arXiv:quant-ph/0607068v2 8 Nov 2006
Self-cooling of a micro-mirror by radiation pressure
S. Gigan
1,2
, H. R. B¨ohm
1,2
, M. Paternostro
2,†
, F. Blaser
1,2
, G. Lange r
3
, J.
B. Hertzberg
4,5
, K. Schwab
4,‡
, D. B¨auerle
3
, M. Aspe lmeyer
1,2∗
, A. Zeilinger
1,2
1
Physics Faculty, Institute for Experimental Physics,
University of Vienna, Boltzmanngasse 5,
A-1909 Vienna, Austria
2
Institute for Quantum Optics and Quantum Information (IQOQI),
Austrian Academy of Sciences,
Boltzmanngasse 3, A-1090 Vienna, Austria
3
Institute for Applied Physics,
Johannes-Kepler-University Linz,
Altenbergerstr. 69, A-4040 Linz, Austria
4
Laboratory for Physical Sciences,
University of Maryland, College Park, MD 20740 USA
5
Department of Physics, University of Maryland,
College Park, MD 20740, USA
†
permanent address: School of Mathematics and Physics,
Queen’s University Belfast, UK
‡
present address: Cornell University, USA
(Dated: February 1, 2008)
We demonstrate passive feedback cooling of a mechanical resonator based on radiation pressure
forces and assisted by photothermal forces in a high-finesse optical cavity. The resonator is a
free-standing high-reflectance micro-mirror (of mass m ≈ 400 ng and mechanical quality factor
Q ≈ 10
4
) that is used as back-mirror in a detuned Fabry-Perot cavity of optical fin esse F ≈ 500.
We observe an increased damping in the dynamics of the mechanical oscillator by a factor of 30 and
a corresponding cooling of the oscillator modes below 10 K starting from room temp erature. This
effect is an important ingredient for recently proposed schemes to prepare quantum entanglement
of macroscopic mechanical oscillators.
INTRODUCTION
Cooling of mechanical resonators is cur rently a hot topic in many fields of physics including ultra-high precision
measurements [1], detection of gravitational waves [2, 3] a nd the study of the transition b etween classical and q uantum
behavior of a mechanica l system [4, 5, 6]. Here, we report the first observation of self-cooling of a micro-mirror by
radiation pressure inside a high-finesse optical cavity. In essence, changes in intensity in a detuned cavity, as caused
by the thermal vibration of the mirror, provide the mechanism for entropy flow from the mirror’s oscilla tory motion
to the low-entro py cavity field [2]. T he crucial coupling betwe e n radiation and mechanical motion was made possible
by producing free-standing micro-mirrors of low mass (m ≈ 400 ng), high re flec tance (>99,6%)and high mechanical
quality (Q ≈ 10
4
). We observe cooling of the mechanical oscillator by a factor of more than 30, i.e. from room
temper ature to be low 10 K. In addition to purely pho tothermal effects [7] we identify radiation-pressure as a relevant
mechanism participating to the c ooling . In contrast to earlier expe riments, our technique does not need any active
feedback [8, 9, 10]. Our results suggest that it should be possible to reach very low temperatures. We expect that
improvements of our method will allow for pure radiatio n pressure cooling, with cooling ratios beyond 1.000, and thus
possibly enable cooling all the way down to the quantum mechanical ground state of the micro-mirror.
Radiation pressur e forces inside optical cavities are known to pose an ultimate limit on the sensitivity of interfero-
metric measurements [11, 12]. However, less known, radiation pressure can also be used for the opposite, namely to
counteract the dy namics of a cavity mirror via dynamical ba ck action [2, 7, 13]. In a recent experiment Metzger and
Karrai [7] presented a passive c ooling mechanism for a micro-mechanical oscillator based on bolometric back action.
Even though this scheme has intrinsically limited cooling capability since it ultimately relies on heating by absorption,
it may allow for a quantitatively significant reduction of the oscillator’s thermal motion. A more powerful scheme
is provided by the use of radiation pressure as a feedback force [2]. In this case, o ptical absorption does not impose
a fundamental limit. The difficulty in utilizing radiation pres sure for this cooling purpose is that it requires stable
∗
Corresponding author: markus.aspelmeyer@quantum.at
2
control of the detuning of a high-finesse c avity, strong optomechanical coupling and a low mass of at leas t one cavity
mirror, hence nano- or micro-mechanical systems of high optical and mechanical quality (characterized by the cavity
finesse F and the mechanical quality factor Q). Although cavity-induced radiation-pressure effects have already been
used to modify elastic pr operties of mirro rs [7, 14, 15, 16] and to enforce mechanical instabilities [14, 17, 18, 19, 20],
none of the previous experiments was able to combine these strict requirements. We have overcome this limitation
by developing a method to produce free-standing micro-mirrors of low mass (Q ≈ 400ng) high r e flec tance (>99.6%)
and high mechanical quality (Q ≈ 10
4
). Using s uch micro-mirrors in a detuned o ptica l cavity allows us to observe
for the first time self-cooling in a regime where, a lthough photother mal effects are still present, radiation pressure
significantly participates in the self-cooling process.
IDEA OF RADIATION-PRESSURE COOLING
Radiation pressure forces in an optical cavity arise due to the momentum transfer of photons reflected from the
mirror surface. For certain cavity detuning, i.e. if the cavity angular frequency ω
c
is off resonance with the frequency
ω
l
of the pump laser, the radiation pressure is highly sens itive to small displacements of the cavity mirror. This is
a conseq uence of the fact that the energy stored in a cavity field varies strongly with detuning. As a conse quence,
the dynamics of an oscilla ting mirror inside a detuned cavity is modified by a mechanical rigidity tha t depends o n
the detuning. For a high- finesse cavity, the radiation-press ure induced back action can act on the mirror motion
in a way to induce low noise damping. This is the general concept o f dynamical back actio n which has first been
introduced by Braginsky [13]. A simple classical description of the dynamics of the mirror shows that both the
resonance frequency ω
M
and the natural da mping rate γ of the mirror motion are modified by radiation pressure to
ω
eff
and γ
eff
, respectively [2, 7]. In particular, within the clas sical framework, the modified damping rate follows
γ
eff
= γ +
β(∆)
2m
2κ
(2κ)
2
+ ω
2
M
(1)
with the cavity decay rate κ = πc/2F L, the cavity finesse F , the cavity length L and the vacuum speed of light c.
Optimum damping is achieved when 1/ κ is of the order of ω
M
, which for ω
M
in the MHz range requires a high finesse
cavity. Equation (1) depends on β(∆), the s patial gradient of the radiation fo rce evaluated at a (spatial) detuning
∆
x
= L∆/ω
l
. Here, ∆ is the effective detuning between cavity and laser frequency, including the effect of radiation
pressure [21]. The contribution β, induced by radiation pressure, can be positive or negative depending on the sign
of ∆. It is straightforward to show that β(∆) is negative for ∆ < 0, corresponding to γ
eff
< γ. In this regime, the
system can enter into instability. The focus of this work is the investigation of the opposite regime (β(∆) > 0) in
which γ
eff
> γ. This low-noise damping results in a reduction of the mirror temperature and hence self-cooling is
achieved. The previous self-cooling experiments based on bolometric forces [7] were operated in the regime of negative
detuning where radiation pressure counteracts the cooling.
To observe the self-cooling effect a read-out scheme of the mirror motion is required. To do that it turns out that
it is sufficient to measure the statistica l properties of the optical field that leaks out of the cavity. In a way, the
output cavity field re presents a ”blank paper” on which the dynamics of the mirror can be written. It is p ossible to
briefly sketch the main idea of our self-cooling read-out pro cess by exploiting a simple (but for our purposes sufficient)
semiclassical picture. A full quantum mechanical framework, which generalizes the classical picture for self cooling
proposed so far in the literature [2, 10], is presented else where [22]. Not only is this (more g e neral) approach in
agreement with the classic al picture taken into account by Eq. 1 but it also paves the way toward the rig orous
study of the limitations imposed to self cooling by the influences of quantum noise [22]. The total energy of a cavity
consisting of a fixed mirror and a movable mirror driven by an input las e r field of power P is given by [21, 23]
E = ~(ω
c
− ω
l
)(X
2
+ Y
2
) − ~
ω
c
2L
(X
2
+ Y
2
− 1)q +
1
2
p
2
m
+ mω
2
M
q
2
+
√
2~EY, (2)
where X and Y are the quadratures of the cavity field, p and q are the momentum and position quadratures of
the oscillating mirror, and E =
p
2κP/~ω
l
is the coupling rate between the cavity and the input laser field. If the
time-scale set by the cavity decay rate is the shortest in the dynamics of the system, i.e. κ ≫ ω
M
, the cavity field
follows the mirror motion adiabatically. As a consequence, the fluctuations δY
out
of the field leaking out of the cavity
are directly related to the fluctuations of the mirror’s position q uadrature as δY (t) = A(∆, κ, E)δq(t)[22, 24], where
we have neglected any noise in the system. For the parameter regime of our experiment the signal-to-noise ratio of
the contribution given by the mirror’s spectrum is as large as 10
7
. The dynamics of the output field quadrature is
3
thus entirely determined by the mirror motion via the function (∆, κ, E). Therefore, a phase-sensitive measurement
of the output field quadrature δY
out
is capable of ”monitoring” the full mirr or dynamics. It is particularly interesting
to measure the power spectrum, since S
Y
out
=
R
dt
′
e
iωt
′
hδY
out
(t)δY
out
(t + t
′
)i = T (∆)S
q
, where S
q
is the spectrum
of the mirror motion. In other words, the quadrature p ower spectrum of the mirror motion S
q
and of the output
cavity field S
Y
out
are directly related via a transfer function T (∆). This correspondence is at the basis of our readout-
scheme. Note that the full tra nsfer function has to take into account the sensitivity of the specific detection scheme
used. This detection strategy allows us to infer the effective temperature of the mirror Brownian motion through
the study of its displac ement power-spectrum, i.e. its frequency-dependent mean square displacement. The power
sp e c trum follows a Lorentzian dis tribution centered around fω
0
=
p
ω
2
eff
− 2γ
2
eff
with a full width at half maximum
(FWHM) w
FWHM
≈ 2γ
eff
(for ω
2
0
≫ γ
2
eff
), thus proportional to the introduced damping. The area of the power
sp e c trum, hx
2
i =
R
+∞
−∞
dωS
q
, is proportional to the mean energy hEi of the vibrational mode and hence, via the
equipartition law, to the effective temperature of the mirror, since hEi = mω
2
M
hx
2
i = k
B
T
eff
. The relative change in
area underneath the power spectrum is therefore a direct measure for the change in effective tempe rature.
EXPERIMENTAL RESULTS AND DISCUSSION
The system under investigation is a doubly clamped cantilever used as the end mirror of a linear optical cavity
driven by an ultra stable Nd:YAG laser (see figure 1). The input mirror of the cavity is attached to a piezo electric
transducer which is fed by a control loop allowing us to lock the pr e cise length of the cavity either at resonance o r
detuned (off resona nce ) w ith respect to the las e r frequency. The error-signal input to the control loop is obtained via
the Pound-Drever-Hall (PDH) technique [25]. It has been shown [24] that the PDH error signal is propor tio nal to the
phase quadrature of the output field Y
out
and hence to the mirror motion (see a bove). An intuitive way to view it is
that the er ror signal is propor tional to the variation of the cavity length. Above the cut-off fr e quency of our control
loop, the fluctuations in the error sig nal are therefore directly related to the thermal noise of the cantilever (the input
mirror is assumed to be fixed).
We measured the PDH power spectrum for different input powers and cavity detunings. The detuning was achieved
by adding an offset to the error signal. With this method, the mechanical damping can be directly measured by
determining the FWHM of the r e sonance peak of the observed mechanical mode. To obta in the e ffective temperature
of the mode one has to calculate the area underneath the resonance pe ak and to account for the sensitivity of the
error signal. This is done by normalizing the measured mirr or amplitudes by the gradient of the PDH signal. The
results are summarized in Figure s 2, 3 and 4.
Figure 2 shows the noise spectrum of the oscillator for two different detunings at 2 mW input laser power. The
width of the peak increases and the area of the peak dec reases, indicative for both overda mping and cooling of the
mechanical mode. This behavior is in full agr eement with the theor e tical model presented above. We investigate the
sp e c ific variation of both mechanical da mping and of self-cooling with detuning for different input laser powers of
1 mW and 2 mW, respectively (Figures 3 and 4). Figure 3 shows the change in width of the mechanical mode. For
positive detunings, the peak is broadened from a natural width o f 32 Hz to well above 800 Hz corresponding to an extra
damping of the mode. At large detuning values the stability of the locking limits the precision of the measurements.
For negative detuning (not shown), we observed a narrowing of the peak, associated with an amplification of the
mirror motio n (i.e. ” negative” damping), which r apidly leads to a self-oscillation region. In Figure 4 the same data
set is use d to obtain the c orresponding cooling ratio from the relative change in area of the p ower spectrum, since the
total peak area is a measure of temperature. As expected, the increas e in damping is accompanied by a cooling of the
mechanical mode. At large detuning, the cooling-effect is slightly enhanced compared to our simple model, which can
be due to the reduced the contribution of ther mal background of other oscillator modes. The best expe rimental cooling
ratio in our detuning range is above 30. Since our experiment is performed at room temperature, this corresponds to
a cooling of the mode from 300 K to below 10 K (Fig. 2).
We explicitly compare the exper imental results for positive detuning with the theore tical predictions obtained if the
effect is due only to radia tio n pressure. To do that, we have independently evaluated the effective mass participating
to the dynamics of the system, which leaves no free parameter for the evaluation of radiation-pressure forces and hence
allows a full quantitative treatment. The effective mass can be much smaller than the total mass of the cantilever
[19, 27]. ]. For our mirror, an independent assessment both via spatial tomography of the v ibrational mode and via a
calibrated reference results in a value of 22 4 ng at the probing point (see Appendix). This results in a theoretically
exp ected cooling less stro ng than the experimentally observed one. To ge t a clear, immediate fig ure of the ”strength”
of the radiation pressure effect required to replicate the experimental data we assume a fixed effective mass and allow
for variation of the input power. . We find tha t, for an effective mass of 18 (26) ng, a power 2.2 (3.3) times larger
4
FIG. 1: Sketch of the experimental setup. (a) A cavity is built between the cantilever and a regular concave mirror of 25 mm
focal length and 99.3% reflectivity. The cavity length was slightly shorter than 25 mm such as t o obtain a waist of app roximately
20 µm at the location of the surface of the cantilever. In this configuration we measured a cavity finesse of 500. To minimize
damping of the mechanical mod e due to gas friction, the cavity is placed in a vacuum chamber which is kept at 10
−5
mbar.
The cavity is pumped with a Nd:YAG laser at 1064 nm. The beam is phase modulated at 19 MHz (MOD) by a resonant
electro-optic modulator (EOM) before it is injected into the cavity via the input mirror (IM). The beam reflected from the
cavity is sent via a b eam splitter (BS) onto a high-speed PIN photodio de (PD). After amplification of the photocurrent, its
AC part is d emodulated with the initial modulation frequency to obtain the PDH error signal. This error signal is then used
to feed a low-frequency control loop (PID) to stabilize the cavity length via a piezo actuator. In addition, the error signal
is fed to a spectrum analyzer (SA) to record the dynamics of the mechanical mode. (b) The cantilever is a doubly clamped
free standing Bragg mirror (520 µm long, 120 µm wide and 2.4 µm thick) that has been fabricated by using U V excimer-laser
ablation in combination with a dry-etching process [26]. The reflectivity of the Bragg mirror is 99.6% at 1064 nm. (c) Power
spectrum of the micro-mirror. We have isolated a mechanical mode at 280 kHz with a natural width of 32 Hz, corresponding
to Q ≈ 9000. All measurements presented in this work have been made on this mode.
-600 -400 -200 0 200 400 600
1E-10
1E-9
1E-8
1E-7
1E-6
300 K
8 K
PDH noise power spectrum [mV
2
/Hz]
Relative fre quency (
ω−ω
c
)/2
π
[Hz]
∆=0
∆=0.22 κ
FIG. 2: Power spectrum of the mechanical mode at two different relative detuning levels ∆ of the cavity for an input power of
2 mW. The data is obtained from the PD H power spectrum, which is directly proportional to the displacement power spectrum
of the micro-mirror. Experimental points are taken with the spectrum analyzer, averaged over 30 consecutive measurement
runs. Solid lines are Lorentzian fits to the data. The areas obtained from the fit correspond to temperatures of 300 K and 8
K, respectively.
5
0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35
0
200
400
600
800
1000
Effective damping (FWHM) [Hz]
Relative detuni ng
∆
/
κ
1mW
2mW
FIG. 3: Radiation-pressure induced damping of mirror dynamics. We show the measured width of the mechanical mode
at 278 kHz at different detuning levels of the cavity and for input laser powers of 1 mW and 2 mW. The data is obtained
directly from Lorentzian fits on the measured power spectra of the PDH error signal. Error bars represent absolute errors
based on experimental uncertainty. Solid lines represent theoretical predictions of purely radiation-pressure effects for F = 500,
Q = 9000, an effective mass of 9 ng and input powers of 1 mW and 2 mW, respectively. The inferred effective mass of 224 ng
indicates the presence of an additional damping force of photothermal nature (see text).
0,0 0,1 0,2 0,3
10
1
30
300
Effective temperature T
eff
[K]
Cooling ratio
Relative detuni ng
∆/κ
1 mW
2 mW
FIG. 4: Self-cooling of the mechanical resonator. We show the cooling ratio on the mechanical mode as a function of detuning
and for inp ut laser powers 1 mW and 2 mW. The data is obtained as normalized area of the measured PDH power spectrum,
compensated for the detuning dependent sensitivity of the PDH cavity response. Error bars represent absolute errors based
on experimental uncertainty. The self-cooling effect increases for increasing laser power and detuning, in agreement with
the theoretical predictions (solid lines). The right axis shows the inferred effective temperature of the mechanical oscillator.
Radiation pressure contributes between 30% and 50% to the overall cooling, which is assisted by photothermal effects.
than the nominal value used in the experiment is required in order to match the theore tical predictions with both the
observed damping and cooling In other words, radiation pressure accounts for at least 30% of the observed cooling but
may be as strong as 50%, i.e. cooling by a factor between 8 and 12. We attribute the additional cooling in our setup to
the presence of photothermal e ffects. Similar to the bolometric forc e s reported in [7], differential heating of the o uter
layers of the dielectric Bragg mirror can result in time-delayed changes of the cavity length eventually introducing a
retarded forc e that can contribute to the self-cooling mechanism. In a thin-layered medium the delayed force induced
by pho tothermal effects can have typical time constants on the order of se veral tens of ns (see Appendix), fast enough
![FIG. 1: Sketch of the experimental setup. (a) A cavity is built between the cantilever and a regular concave mirror of 25 mm focal length and 99.3% reflectivity. The cavity length was slightly shorter than 25 mm such as to obtain a waist of approximately 20 µm at the location of the surface of the cantilever. In this configuration we measured a cavity finesse of 500. To minimize damping of the mechanical mode due to gas friction, the cavity is placed in a vacuum chamber which is kept at 10−5 mbar. The cavity is pumped with a Nd:YAG laser at 1064 nm. The beam is phase modulated at 19 MHz (MOD) by a resonant electro-optic modulator (EOM) before it is injected into the cavity via the input mirror (IM). The beam reflected from the cavity is sent via a beam splitter (BS) onto a high-speed PIN photodiode (PD). After amplification of the photocurrent, its AC part is demodulated with the initial modulation frequency to obtain the PDH error signal. This error signal is then used to feed a low-frequency control loop (PID) to stabilize the cavity length via a piezo actuator. In addition, the error signal is fed to a spectrum analyzer (SA) to record the dynamics of the mechanical mode. (b) The cantilever is a doubly clamped free standing Bragg mirror (520 µm long, 120 µm wide and 2.4 µm thick) that has been fabricated by using UV excimer-laser ablation in combination with a dry-etching process [26]. The reflectivity of the Bragg mirror is 99.6% at 1064 nm. (c) Power spectrum of the micro-mirror. We have isolated a mechanical mode at 280 kHz with a natural width of 32 Hz, corresponding to Q ≈ 9000. All measurements presented in this work have been made on this mode.](/figures/fig-1-sketch-of-the-experimental-setup-a-a-cavity-is-built-24tcz4zq.webp)