Approaching the motional ground state of a 10 kg object
Chris Whittle,
1
Evan D. Hall,
1
Sheila Dwyer,
2
Nergis Mavalvala,
1
Vivishek Sudhir,
1, 3
R. Abbott,
4
A. Ananyeva,
4
C. Austin,
5
L. Barsotti,
6
J. Betzwieser,
7
C. D. Blair,
7, 8
A. F. Brooks,
4
D. D. Brown,
9
A. Buikema,
6
C. Cahillane,
4
J. C. Driggers,
10
A. Effler,
7
A. Fernandez-Galiana,
6
P. Fritschel,
6
V. V. Frolov,
7
T. Hardwick,
5
M. Kasprzack,
4
K. Kawabe,
10
N. Kijbunchoo,
11
J. S. Kissel,
10
G. L. Mansell,
10, 6
F. Matichard,
4, 6
L. McCuller,
6
T. McRae,
11
A. Mullavey,
7
A. Pele,
7
R. M. S. Schofield,
12
D. Sigg,
10
M. Tse,
6
G. Vajente,
4
D. C. Vander-Hyde,
13
Hang Yu,
6
Haocun Yu,
6
C. Adams,
7
R. X. Adhikari,
4
S. Appert,
4
K. Arai,
4
J. S. Areeda,
14
Y. Asali,
15
S. M. Aston,
7
A. M. Baer,
16
M. Ball,
12
S. W. Ballmer,
13
S. Banagiri,
17
D. Barker,
10
J. Bartlett,
10
B. K. Berger,
18
D. Bhattacharjee,
19
G. Billingsley,
4
S. Biscans,
6, 4
R. M. Blair,
10
N. Bode,
20, 21
P. Booker,
20, 21
R. Bork,
4
A. Bramley,
7
K. C. Cannon,
22
X. Chen,
8
A. A. Ciobanu,
9
F. Clara,
10
C. M. Compton,
10
S. J. Cooper,
23
K. R. Corley,
15
S. T. Countryman,
15
P. B. Covas,
24
D. C. Coyne,
4
L. E. H. Datrier,
25
D. Davis,
13
C. Di Fronzo,
23
K. L. Dooley,
26, 27
P. Dupej,
25
T. Etzel,
4
M. Evans,
6
T. M. Evans,
7
J. Feicht,
4
P. Fulda,
28
M. Fyffe,
7
J. A. Giaime,
5, 7
K. D. Giardina,
7
P. Godwin,
29
E. Goetz,
5, 19, 30
S. Gras,
6
C. Gray,
10
R. Gray,
25
A. C. Green,
28
E. K. Gustafson,
4
R. Gustafson,
31
J. Hanks,
10
J. Hanson,
7
R. K. Hasskew,
7
M. C. Heintze,
7
A. F. Helmling-Cornell,
12
N. A. Holland,
11
J. D. Jones,
10
S. Kandhasamy,
32
S. Karki,
12
P. J. King,
10
Rahul Kumar,
10
M. Landry,
10
B. B. Lane,
6
B. Lantz,
18
M. Laxen,
7
Y. K. Lecoeuche,
10
J. Leviton,
31
J. Liu,
20, 21
M. Lormand,
7
A. P. Lundgren,
33
R. Macas,
26
M. MacInnis,
6
D. M. Macleod,
26
S. M ́arka,
15
Z. M ́arka,
15
D. V. Martynov,
23
K. Mason,
6
T. J. Massinger,
6
R. McCarthy,
10
D. E. McClelland,
11
S. McCormick,
7
J. McIver,
4, 30
G. Mendell,
10
K. Merfeld,
12
E. L. Merilh,
10
F. Meylahn,
20, 21
T. Mistry,
34
R. Mittleman,
6
G. Moreno,
10
C. M. Mow-Lowry,
23
S. Mozzon,
33
T. J. N. Nelson,
7
P. Nguyen,
12
L. K. Nuttall,
33
J. Oberling,
10
Richard J. Oram,
7
C. Osthelder,
4
D. J. Ottaway,
9
H. Overmier,
7
J. R. Palamos,
12
W. Parker,
7, 35
E. Payne,
36
R. Penhorwood,
31
C. J. Perez,
10
M. Pirello,
10
H. Radkins,
10
K. E. Ramirez,
37
J. W. Richardson,
4
K. Riles,
31
N. A. Robertson,
4, 25
J. G. Rollins,
4
C. L. Romel,
10
J. H. Romie,
7
M. P. Ross,
38
K. Ryan,
10
T. Sadecki,
10
E. J. Sanchez,
4
L. E. Sanchez,
4
T. R. Saravanan,
32
R. L. Savage,
10
D. Schaetzl,
4
R. Schnabel,
39
E. Schwartz,
7
D. Sellers,
7
T. Shaffer,
10
B. J. J. Slagmolen,
11
J. R. Smith,
14
S. Soni,
5
B. Sorazu,
25
A. P. Spencer,
25
K. A. Strain,
25
L. Sun,
4
M. J. Szczepa ́nczyk,
28
M. Thomas,
7
P. Thomas,
10
K. A. Thorne,
7
K. Toland,
25
C. I. Torrie,
4
G. Traylor,
7
A. L. Urban,
5
G. Valdes,
5
P. J. Veitch,
9
K. Venkateswara,
38
G. Venugopalan,
4
A. D. Viets,
40
T. Vo,
13
C. Vorvick,
10
M. Wade,
41
R. L. Ward,
11
J. Warner,
10
B. Weaver,
10
R. Weiss,
6
B. Willke,
21, 20
C. C. Wipf,
4
L. Xiao,
4
H. Yamamoto,
4
L. Zhang,
4
M. E. Zucker,
6, 4
and J. Zweizig
4
1
LIGO Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139
2
LIGO Hanford Observatory, Richland, WA 99352
3
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139
∗
4
LIGO, California Institute of Technology, Pasadena, CA 91125, USA
5
Louisiana State University, Baton Rouge, LA 70803, USA
6
LIGO, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
7
LIGO Livingston Observatory, Livingston, LA 70754, USA
8
OzGrav, University of Western Australia, Crawley, Western Australia 6009, Australia
9
OzGrav, University of Adelaide, Adelaide, South Australia 5005, Australia
10
LIGO Hanford Observatory, Richland, WA 99352, USA
11
OzGrav, Australian National University, Canberra, Australian Capital Territory 0200, Australia
12
University of Oregon, Eugene, OR 97403, USA
13
Syracuse University, Syracuse, NY 13244, USA
14
California State University Fullerton, Fullerton, CA 92831, USA
15
Columbia University, New York, NY 10027, USA
16
Christopher Newport University, Newport News, VA 23606, USA
17
University of Minnesota, Minneapolis, MN 55455, USA
18
Stanford University, Stanford, CA 94305, USA
19
Missouri University of Science and Technology, Rolla, MO 65409, USA
20
Max Planck Institute for Gravitational Physics (Albert Einstein Institute), D-30167 Hannover, Germany
21
Leibniz Universit ̈at Hannover, D-30167 Hannover, Germany
22
RESCEU, University of Tokyo, Tokyo, 113-0033, Japan.
23
University of Birmingham, Birmingham B15 2TT, UK
24
Universitat de les Illes Balears, IAC3—IEEC, E-07122 Palma de Mallorca, Spain
25
SUPA, University of Glasgow, Glasgow G12 8QQ, UK
26
Cardiff University, Cardiff CF24 3AA, UK
27
The University of Mississippi, University, MS 38677, USA
28
University of Florida, Gainesville, FL 32611, USA
29
The Pennsylvania State University, University Park, PA 16802, USA
arXiv:2102.12665v1 [quant-ph] 25 Feb 2021
2
30
University of British Columbia, Vancouver, BC V6T 1Z4, Canada
31
University of Michigan, Ann Arbor, MI 48109, USA
32
Inter-University Centre for Astronomy and Astrophysics, Pune 411007, India
33
University of Portsmouth, Portsmouth, PO1 3FX, UK
34
The University of Sheffield, Sheffield S10 2TN, UK
35
Southern University and A&M College, Baton Rouge, LA 70813, USA
36
OzGrav, School of Physics & Astronomy, Monash University, Clayton 3800, Victoria, Australia
37
The University of Texas Rio Grande Valley, Brownsville, TX 78520, USA
38
University of Washington, Seattle, WA 98195, USA
39
Universit ̈at Hamburg, D-22761 Hamburg, Germany
40
Concordia University Wisconsin, 2800 N Lake Shore Dr, Mequon, WI 53097, USA
41
Kenyon College, Gambier, OH 43022, USA
The motion of a mechanical object — even a human-sized object — should be governed by the
rules of quantum mechanics. Coaxing them into a quantum state is, however, difficult: the thermal
environment effectively masks any quantum signature of the object’s motion. Indeed, it also masks
effects of proposed modifications of quantum mechanics at large mass scales. We prepare the
center-of-mass motion of a
10 kg
mechanical oscillator in a state with an average phonon occupation
of 10
.
8. The reduction in oscillator temperature, from room temperature to
77 nK
, represents
a 100-fold improvement in the reduction of temperature of a solid-state mechanical oscillator —
commensurate with a 11 orders-of-magnitude suppression of quantum back-action by feedback — and
a 10 orders-of-magnitude increase in the mass of an object prepared close to its motional ground
state.
The apparent classical behavior of tangibly massive
objects is, according to conventional quantum mechan-
ics, the symptom of decoherence. Thermal decoherence,
caused by the interaction of a system with a thermal
environment, is by far the most pervasive. For a me-
chanical oscillator of mass
m
and natural frequency Ω
0
,
thermal decoherence induces motion characterized by
spectral density
S
th
x
[Ω
0
] = (2
n
th
[Ω
0
] + 1)
S
zp
x
[Ω
0
], where
n
th
[Ω
0
]
≈
k
B
T/
~
Ω
0
is the average thermal phonon occu-
pation due to the environment (at temperature
T
) and
S
zp
x
[Ω
0
] = 8
x
2
zp
/
Γ
0
[Ω
0
] is its motional zero-point fluctu-
ation,
x
zp
=
√
~
/
(2
m
Ω
0
)
, concentrated in a frequency
band of width Γ
0
[Ω
0
]. Prosaic as they are, thermal fluc-
tuations obscure signatures of decoherence that allegedly
arise from modifications of quantum mechanics at large
masses [
1
–
3
], and limit the sensitivity of mechanical trans-
ducers in metrology applications [
4
,
5
]. Techniques to
probe both frontiers call for large mass mechanical objects
prepared in pure quantum states.
Over the past decade, progressively larger objects all
the way to nanomechanical oscillators have been prepared
in their motional ground state [
6
–
10
]. A vast major-
ity of these experiments rely on isolating the oscillator
in an elastic or electromagnetic trap in the
&
100 kHz
(or higher) frequency range, embedded in a sideband-
resolved electromagnetic cavity, typically in a cryogenic
environment. These methods do not address a number
of technical challenges unique to mechanical oscillators
above the milli-/gram mass scale. For one, the large opti-
cal power required to trap massive oscillators introduces
extraneous heating and other opto-mechanical nonlinear-
ities. Meanwhile, the low resonant frequency of large
∗
vivishek@mit.edu
suspended oscillators doubly compounds the problem of
thermal decoherence by increasing the intrinsic thermal
motion (
n
tot
∝
1
/
Ω
0
) and precluding efficient cavity side-
band cooling. Therefore a different route is needed to
prepare large-mass oscillators in pure quantum states.
The Advanced LIGO gravitational-wave detectors offer
a unique perspective on this problem. Advanced LIGO
is a pair of Michelson interferometers, each with
4 km
long Fabry–P ́erot arm cavities formed by
40 kg
mirrors
that hang on fused silica fibers (see Fig. 1). The differ-
ential motion of each pair of arm cavity mirrors forms
a mechanical oscillator with a reduced mass of
20 kg
;
the differential motion of each such oscillator in either
arm, sensed by the Michelson interferometer, forms a
mechanical oscillator of effective mass
m
=
10 kg
that
is the object of our attention. The oscillator follows
the pendulum-like motion of the suspended mirror at
a frequency Ω
0
≈
2
π
·
0
.
43 Hz
; gravitational stress dilu-
tion is expected to realize a quality factor of
Q
0
≈
10
8
[
12
,
13
]. Its displacement fluctuates due to the presence
of
n
th
[Ω
0
]
≈
10
13
phonons. The interferometer resonantly
transduces the differential arm motion into optical power
fluctuations at the anti-symmetric port, which is sensed
by homodyne detection (using a local oscillator derived
from a slight offset of the differential arm length [
14
]);
during ordinary operation, these fluctuations encode the
gravitational-wave signals. In this state, the homodyne
photocurrent fluctuations bear the apparent displacement
δx
obs
=
δx
+
δx
imp
; here
δx
is the physical motion of the
differential arm, which contains the displacement of the
oscillator, and
δx
imp
is the measurement imprecision. The
imprecision noise, depicted in Fig. 1b, is
2
·
10
−
20
m
/
√
Hz
around
100–200 Hz
and is largely quantum shot noise —
suppressed by
∼
3 dB
by injection of squeezed light [
15
],
and shaped by the response of the signal recycling cav-
ity — with a secondary contribution from mechanical
3
Laser
Squeezed
source
x
obs
= x
+
x
imp
χ
fb
-1
Test masses
Signal recycling
Powerl recycling
Anti-symmetric port
10
-15
a
b
10
8
6
4
2
0
Displacement impr
ecision [10
-20
m/
Hz]
10
2
10
3
Fr equency [Hz]
Imprecision (squeezed)
Imprecision (no squeezing)
Sum of classical noises
FIG. 1.
Advanced LIGO interferometer.
(a) Laser light is split and recombined at a beam-splitter, forming a Michelson
interferometer. The peak sensitivity of the Michelson is enhanced by the Fabry–P ́erot cavities in the arms and the injection of
squeezed vacuum through the anti-symmetric port; its broadband response is shaped by the signal-recycling mirror, and power
at the beam-splitter is enhanced by a power-recycling mirror [11]. Inset shows the suspension system of each of the four 40 kg
mirrors: the final mass on the forward chain is the
40 kg
mirror, suspended on fused silica wires (purple); they can be displaced
by electrostatic forces due to voltages applied on electrodes (yellow) etched onto the reaction mass suspended behind it; average
human sketched for scale. (b) The displacement sensitivity (red) of the interferometer is
2
·
10
−
20
m
/
√
Hz
at
100–200 Hz
, where
it is largely shot-noise (light red), suppressed by about
3 dB
from injection of squeezed vacuum (red), and a combination of
extraneous technical noises (gray). Blue band shows the frequency interval in which the pendulum mode is trapped and cooled.
dissipation in the mirror coatings [
16
]. This sensitivity
is equivalent to
n
imp
≡
S
imp
x
/
2
S
zp
x
≈
3
.
5
·
10
−
13
phonons
for a 10
kg
oscillator at
∼
150 Hz
— a record low number
(Ref. [
9
] demonstrates
n
imp
≈
10
−
7
) tantamount to resolv-
ing the zero-point motion of the oscillator with
∼
125
dB
signal to squeezed-shot-noise ratio, and comparable to the
requirement to feedback cool the oscillator to its ground
state (
n
imp
∼
1
/
2
n
th
, for a viscously-damped oscillator
[17]).
In order to take advantage of this exquisite impre-
cision, we actively stiffen the pendulum mode by syn-
thesizing a force proportional to the observed displace-
ment (i.e.
∝
Ω
2
fb
δx
obs
) and in-phase with the motion
δx
, trapping the pendulum mode as an oscillator around
Ω
fb
≈
2
π
·
148 Hz
. Two additional sources of decoherence
plague this scheme. First, the exquisite measurement
imprecision comes at the expense of additional quan-
tum back-action on the pendulum mode — equivalent to
about
n
ba
[Ω
0
]
≈
1
.
0
·
10
12
phonons (from the
200 kW
intracavity power and the anti-squeezed intracavity pho-
ton number fluctuations)[
18
]; however, as long as the
measurement record resolves the quantum back-action
at a rate comparable to the thermal decoherence, active
feedback can suppress it [
9
,
17
,
19
,
20
]. Secondly, the feed-
back of amplified imprecision noise leads to an additional
“feedback back-action”,
n
fb
≈
Q
2
0
(Ω
fb
/
Ω
0
)
4
n
imp
(see Sup-
plementary Information), which increases with the trap
frequency. However, this is partially compensated by the
Ω
fb
/
Ω
0
≈
300 fold reduction in both the thermal occu-
pation and decay rate of the trapped oscillator due to
structural damping [4].
To trap and damp the oscillator, we adjust the feed-
back control so that
δF
fb
=
χ
−
1
fb
δx
obs
, with a feedback
filter,
χ
−
1
fb
∝
Ω
2
fb
+ iΩΓ
fb
, between
100–200 Hz
. This is
implemented by careful shaping of the control loop that is
otherwise used to stabilize the interferometer at its linear
operating point. The feedback force is applied on the mir-
ror electrostatically [
22
]: gold electrodes on the reaction
mass (see Fig. 4a) are held at a
400 V
bias, whose fringing
field polarizes the dielectric test mass; control voltages
added on interleaved electrodes produce a proportional
force (extraneous force noise produces
1 phonon of
excess occupation on average, see Supplementary Infor-
mation). The overall feedback gain is adjusted so that the
system’s effective susceptibility takes the form,
χ
eff
[Ω]
∝
(
−
Ω
2
+ Ω
2
eff
+ iΩΓ
eff
[Ω])
−
1
/m
, of that of an oscillator with
frequency Ω
eff
=
√
Ω
2
0
+ Ω
2
fb
≈
Ω
fb
≈
2
π
·
148 Hz
. Delays
in the feedback loop limit the trap frequency and cause
the oscillator to be intrinsically “cold-damped” [
23
]. In
particular, the phase response of the notch filters used to
prevent excitation of the violin modes of the suspension
(at
500 Hz
and harmonics, featuring quality factors
&
10
9
)
in conjunction with the feedback filter leaves the inter-
ferometer’s length control system with a phase margin of
1
◦
for a trap frequency of
148 Hz
. Physical delay in the
loop also cold-damps the trapped oscillator to a quality
4
1
10
|
e
f
f
| [a.u.]
200
180
160
140
120
100
180
135
90
45
0
-45
arg (
e
f
f
) [deg]
200
180
160
140
120
100
Frequency [Hz]
10
-20
10
-19
10
-18
Apparent displacement [m/
√
Hz]
200
180
160
140
120
100
Frequency [Hz]
a
b
10
1
10
2
10
3
0.01
0.1
1
Q
−
1
e
f
f
Phonon number
FIG. 2.
Trapping and cooling of a 10 kg oscillator to 10 quanta.
(a) Effective susceptibility of the oscillator for each setting
of the damping filter, measured by exciting the feedback loop at each frequency and demodulating its response at the same
frequency. The lines show fits to a model of the susceptibility of a damped harmonic oscillator with an additional delay, i.e.
χ
eff
[Ω]
e
i
Ω
τ
; fits to the phase response produce
τ
=
0
.
9 ms
. (b) Displacement spectrum of the oscillator as the damping is
increased. Solid lines show fits to a model of the observed spectrum
S
obs
x
(see text for details) where the effective susceptibility
is determined by the response measurements in panel (a), and only the frequency-dependent imprecision noise and force noise
are variable. Inset shows the inferred average phonon occupation for each of the curves in the main panel, as a function of the
damping quality factor; also shown is a model (black dashed) with model uncertainties (gray band). (The disagreement between
the simple model and data — both the transfer functions and spectra — around
150–155 Hz
arises from a coupling between the
motion of the pendulum and the upper intermediate mass of the suspension [21].)
factor of
≈
50 (see Fig. 2b red trace, and SI for further
details). The oscillator is damped further by modifying
the imaginary part of the feedback filter. Fig. 2a shows
the effective susceptibilities of the trapped and damped
oscillator so realized. The largest damping rate, corre-
sponding to a quality factor of
≈
1, is limited by the
gain margin (
≈
10
−
3
) of the control loop. Around the
trap frequency (
100–200 Hz
), additional force noise on the
oscillator due to feedback is dominated by sub-quantum
fluctuations of the squeezed imprecision noise.
The calibrated in-loop signal, depicted in Fig. 2b, shows
the apparent displacement fluctuations of the trapped
and damped oscillator (
δx
obs
). This can be understood
using a simple model (see Supplementary Information),
δx
obs
=
χ
eff
(
δF
th
+
δF
ba
−
χ
−
1
fb
δx
imp
)+
δx
imp
. It describes
the oscillator — with intrinsic susceptibility
χ
0
— whose
displacement responds via the feedback-modified effective
susceptibility
χ
eff
= (
χ
−
1
0
+
χ
−
1
fb
)
−
1
, to three forces: a
frequency-dependent structural thermal force (
δF
th
), a
white quantum measurement back-action force (
δF
ba
),
and an additional force noise (
∝
χ
−
1
fb
δx
imp
) due to feed-
back of imprecision noise through the feedback filter; and
riding on the imprecision noise (
δx
imp
). The spectra of
the observed displacement
S
obs
x
predicted by this model
are shown as the solid lines in Fig. 2b. In the model,
the effective susceptibility is fully determined by the re-
sponse measurements shown in Fig. 2a, independent of
the frequency-dependent force noise and imprecision noise.
The latter, determined self-consistently amongst the dis-
placement noise in Fig. 2b, shows a variation between
the different feedback settings of less than 1%, consistent
with expected drift in the Advanced LIGO interferometer
over the
∼
2
hr
timescale over which the experiment was
performed. Several sources of uncertainty are accounted
for in this process. Calibration of the displacement spec-
tra contributes
≈
2% uncertainty [
21
,
24
]. Uncertainties
in the effective susceptibility
χ
eff
— from fits to Fig. 2a —
are at the 1% level, limited by the 1 s averaging used per
point in measurements of the response (see Supplemen-
tary Information). The dominant uncertainty is in the fits
to the displacement spectra of Fig. 2b using the model for
S
obs
x
: the frequency-dependence of the imprecision noise
and structurally damped thermal force noise produce a
≈
5% variation between the various spectra in Fig. 2b.
The effective phonon occupation (
n
eff
) of the cooled
oscillator can be defined through,
~
Ω
eff
(
n
eff
+
1
2
) =
〈
p
2
/
(2
m
) +
m
Ω
2
eff
x
2
/
2
〉
, where
x
(
p
) is the physical dis-
placement (momentum) of the oscillator at the trap fre-
quency. Assuming the displacement and momentum to
be zero-mean, their second moments can be estimated
as the integral of their spectral densities. However, in
principle, two factors complicate this procedure: at lower
frequencies, structural damping renders the displacement
5
Feedback
cooling
Sideband
cooling
10
-
32
10
-
27
10
-
22
10
-
17
10
-
12
10
-
7
10
-
2
10
3
10
0
10
3
10
6
10
9
10
12
Oscillator
mass
[
kg
]
Phonon
occupation
n
m
aLIGO
LIGO
'
09
Vinante
'08
Corbitt
'07
Rossi
'18
Wilson
'15
D'Urso
'03
Monroe
'95
Deli
ć
'20
Roll '64
Urvoy
'19
Clark
'17
Hamann
'97
Schleier
-
Smith
'11
3G
FIG. 3.
A selection of oscillator cooling experiments
[
9
,
10
,
17
,
25
–
34
]. The initial occupations mentioned are those of
the relevant oscillator mode as defined by the natural trap
frequency, at its ambient temperature. For atomic physics ex-
periments, this is usually at room temperature in a harmonic
electromagnetic trap; whereas for most solid-state mechanical
oscillators, it is the harmonic mode defined by the Hookean
restoring force of its elastic suspension, and typically at cryo-
genic temperatures (the exception is the recent work from
Deli ́c et al. [
10
] which demonstrated cavity-cooling of an elec-
tromagnetically trapped nano-particle to its ground state).
Our result (“aLIGO”) sets a new record in the macroscopic
mass range, reaching 10
.
8
±
0
.
8 phonons. Experiments with
future gravitational-wave interferometers (“3G”) will achieve
occupations below 1.
variance singular [
4
], while at higher frequencies, feedback
back-action precludes a finite momentum variance [
35
].
In practice, the feedback filter
χ
−
1
fb
∝
Ω
2
fb
+ iΩΓ
fb
is es-
tablished around
100–200 Hz
in an envelope that falls-off
at least as Ω
−
2
(at frequencies below 10 Hz, the inter-
ferometer’s length control loop picks up again), which
regulates both these problems. In this fashion, within
100–200 Hz
, the trapped oscillator approximately satisfies
the equipartition principle, and so an effective phonon oc-
cupation can be assigned using the physical displacement
spectrum:
n
eff
≈
∫
S
x
[Ω]
2
x
2
zp
dΩ
2
π
.
Note that the 100 Hz frequency band in which the oscilla-
tor is established is much larger than the expected deco-
herence rate of the trapped oscillator, (
n
th
[Ω
eff
] +
n
ba
+
n
fb
[Ω
eff
])Γ
0
[Ω
eff
]
≈
2
π
·
10
Hz
. We evaluate the integral
using the physical displacement spectrum reconstructed
from parameters obtained from fits of the observed dis-
placement to a model of
S
obs
x
; the integration interval is
effectively determined by the fall-off of the feedback filter
χ
−
1
fb
outside
100–200 Hz
. The minimum phonon occupa-
tion of the 10 kg oscillator, corresponding to the purple
trace in Fig. 2b, is thus inferred to be 10
.
8
±
0
.
8; this
is equivalent to an effective mode temperature of
77 nK
.
This demonstration sets a new record for the quantum
state purity (
≈
10% ground state fidelity) for an object
of such large mass (see Fig. 3).
The preparation of massive objects progressively nearer
their ground state opens the door for more sophisticated
demonstrations and applications of macroscopic quan-
tum phenomena and quantum metrology. For instance,
feedback based on quantum-nondemolition measurements
of the oscillator’s displacement can simulate interaction
with a squeezed environment, thereby producing squeezed
states of test mass motion [
36
]. Beyond an exploration
into fundamental quantum mechanics at massive scales,
the realization of squeezed oscillator states in Advanced
LIGO and other gravitational-wave detectors could yield
improved narrowband astrophysical sensitivity at frequen-
cies where thermal noise dominates [
37
]. The next in-
cremental upgrade of Advanced LIGO with frequency-
dependent squeezing will enable this in the
50–200 Hz
range [
38
]. Moreover, it has been shown that entangle-
ment between test masses is generated by measuring the
common and differential modes of a interferometer oper-
ating at the standard quantum limit [
39
]. In this way,
feedback-mediated state preparation can be used to pro-
duce entangled states of kg-scale masses separated over
kilometers.
The most intriguing possibility harnesses the ready sus-
ceptibility of kg-scale masses to gravitational forces; with
this work, it becomes possible to prepare them in near-
quantum states. This hints at the tantalizing prospect of
studying gravitational decoherence on massive quantum
systems.
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ACKNOWLEDGMENTS
Funding:
LIGO was constructed by the California Insti-
tute of Technology and Massachusetts Institute of Tech-
nology with funding from the National Science Foun-
dation, and operates under Cooperative Agreement No.
PHY–0757058. Advanced LIGO was built under Grant
No. PHY–0823459, with additional support from the
Science and Technology Facilities Council (STFC) of the
United Kingdom, the Max-Planck-Society (MPS), and
the Australian Research Council. EDH is supported by
the MathWorks, Inc.
Appendix A: Model of measurement and feedback
The displacement of the oscillator (
δx
) responds to a
sum of thermal, back-action, and feedback forces:
χ
−
1
0
δx
=
δF
th
+
δF
ba
+
F
fb
.
(A1)
8
Here, the susceptibility of the oscillator
χ
−
1
0
[Ω] =
m
(
−
Ω
2
+ Ω
2
0
+
i
ΩΓ
0
[Ω]) is well approximated by the
test mass pendulum mode at frequency Ω
0
≈
2
π
·
0
.
43 Hz
,
which is structurally damped, so that its damping rate
is frequency dependent: Γ
0
[Ω] = (Ω
0
/Q
0
)(Ω
0
/
Ω), with a
quality factor
Q
0
≈
10
8
.
The thermal force (
δF
th
) is characterized by its spectral
density,
S
th
F
[Ω] = 4
~
(
n
th
[Ω] +
1
2
)
Im
χ
−
1
0
[Ω]
,
(A2)
where
n
th
[Ω]
≈
k
B
T/
(
~
Ω)
≈
9
·
10
12
(Ω
0
/
Ω) is the average
thermal phonon occupation. The back-action force (
δF
ba
),
arising from radiation pressure quantum fluctuations, is
characterized by
S
ba
F
[Ω] =
16
~
F
λc
P
cav
e
2
r
asqz
,
(A3)
where
P
cav
≈
200 kW
is the mean arm cavity power at
wavelength
λ
=
1064 nm
,
F ≈
45
is the effective finesse of
the signal-recycled arm cavities, and
r
asqz
quantifies the
increase in quantum fluctuations of the intracavity optical
amplitude due to antisqueezing due to the phase-squeezed
vacuum injected at the interferometer’s dark port; here
10
log
10
e
2
r
asqz
=
(8
±
1) dB
[
40
]. The back-action force
can be quantified in terms of an average phonon oc-
cupation
n
ba
via
S
ba
F
≡
4
~
n
ba
Im
χ
−
1
0
[Ω
0
], which gives
n
ba
≈
1
.
0
·
10
12
.
The feedback force
F
fb
is based on a linear estimate of
the oscillator’s position,
δx
est
≡
G
(
δx
+
δx
imp
);
(A4)
here,
δx
imp
is the displacement imprecision (due to sensing
noise), and
G
is the sensing function of the interferometer.
Such an estimate is obtained only when the interferometer
is stabilized at its linear operating point, achieved by a
feedback loop that forces the test mass (modeled by the
actuation function
A
) based on a filtered (by
D
0
) record
of the error signal
x
est
. We create an additional feedback
path consisting of a digital filter
D
T
in series with
D
0
to
produce the trap, and a parallel path consisting of the
digital filter
D
C
to cold-damp the trapped oscillator. The
combined feedback force thus exerted is
F
fb
=
A
(
D
0
D
T
+
D
C
)
δx
est
+
δF
fb
≡−
χ
−
1
fb
(
δx
+
δx
imp
) +
δF
fb
;
(A5)
here,
δF
fb
models extraneous force fluctuations due to
the actuator. Solving for
δx
between Eqs. (A1) and (A5)
gives the
physical
displacement fluctuations,
δx
=
χ
eff
(
δF
tot
−
χ
−
1
fb
δx
imp
)
;
(A6)
here,
χ
eff
≡
(
χ
−
1
0
+
χ
−
1
fb
)
−
1
is the effective susceptibility
of the oscillator, and
δF
tot
≡
δF
th
+
δF
ba
+
δF
fb
is the
total force noise.
The oscillator can be trapped and cooled by synthesiz-
ing an effective susceptibility of the form,
χ
−
1
eff
=
m
(
−
Ω
2
+ Ω
2
eff
+
i
ΩΓ
eff
)
.
(A7)
δF
fb
δF
th
δF
ba
δx
imp
δx
δx
obs
χ
0
G
G
-1
D
0
D
T
D
C
A
FIG. 4. Schematic of the physical system consisting of the
intrinsic mechanical response
χ
0
, the interferometer’s sensing
function
G
, the digital filters
D
0
,
T
,
C
, and the actuation
A
;
G
−
1
denotes the digital filter used to reconstruct the apparent
displacement. The feedback is subject to fluctuations arising
from actuator force noise
δF
fb
, thermal noise
δF
th
, back-action
noise
δF
ba
and imprecision noise
δx
imp
.
We are able to do this by careful design of the effective
loop filter
χ
−
1
fb
, which is switched on in a sequence that
both traps the oscillator, and keeps the interferometer
unconditionally stable.
What we observe is the
apparent
displacement
δx
obs
inferred using an estimate for the inverse sensing function
G
−
1
(that forms part of LIGO’s calibration pipeline [
41
]).
That is,
δx
obs
≡
G
−
1
δx
est
≈
δx
+
δx
imp
; amplitude un-
certainty in this estimate is at the 2% level [
21
]. Using
the known expression for the physical displacement in
Eq. (A6), the apparent displacement is,
δx
obs
=
χ
eff
(
δF
tot
+
χ
−
1
0
δx
imp
)
.
(A8)
This model produces the spectrum of the observed signal,
S
obs
x
[Ω] =
S
tot
F
[Ω]
/m
2
(Ω
2
eff
−
Ω
2
)
2
+ (ΩΓ
eff
)
2
+
(Ω
2
0
−
Ω
2
)
2
+ (ΩΓ
0
[Ω])
2
(Ω
2
eff
−
Ω
2
)
2
+ (ΩΓ
eff
)
2
S
imp
x
[Ω]
,
(A9)
that is used to fit the data in Fig. 2b in the main text.
However the apparent motion — since it contains correla-
tions impressed by the feedback of imprecision — cannot
be directly compared to the spectrum of a physical oscil-
lator that is damped.
The spectrum of the physical motion of the oscillator
(
δx
in Eq. (A6)),
S
x
[Ω] =
S
tot
F
[Ω]
/m
2
(Ω
2
eff
−
Ω
2
)
2
+ (ΩΓ
eff
)
2
+
Ω
4
fb
+ (ΩΓ
fb
)
2
(Ω
2
eff
−
Ω
2
)
2
+ (ΩΓ
eff
)
2
S
imp
x
[Ω]
,
(A10)
can be directly compared against that of an oscilla-
tor trapped at frequency Ω
eff
, and featuring a damped
linewidth Γ
eff
. Writing
x
=
x
zp
(
b
+
b
†
) for the posi-
tion of such an oscillator, with zero-point motion
x
zp
=
9
√
~
/
2
m
Ω
eff
and creation operator
b
, presumed to exist
in a thermal state, we use the identities,
Var[
x
] = 2
x
2
zp
(
〈
b
†
b
〉
+
1
2
)
(A11)
Var[
x
] =
∫
S
x
[Ω]
dΩ
2
π
,
(A12)
to assign an effective phonon number
n
eff
≡〈
b
†
b
〉
:
n
eff
+
1
2
=
∫
S
x
[Ω]
2
x
2
zp
dΩ
2
π
.
(A13)
Here, the integral is understood to be evaluated in the
frequency interval where the oscillator susceptibility is
realized.
When the imprecision noise is white (i.e.
S
imp
x
[Ω]
≈
S
imp
x
[Ω
eff
]), and the frequency-dependence of the
structurally-damped thermal phonon number can be ne-
glected (i.e.
n
th
[Ω]
≈
n
th
[Ω
eff
]) — both inapplicable to
the current experiment, but useful to develop intuition —
the phonon occupation can be explicitly evaluated as,
n
eff
+
1
2
≈
(
n
tot
[Ω
eff
] +
(
Ω
eff
Γ
0
[Ω
eff
]
)
2
n
imp
+
1
2
)
Γ
0
[Ω
eff
]
Γ
eff
[Ω
eff
]
+
n
imp
Γ
eff
[Ω
eff
]
Γ
0
[Ω
eff
]
,
(A14)
where the factor in the parentheses in the first line is the
total initial occupation, consisting of the sum of thermal
and back-action quanta (
n
tot
=
n
th
+
n
ba
), and an addi-
tional contribution (Ω
eff
/
Γ
0
)
2
n
imp
due to fluctuations in
the trap from feedback of imprecision noise due to the
active spring. Here,
n
imp
≡
S
imp
x
[Ω
eff
]
/
2
S
zp
x
[Ω
eff
], is the
phonon-equivalent imprecision (
S
zp
x
[Ω
eff
] = 8
x
2
zp
/
Γ
0
[Ω
eff
]
is the peak zero-point spectrum of the trapped oscillator).
1. Effect of actuator force noise
It has been documented that the electrostatic drive
(ESD) that is used to actuate the test masses produces
excess force noise that arises from a combination of charg-
ing effects and driver voltage noise [
22
]. In the context of
feedback cooling the test mass, this force (termed
δF
fb
in
Eq. (A6)) acts as an excess thermal force that heats the
trapped oscillator, resulting in additional phonons (
n
fb
)
that add to the thermal occupation. From Ref. [
22
], it
can be inferred that
√
S
fb
F
≈
(4
·
10
−
18
N
/
√
Hz)
(
10 Hz
f
)
.
(By comparison, the typical actuation strength used to
keep the interferometer locked is
∼
10
−
6
N
/
√
Hz
at 10
Hz.) An extraneous phonon occupation
n
fb
,
ex
can be as-
sociated with this force noise via,
S
fb
F
≈
4
~
m
Γ
0
Ω
eff
n
fb
,
ex
.
Assuming the oscillator is trapped at Ω
eff
≈
2
π
·
148 Hz,
the equivalent phonon occupation from excess ESD noise
is,
n
fb
,
ex
.
10
−
3
.
2. Effect of filter delay
If the feedback is implemented with an overall delay
τ
— for example arising from delays in the computation
of the digital filter — the trapping and cooling filter is
modified to
χ
−
1
fb
e
i
Ω
τ
=
m
(Ω
2
fb
+
i
ΩΓ
fb
)
e
i
Ω
τ
≈
m
[Ω
2
fb
(1
−
τ
Γ
fb
) +
i
Ω(Γ
fb
+
τ
Ω
2
fb
)]
,
where in going to the second line, we assume that the
delay is small compared to the characteristic frequency at
which it occurs, i.e. Ω
τ
1. Thus, even in the absence of
active damping (i.e. Γ
fb
= 0), delay in the loop manifests
as damping
τ
Ω
2
fb
. This serves to stabilize the trapped
oscillator.
Delay in other parts of the loop manifest as an overall
phase factor in the closed-loop gain,
χ
eff
e
i
Ω
τ
. Fits to the
phase response of the closed-loop gain in the main text
resolve this overall phase shift at the level of
≈
0
.
9 ms
,
consistent with expected delays in the loop.
Appendix B: Data analysis
1. Uncertainty in transfer function fits
The standard deviation in the transfer function estimate
ˆ
G
derived from signals with coherence
C
over
N
averages
is given by [42]
σ
ˆ
G
=
√
(1
−
C
)
2
CN
ˆ
G.
(B1)
Similarly, the coherence estimate
ˆ
C
has standard devia-
tion
σ
ˆ
C
=
√
2
C
N
(1
−
C
)
.
(B2)
Even assuming a worst-case true coherence
C
within this
range, most of the uncertainty in our data arises from the
1 s average duration and corresponding 1 Hz bin width.
We fit to a time-delayed resonator model,
χ
eff
∼
1
Ω
2
0
−
Ω
2
+ iΩ
0
Ω
/Q
e
i(
φ
−
Ω
t
)
,
(B3)
and propagate these uncertainties using orthogonal dis-
tance regression [43].