A Fully Crystalline Cryogenic Reference Cavity
byDhruv Kedar
B.A., University of Chicago, 2013

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
2023

Committee Members:
Jun Ye, Chair
Andrew Ludlow
James K. Thompson
Thomas R. Schibli
Svenja Knappe

ii
Kedar, Dhruv (Ph.D., Physics)
A Fully Crystalline Cryogenic Reference Cavity
Thesis directed by Prof. Jun Ye

To date, neutral atom optical lattice clocks have demonstrated the highest precision measurements of fractional frequency shifts. Frequency noise of the optical local oscillator, used to drive
the clock transition, is the primary limitation for how quickly these clocks can measure a quantity.
Improving the stability of these oscillators is therefore crucial in enabling efficient characterizationof physics at smaller energy scales. State-of-the-art optical oscillators employing cryogenic reference
cavities have been used to push this frontier, enabling record-level stabilities and precision of neutral atom optical clocks. The exceptional stability of these oscillators has resulted in foundational
advances in optical frequency metrology, establishing these systems as worthy scientific pursuits in
expanding the role of measurement science.
The fundamental performance of cryogenic cavities utilizing crystalline spacers and substrates
has been limited by the Brownian thermal noise associated with mechanical dissipation of the mirror
coatings. Recently, crystalline Al1−x Gax As/GaAs coatings have emerged as a promising candidate
for reduced coating thermal noise. We present measurements of the frequency noise of a fully
crystalline cryogenic reference cavity with Al0.92 Ga0.08 As/GaAs optical coatings.
We report on previously unmeasured birefringent noise of crystalline coatings associated with
anti-correlated frequency fluctuations between the two polarization modes of the cavity, and identify
variables that affect its magnitude. We present phenomenological characterizations of this noise andimplement an interrogation scheme that cancels it by simultaneous probing of both polarization
modes. These results challenge our current understanding of noise processes generating length
fluctuations and introduce our discovery of novel light induced frequency noise.

Dedication

To family and friends for their unconditional love and support. You’ve taught me what thatmeans. To J. M. W, M. R. C., and J. H., I wouldn’t be here without you.

Acknowledgements

Like most sections in this thesis, this will be extremely verbose. But, there is a village tothank.
First, Jun and Debbie. Debbie took on the task of accepting me into her group and JILA
when I had few experimental skills. She quickly taught me that there are no parts of the experimentthat one can afford to shy away from, i.e. “black boxes” should not really be black boxes. On several
occasions Debbie came down to the lab to walk me through various parts of the machine that I
could not follow by myself. After the rotating gas machine was decommissioned, Jun accepted
me into his group. I am indebted to Jun and Debbie for resolving to find a happy home for her
remaining students. Both have been some of the greatest advocates for the well being of theirstudents.
Jun is one of the most incredible scientists I have come across; coupled with an infectious
attitude towards stoking our excitement in lab, it is no surprise that the group has developed into a
powerhouse capable of tackling the most ambitious of experiments (or Jun’s words, ‘nontrivial’). I
will miss many of your poetic physics-related metaphors and I cannot thank you enough for taking
the time to nurture me into the scientist I am today. I drifted through undergraduate lab courses
thinking that measurement noise was an irritating byproduct of taking data but I will never forgetyour statement that “without noise, physics would be boring”. Both Debbie and Jun taught methat values and methods of communicating your work effectively and I hope I can do you bothjustice.
The stable lasers team of Wei Zhang, Eric Oelker, John Robinson, and Will Milner, thankv
you all for being steadfast colleagues who are willing to wade through the muddy trenches of cavity
work. Eric Oelker in particular brought in his LIGO expertise and taught me nearly everything I
know about low noise measurements. Thank you for teaching me that most things can be simulated,
and when they can’t you build the experiment. Wei is one of the kindest and hardest workers that
I will likely ever know, I always appreciate you taking the time to relay historical knowledge of Si4.
Si6 has largely been built off the tenacity of grad students past, John and Will. Both are extremely
careful scientists who find pleasure in the minute details and I can always count on you to engage
on peculiarities that others might find trite.
Thanks to the rest of the strontium group members who I’ve interacted with over the years:
Colin Kennedy, Toby Bothwell, Shimon Kolkowitz, Sarah Bromley, Ed Marti, Sara Campell, Aki
Goban, Christian Sanner, Ross Hutson, Lindsay Sonderhouse, Lingfeng Yan, Alex Aeppli, Maya
Miklos, Yee Ming Two, Will Warfield. You’ve all been great mentors and co-workers and I wish I
had the space to single out memorable interactions I’ve had with all of you. The larger Ye groupfamily on the combs, KRb, YO, and OH, I’ve enjoyed your friendships, knowledge, and meaningfulconversations we’ve shared.
Jan Hall has always been a kind presence in S1B20 and some of my fondest memories fromthat room are the long impromptu conversations we would have on cold cavity drift rates or the
merits of particular voltage regulators. The electronics shop has also been invaluable in all theexperiments in the Ye group. It turns out that pushing frontiers of frequency metrology go hand inhand with pushing frontiers of circuit stability. Terry Brown has been an invaluable mentor on thisregard. Everything I know, and do not know with regards to electronics can be attributed to him
so please seek him out if my circuits present issues. Much of the work in this thesis would have beensignificantly more difficult to realize without his magic resonant photodetectors. James Fung-a-Fat
and Carl Sauer have always been reliable resources when it comes to understanding or repairing
the wizardry of JILA PCBs. The machine shop has similarly been invaluable to Ye group research
efforts, and a consistent source of pleasure to interact with. James, Hans, Kyle, Calvin, and Adam
are always available for assistance with any mechanical problem and can be counted on to add

vi
an appropriate level of humor to any issue. Adam, Calvin, and James in particular constructed
some integral components for the Si6 cryogenic environment. Computing gurus JR, Corey, Dan,and Eric provided support necessary to keep the lab work progressing forward. Special thanks
go to the JILA purchasing department who have gracefully handled numerous rush orders withcomplaint and who can always be counted on to reliably advocate on our behalves with respect tonon-CU entities. Krista and Amy also deserve a special recognition for patiently dealing with myadministrative ineptitude. JILA could not ask for a better visionary than Beth, thanks allowing us
to spend most of our efforts focusing on science.
Our cavity counterparts at PTB have been a delight to collaborate with. Anyone will be
hard pressed to find a kinder and more generous set of individuals than Uwe, Thomas, and Fritz- each of whom are treasure troves of knowledge relating to the history and development of thesilicon cavity effort. Recent addition Jialiang You was a close collaborator in understanding thenovel noise mechanisms associated with crystalline coatings. Without his tenacity and propensity
for careful measurements, our current understanding of AlGaAs coating effects would be lacking.
Garrett Cole has also been a great collaborator, always interested in exploring new phenomena
related to his crystalline coatings.
It takes a special environment to be able to call everyone you interact with a friend. By
necessity, JILA has been a home but I could not ask for a better environment to spend a good
portion of my life. Several of my friendships have been formed in this building, and thank you all
for showing me that there is a world outside of lab. To the of dodos of Rocky Mountain Runner,
the wise old sages of Boulder Track Club, and fearless minions of OSMP, thank you for keeping mesane and helping me reassess what is physically and mentally possible. To my family, thank you
for being there and always supporting my interests. And to Boba, Attila, and Sunny, thanks forshowing me that there’s another component to happiness.

Contents

Chapter
1 Introduction
1.1

1

Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Oscillators and stabilization

3
10

2.1

Fabry-Perots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2

Thermal noise in cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.1

Normal mode method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.2

Direct approach

2.2.3

Temperature dependent damping . . . . . . . . . . . . . . . . . . . . . . . . . 23

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3

Cavity locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4

Frequency stability of oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5

Frequency stability of atom-steered oscillators . . . . . . . . . . . . . . . . . . . . . . 36

3 Experimental setup
3.0.1

Silicon in cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.0.2

AlGaAs coatings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.0.3

Cryostation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.0.4

Noise measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4 Technical noise sources
4.1

43

59

Laser stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

viii
4.2

4.3

4.4

Residual amplitude modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2.1

RAM from the modulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2.2

AM and PM on the phase reference . . . . . . . . . . . . . . . . . . . . . . . 72

4.2.3

Etalon effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Vibration sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.3.1

Measuring cavity sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.3.2

Improving the sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.3.3

Feedback to AVI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Thermo-expansive length stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.4.1

Heat flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.4.2

Precision thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.4.3

Cooldown tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.4.4

Thermal time constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.4.5

Temperature stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.5

Vacuum pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.6

Dissemination noise

4.7

Technical noise budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5 Frequency noise of the cavity
5.1

5.2

123

Photo-birefringent noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.1.1

Photothermal effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.1.2

Photothermal effects in crystalline coatings: transient response . . . . . . . . 128

5.1.3

Photothermal effects in crystalline coatings: coherent response . . . . . . . . 133

Frequency instability of the cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.2.1

Mode area dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

5.2.2

Correlation of light induced noise . . . . . . . . . . . . . . . . . . . . . . . . . 149

5.2.3

Dual frequency locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

ix
5.2.4
5.3

5.4

5.5

Residual noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Direct measurement of birefringent noise . . . . . . . . . . . . . . . . . . . . . . . . . 158
5.3.1

Two optical paths in one setup . . . . . . . . . . . . . . . . . . . . . . . . . . 159

5.3.2

Asynchronous detection of cavity error signals . . . . . . . . . . . . . . . . . . 160

5.3.3

Simultaneous detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

Long term instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
5.4.1

Etalon identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

5.4.2

Pointing stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

5.4.3

Top setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Drift rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

6 Novel applications of crystalline cavities

188

6.1

Ratio Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

6.2

Dark matter searches with oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

6.3

Timescale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

6.4

Gravitational wave detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

7 Conclusion

209

Bibliography

213

Appendix
A Dick effect limited instability and laser noise227

B Analog locking

233

x

TablesTable
2.1

Relations between the fractional frequency noise spectrum Sy (f ), variance σy2 (τ ),and modified variance mod σy2 (τ ). Calculation of these relations is taken from [1]. . . 32

3.1

Comparison of bulk substrate and spacer materials. . . . . . . . . . . . . . . . . . . . 46

3.2

Parameters of the AlGaAs/GaAs coatings . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3

Resonances of Si3 identified in the noise model, with center frequency fi , amplitude
ai , and width Γi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.1

Comparison of low frequency AC sensitivities and DC sensitivities. . . . . . . . . . . 84

5.1

Fast and slow time constants (in seconds) of a double exponential fit the data in
Fig.5.7, and analogous data taken when locked to the other cavity mode. . . . . . . . 131

xiFiguresFigure1.1

Left: Plot of the time record of all the SrI systematic shifts. Changes in atom
number, ambient temperature, or magnetic field all result in corrections to the clock
frequency, and their total magnitude is shown over a six hour data campaign. The
clock achieves over 98% uptime over the course of this single comparison day andslight gaps in the data indicate brief periods where the laser is not locked to theatoms. Right: The same data is plotted as a fractional instability normalized to
the Sr clock frequency. The individual contributions of density shift (blue), BBR(red) and second order Zeeman shift (yellow) are shown as the dashed curves. For
operation times up to 104 s, fluctuations in systematic offsets are bounded below
4 × 10−19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2

5

Density shift evaluations. Density shift measurements were performed over fourweeks at the same trap conditions. The weighted mean of all measurements is shownwith a dashed, black line, with a reduced chi-squared of 1.07

1.3

. . . . . . . . . . . . .

5Non-synchronous comparison (i.e. interrogation times of the local oscillator do not
overlap such that there is sensitivity to laser noise) with the JILA 3D optical latticeclock demonstrates that the beat between the two clocks averages below the quoted
total systematic uncertainty. The black line is a white noise τ −1/2 fit to the single
clock instability.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

xii
2.1

Left: Calculation of transverse modes for Si6. The x-axis refers to a total mode
number (m + n). The closest degenerate mode to HG00 is one where m + n = 10,
and this is still 6.4 MHz away. Right: We measure the cavity couplings to some of
the low order modes with m+n ≤ 3. The highest measured coupling is to HG00 with
η = 0.5. The sum of couplings for displayed modes is still less than 1, indicatingthat there is a finite coupling to other higher modes. We were unable to measurethese couplings, indicating that the residual η ∼ 0.4 is likely distributed over very
many modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2

Left: We calculate the coupling efficiency for an incident Gaussian beam to the
HG00 mode of a cavity. Both beams have the same waist w0 but the incident beam
is displaced from the cavity mode in the transverse direction. When the radialdisplacement δr is equivalent to the cavity waist, the coupling has already dropped
to 0.3. Right: We now assume that the incident field is perfectly overlapped withthe cavity mode in the transverse direction, but the waist of the incident mode win
differs from waist of the HG00 mode. We see that the coupling is much less sensitive
to variations in win . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3

We consider an oscillator with resonance at 1 kHz and varying mechanical loss angle
ϕ. Top: Transfer function Re[χ(ω)]. Bottom: Displacement noise spectrum Sx (ω).
In both cases a lower value of ϕ results in a sharper resonance but only in thedisplacement noise can we see an effect of the quality factor on the low frequency
region. Mechanical resonances of the cavity components are always designed to be
above 100 s of Hz so ω ≪ ω0 is the regime of interest. . . . . . . . . . . . . . . . . . 18

2.4

Reproduced from [2], the x’s denote the noise amplitude contribution from the first
100 normal modes of a 10 cm mirror with a 2.2 mm beam spot. The solid line
illustrates the cumulative noise of all these modes and the summation has still notconverged over the 100 modes. Notably the mirror diameter is crucial for this calculation whereas it is unnecessary in the direct approach. . . . . . . . . . . . . . . . 20

xiii
2.5

Evidence of a temperature dependent loss peak in crystalline silicon from [3]. A
feature at 160 K is discovered to be associated with impurities in the silicon. Anacid etch of the ingot appears to remove the loss peak. . . . . . . . . . . . . . . . . . 23

2.6

Comparison of lineshapes of the various locking schemes discussed. The cavity transmission function (red) is used to enact side of line locking. Its derivative is used in
White’s method [4] for modulation the laser frequency. The true dispersive lineshapeis obtained via the PDH [5] and Hänsch-Couillard schemes [6]. . . . . . . . . . . . . 26

2.7

Top left: Relative power in the carrier, first, second, and third order sidebands forvarying modulation index β. The total optical power is normalized to unity. At β =1, roughly half the power is contained in the carrier while each first order sideband
contains 25%. Top right: Discriminator slope D−1 for different modulation depth.
Again, maximum sensitivity is achieved close to β = 1. Note that there are valuesof modulation index where there is nearly no sensitivity to the PDH signal. At
β = 2.3, power in the carrier drops to zero meaning no light enters the cavity. At
β = 3.8, there is still power in the carrier but there is no electric field oscillating at
1Ω. A PDH signal can be recovered by using the 2Ω sidebands as a phase reference.
Bottom: Error signals measured for varying modulation index. Curves are offset
vertical and horizontally for clarity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.8

Calculation of the frequency domain filter functions |Wa (f )| and |Wm (f )| for the Allan and modified Allan variances. The modified Allan variance has less susceptibility
to high frequency noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.9

Calculation of the time domain filter functions wa (t) and wm (t) for the Allan andmodified Allan variances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.10 Simulation that compares the variance and modified variance for different values of
α where Sy (f ) ∝ f α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

xiv
2.11 Top: We simulate datasets with frequency noise spectrum Sy (f ) ∼ f α for α =−2, −1, 0. Discrete frequency jumps of the same magnitude are added to each timeseries. Bottom: Solid circles indicate the modified variance calculated for the corresponding datasets from the plot above. Dashed lines are variances of the unperturbed
white, pink, and brown noise without the discrete frequency jump. The frequency
jump has the effect of deteriorating the stability of all three time series but alsochanges the long term instability to scale as mod σy (τ ) ∼ τ 0.5 . Data that is dominated by white frequency noise and flicker frequency noise can still have variancesthat corresponding to random walk frequency noise if there is a discrete perturbation. 37
2.12 Calculation of the time domain filter functions rRabi (t) and RRamsey (t) for the two
spectroscopic methods. In both cases we assume a cycle time of 5 s, a spectroscopy
time of 3 s, and in the case of Ramsey a π pulse time of 100 s. . . . . . . . . . . . . 40
2.13 We calculate the Dick effect via Eq.2.66 for different laser spectra Sy (f ) = 10−33 f αwith Tc . For reference, Sy (f ) = 10−33 f −1 is the thermal noise limit of Si3. The
corresponding Allan deviation is insensitive to the type of noise that dominates thelaser spectrum and always follows σy (τ ) ∼ τ −1/2 . However the amplitude of the
Dick effect is determined by the slope of the laser spectrum, with a lower varianceassociated with larger values of α. Solid lines indicate variances calculated with the
Rabi sensitivity function and dotted lines are calculated with the Ramsey sensitivity.
As α decreases, Ramsey spectroscopy provides increasing improvements in stability
over Rabi spectroscopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.14 The various amplitude coefficients |gk /g0 |2 are calculated for different spectroscopic
method. All traces use Tc = 5 s, τd = 4 s. The gold curve uses a conservativetπ = 0.2 s whereas the green trace assumes infinitely short pulses of tπ = 0. The
majority of the Rabi sensitivity is concentrated around frequencies of 1/Tc whereas
Ramsey spectroscopy has a higher contribution from the high frequency harmonics
of the cycle time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

xv
3.1

Loss angle ϕ temperature dependence for commonly used cavity spacer materials.
ULE is not shown on this plot but has a measured room temperature loss angle of
1.6 × 10−5 [7]. Crystalline spacer materials have a ϕ so low that the total Brownian
thermal noise limit of crystalline cavities is limited by the loss angle of the opticalcoating. Figure taken from [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2

Coefficients of thermal expansion for commonly used cavity spacer materials. Between the two crystalline materials of sapphire and silicon, only the latter has a
ZCTE. Fused silica can be an attractive option due to a ZCTE around 170 K but
the significantly higher loss angle means that a silicon cavity will still have a superior
thermal noise limited performance. Figure taken from [8]. . . . . . . . . . . . . . . . 45

3.3

Coefficient of thermal expansion of silicon across the three ranges of operation, 124
K, 16 K, and 4 K. At 124 K, α′ = 2 × 10−8 K−2 , similar to the CTE slope aroundthe ULE room temperature ZCTE. The slope is considerably shallower at 17 K withα′ = 1×10−9 K−2 but requires different cooling technology. There is no zero crossingof α at 4 K but values of 10−11 can be achieved, requiring only µK temperaturestability for 1 × 10−17 cavity instability. . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4

Two 55 cm silicon ingots grown at IKZ Berlin. Image courtesy of Thomas Legero. . 47

3.5

Ringdown measurement of Si6 transmitted light with the cavity at vacuum andcooled to 4 K. No difference in finesse is measured between 4 K and 16 K. A finesseof 290,000 is measured. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.6

Brownian noise contributions for the four silicon cavities referenced through thisthesis. Coating thermal noise is consistently the dominant contribution due to the
exceptionally low loss angle of silicon. Coating materials with high mechanical quality factor are therefore instrumental in developing optical oscillators with longer
coherence times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

xvi
3.7

Reflectance of the two crystalline mirrors used on Si6 at a temperature of 4 K. Datais from Garrett Cole. We use a 1542 nm laser to lock to the cavity, centered withinthe ∼ 150 nm passband of the coatings. . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.8

The mode splitting of Si6 is observed to vary with temperature and intracavity power.
We plot the change in mode splitting normalized to the 8.6 kHz cavity linewidth. . . 52

3.9

A capacity map for our cryostation’s cooler displays the cooling power available oneach stage. We operate in the lower right corner of the plot with less than 100 mW
heatload on stage 2 and roughly 1 W heatload on stage 1. . . . . . . . . . . . . . . . 54

3.10 Vibration noise measured on the AVI with vibration isolation enabled for all threeaxes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.11 A solidworks mockup of the Si6 cryostation, cavity, and part of the optical setup. . . 57
4.1

(Left) A comparison of frequency noise between the RIO and NKT obtained via
heterodyne beat with cavity stabilized light. (Right) The closed loop gain necessary
to push the free running laser noise to the Si6 thermal noise at a given offset frequency. 61

4.2

AOM1 is used for both intensity and frequency feedback but it does not exactly
actuate along the two orthogonalized quadratures. A transfer function of the crosscoupling between frequency to optical power incident on the cavity is shown in (a).
Intensity and frequency servo loops are also shaped to suppress any cross couplings.
Feedback to the AOM frequency is highpassed with a 50 Hz corner while the intensity
servo has an integrator enabled at a 100 Hz corner. The measured cross couplingof the AOM between frequency and intensity is small enough that the control signal
from each cannot introduce noise in the opposite quadrature above the Si6 thermal
noise.

4.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Voltage-to-frequency transfer function of the VCO to AOM on the Si3 and Si6 systems. The delay of the Si6 modulator is slightly lower in frequency and consequently
the maximum attainable loop bandwidth is lower.

. . . . . . . . . . . . . . . . . . . 62

xvii
4.4

Topology of the Si6 PDH loop. GC , GP ZT , and GAOM denote the loop filters for thecommon, piezo, and AOM stages. Responses of the actuators are denoted by AP ZT
and AAOM . Plant responses of the cavity filter function and the limit imposed by
the photodetector bandwidth are given by Pcav and PP D . . . . . . . . . . . . . . . . 62

4.5(a) The input amplification stage of the PDH servo board uses a low-noise FETwith only 40dB of gain at low frequency. This pushes the input referred noise of the√board to the 10 nV/ Hz regime in the low frequency regime. Combined with a large
discrimination slope achieved with the typical operation of a few microwatts incident
on the cavity (b), the servo noise can be pushed to a frequency noise limit that isinsignificant compared to the expected Si6 thermal noise. A large input referrednoise can always be alleviated by using a corresponding steeper discrimination, but
comes at the cost of increased sensitivity to photothermal effects. . . . . . . . . . . . 63

4.6(a) The closed loop actuator crossover illustrates the relative loop shaping between
the two actuators. The phase measurement is informative in identifying the first,
and higher order mechanical resonances of the piezo. . . . . . . . . . . . . . . . . . . 63

4.7

Summing an excitation into the closed loop yields the complete transfer function
with all plant responses included. In the phase domain, this is particularly usefulfor identifying plant response poles in the high frequency. Modeling the closed loop
phase reveals an additional measured pole at 50 kHz which is identified as the PDH
photodetector response. This can then be appropriately compensated by adding azero in the loop. The 2π phase wrap from 10 - 2000 Hz identifies the four integrator
stages implemented with corners throughout that band. . . . . . . . . . . . . . . . . 64

4.8

Calculation of the fundamental shot noise limited length resolution on Si6. We
account for the imperfect cavity coupling. Dashed lines indicated the thermal noise
limit of different cavities.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

xviii
4.9

Diagram of the differential optical phase that can be acquired between the carrier
and modulation sidebands after their generation. Red phases are detected in the
RAM and green phases are differential to the PDH path. RAM and PDH feedback
to the phase errors measured on each photodetector are applied to the EOM and
AOM locations respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.10 Top: PDH error signals with Ω = 5 MHz, PM of ϕ+ = ϕi = π/6, and AM of
α+ = 1/α− = 1.5. The Left column displays demodulation phase of to extract thequadrature component, and φ = π/3 is on the Right. Curves are offset horizontallyfor clarity but not vertically. We can see that PM results in a large offset of the baseline signal that is not removed at φ = 0. For AM this is not the case though nonzero
demodulation phase will reveal a large offset. Bottom: We show the magnitude of
the PDH slope in all three cases. Dashed black lines indicate an offset frequency ofhalf the cavity linewidth. At φ = 0 PM has the surprising feature that maximum
sensitivity to the error signal does not occur at an offset of 0 Hz. Additionally theminimum sensitivity is shifted from the expected location at half the cavity linewidth. 74
4.11 Simulating the effect of a 1 MHz linewidth etalon with 20% coupling stationed 3 MHz
away from the cavity resonance. A significant distortion of the error signal occurs,
and the etalon will always decrease the effective discrimination slope. Coupling to
the etalon will typically be lower than the parts per thousand level if all optics
have high quality AR coatings. The parameters are purposely exagerrated here todemonstrate the etalon effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

xix
4.12 Top: Fractional frequency noise of the out of loop RAM characterization. Though
far from the projected Si6 thermal noise, the cavity instability should be limited toa level similar to the Si3 thermal noise. This would mark a significant improvement
over the instability of the Si4 system with the added benefit that the 4 K closed
cycle cryostat makes Si6 a hands-free oscillator. Dark noise corresponds to noise of
the resonant photodetector. A high white noise level is visible at frequencies larger
than 1 Hz. This specific dataset coupled to the TEM01 mode of the cavity without aphase plate. The poor mode coupling leads to a correspondingly higher contributionof shot noise. Right: Modified Allan deviation of traces on the left. . . . . . . . . . 77
4.13 Vertical acceleration sensitivity of the cavity. Si6 is supported at the midplane andwe measure a sensitivity that is roughly centered at zero. We rotate the cavity on
its mount to be near one of these zero crossings. . . . . . . . . . . . . . . . . . . . . 80
4.14 Transfer functions (left) and phase responses (right) of the three acceleration responses to frequency when driving the AVI along a particular axis. Different rows
display the data of shaking the AVI along each direction. The acceleration response
to driving the platform along gravity (bottom row) is largely contained along V̂ .
This is less the case for driving along the horizontal axes where we can see a few
cross couplings at low frequency. This likely suggests that the platform is torquedin the H1-H2 plane, shaking along the AVI’s horizontal axes generates an admixtureof excitation along H1 and H2. In the phase domain, we can identify some sort ofmechanical resonance of the cryostation near 100 Hz, where phase wrappings areevident in all axes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

xx
4.15 The frequency dependence of measured acceleration sensitivities on Si6. The vertical sensitivity is notably higher than the mid-10−12 s value that can be attained byadjusting the spacer rotation. We observed that the values of the horizontal sensitivity had a slight dependence on rotation angle as well. The current orientation of
the spacer was chosen to minimize the largest sensitivity (in this case kH1 ) without
significantly increasing the others. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.16 Vibration induced frequency noise measured along the different axes and their respective couplings to the cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.17 Various cavity mounts that we tested for improved sensitivity to acceleration noise.
Top left: The original folded cup design from Si4, made from G10 composite.
Top right: A simplified structure made from G10. Bottom left: A monolithic
stainless steel support machined a block of 316 stainless. Hemispherical divots havebeen milled out from the top of each post. Spheres of different material can beplaced within each divot, the image shows a set of teflon balls resting within the
indentations. Bottom right: Our final cavity support made entirely from stainlesssteel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

xxi
4.18 Measured sensitivities for the variety of cavity mounts we considered. Label ”NewG10 mount” refers to the green G10 cup from Fig.4.17, ”Fully S316” refers to thestainless steel mount with rounded tips, and ”Divot” is the stainless mount witha hemisphere milled out from the top of the supporting posts. We place stainlesssteel, nylon, and teflon spheres in the divots and rest the cavity on top. All traces
have been smoothed with a 5th-order spline to improve visual clarity. Horizontal
sensitivities are the more meaningful quantity since we know that kV can be zeroedby adjusting the cavity-mount angle. Variation of kV between the different mounts
is due to imperfect placement of the cavity. All ki steadily increase above 100
Hz, indicating that some other component of the cryostation is contributing to anincreased acceleration sensitivity at high frequency. A large rollup in the kH1 isvisible for all the different cavity supports. At the time of these measurements,
we were suspected that this was not representative of the true H1 sensitivity but
were unsure of its source. We later understood that driving the AVI along theH1 axis (collinear to the coldfinger axis) would excite a compressive mode along
the coldhead’s bellows. This generates air currents that modulate the laser phasecoherently with the cavity’s acceleration response and appear as an artificial kH1
signal. We later remove this effect by pulling a vacuum on the cryostation, and its
absence is noted in Fig.4.15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.19 Vibrations measured along H1 with the AVI isolation enabled are shown in blue.
When we use a lower noise external accelerometer to measure the acceleration spectrum, we can feedback to the AVI by modulating its displacement along H1 to further
reduce the measured noise (black). At frequencies below 40 Hz, the black curve islimited by the noise floor of our new in-loop accelerometer. . . . . . . . . . . . . . . 91

xxii
4.20 Plots of the coherent response measured along each axis when driving the AVI along
one direction. In the top plot for example, the AVI is driven along H1 and we showqP
2
the corresponding transfer functions ai /i ai . The vast majority of the coherent
displacement occurs along H1 at low frequency, though in regions around 50 Hz and
80 Hz, an equal response is measured in the vertical axis. Single axis feedback along
H1 would exhibit strong cross coupling to another axis at these frequecies. . . . . . . 92
4.21 Schematic of the thermal circuit that models heat transfer between the 30 K stage
and the cavity. Radiative and conductive pathways are placed in parallel, and thelow pass filter is completed with the inclusion of materials with high thermal capacitance. An additional lowpass should be included to capture the heat from the 300
K vacuum chamber to the cryogenic 30 K stage, but details this support structure
within the Montana Instruments cryostation are proprietary, and are not shared withus. We calculate an active-passive time constant of ∼10000 s and a passive-cavity
time constant of 600 s. Note that the passive-cavity time constant is reduced fromthe 1200 s modeled and measured in Si4. The G10 cavity support in Si4 is replacedwith the aforementioned stainless steel mount which has a two times lower thermal
resistivity. Unlike the 124 K system, the thermal capacitance of silicon is very small
at 4 K and results in a low thermal time constant between the baseplate of thepassive shield and the cavity mirrors. . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.22 Volumetric heat capacities of materials typically used in a cryogenic environment.
Common materials (dark blue curves) have a vanishingly small heat capacity below
10 K, requiring use of more exotic alloys.

. . . . . . . . . . . . . . . . . . . . . . . . 97

4.23 Thermal conductivity of high purity copper with different residual resistivity ratios. 101
4.24 Various gold-plated components for the thermal shields. The radiation shield is on
the left, 4 K active shield in the middle, and 4 K passive shield on the right. A bafflewith a 5 mm aperture on the radiation limits the room temperature radiation loadpassing through the windows to the cavity. . . . . . . . . . . . . . . . . . . . . . . . 101

xxiii4.25 Left: Temperature calibration of the three Cernox thermistors installed on the cryogenic thermal shields. Right: Sensitivity of the thermistors. . . . . . . . . . . . . . . 103
4.26 Left: A copper disk is used as a dummy cavity, a thermistor attached to it allows us
to measure the expected cavity temperature during cooldown. Right: The active
shield with thermisters installed. Temperature sensors from the passive shield passthrough a small slit beneath the lid. A thin layer of vacuum grease between thelid and cylindrical section improves the thermal contact. Cables for the sensors arewrapped around each shield several times for thermal anchoring. This is especially
important on the radiation shield so that heat flowing from the 300 K vacuum
feedthroughs is dumped on the 30 K stage where the coldhead has more coolingpower. Sensors are attached to the shields with VGE-7041 varnish and baked at 150
C for 6-10 hours for curing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.27 During one of the initial cooldown tests, two thermistors were placed on each of
the active and radiation shields to measure thermal gradients. By design, the highpurity OFHC copper should have suppressed gradients to the mK level. Surprisingly,
a residual gradient of 1 K was measured on the radiation shield (left column) and
0.65 K on the 4 K active shield (right column). We were unable to find a proper
explanation for this, it is possible that curing the epoxy that attached each thermistor
to the shield introduced resistance offsets via mechanical stresses. This is anotherargument for securing the thermistors with a mechanical clamp in future systems.
The bottom row displays an Allan deviation of the gradient fluctuations. The active
shield gradient is remarkably static with fluctuations slightly over the 10 µK level
over 1000 s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.28 The cryostation is heated from 4 K to room temperature while we read out thermistors on the top and bottom of the active shield. The difference in measured
temperature (right) varies significantly over the warmup process. . . . . . . . . . . . 107

xxiv
4.29 Temperatures on the thermal shields and cryocooler stages are continuously measured during the final cooldown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.30 Thermal time constants In a test cooldown of the system, the cavity is replacedwith a a copper disk of the same thermal mass. Thermal sensors are attached to thetestmass, inner 4K shield, and outer 4K shield. The temperature of the outer 4K
shield (active) is stepped via a ring heater wound around its base, and we monitor
the subsequent temperature increases of the inner 4K shield (passive) and testmass.
We extract the time constants between the different stages as well as the thermal
gradients between them. The ring heater on the active shield base also allows for
active temperature stabilization to either the active or passive thermistor. . . . . . . 110
4.31 Top left: Time series of the inloop and witness thermistors placed on the 4 K
active shield (loop is closed on the inloop thermistor). Top right: Instability ofthe two thermistor readings. Bottom left: The witness’ temperature fluctuations
are converted to frequency fluctuations via our measured CTE. The brown curve isthe witness variations filtered by the passive platform and cavity mount. Bottom
right: Instability of the traces on the left. . . . . . . . . . . . . . . . . . . . . . . . . 112
4.32 Vapor pressure of most gases as a function of temperature, taken from [9]. Below 20
K, the only residual gas aside from Helium with a pressure above 10−13 Torr is H2 ,which has a vapor pressure of ∼ 10−6 Torr at 4 K. At the UHV range, most curves
are extrapolated with decreasing temperature. . . . . . . . . . . . . . . . . . . . . . . 114

xxv
4.33 Schematic of the modifications for additional pumping on the cryostation. Montana
Instruments provides a control unit connected to the cryostation with a long 2 m,
0.375” ID vacuum bellows. Within the control unit is a small roughing pump thatis valved off from the bellows with a case valve contained within the control unit. A
vent valve can be left open to the environment or connected to a nitrogen canister tovent the system to ambient pressure. We removed this connection by separating itfrom the pumping bellows with a gate valve. Instead, we add a tee that is connected
to a turbo and scroll pump. After operating these two for a few days, we valve themoff and activate a small 2 L/s diode pump. . . . . . . . . . . . . . . . . . . . . . . . . 116
4.34 Left: Frequency noise of Si3-Si6 beatnote when the vacuum pumps are runningand when shut off. There is no additional noise measured due to vibrations of thepumps. Right: Fractional instability of the beatnote data on the left. In the low
frequency regime, we can slowly displace the AVI platform slightly above the ambient
vibration noise and look for corresponding changes in the cavity noise. Below 1 Hz,
accelerations coupling directly to the cavity should generate frequency perturbations
significantly below the thermal noise limit. Any observed increase in cavity noise
could therefore be an indication of poor vacuum within the cryostation. We do not
observe any such effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.35 The fiber noise cancellation servo is locked with high bandwidth of ∼150 kHz. Thereis little noise present at this high offset frequency, but significant frequency noise ispresent at 1 kHz and below. The high servo bandwidth is necessary for providing
the large servo gain needed to suppress the in-loop error to the cavity thermal noise
limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

xxvi
4.36 Schematic of the setup used to measure the out-of-loop fiber noise. Light from the
NKT laser is split into two paths, one going to a fiber noise cancelled interferometer,
and the other serving as a frequency reference. The fiber noise cancelled link, which
typically distributes light to the Si3-Si6 photodetector, is instead sent to a separateout of loop photodetector to generate a heterodyne beatnote with the reference light.
Frequency noise of this beatnote measured on the OOL PD quantifies the frequency
noise associated with the light dissemination. . . . . . . . . . . . . . . . . . . . . . . 120
4.37 Measurement of the dissemination noise using the schematic shown in Fig.4.36. Left:
Fractional frequency noise of the measurement. Right: Fractional Allan deviationof the measurement. Dissemination noise is well below the projected thermal noise
limit of Si6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.38 Compilation of technical noise sources measured on Si6. The frequency noise due tointensity fluctuations is discussed in the following chapter. . . . . . . . . . . . . . . . 122
5.1

Coating and substrate diffusion lengths are plotted as solid red and green lines respectively. Dotted and dash dotted red lines indicate two length scales relevant tothe coating diffusion, i.e. the beam size and the coating thickness. . . . . . . . . . . 125

5.2

Left: Refractive indices of the high and low index GaAs and AlGaAs layers. Right:
Temperature dependence of the refractive indices. . . . . . . . . . . . . . . . . . . . . 127

5.3

Calculated photothermal transfer functions for the elastic spacer, substrates, andcoating responses, and the thermo-refractive coating effect. In the low frequency
regime where we aim to be thermal noise limited, photothermal effects from thespacer are the dominant contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.4

Transient response of dropping the transmitted intensity from 300 nW to 100 nW for
the E1 mode (solid line) and the E2 mode (dotted line). Intensity is dropped at ∼10seconds, and there is a fast transient, followed by a slow settling of the frequency
that exhibits opposite sign for the two polarizations. . . . . . . . . . . . . . . . . . . 129

xxvii
5.5

Transient response of the E1 when changing the transmitted power between different values. A clear asymmetry in time constants is visible between the left andright panels. Increasing laser power results in a much faster frequency responsethan decreasing the power, indicating that there are different photothermal transfer
functions depending on the sign of the intensity change. . . . . . . . . . . . . . . . . 129

5.6

Frequency responses renormalized by the mangitude of the transmitted power change.
Solid lines indicate curves taken when decreasing the optical power, dashed lines indicate an increase. Curves of same color are taken with the same set of initial andfinal power values. It is clear that there is no singular transfer function δν/δP that
captures the frequency response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.7

Left: Single and double exponential fits of the data in Fig.5.5. Right Frequency
residuals of a single exponential fit (pink) and the double exponential fit (grey). . . . 132

5.8

Left: Measured photothermal transfer function of the cavity response when modulating the transmitted power while locked to a single polarization mode of Si6. Datais taken at an operating temperature of 4 K. Right: Renormalizing the left set of√
curves by Ptransmitted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.9

Converting the photothermal transfer function from units of frequency deviation tolength deviation allows us to directly compare the coating response of Si5 (stars andsolid lines) and Si6 (circles and dotted lines) without complications from the different
cavity lengths. Renormalizing by the transmitted power produces a similar set oftransfer functions for the two systems, despite over an order of magnitude span in
power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

xxviii
5.10 Measurements of the cavity frequency noise for different transmitted powers, ranging
from a minimum of 90 nW to 1000 nW transmitted. When this data was taken,
optical scatter off one of the input vacuum optics contributed a large noise humpat 100 mHz offset. Cavity noise at frequencies below this offset is limited by thistechnical source. Below a cavity transmission of 90 nW, electronic noise sources
start to become significant and no further reduction in frequency noise is observed.
A dotted red line provides a 1/f 1.5 guide to eye, well above the projected Si6 thermal
noise (black line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.11 Left: Another set of frequency noise measurements taken with different intracavity√
powers. Right: Curves on the left are normalized by Pc and collapse to a single
noise limit despite the factor of 10 difference in optical power. A dotted red line isdisplayed as a 1/f 1.5 guide to the eye . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.12 Light-induced noise at a cavity temperature of 4 K (solid lines) and 16 K (dottedlines). Curves of same color are measured with same intracavity power. We do notmeasure any changes in coating loss and transmission between the two temperatures. 141
5.13 Top: Light-induced noise measured on Si5 at 124 K for different intracavity powers.
√
Bottom left: Comparison fractional frequency noise of Si5 and Si6 with the Pc
rescaling implemented. The light induced noise in Si5 is over 10 dB lower than thatin Si6 due to the longer cavity length. Bottom right: Renormalizing by the cavity
lengths allows us to compare the absolute coating noise between the two system. . . 143
5.14 Merit factor gm,n for different spatial modes in the rectangular basis is plotted on
the top, and gρ,l for the cylindrical basis is plotted for the bottom. Rectangularmodes HG0,i and HGi,0 have the same scaling factor due to separability along thetwo axes. This is not the case for cylindrical modes where increasing the azimuthalmode index leads to a larger effective mode area. . . . . . . . . . . . . . . . . . . . . 145

xxix
5.15 Generation of higher order modes with phase plates. The top row shows the phase
plate (middle) needed for converting the fundamental mode (left) into the HG11mode (intensity profile on the right). Yellow regions of the mode converter imprint
an additional π phase shift when compared to the purple regions. The bottom row
shows a similar procedure for generating the LG03 mode.

. . . . . . . . . . . . . . . 146

5.16 When using higher order modes, sensitivity to clipping is increased. Not only is themode area greater, beam size will be larger at all locations along the propagation
axis. Here, we compare the beam size for the LG00 , LG20 , and LG02 modes atvarious locations before the cavity. Locations of various optical apertures are shown
as dashed lines at their corresponding distances from the cavity input mirror. A
pink dot indicates the size of the 4K active window at its location 9 cm from thecavity. The significantly larger beam radius of the LG02 mode makes it much more
susceptible to clipping on this optic. . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.17 Mode area effect on birefringent noise. We separately measure the birefringent noiseof the HG00 and HG01 modes by addressing a single polarization of the cavity. The
01 and S 00 for the two modes reveals the dependence
ratio of birefringent noise Sbiref
biref

on mode area. If the birefringent noise is independent of the mode area, we expect
a ratio of 1. We find a ratio of 0.71, consistent with the expectation of 0.75 if thenoise scales with mode area gmn w02 . A large peak visible at 0.057 mHz is the result
01 dataset and was omitted in the fit. . . . 148
of a ground loop that appeared in the Sbiref

5.18 Data from the bottom right of Fig.5.13 is now divided by the mode area of the Si5and Si6 cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

xxx
5.19 (a)Optical layout for probing the Si6 system with active RAM suppression at a modulation frequency (5 MHz) where the cavity error signal is encoded in a traditional
PDH locking setup. Dual-tone probing is achieved by driving AOM2 with two RF
tones separated by the birefringent mode splitting of Si6 (green and purple sets oftones). A half waveplate before the cavity is aligned to couple an equal mixture ofs- and p-polarized light to the two polarization modes of the cavity, labeled Ê1 andÊ2 . The collinear beams are then launched upwards with a 45◦ mirror to couple into
the cavity. The reflected cavity error signal has equal contributions from noise of thetwo polarization modes, and any anti-correlated noise is automatically rejected. (b)Frequency landscape of the tones applied to the Si6 cavity. The two tones applied
to AOM2 separate the carrier and the two sidebands each into two tones separated
by the birefringent mode splitting of the cavity. The purple tones shown address one
polarization mode (Ê1 ) while the green tones simultaneously address the orthogonalmode (Ê2 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.20 Left: At t = 0, the laser is initially dual frequency locked to E1 and E2 modes.
Around t = 10, the tone addressing E2 is removed so that the laser is only lockedto E1. A frequency jump corresponding to ∆ is observed. The E2 tone is thenreintroduced, and the E1 tone is removed. The laser frequency, distributed after
AOM1, is continuously monitored in a beat with Si3. Right: A cartoon indicating
the location of the AOM2 tones relative to the two polarization modes. Colors
correspond to the data on the left. With ∆ = 0, there is no frequency jump whenswitching from the dual mode lock to the single mode lock, indicating that the
spacing of the AOM2 tones is equivalent to fBiref . . . . . . . . . . . . . . . . . . . . . 155
5.21 Residual noise Sresidual + SBrown after cancellation of the birefringent noise at both
4 K and 16 K shows no temperature dependence. . . . . . . . . . . . . . . . . . . . . 157

xxxi
5.22 Fractional frequency instability of Si6 of birefringent noise (red squares) and residual
noise at 16 K (blue circles). The residual noise is measured over a continuous 24000
s dataset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
5.23 The optical scheme of [10] splits the beam across two different paths that are prepared in frequency and polarization to address the birefringent modes. Beams arerecombined on orthogonal ports of a polarizing beam splitter and coupled into thecavity. A Faraday rotator before the cavity swaps the polarizations of each incoming
beams such that reflected light from the cavity fast axis is measured on the same
path as incident light addressing the cavity slow axis. A heterodyne beatnote between the birefringent modes can be detected in cavity transmission after interfering
the two polarizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.24 Left: schematic of a proposed optical setup. Ideally, a switchable polarizing element could rotate the incident light to the fast or slow axis of the cavity in sync
with the data acquisition rate. Right: A cartoon illustrating the concept of theasynchronous birefringence measurement. An AOM switches the laser frequency toalternate between addressing the two modes. The demodulated signal is recordedon digital multimeter and the corresponding phase error of each cavity mode can bereconstructed to yield the information we want. . . . . . . . . . . . . . . . . . . . . . 162
5.25 The Si6-Si3 heterodyne beatnote (blue) where two lases are locked to the two separate
cavities. Probing the Si6 cavity with a laser locked to Si3 recovers the Si6 noise
spectrum with the Si3 noise included. This signal is measured in cavity reflection
and then converted to frequency noise by calibrating the discrimination slope (pink).
Both traces encode the PDH error signal at the same modulation frequency of 5 MHz.164
5.26 Attenuation factor in transmission of power coupled into the cavity. . . . . . . . . . 164

xxxii5.27 Left: Shot noise limited length resolution in a 1 Hz bandwidth for the Si6 cavity.
Reducing the optical power in the carrier will decrease the magnitude of the birefringent noise, but also hinder the ability to measure that noise at high frequency.
Lowering the measurement bandwidth can mitigate this. Right: Johnson noise
equivalent for a particular level of shot noise. If the detector noise floor is limited
by Johnson noise and other op amp noise sources are below it, this yields the noise
equivalent power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.28 Fractional frequency noise of the Si3-Si6 beatnote is computed with an evenly sampled time series of beat (black). The effect of aliasing the data, or introducing
deadtime between measurements can be simulated by extracting a time series ofevery alternate point (e.g. all even timestamps), doubling the effective acquisition
period to account for deadtime, and computing the power spectrum (blue). The low
frequency noise of the beatnote is fully recovered, but an artificial white noise floor isevident at higher frequencies. Increasing the deadtime or sampling even fewer points
has the same effect, but accentuates the magnitude of the white noise. In the asynchronous measurement, datapoints will alternate between addressing the fast andslow axes of the cavity. The degree of common mode rejection for noise present inboth modes can be estimated by taking the difference of two signals aliased from thesame dataset. In orange, we show the difference channel between an aliased signal
composed of all even points of the Si3-Si6 timeseries and an aliased signal composedof all odd points. Low frequency noise in the difference is suppressed to the levelof the white noise which is incoherent to the two datasets. Over 20 dB of common
noise rejection is visible at the lower range of 100 mHz. . . . . . . . . . . . . . . . . 166

xxxiii
5.29 Sensitivity to the AC components (blue) from the Pound-Drever-Hall modulation
scheme is contrasted to sensitivity obtained from the DC components (green) via
the side of fringe interrogation. At a frequency indicated by the dashed magentaline, AC and DC components should yield equivalent sensitivity to cavity noise. The
cavity transmission function is overlaid in red. . . . . . . . . . . . . . . . . . . . . . . 167
5.30 Left: Cavity noise measurements from the AC (modulation at 5 MHz) and DC
signals. The DC signal is slightly smaller in amplitude due a smaller sensitivity
to cavity noise at this particular detuning (at the cavity pole). Right: Detunings
for the AC and DC measurements are now swapped, so that each measurement isinsensitive to the amplitude noise converted by the cavity. . . . . . . . . . . . . . . . 168
5.31 Top: Transimpedance gain of detector for measuring birefringent noise measured by
Terry Brown. X-axis should be in kHz. Bottom: Current noise of the detector. We
use the typical Hamamatsu G12180-003 InGaAs PIN with 1.1 A/W responsivity at
1550 nm. At 770 kHz, noise is equivalent to shot noise of 220 nW. . . . . . . . . . . 170
5.32 Laser-cavity error signals of demodulating the two tone electric field at the birefringent splitting 2∆. Different curves indicated a different extraction phase ϕ. All
curves are shifted horizontally for clarity though there is no vertical offset added.
Left: Error signals when Ẽ1 is transmitted. Right: Error signals when Ẽ2 is transmitted. Since components from the cavity modes oscillate out of phase with eachother, the large residual amplitude modulation is removed. However, the signal retains it symmetry about zero and is three times smaller. Detuning the laser by 2∆
to measure laser-cavity noise of a single mode no longer has any sensitivity when
the polarizer is orientated to transmit Ẽ2 . In both cases, sensitivity to laser noise
can be obtained by detuning the probe to the side of both lines, similar to the sideof-line probing demonstrated in the previous section. Alternatively, locking to bothmodes simultaneously and reducing the splitting of polarization tones to 2∆ − Γcwill naturally shift the absorptive signal by Γc /2 where there is sensitivity to detuning.173

xxxiv
5.33 Left column: Sensitivity to noise on the mode splitting when Ẽ1 is transmitted.
Right column: Sensitivity to noise on the mode splitting when Ẽ2 is transmitted.
Top row: Laser is tuned such that both modes are addressed simultaneously. Bottom row: Laser is detuned by 2∆ so that only one mode is addressed. We regain
sensitivity to the birefringent noise with the laser resonant with a single mode, but
this sensitivity depends strongly on whether Ẽ1 or Ẽ2 is transmitted. . . . . . . . . . 175
5.34 Si3-Si6 beatnote with krypton gas flowed into optics enclosure at different times.
The blue segment highlights the steady state without any deliberate perturbation
to the optical path length. A short burst of gas is released at the beginning of theorange segment. The flow is continuous for the duration of the green segment andthen shut off at the end. A clear chirp in the cavity frequency is visible as the gas
dissipates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
5.35 Top: Frequency response of the cavity to a step of the vacuum chamber temperature.
Bottom: Residuals of an exponential fit to the frequency response above display
a modulation indicative of an etalon in the system. The lack of a significant time
constant between the temperature step and etalon response suggests that one of
the etalon surfaces is not thermally isolated from the 300 K stage. Modeling thevacuum chamber temperature as a phase ramp, we can largely reproduce the cavity
frequency response by fitting the coupling of temperature to phase. In the orange
trace, we correspond a 100 mK vacuum chamber fluctuation to a 2π phase shift.
Active suppression of the 2 Hz peak-to-peak etalon amplitude below the 10−16 level
would require ∼mK stabilization of the window temperature. This is difficult, aneasier approach is to attempt to reduce the etalon finesse. . . . . . . . . . . . . . . . 178
5.36 Sweeping the laser frequency, several transverse modes visible in the scan as well
as a modulation on the baseline indicating the presence of an etalon. A fit to thisreveals the etalon’s free spectral range to be 160 MHz which corresponds to a cavity
length of 0.9 m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

xxxv
5.37 Top: Dedrifted Si3-Si6 beatnote overlaid with the PDHDC signal multiplied by thehorizontal sensitivity coupling. Bottom: Instability of the beat, and PDHDC measurements when propagated through the horizontal and vertical transfer functions.
Some sections of the beatnote instability between 100-1000 s appear to be well described by the horizontal pointing deviations. For longer averaging times, the cavity
instability might result from a linear combination of vertical and horizontal pointing
fluctuations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
5.38 Modulating the voltage applied to the piezo allows us to measure a transfer function
between the cavity frequency response and changes in the PDH DC signal. We
do this for both axes to measure the system’s sensitivity to beam pointing. Left:
The piezo is place in the mirror’s horizontal pin. We drive it with a sawtooth andsimultaenously record the frequency deviation. Red trace is the frequency response,
and the green trace is the PDH DC voltage converted to frequency via our measured
coupling. Right: Same data, except the piezo is now driving the mirror’s vertical pin.182
5.39 Left: The light is coupled to the cavity from the bottom setup and the transmitted
beam is imaged from the top. Right: Light is coupled from the top and imaged on
the bottom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
5.40 Fig.4.12 is reproduced but now we include a measurement of RAM from the top
setup. The modified allan deviation of RAM instability is now consistently better
than the Si3 thermal noise from 0.5-10 s, and the fractional frequency noise shows aconsiderable 6 dB improvement around 100 mHz. No effort was put into improving
the long term instability of RAM in the top setup, so we expect that this can be
improved. The top setup couples to the HG00 mode while the bottom setup couples
to HG01 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

xxxvi5.41 A five day measurement of the Si3-Si6 beatnote with the Si3 drift removed. During
this period the Si3 drift was measured by the SrI lattice clock so its sign and magnitude is know. The dotted line is a fit to the entire dataset and finds a fractional
drift rate of −2.3 × 10−20 /s, whereas the dashed line fits to the latter two thirds
where a more linear behaviour is observed. This segment finds a fractional drift rateof −5.6 × 10−20 /s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
6.1

We simulate a continuous dataset of different types of noise, Sy ∼ f 0 , f −1 , f −2 .
The performance of the periodogram and Lomb-Scargle periodogram estimators iscompared. The Lomb-Scargle periodogram consistently underestimates the noise
spectrum by a factor of 2 in amplitude. A correction therefore needs to be included
in order to obtain a comparable result. . . . . . . . . . . . . . . . . . . . . . . . . . . 193

6.2

Exclusion limits for de measured between the three optical clocks during the BACON
frequency intercomparison data campaign. The dataset is the same as that from [11].
The most stringent bounds on de are set by the Yb/Sr frequency ratio despite therelatively higher sensitivity of the Al+ /Yb comparison. . . . . . . . . . . . . . . . . . 195

6.3

Left: Comparison of the data record of the H-maser measured against the siliconcavity frequency, and the 1D Sr lattice clock measured against the silicon cavity. The
time series of the maser-cavity frequency ratio is decimated by a factor of 720 forclarity. The short term instability of the atom-cavity measurement is far superior tothat of the maser-cavity ratio, but the enhanced sensitivity of the maser to the scalar
DM oscillation will enable it to set a competitive bound. Right: Frequency noise of
the fH /fcav (blue) and the fSr /fcav (red) ratios calculated with the Lomb-Scargle
method. A noise model for each ratio derived from the frequency noise contributionsof the cavity, maser, and 1D Sr lattice is shown in black. . . . . . . . . . . . . . . . . 197

6.4

A cartoon illustrating the dependence of the H maser, silicon cavity, and strontium
clock transition on α and me . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

xxxvii
6.5

The ratio of the observed DM signal compared to the actual signal amplitude when
considering a nonzero phase δ to the DM field. . . . . . . . . . . . . . . . . . . . . . 198

6.6

The constraints placed in this work are the blue curve for the maser-cavity frequency
measurement, and the red curve denoting the atom-cavity measurement. Purple
lines indicated limits set from equivalence principle tests [12, 13]. Black curves are
exclusion bounds determined from atom-atom frequency comparisons [14, 15, 16].

6.7

. 199

Exclusion limit for the electron mass coupling dme obtained from the maser-cavity
frequency comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

6.8Long term instability of the silicon cavity (orange and red points) with our modelfor the oscillator noise (dashed black line). In a single day’s measurement, particular
Si3 datasets have displayed thermal noise limited performance out to 104 seconds.
This low instability is naturally degraded with the inclusion of significantly moredata. A similar effect is apparent in the Si6 long term behaviour, short datasets of
6-8 hours demonstrate instabilities of mod σy < 1 × 10−16 for 103 s, though a weeklong dataset shows mod σy = 1 × 10−16 at 200 s. Our intent is to point out that
length of the dataset is critical when assessing the long term cavity behaviour. . . . 202

6.9

Figure adapted from [17]. Stability of the optical and microwave timescales is evaluated when measuring against an optical Sr standard for two duty cycles. Measurement of the OLO for 1 hour per day is still superior to the microwave timescale whenmeasured against the strontium clock 12 hours/day. This is directly related to thesuperior short term thermal noise limited performance of Si3 and the excellent longterm control. We additionally display the significant advantage enabled by measuring the microwave timescale against an optical reference compared to a microwave
reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

6.10 Top: Bottom: Curves from the top plot are converted to a strain sensitivity bynormalization with the cavity’s mechanical response to a gravitational wave. The
aLIGO strain sensitivity is displayed in red. . . . . . . . . . . . . . . . . . . . . . . . 207

xxxviii
7.1

We plot the total Brownian thermal noise limit for MJM (purple), Si2/Si3 (green),
the limit for a 21 cm silicon cavity at 16 K with dielectric coatings using the measured loss angle from [18], and the limit for a 21 cm silicon cavity at 16 K with
AlGaAs/GaAs coatings using the measured loss angles from [19, 20]. A dashed blackline indicates the instability where an oscillator has experienced a phase excursionof ϕrms = 1 rad. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

A.1 The laser has a white phase noise spectrum where Sy (f ) ∼ f 2 . Top row shows thelaser Sy (f ) and the corresponding modified Allan variance. The bottom row shows
the Dick effect limited 1 s clock stability that can expected for different dead times
and pulse times. Different behaviour will be visible for Rabi spectroscopy (lowerleft) and Ramsey spectroscopy (lower right). . . . . . . . . . . . . . . . . . . . . . 229
A.2 The laser has a white frequency noise spectrum where Sy (f ) ∼ f 0 . . . . . . . . . . . 230
A.3 The laser has a flicker frequency noise spectrum where Sy (f ) ∼ f −1 . . . . . . . . . . 231
A.4 The laser has a random walk frequency noise spectrum where Sy (f ) ∼ f −2 . . . . . . 232
B.1 Amplitude of different sidebands generated via phase modulation on an EOM. . . . 234

Chapter 1

Introduction

For the majority of us, our introduction to physics involves identifying the forces exerted on
an object and collecting these terms into an equation that captures the evolution of that object’s
state. In a closed system where the energy is conserved and phase space is a natural description,
this amounts to identifying the Hamiltonian and allows us to calculate useful observables. For
many classes of problems in classical mechanics, we can exactly calculate all the desired quantities;
a canonical example is an oscillating pendulum where the Hamiltonian consists solely of the mass’
kinetic energy and its potential energy in a gravitational field. The efficacy of this description
depends on how accurately the model captures the physical dynamics. We can often benchmark
our calculations against a priori knowledge of the exact ground state, or the exact equations of
motion.
In the quantum realm, this can be more difficult. We might want understand a particular
process and generate a Hamiltonian as an ansatz for the system under study. If the system is
especially complicated, we might not have exact knowledge of the ground state. Indeed in quantum systems with large interactions (quantum Hall physics, superconductivity, magnetism), thecomplexity is such that these problems may be intractable to solve numerically. An alternative approach has been to use quantum systems as ’simulators’, where a precisely controllable system canbe used to engineer a relevant Hamiltonian and compute the associated state variables [21, 22, 23].
The first production of degenerate gases of neutral atoms heralded the ability to control a system’sinteractions and energy scales to the point where its quantum state can be subjected to carefully

2chosen Hamiltonians [24, 25]. We’ve now developed the capability to reconstruct intractable groundstates by preparing quantum resources and subjecting them this Hamiltonian [26, 27]. Quantum
simulation with degenerate gases has a well established history of lending access to global observables, and the recent advent of quantum microscopes [28, 29] has now also enabled us to extract
local observables.
Despite the power of this method, it can be argued that this is not the most natural way to
extract a system’s ground state. In introductions to quantum mechanics, we usually learn to solve
quintessential problems (e.g. the quantum harmonic oscillator) by diagonalizing the Hamiltonian,providing us with direct access to the energy eigenstates. This may seem to be a preposterous
suggestion for studying real physical systems - how does one actually diagonalize the shifted energy
spectrum of a many body system when an atom interacts with light?
This concept is in fact a central tenet of metrology, where spectroscopy is used as a tool tomeasure these energy shifts in the frequency domain [30]. Clever spectroscopic methods have beendeveloped to enable us to resolve finer and finer interactions, i.e. probe smaller and smaller energy
scales. Of course, as with all measurements, a quantity can only be resolved if the measurement
device does not limit the measurement resolution. In principle, we would like an infinitely smaller
ruler - a delta function of sorts - to be able to study arbitrary systems, investigate interesting
Hamiltonians, and probe new phenomena at the smallest length scales possible.
This is perhaps a peculiar, inverted approach to atomic physics. but frequency metrology
with atoms has enabled us to make extremely high resolution measurements of frequency shifts
and Hamiltonian ground states [31, 32, 33]. Particular atoms support so called ’clock transitions’,
where an excited state lifetime is nearly metastable and optical transitions to the excited state cantherefore support some of the highest natural quality factors available to us [34, 35, 36]. The sametransition makes these atoms suitable as clocks where an atom can be placed in a superpositionof ground and excited states of the clock transition, precess at an optical frequency, and precisely
measure an optical phase shift corresponding to some external potential. This is an interesting feature of these systems - the longer we allow an atom to precess in that superposition and accumulate

3phase, the finer our ability to resolve a smaller phase shift. In effect, our measurement device (inthis case a shift of the clock frequency) becomes more precise the longer we measure for.
It appears that we should simply measure forever, to achieve our goal of letting our measurement precision approach that of a delta function in the frequency domain. In reality, this istechnically demanding and requires exquisite control of our atomic system. An alternative approach is to use many identical copies of the system so that the measurement precision is naturally
improved by incoherent averaging. This reveals the power of a many-body approach where a coherent preparation of these identical copies allows us to leverage improved statistics associated withcoherent averaging of a measurement. Optical clocks are well suited to these problems because thecoherence between atoms exists for a long period of time, tens of seconds in particular demonstrations [37, 38]. Our ability to coherently interrogate these atoms for the same duration however
is a formidable problem, and remains one of the primary hurdles in optical frequency metrology.
Optical transitions must be driven with an optical local oscillator (OLO), in this case a laser. Any
OLO of the correct color can interrogate the atoms and using them as a frequency discriminant
forms the basis of the optical clock. Stability, the capability of the clock to quickly measure small
frequency shifts, is directly tied to the coherence time of the oscillator. The impressive coherence
times afforded in atoms with optical clock transitions can only be utilized if the OLO introduces
a minimal amount of phase noise during clock spectroscopy. This thesis will detail our efforts topush the frontiers of atom-light coherence by developing a next generation oscillator stabilized toa cryogenic, crystalline Fabry-Perot resonator.

1.1

Thesis outline
The first half of my graduate career focused on the atom side of the clock. With the develop-

ment of state-of-the-art silicon cavities [39, 40, 41, 42, 43], it suddenly became feasible to average tothe systematic uncertainty our Strontium lattice clocks in a matter of minutes [44]. Instead of thecavity being the bottleneck for achieving groundbreaking atomic precision, the quantum resource
was now the primary limitation. There was thus a strong motivation to dedicate effort to improving

4
the atom-atom coherence. Identifying limitations to this parameter typically require an evaluationof all systematics that couple to the atomic frequency. For metrology, it is useful to evaluate theentire systematic error budget, quoting an accuracy and uncertainty for each mechanism.
This resulted in a new SrI accuracy evaluation with a fractional uncertainty of 2 × 10−18 [45].
Blackbody radiation and AC stark systematics were addressed in a novel way that had not been
approached in previous Strontium lattice clocks. Active temperature stabilization of the entire
vacuum chamber resulted in a residual gradient of less than 100 mK, that was stable to better than
1 mK over a 10,000 second measurement. The effects of this extended far beyond precise control of
the thermal environment - loading into the optical lattice and consequent mode occupations andsample temperature were well controlled and highly reproducible. The effective optical potential
sampled by the atom therefore allows for an AC stark shift that remains consistent over the courseof weeks. Density shifts coefficients for example no longer varied over the course of a day, andexhibit more consistency on a day to day basis. On a day timescale, we calculate point-by-point
corrections to the atomic frequency to the major systematics that may fluctuate between eachinterrogation, or longer. Over a 6 hour dataset, the total corrections that need to be applied are
displayed in Fig.1.1. The clock is precise to the 10−19 level for the entire fractional Allan deviation
record of 104 seconds. The remarkable point to note is the atomic ensemble and environment are
passively controlled to this level over the course of the measurement, meaning that a comparison
with a more stable frequency reference would presumably allow SrI to average to the 10−19 without
providing any corrections to the strontium atomic frequency.
These are all in-loop measurements of a sense. We measure a coupling to a systematiceffect, and use a detection channel for the corresponding environmental or ensemble property to
calculate the corresponding frequency shift. This is an important component of assembling theclock’s uncertainty budget, but a true evaluation of the level of frequency control can only occur
by measuring against another frequency reference. In this case, we measured against a a 3D opticallattice clock, SrII.
The SrI-SrII frequency comparison was performed asynchronously, meaning that the mea-

Net Systematic
Shifts / Hz

10 18

2.246
2.247

Quad Zeeman
Density
BBR
Total

10 19

2.248
2.249 0Fractional Systematic
Shifts

5

2.245

1

2

3

4

Time / Hr

5

6

10 20

101

102

Time / s

103

104

Figure 1.1: Left: Plot of the time record of all the SrI systematic shifts. Changes in atom number,ambient temperature, or magnetic field all result in corrections to the clock frequency, and their
total magnitude is shown over a six hour data campaign. The clock achieves over 98% uptime overthe course of this single comparison day and slight gaps in the data indicate brief periods where
the laser is not locked to the atoms. Right: The same data is plotted as a fractional instability
normalized to the Sr clock frequency. The individual contributions of density shift (blue), BBR(red) and second order Zeeman shift (yellow) are shown as the dashed curves. For operation times
up to 104 s, fluctuations in systematic offsets are bounded below 4 × 10−19 .

Density Shift per 1000
atoms × 10 17

0.0
0.5
1.0
1.5
2.0

2.5
-1101-1401-1501-2101-2201-2801-3102-0402-0502-0702-08
1
0
19- 19- 19- 19- 19- 19- 19- 19- 19- 19- 19Date
Figure 1.2: Density shift evaluations. Density shift measurements were performed over four weeks
at the same trap conditions. The weighted mean of all measurements is shown with a dashed, blackline, with a reduced chi-squared of 1.07

surement stability was sensitive to the noise of the local oscillator. The level of precision achieved
is characteristic of the excellent systematic control of both clocks but the time taken to average tothis value is a feature of the local oscillator’s low instability. Similarly, an accuracy evaluation can

SrI - SrII Fractional
Frequency Instability

6

10 16
10 17
10 18
10 19

101

102

Time / s

103

104

Figure 1.3: Non-synchronous comparison (i.e. interrogation times of the local oscillator do not overlap such that there is sensitivity to laser noise) with the JILA 3D optical lattice clock demonstrates
that the beat between the two clocks averages below the quoted total systematic uncertainty. The
black line is a white noise τ −1/2 fit to the single clock instability.

only be performed by measuring against another accurate reference. After converting the old SrImachine into an accurate and precise lattice clock, we participated in a three way frequency ratio
intercomparison with the Al+ ion clock and Yb lattice clock at NIST.
These results are more comprehensively detailed in [46] so we won’t delve further into them
here. Instead, we’ll turn our eye to the technology that has enabled all of these measurements ofrecord breaking precision, the local oscillator. With the impressive progress in improving atomatom coherence [38, 37], the development of better flywheels have naturally received less attention.
By themselves lasers stabilized to cryogenic monocrystalline silicon cavities have realized linewidthsof a few mHz at 1.5µm [39, 40, 42, 43]. However, particular research directions are only possible withthe development of better continuous optical phase references [17, 47]. These studies have enshrined
high stability optical oscillators as key components in pushing frontiers of optical metrology.
These cavities also occupy a unique space in frequency metrology. Nearly all areas of researchleverage the quantum properties of atoms for sensing and measurement applications, but here wehave a classical resource that so far has no competitive quantum counterpart. The development

7of quantum based optical oscillators has been an intensive area of study [48, 49, 50, 51, 52] but
optical interferometry has still remained a superior method for low noise length and frequency
measurement. It is in fact remarkable that the leading results in low noise length resolution areachieved by stabilization to a macroscopic, classical crystal. A significant advantage of working
with the classical resource is that error correction between the the oscillator (free running laser)
and the spectroscopic feature (the cavity) is performed at high bandwidth. More importantly, thespectroscopic feature is always present, and can be interrogated continuously without any deadtime
associated with preparing the resource. In concept, the cavity can provide a limitless signal to noise
ratio if available optical power is not a concern.
The field has progressed to the point where the achievable frequency stability is limited byfundamental fluctuations of the cavity materials due to Brownian thermal noise. Crystalline cavities
have refined this to a limitation that originates nearly entirely from the coating Brownian thermal
noise. In our bandwidth of interest, frequency noise of our 10 cm-scale resonator is fundamentally
limited by length fluctuations of the thin 10 µm Bragg stack. A similar issue plagues the LIGO
gravitation wave detector, where detection at 100 Hz is largely limited by coating thermal noise[53]. In this case the contrast is even more stark where the 10µm coating places a constraint on thestrain noise budget of an interfermometer with 4 km long arms. Next generation detectors haveproposed to circumvent this limitation by operating at cryogenic temperatures [54, 55, 56], but thisintroduces more complications than it solves.
On the opposite length scale, experiments with microresonators [57, 58, 59] interrogate displacement noise at the higher bandwidth of 103 − 105 Hz, yet are still hampered by thermal noiseof the mechanical oscillator. Coating thermal noise is therefore a ubiquitous mechanism of concern
within many fields employing optical interferometry techniques. Inventive solutions to reduce its
contributions have been investigated, including the interrogation of higher order spatial modes [60],or the design of resonators that support flat topped mesa modes [61].
These proposals are still difficult to implement and so far the primary research focus has been
the development of optical coatings with low mechanical loss. Crystalline materials in the bulk have8
demonstrated quality factors significantly higher than commonly used amorphous substitutes. The
switch from ultra low expansion glass and fused silica spacers and substrates to ones made of siliconheralded significant advances in low noise interferometry [62]. A similar investigation with opticalcoatings has lead to the development of Al1−x Gax As/GaAs crystalline coatings [19]. Measured
room temperature loss angles in these thin films are ten times lower than those measured withamorphous SiO2 /Ta2 O5 dielectric stacks [19, 43, 63]. Fully crystalline cavities (spacer, substrates,
mirrors) have the potential to realize even lower laser linewidths and enable novel demonstrations
and applications of enhanced atom-light coherence.
The scope of this thesis investigates novel light-induced noise mechanisms associated with
one of the first realizations of a fully crystalline Fabry-Perot resonator. We operate a 6 cm siliconcavity with crystalline Al0.92 Ga0.08 As/GaAs coatings (henceforth referred to as Si6) in a closed cycle
cryostat that enables operation down to base temperatures of 4 K. Reducing our technical noise
contributions and using the measured room temperature mechanical loss angle of these coatings,
we expected a fractional Brownian thermal noise limited performance of 1.3 × 10−17 for Si6. We
instead observed unexpected dynamic and static light-induced frequency shifts that limited thecavity stability to ten times this value. We determined these effects to be unique features of thecrystalline coatings and in comparison with a similar AlGaAs coating silicon cavity operating at
124 K, we identified physical parameters that set the magnitude of this noise, and corresponding
ways to mitigate it. A unique interrogation scheme is implemented to suppress these effects, andwe study the residual noise present after cancellation.
The structure of this work is as follows• In Chapter 2, we’ll present the concepts necessary to understand thermal noise and to
consider when designing a Fabry-Perot cavity. We will lay out the formalism used whentransferring the oscillator stability to an atomic reference.
• In Chapter 3, we revisit the decision to use silicon as a cavity material and the advantagesof crystalline coatings over dielectric coatings. We will provide a brief overview of the9cryogenic environment and the methods we use to measure the Si6 frequency noise.
• Chapter 4 will cover the various noise sources that can reduce the stability of the laser locked
to the resonator. We will detail our approach to mitigating each of these contributions.
• Chapter 5 concerns the actual frequency noise measurements of Si6. We will present ourobservations of static and dynamic light-induced effects identified in the crystalline semiconductor AlGaAs/GaAs coatings• In Chapter 6, we will present novel uses of ultrastable resonators and detail the experiments that were only possible with the excellent long term stability afforded by cryogenic
crystalline resonators.

Chapter 2

Oscillators and stabilization

The laser is an integral part of the optical lattice clock and the silicon cavity in particular
has enabled nearly all of the results previously discussed. Here we will present an introduction
into cavity design, the fundamental noise limit of a resonator, and how the laser noise spectrum
determines clock stability and rotation fidelity.

2.1

Fabry-Perots
The cavity is a linear system, much simpler conceptually than many of the experiments

undertaken in JILA. The formalisms of incident, reflected, transmitted fields, longitudinal andtransverse modes, mode couplings, etc. are all well understood, and the true challenges lie with thetask of precision interferometry. There are many detailed resources on cavities [64, 65] that derive
the wave equation solutions for the electric fields supported by the resonator. Here we will only
present some of the main results and key features that need to be considered when designing andcharacterizing the cavity.
The response of the Fabry Perot cavity is entirely determined by mirror parameters andthe physical distance between mirrors. Any resonator of length L in vacuum will support a modeof frequency ν = c/2L. However, a continuum of longitudinal modes is in fact supported withfrequencies νi = nc/2L. These will be evenly spaced by the free spectral range (FSR) of theresonator
FSR =

c
2L

(2.1)

11
The width of these resonances however is related to both the mirror parameters and thecavity length. The linewidth of the cavity is given as
∆ν =

FSR
F

(2.2)

The cavity finesse F is a measure of the mirror losses. For a single mirror, the electric fieldreflectivity, transmissivity, and loss are ri , ti , li respectively. These parameters share the relation
r2 + t2 + l2 = 1(2.3)

The finesse depends entirely on the reflectivity and is√
π r1 r2F=
1 − r1 r2(2.4)

The linewidth of the cavity determines the storage time of a photon as τs = (2π∆ν)−1 . If the cavity
length is known, the finesse can be measured via cavity ringdown where the exponential decay of
optical power leaving the cavity yields the characteristic storage time τs .
The mirror transmissivities are needed for calculating the circulating and transmitted opticalpower through the cavity. The corresponding electric fields are found to be [64]t1 t2 e−iϕ/2Einc
1 − r1 r2 e−iϕt1 e−iϕ/2
Ecirc =
Einc
1 − r1 r2 e−iϕr1 − r2 (r12 + t21 )e−iϕEref l =Einc
1 − r1 r2 e−iϕEtrans =

(2.5)
(2.6)
(2.7)

Where the round trip phase shift ϕ = kL. A special case of interest is the symmetric cavity wherer1 = r2 and t1 = t2 . When the phase shift is a multiple of 2π, the reflected power drops to zeroand nearly all the incident power is transmitted (limited by the mirror losses l1 , l2 ). In this case,the transmitted field isEtrans =

t1 t2
Einc
t1 t2 + l1 l2(2.8)

When the transmissivity is on the same order as the losses, less than half the incident field is
outcoupled from the cavity. As we will see, this is the case for the crystalline coatings in Si6 where12ti ∼ li . The ratio of circulating power to incident power |Ecirc |2 /|Einc |2 is also referred to as thecavity gain, and describes the build-up of power coupled into the resonator.
This representation, while useful for describing the ratios of fields within and outside theresonator, does not provide a realistic description of the longitudinally varying electric field. The
ansatz for these equations assumed outcoupled and intracavity fields of a traveling plane wave. Fromexperience we know that there is a finite radial extent to these waves so a better approximation
is to retain the fast eiωt oscillations longitudinally while treating the transverse extent as a slowly
varying field.
It is well known that that beams with Gaussian intensity profile are solutions of the paraxial
Helmholtz equation for a wave propagating within a resonator. The general solution for the intensity
profile with azimuthal symmetry in the transverse direction r is [65]I(r, z) =

h
2P
2r2 i
exp
−
πw2 (z)w2 (z)

We can see that the beam size w(z) is dependent on the longitudinal position, given byr z 2w(z) = w0 1 +
zr

(2.9)

(2.10)For Rayleigh range
zr =

πw02
λ

(2.11)

The waist w0 is the primary feature characterizing the beam size and is determined entirely by thecavity length and mirror radius of curvature.
r 
1/4
g1 g2 (1 − g1 g2 )λL
w0 =π (g1 + g2 − 2g1 g2 )2(2.12)

Factor’s gi are calculated for a single mirror as
gi = 1 −dRi(2.13)

Stability of a resonator is determined by the criterion 0 ≤ g1 g2 ≤ 1. For a cavity outside thisparameter range, most light coupled into the resonator will eventually be reflected away from thecavity axis.

13
The wavefront of the Gaussian beam experiences a longitudinally varying phase shift whencompared to the wavefront of a plane wave. This is the Gouy phase shift and is calculated as
ϕ(z) = − arctanhzi(2.14)

zr

The Gaussian beam is not the only solution to the paraxial wave equation. This relies on
the assumption of cylindrical symmetry of the resonator mode and when this is broken, a basis ofsolutions is provided by the orthonormal set of Hermite-Gauss (HG) modes. The spatial profile ofany beam can be decomposed as a linear combination of these modes. The spread in transverse
momentum is larger for these higher order and correspondingly they acquire a larger Gouy phaseshift ofϕmn (z) = −(m + n + 1) arctanhzi
zr(2.15)

Where m, n characterize the order of the mode along the two transverse directions. We can seethat higher order modes acquire a larger phase shift. This manifests as a frequency shift for thesehigher order modes, with mode spacing of∆νmn = FSR(m + n + 1)

h ϕ(z ) − ϕ(z ) i
1

2

2π

(2.16)

Where z1 , z2 are the locations of the two cavity mirrors.
The Si6 cavity has a spacer length of 6 cm and mirrors with 1 m radius of curvature, corresponding to a FSR of 2.5 GHz. The transverse mode spacing is roughly 250 MHz, well outside thecavity linewidth and offset frequency of the modulation sidebands. This is an important consideration since low-order modes near the fundamental HG00 may have finite coupling. If they are close
to the HG00 mode in the frequency domain, the resonant feature being probed might experienceline pulling effects as coupling to these other modes fluctuates with slow drifts in beam pointing.
High coupling to a particular mode is obtained by shaping the beam size, pointing, andlocation to overlap with the corresponding mode supported by the cavity. For the HG00 mode, it
is desirable for the incident beam to have a size corresponding to the size of the cavity mode and14
the input mirror, w(z1 ). This maximizes the overlap in the longitudinal direction. The general
expression for the coupling to a cavity mode with electric field Emn isR∞

2−∞ |Ei Emn | dx dy
R∞22−∞ |Ei | dx dy −∞ |Emn | dx dyη = R∞(2.17)

With incident field Ei . In general, it is difficult to calculate this quantity ab initio. Instead,
it can be easily measured by locking to each higher order mode and extracting the coupling. A
demonstration of this is illustrated in Fig.2.1.

1.0
Coupling efficiency

Transverse mode
frequency (MHz)

2000
1500
1000
500
0

0

2

4

6

8

10

Mode number

12

14

16

0.8
0.6
0.4
0.2
0.0

00

01

02
20
HG mode03

30

Figure 2.1: Left: Calculation of transverse modes for Si6. The x-axis refers to a total mode number(m + n). The closest degenerate mode to HG00 is one where m + n = 10, and this is still 6.4 MHz
away. Right: We measure the cavity couplings to some of the low order modes with m + n ≤ 3.
The highest measured coupling is to HG00 with η = 0.5. The sum of couplings for displayed modes
is still less than 1, indicating that there is a finite coupling to other higher modes. We were unable
to measure these couplings, indicating that the residual η ∼ 0.4 is likely distributed over very many
modes.

2.2

Thermal noise in cavities
We’re well aware that systems in equilibrium can be well described by their statistical aver-ages, and in the strontium experiments we make use of the fact that there are deviations aroundthese averages (e.g. a distribution in the states of an optically trapped atom). Fluctuations ofequilibrium systems are important considerations in the quantum world, but perhaps less noted in
the macroscopic classical world. Nevertheless, we are deeply interested in this.

1.0

10 1

0.8

Coupling efficiency

Coupling efficiency

100

10 2
10 3
10 4
10 5
0.0

0.5

1.0

1.5
r [w0]

2.0

2.5

3.0

15

0.6
0.4
0.2
0.0 1
10

100
win [w0]

101

Figure 2.2: Left: We calculate the coupling efficiency for an incident Gaussian beam to the HG00
mode of a cavity. Both beams have the same waist w0 but the incident beam is displaced fromthe cavity mode in the transverse direction. When the radial displacement δr is equivalent to thecavity waist, the coupling has already dropped to 0.3. Right: We now assume that the incident
field is perfectly overlapped with the cavity mode in the transverse direction, but the waist of theincident mode win differs from waist of the HG00 mode. We see that the coupling is much lesssensitive to variations in win .

A central tenet of statistical mechanics discovered in 1905 by Einstein was the existence offinite RMS motion of particles in a thermal bath. This was refined in the 1950s as the notion thatany coupling between a system and a bath gives rise to a back-action on an observable of the system[66, 67]. Stated another way, there is a fundamental relationship between the behaviour of moleculesin thermal equilibrium with the environment and fluctuations of a macroscopic observable. A
familiar example is the Johnson noise we measure of a resistor in thermal equilibrium with its
environment - dissipation through the resistive material gives rise to a proportional level of current
fluctuations measured across the resistor. Capacitors however don’t contribute to the current noise,so a general formulation for Johnson noise equates the amplitude of the fluctuations to the realpart of the circuit impedance Z.
The general concept associating these two quantities is the fluctuation dissipation theorem
which relates the fluctuations of an energy spectrum to the imaginary component of the system’s
response function.

16

Sx =

4kB T
Re[1/Z](2πf )2(2.18)The left hand side represents the power spectrum of length noise fluctuations which arefamiliar to us, the right side contains the mechanical impedance Z of the system under study.
This will end up setting a fundamental limit for the length resolution that we can resolve in any
experiment using optical interferometry. A demonstration of this relation is provided next.

2.2.1

Normal mode method

Different types of damping lead to different calculations of the impedance, so intuition forthe physical mechanism driving the frication between particles was important. The historicaldistinction was between external velocity damping (viscous drag exerted on a particle in fluid-likeenvironment) vs internal damping (found to obey Hooke’s law). The latter has been accepted as theappropriate formalism describing thermal noise in optical resonators. One of the first applications
to optical interferometry was presented in [68]In these calculations, a particular form of Hooke’s law is used
Fth = −k[1 + iϕ(ω)]x(2.19)The imaginary component iϕ(ω) indicates a phase lag between the driving force and material
response. The existence of a nonzero ϕ implies that the material is not perfectly elastic given the
delay between an applied stress and the resulting strain. Structural damping refers to a frequency
independent ϕ which is what we’ll consider for now. The loss angle ϕ is used interchangeably withthe mechanical quality factor Q = 1/ϕ. Q denotes the ratio of energy stored in the oscillator toenergy lost in a single cycle and the origin of this quantity is nebulous. Not only is it material
dependent, but we will see later that it can vary significantly on the the method of synthesizing amaterial.
Analogous to the γ of the damped driven pendulum, we can see that ϕ quantifies the energy
loss of the system when driven by an external force. As we derive the transfer function of the17
response, we will find that this is also equivalent to the damping ratio γ that quantifies the widthof the system resonance.
Solving the system’s equation of motionmẍ + k(1 + iϕ) = F(2.20)

With a generic normalized solution to the systems motion x(ω) = eiωt , the transfer function
χ(ω) of the oscillator is solved asχ(ω) =ω02 (1 + iϕ)≈1(ω02 − ω 2 ) + iϕω02

In the limit where ω is much less than the resonance frequency(2.21)
pk/m. Note that the response

is flat in this region. The impedance is found to beF(k − mω 2 ) + ikϕ
=
ẋ
iω
1
kϕω + i(kω − mω 3 )=Z(k − mω 2 )2 + k 2 ϕ2Z=

(2.22)(2.23)

Substituting back into Eq.2.18, the fluctuations in displacement are determined as
Sx (ω) =4kB T ω02 ϕ
2
mω[(ω0 − ω 2 )2 + ω04 ϕ2 ](2.24)

in the low frequency limit where ω < ω0 we findSx (ω) =4kB T ϕωk

(2.25)

since the loss angle ϕ ≪ 1. The intuition here is the same, a stiffer spring constant reduces
the overall amplitude of the thermal noise. There is an important point not to be missed - thelow frequency limit of the oscillator’s transfer function is flat and does not depend at all on themechanical damping. The quantity ϕ only affects the resonance width. However, ϕ is enormously
important in determining the amplitude of displacement fluctuations. Unlike the oscillator response,the inherent noise in the oscillator position is dependent on the quality factor of its spring.
The spring constant is a convenient approximation - for a solid material in the elastic regime,
we can substitute k → EA/L, for a Young’s modulus E, cross sectional area A and length L so18

| (f)| [arb.]

101

100

10 1 2
10

Frequency [Hz]Sx(f) [arb.]

10 11

103

1/f guide
1/f5 guide
= 0.6
= 0.4
= 0.2
= 0.2
= 0.1

10 12
10 13
10 14
10 15 2
10

Frequency [Hz]103

Figure 2.3: We consider an oscillator with resonance at 1 kHz and varying mechanical loss angle
ϕ. Top: Transfer function Re[χ(ω)]. Bottom: Displacement noise spectrum Sx (ω). In both cases
a lower value of ϕ results in a sharper resonance but only in the displacement noise can we seean effect of the quality factor on the low frequency region. Mechanical resonances of the cavity
components are always designed to be above 100 s of Hz so ω ≪ ω0 is the regime of interest.

that
Sx (f ) =2kB T ϕL
πf EA

(2.26)

Remarkably this simple picture provides an excellent approximation of the thermal noise
calculated for the cavity spacer [7]:Sx,sp (f ) =

2kB T
L
ϕ
f π 2 (R2 − r2 ) E(2.27)

for a spacer with central bore of radius r and outer radius of R.
The complication arises when we need to consider a system comprised of several coupledsprings. The system’s response to an applied force is determined by the superposition of the19
responses of each of its normal modes. Including contributions from the higher order modes isoften necessary for convergence of the displacement noise.
This is not critical in our particular example of the cavity spacer, but the normal mode
approach is found to underestimate the total thermal noise contribution particularly when theforces are not homogeneously distributed over the material [69]. In our intuitive picture of thespacer behaving as a solid spring, we can convince ourselves that the internal frictional forces are
uniformly distributed across its volume to generate displacement fluctuations. In the case of thecavity mirrors, sections of the coating far from the beam location should not contribute significant
noise.
It is in these nuanced scenarios where the normal mode summation approach is exceedingly
difficult, and a solution to this problem was proposed by Yuri Levin in the oft-called ”direct
approach” [69].

2.2.2

Direct approach

The significance of Levin’s approach to the FDT calculation of thermal noise is well-statedby [70]:
Until recently, the only method for evaluating Z was based on an expansion of thevibrating state of the mirror on a basis of normal modes. However, computation of
the eigenfrequencies, determination of associated eigenmodes and of their couplingwith the main beam, is a very heavy task in terms of computational power.
Gillespie and Raab used a multi-order mode summation to calculate coating thermal noise [2].
In the low frequency limit, the total displacement noise of a surface is calculated via the individual
contributions of the n modes
Sx =

n
X
i

Sx,i =

n
X
4kB T
i

ω

ϕi
2
αi mω0,i(2.28)

Where the ω0,i are the resonances of each individual mode. The coupling to each mode i is parameterized by an effective mass ai (computed independently for each mode i) that weights the relative
amplitude contributions for the total displacement noise.

20

Figure 2.4: Reproduced from [2], the x’s denote the noise amplitude contribution from the first 100
normal modes of a 10 cm mirror with a 2.2 mm beam spot. The solid line illustrates the cumulative
noise of all these modes and the summation has still not converged over the 100 modes. Notably
the mirror diameter is crucial for this calculation whereas it is unnecessary in the direct approach.

The herculean approach necessary to provide an accurate approximation of thermal noisecannot be overstated. Each mode may have a different spring constant due varying resonances ω0 ,correspondingly affecting the amplitude of each noise spectrum Sx,i . Fig.2.4 illustrates the summation over the first 100 mechanical modes of the coating and convergence is still not evident. Eachmode is statistically independent, and requires an additional calculation to determine the mode
amplitude. Additional complications such as displacing the beam away from the mirror require
recomputation of all modes since the relative contributions of the various geometrical eigenmodes
will change. Ultimately one finds that Sx through the summation method is insensitive to theseeffects but that comes at the cost of a highly intensive calculation.
The direct approach is far simpler and uses Levin’s realization that mechanical impedance Z
can be derived by calculating the mirror’s deformation due to an incident pressure with profile of theintracavity optical mode. This pressure is generally considered as P (r) = F0 I(r) with amplitude

21
F0 and profile
I(r) =

 r2 
1
exp
− 2
πw2
w0

(2.29)

This pressure generates a strain U within the material and the displacement fluctuations canbe found as
Sx =

4kB T
Uϕ
πf

(2.30)

Notable distinctions exist between this form and Eq.2.18. The denominator in Eq.2.30 is
now proportional to f −1 rather than f −2 and an explicit dependence on the loss angle ϕ is now
apparent. A direct comparison can be made to Eq.2.25, where the spectrum only scales with f −1
due to a linear frequency dependence of Re[1/Z]. Using our prior knowledge of the FDT applied tothe cavity spacer, we know that we should expect Sx ∝ ϕ/f so we can infer that the elastic energy
stored in the material (U ) will not depend on either of these quantities. Unlike the summation
method, we don’t need to know the mirror size or beam location relative to the coating center.
Levin’s approach provides a much more intuitive picture where we only need to calculate theamplitude response to the applied pressure, not the damping ratio or frequency dependence of thestrain. The calculation is complicated and can be found in [70] but the general idea is to integrate
the strain density u as
Z1U=
udV = −
2
V

Z
P (r)uz (r)dS(2.31)S

Depending on the approximations made, one can find two different solutions and this will be
important as we make the distinction between thermal noise due to the optical coating and thermal
noise due to the cavity substrate. The optical pressure is incident on both surfaces with the sameprofile - all that differs is the consideration of whether or not the beam size is comparable to thedepth of the surface. A nuance to this calculation is that the correlation length of the coating
response must be considered; a force exerted towards the edge of the coating will only generate
phase fluctuations on the optical mode of the coating response is coherent over that distance.

22
Jumping ahead, what we will find [70, 71] is that the notable difference between the substrateand coating thermal noises is a relative factor d/w0 where d is the coating thickness.

2kB T 1 − σ 2 1
ϕf
π 3/2 w0 E2
1 − 2σcoat d
4kB T 1 − σsubϕ
Sx,coat (f ) =2fπ
1 − σcoat w02 E
Sx,sub (f ) =(2.32)(2.33)For a rigid cryogenic cavity, the total thermal noise is therefore the sum of all these terms
Sxtot (f ) = Sx,sp (f ) + 2Sx,sub (f ) + 2Sx,coat (f )(2.34)

Here the Poisson ratio is denoted by σ. When w0 is small compared to the thickness of thesubstrate, the surface is assumed to be an infinite half space and this assumption is used to calculate
the substrate’s displacement fluctuations. This approximation is correct up to a constant but aw
correction accounting for its finite thickness can be applied as [1 + O( D)] for substrate thickness D.

A potential point of confusion is the scaling of coating thermal noise with 1/w02 . This is not basedon the intuition that the applied pressure encompasses a mode area w02 , but rather that coating
noise is suppressed by d/w0 relative to the substrate noise via approximations in the calculation[71].
In real coatings, the mirror is composed of alternating stacks of high and low refractive
index materials. Often the loss angles of these two materials are different so the coating thermal
noise noise must be decomposed into separate contributions for each set of layers. Eq.2.33 would
imply that the coating thermal noise would be d/w0 times smaller than Sx,sub but we will see ina later section on the AlGaAs coatings that this is rarely the case. Contrary to what has been
insinuated so far, loss angle is not innate to a particular material and often varies by orders ofmagnitude depending on its size and the way it is made. The loss angle for coatings is often ordersof magnitude larger than the loss angle of a bulk material, so Sx,coat is generally a significant sourceof length fluctuations.

23
2.2.3

Temperature dependent damping

One point glossed over in Eqs.2.27,2.32,2.33 is the existence of a temperature dependent
loss angle. Reduction of thermal noise is not always as straightforward as reducing the operating
temperature. Particular materials have pronounced peaks in ϕ at low temperature, meaning that
reducing T can lead to a surprising increase in the thermal noise limit. For this reason, temperaturedependent loss angles of a material are often measured in ringdown experiments of mechanical cantilevers. The infrastructure and effort is significantly lower than operating a cavity and measuring
the thermal noise over a continuous spectrum from 4 K to 300 K.

Figure 2.5: Evidence of a temperature dependent loss peak in crystalline silicon from [3]. A feature
at 160 K is discovered to be associated with impurities in the silicon. An acid etch of the ingot
appears to remove the loss peak.

The sources of these temperature dependent losses are not fully understood. Several models
have been developed to explain particular features evident in some materials and the consensus isthat there is not a singular process that gives rise to these effects [72].
Thermally activated transitions have an associated activation energy, and as the ambient
temperature approach that energy, a loss peak can often be observed in the material’s loss angle that24
increases with lower ambient temperature [73]. Phonon-phonon scattering is another process thathas been identified to be a source of additional dissipation in low temperature crystalline sapphire[74]. Chemical impurities have also been discovered to be a source of temperature dependent peaks
in ϕ. The particular impurities can be very difficult to identify but different fabrication methodsof the material have been shown to remove some of these features [3].
The point of these details is to demonstrate the complexities of understanding material loss
angles. A significant amount of research has been devoted to this topic but there is no singular model
to capture temperature dependence of ϕ. Knowledge of a material’s ϕ at a particular temperature
can only be identified through measurement, and the value of ϕ itself depends on the material
choice. It is therefore not a generality that the room temperature value can be extended down tothe cryogenic regime.
Later, we will discuss the application of crystalline coatings to a cryogenic silicon cavity at 4
K (Si6). Room temperature measurements of the coating loss angle have been performed, but few
available results exist for the cryogenic regime. The true thermal noise limit for Si6 is therefore amystery, but we will improve technical noise sources under the assumption that it follows its room
temperature value.
Thermo-elastic noise is another fundamental source we have not discussed, where temperaturefluctuations drive gradients across the mirror surface leading to local deformations. This effect scales
as 1/T 2 and is negligible in cryogenic systems, so we will not cover it here. It is worth noting thatthe the thermo-elastic and thermo-refractive effects that comprise thermo-optic are coherent, andcan therefore be coherently cancelled with clever design of the coating Bragg stack [75]. This isfundamentally different from Brownian thermal noise which is incoherent.

2.3

Cavity locking
Transferring the cavity stability to optical signal that can be distributed and used is a nontriv-

ial task that was a significant barrier to optical interferometry in the late 20th century1 . Locking
1Technical details relating to cavity locking implemented on Si6 will be discussed in the next chapter

25
a laser to a reference requires a dispersive signal that allows a servo to identify the correct signof frequency corrections. Transmission through the cavity linewidth provides an absorptive signal
so locking at the transmission peak does not reveal information about the parity of the frequency
error. The side of the line however can be approximated as linear so the transmitted power when
detuned at the cavity half linewidth can be a suitable locking point. For a while this ”side of line”
locking was a ubiquitous method of frequency locking but it is highly sensitive to fluctuations in
the DC value of the transmitted power. White [4] noted that the derivative of the cavity reflection
function provides a suitable frequency discriminant for locking to the transmission peak, and enacted a scheme where the laser is dithered within the cavity resonance to scan out the derivativeof the cavity transmission. The cavity converts frequency modulation into amplitude modulation
which can be easily recovered through photodetection. Similar to side of line locking, the error
signal quickly drops to zero, making it difficult to correct for frequency excursions larger than ahalf linewidth. The bandwidth of the lock is also limited by the cavity due to the delay associatedwith the intracavity storage time.
Hänsch and Couillard developed a unique method of generating a dispersive error signal
by placing a linear polarizer within a cavity and encoding frequency fluctuations in polarization
ellipticity [6]. This method however is not applicable to rigid cavities where a solid spacer prevents
access between the mirrors. However, this was an important demonstration that an error signal
could be obtained interferometrically between a beam reflected from the cavity input mirror andthe leakage beam that entered the cavity. This method makes use of the imaginary component of
the cavity reflection function - the beam that enters the cavity experiences a frequency dependent
phase shift whereas the beam that does not can serve as a phase reference.
Both of these methods suffer from the issue of detecting a signal at baseband, where technicalnoise of the laser and RF environment can corrupt detection of the cavity signal. A clever technique
was pioneered by Pound [76] for locking an oscillator to microwave cavities, where a phase modulator
imprints sidebands nonresonant with the cavity. The error signal is then encoded high in the RF
regime where technical effects are negligable. This was adapted to optical oscillators by Drever and26
Hall and the resulting Pound-Drever-Hall (PDH) technique [5] has become the standard approach

Signalfor transferring cavity stability to an oscillator.

1.0
0.8
0.6
0.4
0.2
0.0
0.2
0.4

Reflection signal
Derivative reflection signal
Dispersive signal

100

75

50

25
0
25
Detuning [kHz]

50

75

100

Figure 2.6: Comparison of lineshapes of the various locking schemes discussed. The cavity transmission function (red) is used to enact side of line locking. Its derivative is used in White’s method[4] for modulation the laser frequency. The true dispersive lineshape is obtained via the PDH [5]and Hänsch-Couillard schemes [6].

A phase modulator places sidebands at an offset frequency of Ω around a carrier of frequency
ω with modulation index β. The incident electric field becomes
Ei = E0 ei(ωt+β sin(Ωt))(2.35)

Expansion of the phase modulation term to first order with Bessel functions gives
Ei = E0 J0 (β)eiωt + J1 (β)ei(ω+Ω)t − J1 (β)ei(ω−Ω)t(2.36)

Note that the sidebands have an opposite amplitude of electric field. As we will see later, thisensures that the baseline value of the error signal far from the cavity resonance is close to zero.
Additionally, there will be no photocurrent from the cavity signal when locked to the center of theresonant feature.
The field reflected from the cavity Er will be modified by complex components relating to27
the cavity transmission and phase shiftEr = E0 J0 (β)eiωt R(ω) + J1 (β)ei(ω+Ω)t R(ω + Ω) − J1 (β)ei(ω−Ω)t R(ω − Ω)

(2.37)

Our photodetector will measure the optical power Pr = Er∗ Er . Components of the optical power
will be present at DC, and also oscillate at frequencies Ω, 2Ω, etc. This is the key advantage of
the PDH method: the cavity error signal is present at the sideband frequency Ω which is generally
chosen to be much larger than the cavity linewidth so that it serves as the phase reference for thephase shifted intracavity light. Components at DC and 2Ω are filtered out so that the only signal
we receive is the power at ΩPr = E02 J0 (β)J1 (β)

R(ω)R∗ (ω + Ω) − R∗ (ω)R(ω − Ω) e−iΩt∗∗+ R (ω)R(ω + Ω) − R(ω)R (ω − Ω) eiΩt(2.38)
(2.39)

Demodulating the signal at Ω recovers the PDH error signalVr ∝ 2E02 J0 (β)J1 (β) Re χ(ω) cos φ + Im χ(ω) sin φ(2.40)

Where φ is the demodulation phase between the error signal at Ω and the local oscillator driving
the EOM with modulation Ω. The quantity
χ(ω) = R(ω)R∗ (ω + Ω) − R∗ (ω)R(ω − Ω)

(2.41)

Provides the dispersive signal shown in Fig.2.6.
There are a few observations we can make that will provide some intuition into the nature of
the error signal. First, we can see that the signal is maximized when the Bessel product J0 (β)J1 (β)
is maximized. This occurs for a modulation depth of β ∼ 1, where the power in the carrier is 3 dB
higher than the sum of power contained in the two sidebands. Second, the imaginary component of
χ(ω) contains the dispersive signal so the error signal is maximized when ϕ = π/2. This condition
is typically enforced by adding delay lines of appropriate length if the demodulation oscillator isanalogue, or by directly adjusting the signal phase if it is digitally synthesized.

28
The approximate slope of the frequency discriminator via the PDH method can be obtainedby making a few approximations. When the sidebands are placed far outside the cavity transmission
feature, it is reasonable to set R(ω ± Ω) ≈ 1. When the laser is within the cavity linewidth, we canapproximate
R(ω) = −ωBy additionally setting E02 J0 (β)J1 (β) =√ω + iΓ/2
2δω
≈ −i
2
2
(Γ/2) + ωΓ

(2.42)

Pc Ps for carrier power Pc and sideband power Ps , wefind that the slope
√
p
Pc Ps
Pr = 4 Pc Ps Im R(ω) ≈ 8
δω ≡ DδωΓ

(2.43)

This quantity D is often referred to as the discrimination slope and dictates the magnitude of the
PDH signal for cavity locking. More optical power increases the sensitivity but often at the costof increased sensitivity to local heating effects in the mirror coating. A narrower linewidth also
provides a larger slope, which is achieved by making a longer cavity with higher finesse mirrors.
One interesting point of note is the difference in discrimination slopes between side of fringe
locking and PDH. The Lorentzian formula for the power in cavity transmission, and the slope is
P (ω) =Pt
ω 2
1 + Γ/2
 ω 2 i−28ω h
P ′ (ω) = 2 1 +
Pt
Γ
Γ/2

(2.44)(2.45)At the half linewidth where one operates for side of fringe locking
P ′ (ω = Γ/2) =PtΓ

(2.46)

Compare this to the PDH discriminator D, where we operate at the optimal point Pc = 2Ps . Then
we have
√
D=

8P0
Γ

(2.47)

There is a slight advantage in signal with the PDH method due to the fact that only half theincident power is used in side of fringe locking. However, the transmitted power Pt is distinct from

Carrier power
1st sideband power
2nd sideband power
3rd sideband power

0.5
0.0

0

1

2
3
4
Modulation index

5

Discriminator [Hz/V]

Power [P0]

1.0

29

105
104
103

0

1

2
3
4
Modulation index

5

Error signal [arb.]

4
3
2
= 0.1
= 0.9
= 1.7
= 2.5
= 3.3
= 4.1

1
0
4

3

2

1

0
1
Freq. [Omega]

2

3

4

Figure 2.7: Top left: Relative power in the carrier, first, second, and third order sidebands forvarying modulation index β. The total optical power is normalized to unity. At β = 1, roughly
half the power is contained in the carrier while each first order sideband contains 25%. Top right:
Discriminator slope D−1 for different modulation depth. Again, maximum sensitivity is achieved
close to β = 1. Note that there are values of modulation index where there is nearly no sensitivity
to the PDH signal. At β = 2.3, power in the carrier drops to zero meaning no light enters thecavity. At β = 3.8, there is still power in the carrier but there is no electric field oscillating at
1Ω. A PDH signal can be recovered by using the 2Ω sidebands as a phase reference. Bottom:
Error signals measured for varying modulation index. Curves are offset vertical and horizontallyfor clarity.

the incident power P0 as seen in Eq.2.8. Even if the cavity coupling is higher, high mirror loss willlead to a Pt smaller than P0 .
Recall that our calculation of D made the approximation that the laser was well within thelaser linewidth. For frequency excursions outside the cavity linewidth, the cavity pole becomes a30significant feature and has the effect of lowpassing the signal. The frequency dependent optical
gain then becomesD
Df (f ) = p
1 + 4f 2 /Γ2(2.48)

We’ve identified the magnitude of the signal that can be recovered from the cavity, it is also
instructive to see how this relates to the noise that is measured. The signal that arrives on our
photodetector is fundamentally limited by the optical shot noise of the incident light. If the carrier
is fully coupled through the cavity, the shot noise limited spectrum of power fluctuations at thedetector will be
SP = 2hν(2Ps )(2.49)

Shot noise will be indistinguishable from the amplitude fluctuations that comprise the error signal.
This will therefore set a limit on the fundamental level of frequency noise that we can resolve andwe calculate this limit by simply using the magnitude of the cavity’s frequency discriminant
Sf =SP
4hνPs Γ2
hνΓ2
=
=
D2
64 Pc Ps
16Pc

(2.50)The spectrum of frequency fluctuations Sf is set entirely by the cavity linewidth and the power
contained in the carrier. This sets a fundamental limit on the level of frequency noise that thelight allows us to resolve. For a cavity with 10 kHz linewidth at 1542 nm, a carrier of 100 nW only
allows a fractional frequency resolution of 1 × 10−17 at a 1 Hz bandwidth. This is surprisingly not
far from the optical powers we tend to use in the Si3 and Si4 systems, next generation cavities willneed to be mindful of this.

2.4

Frequency stability of oscillators
We’ve discussed the fundamental noise sources that limit the cavity instability, as well as

fundamental sources introduced when transferring the cavity stability to a laser. Lower noise isalways better in terms of making a more stable oscillator, but we should understand why this might

31
be important. In particular, when the laser is distributed to the strontium experiments, what partsof the frequency spectrum are important and which parts should we focus our efforts on improving?
First, let’s consider how we convert the frequency noise spectrum into a more intuitive quantity. Measurement precision is more intuitively thought of in terms of fractional resolution at aparticular timescale, e.g. a standard HP33401 multimeter used throughout the lab can measurewith ppm resolution over 1 second of measuring. How the resolution increases with integration or
averaging time depends on the noise mechanism limiting the measurement stability. More generally, the spectrum of noise will determine whether the measurement stability worsens, improves, orstays constant. The sample variance σy2 (τ ) was developed as a metric to characterize this stability
for an averaging time τ [77]. The Allan variance is a particularly useful form of variance that we
discuss here which does not diverge for particular noise processes in the f → 0 limit. This can bedirectly related to the fractional frequency noise spectrum of a quantity Sy (f ) throughσy2 (τ ) = 2

Z ∞Sy (f )
0sin4 (πf τ )df(πf τ )2

(2.51)

It is difficult to extract an intuitive relation between σy2 (τ ) and Sy (f ) so let’s instead consider
various slopes on the spectrum of Sy (f ) and calculate the corresponding behaviour of the variance.
We can express
Sy (f ) =2X

hα f α(2.52)

α=−2This summation contains the five typical characterizations of frequency noise: random walk (α =−2), flicker frequency noise (α = −1), white frequency noise (α = 0), flicker phase noise (α = 1),
and white phase noise (α = 0). The relation between the variance and frequency noise spectrum
for equivalent values of α is displayed in Table 2.1. Note that we have used the relation between
phase noise and absolute frequency noise Sν (f )Sν (f ) = f 2 Sϕ (f )(2.53)

There are a few interesting points to note from the variance calculation. White frequency noisein variance scales as σy2 (τ ) ∼ 1/τ and can thus be averaged down by measuring longer. While32
fundamental sources like photon shot noise appear to be a stringent limitation in the noise spectrum,
they do not limit the measurement precision - they only limit the amount of time it takes to average
down to a particular precision. A lower white noise floor is nevertheless beneficial for the purposesof reducing the measurement time.
Thermal noise in the other hand which scales as Sy (f ) ∼ 1/f cannot be averted by integrating
the measurement over longer times. We can see that σy2 (τ ) ∼ h−1 and is independent of τ . An
oscillator that is thermal noise limited will therefore never be able to measure quantities below
a particular precision. This highlights the particular need to mitigate thermal noise with cleverstrategies when using oscillators in metrology.
Type of noiseαSy (f )σy (τ )

mod σy (τ )Random walk frequency noise-2h−2 f −22 2
3 π h−2 τ

0.82σy2 (τ )Flicker frequency noise-1

h−1 f −1

2 ln(2)h−1

0.67σy2 (τ )White frequency noise0

h0 f 00.5σy2 (τ )Flicker phase noise1

h1 f 1White phase noise2

h2 f 21−1
2 h0 τ
1+3 ln(2πfH τ )h−1 τ −24π 2
3fH
h τ −24π 2 −26.25−28π h−1 τ3−3
8π h−2 τ

Table 2.1: Relations between the fractional frequency noise spectrum Sy (f ), variance σy2 (τ ), andmodified variance mod σy2 (τ ). Calculation of these relations is taken from [1].

In the high frequency regime the variance does not discriminate between flicker and whitephase noise, both scale as 1/τ −2 . These can originate from entirely different processes and so it isuseful to be able to identify which type of noise may be a limitation. Furthermore, the magnitude
of both noise processes is dependent on the bandwidth of the frequency noise measurement - ahigh pass filter with a corner fH is needed to avoid a divergence in the calculation of the variance(particularly a divergence in the integral of Eq.2.51). The ”modified variance” mod σy2 (τ ) was
developed for this purpose as it does not have these shortfalls. Equivalent conversion from thefrequency noise spectrum is displayed in Table 2.1. The modified Allan variance is calculated as

33
[78]
mod σy2 (τ ) = 2

Z ∞Sy (f )
0sin6 (πf τ )df(πf nτ )2 sin2 (πf τ /n)(2.54)

Where τ = nτ0 and τ0 is the minimum data interval.

Amplitude

100|Wa(f)||Wm(f)|

10 1

10 2

10 3

0

1

2

3

f

4

5

6

Figure 2.8: Calculation of the frequency domain filter functions |Wa (f )| and |Wm (f )| for the Allanand modified Allan variances. The modified Allan variance has less susceptibility to high frequency
noise

Oftentimes a time series of the signal is measured directly so it is desirable to be able tocalculate the variances directly from this data. To see how this follows from Eqs.2.51, 2.54, notethat the quantity within the integral can be interpreted as a filter to the frequency spectrum Sy (f ).
We can define separate filters for the variance and modified variances as Wa (f ), Wm (f ) respectively√ sin2 (πf τ )2(πf nτ )√ sin2 (πf τ )| sin(πf τ )|Wm (f ) = 2(πf nτ )2
Wa (f ) =(2.55)(2.56)Such that the variance is calculated as
mod σy2 (τ ) =Z ∞0

Sy (f )|W (f )|2 df(2.57)

34
We can relate both frequency domain filters to time domain filters via Parseval’s theorem.
The variances can then be obtained directly from the time series of data y(t) by
* Z
 +
2

∞

σy2 (t) =mod σy2 (t) =y(t)wa (t)dt(2.58)

2 +y(t)wm (t)dt(2.59)

−∞

* Z
∞
−∞

In the time domain, we can calculate these filters to look like




1



√
0<t≤τ
−



2τ


1
wa (t) =√τ < t ≤ 2τ


2τ






 0
t ̸∈ (0, 2τ ]




1


− √ t
0<t≤τ



2τ 2





1


 √ (2t − 3τ ) τ < t ≤ 2τwm (t) =
2τ 2




1



√

−
(t − 3τ ) 2τ < t ≤ 3τ


2τ 2






 0
t ̸∈ (0, 3τ ](2.60)(2.61)

A illustration of these filters is given in Fig.2.9. The wa (t) has a flat weighting within theintervals of width τ around t = τ , whereas wm (t) has a sort of triangular sensitivity. These arecorrespondingly called Π and Λ filters after the apparent shape of their sensitivities. The shapeof these filters also provides some intuition as to why the the modified variance may provide lesssensitivity to low frequency noise sources as seen in Table 2.1. The resulting variances for thedifferent types of noise presented in Table 2.1 are compared in Fig.2.10.
These techniques are all useful for characterizing continuous noise processes, i.e. noise thatis consistently present in the frequency noise spectrum. In reality, measurement of the cavity
frequency will occasionally encounter aperiodic perturbations such as frequency jumps. This canhave the effect of deteriorating the stability, but also making it look like there are additional noise

35

1.00
0.75

1
2

0.50
Amplitude

wa(t)
wm(t)

0.25
0.00
0.25
0.50

1
2

0.75
1.00

2

1

0

1
t[ ]

2

3

4

f2 noise

10 15

Fractional m-Variancemod y( )
y( )

10 16
10 17 0
10

101

10 15

Time [s]f0 noise102

mod y( )
y( )

10 16
10 17 0
10

103

101

Tau [s]102

Fractional m-Variance

Fractional m-Variance

Fractional m-Variance

Figure 2.9: Calculation of the time domain filter functions wa (t) and wm (t) for the Allan andmodified Allan variances.

103

f1 noise

10 15

mod y( )
y( )

10 16
10 17 0
10

101

10 16
6 × 10 17
4 × 10 17
3 × 10 17
2 × 10 17
10 17 010

Tau [s]f 1 noise

102

103

102

103

mod y( )y( )101Tau [s]Figure 2.10: Simulation that compares the variance and modified variance for different values of α
where Sy (f ) ∝ f α .

processes at low frequency. An example is shown in Fig.2.11. A simulation of flicker frequency noise

36
has a series of discrete frequency jumps included. The modified variance already departs from theexpected mod σy (τ ) ∼ τ 0 at short averaging times and scales as σy (τ ) ∼ τ 0.5 . Without access tothe time series, one could incorrectly infer that the dataset has a significant amount of random
walk frequency noise. The variance and modified variance do not uniquely determine the nature of
frequency excursions in a dataset. One should be aware that these may not be the correct tools toanalyze non-continuous noise processes.

2.5

Frequency stability of atom-steered oscillators
We spent a considerable length going over the application of filter functions to a dataset,allowing us to calculate variances of the oscillator’s stability. There are many software options thatwill do this for us automatically so it may appear to have been an unnecessary effort. However,the process of filtering a time series to calculate the free running oscillator stability is the exact
process that we will use to determine the stability of an oscillator steered to an atomic reference.
Coherent spectroscopy techniques in clocks place an atom in a superposition between two
states where it can evolve under the Hamiltonian of the laser’s electric field. The spectroscopiclineshape relates fluctuations in the relative excited state - ground state population to a phase error
in the oscillator. This mapping is dependent on the duty cycle Tc and spectroscopic time τd forwhich the atom is interrogated. A ”dead time” during which the atom is not interrogated leads tothe aliasing Dick effect [79, 80], where the laser phase evolves without being tracked by the atom.
The stability of the atom-steered oscillator is dependent on these parameters since the atom is
not continuously used to correct phase errors of the laser and is therefore blind to phase evolution
during the dead time.
This mapping from the free running oscillator stability to the atom-steered oscillator stability
is accomplished through filter functions - in particular, the sensitivity function of the atom. Rabiand Ramsey spectroscopy have different sensitivity functions and weight the f α types of noise
differently (similar to our investigation of variances from above). The Dick effect limited variancewill also be different when the atom is interrogated with the two spectroscopic methods.

37

0.6

Sy 1/f0
Sy 1/f
Sy 1/f2

Frequency [Hz]

0.4
0.2
0.0
0.2

Fractional mod y( )

0

10000

20000

30000
Time [s]

40000

50000

60000

10 16

10 17

10 18 010

Sy 1/f0
Sy 1/f
Sy 1/f2

101

102
Time [s]103104

Figure 2.11: Top: We simulate datasets with frequency noise spectrum Sy (f ) ∼ f α for α =−2, −1, 0. Discrete frequency jumps of the same magnitude are added to each time series. Bottom:
Solid circles indicate the modified variance calculated for the corresponding datasets from the plot
above. Dashed lines are variances of the unperturbed white, pink, and brown noise without thediscrete frequency jump. The frequency jump has the effect of deteriorating the stability of all
three time series but also changes the long term instability to scale as mod σy (τ ) ∼ τ 0.5 . Data
that is dominated by white frequency noise and flicker frequency noise can still have variances that
corresponding to random walk frequency noise if there is a discrete perturbation.

38
The Dick effect limited variance has analogues to the Allan variance from Eq.2.58. The filter
function r(t) for a spectroscopic sequence is multiplied by the time series of the laser frequency(y(t) as discussed earlier). The end result however is not a variance of the laser frequency, rather
a variance of the excitation fraction δfexc during the atomic state readout. Quantitatively, this isexpressed as [81]1
δfexc =
2

Z Tc
y(t)r(t)dt(2.62)

0Note that the integral runs over the cycle time Tc , not the spectroscopic duration τd . Preparation
and readout of the atomic sample takes a finite amount of time and consequently most clocks operatewith τd < Tc . Similar to our depiction of the variance filter functions in Fig.2.9, this implies a lackof sensitivity to laser frequency noise over a duration Tc − τd . The sensitivity function will acquirea value of zero (see Fig.2.12 for an illustration) and this dead time leads to the aliasing effect ofDick noise.
During a cycle of time Tc where interrogation of the oscillator begins at time Tc − τd for
duration τd , the sensitivity functions for Rabi and Ramsey spectroscopy take the form

0 < t ≤ Tc − τd0rRabi (t) =





2.509π 2 t−Tc +τdt−Tcsin(β) − sin
β + sin τd β
Tc − τd < t ≤ Tc
τd
β3




0
0 < t ≤ Tc − τd






 h

i π



Tc − τd < t ≤ Tc − τd + tπ /2sin t − Tc + τdtπrRamsey (t) =




1
Tc − τd + tπ /2 < t ≤ Tc − tπ /2






h

iπ
π



sin t − Tc + tπ /2+
Tc − tπ /2 < t ≤ Tc
tπ 2

(2.63)(2.64)(2.65)In the Rabi sensitivity, β =Ramsey sequence.

√

π 2 + 2.5092 and tπ refers to the time for a π pulse in the39First, consider the case where τd = Tc . We can see that the Rabi sensitivity function does not
uniformly weight all portions of the laser frequency. The consequence of this is a decreased ability
to track the laser phase. Phase excursions in the frequency range where the laser is dominatedby 1/f noise will not be perfectly measured by the atom during Rabi spectroscopy. If it waspossible to constantly interrogate the atom with sequential Rabi pulses, we would still find errors
in our measured laser phase. Ramsey spectroscopy does not have this issue, as we see that thesensitivity is constant over the duration of atomic interrogation. However, the finite time required
to drive an atom to fexc = 0.5 leads to a softening of the curve at the beginning and end of thecoherent spectroscopy. This will also have the effect of reducing the sensitivity to laser noise insome frequency band, but less so than in the case of Rabi spectroscopy.
With the introduction of deadtime (τd < Tc ), the picture becomes even less intuitive. Previously we found that the variance of a signal is expressed as the integration of the sensitivity
function and the laser spectrum Sy (f ). In reality, the signal is sampled by a measurement device
with a particular rate and the variance becomes a summation of the spectral density measured overinteger multiples of each frequency bin.
The Dick effect follows the same formulation. The variance is calculated as a summation overdiscretized frequency bins for integer multiples of 1/Tc . The sampled spectrum is then [82, 83],
Sd (f ) =

∞
X
gk 2
k=1

g0

Sy (k/Tc )(2.66)The laser spectrum is periodically sampled every k/Tc and the bin amplitude gk is given by
1
gk =
Tc

Z Tc

g(t)e−2πikt/Tc dt(2.67)

0

The Dick effect limited variance is related to this sampled spectrum by
σy2 (τ ) =Sdτ(2.68)

This model is insensitive to the type of spectroscopy being performed, only the sensitivity
g(t) will change. Surprisingly one finds that the sampled spectrum Sd is white frequency noise and40
the Dick effect limited variance will then simply scale as σy2 (τ ) ∼ τ −1 . As we show in Fig.2.13,
the shape of the laser spectrum has no determination on the relation between averaging time andvariance. The amplitude of the variance is the only component that varies with the spectroscopy
sensitivity function and the spectral components of the laser.

rRabi(t)rRamsey(t)1.0Amplitude

0.8
0.6
0.4
0.2
0.0
0

1

2

t[ ]

3

4

5

Figure 2.12: Calculation of the time domain filter functions rRabi (t) and RRamsey (t) for the two
spectroscopic methods. In both cases we assume a cycle time of 5 s, a spectroscopy time of 3 s,
and in the case of Ramsey a π pulse time of 100 s.

What does this mean in terms of the impact on the experimenter? If the Dick effect limited
variance always scales as 1/τ , how should the laser spectrum be shaped? Intuition is gained bylooking at the various Fourier components gk . Fig.2.14 calculates a series of these for different spectroscopic sequences with the same cycle and interrogation time. The spectral window around 1/Tc
has the largest amplitude in the summation of Eq.2.66 but harmonics of higher k are weighted differently depending on the spectroscopic sequence. For a given evolution time τd , Rabi spectroscopy
can be sensitive to certain spectral windows where Ramsey spectroscopy has no sensitivity. Within
Ramsey sequence, the pi-pulse time can be intentionally chosen to shift the spectroscopic sensitivity
to windows where the laser spectrum may have less noise, e.g. avoid mechanical resonances of thecavity or large ground loops introduced by electronics.

Sy(f) (1/Hz)

10 29
10 30
10 31
10 32
10 33
10 34
10 35 2
10

41
Sy(f)
Sy(f)
Sy(f)
Sy(f)

10 1

f1
f0
f 1
f 2

101

100
Frequency (Hz)

10 16

102
Sy(f)
Sy(f)
Sy(f)
Sy(f)

y( )

10 17

f1
f0
f 1
f 2

10 18
10 19 010101

Tau (s)102

103

Figure 2.13: We calculate the Dick effect via Eq.2.66 for different laser spectra Sy (f ) = 10−33 f αwith Tc . For reference, Sy (f ) = 10−33 f −1 is the thermal noise limit of Si3. The corresponding
Allan deviation is insensitive to the type of noise that dominates the laser spectrum and always
follows σy (τ ) ∼ τ −1/2 . However the amplitude of the Dick effect is determined by the slope ofthe laser spectrum, with a lower variance associated with larger values of α. Solid lines indicate
variances calculated with the Rabi sensitivity function and dotted lines are calculated with theRamsey sensitivity. As α decreases, Ramsey spectroscopy provides increasing improvements instability over Rabi spectroscopy.

The point of this is to illustrate the importance of tailoring the spectroscopic sequence tointerrogate the ’quieter’ windows of the laser spectrum. Global benchmarks for laser noise such as
laser linewidth do not uniquely determine the noise spectrum and often have little correspondance
to the Dick effect limited variance that is realized in clock spectroscopy. Recent results in qubit
control have come to identify the role that the local oscillator noise plays in determine qubit fidelity
when driving rotations between the qubit states [84]. The physical process is exactly the same aswhat we have described in the past two sections - a filter function samples particular spectral

42components of the local oscillator to determine a variance in excitation probability (2.62). Driving
fast qubit rotations does not make the atom immune to this process, the spectral window forsensitivity is simply shifted to a higher bandwidth.

100

rRabi(t)rRamsey(t)rRamsey, t = 0(t)|gk/g0|2

10 2
10 4
10 6
10 8
10 10

0

10

20

k

30

40

Figure 2.14: The various amplitude coefficients |gk /g0 |2 are calculated for different spectroscopic
method. All traces use Tc = 5 s, τd = 4 s. The gold curve uses a conservative tπ = 0.2 s whereas
the green trace assumes infinitely short pulses of tπ = 0. The majority of the Rabi sensitivity isconcentrated around frequencies of 1/Tc whereas Ramsey spectroscopy has a higher contribution
from the high frequency harmonics of the cycle time.

Chapter 3

Experimental setup

Having presented the primary concepts necessary to design, operate, and characterize a cavity,
we’ll discuss now these realizations in the Si6 system. We’ll detail out choice of silicon as a spacer
material, operation of the 4 K cryostat, unique features of crystalline AlGaAs/GaAs mirrors, andcharacterization of the Si3 cavity which we use as a reference oscillator.

3.0.1

Silicon in cavities

It’s evident that three key parameters that dictating the Brownian thermal noise limit are
E,ϕ, and T . For improved oscillator stability, a tempting option is to reduce the cavity temperature.
The Sy ∼ T scaling indicates that modest reductions below room temperature will have little effect,
rather cooling into the cryogenic regime can lead to substantial improvements. Technical issuesof cryogenics aside, the advantage of ULE cavities are their relatively low insensitivity to room
temperature fluctuations. A material’s length sensitivity δL to temperature changes δT is given as

1∆L/L = α(T )δT + α′ (T )2 δT 2 + O(δT 3 )
2

(3.1)

For a coefficient of thermal expansion (CTE) α. Particular materials have α = 0 at certain temperatures, and ULE has a zero crossing of its CTE at 20◦ C. Residual temperature sensitivity isdominated by the much smaller second order sensitivity of α′ (T ) = 17×10−9 K−2 [85]. Temperaturefluctuations of 0.1 mK, routinely achievable with active temperature stabilization, therefore only44
leads to length fluctuations of 1 × 10−16 , close to the thermal noise limit of most room temperature
ULE cavities. Without a zero crossing of the CTE, equivalent magnitude of thermal fluctuations
would require α(T ) = 1 × 10−12 which is an absent property for room temperature materials.
The existence of the zero CTE (ZCTE) is therefore seen as a soft requirement for thermal
noise limited cavity operation. This is also relevant in the cryogenic regime, where temperaturestabilization is often more difficult due to the added thermal fluctuations of the cryocooler and the
reduced thermal mass of most materials. The ZCTE property is not ubiquitous to all materials rather, few are found to have it.
Fortuitously, silicon is a material that has been identified to have not one but two ZCTEs
located at 124 K and 16 K [86] (Fig.3.3). Operation at these temperatures is achieved with different
cooling technologies: the Si3 cavity in our lab operates at 124 K where temperature stabilizationis accomplished by boiling off 77 K liquid nitrogen gas, and 16 K operation can be achieved with acryocooler using cold helium circulation. The Si6 cavity operates at both 16 K and 4 K, where theCTE is not zero but approaches 1 × 10−11 and has a inconsequential α′ sensitivity. The silicon loss
angle at these temperatures is found to be at the 10−8 level [87], three orders of magnitude lower
than the ULE loss angle of 1.6 × 10−5 [7]. Note that many materials are found to have temperaturedependent ϕ so operation temperature of the cavity is an important point to consider.
A large Young’s modulus is also beneficial for reduced thermal noise. ULE is already fairly
stiff with E = 68 GPa but our typical crystalline materials are again found to have advantageous
properties. Silicon is found to have a Young’s modulus between 130-194 GPa depending on the
crystal orientation [88], while sapphire has a considerably higher value of E = 400 GPa. The
anisotropic structure of the crystal is leveraged in unique ways which will be detailed in the following
chapter.
Ultimately we choose to use silicon as our spacer material and this has proved to be a fruitfulcollaboration with our colleagues at PTB, Germany. The Si6 spacer is grown as a high purity
monocrystalline ingot using the float-zone method. X-ray crystallography identifies the stiffest⟨111⟩ axis along which the ingot is to be grown. A check of the crystal purity is done by measuring

45

Figure 3.1: Loss angle ϕ temperature dependence for commonly used cavity spacer materials.
ULE is not shown on this plot but has a measured room temperature loss angle of 1.6 × 10−5[7]. Crystalline spacer materials have a ϕ so low that the total Brownian thermal noise limit ofcrystalline cavities is limited by the loss angle of the optical coating. Figure taken from [8].

Figure 3.2: Coefficients of thermal expansion for commonly used cavity spacer materials. Between
the two crystalline materials of sapphire and silicon, only the latter has a ZCTE. Fused silica canbe an attractive option due to a ZCTE around 170 K but the significantly higher loss angle meansthat a silicon cavity will still have a superior thermal noise limited performance. Figure taken from
[8].

46
the resistivity along the length of the spacer, and a value of 30,000 Ω cm is found. Subsequently it
is sent to several US and European vendors for machining, polishing, lapping the ends, and etching
the surface.
The spacer is cut with the same geometry as Si4. With the operating temperature andmaterial properties being the same, we can calculate the spacer Brownian noise as Ssp = 8×10−38 /f .
Substrates are cut from the same ingot with same orientation, ⟨111⟩ being normal to thecoating surface. In 2014, Coastline optics was only able to polish substrates to a maximum of 1 m
radius of curvature. With a cavity length of 6 cm, we can calculate the waist of the fundamental
TEM00 mode supported by the cavity and find w0 = 286 µm. At the mirror locations, the spot
size is found to be w(L/2) = 294 µm. We now have all the parameters needed to calculate thesubstrate thermal noise and find Ssb = 2 × 10−35 /f . So far, the total fractional thermal noise limitof the cavity is found to be
Ssp + 2Ssb = 4 × 10−35 /f(3.2)

This is remarkably low and would imply a fractional frequency instability of σy = 6 × 10−18 , farlower than realized in state of the art optical resonators. As we will now see, the ”dominant”
limitation to the realized frequency noise is due to the coating.
ULE
ϕ
E
σ

1.6 × 10−5
68 GPa0.18

SiliconϕE
σ

1 × 10−8
190 GPa
0.29

SapphireϕEσ

2 × 10−8
400 GPa
0.28

Table 3.1: Comparison of bulk substrate and spacer materials.

3.0.2

AlGaAs coatings

A key demonstration of Si2, Si3, and Si4 was thermal noise limited performance with instabilities in the mid 10−17 s. We’ve seen that silicon substrate and spacer noise is incredibly small, so
it stands to reason that the coating thermal noise overshadows these other two. The larger thermal

CTE [1K ] (1 × 10 11)

250
200
150
100
50
0
50

0

50

100

150 200
Temp (K)CTE [1K ] (1 × 10 11)

4

250

300

T = 16 K

2
0
2
4
17.50

17.52

17.54 17.56
Temp (K)

17.58

17.60

5.0
T = 124 K
2.5
0.0
2.5
5.0
7.5
0.006 0.007 0.008 0.009 0.010 0.011 0.012 0.013
Temp (K) + 123.7 K
10CTE [1K ] (1 × 10 11)

CTE (10 8/K)

47

8

T=4K

6
4
2
3.0

3.5

4.0

4.5
5.0
Temp (K)

5.5

6.0

Figure 3.3: Coefficient of thermal expansion of silicon across the three ranges of operation, 124
K, 16 K, and 4 K. At 124 K, α′ = 2 × 10−8 K−2 , similar to the CTE slope around the ULE
room temperature ZCTE. The slope is considerably shallower at 17 K with α′ = 1 × 10−9 K−2 but
requires different cooling technology. There is no zero crossing of α at 4 K but values of 10−11 canbe achieved, requiring only µK temperature stability for 1 × 10−17 cavity instability.

Figure 3.4: Two 55 cm silicon ingots grown at IKZ Berlin. Image courtesy of Thomas Legero.

48
noise of the coating can be attributed to one quantity, the loss angle ϕ.
Loss angles of optical coatings are consistently found to be larger than that of the spacerand 3-4 orders of magnitude higher than silicon. AlGaAs/GaAs thin film coatings have a measured
room temperature loss angle of 2.5 × 10−5 [19, 89]. Compared to silicon, the larger coating loss
angle would imply a 1000 times larger thermal noise when compared to Ssb but recall that coating
thermal noise is modified by a factor of d/w, roughly 1/30 in Si6. It is peculiar that the ϕ of acrystalline coating is slightly larger than that of the ULE spacer when we’ve stated that crystalline
materials should have higher mechanical quality factors than amorphous ones.
This is an issue particular to the thin film nature of the coating. The bulk-melt SiO2 substrates of the LIGO test masses have a measured Q of 107 at room temperature [90] but IBS
sputtered SiO2 films have a ϕ = 4 × 10−5 [91]. The origin of significantly higher ϕ in thin films has
been the source of intensive study. Theories that the majority of dissipation could be occurring
between the interfaces of the many layers were debunked in a study where the total coating thickness was kept constant and the number of layers was varied [92]. Similarly, varying the coating
thickness and observing a proportional increase in coating thermal noise was crucial in removing
the substrate-coating interface as an additional source of dissipation.
In conventional dielectric coatings alternate quarter wave stacks of high index Ta2 O5 and low
index SiO2 materials use interference of the Fresnel reflections to generate a high reflectivity surface.
The Ta2 O5 loss angle is found to be (4.2 ± 0.4) × 10−4 [93], significantly higher than that of thesilica layers. Reducing the coating’s effective loss angle is therefore best accomplished by reducing
the loss of the high index layers, or finding a substitute material with lower loss. Consequently
there is still a dedicated effort to fabricate high-Q amorphous coatings by varying the dopants andannealing process
Titania has been found to be a promising dopant within the tantala-oxygen structure for
reducing the overall loss [94]. Introduction of up to 40% TiO2 has been found to reduce the tantala
loss angle by over a factor of two to ϕ = (1.8 ± 0.1) × 10−4 . Loss angles between samples oftantala from other vendors have differed, suggesting that the fabrication process may be extremely

49
important in engineering low loss [95]. This has led to an interest in investigating loss angles of
coatings with different annealing and sputtering parameters. Extended annealing time of SiO2
coatings has demonstrated a reduction in ϕ [96] but the improvement varies depending on thevendor.
Recent studies of Ta2 O5 mixed with GeO2 and annealed at high temperature have emerged
as promising candidates for the high index layers [97]. Loss angles as low as ϕ = 1 × 10−4 havebeen measured in these samples. Fine tuning annealing parameters has also demonstrated a slightdecrease in Ti2 O5 loss angle [98]. Distinction between high temperature deposition and high temperature annealing and varying the two separately has also been beneficial.
Next generation gravitational wave detectors such as the aLIGO+ upgrade requires a coating
thermal noise corresponding to a loss angle of 0.9×10−4 . Future cryogenic detectors such as Voyager[56] and the Cosmic Explorer [99] operating at 124 K require even lower loss angles of ϕ < 6 × 10−5 .
Exploring the parameter space of doping and material deposition has spawned promising advancesin amorphous coating technology, but still far from the requirements of these future experiments.
A deeper understanding of coating loss angles is needed.
Several studies have now revealed that the emergence or suppression of long range order inamorphous materials is correlated with the magnitude of a material’s loss angle [100, 101]. Bulk
glasses that are close to being perfectly amorphous can still have a remarkably high Q, but this is
reduced in thin film amorphous layers where the emergence of some order makes it an imperfectglass [102].
A similar phenomenon exists in crystalline materials, where the long range order of a crystal
supports a high mechanical quality factor. Deviation from perfectly amorphous or perfectly crystalline material appears to result in a degradation of quality factor. Measurements of bulk GaAs
at cryogenic temperatures on a large 3” diameter by 1” thick disk found a Q ∼ 107 [103]. Notethat this is reduced by a factor of over 200 when epitaxially grown as a thin film. Still, the AlGaAs
coating loss angle is sufficiently low to meet the requirements of next generation interferometers andcavities. Further reductions in the thin film ϕ have not been thoroughly explored for this reason,

50
and it is possible that the high index GaAs and low index AlGaAs layers may also have differing
Q’s.
AlGaAs/GaAs coatings are not the only demonstration of high-Q single crystal structuresthat can be used in mirrors. Measurements of InGaP membranes in cavity optomechanics experiments [104] reveal a Q = 2 × 106 at room temperature and 17 K. Measurements are conducted atoffset frequencies f > 100 kHz far from the interrogation band for ultrastable resonators, but thisdemonstrates that there are other phosphides and nitrides that could be studied as next generation

0.050

Trace #0
CH1
Trace #1
CH1
Trace #2
CH1
Trace #3
CH1
Trace #4
CH1

0.045
0.040

Volts

0.035

Tau [microseconds]

coatings.

Finesse

0.020
0.015

40

20

0
20
Time [microseconds]

40

60

18.6
18.4
18.2

0.030
0.025

18.8

296000294000292000290000288000286000

0.0

0.5

1.0

1.5 2.0 2.5
Run number

3.0

3.5

4.0

0.0

0.5

1.0

1.5 2.0 2.5
Run number

3.0

3.5

4.0

Figure 3.5: Ringdown measurement of Si6 transmitted light with the cavity at vacuum and cooled
to 4 K. No difference in finesse is measured between 4 K and 16 K. A finesse of 290,000 is measured.

The mirror coatings in Si6 are composed of an 11.6 µm-thick multilayer stack of alternating
high-index (n = 3.3) GaAs and low-index (n = 2.9) Al0.92 Ga0.08 As, and are oriented with parallel
coating polarizations. The GaAs ⟨100⟩ axis is normal to the coating, and parallel to the crystalline⟨111⟩ axis of the crystalline silicon. Two near-planar mirrors of 1 m radius of curvature realize cryogenic transmission (T ) of 11.8 ppm and loss (L) of 10 ppm combined for both mirrors, supportinga finesse of 290,000. We note that at ambient temperature and pressure, we measure a finesse of
220,000 which increases to 240,000 upon pulling vacuum on the cavity.

51
Coating parameter
Loss angle (ϕ)
Young’s modulus (E)Poisson ratio (σ)
Coating thickness (d)Mirror radius curvature
Beam waist (w0 )Value
2.5 × 10−5
85 GPa0.32
11.6 µm
1m
286 µmTable 3.2: Parameters of the AlGaAs/GaAs coatings

We measure a 770 kHz birefringent splitting of the polarization modes, well resolved outsideof the 8.6 kHz cavity linewidth. A slight temperature and optical power dependence of the mode
splitting has been observed (Fig.3.8).

10 32

Spacer Brownian
Substrate Brownian
Coating Brownian

Fractional frequency
noise at 1 Hz

10 33
10 34
10 35
10 36
10 37
10 38

Si4
6 cm
4K
SiO2/Ta2O5

Si3
21 cm
124 K
SiO2/Ta2O5

Si6
6 cm
16 K
AlGaAs/GaAs

Si6
6 cm
4K
AlGaAs/GaAs

Si5
21 cm
124 K
AlGaAs/GaAs

Figure 3.6: Brownian noise contributions for the four silicon cavities referenced through this thesis.
Coating thermal noise is consistently the dominant contribution due to the exceptionally low loss
angle of silicon. Coating materials with high mechanical quality factor are therefore instrumental
in developing optical oscillators with longer coherence times.

52

1.0

Mirror 1
Mirror 2

Reflectivity

0.8
0.6
0.4
0.2
0.0
1400

1450

1500

1550
1600
Wavelenth [nm]

1650

1700

Birefringent splitting - 770KHz[linewidths]

Figure 3.7: Reflectance of the two crystalline mirrors used on Si6 at a temperature of 4 K. Data isfrom Garrett Cole. We use a 1542 nm laser to lock to the cavity, centered within the ∼ 150 nm
passband of the coatings.

17K data - TEM00
4K data - TEM00
4K data - TEM01

100
10 1
10 2
100

200

300 400 500 600
Transmitted power [nW]700

Figure 3.8: The mode splitting of Si6 is observed to vary with temperature and intracavity power.
We plot the change in mode splitting normalized to the 8.6 kHz cavity linewidth.

3.0.3

Cryostation

The cavity is housed in a custom-designed cryostation from Montana Instruments capableof reaching base temperatures of 3.5 K. An RDK-101D GM cryocooler from Sumitomo Heavy
Industries provides cooling power to the sample space within the cryostation. Temperatures of

53
the stage 1 and 2 are dependent on the applied heat load, a capacity map of the cooling power
is provided in Fig.3.9. Montana provides thermistors installed on the stage 1 and 2 of the cryo,and measurements of their temperatures provide our best estimate of the heat load on each shield.
With the cavity in place we typically find a stage 1 temperature of 28 K and stage 2 of 2.7 K,
indicating heat loads of roughly 1 W and <30 mW respectively. A total of three thermal shields
are used to successively step down the base temperature and reduce the radiation heat load. The
outermost shield is attached to the 30 K platform, and the two inner shields are both at 4 K. The
outer of the two is used for active stabilization with a 50 Ohm manganin heater wrapped around
its base, and the inner passively damps thermal fluctuations at 4 K. Thermal shields supplied byMontana are made of 6061 aluminum, though these have been substituted with high purity copper
shields constructed by the JILA machine shop. Details on this will be provided in the next chapter.
Thermal sensors on each shield allow for measurement of time constants for heat flow between
stages and as witnesses for temperature fluctuations on each surface. A 20 pin vacuum feedthroughat the cryostation base allows us to read these out with our own measurement devices.
Four sets of UVFS superpolished windows from Coastline optics are placed above and below
the cavity, one for the 300 K vacuum chamber, one for the 30 K radiation stage, one for the 4 K
active stage, and one for the 4 K passive stage. Each is mounted with a 6◦ wedge relative to thecavity substrate, and offset in rotation to prevent low angle scatter and etalons between windows.
A serrated aluminum clamp holds each window in place with a thin layer of Apiezon-N vacuum
grease between the optic and mount to ensure good thermal contact and prevent stresses fromdeveloping across the window during cooldown.
A large brass box with acoustic foam on its interior encloses the optics to lowpass any room
temperature and pressure fluctuations. A secondary box was installed on the AVI around the optics
further dampen pressure fluctuations. Quarter inch thick rubber curtains around the optical table
serve as a final layer for decoupling the optics environment from the lab.
The coldhead generates significant vibrations while operating, so care has to be taken to decouple this from the cavity as much as possible. The cryocooler sits atop a pillar directly mounted

2n

2nd Stage
Heat Load

0W

2.5

27

29

31
33
35
1st Stage Temperature [K]

37

39
54

RDK-101D Cold Head Capacity Map (60 Hz)
5.5
5.0
4.5
4.0
3.5

1st Stage
Heat Load

2nd Stage Temperature [K]With F-20L Compressor and 3 m (10 ft.) Helium Gas Lines

0W

1W

2W

0.2W
0.15W
0.1W
0W

26

27

5W

0.3W

3.0
2.5

0.5W
4W
3W

2nd Stage
Heat Load28 29 30 31 32 33
1st Stage Temperature [K]Specifications subject to change without notice.
Note: Capacity maps for reference only.			

34

35©SHI Cryogenics Group 7/21

Figure 3.9: A capacity map for our cryostation’s cooler displays the cooling power available on eachstage. We operate in the lower right corner of the plot with less than 100 mW heatload on stage 2
and roughly 1 W heatload on stage 1.

onto the concrete foundation of the JILA basement. The cryostation meanwhile sits atop a commercial TS-140 active vibration isolation (AVI) platform that provides acceleration noise feedback
from 5-200 Hz. This is placed atop a long three foot high optical table that is shared by Si3.
Note that there is not strong coherence between acceleration noise measured on the two systems.
Vibrations inevitably will couple through the ground from the coldhead to cavity but this resultin an amplitude reduction is the ideal mechanism if couplings cannot be avoided. The only direct
mechanical link is through the horizontal copper braid that pumps heat from the cryostation to thecooler. The link design is not shared with us, but we suspect it is a braid that can be compressed
and tensioned along its axis. Torquing the AVI has reduced vibration noise measured near thecavity, indicating that the coldfinger is more rigid along its transverse axis.
The helium compressor attached to the coldhead also generates significant vibrations and is aconsiderable heat source. We place this outside the lab and run long 10 m vacuum insulated hoses

g/ Hz10 5
10 6

Cryo onCryo off

10 7
10 8 1
10

g/ Hz

10 5
10 6

100

Freq. [Hz]H1 axis

101

102

101

102

101

102

Cryo onCryo off

10 7
10 8 1
10
10 5

g/ Hz

55

Vertical axis

10 6

100

Freq. [Hz]H2 axis

Cryo onCryo off

10 7
10 8 1
10100

Freq. [Hz]Figure 3.10: Vibration noise measured on the AVI with vibration isolation enabled for all threeaxes.

to the cryo to decouple vibrations between the two sides. The system is entirely closed cycle and56
has a hands-free operation once cooled to cryogenic temperatures. The coldhead needs a routine
maintenance every 20,000 hours of runtime, though this is extended if the system is rarely thermallycycled between 4 K and 300 K. The maintenance procedure involves warming the system back toroom temperature to exchange the coldhead for a new identical model, but the system is not vented
to ambient pressure. Nevertheless, care has to be taken during the warmup to avoid the formationof ice on the colder surfaces of the cavity and 4 K windows.
A beryllium-copper bellows surrounds the coldfinger, connecting the chamber and coldheadat room temperature. Pulling vacuum on the system compresses the bellows so the AVI often needs
to be tensioned away from the coldhead with a mechanical spring.
These all contribute to reducing the vibration noise measured at the cryostation, and further
steps are taken to reduce the cavity length sensitivity to accelerations. Many of these design choices
have been enacted with deliberation and the following chapter will expand on this to explain their
merits. A schematic illustrated in Fig.3.11 displays several of these key features. The vacuum
chamber and thermal shields are hidden from view to reveal the cavity, its support, and charcoal
adsorbers mounted to the lid of the passive shield. The wedged window mounts, visible above thecavity, are oriented to suppress the formation of etalons.

3.0.4

Noise measurements

We’re fortunate to have Si3 in continuous operation at a nearly identical wavelength. A
heterodyne beat with Si6 appears in the RF domain where we can easily measure the frequency
noise with commercial electronics. In [41], Si3 was demonstrated to be thermal noise limited to a
fractional instability of 3.7 × 10−17 from 0.2-1000 s. Later studies [17] extended the thermal noise
performance out to 104 seconds where the phenomenal precision and uptime of the Gen-2 SrI allows
us to continuously measure its long term instability. Short term instability can be measured via
three-cornered hat analysis with the 40-cm 698 nm ULE cavity. Eric Oelker’s fantastic work on

57

Figure 3.11: A solidworks mockup of the Si6 cryostation, cavity, and part of the optical setup.

the Si3 system resulted in a cavity with noise parameters of
Sy =

0
X
i=−1

hi f i +NXai

i=1 1 +

f −fi 2
Γi /2

(3.3)

In a band of 1mHz - 100 Hz. The first term consists of a white noise floor and the 1/f thermal
noise limit with fractional values
h−1 = 1.7 × 10−33 1/Hz

(3.4)

h0 = 4 × 10−34 1/Hz

(3.5)

The second term in Eq.3.3 details various mechanical resonances with parameters given in Table
3.3.
At higher frequency where the 124 K cavity is no longer thermal noise limited, the nearly 10x
lower linewidth of Si3 suggests that its white noise floor should easily be lower than that of Si6.

58
Index

fi (Hz)ai (1/Hz)Γi (Hz)1

5.7

7 × 10−34

1.0

2

12.7

1.5 × 10−34

1.5

3

20.0

4 × 10−34

0.1

30.0

5 × 10−34

0.1

5

40.0

5 × 10−34

0.1

6

45.0

1 × 10−34

4

55.0

4 × 10−34

1.2

4

7

Table 3.3: Resonances of Si3 identified in the noise model, with center frequency fi , amplitude ai ,and width Γi .

This is all to say that we have a robust characterization of the Si3 noise, and direct beat with Si6
will result in a composite spectrum that is likely limited by Si6 at higher frequency, and hopefully
limited by Si3 thermal noise at lower frequency. All noise measurements are made by beating lightfrom the two cavity distributed onto a separate breadboard with fiber phase noise cancelled links.
The Si6 laser is locked to a mode of the 6 cm cavity that minimizes the RF beat with Si3. For the
TEM00 mode, this occurs at 600 MHz and is downconverted to 10 kHz where we use a 4th orderactive lowpass filter with 1 kHz passband to attenuate any high frequency tones. A time series ofthis 10 kHz beat is then logged on a K + K FXE80 deadtime-free frequency counter.

Chapter 4

Technical noise sources

The cavity is designed with the Brownian thermal noise in mind as the ultimate limit to thelaser stability. Numerous technical noise sources can prevent this realization, and each presents its
particular challenges that require particular methods for mitigation. We’ll now discuss the noise
sources present in Si6 and detail our approaches to managing each of them.

4.1

Laser stabilization
The first question is whether, for high-stability experiments, one must use a laser
with special properties, for example a massive invar construction. The answer bothfrom theory and experiment is no, absolutely no. - Jan L. Hall

The main reason for choosing silicon as a spacer material is attributed to its unique physicalproperties, but another part of the choice was predicated on the wide availability of high quality
optics at a telecom band. Narrow linewidth fiber lasers are readily available allowing for direct
locking to the cavity. Not only does this preclude the need for prestabilization of a noisier freerunning laser, but the output mode is cleanly TEM00 so no complex beam shaping is required.
A number of compact fiber components are available at 1550 - the entire setup could in principle be made in fiber were it not for the need to couple into the cavity with free space mirrors.
Nevertheless, we incorporate as much of the optical setup in polarization maintaining (PM) fiber
as possible. Light from the NKT passes through a fiber isolator to prevent backscattered light fromseeding the laser, and then continues to a high bandwidth 40 MHz fiber acousto-optic modulator

60(AOM1). A 90/10 fiber splitter distributes the majority of the light through a fiber phase noise cancelled link to form a heterodyne beatnote with another ultrastable laser in the lab. The remaining
10 percent of light bypasses this setup and passes through an EOspace fiber electro-optic modulator(EOM) with 1 GHz bandwidth and half wave voltage of Vπ = 5 V. A non-polarizing isolator was
installed between this EOM and the free-space collimator after identifying a large etalon between
a free-space optic and an in-fiber surface.
The light is then outcoupled into the free space path where two mirrors are used to align the
beam to the cavity mode. An 8.1mm Schafter-Kirchhoff collimator and 750 mm lens are used tomode match into the cavity given the length of the free space path. A 45 degree mirror launches
the beam vertically into the cavity and a Faraday rotator and PBS distribute the reflected light toa detector for PDH feedback. A half waveplate after the coupling mirrors allows for alignment toeither of the two polarization modes, given the large birefringent splitting of the AlGaAs coatings.
Between the two steering mirrors a plate splitter picks off some light for RAM detection. All mirrors
and lenses are superpolished to λ/10 roughness and cube beamsplitters are two optically contacted
prisms.
While the Si4 system used a RIO external cavity laser with ∼1kHz linewidth, the Si6 system
replaced this with a significantly lower noise NKT distributed feedback fiber laser. The frequency
noise of the two lasers is compared in Fig.4.1. The significantly lower linewidth relaxes the bandwidth constraints on the servo loop locking the laser to the cavity. Rather than relying on theinherently large bandwidth offered with current modulation, the Si6 servo loop actuates only on an
AOM at higher frequency (see Fig.4.2 for AOM transfer function and Fig.4.3 for the modulator’s
AM-FM cross coupling), and a piezo within the DFB package at low frequency. Conveniently, theNKT piezo has a dynamic range of a full free spectral range of Si6 so a TEM00 mode can always
be found within its tuning range.
The JILA school of laser locking involves throwing enough loop gain at the laser to suppress
its frequency noise to a sufficient level. Loop bandwidth often pushes the edges of what is allowablewith actuator or electronics delay, with integration stages tuned to aggressively high corners. This

RIO
NKT

103
Gain (dB)Frequency noise Hz/ Hz

104

102
101
100 0
10

101

104

102
103
Frequency (Hz)

140
135
130
125
120
115
110
105
100 0
10

61
RIO
NKT

101

104

102
103
Frequency (Hz)10 3

Phase [deg.]

Transfer function [Hz/ W]

Figure 4.1: (Left) A comparison of frequency noise between the RIO and NKT obtained via heterodyne beat with cavity stabilized light. (Right) The closed loop gain necessary to push the freerunning laser noise to the Si6 thermal noise at a given offset frequency.

10 4
101

102

103
Frequency [Hz]

104

105

150
100
50
0
50
100
150
101

102

103
Frequency [Hz]

104

105

Figure 4.2: AOM1 is used for both intensity and frequency feedback but it does not exactly actuatealong the two orthogonalized quadratures. A transfer function of the cross coupling between frequency to optical power incident on the cavity is shown in (a). Intensity and frequency servo loops
are also shaped to suppress any cross couplings. Feedback to the AOM frequency is highpassedwith a 50 Hz corner while the intensity servo has an integrator enabled at a 100 Hz corner. The
measured cross coupling of the AOM between frequency and intensity is small enough that thecontrol signal from each cannot introduce noise in the opposite quadrature above the Si6 thermal
noise.

is a nice approach that is generalizable to most problems, but Eric Oelker introduced some finesse
to this problem by introducing us to some of the methods used in the LIGO community. The
narrow linewidth of the NKT laser allows us close the loop with a respectable 150 kHz bandwidth,
but the second integration stage is only enabled at the kHz decade. The loop topology of our
PDH servo is illustrated in Fig.4.4, it consists of a common gain stage with integrators on the two
actuator paths, and then splits into separate loop filters for the piezo and AOM. Canonically, the0Phase [deg.]

Transfer function [MHz/V]

62
Si6
Si3

106
9 × 105
8 × 105
101

102

103
Frequency [Hz]

104

10
20
30
40
101

105

Si6
Si3

102

103
Frequency [Hz]

104

105

Figure 4.3: Voltage-to-frequency transfer function of the VCO to AOM on the Si3 and Si6 systems.
The delay of the Si6 modulator is slightly lower in frequency and consequently the maximum
attainable loop bandwidth is lower.
𝑒"#$
𝑒!
𝑋

+
−

𝑍+

Σ

𝑍,

𝑍.

Σ

𝑍-

𝐺"#$

𝐴"#$
𝑌

𝐺!
𝑍.

𝑃"*

𝐺%&'

Σ

𝐴%&'

𝑃!()

Figure 4.4: Topology of the Si6 PDH loop. GC , GP ZT , and GAOM denote the loop filters for thecommon, piezo, and AOM stages. Responses of the actuators are denoted by AP ZT and AAOM .
Plant responses of the cavity filter function and the limit imposed by the photodetector bandwidth
are given by Pcav and PP D .

piezo provides low frequency feedback with the gain rolled off well below its mechanical resonanceof 30 kHz while the VCO addressing the AOM is highpassed with a 50 Hz corner. Common stage
integrators both have corners at 10 kHz below; laser noise is suppressed well below thermal noise(Fig.4.5), but this is largely due to the fact that the native laser noise is already quite low. Such ascheme would not work for locking a free running ECDL directly to a narrow linewidth cavity, evenwith access to high bandwidth current modulation on the diode. Additionally, this works nicelyon the Si6 system but the limited feedback at high frequency can occasionally make it difficult towork on the optical table with the laser locked to cavity (high frequency mechanical perturbations
occur in a region of low loop gain).

63

10 6

Input referred noise
1k Johnson noise

10 33

Vrms/ HzPSD [1/Hz]

10 7
10 8
10 9 1
10

PDH servo inloop
Si6 projected thermal noise

10 34
10 35
10 36

100

10 37
10 1

102

101
Freq. [Hz]

100

102

101
Frequency [Hz]Figure 4.5: (a) The input amplification stage of the PDH servo board uses a low-noise FET with
only√40dB of gain at low frequency. This pushes the input referred noise of the board to the 10
nV/ Hz regime in the low frequency regime. Combined with a large discrimination slope achieved
with the typical operation of a few microwatts incident on the cavity (b), the servo noise can be
pushed to a frequency noise limit that is insignificant compared to the expected Si6 thermal noise. A
large input referred noise can always be alleviated by using a corresponding steeper discrimination,
but comes at the cost of increased sensitivity to photothermal effects.

100

Data
Model

100
Phase [deg.]

Gain [dB]

50
0
50
101

102

103
Freq. [Hz]

104

105

0
100
101

Data
Model

102

103
Freq. [Hz]

104

105

Figure 4.6: (a) The closed loop actuator crossover illustrates the relative loop shaping between
the two actuators. The phase measurement is informative in identifying the first, and higher ordermechanical resonances of the piezo.

Nonetheless, it is easier to focus on low noise performance if the bandwidth of the servo
does not need to be high. At the input of the servo, noise from the initial amplification stage is
indistinguishable from laser noise, so the board noise needs to be sufficiently low as to not write thisback onto the laser at a significant level. This input referred noise (Fig.4.5 (left)) is easily designed
to be low since we don’t require much amplification in the initial stages of the servo. Generally
the loop should be able to suppress noise arising from any later stages or actuators. One usefuldiagnostic tool was to terminate each amplification stage with a 10 kΩ resistor and measure the

64

250

Data
Model

100
Phase [deg.]

Gain [dB]

200

Data

150
100
50
0
101

102

103
Freq. [Hz]

104

105

0
100
101

102

103
Freq. [Hz]

104

105

Figure 4.7: Summing an excitation into the closed loop yields the complete transfer function
with all plant responses included. In the phase domain, this is particularly useful for identifying
plant response poles in the high frequency. Modeling the closed loop phase reveals an additional
measured pole at 50 kHz which is identified as the PDH photodetector response. This can then be
appropriately compensated by adding a zero in the loop. The 2π phase wrap from 10 - 2000 Hz
identifies the four integrator stages implemented with corners throughout that band.

coherence between the input an output referred noise. No coherence indicates that the amplifier’s
noise surpasses that of the resistors Johnson noise.
Deliberately driving excitations in particular locations of the board can gives access to particular properties of the loop. For example, consider an excitation eP ZT summed in at the beginningof the subloop actuating solely on the piezo. Measuring the transfer function Z4 /Z3 across the
input and output of the excitation sum point yields the closed loop transfer function between the

65
two actuators.
Z3 = Gc X − PP D PCav GC Y
Z4 = Z3 + eP ZTY = Z4 GP ZT PP ZT + Z3 GAOM AAOM
= (Z3 + eP ZT )GP ZT AP ZT + Z3 GAOM AAOM
= Z3 (GP ZT AP ZT + GAOM AAOM ) + eP ZT GP ZT AP ZT

Z3 = Gc X − Z3 (PP D PCav GC )(GP ZT AP ZT + GAOM AAOM )+ eP ZT (PP D PCav GC )GP ZT AP ZT≡ Gc X − Z3 β(GP ZT AP ZT + GAOM AAOM ) + eP ZT βGP ZT AP ZT

Z4
eP ZT
Gc X + eP ZT GAOM AAOM β
=1+
=
Z3
Z3
Gc X − eP ZT GP ZT AP ZT β

For large gain eP ZT GP ZT AP ZT β >> Gc X (where we let β = PP D PCav GC ), so to first order
we find
Z4
GAOM AAOM=−Z3GP ZT AP ZT(4.1)

Note the inclusion of AAOM and AP ZT , indicating that this measurement includes the actuator
response. Without any knowledge of the actuator gains, this allows an empirical method of loop
tuning. The measured loop crossover on Si6 is shown in Fig.4.6.
Similarly, measuring the transfer function across an excitation summed into the servo input
returns the entire closed loop transfer function with all plant responses included. Suppose we sum
an excitation eC and measure the transfer function Z1 /Z2 . All servo stages and actuator responses
can be lumped together as GT for brevity. Then:66

Z2 = Z1 + eC
Z1 = X − PP D Pcav Y = X − eC PP D Pcav GT − Z1 PP D Pcav GTZ1
X − PP D Pcav GT
=
Z2
X +1
≈ −PP D Pcav GT
Measurement of this transfer function yields information about all plant responses, which canbe particularly useful if some are unknown. A measurement of the closed loop transfer function of
the Si6 PDH servo loop is shown in Fig.4.7.

4.1.0.1

Shot noise limit

In the absence of technical noise, infinite gain at DC noes not imply that that the laser
linewidth will be arbitrarily small. There are a number of mechanisms where the servo writes noise
onto the laser, e.g. a noisy photodetector will set a limit on frequency noise measurement at somelevel. Generally the solution is simply to increase the optical power collected from the cavity sinceeach additional photon carries more length information. As such, it is instructive to consider how
much information a single photon will carry, especially since a number of measurements made lateron had to take this into consideration.
For light incident on the cavity with modulation sidebands of power Ps each and carrier Pc ,the photon shotnoise on the detector will beS=phν(2Ps + ϵPc )B

In a bandwidth B with cavity coupling 1 − ϵ and optical frequency ν. From earlier, we see that an
approximation of the discriminator is given by√
8 Ps Pc
D=(1 − ϵ)νc

67
In the limit 2Ps ≫ ϵPc , fractional length resolution per bandwidth is
√
S
hνc
=√
DB
32νPc

(4.2)

In a 1 Hz bandwidth with a maximized discriminator (no loss due to coupling), collecting 1 nW of
power in the carrier over a 1 Hz bandwidth resolves a fractional frequency of 9 × 10−17 . A single
photon over the same bandwidth is enough to resolve the cavity linewidth to 1 part in 5. The noise
equivalent power of most photodetectors often necessitates significantly more optical power for the
shotnoise limit to surpass the technical floor of the detector, but this is all to say that there is anincredible amount of information contained in the collection of each photon. Given the few µW’s
of power typically used in these systems, shot noise limited resolution is very difficult to attain andhaving this as a limitation is typically an indicator of a separate problem. White frequency noisethat is identified in the cavity noise spectrum is likely to be technical in origin.
An alternative method for determining requisite optical power is to push electronic noise to
the thermal noise level. If some voltage noise V0 is measured in the detection chain, the discriminatorslope is chosen to be large enough such that this becomes insignificant.
V0
V0=< SCT N
DV
RGD

(4.3)

For detector responsivity R and transimpedance gain G.

4.2

Residual amplitude modulation
The cavity can be designed to support a pristine optical phase, but transferring its stability to

the laser is often a significant hurdle for state of the art systems. Residual amplitude modulation(RAM) encompasses a variety of parasitic effects that prevent the experimenter from correctly
tracking the phase of the cavity’s optical field. In short, all of these effects introduce amplitude
modulation at the frequency band where the cavity information is contained. We’ve previously
seen that a strength of the PDH technique is to encode the cavity information high in the RF
where technical noise sources are less dominant, thus the residual AM at the modulation frequency

Fractional length resolution(1Hz bandwidth)

68
Si3
Si6

10 16

10 17
100

101
102
Detected power [nW]103

Figure 4.8: Calculation of the fundamental shot noise limited length resolution on Si6. We accountfor the imperfect cavity coupling. Dashed lines indicated the thermal noise limit of different cavities.

is generally independent of the laser’s spectral characteristics. Unfortunately this same methodof shifting the PDH signal to a heterodyne beat opens one up to issues where the light used as alocal oscillator does not provide a clean reference signal. Dispersive effects between the carrier andfrequency offset sidebands generate unwanted amplitude modulation (the RAM signal) which on
cavity resonance, are indistinguishable from the cavity error signal. RAM is the all encompassing
term that captures the issues of having an imperfect phase reference. We’ll now discuss mitigationof these effects.
It’s important to note that there are approaches other than PDH that have been developed
to eliminate some sources of RAM that are common in PDH. A crucial component of the PDH
method is that the modulation sidebands are antiphase, allowing the baseline off-resonant signal
to in principle be centered at zero. Alternative methods of tilt-locking [105] and squash locking[106] have explored using a phase reference that is spatially distinct from the resonant cavity mode,
rather than spectrally distinct as found in PDH. In these schemes, the phase reference is a higher
order nonresonant HG mode, where lobes of the spatial mode contain relative phase differences of π.
Interference between the HG00 mode and the HG01 mode is detected on a quadrant photodetector
and the anti-phase lobes of the phase reference reproduces the zero baseline signal when off resonant
from the cavity. The tradeoff here is that the RAM and cavity signals are encoded at baseband

69
and both are now highly sensitive to optical scatter and alignment drifts. Shot noise limited RAM
detection in a homodyne configuration is very difficult, so we opt for the traditional PDH detection
to lock to the cavity.

4.2.1

RAM from the modulator

The phase modulation device which generates the sidebands is often a major source of RAM.
There is an irony that the modulator allowing us to enjoy heterodyne detection is itself a sourceof parasitic effects, but there are now several developed techniques for dealing with this. A brief
overview of various electro-optic modulation devices can be found in [107] and they can typically begrouped into free space and fiber based components. A LiNbO3 optical waveguide modulator is used
for Si6, where we make use of the low Vπ drive voltage but do not benefit from the spatially separated
polarization components particular to tetragonal 4̄2m crystals like dihydrogen phosphates.
In trigonal crystals of the 3m class (of which LiNbO3 is included), the refractive indices no ,nefor the extraordinary and ordinary waves differ due to different electro-optic coefficients r13 , r33 forthe two polarizations [65]. Taylor expansion for the field dependence of no ,ne returns

1
no (E) = no − n3o r13 E2
1 3
ne (E) = ne − ne r33 E
2

(4.4)
(4.5)

We can see that the refractive index will have a DC value, and an AC term oscillating at theapplied electric field. For LiNbO3 , one finds r13 = 8.6 ppm/V and r33 = 30.8 ppm/V [108] so there
will be significant differences between no (E) and ne (E) in the AC term. The corresponding phaseshift for the two axes differs by a DC phase shift ϕo,e arising from the natural birefringence of thecrystal, and an AC phase shift with the modulation index δo,e for the two polarizations. This is70
expressed as
Φo = ϕo + δo sin Ωt(4.6)

Φe = ϕe + δe sin Ωt(4.7)

In the rotating frame of the laser, the electric field at the crystal output will be
Eout ∝ co exp[i(ϕo + δo sin Ωt)] + ce exp[i(ϕe + δe sin Ωt)]

(4.8)

Keeping terms first order in modulation frequency Ω and using ∆ϕ = ϕe − ϕo , the detected opticalpower will be
Pout ∝ c2e + c2o + 2co ce∞
X

Jk (δe − δo ) cos(∆ϕ + kΩt)

(4.9)

k=−∞

This can be separated into three signals. For k = 0:
P0 = c2e + c2o + 2co ce J0 (δe − δo ) cos(∆ϕ)(4.10)For k of even order, optical power at kΩ is :Pk−even = 2co ce Jk (δe − δo )[cos(∆ϕ + kΩt) + cos(∆ϕ − kΩt)]= 4co ce Jk (δe − δo ) cos ∆ϕ cos kΩt(4.11)(4.12)

Similarly for k of odd order:
Pk−odd = 4co ce Jk (δe − δo ) sin ∆ϕ sin kΩt(4.13)

Phase shifts ϕo,e can be time varying, as well as prefactors co,e which are identified in [109]to be alignment factors between the crystal and an output polarizer. Changes in the ambientenvironment can easily drive rotations of these axes, thus the power oscillating at DC generally
does not share the same noise spectrum as power at the AC components (despite the shared terms).
Though active and passive stabilization of the crystal can help decouple it from the environment, it
is typically not enough to enable sub-ppm control of Pk that is required for the current generationof ultrastable lasers.

71
However, power oscillating at the modulation frequency Ω contains a term sinusoidal in the
differential phase shift δϕ. For δϕ = 0, the RAM signal is effectively suppressed, and this condition
can be actively forced by feeding a DC bias voltage to the EOM. The net effect is to introduce
another DC phase shift ϕDC that modifies ∆ϕ → ∆ϕ + ϕDC . Power at Pk−odd is monitored andthe DC bias is chosen such that ϕDC = −∆ϕ and
Pk−odd ∝ sin(∆ϕ + ϕDC ) = 0(4.14)In this manner, amplitude modulation due to the crystal birefringence can be actively suppressed
below the ppm level. Typically this is enacted by sampling part of the optical field on a photodetector before the light sees the cavity resonance. As long as there are no polarization sensitiveelements between the detector pickoff and cavity, differential RAM due to crystal birefringenceshould be insignificant.
This is a powerful technique for actively suppressing amplitude modulation present around
an offset frequency of Ω [110], but only occurs up to the point of detection before the beam interacts
with the cavity. Any signal after cavity reflection is convoluted by it’s transfer function R(ω). The
weakness of this technique is that feedback cannot easily occur at the same point where the PDH
signal is detected1 .
The location of RAM detection is critical since there is invariably a differential signal thatwill be measured at the PDH detector. Fig.4.9 illustrates the RAM contributions that can beaccumulated after each surface. The signals measured on each detector are
ϕRAM = ϕ1 + ϕ2 + ϕ3 + ϕ4 + ϕ5 + ϕ6 + ϕ7 + ϕ8 + ϕ9

(4.15)

ϕPDH = ϕRAM −(ϕ6 + ϕ7 + ϕ8 + ϕ9 )+ϕ10 + ϕ11 + 2ϕ12 + 2ϕ13 + 2ϕ14(4.16)+ϕ15 + ϕ16 + ϕ17 + ϕ18(4.17)

When the RAM servo is engaged, noise canceled up to its photodetector is removed, so the remaining
1

For highly birefringent resonators where the mode splitting is resolved outside the linewidth, the RAM signal
can be measured after cavity reflection the PDH detector by encoding it in a nonresonant orthogonal polarization.
This possibility is an attractive feature for cavities with crystalline coatings, though we will later discuss why this is
not implemented in Si6

𝜙RAM = 𝜙1 + 𝜙2 + 𝜙3 + 𝜙4 + 𝜙4 + 𝜙5 + 𝜙6 + 𝜙7 + 𝜙8 + 𝜙9
𝜙PDH = −( 𝜙6 + 𝜙7 + 𝜙8 + 𝜙9 ) + 𝜙10 + 𝜙11 + 2𝜙12 + 2𝜙13 + 2𝜙14 + 𝜙15 + 𝜙16 + 𝜙17 + 𝜙18
−𝜙RAM
𝜙2

EOM
𝜙1

Polarizer

𝜙3

𝜙4

RAM
PD

𝜙9
𝜙5

𝜙8
𝜙7
𝜙6

Plate splitter

𝜙13

λ/2

AOM

Polarizer

Mirror 1

Collimator

𝜙14

72

−𝜙PDH

Isolator
Lens

NKT
RAM mirror

𝜙10

Faraday rotator

𝜙11

𝜙12

Mirror 2

𝜙15

45° mirror

PDH mirror

𝜙16

𝜙17

𝜙18

PDH
PD

Isolator

𝜙PDH

Lens

Figure 4.9: Diagram of the differential optical phase that can be acquired between the carrier andmodulation sidebands after their generation. Red phases are detected in the RAM and green phases
are differential to the PDH path. RAM and PDH feedback to the phase errors measured on eachphotodetector are applied to the EOM and AOM locations respectively.

phase error accumulated at the PDH detector location is
ϕPDH = −(ϕ6 + ϕ7 + ϕ8 + ϕ9 )+ϕ10 + ϕ11 + 2ϕ12 + 2ϕ13 + 2ϕ14
+ϕ15 + ϕ16 + ϕ17 + ϕ18(4.18)(4.19)

RAM feedback can even have deleterious effects if noise present in the RAM path is not shared in
the PDH path.

4.2.2

AM and PM on the phase reference

We haven’t specified where these additional phases in ϕPDH can manifest as. We can generally
approximate any small amplitude modulation α or phase modulation ϕ present on the sidebandson the incident electric field as

73



Ei = E0 J0 (β)eiωt + α+ J1 (β)ei(ω+Ω)t eiϕ+ − α− J1 (β)ei(ω−Ω)t eiϕ−(4.20)To understand how this affects the detected signal, we don’t need to involve the cavity.
Instead we can treat the laser as being detuned from the cavity resonance so that we only measurea reflection off the input mirror. The detected signal oscillating at Ω is extracted asPΩ = E02 J0 (β)J1 (β) α+ e−iΩt e−iϕ+ − α− eiΩt e−iϕ− + α+ eiΩt eiϕ+ − α− e−iΩt eiϕ−iihh
∝ Re α+ e−iϕ+ − α− eiϕ− cos(Ωt) + Im α+ e−iϕ+ − α− eiϕ− sin(Ωt)(4.21)(4.22)

Demodulating to baseband, we findiihhV ∝ Re α+ e−iϕ+ − α− eiϕ− cos(φ) + Im α+ e−iϕ+ − α− eiϕ− sin(φ)(4.23)The phase φ is between the local oscillator and PDH signal. We can consider the limiting cases
to gain some intuition into how AM and PM affect the RAM signal. When α+ = α− , the AM is
common to both sidebands and only has a second order contribution to the RAM signal by varying
the discrimination slope. An imbalance of the form α+ ̸= α− results in the traditional feature
where the baseline changes to a nonzero value. In the case of PM, ϕ+ = −ϕ− pushes the electricfield phasors away from each other, but the net vector is only perturbed in magnitude to second
order. Conversely ϕ+ = ϕ− moves both field vectors together, resulting in a rotation of the total
electric field Ei . In-phase PM can therefore have a tremendous effect on the RAM signal. Notably,
active compensation of PM cannot be achieved by deliberately applying AM on the sidebands. The
two quadratures are distinct, and the power of the EOM DC control is that a single actuator canbe used to squash RAM originating from both quadratures.
Once the cavity is introduced, AM and PM have distinctly different ways of modifying the
PDH signal. Fig.4.10 illustrates some examples of how unwanted modulation on the sidebands cancorrupt the sensitivity to the cavity error.
In reality, it is rare that parasitic modulation on the PDH sidebands can have such a correlated
effect. It is more typical that we see |ϕ+ | ≠ |ϕ− | and |α+ | ≠ |α− |. Optical scatter and numerous

=0

0.0
normalwith AM
with PM

0.5
8

6

4

2 0
2
Frequency [MHz]

4

6

8

0.5
0.0
0.5
6

4

2
0
2
Frequency [kHz]

4

1.0

8

6

4

2 0
2
Frequency [MHz]

10 4
10 5

normalwith AM
with PM4202Frequency [kHz]

4

4

6

8

normal error signal
error signal with AM
error signal with PM

0.5
0.0
0.5
6

4

2
0
2
Frequency [kHz]10 3|Slope of Error signal||Slope of Error signal|

0.5

1.0

6

10 3

10 6

0.0

1.0

normal error signal
error signal with AM
error signal with PM

Error signal

Error signal

1.0

1.0

normalwith AM
with PM0.5

0.5

Error signal

Error signal

1.0

74

= /3

4

6
normalwith AM
with PM

10 4
10 5
10 6

4

2

0
2
Frequency [kHz]4

Figure 4.10: Top: PDH error signals with Ω = 5 MHz, PM of ϕ+ = ϕi = π/6, and AM of
α+ = 1/α− = 1.5. The Left column displays demodulation phase of to extract the quadraturecomponent, and φ = π/3 is on the Right. Curves are offset horizontally for clarity but notvertically. We can see that PM results in a large offset of the baseline signal that is not removedat φ = 0. For AM this is not the case though nonzero demodulation phase will reveal a large
offset. Bottom: We show the magnitude of the PDH slope in all three cases. Dashed black lines
indicate an offset frequency of half the cavity linewidth. At φ = 0 PM has the surprising featurethat maximum sensitivity to the error signal does not occur at an offset of 0 Hz. Additionally theminimum sensitivity is shifted from the expected location at half the cavity linewidth.

other effects can generate these imbalances, but a highly common source are parasitic etalons within
the optical path. Issues relating to this have been thoroughly studied in [111], so we’ll just provide
a brief overview here.

75
4.2.3

Etalon effects

An etalon is an unintentional cavity supported by two reflecting surfaces, with finesse typicallyranging from 0.1 to 10. The length between these surfaces is never controlled as carefully as that of
the silicon cavity, and as it drifts in frequency, each of the tones in the PDH triplet will experiencea time varying phase offset. For an etalon centered at frequency δ, the field transmitted will havethe form
Et = E0 J0 (β)eiωt 1 − Re (ω − δ)+ J1 (β)ei(ω+Ω)t 1 − Re (ω + Ω − δ)− J1 (β)ei(ω−Ω)t 1 − Re (ω − Ω − δ)This is identical to the field transmitted through Si6, except we now use the etalon’s reflectance function Re (ω − δ) rather than the cavity’s R(ω). Instead of locking a laser to one cavity,
we will effectively be stabilizing to the composite system of Si6 and the etalon. As this is a parasitic
effect, care is taken to avoid having parallel high reflectivity surfaces. Consequently, its finesse istypically poor, meaning the linewidth will be significantly larger than that of the cavity. This ends
up being a significant issue - since its length is not systematically controlled, drifts in the etalon
length (even far from its resonance peak) will cause a continuous differential offset between the
PDH sidebands. As discussed earlier, this will directly manifest as a RAM effect, degrading thecavity stability. Fortunately this effect can be easily identified by observing a small modulation of
the error signal, evident in Fig.4.11. Observing a full modulation of the etalon fringe allows us to
extract its length, and correspondingly identify the optics generating the reflections.
Given the low thermal noise limit in Si6, we are wary of all these effects. Fiber components
are passively isolated from temperature fluctuations and we actively feedback to our EOM voltage
to stabilize the detected RAM. Despite active stabilization of the EOM temperature, we observe
large levels of free running RAM below 1 Hz when the DC voltage servo is not engaged. The
fractional cavity instability climbs to the 10−15 level without any RAM compensation, the servo
is therefore activated with a ∼300 Hz bandwidth to ensure that there is sufficient gain near DC.

76

1.0
Error signal

0.6
0.4
0.2
0.0

cavity transmission
etalon transmission

20

15

10

Error signal

1.0

5 0 5 10Frequency [MHz]15

normal error signal
error signal with AM

0.5
0.0
0.5
1.0

6

4

2 0
2
Frequency [kHz]46

normalwith etalon

0.50
0.25
0.00
0.25
0.50

20

15

10

5
0
5
Frequency [MHz]10 3|Slope of Error signal|

Transmission

0.8

10

15

normalwith AM

10 4
10 5
10 6

4

2

0
2
Frequency [kHz]4

Figure 4.11: Simulating the effect of a 1 MHz linewidth etalon with 20% coupling stationed 3 MHz
away from the cavity resonance. A significant distortion of the error signal occurs, and the etalon
will always decrease the effective discrimination slope. Coupling to the etalon will typically belower than the parts per thousand level if all optics have high quality AR coatings. The parameters
are purposely exagerrated here to demonstrate the etalon effect.

The vast majority of this appears to originate after the EOM - detecting the RAM right after
the light is outcoupled from a fiber indicated similar levels of noise. In the free space path, allmirrors and lenses are superpolished with a 20-10 scratch-dig to inhibit unwanted optical scatter.
We take care to tilt all optical surfaces so that no etalons can be supported. Despite this, we still
identified a large etalon between one of the optical surfaces within or on the cryostation and an
in-fiber component. Details of this characterization can be found in the next chapter. With theservo engaged, we measure the residual RAM effect on the PDH detector in Fig.4.12.
The RAM measured at the PDH detector is consistent with the Si3 level, far from the
projected thermal noise of Si6. Though the noise is controlled at the 1 ppm level, we’re impairedby the relatively large linewidth of Si6. The measured finesse is nearly 60% larger in Si4, meaningthat an equivalent level of RAM relative to the cavity linewidth would reach the 2-3×10−17 range.

77
For a 21 cm cavity with 500,000 finesse, our measured out of loop RAM would be well below theSi6 thermal noise.
We don’t have a solid idea of where the residual signal comes from. RAM control at the low
1×10−7 has been demonstrated [112], but this is successful when the majority of RAM originates incomponents that can be easily altered or moved. At some point we placed a mirror before the input
vacuum optic and reflect the beam off its surface and the measured RAM was lower, suggestingthat the dominant contribution arises from an in-vacuum surface. In the next chapter, we note thatchanging the input beam pointing to couple to the cavity can lead to a substantial improvementin frequency stability. We suspect that one of the vacuum windows or the input cavity substrate
is dirty or scatters light. The prospect of locking to the cavity from the top is very appealing, as
we may be able to avoid these issues. There are however other technical challenges associated withthat scheme.

10 30
10 31
10 32
10 33
10 34 210

10 15

OOL RAM
Dark noise
Si3 thermal noise

Fractional MDev

Fractional PSD (1/Hz)

10 29

OOL RAM
Dark noise
Si3 thermal noise
Si6 thermal noise

10 16

10 17
10 1
Frequency (Hz)

100

100

101
Time (s)102

103

Figure 4.12: Top: Fractional frequency noise of the out of loop RAM characterization. Though
far from the projected Si6 thermal noise, the cavity instability should be limited to a level similar
to the Si3 thermal noise. This would mark a significant improvement over the instability of the Si4
system with the added benefit that the 4 K closed cycle cryostat makes Si6 a hands-free oscillator.
Dark noise corresponds to noise of the resonant photodetector. A high white noise level is visible
at frequencies larger than 1 Hz. This specific dataset coupled to the TEM01 mode of the cavity
without a phase plate. The poor mode coupling leads to a correspondingly higher contribution ofshot noise. Right: Modified Allan deviation of traces on the left.

78

4.3

Vibration sensitivity
The large stiffness of silicon as a cavity material has already become apparent due to its role

in reducing the thermal noise contribution of the spacer. Our intuition supports the concept that
this should be beneficial when the cavity is in the presence of any compressive force. While thecavity is a rigid body, its length will still be modified as
∆L
F
∝
L
EA

(4.24)

for a force F and cross sectional area A. This simple model treats the cavity as a bulk elastic
spring, with spring constant determined by the material stiffness E. With a stiffer spacer, anapplied force induces a smaller fractional length change. This applies to forces along all three axesof the cavity, though the length changes due to horizontal forces are modified by the Poisson’s ratio
σ which relates the applied stress to an orthogonal strain. Generally the sensitivity to vibrations
can vary between the vertical and two horizontal axes. Unlike thermal noise, this does not impose
any fundamental limit but rather eases our task of seismic isolation and reducing those compressive
forces.
We opted to cut the silicon spacer along the stiffer ⟨111⟩ axis, though not for the purpose of
simply reducing the bulk spring constant or reducing spacer thermal noise. Silicon has a diamondlike cubic lattice structure with atoms arranged tetrahedrally. There is a three-fold rotational
symmetry of the stiffness matrix C when oriented along ⟨111⟩ [113]


0
−13.5
0 
 194.4 54.3 44.8
 54.3 194.4 44.8
0
13.5
0 






44.8 204.0000  44.8
 × 109 Pa
C⟨111⟩ = 



 0
0
0
60.5
0
13.5






−13.5 13.5
0
0
60.5
0 




0
0
0
13.5
0
70.0

(4.25)

When the cavity is suspended or supported with similar rotational symmetry, the anisotropic
structure of the crystal means that the vertical sensitivity will vary sinusoidally in magnitude with

79
a 120◦ period. By varying the angle between the cavity’s horizontal axes and the three points
of support, we can tune the magnitude of the vertical sensitivity making this an effective knob
for reducing the cavity’s frequency coupling to accelerations. This phenomena was previously
measured in the first realizations of cryogenic silicon cavities [39] and it is now well establishedthat rotating the it allows us to reduce the vertical sensitivity. The cavity vent holes are drilled
along the ⟨2 − 1 − 1⟩, ⟨−1 − 12⟩, and ⟨−12 − 1⟩ crystal axes, so the approximate angle between
the cavity mount and ⟨2 − 1 − 1⟩ can be easily determined when viewing the system from thetop. Less consideration has been given as to whether this circumstance of reduced sensitivity canoccur for the two horizontal directions. Our FEM simulations suggest that we should expect thetwo horizontal sensitivities to vary sinusoidally with 360◦ period. Our measurements have clearly
indicated a slight rotational dependence of these couplings, but the magnitude appears to depend
strongly on the geometry and material of the support structure.
This can leave the cavity to be more susceptible to horizontal perturbations. Simulating themechanical eigenmodes of Si6 confirms this as the lowest resonances of the cavity are bending modes.
The resonant frequency is still very high however, with the aforementioned resonances occurring
near 27 kHz. It is more likely that weak low frequency resonances in the support structure could
be a limitation, so a proper design of the mount is crucial. Ideally, mechanical resonances of the
combined mount and cavity should still be high, > 100 Hz. The low form factor of Si6 and the 4
K cryostation is fortunate in this regard, since the total system mass cannot reach the levels of the
21 cm Si2 and Si3 systems2 .
With the constraint of needing to mount the cavity at three points, we orient it verticallyunderneath its supporting ring. Our intuition suggests that we should support it at the midplane,
where vertical forces act oppositely on the two spacer halves and lead to a ∆L = 0 change in thecavity length. This has traditionally been used in ULE cavities [114, 115], where vertically oriented
football-shaped spacers are supported at the midplane at three points to reduce overconstraining
2

Si2 and Si3 measured a low lying mechanical resonance at 40 Hz originating from a tilting eigenmode of their
PEEK cavity mountSensitivity kz [g 1]

1

80

1e 10

0
1
0

20

40

60

80

2-1-1 rotation angle

100

120

Figure 4.13: Vertical acceleration sensitivity of the cavity. Si6 is supported at the midplane andwe measure a sensitivity that is roughly centered at zero. We rotate the cavity on its mount to benear one of these zero crossings.

and the vertical acceleration sensitivity. In the crystalline cavity, this has the additional advantage
of manifesting as a zero crossing of the vertical sensitivity, meaning that the cavity length is not
perturbed when subjected to acceleration along the spacer. As the cavity is supported farther awayfrom the midplane, the sensitivity curve is displaced upwards or downwards in magnitude to thepoint where the desired zero crossing will not appear.
This occurred in the Si2 and Si3 systems where the supporting ring on the spacer was intentionally offset from the midplane for the purpose of placing the zero crossing at the shallowest
sections of the sinusoidal sensitivity. Measurements of the sensitivity do not exactly reproduce
simulated results and it was found that no orientation of the cavity resulted in a nulled verticalsensitivity. The mass imbalance between the top and bottom sections of the cavity relative to themounting plane drive the finite acceleration coupling. A silicon ring placed around the top of thecavity spacer revived the zero crossing [39].
The crystalline structure of the spacer has thus been beneficial for factors unrelated to thelow mechanical loss angle. Careful consideration given to the crystal orientation has allowed us toprecisely control some components of the cavity’s response to accelerations. A significant component
however is also due to the geometry of the cavity mount. This has been extensively examined in
ULE systems [107, 116, 117, 118, 119], but it has been less studied in crystalline systems where we81
are constrained to the tripod. We will next discuss some of our related findings.

4.3.1

Measuring cavity sensitivity

To measure the cavity’s response to vibrations, we want to provide an acceleration to thecavity and measure the frequency response. Along the two horizontal axes, we can easily perform
a DC measurement of the sensitivity k by tilting the optical table along the two directions H1
and H2. This introduces an acceleration due to gravity as g sin θ for tilt angle θ. The resulting
frequency shift ∆f returns the fractional sensitivity ask=∆f 1
f g sin θ(4.26)

There is no trivial analogue to performing a DC measurement of the sensitivity along the Z axis
since tilting the table barely modifies the vertical projection of gravity g → g cos θ ≈ g.
An easy solution is to modulate a force along all three directions, and measure the coherent
cavity frequency response to obtain kH1 , kH2 , kV . Sweeping this modulation frequency, we can map
out the spectrum of the cavity’s AC acceleration sensitivity. The active vibration isolation platform
supporting the cryostation has the option for modulating the table displacement along all threecartesian axes. Driving the AVI along a particular direction can drive accelerations along all threeaxes due to cross couplings of the platform.
Previously we’ve suppressed these complications by identifying a drive frequency where thesecross couplings are minimized and measuring the cavity response. This has the benefit of being
procedurally simple but does not provide access to the full frequency dependence of kH1 , kH2 , kV .
If we instead modulate at an arbitrary frequency regardless of cross couplings and measure theacceleration response aH1 , aH2 , aV along all three directions (3×1 matrix A, the frequency response
will be∆fi=fX

ai eiϕj kj(4.27)

j=H1,H2,V

Each term has an associated phase that will be frequency dependent. We cannot identify any
individual components with this single measurement, but if we drive the AVI along the three82
orthogonal directions we can construct the 3×3 matrix of measured sensitivity and phase responses
k. The 3 × 1 diagonalized vector of acceleration sensitivity kD is calculated as∆fH1 /f kD = P −1 A−1 ∆fH2 /f  P∆fV /f(4.28)

By finding the invertible matrix P .
Performing lock-in detection of the cavity with our modulation frequency allows us to easily
make this measurement over a broad band of frequencies by simultaneously shifting the drive anddemodulation. This requires an analog signal corresponding to the cavity frequency. We phaselock a VCO to the Si3-Si6 beatnote with high bandwidth such that the servo in-loop is well below
the frequency response we aim to resolve. The control signal sent to the VCO is then used as ourmeasure of the cavity noise. A standard set of data is illustrated in Fig.4.14 where we measure thephase and transfer function of our accelerometer signal to the VCO.
It is difficult to extract useful information on the magnitude of the acceleration sensitivity from this plot, but performing the diagonalization of Eq.4.28 returns the significantly moremeaningful kH1 , kH2 , kV (Fig.4.15). Notably, there is a strong frequency dependence of all three
sensitivities. Their magnitudes are nearly ten times larger above 100 Hz, and even in the range of
10-60 Hz they can fluctuate by a factor of 3 or 4. We don’t have a good intuition why this occurs.
Near a mechanical resonance, we expect a peak in the sensitivity but this should monotonicallydecrease away from the resonant frequency with a width related to its quality factor. Montana
instruments has shared that the 300 K to 30 K support structure is largely made from G10, which
could result in a broader resonance. Nevertheless, we have confidence in the AC acceleration sensitivity data since in the low frequency limit our measurements of kH1 and kH2 approach the values
measured with our DC tilt results (Tab.4.1). We naturally expect this to occur, so for the purposesof calculating the contributions of vibrations to frequency noise, we assume that the AC sensitivities
measured at 10 Hz retain the same value down to DC.
These sensitivities can be multiplied by our measured vibration noise along each axis to20

Drive in H1
H1-H1
H1-H2
H1-Z

30
50
60

H1-H1
H1-H2
H1-Z

150
100
Phase [deg.]

Gain [dB]

40

70

50
0
50
100

80
101

150
Freq. [Hz]

101

102

Freq. [Hz]Drive in H2
150
100
Phase [deg.]

Gain [dB]

30
40
50

Freq. [Hz]

0
50

101

Freq. [Hz]Drive in Z

102

Drive in Z
Z-H1
Z-H2
Z-Z

150
100
Phase [deg.]

Gain [dB]

50

150
102

30
40
50

Z-H1
Z-H2
Z-Z

50
0
50
100

60
70
101

H2-H1
H2-H2
H2-Z

100

60

20

102

Drive in H2
H2-H1
H2-H2
H2-Z

20

70
101

83

Drive in H1

150
Freq. [Hz]

102

101

Freq. [Hz]102

Figure 4.14: Transfer functions (left) and phase responses (right) of the three acceleration responses to frequency when driving the AVI along a particular axis. Different rows display thedata of shaking the AVI along each direction. The acceleration response to driving the platform
along gravity (bottom row) is largely contained along V̂ . This is less the case for driving along
the horizontal axes where we can see a few cross couplings at low frequency. This likely suggeststhat the platform is torqued in the H1-H2 plane, shaking along the AVI’s horizontal axes generates
an admixture of excitation along H1 and H2. In the phase domain, we can identify some sort ofmechanical resonance of the cryostation near 100 Hz, where phase wrappings are evident in allaxes.

obtain the vibration induced frequency noise on the cavity. The cavity frequency will be unable todistinguish between noise from orthogonal components, so we calculate the total contribution to84
kV [g −1 ]kH1 [g −1 ]kH2 [g −1 ]DC measurement

AC measurement0.8(3) × 10−11
1.2(6) × 10−11

1.0 × 10−11
0.7 × 10−11
2.2 × 10−11

Table 4.1: Comparison of low frequency AC sensitivities and DC sensitivities.

Fractional frequency
sensitivity (g 1)

10 9

kH1
kH2
kZ

10 10
10 11
10 12 110Frequency (Hz)102

Figure 4.15: The frequency dependence of measured acceleration sensitivities on Si6. The verticalsensitivity is notably higher than the mid-10−12 s value that can be attained by adjusting thespacer rotation. We observed that the values of the horizontal sensitivity had a slight dependence
on rotation angle as well. The current orientation of the spacer was chosen to minimize the largest
sensitivity (in this case kH1 ) without significantly increasing the others.

its fractional noise as
Sy = (aV kV )2 + (aH1 kH1 )2 + (aH2 kH2 )2(4.29)

Note that we are not considering the relative phases between these terms, so Eq.4.29 is only an
upper limit on the true vibration-induced frequency noise. Our projection depicted in Fig.4.16
suggests that the noise contribution due to vibrations should not limit measurement of Brownian
thermal noise at frequencies below 1 Hz.

4.3.2

Improving the sensitivity

The acceleration sensitivities shown in Tab.4.1 and Fig.4.14 were the result of many modifications to the Si6 cavity support, most through trial and error. Initial measurements of the vibration

Vibration-induced fractionalfrequency noise [1/Hz]

10 30
10 31
10 32

85
V
H1
H2
Total
Thermal noise

10 33
10 34
10 35

101

Frequency (Hz)102

Figure 4.16: Vibration induced frequency noise measured along the different axes and their respective couplings to the cavity.

coupling revealed sensitivities ten times worse, identical to the values measured in Si4. We’ll review
some of the more impactful upgrades here.
It’s useful to introduce some historical context - in Si4 the cavity-support rotation angle
was optimized to minimize the vertical sensitivity (kV = 5 × 10−12 g−1 ), but large horizontal
sensitivities of kH1 ∼ kH2 = 2×10−10 g−1 were significant limitations. There was no clear indication
why these values might be so poor. One of the cavity substrates was contacted off-center to thespacer borehole, and a horizontal displacement of the beam from the cavity axis could beget a
4 × 10−11 /g/mm sensitivity. Standard machining tolerances of 100 µm could easily lead to adisplaced substrate curvature, but we would expect a residual sensitivity of 4 × 10−12 /g. The 50
times larger value measured on kH1 and kH2 would be unphysical. This claim also lost its attraction
when we swapped the Si4 cavity for the Si6 cavity and remeasured the exact same sensitivity curves.
It became clear that the cavity itself was not the primary encumbrance.
Retaining the same level of vibration-induced noise on Si6 would have prevented a thermal
noise limited laser performance at 1 Hz, so it was imperative that we understand the source of thishigh value. A number of studies have investigated the design of the cavity spacer with the intent of
reducing the sensitivity. With the Si6 spacer already machined and contacted to substrates, this is

86

Figure 4.17: Various cavity mounts that we tested for improved sensitivity to acceleration noise.
Top left: The original folded cup design from Si4, made from G10 composite. Top right: A
simplified structure made from G10. Bottom left: A monolithic stainless steel support machineda block of 316 stainless. Hemispherical divots have been milled out from the top of each post.
Spheres of different material can be placed within each divot, the image shows a set of teflon balls
resting within the indentations. Bottom right: Our final cavity support made entirely fromstainless steel

not an option. Instead, we suspect there may be gains from modifying the support structure. The
original Si4 cavity support was composed of three thin, nested concentric cups of G10 fiberglass
glued together. Three tiny 1 mm tall flat rods equally spaced on the top of the outer cup support thecavity. The intent was to maximize the effective path for heat flow, lengthening the thermal time
constant. Unfortunately, G10 has a relatively low bulk modulus of 18 GPa that is characteristic ofmost laminates. The low mechanical quality factor of the material combined with its three times

87larger effective length transforms the cup into a lossy mount with a low spring constant.
Our first avenue for improvement was to increase the structural spring constant by substituting the nested layer design for a hollowed out half-cylinder. The cavity points of contact needto maintain a 120◦ symmetry for zeroing the vertical sensitivity, so three longer posts tapered toa point are mounted to the top of the cup. This is hard to visualize so a photo of the structure
installed in the cryostation is depicted in the top right of Fig.4.17. Despite the green color, the support is still made of G10, but the improved design has pushed the mechanical resonant frequencies
much higher. Remeasuring the sensitivity spectrum showed a dramatic improvement, with both
kH1 , kH2 < 1 × 10−10 below 100 Hz.
Clearly the design of the support structure was of significant importance for the acceleration
sensitivities. We were largely concerned with reducing the horizontal sensitivities since we knowthat we can tune kV close to a minimum. Since increasing the mount’s spring constant had aconsiderable impact on the values of kH1 and kH2 , it stands to reason that it could be helpful tostiffen the support even further. We decided to test out an analogous design made from stainlesssteel instead of G10. The larger Young’s modulus of 200 GPa allows us some leeway with thegeometrical details, we opted to increase the post length and minimize the size of the surrounding
cup to increase the thermal time constant.
Si2 and Si3 have had success with resting the cavity on PEEK supports which is less structurally stiff than G10, so we were curious whether the material in contact with the silicon spacer
had a significant effect on the measured sensitivity. In our first design of the stainless mount, we
cut a hemispherical divot into the top of each post so that we could place small spheres of different
material in here, rest the cavity on top, and measure its coupling to accelerations. In principle
this should provide us with a methodical method for studying the importance of this contacting
material. In reality, this process of exchanging the sphere material, replacing the cavity, and remeasuring the sensitivity is extremely tedious. We only ended up performing this test for teflon,
nylon, and stainless steel spheres. We surprisingly observed a small dependence of kH1 , kH2 on
the tip material with teflon balls providing the smallest sensitivity and nylon balls returning the88largest. This ran counter to our recently developed intuition that stiffer materials lead to a lower
coupling.
The stainless steel component of this mount was monolithic and machined from a single
block of SS316 with a computer controlled CNC. The cutting process lead to some warping of thestainless posts and we worried that this may have some adverse effect on the sensitivity. We opted
to build a second stainless mount (posts, tips, and base all made from SS316) by rounding thetips of rolled stainless rods and pressing them into a machined baseplate (lower right of Fig.4.17).
Stresses introduced in the cutting and assembly process are much lower, and undesired tilts of thestainless posts can be controlled with much better precision. Remeasuring the sensitivity with thisnew mount revealed that we could attain much lower values with kH1 , kH2 ∼ 1 × 10−11 . This isan over 20 times improvement in sensitivity over the kH1 , kH2 ∼ 1 × 10−10 measured on Si4. The
corresponding frequency noise contribution from vibrations will be over 20 dB lower than measured
on the previous system. This is more than sufficient for the stricter requirements from the Si6
projected thermal noise so we opt to use this support for all measurements going forward (Fig.4.15
is the sensitivity spectrum measured after cooling the cavity to 4 K). All our intuition here isempirical but the takeaway seems to be that stiffer materials and a properly constructed mountare a path towards a lower vibrational sensitivity.

4.3.3

Feedback to AVIReducing the cavity sensitivities proved to be a fruitful method, but further reduction has
proved to be difficult. There is no quantitative barrier preventing this, historically it is rare thatcavities attain ki < 1 × 10−11 g−1 . If the total vibrations are prohibitive for measuring thermal
noise, trying to reduce the acceleration noise can be a better avenue.
The necessary approach depends on the exact spectrum of vibration noise. In our system,
aV , aH1 , aH2 are roughly the same magnitude, but kH2 is the largest sensitivity. The frequency
noise contribution aH2 kH2 is therefore larger than the other two terms, so reducing aH2 should
lead to a corresponding decrease in the vibration-induced frequency noise. Reducing aH2 means

Fractional Freq Sens [1/g]10 10
Fully S316 mount
Divot - SS balls
Divot - Teflon balls
Divot - Nylon balls
New G10 mount

10 11
10 12 1
10

Freq. [Hz]Fractional Freq Sens [1/g]

102

kH1

10 9
10 10
10 11
10 12 110
10 9Fractional Freq Sens [1/g]

89

kz

10 9

Fully S316 mount
Divot - SS balls
Divot - Teflon balls
Divot - Nylon balls
New G10 mount

Freq. [Hz]

102

kH2

10 10
Fully S316 mount
Divot - SS balls
Divot - Teflon balls
Divot - Nylon balls
New G10 mount

10 11
10 12 1
10

Freq. [Hz]102

Figure 4.18: Measured sensitivities for the variety of cavity mounts we considered. Label ”NewG10 mount” refers to the green G10 cup from Fig.4.17, ”Fully S316” refers to the stainless steel
mount with rounded tips, and ”Divot” is the stainless mount with a hemisphere milled out fromthe top of the supporting posts. We place stainless steel, nylon, and teflon spheres in the divots
and rest the cavity on top. All traces have been smoothed with a 5th-order spline to improvevisual clarity. Horizontal sensitivities are the more meaningful quantity since we know that kV canbe zeroed by adjusting the cavity-mount angle. Variation of kV between the different mounts isdue to imperfect placement of the cavity. All ki steadily increase above 100 Hz, indicating thatsome other component of the cryostation is contributing to an increased acceleration sensitivity athigh frequency. A large rollup in the kH1 is visible for all the different cavity supports. At thetime of these measurements, we were suspected that this was not representative of the true H1sensitivity but were unsure of its source. We later understood that driving the AVI along the H1axis (collinear to the coldfinger axis) would excite a compressive mode along the coldhead’s bellows.
This generates air currents that modulate the laser phase coherently with the cavity’s acceleration
response and appear as an artificial kH1 signal. We later remove this effect by pulling a vacuumon the cryostation, and its absence is noted in Fig.4.15.

90that we need to do a better job of noise feedback than the AVI. The current limitation is the noise
limit of the in-board sensors, meaning that a lower noise accelerometer would enable improved
feedback. Similar to how we would excite the AVI by driving it with a synthesizer, we can perform
broadband noise cancellation by instead feeding it with a servo output’s control signal.
In Fig.4.19, we place a high-sensitivity accelerometer on the cryostation split plate and use
the measurement as an error signal to a servo. The servo output modulates displacement along
the H1 axis, and a second accelerometer near the same location serves as an out of loop measurefor the reduction in vibration noise. We measure a clear reduction in measured acceleration noise,
though we again are limited by the noise floor of our improved detectors and receive a 3dB penalty
in terms of unwanted noise added by the servo.
The principle of this method is simple, but in this test we neglected to account for the
feedback range of the AVI. The platform’s vibration isolation is highpassed with a 2 Hz corneras it is difficult to distinguish between tilt and displacement below this frequency. Extending the
feedback to lower frequency would require a tiltmeter on the optics table to measure the slowly
varying inclination of the cryostation.
This demonstration also assume that feedback to the H1 axis only couples to the vibrationnoise measured along H1. In reality, we previously identified cross couplings between drive andresponse along the cryostation axes. To truly minimize acceleration noise along H1, it is necessary
to apply a control signal to H2 and V as well. The matrix delineating all couplings is our previously
measured as A. Feedforward on the vibration noise requires knowledge of both real and imaginary
components of A, as well as the frequency dependence of all ki . A feedback scheme will be significantly simpler since correction of accelerations measured on all axes can be propagated through∥A∥.
The frequency dependence of all nine terms is illustrated in Fig.4.20. Driving the AVI alonga particular axis clearly couples to the other two, but generally with a small magnitude. If we’re
not trying to reduce the acceleration noise by a significant amount, it is actually appropriate to
only perform single axis feedback, e.g. drive in H1 to reduce noise along H1, drive in H2 to reduce

91
noise along H2, etc. In H2 for example, cross couplings between H1 and V are 10 dB smaller thanthe H2-H2 coupling. If we only need a factor of two reduction in the amplitude spectrum of aH2 ,this can be done without any propagation through A. Vibration noise on Si6 is low enough that
we don’t need to try any of these schemes, but future cavities with a projected thermal noise below
the 10−17 level will likely have to consider this.

g/ Hz

10 6

H1 AVI servo on
H1 additional servo on

10 7

10 8 010101
Freq. [Hz]102

Figure 4.19: Vibrations measured along H1 with the AVI isolation enabled are shown in blue.
When we use a lower noise external accelerometer to measure the acceleration spectrum, we canfeedback to the AVI by modulating its displacement along H1 to further reduce the measured noise(black). At frequencies below 40 Hz, the black curve is limited by the noise floor of our new in-loop
accelerometer.

4.4

Thermo-expansive length stabilization
With the finite ∼ 10−11 K−1 CTE of silicon at 4 K, temperature fluctuations need to belimited to the µK level to enable the thermal noise limited performance of Si6 at ∼ 1 × 10−17 .
Measuring temperature at this level is well below the precision of commercial thermometers. We
instead need to be conscious of temperature couplings in system and try to strategically passively
dampen them by careful design of our cryostation environment. At 4 K, our choice of materials
for the cavity support will become important. Thermal isolation stages can lowpass temperature
variations to the nK level, and this corner frequency will be determined by the thermal resistanceand capacitance of the system. We will now discuss our approach to meeting these requirements,

Normalized cross couplings
Normalized cross couplings
Normalized cross couplings

92

100
10 1
10 2
10 3 110

H1-H1 coupling
H1-H2 coupling
H1-Z coupling

Frequency (Hz)

102

100
10 1
10 2
10 3 1
10

H2-H1 coupling
H2-H2 coupling
H2-Z coupling

Frequency (Hz)

102

100
10 1
10 2
10 3 1
10

Z-H1 coupling
Z-H2 coupling
Z-Z coupling

Frequency (Hz)102

Figure 4.20: Plots of the coherent response measured along each axis when driving the AVI along one
direction. In the top plot
qPfor example, the AVI is driven along H1 and we show the corresponding2
transfer functions ai /i ai . The vast majority of the coherent displacement occurs along H1 at
low frequency, though in regions around 50 Hz and 80 Hz, an equal response is measured in thevertical axis. Single axis feedback along H1 would exhibit strong cross coupling to another axis atthese frequecies.

and begin with a model of the thermal couplings within the Si6 system.

93
4.4.1

Heat flow

In our VHV vacuum, convective heat transfer through gas is minimized. Heat flow canoccur conductively through the cavity support and though radiative exchange between surfaces atdifferent temperature. We’ll consider radiative exchange first.

4.4.1.1

Radiative exchange

Heat exchange between two blackbodies can be calculated via the Stefan-Boltzmann lawq̇ = σB A(To4 − Ti4 )(4.30)For surface area A, Stefan-Boltzmann constant σB , and temperatures To , Ti of the two surfaces.
The description is complicated by the fact that the two surfaces are often not perfect emitters, andare instead ”grey bodies” with emissivity 0 < ϵ < 1. Furthermore, the two surfaces may differ
in their total surface area, e.g. a large vacuum chamber around a smaller thermal shield. The”interaction” between these two bodies cannot be neglected, and it is necessary to calculate theform factor Fo→i that determines the fraction of energy radiation from surface o and received bysurface i [120]. This will be unique to the geometry of the two surfaces. For concentric cylinders
with an inner surface i and outer surface o, one finds
Fo→i = Ao

1
1
1
+
−
Ao ϵi Ai ϵo Ao
(4.31)For this configuration it’s clear that Fo→i ̸= Fi→o so that the corresponding radiation energies
Qo→i ̸= Qi→o . This is due to the ability of the larger concave surface o to radiate back onto itself.
Thus the total shape factor Fo→i + Fo→o + Fi→o = 1. The radiative exchange between grey bodies
is then found asq̇ = σB111+−
Ao ϵo Ai ϵi Ao−1(To4 − Ti4 )(4.32)

Low emissivities and similar temperatures between the two surfaces will minimize the heat
flow. The majority of radiative exchange in our systems occurs between thermal shields due to94
their large surface areas. We use three thermal shields to step down the radiative coupling andmaintaining the two active and passive shields at 4 K suppresses any heat radiative heat flow
between them and the cavity. As we’ll elaborate in a later section, we gold plate all shields toreduce the emissivity and lower the heat flow. The majority of heat exchange occurs between theroom temperature vacuum chamber and the radiation shield due to the large (300 K)4 heat load.
The stage 1 temperature of the cryocooler consequently needs to have a sufficiently large cooling
capacity. Room temperature fluctuations can easily pull the temperature of the radiation shield,
so we heat it to 42-45 K with a 50 Ω manganin heater wrapped around the base3 . The diurnal lab
temperature fluctuations then have a suppressed coupling to the 30 K stage.
The heater has enough capacity to maintain the shield temperature above its steady statevalue, but this doesn’t relax our need to keep this stage cold. A hotter shield temperature of 60 K
will have a five times larger radiative heat exchange with the 4 K active shield which can be large
enough to overwhelm the cryo’s stage 2 cooling capacity. This results in a higher base temperatureof the the system, and the hotter cavity will have a larger CTE, making it more susceptible totemperature fluctuations. In the 21 cm system, multi-layer insulation foil is used between thermal
shields to significantly lower heat flow by introducing intermediate surfaces T 1, T 2, T 3, ..., eachof which have their own inside and outside surface emissivities ϵoi , ϵii . Radiative exchange is thenreduced from ∼ (ϵio To4 − ϵoi Ti4 ) to (ϵio To4 − ϵo1 T14 ) + (ϵi1 T14 − ϵo2 T24 ) + (ϵi2 T24 − ϵo3 T34 ) + .... Insufficientspace between the thermal shields on Si6 prevents us from including this in our system, but this isa powerful method to consider. Proper choice of the ϵoi , ϵii can lead to a significantly lower radiative
heat load.
Lowering the radiative exchange between stages is always beneficial, but it’s useful to think
about this in the context of the overall heat flow, both conductive and radiative. A frequent analogy
is to adapt the language of electrical circuits. The thermal counterpart to Ohm’s law is∆T = QR
3

Despite the cryo stage 1 being at 30 K, the radiation shield attains a lowest temperature of 40 K

(4.33)

95
For heat density q. A thermal resistance R can be used to quantify an impedance to heat flow
through the system. If there are two paths for heat to flow (conductive and radiative exchange),preference is give to whichever has the lower resistance. The effective resistance across both paths
will be
Ref f =

1
Rradiative

+

1

−1
(4.34)

Rconductive

So if Rradiative ≫ Rconductive , there is little to be gained by further suppressing radiative exchange.
Radiative resistance between grey bodies isRradiative =

1
1
1
+
+
ϵi Ai ϵo Ao Ao



−1

σB ∗ (Ti2 + To2 )(Ti + To )(4.35)In Si6, radiative resistance between all surfaces is considerably higher than the resistance throughthe conductive path. To good approximation, heat flow can be understood by only considering the
conductive exchange.

Figure 4.21: Schematic of the thermal circuit that models heat transfer between the 30 K stage
and the cavity. Radiative and conductive pathways are placed in parallel, and the low pass filter
is completed with the inclusion of materials with high thermal capacitance. An additional lowpass
should be included to capture the heat from the 300 K vacuum chamber to the cryogenic 30 K stage,
but details this support structure within the Montana Instruments cryostation are proprietary, andare not shared with us. We calculate an active-passive time constant of ∼10000 s and a passivecavity time constant of 600 s. Note that the passive-cavity time constant is reduced from the 1200s modeled and measured in Si4. The G10 cavity support in Si4 is replaced with the aforementioned
stainless steel mount which has a two times lower thermal resistivity. Unlike the 124 K system,
the thermal capacitance of silicon is very small at 4 K and results in a low thermal time constant
between the baseplate of the passive shield and the cavity mirrors.

96
4.4.1.2

Conductive exchange

Thermal isolation between the various stages with conductive heat flow is modeled as a seriesof low pass filters with a corner frequency determined by the support geometry and material thermal
properties. A rod for instance will have a thermal resistance calculated analogously to the electricalresistance. Rather than use the resistivity ρ, thermal conductivity k yields a thermal resistance of
R = L/(kA) for length L and cross sectional area A. Thermal capacitance is simply the product ofa material’s heat capacity Cp and it’s total volume V (often expressed as ρ · m). A long thermal
constant will maximize the product RC. With our commercial cryostation, we’re clearly limitedwith the total volume available for packing material into the enclosure. It may appear that a morepractical approach would simply be to prioritize the thermal resistance. In the presence of heatflow q, high resistivity will lead to larger thermal gradients as
q = k∆T(4.36)

Within the cryostation, stage 1 and stage 2 of the cryocooler fix a minimum base temperature, so
structures with large thermal resistance will find themselves hotter. This can be troublesome at 4
K operation where we want the cavity temperature to be as low as possible for reduced thermal
noise and thermal expansion sensitivity. We can gather that mindful selection of our materials will
be important for ensuring good thermal isolation.
Unfortunately, the thermal properties of most materials asymptotically approach zero at low
temperatures. The Debye model, which provides an accurate description of phonon excitations insolids, estimates the low temperature thermal dependence of heat capacity to scale as Cp ∝ T 3 .
This makes it extremely difficult for cryogenic systems to match the thermal time constants foundin room temperature systems. The benefits provided by a reduced Brownian thermal noise can beoffset by poor thermal instability, especially if the the temperature control is not precise enough toenable realization of thermal noise limited system.

97
SS
G10
Nylon
Lead
Brass
Nickel
GdRh
Gd06Er04Rh
Er3Ni
ErNi
ErNi2
ErAl2
ErDy08Ni2
KaptonNd
Er3Ni_N
Er09Yb01Ni
Er3Co
Er06Pr04
HoCu2
ErNi09Co01
Ho2Al
ErAg09Al01
HoSb
DySb
GdSb
Er
Er05Pr05
GAP_GdAlO3
GOS_Gd2O2S

Volumetric Cp (cm J3 * K )

100

10 1

10 2

10 3

2

4

6

8
10
12
Temperature (K)

14

16

18

Figure 4.22: Volumetric heat capacities of materials typically used in a cryogenic environment.
Common materials (dark blue curves) have a vanishingly small heat capacity below 10 K, requiringuse of more exotic alloys.

4.4.1.3

Material properties

So how do we engineer a good thermal isolation? Fortunately the NIST cryogenics division
maintains a database of materials suitable for cryocooler applications, as well as several usefulassociated properties. Fig.4.22 displays the heat capacities for most materials that might placedin a cryogenic environment. At 4 K, a few materials stand out as promising choices. First, we cansee that a group of Gadolinium-based materials reveal a drastic change in Cp close to our desired
operation temperature. Particular rare-earth magnetic materials exhibit an antiferromagnetic phasetransition at low temperatures, and an anomalously large heat capacity will be evident close to thecritical point.
Gadolinium falls in this class of materials, and maintains this property when formed as
the compounds Gd2 O2 S (GOS) and GdAlO3 (GAP). Addition of gadolinium into different mixed

98
oxides shifts the temperature of the phase transition, providing a unique method of tuning theheat capacity. Unfortunately neither GOS nor GAP are metallic alloys, instead they are formed
as compounds that are sintered into ceramics [121]. Most of these materials have their thermal
properties measured while being used as ”regenerator materials”, meaning that the compound will
be formed into particulate spheres with ∼ 100µm diameter and used as a heat exchanger for the
helium gas that flows through it. Synthesis via sintering is nontrivial and requires powders of
the oxide and material to be heated and pressurized with particular granule sizes to form large
structures of the compound. At the time of our interest, no commercial vendors were found that
could produce large samples of GOS. It was additionally unclear whether a large ceramic blockof GOS could be easily machined, we found no literature on this. Most of these exotic materials
may be suitable for use as fine powders in cryogenics, but it seems that there is still significant
engineering required to make them useful for our applications.
HoCu2 is a metallic alloy that is easier to synthesize, and has a has consistently large Cp
over a large temperature range. It is generally useful as a high heat capacity material, but more
importantly is produced by our commercial cryostation vendor Montana Instruments. This has
become our high-Cp material of choice for 4 K operation since they produce and machine the alloy
to our specifications.
Commercial vendors are capable of providing arbitrary alloys of most metals via the vacuum
arcing process. This is capable of providing large bulk samples of the alloy, and appeared to
be an attractive means for getting HoCu2 samples that we could then machine into cryostation
components. We ordered a large sample of HoCu2 from ACI alloys that was arc-cast into a few
40 mm × 12 mm cylindrical disks. Our intent was to test capabilities of machining this sample,
though we quickly found this to be much more difficult than expected.
Initial handling and inspection of the disk suggested that the arc casting process formed asample with several non-uniform grain boundaries. Working the sample with an EDM cutter rippedout large chunks of it, and ultimately we did not find a method for consistently shaping the material.
Likely the process used for forming the alloy is not ideal. HoCu2 can also be formed through an

99atomization process which works to combine the individual components at high temperature andthen quickly solidify them. This ensures a more consistent material with the desired properties
from the alloy, the vacuum arc process often leads to an inconsistency in phase of the bulk material
leading to gradient properties. Alloyed samples of HoCu2 were not used in the final redesign of thecryostation.
At 4.5 K HoCu2 has Cp = 0.3 J/cm3 K, so a time constant of 1000 s for a disk with a 10 cm3
volume requires a thermally resistive component with R > 300 W/K. For cavity support structures,
we might often opt for a design where cross sectional area A ≪ L, or L ∼ 10A. For our desired
time constant, the required thermal conductivity would be k = 0.03 W/mK. The point of this is toillustrate that our lack of flexibility in choosing high-Cp materials means we will have to be fairly
stringent with material choice for the thermally resistive components. Copper and steel, which
may typically be used for thermal isolation in room temperature systems, have k = 10 W/mK
and k = 0.3 W/mK respectively. Neither of these would be appropriate for our thermal filter,and generally most metals will not be. The Wiedemann-Franz law provides an empirical relation
between a material’s thermal conductivity and electrical conductivity σ,
κ
= LT
σ

(4.37)For temperature T and proportionality constant L = 2.4 × 10−8 V 2 /K 2 . Metallic compounds
which will naturally be electrically conductive, will not be candidates for providing us with highthermal resistance. Rather, we are better off using materials that are electrically insulating likeplastics or ceramics. PEEK is a polymer with excellent thermal resistivity and is commonly used incavity systems. However, its Young’s Modulus of 3.6 GPa is very low, and would be prohibitive inapplications requiring sharp, high-Q mechanical resonances far from the frequency band of interest.
Composite materials with glass or polymers bonded with an epoxy (e.g. G-10 fiberglass) suffer fromthis same issue, where the epoxy leads to a low thermal conductivity, but is also structurally weak
when compared to metals. Thus a tradeoff exists where we often have to balance good mechanicalproperties with our desired thermal properties.

100
Carbon fiber reinforced plastic (CFRP) was introduced earlier as a suitable material for
optimizing the vibration sensitivity of the cavity support. Its stiffness significantly surpasses thatof G-10 and PEEK, but the thermal conductivity is comparable. Cryogenic measurement of thisvalue is inconsistent, since different carbon fiber-epoxy ratios can lead to different values for k(adding more epoxy will reduce the stiffness and the thermal conductivity). Values of k = 0.03
W/mK at 4 K seem reasonable [122]. The passive platform is supported by a series of G10 rings
epoxied to the active base. We replace these with thinner CFRP rings, gaining a significant increasein active-passive thermal time constant due to the lower thermal conductivity and smaller crosssectional area. The acceleration sensitivity is not compromised due the larger > 100 Gpa Young’s
Modulus.

4.4.1.4

Thermal shields

From Si4, there were existing concerns that large thermal gradients could be present along
the length of the shield. The vendor provided shields were originally 0.5 mm thick aluminum, andJohn Robinson had found significant coupling of the cavity frequency to thermal fluctuations of
the radiation shield. Estimations of the radiation shield temperature suggested that it may be at
80 K, significantly warmer than the stage 1 temperature of 30 K. His solution was to replace thiswith a custom designed shield machined from copper with 2 mm thickness and polished on theoutside to reduce the emissivity. A thermistor placed on its outside surface after the swap recorded
a temperature closer to 50 K. It is unclear whether there were residual gradients, but clearly the
reduction in emissivity had a significant impact on the thermal environment.
Seeing the outcome of this upgrade, we decided to reconstruct all thermal shields for Si6.
We first wanted to increase the thickness to 2 mm for the two 4 K shields, and 3 mm for theradiation shield. We chose 4N oxygen-free high thermal conductivity copper as our material, so theadditional thermal mass from the increased shield volume would be negligible. High purity copper
is characterized by its residual resistance ratio RRR (ratio of resistivity at finite temperature vs 0K), and OFHC with RRR of 150 has k = 2500 W/mK at 30 K. Compared to the original aluminum

101

Thermal conductivity k (mW* K )

104

RRR50
RRR100
RRR150
RRR300
RRR500

103

102

101

Temp (K)102

Figure 4.23: Thermal conductivity of high purity copper with different residual resistivity ratios.

shields, our proposed upgrades should reduce any thermal gradient by over 500 times.

Figure 4.24: Various gold-plated components for the thermal shields. The radiation shield is on
the left, 4 K active shield in the middle, and 4 K passive shield on the right. A baffle with a 5 mm
aperture on the radiation limits the room temperature radiation load passing through the windows
to the cavity.

The Montana cryostation does not support thermal shields of our desired thickness - shields
must be threaded onto a tapped baseplate and the difference in diameter for the active and passive

102
baseplates is slightly larger than 3 mm. We consequently adopt a unique approach in the shielddesign, forming them out of concentric threaded cylinders that can be twisted together. The
complex machining required likely introduces many stresses into the components, so we anneal
them at 500 C for several hours in an argon backfilled chamber.
As the purity of the copper increases, oxidation occurs more quickly and this would reduce
the surface emissivity, leading to a higher radiative coupling. The benefits we gain from improved
thermal conduction through the shield is less helpful if there is an increased heat load, essentiallynegated, so we gold plate all shield components after annealing. The JILA shop process doesn’t
apply a nickel strike as diffusion barrier since that plating process is line of sight only. Direct
plating of gold on copper can reduce the plating emissivity as copper diffuses into the gold but
some studies [123] suggest that a thick enough plating can inhibit this process. Surface emissivity
is expected to be somewhere between 0.03-0.1 depending on the plating. Note that polished copper
can easily reach emissivities in this range, but the process is labor intensive and may not be possible
for complicated geometries.

4.4.2

Precision thermometry

A few types of parasitic effects can corrupt accurate and precise readout of the thermistor.
With Joule heating, measurement of a thermistor with value R with a current I dissipates a power
I 2 R. A lower readout current is beneficial for mitigating this effect, but comes at the cost of
reduced signal-to-noise in the readout. A thermistor can be chosen to have a lower resistance at
the relevant temperature, but this is complicated by the fact that we would like a sufficiently large
resistance from room temperature to 4 K.
Platinum based thermistors (PT-100) are a staple for accurately characterizing the room
temperature environment around the 1D Sr optical lattice clock, and for measuring the thermal
environment around the 124 K cavity. However, they quickly lose sensitivity below 100 K as theresistance drops below 10 Ω, necessitating a different option. Zirconium oxide thin film thermistors
are a robust option that are ideal for thermometry between 1 and 100 K. However, they still have103
a sizeable > 30Ω resistance at room temperature, unlike germanium or carbon based thermistors
which are often used in cryogenic environments. A typical resistance and sensitivity curve is shownin Fig.4.25.

Figure 4.25: Left: Temperature calibration of the three Cernox thermistors installed on the cryogenic thermal shields. Right: Sensitivity of the thermistors.

Self heating effects are more significant below 4K where the thermal mass of the support
structure is greatly reduced compared to all higher temperatures.
Reading out an accurate value of the thermistor is also nontrivial as the readout cables need
to be electrically conductive but thermally resistive. A normal copper wire cannot be used as it will
provide a thermal short from the cryogenic environment to the room temperature feedthrough on
the vacuum chamber. We instead use phosphor-bronze wire as it provides strong thermal isolationWwith a bulk thermal resistivity of k = 1.6 m·K
at 4 K temperature. A conservative estimate for 36

AWG gauge with cross sectional area 0.0127 mm2 and a length of 10 cm will flow 5 µW of heat
between the 4 K and 30 K stages. The stage 1 of the cryo can provide significantly larger coolingpower, so the estimated heat flow can be safely managed. Electrical resistivity appears sufficiently
small with ρ = 11µΩ·cm, but the same gauge of wire yields a resistance per unit length is R/L =8.7 Ω/m. This is sizeable compared to the thermistor resistance as a 10 cm strand of cable willappear as a nearly 1 K temperature offset at 10 K. These resistive contributions of the cables are
eliminated by performing a 4 wire measurement of the thermistor resistance.
Cables were soldered to the thermistors with 60/40 lead/tin and residual flux was cleaned off

104with an acetone-dipped swab. The dissimilar junction between the readout wires and the thermistor
can generate an electromotive force via the Seebeck effect which provides a slowly varying voltage
offset. An upgraded measurement scheme of using an AC bridge to perform the 4-wire readout wasplanned for a next generation upgrade of the thermometry, but was not implemented in this work.
A MKT50 from Anton Paar was used for the majority of low noise resistance measurements.
The Si3 system clamps thermistors to the thermal shields with a small retaining bar screwed
into the copper shield. The small volume of the Si6 cryostat prevented us from designing thermal
shields with > 1 mm of clearance between the stages and we could not use the same technique given
the 0.8 mm vertical profile of the Cernox thermistors. Lakeshore technicians recommend indium
solder as the ideal means of affixing the thermistors to a metal surface for maintaining UHV
conditions but initial attempts to solder the thermistors to the thermal shields were unsuccessful.
Silver based epoxy was suggested as another alternative. The Montana Instruments cryostat makes
liberal use of cryogenic varnish to hold various components together so we decided to use this, giventhat it worked well for Si4 operation. A thin film of Apiezon N-type grease was placed underneaththe thermistor, with IMI-7031 varnish applied to the edges. The readout cables were additionallyclamped to the shields in a few places for stress relief and the structure was cured at 150C for 12hours.

4.4.3

Cooldown tests

With the multitude of design changes implemented on the cryostation, we worried that theadditional material might prevent us from reaching a base temperature of 4 K. We opted to cool
the system down to its lowest achievable base temperature before inserting the Si6 cavity. With thefull array of temperature sensors in place, this would allow us to measure thermal time constants
between various stages as well as identify any thermal shorts or strong couplings to the environment.
Multiple sensors were placed on the active, passive, and radiation shields to characterize
thermal gradients across them. The silicon cavity was replaced with a copper disk designed to givethe same thermal mass at 4 K and a thermistor was also placed on this to characterize the expected105

Figure 4.26: Left: A copper disk is used as a dummy cavity, a thermistor attached to it allows
us to measure the expected cavity temperature during cooldown. Right: The active shield withthermisters installed. Temperature sensors from the passive shield pass through a small slit beneaththe lid. A thin layer of vacuum grease between the lid and cylindrical section improves the thermal
contact. Cables for the sensors are wrapped around each shield several times for thermal anchoring.
This is especially important on the radiation shield so that heat flowing from the 300 K vacuum
feedthroughs is dumped on the 30 K stage where the coldhead has more cooling power. Sensors
are attached to the shields with VGE-7041 varnish and baked at 150 C for 6-10 hours for curing.

cavity temperature and time constants.
The initial cooldown thankfully brought the test mass down to cryogenic temperatures,
though it settled to 5.8 K, slightly hotter than our expected 4 K. The CTE of silicon starts torapidly increase above a temperature of 5 K so some minor improvements were needed in the architecture of the thermal environment. Surprisingly, we identified large, nearly 1 K temperature
gradients across the active shield despite the increased shield wall thickness and the significantly
higher thermal conductivity of high purity OFHC. Fluctuations between the bottom and top of
the shields were highly correlated however, with a negligible thermal time constant. No amount of
numerical simulations could reproduce this gradient and given the strong coherence of thermistors

106
across the shields, it’s likely that the thermistors themselves had a sizeable thermal offset. When
warming the chamber to room temperature, the gradient was clearly non-monotonic (Fig.4.28) andseemed to take unphysical values, e.g. changing sign at various temperatures. It is possible thatthe first order discontinuities in the warmup curve were the result of stress release in the sensor cable junction. In hindsight it would have been useful to perform a huddle test of all thermistors
during a practice cooldown to identify any offsets and jumps in their temperature readings, but theRadiation shield gradient

70.5
70.0
69.5
69.0
68.5
68.0
67.5

Temp - 650 [mK]Temp - 1000 [mK]

mounting stresses would no doubt be different after installing them on their respective surfaces.

0

2000

8000

10 1

10 2

10000

101

102
Time [s]103

Active shield gradient0

Temp instability [mK]Temp instability [mK]

100

4000
6000
Time [s]

0.475
0.450
0.425
0.400
0.375
0.350
0.325
0.300

1000 2000 3000 4000 5000 6000
Time [s]

10 2

10 3

101

102
Time [s]103

Figure 4.27: During one of the initial cooldown tests, two thermistors were placed on each of
the active and radiation shields to measure thermal gradients. By design, the high purity OFHC
copper should have suppressed gradients to the mK level. Surprisingly, a residual gradient of 1
K was measured on the radiation shield (left column) and 0.65 K on the 4 K active shield (rightcolumn). We were unable to find a proper explanation for this, it is possible that curing the epoxy
that attached each thermistor to the shield introduced resistance offsets via mechanical stresses.
This is another argument for securing the thermistors with a mechanical clamp in future systems.
The bottom row displays an Allan deviation of the gradient fluctuations. The active shield gradientis remarkably static with fluctuations slightly over the 10 µK level over 1000 s.

Ultimately, the first cooldown taught us that our careful attempts to thermally anchor ther-Temp [K]Temp difference [K]Active bottom
Active top

102
101
100

0

2

4
Time [hours]

107

5

6

8

Active shield gradient

4
3
2
1
0
1

50

100 150 200 250
Absolute temperature [K]300

Figure 4.28: The cryostation is heated from 4 K to room temperature while we read out thermistors
on the top and bottom of the active shield. The difference in measured temperature (right) varies
significantly over the warmup process.

mistor cables to various stages were insufficient. Stepping temperatures across the radiation shieldrevealed that the testmass temperature would increase with a slightly shorter thermal time constant
than the passive shield, indicating heat flow between the testmass and radiation shield through thesensor wires. We went through a few iterations of cooling down, identifying an issue, warming up,
and making minor improvements. After a few thermal cycles, the cryogenic varnish would peel off
the shield resulting in a sensor dangling in free space, or worse, shorting two shields together. This
technique worked well given the minimal clearance between shields, but in the future it would be
preferable to use a mechanical clamp with the sensor sandwiched between layers of indium foil.
Ultimately, it was crucial to wrap the readout wires around each thermal shield several times overto provide a proper thermal anchoring of the sensor.
As we learnt more through this process, the active-passive time constant increased by a factorof two, from 5000 s during the initial test to 11000 s on the penultimate cooldown. Our model ofthe thermal resistance circuit predicts a ∼10000 s time constant between the active and passive
stages. This seemed to indicate success in removing unwanted thermal links and the diminishingreturns dissuaded us from further tests. At this point, we decided to cooldown the cavity rather
than the testmass. In our final configuration, we ended up installing a single sensor on each of theradiation, active, and passive shields.

108Stage 1
Stage 2
Radiation shield
Platform
Active bottom
Passive bottom

Temp [K]

102

101

100

0

20

40

Time [hours]

60

80

100

Figure 4.29: Temperatures on the thermal shields and cryocooler stages are continuously measured
during the final cooldown.

4.4.3.1

Cooldown procedure

Cooling down the cryostation with the cavity in place is a religious experience. The relatively
large linewidth of Si6 already makes RAM control difficult and any further reduction would be disastrous. We try to apply standard cleanroom procedures, with 1/4-inch rubber curtains installedaround optics enclosure and a HEPA operated inside. The optical table and cryostation are thoroughly cleaned of any dust and everything but the cavity mount is covered with UHV foil. Kapton
tape covering the vent holes of the spacer is quickly removed and the cavity needs to be set on its
support at the measured angle that minimized total acceleration couplings. The passive shield andits lid are then immediately placed over the cavity, hopefully giving some protection from dust.
The active shield is threaded into its mount, cables for the passive sensor are fed through notches
in the lid, and then thermally anchored to exterior of the active shield. The same procedure is done
for the radiation shield, with thermal anchoring occurring in several locations to impede direct heatflow from room temperature to 4 K. With the cryostation closed, we quickly measure all thermistors
and heater resistances to confirm that there are no electrical shorts. After realigning to the cavity,
the finesse is remeasured to ensure that there is no contamination of the optical coatings. Vibration
sensitivity is also measured to ensured that the cavity is placed at the correct angle relative to the

109
support structure. Vacuum can then be pulled on the cavity, and the cooldown to 4 K is initiated4
.

4.4.4

Thermal time constants

With the cavity at 4 K, we can verify thermal time constants between various stages byperforming a discrete temperature step and recording the cavity frequency response. With theactive shield held at a fixed value after a temperature step, the thermal behaviour of the passive
shield and cavity follows
Cact Ṫpass =Cpass Ṫcav =1Ract−pass
1
Rpass−cav(Tact − Tpass ) −(Tpass − Tcav )1
Rpass−cav(Tpass − Tcav )(4.38)(4.39)

Transforming the above ODEs into Laplace space, we find that the cavity response is well describedby a second order low-pass filter with poles corresponding to the active-passive and passive-cavity
time constants (A step response with the dummy cavity shown in Fig.4.30 highlights this effect).
Our final measurements of the time constants between the various 4 K stages were τactive−passive =12000 s, τpassive−cavity = 300 s. The amplitude of the cavity’s frequency response tells us its
sensitivity to thermal fluctuations, and we find a CTE of 4.5 × 10−11 K−1 . Notably this is larger
than the 2 × 10−11 found in Si4, indicating that Si6 spacer is sitting at a higher temperature. Our
cooldown tests indicated that the cavity equilibrated close to 4.9 K as opposed to the lower 4 K in
Si4. The cavity’s length response to thermal fluctuations will be 2.5 times larger, but the thermal
time constant between the active platform and cavity has been increased nine-fold. There is still a
net gain in suppressing temperature variations at 4 K.
A similar step response can be performed with the radiation shield, though in this case theactive shield is actively servoed to prevent any confounding coupling between the radiation andactive platforms. At 4 K, a radiation shield - cavity coupling of 10 Hz/K was measured with a time
4

The cavity frequency will move by several MHz as it cools to 4 K. As the cavity warms from 4 K and 16 K, we
measure a frequency shift of 800 kHz. This contrasts with the 1.6 MHz that would be expected from integrating thesilicon CTE with data from [124]Temp [K]

110

0.200
0.175
0.150
0.125
0.100
0.075
0.050
0.025
0.000

activepassivetestmass

0

2000

4000

6000
8000
Time [s]

10000

12000

14000

Figure 4.30: Thermal time constants In a test cooldown of the system, the cavity is replacedwith a a copper disk of the same thermal mass. Thermal sensors are attached to the testmass,
inner 4K shield, and outer 4K shield. The temperature of the outer 4K shield (active) is stepped
via a ring heater wound around its base, and we monitor the subsequent temperature increases of
the inner 4K shield (passive) and testmass. We extract the time constants between the different
stages as well as the thermal gradients between them. The ring heater on the active shield basealso allows for active temperature stabilization to either the active or passive thermistor.

constant of 3000 s. Upon warming up to 16 K, we were unable to identify any frequency responseto the temperature change, reflecting the lower CTE of the cavity near it’s zero crossing5 .
A step response on the vacuum chamber revealed a frequency coupling of 200 Hz/K, significantly higher than the 4 Hz/K measured in Si4. Our radiation shield was measured to be below 50
K, so radiative coupling to the vacuum chamber could not explain our significantly higher couplingmeasured. In [43], it was speculated that the reduced room temperature coupling of 4 Hz/K waslargely due to the thermal upgrades that resulted in a significantly lower radiation shield temperature. One key improvement is that the manufacturer supplied optical windows were exchanged
with superpolished BK-7 windows in the hopes of reducing optical scatter. During the Si6 upgrades, these were replaced with thinner superpolished UVFS windows due to concerns over the
BK-7 performance deep in the cryogenic regime. UVFS transmits more of the 300 K blackbody
spectrum than BK-7 but not enough to explain the 50 times larger room temperature coupling. It
5
At the time of this measurement we were limited by an etalon in the system that modulated the beatnote by anamplitude of 2 Hz, it was difficult to identify any exponential setting with this present

111
is more likely that heat flows through the temperature sensor cables at the 300 K feedthroughs tothe cryogenic stages. The exact path is unclear, but suggests a possible mechanism for the higher
coupling to room temperature fluctuations.

4.4.5

Temperature stabilization

With our couplings identified, we now know the level of stabilization needed on each stage toattain thermal noise limited performance. Coupling from the 30 K stage only needs temperaturestabilization to 0.5 mK. We accomplish this by feeding back to the power applied to a resistive heaterwrapped around the base of the radiation shield. This loop is closed with ∼ 1 Hz bandwidth, andalong with the 3000 s time constant, we are confident that this does not contribute any instability
above thermal noise out to 1000 s. In Fig.4.27, we identified that the gradient along the shield isstable below the 0.5 mK for all measurement times.
Stabilization at 4 K is a more daunting task due to the larger coupling via the cavity CTE.
The 4 K active shield also has a resistive heater wrapped around the platform base, and we canuse this to close a loop on a thermistor placed on this shield. In one of our test cooldowns, weplaced a second witness thermistor on this shield to view out of loop temperature fluctuations with
4 K temperature stabilization enabled. The witness performance, visible in Fig.4.31, closely follows
the inloop stability though it seems to saturate at 10 µK temperature variations. We’re unsure
whether this is due to the noise floor of our measurement device or if this was an actual differential
fluctuation. The witness stability is insufficient for the cavity performance we require, limiting it at
the 4 × 10−16 level. We can however make use of the long thermal time constant between active andpassive stages, and calculate the lowpassed temperature fluctuations that we would expect at thecavity. Using our measured CTE, we can calculate the technical contribution of 4 K fluctuations
and find that we would expect the thermal noise limited performance to 200 s.
The cavity can still experience temperature fluctuations that are not entirely captured by thevariations on the active shield. On the Si4 system, the 4 K platform was stabilized to the passive
shield temperature and saw improved frequency stability. On Si6, a thermistor was placed on the

0.05
0.00
0.05
0.10
0

2000

4000
6000
Time [s]6 × 10 3
4 × 10 3
3 × 10 3
101

8000

Fractional MDev

PSD [1/Hz]

102
Time [s]

103

10 15

10 30
10 31
10 32
10 34 310

inloopoutloop

10 2

10 29

10 33

112

2 × 10 2
Temp instability [mK]Temp [mK]0.10

inloopoutloop

outloop active
lowpassed active
Thermal noise

10 2
Freq. [Hz]

10 1

10 16
10 17
10 18
10 19

outloop active
lowpassed active
Si6 projected thermal noise

101

102
Time [s]103

Figure 4.31: Top left: Time series of the inloop and witness thermistors placed on the 4 K active
shield (loop is closed on the inloop thermistor). Top right: Instability of the two thermistor readings. Bottom left: The witness’ temperature fluctuations are converted to frequency fluctuations
via our measured CTE. The brown curve is the witness variations filtered by the passive platform
and cavity mount. Bottom right: Instability of the traces on the left.

passive shield and we could feedback to variations on its reading with a slow < mHz loop bandwidth.
For the experiments described in the next chapter, we never observed any coherence between thepassive shield temperature and the cavity instability (perhaps due to the larger active-passive time
constant). We therefore stabilize the 4 K platform to the active shield temperature, and do notread out the thermistor placed on the passive shield. Excitation with 100 µA of current was enoughto slightly heat the cavity, so we disable the readout to remove an additional heat source to thecavity. The potential for stabilizing the passive shield temperature is attractive when the cavity
is operated at 16 K, but in studies of the long term instability we again never saw evidence that
thermal variations were a dominant noise source.

113

4.5

Vacuum pressure
So far we’ve discussed physical effects that can vary the cavity length by coupling to thespacer. Our true intent is to keep the optical path between the spacer to be constant. Gases within
the spacer borehole vary the vacuum index of refraction n0 , and will easily modify it by coupling
to pressure and temperature.
Variations of air’s refractive index have been well characterized with the Edlen equation [125],
so we can get a sense of what vacuum stability we may need within the cavity to achieve thermal
noise limited performance. The absolute refractivity n of air at 15◦ C and 1 atm varies dispersively
with wavenumber σ[µm]−1 as [126](n − 1)s × 108 = 8342.13 + 2406147(130 − σ 2 )−1 + 15997(38.9 − σ 2 )−1(4.40)The 3σ uncertainty of this equation applies to visible radiation, but for just attaining a roughquantity we can extend it to the IR. At 1.54 µm, (n − 1)s = 27632 × 10−8 . The pressure andtemperature dependence of the refractive index is modified as(n − 1)T P =(P/Pa)(n − 1)s 1 + 10−8 (0.601 − 0.00972 T /◦ C)(P/Pa)
×96095
1 + 0.003661 T /◦ C

(4.41)The rough scaling of n with temperatures T and pressures P at ambient conditions goes as(n − 1)P ≈ 2.8 × 10−7 (P/mbar) = 2.8 × 10−4 (P/atm)(4.42)(n − 1)T ≈ 1.1 × 10−6 (T /◦ C)(4.43)

Limiting pressure induced frequency variations to the 10−17 level requires the optical pathwithin the cavity to be this stableδν= (2.8 × 10−7 /mbar) · δPν

(4.44)

The constraints on the intracavity pressure are stringent, requiring a UHV environment tobe maintained with the cavity spacer. Vacuum conductance to the spacer interior is purposely kept
to be poor so that it is difficult for dust to enter this region before pulling a vacuum.

114
However, the cryogenic environment of around the cavity greatly assists us in lowering thebackground pressure at 4 K by adsorbing any molecules that strike a cold surface. The saturated
vapor pressure of most gases decreases dramatically with temperature. The vapor pressure for mostcommon gases is illustrated in Fig.4.32. Aside from hydrogen and helium, there is a negligableamount of other molecules at 4 K. The large surface area of the thermal shields effectively acts asa cryo pump by condensing background gases that collide with them. Small charcoal adsorbers
mounted above the cavity serve to capture any remaining molecules that could enter the cavity
borehole.

Figure 4.32: Vapor pressure of most gases as a function of temperature, taken from [9]. Below
20 K, the only residual gas aside from Helium with a pressure above 10−13 Torr is H2 , which has
a vapor pressure of ∼ 10−6 Torr at 4 K. At the UHV range, most curves are extrapolated withdecreasing temperature.

The variation of the index of refraction at cryogenic temperature is negligible for the cavity
frequency, but it’s interesting to consider how the dispersion of n from Eq.4.40 can generate afrequency shift between the carrier and modulation sidebands, i.e. do we need to worry about the
air’s dispersion generating RAM? For modulation tones 5 MHz offset from the carrier, it turns outthat this effect contributes a frequency shift δν/ν < 1 × 10−21 .

115
Despite the cryo-pumping being sufficient at 4 K, it is naturally less effective between the 300
K vacuum chamber and the 30 K shield. Montana instruments advertises the vacuum environmentof their cryostation as being suitable for most applications that may be insensitive to pressure
fluctuations outside the sample space. For our system, these effects vary the optical path length andcan appear as a frequency shift of the cavity. The residual pressure fluctuations in this environment
were sufficient for Si4 to realize its thermal noise limit, but we discovered an important coupling toacceleration noise that could be critical for measuring improved stabilities. In Si4, measurements ofthe vibration sensitivity always produced unphysical results at low frequency. Measurements of the
DC sensitivity via tilting the optical table and AC measurements from the coherent displacementof the AVI would return horizontal sensitivities of ∼ 1 × 10−9 . Multiplying this by the ambient
acceleration noise would result in a frequency noise significantly higher than the thermal noise limit.
At the time, we attributed this to some flaw in the measurement that we could not pinpoint.
The same phenomena was observed in Si6, but an improved acceleration sensitivity wascritical for achieving thermal noise limited stability. Converse to Si4, our measurements were
performed with the cryostation at room temperature and pressure and the large rollup of the kH1
sensitivity that was present in Si4 was still visible (Fig.4.18). By accident, we discovered that theabnormally large horizontal sensitivities could be reduced by removing the lid of the cryostation’s
vacuum chamber, opening the radiation shield directly to the environment. This does not change
the steady state atmospheric environment around of the thermal shields or cavity but we inferredthat shaking the AVI generated some modulation of the refractive index of air within the chamber.
These pressure fluctuations were coherent with the dynamic displacement of the AVI, and could
therefore appear as a coherent frequency response of the cavity. Opening the vacuum chamber tothe environment eliminated the drum-like modes of the cryostation lid that were being excited and
we instead found the cavity’s acceleration sensitivity to be largely flat at low frequency (Fig.4.15).
We hypothesized that driving the AVI platform compresses the bellows connecting the vacuum
chamber to the coldhead and drives air around the thermal shields.
Removing any air from the cryostation should eliminate this effect, yet when we remeasured

116

Figure 4.33: Schematic of the modifications for additional pumping on the cryostation. Montana
Instruments provides a control unit connected to the cryostation with a long 2 m, 0.375” ID vacuum
bellows. Within the control unit is a small roughing pump that is valved off from the bellows witha case valve contained within the control unit. A vent valve can be left open to the environment
or connected to a nitrogen canister to vent the system to ambient pressure. We removed thisconnection by separating it from the pumping bellows with a gate valve. Instead, we add a tee thatis connected to a turbo and scroll pump. After operating these two for a few days, we valve themoff and activate a small 2 L/s diode pump.

the acceleration sensitivity after the final installation of the cavity (involved pulling vacuum on
the system), the same rollup in acceleration sensitivity was observed. The process of pumping thevacuum chamber was directed entirely by a control unit provided by Montana Instruments. A smallroughing pump within the unit would run for a few minutes before valving off the system, uponwhich it could be cooled to cryogenic temperatures. Exposing some of the vacuum connections
within the control unit revealed that a 2.3 m bellows with a 3/8 inch inner diameter connects
Montana’s roughing pump to the coldhead. Pumping to vacuum chamber was therefore severely
conductance limited, with an estimated volume of 5 mL/s removed. Achieving a pressure of 1
mTorr at the cryostation would require more than 3 hours of continuous pumping, far longer thanthe control unit determined.
We opted to disconnect the control unit from the long vacuum bellows and instead attach a

117
turbo backed by a roughing pump. Running this continuously for a three days brings the cryostation
to a pressure close to the minimum of the turbo’s vacuum range. We then close a valve placed after
the turbo (see Fig.4.33 for a schematic) and activate a small 2 L/s Vaclon diode pump. After thisprocess, intentionally shaking the AVI platform results in no changes in the acceleration sensitivity.
Furthermore, the current on the diode pump does not show any fast or slow variations. In thelimit where vibrations dominate the noise budget of the cavity, the measured frequency noise is
now consistent with our measurements of ambient accelerations propagated through the cavity’s
acceleration sensitivity function. We conclude that refractive index changes due to vibration-driven
pressure fluctuations do not contribute to the measured cavity instability.

10 29

Si3-Si6 beatnote

10 14

Normal trace (no pumps on)
Pump chamber with roughing
Pump chamber with roughing+turbo

Fractional ADev

Fractional PSD (1/Hz)

10 28

10 30
10 31

Normal trace (no pumps on)
Pump chamber with roughing
Pump chamber with roughing+turbo

10 15

10 32
10 33

100

101
Frequency (Hz)102

10 16 3
10

10 2

10 1
Time [s]100101Figure 4.34: Left: Frequency noise of Si3-Si6 beatnote when the vacuum pumps are running andwhen shut off. There is no additional noise measured due to vibrations of the pumps. Right:
Fractional instability of the beatnote data on the left. In the low frequency regime, we can slowly
displace the AVI platform slightly above the ambient vibration noise and look for corresponding
changes in the cavity noise. Below 1 Hz, accelerations coupling directly to the cavity should generatefrequency perturbations significantly below the thermal noise limit. Any observed increase in cavity
noise could therefore be an indication of poor vacuum within the cryostation. We do not observe
any such effects.

4.6

Dissemination noise
When all technical noise sources are below the thermal noise limit, a laser locked to the cavity

fringe will inherit the thermal noise limited instability. However, the laser has to be distributed
to somewhere for characterization, either via beatnote with another laser or by measurement with

118
an atomic reference. In our case, we analyze a heterodyne beatnote of Si3-stabilized light and Si6
stabilized light. As we clarified in Fig.4.9 within our discussion of residual amplitude modulation,
the location of our detector is crucial for determining what noise gets removed. Similarly, thelocation we use to pick off light for distribution can have an impact on the frequency noise wemeasure. Phase shifts picked up in the optical path before the resonator are indistinguishable fromcavity noise when measured at the PDH detector. The PDH servo will apply a frequency correction
to the laser to nullify this effect, but measuring the light before it encounters these perturbations
will effectively write this noise onto it. These undesired effects typically appear in the RAM signal
as differential shifts between the carrier and modulation sidebands due to dispersion or etalons,and we’ve established that active RAM stabilization cannot be implemented on the cavity signal.
A typical solution is to pick off the distribution light as close to the PDH detector as possible
so that differential shifts are minimized. The distribution light will typically be 100s of µW in
power, as compared to the few µW incident on the cavity. Splitting the light at the PDH detector
location is therefore problematic since it will generate huge photothermal effects. In the Si6 system,
we associate this with significant frequency perturbations, so this is not feasible. Modulators andfiber components at 1542 nm are plentiful, so we instead opt to pick off light in the fiber path of
our optical setup, before it is outcoupled into free space.
A 90/10 fiber splitter after AOM1 (Fig.4.36) sends the majority of the laser’s optical power
through a 5 m fiber to beat with a laser stabilized to Si3. Significant frequency noise is pickedup through environmental perturbations to the fiber, so we implement noise cancellation on thislink via feedback to an AOM placed before the 5 m fiber. An in-fiber Faraday mirror forms thephase reference for one of the arms in the noise stabilization interferometer, and is sealed within
an acoustic enclosure. The inloop performance of the servo is well below the cavity thermal noise
floor (Fig.4.35), but this doesn’t equate to the actual frequency noise that will be added to the
distributed light.
Extraneous noise encoded at the fiber noise beatnote, or length fluctuations of the reference
arm will result in an additional frequency noise contribution. It’s therefore important to perform

119

Fractional frequency
noise [1/ Hz]

10 14
10 15
Free running noise
Servo in-loop
Projected Si6 thermal noise

10 16
10 17
10 18
10 19
10 20 010101
Freq. [Hz]102

Figure 4.35: The fiber noise cancellation servo is locked with high bandwidth of ∼150 kHz. Thereis little noise present at this high offset frequency, but significant frequency noise is present at 1
kHz and below. The high servo bandwidth is necessary for providing the large servo gain needed
to suppress the in-loop error to the cavity thermal noise limit.

an out of loop measurement to identify that the servo is actually removing the noise we would like.
One way to quantify this is to perform a loopback measurement, where an additional fiber link at
the remote location is noise cancelled and distributes light back to the initial location. This lightbeats with a reference arm at the initial location, so the heterodyne beat is encoded at the sum
frequency of the two AOMs used for each fiber link. This requires twice the infrastructure of asingle link, and also involves perturbing the setup at the remote end. In the specific case of thesilicon cavities, this beatnote is formed on the frequency comb table and could potentially perturb
the comb stabilization to the Si3 1542 nm light.
An easier and equally valid method is to simply take the fiber distributing light to theremote location, and instead bringing it back to the initial location to beat with the reference arm.
A diagram detailing this setup is shown in Fig.4.36. The fiber noise beatnote between the reference
arm and the distributed light (offset by the FNC AOM frequency of 77 MHz) is measured on a
separate out-of-loop photodetector. Note that we distribute light to Si3 before it passes throughthe fiber EOM. This is to avoid extra modulation tones on the Si3-Si6 beatnote, as well as to avoid
the photon shot noise written onto the laser by the RAM servo. There is a significant path length
difference between the PDH detector and the distribution fiber, so we may worry about additional

120
frequency fluctuations as explained earlier. To properly characterize this, we would ideally beatthe distributed light with light from the cavity transmission to identify any differential noise. We
will discuss this in a later section.
The instability of this beatnote is displayed in Fig.4.37. No deviations above the projected Si6
thermal noise level were observed from averaging times of 100 ms to 1000 s. Metrology applications
requiring an accuracy evaluation would also need to know whether there a frequency offset identified,our measurement only indicates that any offset is stable. We find that the light dissemination
introduces a fractional frequency offset of < 1 × 10−21 .
FNC PD

Faraday mirror

FNC
AOM

NKT

AOM

Faraday mirror

OOL PD

Figure 4.36: Schematic of the setup used to measure the out-of-loop fiber noise. Light from the
NKT laser is split into two paths, one going to a fiber noise cancelled interferometer, and the otherserving as a frequency reference. The fiber noise cancelled link, which typically distributes lightto the Si3-Si6 photodetector, is instead sent to a separate out of loop photodetector to generate aheterodyne beatnote with the reference light. Frequency noise of this beatnote measured on the
OOL PD quantifies the frequency noise associated with the light dissemination.

4.7

Technical noise budget
All the mechanisms described so far will contribute frequency noise that can degrade the cav-

ity stability. Fig.4.38 compiles these to calculate the frequency noise we can expect from technicalsources. Ideally these would all be below the thermal noise limit, and the majority of them arewith the exception of RAM. We’ve discussed that the RAM contribution may be generated fromsome in vacuum scatter, and the only way to rectify this would be to warm the cavity to room
temperature, identify and then clean the offending optic. This is an invasive procedure and since

OOL fibernoiseSi6 projected thermal noise

10 33
10 34
10 35
10 36 3
10

10 2

10 1
100
Frequency (Hz)

121

10 16
Fractional ADev

Fractional PSD (1/Hz)

10 32

101

OOL FibernoiseSi6 projected thermal noise

10 17

10 18
10 1

100

101
Time [s]102103

Figure 4.37: Measurement of the dissemination noise using the schematic shown in Fig.4.36. Left:
Fractional frequency noise of the measurement. Right: Fractional Allan deviation of the measurement. Dissemination noise is well below the projected thermal noise limit of Si6.

the current level RAM would already allow for a stability improvement over Si4, we thought itwould be best to proceed and measure the frequency noise of Si6.

122

Sy [1/Hz]

10 30
10 32
10 34
10 36

Modified Allan deviation

10 38 3
10

10 1
100
Frequency [Hz]

10 2

101

102

10 16
10 17
10 18 1
10100Time [s]101102

Si6 thermal noise
Vibrations
Out-of-loop fibernoise
Intensity
RAM
PDH servo
Temp fluctuations
Figure 4.38: Compilation of technical noise sources measured on Si6. The frequency noise due tointensity fluctuations is discussed in the following chapter.

Chapter 5

Frequency noise of the cavity

Of course, the real test of the achieved performance can only be accomplished witha second, independent detector system. Disappointment is the experimenters’ first
reward for this measurement. - Jan L. Hall
The objective of the Si6 system was to distribute an improved local oscillator to the strontium
experiments where the lower instability could be used to realize record levels of atom-light coherence.
Silicon cavities I-IV followed a standard protocol of reducing technical noise sources below thethermal noise limit to achieve a thermal noise limited instability. Recent progress on ultrastable
resonators has not yielded any significant surprises in this regard. The work is difficult, but a recipe
exists for reducing the oscillator linewidth. Hints that we did not fully understand dielectric effects
were noted in Si4 [43] with the surprising observation of a power dependent drift rate. Nevertheless,coating effects were thought to be well understood regarding the laser linewidth.
Cavities with crystalline coatings demonstrated that this was not the case. Si5 and Si6
observed a multitude of bizarre effects, power dependent frequency noise and correlated mode
fluctuations that had never been measured in dielectric, amorphous coatings. It took months to even
identify these phenomena as coating effects, and only recently has there been an understanding thata dedicated effort is necessary to understand these processes [127]. In this chapter we’ll detail our
measurements of these novel coating effects and present new methods for studying and suppressing
them.

124

5.1

Photo-birefringent noise

5.1.1

Photothermal effect

In the technical noise budget, we mentioned that intensity noise can drive frequency fluctuations, necessitating stabilization of the laser intensity. This mechanism is not due variations in thediscrimination slope, but rather to the heat-induced length change of the cavity.
This was first noticed in the LIGO collaboration where it was discovered that the optical shotnoise of an incident beam was driving surface fluctuations on the test mass coatings [128]. Coined
the photothermal effect, photons absorbed in the coating would locally heat and deform the surface,
generating a displacement that could be read out in the interferometer. The same mechanism isapparent in our Fabry-Perot resonator, photons absorbed within the coating generate an elastic
contribution where the surface expands due to a nonzero coefficient of thermal expansion, and arefractive contribution where the nonzero dn/dT of the dielectric stack also changes the cavity
length. Eventually the absorbed heat propagates to the substrate and cavity spacer, each of which
has their own photothermal contribution.
A formalism from [129] details the photothermal contributions of the cavity, allowing us tocalculate length changes originating from each component, and different mechanisms. There are
two explicit effects we will consider, the first involves the coating, substrate, and spacer’s material
thermal expansion. Intuitively, we can expect this elastic response to depend on a few material
parameters for each component i: the coefficient of thermal expansion αi , the thermal conductivity
λi , density ρi , and the heat capacity Ci . For a given beam radius r0 , we can define a wavenumberrki =iωρi Ci
λi(5.1)

Where ki−1 denotes a penetration depth for the substrate and coatings. This is a roundabout way
to indicate that there is a physically relevant timescale when the penetration depth is equivalent
to the beam radius. We define this as the critical relaxation frequency
ωi =λi
ρi Ci r02

(5.2)

125
For the substrate and coatings on Si6, we find ωc = 1700 Hz, and ωs to be orders of magnitude
higher due to the high thermal conductivity of silicon1 . A plot of the two coating and substratediffusion lengths in Fig.5.1 compares them with relevant length scales of the cavity.

100

Length (m)

10 1
10 2

Substrate Diffusion length
Coating Diffusion length
Beamspot size on mirror
Cavity length
Coating thickness

10 3
10 4
10 5
10 6 410
10 3 10 2 10 1

100

101

Frequency (Hz)

102

103

104

Figure 5.1: Coating and substrate diffusion lengths are plotted as solid red and green lines respectively. Dotted and dash dotted red lines indicate two length scales relevant to the coating diffusion,i.e. the beam size and the coating thickness.

I’ll regurgitate the relevant expressions for the substrate and coating responses from [129].
The photothermal response can also be calculated in the linear regime as a transfer function, with alength change originating from a modulation of the optical power. First, let’s define two parametersthat we’ll re-use in the coating and substrate responses. Making the substitution ξc = kc r0 withcoating thickness d, the filter function F attenuates any response at a diffusion length larger thanthe coating thickness
F =11
cosh(ξc d/r0 ) + R sinh(ξc d/r0 )I don’t quote a number since it’s hard to find data on the cryogenic thermal parameters for silicon

(5.3)

126
The coating elastic response is then calculated as
"
Z
2
δXc
αc ωc ∞
− ξ2
δ X̃c (ω) =
dξξe
F
=
δP
πλc iω 0
γ1



(5.4)


ξ

cosh(ξd/r0 ) − cosh(ξc d/r0 ) + R
sinh(ξd/r0 ) − sinh(ξc d/r0 )
ξc

!


 ξ

− γ2 R cosh(ξd/r0 ) − cosh(ξc d/r0 ) +sinh(ξd/r0 ) − sinh(ξc d/r0 )
ξc

(5.5)
!#

(5.6)
With coefficients γ1 , γ2 defined as
1 1 + σc
1 + σs Ec
1 + (1 − 2σs )2 a − σc
1 + σc Es
2
1 − σs Ecγ2 =1 − σc Es
γ1 =

(5.7)
(5.8)

The substrate elastic response isδXs
αs (1 + σs ) ωs
δ X̃s (ω) = −
=
δP
πλs
iω

Z ∞
0

h
ξ2
ξi
dξξe− 2 F 1 −
ξs

(5.9)

The coating also has a thermorefractive response due to the temperature dependence of the layers’
refractive indices
λβ
δXtr
=
δ X̃tr (ω) =δP
2πλc r0Z ∞0ξ2dξe− 2

ξ sinh(ξc d/r0 ) + R cosh(ξc d/r0 )ξc cosh(ξc d/r0 ) + R sinh(ξc d/r0 )(5.10)

Where the parameter β specifically captures the optical length change with temperature
β=

∂nL
H
1 ( ∂n∂T + αH nH ) + ( ∂T + αL nL )4
n2H − n2L

(5.11)The total photothermal response of the cavity, elastic and thermorefractive effects included, iswritten asδ X̃T = δ X̃spacer + δ X̃s + δ X̃c + δ X̃tr

(5.12)

These expressions are intimidating, but there is some useful intuition to be gained fromstaring at them. Parameters γ1 and γ2 depend solely on properties relating to the deformation and127
modulus of the materials. If the coating and substrate have matched σs = σc and Es = Ec , wewill find that γ1 = γ2 = 1. Parameter R depends on the thermal properties of the coating andsubstrate and similarly reduces to unity of they share the same thermal parameters. In this case,the elastic response of the coating will be miniscule. It seems surprising that the thin film response
will depend so strongly on properties of the substrate but recall that we’ll be operating in the low
frequency regime where the heat has diffused to a length scale larger than the coating thickness.
It’s clearly a tall order to have a coating and substrate share the same material properties
if we wish to minimize the coating elastic response. There exists another method - note that thesubstrate and coating elastic response have opposite sign. If we have the freedom to choose ouroperating temperature, we can find a regime where the different thermal expansion coefficients
αc and αs yield δ X̃s ≈ −δ X̃c . This will also suppress the joint photothermal response of thecoating and substrate. However, there will still remain the coating’s thermorefractive response
which additionally depends on the optical properties of the bragg stack. The most straightforward
method of cancelling this effect requires the temperature dependence of the refractive index to have
opposite sign for the high and low index layers, i.e. dnH /dT + dnL /dT ∼ 0. The physical picture
is complicated with the consideration that the intensity is reduced in each following layer of thestack by the attenuation factor nL /nH .
3.4

2.5
2.0

3.2

GaAs
AlGaAs

3.1

1.5
1.0
0.5

3.0
2.9
100

dn/dT ×104

Refractive index

3.3

GaAs
AlGaAs0.0101
Temp [K]

102

100

101
Temp [K]102

Figure 5.2: Left: Refractive indices of the high and low index GaAs and AlGaAs layers. Right:
Temperature dependence of the refractive indices.

I only point this out to illustrate that it is highly complicated to assemble a traditional

128
cavity with no response to optical power fluctuations. Photothermal effects are hard to cancel, andan easier route is always to stabilize the laser power. Of course, this task becomes easier if theamplitude of the total photothermal response δ X˜T is small.

Amplitude (Hz/W)

109

Coating Photothermal Elastic
Substrate Photothermal Elastic
Coating photothermo-refractive
Spacer photothermal
Total Photothermal

108

107

106 4
10

10 3

10 2

10 1

100

101

Frequency (Hz)

102

103

104

Figure 5.3: Calculated photothermal transfer functions for the elastic spacer, substrates, and coating responses, and the thermo-refractive coating effect. In the low frequency regime where we aim
to be thermal noise limited, photothermal effects from the spacer are the dominant contribution

5.1.2

Photothermal effects in crystalline coatings: transient response

The formalism presented above represents the best understanding we currently have for modeling photothermal effects in coatings. These models have been benchmarked and verified against
measurements on interferometers and cavities employing traditional dielectric coatings. In the Si6
cavity, we made several anomalous measurements that defied some of the key conclusions drawn
from the theory of photothermal effects.
The large polarization mode splitting of the cavity enables us to address and lock to either of
the two polarizations (I’ll arbitrarily call these E1 and E2 ). The sign of the photothermal responseis expected to be determined entirely by the sign of δ X˜T . Our first measurements on the cavity
however demonstrated that there may be additional effects arising with these crystalline coatings.
In Fig.5.4 we show a typical set of measurements where we drop the transmitted intensity of129

Frequency [Hz]

150
100

300nW -> 100nW: E1
300nW -> 100nW: E2

50
0
50

0

25

50

75

100
125
Time [s]

150

175

200

Figure 5.4: Transient response of dropping the transmitted intensity from 300 nW to 100 nW for
the E1 mode (solid line) and the E2 mode (dotted line). Intensity is dropped at ∼10 seconds, andthere is a fast transient, followed by a slow settling of the frequency that exhibits opposite sign for
the two polarizations.

Decrease power
200
150

Frequency [Hz]Frequency [Hz]Increase power
0300nW -> 100nW
850nW -> 300nW
850nW -> 100nW

100
50

50
100
150
100nW -> 300nW
300nW -> 850nW
100nW -> 850nW

200

0
0

50

100

150
200
Time [s]

250

300

0

25

50

75

100 125 150 175 200
Time [s]Figure 5.5: Transient response of the E1 when changing the transmitted power between different
values. A clear asymmetry in time constants is visible between the left and right panels. Increasing
laser power results in a much faster frequency response than decreasing the power, indicating that
there are different photothermal transfer functions depending on the sign of the intensity change.

the cavity from 300 nW to 100 nW by stepping the RF power applied to an AOM. The two curves
show the frequency response when locked to either of the two polarization modes. Notably, there
is a fast transient response where the laser frequency instantly increases, and then slowly increases
or decreases with an opposite amplitude for the two polarizations. Nothing in our photothermal

130model captures the behaviour where orthogonal polarization modes exhibit opposite frequency
responses. Given that this was not observed in Si4 and the only change between the cavities was
the choice of optical coating, it seems safe to assume that this peculiarity is specific to the switch
to a semiconductor, crystalline material.

500

300nW -> 100nW
100nW -> 300nW
300nW -> 850nW
850nW -> 300nW
850nW -> 100nW
100nW -> 850nW/ P [Hz/ W]

400
300
200
100
0
0

25

50

75

100
Time [s]

125

150

175

200

Figure 5.6: Frequency responses renormalized by the mangitude of the transmitted power change.
Solid lines indicate curves taken when decreasing the optical power, dashed lines indicate an increase. Curves of same color are taken with the same set of initial and final power values. It isclear that there is no singular transfer function δν/δP that captures the frequency response.

We can try to simplify the picture by simply stepping the laser power by different amounts
when locked to a single polarization mode and observing the frequency response. Transient responses of the E1 for different increases in transmitted power (Fig.5.5) display a nonlinear behaviour where the time constant of the frequency response appears to vary with the initial andfinal values of the intensity step. Furthermore, changing the sign of the intensity step (decreasing
the transmitted power rather than increasing) shows a settling behaviour with significantly fastertime constants. It seems that there is not a single photothermal transfer function that can capture
the frequency response for any arbitrary modulation of the laser power. We can at least conclude
that the transfer function depends on the sign of the intensity modulation.
To accentuate this point, we renormalize all the frequency responses of Fig.5.5 by the magni-

131tude of the optical power step δP . The resulting set of curves in Fig.5.6 show drastically different
amplitudes and time constants depending on the sign of the power change, initial value, and final
value. We cannot identify a single transfer function δ X˜T that can produce a universal set of curves
for the parameters of Fig.5.5 data.
A more complicated model would also be required to explain the varying time constants
observed in all transient responses. In Fig.5.7, attempts to fit an exponential decay to each transient
results in large frequency residuals and does not appear to capture the trend of the data. We instead
find that a double exponential fit of the form




t+b
t+b 
f (t) = a exp −+ exp −+cτ1
τ2
(5.13)

provides a better agreement to the data. This isn’t to say that we’ve proposed this model withsome scientific insight, rather it’s evident that there is more than one time constant at play. While
the fit residuals are smaller, there still appears to be some remaining oscillation around the dataso this heuristic model is incomplete.
Time constants returned from the double exponential fit are presented in Table 5.1. The
fast transient occurs on timescales of seconds while the longer settling takes roughly an orderof magnitude longer. Data appears to be roughly consistent between measurements on the twopolarizations, though τslow varies significantly depending on the conditions of the intensity step.

100 nW → 300 nW
300 nW → 850 nW
100 nW → 850 nW

E1
τf ast τslow
1.9
6.4
2.6
25.6
1.5
12.1

E2
τf ast τslow
2.7
9.3
2.3
21.4
2.0
16.6

100 nW ← 300 nW
300 nW ← 850 nW
100 nW ← 850 nW

8.9
0.27
12.3

10.5
3.6
15.3

28.5
17.2
59.5

37.6
47.6
83.7

Table 5.1: Fast and slow time constants (in seconds) of a double exponential fit the data in Fig.5.7,
and analogous data taken when locked to the other cavity mode.

132
300nW -> 100nW

Frequency
residuals [Hz]Frequency [Hz]

102
101
100

101

Time [s]

101
100
10 1

101

Time [s]101Time [s]

175

200

50

75

100 125
Time [s]

150

175

200

101

Time [s]

10
20
50

75

100 125
Time [s]

150

175

200

Residuals single exp
Residuals double exp

10
0
25

Frequency
residuals [Hz]101

Residuals single exp
Residuals double exp

20102850nW -> 100nW

102

0

25

850nW -> 300nW

100

Frequency [Hz]50

75

100 125
Time [s]

150

175

200

Residuals single exp
Residuals double exp

20
10
0
10

101

Time [s]

102

25

50

75

100 125
Time [s]100nW -> 850nW

102Frequency
residuals [Hz]Frequency [Hz]150

Residuals single exp
Residuals double exp

25

102

101

101
100
10 1

100 125
Time [s]5.0Frequency
residuals [Hz]Frequency [Hz]

102

100

75

300nW -> 850nW

100

10 1

50

2.5

102

101

10 1

25

0.0

Frequency
residuals [Hz]Frequency [Hz]102

Residuals single exp
Residuals double exp

2.5

100nW -> 300nW

Frequency
residuals [Hz]Frequency [Hz]

102

102

5.0
2.5
0.0
2.5
5.0

101

Time [s]

102

150

175

200

Residuals single exp
Residuals double exp

0
20
25

50

75

100 125
Time [s]

150

175

200

Figure 5.7: Left: Single and double exponential fits of the data in Fig.5.5. Right Frequency
residuals of a single exponential fit (pink) and the double exponential fit (grey).

133
5.1.3

Photothermal effects in crystalline coatings: coherent response

It’s clear that there is no universal transfer function that easily describes the photothermal
response of the coating to an intensity step. Despite the several complications that arise when wetry to extract a transfer function, we’re going to go ahead and measure one to see what else welearn about the coating response. In the previous section, we instantaneously step the intracavity
intensity and look for the steady-state frequency response. Now, we’ll modulate the intracavity
intensity at a particular frequency, and look at the coherent frequency oscillation that appears atthat same value.
Ideally we expect to measure something similar to the curves calculated in Fig.5.3. However,
we know that the transfer function has an asymmetric response depending on the sign of theintensity change indicating some form of hysteresis, so it is difficult to extract a pure sinusoidal
frequency response when given a sinusoidal drive. We attempt to minimize this effect by modulating
the optical power with a depth large enough to resolve the signal above the residual frequency noiseof the Si3-Si6 beatnote. Depending on the measurement bandwidth, this equates to an amplitudeof a few Hz in the frequency response. The optical power through the cavity appears as
Ptot (t) = P0 + δP (t)(5.14)

Where the transmitted power P0 is significantly larger than the modulated power δP (t). The
transfer function δν(ω0 )/δP (ω0 ) is then a measure of the cavity resonance frequency coherently
modulated at frequency ω0 .
A summary of our measurements is displayed in Fig.5.8. We address a single polarization
mode of the cavity, and measure the cavity response as we modulate the optical power at different
frequencies. In all these measurements we ensure that we are in the linear response regime where amodulation of the optical power always corresponds to the same proportional frequency response.
Curiously, we see that the transfer function increases in magnitude with lower frequency for asingle curve. Calculations in Fig.5.3 indicate that we should expect a flat frequency response
until the spacer contribution becomes significant. It appears possible that the calculation of the

108

10 2

10 1
100
Frequency [Hz]101

Photothermal amplitude Pt[Hz/W W]

Photothermal amplitude [Hz/W]

134
126nW transmitted
248nW transmitted
542nW transmitted
843nW transmitted
1208nW transmitted

126nW transmitted
248nW transmitted
542nW transmitted
843nW transmitted
1208nW transmitted

106

10 2

10 1
100
Frequency [Hz]101

Figure 5.8: Left: Measured photothermal transfer function of the cavity response when modulating
the transmitted power while locked to a single polarization mode of Si6. Data
√ is taken at an
operating temperature of 4 K. Right: Renormalizing the left set of curves by Ptransmitted .

spacer photothermal transfer function could be underestimated, but we see this is not the caseas we remeasure the transfer function for different transmitted powers (additionally the frequency
dependence of the data does not match the 1/f scaling expected from theory). As we increase theoptical power the overall amplitude of the TF decreases, though it maintains the same qualitative
shape. The observed dependence on the optical power P0 is again unexpected.
One possibility is that local heating of the coating is significantly changing some material
parameters such that the coating response increases. Without a theory that captures the frequency
dependence of our observed response, it is difficult to identify coupling of the optical power to thecorrect material parameters. The response has an inverse dependence with thermal conductivity,
increases in optical power could heat the coating and thereby increase k, which would result ina lower overall coating response. Increases in coating heat capacity Cc and coefficient of thermal
expansion αc would have the opposite effect and increase the magnitude of the photothermal transfer
function. For this to be the dominant mechanism, thermal conductivity must have a stronger
temperature dependence than the other two material parameters. Low temperature phonon models
approximate a similar T 3 dependence for both kc and Cc , so it is unclear how likely this may be.
We can summarize our findings into three key points:• We measure a photothermal transfer function that changes magnitude depending on the135power P0 transmitted through the cavity.
• All measured transfer functions share the same approximate f −1/3 frequency dependence• The magnitude of the transfer function is the same for both polarizations, though has
opposite phase for the two.
Given that none of these behaviours were observed in Si4, this strongly suggests that a
new mechanism is responsible for the observed photothermal response of the AlGaAs coating. All
these observations complement our previous measurements of the cavity’s photothermal transient
response. A crucial point is that we now have a more quantitative measurement of the opticalpower - frequency coupling, and we can try to heuristically extract some relation between theseparameters. In the right panel of Fig.5.8, we renormalize the measured transfer functions by theroot of the optical power transmitted through the cavity for those measurements, P0 . There is no
theoretical justification for this, instead we simply notice that this appears to reveal a universal
behaviour in the frequency response. To rephrase this, it appears that our observed photothermaltransfer function should have an inverse scaling with the optical power of the cavity asδ X̃measured ∝ √1
Ptransmitted(5.15)

With these being the first measurements made of photothermal effects in crystalline coatings,
it is difficult to ensure that this is a reproducible effect and not marred by an anomaly such as alarge defect sampled by the optical mode. Fortunately, a similar set of data has been measured on
Si5 at PTB. Comparing results allows us to form a picture of the mechanism affecting the transfer
function by identifying the variables that might couple to it.
First, we need to remove the confounding effects due to the different spacer lengths of thetwo cavities. We convert the transfer function of frequency response into length response so that weonly consider photothermally induced length changes of the coating. We next treat the Si5 data in√a similar manner as the Si6 PT by rescaling all curves with Ptransmitted . A comparison to the Si6
measurements is seen in Fig.5.9. One of the measured Si5 curves remains displaced from the others,

136

126nW transmitted
542nW transmitted
1208nW transmitted

Photothermal amplitude Pt[m/W W]

10 6

10 2

380nW transmitted
2.34uW transmitted
4.17uW transmitted

10 1
Frequency [Hz]100

Figure 5.9: Converting the photothermal transfer function from units of frequency deviation tolength deviation allows us to directly compare the coating response of Si5 (stars and solid lines) andSi6 (circles and dotted lines) without complications from the different cavity lengths. Renormalizingby the transmitted power produces a similar set of transfer functions for the two systems, despiteover an order of magnitude span in power.

but the rest all share a similar magnitude and frequency dependence. There may easily be othervariables that weakly couple to these transfer functions, but it is remarkable that we only need
one experimental parameter to find a universal trend between different datasets on two different
systems that share a similar coating (albeit at different temperatures).
The presence of this coating response means that we need to be mindful of laser power fluctuations. For cavities with dielectric coatings, a common method for mitigating photothermal effects
is to simply reduce the optical power so that the product of the photothermal transfer function andintensity noise is below the fundamental cavity instability. With the crystalline coatings, we seethat the magnitude of the transfer function increases as we reduce the optical power. Intensity noise−1/2

will scale with Pt while the transfer function scales as Pt, so this method still works in principle

but its efficacy is reduced. A better practice is to stabilize the optical power in transmission sothat the resulting intensity fluctuations produce a negligible frequency shift. This is the approach

137
that we enact on Si6.
One point which we have not discussed is the polarization dependence of the photothermal
response. We find that
E1
E2
δ X̃meas= −δ X̃meas(5.16)

One could imagine that schemes which simultaneously excite both polarization modes of the cavity
would have a significantly suppressed photothermal response. The residual effect should be limitedby the coating elastic and thermo-refractive perturbations which have an amplitude independentof the light polarization. In the next section, we will discuss methods that enable us to probe thecavity in this manner.

138

5.2

Frequency instability of the cavity
Our photothermal measurements from the previous section were the first indication that there

may be novel noise mechanisms associated with the use of crystalline coatings in the cavity. So
far we have only considered responses of the cavity frequency to changes in laser power. Now, weturn our eye to the steady state and study the frequency noise measured on Si6. The majority ofmeasurements detailed in this section are the subject of [63].
Our first measurements of the cavity noise returned instabilities over 20 dB higher than the
expected thermal noise limit, with mod σy ∼ 2×10−16 . Technical noise sources were suppressed well
below this level, so there was no clear indication why the cavity stability was so poor. Consistently
remeasuring the cavity noise budget did not yield any new insights. While it was tempting to
speculate whether this could in fact be the true thermal limit, it would require the mechanical loss
angle of the coating to be one hundred times worse at cryogenic temperatures than the measured
room temperature value. Another point of speculation was whether additional loss could arise fromthe substrate-coating interface due to the CTE mismatch and the transfer from the original GaAs
growth wafer.
At PTB, Si5 had already been operating at cryogenic temperatures for nearly a year and thecavity instability was also above the expected thermal noise floor, though it realized a much lower
limit of mod σy = 4.5 × 10−17 . We coincidentally noticed that this instability was lower than thatof Si6 by nearly the ratio of the two cavity lengths, suggesting that some coating effect could be inplay.
A critical moment was the observation that cavity’s frequency noise would vary with thelevel of power transmitted through the cavity. Increasing the transmitted power would result ina highly repeatable increase in frequency noise, and reducing it would return it to its previous
instability. We ruled out any possible saturation effects and contributions due to limited signal tonoise typically scale in the opposite direction, where increasing the cavity power and discrimination
slope result in a lower electronic noise floor. Varying the modulation depth of the EOM can directly

139change the discrimination slope for a constant carrier power. Furthermore, if the carrier power was
kept constant, varying the optical power stored in the sidebands had no effect on the observed
cavity noise. There is additionally no settling time that we observe with this frequency noise,
instantaneous reduction of power appears to instantly reduce the cavity instability. From theseobservations, we determine that the frequency noise measured on Si6 is not a transient effect, it
only cares about the steady state rate of photons entering the cavity.

10 29

90nW
180nW
360nW
500nW

Sy (1/Hz)

10 30

1000nW
1/f guide
1/f 1.5 guide
Expected Si6 thermal noise

10 31
10 32
10 1

Frequency (Hz)100

Figure 5.10: Measurements of the cavity frequency noise for different transmitted powers, ranging
from a minimum of 90 nW to 1000 nW transmitted. When this data was taken, optical scatter off
one of the input vacuum optics contributed a large noise hump at 100 mHz offset. Cavity noiseat frequencies below this offset is limited by this technical source. Below a cavity transmission of
90 nW, electronic noise sources start to become significant and no further reduction in frequency
noise is observed. A dotted red line provides a 1/f 1.5 guide to eye, well above the projected Si6
thermal noise (black line).

This was reminiscent of our measurements of the photothermal transfer function, but now
the steady state frequency noise we observe has an optical power dependence opposite of what
we measured when modulating the intensity. In the latter measurement, increasing the opticalpower resulted in a reduced photothermal coupling. With the steady state noise, increasing thecoupled cavity power results in a higher instability. We can additionally confirm that we are not140limited by photothermal-induced noise by reducing intensity fluctuations via feedback, and seeing
no corresponding reduction in frequency noise.
Thus it appears that we are observing a different coating mechanism that now produces
extra noise in the steady state cavity response. This clear trend that we observe suggests a strict
monotonic relationship between frequency noise and cavity coupled power. Again, we try to rescale
the sets of curves in Fig.5.10 to extract the dependency of Sy on transmitted power Pt . Surprisingly,
we again find a square root Pt dependence such that
Sy ∝

p
Pt

(5.17)

This empirical relation suggests a potential mitigation method: significantly lowering the opticalpower might allow us to reduce these power dependent frequency fluctuations to a level below
the thermal noise limit. Unfortunately, an estimation of the necessary optical power proves that
this is an unfeasible strategy. At a transmitted power of 100 nW, we measure a lower boundof Sy = 2 × 10−32 /f . Supposing we want to reduce this to a level comparable to the projected
thermal noise floor of SBrown = 1.7 × 10−34 /f , we would need a 104 reduction in optical power, or

10 29

10 28

10 30

10 29

Sy
( 1 )
Pc Hz W

Sy (1/Hz)a transmitted power of 10 pW.

10 31
10 32 21010 30

145 mW
59 mW
18 mW

10 1
Frequency (Hz)100

10 31 210145 mW
59 mW
18 mW
1/f 1.5 guide

10 1Frequency (Hz)100

Figure 5.11: Left: Another set of frequency noise measurements
taken with different intracavity√
powers. Right: Curves on the left are normalized by Pc and collapse to a single noise limit
despite the factor of 10 difference in optical power. A dotted red line is displayed as a 1/f 1.5 guideto the eye

Detection and signal to noise restrictions aside, the shot noise limited length resolution of 10

141
pW corresponds to a white noise floor of Sy ≈ 10−30 . Only at bandwidths below 100 µHz will thethermal noise limit exceed the shot noise limited resolution and this is clearly an impractical lengthof measurement time. This strategy is also unfeasible as it is clear from Fig.5.10 and Fig.5.11 thatthe cavity noise is not 1/f . We can identify a lower 1/f bound which the noise never drops below,
but it appears to instead have a slope Sy ∼ 1/f 1.5 . This steeper slope indicates that the cavity
instability degrades with longer averaging times, so reducing the measurement bandwidth reduces
the relative thermal noise contribution.

Sy (1/Hz)

10 29

Si6: 4K
Si6: 16K

10 30
10 31
10 32 210145 mW
59 mW
18 mW

10 1
Frequency (Hz)100

Figure 5.12: Light-induced noise at a cavity temperature of 4 K (solid lines) and 16 K (dottedlines). Curves of same color are measured with same intracavity power. We do not measure anychanges in coating loss and transmission between the two temperatures.

If this noise originates from the fluctuation-dissipation relation, we would expect to see astrong temperature dependence. For the Brownian thermal noise limit scales with temperature, asimilar process driving the Si6 noise could also serve as a means for managing it. Unfortunately, it
is not an option to lower the temperature of the Si6 cavity below 4K. The 2.5 K stage 2 temperatureof the coldhead places a limit on achievable base temperatures, but it is trivial to heat the cavity
up. Less than a watt of power on the 4 K active shield is needed to raise the cavity temperature
to 16 K, where silicon exhibits zero crossing of the CTE. Comparing this light-induced noise of

142
the cavity between 4 K and 16 K should provide a large signal to discriminate any temperaturedependence.
In Fig.5.12, we compare noise measurements at the two different operating temperatures
at identical intracavity powers. To our surprise, we we cannot discern a difference, suggestingthat in this temperature range light induced noise has no strong temperature dependence. It is
possible that there is no explicit dependence on temperature, but rather that the magnitude of thenoise is set by some material parameter which has a weak relation with T (the thermo-refractive
coating transfer function for example depends on dn/dT ). Regardless, our current understandingof fundamental cavity noise terms originates with fluctuation-dissipation, where noise sources arethermodynamically driven. The peculiarity of this light induced noise cannot be overstated as we
do not observe the typical ∝ T or ∝ T 2 scaling that is characteristic to known dissipation-induced
processes.
Again, as a consistency check, we can compare our results to another system to ensure that
this is a reproducible feature. The same effect was measured on Si5, though at a much lower
magnitude in fractional units. The fractional frequency light induced noise was measured to beover 10 dB lower (Fig.5.13) than what we measured in Si6. We note that the absolute lengthnoise Sx is the proper quantity to compare, since the noise originates in the coating and the longer
spacer length of the 21 cm cavity converts this into a smaller fractional frequency noise. As with
fundamental noise mechanisms, this indicates that even longer cavities could advantageously reducethe contribution of this noise mechanism, though technical issues become significant over spacer
lengths of 10s of cm. Rescaling the length noise in the bottom right panel of Fig.5.13 brings thetwo systems into much closer agreement, though there is still a slight discrepancy between the
Si5 and Si6 sets of curves. It is reasonable to surmise whether this could be a residual direct
temperature dependence, but this would be an odd behaviour. Interpreted in this manner thedata would suggest that lower temperature results in an increased noise, contrary to our intuition.
Furthermore, the lack of any difference between data at 4 K and 16 K suggests that this is not atypical thermodynanically driven mechanism.

143

10 29
10 31
10 33 4
10

Sy
( 1 )
Pc Hz W

Si5: 124K

10 28
10 29
10 30
10 31
10 32
10 33 2101600 mW
287 mW
71 mW

10 3
Si6 4K Pc: 18mW
Si6 16K: 56 mW

10 2Frequency (Hz)10 30
Sx ( m2 )Pc Hz W

Sy (1/Hz)

10 27

Si5 124K: 287 mW
Si5 124K: 71 mW

10 1
Frequency (Hz)

100

100

10 1

Si6 4K Pc: 18mW
Si6 16K: 56 mW

10 31
10 32
10 33
10 34 210

Si5 124K: 287 mW
Si5 124K: 71 mW

10 1
Frequency (Hz)100

Figure 5.13: Top: Light-induced noise measured on Si5 at 124 K for different√intracavity powers.
Bottom left: Comparison fractional frequency noise of Si5 and Si6 with the Pc rescaling implemented. The light induced noise in Si5 is over 10 dB lower than that in Si6 due to the longer cavity
length. Bottom right: Renormalizing by the cavity lengths allows us to compare the absolutecoating noise between the two system.

It is more likely that there are other parameter differences between Si5 and Si6 that we have
not yet considered. The crystalline coatings were grown in the same run, so it seems improbablethat stresses within the AlGaAs and GaAs could be any different. The similar static birefringenceof the multilayers corroborates this, so the only remaining disparity is the beam waist supportedby the cavity. Whereas Si6 uses two mirrors with 1 m radius of curvature, Si5 has a planar mirror
on the input end to the cavity. In materials with large electro-optic effects where the optical index
varies with applied electric field, intensity - not power - is the proper variable to describe thematerial response. It stands that a similar relation may apply here, and that we should instead be
normalizing the noise curves by intensity. Therefore we need to verify that there is in fact a scaling
with mode area.

144
5.2.1

Mode area dependence

Once the cavity is constructed and the thermal environment is kept constant, the only method
for reducing the fundamental Brownian noise limit is through reducing the coating thermal noise
contribution. As demonstrated earlier, coating thermal noise has a 1/w02 beam size dependence.
The beam waist itself cannot be modified once the mirrors are chosen to support a particular radiusof curvature, but higher order spatial modes of the cavity can be interrogated to reduce the effective
beam size. Recalling that thermal noise is calculated via the fluctuation-dissipation theorem as
SBrown =2kB T Wdiss
π 2 f 2 F0

(5.18)The strain energy dissipated across the coating Wdiss follows the pressure profile of the beam’s
amplitude distribution. This will be none other than the spatial profile of the beam which can be
described in either the cylindrical Laguerre-Gauss basis, or the cartesian Hermite-Gauss modes. By
calculating the effective size increase for a fixed optical power, a merit factor gm,n can be defined
to modify the strain energy in the form
Wdiss ∝

gm,nw02

(5.19)The merit factor will depend on the mode indices m, n for HG modes, or cylindrical indices ρ, l for
LG modes and will have a value less than unity for any non-fundamental modes. For the gaussian0,0m,n̸=0< Wdissand the thermal noise limit is reduced as the mode
00 mode, g0,0 = 1 so that Wdiss

index increases. Formalism for calculating the merit factor is detailed in [130], and the merit factor
differs for coating Brownian and substrate Brownian noises. We only consider the merit factor forcoating thermal noise, given that we are only interested in effects originating within the crystalline
coating. For coatings, one finds

gm,n =

4
π

Z ∞

Z ∞
dp

0

2
2
2
dqe−p +q Lm (p2 )Ln (q 2 )(5.20)

0for the HG basis, where the various Lm (x) are the ordinary Laguerre polynomials of the n-th

145order. In the Laguerre-Gauss basis,
r Z ∞
2
2
G0,ρ,l = 2
dxe−2x Ll (x2 )2 Lρ+l (x2 )2
π
r 0
Z
2
ρ! 2 ∞
2Gρ,l = 2
dxe−2x x4l Llr (x2 )4π (ρ + l)!
0(5.21)(5.22)For modes with no angular momentum l = 0,gρ,l=0 = G0,ρ,l(5.23)

1gρ,l>0 = G0,ρ,l + Gρ,l2(5.24)For higher angular momentum modes,

1.0

HG0, *
HG*, 0

Scaling gm, n

0.8
0.6
0.4
0.2
0.0

0

1

1.0

2
3
HG mode number

4

LG0, *
LG*, 00.8
Scaling g , l

5

0.6
0.4
0.2
0.0

0

1

2
3
LG mode number

4

5

Figure 5.14: Merit factor gm,n for different spatial modes in the rectangular basis is plotted on
the top, and gρ,l for the cylindrical basis is plotted for the bottom. Rectangular modes HG0,i and
HGi,0 have the same scaling factor due to separability along the two axes. This is not the case forcylindrical modes where increasing the azimuthal mode index leads to a larger effective mode area.

Values of the merit factors for the two bases are displayed in Fig.5.14. In the rectangularbasis, separability of the two mode indices implies that gm,n = gn,m . A sizeable 25% reduction inthermal noise can be obtained by simply interrogating the first non-fundamental mode, HG01 or
HG10 . This is typically also in the regime where reasonable coupling to a higher order mode can

146
be achieved simply by adjusting the beam pointing into the cavity. Higher order modes suffer fromreduced coupling efficiency, and offer marginal returns in reducing the coating thermal noise. High
coupling can be regained by using a mode converter, but alignment of a phase plate is nontrivial forrectangular modes with index larger than 2. One can instead use vortex phase plates to generate LGmodes with angular momentum. For mode indices larger than 1, LG modes offer a larger reduction
in coating noise when compared to HG modes and the spiral plate also relaxes constraints onalignment. A comparison of phase plates for the HG11 and LG3,0 mode is displayed in Fig.5.15.

Figure 5.15: Generation of higher order modes with phase plates. The top row shows the phase
plate (middle) needed for converting the fundamental mode (left) into the HG11 mode (intensity
profile on the right). Yellow regions of the mode converter imprint an additional π phase shift
when compared to the purple regions. The bottom row shows a similar procedure for generating
the LG03 mode.

To test mode dependence of the light induced noise, we considered a scheme of generating the
LG03 with a phase plate. This would in principle easily reveal mode area dependence of birefringent
noise since we would be looking for a 50% reduction in Smeas . We never ended up implementing thissince we had fears that the strong birefringence of the cavity might break the cylindrical symmetryof spatial modes, and the cavity might only support the HG basis. The easiest method would be
to couple to the HG01 mode by adjusting the pointing, so we opted for this route.
We’ll provide a more thorough description in the next section, but our intuitive understandingof the light induced effects paints a picture where the frequency noise is anti-correlated between

Beam radius (mm)

147

3.5
3.0
2.5
2.0
1.5
1.0
0.5

LG00
LG20
LG02
Distance to 4K active window
Distance to 45 degree mirror
Distance to Faraday

0

10

20
30
Distance from cavity bottom (cm)40

50

Figure 5.16: When using higher order modes, sensitivity to clipping is increased. Not only is themode area greater, beam size will be larger at all locations along the propagation axis. Here, we
compare the beam size for the LG00 , LG20 , and LG02 modes at various locations before the cavity.
Locations of various optical apertures are shown as dashed lines at their corresponding distancesfrom the cavity input mirror. A pink dot indicates the size of the 4K active window at its location
9 cm from the cavity. The significantly larger beam radius of the LG02 mode makes it much more
susceptible to clipping on this optic.

the two birefringent modes of the cavity. We’ll consequently call this measured noise ”birefringent
noise”. Given that it significantly dwarfs our technical sources and the projected Brownian thermal
noise limit, we can look for dependencies on mode area by directly locking to a single polarizationof the cavity. We expect that our measured noise Smeas mainly consists of the birefringent noise
Sbiref .
01 /S 00 ,By looking at the corresponding ratio of light induced noise of the two modes Sbiref
biref

we can identify any dependence on mode area. Recall that for the higher HG01 mode, the scaling
factor is calculated as g01 = 0.75. A direct dependence on the mode area w02 would therefore return01
Sbiref
g01
0.75
=
=
00
g00
1
Sbiref(5.25)In Fig.5.17 we measure the birefringent noise of the HG00 and HG01 modes at the same
intracavity power and temperature by interrogating a single polarization in the Si6 system. We finda ratio of 0.71±0.02, similar to the expectation of g01 /g00 = 0.75 for birefringent noise dependent onmode area w02 . Varying the optical power returns consistent results, suggesting that the birefringent
noise scales with w02 . We can then return to our traces from Fig.5.13 and rescale them by thecorresponding mode areas for the two resonators (450µm for Si5, 300µm for Si6). Shown in Fig.5.18,

148

2.0

Calculated g01/g00
Fit

01 / S 00
Sbiref
biref

1.5
1.0
0.5
0.0 2
10

10 1
Frequency (Hz)100

Figure 5.17: Mode area effect on birefringent noise. We separately measure the birefringent noise of
the HG00 and HG01 modes by addressing a single polarization of the cavity. The ratio of birefringent
01
00
noise Sbiref
and Sbiref
for the two modes reveals the dependence on mode area. If the birefringent
noise is independent of the mode area, we expect a ratio of 1. We find a ratio of 0.71, consistent
with the expectation of 0.75 if the noise scales with mode area gmn w02 . A large peak visible at 0.057
01 dataset and was omitted in the fit.
mHz is the result of a ground loop that appeared in the Sbiref

both sets of curves appear to be in better agreement.

x
w02 SBiref
m4 )
(
Hz W
Pc

10 37
10 38
10 39
10 40
10 41 210

Si6 4K Pc: 18mW
Si6 16K: 56 mW

Si5 124K: 287 mW
Si5 124K: 71 mW
10 1

Frequency (Hz)100

Figure 5.18: Data from the bottom right of Fig.5.13 is now divided by the mode area of the Si5and Si6 cavity.

149
5.2.2

Correlation of light induced noise

For fundamental noise mechanisms, identifying the parameters that set these limits is necessary for devising strategies to reduce their contributions. Both systems see this noise as a limitation,
but reducing the intracavity intensity is not practical for either. Ideally we would hope to suppress
this effect entirely, but that requires us to coherently cancel this with an identical copy of the noise.
We see that this is possible with thermal dissipation noise sources, where thermo-refractive andthermo-elastic terms are engineered to coherently cancel and reduce the overall thermo-optic noise[75, 131]. In this case a second physically distinct mechanism is needed, and our poor understandingof the anomalously high frequency noise in Si6 is already barrier.
However, another distinct noise process is not necessary for coherent cancellation. A clue tothe nature of the crystalline coating noise lies in our previous set of photothermal measurements.
From the coherent intensity modulation and the transient frequency responses, we saw that themagnitude of the photothermal transfer function was the same for both birefringent modes. However
the amplitude was opposite, suggesting that averaging the two could suppress that photothermal
effect. It provokes the question of whether the same effect could be present in the steady state noise.
The frequency noise we measure for the two polarization modes is identical, but the measurement
does not discriminate the relative phase between them. If the birefringent length fluctuations areout of phase, simultaneously probing both modes should suppress this noise. To be explicit, ourhypothesis is that length fluctuations of the two modes look like1dE
tot (t) = dBrown (t) + dresidual (t) + dbiref (t)(5.26)

2dE
tot (t) = dBrown (t) + dresidual (t) − dbiref (t)(5.27)

We assign a term to length fluctuations originating from different sources: dBrown encapsulates thelength noise from the total Brownian thermal noise limit, and technical noise sources (characterized
and uncharacterized) are contained in dresidual . Both modes share these two common terms. A third,
dbiref , refers to the birefringent noise that is associated with the orthogonal polarization of the twomodes. Notably the magnitude of this term is the same in both expressions, but the amplitude is150
opposite between them. Our noise characterization of the cavity suggests that the technical sources
are suppressed to a level near the thermal noise, so we expect that
1dE
tot (t) ∼ dbiref (t)(5.28)

2dE
tot (t) ∼ −dbiref (t)(5.29)

E1
1
If this were true, a simultaneous time series of the length noise dE
tot (t) and dtot (t) would

appear to be largely anticorrelated. Asynchronous measurements would not show this effect, andsince phase information is lost when transforming to the frequency domain, measurements of Sy
for the two modes would show an identical result:
SyE1 = SyE2 = SBrown + Sresidual + Sbiref(5.30)

Only by simultaneously addressing both modes, we would see
1 E2
dE
tot (t) = dBrown (t) + dresidual (t) + dbiref (t) − dbiref (t)= dBrown (t) + dresidual (t)(5.31)(5.32)

And correspondingly
SyE1 E2 = SBrown + Sresidual(5.33)To test this, we need to develop a scheme that can simultaneously interrogate both modes tocancel the birefringent noise.

5.2.3

Dual frequency locking

PTB came up with a unique method that involves two optical setups. One probes the cavity
from the top, another from the bottom, and the two are locked on orthogonal modes. A signal
stabilized to the sum of the two error signals could be synthesized digitally and fed back to a
separate actuator. This proved to a highly versatile setup, but requires two lasers, two sets oflocking and detection electronics, and also requires the two setups to have their individual noise

151

Figure 5.19: (a)Optical layout for probing the Si6 system with active RAM suppression at a
modulation frequency (5 MHz) where the cavity error signal is encoded in a traditional PDH
locking setup. Dual-tone probing is achieved by driving AOM2 with two RF tones separated by
the birefringent mode splitting of Si6 (green and purple sets of tones). A half waveplate before
the cavity is aligned to couple an equal mixture of s- and p-polarized light to the two polarization
modes of the cavity, labeled Ê1 and Ê2 . The collinear beams are then launched upwards witha 45◦ mirror to couple into the cavity. The reflected cavity error signal has equal contributions
from noise of the two polarization modes, and any anti-correlated noise is automatically rejected.
(b) Frequency landscape of the tones applied to the Si6 cavity. The two tones applied to AOM2
separate the carrier and the two sidebands each into two tones separated by the birefringent mode
splitting of the cavity. The purple tones shown address one polarization mode (Ê1 ) while the green
tones simultaneously address the orthogonal mode (Ê2 ).

budgets suppressed below the level of the birefringent noise. This is an intensive task, so we would
like to a enact a solution that has less redundancy built into it.
The principle of our desired method is similar to PTB’s approach - we can treat the birefringent modes as two distinct cavities, and we want a signal corresponding to their center-of-mass152
motion. Our chosen approach is illustrated in Fig.5.19. The optical elements AOM1 and the EOM
are used to enact traditional PDH feedback, the EOM generating modulation tones at our typical sideband offset of 5 MHz. A second AOM (AOM2) generates a doublet (two tones generated
via DDS) with a splitting between the tones corresponding to the birefringent mode splitting ofroughly 770 kHz. These two tones are used to address both modes of the cavity simultaneously inthe frequency domain. However, the birefringent modes will only transmit orthogonal polarizationsof light, so a half-waveplate after the Faraday element is used to rotate the in-plane p-polarization
to project equally along the polarization axes Ẽ1 and Ẽ2 of the cavity.
A photodiode in cavity transmission can be used to monitor the sum power transmitted
for both polarizations, and feedback to the sum (note that this method does not allow us to
identify differential amplitude noise of the two tones). In cavity reflection, sidebands and cavity
leakage signals along the Ẽ1 and Ẽ2 and polarizations are recombined on the half waveplate toan identical projection, and then rotated to s-polarization after transmission through the Faraday.
Light detected on the PDH photodetector contains the sum of error signals relating to the two
distinct modes. If the modes are identical (i.e. we space the two tones by a value corresponding
to the cavity free spectral range), we simply recover an error signal twice as large as we would fora single cavity. However if there is any anti-correlated noise between the two modes, we opticallyaverage this effect in our proposed scheme. We collect error signals of the two modes correspondingtoVE1 (t) = VBrown (t) + Vresidual (t) + Vbiref (t)(5.34)

VE2 (t) = VBrown (t) + Vresidual (t) − Vbiref (t)(5.35)

Vtot = VBrown (t) + Vresidual (t)(5.36)

So that

Note that this is identical to the condition from Eq.5.32. There are a few caveats to consider.
First, for the quantity Vbiref (t) to be identical in the two equations above, we know that the
intracavity power within each birefringent mode needs to be identical. Second, the discrimination

153
slope for each mode needs to also be similar, i.e. cavity couplings and sideband powers for the twomodes must also be the same. Violation of the first condition generates a non-common level ofbirefringent noise for the two modes, and violation of the second leads to poor rejection of Sbiref inthe optical cancellation.
The advantage of this method is that optical, mechanical, and thermal noise sources arecommon to both modes. If the half waveplate is set to couple an equal mixture of Ẽ1 and Ẽ2 ,cavity couplings will necessarily be the same. Amplitude modulation due to etalon effects will benearly identical, especially if the etalon free spectral range is significantly larger than the birefringent
mode splitting (generally the case). Thus, the technical noise suppression is practically free of extra
effort when compared to the dual optical setup scheme.
A disadvantage is that there is no method to directly extract the birefringent noise signal.
The optical averaging automatically cancels the antiphase birefringent contributions. One could
demodulate the PDH signal at the 770 kHz mode splitting in an attempt to recover the birefringent
noise, but we will demonstrate in a later section that this is not in fact possible. A more practicalissue is that our scheme generates an additional shot noise contribution due to the fact that anonresonant tone of light is encoded in each polarization. The electric field projected along Ẽ1 seesa copy of the PDH modulation triplet displaced by the mode splitting in the frequency domain, so
the effect cavity contrast for each mode is reduced significantly. Avoiding this issue would require us
to generate the two polarizations in distinct optical paths and then prepare them in the frequency
domain to be resonant with different modes. This negates the rejection of common mode technicalnoise that we value, so we make do with the reduced signal to noise.
Identifying the mode splitting that separates the dual tones applied to AOM2 can by done bylocking to each mode and measuring the beat offset from Si3. An insitu method is also demonstrated
in Fig.5.20. Suppose that the AOM2 tones are offset from the actual mode splitting fBiref by thequantity 2∆ and that dual frequency locking is used to lock to the center of the two polarization
modes. We can remove one of the tones applied to AOM2 so that the laser is locked to only the E1mode or the E2 mode. Depending on the mode, the laser frequency will be shifted by ±∆ nearly

154
instantaneously limited by the locking servo’s bandwidth. The splitting applied to AOM2 can be
adjusted to minimize the frequency jump ∆. A slow frequency change with sign dependent on
the intracavity polarization will be visible due to opposite parity of the E1 and E2 photothermal
effects.
A natural question is to ask whether a static offset between the tones addressing E1 and E2
is sufficient, i.e. should this be dynamically varied to track the center of both cavity modes. With
perfectly balanced phase error contributions from each mode this effect should be negligible, but
we know this is not the reality. As mentioned earlier, our method of dual frequency locking does
not allow us to access the difference channel for frequency noise between both modes. Without
this information we do not have an error signal to allow us to feed back to the mode splitting.
Fortunately PTB verified that this is unnecessary, as locking the two modes independently withdifferent laser produces an identical instability as using the dual frequency locking.

5.2.4

Residual noise

Implementing the dual frequency lock provided us with an immediate improvement in thecavity instability. There were several notable features. First, the near 10 dB reduction of themeasured frequency noise now displays a low frequency noise spectrum that is consistent with
1/f behaviour. There are no additional improvements we made to the technical noise budget
indicating that the peculiar 1/f 1.5 components observed on a single polarization mode must havebeen characteristic of the birefringent noise.
The birefringent noise power dependence of that we had clearly measured when addressing
a single polarization is no longer evident in the residual noise. Varying the intracavity power overan order of magnitude consistently returns the same measurements of Sy .
Similar to our studies of the birefringent noise, we can examine the temperature dependenceof the residual noise (Sresidual ) by operating the system at 4 K and the lower CTE zero crossing at
16 K. No change in residual noise was observed, indicating that we are unlikely to be limited bysome form of thermally driven fluctuations.

155

300

= -80 Hz= 0 Hz= +120 Hz= +220 Hz

Frequency [Hz]

200
100

E2 lock

0100

E1 lock

200
300

0

10

20

30
Time [s]

40

50

60

Figure 5.20: Left: At t = 0, the laser is initially dual frequency locked to E1 and E2 modes. Aroundt = 10, the tone addressing E2 is removed so that the laser is only locked to E1. A frequency jumpcorresponding to ∆ is observed. The E2 tone is then reintroduced, and the E1 tone is removed.
The laser frequency, distributed after AOM1, is continuously monitored in a beat with Si3. Right:
A cartoon indicating the location of the AOM2 tones relative to the two polarization modes. Colors
correspond to the data on the left. With ∆ = 0, there is no frequency jump when switching fromthe dual mode lock to the single mode lock, indicating that the spacing of the AOM2 tones is
equivalent to fBiref .

156
A large hump at 100 mHz in the measured frequency noise was identified to be related to some
optical scatter within the vacuum. Varying the beam pointing into the cavity could significantly
reduce the amplitude of this feature, despite the reduced cavity contrast. By fortune, we discoveredthat optimized coupling to the HG01 mode via beam pointing would heavily suppress this scattering
shelf. Our hypothesis was that some vacuum grease within the cryogenic environment had migrated
to one of the 4 K windows. Evidence for this mechanism was observed in the Si4 system, several of
the cryogenic windows were observed to have small specks of grease across the top surface. In thiscase, there was no significant impact observed in the cavity instability but we were not so luckywith Si6.
At the cost of reduced cavity coupling and a higher shot noise contribution, we measured
an improved long term instability with the HG01 mode. This test had the additional benefit ofallowing us to identify any dependence of this residual noise on mode area. We detected no such
01
00
, allowing us to conclude that the residual noise in Si6 did
and Sresidual
difference between Sresidual

not share the same local properties that we measured with the birefringent noise. Whatever noisemechanism is currently limiting the cavity instability will therefore have a correlation length larger
than the spot size since we cannot average any frequency fluctuations by increasing the area probedon the coating.
Together with the lack of dependence on spot size and temperature, we can conclude thatthe residual noise we measure is not the system’s thermal noise limit. Though we have not directly
measured the coating loss angle, we can place an upper bound on its magnitude at each temperatureby equating a loss angle limit to our measured Sy .
In the frequency band of 0.01-0.3 Hz, we identify an instability consistent with Sy = 3.37(9)×10−33 /f from a continuous 30,000 second measurement against Si3. This performance is slightly
better than the thermal noise limited instability measured on Si4 of Sy = 4.12(5) × 10−33 /f . Using
the formalism of ?? for calculating the coating thermal noise limit, we therefore place constraints

157

Sy (1/Hz)

10 30
10 31
10 32
10 33 2
10

Cavity temp. 16K
Cavity temp. 4K

Si3 1/f
Fitted 1/f
10 1

Frequency (Hz)100

Figure 5.21: Residual noise Sresidual + SBrown after cancellation of the birefringent noise at both 4
K and 16 K shows no temperature dependence.

on the maximum value of the AlGaAs/GaAs coating loss angle ϕ:
ϕ4.7 K < 4.1 × 10−4

(5.37)

ϕ16.7 K < 1.2 × 10−4

(5.38)

Note that measurements from [18] extract a loss angle of SiO2 /Ta2 O5
ϕ4 K < 5.6 × 10−4

(5.39)

ϕ16 K < 3.2 × 10−4

(5.40)

These bounds already imply that these crystalline coatings can yield a sensitivity improvement overconventional dielectric coatings at 4 K, and a significant improvement at 16 K.
The origin of the residual noise is not clear. It is natural to ask whether this could be unsuppressed birefringent noise. However, we have already laid out a number of observed properties thatare atypical of the birefringent fluctuations. Additional measurements where we introduce a slight
imperfect cancellation of birefringence (i.e. improper matching of intracavity powers and discrimination slopes for the two modes) further confirms this, as the residual noise does not increase.From
our mode area measurements, we also can conclude that the residual noise is not limited by local
fluctuations of the optical coating. It remains possible that the global noise dominating Sy is an

158

mod y

10 15

10 16

10 1710 1

100

Si6 instability - single polarization
Si6 instability - dual polarization
104
102
103

101
Time (s)Figure 5.22: Fractional frequency instability of Si6 of birefringent noise (red squares) and residual
noise at 16 K (blue circles). The residual noise is measured over a continuous 24000 s dataset.

uncharacterized technical source that we have not yet identified.

5.3

Direct measurement of birefringent noise
We had the motivation to directly measure birefringent noise, independent of technical noise

contributions. PTB developed a method for this involving two separate optical setups, one probing
the cavity from the top and another from the bottom. Two separate lasers could be locked to thetwo orthogonal polarizations of the cavity and then independently beat with Si2. A simultaneoustime series record of the frequency noise of the two modes then allows one to calculate all linear
combinations with the noise of each mode. Noise between the two modes can be calculated directly,
and technical noise sources common to both can be strongly suppressed. The power of this method
is clear - high frequency information often swamped by vibration noise should now be only limitedby optical shot noise. Similarly, thermal fluctuations that move the cavity frequency are common
to both modes, so birefringent noise can be measured at temperatures away from the silicon zerocrossing.
In principle, measurements made in this fashion on Si6 can identify the temperature depen-

159
dence of this noise from 4K to nearly 100K. Unfortunately, significant difficulties are associatedwith this dual optical setup scheme. Technical sources need to be suppressed below the level ofthe birefringent noise in both setups, and it is clear that we have already struggled with this on asingle optical setup. Etalons present only in one setup for example, will appear in the birefringent
noise channel when calculating the difference frequency of the lasers stabilized to the orthogonalmodes. We therefore attempted a few measurement schemes that I will detail below. Ultimately
all were unsuccessful, but they have some interesting ideas associated with them so they are worth
recounting.

5.3.1

Two optical paths in one setup

The issue of measuring both polarization modes independently is not a problem unique tobirefringence in crystalline coatings. A JILA experiment in 2000 demonstrated a measurement of
mirror birefringence between two cavity modes by simultaneously coupling both polarization modes[10]. This, and recent proposals from the LIGO community [132] suggest that measurements of themode splitting can place constraints on coupling strengths of axion-like dark matter by using thecavity to identify phase shifts between orthogonal light polarizations.
A schematic for this scheme taken from [10] is illustrated in Fig.5.23. While the beam paths
share as many optical elements as possible, there is still room to pick up differential phase shifts. It
would be prudent to feed back to each EOM for residual amplitude modulation accumulated alongeach path so a total of four detectors before the cavity are needed in this scheme. An interestingalternative is alluded to in the diagram, with different modulation tones at Ω1 and Ω2 fed to EOM1
and EOM2 respectively. Rather than spatially separate the two paths, information from the twocavity modes can be encoded in the frequency domain with different modulation for the two tones.
Both tones can be recombined in fiber after EOM1 and EOM2 and outcoupled to share the same
optical elements up to the cavity, where a half-waveplate can project the incident light along thetwo cavity axes. Similar to our current scheme of cancelling birefringent noise, this results in
wasted light and higher photon shot noise as each polarization incident on the cavity contains an

160

Figure 5.23: The optical scheme of [10] splits the beam across two different paths that are preparedin frequency and polarization to address the birefringent modes. Beams are recombined on orthogonal ports of a polarizing beam splitter and coupled into the cavity. A Faraday rotator before thecavity swaps the polarizations of each incoming beams such that reflected light from the cavity fastaxis is measured on the same path as incident light addressing the cavity slow axis. A heterodyne
beatnote between the birefringent modes can be detected in cavity transmission after interfering
the two polarizations.

off resonant tone. Crucially, this no longer locks lasers separately to each mode since the cavity
reflected light reaches a photodetector shared for both polarizations. Error signals for the twopolarizations are individually obtained by demodulating at their separate sideband frequencies andthe sum signal can then yield a signal proportional to the birefringent noise. Though not an ideal
method to measure the mode splitting signal, this will save some optics, two photodetectors, andsome extra time aligning a second beampath.
We do not implement this on the Si6 system since the use of high-Q resonant photodetectors
makes it very difficult to detect the two tones with high gain and low noise. Note that both tones
have to be sufficiently spaced apart so that signals encoded on one tone cannot be detected on theother (e.g. in-loop error of the PDH servo).

5.3.2

Asynchronous detection of cavity error signals

The method above requires simultaneous addressing of the cavity with a second frequency
tone of opposite polarization, resonant with the other polarization mode.
An analogy can be drawn to the atomic servo that steers a laser to unperturbed clock tran-

161
sition in an atom. The fundamental picture is identical - a magnetic field is applied to strontium
atoms trapped in an optical lattice to define a quantization axis. Spin states in the ground 1 S0 and
3P

0 excited states are perturbed in frequency by the first order Zeeman effect:∆νZ1 = −

µB δg
mF B
h

(5.41)

Opposite mF spin states can be addressed such that the sum channelδνZeeman = ∆νZ1,mF =−9/2 + ∆νZ1,mF =+9/2 + ...

(5.42)Is independent of magnetic field to first order. Higher order terms in B 2 persist, but these can becalculated and corrected for since the difference channel ∆νZ1,mF =−9/2 − ∆νZ1,mF =+9/2 directly
identifies the magnetic field experienced by the atoms.
Technical complications of simultaneously preparing and interrogating both stretched states
mF = ±9/2 prevents the experimenter from synchronously cancelling the first order Zeeman shift.
Rather, atoms are prepared and interrogated in one spin state, detected, and then prepared andinterrogated in the opposite spin state. Key to this method is the assumption that the magnetic field
is not drastically drifting between these measurements on a timescale similar to the duty cycle of theexperiment, i.e. the measurement is performed faster than the relevant quantity changes. If noiseof the clock laser is significantly faster than the fluctuations in magnetic field, this asynchronous
measurement is sufficiently fast to identify the measured frequency noise to originate from laser
phase noise. The deadtime between these measurements still has consequences in the form ofgenerating additional measurement noise via the Dick effect. The approximate effect of this is tointroduce an artificial white noise floor in the power spectrum of the measured signal, and theamplitude of that noise floor will increase with longer deadtimes.
The metaphor between the atomic servo and the birefringent noise is straightforward. Insteadof measuring magnetic field noise, we’re interested in the noise associated with the splitting of thetwo cavity polarization modes. The difference channel of interrogating both modes returns thisnoise, while the sum channel cancels it and reveals the underlying noises. Similar to the atomic

162

Figure 5.24: Left: schematic of a proposed optical setup. Ideally, a switchable polarizing element
could rotate the incident light to the fast or slow axis of the cavity in sync with the data acquisition
rate. Right: A cartoon illustrating the concept of the asynchronous birefringence measurement. An
AOM switches the laser frequency to alternate between addressing the two modes. The demodulated
signal is recorded on digital multimeter and the corresponding phase error of each cavity mode canbe reconstructed to yield the information we want.

163
servo, the protocol is to measure frequency noise of one polarization mode, jump the laser frequency
and polarization to address the other one, measure it’s noise, and stitch the data together to obtain
both measurement channels. We’re interested in measuring birefringent noise on timescales of 1
Hz and below, so switching between modes at a 1 kHz rate should be sufficient. There will be a
Dick-style aliasing effect but sampling these two polarizations quickly enough will give a faithfulrepresentation of the noise at low frequencies.
Maximized coupling to a single cavity polarization requires jumping both the laser frequency
and polarization. The former can easily be done, a potential option for the latter is to use a liquid
crystal waveplate to rotate the polarization. At visible wavelengths, the rise time can be in the
millisecond range but at 1550 nm, all options unfortunately have switching times on the scale of
100 ms. A better option is to use a Pockels cell, which has a significantly higher switching rate
but requires a kilovolt drive for a quarter wave phase shift. A proposal is illustrated in 5.24 where
the switching element is placed directly before the cavity. We ultimately decided on using a static
half waveplate to equally couple to both polarization modes and only jump the laser frequency toaddress a single mode.
Jumping the laser at a kHz rate to lock to the two modes has the unfortunate consequencethat the laser servo will receive a large perturbation at this rate, and attempt to correct for it witha transient that depends on the servo bandwidth. Measurements made in that kHz bandwidth
will therefore contain the contribution of the transient and will not be an accurate measure of thecavity frequency. Instead, we develop a method to passively probe the cavity frequency without
any servos feeding back information related to it. The Si3 local oscillator has lower frequency noise
than a laser locked to Si6 in the entire bandwidth of interest. From the perspective of the Si6 cavity,
probing with Si3 stabilized light is akin to probing with a frequency delta function, or performing
spectroscopy on Si6 by using Si3.
With the laser locked to Si3, there is no need to encode information from the cavity highin the RF. Clearly we don’t want to do a pure homodyne readout of the cavity at DC, but since
the phase lock to Si3 is independent of modulation frequency Ω the modulation sidebands can beFractional PSD (1/Hz)

10 29
10 30

164

Lock to Si6, beat with Si3
Lock to Si3, probe Si6
Si3 1/f

10 31
10 32
10 33

101
Frequency (Hz)

100

102

Figure 5.25: The Si6-Si3 heterodyne beatnote (blue) where two lases are locked to the two separate
cavities. Probing the Si6 cavity with a laser locked to Si3 recovers the Si6 noise spectrum with the
Si3 noise included. This signal is measured in cavity reflection and then converted to frequency
noise by calibrating the discrimination slope (pink). Both traces encode the PDH error signal at
the same modulation frequency of 5 MHz.

lowered considerably below a MHz. Amplitude modulation picked up at the sideband frequency canbe decreased as Ω is lowered. The only constraints are that the modulation needs to be significantly
faster than polarization switching rate, and that the frequency is high enough such that power inTransmission attenuation

sidebands doesn’t transmit through the cavity.
100
10 1
10 2
10 3
100

101

102

Freq [Hz]

103

104

105

Figure 5.26: Attenuation factor in transmission of power coupled into the cavity.

A minimum modulation frequency might be 10 kHz for a 1 kHz acquisition rate, but estimation of the required sideband power is a more complicated problem. With the power dependenceof the birefringent noise, we want to operate in the ideal situation where nearly all of the power

165
is concentrated in the sidebands. Unfortunately, more of this is transmitted through the cavity as
the modulation frequency is decreased. A worthwhile place to start is by identifying the minimum
length noise we can resolve when limited by photon shot noise (in the limit of a weak carrier, strong

Shot noise equivalent
optical power (nW)Fractional length resolution(1Hz bandwidth)sideband). Results of a calculation are shown in Fig.5.27.

10 16

10 17
100

101
102
Carrier power (nW)

103

104
103
102
103

104
105
Resistor (Ohm)106

Figure 5.27: Left: Shot noise limited length resolution in a 1 Hz bandwidth for the Si6 cavity.
Reducing the optical power in the carrier will decrease the magnitude of the birefringent noise,
but also hinder the ability to measure that noise at high frequency. Lowering the measurement
bandwidth can mitigate this. Right: Johnson noise equivalent for a particular level of shot noise.
If the detector noise floor is limited by Johnson noise and other op amp noise sources are below it,
this yields the noise equivalent power.

Birefringent noise was measured to have a 1 × 10−16 fractional frequency amplitude at 1
Hz, and if we hope to resolve the noise at this level, the minimum power can detect is around 10
nW. The photodetector used cannot have a transimpedance gain too high, else the optical shotnoise of our 10 nW will be hidden under electronic noise of the detector. Supposing we choose a√transimpedance gain of 100 kΩ, the feedback resistor’s Johnson noise of 40 nV/ Hz needs to bepushed below the birefringent noise with a sufficiently large discriminator slope. With a fractionalfrequency birefringent noise of 2 × 10−32 at 1 Hz, we can calculate the necessary sideband opticalpower that will push the detector noise well below this quantity:
Ps =




∆νc
· 40nVGPc (2 × 10−32 )(5.43)For transimpedance gain G and cavity linewidth ∆νc . For our minimum choice of carrier power,
this returns Ps ≈ 300 nW so we choose a modulation frequency of 40 kHz to prevent more than a
few nW from transmitting through the cavity.

166

10 28
10 29

Si6-Si3
Si6-Si3: aliased x2
Si6-Si3: diff. aliased x2
Si3 thermal noise

Sy [1/Hz]

10 30
10 31
10 32
10 33
10 1

100

Frequency [Hz]

101

102

Figure 5.28: Fractional frequency noise of the Si3-Si6 beatnote is computed with an evenly sampled
time series of beat (black). The effect of aliasing the data, or introducing deadtime between
measurements can be simulated by extracting a time series of every alternate point (e.g. all eventimestamps), doubling the effective acquisition period to account for deadtime, and computing thepower spectrum (blue). The low frequency noise of the beatnote is fully recovered, but an artificial
white noise floor is evident at higher frequencies. Increasing the deadtime or sampling even fewer
points has the same effect, but accentuates the magnitude of the white noise. In the asynchronous
measurement, datapoints will alternate between addressing the fast and slow axes of the cavity. The
degree of common mode rejection for noise present in both modes can be estimated by taking thedifference of two signals aliased from the same dataset. In orange, we show the difference channel
between an aliased signal composed of all even points of the Si3-Si6 timeseries and an aliased signal
composed of all odd points. Low frequency noise in the difference is suppressed to the level of thewhite noise which is incoherent to the two datasets. Over 20 dB of common noise rejection is visibleat the lower range of 100 mHz.

With these parameters, we can begin our study of the Si6 birefringent noise. A compilationof some of these measurements is displayed in Fig.5.30. It is worth noting that the phase lock to Si3
allows us to easily park the probing laser on various parts of the Si6 cavity lineshape. Sensitivity
to different parts of the cavity error signal can be extracted by tuning this offset frequency around.
In PDH schemes, the error signal is linearized at the point of maximized cavity transmission

167
meaning power oscillating at the modulation frequency will carry the largest signal to noise ration.
Conversely, the homodyne component will also contain information about the cavity frequency as
the laser is detuned from the transmission maximum. A calculation of these different sensitivities
in Fig.5.29 indicates that the probe-cavity detuning can in principle be used to extract the same
information of the reflected cavity signal at AC and DC. In practice, homodyne measurement issignificantly harder since just measuring optical shot noise of the off-resonant probe is a difficult
endeavor below a few Hz.

|Slope|

100

10 1

10 2
10000

|Slope of Error signal||Slope of Cavity reflection|
Transmission

7500

5000

2500
0
2500
Frequency [Hz]

5000

7500

10000

Figure 5.29: Sensitivity to the AC components (blue) from the Pound-Drever-Hall modulation
scheme is contrasted to sensitivity obtained from the DC components (green) via the side of fringeinterrogation. At a frequency indicated by the dashed magenta line, AC and DC components should
yield equivalent sensitivity to cavity noise. The cavity transmission function is overlaid in red.

As a demonstration, we detect cavity noise with our typical scheme of modulating the probe
light at 5 MHz, where we can detect an AC PDH signal and simultaneously look at the DC
component on the same photodetector. In the left panel of Fig.5.30, we compare the AC PDH
signal taken with the probing laser parked at the peak of cavity resonance (red curve) to the DC
signal taken with the laser detuned to half the cavity linewidth (indicated with dashed black line in
Fig.5.29), where the cavity reflection function provides strong sensitivity but the PDH signal is very
weak (blue curve). Comparable features of the cavity noise can be seen in both signals, though asignificantly higher white noise floor in the DC signal obscures features with lower frequency noise.
This is simply a result of detecting a DC signal while modulating the laser with 5 MHz sidebands,

168
which provide no frequency noise information at DC and only increase the observed optical shotnoise.
10 4

AC signal, detuning = 0 Hz
DC signal, detuning = c/2

Voltage noise (V/ Hz)

Voltage noise (V/ Hz)

10 4

10 5

10 6

100

101
Frequency (Hz)

102

10 5

10 6

100

AC signal, detuning = c/2
DC signal, detuning = 0 Hz

101
Frequency (Hz)102

Figure 5.30: Left: Cavity noise measurements from the AC (modulation at 5 MHz) and DC
signals. The DC signal is slightly smaller in amplitude due a smaller sensitivity to cavity noise at
this particular detuning (at the cavity pole). Right: Detunings for the AC and DC measurements
are now swapped, so that each measurement is insensitive to the amplitude noise converted by thecavity.

We can then reduce the modulation frequency to 40 kHz in accordance with our previously
detailed scheme. We were unexpectedly unable to measure cavity noise in a manner similar towhat is demonstrated in Fig.5.30 with the modulation at 5 MHz. With the probe off resonant fromthe cavity, we measure a level of noise consistent with shot noise. Upon tuning this to the cavity
resonance, a significant amount of intensity noise was observed at a level higher than any of thenoise features of the cavity. Lowering the modulation frequency seemed to have an unintended
consequence of generating a significant amount of AM in the measurement. To ensure that thiswasn’t frequency noise generated in the phase lock to the Si3 light, we directly outcoupled lightfrom the laser stabilized to Si3 and transferred it to the Si6 optical table over a fiber phase noise
cancelled link. This has the benefit of removing any contributions of the Si6 laser servo since weare now directly using light from the Si3 stabilized laser.
The same level of excess AM was observed when measuring with the Si3 laser, indicating that
it wasn’t originating in our control loop. We noticed instead that it was reduced when increasing thegain of the Si3 PDH loop. Since Si3 achieves thermal noise limited performance, clearly the servo

169
noise was not a limitation so it was strange that a high level of amplitude modulation would only
appear when probing the Si6 system on cavity resonance. Additionally, this was only noticeable
when detecting in homodyne with the cavity reflected light, or in heterodyne with sidebands placeda few kHz away from the carrier. Modulating the Si3 probe light at a high MHz-level frequency
would recover the typical cavity noise we measure when directly locking to Si6. We ultimately
discarded this technique to move on to other measurements, but it is useful as a means to probe
another system without requiring additional stabilization electronics.

5.3.3

Simultaneous detection

When we realized that we could cancel the birefringent noise with the dual mode - dual
polarization scheme, a natural question to ask was whether we could measure the birefringent noisewith a similar scheme. We again ended up spending some time to investigate this but realizedthat it still would not provide us with the same information gained from probing the orthogonal
polarization modes from opposite sides of the cavity. Here we’ll discuss the main points of themeasurement.
In the previous section, we attempted to asynchronously measure the birefringent noise, but
discovered that this generates additional white noise due to the Dick effect aliasing. However, thisdoesn’t occur with birefringent cancellation since the two modes are probed simultaneously. Ourthought was that in addition to detecting a PDH signal with the cancelled noise, we could also
directly measure the birefringent noise by detecting the power oscillating at the 770 kHz mode
splitting.
In principle, this would be as easy as taking the photodetector signal and splitting it into two
components, one at the sideband modulation frequency and the other at the mode splitting. Unfortunately the narrow resonance of the photodetector’s LC circuit provides very low transimpedance
gain at frequencies over a MHz away from the PDH modulation. We instead split the signals in the
optical domain, employing Terry’s wizardry to build another resonant photodetector with a gain
resonance near the mode splitting (Fig.5.31).

170

Figure 5.31: Top: Transimpedance gain of detector for measuring birefringent noise measured by
Terry Brown. X-axis should be in kHz. Bottom: Current noise of the detector. We use the typical
Hamamatsu G12180-003 InGaAs PIN with 1.1 A/W responsivity at 1550 nm. At 770 kHz, noise
is equivalent to shot noise of 220 nW.

Again, we unfortunately spent some time trying to measure this signal before realizing it
was a futile effort. We illustrate this by calculating the sensitivity to noise apparent at the mode
splitting. First, consider the electric field incident upon the cavity when we use our dual polarization
locking scheme.

Ein = E0 c1 ei(ω+∆)t + c2 e(iω−∆)t x̂ + E0 c1 ei(ω+∆)t + c2 e(iω−∆)t ŷ

(5.44)The reflected field then contains information from the two cavity polarizations

Eref = E0 c1 R1 (ω + ∆)ei(ω+∆)t + c2 R1 (ω − ∆)e(iω−∆)t x̂+ E0 c1 R2 (ω + ∆)ei(ω+∆)t + c2 R2 (ω − ∆)e(iω−∆)t ŷ
The signal detected at the photodetector signal will be(5.45)(5.46)

171h Pref = E02 c21 |R1 (ω + ∆)|2 + |R1 (ω − ∆)|2
+ c22 |R2 (ω + ∆)|2 + c22 |R2 (ω − ∆)|2+ c1 c2 R1 (ω + ∆)R1∗ (ω − ∆) + R2 (ω + ∆)R2∗ (ω − ∆) e2i∆ti+ c1 c2 R1 (ω − ∆)R1∗ (ω + ∆) + R2 (ω − ∆)R2∗ (ω + ∆) e−2i∆t(5.47)(5.48)(5.49)(5.50)The signal oscillating at 2∆ is entirely even, meaning we unfortunately recover no sensitivity
to the birefringent noise. This is a natural consequence of the fact that the information of thetwo cavity modes are encoded in different polarizations and will not interfere naturally at thephotodetector. Interference between these modes is necessary. We can place a polarizer at a 45◦
angle to the x̂ and ŷ axes to define a new polarization basis Ê1 and Ê2 (rotated 45◦ from x̂, ŷ). In
this frame, the reflected field becomes
Eref = Ẽ0 c1 R1 (ω + ∆)ei(ω+∆)t + c2 R1 (ω − ∆)e(iω−∆)t Ê1
+ Ẽ0 c1 R1 (ω + ∆)ei(ω+∆)t + c2 R1 (ω − ∆)e(iω−∆)t Ê2+ Ẽ0 c1 R2 (ω + ∆)ei(ω+∆)t + c2 R2 (ω − ∆)e(iω−∆)t Ê1
− Ẽ0 c1 R2 (ω + ∆)ei(ω+∆)t + c2 R2 (ω − ∆)e(iω−∆)t Ê2(5.51)(5.52)(5.53)(5.54)

There are two scenarios for generating the electric field interference depending on how weorient the polarizer. First suppose it passes Ẽ1 . Then the transmitted field ishiEref = Ẽ0 c1 R1 (ω + ∆) + c1 R2 (ω + ∆) ei(ω+∆)t Ê1hi+ Ẽ0 c2 R1 (ω − ∆) + c2 R2 (ω − ∆) e(iω−∆)t Ê1(5.55)(5.56)

172
Detected power ishihiPref = c21 E˜02 R1 (ω + ∆) + R2 (ω + ∆) R1∗ (ω + ∆) + R2∗ (ω + ∆)hihi
+ c22 E˜02 R1 (ω − ∆) + R2 (ω − ∆) R1∗ (ω − ∆) + R2∗ (ω − ∆)h+ c1 c2 E˜02 R1 (Ω + ∆)R1∗ (Ω − ∆) + R1 (Ω + ∆)R2∗ (Ω − ∆)i
+ R2 (Ω + ∆)R1∗ (Ω − ∆) + R2 (Ω + ∆)R2∗ (Ω − ∆) e2i∆th+ c1 c2 E˜02 R1 (Ω − ∆)R1∗ (Ω + ∆) + R1 (Ω − ∆)R2∗ (Ω + ∆)i
+ R2 (Ω − ∆)R1∗ (Ω + ∆) + R2 (Ω − ∆)R2∗ (Ω + ∆) e−2i∆t(5.57)(5.58)(5.59)(5.60)(5.61)(5.62)

Cross terms between R1 (ω) and R2 (ω) now appear at both AC and DC indicating interference
generated between the two tones. Setting
χEˆ1 (ω) = R1 (ω + ∆)R1∗ (ω − ∆) + R1 (ω + ∆)R2∗ (ω − ∆)(5.63)+ R2 (ω + ∆)R1∗ (ω − ∆) + R2 (ω + ∆)R2∗ (ω − ∆)(5.64)

We can extract the signal oscillating at the mode splitting frequency 2∆ and recover thefamiliar form of the error signalhiPref = c1 c2 E˜02 Re[χ(ω)] cos(ϕ) + Im[χ(ω)] sin(ϕ)(5.65)In the other scenario where the polarizer is oriented to transmit Ẽ2 , the reflected field changestohiEref = Ẽ0 c1 R1 (ω + ∆) − c1 R2 (ω + ∆) ei(ω+∆)t Ê2hi+ Ẽ0 c2 R1 (ω − ∆) − c2 R2 (ω − ∆) e(iω−∆)t Ê2(5.66)(5.67)The equivalent χ(ω) quantity will then appear asχEˆ2 (ω) = R1 (ω + ∆)R1∗ (ω − ∆) − R1 (ω + ∆)R2∗ (ω − ∆)(5.68)− R2 (ω + ∆)R1∗ (ω − ∆) + R2 (ω + ∆)R2∗ (ω − ∆)(5.69)

173
Plots of Pref for polarizers transmitting Ê1 , Ê2 with varying the laser-cavity detuning ω aredisplayed in Fig.5.32. Notably there is the presence of a large residual amplitude modulation thatcan be eliminated by choosing an optimal orientation, but no orientation will recover a dispersive

4
3
2
1
0
1
2
3

= 0 /6= 1 /6= 2 /6= 3 /6= 4 /6= 5 /61.5

1.0

0.5 0.0
0.5Frequency [MHz]

1.0

1.5

Error signal

Error signal

signal that provides maximum sensitivity at zero detuning.

0.75
0.50
0.25
0.00
0.25
0.50
0.75
1.00

= 0 /6= 1 /6= 2 /6= 3 /6= 4 /6= 5 /61.5

1.0

0.5 0.0
0.5Frequency [MHz]

1.0

1.5

Figure 5.32: Laser-cavity error signals of demodulating the two tone electric field at the birefringent
splitting 2∆. Different curves indicated a different extraction phase ϕ. All curves are shiftedhorizontally for clarity though there is no vertical offset added. Left: Error signals when Ẽ1 istransmitted. Right: Error signals when Ẽ2 is transmitted. Since components from the cavity
modes oscillate out of phase with each other, the large residual amplitude modulation is removed.
However, the signal retains it symmetry about zero and is three times smaller. Detuning thelaser by 2∆ to measure laser-cavity noise of a single mode no longer has any sensitivity when thepolarizer is orientated to transmit Ẽ2 . In both cases, sensitivity to laser noise can be obtained bydetuning the probe to the side of both lines, similar to the side-of-line probing demonstrated in theprevious section. Alternatively, locking to both modes simultaneously and reducing the splittingof polarization tones to 2∆ − Γc will naturally shift the absorptive signal by Γc /2 where there issensitivity to detuning.

These calculations illustrated in Fig.5.32 display the sensitivity to laser-cavity frequency noise
when detecting at 770 kHz, but not the noise associated with the birefringence. To calculate that,
we are instead interested in measuring the signal that appears as fluctuations of the mode splitting
2∆. Fig.5.33 displays this by calculating the sensitivity to noise ξ that is anticorrelated for thetwo modes (i.e. the cavity transfer functions shift to R1 (ω + ∆ + ξ) and R1 (ω − ∆ − ξ)). The
upper row depicts the error signal when probing both polarization modes simultaneously for thetwo different choices in polarizer orientation. It is immediately apparent that the birefringent signal
is not encoded at the splitting 2∆. Similar to the upper set of curves in Fig.5.32 however, there is174sensitivity to the birefringent noise if we detune the laser by 2∆, meaning only one of the cavity
modes is addressed (bottom row of Fig.5.33). Technical noise affecting the cavity stability can still
be a significant contribution to the measurement since we are no longer measuring differentiallybetween the two polarizations. This same information can be obtained much more easily by simply
locking to one of the polarization modes with a single frequency tone.
The power of the simultaneous birefringent measurement scheme was that any signal common
to the instability of the two polarization modes would be suppressed. Unfortunately it does notappear that there is an easy way to enact this by playing clever tricks with modulation and detection.
The optimal scheme may indeed require two separate lasers addressing individual modes, similar
to what is proposed in [10]. Birefringent noise independent of common technical sources can thenbe easily reconstructed.

5.4Long term instability
So far we’ve discussed our approach to dealing with the unique issues presented with thelength noise of crystalline coatings. However, we were pleased to discover some positive attributes,
namely a significantly smaller long term frequency drift. A simultaneous small, highly linear drift
and good long term instability could make the system a promising component for an all opticaltimescale[17]. Our proof of principle study used Si3 as a local oscillator but constant refills of theliquid nitrogen dewar prevented this from operating as a hands-free system. Conversely, the closed
cycle environment of Si6 allows us to run it fully autonomously. Choosing to operate Si6 at the 16
K zero crossing relaxes the stringent temperature stabilization environments we encounter on the
21 cm systems. So despite the difficulties of improving the short term instability due to the larger
cavity linewidth, the 16 K operation offers some unique advantages for chasing record setting longterm stabilities.
Preliminary measurements indicated that the Si3 drift of -26 µHz/s dwarfed that of Si6.
Direct measurement of the Si3-Si6 beatnote required us to actually measure the cavity frequency
against Si3 for several days in order to be able to clearly resolve the long term behaviour. New sets

1.5

1.0

1.0

0.5

0.5

0.0
= 0 /6= 1 /6= 2 /6= 3 /6= 4 /6= 5 /6

0.5
1.0
1.5
3

Error signal

Error signal

1.5

0.5

1.5
0.002
0.001

0
= 0 /6= 1 /6= 2 /6= 3 /6= 4 /6= 5 /61
2

31000 750 500 250 0 250 500 750 1000
Change in splitting [Hz]Error signal

Error signal

0.0
1.0

2
1

175
= 0 /6= 1 /6= 2 /6= 3 /6= 4 /6= 5 /60.000= 0 /6= 1 /6= 2 /6= 3 /6= 4 /6= 5 /6

0.001
0.002
0.003
1000 750 500 250 0 250 500 750 1000
Change in splitting [Hz]Figure 5.33: Left column: Sensitivity to noise on the mode splitting when Ẽ1 is transmitted.
Right column: Sensitivity to noise on the mode splitting when Ẽ2 is transmitted. Top row:
Laser is tuned such that both modes are addressed simultaneously. Bottom row: Laser is detunedby 2∆ so that only one mode is addressed. We regain sensitivity to the birefringent noise with thelaser resonant with a single mode, but this sensitivity depends strongly on whether Ẽ1 or Ẽ2 istransmitted.

of technical noise sources which were insignificant in our measurements of the residual noise now
became important. These had to be minimized to identify a clear linear behaviour of the cavity
frequency. Here, we’ll detail some of the principle sources and the methods we used for reducing
their impact.

5.4.1

Etalon identification

A significant hurdle to good long term stability of the cavity has been an etalon present in
the system. Sudden changes to the thermal environment of the optical setup such as opening the
primary enclosure would result in a slow, sinusoidal modulation of the cavity frequency. Eventually
this would dissipate, though a diurnal modulation would often still be visible. The amplitude of

176
this fringe was a few Hz, and even one or two oscillations in a day would be enough to bury thesignal of a smaller linear drift less than that of Si3.
It is not immediately obvious that this was an etalon, possible causes could be thermally
driven beam pointing, or coupling of the cavity to room temperature. There are systematic ways
to eliminate these theories, but the former was invalidated by realigning the entire optical setup
several times with no reduction in the amplitude of the frequency modulation. For the latter
case, we could measure the cavity coupling to room temperature by heating the vacuum chamberand measuring the corresponding frequency response. Multiplying this transfer function by room
temperature fluctuations did not produce any signal coherent with the fringe that we were trying
to eliminate on the cavity. Changing the cavity CTE by heating away from 4 or 16 K should also
modify this transfer function, and we were unable to detect any difference in the fringe amplitude.

Frequency [Hz]

10260
10258
10256
10254
0

250

500

750

1000 1250
Time [s]

1500

1750

2000

Figure 5.34: Si3-Si6 beatnote with krypton gas flowed into optics enclosure at different times.
The blue segment highlights the steady state without any deliberate perturbation to the optical
path length. A short burst of gas is released at the beginning of the orange segment. The flow is
continuous for the duration of the green segment and then shut off at the end. A clear chirp in thecavity frequency is visible as the gas dissipates.

While trying to measure the sensitivity of the system to air pressure, we performed tests offlowing krypton gas into the optics enclosure. The heavier, inert gas should be less sensitive toambient turbulence and a lower frequency noise would indicate that Si6 is limited by air pressure
fluctuations. While we observed no change in cavity instability, we noticed that flowing the gas into

177
the Si6 box would drive the same fringe that we were trying to eliminate. This strongly suggeststhat an etalon was present in the system and that it’s length was being chirped as the larger index
of refraction increased the optical path length.
If the frequency modulation was indeed an etalon, this signal should appear as an amplitude
modulation between the carrier and modulation sidebands. We confirmed this in a long measurement of the out of loop RAM, and the next task was to identify the two reflecting surfaces
supporting a parasitic cavity. As mentioned earlier, complete realignments of the system were unable to remove this effect so it was unlikely to be formed between two surfaces that we had easy
access to. A possibility was that one of the surfaces could be within the cryogenic environment.
Intuitively this would not be likely since all vacuum and cryo windows are mounted with a 6◦ wedge
against gravity.
Identifying the troublesome optic is nontrivial since we cannot open the vacuum enclosure tomodify its orientation. This is not necessary though, we simply need to deterministically vary the
optical path length. This can be done by affecting the temperature of the optic and the nonzero CTE
will generate a length change that should allow us to observe a frequency modulation. Recallingthat we have heaters on the 300 K, 30 K, and 4K stages, we can step the temperature of eachthermal shield and look for a corresponding change in the cavity frequency. Crucially, this change
in frequency should occur within the time constant between various stages in order to be associatedwith the surface that was thermally stepped. For example if we change the setpoint of the 30 K
radiation shield and the frequency modulation occurs 15,000 seconds later, the likely scenario isthat the heat has diffused to the 4 K stages through the measured 12,000 time constant and is now
perturbing the 4 K active window. Conversely if we measure an immediate frequency response, we
can pinpoint the 30 K window as the problematic surface.
Iterating through the different thermal stages of the cryostation, we determined the bottom
300 K vacuum window to be one of the etalon surfaces (Fig.5.35). Unfortunately this is the difficult
to manage of the cryostation’s optical surfaces since it is directly coupled to ambient pressure andtemperature fluctuations. It is difficult to access and a heater cannot be placed close enough to theFrequency [Hz]

178

10260
10240
10220
0

1

2

3

4
Time [hour]

5

6

7

6

7

Si6-Si3 beat
Temp of vacuum chamber [arb. units]
Frequency noise conversion of temperature ramp2
Frequency [Hz]

3

1
0
1
2
3

0

1

2

3

4
Time [hour]

5

Figure 5.35: Top: Frequency response of the cavity to a step of the vacuum chamber temperature.
Bottom: Residuals of an exponential fit to the frequency response above display a modulation
indicative of an etalon in the system. The lack of a significant time constant between the temperature step and etalon response suggests that one of the etalon surfaces is not thermally isolated
from the 300 K stage. Modeling the vacuum chamber temperature as a phase ramp, we can largely
reproduce the cavity frequency response by fitting the coupling of temperature to phase. In theorange trace, we correspond a 100 mK vacuum chamber fluctuation to a 2π phase shift. Active
suppression of the 2 Hz peak-to-peak etalon amplitude below the 10−16 level would require ∼mKstabilization of the window temperature. This is difficult, an easier approach is to attempt toreduce the etalon finesse.

optic to close a loop with our desired bandwidth. We tried shielding it from changes in air pressure
with lens tubes but had little success.
Since active stabilization is hard, a better method is to lower the etalon finesse by addressing

Voltage [V]

179

0.02
0.01
0.00
0.01
0.02
0.03
400

300

200

100
0
100
Frequency [MHz]

200

300

400

400

300

200

100
0
100
Frequency [MHz]

200

300

400

0.002

Voltage [V]

0.001
0.000
0.001
0.002
0.003

Figure 5.36: Sweeping the laser frequency, several transverse modes visible in the scan as well as amodulation on the baseline indicating the presence of an etalon. A fit to this reveals the etalon’s
free spectral range to be 160 MHz which corresponds to a cavity length of 0.9 m.

the second surface. Precise knowledge of the etalon length is required, but fortunately any etalon
seen by the cavity will be present as amplitude modulation at the PDH detector. In Fig.5.36, we
scan the laser frequency by nearly 1 GHz and observe a modulation on the baseline amplitude. Notethat this modulation amplitude compared to the PDH signal is roughly 400 ppm, fairly consistent
with the relative amplitude of the etalon observed in Fig.5.34. Given that these signals roughlymatch, we have some confidence that the etalon presence is fully captured in the RAM measurement
and that we are measuring the same parasitic fringe.
The frequency of the baseline modulation is fitted as 160 MHz, corresponding to an etalon

180
free spectral range of 0.9 m. Now that we have identified one surface of the parasitic cavity, it isstraightforward to identify the next optical element 0.9 away. The free space path is less than 0.4
m long, so somewhat surprisingly it must be the case that the second reflecting surface is a fiber
element. We narrowed this down to one of the surfaces of our fiber EOM, presumably the faces are
angle cut so its unclear why this was supporting a large reflection. Regardless, we placed a fiber
isolator after the EOM and reduced the etalon amplitude by 20 dB.

5.4.2

Pointing stability

With the etalon oscillation suppressed, the next goal was to identify the source of the remaining long term instability. Electronic offsets varying on the timescale of hours or days canbe painstaking to pinpoint. Conscientious grounding can help minimize these effects but slowly
varying lab temperatures can easily drive voltage offsets or affect the setpoint of voltage references.
We determine the sensitivities of our electronics to these effects by locally heating our mixers andservos by a few K (larger than the day-to-day thermal variation within S1B20) and looking at
the resulting frequency excursion. With proper thermal insulation and improved grounding, weestimate that we minimize these effects to below the 10−16 level.
The cavity circulating power is always stabilized in transmission, but we still observed fluctuations in the DC voltage collected at the PDH photodetector. Correlations between the beatnote
and this signal over long time scales (5.37) were evident, suggesting that this could be a reasonable detection channel for identifying the source of the long term beat instability. Our assumption
is that there are likely small fluctuations in the beam pointing that either generate some opticalscatter or modify the cavity contrast.
While pointing variations likely originate with small changes in the thermal environment,
it is easier to actively stabilize the beam position than to temperature stabilize all the optics. A
quadrant photodetector can measure horizontal and vertical variations of the beam position at the
PDH photodetector location. To quantify our sensitivity to this effect, we can measure the transfer
function between changes in the PDH DC voltage and cavity frequency when varying the beam

181

Frequency [Hz]2

Si6-Si3 beat
PDHDC - H coupling

1
0
1

Fractional MDev

20

10 15

40

60

Time [hour]

80

100

Si6
PDHDC - H coupling
PDHDC - V coupling

10 16
10 17

100

101

102
Time [s]

103

104

105

Figure 5.37: Top: Dedrifted Si3-Si6 beatnote overlaid with the PDHDC signal multiplied by thehorizontal sensitivity coupling. Bottom: Instability of the beat, and PDHDC measurements when
propagated through the horizontal and vertical transfer functions. Some sections of the beatnote
instability between 100-1000 s appear to be well described by the horizontal pointing deviations.
For longer averaging times, the cavity instability might result from a linear combination of verticaland horizontal pointing fluctuations.

pointing. We place a piezo between the pins of one of the steering mirrors used for coupling into
the cavity and record the corresponding frequency deviation (Fig.5.38).
We measure vertical and horizontal couplings of the PDHDC to frequency of 300 Hz/V and100 Hz/V respectively. A desired 0.1 Hz frequency stability (fractionally 5 × 10−16 ) requires 300
ppm control in the vertical direction, and 1000 ppm control in the horizontal direction. Based on
our measured long term instability, it appears that the passive control is already near this level. A

3

1.0

2

0.5

1

Frequency [Hz]Frequency [Hz]

1.5

0.0
0.5
1.0
1.5

25

50

75

100 125
Time [s]012

Si6-Si3 beat - H piezo drive
PDHDC

0

182
Si6-Si3 beat - V piezo drive
PDHDC

150

175

200

3

0

25

50

75

100 125
Time [s]

150

175

200

Figure 5.38: Modulating the voltage applied to the piezo allows us to measure a transfer function
between the cavity frequency response and changes in the PDH DC signal. We do this for bothaxes to measure the system’s sensitivity to beam pointing. Left: The piezo is place in the mirror’s
horizontal pin. We drive it with a sawtooth and simultaenously record the frequency deviation. Red
trace is the frequency response, and the green trace is the PDH DC voltage converted to frequency
via our measured coupling. Right: Same data, except the piezo is now driving the mirror’s verticalpin.

feedback loop for further stabilization would only require a gain of 10 dB at an averaging time of
104 seconds.

5.4.3

Top setup

The increased sensitivity to beam pointing is largely due to the need to couple to the HG01
mode of the cavity. That in turn was necessitated by the observation that the frequency noisefrom 0.01-0.1 Hz was highly dependent on the input beam angle, and could be reduced by nearly
20 dB when aligning to maximize transmission on HG01 . There are two routes to approach thisproblem - one is to build a quadrant photodetector, measure the beam wander by picking off light
before it is focused on to the PDH photodetector and servoing a piezo-controlled mirror. This isrelatively straightforward and we designed and assembled a circuit to provide the quadrant sum
and difference ports. The quadrant photodiode was unfortunately on backorder for a few months
and it wasn’t feasible to wait for this to arrive.
Instead we exercised our second route forward; we decided to construct a setup above thecavity (called the ”top breadboard”) that could launch the beat downwards through the top sub-

183
strate and mirror. The general idea is that we could avoid the bottom optic or substrate thatwas responsible for the poor RAM signal that we would measure. Of course, this is a significantly
larger undertaking and requires constructing an entirely separate setup that still needs to have no
additional technical noise contributions.
The majority of measurements made on the top breadboard have been tirelessly carried out
by our recent postdoc Zhibin Yao. Already, we have found some promising properties of this scheme.
In the bottom setup, we struggled to achieve more than 40% mode coupling to the cavity but fromthe top we can now achieve closer to 60%. Imaging the beam before it enters the vacuum chambergdoes not reveal any significant differences in the spatial mode. Rather, this suggests that one ofg
the bottom vacuum
optics may be adding a speckle onto the wavefront that decreases its overlap

with the cavity’s TEM00 mode. This is corroborated by images we take of a beam transmitted
from both sides of a cavity. A clear gaussian mode is imaged when coupling light from the bottom,
whereas coupling from the top results in a highly lopsided transmission profile (Fig.5.39)

Figure 5.39: Left: The light is coupled to the cavity from the bottom setup and the transmitted
beam is imaged from the top. Right: Light is coupled from the top and imaged on the bottom.

We can additionally measure the residual RAM from the top setup after active stabilization on
the RAM photodetector. Recall that this was the dominant technical noise contribution measured in
Fig.4.38 and was only slightly below our measured instability of Si6 with birefringent noise cancelled.
A reduction of this noise source could very well improve the cavity stability. Surprisingly our initial
measurements after aligning to cavity showed a dramatic improvement over the out of loop RAM
measured in the bottom setup. Without significant optimization the fractional frequency instability

184
dips below the Si3 thermal noise for over an order of magnitude (Fig.5.40). The fractional frequency
noise is also dramatically improved across the entire spectrum when compared to the RAM in thebottom setup. A 10 dB reduction of white noise is observed due to the increased mode coupling
and our ability to use the HG00 mode.
Our method of mounting the top breadboard on long optical posts above the cryostation opens
us up to issues of relative motion between the optical beam and the cavity. Differential displacement
noise can appear as refractive index changes while also modulating the cavity coupling. Currently
we measure a higher frequency instability from the top breadboard and we suspect that differential
motion is the likely cause.
The bottom setup has also been left intact during the installation of the top breadboard.
A second set of resonant PDs is required as we simply used the ones from the bottom, but thisleaves open the possibility of directly measuring birefringent noise using the PTB technique [20].
Two separate lasers can be locked to the independent birefringent modes of the cavity, and thefrequency difference (assuming all technical noise sources are below this level) will yield the direct
birefringent noise.

5.5

Drift rate
Low drift rates are coveted in cavities and in ULE spacers, cavity drift was thought to be

associated with the spacer slowly sagging with time. The development of cavities with crystalline
spacers and substrates held an additional attraction, namely that crystals are expected to have afixed length dictated by the crystal lattice constant. Drift should be a bygone property, nonexistentin the JILA and PTB silicon cavities.
Surprisingly a nonzero drift was observed; while orders of magnitude smaller than its ULE
counterparts, Si3 measured an absolute drift rate of -6 Hz/day, or fractionally −1.4 × 10−19 /s.
Over time this has exhibited a long term settling from −30 µHz/s to −25 µHz/s. Intermittent
interruptions where the laser falls out of lock have not affected this long term behaviour, suggestingthat this might be endemic to the cavity.

185

Fractional PSD (1/Hz)

10 29

OOL RAM - bottom
OOL RAM - top
Dark noise
Si3 thermal noise

10 30
10 31
10 32
10 33
10 34 2
10

Fractional MDev

10 15

10 16

10 1
Frequency (Hz)100

OOL RAM - top
OOL RAM - bottom
Dark noise
Si3 thermal noise
Si6 thermal noise

10 17
100

101
Time (s)102

103

Figure 5.40: Fig.4.12 is reproduced but now we include a measurement of RAM from the top setup.
The modified allan deviation of RAM instability is now consistently better than the Si3 thermal
noise from 0.5-10 s, and the fractional frequency noise shows a considerable 6 dB improvement
around 100 mHz. No effort was put into improving the long term instability of RAM in the top
setup, so we expect that this can be improved. The top setup couples to the HG00 mode while thebottom setup couples to HG01 .

Si4 however added some new information, the identification that the drift rate was dependent
on the intracavity power. With transmitted powers of 700 nW, the drift rate was over 50 times
than what was measured in Si3 at −6 × 10−19 . Significantly reducing the transmitted power to 40
nW resulted in a measured drift of −3 × 10−19 , only two times higher than Si3. Si3 maintains ahigher transmitted power of order 100 nW so it’s unclear why the behaviour is so different between

186
the two systems. Power dependent drift was unfortunately not measured in Si3. The Si4 results
suggest that some mechanism in the cavity favors few photons circulating between the mirrors. A
fit through the Si4 drift measurements suggests a zero drift at zero intracavity power but there isno clear theory as to why this might be the case.
Si6 coming online was exciting from the perspective of adding another data point to thesemeasurements. To our surprise, long term drift measurements of Si6 (after implementing changes
from section 5.4 to improve the long term instability) indicated a drift rate substantially lower thananything measured in Si4. Fractional drift rates below 1 × 10−19 /s were measured over several
one week periods. A particular measurement is shown in Fig.5.41. The Si3 drift was frequently
measured by the SrI lattice clock during an AC stark evaluation so we have accurate knowledge ofthe average drift over this period. We find a fractional drift rate below −6 × 10−20 /s, a factor three
lower than measured in Si3. It is not clear that this is even the true drift rate of the cavity, Fig.5.37
shows the same dataset and suggests that the residual instability could be due to slow variations
in beam pointing. The drift innate to the cavity material could be lower.
Regardless, this is significantly smaller than anything measured in Si4. Both cavity spacers
and substrates were cut from the same ingot so it follows that the silicon purity should be identical.
One conclusion we can infer is that the residual drift rate of Si3 and Si6 could have been due toa drift of the amorphous coatings. In the glassy material, there is no lattice constant to define
the intramolecular spacing so these may have been changing length over time. This doesn’t to be
an unreasonable statement - ULE spacers in our lab are found to have a drift rate of 10 kHz aday. The Si3 drift is over 1000 times smaller. The coating thickness is 10 µm, so fractionally thecoatings are 0.212/(2 · 10 × 10−6 ) = 104 smaller than the spacer. The reduction in drift measuredin Si3 is 103 , not quite commensurate with the switch from a glassy spacer to a crystalline spacer.
This speculation is heuristic and doesn’t take into account power dependent effects which
haven’t been deeply studied in either the Si3 or our ULE systems. Nevertheless, it does appear thatthe drift rate is significantly suppressed in fully crystalline cavities and a true measurement would
require a measurement campaign of several weeks while directly measuring the cavity frequency

187
with an optical clock.

3

Si6-Si3 beat - Si3 drift removed

Frequency [Hz]

4
5
6
7
8

0

20

40

60
Time [hour]

80

100

120

Figure 5.41: A five day measurement of the Si3-Si6 beatnote with the Si3 drift removed. During
this period the Si3 drift was measured by the SrI lattice clock so its sign and magnitude is know.
The dotted line is a fit to the entire dataset and finds a fractional drift rate of −2.3 × 10−20 /s,
whereas the dashed line fits to the latter two thirds where a more linear behaviour is observed.
This segment finds a fractional drift rate of −5.6 × 10−20 /s.

Chapter 6

Novel applications of crystalline cavities

Despite the birefringent noise identified in Si6, the capabilities of the cavity are still powerful.
This is not particular to crystalline coatings, but rather the unprecedented long term stability
attained with cryogenic silicon cavities. With Si3, several demonstrations highlighted the role ofthe cavity in future research directions. Here, we will explore some of those experiments. We
will identify the role of the cavity as not a technical upgrade, but rather a critical tool that hasenabled key advances in frequency metrology, fundamental physics tests, and searches for emergent
phenomena [47, 17, 133, 134].

6.1

Ratio Comparison
Searches for physics beyond the Standard Model description have often been associated witha top down approach where massive detectors sensitive to energy at the GeV level attempt direct
or indirect observation of new physics. Dark matter searches in particular have often followed thisapproach; a leading candidate for dark matter, the weakly interacting massive particle (WIMP), hasbeen the subject of intensive experimental efforts at accelerators and other large facilities without
conclusive detection [135]. The primary method for direct detection relies on energy deposition,i.e. where the interaction of WIMPs with a large volume of material and a useful signal is obtained
from the observation of a few events. With a low event rate, bubble chambers and scintillators will
employ 100s to 10000s of liters of a particular material in a sterile, underground environment. The
manpower and complexity of these experiments is significant and often prohibitive.

189
These experiments have succeeded in setting stringent limits on the WIMP candidate mass,
but generally in the high energy range. Other candidates for dark matter have been proposed in asignificantly lower mass range, including ultralight scalar dark matter (the dilaton). This class ofmodels proposes that dark matter is composed of a light bosonic field where the particle mass could
fall in the range of 10−24 − 1 eV [136, 137]. The coupling to Standard Model fields is theorized tolead to a coherent oscillation of the fine-structure α and the electron mass me [138]. The oscillation
frequency of this scalar field can be directly related to oscillations in the fine structure constant, withthe oscillation amplitude dictating the magnitude of the coupling to the electromagnetic gauge withmodulus de and to the electron mass with dme . In particular, the Lagrangian of the dark matterfield ϕ is given as [139]deLϕ = κϕFµν F µν − dme me ψ̄e ψe
4


(6.1)

The overall constant κ is static but the first term captures the coupling of the DM field to theelectromagnetic tensor Fµν and the second term describes the coupling to the electron mass withelectron field ϕe .
We’ve touched on this point several times but optical frequency metrology has demonstrated
some of the highest precision in frequency fluctuations. Optical and microwave transitions in atomic
clock transitions are known to demonstrate a dependence on the fine structure constant and electronmass. A differential dependence on α is evident between optical and microwave transitions In opticalclocks, the transition frequency exhibits the proportionality
foptical ∝ α2+ϵ me c2(6.2)

The factor ϵ originates from the relativistic correction to the atomic energy levels and differs between
atoms. In the nonrelativistic approximation, frequency ratios of different electronic transitions
between atoms would not exhibit α sensitivity [140, 139].
The enhancement factor ϵ determines the sensitivity to fluctuations in alpha. The variation

190
of the electronic transition frequency to fine structure oscillations can then be expressed as
δE
δα
=ϵ
E0
α

(6.3)

A frequency ratio intercomparison between optical clocks with different values of ϵ can therefore be sensitive to scalar DM oscillations. Between two clocks A and B, fluctuations in thefrequency ratio can be related to the variations in the fine structure constant via
δfA
fA




 δα
δfB
∝ ϵA − ϵB
fB
α

(6.4)

It is therefore the case that precise measurements of frequency ratios between optical clocks canplace bounds on the electromagnetic gauge modulus de [141, 142, 143]. In the BACON collaboration, we measured three frequency ratios between the Al+ ion clock, and the Yb and Sr latticeclocks. We use this dataset to perform one of the highest precision measurements of three fundamental constants of nature. In principle if we want to investigate the possibility of fluctuations
in α, i.e. fluctuations in these frequency ratios, the same dataset can be used to place bounds on
this oscillation. This of course relies on the electronic transition of each clock having a slightly
different relativistic correction to α. As it turns out, this is indeed the case and atomic structure
calculations reveal sensitivity factors [143]ϵAl+ = 0.008(6.5)

ϵY b = 0.31

(6.6)

ϵSr = 0.06

(6.7)

From Eq.6.5 we see that the largest sensitivity to α variation is obtained in the fY b /fAl+ frequency
intercomparison. However, this does not necessarily imply that frequency intercomparison between
these two clocks can lead to the most stringent exclusion bound. To clarify this point, we need tounderstand how to convert the ratio data to a bound on the electromagnetic gauge modulus de .
The ansatz for the coherent DM field is an oscillation [139, 144]
ϕ(t) = ϕ0 cos(mϕ t − mϕ ⟨v⟩ · r)

(6.8)

191
The first term concerns the time dependent oscillation of the scalar field and we see that the DM
mass is equated to the frequency of oscillation. From the perspective of metrology, the dark mattercandidate mass mϕ is related to the measurement bandwidth:mϕ = 4.13 × 10−15 f(6.9)

Note that this this relation implies that a longer data record allows us to place constraints on alower candidate mass range. In the case of the three-way frequency ratio measurement, the upper
end of the mass range is set by the cycle time of the involved clocks. For lattice clocks, a 4 point
locking sequence may be used to reject magnetic field noise by probing opposite stretched states,
meaning each measurement of the optical transition frequency occurs in ∼ 4 s. This corresponds
to the upper limit of mϕ ∼ 10−15 in our exclusion plot.
The second term identifies a spatial variation of the dark matter field between the measurement locations. The theory considers a slow moving mass of dark matter where v is the virial
velocity of dark matter in our galaxy and is roughly 5 × 10−4 c. Given that the clocks are notco-located, differential phase shifts due to the spatially varying DM field could be an important
contribution.
We can calculate the de Broglie wavelength of the coherent oscillation as λDB ≈ h/(mϕ v).
In the mass range of 10−21 < mϕ < 10−17 eV, the DM wavelength varies from 10-10,000 parsecs,
significantly larger than the O(1 km) separation between clocks. We can approximate this effectto be spatially homogeneous on the 1 km length scale, so we neglect it in our analysis and only
consider the time dependent oscillation. We can rewrite Eq.6.4 in terms of the coupling to the DM
field from the Lagrangian as
δfA
fA






δfB
= de ϵA − ϵB κϕ0 cos 4.13 × 10−15 f t
fB(6.10)The quantity ϕ0 is evaluated from the energy density ρDM = 21 m2ϕ ϕ20 , where it is measured thatρDM = 0.3 GeV/cm3 . We then identify the scaling
√
ϕ0 =

2ρDM
mϕ

(6.11)

192
The magnitude of the coupling de is then calculated from

de =

δfA
fA



δfB
fB



1
4.13 × 10−15 f√(ϵA − ϵB ) κ 2ρDM

(6.12)

Note the frequency dependence in our calculation of de . Measurement of a frequency ratio between√
two atomic clocks will typically scale with stability σy ∼ 1/ τ if not limited by any fluctuation
systematic effects. The power spectrum of this frequency measurement will look white (f 0 ), meaning that the de exclusion will scale as f 1 . A longer dataset will not only allow for exclusions to beplaced at a lower mass range, but will also place a lower limit on the magnitude of de . Ultimately,
constraints on the low mass range are limited by coherence lengths on the order of galaxies.
The transformation from a ratio dataset to an exclusion limit is therefore conceptually simple.
However in practice, the data is often taken in a non-continuous matter. Clocks may be operational
for a short period of time, then experience an extended down time, and then continue taking data
again. The existence of these time gaps complicates the calculation of the ratio’s power spectral
density. Traditional estimators for the power spectrum such as the periodogram are not appropriate
tools for this situation. We instead use a particular method intended as an improved estimator for
gapped data, the Lomb-Scargle periodogram.
The Lomb-Scargle periodogram may be unfamiliar to those primarily concerned with shortterm clock and oscillator instability since these are typically continuous data records. The periodogram is an appropriate estimator of the continuous Fourier spectrum that is no longer evenly
sampled but rather discretely sampled. For long datasets, the latter constraint may not hold andthe Lomb-Scargle is an improved estimator for datasets that are not evenly sampled. A confidence
interval is required to determine whether the LS periodogram actually contains a periodic signal.
Probabilities of these false positives can be obtained by bootstrapping of the data.
We now have all the tools necessary to produce a de exclusion limit, and a plot of ourcalculation is shown in Fig.6.2. Bounds are placed in the mass range of 10−21 − 10−15 eV.
In the ratios involving the Yb lattice clock, a deviation from de ∼ f is observed. This√originates from a deviation in the ratio stability from σy ∼ 1/ τ at long times. At this point,Fractional Frequency Noise (1/Hz)

100

193

White noise

10 1
10 2
103
101
10 1
10 33
10

1/f Noise

Random walk noise

100
10 3
6
10 10
5

Scipy periodogram
Lomb-Scargle
10 4
10 3

10 2

Frequency (Hz)10 1

Figure 6.1: We simulate a continuous dataset of different types of noise, Sy ∼ f 0 , f −1 , f −2 . The
performance of the periodogram and Lomb-Scargle periodogram estimators is compared. The
Lomb-Scargle periodogram consistently underestimates the noise spectrum by a factor of 2 inamplitude. A correction therefore needs to be included in order to obtain a comparable result.

the power spectrum of the Yb/Sr and Al+ /Yb ratios flicker, likely due to a fluctuating systematic
offset. The Al+ /Sr ratio has a shorter dataset and is not long enough to resolve this feature.
A more important feature is the comparison between the exclusion limits set by the Yb/Sr194
ratio and the Al+ /Yb ratio. From Eq.6.5, we expect
ϵY b − ϵAl+ = 0.302(6.13)ϵY b − ϵSr = 0.25

(6.14)

Sensitivity to a dark matter oscillation is larger in the Al+ /Yb ratio, yet the exclusion bound is not
as strong as that between the lattice clock ratios. This is due to the superior stability demonstrated
in lattice clocks, largely due to the fact that neutral atom clocks can interrogate several atoms at
once and have much smaller quantum projection noise of the measurement. The magnitude ofthis limit is directly related to the Dick effect - longer interrogation times and shorter deadtimes
can improve the de exclusion, just as it enables shorter averaging times to reach a particularlevel of frequency precision. Nevertheless, particular ion-neutral atom frequency comparisons suchas 171 Yb+ /87 Sr which are currently operating have such a strong alpha sensitivity that the highinstability of the ratio measurement can still lead to superior de exclusion limits [145]. Proposed
neutral atom comparisons [146] leverage certain transitions that can provide greater sensitivity withthe significantly improved instability of frequency comparisons between optical lattice clocks.
A significant variable for improvement is the stability of the local oscillator. Better cavity
stabilized lasers with lower thermal noise limited performance will enable a reduction in the delimit for all frequency intercomparisons. The cavity performance is therefore a crucial component
in the role of clocks probing fundamental physics.

6.2

Dark matter searches with oscillators
Frequency ratio comparisons are clearly a powerful tool for investigating α variations, andthe role of a stable oscillator is evident. However, there is another opportunity presented when theoscillator in question is a monocrystalline cavity. In the ratio comparison, sensitivity to de is only
attained due to the differential dependence on α between electronic transitions in different clock
atoms. We saw that this originates from the relativistic correction to the transition energy. In a
crystal, the length is classically determined by the number of atoms N and the Bohr radius aB ;

195

Stochastic Amp. de Limit100

Al + /Sr
Al + /Yb
Yb/Sr

10 1
10 2
10 3
10 4
10 5
10 6
10 7

10 21

10 20

10 19

Mass m (eV)

10 18

10 17

Figure 6.2: Exclusion limits for de measured between the three optical clocks during the BACON
frequency intercomparison data campaign. The dataset is the same as that from [11]. The moststringent bounds on de are set by the Yb/Sr frequency ratio despite the relatively higher sensitivityof the Al+ /Yb comparison.

there is no room for a significant relativistic contribution. In a cavity with a crystalline spacer, thephase shift defined by the cavity length L is simply Φ = ωL/c. The frequency ω can be arbitrary
but is typically chosen to be the optical transition frequency of the clock atom, so that ω ∝ e2 /aB ℏ.
Then,
Φ=

ωL
e2 N aB∝= NαcaB ℏ c(6.15)

A variation in the fine structure constant then appears as a proportional variation of the laser phase
δΦ
δα
∼
Φ
α

(6.16)The sensitivity of the cavity frequency to fine structure oscillations scales as α1 . If we now compare
this to the electronic transition frequency of the strontium atom, we can obtain a significantly

196
enhanced alpha sensitivity as
δfSr
fSr



δfcav
δα
∝ (2.06 − 1)fcav
α

(6.17)

Remarkably, measuring the cavity frequency (which is already required in clock operation) can
provide a 4 fold enhancement in sensitivity when compared to the Yb/Sr ratio (∆ϵ = 1.06 comparedto ∆ϵ = 0.25). The crystalline cavity is no longer a factor in the magnitude of the exclusion limit,
but is now directly used as a probe for variations in fundamental constants [47].
We can now perform a calculation of the de exclusion limit in the same manner as we didwith the ratio data. In both studies, it is important to note that the observation run extends fora time less than the coherence time of the dark matter oscillation. Were this not the case, further
collection of data would not improve the signal to noise of the DM signal. The dataset for the
fSr /fcav comparison is a consecutive measurement over 12 days using the 124 K Si3 cavity. Despite
the earlier assertion that the ratio dataset could be used for the cavity-Sr atom analysis, technicalissues prevent us from doing so. At the time of the three-way ratio data campaign, measurementof the cavity frequency was corrupted by additional path length noise that we had not identified in
the distribution of the cavity stabilized light to the Sr atom. Measurement of the cavity frequency
needs to be verified by the noise model of the cavity instability. Otherwise, additional noise may
be misconstrued as a DM oscillation. The silicon cavity noise spectrum is fit to a model of
Sy = 4 × 10−33 f 0 + 1.7 × 10−33 f −1 + 9 × 10−38 f −2

(6.18)

During this measurement campaign, a hydrogen maser at NIST was distributed to JILA
across a fiber link, allowing for a direct comparison of the Sr and silicon cavity frequencies to themaser frequency. The maser has a significantly higher instability when compared to the opticaloscillator and the optical atomic transition, however it operates on a hyperfine manifold which
exhibits a larger sensitivity to α. The transition frequency varies as2

2

fH ∝ (α me )α µp



me
mp


(6.19)

1.0

10 25

0.5

PSD (1/Hz)

Fractional Dev. (10 14)

197

10 22

0.0
0.5

H/Si
Sr/Si

1.0

0

10 28
10 31
10 34
10 37

10
20
30
Elapsed Time (Days)

H/Si Data
Sr/Si Data
Sr/Si Model
H/Si Model

10 5 10 3 10 1
Frequency (Hz)Figure 6.3: Left: Comparison of the data record of the H-maser measured against the silicon cavity
frequency, and the 1D Sr lattice clock measured against the silicon cavity. The time series of themaser-cavity frequency ratio is decimated by a factor of 720 for clarity. The short term instabilityof the atom-cavity measurement is far superior to that of the maser-cavity ratio, but the enhanced
sensitivity of the maser to the scalar DM oscillation will enable it to set a competitive bound.
Right: Frequency noise of the fH /fcav (blue) and the fSr /fcav (red) ratios calculated with theLomb-Scargle method. A noise model for each ratio derived from the frequency noise contributionsof the cavity, maser, and 1D Sr lattice is shown in black.

(a)

(b)

(c)

0.5

10

Sr/Si

PSD (1/Hz)

Fractional Dev. (10 14)

Figure 6.4: A cartoon illustrating the dependence of the H maser, silicon cavity, and strontium
1.0on α andH/Si
clock transition
me .
25

0.0

10 29

Dataand mp is the proton mass.
Where µp is the proton magnetic moment, me is the electronH/Si
mass,
Sr/Si Data
10 33
0.5
Sr/Sidue
Model
Comparison with the Sr atom will not provide a significant boundto the relatively similar α2+ϵ

1.0

0
20
Elapsed Time (Days)

10 37

H/Si Model

10 5 10 3 10 1
Frequency (Hz)198
dependence, but frequency ratio between the H maser and silicon cavity will appear asδfH δfcav
m2
m2 δα∝ α 4 µ p e N α ∼ α 3 µp e
fH
fcav
mp
mp α

(6.20)The scaling with α3 is significantly more favourable than the α1.06 from the cavity-atom comparison.
The frequency noise visible in Fig.6.3 indicates an H/Si frequency ratio dominated by the white noiseof the maser at nearly all frequencies, but the higher α sensitive will allow the maser comparison
to place a competitive de bound at low measurement bandwidth.

Correction

1.0
0.5
0.0 7
10

10 5
10 6
Frequency (Hz)10 4

Figure 6.5: The ratio of the observed DM signal compared to the actual signal amplitude when
considering a nonzero phase δ to the DM field.

Before producing an exclusion plot, there are a few points to consider. Unlike the frequency
ratio data, both the cavity and H maser experience a linear drift which is removed in the analysis.
The impact of this drift on the generated Lomb-Scargle spectrum needs to be carefully quantified.
We simulate a DM oscillation over a range of frequencies and compare the peak amplitude recovered
through the LS periodogram with the known amplitude. The obtained amplitude is underestimatedby a factor of up to 0.7 at our lowest measurement bandwidth of 0.7 µHz. A similar correctionneeds to be applied in consideration of a slowly varying phase of the DM field. Our expression forthe scalar field should in fact contain a phase term δ as ϕ(t) ∝ cos(mϕ t + δ). Particular valuesof δ can lead to an underestimation of the field amplitude. At high bandwidth, the DM fieldcompletes several oscillations over the data campaign so that the observed amplitude is equivalent

199
to the actual signal amplitude. This is not the case for lower measurement bandwidth, and again
a correction needs to be applied. A simulation of the necessary correction is illustrated in Fig.6.5.
With these points in mind, we can generate the de exclusion plot which is shown in Fig.6.6.
The cavity-atom comparison demonstrates a significant improvement in the exclusion bound within
the mass range of 10−19 < mϕ < 10−17 . Below this range, the maser-cavity comparison sets a lower
limit due to the improved long term instability of the maser. This is comparable to the constraints
placed by a Rb/Cs atom-atom frequency comparison [15].

10 2

Exclusion Limit (de)

10 3
10 4
10 5
10 6

H/Si Limit
Sr/Si Limit
Sr/Si Projected

Dy/Dy
Rb/Cs
Network
UW
MICROSCOPE

10 20 10 19 10 18 10 17 10 16
Dark Matter Candidate Mass: m c2(eV)Figure 6.6: The constraints placed in this work are the blue curve for the maser-cavity frequency
measurement, and the red curve denoting the atom-cavity measurement. Purple lines indicatedlimits set from equivalence principle tests [12, 13]. Black curves are exclusion bounds determinedfrom atom-atom frequency comparisons [14, 15, 16].

It is evident that tabletop atomic physics experiments can provide meaningful contributions
to the search for varying fundamental constants. More important is the fact that exclusion bounds

200
placed by this experiment use an entirely different platform. The cavity stability is directly related
to the magnitude of the fSr /fcav bound that we can set. An improved long term instability ora lower thermal noise limit enables a more stringent exclusion of de . The current bound set in
Fig.6.6 does not even use the full capability of the local oscillator stability. Uptime of the Srclock during measurement of the cavity was only 30% of the 11 day data campaign. The deadtime introduced by the time gaps is significant; a calculation of the bound that could be set in
the fSr /fcav measurement with a continuous operation is calculated as the dotted green curve of
Fig.6.6.

Exclusion Limit (dme)

10 2
10 3
10 4
10 5

H/Si
UW
MICROSCOPE

10 20
10 19
10 18
Dark Matter Candidate Mass: m c2(eV)Figure 6.7: Exclusion limit for the electron mass coupling dme obtained from the maser-cavity
frequency comparison.

There is one additional point of interest that we have not yet touched on. The second term of
the Lagrangian in Eq.6.1 reveals a sensitivity to the electron mass with the electron mass modulus
dme . Oscillators or references with differential me sensitivity could contribute to an exclusion limit
for this coupling. The cavity and Sr atomic frequency both scale with me so their frequency ratio
will be insensitive to fluctuations of this term. The H maser frequency however scales with m2e so
a frequency comparison with either the cavity or Sr atom could constrain the value of dme withsensitivity ∼ me . The H/Si comparison has lower deadtime and probes a larger mass range due to201
the longer extent of the data campaign, so we use this to evaluate the dme exclusion.
The exclusion is displayed in Fig.6.7. The maser-cavity measurement results in a significantly
lower bound in the range of 2 × 10−20 to 3 × 10−19 . The exclusion magnitude is limited by the H
maser instability in the majority of this mass range. Nevertheless, the measurement precision isgood enough that an extended data campaign could still place competitive bounds in a lower mass
range.

6.3

TimescaleIn the previous section we leveraged the excellent long term stability of the silicon cavity toplace competitive bounds at the low mass range of de and dme . For the de limit, a particularly
important feature is the lower frequency noise of the silicon cavity when compared to the maser. In
Fig.6.3(b) the silicon cavity displays a superior performance at frequencies down to 1 µHz. Statedanother way, the silicon cavity is a superior oscillator at averaging times up to nearly 105 seconds,
coincidentally the timescale of a day.
This begs the question of whether the silicon cavity as a free running oscillator can act as atimescale superior to the maser. Over the course of a single day, this is clear from the improved
optical local oscillator (OLO) instability. The maser instability is well known from an N-cornered
hat analysis of the NIST AT1 maser array. Measurement against the silicon cavity frequency
therefore allows us to ascribe features above the maser noise model to the optical cavity instability.
In Fig.6.8, we see that measurement of the silicon cavity frequency by the AT1 timescale is only
limited by the optical oscillator at 2 × 105 s.
At longer averaging times it may appear that the optical oscillator has already lost thecapability to serve as an improved timescale, but instability is not necessarily the most important
metric. Eric Oelker and Will Milner vested considerable time with the NIST timescale division to
apply the usual predictive model used for microwave standards to the optical oscillator. The detailsof this study are more thoroughly discussed in [17] so we’ll only cover the key conceptual points
needed to illustrate the unique space that these silicon cavities occupy. For microwave timescales,

202

Figure 6.8: Long term instability of the silicon cavity (orange and red points) with our model forthe oscillator noise (dashed black line). In a single day’s measurement, particular Si3 datasets havedisplayed thermal noise limited performance out to 104 seconds. This low instability is naturally
degraded with the inclusion of significantly more data. A similar effect is apparent in the Si6 longterm behaviour, short datasets of 6-8 hours demonstrate instabilities of mod σy < 1 × 10−16 for 103
s, though a week long dataset shows mod σy = 1 × 10−16 at 200 s. Our intent is to point out that
length of the dataset is critical when assessing the long term cavity behaviour.

a Kalman filter is typically used to estimate the oscillator frequency by weighting the accumulated
timing errors. This is achieved by measuring the oscillator against a frequency standard. In our case
we use our 1D Sr optical lattice clock which is unconventional in the benchmarking of microwave
timescales. However recall that we established this as an appropriate choice of standard in the threeway optical frequency ratio intercomparison of [11]; measurements of the Yb/Sr optical frequency
ratio over a five month period resulted in a series of measurements spread within a 1 × 10−17 range,
each with total statistical and systematic uncertainties at the mid 10−18 level. Accuracy aside, thecapability of the neutral atom Sr clock to consistently measure an optical frequency is evident. More
importantly, the ratio evaluation is performed at a level much lower than the OLO and microwave
stabilities across all averaging times. This makes the Sr clock suitable as a frequency reference.

203
It is crucial to note that the prospect of using Si3 as a timescale is inherently tied to thepredictable drift rate of crystalline cavities [147]. Predictable not in the sense that the drift rate
can be calculated ab initio, but rather that it is consistent over several months. To date, no ULE
cavity has exhibited such performance and likely never will; changes in the drift rate of our 40 cm
ULE cavity already require daily dedrifting for use as an OLO in the 1D optical lattice clock, letalone its use in a timescale. The Si3 fractional drift rate of 1.4 × 10−19 s−1 exceeds that of NISTs
older masers (5 × 10−22 s−1 ) but is consistently stable over the span of a year, whereas the latter
is not stable over intervals of months. A lower drift rate therefore does not establish a superior
timescale, a high level of predictability of the oscillator frequency is more important.
Furthermore, a true timescale is generated by steering to an atomic standard, not simply
evaluating the LO frequency deviations. At this point we’ve discussed limitations due to deadtime
several times over. The situation here is no different - a superior shot term stability will always
lead to a lower Dick effect-limited instability. In the context of a timescale, steering a microwave
standard with a fixed duty cycle to a precise optical reference will not outperform the steered OLO.
This carries the caveat that the reference uptime is not so intermittent that the OLO instability has
surpassed the microwave standard, but we see from Fig.6.8 that this simply requires us to operate
the clock once per day.
A calculation of this effect is presented in Fig.6.9. The ability of the OLO to maintain low
frequency instability is critical for the superior performance of this all-optical timescale. The longterm predictive behaviour is equally important in operating this timescale for several months; thefirst quality has so far only been realized with cryogenic crystalline reference cavities and the second
quality is a feature unique to them. A free running optical oscillator oscillator has been used toestablish improved timekeeping when compared to conventional microwave standards. Again, we
make the point that these crystalline resonators directly enable advances in metrology rather thanindirectly aiding clock stability.

204

Modified Allan deviation

10 15

10 16

10 17

10 18

Maser Steered 1 Hour/Day
Maser Steered 12 Hours/Day
Cavity Steered 1 Hour/Day
Cavity Steered 12 Hours/Day
103
104
105

106

Averaging time (s)107

Figure 6.9: Figure adapted from [17]. Stability of the optical and microwave timescales is evaluatedwhen measuring against an optical Sr standard for two duty cycles. Measurement of the OLO for
1 hour per day is still superior to the microwave timescale when measured against the strontium
clock 12 hours/day. This is directly related to the superior short term thermal noise limited
performance of Si3 and the excellent long term control. We additionally display the significant
advantage enabled by measuring the microwave timescale against an optical reference compared toa microwave reference.

6.4

Gravitational wave detection
Given the small displacement fluctuations that can be measured with our silicon cavities, it

is interesting to consider the strain sensitivity of two orthogonally oriented resonators. This is nota novel concept - resonant mass detection of gravitational waves (GWs) was first investigated inWeber bars [148], and later refined in resonant detectors such as AURIGA [149], miniGRAIL [150],
and ALLEGRO [151]. These generally use capacitive transduction to measure the stretching andcontraction of the bar detector in response to an incident GW. Crystalline resonators straddle aunique space between GW interferometers and resonant detectors, where optical interferometry isused to read out length variations of the Fabry-Perot but the rigid spacer between the mirrors turns
this into a resonant device. The cavity therefore becomes a transducer of mechanical displacement
into changes of optical phase, which we are proficient at measuring.

205
The sensitivity limit to length fluctuations is determined by the measured noise spectrumof the oscillator, and fundamentally limited by the thermal noise of the resonator. We omitted
one caveat when discussing this in Chapter 2 primarily because we aim to interrogate the cavity
thermal noise at the low bandwidth ≲ 1 Hz regime. In reality, the high frequency thermal noise ofeach component will be lowpassed above the resonator’s resonance; this same principle is present
in LIGO’s ability to lowpass the suspension thermal noise with a steep Sy ∼ f −4 character. The
total thermal noise of the cavity is more accurately expressed as [152]Sx,Brownian =

ϕsp
4kB T
f04
2
2
2π 2 Esp (Ro2 − Ri ) (f0 − f 2 )2 + f04 ϕ2sp f

2 )4kB T (1 − σsb12d(1 − 2σsb ϕcoat+
ϕsb 1 + √2π 1.5 Esb w0 f
w0 π(1 − σsb ) ϕsb(6.21)(6.22)The spacer Brownian noise is lowpassed above the resonant frequency f0 , and enhanced
around this region by the material’s quality factor. Note that similar lowpassing of the substrateand coating noise terms also occurs, but the resonant frequency of these components are significantly
higher than that of the spacer.
The interferometer’s strain sensitivity Sh (f ) is determined from the fractional length noiseas [153]
Sh (f ) =Sy (f )
|G(f )|2(6.23)

Here G(f ) is the cavity transfer function to a length perturbation. This can be modeled as adamped harmonic oscillator driven by a gravitational wave of frequency f . As we discussed in
the thermal noise section, the transfer function of the system displays a lowpassing effect withenhanced sensitivity at the resonance frequency f0 , with amplitude determined by the mechanicalquality factor of the resonating material. The effect of the transfer function is therefore to resonantly
amplify a signal at the spring’s resonant frequency f0 . Maximal sensitivity to a GW signal in a
local frequency band will occur at f0 . The efficacy of using cavities for GW detection amounts tomonitoring signals around the primary internal mode of vibration.

206
However there is one important point to note, the driving force, the gravitational wave inthis instance, scales with frequency as [154]FGW ∝ h(2πf )2 cos(2πf t)(6.24)The effect of this is to scale the typical damped harmonic oscillator transfer function by f 2 , effectively treating it as a highpass filter below the resonance. From Eq.6.23, we can now see thatthe strain sensitivity of the Fabry-Perot is highly suppressed as the bandwidth decreases. Unfortunately, this is the exact frequency band where these oscillators attain the best length instability.
Note that this measurement, unlike the others presented in this chapter, does not rely on theuse of crystalline cavities. Rather, cryogenic silicon cavities provide a measurement advantage dueto the lower thermal limit, and the enhanced detection sensitivity at f0 due to the significantly lower
loss angle of silicon compared to ULE. This is important in the context that cavity measurementof gravitational waves does not provide a competitive sensitivity limit in other practical frequency
bands. Unlike normal optical interferometry, the resonator geometry design choice is inverted: themechanical resonance is no longer placed above the bandwidth of interest, but rather is purposely
placed at a frequency where the gravitational wave signal power is expected to be high.
We illustrate this effect in Fig.6.10. In Si6, a spacer resonance appears at 10s of kHz, manifesting as a sharp peak in the thermal noise limit of the cavity. In an actual experiment, there
will be a white noise limitation due to the shot noise limit which can be reduced by coupling more
optical power into the cavity. In our current realization of the Si6 stability, we operate with lowlevels of light to reduce any birefringent effects and the shot noise contribution at the spacer resonant frequency is much higher than the resonance peak amplitude. In practice, it is not feasible toreduce the shot noise limit by the five orders of magnitude necessary to have the Si6 total thermal
noise be dominant at 30 kHz. We consider all cases however, simply for the purpose of providing
some intuition.
The cavity’s mechanical transfer function also has a resonance at the same mechanical eigenfrequency of the cavity. The resulting strain sensitivity displays a sharp, narrow enhanced sensitivity

207

Sy(f) [1/Hz]

10 32
10 35
10 38
10 41

Sh(f) [1/ Hz]

10 44 3
10
10 12
10 14
10 16
10 18
10 20
10 22
10 24 2
10

Si6 - measured noiseSi6 - total thermal noiseSi6 - spacer thermal noise

10 2

10 1

101
100
102
Frequency [Hz]

103

104

105

Si6 - measured noiseSi6 - total thermal noiseSi6 - spacer thermal noise
aLIGO

103

Frequency [Hz]

104

105

Figure 6.10: Top: Bottom: Curves from the top plot are converted to a strain sensitivity bynormalization with the cavity’s mechanical response to a gravitational wave. The aLIGO strain
sensitivity is displayed in red.

at this region for a cavity with noise spectrum dominated by technical effects, or coating/substrate
thermal noise. If instead the spacer thermal noise is dominant, the strain Sh (f ) will instead bemonotonically decreasing with frequency.
At the highest sensitivity of aLIGO in the 100-1000 Hz band, strain measurements with athermal noise limited Si6 cavity is nearly 10 orders of magnitude higher in amplitude. State of theart Fabry-Perot cavities have no chance at providing useful bounds on GW detection. Note that
this comparison is strictly related to the detection of gravitational waves, meaning that the rigid

208
spacer of the cavity is a significant inhibitor to detection. If we are instead concerned with the√√amplitude spectral density of length fluctuations Sx (m/ Hz), thermal noise limited cryogenic
reference cavities can provide a better resolution. This again requires the shot noise limit to be√
negligible, but Si6 for example achieves a length noise spectral resolution of 3 × 10−19 m/ Hz at√
100 Hz, compared to ∼ 2 × 10−20 m/ Hz for aLIGO.
The only promising advantage of these systems occurs at high frequencies > 10 kHz, outside
the band of GW interferometer operation. There are of course many technical difficulties associatedwith achieving this level of strain sensitivity with tabletop optical cavities, and it is only useful if
there are interesting GW sources emitting at these frequency bands. Nevertheless it is interesting toconsider the whether these systems can serve as useful detectors for astrophysical phenomena.

Chapter 7

Conclusion

We begin our discussion of ultrastable lasers by remarking that the usual and reallasers which we can buy and use in the lab do not in fact emit coherent light. The
critical issue here is phase stability, required for any meaningful claim of coherence. We use lasers to allow us to excite transitions in the optical domain, since
the higher frequency can lead to a higher line Q. However it also immediately brings
us the expectation of phase incoherence within the few millisecond domain due tovibrations imposed on the resonator structure. Our position is that many of themost interesting physical measurements using lasers will only be possible if the fullpotential coherence time is available – limited only by inescapable fundamental processes. Thus it is interesting work to identify and eliminate non- essential noise
sources so that one may begin to appreciate what the real limits may be. - Jan L.
Hall
At this point, we’ve hopefully presented a convincing case as to why developing better local oscillators are imperative to pushing frontiers of optical frequency metrology. However, we’ve also
demonstrated that a tremendous amount of work is necessary to achieve the levels of stability realized in this thesis. Promising candidates for reducing Brownian thermal noise such as AlGaAs/GaAs
crystalline coatings have not lead to a straightforward reduction in oscillator instability. In this firstinvestigation of the spectral noise of a silicon cavity with crystalline coatings, we observed several
static and dynamic light-induced effects that affect the stability of the cavity’s spectral feature. To
date, no clear theoretical framework has been presented which will be important for future use of
these coatings in ultrastable oscillators, gravitational wave detectors, and quantum optomechanics
experiments.
So what is next? It appears that the future of low noise optical interferometry may be at an
impasse. Significant research efforts have been dedicated to developing new dielectric coatings that210
can exhibit higher mechanical Q than conventionally used SiO2 /Ta2 O5 coatings, but this may not
yield the factor of 10 in loss angle that we hope to achieve. Making longer cavities is not prohibitive,
but does become technically demanding when placing the resonator in a cryogenic environment.
A more meaningful approach is to consider what levels of stability we may need. It would
be a delusion to expect that similar approaches to developing silicon resonators could realize an
oscillator with 10−19 level instability. The demands of reducing technical noise sources are far too
high. Fig.7.1 shows us the optical coherence time Tcoh that can be expected for different cavity
stabilized lasers that we have realized in our lab, or have considered for future systems. Different
laser coherence times can be obtained with different measurement methods, here we integrate theBrownian thermal noise limit until the the RMS phase excursion of the laser is 1 rad. This scales
as Tcoh ∼ τ −1 so an improved cavity stability will lead to an improved coherence time. What isnoteworthy is that particular systems that we could consider (21 cm cavities at 16 K) provide a
dramatic improvement in the optical phase coherence. Issues related to crystalline coatings aside,a fully crystalline 21 cm cavity at 16 K could coherently drive a transition for the entire lifetime of
the 87 Sr 3 P0 clock state.
This is a metrologically useful ultimate goal for oscillator stability, i.e. coherently interrogate
the optical clock transition over the maximum possible spectroscopic duration. The technicaldemands are also not outrageous. In Si6 we reduce all noise sources to the 10−17 level or below(with the exception of RAM, which we identify as an optical scatter issue). Electronic contributions
will be reduced proportionally with the cavity linewidth, meaning a three times longer cavity will
make these sources three times less relevant. Technical effects that couple to the cavity length canbe proportionally suppressed with additional thermal mass in the case of temperature fluctuations,or reduced via more aggression active stabilization in the case of vibration noise.
There will certainly be challenges associated with realizing a Brownian thermal noise limited
performance of 4 × 10−18 , but this is not out of the realm of imagination, considering what we havealready demonstrated in the JILA-PTB collaboration. With the rapid progress still being madewith cryogenic reference cavities, we could very well reach the point where a classical resource

211

mod y( )10 15

40 cm ULE
Si2/Si3
21 cm, dielectric coatings, 16 K
21 cm, crystalline coatings, 16 K

10 16
10 17
10 18

100

101

Averaging time [s]102

Figure 7.1: We plot the total Brownian thermal noise limit for MJM (purple), Si2/Si3 (green),
the limit for a 21 cm silicon cavity at 16 K with dielectric coatings using the measured loss angle
from [18], and the limit for a 21 cm silicon cavity at 16 K with AlGaAs/GaAs coatings using themeasured loss angles from [19, 20]. A dashed black line indicates the instability where an oscillator
has experienced a phase excursion of ϕrms = 1 rad.

provides us with all the capabilities we would ever require for an optical oscillator. The record
breaking long term instability that we have demonstrated has no immediate competitor so far.
Low noise optical oscillators have come far from the days where Hz-level laser linewidth wasa technical feat in the 2000-2010 epoch. We have now afforded ourselves the right to speculate
on capabilities of true mHz linewidth oscillators. Future oscillators could remain phase coherent
over a significant span of the distance between the earth and the sun. Again, we stress that thiswill enable new approaches to the way we investigate fundamental physics. Cavity stabilized lasers
have already been used to search for Lorentz violations [155, 133], but space-based clocks [156] andinterferometers will directly benefit from the enhanced sensitivity afforded by highly phase stableoscillators.
Working in low noise optical interferometry can often be a thankless task, but the importanceof its outcomes cannot be disputed. The unofficial patron saint of cavity stabilization summarizes
this nicely:

212
Cheers to all ye who wonder about these kinds of beautiful things - Jan L. Hall

Bibliography[1] S. T. Dawkins, J. J. McFerran, and A. N. Luiten, “Considerations on the measurementof the stability of oscillators with frequency counters,” IEEE Transactions on Ultrasonics,
Ferroelectrics, and Frequency Control, vol. 54, no. 5, pp. 918–925, 2007.
[2] A. Gillespie and F. Raab, “Thermally excited vibrations of the mirrors of laser interferometer
gravitational-wave detectors,” Phys. Rev. D, vol. 52, pp. 577–585, Jul 1995.
[3] C. C. Lam and D. H. Douglass, “Observation of oxygen impurities in single-crystal silicon bymeans of internal friction,” Journal of Low Temperature Physics, vol. 44, pp. 259–264, 1981.
[4] A. White, “Frequency stabilization of gas lasers,” IEEE Journal of Quantum Electronics,
vol. 1, no. 8, pp. 349–357, 1965.
[5] R. Drever, J. L. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, and H. Ward, “Laser phaseand frequency stabilization using an optical resonator,” Applied Physics B, vol. 31, no. 2,
pp. 97–105, 1983.
[6] T. Hansch and B. Couillaud, “Laser frequency stabilization by polarization spectroscopy ofa reflecting reference cavity,” Optics communications, vol. 35, no. 3, pp. 441–444, 1980.
[7] K. Numata, A. Kemery, and J. Camp, “Thermal-noise limit in the frequency stabilization oflasers with rigid cavities,” Phys. Rev. Lett., vol. 93, p. 250602, Dec 2004.
[8] G. Harry, T. P. Bodiya, and R. DeSalvo, Optical Coatings and Thermal Noise in Precision
Measurement. 2012.
[9] V. Baglin, “Cryopumping and vacuum systems,” 2020.
[10] J. L. Hall, J. Ye, and L.-S. Ma, “Measurement of mirror birefringence at the sub-ppm level:
Proposed application to a test of qed,” Phys. Rev. A, vol. 62, p. 013815, Jun 2000.
[11] B. A. C. O. N. B. Collaboration, “Frequency ratio measurements at 18-digit accuracy using
an optical clock network,” Nature, vol. 591, no. 7851, pp. 564–569, 2021.
[12] S. Schlamminger, K.-Y. Choi, T. A. Wagner, J. H. Gundlach, and E. G. Adelberger, “Testof the equivalence principle using a rotating torsion balance,” Phys. Rev. Lett., vol. 100,
p. 041101, Jan 2008.
[13] J. Bergé, P. Brax, G. Métris, M. Pernot-Borràs, P. Touboul, and J.-P. Uzan, “Microscope
mission: First constraints on the violation of the weak equivalence principle by a light scalardilaton,” Phys. Rev. Lett., vol. 120, p. 141101, Apr 2018.

214[14] K. Van Tilburg, N. Leefer, L. Bougas, and D. Budker, “Search for ultralight scalar dark
matter with atomic spectroscopy,” Phys. Rev. Lett., vol. 115, p. 011802, Jun 2015.
[15] A. Hees, J. Guéna, M. Abgrall, S. Bize, and P. Wolf, “Searching for an oscillating massive
scalar field as a dark matter candidate using atomic hyperfine frequency comparisons,” Phys.
Rev. Lett., vol. 117, p. 061301, Aug 2016.
[16] P. Wcislo, P. Ablewski, K. Beloy, S. Bilicki, M. Bober, R. Brown, R. Fasano, R. Ciurylo,
H. Hachisu, T. Ido, et al., “New bounds on dark matter coupling from a global network of
optical atomic clocks,” Science Advances, vol. 4, no. 12, p. eaau4869, 2018.
[17] W. R. Milner, J. M. Robinson, C. J. Kennedy, T. Bothwell, D. Kedar, D. G. Matei, T. Legero,U. Sterr, F. Riehle, H. Leopardi, T. M. Fortier, J. A. Sherman, J. Levine, J. Yao, J. Ye, andE. Oelker, “Demonstration of a timescale based on a stable optical carrier,” Phys. Rev. Lett.,
vol. 123, p. 173201, Oct 2019.
[18] J. M. Robinson, E. Oelker, W. R. Milner, D. Kedar, W. Zhang, T. Legero, D. G. Matei,
S. Häfner, F. Riehle, U. Sterr, and J. Ye, “Thermal noise and mechanical loss of sio2/ta2o5
optical coatings at cryogenic temperatures,” Opt. Lett., vol. 46, pp. 592–595, Feb 2021.
[19] G. D. Cole, W. Zhang, M. J. Martin, J. Ye, and M. Aspelmeyer, “Tenfold reduction ofbrownian noise in high-reflectivity optical coatings,” Nature Photonics, vol. 7, no. 8, pp. 644–
650, 2013.
[20] J. Yu, D. Kedar, S. Häfner, T. Legero, F. Riehle, S. Herbers, D. Nicolodi, C. Y. Ma, J. M.
Robinson, E. Oelker, J. Ye, and U. Sterr, “Excess noise in highly reflective crystalline mirror
coatings,” 2022.
[21] I. Bloch, J. Dalibard, and S. Nascimbène, “Quantum simulations with ultracold quantum
gases,” Nature Physics, vol. 8, pp. 267–276, Apr. 2012.
[22] S. Lloyd, “Universal quantum simulators,” Science, vol. 273, no. 5278, pp. 1073–1078, 1996.
[23] C. Gross and I. Bloch, “Quantum simulations with ultracold atoms in optical lattices,”
Science, vol. 357, no. 6355, pp. 995–1001, 2017.
[24] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of bose-einstein condensation in a dilute atomic vapor,” Science, vol. 269, no. 5221,
pp. 198–201, 1995.
[25] B. DeMarco and D. S. Jin, “Onset of fermi degeneracy in a trapped atomic gas,” Science,
vol. 285, no. 5434, pp. 1703–1706, 1999.
[26] P. Törmä and K. Sengstock, eds., Quantum Gas Experiments - Exploring Many-Body States.
Cold Atoms Series, Imperial College Press, 2014.
[27] J. L. Bohn, A. M. Rey, and J. Ye, “Cold molecules: Progress in quantum engineering ofchemistry and quantum matter,” Science, vol. 357, no. 6355, pp. 1002–1010, 2017.
[28] W. S. Bakr, J. I. Gillen, A. Peng, S. Fölling, and M. Greiner, “A quantum gas microscopefor detecting single atoms in a hubbard-regime optical lattice,” Nature, vol. 462, no. 7269,
pp. 74–77, 2009.

215[29] L. W. Cheuk, M. A. Nichols, M. Okan, T. Gersdorf, V. V. Ramasesh, W. S. Bakr, T. Lompe,and M. W. Zwierlein, “Quantum-gas microscope for fermionic atoms,” Phys. Rev. Lett.,
vol. 114, p. 193001, May 2015.
[30] J. Stewart, J. Gaebler, and D. Jin, “Using photoemission spectroscopy to probe a stronglyinteracting fermi gas,” Nature, vol. 454, no. 7205, pp. 744–747, 2008.
[31] M. Bishof, Y. Lin, M. D. Swallows, A. V. Gorshkov, J. Ye, and A. M. Rey, “Resolved atomic
interaction sidebands in an optical clock transition,” Phys. Rev. Lett., vol. 106, p. 250801,
Jun 2011.
[32] G. E. Marti, R. B. Hutson, A. Goban, S. L. Campbell, N. Poli, and J. Ye, “Imaging optical frequencies with 100 µHz precision and 1.1 µm resolution,” Phys. Rev. Lett., vol. 120,
p. 103201, Mar 2018.
[33] A. Goban, R. Hutson, G. Marti, S. Campbell, M. Perlin, P. Julienne, J. D’incao, A. Rey, andJ. Ye, “Emergence of multi-body interactions in a fermionic lattice clock,” Nature, vol. 563,
no. 7731, pp. 369–373, 2018.
[34] J. L. Hall, “Nobel lecture: Defining and measuring optical frequencies,” Rev. Mod. Phys.,vol. 78, pp. 1279–1295, Nov 2006.
[35] T. W. Hänsch, “Nobel lecture: Passion for precision,” Rev. Mod. Phys., vol. 78, pp. 1297–
1309, Nov 2006.
[36] M. M. Boyd, T. Zelevinsky, A. D. Ludlow, S. M. Foreman, S. Blatt, T. Ido, and J. Ye, “Opticalatomic coherence at the 1-second time scale,” Science, vol. 314, no. 5804, pp. 1430–1433, 2006.
[37] T. Bothwell, C. J. Kennedy, A. Aeppli, D. Kedar, J. M. Robinson, E. Oelker, A. Staron, andJ. Ye, “Resolving the gravitational redshift across a millimetre-scale atomic sample,” Nature,
vol. 602, no. 7897, pp. 420–424, 2022.
[38] A. W. Young, W. J. Eckner, W. R. Milner, D. Kedar, M. A. Norcia, E. Oelker, N. Schine,
J. Ye, and A. M. Kaufman, “Half-minute-scale atomic coherence and high relative stability
in a tweezer clock,” Nature, vol. 588, no. 7838, pp. 408–413, 2020.
[39] D. G. Matei, T. Legero, C. Grebing, S. Häfner, C. Lisdat, R. Weyrich, W. Zhang, L. Sonderhouse, J. M. Robinson, F. Riehle, J. Ye, and U. Sterr, “A second generation of low thermal
noise cryogenic silicon resonators,” Journal of Physics: Conference Series, vol. 723, p. 012031,
jun 2016.
[40] D. G. Matei, T. Legero, S. Häfner, C. Grebing, R. Weyrich, W. Zhang, L. Sonderhouse, J. M.
Robinson, J. Ye, F. Riehle, and U. Sterr, “1.5 µm lasers with sub-10 mhz linewidth,” Phys.
Rev. Lett., vol. 118, p. 263202, Jun 2017.
[41] E. Oelker, R. Hutson, C. Kennedy, L. Sonderhouse, T. Bothwell, A. Goban, D. Kedar, C. Sanner, J. Robinson, G. Marti, D. Matei, T. Legero, M. Giunta, R. Holzwarth, F. Riehle, U. Sterr,and J. Ye, “Demonstration of 4.8 × 10−17 stability at 1 s for two independent optical clocks,”
Nature Photonics, vol. 13, p. 714–719, 10 2019.

216[42] W. Zhang, J. M. Robinson, L. Sonderhouse, E. Oelker, C. Benko, J. L. Hall, T. Legero, D. G.
Matei, F. Riehle, U. Sterr, and J. Ye, “Ultrastable silicon cavity in a continuously operatingclosed-cycle cryostat at 4 k,” Phys. Rev. Lett., vol. 119, p. 243601, Dec 2017.
[43] J. M. Robinson, E. Oelker, W. R. Milner, W. Zhang, T. Legero, D. G. Matei, F. Riehle,
U. Sterr, and J. Ye, “Crystalline optical cavity at 4 k with thermal-noise-limited instability
and ultralow drift,” Optica, vol. 6, pp. 240–243, Feb 2019.
[44] B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang,
S. L. Bromley, and J. Ye, “An Optical Lattice Clock with Accuracy and Stability at the 10−18
Level,” Nature, vol. 506, pp. 71–75, 2014.
[45] T. Bothwell, D. Kedar, E. Oelker, J. Robinson, S. Bromley, W. Tew, J. Ye, and C. Kennedy,“Jila sri optical lattice clock with uncertainty of 2.0 × 10−18 ,” Metrologia, vol. 56, p. 065004,
2019-10 2019.
[46] T. Bothwell, A Wannier-Stark Optical Lattice Clock With Extended Coherence Times. PhD
thesis, 2022.
[47] C. J. Kennedy, E. Oelker, J. M. Robinson, T. Bothwell, D. Kedar, W. R. Milner, G. E. Marti,
A. Derevianko, and J. Ye, “Precision metrology meets cosmology: Improved constraints on
ultralight dark matter from atom-cavity frequency comparisons,” Phys. Rev. Lett., vol. 125,
p. 201302, Nov 2020.
[48] D. Meiser, J. Ye, D. R. Carlson, and M. J. Holland, “Prospects for a millihertz-linewidthlaser,” Phys. Rev. Lett., vol. 102, p. 163601, Apr 2009.
[49] J. Olson, R. W. Fox, T. M. Fortier, T. F. Sheerin, R. C. Brown, H. Leopardi, R. E. Stoner,C. W. Oates, and A. D. Ludlow, “Ramsey-bordé matter-wave interferometry for laser frequency stabilization at 10−16 frequency instability and below,” Phys. Rev. Lett., vol. 123,
p. 073202, Aug 2019.
[50] J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A
steady-state superradiant laser with less than one intracavity photon,” Nature, vol. 484,
pp. 78–81, April 2012.
[51] M. Thorpe, T. Fortier, M. Kirchner, T. Rosenband, and L. Rippe, “Frequency-stabilization
to 6 × 10−16 via spectral-hole burning,” 2011-09-11 2011.
[52] K. G, D. S., F. F., Z. S., B. S. C., S. S.A., B. S., and S. F., “Towards an active optical clock using an optical conveyor within a ring cavity,” in 2021 Joint Conference of theEuropean Frequency and Time Forum and IEEE International Frequency Control Symposium(EFTF/IFCS), pp. 1–5, 2021.
[53] The LIGO Scientific Collaboration, “Advanced LIGO,” Classical and Quantum Gravity,
vol. 32, p. 074001, mar 2015.
[54] The Einstein Telescope Collaboration, “The einstein telescope: a third-generation gravitational wave observatory,” Classical and Quantum Gravity, vol. 27, p. 194002, sep 2010.
[55] The KAGRA Collaboration, “Large-scale cryogenic gravitational-wave telescope in japan:
Kagra,” Journal of Physics: Conference Series, vol. 610, no. 1, p. 012016, 2015.

217[56] The LIGO Scientific Collaboration, “Exploring the sensitivity of next generation gravitational
wave detectors,” Classical and Quantum Gravity, vol. 34, p. 044001, Feb. 2017.
[57] J. Cripe, T. Cullen, Y. Chen, P. Heu, D. Follman, G. D. Cole, and T. Corbitt, “Quantum
backaction cancellation in the audio band,” Phys. Rev. X, vol. 10, p. 031065, Sep 2020.
[58] M. T. Nery, J. R. Venneberg, N. Aggarwal, G. D. Cole, T. Corbitt, J. Cripe, R. Lanza, and
B. Willke, “Laser power stabilization via radiation pressure,” Opt. Lett., vol. 46, pp. 1946–
1949, Apr 2021.
[59] N. Aggarwal, T. J. Cullen, J. Cripe, G. D. Cole, R. Lanza, A. Libson, D. Follman, P. Heu,
T. Corbitt, and N. Mavalvala, “Room-temperature optomechanical squeezing,” Nature
Physics, vol. 16, no. 7, pp. 784–788, 2020.
[60] P. Fulda, PRECISION INTERFEROMETRY IN A NEW SHAPE: HIGHER-ORDERLAGUERRE-GAUSS MODES FOR GRAVITATIONAL WAVE DETECTION. PhD thesis, 2012.
[61] E. D’Ambrosio, R. O’Shaugnessy, K. Thorne, P. Willems, S. Strigin, and S. Vyatchanin,“Advanced ligo: non-gaussian beams,” Classical and Quantum Gravity, vol. 21, no. 5, p. S867,
2004.
[62] T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. Martin, L. Chen,
and J. Ye, “A sub-40-mhz-linewidth laser based on a silicon single-crystal optical cavity,”
Nature Photonics, vol. 6, no. 10, pp. 687–692, 2012.
[63] D. Kedar, J. Yu, E. Oelker, A. Staron, W. R. Milner, J. M. Robinson, T. Legero, F. Riehle,
U. Sterr, and J. Ye, “Frequency stability of cryogenic silicon cavities with semiconductor
crystalline coatings,” 2022.
[64] E. Hecht, Optics. Pearson, 2012.
[65] B. E. A. Saleh and M. C. Teich., Electro-Optics, ch. 18, pp. 696–736. John Wiley Sons, Ltd,
1991.
[66] H. B. Callen and T. A. Welton, “Irreversibility and generalized noise,” Phys. Rev., vol. 83,
pp. 34–40, Jul 1951.
[67] R. Kubo, “The fluctuation-dissipation theorem,” Reports on Progress in Physics, vol. 29,
p. 255, jan 1966.
[68] P. R. Saulson, “Thermal noise in mechanical experiments,” Phys. Rev. D, vol. 42, pp. 2437–
2445, Oct 1990.
[69] Y. Levin, “Internal thermal noise in the ligo test masses: A direct approach,” Phys. Rev. D,
vol. 57, pp. 659–663, Jan 1998.
[70] F. Bondu, P. Hello, and J.-Y. Vinet, “Thermal noise in mirrors of interferometric gravitational
wave antennas,” Physics Letters A, vol. 246, no. 3, pp. 227–236, 1998.
[71] Y. T. Liu and K. S. Thorne, “Thermoelastic noise and homogeneous thermal noise in finite
sized gravitational-wave test masses,” Phys. Rev. D, vol. 62, p. 122002, Nov 2000.

218[72] Optical Coatings and Thermal Noise in Precision Measurement. Cambridge University Press,
2012.
[73] I. Martin, H. Armandula, C. Comtet, M. M. Fejer, A. Gretarsson, G. Harry, J. Hough, J.M. M. Mackowski, I. MacLaren, C. Michel, J.-L. Montorio, N. Morgado, R. Nawrodt, S. Penn,
S. Reid, A. Remillieux, R. Route, S. Rowan, C. Schwarz, P. Seidel, W. Vodel, and A. Zimmer,“Measurements of a low-temperature mechanical dissipation peak in a single layer of ta2 o5
doped with tio2 ,” Classical and Quantum Gravity, vol. 25, p. 055005, feb 2008.
[74] V. B. Braginsky, V. P. Mitrofanov, V. I. Panov, and R. V. Krotkov, “Systems with small
dissipation,” 1986.
[75] T. Chalermsongsak, E. D. Hall, G. D. Cole, D. Follman, F. Seifert, K. Arai, E. K. Gustafson,
J. R. Smith, M. Aspelmeyer, and R. X. Adhikari, “Coherent cancellation of photothermal
noise in gaas/al0 .92ga0 .08as bragg mirrors,” Metrologia, vol. 53, p. 860, mar 2016.
[76] R. V. Pound, “Electronic frequency stabilization of microwave oscillators,” Review of
Scientific Instruments, vol. 17, no. 11, pp. 490–505, 1946.
[77] L. Cutler and C. Searle, “Some aspects of the theory and measurement of frequency fluctuations in frequency standards,” Proceedings of the IEEE, vol. 54, no. 2, pp. 136–154, 1966.
[78] J. Rutman and F. Walls, “Characterization of frequency stability in precision frequency
sources,” Proceedings of the IEEE, vol. 79, no. 7, pp. 952–960, 1991.
[79] G. J. Dick, “Local oscillator induced instabilities in trapped ion frequency standards,” inProceedings of the 19th Annual Precise Time and Time Interval Systems and Applications
Meeting, pp. 133–147, 1989.
[80] G. Santarelli, C. Audoin, A. Makdissi, P. Laurent, G. Dick, and A. Clairon, “Frequency
stability degradation of an oscillator slaved to a periodically interrogated atomic resonator,”
IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 45, no. 4,
pp. 887–894, 1998.
[81] G. J. Dick, J. D. Prestage, C. A. Greenhall, and L. Maleki, “Local oscillator induced degradation of medium-term stability in passive atomic frequency standards,” in Proceedings of the22th Annual Precise Time and Time Interval Systems and Applications Meeting, pp. 487–508,
1990.
[82] C. Greenhall, “A derivation of the long-term degradation of a pulsed atomic frequency standard from a control-loop model,” IEEE Transactions on Ultrasonics, Ferroelectrics, andFrequency Control, vol. 45, no. 4, pp. 895–898, 1998.
[83] C. Audoin, G. Santarelli, A. Makdissi, and A. Clairon, “Properties of an oscillator slaved to aperiodically interrogated atomic resonator,” IEEE Transactions on Ultrasonics, Ferroelectrics,
and Frequency Control, vol. 45, no. 4, pp. 877–886, 1998.
[84] M. L. Day, P. J. Low, B. White, R. Islam, and C. Senko, “Limits on atomic qubit control
from laser noise,” npj Quantum Information, vol. 8, no. 1, pp. 1–10, 2022.
[85] R. W. Fox, “Temperature analysis of low-expansion fabry-perot cavities,” Opt. Express,
vol. 17, pp. 15023–15031, Aug 2009.

219[86] E. Wiens, Q.-F. Chen, I. Ernsting, H. Luckmann, U. Rosowski, A. Nevsky, and S. Schiller,“Silicon single-crystal cryogenic optical resonator,” Opt. Lett., vol. 39, pp. 3242–3245, Jun
2014.
[87] R. Nawrodt, A. Zimmer, T. Koettig, C. Schwarz, D. Heinert, M. Hudl, R. Neubert, M. Thürk,
S. Nietzsche, W. Vodel, P. Seidel, and A. Tünnermann, “High mechanical q-factor measurements on silicon bulk samples,” Journal of Physics: Conference Series, vol. 122, p. 012008,
jul 2008.
[88] J. J. Wortman and R. A. Evans, “Young’s modulus, shear modulus, and poisson’s ratio insilicon and germanium,” Journal of Applied Physics, vol. 36, no. 1, pp. 153–156, 1965.
[89] S. D. Penn, M. M. Kinley-Hanlon, I. A. O. MacMillan, P. Heu, D. Follman, C. Deutsch, G. D.
Cole, and G. M. Harry, “Mechanical ringdown studies of large-area substrate-transferred
gaas/algaas crystalline coatings,” J. Opt. Soc. Am. B, vol. 36, pp. C15–C21, Apr 2019.
[90] A. Schroeter, R. Nawrodt, R. Schnabel, S. Reid, I. Martin, S. Rowan, C. Schwarz, T. Koettig,
R. Neubert, M. Thürk, W. Vodel, A. Tünnermann, K. Danzmann, and P. Seidel, “On themechanical quality factors of cryogenic test masses from fused silica and crystalline quartz,”
2007.
[91] S. D. Penn, P. H. Sneddon, H. Armandula, J. C. Betzwieser, G. Cagnoli, J. Camp, D. R. M.
Crooks, M. M. Fejer, A. M. Gretarsson, G. M. Harry, J. Hough, S. E. Kittelberger, M. J.
Mortonson, R. Route, S. Rowan, and C. C. Vassiliou, “Mechanical loss in tantala/silica
dielectric mirror coatings,” Classical and Quantum Gravity, vol. 20, p. 2917, jun 2003.
[92] S. D. Penn, P. H. Sneddon, H. Armandula, J. C. Betzwieser, G. Cagnoli, J. Camp, D. R. M.
Crooks, M. M. Fejer, A. M. Gretarsson, G. M. Harry, J. Hough, S. E. Kittelberger, M. J.
Mortonson, R. Route, S. Rowan, and C. C. Vassiliou, “Mechanical loss in tantala/silica
dielectric mirror coatings,” Classical and Quantum Gravity, vol. 20, p. 2917, jun 2003.
[93] D. R. M. Crooks, G. Cagnoli, M. M. Fejer, A. Gretarsson, G. Harry, J. Hough, N. Nakagawa,
S. Penn, R. Route, S. Rowan, and P. H. Sneddon, “Experimental measurements of coating
mechanical loss factors,” Classical and Quantum Gravity, vol. 21, p. S1059, feb 2004.
[94] G. M. Harry, M. R. Abernathy, A. E. Becerra-Toledo, H. Armandula, E. Black, K. Dooley,
M. Eichenfield, C. Nwabugwu, A. Villar, D. R. M. Crooks, G. Cagnoli, J. Hough, C. R. How,
I. MacLaren, P. Murray, S. Reid, S. Rowan, P. H. Sneddon, M. M. Fejer, R. Route, S. D.
Penn, P. Ganau, J.-M. Mackowski, C. Michel, L. Pinard, and A. Remillieux, “Titania-doped
tantala/silica coatings for gravitational-wave detection,” Classical and Quantum Gravity,
vol. 24, p. 405, dec 2006.
[95] P. G. Murray, Measurement of the mechanical loss of test mass materials for advanced
gravitational wave detectors. PhD thesis, 2008.
[96] M. Granata, E. Coillet, V. Martinez, V. Dolique, A. Amato, M. Canepa, J. Margueritat,
C. Martinet, A. Mermet, C. Michel, L. Pinard, B. Sassolas, and G. Cagnoli, “Correlated
evolution of structure and mechanical loss of a sputtered silica film,” Phys. Rev. Mater.,vol. 2, p. 053607, May 2018.

220[97] G. Vajente, L. Yang, A. Davenport, M. Fazio, A. Ananyeva, L. Zhang, G. Billingsley, K. Prasai, A. Markosyan, R. Bassiri, M. M. Fejer, M. Chicoine, F. m. c. Schiettekatte, and C. S.
Menoni, “Low mechanical loss tio2 : geo2 coatings for reduced thermal noise in gravitational
wave interferometers,” Phys. Rev. Lett., vol. 127, p. 071101, Aug 2021.
[98] M. Granata, A. Amato, L. Balzarini, M. Canepa, J. Degallaix, D. Forest, V. Dolique,
L. Mereni, C. Michel, L. Pinard, et al., “Amorphous optical coatings of present gravitationalwave interferometers,” Classical and Quantum Gravity, vol. 37, no. 9, p. 095004, 2020.
[99] M. Evans et al., “A Horizon Study for Cosmic Explorer: Science, Observatories, and Community,” 9 2021.
[100] B. Shyam, K. H. Stone, R. Bassiri, M. M. Fejer, M. F. Toney, and A. Mehta, “Measurement
and modeling of short and medium range order in amorphous ta2 o5 thin films,” Scientificreports, vol. 6, no. 1, pp. 1–7, 2016.
[101] L. Yang, M. Fazio, G. Vajente, A. Ananyeva, G. Billingsley, A. Markosyan, R. Bassiri, M. M.
Fejer, and C. S. Menoni, “Structural evolution that affects the room-temperature internal
friction of binary oxide nanolaminates: implications for ultrastable optical cavities,” ACS
Applied Nano Materials, vol. 3, no. 12, pp. 12308–12313, 2020.
[102] K. Prasai, J. Jiang, A. Mishkin, B. Shyam, S. Angelova, R. Birney, D. Drabold, M. Fazio,
E. Gustafson, G. Harry, et al., “High precision detection of change in intermediate rangeorder of amorphous zirconia-doped tantala thin films due to annealing,” Physical review
letters, vol. 123, no. 4, p. 045501, 2019.
[103] G. Cole. private communication, Dec. 2022.
[104] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod.
Phys., vol. 86, pp. 1391–1452, Dec 2014.
[105] B. Slagmolen, D. Shaddock, M. Gray, and D. McClelland, “Frequency stability of spatial
mode interference (tilt) locking,” IEEE Journal of Quantum Electronics, vol. 38, no. 11,
pp. 1521–1528, 2002.
[106] F. Diorico, A. Zhutov, and O. Hosten, “Laser-cavity locking at the 10−7 instability scaleutilizing beam elipticity,” 2022.
[107] M. J. Martin, Quantum Metrology and Many-Body Physics: Pushing the Frontier of the
Optical Lattice Clock. PhD thesis, 2013.
[108] N. Courjal, M.-P. Bernal, A. Caspar, G. Ulliac, F. Bassignot, L. Gauthier-Manuel,and M. Suarez, “Lithium niobate optical waveguides and microwaveguides,” in Emerging
Waveguide Technology (K. Y. You, ed.), ch. 8, Rijeka: IntechOpen, 2018.
[109] N. C. Wong and J. L. Hall, “Servo control of amplitude modulation in frequency-modulation
spectroscopy: demonstration of shot-noise-limited detection,” J. Opt. Soc. Am. B, vol. 2,
pp. 1527–1533, Sep 1985.
[110] W. Zhang, M. J. Martin, C. Benko, J. L. Hall, J. Ye, C. Hagemann, T. Legero, U. Sterr,
F. Riehle, G. D. Cole, and M. Aspelmeyer, “Reduction of residual amplitude modulation to 1
×10−6 for frequency modulation and laser stabilization,” Opt. Lett., vol. 39, pp. 1980–1983,
Apr 2014.

221[111] H. Shen, L. Li, J. Bi, J. Wang, and L. Chen, “Systematic and quantitative analysis of residual
amplitude modulation in pound-drever-hall frequency stabilization,” Phys. Rev. A, vol. 92,
p. 063809, Dec 2015.
[112] J. Bi, Y. Zhi, L. Li, and L. Chen, “Suppressing residual amplitude modulation to the 1 × 10−7level in optical phase modulation,” Appl. Opt., vol. 58, pp. 690–694, Jan 2019.
[113] M. A. Hopcroft, W. D. Nix, and T. W. Kenny, “What is the young’s modulus of silicon?,”
Journal of Microelectromechanical Systems, vol. 19, no. 2, pp. 229–238, 2010.
[114] M. Notcutt, L.-S. Ma, J. Ye, and J. L. Hall, “Simple and compact 1-hz laser system via an
improved mounting configuration of a reference cavity,” Opt. Lett., vol. 30, pp. 1815–1817,
Jul 2005.
[115] A. D. Ludlow, The Strontium Optical Lattice Clock. PhD thesis, 2008.
[116] M. D. Swallows, M. J. Martin, M. Bishof, C. Benko, Y. Lin, S. Blatt, A. M. Rey, andJ. Ye, “Operating a 87sr optical lattice clock with high precision and at high density,” IEEE
Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 59, no. 3, pp. 416–
425, 2012.
[117] S. A. Webster, M. Oxborrow, and P. Gill, “Vibration insensitive optical cavity,” Phys. Rev.
A, vol. 75, p. 011801, Jan 2007.
[118] T. Nazarova, F. Riehle, and U. Sterr, “Vibration-insensitive reference cavity for an ultranarrow-linewidth laser,” Applied Physics B, vol. 83, no. 4, p. 531, 2006.
[119] J. Keller, S. Ignatovich, S. A. Webster, and T. E. Mehlstäubler, “Simple vibration-insensitive
cavity for laser stabilization at the 1016 level,” Applied Physics B, vol. 116, no. 1, pp. 203–210,
2014.
[120] M. F. Modest, “Chapter 4 - view factors,” in Radiative Heat Transfer (Second Edition) (M. F.
Modest, ed.), pp. 131–161, Burlington: Academic Press, second edition ed., 2003.
[121] X. Zhi, R. Cao, C. Huang, K. Wang, and L. Qiu, “Feasibility analysis of hocu2 and gd2 o2 s
as regenerative materials around 4–20 k,” Applied Thermal Engineering, vol. 192, p. 116921,
2021.
[122] K. Shinozaki, T. Mizutani, T. Fujii, T. Onaka, T. Nakagawa, and H. Sugita, “Thermal
property measurements of critical materials for spica payload module,” Physics Procedia,
vol. 67, pp. 270–275, 2015. Proceedings of the 25th International Cryogenic Engineering
Conference and International Cryogenic Materials Conference 2014.
[123] S. Pucic, “Diffusion of copper into gold plating,” in 1993 IEEE Instrumentation andMeasurement Technology Conference, pp. 114–117, 1993.
[124] K. G. Lyon, G. L. Salinger, C. A. Swenson, and G. K. White, “Linear thermal expansion
measurements on silicon from 6 to 340 k,” Journal of Applied Physics, vol. 48, no. 3, pp. 865–
868, 1977.
[125] B. Edlén, “The refractive index of air,” Metrologia, vol. 2, no. 2, p. 71, 1966.

222[126] K. P. Birch and M. J. Downs, “Correction to the updated edlén equation for the refractive
index of air,” Metrologia, vol. 31, p. 315, jan 1994.
[127] G. Cole, S. Ballmer, G. Billingsley, S. Cataño-Lopez, M. Fejer, P. Fritschel, A. Gretarsson,
G. Harry, D. Kedar, T. Legero, C. Makarem, S. Penn, D. Reitze, J. Steinlechner, U. Sterr,S. Tanioka, G.-W. Truong, J. Ye, , and J. Yu, “Substrate-transferred gaas/algaas crystalline
coatings for gravitational-wave detectors: A review of the state of the art.” 2022.
[128] V. Braginsky, M. Gorodetsky, and S. Vyatchanin, “Thermodynamical fluctuations and photothermal shot noise in gravitational wave antennae,” Physics Letters A, vol. 264, no. 1, pp. 1–
10, 1999.
[129] A. Farsi, M. Siciliani de Cumis, F. Marino, and F. Marin, “Photothermal and thermorefractive effects in high reflectivity mirrors at room and cryogenic temperature,” Journalof Applied Physics, vol. 111, no. 4, p. 043101, 2012.
[130] J.-Y. Vinet, “Thermal noise in advanced gravitational wave interferometric antennas: A
comparison between arbitrary order hermite and laguerre gaussian modes,” Phys. Rev. D,
vol. 82, p. 042003, Aug 2010.
[131] C. Panuski, J. Goldstein, D. Englund, and R. Hamerly, “Coherent thermo-optic noise cancellation in an optical microcavity,” in Frontiers in Optics + Laser Science 2021, p. FTh6B.3,
Optica Publishing Group, 2021.
[132] H. Liu, B. D. Elwood, M. Evans, and J. Thaler, “Searching for axion dark matter withbirefringent cavities,” Phys. Rev. D, vol. 100, p. 023548, Jul 2019.
[133] H. Müller, S. Herrmann, A. Saenz, A. Peters, and C. Lämmerzahl, “Optical cavity tests oflorentz invariance for the electron,” Phys. Rev. D, vol. 68, p. 116006, Dec 2003.
[134] E. Wiens, A. Y. Nevsky, and S. Schiller, “Resonator with ultrahigh length stability as a probe
for equivalence-principle-violating physics,” Phys. Rev. Lett., vol. 117, p. 271102, Dec 2016.
[135] M. Aaboud, G. Aad, B. Abbott, B. Abeloos, S. Abidi, O. AbouZeid, N. L. Abraham,
H. Abramowicz, H. Abreu, R. Abreu, et al., “Search for dark matter and other new phenomena in events with an energetic jet and large missing transverse momentum using theatlas detector,” Journal of High Energy Physics, vol. 2018, no. 1, pp. 1–53, 2018.
[136] E. G. Ferreira, “Ultra-light dark matter,” The Astronomy and Astrophysics Review, vol. 29,
no. 1, pp. 1–186, 2021.
[137] D. Antypas, A. Banerjee, C. Bartram, M. Baryakhtar, J. Betz, J. J. Bollinger, C. Boutan,
D. Bowring, D. Budker, D. Carney, G. Carosi, S. Chaudhuri, S. Cheong, A. Chou, M. D.
Chowdhury, R. T. Co, J. R. C. López-Urrutia, M. Demarteau, N. DePorzio, A. V. Derbin,
T. Deshpande, M. D. Chowdhury, L. Di Luzio, A. Diaz-Morcillo, J. M. Doyle, A. DrlicaWagner, A. Droster, N. Du, B. Döbrich, J. Eby, R. Essig, G. S. Farren, N. L. Figueroa, J. T.
Fry, S. Gardner, A. A. Geraci, A. Ghalsasi, S. Ghosh, M. Giannotti, B. Gimeno, S. M. Griffin,
D. Grin, D. Grin, H. Grote, J. H. Gundlach, M. Guzzetti, D. Hanneke, R. Harnik, R. Henning,
V. Irsic, H. Jackson, D. F. J. Kimball, J. Jaeckel, M. Kagan, D. Kedar, R. Khatiwada,
S. Knirck, S. Kolkowitz, T. Kovachy, S. E. Kuenstner, Z. Lasner, A. F. Leder, R. Lehnert,
D. R. Leibrandt, E. Lentz, S. M. Lewis, Z. Liu, J. Manley, R. H. Maruyama, A. J. Millar, V. N.

223
Muratova, N. Musoke, S. Nagaitsev, O. Noroozian, C. A. J. O’Hare, J. L. Ouellet, K. M. W.
Pappas, E. Peik, G. Perez, A. Phipps, N. M. Rapidis, J. M. Robinson, V. H. Robles, K. K.
Rogers, J. Rudolph, G. Rybka, M. Safdari, M. Safdari, M. S. Safronova, C. P. Salemi, P. O.
Schmidt, T. Schumm, A. Schwartzman, J. Shu, M. Simanovskaia, J. Singh, S. Singh, M. S.
Smith, W. M. Snow, Y. V. Stadnik, C. Sun, A. O. Sushkov, T. M. P. Tait, V. Takhistov,D. B. Tanner, D. J. Temples, P. G. Thirolf, J. H. Thomas, M. E. Tobar, O. Tretiak, Y. D.
Tsai, J. A. Tyson, M. Vandegar, S. Vermeulen, L. Visinelli, E. Vitagliano, Z. Wang, D. J.
Wilson, L. Winslow, S. Withington, M. Wooten, J. Yang, J. Ye, B. A. Young, F. Yu, M. H.
Zaheer, T. Zelevinsky, Y. Zhao, and K. Zhou, “New horizons: Scalar and vector ultralightdark matter,” 2022.
[138] A. Derevianko and M. Pospelov, “Hunting for topological dark matter with atomic clocks,”
Nature Physics, vol. 10, no. 12, pp. 933–936, 2014.
[139] A. Arvanitaki, J. Huang, and K. Van Tilburg, “Searching for dilaton dark matter with atomic
clocks,” Phys. Rev. D, vol. 91, p. 015015, Jan 2015.
[140] M. S. Safronova, D. Budker, D. DeMille, D. F. J. Kimball, A. Derevianko, and C. W. Clark,“Search for new physics with atoms and molecules,” Rev. Mod. Phys., vol. 90, p. 025008, Jun
2018.
[141] V. A. Dzuba, V. V. Flambaum, and J. K. Webb, “Calculations of the relativistic effects
in many-electron atoms and space-time variation of fundamental constants,” Phys. Rev. A,
vol. 59, pp. 230–237, Jan 1999.
[142] V. A. Dzuba, V. V. Flambaum, and J. K. Webb, “Space-time variation of physical constants
and relativistic corrections in atoms,” Phys. Rev. Lett., vol. 82, pp. 888–891, Feb 1999.
[143] V. Flambaum and V. Dzuba, “Search for variation of the fundamental constants in atomic,
molecular, and nuclear spectra,” Canadian Journal of Physics, vol. 87, no. 1, pp. 25–33, 2009.
[144] Y. V. Stadnik and V. V. Flambaum, “Enhanced effects of variation of the fundamental
constants in laser interferometers and application to dark-matter detection,” Phys. Rev. A,
vol. 93, p. 063630, Jun 2016.
[145] M. Filzinger, S. Dörscher, R. Lange, J. Klose, M. Steinel, E. Benkler, E. Peik, C. Lisdat,
and N. Huntemann, “Improved limits on the coupling of ultralight bosonic dark matter tophotons from optical atomic clock comparisons,” 2023.
[146] M. S. Safronova, S. G. Porsev, C. Sanner, and J. Ye, “Two clock transitions in neutral yb forthe highest sensitivity to variations of the fine-structure constant,” Phys. Rev. Lett., vol. 120,
p. 173001, Apr 2018.
[147] C. Hagemann, C. Grebing, C. Lisdat, S. Falke, T. Legero, U. Sterr, F. Riehle, M. J. Martin,
and J. Ye, “Ultrastable laser with average fractional frequency drift rate below 5 × 10−19 /s,”
Opt. Lett., vol. 39, pp. 5102–5105, Sep 2014.
[148] J. Weber, “Gravitational-wave-detector events,” Phys. Rev. Lett., vol. 20, pp. 1307–1308,
Jun 1968.
[149] Prodi, G. A. and others, “The ultracryogenic gravitational wave detector auriga,” inInternational Conference on Gravitational Waves: Sources and Detectors, 1996.

224[150] A. de Waard, L. Gottardi, J. van Houwelingen, A. Shumack, and G. Frossati, “Minigrail, thefirst spherical detector,” Classical and Quantum Gravity, vol. 20, p. S143, apr 2003.
[151] B. Abbott, R. Abbott, R. Adhikari, J. Agresti, P. Ajith, B. Allen, R. Amin, S. B. Anderson, W. G. Anderson, M. Arain, M. Araya, H. Armandula, M. Ashley, S. Aston, P. Aufmuth, C. Aulbert, S. Babak, S. Ballmer, H. Bantilan, B. C. Barish, C. Barker, D. Barker,
B. Barr, P. Barriga, M. A. Barton, K. Bayer, K. Belczynski, J. Betzwieser, P. T. Beyersdorf,
B. Bhawal, I. A. Bilenko, G. Billingsley, R. Biswas, E. Black, K. Blackburn, L. Blackburn,D. Blair, B. Bland, J. Bogenstahl, L. Bogue, R. Bork, V. Boschi, S. Bose, P. R. Brady, V. B.
Braginsky, J. E. Brau, M. Brinkmann, A. Brooks, D. A. Brown, A. Bullington, A. Bunkowski,
A. Buonanno, M. Burgamy, O. Burmeister, D. Busby, R. L. Byer, L. Cadonati, G. Cagnoli,
J. B. Camp, J. Cannizzo, K. Cannon, C. A. Cantley, J. Cao, L. Cardenas, M. M. Casey,
G. Castaldi, C. Cepeda, E. Chalkey, P. Charlton, S. Chatterji, S. Chelkowski, Y. Chen,
F. Chiadini, D. Chin, E. Chin, J. Chow, N. Christensen, J. Clark, P. Cochrane, T. Cokelaer,
C. N. Colacino, R. Coldwell, R. Conte, D. Cook, T. Corbitt, D. Coward, D. Coyne, J. D. E.
Creighton, T. D. Creighton, R. P. Croce, D. R. M. Crooks, A. M. Cruise, A. Cumming,
J. Dalrymple, E. D’Ambrosio, K. Danzmann, G. Davies, D. DeBra, J. Degallaix, M. Degree, T. Demma, V. Dergachev, S. Desai, R. DeSalvo, S. Dhurandhar, M. Dı́az, J. Dickson,
A. Di Credico, G. Diederichs, A. Dietz, E. E. Doomes, R. W. P. Drever, J.-C. Dumas, R. J.
Dupuis, J. G. Dwyer, P. Ehrens, E. Espinoza, T. Etzel, M. Evans, T. Evans, S. Fairhurst,
Y. Fan, D. Fazi, M. M. Fejer, L. S. Finn, V. Fiumara, N. Fotopoulos, A. Franzen, K. Y.
Franzen, A. Freise, R. Frey, T. Fricke, P. Fritschel, V. V. Frolov, M. Fyffe, V. Galdi, J. Garofoli, I. Gholami, J. A. Giaime, S. Giampanis, K. D. Giardina, K. Goda, E. Goetz, L. Goggin,
G. González, S. Gossler, A. Grant, S. Gras, C. Gray, M. Gray, J. Greenhalgh, A. M. Gretarsson, R. Grosso, H. Grote, S. Grunewald, M. Guenther, R. Gustafson, B. Hage, W. O.
Hamilton, D. Hammer, C. Hanna, J. Hanson, J. Harms, G. Harry, E. Harstad, T. Hayler,
J. Heefner, I. S. Heng, A. Heptonstall, M. Heurs, M. Hewitson, S. Hild, E. Hirose, D. Hoak,
D. Hosken, J. Hough, E. Howell, D. Hoyland, S. H. Huttner, D. Ingram, E. Innerhofer,
M. Ito, Y. Itoh, A. Ivanov, D. Jackrel, B. Johnson, W. W. Johnson, D. I. Jones, G. Jones,
R. Jones, L. Ju, P. Kalmus, V. Kalogera, D. Kasprzyk, E. Katsavounidis, K. Kawabe,
S. Kawamura, F. Kawazoe, W. Kells, D. G. Keppel, F. Y. Khalili, C. Kim, P. King, J. S.
Kissel, S. Klimenko, K. Kokeyama, V. Kondrashov, R. K. Kopparapu, D. Kozak, B. Krishnan, P. Kwee, P. K. Lam, M. Landry, B. Lantz, A. Lazzarini, B. Lee, M. Lei, J. Leiner,
V. Leonhardt, I. Leonor, K. Libbrecht, P. Lindquist, N. A. Lockerbie, M. Longo, M. Lormand, M. Lubinski, H. Lück, B. Machenschalk, M. MacInnis, M. Mageswaran, K. Mailand,
M. Malec, V. Mandic, S. Marano, S. Márka, J. Markowitz, E. Maros, I. Martin, J. N. Marx,
K. Mason, L. Matone, V. Matta, N. Mavalvala, R. McCarthy, B. J. McCaulley, D. E. McClelland, S. C. McGuire, M. McHugh, K. McKenzie, J. W. C. McNabb, S. McWilliams,
T. Meier, A. Melissinos, G. Mendell, R. A. Mercer, S. Meshkov, E. Messaritaki, C. J. Messenger, D. Meyers, E. Mikhailov, P. Miller, S. Mitra, V. P. Mitrofanov, G. Mitselmakher,
R. Mittleman, O. Miyakawa, S. Mohanty, V. Moody, G. Moreno, K. Mossavi, C. MowLowry,
A. Moylan, D. Mudge, G. Mueller, S. Mukherjee, H. Müller-Ebhardt, J. Munch, P. Murray,
E. Myers, J. Myers, T. Nash, D. Nettles, G. Newton, A. Nishizawa, K. Numata, B. O’Reilly,
R. O’Shaughnessy, D. J. Ottaway, H. Overmier, B. J. Owen, H.-J. Paik, Y. Pan, M. A.
Papa, V. Parameshwaraiah, P. Patel, M. Pedraza, S. Penn, V. Pierro, I. M. Pinto, M. Pitkin,
H. Pletsch, M. V. Plissi, F. Postiglione, R. Prix, V. Quetschke, F. Raab, D. Rabeling, H. Radkins, R. Rahkola, N. Rainer, M. Rakhmanov, M. Ramsunder, K. Rawlins, S. Ray-Majumder,225
V. Re, H. Rehbein, S. Reid, D. H. Reitze, L. Ribichini, R. Riesen, K. Riles, B. Rivera, N. A.
Robertson, C. Robinson, E. L. Robinson, S. Roddy, A. Rodriguez, A. M. Rogan, J. Rollins,
J. D. Romano, J. Romie, R. Route, S. Rowan, A. Rüdiger, L. Ruet, P. Russell, K. Ryan,
S. Sakata, M. Samidi, L. Sancho de la Jordana, V. Sandberg, V. Sannibale, S. Saraf, P. Sarin,
B. S. Sathyaprakash, S. Sato, P. R. Saulson, R. Savage, P. Savov, S. Schediwy, R. Schilling,
R. Schnabel, R. Schofield, B. F. Schutz, P. Schwinberg, S. M. Scott, A. C. Searle, B. Sears,
F. Seifert, D. Sellers, A. S. Sengupta, P. Shawhan, D. H. Shoemaker, A. Sibley, J. A. Sidles,
X. Siemens, D. Sigg, S. Sinha, A. M. Sintes, B. J. J. Slagmolen, J. Slutsky, J. R. Smith,
M. R. Smith, K. Somiya, K. A. Strain, D. M. Strom, A. Stuver, T. Z. Summerscales, K.-X.
Sun, M. Sung, P. J. Sutton, H. Takahashi, D. B. Tanner, M. Tarallo, R. Taylor, R. Taylor,
J. Thacker, K. A. Thorne, K. S. Thorne, A. Thüring, K. V. Tokmakov, C. Torres, C. Torrie,
G. Traylor, M. Trias, W. Tyler, D. Ugolini, C. Ungarelli, K. Urbanek, H. Vahlbruch, M. Vallisneri, C. Van Den Broeck, M. Varvella, S. Vass, A. Vecchio, J. Veitch, P. Veitch, A. Villar,
C. Vorvick, S. P. Vyachanin, S. J. Waldman, L. Wallace, H. Ward, R. Ward, K. Watts,
J. Weaver, D. Webber, A. Weber, A. Weidner, M. Weinert, A. Weinstein, R. Weiss, S. Wen,
K. Wette, J. T. Whelan, D. M. Whitbeck, S. E. Whitcomb, B. F. Whiting, C. Wilkinson,
P. A. Willems, L. Williams, B. Willke, I. Wilmut, W. Winkler, C. C. Wipf, S. Wise, A. G.
Wiseman, G. Woan, D. Woods, R. Wooley, J. Worden, W. Wu, I. Yakushin, H. Yamamoto,Z. Yan, S. Yoshida, N. Yunes, M. Zanolin, J. Zhang, L. Zhang, P. Zhang, C. Zhao, N. Zotov,
M. Zucker, H. zur Mühlen, and J. Zweizig, “First cross-correlation analysis of interferometricand resonant-bar gravitational-wave data for stochastic backgrounds,” Phys. Rev. D, vol. 76,
p. 022001, Jul 2007.
[152] G. González, “Suspensions thermal noise in the ligo gravitational wave detector,” Classicaland Quantum Gravity, vol. 17, p. 4409, nov 2000.
[153] L. A. Wainstein and V. D. Zubakov, “Extraction of signals from noise,” American Journal of
Physics, vol. 31, no. 1, pp. 68–68, 1963.
[154] Gravitational-Wave Detectors, ch. 6, pp. 197–267. John Wiley & Sons, Ltd, 2011.
[155] C. Braxmaier, H. Müller, O. Pradl, J. Mlynek, A. Peters, and S. Schiller, “Tests of relativity
using a cryogenic optical resonator,” Phys. Rev. Lett., vol. 88, p. 010401, Dec 2001.
[156] S. Kolkowitz, I. Pikovski, N. Langellier, M. D. Lukin, R. L. Walsworth, and J. Ye, “Gravitational wave detection with optical lattice atomic clocks,” Phys. Rev. D, vol. 94, p. 124043,
Dec 2016.
[157] R. Nawrodt, C. Schwarz, S. Kroker, I. W. Martin, R. Bassiri, F. Brückner, L. Cunningham, G. D. Hammond, D. Heinert, J. Hough, T. Käsebier, E.-B. Kley, R. Neubert, S. Reid,
S. Rowan, P. Seidel, and A. Tünnermann, “Investigation of mechanical losses of thin silicon
flexures at low temperatures,” Classical and Quantum Gravity, vol. 30, p. 115008, may 2013.
[158] K. Craig, J. Steinlechner, P. G. Murray, A. S. Bell, R. Birney, K. Haughian, J. Hough,I. MacLaren, S. Penn, S. Reid, R. Robie, S. Rowan, and I. W. Martin, “Mirror coating
solution for the cryogenic einstein telescope,” Phys. Rev. Lett., vol. 122, p. 231102, Jun 2019.
[159] J. T. VanderPlas, “Understanding the lomb–scargle periodogram,” The Astrophysical Journal
Supplement Series, vol. 236, p. 16, may 2018.

226[160] P. G. Westergaard, J. Lodewyck, and P. Lemonde, “Minimizing the dick effect in an optical lattice clock,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control,
vol. 57, no. 3, pp. 623–628, 2010.

Appendix A

Dick effect limited instability and laser noise

In the context of Dick effect limited stability, a distinction between Ramsey and Rabi spectroscopy is the unique feature that increased spectroscopic time does not always result in an improved clock stability. As we discussed in chapter 2, this is related to the spectroscopic sensitivity
function of Rabi interrogation where laser phase evolution at the interrogation time τ /2 is weighted
more heavily than the beginning and end of spectroscopy. This concept is familiar to us in lab, and
it is not difficult to measure a reduction in stability for longer Rabi interrogation times. However,
it is a misconception to think that this is a universal feature of Rabi spectroscopy. This feature
is instead dependent on the spectrum of the local oscillator. For particular types of frequency
noise (e.g. white PM), sampling laser noise with the Rabi spectroscopic sensitivity function has
the feature where increased spectroscopic time will result in am improved stability. Here we willshow the dependence of clock stability for different Tc and τ when the local oscillator has different
spectrums Sy (f ) ∼ hα f α .
Each figure displays a contour plot for the 1 s Dick-effect limited instability for a given
spectroscopy time and dead time. We show the power spectral density of the laser’s fractionalfrequency noise as well as the mod σy (τ ). In optical clocks, we are most familiar with the thermal
noise limited laser spectra of Fig.A.3. We can see the clear feature where increased Rabi pulse
time results in a reduced clock stability, whereas this is not the case with Ramsey interrogation.
However, lasers with excess high frequency noise show the opposite behaviour. Fig.A.1 calculates
stabilities for a laser with a white phase noise spectrum, Sy ∼ f 2 . The relative insensitivity of228
Rabi spectroscopy to high frequency noise (when compared to Ramsey), results in a superior clockstability for equivalent deadtimes and spectroscopic times. The point here is that optimizationof clock stability requires us to give careful thought to the spectroscopic sequence and depends
strongly on the interrogation oscillator’s frequency noise spectrum.

229

10 30

10 16

Laser noise

Laser noise

10 32

mod y( )PSD (1/Hz)

10 31
10 33

10 17
10 18

10 34
10 1

101
100
102
Frequency (Hz)Dick Effect Rabi (×10 17)

10 19 1
10
0.7

0.5

0.5

Dead time (s)3.2
00 4.00
2.4
1.60
0.80

Dead time (s)

102

0.4
0.3
0.2

0.4

2.00

0.6

4.80

0.6

0.1

101
Tau (s)Dick Effect Ramsey (×10 17)
100

12.00

0.7

103

8.00 10.00
6.00
4.00

10 35 2
10

0.3
0.2

1

2
3
Rabi Pulse Time (s)

4

0.1

1

2
3
4
Ramsey Dark Time (s)Figure A.1: The laser has a white phase noise spectrum where Sy (f ) ∼ f 2 . Top row shows the laser
Sy (f ) and the corresponding modified Allan variance. The bottom row shows the Dick effect limited
1 s clock stability that can expected for different dead times and pulse times. Different behaviour
will be visible for Rabi spectroscopy (lower left) and Ramsey spectroscopy (lower right).

230

10 30

10 16

Laser noise

Laser noise

10 32

mod y( )PSD (1/Hz)

10 31
10 33

10 17
10 18

10 34
10 1

101
100
102
Frequency (Hz)Dick Effect Rabi (×10 17)

10 19 1
10
0.7
0.5

0.4

5

0.4

0.3
0.2
0.1

1.20
0.90 1.05
00.75
.60

0.5

Dead time (s)

0.6
0.60

0.6

1.3
1.05 1.250
00.90
.75

Dead time (s)

0.7

103

101
Tau (s)Dick Effect Ramsey (×10 17)
100

0.45

10 35 2
10

0.4
0.3
0.2

1

2
3
Rabi Pulse Time (s)

4

0
0.3

0.1

0.15

1

2
3
4
Ramsey Dark Time (s)Figure A.2: The laser has a white frequency noise spectrum where Sy (f ) ∼ f 0 .

102

231

10 30

10 16

Laser noise

Laser noise

10 32

mod y( )PSD (1/Hz)

10 31
10 33

10 17
10 18

10 34
10 1

101
100
102
Frequency (Hz)Dick Effect Rabi (×10 17)

0.7
Dead time (s)

0.4

0.90

0.3
0.1

0.7

0.75

0.60

1

2
3
Rabi Pulse Time (s)101
Tau (s)Dick Effect Ramsey (×10 17)
100

0.6

1.05
0.90
0.7
5

0

1.05

0.5

0.2

10 19 1
10

1.20

1.2

0.6

103

Dead time (s)10 35 2
10

0.5

0.45

0.4
0.3

0.30

0.2
4

0.1

0

0.6

1

0.15

2
3
4
Ramsey Dark Time (s)Figure A.3: The laser has a flicker frequency noise spectrum where Sy (f ) ∼ f −1 .

102

232

10 30

10 15

Laser noise

Laser noise

10 32

mod y( )PSD (1/Hz)

10 31
10 33

10 16
10 17

10 34
10 1

101
100
102
Frequency (Hz)Dick Effect Rabi (×10 17)

0.7

2.0
0

0
1.6

0

1.2

0.4
0
0.8

Dead time (s)

0

2.4

0.5
0.3
0
0.4

0.2
0.1

10 18 1
10
0.7

1

2
3
Rabi Pulse Time (s)4101
Tau (s)Dick Effect Ramsey (×10 17)
100

0.90

0.5

0.75

0.4

0.60

0.3

0.45

0.2

0.30

0.1

102

1.05

0.6

0
2.8

0.6

103

Dead time (s)10 35 2
10

1

2
3
4
Ramsey Dark Time (s)Figure A.4: The laser has a random walk frequency noise spectrum where Sy (f ) ∼ f −2 .

Appendix B

Analog locking

We’ve discussed a few cases where reduced optical power within the cavity has associatedbenefits, e.g. lower drift rate in Si4 and lower birefringent noise in Si6. An attractive scenario
would be to operate the cavity with optical power just above the shot noise limited length resolution without any detrimental effects. Lowering the overall power incident on the cavity can beappropriate method at the cost of a reduced cavity signal. Photodetector noise and other technicalnoise sources will often become consequential at these levels. A useful method employed in Si6
is to increase the modulation index of the EOM sidebands. Power in the carrier, the important
metric for the aforementioned effects, is reduced while power in the sidebands increases. However,this is limited to a practical sideband/carrier ratio of ≈ 10 as illustrated in Fig.B.1. The Bessel
function dictating sideband amplitude restricts the maximum power we can store in the first ordermodulation tones. Driving β > 3 generates stronger second order harmonics which can be harmful
in terms of RAM control.
The ideal situation would be to couple only a few photons into the cavity (again, just above
the length resolution limited by optical shot noise) while maintaining massive sidebands for astrong heterodyne beat. This would ensure that technical issues related to the servo lock andphotodetection never become significant problems. Rather than relying on the phase modulation
generated via an EOM, we can try to synthesize the PDH triplet by applying a set of independently
generated tones to an AOM. Decoupled control to each tone can allow us to arbitrarily adjust theirphase and amplitudes without any effect on the other two. Generally the most significant sourcePower [P0]

234

1.0
0.8
0.6
0.4
0.2
0.0

Carrier power
1st sideband power
2nd sideband power
3rd sideband power

0

1

2
3
Modulation index

45Power [P0]

100
10 1
10 2
10 3 1
10

Carrier power
1st sideband power
2nd sideband power
3rd sideband power

100
Modulation index

Figure B.1: Amplitude of different sidebands generated via phase modulation on an EOM.

of measured RAM is generated by the EOM. Triplet synthesis within the AOM is insensitive topolarization fluctuations, so it possible that the free running RAM introduced with this method ismuch lower.
This sounds like a novel scheme but we’ve already been playing around with this concept
in our dual frequency locking. In that case there is no dynamic readjustment of the electric fieldphasors but we are able to independently control the amplitude and phase. To generate PDH• The two sidebands need to maintain a relative phase of 180◦
• We still want to implement active RAM cancellation so need to be able to feed back to the235phase and amplitude of the sidebands• As discussed in the RAM section of chapter 3, this requires applying common phase offsets
ϕ+ = ϕi to the two tones, and inverse amplitude offsets a+ = −a−
Triplet synthesis via a DDS allows us direct access to the amplitude and phase of the electric fields,
a unique capability that we do not enjoy when using active RAM stabilization on EOM generated
sidebands. These are truly orthogonal quadratures but it is worth considering whether our in-phaseand quadrature detection channels are aligned to the parameters we can modify (i.e. does the inphase error signal solely depend on the sideband amplitude). We’ll revisit a derivation of the error
signals, and also demonstrate that this is indeed the case when we apply a correction signal.
In section 3.2.2 we found that the generalized electric field generated by the EOM isEi = E0 J0 (β)eiωt + α+ J1 (β)ei(ω+Ω)t eiϕ+ − α− J1 (β)ei(ω−Ω)t eiϕ−(B.1)

and the corresponding detected RAM signal in the presence of AM and PM isihihV ∝ Re α+ e−iϕ+ − α− eiϕ− cos(φ) + Im α+ e−iϕ+ − α− eiϕ− sin(φ)(B.2)

A particular issue is that we will not be identify all four αi , ϕi components. In AM for example,
a single stage of demodulation at the sideband frequency Ω will not yield information about theindividual components α+ and α− . The correction we apply will contain some combination of thetwo and we should consider how bad of an approximation this is - can this suppress the AM (andsimilarly the PM) to necessary levels?
First, let’s consider quadrature demodulation of the error signal. The in-phase component isfound to be
h
i


ϕ2 
ϕ2 
VI ∝ Re α+ e−iϕ+ − α− eiϕ− ≈ α+ 1 + + − α− 1 + −
2
2

 1

α+ ϕ2+ − α− ϕ2−= α+ − α− −
2

(B.3)
(B.4)

236
We can expand the overall field amplitudes of the two sidebands asα+ = 1 + ϵ+

(B.5)α− = 1 + ϵ−

(B.6)


 1
 1

VI ∝ ϵ+ − ϵ− −ϕ2+ − ϕ2− +ϕ2+ ϵ+ − ϕ2− ϵ−
2
2

(B.7)

The corresponding in-phase signal is then≈ ϵ+ − ϵ−(B.8)

For the quadrature signal, we findhi
VQ ∝ Im α+ e−iϕ+ − α− eiϕ− ≈ α+ ϕ+ + α− ϕ− ≈ ϕ+ + ϕ− − ϕ+ ϵ+ + ϕ− ϵ−≈ ϕ+ + ϕ−(B.9)(B.10)(B.11)

Crucially, this tells us how to properly apply feedback to the RAM signal to suppress bothcomponents. Recall that ϕ± are defined as common mode rotations of the two sideband phasors,i.e. will result in upper and lower sidebands with respective phases of ϕ+ and −π + ϕ− . In our firstorder approximation in the presence of zero residual offset, ϕ+ = −ϕ− will cause no quadrature
RAM signal. Unless these terms are zero, there will still be a contribution apparent at the in-phasechannel VI .
By directly modifying the phase of the DDS-synthesized sidebands, we can apply a phasecorrection to the upper and lower sideband electric fields as ϕc = −(ϕ+ + ϕ− )/2. Each sideband
therefore acquires an additional phase shift as
Eupper,lower = J1 (β)ei(ω±Ω)t ei(ϕc +ϕ± )(B.12)

Consider how this modifies the in-phase signal. We can substituteϕ+ → ϕ+ + ϕc = (ϕ+ − ϕ− )/2(B.13)

ϕ− → ϕ− + ϕc = (ϕ− − ϕ+ )/2(B.14)

237
Then this results in
VQ = 0(B.15)

VI ∝ ϵ+ − ϵ− − 0 + O(ϵ± )O(ϕ2± )(B.16)

Substitution into the in-phase signal, we find

= ϵ+ − ϵ−(B.17)

Now the first order approximation we made is an exact reduction to the AM components. In
practice, it may be easier to apply the correction phase shift only to a single sideband, such thatϕ+ → ϕ+ + ϕc , ϕ− is unchanged, and ϕc = −(ϕ+ − ϕ− ). It’s therefore instructure to consider
whether actuating on a single sideband produces an appreciable difference from actuating on bothsidebands. We find that the quadrature signal VQ will be nulled but the second order in-phasesignal will be modified as ϵ − ϵ +−→ ϕ2+ − 2ϕ+ ϕ− + ϕ2− ϵ+ − ϕ2− ϵ−O(ϵ± )O(ϕ2± ) = ϕ2+ − 2ϕ+ ϕ− + ϕ2−
2(B.18)For small amounts of AM and PM, this is a negligable correction.
Unlike PM, a common mode amplitude change to both sidebands ϵ+ = ϵ− results in a null
RAM signal for the in-phase quadrature (to first order). An AM correction can be made by scaling
the two amplitudes. Given our error signal, we apply a correction
αc = −(ϵ+ − ϵ− )(B.19)

Actuating on a single sideband, we can then perform the transformation
α+ → αc + α+ = 1 + ϵ−

(B.20)

1 
VI = 0 + α− ϕ2+ − ϕ2− .
2(B.21)Such that

Notably, this section is titled ’Analog locking’ rather than ’Digital locking’. The initial realizationof a DDS-generated triplet found that analog feedback to the sideband amplitudes through variable
voltage attenuators was more straightforward than writing digital words to the DDS.