Coupled Mode Theory of Optomechanical Crystals

Abstract

Abstract-Acousto-optic interaction in optomechanical crystals allows unidirectional control of elastic waves over optical waves. However, as a result of this nonlinear interaction, infinitely many optical modes are born. This article presents an exact formulaion of coupled mode theory for interaction between elastic and photonic Bloch waves moving along an optomechanical waveguide. In general, an optical wavefront is strongly diffracted by an elastic wave in frequency and wavevector, and thus infinite modes with different frequencies and wavevectors appear. We discuss resonance and mode conversion conditions, and present a rigorous method to derive coupling rates and mode profiles. We also find a conservation law which rules over total optical power from interacting individual modes. We present application examples to the theory to optomechanical waveguides and cavities, as well as non-reciprocal transmission of light and optomechanical switches.

Higher‐Order Interactions in Quantum Optomechanics: Revisiting Theoretical Foundations

“…The optomechanical interaction ℍ OM is inherently nonlinear by its nature, which is quite analogous to the third-order Kerr optical effect in nonlinear optics [6,7]. These for instance include the optomechanical arrays [8][9][10][11][12][13][14][15][16][17], squeezing of phonon states [18][19][20], Heisenberg's limited measurements [21], non-reciprocal optomechanical systems [22][23][24][25][26][27][28], sensing [29][30][31], engineered dissipation [32], engineered states [33], and non-reciprocal acousto-optical effects in optomechanical crystals [34][35][36].…”

Operator approach in nonlinear stochastic open quantum physics

“…The optomechanical interaction ℍ OM is inherently nonlinear by its nature, which is quite analogous to the third-order Kerr optical effect in nonlinear optics [25,26]. These for instance include the optomechanical arrays [27][28][29][30][31][32][33][34][35][36], squeezing of phonon states [37][38][39], Heisenberg's limited measurements [40], non-reciprocal optomechanical systems [41][42][43][44][45][46], sensing [47][48][49], engineered dissipation [50], engineered states [51], and non-reciprocal acousto-optical effects in optomechanical crystals [52][53][54].…”

“…Under a rather idealistic case of when an infinite number of optical modes are involved in optomechanical interaction, and both summations are let to extend into infinity, it would be possible to demonstrate that (57) and ( 59) are actually the same. For this to occur, we need first to show (52) and (54) are the same representations of the same identities, too. This is not too difficult to prove, indeed.…”

Higher-Order Interactions in Quantum Optomechanics: Analytical Solution of Nonlinearity

“…In quantum optomechanics the standard interaction Hamiltonian is simply the product of photon number n = â † â and the position x zp ( b + b † ) operators [1][2][3][4][5][6], where x zp is the zero-point motion, and â and b are respectively the photon and phonon annihilators. This type of interaction can successfully describe a vast range of phenomena, including optomechanical arrays [7][8][9][10][11][12][13], squeezing of phonon states [14][15][16], non-reciprocal optomechanics [17][18][19][20], Heisenbergs limited measurements [21], sensing [22][23][24], engineered dissipation and states [25,26], and non-reciprocal acousto-optics [27]. In all these applications, the mathematical toolbox to estimate the measured spectrum is Langevin equations [29][30][31][32].…”