page_title: Cavity optomechanics
section_title: Equations of motion
hierarchical_section_title: Cavity optomechanics / Concepts of cavity optomechanics / Mathematical treatment / Equations of motion
caption_attribution_description: English: A plot of the optically induced damping of a mechanical oscillator in an optomechanical system.
context_page_description: Cavity optomechanics is a branch of physics which focuses on the interaction between light and mechanical objects on low-energy scales. It is a cross field of optics, quantum optics, solid-state physics and materials science. The motivation for research on cavity optomechanics comes from fundamental effects of quantum theory and gravity, as well as technological applications.
  The name of the field relates to the main effect of interest, which is the enhancement of radiation pressure interaction between light and matter using optical resonators. It first became relevant in the context of gravitational wave detection, since optomechanical effects have to be taken into account in interferometric gravitational wave detectors. Furthermore, one may envision optomechanical structures to allow the realization of Schrödinger's cat. Macroscopic objects consisting of billions of atoms share collective degrees of freedom which may behave quantum mechanically, e.g. a sphere of micrometer diameter being in a spatial superposition between two different places.
context_section_description: From the linearized Hamiltonian one can, adding dissipation and noise terms to the Heisenberg equations of motion, derive the so-called linearized quantum Langevin equations, which govern the dynamics of the optomechanical system.
  Here  and  are the input noise operators (either quantum or thermal noise) and  and  are the corresponding dissipative terms. Note that for optical photons, thermal noise can be neglected due to the high frequencies, such that the optical input noise can be described by quantum noise only (this does not hold for microwave implementations of the optomechanical system). For the mechanical oscillator thermal noise has to be taken into account and is the reason why many experiments are placed in additional cooling environments to lower the ambient temperature.
  Rewritten in frequency space (i.e. a Fourier transform is applied), these first order differential equations become easily solvable.
  
  Two main effects of the light on the mechanical oscillator can then be expressed in the following way:
  The former is termed the optical-spring effect and may lead to significant frequency shifts in the case of low-frequency oscillators, such as pendulum mirrors. In the case of higher resonance frequencies,  MHz, it does not significantly alter the frequency. For a harmonic oscillator, the relation between a frequency shift and a change in the spring constant originates from Hooke's law.
  The latter equation shows optical damping, i.e. the intrinsic mechanical damping  becomes stronger (or weaker) due to the optomechanical interaction. From the formula one can see that for negative detuning and large coupling, the mechanical damping can be greatly increased, which corresponds to cooling of the mechanical oscillator. In the case of positive detuning the optomechanical interaction leads to a negative contribution to the effective damping. This can lead to instability, when the effective damping drops below zero, , which means that it turns into an overall amplification rather than a damping of the mechanical oscillator.
caption: The optically induced damping of the mechanical oscillator that adds to the intrinsic mechanical damping.
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