# Narrow-line absorption at 689 nm in an ultracold strontium gas

Fachao Hu  These authors contributed equally to this work.  Canzhu Tan  These
authors contributed equally to this work.  Hefei National Laboratory for
Physical Sciences at the Microscale and Shanghai Branch, University of Science
and Technology of China, Shanghai 201315, China  CAS Center for Excellence and
Synergetic Innovation Center in Quantum Information and Quantum Physics,
University of Science and Technology of China, Shanghai 201315, China  Yuhai
Jiang<EMAIL_ADDRESS>Shanghai Advanced Research Institute, Chinese
Academy of Sciences, Shanghai 201210, China  CAS Center for Excellence and
Synergetic Innovation Center in Quantum Information and Quantum Physics,
University of Science and Technology of China, Shanghai 201315, China
Matthias Weidemüller<EMAIL_ADDRESS>Hefei National Laboratory
for Physical Sciences at the Microscale and Shanghai Branch, University of
Science and Technology of China, Shanghai 201315, China  CAS Center for
Excellence and Synergetic Innovation Center in Quantum Information and Quantum
Physics, University of Science and Technology of China, Shanghai 201315, China
Physikalisches Institut, Universität Heidelberg, Im Neuenheimer Feld 226,
69120 Heidelberg, Germany  Bing Zhu<EMAIL_ADDRESS>Physikalisches Institut, Universität Heidelberg, Im Neuenheimer Feld 226,
69120 Heidelberg, Germany  Hefei National Laboratory for Physical Sciences at
the Microscale and Shanghai Branch, University of Science and Technology of
China, Shanghai 201315, China  CAS Center for Excellence and Synergetic
Innovation Center in Quantum Information and Quantum Physics, University of
Science and Technology of China, Shanghai 201315, China

###### Abstract

We analyse the spectrum on the narrow-line transition
$5\mathrm{s^{2}}\,^{1}\textrm{S}_{0}-5\mathrm{s}5\mathrm{p}\,^{3}\textrm{P}_{1}$
at 689 nm in an ultracold gas of 88Sr via absorption imaging. In the low
saturation regime, the Doppler effect dominates in the observed spectrum
giving rise to a symmetric Voigt profile. The atomic temperature and atom
number can accurately be deduced from these low-saturation imaging data. At
high saturation, the absorption profile becomes asymmetric due to the photon-
recoil shift, which is of the same order as the natural line width. The line
shape can be described by an extension of the optical Bloch equations
including the photon recoil. A lensing effect of the atomic cloud induced by
the dispersion of the atoms is also observed at higher atomic densities in
both the low and strong saturation regimes.

##  I Introduction

The existence of metastable states and narrow-line transitions among the
alkaline-earth and alkaline-earth-like atoms brings new opportunities for
studying cold and ultracold atoms, for example the optical-lattice clocks Jun
Ye (2008); Ludlow  _et al._ (2015), the time variation of fundamental
constants Safronova  _et al._ (2018a, b); Kennedy  _et al._ (2020), atom
interferometersHu  _et al._ (2017, 2019); Rudolph  _et al._ (2020), nonlinear
quantum optics Ye  _et al._ (1998); Christensen  _et al._ (2015); Westergaard
_et al._ (2015), and strongly correlated Rydberg gases Dunning  _et al._
(2016); Madjarov  _et al._ (2020). While many of these applications rely on
the clock transition ${}^{1}\textrm{S}_{0}-^{3}\textrm{P}_{0}$ with a
linewidth on the level of mHz, the other narrow one
${}^{1}\textrm{S}_{0}-^{3}\textrm{P}_{1}$ with a $(1\sim 100)$-kHz linewidth
enable the cooling of the atoms down to the photon-recoil-limited regime
Curtis  _et al._ (2001); Loftus  _et al._ (2004); Guttridge  _et al._ (2016)
and a direct laser cooling to quantum degeneracy was demonstrated Stellmer
_et al._ (2013). Very recently, the kHz-transitions play an essential role in
the realizations of optical tweezer arrays of alkali-earth Norcia  _et al._
(2018); Cooper  _et al._ (2018) and alkali-earth-like Saskin  _et al._ (2019)
atoms.

Benefiting from the narrow line width of the
${}^{1}\textrm{S}_{0}$-${}^{3}\textrm{P}_{1}$ transitions in alkali-earth or
alkali-earth-like systems, fluorescence signals from these transitions can be
employed for studying collective atomic scattering and motional effects
Bromley  _et al._ (2016), measuring atomic transition properties Ferrari  _et
al._ (2003); Ido  _et al._ (2005); Schmitt  _et al._ (2013), and detecting
single atoms with high fidelities Saskin  _et al._ (2019). On the other hand,
absorption imaging using broad dipole-allowed transitions ($\sim$ 10 MHz) may
be by far the most widely-used method in diagnosing ultracold-atom systems,
providing accurate information on the spatial distribution of atoms, the atom
number, and the atomic temperature Ketterle  _et al._ (1999); Ketterle and
Zwierlein (2008). However, absorption with narrow-line transitions were rarely
studied in the ultracold regime, where the photon recoil energy is comparable
to the absorption linewidth including the Doppler effect. Oates _et al._
studied the atomic-recoil-induced asymmetries in a form of saturation
spectroscopy with a Ca optical-clock apparatus Oates  _et al._ (2005), and the
photon-recoil effect on the dispersion was observed in a Yb vapor cell in Ref.
Grimm and Mlynek (1988). Stellmer _et al._ have implemented the absorption
imaging on the 7.5-kHz transition at 689 nm to resolve the hyperfine structure
of the fermionic 87Sr at a magnetic field of about 0.5 G Stellmer  _et al._
(2011). They observed a Lorentzian lineshape with a full width at half maximum
(FWHM) of about 40 kHz, without discussing further details on the spectrum.

In this work, we study in detail the absorption spectrum on the narrow
transition
$5\mathrm{s^{2}}\,^{1}\textrm{S}_{0}-5\mathrm{s}5\mathrm{p}\,^{3}\textrm{P}_{1}$
at 689 nm with an ultracold 88Sr atomic cloud. We measure the spectrum in both
the weak and strong saturation regimes. At low saturations, the absorption
lineshape is close to a Gaussian shape essentially determined by the Doppler
effect in the temperature range studied here. Thus, this regime can be
exploited for thermometry of the atomic sample, which is confirmed by a
comparison to the temperature obtained by the standard time-of-flight (TOF)
method Ketterle  _et al._ (1999); Ketterle and Zwierlein (2008) using the
broadband transition
$5\mathrm{s^{2}}\,^{1}\textrm{S}_{0}-5\mathrm{s}5\mathrm{p}\,^{1}\textrm{P}_{1}$
. The narrow-line absorption imaging at low saturation also provides
information on the atom numbers and atomic densities with a comparable
accuracy to detection methods based on the broad (blue) line. In the strong
saturation regime, an asymmetric lineshape is observed. We have performed a
theoretical simulation based on the optical Bloch equations (OBEs) involving
the momentum transfers during the imaging process and confirmed that the
photon recoil has important influence on the line shape. We also observe a
density-dependent lensing effect in the absorption images at large detunings
of the imaging light.

The article is organized as follows: We show our experimental setup in Sec.
II. The low- and high-saturation absorption spectra are described in Secs.
III.1 and III.2, respectively. The theoretical simulation and comparison to
experiments in the high-saturation regime are discussed in Sec. III.3. The
observation of lensing effect is presented in Sec. IV. Sec. V concludes the
paper.

##  II Experimental setup

Figure 1: (a) Schematic of the top view of experimental setup. HWP: half wave-
plate; PBS: polarizing beam-splitter. g and B represent the gravity and
magnetic field, respectively. See text for more details. (b) Time sequence for
absorption imaging. See text for explanations of $\tau_{\textrm{TOF}}$ and
$\tau_{\textrm{exp}}$. (c) Absorption spectrum showing all three Zeeman
sublevels of ${}^{3}\textrm{P}_{1}$ state when the imaging light polarization
is tuned to about 45∘ angled to the residual magnetic field. Black points are
the measured peak OD, and the red curve is the fit to a multi-peak Gaussian
function. The obtained Zeeman splitting is 167.7(1.2) kHz, corresponding to a
magnetic field of 79.9(6) mG.

Fig. 1(a) shows the experimental setup. The 88Sr atoms are first loaded into a
two-stage magneto-optical trap (MOT) for the laser cooling and trapping Nosske
_et al._ (2017); Qiao  _et al._ (2019), operated on the broad
$5\mathrm{s^{2}}\,^{1}\textrm{S}_{0}-5\mathrm{s}5\mathrm{p}\,^{1}\textrm{P}_{1}$
and narrow
$5\mathrm{s^{2}}\,^{1}\textrm{S}_{0}-5\mathrm{s}5\mathrm{p}\,^{3}\textrm{P}_{1}$
transitions, respectively. We could create an atomic cloud of $10^{6}$ atoms
with a density of about $10^{10}$ cm-3 and a temperature around 1 $\mu$K. A
cigar-shaped optical dipole trap (ODT) formed by two horizontally propagating
beams at the wavelength of 532 nm, is simultaneously switched on at the
second-stage MOT. The two ODT beams both have a waist of about 60 $\mu$m and
cross at an angle of 18∘. Holding atoms in the ODT for 200 ms to reach
equilibrium after switching off the MOT, we obtain about $(0.5\cdots 5)\times
10^{5}$ atoms at a temperature of $0.7\cdots 6$ $\mu$K depending on the ODT
power. At a power of 0.6 W for each beam the trap depth of the ODT is about
$6\mu$K and the trap frequencies are $2\pi\times$(217, 34, 217) Hz along the
$x$, $y$, and $z$ directions [see Fig. 1(a)], respectively, resulting in cloud
radii of (27, 69, 27) $\mu$m and a peak density of $7\times 10^{11}$cm-3. The
temperatures along the $y$ and $z$ directions are mapped out by the standard
TOF method. The above-mentioned atom numbers, cloud sizes, and temperatures
are measured using absorption imaging with the broad
$5\mathrm{s^{2}}\,^{1}\textrm{S}_{0}-5\mathrm{s}5\mathrm{p}\,^{1}\textrm{P}_{1}$
transition. The lifetime of the atomic clouds in the ODT is about 2 s, limited
by the collisions with background gas.

The imaging light at 689 nm is delivered from a commercial tapered amplifier
seeded by an external-cavity diode laser (Toptica TApro), used also for the
narrow-line MOT cooling, which is frequency-stabilized to a passive ultra-low
expansion cavity with a short-term noise of 1 kHz level and a long-term drift
of 8 kHz/day Qiao  _et al._ (2019). As shown in Fig. 1(a), the imaging beam
propagates along the $z$ direction with a tunable linear polarization and has
a $1/e^{2}$ diameter of 4.2 mm. The imaging pulse length and intensity are
controlled by an accousto-optic modulator (not shown in the figure). The
imaging system consists of two achromatic lenses with focal lengths of +200 mm
and +300 mm, and maps the absorption to an EM-CCD camera from Andor with a
magnification factor of 1.5. We have an imaging resolution of about 12 $\mu$m.

The imaging sequence is described in Fig. 1(b). The absorption imaging on the
narrow-line transition is performed after rapidly switching off the ODT to
avoid the differential AC Stark shifts on the energy levels. A quantization
magnetic field along the vertical direction is applied (rising time 2 ms)
before the imaging pulse to split the Zeeman sublevels of
${}^{3}\textrm{P}_{1}$ state, as seen in Fig. 1(c). After a given time-of-
flight (TOF) time $\tau_{\textrm{TOF}}$, the atoms are shined by the imaging
light with an exposure time $\tau_{\textrm{exp}}=200$ $\mu$s. By tuning the
$\tau_{\textrm{TOF}}$ we can tune the atomic density during the absorption,
which plays an important role in observing the dispersive lensing effect
discussed in Sec. III. As done in a standard absorption imaging sequence, two
additional images with and without the imaging light are taken after the first
pulse. The three images are then processed (see, e.g., Lewandowski  _et al._
(2003)) to obtain the two-dimensional optical density (OD) distribution (see
the insets of Fig. 5).

By changing the linear imaging polarization angle in the $x-y$ plane, all
three Zeeman sublevels of the ${}^{3}\textrm{P}_{1}$ state are addressable. An
example is shown in Fig. 1(d). The peak OD is measured as a function of the
imaging detuning showing three peaks at a magnetic field of about 80 mG. The
relative line strengths are determined by the polarization and the different
coupling strengths of the three corresponding transitions (see Fig. 1(d)). We
have used this measurement to optimize the compensation of the background
magnetic field to be better than 5 mG in our setup and to calibrate the
quantization fields. For the absorption studies, we apply a field of 4 G to
split the sublevels and the imaging polarization is tuned parallel to the
quantization axis, so that the system is subjected only to the closed $\pi$
transition ($m_{j}=0\rightarrow m_{j^{\prime}}=0$), which can be treated as a
perfect two-level system.

##  III Measurements and analysis

Thanks to the high sensitivity and large dynamical range of our imaging camera
(Andor iXon 897) at 689 nm, we can study the absorption spectrum on the
narrow-line transition with a saturation parameter $s$ ranging from 0.01 to
more than 100. Meanwhile, the cloud temperature and the atomic density can be
controlled via the ODT depth and the TOF time $\tau_{\textrm{TOF}}$
independently.

Figure 2: (a) Low-saturation absorption spectra at temperatures of 1.3 $\mu K$
(black) and 5.7 $\mu K$ (red). The integrated absorption signal over the
atomic cloud region is plotted as a function of the imaging detuning. The
solid curves are fits to the Voigt profile. See text for more details. (b)
Spectroscopic thermometry. The fitted Doppler widths from (a) are used to
estimate the temperatures $T_{\textrm{Fit}}$, which are plotted against the
TOF measurement results $T_{\textrm{TOF}}$ in the lower panel of (b). A linear
fit to the data (black dashed line) gives a slope of 1.05(3). The red dashed
line represents $T_{\textrm{Fit}}=T_{\textrm{TOF}}$. In the upper panel we
also show the ratio (black open circles) between the atom numbers obtained
from the narrow- ($N_{\textrm{red}}$) and broad-linewidth
($N_{\textrm{blue}}$) imaging, which is a constant of 1.06(2) (gray solid
line).

###  III.1 Low-saturation absorption

In Fig. 2(a), we show two measured absorption spectra at temperatures of 1.3
$\mu$K (black points) and 5.7 $\mu$K (red points) with a saturation parameter
of $s=0.1$. The TOF time $\tau_{\textrm{TOF}}$ (see Fig. 1(b)) is chosen to be
3.1 ms to minimize the lensing effect (see Sec. IV) as well as to keep large
enough signal-to-noise ratios (SNRs) in the OD images. The plotted signals in
Fig. 1(b) are the OD integrals over the whole atomic cloud region divided by
the peak cross section $\sigma_{0}=3\lambda^{2}/2\pi$, which is the standard
way to calculate the atom number in the absorption imaging (see the following
paragraph for a correction).

Symmetric lineshapes are observed in both cases and the linewidth increases
with the increasing temperature. The spectra fit well to Voigt profiles with a
fixed Lorentzian width of $v_{L}=10.01$ kHz, resulted from the power
broadening $\Gamma\sqrt{1+s}/2\pi$ and the detection bandwidth
$0.9/\tau_{\textrm{exp}}=4.5$ kHz due to the finite length of the square-shape
imaging pulse (see Fig. 1(b)), where $\Gamma/2\pi=7.5$ kHz is the natural
linewidth. The FWHM Gaussian width $v_{G}$ obtained from the Voigt profile
fitting is used to deduce the temperature $T_{\textrm{Fit}}$ along the imaging
propagation direction, from the relation $v_{G}=\frac{2}{\lambda}\sqrt{2\ln
2k_{b}T_{\textrm{Fit}}/m}$. Here $k_{b}$ is the Boltzmann constant, $\lambda$
is the transition wavelength, and $m$ is the atomic mass. $T_{\textrm{Fit}}$
obtained in this way are compared to those measured by the TOF method in the
lower panel of Fig. 2(b). The linear fit between $T_{\textrm{Fit}}$ and
$T_{\textrm{TOF}}$ (black dashed line) results in a slope of 1.05(3), which
agrees excellently with the ideal case of $T_{\textrm{Fit}}=T_{\textrm{TOF}}$
(red dashed line). We also notice the empirical density broadening in the
saturation fluorescence spectroscopy reported in Ido  _et al._ (2005). The
linear slope is only modified slightly to 1.05(6) even if we take the
empirical density relation following Ref. Ido  _et al._ (2005).

Figure 3: High-saturation absorption. (a) The measured absorption lineshapes
at low (blue dots) and high (red dots) saturations. The data in the high-
saturation case ($s=35.8$) showing asymmetric profile is fitted to the
numerical solution of Eq. (5) (red curve) and magnified by 6 times to have a
better visualization. As a comparison, the low-saturation ($s=0.09$) data is
symmetric and fits well to the Voigt profile (blue curve). (b) - (d), the
population difference $\Delta\rho(p)$ obtained from the OBE solutions at there
different detunings [$0,\pm 5\Gamma$, as marked by the grey vertical lines in
(a)] after an exposure time of 100 $\mu$s and 200 $\mu$s, respectively. As a
reference, we also show the initial distribution at $t=0$, which is the
Maxwell-Boltzmann one determined by the cloud temperature. The black solid
vertical lines mark the resonant momentum positions, where the probe detuning
is compensated by the Doppler effect. The inset images show measurements of
the two-dimensional OD distributions at low (upper) and high (lower)
saturations for their respective detunings.

In addition to the temperature, the atom number and atomic density can also be
extracted from the narrow-linewidth absorption imaging in the low saturation
regime. The broad (blue) transition typically used in determining the atom
number and atomic density has a natural linewidth on the order of 10 MHz, much
broader than the Doppler width. The absorption cross-section in the broad-
transition imaging can hence be regarded as temperature-independent. However,
for the narrow transition with a natural linewidth smaller than the Doppler
width ($\Gamma/2\pi v_{G}<1$), the Doppler effect has to be considered when
calculating the atom number Foot (2004). This is done by convolving the
velocity-dependent Lorentzian absorption profile with the Maxwell-Boltzmann
velocity distribution in the atomic sample (see Appendix Appendix). The
convolution results in a relationship between the measured OD and the atom
number similar to the broadband absorption imaging case, modified by a
coefficient depending on the ratio between the Doppler-broadened width and the
natural linewidth,

$\displaystyle OD_{0}(x,y)$ $\displaystyle=n(x,y)\sigma_{0}\times
C(\Gamma,v_{G})\,,$ (1)

where $C(\Gamma,v_{G})=\sqrt{\pi}\alpha e^{\alpha^{2}}\textrm{Erfc}(\alpha)$
is the coefficient with $\alpha=\sqrt{\ln 2}\Gamma/2\pi v_{G}$, $OD_{0}(x,y)$
are the on-resonance OD spatial distribution, and $n(x,y)$ is the atomic
column density. Erfc(x) is the complementary error function. The derivation of
the coefficient is presented in the Appendix Appendix. With the on-resonance
OD and the temperature-dependent $v_{G}$ determined from the spectrum fitting,
the atom number and atomic density can be obtained with Eq. (1). The upper
panel of Fig. 2(b) shows the ratio of the atom number determined by absorption
imaging with the narrow
$5\mathrm{s^{2}}\,^{1}\textrm{S}_{0}-5\mathrm{s}5\mathrm{p}\,^{3}\textrm{P}_{1}$
($N_{\textrm{red}}$) and broad
$5\mathrm{s^{2}}\,^{1}\textrm{S}_{0}-5\mathrm{s}5\mathrm{p}\,^{1}\textrm{P}_{1}$
($N_{\textrm{blue}}$) transitions, which lies close to 1 (gray solid line).

###  III.2 Strong-saturation absorption

In this section we study the narrow-line absorption spectrum at strong
saturations with $s\gg 1$. The question arises, to which extent the photon
recoils impacts the absorption profile, as the the recoil shift is comparable
to the natural linewidth ($\sim 4.78$ kHz vs. $7.5$ kHz for the narrow
transition studied here). During the absorption process, absorption and
spontaneous emission events give rise to the change of momentum distribution,
and hence affecting the subsequent absorption. For narrow-line transitions,
few photon-recoil events are enough to drive the atom out of resonance with
the imaging laser. This is in stark contrast to broadband transitions, since
the natural linewidth is usually much larger than the Doppler shifts induced
by photon recoils in a cold atom sample. Nevertheless, the photon-recoil
effect was already considered when imaging light species like Li using broad
transitions Horikoshi  _et al._ (2017), where the recoil-induced detuning and
blurring place strong constraints on the proper imaging conditions.

In Fig. 3(a) we show two absorption spectra at saturation parameters of
$s=0.09$ and $s=35.8$, respectively. We observe a decrease of the integrated
OD signal at all detunings due to the saturation effect (note that the data at
higher saturation is magnified by a factor of 6 for a better view). More
importantly, the lineshape is asymmetric at the high saturation, namely the
integrated OD approaches zero more slowly on the negative-detuning side than
that on the positive one, and the absorption peak is shifted by a few kHz to
the positive detuning. At high saturation, differences can already be seen in
the OD images at the two detuning sides [see lower rows in the insets of Figs.
3(b-d)], namely a wider spatial extension for the position detuning than that
for the negative one. In this series of experiments, the influence of the
lensing effect on the OD measurement(see Sec. IV) is negligible due to the low
atomic densities involved here.

The observed asymmetry and peak shift can be interpreted qualitatively by
considering the absorption process including the influence of the photon
recoil. The photon recoil associated with each absorption-spontaneous emission
cycle redistributes the momentum of atoms, which depends strongly on the light
detuning Stenholm (1978). Consequently, an asymmetric lineshape and the shift
of the maximum of absorption emerges when more and more photons are scattered
due to the momentum redistribution in the atomic cloud. In order to resolve
such effects, the Doppler width has to be comparable to the power-broadened
line width. In the case of the strong saturation in Fig. 3 (a), the power-
broadened Lorentzian width $\Gamma\sqrt{1+s}\sim 45$ kHz is close to the
Doppler one of $\sim 40$ kHz. In the following subsection a quantitative
description is presented incorporating the photon-recoil effect in an OBE
formalism.

###  III.3 Spectrum lineshape simulation

Figure 4: Absorption peak position shift. The relative position of the
absorption peak at different saturation parameters $s$ are compared for
imaging times of 100 $\mu$s (red circles and curve) and 200 $\mu$s (black
circles and curve). The black and red curves are the calculated results
without any free parameters, while the black and red circles are the fitted
results from measurements by using the peak position and height as the two
free fitting parameters. See text for more discussions.

We take an OBE formalism including the method of so-called ’momentum families’
from Ref. Castin  _et al._ (1989), originally developed to understand laser
cooling on a narrow-line transition. The model considers a two-level atom
system with an initial Maxwell-Boltzmann thermal distribution, interacting
with a single near-resonant monochromatic homogeneous probe beam. The state of
an atom with momentum $\bm{p}$ is expressed in the form of
$\\{\ket{g,\bm{p}},\ket{e,\bm{p}}\\}$, where $\ket{g(e)}$ corresponds to the
atomic ground (excited) state. The system Hamiltonian driven under a laser
beam propagating along the $z$ axis is,

$\centering
H_{0}=\frac{\hat{\bm{p}}^{2}}{2m}+\hbar\omega_{0}\ket{e}\bra{e}-\hat{\bm{D}}\cdot\hat{\bm{E}}\@add@centering$
(2)

where $\omega_{0},\hat{\bm{D}},\hat{\bm{E}}$ are the transition frequency,
dipole moment operator, and laser electric field, respectively. In our case,
only the $\pi$-transition branch $m_{j}=0\rightarrow m_{j^{\prime}}=0$ is
considered, and only momentum along the light propagation axis $p=p_{z}$ is
preserved, with the other two components $p_{x},p_{y}$ traced over. The system
Hamiltonian under the rotating-wave approximation becomes,

$\centering
H_{S}=\frac{\hat{p}^{2}}{2m}-\hbar\delta\ket{e}\bra{e}+\frac{\hbar\Omega}{2}(e^{ikz}\ket{e}\bra{g}+\ket{g}\bra{e}e^{-ikz})\@add@centering$
(3)

where $\delta,\Omega$ are the bare detuning and Rabi frequency.

The evolution of states $\ket{g,p},\ket{e,p+\hbar k}$ with any momentum p
remains globally closed under $H_{S}$ when the spontaneous emission is not
considered, for which reason the states $\ket{g,p}$, $\ket{e,p+\hbar k}$ are
grouped as a family $\mathcal{F}(p)$. The system density matrix $\rho$
expanded in this basis is,

$\centering\begin{aligned} \rho_{gg}(p)&=\braket{g,p}{\rho}{g,p}\\\
\rho_{ee}(p)&=\braket{e,p+\hbar k}{\rho}{e,p+\hbar k}\\\
\rho_{ge}(p)&=\rho_{eg}^{*}(p)=\braket{g,p}{\rho}{e,p+\hbar k}\\\
\end{aligned}\,.\@add@centering$ (4)

The equations of evolution under $H_{S}$ together with the spontaneous
emission processes are,

$\displaystyle\dot{\rho}_{gg}(p)$ $\displaystyle=\Gamma\bar{\pi}_{e}(p-\hbar
k)-\frac{i\Omega}{2}(\rho_{eg}(p)-\rho_{ge}(p))\,,$ (5)
$\displaystyle\dot{\rho}_{ee}(p)$
$\displaystyle=-\Gamma\bar{\pi}_{e}(p)+\frac{i\Omega}{2}(\rho_{eg}(p)-\rho_{ge}(p))\,,$
$\displaystyle\dot{\rho}_{ge}(p)$ $\displaystyle=\dot{\rho}_{eg}^{*}(p)$
$\displaystyle=-(i(\bar{\delta}-\frac{kp}{m})+\frac{\Gamma}{2})\rho_{ge}(p)+\frac{i\Omega}{2}(\rho_{gg}(p)-\rho_{ee}(p))\,,$

where $\bar{\delta}=\delta-\hbar k^{2}/(2m)$ and the term $\bar{\pi}_{e}$
represents the impact of spontaneous decay on the system evolution, defined as

$\displaystyle\bar{\pi}_{e}(p)=$
$\displaystyle\int\limits_{-\infty}^{+\infty}dp_{x}\int\limits_{-\infty}^{+\infty}dp_{y}\int\limits_{-\hbar
k}^{+\hbar k}dp^{\prime}\mathcal{N}(p^{\prime})$ (6)
$\displaystyle\braket{e,p_{x},p_{y},p_{z}=p+p^{\prime}}{\rho}{e,p_{x},p_{y},p_{z}=p+p^{\prime}}\,.$

Here $\mathcal{N}(p^{\prime})=\frac{3}{4\hbar k}(1-p^{\prime
2}/\hbar^{2}k^{2})$ results from the classical dipole radiation pattern Castin
_et al._ (1989) of the $\pi$ transition. With all the atoms initially at the
ground state $\ket{g}$ with a Maxwell-Boltzmann distribution of temperature
$T$, we numerically integrate the equations (5) to get the system evolution.
The solution of the off-diagonal elements $\rho_{eg}(p)$ results in the
susceptibility $\chi(p)\propto n\rho_{eg}(p)$ with the atomic density $n$. The
absorption profile is then calculated by tracing the imaginary part of the
susceptibility over all momenta, i.e. $\sum_{p}\mathrm{Im}\chi(p)$, and then
integrating over the interaction duration.

For the solid curves in Figs. 3(a), we fit the experimental data to the
calculated profiles with the maximum integrated OD and the peak position as
the only free parameters. Both the lineshape asymmetry and the shift of the
absorption peak at high saturation can be reproduced very well by Eq. (5)
including the momentum transfer due to the photon-scattering events. While the
model predicts a significant shift of the absorption peak, its position is
still used as a free parameter in the fits to account for the possible
deviation between the measurements and the calculations, as discussed in more
details in Fig. 4.

One can gain further insight into the photon-recoil effects by considering the
quasi-steady solution of the off-diagonal element in Eq. (5),
$\mathrm{Im\rho_{eg}(p)}\propto\Delta\rho(p)=\rho_{gg}(p)-\rho_{ee}(p)$. We
show from Fig. 3(b) to 3(d) the calculated distribution of the population
difference $\Delta\rho(p)$ at two saturation parameters of $s\approx 0.09$
(blue curves) and $s=35.8$ (red curves) after 200-$\mu$s atom-light
interaction time (about $10/\Gamma$, the imaging pulse length in this
measurement), when the probe laser is detuned by $-5\Gamma,0,+5\Gamma$ from
left to right. At the low saturation ($s\approx 0.09$), $\Delta\rho(p)$ is
only slightly modified compared to the initial Maxwell-Boltzmann distribution
(black dot-dashed lines), remaining almost Gaussian even after long
interaction time, such that the convolution between the velocity-dependent
Lorentzian profile. The momentum distribution results in a lineshape nearly
the Voigt one, as the blue curve seen in Fig. 3(a). When highly saturated
($s=35.8$), however, the $\Delta\rho(p)$ distribution is strongly modified and
depleted near the resonant momentum (marked by vertical dashed lines) where
the Doppler shift compensates the bare imaging detuning. In Fig. 3(b) with a
detuning of $-5\Gamma$, the distribution maintains a Gaussian shape with the
center shifted by $\sim 1.2\hbar k$ after 200 $\mu$s. While at a detuning of
$+5\Gamma$ in Fig. 3(d), two peaks appear on the opposite sides of the
resonant momentum. Such a strong dependence on the detuning leads to the
observed asymmetric lineshape and the peak shift.

The effects of the photon recoil can also be revealed by studying the time
evolution of the momentum distribution. In Figs. 3(b-d) the $\Delta\rho(p)$ at
$s=35.8$ after 100-$\mu$s interaction (red dash-dot curves) are shown as a
comparison to the 200-$\mu$s case. Small but clear differences of
$\Delta\rho(p)$ are observed for all three detunings indicating that the
momentum distribution undergoes some time evolution, which may result in a
time-dependent absorption lineshape. This is actually demonstrated in Fig. 4
by comparing the saturation-dependent shift of the absorption peak position
for the 100- and 200-$\mu$s imaging durations. The peak position is shifted
towards the positive detuning when increasing the imaging intensity and such a
shift becomes larger in the case of a longer exposure, i.e. more photons are
scattered. The solid curves represent the calculated results without any free
parameters, while the solid dots are from fits with the peak position and
height as the free fitting parameters [see Fig. 3(a)]. Overall, the fitted
shifts agree well with the calculations without free parameters, while
deviations are seen for some points coming from fluctuations of experimental
conditions like laser power and atom number, as well as the low SNR for large
saturation parameters.

##  IV The lensing effect

Figure 5: Absorption spectrum with $s=17$ at two different atomic densities of
$8.9\times 10^{10}$ cm-3 (red circles) and $2.8\times 10^{11}$ cm-3 (blue
diamonds). The red curve is a fit to the numerical solution of Eq. (5). We
obtain negative peak ODs at some large positive detunings. In the right inset,
the lower OD image measured at a large positive detuning with the high atomic
density has a dark hole instead of a bright peak in the cloud center, caused
by the lensing effect. At the large negative detuning, the dark position
appears at the edges of the cloud (left inset). As a comparison, we also show
an example of the OD images for the low-density case with a normal Gaussian
distribution in the upper panel of the inset.

As shown in Fig. 5, we have also experimentally observed another phenomenon in
the absorption spectrum at high atomic densities, the so-called lensing effect
which is well known in standard absorption imaging. The absorption spectra at
two different atomic densities are compared at a saturation of $s=17$. In the
low-density case ($n\sim 8.9\times 10^{10}$ cm-3, red dots in the figure), we
find a similar asymmetry as that in Fig. 3(a) for the high saturation. With a
3-fold higher density ($n\sim 2.8\times 10^{11}$ cm-3, blue diamonds in the
figure), a negative peak OD is obtained from the two-dimension Gaussian fit at
some large positive detunings. Checking the OD images there (one example shown
as the right inset in Fig. 5), a dark hole instead of a bright peak is seen at
the central region of the atomic cloud for the large positive detuning, while
at the negative one a dark edge is observed. This phenomonon is related to the
microscopic lensing effect studied in e.g. Refs. Labeyrie  _et al._ (2003);
Wang and Saffman (2004); Labeyrie  _et al._ (2007); Roof  _et al._ (2015); Han
_et al._ (2015); Noaman  _et al._ (2018); Gilbert  _et al._ (2018), where a
spatial-dependent index of refraction leads to a focusing or defocusing effect
on the imaging beam depending on the detuning.

The observed lensing effect can be understood from the following equation for
describing the phase shift of the imaging field in the transverse plane
propagating through a cloud of two-level atoms Labeyrie  _et al._ (2007),

$d\phi(x,y)=-\sigma_{0}n(x,y,z)dz\frac{\delta/\Gamma}{1+4(\delta/\Gamma)^{2}+s(x,y)}\,.$
(7)

Here $dz$ is the thickness of the atomic cloud along the light propagation
direction, $\delta$ is the detuning, and $s(r)$ has a spatial dependence due
to the intensity distribution of a Gaussian probe beam. Spatial inhomogeneity
of the index of refraction can be induced by the spatial distribution of the
atomic density, or the probe intensity, or both. For negative (positive)
detuning, Eq. (7) leads to a focusing (defocusing) of the imaging beam.

The observed lensing effect here mainly stems from the density inhomogeneity
as indicated by the density-dependence (see Fig. 5) and the fact that the
imaging beam is much larger than the atomic cloud ($\sim 200$ times). The
lensing induced by such densitiy inhomogeneity was observed in both the weak-
Roof  _et al._ (2015) and strong-saturation Labeyrie  _et al._ (2003); Wang
and Saffman (2004); Labeyrie  _et al._ (2007) regimes. The lensing effect
shown in Fig. 5 with strong saturation is also observable in the weak-probe
case in our experiment. However, to quantitatively explain our observation,
detailed calculations on the light propagation are needed like in Refs. Han
_et al._ (2015); Gilbert  _et al._ (2018), even including the atom dipolar
interactions or multiple scattering events (e.g. Bromley  _et al._ (2016); Zhu
_et al._ (2016); Chabé  _et al._ (2014)), which is beyond the scope of this
paper.

##  V Conclusion

In conclusion, we have studied both experimentally and theoretically the
absorption spectrum of a narrow-line transition at 689 nm in an ultracold 88Sr
gas. The atomic cloud temperature down to 1 $\mu$K can be inferred from the
measured absorption lineshape at low probe saturations ($s\ll 1$) if the
Doppler width dominates over other line-broadening effects. Information on the
atom number can also be reliably extracted from the low-saturation absorption.
In the strongly saturated regime, we observed the photon-recoil-induced
asymmetry in the absorption spectrum, which can be described by two-level OBEs
involving the photon recoils. We also showed a lensing effect when probing a
high-density sample, which is due to the spatial-dependent dispersive response
of the atomic cloud to the imaging field. It is of strong interest in studying
further the weak-probe high-density regime because of the collective and
cooperative effects that are predicted theoretically Bienaimé  _et al._
(2013); Zhu  _et al._ (2016); Kupriyanov  _et al._ (2017); Bettles  _et al._
(2020). The narrow-line absorption can also be employed as sensitive probe for
other cold atom systems with similar narrow-line transitions, like, e.g., Yb.
The good resolution also makes the narrow-line absorption applicable to
detection of interactions in more complicated systems, e.g. the spatial
correlation Günter  _et al._ (2012) due to Rydberg blockade.

## Acknowledgements

We acknowledge C. Qiao, L. Couturier, and I. Nosske for their contributions on
setting up the experiment at the early stage of project. F.H acknowledges
Yaxiong Liu for helpful discussions on numerical algorithms. M.W.’s research
activities in China are supported by the 1000-Talent-Program. The work was
supported by the National Natural Science Foundation of China (Grant Nos.
11574290 and 11604324) and Shanghai Natural Science Foundation (Grant No.
18ZR1443800). Y.H.J. also acknowledges support under Grant No. 11827806.

## Appendix

The low-saturation ($s\ll 1$) OD spatial distribution is represented as,

$\centering
OD(x,y)=\int_{-\infty}^{+\infty}\sigma_{0}n(x,y)f(v)L(\delta,v,\Gamma)dv\@add@centering$
(8)

where $L(\delta,v,\Gamma)=\frac{\Gamma^{2}/4}{(\delta-kv)^{2}+\Gamma^{2}/4}$
is the Lorentzian profile with $\delta$ the bare laser detuning, $\Gamma$ the
natural linewidth, $k=2\pi/\lambda$ the laser wavenumber, and $v$ the atom
velocity, $f(v)=\frac{1}{u\sqrt{\pi}}e^{-v^{2}/u^{2}}$ is the Gaussian
velocity distribution with $u=\sqrt{2k_{B}T/m}$ the most probable speed. The
Doppler width $v_{G}$ is related to $u$, $v_{G}=ku\sqrt{\ln 2}/\pi$. Then Eq.
(8) reads

$\displaystyle OD(x,y)$
$\displaystyle=\sigma_{0}n(x,y)\int_{-\infty}^{+\infty}\frac{1}{u\sqrt{\pi}}e^{-(v/u)^{2}}\frac{\Gamma^{2}/4}{(\delta-
kv)^{2}+\Gamma^{2}/4}dv$ (9)
$\displaystyle=\sigma_{0}n(x,y)\frac{\alpha^{2}}{\sqrt{\pi}}\int_{-\infty}^{+\infty}\frac{e^{-(x^{\prime}+\delta/ku)^{2}}}{x^{\prime
2}+\alpha^{2}}dx^{\prime}$

The substitution $x^{\prime}=kv-\delta$ is used in the second step. Here
$\alpha=\frac{\sqrt{\ln 2}\Gamma}{2\pi v_{G}}$ represents the ratio between
the natural linewidth and the Doppler width. At the on-resonance condition
($\delta=0$) we have the Eq. (1). The coefficients
$C(\Gamma,v_{G})=\sqrt{\pi}\alpha e^{\alpha^{2}}\textrm{Erfc}(\alpha)$ for
correcting the on-resonance absorption cross section are plotted in Fig. for
the
$5\mathrm{s^{2}}\,^{1}\textrm{S}_{0}-5\mathrm{s}5\mathrm{p}\,^{1}\textrm{P}_{1}$
(black dashed line),
$5\mathrm{s^{2}}\,^{1}\textrm{S}_{0}-5\mathrm{s}5\mathrm{p}\,^{3}\textrm{P}_{1}$
(blue dashed line) transitions of 88Sr and the D2 transition of 87Rb (red
dotted line) as a comparison.

Figure 6: The coefficient for correcting the on-resonance absorption cross
section due to the Doppler effect. The plotted temperature range is $0.01-10$
$\mu$K. Three atomic transitions are compared: the broad
$5\mathrm{s^{2}}\,^{1}\textrm{S}_{0}-5\mathrm{s}5\mathrm{p}\,^{1}\textrm{P}_{1}$
(black dashed line) and narrow
$5\mathrm{s^{2}}\,^{1}\textrm{S}_{0}-5\mathrm{s}5\mathrm{p}\,^{3}\textrm{P}_{1}$
(blue dashed line) transitions in 88Sr, and the D2 line of 87Rb (red dotted
line). This coefficient is 1 for broad transitions ($\Gamma\gg 2\pi v_{G}$)
and strongly modified for narrow ones ($\Gamma\lesssim 2\pi v_{G}$) in the
ultracold range.

## References

  * Jun Ye (2008) H. K. Jun Ye, H. J. Kimble, Science (2008), 10.1126/science.1148259.
  * Ludlow  _et al._ (2015) A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. Schmidt, Reviews of Modern Physics 87, 637 (2015).
  * Safronova  _et al._ (2018a) M. Safronova, D. Budker, D. DeMille, D. F. J. Kimball, A. Derevianko, and C. W. Clark, Reviews of Modern Physics 90 (2018a), 10.1103/RevModPhys.90.025008.
  * Safronova  _et al._ (2018b) M. S. Safronova, S. G. Porsev, C. Sanner, and J. Ye, Physical Review Letters 120 (2018b), 10.1103/physrevlett.120.173001.
  * Kennedy  _et al._ (2020) C. J. Kennedy, E. Oelker, J. M. Robinson, T. Bothwell, D. Kedar, W. R. Milner, G. E. Marti, A. Derevianko, and J. Ye, Physical Review Letters 125 (2020), 10.1103/physrevlett.125.201302.
  * Hu  _et al._ (2017) L. Hu, N. Poli, L. Salvi, and G. M. Tino, Physical Review Letters 119 (2017), 10.1103/physrevlett.119.263601.
  * Hu  _et al._ (2019) L. Hu, E. Wang, L. Salvi, J. N. Tinsley, G. M. Tino, and N. Poli, Classical and Quantum Gravity 37, 014001 (2019).
  * Rudolph  _et al._ (2020) J. Rudolph, T. Wilkason, M. Nantel, H. Swan, C. M. Holland, Y. Jiang, B. E. Garber, S. P. Carman, and J. M. Hogan, Physical Review Letters 124 (2020), 10.1103/physrevlett.124.083604.
  * Ye  _et al._ (1998) J. Ye, L.-S. Ma, and J. L. Hall, Journal of the Optical Society of America B 15, 6 (1998).
  * Christensen  _et al._ (2015) B. T. R. Christensen, M. R. Henriksen, S. A. Schäffer, P. G. Westergaard, D. Tieri, J. Ye, M. J. Holland, and J. W. Thomsen, Physical Review A 92 (2015), 10.1103/physreva.92.053820.
  * Westergaard  _et al._ (2015) P. G. Westergaard, B. T. Christensen, D. Tieri, R. Matin, J. Cooper, M. Holland, J. Ye, and J. W. Thomsen, Physical Review Letters 114 (2015), 10.1103/physrevlett.114.093002.
  * Dunning  _et al._ (2016) F. B. Dunning, T. C. Killian, S. Yoshida, and J. Burgdörfer, Journal of Physics B: Atomic, Molecular and Optical Physics 49, 112003 (2016).
  * Madjarov  _et al._ (2020) I. S. Madjarov, J. P. Covey, A. L. Shaw, J. Choi, A. Kale, A. Cooper, H. Pichler, V. Schkolnik, J. R. Williams, and M. Endres, Nature Physics (2020), 10.1038/s41567-020-0903-z.
  * Curtis  _et al._ (2001) E. A. Curtis, C. W. Oates, and L. Hollberg, Phys. Rev. A 64, 031403 (2001).
  * Loftus  _et al._ (2004) T. H. Loftus, T. Ido, A. D. Ludlow, M. M. Boyd, and J. Ye, Physical Review Letters 93 (2004), 10.1103/physrevlett.93.073003.
  * Guttridge  _et al._ (2016) A. Guttridge, S. A. Hopkins, S. L. Kemp, D. Boddy, R. Freytag, M. P. A. Jones, M. R. Tarbutt, E. A. Hinds, and S. L. Cornish, Journal of Physics B: Atomic, Molecular and Optical Physics 49, 145006 (2016).
  * Stellmer  _et al._ (2013) S. Stellmer, B. Pasquiou, R. Grimm, and F. Schreck, Physical Review Letters 110 (2013), 10.1103/physrevlett.110.263003.
  * Norcia  _et al._ (2018) M. A. Norcia, A. W. Young, and A. M. Kaufman, Physical Review X 8, 041054 (2018).
  * Cooper  _et al._ (2018) A. Cooper, J. P. Covey, I. S. Madjarov, S. G. Porsev, M. S. Safronova, and M. Endres, Physical Review X 8 (2018), 10.1103/physrevx.8.041055.
  * Saskin  _et al._ (2019) S. Saskin, J. Wilson, B. Grinkemeyer, and J. Thompson, Physical Review Letters 122 (2019), 10.1103/physrevlett.122.143002.
  * Bromley  _et al._ (2016) S. L. Bromley, B. Zhu, M. Bishof, X. Zhang, T. Bothwell, J. Schachenmayer, T. L. Nicholson, R. Kaiser, S. F. Yelin, M. D. Lukin, A. M. Rey, and J. Ye, Nature Communications 7 (2016), 10.1038/ncomms11039.
  * Ferrari  _et al._ (2003) G. Ferrari, P. Cancio, R. Drullinger, G. Giusfredi, N. Poli, M. Prevedelli, C. Toninelli, and G. M. Tino, Physical Review Letters 91 (2003), 10.1103/physrevlett.91.243002.
  * Ido  _et al._ (2005) T. Ido, T. H. Loftus, M. M. Boyd, A. D. Ludlow, K. W. Holman, and J. Ye, Physical Review Letters 94 (2005), 10.1103/physrevlett.94.153001.
  * Schmitt  _et al._ (2013) M. Schmitt, E. A. L. Henn, J. Billy, H. Kadau, T. Maier, A. Griesmaier, and T. Pfau, Opt. Lett. 38, 637 (2013).
  * Ketterle  _et al._ (1999) W. Ketterle, D. S. Durfee, and D. Stamper-Kurn, arXiv preprint cond-mat/9904034 (1999).
  * Ketterle and Zwierlein (2008) W. Ketterle and M. W. Zwierlein, arXiv preprint arXiv:0801.2500 (2008).
  * Oates  _et al._ (2005) C. Oates, G. Wilpers, and L. Hollberg, Physical Review A 71 (2005), 10.1103/physreva.71.023404.
  * Grimm and Mlynek (1988) R. Grimm and J. Mlynek, Physical Review Letters 61, 2308 (1988).
  * Stellmer  _et al._ (2011) S. Stellmer, R. Grimm, and F. Schreck, Physical Review A 84 (2011), 10.1103/physreva.84.043611.
  * Nosske  _et al._ (2017) I. Nosske, L. Couturier, F. Hu, C. Tan, C. Qiao, J. Blume, Y. H. Jiang, P. Chen, and M. Weidemüller, Physical Review A 96 (2017), 10.1103/physreva.96.053415.
  * Qiao  _et al._ (2019) C. Qiao, C. Z. Tan, F. C. Hu, L. Couturier, I. Nosske, P. Chen, Y. H. Jiang, B. Zhu, and M. Weidemüller, Applied Physics B 125 (2019), 10.1007/s00340-019-7328-3.
  * Lewandowski  _et al._ (2003) H. J. Lewandowski, D. Harber, D. L. Whitaker, and E. A. Cornell, Journal of low temperature physics 132, 309 (2003).
  * Foot (2004) C. Foot,  _Atomic Physics_ (Oxford University Press, 2004).
  * Horikoshi  _et al._ (2017) M. Horikoshi, A. Ito, T. Ikemachi, Y. Aratake, M. Kuwata-Gonokami, and M. Koashi, Journal of the Physical Society of Japan 86, 104301 (2017).
  * Stenholm (1978) S. Stenholm, Applied Physics 15, 287 (1978).
  * Castin  _et al._ (1989) Y. Castin, H. Wallis, and J. Dalibard, Journal of the Optical Society of America B 6, 2046 (1989).
  * Labeyrie  _et al._ (2003) G. Labeyrie, T. Ackemann, B. Klappauf, M. Pesch, G. Lippi, and R. Kaiser, The European Physical Journal D-Atomic, Molecular, Optical and Plasma Physics 22, 473 (2003).
  * Wang and Saffman (2004) Y. Wang and M. Saffman, Phys. Rev. A 70, 013801 (2004).
  * Labeyrie  _et al._ (2007) G. Labeyrie, G. Gattobigio, T. Chanelière, G. Lippi, T. Ackemann, and R. Kaiser, The European Physical Journal D 41, 337 (2007).
  * Roof  _et al._ (2015) S. Roof, K. Kemp, M. Havey, I. M. Sokolov, and D. V. Kupriyanov, Opt. Lett. 40, 1137 (2015).
  * Han  _et al._ (2015) J. Han, T. Vogt, M. Manjappa, R. Guo, M. Kiffner, and W. Li, Physical Review A 92 (2015), 10.1103/physreva.92.063824.
  * Noaman  _et al._ (2018) M. Noaman, M. Langbecker, and P. Windpassinger, Opt. Lett. 43, 3925 (2018).
  * Gilbert  _et al._ (2018) J. R. Gilbert, C. P. Roberts, and J. L. Roberts, J. Opt. Soc. Am. B 35, 718 (2018).
  * Zhu  _et al._ (2016) B. Zhu, J. Cooper, J. Ye, and A. M. Rey, Physical Review A 94 (2016), 10.1103/physreva.94.023612.
  * Chabé  _et al._ (2014) J. Chabé, M.-T. Rouabah, L. Bellando, T. Bienaimé, N. Piovella, R. Bachelard, and R. Kaiser, Physical Review A 89 (2014), 10.1103/physreva.89.043833.
  * Bienaimé  _et al._ (2013) T. Bienaimé, R. Bachelard, N. Piovella, and R. Kaiser, Fortschritte der Physik 61, 377 (2013).
  * Kupriyanov  _et al._ (2017) D. Kupriyanov, I. Sokolov, and M. Havey, Physics Reports 671, 1 (2017).
  * Bettles  _et al._ (2020) R. J. Bettles, M. D. Lee, S. A. Gardiner, and J. Ruostekoski, Communications Physics 3, 1 (2020).
  * Günter  _et al._ (2012) G. Günter, M. R. de Saint-Vincent, H. Schempp, C. S. Hofmann, S. Whitlock, and M. Weidemüller, Physical Review Letters 108 (2012), 10.1103/physrevlett.108.013002.


Capacity Bounds for Optical WDM Channels with Peak Power Constraints

Viswanathan Ramachandran, Gabriele Liga, Astrid Barreiro, and Alex Alvarado

Eindhoven University of Technology, 5600 MB Eindhoven, the Netherlands.

<EMAIL_ADDRESS>
###### Abstract

We investigate optical WDM transmission from the standpoint of an information-
theoretic interference channel. Achievable rates that outperform treating
interference as noise are presented and validated using split-step Fourier
method simulations.

##  1 Introduction

In optical transmission, data from different users are typically multiplexed
into a single optical fiber using wavelength division multiplexing (WDM). The
Kerr effect causes a signal in one wavelength to interfere with signals in
other wavelengths, resulting in a stochastic nonlinear channel with memory
when combined with chromatic dispersion. This channel is described by the
nonlinear Schrödinger equation (NLSE). Despite the multiuser nature of optical
WDM channels, most papers in the literature resort to a single-user approach
for their information-theoretic analysis. This usually involves the study of
the capacity of a given user under different statistical assumptions for the
interfering users [1]. Multi-user information theoretic studies exist, but
only for highly simplified models [2, 3]. In [4], we investigated the set of
_simultaneously achievable rates_ for a realistic channel model for the first
time, based on multi-user information theory. However, the channel model in
[4] ignored the frequency separation between adjacent channels while modelling
the strength of cross-channel interactions. Furthermore, the results in [4]
were not validated via split-step Fourier (SSFM) simulations. The main
contributions of this paper are: (i) a correction of the model in [4] to allow
separation-dependent coupling between adjacent channels, and (ii) a validation
of the results in [4] based on SSFM simulations. While the obtained results
show improved lower bounds compared to treating interference as noise (TIN),
the rates from simulations are smaller than that predicted by the model.

##  2 Channel Model

${E_{1}}$${E_{2}}$${E_{K}}$E-OE-OE-OMFMFMFO-EO-EO-EDSPDSPDSPWDM MUXWDM DE-
MUX${D_{1}}$${D_{2}}$${D_{K}}$SSMFEDFA$X_{1}^{n}$$X_{2}^{n}$$X_{K}^{n}$$Y_{1}^{n}$$Y_{2}^{n}$$Y_{K}^{n}$Interference
Channel
(2)$p(Y_{1}^{n},\cdots,Y_{K}^{n}|X_{1}^{n},\cdots,X_{K}^{n})$$M_{1}$$M_{2}$$M_{K}$$\hat{M}_{1}$$\hat{M}_{2}$$\hat{M}_{K}$

Fig. 1: WDM interference channel model under consideration.

We study the system in Fig. 1, where the interference channel encompasses the
electro-optical conversion, WDM multiplexing, channel, demultiplexing,
receiver DSP, matched filtering and sampling. We assume single-polarization
transmission and study a single span of standard single mode fiber (SSMF). A
first-order regular perturbative discrete-time model [5] can be used to
express the output of user $k\in\\{1,2,\ldots,K\\}$:

$\displaystyle Y_{k}[i]$
$\displaystyle=X_{k}[i]+N_{k}[i]+j\gamma\sum_{t=-\infty}^{\infty}X_{k}[i-t]\sum_{l=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}S_{k,w}^{l,m,t}\sum_{\begin{subarray}{c}w=1\\\
w\neq k\end{subarray}}^{K}X_{w}[i-l]X_{w}[i-m]^{*},$ (1)

where $X_{k}[i]$ is the input of user$-k$ at instant $i\in\\{1,2,\ldots,n\\}$,
$Y_{k}[i]$ is the output at time $i$, while $\gamma$ is the fiber nonlinearity
parameter. The channel coefficients $S_{k,w}^{l,m,t}$ given in [5, eq. (7)]
are computed numerically. The term $N_{k}[i]$ models additive complex Gaussian
noise from the amplifier, assumed to be zero mean and variance
$2\sigma_{k}^{2}$. Length-$n$ codewords are assumed with peak power
constraints, i.e., $\max_{i\in\\{1,2,\ldots,n\\}}|x_{k}[i]|^{2}\leq
P_{k},\>\forall k\in\\{1,2,\ldots,K\\}.$

The largest contribution to the NLI for few-span systems of relatively short
lengths utilizing lumped amplification comes from the $S_{k,w}^{m,m,0}$ terms
in (1), as observed in [6, Figs. 4 and 5] and [5, eq. (8)]. We also truncate
the sums in (1) to $-M\leq t,l,m\leq M$ to get the following approximate
model:

$\displaystyle Y_{k}[i]$ $\displaystyle\approx
X_{k}[i]\Biggl{(}1+j\sum_{m=-M}^{M}\sum_{\begin{subarray}{c}w=1\\\ w\neq
k\end{subarray}}^{K}c_{k,w}^{m}|X_{w}[i-m]|^{2}\Biggr{)}+N_{k}[i],$ (2)

where $\gamma S_{k,w}^{m,m,0}\triangleq c_{k,w}^{m}$ for brevity. In [4], the
XPM terms corresponding to $c_{k,w}^{m}$ were assumed to be independent of
$w$, which we correct here, thus allowing for frequency separation-dependent
coupling between the channels.

Now let $\mathcal{C}_{K}$ denote the capacity region of the given system,
i.e., the maximal set of jointly achievable rates $(R_{1},R_{2},\ldots,R_{K})$
of all the users. The single-user capacity is
$C_{k}\triangleq\max\\{R_{k}|(R_{1},R_{2},\ldots,R_{K})\in\mathcal{C}_{K}\\}$.
We now have the following theorem, whose proof is similar to the one given in
[7, App. A,B].

######  Theorem 1.

The single-user capacity $C_{k}$ is bounded as

$\displaystyle\log_{2}\Biggl{(}\\!\\!1+\frac{P_{k}}{2\sigma_{k}^{2}e}\Biggl{(}\\!\\!1+\Biggl{(}\sum_{\begin{subarray}{c}w=1\\\
w\neq
k\end{subarray}}^{K}P_{w}\sum_{m=-M}^{M}c_{k,w}^{m}\\!\Biggr{)}^{2}\Biggr{)}\\!\\!\\!\Biggr{)}\leq
C_{k}\leq\log_{2}\Biggl{(}\\!\\!1+\frac{P_{k}}{2\sigma_{k}^{2}}\Biggl{(}\\!\\!1+\Biggl{(}\sum_{\begin{subarray}{c}w=1\\\
w\neq
k\end{subarray}}^{K}P_{w}\sum_{m=-M}^{M}c_{k,w}^{m}\\!\Biggr{)}^{2}\Biggr{)}\\!\\!\\!\Biggr{)},\>\forall\>k\in\\{1,2,\ldots,K\\}.$
(3)

$-30$$-25$$-20$$-15$$-10$$-5$$0$$5$$10$$0$$2$$4$$6$$8$$10$$12$$14$Inadmissible
Region?$P_{1}~{}\text{(dBm)}$$R_{1}$ (bits/sym)Upper bound (Theorem 1)Lower
bound (Theorem 1)TIN

(a) Capacity bounds in Theorem 1 vs. input power

Parameter | Value
---|---
Memory Length $M$ (eq. (2)) | 5
Number of WDM users | 3
Distance | $250\>\textrm{km}$
Nonlinearity parameter $\gamma$ | $1.2\>\textrm{W}^{-1}\textrm{km}^{-1}$
Signalling Rate | $32\>\textrm{Gbaud}$
Fiber attenuation $\alpha$ | $0.2\>\textrm{dB/km}$
Group velocity dispersion $\beta_{2}$ | $-21.7\>\textrm{ps\textsuperscript{2}/km}$
RRC pulse-shaping roll-off | 0.1
|
(b) Model Parameters

Fig. 2: Bounds versus peak input power, and model parameters

$-30$$-25$$-20$$-15$$-10$$-5$$0$$5$$10$$15$$20$$25$$30$$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$P_{1}~{}\text{(dBm)}$$R_{1}$
(bits/sym)TIN - simulationTIN - modelLower bound - simulationLower bound -
modelUpper bound (Theorem 1)

Fig. 3: Capacity lower bounds based on the model (2) versus SSFM.

##  3 Numerical Results

In the table shown as Fig. 2(b), we summarize the parameters used in our
numerical results. For $P_{1}=P_{2}=P_{3}$, the bounds on user$-1$ in Theorem
1 are plotted versus the input power in Fig. 2(a). Any rate below the lower
bound in (3) is achievable, which gives the shaded blue area, while the upper
bound in (3) gives an inadmissible region (shaded gray). The TIN bound is
achieved by treating $jX_{k}[i]\sum_{m=-M}^{M}\sum_{w\neq
k}c_{k,w}^{m}|X_{w}[i-m]|^{2}$ in (2) as Gaussian noise, which is shown as the
shaded purple area. The achievability of the shaded orange area is unknown.
Our lower bound outperforms TIN for higher powers, but coincides with TIN for
$P_{1}<-6$ dBm.

In Fig. 3, we compare the capacity bounds in Theorem 1 & TIN obtained from the
model (2) with SSFM simulations. The simulations are performed with Gaussian
inputs for all the users with variance $P_{1}$ and mismatched decoding. It is
observed that the power at which the TIN curves peak differs between the model
and simulations. Furthermore, it is observed that beyond a certain power ($0$
dBm in Fig. 3), the capacity lower bound from simulations is smaller than that
predicted by the model. This is chiefly due to the inaccuracy of the adopted
model at higher powers.

##  4 Conclusions

Obtaining tighter lower bounds as well as designing schemes that can achieve
the presented capacity bounds in practice, are among the avenues for further
research.

The work of V. Ramachandran, A. Barreiro and A. Alvarado has received funding
from the ERC under the EU’s Horizon 2020 research and innovation programme via
the Starting grant FUN-NOTCH (grant ID: 757791) . The work of G. Liga is
supported by the EuroTechPostdoc programme under the EU’s Horizon 2020
research and innovation programme (grant ID: 754462).

## References

  * [1] E. Agrell, M. Karlsson, “Influence of behavioral models on multiuser channel capacity,” J. Light. Technol. 33, (2015).
  * [2] M. Taghavi et al., “On the multiuser capacity of WDM in a nonlinear optical fiber,” IEEE Trans Info. Theory 52, (2006).
  * [3] H. Ghozlan, G. Kramer, “Models and information rates for multiuser optical…,” IEEE Trans Info. Theory 63, (2017).
  * [4] V. Ramachandran et al., “Capacity Region Bounds for K-User Optical WDM Channels…,” OFC 2022, Tu3B.4, (2022).
  * [5] R. Dar et al., “Properties of nonlinear noise in long, dispersion-uncompensated fiber links,” Opt. Express 21, (2013).
  * [6] R. Dar et al., “Pulse collision picture of inter-channel nonlinear interference…,” J. Light. Technol. 34, (2016).
  * [7] V. Ramachandran et al., “Capacity Bounds for the $K$-User Dispersive Nonlinear…,” arXiv: 2110.09405, (2022).


# Visualization for Petrov’s odd unitary group

A.A. Ambily  and  V.K. Aparna Pradeep
Department of Mathematics Department of Mathematics
Cochin University of Science and Technology Cochin University of Science and
Technology
Cochin University P.O. Cochin University P.O.
682022, Kerala 682022, Kerala
India India
Email<EMAIL_ADDRESS><EMAIL_ADDRESS>Email<EMAIL_ADDRESS>
###### Abstract.

In this article we define a set of matrices analogous to Vaserstein-type
matrices which was introduced in the paper ‘Serre’s problem on projective
modules over polynomial rings and algebraic $K$-theory’ by Suslin-Vaserstein
in 1976. We prove that these are elementary linear matrices. Also, under some
conditions, these matrices belong to Petrov’s odd unitary group which is a
generalization of all classical groups. We also prove that the group generated
by these matrices is a conjugate of the Petrov’s odd elementary hyperbolic
unitary group when the ring is commutative.

###### Key words and phrases:

Quadratic modules, Odd unitary groups, Elementary hyperbolic unitary groups,
ESD transvections, Vaserstein-type matrices

###### 2010 Mathematics Subject Classification:

19G99, 13C10, 11E70, 20H25

##  1\. Introduction

In [1], L.N. Vaserstein and A.A. Suslin studied the freeness of projective
modules over polynomial rings. They proved that a positive solution exists for
Serre’s problem for polynomial rings in five variables over an arbitrary field
and also for polynomial rings in four variables over a principal ideal domain.
For a given alternating matrix $\varphi$ of size $2n$, Vaserstein proved the
existence of two elementary matrices of size $2n-1$ which can be modified to
get symplectic matrices with respect to $\varphi$.

In [2], P. Chattopadhyay and R.A. Rao used Vaserstein’s construction for
defining relative elementary symplectic group with respect to a given
invertible alternating matrix. They proved dilation principle and local-global
principle for the relative elementary symplectic group and used these results
for the study of symplectic transvection groups. They have also established
the equality of the orbit space of a unimodular element under the action of
the linear group, the symplectic group with respect to the standard symplectic
form, and the symplectic group with respect to an invertible alternating
matrix.

In [3], V.A. Petrov introduced a new type of classical-like group known as odd
unitary groups which generalizes all the classical Chevalley groups, the
groups ${\rm U}_{2n+1}(R)$ of E. Abe [4], and the classical-like groups such
as Bak’s hyperbolic unitary groups [5] and G. Tang’s Hermitian groups [6]. The
importance of this group is that it include the odd dimensional orthogonal
group ${\rm O}_{2n+1}$ as well as the groups ${\rm U}_{2n+1}(R)$ of E. Abe
which was not covered in the theory of quadratic or Hermitian groups. Later A.
Bak and R. Preusser [7] studied a subclass of Petrov’s odd unitary groups
which contain all the classical Chevalley groups, and classified the E-normal
subgroups of the members of this subclass. The E-normal subgroups of odd
unitary groups are also studied by W. Yu, Y. Li and H. Liu in [8] and also by
R. Preusser in [9].

In a recent preprint [10], A.A. Ambily and R.A. Rao have defined the
Vaserstein-type matrices for DSER elementary orthogonal transformations and
proved that the group generated by these matrices is a congruent of the DSER
group. In this paper, we do an analogous result for Petrov’s odd elementary
hyperbolic unitary group. Since Petrov’s odd unitary group is a generalization
of all classical groups and it also coincides with the DSER elementary
orthogonal group in the commutative ring case with pseudo involution
$a\mapsto-a$, the result in this paper generalizes analogous results in
literature.

Here we define two matrices, $L(v)$ and $L(v)^{*}$, similar to the
construction by Vaserstein which reduces to Vaserstein-type matrices and the
matrices defined by Ambily-Rao in certain special cases. We prove that for
arbitrary $v\in R^{n+2m-1}$, these matrices belong to ${\rm E}_{n+2m}(R)$. We
also prove that if $v=(a_{1},\ldots,a_{n+2m-1})\in R^{n+2m-1}$ satisfies the
condition

$\overline{a}_{1}-a_{1}=\bar{1}^{-1}\overline{(a_{2},\ldots,a_{n+2m-1})}\varPsi(a_{2},\ldots,a_{n+2m-1})^{t}=q(a_{2},\ldots,a_{n+2m-1}),$

then $L(v)\in{\rm U}_{2m}(R,\mathfrak{L}_{\max})$ and for those $v$ satisfying

$\overline{\bar{1}a_{1}}-\bar{1}a_{1}=\overline{(a_{2},\ldots,a_{n+2m-1})}(\widetilde{\psi}^{\prime}_{m-1}\perp(\bar{1}^{-1})^{2}\varphi)^{t}(a_{2},\ldots,a_{n+2m-1})^{t},$

$L(v)^{*}$ is an isometry, but need not belongs to the odd unitary group.
These results involve more calculations than the case of Bak’s unitary group
because of the fact that, for a transvection to be an element in Petrov’s odd
unitary group they have to be an isometry as well as congruent to the identity
modulo the odd form parameter taken. The latter condition is an additional
requirement in this group.

The following is the main result proved in this article.

######  Theorem 1.1.

Let $R$ be a commutative ring with unity. The group generated by $L(v)$ and
$L(v)^{*}$ for $v\in R^{2m+n-1}$ is congruent to the Petrov’s odd elementary
hyperbolic unitary group ${\rm EU}_{2m}(R,\mathfrak{L}_{\max})$.

We can study the relative version of Petrov’s odd unitary group using this
result analogous to the study done by Chattopadhyay and Rao in [2]. This
theorem will help us to visualize the odd elementary hyperbolic unitary group
in a simpler form and enable us to study the structure of that group in more
depth.

##  2\. Preliminaries

###  2.1. Odd Unitary Groups

Odd unitary groups were defined by V.A. Petrov in [3].

######  Definition 2.1.

Let $R$ be an associative ring with 1. An additive map $\sigma:R\to R$ defined
by $r\mapsto\bar{r}$ satisfying the properties
$\overline{r_{1}r_{2}}=\bar{r}_{2}\bar{1}^{-1}\bar{r}_{1}$ and
$\bar{\bar{r}}=r$ for all $r,r_{1},r_{2}\in R$ is called a pseudoinvolution on
$R$.

Let $R$ be a ring with pseudoinvolution and $V$ be a right $R$-module. A map
$\langle,\rangle:V\times V\to R$ is called a sesquilinear form on $V$ if it is
biadditive and satisfy the equation

$\langle ur,vs\rangle=\bar{r}\bar{1}^{-1}\langle u,v\rangle s,$

for all $u,v\in V$ and $r,s\in R$.

A sesquilinear form is anti-Hermitian if it also satisfies $\langle
u,v\rangle=-\overline{\langle v,u\rangle}$ for all $u,v\in V$.

######  Definition 2.2.

Let $V$ be a right $R$-module and $\langle,\rangle$ be an anti-Hermitian form
on it. The set $\mathfrak{H}=V\times R$ is a group with composition defined
as:

$(u,r)\dot{+}(v,s)=(u+v,r+s+\langle u,v\rangle).$

Here the identity element is $(0,0)$ and the inverse of $(u,r)$ is
$(-u,-r+\langle u,u\rangle)$ which is denoted by $\dot{-}(u,r)$. This group is
called the Heisenberg group of the form $\langle,\rangle$.

Consider the right action of $R$ on $\mathfrak{H}$ defined by
$(u,r)\leftharpoonup s=(us,\bar{s}\bar{1}^{-1}rs).$ The trace map on
$\mathfrak{H},~{}tr:\mathfrak{H}\to R$ is a group homomorphism defined by
$tr((u,r))=r-\bar{r}-\langle u,u\rangle.$

Define $\mathfrak{L}_{\min}$ and $\mathfrak{L}_{\max}$ as follows.

$\mathfrak{L}_{\min}=\\{(0,r+\bar{r}):r\in R\\},\hskip
5.69054pt\mathfrak{L}_{\max}=\\{\zeta\in\mathfrak{H}:tr(\zeta)=0\\}.$

The subsets $\mathfrak{L}_{\min}$ and $\mathfrak{L}_{\max}$ are subgroups of
$\mathfrak{H}$. These subgroups are stable under the action of $R$ and
satisfies $\mathfrak{L}_{\min}\leq\mathfrak{L}_{\max}$.

######  Definition 2.3.

An _odd form parameter_ is a subgroup $\mathfrak{L}$ of $\mathfrak{H}$
satisfying $\mathfrak{L}_{\min}\leq\mathfrak{L}\leq\mathfrak{L}_{\max}$ and is
stable under the action of $R$.

If $\mathfrak{L}$ is an odd form parameter corresponding to the sesquilinear
form $\langle,\rangle$, then the pair $q=(\langle,\rangle,\mathfrak{L})$ is
called an odd quadratic form and the pair $(V,q)$ is called an odd quadratic
space.

Let $(V,q)$ be an odd quadratic space. The even part of the odd form parameter
$\mathfrak{L}$ is defined as $\mathfrak{L}_{ev}=\\{a\in
R:(0,a)\in\mathfrak{L}\\}$ and the even part of the quadratic space is defined
as

$V_{ev}=\\{u\in V:(u,a)\in\mathfrak{L}\mbox{ for a certain $a\in R$}\\}.$

Let $V$ and $V^{\prime}$ be two $R$-modules equipped with form parameters
$\mathfrak{L}$ and $\mathfrak{L}^{\prime}$ respectively. An isometry $f:V\to
V^{\prime}$ preserves the form parameters if
$(f(v),r)\in\mathfrak{L}^{\prime}$ for all $(v,r)\in\mathfrak{L}$. Two such
isometries $f$ and $g$ are said to be equivalent modulo
$\mathfrak{L}^{\prime}$ if $(f(v)-g(v),\langle
g(v)-f(v),g(v)\rangle)\in\mathfrak{L}^{\prime}$ for all $v\in V$. It can be
verified that this relation is an equivalence relation on the set of all
isometries from $V$ to $V^{\prime}$. If $f$ and $g$ are equivalent modulo
$\mathfrak{L}$, we denote it by $f\cong g({\rm mod}~{}\mathfrak{L})$.

######  Definition 2.4.

The _odd unitary group_ denoted by ${\rm U}(V,q)$ is the group of all
bijective isometries $f$ on $V$ such that $f\cong e({\rm
mod}~{}\mathfrak{L}),$ where $e$ is the identity map on $V$.

Let $u,v$ be elements of an odd quadratic space $V$ and $r$ be an element of
$R$ such that $\langle u,v\rangle=0,~{}(u,0)\in\mathfrak{L}$ and
$(v,r)\in\mathfrak{L}$. The transformations on $V$ of the form $T_{u,v}(a)$
defined by

$T_{u,v}(r)(w)=w+u\bar{1}^{-1}(\langle v,w\rangle+r\langle
u,w\rangle)+v\langle u,w\rangle,$

are known as Eichler-Siegel-Dickson transvections.

######  Lemma 2.5 ([3], Lemma 1).

The transvections $T_{u,v}(a)$ lie in ${\rm U}(V,q)$.

A pair of vectors $(u,v)$ such that $\langle
u,v\rangle=1,~{}(u,0)\in\mathfrak{L},~{}(v,0)\in\mathfrak{L}$ is called a
hyperbolic pair. Let $\mathbb{H}$ be odd quadratic space spanned by vectors
$e_{1}$ and $e_{-1}$ such that $\langle e_{1},e_{-1}\rangle=1$, equipped with
the odd form parameter $\mathfrak{L}$ generated by $(e_{1},0)$ and
$(e_{-1},0)$. Denote the orthogonal sum of $m$ copies of $\mathbb{H}$ by
$\mathbb{H}^{m}$.

Consider an odd quadratic space $V_{0}$ equipped with an odd quadratic form
$q_{0}=(\langle,\rangle_{0},\mathfrak{L}_{0})$. The orthogonal sum
$V=\mathbb{H}^{m}\oplus V_{0}$ is called the odd hyperbolic unitary group and
is denoted by ${\rm U}_{2m}(R,\mathfrak{L})$.

######  Definition 2.6.

The greatest number $n$ satisfying the condition that there exist $n$ mutually
orthogonal hyperbolic pairs in $(V,q)$ is called the Witt index of $(V,q)$,
denoted by ${\rm ind(V,q)}$.

Let $V$ be an odd quadratic space that has Witt index at least $n$. Then we
can choose an embedding of $\mathbb{H}^{n}$ to $V$. Fix any such embedding.
Then we have elements $\\{e_{i}\\}_{i=1,\ldots,n,-n,\ldots,-1}$ in $V$ such
that $\langle e_{i},e_{j}\rangle=0$ for $i\neq-j,$ $\langle
e_{i},e_{-i}\rangle=1$ for $i=1,\ldots,n$ and $(e_{i},0)\in\mathfrak{L}$. Let
$V_{0}$ denote the orthogonal complement of $\langle
e_{1},\ldots,e_{n},e_{-n},\ldots,e_{-1}\rangle$ in $V$ and
$\langle,\rangle_{0}$ and $\mathfrak{L}_{0}$ be the restrictions of
$\langle,\rangle$ and $\mathfrak{L}$ respectively to $V_{0}$. Then $V$ is
isometric to the odd hyperbolic space $\mathbb{H}^{n}\oplus V_{0}$ with the
quadratic form $(\langle,\rangle_{0},\mathfrak{L}_{0})$. Thus ${\rm
U}(V,q)={\rm U}_{2n}(R,\mathfrak{L}_{0})$.

Consider a non-reduced root system of type $BC_{l}$ consisting of vectors
$\chi_{i}\pm\chi_{j},~{}\pm\chi_{i}$ and $\pm 2\chi_{i}$, where
$\\{\chi_{i}\\}_{i=1,\ldots,n}$ is an orthonormal basis for the
$n$-dimensional Euclidean space. Corresponding to the roots we can define
elementary transvections as follows.

$\displaystyle
T_{\chi_{i}-\chi_{j}}(r)={T_{i,j}(r)=T_{e_{-j},-e_{i}r\varepsilon_{j}}(0),~{}r\in
R},$ $\displaystyle
T_{\chi_{i}}(u,r)={T_{i}(u,r)=T_{e_{i},u\varepsilon_{-i}}(-\bar{\varepsilon}_{-i}\bar{1}^{-1}r\varepsilon_{-i}),~{}(u,r)\in\mathfrak{L}},$
$\displaystyle T_{2\chi_{i}}(r)=T_{i}(0,r),~{}r\in\mathfrak{L}_{ev},$

where $i,j\in\\{-m,\ldots 1,-1,\ldots,m\\}$ and
$\varepsilon_{i}=\begin{cases}\bar{1}^{-1},~{}i>0\\\ -1,~{}i<0.\end{cases}$

The odd elementary hyperbolic unitary group ${\rm EU}_{2m}(R,\mathfrak{L})$ is
the subgroup of ${\rm U}_{2m}(R,\mathfrak{L})$ generated by all elementary
transvections.

##  3\. Matrix form of elementary transvections

Let $R$ be a commutative ring with identity. Here we write the matrices for
the generators of ${\rm EU}_{2m}(R,\mathfrak{L})$ when the odd quadratic space
$V_{0}$ is free of rank $n$. Let
$\\{v_{1},\ldots,v_{n},e_{1},e_{-1},\ldots,e_{m},e_{-m}\\}$ be a basis for the
odd hyperbolic unitary space $V=\mathbb{H}^{m}\oplus V_{0}$. Let the unitary
space $V$ be equipped with anti-Hermitian form $\langle,\rangle$, i.e.,
$\langle u,v\rangle=-\overline{\langle v,u\rangle}$ for all $u,v\in V$. We use
the following important result proved by Petrov in [3].

######  Proposition 3.1 ([3], Proposition 1).

The group ${\rm EU}_{2l}(R,\mathfrak{L})$ coincides with the group generated
by all elements of the form $T_{e_{\pm 1},v}(a),$ where $\langle
e_{1},v\rangle=\langle e_{-1},v\rangle=0$ and $(v,a)\in\mathfrak{L}$.

Let
$u=t_{1}v_{1}+\ldots+t_{n}v_{n}+b_{1}e_{1}+b_{-1}e_{-1}+\ldots+b_{m}e_{m}+b_{-m}e_{-m}$
be an arbitrary element of $V$. For $i\in\\{1,\ldots,m\\}$, we have
$\varepsilon_{-i}=-1$ and therefore $T_{i}(u,a)=T_{e_{i},-u}(-a)$. Also for
$i\in\\{-m,\ldots,-1\\}$, we have $\varepsilon_{-i}=\bar{1}^{-1}$ and
therefore $T_{i}(u,a)=T_{e_{i},u\bar{1}^{-1}}(-a)$. Thus for arbitrary
$(v,a)\in\mathfrak{L}$ with $\langle e_{1},v\rangle=\langle e_{-1},v\rangle=0$
we have $T_{e_{1},v}(a)=T_{1}(-v,-a)$ and
$T_{e_{-1},v}(a)=T_{-1}(v\bar{1},-a)$. Thus we can restate the above
proposition as follows.

######  Proposition 3.2.

The group ${\rm EU}_{2l}(R,\mathfrak{L})$ coincides with the group generated
by all elements of the form $T_{\pm 1}(v,a)$, where $\langle
e_{1},v\rangle=\langle e_{-1},v\rangle=0$ and $(v,a)\in\mathfrak{L}$.

######  Note 3.3.

The pseudoinvolution we have taken satisfies the properties
$\overline{r_{1}r_{2}}=\bar{r}_{2}\bar{1}^{-1}\bar{r}_{1}$ and
$\bar{\bar{r}}=r$ for all $r,r_{1},r_{2}\in R$. Therefore we have
$\bar{1}\cdot\bar{1}=\overline{\bar{1}^{-1}}$.

Now we write the matrices for the generators $T_{\pm 1}(v,a)$, of the Petrov’s
odd elementary hyperbolic unitary group.

For computing $T_{1}(u,a)=T_{e_{1},-u}(-a)$, the element $u$ should satisfy
$\langle e_{1},-u\rangle=0$ which implies that $b_{-1}=0$. Therefore $u$ has
the form

$u=t_{1}v_{1}+\ldots+t_{n}v_{n}+b_{1}e_{1}+b_{2}e_{2}+b_{-2}e_{-2}+\ldots+b_{m}e_{m}+b_{-m}e_{-m}.$

We have
$T_{1}(u,a)(w)=T_{e_{1},-u}(-a)(w)=w+e_{1}\bar{1}^{-1}(\langle-u,w\rangle-a\langle
e_{1},w\rangle)-u\langle e_{1},w\rangle$. Thus we get

$\displaystyle T_{1}(u,a)(v_{k})$
$\displaystyle=v_{k}+e_{1}\bar{1}^{-1}\langle-v_{1}t_{1}-\dots-
v_{n}t_{n},v_{k}\rangle$
$\displaystyle=v_{k}-e_{1}(\bar{1}^{-1})^{2}(\overline{t}_{1}\langle
v_{1},v_{k}\rangle+\ldots+\overline{t}_{n}\langle v_{n},v_{k}\rangle),\mbox{
for }k\in\\{1,\ldots n\\}.$

Also $T_{1}(u,a)(e_{1})=e_{1}$ and

$T_{1}(u,a)(e_{-1})=-t_{1}v_{1}-\ldots-
t_{n}v_{n}-e_{1}(\overline{b_{1}}(\bar{1}^{-1})^{2}+a\bar{1}^{-1}+b_{1})+e_{-1}-b_{2}e_{2}-b_{-2}e_{-2}-\ldots-
b_{m}e_{m}-b_{-m}e_{-m}.$

For $k\in\\{2,\ldots,m\\}$, we get
$T_{1}(u,a)(e_{k})=e_{k}+e_{1}\bar{1}^{-1}\overline{b_{-k}}$ and for
$k\in\\{-m,\ldots,-2\\}$,
$T_{1}(u,a)(e_{k})=e_{k}-e_{1}(\bar{1}^{-1})^{2}\overline{b_{-k}}$.

Therefore the matrix of $T_{1}(u,a)$ is of the form

${\scriptsize\begin{pmatrix}1&\ldots&0&0&-t_{1}&0&0&\ldots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&\ldots&1&0&-t_{n}&0&0&\ldots&0&0\\\
-(\bar{1}^{-1})^{2}(\overline{t}_{1}\langle
v_{1},v_{1}\rangle&\ldots&-(\bar{1}^{-1})^{2}(\overline{t}_{1}\langle
v_{1},v_{n}\rangle&1&-(\overline{b}_{1}(\bar{1}^{-1})^{2}&\bar{1}^{-1}\overline{b}_{-2}&-(\bar{1}^{-1})^{2}\overline{b}_{2}&\ldots&\bar{1}^{-1}\overline{b}_{-m}&-(\bar{1}^{-1})^{2}\overline{b}_{m}\\\
+\ldots+\overline{t}_{n}\langle
v_{n},v_{1}\rangle)&&+\ldots+\overline{t}_{n}\langle
v_{n},v_{n}\rangle)&&\hskip 22.76219pt+a\bar{1}^{-1}+b_{1})&&&&&\\\
0&\ldots&0&0&1&0&0&\ldots&0&0\\\ 0&\ldots&0&0&-b_{2}&1&0&\ldots&0&0\\\
0&\ldots&0&0&-b_{-2}&0&1&\ldots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&\ldots&0&0&-b_{m}&0&0&\ldots&0&0\\\
0&\ldots&0&0&-b_{-m}&0&0&\ldots&0&1\end{pmatrix}}.$

Now for computing $T_{-1}(u,a)=T_{e_{-1},u\bar{1}^{-1}}(-a)$, the element $u$
should satisfy $\langle e_{-1},u\bar{1}^{-1}\rangle=0$ which implies
$b_{1}=0$. Therefore $u$ will be of the form

$u=t_{1}v_{1}+\ldots+t_{n}v_{n}+b_{-1}e_{-1}+b_{2}e_{2}+b_{-2}e_{-2}+\ldots+b_{m}e_{m}+b_{-m}e_{-m}.$

We have

$T_{-1}(u,a)(w)=T_{e_{-1},u\bar{1}^{-1}}(-a)(w)=w+e_{-1}\bar{1}^{-1}(\langle
u\bar{1}^{-1},w\rangle-a\langle e_{-1},w\rangle)+u\bar{1}^{-1}\langle
e_{-1},w\rangle.$ $\displaystyle\mbox{Thus }~{}~{}T_{-1}(u,a)(v_{k})$
$\displaystyle=v_{k}+e_{-1}\bar{1}^{-1}\langle(v_{1}t_{1}+\dots+v_{n}t_{n})\bar{1}^{-1},v_{k}\rangle$
$\displaystyle=v_{k}+e_{-1}\bar{1}^{-1}(\overline{t}_{1}\langle
v_{1},v_{k}\rangle+\ldots+\overline{t}_{n}\langle v_{n},v_{k}\rangle),\mbox{
for }k\in\\{1,\ldots n\\},$ $\displaystyle T_{-1}(u,a)(e_{1})=-t_{1}v_{1}$
$\displaystyle-\ldots-
t_{n}v_{n}+e_{1}+e_{-1}(-\overline{b_{-1}}+a\bar{1}-b_{-1})-b_{2}e_{2}-b_{-2}e_{-2}$
$\displaystyle-\ldots-b_{m}e_{m}-b_{-m}e_{-m}$

and $T_{-1}(u,a)(e_{-1})=e_{-1}$.

Also $T_{-1}(u,a)(e_{k})=\begin{cases}e_{k}-e_{-1}\overline{b}_{-k},\mbox{ for
}k\in\\{2,\ldots,m\\},\\\ e_{k}+e_{-1}\bar{1}^{-1}\overline{b}_{-k}\mbox{ for
}k\in\\{-m,\ldots,-2\\}.\end{cases}$

Therefore the matrix of $T_{-1}(u,a)$ is of the form

${\scriptsize\begin{pmatrix}1&\ldots&0&-t_{1}&0&0&0&\ldots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&\ldots&1&-t_{n}&0&0&0&\ldots&0&0\\\ 0&\ldots&0&1&0&0&0&\ldots&0&0\\\
\bar{1}^{-1}(\overline{t}_{1}\langle
v_{1},v_{1}\rangle&\ldots&\bar{1}^{-1}(\overline{t}_{1}\langle
v_{1},v_{n}\rangle&-\overline{b}_{-1}+a-b_{-1}&1&-\overline{b}_{-2}&\bar{1}^{-1}\overline{b}_{2}&\ldots&-\overline{b}_{-m}&\bar{1}^{-1}\overline{b}_{m}\\\
+\ldots+\overline{t}_{n}\langle
v_{n},v_{1}\rangle)&&+\ldots+\overline{t}_{n}\langle
v_{n},v_{n}\rangle)&&&&&\\\ 0&\ldots&0&-b_{2}&0&1&0&\ldots&0&0\\\
0&\ldots&0&-b_{-2}&0&0&1&\ldots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&\ldots&0&-b_{m}&0&0&0&\ldots&0&0\\\
0&\ldots&0&-b_{-m}&0&0&0&\ldots&0&1\end{pmatrix}}.$

Now we consider the case when $R$ is a commutative ring and the involution is
$a\mapsto\bar{a}=-a$. In this case, the above matrices reduces to the
following form.

${\scriptsize T_{1}(u,a)=\begin{pmatrix}1&\ldots&0&0&-t_{1}&0&0&\ldots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&\ldots&1&0&-t_{n}&0&0&\ldots&0&0\\\ t_{1}\langle
v_{1},v_{1}\rangle&\ldots&t_{1}\langle
v_{1},v_{n}\rangle&1&a&b_{-2}&b_{2}&\ldots&b_{-m}&b_{m}\\\
+\ldots+t_{n}\langle v_{n},v_{1}\rangle&&+\ldots+t_{n}\langle
v_{n},v_{n}\rangle&&&&&&&\\\ 0&\ldots&0&0&1&0&0&\ldots&0&0\\\
0&\ldots&0&0&-b_{2}&1&0&\ldots&0&0\\\ 0&\ldots&0&0&-b_{-2}&0&1&\ldots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&\ldots&0&0&-b_{m}&0&0&\ldots&0&0\\\
0&\ldots&0&0&-b_{-m}&0&0&\ldots&0&1\end{pmatrix}},\mbox{ and }$

${\scriptsize T_{-1}(u,a)=\begin{pmatrix}1&\ldots&0&-t_{1}&0&0&0&\ldots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&\ldots&1&-t_{n}&0&0&0&\ldots&0&0\\\ 0&\ldots&0&1&0&0&0&\ldots&0&0\\\
t_{1}\langle v_{1},v_{1}\rangle&\ldots&t_{1}\langle
v_{1},v_{n}\rangle&a&1&b_{-2}&b_{2}&\ldots&b_{-m}&b_{m}\\\
+\ldots+t_{n}\langle v_{n},v_{1}\rangle&&+\ldots+t_{n}\langle
v_{n},v_{n}\rangle&&&&&&&\\\ 0&\ldots&0&-b_{2}&0&1&0&\ldots&0&0\\\
0&\ldots&0&-b_{-2}&0&0&1&\ldots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&\ldots&0&-b_{m}&0&0&0&\ldots&0&0\\\
0&\ldots&0&-b_{-m}&0&0&0&\ldots&0&1\end{pmatrix}}.$

##  4\. Visualization of odd elementary hyperbolic unitary group

Let $\varphi$ be an invertible alternating matrix of size $2n$ of the form
$\begin{pmatrix}0&-c\\\ c^{t}&\mu\end{pmatrix}$ and $\varphi^{-1}$ is of the
form $\begin{pmatrix}0&d\\\ -d^{t}&\rho\end{pmatrix}$, where $c,d\in
R^{2n-1}$. For $v\in R^{2n-1}$, Vaserstein (in [1], Lemma 5.4) considered the
matrices

$\alpha=I_{2n-1}+d^{t}v\mu\mbox{ and }\beta=I_{2n-1}-\rho v^{t}c$

and proved that the matrices

$\begin{pmatrix}1&0\\\ v^{t}&\alpha\end{pmatrix}\mbox{ and
}\begin{pmatrix}1&v\\\ 0&\beta\end{pmatrix}$

belongs to ${\rm E}_{2n}(R)\cap{\rm Sp}_{\varphi}(R)$, where ${\rm
Sp}_{\varphi}(R)$ is the group of all symplectic matrices with respect to the
form $\varphi$.

Here we define analogous matrices for Petrov’s odd elementary hyperbolic
unitary group. The form matrix for Petrov’s group is
$\varPsi=\widetilde{\psi}_{m}\perp\varphi$, where $\varphi$ is the form matrix
for $V_{0}$ and $\widetilde{\psi}_{r}$ is defined as follows.

$\widetilde{\psi_{1}}=\begin{pmatrix}0&1\\\ -\bar{1}&0\end{pmatrix},\mbox{ and
}\widetilde{\psi}_{r}=\widetilde{\psi}_{1}\perp\widetilde{\psi}_{r-1}\mbox{
for }r>1.$

Then $\varPsi$ can be written in the form $\begin{pmatrix}0&c\\\
-\bar{1}c^{t}&\mu\end{pmatrix}$, where
$c=\begin{pmatrix}1&\ldots&0\end{pmatrix}\in R^{n+2m-1}$ and $\mu$ is the
$(n+2m)\times(n+2m)$ matrix given by $\mu=\begin{pmatrix}0&0&0\\\
0&\widetilde{\psi}_{m-2}&0\\\ 0&0&\varphi\end{pmatrix}$. Also we have
$\varPsi^{-1}$ is of the form $\begin{pmatrix}0&d\\\
-\bar{1}d^{t}&\rho\end{pmatrix}$, where
$d=\begin{pmatrix}-\bar{1}^{-1}&\ldots&0\end{pmatrix}\in R^{n+2m-1}$ and
$\rho=\begin{pmatrix}0&0&0\\\ 0&\widetilde{\psi}^{\prime}_{m-2}&0\\\
0&0&\varphi^{-1}\end{pmatrix}$, where $\widetilde{\psi}^{\prime}_{m}$ is
defined as follows.

$\widetilde{\psi}^{\prime}_{1}=\begin{pmatrix}0&-\bar{1}^{-1}\\\
1&0\end{pmatrix}\mbox{ and
}\widetilde{\psi}^{\prime}_{r}=\widetilde{\psi}^{\prime}_{1}\perp\widetilde{\psi}^{\prime}_{r-1},\mbox{
for }r>1.$

For $v=(a_{1},\ldots,a_{n+2m-1})\in R^{n+2m-1}$, define the Vaserstein type
matrices $L(v)$ and $L(v)^{*}$ as

$L(v)=\begin{pmatrix}1&0\\\ v^{t}&\alpha\end{pmatrix}\mbox{ and
}L(v)^{*}=\begin{pmatrix}1&v\\\ 0&\beta\end{pmatrix},$

where $\alpha=I_{n+2m-1}+d^{t}\overline{v}\mu$ and
$\beta=I_{n+2m-1}-\bar{1}^{-1}\rho\overline{v}^{t}c$.

By direct computations, we get the matrices $\alpha$ and $\beta$ corresponding
to the form $\Psi$ as follows:

$\alpha={\scriptsize{\begin{pmatrix}1&\overline{a}_{3}&-\bar{1}^{-1}\overline{a}_{2}&\ldots&\overline{a}_{2m-1}&-\bar{1}^{-1}\overline{a}_{2m-2}&-\bar{1}^{-1}(\overline{a}_{2m}\varphi_{11}+\overline{a}_{2m+1}\varphi_{21}&\ldots&-\bar{1}^{-1}(\overline{a}_{2m}\varphi_{1n}+\overline{a}_{2m+1}\varphi_{2n}\\\
&&&&&&+\ldots+\overline{a}_{2m-1+n}\varphi_{n1})&&+\ldots+\overline{a}_{2m-1+n}\varphi_{nn})\\\
0&1&0&\ldots&0&0&0&\ldots&0\\\ 0&0&1&\ldots&0&0&0&\ldots&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&0&0&\ldots&0&0&0&\ldots&1\end{pmatrix}}},$

$\mbox{ and }\beta=\scriptsize{\begin{pmatrix}1&0&0&\ldots&0\\\
(\bar{1}^{-1})^{2}\overline{a}_{3}&1&0&\ldots&0\\\
-\bar{1}^{-1}\overline{a}_{2}&0&1&\ldots&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\
(\bar{1}^{-1})^{2}\overline{a}_{2m-1}&0&0&\ldots&0\\\
-\bar{1}^{-1}\overline{a}_{2m-2}&0&0&\ldots&0\\\
-\bar{1}^{-1}(\overline{a}_{2m}(\varphi^{-1})_{11}+\overline{a}_{2m+1}(\varphi^{-1})_{12}+\ldots+\overline{a}_{2m-1+n}(\varphi^{-1})_{1n})&0&0&\ldots&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\
-\bar{1}^{-1}(\overline{a}_{2m}(\varphi^{-1})_{n1}+\overline{a}_{2m+1}(\varphi^{-1})_{n2}+\ldots+\overline{a}_{2m-1+n}(\varphi^{-1})_{nn}&0&0&\ldots&1\end{pmatrix}}.$

Therefore, we have

$L(v)={\scriptsize{\begin{pmatrix}1&0&0&0&\ldots&0&0&0&\ldots&0\\\
a_{1}&1&\overline{a}_{3}&-\bar{1}^{-1}\overline{a}_{2}&\ldots&\overline{a}_{2m-1}&-\bar{1}^{-1}\overline{a}_{2m-2}&-\bar{1}^{-1}(\overline{a}_{2m}\varphi_{11}+\overline{a}_{2m+1}\varphi_{21}&\ldots&-\bar{1}^{-1}(\overline{a}_{2m}\varphi_{1n}+\overline{a}_{2m+1}\varphi_{2n}\\\
&&&&&&&+\ldots+\overline{a}_{2m-1+n}\varphi_{n1})&&+\ldots+\overline{a}_{2m-1+n}\varphi_{nn})\\\
a_{2}&0&1&0&\ldots&0&0&0&\ldots&0\\\ a_{3}&0&0&1&\ldots&0&0&0&\ldots&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
a_{n+2m-1}&0&0&0&\ldots&0&0&0&\ldots&1\end{pmatrix}}},$

$\mbox{ and
}L(v)^{*}={\scriptsize{\begin{pmatrix}1&a_{1}&a_{2}&0&\ldots&a_{n+2m-1}\\\
0&1&0&0&\ldots&0\\\ 0&(\bar{1}^{-1})^{2}\overline{a}_{3}&1&0&\ldots&0\\\
0&-\bar{1}^{-1}\overline{a}_{2}&0&1&\ldots&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&(\bar{1}^{-1})^{2}\overline{a}_{2m-1}&0&0&\ldots&0\\\
0&-\bar{1}^{-1}\overline{a}_{2m-2}&0&0&\ldots&0\\\
0&-\bar{1}^{-1}(\overline{a}_{2m}(\varphi^{-1})_{11}+\overline{a}_{2m+1}(\varphi^{-1})_{12}+\ldots+\overline{a}_{2m-1+n}(\varphi^{-1})_{1n})&0&0&\ldots&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&-\bar{1}^{-1}(\overline{a}_{2m}(\varphi^{-1})_{n1}+\overline{a}_{2m+1}(\varphi^{-1})_{n2}+\ldots+\overline{a}_{2m-1+n}(\varphi^{-1})_{nn}&0&0&\ldots&1\end{pmatrix}}}.$

By an argument analogous to Lemma 5.4 in [1], we can prove that both $L(v)$
and $L(v)^{*}$ are elementary matrices.

######  Theorem 4.1.

For $v=(a_{1},\ldots,a_{n+2m-1})\in R^{n+2m-1}$, we have
$L(v),~{}L(v)^{*}\in{\rm E}_{n+2m}(R).$

###### Proof.

By direct computation, we can verify that $\bar{v}\mu d^{t}=0$. Therefore
$1+\bar{v}\mu d^{t}=1$. By [1, Lemma 2.2], we have
$\alpha=1+d^{t}\bar{v}\mu\in{\rm GL}_{n+2m}(R)$ and $\begin{pmatrix}1&0\\\
0&\alpha\end{pmatrix}\in{\rm E}_{n+2m}(R)$. Now by using [1, Lemma 2.1], we
get $\begin{pmatrix}1&0\\\ v^{t}&I_{n+2m-1}\end{pmatrix}\in{\rm E}_{n+2m}(R)$.
Therefore we have

$\begin{pmatrix}1&0\\\ v^{t}&\alpha\end{pmatrix}=\begin{pmatrix}1&0\\\
v^{t}&I_{n+2m-1}\end{pmatrix}\begin{pmatrix}1&0\\\
0&\alpha\end{pmatrix}\in{\rm E}_{n+2m}(R).$

∎

The matrices $L(v)$ and $L(v)^{*}$ are defined for arbitrary
$v=(a_{1},\ldots,a_{n+2m-1})\in R^{n+2m-1}$. Under some condition on the
components of $v$, we get the following result.

######  Theorem 4.2.

We have $L(v)\in{\rm U}_{2m}(R,\mathfrak{L}_{\max})$, if
$v=(a_{1},\ldots,a_{n+2m-1})\in R^{n+2m-1}$ satisfies the condition

$\overline{a}_{1}-a_{1}=\bar{1}^{-1}\overline{(a_{2},\ldots,a_{n+2m-1})}\varPsi(a_{2},\ldots,a_{n+2m-1})^{t}=q(a_{2},\ldots,a_{n+2m-1}).$
(1)

Also, $L(v)^{*}$ preserves the form matrix $\varPsi$, for $v$ satisfying the
condition

$\overline{\bar{1}a_{1}}-\bar{1}a_{1}=\overline{(a_{2},\ldots,a_{n+2m-1})}(\widetilde{\psi}^{\prime}_{m-1}\perp(\bar{1}^{-1})^{2}\varphi)^{t}(a_{2},\ldots,a_{n+2m-1})^{t}.$
(2)

###### Proof.

By direct calculation we can verify that under the condition given in Equation
(1),

$\bar{1}^{-1}\overline{L(v)}^{t}\varPsi L(v)=\varPsi.$

That is, $L(v)$ preserves the form matrix for Petrov’s group. Now to prove
that $L(v)$ belongs to ${\rm U}_{2m}(R,\mathfrak{L}_{\max})$, it suffices to
prove that $L(v)\cong I_{n+2m}$ (mod $\mathfrak{L}_{\max}$). That is
$(L(v)(w)-w,\langle w-L(v)(w),w\rangle)\in\mathfrak{L}_{\max}$, for all $w\in
V$.

For $w=(w_{1},\ldots,w_{n+2m})\in V$, we have

$L(v)(w)=\begin{pmatrix}w_{1}\\\ x\\\ a_{2}w_{1}+w_{3}\\\ a_{3}w_{1}+w_{4}\\\
\vdots\\\ a_{n+2m-1}w_{1}+w_{n+2m}\end{pmatrix},$ $\displaystyle\mbox{where
}x$
$\displaystyle=a_{1}w_{1}+w_{2}+\overline{a}_{3}w_{3}-\bar{1}^{-1}\overline{a}_{2}w_{4}+\ldots+\overline{a}_{2m-1}w_{2m-1}$
$\displaystyle\hskip
28.45274pt-\bar{1}^{-1}\overline{a}_{2m-2}w_{2m}-\bar{1}^{-1}(\overline{a}_{2m}\varphi_{11}+\overline{a}_{2m+1}\varphi_{21}+\ldots+\overline{a}_{2m-1+n}\varphi_{n1})w_{2m+1}$
$\displaystyle\hskip
56.9055pt-\ldots-\bar{1}^{-1}(\overline{a}_{2m}\varphi_{1n}+\overline{a}_{2m+1}\varphi_{2n}+\ldots+\overline{a}_{2m-1+n}\varphi_{nn})w_{n+2m}.$

Let $u=L(v)(w)-w$. Then

$u=\begin{pmatrix}0\\\ x_{1}\\\ a_{2}w_{1}\\\ a_{3}w_{1}\\\ \vdots\\\
a_{n+2m-1}w_{1}\end{pmatrix},$

where $x_{1}=x-w_{2}.$ Now, let $a=\langle w-L(v)(w),w\rangle.$ By
computations we can see that

$\displaystyle a$
$\displaystyle=\bar{1}^{-1}\overline{w}_{1}\overline{a}_{1}w_{1}+(s_{1}+\overline{s}_{1})-(s_{2}+\overline{s}_{2})+\ldots+(s_{2m-3}+\overline{s}_{2m-3})-(s_{2m-2}+\overline{s}_{2m-2})$
$\displaystyle\hskip
56.9055pt-(s_{2m-1}+\overline{s}_{2m-1})-(s_{2m}+\overline{s}_{2m})-\ldots-(s_{2m+n}+\overline{s}_{2m+n}),$

where $s_{k}=\begin{cases}\bar{1}^{-1}\overline{w}_{k+2}a_{2k-1}w_{1},\mbox{
for }1\leq k\leq 2m-2\mbox{ and $k$ is odd},\\\
\overline{w}_{k+2}a_{k}w_{1},\mbox{ for }1\leq k\leq 2m-2\mbox{ and $k$ is
even},\\\
\bar{1}^{-1}\overline{w}_{k+2}(\overline{\varphi}_{1,k-(2m-2)}a_{2m}+\overline{\varphi}_{2,k-(2m-2)}a_{2m+1}+\ldots+\overline{\varphi}_{n,k-(2m-2)}a_{n+2m-1})w_{1},\\\
\hskip 170.71652pt\mbox{ for }2m-1\leq k\leq 2m+n-2.\end{cases}$

Thus
$a-\overline{a}=\bar{1}^{-1}\overline{w}_{1}(\overline{a}_{1}-a_{1})w_{1}.$
Also we can verify that

$\langle
u,u\rangle=\bar{1}^{-1}\overline{w}_{1}q(a_{2},\ldots,a_{n+2m-1})w_{1}=\bar{1}^{-1}\overline{w}_{1}(\overline{a}_{1}-a_{1})w_{1}.$

Therefore $a-\overline{a}=\langle u,u\rangle$, which implies
$(u,a)\in\mathfrak{L}_{\max}$. Thus $L(v)\in{\rm
U}_{2m}(R,\mathfrak{L}_{\max}).$

Now assume that $v$ satisfies the condition given in Equation (2). Then we can
see that

$\bar{1}^{-1}\overline{L(v)^{*}}^{t}\varPsi L(v)^{*}=\varPsi,$

which shows that $L(v)^{*}$ preserves the form matrix $\varPsi.$ ∎

######  Remark 4.3.

By assuming the condition given in Equation (2) and doing similar calculations
as in the proof of Theorem 4.2, for $u_{1}=L(v)^{*}(w)-w$ and $b=\langle
w-L(v)^{*}(w),w\rangle$, we get

$\langle
u_{1},u_{1}\rangle=\overline{w}_{2}(a_{1}-(\bar{1}^{-1})^{2}\overline{a}_{1})w_{2}~{}~{}\mbox{
and
}~{}~{}b-\bar{b}=\overline{w}_{2}(a_{1}-(\bar{1}^{-1})^{2}\overline{a}_{1})w_{2}+(1-(\bar{1}^{-1})^{2})(s+\overline{s}),$

where
$s=\overline{w}_{2}\begin{pmatrix}a_{2m}&a_{2m+1}&\ldots&a_{n+2m-1}\end{pmatrix}\begin{pmatrix}w_{2m+1}\\\
w_{2m+2}\\\ \vdots\\\ w_{n+2m}\end{pmatrix}$.

Thus $(u_{1},b)\notin\mathfrak{L}_{\max}$ and hence $L(v)^{*}\notin{\rm
U}_{2m}(R,\mathfrak{L}_{\max})$ in general. In the special case in which the
involution satisfies $(\bar{1}^{-1})^{2}=1$, we have both $L(v)$ and
$L(v)^{*}$ belongs to ${\rm U}_{2m}(R,\mathfrak{L}_{\max})$.

Next, we shall consider the group generated by these two type of matrices and
we have the following theorem.

######  Theorem 4.4.

Let $R$ be a commutative ring with unity. The group generated by $L(v)$ and
$L(v)^{*}$ for $v\in R^{2m+n-1}$ is congruent to the Petrov’s odd elementary
hyperbolic unitary group ${\rm EU}_{2m}(R,\mathfrak{L}_{\max})$.

Let $P$ be the matrix $P=\begin{pmatrix}0_{2m\times n}&I_{2m}\\\
I_{n}&0_{n\times 2m}\end{pmatrix}$. Then we have

$\displaystyle P^{t}L(v)P$
$\displaystyle={\scriptsize\begin{pmatrix}1&\ldots&0&a_{2m}&0&0&0&\ldots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&\ldots&1&a_{n+2m-1}&0&0&0&\ldots&0&0\\\ 0&\ldots&0&1&0&0&0&\ldots&0&0\\\
-\bar{1}^{-1}(\overline{a}_{2m}\varphi_{11}&\ldots&-\bar{1}^{-1}(\overline{a}_{2m}\varphi_{1n}&a_{1}&1&\overline{a}_{3}&-\bar{1}^{-1}\overline{a}_{2}&\ldots&\overline{a}_{2m-1}&-\bar{1}^{-1}\overline{a}_{2m-2}\\\
+\ldots+\overline{a}_{n+2m-1}\varphi_{n1})&&+\ldots+\overline{a}_{n+2m-1}\varphi_{nn})&&&&&&&\\\
0&\ldots&0&a_{2}&0&1&0&\ldots&0&0\\\ 0&\ldots&0&a_{3}&0&0&1&\ldots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&\ldots&0&a_{2m-2}&0&0&0&\ldots&0&0\\\
0&\ldots&0&a_{2m-1}&0&0&0&\ldots&0&1\end{pmatrix}}$
$\displaystyle=T_{-1}(-a_{2m}v_{1}-a_{2m+1}v_{2}-\ldots-
a_{n+2m-1}v_{n}-a_{2}e_{2}-a_{3}e_{-2}-\ldots-
a_{2m-2}e_{m}-a_{2m-1}e_{-m},a_{1}),$

and

$\displaystyle P^{t}L(v)^{*}P$
$\displaystyle={\scriptsize\begin{pmatrix}1&\ldots&0&0&-\bar{1}^{-1}(\overline{a}_{2m}(\varphi^{-1})_{11}+\overline{a}_{2m+1}(\varphi^{-1})_{12}&0&0&\ldots&0&0\\\
&&&&+\ldots+\overline{a}_{2m-1+n}(\varphi^{-1})_{1n})&&&&&\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&\ldots&1&0&-\bar{1}^{-1}(\overline{a}_{2m}(\varphi^{-1})_{n1}+\overline{a}_{2m+1}(\varphi^{-1})_{n2}&0&0&\ldots&0&0\\\
&&&&+\ldots+\overline{a}_{2m-1+n}(\varphi^{-1})_{nn}&&&&&\\\
a_{2m}&\ldots&a_{n+2m-1}&1&a_{1}&a_{2}&a_{3}&\ldots&a_{2m-2}&a_{2m-1}\\\
0&\ldots&0&0&1&0&0&\ldots&0&0\\\
0&\ldots&0&0&(\bar{1}^{-1})^{2}\overline{a}_{3}&1&0&\ldots&0&0\\\
0&\ldots&0&0&-\bar{1}^{-1}\overline{a}_{2}&0&1&\ldots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&\ldots&0&0&(\bar{1}^{-1})^{2}\overline{a}_{2m-1}&0&0&\ldots&0&0\\\
0&\ldots&0&0&-\bar{1}^{-1}\overline{a}_{2m-2}&0&0&\ldots&0&1\end{pmatrix}}$
$\displaystyle=T_{1}(\bar{1}^{-1}(\overline{a}_{2m}(\varphi^{-1})_{11}+\overline{a}_{2m+1}(\varphi^{-1})_{12}+\ldots+\overline{a}_{2m-1+n}(\varphi^{-1})_{1n})v_{1}$
$\displaystyle\hskip
56.9055pt+\ldots+\bar{1}^{-1}(\overline{a}_{2m}(\varphi^{-1})_{n1}+\overline{a}_{2m+1}(\varphi^{-1})_{n2}+\ldots+\overline{a}_{2m-1+n}(\varphi^{-1})_{nn})v_{n}$
$\displaystyle\hskip
85.35826pt-(\bar{1}^{-1})^{2}\overline{a}_{3}e_{2}+\bar{1}^{-1}\overline{a}_{2}e_{-2}-\ldots-(\bar{1}^{-1})^{2}\overline{a}_{2m-1}e_{m}+\bar{1}^{-1}\overline{a}_{2m-2}e_{-m}),-\bar{1}a_{1}).$

Also by direct computations, we can verify that $(u_{1},a_{1})$ and
$(u_{2},-\bar{1}a_{1})$ belongs to $\mathfrak{L}_{\max}$, where

$u_{1}=-a_{2m}v_{1}-a_{2m+1}v_{2}-\ldots-
a_{n+2m-1}v_{n}-a_{2}e_{2}-a_{3}e_{-2}-\ldots-
a_{2m-2}e_{m}-a_{2m-1}e_{-m}\mbox{ and }$ $\displaystyle u_{2}$
$\displaystyle=\bar{1}^{-1}(\overline{a}_{2m}(\varphi^{-1})_{11}+\overline{a}_{2m+1}(\varphi^{-1})_{12}+\ldots+\overline{a}_{2m-1+n}(\varphi^{-1})_{1n})v_{1}$
$\displaystyle\hskip
56.9055pt+\ldots+\bar{1}^{-1}(\overline{a}_{2m}(\varphi^{-1})_{n1}+\overline{a}_{2m+1}(\varphi^{-1})_{n2}+\ldots+\overline{a}_{2m-1+n}(\varphi^{-1})_{nn})v_{n}$
$\displaystyle\hskip
85.35826pt-(\bar{1}^{-1})^{2}\overline{a}_{3}e_{2}+\bar{1}^{-1}\overline{a}_{2}e_{-2}-\ldots-(\bar{1}^{-1})^{2}\overline{a}_{2m-1}e_{m}+\bar{1}^{-1}\overline{a}_{2m-2}e_{-m}).$

Thus $P^{t}L(v)P$ and $P^{t}L(v)^{*}P$ belongs to ${\rm
EU}_{2m}(R,\mathfrak{L}_{\max})$.

Now to prove the reverse inclusion take an arbitrary element in ${\rm
EU}_{2m}(R,\mathfrak{L}_{\max})$. By Proposition 3.2, the elements of ${\rm
EU}_{2m}(R,\mathfrak{L}_{\max})$ are generated by elements of the form $T_{\pm
i}(v,a)$ where $(v,a)\in\mathfrak{L}_{\max}$ and $\langle
e_{1},v\rangle=\langle e_{-1},v\rangle=0$. Therefore $v$ has the form

$v=t_{1}v_{1}+\ldots+t_{n}v_{n}+b_{2}e_{2}+b_{-2}e_{-2}+\ldots+b_{m}e_{m}+b_{-m}e_{-m}.$

Then $T_{1}(v,a)=P^{t}R(u)P$, where

$\displaystyle u$
$\displaystyle=(-\bar{1}^{-1}a,\bar{1}^{-1}\overline{b}_{-2},-(\bar{1}^{-1})^{2}\overline{b}_{2},\ldots,\bar{1}^{-1}\overline{b}_{-m},-(\bar{1}^{-1})^{2}\overline{b}_{m},(\bar{1}^{-1})^{2}(\overline{t}_{1}\varphi_{11}+\ldots+\overline{t}_{n}\varphi_{1n}),$
$\displaystyle\hskip
170.71652pt\ldots,(\bar{1}^{-1})^{2}(\overline{t}_{1}\varphi_{n1}+\ldots+\overline{t}_{n}\varphi_{nn}))$

and $T_{-1}(v,a)=P^{t}L(u)P$, where
$u=(a,-b_{2},-b_{-2},\ldots,-b_{m},-b_{-m},-t_{1},\ldots,-t_{n}).$

Therefore the group generated by $L(v)$ and $L(v)^{*}$, for $v\in R^{2m+n-1}$
is congruent to the Petrov’s odd elementary hyperbolic unitary group ${\rm
EU}_{2m}(R,\mathfrak{L}_{\max})$.

######  Remark 4.5.

The matrix $P$ we got in Theorem 1.1 is invertible and $P^{-1}=P^{t}$. Thus
the two groups considered here are conjugate subgroups of $\rm{GL}_{n+2m}(R).$

######  Example 4.6.

Here we illustrate Theorem 1.1 in which in the special case where $R$ is a
commutative ring and the involution is $a\mapsto\bar{a}=-a$. In this case, the
anti-Hermitian form becomes Hermitian. The form matrix for Petrov’s group in
this case is $\varPsi=\widetilde{\psi}_{m}\perp\varphi$, where $\varphi$ is
the form matrix for $V_{0}$ and $\widetilde{\psi}_{r}$ for $r>1$ is defined as
follows.

Let $\widetilde{\psi_{1}}=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$ and
$\widetilde{\psi}_{r}=\widetilde{\psi}_{1}\perp\widetilde{\psi}_{r-1}$ for
$r>1$. The form matrix $\varPsi$ is of the form $\begin{pmatrix}0&c\\\
c^{t}&\mu\end{pmatrix}$, where $c=\begin{pmatrix}1&\ldots&0\end{pmatrix}\in
R^{n+2m-1}$ and $\mu$ is the $(n+2m)\times(n+2m)$ matrix given by
$\mu=\begin{pmatrix}0&0&0\\\ 0&\widetilde{\psi}_{m-2}&0\\\
0&0&\varphi\end{pmatrix}$, and $\varPsi^{-1}$ is of the form
$\begin{pmatrix}0&d\\\ d^{t}&\rho\end{pmatrix}$, where
$d=\begin{pmatrix}1&\ldots&0\end{pmatrix}\in R^{n+2m-1}$ and
$\rho=\begin{pmatrix}0&0&0\\\ 0&\widetilde{\psi}_{m-2}&0\\\
0&0&\varphi^{-1}\end{pmatrix}$.

Thus for $v=(a_{1},\ldots,a_{n+2m-1})\in R^{n+2m-1}$, we get the matrices
$\alpha$ and $\beta$ as follows:

$\alpha=I_{n+2m-1}-d^{t}v\mu\mbox{ and }\beta=I_{n+2m-1}-\rho v^{t}c.$

The matrices $\alpha$ and $\beta$ are of the following form.

$\alpha={\scriptsize{\begin{pmatrix}1&-a_{3}&-a_{2}&\ldots&-a_{2m-1}&-a_{2m-2}&-a_{2m}\varphi_{11}-a_{2m+1}\varphi_{21}&\ldots&-a_{2m}\varphi_{1n}-a_{2m+1}\varphi_{2n}\\\
&&&&&&\hskip 14.22636pt-\ldots-a_{2m-1+n}\varphi_{n1}&&\hskip
14.22636pt-\ldots-a_{2m-1+n}\varphi_{nn}\\\ 0&1&0&\ldots&0&0&0&\ldots&0\\\
0&0&1&\ldots&0&0&0&\ldots&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&0&0&\ldots&0&0&0&\ldots&1\end{pmatrix}}},$

$\mbox{ and }\qquad\beta={\scriptsize{\begin{pmatrix}1&0&0&\ldots&0\\\
-a_{3}&1&0&\ldots&0\\\ -a_{2}&0&1&\ldots&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\ -a_{2m-1}&0&0&\ldots&0\\\
-a_{2m-2}&0&0&\ldots&0\\\
a_{2m}(\varphi^{-1})_{11}+a_{2m+1}(\varphi^{-1})_{12}+\ldots+a_{2m-1+n}(\varphi^{-1})_{1n}&0&0&\ldots&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\
a_{2m}(\varphi^{-1})_{n1}+a_{2m+1}(\varphi^{-1})_{n2}+\ldots+a_{2m-1+n}(\varphi^{-1})_{nn}&0&0&\ldots&1\end{pmatrix}}}.$

Therefore, we have the matrices $L(v)$ and $L(v)^{*}$ of the form

$L(v)={\scriptsize{\begin{pmatrix}1&0&0&0&\ldots&0&0&0&\ldots&0\\\
a_{1}&1&-a_{3}&-a_{2}&\ldots&-a_{2m-1}&-a_{2m-2}&-a_{2m}\varphi_{11}-a_{2m+1}\varphi_{21}&\ldots&-a_{2m}\varphi_{1n}-a_{2m+1}\varphi_{2n}\\\
&&&&&&&-\ldots-a_{2m-1+n}\varphi_{n1}&&-\ldots-a_{2m-1+n}\varphi_{nn}\\\
a_{2}&0&1&0&\ldots&0&0&0&\ldots&0\\\ a_{3}&0&0&1&\ldots&0&0&0&\ldots&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
a_{n+2m-1}&0&0&0&\ldots&0&0&0&\ldots&1\end{pmatrix}}}$

$\mbox{ and
}L(v)^{*}={\scriptsize{\begin{pmatrix}1&a_{1}&a_{2}&\ldots&a_{n+2m-1}\\\
0&1&0&\ldots&0\\\ 0&-a_{3}&1&\ldots&0\\\ 0&-a_{2}&0&\ldots&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\ 0&-a_{2m-1}&0&\ldots&0\\\
0&-a_{2m-2}&0&\ldots&0\\\
0&a_{2m}(\varphi^{-1})_{11}+a_{2m+1}(\varphi^{-1})_{12}+\ldots+a_{2m-1+n}(\varphi^{-1})_{1n}&0&\ldots&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&a_{2m}(\varphi^{-1})_{n1}+a_{2m+1}(\varphi^{-1})_{n2}+\ldots+a_{2m-1+n}(\varphi^{-1})_{nn}&0&\ldots&1\end{pmatrix}}}.$

By Remark 4.3, both the matrices $L(v)$ and $L(v)^{*}$ in this example belongs
to ${\rm U}_{2m}(R,\mathfrak{L}_{\max})$.

Now let $P$ be the matrix $P=\begin{pmatrix}0_{2m\times n}&I_{2m}\\\
I_{n}&0_{n\times 2m}\end{pmatrix}$. We have

$\displaystyle P^{t}L(v)P$
$\displaystyle={\scriptsize\begin{pmatrix}1&\ldots&0&a_{2m}&0&0&0&\ldots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&\ldots&1&a_{n+2m-1}&0&0&0&\ldots&0&0\\\ 0&\ldots&0&1&0&0&0&\ldots&0&0\\\
a_{2m}\varphi_{11}+\ldots&\ldots&a_{2m}\varphi_{1n}+\ldots&a_{1}&1&-a_{3}&-a_{2}&\ldots&-a_{2m-1}&-a_{2m-2}\\\
+a_{n+2m-1}\varphi_{n1}&&+a_{n+2m-1}\varphi_{nn}&&&&&&&\\\
0&\ldots&0&a_{2}&0&1&0&\ldots&0&0\\\ 0&\ldots&0&a_{3}&0&0&1&\ldots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&\ldots&0&a_{n+2m-2}&0&0&0&\ldots&0&0\\\
0&\ldots&0&a_{n+2m-1}&0&0&0&\ldots&0&1\end{pmatrix}}$
$\displaystyle=T_{-1}(-a_{2m}v_{1}-\ldots-
a_{n+2m-1}v_{n}-a_{2}e_{2}-a_{3}e_{-2}-\ldots-
a_{2m-2}e_{m}-a_{2m-1}e_{-m},a_{1}),\mbox{ and }$

$\displaystyle P^{t}L(v)^{*}P$
$\displaystyle={\scriptsize\begin{pmatrix}1&\ldots&0&0&-(a_{2m}(\varphi^{-1})_{11}+\ldots
a_{n+2m-1}(\varphi^{-1})_{n1})&0&0&\ldots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&\ldots&1&0&-(a_{2m}(\varphi^{-1})_{1n}+\ldots-
a_{n+2m-1}(\varphi^{-1})_{nn})&0&0&\ldots&0&0\\\
a_{2m}&\ldots&a_{n+2m-1}&1&a_{1}&a_{2}&a_{3}&\ldots&a_{2m-2}&a_{2m-1}\\\
0&\ldots&0&0&1&0&0&\ldots&0&0\\\ 0&\ldots&0&0&-a_{3}&1&0&\ldots&0&0\\\
0&\ldots&0&0&-a_{2}&0&1&\ldots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&\ldots&0&0&-a_{2m-1}&0&0&\ldots&0&0\\\
0&\ldots&0&0&-a_{2m-2}&0&0&\ldots&0&1\end{pmatrix}}$
$\displaystyle=T_{1}((a_{2m}(\varphi^{-1})_{11}+\ldots
a_{n+2m-1}(\varphi^{-1})_{n1})v_{1}+\ldots+(a_{2m}(\varphi^{-1})_{1n}$
$\displaystyle\hskip
22.76219pt+\ldots+a_{n+2m-1}(\varphi^{-1})_{nn})v_{n}+a_{3}e_{2}+a_{2}e_{-2}+\ldots+a_{2m-1}e_{m}+a_{2m-2}e_{-m},a_{1}).$

Also, $(u_{1},a_{1})$ and $(u_{2},a_{1})$ belongs to $\mathfrak{L}_{\max}$,
where

$u_{1}=-a_{2m}v_{1}-\ldots-a_{n+2m-1}v_{n}-a_{2}e_{2}-a_{3}e_{-2}-\ldots-
a_{2m-2}e_{m}-a_{2m-1}e_{-m}~{}\mbox{ and }$ $\displaystyle u_{2}$
$\displaystyle=a_{2m}(\varphi^{-1})_{11}+\ldots+a_{n+2m-1}(\varphi^{-1})_{n1})v_{1}+\ldots+(a_{2m}(\varphi^{-1})_{1n}+\ldots+a_{n+2m-1}(\varphi^{-1})_{nn})v_{n}$
$\displaystyle\hskip
56.9055pt+a_{3}e_{2}+a_{2}e_{-2}+\ldots+a_{2m-1}e_{m}+a_{2m-2}e_{-m}.$

Thus $P^{t}L(v)P$ and $P^{t}L(v)^{*}P$ belongs to ${\rm
EU}_{2m}(R,\mathfrak{L}_{\max})$.

Now for the reverse inclusion take an arbitrary element in ${\rm
EU}_{2m}(R,\mathfrak{L}_{\max})$. By Proposition 3.2, the elements of ${\rm
EU}_{2m}(R,\mathfrak{L}_{\max})$ are generated by elements of the form $T_{\pm
i}(v,a)$ where $(v,a)\in\mathfrak{L}_{\max}$ and $\langle
e_{1},v\rangle=\langle e_{-1},v\rangle=0$. Therefore $v$ has the form
$v=t_{1}v_{1}+\ldots+t_{n}v_{n}+b_{2}e_{2}+b_{-2}e_{-2}+\ldots+b_{m}e_{m}+b_{-m}e_{-m}$.
Then we have

$\displaystyle T_{1}(v,a)$
$\displaystyle=P^{t}L((a,b_{-2},b_{2},\ldots,b_{-m},b_{m},t_{1},\ldots,t_{n}))P,~{}\mbox{
and }$ $\displaystyle\vspace{2mm}T_{-1}(v,a)$
$\displaystyle=P^{t}R((a,b_{2},b_{-2},\ldots,b_{m},b_{-m},\varphi_{11}t_{1}+\ldots+\varphi_{n1}t_{n},\ldots,\varphi_{1n}t_{1}+\ldots+\varphi_{nn}t_{n}))P.$

Therefore the group generated by $L(v)$ and $L(v)^{*}$ for $v\in R^{2m+n-1}$
is congruent to the Petrov’s group ${\rm EU}_{2m}(R,\mathfrak{L}_{\max})$.

## Acknowledgements

The first author would like to acknowledge the support by “Seed Money for New
Research Initiatives [2017-18]”, Cochin University of Science and Technology.
The second author thank Department of Science and Technology (DST) for the
INSPIRE fellowship which supported this work.

## References

  * [1] L. N. Vaseršteĭn and A. A. Suslin, Serre’s problem on projective modules over polynomial rings and algebraic $K$-theory. Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), 993–1054, 1199.
  * [2] P. Chattopadhyay and R. A. Rao. Equality of elementary linear and symplectic orbits with respect to an alternating form. J. Algebra. 451 (2016), 46–64.
  * [3] V. A. Petrov. Odd unitary groups. J. Math. Sci. 130(3) (2005), 4752-4766.
  * [4] Abe E.. Coverings of twisted Chevalley groups over commutative rings Sci. Rep. Tokyo Kyoiku Daigaku Sect. A. 366-382(13) (1977), 194–218.
  * [5] A. Bak. The stable structure of quadratic modules. PhD thesis. New York (NY), USA: Columbia University (1969).
  * [6] G. Tang, Hermitian groups and $K$-theory. $K$-Theory. 13 (1998), 209–267.
  * [7] A. Bak and R. Preusser. The E-normal structure of odd dimensional unitary groups. J. Pure Appl. Algebra.222(9) (2018), 2823–2880.
  * [8] W. Yu, Y. Li and H. Liu. A classification of subgroups of odd unitary groups. Comm. Algebra. 46 (2018), 3795–3805.
  * [9] R. Preusser. The E-normal structure of Petrov’s odd unitary groups over commutative rings. Comm. Algebra. 48 (2020), 1114–1131.
  * [10] A. A. Ambily and R. A. Rao. Visualizing the DSER elementary orthogonal group. Preprint (2021).