Quantum correlations between the light and kilogram-mass mirrors of LIGO
Haocun Yu,
1
L. McCuller,
1
M. Tse,
1
L. Barsotti,
1
N. Mavalvala,
1
J. Betzwieser,
2
C. D. Blair,
2
S. E. Dwyer,
3
A. Effler,
2
M. Evans,
1
A. Fernandez-Galiana,
1
P. Fritschel,
1
V. V. Frolov,
2
N. Kijbunchoo,
4
F. Matichard,
5, 1
D. E. McClelland,
4
T. McRae,
4
A. Mullavey,
2
D. Sigg,
3
B. J. J. Slagmolen,
4
C. Whittle,
1
A. Buikema,
1
Y. Chen,
6
T. R. Corbitt,
7
R. Schnabel,
8
R. Abbott,
5
C. Adams,
2
R. X. Adhikari,
5
A. Ananyeva,
5
S. Appert,
5
K. Arai,
5
J. S. Areeda,
9
Y. Asali,
10
S. M. Aston,
2
C. Austin,
7
A. M. Baer,
11
M. Ball,
12
S. W. Ballmer,
13
S. Banagiri,
14
D. Barker,
3
J. Bartlett,
3
B. K. Berger,
15
D. Bhattacharjee,
16
G. Billingsley,
5
S. Biscans,
1, 5
R. M. Blair,
3
N. Bode,
17, 18
P. Booker,
17, 18
R. Bork,
5
A. Bramley,
2
A. F. Brooks,
5
D. D. Brown,
19
C. Cahillane,
5
K. C. Cannon,
20
X. Chen,
21
A. A. Ciobanu,
19
F. Clara,
3
S. J. Cooper,
22
K. R. Corley,
10
S. T. Countryman,
10
P. B. Covas,
23
D. C. Coyne,
5
L. E. H. Datrier,
24
D. Davis,
13
C. Di Fronzo,
22
K. L. Dooley,
25, 26
J. C. Driggers,
3
P. Dupej,
24
T. Etzel,
5
T. M. Evans,
2
J. Feicht,
5
P. Fulda,
27
M. Fyffe,
2
J. A. Giaime,
7, 2
K. D. Giardina,
2
P. Godwin,
28
E. Goetz,
7, 16
S. Gras,
1
C. Gray,
3
R. Gray,
24
A. C. Green,
27
Anchal Gupta,
5
E. K. Gustafson,
5
R. Gustafson,
29
J. Hanks,
3
J. Hanson,
2
T. Hardwick,
7
R. K. Hasskew,
2
M. C. Heintze,
2
A. F. Helmling-Cornell,
12
N. A. Holland,
4
J. D. Jones,
3
S. Kandhasamy,
30
S. Karki,
12
M. Kasprzack,
5
K. Kawabe,
3
P. J. King,
3
J. S. Kissel,
3
Rahul Kumar,
3
M. Landry,
3
B. B. Lane,
1
B. Lantz,
15
M. Laxen,
2
Y. K. Lecoeuche,
3
J. Leviton,
29
J. Liu,
17, 18
M. Lormand,
2
A. P. Lundgren,
31
R. Macas,
25
M. MacInnis,
1
D. M. Macleod,
25
G. L. Mansell,
3, 1
S. M ́arka,
10
Z. M ́arka,
10
D. V. Martynov,
22
K. Mason,
1
T. J. Massinger,
1
R. McCarthy,
3
S. McCormick,
2
J. McIver,
5
G. Mendell,
3
K. Merfeld,
12
E. L. Merilh,
3
F. Meylahn,
17, 18
T. Mistry,
32
R. Mittleman,
1
G. Moreno,
3
C. M. Mow-Lowry,
22
S. Mozzon,
31
T. J. N. Nelson,
2
P. Nguyen,
12
L. K. Nuttall,
31
J. Oberling,
3
Richard J. Oram,
2
C. Osthelder,
5
D. J. Ottaway,
19
H. Overmier,
2
J. R. Palamos,
12
W. Parker,
2, 33
E. Payne,
34
A. Pele,
2
C. J. Perez,
3
M. Pirello,
3
H. Radkins,
3
K. E. Ramirez,
35
J. W. Richardson,
5
K. Riles,
29
N. A. Robertson,
5, 24
J. G. Rollins,
5
C. L. Romel,
3
J. H. Romie,
2
M. P. Ross,
36
K. Ryan,
3
T. Sadecki,
3
E. J. Sanchez,
5
L. E. Sanchez,
5
T. R. Saravanan,
30
R. L. Savage,
3
D. Schaetzl,
5
R. M. S. Schofield,
12
E. Schwartz,
2
D. Sellers,
2
T. Shaffer,
3
J. R. Smith,
9
S. Soni,
7
B. Sorazu,
24
A. P. Spencer,
24
K. A. Strain,
24
L. Sun,
5
M. J. Szczepa ́nczyk,
27
M. Thomas,
2
P. Thomas,
3
K. A. Thorne,
2
K. Toland,
24
C. I. Torrie,
5
G. Traylor,
2
A. L. Urban,
7
G. Vajente,
5
G. Valdes,
7
D. C. Vander-Hyde,
13
P. J. Veitch,
19
K. Venkateswara,
36
G. Venugopalan,
5
A. D. Viets,
37
T. Vo,
13
C. Vorvick,
3
M. Wade,
38
R. L. Ward,
4
J. Warner,
3
B. Weaver,
3
R. Weiss,
1
B. Willke,
18, 17
C. C. Wipf,
5
L. Xiao,
5
H. Yamamoto,
5
Hang Yu,
1
L. Zhang,
5
M. E. Zucker,
1, 5
and J. Zweizig
5
1
LIGO, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
2
LIGO Livingston Observatory, Livingston, LA 70754, USA
3
LIGO Hanford Observatory, Richland, WA 99352, USA
4
OzGrav, Australian National University, Canberra, Australian Capital Territory 0200, Australia
5
LIGO, California Institute of Technology, Pasadena, CA 91125, USA
6
Caltech CaRT, Pasadena, CA 91125, USA
7
Louisiana State University, Baton Rouge, LA 70803, USA
8
Universit ̈at Hamburg, D-22761 Hamburg, Germany
9
California State University Fullerton, Fullerton, CA 92831, USA
10
Columbia University, New York, NY 10027, USA
11
Christopher Newport University, Newport News, VA 23606, USA
12
University of Oregon, Eugene, OR 97403, USA
13
Syracuse University, Syracuse, NY 13244, USA
14
University of Minnesota, Minneapolis, MN 55455, USA
15
Stanford University, Stanford, CA 94305, USA
16
Missouri University of Science and Technology, Rolla, MO 65409, USA
17
Max Planck Institute for Gravitational Physics (Albert Einstein Institute), D-30167 Hannover, Germany
18
Leibniz Universit ̈at Hannover, D-30167 Hannover, Germany
19
OzGrav, University of Adelaide, Adelaide, South Australia 5005, Australia
20
RESCEU, University of Tokyo, Tokyo, 113-0033, Japan.
21
OzGrav, University of Western Australia, Crawley, Western Australia 6009, Australia
22
University of Birmingham, Birmingham B15 2TT, UK
23
Universitat de les Illes Balears, IAC3—IEEC, E-07122 Palma de Mallorca, Spain
24
SUPA, University of Glasgow, Glasgow G12 8QQ, UK
25
Cardiff University, Cardiff CF24 3AA, UK
26
The University of Mississippi, University, MS 38677, USA
27
University of Florida, Gainesville, FL 32611, USA
28
The Pennsylvania State University, University Park, PA 16802, USA
29
University of Michigan, Ann Arbor, MI 48109, USA
arXiv:2002.01519v1 [quant-ph] 4 Feb 2020
2
30
Inter-University Centre for Astronomy and Astrophysics, Pune 411007, India
31
University of Portsmouth, Portsmouth, PO1 3FX, UK
32
The University of Sheffield, Sheffield S10 2TN, UK
33
Southern University and A&M College, Baton Rouge, LA 70813, USA
34
OzGrav, School of Physics & Astronomy, Monash University, Clayton 3800, Victoria, Australia
35
The University of Texas Rio Grande Valley, Brownsville, TX 78520, USA
36
University of Washington, Seattle, WA 98195, USA
37
Concordia University Wisconsin, 2800 N Lake Shore Dr, Mequon, WI 53097, USA
38
Kenyon College, Gambier, OH 43022, USA
Measurement of minuscule forces and displace-
ments with ever greater precision encounters a
limit imposed by a pillar of quantum mechanics:
the Heisenberg uncertainty principle. A limit
to the precision with which the position of an
object can be measured continuously is known
as the standard quantum limit (SQL) [1–4].
When light is used as the probe, the SQL
arises from the balance between the uncertainties
of photon radiation pressure imposed on the
object and of the photon number in the
photoelectric detection.
The only possibility
surpassing the SQL is via correlations within
the position/momentum uncertainty of the object
and the photon number/phase uncertainty of the
light it reflects [5].
Here, we experimentally
prove the theoretical prediction that this type
of quantum correlation is naturally produced
in the Laser Interferometer Gravitational-wave
Observatory (LIGO). Our measurements show
that the quantum mechanical uncertainties in the
phases of the 200 kW laser beams and in the
positions of the 40 kg mirrors of the Advanced
LIGO detectors yield a joint quantum uncertainty
a factor of 1.4 (3 dB) below the SQL. We
anticipate that quantum correlations will not only
improve gravitational wave (GW) observatories
but all types of measurements in future.
The Heisenberg uncertainty principle dictates that
once an object is localized with sufficient precision,
the momentum of that object must become accordingly
uncertain. In a one-off measurement, this does not pose a
problem. But in the case where the position of an object
must be measured continuously, as in gravitational wave
(GW) detectors, the momentum uncertainty from the act
of measuring position evolves into position uncertainty
for future position measurements – a process known as
quantum backaction. In striking a balance between the
precision of position measurements and the imprecision
caused by quantum backaction, an apparent maximum
precision for a continuous position measurement is
reached. This is the SQL, and for an interferometric
measurement, as long as the shot noise and QRPN are
uncorrelated, the SQL is indeed the limit.
The SQL was first introduced by Braginsky et al. [2, 3]
as a fundamental limit to the sensitivity of gravitational
wave detectors.
It should be possible to reach the
SQL with objects that are
macroscopic
or even human-
scale, because it is the quantization of the probe light
that enforces the SQL (see, e.g., footnote 1 of [4]). In
principle, the SQL can be surpassed when the shot noise
and QRPN are correlated. Such correlations already
exist in the interferometer, because incoming quantum
fluctuations entering from its output port drive both the
shot noise and the QRPN, giving rise to ponderomotive
squeezing. An injected squeezed state, when combined
appropriately with ponderomotive squeezing, enables
surpassing the SQL (see Sec. IVB of [4]). Alternative
methods for surpassing the SQL are presented in [4], and
extended to include optical spring effects in [6].
Here, we inject a laser mode in a squeezed vacuum
state in a laser interferometric GW detector with
40 kg mirrors, and use the optomechanically-induced
correlations of ponderomotive squeezing to surpass the
free-mass SQL. This measurement marks two significant
milestones of quantum measurement. First, we directly
observe QRPN contributing to the motion of kg-mass
objects, providing evidence that quantum backaction
imposed by the Heisenberg uncertainty principle persists
even at human scales. Second, we surpass the SQL,
proving the existence of quantum correlations involving
the position uncertainty of the 40 kg mirrors.
This
measurement is an important step toward further
improvements in GW sensitivity through quantum
engineering techniques [4, 6–10].
A significant barrier to revealing quantum correlations
between light and macroscopic objects is the ubiquitous
presence of thermal fluctuations that drive their
motion.
Previous demonstrations of QRPN have
involved cryogenically pre-cooled, pico- to micro-gram
scale mechanics [10–14], with two exceptions [15,
16].
Similarly, previous sub-SQL measurements of
displacements have also been performed on cryogenically
pre-cooled mechanical oscillators at the pico- [17] to
nano-gram [18] mass scale. The present measurements
are performed on the room-temperature, 40 kg mirrors
of Advanced LIGO using 200 kW of laser light, and are
enabled by injection of squeezed states and subtraction
of classical noise to reveal quantum noise below the SQL.
We performed this experiment using the Advanced
LIGO detector in Livingston, Louisiana.
For the
third astrophysics observing run, squeezed vacuum is
injected into the interferometer with squeezing level and
squeezing quadrature angle tuned to maximize the GW
sensitivity [19]. In this experiment, the interferometer is
maintained in the observing configuration [20], except
3
data is taken with an increased squeezing level and over
a range of squeezing angles, in order to fully characterize
the quantum noise.
FIG. 1.
Simplified schematic of the experimental setup.
Squeezed vacuum (dotted red) is injected through the output
Faraday isolator, and co-propagates with the 1064 nm light
(solid red) of the main interferometer. A frequency-shifted
control field (orange) is used to sense and tune the squeeze
angle.
The Advanced LIGO detector is a Michelson
interferometer with two 4-km Fabry-Perot arms, as well
as power- and signal- recycling cavities at the input
and output ports of the beamsplitter, respectively (see
Fig. 1). The arm-cavity optics are 40 kg fused silica
mirrors, suspended as pendulums inside an ultrahigh
vacuum envelope [21]. During the measurement, 200
±
10
kW of 1064 nm laser power circulates in each arm
cavity. After passing through an output mode cleaner,
the differential arm displacement signal (∆
x
) is detected
as modulations of a small static field at the output
due to a deliberate mismatch in the interferometer arm
lengths [21]. The displacement signal ∆
x
is part of a
closed servo loop, which is monitored by a continuous
calibration procedure that also extracts the instrument
sensing function by driving ∆
x
motion and measuring the
optical response. Details of the squeezed light source and
its operation, including the control method for adjusting
squeezing angle, are found in [19]. For this measurement,
injected squeezing results in 3.3 dB of squeezing and 7.7
dB of antisqueezing measured at the GW readout.
An analytic model of the displacement sensitivity in
an idealized LIGO interferometer illustrates how the
combination of ponderomotive squeezing and injected
squeezing allows us to surpass the SQL. A model which
builds on methods developed in [4, 6], with extensions to
account for losses and off-resonance cavities, is provided
in the Methods section. Here, the idealized model is
used for clarity. The Heisenberg uncertainty principle
applied to interferometric measurement of differential
displacement, ∆
x
, sets a limit to the one-sided spectral
density of:
∆
x
2
(Ω) =
S
(Ω
,φ
)(1 +
K
2
(Ω))
~
c
8
k
|
G
(Ω)
|
2
P
arm
(1)
with
K
(Ω) =
32
k
|
G
(Ω)
|
2
P
arm
m
Ω
2
c
G
(Ω)
≡
√
γc
2
L
1
γ
+
i
Ω
(2)
Here
P
arm
is the circulating arm power,
k
the laser
wavenumber, Ω
/
2
π
the sideband frequency of the GW
readout, and
m
each mirror mass.
L
is the arm length of
3995 m and
γ
the signal bandwidth of 2
π
·
450 Hz in LIGO.
G
(Ω) is the optical field transmissivity between the arm
cavities and readout detector, making 2
kG
(Ω)
√
P
arm
the
sensing function relating
δx
to the emitted optical field
that modulates the homodyne readout power.
The factors
S
(Ω
,φ
) and (1 +
K
2
(Ω)) capture the
radiation pressure interaction whereby the mirror oscil-
lator motion correlates the injected optical amplitude
quadrature to the output phase quadrature, with
K
(Ω)
the pondermotive interaction strength. The theory of
pondermotive squeezing is detailed in Sec. IVA-B of
[4].
S
(Ω
,φ
) accounts for injection of squeezed states.
Without injected squeezing,
S
=1, in which case the
arm power
P
arm
may be chosen to minimize ∆
x
(Ω) by
balancing shot noise and radiation pressure noise. The
resulting minimum ∆
x
(Ω) is the free-mass SQL for a
Michelson interferometer with a Fabry-Perot cavity in
each arm [4]:
∆
x
2
(Ω)
≥
∆
x
2
SQL
(Ω)
≡
8
~
m
Ω
2
(3)
When injecting squeezed states at squeeze angle
φ
with
squeeze factor
r
, the squeezing measured at the readout,
S
(Ω
,φ
), becomes:
S
(Ω
,φ
) =
e
−
2
r
cos
2
(
φ
−
θ
(Ω)
)
+
e
2
r
sin
2
(
φ
−
θ
(Ω)
)
(4)
θ
(Ω) = arctan(
K
(Ω))
(5)
φ
=0 is defined as the squeezing angle that reduces the
shot noise power spectral density, where
θ
→
0, by a factor
of
e
−
2
r
.
The expression
φ
−
θ
(Ω) characterizes the frequency-
dependent interaction between pondermotive and in-
jected squeezing. Eqn. 4 indicates that at frequencies
where
θ
(Ω)=
φ
, the two conspire to produce a minimum
in the quantum noise spectrum, appearing as a “dip” in
the curves of Fig. 2. Whereas the
S
= 1 case led to the
SQL in Eqn 3, injecting squeezed states allows the SQL
to be surpassed at measurement frequencies for which
S
(Ω
,φ
)
<
1.
Fig 2 shows amplitude spectral densities of differential
displacement.
Exposing the sub-SQL dip requires
reliably estimating and subtracting classical noise around
40 Hz. The data are acquired as three sets of spectral
4
FIG. 2.
Top: Differential displacement (
∆
x
) noise spectral density of the interferometer.
The grey trace shows the
measured total noise level of the interferometer with unsqueezed vacuum state (i.e. the reference). The blue trace is the model
of quantum noise during the reference measurement. The green trace shows the inferred quantum noise of the interferometer
with injected squeezing at 35
◦
, and its corresponding model is the purple trace. The notch feature, or “dip,” results from the
ponderomotive squeezing affecting the injected optical squeezed states. It reaches -3 dB of the free-mass SQL (red dashed trace,
given by Eqn. 3) at 40 Hz. Bottom: Phase-space representation of the modeled quantum states entering through the dark port
of the interferometer (left) and the output states (right), which are indexed to indicate their frequency dependence. Drawn are
the unsqueezed vacuum (dotted blue) and squeezing at
φ
=35
◦
(solid purple). In the unsqueezed vacuum case, ponderomotive
squeezing distorts the ellipse for frequencies below 100 Hz, increasing QRPN in the readout quadrature (blue arrows). In the
injected squeezing case, the same physical process creates a state with reduced noise at 40 Hz (purple arrows).
measurements in each of two operating modes – with and
without squeezing injection. By alternating operation
between the two modes, we establish that the noise is
consistent within statistical variations, confirming that
it is stationary over the duration of the experiment.
To further address the concern that the classical noise
between modes of operation may be changing, additional
data at a range of squeezing angles are obtained, as shown
in Fig. 3 .
In Fig. 2, the black trace is the measured total noise
at the readout with squeezing disengaged, including
both quantum and classical noise contributions. It is
generated from a 90-minute average split across three
non-contiguous time periods where the squeezer cavity is
set off-resonance [19], allowing the unsqueezed vacuum
state to enter the interferometer.
The blue trace is
the modeled quantum noise contribution to the total
noise measurement of the black trace.
Subtracting
the blue trace from the black trace gives the total
classical noise contribution. We verify that this classical
noise component is stationary, and independent of
squeezer status (see discussion of Figure 3 below and
details in Methods). The model shows that quantum
noise dominates the interferometer sensitivity at high
frequencies (Ω
> γ
≈
2
π
·
450 Hz), and accounts for
28% of the total measured noise power at 40 Hz. Of
the remaining non-quantum noise, 24% is estimated
to be coating and thermooptic noise, with the rest
unidentified [20].
The green trace of Fig. 2 shows the inferred quantum
noise spectrum with squeezing injected at
φ
=35
◦
. This
angle, determined from the model fit, places the dip
in the frequency region where the ratio of the total
noise in the reference spectrum and the SQL curve
is minimized.
The green trace is calculated as the
total measured displacement spectrum while the squeezer
is engaged, minus the classical noise contribution
previously established from the reference measurement.
5
FIG. 3. Quantum noise spectra at additional squeezing angles of 7
◦
(magenta), 24
◦
(blue), 46
◦
(orange). Each data set is
plotted with the same classical noise subtraction as Fig. 2, and with a corresponding quantum noise model curve (brown).
The model without injected squeezing (blue) is plotted for comparison. The differences between the squeezed data sets and the
reference model show that QRPN contributes to the motion of the Advanced LIGO mirrors. The QRPN contribution can be
increased and decreased as the injected state is varied. These data use less observing time than Fig. 2 and have correspondingly
larger statistical fluctuations.
The purple trace shows the quantum noise model
corresponding to
φ
=35
◦
squeezing, featuring a dip in the
quantum noise that reaches down to 70% or 3 dB of the
SQL at 40 Hz.
Squeezing measurements at three additional
φ
’s are
presented in Fig.3 . They show that QRPN contributes
to the motion of the Advanced LIGO mirrors. At each
φ
, the quantum noise trace is calculated by subtracting
the same classical noise contribution (determined from
the reference data) from the measured displacement
spectrum. We note that the modeled quantum noise
plotted here requires the full functional form of
S
(Ω
,φ,ψ
)
in Eqn. 9 in Methods, rather than the simplified version
of Eqn. 4. These additional measurements characterize
contributions from an unwanted phase shift due a slight
detuning of the signal cavity, which manifests as a
squeeze angle shift of
ψ
=8
◦
accumulating across the
frequency region where Ω
∼
γ
. A total of 12 squeezing
measurements are combined to plot
S
(Ω
,φ,ψ
) in the
Extended Data.
Uncertainty in both data and model are calculated
here, with additional details in Methods. The statistical
error in the power spectrum measurement of the quantum
noise, after subtraction, is 8% at 40 Hz (for a 0.5
Hz bin width).
We test for discrepancies between
the three reference datasets, and find that the relative
uncertainty in the classical noise stationarity is bounded
by the same statistical error.
Errors in the optical
sensing function 2
kG
(Ω)
√
P
arm
, along with the ∆
x
servo
loop compensation, are determined from the online
interferometer calibration procedure [20], and bounded
to be
±
3% [22]. Uncertainties in arm cavity power is 5%.
Aside from the reference, the model curves of Figs. 2 and
3 require the squeeze factor
r
and interferometer losses
[19], which are determined from fits across all datasets,
along with the signal-recycling cavity detuning
ψ
=8
◦
.
Optical spring effects are accounted in the calibration
but, at this
ψ
, are insignificant for the quantum noise
model.
The measurements presented here represent long-
awaited milestones in verifying the role of quantum
mechanics in limiting the measurement of small
displacements generally, and in the sensitivity of GW
detectors in particular.
First, we observe that QRPN contributes to the motion
of the kilogram-scale mirrors of LIGO. This observation
is also made in the Advanced Virgo GW detector [23].
It is remarkable that quantum vacuum fluctuations can
influence the motion of these macroscopic, human-scale
objects, and that the effect is measured. This is quantum
mechanics at its experimentally most macroscopic scale.
Second, revealing quantum noise below the SQL in
the Advanced LIGO detector is the first realization of a
quantum nondemolition technique in GW detectors [2, 3],
where quantum correlations prevent the measurement
device from demolishing the same information one is
trying to extract. Exploiting quantum correlations allows
a fundamental quantum limit to be manipulated to
improve measurement precision.
Finally, we must not forget the foremost scientific
objectives of the Advanced LIGO detectors: they are
designed for astrophysical observations of GWs from
violent cosmic events. During the third observing run,
the squeezing angle is set to optimize the sensitivity to
GWs from binary neutron star mergers [19]. This is
not the squeeze angle where shot noise is minimized,
but where the combination of shot noise and QRPN
are minimized, implying that backaction evasion plays
a role in optimizing the sensitivity of the Advanced
LIGO detector.
This is one of the factors that has
allowed Advanced LIGO to go from detecting roughly one
astrophysical event per month in observing runs 1 and 2,
to about one astrophysical trigger per week in the third
observing run. In the future, with further mitigation of
classical noise, sub-SQL performance of GW detectors
promises ever greater astrophysical reach.
6
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7
METHODS
This section expands on four topics related to the
measurement:
a) the augmented model for a non-
ideal interferometer, b) measurement uncertainty, c)
quantum noise model uncertainty, d) non-stationary
noise uncertainty, and e) the additional plots in Extended
Data.
The model curves present in Figures 2-5 are calculated
from the full coupled-cavity equations of [6], which are
exact and omit only effects from high-order transverse
optical modes. The model provided by equations 1-5
represents an idealized interferometer with all cavities
on resonance and no optical losses. Here we extend the
model to consider the dominant experimental deviations
from the ideal case, without the complexity of the exact
equations. This extension includes imperfect input and
output efficiency, as well as the additional frequency-
dependent effect on the squeezing angle from the small,
unintended phase shift within the signal-recycling cavity.
For the parameters of this paper, the following model
is accurate to 5% of the exact model quantum power
spectral density between 10Hz to 100Hz.
The input and output efficiency of the interferometer
are introduced using two new parameters,
η
i
and
η
o
respectively. The input efficiency represents the total
fractional coupling of optical power between the squeezer
cavity and the interferometer, and the output efficiency
is the total from the interferometer to the readout
homodyne detector. They must be considered separately
due to differences in their interaction with QRPN,
leading to the expressions:
∆
x
2
(Ω) =
S
∗
·
(
1 +
η
o
K
2
(Ω)
)
~
c
η
o
8
k
|
G
(Ω)
|
2
P
arm
(6)
(1
−
η
e
) = (1
−
η
i
) +
1
1 +
K
2
(Ω)
(1
−
η
o
)
(7)
S
∗
(Ω
,φ,ψ
) =
η
e
S
(Ω
,φ,ψ
) + (1
−
η
e
)
(8)
S
(Ω
,φ,ψ
) =
e
−
2
r
cos
2
(
φ
−
θ
∗
)
+
e
2
r
sin
2
(
φ
−
θ
∗
)
(9)
θ
∗
= arctan(
K
(Ω)) +
Ω
2
γ
2
+ Ω
2
ψ
(10)
External output loss does not change the dark-port to
arm cavity optical field transmissivity
G
(Ω), but it does
modify the dark-port to readout transmissivity, lowering
the sensing function to be 2
kG
(Ω)
√
η
o
P
arm
. This leads
to the
η
o
terms in Eqn. 6, where shot-noise scales as
1
/η
o
, but the QRPN term does not. QRPN pertains
to real motion, and its reduced influence on the optical
quantum noise is compensated by the ∆
x
calibration.
A frequency-dependent effective efficiency,
η
e
, accounts
for the output loss 1
−
η
o
not being able to affect the real
motion of the masses due to radiation pressure, while
the squeezed state is degraded by both input and output
losses. The form of Eqn. 7 reflects the relation of the
input, output and effective losses rather than efficiencies,
and it is accurate for small losses.
The total squeezing angle shift due to the signal
recycling cavity is encoded in the parameter
ψ
.
It
appears alongside the pondermotive effect on the
squeezing angle in Eqn. (10), except it accumulates
through the cavity pole transition. This formulation is
accurate for small detunings of the interferometer signal
cavity, and is related to the physical phase shift
ξ
within
the signal recycling cavity by
ψ
=10
.
7
ξ
, calculated for
the LIGO Livingston mirror parameters. Notably absent
from this non-ideal model but present in [6], is the
contribution of the optical-spring effect due to
ξ
6
=0. We
note that the above non-ideal model is accurate to 1%
in the zero detuning case
ψ
=
ξ
=0. While strong optical
springs are an alternative method of achieving sub-SQL
quantum noise sensitivity, the accuracy of the above
augmented model indicates that the spring contribution
is mostly negligible for this measurement.
Figure 2 shows that quantum noise accounts for only
28% of the total interferometer noise power at 40 Hz.
For this reason, classical noises must be subtracted
in order to reveal the quantum limited displacement
sensitivity. The interferometer is a complex instrument
with such sensitivity that the following considerations
must be addressed to validate the subtraction. First, the
fiducial quantum noise model of the reference dataset
and the parameters it relies on must be established
and the data must be calibrated. Second, the classical
noise established for the reference operating mode
must be representative of the classical noise during
squeezing operation. In particular, the classical noise
during the reference period must not be higher than
during squeezing, which would bias our inference to
underestimate the quantum noise contribution during
squeezing.
The reference and squeezing datasets are
taken in multiple, alternating segments and we test
for variations arising from non-stationary (time-varying)
noise.
Furthermore, the non-stationary noise power
contributions are mitigated by using a statistically
“robust” median based computation to calculate the
sampled power spectra.
The following paragraphs proceed by detailing
the measurement sequence used to characterize the
stationarity, then describe how uncertainty propagates
through the data analysis for the post-subtraction
quantum noise curves. We then show how the calibration
and interferometer data outputs are combined with
external measurements to establish our quantum noise
model.
The statistical uncertainty is then outlined,
followed by the methodology for characterizing the noise
stationarity between squeezing and reference datasets.
Finally, the spectral density estimator is described.
The data shown in Fig.
2 were taken over a 5
hour period on the advanced LIGO detector.
To
avoid variations of classical noise and calibration,
the interferometer power is held constant across all
measurements.
To minimize statistical error, the
majority of the measurement time is spent in the two
modes plotted: three 30-minute “reference” segments
8
with the squeezer disabled, alternating with three 30-
minute segments with squeezing at
φ
=35
◦
.
Each
reference segment is following by a squeezing segment,
alternating three times to establish that the classical
noise contribution is constant across the total duration.
The remaining time is split across nine additional
segments at varying input squeezing angles, and the final
segment is a fourth reference without squeezing.
Here we describe how the uncertainty propagates
through the subtraction in our measured quantum noise
curves.
The symbols for the frequency dependent
reference and squeezing data are
D
r
,
D
s
, and
M
r
,
M
s
for the models. The post-subtraction inferred quantum
noise is given as
Q
in the expression
Q
(Ω) =
D
s
(Ω)
−
(
D
r
(Ω)
−
M
r
(Ω)
)
(11)
The relative error of the post-subtraction squeezed
quantum noise is given by
δQ
, composed of the
quadrature sum of relative errors due to the optical
sensitivity calibration,
δG
; the servo loop calibration,
δC
;
the modeling uncertainty,
δM
r
; statistical fluctuations
δD
r
,
δD
s
; and relative stationarity uncertainty terms,
δN
t
, and
δN
m
. All of these uncertainties are frequency-
dependent, but the argument Ω is suppressed for space.
The definitions of these components are clarified in the
text following, but contribute to the expression:
δQ
2
=
δG
2
+
1
Q
2
(
M
2
r
·
δM
2
r
+ (
D
r
−
D
s
)
2
·
δC
2
+
D
2
r
·
δD
2
r
+
D
2
s
·
δD
2
s
+ (
D
r
−
M
r
)
2
·
(
δN
2
t
+
δN
2
m
)
)
(12)
The lines of the above relation represent terms with
different magnitudes of scaling terms.
Given that
Q
≈
M
s
∼
D
r
−
D
s
, the top line for the calibration
and model error has terms with order-1 coefficients,
indicating that the relative errors quoted in the main text
remain small for the comparison to the dip model. The
lower two lines of eq. 12 show that the relative statistical
fluctuations and stationarity uncertainties are magnified
by the ratio,
V
, of the total classical PSD to the squeezed
quantum PSD, approximately a factor of
V
=7
.
2, at 40
Hz.
The first line of Eqn. 12 includes the calibration and
unsqueezed reference quantum noise model uncertainty
terms,
δG
,
δC
,
δM
r
.
The LIGO online calibra-
tion system determines the optical sensing function
2
kG
(Ω)
√
η
o
P
arm
which affects both the model and
calibration uncertainties. To prevent double-counting
in the incoherent sum, this optical gain has been
isolated to the factor
δG
and should not be considered
in
δC
or
δM
r
.
The sensing function is monitored
continuously by injecting displacement signals at several
frequencies. Some of these appear as narrow lines in the
measured spectra of Figure 2. From these continuous
injections, the bandwidth
γ
and the product
η
o
P
arm
,
are determined. In addition, parameters related to the
optical spring are measured [22], but primarily affect the
sensing function at frequencies
<
10Hz for the measured
detuning. Additional lines monitor the ∆
x
servo loop
actuators to apply the frequency-dependent correction
for the servo closed loop response, which is contained in
δC
. The quoted calibration uncertainty of
±
3% is the
incoherent sum
√
δG
2
+
δC
2
.
Having factored
δG
out of
δM
r
, any error in
subtracting the classical noise estimate between reference
data and model can only arise from estimating the
shot noise and QRPN components represented by the
term
g
(1 +
η
o
K
2
(Ω)). Here,
g
is a scale factor relating
homodyne power to optical field. It is unknown because
the calibration system exports its sensing function
in an end-to-end fashion with the photodectors in
arbitrary voltage digitization units; however, the
g
may be well estimated using a cross-correlation method
detailed below.
The remaining
g
K
2
(Ω) contribution
may be estimated from the factors
|
G
(Ω)
|
2
√
P
arm
.
Independent measurements establish the quoted arm
power
P
arm
=200
±
10 kW, and this, combined with the
optical sensing gain calibration, allows us to determine
the output efficiency,
η
o
. The squeezing level at high
frequencies is determined by
r
and
η
o
·
η
i
(see Eqns. 7-
8), and using the extended datasets with
φ
= 0
◦
, the
input efficiency
η
i
may be determined from the observed
readout squeezing level.
The parameters describing the status of the interfer-
ometer and squeezer during the experiment are listed in
Table 1 of Extended Data with uncertainties. They are
also the values used in the modeling of quantum noise
calculation. Immediately before the 5 hour dataset, the
nonlinear parameter of the squeezer was measured to
calculate
r
. The squeezing angle is determined ultimately
through a model fit, but it agrees with our knowledge
of the nonlinear conversion from the coherent control
field demodulation angle to the observed squeezing angle
and the settings during the shot-noise squeezing (
φ
=0
◦
)
and antisqueezing (
φ
=90
◦
) datasets.
The frequency-
dependent contributions of the squeezing and arm power
modeling uncertainties are shown in Fig. 4, and they do
not strongly influence the model at the sub-SQL dip.
The following cross-correlation method [24] is used
to determine the factor
g
, that relates the arbitrary
experimental photodetector units back to the physical
optical field units. Two photodetectors are located at the
readout port of the LIGO interferometer (see figure 1).
When squeezing is not injected, shot-noise and readout
electronics noise (i.e.
dark noise) are uncorrelated
between the two photodetectors, while QRPN and all of
the classical noises are correlated. If the cross correlation
and dark noise is subtracted from total noise power for
the reference dataset, then only the shot noise remains,
calibrated to displacement. This precisely determines
the optical sensing gain in physical units, up to the
uncertainty
δG
.
The dark noise is only 1% of the
shot noise power and so contributes negligibly to the
uncertainty in this subtraction.
9
The statistical uncertainty arises in that the
fluctuations intrinsic to noise also limit our ability
to estimate it. With total measurement time
T
i
for
a given dataset
i
, and bin width of ∆
F
= 0
.
5Hz in
the spectral density calculation, the relative statistical
uncertainty of the inferred quantum noise power is
δD
i
= (
ET
i
∆
F
)
−
1
/
2
, with
E
the statistical efficiency
accounting for the spectral estimation method. For the
median method detailed below, we determine through
numerical experiments on white noise that
E
= 1
.
0 for
single-bin error bars. The bin-bin covariance due to the
apodization window causes
E
= 60% when averaging
multiple adjacent datapoints.
The total statistical
uncertainty of 8% includes both datasets
δD
r
and
δD
s
and their scaling by
V
in Eqn. 12.
Here we describe and characterize the terms
δN
t
,
δN
m
in the uncertainty budget of Eqn. 12.
We
label these terms together the stationarity uncertainty,
and they are intended to quantify potential variations
between the classical noise power as estimated from
the unsqueezed reference dataset and the classical noise
power actually present in the squeezing measurements.
Under the presupposition that the models,
M
r
, and
M
s
are perfect, and the statistical noise is small,
these uncertainties are defined as the relative difference
D
s
−
M
s
≡
(
D
r
−
M
r
)
·
(1+
δN
t
+
δN
m
).
The two are
distinguished as the changes to classical noise arising
from variations in time,
δN
t
, and from switching the
physical operating mode between the reference and
squeezing,
δN
m
.
The time variation contribution to non stationarity,
δN
t
, is mitigated both through the spectral density
estimation method and the use of three alternating
segments for the reference and squeezed data. The aim
of the alternating segments is for the operating mode
to switch on a timescale faster than the environmental
variation. The environmental timescale is not known or
even well-defined, so instead the discontiguous segments
of reference time are compared, setting a limit to
the non-stationarity of the squeezing segment between
them. This is done likewise for the squeezing segments
surrounding a reference segment. We define a metric
for the relative non-stationarity between two such
discontiguous segments to be
N
ij
= 2
D
i
−
D
j
D
i
+
D
j
(13)
Each pair of datasets makes an estimate of the
noise contribution varying at and below the separation
timescale of the datasets, here 1 hour. Each estimate
N
ij
is limited by the statistical error of the constituent
datasets, and they are shown in Fig. 6.
Since each
pair only constitutes a fraction of the full data, the
multiple estimates are combined to reduce the statistical
uncertainty.
N
2
Σ
=
1
6
(
N
2
R
12
+
N
2
R
23
+
N
2
R
31
+
N
2
S
12
+
N
2
S
23
+
N
2
S
31
)
(14)
Finally, these metrics must be related to the stationarity
term
δN
t
.
The averaged nonstationary power
N
2
Σ
represents an estimate of the time-varying contibution
between adjacent reference and squeezing segments, of
which there are three. For many such segments, assuming
random fluctuations to the environmental noise level
at the alternation time scale, the contributions add in
quadrature to give
δN
2
t
.
N
2
Σ
/
3. We then propagate
the statistical noise limits for segments one third the
length of the total reference time
T
. This arrives at
the statistical limit to our stationarity uncertainty of
δN
t
≈
√
2(
ET
∆
F
)
−
1
/
2
. Because the total squeezing data
time is also
T
, our limit to the time variation contribution
to non-stationarity evaluates to be the same as the
total statistical uncertainty from both the squeezing and
unsqueezed datasets,
δN
2
t
≈
δD
2
r
+
δD
2
s
. In addition
to the individual pairs, Fig. 6 also shows the combined
estimate
N
2
Σ
.
The operating-mode varying component
δN
m
of
non-stationary noise is constrained by the following
arguments.
The first is that it is quantitatively
constrained by the data at additional squeezing angles
depicted in Fig. 3
of the main text and Fig. 5 of
the Extended Data.
There, the same classical noise
estimate is subtracted and the model curves maintain
their agreement with the inferred quantum noise at
alternate squeezing angles. Those datasets however have
limited statistical bounds due to their short duration.
The term
δN
m
may be considered small for the following
physical reasons. The primary reason is that during the
without-squeezing time, the optical path is not changed,
only the squeezer OPO is operated off of resonance to
stop its nonlinear parametric interaction. This means
that environmental scatter noise - the very low-power
light leaking from the interferometer to the squeezer
system - does not impinge on different scattering surfaces
between the two modes. In the event that such scatter
does matter, the fourth reference taken at the end of
the entire measurement period uses an in-vacuum beam
diverter to block the path to the squeezer.
Testing
that fourth reference against the other three through the
N
ij
method shows no significant changes to the classical
noise.
In the event that the classical noise does change from
the switch to squeezing, we argue that the addition of
the nonlinear parametric interaction from the squeezer
on this scattered light is more likely to increase the noise
only during the squeezing segments. This implies that
the measurement should not be biased low and will not
over-estimate how much we have surpassed the SQL.
Indeed, the few data points in Figs. 2 and 4 that exceed
the model beyond the statistical fluctuations may be due
to such a squeezer-specific noise source. We attribute
10
the minimal classical noise contribution to the use of a
traveling wave OPO cavity, in-vacuum suspended layout
and coherent control implementation [19].
Finally, we describe the median method used for our
spectral density estimation.
We claim through the
above arguments that the classical noise is established
to be stationary in these datasets, however it is known
from astrophysical analysis that these complex detectors
have intermittant time-resolved glitches and artifacts of
varying strengths. Intervals of excess noise are nontrivial
to identify due to the inherently random nature of
noise, and time-resolved noise power vetoes can introduce
selection bias.
We use the Welch - Bartlett overlap
method to estimate the power spectral density with no
selection vetoes.
Instead, rather than averaging the
individual spectra independently at each frequency, the
sample median at each frequency is taken. This generates
a bin-by-bin median strain spectral power density.
Initially, the entire period for a given spectral density
estimate is split into N 2-second segments, where
each segment overlaps the segment before it by 50%,
implementing the Welch method. For each segment,
the time-series is linear detrended and a Hann window
is applied, then converted to a displacement spectrum
by a Fourier Transform.
The collection of segments
gives N estimates of the power density in each frequency
bin, each nominally following a chi-square distribution
on two variables (the real and imaginary parts of the
Fourier transform), but the distribution has an extended
tail due to glitches and transients of the detector.
The median is picked for each frequency bin, and
then a computed scale factor is applied to convert the
distribution median to the mean noise power.
This
technique is unbiased for stationary noise, and greatly
improves the robustness to glitches and nonstationary
contributions, without selection bias from time-domain
band-limited noise vetos.
The downside is that the
statistical efficiency is approximately
√
2 worse than the
typical Welch method for a given spectrum averaging
time.
Fig. 4 of Extended Data shows a variation of Figure
2 spanning a wider frequency range. The figure includes
the frequency-dependent uncertainties of Eqn. 12 in its
model curves and subtracted quantum noise plots.
Fig. 5 shows a measurement of (upper) and model
of (lower) the squeezing term
S
∗
(Ω
,φ,ψ
) of the
augmented model. The quantum noise spectrum at ten
additional
φ
’s is determined by subtracting the classical
noise contribution (previously established through the
reference measurement) from the measured displacement
spectrum at each
φ
.
Each inferred quantum noise
spectrum is then divided by the modeled quantum
noise spectrum without injected squeezing (blue trace in
Fig.2) to obtain the observed squeezing term
S
∗
(Ω
,φ,ψ
).
The dashed lines indicate cross-sections in other figures.
Green is
φ
=35
◦
in Fig. 2, and yellow blue and purple
correspond to the angles of Fig. 3 .
Acknowledgements:
LIGO was constructed by the
California Institute of Technology and Massachusetts
Institute of Technology with funding from the National
Science Foundation, and operates under Cooperative
Agreement No.
PHY-0757058.
Advanced LIGO
was built under Grant No.
PHY-0823459.
The
authors also gratefully acknowledge the support of the
Australian Research Council under the ARC Centre of
Excellence for Gravitational Wave Discovery, grant No.
CE170100004 and Linkage Infrastructure, Equipment
and Facilities grant No. LE170100217 and Discovery
Early Career Award No. DE190100437; the National
Science Foundation Graduate Research Fellowship under
Grant No.
1122374; the Science and Technology
Facilities Council of the United Kingdom, and the LIGO
Scientific Collaboration Fellows program.