of 14
science.sciencemag.org/content/
372/6548/
1333
/suppl/DC1
Supp
lementary
Material
s for
Approaching the motional ground state of a 10
-
kg object
Chris
Whittle
,
Evan D.
Hall
,
Sheila
Dwyer
,
Nergis
Mavalvala
,
Vivishek
Sudhir
*
,
R.
Abbott
,
A.
Ananyeva
,
C.
Austin
,
L.
Barsotti
,
J.
Betzwieser
,
C. D.
Blair
,
A. F.
Brooks
,
D. D.
Brown
,
A.
Buikema
,
C.
Cahillane
,
J. C.
Driggers
,
A.
Effler
,
A.
Fernandez
-
Galiana
,
P.
Fritschel
,
V. V.
Frolov
,
T.
Hardwick
,
M.
Kasprzack
,
K.
Kawabe
,
N.
Kijbunchoo
,
J. S.
Kissel
,
G. L.
Mansell
,
F.
Matichard
,
L.
McCuller
,
T.
McRae
,
A.
Mullavey
,
A.
Pele
,
R. M. S.
Schofield
,
D.
Sigg
,
M.
Tse
,
G.
Vajente
,
D. C.
Vander
-
Hyde
,
Hang
Yu
,
Haocun
Yu
,
C.
Adams
,
R. X.
Adhikari
,
S.
Appert
,
K.
Arai
,
J. S.
Areeda
,
Y.
Asali
,
S. M.
Aston
,
A.
M.
Baer
,
M.
Ball
,
S. W.
Ballmer
,
S.
Banagiri
,
D.
Barker
,
J.
Bartlett
,
B. K.
Berger
,
D.
Bhattacharjee
,
G.
Billingsley
,
S.
Biscans
,
R. M.
Blair
,
N.
Bode
,
P.
Booker
,
R.
Bork
,
A.
Bramley
,
K. C.
Cannon
,
X.
Chen
,
A. A.
Ciobanu
,
F.
Clara
,
C. M.
Compton
,
S. J.
Cooper
,
K. R.
Corley
,
S. T.
Countryman
,
P. B.
Covas
,
D.
C.
Coyne
,
L. E. H.
Datrier
,
D.
Davis
,
C.
Di Fronzo
,
K. L.
Dooley
,
P.
Dupej
,
T.
Etzel
,
M.
Evans
,
T. M.
Evans
,
J.
Feicht
,
P.
Fulda
,
M.
Fyffe
,
J. A.
Giaime
,
K. D.
Giardina
,
P.
Godwin
,
E.
Goetz
,
S.
Gras
,
C.
Gray
,
R.
Gray
,
A. C.
Green
,
E. K.
Gustafson
,
R.
Gustafson
,
J.
Hanks
,
J.
Hanson
,
R. K.
Hasskew
,
M. C.
Heintze
,
A. F.
Helmling
-
Cornell
,
N. A.
Holland
,
J. D.
Jones
,
S.
Kandhasamy
,
S.
Karki
,
P. J.
King
,
Rahul
Kumar
,
M.
Landry
,
B. B.
Lane
,
B.
Lantz
,
M.
Laxen
,
Y. K.
Lecoeuche
,
J.
Leviton
,
J.
Liu
,
M.
Lormand
,
A. P.
Lundgren
,
R.
Macas
,
M.
MacInnis
,
D. M.
Macleod
,
S.
Márka
,
Z.
Márka
,
D. V.
Martynov
,
K.
Mason
,
T. J.
Massinger
,
R.
McCarthy
,
D. E.
McClelland
,
S.
McCormick
,
J.
McIver
,
G.
Mendell
,
K.
Merfeld
,
E. L.
Merilh
,
F.
Meylahn
,
T.
Mistry
,
R.
Mittleman
,
G.
Moreno
,
C. M.
Mow
-
Lowry
,
S.
Mozzon
,
T. J. N.
Nelson
,
P.
Nguyen
,
L. K.
Nuttall
,
J.
Oberling
,
Richard J.
Oram
,
C.
Osthelder
,
D. J.
Ottaway
,
H.
Overmier
,
J. R.
Palamos
,
W.
Parker
,
E.
Payne
,
R.
Penhorwood
,
C. J.
Perez
,
M.
Pirello
,
H.
Radkins
,
K. E.
Ramirez
,
J. W.
Richardson
,
K.
Riles
,
N. A.
Robertson
,
J. G.
Rollins
,
C. L.
Romel
,
J. H.
Romie
,
M. P.
Ross
,
K.
Ryan
,
T.
Sadecki
,
E. J.
Sanchez
,
L. E.
Sanchez
,
T. R.
Saravanan
,
R. L.
Savage
,
D.
Schaetz
,
R.
Schnabel
,
E.
Schwartz
,
D.
Sellers
,
T.
Shaffer
,
B. J. J.
Slagmolen
,
J. R.
Smith
,
S.
Soni
,
B.
Sorazu
,
A. P.
Spencer
,
K. A.
Strain
,
L.
Sun
,
M. J.
Szczepańczyk
,
M.
Thomas
,
P.
Thomas
,
K. A.
Thorne
,
K.
Toland
,
C.
I.
Torrie
,
G.
Traylor
,
A. L.
Urban
,
G.
Valdes
,
P. J.
Veitch
,
K.
Venkateswara
,
G.
Venugopalan
,
A. D.
Viets
,
T.
Vo
,
C.
Vorvick
,
M.
Wade
,
R. L.
Ward
,
J.
Warner
,
B.
Weaver
,
R.
Weiss
,
B.
Willke
,
C. C.
Wipf
,
L.
Xiao
,
H.
Yamamoto
,
L.
Zhang
,
M. E.
Zucker
,
J.
Zweizig
*Corresponding author. E
-
mail: vivishek@mit.edu
Published
18 June
2021,
Science
372
,
1333
(2021)
DOI:
10.1126/science.
a
bh2634
This PDF file
includes:
Supplementary
Text
Fig
. S1
References
S1 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
.
(S1)
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
≈
8
·
10
7
.
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]
,
(S2)
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
,
(S3)
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
(
15
). The back-action force can be quantified in terms of an average phonon occupation
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
);
(S4)
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
;
(S5)
here,
δF
fb
models extraneous force fluctuations due to the actuator. Solving for
δx
between
Eqs. (S1) and (S5) gives the physical displacement fluctuations,
δx
=
χ
eff
(
δF
tot
−
χ
−
1
fb
δx
imp
)
;
(S6)
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.
2
The oscillator can be trapped and cooled by synthesizing an effective susceptibility of the
form,
χ
−
1
eff
=
m
(
−
Ω
2
+ Ω
2
eff
+
i
ΩΓ
eff
)
.
(S7)
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 (
31
)). That is,
δx
obs
≡
G
−
1
δx
est
≈
δx
+
δx
imp
; amplitude uncertainty in this estimate is at the
2%
level (
18
).
Using the known expression for the physical displacement in Eq. (S6), the apparent displacement
is,
δx
obs
=
χ
eff
(
δF
tot
+
χ
−
1
0
δx
imp
)
.
(S8)
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]
,
(S9)
that is used to fit the data in Fig. 2B in the main text. However the apparent motion — since it
contains correlations impressed by the feedback of imprecision — cannot be directly compared
to the spectrum of a physical oscillator that is damped.
The spectrum of the physical motion of the oscillator (
δx
in Eq. (S6)),
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]
,
(S10)
can be directly compared against that of an oscillator trapped at frequency
Ω
eff
, and featuring
a damped linewidth
Γ
eff
. Appealing to the equipartition principle — under the assumptions
mentioned in the main text — we may assign an effective phonon number
n
eff
via,
n
eff
+
1
2
=
∫
S
x
[Ω]
2
x
2
zp
dΩ
2
π
.
(S11)
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 neglected (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
]
,
(S12)
3
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 additional 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).
S1.1 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 charging effects and driver voltage
noise (
32
). In the context of feedback cooling the test mass, this force noise (termed
δF
fb
in
Eq. (S6)) acts to heat the trapped oscillator, resulting in additional phonons that add to the
thermal occupation. From Ref. (
32
), 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 associated 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
.
S1.2 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.
S2 Data analysis
S2.1 Uncertainty in transfer function fits
The transfer function measurements in Figure 2A are fit to a time-delayed resonator model,
χ
eff
[Ω]
∝
1
Ω
2
0
−
Ω
2
+ iΩ
0
Ω
/Q
e
i(
φ
−
Ω
t
)
.
(S13)
4
In doing so, we account for statistical uncertainities in the estimate
ˆ
χ
eff
: the standard deviation
in such an estimate, when it is measured using excitation and response signals with coherence
C
over
N
statistically independent averages is given by (
33
),
σ
ˆ
χ
eff
=
√
(1
−
C
)
2
CN
ˆ
χ
eff
.
(S14)
Similarly, the coherence estimate
ˆ
C
has standard deviation
σ
ˆ
C
=
√
2
C
N
(1
−
C
)
.
(S15)
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.
Uncertainities arising from the fit are propagated through the analysis using orthogonal
distance regression (
34
).
5
δF
fb
δF
th
δF
ba
δx
imp
δx
δx
obs
χ
0
G
G
-1
D
0
D
T
D
C
A
Fig. S1
Schematic of the physical system consisting of the intrinsic mechanical response
χ
0
, the in-
terferometer’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
.
6
7
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