Frequency-entangled <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>W</mml:mi></mml:math> states and quantum frequency translation protocols via forward Brillouin interactions

Complex quantum states of light are not only central to advancing our understanding of quantum mechanics, but are also necessary for a variety of quantum protocols. High-dimensional, or multipartite, quantum states are of specific interest, as they can exhibit unique properties both fundamentally and in application. The synthesis of high-dimensional, entangled photonic states can take the form of various schemes, which result in varying forms of entanglement. Frequency entanglement is specifically attractive due to compatibility with integrated systems and resistance to decoherence in fiber transportation; however, increasing the dimension of frequency-entangled states requires a system that offers quantum interactions between a large set of distinct frequencies. Here, we show how the phonon-photon interactions of forward Brillouin scattering, which offer access to a ladder of optical resonances permitted by a single mechanical mode, can be used for fast synthesis of frequency-entangled, single-photon $W$ states. In our proposed system, simultaneous laser pulses of different frequencies dynamically evolve either an injected single photon or a heralded single phonon, generating $W$ states of selected dimension and output frequency. This method enables the synthesis of ``perfect'' $W$ states by adjusting the pulse amplitudes. In addition, we show how this system can be used for quantum frequency translation.