A general scheme for trust-region methods on Riemannian manifolds is proposed and analyzed. Among the various approaches available to (approximately) solve the trust-region subproblems, particular attention is paid to the truncated conjugate-gradient technique. The method is illustrated on problems from numerical linear algebra.

Abstract:Optomechanical coupling between a mechanical oscillator and light trapped in a cavity increases when the coupling takes place in a reduced volume. Here we demonstrate a GaAs semiconductor optomechanical disk system where both optical and mechanical energy can be confined in a sub-micron scale interaction volume. We observe giant optomechanical coupling rate up to 100 GHz/nm involving picogram mass mechanical modes with frequency between 100 MHz and 1 GHz. The mechanical modes are singled-out measuring their dispersion as a function of disk geometry. Their Brownian motion is optically resolved with a sensitivity of 10 -17 m/√Hz at room temperature and pressure, approaching the quantum limit imprecision.Optomechanical systems generally consist of a mesoscopic mechanical oscillator interacting with light trapped in a cavity [1][2][3]. These systems have attracted a growing interest since first experimental evidences that cavity light can be used to optically self-cool the oscillator towards its quantum regime [4][5][6][7][8][9]. They are now studied in an increasing number of geometries and compositions, with the common purpose of coupling photons and phonons in a controlled way. Beyond the mere goal of reaching the quantum ground state of a mechanical oscillator, today concepts developed in optomechanics find applications in very different fields such as cold atoms physics [10-11], mechanical sensing [12] or Josephson circuitry [13]. High-frequency nanomechanical oscillators are generally welcome in these applications, to ease the access to the quantum regime or to develop highspeed sensing systems. However since their sub-wavelength size generally imply a weak interaction with light, these oscillators need to be inserted in a cavity to enhance the optical/mechanical interaction [14]. The typical optomechanical coupling obtained using this approach is of 10 MHz/nm for visible photons [15][16] or 10 kHz/nm in the microwave range [17]. A coupling enhancement can be obtained by further confining mechanical and optical modes in a small interaction volume, as recently achieved in nano-patterned photonic crystals [18]. However these structures are complex to design and fabricate, and being based on silicon technology, they do not allow the insertion of an optically active medium. This precludes exploring novel situations where a (quantum) mechanical oscillator would be coupled to a (quantum) photon emitter embedded in the host material. In this paper we present a gallium arsenide (GaAs) nano-optomechanical disk resonator, a system at the crossroads with III-V semiconductor nano-photonics. This resonator combines the assets of both nano-scale mechanical systems (high frequency and low mass in the pg range) and semiconductor optical microcavities, with optical quality factor above 10 5 . The high refractive index of GaAs enables storing light in a sub-micron mode-volume whispering gallery mode of the disk, where it couples to high frequency (up to the GHz) vibrational modes of the structure. Thanks to the miniatu...

Superfluidity is an emergent quantum phenomenon which arises due to strong interactions between elementary excitations in liquid helium. These excitations have been probed with great success using techniques such as neutron and light scattering. However measurements to-date have been limited, quite generally, to average properties of bulk superfluid or the driven response far out of thermal equilibrium. Here, we use cavity optomechanics to probe the thermodynamics of superfluid excitations in real-time. Furthermore, strong light-matter interactions allow both laser cooling and amplification of the thermal motion. This provides a new tool to understand and control the microscopic behaviour of superfluids, including phonon-phonon interactions, quantised vortices and two-dimensional quantum phenomena such as the Berezinskii-Kosterlitz-Thouless transition. The third sound modes studied here also offer a pathway towards quantum optomechanics with thin superfluid films, including femtogram effective masses, high mechanical quality factors, strong phonon-phonon and phonon-vortex interactions, and self-assembly into complex geometries with sub-nanometre feature size.Comment: 6 pages, 4 figures. Supplementary information attache

Vibrating nano-and micromechanical resonators have been the subject of research aiming at ultrasensitive mass sensors for mass spectrometry, chemical analysis and biomedical diagnosis.Unfortunately, their merits diminish dramatically in liquids due to dissipative mechanisms like viscosity and acoustic losses. A push towards faster and lighter miniaturized nanodevices would enable improved performances, provided dissipation was controlled and novel techniques were available to efficiently drive and read-out their minute displacement. Here we report on a nanooptomechanical approach to this problem using miniature semiconductor disks. These devices combine mechanical motion at high frequency above the GHz, ultra-low mass of a few picograms, and moderate dissipation in liquids. We show that high-sensitivity optical measurements allow to direct resolve their thermally driven Brownian vibrations, even in the most dissipative liquids.Thanks to this novel technique, we experimentally, numerically and analytically investigate the interaction of these resonators with arbitrary liquids. Nano-optomechanical disks emerge as probes of rheological information of unprecedented sensitivity and speed, opening applications in sensing and fundamental science.

We analyze the magnitude of the radiation pressure and electrostrictive stresses exerted by light confined inside GaAs semiconductor WGM optomechanical disk resonators, through analytical and numerical means, and find the electrostrictive stress to be of prime importance. We investigate the geometric and photoelastic optomechanical coupling resulting respectively from the deformation of the disk boundary and from the strain-induced refractive index changes in the material, for various mechanical modes of the disks. Photoelastic optomechanical coupling is shown to be a predominant coupling mechanism for certain disk dimensions and mechanical modes, leading to total coupling gom and g(0) reaching respectively 3 THz/nm and 4 MHz. Finally, we point towards ways to maximize the photoelastic coupling in GaAs disk resonators, and we provide some upper bounds for its value in various geometries.

We report on wavelength-sized GaAs optomechanical disk resonators showing ultra-strong optomechanical interaction. We observe optical transduction of a disk mechanical breathing mode with 1.4 GHz frequency and effective mass of ~ 2 pg. The measured vacuum optomechanical coupling rate reaches g 0 = 0.8 MHz, with a related differential optomechanical coupling factor g om = 485 GHz/nm. The disk Brownian motion is optically resolved with a sensitivity of 10 -17 m/√Hz at room temperature and pressure.Optomechanical systems 1-3 combining a mechanical oscillator and an optical cavity find now applications in very different fields of physics, from mesoscopic quantum physics to cold atoms 4-5 and mechanical sensing 6 . GHz mechanical oscillators can help accessing the quantum regime of optomechanics, can allow developing ultra-fast sensing systems, or can match hyperfine transitions of (artificial) atoms interfaced with the mechanical system. The difficulty lies generally in coupling such GHz oscillators to photons efficiently, in order to offer optical control over the oscillator motion, together with fine optical read-out sensitivity. 7 A useful way a)

Anasazi is a package within the Trilinos software project that provides a framework for the iterative, numerical solution of large-scale eigenvalue problems. Anasazi is written in ANSI C++ and exploits modern software paradigms to enable the research and development of eigensolver algorithms. Furthermore, Anasazi provides implementations for some of the most recent eigensolver methods. The purpose of our article is to describe the design and development of the Anasazi framework. A performance comparison of Anasazi and the popular FORTRAN 77 code ARPACK is given.

Collective phenomena emerging from non-linear interactions between multiple oscillators, such as synchronization and frequency locking, find applications in a wide variety of fields. Optomechanical resonators, which are intrinsically non-linear, combine the scientific assets of mechanical devices with the possibility of long distance controlled interactions enabled by travelling light. Here we demonstrate light-mediated frequency locking of three distant nano-optomechanical oscillators positioned in a cascaded configuration. The oscillators, integrated on a chip along a coupling waveguide, are optically driven with a single laser and oscillate at gigahertz frequency. Despite an initial frequency disorder of hundreds of kilohertz, the guided light locks them all with a clear transition in the optical output. The experimental results are described by Langevin equations, paving the way to scalable cascaded optomechanical configurations.Synchronization and frequency locking have been observed in a large variety of contexts ranging from physics to biology, e.g. in classical coupled pendula [1], in coupled lasers [2], and in the rhythmic beating of pacemaker cells [3]. These phenomena have found practical applications in RF communication [4], signal-processing [5], novel computing and memory concepts [6,7], clock synchronization and navigation [8], as well as in phased locked loop circuits [9]. For these applications, micro-and nano-mechanical devices are known to present opportunities of integration and scalability [10][11][12][13][14][15] but more recently, optomechanical systems further emerged as new appealing candidates. Indeed they support non-linearly coupled optical and mechanical modes [16,17], and add to the mechanics the assets of optical techniques in terms of precision and long-distance communications [18][19][20][21][22][23][24][25].Light injected in an optomechanical cavity can deform it under the action of optical forces, and in the dynamical backaction regime can amplify its mechanical motion. When amplification overcomes mechanical dissipation, the system transits to a stable limit cycle, often referred to as optomechanical self-oscillation [26,27]. In the last years, several studies investigated the synchronization of such optomechanical oscillators [20][21][22]25]. The synchronization of two oscillators placed close to contact and sharing a common optical mode was reported in [20]. Recently, the same configuration was pushed up to 7 resonators [25]. Two spatially-separated oscillators integrated in a common optical racetrack cavity were also synchronized in [22]. The possibility of locking two optomechanical systems without sharing a common optical mode was implemented as well in [21], in two steps and with two lasers. The optical output of a first laser-driven optomechanical oscillator was transduced into an electrical signal, which was carried away to fed a distant electro-optic modulator. The latter modulated a second laser driving a second optomechanical oscillator, ultimately insuring phase lo...