Quantum measurement backaction and upconverting
microwave signals with mechanical resonators
by
R. W. Peterson
B.S., University of Minnesota, 2011
M.S., University of Colorado, 2014

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2017This thesis entitled:
Quantum measurement backaction and upconverting microwave signals with mechanicalresonators
written by R. W. Peterson
has been approved for the Department of Physics

Prof. Cindy Regal

Prof. Konrad Lehnert

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.

iiiPeterson, R. W. (Ph.D., Physics)
Quantum measurement backaction and upconverting microwave signals with mechanical resonators
Thesis directed by Prof. Cindy Regal

The limits of optical measurement and control of mechanical motion are set by the quantum
nature of light. The familiar shot noise limit can be avoided by increasing the optical power, but athigh enough powers, the backaction of the randomly-arriving photons’ radiation pressure can growto become the dominant force on the system. This thesis will describe an experiment showing how
backaction limits the laser cooling of macroscopic drumhead membranes, as well as work on how
these membranes can be used to upconvert microwave signals to optical frequencies, potentiallypreserving the fragile quantum state of the upconverted signal.

Dedication

To my parents Wendy and Bruce, and my sister Amelia

v

Acknowledgements

My years in graduate school at JILA have been both tremendously challenging and rewarding.
I have many to thank for their support, guidance, and friendship over the years, extending far
beyond when I first came to Boulder six years ago. I would begin by thanking my parents forteaching me the virtues of honesty and patience, and allowing me the profound freedom to makemy own choices as a child. My family has always supported my goals and accomplishments—I havealways felt that I’ve had the wind at my back. Of my years in high school, I am most grateful tomy teachers Charlie Leibfried and Jim Stellmaker, who showed me the beauty of music, as well as
the power of consistent practice to transform yourself.
I owe my initial development as a researcher primarily to my undergraduate advisor, Shaul
Hanany, and his students. From my time in his group, I learned how to work independently, how
to be uncompromising on details, and how to radically reject Minnesota nice. I also thank Tamara
Diedrich for the chance to try geology research and fieldwork at NRRI in Duluth, and Edwin Kan
for the introduction to nanofabrication at the wonderful Cornell cleanroom.
I could not be happier with my decision to come to Boulder, and the opportunity to work in
Cindy’s lab. Besides learning optics, I’ve learned from her how to tell a story with data, to bothclearly explain results and speak to their broader relevance. I appreciate the trust and freedom
she has given me and the rest of the lab, as well as the support we have inevitably required.
Tom patiently spent years of his postdoc sharing his arcane knowledge of cavity design, analog
electronics, and quantum measurement with me. I am also grateful to the rest of the lab—in theearly years Adam, Ben, Brian, and the rest of us fit on one bench. As the lab has grown, seating at

virestaurants has become more difficult, though far offset by the pleasure of working with Nir, Tom,
Ran, Oliver, and Max, as well as Mark, Yiheng, Tobias, and the rest on the atom experiment. It
has also been a pleasure to know and work with some of the undergraduates in the lab, among
them Christine, Alec, Josh, Randall, and Gabriel.
My work on microwave upconversion has also given me the opportunity to work closely with
Konrad and his group, from whom I have received an appreciated second dose of wisdom. I thankReed, Brad, Pete, Tim, and Andrew for teaching me the nuances of microwave photons, fillingthe fridge over the weekend, and above all being a pleasure to work with. The JILA shops haveprovided essential help throughout my PhD; the upconversion experiment has provided them withperhaps the most unresonable and technically demanding projects of my time here, which they
have nevertheless completed with their renowned skill.
The years in graduate school would not have been superlative without my friends and theircompanionship, both in Boulder and Duluth. I cherish our times together, including the night
hikes, clandestine home renovation, daily lunch in the H-Bar, sailing Lake Superior, and canoeing
the Boundary Waters. I would especially single out those that suffered the most: my roommates in
Beefcake Manor during the four years I spent there. Finally, I thank my wife Charlie for her support,
especially in the final months I spent writing up while while we planned our future.

vii

Contents

Chapter
1 Introduction

1

2 Optomechanical theory8

2.1

Derivation of the equations of motion. . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.2State-space representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3

Practical calculations: sideband cooling and RPSN . . . . . . . . . . . . . . . . . . . 13

2.4

Practical calculations: Microwave-to-optical converter . . . . . . . . . . . . . . . . . 19

2.5Classical laser noise analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.6

List of vectors and matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6.1

Single-port optomechanical system . . . . . . . . . . . . . . . . . . . . . . . . 22

2.6.2

Microwave-to-optical converter . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Optomechanical cavities - calculations and characterization

26

3.1

Cavity characterization - empty cavity . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2

Cavity characterization - with membrane . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Optomechanical cavities

40

4.1

First cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2

Our canonical optomechanical cavity design: the RPSN cavity

4.3

Later designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

. . . . . . . . . . . . 43

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4.4

Cavity components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.5

Cavity construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.6

Epoxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.7

Lab environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Electro-optomechanical cavities

57

5.1

Microwave circuits and the flip-chip electromechanical device . . . . . . . . . . . . . 57

5.2

First converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2.1

5.3

Design and control loop resonances . . . . . . . . . . . . . . . . . . . . . . . . 62

Wireless converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6 Optomechanical experiments

74

6.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.2

Cooling to the quantum backaction limit . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.3

Classical laser noise and beam preparation . . . . . . . . . . . . . . . . . . . . . . . . 80

6.4

Further analysis of classical and substrate noise effects on sideband thermometry . . 82

6.5

Characterizing the mechanical mode’s thermal bath using RPSN . . . . . . . . . . . 84

6.6

Determining the quantum efficiency of detection by measuring squeezed light . . . . 87

6.7

Historical trajectory of device performance

6.8

Comment on operation in an optical-access dilution refrigerator . . . . . . . . . . . . 91

7 Electro-optomechanical experiments. . . . . . . . . . . . . . . . . . . . . . . 90

92

7.1

Description via transfer matrix formalism, scattering parameter measurement . . . . 92

7.2

Comments on modematching in heterodyne . . . . . . . . . . . . . . . . . . . . . . . 97

7.3

Original results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7.4

Ongoing progress towards the quantum regime . . . . . . . . . . . . . . . . . . . . . 98

7.5

Future advances in detection of upconverted quantum states . . . . . . . . . . . . . . 101

ix
Bibliography

106

x

TablesTable
6.1

Ratio of radiation pressure to thermal forces in selected publications, in order ofpublishing date. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

xiFiguresFigure1.1

The 0.5 mm-sized gray square in the center of the Si chip is the transparent, 50 nm
thick Si3 N4 membrane typical of those whose motion is studied in this thesis. Photo
credit: Tom Purdy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2

Bug inside wooden replica of the great seal. Photo credit: Austin Mills, CC BY-SA .

5

2.1

Basic diagram of an optomechanical system. The incoming and outgoing light fields
are denoted by ain and aout , while the light field in the cavity (confined by thetwo mirrors) is denoted by a. The membrane’s vibrational mode is denoted by c,while the optomechanical coupling between light intensity and mechanical position
is denoted by g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2

Using parameters ωm = 2π × 10 MHz, κ = 2π × 0.4 MHz, Γm = 2π × 32 Hz,√
g = 2π × 200 N Hz, ∆ = −2π × 10 MHz, and nth = 40, three selected spectra
for N = {1, 105.5 , 106 } demonstrating undamped thermomechanical motion (red),damping near the ground state (green), and damping into the strong coupling regime(blue), where the mode hybridization is visible. . . . . . . . . . . . . . . . . . . . . . 15

2.3

Using parameters ωm = 1.48π × 10 MHz, κ = 2π × 2.6 MHz, Γm = 2π × 0.1 Hz,√
g = 2π × 15 N Hz, ∆ = −2π × 1.62 MHz, and nth = 4200, a cooling curve iscalculated, clearly showing saturation of the mechanical mode’s phonon occupationn̄ at the backaction limit nba = 0.18, replicating the theory curve in Fig. 6.7. . . . . 16

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2.4

Same parameters as in Fig. 2.3, except the pump is now on resonance (∆ = 0) andthe mode is precooled to nth = 0.6. The correspondence between N and lock powerin µW in Fig. 6.9 is not exact, influenced by the uncalibrated detuning, among otherfactors.

3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Left) Schematic drawing of a membrane-at-the-end cavity, with characteristic widthof transverse mode outlined in red. Right) Transverse mode of the optical cavity(white gaussian), with nodal lines of the (4,4) mechanical mode of the membrane
overlaid. Optomechanical coupling g is proportional to the overlap integral between
the optical transverse modeshape and the mechanical modeshape. The optical mode
has a 180 µm 1/e2 diameter, and the membrane is 500 µm square. Taken from
Fig. 1a in Ref. [19]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2

Hermite-Gaussian modes umn , which form an orthogonal basis set for transverse
modes in an optical cavity. (Image: DrBob at en.wikipedia) . . . . . . . . . . . . . . 29

3.3

Figure taken from Ref. [71]. a) Speed ∆0 of the change in cavity length/resonant frequency compared to κ. Note the ∆0 = 0.001 regime corresponding to quasistaticallytracing the cavity lineshape, the ∆0 = 1000 regime corresponding to a cavity ringdown with exponential decay, as well as the intermediate regime. b) Experimental
data motivating the investigation of fitting ringdowns in the intermediate regime. . . 32

3.4

Experimental determination of G (arbitrary units) (green, blue circles) and finesse(red triangles) for a ` = 6.3 mm, `2 = 0.5 mm optical cavity comprised of two 96 ppm
transmission mirrors and a 40 nm thick membrane. Because of systematic drift in theexperiment, first G (green), then finesse (red), then G again (blue) was measured.
The average of the two G curves’ positions corresponds to the expectations of themodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

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3.5

Model of the cavity in Fig. 3.4 where the input mirror close to the membrane isadjusted to bring the cavity into resonance. Shown are G (green), Gwrong (dashedgreen), and finesse (red). Note that the high-finesse peak is skinnier than the lowfinesse trough. Also note the alternating absolute maxima of G and Gwrong . . . . . . 38

3.6

Model of the cavity in Fig. 3.4 where the output mirror far from the membrane isadjusted to bring the cavity into resonance. Shown are G (green), Gwrong (dashedgreen), and finesse (red). Note that the high-finesse peak is wider than the lowfinesse trough. Also note that the absolute maxima of G and Gwrong occur together.
Both situations are the opposite of the simulation in Fig. 3.5 . . . . . . . . . . . . . 39

4.1

Membrane-at-the-end cavity. The membrane is positioned along the desired pointof the optical intensity standing wave, and one end mirror’s position is adjustable tobring the cavity into resonance with a fixed-frequency laser. . . . . . . . . . . . . . . 42

4.2

Early cavity designs. (top) Immediate predecessor to the device used in Ref. [19].
Cavity length ` = 10 mm, mirror radii of curvature R = 5 cm and plano. (bottom)
Device used in Ref. [19]. Note the inset curved mirror relative to (top) left, which
reduced the cavity length to ` = 5 mm. Mirror radii of curvature R = 5 cm and plano. 44

4.3

RPSN cavity. The principal design change since Fig. 4.2 is the open interior of thecavity, allowing for nondestructive replacement of the membrane inside the cavity.
Cavity length ` = 5 mm, mirror radii of curvature R = 5 cm and plano. . . . . . . . 46

4.4

Mini-mirror cavity. A realization of a cavity made from few-mm diameter mirrors
machined down from the 7.75 mm diameter substrates we bought commercially. . . . 46

xiv
4.5

Backaction limit cavity, shown in copper enclosure used in dilution refrigerator. Bothviews show an end mirror epoxied using the back glue joint (four radial holes around
optical axis), as well as a copper foil providing a thermal connection between themembrane stack and the copper enclosure, which thermally shorts the bulk of thecavity spacer. Cavity length ` = 1.65 mm, mirror radii of curvature R = 2.5 cm andplano. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.6

Cavity schematic (not to scale). Representative of the RPSN cavity, but some glue
joints have been schematically simplified. Optical axis is indicated by dashed red line.
The invar cavity spacer (gray), along with the thicknesses of the mirrors and mirror
stack components, defines the cavity length. The flat mirror stack (left) consists ofa piezo (beige), an invar ring (gray), and the flat mirror (light blue). The stack is
assembled separately and centered to the hole in the cavity spacer with an alignment
jig (not shown). The membrane stack is as depicted in Fig. 4.7. The curved mirror(light blue) is glued directly to the right side of the cavity spacer, using a glue jointthat shares characteristics of the edge glue and the back glue joints in Fig. 4.8. . . . 48

4.7

Membrane stack. (a) The membrane (blue, top) is epoxied to a machined siliconholder (blue, bottom), which is manipulated by a tapped invar holder (gray, rotation
symbol) attached to a five-axis stage. It is interferometrically aligned to the cavity
mode before being epoxied to the rest of the membrane stack. The bottom of thestack consists of the piezo (beige), as well as two invar plates (gray) that are attachedwith screws. (b) Complete membrane stack after membrane alignment and epoxying.
The five-axis stage is unscrewed from the invar holder and backed off, leaving acomplete optomechanical cavity. (c) When removal of the membrane is desired, thescrews connecting the two invar plates are removed, and the assembly from the top
invar plate up (including the membrane) is removed by hand. . . . . . . . . . . . . . 49

xv
4.8

A generic taxonomy of the glue joints used to epoxy cavity components together.
(a) The edge glue joint is the most robust, and allows for the two surfaces to bein planar contact. (b) The face glue joint is often used where radial clearance foran edge glue is not possible or ideal. It is more susceptible to delamination due
to thermal cycling or applied mechanical stress, and the layer of glue between thesurfaces can be uneven. (c) The back glue joint is an alternative to the face glue
joint that allows for planar contact between the two surfaces. It is also susceptible todelamination, and care must be taken to ensure the channels are filled with epoxy to
make full contact with the second surface. Moving the gluing surface to the backsideof one of the components can simplify assembly and reduce the complexity of thecavity interior geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.1

Photograph of the first converter, depicting the optical cavity enclosed in an open
copper box (left), and with the lid closed (right). A hole in the left side of thebox allows transmission of the free-space optical light, while the SMA connector on
the right side of the box couples microwave signals to the flip chip via a flexible
transmission line. The membrane stack is covered by the Nb shield. . . . . . . . . . 59

5.2

Rendering of the design of the first converter, depicting the assembled converter(left) and an exploded view (right) of its components. Parts/materials depicted:
invar (gray), piezo (yellow), mirror (translucent), Cu-coated invar (copper), flexible
transmission line (beige), flip chip (green), Nb shield (gray). . . . . . . . . . . . . . . 60

xvi
5.3

Detail of the first converter membrane stack (omits piezo and affixed invar plates).
The Cu-coated invar mount (right) locates the flexible microwave transmission line
(beige) and the flip-chip (green) for wirebonding necessary to complete the microwave
circuit. The Nb shield (gray) is registered to the invar mount to align its aperture to
the optical axis position. While the aperture is depicted as a circle in this rendering,by using wire electrical discharge machining techniques, more complex shapes canbe cut to precisely conform to the optical clear aperture through the membrane,
limiting scattering outside this surface. . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.4First converter schematic. (Not to scale.) The design of the first converter was
adapted from the design of the RPSN cavity in Chapter 4. The flat mirror stack (left)
consists of a piezo (beige), invar spacer (gray), and the flat mirror (light blue), while
the curved mirror is epoxied directly to the cavity spacer (gray). The membranestack is similar to the RPSN cavity in that the piezo (beige) supports two invar
plates (gray) which allow nondestructive removal of the membrane, here a flip-chipelectromechanical resonator. The complete membrane package is detailed in Fig. 5.5. 61

5.5Detailed schematic of the membrane stack for the first converter. Optic axis isindicated by dashed red line. The base is a copper-coated invar plate (orange) withraised features to precisely position the flexible microwave transmission line (red)and the flip-chip (blue). The transmission line ends in a SMA connector (darkgray). The flip-chip is covered by a Nb shield (light green) that blocks scattered
light from hitting the section of the membrane coated with capacitive Nb pads (left). 62

5.6

Transfer function of the cavity mirror assembly. Excitation is added to lockbox
output and amplified before piezoelectrically exciting the cavity end mirror stack(while cavity is locked). Excitation is read out optically via a 780 nm beam (where
the cavity has a very small finesse). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

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5.7

Transfer function of the membrane assembly. Excitation is added to the bias voltage
for the membrane piezo and piezoelectrically excites the membrane stack. Excitationis read out optically via a 780 nm beam (where the cavity has a very small finesse).

5.8

65

Design (top) and measured transfer function (bottom) of the elliptic filter tuned tonotch out a specific frequency in the stopband. The design spec is for 2.7 kHz as
shown, but typically a tunable capacitor in the LC is used to exactly match thedesired frequency. Filter designed in AADE Filter Design program (Neil Heckt). . . 66

5.9

Design concept (top) and measured transfer function (bottom) of a bandpass/notch
filter design using resonant LC shunts to filter noise at ±ωm from rf carrier signals.
Used in Fig. 6.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.10 Rendering of the wireless converter design. Parts A and B form the optical cavity,and correspond to the flat and curved mirrors, respectively. The membrane is glued
directly to C, the second half of the bottom of the microwave 3d cavity is formed
by D, and the parts E and F (together the “hat”) completely define the microwave
3d cavity. The remaining parts affix coaxial lines terminated in loops that couple
inductively to the field inside the 3d cavity. The design name comes from these
coaxial lines being wirelessly coupled to the LC circuit. Rendering from Ref. [93]. . . 69
5.11 Wireless converter with only optical cavity spacer and half of the microwave cavity
bottom installed. Optical cavity mirrors are visible, as is the circular pocket where
the flip chip is inserted and directly epoxied according to previous alignment procedures. All parts made of invar, but the microwave cavity parts are Au-coated forhigher electrical conductivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.12 The first demonstration of membrane alignment and epoxying in the wireless converter design. The microwave cavity bottoms are test pieces made of Al, not invar.
After epoxying the membrane, the curved mirror (front) was removed so the second
half of the microwave cavity bottom could be installed. After replacing the curved
mirror, the converter is ready for installation of the microwave hat. . . . . . . . . . . 71

xviii5.13 The complete wireless converter, with all optical and microwave components assembled. The two coaxial lines are for reflection and transmission measurements. . . . . 72
5.14 Wireless converted installed at the base plate of an optical access dilution refrigerator. Visible in the center of the optical cavity spacer is a lens which limits divergenceof the free-space beam coupled to the cavity mode. Author’s reflection on the Aucoated Cu base plate.
6.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Frequency-space diagram depicting optomechanical sideband cooling. The coolinglaser (orange) excites both Stokes (red) and anti-Stokes (blue) sidebands separated
by the mechanical frequency ωm , whose amplitudes are suppressed or enhanced by
the cavity susceptibility (gray, real part, FWHM κ). Due to the quantum natureof the scattering process, the red sideband rate is proportional to n + 1, while theblue sideband is proportional to n. This asymmetry is the source of the quantum
backaction limit, where the red sideband remains at a constant finite amplitude,
defining a synthetic thermal bath of the light whose temperature the mechanicalmode cannot be cooled below. Adapted from Ref. [3] . . . . . . . . . . . . . . . . . . 75

6.2

Successive regimes of optomechanical sideband cooling. While Γopt < Γm , sideband amplitude is given by cavity susceptibility while no cooling takes place. For
Γopt > Γm , the ratio of sideband amplitudes remains fixed but the mode temperature
decreases (classical regime). When Γopt = nth Γm ,, the mechanical mode approaches
the ground state (n̄ = 1). Finally, at Γopt = (nth /nba )Γm backaction and thermal forces on the system are equal. Beyond this is the quantum backaction limit,
where the equal red and blue sidebands imply the mechanical mode is in thermal
equilibrium with the synthetic thermal bath provided by the shot noise of the light.
Adapted from Ref. [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

xix
6.3

Detailed rf and laser schematic. Top: The 1064nm laser light is produced, sent
through a filter cavity, and sent through the lock AOM (acousto-optic modulator)
in a double-pass configuration to stabilize it to the optomechanical cavity. After
this, it is split into several parts which form the LO, cooling laser, and locking laser
after passing through a series of AOMs. Bottom: rf setup for producing the tones
driving AOMs 1, 2, and 3. An 86 MHz tone is filtered and split, where two paths are
modulated by ∼10 MHz tones whose difference is the cooling-locking laser detuning.
The frequency of modulation of the AOM 2 is the heterodyne frequency. This setup
has low noise near ωm , and can be easily converted to homodyne operation. Because
AOMs 2 and 3 select the opposite order of AOM 1, noise or drift in the VCO iscommon-mode and is not written onto the light. . . . . . . . . . . . . . . . . . . . . 78

6.4

Essential experimental setup. The cooling laser is injected into the optomechanicalcavity. The transmitted light is sent into a heterodyne receiver along with a local
oscillator (LO). The locking laser is injected in the opposite direction and measured
in reflection to perform Pound-Drever-Hall stabilization of the cavity. Adapted from
Ref. [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.5

Detailed optical schematic representative of both the backaction-limit cooldown, as
well as ongoing microwave to optical conversion experiments (Chapter 7). Beams
are denoted as follows: the lock beam is green, the LO beam is blue, the damping
beam (in reflection, used now for microwave to optical conversion) in red, and thesignal beam (either the damping beam moved to be injected in the rear, or a separate
beam made from splitting the LO) in yellow. Dashed mirrors are flip mirrors used
for alignment or auxiliary detection. The optical setup is on two breadboards tofacilitate access to the cryostat. The bottom breadboard is routinely removed andrelocated via position retainers on the posts that support it. . . . . . . . . . . . . . . 83

xx
6.6

Red and blue sideband amplitudes captured in heterodyne detection. Sideband amplitudes are normalized to the off-resonant level, which is assumed to be shot noise.
At low Γopt , the amplitudes differ due to the ratio of cavity susceptibility at eachfrequency, while at high Γopt they become equal when the mechanical mode is inthermal equilibrium with the light field. The last two points are below the expectation because of off-resonant excess mechanical noise which distorts the normalization.
Adapted from Ref. [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.7

A cooling curve near optimal ∆, demonstrating cooling of the mechanical mode to
the quantum backaction limit. Insets show superimposed spectra of red and blue
sidebands fro the third (left) and third-to-last (right) points on the cooling curve.
Adapted from Ref. [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.8

The minimum phonon occupation reached as ∆ is varied from near resonance to its
optimal value. Near resonance, nba diverges. Adapted from Ref. [3]. . . . . . . . . . 86

6.9

An uncalibrated measurement of RPSN. For a fixed cooling beam power (here justunder n̄ = 1), increasing the near-resonant locking beam power appears to “heat” theresonator as its backaction force on it increases. Typically, the y-axis is calibrated
in units of position spectral density, and the x-axis is calibrated relative to the SQL
power. Without calibration, it is clear that the RPSN grows to 10 times the thermal
motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.10 Cooling to the quantum backaction limit with two bath temperatures. The orange
curve was taken under the same conditions as the main data result. By increasing thelock power, the additional RPSN increases the bath temperature. By comparing thefractional increase of the lock power to the fractional increase of the bath temperature
nth , the component of the bath temperature due to RPSN can be determined. . . . . 88

xxi
6.11 Superimposed spectra of optomechanically squeezed light produced in the backaction
limit experiment. This light was collected via direct photodetection, as opposed toheterodyne detection as used in the rest of the experiment. Increasing the powerof the locking beam saturates the amount of squeezing, indicating the regime where
measurement quantum efficiency 0 dominates. By fitting these curves, 0 is determined to be 0.50, which can be related to the corresponding value  in the main
experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
7.1a.) Schematic of converter, consisting of an LC circuit and an optical cavity coupled
to the same Si3 N4 drumhead resonator. b.) Frequency-domain picture of the twoelectromagnetic resonators, the electrical and optical pump tones, and the uppersideband tone to be upconverted or downconverted. c.) Rendering of the system,
showing the optical cavity mirrors, which contain a membrane chip, and a second chip
which capacitively coupled to membrane motion via a patterned superconducting Nb
circuit. Adapted from Ref. [42]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

7.2a.) Transfer function elements denoting transmission and reflection. Gain and loss
in the measurement chain are denoted by α, β, γ, and δ. b.) Simplified experimental
schematic. Adapted from Ref. [42] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.3

Detailed schematic of the converter measurement setup. This version can be compared to the original in the supplement to Ref. [42]: this setup now has optical
heterodyne detection and a dilution refrigerator instead of a 4 K cryostat. For a
more detailed representation of the optical setup, see Fig. 6.5. . . . . . . . . . . . . . 96

7.4a.) Mechanical modeshapes of the modes used for conversion. b.) Power scan of theconversion sweeps, showing the impedance matching condition to maximize efficiency.
c.) Individual upconversion and downconversion spectra showing amplitude andphase response. Adapted from Ref. [42]. . . . . . . . . . . . . . . . . . . . . . . . . . 99

xxii
7.5

Added noise during upconversion at 4 K. Inset: Optical spectrum of upconverted
microwave signal on top of damped thermomechanical motion. Adapted from Ref. [42].100

7.6

Power sweep for first cooldown with the wireless converter design. This achieved
similar efficiency to our original converter, at approximately 10%. . . . . . . . . . . . 102

7.7Power sweep for third cooldown with the wireless converter design. This achieved
our best efficiency to date, approximately 20%. . . . . . . . . . . . . . . . . . . . . . 102

7.8

Added noise during the first cooldown of the wireless converter design. While much
improved from previous results due to operation in a dilution refrigerator, the added
noise is limited by technical noise particular to this cooldown, and is not in thequantum regime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.9

Speculative design for a double-barreled cavity, where one spatial mode (red) can be
actively stabilized via a piezo actuator common to both cavity modes, thus stabilizing
the other spatial mode (magenta) which can be scanned to the desired filter frequency
with an additional piezo actuator.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Chapter 1

Introduction“Yet even though light is so weightless we have given its name to that condition,
it presses against what it falls on, just as wind, which we cannot see, pushes thearms of a mill.
The Book of the New Sun, Gene Wolfe
The past several years have seen tremendous progress in cavity optomechanics, the field ofstudy concerning the effect of light’s radiation pressure on mechanical motion. One of the major
themes underlying this progress is the arrival of optomechanical systems in the quantum regime.
This encompasses both the quantum nature of the optical measurement process itself, as wellas quantum motion of the mechanical resonators. The arrival of optomechanical systems withparameters sufficient to encounter these regimes allowed for experimental demonstration of many
longstanding predictions of how quantum mechanics can both limit and enhance measurement [1,2]. There has been advancement along several fronts, from fundamental studies of the quantum
mechanics of position measurement, to applications of quantum measurement in sensitive position
detection such as gravitational wave observatories, to the use of mechanical motion as a link between
quantum systems across the electromagnetic spectrum, potentially linking future quantum networksof superconducting qubits, atoms, and ions.
From one point of view, the recent achievements of cavity optomechanics bring the longstanding and exquisite quantum control over atoms and ions to a new regime: tangible, macroscopic objects whose vibrational modes behave in ways that are manifestly quantum. One of the2membrane resonators used in our experiments is shown in Fig. 1.1, illustrating its visible size. In
one of the experiments discussed in this thesis, a vibrational mode of a drumhead resonator was
coupled so strongly to the light field controlling it, that its link to its physical environment became
negligible as it entered into thermal equilibrium with the quantum noise of the light bouncing off
of it [3]. In atomic physics, this corresponds to Doppler-limited cooling, something experimentallydemonstrated 31 years previously [4]. The relation of the field of optomechanics to earlier workin atomic physics on photon recoil and position measurement is explored in detail in Ref. [5]. In
addition to bringing quantum control to macroscopic objects, a unique feature of radiation pressureis that it is fundamentally a non-resonant process. Photons, no matter their frequency, all carry
momentum. Though visible or near-infrared photons may carry tens of thousands of times as muchenergy than a microwave-frequency photon, both optical and microwave light have been used in
the most state-of-the-art experiments in optomechanics. Practically speaking, both the mechanicalmode and the light are engineered into resonant structures to enhance the interaction, but the
freedom to choose the nature of these structures allows for many opportunities. This is in contrast
to atoms and ions, whose resonances are set by nature. The uniformity of these resonances allow
for applications like ultraprecise clocks, explorations of many-body physics, and building near-ideal
quantum systems from the ground up [6, 7, 8], but these experiments must use wavelengths particular to the atom or ion species to address them. In our case, the choice of wavelength was partiallybased on existing availability of low-noise laser components at 1064 nm. Different priorities mightsuggest other wavelengths, such as the broad tunability of a Ti:Sapphire laser from 700 to 1000
nm, or operation in the telecom band at 1550 nm to benefit from the absorption minimum insilica fiber-optic cables for networking applications. In this thesis, we also describe work where ourmembrane resonator is coupled to microwave light at 7 GHz, which is in the band of operation ofsuperconducting circuits, a promising platform for quantum information [9, 10, 11].
The story of optomechanics in its current form begins in early studies of gravitational wave
detection. The so-called ponderomotive effect of light, that is, its ability to damp the periodic
motion of a mechanical object reflecting or absorbing electromagnetic radiation, was first explored3

Figure 1.1: The 0.5 mm-sized gray square in the center of the Si chip is the transparent, 50 nm
thick Si3 N4 membrane typical of those whose motion is studied in this thesis. Photo credit: Tom
Purdy

4
and experimentally measured in the late 1960s [12, 13]. However, an example of an optomechanicalresonator is found earlier in history. In 1945, the Soviets presented a wooden carving of the Great
Seal of the United States to the American ambassador in Moscow (Fig. 1.2). Inside it, a diaphragm
coupled to a UHF cavity, designed by Leon Theremin, was used as a passive bug to record theambassador’s conversations [14]. The vibrations of the diaphragm modulated the cavity’s resonant
frequency, thereby transmitting sounds in the room via radio waves if the cavity was driven by adistant UHF antenna. It is an interesting coincidence that the first scientifically published workin the 1960s also used UHF resonators. Today, little work is done below microwave frequencies,
because even at the lowest practical cryogenic temperatures (¡100 mK in a dilution refrigerator),
the electromagnetic modes would have appreciable thermal excitation, precluding any quantum
effects.
Perhaps the highest-profile results of optomechanics in the quantum regime were the demonstrations of sideband cooling of a mechanical mode into its quantum ground state, using microwave
and optical cavities, respectively [15, 16]. By lowering the average temperature of the mechanicalmode, both experiments prepared a mechanical resonator in its ground state of motion with greaterthan 50% probability. These results did not approach the quantum limit of the cooling processitself, which is limited by the ever-present shot noise fluctuations of light. My work demonstrated
cooling to this quantum backaction limit, where the light’s coherent interaction with mechanicalmotion has a minimum effective temperature [3].
In general, the results during my time as a graduate student demonstrate interferometric measurements made at the quantum limit. The observation of radiation pressure shot noise(RPSN)—the fluctuating force of the randomly-arriving photons that drives additional motion
during a position measurement—represented the resolution of a series of theoretical questions into
the quantum limits of optical interferometers long thought to be relevant to gravitational wave detectors [17, 18, 19]. Subsequent work in our group showed that the correlations naturally inducedby RPSN could be used to reduce the quantum noise of light below the shot noise limit, generating
so-called squeezed light [20, 21, 22]. The first demonstrations of optomechanically squeezed light5

Figure 1.2: Bug inside wooden replica of the great seal. Photo credit: Austin Mills, CC BY-SA

6
demonstrated noise a few percent below the usual vacuum noise level, while our demonstration
increased this to 32% below vacuum noise. Squeezed light can be used as a resource to improvesensitivity beyond usual quantum limits, as has been demonstrated in gravitational wave observatories [23, 24]. These demonstrations used nonlinear media such as KTP to generate squeezed light,
the technology where the highest levels of squeezing (15 dB, or 97% below vacuum noise) have beenreported [25, 26]. However, optomechanically squeezed light is uniquely powerful because it canenhance the measurement of the same mechanical object that generated it, a situation relevant togravitational wave detectors limited by RPSN [27, 28, 29]. Our most recent result demonstrates how
such correlations can be exploited by modifying the readout phase of the measurement to improvesensitivity, a technique called variational readout [30, 28, 31]. Chapter 6 describes my measurement of the quantum backaction limit of laser cooling, and summarizes the utility of measurementsof RPSN, squeezed light, and other previous quantum measurement results for calibrating andunderstanding optomechanical systems in the quantum regime.
Another focus during my time in graduate school was developing an integrated device couplingmechanical motion to both microwave and optical fields, and the progress towards making it operate
in the quantum regime. A quantum link between microwave and optical frequencies would enablelong-distance quantum networks of superconducting circuits, since quantum states at microwave
frequencies cannot survive outside of a dilution refrigerator’s base temperature of 10 mK. Opticallyencoded quantum states, such as single photons, comfortably exist at room temperature, andapplications can leverage the technologies available in visible and near-infrared wavelengths. In
particular, the minimum of absorption in silica at 1550 nm and low-noise photonic technologies
enable long-distance transmission and detection [32], while atomic and ionic systems could form
long-lived quantum memories [33, 34]. These technologies have come together in small initialdemonstrations, and could one day constitute more ambitions quantum networks [35, 36, 37, 38, 39].
Mechanical resonators form a unique link between microwave and optical frequencies because theradiation pressure force of light works equally in either regime. Unlike nonlinear materials used
as electro-optic modulators (a common way to convert classical electrical signals into light), high7efficiencies are readily attainable [40, 41, 42, 43]. Our approach of integrating existing microwave
and optical experimental systems to share a common mechanical element led to quick progress andis described in Chapter 7.
Aside from detailing experimental results, this thesis begins with theoretical backgroundon the optomechanical interaction (Chapter 2), employing a matrix formalism to conveniently andelegantly solve the equations of motion, especially in the case of the microwave to optical converter,
where many input and output modes intractably multiply the number of equations to keep track of.
Following that is a discussion of optical cavities, including relevant calculations and measurements(Chapter 3). A one-dimensional model for calculating the effects of our vibrating membrane is alsopresented, with representative calculations and measurements.
An important aspect of my graduate work has been the design and construction of the optomechanical devices used in our experiments. They are high-finesse optical cavities with membraneresonators interferometrically aligned and epoxied into place between the cavity mirrors. Because
our experiments operate inside cryostats at temperatures between 30 mK and 4 K, these cavities
must be designed to survive thermal cycling while maintaining precise alignment between the twomirrors and the membrane. This thesis devotes two chapters to the history and evolution of our optomechanical cavity designs. The first, Chapter 4, describes the cavities used in our purely opticalexperiments, which incorporate the newest design concepts. The second, Chapter 5, describes the
optical design of the devices used to couple microwave and optical frequency light, which consistof membrane resonators coupled to both optical light and microwave fields. Benefiting from earlier
experimentation in the purely optical designs, these hybrid devices also incorporate design featuresunique to their purpose, such as coupling the microwave circuit to transmission lines for excitation
and readout without sacrificing interferometric alignment of the membrane resonator.

Chapter 2

Optomechanical theory

My goal is to present a minimally self-consistent description of the optomechanical theory
needed to explain the results in this thesis. In particular, I do not wish to compete with theseveral references available at various levels of technical detail that I have personally found useful [44, 45, 46]. I will present a derivation in truncated form, via the use of matrix techniques torepresent and solve the differential equations. This contrivance allows for rapid calculation using
symbolic or numerical computer packages. It produces exact (up to the assumptions made) expressions, which are typically much more unwieldly than the simplified expressions found in mostpapers, reviews, and theses. However, results such as the quantum backaction limit to laser coolingemerge without any additional derivations. A few demonstrations of this, and the references tocorresponding literature results should serve to demonstrate the power of this technique. I would
like to acknowledge Joe Kerckhoff for introducing this representation to those of us working on
optomechanics in Boulder, and demonstrating its power [47, 48]. It was essential for our workon microwave to optical upconversion, if only because the algebra describing so many modes was
obviously intractable [42].
While there are multiple approaches to deriving the optomechanical interaction and theresults of dynamical backaction, including classical descriptions, I will present the standard Hamiltonian formalism. This can be used to describe the dynamics of optomechanical systems in bothclassical and quantum regimes. The optical and mechanical modes used in this derivation are completely generic. While initially referring to an optical cavity with a vibrating dielectric membrane,

9
they apply equally well to microwave LC circuits with mechanically compliant components, as will
be discussed later in the chapter.

2.1

Derivation of the equations of motion
An optomechanical system is described by two harmonic oscillator modes, describing thelight field in the cavity, and the mechanical mode of the vibrating object. Canonically, one of
the end mirrors is assumed to move, although the formalism is identical to our case of a thindielectric membrane, or indeed one of the many optomechanical systems. A schematic diagram of
the following modes are shown in Fig. 2. Let the annihilation (creation) operators for these fields
be a (a† ) and c (c† ), respectively. The radiation pressure of light is proportional to the intensity,
thus the interaction Hamiltonian is the number operator a† a multiplied by the position operatorq~
x = xzp (c + c† ), where xzp =2mωm is the zero-point motion of the mode of effective mass m
and frequency ωm . The strength of the interaction is the optomechanical coupling, which describes
how much the resonant frequency of the optical cavity ωc changes with respect to motion of thecmechanical mode, G = ∂ω∂x . G has a typical scale of MHz/nm in our system. Combined with

xzp , the optomechanical coupling takes on the more general form g0 = Gxzp , which, with units of
Hz, can be directly compared to other coupling or dissipation rates in the system (in our system
g0 ∼ 10 Hz). We can write the Hamiltonian of the system (without dissipation) as
H = ~ωc a† a + ~ωm c† c + ~g0 a† a(c + c† )(2.1)

where constant terms such as the 21 in ~ωc (a† a + 12 ) are omitted [49].
To simplify the interaction, the optical field is linearized by splitting it into a classical amplitude and a remaining fluctuating term, a → α + δa. Conceptually, we anticipate that the classicalcomponent of the pump will set the scale for the magnitude of the coherent interaction, while thefluctuations will contain the effects of radiation pressure and dynamical backaction on the light.
To this end, interaction terms proportional to |α|2 represent dc shifts and can be dropped; terms
proportional to αδa capture the desired interaction and will be kept; and higher-order terms such

10

ain
aout

a
c

g

Figure 2.1: Basic diagram of an optomechanical system. The incoming and outgoing light fields aredenoted by ain and aout , while the light field in the cavity (confined by the two mirrors) is denoted bya. The membrane’s vibrational mode is denoted by c, while the optomechanical coupling between
light intensity and mechanical position is denoted by g.

11
as δa2 are negligible. It is also convenient to move into a rotating frame at the laser frequency ωL ,giving rise to a detuning ∆ = ωc − ωL . After these adjustments, the Hamiltonian takes the form√
H = ~∆δa† δa + ~ωm c† c + ~g0 N (δa† + δa)(b + b† )(2.2)

where we introduce the intracavity photon number N = |α|2 . At this point, the effects of dissipation,
driving, and fluctuations must be taken into account. This is done by moving to the Heisenberg
picture, where operators are time-dependent. The resulting Heisenberg-Langevin equations of
motion contain input and output operators, representing input fields of vacuum noise, coherent
excitation (and for the mechanical resonator, thermal noise), and output fields for modes which
will be detected. This yields the equations√√
δ ȧ(t) = −(κ/2 + i∆)δa(t) − ig N (c + c† ) − κδain (t)

(2.3)

√
ċ(t) = −(Γm /2 + iωm )c(t) − ig N (δa + δa† ) − Γm cin (t)(2.4)(as well as their complex conjugates). This forms a complete set of equations of motion, along with
input-output relations of the form
δaout (t) =√κδa(t) + δain (t)

(2.5)
(2.6)

which states that the input field is almost entirely promptly reflected with R ≈ 1, and is superimposed with the cavity leakage field. An equivalent relation can be written for cout , however for
most mechanical systems it is implausible to imagine measuring incoming and outgoing mechanicalspectra directly. An exception might be engineered acoustic structures like those used in surface
acoustic wave (SAW) devices.

2.2

State-space representation
This set of equations can be written in a condensed matrix representation unimaginatively

known as a state space model in the jargon of control theory. (Admittedly, it could be worse. It

12could be called ABCD matrix analysis.) Specifically, writing the column vectors
a(t) = (a(t), c(t), a† (t), c† (t))T(2.7)

for the internal modes of the system,ain (t) = (ain (t), cin (t), a†in (t), c†in (t))T(2.8)

aout (t) = (aout (t), cout (t), a†out (t), c†out )T(2.9)

for the input modes, andfor the output modes allows us to represent the equations of motion in the formȧ(t) = Aa(t) + Bain (t)(2.10)

aout (t) = Ca(t) + Dain (t)(2.11)

where the matrices A, B, C, and D together constitute the system. Their form is given in the lastsection of the chapter, since we can trivially extend them from the case above (a single-port opticalcavity coupled to a single mechanical mode) to a more complicated system of multiport opticaland microwave cavities coupled to the same mechanical mode, which will also be discussed below.
With the details aside, the following discussion is general to any optomechanical system making
the assumptions above.
To proceed, we begin by taking Fourier transforms of the various column vectors, for example
Z ∞a[ω] =

eiωt a(t)dt(2.12)−∞(2.13)
There are two approaches to use at this point: sticking with the state space representation allows
calculation of a susceptibility
χ[ω] = (−iωI − A)−1 B(2.14)

13
satisfying the relation a = χ[ω]ain , or moving to a transfer matrix representation to calculate atransfer function
ξ[ω] = C(−iωI − A)−1 B + D(2.15)

satisfying the relation a = ξ[ω]aout .
Now, calculation of operator spectra are straightforward. The spectrum of the internal modes,
S[ω], is given byS[ω] = ha[−ω]] a[−ω]T i(2.16)

where ] denotes elementwise conjugation (and T , as usual, transposition without conjugation). By
using the relations between vectors above, this can be written
S[ω] = χ[−ω]] hain [−ω]] ain [−ω]T iχ[−ω]T = χ[−ω]] Σχ[−ω]T(2.17)

where we introduce the covariance Σ, containing expectation values of the input noise operators.
For example, the mechanical bath, at temperature nth , gives hcin [ω0]† cin [ω]i = 2πδ(ω − ω0)nth andhcin [ω 0 ]cin [ω]† i = 2πδ(ω − ω 0 )(nth + 1). For the optical modes, an equivalent expression is obtainedby setting nth = 0, corresponding to the situation where only vacuum noise is present. We cancalculate the spectrum of the output modes, Sout [ω], equivalently,
Sout [ω] = ξ[−ω]] Σξ[−ω]T(2.18)

Individual elements correspond to the spectra of the various internal or output operators. For
example, in the system described for the derivation, S2,2 /2π = c[ω]† c[ω] is the phonon spectrum of
the mechanical mode, which can be integrated over to give the phonon occupation n̄.

2.3

Practical calculations: sideband cooling and RPSN
At this point, we can replicate two representative results that illustrate the power of thismethod. The first is the phonon spectrum for a device in the resolved sideband regime (2ωm > κ/2),
showing the phonon spectrum as it is damped into its ground state, and continuing until the14√strong coupling regime (g0 N > κ). Using parameters representative of the first experimental
demonstration of ground state cooling (Ref. [15]), we can recreate Fig. 3b in that work (Fig. 2.2).
While the specific physics of strong coupling are included in the model, they will not be explainedsince they are of no further relevance to this thesis. Satisfaction will thus necessarily focus on how
straightforward it was to calculate unrelated results using versatile machinery.
The second result is the quantum backaction limit of sideband cooling. By selecting parameters of my result in Ref. [3], which is in the intermediate regime of sideband resolution, thebackaction limit clearly appears in a cooling curve (Fig. 2.3) which replicates the result in Fig. 6.7.
Both examples represent situations where the classical sideband cooling regime is complicated byadditional effects. While they are both, of course, logical consequences of the relatively straightforward theory above, discussing them usually involves bringing in “extra” arguments on top of theclassical sideband cooling picture that is typically used to heuristically explain results. Which ofthese pictures is more useful depends on context. It is fair to observe that when using the matrix
formalism, it is difficult to collect and condense analytical expressions which are simplified enoughto meaningfully explain what is happening on their own.
For the sake of clarity and explanation, the typical analytical results will also be presented.
Perhaps the most relevant to this thesis is the mechanical mode’s response to optical damping,
resulting in the rate equationn̄(Γopt ) =n0 Γm + nba Γopt
Γm + Γopt(2.19)

where the backaction limit nba is given by the expressionnba (∆) = −(ωm + ∆)2 + (κ/2)24ωm ∆(2.20)

which is solely a function of sideband resolution of the cavity. However, this statement belies
its origin in shot noise fluctuations which the cavity merely filters to set an effective temperature
nba [50, 51]. In general, one can add additional optical damping terms to the rate equation (Eq. 2.19)
if more lasers are incident on the cavity. A prime example is in the case of our radiation pressure shotPhonon spectrum (quanta Hz- 1 )

15

1000

1

10 - 3

- 12

- 11

- 10

-9

-8

Frequency (MHz)Figure 2.2:
√ Using parameters ωm = 2π × 10 MHz, κ = 2π × 0.4 MHz, Γm = 2π × 32 Hz, g =2π × 200 N Hz, ∆ = −2π × 10 MHz, and nth = 40, three selected spectra for N = {1, 105.5 , 106 }demonstrating undamped thermomechanical motion (red), damping near the ground state (green),
and damping into the strong coupling regime (blue), where the mode hybridization is visible.

16

Phonon occupation n

10

1
0.5

0.1
100

1000

10 4

Optomechanical damping rate Γopt (Hz)Figure 2.3: Using
parameters ωm = 1.48π × 10 MHz, κ = 2π × 2.6 MHz, Γm = 2π × 0.1 Hz,√
g = 2π × 15 N Hz, ∆ = −2π × 1.62 MHz, and nth = 4200, a cooling curve is calculated,clearly showing saturation of the mechanical mode’s phonon occupation n̄ at the backaction limit
nba = 0.18, replicating the theory curve in Fig. 6.7.

17
noise (RPSN) experiment, where an additional near-resonant beam was used to drive mechanicalmotion via a fluctuating radiation pressure force [22]. Despite nba diverging as ∆ → 0, the productnba Γopt does not, leaving this as a useful heuristic to also understand RPSN, at least in the casewhere there is an additional red-detuned laser to damp and readout the mechanical motion. For
technical reasons, this is almost always the case in our experiments, as it provides a constantamount of damping that overcomes any instability that may arise if the near-resonant laser isslightly blue-detuned, and thus anti-damps or amplifies mechanical motion to some degree.
Another notable calculation is the effect of RPSN on the mechanical mode. Increasing thepower of a resonant pump will eventually drive the mechanical motion, increasing its apparent
temperature. By setting ∆ = 0 in the model, we can readily calculate a situation similar to
Fig. 6.9, where the mechanical mode temperature is monitored as the resonant pump power isincreased until its effect on the resonator is roughly ten times that of the original thermal motion(Fig. 2.4).
In both cases, the mechanical spectrum has been calculated directly. Of course, the only
experimentally accessible way to infer knowledge of the mechanical state is to measure an optical
output mode. After accounting for realistic conditions such as being coupled to an internal loss inthe cavity or a second cavity mirror, the simple expressions for the output spectrum can be readily
modified to yield the appropriate result. While interpreting this output spectrum in general requires
some form of calibration to infer the mechanical state (our old default approaches are detailed inRef. [19]), the recent emergence of sideband asymmetry as a measurement technique simplifies theconceptual considerations greatly [52, 53, 54, 55, 56, 3]. In the classical cooling regime, wheren̄ > {1, nba }, the ratio of red and blue mechanical sidebands, regardless of the attenuation of theoutput signal, is given simply by the ratio of the cavity susceptibility evaluated at each sideband
frequency. While in direct photodetection and homodyne the upper and lower mechanical sidebands
are folded together, in heterodyne they are separately visible and their amplitudes can be measured
to perform mechanical thermometry. Specifically, one monitors the shift in sideband amplitudes
that occurs as the system enters the quantum regime. At this point, the red and blue sidebands,

Phonon occupation n

18

5

2

1

0.5
1

10

100

1000

10 4

Signal beam photon number N
Figure 2.4: Same parameters as in Fig. 2.3, except the pump is now on resonance (∆ = 0) and themode is precooled to nth = 0.6. The correspondence between N and lock power in µW in Fig. 6.9
is not exact, influenced by the uncalibrated detuning, among other factors.

19
whose scattering rates depend on n̄ + 1 and n̄, respectively, will change. For a system with thelaser at ∆ = 0, the previously equal sidebands will become asymmetric, directly proportional to then̄ratio n̄+1
. For a detuned laser with ∆ < 0, the ratio of the new asymmetry to the old asymmetryn̄
is proportional to n̄+1
. Additionally, sideband asymmetry provides information on the quantum

efficiency of detection  is available, since the magnitude of the asymmetry induced by the 1 inn̄n̄+1 , compared to the amplitude of shot noise in the heterodyne receiver, is a direct measure ofwhat fraction of the intracavity signal was detected. In sum, sideband asymmetry, when available,
is a straightforward tool to infer the state of the mechanical mode that is insensitive to loss in thedetected signal requires little additional calibration.

2.4

Practical calculations: Microwave-to-optical converter
One of the most technologically relevant applications of optomechanics may be to convert

between microwave and optical signals at the quantum level, using the coherent optomechanical dissipation to overwhelm mechanical decoherence that might otherwise spoil fragile quantum information. The mechanisms for this conversion have undergone substantial theoretical
study [57, 58, 59, 60, 61, 62, 63, 64, 65, 66], and are related to similar approaches using theelectro-optic effect [67, 40, 41]. In order to describe such a converter, we modify the derivations
above to a more general case, where the mechanical mode of interest is separately coupled to twoelectromagnetic resonators. In our case, they will be an optical mode at 1064 nm and a microwave
resonant circuit at 7 GHz, but for the purposes of this calculation that distinction is not relevant.
We begin by briefly writing a new Hamiltonian, and anticipating how the vectors and matrices
describing this system will differ from before. We note that this particular derivation was where
the state space model was first of immense benefit to us. This derivation follows one given in the
supplementary material to Ref. [42]. The expanded Hamiltonian is given by
H = ~ωo a† a + ~ωe b† b + ~ωm c† c + ~Go (a† a)x + ~Ge (b† b)x(2.21)

20
where we use the reserved letter b to denote the annihilation operator of the microwave cavity.
Switching to a more uniform system of labels, the mode frequencies ωi (and other quantities) areindexed by i = {o, e, m} to denote optical, electrical, and mechanical. We also introduce additional
ports in the optical and microwave resonators: let the total linewidth of the optical resonator beκo = κo,L +κo,R +κo,int , comprising the left mirror (the port that is used during measurement of theconverter), right mirror, and internal loss. The microwave resonator has two ports with linewidthκe = κe,R + κo,int , and the mechanical resonator has a single port with linewidth κm (which forsymmetry in this context we have used instead of Γm ).
The (now more numerous) internal modes of the system are described by the vector a =(â, b̂, ĉ, â† , b̂† , ĉ† ). The much more numerous input and output modes ain and aout , along with thematrices A, B, C, and D are detailed in the last section of this chapter. In this case, using thetransfer function formalism will be the focus of this analysis. This formalism reflects our measurement, where we can apply fields (i.e. modify ain ) and measure fields (i.e. measure components ofaout ). The upconversion transmission coefficient of our converter t1 (ω) corresponds to the transfer
function element ξaout,L ,bin,R (ω) = aout,L (ω)/bin,R (ω). In the weak-coupling (κe , κo  Γe , Γo ) limit(which our experiment satisfies), it can be simplified to√r√κe,R κo,LΓe Γot1 (ω) = ξaout,L ,bin,R (ω) = −
A exp(iφ)
0κe κo −i(ω − ωm ) + (Γe + Γo + κm )/2(2.22)

0 = ω + Ω + Ω , where Ω and Ω are the electro- and optomechanical frequency shifts
Here, ωmmeoeo
0 as simply ω ). Γ and Γ are the electro- and optome(which we will otherwise omit, writing ωmmeochanical damping rates (see last section). The gain term A and the associated phase shift φ aregiven by 0 )2 + (κ /2)2
0 )2 + (κ /2)2(∆e − ωm(∆o − ωmeo×(2.23)
00−4∆e ωm−4∆o ωm




κe /2
κo /2
φ = arctan+ arctan(2.24)
00∆e + ωm∆ o + ωm
2  

2 
κe
κo
0
For ∆e = ∆o = −ωm , A simplifies to 1 + 4ω0
1 + 4ω0, giving it a simple interpretation


A = Ae × A o =min terms of the resolved-sideband parameter.

m21
Reflection measurements are described generically by the following transfer function element:ξaout,L ,ain,L (ω) = − 1 −κo,L−i(ω − ωm − Ωe − Ωo B) + (Γe + Γo B + κm )/2−i(ω + ∆o ) + κo /2−i(ω − ωm − Ωe − Ωo ) + (Γe + Γo + κm )/2(2.25)

where B = (1 −κo,Liκo,L /2∆o )/(1 − −i(ω+∆o )+κo /2 ) is a resonant enhancement factor of order unity.

Equivalent expressions for other electromagnetic output fields can be obtained via substitution of
appropriate parameters. As an aside, this is a case where algebraic simplification of the exact result
does not quite give a satisfactory answer—in particular, the factor B and its associated expression
defy useful interpretation.
Another path through the system—used for signal-to-noise measurements—is from the mechanical bath input to an optical output:
√



r
p
κo,L
κo /2
Γo κm
ξaout,L ,cin (ω) = −i 
Ao exp i arctan(2.26)
0 ) + (Γ + Γ + κ )/20
κo −i(ω − ωm∆o + ωmeomFrom these, and consideration of how to calculate the operator output spectrum in earlier
sections, we can write the signal to noise ratio (SNR) of an upconverted signal:
!
0)
2πδ(ω
−
ω
SNR = Sin (ω) |ξaout,L ,bin,R (ω)|2 †haout,L aout,L i(2.27)

where Sin is the spectral density of the input microwave field and the second term is noise at theoutput of the converter referred to the input. If we assume all fields in ain except ĉin and ĉ†in are that
0 and Γ = Γ , the expression for SNR simplifies
of vacuum, and additionally use ∆e = ∆o = −ωm
e
o

to:
"



 #−1
κe,R κm nenv
κo 2
κe 2
SNR = Ae++× Sin (ω)
0
0
κe
Γe
4ωm
4ωm

−1
κm nenv= Ae ηe+ (Ae − 1) + (Ao − 1)× Sin (ω)Γe
In this case the added vibrational noise, nvib , is found through SNR = Sin /nvib .

(2.28)(2.29)

22

2.5

Classical laser noise analysis
An important check on the results of any experiment claiming to see quantum correlations andeffects in continuous measurement is to assess the potential impact of classical amplitude and phasenoise on the measurement. Classical noise on the laser can drive an optomechanical system via
radiation pressure as readily as quantum noise, and in many systems can be large enough to swamp
any quantum signal. Calculations to capture the effect of classical laser noise are conceptuallystraightforward: add classically-correlated amplitude and phase noise terms to the description of
the incoming pump laser alongside the quantum noise operators. The difficulty lies in the more
complicated algebra, and in the end identifying and collecting terms in the calculation which clearly
elucidate classical noise’s effect on both mechanical motion and the optical measurement of that
motion. A history of useful published calculations and discussion of classical noise in our group
includes Refs. [19, 22, 68, 54, 3], including supplemental materials. Other especially pertinent
sources include Refs. [69, 70, 21, 53]. I apologize for not providing additional technical detail on
the effects of classical laser noise, but humbly submit that writing an additional technical documentwith insidious factor-of-two and transcription errors may be of little benefit to the community.

2.6

List of vectors and matrices

2.6.1

Single-port optomechanical system





−ig
0
−ig
 iωm − Γm /2





−ig
i∆ − κ/2
−ig
0




A=



0
ig
−iωm − Γm /2
ig




ig
0
ig
−i∆ − κ/2

(2.30)p√ p√
B = Diag( Γm , κ, Γm , κ)

(2.31)

23

2.6.2

C=B

(2.32)

D = −I(2.33)

Σ = Diag(nth , 0, nth + 1, 1)(2.34)

Microwave-to-optical converter√√
In the expressions presented below, ge = Ge xzp ne and go = Go xzp no are electro- andoptomechanical couplings in Hz. The electrical pump produces a classical intracavity field describedby β, where |β| =√ne and Arg[β] = θp . Likewise, the optical pump produces a classical intracavity

field described by α, where |α| =√no and Arg[α] = φp . We also use nth,m to describe the mechanicalbath phonon occupation, and use nth,e (nth,o ) to describe the average photon occupation of theelectrical (optical) input fields, which should both be zero at dilution refrigerator temperatures,
but are included for completeness and for comparison to previous results at 4 K.

∆ e + ωm∆e − ωm+(∆e + ωm )2 + (κe /2)2 (∆e − ωm )2 + (κe /2)2
∆o + ωm∆o − ωm2Ωo = go+(∆o + ωm )2 + (κo /2)2 (∆o − ωm )2 + (κo /2)2


κe
κe
Γe = ge2−(∆e + ωm )2 + (κe /2)2 (∆e − ωm )2 + (κe /2)2


κo
κo
2
Γo = go−(∆o + ωm )2 + (κo /2)2 (∆o − ωm )2 + (κo /2)2Ωe = ge2(2.35)(2.36)(2.37)(2.38)24i∆o − κ2o−ieiφp g

0
0
o




0
i∆e − κ2e−ieiθp ge0 −ie−iφp go −ie−iθp ge − κ2m − iωm −ieiφp go
A=


0
0
ie−iφp go−i∆o − κ2o




0
0
ie−iθp ge0ie−iφp goie−iθp ge
0
ieiφp go





M =



√

κo,L

√

κo,R

√

−ieiθp ge−ieiθp ge00ie−iφp go−i∆e − κ2eie−iθp geieiθp geiωm − κ2m0 Mκo,int

0

0

0

0

0

√

0



















(2.39)(2.40)



0



0



 M
B=
0


−ieiφp go0 κo,L √ κo,R
0


√

0
κe,R


C=

0
0




0
0


0
0

0
√

κe,R

0
√

0

κe,int

0

0

√
κm

0








(2.41)


0

0

0

0

0

0

0

0

0

0
0
0

√
√

κo,L

0

κo,R

0

0

√

κe,R

0 


0 



0 


0 



0 


0


0 0 0
 −1 0 0 0 0 0 0 0 −1 0 0 0 0 0
0 0 0



0 0 −1 0 0 0
0 0 0
 0
D=

 0
0 0 0 0 0 −1 0 0 0



 0
0 0 0 0 0 0 −1 0 0

0
0 0 0 0 0 0
0 0 −1(2.42)


0 0 


0 0 



0 0 


0 0 



0 0 


0 0

(2.43)

25


 N
Σ=
0


0
N +I

(2.44)








0
0
0
0
0 
 nth,o




 0

n
0
0
0
0
th,o






0
0
n
0
0
0


th,o


N =

 0
0
0
nth,e
0
0 






 0

0
0
0
n
0
th,e




0
0
0
0
0
nth,m


ain =

âin,L âin,R âin,int b̂in,R b̂in,int ĉinaout =âout,L âout,R b̂out,Râ†out,Lâ†out,Râ†in,Lb̂†out,Râ†in,Râ†in,intb̂†in,R(2.45)b̂†in,int

ĉ†in

T
(2.46)

T(2.47)

Chapter 3

Optomechanical cavities - calculations and characterization

The simplest one-dimensional description of an optical cavity has the end mirrors serve as
nodes of the electric field. Since the light inside the cavity is the sum of both left and rightpropagating waves, a standing wave is formed whose intensity is periodic every half-wavelength.
Thus an optical cavity is “on resonance” if its length ` is exactly an integer number of halfwavelengths long. Off resonance, no mode is supported. This suggests two possible ways of making
an optical cavity in practice: The first is to adjust the length of the cavity via some feedback
mechanism until the cavity is in resonance with a fixed-frequency laser. The second is to tune thefrequency of the laser until its wavelength is such that the fixed cavity length is now an integer
number of half-wavelengths. This chapter will present the properties of optical cavities and themethods used to characterize them in the lab.
The simple one-dimensional model in the introductory paragraph does not tell us about
the transverse profile of the cavity mode. However, coupling light to mechanical motion requiresthat the transverse mode spot probe specific parts of the moving material. Otherwise, differential
mechanical motion in the transverse plane can cancel out the optical signal. Fig. 3.1 shows aschematic view of the transverse mode inside a cavity, as well as a typical optical transverse mode
imaged in transmission. To describe this mode, we will consider propagation of light in the paraxial(or small angle of divergence) limit.
The geometry of an optical cavity mode is described by the solution to the paraxial Helmholtz

27

Figure 3.1: Left) Schematic drawing of a membrane-at-the-end cavity, with characteristic width oftransverse mode outlined in red. Right) Transverse mode of the optical cavity (white gaussian), withnodal lines of the (4,4) mechanical mode of the membrane overlaid. Optomechanical coupling g is
proportional to the overlap integral between the optical transverse modeshape and the mechanicalmodeshape. The optical mode has a 180 µm 1/e2 diameter, and the membrane is 500 µm square.
Taken from Fig. 1a in Ref. [19].

28equation, propagating in the z direction. Here I take the lowest-order Gaussian transverse profile,

 


r2
w0
−r
exp −i kz − k
2R(z) − φ(z)(3.1)
E = E0 x̂
exp
w(z)
w(z)2
w(z)2
where k = 2π/λ is the wavenumber, w(z) is the 1/e2 power radius of the beam, with the waist
w0 = w(0), R(z) the radius of curvature of the wavefronts, and φ(z) the Gouy phase. An opticalcavity can be thought of as the placement of mirrors to act as nodes of the electric field along
two phase front surfaces of this mode, in analogy with the simple construction at the start of thechapter. For example, a plano mirror can be placed at z = 0, where the phase fronts are planar,
and a curved mirror with radius of curvature equal to R(z) can be placed a distance z away. The
volume between them is a cavity mode whose intensity profile is proportional to E∗ · E.
Stable optical cavities are described by a set of TEM modes, which describe both the longitudinal and transverse dimensions. For our purposes, it is convenient to describe the transverse
modes using Hermite-Gaussian modes umn , shown in Fig. 3.2. The lowest-order transverse mode issimply the gaussian profile of the TEM00 mode described in Eqn. 3.1. The rectangular symmetryof these modes corresponds to the higher-order modeshapes observed in our cavities due to tiltingof the membrane, which breaks cylindrical symmetry. The angular frequencies ωqnm of the cavity
mode resonances are given by√
arccos ± g1 g2
πcωqnm = q + (n + m + 1)
×
π
`

(3.2)

Here, q denotes the longitudinal mode, which for the lowest-order transverse mode m = 0, n = 0, isincremented in angular frequency units of πc` , called the free spectral range (FSR). In non-angularcunits (as it is unfortunately often quoted), FSR = 2`
can be clearly seen to be the inverse of

the round-trip time for light to traverse the cavity. The Gouy phase shift means that the higherorder (m, n) transverse modes are in general nondegenerate in frequency. This spacing, in units of√arccos ± g1 g2
× FSR, is the transverse mode spacing or TMS. Here, g1 = 1 − R`1π

and g2 = 1 − R`2describe the cavity geometry. In the regime discussed in this thesis, cavities are relatively near theplanar-planer limit (Ri  `), and one mirror is plano (R2 = ∞, g2 = 1), so transverse modes arealways nondegenerate.

29

Figure 3.2: Hermite-Gaussian modes umn , which form an orthogonal basis set for transverse modes
in an optical cavity. (Image: DrBob at en.wikipedia)

30
The specific equations describing a Gaussian beam are useful for routine calculations. The
πw2

2 /z, where z =0radius of curvature is given by R(z) = z + zRR
λ is the Rayleigh range. Theq1/e2 intensity beam radius is w(z) = w0 1 + ( zzR )2 . In practice, a cavity’s waist can be calculated

by solving for w0 in the equation R(`) = Rmirror , since typically the radius of curvature of thecurved mirror Rmirror is well-known. The cavity length can be measured directly via examining thetransverse optical modes of the cavity. With a known radius of curvature mirror, the length canbe directly computed from the ratio of TMS to FSR, using Eqn. 3.2.

3.1

Cavity characterization - empty cavity
The most basic property of an optical cavity is its FWHM linewidth κ, the inverse of the cavity

decay time τ . Practical measurements of κ involve either slowly scanning across the cavity resonancein frequency to trace out the lineshape, or from a cavity ringdown to observe the exponential decay.
The ringdown is the simplest, most direct measurement, because it doesn’t require an externalfrequency reference, being a time-domain measurement. The cavity is probed with light, and thenthe light is cut off either with a fast switch or by moving the laser frequency off cavity resonancevery quickly. Scanning an end-mirror piezo at kilohertz frequencies with sufficient amplitude istypically able to achieve a fast enough velocity to clearly see and fit to the exponential decay of thelight in the cavity. In Fig. 3.3, calculations from Oliver Wipfli’s Master’s thesis (Ref. [71]) show
the lineshapes from the two regimes discussed above. For determination of κ in the slow scanningregime, known frequency markers (usually from an electro-optic modulator) must be placed on thelight to scale a fit to the lorentzian lineshape. Fully in the ringdown regime, a simple exponential
fit will suffice. The use of the fit parameter τ = 1/κ from either method will be discussed in
the following paragraph. For data in the intermediate regime of Fig. 3.3, consult Ref. [71] for thederivation of the nonlinear equation of motion and comments on fitting it to experimental data.
The cavity linewidth varies with cavity length, as the round-trip time increases. The cavity

31
finesse F, proportional to the power build up, is given by
F=

2π
ΣLi

(3.3)

where the Li describe loss channels in the cavity, which includes mirror transmissions as well as
any internal loss due to scattering or absorption in the mirrors or objects inside the cavity such asa membrane. The finesse is related to linewidth via the free spectral range (FSR)
F=

F SR
πc 1
=
κ
` κ

(3.4)

where I will now insist that the FSR and κ are given in angular units. For an empty cavity,
the total cavity loss is likely to be dominated by the mirror transmissions as long as the mirrors
are clean, that is, ΣLi = TL + TR = 2T , in the case of a symmetric cavity whose left and rightmirrors have transmissions (TL , TR = T ). In our case, T = 100 parts per million (ppm) is atypical value, corresponding to F = 31, 000. Via repeated measurement of cavity linewidth after
successive cleaning procedures (using First Contact from Photonic Cleaning Solutions), the mirror
transmission can be inferred with confidence without requiring the more involved measurements
described below.
While the total linewidth is straightforward to measure, the determination of the loss rates
through individual mirrors and internal loss is more complicated to experimentally investigate. In
general, one wishes to know the transmission rates of each mirror, TL and TR , and the internal loss
L, corresponding to scattering or absorption due to substrate surface roughness or imperfections in
the dielectrics of either the mirror coating stacks or (if present) the membrane’s dielectric material.
A method for determining these quantities for an optical cavity with two identical mirrors using
only reflection and transmission measurements is given in Ref. [72]. A rederivation reproduced here
from the supplementary material of our work in Ref. [42] describes the additional measurements
to be made if the mirrors have unequal transmissions. (Note that even for a symmetric cavity
with identical mirrors, the presence of a membrane dispersively shifts the cavity dynamics, which
can result in separate effective mirror transmissions for each mirror.) With nonidentical mirrors,

32

Figure 3.3: Figure taken from Ref. [71]. a) Speed ∆0 of the change in cavity length/resonant
frequency compared to κ. Note the ∆0 = 0.001 regime corresponding to quasistatically tracing thecavity lineshape, the ∆0 = 1000 regime corresponding to a cavity ringdown with exponential decay,
as well as the intermediate regime. b) Experimental data motivating the investigation of fitting
ringdowns in the intermediate regime.

33
reflection and transmission measurements must be taken from both directions. For each directionthat the cavity is probed, there is an additional modematching factor L (R ) describing the overlapof the left (right) input beam with the cavity’s TEM00 mode. To solve for all five unknowns, fivemeasurements will be made, namely the cavity finesse, the relative transmission and reflection fromprobing the left mirror of the cavity, and the relative transmission and reflection from probing theright mirror of the cavity.
The power in transmission probed from the left side, Pt,L , can be written normalized to themodematched input power L Pin,L as
Pt,L= 4TL TRL Pin,L1
TL + TR + L


(3.5)

while the useful power in reflection, which is the reflected power of the input beam Pr,L minus
the noninteracting unmodematched portion of the input beam (1 − L )Pin,L , all normalized tomodematched input power, can be written
Pr,L − (1 − L )Pin,L=L Pin,L

L + TR − TL
L + TR + TL

2
(3.6)

Now, with the trick of dividing Eqn. 3.5 by Eqn. 3.6, we are left withPt,L
4TL TR=
Pr,L − Pin,L(L + TR − TL )2 − (L + TR + TL )2(3.7)

which conveniently only has directly measurable optical powers on the left hand side, and themodematching L has dropped out. With another equation describing measurements taken probingfrom the right side (interchange L and R subscripts in Eqn. 3.7), along with a measurement of the
2πcavity finesse F = TL +T, the system of equations can be solved to yield TL , TR , and L. With
R +L

substitution of those values into Eqn. 3.5, L and R can be determined as well.

3.2

Cavity characterization - with membrane
A simple but effective model of the membrane-at-the-end cavity that reproduces key opticalcharacterization measurements is a one-dimensional transfer-matrix formalism. In the context of34
optomechanical cavities, descriptions of the model are given in graduate student theses (Refs. [73,71]). The former work describes the details of the model in general, while the latter covers aspectsof the calculations used in our group. By modeling the cavity as a stack of dielectric materials in one
dimension, the cavity’s linewidth κ, as well as the optomechanical coupling G, can be determined.
However, some realistic scenarios such as the membrane being tilted with respect to the cavity axis
cannot be modeled in this way.
A model of our membrane-at-the-end optomechanical system can be constructed via matrix
multiplication of the transfer matrices for mirrors, vacuum, and dielectric plates [73, 72]. Mirrors
are represented by

i/Y0 n 0Mmirror (n) = 
(3.8)

iY0 n
0
q
p
where Y0 = 0 /µ0 is the vacuum conductance, and n = 1−r1+r is the effective index of a mirror√
with amplitude reflectivity r = 1 − T . This models the mirror as the reflection resulting fromthe index contrast between vacuum and an unphysically very high index material. Without thissimplification, the entire mirror stack would need to be modeled, which is straightforward but notnecessary for our purposes in this chapter and often not possible since coating houses may notdisclose the stack. Instead, the mirror reflectivity can be obtained via the coating curves or ameasurement of κ. Next, vacuum is represented by
i sin(kL)/Y0  cos(kL)
Mvacuum (L) = 

iY0 sin(kL)
cos(kL)(3.9)

where k is the wavevector of the light, and L is the length of the vacuum element. A dielectricplate is represented by
Mdielectric (n, Ld ) = cos(knLd )i sin(knLd )/nY0 
inY0 sin(knLd )cos(knLd )(3.10)

where n is the index of the dielectric, which can model absorption via an imaginary index (n =nR − inI ), and Ld is the thickness of the dielectric. The absorption can also be expressed as a perpass absorption δ via Beer’s law, δ = 1 − exp(−4πLd nI /λ). We can now write the characteristic

35
matrix of our membrane-at-the-end optomechanical cavity, denoting the lengths of the parts of thecavity to the left and right of the membrane as `1 and `2 :M = Mmirror (n1 )Mvacuum (`1 )Mdielectric (nSi3 N4 , Ld )Mvacuum (`2 )Mmirror (n2 )(3.11)

and, from this point, solve the system for the appropriate resonance condition. The resonancecondition used in these calculations is that the transmitted phase is an integer multiple of π, or=(M11 + M12 + M21 + M22 ) = 0, a condition that can be calculated numerically [73].
The details of how to set up the problem and solve for resonance are very important, especiallyin our setup. Because we operate with a fixed-frequency laser, the end mirror positions must be
adjusted, for example, by changing `1 to bring the cavity into resonance with the laser. Changing
the membrane position is a differential operation that preserves ` = `1 + `2 , by increasing `2 anddecreasing `1 by the same amount δx. For situations such as determining a cavity parameter like κof G as a function of membrane position, keeping track of what exactly is moving is critical. In fact,depending on which of the model lengths is adjusted, such calculations can give different results.
An example that illustrated this point to us is the measurement of the optomechanical coupling G as a function of membrane position. With G = ∂ω∂xcav , we though that a simple measurementin transmission of the cavity while scanning the end mirror would let us measure a signal proportional to G as a function of membrane position. For a constant scan amplitude on the end mirror
piezo, we thought that the shift in time of the cavity modes’ resonances would correspond tofrequency and thus G via the conversion factor FSR/532 nm. Modeling this situation, however,revealed that we were not measuring G. Instead we were measuring a subtly different but relatedparameter Gwrong . Because G is fundamentally a derivative with respect to frequency, a position
scan of the end mirror does not directly probe it. And while a correspondence exists between G
and our Gwrong via the chain rule, it is instructive to determine how to properly measure G with afixed-frequency laser. By applying 500 MHz fm sidebands to the laser via an electro-optic modulator, a direct frequency scale can be used for measurement. With a constant-amplitude high-voltage
sine wave added to the membrane piezo bias voltage, a constant amount of motion is added to36
the membrane. By capturing the jitter of the cavity resonance (by using infinite persistence on an
oscilloscope) and comparing it to the frequency markers, a signal proportional to G is obtained.
Data for a test cavity illustrating this measurement of G (green, blue) along with finesse (red) are
shown in Fig. 3.4. Because of systematic drift in the experiment due to thermal gradients as thelab heats during the day, the data are sequentially shifted. However, since the finesse was measured
in between the green and blue traces for G, the horizontal average of the green and blue curves
matches the expectation of the model, shown in Fig. 3.5, corresponding to the experimental setup
where the mirror nearest the membrane was scanned. An alternative situation modeled in Fig. 3.6
shows the different behavior when the far mirror is scanned. The subtle changes in relative widthof the maxima and minima, as well as the phase change of Gwrong illustrate the importance of thedetails of cavity operation, both experimentally and theoretically. For this cavity, the experimental
data in Fig 3.4 show a maximum finesse of 30,000, while the model in Fig. 3.5 has a maximum
finesse of 41,000. To obtain agreement on this point between the data and the model, introductionof an imaginary index of refraction as described above could be used to model the loss resulting inlower than ideal maximum finesse. Typically, values of nI ≈ 10−5 are used to capture per-pass lossof 1-10 ppm in our cavities.

37

3

28x10

0.5

26

24

22

0.3

FinesseAbs[g/g0]

0.4

20
0.2
18
0.1

16
0

200

400
Membrane position (m)

600

-9

800x10Figure 3.4: Experimental determination of G (arbitrary units) (green, blue circles) and finesse (redtriangles) for a ` = 6.3 mm, `2 = 0.5 mm optical cavity comprised of two 96 ppm transmission
mirrors and a 40 nm thick membrane. Because of systematic drift in the experiment, first G (green),
then finesse (red), then G again (blue) was measured. The average of the two G curves’ positions
corresponds to the expectations of the model.

38

40 00035 000Finesse0.6
0.4

30 000Abs[g/g0 ]

0.8

0.2
25 000
0.0
-400

-200

0

200

400

Membrane position (nm)Figure 3.5: Model of the cavity in Fig. 3.4 where the input mirror close to the membrane is adjusted
to bring the cavity into resonance. Shown are G (green), Gwrong (dashed green), and finesse (red).
Note that the high-finesse peak is skinnier than the low-finesse trough. Also note the alternating
absolute maxima of G and Gwrong .

39

40 00035 000Finesse0.6
0.4

30 000Abs[g/g0 ]

0.8

0.2
25 000
0.0
-400

-200

0

200

400

Membrane position (nm)Figure 3.6: Model of the cavity in Fig. 3.4 where the output mirror far from the membrane isadjusted to bring the cavity into resonance. Shown are G (green), Gwrong (dashed green), andfinesse (red). Note that the high-finesse peak is wider than the low-finesse trough. Also note thatthe absolute maxima of G and Gwrong occur together. Both situations are the opposite of thesimulation in Fig. 3.5

Chapter 4

Optomechanical cavities

How foolish to call them mirrors. They are to mirrors as the enveloping firmament is to a child’s balloon. They reflect light indeed; but that, I think, is nopart of their true function. They reflect reality, the metaphysical substance that
underlies the material world.
The Book of the New Sun, Gene Wolfe
Chapters 4 and 5 will discuss the optomechanical cavities designed and used for these experiments. This chapter will be concerned with the pure optomechanical cavities used in experiments
since the lab’s inception [19, 22, 68, 54, 3, 31]. The cavity designs covered in this chapter are not
an exhaustive list of everything ever built, but a history reflecting those cavities used in our lab’s
publications, with additional designs that shaped our thinking or demonstrated important technological advances. The cavities for the electro-optomechanical experiments, although developed inparallel, will be discussed separately in Chapter 5.
My contributions to optomechanical cavity design started when I joined the lab. The optomechanical cavities discussed here were principally designed by Tom Purdy and subsequently by
Nir Kampel, with my advice and assistance, and built in whole or in part by all of us at various
times. The optical design of the electro-optomechanical devices in Chapter 5 is the area where I
made more central contributions. As the only student who has continuously participated in cavity
design and construction from (nearly) the beginning, this thesis is an opportunity to document thehistory of cavity design in the lab up until the end of 2016.
Our optomechanical cavities consist of a dielectric membrane coupled to the cavity mode

41
defined by two mirrors, the so-called membrane-at-the-end concept shown in Fig. 4.1. Compared toother optomechanical systems, those using a Fabry-Pérot cavity to confine the light typically exhibit
power handling capabilities closer to typical optical cavities, and are often associated with lowerfrequency, higher-mass, higher-quality-factor mechanical modes [19]. Many early optomechanicalsystems were designed with this general approach, with more recent work demonstrating the design
we adopted, of two cavity mirrors and a separate mechanically-compliant membrane placed in
between [74, 75, 76, 77, 78, 79, 80, 81, 82]. The principal design challenge is to define and maintaininterferometric alignment between all three optical elements, despite the thermal contraction ofmaterials as the cavity is cooled to cryogenic temperatures. Additionally, piezoelectric actuationof two optical elements is required to bring the cavity into resonance with a fixed-frequency laser,
as well as to position the membrane in the optimal position along the standing wave of opticalintensity inside the cavity (Fig. 4.1). While actuation of the mirror and membrane prevent a truly
monolithic design, we strive to make the cavity as rigid as possible. In particular, we do not include
the ability to adjust other degrees of freedom, such as the transverse position or tip-tilt of opticalelements, which we assume will remain aligned as the cavity is cooled.
The constraints of fast iteration on designs early on, as well as the complicated geometries
required for aligning and positioning the membrane inside the optical cavity, made us select invar as
the cavity spacer material. Invar has the lowest thermal expansion of a metal alloy; its discovery in
the early 20th century and subsequent impact on precision measurement merited a Nobel Prize [83].
Selecting the appropriate amorphous or crystalline materials would allow us to match thermal
expansion coefficients of the mirror substrates or membrane substrate [84], but machining thesematerials is beyond the scope of what a researcher in a machine shop can do. Invar is a difficult
to machine material, and the intricate geometries we designed are possibly beyond the skill of theless patient graduate student or postdoc, but with sharp tooling, slow cutting speeds, and above all
patience, it is possible to machine cavity spacers yourself, as we did for the first several years. Withless variability in design, cavities are now made by the professional machinists in the instrument
shop, who can achieve tighter tolerances, better parallelism, and can properly anneal the invar

42

Figure 4.1: Membrane-at-the-end cavity. The membrane is positioned along the desired point of
the optical intensity standing wave, and one end mirror’s position is adjustable to bring the cavity
into resonance with a fixed-frequency laser.

during machining, improving its dimensional stability.

4.1

First cavities
The first cavities in the lab explored a variety of design concepts, first among them how toalign the elements and how to make the cavities survive cooling to 5 K without the mirrors or
membrane literally falling off. As it turns out, simply adding more epoxy is not the answer, since
it strongly clamps the membrane chip, placing a large amount of stress on it which can drasticallylower the Q [85, 86].
Fig. 4.2 depicts the design that resulted in our first optomechanics publication. Two devices
are pictured, the top is the immediate predecessor to the device used in Ref. [19], while the bottom
is the device used in Ref. [19]. They are largely identical, except for the inset curved mirror(bottom right) which reduced the cavity length from 10 mm to 5 mm. From our current design
perspective, these cavities reflect the beginnings of several design features that we have used since.
Namely, the construction of the spacer out of invar, the geometric clearance to operate in the 1
in diameter bore of the flow cryostat used in early experiments, and the use of piezo actuators on
the membrane and one end mirror as the only available degrees of freedom. However, from ourcurrent point of view, we can identify several critical features and design choices that are absent

43from this design. First among them is accessibility of the membrane. It is often the case that themembrane tilts relative to the cavity mode, gets dirty, or just has a low Q once it’s been cooled
down, all of which necessitate replacement. In these early designs, everything must be discarded
and built anew, since the membrane is glued in place and then entombed by the 1/2 in diameterflat mirror (Fig. 4.2 top and bottom, right side). All subsequent designs allow for nondestructive
removal and reinsertion of a new membrane, in some form consistent with other evolving design
constraints. There is also a trend towards shorter cavity lengths, starting from this “baseline” of 5mm, which leads to designs with smaller diameter mirror substrates, and more concentrically-inlaid
structures to allow the membrane to sit on a stack of spacers and a piezo whose height is greaterthan the cavity length. The shift towards smaller mirror radii of curvature to reduce the opticalcavity waist also has influenced cavity design, since radii less than 5 mm are not compatible withstandard superpolished fused silica substrates, which further complicates the design.

4.2

Our canonical optomechanical cavity design: the RPSN cavity
The next prominent cavity design was first used in Refs [22], shown in Fig. 4.3. It continued

to be used through Refs. [68, 54], though in honor of the first result it produced, I will refer
to it as the “RPSN cavity.” As a single design used in several experiments, it is in many ways
the canonical reference that solidified our design concept, and was the point of departure forthe electro-optomechanical devices in Chapter 5. The open structure allows for nondestructive
replacement of the membrane, which is placed on a multilayer “membrane stack.” The membranestack operationally allows for interferometric alignment and epoxying of the membrane into place,while it can be removed by unscrewing the plate it’s epoxied to, as depicted in Fig. 4.7. This allows
removal of the membrane and epoxy without dust generation inside the cavity.

4.3

Later designs
A longstanding challenge in cavity design has been the relatively large size of the mirror

substrates. The standard size of 7.75 mm is too wide to make the increasingly short cavities we44

Figure 4.2: Early cavity designs. (top) Immediate predecessor to the device used in Ref. [19].
Cavity length ` = 10 mm, mirror radii of curvature R = 5 cm and plano. (bottom) Device used
in Ref. [19]. Note the inset curved mirror relative to (top) left, which reduced the cavity length to` = 5 mm. Mirror radii of curvature R = 5 cm and plano.

45
were planning (` ≈ 2 mm) while still providing comfortable access to insert a membrane in between
the mirrors. Tom Purdy explored machining these mirror substrates to both a smaller diameterand shorter thickness, while protecting the already-coated supermirror coating. The mini-mirror
cavity in Fig. 4.4 demonstrated an optical cavity using these mirrors, without an increase in cavity
linewidth due to surface damage of the dielectric coating. This technique is now used routinely forthe devices in Chapter 5, where integration of microwave-frequency electromagnetic modes placeseven more stringent constraints on the geometry of the cavity mirror substrates.
The final design to be discussed in this chapter is the device used in Refs. [3, 31], the “backaction limit” cavity, shown in Fig. 4.5. The largest change is that the two halves of the cavity—eachholding one of the cavity mirrors—separate from each other. This is since the shorter cavity length(` ≈ 2 mm) cannot accommodate the membrane stack concept in Fig. 4.7. In particular, the removal of the membrane is no longer possible, since the thickness of the membrane, the membraneholder, and the top invar plate is greater than the cavity length. While the same membrane stackis used in the backaction limit cavity, the membrane can only be removed after unscrewing the top
invar plate if the cavity spacer itself is first unscrewed and separated. We found good repeatability for reassembling the empty optical cavity. In fact, the repeatability of cavity separation andrejoining was good enough to maintain membrane alignment if one was epoxied to the membranestack. This modularity was a very important test, since these techniques are critical to the latest
electro-optomechanical cavity design in Chapter 5.

4.4

Cavity components
Before describing the assembly procedure, I will provide an inventory of components, and an

outline their importance in our devices.
The most important aspect of any optical cavity is the mirror. We use superpolished dielectricsupermirrors on a fused silica substrate, which are available commercially from local vendors. Cavity
mirrors are characterized by their transmission and scattering/absorption, typically in parts-permillion or ppm. The RPSN cavity uses one planar and one 5 cm radius of curvature (ROC)46

Figure 4.3: RPSN cavity. The principal design change since Fig. 4.2 is the open interior of thecavity, allowing for nondestructive replacement of the membrane inside the cavity. Cavity length` = 5 mm, mirror radii of curvature R = 5 cm and plano.

Figure 4.4: Mini-mirror cavity. A realization of a cavity made from few-mm diameter mirrors
machined down from the 7.75 mm diameter substrates we bought commercially.

47

Figure 4.5: Backaction limit cavity, shown in copper enclosure used in dilution refrigerator. Bothviews show an end mirror epoxied using the back glue joint (four radial holes around optical axis),
as well as a copper foil providing a thermal connection between the membrane stack and the copper
enclosure, which thermally shorts the bulk of the cavity spacer. Cavity length ` = 1.65 mm, mirror
radii of curvature R = 2.5 cm and plano.

48

Figure 4.6: Cavity schematic (not to scale). Representative of the RPSN cavity, but some glue
joints have been schematically simplified. Optical axis is indicated by dashed red line. The invar
cavity spacer (gray), along with the thicknesses of the mirrors and mirror stack components, defines
the cavity length. The flat mirror stack (left) consists of a piezo (beige), an invar ring (gray), andthe flat mirror (light blue). The stack is assembled separately and centered to the hole in the cavity
spacer with an alignment jig (not shown). The membrane stack is as depicted in Fig. 4.7. The
curved mirror (light blue) is glued directly to the right side of the cavity spacer, using a glue jointthat shares characteristics of the edge glue and the back glue joints in Fig. 4.8.

49

a

b

c

Figure 4.7: Membrane stack. (a) The membrane (blue, top) is epoxied to a machined silicon holder(blue, bottom), which is manipulated by a tapped invar holder (gray, rotation symbol) attached toa five-axis stage. It is interferometrically aligned to the cavity mode before being epoxied to therest of the membrane stack. The bottom of the stack consists of the piezo (beige), as well as twoinvar plates (gray) that are attached with screws. (b) Complete membrane stack after membranealignment and epoxying. The five-axis stage is unscrewed from the invar holder and backed off,
leaving a complete optomechanical cavity. (c) When removal of the membrane is desired, thescrews connecting the two invar plates are removed, and the assembly from the top invar plate up(including the membrane) is removed by hand.

50

a

b

c

Figure 4.8: A generic taxonomy of the glue joints used to epoxy cavity components together. (a)The edge glue joint is the most robust, and allows for the two surfaces to be in planar contact. (b)The face glue joint is often used where radial clearance for an edge glue is not possible or ideal. It is
more susceptible to delamination due to thermal cycling or applied mechanical stress, and the layerof glue between the surfaces can be uneven. (c) The back glue joint is an alternative to the face glue
joint that allows for planar contact between the two surfaces. It is also susceptible to delamination,and care must be taken to ensure the channels are filled with epoxy to make full contact with thesecond surface. Moving the gluing surface to the backside of one of the components can simplify
assembly and reduce the complexity of the cavity interior geometry.

51
mirror, both with 96 ppm transmission and approximately 1 ppm scattering/absorption, which
are representative values for the rest of our devices as well. It is apparent in many other sectionsof this thesis that having different mirror transmissions can be useful in an optical cavity. The
principal constraint is that it’s a significant monetary investment to have a coating run done
by a vendor, so we initially had only one batch of mirrors with the same mirror transmission
value. Another constraint is that the cavity geometry, set by the cavity length and mirror ROC,
defines a cavity waist, whose size is ideally matched to the mechanical wavelength of the membraneresonator (Chapter 3). The practical limitation here is the commercial availability of small ROC
mirror substrates. Local vendors have supplied us with substrates down to 2.5 cm (Advanced Thin
Films) and 1.0 cm, and potentially 5 mm (Perkins Precision Developments). Mirror substrate
ROC less than 5 mm are still of interest to us, but they are technologies being developed byresearchers [87, 88, 89]. As a general statement, our experiments have been enabled by the highquality of commercially available supermirrors, but desired improvements in substrate geometry
have forced us to consider different approaches.
Mirror cleaning is also an important aside. The optical losses of these mirrors are low enoughthat using lens paper or a solvent wash are inappropriate. We use a drying polymer solution suchas First Contact (Photonic Cleaning Solutions) to clean the mirrors, as well as to protect themduring handling and storage. We define mirrors to be dirty using the most sensitive metric: Whenthe linewidth of a cavity formed by these mirrors is appreciably higher than what the cavity lengthand mirror coating curves imply it should be. For us this can be a change in linewidth under
100 kHz, for linewidths typically of order 1 MHz. Aside from the manual technique to apply andremove the polymer solution, consideration should be given to the problem of dust accumulation
after removal. The polymer peel-off can leave the surface charged, possibly attracting particles
in the air unless precautions are taken [90]. Ion-generating guns using mechanical compression ofa piezo are relatively inexpensive and a reasonable way to neutralize surface charge, if that is aconcern.
The membrane resonator is the second key component. Many of our experiments have been

52
performed using commercially available Si3 N4 membranes (Norcada). A typical chip has a 50 nm
thick film suspended over a 0.5 mm x 0.5 mm square hole in a Si frame. They exhibit remarkablemechanical properties, with Q > 106 at 4 K, and fundamental frequencies ωm ≈ 2π × 1 MHz due tothe very high stress of the Si3 N4 film [78, 86]. A key aspect of cavity design is the reproducibility ofthe membrane Q due to stress on the chip from mounting [85]. Our approach in most designs is toplace the membrane chip on a thermally matched Si holder, machined to support three corners of themembrane, and attached with three very small dabs of epoxy on those corners. Subsequent designs
have accommodated more sophisticated membranes, shielded from the substrate by a phononic
band gap structure [91]. These were fabricated by collaborators, and eventually ordered as acustom membrane run from the same vendor.
Because we use a fixed-frequency laser, it is necessary to scan the cavity length to bringit into resonance with the laser. It is also necessary to position the membrane along the linear
portion of the cavity standing wave, where linear optomechanical coupling is maximized. For thispurpose, two piezo actuators are used. We use multilayer ring piezos from Noliac, which are able
to scan across multiple half wavelengths at room temperature, and about one half wavelength atcryogenic temperatures. They are also notable for surviving thermal cycling as well as many othercomponents. At cryogenic temperatures, piezos can be operated harmlessly with reverse polarity,
giving extra range.

4.5

Cavity construction
For the sake of clarity, discussion of the construction and assembly procedures for our cavities

will be condensed to focus on the RPSN cavity. The more complicated procedures in Chapter 5 willagain follow this generic prescription, as do later optomechanical cavity designs derived from the
RPSN cavity. A schematic depiction of the cavity in Fig. 4.3 is shown in Fig. 4.6, highlighting thecomponents and epoxy joints necessary to construct it. Before discussing the assembly procedure
for the RPSN cavity, an aside on how to build an “empty” cavity will serve to illustrate how adding
the third optical element complicates the process.

53
A optical cavity is formed whenever two mirrors are oriented towards each other in a geometrythat satisfies a stability condition



`
`
0≤ 1−
1−
≤1
R1
R2

(4.1)

where R1 and R2 are the two radii of curvature. In practice, we take R1 = R and R2 = ∞ for asingle curved mirror and a plano mirror, with `  R, near the plane-parallel cavity limit. Moredetails of cavity stability and mode structure are given in Chapter 3. An interesting practical result
arises: The two mirrors do not need to be pointed exactly at each other, that is, the flat backside
surfaces of the substrate need not be parallel to each other. As long as the angle is small enoughthat any part of the curved mirror’s surface is locally parallel to the flat mirror’s surface, a cavity
mode is formed. This effectively means that an optical cavity doesn’t need to be aligned to bebuilt. What I will call the “bulk geometry” of the cavity spacer is enough to define a cavity mode.
Machining tolerance and mirror placement still need to be good enough to ensure that the cavity
mode forms roughly in the right spot, but there is no interferometric constraint. Essentially, twomirrors can be positioned just according to bulk geometry “for free,” while any additional opticalelements in the cavity mode must be interferometrically aligned. In practice, we epoxy together
mirror stacks first, consisting of a piezo for mirror positioning, an invar ring, and finally the mirror
on top. The invar ring serves to absorb the stress due to thermal mismatch of the piezo, which
shrinks much more than invar and fused silica. The mirror stack, along with a bare mirror, areepoxied to a cavity spacer as schematically shown in Fig. 4.6. This defines a cavity mode, which
we actively probe during the rest of the assembly by scanning the mirror piezo and measuringin transmission, both on a photodetector and on a camera. The camera allows for coupling into
the TEM00 mode of the cavity, while minimizing coupling to higher-order modes. At this point,
the membrane can be aligned and epoxied. It is held to a 5-axis stage providing XYZ positioning
and tip-tilt angle orientation. It is placed vertically above the position in the cavity where it is
to be epoxied. The cavity, whose post on the optical table is located by a post position retainer,
is temporarily removed while the membrane is moved down to the optical axis. The membrane is54aligned via the tip-tilt stages to make it retroreflect the incident laser. It is then raised, and thecavity is returned and realigned by twisting the post in the position retainer until the transmission
is maximized. This step coarsely aligns the membrane to the cavity mode, since in the next step, themembrane is inserted into the cavity a small distance from the membrane stack. Until it is aligned,
no light will be transmitted through the cavity. This coarse alignment makes it much easier to findthe correct alignment by fine-tuning. Exactly aligning the membrane to the cavity mode is done
via monitoring the image of the membrane and the optical spot in transmission. If the membrane
is tilted, the TEM modes hybridize to form eigenmodes with higher-order TEM components inthe direction of the tilt. By ensuring that the membrane can be scanned along the cavity length(using an additional piezo actuator on the XYZ stage) without periodically hybridizing beyond
the TEM00 mode, alignment is achieved. Finally, the membrane must be brought into contact
with the membrane stack for epoxying. It is here that the concepts of bulk geometry and microgeometry diverge. The membrane cannot be simply pushed into planar alignment with the top of
the membrane stack, since this is not interferometrically aligned to the optical mode. The mostthat can be done is for a corner or an edge of the membrane to come to the threshold of contact
with the membrane stack. This is done by scanning the membrane stack piezo. If the membrane is
not in contact with the membrane stack, there is no effect to scanning this piezo. If the membrane
has made contact, the piezo will begin to push it during the fraction of its sweep in which themembrane stack is in contact with the membrane. Once found, the membrane should be backed
off until it’s just beyond contact. At this point, small dabs of epoxy can be used to attach themembrane (really, the silicon holder beneath it, shown in Fig. 4.6).

4.6

Epoxy
Gluing together the components of the cavity requires both suitable epoxy and suitable

technique. Our epoxy of choice is Stycast 2850 FT, as it is cryogenically compatible, somewhat
thermally conductive, and not too dissimilar in thermal expansion to materials we use to crack due
to stress. While it is nominally designed to be thermally matched to aluminum (set by the filler

55material), it has reliably connected fused silica to invar, and piezos to invar in our applications. It
is a two-part epoxy consisting of a resin and a hardener catalyst. The resin compound (properly,
the Stycast 2850 FT itself) consists of a chemical resin as well as a filler material, in this case finealuminum oxide fibers. We typically use catalyst 9 to prepare the epoxy, as its higher viscosity
than catalyst 23LV makes it more workable to “paint on” a fillet of glue around edges, as shownin Fig. 4.8. For back glue joints with narrow aspect-ratio access holes or face joints, catalyst 23LV
may be more appropriate, as the thinner epoxy can work into small spaces and be spread uniformly
over a surface more easily.
Preparing components for epoxying involves proper surface treatment and cleaning. Cavity
mirrors can only be cleaned with a polymer solution such as First Contact, and membrane chips
typically should not be cleaned so much as replaced if they become dirty. In these cases, it is best
to never get them dirty in the first place. For metal components, we typically sand the surfaces
with 600 grit sandpaper and then clean them according to a standard protocol: manual cleaning
with Simple Green and white (nonabrasive) Scotch Brite, ultrasonic cleaning in a detergent solution,
deionized water rinse, and a solvent rinse with acetone, isopropanol, and methanol. Other materials
like piezos can be swabbed or rinsed with isopropanol to clean the surfaces prior to gluing.
For epoxy preparation, it is vital to use the recommended vacuum deairing to remove air
incorporated during mixing. For application, I recommend using lengths of small-diameter spring
steel held freehand, or music wire held in a pin vise. For the smallest of glue joints, aluminum wirefrom a wirebonder can be used as the applicator, since the stereoscope and the geared-down motionof the needle makes application of tiny amounts of epoxy easier. We do not cure at an elevatedtemperature out of a concern that it could be harmful to the cavity mirrors, although curing in anoven or a hot plate is common for many cryogenic applications of Stycast 2850. Without elevatedtemperatures, curing takes approximately 24 hours.
Replacement of the membrane in a membrane stack assembly such as in Fig. 4.7 requires removal of cured epoxy from a reusable invar plate, as do other “undo” operations when a component
has failed. I typically use a single-sided razor blade as a chisel; with a light tap the epoxy can be

56
delaminated from the metal surface. Any remainder is easily removed with a fresh xacto knife ora dental scaler. Light sanding with 600 grit sandpaper removes any residual epoxy or metal flashfrom the removal process. After cleaning, the part is ready for reuse.

4.7

Lab environment
Cavity construction is best carried out in a clean environment. The current solution is toenclose the cavity construction optical setup and place a HEPA filter above it. The JILA-standard
canopies above the optical table readily serve this purpose, since one of the standard 2’ x 4’ melamine
panels can be removed and a HEPA filter put in its place. The canopy framing can support either
flexible antistatic vinyl sheeting or a track made from T-slotted framing (80/20 Inc.) with sliding
plexiglass panels. A quieter fan makes it psychologically easier to keep the filter for the clean area
on continuously, which is the best option. Note that later designs, such as the mini-mirror cavity
and the backaction limit cavity (Figs. 4.4, 4.5) have covers to enclose the cavity volume, preventing
dust from entering the cavity after assembly and damaging its optical or mechanical properties.

Chapter 5

Electro-optomechanical cavities

The development of electro-optomechanical cavities began in parallel with the developmentsof Chapter 4, with the original optical design principally inspired by the RPSN cavity. The cavities
incorporate both an optical cavity and a microwave-frequency LC circuit, each coupled to the samemechanical resonator. Our success in developing this platform was due to the collaborative effort inits design and fabrication. Design and fabrication of the LC circuit, which is on the same substrateas the membrane, was the result of the work of several collaborators. In this thesis, I will only
give a brief outline of the microwave design. References are available which explore this topic inmore detail [42, 92, 93]. In particular, Reed Andrews’s thesis (Ref. [92]) contains the parallel setof content describing the design and integration of the device from the microwave perspective.
Because the experiments performed with these electro-optomechanical cavities are focused on
the application of quantum state conversion between microwave and optical frequencies, the cavities
in this chapter will be referred to as converters, emphasizing the two vastly different frequencies ofthe electromagnetic modes that they bridge.

5.1

Microwave circuits and the flip-chip electromechanical device
While a complete understanding of the converter design requires equal discussion of the op-tical, mechanical, and microwave components, my principal contribution was in the optical design.
In this section, I will provide an overview of the electrical and mechanical design of the converters.
For more details, I refer to our collaboration’s publications focusing on the mechanical and electrical

58
design [42, 92, 93].
The basic design principal for the electromechanical portion of the design is the flip-chip.
This structure is a stack, formed by a bottom chip and a top chip. The bottom chip contains an
inductive loop and capacitor pads formed by deposition of superconducting Nb next to a hole inthe chip that allows an optical beam to pass through. These form part of the LC circuit that is
electromechanically coupled to motion of the membrane. In the first converter, it also contains a
microwave transmission line inductively coupled to the partial LC circuit that allows for an electricalconnection to excite and readout the circuit. The top chip contains the membrane; its principal
difference from the commercially available Si3 N4 chips used in some of the designs in Chapter 4 isthat part of the membrane surface is metallized with Nb. The top and bottom chips are aligned
and placed one on top of the other, and epoxied in place. The separation of the capacitive pads oneach chip, which forms a parallel plate capacitor, is defined by pillars on the bottom chip to be a
few hundreds of nanometers.
The complete electromechanical flip-chip is almost entirely self-contained, and most importantly can be manipulated, aligned, and epoxied into place in an almost identical manner to themembrane chips used in Chapter 4 for the optomechanical cavities. This conformance to establishedoptical assembly procedures and expertise allowed for fast progress in developing the first converter
concept. The one aspect that distinguishes the flip chip is the necessity to couple microwave signals
to the LC circuit. Our two approaches to this problem define the different design approaches of
our two converters. The first converter uses a wired microwave connection, while the second uses
wireless coupling to the LC chip.

5.2

First converter
The first converter design was our initial attempt at combining existing electromechanical andoptomechanical technologies developed at JILA. Instead of suspended aluminum membranes likethose used in preceding electromechanics experiments [94, 95, 15], colleagues developed the flip-chipelectromechanical resonator. Its Nb (instead of Al) circuit superconducts at 4 K, allowing operation

59

Figure 5.1: Photograph of the first converter, depicting the optical cavity enclosed in an open copperbox (left), and with the lid closed (right). A hole in the left side of the box allows transmission of
the free-space optical light, while the SMA connector on the right side of the box couples microwave
signals to the flip chip via a flexible transmission line. The membrane stack is covered by the Nbshield.

of the device in simpler and more accessible LHe cryostats as opposed to a dilution refrigerator. As
discussed in the previous section, the flip chip is also compatible with existing membrane-at-the-end
optomechanical designs and assembly procedures in our lab. For more details on its design andconstruction, refer to Reed’s thesis, Ref. [92].
The primary visual difference between the first converter and its predecessor optical designs
is that it was not designed to fit in a small flow cryostat, leaving the design free to use simple
rectilinear symmetry rather than the upright cylinder shape of all previous designs. A renderingof the converter design is shown in Fig. 5.2, illustrating the rectilinear symmetry and open design.
The flexible microwave transmission line that connects the flip chip to the microwave coax wiring in
the cryostat requires room for access, so the cavity spacer has a more pronounced U shape, whereas
the predecessor optical designs kept circumferential support to add rigidity to the spacer.
A schematic cross-section of the design is shown in Fig. 5.4, which clearly depicts the design
heritage from the RPSN cavity in Fig. 4.6. The concept of operationally isolating the electromechanical circuit in the membrane package allowed for immediate integration into existing opticaldesign concepts, which accelerated initial progress on the converter project. The necessarily morecomplicated membrane stack is depicted schematically in Fig. 5.5, while a detail exploded view

60

Figure 5.2: Rendering of the design of the first converter, depicting the assembled converter (left)
and an exploded view (right) of its components. Parts/materials depicted: invar (gray), piezo(yellow), mirror (translucent), Cu-coated invar (copper), flexible transmission line (beige), flip chip
(green), Nb shield (gray).

Figure 5.3: Detail of the first converter membrane stack (omits piezo and affixed invar plates). The
Cu-coated invar mount (right) locates the flexible microwave transmission line (beige) and the flipchip (green) for wirebonding necessary to complete the microwave circuit. The Nb shield (gray) isregistered to the invar mount to align its aperture to the optical axis position. While the apertureis depicted as a circle in this rendering, by using wire electrical discharge machining techniques,more complex shapes can be cut to precisely conform to the optical clear aperture through themembrane, limiting scattering outside this surface.

61

Figure 5.4: First converter schematic. (Not to scale.) The design of the first converter was adaptedfrom the design of the RPSN cavity in Chapter 4. The flat mirror stack (left) consists of a piezo(beige), invar spacer (gray), and the flat mirror (light blue), while the curved mirror is epoxied
directly to the cavity spacer (gray). The membrane stack is similar to the RPSN cavity in that thepiezo (beige) supports two invar plates (gray) which allow nondestructive removal of the membrane,
here a flip-chip electromechanical resonator. The complete membrane package is detailed in Fig. 5.5.

62

Figure 5.5: Detailed schematic of the membrane stack for the first converter. Optic axis is indicatedby dashed red line. The base is a copper-coated invar plate (orange) with raised features to precisely
position the flexible microwave transmission line (red) and the flip-chip (blue). The transmission
line ends in a SMA connector (dark gray). The flip-chip is covered by a Nb shield (light green) thatblocks scattered light from hitting the section of the membrane coated with capacitive Nb pads
(left).

render in Fig. 5.3 depicts the portion of the membrane package that is aligned and epoxied into theoptical cavity as a unit. The package is manipulated by a five-axis stage via the two 0-80 tappedholes on each side of the invar mount’s top. While the flexible microwave transmission line presents
an awkward geometry, the heavy SMA connectorization does not need to be connected until after
assembly.
Concern that stray light from the optical cavity could hit the superconducting circuit anddegrade its performance led us to design a Nb cover to be placed over the flip-chip. Since the flipchip is precisely located by the final Cu-coated invar plate to align it with the flexible transmissionline, the cover can be attached to the invar plate while still covering the exposed LC circuit to the< 50µm tolerance to prevent obscuring the optical access.

5.2.1

Design and control loop resonances

The active stabilization required to operate our optomechanical cavities, while not unusualfor optical cavities, is complicated by the difficulties of cryogenic operation. The quality factor
of material resonances can increase, both in the cavity spacer itself, as well as the entire cryostat
system. While each cavity design (and indeed, each cryostat) requires tweaking of the feedback
loop for stability, this particular design iteration provides an instructive illustration of some of the

63
problems encountered in cavity operation, as well as one strategy to overcome them.
The first hybrid device was operated in a 4 K bath cryostat (HDL-12, by IRLabs, Inc.). The
cavity is mounted on a cold plate attached to the LHe stage, which is shrouded by a LN2 stage,
all of which is suspended from the top of the cylindrical cryostat. The largest initial challenge to
optical operation of the converter was the inability to lock the cavity using our Pound-Drever-Hall
setup. A 2.7 kHz resonance in the feedback loop was the primary cause. Practically speaking, theaddition of an elliptic filter designed to notch out the feedback loop’s response at this frequency
solved the problem (Fig. 5.8). In general, one might expect that active feedback could also help
reduce oscillation at this frequency. However, this depends on having sufficient phase and amplitude response at a given frequency servo out an unwanted resonance. Without those conditions
being met, simply attenuating the servo response at the problem frequency can prevent the feedback circuit from doing more harm than good. The design of these rf filters have inspired otherapplications, such as filtering noise at the mechanical frequency on rf carrier tones (Fig. 5.9).
In the subsequent cooldown (where the data in Ref. [42] were taken), we observed similar
oscillation at 2.5 kHz. In an attempt to learn more about the problem, network analysis was
performed on both the cavity mirror piezo and the membrane piezo. The electronic excitationfrom the network analyzer is added to the lockbox output to the cavity piezo, and to the biasline to the membrane piezo. In each case, 780 nm laser light is used to read out its effect on
the cavity. At this wavelength, the cavity mirrors form only a low-finesse cavity, eliminating any
optomechanical effects. The transfer functions of the two piezo stacks are shown in Figs. 5.6, 5.7.
They show many similar resonances, though they are all shifted slightly in frequency from their“partner” in the opposite figure. In particular, the 2.480 kHz peak in Fig. 5.6 seems to correspondto the oscillation seen trying to lock the cavity. The piezos themselves all are specified to have afundamental resonance above approximately 500 kHz (Noliac, Inc.), so even loaded by the mirror
or membrane stacks, they are not themselves suspect. A simple argument suggests an explanation:
the speed of sound c in metals is typically of order 5000 m/s. The wavelength of a corresponding 2.5
kHz vibrational mode is 2 m. With the cryostat for this experiment standing 1 m tall, one might

64
suspect that a resonance of the entire cryostat could be responsible. Given the small mechanicalconnections between temperature stages designed to minimize thermal contact, this argument seems
plausible.

5.3

Wireless converter
The second design for our microwave to optical converter set out to solve some of the prob-

lems encountered with the first. Namely, coupling to the LC circuit using a flexible transmission
line can induce lateral forces on the membrane stack, potentially contributing to interferometricmisalignment of the membrane during cryogenic operation. The main design change was to enclose the flip chip in a microwave 3D cavity to enable wireless inductive coupling. A rendering of
the design is shown in Fig. 5.10. The design and details of the wireless coupling are described inRef. [93], whereas this section will focus on the changes in optical design and construction.
The microwave package, previously consisting of the flip chip, its invar mount, and accessories,
now properly consists of the flip chip and the enclosing microwave cavity. The microwave 3d cavity
is formed by two top and two bottom pieces which fit together. The tops, together called the “hat”,
can be separately selected and attached to each other to adjust the resonant frequency of the 3d
cavity to properly correspond to the resonant frequency of individual flip chips. The microwave
cavity bottoms change the optical alignment procedure in that they must be sequentially assembled
during membrane alignment. However, the previous cavity and membrane assembly procedure canbe largely reproduced. The empty optical cavity is a two-piece spacer similar in concept to thebackaction cavity design in Chapter 4, but geometrically enlarged to contain the microwave 3D
cavity. Additionally, both mirrors are on longer, thinner stacks to achieve a cavity length ` ≈ 2
mm. The ends of the stacks penetrate the microwave 3d cavity through apertures whose diameteris small enough to be a waveguides below cutoff for the microwave fields. Because of this, they
are made of microwave-compatible materials only (in particular not invar). In this design, just themirror substrates and fused silica spacers, along with epoxy, penetrate the microwave cavity. With
the empty cavity assembled, the microwave cavity cannot be installed. By inserting the half ofMirror piezo transfer function (arb)

65

-3

10

-4

10

-5

10

1

2

3

4
5
6
Frequency (kHz)

7

8

9

10

Membrane piezo transfer function (arb)

Figure 5.6: Transfer function of the cavity mirror assembly. Excitation is added to lockbox output
and amplified before piezoelectrically exciting the cavity end mirror stack (while cavity is locked).
Excitation is read out optically via a 780 nm beam (where the cavity has a very small finesse).

-2

10

-3

10

-4

10

1

2

3

4
5
6
Frequency (kHz)

7

8

9

10

Figure 5.7: Transfer function of the membrane assembly. Excitation is added to the bias voltage
for the membrane piezo and piezoelectrically excites the membrane stack. Excitation is read out
optically via a 780 nm beam (where the cavity has a very small finesse).

66

8.7 mH5 uF

6.2 uF305 nF0

Magnitude (dB)

-10
-20
-30
-40
-50
-60
1

2

3
Frequency (kHz)

4

5

6

1

2

3
Frequency (kHz)

4

5

6

Phase (degrees)

-45

-90

-135

-180

-225

Figure 5.8: Design (top) and measured transfer function (bottom) of the elliptic filter tuned tonotch out a specific frequency in the stopband. The design spec is for 2.7 kHz as shown, but
typically a tunable capacitor in the LC is used to exactly match the desired frequency. Filter
designed in AADE Filter Design program (Neil Heckt).

67

0Magnitude (dB)

-10

-20

-30

-40
8

10

12

14
16
Frequency (MHz)

18

20

8

10

12

14
16
Frequency (MHz)18

20

Phase (degrees)

45

0

-45

-90

Figure 5.9: Design concept (top) and measured transfer function (bottom) of a bandpass/notch
filter design using resonant LC shunts to filter noise at ±ωm from rf carrier signals. Used in Fig. 6.3.

68
the microwave cavity bottom to which the flip chip is epoxied, and then replacing the removable(curved) mirror, the configuration is analogous to previous membrane alignment and epoxying
procedures (Fig. 5.11). Because the flip chip is directly epoxied without additional spacers, thebottom chip is directly grasped and manipulated with a pair of tweezers mounted to a five-axis
stage. After alignment and epoxying of the flip chip, the curved mirror must be temporarily removed
while the second half of the microwave bottom is attached (Fig. 5.12). Replacement of the curved
mirror allows a small amount of freedom to reposition the cavity’s transverse mode location, making
the remove/replace procedure more resilient and also allowing for small corrections due to imperfect
positioning of the membrane chip during its original installation. In summary, the principal change
in this design from an optical point of view is that the optical cavity must be disassembled andreassembled before operation, and the membrane (here a flip-chip electromechanical resonator) is
mounted directly on the cavity spacer.
The completed wireless converter is shown on the assembly table in Fig. 5.13. Portions of
the converter that define the microwave 3d cavity are Au-coated to increase conductivity, though
the entire assembly is made of invar. Torquing the screws holding the microwave bottoms or hat
produces only a small visible shift of the optical alignment (in particular its transverse position),but can be countered by evenly and lightly tightening the screws for each subassembly. This degreeof freedom was not previously available; future work could quantify how useful tightening screws
post-epoxying could be for adjusting optical alignment. The wireless converter installed in the baseof our optical-access dilution refrigerator is shown in Fig. 5.14.
Familiarity with disassembling cavities before use suggests additional possibilities for cavity
design, in particular the partial disassembly of the microwave cavity bottom to insert a thin cover
to shield the part of the LC circuit exposed to stray optical light. Due to the different assembly
concept, the previous Nb shield concept does not transfer neatly. Instead, a spare bottom chipwith a machined hole for optical access (whose profile is the exact shape we want to use to block
scattered light) can be used.

69

Figure 5.10: Rendering of the wireless converter design. Parts A and B form the optical cavity,and correspond to the flat and curved mirrors, respectively. The membrane is glued directly to C,
the second half of the bottom of the microwave 3d cavity is formed by D, and the parts E and F(together the “hat”) completely define the microwave 3d cavity. The remaining parts affix coaxial
lines terminated in loops that couple inductively to the field inside the 3d cavity. The design namecomes from these coaxial lines being wirelessly coupled to the LC circuit. Rendering from Ref. [93].

70

Figure 5.11: Wireless converter with only optical cavity spacer and half of the microwave cavity
bottom installed. Optical cavity mirrors are visible, as is the circular pocket where the flip chip isinserted and directly epoxied according to previous alignment procedures. All parts made of invar,
but the microwave cavity parts are Au-coated for higher electrical conductivity.

71

Figure 5.12: The first demonstration of membrane alignment and epoxying in the wireless converter
design. The microwave cavity bottoms are test pieces made of Al, not invar. After epoxying themembrane, the curved mirror (front) was removed so the second half of the microwave cavity bottom
could be installed. After replacing the curved mirror, the converter is ready for installation of the
microwave hat.

72

Figure 5.13: The complete wireless converter, with all optical and microwave components assembled. The two coaxial lines are for reflection and transmission measurements.

73

Figure 5.14: Wireless converted installed at the base plate of an optical access dilution refrigerator.
Visible in the center of the optical cavity spacer is a lens which limits divergence of the free-space
beam coupled to the cavity mode. Author’s reflection on the Au-coated Cu base plate.

Chapter 6

Optomechanical experiments

This chapter will cover my main experimental work on optomechanical systems not related
to microwave to optical conversion. In particular, it will focus on our demonstration of sideband
cooling a mechanical mode’s temperature to the quantum backaction limit [3]. This demonstration
has broader relevance because it signifies a regime where the mechanical resonator’s dynamics are
dominated by the coherent coupling to the light field and its quantum noise, thus the decoherence
from its connection to a thermal bath becomes negligible. Such a regime enables future work on
the microwave to optical conversion experiment, demonstrating that the converter could operate
in a quantum-enabled regime to perform tasks such as transferring quantum states or generating
entanglement between microwave and optical light. A perspective of past work will also be given,
with an emphasis on how previous scientific results, such as observing RPSN, generating squeezing
light, or performing sideband thermometry, can now be used as a suite of powerful calibration techniques for subsequent experiments. Our work on reaching the backaction limit used measurementsof all of these phenomena to directly calibrate or cross-check measurements, making discussion ofthese past results a natural part of the story.

6.1

BackgroundThe concept of using radiation pressure as a tool to manipulate mechanical motion arose

decades ago in the context of gravitational wave detection[12, 13]. This initial work demonstrated
that a harmonically-bound object buffeted by electromagnetic radiation (there, a UHF field inside

75

∆

κ

ωm
Γ+∝ n+1 Γ- ∝ n

Sideband amplitude

Figure 6.1: Frequency-space diagram depicting optomechanical sideband cooling. The cooling laser(orange) excites both Stokes (red) and anti-Stokes (blue) sidebands separated by the mechanicalfrequency ωm , whose amplitudes are suppressed or enhanced by the cavity susceptibility (gray, realpart, FWHM κ). Due to the quantum nature of the scattering process, the red sideband rate is
proportional to n + 1, while the blue sideband is proportional to n. This asymmetry is the sourceof the quantum backaction limit, where the red sideband remains at a constant finite amplitude,
defining a synthetic thermal bath of the light whose temperature the mechanical mode cannot becooled below. Adapted from Ref. [3]n ≈ n0n≈1

Γ0n ≈ nbann0Γ0 n0 Γ0Cooling rate Γoptba

Figure 6.2: Successive regimes of optomechanical sideband cooling. While Γopt < Γm , sideband
amplitude is given by cavity susceptibility while no cooling takes place. For Γopt > Γm , theratio of sideband amplitudes remains fixed but the mode temperature decreases (classical regime).
When Γopt = nth Γm ,, the mechanical mode approaches the ground state (n̄ = 1). Finally, at
Γopt = (nth /nba )Γm backaction and thermal forces on the system are equal. Beyond this is thequantum backaction limit, where the equal red and blue sidebands imply the mechanical mode isin thermal equilibrium with the synthetic thermal bath provided by the shot noise of the light.
Adapted from Ref. [3]76a cavity) can be subject to additional engineered dissipation. Subsequent theoretical investigation
focused on the quantum nature of this process: does the random arrival of individual photons (shotnoise) disturb the measurement or control process? In fact, this random backaction force, calledradiation pressure shot noise (RPSN), is a fundamental limit of optical position measurement [18].
In the context of optomechanical sideband cooling, backaction is better conceptualized as a temperature, defined by the shot noise fluctuations inside the optical cavity, below which the mechanicalmode cannot be cooled. More specifically, cooling to the quantum backaction limit is the end of theline for sideband cooling with a coherent state, in that cooling below nba is not possible without
employing more sophisticated techniques such as using squeezed light [96].
To briefly review the relevant theory, a red-detuned cooling laser will contribute additional
optomechanical damping Γopt to the mechanical mode, thus lowering its temperature from that ofkTm
its thermal bath nth = ~ω(to which it has a coupling Γm ). The damped resonator’s temperaturemn̄ is given byn̄ =nth Γm + nba Γopt
Γm + Γopt(6.1)

where nba is the backaction limit, describing the effective thermal bath of the shot noise fluctuationsof the light inside the optical cavity. Sideband cooling is the result of a balance between the red
sideband, which heats the resonator but is suppressed by the off-resonant cavity susceptibility, andthe blue sideband, which cools the resonator and is resonantly enhanced by the cavity susceptibility(Fig. 6.1). The backaction limit is given by
nba = −

(ωm + ∆)2 + (κ/2)2
4ωm ∆

(6.2)

which is simply a function of sideband resolution, as clearly illustrated in the case of optimal
detuning, where nba = (κ/4ωm )2 [50, 51]. While the optical damping Γopt is dependent on detuning,
it is operationally trivial to achieve the same amount of damping for a variety of ∆ by adjusting
the optical power of the cooling laser.
While other methods of detection allow determination of the mechanical mode temperaturen̄ with proper calibration, using heterodyne detection to independently measure red and blue

77
mechanical sidebands is convenient since it allows for direct thermometry of the mechanical mode
via sideband asymmetry when the temperature is near the quantum regime n̄ ≈ 1 [97, 52, 54, 53, 55,56]. The various regimes of sideband behavior are shown in Fig. 6.2. In the first two, Γopt is either
smaller or larger than Γm ; in both cases the ratio of sideband amplitudes is given by the cavity
susceptibility, and sideband cooling is present in the latter case. As Γopt = nth Γm , the mode is nearΓm
the ground state as the sideband ratio begins to change. At Γopt = nth
nba , backaction and thermal

motion contribute equally, while beyond this is the quantum backaction limit, where backaction isthe dominant force on the system. Sideband amplitudes are equal because the mechanical mode isin thermal equilibrium with the synthetic bath provided by the shot noise of the light.

6.2

Cooling to the quantum backaction limit
The simplified experimental setup is shown in Fig. 6.4 (a more detailed schematic is given

below). A red-detuned cooling laser is injected into the 10 ppm-transmission mirror of a κ = 2π×2.6
MHz optical cavity coupled to the motion of the ωm = 2π × 1.48 MHz mechanical mode of a Si3 N4drumhead resonator (500 µm x 500 µm x 40 nm), and collected in transmission from a 100-ppm
mirror by a heterodyne receiver. A second locking laser is injected in the orthogonal polarizationmode, and actively stabilizes the cavity length and laser frequency. The optomechanical cavity,
described in detail in Chapter 4, is mounted to the base plate of a dilution refrigerator. The
mechanical mode is coupled to a thermal bath nth , whose temperature need not correspond tothe bulk temperature of the base plate (but can be directly measured in this experiment), with avery small mechanical decay rate Γm = 2π × 0.18 Hz, measured via observing the ringdown of amechanical excitation driven by a piezo actuator. This measurement is performed optically witha 904 nm diode laser. At this wavelength, the cavity mirrors do not have a high finesse, meaning
optomechanical effects such as damping the mechanical motion are absent.
By collecting the transmitted light in heterodyne detection, the red and blue sidebands are
directly visible (Figs. 6.4, 6.7). By performing a simultaneous fit to both sidebands at ωhet ± ωm(where ωhet is the heterodyne frequency), and using the off-resonant shot noise as an absolute cal-78

Innolight
Mephisto

κ/2 = 20 kHz

~
LO
VCO
ZX95-200A+

ZHL-2-1W+

~

Lock
AOM

Freq. tune
Fast lock

AOM 1

AOM 2

Cooling laser

AOM 3

Locking laser

λ/4

VCO
ZX95-78+~

PX1002-BLK
ERA-4+ 86.85 MHz SAW HELA-10D+

A
B
C

A

AOM 1
ZP-5H+B, C

L
Agilent
33520B~

I

R

ZHL-2-1W+

BPF-A76+
VVA

AOM 2, 3

Elliptic
bandpass

Figure 6.3: Detailed rf and laser schematic. Top: The 1064nm laser light is produced, sent througha filter cavity, and sent through the lock AOM (acousto-optic modulator) in a double-pass configuration to stabilize it to the optomechanical cavity. After this, it is split into several parts which
form the LO, cooling laser, and locking laser after passing through a series of AOMs. Bottom: rf
setup for producing the tones driving AOMs 1, 2, and 3. An 86 MHz tone is filtered and split,
where two paths are modulated by ∼10 MHz tones whose difference is the cooling-locking laser
detuning. The frequency of modulation of the AOM 2 is the heterodyne frequency. This setup has
low noise near ωm , and can be easily converted to homodyne operation. Because AOMs 2 and 3
select the opposite order of AOM 1, noise or drift in the VCO is common-mode and is not written
onto the light.

79

10 ppm100 ppm
Locking laser

Cooling laser
LO
Dilution refrigerator-

Figure 6.4: Essential experimental setup. The cooling laser is injected into the optomechanicalcavity. The transmitted light is sent into a heterodyne receiver along with a local oscillator (LO).
The locking laser is injected in the opposite direction and measured in reflection to perform PoundDrever-Hall stabilization of the cavity. Adapted from Ref. [3].

80
ibration of sideband amplitude, sideband thermometry can immediately be performed. By taking
the ratio of red to blue sideband amplitudes, R, as a function of the increasing optical damping Γopt ,the amplitude ratio due to cavity susceptibility only, s, can be extrapolated at Γopt = 0. (Equivalently, the detuning ∆ could be determined directly and s calculated for each cooling experiment.)
The temperature of the mode is then n̄−1 = (R/s) − 1. Thus, for each cooling experiment, s
determines ∆ (with κ and ωm independently measured).
An exceptional cooling experiment is shown in Figs. 6.4, 6.7, where the extracted sideband
heights and the inferred mode temperature n̄ are shown, respectively. The signature of the quantum
backaction limit is clear in Fig. 6.4, which parallels the expectation given in Fig. 6.2 as the sideband
amplitudes shift from being in ratio according to the cavity susceptibility to being equal. By
inferring the mode temperature n̄, it is clear in Fig. 6.7 that the temperature asymptoticallyapproaches nba as expected. The final point in the cooling curves corresponds to n̄ = 0.20±0.02, for
an expected nba = 0.18. The cooling experiment can be repeated at a variety of detunings (Fig. 6.8),
which demonstrates the nature of the quantum backaction limit. Increased sideband resolution as
the cooling laser moves from near resonance to further red detuned lowers nba , consistent with the
expectation of Eqn. 6.2.

6.3

Classical laser noise and beam preparation
The essential problem of the optical and rf design of this experiment is to prepare laser lightwith as little classical amplitude and phase noise near ωm as possible. With lasers required onresonance, red detuned from resonance, and as a heterodyne LO, the technical task is to produce
clean laser tones with 1 MHz and 10 MHz detunings from each other. Our approach is illustrated
in Fig. 6.3, where the optical and rf subsystems to produce the lasers used in the experiment are
detailed. All laser beams are sourced from a InnoLight Mephisto 1064nm continuous-wave source,which is sent through a 40 kHz filter cavity (filtering the noise at 20 kHz), all but ensuring shot-noise
limited performance at noise frequencies from the carrier near ωm . Unfortunately, the light mustbe further manipulated, which can add noise. All the light is sent through an AOM in double-pass81
configuration to shift the frequency to follow the optomechanical cavity. From this point, the lightis split to form the several lasers used in the experiment. A copy is kept unaltered to serve as an
LO (its noise performance being critical to good detection). From this point, the remainder of thelight is sent through a series of AOMs to implement the following scheme: By passing the lightthrough two AOMs in series, selecting opposite scattering orders in each crystal, shifts in frequency
from dc to 10 MHz are possible. This is done to two beams (with both using the same AOM 1
for the first shift) in the following scheme. The cooling laser’s frequency relative to the LO (theheterodyne frequency fhet ) is set by fhet = fAOM2 − fAOM1 , while the detuning of the cooling beamfrom resonance (more precisely, from the locking laser, which can be within 10s of kHz of resonancedepending on how the lock is tuned) is given by fAOM3 − fAOM2 . By looking at the bottom of
Fig. 6.3, the procedure for generating the rf tones that run the AOMs is detailed. They use an
86 MHz tone from a VCO as a common-mode source, filtered by a commercially available SAW
filter built for the obsolete 2G wireless phone standard. As an aside, in the VHF band in which
our AOMs operate, there is typically only one SAW filter commercially available in any usefulfrequency range. Despite their very narrow bandwidth (25 kHz), the necessity of multiple tones
with 1-10 MHz spacing requires other solutions. Discrete inductors and capacitors cannot make ahigh-enough Q filter to appreciably attenuate noise at ωm from the carrier, and VHF cavity filters,
notwithstanding their technical design considerations, are roughly the size of a beer keg and are
extremely unwieldy. Our solution has been to dodge the problem by mixing clean signals near 10
MHz up to the VHF band, where other than filtering mixer images there is no need for additional
filtering. The generation of 10 MHz signals is done using digital function generators, typically
Agilent 33520Bs. While they have many undesirable noise characteristics, including harmonic andnon-harmonic spurs, these can be filtered using hand-built elliptic bandpass filters resonantly tuned
to attenuate exactly at ωm to either side of the carrier. We achieve 20 dB of attenuation at themechanical sidebands, with 3 dB of insertion loss for the carrier (Fig. 5.9). Once mixed onto the
VCO signal, and after filtering the image sideband and residual carrier, the tone is sent througha voltage-variable attenuator (optional for laser intensity stabilization; our preferred lower-noise

82
technique is to instead feed back to an EOM crystal to modulate the polarization going into apolarizing beamsplitter near 45◦ . After a power amplifier, the signal is directly sent to the AOM.
This beam preparation scheme produces light with amplitude (phase) noise 0.2% (2%) of shot noiseat 5 µW, a representative power needed to reach n̄ = nba .
A corresponding more-detailed optical schematic describing the optical setup of the experiment itself is shown in Fig. 6.5. It corresponds to a slightly idealized version of the setup that canbe used for both the current backaction cooling experiment, as well as ongoing microwave to opticalexperiments. In the experiment described in this chapter, the damping beam is denoted in yellow,
and operates in transmission. For other experiments where operation in reflection is desired, thebeampath in red can be used, which also is detected by the heterodyne receiver. A corresponding
version of this setup has been used in subsequent work employing homodyne detection [31]. In thisscheme, the rf electronics detailed in Fig. 6.3 are modified by adding a mixing stage to channel A
equivalent to those in channels B and C. This allows for cooling and locking laser frequencies ∼ 1
MHz from the LO, and a copy of the LO (called the signal beam) can be used as an on- or nearresonant probe. For quantum measurement applications where a large total collection efficiency is
critical, the factor of two greater efficiency of homodyne detection is very useful. Like the coolinglaser in the backaction cooling experiment, the signal beam is injected into the cavity from behind(yellow beampath in Fig. 6.5) to obtain a clean signal in transmission. Now, with the cooling laser
moved to operate in reflection for microwave to optical experiments (Chapter 7), the signal beam
also serves to assist in alignment and calibration of the optical cavity operated in reflection.

6.4

Further analysis of classical and substrate noise effects on sideband
thermometry
A qualitative suggestion that classical noise is not significant is the fact that the coolingcurves saturate at the expected nba . Given the levels of classical laser noise on the cooling laser,an analysis of the sideband thermometry procedure used to calculate n̄ was performed including
the effects of classical nose [54, 53, 70, 97]. It was found that the contribution was ∆n̄laser = 0.006,

83

SignalLock Trans.

Faraday Isolator

λ/4
λ/2

λ/2
Signal IS

Damp IS

Damp

Breadboard

30 mK
1K
4K
50 K
100 K
Cryostat 300 K

Signal Trans.

λ/4
λ/2

λ/2
λ/2

Heterodyne detector

λ/2
LO IS

Lock IS

Faraday Isolator

PDH

λ/2

λ/2

Lock
LO

Breadboard

λ/2

λ/2

Faraday Isolator

Figure 6.5: Detailed optical schematic representative of both the backaction-limit cooldown, as
well as ongoing microwave to optical conversion experiments (Chapter 7). Beams are denoted asfollows: the lock beam is green, the LO beam is blue, the damping beam (in reflection, used now for
microwave to optical conversion) in red, and the signal beam (either the damping beam moved to
be injected in the rear, or a separate beam made from splitting the LO) in yellow. Dashed mirrors
are flip mirrors used for alignment or auxiliary detection. The optical setup is on two breadboards
to facilitate access to the cryostat. The bottom breadboard is routinely removed and relocated via
position retainers on the posts that support it.

84
smaller than the statistical error in our fits of ∆n̄ = 0.02. An additional source of noise is offresonant mechanical modes whose noise power rides on top of shot noise [19, 98, 91], affecting
the normalization of sideband amplitudes to the (assumed) off-resonant shot noise level. This isvisible in the last points of Fig. 6.7, where the higher off-resonant levels result in artificially lower
normalized sideband heights. But because sideband thermometry relies on the ratio of sideband
amplitudes, to first order this has a small effect on determination of n̄. An analysis finds a systematic
error of ∆n̄sub = 0.006, of the same magnitude as classical laser noise and thus not a significant
source of error. An additional check on the magnitude of possible systematic error is that themechanical sidebands (Fig. 6.7 inset) retain a lorentzian lineshape [70]. That these two main sourcesof systematic error are small suggests that the simple calculations of sideband thermometry areappropriate, and that this error analysis need not be represented elsewhere in the Chapter to give
context to results.

6.5

Characterizing the mechanical mode’s thermal bath using RPSN
Characterization of the membrane mode’s thermal bath was an important step in transition-

ing to dilution refrigerator operation. In previous work at 4 K, it was found that the membrane
effectively thermalized to the bulk temperature of the cryostat cold finger. This assumption would
not necessarily hold true at millikelvin temperatures, as thermal conductivities are much lower
and the cooling power of the cryostat is far less. Sideband asymmetry measurements allow forstraightforward measurement of the mechanical decoherence rate nth Γm , which in conjuction witha mechanical ringdown measurement to determine Γm , directly determines the bath temperature.
Comparison to the bulk temperature, as measured by a ruthenium oxide thermometer on the mixing chamber of the dilfridge, is a first-order answer to whether the optomechanical cavity, and inparticular the membrane, has thermalized. However, the locking laser, even at injected powers ofhalf a microwatt, can still contribute enough RPSN to be a significant component of the membrane’s thermal bath. By varying the lock power (Fig. 6.9), the effect of RPSN on temperature
can be assessed. Specifically, the linear dependence of the bath temperature nth on lock power can

Sideband amplitude (SN)

85

1

0.1
10 2

10 3
10 4
Cooling rate Γ opt (Hz)Figure 6.6: Red and blue sideband amplitudes captured in heterodyne detection. Sideband amplitudes are normalized to the off-resonant level, which is assumed to be shot noise. At low Γopt , theamplitudes differ due to the ratio of cavity susceptibility at each frequency, while at high Γopt they
become equal when the mechanical mode is in thermal equilibrium with the light field. The lasttwo points are below the expectation because of off-resonant excess mechanical noise which distorts
the normalization. Adapted from Ref. [3].

Phonon occupation n

10

1.4

1.08

1.2

1.04

1
-6

0

Freq. (kHz)

6

1
-60

0

Freq. (kHz)

60

1

0.1
10 2

10 3
10 4
Cooling rate Γ opt (Hz)Figure 6.7: A cooling curve near optimal ∆, demonstrating cooling of the mechanical mode to thequantum backaction limit. Insets show superimposed spectra of red and blue sidebands fro thethird (left) and third-to-last (right) points on the cooling curve. Adapted from Ref. [3].

86

Minimum phonon occupation

100

10

1

0.1

-2

-1.5
-1
-0.5
Cooling laser detuning ∆ (MHz)0

Figure 6.8: The minimum phonon occupation reached as ∆ is varied from near resonance to its
optimal value. Near resonance, nba diverges. Adapted from Ref. [3].

87

Phonon occupation

10
7
6
5
4
3
2

1
7
6
2

3

4

5

6 7 8 92

3

1
Lock input power (uW)45

6 7 8 9102

Figure 6.9: An uncalibrated measurement of RPSN. For a fixed cooling beam power (here justunder n̄ = 1), increasing the near-resonant locking beam power appears to “heat” the resonator as
its backaction force on it increases. Typically, the y-axis is calibrated in units of position spectral
density, and the x-axis is calibrated relative to the SQL power. Without calibration, it is clear that
the RPSN grows to 10 times the thermal motion.

be fit and used to determine the fraction of the bath temperature due to RPSN. It was found thatof the total effective bath temperature of 360 mK, the thermometer accounted for 70 mK, RPSN
of the locking laser accounted for 166 mK, with 96 mK left over. In fact, RPSN was the dominant
thermal bath of the membrane. Subsequent work to improve optical aligment and detection allowed
for operation with much lower locking laser powers in the hundreds of nanowatts, where the RPSN
contribution is less substantial.
A check on the main result is to use RPSN to increase the bath temperature of the mechanicalmode while still attempting to cool to the quantum backaction limit. In Fig. 6.10, a cooling curve istaken under nominal conditions (orange), and with additional locking laser power (green). Thoughtechnical noise prevented increasing Γopt much further, the cooling curve still follows a trend of
asymptotically approaching nba .

6.6

Determining the quantum efficiency of detection by measuring squeezedlight
It should not be surprising that a cavity that can be cooled to quantum backaction limit

would be able to generate strong ponderomotive squeezing. A thought experiment suggests red

882

0.6 uW lock power
4.8 uW lock power
Quantum backaction limit

Phonon occupation

10
6
5
4
3
2

1
6
5
4
3
2
2

3

4

5

6 7 8 9

2

3

4

3

5

6 7 8 910
Optomechanical damping rate (Hz)

2

3

4

4

10

Figure 6.10: Cooling to the quantum backaction limit with two bath temperatures. The orange
curve was taken under the same conditions as the main data result. By increasing the lock power,
the additional RPSN increases the bath temperature. By comparing the fractional increase of
the lock power to the fractional increase of the bath temperature nth , the component of the bathtemperature due to RPSN can be determined.

1.6

2.5V & −50kHz
5.0V & −50kHz
10V & −50kHz
12.5V & −50kHz
10V & −250kHz

shot noise normalized spectrum

1.5
1.4
1.3
1.2
1.1
1
0.9

∆=−170.0kHz

0.8

nth = 13.0

0.7

N(2.5V) = 7.0x106

0.6

ε’ = 50.0%

0.5
1.44

1.46

1.48
1.5
frequency [MHz]Γ = 200.0Hz
1.52

1.54

Figure 6.11: Superimposed spectra of optomechanically squeezed light produced in the backaction
limit experiment. This light was collected via direct photodetection, as opposed to heterodyne
detection as used in the rest of the experiment. Increasing the power of the locking beam saturates
the amount of squeezing, indicating the regime where measurement quantum efficiency 0 dominates.
By fitting these curves, 0 is determined to be 0.50, which can be related to the corresponding value
 in the main experiment.

89
and blue sidebands ought to be highly correlated: Imagine that the resonator is cooled into its
ground state. A shot noise fluctuation drives a Stokes scattering process, where the resonator heats
up by one phonon, and a red photon is emitted from the cavity. Because of the very strong coherent
coupling, the resonator is very quickly cooled back into its ground state, and the cavity emits ablue photon in short succession. The average temperature of the resonator is essentially the duty
cycle of this process, and if it is sufficiently smaller than n̄ = 1, the two photons are emitted withvery strong correlations. While this describes the correlations in the sidebands of the transmitted
cooling laser, the same regime should be accessible with the near-resonant laser used in squeezing
experiments.
Detection of optomechanically squeezed light can be a powerful calibration tool, since theamount of squeezing is directly related to the total quantum efficiency of the detection. In principle,
squeezing could be measured with the heterodyne receiver, however, the experimental setup did
not have the heterodyne phase locked (since when collecting power in red and blue sidebands, it issufficient to let the phase drift and average over all values). The simplest way to detect squeezed
light is in direct photodetection. With the near-resonant laser (operationally the locking laser)slightly red detuned, the squeezing, which normally occurs in the phase quadrature of the light,
is rotated into the amplitude quadrature which can be measured in direct detection. This entails
a slightly modified readout chain, since at some point before the usual heterodyne receiver, thetransmitted light from the cavity is split off onto a photodetector. This can be useful in practicesince it separates the heterodyne visibility, and any differences in propagation loss can be easily
calibrated out. By using the locking beam to drive the cavity to produce squeezed light, we candirectly calibrate the detection efficiency . In Fig. 6.11, with a fixed cooling laser power, the lockpower is increased, giving the successive superimposed lineshapes measured in direct detection.
With knowledge of the mechanical resonator’s damped linewidth and temperature from sideband
asymmetry measurements, fits to a model can extract the collection efficiency of the squeezing
measurement, 0 . In this regime, thermal effects are small compared to collection efficiency, whichquickly saturates in almost all of the traces in Fig. 6.11. We find 0 = 0.50, consistent with our90
Table 6.1: Ratio of radiation pressure to thermal forces in selected publications, in order of publishing date.
Paper Ref.
Rs

NJP [19]
0.091

RPSN [22]1.2Squeezing [68]5.1

Backaction limit [3]14Variational readout [31]3.4

determination of  = 0.04 from the sideband asymmetry measurements in the main experiment.
With additional propagation losses of 5% to the heterodyne receiver, a visibility of 0.42, and afactor of 0.5 for a heterodyne measurement, we have 0.5 × 0.95 × 0.422 × 0.5 = 0.04. This verifies
that extracting  from the sideband asymmetry measurement is viable and accurate.

6.7

Historical trajectory of device performance
By quantifying the performance of successive devices, a big-picture trend can be seen which

informs understanding of current and future capabilities. Table 6.1 shows the performance ofnotable results in terms of the ratio of radiation pressure to thermal forces, Rs ,
Rs =

C
1
nth 1 + (2ωm /κ)2

(6.3)

The trend is that after publication of initial work in the radiation-pressure dominated regime, themost important quantum measurement publications demonstrate an increasing level of radiationpressure forces dominating the dynamics. Work such as the initial publication on efficient microwave
to optical conversion is not in the quantum regime (Rs = 0.0011 for noise measurements), although
as an initial demonstration of a new technology, one expects it not to be as developed in performanceas concurrent work on well-understood systems [42]. As a single metric, it does not tell the entire
story of the relevance of individual works. For example, one of our latest publications reports ameasurement sensitivity of 5.3 times the SQL, the lowest to our knowledge, and also demonstrates
sensitivity off the mechanical resonance below the SQL for its quantum efficiency [31]. These claims,
which benefit from both a large Rs as well as a large  are not fully represented in our choice of Rs
as a metric.

91

6.8

Comment on operation in an optical-access dilution refrigerator
This experiment benefited from the much lower bath temperature provided by a dilution

refrigerator. Even though this first attempt exhibited a higher bath temperature than that of
the bulk cryostat temperature, subsequent work has reduced the magnitude of the RPSN of thelocking laser and other auxiliary effects to a level where the mechanical mode can now thermalize
to a temperature very close to the bulk as read out via standard cryogenic thermometry. However,choice of our dilution refrigerator design involved much consideration and design work. Initial work
I performed on characterizing the 4 K system used for the microwave to optical conversion project
informed us that LN2-jacketed bath cryostats produce significant vibrational noise power, even at
MHz frequencies, due to turbulence in the liquid as it boils off. At the time, low-vibration pulse-tube
technology was in its early stages (and to this day a sub-kelvin low-vibration pulse-tube cryostat
simply does not exist). We selected a cryostat built by Cryoconcept that was engineered for opticalaccess by tipping the conventional dilution refrigerator design on its side [99]. The first model wasbuilt for a group at LKB in Paris, and its use is described in Ref. [100]. The design allows forthermal contraction to occur along the optic axis, nearly eliminating optical misalignment during
cooldown. A side port allows transmission measurements via a 45◦ mirror mounted at the 4 K stage.
A LHe bath and vapor-cooled shields allow for very low vibration compared to other options. The
windows were custom-designed by us to achieve the desired thermal performance, using a thermal
model to calculate the attenuated 300 K blackbody heat load after passing through the fused silica
substrates used for our cold windows, which are placed at the still, 4 K, and both vapor-cooled
stages [101]. The use of 1 cm diameter apertures in the windows restricts the solid angle to a level
where, combined with the partial attenuation of the fused silica, the heat load at base is only a
few microwatts. Moving a shutter that blocks the rear optical access (and associated blackbody
radiation) while cold resulted in no increase of the 30 mK base temperature of the refrigerator.

Chapter 7

Electro-optomechanical experiments

7.1

Description via transfer matrix formalism, scattering parameter measurement
The derivation of transfer functions between microwave, optical, and mechanical inputs/ouputs

in Chapter 2 establishes a measurement and characterization concept that is directly translatable
to experiment. Specifically, by producing coherent excitations with a network analyzer, the input
ports of the converter can be probed, and the desired output field’s response can be detected andcompared to a reference in the network analyzer. By using upper sideband modulation on either
the microwave or optical pump, and heterodyne detection of the desired output field to isolate theupper sideband response, the measurement directly corresponds to the transfer function element
such as t1 = ξaout,L ,bin,R linking the annihilation operators of the particular elements in ain and aout
(here the microwave input and the optical output). This is the process responsible for potentiallynoiseless transduction between the input and output modes. The transfer fuction’s particular form
is given by Eq. 2.22, which looks like an optical cavity in transmission, with the relative strengths ofthe coherent coupling rates Γe and Γo determining an impedance-matching condition, and their sum
determining the bandwidth of conversion. The transfer function linking lower-sideband excitations(creation operators) represents the amplification or entanglement mode of the device, while crossterms capture nonideal performance. In particular, the element ξa†out,L ,bin,R, representing upconver-sion of the upper-sideband signal to the lower sideband of the optical pump, corresponds to gain (orequivalently added noise) in the conversion process, placing a fundamental limit on how noiseless93

Figure 7.1: a.) Schematic of converter, consisting of an LC circuit and an optical cavity coupled
to the same Si3 N4 drumhead resonator. b.) Frequency-domain picture of the two electromagnetic
resonators, the electrical and optical pump tones, and the upper-sideband tone to be upconverted
or downconverted. c.) Rendering of the system, showing the optical cavity mirrors, which contain amembrane chip, and a second chip which capacitively coupled to membrane motion via a patterned
superconducting Nb circuit. Adapted from Ref. [42].

94
the converter can be. This process is suppressed in the resolved sideband regime (2ωm > κ{e,o} /2)
by the two electromagnetic cavities, but plays a finite role in any practical experiment. The effective
gain implied by this process is given by Eq. 2.23, where in the following text useful simplifications
are provided.
The measurement procedure consists of modulating the signal from a network analyzer (at rf
frequencies near ωm ) as an upper sideband on the microwave or optical pumps, and demodulating
the reflected pump signals to separate the upper and lower sidebands in detection (Fig. 7.2). The
transfer functions of the converter are here referred to simply as t1 , t2 , r1 , and r2 for reflection andtransmission measurements, with i = {1, 2} referring to probes originating from the microwave andoptical ports, respectively. The question of calibrating out any loss or gain in the experimental
setup leads to a interesting result, in that for measurements of t1 and t2 , there is a calibrationfree way to measure the symmetric transmission product t1 t2 via making four measurements. By
0moving the pumps far off cavity resonances ωe and ωo and making reflection measurements r1,off
0, where the prime denotes the raw, uncalibrated measurement, the electromagnetic cavities
and r2,off

both have a reflectivity of 1 and we just measure any loss and gain in the measurement circuit.
Then, we havet1 t2 =

t01 t02
00
r2,offr1,off(7.1)

where by dividing out by the losses and gains, t1 t2 is obtained immediately. Reflection measurements do not benefit from such a trick, but by using a wide probe scan, the relative size of any
reflection feature is easily obtained.
Our measurement system is now more sophisticated than our original work in Ref. [42]. In
particular, it now has optical heterodyne detection, which can directly measure the desired upper
sideband of the pump (rather than inferring it from a measurement of the photon number spectrum
in direct photodetection). The converter has also been moved into a dilution refrigerator, substantially reducing the thermal environment nth and potentially enabling quantum-limited performance.
This new setup is reflected in Fig. 7.2, and shown in more detail in Fig. 7.3.

95

Figure 7.2: a.) Transfer function elements denoting transmission and reflection. Gain and loss
in the measurement chain are denoted by α, β, γ, and δ. b.) Simplified experimental schematic.
Adapted from Ref. [42]96

Figure 7.3: Detailed schematic of the converter measurement setup. This version can be comparedto the original in the supplement to Ref. [42]: this setup now has optical heterodyne detection anda dilution refrigerator instead of a 4 K cryostat. For a more detailed representation of the opticalsetup, see Fig. 6.5.

97

7.2

Comments on modematching in heterodyne
While our original results required a calibration of the effective upper sideband amplitude due

to measuring in direct detection, it turns out that this measurement scheme was ideal for measuring
the modematching to the cavity . The modematching factor naturally appears symmetrically in theequations in Chapter 2 because it appears in optical probing and detection of the cavity. Specifically,
only the modematched portion of the light drives the cavity, while in direct photodetection themodematching appears again due to the signal fluctuations beating only against the modematched
portion of the optical carrier (see supplemental material in Ref. [92]). In heterodyne detectionof the optical pump, the measurement of the conventional visibility η applies only to the total
optical carrier, while the signal is (in general) emitted into a different spatial mode than the
carrier. In effect, this breaks the parallelism of  in the case of direct detection, which allows forthe convenient calibration procedure given in Eq. 7.1. While  still appears in the driving term,
detection of the emitted signal is set by the overlap of the cavity mode to the local oscillator, notthe pump laser. This value, distinct from the conventional η, we can call η 0 , and can be determinedby measuring the visiblity of a separate laser transmitted through the cavity from the other side√against the local oscillator. Practically speaking, the correspondence  → η 0 should apply toheterodyne measurements of an imperfectly-modematched pump in reflection, although this requiresa manual modification of the otherwise self-calibrating upconversion and downconversion efficiency
measurements.

7.3

Original results
In this section, I will briefly review the results from our original experiment, which aredetailed in both a paper and Reed Andrews’s thesis [42, 92]. To clarify the differences between
this original experiment and subsequent efforts, I will note that this used the original converter
design detailed in Chapter 5 (as opposed to the wireless design), operated at 4 K (instead of
30 - 100 mK), and did not have optical heterodyne detection (though as I noted, the inferred

98
upper sideband amplitude was calculated and presented in the results). By scanning the probe
tone ω across several mechanical modes (ωm = {380, 560, 1240} kHz), the different responses could
be measured (Fig. 7.4). In particular, they differ in apparent efficiency of |t1 t2 | since they haveappreciably different resolved sideband parameters, and thus different amounts of (undesirable)
gain A (see Eq. 2.23). After accounting for A, we infer a bidirectional efficiency of 0.086 ± 0.007,
and a coherent transfer bandwidth of 30 kHz. Operating at a bath temperature of 4 K, we did not
expect it to operate in the quantum regime. Its noise performance is shown in Fig. 7.5, where as
the coherent couplings Γe and Γo are increased, the signal to noise increases (and thus added noise
nvib decreases).

7.4

Ongoing progress towards the quantum regime
Current work in the group is focused on bringing our converter devices into the quantum

regime. The main improvement in this regard is to operate the converter in a dilution refrigerator,
bringing the bath temperature from 4 K to 100 mK. My demonstration of backaction-limited cooling
was the first time one of our optomechanical cavities was used in a dilution refrigerator; reaching
the backaction limit demonstrates the same parameters necessary to convert quantum states. Thatsuggests that reaching the quantum regime is possible for our devices. This section will describe
ongoing progress towards that goal, including operation in a dilution refrigerator, incorporating
new wireless cavity designs, and improving optical measurement via heterodyne detection.
The wireless converter has been operated a handful of times as the new design is being
tested and characterized. Preliminary results show that improvements in conversion efficiency havebeen successful. The initial cooldown of the converter showed roughly the same ∼ 10% efficiency,
while the third cooldown showed a factor of two improvement, to 20%. This is primarily due toimprovements in the optical cavity loss, and modematching of the pump beam to the optical cavity’s
spatial mode, both of which limited previous efficiency. The optical cavity, by using asymmetricmirror transmissions (7 ppm and 96 ppm instead of two 96 ppm mirrors in the first converter),has an improved collection fraction of 0.32, compared to 0.20 (internal loss still plays a large role),

99

Figure 7.4: a.) Mechanical modeshapes of the modes used for conversion. b.) Power scan of theconversion sweeps, showing the impedance matching condition to maximize efficiency. c.) Individual upconversion and downconversion spectra showing amplitude and phase response. Adaptedfrom Ref. [42].

100

Figure 7.5: Added noise during upconversion at 4 K. Inset: Optical spectrum of upconverted
microwave signal on top of damped thermomechanical motion. Adapted from Ref. [42].

101while the optical modematching has been increased from 0.5 to 0.85.
The noise properties of the wireless converter are also much improved (due to operation
in a dilution refrigerator), but are still not in the quantum regime. While the best parametersfrom various cooldowns would together allow for quantum-limited performance at Γe + Γo ∼ 10
kHz, technical noise larger than the damped thermomechanical motion causes nvib to saturate(Fig. 7.8). At this point, it is fair to say that ongoing efforts to test and refine the design of thewireless converter stand a good chance of overcoming these obstacles soon. Potential problems
include scattered optical light causing loss (or noise) in the superconducting LC circuit, blackbody
radiation, or parameter noise in the microwave circuit of other origin.

7.5

Future advances in detection of upconverted quantum states
The ultimate goal of a converter which can upconvert microwave quantum states to opticalfrequencies will require additional experimental tools to fully realize. While there is progress on
many fronts, in particular the creation of on-demand microwave single photons that could serve as
the input to the converter [102], the optical setup for detection at the single-photon level has not yet
been demonstrated in our lab. The principal problem is to filter the large, coherent pump—which
could have as many as 108 photons yet is only ωm detuned in frequency—to allow for detection of
single photons. With a frequency spacing in the MHz, an optical filter cavity is perhaps the only
practical way to sufficiently filter out such a large signal. With high-quality mirrors commerciallyavailable at the < 10 ppm level for cavity ringdown spectroscopy applications [103, 104], suchan application is feasible. Using 1 m ROC mirrors in a concentric configuration (` ≈ 2 m), a
F = 450, 000 cavity has a linewidth of κ = 167 Hz. Far off resonance, the suppression goes as(δ/κ)2 = 3 × 108 for δ = 1 MHz. This provides sufficient filtering of the pump, which would
be reflected from the input mirror and, if necessary, prevented from propagating back into theconverter using a Faraday isolator. The crosstalk of the isolator need not be as good as the cavity
suppression, since it just needs to attenuate the reflected pump to be a small fraction of the pumpdriving the converter.

102

apparent efficiency

1
6
4
2

0.1
6
4
2

optomechanicalrate = 1 kHz

0.01
6
2

3

10
10
10
electromechanical conversion rate (Hz)4

Figure 7.6: Power sweep for first cooldown with the wireless converter design. This achieved similar
efficiency to our original converter, at approximately 10%.

Figure 7.7: Power sweep for third cooldown with the wireless converter design. This achieved ourbest efficiency to date, approximately 20%.

100

32

80

40

60

53

40

80

500

1000

Γe/2� (Hz)1500

Added vibrational noise(photons s-1 Hz-1)

SNR (linear units)

103

2000

Figure 7.8: Added noise during the first cooldown of the wireless converter design. While much
improved from previous results due to operation in a dilution refrigerator, the added noise is limited
by technical noise particular to this cooldown, and is not in the quantum regime.

104
The more troublesome question in this scheme is how to stabilize the filter cavity without
introducing additional photons that would be transmitted on cavity resonance. In the typicalPound-Drever-Hall technique, an fm-modulated laser on cavity resonance is used [105, 106]. Whilein principle the modulation depth could be set to exactly suppress the carrier, this would also
reduce the amplitude of the error signal to zero. To reduce the effects of RPSN, we have worked
to decrease the power necessary to lock our optomechanical cavities. Yet we still operate at the
100 nW level, which is equivalent to a pump of as many as 105 intracavity photons. Suppression
at this level via polarization is beyond our usual level of control, but is possible in principle with
Glan-Thompson polarizers, which claim an extinction ratio of 105 . A dual-coating mirror could be
used to lock the cavity using a different wavelength of light. This complicates the mirror coating
design, although dichroic cleanup filters are commercially available with similar extinction to thebest polarizers. A speculative idea to get around this problem is to use a double-barreled cavity
which forms two adjacent spatial modes: one for actively stabilizing the cavity length, the second
to serve as a seemingly passively-stable filter (Fig. 7.9). This has the benefit of reducing the opticalcrosstalk to zero, if it proves to be a practical idea.

105

Figure 7.9: Speculative design for a double-barreled cavity, where one spatial mode (red) can be
actively stabilized via a piezo actuator common to both cavity modes, thus stabilizing the otherspatial mode (magenta) which can be scanned to the desired filter frequency with an additionalpiezo actuator.

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