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Cavity optomechanics with variable polarizability mirrors

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Cavity optomechanics with frequency-independent high-reflectivity mirrors shows different operation regimes distinguished by the ratio of the mechanical frequency and the photon loss rate. Working in the resolved-sideband regime thus enables cooling or amplification of the mechanical motion while optomechanical systems in the bad-cavity limit can be used to efficiently measure the mechanical motion. The use of mirrors with frequency-dependent reflectivity can bring new, interesting effects, such as Doppler cooling of the mechanical motion or modification of the sideband ratio. Here, we develop a full quantum theory of cavity optomechanics where the mechanically compliant mirror has reflectivity that strongly depends on the frequency of the incident light and identify regimes where these new optomechanical effects can be observed. These results are relevant for mirrors formed by self-assembled two-dimensional atomic layers, where the reflectivity is sharply peaked around the internal resonance of the atoms, or for structured membranes with engineered spatial defects.

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Cavity optomechanics with variable polarizability mirrors

  1. 1. Cavity optomechanics with variable polarizability mirrors 1 Max Planck Institute for the Science of Light, Erlangen, Germany 2 Department of Physics and Astronomy, Aarhus University, Aarhus, Denmark Ondřej Černotík,1 Aurélien Dantan,2 and Claudiu Genes1 Variable reflectivity Motivation The sideband ratio is one of the most crucial parameters of optomechanical systems. Especially with microcavities, it is challenging to reach the resolved- sideband regime—characterized by the cavity linewidth being smaller than the mechanical frequency—owing to short roundtrip time for the cavity field. We show that the sideband resolution can be improved by using mirrors with internal resonances. The internal resonance (caused by, for example, an array of emitters inside the reflector) results in strongly frequency-dependent reflection which leads to a drastically improved cavity linewidth. [1] R. Bettles et al., Phys. Rev. Lett. 116, 103602 (2016); E. Shahmoon et al., Phys. Rev. Lett. 118, 113601 (2017). [2] S. Fan and J. Joannopoulos, Phys. Rev. B 65, 235112 (2002). [3] SEM image of a high-contrast grating (50 μm2 ) patterned on a 500 μm2 suspended SiN membrane from Aarhus University. [4] R. Lang et al., Phys. Rev. A 7, 1788 (1973). Future directions Description of cavities with lossy mirrors Dispersive and dissipative optomechanical effects Non-Markovian dynamics in optomechanical systems We consider a one-sided cavity and quantize the electromagnetic field using the modes of the universe [4]. We write the positive-frequency component of the electric field as with field modes . The mode functions satisfy the Helmholtz equation where is the dielectric function. We can now use the electric field to recover the total Hamiltonian, Modes of the universe Cavity and external fields We can separate the integration into three regions—inside the cavity, outside, and at the boundary between them. We can then identify the cavity and external fields and their coupling From the input–output relation, we can obtain the linewidth for the cavity modes, , which agrees with the spectral properties of the mode functions inside the cavity. Optomechanical interaction We can repeat the analysis for a cavity with a displaced input mirror. The mode functions will be modified but one can still find the cavity and external fields and their coupling. The dispersive and dissipative optomechanical effects are captured by the Heisenberg–Langevin equation for the cavity field, where is the optical frequency shift per displacement. We consider structured membranes [2] in which dispersion does not lead to loss. The coupling rates and are not independent since the coupling of the resonance to the modes of the universe on both sides is symmetric. The dynamics of the cavity field can be obtained by eliminating the external field and the guided resonance from the equations of motion. Internal resonances Internal resonances occur when a mirror is doped (or formed) by regularly spaced quantum emitters [1] or when it is structured by a periodic grating [2,3]. The reflectivity around such a resonance is strongly enhanced by interference. Using such reflectors in optical cavities can lead to narrow, non-Lorentzian lineshapes. How can such systems be described? What optomechancial properties do they have? Detuning Cavity transmission Normal cavity With internal resonance Mechanical sidebands Displacement Description of strongly dispersive membranes is more difficult. Dispersion and absorption are related via the Kramers–Kronig relations; loss must be accompanied by noise. These effects have to be included in any rigorous model. [2,3]. The reflectivity around such a resonance is strongly enhanced by interference. Using such reflectors in optical cavities can lead to narrow, non-Lorentzian lineshapes.

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