Coherent scattering has recently attracted attention as a means of controlling the motion of levitated particles in three dimensions using a single optical cavity. In these systems, scattering of photons from the trapping field to a cavity mode has been used to cool all three modes of the centre-of-mass motion of levitated particles. The possibility of employing coherent scattering for more general quantum control has, however, not yet been discussed in the literature. Here, we present strategies for generating nonclassical correlations and for engineering interactions between motional modes of levitated particles using coherent scattering. We expand the theory developed by Gonzalez-Ballestero et al. to realize more general bilinear interactions in levitated optomechanics with coherent scattering. Going beyond the simple stationary picture, we introduce amplitude modulation as an important tool to modify the optomechanical interaction and discuss how it can be used to resonantly enhanced certain parts of the interaction, allowing, for example, strong one- and two-mode squeezing of motion. Our results thus show the potential of using coherent scattering for full quantum control of the motion of levitated particles.

1. Motional Gaussian states and gates for a levitating particle
Ondřej Černotík and Radim Filip
Department of Optics, Palacký University Olomouc,
17. listopadu 1192/12, 77146 Olomouc, Czechia
ondrej.cernotik@upol.cz
Squeezing in one dimension
Comparison to conventional optomechanics
Convetional optomechanics
relies on a dispersive shift of
the cavity resonance, using
tweezer only for trapping.
Conventional opto-
mechanics requires one
cavity per motional
mode for control. Full
control over the motion
needs also interactions
between mechanical
modes which are
difficult to engineer.
Coherent scattering
Optomechanical interaction is mediated by tweezer
photons scattered off the particle [1]. This coupling
enables control of all three centre-of-mass modes using
a single cavity mode [2,3].
Tunability of interactions
intracavity intensity
radial coupling
axial coupling
The coupling strength for axial and radial
modes is controlled by positioning the particle
within the cavity standing wave. The relative
strength for the two radial modes is set by the
polarization of the cavity and tweezer fields.
Highlights
Future plans
Multidimensional motion
Applications
creating genuine three-mode entanglement
novel states for quantum sensing below the standard
quantum limit and testing quantum mechanics
ultimate goal: full quantum control of particle motion
Nonlinear motion
higher-order optomechanical coupling [4]
nonlinear potential with non-Gaussian tweezers [5]
coupling to two-level systems [6]
cavity-mediated interactions
Radial and axial mode:
Two radial modes:
free oscillations and thermal noise cavity-mediated collective dissipation
The radial and axial modes couple to orthogonal field quadratures which has important implications for the effective interactions.
Interactions between mechanical modes can be induced by virtual photons (two radial modes), collective dissipation, or
measurement-based feedback. This is not straightforward with standard dispersive optomechanics.
References
[1] C. Gonzalez-Ballestero et al., PRA, 100, 013805 (2019).
[2] U. Delić et al., PRL 122, 123602 (2019).
[3] D. Windey et al., PRL 122, 123601 (2019).
[4] U. Delić et al., arXiv:1902.06605.
[5] M. Šiler et al., PRL, 121, 230601 (2018).
[6] G.P. Conangla et al., Nano Lett. 18, 3956 (2018).
10 2
10 1
100
Modulation depth
10 6
10 4
10 2
100
Squeezingdegree
100
101
SqueezedvarianceVsq
101
104
107
Thermal noise n
100
101
102
10 2
10 1
100
Sideband ratio / m
0.5
1.0
1.5
Basic Hamiltonian
Equations of motion
Adiabatic elimination of cavity dynamics
Assuming a large detuning between the tweezer and cavity
frequencies, , we get the effective mechanical dynamics
with the effecitve rates
10 3
10 2
10 1
100
Modulation depth
10 3
10 2
10 1
Effectiveparameters
|eff|/m,eff/m
0 250 500 750 1000
Time mt
10 6
10 3
100
Squeezingdegree
100
10 4
10 2
100
102
104
Initial temperature n0
10 1
100
101
102
SqueezedvarianceVsq
0 /2 3 /2 2
Modulation phase
10 1
100
Amplitude modulation of the tweezer
Modulation of the tweezer field results in parametric amplification of the motion and modulation of the optomechanical
coupling according to
with modulation depth and phase .
Adiabatic elimination of the cavity field now leads to the
effective mechanical dynamics
with
, leading to a lower instability
threshold and stronger squeezing.
When we choose the detuning ,
the cavity field couples to the Bogoliubov mode
via the Hamiltonian
where .
Combination of parametric and dissipative squeezing
leads to stronger squeezing than either technique
alone.
parametric
squeezing dissipative
squeezing
this approach opens the way towards full control of particle
motion via coherent scattering
strong mechanical squeezing possible with parametric
amplification and coherent scattering
prospect of interactions between all centre-of-mass modes
, stable
, unstable
state-of-the-art parameters [2]
state-of-the-art parameters [2]
dissipative squeezing alone
state-of-the-art parameters [2] (top)
and enhanced coupling (bottom)
0 5 10 15 20
0.90
0.95
1.00
Squeezingdegree
1.0
1.2
1.4
0 100 200 300 400 500
Time mt
10 6
10 3
100
100
101
10 3
100
103
Initial temperature n0
10 1
100
101
102
103
SqueezedvarianceVsq
state of the art
unstable
unstable
stable