[Paper Review] Effective mass in quantum effects of radiation pressure

M. Pinard, Yassine Hadjar+1 more|arXiv (Cornell University)|Jan 19, 1999

Mechanical and Optical Resonators1 references79 citations

TL;DR

This paper introduces an effective mass concept in optomechanical systems where radiation pressure couples light to mechanical vibrations in a high-finesse cavity. By modeling the mirror as a mechanical resonator with distributed acoustic modes, the authors derive an effective susceptibility that reduces the effective mass below the physical mass, especially for small optical waists in plano-convex mirrors, thereby enhancing quantum optomechanical effects such as radiation pressure-induced phase shifts and noise reduction.

ABSTRACT

We study the quantum effects of radiation pressure in a high-finesse cavity with a mirror coated on a mechanical resonator. We show that the optomechanical coupling can be described by an effective susceptibility which takes into account every acoustic modes of the resonator and their coupling to the light. At low frequency this effective response is similar to a harmonic response with an effective mass smaller than the total mass of the mirror. For a plano-convex resonator the effective mass is related to the light spot size and becomes very small for small optical waists, thus enhancing the quantum effects of optomechanical coupling.

Motivation & Objective

• . To model optomechanical coupling beyond the single harmonic oscillator approximation by including internal acoustic modes of the mirror.
• . To derive an effective susceptibility that captures the collective response of all acoustic modes to radiation pressure.
• . To show that the effective mass in this model is significantly smaller than the physical mirror mass, especially for small optical beam waists.
• . To demonstrate that this reduced effective mass enhances quantum effects such as phase shifts and noise reduction in cavity optomechanics.
• . To unify the description of thermal and radiation pressure effects via a single effective response function.

Proposed method

• . Use of a one-dimensional optomechanical model to establish baseline coupling between mirror motion and light phase shift via ψ(t) = 2kz(t).
• . Derivation of the mechanical susceptibility χ[Ω] for a harmonic oscillator model with mass M and damping characterized by Q.
• . Extension to distributed systems by solving the elastodynamic equations for a resonator under radiation pressure, using Hooke's law and boundary conditions.
• . Decomposition of the mirror's displacement into acoustic modes, with each mode treated as a forced harmonic oscillator coupled to radiation pressure.
• . Calculation of the effective susceptibility by integrating the overlap between radiation pressure force and each acoustic mode's spatial profile.
• . Use of the fluctuation-dissipation theorem to relate thermal noise to the effective susceptibility, enabling unified treatment of thermal and quantum effects.

Experimental results

Research questions

• RQ1 . How does the inclusion of internal acoustic modes affect the effective optomechanical coupling in a cavity with a movable mirror?
• RQ2 . What is the effective mass of the mirror when all acoustic modes are considered, and how does it compare to the physical mass?
• RQ3 . How does the optical beam waist influence the effective mass in a plano-convex resonator geometry?
• RQ4 . Can the same effective susceptibility describe both thermal fluctuations and radiation pressure effects?
• RQ5 . To what extent does a reduced effective mass enhance quantum optomechanical effects such as phase shifts and noise reduction?

Key findings

• . The effective mass of the mirror in the optomechanical system is significantly smaller than its physical mass, especially for small optical beam waists.
• . For a plano-convex resonator, the effective mass scales inversely with the optical waist size, becoming very small when the waist is small.
• . The effective susceptibility derived from the full acoustic mode structure accurately describes both thermal and radiation pressure effects in a unified framework.
• . The effective response can be approximated as a harmonic oscillator at low frequencies, with the effective mass dominating the quantum coupling strength.
• . Quantum effects such as radiation pressure-induced phase shifts and noise reduction are enhanced when the effective mass is reduced, as the coupling strength is inversely proportional to effective mass.
• . The model predicts that quantum optomechanical effects are most pronounced when the effective mass is minimized, which occurs for tightly focused beams in suitable resonator geometries.