# Cold-atom sources for the Matter-wave laser Interferometric Gravitation
Antenna (MIGA)

Quentin Beaufils  LNE–SYRTE, Observatoire de Paris, Université PSL, CNRS:UMR
8630, Sorbonne Université, 61 avenue de l’Observatoire, F–75014 Paris, France
<EMAIL_ADDRESS>Leonid A. Sidorenkov  LNE–SYRTE, Observatoire de
Paris, Université PSL, CNRS:UMR 8630, Sorbonne Université, 61 avenue de
l’Observatoire, F–75014 Paris, France  Pierre Lebegue  LNE–SYRTE, Observatoire
de Paris, Université PSL, CNRS:UMR 8630, Sorbonne Université, 61 avenue de
l’Observatoire, F–75014 Paris, France  Bertrand Venon  LNE–SYRTE, Observatoire
de Paris, Université PSL, CNRS:UMR 8630, Sorbonne Université, 61 avenue de
l’Observatoire, F–75014 Paris, France  David Holleville  LNE–SYRTE,
Observatoire de Paris, Université PSL, CNRS:UMR 8630, Sorbonne Université, 61
avenue de l’Observatoire, F–75014 Paris, France  Laurent Volodimer  LNE–SYRTE,
Observatoire de Paris, Université PSL, CNRS:UMR 8630, Sorbonne Université, 61
avenue de l’Observatoire, F–75014 Paris, France  Michel Lours  LNE–SYRTE,
Observatoire de Paris, Université PSL, CNRS:UMR 8630, Sorbonne Université, 61
avenue de l’Observatoire, F–75014 Paris, France  Joseph Junca  LP2N,
Laboratoire Photonique, Numérique et Nanosciences, Université
Bordeaux–IOGS–CNRS:UMR 5298, rue F. Mitterrand, F–33400 Talence, France
Xinhao Zou  LP2N, Laboratoire Photonique, Numérique et Nanosciences,
Université Bordeaux–IOGS–CNRS:UMR 5298, rue F. Mitterrand, F–33400 Talence,
France  Andrea Bertoldi  LP2N, Laboratoire Photonique, Numérique et
Nanosciences, Université Bordeaux–IOGS–CNRS:UMR 5298, rue F. Mitterrand,
F–33400 Talence, France  Marco Prevedelli  Dipartimento di Fisica e
Astronomia, Università di Bologna, Via Berti-Pichat 6/2, I-40126 Bologna,
Italy  Dylan O. Sabulsky  LP2N, Laboratoire Photonique, Numérique et
Nanosciences, Université Bordeaux–IOGS–CNRS:UMR 5298, rue F. Mitterrand,
F–33400 Talence, France  Philippe Bouyer  LP2N, Laboratoire Photonique,
Numérique et Nanosciences, Université Bordeaux–IOGS–CNRS:UMR 5298, rue F.
Mitterrand, F–33400 Talence, France  Arnaud Landragin  LNE–SYRTE, Observatoire
de Paris, Université PSL, CNRS:UMR 8630, Sorbonne Université, 61 avenue de
l’Observatoire, F–75014 Paris, France  Benjamin Canuel  LP2N, Laboratoire
Photonique, Numérique et Nanosciences, Université Bordeaux–IOGS–CNRS:UMR 5298,
rue F. Mitterrand, F–33400 Talence, France  Remi Geiger  LNE–SYRTE,
Observatoire de Paris, Université PSL, CNRS:UMR 8630, Sorbonne Université, 61
avenue de l’Observatoire, F–75014 Paris, France

###### Abstract

The Matter-wave laser Interferometric Gravitation Antenna (MIGA) is an
underground instrument using cold-atom interferometry to perform precision
measurements of gravity gradients and strains. Following its installation at
the low noise underground laboratory LSBB in the South-East of France, it will
serve as a prototype for gravitational wave detectors with a horizontal
baseline of 150 meters. Three spatially separated cold-atom interferometers
will be driven by two common counter-propagating lasers to perform a
measurement of the gravity gradient along this baseline. This article presents
the cold-atom sources of MIGA, focusing on the design choices, the realization
of the systems, the performances and the integration within the MIGA
instrument.

###### keywords:

Atom interferometry and Gravitational wave detection and Cold-atoms and
Gravity gradiometry and Instrumentation.

##  1 Introduction

Gravitational wave astronomy requires the development of detectors addressing
the various frequency bands of interest to the astrophysics and fundamental
physics communities [1, 2]. Besides the current ground-based laser
interferometers LIGO [3], VIRGO [4] and KAGRA [5] operating in the $\sim 10$
Hz to 2 kHz frequency band, and the Pulsar timing array probing the nHz region
of the spectrum, the space-based laser interferometer LISA is planned to be
launched in 2034 to probe the $\sim 1-100$ mHz frequency band. Beyond these
detectors, several instruments are being designed to probe the intermediate
frequency band $0.1-10$ Hz on the ground [6, 7] or in space [8, 9]. Ground
based cold-atom interferometers represent a promising candidate solution owing
to their large immunity to seismic noise [10] and their potential to reject
the gravity gradient noise to a sufficient extent [11] in that frequency band.

The Matter-wave laser Interferometric Gravitation Antenna (MIGA) aims at
providing the first platform to study the key techniques for GW detection with
matter wave interferometry. The project is described in details in Ref. [12].
An overview of the instrument is also shown on figure 1. It will consist of
two perpendicular arms of 150-meter long vacuum tubes (diameter of 50 cm)
containing two parallel laser beams propagating at different heights
(separated by 30 cm) in order to drive a cold-atom interferometer that
measures the local, horizontal, component of the gravitational field. Three
Bragg atom interferometers (AIs) will be distributed in each arm with a mutual
interval distance of nearly 75 m. The differential measurements of gravity on
various baselines will allow to extract the local gravity gradient.

The characteristics of the MIGA instrument (horizontal interrogation
direction, Bragg diffraction) require a specific development for the atomic
preparation and detection. An important feature of the instrument is the
ability to produce cold atom clouds with the appropriate quantum state which
are injected in the AI region, and to detect the state of the atoms at the
output of the interferometer. A Bragg interferometer requires a narrow
velocity selection for the input state, and the ability to detect the output
velocity state. This article presents the working principle of the cold-atom
systems (excluding the interferometer part), their design and realization, and
their performances.

##  2 Overview of the future MIGA instrument

The future system relies on horizontal Bragg interferometers where three light
pulses are used to split, deflect and recombine the atomic waves. The cold
atoms enter the interferometer from the bottom and interact with two
horizontal pairs of counterpropagating Bragg lasers separated by a distance of
30.6 cm in the vertical axis. The splitting and recombination ($\pi/2$) pulses
are done with the bottom beam while the deflection ($\pi$) pulse is performed
with the top beam. The separation between the 2 beams corresponds to an
interferometer with a pulse separation time $T=250$ ms. The Bragg transitions
at each position are based on the interaction of the atom with two counter-
propagating laser beams of the same frequency, which couple the momentum
states $|\vec{p}-\hbar\vec{k}\rangle$ and $|\vec{p}+\hbar\vec{k}\rangle$,
where $\vec{p}$ is the momentum of the atom and
$\vec{k}=\frac{2\pi}{\lambda}\hat{u}$ is the wave vector of the lasers
($\lambda\simeq 780$ nm, $\hat{u}$ unit vector of the laser propagation). With
a vertical launch velocity $v_{0}=4.45$ m$/$s, the time spent from the launch
to the apogee ($\pi$ Bragg pulse) is $t_{ap}=454$ ms.

The experimental sequence starts with a cold atom production phase during
which the Rubidium 87 atoms are collected in a magneto-optical trap (MOT),
laser cooled using polarization gradient cooling and vertically launched. On
their way up before the interferometer, the atoms are prepared in the desired
quantum state (internal state and velocity). On the way down, the probability
for an atom to be in a given output momentum state is detected using a
momentum-to-internal state labelling technique based on a Raman transition,
followed by a fluorescence detection.

Figure 1:  Overview of the MIGA instrument. (a) CAD view of the turret at the
extremity of one of the two orthogonal 150-m long vacuum tubes of MIGA; one
cold-atom system is shown on the right without its magnetic shields. The
vertical panels represent valves to isolate the tubes from the chamber hosting
the optics. (b) CAD drawing of the cold-atom source with a section of the
vacuum tube (diameter: 500 mm); the two red cylinders represent the Bragg
beams used to drive the atom interferometer and separated by $30.6$ cm. The
distance from the ground to the bottom of the vacuum tube is about 1 meter.
(c) Sectional view of the core of the cold-atom source: a 2D MOT (right) loads
a 3D MOT (center) from which the atoms are launched vertically. On their way
up, they pass through various light beams to prepare their quantum state (see
text). The interferometer region doesn’t appear on this picture.

##  3 Description of the cold atom head

###  3.1 The cold atom production

A two-dimensional magneto-optical trap (2D MOT) fed by a rubidium alloy
dispenser is realized in a vacuum chamber connected to the atom preparation
chamber by a differential vacuum tube (see figure 1 (b) and 2), allowing to
reach a loading rate of $6\times 10^{8}$ atoms$/$s in a three-dimensional MOT
(3D MOT).

The atoms from the 3D MOT are then simultaneously cooled down and vertically
accelerated to a velocity of $v_{0}=4.45$ m$/$s in a moving molasses. Two
acousto-optic modulators respectively driving three top and three bottom MOT
beams allow to apply a detuning $\delta$ between each pair of
counterpropagating beams. The three pairs of beams are mutually perpendicular
and form a angle of $arccos(\frac{1}{\sqrt{3}})$ with the vertical axis,
yielding an equilibrium velocity in the molasses of
$v_{0}=\frac{\sqrt{3}}{2}\delta\lambda$, where $\lambda$ is the laser
wavelength. After shutting off the MOT magnetic field, the global laser
detuning from atomic transition is increased from 12 to 150 MHz and the laser
intensity is decreased to zero in a 2 ms ramp, allowing the atomic cloud to
reach a temperature of $\approx 2$ $\mu$K. The molasses duration is limited to
a few ms by the transit time of the accelerated cloud through the MOT beams.

Figure 2:  Photos of the cold atoms unit. Left: MOT and preparation/detection
chamber with optics and magnetic coils. Top right: the complete vacuum system
of the atom head without optics. Bottom right: The 2D MOT chamber with optics
and magnetic coils.

###  3.2 The laser system

The laser system used for the cold-atom sources is described in [13]. We give
a short description here for consistency. The different lasers used to cool
and manipulate the atoms are delivered from an all-fibered laser module. The
laser architecture is based on frequency doubled telecom lasers. A Master
laser is locked using a Rubidium 85 saturated absorption spectroscopy signal
and references 3 slave lasers which are respectively used for the 2D MOT
cooling laser, the 3D MOT cooling and Raman 2 laser, and the 3D MOT repumper
and Raman 1 laser. The 3 slave lasers are all phase locked to the Master laser
via an adjustable frequency offset. The repumping light for the 2D MOT is
generated by a fiber electro-optic phase modulator at 1560 nm fed with the
appropriate microwave frequency.

After amplification of the 1560 nm light emitted by laser diodes in Erbium
doped fiber amplifiers and second harmonic generation in PPLN waveguide
cristals, the 780 nm light is sent to optical splitters and guided to the
experiment chamber using several optical fibers. The laser module nominally
delivers 300 mW total power for the 2D MOT, 150 mW total power for the 3D MOT,
350 mW in the Raman 1 beam and up to 700 mW in the Raman 2 beam. The relative
power and polarization fluctuations at the fiber outputs have been measured to
be below one percent rms on a timescale of 100 hours. The phase lock signals
are controlled by radio and microwave frequency sources all referenced to a
stable 100 MHz quartz oscillator. The full laser system is hosted in a
$1.7\times 0.5\times 0.5\ \text{m}^{3}$ transportable rack. The control of the
laser system and of the experiment is provided by home-made electronics [14].

###  3.3 The magnetic shield and bias field

The three AI parts of the instrument will be enclosed in mu-metal in order to
maintain the stray magnetic field to a low level (see figure 3). The
interferometer regions will be enclosed in a $2404\times 598\times 720$ mm
magnetic shield in order to suppress the measurement bias due to the second
order Zeeman effect. The atoms preparation region is also shielded in order to
keep a low magnetic field during the moving molasses sequence. After turning
off the MOT magnetic field, residual magnetic field at the position of the 3D
MOT of $\approx 7$ mG was measured in-situ by microwave spectrocopy of the
atomic hyperfine transition.

A homogeneous horizontal bias magnetic field of $\approx 200$ mG is produced
by 4 vertical copper rods disposed around the atom head in order to define a
quantization axis during the state preparation phase and the detection phase.

Figure 3: Magnetic shields. (a) Technical drawing of the outer shield
surrounding a part of the vacuum tube above the cold-atom source. (b)
Photograph of the shield with an open part. (c) Inner shield surrounding the
cold-atom source

###  3.4 Preparation of the atoms before the interferometer

In order to inject the atoms with the right velocity in the Bragg
interferometer, the quantum state (momentum and spin state) of the atoms is
prepared on their way up, before the interrogation region. A first counter-
propagating velocity-selective Raman pulse (beam ’Raman 1’ in Fig. 4) is used
to select the atoms from a mix of the Zeeman sub-levels of $|F=2\rangle$ to
the $m_{F}=0$ Zeeman sub-level of the $|F=1\rangle$ hyperfine state, with a
relatively narrow velocity class in the direction of the Raman lasers (e.g.
corresponding to a temperature of $\approx 100$ nK). The atoms which are not
transferred to $|F=1\rangle$ by the Raman transition are then pushed by a
laser tuned on resonance with the cycling transition (’push 1’ in Fig. 4). A
similar Raman/push procedure is then applied a second time to clean the
remaining unwanted $F=1$ atoms produced by spontaneous emission during the
first Raman selection pulse. For this purpose, we use a second Raman light
pulse (’Raman 2’, $1/e^{2}$ beam waist of $w=30$ mm) to transfer the atoms
back to the $F=2$ state. The remaining atoms in the $F=1$ state are pushed
with an orthogonal beam tuned on the $F=1\rightarrow F^{\prime}=0$ transition
(’push 2’).

Figure 4:  Atom preparation (left) and detection system (right). In the
preparation, the atom cloud moving upward and initially containing atoms in
all sublevels of $F=2$ goes through a series of Raman transitions and pushing
beams, which leaves the remaining atoms in the $F=2,m_{F}=0$ state with a much
reduced velocity spread in the x direction. For the detection, atoms are
leaving the interferometer in a superposition of two momentum states. The
population in one of them is transferred to $F=1$ with a Doppler sensitive
Raman transition. State-selective fluorescence detection then allows to
retrieve the output state and thus the interferometric phase.

The angle of the Raman beams with respect to the horizontal plane is set to
few degrees in order to introduce a Doppler effect and thus lift the
degeneracy between the $|p\rangle\rightarrow|p+2\hbar k\rangle$ and
$|p\rangle\rightarrow|p-2\hbar k\rangle$ transitions. The atoms must enter the
interrogation region with a well defined velocity in order to fulfill the
resonance condition in the Bragg interferometer where the laser beams are
horizontal (wavevector $\vec{k}=k\vec{e}_{x}$). For a $2n\hbar k$ transition
coupling the momentum states
$|\vec{p}_{0}\rangle\rightarrow|\vec{p}_{0}+2n\hbar\vec{k}\rangle$, the input
momentum must fulfill $p_{0x}=-n\hbar k$. The vertical velocity at the first
Bragg pulse is $gT\simeq 2.5$ m.s-1, resulting in an angle $\theta=n\hbar
k/MgT\simeq n\times 3.2$ mrad of the atom trajectory with respect to gravity.
A rough control of this angle can be obtained by adjusting the directions of
the 3D MOT Raman beams. Then, a more controlled step consists in selecting the
right horizontal velocity class with the two step Raman selection by using
different two-photon detunings for the ’Raman 1’ and ’Raman 2’ beams.

###  3.5 Detection of the atomic state

Figure 5:  CAD drawing of the atomic preparation and detection system. The
bottom panel shows how the 3 elliptical light sheets used for the detection
are produced from the light originating from two elliptical collimators.

After their interrogation by the Bragg beams in the interferometer, the two
different momentum states $|\pm n\hbar k\rangle$ of the atoms are labeled to
two different internal states with the Raman 2 laser beam (see figure 4). More
precisely, the velocity selective feature of the Raman transition is used to
transfer the $|F=2,-n\hbar k\rangle$ atoms to the $F=1$ internal state, while
the $|F=2,n\hbar k\rangle$ atoms remain in the $F=2$ internal state. The
internal state of the atoms can then be resolved by state selective
fluorescence detection. The falling atomic cloud crosses a series of
horizontal light sheets. Detection of the population in $F=2$ (that
corresponds to the $|F=2,n\hbar k\rangle$ output state of the interferometer)
is first realized by measuring the fluorescence induced by a first light sheet
beam tuned on resonance with the $F=2\rightarrow F^{\prime}=3$ transition. The
retro-reflected light sheet is partially blocked at the reflection mirror so
that the $F=2$ atoms are pushed away in the process. The $F=1$ atoms are then
re-pumped to the $F=2$ state using a thinner intermediate light sheet tuned to
the $F=1\rightarrow F^{\prime}=2$ transition. Those atoms are then detected in
a third light sheet similar to the first one, measuring the population
initially in $F=1$ (that corresponds to the $|F=2,-n\hbar k\rangle$ output
state of the interferometer). The preparation and detection systems CAD
drawings are shown in figure 5.

Figure 6:  Time of flight fluorescence signal of the upper light sheet (atoms
in $F=2$) recorded by the detection system.

The fluorescence light emitted by the atoms in the two light sheets is
collected by a lens with a $2\%$ collection efficiency and imaged on a two-
quadrant photodiode. Figure 6 shows a typical time of flight fluorescence
signal. The atomic population in a hyperfine state is retrieved by integrating
the signal in the corresponding photodiode. The transition probability is
deduced from the signals from the two light sheets (see section 2.6) after
correction of systematic cross-talks between the two optical systems. The
signal to noise ratio of the detection system is limited by electronic noise
on the photodiode due to its dark current. The contribution from the
electronic noise is $\sigma_{elec}=25$ $\mu V/\sqrt{Hz}$. From a mean atomic
signal of $S\simeq 100$ mV we deduce a maximal signal to noise ratio of about
1000.

##  4 Characterization of the system

###  4.1 State preparation

Figure 7: Left: Raman spectroscopy. Transition probability as a function of
Raman detuning, for a Rabi frequency of $\Omega=2\pi\times 25$ kHz. The
gaussian fit (solid line) indicates a full width at half maximum (FWHM) of
$90$ kHz limited by the atomic velocity spread. Inset: Transition probability
versus duration showing Rabi oscillations for different Raman light
intensities. Right: Raman spectrocopy FWHM of the peak for different Raman
Rabi frequencies. Each point was extracted from a resonance peak obtained with
a Raman pulse adjusted to half the Rabi period. The width of the peak is
Fourier limited for large Rabi frequencies but limited to $\approx 90$ kHz by
velocity spread for lower values. The solid line shows the expected FWHM as
the result of numerical integration of the convolution between a gaussian
velocity distribution for a temperature $T=2$ $\mu$K and the profile of a
square transition pulse.

In order to characterize the velocity selection during the state preparation
phase, we performed Raman spectroscopy with the bottom Raman beam (Raman 1,
see figure 7). Due to the Doppler effect, the observed resonance is at a Raman
detuning of $810$ kHz from the hyperfine transition. This corresponds to a
component of the velocity of $0.31$ m/s for the atoms along the Raman wave
vector axis. Such value is consistent with an angle of 4.6 degrees between the
Raman wave vector and the horizontal plane, given a vertical velocity for the
atoms of $4.0$ m/s when they cross the Raman beam. The inset of figure 7 (a)
shows Rabi oscillations on the Raman 1 transition for different Raman light
intensities. We observe a maximum transition probability of 10$\%$ for a Rabi
frequency of $\Omega_{R}\approx 2\pi\times 100$ kHz. This is mostly limited by
atoms from the molasses are distributed among all the different spin
projection states and only a fraction of the atoms being initially in $F=2$,
$m_{F}=0$. Moreover, transverse inhomogeneities of the light intensity lead to
the damping of the oscillations and limit the contrast.

Figure 8: (a) Raman spectroscopy performed with the upper Raman beam (Raman
2), for three different mean velocities selected with the lower Raman beam
(Raman 1). The peaks correspond to three different Raman 1 frequencies
incremented by 40 kHz steps. Inset: Raman 2 resonance as a function of Raman 1
frequency. The offset between the two frequencies is due to a difference in
vertical velocity when the atoms cross the beam leading to a different Doppler
shift. (b) Raman spectroscopy performed with the upper Raman beam (Raman 2),
for four different velocity spreads selected with the lower Raman beam (Raman
1). The legend gives the corresponding effective temperatures deduced from a
gaussian fit of the peak.

Raman velocimetry allows to estimate the temperature of the cloud. We
performed Raman spectroscopy for different Rabi frequencies, adjusting the
pulse duration to half a Rabi period. Figure 7 (b) shows the FWHM extracted
from a Gaussian distribution adjustment as a function of the Rabi frequency.
For large Rabi frequency the width of the peak is Fourier limited, whereas it
reaches a constant value of $\approx 90$ kHz when the Rabi frequency is below
$50$ kHz. This value is limited by the Doppler effect induced by the atomic
velocity spread, which corresponds to an atomic temperature of $\approx 2$
$\mu$K.

The main function of the Raman 1 beam is to allow for a narrow horizontal
velocity selection in order to couple the atoms to the Bragg interferometer
with the right velocity. Figure 8 (a) shows Raman spectroscopy performed with
the top Raman beam (Raman 2), for different horizontal velocity distributions
selected using Raman 1. For each peak we performed a Raman 1 transition with a
different frequency, and a light intensity and pulse duration corresponding to
a Fourier limited Gaussian FWHM of $\approx 23$ kHz in order to select a
velocity spread corresponding to an effective temperature of $150$ nK along
this direction. The spectroscopy was performed with a longer Raman 2 pulse in
order to resolve the Doppler induced frequency width related to this
temperature. Gaussian adjustments of the spectra are in good agreement with an
effective temperature of $150$ nK. The Figure 8 (b) shows the same experiment,
but for a single mean velocity selected and various effective temperatures.
These results demonstrate that we have a good control on the momentum
distribution that will enter the interferometer.

Figure 9 shows Rabi oscillations for the upper Raman transition, after
velocity selection with the bottom Raman beam. That transition is the last
step of the state preparation sequence, that allows to enter the Bragg
interferometer in the state $F=2$, $m_{F}=0$ with a narrow velocity spread
along the interferometer axis, and the right velocity for the first Bragg
transition. In the process of selecting internal state and velocity, 98$\%$ of
the atoms are removed and we obtain an effective flux of $1.2\times 10^{7}$
atoms$/$s in the interferometer.

###  4.2 Output state measurement

Figure 9:  Transition probability versus pulse duration for Raman 2
transitions, performed when the atoms are going up (blue dots) and on their
way down (orange diamonds).

The measurement of the output state of the interferometer relies on the
combination of external state labelling by a Raman transitions on the way
down, and state selective fluorescence detection (see figure 4). We evaluated
the contrast of the Raman 2 transition on the way up and on the way down by
recording Rabi oscillations visible on figure 9. The selection pulse on the
way up occurs at $T=80$ ms after launch, on the way down it occurs at $t=853$
ms. At the latter time we estimate the atom cloud diameter (FWHM of the atomic
distribution) to be $3.3$ cm. We obtain a transfer efficiency of 55$\%$,
limited by light intensity inhomogeneity across the laser beam due to the
limited size of the gaussian profile of the beam (3.5 cm FWHM). The use of a
top-hat laser profile is expected to increase this contrast to more than
70$\%$ [15].

Figure 10: Standard deviation of the measured transition probability, as a
function of the number of atoms detected. The green line corresponds to $1/N$
the noise limit associated to the electronic noise from the photodiodes shown
in figure 6. The red dashed line corresponds to the quantum projection noise
proportionnal to $1/\sqrt{N}$ .

Finally, we investigated the detection noise by measuring the standard
deviation of the measured transition probability, as a function of the number
of atoms ($N$) detected, for a sample in a state of equal repartition between
the two hyperfine levels (see figure 10). This corresponds to the $1/2$
transition probability state often used as the output state of an
interferometer. For low atom numbers, the noise follows a $1/N$ behaviour due
to the electronic noise of the detection photodiodes. For high atom numbers,
the noise saturates at a floor independent of the atom number, limited by
fluctuations in the balance of the detection efficiency between the two
hyperfine states. This noise floor level can be decreased by improving the
power stability of the detection lasers.

##  5 Conclusion

We presented a characterization of the cold-atom sources developed for the
MIGA interferometer. They allow the production of $10^{7}$ atoms per second
selected in the desired velocity class for the Bragg interferometer, with a
tunable effective temperature in the $100$ nK range along the interferometer
axis. The interferometer’s output state can be measured by a detection system
comprising velocity selective Raman spectroscopy coupled to hyperfine state
sensitive fluorescence detection. We reach a signal to noise ratio of 1000
with atoms in the desired state for interferometry, and a shot-to-shot
standard deviation on the transition probability of $5\times 10^{-4}$, which
fulfills the requirements for the future gravitation antenna.

To date, five atom source units have been produced, three of which will be
part of the 150 m long gradiometer in Rustrel and two others will be part of a
$1.5$ m long gradiometer in LP2N in Bordeaux. More atom sources can be
produced for potential extensions of the MIGA instrument (second horizontal
axis, or extension of the baseline with more atom interferometers). Further
characterisation and optimisation will be carried out during integration of
the subsystem to the instrument. A possible design evolution may include the
use of top-hat beam shapers to increase the contrast of Raman transitions
[15], adaptation of the detection system to large momentum transfer in the
interferometer, or the use of interleaved interferometry [16].

## References

  * [1] Sesana, A.  Prospects for multiband gravitational-wave astronomy after GW150914.  _Phys. Rev. Lett._ 116, 231102, DOI: 10.1103/physrevlett.116.231102 (2016).
  * [2] Auger, G. & Plagnol, E.  _An Overview of Gravitational Waves: Theory, Sources and Detection_ (World Scientific Publishing, 2017).
  * [3] Aasi, J. _et al._ Advanced LIGO.  _Class. Quantum Grav._ 32, 074001, DOI: 10.1088/0264-9381/32/7/074001 (2015).
  * [4] Acernese, F. _et al._ Advanced virgo: a second-generation interferometric gravitational wave detector.  _Class. Quantum Grav._ 32, 024001, DOI: 10.1088/0264-9381/32/2/024001 (2014).
  * [5] Akutsu, T. et al.  Kagra: 2.5 generation interferometric gravitational wave detector.  _Nature Astronomy_ 3, 35–40 (2019).
  * [6] Einstein Telescope Design Study, https://www.et-gw.eu/.
  * [7] Shimoda, T. _et al._ Torsion-Bar Antenna: A ground-based mid-frequency and low-frequency gravitational wave detector.  _International Journal of Modern Physics D_ 29, 1940003, DOI: 10.1142/S0218271819400030 (2020).  _eprint: https://doi.org/10.1142/S0218271819400030.
  * [8] Shuichi Sato et al.  The status of decigo.  _Journal of Physics: Conference Series_ 840, 012010, DOI: 10.1088/1742-6596/840/1/012010 (2017).
  * [9] Lacour, S. _et al._ SAGE: using CubeSats for gravitational wave detection.  In _Proceedings of SPIE_ , vol. 10699 of _Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series_ , 106992R, DOI: 10.1117/12.2313639 (2018).  1806.08106.
  * [10] Dimopoulos, S., Graham, P., Hogan, J., Kasevich, M. & Rajendran, S.  Atomic gravitational wave interferometric sensor.  _Phys. Rev. D_ 78, 122002 (2008).
  * [11] Chaibi, W. _et al._ Low frequency gravitational-wave detection with ground-based atom interferometer arrays.  _Phys. Rev. D_ 93, 021101, DOI: 10.1103/PhysRevD.93.021101 (2016).
  * [12] Canuel, B. _et al._ Exploring gravity with the MIGA large scale atom interferometer.  _Sci. Rep._ 8, 14064, DOI: 10.1038/s41598-018-32165-z (2018).
  * [13] Sabulsky, D. O. _et al._ A fibered laser system for the miga large scale atom interferometer.  _Sci. Rep._ 10, 3268 (2020).  1911.12209.
  * [14] (2019).  A. Bertoldi et al, A real-time control system for experiments in amo physics, Submitted to RSI.
  * [15] Mielec, N. _et al._ Atom interferometry with top-hat laser beams.  _Appl. Phys. Lett._ 113, 161108, DOI: 10.1063/1.5051663 (2018).
  * [16] Savoie, D. _et al._ Interleaved atom interferometry for high-sensitivity inertial measurements.  _Sci. Adv._ 4, eaau7948, DOI: 10.1126/sciadv.aau7948 (2018).

## Acknowledgements

This work was realized with the financial support of the French State through
the “Agence Nationale de la Recherche” (ANR) within the framework of the
“Investissement d’Avenir” programs Equipex MIGA (ANR-11- EQPX-0028) and IdEx
Bordeaux - LAPHIA (ANR-10-IDEX-03-02). This work was also supported by the
region d’Aquitaine (project IASIG-3D and USOFF). We also acknowledge support
from the CPER LSBB2020 project; funded by the “region PACA”, the “departement
du Vaucluse”, and the “FEDER PA0000321 programmation 2014-2020”. We
acknowledge the financial support from Ville de Paris (project HSENS-MWGRAV),
DIM SIRTEQ, and Agence Nationale pour la Recherche (project PIMAI,
ANR-18-CE47-0002-01 and project EOSBCMR, ANR-18-CE91-0003-01). X.Z. thanks the
China Scholarships Council (No 201806010364) program for financial support.
J.J. thanks “Association Nationale de la Recherche et de la Technologie” for
financial support (No 2018/1565).

We acknowledge the contribution of Louis Amand to the design and assembly and
of Bess Fang to preliminary tests of the system. We thank José Pinto for his
support with the electronic systems.

## Author contributions statement

The project was designed by R.G. B.C. A.L. A.B and P.B. .The design of the
atom heads was done by R.G, D.H, B.V., B.C. and A.L.. The systems were built
and tested by R.G, Q.B, B.V., D.H., L.V.,M.L.,M.P., L.S., J.J., A.B., D.S.,
X.Z. and P.L. .The data were acquired and analysed by Q.B. and P.L. .The
manuscript was written by Q.B and R.G. . All the authors have read,
contributed to, and approved the final manuscript.


its consequences for the blow-up process described at §3.3, is an important
reason why in this article we restrict our attention to length two. The
analogous statement in higher length $\ell$ is that there is a filtration

$0=\operatorname{End}^{\ell}(F)\subsetneq\dots\subsetneq\operatorname{End}^{1}(F)\subsetneq\operatorname{End}(F)$

where

$\operatorname{End}^{k}(F)\mathrel{\mathop{\mathchar
58\relax}}=\\{\varphi\in\operatorname{End}(F)\mid\varphi(F^{i})\subseteq
F^{i-k}\quad\forall i=1,\dots\ell\\},$

and we have
$\operatorname{Aut}_{U}(F)\cong\operatorname{id}_{F}+\operatorname{End}^{1}(F)$.
For the graded sheaves it follows, using the standard property ([HL10] Prop
1.2.7) that there are no homomorphisms ‘forwards’ in the HN filtration, that
we have

$\operatorname{Aut}(\mathrm{gr}F)\cong\left(\prod_{i=1}^{\ell}\operatorname{Aut}(F_{i})\right)\ltimes\bigoplus_{i>j}\operatorname{Hom}(E_{i},E_{j}),$

and hence that

$\operatorname{Aut}_{U}(\mathrm{gr}F)\cong\operatorname{id}_{\mathrm{gr}F}+\bigoplus_{i>j}\operatorname{Hom}(E_{i},E_{j}).$

Similarly to the above, we can get an injective map

$\operatorname{End}^{1}(F)\hookrightarrow\bigoplus_{i>j}\operatorname{Hom}(E_{i},E_{j}).$

However, there is no reason to suppose that this map will be surjective,
meaning that the analogue of Condition
$\dim\operatorname{Stab}_{U}(x)=\dim\operatorname{Stab}_{U}(p(x))$ need not
hold.

### 7.3. Proof of the main theorem

Now that all of the necessary ideas are in place, we are ready to prove the
main result. The basic idea of the proof is this: beginning with a suitable
subset

$Y^{\prime}\subset\mkern
1.5mu\overline{\mkern-1.5muY_{\tau}\mkern-1.5mu}\mkern 1.5mu$

of the projective closure of $Y_{\tau}$, one performs the blow-up process
described above in §3.3, as in Theorem 3.1.8. This corresponds to refining the
instability stratification, and hence to refining the Harder-Narasimhan
stratification of the Quot scheme. This results in a space
$\pi\mathrel{\mathop{\mathchar 58\relax}}\widehat{Y}^{\prime}\rightarrow
Y^{\prime}$ satisfying the condition $\star_{d}$. An open subset of
$\widehat{Y}^{\prime}$ therefore has a projective categorical quotient by the
$P_{\beta}$ action, and these quotients will be the projective completions
mentioned in Theorem 7.1.3. To prove the theorem, we must show that certain
loci in this projective completion can be identified with the right moduli
spaces of sheaves. This will be done via the isomorphism over the complement
of the exceptional divisor given by restricting the blow-down map, as in
Corollary 3.3.4. The crucial point is that, as a result of the first part of
Corollary 7.2.6, we are in the good situation discussed in §3.3, and so the
locus we blow up will not at any stage contain the minimal weight space. This
allows us to explicitly determine a locus that remains stable and untouched
throughout the blow-up process, whose quotient is the quasi-projective moduli
space of the theorem. Before moving on to the proof of the theorem, let us
make a few remarks.

######  Remark 7.3.1.

  1. 1)

It may happen for a given value of $\tau$ and $d$ that $M_{\tau,d}$ is empty:
for example, if $X$ is a smooth projective curve and we choose $\tau$ to be
such that $\operatorname{Ext}^{1}(F_{2},F_{1})$ is forced to vanish for degree
reasons.

  2. 2)

As observed above, the situation is particularly good if the Harder-Narasimhan
type in question is coprime. Similarly to the reductive-semistable case, this
is a consequence of the fact that, for these HN types, semistability and
stability (in the sense of GIT) coincide for the action of $\mkern
1.5mu\overline{\mkern-1.5muL_{\beta}\mkern-1.5mu}\mkern 1.5mu$. And, again as
in the reductive case [Kir85] if this does not happen one can perform a
sequence of blow-ups according to the dimensions of reductive stabilisers, to
obtain a space on which this condition is satisfied, yielding partial
desingularisations of the projective completions in the theorem above, and
hence an alternative compactification of $M_{\tau,d}$.

###### Proof of Theorem 7.1.3.

First, we note that the unipotent radical is abelian, by its description at
Remark 6.2.6. Thus, we blow up according to the dimensions of stabilisers in
$U$. Now, let $d_{\min}$ and $d_{\max}$ be respectively the minimal and
maximal values of $\dim\operatorname{Stab}_{U}(y)$ across $y\in Y_{\tau}$.

By Corollary 7.2.6, since we consider only $q\in Y_{\tau}^{ss}$, corresponding
to a point in the Quot scheme $[q\mathrel{\mathop{\mathchar
58\relax}}V\otimes\mathscr{O}(-n)\rightarrow\mathcal{F}]$, this is equivalent
to blowing up according to the dimension of the group
$\operatorname{Hom}(\mathcal{F}_{2},\mathcal{F}_{1})$.

_Case: $d=d_{\min}$_

Consider the locus

$(Y^{s}_{\tau})^{d_{\min}}\coloneqq\\{y\in
Y^{s}_{\tau}\mid\dim\operatorname{Stab}_{U}(y)=d_{\min}\\}.$

Then by Proposition and Corollary 7.2.6, orbits under the $P_{\beta}$ action
on this space correspond to isomorphism classes of sheaves $\mathcal{F}$ of
Harder-Narasimhan type $\tau$, with
$\dim\operatorname{Hom}(\mathcal{F}_{2},\mathcal{F}_{1})=d_{\min}$, and
$F_{1}$, $F_{2}$ stable sheaves. Furthermore, the sheaves in

$P_{\beta}(Z_{\tau}^{s})^{d_{\min}}=U_{\beta}(Z^{s}_{\tau})^{d_{\min}}=U_{\beta}Z^{s}_{\tau}\cap(Y^{s}_{\tau})^{d_{\min}}$

are precisely those sheaves which are in addition Harder-Narasimhan split.
Hence, by Corollary 7.1.7, to construct the moduli space $M_{\tau,d}$ it
suffices to exhibit a geometric quotient of the locus
$(Y^{s}_{\tau})^{d_{\min}}\setminus U_{\beta}(Z^{ss}_{\tau})^{d_{\min}}$ by
its $P_{\beta}$ action.

Taking the projective closure $\mkern
1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu_{\tau}$ in the ambient
projective space, we first blow up $\mkern
1.5mu\overline{\mkern-1.5muY_{\tau}\mkern-1.5mu}\mkern 1.5mu$ along the
closure of the subset

$C_{d_{\max}}(Y_{\tau})=\\{y\in
Y_{\tau}\mid\dim\operatorname{Stab}_{\hat{U}}(y)=d_{max}\\}=U\\{y\in
Z_{\min}\mid\dim\operatorname{Stab}_{U}(y)=d_{\max}-1\\}$

This results in a space $\mkern
1.5mu\overline{\mkern-1.5muY_{\tau}\mkern-1.5mu}\mkern 1.5mu^{(1)}$ with an
action of $P_{\beta}$ lifting that on $\mkern
1.5mu\overline{\mkern-1.5muY_{\tau}\mkern-1.5mu}\mkern 1.5mu$. Since the
centre of the blow-up did not contain $Z_{\tau}$, the minimal weight space for
$\lambda_{\beta}$ acting on $\mkern
1.5mu\overline{\mkern-1.5muY_{\tau}\mkern-1.5mu}\mkern 1.5mu^{(1)}$, which we
denote by $Z_{\tau}^{(1)}$, is the proper transform of the minimal weight
space $Z_{\tau}$ of $\mkern
1.5mu\overline{\mkern-1.5muY_{\tau}\mkern-1.5mu}\mkern 1.5mu$, as in §3.3 case
$(1)$. Let us simplify notation by denoting by $Y_{\tau}^{(1)}$ the open
subset

$(\mkern 1.5mu\overline{\mkern-1.5muY_{\tau}\mkern-1.5mu}\mkern
1.5mu^{(1)})^{0}_{\min}\subset\mkern
1.5mu\overline{\mkern-1.5muY_{\tau}\mkern-1.5mu}\mkern 1.5mu^{(1)}$

consisting of the points flowing down to $Z_{\tau}^{(1)}$ under
$\lambda_{\tau}$ as $t\rightarrow 0$. Finally, we linearise the $P_{\beta}$
action on $\mkern 1.5mu\overline{\mkern-1.5muY_{\tau}\mkern-1.5mu}\mkern
1.5mu^{(1)}$ by replacing the linearisation $\mathscr{L}_{\tau}^{per}$ with
its perturbation by a small multiple of the exceptional divisor of the blow-
up. Because all of these blow-ups may be performed in the ambient space, we
may assume that at each stage of the blow-up process the exceptional divisor
is just the projectivised normal bundle to the centre of the blow-up, and the
corresponding weights are just certain small perturbations of the weights for
$Y$, in the way described in Remark 3.3.2.

This process strictly reduces the dimensions of the stabilisers in $U_{\beta}$
over the centre of the blow-up (see [BDHK20] and also [HJ] §2.3.7), thus
guaranteeing that the dimensions

$\dim\operatorname{Stab}_{U_{\beta}}(y),\text{ for }y\in Z_{\tau}^{(1)}$

are all strictly less than $d_{\max}$. We now blow up $\mkern
1.5mu\overline{\mkern-1.5muY_{\tau}\mkern-1.5mu}\mkern 1.5mu^{(1)}$ along the
closure of the locus of points where the $\hat{U}_{\beta}$-stabiliser is
maximal, and we repeat this process until it terminates with a space
$\pi\mathrel{\mathop{\mathchar 58\relax}}\widehat{Y}_{\tau}\rightarrow\mkern
1.5mu\overline{\mkern-1.5muY_{\tau}\mkern-1.5mu}\mkern 1.5mu$. This space
carries a $P_{\tau}$ action lifting that on $\mkern
1.5mu\overline{\mkern-1.5muY_{\tau}\mkern-1.5mu}\mkern 1.5mu$, and this action
has a well-adapted linearisation $\mathscr{\widehat{L}}^{per}_{\tau}$ obtained
as described, by perturbing at each stage of the blow-up process with a small
multiple of the exceptional divisor. Let us denote the minimal
$\lambda_{\tau}$ weight space in $\widehat{Y}_{\tau}$ by $\widehat{Z}_{\tau}$.
This has basin of attraction $(\widehat{Y}_{\tau})^{0}_{\min}$. Then we have
arrived at a situation where

$\dim\operatorname{Stab}_{U_{\beta}}(y)=d_{\min}\text{ }\forall\text{
}y\in(\widehat{Y}_{\tau})^{0}_{\min}.$

In other words, condition $\star_{d}$ is satisfied, so by Theorem 3.1.8 there
is a projective categorical quotient of the locus
$\widehat{Y}_{\tau}^{P_{\beta}-ss}$, given by composing a geometric quotient
by $\hat{U}_{\beta}$ with a categorical quotient by the residual action of the
reductive group $(\operatorname{Stab}\beta)/\lambda_{\beta}(\mathbb{G}_{m})$.
We define

$\mkern 1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{\tau,d_{\min}}\coloneqq\widehat{Y}_{\tau}\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}_{\mathscr{\widehat{L}}^{per}_{\tau}}P_{\beta}.$

In addition, the complement $U_{\beta}\widehat{Z}_{\min}$ also has a
categorical quotient by the $P_{\beta}$ action. Indeed, the geometric quotient
of $U_{\beta}\widehat{Z}_{\min}$ by $\hat{U}$ is of course
$\widehat{Z}_{\min}$, and the open subset
$\widehat{Z}_{\min}^{\operatorname{Stab}\beta-ss}$ has a categorical quotient
given by the usual GIT quotient with respect to the canonical linearisation,
as described in §2.3. Hence we observe that
$\widehat{Z}_{\min}\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}_{\mathscr{L}_{\beta}}\operatorname{Stab}\beta$
is a categorical quotient of $U\widehat{Z}_{\min}$. Moreover, the blow-down
map $\pi\mathrel{\mathop{\mathchar
58\relax}}\widehat{Y}_{\tau}\rightarrow\mkern
1.5mu\overline{\mkern-1.5muY_{\tau}\mkern-1.5mu}\mkern 1.5mu$ restricts to
give a blow down

$\widehat{Z}_{\min}\rightarrow Z_{\tau},$

again because the new minimal weight space is the proper transform of the old
one. Moreover, under this map, the image of the semistable locus for
$\operatorname{Stab}\beta\curvearrowright\widehat{Z}_{\min}$ with respect to
$\widehat{\mathscr{L}}_{\beta}$ is contained in $Z_{\tau}^{ss}$ \- that is,
the semistable locus for $\operatorname{Stab}\beta\curvearrowright Z_{\tau}$
with respect to $\mathscr{L}_{\beta}$. This follows by the same argument as in
[Kir85] because the linearisation $\widehat{\mathscr{L}}_{\beta}$ may be
chosen to be an arbitrarily small peturbation of the pullback
$\pi^{*}\mathscr{L}_{\beta}$ by the exceptional divisor.

Since the blow-up is equivariant, we thus get a map of their quotients

$\widehat{Z}_{\min}\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}_{\widehat{\mathscr{L}}_{\beta}}\operatorname{Stab}\beta\rightarrow
Z_{\tau}\mathbin{\mathchoice{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}{\mkern-5.0mu/\mkern-6.0mu/\mkern-4.0mu}}_{\mathscr{L}_{\beta}}\operatorname{Stab}\beta\cong
M_{P_{1}}^{ss}\times M_{P_{2}}^{ss},$

where the latter is a product of Simpson moduli spaces of semistable sheaves.
It remains to identify a suitable quasi-projective subscheme of $\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu_{\tau,d_{\min}}$ with
the desired moduli space of sheaves.

Let $E$ be the exceptional locus of the map $\pi\mathrel{\mathop{\mathchar
58\relax}}\widehat{Y}_{\tau}\rightarrow\mkern
1.5mu\overline{\mkern-1.5muY_{\tau}\mkern-1.5mu}\mkern 1.5mu$; then $\pi$
gives an identification

$\widehat{Y}_{\tau}\setminus E\cong\mkern
1.5mu\overline{\mkern-1.5muY_{\tau}\mkern-1.5mu}\mkern 1.5mu\setminus\pi(E).$

The locus $(Y^{ss}_{\tau})^{d_{\min}}$ is untouched by the blow-up, and hence
this isomorphism restricts to an identification of
$(Y^{ss}_{\tau})^{d_{\min}}$ with a subset of $\widehat{Y}_{\tau}\setminus E$.
And, since at every stage $\pi_{i}\mathrel{\mathop{\mathchar 58\relax}}\mkern
1.5mu\overline{\mkern-1.5muY_{\tau}\mkern-1.5mu}\mkern
1.5mu^{(i+1)}\rightarrow\mkern
1.5mu\overline{\mkern-1.5muY_{\tau}\mkern-1.5mu}\mkern 1.5mu^{(i)}$ of the
blow-up process the centre of the blow-up did not contain the minimal
$\lambda_{\tau}$ weight space of $\mkern
1.5mu\overline{\mkern-1.5muY_{\tau}\mkern-1.5mu}\mkern 1.5mu^{(i)}$, the blow
down map $\pi\mathrel{\mathop{\mathchar
58\relax}}\widehat{Y}_{\tau}\rightarrow\mkern
1.5mu\overline{\mkern-1.5muY_{\tau}\mkern-1.5mu}\mkern 1.5mu$ also restricts
to an isomorphism $\widehat{Z}_{\tau}\setminus E\cong(Z_{\tau})^{d_{\min}}$.
Futhermore, all of these identifications are $P_{\beta}$-equivariant, since
$\pi$ is $P_{\beta}$ equivariant.

Proceeding analogously to §7.1, let
$\widehat{Z}^{s}_{\tau}\subset\widehat{Z}_{\tau}$ denote the stable locus for
the action of $\operatorname{Stab}\beta$ on $\widehat{Z}_{\tau}$ with respect
to the canonical linearisation $\widehat{\mathscr{L}}_{\beta}$ for
$\widehat{Y}_{\tau}$. Then let $\widehat{Y}^{s}_{\tau}$ be the inverse image
$\widehat{p}_{\beta}^{-1}(\widehat{Z}^{ss}_{\tau})$ under the flow

$\widehat{p}_{\beta}\mathrel{\mathop{\mathchar
58\relax}}(\widehat{Y}_{\tau})^{0}_{\min}\rightarrow\widehat{Z}_{\tau}$

given by taking $t\rightarrow 0$ in the $\lambda_{\tau}$ action. By the
Hilbert-Mumford criterion given in Theorem 3.2.3 applied to the blow-up space
$\widehat{Y}_{\tau}$, one has a quasi-projective geometric
$P_{\beta}$-quotient

$\widehat{Y}_{\tau}^{ts}\rightarrow\widehat{Y}_{\tau}^{ts}/P_{\beta}\subseteq\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu_{\tau,d_{\min}}.$

Then, by the classical Hilbert-Mumford criterion for reductive groups, we may
identify $\widehat{Z}^{s}_{\tau}\setminus E$ with $(Z^{s}_{\tau})^{d_{\min}}$,
via $\pi$. It follows that we may identify $(Y^{s}_{\tau})^{d_{\min}}$ with
$\widehat{Y}^{s}_{\tau}\setminus E$, and hence we obtain via the blow-down map
a $P_{\beta}$-equivariant isomorphism

$\widehat{Y}^{ts}_{\tau}\setminus E\cong(Y_{\tau}^{ts})^{d_{\min}}.$

Taken together, this tells us we can restrict the $P_{\beta}$-quotient map

$\widehat{Y}_{\tau}^{P_{\beta}-ss}\rightarrow\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern 1.5mu_{\tau,d_{\min}}$

to get a quasi-projective geometric $P_{\beta}$-quotient:

$M_{\tau,d_{\min}}\coloneqq(Y^{ts}_{\tau})^{d_{\min}}/P_{\beta}.$

This proves the first part of the theorem in the case $d=d_{\min}$.

For the second part, observe that that the map

$\pi\circ\widehat{p}_{\tau}\mathrel{\mathop{\mathchar
58\relax}}(\widehat{Y)}^{0}_{\min}\rightarrow\widehat{Z}_{\tau}\rightarrow
Z_{\tau}$

is $P_{\beta}$-equivariant, and that we have
$\widehat{Y}_{\tau}^{ts}\subset(\widehat{Y}_{\tau})^{0}_{\min}$. Hence the
above map, may be restricted to $\widehat{Y}_{\tau}^{ts}$, and then induces a
map of quotients by the universal property of categorical quotients: $\mkern
1.5mu\overline{\mkern-1.5muM\mkern-1.5mu}\mkern
1.5mu_{\tau,d_{\min}}\rightarrow M_{P_{1}}^{ss}\times M_{P_{2}}^{ss},$ which
extends the map taking a sheaf to its associated graded.

For the final part of the theorem, we use the last part of Lemma 7.1.4.

_Case: $d>d_{min}$_

Supposing inductively that the theorem holds for some $d-1$, we return to our
original $\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern
1.5mu_{\tau}$. By upper semi-continuity of stabiliser dimension, the subset

$\mkern 1.5mu\overline{\mkern-1.5muY^{d}_{\tau}\mkern-1.5mu}\mkern
1.5mu\coloneqq\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern
1.5mu_{\tau}\setminus\\{y\in\mkern
1.5mu\overline{\mkern-1.5muY_{\tau}\mkern-1.5mu}\mkern
1.5mu\mid\dim\operatorname{Stab}_{U}(y)<d\\}$

is closed in $\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern
1.5mu_{\tau}$. Furthermore, condition
$\dim\operatorname{Stab}_{U}(x)=\dim\operatorname{Stab}_{U}(p(x))$ ensures
that, whenever the locus

$(Y_{\tau}^{s})^{d})=Y_{\tau}^{d}\cap Y_{\tau}^{s}$

is non-empty, the locus $(Z_{\tau}^{s})^{d}=Z_{\tau}^{d}\cap Z_{\tau}^{s}$
will also be non-empty. We then simply repeat the process described above,
replacing $\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern
1.5mu_{\tau}$ by $\mkern
1.5mu\overline{\mkern-1.5muY^{d}_{\tau}\mkern-1.5mu}\mkern 1.5mu$ and
$d_{\min}$ by $d$.

∎

In particular, as a special case we recover a result obtained in [BPMG13] by
different means. It is interesting to note that the same discrete invariant
appears there as when one approaches the problem using non-reductive GIT.

######  Corollary 7.3.2.

[BPMG13] Let $X$ be a nonsingular projective curve, and $\tau=(P_{1},P_{2})$ a
pure HN type corresponding to coherent sheaves of rank 2. Then the space
$M_{\tau,d}$ above is a quasi-projective moduli space of indecomposable vector
bundles $F$ of Harder-Narasimhan type $\tau$ with
$\dim\operatorname{End}(\mathcal{F})=d+1$.

## References

  * [Ati56] M.F. Atiyah.  On the krull-schmidt theorem with application to sheaves.  Bulletin de la Société Mathématique de France, 84:307–317, 1956\.
  * [BDHK] G. Bérczi, B. Doran, T. Hawes, and F. Kirwan.  Geometric invariant theory for graded unipotent groups and applications.  Journal of Topology, 11(3):826–855.
  * [BDHK20] G. Bérczi, B. Doran, T. Hawes, and F. Kirwan.  Projective completions of graded unipotent quotients.  ArXiv e-prints, July 2020.
  * [BDK] G. Bérczi, B. Doran, and F. Kirwan.  Graded linearisations.  arXiv preprint 1703.05226.
  * [BHJK] G. Bérczi, V. Hoskins, J. Jackson, and F. Kirwan.  Moduli of unstable sheaves.  In preparation.
  * [BHK18] G. Bérczi, V. Hoskins, and F. Kirwan.  Stratifying quotient stacks and moduli stacks.  Abel Symposia: Geometry of Moduli (14), 2018.
  * [BHKD18] G. Bérczi, T. Hawes, F. Kirwan, and B. Doran.  Constructing quotients of algebraic varieties by linear algebraic group actions.  In Handbook of group actions. Vol. IV, volume 41 of Adv. Lect. Math. (ALM), pages 341–446. Int. Press, Somerville, MA, 2018.
  * [BJK18] G. Bérczi, J. Jackson, and F. Kirwan.  Variation of non-reductive geometric invariant theory.  In Surveys in differential geometry 2017. Celebrating the 50th anniversary of the Journal of Differential Geometry, volume 22 of Surv. Differ. Geom., pages 49–69. Int. Press, Somerville, MA, 2018.
  * [BK17] G. Bérczi and F. Kirwan.  Graded unipotent groups and grosshans theory.  Forum of Mathematics, Sigma, 5, 2017.
  * [BPMG13] L. Brambila-Paz and O. Mata-Gutiérrez.  Un teorema de Torelli para haces vectoriales inestables.  Aportaciones Matemáticas - Memorias de la Sociedad Matemática Mexicana, 46, pp. 03-16, 2013.
  * [Bri] M. Brion.  Introduction to actions of algebraic groups.  Online lecture notes: https://www-fourier.ujf-grenoble.fr/ mbrion/notes_luminy.pdf.
  * [DH98] I. V. Dolgachev and Y. Hu.  Variation of geometric invariant theory quotients.  Publications Mathématiques de L’IHÉS, 87(1):5–51, 1998\.
  * [DK07] B. Doran and F. Kirwan.  Towards non-reductive geometric invariant theory.  Pure and Applied Mathematics Quarterly, 3, 04 2007.
  * [Dol03] I. Dolgachev.  Lectures on Invariant Theory.  London Mathematical Society Lecture Note Series. Cambridge University Press, 2003.
  * [FGI+05] B. Fantechi, L. Göttsche, L. Illusie, S. Kleiman, N. Nitsure, and A. Vistoli.  Fundamental Algebraic Geometry: Grothendieck’s FGA Explained.  Mathematical surveys and monographs. American Mathematical Society, 2005\.
  * [Hes78] W.H. Hesselink.  Uniform instability in reductive groups.  Journal für die reine und angewandte Mathematik, 0303_0304:74–96, 1978.
  * [HJ] V. Hoskins and J. Jackson.  Quotients by parabolic groups and moduli spaces of unstable objects.  arxiv eprint.
  * [HK12] V. Hoskins and F. Kirwan.  Quotients of unstable subvarieties and moduli spaces of sheaves of fixed Harder–Narasimhan type.  Proceedings of the London Mathematical Society, 105(4):852–890, 2012.
  * [HL10] D. Huybrechts and M. Lehn.  The geometry of moduli spaces of sheaves.  Cambridge University Press, 2010.
  * [HN75] G. Harder and M. S. Narasimhan.  On the cohomology groups of moduli spaces of vector bundles on curves.  Math. Ann., 212:215–248, 1975.
  * [Hos11] V. Hoskins.  Moduli spaces of complexes of sheaves.  PhD thesis, Oxford University, UK, 2011.
  * [Hos18] V. Hoskins.  Stratifications for moduli of sheaves and moduli of quiver representations.  Algebraic Geometry, 5 (6) (2018) 650–685, 2018.
  * [Kem78] G.R. Kempf.  Instability in invariant theory.  Annals of Mathematics, 108(2):299–316, 1978.
  * [Kir84] F. Kirwan.  Cohomology of Quotients in Symplectic and Algebraic Geometry.  Mathematical Notes - Princeton University Press. Princeton University Press, 1984.
  * [Kir85] F. Kirwan.  Partial desingularisations of quotients of nonsingular varieties and their Betti numbers.  Annals of Mathematics, 122(1):41–85, 1985.
  * [KL89] F. Kirwan and R. Lee.  The cohomology of moduli spaces of k3 surfaces of degree 2 (i).  Topology, 28(4):495 – 516, 1989.
  * [MFK94] D. Mumford, J. Fogarty, and F.. Kirwan.  Geometric Invariant Theory.  Springer-Verlag, Berlin, 1994.
  * [Mum66] D. Mumford.  Lectures on curves on an algebraic surface.  With a section by G. M. Bergman. Annals of Mathematics Studies, No. 59\. Princeton University Press, Princeton, N.J., 1966.
  * [Nag60] M. Nagata.  On the fourteenth problem of Hilbert.  In Proc. Internat. Congress Math. 1958, pages 459–462. Cambridge Univ. Press, New York, 1960.
  * [New78] P. E. Newstead.  Introduction to Moduli Problems and Orbit Spaces.  T.I.F.R. Lecture Notes. Springer-Verlag, 1978.
  * [Nit11] N. Nitsure.  Schematic Harder–Narasimhan filtration.  Internat. J. Math, 22(10):1365–1373, 2011.
  * [NM84] L. Ness and D. Mumford.  A stratification of the null cone via the moment map.  American Journal of Mathematics, 106(6):1281–1329, 1984.
  * [Rob90] P. Roberts.  An infinitely generated symbolic blow-up in a power series ring and a new counterexample to Hilbert’s fourteenth problem.  Journal of Algebra, 132(2):461 – 473, 1990.
  * [Rud97] A. Rudakov.  Stability for an abelian category.  Journal of Algebra, 197(1):231 – 245, 1997.
  * [Sha77] S. Shatz.  The decomposition and specialization of algebraic families of vector bundles.  Compositio Math., 35:163–187, 1977.
  * [Sim94] C. T. Simpson.  Moduli of representations of the fundamental group of a smooth projective variety. ii.  Publications mathématiques de l’IHÉS, 80(1):5–79, 1994\.
  * [Tha96] M. Thaddeus.  Geometric invariant theory and flips.  J. Amer. Math. Soc., 9(3):691–723, 1996.