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2Physics

2Physics Quote:
"Taking advantage of the ambipolarity, one could think of electrostatically defining a PN junction in a few-layer black phosphorus flake, as already pioneered in single-layer tungsten diselenide (WSe2). A PN junction could be used to boost the photoresponse and generate electrical power via the photovoltaic effect. Given the small and direct bandgap of few-layer black-phosphorus, it would be possible to harvest photons also in the near-infrared part of the spectrum." -- Michele Buscema, Dirk J. Groenendijk, Sofya I. Blanter, Gary A. Steele, Herre S.J. van der Zant, Andres Castellanos-Gomez
(Read Full Article: "Few-layer Black Phosphorus Phototransistors for Fast and Broadband Photodetection" )

Sunday, August 31, 2014

A True Randomness Generator Exploiting a Very Long and Turbulent Path

From Left to Right: Paolo Villoresi,  Davide Marangon, Giuseppe Vallone

Authors:
Davide G. Marangon, Giuseppe Vallone,  Paolo Villoresi

Affiliation:
Department of Information Engineering, University of Padova, Italy.

Random numbers are the main ingredients of cryptographic protocols for both Classical and Quantum Information. However, it is well known that to rely on random numbers produced with deterministic algorithms can be very risky and it is of fundamental priority to discover physical processes to generate "pure" random numbers. Usually True Random Number Generators (TRNG) are implemented by exploiting classical or quantum microscopical processes. However it can be shown that random numbers can be extracted from macroscopic physical systems.


In the 60s, the famous "Butterfly Effect" captured the idea that when one deals with the terrestrial atmosphere, very tiny perturbations such as the air moved by the tail strokes of a butterfly can lead to very huge consequences as a hurricane in some other place in the world. Terrestrial atmosphere indeed may be seen as a physical system ruled by a chaotic dynamic. Moreover, while statistical models are available for average trends, the prediction of the instantaneous motion of the air mass in a spot is out of reach.

From the textbooks we know that the propagation of light through an inhomogeneous medium is strongly influenced by the refractive index distribution. We experimentally investigated this phenomenon with the purpose of realizing if such propagation along a free-space path may induce a useful randomness. An intuition of such effect manifested during the campaigns for the experiments we carried out at the Canarias on the quantum Communications along extremely long links [1, 2]. The atmospheric turbulence in the path is very strong, preventing for example the direct application of interferometry [3]. However, the effect of turbulence is crucial there for the application of a method that exploits brief moments of high transmissivity for good communication [2]. We tried to turn it here instead into a useful resource for randomness.
FIG. 1. The experiment was set up between the islands of La Palma and Tenerife where a laser beam (with λ = 810 nm) was exchanged between the two islands. After propagating across a 143 km Free Space Optical link, the wavefront of the beam features a randomly composed speckle pattern as consequence of the distortions induced by the atmosphere.

The experiment we describe here was set between the two islands of La Palma and Tenerife: on the rooftop of the Jacobus Kaptein Telescope (JKT) building at an altitude of 2360 m; our transmitting telescope for the Quantum Communication was aimed to send a continuous laser beam towards the ESA Optical Ground Station (OGS) 143 km far away, at Izana, near the mount Teide, see Figure 1. The telescope -- that was designed and realized in Padova -- is a refractor based on a 230 mm aspheric singlet. In the path, the turbulent atmosphere is comparable to a dynamic volumetric scatterer and the electromagnetic field is subjected to phase delays and amplitude fluctuations, induced by the inhomogeneities of the refractive index of the air [4–6]. The receiver then observes a beam profile which does not feature the typical intensity Gaussian distribution, rather a collection of clear and dark spots of irregular shape, the so-called speckle pattern. The speckle pattern evolves according the unpredictable dynamic of the turbulence as consequence of the random walks the electromagnetic field suffers while propagating. Therefore at the receiving plane a continuously and randomly evolving distribution of speckles was acquired with a CMOS camera and for every frame one has a variable number of spots randomly taking different spatial configurations [7].

Randomness is then extracted by using the geometrical complexity of the frames evaluating the centers of mass, the so-called centroids, of those speckle areas with the same intensity. For the implementation of the method the relevant pixels in CCD are labelled sequentially with an index s, s ∈ {1, . . . , N}, the nf  speckle centroids of the frame f are elaborated, an ordered sequence Sf = {s, s, . . . , snf } with s1 < s2 < · · · < snf  is formed, by considering then the pixels where a centroid falls in. The pixel grid can be regarded as the classical collection of urns where the turbulence randomly throws balls (the centroids) in, see Figure 2. Because of the random nature of the process, the centroids visit every part of the grid with the same probability. A given frame f  “freezes” one Sf  out of the
possible and equally likely sequences of nf centroids. Among all of combinations, a given Sf can be univocally identified with its lexicographic index I (Sf )
with 0 ≤ I (Sf ) ≤ Tf  - 1. Basically, (2) enumerates all the possible arrangements which succeed a given centroids configuration. As an uniform RNG is supposed to yield numbers identically and independently distributed (i.i.d.) in a range [X,Y ], as this method generates a random integer in the range [0, Tf - 1]. In order then to optimize the conversion from integer to random bits without introducing any bias, an efficient algorithmic procedure was applied to the bits [8].
FIG. 2. (click on the figure to view higher resolution) The figure represents a scheme of the mechanism employed to extract randomness from the frames of the captured video. Every frame features a different spatial disposition of centroids (the yellow crosses). To every centroid configuration, a univocal lexicographic index is associated. The lexicographic index then is converted in random bits.

In this proof of principle, a generation rate of 400 kbit/s was achieved but it can be easily enhanced by using cameras with higher resolutions. Another point, worth to be stressed, is that this method does not rely on sensitive and hardly detectable processes which require extremely tuned hardware: indeed unavoidable hardware non-idealities can induce bits dependencies and bias. In addition, from the theoretical point of view, the strength of the method lies in the fact the dynamic of turbulent atmosphere on such a long link represents a physical process which is practically impossible to be predicted, both analytically (at the present time only statical models are given) and numerically (it would require an unbounded computational power).

In addition to a sound knowledge of the physical process employed, it is necessary to apply statistical tests in order to exclude the presence of defects caused by a faulty hardware. This has been done by applying the most stringent test batteries for randomness such as the Alphabit and Rabbit batteries belonging to the TESTU01, the NIST SP-800-22 suite and the AIS31 suite. All the tests were successfully passed.

The presented procedure then could be an efficient method to generate random numbers to be employed in long range QC setups. More in detail, bits generated in this way could be used in connection with other protocols involving Quantum Random Number Generator: for example in the first well known experiment of randomness expansion by means of non-locality [9] the initial seed was obtained by mixing numbers obtained with several generators including atmospheric radio electromagnetic noise. Finally, the extraction algorithm can be easily adapted to other paradigms involving spatial random complex patterns.

References:
[1] Ivan Capraro, Andrea Tomaello, Alberto Dall’Arche, Francesca Gerlin, Ruper Ursin, Giuseppe Vallone, Paolo Villoresi, "Impact of turbulence in long range quantum and classical communication". Physical Review Letters, 109, 200502, (2012). Abstract.
[2] Giuseppe Vallone, Davide Marangon, Matteo Canale, Ilaria Savorgnan, Davide Bacco, Mauro Barbieri, Simon Calimani, Cesare Barbieri, Nicola Laurenti, Paolo Villoresi, "Turbulence as a Resource for Quantum Key Distribution in Long Distance Free-Space Links". arXiv:1404.1272 [quant-ph] (2014).
[3] Cristian Bonato, Alexander V. Sergienko, Bahaa E. A. Saleh, Stefano Bonora, Paolo Villoresi, "Even-Order Aberration Cancellation in Quantum Interferometry". Physical Review Letters, 101, 233603 (2008). Abstract.
[4] Larry C. Andrews and Ronald L. Phillips, "Laser beam propagation through random media", volume 152 (SPIE press, 2005). 
[5] R. L. Fante, "Electromagnetic beam propagation in turbulent media". Proceedings of the IEEE, 63, 1669,(1975). Abstract.
[6] R. L. Fante, "Electromagnetic beam propagation in turbulent media - An update". Proceedings of the IEEE, 68, 1424 (1980). Abstract.
[7] Davide G. Marangon, Giuseppe Vallone, Paolo Villoresi, "Random bits, true and unbiased, from atmospheric turbulence". Scientific Reports, 4 : 5490 (2014). Full Article.
[8] Peter Elias. "The efficient construction of an unbiased random sequence". Annals of Mathematical Statistics, 43, 865 (1972). Full Article.
[9] S. Pironio, A. Acín, S. Massar, A. Boyer de la Giroday, D.N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T.A. Manning,  C. Monroe. "Random numbers certified by Bell’s theorem". Nature, 464, 1021 (2010). Abstract.

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Sunday, August 24, 2014

Quantum Information in the Service of Metrology

The spin-spin measurement team at Weizmann Institute of Science, Israel. From left-to-right: Nir Navon, Nitzan Akerman, Roee Ozeri, Shlomi Kotler and Yinnon Glickman

Authors: Roee Ozeri1, Shlomi Kotler1,2

Affiliation:
1Department of Physics of Complex Systems, Weizmann Institute of Science, Israel.
2Current address: Physical measurement Laboratory, National Institute of Standards and Technology, Boulder, USA.

Quantum systems have been extensively studied in the past few years as possible ultra-efficient computers. Such systems have to be as isolated as possible from their environment in order to prevent it from “measuring” the state of the quantum computer, a process which would render the computer classical. Candidate systems that were experimentally studied thus had to be sufficiently isolated from their environment while allowing for a high degree of controllability.

In addition, active methods were developed, in which special quantum states and control techniques were engineered and helped mitigate the effect of noise. Many such techniques, such as dynamic-decoupling or decoherence-free subspaces, were experimentally implemented with great success, increasing the coherence times of quantum systems by many orders of magnitude. Other methods, such as Quantum error-correction codes, have had proof-of principle demonstrations but hold the promise of being able to reject noise in a way where quantum super-positions of very large systems will be maintained coherent for as long as necessary – one of the key requirement from a quantum computer.

A different quantum technology which has seen great progress in recent years is that of quantum metrology. At face value, quantum sensors seem to be exactly antipodal to quantum computers. Here, quantum systems do have to couple to their environment in order to sense some aspect of it. However, as in many seemingly contradicting concepts there can also be a lot of common ground. For example, environmental noise (i.e. that part of the environment which you don’t want to measure!) is a foe of both quantum sensors and quantum computers. Can we therefore use the techniques that were developed to help quantum computers overcome the harmful effects of noise to improve on the measurement precision of quantum sensors? The answer is yes, and with great success! Along these lines, different dynamic modulations schemes, originally used in dynamic decoupling were used for the measurement of alternating signals. This way quantum lock-in amplifiers as well as quantum noise spectrum analyzers were demonstrated. In a recent experiment in our lab at the Weizmann Institute of Science, we used the powerful technique of decoherence-free subspaces in order to measure the very weak magnetic interaction between two electrons that were separated by more than two microns [1].

Figure 1: An artist impression of the spin-spin experiment. Two electrons are placed two microns away from each other. The magnetic field emanating from one electron interacts with the spin of the other electron, resulting in a change of the spin-orientations.

Electrons, like many fundamental particles have an intrinsic magnetic dipole moment which is aligned with their spin. These tiny magnets have a magnetic field that decays as the cube of the distance from the electron. To illustrate, the magnetic field of a single electron two microns away from it, is as small as the earths’ magnetic field at 10 times the distance to the moon. When two electrons feel each other’s magnetic fields their spins interact as magnets do: their opposite poles will attract, their identical poles repel in a way that torques will be applied and the two spins will respectively rotate due to this interaction.

The magnetic interaction between two electrons was never directly observed before. At short atomic distances, such as that between the two electrons of a Helium atom, the magnetic interaction is large enough to be easily measured. Unfortunately, at these distances, it is overwhelmed by the much larger exchange interaction between them which is the result of the interplay between the strong Coulomb interaction between the electron charges and Fermi’s exclusion principle. At large distances, where the exchange interaction is negligible, the magnetic interaction between the electrons is also very small. At a distance of two microns for example, the rotation rate the two spins impose on each other is on the order of one rotation every four minutes. This interaction is way too small to be measured due to typical magnetic noise in labs.
Figure 2: An image of the trap in which the ions were trapped for the duration of the measurement. The image is taken through one of the ultra-high vacuum chamber view-ports.

This is where techniques, borrowed from quantum computing science come to the rescue. We have placed the spins of two trapped Sr+ ions in a decoherence-free subspace that was completely immune to the effect of magnetic field noise. While being immune to noise, this subspace still allowed for the slow and gentle two-spin correlated dance to be performed without interference. Under the protection provided by this technique we could allow the electronic spins to rotate coherently for 15 seconds, after which we measured their collective rotation of more than 20. We also changed the distance between the electrons and verified that the interaction between them varies inverse cubical with their separation. Thus, almost a 100 years after the discovery of the electronic spin, we were able to cleanly observe the interaction between two such tiny magnets.

This measurement bears importance that reaches beyond its demonstrative nature. This is because some hypothetical anomalous spin forces are speculated to modify the interaction between electronic spins at large distances. The motivation for the introduction of these anomalous forces is due to their ability to explain the weakness with which certain symmetries are broken in nature. The experimental bound on the strength and range of these hypothetical fields is therefore important.

The use of quantum error-suppression schemes for the benefit of precision measurements is a fast developing area of research. With the advent of experimental quantum error-correction codes, another opportunity will emerge to apply these codes towards the detection of small and highly correlated signals [2-5].

References:
[1] Shlomi Kotler, Nitzan Akerman, Nir Navon, Yinnon Glickman, Roee Ozeri, "Measurement of the magnetic interaction between two bound electrons of two separate ions". Nature, 510, 376 (2014). Abstract.
[2] Roee Ozeri, "Heisenberg limited metrology using Quantum Error-Correction Codes". arXiv:1310.3432 [quant-ph].
[3] G. Arrad, Y. Vinkler, D. Aharonov, A. Retzker, "Increasing Sensing Resolution with Error Correction". Physical Review Letters, 112, 150801 (2014). Abstract.
[4] E. M. Kessler, I. Lovchinsky, A. O. Sushkov, M. D. Lukin, "Quantum Error Correction for Metrology". Physical Review Letters, 112, 150802 (2014). Abstract.
[5] W. Dür, M. Skotiniotis, F. Fröwis, B. Kraus, "Improved Quantum Metrology Using Quantum Error Correction". Physical Review Letters, 112, 080801 (2014). Abstract.

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Sunday, August 17, 2014

Comparing Matter Waves in Free Fall

[From Left to Right] J. Hartwig, D. Schlippert, E. M. Rasel

Authors: J. Hartwig, D. Schlippert, E. M. Rasel

Affiliation: Institut für Quantenoptik and Centre for Quantum Engineering and Space-Time Research (QUEST), Leibniz Universität Hannover, Germany

Introduction to Einstein’s Equivalence Principle

Einstein’s general relativity is based on three fundamental building blocks: local Lorentz invariance, the universality of the gravitational redshift and the universality of free fall. The enormous importance of general relativity in modern science and technology merits a continuous effort in improving experimental verification of these underlying principles.

The universality of free fall is one of the oldest mechanical theories originally proposed by Galileo. Testing can be done by so called free fall experiments, where two bodies with different composition are freely falling towards a third gravitating body.
Figure 1: Goddard Spaceflight Center Laser Ranging Facility. Source: NASA

Amongst the most sensitive measurements of this principle is the Lunar Laser Ranging experiment, which compares the free fall of earth and moon in the solar gravitational potential [1]. This measurement is only surpassed by torsion balance experiments based on the design of Eötvös [2]. In addition, exciting new insights are expected from the MICROSCOPE experiment [3] that is planned to launch in 2016.

Figure 2: Torsion balance experiment as used in the group of E. Adelberger, University of Washington. Source: Eöt-Wash-Group

The emergence of quantum physics and our improved understanding of the basic building blocks of matter increases the interest scientists have in the understanding of gravity. How do gravity and quantum mechanics interact? What`s the connection between different fundamental particles and their mass? Is there a deeper underlying principle combining our fundamental theories? To comprehensively approach these questions a wide array of parameters must be analyzed. The way how certain test materials may act under the influence of gravity can either be parametrized using a specific violation scenario, like the Dilaton scenario by T. Damour [4], or by using a test theory such as the the extended Standard Model of particles (SME) [5]. Since the SME approach is not based on a specific mechanism of violating UFF it also does not predict a level to which a violation may occur. Instead, it delivers a model-independent approach to compare methodically different measurements and confine possible violation theories.

Table 1 states possible sensitivities for violations based on the SME framework for a variety of test masses and underlines the importance of complementary test mass choices are. Hence in comparison to classical tests, the use of atom interferometry opens up a new field of previously inaccessible test masses with perfect isotopic purity in a well-defined spin state. Quantum tests appear to differ from previous test also in a qualitative way. They allow to perform test with new states of matter, such as wave packets by Bose Einstein condensates being in a superposition state. The work presented here is just another early step in a quest to understand the deeper connections between the quantum and classical relativistic world.
Table 1: Sample violation strengths for different test masses linked to “Neutron excess” and the “total Baryon number” based on the Standard Model Extension formalism. The test mass pairs are chosen according to the best torsion balance experiment [6] and existing matter wave tests [7]. An anomalous acceleration would be proportional to the stated numerical coefficients. Source: D. Schlippert et al., Phys. Rev. Lett. 112, 203002 (2014).

Measuring accelerations with atom interferometry

Measuring accelerations with a free fall experiment is always achieved by tracking the movement of an inertial mass in free fall in comparison to the lab frame of reference. This is even true when the inertial mass in question needs to be described by a matter wave operating on quantum mechanics. Falling corner cube interferometers operating on this principle are among the most accurate measurements of gravity with classical bodies. They use a continuous laser beam to track the change of velocity of a corner cube reflector due to gravity in a Michelson interferometer with the corner cube changing one of the arm lengths. Acceleration sensors based on free falling matter waves use a similar principle.
Figure 3: A side view of the experimental setup with the two-dimensional (left side) and three-dimensional (right side) magneto-optical traps employed in [Phys. Rev. Lett. 112, 203002 (2014)].

The first demonstration of a true quantum test of gravity with matter waves was performed 1975 with neutrons in the Famous COW experiment [8]. We will focus on atom interferometers using alkaline atoms, since they are most commonly used for inertial sensing and are also employed in the discussed experiment. Experiments of this kind were first used for acceleration measurements in 1992 [9] and have improved in their performance ever since. The first test of the equivalence principle comparing two different isotopes was then performed in 2004 [10]. Research on quantum tests is for example proposed at LENS in Italy [11], in Stanford in an already existing large fountain [12] and in the scope of the French ICE mission in a zero-g plane [13]. All these initiatives show the high interest of testing gravity phenomena with quantum matters as opposed to classical tests.

In the case of atom interferometers, coherent beam splitting is performed by absorption and stimulated emission of photons. Which atomic transitions are used is dependent on the specific application but, in the case of alkaline atoms two photon transitions coupling either two hyperfine and respective momentum states (Raman transitions) or just momentum states (Bragg transitions) are employed. The point of reference for the measurement is then given by a mirror reflecting the laser beams used to coherently manipulate the atoms, since the electromagnetic field is vanishing at the mirror surface. This results in a reliable phase reference of the light fields and constitutes the laboratory frame. The role of the retroreflecting mirror is similar to the one of the mirror at rest in in the case of the falling corner cube experiment.

The atomic cloud acts as the test mass, which in an ideal case, is falling freely without any influence by the laboratory, except during interaction with the light fields employed as beam splitters or mirrors. During the interaction, the light fields drive Rabi-oscillations in the atoms between the two interferometer states |g> and |e> with a time 2τ needed for a full oscillation. This allows for the realization of beam splitters with a τ/2 pulse length resulting in an equal superposition of |g> and |e>. Mirrors can be realized the same way by applying the beam splitter light fields for a time of τ which leads to an inversion of the atomic state. These pulses are generally called π/2 (for the superposition) and π (for the inversion pulses) in accordance with the Rabi-oscillation phase. The simplest geometry used to measure acceleration is a Mach- Zehnder-like geometry. This is produced by applying a π/2-π-π/2 sequence with free evolution times T placed between pulses. The resulting geometry can be seen in Figure 4.

Figure 4: Space-time diagram of a Mach-Zehnder-like atom interferometer. An atomic ensemble is brought into a coherent superposition of two momentum states by a stimulated Raman transition (π/2 pulse). The two paths I+II propagate separated, are reflected by a pi-pulse after a time T and superimposed and brought to interference with a final π/2 pulse after time 2T. The phase difference is encoded in the population difference of the two output states.

During the interaction with the light fields, the lattice formed by the two light fields imprints its local phase onto the atoms. This results in an overall phase scaling with the relative movement between the atomic cloud and the lattice. Calculating the overall phase imprinted on the atoms results in first order term, Φ=a*T2*keff, where keff is the effective wave vector of the lattice and a is the relative acceleration between lattice and atoms. This immediately shows the main feature of free fall atom interferometry: the T2 scaling of the resulting phase. This is of particular interest for future experiments aiming for much higher free evolution times than currently possible. The phase Φ also shows another key feature. As the acceleration between atoms and lattice approaches zero, the phase also goes to zero, independently of the interferometry time T. This yields a simple way to determine the absolute acceleration of the atomic sample by accelerating the lattice until the lattice motion is in the same inertial reference frame as the freely falling atoms.

Lattice acceleration is achieved by chirping the frequency difference between the two laser beams used for the two photon transition. This transforms the measurement of a relatively large phase, spanning many thousand radians, to a null measurement. The signal produced is the population difference between the interferometer states |g> and |e> as a function of lattice acceleration and thus frequency sweep rate, α. The sweep rate corresponding to a vanishing phase directly leads to the acceleration experienced by the atoms according to lattice acceleration formula a=α/keff. Taking into account Earth’s gravitational field and a lattice wavelength of 780/2 nm (the factor of ½ is introduced due to the use of a two-photon transition) this leads to a sweep rate of around 25 MHz/s. The advantage of this method is that the acceleration measurement is now directly coupled to measurement of the wavelength of the light fields and frequencies in the microwave regime, which are easily accessible.

Our data
Figure 5: Determination of the differential acceleration of rubidium and potassium. The main systematic bias contributions do not change their sign when changing the direction of momentum transfer. Hence, the mean acceleration of upward and downward momentum transfer direction greatly suppresses the aforementioned biases. Source: D. Schlippert et al., Phys. Rev. Lett. 112, 203002 (2014)

Figure 6:  the wave nature of 87Rb and 39
atoms and their interference are exploited 
to measure the gravitational acceleration.
In order to test the universality of free fall, we simultaneously chirp the Raman frequencies to compensate for the accelerations a(Rb,±)(g) and a(K,±)(g) experienced by rubidium and potassium that were previously identified (Figure 6). Here, the observed phase shift exhibits contributions due to additional perturbations, such as magnetic field gradients. We make use of a measurement protocol based on reversing the transferred momentum (upward and downward directions ±). This technique makes use of the fact that many crucial perturbations do not depend on the direction of momentum transfer. Thus, by computing the half-difference of the phase differences determined in a single momentum direction, phase shifts induced by, e.g., the AC-Stark effect or Zeeman effect, can be strongly suppressed [14].
Figure 7: Allan deviations of the single species interferometer signals and the derived Eötvös ratio. Source: D. Schlippert et al., Phys. Rev. Lett. 112, 203002 (2014)

The data presented in this work [15] was acquired in a data run that was ~4 hours long. By acquiring 10 data points per direction of momentum transfer, and species and then switching to the opposite direction, we were able to determine the Eötvös ratio of rubidium and potassium to a statistical uncertainty of 5.4 x 10-7 after 4096s; the technical noise affecting the potassium interferometer is the dominant noise source.

Taking into account all systematic effects, our measurement yields η(Rb,K)=(0.3 ± 5.4) x 10-7.

Outlook

In our measurement, the performance was limited both by technical noise and the limited free evolution time T. In order to improve these parameters, we are currently extending the free fall time in our experiment. Furthermore, in an attempt to increase the contrast of our interferometers and thus the signal-to-noise ratio, we are working on implementing state preparation schemes for both species.

We expect to constrain our uncertainty budget (which currently is on the 10 ppb level for the Eötvös ratio) on the ppb level and below through the use of a common optical dipole trap applied to both species. By using Bose-Einstein-condensed atoms, we gain the ability to precisely calculate the ensembles, as well as carefully control the input state. This technique will also be able to reduce uncertainty factors linked to the transverse motion of the cloud, in addition to spatial magnetic field and gravitational field gradients.

Improving the precision of a true quantum test into the sub-ppb regime is the focus of current research. For example we are currently planning a 10m very long baseline atom interferometer (VLBAI) in Hannover [16]. In the framework of projects funded by the German Space Agency (DLR), we moreover develop experiments that are suitable for microgravity operation in the ZARM drop tower in Bremen and on sounding rocket missions [17].

Parallel to the development done in the LUH and at a national level, we are also involved in projects on an international level looking into extending the frontier of atom interferometry and especially the test of the equivalence principle. A major project investigating the feasibility of a space borne mission is the STE-Quest Satellite Mission proposed by a European consortium including nearly all major research institutions working in the field of inertial sensing with atom interferometry, as well as a variety of specialist of other fields [18]. This mission is aimed towards doing a simultaneous test of the equivalence principle with two rubidium isotopes and a clock comparison with several ground based optical clocks, pushing the sensitivity to the Eötvös ratio into the 10-15 regime.

References:
[1] James G. Williams, Slava G. Turyshev, Dale H. Boggs, "Progress in Lunar Laser Ranging Tests of Relativistic Gravity". Physical Review Letters, 93, 261101 (2004). Abstract.
[2] S. Schlamminger, K.-Y. Choi, T. A. Wagner, J. H. Gundlach, E. G. Adelberger, "Test of the Equivalence Principle Using a Rotating Torsion Balance". Physical Review Letters, 100, 041101 (2008). Abstract.
[3] P. Touboul, G. Métris, V. Lebat, A Robert, "The MICROSCOPE experiment, ready for the in-orbit test of the equivalence principle". Classical and Quantum Gravity, 29, 184010 (2012). Abstract.
[4] Thibault Damour, "Theoretical aspects of the equivalence principle". Classical Quantum Gravity, 29, 184001 (2012). Abstract.
[5] M.A. Hohensee, H. Müller, R.B. Wiringa, "Equivalence Principle and Bound Kinetic Energy". Physical Review Letters, 111, 151102 (2013). Abstract.
[6] S. Schlamminger, K.-Y. Choi, T.A. Wagner, J.H. Gundlach, E.G. Adelberger, "Test of the Equivalence Principle Using a Rotating Torsion Balance". Physical Review Letters, 100, 041101 (2008). Abstract.
[7] A. Bonnin, N. Zahzam, Y. Bidel, A. Bresson, "Simultaneous dual-species matter-wave accelerometer". Physical Review A, 88, 043615 (2013). Abstract ; S. Fray, C. Alvarez Diez, T. W. Hänsch, M. Weitz, "Atomic Interferometer with Amplitude Gratings of Light and Its Applications to Atom Based Tests of the Equivalence Principle". Physical Review Letters, 93, 240404 (2004). Abstract ; M. G. Tarallo, T. Mazzoni, N. Poli, D. V. Sutyrin, X. Zhang, G. M. Tino, "Test of Einstein Equivalence Principle for 0-Spin and Half-Integer-Spin Atoms: Search for Spin-Gravity Coupling Effects". Physical Review Letters, 113, 023005 (2014). Abstract.
[8] R. Colella, A. W. Overhauser, S. A. Werner, "Observation of Gravitationally Induced Quantum Interference". Physical Review Letters, 34, 1472 (1975). Abstract.
[9] M. Kasevich, S. Chu, "Measurement of the gravitational acceleration of an atom with a light-pulse atom interferometer". Applied Physics B, 54, 321–332 (1992). Abstract.
[10] Sebastian Fray, Cristina Alvarez Diez, Theodor W. Hänsch, Martin Weitz, "Atomic Interferometer with Amplitude Gratings of Light and Its Applications to Atom Based Tests of the Equivalence Principle". Physical Review Letters, 93, 240404 (2004). Abstract.
[11] M. G. Tarallo, T. Mazzoni, N. Poli, D. V. Sutyrin, X. Zhang, G. M. Tino, "Test of Einstein Equivalence Principle for 0-Spin and Half-Integer-Spin Atoms: Search for Spin-Gravity Coupling Effects". Physical Review Letters, 113, 023005 (2014). Abstract.
[12] Susannah M. Dickerson, Jason M. Hogan, Alex Sugarbaker, David M. S. Johnson, Mark A. Kasevich, "Multiaxis Inertial Sensing with Long-Time Point Source Atom Interferometry". Physical Review Letters, 111, 083001 (2013). Abstract. 2Physics Article.
[13] G Varoquaux, R A Nyman, R Geiger, P Cheinet, A Landragin, P Bouyer, "How to estimate the differential acceleration in a two-species atom interferometer to test the equivalence principle". New Journal of Physics, 11, 113010 (2009). Full Article.
[14] J. M. McGuirk, G. T. Foster, J. B. Fixler, M. J. Snadden, and,M. A. Kasevich, "Sensitive absolute-gravity gradiometry using atom interferometry". Physical Review A, 65, 033608 (2002). Abstract; Anne Louchet-Chauvet, Tristan Farah, Quentin Bodart, André Clairon, Arnaud Landragin, Sébastien Merlet, Franck Pereira Dos Santos, "The influence of transverse motion within an atomic gravimeter". New Journal of Physics, 13, 065025 (2011). Full Article.
[15] D. Schlippert, J. Hartwig, H. Albers, L. L. Richardson, C. Schubert, A. Roura, W. P. Schleich, W. Ertmer, E. M. Rasel, "Quantum Test of the Universality of Free Fall". Physical Review Letters, 112, 203002 (2014). Abstract.
[16] http://www.geoq.uni-hannover.de/350.html
[17] http://www.iqo.uni-hannover.de/quantus.html
[18] D N Aguilera, H Ahlers, B Battelier, A Bawamia, A Bertoldi, R Bondarescu, K Bongs, P Bouyer, C Braxmaier, L Cacciapuoti, C Chaloner, M Chwalla, W Ertmer, M Franz, N Gaalou, M Gehler, D Gerardi, L Gesa, N Gürlebeck, J Hartwig, M Hauth, O Hellmig, W Herr, S Herrmann, A Heske, A Hinton, P Ireland, P Jetzer, U Johann, M Krutzik, A Kubelka, C Lämmerzah, A Landragin, I Lloro, D Massonnet, I Mateos, A Milke, M Nofrarias, M Oswald, A Peters, K Posso-Trujillo, E Rase, E Rocco, A Roura, J Rudolph, W Schleich, C Schubert, T Schuldt, S Seide, K Sengstock, C F Sopuerta, F Sorrentino, D Summers, G M Tino, C Trenkel, N Uzunoglu, W von Klitzing, R Walser, T Wendrich, A Wenzlawski, P Weßels, A Wicht, E Wille, M Williams, P Windpassinger, N Zahzam,"STE-QUEST—test of the universality of free fall using cold atom interferometry". Classical Quantum Gravity, 31, 115010 (2014), Abstract.

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Sunday, August 10, 2014

Over 100-bit Integrated All-Optical Memory on a Photonic Crystal Chip is Demonstrated

[From Left to Right] Eiichi Kuramochi, Kengo Nozaki, Akihiko Shinya, Masaya Notomi

Authors: Eiichi Kuramochi, Kengo Nozaki, Akihiko Shinya, Masaya Notomi

Affiliation: NTT Nanophotonics Center and NTT Basic Research Laboratories, NTT Corporation, Atsugi-shi, Kanagawa, Japan.

Random access memory (RAM) is used extensively in a wide variety of instruments. It is based on the bistable operation of electronic transistors and memorizes bit information. Recently, its optical counterpart, optical RAM (o-RAM), has become highly desirable in high-speed network processing, especially for routers, because it is more efficient to manage network information all-optically without power-consuming electric-to-optical (EO) or OE conversions. Various o-RAM devices have been proposed and fabricated, but so far they have been too big, too power-consuming, or too difficult to integrate.

In 2012, we demonstrated ultralow-power and ultra-small o-RAMs using a photonic crystal nanocavity [1]. Photonic crystals enable strong light confinement, so these o-RAMs were able to achieve a huge reduction of footprint and power consumption. However, the size of these o-RAMs was limited to just four bits. In the June 2014 issue of 'Nature Photonics' [2], we reported our successful integration of wavelength-addressable 105-bit o-RAMs. This is the first realization of an integrated o-RAM with more than four bits.

We implemented a 105-bit o-RAM in a small silicon photonic crystal chip less than 1.1 mm long (Fig. 1). This chip contained 128 nanocavities, each of which can serve as an o-RAM. This novel configuration enables wavelength-addressable o-RAM operation.
Figure 1: (a) Electron microscope images of large-scale nanocavity array integrated in a Si photonic crystal. (b) Transmission spectrum of nanocavity array coupled to an in-plane input/output waveguide. 105 cavities can be operated as bistable optical memories. (c) Schematic of o-RAM operation using the bistable output bias power of a nanocavity side-coupled to a bus waveguide. (d) o-RAM operation of 105 cavity modes in the nanocavity array.

First we explain how an optical memory works. It is well-known that a cavity with optical nonlinear medium exhibits optical bistability at a certain condition. In the present device, we make use of this phenomenon for bit memory operation. Figure 1(c) explains operation mechanism. When one injects light into a nonlinear cavity, the output light intensity shows hysteresis with two bistable states. We can switch between the two bistable states by applying an optical pulse or cutting the bias light. This operation can be used as a bit memory where the high output state (OFF) and low output state (ON) correspond to binary information of '0' and '1'. It should be noted that in a photonic crystal nanocavity, the optical nonlinearity effect is greatly enhanced by high quality factor (Q: ~105) and ultra-small mode volume (< 1 (λ/n)3).

In the present study, we were able to achieve bistable o-RAM operation of 105 nanocavities integrated monolithically in a photonic crystal chip as shown in Fig. 1(d). The optical transmission spectrum is shown in Fig. 1(b). Here each dip corresponds to each cavity. Fabricated cavities have different resonant wavelengths with an averaged spacing of 0.23 nm, which was realized by precisely changing the lattice constant of photonic crystal by 0.125 nm. Such high level of nanofabrication accuracy was achieved by cutting-edge electron beam lithography.

In order to realize wavelength-addressable o-RAMs, each cavity should have a sharp resonance and a wide mode spacing to avoid undesired mode overlap. Conventional high quality factor (Q) cavities in a photonic crystal do not have sufficiently wide mode spacing. It is well known that a three-missing-hole defect cavity (so-called L3 cavity) in a photonic crystal has a wide mode spacing, but suffers from relatively low Q [3]. To overcome this problem, here we employed a modified L3 nanocavity as shown in Fig. 2, where we systematically tuned the 6 sets of the holes denoted A-F.
Figure 2: (a) New tuning design of an L3 nanocavity that improves Q over 10 times. (b) Spectrum of the fundamental cavity mode of an L3 nanocavity. The full-width half-maximum (FWHM) line-width (1.5 pm) corresponds to the experimental Q of 1.0 X 106.

This novel design allows ~10 times enhancement of Q to the conventional L3 cavities with keeping the volume of the cavity mode. Not only the theoretical Q but also the experimental Q was enhanced to ~106 which is 10 times higher than previously reported value [3]. The above-mentioned highly-dense wavelength division multiplexing scheme was enabled by this high performance of modified L3 cavities.

Although we employed Si for o-RAMs discussed above, we also employed InGaAsP for constructing multi-bit o-RAMs, which showed even small operation power (~100 nW) [2] owing to the efficient optical nonlinearity.

We are expecting that the present technology will be employed in an all-optical router as we mentioned. In this application, gigantic RAMs are not necessary, but kilobyte-sized RAMs would be sufficient. To realize this, we plan to combine the present wavelength addressing memory integration scheme with the parallel integration scheme. We have already demonstrated the parallel integration of o-RAM with an all-optical addresser in four-bit o-RAM [1]. The appropriate combination of two methods would be sufficient for our target. Of course, high-speed optical RAMs should also play various important roles in all-optical logical processing other than routers. In addition to that, the demonstrated wavelength addressing nanocavity integration technologies for o-RAM can also be applied for other types of devices such as multi-channel all-optical switches [4].

References: 
[1] Kengo Nozaki, Akihiko Shinya, Shinji Matsuo, Yasumasa Suzaki, Toru Segawa, Tomonari Sato, Yoshihiro Kawaguchi, Ryo Takahashi, and Masaya Notomi, “Ultralow-power all-optical RAM based on nanocavities”. Nature Photonics, 6, 248 (2012). Abstract.
[2] Eiichi Kuramochi, Kengo Nozaki, Akihiko Shinya, Koji Takeda, Tomonari Sato, Shinji Matsuo, Hideaki Taniyama, Hisashi Sumikura and Masaya Notomi, “Large-scale integration of wavelength-addressable all-optical memories on a photonic crystal chip”. Nature Photonics, 8, 474 (2014). Abstract.
[3] Yoshihiro Akahane, Takashi Asano, Bong-Shik Song, and Susumu Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal”. Nature, 425, 944 (2003). Abstract.
[4] Kengo Nozaki, Eiichi Kuramochi, Akihiko Shinya, and Masaya Notomi, "25-channel all-optical gate switches realized by integrating silicon photonic crystal nanocavities". Optics Express, 22, 14263 (2014). Abstract.

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Sunday, August 03, 2014

Milestones in a Continuing Tale of Big and Small : Large Magnitude Squeezed Light at 100 Hz, and a Squeezed 4km Gravitational-wave Detector

Sheon Chua

[Sheon Chua is the recipient of the 2013 GWIC (Gravitational Wave International Committee) Thesis Prize for his PhD thesis “Quantum Enhancement of a 4km Laser Interferometer Gravitational-Wave Detector” (PDF). -- 2Physics.com]


Author: Sheon Chua

Affiliation:

Currently at: Laboratoire Kastler Brossel, University of Pierre and Marie Curie (UPMC), Paris, France.

PhD research performed at: Centre for Gravitational Physics, Australian National University (ANU), Canberra, Australia.

Gravitational-wave sources of astronomical size from our Universe. Gravitational-wave displacement signals expected at one thousandth of the diameter of a single proton. Interferometric instruments with kilometre-long arms. Light ‘squeezed’ on the quantum scale.

The construction and implementation of second-generation laser-interferometric gravitational-wave detectors [1] are rapidly progressing [2], forming a detector network expected to be online over the next few years. These amazing instruments will use state-of-the-art isolation systems, optics, and hundred watt input lasers, and have kilometre-scale arms. For these detectors, the effect of a passing gravitational wave causes a relative displacement change between the interferometer arm end mirrors, which is encoded in the relative phase of the light beams propagating in the arms [3]. The relative displacement sensitivities will be of order of 10-19 m in the 10 Hz to 10 kHz Fourier frequency band, achieved after monumental efforts in research and development across many fields of physics and engineering.

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

Paul Fulda (2012): "Precision Interferometry in a New Shape: Higher-order Laguerre-Gauss Modes for Gravitational Wave Detection"
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.

However, even after these impressive technological efforts, there remain fundamental noise sources that limit measurement sensitivity that arise from the underlying physics of the instrument itself. One such noise source is the quantum nature of light [4], that comes from a non-zero commutation relationship between a light beam’s phase (ϕ) and amplitude (A) [5]. The Heisenberg Uncertainty Principle for this specific pair of quantities is given by ∆ϕ ≥ 1. Figure 1(a) shows this relation diagrammatically as a noise ‘ball’. As the gravitational-wave signal is encoded in the relative phase, with quantum phase noise Δϕ present we reach a signal-to-noise level where we can no longer distinguish a passing gravitational wave in the measurement. Therefore, the quantum noise of light is a limitation to achievable sensitivity.

However, the Heisenberg Uncertainty Principle relation is multiplicative. This means that one of the uncertainties can be below the quantum level, or ‘squeezed’, if the other uncertainty is above the level, or ‘antisqueezed’. This is illustrated in Figure 1(b), where the overall uncertainty is the same, but the individual uncertainties have been ‘rearranged’. The amount of squeezing has units of decibels [dB], referenced to the unsqueezed quantum noise level amplitude, given by [dB]=20 log10[(anti)squeezed noise / unsqueezed noise].

Figure 1 (a) Quantum noise ‘ball’, showing the even distribution of uncertainty between the two quantities amplitude (A) and phase (ϕ). (b) Squeezed noise, where the uncertainty in one quantity is less than quantum noise, while the other uncertainty is greater than quantum noise.

As an example, if we have 6 dB of squeezing, we mean that the noise is squeezed to about half the value of the quantum noise level, or that the noise is reduced by a factor of 2. It follows that if we inject squeezed light into an interferometer so that it results in the phase uncertainty being reduced, the measurement sensitivity limited by quantum phase noise will be improved.

The tale of squeezed light for enhancing gravitational-wave detectors is now three decades young, with theoretical proposals for injecting squeezed light into interferometers published in the early 1980s [6], a few years before first experimental measurement of squeezing [7] took place. Since then, there has been a steady advancement in techniques and technologies to generate squeezed light within the 10 Hz to 10 kHz detection band [8-10], as well as to implement squeezed light with interferometers [11-14]. The GEO600 detector is now routinely using squeezed light, with ever-increasing timescales and duty cycles [15].
Figure 2: First measurement of greater than 10 dB squeezing across the audio gravitational-wave detection band, with 11.6 dB from 200 Hz and above. The degradation of squeezing level below 100 Hz is due to remaining residual classical noise entering the squeezing detector. Adapted from [16], and includes resolution bandwidth and window information.

The first milestone recently added to this story is the measurement of greater than 10 dB squeezing across the 10 Hz – 10 kHz frequency band [16]. This measurement was achieved by a team at the Australian National University, with valuable input from the Albert Einstein Institute. Figure 2 shows the result, with a maximum of 11.6 dB measured at 100 Hz and above. This was achieved after a detailed study characterizing and minimizing classical noise sources that impacted the squeezing measurement. This result represents the current record for squeezing in the 10 Hz – 10 kHz band, and further demonstrates the availability of large squeezing magnitude applicable for gravitational-wave detector enhancement.

The second milestone recently achieved is realising a squeezed 4 km interferometric gravitational-wave detector [17]. This was an experiment completed on the Enhanced LIGO 4 km interferometer in Washington State USA, performed by scientists from across the LIGO Scientific Collaboration, with LIGO Hanford Observatory, LIGO Massachusetts Institute of Technology, Australian National University and the Albert Einstein Institute being the lead institutions.
Figure 3: Enhanced LIGO interferometer with squeezing. (a) The Reference trace shows the displacement sensitivity of the interferometer without squeezing being injected, while the Squeezing trace shows the interferometer with squeezing injected. (b) Squeezing enhancement in LIGO’s most sensitive frequency band, at a lesser level due to significant contributions from noise sources other than quantum noise. Adapted from [17].

Figure 3(a) shows the interferometer displacement sensitivity curve with and without squeezed light. Up to 2.15 dB of squeezing enhancement is measured in the quantum noise limited regime (above 150 Hz). This is in line with the expected experiment parameters. Furthermore, as shown in Figure 3(b), in the most sensitive band between 150 Hz and 300 Hz, there is enhancement gained by squeezing. This result confirmed the compatibility of squeezing at lower detection frequencies where future gravitational-wave detectors will have their best sensitivity.

Squeezed light is a tool that is now available for, and being used for enhancing interferometric gravitational-wave detectors [15]. Third generation detector designs, such as the Einstein Telescope [18], have squeezed light injection as part of baseline technology. To realise maximum benefit from squeezed light injection, further improvements and refinements are needed, such as for improved parameters for squeezing injection and for minimizing adverse impacts on future detectors with more stringent requirements. This development work continues on as I write. It is safe to say that there are many more milestones to come in this continuing tale of big and small.

This article is a ‘synopsis’ of the squeezed light story and the two milestone results. For an in-depth review of squeezed light, squeezed light technologies and injection experiments up to 2013 (including both of these recent milestones), a Topical Review article is to be published soon [19]. I also recommend the LIGO Magazine, Issue 3 [20], which is focussed on squeezed light.

References:
[1] Advanced LIGO website: www.advancedligo.mit.edu ; Advanced Virgo website: wwwcascina.virgo.infn.it/advirgo ; KAGRA website: gwcenter.icrr.u-tokyo.ac.jp/en/; GEO600 website: www.geo600.org
[2] For example: www.advancedligo.mit.edu/adligo_news.html .
[3] For an expanded introduction to interferometric gravitational-wave detector measurement, I recommend this short video: www.youtube.com/watch?v=RzZgFKoIfQI .
[4] P.R. Saulson, "Fundamentals of interferometric gravitational wave detectors". World Scientific, Singapore (1994).
[5] D.F. Walls and G. Milburn, "Quantum Optics". Springer-Verlag, 2nd edition, Berlin (2008).
[6] Carlton M. Caves, "Quantum-mechanical noise in an interferometer". Physical Review D, 23, 1693 (1981) . Abstract.
[7] R.E. Slusher, L.W. Hollberg, B. Yurke, J.C. Mertz, J.F. Valley, "Observation of Squeezed States Generated by Four-Wave Mixing in an Optical Cavity", Physical Review Letters, 55, 2409 (1985). Abstract.
[8] Kirk McKenzie, Nicolai Grosse, Warwick P. Bowen, Stanley E. Whitcomb, Malcolm B. Gray, David E. McClelland, Ping Koy Lam, "Squeezing in the Audio Gravitational-Wave Detection Band". Physical Review Letters, 93, 161105 (2004). Abstract.
[9] Roman Schnabel and Henning Vahlbruch, "Squeezed Light – the first real application starts now". 2Physics : April 03, 2008.
[10] Tobias Eberle, Sebastian Steinlechner, Jöran Bauchrowitz, Vitus Händchen, Henning Vahlbruch, Moritz Mehmet, Helge Müller-Ebhardt, Roman Schnabel, "Quantum Enhancement of the Zero-Area Sagnac Interferometer Topology for Gravitational Wave Detection", Physical Review Letters, 104, 251102 (2010). Abstract.
[11] Kirk McKenzie, Daniel A. Shaddock, David E. McClelland, Ben C. Buchler, and Ping Koy Lam, "Experimental Demonstration of a Squeezing-Enhanced Power-Recycled Michelson Interferometer for Gravitational Wave Detection", Physical Review Letters, 88, 231102 (2002). Abstract.
[12] Henning Vahlbruch, Simon Chelkowski, Boris Hage, Alexander Franzen, Karsten Danzmann, Roman Schnabel, "Demonstration of a Squeezed-Light-Enhanced Power- and Signal-Recycled Michelson Interferometer", Physical Review Letters, 95 211102 (2005). Abstract.
[13] Keisuke Goda, Alan Weinstein, Nergis Mavalvala, "Beating the Quantum Limit in Gravitational Wave Detectors". 2Physics : May 10, 2008.
[14] Hartmut Grote, Roman Schnabel, Henning Vahlbruch, "A Gravitational Wave Observatory Operating Beyond the Quantum Shot-Noise Limit". 2Physics : September 25, 2011.
[15] H. Grote, K. Danzmann, K. L. Dooley, R. Schnabel, J. Slutsky, H. Vahlbruch, "First Long-term Application of Squeezed States of Light in a Gravitational-Wave Observatory". Physical Review Letters, 110, 181101 (2013). Abstract.
[16] M S Stefszky, C M Mow-Lowry, S S Y Chua, D A Shaddock, B C Buchler, H Vahlbruch, A Khalaidovski, R Schnabel, P K Lam, D E McClelland, "Balanced Homodyne Detection of Optical Quantum States at Audio-Band Frequencies and Below". Classical and Quantum Gravity, 29 145015 (2012). Abstract.
[17] The LIGO Scientific Collaboration, "Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light". Nature Photonics 7,  613 – 619 (2013). Abstract.
[18] Einstein Telescope: www.et-gw.eu .
[19] S. Chua et al, "Quantum Squeezed Light for Advanced Gravitational-wave Detectors". Classical and  Quantum Gravity Topical Review, accepted for publication (2014).
[20] LIGO Magazine, Issue 3: www.ligo.org/magazine/LIGO-magazine-issue-3.pdf .

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