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.

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.