Cavity optomechanics with variable polarizability mirrors
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This document discusses cavity optomechanics, emphasizing the significance of improving the sideband resolution through the use of variable polarizability mirrors with internal resonances. These mirrors create frequency-dependent reflection that enhances cavity linewidth, which is essential for achieving the resolved-sideband regime in optomechanical systems. The work also addresses future directions, including the effects of lossy mirrors and non-Markovian dynamics in optomechanical interactions.
![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.](https://image.slidesharecdn.com/03postergrc-191113144214/85/Cavity-optomechanics-with-variable-polarizability-mirrors-1-320.jpg)
Cavity optomechanics with variable polarizability mirrors
- 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.