Topological states of matter are particularly robust, since they exploit global features of a material's band structure. Topological states have already been observed for electrons, atoms, and photons. It is an outstanding challenge to create a Chern insulator of sound waves in the solid state. In this work, we propose an implementation based on cavity optomechanics in a photonic crystal. The topological properties of the sound waves can be wholly tuned in situ by adjusting the amplitude and frequency of a driving laser that controls the optomechanical interaction between light and sound. The resulting chiral, topologically protected phonon transport can be probed completely optically. Moreover, we identify a regime of strong mixing between photon and phonon excitations, which gives rise to a large set of different topological phases and offers an example of a Chern insulator produced from the interaction between two physically distinct particle species, photons and phonons.

Recently, there has been growing interest in the creation of artificial magnetic fields for uncharged particles, such as cold atoms or photons. These efforts are partly motivated by the resulting desirable features, such as transport along edge states that is robust against backscattering. We analyze how the optomechanical interaction between photons and mechanical vibrations can be used to create artificial magnetic fields for photons on a lattice. The ingredients required are an optomechanical crystal, i.e., a free-standing photonic crystal with localized vibrational and optical modes, and two laser beams with the right pattern of phases. One of the two schemes analyzed here is based on optomechanical modulation of the links between optical modes, while the other is a lattice extension of optomechanical wavelengthconversion setups. We analyze both schemes theoretically and present numerical simulations of the resulting optical spectrum, photon transport in the presence of an artificial Lorentz force, edge states, and the photonic AharonovBohm effect. We discuss the requirements for experimental realizations. Finally, we analyze the completely general situation of an optomechanical system subject to an arbitrary optical phase pattern and conclude that it is best described in terms of gauge fields acting in synthetic dimensions. In contrast to existing nonoptomechanical approaches, the schemes analyzed here are very versatile, since they can be controlled fully optically, allowing for time-dependent in situ tunability without the need for individual electrical addressing of localized optical modes.

It is now well established that photonic systems can exhibit topological energy bands. Similar to their electronic counterparts, this leads to the formation of chiral edge modes which can be used to transmit light in a manner that is protected against backscattering. While it is understood how classical signals can propagate under these conditions, it is an outstanding important question how the quantum vacuum fluctuations of the electromagnetic field get modified in the presence of a topological band structure. We address this challenge by exploring a setting where a nonzero topological invariant guarantees the presence of a parametrically unstable chiral edge mode in a system with boundaries, even though there are no bulkmode instabilities. We show that one can exploit this to realize a topologically protected, quantum-limited traveling wave parametric amplifier. The device is naturally protected against both internal losses and backscattering; the latter feature is in stark contrast to standard traveling wave amplifiers. This adds a new example to the list of potential quantum devices that profit from topological transport.

It has been predicted and experimentally demonstrated that by injecting squeezed light into an optomechanical device it is possible to enhance the precision of a position measurement. Here, we present a fundamentally different approach where the squeezing is created directly inside the cavity by a nonlinear medium. Counterintuitively, the enhancement of the signal to noise ratio works by de-amplifying precisely the quadrature that is sensitive to the mechanical motion without losing quantum information. This enhancement works for systems with a weak optomechanical coupling and/or strong mechanical damping. This could allow for larger mechanical bandwidth of quantum limited detectors based on optomechanical devices. Our approach can be straightforwardly extended to Quantum Non Demolition (QND) qubit detection.Recent progress in cavity optomechanics [1,2] has been so exceptional that the precision of a position measurement has been pushed until the limit set by the principles of quantum mechanics, the so-called Standard Quantum Limit (SQL) [3][4][5]. A measurement precision close to the SQL has been demonstrated in optomechanical devices with cavities both in the optical [6][7][8] and in the microwave [9] domain. Optomechanical position detection is not only of fundamental interest but finds also application in acceleration [10,11], magnetic field [12,13], and force detectors [14? ]. Thus, an important goal for the future is to develop new techniques to enhance its precision on different optomechanical platforms. Seminal efforts have focused on gravitational wave detection in optomechanical interferometers [16][17][18][19][20]. The standard route to enhance the detection precision consists in injecting squeezed light into the interferometer [16,17,21]. This technique has recently been demonstrated in the Laser Interferometer Gravitational Wave Observatory (LIGO) [22] and in a cavity optomechanics setup [23]. Externally generated squeezed light could also find application in QND qubit state detection [24,25]. Alternatively, one can enhance a dispersive quantum measurement by generating the appropriate squeezing directly inside the cavity by means of a Kerr nonlinearity [18,20,21,[26][27][28], by the dissipative optomechanical interaction [29], or potentially, by exploiting the ponderomotive squeezing [30][31][32].In this letter, we propose a new pathway to precision enhancement in optomechanical detection. In our approach, a nonlinear cavity is operated as a phase-sensitive parametric amplifier, as shown in Fig. 1. It amplifies a seed laser beam and its intensity fluctuations. Simultaneously, it de-amplifies the phase quadrature where the mechanical vibrations are imprinted. At first sight it might appear counter-intuitive that de-amplification can improve a (quantum) measurement. Here, we suggest that it might be worth to de-amplify a signal if the noise is suppressed by a larger factor thus obtaining a net enhancement of the signal to noise ratio. Indeed, our analysis shows that for optomechanical position detect...

There is enormous interest in engineering topological photonic systems. Despite intense activity, most works on topological photonic states (and more generally bosonic states) amount in the end to replicating a well-known fermionic single-particle Hamiltonian. Here we show how the squeezing of light can lead to the formation of qualitatively new kinds of topological states. Such states are characterized by non-trivial Chern numbers, and exhibit protected edge modes, which give rise to chiral elastic and inelastic photon transport. These topological bosonic states are not equivalent to their fermionic (topological superconductor) counterparts and, in addition, cannot be mapped by a local transformation onto topological states found in particle-conserving models. They thus represent a new type of topological system. We study this physics in detail in the case of a kagome lattice model, and discuss possible realizations using nonlinear photonic crystals or superconducting circuits.

We investigate the nonlinear response of a vibrating suspended nanomechanical beam on external periodic driving. The amplitude of the fundamental transverse mode behaves thereby like a weakly damped quantum particle in a driven anharmonic potential. Upon using a Born-Markovian master equation, we calculate the fundamental mode amplitude for varying driving frequencies. In the nonlinear regime, we observe resonances which are absent in the corresponding classical model. They are shown to be associated with resonant multi-phonon excitations. Furthermore, we identify resonant tunneling in a dynamically induced bistable effective potential.

We analytically investigate the nonlinear response of a damped doubly clamped nanomechanical beam under static longitudinal compression which is excited to transverse vibrations. Starting from a continuous elasticity model for the beam, we consider the dynamics of the beam close to the Euler buckling instability. There, the fundamental transverse mode dominates and a quantum mechanical time-dependent effective single particle Hamiltonian for its amplitude can be derived. In addition, we include the influence of a dissipative Ohmic or super-Ohmic environment. In the rotating frame, a Markovian master equation is derived which includes also the effect of the time-dependent driving in a non-trivial way. The quasienergies of the pure system show multiple avoided level crossings corresponding to multiphonon transitions in the resonator. Around the resonances, the master equation is solved analytically using Van Vleck perturbation theory. Their lineshapes are calculated resulting in simple expressions. We find the general solution for the multiple multiphonon resonances and, most interestingly, a bath-induced transition from a resonant to an antiresonant behavior of the nonlinear response.

We show how the snowflake phononic crystal structure, which recently has been realized experimentally, can be turned into a topological ins ulator for mechanical waves. This idea, based purely on simple geometrical modifications, could be readily implemented on the nanoscale.