Topological Phases of Sound and Light

Topological states of matter are particularly robust, since they exploit global features insensitive to local perturbations. In this work, we describe how to create a Chern insulator of phonons in the solid state. The proposed implementation is based on a simple setting, a dielectric slab with a suitable pattern of holes. Its topological properties 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 along the edges 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. This would be an example of a Chern insulator produced from the interaction between two physically very different particle species, photons and phonons.

Possible design based on "snowflake" 2D optomechanical crystal (Painter group), here: with suitable defects forming a superlattice (array of cells) Modeling an optomechanical array modes the opsic mer hand, noise) echaneads to nsition. l model and the for a metry, ards a ordered hase in regard to phase transitions recently include cold atomic gases [18][19][20][21][22][23], nonlinear cavity arrays [24,25] and optical fibres [26]. In a very recent work and along the lines of [18], the preparation of long-range order for photonic modes was proposed using the linear dissipative e⇤ects in an optomechanical array [6]. Our work adds the novel aspect of a mechanical phase transition to the studies of driven dissipative many-body systems.
Model. -We study the collective quantum dynamics of a two-dimensional homogeneous array of optomechanical cells (Fig. 1). Each of these cells consists of a mechanical mode and a laser driven optical mode that interact via the radiation pressure coupling at a rate g 0 (~= 1): The mechanical mode (b j ) is characterized by a frequency ⇤. The cavity mode (â j ) is transformed into the frame rotating at the laser frequency (⇥ = ⇧ laser ⇧ cav ) and driven at the rate L . In the most general case, both photons and phonons can tunnel between neighboring sites ⌃ij⌥ at rates J/z and K/z, where z denotes the coordination number. The full Hamiltonian of the array is given byĤ = ⇤ jĤ om,j +Ĥ int , witĥ To bring this many-body problem into a treatable form, we apply the Gutzwiller ansatzÂ † iÂ j ⇥ ⌃Â † i ⌥Â j + optical coupling strength

Phonon Topological Materials
What about topological transport of phonons?

Features
• Topologically protected transport of phonons in the solid state  Hofstadter butterfly spectrum arbitrary optical reconfiguration of magnetic field distribution

Dynamical Gauge Fields
Mechanical oscillation: now self-oscillations (pumped by blue-detuned laser), instead of externally driven the optomechanical interaction: A photon a in a lattice hopping from site to site is accompanied by the coherent emission or absorption of a phonon b. For a photon tunneling between sites j and j+1, this is described by b j,j+1 a † j+1 a j (and the opposite process). Such a setting can be described by the Hamiltonian We set~= 1. Here, j is a lattice site index (n, m), l = i, j is an index for a directed link between two lattice site i and j, and the summation over hi, ji is over nearest neighbors of the underlying lattice. Photons (phonons) have frequencies ⌫ j (! l ). The J ij = J ji are the coupling constants for the phononassisted photon tunneling process whose implementation we will describe below. First, we will study the case for three sites. However, it is the sin to have such a betwee In the tor to b limit-c We no For ex couple worked would discuss The Hamilt obtaine will fo limit o phonon-assisted photon hopping Optomechanical Hamiltonian: Amplitude B x(t) FIG. 1. (color online). a chanical implementation coupling to phonons b. so-called "modulated lin blue dots depict the opti sites). Green dots denote optomechanically couple an optomechanical cell (d lattice. The blue dots de ellipses represent the pha directed links between la