Table Of ContentOptical phase-space reconstruction of mirror motion at the attometer level
T. Briant, P.F. Cohadon, M. Pinard, and A. Heidmann
Laboratoire Kastler Brossel, Case 74, 4 place Jussieu, F75252 Paris Cedex 05, France∗
(Dated: 20 September2002)
We describe an experiment in which the quadratures of the position of an harmonically-bound
mirror are observed at the attometer level. We have studied the Brownian motion of the mirror,
both in the free regime and in the cold-damped regime when an external viscous force is applied
byradiation pressure. We havealso studied the thermal-noise squeezing when theexternal force is
parametrically modulated. We have observed both the 50 % theoretical limit of squeezing at low
gain and the parametric oscillation of themirror for a large gain.
PACSnumbers: 42.50.Lc,05.40.Jc,04.80.Nn
3
0 I. INTRODUCTION dictionsofthefluctuations-dissipationtheorem[22]. The
0
experimental setup has also been modified to include an
2
externalviscousforceappliedtothemirror,andasimilar
Optomechanical coupling, that is the cross-coupling
n analysis is given in Sec. IV in the corresponding case of
betweenthe motionofamirrorandalaserfieldreflected
a thecold-dampedregime. Ourresultsdemonstratethata
J upon it, first appeared in the context of interferometric
gravitational-wave detection [1, 2] with the existence of new equilibrium is obtained with a reduction of thermal
9
noise.
the so-called Standard Quantum Limit [3, 4, 5]. The
InSec. Vwepresentanexperimentofparametricam-
2 unique sensitivity of interferometric techniques has since
v then been used for other high-sensitivity measurements, plificationofthe thermalnoise[23], belowthe oscillation
9 such as AFM [6] or optical transducers [7, 8]. The field threshold. Theexperimentalsetupismodifiedinorderto
4 modulatethestrengthoftheviscousforce,andsqueezing
has recently encountered much attention and has gained
0 ofthe thermalnoise has been observed. The observation
a life of its own in the quantum optics community: sev-
7 ofparametricoscillationsofthemirrorisfinallydescribed
eral schemes involving a cavity with a movable mirror
0
in Sec. VI.
2 have been proposed either to create non-classical states
0 of both the radiation field [9, 10] and of the motion of
/ themirror[11,12,13],ortoperformQNDmeasurements
h II. EVOLUTION IN PHASE-SPACE
p [14]. Recent progress in low-noise laser sources and low-
- lossmirrorshavemadethefieldexperimentallyaccessible
nt [15, 16, 17]. Themirrormotionismonitoredbyanoptomechanical
a Arecenttrendinquantumopticsistofullyreconstruct displacement sensor. It relies on the sensitivity of the
u the quantumstate ofeither a mode ofthe radiationfield phase of a reflected light beam to mirror displacements.
q throughthequantumtomographytechnique[18,19]ora Monitoring such a phase-shift allows to reconstruct the
:
v trapped atom[20], but to our knowledge,no suchexper- dynamical evolution of the mirror. This phase-shift can
i iment has yet been performed at the quantum level on a be induced by various kinds of mirror motions, either
X
mechanical oscillator. external [17, 24] or internal [15, 25, 26]. The former is
r important for suspended mirrors since the excitation of
a In this paper, we present an experiment which com-
pendulummodesofthesuspensionsystemleadstoglobal
pletes the analysis of the motion of a plano-convex mir-
displacementsofthemirror.Thelattercorrespondstode-
ror given in [15, 16, 21], reconstructing the phase-space
formations of the mirror surface due to the excitation of
distributionofthemotionthroughthesimultaneousclas-
internal acoustic modes of the substrate. These various
sicalmeasurementof both quadraturesof the mirror po-
degrees of freedom have however different resonance fre-
sition. The technique is applied to a variety of states
quencies andone canselectthe mechanicalresponseof a
of motion: Brownian motion and its cold-damped and
particular mode by using a bandpass filter.
squeezed counterparts.
In our experiment we detect only frequencies around
In Sec. II we present the experimental setup used to
thefundamentalacousticresonanceofthemirrorsothat
monitor the motion of the mirror and we describe how
themechanicalresponseismainlyruledbythebehaviour
the motion in phase-space can be reconstructed. In Sec.
of this particular internal mode. The mirror motion can
III we consider the case of a free mirror at thermal equi-
then be approximated as the one of a single harmonic
librium andwe compareour experimentalresults to pre-
oscillator characterized by its resonance frequency Ω ,
M
its quality factor Q, andits mass M. This mass actually
corresponds to an effective mass which describes the ef-
∗Unit´emixtederechercheduCentreNationaldelaRechercheSci- fective motion of the mirror as seen by the light, that is
the deformation of the mirror surface averaged over the
entifique, de l’Ecole Normale Sup´erieure et de l’Universit´e Pierre
etMarieCurie;URL:www.spectro.jussieu.fr/Mesure beam spot [25, 26, 27].
2
The temporal evolution of the mirror position x(t) is sideof100mm. Suchdimensionsleadtoaresonancefre-
given by linear response theory [28]. Assuming that a quencyofthe fundamentalmodeofthe orderof2MHz.
force F is applied to the mirror, the displacement x[Ω] Theoscillator’sparametershavethefollowingvalues[15],
in Fourierspace at frequency Ω is relatedto the force by
Ω =2π 1859 kHz, M =230 mg, Q=44000. (8)
M
×
x[Ω]=χ[Ω]F [Ω], (1)
Themirrorcoatedontheflatsideoftheresonatorisused
as the end mirror of a single-ended Fabry-Perot cavity
where χ[Ω] is the mechanical susceptibility of the mir-
witha Newport high-finesse SuperMirror asinput mirror
ror. For a viscously-damped harmonic oscillator it has a
(figure 1). The whole provides a 1 mm-long cavity with
Lorentzian behaviour given by
an optical finesse = 37000. The light beam entering
F
1 the cavity is provided by a frequency-stabilized titane-
χ[Ω]= , (2)
M(Ω2 Ω2 iΓΩ) sapphirelaserworkingatλ=810nm. Thelightbeamis
M − − also intensity-stabilized and spatially filtered by a mode
where Γ=Ω /Q is the damping rate [29]. cleaner.
M
Depending on the nature of the applied force F, the
spectrum of the mirror motion will typically be peaked
around the mechanical resonance frequency Ω over a
M
frequency span Γ, which is much smaller than Ω for a
M
high-Q harmonic oscillator. To study the temporal evo-
lution of the motion in phase-space, it is preferable to
describe the motion in the rotating frame in order to re-
movetheintrinsicoscillatorydependencewithtime. This
leads to the two quadratures X1 and X2, defined by
x(t)=X1(t)cos(ΩMt)+X2(t)sin(ΩMt). (3)
FIG. 1: Experimental setup. A plano-convex mirror is used
This equation yields the expression of the two quadra- as the end mirror of a high-finesse cavity. A frequency and
tures in Fourier space, intensity-stabilized laser beam issent intothecavityandthe
phase of the reflected beam is measured by a homodyne de-
X1[ω] = x[ΩM +ω]+x[ ΩM +ω], (4) tection. A demodulation system working at the mechanical
− resonancefrequencyΩM extractsbothquadraturesofthesig-
X2[ω] = i(x[ΩM +ω] x[ ΩM +ω]). (5) nal.
− − −
QuadraturesX1(t)andX2(t)varyveryslowlywithtime
Near an optical resonance of the cavity the intracav-
(over a 1/Γ timescale), and ω which represents the fre-
ity intensity shows an Airy peak when the cavity length
quencymismatchΩ Ω betweentheanalysisfrequency
− M is scanned through the resonance, and the phase of the
andthemechanicalresonancefrequencyisalwaysconsid-
reflected field is shifted by 2π. The slope of this phase-
ered small as compared to Ω .
M shift strongly depends on the cavity finesse and for a
Expressions of the quadratures in presence of an ap-
displacement δx of the end mirror one gets at resonance
plied force can be derived from eq. (1) by inserting the
a phase-shift δϕ on the order of
expressionsofx[Ω +ω]andx[ Ω +ω]inthedefini-
M M
−
tions of X1[ω] and X2[ω]. We find after simplification, δx
δϕ 8 +δϕ , (9)
n
≃ F λ
1 1
X1[ω] = F2[ω], (6) where λ is the optical wavelength and δϕ the phase
−2MΩ (cid:18) iω+Γ/2(cid:19) n
M
− noise of the reflected beam. At high frequency, all tech-
1 1
X2[ω] = F1[ω], (7) nicalnoisescanbe suppressedandthe phasenoisecorre-
2MΩM (cid:18) iω+Γ/2(cid:19) sponds tothe quantumnoiseofthe incidentbeamwhich
−
is inversely proportional to the square root of the light
wherewehaveintroducedthetwoquadraturesF1 andF2 power.
of the applied force, defined in Fourier space by similar
The phase of the reflected field is monitored by a ho-
expressions as for X1 and X2 (eqs. 4 and 5). Both posi- modyne detection working at the quantum level. For a
tion quadratures have a Lorentzian response centered at
100µW incidentlaserbeam,wehaveshownthatthesen-
zero frequency and of width Γ.
sitivitytomirrordisplacementsislimitedbythequantum
The harmonic oscillator we consider throughout the
noise of light and it has been measured to
paperisthe high-Qfundamentalacousticmodeofamir-
ror coated on a plano-convex resonator made of fused δxmin =2.810−19 m/√Hz, (10)
silica. The resonator is 1.5 mm thick at the center with
adiameterof14mmandacurvatureradiusoftheconvex at an analysis frequency of 2 MHz [15].
3
To extract both quadratures of the mirror motion we where k is the Boltzmann constant and T the temper-
B
use a demodulation system at frequency Ω (figure 2). ature of the thermal bath.
M
Abandpassfilterofwidth10kHzfirstselectsfrequencies Thesensitivityofourexperimentishighenoughtode-
around the fundamental resonance frequency in the sig- tecttheBrownianmotion.Wehavealreadyobservedthe
nal given by the homodyne detection. The signal is then noise spectrum of the mirror displacement, which has a
mixed with two square signals at frequency Ω , one de- Lorentzian shape centered at the mechanical resonance
M
phasedby π/2 with respectto the other one. Harmonics frequencyΩ andawidthΓ[15].Theheightofthepeak
M
of the output signals are cancelled out by two electronic isatleast4ordersofmagnitudelargerthanthe sensitiv-
low-passfilters. The filters also have to cancel the signal ity defined by the quantum phase noise of the reflected
duetootheracousticmodesofboththeresonatorandthe beam (eq. 10).
couplingmirror. Theexpectedwidthofthefundamental Wehaveobservedthe temporalevolutionoftheBrow-
mode being of the order of 43 Hz and the closest mode nianmotionbysending the outputs ofthe demodulation
3 kHz above, we use second-order filters with a 500 Hz system in a digital oscilloscope. The resulting traces of
cut-off frequency. We have checkedthe transfer function the twoquadraturesX1 andX2 areplotted in the upper
of each filter in order to ensure that the dissymmetry curve of figure 3 for an acquisition time of 500 ms. The
between both channels is less than 1%. temporal trajectory in phase-space appears as a random
walk around the center (X1 =0,X2 =0).
FIG. 2: Demodulation of the position signal: the signal ϕ(t)
given by the homodyne detection is filtered and mixed with
twoπ/2-dephasedsquaresignalsatthemechanicalresonance
frequency ΩM. Low-pass filters are used to extract the low-
frequency component of thequadratures.
Note that X1 and X2 are conjugate quantum observ-
ables and therefore cannot be measured simultaneously
at the quantum level. This is not in contradiction with
our experimental setup since we extract the position
quadraturesX1 andX2 fromthephaseϕofthereflected
fieldwhichalsocontainsthe quantumphasenoiseδϕ of
n
light (eq. 9). A measurement of the quadratures is thus
contaminated by the quantum noise of light so that it
is not possible to measure simultaneously both quadra-
tures at the quantum level. For this paper this does not
constitute a limitation since the signals studied in next FIG. 3: Brownian motion of the plano-convex mirror in the
sections(thermalnoiseandexternalforce)arewayabove (X1,X2) phase-space. Full scale corresponds to±210−15 m.
the Standard Quantum Limit imposed by the measure- Top: temporal acquisition over 500 ms. Bottom: histogram
ment and back-action noises [3, 4, 5]. of the distribution acquired over 10 minutes. The numberof
cellsis256×256,thelimitofsensitivity(quantumphasenoise
of light) corresponding approximately to 1 cell.
III. OBSERVATION OF BROWNIAN MOTION
This plot can be compared to the Brownian motion
of a mirror coated on a torsion oscillator, as observed in
We present in this section the results obtained when
ref. [17] by a similar optical interferometry technique.
themirrorisfree(noexternalforce)andatroomtemper-
The largerdensity of points and the more circularshape
ature. The thermal equilibrium can be described as the
of our results is due to the larger ratio between the ac-
result of a coupling with a thermal bath via a Langevin
quisition time τ and the evolution time ( 2π/Γ) of
force FT applied on the mirror. The resulting Brownian the Brownianmaoctqioninphase-space. Inoure≃xperiment,
motion is the responseof the acoustic mode to this force
τ = 500 ms and Γ/2π = 43 Hz, so that the phase
according to eq. (1). The fluctuations-dissipation theo- acq
space is mapped approximately 20 times. In the case
remrelatesthespectrumS oftheLangevinforcetothe
T ofthe torsionoscillator,asthemechanicalresonancefre-
dissipative part of the mechanical susceptibility [22],
quencyistwoordersofmagnitudelower,Γ/2π 20mHz
≃
2k T and the phase space is mapped just once even for a
B
ST [Ω]=− Ω Im(1/χ[Ω]), (11) τacq = 80 s acquisition time. This explains why results
4
in ref. [17] look similar to the Brownianmotion of a free Acquiring data during a longer time allows to recon-
particle: ergodicityisirrelevantonsuchshorttimescales. struct the distribution in phase-space. The full voltage
Ourresultsaremorelikelytobecomparedtothether- scaleattheoutputofthedemodulationsystemisdivided
momechanical noise observed with an AFM cantilever into 256 cells so that one cell approximatelycorresponds
[23],althoughthesimplertwo-wavetechniqueusedinthis to the limit of sensitivity ∆xmin. We then perform an
workyieldedalowersensitivity,ontheorderof10−12 m. histogram by accumulating the temporal traces of the
In this experiment the resonance frequency also lay in quadratures in the 256 256 cells. For an acquisition
×
the 20 kHz range but the lower mechanical quality fac- time of 10 minutes the total number of points delivered
tor of the cantilever drastically lowered the timescale of by the digital oscilloscope is of the order of 6 105 and
the Brownianmotion,enablingtodisplayresultscompa- the phase-space is mapped approximately 25000 times.
rable to ours. One then gets a good statistics for the phase-space dis-
We have calibrated the observed displacements by us- tribution as shown in the bottom curve of figure 3. The
ing a frequency modulation of the laser beam. As shown distributionhas a gaussianshape with a revolutionsym-
by eq. (9) we measure a displacement with a reference metry verified with an agreement better than 1%. The
correspondingtothe opticalwavelength.Adisplacement width of the distribution is
∆x of the mirror is thus equivalent to a frequency varia-
−17
tion ∆ν of the laser related to ∆x by ∆X1 =∆X2 =36.610 m. (16)
∆ν ∆x This result can be compared to the theoretical value
= , (12)
ν L expected for the Brownianmotion of a harmonic oscilla-
tor. The two quadratures F and F of the Langevin
T1 T2
whereν is the opticalfrequency andL the cavitylength. force are uncorrelated and have a flat spectrum (S =
T1
Plots of figure 3 correspond to a full scale of the os-
S = 2S ). From eqs. (6), (7) and (11) one gets the
T2 T
cilloscope of 100 mV. We have applied a frequency spectrum for the two position quadratures
±
modulation of calibrated amplitude ∆ν = 200 Hz and
obtained a deviation in phase-space of 27 mV. From Γ
S [ω]=S [ω]= k T. (17)
eq. (12) this corresponds to an equivalent displacement X1 X2 MΩ2 (ω2+Γ2/4) B
∆x=5.410−16 m,sothat1mV isequivalentto210−17 M
m and full scale in figure 3 corresponds to 2 10−15 m. Both noise spectra are centered around zero frequency
±
Thiscalibrationalsoallowstodeterminethesensitivity with a Lorentzian shape of width Γ. Integration over
of the measurement in phase-space. The quantum phase frequency leads to the variances
noise of light induces a rms voltage noise at the output
k T
of the demodulation system of 0.86 mV. The smallest 2 2 B
∆X =∆X = . (18)
observable displacement in phase-space is then 1 2 MΩ2
M
∆xmin =1.710−17 m, (13) According to the characteristics of the harmonic oscilla-
tor (eq. 8) one gets dispersions for the two quadratures
andcorrespondstoafewattometers. Onecanrelatethis of 36.3 10−17 m in excellent agreement with the experi-
sensitivity to the one expressed in term of spectral am- mental value (eq. 16).
plitude (eq. 10). Since the phase noise of the reflected Finallywehavedeterminedthe correlationfunctionof
beam is a white noise on the frequency scale of the mea- themotioninphase-spacedefinedforquadraturesXi,Xj
surement,the minimummeasurabledisplacement∆xmin (i,j =1,2) as
in phase-space is equal to the spectral sensitivity δxmin
integrated over the measurement bandwidth, that is Cij(τ)=hXi(t)Xj(t+τ)it, (19)
∆x2min =(δxmin)2Z |H[ω]|2dω, (14) wovheerrtehtehmeebarsaucrkeemtsenh.t..tiitmset.anCdurfvoerstaheintfiemguproer4alshaovwerathgee
experimentalresultobtainedforanacquisitiontimeof10
where H[ω] is the transfer function of the low-pass fil- minutes. As expected for a harmonic oscillator in ther-
tersattheoutputofthedemodulationsystem. Fromthe malequilibrium,thereisnocross-correlationbetweenthe
value of the spectral sensitivity (eq. 10) and the char- two quadratures (C12 = C21 = 0) and auto-correlation
acteristics of the filter (second-order filter with cut-off functionsC11 andC22 areequalandexhibitanexponen-
frequency 460 Hz and quality factor 2200), one gets tial decay from their initial values corresponding to the
variances,
−17
∆xmin 1.6510 m, (15)
≃ 2
C (τ)=∆X exp( Γτ/2). (20)
in excellent agreementwith the measuredvalue (eq. 13). ii i −
As shown by eq. (14) the sensitivity depends on the TheslopeofthecurveleadstoadampingconstantΓ/2π
frequency cut-off of the filters and can be increasedwith of 48 Hz in good agreement with the expected value
a smaller measurement bandwidth. (Γ/2π 43 Hz).
≃
5
FIG. 5: Cold-damping experimental setup. The mirror is
submittedtotheradiationpressureofanauxiliarylaserbeam
FIG.4: Plotsinlin-logscaleofthetemporalcorrelationfunc-
reflectedfromthebackonthemirrorandintensity-modulated
tionsC11(τ),C22(τ)forbothquadraturesasafunctionofthe
delayτ. a: freemotionatroomtemperature. b: colddamped by an acousto-optic modulator (AOM). The feedback loop
mirror (section IV). cand d: parametric cooling (section V). allows to apply an additional viscous force.
e: phase noise of light.
to a thermal equilibrium but at a different temperature,
Curve e in figure 4 represents the correlation function depending on the gain g of the loop. The fluctuations-
dissipation theorem allows to define an effective temper-
for the phase noise of the reflected beam obtained when
ature T which is smaller than the room temperature
thecavityisoutofresonancewiththeincidentlaser.One eff
T for a positive gain g,
getsanexponentialdecayrelatedtothecut-offfrequency
of the low-pass filters of the demodulation system. This
T
curveclearlyshowsthatthemeasurementnoiseisalmost Teff = . (22)
1+g
negligibleascomparedtothethermalnoiseofthemirror.
The freezing of the mirror of course goes with a reduc-
tion of the Brownian motion which would appear as a
IV. COLD-DAMPED REGIME
shrinking of the distribution in phase-space. The vari-
ancesofquadraturesX1 andX2 canbecalculatedinthe
We now study the temporal evolution of the mirror in samewayasinprevioussectionsandonegetsexpressions
the cold-dampedregimeobtainedbyfreezing the motion similarto eq. (18)withtemperatureT replacedbyT ,
eff
withanadditionalradiationpressureappliedonthemir-
k T
ror. Theprincipleofthe colddampingmechanism[30]is 2 2 B
∆X =∆X = . (23)
to monitor the thermal motion and to use an electronic 1 2 MΩ2 (1+g)
M
feedback loop which correctsthe displacement by apply-
The change in the mechanical susceptibility also affects
ing an appropriate damping force [16].
As shown in figure 5 an acousto-optic modulator and the spectra of X1 and X2. They still have a Lorentzian
shape but with a width equal to the effective damping
anintensity-modulatedbeamis addedtothe experimen-
(1+g)Γ and a power at resonance reduced by a factor
tal setup. This allows to apply a controlled radiation
2
(1+g) .
pressure on the mirror. An electronic differentiator with
Thecolddampingofamirrorhasalreadybeendemon-
a variable gain drives the modulator to create a viscous
strated by studying the thermal noise spectrum of the
force,that is a force proportionalto the measuredveloc-
mirror [16, 21]. Noise reductions larger than 30 dB at
ity of the mirror. The incident beam on the modulator
the mechanicalresonancefrequency andtemperature re-
is intensity-stabilized in order to apply a well calibrated
duction factors larger than 10 have been obtained. We
force.
present here the results obtained in phase-space for a
Theeffectofthisforceistomodifythedampingofthe
moderate gain g 3.
mirror. Itcanbedescribedasachangeinthemechanical
≃
The motion of the cold-damped mirror is shown in
susceptibility which now becomes [16]
figure 6 (top curve) which also displays the related his-
1 togram (bottom curve). As expected this distribution
χ = , (21)
fb M(Ω2 Ω2+(1+g)ΓΩ) hasagaussianshapewithcylindricalsymmetry,butwith
M − a reduced width as compared to the initial thermal dis-
where g is a dimensionless parameter characterizing the tributionoffigure3.Ratiosofthevarianceofthequadra-
gain of the feedback loop and proportional to the elec- tures between the two situations are in agreement with
tronicgainandtotheintensityoftheauxiliarybeam. As eq. (23) and with the value of the feedback gain.
noadditionalthermalnoiseisassociatedwiththecooling Curvesbinfigure4showtheauto-correlationfunctions
process, the mirror motion is still described by the evo- for the two quadratures. They correspond to the ones of
lution equation (1) where the mechanical susceptibility a harmonic oscillator in thermal equilibrium at a lower
χ is replaced by χ . The resulting motion is equivalent temperatureT .Comparedtothe Brownianmotionat
fb eff
6
roomtemperature,the initialvalueatτ =0is decreased controlitisevenpossibletodifferentiatethesignalgiven
asthevariancesandthetimeconstantoftheexponential by the homodyne detection as in the case of cold damp-
decay is reduced. This is associated with the increase of ing, so that we apply a modulated force proportional to
the effective damping by the cold damping mechanism the velocity x˙ of the mirror,
((1+g)Γ/2π 170 Hz).
≃ F(t)=2gMΓcos(2Ω t)x˙(t). (25)
M
This corresponds to the less usual case of a parametric
amplification via a modulation of the relaxation rate of
the mechanical oscillator.
For such a viscous force one gets from eqs. (4) and (5)
the two quadratures F1 and F2 of the applied force,
F1[ω] = gMΓΩMX2[ω], (26)
F2[ω] = gMΓΩMX1[ω]. (27)
Accordingtotheevolutionequations(6)and(7)onefinds
that the two quadratures X1 and X2 of the mirror posi-
tion are decoupled and obey the following equations,
1 1
X1[ω] = −2MΩM (cid:18) iω+Γ1/2(cid:19)FT2[ω], (28)
−
1 1
X2[ω] = 2MΩM (cid:18) iω+Γ2/2(cid:19)FT1[ω], (29)
−
where F and F are the two quadratures of the
T1 T2
Langevinforce FT andwhere Γ1, Γ2 are the quadrature-
dependent effective dampings in presence of the modu-
FIG. 6: Cold-damped motion with a gain g = 3, in the
lated feedback force,
(X1,X2)phase-space(samescale asinfigure3). Theshrink-
ing of the motion is due to the cooling mechanism. Top:
temporal acquisition over 500 ms. Bottom: corresponding Γ1 = Γ(1+g), (30)
histogram acquired over 10 minutes. Γ2 = Γ(1 g). (31)
−
As for the cold damping mechanism, the parametric
amplification changes the damping of the oscillator but
V. PARAMETRIC COOLING the effect now depends on the quadrature, one damping
being increased while the other one is decreased. As a
result,onequadratureofthemotionisamplifiedwhereas
Distributions obtained in previous sections are sym-
theotheroneisattenuated. Fromeqs. (28)and(29)one
metric since there is no privileged quadrature. We now
gets the variances,
explore the possibility to obtain asymmetric distribu-
tions, that is to squeeze the thermal noise. By anal- k T Γ
2 B
ongieyntwwitahyqtouaancthuiemvessquucehezsetdatestsaitsetsooufsleigphatr,amoneetrciconavme-- ∆X1 = MΩ2M Γ1, (32)
k T Γ
plification. One can take advantage of the presence of ∆X2 = B , (33)
an intensity-controlledauxiliary laser beam to modulate 2 MΩ2M Γ2
someparametersofthemechanicaloscillatorattwicethe
which have to be compared to eq. (18).
resonance frequency Ω .
M
In the case of a modulated restoring force (eq. 24)
Parametricamplificationofamechanicalharmonicos-
cillatorisusuallydonebymodulatingitsspringconstant the two quadratures F1 and F2 of the force respectively
[23, 31]. An equivalent mechanism is obtainedfor a mir- depend on X1 and X2 so that the evolution equation of
ror by applying a modulated force proportional to the X1 and X2 are no longerdecoupled. One has to consider
◦
new quadratures obtained by a rotation of 45 in phase-
position x of the mirror,
space in order to obtain decoupled equations similar to
F (t)=2gMΓΩ cos(2Ω t)x(t), (24) eqs. (28) and (29), with the same effective dampings Γ1
M M
andΓ2. Bothforcesthusgivethesameresultsexceptfor
whereg is the gainofthe parametricamplification. This a global rotation in phase-space.
can easily be done in our experiment by monitoring the The feedback loop in our experimental setup is modi-
mirror position and accordinglycontrolling the force ap- fied in order to modulate the amplitude of the force ap-
plied by the auxiliary beam. Since we use an electronic plied by the auxiliary laser beam at twice the resonance
7
frequency Ω (figure 7). A reference signal is used to Compared to the dispersion obtained at room tempera-
M
synchronize this modulation with the demodulation of ture (eq. 16) the variance of quadrature X1 is reduced
the quadratures. We can change the phase between the by a factor 1/(1+g) whereas the one of quadrature X2
two signals at Ω and 2Ω in order to select the de- is increased by a factor 1/(1 g). One quadrature is
M M
−
coupled quadratures, either X1 and X2 for a modulated thus cooled while the other one is heated and the state
◦
viscous force (eq. 25) or quadratures rotated by 45 for corresponds to a thermal squeezed state.
a modulated restoring force (eq. 24). Curves c and d of figure 4 show the auto-correlation
functions for the two quadratures X2 and X1, respec-
tively. Initial values (τ =0) correspond to the variances
deduced from eq. (34) and the time constants of the ex-
ponentialdecaysarerelatedtothe effectivedampingsΓ2
forcurvecandΓ1forcurved,inagreementwitheqs. (30)
and(31). Notethatwehavealignedthetwoquadratures
X1 and X2 with the axes of the squeezed thermal state
by a proper choice of the phase between the two refer-
encesignalsatΩ and2Ω . Wehavethencheckedthat
M M
thereisnocross-correlationbetweenthetwoquadratures
(C12 =C21 =0).
FIG. 7: Experimental setup of parametric amplification. An The treatment presented here is based on linear re-
electronic mixermodulates thesignal of thefeedback loop at sponse theory and is only valid for a gain g smaller than
twice the mechanical resonance frequency ΩM. The demod- 1. For a gain equal to 1, the effective damping Γ2 van-
ulation system is synchronized with the reference signal at ishes(eq.31)andthevarianceofquadratureX2 becomes
2ΩM. infinite (eq. 33). This actually corresponds to the oscil-
lation threshold of the parametric amplification and will
be studied in next section.
FIG.8: Parametric amplification oftheBrownian motionfor
aingfiaginurge =3).0.8T,hienstqhueee(zXin1g,Xo2f)thpehatshee-srpmaaclen(osiasmeeisscdaulee taos FanIGd.va9r:iaRnecleasti∆veXv12a(luc)e,s∆ofXe22ff(edc)tinvoerdmaamlipzeindgtsoΓt1he(iar)v,aΓlu2e(sbi)n,
thermal equilibrium at room temperature, for different gains
thequadrature-dependentcooling. Top: temporalacquisition
of amplification. Solid lines are theoretical predictions.
over500ms. Bottom: correspondinghistogramacquiredover
10 minutes.
Wehavecheckedthetheoreticalbehaviourofthepara-
Theresultingmotioninphasespaceisshowninfigure8
metric amplification as a function of the gain g in the
for a parametric gain g = 0.8. The distribution still has
linear regime (g <1). We have repeated the experiment
a gaussian shape but with different widths for the two
for different values of the gain and reported the effective
quadratures. The distribution is no longer symmetric dampings Γ and the variances∆X2 for the two quadra-
and dispersions of both quadratures are found equal to i i
tures (i=1,2). Points in figure 9 show the experimental
−17 −17
∆X1 =2710 m, ∆X2 =7810 m. (34) results for these parameters, normalized to their respec-
8
tive values in thermal equilibrium at room temperature. stayedalongertimenearthenegativepositionthannear
The gain g is estimated by an averageover the indepen- the positive one. This explains why the negative peak is
dent values of the effective dampings and the widths of higherthanthepositiveone.Foralongeracquisitiontime
the distributions in phase-space. one would obtain symmetric peaks.
The measurements are in very good agreement with Foralargergainthesystemcanstayafewhourswith-
thetheoreticalexpressionsgivenbyeqs. (30)to(33)and out any jump. During the acquisition time the mirror
shown as solid lines in figure 9 without any adjustable thenstaysnearonepositionandthedistributionexhibits
parameter. Both dampings have linear and opposite de- only one peak as shown in figure 11.
pendence with the gain. The variance of quadrature X2
increases and diverges at the oscillation threshold g =1,
whereastheoneofquadratureX1 goesdownto0.5. The
efficiency ofparametricamplificationwhichis mainlyre-
lated to the power of the auxiliary laser beam is large
enoughtoobtainagaingclosetounity.Onethenreaches
the 50%theoreticallimit ofsqueezing for the varianceof
quadrature X1.
VI. PARAMETRIC OSCILLATION
As in usual parametric amplification the system oscil-
lates above threshold, that is for a gain g larger than 1.
This threshold is actually a consequence of the fact that
the gain gΓ of parametric amplification becomes equal
to the mechanical losses which are related to the damp-
ing Γ. The damping Γ2 then vanishes and the noise of
quadrature X2 becomes infinite. Above threshold the
mirror oscillates at frequency Ω in phase with quadra-
M FIG.10: Parametricoscillationofthemirrorforagaing>1,
ture X2. The amplitude of oscillation depends both on in the (X1,X2) phase-space. Scale is 4 times larger than in
the gainandonsaturationmechanisms whichtake place
figure 3. Top: temporal acquisition over 500 ms. Bottom:
inthe parametricoscillationregime.Inourcasetheyare corresponding histogram acquired over 10 minutes. The dis-
mainlyrelatedtothe saturationofthe intensitymodula- tribution appears as two opposite squeezed peaks with non-
tion of the auxiliary laser beam. zero equal mean amplitudes.
The distribution of the mirror motion no longer ap-
pears as a peak located at the center of phase-space
(X1 = 0, X2 = 0). It rather looks like a squeezed dis-
tribution centered at a non-zero position along the X2
axis,the amountof squeezing depending onthe distance
to threshold. As the modulation of the feedback loop is
at frequency 2Ω , the system is invariant by a π phase-
M
shift and there are two possible positions for the center
of the distribution, one for a positive value of X2 , the
h i
other one for a negative and opposite value X2 .
−h i
Figures10and11showtheexperimentaltemporalevo-
lutions and distributions obtained above threshold, re-
spectively for a gain close to unity and for a larger gain.
As expected the distribution in figure 10 exhibits two
peaks along the X2 axis, symmetrically located with re-
spect to the center. The temporal evolution (top curve
infigure10)alsoshowsaconcentrationofthemirrormo-
tionaroundthetwooppositepositions.Onecanhowever
note that the motion sometimes explores the neighbour-
hoodofthecenterofphase-spaceandcanjumpfromone
position to the other one. Delay between two jumps can
beassmallasafewsecondsforagainveryclosetounity FIG.11: Parametricoscillationofthemirrorforagaingmuch
andcorrespondstoafewminutesinthecaseoffigure10. larger than 1, in the (X1,X2) phase-space (same scale as in
Over the 10-minutes acquisition time of the distribution figure 10). Top: temporal acquisition over 500 ms. Bottom:
the system has actually made only one jump and it has corresponding histogram acquired over10 minutes.
9
Parametricoscillationalsosqueezesthe mirrormotion For a 500 mW beam, this corresponds to a mean dis-
alongtheX1 quadrature.Asusualthesqueezingeffectis placement of 6 10−15 m, in very good agreement with
larger close to threshold and the distribution goes back the one observed in figure 11.
toadiskforlargegain(figure11).Incontrastwithpara-
metric oscillation in optics [32, 33], the fluctuations are
of the same order as the mean value of the oscillation,
that is the distance between the two peaks in figure 10
is of the same order as their widths. As a consequence
one cannot use a linear approach to describe the evolu-
tion of fluctuations and the distribution does notappear VII. CONCLUSION
as an ellipse but as a more complex and distorted shape
[34]. Itishoweverpossibletoevaluatethe variance∆X2
1
and one obtains in the case of figure 10 a variance equal We have presented an experiment of high-sensitivity
to 0.55 times the variance of the thermal noise at room displacement measurement, reconstructing the phase-
temperature. space distribution of a mechanical oscillator with a
Letusfinallynotethattheamplitudeofthepeaks(dis- thousand-foldincrease in sensitivity upon what had pre-
tance to the center of phase-space)depends on the gain. viously been reported. The sensitivity of our experimen-
One gets a larger amplitude for a higher gain. There is talsetupiscurrentlylimitedto1.710−17mbythedamp-
howevera limit related to the saturationof the feedback ing timescale of the oscillator, but it could easily be ex-
loop. Foraverylargegaintheradiationpressureapplied tended to the attometer level by reducing the analysis
on the mirror is limited by the modulation capabilities bandwidth.
of the laser beam intensity. One can easily estimate this Both feedback schemes studied in this paper are able
limit. When the feedback loop saturates, the intensity to reduce the thermal noise of the mirror. Their effects
modulation tends towards a square signal with a 100% however are different. In the case of cold damping the
modulationdepth.The componentofthe radiationpres- resulting state is still a thermal equilibrium but at a
sure force at frequency Ω is thus on the order of lower effective temperature, in principle down to a zero
M
temperature for a very large gain [35]. One then gets a
P 4 symmetric gaussian distribution in phase-space. In con-
F (t) 2 cos(Ω t), (35)
rad M
≃ c π trast, parametric amplification only cools one quadra-
ture of the mirror motion, the other one being heated.
whereP isthelightpower.Inthisexpression2P/crepre-
Onethengetssqueezedthermalstateswithanasymmet-
sentstheconstantforceexertedbylightwithoutmodula-
ric gaussian distribution in phase-space. This situation
tionand4/πisacorrectionfactorduetothefactthatthe
is somewhat equivalent to the one which would be ob-
modulationcorrespondstoasquaresignalratherthanto
tained with quadrature-dependent feedback loops where
asine one. The responseofthe mirrorisa forcedoscilla-
thezero-temperaturequantumstateofthemirrorshould
tion with a non-zero mean amplitude hX2i obtained by be reached via squeezed thermal states [36].
identifying the intrinsic damping force to the radiation
pressure, These results show that the observation of motion in
phase-space gives a better understanding and control of
optomechanical coupling, on the way to experimental
P 4
MΓΩM X2 2 . (36) demonstration of related quantum effects.
h i≃ c π
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to the mechanical resonance, there is no observable dif-
