ii) Job File: _jb120913.MemLC.rwa___________________________________
iii) Layer ID: _B3___________________________________________________
iv) Exposure Dose:__________ (250) mJ/cm2, Focus: _________(0.0) µme) Place wafer in hotplate at 110 C for 60 sec (load into hotplate and run program 660-PEB).
Spin-develop with MF-26A for ______________(30/30 DSP) s. Alternatively hand-develop in MF-26A, 60
sec, DI rinse 2 min, spin dry
Inspect under microscope___________________________________________Dry Etch SiO2 with Trion Fl (created 08/25/2011 by KC)
Date/Initials________________

•
•
•

•
•

•

RecipePreconditon machine by running plasma for 5-10 min without wafer
Etch wafer:
O2 ________(18) sccm
Power f:______(30)W,
CF4 _______(42) sccm
r:______( 0)W
press:______(100) mTorr
Voltage:______(-70)V

Total etch time_____(4:00)min

Inspect under microscope: ____________________________________________________
Strip resist:• SVC-150 at ~70C for 15 to 20 minutes• DI Water Rinse• Ultrasound “dirty” acetone (2 min), Ultrasound “clean” acetone (2 min), dip IPA, dip DI water, then
immediately place in rinser dryer and run.
Inspect under microscope: ____________________________________________________

Wet Etch NbTiN (created 3/01/2012 by KC)•

Wet-etch NbTiN in:(30%): 20 ml

H2O: 100 ml

H2O2 (30%): 40 ml

NH4OH

124

•
•
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•
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•

At 62 CTime:

s clear +s

Inspect under microscope: ____________________________________________________
Clean in acetone in ultrasonic bath (power setting 70): In “dirty” acetone 3 min, in clean acetone 3 min,rinse with IPA and spin dry.
Inspect under microscope: _______________________________________________________
Measure etched thickness using profilometer: ____________________________________(~100) nm

HF Etch SiO2
Date/Initials________________
Use BOE
Details:

Pattern SPR 660L Resist (created 6/10/2011 by KC)
Date/Initials________________

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•
•

•
•

Clean masks in automated washer/drier 100s wash, 200s dry
Clean wafer in automated washer/drier
Pattern ____1____-layer resist on a wafer:a) Rinse spinner nozzle with acetone, dry with a wipe, and run program PRG-660 a few times on dummy
wafer to purge resist line.
b) Load wafer and spin P-20 primer and then SPR660L resist at 2600 rpm for 40s (run program 3A26P )c) Place wafer in hotplate at 95 C for 60 sec (load into hotplate and run program 660-115).
d) Expose on stepper:i) Masks: _ ___ii) Job File: _ ___________________________________
iii) Layer ID: _B2___________________________________________________
iv) Exposure Dose:__________ (250) mJ/cm2, Focus: _________(0.0) µme) Place wafer in hotplate at 110 C for 60 sec (load into hotplate and run program 660-PEB).
Spin-develop with MF-26A for ______________(30/30 DSP) s. Alternatively hand-develop in MF-26A, 60
sec, DI rinse 2 min, spin dry
Inspect under microscope___________________________________________

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Dry Etch Si3N4 with Axic

(created 8/25/2011 by KC)

Date/Initials________________•••

Recipe JM_Nitride
Precondition machine for 5-10 minutes
Etch wafer:
Endpoint Filename (if saved):______________
CF4 _________(42) sccm
RIE forward______(150),
O2___________(2) sccm
RIE refl_______(0) W
p = ____________(40) mTorr
DC bias ________(380) V

Total time _____(240)s

Inspect under microscope: ____________________________________________________
•
•

•
•

Inspect all under microscope:_________________________________________________________
Strip resist:• Clean Asher without wafer: 50 sccm O2 50 W (subtract any offset) 3:00 min• Use Asher with wafer:
50 sccm O2 50 W (subtract any offset) 3:00 min• Ultrasound “dirty” acetone (2 min), Ultrasound “clean” acetone (2 min), dip IPA, dip DI water, then
immediately place in rinser dryer and run.
Inspect under microscope:_______________________________________________
Deck-tech to measure Nitride thickness etched: Center of wfr: _________(____) nm, Edge:________(____)
nm

Pattern 7 micron SPR 220-3.0 Resist(created 8/25/2011 by KC)

Date/Initials________________

•
•
•
•

•
•
•

Rinse spinner nozzle with acetone, dry with a wipe, and run program PRG-220-3 a few times on dummy
wafer to purge resist line.
Load wafer and spin P-20 primer and then SPR220-3.0 resist at 2500 rpm (run program 3B25P)
Place wafer in hotplate at 115 C for 90 sec (load into hotplate and run program 220-3).
Expose on stepper:o Masks: __o Job File: _ ___________________________________o Layer ID: ____TR________________________________________________o Exposure Dose:__________ (275) mJ/cm2, Focus: _________(0.0) µm
NO Post-bake!
Spin-develop on Solitec with MF-26A for ______________(60) s. Alternatively hand-develop in MF-26A,
60 sec, DI rinse 2 min, spin dry
Inspect under microscope___________________________________________

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Dry Etch Si Wafer with RIE(created 1/13/2011 by KC)

Date/Initials________________

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•

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Deep RIE:
Recipe______________ (SPECBNOD)
Etch Cycles_____________( ~25 cycles)
He leak rate______________(<5 mT/min)
Inspect under microscope: ____________________________________________________
Strip resist:• Clean Technics PE-IIA asher without wafer: 50 sccm O2 100 W (subtract any offset) 3:00 min• Etch wafer in the asher:
50 sccm O2 100 W (subtract any offset) 5:00 min• Ultrasound “dirty” acetone (2 min), Ultrasound “clean” acetone (2 min), dip IPA, dip DI water, then
immediately place in rinser dryer and run.
Inspect under microscope: ____________________________________________________
Etch depth (either dek-tek or use focus on microscope)_____________________________

Glue Wafer to Sapphire Backing Wafer with Wax for DRIE

(created 3/2/2014 KC; modified 12/16/2014 KC)

Date/Initials________________

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Clean DSP sapphire backing wafer and Si sample wafer in automated washer/drier
Ash (“wax side”) DSP sapphire backing wafer and Si sample wafer in O2 plasma, 50 W, 3 min.
Consult Dan Schmidt’s recipe cartoon for “Double spun-on wax” (available on wiki) for the following:
Spin thick wax 2:1 = wax:EBR by weight (_______rpm, _____(30) sec) using the wax spinner in bay 5(Notes: Wax should not be more than ~1 month old; Squirt wax with plastic disposable droppers (withtapered tips cut off!) to cover 2/3-3/4 wafer, then spin) on:a) sapphire backing wafer (ashed side)b) your wafer (side to be buried in wax) and
Put each wafer in its own waxer disc and bake both in vacuum oven (~25 in Hg) at _____ C (134C, ovenset to 148C) for ____(20) min
Let wafers cool (2 min) propped out of the waxer disc, than sandwich together in one of the waxer discs
Top with the weight disc and bake in vacuum (~25 in Hg) oven at ______C (134C, oven set to 148C) for_________(20min - 60 min)If applicable inspect the front side of your wafer for cleanliness and that the previously patterned resist still
looks good.
If applicable deck-tech the thickness of the previously patterned resist to make sure wax baking did notchange it: center__________________________, edge_____________________________
Spin clean Si wafer top with acetone and IPA. Ready for lithography or etching.

Dry Etch Si Wafer with DRIE

(created 1/13/2011 by KC, modified 12/29/14 by KC)

Date/Initials________________•

If etching through the Si wafer, you need a sapphire backing wafer as etch stop (see “Glue Wafer to
Sapphire Backing Wafer with Wax for Deep RIE”)

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•

Load wafer (face up) into STS Deep RIE
Set/check that T = 20 C.
Recipe specbnod:• Etch:12 sec, 130 sccm SF6, 13 sccm O2, 36 mT, Platen: 295V, 12 W, Coil: 600W
• Passivate
8 sec, 85 sccm C4F4,18 mT, Platen: 0 V, 0 W, Coil: 600W

Etch rate of Si wafer ~1 um/cycle, ~3 cycles/min(275 um wafer takes ~90 min, 425 um wafer takes ~2hr 20 min)Half-way through the etch rotate wafer by 180 deg. for more even etch.
Run
_____
_____
_____
_____
_____
_____
_____
_____
_____

Cycles (Total)_____(____)_____(____)_____(____)_____(____)_____(____)_____(____)_____(____)_____(____)_____(____)He leak rate________________________________________________________________________

Thickness left side_____________________________________________________________________________________________________________________•

Inspect and measure etched thickness with microscope:••

See “Remove Resist, Backing Wafer and Wax after DRIE”.
Also see “Remove Resist Residue after DRIE”

Remove Resist, Backing Wafer, and Wax after DRIE

Thickness center_____________________________________________________________________________________________________________________

Thickness right___________________________________________________________________________________________________(created 1/13/2011 by KC, modified 6/2/14 by KC)

128

Date/Initials________________••

Strip Deep RIE resist:• Clean Technics PE-IIA asher without wafer: 50 sccm O2 50 W (subtract any offset) 5:00 min• Etch wafer sandwich in the asher:
50 sccm O2 50 W (subtract any offset) 5:00 min• Spin clean resist side with Acetone and IPA
Inspect under microscope: ____________________________________________________

Wet Etch Silicon in KOH-Based Etchant (created 4/23/2013 by KC, modified 1/2/15 KC)
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•

Dip chips/wafer in IPA to wet them (helps brake surface tension in KOH later)
Etch chips/wafer (etching side down in beaker) in heated KOH-based etchant (KOH mix or PSE-200)* at_____(80)C for ______hr:min(etch rate of <100> wafer is about ~1 um/min at 80 C).
Dip into heated ___(55)C DI water, then dilute with cold DI water slowly and rinse/soak for ___(~5)min.
Do not dry (keep in DI water until next step.)
Soak in DI water for _______(5)min.
Then rinse/dry on spinner with DI water and IPA (can reverse order)
Inspect under microscope: ____________________________________________________
Ultrasound in dirty acetone to remove koh etch residue

Remove Resist, Backing Wafer, and Wax after DRIE

(created 1/13/2011 by KC, modified 6/2/14 by KC)

Date/Initials________________••

Remove backing wafer and clean wax:
• Heat on hotplate 125-130 C and slide off wafer/pieces from backing wafer• Soak in first acetone bath _____(20) min, then ultrasonicate in second acetone bath _____(2) min.
(Alternative: soak in acethone bath overnight, then soak in 2nd acetone bath for ____(20) min.)• Spray with ACETONE then IPA while spinning dry.
(Alternative: dip in IPA and let air dry.)
Inspect under microscope: ____________________________________________________

Paint Wet Wax in Membrane hole (created 1/2/15 by KC)
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•
•

Flip the wafer top-side down into the waxer holder (so membrane windows are up).
Dab the wax mixture into the membrane windows (back of wafer) under microscope. Try to put in justenough wax to fill the window trench. Do not overfill to avoid spilling wax over the wafer surface.
Bake with waxer holder in vacuum oven at ______(~135 C when set to 148 C) for _____(20 min). Wax
should be dry after cooling.
Inspect both sides under microscope: ____________________________________________________
Clean off wax from the alignment marks.

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Glue Wafer to Sapphire Backing Wafer with Wax for DRIE

(created 3/2/2014 KC; modified 12/16/2014 KC)

Date/Initials________________

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•
•

Clean DSP sapphire backing wafer and Si sample wafer in automated washer/drier
Ash (“wax side”) DSP sapphire backing wafer and Si sample wafer in O2 plasma, 50 W, 3 min.
Consult Dan Schmidt’s recipe cartoon for “Double spun-on wax” (available on wiki) for the following:
Spin thick wax 2:1 = wax:EBR by weight (_______rpm, _____(30) sec) using the wax spinner in bay 5(Notes: Wax should not be more than ~1 month old; Squirt wax with plastic disposable droppers (withtapered tips cut off!) to cover 2/3-3/4 wafer, then spin) on:a) sapphire backing wafer (ashed side)b) your wafer (side to be buried in wax) and
Put each wafer in its own waxer disc and bake both in vacuum oven (~25 in Hg) at _____ C (134C, ovenset to 148C) for ____(20) min
Let wafers cool (2 min) propped out of the waxer disc, than sandwich together in one of the waxer discs
Top with the weight disc and bake in vacuum (~25 in Hg) oven at ______C (134C, oven set to 148C) for_________(20min - 60 min)If applicable inspect the front side of your wafer for cleanliness and that the previously patterned resist still
looks good.
If applicable deck-tech the thickness of the previously patterned resist to make sure wax baking did notchange it: center__________________________, edge_____________________________
Spin clean Si wafer top with acetone and IPA. Ready for lithography or etching.

Pattern 7 micron SPR 220-7.0 Resist (BMF) (created 5/29/2014 by KC; modified 12/24/14 KC)
Date/Initials________________•••

Clean masks in automated washer/drier 100s wash, 200s dry
Clean wafer in automated washer/drier
On bench Wafer Coat 2:o On a dummy wafer two times spin-purge SPR220-7.0 resist and P-20:
o On your wafer spin SPR220-7.0 resist with recipe ____________________________________(3IN COAT\ 3IN-SPR220_7-P20-2500-SLOWSPINUP_KAT3)
o Bake your wafer on a hotplate with recipe ____________________________________(SPR220-7 3IN NO TOUCH KAT)o If needed remove edge bead with acetone-soaked wipe, or Q-tip while spinning on a spinner

Dry Etch Si Wafer with DRIE
Date/Initials________________(created 1/13/2011 by KC, modified 12/29/14 by KC)

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•

If etching through the Si wafer, you need a sapphire backing wafer as etch stop (see “Glue Wafer to
Sapphire Backing Wafer with Wax for Deep RIE”)•••

Load wafer (face up) into STS Deep RIE
Set/check that T = 20 C.
Recipe specbnod:• Etch:12 sec, 130 sccm SF6, 13 sccm O2, 36 mT, Platen: 295V, 12 W, Coil: 600W
• Passivate
8 sec, 85 sccm C4F4,18 mT, Platen: 0 V, 0 W, Coil: 600W

Etch rate of Si wafer ~1 um/cycle, ~3 cycles/min(275 um wafer takes ~90 min, 425 um wafer takes ~2hr 20 min)Half-way through the etch rotate wafer by 180 deg. for more even etch.
Run
_____
_____
_____
_____
_____
_____
_____
_____
_____

Cycles (Total)_____(____)_____(____)_____(____)_____(____)_____(____)_____(____)_____(____)_____(____)_____(____)He leak rate________________________________________________________________________

Thickness left side_____________________________________________________________________________________________________________________•

Inspect and measure etched thickness with microscope:••

See “Remove Resist, Backing Wafer and Wax after DRIE”.
Also see “Remove Resist Residue after DRIE”

Thickness center_____________________________________________________________________________________________________________________

Thickness right___________________________________________________________________________________________________

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Remove Resist, Backing Wafer, and Wax after DRIE

(created 1/13/2011 by KC, modified 6/2/14 by KC)

Date/Initials________________

•

•
•

•

Strip Deep RIE resist:• Clean Technics PE-IIA asher without wafer: 50 sccm O2 50 W (subtract any offset) 5:00 min• Etch wafer sandwich in the asher:
50 sccm O2 50 W (subtract any offset) 5:00 min• Spin clean resist side with Acetone and IPA
Inspect under microscope: ____________________________________________________
Remove backing wafer and clean wax:
• Heat on hotplate 125-130 C and slide off wafer/pieces from backing wafer• Soak in first acetone bath _____(20) min, then ultrasonicate in second acetone bath _____(2) min.
(Alternative: soak in acethone bath overnight, then soak in 2nd acetone bath for ____(20) min.)• Spray with ACETONE then IPA while spinning dry.
(Alternative: dip in IPA and let air dry.)
Inspect under microscope: ____________________________________________________

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Planar circuits and Bottom chip
Date/Initials:_____________

wfr_________________________________________________
Project name: fullwafer_param.x-cut.gds

Circle Wafer Type: Oxidized Si,

Unoxidized Si,

Sapphire, Other___________

Stepper job file:jb110815.MemLC.rwa
Layer / Image
PM
Note
->
|
V
SUSSPM—W1
PM
MLA150Nb/NbTiN
B1
MLA150Al (separate job for each layer)
B2

xic Image ()

Process Outline
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Sputter 200nm NbTiN
Etch bottom layer of metal NbTiN (etch in Oxford Fl)
Pattern 1 um resist
Sputter 200nm Nb lesker high temp
Sputter Al 150nm in SIS
Pattern 1um resist
Pattern Al with mF-26A
Pattern 1 um resist
Etch Nb in IPE RIE
Pattern 3um resist
Etch trenches into Si around metal (DEEP RIE)
Glue to Saph wafer
Pattern 7um resist
Dice with DRIE
Final cleans

Thickness of:Wafer center:
Wafer edge:
Bottom layer (Al):_________________________________________
Top layer (NbTiN):____________________________________________

Sputter NbTiN in big PVD System
Date/Initials________________

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Stress data:____________________________________________________
Inspect wafer surface under microscope:_________________________________________________
Clean wafer in automated washer/drier
Load wafer into
Load Lock p = _____________________(< 5x10-6 ) torr
Main chamber base p = ______________(< 5x10-8 torr)
Recipe:Ti Power:
Nb Power:Time:
Pressure:
N2 Flow:
Record parameters:
RF Clean:__________(10)mTorr, _______(45)W, _______(-298)V, ________(120)sec
pre-dep: ________( )mTorr, Nb: _______( )W, _______( )V, ________( )A, ________( )sec
Ti: _______( )W, _______( )V, ________( )A,
dep: ________( )mTorr, Nb: _______( )W, _______( )V, ________( )A, ________( )sec
Ti: _______( )W, _______( )V, ________( )A,

Inspect under microscope: ____________________________________________________

Pattern SPR 660L Resist (created 6/10/2011 by KC)
Date/Initials________________

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•

•
•

Clean wafer in automated washer/drier
Pattern ____1____-layer resist on a wafer:a) Purge if first to use spinner in a few hoursb) Load wafer and spin P-20 primer and then SPR660L resist at 2600 rpm forc) Place wafer in hotplate at 95C for 60 sec).
d) Expose on MLA150:
i) Job File: _ ___________________________________
ii) Layer ID: ____________________________________________________
iii) Exposure Dose:__________ (250) mJ/cm2, Focus: _________(0.0) µme) Place wafer in hotplate at 110 C for 60 sec).
Spin-develop in Brewer with MF-26A for ______________(30/30 DSP) . Alternatively hand-develop in
MF-26A, 60 sec, DI rinse 2 min, spin dry
Inspect under microscope___________________________________________
Film on Wafer Spin speed Thickness

Dose

Calibration

Nb

3000 rpm

1.01 μm

Nb

2600 rpm

1.12 μm 250 mJ/cm2 LRV

2270 mJ/cm LRV

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Dry Etch NbTiN with Oxford Fl (vertical wall) (created 8/25/2011 by KC)
Date/Initials________________•

Load cleaning wafer into Oxford Fluorine RIE and run recipes:__________________(10) min
CMB Ti undr less O2.prc______(1-2) min to condition
then load your wafer, use endpoint detector and run:

Recipe _________________ (CMB Ti undr less O2.prc) Endpoint Filename (if saved):______________
CHF3__________________(25)sccm~_ V
Ar__________________(100)sccm
O2__________________(2.5)sccm
For 200 nm NbTiN
p = ____________( ) mTorr
RIE = f______( ), r_______( ) W(overetch by 10%)ICP = f______( ), r_______( ) W(200)
DC bias = _____________( ) Vs(150)
T = _________( )CsTotal time = _____( )s
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•
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•

Inspect all under microscope:_________________________________________________________
Strip resist:
Clean Asher without wafer:
50 sccm O2 50 W (subtract any offset) 3:00 min
Use Asher with wafer: 50 sccm O2 50 W (subtract any offset) 3:00 min
Ultrasound “dirty” acetone (2 min), Ultrasound “clean” acetone (2 min), dip IPA, dip DI water, then
immediately place in rinser dryer and run.
Inspect under microscope:_______________________________________________
Deck-tech to measure Nb thickness etched: Center of wfr: _________(____) nm, Edge:________(____) nm

Pattern SPR 660L Resist (created 6/10/2011 by KC)
Date/Initials________________

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•

•
•

Clean wafer in automated washer/drier
Pattern ____1____-layer resist on a wafer:a) Purge if first to use spinner in a few hoursb) Load wafer and spin P-20 primer and then SPR660L resist at 2600 rpm forc) Place wafer in hotplate at 95C for 60 sec).
d) Expose on stepper:i) Job File: _ ___________________________________
ii) Layer ID: ____________________________________________________
iii) Exposure Dose:__________ (250) mJ/cm2, Focus: _________(0.0) µme) Place wafer in hotplate at 110 C for 60 sec).
Spin-develop in Brewer with MF-26A for ______________(30/30 DSP) . Alternatively hand-develop in
MF-26A, 60 sec, DI rinse 2 min, spin dry
Inspect under microscope___________________________________________

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Sputter Nb in Lesker High Temp
Date/Initials________________

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•

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•

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Stress data:____________________________________________________
Inspect wafer surface under microscope:_________________________________________________
Clean wafer in automated washer/drier
Load wafer into
Load Lock p = _____________________(< 5x10-6 ) torr
Main chamber base p = ______________(< 5x10-8 torr)
Recipe:Nb Power:Time:
Pressure:
N2 Flow:
Record parameters:
RF Clean:__________(10)mTorr, _______(45)W, _______(-298)V, ________(120)sec
pre-dep: ________( )mTorr, Nb: _______( )W, _______( )V, ________( )A, ________( )sec
Ti: _______( )W, _______( )V, ________( )A,
dep: ________( )mTorr, Nb: _______( )W, _______( )V, ________( )A, ________( )sec
Ti: _______( )W, _______( )V, ________( )A,
Strip resist:o Ultrasound in “Liftoff” for 10 minutes
o Ultrasound “dirty” acetone (5 min), Ultrasound “clean” acetone (2 min), dip IPA, dip DI water,then immediately place in rinser dryer and run.
Inspect under microscope: ____________________________________________________
Deck-tech to measure Nb thickness dep’d: Center of wfr: _________(____) nm, Edge:________(____) nm

Pattern SPR 660L Resist (created 6/10/2011 by KC)
Date/Initials________________

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•

•
•

Clean wafer in automated washer/drier
Pattern ____1____-layer resist on a wafer:a) Purge if first to use spinner in a few hoursb) Load wafer and spin P-20 primer and then SPR660L resist at 2600 rpm forc) Place wafer in hotplate at 95C for 60 sec).
d) Expose on MLA150:
i) Job File: _ ___________________________________
ii) Layer ID: ____________________________________________________
iii) Exposure Dose:__________ (250) mJ/cm2, Focus: _________(0.0) µme) Place wafer in hotplate at 110 C for 60 sec).
Spin-develop in Brewer with MF-26A for ______________(30/30 DSP) . Alternatively hand-develop in
MF-26A, 60 sec, DI rinse 2 min, spin dry
Inspect under microscope___________________________________________

136

Dry Etch in Nb with IPE RIE (created 08/25/2011 by KC)
Date/Initials________________•••

Recipe nbsf6.prc
Preconditon machine by running plasma for 5-10 min without wafer
Etch wafer:
Power f:______(____)W,
SF6 _______(____) sccm
r:______( 0)W
press:______(____) mTorr Voltage:______(____)V

Total etch time_____(____)min

Rough etch rate ~ ____________ (?)•••

Inspect under microscope: ____________________________________________________
Strip resist:• Clean Asher without wafer: 50 sccm O2 50 W (subtract any offset) 3:00 min• Etch in Asher with wafer:
50 sccm O2 50 W (subtract any offset) 3:00 min• Ultrasound “dirty” acetone (2 min), Ultrasound “clean” acetone (2 min), dip IPA, dip DI water, then
immediately place in rinser dryer and run.
Inspect under microscope: ____________________________________________________

Sputter 100nm Al in SIS System (created 3/1/2012 by KC)
Date/Initials________________(Check recent deposition rates)• Inspect wafer surface under microscope:_________________________________________________• Clean wafer in automated washer/drier• Load wafer into SIS-system:
Load Lock p = _____________________(< 5x10-6 ) torr ~15 min pump
Main chamber base p = ______________(< 5x10-8 torr)• Recipe: QubitBaseCleanAldep: qb_sub_Clean60W (60 W rf clean, 60 sec) (for cleaning blank wfr) OR
qb_sub_Clean110W_junction (420 sec) (for removing AlOx)
Data at 300 W, 5 mT:
qb_sub_Aldep100nm:
line value
380 s  ~100 nm
82
5 mT
1140 s  ~362 nm (7/12/11 kc)
230
gun 2 (Al)313 s  ~100 nm (7/13/11 kc)
240
300 W
385 s  110 nm (8/6/11 dl)
825
380 sec  ~100 nm (0.26 nm/sec)
525 s  150 nm (8/11/11 dl)
350 s  102 nm (9/3/11 jdw)
315 s  81 nm (2/1/12 fql)
317 s  101 nm (2/10/12 fql)•

Record parameters:
RF clean: _________(15)mTorr, ________(60)sec, ______/_____(60/0) forw/refl W, __________(289)V
Al pre-dep: ________(5.0)mTorr, _______(300)W, _______(280)V, ________(1.0)A, ________(60)sec
Al dep:____________(5.0)mTorr, _______(300)W, _______(280)V, ________(1.0)A, _______(350)sec•

Inspect wafer under microscope:___________________________________________________________

137

Etch Al MF-26A (created 3/01/2012 by KC)
Date/Initials________________

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Spin-develop in Brewer with MF-26A for ______________(4x45 DSP) .
Strip Resisto Sonicate in “dirty” acetone 3 min, in clean acetone 3 min, dip IPA, dip DI water, then immediately placein rinser dryer and run.
Inspect under microscope: _______________________________________________________
Measure etched thickness using profilometer: ____________________________________

Pattern SPR 660L Resist (created 6/10/2011 by KC)
Date/Initials________________

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Clean wafer in automated washer/drier
Pattern ____1____-layer resist on a wafer:a) Purge if first to use spinner in a few hoursb) Load wafer and spin P-20 primer and then SPR660L resist at 2600 rpm forc) Place wafer in hotplate at 95C for 60 sec).
d) Expose on stepper:i) Job File: _ ___________________________________
ii) Layer ID: _ ___________________________________________________
iii) Exposure Dose:__________ (250) mJ/cm2, Focus: _________(0.0) µme) Place wafer in hotplate at 110 C for 60 sec).
Spin-develop in Brewer with MF-26A for ______________(30/30 DSP) . Alternatively hand-develop in
MF-26A, 60 sec, DI rinse 2 min, spin dry
Inspect under microscope___________________________________________Dry Etch Si Wafer with Deep RIE(created 1/13/2011 by KC)

Date/Initials________________

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Deep RIE:
Recipe______________ (SPECZNOD)
Etch Cycles_____________( ~10 cycles)
He leak rate______________(<5 mT/min)
Inspect under microscope: ____________________________________________________
Strip resist:• Clean Technics PE-IIA asher without wafer: 50 sccm O2 100 W (subtract any offset) 3:00 min• Etch wafer in the asher:
50 sccm O2 100 W (subtract any offset) 5:00 min• Ultrasound “dirty” acetone (2 min), Ultrasound “clean” acetone (2 min), dip IPA, dip DI water, then
immediately place in rinser dryer and run.
Inspect under microscope: ____________________________________________________

138

•

Etch depth (either dek-tek or use focus on microscope)_____________________________

Glue Wafer to Sapphire Backing Wafer with Wax for Deep RIE(created 1/6/2011 by KC)

Date/Initials________________

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Spin thick wax (2:1 wax:EBR by weight) on a) the back of your wafer and b) sapphire backing wafer(__________ (MAX) rpm, _____________ (60-90) sec) on the wax spinner
Bake both wafers in vacuum oven, 20 min, 125 C in waxer discs
Let wafer cool, than sandwich together in the waxer disc
Top with the weight disc and bake in vacuum oven for _________(20min - 60 min), 125 C

Pattern 7 micron SPR 220-7.0 Resist On Backside (created 8/25/2011 by KC)
Date/Initials________________

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Purge line if first to use in a few hours
Load wafer and spin P-20 primer and then SPR220-7.0 resist at 2500 rpm
Remove edge bead with acetone-soaked wipe
Place wafer in hotplate at 115 C for 90 sec (load into hotplate and run program 220-7).
Expose on MLA150:
 Job File: _ ___________________________________
 Layer ID: ____________________________________________________
 Exposure Dose:__________() mJ/cm2, Focus: _________(0.0) µm
NO Post-bake! WAIT ~20 minutes for water to be reabsorbed into resist
Spin-develop on Solitec with MF-26A for ______________(80) s. Alternatively hand-develop in MF-26A,
80 sec, DI rinse 2 min, spin dry
Inspect under microscope___________________________________________
Measure resist thickness____________________________________Dry Etch Si Wafer with Deep RIE(created 1/13/2011 by KC)

Date/Initials________________

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Deep RIE:
Recipe______________ (SPECZNOD)
Etch Cycles_____________( ~400 cycles)
He leak rate______________(<5 mT/min)
Inspect under microscope: ____________________________________________________
Strip resist:• Clean Technics PE-IIA asher without wafer: 50 sccm O2 100 W (subtract any offset) 3:00 min• Etch wafer in the asher:
50 sccm O2 100 W (subtract any offset) 5:00 min• Place on spinner sapphire side down, spray with acetone then IPA then DI water
Inspect under microscope: ____________________________________________________
Etch depth (either dek-tek or use focus on microscope)_____________________________

139

Remove chips from sapphire backing wafer(created 1/6/2011 by KC)

Date/Initials________________

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Soak (typically overnight) in acetone with wafer held so that die fall onto cleanroom wipe
Place die in clean acetone bath
Sonicate 5 minutes high power
Dip IPA
Blow Dry
Inspect under microscope: ____________________________________________________

Final Clean(created 1/6/2011 by KC)

Date/Initials________________

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Soak die for 20 minutes in EKC 265 at 70C
Dip IPA bath
Dip IPA bath
Dip DI water
Place on spinner and spray with DI water, let dry (3k rpm for 4 minutes is usually sufficient)• Alternatively, spray DI water and end with a light squirt of IPA and spin dry (takes a few seconds)