Thermal Noise of Solids in Non Equilibrium Steady States

Jennifer Sanchez, Livia Conti, and Alex FontanaCalifornia State University of FullertonNational Institute for Nuclear Physics

In the past years Advanced LIGO-Virgo have detected ten binary black hole mergers and onebinary neutron star merger. While the LIGO and Virgo detectors have made upgrades with newinstrumentation that has allowed them to be able to make their precision better than one-thousandththe diameter of a proton, they still lack sensitivity. Their search for gravitational waves has beenhindered by noises that mask or mimic the signal. The detectors have been designed to measurethe most faintest noises since gravitational wave signals are extremely quiet; however, this leads toother unwanted noises that are equally quiet. One of the main noise sources in advanced LIGO andVirgo’s sensitivity curve is thermal noise. A standard way to measure thermal noise is by usingthe Fluctuation Dissipation Theorem; however, it is important to note that this theorem assumesthermal equilibrium. Most systems are not found in thermal equilibrium, and therefore we cannotuse this form of measurement. Thermal noise in non equilibrium systems has not been fully explored.

I. INTRODUCTION

Black hole binaries, along with other dense bodied bi-naries, are responsible for the gravitational waves thathave been detected by LIGO and Virgo. Since 2015, theyear in which the first gravitational wave (GW) was de-tected, there have been nine additionally confirmed gravi-tational waves from binary black hole (BBH) mergers andone from a binary neutron star (BNS) merger. The cur-rent operational interferometers are only able to observethe closest and loudest sources that exist in the local uni-verse. Now that gravitational waves can be detected andanalyzed, this is only the beginning for GW science.

A. Gravitational Wave Interferometery

In 2015, the Laser Interferometer Gravitational WaveObservatory (LIGO) detected distortions in spacetimethat were produced by the gravitational waves of twocolliding black holes nearly 1.3 billion light years away.This event revolutionized the way we observe our uni-verse and will be recorded as one of our greatest scien-tific advancements. In 1916, Einstein had predicted theexistence of gravitational waves in his theory of generalrelativity. It was not until the 1960s when Joseph Weber,an American physicists, developed the first gravitationalwave detector.

Weber understood that gravitational waves distortspacetime which meant that as they pass through an ob-ject, the object stretches and squeezes then returns backto its original shape. When the object returns to its orig-inal shape, it creates small sound waves within the ob-ject. These waves are extremely small; however, Weberbelieved they could be observed using piezoelectric sen-sors which would be mounted on a large aluminum cylin-der. When the longitudinal mode of the bar is excited,the detector is able to respond to the energy depositionby the wave. Despite the fact that Weber’s experiment

was not successful, it greatly influenced efforts to detectgravitational waves.

It was not until the late 19th century when interferom-etry was actually invented. Interferometers are a specialtool that allow us to make extremely discrete measure-ments like no other tool. Interferometers are able to cre-ate an interference pattern when the detector merges twoor more light sources. This interference pattern containsembedded messages from the source. The figure belowshows the fundamental set up of a LIGO-Virgo’s inter-ferometer [5].

FIG. 1: A fundamental Michaelson laser interferometerlayout. The laser passes through a 50/50 beam splitterwhere the two split lasers each interact with a series of

mirrors (not shown for simplicity) which are thenreflected back to the beam splitter. The resulting fringe

pattern reaches the photodiode (the black circle).

Since gravitational waves cause spacetime distortions,this means space in one direction will be stretched whileits perpendicular direction is simultaneously compressed.

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Applying this to Advanced LIGO-Virgo’s interferometermeans that one arm of the detector will become longer(stretched) while the other becomes shorter (squeezed)and vice versa. As a result, this oscillation allows thelight of the laser to return and merge together in an outof phase wave. This wave interference is then taken in bythe photodiode [2].

The detectors are able to measure such faint signalssince they are highly sensitive. This, however, also meansthat they are reading in other sources of ”noise” such astraffic nearby or crows pecking on the interferometer’sarms. In order to obtain a clear signal, these detectorsneed to filter out as much noise as possible. The fol-lowing figure shows LIGO and Virgo’s independent noisesources.

FIG. 2: Virgo Noise Budget

FIG. 3: aLIGO Noise Budget

Looking at Fig. 2 and 3, it is apparent that the detec-tors are limited in the mid and low frequency band bythermal and quantum noise. The total thermal noisebudget includes contributions from Brownian thermalnoise which is caused by mechanical losses and thermoe-lastic noise which is caused by temperature fluctuationsin the material.

B. Thermal Noise

Thermal noise is one of the main noise sources thatlimits the astrophysical reach of current and future grav-itational wave detectors. Thermally excited vibrations ofthe mirrors internal modes contribute significantly to thethermal noise within this frequency region. Since thermalnoise is the average thermal kinetic energy of the atomswithin a given material, it has been suggested to use cryo-genic temperatures on the object of study. Substantialresearch efforts have been made to study thermal noiseexperimentally and theoretically; however, these mod-els typically rely on the fluctuation dissipation theoremwhich assumes that the system is in thermodynamic equi-librium. The fluctuation dissipation theorem relates thedissipated power in the auxiliary elastic problem to thespectral density of the mirrors thermal fluctuations [4].The spectral density of the thermal driving force is givenby

F 2th(ω) = 4kBTR(ω) (1)

Where the mechanical resistance of the system is R(ω).There is a restoring force and thermal noise for a particlesubjected to velocity dampening where its equation ofmotion is

mx+ fx+ kx = Fth (2)

In this equation of motion, m represents the mass ofthe particle, f represents the frictional force describedin the Langevin equation, and k is the springs constant.In this experiment, the dampening is modeled as inter-nal dampening which requires a complex spring constant.This will extend Hooks law which will ultimately yield us

Fs = −k[1 + iφ(ω)]x. (3)

There is a lag between the springs force and responsedue to the imaginary part of the spring constant. As aresult, each cycle dissipates some of the energy in thespring. The PSD of the position of a particle subject tointernal damping results is

x2(ω) =4kBTkφ(ω)

ω[(k −mω2)2 + k2φ2)](4)

The experiment begins with an mechanical resonatorwith low frequency acoustic modes (Fig. 4). The alu-minum rod (a) is placed in a high vacuum to exclude theunwanted noises and perturbations. There is a thermalsource that acts on the aluminum mass which producesa dominant noise in the longitudinal (b) and transverse(c) low frequency mode. In order to measure the inducedthermal difference, the temperature at the top of the rodis measured as well as the bottom of the mass.

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FIG. 4: Basic setup of the experiment. There is a fixedrod that is attached to a movable mass. (a) We measurethe temperature difference from T1 located at the bot-tom of the mass and T2 at the top of the rod. Initially,the heater heats the mass while the other componentsof the experiment are in room temperature. Due to thethermal excitement, the rod will begin to resonate in the(b) longitudinal mode around 1.4 kHz and (c) transversemode around 300 Hz.

The chamber suspends the oscillator over a cascade ofmechanical filters. These mechanical filters are crucial forthe experiment since mechanical noise is the leading noisesource. Mounted under the oscillator is an interferometerbench. In addition, the base of the oscillator has beenpolished to allow the infrared laser to be reflected.

FIG. 5: The oscillator is the cubic mass with the rod.Two metal plates hold the interferometer bench, but donot touch the oscillator.

The additive object in front of the cubic mass is thecontactless thermopile sensor. Mounted under the oscil-lator is the interferometer bench. This entire system isprotected with mechanical filters which is then placed ina high vacuum chamber [1].

C. High Vacuum System

In order to measure thermal fluctuations, it is crucialto reduce external perturbations. This will allow us tohave higher sensitivity thus allowing us to have a clearsignal of thermal noise. By placing our experiment in ahigh vacuum chamber, we able to achieve such sensitivity.Our experiment included three different vacuum pumpsthat led to the chamber. The first vacuum pump drivesout the air in the system until it reaches a pressure of10−2 mbar. The second vacuum pump, Turbo, continuesto lower the pressure until it reaches our desired pressureof 10−5 mbar. Lastly, the ion pump silently lowers thepressure of the chamber up to 10−11 mbar. Attachedto the vacuum system are three vacuum gauges that areplaced on the head of the primary, Turbo, and finally thechamber. We are able to collect the readings from theion pump using the acquisition system.

FIG. 6: Basic setup of the high vacuum system andchamber. Inside the chamber is the oscillator which issurrounded by three layers of mechanical filters.

While it is possible to simply connect the turbo directlyto the chamber, this is not advised. The chamber wouldreach a pressure 10−6 mbar; however, it would inducesuch high mechanical noise to the optical table.

Throughout the experiment, we faced a problem withthe vacuum system. There was a small leak within thesystem and we could not locate it. We measured the pres-sure of the vacuum chamber over time and noted thatwhile the pressure does not drastically decrease, there isa clear sign of a leak. Suggestions were made to unmountthe vacuum system. Since the leak was fairly small, wewere worried that such an attempt would worsen the leak-age rate.

D. Interferometry

An infrared laser with a wavelength of 1064 nm passesthrough a half-wave plate (λ2 ) which rotates the polariza-tion angle to 45◦. In other words, when the light emergesfrom the half-wave plate, the polarized component will

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lag by π or 180◦. This will shift the sine wave by π. Thisrotated wave then goes straight across a beam splitter(BS) where it reaches a polarizing beam splitter (PBS).A beam splitter will split the laser accordingly so theother portion of the beam is sent out of the optics setup.This PBS separates the s and p light. The s wave reachesthe fixed reference mirror while p reaches the bottom ofthe cubic mass and is reflected back.

It is important to note that when the p waves returns,it carries information regarding the displacement of themass. The reflected s and p light return to the PBS thento the BS. From the BS, s and p meet the half-wave platewhich induces a phase shift. This means that now s andp both obtain information regarding the displacement ofthe mass in a cosine wave. The BS 50:50 splits the in-tensity of both s and p ideally equally into two differentarms. Half of the intensity of the wave reach a PBS whichseparates s and p and then absorbed by photodiode (PD)A and B respectively.

The other 50% of the wave reach a quarter-wave (λ4 )plate which translates the incoming cosine wave into asine wave. This is because the quarter-wave plate shiftsthe polarized wave by π

2 or 90◦. The two componentsboth have the same amplitude; however, the 90◦ phaseshift will change linear polarization to circular polariza-tion. They then reach a PBS where s and p are absorbedby PD C and D.

FIG. 7: Scheme of the interferometers setup that is fo-cused on the cubic mass.

Light is a transverse electromagnetic wave. The direc-tion of propagation is along the +z axis, so the electric

field vector, ~E, must lie in the plane formed by the x andy axis. The electric field of the propagating p is,

~E(z, t) = p√

2E0ei(kz−ωt) (5)

where p (or s) is the amplitude of the electric field vector,ω is the angular frequency which is equal to 2πf , k is thewavenumber or 2π

λ . By looking at the beam received bythe photodiodes, the electric field vector should move ina circle with a constant radius. When the wave reachesthe PBS, the intensity is ideally split evenly upon thetwo components, s and p. This yields the electric field

vector,

~E(z, t) = (p+ s)E0ei(kz−ωt) (6)

Ultimately, the product of the second half-wave plateproduces a phase shift which allows the two beams (ref-erence and return beam) to share the information aboutthe interference of the p beam as well as the s beam. Thisresults in,

E(z, t) =Eoe

i(kz−ωt)

2√

2[p(ei(k2L2(t)+π) + eik2L1(t))

+ s(ei(k2L2(t)+π) + ei(k2L1(t)+π2 ))].

(7)

The instruments in this experiment are placed on anoptical bench has its own suspension system to reduceexternal perturbations such as seismic noise. The infor-mation the photodiodes receive are then transferred tothe acquisition system. The acquisition system allows usto plot the the contrast of the ellipse. Since the photo-diodes from the two interferometers produce a sine andcosine pattern, we can plot the results which yield a cir-cle as expected. This contrast allows us to understandthe eccentricity of our system. In order to maximize oursystem, we aim to reduce the eccentricity of the ellipse.

E. More Noise

The fringe patterns that we obtain from the photodi-odes are then fed to our acquisition system (Analog toDigital Converter, ADC). The acquisition system con-sists of software and hardware which allows us to collectdata from the photodiodes, and convert it into practicalunits. As a result, this produces two kinds of mechanicalnoises: acquisition and dark noise. Acquisition noise iselectrical noise that is independent of the oscillator.

Dark noise is fairly small electrical current that flowsthrough our photodiodes even if no photons are beingabsorbed. This dark noise cannot be avoided, and inorder to reach maximum sensitivity, obtaining a minimaldark noise is required.

The white noise (random signal having equal intensityat different frequencies, giving it a constant power spec-tral density) remains consistence to all four photodiodesused. Obtaining low dark and acquisition noise is im-portant because these are the quietest sources of noises.Since these noises are inevitable, other sources of noisesare just added to this noise curve. The behavior of thisnoise curve should resemble white noise; linear with equalintensities at different frequencies. As a result, the darknoise should be located at a much lower frequency thanthe background spectra [1].

II. PHOTODIODE RESULTS

In order to measure the phase offset of our signals, weplot one signal of the interferometer against the other.

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We name the results from photodiodes A and B, Cx whilethe results from photodiodes C and D are called Cy. Thetwo are calculated as:

Cx =(A−B)

A+B(8)

and

Cy =(C −D)

C +D)(9)

We are able to build the complete contrast, C = Cx+iCywhere Cx contains the phase, Acos(φ) and Cy as sin(φ).When we calibrate, we are constructing Cx and Cy for awhole interference fringe. Calibrating allows us to con-vert our reading from the photodiodes which is recordedin volts to be converted in practical measurements suchas meters. By plotting Cx and Cy, we should expect aunit circle. We are able to create this interference fringeby either: 1) exciting the optical table (tapping on thetable) for a full measurement or 2) by using the full dataitself to create an ellipse fit.

FIG. 8: Comparison of a calibration when exciting theoptical table (hit) and calibrating the ellipse using a fitwith all the data points.

This contrast plot shows a strange inspiral behaviorwhich we believe to be a result of tapping on the opticaltable. By doing so, we excite the oscillator too harshly,and it is not being excited in its natural resonant. Tocorrect this, we decided to use a new technique. Insteadof tapping the optical table to excite the oscillator, weapply some voltage to the heater then return the heaterto its original voltage. This allows the oscillator to beexcited enough to create a full fringe on its own (Fig 9).

This new technique is compared with another tech-nique where we let the oscillator naturally create a fullfringe; however, this method takes a longer time to record(Fig. 10). Once we have collected our measurements andcompared our techniques, we select the calibration with

FIG. 9: Calibrating data using the heater.

FIG. 10: Calibrating data using the decimated data

the best fit. Calibrating the interferometer is importantbecause this allows us to determine the phase offset whichis then used in our analysis.

We then begin recording measurements in a dark quietroom in order to reduce as much noise as possible. Atypical measurement collection ranges from 24-48 hours.When the measurements have been collected, we take alook at the amplitude spectra density (ASD) and focuson our oscillator’s longitudinal resonant frequency.

Looking at the ASD, we are able to observe the trans-verse and longitudinal resonant modes at ∼ 300Hz and∼1400 Hz. These peaks are then fitted using a Lorentziancurve,

y(f) =2

πA

ω

4(f − f0)2 + (ω)2+ y0 (10)

where the resonant frequency is represented by f0, the

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FIG. 11: Longitudinal mode of oscillator.

full width at half maximum is ω, the area under the curveis A, and y0 is the parameter constant that fits the ad-ditive background noise. The area under this curve ishelpful when estimating the time averaged mean squarevibration 〈x1(t)2〉 of the oscillator. At thermal equilib-rium, the equipartition theorem states that,

〈x1(t)2〉 =kBt

m1ω21

(11)

where m represents the mass of the longitudinal resonantmode at the angular frequency, ω1. By measuring thearea under the Lorentzian curve, we can estimate theeffective temperature, Teff :

Teff =m1ω

21〈x1(t)2〉kB

(12)

The longitudinal mode of the oscillator is seen in the1393-1398 Hz frequency band; however, there is a clearpeak of noise from 1400 to 1403 Hz. This peak is un-known and current studies are being conducted to try andunderstand its origin. Regardless, once we have producedthe ASD, we look for a white noise frequency band nearthe resonant mode to add to our Lorentzian fit as wellas four Lorentzian parameters: the resonant frequencyband, its amplitude, χ2R, the area under the the reso-nant mode, and the full width at half maximum band.Once we have added our parameters, we look at the errorbars of our fit.

FIG. 12: Fit results and error bars.

The first plot describes the white noise around the res-onant mode. The results show that the white noise is

consistent within the frequency band. The second plotshows the area under the resonant mode in m2.

FIG. 13: Fit results and error bars.

The third plot shows resonant frequency as time pro-gresses. The plot shows that mode drifts slightly; how-ever, this is expected due to ambient temperature fluctu-ations. The fourth plot shows the χ2R which is a test thatallows one to see how well their data fits their model. Asa good rule of thumb, an optimal χ2R value is 1 and any-thing significantly higher than it indicated a poor modelfit while anything lower suggests the model is over-fittingthe data. To continue, we then create more cuts that ex-clude data outliers.

FIG. 14: Additional cuts.

In red we have data that has been selected for ourmodel while the blue points represent the total points.Our experiment begins by keeping the cubic mass in ther-mal equilibrium and making sure those results are con-sistent with models that describe thermal noise such asthe dissipation fluctuation theorem. This is initially donein order to confirm that our set up is correct. If our ex-periment has been arranged correctly, our Teff shouldread 300K. Once this has been checked off, we proceedby adding a thermal gradient between the rod and thecubic mass. By introducing this thermal gradient, weare able to observe how the transverse and longitudinal

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resonant modes behave. Since angular frequency, ω, is di-rectly proportional to average temperature, and ω2 = k

m ,we know that the stiffness (k) is directly proportional totemperature. Focusing on the two resonant modes, weobserve that when a thermal gradient is introduced, thetwo modes shifted to the a lower frequency. Again, thisis expected because ω ∝ T .

As we provide the heater with more voltage, the modesdrift further and further from their normal resonant fre-quencies. We proceed by converting the area under thecurve to units of kelvin as shown below:

FIG. 15: Histogram of average Teff .

Since our measuremnts were in thermal equilibrium,we expect our Teff to be around 300K. From the plot, wenote that the Teff mean is 256.8K. Our results show thatthis was a huge success considering that every other at-tempt in this experiment had read somewhere from 190-205K.

III. IMPROVEMENTS

We have implemented a new calibration technique inthe experiment which has improved the way we analyzeour data measurements. Initially, we would excite the os-cillator by lightly tapping on the suspended optical tablewhich would allow us to obtain a whole fringe. However,by doing so, our contrast plot would show an exagger-ated ellipse with spiraling arms. We believe this ellipsebehaves so because when we tap on the suspended opticaltable, we are exciting the oscillator in a random directionand not with its natural resonant frequency. This cali-bration is ultimately used in further analysis of our mea-surements so one can see how calibration is important.As a result, we began performing a new technique withmore promising results. We heat or cool the oscillatorwith the heater inside the vacuum chamber which allowsthe ellipse to either expand or contract then return to itsnatural shape. The next step is to use this new calibra-tion technique on our previous analysis measurements to

compare our results.Throughout the experiment, reducing the background

noise was a main goal. We mainly focused on the longi-tudinal mode of the oscillator. There was a great confu-sion as to where the adjacent frequency peak was comingfrom. A suggestion had been made to recreate the sameinterferometer bench layout as the original setup. By do-ing so, we hope to observe how the adjacent frequencypeak behaves as the mass changes. The acquisition sys-tem is also limiting the sensitivity of the measuring ther-mal noise. By obtaining a quieter signal from the system,we would obtain a clearer signal of the transverse and lon-gitudinal resonant modes. An obvious way to reduce thisnoise would be to lower the intensity of the laser beam;however, this would also lower the overall data from thephotodiodes. Another way to reduce the noise would tofind a way to lower the vibrations of the vacuum pumps.Since ion pumps do not contain coils or require oil, theyproduces no vibrations thus no external noises. The ionpump can reach pressures up to 10−11mbar which wouldincrease the chamber’s sensitivity. An ion pump was in-stalled within the vacuum system, however, it was notlowering the chambers pressure after a given time. Thiswould occur because when the ion pump chamber wasopened; the ion pump would automatically change itssetting into safety mode and stop pumping. We believethere is some sort of out-gassing within the system andfurther investigation must be made.

IV. CONCLUSION

The ability to detect gravitational waves in physics isa current challenge. The main obstacle that the detec-tors face is reducing their noise in order to obtain a cleargravitational wave signal. Thermal noise is one of theleading sources of noise that is hindering the detectorssensitivity, and we can expect that future detectors willencounter this problem as well. Currently, thermal noiseis being measured using the fluctuation-dissipation theo-rem; however, this relies on the system remaining in ther-mal equilibrium. It is important to note that the currentdetectors are not working in thermodynamic equilibriumand most natural systems do not work under such con-ditions. There is no complete model for thermal noisein non equilibrium states, and such model is needed todetermine how this affects the detectors.

V. ACKNOWLEDGEMENTS

I would like to especially thank my advisor, Dr. LiviaConti for introducing me to experimental physics. Notonly was I exposed to new theoretical topics in physics,but Dr. Conti was also kind enough to thoroughly ex-plain the mechanics of her project. Dr. Giacamo Cianiand Dr. Marco Bazzano were also instrumental in help-ing me solidify my understanding within the experiment.

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Additionally, I worked closely with a visiting PhD stu-dent, Alex Fontana, for the month of June. Alex allowedme to perform tasks within the experiment that were be-yond the scope of an average undergraduate student. I

would lastly like to thank the National Science Founda-tion and the University of Florida for funding this stellarexperience.

[1] Conti, Livia, Paolo De Gregorio, Gagik Karapetyan, Clau-dia Lazzaro, Matteo Pegoraro, Michele Bonaldi, and Lam-berto Rondoni. ”Effects of Breaking Vibrational EnergyEquipartition on Measurements of Temperature in Macro-scopic Oscillators Subject to Heat Flux.” Journal of Statis-tical Mechanics: Theory and Experiment 2013.12 (2013):n. pag. Print.

[2] LIGO. ”What Is an Interferometer?” GravitationalWaves. N.p., n.d. Web. 11 Aug. 2019.

[3] Lovelace, Geoffrey. Numerically Modeling Brownian Ther-mal Noise in Amorphous and Crystalline Thin Coatings.Classical and Quantum Gravity, vol. 35, no. 2, 2017, p.025017.,doi:10.1088/1361-6382/aa9ccc.

[4] Fejer, Martin M. Thermal Noise in Mirror Coatings forGravitational Wave Detection. Optical Interference Coat-ings 2016, 2016, doi:10.1364/oic.2016.mb.1.

[5] B. P. Abbott et al. (LIGO Scientific Collaboration andVirgo Collaboration), Observation of gravitational wavesfrom a binary black hole merger, Phys. Rev. Lett. 116,061102 (2016).