Precision Interferometry in a New Shape: Higher-order Laguerre-Gauss Modes for Gravitational Wave Detection

[Paul Fulda is the recipient of the 2012 GWIC (Gravitational Wave International Committee) Thesis Prize for his PhD thesis “Precision Interferometry in a New Shape: Higher-order Laguerre-Gauss Modes for Gravitational Wave Detection” (PDF). -- 2Physics.com]

Author: Paul Fulda

Affiliation:

Currently at Department of Physics, University of Florida, Gainesville, USA

PhD research performed at School of Physics and Astronomy, University of Birmingham, UK.


With the approach of advanced detector science runs, the field of gravitational wave detection stands on the brink of finally making the much sought after first direct detection of gravitational waves. Never a bunch to rest on our laurels, however, we are always trying to think of ways to push the sensitivity of these kilometer-scale interferometers ever higher. A single detection, with its ramifications in the areas of astrophysics, cosmology and General Relativity, would be an incredible achievement for a scientific field long in the making.

However, since the observable astrophysical event rate scales as the cube of the observable event distance (and hence roughly as the cube of the broadband sensitivity), any sensitivity increase we can make will bring with it an ever greater number of detected gravitational wave events. The more events we have, the better statistics we can do with them, and the better we can refine our model of the universe itself. With that in mind, in this article I will focus on one of many ideas for improving the sensitivity of gravitational wave detectors beyond the 'advanced' generation; the idea of using higher-order Laguerre-Gauss modes to reduce the thermal noise of the test-masses of the interferometers.

2Physics articles by past winners of the GWIC Thesis Prize:

Rutger van Haasteren (2011): "Pulsar Timing Arrays: Gravitational-wave detectors as big as the Galaxy"
Haixing Miao (2010): "Exploring Macroscopic Quantum Mechanics with Gravitational-wave Detectors"
Holger J. Pletsch (2009): "Deepest All-Sky Surveys for Continuous Gravitational Waves"
Henning Vahlbruch (2008): "Squeezed Light – the first real application starts now"
Keisuke Goda (2007): "Beating the Quantum Limit in Gravitational Wave Detectors"
Yoichi Aso (2006): "Novel Low-Frequency Vibration Isolation Technique for Interferometric Gravitational Wave Detectors"
Rana Adhikari (2003-5)*: "Interferometric Detection of Gravitational Waves : 5 Needed Breakthroughs"
*Note, the gravitational wave thesis prize was started initially by LIGO as a biannual prize, limited to students of the LIGO Scientific Collaboration (LSC). The first award covered the period from 1 July 2003 to 30 June 2005. In 2006, the thesis prize was adopted by GWIC, renamed, converted to an annual prize, and opened to the broader international community.

The rest of this article won't make a lot of sense if I don't first explain what a higher-order laser mode is, but in order to understand this concept we also need to know a little bit about optical cavities. An optical cavity exists wherever there is an arrangement of mirrors that form a closed optical path. Perhaps the simplest case is a linear cavity, which consists of two partially transmissive mirrors, one flat and one curved, orientated with the reflective surfaces parallel to each other (see Fig. 1). When the length of the cavity is tuned just right relative to the frequency of the light, light entering through one partially transmissive mirror creates a standing wave inside the cavity. This condition is known as a resonance of the cavity, and it creates a higher circulating light power inside the cavity than the light power entering. The light is 'stored' inside the cavity for some time before it leaks out again.

Fig 1: A cartoon picture of an optical cavity. Light enters the cavity through the input mirror from the left, and resonates inside the cavity if the cavity length is divisible by an integer number of half wavelengths.

There is another condition on top of the length/frequency condition in order for light to resonate in an optical cavity; the amplitude cross-section of the light (i.e. the beam shape) must be unchanged after several reflections from the mirrors that make up the cavity. If this is not the case, the light will not properly add up after each reflection to make a standing wave. For a cavity made up of spherically curved mirrors, this condition is satisfied by sets of so-called "spatial modes". There are two main families of these spatial modes - Hermite-Gauss (HG) modes and Laguerre-Gauss (LG) modes (see Fig. 2). In a well designed optical cavity, each higher mode order (the order being n+m for HG modes and 2p+l for LG modes) is resonant at a different cavity tuning. These modes are mathematical solutions to the wave equation in the beam-like or 'paraxial' limit. However, far from being purely abstract mathematical constructs, they really exist and show up in interferometers all the time, as we shall soon see!

Fig 2: [click on the image to see better resolution] Intensity patterns for ideal higher-order HG (top left) and LG (top right) modes. The lower half of the image shows the higher-order LG modes generated in the lab using an SLM (in negative to show detail).

Advanced interferometric gravitational wave detectors make use of optical cavities in many places. Advanced LIGO for example has a pre- mode-cleaner cavity, an input mode cleaner cavity and an output mode cleaner cavity, in addition to the 4 coupled cavities of the central interferometer (see Fig. 3) [1]. These optical cavities are used outside the central interferometer as 'mode cleaners' for their ability to filter the frequency and spatial properties of light, and inside the central interferometer in order to increase the circulating light power and the interaction time between the light and passing gravitational waves. LIGO would not have been able to achieve the unprecedented strain sensitivity it did without using these crucial interferometric tools.

Fig 3: [click on the image to see better resolution] The optical layout of an Advanced LIGO interferometer, showing the input and output mode cleaner cavities, the two 4km long arm cavities and the two recycling cavities.

Usually one aims to use only the lowest order 'fundamental' mode (the HG00/LG00 mode), for precision interferometric metrology. The presence of any of the higher-order modes commonly signifies an imperfection in the optical setup, for example a misalignment or a discrepancy in the characteristic size of two interfering beams. As a result, there are several activities directed towards reducing the presence of these modes in interferometers [2, 3].

Recently, however, it was shown that by using one of the higher-order Laguerre-Gauss modes instead of the funamental mode it may be possible to reduce the effects of mirror thermal noise, which is expected to be one of the limiting noise sources in gravitational wave detectors [4]. Mirror thermal noise appears as random fluctations of the reflecting surfaces, which can mask any gravitational wave passing through the interferometer. Higher-order LG modes are able to reduce the effect of this noise by averaging better over the mirror surface, effectively smoothing out the fluctuations. Higher-order LG modes also have the interesting property of having spiral phasefronts, thereby carrying angular momentum - a property which has made them useful in the fields of cold atoms and bio-photonics [5].

Could it then be a case of "from villain to hero" for higher-order modes? This is the question we sought to answer with a program of research into the application of LG modes in precision interferometry. Although the thermal noise benefits of using LG modes made them seem like an attractive option, it was crucial to test their compatibility with all of the other technologies that are used in gravitational wave detectors. At the University of Birmingham we started this investigation using simulations of advanced interferometer where the main fundamental mode beam was exchanged for a LG33 beam (top right in each panel of Fig. 1) [6]. The LG33 beam performed very favorably in the simulations, giving us the motivation to move forward to table-top lab experiments.

We used a device called a spatial light modulator (SLM) to generate the higher-order LG modes in the lab. This device works somewhat like an LCD screen, except that each pixel can change the phase of the light reflected from it instead of the intensity. By displaying a specific phase pattern on the SLM, we were able to convert a fundamental mode beam into almost any higher-order LG mode we wanted (see the lower half of Fig. 2). We confirmed the spiral phase front of the beams by interfering them with a fundamental mode beam, creating the spiral intensity patterns shown in Fig. 4. However, the mode purity we could achieve with the SLM alone was not high enough for use with gravitational wave detectors.

Fig. 4: Images of the interference pattern between various SLM generated higher-order LG beams and a fundamental mode beam. The spiral pattern indicates the presence of a spiral phasefront in the LG beams. From left: LG22, LG33, LG44, LG55.

The positive aspect of this was that it gave us the perfect opportunity to make one of the most crucial compatibility tests for the LG33 mode - if the LG33 mode was to be effective in gravitational wave detectors, it would have to be compatible with mode cleaner optical cavities and the control schemes that are commonly used with them. If it worked, we would have the added bonus of a `cleaned' LG33 mode after the mode cleaner. As predicted by the earlier simulations, the LG33 mode was compatible with a mode cleaner cavity, and we achieved mode purity in excess of 99%. Figure 5 shows images of two kinds of LG33 mode before and after the mode cleaner cavity [7].

Fig. 5: Images of the SLM generated LG33 mode before (top) and after (bottom) transmission through a linear optical cavity.

With these positive results we decided to move on to the next level of compatibility testing: using LG modes with a suspended 10m prototype interferometer in Glasgow. The specific aim here was to address a tricky problem with LG modes that was identified through simulations; with larger beam sizes, the imperfections in the mirrors that form an optical cavity can cause a significant reduction in the mode purity [8,9].

We set up an LG mode conversion path on the Glasgow prototype input bench (see Fig. 6), and directed the beam into the vacuum tank towards a 10m long optical cavity. We employed a sophisticated method to precisely tune the length of the cavity, while observing the transmitted light power with a high-speed infra-red camera. Since different modes are resonant in the cavity at different tunings, the high-speed camera measurements gave us a good picture of the higher-order mode content in the cavity.

Fig. 6: [click on the image to see better resolution] The LG mode conversion path for the Glasgow 10m prototype interferometer experiment. The red line shows the LG33 conversion path, the green line shows the fundamental mode 'control' path, and the purple line shows the original laser path.

We saw that even when we injected a LG33 mode, we could only observe the rectangular shaped HG modes resonating in the cavity (see Fig. 7). The work reported in [8] and [9] suggested that this behavior might be expected in a cavity where the mirrors are astigmatically curved, rather than the spherically curved ideal case. This was a setback for the use of LG modes in gravitational wave interferometers, as it was the first experimental evidence that mirror surface imperfections might have a stronger impact on mode purity for the LG33 mode than for the fundamental mode. The cavity mirrors were taken out of the prototype and independent measurements of the surface figures were made. The measured astigmatism values were used as inputs for a simulation of the 10m cavity, which reproduced the resonance structures seen in Fig. 7 very well. This was the confirmation that was needed to show that the LG33 mode really was more sensitive to mirror surface imperfections that the fundamental mode [10].

Fig. 7: [click on the image to see better resolution] A trace showing reflected power from the 10m cavity as a function of time as the cavity is swept across a resonance with the LG33 input beam. The resonance is split into several peaks, as predicted in [8] and [9]. Images from the high-speed camera in transmission of the cavity show the resonant mode shape around several of the separate resonance peaks.

From here it was possible to derive tolerances on the mirror imperfections that would be allowable if the LG33 mode was to be used as the readout beam in a real gravitational wave detector. The requirements that the LG33 mode puts on the quality of the mirror surfaces are beyond what is achievable with the current technology, and as such they do look very tough to meet. However, coating and polishing technology is always moving forward with the demands of ever more precise experiments, so at some time in the future we may yet see the LG33 mode flashing on the projector screens in a gravitational wave detector installation.

Those of us who work in the business of laser interferometry become quite accustomed to the sight of higher-order spatial laser modes. It can be easy to take their aesthetic qualities for granted, especially when a lot of our efforts are directed towards minimizing their presence. I still remember the first time I aligned an optical cavity though, and how I was mesmerized by the small CCTV camera picture showing the strange and beautiful patterns in the cavity transmitted light. I'm very glad that my PhD research took me in the direction of studying a particular one of these higher-order laser modes in great detail. By the end of my studies, the higher-order modes weren't just obscure patterns of light to me any more; rather they seemed like cryptic hieroglyphs which when translated could provide a wealth of knowledge about the state of an optical system.

References:
[1] Gregory M. Harry and the LIGO Scientific Collaboration. "Advanced LIGO: the next generation of gravitational wave detectors". Classical and Quantum Gravity, 27(8):084006 (2010). Abstract.
[2] Euan Morrison, Brian J. Meers, David I. Robertson, and Henry Ward. "Automatic alignment of optical interferometers". Applied Optics, 33:5041–5049 (1994). Abstract.
[3] Guido Mueller, Qi-ze Shu, Rana Adhikari, D. B. Tanner, David Reitze, Daniel Sigg, Nergis Mavalvala, and Jordan Camp. "Determination and optimization of mode matching into optical cavities by heterodyne detection". Optics letters, 25(4):266-268 (2000). Abstract.
[4] Benoît Mours, Edwige Tournefier and Jean-Yves Vinet. "Thermal noise reduction in interferometric gravitational wave antennas: using high order TEM modes". Classical and Quantum Gravity, 23:5777-5784 (2006). Abstract.
[5] H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop. "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity". Physical Review Letters, 75(5):826–829 (1995). Abstract.
[6] Simon Chelkowski, Stefan Hild, and Andreas Freise. "Prospects of higher- order Laguerre-Gauss modes in future gravitational wave detectors". Physical Review D, 79(12):122002 (2009). Abstract.
[7] Paul Fulda, Keiko Kokeyama, Simon Chelkowski, and Andreas Freise. "Experimental demonstration of higher-order Laguerre-Gauss mode interferometry". Physical Review D, 82(1):012002 (2010). Abstract.
[8] Charlotte Bond, Paul Fulda, Ludovico Carbone, Keiko Kokeyama, and Andreas Freise. "Higher order Laguerre-Gauss mode degeneracy in realistic, high finesse cavities". Physical Review D, 84(10):102002 (2011). Abstract.
[9] T. Hong, J. Miller, H. Yamamoto, Y.Chen, R. Adhikari. "Effects of mirror aberrations on Laguerre-Gaussian beams in interferometric gravitational wave detectors". Physical Review D, 84(10):102001 (2011). Abstract.
[10] B. Sorazu, P. Fulda, B.W. Barr, A.S. Bell, C. Bond, L. Carbone, A. Freise, S.H. Huttner, J. Macarthur, K.A. Strain. "Experimental test of higher-order Laguerre–Gauss modes in the 10m Glasgow prototype interferometer". Classical and Quantum Gravity, 30(3):035004 (2013). Abstract.

The World's Most Stable Laser : Impact on Atomic Clocks, Precision Tests in Fundamental Physics, Gravitational Wave Detection

Jun Ye [photo courtesy: JILA/University of Colorado, USA]

No more than only 2 parts in 10,000 trillion -- that's the new world record set for the frequency variation by the most stable laser as reported in an advance on-line publication in Nature Photonics [1]. The laser is developed and tested by an international collaboration of scientists at NIST/JILA in Boulder, Colorado, USA and a group at Physikalisch-Technische Bundesanstalt (PTB), the German counterpart of the National Institute of Standards and Technology (NIST).

Their work represents a new approach for constructing high-quality optical cavities that will bring more than an order of magnitude improvement over prior designs. In particular, it will accelerate progress in development of optical clocks, which operate at frequencies more than 10,000 times higher than the approximately 9.2 GHz microwaves used as the basis of the current worldwide time standard.

In addition, the novel laser design is expected to usher in a new level of precision to research in gravitational wave detection on Earth and in space, and precision tests of relativity as well as fundamental physics research in cavity quantum electrodynamics and quantum optomechanics.

“The previous stability limit of about 2 X 10^-16 was good,” says study co-author Jun Ye of JILA and PML’s Quantum Physics Division. “But it prevented us from exploring the full potential of modern optical atomic clocks where the atomic coherence time can be exceedingly long. The potential of pushing the laser stability better by an order of magnitude will allow us to realize atomic clocks that have unprecedented stability approaching 1 x 10^-17 over 1 second.

“It will also have a dramatic impact on large-scale precision measurement instruments such as space-borne optical interferometers for gravitational-wave detection with baselines of many thousands of kilometers. And it may be of intense interest to the communications industry, since our system was tested at a familiar telecomm wavelength of 1.5 micrometers.” That is the wavelength with the lowest loss in fiber-optic networks.

This shows the size of the new silicon resonator compared to the size of a coin [image courtesy: Physikalisch-Technische Bundesanstalt, Germany]

Research on ultra-stable lasers typically employs some kind of optical-cavity interferometer, comprising a spacer with mirrors at each end. That design severely restricts the range of optical frequency which can resonate in the cavity. By superimposing the cavity output beam on another highly controlled reference beam, the interference effects (periodic reinforcement “beats”) can reveal stability with exquisite sensitivity. Such systems, however, have historically been subject to thermal fluctuations that alter the cavity dimensions, and hence reduce frequency stability.

“We addressed that problem in several ways,” says Ye, who began to collaborate on the project in 2007 while visiting PTB in Germany on a grant from the Alexander von Humboldt Foundation. The German team was headed by Uwe Sterr and Fritz Riehle. Ye also recruited a student, Michael Martin, and a visitor in his lab, Lisheng Chen. “By far the most significant factor was our decision to substitute single-crystal silicon for the ultralow expansion glass (ULE) or fused silica customarily employed in the cavity mirrors and spacers.”

Single-crystal silicon has a coefficient of thermal expansion that approaches zero at 124 K, “so an all-silicon interferometer can be made insensitive to temperature fluctuations” at that point, the authors write in Nature Photonics. In addition, at 124 K the crystal has a much lower mechanical loss compared to conventional optical glass, and a much higher Young’s modulus. Both mechanical properties combine to minimize the fundamental thermal fluctuations. The group used extensive computer modeling to create a design that reduced the effects of environmental vibrations on the 21 cm long resonator, which they mounted in a vertical configuration after testing its response to external forces in all three dimensions.

Finally, the German researchers devised a novel and simple cryostat design using evaporated liquid nitrogen gas as the coolant. The gas flows through superinsulated vacuum tubes to an outer heat shield, and careful control of the flow limits temperature deviations in the system to about 1 mK from the 124 K target.

Diagram of the device interior [image credit: Thomas Kessler, Physikalisch-Technische Bundesanstalt, Germany].

The new all-silicon unit was tested for 24 hours against two of the best-performing conventional ULE-based optical cavity-stabilized lasers – one from JILA and one from PTB, with thermal noise variation in the range of 6 X 10^-16 and 2 X 10^-16 respectively. The results, Ye says, “show that the all-silicon system surpasses the performance of any other optical cavities ever reported.” As a long-term stable frequency reference, a preliminary test has shown that it is equivalent to the stability of a hydrogen maser at time intervals up to 1,000 seconds.

The scientists are already at work to improve those figures. Among other modifications, they will attempt to reduce feedback errors due to spurious amplitude modulation, suppress noise from the cryostat, and experiment with different optical coatings on the silicon mirrors. The very thin optical coatings now remain the only significant contribution to the thermal noise of the cavity and new approaches are already being investigated jointly with Markus Aspelmeyer’s group at the University of Vienna.

Graph comparing average instability over time of the new design (green) to two ultra-stable ULE units: Ref. 1 (blue), a PTB laser, and Ref. 2 (orange), a JILA laser [image courtesy: NIST, USA].

Many of most critical experiments in atomic physics require lasers with extremely narrow linewidth and extremely stable output to interrogate clusters of ultracold atoms or single trapped ions. “Stable lasers such as the one reported are already unlocking some of the mysteries of minute atomic interactions that are otherwise hidden,” Ye says.

Reference
[1] T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. J. Martin, L. Chen, J. Ye, "A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity", Nature Photonics (Published online September 9th, 2012). doi:10.1038/nphoton.2012.217. Abstract.

Pulsar Timing Arrays: Gravitational-wave detectors as big as the Galaxy

Rutger van Haasteren

[Rutger van Haasteren is the recipient of the 2011 GWIC (Gravitational Wave International Committee) Thesis Prize for his PhD thesis “Gravitational Wave detection and data analysis for Pulsar Timing Arrays” (PDF). -- 2Physics.com]

Author: Rutger van Haasteren 

Affiliation: 
Currently at Albert-Einstein-Institute (Max-Plack Institute for Gravitational Physics) in Hannover, Germany;
PhD research done at Leiden Observatory, Leiden University, The Netherlands.

Pulsars, rapidly rotating neutron stars that send an electromagnetic pulse towards the Earth with each revolution, are intimately connected to gravitational research and testing of Einstein's general theory of relativity. Besides the fact that neutron stars have very strong gravitational fields -- which is interesting from a general relativity point of view, their use as accurate clocks allows for a whole range of new gravitational experiments. Especially millisecond pulsars, recycled pulsars that have been spun-up by a companion star (see this video by John Rowe Animation/Australia Telescope National Facility, CSIRO, Australia), are very stable rotators due to their high spin frequency, relatively low magnetic field, and high mass. This makes millisecond pulsars most suitable as nearly-perfect Einstein clocks.

2Physics articles by past winners of the GWIC Thesis Prize:

Haixing Miao (2010): "Exploring Macroscopic Quantum Mechanics with Gravitational-wave Detectors"
Holger J. Pletsch (2009): "Deepest All-Sky Surveys for Continuous Gravitational Waves"
Henning Vahlbruch (2008): "Squeezed Light – the first real application starts now"
Keisuke Goda (2007): "Beating the Quantum Limit in Gravitational Wave Detectors"
Yoichi Aso (2006): "Novel Low-Frequency Vibration Isolation Technique for Interferometric Gravitational Wave Detectors"
Rana Adhikari (2003-5)*: "Interferometric Detection of Gravitational Waves : 5 Needed Breakthroughs"
*Note, the gravitational wave thesis prize was started initially by LIGO as a biannual prize, limited to students of the LIGO Scientific Collaboration (LSC). The first award covered the period from 1 July 2003 to 30 June 2005. In 2006, the thesis prize was adopted by GWIC, renamed, converted to an annual prize, and opened to the broader international community.

We can accurately track a pulsar's trajectory with respect to the Earth by monitoring the arrival times of their pulses; given that for the most well-timed millisecond pulsars we can determine the time of arrival of a single averaged pulse up to 50 nanoseconds, we are effectively sensitive to variations in the Earth-pulsar distance up to only several dozens of meters. This is because light can only travel about one foot in a nanosecond. Because some pulsars are in very tight binary systems, such an accurate measurement of the orbit of a pulsar around a companion can be used to verify/falsify the predictions of general relativity. This was first done with the binary pulsar, PSR J1915+1606, also called the Hulse-Taylor binary, which was discovered in 1974 [1]. This system is a double neutron star system of which one of the two bodies is a pulsar. The two stars are close together: a full orbital period only takes 7.75 hours. For two such massive bodies in such a tight orbit general relativity predicts that the emission of gravitational waves is significant, which would cause a decrease in the orbital period due to loss of energy of the system (see see this video by John Rowe Animation/Australia Telescope National Facility, CSIRO, Australia). By closely tracking the dynamics of the Hulse-Taylor binary, this decrease in the orbital period was confirmed exactly (figure 1), confirming the existence of gravitational waves. This has resulted in Hulse and Taylor been awarded the Nobel prize in physics in 1993 [2].

Figure 1: Decreasing period of rotation of binary pulsar PSR J1915+1606

The confirmation of the existence of gravitational waves with the Hulse-Taylor binary is considered an indirect detection of gravitational waves, because it has been shown that the energy loss of a system is consistent with gravitational-wave emission. A direct detection would have to consist of evidence that the gravitational waves are present elsewhere than at the point of emission: by using a gravitational-wave detector.

Generally speaking, two approaches exist to directly detect gravitational waves:
1) A large body of mass is used as a resonator, where the gravitational waves are expected to excite the resonant frequencies of such a so-called resonant-mass detector.
2) A signal is sent from one place to another, where the gravitational waves are expected to perturb the propagation of the signal such that its arrival time slightly changes. In laser interferometry detectors (e.g. LIGO) this results in a changing interference pattern at the point of recombination of two laser beams.

Figure 2: Concept of a pulsar timing array [Image credit: David J. Champion]

As it turns out, millisecond pulsars can be used to 'construct' a gravitational-wave detector of the second kind, where the pulse propagation from the pulsar to the Earth is perturbed by astrophysical gravitational waves [3]. The very regular arrival times of a millisecond pulsar will arrive slightly early or late due to gravitational waves that pass through the Earth-pulsar system, which in principle makes the gravitational waves detectable. Because the typically observed millisecond pulsars for these purposes are several kpc away, a pulsar timing array is basically a gravitational wave detector of galactic scale (figure 2; Also see this video by John Rowe Animation/Australia Telescope National Facility, CSIRO, Australia)).

A pulsar timing array is sensitive to gravitational waves with frequencies of a few dozen to a few hundred nHz [4], which is the frequency range where tight supermassive black-hole binaries (SMBHB) are expected to be the dominant sources of continuous gravitational waves. A canonical SMBHB system that would contribute to the gravitational-wave signal would consist of two supermassive black holes with masses of close to one billion solar masses at a distance of a Gpc, with an orbital period of several months to years. Many such systems are expected to exist in the universe, which would results in an isotropic superposition usually called a stochastic background of gravitational waves [5]. Large-scale computer simulations of the evolution of the universe suggest that some of the individual sources might be eventually uniquely detectable, but the bulk of the signal would consist of such an isotropic stochastic background of gravitational waves [6]. Because the evolution of the universe is intimately connected to the SMBHB gravitational-wave signal, it is thought that measuring the stochastic gravitational-wave background and possibly single SMBHB sources would contribute greatly to our understanding of cosmology. This is a frequency band that is unreachable for any other type of gravitational-wave detector, which makes pulsar timing arrays a unique and complementary tool next to the other gravitational-wave detection programmes like the ground-based gravitational observatories.

Pulsar timing array science is still relatively new, and a new international pulsar timing array (IPTA, [7]) collaboration has only recently been formed as an alliance between three ongoing pulsar timing array efforts: the European Pulsar Timing Array (EPTA, [8]), the North American Nanohertz Observatory for Gravitational waves (NANOGrav, [10]), and the Australian Parkes Pulsar Timing Array (PPTA, [9]). The PPTA uses a single radio telescope based in Parkes, Australia, with a 64m dish. NANOGrav uses the worlds two largest single-dish radio telescope: the 100m Green Bank Telescope, and the 305m Arecibo Observatory. The EPTA uses five radio telescopes spread throughout Europe: the Westerbork synthesis radio telescope in the Netherlands, the Lovell telescope in the UK, the Effelsberg telescope in Germany, the Nancay radio telescope in France, and the Sardinia radio telescope in Italy. These five European radio telescopes are currently being linked together to coherently combine their signals, effectively forming one big phased array called the Large European Array for Pulsars (LEAP, [8]). This improvement should boost sensitivity for pulsar timing array purposes.

The Effelsberg Radio Telescope near Effelsberg, Germany. This is the worlds second-largest fully steerable radio telescope, with a diameter of 100 meters. [Image credit: Gemma Janssen]

Between the ground-based gravitational-wave detectors and pulsar timing arrays, it is basically a scientific race focused on who will make the first detection, with both projects having good chances of being the first. Pulsar timing arrays have the advantage that the signal rms is expected to increase sharply with time. Even if pulsar timing arrays cannot reduce their noise with their always ongoing efforts, sensitivity will still gradually increase over time, making a detection possible. However, the theoretical predictions about the stochastic background amplitude and the event rates of single sources are less certain than for ground-based detectors. The big ground-based detectors are currently upgrading their instruments, which are expected to become operational somewhere in 2015.

The Parkes Radio Telescope near Parkes, Australia. This radio telescope with a diameter of 64 meters is the worlds leading radio telescope in discovery of radio pulsars. [Image credit: Aristeidis Noutsos]

Even without upgrading instruments, sensitivity of both types of gravitational-wave detectors can be increased with better data analysis methods which would allow more information to be extracted from the data. In order to do that, a Bayesian data analysis method for pulsar timing arrays has been developed that can theoretically extract all information of the signal that is present in the data. General relativity describes the gravitational-wave signal of the stochastic background as a both time correlated and spatially correlated signal between all the pulsars, which means that the data of the different pulsars cannot be treated individually. Extracting such a signal from the data is non-trivial, especially for non-uniformly sampled data with ill-understood noise like that of millisecond pulsars. The Bayesian analysis is suitable for such an analysis, and has been shown to work well for both stochastic background signals [11], and single sources like the gravitational-wave memory effect [12]. The developed Bayesian analysis has resulted in the most stringent upper-limit on the stochastic gravitational-wave background to date [11].

The Westerbork Synthesis Radio Telescope near Westerbork, The Netherlands. This radio telescope is composed of 14 dishes with a diameter of 25 meter, which combine into a radio telescope of similar sensitivity to that of the Effelsberg Radio Telescope. [Image credit: Cees Bassa]

In the coming years, the Chinese five hundred meter aperture spherical telescope (FAST, [13]), and the planned Square Kilometre Array (SKA, [14]) will provide a major leap in sensitivity. Especially the SKA, built by a collaboration of 20 countries, will dramatically change pulsar timing array science. It will be a phased array of many dishes located across South Africa, Australia and New Zealand [15]. With its vast collecting area of one million square meter it is expected to find nearly all the pulsars in the Galaxy. With all those pulsars a very sensitive gravitational-wave detector with possibly up to one hundred arms can be constructed. This should open up a new window to observe the universe, and provide unique insights into cosmology.

References:
[1] R.A. Hulse, J.H. Taylor, "Discovery of a pulsar in a binary system", The Astrophysical Journal, 195, L51 (1975). Full text.
[2] http://www.nobelprize.org/nobel_prizes/physics/laureates/1993/
[3] Frank B. Estabrook and Hugo D. Wahlquist, "Response of Doppler spacecraft tracking to gravitational radiation", General Relativity and Gravitation, 6, 439 (1975). Abstract.
[4] R.S. Foster, D.C. Backer, "Constructing a pulsar timing array", The Astrophysical Journal, 361, 300 (1990). Abstract.
[5] E.S. Phinney, "A Practical Theorem on Gravitational Wave Backgrounds", eprint arXiv:astro-ph/0108028v1 (2001).
[6] A. Sesana, A. Vecchio, C. N. Colacino, "The stochastic gravitational-wave background from massive black hole binary systems: implications for observations with Pulsar Timing Arrays", Monthly Notices of the Royal Astronomical Society, 390, 192 (2008). Abstract.
[7] G Hobbs, A Archibald, Z Arzoumanian, D Backer, M Bailes, N D R Bhat, M Burgay, S Burke-Spolaor, D Champion, I Cognard, W Coles, J Cordes, P Demorest, G Desvignes, R D Ferdman, L Finn, P Freire, M Gonzalez, J Hessels, A Hotan, G Janssen, F Jenet, A Jessner, C Jordan, V Kaspi, M Kramer, V Kondratiev, J Lazio, K Lazaridis, K J Lee, Y Levin, A Lommen, D Lorimer, R Lynch, A Lyne, R Manchester, M McLaughlin, D Nice, S Oslowski, M Pilia, A Possenti, M Purver, S Ransom, J Reynolds, S Sanidas, J Sarkissian, A Sesana, R Shannon, X Siemens, I Stairs, B Stappers, D Stinebring, G Theureau, R van Haasteren, W van Straten, J P W Verbiest, D R B Yardley and X P You,"The International Pulsar Timing Array project: using pulsars as a gravitational wave detector", Classical and Quantum Gravity, 27, 084013 (2010). Abstract.
[8] R D Ferdman, R van Haasteren, C G Bassa, M Burgay, I Cognard, A Corongiu, N D'Amico, G Desvignes, J W T Hessels, G H Janssen, A Jessner, C Jordan, R Karuppusamy, E F Keane, M Kramer, K Lazaridis, Y Levin, A G Lyne, M Pilia, A Possenti, M Purver, B Stappers, S Sanidas, R Smits and G Theureau, "The European Pulsar Timing Array: current efforts and a LEAP toward the future", Classical and Quantum Gravity, 27, 084014 (2010). Abstract.
[9] G. Hobbs, D. Miller, R. N. Manchester, J. Dempsey, J. M. Chapman, J. Khoo, J. Applegate, M. Bailes, N. D. R. Bhat, R. Bridle, A. Borg, A. Brown, C. Burnett, F. Camilo, C. Cattalini, A. Chaudhary, R. Chen, N. D’Amico, L. Kedziora-Chudczer, T. Cornwell, R. George, G. Hampson, M. Hepburn, A. Jameson, M. Keith, T. Kelly, A. Kosmynin, E. Lenc, D. Lorimer, C. Love, A. Lyne, V. McIntyre, J. Morrissey, M. Pienaar, J. Reynolds, G. Ryder, J. Sarkissian, A. Stevenson, A. Treloar, W. van Straten, M. Whiting and G. Wilson, "The Parkes Observatory Pulsar Data Archive", Publications of the Astronomical Society of Australia, 26, 103 (2009). Full Text.
[10] P. B. Demorest, R. D. Ferdman, M. E. Gonzalez, D. Nice, S. Ransom, I. H. Stairs, Z. Arzoumanian, A. Brazier, S. Burke-Spolaor, S. J. Chamberlin, J. M. Cordes, J. Ellis, L. S. Finn, P. Freire, S. Giampanis, F. Jenet, V. M. Kaspi, J. Lazio, A. N. Lommen, M. McLaughlin, N. Palliyaguru, D. Perrodin, R. M. Shannon, X. Siemens, D. Stinebring, J. Swiggum, W. W. Zhu, "Limits on the Stochastic Gravitational Wave Background from the North American Nanohertz Observatory for Gravitational Waves", eprint arXiv:1201.6641 (2012)
[11] R. van Haasteren, Y. Levin, G. H. Janssen, K. Lazaridis, M. Kramer, B. W. Stappers, G. Desvignes, M. B. Purver, A. G. Lyne, R. D. Ferdman, A. Jessner, I. Cognard, G. Theureau, N. D’Amico, A. Possenti, M. Burgay, A. Corongiu, J. W. T. Hessels, R. Smits and J. P. W. Verbiest, "Placing limits on the stochastic gravitational-wave background using European Pulsar Timing Array data", Monthly Notices of the Royal Astronomical Society, 414, 3117 (2011). Abstract.
[12] Rutger van Haasteren and Yuri Levin, "Gravitational-wave memory and pulsar timing arrays", Monthly Notices of the Royal Astronomical Society, 401, 2372 (2010). Abstract.
[13] R. Smits, M. Kramer, B. Stappers, D.R. Lorimer, J. Cordes, and A. Faulkner, "Pulsar searches and timing with the square kilometre array", Astronomy & Astrophysics, 505, 919 (2009). Abstract.
[14] T. Joseph W. Lazio, "The Square Kilometre Array", in Panoramic Radio Astronomy: Wide-field 1-2 GHz Research on Galaxy Evolution (2009). Full Text.
[15] http://www.skatelescope.org/news/dual-site-agreed-square-kilometre-array-telescope/

A Gravitational Wave Observatory Operating Beyond the Quantum Shot-Noise Limit

[L to R] Hartmut Grote, Roman Schnabel, Henning Vahlbruch













Authors: Hartmut Grote, Roman Schnabel, Henning Vahlbruch

Affiliation: Institute for Gravitational Physics, Leibniz Universität Hannover and Max-Planck-Institute for Gravitational Physics (Albert-Einstein-Institute, AEI), Hannover, Germany

Quantum-squeezed light has now gone beyond the development-phase in the confinement of laboratory, and is now improving the sensitivity of the German/British gravitational-wave (GW) observatory GEO600 located close to Hannover (Germany). This is what our recent publication in Nature Physics reports [1]. Arguably -- for the first time -- a technology, which exploits the special findings of quantum mechanics, is put into a real application in metrology. The idea that squeezed states of light might be valuable in the field of GW detection is already 30 years old [2], but only now realized.

Past 2Physics article by this group:
April 03, 2008: "Squeezed Light – the first real application starts now"
by Roman Schnabel and Henning Vahlbruch

Gravitational waves are predicted by Einstein’s general theory of relativity, and are generated, for example, by black-hole binary systems. In principle, they can even be observed on earth by kilometer-scale Michelson-type laser interferometers measuring the changes in distance between mirrors suspended in vacuum. However, so far they have not been observed directly. Fig. 1 shows one of the suspended mirrors of the GW observatory GEO600 (mirror at bottom right). More details about GEO600 can be found in Ref. [3].

Fig. 1: One of four suspended 5 kg test-masses of the GEO600 gravitational wave detector (mirror at bottom right) together with reaction and suspension point masses. Changes in the 600 m distance between the test-masses are now measured with new quantum-squeezed laser light (image by courtesy of Harald Lück, AEI).

In the past, its measurement sensitivity at frequencies above several hundred hertz has been limited by the vacuum (zero-point) fluctuations of the electromagnetic field. Now, GEO600 incorporates an additional laser – a squeezed light laser – which has previously been presented in Ref. [4]. This new laser operates below its laser threshold and is based on parametric down-conversion of 532nm light thereby producing quantum correlated quasi-monochromatic photon-pairs at 1064 nm.































Fig. 2: View into the GEO600 central building. The squeezed-light laser is shown in the front. Its optical table is surrounded by several vacuum chambers containing suspended interferometer optics such as the mirror shown in Fig.1.

This rather dim laser mode is matched into the GEO600 laser interferometer where it interferes with the observatory’s ordinary 3 kW laser beam. As a result, the vacuum fluctuations at the photo-diode in GEO600’s output port are reduced (“squeezed”). GEO600 now operates with its best ever sensitivity being 50% higher than before at signal frequencies above 1 kHz, as shown in Fig. 3. Our success has finally proven the qualification of squeezed light as a key technology for future GW astronomy.





















 
Fig. 3: Nonclassical reduction of the GEO600 instrumental noise (calibrated to a space strain) using squeezed vacuum states of light. The black trace shows the observatory noise spectral density without the injection of squeezed light. An injection of squeezed vacuum states into the interferometer leads to a broadband noise reduction of up to 50% (3.5 dB in power, red trace). The peaks are not due to gravitational waves. They appear at well-known frequencies and are mainly due to violin modes of the mirror’s pendulum suspensions.

During the past years GEO600 has been made one of the most sensitive measuring devices ever built. Up to one hundred scientists from Germany, UK and other countries have contributed [5]. All of them are also members of the LIGO Scientific Collaboration (LSC) [6]. A number of new technologies have been developed, some of which are by now also implemented in the other gravitational wave observatories, namely the US LIGO and the Italian/French Virgo project. In this course of steady improvements, “classical” techniques have been driven to its extremes and GEO600 eventually became so sensitive that the squeezing technology became worth the effort.

Squeezed light has been generated in several research laboratories in the world before, however, it is a rather involved technique and leaving the conditions of a laboratory to an environment of continuous operation is difficult. All these aspects explain why only now the squeezed light is used for the first time in a GW observatory. In the past 5 years about a dozen of physicists have been working on the squeezed laser development in Hannover to enable this leap [7]. A recent review article summarizes the progress on squeezed light generation over the past years [8].

We are convinced that squeezed light will be used in all GW observatories around the globe in near future. The squeezing technology is certainly not exhausted yet. We believe that improvements of up to 200% are feasible with current technology.

References
[1] The LIGO Scientific Collaboration, "A gravitational wave observatory operating beyond the quantum shot-noise limit", Nature Physics, doi:10.1038/nphys2083 (Published online September 11, 2011). Abstract. Free Download.
[2] C. M. Caves, "Quantum-mechanical noise in an interferometer". Phys. Rev. D 23, 1693 (1981). Abstract.
[3] Willke, B. et al., "The GEO 600 gravitational wave detector", Class. Quantum Grav. 19, 1377 (2002). Abstract; Grote, H. et al., "The GEO 600 status", Class. Quantum Grav. 27, 084003 (2010). Abstract.
[4] H. Vahlbruch, A. Khalaidovski, N. Lastzka, C. Gräf, K. Danzmann, R. Schnabel, "The GEO600 squeezed light source", Class. Quantum Grav. 27, 084027 (2010). Article.
[5] The GEO600 team: www.geo600.org
[6] The LIGO scientific collaboration: www.ligo.org
[7] The Quantum Interferometry Group at the AEI: www.qi.aei-hannover.de
[8] R. Schnabel, N. Mavalvala, D. E. McClelland, P. K. Lam, "Quantum metrology for gravitational wave astronomy", Nature Communications, 1:121, doi: 10.1038/ncomms1122 (2010). Abstract.

Exploring Macroscopic Quantum Mechanics with Gravitational-wave Detectors

Haixing Miao

[Haixing Miao is the recipient of the 2010 GWIC (Gravitational Wave International Committee) Thesis Prize for his PhD thesis “Exploring Macroscopic Quantum Mechanics in Optomechanical Devices" (PDF). -- 2Physics.com]

Author: Haixing Miao

Affiliation:
Australian International Gravitational Research Centre (AIGRC), University of Western Australia, Perth, Australia;
Theoretical AstroPhysics Including Relativity (TAPIR), California Institute of Technology, Pasadena, USA.

Do macroscopic objects have wavy behaviors predicted by quantum mechanics, the same as microscopic atoms? Is there any boundary or transition between the quantum world and the classical world which we experience daily? Interestingly, advanced laser interferometer gravitational-wave (GW) detectors may give answers to these fundamental questions.

Fig. 1 LIGO detector at Livingston (left). It consists of a Michelson interferometer which uses a highpower laser to measure differential motions of input test masses (ITM) and end test masses (ETM) caused by gravitational waves.

It might seem unlikely, at least from the first sight, that a GW detector (e.g., LIGO detector [1] shown in Fig. 1) can study something quantum, given its apparent classical features: (i) using a high-power laser and kilogram-scale mirrors as test masses, (ii) having these test masses widely separated by kilometers, and (iii) operating at the room temperature. How can we probe delicate quantum mechanics with such a giant? The answer lies in the fact that, to detect weak GWs from the distant universe, the GW detector has to be extremely sensitive to the tiny displacement of the kilogram test mass, even sensitive enough to probe the quantum zero-point motion of macroscopic test masses.

2Physics articles by past winners of the GWIC Thesis Prize:
Holger J. Pletsch (2009): "Deepest All-Sky Surveys for Continuous Gravitational Waves"
Henning Vahlbruch (2008): "Squeezed Light – the first real application starts now"
Keisuke Goda (2007): "Beating the Quantum Limit in Gravitational Wave Detectors"
Yoichi Aso (2006): "Novel Low-Frequency Vibration Isolation Technique for Interferometric Gravitational Wave Detectors"
Rana Adhikari (2003-5)*: "Interferometric Detection of Gravitational Waves : 5 Needed Breakthroughs"
*Note, the gravitational wave thesis prize was started initially by LIGO as a biannual prize, limited to students of the LIGO Scientific Collaboration (LSC). The first award covered the period from 1 July 2003 to 30 June 2005. In 2006, the thesis prize was adopted by GWIC, renamed, converted to an annual prize, and opened to the broader international community.

Fig. 2 Plot showing the sensitivity of LIGO Hanford detector compared with the standard quantum limit (SQL) -- a benchmark for quantumness. This figure is adopted from Ref.[2], which reports cooling of kilogram test masses down to an effective temperature of 1.4μK (Experiment is led by Nergis Mavalvala and Thomas Corbitt from MIT).

Indeed, with state-of-the-art technology, initial LIGO detector is only a factor of 10 away from the Standard Quantum Limit (SQL) that is imposed by the Heisenberg Uncertainty Principle [3] (illustrated in Fig. 2). Currently, in the GW community, significant efforts have been put into improving the detector sensitivity by reducing the classical thermal noises that cause random jittering of test masses. The future AdvLIGO [4] and other advanced GW detectors [5] under construction are anticipated to be operating at or beyond the SQL, with their sensitivities limited by noises that have purely quantum origin. To further increase the detector sensitivity, we need to manipulate the light at the quantum level, e.g. the use of quantum squeezed light [6-7]. Advanced GW detectors can be viewed as quantum devices, regardless of their bulky appearance.

Fig. 3 Figure showing schematically the creation of quantum superposition of macroscopic test masses by coherently amplifying the momentum of a single photon with advanced GW detectors. Please refer to Ref. [10] for more details of the experimental protocol.

With a sequence of studies [8-11], it is shown that, by using appropriate protocols, advanced GW detectors allow us to prepare kilogram test masses in different quantum states, and to study their quantum dynamics. For example, by superimposing a single photon―the light quantum―onto a strong light field in the GW detector, the momentum of the photon can be coherently amplified, and can even place the macroscopic test masses into a quantum superposition, as depicted schematically in Fig. 3. The GW detector, in some sense, acts as a “quantum amplifier”, and brings the quantumness of the microscopic photon into the macroscopic world.

Besides, if we simultaneously measure the common and differential motions of test masses, we can create Einstein-Podolsky-Rosen type quantum entanglement among widely separated test masses [9]. Furthermore, we can study the dynamics of such a macroscopic quantum entanglement, which allows us to explore some interesting decoherence effects that could be unique to macroscopic objects [12].

Future advanced GW detectors can, therefore, not only detect tiny ripples in the spacetime and open up a new window into observing our universe, but also help us to gain deeper understanding of quantum behaviors of macroscopic objects, which might reveal exciting new phenomena.

References:
[1] LIGO website: ligo.caltech.edu and a recent review article by the LIGO Scientific Collaboration (LSC), “LIGO: the Laser Interferometer Gravitational-wave Observatory”, Rep. Prog. Phys. 72, 076901 (2009). Abstract.
[2] LIGO Scientific Collaboration (LSC), “Observation of a kilogram-scale oscillator near its quantum ground state”, New J. Phys. 11 073032 (2009). Abstract.
[3] V. B. Braginsky and F. Y. Khalili, "Quantum Measurement", publisher: Cambridge
University Press (1992).
[4] Advanced LIGO website: advancedligo.mit.edu
[5] Advanced VIRGO website: cascina.virgo.infn.it/advirgo ; Large-scale Cryogenic Gravitational wave Telescope (LCGT) website: gw.icrr.u-tokyo.ac.jp/lcgt/
[6] H. Vahlbruch, M. Mehmet, N. Lastzka, B. Hage, S. Chelkowski, A. Franzen, S. Gossler, K. Danzmann, and R. Schnabel, “Observation of squeezed light with 10dB quantum noise reduction”, Phys. Rev. Lett. 100, 033602 (2008). Abstract.
[7] K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, Nature Physics 4, 472 (2008). Abstract.
[8] Helge Müller-Ebhardt, Henning Rehbein, Chao Li, Yasushi Mino, Kentaro Somiya, Roman Schnabel, Karsten Danzmann, and Yanbei Chen, “Quantum-state preparation and macroscopic entanglement in gravitational-wave detectors”, Phys. Rev. A 80, 043802 (2009). Abstract.
[9] Helge Müller-Ebhardt, Henning Rehbein, Roman Schnabel, Karsten Danzmann, and Yanbei Chen, “Entanglement of Macroscopic Test Masses and the Standard Quantum Limit in Laser Interferometry”, Phys. Rev. Lett. 100, 013601 (2008). Abstract.
[10] Farid Ya. Khalili, Stefan Danilishin, Haixing Miao, Helge Müller-Ebhardt, Huan Yang, and Yanbei Chen, “Preparing a Mechanical Oscillator in Non- Gaussian Quantum States”, Phys. Rev. Lett. 105, 070403 (2010). Abstract.
[11] Haixing Miao, Stefan Danilishin, Helge Müller-Ebhardt, Henning Rehbein, Kentaro Somiya, and Yanbei Chen, “Probing macroscopic quantum states with a sub-Heisenberg accuracy”, Phys. Rev. A 81, 012114 (2010). Abstract.
[12] Lajos Diósi, “A universal master equation for the gravitational violation of the quantum mechanics”, Phys. Lett. A 120, 377 (1987). Abstract ; Roger Penrose, The Road to Reality: A Complete Guide to the Laws of the Universe, Publisher: Alfred A. Knopf (2005).

Deepest All-Sky Surveys for Continuous Gravitational Waves

Holger J. Pletsch

[This is an invited article from Dr. Holger J. Pletsch who is the recipient of the 2009 GWIC (Gravitational Wave International Committee) Thesis Prize for his PhD thesis “Data Analysis for Continuous Gravitational Waves: Deepest All-Sky Surveys” (PDF). The thesis also received the 2009 Dieter Rampacher Prize of the Max Planck Society in Germany -- awarded to its youngest Ph.D. candidates usually between the ages of 25 and 27 for their outstanding doctoral work. -- 2Physics.com]

Author: Holger J. Pletsch
Affiliation: Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut) and Leibniz Universität Hannover

Besides validating Einstein's theory of General Relativity, direct detection of gravitational waves will also constitute an important new astronomical tool. Prime target sources of continuous gravitational waves (CW) for current Earth-based laser-interferometric detectors as LIGO [1] are rapidly spinning compact objects, such as neutron stars, with nonaxisymmetric deformations [2].

Very promising searches are all-sky surveys for prior unknown CW emitters. As most neutron stars are electromagnetically invisible, gravitational-wave observations might allow to reveal completely new populations of neutron stars. Therefore, a CW detection could potentially be extremely helpful for neutron-star astrophysics. Even the null results of today's search efforts, yielding observational upper limits [3], already constrain the physics of neutron stars.

2Physics articles by past winners of the GWIC Thesis Prize:
Henning Vahlbruch (2008): "Squeezed Light – the first real application starts now"
Keisuke Goda (2007): "Beating the Quantum Limit in Gravitational Wave Detectors"
Yoichi Aso (2006): "Novel Low-Frequency Vibration Isolation Technique for Interferometric Gravitational Wave Detectors"
Rana Adhikari (2003-5)*: "Interferometric Detection of Gravitational Waves : 5 Needed Breakthroughs"
*Note, the gravitational wave thesis prize was started initially by LIGO as a biannual prize, limited to students of the LIGO Scientific Collaboration (LSC). The first award covered the period from 1 July 2003 to 30 June 2005. In 2006, the thesis prize was adopted by GWIC, renamed, converted to an annual prize, and opened to the broader international community.

The expected CW signals are extremely weak, and deeply buried in the detector instrument noise. Thus, to extract these signals sensitive data analysis methods are requisite. A powerful method is coherent matched filtering, where the signal-to-noise ratio (SNR) increases with the square root of observation time. Hence, detection is a matter of observing long enough, to accumulate sufficient SNR.

The CW data analysis is further complicated by the fact that the terrestrial detector location Doppler-modulates the amplitude and phase of the waveform, as the Earth moves relative to the solar system barycenter (SSB). The parameters describing the signal's amplitude variation may be analytically eliminated by maximizing the coherent matched-filtering statistic. The remaining search parameters describing the signal's phase are the source's sky location, frequency and frequency derivatives. The resulting coherent detection statistic is commonly called the F-statistic [4].

However, what ultimately limits the sensitivity of all-sky surveys for unknown CW sources using the F-statistic is the finite computing power available. Such searches are computationally very expensive, because for maximum sensitivity one must convolve the full data set with many signal waveforms (templates) corresponding to all possible sources. But the number of templates required for a fully coherent F-statistic search increases as a high power of the coherent observation time. For a year of data, the computational cost to search a realistic range of parameter space exceeds the total computing power on Earth [4,5]. Thus a fully coherent search is limited to much shorter observation times.

Searching year-long data sets is accomplished by less costly hierarchical, so-called “semicoherent” methods [6,7]. The data is broken into segments, which are much smaller than one year. Just every segment is analyzed coherently, computing the F-statistic on a coarse grid of templates. Then the F-statistics from all segments (or statistics derived from F) are incoherently combined using a common fine grid of templates. This way phase information between segments is discarded, hence the term “semicoherent”.

A central long-standing problem in these semicoherent methods was the design of, and link between, the coarse and fine grids. Previous methods, while creative and clever, were arbitrary and ad hoc constructions. In most recent work [8], the optimal solution for the incoherent combination step has been found. The key quantity is the fractional loss, called mismatch, in expected F-statistic for a given signal at a nearby grid point. Locally Taylor-expanding the mismatch (to quadratic order) in the differences of the coordinates defines a positive definite metric. Previous methods considered parameter correlations in the F-statistic to only linear order in coherent integration time, discarding higher orders from the metric.

The F-statistic has strong "global" (large-scale) correlations in the physical coordinates, extending outside a region in which the mismatch is well approximated by the metric. In recent work [9], an improved understanding of the large-scale correlations in the F-statistic was found. Particularly, for realistic segment durations (a day or longer) it turned out to be also crucial to consider the fractional loss in F to second order in coherent integration time.

Exploiting these large-scale correlations in the coherent detection statistic F has lead to a significantly improved semicoherent search technique for CW signals [8]. This novel method is optimal if the semicoherent detection statistic is taken to be the sum of one coarse-grid F-statistic value from each data segment.

More precisely, the improved understanding of large-scale correlations yields new coordinates on the phase parameter space. In these coordinates the first fully analytical metric for the incoherent combination step is obtained, accurately approximating the mismatch. Hence, the optimal (closest) coarse-grid point from each segment can be determined for any given fine-grid point in the incoherent combination step. So the new method combines the coherent segment results much more efficiently, since previous techniques did not use metric information beyond linear order in coherent integration time.

Fig.1: The Einstein@Home screensaver

The primary application area of this new technique is the volunteer distributed computing project Einstein@Home [10]. Members of the public can sign up their home or office computers (hosts) through the web page, and download a screensaver. When idle, the screensaver displays (Fig.1), and in the background hosts automatically download small chunks of data from the servers, carry out analysis, and report back results. While more than 250k individuals have already contributed, the computational power (0.25 PFlop/s) achieved is in fact competitive with the world's largest supercomputers.

What improvement to expect from the new search technique for Einstein@Home? Via Monte Carlo simulations an implementation of this new method has been compared to the conventional Hough transform technique [7] that has been previously used on Einstein@Home. To provide a realistic comparison, simulated data covered the same time intervals as the input data of a recent Einstein@Home search run, which employed the conventional Hough technique. Those data, from LIGO’s fifth science run (S5), included 121 data segments of 25-hour duration. The false alarm probabilities were obtained using many simulated data sets with different realizations of stationary Gaussian white noise. To find the detection probabilities, different CW signals with fixed source gravitational-wave amplitude were added. Other source parameters were randomly drawn from uniform distributions.














Fig.2: (right click on the image to see higher resolution version) Performance demonstration of the new search method. Left panel: Receiver operating characteristic curves for fixed source strain amplitude. Right panel: Detection probability as a function of source strain amplitude, at 1% false alarm probability.

The results of this comparison are illustrated in Fig. 2. The right panel of Fig. 2 shows the detection efficiencies for different values of source gravitational-wave amplitude (strain), at a fixed 1% false alarm probability. The new method has been applied in two modes of operation: first, F-statistics were simply summed across segments; second, only ones or zeros were summed (number counts) depending upon whether F exceeds a predefined threshold in a given segment. In both modes of operation, the new technique performs significantly better than the conventional Hough method. For instance, 90% detection probability with the new method (in number-count operation mode) is obtained for a value of source strain amplitude about 6 times smaller as needed by the Hough method (which is also based number counts): thus the "distance reach" of the new technique is about 6 times larger. This increases the number of potentially detectable sources by more than 2 orders of magnitude, since the "visible" spatial volume increases as the cube of the distance, as illustrated in Fig. 3.

Fig.3: Artist’s illustration of increased "visible" spatial volume due to the novel search technique.

The current Einstein@Home search run [10] in fact deploys this new technique for the first time, analyzing about two years of LIGO’s most sensitive S5 data. The combination of a better search technique, plus more and more sensitive data, greatly increases the chance of making the first gravitational wave detection of a CW source. In the long term, the detection of CW signals will provide new means to discover and locate neutron stars, and will eventually provide unique insights into the nature of matter at high densities.

References:
[1] B. Abbott et al. (LIGO Scientific Collaboration), "LIGO : the Laser Interferometer Gravitational-wave Observatory", Rep. Prog. Phys. 72, 076901 (2009), Abstract.
[2] R. Prix (for the LIGO Scientific Collaboration), in “Neutron Stars and Pulsars”, Springer, (2009).
[3] B. Abbott et al. (LIGO Scientific Collaboration), "Beating the spin-down limit on gravitational wave emission from the Crab pulsar", Astrophys. J. Lett. 683, L45 (2008), Abstract; B. Abbott et al. (LIGO Scientific Collaboration), "All-Sky LIGO Search for Periodic Gravitational Waves in the Early Fifth-Science-Run Data", Phys. Rev. Lett. 102, 111102 (2009), Abstract; B. Abbott et al. (LIGO Scientific Collaboration), "Einstein@Home search for periodic gravitational waves in early S5 LIGO data", Phys. Rev. D 80, 042003 (2009), Abstract.
[4] P. Jaranowski, A. Królak and B. F. Schutz, "Data analysis of gravitational-wave signals from spinning neutron stars: The signal and its detection", Phys. Rev. D 58, 063001 (1998), Abstract; P. Jaranowski and A. Królak, "Gravitational-Wave Data Analysis. Formalism and Sample Applications: The Gaussian Case", Living Reviews in Relativity, 8 (2005), Link.
[5] P. R. Brady, T. Creighton, C. Cutler and B. F. Schutz, "Searching for periodic sources with LIGO", Phys. Rev. D 57, 2101 (1998), Abstract.
[6] P. R. Brady and T. Creighton, "Searching for periodic sources with LIGO. II. Hierarchical searches", Phys. Rev. D 61, 082001 (2000), Abstract.
[7] B. Krishnan, A. M. Sintes, M. A. Papa, B. F. Schutz, S. Frasca and C. Palomba, "Hough transform search for continuous gravitational waves", Phys. Rev. D 70, 082001 (2004), Abstract.
[8] H. J. Pletsch and B. Allen, "Exploiting Large-Scale Correlations to Detect Continuous Gravitational Waves", Phys. Rev. Lett. 103, 181102 (2009), Abstract.
[9] H. J. Pletsch, "Parameter-space correlations of the optimal statistic for continuous gravitational-wave detection", Phys. Rev. D 78, 102005 (2008), Abstract.
[10] Einstein@Home: http://einstein.phys.uwm.edu/.