Parametric feedback cooling of levitated optomechanics in a parabolic
mirror trap
Vovrosh, J., Rashid, M., Hempston, D., Bateman, J., Paternostro, M., & Ulbricht, H. (2017). Parametric feedback
cooling of levitated optomechanics in a parabolic mirror trap.
Journal of the Optical Society of America B: Optical
Physics
,
34
(7), 1421-1428. https://doi.org/10.1364/JOSAB.34.001421
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Journal of the Optical Society of America B: Optical Physics
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Download date:10. Aug. 2022
Parametric Feedback Cooling of Levitated Optomechanics in a Parabolic Mirror Trap
Jamie Vovrosh,
1
Muddassar Rashid,
1
David Hempston,
1
James
Bateman,
1, 2
Mauro Paternostro,
3
and Hendrik Ulbricht
1, ∗
1
Department of Physics and Astronomy, University of Southampton, SO17 1BJ, United Kingdom
2
Department of Physics, College of Science, Swansea University, Swansea SA2 8PP, UK
3
Centre for Theoretical Atomic, Molecular, and Optical Physics,
School of Mathematics and Physics, Queen’s University, Belfast BT7 1NN, United Kingdom
Abstract: Levitated optomechanics, a new experimental physics platform, holds promise for
fundamental science and quantum technological sensing applications. We demonstrate a simple and
robust geometry for optical trapping in vacuum of a single nanoparticle based on a parabolic mirror
and the optical gradient force. We demonstrate rapid parametric feedback cooling of all three
motional degrees of freedom from room temperature to a few mK. A single laser at 1550nm, and a
single photodiode, are used for trapping, position detection, and cooling for all three dimensions.
Particles with diameters from 26nm to 160nm are trapped without feedback to 10
−5
mbar and with
feedback-engaged the pressure is reduced to 10
−6
mbar. Modifications to the harmonic motion in
the presence of noise and feedback are studied, and an experimental mechanical quality factor in
excess of 4×10
7
is evaluated. This particle manipulation is key to build a nanoparticle matter-wave
interferometer in order to test the quantum superposition principle in the macroscopic domain.
Introduction: Levitated optomechanics, where light
is used to trap a single particle and manipulate its centre-
of-mass motion, holds promise for unprecedented pre-
cision in the control of massive nanoparticles in vac-
uum [1]. Such precision will be key for the development
of applications in sensing of magnetic spin resonance [2],
investigating gravitational effects and detecting inertial
forces. Background-free spectroscopy experiments with
single particles will benefit from such exquisite control.
Moreover, new avenues through which to test fundamen-
tal science will become possible, such as interferometric
and non-interferometric tests of the quantum superposi-
tion principle [3] as well as the test of the interplay be-
tween quantum theory and gravity [4–7]. Recently the
motion of single micro/nanoparticles has been optically
detected and cooled by linear [8] and parametric feed-
back [9]. One experiment has already reached the single
photon recoil limit [10]. The motion of neutral particles
has been cooled by cavity-assisted methods [11–13]. For
charged particles, the use of Paul traps combined with
optical cavities has led to temperatures below one Kelvin
in high vacuum (' 10
−6
mbar) [14]. Elsewhere, the ro-
tation and torsional motion of nanostructures has been
studied and controlled [15–17]. Further the first demon-
stration of squeezing the classical thermal state has been
reported [18].
In this paper, we demonstrate a single-mirror trap ge-
ometry which has sufficient position resolution to ap-
proach a low-phonon state of the centre-of-mass motion
of the trapped nanoparticle by parametric feedback cool-
ing alone. Through this mechanism, we have been able
to reduce the amplitude of the motion of the centre-
of-mass of the particle to energies corresponding to ef-
fective temperatures of just 1mK. Correspondingly, our
trapped nanoparticle is prepared in motional states ther-
mally populated by ∼ 150 phonons in average.
Results: The particle is trapped by the optical gradi-
ent force, and detected by light scattered in the Rayleigh
regime (cf. Fig. 1)) for an illustration of the experi-
mental setting). The reflective geometry used in our
experiment allows for very tight focusing via a large
numerical aperture, which minimises the optical scat-
tering force (radiation pressure) for sub-micron parti-
cles. In the limit where the particle is undergoing small-
amplitude oscillations, the trapping potential is harmonic
and the motion along the three spatial directions (la-
belled as x, y, and z) is decoupled. Under these condi-
tions, motion along a given spatial direction can be char-
acterised by a trap frequency ω
0,{z,x,y}
=
p
k
0,{z,x,y}
/m,
which is defined by the particle mass m and the optical
spring constant k
0,z
= 2αP /(cπ
0
w
6
f
/λ
2
) (z-direction)
and a different spring constant in the x, y directions
due to the different gradient of the Gaussian light fo-
cus: k
0,{x,y}
= 8αP /(cπ
0
w
4
f
) for a polarisable particle
that is dipole-trapped in a focused Gaussian laser beam.
Here P is the trapping laser power, α the polarisability
of the particle, w
f
the waist of the trapping laser beam
at the focus, c the speed of light, λ is the wavelength
of the trapping laser and
0
the permittivity of vacuum.
For large amplitude oscillations, behaviour is anharmonic
and motional modes are coupled, as has been observed
in both low vacuum and high vacuum [19].
We control the particle motion by feedback modula-
tion of the optical spring constant, which is the control
parameter of the motion, via variation of the trapping
laser power. We define the depth of this modulation as
η = I
fb
/I
0
where I
0
is the laser intensity without feed-
back and I
fb
is the amplitude of the feedback modulation.
arXiv:1603.02917v3 [quant-ph] 23 Jan 2017
2
Vacuum Chamber
Parabolic
Mirror
λ/2
λ/4
3 Lock in amplifiers
X Y Z
Photodiode
1550nm
Laser
AOM
EDFA
FIG. 1. Trapping and controlling the motion of the particle. a) Time trace of the particle’s three-dimensional motion
in the trap, the total signal measured by a single photodiode detector. The particle’s motion along the (x,y,z)-directions
modulates the phase of the backscattered light re-collimated by the parabolic mirror. The modulated light is measured by a
single photodiode and the position is calculated from the fit in Fig.1c) (see supplement S1 for details). No feedback is applied
and the pressure is 7×10
−2
mbar. While the spring constant for x and y are symmetric, the respective spectral peaks are
separated by manipulation of the polarisation of the trapping laser. b) Optical setup for parametric feedback cooling. The
three-dimensional position of a trapped nanoparticle is measured using a single photodiode. The signal is frequency doubled
and phase shifted using three lock-in amplifiers. The sum of these signals is used to modulate the intensity of the trapping laser
by an acusto optical modulator (AOM). The modulated signal is amplified in an optical Erbium doped fibre amplifier (EDFA)
(see supplement S6 for more details). c) Power spectral density (PSD) of time trace shown in a). The plot shows the zoom
into the motion in z-direction. The PSD is fitted to Eq.(2), which is used to extract information about the trapped particle
and its motion (see supplement S1 for details). d) We show the incident and outgoing light fields at the parabolic mirror. The
detection of the position of the particle is using optical interferometry between the light scattered off the particle E
scat
and
the diverging reference E
div
. Here the position of the detector is chosen so that the amplitudes of both fields are comparable,
unlike the standard homodyne detection scheme (c.f. supplement S2 for details on position resolution of this interferometric
scheme).
Such intensity modulation provides a time-varying, non-
conservative motional damping resulting in small oscilla-
tion amplitudes. The equation of motion for the trapped
particle in the x-direction is
¨x(t) + Γ
0
˙x(t) +
k
0
+ k
fb
(t)
m
x(t) =
F
th
(t)
m
, (1)
where Γ
0
is the damping rate of the particle’s motion. We
have introduced F
th
as a stochastic driving term arising
from those same sources origin to the motional damping
Γ
0
- at the parameter range of this experiment dominated
by collisional mechanisms - and k
fb
(t) as the variation
in trap stiffness caused by parametric feedback. Simi-
lar equations hold for the particles motion in y and z
directions.
To achieve parametric feedback, unlike previous exper-
3
a) b)
c)
d)
FIG. 2. Analysing cooling rate δΓ and damping rate Γ
0
. a) We show an instance of PSD with x, y, z peaks at
temperatures T
cm
taken simultaneously. The red data represents a PSD without feedback at 300 K at a pressure of 7 × 10
−2
mbar. The blue data is a PSD of a feedback cooled particle at 6 × 10
−6
mbar. The lowest temperatures reached in this
experiment are indicated in the blue graph. The motion in x, y, z-directions are cooled with independent feedback parameters.
The grey data represents the noise floor of our system. b) Shows a more detailed variation of the feedback cooling of y at
different pressures. Feedback parameters are optimised to maximise the cooling effect. We observe a frequency shift for PSDs
taken at different motional temperature T
cm
and background pressure, p. c) Shows the translational temperature T
cm
for the
motion in x, y, z-direction vs pressure, p. An extrapolation of the fit (shown in dashed-lines) through the cooling rate shows
that for this data set the ground state temperature (intercept with horizontal dashed lines) could be reached for the pressures
of 10
−8
to 10
−9
mbar; if photon recoil and other sources of noise can be overcome [10]. d) The damping rate Γ
0
as extracted
from fitting a Lorentzian to the PSD (see supplement S1 for details), here is shown for x, y and z-motion for the same data
set as in Fig.2c). The dashed-lines with associated colours are fitting of environmental damping being linearly proportional
pressure. The lowest damping rate obtained is 2 mHz at 6 × 10
−6
mbar. The linear fit includes the pressure dependency only,
while other effects may play a role at low pressure.
iments [8, 9] we lock an external oscillator to the parti-
cle’s motion by means of a phase-locked loop (PLL) as
also done in [10]. For a system with a high mechanical
quality factor but with significant noise by damping fac-
tor Γ
0
, this permits an accurate tracking of the motional
phase of the mechanical harmonic oscillator, and allows a
sinosoidal feedback to be applied relative to the noise in-
duced phase jumps. In more detail, our system is driven
by white noise, yet it can respond only to frequencies
that fall within its bandwidth. Our capabilities of track-
ing the individual trajectories of the trapped nanopar-
ticle as it undergoes collisions with the surrounding gas
opens up the possibility for the assessment of the stochas-
tic thermodynamics of the system. Therefore the phase
of the mechanical oscillator can be extracted from the
externally locked oscillator, and we implement paramet-
ric feedback with a controlled phase shift relative to the
particle motion, in close analogy with early work on mi-
crocantilevers [20]. The trapped particle, which is driven
strongly out of equilibrium by our driving and feedback
mechanism embodies a very promising candidate for the
study of dynamical processes and (quantum) fluctuation
4
theorems [21–23], while classical studies have been done
already [19]. Such a study could turn out to be very in-
formative for the characterisation of the properties of the
environment into which the particle is effectively embed-
ded [24].
The corresponding form of the power spectral density
(PSD) of the oscillation is then straightforward to cal-
culate from Eq.(1), leading to the modified Lorentzian
response
S
x
(ω) =
k
B
T
0
πm
Γ
0
([ω
0
+ δω]
2
− ω
2
)
2
+ ω
2
[Γ
0
+ δΓ]
2
, (2)
with k
B
the Boltzmann constant. The interpretation of
this result is then straightforward: by tuning the modu-
lation depth of the trap and the relative phase between
feedback signal and particle’s motion, we are able to
change the sign and amplitude of both δΓ and δω, thus
tuning the effective dynamics of the particle from an un-
derdamped motion to an overdamped one, and blue or
red-detuning its oscillation frequency with respect to that
of the trap. This demonstrates the full control that we
have managed to achieve over the motion of the nanopar-
ticle.
Assuming equilibration at the steady state of the mo-
tion, the equipartition theorem provides an effective tem-
perature of the centre-of-mass motion given by T
cm
=
m(ω
0
+ δω)
2
hx
2
(t)i/k
B
. As δω ω
0
for all the exper-
imental conditions of our work, we can write, T
cm
=
T
0
Γ
0
/(Γ
0
+ δΓ), where T
0
is the equilibrium tempera-
ture in the absence of parametric feedback cooling, which
is here assumed to be 300K. We have experimentally
checked the motion of the particle for different trapping
laser powers and observed no dependency on such param-
eter. We instead observe a constant motional tempera-
ture, which verifies our assumption that the motion of the
particle without active feedback is thermalised at room
temperature. We use the equation for T
cm
to evaluate the
centre of mass temperature from fitting the Lorentzian
Eq.(2) to the power spectral densities (see Supplement
S1 and S7 for more details).
Discussion: Particles can be trapped at atmospheric
pressure and kept optically trapped without feedback
down to pressures on the order of 10
−5
mbar, see Fig. 3c).
This indicates that using 1550nm trapping light and par-
ticles with presumably low absorption cross section min-
imises the effects of internal heating and ionisation, which
would ultimately result in trap loss. With feedback the
pressure could be reduced to the limits of the present
setup. We trap particles of diameter ranging from 26nm
to 160nm at comparable trapping laser powers.
We evaluate the radius r and mass m of the particle
from fitting Eq.(2) to PSDs. For typical data we get
m=3.3(±0.7) × 10
−19
kg and r = 32(±6)nm (see supple-
ment S1 and S3 for details).The large error bars result
from a large uncertainty in the measurement of the pres-
sure in the vacuum chamber. To circumvent such limita-
tions and to perform a parameter independent measure-
ment we perform a further experiment to directly mea-
sure the amplitude of particle motion by scanning the
wavelength of the trapping laser to modulate the relative
amplitudes of the first and second harmonic PSD peaks
of the z-motion of a trapped particle (see supplementary
section S2 for more details). By comparing the change
in amplitude between the first and second harmonic we
are able to measure the amplitude of particle motion
within the trap to be 119(±10)nm. From the equipar-
tition theorem: m = k
b
T
cm
/ω
2
0
z
2
0
, with T
cm
= 300K we
can obtain a pressure independent measure of the par-
ticles mass m = 3(±0.5) × 10
−19
kg and thus a particle
radius r=30(±2)nm. We evaluate the position resolution
to be S
x,exp
= 200fm/
√
Hz(±8%) for the same 60nm di-
ameter particle as used for the wavelength scan. This
position resolution is sufficient to measure a zero-point
motion ∆x = 2.6pm with an integration time of one sec-
ond.
The feedback compensates the thermalisation effect by
gas collisions and stabilises the motion at a lower tem-
perature. However, the minimum temperature that can
be reached is dependent on the pressure of the system as
shown in Fig. 2c). This dependence is because the pres-
sure damping from gas collisions acts as the dominant
heating mechanism in our system. The motion of the
particle can be cooled and the lowest temperatures T
cm
achieved are on the order of 1mK for x,y and z simul-
taneously at the lowest achieved pressure in the vacuum
chamber of 1×10
−6
mbar. The temperature reached is
limited by both the pressure and the overall noise floor
present in the experiment. An extrapolation of this pres-
sure dependence indicates that motional ground state can
be achieved at a pressure on the order of 10
−9
mbar, which
is technically feasible although not implemented in our
current setup.
The collisional damping rate Γ
0
is analysed for differ-
ent pressures in Fig. 2d), and found to be dominating
the width of the PSD ∆ω and used to analyse the me-
chanical factor as Q
m
= ω
0
/∆ω. Q
m
is expected to be
exceptionally high for the motion of a trapped particle
[19], assuming the relation, Q
m
= ω
0
/Γ
0
, to hold and a
linear decrease of Γ
0
with decreasing pressure. Experi-
mentally we find Q
m
=4.4×10
7
from the respective colli-
sional damping data at 6×10
−6
mbar, where the damping
rate is measured to be Γ
0
/(2π)=2mHz. Assuming that
this scaling can be extrapolated to ultra-high vacuum, we
find a maximum Q
m
=10
12
at 10
−9
mbar, in agreement
with previous studies [19].
Feedback cooling rates achieved are on the order of
100Hz depending on the feedback modulation depth η
and phase φ. Parametric feedback is by definition the
modulation of the optical spring constant k
trap
at twice
the trap frequency. Here we report on an effect of the
phase of the feedback on the temperature and frequency
of the motion of the trapped particle, see Figs. 3a) and
