Quantum Optomechanics with Engineered Membrane
Resonators
byPen-Li Yu
B.S., National Taiwan University, 2007

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
2016This thesis entitled:
Quantum Optomechanics with Engineered Membrane Resonators
written by Pen-Li Yu
has been approved for the Department of Physics

Cindy Regal

Konrad Lehnert

Date

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

iii
Yu, Pen-Li (Ph.D., Physics)
Quantum Optomechanics with Engineered Membrane Resonators
Thesis directed by Prof. Cindy Regal

Mechanical oscillators coupled to an electromagnetic cavity have emerged as a newfrontier in quantum optics. This coupling presents an opportunity for manipulating thequantum state of light, connecting different quantum resources, and ultra-sensitive force
detectors. A particular enabling platform utilizes high-stress silicon-nitride membrane resonators. Because of its large tensile stress, the membrane exhibits remarkable mechanicalquality factors (Qs) that are even higher than that of single crystalline silicon. The membrane with such high Q combined with robust cryogenic Fabry-Perot cavity enables a varietyof quantum optics experiments, such as squeezed light generation. Improving upon theseencouraging results requires understanding and engineering the membrane resonators. The
membrane dissipation can be classified as the internal loss and the external loss. We areable to manage the internal loss of a hybrid membrane with a curvature map, and control
the external loss of a membrane with a phononic crystal (PnC) shield. In this dissertation, I
will present squeezed light generation, our studies and engineering of high-stress membranemechanics, and optomechanical Raman-ratio thermometry with a PnC-isolated membrane.

Dedication

To my best friend and dearest wife, Bethany Huo.

v

Acknowledgements

I am very grateful to my advisor Cindy Regal, who thoughtfully, wisely, and efficiently
guided the overall development of my thesis work. I also thank Thomas Purdy for his humor
and hands-on help. I thank Nir Kampel for valuable insights. I am grateful for Katrina
Cicak’s awesome fabrication. I especially wish to thank Yeghishe Tsaturyan for creative
input. I also want to thank Robert Peterson for delightful discussions. I thank Adam
Kaufman and Brian Lester for creating a relaxed community. I thank every staff in JILA forbeing so supportive. David Alchenberger, Hans Green, Blaine Horner, and J.R. Raith areenormously appreciated. I am very proud to work in this vibrant and creative environment.
I will always be a JILAn!

Contents

Chapter
1 Introduction
1.1

1Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.1.1

High Stress Membrane Mechanics . . . . . . . . . . . . . . . . . . . .

2

1.1.2

Generation of Squeezed Light . . . . . . . . . . . . . . . . . . . . . .

3

1.1.3

Raman-Ratio Thermometry . . . . . . . . . . . . . . . . . . . . . . .

42 Internal Loss of High-Stress Membrane Resonator5

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.2

Quality Factors of High-Stress, Fully Metalized Membranes . . . . . . . . . .

6

2.3

Discussion: High-Stress Membrane Mechanics . . . . . . . . . . . . . . . . .

10

2.4

Ultrahigh-Q Metallized Membrane by Avoiding the Edge Curvature . . . . .

113 External Loss of High-Stress Membrane Resonators

15

3.1

Picture I: Radiation Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

3.2

Picture II: Hybridization of the Membrane Modes and the Substrate Modes .

17

3.3

Experimental Evidence I: Qs Depend on the Mode Shape Symmetry. . . . .

18

3.4

Experimental Evidence II: Qs Depend on the Frequency of the Substrate

3.5

Mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

FEM Simulation of the External Loss . . . . . . . . . . . . . . . . . . . . . .

20

vii
4 A Phononic Bandgap Shield for High-Q Membrane Microresonators

22

4.1

Geometry of the PnC Devices . . . . . . . . . . . . . . . . . . . . . . . . . .

23

4.2

Fabrication of the PnC Devices . . . . . . . . . . . . . . . . . . . . . . . . .

24

4.3

Isolation Profile: Externally Driven Mechanical Spectrum . . . . . . . . . . .

25

4.4

Probing the Non-Membrane Modes . . . . . . . . . . . . . . . . . . . . . . .

27

4.5

Simulation of the Membrane Modes and the Non-Membrane Modes . . . . .

28

4.6

Loss Factor Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

4.7

FEM Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

5 Ponderomotive Squeezing with a Membrane Mechanical Resonator

34

5.1

Introduction to Ponderomotive Squeezing . . . . . . . . . . . . . . . . . . . .

34

5.2

Quantum Descriptions of Light . . . . . . . . . . . . . . . . . . . . . . . . .

36

5.2.1

Quadrature Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . .

36

5.2.2

Coherent States and Squeezed States . . . . . . . . . . . . . . . . . .

38

Squeeze Light by Correlating Two Quadratures . . . . . . . . . . . . . . . .

39

5.3.1

Kerr Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

5.3.2

Optomechanical Kerr Effect . . . . . . . . . . . . . . . . . . . . . . .

40

Experimental Requirements for Observing Ponderomotive Squeezing . . . . .

44

5.4.1

RPSN Needs to be Greater than the Thermal Noise . . . . . . . . . .

45

5.4.2

Quantum Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

5.4.3

Maximum Squeezing with Sufficiently Large Nonlinearity . . . . . . .

46

Our System and Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

5.5.1

Direct Photodetection . . . . . . . . . . . . . . . . . . . . . . . . . .

47

5.5.2

Balanced Homodyne Detection . . . . . . . . . . . . . . . . . . . . .

50Discussion, Conclusions, and Future Aspects . . . . . . . . . . . . . . . . . .

52

5.6.1

A Kerr Medium with Finite Response Time . . . . . . . . . . . . . .

52

5.6.2

Conclusions and Future Aspects . . . . . . . . . . . . . . . . . . . . .

54

5.3

5.4

5.5

5.6

viii
6 Optomechanical Raman-Ratio Thermometry

55

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

6.2

Principle of Raman-Ratio Thermometry . . . . . . . . . . . . . . . . . . . .

57

6.3

Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

6.4

Optical Damping and Effective Mode Temperature Measurement . . . . . . .

59

6.5

Physical Temperature Measurement . . . . . . . . . . . . . . . . . . . . . . .

61

6.6

Systematic Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

6.7

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

7 Future Directions and Conclusions

65

Appendix
A High-Stress Membrane Mechanics

66

A.1 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

A.1.1 Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

A.2 Normal Modes of a High-Tension Plate . . . . . . . . . . . . . . . . . . . . .

67

A.2.1 1D Eigensolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

A.2.2 Zero-Order Solution. . . . . . . . . . . . . . . . . . . . . . . . . . .

69

A.2.3 First Order Correction . . . . . . . . . . . . . . . . . . . . . . . . . .

69

A.2.4 1D Eigensolution in the Small λm Limit . . . . . . . . . . . . . . . .

70

A.2.5 Mode Functions of Bilayer Plate . . . . . . . . . . . . . . . . . . . . .

70

A.3 Anelastic Loss of a High-Stressed Plate . . . . . . . . . . . . . . . . . . . . .

70

A.3.1 Stored Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

A.3.2 Effective Loss Modulus . . . . . . . . . . . . . . . . . . . . . . . . . .

73

A.4 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

A.4.1 Isotropic Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

A.4.2 Curvature Induced near the Clamped Edge and around the Antinodes

74

ix
A.4.3 Edge Loss and the Frequency Dependence of the Quality Factor . . .

74

A.4.4 Physical Meaning of λ . . . . . . . . . . . . . . . . . . . . . . . . . .

77

B Calculation of Optical Spectra of the Squeezed Light78

B.0.1 Heisenberg-Langevin Equations . . . . . . . . . . . . . . . . . . . . .

78

B.0.2 Optical Output Spectrum . . . . . . . . . . . . . . . . . . . . . . . .

80

C Laser Classical Noise Measurement

84

C.1 Mathematical Description of the Laser Noise . . . . . . . . . . . . . . . . . .

85

C.2 Classical Amplitude Noise Measurement . . . . . . . . . . . . . . . . . . . .

86

C.2.1 Mathematical Description of the Direct Detection . . . . . . . . . . .

87

C.2.2 Mathematical Description of the Balanced Detection . . . . . . . . .

88

C.3 Classical Frequency Noise Measurement . . . . . . . . . . . . . . . . . . . . .

89

C.3.1 Mathematical Description of the Delayed-Line Heterodyne (Homodyne) with Balanced Detection . . . . . . . . . . . . . . . . . . . . .

91

C.3.2 Heterodyne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

C.3.3 Homodyne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

C.4 Data and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

C.4.1 Classical amplitude noise Cxx measurement . . . . . . . . . . . . . . .

93

C.4.2 Classical frequency noise Cyy measurement . . . . . . . . . . . . . . .

93

D Light and Mechanics Spectra with Laser Classical Noise96

D.1 Three effects from the classical noise . . . . . . . . . . . . . . . . . . . . . .

96

D.2 Error analysis in ratio thermometry . . . . . . . . . . . . . . . . . . . . . . .

99

Bibliography102

Chapter 1
Introduction

Mechanical oscillators coupled to an electromagnetic cavity have emerged as a new frontier in quantum optics. This coupling presents an opportunity for manipulating the quantum
state of light, connecting different quantum resources, studying quantum measurement backaction in the optical detection of macroscopic objects, and ultra sensitive force detectors.
A diverse suite of quantum optomechanical systems have been developed, covering nearly
20 orders of magnitude in effective mass and 10 orders of magnitude in frequency [1]. A
particular enabling platform utilizing high-stress silicon-nitride membrane resonator was introduced by the Harris group in 2008 [2, 3]. In this realization, a membrane is placed in astandard high-finesse Fabry-Perot cavity. The silicon nitride membrane exhibits remarkablemechanical quality factors (Qs) that are even higher than that of single crystalline silicon.
Because of their large tensile stress, Si3 N4 membranes can have Q-frequency products exceed
1014 Hz.
Membranes with such high Q combined within robust cryogenic Fabry-Perot cavity [4]have proven fruitful. Our lab focuses on making cryogenically compatible membrane optomechanics devices where thermal effects can be reduced and quantum aspects of the optomechanical system revealed. Our experiments can often be discussed using a common
metric for ones ability to reach a quantum limit. We define an optomechanical cooperativity
as C = 4G2 /κγm , where G is the many-photon optomechanical coupling, κ is the cavity
linewidth, and γm is the mechanical dissipation (these parameters will be defined in more2detail later). Cooperativity greater than the thermal phonon occupancy (nth ) enables avariety of experiments: observation of quantum measurement backaction [5], squeezed light
generation [6], cooling the mechanical mode to its motional ground state [7], and microwave
to optical frequency conversion [8].
Improving upon these encouraging results and exploring new directions requires understanding and engineering the membrane resonators. Early work in my dissertation focusedon understanding tensioned membrane resonators whose quality factors and their dependences were not clear to the optomechanics community [9, 10, 11, 12, 13]. These studies
focused on investigating both internal loss and external loss-mechanisms, and have resulted
in important predictive power. In this thesis, I show how we are able to manage the internal
loss of a hybrid membrane by understanding a spatial map of the curvature of a membrane [14], and we control the external loss of a membrane by creating a phononic crystal(PnC) shield [15, 16]. I have also been involved in the optomechanics experiments within thelarger group; I worked on our study of squeezed light generation [6], and in this dissertation
I elaborate on how to understand the ponderomotive squeezing we realized. As a final goalof my dissertation work, I wanted to show that engineered membranes can have a positive
impact on quantum cavity optomechanics experiments. This was realized in our experiments
on optomechanical Raman-Ratio thermometry with a PnC-isolated membrane that I present
in this dissertation [7, 4]. Lastly, as I discuss in the outlook, the PnC-isolated membranes
are seeing application in a number of new experiments in progress.
1.1

Outline1.1.1

High Stress Membrane Mechanics

Tension in the Si3 N4 membrane strongly alters its dynamics. A variety of loss mechanisms considered extensively for untensioned resonators in the fields of micro- and nanoelectromechanical systems (MEMS and NEMS) required analysis in the large stress regime.

3
Around 2010, the internal loss in this regime (loss associated with the elasticity of the resonator), is first analyzed by scientists in the context of silicon nitride strings [11, 12]. In
Chapter 2, we develop a theoretical model to interprets our results of the 2-dimensional
Si3 N4 membranes with and without a metallic layer. We focus on the spatial dependence of
the internal loss and experimentally verify the prediction from the model that the damping
can be reduced significantly by patterning the metal film.
In Chapter 3, we investigate the external loss, loss associated with the surrounding
environment of the Si3 N4 membrane. I provide two intuitive physical pictures, and support
them with experimental and simulation results. In Chapter 4, I extend these pictures todesign, measure, and understand a membrane mechanical resonator in a PnC I design. A
unique aspect of our PnC design is it allows us to preserve the high tension of the membrane.

1.1.2

Generation of Squeezed Light

In Chapter 5 we investigate optomechanical squeezed light. The squeezing experiment
would be an important demonstration of the capabilities of manipulating the quantum stateof light. It is a natural extension of the RPSN work described in Ref. [5]. We first discuss thephysics of ponderomotive squeezing in the context of Kerr squeezing. We show that the nonlinearity, described by the magnitude of the complex Kerr coefficient, can be approximatedas the optomechanical cooperativity. We experimentally realize C/nth ∼ 5 and demonstratestrong and continuous optomechanical squeezing of 1.7 ± 0.2 dB below the shot-noise level.
The observed squeezing spectra suffer from the thermal noise of the substrate modes, andhence the results are a good example of experiments that will be improved in the future by
PnC membrane devices. The peak level of squeezing measured near the mechanical resonance is well described by a model whose parameters are independently calibrated and that
includes thermal motion of the membrane with no other classical noise sources.

4
1.1.3

Raman-Ratio Thermometry

In Chapter 6, with a PnC-isolated membrane, we investigate using Raman sideband
asymmetry observed for a membrane with low phonon occupation for the measurement ofeffective and physical temperatures of the membrane. Exceptionally good isolation from theenvironment by the PnC allows us to perform this thermometry at variety of environment
temperatures. In addition to the thermometry, this experiment also demonstrates nearground state cooling of a membrane mechanical resonator.

Chapter 2
Internal Loss of High-Stress Membrane Resonator

2.1

Introduction
The dynamics of the high-stress membranes have many qualitative differences from themore typical flexural mode resonators. For examples, the resonant frequency of our highstress Si3 N4 membrane is 1000 times larger than a non-stressed plate resonator with thesame dimension. These high-stress resonators are found to have remarkable mechanical Qs,
but their dependences on the stress was not always clear to the MEMS and optomechanics
community [9, 10]. For the case of Si3 N4 string, an elastic model including the stress effectwas verified quantitatively in the experiments of Ref. [11, 12]. The loss energy was foundto be proportional to the bending energy and observed to change much less than the totalenergy as the stress increased.
In this chapter, I investigate the internal loss of the 2D high-stress membranes. As
discussed in Chapter 3, we believe that in much of our work with pure Si3 N4 membranes weare limited by external loss, but internal loss is also imperative to understand quantitatively.
By coating ultrahigh-Q Si3 N4 membranes with a more lossy metal, we can precisely measure
the effect of material loss on Q’s of mechanical modes over a large range of frequencies. We
can also examine the spatial dependence of loss by patterning the coated metal. We developa theoretical model that interprets our results and predicts the damping can be reduced
significantly by patterning the metal film. Using such patterning, we fabricate Al-Si3 N4membranes with ultrahigh Q at room temperature. Our work elucidates the role of material

6
loss in the Q of high-stress membrane resonators.
In addition, this work informs the design of hybrid mechanical oscillators for opticalelectrical-mechanical quantum interfaces. A variety of proposed cavity mechanics experiments would be enabled by the addition of a metallic layer to Si3 N4 while maintaining
high Q [17, 18, 19, 20]. Foremost, a metallic membrane section could form a capacitor platethat couples to a microwave LC resonator; in fact, pure metallic drums have recently been
ground-state cooled using a combination of cryogenic and microwave cavity cooling [17].
With a hybrid dielectric/metallic membrane, one could couple mechanical motion simultaneously to optical light and a microwave electrical circuit in the quantum regime [18]. Such a
device could solve the difficult, yet crucial, problem of transferring quantum states between
microwave and optical photons. It could also enable enhanced detection of excitations ina room-temperature electrical circuit via photodetection [19]. Further, a magnetic metallic
film could be used for magnetic coupling of spins to membrane motion [20]. However, thesuccess of these applications will hinge on creating hybrid membranes with a sufficiently highquality factor at relevant temperatures.
2.2

Quality Factors of High-Stress, Fully Metalized Membranes
To distinguish the material loss and the external loss of a metalized membrane, we firstmeasured the Q’s of pure Si3 N4 membranes. We use 50 nm-thick stoichiometric LPCVD
nitride membranes that are supported by a 200 µm-thick silicon frame (from Norcada Inc.).
The membranes are in a square geometry of side length l =0.5 mm or 1 mm with tensilestress σ∼0.9 GPa and mass density ρ∼2.7 g/cm3 . The membrane mode shapes are given byapproximately sinusoidal functions like those shown in Fig. 2.1(b) with resonant frequenciespfmn ∼ σ(m2 + n2 )/4ρl2 , where m, n are the integer mode indices representing the numberof antinodes. The silicon frame is glued at three corners to a metal form on a piezoelectric
actuator. To probe the mechanical displacement, we position the membrane at the end of
one arm of a Michelson interferometer. We characterize the mechanical quality factor by

7

Figure 2.1: Geometry of membrane modes. (a) Image of a 1 mm membrane in its siliconframe. (b) Illustrations of the (m, n) = (2, 2) and (15, 15) modes. (c) Image of a patterned
Al film on top of Si3 N4 . The central square is Al and the rim is Si3 N4 suspended on a Si
frame. (d) Schematic diagram of the Al/Si3 N4 bilayer membrane (50 nm Al and 50 nm
Si3 N4 ).

monitoring the ringdown of the mechanical excitation in vacuum of less than 10−6 torr.
We deposit 50 nm of Al using e-beam evaporation on top of the pure Si3 N4 membranemeasured in Fig. 2.2(a) (green points). The membrane remains under large tensile stress,
but adding the additional film does decrease the effective stress to σeff = 0.35 GPa. With theaddition of the metal, we see a drop in Q to a maximum of ∼ 2 × 105 as shown in Fig. 2.2(a).
By drawing on our knowledge of the Q’s of the pure Si3 N4 membrane [Fig. 2(b)], we canclearly distinguish material loss and external loss (See Chapter 3). The open squares inFig. 2.2(a) represent the asymmetric modes found to be radiation-loss limited for the pure
Si3 N4 membrane. If we remove these points from the Al-Si3 N4 membrane measurements,
we arrive at a clean set of points (closed squares) representing the material loss-limited Q
as a function of frequency. Damping rates γ = 2πf /Q for two datasets obtained using thismethod are shown in Fig. 2.2(b).
We have developed a theoretical framework to describe the frequency dependence of

8

107

•

a
••• • • ••••••• •••• •••• • •• •• • • •• •                 


  


Q

106

105

104
0

1

Γ2Π Hz 

50
20

5

3

4

b

100

10

2

f MHz    
























 


  
  

2
1
0

1

2

f MHz 

3

4

Figure 2.2: Extracting the material loss-limited Q. (a) Measured quality factors of a square
Si3 N4 membrane before (green circles) and after (blue squares) adding a 50 nm film of Al.
The modes limited (not limited) by radiation loss are marked by open (closed) circles. The
data are plotted as a function of frequency measured after adding the Al. (b) Mechanicallinewidth of the modes limited by the material loss of Al for 0.5 × 0.5 mm (red diamonds)
and 1 × 1 mm (blue squares) membranes. To compare to theory, we calculate the damping
rate γmn /2π for each mode, and the points are connected with the displayed lines.

the material loss-limited quality factors of our two-dimensional structures. We model themembrane as an anelastic plate that dissipates mechanical energy under cyclic loading [21].
Under oscillation, the material’s strains and stresses are not in phase, and the energy supplied by the out-of-phase stresses is converted irreversibly to heat. This picture has beensuccessfully developed to understand damping in one dimensional Si3 N4 strings [11, 12].
In our case, we start by applying standard plate theory with an in-plane force [22],i.e., under tensile stress, to determine the normal modes. The modes must satisfy theboundary conditions of the clamped plate W =(∂/∂x)W =0 or W = (∂/∂y)W = 0 for all fouredges. We express the 2D mode function Wmn (x, y) as a product of stressed-beam functions

9
um (x)un (y). We have verified the accuracy of this description via perturbation theory (seeappendix A). We use a closed-form expression for the function un (x):vm (x),
0 ≤ x ≤ 2l
um (x) =(−1)m+1 vm (l − x), l < x ≤ l
2
 h

i
h −x ih mπx i
√
mπx
λmπ
+
exp
− cos
vm (x) = 2a sin
l
2
λl/2
l
r
mπ σ
ωm ∼
.
l
ρ

(2.1)

(2.2)
(2.3)

It is a sinusoid with an exponential correction near the edge for the clamped boundary
condition.
For each mode, we can calculate the loss due to anelasticity. The oscillation of theplate induces oscillating strains εxx eiωt , εyy eiωt , and εxy eiωt , and the accompanying stresses
are given by the usual constitutive equation of classical plate [23] with the complex Young’s
modulus Ẽ = E1 + iE2 , where E2 is called the loss modulus. During one cycle, the full
expression for the energy lost is
Z
∆U =

o
2πE2 (x, y) n (εxx + εyy )2 ε2xy+
− εxx εyy dV
1+ν
2(1 − ν)4

(2.4)

where ν is the Poisson’s ratio (see appendix A). Note, the strain term εxx = −z(∂ 2 W/∂x2 )is proportional to the curvature of the mode function. To calculate the quality factor, we
also need an expression for the total stored energy. It can be obtained from the maximumRkinetic energy U = 2ρπ 2 f 2 W (x, y)2 dV . The quality factor for a particular mode Wmn isthen given by Qmn = 2πUmn /∆Umn .
We start by using our theory to calculate the damping of fully-metalized membranes.
We apply a least-squares fit to the two datasets (two different-sized membranes) in Fig. 2.2(b)assuming a single frequency-independent loss modulus. This reveals an effective bilayer
E2 = 0.55 GPa. The corresponding Al loss modulus is consistent with typical values forthin-film polycrystalline Al at room temperature, as measured, for example, via depositing

10
Al on a low-loss Si cantilever [24]. The presumed microscopic origin of the loss is related to
crystallographic defects such as grain boundary sliding [25, 26] or kinks on dislocations [27].
Despite this underlying complexity, our model assumes very little about the microscopic
origin of the loss. Namely, we assume that the defects are uniformly distributed within thedeposited metal in the x and y directions. We also assume the temperature stays sufficiently
constant in our measurements so as not to affect the loss modulus. We have verified thatthe heating due to our measurement laser of power 150 µW is not a significant effect bymeasuring constant quality factors as the power is varied from 10 to 900 µW.
2.3

Discussion: High-Stress Membrane Mechanics
With continued analysis of the theory we can not only model, but understand the Q

dependencies seen in Fig. 2.2, and put our observations in the context of other studies in
1D and 2D [28, 29, 11, 12]. We would like to understand: (1) The frequency dependencies,
i.e., why an extremely corrugated mode has only a slightly lower Q than the fundamental
mode in our measurements (2) The geometry dependence, i.e., how damping should scalewith resonator size. First, we address the frequency dependence. As noted above in thediscussion of Eq. (1), the loss is given by an integral of terms proportional to the mode
curvatures. We identify two contributions to the curvature, namely that induced at theclamped edge and that near the antinodes in the interior of the membrane. If the curvature
at the edge dominates we expect a flat Q as a function of frequency, or if the antinodecontribution dominates we expect a decreasing Q as the frequency (and correspondingly
the number of antinodes) increases. We quantify these statements by deriving a simplified
expression for Q as a function of mode indices m and n for the limit of (1) an isotropic
membrane, i.e. constant E2 in x and y and (2) high-stress quantified by small λm and λnpwhere λ = E 0 h2 /3σl2 is a dimensionless stress parameter. Here E 0 = E1 /(1 − ν 2 ) and h,l, and σ are the height, length, and stress of the membrane respectively. In these limits,
Eq. (1) becomes an integral over squared sinusoidal terms (antinode contribution) and an

11exponential term (edge contribution) to give a total Q of (see appendix A)

−1
1 E1
(m2 + n2 )π 2
Qmn ∼
1 + λ
λ E2 |{z} |
{z4
}
edge
antinode(2.5)

The term λ(m2 + n2 )π 2 /4 determines whether there will be a frequency-dependent Q. For
our experiments, and similar experiments with large membranes [28, 29], λ ∼ 10−4−10−3 ,and hence we expect a relatively flat Q. However, if the edge length is decreased, λ increases
and the antinode contribution can become large. Hence, a frequency dependence appears
for experiments such as those in Ref. [11] where shorter strings (< 35 µm) are used.
Further, the prefactor 1/λ in Eq. (2) determines the geometry and stress dependence for
the Q of the fundamental mode. Physically, λ can be written as the ratio of bending energy
to elongation energy (see appendix A), and as discussed in Ref. [11], exciting energy in theform of elongation energy rather than bending energy leads to higher Q. More concretely,
based upon Eq. (2), we predict that if the membrane side length l is doubled, Q of thefundamental mode will double for the same loss modulus, and this is exactly what is observedin Fig. 2.2(b); the analogous effect in 1D was observed in Ref. [11]. We also see that as the√stress is varied, Q scales with σ, and hence the linewidth γ = 2πf /Q remains constant.
While Eq. (2) only holds in the stressed limit, a calculation in the zero stress (flexural) limit
reveals the linewidth increases by a only a few factors from the highly-stressed case.
2.4

Ultrahigh-Q Metallized Membrane by Avoiding the Edge Curvature
Our analysis above indicates that by making the loss modulus near the membraneedge small, we can reduce the loss significantly. Using our ability to control the addition ofmaterial loss with the Al film, we can directly test this prediction. The inset to Fig. 2.1(c)shows a 1 × 1 mm membrane where we deposited Al nearly everywhere except in a small∼50 µm region near the edge. The quality factors of this membrane were measured tobe dramatically higher (blue circles) than a control experiment (red squares) in which an12identical layer of Al was deposited everywhere on a separate membrane (Fig. 2.3). For thesedata we show the Q’s measured for all modes, but identify the lower-Q asymmetric modesby open circles or squares. For both datasets in Fig. 2.3 we anneal the membranes at 340o C
after depositing the Al film resulting in an effective stress of σeff = 0.6 GPa. While annealing
was not necessary for studying the fully-metallized membranes of Fig. 2.2, the unequal stressof the Al film in the partially-metallized membrane makes the higher-order modes difficult
to identify. The annealing mitigates this problem, but we still cannot identify modes past
3 MHz. Hence past this point we analyze the Q envelope by measuring the highest Q mode

Q

107



106

105

104
0

curvature 2 arb .

in every 50 kHz window.

• •• •• • • ••• •01
2
 •••••••• •••••••••••••••• ••••x Μm  • ••• •• • •••••••••• • •   •   • ••••••••••••••••••
          •••••••• ••• 






1

2

3

4

f MHz 

5

67

Figure 2.3: Ultrahigh-Q metal-covered membranes. Measured quality factors of two Si3 N4membranes with Al everywhere but near the edge of the membrane (blue circles) [seeFig. 2.1(c)] and a full film of Al as a control experiment (red squares). Asymmetric modes(with one mode index less than or equal to two) are marked by open circles or open squares.
Calculated quality factors for each geometry are shown by the two lines; a single loss modulus is used for both. (Inset) The square of the curvature of a stressed mode as a functionof distance along one coordinate of the membrane. This function decays exponentially near
the membrane edge.

We can again apply our theory to quantitatively predict the Q for this new geometry.
Since E2 now becomes a function of position on the membrane, we return to using the full
expression of Eq. (1). We assign a finite loss modulus for the Al region and zero loss for the
Si3 N4 rim. We approximate the mode functions as those expected for a uniformly stressed13
membrane. This reveals the solid line in Fig. 2.3, which we find scales as 1/n2 for diagonalmodes, as expected from antinode contributions. The dashed line shows the corresponding
result for the fully metallized membrane using the same loss modulus. (The loss modulus
found here is E2 = 0.3 GPa, which is a smaller value than for Fig. 2.2 due to the annealing.)
We find the theory successfully traces out the envelope of the measured Q’s. The lowest
order modes of the partially-metallized membrane reach as high as Q = 6.5 × 106 ; this falls
short of the predicted Q just from Al material loss (blue line) likely because these modes are
now again limited by radiation loss (see Chapter 3). In comparison, recent measurementsof metal microstrings at room temperature revealed Q’s of 103 − 105 [30, 31], and even at
cryogenic temperatures, where the metal’s material loss is significantly reduced, observed Q
values for tensioned microresonators are typically 105 −106 [32, 33, 27, 17].
It is elucidating to understand what would happen to the Q trends we observe for thepartially-metallized membrane upon varying the membrane stress. This requires analysisof the spatial dependence of the mode curvature in the membrane plane. In the inset toFig. 2.3 we see the high-curvature area only occupies a very small ∼1 µm region near theedge of a high-tension membrane (here we use our lowest σeff = 0.35 GPa); specifically, thedecay length is λl/4 (see appendix A). As the stress is reduced (and hence λ is larger), themode curvature becomes more uniformly distributed over the membrane plane. Hence wewould not expect a dramatic difference in Q for a purely flexural mode when avoiding lossy
material only at the edge.
The localized mode curvature of the tensioned membrane that we observe provides insight into a variety of membrane applications. Note for higher-order two-dimensional modes
the curvature varies along the edge of the membrane, i.e. there are low-curvature regions
near the nodes at the membrane edge (see appendix A). Thus, to create an electrical link
between a central metallized patch and external circuits, and maintain high-Q performance,
one could tailor metal connections to match up with the low-curvature regions near nodes at
the membrane edge (see appendix A). Further, membrane patterning via holes is a promis-

14
ing technique to increase reflectivity of membranes for optomechanics experiments, but likemetal deposition, also has potential to introduce defects. A full understanding of the mode
curvature of the two-dimensional membrane is important for understanding the change in
Q, or a lack of a decrease in Q, in recent patterning experiments [34, 35].

Chapter 3
External Loss of High-Stress Membrane Resonators

External loss is loss associated with the structure surrounding the resonator. Specifically, in our system, the structure is the supported silicon frame and the epoxy [see Fig. 3.1(a)].
The finite elastic coupling between the membrane and the silicon frame enables a dissipation
channel and extraneous noise. It is an important limitation to the mechanical dissipation,and hence the optomechanical cooperativity. These experiments include the ponderomotive
squeezing studied in chapter 5, optomechanical cooling, and optomechanical Raman-ratio
thermometer discussed in chapter 6. In this chapter, I describe two physical pictures of theexternal loss [Fig. 3.1(b)-(c)], and provide theory, simulation, and experimental evidences.
These two pictures are equivalent to some extent and complimentary to each other. The
understanding and observation of the external loss led to the project of phononic crystal(PnC) that is discussed in chapter 4.
3.1

Picture I: Radiation Loss
The most intuitive picture of the external coupling is radiation loss. In this picture, wehave a vibrating membrane and an infinitely large surrounding structure. As the membrane
oscillates, it radiates acoustic energy through generating elastic waves on the surrounding
structure. The energy is lost because the waves never return. They go off to infinity or realistically are absorbed by lossy material (such as epoxy) surrounding the finite-sized supporting
structure.

(a)

16

(c)

(b)

mem
g

sub

Figure 3.1: Basics of the external loss. (a) Photograph of a glued Si3 N4 device. The semitransparent square in the center is the Si3 N4 membrane. The black chip is the silicon frame.
The white globules in the three corners are epoxy. (b) Cartoon picture of the radiation
loss. Oscillating membrane (red) radiates elastic waves (yellow) on the supporting substrate(purple). (c) Coupled-modes model. The energy of the membrane mode (labeled as ‘mem’)is dissipated through coupling to the substrate mode (labeled as ‘sub’). The spring between
them represent the coupling between the two modes.

The radiation loss is expected to be low for symmetric modes (n = m) and high forasymmetric modes (n 6= m and n = 1 or m = 1). Therefore the radiation-loss-limited Qs as
a function of mode frequencies are non-monotonic. This trend was first studied by Ignacio
Wilson-rae and the Parpia group in Cornell [28]. This expectation can be understood by
the destructive interference of the elastic waves. We can think of the antinodes near the
periphery as point radiators with alternating π phase. Therefore, a more symmetric mode
generates higher-order destructive interference. One expectation from this picture is thatthe symmetric modes with even number indices have even lower radiation loss because thetotal force applied on the surrounding environment is zero. Specifically, (2, 2) mode and (4, 4)
mode are expected to have higher Q compared with (3, 3) and (5, 5). This parity dependence
is roughly consistent with our observations when the membrane are glued at four corners.
An analytic expression of the radiation loss for high-stress, square membrane resonators
is given by [28, 36, 37]:
Qrad
mn = α116π√

n2 + m2 h 1
1 i−1 l ρs+,
n2 m 2
n4 m4
h ρr

(3.1)

where ρs and ρr are the mass densities of the substrate and the membrane, respectively. The

17
prefactor α is a geometry-independent constant representing the acoustic mismatch between
the membrane and the substrate. As expected, the radiation loss is minimal at n = m. It isis maximal when n = 1 or m = 1 and n 6= m. However, this model predicts that Qnn ∝ n,regardless of the parity. It is worthwhile to note that in this expression, the radiation-loss
limited Q scales with the aspect ratio l/h.
3.2

Picture II: Hybridization of the Membrane Modes and the Substrate
Modes
Another picture of the external loss employs a coupled-oscillator model [Fig. 3.1(c)].

The two coupled modes are the membrane mode and the mechanical mode of the substrate.
The coupling enable the two modes to exchange energy when their resonant frequencies areclose to each other [29]. The energy of the membrane mode can then dissipated through themechanical loss of the substrate. The loss could be large if a fair amount of epoxy is applied
to the substrate. This coupled-modes picture is equivalent to the radiation-loss picture
because mechanical modes of the substrate are needed to carry radiated elastic waves. The
symmetry of the membrane modes determine the coupling strength between the substratemode and the membrane mode
In addition to an extra loss channel, the hybridization leads to extra mechanical modesin the spectrum of the membrane. These non-membrane modes have been observed andlimited optomechanical effects such as ground state cooling and ponderomotive squeezing
discussed in chapter 5 and Ref. [5]. The non-membrane modes have a large effective massand hence can always clearly distinguish between low-mass membrane modes and modes
that have a substrate component. The non-membrane modes are discussed in more detail in
Chapter 4.

18
3.3

Experimental Evidence I: Qs Depend on the Mode Shape Symmetry.
To probe the mechanical Q, we position the membrane at the end of one arm of aMichelson interferometer as described in Chapter 2. As shown in Fig. 3.2, we have the ability
to measure the quality factors of many modes (up to 150) with different symmetries and to
confidently assign a mode (m, n) to all measured points. When the data are plotted versus
resonant frequency [green circles in Fig. 3.2(c)], the Q is non-monotonic. However, when thedata are plotted as a function of mode index in each dimension [Fig. 3.2(a)-(b)], we see thatthe asymmetric modes (indices n and m dissimilar) have strikingly smaller quality factors
than do the more symmetric modes (n and m nearly equal). This observation is consistent
with expected trends for radiation loss of elastic waves through the membrane clamp. While
we see consistently low Q’s with highly asymmetric modes, we see some variability due to themembrane mounting structure, especially among the lowest order modes [38], as expected
for a radiation loss mechanism. We note that, for the high-order symmetric modes thatasymptote to Q’s over a million, it becomes unclear whether radiation or material loss is thedominant effect.

(a)

19

(b)

(c)

Figure 3.2: Radiation loss for a square membrane. Measurements of quality factors for manydifferent modes of an (a) 0.5 × 0.5 mm and (b) 1 × 1 mm Si3 N4 membrane. The symmetricmodes generally have higher Q than asymmetric modes, as predicted by a radiation loss
model [28].

3.4

Experimental Evidence II: Qs Depend on the Frequency of the Substrate
Mode.
In this section, I describe an experimental observation of the hybridization between amembrane mode and a substrate mode. The experiment is similar to that in the last section,
except here I measure the displacement spectrum driven by the piezo actuator. In a narrowfrequency range, I identified a membrane mode (the sharp features) and a non-membranemode (the bumps in the center) that are very close in frequency. I monitor them over a
period of more than one week. The resonant frequency of the membrane is drifting toward

20
the non-membrane mode at a speed of -0.3 kHz/day. The drifting may be the result of astress reduction due to stress relaxation in membrane or mounting epoxy, or difference inmass loading from surface contamination. I find that the Q of the membrane mode decreases
monotonically as their frequency separation (∆f = fmem − fsub ) is reduced. The Q decreases
from a initial value of ∼ 3 × 106 to < 4 × 105 when ∆f < 0.2 kHz.

(a)slow drift(b)Figure 3.3: Hybridization of a membrane mode and a non-membrane mode. (a) A membrane mode (sharp features) drifts toward a non-membrane mode (bump features around
1748.8 kHz). (b) The membrane Q decreases monotonically as their frequency separation(∆f = fmem − fsub ) is reduced.

3.5

FEM Simulation of the External Loss
In this section, I employ a finite-element-method (FEM) software COMSOL to simulate

the whole membrane device in order to visualize the external loss. See Chapter 4.7 forsimulation outline. I found many membrane modes and non-membrane modes of the device.

21
The membrane modes can be identified by the energy distribution: more than 99% of theenergy is stored in the membrane. Even through there is little energy stored in the substrate,
the energy distributions on the substrate show the interference patterns (see Fig. 3.4), as
expected from the radiation loss.
(a)

0

(b)

-50

(c)

(Energy in dB)

-100

-150
-170

Figure 3.4: Simulated interference patterns. (a) (1,1) mode. (b) (1,2) mode with directional
interference. (c) (2,2) mode with a higher-order interference. The energy stored in themembrane is higher than the other two modes, as indicated by the color of the energy
distribution.

Chapter 4
A Phononic Bandgap Shield for High-Q Membrane Microresonators

Before the work described in this chapter, the external loss in the membrane devices inour group and others has been addressed mainly by varying the techniques for grasping thesilicon frame [38, 39, 4]. Specifically, in our experiment we minimize the amount of epoxy
and the contact area between the silicon frame and its holder. However more sophisticatedtechniques for control of acoustic waves have been extensively studied in the field of acoustic
metamaterials. In particular, a phononic crystal (PnC) with acoustic bandgaps can be used
to filter or confine acoustic waves [40, 41, 41, 42, 43, 44]. The use of PnC bandgaps tosuppress the radiation loss of gigahertz, in-plane resonators has been demonstrated in thefields of microelectromechanical systems (MEMS) and optomechanics [45, 46].
This chapter describes our first attempt to implement the PnC technique to isolate ourmembrane. This work is originated from my experimental and theoretical work on the substrate engineering and Yeghishe Tsaturyan’s work on the PnC simulation. We demonstratea high-tension membrane inside of a silicon PnC structure that provides a shield for acoustic
modes at megahertz frequencies. We probe the membrane modes and the non-membranemodes by measuring the displacement spectra of the membrane and different componentsof the support structure. We find that inside the observed bandgaps the density and theamplitude of the non-membrane modes are greatly suppressed. In addition, the membranemodes inside the observed bandgap are shielded from an external mechanical drive by up to
30 dB.

23
The Qs from these first attempt devices were not particularly high; Q ∼ 3 × 106 at atemperature of 5K. This is possibly because these membranes are limited by the material loss
from defects generated in this new fabrication process. Nevertheless, based on the isolation
provided by one device described in this chapter, we have demonstrated a wide-range RamanRatio thermometer over a wide temperature range. With continuing engineering, we haveobserved a Q of (17±2)×106 for the (2,2) mode of a 500 µm × 500 µm × 100 nm membrane
in a dilution fridge with a base temperature of ∼ 500 mK. The device was fabricated in the
NIST cleanroom.
4.1

Geometry of the PnC Devices
The device consists of a patterned silicon substrate with a center island that contains

the high-tension square film of Si3 N4 suspended across a mm-scale frame [Fig. 4.1(a)]. The
fabricated membranes in the PnCs are experimentally confirmed to be under a high tensilestress of 1 GPa: The fundamental membrane frequency for devices A and B is 1.1 MHz.
The unit cell length scale required to create a bandgap centered at a frequency f can be
estimated by λ/2 = v/2f ∼ 1 mm, where λ and v are the acoustic wavelength and velocity
in silicon, respectively. For bandgaps centered at megahertz frequencies, we can fit 3 to 4
unit cells around the membrane with a 1 cm square chip. Our unit cell is composed of asquare block with four bridges [Fig. 4.1(c)] [47, 48].
We study two different devices (A and B) with different PnC shields [Fig. 4.1(a)],and a reference device (C) without the PnC shield. (See Table 4.1 for measured geometry
parameters.) In Figs. 4.3(a) and (b), we display the band diagrams for the two different PnCs
with a infinite number of unit cells; this calculation is completed with the finite-elementmethod (FEM) software COMSOL using the measured device parameters.

(b)

(a)

membrane

24

(c)

w
b
a

t

0.5 cm(d)membraneframedeliever energy

PnC

radiate energy

mem framepiezo& epoxy

Figure 4.1: (a) Photograph of device A. The outermost chip frame (CF) is connected to thepiezoelectric actuator at four corners (the blue dashed region). (b) Expanded view of thewhite-dashed regions shows a square membrane (M, yellow) surrounded by a membrane frame(MF, light blue) and a PnC unit cell (PnC, light blue). The red, green, and purple spots in(b) and the blue spot in (a) are locations of displacement measurements in Fig. 4.3 (c)-(f).
(c) Schematic of the PnC unit cell and definitions of the geometry parameters. See Table 4.1
for the values of these parameters for the devices A and B.

4.2

Fabrication of the PnC Devices
This section outlines our first fabrication recipe developed by Katarina Cicak (Fig. 4.2).

The fabrication was carried out by Kat and me. The process begins with the growth of a
100-nm-thick Si3 N4 film by low-pressure chemical vapor deposition on both sides of a 300µm thick Si wafer. The membrane and PnC structure are created in two sequential steps;
each starts with the patterned removal of the back Si3 N4 layer followed by deep reactive-ion
etching (DRIE) for bulk Si machining. In the first step, the DRIE stops 10s of microns
short of etching fully through the wafer, and a KOH wet etch completes the release of theTable 4.1: Measured geometry parameters of the devices
Definition [µm]number of unit cellsb
unit cell size
block length
bridge width
wafer thickness
membrane length
membrane frame size
membrane thicknessab

symbolabwtlDevice A
31100 µm
686
97
300
372
786
0.1

Device B
4.5
800 µm
542
96
300
367
783
0.1

25

Device Ca
500
104
0.04

Norcada Inc.
between the center and the edge of the chipsquare Si3 N4 membrane on the front of the wafer. In the second step, the PnC crosses aremicromachined with DRIE all the way through the wafer (resulting in PnC holes that arevertical to ∼ 1◦ ). During fabrication (except for the KOH step), the front side of the wafer
is glued with processing adhesive to a protection substrate, and the final devices are released
from the protection substrate and cleaned using solvents and a sulfuric-acid-based solution.
4.3

Isolation Profile: Externally Driven Mechanical Spectrum
To characterize the mechanical properties of the devices, we employ the reciprocity

principle: delivering energy from the chip frame to the membrane is the reverse processof radiating energy from the membrane to the chip frame. Therefore the displacement
spectrum of the membrane under an external drive is a measure of the isolation profile. We
excite the chip at different frequencies through a piezoelectric ring actuator connected to all
four frame corners with double-sided tape and measure displacement using a Mach-Zehnderinterferometer. This interfermetry scheme allows us to easily probe displacement at different
chip locations. In addition, compared with the detection system used in previous chapters,
we improve the ground loops to reduce the noise floor and enhance displacement sensitivity.
First we present studies in which we probe the displacement of the Si3 N4 membrane.
We position the optical spot slightly off the membrane center to allow a variety of modes

26
ResistSilicon Nitride
Silicon
Wax
Sapphire

a
b
c
d
e
f
g

Figure 4.2: Fabrication process of the PnC-shielded membrane. (a) 100 nm of high-stress
Si3 N4 film is grown by LPCVD process on a 300-µm-thick Si wafer. (b) The sample wafer
is attached to a mounting sapphire substrate with processing wax. (c) Membrane window
is lithographically defined in a resist mask on the sample, followed first by a CF4 + O2
plasma etch to remove the top Si3 N4 , then by deep reactive ion etching (DRIE) into the Si
wafer. The etch is stopped with tens of micrometers of Si left in the trench. (d) The sample
wafer is separated from the mounting substrate by heating to reflow the wax and soaking in
acetone. The remaining Si in the trench is removed by a KOH wet etch, finally releasing the
Si3 N4 membrane. (e) Next, the sample wafer is carefully attached to a mounting substrate
again. (f) Phononic crystal pattern is lithographically defined in a resist mask on the sample,
followed first by a CF4 + O2 plasma etch to remove the top Si3 N4 , then by a DRIE etch all
the way through the Si wafer and bottom Si3 N4 film forming cross-shaped phononic crystal
holes. (g) Individual sample chips are separated from the sapphire substrate by soaking themin acetone. Chips are cleaned by a sequence of aggressive solvents and a sulfuric-acid-based
solution to remove any processing debris.

to be probed. The driven displacements as a function of frequency for devices A and B arecompared with that of a control device C in Fig. 4.3(c) and (d), respectively. We find that27
the displacement is clearly suppressed in the frequency ranges of 1.5-2.75 MHz and 4.054.45 MHz (2.65-3.25 MHz and 3.5-4.5 MHz) for device A (B), resulting in a flat responsethat is limited by the shot noise of optical detection. These “observed bandgaps” roughly
overlap with the calculated bandgaps [grey regions in both Fig. 4.3(a),(b) and (c),(d)]. The
center frequencies of the observed and predicted bandgaps are consistent within ∼10%.
í

(a)

(c) 1022
11(4,4) Device A Device Cí(e) 10

í

10

í

10

í

10

M
MF
PnC
CF

(3,2)

MF modes

piezo
modes

í

10

í

X

M

k space

*

1

(d)

66

1.5

2

(2,1)

(1,1)

2.5

3

3.5

4

4.5

frequency [MHz](2,2) (3,1) (3,2)(1,4)(3,3)(2,4)

(3,4)

5

2800

5.5

(4,4) Device B Device C

33
22
11

displacement [m]displacement [m]

44

í

10

í

10

2900

2950

frequency [kHz]

í

10

55

2850

(f) 10í

freq.+MHz/

frequency [MHz](1,4) (3,3) (2,4) (3,4)

10

10

*

(b) 7700(2,2) (3,1) (3,2)

displacement [m]displacement [m]

33

00

(2,1)

í

freq.+MHz/

frequency [MHz]

44

(1,1)

3000

M
MF
PnC
CF

piezo
modes
í

10

í

10
í

10
*

X

M

k space

*

1

1.5

2

2.5

3

3.5

4

frequency [MHz]

4.5

5

5.5

2210

2215

2220

2225

frequency [kHz]

2230

2235

Figure 4.3: Frequency-dependent mechanical response. (a)-(b) Simulated band diagramsfor infinite number of the unit cells used in devices A and B. Bandgap ranges are shownin grey. (c)-(d) Measured membrane displacement spectra of devices A, B, and C. The
data are smoothed with a 4 kHz bandwidth. The ranges of ideal bandgaps are shown ingrey. Membrane modes predicted based on the observed fundamental mode frequencies ofdevices A and B (up to the (4,4) mode) are shown by dashed lines. (e)-(f) Probing thenon-membrane modes via measuring displacement spectra at different locations on device B:
At the membrane (red), the membrane frame (MF, green), the PnC (purple), and the cornerof chip frame (CF, blue). (e) An example spectral region in an observed bandgap in device
B. (f) An example of two non-membrane modes of device B outside the observed bandgaps.

4.4

Probing the Non-Membrane Modes
Most of the modes we see in Fig. 4.3(c) and (d) are non-membrane modes; the finite

number of membrane modes are shown by dashed lines. To further understand the spectrum,

28
we meassurent the displacement spectra of different locations on the device. Physically, thechip consists of (1) the membrane (M, red), (2) the membrane frame (MF, green), (3) the
PnC structure (PnC, purple), and (4) the chip frame (CF, blue) [Fig. 4.1(d)]. The membrane
and the MF together form a “defect” embedded in the PnC lattice. We optically probe theMF, the PnC, and the CF by focusing on the three different locations indicated in Fig. 4.1(b).
Looking at these spectra in conjunction with the membrane displacement, we can understandthe origin of the non-membrane modes. The piezoelectric actuator itself has a frequencydependent structure, and measuring at the CF reveals information about this structure.
Mainly the displacement measured on the corner of CF is limited by the detection noise, butsome “piezo-modes” are clearly identifiable [see Fig. 4.3(e) and (f) for two examples].
We find that the combined spectra have distinct features inside and outside the observed bandgaps. Inside the observed bandgap, the spectra of the PnC, the MF, and themembrane are flat except a couple of “defect modes” observed in the spectra of the MF
and the membrane [see Fig. 4.3(e) for one example]. While the mechanical modes of the
MF cannot be completely avoided in the bandgap, they only occur sparsely and are clearly
separable from the membrane modes. Outside the observed bandgaps, most modes, except
for the membrane modes, have comparable motion in the membrane, the MF, and the PnC[see Fig. 4.3(f) for one example]. We also find that piezo modes greatly enhance the motionof other components. While inside the observed bandgaps, the piezo modes do not induce
any of the observed motion of other components [compare Fig. 4.3(f) and (e)].
4.5

Simulation of the Membrane Modes and the Non-Membrane Modes
The observed eigenmodes include admixtures of modes created by the membrane, theMF, the PnC, and the CF. We use a FEM to simulate the whole device in order to visualize
and characterize the expected frequency-dependent structure of all the modes. The boundary
conditions for the simulation fix the corners of the back side of the chip. We find all the
eigenmodes between 1 and 5 MHz. To estimate the motion that will be observed on the

29
Si3 N4 membrane [as measured in Fig. 4.3(c) and (d)], for each mode we calculate a “partition
coefficient” defined by the ratio of the energy stored in the membrane to the energy stored
in the whole device:R%(x)|u(x)|2 d3 x
Emem ≡ R mem,%(x)|u(x)|2 d3 x
whole

(4.1)

where u(x) is the simulated displacement field and %(x) is the mass density field.
The partition coefficient Emem is plotted in Fig. 4.4 as a function of mode frequency
using the parameters for devices B and C. The membrane modes are clearly identifiable as
the Emem ' 0 dB; these modes have the small effective mass associated with the Si3 N4 membrane. A majority of the non-membrane modes of device C have an Emem between −40 to−60 dB; these modes have a much larger effective mass associated with the silicon substrate.
For device B, there are two ranges with reduced Emem that roughly overlap with the ideal calculated bandgaps. The reductions are finite (Emem between −70 to −130 dB) and smoothly
degraded because the simulation takes into account the finite number of unit cells. There
are also a finite number of non-membrane modes with Emem > −40 dB. Inside the device-B
bandgap, these modes can be classified as defect modes with Emem < −30 dB. Outside of thedevice-B bandgap and in device C, there is a larger number of modes (∼ 4% of the modes)
with Emem > −40 dB. These modes with the largest Emem tend to be clustered near the
expected membrane mode frequencies.
In Fig. 4.4(b)-(f) we also show the displacement profile of example modes on a logarithmic scale. We see that the non-membrane modes inside the bandgaps are dominated by the
MF or the CF, and the displacement field decays exponentially in the PnC [Fig. 4.4(c),(d)].
On the contrary, the non-membrane modes outside the bandgaps have a uniformly distributed displacement field [Fig. 4.4(b)]. We also find that for the membrane modes insideand outside the bandgaps [Fig. 4.4(e) and (f)], the displacement fields in the PnC behave
the same as the non-membrane modes inside and outside the bandgaps. In other words,
inside the bandgap, the PnC acts as a passive mechanical filter that decouples the CF and

30

(a) 0
Emem (dB)

-30
-60
-90

-120

Dev. B
Dev. C

frequency (MHz)

(b)

(e)

(c)

f= 1.27 MHz
Emem = -52 dB

(f)

(d)

f= 3.09 MHz
Emem = -37 dB

f= 3.02 MHz
Emem = -127 dB
0 dB-50 dB

f= 3.18 MHz
Emem = -0.0029 dB(0.99934)

f= 1.09 MHz
Emem = -0.0093 dB(0.99786)-100 dB

Figure 4.4: Simulated membrane and non-membrane modes for devices B and C. (a) Simulated partition coefficient Emem of devices B and C. Data of device B (C) are red (blue).
Data of each device are connected by lines to see the trend. Ideal calculated bandgaps areshown in grey. (b)-(f) Simulated displacement field for four kinds of modes. Color scheme
represents the amplitude of displacement in a logarithmic scale. (b) An example of a nonmembrane mode outside the bandgap. (c) An example of a MF mode inside the bandgap.
(d) An example of a CF mode inside the bandgap. (e)-(f) Two examples of membrane modes
inside/outside the bandgap.

31
the “defect”; outside the bandgap, the PnC moves with all the other components together,i.e., they can be strongly coupled.

(a)

È

(3,3)
È

(2,4)

60 (1,2)

È

ÈÈ

È

(1,1)

(3,4)

(1,3)

Â (4,4)

40

(4,4)

80

È

Â
È

RC #dB'80

R A #dB'

(b)

(1,4)

È

È

È

È

60

Â

È

5

10

(2,2)

40

Â (2,2)

20

0

5
10
15
local
densityofmodes
local density of modes

20

0

15

local
of modesmodes
local density
density of

Figure 4.5: The ratio of the actuated energy of the membrane modes provided by the piezoelectric actuator to that by the thermal fluctuating force as a function of a local density ofmodes. The density is determined from the data in Fig. 4.1(c) by counting the number ofobserved modes in a 50 kHz range centered at each membrane mode. (a) Data for device A.
The membrane mode indices are labeled. (b) Data for device C. The corresponding modes
with lowest R in (a) are shown with square symbols.

4.6

Loss Factor Ratio
In the previous sections we have discussed the mechanical properties of the devices

over a large frequency range and gained understanding of the non-membrane modes. In
this section we focus on the narrow spectra near the membrane modes. We quantitatively
analyze the isolation of the membrane modes by recording the piezo actuated energy of the(1,1) through (4,4) membrane modes of devices A and C. To obtain a calibrated measure of
the relative actuated energy, we also measure the thermal fluctuating energy for each mode,
which is not shielded by the PnC. We define the ratio of the driven to thermal energies:
R = R(p, f, Bw ) ∝|D(p, f )/η|2|D(p, f )|2=,[Sd (f )/η 2 ]Bw
Sd (f )Bw

(4.2)

32
where D(p, f ) is the driven displacement amplitude measured with a network analyzer under external driving power p, Sd (f ) is the displacement spectral density measured with aspectrum analyzer without external driving power, η is the overlap factor between the optical spot and the membrane mode shape, and Bw =2 Hz is the resolution bandwidth of thespectrum analyzer. The lower the R, the higher the mode is isolated.
Another useful measure is the local mode density near each membrane mode. The
piezoelectric actuator does not directly drive the membrane; it drives the membrane throughthe chip frame, the PnC, and the membrane frame. In other words, the piezoelectric actuator
actuates the membrane mode through the non-membrane modes. This measure provides anestimate of the expected driving efficiency, but not necessarily a rigorous correspondence,
because the set of optically measured modes will not necessarily correspond to the set ofmodes that couples best to a particular membrane mode.
In Fig. 4.5 we plot R as a function of the local mode density near each membrane mode.
We see a positive correlation between the driven motion and this mode density for device A.
We also observe a much larger dynamic range in R for device A than for device C, which is
as expected because the phononic crystal structure introduces a nonuniformity to the local
mode structure. A direct comparison between devices A and C shows the smallest R in
device A is 30 dB smaller than the smallest R in device C, indicating that in the bandgap
membrane modes can be significantly isolated from the chip frame.
4.7

FEM Simulation
In this section, I outline the simulations of the band diagrams and the mechanicalmodes of our devices with COMSOL. The first task is to simulate the eigenmodes of an
infinite crystal, while the second task is to simulate the eigenmodes of the physical devices.
The intial studies of these simulations were carried out by Yegeishe Tsaturyan [49].
To simulate the band diagram of the infinite crystal, we implement a unit cell with a
Periodic Condition. We select the boundary and set the periodicity to Floquet periodicity.

33
We need to specify a k-vector that defines the phase difference between the adjcent unit cells
in the infinite crystal. To generate the band diagram, we have to calculate the eigenmodesof the crystal with the representative k-wave vectors. We simulate the eigenmodes with aparametric sweep to vary the k-vector along the boundary of the first Brillouin zone. Thenthe eigenfrequecies as a function of these k-wave vectors are plotted as the band diagram.
Note that the meshes on the peridoc boundaries have to be matched.
To simulate the mechanical modes of the physical devices, we include the membrane,
the silicon frame, and even the epoxy. The simulation of the mechanical modes is straightfoward, but extra care is needed to mesh to the objects. The default tetrahedral mesh willtake us infinite amount of time to complete the simulation. The reason is that the size oftetrahedral mesh is defined by the smallest dimension of our device - the membrane thickness. Hence we use a swept technique for the membrane, and define the smallest dimension
with the distribution setting.

Chapter 5
Ponderomotive Squeezing with a Membrane Mechanical Resonator

In this chapter, I discuss the physics of ponderomotive squeezing and our experimental
results. The experiment is a natural extension of the RPSN work described in Ref. [5].
For this experiment, I built the balanced homodyne detection, contributed to data taking
and analysis, and helped with the manuscript. Later on I developed the conceptual andtheoretical framework discussed in Sec. 5.3. A number of the figures and some of the text
are taken directly from Ref. [6] written in conjunction with Thomas Purdy, Robert Peterson,
and Nir Kampel.
5.1

Introduction to Ponderomotive Squeezing
Coherent light is a ubiquitous tool for metrology. However, as dictated by quantum

mechanics, any measurement of the properties of the coherent state will be limited by quantum noise. Quantum noise, unlike classical noise, cannot be eliminated by any technical trick(Fig. 5.1). However, it is possible to generate light with less noise in a selected quadraturethan the noise of a coherent state. Such light is called squeezed light.
The history of squeezed light is intimately linked to quantum limited displacement
sensing [50], owing to proposals to increase the displacement sensitivity of large scale gravitational wave observatories with squeezed light [51, 52, 53, 54]. Squeezed light was first
produced using atomic sodium as a nonlinear medium [55], and was soon followed with experiments employing optical fibers [56] and nonlinear crystals [57]. Substantial squeezing

35
has been achieved in modern experiments (up to 12.7 dB [58]), and using squeezing has
allowed a enhanced sensitivity in gravitational wave detectors [59] and in biological measurements [60]. Squeezed microwave fields, which have now been demonstrated with up to
10 dB of noise suppression [61], are an important tool in quantum information processing
with superconducting circuits.
Early on, searches for ever-better squeezing materials led to suggestions that an optomechanical cavity, in which radiation pressure changes the cavity length proportional to
optical intensity, could act as a low-noise Kerr nonlinear medium [62, 63, 64], and hence could
be a useful source of squeezed light [65]. This optomechanical method of manipulating thequantum fluctuations has historically been termed ponderomotive [66] squeezing. A uniqueadvantage of utilizing optomechanical nonlinearity is that correlations induced by a mechanical object can be used to enhance displacement sensitivity for that same object [54, 67].
However, experimentally it has proven difficult to realize the substantial interplay between mechanical motion and quantum fluctuations of light required for ponderomotive
squeezing. Early on, radiation pressure induced optical non-linearity (bistability) was experimentally demonstrated in a cavity with a pendulum-suspended end mirror [68]. Recently,
novel optomechanical systems have been developed to manipulate mechanical motion withradiation pressure. Ponderomotively squeezed light at the few percent level has been demonstrated using a mechanical mode of an ultracold atomic gas inside an optical cavity [69], andvery recently using a silicon micromechanical resonator [70]. The former experiment waslimited by nonlinearities in this interaction and the latter by excess mechanical thermal motion. In our experiment, we observe ponderomotive squeezing at 1.7±0.2 dB below (32%
below) the shot noise level and optical amplification of quantum fluctuations by over 25 dB.
The squeezing is realized on light transmitted through a Fabry-Perot optical cavity with
an embedded, mechanically-compliant, dielectric membrane. The squeezing attained here isdramatically larger, and the spectrum contains considerably less technical noise in the data
than previous studies.

36

(a)
laser

PD

PD

(b)

-

Modulator
laser

PD

PD

Figure 5.1: The shot noise cannot be eliminated by any technical trick (see Ref. [71] andappendix C) (a) An attempt to kill quantum noise by difference detection. The laser beam
is divided into two beams with equal power and detected by two photodetectors. The
two detectors will simultaneously sense any technical noise of the laser. Using a difference
amplifier, the common technical noise will be subtracted. However, the scheme fails tokill quantum noise. (b) Another attempt to kill quantum noise by feedback control. An
intensity modulator and two photodetectors are used. A fraction of laser beam is directed tothe detector B. The detected signal is feedback to the intensity modulator. This techniquefails to kill quantum noise in detector A.

5.2

Quantum Descriptions of LightIn this section we develop the necessary language to describe quantum properties ofsqueezed light.

5.2.1

Quadrature Amplitudes

In this subsection, we first define the quadrature amplitudes and phasor diagram inclassical optics. Then we show their quantum version.
Light, an electromagnetic wave, can be described byE(r, t) = E0 [α(r, t)eiωt + α∗ (r, t)e−iωt ]p(r, t),(5.1)

where E0 is the electrical field amplitude and p(r, t) is the polarization vector. The dimen-

(a)

(b)

37

(c)

coherent statesqueezed state

Figure 5.2: Phasor representation of light. (a) Noiseless classical light. (b) A coherent state.
(c) A squeezed state.

sionless complex amplitude α(r, t) of the wave can be written as a magnitude α0 (r, t) and aphase φ(r, t).
α(r, t) = α0 (r, t)eiφ(r,t)(5.2)

As a complex number, we can also represent α(r, t) in a complex plane (Fig. 5.2(a)). In optics,
this geometrical representation is called the phasor diagram. The x-axis (real part) is called
the amplitude quadrature, and the y-axis (imaginary part) is called the phase quadrature:
X1 (r, t) = 2Re[α(r, t)] = α(r, t) + α∗ (r, t) = 2α0 (r, t) cos[φ(r, t)](5.3)

X2 (r, t) = 2Im[α(r, t)] = i[α(r, t) − α∗ (r, t)] = 2α0 (r, t) sin[φ(r, t)](5.4)

We can also define an arbitrary quadrature byXθ (r, t) = X1 (r, t) cos(θ) + X2 (r, t) sin(θ)(5.5)

In addition, the fluctuations or modulations in both amplitude and phase can be represented
as fluctuations in the quadrature values:
α(t) = α + δX1 (t) + iδX2 (t)(5.6)

In the quantum mechanical model of light, the complex amplitude α is quantized to be â,38
hence the quadrature amplitudes are quantized to be:X̂1 = â + â†(5.7)

X̂2 = i(â† − â)(5.8)

X̂θ = X̂1 cos(θ) + X̂2 sin(θ)(5.9)= âe−iθ + â† eiθ

(5.10)

Now the quantum mechanics comes into play. As a result of non-zero commutation relation
between X̂1 and X̂2 , the variances of the operators in any state have to obeyh|∆X̂1 |2 ih|∆X̂2 |2 i ≥ 1(5.11)h|∆X̂θ |2 ih|∆X̂θ+π/2 |2 i ≥ 1.

(5.12)

Therefore, we will always have quantum noise in all quadratures. Note that we use a convention that the minimum product is 1. Some references use 1/4.
5.2.2

Coherent States and Squeezed States

The coherent state is a minimum uncertainty state andh|∆X̂1 |2 i = h|∆X̂2 |2 i = h|∆X̂θ |2 i = 1.

(5.13)The limit imposed by quantum noise affects all the quadratures of a coherent state evenly.
In other words, at all quadratures the light is shot noise limited. Since the coherent stateis a Gaussian state, we can present it in the phasor diagram as an area centered around its
expectation value (hX̂1 i, hX̂2 i). The area describes the uncertainty distribution and has adiameter of 1 (Fig. 5.2(b)).
On the other hand, the noise of a squeezed state is less than the shot noise in a selective
quadrature h|∆X̂φ |2 i < 1. As dictated by uncertainty principle, the noise is “squeezed” into
the orthogonal quadrature h|∆X̂φ+π/2 |2 i > 1. In the phasor diagram, its uncertainty area isan ellipse with major radius h|∆X̂φ+π/2 |2 i and minor radius h|∆X̂φ |2 i (Fig. 5.2(c)).

39
5.3

Squeeze Light by Correlating Two Quadratures
To generate a squeezed state out of a normal coherent state, a key is to correlate

two quadratures. One way to generate such a correlation is to pass a light beam througha nonlinear medium such as a Kerr medium. A Kerr medium has an intensity-dependent
refractive index. The amplitude fluctuation will modulate the refractive index, which, inturn, will modulate the phase of the transmitted light. Therefore, it generates correlation
between the amplitude quadrature and the phase quadrature.
An optomechanical system can be thought of as an effective Kerr medium, and hence
ponderomotive squeezing can be understood using many of the same ideas as typical nonlinear media. To understand this, consider a light bouncing off a flexible mirror. Light carries
radiation pressure that is proportional to its amplitude. The radiation pressure produced by
the amplitude fluctuation drives the flexible mirror. As the mirror moves, the phase of thereflected light is modulated. This creates the correlation between the amplitude quadratureand the phase quadrature.
In the following subsection, we describe the mathematical models for a regular Kerr
medium [72] and an optomechanical system. We point out their features and compare theirdifferences.

5.3.1

Kerr Medium

Consider a beam traveling through a medium with an intensity-dependent refractive
index. The input beam and output beam can be described by Eq. 5.6. Specifically, αin (t) =αin + δX1in (t) + iδX2in (t) and αout (t) = αout + δX1out + iδX2out . The nonlinear refractive index
can be described by n(α) = n0 (1 + n2 α2 ), where n0 is its linear refractive index and n2 is thesecond-order nonlinear refractive index. The modulated output phase can be expressed as:
φout = φin +

2πL 
2πnL
=
n0 + n2 (αin (t) + δX1in (t))2 ,λλ(5.14)

40
where L is the length of the medium. On the other hand, the amplitude of the light does
not change.
αout = αin = α(5.15)

We can write the changes in terms of the quadrature amplitudes:
δX1out = δX1in(5.16)

δX2out = δX2in +4πn0 n2 L 2 inα δX1 (t) ≡ δX2in + 2rKerr δX1in (t)λ(5.17)

where rKerr is a parameter describe the degree of correlation. It includes the nonlinearity
and the amplitude of the input field.
When the input beam is a coherent state, the variance of the output quadrature,2h|∆X̂θ |i = h|δXˆ1out cos(θ) + δXˆ2out sin(θ)|2 i, is given by22
V (θ) = h|∆X̂θ |i = 1 + 2rKerr sin(2θ) + 4rKerr
sin2 (θ).

(5.18)The squeezing is achieved when V (θ) < 1. The optimal squeezing and quadrature angle are
given byV (θs ) = 1 − 2rKerr

q2
2+ 2rKerr
1 + rKerr
h 1 i

1θs = − arctan2
rKerr(5.19)(5.20)The level of squeezing increases steadily with the Kerr coefficient [Fig. 5.3(b)]. The optimal
angle asymptotically approaches the value 0, the amplitude quadrature [Fig. 5.3(a)]. But
note that the variance of amplitude quadrature, V (θ = 0), is always equal to the shotnoise limit. This feature is independent of the nonlinearity and is the signature of the Kerrsqueezing.
5.3.2

Optomechanical Kerr Effect

While the full calculation of squeezed light had been studied in our group [6], I wanted
to gain a more complete understanding of the parallels betweens our devices and typical

(a)

(b)

41

Figure 5.3: (a) The variance as a function of quadrature angle for three different Kerr
coefficient values. (b) The minimum variance as a function of the Kerr coefficient.

Kerr media. To investigate this, I formulate the optomechanical interaction into a Kerr-likeinteraction. I consider the canonical cavity optomechanical system where the mechanicaloscillator is a mirror of a one-sided cavity. The cavity enhances the amplitude of the lightand the phase shift. The interaction Hamiltonian is ~gN z, where z is the operator of theeffective mechanical coordinate and N is the intracavity photon number operator [64, 63].

5.3.2.1

The Complex Kerr Coefficient

Here, for simplicity, the input light is the shot noise entering the cavity, and theoutput light is the modulated light inside the cavity. We start with the equation of motion
for intracavity light fluctuations in the frequency domain [4]:


√
ˆ
d(ω)
= χc (ω) −igāx̂(ω) + κξˆin (ω) ≡ −igāχc (ω)x̂(ω) + µ(ω)(5.21)

where µ(ω) is the input shot noise operator in the cavity:√µ(ω) ≡ χc (ω) κξˆin (ω)(5.22)The intracavity light consists of the phase fluctuation due to the position fluctuation of themechanical oscillator and the shot noise. The mechanical oscillator is driven by the radiation

42
pressure of the intracavity shot noise and the thermal fluctuating force:√2ωmΓx̂(ω) = −gā(µ̂† + µ̂) +η̃+ (ω)N(ω)N(ω)√
i Γ eff
eff†= −igāχm (ω)(µ̂ + µ̂) +χ (ω)η̃+ (ω)
2ωm m(5.23)

where
η̃+ =

η̂ † (ω)η̂(ω)+ ∗χm (ω) χm (−ω)(5.24)

N(ω) = N0 (ω) + 2ωm (gā)2 π− (ω) = i

2ωm
χeff
m (ω)

(5.25)

Now I substitute Eq. 5.23 into Eq. 5.21 and define the input and output operators of theamplitude and phase quadratures to be

†
in
†
ûin
+ = µ̂ + µ̂ & û− = −i µ̂ − µ̂ˆ† + dˆ & ûout = −i dˆ† − dˆ
ûout
=
d
+
−

(5.26)(5.27)We arrive at√Γ
π− (ω)χeff
m (ω)η̃+ (ω)
2ωm
√
Γ
out
in
2
eff
in
π+ (ω)χeffû− = û− − i(gā) π+ (ω)χm (ω)û+ + igām (ω)η̃+ (ω)
2ωm

ûout
+ =

2

1 − i(gā) π− (ω)χeff
m (ω)



ûin
+ + igā(5.28)(5.29)

where we define
π+ (ω) ≡ χ∗c (−ω) + χc (ω) = 2
π− (ω) ≡ i [χ∗c (−ω) − χc (ω)] =

κ
− iω
2
2

κ
− iω
2

+ ∆2

2∆
2
κ
− iω + ∆2
2(5.30)(5.31)In the limit of negligible thermal noise and zero detuning, the equations are simply
inûout+ = û+ (ω)(5.32)ininûout− = û− (ω) + r̃(ω)û+ (ω)(5.33)

43
where I definer̃(ω) = −i(gā)2 π+ (ω)χeffm (ω)(5.34)

Therefore, the cavity optomechanical system is equivalent to a Kerr medium with a complex Kerr coefficient r̃(ω). Not surprisingly, the Kerr coefficient is proportional to the optomechanical coupling strength and weighted by the mechanical and cavity response. The
magnitude of r̃(ω) is peaked at the mechanical resonance ω = ωm and can be expressed as|r̃(ω)|2 = (gā)44

1

2 (Γ/2)2(κ/2)2 + ωm

= 4C 2 (1 + (2ωm /κ)2 )−1 ,(5.35)

where C = 4(gā)2 /κΓ = 4N̄ g 2 /κΓ is the optomechanical cooperativity, and N̄ = ā2 is theaverage intracavity photon occupation. That is, the magnitude of the complex Kerr coefficient is about equal to the optomechanical cooperativity in the unresolved sideband regime.
In other words, a small cavity and mechanical linewidth are desired for large nonlinearity. We note that the cavity linewidth cannot be too small compared to the mechanicalfrequency. The resolved sideband factor (1 + (2ωm /κ)2 )−1 is in the equation because thequantum fluctuation at the mechanical frequency need to be large inside the cavity.
(a)

(b)

(c)

Figure 5.4: Frequency dependence of the optomechanical Kerr coefficient r̃(ω) = r(ω)eiφr (ω) .
(a) The magnitude square |r(ω)|2 follows the mechanical response function |χ(ω)|2 . The
bandwidth of nonlinearity is the mechanical bandwidth. (b) The phase provided by r̃(ω).
The Orange line is the phase response provided by χeffm (ω). (c) The real part of the complex
Kerr coefficient. Re[r̃(ω)] = |r̃(ω)| cos[φr (ω)]. It has a fano-shape.
out
2
The variance of the output quadrature, h|ûθ |2 i = h|ûout+ sin(θ) + û− cos(θ)| i, is given

44bySθ (ω) = 1 + Re[r̃(ω)] sin(2θ) + |r̃(ω)|2 sin2 (θ)= 1 + Re[r̃(ω)] sin(2θ) + Re[r̃(ω)]2 sin2 (θ) + Im[r̃(ω)]2 sin2 (θ)(5.36)(5.37)

Compared with Eq. 5.18, we see that the they are basically the same except the extra termIm[r̃(ω)]2 sin2 (θ), which comes from the imaginary Kerr coefficient. This term reduces theamount of squeezing and enhances the amount of anti-squeezing.
I can draw a number of conclusions based on the frequency dependence of Re[r̃(ω)](Fig. 5.4). First, Re[r̃(ω)] is concentrated around the mechanical frequency with a frequency
range characterized by the effective mechanical linewidth. Second, Re[r̃(ω)] has a fano-shape.
Hence, the squeezing spectrum will be a fano-shape, as well. Third, Re[r̃(ω)] is negative below
resonance and positive above the mechanical frequency. That means the squeezed angle isbelow 0 when the frequency is below the mechanical resonance and above zero when thefrequency is above the resonance. In other words, if we look at negative quadrature angles,
the squeezing will happen above the resonance; if we look at positive quadrature angles,
the squeezing will happen below the resonance (See Fig. 5.5). Fourth, Re[r̃(ω)] vanishes asω → ∞ but remain finite at ω = 0. With the experimental parameters described in the later
section, we have Re[r̃(0)] ∼ 0.2 and Re[r̃(ω)] ∼ 0. Hence, in principle, the mechanism allows
squeezing at DC [65].
5.4

Experimental Requirements for Observing Ponderomotive Squeezing
To observe ponderomotive squeezing, like other-types of squeezing experiments, weneed a large nonlinearity and a detection system with a good quantum efficiency. Any
optical loss in the detection will attenuate the squeezing. A large nonlinearity requires a large
optomechanical cooperativity, specifically, large optomechanical coupling, small mechanicaldissipation, and small optical linewidth. The necessary size of nonlinearity is determined by
the size of the mechanical thermal noise.

(a)no squeezing45(b) 











í
í
í




í





I [degrees]

Figure 5.5: Phasor diagrams of the signal beam for real parts of r̃, the complex Kerr parameter. The dashed circle represents the variance of the Gaussian noise distribution of thevacuum state. Distributions inside the dashed circle represent squeezed states. (d) Calculated signal-beam quadrature spectrum from Heisenberg-Langevin equations. We set ∆ − 0
in the idealized case of zero temperature and no optical loss . Otherwise, parameters areeffset√ to experimental values: κ/2π = 1.7 MHz, Γ0 /2π = 2.6 MHz, ωm /2π = 1.53 MHz, andN̄ g0 /2π = 350 kHz. The spectrum is displayed on a logarithmic scale. The region between
the white contours is squeeze.

5.4.1

RPSN Needs to be Greater than the Thermal Noise

Thermal motion of the membrane imprints excess noise onto the phase quadratureof the light (Eq. 5.23), which is uncorrelated with the quantum noise in the amplitude
quadrature and hence can limit the level of squeezing. To obtain significant squeezing near
the mechanical resonance, the level of the optical force, termed radiation pressure shot noise(RPSN), relative to the thermal force driving the membrane should be large [5]. For a beamwith laser-cavity detuning near zero, this ratio is given by R = C/nth ×(1+(2ωm /κ)2 )−1 where
C is the optomechanical cooperativity and nth is the thermal occupation of the mechanicalstate.

46
5.4.2

Quantum Efficiency

Suppose we have an input beam with variance V (θ). When it passes through a detection
system with a quantum efficiency of , the output variance Vd (θ) is given by
Vd (θ) = V (θ) + (1 − )

(5.38)

If the input beam is squeezed and described by V (θs ) = 1 − s where s > 0, we have
Vd (θs ) = (1 − s) + (1 − ) = 1 − s

(5.39)The degree of squeezing is attenuated by the quantum efficiency.

5.4.3

Maximum Squeezing with Sufficiently Large Nonlinearity

With sufficient large nonlinearity, the level of squeezing is purely determined by theratio of RPSN to thermal noise R and the quantum efficiency . It is found through analytical
approximation and numerical comparison that the maximum squeezing one could get is given
by


R
1+R

(5.40)The level of squeezing gives lower bounds on R and the total quantum efficiency . In thelimit of R  1, the level of squeezing can be used to calibrate the total quantum efficiency.
5.5

Our System and Experiment
Our optomechanical cavity (see Fig. 5.6 and Ref. [4]) consists of a 40 nm thick by 500µm square silicon nitride membrane inside of a 3.54 mm long Fabry-Perot optical cavity [3].
We work with the (2,2) drum-head mode of the membrane, with two antinodes along eachtransverse direction, yielding a mechanical resonance frequency of ωm = 2π × 1.524 MHz andmechanical dissipation rate Γ0 = 2π × 0.22 Hz. With g = 2π × 33 Hz, κ = 2π × 1.7 MHz, andin a helium flow cryostat with a base temperature of 4.6 K, we achieve R > 5 when operating

(a) Input:

Membrane

Coherent
State

Output:Squeezed
State

PBS

PD

PBS

47

Balanced (b)Homodyne
Detection

N '

'

d

PD

SignalBeam
Damping
BeamPD

MirrorMirrorBalanced
DirectDetection

PD
PD

LO
I adjust

Figure 5.6: Experimental Diagram for Optomechanical Squeezing. (a) The signal beam
(blue) enters the optomechanical cavity as a coherent state. After the optomechanical interaction, a squeezed state emerges. The signal beam is detected, either with balanced direct
photodetection or with balanced homodyne detection by mixing with an optical local oscillator (LO) (green). A weaker damping beam (red) orthogonally polarized to the signal beamis also injected into the cavity. The two beams are combined before the cavity and separated
after the cavity with polarizing beamsplitters (PBS) and detected with photodetectors (PD).
(b) Signal and damping beam detunings from the cavity resonance.

with N̄ ∼ 108 . This ratio is much larger than achieved in previous work [5], mainly due toincreased optomechanical coupling. In addition to our main signal beam, we inject anotherlaser into the orthogonal polarization cavity mode. This damping beam has a much weaker
power than the signal beam but is detuned by ∆d ∼ −ωm from the cavity resonance. The
damping allows us to avoid parametric instability and work with a mechanical mode with
an effective mechanical linewidth of 2π × 2.6 kHz.
5.5.1

Direct Photodetection

In our first set of experiments, we use direct photodetection to measure the power
spectrum of the amplitude quadrature, SI (ω), which is normalized such that the detected
shot noise is unity. Although a Kerr mechanism does not generate squeezing at the amplitude
quadrature, we can rotate the quadrature by a single beam with a finite detuning ∆. Offresonant phase fluctuations are partially converted into intracavity amplitude fluctuationsand vice versa. The rotated angle is given by φc = tan−1 (2∆/κ). Besides, with sufficientlarge nonlinearity, the squeezed angle is close to the amplitude quadrature (see Eq. 5.19 andFig. 5.3). Hence only a small amount of detuning is needed in our experiment.

48

1.0
40

1.0

30

Δ/2𝜋 = −3 kHz
𝜙𝑐 = −0.0035

0.6

𝑆𝐼 (𝜔)

0.8

0.8

10

1.0

0.6
1.0
0.8

c

Quantum Efficiency

Thermal Noise

0.2
Δ/2𝜋 = −21 kHz

1.50

𝜙𝑐 = −0.025

1.52

1.54

Frequency [MHz]

1.56

1.56

0

d

6

2

b

0.0

10

𝑆𝑧

a

1.54

0.4

1.0
0.8

1.52

Frequency [MHz]Δ/2𝜋 = −13 kHz
𝜙𝑐 = −0.015

0.6

0
1.50

0.6

× 10−17 m/ Hz

Δ/2𝜋 = −6 kHz
𝜙𝑐 = −0.0071

𝑆𝐼 (𝜔opt )𝑆𝐼 (𝜔)

0.8

0.6

20

20

40

Δ/2𝜋 [kHz]

60

1.50

1.52

1.54

1.56

Frequency [MHz]Figure 5.7: Quantum Intensity Noise Suppression. (a) Directly detected optical intensity
noise signal beam spectra for several signal beam detunings. Also displayed are the measured
shot noise level (gray) and the theoretical predictions (black). A 200 Hz bandwidth is used.
The ratio of RPSN relative to thermal drive, R, is fixed at 5.1. The damping beam provides
effΓeff
0 /2π = 2.7 kHz and ωm /2π = 1.524 MHz. However, the total mechanical damping
rate and resonance frequency change with the signal beam detuning ∆. (b) The minimum
value of SI for spectra as displayed in (a) (blue circles). Statistical error bars indicate thestandard deviation. The frequency where the minimum occurs, ωopt , shifts with ∆ due
to the optical spring effect. Also displayed are the mechanical thermal noise floor for ourcurrent parameters 1/R (dashed green), limit set by finite detection efficiency 1 −  (dotdashed green), sum of thermal and detection efficiency limit (dot-dot-dashed green), andexpected squeezing in the absence of optical loss and thermal motion (dotted gray). (c) The
directly detected optical intensity noise signal beam spectrum with intentionally added white,classical amplitude noise (red), theoretical prediction (black), and level of added amplitude
noise (gray). (d) The mechanical displacement spectrum inferred from the damping beamtransmission spectrum (orange), and Lorentzian fit (dotted black). Detection noise floor isalso shown (dotted gray). One additional peak due to excess noise is visible in the bottom
panel of (a) and in (d) at frequencies ∼ 1.545 MHz due to a mechanical mode of the cavity
support structure. The mode may be thermally occupied or driven by piezo noise.

Figure 5.7(a) shows SI for several values of ∆, all at an average transmitted signal
beam power of 110 µW corresponding to N̄ = 1.1 × 108 or R = 5.1. A dip in noise
below the shot noise level is visible in the vicinity of ωm , a clear signature of squeezed
light. The squeezing becomes more pronounced as |∆| is increased because the maximallysqueezed quadrature is rotated toward the amplitude quadrature. The data shows excellent

49
agreement over most of its frequency range, with a Heisenberg-Langevin model including
quantum-noise-limited input optical fields, a thermally occupied mechanical bath coupled tothe membrane, and no other classical noise sources (see appendix B). However, a small excess
of classical noise is visible at the largest detuning, a few tens of kHz above the mechanicalresonance. Here, cavity frequency noise induced from a thermally occupied mechanical modeof the optomechanical cavity support structure [4] is increasingly converted in amplitude
noise at larger |∆|. All of the system parameters used to generate the theory curves ofFig. 5.7(a) are independently measured, except ∆ is calibrated, in part, using the displayeddata.
The shot noise level for the data of Fig. 5.7 is calibrated using balanced direct detection(see appendix C). The transmitted signal is split into two equal power beams and directed
onto a pair of nearly identical photodetectors. Taking the sum of the detected signals is
equivalent to single detector direct detection. However, taking the difference of the detectedsignals removes classical and quantum correlations. We achieve up to 20 dB common mode
suppression. The difference signal consists of only the uncorrelated shot noise level and asmall ∼ 5% contribution from the photodetector dark noise. SI is computed by taking theratio of the power spectra of the sum and difference signals, after subtracting the measured
photodetector dark noise.
The limits of the detected squeezing (Eq. 5.40) are illustrated in Fig. 5.7(b) where
the minimum measured value of SI is plotted as a function of ∆. The finite quantum
efficiency of our detection system is the largest limit to the detected squeezing. Including
losses associated with the cavity c = 0.6, propagation to the photodetector p = 0.8, andphotodetector conversion efficiency d = 0.87, we estimate an overall quantum efficiency of = c p d = 0.42.
Although the transmitted signal beam intensity spectrum is decidedly non-Lorentzian,the mechanical displacement still follows a simple Lorentzian form. The damping beamtransmitted intensity spectrum acts as a probe of mechanical motion uncomplicated by50
strong quantum correlations because its intensity, and thus RPSN effects on the membrane are small [5]. The mechanical displacement spectrum derived from the damping beam
[Fig. 5.7(d)] shows the mechanics still retains a Lorentzian response to locally white force
fluctuations.
1.64 1.55>1.75

1.62 1.54

1.50

1.6

1.25

a

Frequency [MHz]

100

1.53

1.58

1.00

1.52
1.56

1.51
0.75

1.54

−10

0

10

10

20

1.52

1.5

1.48
1
1.46

−120

−100

−80

−60

−40

−20

0

20

b
Sφ

100

10

1
−120

−100

−80

−60

−40

φ [degrees]

−20

0

20

Figure 5.8: Optical Quadrature Spectrum. (a) Color map of Sφ . The white contour is at
the shot noise level, and the region inside this contour is squeezed. Several additional noise
peaks are evident at frequencies away from the mechanical resonance, due to the motion ofthermally occupied modes of the support structure. The inset shows the squeezed region in
more detail and also includes a theoretical prediction (dashed black) of the expected shotnoise contour. The experimental parameters are the same as in Fig. 5.7, except ∆/2π =−42 kHz. (b) Cuts through quadrature phase at three different frequencies 1.517 MHz(blue circles), 1.526 MHz (green stars), and 1.535 MHz (red triangles), averaged over a 1
kHz bandwidth, and corresponding theoretical models with no free parameter (colored solid
curves).

5.5.2

Balanced Homodyne Detection

We next explore all quadratures of the transmitted signal beam with balanced homodyne detection. We interfere the transmitted signal beam with an optical local oscillator

51
whose phase is stabilized relative to the signal beam. Different quadratures can be detected by adjusting the relative phase between the local oscillator and the signal beam. In
Fig. 5.8(a), optical quadrature spectra, Sφ (ω), over varying quadrature phase, φ, are displayed. Sφ is normalized such that the measured shot noise level is unity. The phase φ = 0
corresponds with the amplitude quadrature and gives information equivalent to that obtainedin the direct detection discussed above. (Note, for these measurements ∆ = −2π × 42 kHz,
allowing some squeezing to be rotated into the amplitude quadrature.) Regions with noise
spectral density below the shot noise level are visible over a frequency range of∼ 100

kHz, and φ ranging over tens of degrees. The range of observed squeezing is limited to aregion smaller than predicted by theory because of the thermal motion of the cavity mirrors
and support structure mentioned above. The depth of the observed squeezing in homodyne
detection is also lower than that observed in direct detection. This is partially accounted forby imperfect overlap between the probe beam and homodyne local oscillator, contributing an
additional effective optical loss, m = 0.8. Also, we operate in a regime where the homodyne
local oscillator power is only a factor of less than 10 larger than the probe power, making
the measurement slightly susceptible to the noise of the local oscillator (see appendix B andC).
Homodyne detection also allows us to quantify the coherent amplification of opticalquantum fluctuations in our measurement. In Fig. 5.8(b), we compare our data to a theoretical calculation based upon a Heisenberg-Langevin model. The agreement between themodel and the data allows us to interpret the large noise spectral density (Sφ ∼ 330) near
the phase quadrature at φ = ±90◦ as arising mainly from coherent amplification of quantum
noise or so-called anti-squeezing. This amplification persists despite the large imaginary
component of the mechanical response, which has the potential to limit squeezing and add
thermal noise.

52
5.6

Discussion, Conclusions, and Future Aspects

5.6.1

A Kerr Medium with Finite Response Time

It is worthwhile to discuss the meaning of the complex Kerr coefficient. A consequence of the imaginary component of the Kerr coefficient is that our system has a finite
response time. The delay time can be calculated by differentiating the phase response versus
frequency:
τ (ω) = −i∂ h∂φr (ω)arg[r̃(ω)] = −,∂ω∂ω(5.41)

where r̃(ω) = r(ω)eiφr (ω) . Now we separate the discussion by the resonance case and offresonance case. On mechanical resonance, it can be found that the delay time is just the
response time of the mechanical resonator:
τ (ωm ) ∼ −i∂ harg[χm (ω)]= 2/Γeff
0 ,
∂ω
ω=ωm

(5.42)

provided that κ  Γeff
0 . The response time is inversely proportional to the effective mechanical linewidth. While the response time can be reduced by optomechanical damping, theoptomechanical Kerr coefficient will reduced accordingly. In fact,2
peak nonliearity × (response time)−1 ∼ |r(ωm )|/τ ∼ 2C/(2/Γeff0 ) = 4(gā) /κeff
2
peak nonliearity × squeezing bandwidth ∼ |r(ωm )|Γeff
0 ∼ 2CΓ0 = 8(gā) /κ

(5.43)(5.44)

In other words, the resonant-enhanced optomechanical Kerr nonlinearity comes with theprice of the finite response time and a finite bandwidth. We typically operate the systemwith an effective mechanical linewidth of 1 kHz. This corresponds to a Kerr coefficient of
100 and a response time of 1 ms.
For the off resonance case, I plot |r̃(ω)| and τ (ω) in Fig. 5.9. While both functions
peak at mechanical resonance and decay at off-resonance, they have different lineshapes. It
turns out that the maximum of |r̃(ω)|/τ (ω) occurs at a few mechanical linewidth away fromthe mechanical resonance.

(a)

(b)

53
(c)

Figure 5.9: Optomechanical Kerr coefficient |r̃(ω)| and delayed time τ (ω) from DC to mechanical frequency. (a) |r̃(ω)| as a function of frequency. It peaks at the mechanical resonance(1500 kHz) and flatten at DC. (b) The delayed time τ (ω) as a function of frequency. It also
peaks at the mechanical resonance, but a different lineshape. (c) The ratio of |r̃(ω)| to τ ω.
We normalize it by photon number in the cavity ā2 ∼ 108 . The maximum occurs at a few
mechanical linewidth away from the mechanical resonance.

The Kerr mechanism is a third order nonlinearity. Here we compare the optomechanical nonlinearity achieved in this squeezing experiment to other χ(3) mechanisms. Whilethe nonlinearity strength of different mechanism can vary by ten orders of magnitude, thenonlinear strength divided by the response time is roughly the same [73]. In the following
table, I calculate the Kerr coefficient per photon (rKerr /ā2 ) for a few nonlinear mechanisms
and divide it by their response time (rKerr /ā2 τ ). Here I assume interaction length L for eachmechanism is 1 cm (see Eq. 5.14), except for the optomechanical Kerr nonlinearity. With theexperimental parameters in the chapter, the optomechanical Kerr coefficient is large compared with that of other mechanisms, but its Kerr coefficient divided by its response time iscomparable.

54
Mechanism

rKerr /ā2

rKerr /(ā2 τ )n2Response time

(cm2 /W)(sec)

Electronic polarization

10−16

10−15

10−15

1

Molecular orientation

10−14

10−12

10−13

0.1

Electrostriction

10−14

10−9

10−13

10−4

Saturated atomic absorption

10−10

10−8

10−9

0.1

Thermal effects

10−6

10−3

10−5

10−2

optomechanics

NA

0.1-10−3

10−6

10−3

5.6.2

(sec−1 )Conclusions and Future Aspects

In conclusion, we have experimentally demonstrated that an optomechanical systemwell into the RPSN dominated regime is capable of creating squeezed light. The 1.7 dB
strength of optomechanical squeezing we achieve is significantly larger than previous optomechanical realizations [69, 70]. However, stronger squeezing has, of course, been realizedwith more developed techniques [58]. Increasing detection efficiency and reducing thermal
noise will be required to study the ultimate limits to deeply ponderomotively-squeezed light.
Employing the PnC-shield membrane with ultrahigh Q described in Chapter 4 will
reduce thermal noise. In addition, the phononic bandgap will eliminate the extraneous noisedue to the substrate modes [Fig. 5.7(d) and Fig. 5.8].

Chapter 6
Optomechanical Raman-Ratio Thermometry

In this chapter, I discuss the first optomechanical experiment that utilized the PnCshielded membrane described in Chapter 4. Exceptionally good isolation from the environment by the PnC allows us to perform optomechanical Raman-ratio thermometry at
variety of environment temperatures. For this experiment, I integrated the PnC-shield membrane with the optical cavity, contributed to data taking and analysis, and helped with themanuscript. A number of the figures and much of the text is taken directly from Ref. [7]written in conjunction with Thomas Purdy, Robert Peterson, Nir Kampel, Katarina Cicak,
and Raymond Simmonds.
6.1

Introduction
Raman light scattering has proven to be a robust and powerful technique for in situ

thermometry. Many material-specific properties governing Raman transitions, such as the
Stokes shift, spectral linewidth, and scattering rate vary with temperature. However, for any
Raman system the ratio of spontaneously scattered Stokes versus anti-Stokes photons is a direct measure of the initial population of the motional state. For example, at zero temperature
the process of anti-Stokes scattering, which attempts to lower the motional state below theground state, is entirely suppressed; whereas the Stokes scattering, which raises the motional
state, is allowed. For thermally occupied states, an absolute, self-calibrating temperature
measurement is possible by measuring this asymmetry in Raman scattering. Distributed

56
optical fiber sensors [74] and solid state systems [75, 76, 77] make use of spontaneous Raman
scattering between optical phonon levels for temperature measurements, and combustion
chemistry diagnostics use rotational-vibrational molecular levels in a similar fashion [78].
In principle, Raman scattering from any mechanical degree of freedom in a solid-state
environment has the potential to provide local temperature measurements, but for lowfrequency micromechanical resonances, this is a considerable challenge. Recent experimentsin the field of quantum cavity optomechanics [79, 80, 81] use cavity enhancement to collect
Raman-scattered light from localized acoustic resonances demonstrating the Stokes/antiStokes asymmetry. However, the potential for using this asymmetry to perform absolute,
self-calibrated thermometry remains relatively unexplored. Here we measure the asymmetryof Raman scattering in a membrane-in-cavity optomechanical system (Fig. 6.1) over a wide
range of physical temperatures between 4.8 K and 50 K. We find agreement with conventional
thermometry over this range to within ten percent, and we discuss the sensitivity of themeasurement to a variety of parameters.
In our optomechanical system, the motional states are the MHz frequency vibrational
levels of a membrane mechanical resonator, and an optical resonance is provided by an opticalcavity surrounding the membrane. Departing from previous measurement schemes [79, 81],
the Stokes and anti-Stokes peaks are simultaneously derived from a single, resonant laser
tone. The Raman asymmetry becomes more pronounced in certain mechanical normal modesof the resonator when they are optically cooled near their ground state with a separate laser
tone. We are able to realize a damped displacement spectral density near the membraneresonance frequency equal to that expected from a resonator occupied with as low as n̄ ∼ 2
vibrational quanta (150 µK effective temperature) [82, 83, 84, 85]. We then measure thephysical temperature of our device by extrapolating the Raman sideband asymmetry to zero
optical damping.

57
6.2

Principle of Raman-Ratio Thermometry
Raman-ratio thermometry is based on the idea that at a finite temperature the groundstate manifold of the Raman levels is occupied according to a well understood statisticalweighting. The ratio of Stokes to anti-Stokes Raman transitions is equal to Rsa = e~ωm /kb T =(n̄ + 1)/n̄, where ωm is the mechanical resonance frequency and T is the temperature [86,87]. The spectrum of Raman scattered light transmitted through an optomechanical cavity(Fig. 6.1(b)) is given by:S(ω) ∝

n̄

+

Γm 2
2
+
(ω
−
ω)
m
2

n̄ + 1

.

Γm 2
2
+
(ω
+
ω)
m
2

This assumes the laser is resonant with the optical cavity, and the optical cavity linewidth ismuch larger than Γm . The first (second) term corresponds to anti-Stokes (Stokes) scatteringpeak shifted by ωm (−ωm ) from the input laser frequency, and ω is the frequency relative
to the input laser frequency. The peaks are broadened by the mechanical linewidth, Γm .
Taking the ratio of the amplitude of the Stokes and anti-Stokes peaks directly yields themechanical occupation, n̄ = 1/(Rsa − 1).
6.3

Experimental Setup
We use the membrane-in-cavity optomechanical system in the helium flow cryostat

described in Chapter 5 [Fig. 6.1(a)]. The cryostat temperature, T0 , is set in the range of 4.8
to 50 K. We have employed two devices for our measurements. The data shown in 6.2(a)focus on the (2, 2) mode of a 500 µm square by 40 nm thick membrane, with resonance(2,2)
frequency ωm/2π = 1.509 MHz and intrinsic linewidth Γ0(2,2) /2π = 0.46 Hz. The data

shown in Figs. 6.2(b) and 6.3 are primarily from the (3, 2) mode of a 375 µm square by(3,2)
100 nm thick membrane, ωm/2π = 2.637 MHz, Γ(3,2)
0 /2π = 0.84 Hz, which is shielded by a

PnC substrate described in Chapter 4. The acoustic band structure is engineered to provide

58
(a)

cryostat cold finger

silicon diodethermometercopper heat shield-invar cavity spacer
membraneprobe
beam PBS
PBS
damping
beam

PZT

local
oscillator

mirror

(b)

BS

(c)
probe
beam

(2,2)

𝜅
Stokes∝𝑛+1

anti-Stokes
∝𝑛

−𝜔𝑚

(3,2)

𝜔𝑚

Figure 6.1: Optomechanical Raman-ratio thermometry. (a) Two orthogonally polarizedlaser beams are coupled into a cryogenic membrane-in-cavity optomechanical system. Raman
scattered light from the resonant probe beam is analyzed with balanced heterodyne detection.
The red detuned damping beam Raman sideband cools membrane motion. Beam splitter(BS), polarizing beam splitter (PBS), piezoelectric transducer (PZT). (b) The spectrum oftransmitted probe beam light shows Raman scattering peaks shifted by ±ωm , where ωm is
the mechanical resonance frequency. The asymmetry in the spectral peaks, dependent on theeffective thermal occupation of the mechanical mode, forms the basis for a self-calibrating
thermometer. Also indicated is the spectral response of the optical cavity (dashed curve).
(c) Spatial profile of the (2, 2) and (3, 2) drumhead mode of the membrane.

a gap in the substrate mechanical mode density around the (3, 2) mode1 , which diminishes
noise from thermally occupied modes of the substrate [16, 88].
The optical cavity consists of two mirrors separated by 3.5 mm, each with 10−4 fractional transmission. The optical linewidth, κ, is dependent on the location of the membranein the cavity [38], and is on the order of a few Megahertz. Optomechanical coupling isachieved as the optical resonance frequency is modulated by the displacement of the vibrating membrane along the optical standing wave. This interaction is characterized by a single
1

We note that measured mechanical resonances of all of the membrane modes are shifted lower infrequency compared to those reported in this reference. This shift moves the (3,2) resonance closer to thecenter of mechanical band gap. The change may be the result of a stress reduction due to differences insample mounting technique, mass loading from surface contamination, or device aging.

59
photon coupling rate of g0(2,2) /2π = 33 Hz for the data in Fig. 6.2(a) or g0(3,2) /2π = 18 Hz forthe data in Fig.6.2(b) and Fig. 6.3. The cavity is driven with two orthogonally polarizedlaser beams derived from the same 1064 nm laser source (Fig. 6.1(a)). The probe beam is
actively stabilized to be resonant with the optical cavity. The transmitted Raman scattered
light from this beam is analyzed with an optical heterodyne detection system. The orthogonal polarization mode is driven by the damping beam, which is tuned to a frequency lower
than the optical resonance.
6.4

Optical Damping and Effective Mode Temperature Measurement
For the red-detuned damping beam, the anti-Stokes scattering rate is resonantly en-

hanced by the cavity [86, 87]. Each anti-Stokes scattered photon that exits the cavity carries
one vibrational quantum of energy out of the system. The mode reaches an equilibrium
when the optical cooling rate is matched by the rate at which thermal excitations enter thesystem. The effective temperature of a mode is approximately Teff = T0 ΓΓm0 , in the experimentally relevant case where Γ0  Γm  κ, here Γm is the optically induced mechanicaldamping rate, and Γ0 is the intrinsic mechanical damping rate. The cooling is more efficientwhen the resolved sideband parameter ωm /κ  1. However, such a ratio will suppress theRaman peaks generated by the resonant probe beam. We build our system with ωm /κ ∼ 1
so that we can maintain cooling efficiency and detect the Raman signals generated by one
probe beam.
By varying the power of the damping beam, we are able to damp motion and reduce
the effective mode temperature by up to a factor of 3 × 104 . We take the Stokes/anti-Stokes
peak amplitudes generated by the resonant probe beam as the damping beam power, Pd ,is increased. The power of the transmitted probe beam, Pp , is held at constant. As expected, the height of the anti-Stokes peak decreases with damping beam power as the mode
is cooled. For weak damping the two Raman peaks are equal in amplitude, whereas atstrongest damping the Stokes peak is 50% larger than the anti-Stokes peak as the mode

60
approaches its quantum ground state. From the Stokes/anti-Stokes ratio we compute theeffective temperature of the mode, as displayed in Fig. 6.2. Raman-ratio thermometry indicates that for frequencies near ωm , the (2,2) mode reaches n̄(ωm ) = 2.1 ± 0.2. However,for this mode, the occupations measured over a wide range of damping beam power aresystematically ∼15% larger than predicted by Raman sideband cooling. This discrepancy,
which is absent in the PnC-isolated (3,2) mode [Fig. 6.2(b)], is likely due to the influence of
substrate motion and is discussed below.
To compute the effective temperature of the mode, we numerically integrate the Raman peaks over a 4 kHz span around the mechanical resonance and calculate their ratios.
This narrowband analysis minimizes the effect of noise from the thermally-occupied mechanical modes of the non-PnC substrate [16, 4]. Besides, the mechanical occupation near the
resonance is a relevant figure of merit for many applications of optomechanics, including resonant force sensing and the generation of optomechanically-squeezed light [6]. The analysis
indicates that the (2,2) mode reaches n̄(ωm ) = 2.1 ± 0.2. We stop at this number because
the extraneous noise is significant. In addition, for this mode, the occupations measured
over a wide range of damping beam power are systematically ∼15% larger than predicted by
Raman sideband cooling. This discrepancy, which is absent in the PnC-isolated (3,2) mode
(Fig. 6.2(b)), is likely due to the influence of substrate motion.
It is worthwhile to discuss the mechanical occupation one can reliably measure with
Raman-ratio thermometry. The sensitivity is ultimately limited by the Heisenberg measurementdisturbance uncertainty principle. As we increase the probe beam power to increase the signal
to noise ratio, the optical forces from the shot noise (radiation pressure shot noise, RPSN) [5]will increase as well. The RPSN-driven motion set the lowest temperature one can measure.
In the data of n̄(ωm ) = 2.1, the signal to noise ratio is ∼1 dB. The contribution from RPSN
is 0.2 quantum.

0.1

1
1

10

damping beam power, 𝑃𝑑 [mW]

3
10

1

𝑛(𝜔𝑚 )𝑛(𝜔𝑚 )

10

61

(b)

𝑇eff 𝜔𝑚 [mK]1

𝑇eff 𝜔𝑚 [mK]

(a)

0.3
1
3

10

30

damping beam power, 𝑃𝑑 [mW]

Figure 6.2: Effective mode occupation at mechanical resonance. Numerically integrating
the Raman peaks over a 4 kHz span around the mechanical resonance and taking theirratio yields a measure of the mode effective temperature (orange circles) for (a) the (2,2)
mode of the non-PnC device and (b) the (3,2) mode of the PnC device. Due to decreased
optomechanical coupling, the (3,2) mode does not reach as low a mechanical occupation as
the (2,2) mode. Gray bands are the prediction of Raman sideband cooling whose widthstems from uncertainty in T0 as measured by the diode thermometer. Vertical error bars
represent estimated statistical standard deviation.

6.5

Physical Temperature Measurement
We next consider Raman-ratio thermometry as a means to determine the physical tem-

perature of the device. Here we employ a membrane resonator embedded in a PnC substrate,
which reduces the effects of substrate motion to a level below the statistical noise. The Raman asymmetry of undamped modes is small (∼ 10−4 ) and precludes directly ascertaining
the physical device temperature. Nevertheless, with our ability to perform strong optomechanical cooling, we can extrapolate the temperature of the undamped membrane modes
from measurements of the asymmetry as a function of optical damping (Fig. 6.3(a)). The
undamped occupation of the mechanical mode is given by (dRsa /dPd )−1 × dΓm /dPd × 1/Γ0 .
Here we employ a membrane resonator embedded in a PnC substrate, which reduces theeffects of substrate motion to a level below the statistical noise. Thermometry is performedon the (3, 2) mode with the cryostat held at several different temperatures between 4.8 K
and 50 K. In Fig. 6.3(c) the extrapolated temperatures are compared to the temperature asmeasured by a silicon diode thermometer attached to the cryostat. The results agree within

62
the statistical uncertainty with an average deviation of less than 10%. As an additional
confirmation, we perform the same thermometry procedure on the (5, 2) mode, which isalso located in a phononic band gap of the substrate and find agreement with the previous
measurements. However, the statistical error on the (5, 2) mode measurement is much larger
because the optomechanical coupling is smaller than to the (3, 2) mode.

18 K

102

50 K
1.0
0

10

20

30

damping beam power, 𝑃𝑑 [mW]-2000

0 2000( 𝜔 − 𝜔𝑚 )/2𝜋
[Hz]

(2,3)

1000

𝑆𝑧 [am2 /Hz]

10 K
1.1

(3,2)
2.62

(e)

10

𝑇0 [K]

1.2

(d)
50

100

(c)
cryostat temperature, 𝑇0(Raman-ratio thermometry) [K](b) 104
4.8 K

𝑆𝑧 [am2/Hz]

Stokes/anti-Stokes Ratio, 𝑅𝑠𝑎

(a)

5
5

10

cryostat temperature, 𝑇0(silicon diode thermometer) [K]

2.66

6

4
2
0

50

2.64

𝜔/2𝜋 [MHz]

2

10

20

probe beam power, 𝑃𝑝[mW]

Figure 6.3: (a) Stokes/anti-Stokes peak ratio for the (3, 2) mode at four cryostat temperatures. Solid curves are linear fits to the data. Pp = 26 µW and κ/2π = 2.7 MHz. (b)Stokes (red) and anti-Stokes (blue) spectra near the (3,2) mode for Pd of 3.2 µW (top),
14 µW (middle), and 28 µW (bottom). Resolution bandwidth is 100 Hz. (c) Comparisonof Raman-ratio thermometry with silicon diode thermometer for the (3, 2) mode (brownsquares) and (5, 2) mode (green diamonds). Dashed line indicates agreement between thetwo methods. (d) Raman spectra of (2,3) and (3,2) mode at Pd = 25 µW. Black curves are fit
Lorentzians excluding the region around the (2,3) mode. (e) Extracted temperature, T0 , forvarious probe powers. Vertical error bars in (a), (c), and (e) represent estimated statisticalstandard deviation. Horizontal error bars in (c) and width of gray band in (e) represent
a combination of the diode thermometer accuracy (±0.5 K) and inaccuracy introduced bydrift and oscillations in the temperature over the span of the measurement.

6.6

Systematic Error
Thermometry relying on sideband asymmetry does not require accurate knowledge

of often-difficult-to-measure system parameters such as the optomechanical coupling rate,
effective modal mass, circulating optical power, or optical detection efficiency. Additionally, the single-resonant-probe method [80] employed in this work reduces the sensitivity to63many systematic errors as compared to techniques that employ multiple off-resonant probe
tones [79, 81]. Such effects include drift in the relative amplitude of the probes, optomechanical modification of the mechanical susceptibility, coherent optomechanically induced
interference between the probes [89], and increased sensitivity to classical laser noise [90].
However, many systematic effects in our system can lead to in temperature determination,
and we now discuss several of these potential error sources.
A necessary external input parameter to extract the physical temperature is the intrinsic mechanical linewidth, which is measured via mechanical ringdown with an uncertainty ofa few tenths of a percent. However, long term drift in the mechanical linewidth can be at thepercent level. The intrinsic linewidth is recorded at each cryostat temperature, and foundto increase from 0.84 Hz at 4.8 K to 1.07 Hz at 50 K. The optically-damped mechanicallinewidths are obtained by Lorentzian fits to Raman spectra.
Heating induced by absorbed laser light can cause thermal gradients between the membrane and silicon diode thermometer and increase n̄. We see no evidence for absorptive heating, which would cause deviations from linearity in the data of Fig. 6.3(a).Another, more
fundamental, systematic effect is the RPSN drive motion, as discussed above. It contributes
0.2 K to T0 for Pp = 26 µW probing the (3, 2) mode. We apply this correction to the extracted physical temperature data in Fig. 6.3(c) and the theoretical expectation bands ofFig. 6.2.
In addition to the quantum noise of the probe beam, optomechanically mediated correlations in the classical laser noise can also lead to systematic error. Such noise-squashingor noise-cancellation effects can change the relative heights of the Raman peaks leading toan underestimate of the effective mode occupation [90, 91, 81]. To assess this systematic, wehave independently measured the classical phase and amplitude noise as well as the detuning of the input probe beam (see appendix C). These measurements constrain the potential
error [90] on temperature due to classical laser noise correlations to be at the percent level orless under all operating conditions. Additionally, we have reduced the probe beam power by

64
an order of magnitude and observed only a weak trend in extracted temperature (Fig. 6.3(e)),
which confirms that the effects of both classical laser noise and absorptive heating are small.
The (3, 2) membrane mode is nearly degenerate with the (2, 3) mode, with a frequency
difference between the two modes of 7.2 kHz (Fig. 6.3(d)). For all values of employed optical
damping, the (3,2) mode and (2,3) mode are well resolved in the spectra. For thermometry,
Raman peaks are always compared within ±2 kHz of the peak of (3, 2) mode resonance. Thus
we believe the presence of the (2, 3) mode is a negligible perturbation on the thermometry
data shown in Fig. 6.3. Increasing the optical damping well beyond the level presented here
could cause mode hybridization and significantly complicate the interpretation of the Raman
spectra [92].
6.7

Conclusion
In conclusion, we have shown that Raman asymmetry is a viable technique to diag-nose both the effective mode temperature of an optically damped membrane resonator andthe physical temperature of the same device. These measurements demonstrate the selfcalibrating nature of the method and elucidate many of the systematic uncertainties. Our
results show that the quantum effects governing the asymmetry [93, 81] are visible in amembrane-in-cavity optomechanical system operated within an order of magnitude of room
temperature.
The PnC substrate has reduced our susceptibility to noise from thermally occupied
mechanical modes of the substrate [16]. Without PnC isolation, we have observed thatfrequency noise of the optical resonance due to substrate motion can contribute a systematic
uncertainty in Raman-ratio thermometry at the tens of percent level.

Chapter 7
Future Directions and Conclusions

We see a bright future for the quantum optomechanics with PnC-shielded, high-stressmembranes. For the first generation of the devices, we observed sideband asymmetry evenat 50K. In the course of writing this thesis, the third generation of the PnC devices entereda deep RPSN regime. We operated a PnC-shielded membrane in a 1.7 mm-long FabryPerot cavity in a dilution refrigerator. We observed a mode of 17 million Q and cooled
it to the ground state with a phonon occupation of less than 0.2. The experiment paves
the way for evading measurement backaction, beating the standard quantum limit, andrealizing mechanical squeezed states. For future devices we now collaborate with Norcada
Inc. to optimize the device quality and yield. It would also be interesting to investigate
the highest Q-frequency product one can get with Si3 N4 membrane shielded by the PnC.
We also learned that the membrane is highly functionalizable. For example, the device formicrowave to optical frequency conversion in our group utilize the metalized membrane. Its
mechanical dissipation can be engineered with the curvature map and the PnC developed inthis thesis.

Appendix A
High-Stress Membrane Mechanics

In the world of elasticity, a membrane is a thin film under high tension. The restoring
force of a membrane mode is dominated by tensile rather than internal elastic stress. As
such, the elastic contribution (i.e. the term associated with the Young’s modulus) is ignoredin the equation of motion for a vibrating membrane. However, we find that even though
the elastic contribution in membrane resonant frequencies is negligible, it plays a significant
role in the mechanical dissipation of our membrane. Therefore, our equation of motion must
include both the elastic and the tensile stress contribution. The proper jargon of such object
in elasticity is “a plate with uniform boundary tension” or “a plate with hydrostatic inplane
force”.
A.1

Equation of Motion
For an isotropic plate with uniform boundary tension, the differential equation for atransverse mode w(x, y, t) is [22]D∇4 w − σh∇2 w + ρ

∂ 2w
= 0,
∂t2

(A.1)

where D is the flexural rigidity defined by D = Eh3 /[12(1 − ν 2 )], E is the Young’s modulus,
h is the plate thickness, ν is the Poisson’s ratio, σ is the inplane stress, and ρ is the mass
per unit volume. Assuming sinusoidal oscillation, Eq. A.1 becomesD∇4 W − σh∇2 W − ρω 2 W = 0.

(A.2)

67
In Cartesian coordinates it reads
 4

 2

∂ W
∂ W
∂ 4W
∂ 4W
∂ 2W
D
= ρω 2 W.
++ 2 2 2 − σh
+
∂x4
∂y 4
∂x ∂y
∂x2
∂y 2

(A.3)

It is convenient to express the equation in dimensionless form. For a square plate with sideplength l, let ξ = x/l, η = y/l, and Ω = ω ρl2 /σ:

  2

D ∂ 4W
∂ W
∂ 4W
∂ 4W
∂ 2W
= Ω2 W.
(A.4)++2 2 2 −+σhl2 ∂ξ 4
∂η 4
∂ξ ∂η
∂ξ 2
∂η 2
A.1.1

Boundary Condition

There are three types of boundary conditions for the modes of a vibrating plate:clamped, simply-supported (pinned), and free boundary conditions (see Fig. X). The correctboundary condition for all four sides are the clamped boundary condition. That is
W (ξ, η) all edges =∂W (ξ, η)∂nall edges = 0,

(A.5)

where n is the axis perpendicular to the edge.
A.2

Normal Modes of a High-Tension Plate
To solve Eq. (A.6) with boundary condition Eq. (A.5), note that the coefficient of thecross term 2∂ 4 W/∂ξ 2 ∂η 2 is small for our devices: 2D/(σhl2 ) ∼ 10−6  1. Therefore, we
can apply standard perturbation theory with  ≡ 2D/(σhl2 ) as the “small parameter” and∂ 4 /∂ξ 2 ∂η 2 as the perturbation operator.

  2

∂ 4W
∂ W
∂ 2W
2D ∂ 4 W
D ∂ 4W+−++= Ω2 W.
σhl2 ∂ξ 4
∂η 4
∂ξ 2
∂η 2
σhl2 ∂ξ 2 ∂η 2
|
{z
}

(A.6)

perturbation term

Then the unperturbed equation can be solved by separation of variables. Let W (ξ, η) =X(ξ)Y (η), Ω2 = Ω2ξ + Ω2η , and E 0 = E/(1 − ν 2 ). We have
E 0 h2 ∂ 4 X∂ 2X−= Ω2ξ X,
12σl2 ∂ξ 4
∂ξ 2
E 0 h2 ∂ 4 Y∂ 2Y−= Ω2η Y,
12σl2 ∂η 4
∂η 2∂X
ξ=0,1 = 0,
∂ξ∂YY η=0,1 =η=0,1 = 0.
∂ηX

ξ=0,1 =

(A.7)

68
These are just the dimensionless equations of doubly clamped beams with Young’s modulus
E 0 and pre-stress σ .

A.2.1

1D Eigensolutions

The eigensolutions um (ξ) and the corresponding frequencies Ωm of Eq. A.7 are givenby [94]um (ξ) =vm (ξ),0 ≤ ξ ≤ 21(−1)m+1 vm (1 − ξ), 1 < ξ ≤ 1
2


√
βm
vm (ξ) = 2a sin[βm ξ] +(exp[−αm ξ] − cos[βm ξ])αmΩm ∼ mπ.

(A.8)

(A.9)
(A.10)In Eq. (A.9) a  l is the vibration amplitude, and αm and βm are defined as 1/2 
1/2
p
1
αm =
1 + 1 + 2Ω2m
 1/2 
1/2
p
1
.
βm =
−1 + 1 + 2Ω2m
(A.11)

Note that βm /αm  1 if the mode index is not too large. The mode um (ξ) have a sinusoidal
shape with an small exponential correction near the edge for the clamped boundary condition.
The solutions with intermediate  are given in Ref. [94].

69
A.2.2

Zero-Order Solution

The products {um (ξ)un (η)} are then the solutions of the unperturbed equation. The
normal modes of the plate, however, must be symmetrized because of the degeneracy:







√1 um (ξ)un (η) + un (ξ)um (η) , m > n


2


 

(0)
(ξ, η) = √1 u (ξ)u (η) − u (ξ)u (η) , m < n
Wmn(A.12)
m
n
n
m

2







um (ξ)um (η),
m=n
2
2
2
Ω(0)
mn = Ωm + Ωn .

(A.13)The orthogonality condition isZ Z
1 1 1 (0)(0)
Wmn (ξ, η)Wm0 n0 (ξ, η)dξdη = δmm0 δnn0 .
2
a 0 0
A.2.3

(A.14)First Order Correction

It turns out the zero-order solution is already an excellent approximation. To see that,
we now calculate the first-order correction due to the perturbation term. Note that for
(0)

(0)

(0)

(0)

m 6= n, Ωmn = Ωnm , i.e., the modes Wmn (ξ, η) and Wnm (ξ, η) are degenerate. Hence weneed degenerate perturbation theory. Note that the perturbation operator is diagonalized in(0)

the basis set {Wmn (ξ, η)}. Thus, we can easily calculate the eigenfrequencies to first orderΩmn and the mode shapes with the first order correction Wmn (ξ, η). We can estimate thecontribution from the first-order correction by the following calculation:(0) 2Ωmn 2 − Ωmn
(0) 2

1
a2

Ωmn
Z 1Z 100∼π 2 m2 n2
1
m2 + n2(0)

Wmn (ξ, η)Wm0 n0 (ξ, η)dξdη ≤2π 2 3/2 mm0 nn0 1,|m2 + n2 − m02 − n02 |

(0)

Ω(0)
mn 6= Ωm0 n0 ,(A.15)

assuming βp /αp  1 for p ∈ {m, n, m0 , n0 }, or equivalently π 2 max{m, m0 , n, n0 }2 /2  1.
(0)We see that these two quantities are much smaller than 1. Therefore, we use {Wmn (ξ, η)}as the approximated solution.

70
A.2.4

1D Eigensolution in the Small λm Limit

I define a dimensionless parameter λ =p√2 = E 0 h2 /3σl2 . In the limit of small λm,

which holds for our membranes and other high-tension resonators [11, 28, 29, 95], Eq. (A.11)
becomes 1/2 1/2  2 1/2
p
1
2
2
αm =
1 + 1 + 2Ωm
≡
∼


λ
 1/2 

p
1/2
1
βm =
−1 + 1 + 2Ω2m∼ mπ.
The beam equations Eqs. (A.8)-(A.10) become (back to coordinate x)vm (x),
0 ≤ x ≤ 2l
um (x) =(−1)m+1 vm (l − x), l < x ≤ l
2
 h

i
h mπx ih −x i√λmπmπxvm (x) = 2a sin
+
− cos
exp
l
2
λl/2
l
r
mπ σ
.
ωm ∼
l
ρ
A.2.5

(A.16)(A.17)(A.18)(A.19)

Mode Functions of Bilayer Plate

For the fully or near fully metallized plates, we calculate mode functions by using asingle-layer plate with a thickness of 100 nm, an effective stress, Young’s modulus, massdensity, and Poisson’s ratio. The effective Young’s modulus E = E1 = 135 GPa is theaverage of the Young’s modulus of Si3 N4 and Al weighted by their thickness. The massdensity ρ of Al and Si3 N4 are very similar, and hence we use an effective mass density ofρ ∼ 2.7 g/cm3 . We use an effective Poisson’s ratio of ν = 0.31.
A.3

Anelastic Loss of a High-Stressed Plate
We place the xy plane at the middle plane of the plate and let the plate oscillate with asmall amplitude at an angular frequency ω in the z-direction. The oscillation of plate W eiωt

71
induces a variation of local bending and overall elongation [11], and thus gives oscillatingstrains of






2
∂
 ε̃xx (t)   −z ∂x2 W eiωt   12 ∂x
W eiωt 

 

 

  
 
2 
2





ε̃ε(t) ≡  ε̃yy (t)  =  −z ∂ 2 W eiωt  +  1 ∂ W eiωt ∂y  2 ∂y

 

 
 

2
∂
−2z ∂x∂yW eiωtε̃xy (t)
0
{z
} |
{z
}
|
local bending∼O(W )elongation∼O(W 2 )










∂2

(A.20)

2 −z ∂∂xW2  εxx  iωt 
 iωt
iωt


∂2W  e≈≡ −z ∂y2  εyy  e ≡ ε0 e .








2
∂ W
−2z ∂x∂yεxy(A.21)The strain variation is mostly induced by pure bending, as shown in Eq. (A.20). The
accompanying variation of stresses are given by the usual constitutive equation of a classicalplate [22] with the complex Young’s modulus Ẽ = E1 + iE2 :





 1 ν
 σ̃xx (t) 
0








Ẽ
 ε̃ε(t) ≡ ẼMε̃ε(t).

σ (t) ≡ σ̃0 σ̃yy (t)  = 1 − ν 2  ν 1








0 0 (1 − ν)/2
τ˜xy (t)(A.22)

Note that the stresses can be separated into in-phase and out-of-phase terms as follows:
σ (t)] = <[ẼMε̃ε(t)] = <[ẼMεε0 eiωt ]<[σ̃= E1 Mεε0 cos(ωt) − E2 Mεε0 sin(ωt) .
|
{z
} |
{z
}
in-phase
out-of-phase(A.23)The mechanical work done per unit volume per oscillation is then given byIdε̃εσ T ]<[ ]dt∆w = <[σ̃
dt
Z 2π
ω 

deiωt
=
E1 cos(ωt)εε0 T M − E2 sin(ωt)εε0 T M <[εε0]dtdt0= πE2ε0 T Mεε0 ≥ 0.

(A.24)

72
This leads to dissipation. Meanwhile, the bending energy supplied by the in-phase stresses
is stored in the system.
Z π/2ω
ubending =0

σ (t)] · <[
<[σ̃

dε̃ε
E1 ∆w
]dt = E1ε0 T Mεε0 /2 =
.
dt
E2 2π

(A.25)

Eq. (A.25) is consistent with the strain energy density of a pure bending plate derived inRef. [23]. We can see that the local dissipation ∆w is proportional to the density of stored
bending energy. In other words, the bending energy supplied by the out-of-phase stresses isconverted irreversibly to heat.
To find the total dissipation per cycle, we integrate Eq. (A.24) over the entire volumeof the plate and insert the strain-displacement relations [Eq. (A.20)]. We have
Z

Z

∆wdV = πE2ε0 T Mεε0 dV
ZZZo
πE2 (x, y) n22(εxx + εyy ) − 2(1 − ν)(εxx εyy − εxy /4) dxdydz(A.26)=(1 − ν 2 )Zo
2πE2 (x, y) n (εxx + εyy )2 ε2xy=+
− εxx εyy dV(A.27)(1 + ν)2(1 − ν)4"(!2
 2 2 # )
Z
ZZ
2
2
2
2
∂ W∂ W
πE2 (x, y)
∂ W
∂ W
∂ W
−2(1 − ν)= z 2 dz
+
−
dxdy.
2
2
2
2
2
(1 − ν )
∂x
∂y
∂x ∂y
∂x∂y
{z
}
{z
}
|
|
2
(mean curvature)
Gaussian curvature∆U =(A.28)The total loss is given by the integration of curvatures over the plate plane.

A.3.1

Stored Energy

In order to evaluate the quality factor Q = 2πU/∆U , we need to calculate the stored
energy U and energy loss ∆U . We calculated ∆U above, and here we analyze the stored
energy, which can be obtained from the maximum kinetic energy or the maximum elastic
energy. The maximum kinetic energy isρhω 2
Ukinetic =
2

ZZ

2

2 2

W (x, y) dxdy = 2ρπ f

Z

W (x, y)2 dV.

(A.29)

73
The maximum elastic energy for a tensioned resonator is the sum of the displacement-induced
elongation energy and the bending energy [11]. These two energies are given by
2 
2 )
Z Z (
∂W
σh
∂W
dxdy
Uelongation =
+
2
∂x
∂y
Z
Z
E1 ∆w
Ubending = ubending dV =
dV.
E2 2π
A.3.2

(A.30)(A.31)

Effective Loss Modulus

Using the expressions for Umn and ∆Umn above, and assuming E2 (x, y) is constantover the metallized area, we can evaluate the quality factor or damping rate for each modewith only one unknown quantity E2 . In our data analysis we extract an effective E2 for the
100 nm bilayer membrane from the data using a least-squares fit, as discussed in the maintext. The loss modulus of Al can then be estimated by E2 (hSi3 N4 + hAl )/hAl , where hSi3 N4and hAl are the thickness of the Si3 N4 and the Al, respectively.
A.4

Interpretation

A.4.1

Isotropic Plates

For a plate with uniform thickness and constant E2 , we can simplify Eq. (A.28) using
Green’s theorem. The integration of Gaussian curvature is zero for a clamped plate (CCCC): 2 2 #ZZ " 2
I 2
I 2∂ W ∂ 2W
∂ W
∂ W ∂W∂ W ∂W−
dxdy =dy −dx = 0.
2
2
2
∂x ∂y
∂x∂y
∂y ∂x
∂x∂y ∂x
Thus Eq. (A.28) becomes
πE2 h3∆U =12(1 − ν 2 )2ZZ  2
∂ W
∂ 2W
+
dxdy.
∂x2
∂y 2

(A.32)

Note ∂ 2 W/∂x2 is the curvature of the mode in the x direction, and ∂ 2 W/∂y 2 is the curvaturein the y direction. The sum of the two curvatures is the mean curvature.

74
A.4.2

Curvature Induced near the Clamped Edge and around the Antinodes

In our anelastic model, the local dissipation of a clamped plate is proportional to thesquare of the mean curvature, as shown in Eq. (A.32). Hence, it is important to understandits spatial dependence. Since our 2D normal modes are the product of 1D modes, we startby calculating the curvature of a 1D mode un . In the limit of λn  1, we can approximate
the square of the curvature as
 2
2

h
i
h −x id un (x)2n2 π 2 a 4
2 2
2 nπx
+ n π sin
,
∼
exp
dx2
l4
λ2
λl/4
l

l
0≤x≤ .
2(A.33)The equation above describes the x dependence of the curvature in the x direction. Notethat the coefficient of the first term is much larger than that of the second term. This firstp
exponential term is concentrated in a small range λl/4 = h E 0 /48σ, leading to significant loss near the edge. The second term distributes over the entire length with sinusoidal
dependence, just like the curvature of a simply-supported beam.
For the 2D case, in addition to the above-mentioned features, the large curvatureinduced near the edge oscillates sinusoidally along the plate edge. To see that, we calculate
the mean curvature square of the diagonal mode (n, n) at the edge x = 0:
2
 2
∂ Wnn ∂ 2 Wnn
+
∂x2
∂y 2

h
i
16n2 π 2 a2
2 nπy
sin
.
x=0 ∼
λ2 l4
l

(A.34)

See Fig. A.1 for an illustration of the square of the mean curvature over the entire plate
plane.

A.4.3

Edge Loss and the Frequency Dependence of the Quality Factor

To understand the role of loss induced near the clamped edge, here we study Q ofa mode (m, n) in the limit of small λn and λm. First, we calculate the maximum kinetic
energy. Applying Eq. (A.14) to Eq. (A.29), we have2
Umn ∼ ρha2 l2 (ωm+ ωn2 )/2 ∼ σha2 (m2 + n2 )π 2 /2.

(A.35)

75
Second, we calculate the loss per cycle drawing on our earlier curvature analysis. Here we
use Wmn (x, y) = um (x)un (y). We have verified that the result is the same as using thesymmetrized mode functions in Eq. (A.12):
2
ZZ  2∂ Wmn ∂ 2 Wmn
dxdy
z dz
+
∂x2
∂y 2
2
ZZ  2πE2 h3d um (x)d2 un (y)=un (y) + um (x)dxdy
12(1 − ν 2 )
dx2
dy 2
(Z 
2 Z
2
Z
Z  2
πE2 h3d2 um (x)d un (y)22=dx un (y) dy + um (x) dxdy12(1 − ν 2 )
dx2
dy 2

Z 2
Z 2
d um (x)d un (y)um (x)dx
un (y)dy
+2
dx2
dy 2
( Z
2
2
Z l/2  2l/2  2πE2 h3d um (x)d un (y)=2aldx + 2aldy12(1 − ν 2 )
dx2
dy 2
0
0
)
Z l/2 2
Z l/2 2d un (y)d um (x)um (x)dx
un (y)dy
+8
dx2
dy 2
0
0

Z
Z
h −x ih −y iπE2 h316n2 π 2 a2 l/2
16m2 π 2 a2 l/2
∼
exp
dx
+
exp
dy
12(1 − ν 2 )l 3 λ2λl/4l 3 λ2
λl/4
0
0
|
{z
}
“edge loss”
Zhi
4 4 2 Z l/24m π a4n4 π 4 a2 l/2 2 h nπy i
2 mπx
+
sin
dx +
sin
dy
l3
l
l3
l
0
0
{z
}
|
“antinode loss”
hi  2nπ 2 a Z l/2hi  2mπ 2 a Z l/2
2 mπx
2 nπy
sin
dx −
sin
dy
+8 −
l2
l
l2
l
0
0
|
{z
}
“antinodeloss”

πE2 h34(m2 + n2 )π 2 a2 (m2 + n2 )2 π 4 a2
(A.36)∼+2λ12(1 − ν 2 ) |
l{z
l2
} |
{z
}
edge
antinode


2
2
3
3 2
2
(m + n )π E2 h a(m + n2 )π 2∼
1 +λ.
(A.37)
3(1 − ν 2 )l2 λ|{z} |{z4}edge
antinode

πE2∆Umn =
1 − ν2Z

2

76

Figure A.1: The spatial dependence of the mean curvature square of the mode (2, 2) over theentire plate plane. A low stress of 0.1 MPa is used, and the values of curvature are scaled
nonlinearly to make the structure more visible.

We see that the ratio of the antinode loss to the edge loss is given by λ(m2 + n2 )π 2 /4.
Combining Eq. (A.35) and (A.37), we obtain
2πUmn
3(1 − ν 2 )σl2 λ
Qmn =
∼
∆Umn
E2 h2



(m2 + n2 )π 2
1+λ
4

−1


−1
1 E1
(m2 + n2 )π 2∼
1 +λ.
λ E2 |{z} |
{z4
}
edge
antinode(A.38)

Similarly, we can derive the formula for a 1D string (with ν = 0)
1 E1
Qn ∼
λ E2



n2 π 2
1+λ
4

−1
.

(A.39)For small λ as low as 10−4 −10−3 in our devices and Ref. [28, 29, 95], Eq. (A.38) and Eq. (A.39)
correspond to a slightly decreasing Q as a function of frequency. For λ ∼ 2 × 10−2 as inRef. [11], the edge loss and the antinode loss are comparable when n = 4. This gives astronger dependence on n, and hence frequency.

77
A.4.4

Physical Meaning of λTo investigate the physical meaning of λ, we calculate the elongation energy of the(m, n) mode in the limit of small λn:
2

2 )
Z Z (
dum (x)
du
(y)
σh
n
dxdy
Uelongation =un (y)2 + um (x)2
2
dx
dy
(Z
2
2 )
Z l/2 l/2 
dvm (x)dvn (y)= σhla
dx +
dy
dx
dy
0
0
( Z
)
Z l/2
h
i
h
i
l/2
2π 2 σha2
2
2 mπx
2
2 nπy
∼
m
cos
dx + n
cos
dy
l
l
l
0
0
= π 2 σha2 (m2 + n2 )/2 = Ukinetic = Uelastic .

(A.40)

This means that the maximum elastic energy is dominated by the elongation energy. Thus,
Qmn =

2πUmn
2πUelongation
E1 Uelongation
∼
=
.
∆Umn
2π(E2 /E1 )Ubending
E2 Ubending(A.41)

Using Eq. (A.38) and Eq. (A.41), we find−1Uelongation
1(m2 + n2 )π 2=
1+λUbending
λ4
1∼ , for (n, m) = (1, 1).
λ(A.42)

Thus, λ is the ratio of the bending energy to the elongation energy for the fundamental
mode.

Appendix B
Calculation of Optical Spectra of the Squeezed LightIn this appendix we describe our solution to the Heisenberg-Langevin equations of motion for our optomechanical system. We then compute the expected output optical quadrature spectrum, and the spectrum obtained from balanced homodyne detection and direct
photodetection.

B.0.1

Heisenberg-Langevin Equations

We begin with the following Hamiltonian, H = H0 + Hκ + HΓ , where H0 describes theintracavity coherent dynamics, Hκ represents the coupling of the optical system to external
fields, and HΓ represents the external thermal coupling to the mechanics [86, 63, 64, 5, 96].
H0 = ~ωm c† c + ~ωc a† a + ~GZzp (c + c† )a† a(B.1)

where ωm is the mechanical resonance frequency, c (c† ) is the mechanical annihilation (creation) operator, ωc is the optical resonance frequency, a (a† ) is the optical intracavity annihip
lation (creation) operator, G is the optomechanical coupling constant, and Zzp = ~/2mωm
is the mechanical zero point motion, with m the mechanical resonator effective mass. We
define a single photon optomechanical coupling rate g = GZzp , and a dimensionless mechan
ical displacement operator z = c + c† −hc+c† i. The Hamiltonian is linearized by assuming
a large optical coherent state amplitude compared to the vacuum level, a = (ā + d(t))eiωL t ,where ωL is the optical drive frequency, ā = hai is the intracavity coherent state amplitude,

79
assumed to be real, and d(t) is an operator containing the quantum and classical noise on the
optical field. Terms of order d2 are neglected. The linearized Hamiltonian that encapsulates
the basic interaction is
H0 =~ωm c† c + ~ωc a† a+~GZzp (c + c† )ā∗ ā + HBS + HT M S
HBS =~GZzp ā∗ c† d + ācd†
HT M S =~GZzp ā∗ cd + āc† d†

(B.2)
(B.3)
(B.4)

The resulting linearized Hamiltonian contains both the beam-splitter (HBS ) and the twomode squeezing (HT M S ) Hamiltonians.
We solve the Heisenberg-Langevin equations of motion for this system in the freR∞
quency domain, using the Fourier transformation convention f (ω) ≡ −∞ eıωt f (t)dt, f † (ω) ≡†R ∞ ıωt †e f (t)dt, f † (ω) = f (−ω). We assume thermally driven mechanical motion, with−∞mechanical damping rate Γ0 and initial thermal occupation nth . We include the effects of
the additional optical damping beam in an orthogonal cavity mode, by defining effective
values for ωm , Γ0 , and nth for the motion of the mechanical resonator in the presence of the
optomechanical damping, spring, and cooling induced by the damping beam [86]. The optomechanical effects of the signal beam are intrinsic in the equations of motion. The opticalloss rate to the input port, output port, and internal loss are κL , κR , and κint respectively,
yielding a total cavity damping rate of κ = κL + κR + κint . The external optical input fields
consist of a coherent state, of frequency ωL , incident on the input port of the two-sided
Fabry-Perot cavity, and vacuum incident on the output port. An effective detuning of the
input signal field from the average value of the optomechanically shifted cavity resonanceis given by ∆. The optical output operator, aout = (āout + dout (t))eiωL t , is computed using√√
the cavity input-output relations āout = κR ā, dout + din = κR d, where din is the noise
operator representing the vacuum field incident on the output port [97].

80
B.0.2

Optical Output Spectrum

The quadrature output operator is defined as Xφ (ω) = aout (ω)eiφ + a†out (ω)e−iφ , where
φ is the quadrature phase angle. Because we have assumed ā to be real, the input-output
relation indicate āout is also real, and φ = 0 (φ = 90) corresponds to the amplitude (phase)quadrature. The symmetrized power spectrum of the quadrature operator is SXX (ω).
SXX (ω) =hXφ (−ω)Xφ (ω)is1= (hXφ (−ω)Xφ (ω)i + hXφ (ω)Xφ (−ω)i)
2=Aζζ (ω) + Azz (ω) + Aζz (ω)(B.5)

The spectrum consist of three terms. Aζζ is the shot noise on the output. Azz represents
the noise imprinted by the actual mechanical motion. The cross term Aζz contains thecorrelations between shot noise and motion driven by radiation pressure from the shot noise.
This term is responsible for any squeezing.

2 2

Azz (ω) = κR |ā| g



2

2

|χc (ω)| +|χc (−ω)| −χc (ω)χc (−ω)e2iφ−χ∗c (ω)χ∗c (−ω)e−2iφhz(−ω)z(ω)is√Aζz (ω) = i κR āg (−χc (−ω)e2iφ + χ∗c (ω))hz(−ω)ζ(ω)is + (−χc (ω)e2iφ + χ∗c (−ω))hζ(−ω)z(ω)is +(χ∗c (ω)e−2iφ − χc (−ω))hz(−ω)ζ † (ω)is + (χ∗c (−ω)e−2iφ − χc (ω))hζ † (−ω)z(ω)isAζζ (ω) = hζ † (−ω)ζ(ω)is + hζ(−ω)ζ † (ω)is = 1√√
The output shot noise operator is ζ(ω) = χc (ω) κL κR ξL (ω)+χc (ω) κint κR ξint (ω)+(χc (ω)κR −1)ξR (ω). ξL , ξR , and ξint are the Langevin vacuum noise operators for the input port, output
port, and internal loss of the cavity respectively.
 



1
nth + 1/2
nth + 1/22 2222hz(−ω)z(ω)is =Γ0++ 2ωm g κ|ā| |χc (ω)| + |χc (−ω)||N (ω)|2|χm (ω)|2|χm (−ω)|2√√−ωm āg κR−ωm āg κRhz(−ω)ζ(ω)is =χc (ω);hζ(−ω)z(ω)is =χc (−ω)N (−ω)N (ω)

81hζ † (−ω)ζ(ω)is = hζ(−ω)ζ † (ω)is =1
2

We have introduced the following notation. The cavity susceptibility is χc (ω) =(κ/2 − i(∆ + ω))−1 . The mechanical susceptibility is χm (ω) = (Γ0 /2 − i(ω − ωm ))−1 . The
optomechanical damping and spring effects are encompassed in N (ω) = (χm (ω)χ∗m (−ω))−1 −i2ωm g 2 |ā|2 (χc (ω) − χ∗c (−ω)). We also assume the mechanical thermal and optical vacuum
baths are uncorrelated at different times hξ(−ω 0 )ξ(ω)i = δ(ω − ω 0 ), for noise operator ξ, andwe assume integration over ω 0 for physically relevant quantities.
Any loss in the optical detection system, including propagation losses between thecavity and detector, imperfect mode matching to the homodyne detector, or finite photodetector conversion efficiency can be modeled by a single effective loss port with fractional loss√
ext . The loss port attenuates the signal reaching the detector āout → ext āout , and injects
vacuum noise leading an effective quadrature spectrum of SXX (ω) → ext SXX (ω) + (1 − ext ).
The homodyne detection consists of combining a strong optical local oscillator with theoutput from the cavity on a beam splitter. Both outputs of the beam splitter are recordedon photodetectors and the two photocurrents are subtracted. Assuming an equal splitting
on the beam splitter, the subtracted photocurrent signal is proportional to (āout āLO eiφ −āout āLO e−iφ ) + āout (dLO + d†LO ) + āLO (dout eiφ + d†out e−iφ ), where the annihilation operator of
the local oscillator is aLO = (āLO + dLO (t))eiωL t+φ , and we have neglected terms of order d2 .
The third terms is proportional to Xφ . The second term which represents the local oscillator
vacuum noise beating against the coherent portion of the cavity output field is negligible whenāLO  āout , and is typically ignored. However, in the homodyne detection system describedin the main text, we are limited to a local oscillator power which is less than 10 times larger
than the signal beam power in order to ensure the photodetectors to remain in their linear
range. In this case, the local oscillator noise term must be included to quantitatively model
the homodyne data. The one-sided, symmetrized, shot noise normalized spectra, Sφ (ω) in

82

1.65

Frequency [MHz]

a
1.60

25 dB

1.55

1.50

-1.5 dB

𝜟/𝟐𝝅 = 𝟎

1.65Frequency [MHz]

b
1.60

25 dB

1.55

1.50

-1.5 dB

𝜟/𝟐𝝅 = −𝟒𝟐 kHz

Frequency [MHz]

1.65

c

1.60

25 dB

1.55

1.50

-1.5 dB

𝜟/𝟐𝝅 = −𝟏𝟎𝟎 kHz
-135

-90

-45

0

45

𝜙 [degrees]

Figure B.1: Calculated Homodyne Spectrum for finite signal beam detuning. Homodyne
transmission spectra, Sφ , are calculated for three signal beam-cavity detunings: (a) ∆/2π =0, (b) ∆/2π = −42 kHz, (c) ∆/2π = −100 kHz. The other parameters used are g/2π =eff
eff
33 Hz, m = 6.75 × 10−12 kg, ωm/2π = 1.5243 MHz, Γeff
= 3.8 × 10−4 K,
0 /2π = 2560 Hz, T
82ext = 0.55, κ/2π = 1.7 MHz, κR = 0.6κ, N̄ = 1.1 × 10 , ext |āout | /|āLO |2 = 0.1. The
parameters of panel (b) match the parameters of the measured spectrum presented in Fig. 5.8.
With those parameters, the mechanical damping from the signal beam is 6 kHz. Calculated
spectra are displayed on a logarithmic scale. The region between the white 0 dB contours issqueezed.

the main text are then given by:Sφ (ω) =2 (|āLO |2 (ext SXX (ω) + (1 − ext )) + ext |āout |2 )
2 (|āLO |2 + ext |āout |2 )(B.6)

Using this full expression, one sees the level of perceived squeezing is reduced by the additional uncorrelated noise floor of the local oscillator.
The one-sided, symmetrized, shot-noise-normalized, direct photodetection spectrum,

83
discussed in the main text requires SXX (ω) to be evaluated at φ = 0.
SI (ω) = ext SXX (ω)|φ=0 + (1 − ext )(B.7)

Appendix C
Laser Classical Noise Measurement

Laser noise has two different origins: quantum noise and classical noise. The laser
quantum noise is fundamental; it is a consequence of the Heisenberg Uncertainty Principle.
It cannot be eliminated by any technical trick. The laser classical noise is technical; it isarising, for example, from excess noise of the pump source or from vibrations of the laser
resonator. It can be reduced by methods such as passing the laser beam through a filter
cavity or employing a difference detection scheme.
Many quantum optomechanics experiments concerns the study of optomechanical effects from quantum shot noise of the laser. The radiation pressure of the quantum fluctuations produces many interesting quantum phenomena such as optomechanically squeezed
light described in Chapter 5, sideband asymmetry and ratio thermometry described in Chapter 6, quantum measurement backaction, and quantum backaction limit of optomechanicalsideband cooling. However, the amplitude fluctuation and phase fluctuation from the laser
classical noise, could produce similar optomechanical effects as well. To isolate interesting
quantum effects from the these classical noise effects, we need to minimize the laser classicalnoise. My goal in this appendix is to provide methods and theories for detecting the laser
classical noise, and summarize the measurements of classical noise in Chapter 6 and recent
experiments in our lab.
I first present the methods for detecting the laser classical noise. For each method, I
provide a simple description, mathematical formulation, and the formula for the detected

85
spectrum. In the end, I summarize the measurements of classical laser noise in the labs B236
and X1B40 in the year of 2014. The measurements are done by me and Bob, respectively.
The optical setup in the lab B236 did the sideband asymmetry thermometry measurements,
while that in the lab X1B40 did the measurement of quantum backaction limit of laser
cooling. For each measurement, I show the data and specify the detection method. Then I
describe the data analysis for extracting the values of the laser classical noise. The extracted
numbers are summarized in Table C.1.
C.1

Mathematical Description of the Laser NoiseIn this section, I provide the mathematical description of the laser noise and define

the measurable quantity that represent the size of the laser classical noise. The optical fieldwith laser noise can be described by [90]:â(t) =

1
ˆā + ξl (t) + (δx(t) + iδy(t)) eiωl t ,2where â(t) is the quantized complex amplitude of the laser electric field, ā =(C.1)
pP/~ωl , P is

the total laser power, ωl is the laser frequency, ξˆl (t) is the vacuum noise operator in whichˆ ξˆ† (t0 )i = δ(t − t0 )hξ(t)ˆ 0 )i = 0.
hξˆ† (t)ξ(t(C.2)ˆ ξˆ† (t0 )i = δ(t − t0 ) and hξˆ† (t)ξ(tˆ 0 )i = 0. δx(t) and δy(t) are classical laser amplitudehξ(t)and frequency noise variables, respectively. Since we are only concerned with noise near themechanical frequency, a white noise model can be used:hδx(t)δx(t0 )i ≡ Cxx δ(t − t0 )hδy(t)δy(t0 )i ≡ Cyy δ(t − t0 )hδx(t)δy(t0 )i ≡ Cxy δ(t − t0 ),

(C.3)

86
2
where Cxx , Cyy ≥ 0 and Cxy are real numbers. Cxy is bounded by Cxy≤ Cxx Cyy , as dictatedby the Cauchy-Bunyakovsky-Schwarz inequality. They represent the laser amplitude andfrequency noise. We assume they are proportional to laser power.
C.2

Classical Amplitude Noise Measurement
In this section, we describe two nearly-calibration-free methods to measure the classicalamplitude noise.
The key to measure Cxx is to measure the intensity noise spectrum and the shot noise
spectrum. The first spectrum has both the classical amplitude noise and the shot noise,
while the second spectrum has only the shot noise. By dividing the two spectra, we canextract Cxx . The only requirement is the quantum efficiency of the photodiode.
The first method we employ is a balanced detection (Fig. C.1). The laser beam isdivided into two beams of equal power with a beam splitter and detected by two identicalphotodiodes. The photo-currents on the two photodiodes are added or subtracted electronically. The power spectral density of the added (subtracted) signal, called the sum (difference)
spectrum, is calculated from a digitalized signal with a computer. Several thousand spectra
are averaged to reduce the statistical fluctuation. The sum spectrum is the intensity noise
spectrum and the difference spectrum is the shot noise spectrum.
Alternatively, one could use a direct detection that only requires a single photodiodeand no electronics. That is, one directly shine the laser beam or a flashlight on a photodiodeand calculate the power spectral density of the photocurrent. In this case, the laser spectrum
has both the classical laser noise and the shot noise, while the flashlight spectrum has only
the shot noise. If the power of the laser beam and the flashlight are the same, by dividing
the laser spectrum with the flashlight spectrum we can extract Cxx . The drawback is thatthe flashlight power can not be stabilized and may decrease during the measurement as thevoltage of the battery decays.
We note that we can also mix the two methods. For example, we can measure the

87

+\-

Figure C.1: Optical Setup for balanced detection.

intensity noise spectrum with the direct detection and the shot noise spectrum with thebalanced detection. We can use this method if we don’t have the electronics to do the sum
but can do the difference of the photocurrents.

C.2.1

Mathematical Description of the Direct Detection

Here we calculate the power spectral density in the direct detection. We first calculate
the photocurrents on the diodes, and then calculate their auto-correlation functions. The
spectrum is calculated by performing Fourier transform to the auto-correlation functions.
1ˆWe start with a mode described by â(t) = ā + ξl (t) + 2 (δx(t) + iδy(t)) eiωl t . The
mode experienced by the photodiode is attenuated and added by a new vacuum noise becauseof the finite quantum efficiency σp of the photodiode:



 
p
√
σp
i 1 − σp  â(t)
Â(t) 

=





 p
√
ˆ
i 1 − σp
σp
ξ(t)
loss

(C.4)

Let the gain of the photodetector be G. The photocurrent is given bypˆ + h.c.
ˆ = G[Â(t)]† Â(t) = Gā2 σp + Gāσp δx(t) + Gā√σp √σp ξˆl (t) + i 1 − σp ξ(t)
I(t)

†
√ ˆ
2
ˆ
≡ Gā σp + Gāσp δx(t) + Gā σp ξs (t) + ξs (t) ,(C.5)
where ξˆs (t) ≡√σp ξˆl (t) + ipˆ is the incoherent sum of the vacuum operators. Note
1 − σp ξ(t)ˆ are not correlated, ξˆs (t) still obeys Eq. (C.2). We next calculatethat because ξˆs (t) and ξ(t)88
the auto-correlation:
ˆ I(tˆ + t0 )i = G2 ā4 σ 2 + G2 ā2 σ 2 Cxx δ(t0 ) + G2 ā2 σp δ(t0 )R(t0 ) = hI(t)
p
p

(C.6)

The single-sided power spectral density is then:Z ∞S(ω) = 2SII (ω) = 20

R(t0 )e−iωt dt0 = 2G2 ā2 σp (1 + σp Cxx )

(C.7)

−∞

For the flashlight we expect Cxx = 0. Therefore for the same light power (same voltage on
the photodetector)
Slaser (ω)/Sflashlight (ω) = 1 + σp Cxx .
C.2.2

(C.8)

Mathematical Description of the Balanced Detection

Here we calculate the sum and the difference spectrum in the balanced detection.
Again we start with an input laser mode â(t). The two beams on the two photodiodes,aˆ1 (t) and aˆ2 (t), can be expressed as follows: aˆ1 (t)  √12
=

 

√i
aˆ2 (t)2√i  â(t)
2 

√1
2



ˆ
ξ(t)

.

(C.9)

The modified modes due to the finite quantum efficiency σp of the photodiode are Â1 (t) andÂ2 (t). The sum and the difference current areIˆsum (t) = Iˆ1 (t) + Iˆ2 (t) = G[Â1 (t)]† Â1 (t) + G[Â2 (t)]† Â2 (t)†√ = Gā2 σp + Gāσp δx(t) + Gā σp ξˆs (t) + ξˆs (t)(C.10)

Iˆdiff (t) = Iˆ2 (t) − Iˆ1 (t) = G[Â2 (t)]† Â2 (t) − G[Â1 (t)]† Â1 (t)†√ = Gā σp ξˆd (t) + ξˆd (t) .

(C.11)

Here we rewrite the vacuum noise terms as ξˆs and ξˆd . They both obey Eq. C.2. The one-sided

89
power spectral densities are therefore
Z ∞Ssum (ω) =2
Z−∞∞
Sdiff (ω) =2−∞0hIˆsum (t)Iˆsum (t + t0 )ie−iωt dt0 = 2G2 ā2 σp (1 + σp Cxx )(C.12)

0hIˆdiff (t)Iˆdiff (t + t0 )ie−iωt dt0 = 2G2 ā2 σp .

(C.13)

We can extract the Cxx via
Ssum (ω)/Sdiff (ω) = 1 + σp Cxx .
C.3(C.14)Classical Frequency Noise Measurement
In this section, I describe the detection method for measuring classical frequency noise.

Similar to the measurement of classical amplitude noise, we want a measurement scheme
that, ideally, only detects the classical frequency noise and the shot noise. And we divide
the spectrum by the shot-noise-only spectrum to extract Cyy .
Here we employ a delayed-line technique (Fig. C.2). The delayed-line converts frequency fluctuations of a beam to frequency-dependent phase fluctuations. These phasefluctuations are sensed by the heterodyne or the homodyne detection. The optical setup(Fig. C.2) starts with a laser beam that splits into a signal beam and a strong local oscillator (LO). The signal beam passes through a 8 m fiber that provides a time delay τ ofapproximately ∼ 50 ns. To perform heterodyne detection, we shift the frequency of the
LO by ωLO = 17 MHz via an AOM. The signal beam and the LO are then recombined,
co-propagated, and split equally to make a balanced detection. The difference spectrum of
the photo-currents is then measured.
The shot noise spectrum can be measured with a balanced detection. We can easily
have that by blocking the light on the arm with a delayed line. Note that we can also employ
homodyne detection. In that case, we do not need an AOM to shift the LO frequency.

90

signal beamLO
adjust(optional)

AOM
17 MHz-Figure C.2: Setup for measuring laser classical frequency noise. The beam to be measured
is split into a signal beam and a local oscillator (LO). The signal beam passes a 8 m fiber sothat it is delayed by approximately 50 ns compared with the LO. Then the two beams arerecombined and split equally to make a balanced heterodyne detection.

91
C.3.1

Mathematical Description of the Delayed-Line Heterodyne (Homodyne) with Balanced Detection

The optical modes which shine on the two photodiodes, aˆ1 (t) and aˆ2 (t), can be described by








aˆ1 (t)  √12

=

 
√i
aˆ2 (t)2√i  eiωLO t
2 
√1
2



0





0

  α iβ  â(t)


,



ˆ
eiφ(t) e−iωτ T̂ (−τ )
iβ α
ξ(t)

(C.15)

where T̂ (−τ ) is the time-delay operator. For any function g(t), T̂ (−τ )g(t) = g(t − τ ). α2(β 2 ) is the ratio of the signal beam (LO) power to the total power, α2 + β 2 = 1, and φ(t) is
the relative phase fluctuation between the two arms (derived from fluctuations of the fiber
length, mirror position, etc.). Note that the formula describes both heterodyne (ωLO 6= 0)
and homodyne (ωLO = 0).
Following the same procedures in the previous section, we calculate the difference
spectrum for the case of homodyne and heterodyne separately.

C.3.2

Heterodyne

In this section we derive the formula without locking the phase. Without the phase lock,
R 2π
1
Sdiff (ω)dφ:
we should calculate the phase-averaged, one-sided spectrum S̄diff (ω) = 2π0n2G2 ā2 σ + 2α2 β 2 G2 ā2 σ 2 Cxx (1 + cos[ωτ ] cos[ωLO τ ])+oCyy (1 − cos[ωτ ] cos[ωLO τ ])+2Cxy cos[ωτ ] sin[ωLO τ ] ,(C.16)

where σ = σp 2 is the total quantum efficiency, and  is the visibility of the interferometer.
The ratio of the difference spectrum to the shot noise spectrum under the same laser power

92
is thennr = 1 + α2 β 2 σ Cxx (1 + cos[ωτ ] cos[ωLO τ ])+(C.17)oCyy (1 − cos[ωτ ] cos[ωLO τ ]) − 2Cxy cos[ωτ ] sin[ωLO τ ](C.18)no= 1 + α2 β 2 σ(Cxx + Cyy ) + cos[ωτ ]α2 β 2 σ (Cxx − Cyy ) cos[ωLO τ ] − 2Cxy sin[ωLO τ ] .
(C.19)
The function r has a constant offset plus an oscillating term with the frequency period
2π/τ . From the constant offset one can measure Cxx + Cyy . To measure Cxx , Cyy , and Cxy
separately, in principle one can set ωLO such that sin[ωLO ] = 1 or sin[ωLO ] = 0. However,this is hard to do in our case because we need 30 MHz to rotate π/2 in phase, and thedetection bandwidth is only about 20 MHz. The ideal length of delayed line is > 100 m sothat the we only need < 3 MHz.
We note that given the same laser power, the signal is maximized when the power ofthe signal beam and the LO are equal, α2 = β 2 = 1/2. This is also true for the homodyne
delay line (see below for the formula).
C.3.3

Homodyne

The difference spectrum with phase φ between the LO and the signal beam can be
expressed as


ωτ
ωτ
Sφ (ω) = 2G2 ā2 σ + 8α2 β 2 G2 ā2 σ 2 Cxx cos2 [ ] cos2 [φ] + Cyy sin2 [ ] sin2 [φ] .
2
2(C.20)

We can lock the phase at the FM quadrature φ = π/2:Sφ (ω) = 2G2 ā2 σ + 8α2 β 2 G2 ā2 σ 2 Cyy sin2 [

ωτ
].
2

(C.21)The ratio of the difference spectrum to the shot noise spectrum is therefore
r = 1 + 4α2 β 2 σCyy sin2 [

ωτ
].
2

(C.22)

This technique gives a pure frequency noise measurement if we have the locking capability.

93
C.4

Data and Analysis
In this section, we summarize the measurements of classical laser noise in the labs

B236 and X1B40 in the year of 2014. The optical setup in the lab B236 did the sideband
asymmetry thermometry measurements, while that in the lab X1B40 did the measurementof quantum backaction limit of laser cooling. We summarize the method, data, and extractedclassical noise values (Table C.1).

C.4.1

Classical amplitude noise Cxx measurement

For B236, we employ the balanced detection discussed in section C.2. The data isshown in Fig. C.3 (a) and (b). Using Eq. C.14, we find Cxx < 0.01 at 25 µW. For X1B40,
we use the direct detection discussed in section C.2. The data is shown in Fig.C.3 (c) and(d). Using Eq. C.8, we extract Cxx < 0.004 at 10 µW.
C.4.2

Classical frequency noise Cyy measurement

For B236, we employ the heterodyne delayed-line technique with the balanced detection
discussed in section C.3. We measure the ratio of the difference spectrum and the shot noise
spectrum. The data is shown in Fig. C.4(a). The spectrum of the data is flat over the 6 MHz
frequency range. Comparing with Eq. C.19, the flatness of the spectrum indicates that theoscillation term is small compared with the constant term, Cxx + Cyy . From this constantterm, we find that Cyy < Cxx + Cyy < 0.02 at 25 µW.
For X1B40, we employ the homodyne delayed-line technique with the balanced detection discussed in section C.3. We data of the ratio of the difference spectrum and the shotnoise spectrum is shown in Fig.C.4(b). We see a peak around 6 MHz. Assuming the classicalnoise is white, this peak corresponds to sin2 [ ωτ
] = 1 in Eq. C.22. From the magnitude of the2
peak, we estimate that Cyy < 0.04 at 10 µW.

(a) -80

1

P_sum/P_diff (dB)

PSD (dBm)

-81

-82

-83

0

-1

-84

Difference
Sum

-85

-2

1.0

1.5

2.0

2.5

frequency (MHz)

3.0

3.5

4.0

(d) 2

-90

1

P_laser/P_flashlight (dBm)(c) -88

PSD (dBm)

94

(b) 2

-92

-94

1.0

1.5

2.0

1.0

1.5

2.0

2.5

3.0

3.5

4.0

2.5
frequency (MHz)

3.0

3.5

4.0

frequency (MHz)0-1flash light
laser
-2

-96
1.0

1.5

2.0

2.5

frequency (MHz)

3.0

3.5

4.0

Figure C.3: Measurements of classical amplitude noise Cxx . (a) and (b) show the data oflaser beam with a power of 25 µW in B236 via the balanced detection. (c) and (d) show
the data of the laser beam in X1B40 with the direct detection. (a) The sum and difference
spectrum. (b) The ratio of the sum and difference spectrum. (c) The laser and flash light
spectrum. (d) The ratio of the laser and flash light spectrum.

C.4.2.1

Cxy estimation

We estimate the upper bound of Cxy using our knowledge of Cxx and Cyy and CauchyBunyakovsky-Schwarz inequality.
Table C.1: Measured classical noise in B236 and X1b40
power

Cxx

Cyy

Cxy

B236

25 µW

< 0.01< 0.02 < 0.01

X1B40

10 µW < 0.004< 0.04 < 0.01

95(a) 1.0(b)

10

PSDdiff/PSDshot (dB)

0.5

PSD (dBm)

2

0.0

-0.5

8
7
6
5
4
3
2

1
8
7
6
5
4

-1.0

3

14

16

18

frequency (MHz)

20

0

2

4

6

8

frequency (MHz)

10

12

14

Figure C.4: Measurements of classical amplitude noise Cyy . (a) B236 Data from delayed-Lineheterodyne technique with the balanced detection. The detected spectrum is normalized tothe shot noise spectrum with the balanced detection.. (b) X1B40 Data from delayed-Line
homodyne technique. The detected spectrum is normalized to the shot noise spectrum.

Appendix D
Light and Mechanics Spectra with Laser Classical Noise

Following the goal of previous appendix, bounding the size of laser classical noise
effects, my goal here is to investigate the optomechanical spectra with laser classical noise.
D.1

Three effects from the classical noise
Like the quantum shot noise effects discussed in Appendix B (Eq. B.5), classical am-

plitude and phase noise of the laser incident on the cavity give rise to three different noise
terms in the output optical spectrum: 1) Sc,ln (ω), the transmitted laser classical noise itself, 2) Sc,disp (ω), a component of the measured mechanical motion driven by the radiationpressure of laser classical noise, and 3) Sc,corr (ω), the interference of the previous two terms.
The expression for the full spectrum therefore has lots of terms. Fortunately, the linear
superposition principle allows us to divide the derivation. The classical noise contribution canbe calculated without considering the shot noise and the mechanical thermal noise because
these noise source are not correlated. Therefore, I can write the output light operator dˆoand membrane position operator x̂ as:
n1
o
√ √
ˆ
do = κL κR χc (ω) (δx + iδy) − igāx̂
2o
1 2ωm nx̂ = −gā δxπ+ (ω) + δyπ− (ω)
2 N(ω)noi= − χeff
(ω)
δxπ
(ω)
+
δyπ
(ω)
+
−
2 m

(D.1)
(D.2)
(D.3)

97
With these two equations, we are ready to calculate the output spectrum. For the heterodyne
detection without phase locking, the red sideband spectrum Sred (ω) can be calculated byDEDEdˆo (−ω)dˆ†o (ω) , while the blue sideband Sblue (ω) can be calculated by dˆ†o (−ω)dˆo (ω) . To
distinct the three effects in both sidebands, we define
E
red
red
red
(ω)
(ω) + Sc,corr(ω) + Sc,dispdˆo (−ω)dˆ†o (ω) = Sred (ω) = Sc,ln
D
E
blue
blue
blue
dˆ†o (−ω)dˆo (ω) = Sblue (ω) = Sc,ln(ω) + Sc,disp(ω) + Sc,corr
(ω).
D

(D.4)
(D.5)

Note that here we assume the local oscillator beam has negligible classical noise per powercompared with the signal beam. It follows that the classical noise floor are given by


1
1
red
Sc,ln
(ω) = d κL κR |χc (−ω)|2 (Cxx + Cyy ) ≡ f (−ω) C̃xx + C̃yy
4
4


1
1
2
blue
Sc,ln (ω) = d κL κR |χc (ω)| (Cxx + Cyy ) ≡ f (ω) C̃xx + C̃yy ,
4
4

(D.6)

where f (ω) ≡ d κR |χc (ω)|2 = κ |χc (ω)|2 , C̃xx = κL Cxx , and C̃yy = κL Cyy . Note that Cxx ,Cyy , and Cxy is a function of laser power or ā. The spectra of the mechanical motion driven
by the radiation pressure of the laser classical noise are:red
2
Sc,disp(ω) =f (−ω)(gā)2 |χeffm (ω)| Γnc,ba

(D.7)

blue
2
Sc,disp(ω) =f (ω)(gā)2 |χeffm (ω)| Γnc,ba(gā)2 |π+ (ω)|2 C̃xx + |π− (ω)|2 C̃yy − 4Im [χc (−ω)χc (ω)] C̃xy .
nc,ba =
4Γ

(D.8)
(D.9)

The interference spectra are:


 eff

1
red
Sc,corr(ω) = f (−ω)(gā)2 {C̃xx Im iχeffm (ω)π+ (ω) − C̃xy Im iχm (ω)π− (ω)
2


 eff

+ C̃xy Re iχeffm (ω)π+ (ω) − C̃yy Re iχm (ω)π− (ω) }


 eff

1
blue
Sc,corr(ω) = f (ω)(gā)2 {−C̃xx Im iχeffm (ω)π+ (ω) + C̃xy Im iχm (ω)π− (ω)
2



 eff
+ C̃xy Re iχeffm (ω)π+ (ω) − C̃yy Re iχm (ω)π− (ω) }.

(D.10)

Fig. D.1 show all three classical noise effects on sideband spectra under excessive phase noise.
See the caption for their qualitative features.

98
Red sideband (itemized)
Cyy=1
ε=1

Blue sideband (itemized)
Cyy=1
ε=1

shot noiseS(ω)

classical noise

S'dispSc,disp

Sc,corr

frequency
Red sideband (sum)Blue sideband (sum)

S(ω)

Cyy=1
ε=1
one mechanicalquanta

Cyy=1
ε=1

total
one mechanicalquantasum of thefive terms

frequency

Figure D.1: (Color) Illustration of classical noise effects on red and blue sideband spectra.
Here we model a red-detuned laser with excessive classical phase noise. The quantum efficiency is assumed to be perfect, 1. The upper two plots show each classical noise effects
on the spectra, while the lower two plots show the sum of classical noise effects and the0
overall shapes. Sdisp: the spectrum of the mechanical motion if there is no classical noise.
The classical noise floor (as well as the size of other classical noise effects) is asymmetric on
both sidebands due to cavity susceptibility. The shot noise floor remain symmetric. Sc,disp isalways positive. Sc,corr is negative when there is excessive classical phase noise. In general,
Sc,corr can be positive or negative (Eq. D.10). For the overall shapes, spectrum with one
mechanical quanta noise is added to the red sideband due to the “sideband asymmetry”
arising from the zero-point motion and the correlation of RPSN and shot noise [93]. The
parameters are: Cxx = 0.002 and Cyy = 1 at laser power of 5 µW,  = 1, gā/2π = 1.4 × 105 ,ωm /2π = 1.48 MHz, Qm = 8 × 106 , κ/2π=2.6 MHz, ∆/2π = −1.62 MHz, Tbath = 360 mK.

99
D.2

Error analysis in ratio thermometry
In this section, we analysis the systematic error in sideband-asymmetry thermometry

due to laser classical noise. As shown in Fig. D.1, the three effects described in the previous
section modify the sideband ratio R. This is a source of systematic measurement error to
the phonon occupation measurement. Namely,
1
1−,
Rest /s − 1 R/s − 1∆n̄ =(D.11)

where Rest is the modified sideband ratio and s is the sideband ratio due to cavity susceptibility only.
The first noise term contributes local white noise to both sidebands with different sizeon top of the shot noise floor. This causes a systematic error due to the normalization in
our data analysis. We normalize the spectra such that the noise floor is 1, the value of shotnoise in our theory. Therefore we introduce an extra factor in the sideband-ratio:n̄ + 1R /s =
n̄
est

,

red
1 + Sc,ln= R/s
red
1 + Sc,ln,

red
1 + Sc,ln.
red
1 + Sc,ln(D.12)The second and the third noise terms comprise the majority of the laser noise error.
They add or subtract the sideband height. Here we convert their contributions into unit ofmechanical quanta and express their effects on modified sideband ratio as:
Rest /s =n̄ + 1 + n̄c,ba + n̄redc,corr (ωm )bluen̄ + n̄c,ba + n̄c,corr (ωm )(D.13)

blue
where nc,ba come from the second noise, and n̄redc,corr (ωm ) and n̄c,corr (ωm ) are defined by
red
red
2
n̄red
(ω)(gā)2 |χeffc,corr (ωm ) ≡Sc,corr (ωm )/{fm (ω)| Γ}
blue
blue
2
n̄blue(ω)(gā)2 |χeffc,corr (ωm ) ≡Sc,corr (ωm )/{fm (ω)| Γ}(D.14)

Here we apply the theory to analyze the effect of potential classical noise sources on themeasurement of quantum backaction limit of optomechanical cooling in Ref. [98]. I plot the100classical noise effects in Fig. D.2 with the experimental parameters. Using Eq. D.11 andD.13,
the error due to the classical noise of the cooling laser is bounded by ∆n̄laser ≈ 0.006. Anothersystematic error is the presence of o-resonant substrate mechanical modes that rise above
the shot noise oor [4, 99, 16]. The normalization of sideband amplitude in our analysis tothe o-resonant shot noise level is aected by this noise. However, since n̄ is a function of the
sideband ratio, not their absolute amplitudes, it is not strongly aected, and again leads to asmall ∆n̄sub ≈ 0.006. Additional confirmation of the substrate noise’s small effect is that themeasured mechanical sidebands retain a Lorentzian lineshape [90] as shown by comparing
Fig. D.1 and Fig. D.2. Because both laser noise and substrate noise are small, we otherwise
do not include the eects of classical noise directly in the data presentation in Ref. [98].

101

Red sideband (itemized)
Cyy=0.02
ε=0.04

Blue sideband (itemized)
Cyy=0.02
ε=0.04shot noiseS(ω)

classical noise

S'dispSc,disp

Sc,corr

frequencyS(ω)

Red sideband (sum)one mechanicalquantaCyy=0.02
ε=0.04

Blue sideband (sum)Cyy=0.02
ε=0.04total
one mechanicalquantasum of thefive terms

frequency

Figure D.2: (Color) An example of sideband spectra with small laser classical noise andsmall quantum efficiency. Here we plot each classical noise effect and the sum of them as inFig. D.1. The effect due to laser classical noise floor is negligible. The other two effects are
small and have opposite signs. These plots indicate that classical amplitude and phase noise0
are not significant systematic errors in the measurement of quantum backaction noise (Sdisp).
The parameters are set to: Cxx = 0.002 and Cyy = 0.02 at laser power of 5 µW,  = 0.04,
gā/2π = 1.4 × 105 , ωm /2π = 1.48 MHz, Qm = 8 × 106 , κ/2π=2.6 MHz, ∆/2π = −1.62 MHz,
Tbath = 360 mK.

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