# Bloch oscillations in supersolids

Muhammad. S. Hasan1, J. Polo1,2, J.C. Pelayo1 and Th. Busch1 1Quantum Systems
Unit, Okinawa Institute of Science and Technology Graduate University,
Okinawa, Japan 904-0495  2Quantum Research Centre, Technology Innovation
Institute, Abu Dhabi, UAE<EMAIL_ADDRESS>
###### Abstract

We show that the motion of an accelerated atomic impurity immersed in a spin-
orbit coupled Bose-Einstein condensate in the supersolid stripe phase
undergoes oscillations, similar to the well-known phenomenon of Bloch
oscillations in solids. While the back-action of the oscillatory movement onto
the condensate excites phonon modes inside the supersolid, it does not affect
the position of the roton minimum and therefore not the periodicity of the
matter wave lattice. The ultimate decay of the oscillations is mostly due to
the dispersion of the wavepacket and we show that this can be counteracted to
a large extent by assuming that the impurity is a bright soliton.

##  1 Introduction

Ultracold atoms in optical lattices have, in the last two decades, been
fruitful systems to study the physics of periodic quantum many-body systems in
clean and highly controllable settings [1, 2]. These range from the
paradigmatic realisation of the bosonic [3] and fermionic [4, 5] Mott
insulator transitions, to the creation of systems with topological quantum
matter [6]. However, as optical lattices are static potentials, they do not
allow for phonon modes and therefore any possible connection to condensed
matter systems is incomplete. While Hubbard models with phonons have been
suggested to be in principle realisable in systems made from self-assembled
crystals of polar molecules [7], these have not been experimentally observed
yet. At the same time, the creation of optical lattices that are capable of
sustaining phonon modes is a highly active topic as well [8].

Another way to mimic lattice phonons is to engineer atomic mixtures, where one
species is trapped in an optical lattice, while spatially overlapping with a
second species that is Bose condensed. Scattering between these two components
would allow the atoms trapped in the lattice to effectively couple to the
Bogoliubov excitations of the Bose-Einstein condensate (BEC), which play the
role of phononic degrees of freedom [9, 10]. However, this is a very complex
setup and in this work we consider a significant simplification by replacing
the periodic optical potential by a standing matter wave formed by the BEC
itself.

Periodic matter waves can be created through interference in BECs [11] and
scattering from standing matter waves has been observed before [12, 13].
However, in these experiments the interference patterns are just transitory
and only recently has the realisation of the supersolid phase in BECs provided
a way to create stationary matter waves with broken, but periodic spatial
symmetry. Such phases can be reached by using spin-orbit coupled two-component
condensates [14, 15] or gases in which the atoms have large dipole moments
[16, 17, 18]. An impurity atom immersed in such a system will experience a
periodic mean-field potential, and therefore also the associated band-
structure spectrum. However, as the coupling is mediated by scattering, any
dynamics of the impurity will feedback onto the condensate density and lead to
the excitation of phonon modes in the mean-field lattice which, in turn, can
have an effect on the dynamics of the impurity itself [19, 20].

In this work we consider a paradigmatic example of lattice physics by
exploring the dynamics of an impurity that is accelerated through the
supersolid phase of a spin-orbit coupled BEC. Despite the atomic scattering
present in this dynamics, which can lead to the excitation of phonon modes, we
show that the impurity is able to undergo Bloch-like oscillations, which can
be quite stable and should therefore be observable. In fact, in the weakly
interacting regime considered in this work, the main reason for the ultimate
decay of the Bloch oscillations is the dispersion of the impurity wavepacket
oscillating in the matter wave lattice. We show that this can be mitigated by
considering an impurity that has the form of a non-dispersing bright soliton.

##  2 Model

We consider an atomic Bose-Einstein condensate of $N$ particles trapped in an
external box potential of length $L$. Two different hyperfine states of these
atoms represent a pseudo-spin that is used to implement the spin-orbit
coupling interaction through a two-photon Raman pulse from two counter-
propagating laser beams [21]. The system can then be described as a pseudo
spin $\frac{1}{2}$, where the standard kinetic term in the Hamiltonian is
modified to account for the coupling of the pseudo-spins to their linear
momentum. In the rotating frame of the laser, the single particle Hamiltonian
reads (considering one spatial dimension in the direction of the spin-orbit
coupling only) [22]

$H_{SO}=\frac{\hat{p}_{x}^{2}}{2m}-\frac{\hbar
k_{0}\hat{p}_{x}}{m}\hat{\sigma}_{z}+V_{{box}}\hat{I}+\frac{\hbar\delta}{2}\hat{\sigma}_{z}+\frac{\hbar\Omega}{2}\hat{\sigma}_{x},$
(1)

where ${p}$ and $m$ are the momentum and mass of the atom respectively,
$\Omega$ is the strength of the laser beams which couple the two spin states,
$\hat{\sigma}_{i}(i=x,y,z)$ are the Pauli spin operators and the external box
potential is given by
$V_{box}=\\{0~{}\textrm{for}~{}{-L/2}~{}\leq~{}x~{}\leq~{}L/2~{};~{}\infty~{}\textrm{for}~{}|x|>L/2\\}$.
The detuning is given by $\delta=\Delta\omega_{L}-\omega_{Z}$, where
$\omega_{Z}$ is the energy difference between the two pseudo-spin states and
$\Delta\omega_{L}$ is the frequency difference between the counter-propagating
laser beams and $k_{0}$ is the modulus of the wave number difference between
the two Raman lasers.

To consider a many particle system in the mean field limit, one needs to
include the spin-dependent interactions, which for point-like atomic
interaction potentials can be quantified by the three parameters
$g_{\uparrow\uparrow},g_{\downarrow\downarrow}$ and
$g_{\uparrow\downarrow}=g_{\downarrow\uparrow}$. Taking $\delta=0$ and
$g_{\uparrow\uparrow}=g_{\downarrow\downarrow}=g$ for simplicity, the
functional for the interaction energy is given by

$\displaystyle E_{int}=\frac{1}{4}\int dx\Big{[}$
$\displaystyle(g+g_{\uparrow\downarrow})(|\psi_{\uparrow}(x)|^{2}+|\psi_{\downarrow}(x)|^{2})$
(2)
$\displaystyle+(g-g_{\uparrow\downarrow})(|\psi_{\uparrow}(x)|^{2}-|\psi_{\downarrow}(x)|^{2})\Big{]},$

where $\psi_{\uparrow}$ and $\psi_{\downarrow}$ are the mean-field
wavefunctions of the condensate atoms in the spin-up and spin-down states,
respectively. Using the paradigmatic ansatz given by Li et al. [23] and
minimising the total energy functional in the fully repulsive regime reveals
the possibility to reach different ground states depending on the value of the
Raman coupling $\Omega$ and the recoil energy of the two photon Raman process
given by $E_{rec}=\frac{\hbar^{2}k_{0}^{2}}{2m}$. In this work we concentrate
on the region where $\Omega<4E_{rec}$ in which the ground state of the
condensate is given by a combination of two plane waves with equal and
opposite momenta [22]. Their interference results in modulations in the
density distribution (see Fig. 1) and this spontaneous breaking of the
continuous translation symmetry, along with the coherent nature of the BEC,
defines a supersolid state [15].

Next we consider an impurity atom immersed into the supersolid phase. In the
regime where the mean free path of the impurity is of the order of or larger
than its de Broglie wavelength, i.e. in the weakly interacting regime [19, 20,
24], the interaction with the condensate can be described by a density-density
coupling [25], in contrast to the regime in which Bose-polarons were recently
reported [9, 10]. In such a set-up the mean-field density of the BEC acts as
an effective potential for the impurity, however any dynamics of the impurity
will also have an effect on the condensate density.

Figure 1: A schematic of the system showing the density of the spin-orbit
coupled BEC (blue) and the impurity (red), which is accelerated through the
matter-wave lattice.

All three components are therefore coupled and the full system can be
described by

$\displaystyle i\frac{\partial\psi_{\uparrow}}{\partial t}$
$\displaystyle=\left[-\frac{1}{2}\frac{\partial^{2}}{\partial
x^{2}}-i\gamma\frac{\partial}{\partial
x}+V_{box}(x)+g|\psi_{\uparrow}|^{2}+g_{\downarrow\uparrow}|\psi_{\downarrow}|^{2}+{g}_{c}|\chi|^{2}\right]\psi_{\uparrow}+\frac{\Omega}{2}\psi_{\downarrow},$
(3) $\displaystyle i\frac{\partial\psi_{\downarrow}}{\partial t}$
$\displaystyle=\left[-\frac{1}{2}\frac{\partial^{2}}{\partial
x^{2}}+i\gamma\frac{\partial}{\partial
x}+V_{box}(x)+g_{\uparrow\downarrow}|\psi_{\uparrow}|^{2}+g|\psi_{\downarrow}|^{2}+{g}_{c}|\chi|^{2}\right]\psi_{\downarrow}+\frac{\Omega}{2}\psi_{\uparrow},$
(4) $\displaystyle i\frac{\partial\chi}{\partial t}$
$\displaystyle=\left[-\frac{1}{2}\frac{\partial^{2}}{\partial
x^{2}}+g_{c}(|\psi_{\uparrow}|^{2}+|\psi_{\downarrow}|^{2})+V_{{ext}}^{{imp}}(x)\right]\chi,$
(5)

where $\chi$ is the wavefunction of the impurity atom. To have a more general
picture of the ongoing dynamics, we will in the following use dimensionless
equations by scaling all energies with respect to the energy scale of the box
potential $E_{box}=\hbar^{2}/mL^{2}$ by transforming $x=\frac{x_{0}}{L}$ and
$t=t_{0}\frac{\hbar}{mL^{2}}$. To accelerate the impurity through the
condensate, Eq. (5) also contains a linear potential $V_{ext}^{imp}$ that is
only seen by the impurity. For simplicity we consider the interactions between
the impurity atom and the BEC atoms to be species independent and
characterised by $g_{c}$. All coupling constants are related to their
respective 3D scattering lengths, $a_{s}^{q}$, by the expression
$g^{\prime}_{q}=\frac{4\pi\hbar^{2}a_{s}^{q}}{m},(q=\uparrow\uparrow,\uparrow\downarrow,\downarrow\uparrow,\downarrow\downarrow,c)$,
where $m$ is the mass of the scattering atom and
$g_{q}=\frac{g^{\prime}_{q}}{LE_{{box}}}$. We assume all scattering lengths to
be positive, which corresponds to a repulsive interaction between all atoms
and components. The rescaled spin-orbit coupling constant is given by
$\gamma=\frac{\gamma^{\prime}}{LE_{box}}$, where $\gamma^{\prime}=\hbar
k_{0}/m$. The wavefunctions are normalised as
$\int|\psi_{\uparrow}(x,t)|^{2}dx=\int|\psi_{\downarrow}(x,t)|^{2}dx=\frac{N}{2}$
and $\int|\chi(x,t)|^{2}dx=1$. In the following section we will solve Eqs.
(3)-(5) and explore the dynamics of the impurity inside the supersolid phase
of spin-orbit coupled BEC for various values of the system parameters.

##  3 Bloch Oscillations

To create a well-defined initial state, we assume that the impurity is
initially localised at a fixed position inside the supersolid and held by a
species selective external harmonic trap of frequency $\omega_{x}$, with
$\pi/\tilde{k}\ll\sqrt{\hbar/(m\omega_{x})}\ll L$. Here $\pi/\tilde{k}$ is the
lattice constant of the supersolid, with
$\tilde{k}=\gamma\sqrt{1-\frac{\Omega^{2}}{4(\gamma^{2}+\frac{\tilde{n}(g+g_{\uparrow\downarrow})}{4})}}$
and $\tilde{n}$ denotes the density of the condensate [22].

(a)(b)(c)(d)

Figure 2: Time evolution of the density distribution of the impurity (top) and
density of the spin-orbit coupled BEC at $t=0$ (bottom) for the three
different cases of (a) $\Omega=0$, $\frac{g_{c}}{L}=3.6$, (b)
$\Omega=6.4\times 10^{4}$, $\frac{g_{c}}{L}=3.6$, (c) $\Omega=6.4\times
10^{4}$, $\frac{g_{c}}{L}=5.2$ and (d) in the artificial setting where the
lattice is frozen in time leading to non-dispersive oscillations of the
impurity with $\Omega=6.4\times 10^{4}$, $\frac{g_{c}}{L}=5.2$. Other
parameters of the system are
$\frac{g_{\downarrow\downarrow}}{L}=\frac{g_{\uparrow\uparrow}}{L}$ =
$1.6\times 10^{3}$ and
$\frac{g_{\downarrow\uparrow}}{L}=\frac{g_{\uparrow\downarrow}}{L}=8\times
10^{2}$, $N=5\times 10^{3}$, $\frac{\gamma}{L}=2.8\times 10^{2}$ and $\alpha
L=8.32\times 10^{4}$ and the lattice spacing $\frac{d}{L}=1.23\times 10^{-2}$
in the box units.

The impurity therefore extends over multiple lattice sites of the supersolid
and its energy is larger than the fringe depth of the lattice, cf. [26]. The
coupled ground state of the system is then obtained using a split-step Fourier
method and imaginary time propagation [27, 28]. Since the interaction between
the BEC and the impurity is assumed to be repulsive, the initial state of the
BEC has a density dip at the position of the impurity (see lower row in Fig.
2) and the density of the impurity is weakly corrugated with the matter wave
lattice periodicity. To accelerate the impurity and start the dynamics at
$t=0$, the harmonic potential is switched off and replaced by a linear ramp
$V_{ext}^{imp}(x,t>0)=-\alpha x$. As a reference case, we first consider the
BEC in the absence of the supersolid modulation, $\Omega=0$, when the
condensate density is flat. As one would expect, in this situation the
acceleration leads to the impurity simply getting expelled from the gas (see
Fig. 2(a)).

Contrary to this, in the presence of a lattice structure and for small, but
finite values of coupling between the impurity and the BEC one can see from
Fig. 2(b) that the impurity remains trapped within the condensate by
undergoing oscillations. These oscillations can be identified as Bloch
oscillations, as their period matches closely with the standard textbook
prediction for this effect given by $\tau_{B}=\frac{2\pi}{\alpha d}=6.14\times
10^{-3}$ in the box units [29]. Here $\alpha$ is the strength of the linear
force and $d$ is the lattice spacing. These oscillations are similar to the
Bloch oscillations known for electrons in solid state systems [30, 31] or for
atoms in optical lattices [26, 32, 33, 34, 35], but here they are due to the
periodic structure of the supersolid matter wave potential. Once the impurity
has picked up enough momentum, it gets Bragg reflected at the band edge and
starts running up the linear potential again until its approximate starting
point. There it gets reflected due to the linear potential and is accelerated
through the supersolid again. Let us stress that the numerical simulations are
performed on a grid large enough that boundary effects do not affect the
results shown.

However, one can also see from Fig. 2(b) that over time these oscillations
dephase and that the impurity wavefunction disperses. This is partly due to
the initial state at $t=0$ not being an eigenstate of the system (i.e. once
its harmonic oscillator potential is being removed) and partly due to the
interaction between the impurity and the BEC. The latter leads to the creation
of phonons in the condensate and therefore changes the energy and the momentum
distribution of the impurity. The dephasing of the oscillations and the
dispersing of the impurity wavefunction is even faster for increased
scattering with the matter wave, see Fig. 2(c). For $\frac{g_{c}}{L}=5.2$ the
impurity wavefunction very quickly delocalises across the whole condensate,
however most of it is still located in between the two turning points of the
oscillation. To show that this dephasing of the impurity wavefunction is due
to the scattering with the matter-wave lattice and not due to, say, tunnelling
into higher bands, we show in Fig. 2(d) the situation where the feedback of
the impurity motion onto the condensate is artificially turned off. As can be
seen, in the absence of induced phonon modes in the BEC the impurity
oscillates almost perfectly over a long time.

(a)(b)

Figure 3: The momentum distribution of the impurity as a function of time
(top) corresponding to $\frac{g_{c}}{L}=3.6$ and $\frac{g_{c}}{L}=5.2$. The
bottom panels show the momentum peaks at three different times, namely
$t_{0}=0$, $t_{1}=0.508\tau_{B}$ and $t_{2}=2.341\tau_{B}$ in the box units.

The redistribution of momentum of the impurity due to scattering is shown in
Fig. 3 for the same weak and strong BEC-impurity coupling used in Figs. 2(b)
and (c). One can see that the momentum distribution,
$\chi(k,t)=\frac{1}{2\pi}\int dx\,e^{-ikx}\chi(x,t)$, is initially localised
around zero and then evolves due to the linear potential, until it hits the
point where it gets Bragg reflected. The momentum for which this happens is,
for a plane wave, given by the Bragg reflection criterion,
$2d\sin\theta=\lambda$, where $d$ is the lattice spacing. Setting
$\sin\theta=1$ and rewriting $\lambda=\frac{2\pi}{k}$, we have the standard
condition $k=\frac{\pi}{d}$, which gives $k\approx 2.6\times 10^{2}$ in box
units and which is close to the numerically observed value for our Gaussian
wavepacket.

In addition Fig. 3 shows that the momentum distribution of the impurity
delocalises over time due to scattering with the matter-wave atoms. The top
row in Fig. 3 shows the momentum distribution as a function of time while the
bottom row displays snapshots for the state at the initial time ($t_{0}=0$),
around the time of the first Bragg reflection ($t_{1}=0.497\tau_{B}$) and at a
later stage after two reflections on each side have happened
($t_{2}=2.28\tau_{B}$). One can see that for weaker interactions the momentum
distribution of the impurity starts to de-phase after a few reflections,
however it never becomes fully delocalised on the scale displayed. In contrast
to this, the momentum distribution for the stronger interacting case
$\frac{g_{c}}{L}=5.2$ very quickly delocalises and while the impurity is
mostly trapped within the supersolid, it does no longer behave like one
oscillating particle.

Let us next look at the effect the scattering with the impurity has on the
supersolid, by examining the density distribution. The destruction of the
initial perfect lattice for $t>0$ can be quantified by looking at the average
of the lattice density maxima, $\overline{\rho}_{max}$, and the corresponding
variance, $\pm\sigma_{\overline{\rho}_{max}}$, as a function of time (see Fig.
4). The average lattice maximum is defined as the mean value of all local
maxima of
$\left(|\psi_{\uparrow}(x,t)|^{2}+|\psi_{\downarrow}(x,t)|^{2}\right)$ within
the central region of the spin-orbit coupled BEC, that is
$\frac{x}{L}\in(-1/3,1/3)$, and its variance is
$\sigma_{\overline{\rho}_{max}}^{2}=\langle\overline{\rho}_{max}^{2}\rangle-\langle\overline{\rho}_{max}\rangle^{2}$.
We highlight the changes over time by plotting
$\zeta(t)=\overline{\rho}_{max}(t)-\overline{\rho}_{max}(t=0)$ in Fig. 4 for
the two different cases $\frac{g_{c}}{L}=3.6$ (a) and $\frac{g_{c}}{L}=5.2$
(b). The upper panels show the lattice structure at two instants of time,
namely at $t_{1}=0$ and $t_{2}=3.97\tau_{B}$, with the horizontal black line
indicating the average and the dashed red line the variance. Note that the
variance at $t_{1}=0$ is finite as the lattice density has a dip where the
impurity is located.

(a)(b)

Figure 4: The lattice maxima as a function of time for the cases of
$\frac{g_{c}}{L}=3.6$ and $\frac{g_{c}}{L}=5.2$ respectively. The upper panels
show the condensate density as a function of position at $t_{1}=0$ and
$t_{2}=3.97\tau_{B}$ with the average lattice maximum indicated by a black
line and the variance by red, dashed lines. The lower panels show the changes
in these two quantities over time. The other system parameters are the same as
in Fig. 2

One can see that the phonon modes that are excited in both cases are very
similar at these low energies and on these timescales, and mostly differ in
magnitude. This can also be seen in the detailed plots in the lower row of
Fig. 4, however a large number of differences in the details are visible as
well. While the elementary excitations in spin-orbit coupled supersolid are
given by a hybridisation of the low lying branches corresponding to phonon and
spin excitations [36], the symmetric coupling of the impurity to both
condensate components (see eqs. (3) - (5)) prevents the spin mode from being
excited [37] and all excitations seen in Fig. 4 come from the standard phonon
branch. It is also worth noting that the movement of the impurity does not
lead to any significant variation in the lattice periodicity i.e. there are no
breathing modes in the system. At the energies available in this system, the
scattering processes play a role only in the linear phononic part of the
spectrum and while the excitation of the roton mode is possible once the
impurity has reached a momentum $\hbar k$, we do not observe any significant
sign of it.

Since the decay of observable oscillations can be mainly attributed to the
dispersion and the scattering with the matter wave lattice, it is interesting
to consider this situation for a non-dispersing impurity wavepacket. This can
be realised by assuming that the impurity is a small condensate itself, where
an intrinsically attractive interaction leads to a bright soliton state [38,
39, 40]. Adding this non-linear interaction to the evolution equation (5) of
the impurity leads to

$i\frac{\partial\phi}{\partial
t}=\left[-\frac{1}{2}\frac{\partial^{2}}{\partial
x^{2}}+g_{c}(|\psi_{\downarrow}|^{2}+\psi_{\uparrow}|^{2})+V_{{ext}}^{{imp}}(x)-g_{s}|\phi|^{2}\right]\phi,$
(6)

where the normalisation is now given by $\int|\phi(x,t)|^{2}dx=N_{s}$. Here
$N_{s}$ is the number of atoms in the soliton component and $g_{s}>0$ is the
strength of the self interaction.

The dynamics in this situation is shown in Fig. 5 for the values
$\frac{g_{c}}{L}=3.6$ and $\frac{g_{s}}{L}=80$. Comparing to the situation
shown in Figs. 2(b) and 3(a), it is immediately obvious that the impurity
undergoes many more well defined oscillations and that the momentum
distribution is not spreading out as much. Yet, the soliton still undergoes
scattering with the matter wave lattice and loses energy to it, which can be
seen from the fact that the upper turning point moves down the linear
potential over time. These collisions excite phonon modes in the matter-wave
lattice, which leads to amplitude fluctuations only, see Fig. 5(c). Note that
the the ratio of the interaction strengths needs to be $\frac{g_{s}}{g_{c}}\gg
1$, in order for the impurity to maintain its solitonic properties.

(a)(b)(c)

Figure 5: (a) Density of the soliton component as a function of time. One can
clearly see that the oscillations are much more pronounced when compared to
the single atom case, see Fig. 2(b). (b) Momentum distribution of the soliton
as a function of time. The momentum does not break into smaller peaks in
contrast to the case in 3(a). (c) Lattice density showing the phononic
amplitude fluctuations at $t=3.97\tau_{B}$.

##  4 Conclusions and Outlook

In this work we have considered the accelerated motion of a localised atomic
impurity that is repulsively coupled to a spin-orbit coupled BEC in the
supersolid phase. In this phase the BEC possesses a periodic density
distribution, which is experienced by the impurity as a mean-field lattice
potential. We have shown that once the momentum of the accelerated impurity
fulfils the Bragg condition, it gets reflected and decelerates until coming to
a stop at (almost) its initial position again. There the cycle starts over,
leading to Bloch-like oscillations.

However, for an initially localised single-atom impurity the momentum
distribution disperses over time and the repulsive scattering with the BEC
leads to a redistribution of the individual components. Due to both of these
effects the initially clearly observable oscillations dephase for longer times
and the impurity wavefunction delocalises across the whole mean-field
potential. We have shown that this effect can be partly counteracted by
considering an impurity that has the form of a bright soliton.

The scattering between the impurity and the BEC also leads to the excitation
of phonon modes inside the supersolid. However, due to the low energy of the
collisions we consider these only effect the amplitude of the lattice and not
disturb the periodicity. In fact the periodicity is determined by the position
of the roton-minimum and depends on the details of the spin-orbit coupling
term, which requires higher energies to be affected.

The systems we have considered here is an example of a cold atom analogue of a
well known condensed matter system. While cold atoms have been used to explore
many analogies in the recent two decades using optical lattice settings [1],
the system considered here adds the fundamental aspect of scattering with the
lattice component and the excitation of phonon modes. Even though the
experimental implementation of the system considered above is certainly
challenging, all ingredients required have been independently demonstrated:
Bose-Einstein condensates can be created in flat-bottom box potentials [41],
the striped phase in the presence of spin-orbit coupling has been observed
[15], and single impurities can be introduced into condensates in a controlled
way [42]. We believe that this system will be a feature-rich addition to the
area of quantum simulators in ultracold atom systems with strong connections
to condensed matter physics.

## Acknowledgements

We thank Prof. P. Engels, Dr. K. Gietka, Dr. T. Fogarty and L. Ruks for
fruitful discussions. This work was supported by the Okinawa Institute of
Science and Technology Graduate University. JP also acknowledges the JSPS
KAKENHI Grant Number 20K14417. The authors are also grateful for the the
Scientific Computing and Data Analysis (SCDA) section of the Research Support
Division at OIST.

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