# Lattice control of non-ergodicity in a polar lattice gas

H. Korbmacher  Institut für Theoretische Physik, Leibniz Universität Hannover,
Germany  P. Sierant  ICFO-Institut de Ciències Fotòniques, The Barcelona
Institute of Science and Technology, Av. Carl Friedrich Gauss 3, 08860
Castelldefels (Barcelona), Spain  W. Li  Institut für Theoretische Physik,
Leibniz Universität Hannover, Germany  X. Deng  Leibniz-Rechenzentrum, D-85748
Garching b. München, Germany  J. Zakrzewski  Instytut Fizyki Teoretycznej,
Uniwersytet Jagielloński, Łojasiewicza 11, PL-30-348 Kraków, Poland  Mark Kac
Complex Systems Research Center, Uniwersytet Jagielloński, PL-30-348 Kraków,
Poland  L. Santos  Institut für Theoretische Physik, Leibniz Universität
Hannover, Germany

###### Abstract

Inter-site interactions in polar lattice gases may result, due to Hilbert-
space fragmentation, in a lack of ergodicity even in absence of disorder. We
show that the inter-site interaction in a one-dimensional dipolar gas in an
optical lattice departs from the usually considered $1/r^{3}$ dependence,
acquiring a universal form that depends on the transversal confinement and the
lattice depth. Due to the crucial role played by the nearest- and next-to-
nearest neighbors, the Hilbert-space fragmentation and particle dynamics are
very similar to that of a power-law model $1/r^{\beta_{\mathrm{eff}}<3}$,
where $\beta_{\mathrm{eff}}$ is experimentally controllable by properly
tailoring the transversal confinement. Our results are of direct experimental
relevance for experiments on dipolar gases in optical lattices, and show that
the particle dynamics may be remarkably different if the quasi-1D lattice
model is realized in a strong 3D lattice, or by means of a strong transversal
harmonic confinement.

##### Introduction.–

Ultra-cold atoms and molecules in optical lattices constitute an exceptional
system for studying many-body quantum systems in general Bloch  _et al._
(2008), and out-of-equilibrium dynamics in particular Langen  _et al._ (2015).
The impressive degree of experimental control of state preparation, imaging,
lattice geometry and strength, disorder, and interactions, allows for a
relatively easy modification of parameters in real time compared to other
condensed-matter systems. This has led in recent years to major breakthroughs,
including seminal experiments on many-body localization Schreiber  _et al._
(2015); Rispoli  _et al._ (2019); Lukin  _et al._ (2019).

Most of the present experiments on optical lattices are performed with
particles that interact via contact-like potentials. In the tight-binding
regime, those experiments simulate different forms of the Hubbard model, with
only on-site interactions Jaksch  _et al._ (1998); Greiner  _et al._ (2002).
Weak nearest-neighbor (NN) interactions may result from super-exchange
processes, crucial for long-range antiferromagnetism Mazurenko  _et al._
(2017). Interestingly, recent experiments are starting to explore the
fascinating physics of polar lattice gases, in which particles present
significant or even dominant dipole-dipole interactions. This includes, in
particular, magnetic atoms de Paz  _et al._ (2013); Baier  _et al._ (2016);
Patscheider  _et al._ (2020), polar molecules Yan  _et al._ (2013), and
Rydberg gases Guardado-Sanchez  _et al._ (2021); Scholl  _et al._ (2021).
Polar lattice gases are characterized by relatively strong inter-site
interactions, and hence allow for the realization of different spin models and
extended Hubbard models (EHMs) Dutta  _et al._ (2015). Spin models have been
realized in magnetic atoms de Paz  _et al._ (2013); Patscheider  _et al._
(2020), polar molecules Yan  _et al._ (2013), and Rydberg atoms Scholl  _et
al._ (2021). Seminal EHM experiments have been realized using magnetic Baier
_et al._ (2016) and Rydberg atoms Guardado-Sanchez  _et al._ (2021). An
equivalent realization with polar molecules remains elusive, due to inelastic
collisions, although recent experiments on shielding Valtolina  _et al._
(2020); Anderegg  _et al._ (2021); Schindewolf  _et al._ (2022) open
optimistic perspectives.

Inter-site interactions result in novel physics in polar lattice gases
compared to their non-dipolar counterparts, including new ground-states, as
crystals and lattice supersolids Baranov  _et al._ (2012), and the Haldane
insulator Dalla Torre  _et al._ (2006). The out-of-equilibrium dynamics is
also qualitatively different. In particular, the combination of energy
conservation, finite band-width, and dipolar interactions is expected to
result in Hilbert-space shattering and disorder-free localization for strong-
enough dipolar interactions Li  _et al._ (2021).

Polar lattice gases present interactions not only to the NNs, but potentially
well beyond. Due to the form of the dipolar interactions, this dipolar tail is
typically assumed to decay as $1/r^{3}$, with $r$ the inter-site distance.
However, this assumption must be carefully reconsidered Wall and Carr (2013),
since inter-site interactions are affected by the geometry of the on-site
Wannier functions. This is particularly relevant in low-dimensional models.
Low-dimensionality may be achieved by using a three-dimensional (3D) lattice,
which is much stronger in one (or two) directions. This prevents transverse
hopping, but also compresses the on-site wavefunction, resulting in a modified
dipolar tail. However, this modification remains relatively small Wall and
Carr (2013).

In this Letter, we show that the deviation from the $1/r^{3}$ dependence is
much more relevant if low-dimensionality is achieved by harmonically confining
the transversal directions. We find that the tail acquires a universal
analytic dependence on the transversal confinement, which is approximately
(but not identically) given by a $1/r^{3}$ dependence in a 3D lattice, but it
departs under general conditions very strongly from that dependence if the
transversal confinement is harmonic. The modified tail has very significant
consequences for both the Hilbert-space fragmentation and the dynamics of one-
dimensional (1D) polar lattice gases. In particular, we show that due to the
dominant role played by the NN and next-to-NN interactions, the particle
dynamics is very similar to that of a model with power-law interactions
$1/r^{\beta_{\mathrm{eff}}<3}$, where the transversal confinement controls the
exponent $\beta_{\mathrm{eff}}$, and with it the critical dipolar strength
that marks the threshold for particle localization.

##### Extended Hubbard model.–

We consider dipolar bosons of mass $m$ in a 1D optical lattice,
$V_{0}\sin^{2}(\pi z/\lambda)$, transversally confined by an isotropic
harmonic potential $\frac{1}{2}m\omega_{\perp}^{2}(x^{2}+y^{2})$. The dipole
moments are assumed to be oriented by an external field on the $xz$ plane
forming an angle $\alpha$ with the lattice axis $z$. For a sufficiently deep
lattice, the system is well described by the EHM:

$\hat{H}=-t\sum_{i}\left(\hat{b}_{i}^{\dagger}\hat{b}_{i+1}+\mathrm{H.c.}\right)+\sum_{i}\sum_{j>0}V_{j}\hat{n}_{i}\hat{n}_{i+j},$
(1)

where $\hat{b}_{i}$ ($\hat{b}_{i}^{\dagger}$) is the annihilation (creation)
operator at site $i$, $\hat{n}_{i}=\hat{b}_{i}^{\dagger}\hat{b}_{i}$, and we
impose the hard-core constraint $(\hat{b}_{i}^{\dagger})^{2}=0$, assuming
sufficiently strong on-site interactions. This implies negligible
collisionally-assisted hops, which may be relevant in the soft-core regime
Maik  _et al._ (2013); Kraus  _et al._ (2020).

The inter-site interaction for dipoles $j$ sites apart is characterized by the
coupling constant exc :

$V_{j}=\int d^{3}r\int
d^{3}r^{\prime}V(\vec{r}-\vec{r}\,^{\prime})|\varphi(\vec{r})|^{2}|\varphi(\vec{r}\,^{\prime}-j\lambda\vec{e}_{z})|^{2}$
(2)

where $V(\vec{r})=\frac{C_{dd}}{4\pi
r^{3}}\left(1-3\frac{\left(x\sin\alpha+z\cos\alpha\right)^{2}}{r^{2}}\right)$
is the dipole-dipole interaction. For magnetic dipoles,
$C_{dd}=\mu_{0}\mu^{2}$, with $\mu_{0}$ the vacuum permeability and $\mu$ the
magnetic moment. For electric dipoles, $C_{dd}=\frac{d^{2}}{\epsilon_{0}}$,
with $\epsilon_{0}$ the vacuum dielectric constant, and $d$ the electric
dipole moment. We characterize below the dipole strength by the dipolar
length, $a_{dd}=\frac{mC_{dd}}{12\pi\hbar^{2}}$. The on-site wave function,
$\varphi(\vec{r})=\phi_{0}(x,y)W(z)$, is given by the Wannier function $W(z)$
associated to the lowest-energy band, and by the ground-state of the
transversal trap,
$\phi_{0}(x,y)=\frac{e^{-(x^{2}+y^{2})/2l_{\perp}^{2}}}{\sqrt{\pi}l_{\perp}}$,
with $l_{\perp}^{2}=\hbar/m\omega_{\perp}$ (we assume that
$\hbar\omega_{\perp}$ is much larger than other energies involved in the EHM).

Figure 1: Power $\beta_{\mathrm{eff}}$ for $s=30$ and different $\chi$. The
dots indicate the results obtained employing Eq. (2) with the exact Wannier
functions. The solid line depicts $-\log(G_{2})/\log(2)$. The inset shows the
results for a wider range of $\chi$ values. In both graphs the dashed line
indicates $\beta_{\mathrm{eff}}=3$.

##### The dipolar tail.–

For deep-enough lattices, we may approximate
$W(z)\simeq\frac{e^{-z^{2}/2l^{2}}}{\sqrt{\sqrt{\pi}l}}$, with
$l=\frac{\lambda}{\pi s^{1/4}}$, where $s=\frac{V_{0}}{E_{\mathrm{R}}}$ and
$E_{\mathrm{R}}=\frac{\pi^{2}\hbar^{2}}{2m\lambda^{2}}$ is the recoil energy.
We then obtain for $l_{\perp}>l$ SM :

$\frac{V_{j}}{E_{R}}=\frac{3B^{3/2}}{2\pi^{2}}\left(3\cos^{2}\alpha-1\right)\left(\frac{a_{dd}}{\lambda}\right)f(\sqrt{B}j),$
(3)

where $B=\frac{\pi^{2}}{2}\frac{\chi}{1-\frac{\chi}{2\sqrt{s}}}$,
$\chi=\frac{\hbar\omega_{\perp}}{E_{\mathrm{R}}}$, and

$f(\xi)=2\xi-\sqrt{2\pi}(1+\xi^{2})e^{\xi^{2}/2}\mathrm{erfc}\left(\xi/\sqrt{2}\right).$
(4)

Denoting $V\equiv V_{1}$, we can write $V_{j}=VG_{j}(B)$, with
$G_{j}(B)=f(\sqrt{B}j)/f(\sqrt{B})$, and hence the decay of the interaction
tail has a universal dependence on $B$.

This tail must be compared to the $1/j^{3}$ decay typically assumed in polar
lattice gases. Such a decay is only exact for isotropic on-site functions,
$l=l_{\perp}$ ($\chi=2\sqrt{s}$) and large $s$. It is also recovered for
large-enough distances, $G_{j\to\infty}(B)\to 1/j^{3}$. With a $1/j^{3}$
dependence, the physics of hard-core bosons only depends on $V/t$. In
contrast, actual hard-core lattice dipoles have a universal dependence on both
$V/t$ and $B$, that characterize, respectively, the dipole strength and the
dipolar tail.

##### Next-to-nearest neighbor interactions.–

As shown below, the ratio $G_{2}$ between next-to-NN and NN interactions is
crucial for the Hilbert-space fragmentation and dynamics in a polar lattice
gas. Hence, we introduce at this point the effective power
$\beta_{\mathrm{eff}}(B)$, see Fig. 1, such that
$G_{2}(B)=1/2^{\beta_{\mathrm{eff}}(B)}$. Whereas for $l>l_{\perp}$
($\chi>2\sqrt{s}$), $\beta_{\mathrm{eff}}(B)>3$, approaching
$\beta_{\infty}\simeq 3+\frac{1}{\mathrm{ln}(2)}\frac{9}{2\pi^{2}\sqrt{s}}$ SM
for large $\chi$ and $s$, for $l<l_{\perp}$ ($\chi<2\sqrt{s}$),
$\beta_{\mathrm{eff}}(B)$ may become significantly smaller than $3$. The
markedly larger $G_{2}$ results in a strongly modified dynamics, as shown
below.

##### Hilbert-space fragmentation.–

For the model with just NN interactions (NN model), $V_{j}=V\delta_{j,1}$,
increasing $V/t$ results in an emerging dynamical constraint, given by the
conservation of the number of NN bonds $N_{\mathrm{NN}}=\sum_{j}\langle
n_{j}n_{j+1}\rangle$. This constraint leads to Hilbert-space fragmentation
into dynamically unconnected blocks De Tomasi  _et al._ (2019). The presence
of a $1/j^{3}$ tail results for large-enough $V/t$ in a strong fragmentation
(shattering) of the NN blocks due to the emerging conservation of the number
of next-to-NN bonds, $N_{\mathrm{NNN}}=\sum_{j}\langle n_{j}n_{j+2}\rangle$ Li
_et al._ (2021). As shown below, the controllable modification of $V_{2}/V$
significantly affects this shattering, and with it the particle dynamics.

Figure 2: Shattering of the NN blocks. (top) Average $D$ for $V/j^{\beta}$
interactions, as a function of $V/t$ and $\beta$. (bottom) Same for a polar
lattice gas, as a function of $V/t$ and $\beta_{eff}$ (right axis) and $\chi$
(for $s=30$) (left axis). The color plot is obtained for $N=8$ particles in
$L=16$ sites with open boundary conditions. The inset shows $D$ as a function
of $V/t$ for a dipolar gas with $B=11.5$ ($\beta_{\mathrm{eff}}\simeq 2.6$)
for $N=7$ and $L=14$ and $N=8$ and $L=16$. The crossing point provides an
estimation of the shattering transition. The white curves in both panels show
the crossing points of the $D$ curves for $N=7$ and $L=14$ and $N=8$ and
$L=16$.

In order to study Hilbert-space fragmentation, we employ exact diagonalization
to obtain the eigenstates $|\alpha\rangle$ of $N$ particles in $L$ sites with
open boundary conditions. We then express the Fock states
$|f\rangle=\prod_{l=1}^{L}|n_{l}(f)\rangle$ with population $n_{l}(f)=0,1$ in
site $l$, in the basis of eigenstates,
$|f\rangle=\sum_{\alpha}\psi_{f}(\alpha)|\alpha\rangle$. For the NN model, a
sufficiently large $V/t>10$ results in Hilbert space fragmentation into
unconnected blocks (NN blocks) with a size much smaller than the overall
Hilbert space dimension. Further fragmentation of the NN blocks due to beyond-
NN interactions is characterized for each Fock state $|f\rangle$ by the
fractal dimension, $D_{f}=-\ln(\mathrm{IPR}_{f})/\ln(\Lambda_{f})$ Macé  _et
al._ (2019), where $\Lambda_{f}$ is the size of the NN block to which
$|f\rangle$ belongs, and
$\mathrm{IPR}_{f}=\sum_{\alpha}|\psi_{\alpha}(f)|^{4}$ is the inverse
participation ratio. $D_{f}$ approaches zero when the NN block shatters. The
average, $D$, of $D_{f}$ over the whole Fock basis provides a good
quantitative estimation of the shattering.

Figure 2 shows $D$, as a function of $V/t$, for (a) $V_{j}=1/j^{\beta}$, and
(b) $V_{j}=VG_{j}(B)$, characterized by $\beta_{\mathrm{eff}}(B)$. The
comparison of both graphs shows that, due to the dominant role played by the
next-to-NN interactions, Hilbert-space shattering in an actual dipolar gas may
closely mimic that of a system with modified power-law interactions
$1/j^{\beta=\beta_{\mathrm{eff}}(B)}$. Note also the potentially very large
deviation from the results expected for a $1/j^{3}$ tail.

Figure 3: Homogenization of an initial density wave. Average density at
different sites as a function of time for (a) $V_{j}=V/j^{3}$ and (b)
$V_{j}=G_{j}(B)$, for $B=2.54$ ($\beta_{\mathrm{eff}}\simeq 2$). In both cases
$V/t=16$. Results obtained from Time-Dependent Variational Principle (TDVP)
calculations for $L=60$ and $N=30$ SM . Panels (c) and (d) show the
inhomogeneity parameter averaged in the time interval $190/t<\tau<210/t$,
evaluated for a system with $L=24$, $N=12$ and an initial density wave as in
(a) and (b). Panel (c) shows the results for a model $V/j^{\beta}$ as a
function of $\beta$ and $V/t$, which should be compared with (d) where the
results for a polar gas and different values of $\beta_{\mathrm{eff}}$ and
$V/t$ are presented. The results were obtained using Chebyshev time
propagation SM .

##### Dynamics of an initial density wave.–

The deviation from the $1/j^{3}$ decay has relevant consequences for the
dynamics of particles, well illustrated by the relatively simple case of an
initial density wave:

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with open boundary conditions, which may be prepared using a superlattice
(similar initial conditions have been recently studied in Rydberg gases
Guardado-Sanchez  _et al._ (2021)). Figure 3 compares the dynamics for a
$1/j^{3}$ decay, and the actual evolution for $B=2.54$
($\beta_{\mathrm{eff}}\simeq 2$) for $V/t=16$. Although in both cases we
observe delocalization at long times, the homogenization is approximately four
times slower in the actual polar lattice gas. Density homogenization is well
characterized by the inhomogeneity parameter,

$\eta=\frac{1}{2L\rho(1-\rho)}\sum_{j=1}^{L}\left|\langle\hat{n}_{j}\rangle-\rho\right|,$
(5)

with $\rho=N/L$. Note that $\eta$ ranges from 1 for a maximally inhomogeneous
state (i.e. for a Fock state), to $0$ for a fully homogeneous density, with
$\langle\hat{n}_{i}\rangle=\rho$, for all sites $i$. Figures 3 (c,d) show
$\eta$, after a time $\tau=200/t$ for, respectively, $V_{j}=V/j^{\beta}$ and a
polar gas with different $\chi$ (and hence different $\beta_{\mathrm{eff}}$).
In both cases a marked jump in $\eta$ as a function of $V/t$ characterizes the
onset of strong localization. Note as well the remarkable similarity down to
$\beta_{\mathrm{eff}}\simeq 1$ ($B\simeq 0.3$), of both graphs as a function
of, respectively, $\beta$ and $\beta_{\mathrm{eff}}$. Hence, for $B>0.3$, the
dynamics and the localization threshold at a given time in a polar lattice gas
are basically indistinguishable from those in a system with power-law
interactions $1/j^{\beta_{\mathrm{eff}}}$.

For $\beta_{\mathrm{eff}}<1$, the dynamics of a polar lattice gas departs
significantly from that of a $1/j^{\beta_{\mathrm{eff}}}$ model (see Fig. 3).
In particular, the polar gas presents a marked resonance for
$\beta_{\mathrm{eff}}\simeq 0.8$ ($B\simeq 0.17$), at which the gas becomes
quickly homogeneous even for large $V/t$. Such a resonance is absent in the
corresponding power-law model, which presents a steady re-entrance of the
delocalized regime. This is easy to understand, since for $\beta\to 0$, the
inter-site interactions become a constant of motion, $\frac{V}{2}\sum_{i\neq
j}n_{i}n_{j}=VN(N-1)/2$, with $N$ the total particle number, and hence the
system is formed effectively by non-interacting hard-core bosons. This steady
growth of the extended regime is not present in the actual polar lattice gas,
because for long distances $G_{j}\to 1/j^{3}$. Only for
$\beta_{\mathrm{eff}}\to 0$ delocalization extends to large $V/t$ values.

##### Dynamics for general initial Fock states.–

The previous conclusions are, for $\beta_{\mathrm{eff}}>1$, largely
representative of the dynamics for more general initial Fock states. We have
evaluated the dynamics of all possible initial Fock states for $L=16$ and
open-boundary conditions, fixing only $N=8$ and an initial $N_{NN}=4$. Note
that this set includes $4410$ states with different number of clusters of
various lengths. Figure 4 shows for $V_{j}=V/j^{\beta}$ and for a polar
lattice gas, the value of $\eta$ (averaged over all possible initial
conditions) after an evolution time $\tau=200/t$. Also for this more general
case, there is a marked transition between localization and delocalization.
However, the behavior of the averaged $\eta$ is less abrupt, due to the
difference in the degree of localization between different initial conditions.
The results for both models are again remarkably similar, down to
$\beta_{\mathrm{eff}}\simeq 1$ ($B>0.3$). For lower $\beta_{\mathrm{eff}}$ the
results are markedly different. Note the absence of a resonance in the polar
gas, which is a specific feature of an initial density wave. Note that also
for general initial conditions, the power-law model presents at low $\beta$ a
marked re-entrance of the extended regime, absent in polar lattice gases at
low $B$.

Figure 4: Inhomogeneity $\eta$ after a time $\tau=200/t$, evaluated for a
system with $L=16$ sites, and averaged over all initial Fock states with $N=8$
and initial $N_{NN}=4$. Panels (a) and (b) show, respectively, the case of
$V_{j}=V//j^{\beta}$ and of a polar lattice gas. The thick black curve
indicates the line with $\eta=0.5$. The results were obtained using exact time
evolution. The dashed green (dotted yellow) curves indicate the values in the
($V/t$,$\beta_{\mathrm{eff}})$ space achieved for different $\omega_{\perp}$
for 164Dy in an UV lattice of $\lambda=180\,\mathrm{nm}$ (for NaK, with
$d=0.8\,$Debye, in a lattice with $\lambda=500\,\mathrm{nm}$) with a fixed
lattice depth $s=8$. For the dashed-green (dotted-yellow) curve, from left to
right, $\chi$ varies from $0.0167$ to $4.68$ (from $0.0045$ to $0.21$).

##### Experimental relevance.–

Our results are directly relevant for on-going and near-future experiments
with low-dimensional dipolar gases in optical lattices, including magnetic
atoms, polar molecules, and Rydberg gases. Our results indicate that for a
polar lattice gas there is a very significant difference between the case in
which low-dimensionality is achieved in a 3D lattice, and the case in which it
is obtained by means of transversal harmonic confinement. In the former case,
the suppression of transversal hopping requires a very strong lattice on the
transversal directions, resulting in $l_{\perp}<l$. The $1/j^{3}$ dependence
should be hence approximately reached for a sufficiently large lattice depth
$s$. In the latter case, in contrast, even tight transversal confinements may
result in strongly modified properties. For example, a lattice depth of $s=20$
and a transversal confinement with $\chi\simeq 0.5$ results in
$\beta_{\mathrm{eff}}(B)\simeq 2$. For 164Dy in an UV lattice with
$\lambda=180$nm this would require $\omega_{\perp}/2\pi\simeq 4.6\,$kHz,
whereas a frequency $1.58\,$kHz will be needed for a NaK molecule in a lattice
with $\lambda=500$nm. Note that changing $\omega_{\perp}$, modifies both $V/t$
and $\beta_{\mathrm{eff}}$. In Fig. 4(b) we show the trajectories in
$(V/t,\beta_{\mathrm{eff}})$ for $s=8$ obtained for different $\chi$ for Dy
and NaK (for $s=8$ the hopping times are respectively $10\,\mathrm{ms}$ and
$31\,\mathrm{ms}$). Note that in both cases the localization-to-delocalization
transition may be crossed by changing $\omega_{\perp}$ at a fixed lattice
depth.

##### Conclusions.–

Hard-core polar lattice gases are typically described by a model with
$1/r^{3}$ interactions, whose properties are solely given by the ratio $V/t$.
In contrast, we have shown that an actual polar gas in an optical lattice
presents an interaction decay whose form is universally given by a single
parameter, $B$, given by the transversal confinement and the lattice depth.
The actual decay may depart very significantly from $1/r^{3}$ strongly
modifying the Hilbert-space fragmentation and the dynamics and localization of
particles. Interestingly, fragmentation and particle dynamics become basically
identical to a model with $1/j^{\beta_{\mathrm{eff}}(B)}$, as long as
$\beta_{\mathrm{eff}}>1$. Hence dipolar gases behave to a large extent as a
system with a power-law tunable by a proper choice of the transversal
confinement.

###### Acknowledgements.

J.Z. thanks Titas Chanda for help with TDVP implementation. We acknowledge
support of the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy – EXC-2123 QuantumFrontiers –
390837967, and FOR 2247. Some of the numerical computations have been possible
thanks to PL-Grid Infrastructure. The work of J.Z. has been realized within
the Opus grant 2019/35/B/ST2/00034, financed by National Science Centre
(Poland). X.D. aknowledges support of BMBF through DAQC. PS. acknowledges
support from Ministerio de Ciencia y Innovation Agencia Estatal de
Investigaciones (R & D project CEX2019-000910-S, AEI/10.13039/501100011033,
Plan National FIDEUA PID2019-106901GB-I00, FPI), Fundació Privada Cellex,
Fundació Mir-Puig, and from Generalitat de Catalunya (AGAUR Grant No. 2017 SGR
1341, CERCA program).

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##  I Supplementary Material

###  I.1 Details on the inter-site interactions

####  I.1.1 Derivation of Eq. (3) of the main text

As discussed in the main text, the dipole-mediated inter-site interaction
between particles separated by $j$ sites is characterized by the coupling
constant:

$V_{j}=\int d^{3}r\int
d^{3}r^{\prime}V(\vec{r}-\vec{r}\,^{\prime})|\varphi(\vec{r})|^{2}|\varphi(\vec{r}\,^{\prime}-j\lambda\vec{e}_{z})|^{2},$
(S1)

where we neglect the effect of exchange terms, since they are negligible small
for a sufficiently deep lattice. In the following calculation, we assume that
the dipole is oriented along the lattice axis $z$ ($\alpha=0$ in the notation
of the main text), but as discussed below the calculation can be easily
generalized to any dipole orientation.

As discussed in the main text, we may approximate the on-site wavefunction by
a Gaussian:

$|\varphi(\vec{r})|^{2}=\frac{e^{-z^{2}/l^{2}}}{\sqrt{\pi}l}\frac{e^{-\rho^{2}/l_{\perp}^{2}}}{\pi
l_{\perp}^{2}},$ (S2)

where $\rho=\sqrt{x^{2}+y^{2}}$. Then the Fourier transform of the density is
of the form: $\tilde{n}_{j}(\vec{k})=\tilde{n}(\vec{k})e^{ik_{z}j\lambda}$,
and

$\tilde{n}(\vec{k})=e^{-k_{z}^{2}l^{2}/4}e^{-k_{\rho}^{2}l_{\perp}^{2}/4}.$
(S3)

Using the convolution theorem we may then re-express $V_{j}$ in the form:

$V_{j}\simeq\int\frac{d^{3}k}{(2\pi)^{3}}\tilde{V}(\vec{k})\tilde{n}_{0}(\vec{k})\tilde{n}_{j}(\vec{k})$
(S4)

with $\tilde{V}(\vec{k})=\frac{4\pi
d^{2}}{3}\left[\frac{3k_{z}^{2}}{|\vec{k}|^{2}}-1\right]$, the Fourier
transform of the dipole-dipole interaction potential. Using the form of
$\tilde{n}_{j}(\vec{k})$, we may then re-write:

$\displaystyle V_{j}$ $\displaystyle\simeq$ $\displaystyle\frac{2d^{2}}{3\pi
ll_{\perp}^{2}}\int_{0}^{1}du\left[\frac{3u^{2}}{\Lambda^{2}+(1-\Lambda^{2})u^{2}}-1\right]$
(S5) $\displaystyle\times$
$\displaystyle\int_{0}^{\infty}dqq^{2}e^{-q^{2}/2}\cos\left(q\frac{z_{j}}{l}u\right).$

with $\Lambda=l/l_{\perp}$. For the specific case of a harmonic confinement
with frequency $\omega_{\perp}$ and an optical lattice along $z$ with depth
$V_{0}$, we may define, as in the main text,
$\chi=\frac{\hbar\omega_{\perp}}{E_{\mathrm{R}}}$ and
$s=\frac{V_{0}}{E_{\mathrm{R}}}$, and re-express
$\Lambda(\chi,s)=\frac{1}{\sqrt{2}}\sqrt{\chi}\frac{1}{s^{1/4}}$. We can then
write:

$V_{j}\simeq\frac{2d^{2}}{3\sqrt{2\pi}ll_{\perp}^{2}}F_{j}(\chi,s)$ (S6)

with

$\displaystyle F_{j}(\chi,s)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}du\left[\frac{3u^{2}}{\Lambda^{2}+(1-\Lambda^{2})u^{2}}-1\right]$
(S7) $\displaystyle\times$
$\displaystyle\left[1-\tilde{z}_{j}^{2}u^{2}\right]e^{-\frac{1}{2}\tilde{z}_{j}^{2}u^{2}}$

with $\tilde{z}_{j}=j\pi s^{1/4}$. Changing the integration variable into
$\tilde{u}=\tilde{z}_{j}u$, and re-organizing the integrand we can re-write
the integral in the form:

$F_{j}(\chi,s)=\frac{1}{\tilde{z}_{j}}\int_{0}^{\tilde{z}_{j}}du\left[\left(\frac{2+\Lambda^{2}}{1-\Lambda^{2}}\right)-\left(\frac{3}{1-\Lambda^{2}}\right)\frac{1}{1+\frac{1-\Lambda^{2}}{\Lambda^{2}}\frac{\tilde{u}^{2}}{\tilde{z}_{j}^{2}}}\right](1-\tilde{u}^{2})e^{-\tilde{u}^{2}/2}$
(S8)

For a sufficiently large $s$, due to the rapidly decaying exponential we can
safely move the integral boundary to infinity. Note that this is only possible
if $\Lambda<1$ ($l_{\perp}>l$). We can then perform the integral analytically,
obtaining:

$\frac{V_{j}}{E_{\mathrm{R}}}=\frac{3}{\pi^{2}}B^{3/2}\left(\frac{a_{dd}}{\lambda}\right)f(\sqrt{B}j)$
(S9)

where $f(\xi)$ is the expression of Eq. (4) of the main text. The procedure
for other dipole orientations is identical, and we may obtain the general
expression of Eq. (3) of the main text.

As a side remark, we compare the result of Eq. (S9) and the known result for
the interaction between two dipoles in a 1D system (in absence of lattice)
when the dipole is oriented along the system axis Sinha and Santos (2007). We
can re-write Eq. (S9) in the form:

$\frac{V_{j}}{E_{\mathrm{R}}}=3\pi\left(\frac{\chi}{2}\right)^{3/2}\left(\frac{\tilde{a}_{dd}}{\lambda}\right)f\left(\tilde{j}\pi\sqrt{\frac{\chi}{2}}\right),$
(S10)

with $\tilde{a}_{dd}=a_{dd}/\left(1-\frac{\chi}{2\sqrt{s}}\right)^{3/2}$ and
$\tilde{j}=j/\sqrt{1-\frac{\chi}{2\sqrt{s}}}$. Comparing to the result of Ref.
Sinha and Santos (2007), we note that Eq. (S10) acquires the same form as the
interaction between two dipoles with a regularized dipole length
$\tilde{a}_{\mathrm{dd}}$ separated by an effective distance
$\tilde{j}\lambda$.

####  I.1.2 Asymptotic expression for large $\Lambda$

Let us consider at this point the case $\Lambda=l/l_{\perp}\gg 1$, which is
the typical case in strong 3D optical lattices. In that case, we may
approximate for sufficiently deep lattices:

$V_{j}=\frac{2d^{2}}{3\sqrt{2\pi}ll_{\perp}^{2}}\frac{3}{\Lambda^{2}}\int_{0}^{1}du\frac{u^{2}}{(1-u^{2})}\left[1-\tilde{z}_{j}^{2}u^{2}\right]e^{-\frac{1}{2}\tilde{z}_{j}^{2}u^{2}}.$
(S11)

Since only small $u$ contribute for large $s$, we may expand:

$\displaystyle V_{j}$ $\displaystyle\simeq$
$\displaystyle\frac{2d^{2}}{3\sqrt{2\pi}ll_{\perp}^{2}}\frac{3}{\Lambda^{2}}\int_{0}^{1}du\left(u^{2}+(1-\tilde{z}_{j}^{2})u^{4}+\cdots\right)e^{-\frac{1}{2}\tilde{z}_{j}^{2}u^{2}}$
(S12) $\displaystyle\simeq$
$\displaystyle\frac{2d^{2}}{3\sqrt{2\pi}ll_{\perp}^{2}}\frac{3}{\Lambda^{2}}\frac{\sqrt{\pi}}{2}\left[\frac{\sqrt{2}}{\tilde{z}_{j}^{3}}+(1-\tilde{z}_{j}^{2})\frac{3\sqrt{2}}{\tilde{z}_{j}^{5}}+\cdots\right]\mathrm{erf}\left(\frac{\tilde{z}}{\sqrt{2}}\right)$
$\displaystyle=$
$\displaystyle\frac{-2d^{2}}{\lambda^{3}}\frac{1}{j^{3}}\left[1+\frac{6}{j^{2}\pi^{2}\sqrt{s}}+\cdots\right]$

Then

$\frac{V}{V_{2}}\simeq 8\left[1+\frac{9}{2\pi^{2}\sqrt{s}}+\cdots\right]$
(S13)

Now, writing $\frac{V}{V_{2}}=2^{\beta_{\mathrm{eff}}}$, and anticipating that
$\beta_{\mathrm{eff}}\ll 1$, we may expand:

$2^{\beta_{\mathrm{eff}}}\simeq
8\left[1+\mathrm{ln}(2)(\beta_{\mathrm{eff}}-3)\right]$ (S14)

Comparing both expressions we get the final result:

$\beta_{\infty}\simeq 3+\frac{1}{\mathrm{ln}(2)}\frac{9}{2\pi^{2}\sqrt{s}}$
(S15)

written in the main text.

###  I.2 Time-Dependent Variational Principle algorithm implementation

Time-Dependent Variational Principle algorithm (TDVP) Haegeman  _et al._
(2011); Koffel  _et al._ (2012); Haegeman  _et al._ (2016); Goto and Danshita
(2019) allows one to study time evolution of systems comprised of hundreds of
sites for a limited time. In the algorithm, the time evolved state is
represented as a matrix product state (MPS) (for a review see, e.g.
Schollwoeck (2011)). Such states are represented by tensors which, apart, from
physical indices (the number of sites, Hilbert space dimension on a given
site) are characterized, at each site, by an auxiliary index running over the
space of dimension $\chi$. The higher $\chi$ the more entangled states can be
typically faithfully represented by the MPS representation. During the time
evolution with a many-body Hamiltonian the entanglement in the time evolved
state typically grows with the rate dependent on the properties of the system.
In The rapid growth of the entanglement of an initially low entangled state
prevents one from evolving the state for too long, say beyond tens of
tunneling times in the ergodic regime. Once the motion becomes nonergodic, and
in particular localized, the growth of entanglement in time is much slower
which allows one to faithfully trace the time evolution even up to thousands
of tunneling times (for recent implementations for short range Hamiltonians
see Chanda  _et al._ (2020a, c, b); Sierant and Zakrzewski (2022) where all
the details are discussed). We use the same implementation, extended to the
dipolar-like long range terms using the so called matrix product operator
representation of the Hamiltonian implemented within the Itensor library
Fishman  _et al._ .

Figure S1: Homogenization plot for $B=2.54$ $V/t=16$ obtained with TDVP for
$\chi=256$ (left) and $\chi=384$ (right). Observe that more accurate results
for larger $\chi$ value undergo homogenization in longer times. This is a
typical behavior fornot fully converged TDVP algorithm.

Due to the long range coupling as well as large value of $V/t$, the algorithm
requires quite large amount of CPU time for propagation. For the data
presented in Fig. 3 of the letter, the propagation took more than a month on a
single thread of a fast workstation for $\chi=384$ (less than a week for
$\chi=256$). While the results were different indicating lack of convergence
in the delocalized regime, the qualitative time-dependence obtained was the
same, with homogenization (delocalization) occuring slightly slower for a
larger (more accurate) $\chi$ value confirming the claims expressed in the
main text.

###  I.3 Chebyshev propagation implementation

We use Chebyshev propagation scheme as described in detail in Fehske and
Schneider (2008) for $L=24$ at half filling. In this approach, the time
evolution operator $U(\Delta t)=\exp(-iH\Delta t)$ over time period $\Delta t$
is expanded as

$U(\Delta t)\approx\mathrm{e}^{-\mathrm{i}b\Delta t}\left(J_{0}(a\Delta
t)+2\sum_{k=1}^{N}(-i)^{k}J_{k}(a\Delta
t)T_{k}\left(\mathcal{H}\right)\right),$ (S16)

where $a=(E_{\rm max}-E_{\rm min})/2$, $b=(E_{\rm max}+E_{\rm min})/2$ and
$E_{\rm min}$/$E_{\rm max}$ is the lowest/highest eigenstate energy of the
Hamiltonian $H$. $\mathcal{H}=\frac{1}{a}(H-b)$ is the rescaled Hamiltonian
with the spectrum in the $[-1,1]$ interval, $J_{k}(t)$ is the Bessel function
of the order $k$ and $T_{k}(x)$ is the Chebyshev polynomial of order $k$. The
order of the expansion $N$ and the time step $\Delta t$ are controlled by the
preservation of the unitarity of the evolution, allowing for numerically exact
results, for further details see Ref. Sierant and Zakrzewski (2022).


2021 outburst of RS Oph 29 August 2021

The 2021 outburst of RS Oph. A pictorial atlas of the spectroscopic evolution:
the first 18 days

Ulisse Munari1 and Paolo Valisa2

1: INAF National Institute of Astrophysics, 36012 Asiago, Italy
2: ANS Collaboration, c/o Astronomical Observatory, 36012 Asiago, Italy

Abstract. A pictorial atlas of the spectroscopic evolution at optical
wavelengths is presented for the first 18 days of the 2021 outburst of the
recurrent nova RS Oph, prior to the emergence of high ionization emission
lines. The spectra presented here have been obtained at daily cadence with the
Asiago 1.22m + B&C (3200-7900 Ang, 2.3 Å/pix) and Varese 0.84m \+ Echelle
telescopes (4250-8900 Å, resolving power 18,000). The spectra have been fully
calibrated in IRAF, absolutely fluxed, and heliocentric corrected. The Echelle
spectra have been also corrected for telluric absorptions.

INTRODUCTION

The recurrent nova and symbiotic binary RS Oph underwent a new outburst in
August 2021, following previous ones recorded in 1898, 1933, 1958, 1967, 1985
and 2006. At each new event a better coverage of the eruption was obtained
with observations progressively expanding to cover the whole electromagnetic
spectrum. For the 2006 event, an impressive amount of data has been gathered,
and the book RS Ophiuchi (2006) and the Recurrent Nova Phenomenon edited by
Bode, O’Brien & Darnley (2008, ASPC 401) provides a detailed summary of the
main results.

Yet, a comprehensive and detailed mapping of the spectral evolution at optical
wavelengths seems missing, at leat at the level of completeness, detail and
homogeneity of the present atlas. It aims to fill the gap, by providing a
quick pictorial guide to the spectral evolution of RS Oph during the first 18
days of the 2021 outburst. To make the atlas promptly available, we just
describe how the observations have been collected and how the reduced spectra
have been incorporated in the atlas. This is of course no substitute for a
proper analysis, to be carried out in due time. The latter will benefit from
continued observations covering the rest of the evolution of RS Oph as well as
comparison with the results of observations carried out at other wavelengths
(and not yet published).

While presenting this first batch of spectra, our monitoring of RS Oph will
continue at daily cadence for as long as possible, and updates to this atlas
are planned for later epochs.

The current atlas covers the first 18 days of the eruption (August 8 to 26),
those characterized by low ionization conditions. From day +19 (August 27),
the evolution has progressed to the emergence of higher ionization lines
(HeII, [FeVII]) as reported by Shore et al. (2021b,c).

The current outburst has triggered observations of RS Oph over the whole
electromagnetic spectrum. Preliminar reports have been presented by Cheung et
al. (2021a,b) and Wagner & HESS Collaboration (2021a,b) for $\gamma$-rays;
Enoto et al. (2021a,b), Ferrigno et al. (2021), Luna et al. (2021), Page
(2021), Page et al. (2021), Rout et al. (2021) and Shidatsu et al. (2021) for
the X-rays; Mikolajewska et al. (2021), Munari U., Valisa P., (2021a,b), Shore
et al. (2021a,b,c), and Taguchi et al. (2021a,b) for the optical; Woodward et
al. (2021) for the infrared; and Sokolovsky et al. (2021) and Williams et al.
(2021) for the radio. The results of the search for neutrino emission has been
given by Pizzuto et al. (2021), and on the polarization of optical light by
Nikolov & Luna (2021). A great summary of the main results obtained on the
previous outburst of RS Oph in 2006 is the conference proceedings edited by
Evans et al. (2008).

OBSERVATIONS

RS Oph has been observed each day for the first 18 days of its 2021 eruption
with the Echelle slit-spectrograph mounted on the Varese 0.84m telescope. The
spectra have been recorded over the range 4250-8900 Å, at a 18,000 resolving
power. The data reduction has been performed in IRAF and has included all
usual steps of correction for bias, dark and flat, sky subtraction, wavelength
calibration (via Thorium lamp), and heliocentric correction.
Spectrophotometric standards, located close on the sky to RS Oph, have been
observed each night soon before and after the nova to achieve an optimal flux
calibration, which was also essential to accurately merge the individual 30
Echelle orders into a single 1D fluxed spectrum not affected by dents at the
points of jointure. Telluric dividers were observed each night similarly to
the spectrophotometric standards, to properly remove from the spectra of RS
Oph the presence of telluric absorption lines. A journal of the Echelle
observations is provided in Table 1. The poor quality of the spectrum for Aug
12 is due to cloud cover persisting all night.

RS Oph was also observed on almost each day for the first 18 days of its 2021
eruption with the B&C spectrograph attached to the Asiago 1.22m telescope,
allowing to cover the 3200-7900 Å range at 2.3 Å/pix dispersion. The B&C
spectra have been equally and fully reduced in IRAF and flux calibrated
against spectrophotometric standards observed each night at a similar height
on the horizon as RS Oph. A journal of the B&C observations is presented in
Table 2.

At both telescopes a great emphasis has been put on maintaining the highest
homogeneity throughout the observing campaign on the instrument set-up, choice
of standards, observation and data reduction procedures. Any difference
between the spectra presented in this atlas is more intrinsic to RS Oph than
due to any stage of the observing process.

THE ATLAS

Given the huge intensity of the emission lines displayed by RS Oph and the
necessity to compact many different spectra on the same picture, we adopt to
plot the spectra on all figure as the logarithm of the flux (in erg cm-2 s-1
Å-1). Multi-epoch observations within the same night have been averaged into
one single spectrum per night.

The sequence of B&C spectra is presented first, with Figure 1 covering the
whole wavelength interval, and Figures 2 and 3 zooming on the blue and red
portions respectively, with identification of the principal lines. A list of
the emission lines identified in the spectrum for Aug 23 ($t-t_{\circ}$=+15.28
days) is given in Table 3.

Selected portions of the Echelle spectra then follow. In Figure 4 we present
the evolution of H$\alpha$, and that of HeI 5876 follows in Figure 5. Figures
6 and 7 are devoted to cover H$\beta$ and the nearby FeII multiplet 42 and HeI
lines at 4921 and 5015 Å. Finally, Figure 8, zooms on the sharp components
displayed by H$\alpha$, H$\beta$, FeII #42 5018 and HeI 5015 on top of the
much wider and stronger emission lines, and to the NaI 5890, 5896 Å doublet.

REFERENCE EPOCH

To identify the spectra through the various figures, we have labelled them
according to the observing UT date as listed in Tables 1 and 2. In the same
tables the time elapsed since the onset of the outburst ($t-t_{\circ}$) and
since maximum brightness ($t-t_{\rm max}$) in the $V$-band are listed.

The very quick rise to maximum has been covered by L.P. Lou and B. Wang with
observations obtained with a Sony DSLR camera and reported to AAVSO (observer
code WPIA). Extrapolating back their linear rise to intercept the $V$=11.1 mag
quiescence brightness that RS Oph displayed on preceding days, set the time of
eruption to:

$t_{\circ}=2459435.00~{}{\rm JD}~{}~{}~{}~{}~{}~{}2021~{}{\rm
Aug}~{}08.50~{}~{}~{}~{}(\pm 0.01)$ (1)

The time of passage at maximum $V$-band magnitude is less accurately
determined. Again from the AAVSO light-curve, we estimate:

$t^{V}_{\rm max}=2459436.18~{}{\rm JD}~{}~{}~{}~{}~{}~{}2021~{}{\rm
Aug}~{}09.58~{}~{}~{}~{}(\pm 0.05)$ (2)

ACKNOWLEDGMENTS

We express our gratitude to P. Ochner (Univ. Padova) and A. Vagnozzi (ANS
Collaboration) for their help with the acquisition of some of the spectra used
in this paper.

REFERENCES

Cheung C. C., Ciprini S., Johnson T. J., 2021a, ATel, 14834

Cheung C. C., Johnson T. J., Mereu I., et al., 2021b, ATel, 14845

Enoto T., Maehara H., Orio M., et al., 2021a, ATel, 14850

Enoto T., Orio M., Fabian A., et al., 2021b, ATel, 14864

Evans A., Bode M. F., O’Brien T. J., Darnley M. J., eds., 2008, RS Ophiuchi
(2006) and the Recurrent Nova

Phenomenon, ASPC, 401

Ferrigno C., Savchenko V., Bozzo E., et al., 2021, ATel, 14855

Luna G. J. M., Jimemez-Carrera R., Enoto T., et al., 2021, ATel, 14872

Mikolajewska J., Aydi E., Buckley D., et al., 2021, ATel, 14852

Munari U., Valisa P., 2021a, ATel, 14840

Munari U., Valisa P., 2021b, ATel, 14860

Nikolov Y., Luna G. J. M., 2021, ATel, 14863

Page, K. L. 2021, ATel, 14885

Page K. L., Osborne J. P., Aydi E., 2021, ATel, 14848

Pizzuto A., Vandenbroucke J., Santander M., IceCube Collaboration, 2021, ATel,
14851

Rout S. K., Srivastava M. K., Banerjee D. P. K., et al., 2021, ATel, 14882

Shidatsu M., Negoro H., Mihara T., et al., 2021, ATel, 14846

Shore S. N., Allen H., Bajer M., et al., 2021a, ATel, 14868

Shore S. N., Teyssier F., Thizy O., 2021b, ATel, 14881

Shore S. N., Teyssier F., Guarro J., et al., 2021c, ATel, 14883

Sokolovsky K., Aydi E., Chomiuk L., et al., 2021, ATel, 14886

Taguchi K., Ueta, T., Isogai, K., 2021a, ATel, 14838

Taguchi K., Maheara H., Isogai K., et al., 2021b, ATel, 14858

Williams D., O’Brien T., Woudt P., et al., 2021, ATel, 14849

Wagner, S. J., HESS Collaboration, 2021a, ATel, 14844

Wagner, S. J., HESS Collaboration, 2021b, ATel, 14857

Woodward C. E., Evans A., Banerjee D. P. K., et al., 2021, ATel, 14866

Table 1: Journal of Echelle spectroscopic observations obtained with the Varese 0.84m telescope. date UT | expt | HJD | $t-t_{\circ}$ | $t-t_{\rm max}$
---|---|---|---|---
yyyy-mm-dd | hh:mm | (sec) | (-2459400) | (days) | (days)
2021-08-09 | 19:30 | 900 | 36.316 | 1.32 | 0.14
2021-08-09 | 20:03 | 540 | 36.339 | 1.34 | 0.16
2021-08-09 | 20:55 | 540 | 36.375 | 1.38 | 0.20
2021-08-09 | 21:41 | 360 | 36.407 | 1.41 | 0.23
2021-08-10 | 19:41 | 720 | 37.324 | 2.32 | 1.14
2021-08-10 | 21:21 | 720 | 37.393 | 2.39 | 1.21
2021-08-11 | 19:23 | 720 | 38.311 | 3.31 | 2.13
2021-08-11 | 20:11 | 1200 | 38.344 | 3.34 | 2.16
2021-08-12 | 19:50 | 1200 | 39.330 | 4.33 | 3.15
2021-08-13 | 19:31 | 900 | 40.316 | 5.32 | 4.14
2021-08-14 | 20:40 | 1440 | 41.364 | 6.36 | 5.18
2021-08-15 | 19:40 | 1200 | 42.323 | 7.32 | 6.14
2021-08-16 | 20.05 | 1500 | 43.338 | 8.34 | 7.16
2021-08-17 | 19:20 | 900 | 44.309 | 9.31 | 8.13
2021-08-18 | 19:30 | 1500 | 45.315 | 10.32 | 9.14
2021-08-19 | 20:31 | 1500 | 46.358 | 11.36 | 10.18
2021-08-20 | 19:28 | 1500 | 47.314 | 12.31 | 11.13
2021-08-21 | 19:18 | 1500 | 48.307 | 13.31 | 12.13
2021-08-22 | 19:20 | 1500 | 49.308 | 14.31 | 13.13
2021-08-23 | 19:45 | 2400 | 50.325 | 15.33 | 14.15
2021-08-24 | 19:41 | 1800 | 51.323 | 16.32 | 15.14
2021-08-25 | 19:15 | 1800 | 52.304 | 17.30 | 16.12
2021-08-26 | 19:10 | 2400 | 53.301 | 18.30 | 17.12
Table 2: Journal of B&C spectroscopic observations obtained with the Asiago 1.22m telescope. date UT | expt | HJD | $t-t_{\circ}$ | $t-t_{\rm max}$
---|---|---|---|---
yyyy-mm-dd | hh:mm | (sec) | (-2459400) | (days) | (days)
2021-08-09 | 18:49 | 100 | 36.284 | 1.28 | 0.10
2021-08-09 | 19:56 | 150 | 36.331 | 1.33 | 0.15
2021-08-10 | 19:11 | 120 | 37.300 | 2.30 | 1.12
2021-08-10 | 21:11 | 150 | 37.383 | 2.38 | 1.20
2021-08-11 | 19:09 | 90 | 38.298 | 3.30 | 2.12
2021-08-11 | 22:03 | 130 | 38.419 | 3.42 | 2.24
2021-08-12 | 18:44 | 90 | 39.281 | 4.28 | 3.10
2021-08-12 | 19:05 | 170 | 39.295 | 4.30 | 3.12
2021-08-12 | 20:45 | 120 | 39.365 | 4.37 | 3.19
2021-08-13 | 19:39 | 120 | 40.319 | 5.32 | 4.14
2021-08-13 | 21:04 | 120 | 40.378 | 5.38 | 4.20
2021-08-14 | 20:24 | 100 | 41.351 | 6.35 | 5.17
2021-08-15 | 20:11 | 120 | 42.342 | 7.34 | 6.16
2021-08-15 | 21:16 | 90 | 42.386 | 7.39 | 6.21
2021-08-17 | 19:13 | 120 | 44.301 | 9.30 | 8.12
2021-08-18 | 19:24 | 120 | 45.309 | 10.31 | 9.13
2021-08-19 | 18:44 | 110 | 46.281 | 11.28 | 10.10
2021-08-19 | 19:54 | 90 | 46.330 | 11.33 | 10.15
2021-08-20 | 19:22 | 120 | 47.307 | 12.31 | 11.13
2021-08-20 | 20:51 | 90 | 47.369 | 12.37 | 11.19
2021-08-22 | 19:42 | 150 | 49.321 | 14.32 | 13.14
2021-08-23 | 18:41 | 150 | 50.279 | 15.28 | 14.10
Table 3: Identification of emission lines in the Asiago 1.22m + B&C spectrum of RS Oph for Aug 23 ($t-t_{\circ}$=+15.28 days). $\lambda$ | line | $\lambda$ | line | $\lambda$ | line | $\lambda$ | line
---|---|---|---|---|---|---|---
3721 | H14 | 4176 | FeII 27+28 | 4823 | HeI, FeII 42 | 5676 | NII
3733 | H13 | 4233 | FeII 27 | 4949 | [OIII] | 5755 | [NII]
3750 | H12 | 4273 | FeII 27 | 5007 | [OIII] | 5876 | HeI
3770 | H11 | 4300 | FeII 27+28 | 5019 | HeI+FeII 42 | 5991 | FeII 46
3798 | H10 | 4340 |  H$\gamma$ | 5048 | HeI | 6148 | FeII 74
3837 | H9 | 4363 | [OIII] | 5169 | FeII 42 | 6248 | FeII 74
3856 | HeI | 4388 | HeI | 5185 | FeII | 6318 | FeII
3868 | HeI | 4417 | FeII 27 | 5198 | FeII 49 | 6369 | blend
3889 | H8+HeI | 4471 | HeI | 5234 | FeII 49 | 6433 | FeII 40
3927 | HeI | 4489 | FeII 37 | 5255 | FeII 49 | 6456 | FeII 74
3936 | HeI | 4520 | FeII 37+38 | 5265 | FeII 48 | 6563 |  H$\alpha$
3970 |  H$\epsilon$ | 4555 | FeII 37+38 | 5276 | FeII 49 | 6678 | HeI
4026 | HeI | 4584 | FeII 38 | 5316 | FeII 48+49 | 7065 | HeI
4070 | FeII 22 | 4634 | NIII+FeII 37 | 5363 | FeII 48 | 7281 | HeI
4101 |  H$\delta$ | 4647 | CIII | 5426 | FeII 49 | 7711 | FeII 73
4124 | FeII 22 | 4713 | HeI | 5496 | FeII | 7774 | OI
4144 | HeI | 4861 |  H$\beta$ | 5535 | FeII 55 |  |

Figure 1: Sequence of Asiago 1.22m spectra showing the whole recorded range.
The logarithm of the flux is adopted as the ordinates and a shift is applied
to avoid overplotting.

Figure 2: An expanded view of the 3650-5450 Å interval for the same spectra of
Figure 1. The logarithm of the flux is adopted as the ordinates and a shift is
applied to avoid overplotting.

Figure 3: An expanded view of the 5450-7850 Å interval for the same spectra of
Figure 1. The logarithm of the flux is adopted as the ordinates and a shift is
applied to avoid overplotting.

Figure 4: Evolution of the H$\alpha$ profile from Varese 0.84m Echelle
spectra. The abscissae at the top give the heliocentric radial velocity (in
km/s). The logarithm of the flux is adopted as the ordinates and a shift is
applied to avoid overplotting.

Figure 5: Evolution of the HeI 5876 profile and superimposed NaI doublet from
Varese 0.84m Echelle spectra. The abscissae at the top give the heliocentric
radial velocity (in km/s). The logarithm of the flux is adopted as the
ordinates and a shift is applied to avoid overplotting.

Figure 6: Series of Varese 0.84m Echelle spectra covering the H$\beta$, FeII
multiplet 42 and HeI 4941, 5015 emission lines for the first eight days of the
monitoring (the remaining dates are covered by the following Figure 7). The
logarithm of the flux is adopted as the ordinates and a shift is applied to
avoid overplotting.

Figure 7: Series of Varese 0.84m Echelle spectra covering the H$\beta$, FeII
multiplet 42 and HeI 4941, 5015 emission lines for the last ten days of our
monitoring (the previous eight days are covered by preceeding Figure 6). The
logarithm of the flux is adopted as the ordinates and a shift is applied to
avoid overplotting.

Figure 8: Evolution of the narrow components for H$\alpha$, H$\beta$, HeI 5015
and FeII 5018, as well as the NaI doublet from Varese 0.84m Echelle spectra.
The abscissae at the top give the heliocentric radial velocity (in km/s). The
logarithm of the flux is adopted as the ordinates and a shift is applied to
avoid overplotting.