237b: Ph Gravitational Waves
24 April 2002
WEEK 14: LIGO as a Large Science Project; Quantum Optical Noise in Advanced LIGO Interferometers Lecture 25 by Barry Barish [LIGO-II as a Large Science Project]; Lecture 26, by Alessandra Buonanno and Yanbei Chen [Quantum Optical Noise in Advanced LIGO]
Reading Related to These Lectures:
Items in bold are recommended; others are supplementary. All these references (except [6]) are quite sophisticated and may be hard to follow without some preparation. The exercises are designed to provide that preparation: It may be helpful to work the exercises before doing the reading! A. References relevant to the Buonanno-Chen lecture: Quantum optical noise and beating the standard quantum limit 1. A. Buonanno and Y. Chen, Optical noise correlations and beating the standard quantum limit in advanced gravitational wave detectors, Classical and Quantum Gravity. 18, L95L101 (2001). Available on the web at http://www.iop.org/EJ/S/1/NCA143559/ , and at http://lanl.arXiv.org/abs/gr-qc/0010011 . 2. A. Buonanno and Y. Chen, Laser-interferometer gravitational-wave optical-spring detectors, Classical and Quantum Gravity, 19, 15691574 (2002). Available on the web at http://www.iop.org/EJ/S/1/NCA143559/ , or at http://lanl.arXiv.org/abs/gr-qc/0201063 . 3. A. Buonanno and Y. Chen, Quantum noise in second generation, signal-recycled laser interferometric gravitational-wave detectors, Physical Review D, 64, 042006 (2001); especially Sections I, II, III. Available on the web at http://prd.aps.org/ , and at http://lanl.arXiv.org/abs/grqc/0102012 . 4. A. Buonanno and Y. Chen, Signal recycled laser-interferomter gravitational-wave detectors as optical springs, Physical Review D, 65, 042001 (2002); especially Sections I, III, IV. Available on the web at http://prd.aps.org/ , and at http://lanl.arXiv.org/abs/gr-qc/0107021 . 5. H.J. Kimble, Yu. Levin, A. Matsko, K.S. Thorne and S. Vyatchanin, Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics, Physical Review D 65, 022002 (2001); Sections I, II, III, IV.A, and IV.C. Available on the web at http://prd.aps.org/ , and at http://lanl.arXiv.org/abs/grqc/0008026 . 6. R.D. Blandford and K.S. Thorne, Applications of Classical Physics, Section 5.3 of Chapter 5; available on the web at http://www.pma.caltech.edu/Courses/ph136/ph136.html .
1
B. References relevant to Barishs lecture: The scientic management of large projects. 7. Jain, R.K. and Triandis, H.C. Management of Research and Development Organizations: Managing the Unmanageable (Second Edition), John Wiley & Sons, Inc., 1997, ISBN 0471-14613 8. Sapienza, A.M., Managing Scientists: Leadership Strategies in Research and Development, John Wiley & Sons, Inc., 1995, ISBN 0-471-04367-2 9. Dinsmore, P.C., Human Factors in Project Management (Revised Edition), American Management Association 10. Lewis, J.P., The Project Managers Desk Reference: A Comprehensice Guide to Project Planning, Scheduling, Evaluation, and Systems (Second Edition), McGraw-Hill 11. Traweek, S., Beamtimes and Lifetimes: The World of High-Energy Physicists, Harvard University Press, 1988, ISBN 0-674-06347-3. 12. National Collaboratories: Applying Information Technology for Scientic Research, Computer Science and Telecommunications, Board of the National Academy of Sciences, 118 pages, 1993, ISBN 0-309-04848-6, Library of Congress Catalog #93-083795. EXERCISES Note: Exercise 1 provides an elementary introduction to some of the key ideas and notation used by Buonanno and Chen. Exercises 25, devised by Buonanno and Chen (with minor changes by Kip) provide insights into the Buonanno-Chen lecture and reading. These exercises do not involve extensive or difcult calculations. Rather, they are designed for pedagogy. 1. Shot noise as a beating of vacuum uctuations against classical light Consider a beam of monochromatic light propagating in the +z direction. The beam has some cross-sectional prole (typically Gaussian) which does not interest us in this problem set, so we shall idealize it as a plane wave with a nite cross-sectional area A that is constant. The beams electric eld then has the following form
vac vac E = [D + E1 (t z/c)] cos[0 (t z/c)] + E2 (t z/c) sin[0 (t z/c)] .
(1)
vac Here D is a real number (the amplitude of the light), c is the speed of light, and E1,2 (t z/c) vac are uctuations forced onto the light by the laws of quantum mechanics; E1 is the uctuating electric eld amplitude associated with the beams cosine quadrature (the quadrature vac in which all the light except its uctuations resides), and E2 is that associated with the vac sine quadrature. Because the uctuations E1,2 remain even when D is set to zero, they are called vacuum uctuations. It is conventional to introduce dimensionless vacuum uctuations a1,2 (t z/c) related to the electric-eld vacuum uctuations by
vac E1,2 (t z/c) =
4 0 h a1,2 (t z/c) . Ac 2
(2)
Here h is Plancks constant (divided by 2), 0 is the light beams monochromatic fre vac quency, and A is its cross sectional area. In quantum electrodynamics, E1,2 (t z/c) and a1,2 (t z/c) are Hermitian operators (observables); but we can equally well regard them as classical random processes whose spectral densities are deducible from the laws of quantum electrodynamics and depend on the lights quantum state. We shall adopt this (semiclassical) viewpoint throughout almost all of this problem set. If the light is as quiet as quantum mechanics allows, then these operators are in the standard vacuum state (1/2 quantum of uctuation in each mode), and the light [Eq. (1)], with its nonzero classical amplitude D, is said to be in a coherent state. In other words, a coherent state of light is a superposition of classical (monochromatic) light and standard vacuum uctuations. This is the type of light that an ideally quiet laser produces, and the type that LIGO-I scientists try to achieve in their interferometers. [In some designs for advanced interferometers, the light is in some other quantum state e.g., a squeezed state, which is a superposition of the classical light beam D on squeezed vacuum uctuations, for which vac E1,2 (t z/c) and a1,2 (t z/c) are in a so-called squeezed vacuum state.] In this problem set we shall deal only with coherent input light, i.e. with light whose uctuations are in the standard vacuum state. (a) Show that the light beams mean power i.e. its power when one ignores the vacuum uctuations is A D2 . (3) I= 8 Here and throughout we use Gaussian units, not SI units. (b) Assume that this mean power is large compared to the uctuations. Then the uctuations of the power I(t) arise from a beating of the classical light amplitude D against vac the vacuum uctuations E1 . Derive an equation for the spectral density of these power uctuations SI (f ) in terms of the mean power I, the energy of a carrier photon h0 , and the spectral density of the dimensionless uctuations Sa1 (f ). (c) In his Lecture 20, Kip derived the spectral density of the power uctuations SI (f ) relying on two simple features of the light: that it consists of photons with individual energies h0 , and that these photons arrive randomly, in an uncorrelated way. These two properties, in fact, are the dening measurable features of light that is in a quantum mechanical coherent state. From Kips result, h SI (f ) = 2I 0 (4)
show that the dimensionless amplitude a1 has unit spectral density. If the light had been excited in the sin[(t z/c)] quadrature, then this argument would have implied a unit spectral density for a2 . Therefore, for any light beam in a coherent state, Sa1 (f ) = Sa2 (f ) = 1 . (5)
(d) To make contact with the lecture of Buonanno and Chen, and with the reading for this week, Fourier analyze the dimensionless vacuum uctuation operators:
aj (t) = 3
aj ()eit
d . 2
(6)
[Note that /2 = f is the side-band frequency, since in the electric eld aj (t) is multiplied by the cosine or sine of 0 t in Eq. (1).] Since aj (t) is real, aj () is the complex conjugate (or, quantum mechanically, the Hermitian conjugate) of aj (+). This dictates the more conventional way of writing the Fourier transform (7):
aj (t) =
0
aj ()eit + a ()e+it . j
(7)
In the Buonanno-Chen reading for this week, the spectral density of aj (t) (or of any other quantity) is expressed in terms of its Fourier transform by the following relation: ( )Saj () = |aj ()a ( )| , j (8)
where | is the quantum state of the vacuum uctuations. An equation analogous to this was encountered in your reading in Week 11(Eq. (5.46) of Blandford and Thorne, Ref. [6]), where it was a consequence of the Wiener-Khintchine theorem. From that Blandford-Thorne equation, derive Eq. (8). We shall not need Eq. (8) in this problem set. 2. Quantum-optical noise in one arm of a simple GW interferometer Slides 712 of the Buonanno-Chen Lecture 24 sketch the derivation of the optical noises (shot noise and radiation pressure noise) in a conventional GW interferometer (e.g., LIGO-I) without corner mirrors (input mirrors). Here in this exercise, we shall explore some details of that derivation. To simplify the derivation, we focus on just one of the interferometers arms, which has length L and has a perfectly reecting mirror with mass m at its end as shown below: Mirror Beam splitter Light beam
z=0
z=L+X(t)
(a) The input light is propagating in the +z direction. At z = 0 (the beam splitters location) it has the form explored in Exercise 1:
vac vac E in (t) = [D + E1 (t)] cos 0 t + E2 (t) sin 0 t ,
(9)
vac vac where D is the (absolutely constant) carrier amplitude, and E1 and E2 are the vacuum uctuations. Show that the output light that arrives back at z = 0 at time t has actually traveled a length 2L + 2X(t L/c) inside the arm (where X(t ) is the gravitywave-induced displacement of the mirror at time t ), and that therefore
E out (t) = E in [t 2L/c 2X(t L/c)] vac [D + E1 (t 2L/c)] cos 0 t 20 D vac + E2 (t 2L/c) + X(t L/c) sin 0 t . c 4
(10)
(Here and throughout we use the fact that the mirrors speed X is small compared to the speed of light.) Estimate the fractional error made in the of this equation. (It is an exceedingly small error). Notice that the GW signal (the X term) modulates vac the phase of the carrier. This phase is also modulated by the E2 term, which thereby gives rise to the shot noise in the gravitational-wave signal. Using Eq. (10), explain why the amplitude of that shot noise in the GW signal will be inversely proportional to Iarm , where Iarm is the mean power of the light in the arm; i.e., the spectral density of the GW shot noise will be inversely proportional to Iarm . (b) The light beam exerts a radiation-pressure force on the mirror. Explain why this radiation-pressure force has the magnitude F = 2Iarm /c, where Iarm is the full light power (including shot-noise uctuations) in the interferometer arm. As in Exercise 1, the mean light power is Iarm = D2 A/8 so the mean force is F = D2 A/4c. This mean force is balanced by a steady counter force applied by the wires or bers from which the mirror hangs, and it does not interest us. Rather, we are interested in the uctuating part of this radiation-pressure force the part produced a by beating of the vacuum uctuations against the classical electric eld. Show that this uctuating force is given by vac DAE1 (t L/c) FBA (t) = . (11) 2 The subscript BA arises from a valuable viewpoint on the measurement process: The light is being used to measure the mirror position, and in the act of making its measurement the light exerts the back-action force F BA (t) on the measured object, the mirror. The mirrors position X evolves, in response to the combined inuence of the gravitational waves and this back-action force, as follows: 1 X(t) = LhGW (t) + XBA (t) 2 FBA XBA = . m Iarm . (12)
Show that the mirrors back-action noise XBA (t) is proportional to
(c) From Eqs. (10), (11) and (12), we can derive an input-output relation for the light in the arm. This is usually done in the frequency domain. Using the same decomposition as in Exercise 1, we write the input eld (at the input point z = 0) in the following form: E in (t) = [D + E1 (t z/c)] cos[0 (t z/c)] + E2 (t z/c) sin[0 (t z/c)] , (13) where 4 0 + d arm it h a1,2 e + aarm eit ; (14) 1,2 Ac 0 2 cf. Eqs. (1), (2), and (7). We write E out (t) at z = 0 in this same form but with the arms input dimensionless amplitudes aarm replaced by output dimensionless amplitudes de1,2 noted barm . Show that 1,2 E1,2 (t) = barm = e2i aarm 1 1 barm = e2i (aarm Kaarm ) + 2 2 1 5 2K hGW i e , hSQL
(15)
L 8Iarm 0 8 h , K= , hSQL = . (16) 2 c2 c m m2 L2 (d) The amplitude of the phase quadrature barm is measured (e.g. by recombining the 2 two arms light in the beam splitter and performing photodetection in the manner of Exercise 5 of Week 11). Identify, in this measured quantity, the terms that represent the GW signal, the shot noise and the radiation pressure noise. We could easily compute and study the spectral densities of the shot noise and radiationpressure noise, but we shall not do so, since this interferometer is actually too idealized to be of practical interest. Instead, we shall compute those spectral densities for a conventional interferometer with corner mirrors such as LIGO-I (next exercise). = 3. Quantum optical noise in a conventional interferometer and the standard quantum limit (SQL) In a real, conventional interferometer such as LIGO-I, the arms are Fabry Perot cavities with corner (input) mirrors, and the light from the two arms is recombined in the beamsplitter in the standard manner depicted in Slide 14 of the Buonanno-Chen lecture and in the following drawing:
where
(a) Such an interferometer is operated with mirror positions such that the laser light (almost) all returns toward the laser; almost none goes toward the photodetector (toward the dark port). Explain why this means that vacuum uctuations, with side-band frequencies f = /2 in the GW band behave as follows: Those that accompany the laser light into the interferometer from the bright port (almost) all return out the bright port, and those that enter from the dark port (almost) all return out the dark port. Since the GW signal is measured by the photodetector at the dark port, this means that the origin of the vacuum uctuations that accompany the signal is the dark port. We denote those input vacuum uctuations by a1,2 and denote the output dimensionless electric eld amplitudes at the dark port by b1,2 ; see the gure above. These dimensionless amplitudes are related to the dark-port input and output electric elds by the analog of Eqs. (13) and (14). 6
(b) It turns out that the input-output relations for this conventional interferometer have the same form (15) as those for a single arm without a corner mirror, but with the quantities K and modied due to the storage of the light in the arms Fabry-Perot cavities: hGW i b1 () = a1 () e2i , b2 () = [a2 () K a1 ()] e2i + 2K e , (17) hSQL with , (18) where Io is the laser power entering the beamsplitter, o is the laser frequency, m is the mirror mass, L is the length of the arm cavities, T is the power transmissivity of the input mirrors and is the sideband frequency. Explain in intuitive terms why the modications of Eqs. (16) have the form (18); most especially, explain the origin of the factor 1/( 2 + 2 ) in the expression for K. K= h2 = SQL = 2 = 2 arctan
out (c) In a conventional interferometer the quadrature phase E2 of the output electric eld is measured (e.g. in the manner analyzed in Exercise 5 of Week 11). This is equivalent to measuring the dimensionless output eld b2 (t), which contains the GW signal hGW (t). The Fourier transform of this output eld is given by the input-output relation (17). Renormalize this measured eld in such a way that the coefcient in front of hGW () is unity. Then the other terms represent the noise hn that contaminates the measurement of hGW . Show that the Fourier transform of this GW noise is
8Io o , 2 ( 2 + 2 )mL2
8 h , m2 L2
Tc , 4L
hn =
hSQL ei 2
a1 + K a2 K
.
(19)
The input vacuum uctuations a1 (t) and a2 (t) are in the standard quantum state, since nothing special is done in a conventional interferometer to put them into any other state. Therefore, they have unit spectral densities (Exercise 1). Show that the spectral density of the GW noise is given by h2 Sh (f ) = SQL 2 1 +K K . (20)
Which term in this spectral density is due to photon shot noise, and which term is due to radiation pressure noise? (d) Plot this spectral density for various choices of the laser power Io . On your plots indicate the shot noise and the radiation-pressure noise. Show that the sum of the two noises is always limited by the standard quantum limit (SQL)
SQL Sh = h2 = SQL
8 h . m2 L2
(21)
4. Beating the SQL via the variational output technique Vyatchanin, Matsko and Zubova have invented a method to beat the SQL that entails nothing more than a change in the interferometers readout, and Kimble et. al. [5] have devised a practical way of implementing their so-called variational-output interferometer. 7
(a) The Vyatchanin-Matsko-Zubova idea is to measure an appropriate mixture b = b1 sin + b2 cos (22)
(b)
(c) (d)
(e)
of the output electric elds amplitude and phase quadratures b1 and b2 . Can you gure out a way to do this, when the chosen phase is independent of sideband frequency? [Hint: try mixing the output light with another, reference light beam and then performing photodetection. How can this mixing be achieved? This technique is called homodyne detection.] In a variational-output interferometer one performs this homodyne detection, but with a homodyne phase angle that depends on the (GW) sideband frequency, = (). Kimbles key idea was that this can be done, in practice, by sending the output light through appropriate lters and then performing conventional homodyne detection (at xed homodyne angle). Derive the Fourier transform of the GW noise hn () [the analog of Eq. (19)] that plagues this measurement of b . Derive the spectral density of the GW noise Sh (f ) for this variational-output interferometer. Find the frequency-dependent form for that minimizes Sh (f ), and compute the resulting minimized noise. Show that by this technique, at least in principle, the back-action noise can be removed completely from the GW signal. In practice, ones ability to do this is seriously limited by optical losses in the interferometer. This is because, in the measured quadrature b one is using correlations between the shot noise and the radiation-pressure noise to cancel out the radiation-pressure (back-action) noise; and optical losses (e.g. due to the scattering of light off mirrors, with vacuum leaking into the light beam to replace the scattered light) destroy those correlations. See Ref. [5]. This is the rst time we have met correlations in random processes. One can quantify the correlations using a quantity called the cross spectral density. Read about cross spectral densities in Section 5.3 of Blandford and Thorne, Ref. [6]. Then do the fol lowing: Identify in hn [exercise 4b] the terms proportional to Io as radiation-pressure noise and denote them by [4/(m2 )]F. [Here we are to think of F as the backaction force; m/4 is the reduced mass associated with the measured difference of the four mirror positions, and m2 /4 is the quantity that you multiply against the mirrors position to get the force acting on the mirrors.] Identify the terms in hn proportional to 1/ Io as shot noise and denote them by Z. Evaluate the spectral densities SF and SZ (denoted SF F and SZZ by Buonanno and Chen), and the cross spectral densities SZF and SFZ . (In these evaluations, use the fact that for the standard vacuum state of the input elds a1,2 , there are no correlations between a1 and a2 ; i.e., their cross spectral densities vanish, Sa1 ,a2 = Sa2 ,a1 = 0.) Show that SZ SF SZF SF Z = h2 . (23)
It turns out that the Heisenberg uncertainty principle, when applied to this measurement process, gaurantees that the quantity on the left can never be less than h2 . (f) Show that when = 0 (conventional interferometer) the correlation between shot noise and radiation-pressure noise is zero, i.e., SZF = 0. The resulting uncertainty relation SZ SF h2 enforces the SQL. 8
5. Optomechanical resonances in a signal recycled interferometer In this problem we shall derive the frequencies of the resonances in a signal recycled (SR) interferometer, in the limit of a perfectly reecting signal-recycling mirror. (a) Consider the propagation of a light beam over a distance l, from z = 0 to z = l. We know that E(t, z = l) = E(t l/c, z = 0) . (24) Decompose E(t, z = l) and E(t, z = 0) into quadrature elds and show that, the Fourier components of the quadrature elds, a1,2 () (at z = 0) and b1,2 () (at z = l), are related by the following rotation and phase shift b1 () b2 () where = 0 l/c , = l/c . (26) (b) Consider the conventional interferometer in the gure above, and add a perfectly reecting signal recycling mirror at the dark port. Assume that the distance from the dark port [the place where the input-output relations (17) are valid] to the mirror is l, with 0 l/c = 2N + and N an integer. The quantity is called the detuning phase. Assume also that l is so small that l/c can be neglected for all frequencies in the GW frequency band. Explain why the eigenfrequencies of the closed opticalmechanical system can be evaluated by imposing det I cos 2 sin 2 sin 2 cos 2 e2i 1 0 K 1 = 0, (27) = ei cos sin sin cos a1 () a2 () (25)
where K and are given by Eqs. (18). Show that Eq. (27) simplies to the following eigenequation, K (28) cos(2) cos(2) sin(2) = 0 , 2 which can be further simplied (e.g. using Mathematica) to 16Iarm 0 tan . mL2 c Here m is the mirror mass, c is the light speed, L is the arm-cavity length, and 2 (2 2 tan2 ) + 2 Iarm = I0 , T is the circulating power inside the arm cavities. (29)
(30)
(c) Explain using Eq. (28) or Eq. (29) why, when the laser power is low, there is a pure optical resonance and a pure mechanical resonance. As I0 gradually increases from 0, the two resonances become coupled and shifted from their original positions. In our simple example, the four roots of Eq. (29) can be derived analytically. Identify the regime where the system exhibits an optical spring behavior. In particular, what circulating power is required for this? In what region should the detuning-phase lie? 9