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