Thermal noise Kazuhiro Yamamoto

Thermal noise Kazuhiro Yamamoto
Department of Physics, Faculty of Science, University of Toyama First of all, I thank Kentaro for giving such a chance. Ok, let’s start ! This is the title of my talk of today. Thermal noise, History from Brownian motion till interferometer. In this lecture, I will show the outline of thermal noise, from the Brownian noise, which is the thermal motion investigated firstly, until the latest knowledge about thermal noise in interferometer . Lecture about gravitational wave detector for fresh persons 12 July 2017 @Gofuku campus, University of Toyama, Toyama, Japan

References Peter R. Saulson, Thermal noise in mechanical experiments, Physical Review D 42 (1990) Good review before revolution (on the end of 20th century) S. Rowan, J. Hough, and D.R.M. Crooks, Thermal noise and material issues for gravitational wave detectors Physics Letters A 347 (2005) 25. One of the special articles for 100th anniversary of Annus Mirabilis (year of miracle) of A. Einstein Although I will review thermal noise, there are many excellent review. Some of them are introduced here. The first one is Peter Saulson’s famous paper. I think this is the good review before revolution. On the end of the 20th century, there was the drastic progress in research of thermal noise. It is called “revolution” in this lecture. All people who begin to study thermal noise must read it. The second one is written by Sheila et al. This is one of the special article for 100th anniversary of Annuns Mirabilis which means year of miracle in Latin of Einstein.

References Erick D. Black Notes on thermal noise, with a bibliography
LIGO-T R (2004) From Brown and Einstein to Yamamoto G.M. Harry, T. Bodiya, and R. DeSalvo (Editors) Optical Coatings and Thermal Noise in Precision Measurements Cambridge University Press, Cambridge (2012) The third one is the technical note by Eric Black. The reference list includes many papers. You can find the name, Brown and Einstein, at the beginnings of the reference list. The last person of the reference is me ! The last one is the book , Optical Coatings and Thermal Noise in Precision Measurements. Now it is in press and will appear January of next year. This book includes many topics about thermal noise and coatings.

0.Abstract I would like to explain … (1) History until Fluctuation-Dissipation Theorem What is the Fluctuation-Dissipation Theorem ? (2) Thermal noise of interferometric gravitational wave detector before revolution (drastic progress in research of thermal noise) on the end of 20th century (3) Thermal noise of interferometric gravitational wave detector after revolution on the end of 20th century This is abstract. I would like to explain history until Fluctuation-Dissipation Theorem at first. Next, I will show the thermal noise in resonant and interferometric gravitational wave detector. On the end of 20th century, there were drastic progress of research of thermal noise. It is called “revolution” in this lecture. I will explain the outlines before and after revolution.

Contents 1. Until Fluctuation-Dissipation Theorem 2. Interferometer before revolution 3. Interferometer after revolution 4. Summary This is the contents. I would like to explain history until Fluctuation-Dissipation Theorem. Next, I will show the thermal noise in resonant detector. The research of thermal noise in interferometric gravitational wave detector before and after revolution will be talked. At last, I summarize my lecture.

1. Until Fluctuation-Dissipation Theorem
Robert Brown investigated random motion of small particles (~1 mm) in water. R. Brown, Philosophical Magazine 4 (1828) 161. At first, he thought that this motion of particles from pollens stems from activity of life. However, he discovered that particles from inorganic materials also move at random. OK, let’s start review of thermal noise. The first topic is Brownian motion, of course. As you know, Robert Brown investigated random motion of small particle, let say, about 1 micro meter, in water. His first paper of this topic was published 1828, about 200 years ago. At first, he thought that this motion of particles from pollens stems from activity of life. However he discovered that particles from inorganic materials also move at random. Here, I introduce a trivia. I hope such a stupid misunderstanding is spread in only Japan, not Germany. Anyway, Brown observed motion of small particles from pollen, not pollens themselves. In fact, since pollens are too large, let say, between 25 micro meter and 100 micro meter, it is difficult to observe Brownian motion. Trivia : R. Brown observed motion of small particles from pollens, not pollens themselves ! Since pollens are too large (25 mm~100 mm), it is difficult to observe Brownian motion.

Robert Brown investigated random motion of small particles (~1 mm) in water. R. Brown, Philosophical Magazine 4 (1828) 161. Mechanism was unknown. Many ideas were proposed and rejected. Random collisions with atoms of water ? For example, G. Cantoni, Nuovo Ciment 27 (1867) 156. J. Delsaulx They are not proof but guesses. Although Brown found random motion of small particles, mechanism of this motion was unknown. Many ideas were proposed and rejected. We know that Brownian motion is caused by the random collisions with atoms of water. Some people showed this correct idea. But they only provided only their guesses, not proof.

Albert Einstein showed theory of Brownian motion. A. Einstein, Annalen der Physik 17 (1905) 549. Why is so important this result ? (1)Evidence of existence of atom Avogadro constant derived from observation and his theory is consistent with those from other methods. About 80 years later, the correct theory came. In 1905, Albert Einstein showed theory of Brownian motion. This is the one of the most important his work. Why ? The first reason is that it showed the evidence of existence of atom. Avogadro constant derived from observation and his theory is consistent with those from other methods. Wikipedia (A. Einstein, English)

(2) Relation between diffusion (thermal motion) of particles and viscosity (dissipation) of water He assumed that the law of physics of macroscopic body is the same as that of microscopic one. Einstein’s relation diffusion Here, I would like to show the second reason and emphasis that Einstein proved the relationship between diffusion, thermal motion, of particles and viscosity, dissipation, of water. In this paper, he assumed that the low of physics of macroscopic body is the same as that of microscopic one. Avogadro constant viscosity Wikipedia (A. Einstein, English) 9

Jean Baptiste Perrin’s experiment proved that Einstein’s theory is correct. J. Perrin, Ann. Chim. Phys. 18 (1909) 1. Perrin checked and confirmed Einstein’s assumption (the law of physics of macroscopic body is the same as that of microscopic one) experimentally. Perrin observed Brownian motion and derived Avogadro constant using Einstein’s theory. The result is consistent with those of other methods. In order to check Einstein’s theory, Perrin tried experiment. His result is published on At first Perrin checked Einstein’s assumption, I mean that the law of physics of macroscopic body is the same as that of microscopic one, experimentally and confirmed that it is correct. Next, Perrin observed Brownian motion and derived Avogadro constant from the experimental result using Einstein’s theory. The result is consistent with those of other methods.

Albert Einstein won Nobel prize in Physics for The citation is “for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect”. Brownian motion does not appear. Web site of Nobel foundation

Presentation speech of Nobel prize in Physics 1921 (Laureate is A. Einstein) Throughout the first decade of this century the so-called Brownian movement stimulated the keenest interest. In 1905 Einstein founded a kinetic theory to account for this movement by means of which he derived the chief properties of suspensions, i.e. liquids with solid particles suspended in them. This theory, based on classical mechanics, helps to explain the behaviour of what are known as colloidal solutions, a behaviour which has been studied by Svedberg, Perrin, Zsigmondy and countless other scientists within the context of what has grown into a large branch of science, colloid chemistry. Nevertheless, in presentation speech of Nobel prize ceremony, his work for Brownian motion was introduced. This slide shows this part of presentation speech. Throughout the first decade of this century the so-called Brownian movement stimulated the keenest interest. In 1905 Einstein founded a kinetic theory to account for this movement. You can find Perrin in the second half of this paragraph. Web site of Nobel foundation

Perrin also won Nobel prize in 1926. Web site of Nobel foundation

Thermal fluctuation of electrical voltage (or current) J.B. Johnson, Physical Review 32 (1928) 97. Measurement H. Nyquist, Physical Review 32 (1928) 110. Theory Nyquist’s theorem Thermal fluctuation electrical voltage or current or charge is also famous example of thermal fluctuation. In 1928, important papers about this topic appeared. They were written by Johnson and Nyquist. Johnson explained his measurement of thermal fluctuation in electrical circuit. Nyquist provided his theory about circuit. Anyway, they proved the relations between electrical fluctuation and resistance, I mean, this famous formula. This left hand side is power spectrum of electrical voltage fluctuation. The right hand side is the product of temperature and resistance. So this is the relation between voltage fluctuation and resistance. Relation between electrical voltage fluctuation and resistance

Thermal fluctuation of electrical voltage (or current) J.B. Johnson, Physical Review 32 (1928) 97. He measured thermal current of resistance using (resonant type or band pass type) amplifier. Johnson measured thermal current of resistance using resonant type or band pass type amplifier. This is the schematic view of experiment. Z on left hand side is the resistance A on center is amplifier. J on the right hand side is current meter.

Thermal fluctuation of electrical voltage (or current) J.B. Johnson, Physical Review 32 (1928) 97. He confirmed formula for thermal fluctuation. Johnson’s measurement show that thermal voltage fluctuation formula. The horizontal axis is resistance and vertical axis is the fluctuation of voltage. Bolztmann constant can be derived from this measured proportional coefficient. The result agrees with those by other methods. 16

Typical thermal voltage fluctuation (100 ohm, 300 K) Typical thermal current fluctuation (100 ohm, 300 K) Here, I would like to show useful values, typical thermal fluctuation of voltage and current. If the resistance is 100 ohm at room temperature, the power spectrum of voltage and current is about 1 nV/rtHz and 10 pA/rtHz.

Thermal fluctuation of electrical voltage (or current) H. Nyquist, Physical Review 32 (1928) 110. His theory is based on Principle of energy equipartition Assumption that ohm law is correct even if we consider voltage (current) fluctuation. (Law for small fluctuation is the same as that of macroscopic voltage or current) This assumption is similar to Einstein’s. Let us consider Nyquist’s paper slightly. This is the paper of theory. His theory is based on the principle of energy equipartition and assumption that ohm law is correct even if we consider voltage or current fluctuation. I mean that law for small fluctuation is the same as that of macroscopic voltage or current. I should say that Nyquist did not mention the second item clearly in his paper and this assumption is similar to Einstein’s. 18

Trivia We can found three technical terms named after Nyquist. Nyquist’s Theorem : Thermal noise Nyquist criterion of stability : Stability of control Nyquist sampling theorem : Sampling rate of measurement 　　　　　　　Are all of them work by same person ? 　　　　　　 The answer is yes ! Here is the second trivia. We can find three technical terms named after Nyquist. Nyquist’s theorem, this is the formula of thermal noise, Nyquiest criterion of stability, this describes stability of control, Nyquist sampling theorem , this shows the smallest sampling rate of measurement. I always wondered whether all of them are work by same person. Amazingly, the answer is yes ! Wikipedia (H. Nyquist, English)

Thermal fluctuation of mechanical harmonic oscillator Many people measured and analyzed fluctuation of angle of torsion pendulum using optical lever around 1925. W. Einthoven et al., Physica 5 (1925) 358. J. Tinbergen, Physica 5 (1925) 361. W.J.H. Moll et al., Philosophical Magazine 50 (1925) 626. G. Ising, Philosophical Magazine 1 (1926) 827. F. Zernike, Zeitschrift fuer Physik 40 (1926) 628. A.V. Hill, Journal of Scientific Instruments 4 (1926) 72. Probably, this is not perfect list. Let us consider the other example, thermal fluctuation of mechanical harmonic oscillator. Around 1925, many people measured and analyzed fluctuation of angle of torsion pendulum using optical lever. This is the list of papers. I am afraid that it is not perfect.

Thermal fluctuation of mechanical harmonic oscillator E. Kappler, Annalen der Physik 11-3 (1931) 233. Torsion pendulum He evaluated Avogadro constant and it is consistent with those of other experiment. Measurement Kappler also measured thermal fluctuation of torsion pendulum. This beautiful Gaussian profile is his experiment result. The horizontal axis is the angle. This width is the root mean square of angle fluctuation. He evaluated Avogadro constant and it is consistent with those of other method.

Onsager reciprocity theorem L. Onsager, Physical Review 37 (1931) 405. Fourier’s law In general case, k is tensor. According to Onsager reciprocity theorem, this tensor should be symmetric even if the material is not isotropic (like sapphire !). Thermal conductivity Heat flow Temperature gradient The next topic is the one of the most important theorems of thermal fluctuation, Onsager reciprocity theorem. This is an example. Fourier’s law show that heat flow is the product of thermal conductivity and temperature gradient. In general case, this conductivity is tensor. According to reciprocity theorem, this tensor should be symmetric even if the material is not isotropic, for example, sapphire.

Onsager reciprocity theorem L. Onsager, Physical Review 37 (1931) 405. Onsager’s assumption (1)Microscopic reversibility Symmetry of cross correlation function in time reflection Although I do not explain the details of this famous theorem, I would like to show the Onsager’s assumption to derive this theorem. The first one is microscopic reversibility. He assumed cross correlation function has symmetry in time reflection. The sign of tau is flipped in the first expression. This expression could be rewritten as like this. The number of alpha is exchanged.

Onsager reciprocity theorem L. Onsager, Physical Review 37 (1931) 405. Onsager’s assumption (2)The average decay of fluctuations will obey the ordinary laws. Law for average decay of small fluctuation is the same as that of macroscopic motion (ordinary law). This assumption is similar to Einstein’s and Nyquist’s. The second assumption is that the average decay of fluctuations will obey the ordinary laws. It implies that law for average decay of small fluctuation is the same as that of macroscopic motion, I mean ordinary law. So, this assumption is similar to Einstein’s and Nyquist’s. 24

This Onsager’s paper brought his Nobel prize in Chemistry on 1968. Web site of Nobel foundation

Finally, general theorem appeared. Fluctuation-Dissipation Theorem (FDT) H.B. Callen and R.F. Greene, Physical Review 86 (1952) 702. R.F. Greene and H.B. Callen, Physical Review 88 (1952) 1387. Relation between thermal fluctuation and dissipation Finally, general theorem of thermal noise appeared in Its name is Fluctuation Dissipation Theorem. This theorem describes relationship between thermal fluctuation and dissipation. Why are they related with each other ? Fluctuation is the energy flow from heat bath. On the other hand, dissipation is energy flow to heat bath. So, they are interaction between system and heat bath. So, there are their relation. Fluctuation : Energy from heat bath Dissipation : Energy to heat bath Interaction between system and heat bath

Fluctuation-Dissipation Theorem is valid if (1)system is linear. (2)system is in thermal equilibrium. The fluctuation dissipation theorem is extremely useful because it is valid if the system is linear and in thermal equilibrium. The expression is as follows. G in the left hand side is the power spectrum of thermal fluctuation. Imaginary part of H in the right hand side includes the imaginary part of susceptibility, which is the dissipation. So, here is the relation between fluctuation and dissipation. Power spectrum of thermal fluctuation Imaginary part of susceptibility (dissipation)

Fluctuation-Dissipation Theorem is valid if (1)system is linear. (2)system is in thermal equilibrium. In fact, fluctuation dissipation theorem predicts not only power spectrum, but also cross correlation spectrum of thermal fluctuation. Cross correlation spectrum of thermal fluctuation 28

(a)Einstein’s relation Relation between Brownian motion (fluctuation) and viscosity (dissipation) of water FDT in the case with free mass with viscous damping at low frequency (b)Nyquist’s theorem Relation between thermal voltage fluctuation and resistance (dissipation) FDT in electric circuit (c)Onsager reciprocity theorem Cross correlation spectrum at low frequency in FDT All these formulae are examples of FDT ! Let us review formulae which appear in this lecture. Einstein’s relation predicts relation between Brownian motion, fluctuation and viscosity, dissipation of water. In fact, this relation could be derived from FDT when we consider the case with free mass with viscous damping at low frequency. Nyquist’s theorem shows relation between thermal voltage fluctuation and resistance, dissipation. Nyquist’s theorem also could be derived from FDT if we consider electric circuit. Onsager reciprocity theorem show the cross correlation spectrum at low frequency in FDT. So, all of formulae about thermal fluctuation are examples of FDT !

Assumption of fluctuation dissipation theorem Onsager’s assumption The average decay of fluctuations will obey the ordinary laws. Law for average of small fluctuation is the same as that of macroscopic motion with dissipation. Relation between fluctuation and dissipation is assumed, not proved ! Fluctuation dissipation theorem is based on the Onsager’s assumption, the average decay of fluctuation will obey the ordinary laws. This implies law for small fluctuation is the same as that of macroscopic motion with dissipation. So, this is the relation between fluctuation and dissipation. I must say that relation between fluctuation and dissipation is assumed, not proved when fluctuation dissipation theorem is derived.

Fluctuation : Energy from heat bath Dissipation : Energy to heat bath Interaction between system and heat bath In the case of Brownian motion …. Fluctuation : Random collision of atoms Dissipation : Collision of atoms Even in dissipation, collision is at random. In some cases of dissipation process, atoms give particle energy. However, in average, particle gives atoms energy. Therefore, the dissipation process is the average of fluctuation. Onsager’s assumption So, is Onsager’s assumption correct ? OK, let us remember what fluctuation and dissipation are. The fluctuation is caused by the energy from heat bath. The dissipation implies energy to heat bath. Both of them are interaction between system and heat bath. In the case of Brownian motion, fluctuation is caused by random collision of atoms. The dissipation also stems from collision of atoms. Atoms get energy via collision with particle. Even in dissipation, collision is at random. So in some case of dissipation process, atoms can give particle energy in dissipation process. However, in average, particle gives atoms energy. Therefore, the dissipation process is the average of fluctuation. This is the Onsager’s assumption.

How to derive fluctuation dissipation theorem ? Onsager’s assumption implies that time development of auto (or cross) correlation function of thermal fluctuation is the same as that of step function response which is the decay motion to new equilibrium position after applied constant force vanished. The amplitude of auto correlation function is derived from principle of energy equipartition. Power (or cross correlation) spectrum is Fourier transform of auto (or cross) correlation function. Wiener-Khinchin relation So, how did Callen and Greene derive fluctuation dissipation theorem ? At first, they claimed as follows. Onsager’s assumption implies that time development of auto or cross correlation function of thermal fluctuation is the same as that of step function response. What is the step function ? Let us imagine some physical system. The constant force is applied on this system. But suddenly, this force disappears. So, system moves to new equilibrium position. This is one kinds of decay motion. OK, now we know time development. But how about amplitude of correlation function ? In fact, this amplitude is derived from principle of energy equipartition. Finally, we remember Wiener-Khinchin relation, power or cross correlation spectrum is Fourier transform of auto or cross correlation function. So we can derive fluctuation dissipation theorem.

2. Interferometer before revolution
Interferometric gravitational wave detector Mirrors must be free and are suspended. Now we start to discuss the thermal noise of interferometer. As I have already mentioned, there was the drastic progress of research of thermal noise on the end of 20th century, revolution. Here, I explain the research of thermal noise before this revolution. This is the schematic view of interferometric gravitational wave detector. When gravitational wave comes, for example, this arm becomes longer, this arm becomes shorter. So, the power at photo detector changes. We can detect the gravitational wave. For detection of gravitational wave, mirrors must be free. So, they are suspended. We need suspension systems for mirrors. S. Kawamura, Classical and Quantum Gravity 27 (2010)

Typical example of sensitivity of interferometer (Old version of KAGRA(LCGT)) This is a typical example of sensitivity of interferometer, old version of LCGT. This axis is frequency between 1 Hz and 10 kHz. The vertical axis show sensitivity. If this value is small, it implies good sensitivity. This black line is the total noise. Around 100 Hz, the sensitivity is best. There are three kinds of fundamental noise, seismic noise, thermal noise, and quantum noise. These green and blue lines are thermal noise of suspension and mirror.

Thermal noise of suspension and mirror There are two kinds of thermal noise, suspension and mirror. As I have already mentioned, the mirror is suspended to keep it free. So, this is a pendulum. Pendulum motion is excited thermally and the center of mirror fluctuates. This is the thermal noise of suspension. The elastic vibration of mirror itself is also excited thermally. So, the surface of mirror fluctuates. This is the thermal noise of mirror.

Suspension and mirror : Mechanical harmonic oscillator Resonant frequency : suspension : ~ 1 Hz mirror : > 10 kHz Target frequency of gravitational wave : ~ 100 Hz Off resonance thermal fluctuation of displacement Pendulum motion of mirror and elastic vibration of mirror are similar to the harmonic oscillator. So, we must consider the thermal motion of harmonic oscillator. Resonant frequency of suspension is about 1 Hz. The elastic resonance of mirror is higher than about 10 kHz. However, the frequency of target gravitational wave is about 100 Hz. So, we must discuss the thermal noise in off resonance region, not on resonance. Let us review resonant detector and experiment of torsion pendulum. In the case of resonator, we should consider force on resonance. In the experiment with torsion pendulum, we monitor displacement on resonance. If the interferometer is in air , the viscosity of air is important issue for thermal noise. However, interferometer is in vacuum because air disturbs the propagation of light. Since the pressure is less than 10^(-7) mbar, residual gas damping does not matter. So, mechanical loss in suspension and mirror is crucial. Residual gas damping is not a problem because interferometer in vacuum (< 10-7 mbar). Mechanical loss in suspension and mirror is crucial.

Spectrum density of thermal noise of harmonic oscillator Viscous damping : Friction force is proportional to velocity. Structure damping : Loss is independent of frequency. This is an example of power spectrum of thermal noise of harmonic oscillator. The horizontal axis is frequency. This is the resonant peak. This graph shows the two kinds of thermal noise, caused by viscous damping, solid line, and structure damping, dashed line. Viscous damping is famous. Residual gas damping is an example. The friction force is proportional to frequency. The other one is structure damping. The magnitude of loss is independent of frequency. Many experiments show dissipation in many materials is structure damping. Loss in many materials are structure damping.

Spectrum density of thermal noise of harmonic oscillator Q-value : Magnitude of loss Higher Q implies smaller loss. Higher Q means smaller off resonance thermal noise and better. As I said, to evaluate thermal noise, strength of loss is one of the most important factor. Q-value represents the magnitude of loss. Higher Q is smaller loss. This graph shows how thermal noise depends on Q-values. Q-value of blue lines is 100 times larger than that of red lines. Around resonance, the thermal noise with high q-value is large. On the other hand, off resonance region, high Q implies smaller thermal noise. So, in the case of interferometer, Q must be as high as possible

Measurement of Q-value (Width of resonance peak) So, we need to measure Q-value. But how ? One of the methods is measurement of width of resonance peak. This is the mechanical response of harmonic oscillator. The horizontal axis is frequency. This is the width of resonance peak. Narrower width means higher Q. In our case, Q is much larger than unity. So, resonance width it too narrow to measure it. This method is not so easy, unfortunately. If Q-value is too high, measurement is difficult.

Measurement of Q-value (Decay time of resonance motion) Higher Q imples lower loss. The other method to measure Q is decay time of resonance motion. The horizontal axis is time. At this time, somebody kicks oscillator. The resonant motion is excited. However, amplitude of resonance decays. The longer decay time is higher Q. In our usual case, we adopt this method. In (our) usual cases, we adopt this method.

Recoil loss (problem in measurement of decay time) The principle of measurement of decay time is so simple. However, there is a problem. One of them is recoil loss. Now, we try to measure the Q-values of pendulum. The mass is suspended from this support system. We kick this mass and measure the Q-values. However, owing to back action, support system also vibrate as this figure. It implies that we measure Q-values of the combined system, pendulum and support system. In short, measured Q depends on not only loss in pendulum but also that in support. But, we would like to know only Q of pendulum ! This is called recoil loss. This is the contamination of loss in support. So, in the case of pendulum, this support system should be as rigid and heavy as possible. Contamination of loss in support system Suspension : Rigid and heavy support system

Measurement of Q-value of pendulum (monolithic fused silica) Q = 2.3 * 107 Some groups measured Q-value of pendulum. This is one of the measurements and their Q is one of the highest ones. In order to increase Q, they made monolithic suspension. In usual interferometric gravitational wave detector, mirror is made from fused silica. In this measurement, these fibers are also made from fused silica. Q is the order of 10^7. In order to reduce recoil loss, the support system is heavy and rigid. G. Cagnoli et al., Physical Review Letters 85 (2000) 2442.

Measurement of Q-value of fiber itself It is easier, but this Q-value does not include loss in clamp and so on. Moreover, … Potential energy of total pendulum consists of (1)gravitational one (2)elastic one (1) is much larger than (2). But (1) has no loss. Only (2) has loss. (2) Some groups measured Q-value of pendulum. This is one of the measurements and their Q is one of the highest ones. In order to increase Q, they made monolithic suspension. In usual interferometric gravitational wave detector, mirror is made from fused silica. In this measurement, these fibers are also made from fused silica. Q is the order of 10^7. In order to reduce recoil loss, the support system is heavy and rigid. (1) (2)

Measurement of Q-value of fiber itself Loss of total pendulum (Inverse number of Q) Loss of elastic energy of wire (Inverse number of Q) Some groups measured Q-value of pendulum. This is one of the measurements and their Q is one of the highest ones. In order to increase Q, they made monolithic suspension. In usual interferometric gravitational wave detector, mirror is made from fused silica. In this measurement, these fibers are also made from fused silica. Q is the order of 10^7. In order to reduce recoil loss, the support system is heavy and rigid. Dilution factor : 10-3 or 10-4 in typical cases (Almost unity in KAGRA). Q-values of pendulum is much higher than Q-values of wires.

Recoil loss (problem in measurement of decay time) If you want to measure the Q-value of mirror, I can introduce the excellent support system with extremely small recoil loss. It is called nodal support system. In usual case, the mirror is a cylinder. The center of flat surface is node for many modes. So, this support system can not be excited even if the mirror is excited. Mirror : Nodal support system (Center of flat surface is node for many modes) K. Numata et al., Physics Letters A 276 (2000) 37.

Measurement of Q-value of fused silica mirror Q = 4 * 107 Structure damping Not structure damping How about Q of fused silica mirror ? This is one of results. The horizontal axis represents the frequency of resonant modes. Numata found that Q depends on the grade of silica. For example, result in the left side graph is not so good. And Q is about 10^6. On the other hand, the Q in the right hand side graph is good. Q is a few times 10^7. Moreover, if Q is not so high, Q is independent of frequency. So, the loss in silica is structure damping. On the contrary, if the Q is enough high, it depends on frequency ! This is not structure damping and we must take this fact into account when we calculate thermal noise. K. Numata et al., Physics Letters A 327 (2004) 263.

Evaluated thermal noise based on Q-value measurement OK, now, we know the Q-values of pendulum and silica mirror. I calculated thermal noise. This black line is the goal sensitivity of Advanced LIGO, which is the American future project. This sensitivity is not latest one. Yes, thermal noise is not larger than goal sensitivity. It is fine. But, after this, revolution brought new issues. This sensitivity is not latest one.

3. Interferometer after revolution
Thermoelastic noise of mirror (2) Thermal noise caused by coating mechanical loss (3) Reduction of thermal noise (4) Impact on other fields On the end of 20th century, revolutions came in the field of research of thermal noise. I would like to introduce drastic progress of research. There are 6 topics.

Thermoelastic noise of mirror Thermoelastic damping : a kind of loss Inhomogeneous strain Thermal expansion coefficient Temperature gradient Heat flow We can calculate thermoelastic noise using only material properties. The first revolution is the thermoelastic noise. The thermoelastic damping is a kind of loss. This figure is a simple example. This rod is deformed and we can find inhomogeneous strain in it. The right hand side part stretches, the left hand side part shrinks. Since the usual material has thermal expansion coefficient, temperature gradient is caused by inhomogeneous strain. So heat flows in horizontal direction. It implies dissipation. This is the thermoelastic damping. It must be noted that we can calculate thermoelastic noise using only material properties. In usual cases, mechanism of dissipation is unknown. So, it is a rare case. So, we can obtain the lower limit of the dissipation theoretically. Drawing by Tobias Westphal

Thermoelastic noise of mirror Thermoelastic noise : thermal noise by thermoelastic damping Other interpretation Temperature fluctuation Thermal expansion coefficient Deformation of elastic body This thermoelastic noise corresponds to thermoelastic noise. In fact, there is the other interpretation of thermoelastic noise. Temperature itself fluctuates. Owing to the thermal expansion coefficient, this temperature fluctuation makes elastic body deformed. This is the other interpretation.

Thermoelastic noise of mirror Long, long history of research of thermoelastic damping 1 dimension (bar, ribbon) C. Zener, Physical Review 52 (1937) 230; 53 (1938) dimension (disk) L.D. Landau and E.M. Lifshitz “The Theory of Elasticity” , A.S. Nowick and B. Berry “Anelastic Relaxation in Crystalline Solids” , dimension (mirror) 1999 History of research of thermoelastic damping is so long. The first paper about this topic appeared about 70 years ago. In these paper, 1 dimension systems, bar and ribbon, were discussed. After that, 2 dimension system, disk, was considered. However, the mirror is the 3 dimension system. And, the thermoelastic noise of mirror was evaluated about 10 years ago.

Thermoelastic noise of mirror Thermoelastic noise of 3 dimension mirror without coating V. B. Braginsky et al., Physics Letters A 264 (1999) 1. We can calculate thermoelastic noise using only substrate material properties. This noise is larger than expectation ! This is the first paper about thermoelastic noise of 3 dimension mirror in I must say that, in this paper, only mirror substrate material are taken into account, reflective coating is neglected. Anyway, now, we can evaluate thermoelastic noise using only material properties. The result is amazing. This noise is larger than expectation !

Thermoelastic noise of mirror Fused silica vs. Sapphire Current interferometric gravitational wave detector Fused silica mirror Future interferometer Sapphire was a candidate. Optical properties of fused silica is better. It was expected that thermal noise of sapphire is smaller. However … Before this Braginsky’s paper, there was a discussion about mirror material for interferometric gravitational wave detector. Which is the better, fused silica or sapphire ? In all current interferometers, fused silica mirrors are adopted. However, for future interferometer, sapphire was a candidate. Why ? In fact, the optical properties of fused silica is better. But, it was expected that thermal noise of sapphire mirror is smaller. However, the paper about thermoelastic noise in 1999 proved it was completely wrong.

Thermoelastic noise of mirror This graph shows the amplitude of thermal noise. This black line is an example of future interferometer sensitivity. Now, we can calculate the thermoelastic noise. These thick lines are results. This blue line is thermoelastic noise of sapphire, not fused silica. So, the contribution of thermoelastic noise of sapphire is larger than the future goal sensitivity! One of the advantages of sapphire was lost. Future room temperature interferometer will use fused silica mirror. One of advantages of sapphire was lost. Future (room temperature) interferometer will use fused silica mirror.

(2) Thermal noise caused by coating mechanical loss Mirror consists of not only bulk ! Reflective coating In the previous section, I explained the Q-values of mirror substrate. However, mirror consists of not only bulk ! At first, reflectivity of mirror must be high. So, reflective coating is necessary as figure on the left hand side. Moreover, mirror position must be adjusted to keep sensitivity high. The most popular actuator is coil-magnet actuator. So, we must glued the magnets on mirror as figure on the right hand side. Coating and glued magnet have loss. Let us consider them. Thickness of coating is about 5 micro m. On the contrary, thickness of mirror is 10 cm. So, coating is times thinner. So, even if it has large loss, Q-value of mirror does not so change. So, nobody cared this issue. On the other hand, Glued magnets decrease Q-values. OK, this is the serious problem.

(2) Thermal noise caused by coating mechanical loss Reflective coating : Dielectric multilayer (amorphous) coating for high reflectivity Two kinds of layers (Ta2O5/SiO2) Optical thickness is quarter of wavelength Reflected light : Constructive interference Transmitted light : Destructive interference In the previous section, I explained the Q-values of mirror substrate. However, mirror consists of not only bulk ! At first, reflectivity of mirror must be high. So, reflective coating is necessary as figure on the left hand side. Moreover, mirror position must be adjusted to keep sensitivity high. The most popular actuator is coil-magnet actuator. So, we must glued the magnets on mirror as figure on the right hand side. Coating and glued magnet have loss. Let us consider them. Thickness of coating is about 5 micro m. On the contrary, thickness of mirror is 10 cm. So, coating is times thinner. So, even if it has large loss, Q-value of mirror does not so change. So, nobody cared this issue. On the other hand, Glued magnets decrease Q-values. OK, this is the serious problem.

(2) Thermal noise caused by coating mechanical loss Reflective coating : Dielectric multilayer coating for high reflectivity Advantages (1)Low optical loss (Absorption and scattering) (2)No break with high power In the previous section, I explained the Q-values of mirror substrate. However, mirror consists of not only bulk ! At first, reflectivity of mirror must be high. So, reflective coating is necessary as figure on the left hand side. Moreover, mirror position must be adjusted to keep sensitivity high. The most popular actuator is coil-magnet actuator. So, we must glued the magnets on mirror as figure on the right hand side. Coating and glued magnet have loss. Let us consider them. Thickness of coating is about 5 micro m. On the contrary, thickness of mirror is 10 cm. So, coating is times thinner. So, even if it has large loss, Q-value of mirror does not so change. So, nobody cared this issue. On the other hand, Glued magnets decrease Q-values. OK, this is the serious problem.

(2) Thermal noise caused by coating mechanical loss Mirror consists of not only bulk ! Reflective coating Thickness of coating : ~ 5 mm Thickness of mirror : ~ 10 cm Why can coating contribute thermal noise ? Nobody cared until 1998. In the previous section, I explained the Q-values of mirror substrate. However, mirror consists of not only bulk ! At first, reflectivity of mirror must be high. So, reflective coating is necessary as figure on the left hand side. Moreover, mirror position must be adjusted to keep sensitivity high. The most popular actuator is coil-magnet actuator. So, we must glued the magnets on mirror as figure on the right hand side. Coating and glued magnet have loss. Let us consider them. Thickness of coating is about 5 micro m. On the contrary, thickness of mirror is 10 cm. So, coating is times thinner. So, even if it has large loss, Q-value of mirror does not so change. So, nobody cared this issue. On the other hand, Glued magnets decrease Q-values. OK, this is the serious problem.

(2) Thermal noise caused by coating mechanical loss Mirror consists of not only bulk ! Reflective coating Thickness of coating : ~ 5 mm Thickness of mirror : ~ 10 cm Loss angle f (1/Q) Bulk : 10-8 Coating : 10-4 Comparable ? This is not end of story … In the previous section, I explained the Q-values of mirror substrate. However, mirror consists of not only bulk ! At first, reflectivity of mirror must be high. So, reflective coating is necessary as figure on the left hand side. Moreover, mirror position must be adjusted to keep sensitivity high. The most popular actuator is coil-magnet actuator. So, we must glued the magnets on mirror as figure on the right hand side. Coating and glued magnet have loss. Let us consider them. Thickness of coating is about 5 micro m. On the contrary, thickness of mirror is 10 cm. So, coating is times thinner. So, even if it has large loss, Q-value of mirror does not so change. So, nobody cared this issue. On the other hand, Glued magnets decrease Q-values. OK, this is the serious problem.

(2) Thermal noise caused by coating mechanical loss Y. Levin, Physical Review D 57 (1998) 659. Laser beam This is qualitative discussion, but conclusion is amazing. Q-value of loss at A is the same as that at B. Loss at A can shake illuminated surface more largely owing to conservation of momentum. Contribution of loss near beam spot is quite larger ! However, the previous discussion was perfectly kicked out by Levin in He showed simple example. This is the 1 dimension system, like a bar. This is the laser beam. So, we observe the motion of this surface. He consider the two cases. In the first case, the loss is concentrated at A. In the second case, the loss is at B. The Q-value of loss at A is the same as that at B because of the symmetry of system. Nevertheless, loss at A can shack illuminated surface more largely owing to conservation of momentum. So, thermal noise caused by loss at A is larger. The thermal noise depends on not only Q-values but also spatial distribution of loss.

(2) Thermal noise caused by coating mechanical loss Quantitative evaluation Analytical formula about coating thermal noise G. Harry et al., Classical and Quantum Gravity 19 (2002) 897. N. Nakagawa et al., Physical Review D 65 (2002) Numerical evaluation (Finite Element Method) K. Yamamoto et al., Physics Letters A 305 (2002) 18. Conclusion Coating thermal noise is larger than bulk thermal noise. I must mention that, on the same year, 2002, two papers, which describe the formula of coating thermal noise, by Gregg and Nakagawa, were published. The details about coating thermal noise are in this book.

(2) Thermal noise caused by coating mechanical loss Old summary of coating mechanical loss Similar results (same order of magnitude) Structure damping So, investigation of coating mechanical loss started. This graph show the summary of coating mechanical loss. Sorry, this is not new, in 2006, but, the situation does not so change. The horizontal axis mean the loss angle of coating. Loss angle is the inverse number of Q. So, larger loss angle is larger thermal noise. These red and green circles coating loss at room temperature. I must say that horizontal axis is not log but linear scale. So, they are similar values, I mean that they are on the same order of magnitude. And measurement show the coating loss is almost independent of frequency. So, we can use structure damping model. Loss angle : 1/Q K. Yamamoto et al., Physical Review D 74 (2006)

(2) Thermal noise caused by coating mechanical loss Evaluated thermal noise based on measurement OK, I calculated thermal noise caused by coating loss. This is blue solid line is coating thermal noise. This dashed blue line is the contribution of mirror substrate. This black line is the goal sensitivity of advanced LIGO as example of future detector. Yes, substrate thermal noise is smaller than the goal sensitivity. But, coating thermal noise is larger. The strategy of future detector for this problem will be introduced after. Coating thermal noise is the most serious problem in future!

(2) Thermal noise caused by coating mechanical loss How to reduce coating mechanical loss So far, no silver bullet … (We have to consider not only mechanical but also optical properties of coating) Two promising candidates with low mechanical loss (Somebody could give other candidates …) (a) AlGaAs/AlGaP : Crystal coating Problem : Is it feasible for large mirror for long arm interferometer ? (b) Amorphous Silicon Problem : Large absorption I must mention that, on the same year, 2002, two papers, which describe the formula of coating thermal noise, by Gregg and Nakagawa, were published. The details about coating thermal noise are in this book.

(3) Reduction of thermal noise Coating thermal noise is the most serious problem ! Coating loss reduction is not so easy … Second generation interferometer Advanced LIGO (U.S.A.) and Virgo (Italy and France): Larger beam -> Beyond AdLIGO : Voyager (Cooled silicon mirror) The next topic is reduction of thermal noise. Now, we have new knowledge about thermal noise. Coating thermal noise is the most serious problem. Many people try to reduce coating mechanical loss, but it seems to be not so easy. How do future interferometer project reduce the thermal noise ? The second generation interferometers, Advanced LIGO and Virgo will use larger beam. In LCGT, the other second generation interferometer, cooled sapphire mirror will be used. In Europe, there is the third generation project, Einstein Telescope. Since the sensitivity of ET is 10 times better than that of second generation, both technique will be adopted. The material of mirror is silicon or sapphire. KAGRA (Japan) : Cooled sapphire mirror Third generation interferometer (10 times better sensitivity) Einstein Telescope (ET, Europe) : Cooled silicon or sapphire mirror and larger beam

(3) Reduction of thermal noise Larger beam Correlation between thermal fluctuation at two points is smaller when distance between these points is larger. When we observe wider mirror surface, integral of fluctuation of surface is smaller owing to weaker correlation. Mirror diameter limits beam size. Larger mirror or other shape beam (not Gaussian beam) ? Larger beam is one of the method to reduce thermal noise. However, mirror radius should be 3 times larger than Gaussian beam radius to avoid large clipping loss. So, beam could not be so large. How about other kinds of beam shape ? For example, mesa hat, higher modes …Sorry, I am not so familiar with large beam technique and have not so long time, I will skip it. The details are described by Andrease. Here, only cryogenic technique is introduced.

(3) Reduction of thermal noise Larger beam Mirror radius should be 3 times larger than Gaussian beam radius to avoid large clipping loss. How about other kinds of beam shape ? Mesa hat, higher modes… The details are in Chapter 13 (A. Freise) of G.M. Harry, T. Bodiya, and R. DeSalvo (Editors) Optical Coatings and Thermal Noise in Precision Measurements Cambridge University Press, Cambridge (in press) Larger beam is one of the method to reduce thermal noise. However, mirror radius should be 3 times larger than Gaussian beam radius to avoid large clipping loss. So, beam could not be so large. How about other kinds of beam shape ? For example, mesa hat, higher modes …Sorry, I am not so familiar with large beam technique and have not so long time, I will skip it. The details are described by Andrease. Here, only cryogenic technique is introduced.

(3) Reduction of thermal noise One of the simplest solutions : Cooling mirrors First feasibility study T. Uchiyama et al., Physics Letters A 242 (1998) 211. So, let us start cooling mirror. The cooling is one of the simplest solutions to reduce thermal noise. The first feasibility study for interferometric gravitational wave detector was done by Takashi Uchiyama et al. This paper was published on 1998 and it is the basis of LCGT project.

(3) Reduction of thermal noise One of the simplest solutions : Cooling mirrors In KAGRA, sapphire mirror is suspended by sapphire fibers. Small mechanical loss (small thermal noise) T. Uchiyama et al., Physics Letters A 273 (2000) 310. Large thermal conductivity (effective cooling) T. Tomaru et al., Physics Letters A 301 (2002) 215. In ET(Voyager), silicon is also (only) candidate. S. Hild et al., Classical and Quantum Gravity 28 (2011) R. Nawrodt et al., General Relativity and Gravitation 43 (2011) 593. In LCGT, the sapphire mirror is suspended by sapphire fiber because it has low mechanical dissipation and high thermal conductivity. Low mechanical loss implies the small suspension thermal noise. High thermal conductivity implies effective heat transfer for cooling. The cryogenic technique will be also adopted in ET. In ET, not only sapphire but also silicon is candidate material.

(3) Reduction of thermal noise Amplitude of thermal noise is proportional to (T/Q)1/2 In general, Q-value depends on T (temperature). Cryogenic technique is promising. But the problem is how thermal noise depends on temperature. According to fluctuation dissipation theorem, amplitude of thermal noise is proportional to square root of ratio of temperature to Q-value. In general, Q-value also depends on temperature. So, we must investigate how dissipation depends on temperature in cryogenic region. We must investigate how dissipation depends on temperature in cryogenic region.

(3) Reduction of thermal noise Structure damping in substrate At first, we consider structure damping in substrate. So, Q-values of substrate were measured at low temperature. This is the result. The horizontal axis is temperature. This red line is the sapphire, which is the substrate of LCGT. This blue line is silicon, which is the substrate of ET. These Q-values increase when mirror is cooled. Fine. Thermal noise reduce effectively. On the other hand, Q-value of fused silica decreases drastically. This is bad news. T. Uchiyama et al., Physics Letters A 261 (1999) 5-11. R. Nawrodt et al., Journal of Physics: Conference Series 122 (2008) C. Schwarz et al., 2009 Proceedings of ICEC22-ICMC2008.

(3) Reduction of thermal noise Structure damping in substrate This graph shows the temperature dependence of substrate structure noise. Sapphire and silicon thermal noise is smaller in lower temperature region. However, because of decreasing of Q, silica thermal noise becomes larger. Although silica is popular material for room temperature interferometer, it can not be used in cryogenic interferometer. Fused silica can not be used !

(3) Reduction of thermal noise Thermoelastic damping in substrate OK, so how about substrate thermoelastic noise ? Fortunately, we know how substrate material properties depend on temperature. So, we can calculate it. This is the result. Sapphire, red line and silicon, blue line is similar result. Above, a few tens Kelvin, noise depends on temperature weakly. On the contrary, below a few tens Kelvin, noise becomes extremely smaller if mirror is cooled. At 120 K and 20 K, thermoelastic noise of silicon vanishes because thermal expansion coefficient is zero. This black line is fused silica. Around room temperature, this noise small because of small thermal expansion. Especially, about at 170 K, it also vanishes because thermal expansion is zero. But below 10K, silica thermoelastic noise is comparable with that of sapphire and silicon.

(3) Reduction of thermal noise Structure damping in coating Let’s consider structure damping in coating. Unfortunately, coating loss at low temperature was not so investigated. This is the first measurement result. In fact, this is our results. Loss angle is almost independent of temperature. Loss angle is almost independent of temperature. K. Yamamoto et al., Physical Review D 74 (2006)

(3) Reduction of thermal noise Structure damping in coating Peak at 20 K ? However, recently, the other results were reported by Glasgow university. They claimed that there is a peak around 20K. I. Martin et al., Classical and Quantum Gravity 25(2008)

(3) Reduction of thermal noise Structure damping in coating Peak at 20 K ? Annealing suppresses peak (not perfectly). It is assumed that loss is constant. Kazuhiro’s comment : Mystery still remains. However, in the latest paper, they found that annealing suppresses peak. This suppression is not perfect, but, here, it is assumed that loss is constant. I. Martin et al., Classical and Quantum Gravity 27(2010) 76

(3) Reduction of thermal noise This is the calculated result of coating Brownian noise. This dashed line is coating Brownian noise. The thermo-optic noise also is calculated. This solid line is the thermo-optic noise. So, at least, it is expected that thermo-optic noise is less serious.

(3) Reduction of thermal noise Structure damping in coating Goal temperature KAGRA : 20 K This graph show the temperature dependence of thermal noise of sapphire mirror. Solid, dashed and dotted lines are coating structure, substrate structure, substrate thermoelastic noise. Coating noise is the most serious issue again because the coating loss does not so depend on temperature. On the other hand, loss in substrate becomes smaller at lower temperature. Thermo-optic noise, this thin line, does not matter. Goal temperature of LCGT is about 20K. At this temperature, thermal noise is several times smaller than that at room temperature. Coating thermal noise is the most serious problem !

(3) Reduction of thermal noise How can we cool mirrors ? Sapphire or silicon fiber (high thermal conductivity and high Q-value) OK, we know the mirror thermal noise becomes small at low temperature. But, how do we cool mirrors ? This is the schematic view of cryostat. The top part is the vibration isolation system. The vibration isolation system is outside of cryostat. In short, the temperature of this part is 300 K. The parts below isolation system are cooled. The cooled part is surrounded by radiation shield. The radiation shields are cooled by cryocooler. On the bottom, the main sapphire mirror is. The temperature is 20K in LCGT. This mirror suspended sapphire in LCGT or silicon in ET fiber from this upper stage. The upper mass is connected by soft aluminum heat link with radiation shield. So, the heat in the sapphire mirror goes to the upper mass through the fiber and to the radiation shield through the heat link and cryocooler. (in KAGRA)

(3) Reduction of thermal noise Cryocooler Some kinds of heat engine According to thermodynamics … (a)Hot source gives engine heat (QH). (b)Part of heat (QL) is give to cold sink. (c)Other part of heat is converted to work (W). We use the cryocooler in order to cool the cryostat. In usual cases, liquid nitrogen and helium are used. However, please remember that LCGT and ET interferometers in underground site. Owing to safety and maintenance in underground site, we chose the cryocooler. The Gifford-McMahon cryocooler is the one of the most popular cryocoolers, but, the vibration of this type cryocooler is large. So, we choose the pulse-tube cryocooler. Since it has no solid piston, the vibration is small. But, unfortunately, the vibration of commercial one is no enough small. Wikipedia : heat engine

(3) Reduction of thermal noise Cryocooler Some kinds of heat engine In the case of cryocooler … (a)We apply work (W) to engine. (b)Cold sink gives heat (QL) to engine. (c)Engine gives heat (QH) to Hot source. We use the cryocooler in order to cool the cryostat. In usual cases, liquid nitrogen and helium are used. However, please remember that LCGT and ET interferometers in underground site. Owing to safety and maintenance in underground site, we chose the cryocooler. The Gifford-McMahon cryocooler is the one of the most popular cryocoolers, but, the vibration of this type cryocooler is large. So, we choose the pulse-tube cryocooler. Since it has no solid piston, the vibration is small. But, unfortunately, the vibration of commercial one is no enough small. Wikipedia : heat engine

(3) Reduction of thermal noise Cryocooler Why ? Usual case : Liquid nitrogen and helium Safety and maintenance in mine Usual cryocooler : Gifford-McMahon cryocooler Large vibration (Heat engine has piston !) Pulse-tube cryocooler (without solid piston) But, vibration of commercial one is not enough small. We use the cryocooler in order to cool the cryostat. In usual cases, liquid nitrogen and helium are used. However, please remember that LCGT and ET interferometers in underground site. Owing to safety and maintenance in underground site, we chose the cryocooler. The Gifford-McMahon cryocooler is the one of the most popular cryocoolers, but, the vibration of this type cryocooler is large. So, we choose the pulse-tube cryocooler. Since it has no solid piston, the vibration is small. But, unfortunately, the vibration of commercial one is no enough small.

(3) Reduction of thermal noise Schematic view of silent pulse-tube cryocooler We developed low vibration pulse-tube cryocooler. The details are in these papers. We got the patents. You can find 4 K cold head . This parts vibrates. So, we prepared the cold stage fixed rigidly. Then, cold head and stage are connected by heat link. Cold stage is cooled, but vibration free.

(4) Impact on other fields Contents of G.M. Harry, T. Bodiya, and R. DeSalvo (Editors) Optical Coatings and Thermal Noise in Precision Measurements Cambridge University Press, Cambridge (in press) At last, I will show impact of the other fields. This is the last part of contents of book “optical coatings and thermal noise in precision measurements”. Of course, gravitational wave detection is introduced. However, we can find the name of other fields, high precision laser stabilization via optical cavity, quantum optomechanics, cavity quantum electrodynamics. So, now, thermal noise is not problem in gravitational wave detections. In the other fields, it is also a problem. Thermal noise is also sensitivity limit on the other fields in precision measurement.

(4) Impact on other fields For example … Cavity as reference for laser frequency stabilization Current best laser frequency stabilization with rigid cavity at room temperature is limited by thermal noise of mirrors. I introduce only one example, cavity as reference for laser frequency stabilization. Kenji pointed out current best laser frequency stabilization with rigid cavity at room temperature is limited by thermal noise of mirrors. K. Numata et al., Physical Review Letters 93 (2004)

(4) Impact on other fields This is the result of Kenji. This thick red line is calculated value. This thin black line in the left hand side is the best result of NIST. The dashed line on the right hand side is the experimental result of Virgo. They are comparable with Kenji’s evaluation for thermal noise. So, now, we reduce thermal noise to improve laser frequency stability. 0.1Hz/rtHz*(1Hz/f)1/2 (10mHz-1Hz) Allan deviation :4*10-16 (10mHz-1Hz) 86

(4) Impact on other fields Hot paper (ISI Web of Knowledge) This Kenji’s paper is very hot. I checked how many times Kenji’s paper is referred using Thomson search system. In 6 and half year, it is 106 times referred ! In average, one paper which refers Kenji’s paper is published every three weeks. 1 referred journal paper every 2 weeks ! (until 11 July 2017) 87

4. Summary (1)Long history of research of thermal noise (200 years !)
General theorem for thermal noise, Fluctuation- Dissipation Theorem, appears only 60 years ago. (2)Interferometric detector It is essential to measure Q-values. Pendulum : Rigid and heavy support system Mirror : Nodal support system So, I would like to summarize my long talk. There is a long history of research of thermal noise, about 200 years ! General theorem, fluctuation dissipation theorem appears only 60 years ago. To reduce the thermal noise of resonant detector, cooling using liquid helium or dilution refrigerator and low mechanical loss material, for example, Al5056 are necessary. In interferometric detector, it is essential to measure Q-values. For Q measurement, rigid and heavy support system or nodal support system are needed.

There are open questions and this field will be hot in future.
4. Summary (3)Interferometric detector Drastic progress of research on the end of 20th century New kinds of thermal noise Thermoelastic noise and coating thermal noise Reduction of thermal noise (larger beam and cryogenic techniques) Impact on other fields On the end of 20th century, there was drastic progress of research of thermal noise in interferometric detector. We discovered new kinds of thermal noise, thermoelastic noise and coating thermal noise. Amazingly, now, we can observe the thermal noise directly and direct measurement of off resonance dissipation will be tried. Research and development for reduction of thermal noise, larger beam and cryogenic techniques are in progress. Moreover, now, thermal noise is sensitivity limit not only of gravitational wave detection, but also in other fields. Anyway, there are open questions and this filed will be hot in future. There are open questions and this field will be hot in future.

Thank you for your attention !
That’s all. Thank you for your attention.

FDT in quantum mechanics H.B. Callen and T.A. Welton, Physical Review 83 (1951) 34. When we should take quantum mechanics into account ? Smallest energy in quantum mechanics Average energy in classical statistical mechanics Up to now, all discussions are in classical mechanics, not quantum mechanics. I will show quantum effect briefly. If this expression is correct, we should take quantum mechanics into account. Hbar omega in the left hand side is the smallest energy in quantum mechanics. kBT on the right hand side is the average energy in classical statistical mechanics. At room temperature, if the frequency is more than 6 THz, we should consider quantum mechanics. But it is too high for gravitational detection. So, fortunately, it is not necessary for us to consider quantum formula. At room temperature, if the frequency is more than 6*1012 Hz, we should consider quantum mechanics.

Fluctuation Dissipation Theorem in Quantum mechanics Kubo formula R. Kubo, Journal of the Physical Society of Japan 12 (1957) 570. As the last of the history of fluctuation dissipation theorem, I would like to mention that Ryogo Kubo reported the full formula of fluctuation dissipation theorem in quantum mechanics in This is the final version of FDT. However, I do not know details, so, let us go to the next section.

2. Resonant detector Resonant detector
Gravitational wave excites resonant motion of elastic body. Weber bar (most popular one) The next section is the thermal noise of resonant gravitational wave detector. This is the schematic view of resonator. In many case, resonators are bar a few meters in length and several tens cm in diameter. If the gravitational wave whose frequency is the same as the resonant frequency of detector, which is about 1 kHz, elastic motion of detector is excited and this transducer generates electric signal and this signal is enhanced by this amplifier.. “300 years of gravitation” (1987) Cambridge University Press Fig. 9.8 Diameter : several tens cm Length : a few meters Resonant frequency : about 1 kHz 93

2. Resonant detector Resonator : tidal force of gravitational wave
Thermal fluctuation force must be considered. Observation of thermal fluctuation of torsion pendulum Displacement, not force, was monitored. Formula of thermal fluctuation force (on resonance) T/Q should be small. Low temperature (low T), Small mechanical loss (large Q) We have already discussed the observation of the thermal fluctuation of torsion pendulum. However, we must be careful. There is a difference between resonant gravitational wave detector and torsion pendulum. Resonant detector detect the tidal force of gravitational wave. So, thermal fluctuation force must be consider. On the other hand, in the experiment of torsion pendulum, the displacement, not force, was monitored. The formula of thermal fluctuation force is describe as like this. So, the ratio of temperature and Q-value must be small. So, detector is cooled and we must made resonator with high Q-value, which implies small mechanical loss.

2. Resonant detector First generation (room temperature)
Weber bar (University of Maryland, U.S.A.) … Second generation (4 K) Liquid helium Explorer (Italy, CERN), Allegro (U.S.A.), Niobe (Australia), Crab (Japan) … Third generation (< 100 mK) Dilution refrigerator Nautilus (Italy), Auriga (Italy), Mini-Grail (Netherlands), Mario Schenberg (Brazil) … So, how are resonator cooled ? This is the list of resonator. I must say that this is not a perfect list. There are three generations. The number of generation depends on the temperature. First generation detectors are operated at room temperature. The most famous one is by Wever in university of Maryland, U.S.A. This is the first resonator. Second generation resonator are operated at 4 K using liquid helium. Explorer is Italian one. It is put in CERN, near boarder line between Switzerland and France. Allegro was in U.S.A. Niobe and Crab were in Australia and Japan. Temperature of third generation detectors is less than 100 mK. The dilution refrigerator is adopted. There are two third generation detector in Italy, Nautilus and Auriga. They are at Frascati, suburb of Rome and Padova. In Netherlands and Brazil, Mini-Grail and Mario Schenberg are. This is not a perfect list ! 95

G. Pizzella, ET first general meeting (2008)
NAUTILUS INFN - LNF This is the Nautilus, third generation resonator, near Rome. The long cylinder on the center is bar. This man’s hand is on the transducer. This green one is the cryostat. This is cooled by dilution refrigerator. G. Pizzella, ET first general meeting (2008) 96

2. Resonant detector High Q-value (low mechanical loss) material
Small dissipation at low temperature Sapphire and Silicon (Moscow) Niobium (Australia) CuAl6% (Mini-Grail (Netherlands), Mario Schenberg (Brazil)) K.S. Thorne, Chapter 9 of “300 years of gravitation”(1987) Cambridge University Press p409. A de Waard et al., Classical and Quantum Gravity 21 (2004) S465. O.D. Aguiar et al., Classical and Quantum Gravity 25 (2008) Almost all resonators (Weber bar also) are made from aluminum. Large bulk, low cost ... OK, resonator should be cooled. Moreover, it must consists of high Q-values, in short small mechanical loss material at cryogenic temperature. What do kinds of material have small dissipation ? Sapphire and silicon are adopted in Moscow. At Australia, niobium is used. CuAl6% is adopted in Netherlands and Brazil. However, almost all resonators, Weber bar also, are made from aluminum. The reasons are as follows. It is possible to make large, and low cost and so on.

with Small Dissipation” (1986) University of Chicago Press.
V.B. Braginsky, V.P. Mitrofanov, K.S. Thorne “Systems with Small Dissipation” (1986) University of Chicago Press. This graph shows that Q-values of various materials for resonators. The marks on right upper side are sapphire and silicon. These open circles are aluminum. Yes, Q of aluminum is lower than silicon and sapphire, but it is quite high, on the order of 10^7.

2. Resonant detector What is kinds of aluminum alloy best ?
T. Suzuki et al. discovered that Al5056 has high Q-value. Almost all resonators are made from Al5056. T. Suzuki et al., Physics Letters 67A (1978) 2. OK, but there are a lot of kinds of aluminum alloy. Which is it the best ? Toshikazu Suzuki et al. discovered Al5056 has high Q-values. This graph shows their result. The horizontal axis is temperature, the vertical axis is Q. Only Al5056 has Q which is on the order of 10^7 below nitrogen temperature. Thus, almost all resonators are made from this aluminum alloy. This alloy includes Mg, Mn, and Cr.

Thermoelastic noise of mirror Thermo-optic noise Temperature fluctuation in reflective coating Thermal expansion (a) Temperature coefficient of refractive index (b) Fluctuation of phase of reflected light In the previous discussion about thermoelastic noise, we do not take reflective coating into account. Of course, we must consider it. Temperature fluctuation occurs in reflective coating. Owing to thermal expansion and temperature coefficient of coating, this temperature fluctuation is converted to that of phase of reflected light. This is called thermo-optic noise. In these papers, the details are discussed. The problem is that we do not know material properties of coating well. We must investigate them. Material properties of reflective coating are important issues. V. B. Braginsky et al., Physics Letters A 312 (2003) 244. V. B. Braginsky et al., Physics Letters A 271 (2000) 303. M. Evans et al., Physical Review D 78 (2008)

(2) Thermal noise caused by inhomogeneous loss Mirror is not a harmonic oscillator. It has a lot of resonant modes. Modal expansion Thermal noise of mirror is the summation of those of resonant modes. Thermal noise of resonant mode is the same as that of a harmonic oscillator . The next topic is about thermal noise caused by inhomogeneous loss. Before we start discussion, I will introduce modal expansion. I have already explained the thermal noise of a harmonic oscillator. But, mirror itself is not a harmonic oscillator because it has a lot of resonant modes. In such a case, modal expansion is useful. This method is summarized in this famous paper. Thermal noise of mirror is the summation of those of resonant modes. Thermal noise of resonant mode is the same as that of a harmonic oscillator. Thus, same Q-values implies same thermal noise. Peter R. Saulson, Physical Review D 42 (1990) 2437. Same Q-values implies same thermal noise.

(3) Direct measurement of thermal noise Are formulae of thermal noise correct ? Direct measurement of thermal noise of mirror University of Tokyo, California Institute of Technology Small beam radius (about 0.1 mm) to enhance thermal noise Fabry Perot cavity length ~ 1 cm Direct measurement of thermal noise of suspension University of Tokyo Underground site with small seismic motion Cavity length ~ 100 m The third revolution is direct measurement of thermal noise. OK, many new theory about thermal noise appeared. But, are they correct ? This is the problem. It is essential to check them experimentally. However, it is extremely difficult to measure the thermal noise because it is so small. Nevertheless, the direct measurement of thermal noise of mirror and suspension have already been done ! Measurement for mirror was done in university of Tokyo and Caltech. The experiment of suspension was done in University of Tokyo, again. In the measurement of mirror, beam radius is small, let’s say, about 0.1 mm. This is because thermal noise is larger with smaller beam radius. And, they used short cavity, about 1cm. In the measurement of suspension, seismic motion is serious problem. In this experiment, interferometer is put in underground site because the seismic motion is 100 times smaller than typical one. Cavity length of this experiment is 100 m.

(2) Thermal noise caused by inhomogeneous loss Modal expansion predicts that thermal noise is the same if Q-values are same. But Levin’s discussion proved that it is invalid. What is wrong ? Although some people investigated, K. Yamamoto’s four papers provide clear solution for this problem. Modal expansion predicts that thermal noise is the same if Q-values are same. But Levin’s discussion proved that it is invalid. So, what is wrong ? Although some people investigated this problem, my four papers provide clear solutions for this problem.

(2) Thermal noise caused by inhomogeneous loss What is wrong in modal expansion ? Inhomogeneous loss causes couplings between modes. Inhomogeneous loss destroys the shape of a mode. Other modes appear. These couplings generate correlations between thermal fluctuations of resonant modes. This correlation is not taken into account in modal expansion. If we consider these correlations, modal expansion can provide correct evaluation (advanced modal expansion). K. Yamamoto et al., Physical Review D 75 (2007) What is wrong in modal expansion ? In fact, inhomogeneous loss causes couplings between modes. Please imagine only one resonant mode is excited. This resonant motion decays owing to dissipation. If this loss is distributed inhomogeneously, the shape of mode is destroyed. It implies that other modes appear. This is the couplings between the modes. These couplings generate correlations between thermal fluctuation of resonant modes. This correlation is not taken into account in modal expansion. This is the reason why modal expansion is not correct with inhomogeneous loss. But if we consider these correlations, modal expansion can provide correct evaluation. This new method is called advanced modal expansion. The details are in this paper.

(2) Thermal noise caused by inhomogeneous loss Experimental checks : (Traditional) modal expansion breaks down and advanced modal expansion and Levin’s method are correct if loss is inhomogeneous. Our experiment proved that traditional modal expansion breaks down and advanced modal expansion and Levin’s method are correct if loss is inhomogeneous. The details of these experiments are in these papers. K. Yamamoto et al., Physics Letters A 280 (2001) 289. K. Yamamoto et al., Physics Letters A 321 (2004) 79.

(2) Thermal noise caused by inhomogeneous loss Quantitative discussion K. Yamamoto et al., Physics Letters A 305 (2002) 18. Magnet Reflective coating Levin’s discussion show the important point clearly, but is not quantitative discussion. The quantitative discussion appeared in In fact, this is my paper. The conclusion is perfectly opposite to previous discussion. Although thickness of coating is too small, but, it is on the illuminated surface. So, coating mechanical loss can be a problem. Although magnets decrease Q, they are far from the laser beam spot. So, it is not serious problem. OK, we must change the strategy of research of thermal noise. It can be a problem. No problem The previous expectation is perfectly wrong. The strategy of research of thermal noise must be changed.

(3) Direct measurement of thermal noise BK7 : Structure damping in substrate OK, I show the excellent experimental results. At first, mirror thermal noise measurement by Kenji Numata in university of Tokyo. This is the result of BK7 mirror. The loss of BK7 mirror is dominated by the structure damping in substrate. This black line is measurement result. This grey line is the contribution of substrate structure damping. Measurement agrees with theoretical prediction well. K. Numata et al., Physical Review Letters 91 (2003)

(3) Direct measurement of thermal noise Calcium fluoride : Thermoelastic damping in substrate This is also Kenji’s result. calcium fluoride mirror. In this case, the loss of mirror is dominated by the thermoelastic damping in substrate. This black line is measurement result. This grey line is the contribution of substrate thermoelastic damping. Measurement agrees with theoretical prediction well again. K. Numata et al., Physical Review Letters 91 (2003) 108

(3) Direct measurement of thermal noise Fused silica : Structure damping in coating This is Kenji’s result, fused silica mirror. In this case, the contribution of coating structure damping is largest. This black line is measurement result. This grey line is the contribution of coating structure damping. Measurement agrees with theoretical prediction well again. K. Numata et al., Physical Review Letters 91 (2003) 109 109

(3) Direct measurement of thermal noise Sapphire : Thermoelastic damping in substrate This is the mirror thermal noise measurement which came from Caltech. Their mirror is sapphire. This line A is measured spectrum. This line B is the contribution of thermoelastic damping in substrate. OK, their result also is consistent with theoretical prediction. E.D. Black et al., Physical Review Letters 93 (2004) 110 110 110 110

(3) Direct measurement of thermal noise Direct measurement of thermal noise of suspension Measurement of mirror thermal noise provided excellent and beautiful result. How about suspension thermal noise ? It has already measured, but, I must say that the loss in this experiment is not dominated by that in suspension itself, but coil-magnet actuator. As I have already mentioned, the coil-magnet actuator is used to control the position of mirror. The electrical circuit of actuator has resistance. This is also loss and the effect of this resistance is larger than loss in suspension. But, I would like to emphasis that Q with actuator is still enough high, which is about 10^5. Moreover, we can change this resistance. So, if the measured sensitivity changes when we change the resistance, we are quit sure that we observed thermal noise. Loss : Resistance of coil-magnet actuator (not material or clamp loss) We can change this resistance. Q-value is still enough high (~105). K. Agatsuma et al., Physical Review Letters 104 (2010)

(3) Direct measurement of thermal noise Cryogenic Laser Interferometer Observatory (CLIO,Japan) 100 m, Kamioka (Japan) The direct measurement of suspension thermal noise was done in CLIO interferometer. CLIO stands for cryogenic laser interferometer observatory . This is in Japan and baseline is 100m. This is the prototype for LCGT, cryogenic interferometer in underground site with small seismic motion. Prototype for LCGT ; cryogenic interferometer in underground site with small seismic motion S. Kawamura, Classical and Quantum Gravity 27 (2010) 112

(3) Direct measurement of thermal noise Change of resistance Change of observed sensitivity as theoretical prediction This is the result. There are three lines, red, green and blue ones. The difference between them is caused by the change of resistance. Moreover, change of observed sensitivity perfectly follows theoretical prediction ! Excellent ! The next step is the measurement of thermal noise caused by the loss in suspension itself. Next step : Thermal noise by loss in suspension itself K. Agatsuma et al., Physical Review Letters 104 (2010)

(3) Direct measurement of thermal noise I expect that the sensitivities of LIGO(U.S.A.), Virgo (Italy and France) and GEO (Germany and U.K.) are comparable with suspension thermal noise. However, I can not find refereed papers which claim observation of suspension thermal noise. I expect that the sensitivities of LIGO, Virgo and GEO are comparable with suspension thermal noise. However, I can not find refereed papers which claim observation of suspension thermal noise. But, I hope such papers appear in near future.

(4) Direct measurement of off resonance dissipation Now we observe thermal fluctuation directly. Fluctuation dissipation theorem The next topic is the direct measurement of off resonance dissipation. Now we can observe thermal fluctuation directly. And we remember fluctuation dissipation theorem. It shows the relation between fluctuation and dissipation. So, how about direct measurement of dissipation, which is the imaginary part of susceptibility in the right hand side of fluctuation dissipation theorem ? If possible, we can evaluate thermal noise. Power spectrum of thermal fluctuation Imaginary part of susceptibility (dissipation) How about direct measurement of dissipation (imaginary part of susceptibility) ?

(4) Direct measurement of off resonance dissipation How about dissipation (imaginary part of susceptibility) ? It is not so easy. Imaginary part is much smaller than real part. Unfortunately, this measurement is not so easy. Because imaginary part of is much smaller than real part. In off resonance frequency region, the ratio of imaginary part to real part is described as this formula. So, we need extreme precise measurement. The relative error should be smaller than 10^(-6) ! Off resonance region Extreme precise measurement is necessary. Relative error should be smaller than 10-6 ! 116

(4) Direct measurement of off resonance dissipation On anti resonance Re[H] vanishes. |H| is |Im[H]]| (dissipation). Requirement of relative error is not so serious. However, Ohishi found that trick for direct measurement. This is the schematic view of susceptibility, H in this graph, of interferometer. This is the resonance of suspension. This is the resonance of mirror itself. Between them, there is anti resonance. Why H is so small at anti resonance ? Because at this frequency, real part of H vanishes. So, here, absolute value of H is exactly same as imaginary part. So, if this mechanical system has no loss, H is zero at anti resonance. But if there is dissipation, H at anti resonance is not zero. So, we can derive imaginary part of H from measurement at anti resonance. Since real part is zero, requirement of relative error is not so serious. N. Ohishi et al., Physics Letters A 266 (2000) 228.

(4) Direct measurement of off resonance dissipation Ohishi confirmed that this method works well using small 2 modes oscillator. Ohishi has already confirmed that this method works well in experiment with small 2 modes oscillator as this right hand figures. N. Ohishi et al., Physics Letters A 266 (2000) 228.

(4) Direct measurement of off resonance dissipation In gravitational wave detector, actuator is a problem. Probably, photon pressure actuator is the best one. However, if we apply this method on gravitational wave detector, there are some problems. One of them is actuator. Actuator must apply force on the beam spot. Probably, photon pressure actuator is the best one. This is the schematic view of photon pressure actuator. This is the main beam of interferometer. Behind mirror, there is the laser diode. The light from this laser diode goes along this line and reflected by coating of mirror. So, this light gives momentum to mirror, in short, applied force on this spot. The spot of this beam is the same as that of the interferometer beam. If the amplitude of this light is modulated, periodic force is applied on mirror. S. Hild et al., Classical and Quantum Gravity 24 (2007) 5681.

(4) Direct measurement of off resonance dissipation Measurement with photon pressure actuator in GEO600 Stefan Hild et al measured susceptibility of GEO600 using this photon pressure actuator. I must mention that Stefan did not intend to try Ohishi’s method. Anyway, although the measurement at low frequency region is excellent, susceptibility could not be measured at anti resonance. The progress is necessary. But I hope somebody will try it. Susceptibility could not be measured at anti resonance. The progress is necessary. S. Hild et al., Classical and Quantum Gravity 24 (2007) 5681.

(5) Reduction of thermal noise 100 m, Kamioka (Japan) Cryogenic Laser Interferometer Observatory (CLIO,Japan) Here, please remember Japanese cryogenic interferometer, CLIO. CLIO can provide some results fro cryogenic techniques. So, I would like to introduce them.　But, it is noted that the size of CLIO cryostat is different from that of LCGT. Result of CLIO for cryogenic techniques are introduced. CLIO cryostat have already been installed (different scale from those of LCGT and ET). S. Kawamura, Classical and Quantum Gravity 27 (2010) 121 121

(5) Reduction of thermal noise This is the result of the mirror cooling test. This axis shows the time. The zero is the time when the cryocooler operation started. This axis is the temperature. This blue line is the mirror temperature. Within a week, mirror temperature became 14 K. Since, the goal temperature is 20K, this is the good result. Within about a week, mirror temperature became about 14 K (mirror temperature must be below 20 K).

(5) Reduction of thermal noise Measurement of vibration in cryostat with silent pulse-tube cryocooler at silent site Cryocoolers do not increase vibration even if they are put on site with extremely small seismic motion. This graph shows that measured vibration in cryostat with silent pulse-tube cryocooler. The seismic motion at this site is extremely small. This axis shows the frequency. This axis shows the power spectrum. This red line is the vibration when cryocooler works. This blue line shows vibration when cryocooler rests. Obviously, our cryocooler does not increase the vibration. Moreover, this black line is the seismic motion at Kashiwa, near Tokyo. So, our cryocooler does not increase the motion even if it is put on site with extremely small seismic motion. The details are in my proceedings. K. Yamamoto et al., Journal of Physics: Conference Series 32 (2006) 418.

Thermal noise Kazuhiro Yamamoto

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