# Superfluid fraction of interacting bosonic gases

Daniel Pérez-Cruz  Departamento de Física and IUdEA, Universidad de La Laguna,
La Laguna, Tenerife, Spain  Grigori E. Astrakharchik  Departament de Física,
Universitat Politècnica de Catalunya, Campus Nord B4-B5, E-08034 Barcelona,
Spain  Pietro Massignan  Departament de Física, Universitat Politècnica de
Catalunya, Campus Nord B4-B5, E-08034 Barcelona, Spain

###### Abstract

The superfluid fraction $f$ of a quantum fluid is defined in terms of the
response of the system to a weak and constant drag. Notably, Leggett long ago
derived two simple expressions providing a rigorous upper bound and a
heuristic lower bound for $f$. Here we study the superfluid fraction of
bosonic gases in various two-dimensional potentials, such as regular optical
lattices and disordered speckles, by solving the Gross-Pitaevskii equation and
performing Diffusion Monte Carlo simulations. We show that under conditions
relevant for most ultracold experiments the bounds proposed by Leggett provide
a surprisingly narrow bracketing of the exact value of the superfluid
fraction.

Superfluidity is one of the most striking effects of quantum mechanics. This
phenomenon has been observed in a wide variety of systems, encompassing liquid
helium, dilute Bose gases, strongly-interacting fermionic mixtures and even
neutron stars Kapitza (1938); Baym  _et al._ (1969); Pethick and Smith (2008);
Pitaevskii and Stringari (2016). In the superfluid state, a many-body ensemble
is characterized by a macroscopic wavefunction, and it features remarkable
properties such as frictionless flow and quantized vorticity. Generally,
however, only a portion of a system is superfluid. Following Landau’s two-
fluid hydrodynamics Landau (1941), the total mass density is written as
$\rho=\rho_{n}+\rho_{s}$, where $\rho_{n}$ and $\rho_{s}$ are the normal and
superfluid densities, respectively. At finite temperatures, a normal component
arises naturally due to thermal excitations, and its interplay with the
superfluid component yields second sound Sidorenkov  _et al._ (2013). But a
sizable $\rho_{n}$ can be present even at zero temperature. This happens, for
example, in the presence of an external field, which can even fully destroy
the superfluidity, as in Mott insulators Fisher  _et al._ (1989); Morsch and
Oberthaler (2006); Baym and Pethick (2012).

In two remarkable papers, A. Leggett noticed that a key property prompting the
reduction of the superfluid fraction of a many-body system at zero temperature
is the presence of (spontaneous, or induced) density modulations, which break
translation or Galilean symmetries Leggett (1970, 1998). The superfluid
fraction $f=\rho_{s}/\rho$ is a dynamical property, but surprisingly Leggett
derived simple lower and upper bounds directly from the ground state density
$n=|\Psi|^{2}$, which is real and carries no information about the velocity
field of the condensate and therefore, a priori, about dynamical properties.

Ultracold quantum gases are excellent platforms for exploring the accuracy of
the Leggett’s bounds. Recent works proved that the bounds are exact for dilute
Bose gases in the presence of one-dimensional (1D) optical lattices Chauveau
_et al._ (2023); Tao  _et al._ (2023). Here we demonstrate by performing
Diffusion Monte Carlo and Gross-Pitaevskii simulations, that quite
surprisingly the bounds bracket closely the exact value of the superfluid
density of dilute Bose gases in a broad class of two-dimensional (2D) regular
and disordered potentials. Furthermore, we discuss the applicability of such
bounds to other many-body systems.

Figure 1:  Sketches illustrating Leggett’s lower and upper bounds to the
superfluid fraction $f$. The superflow (blue arrows) may be partially
obstructed by barriers (yellow boxes), or entirely blocked by “transverse
nodal surfaces” (red box).

Superfluid fraction and Leggett’s bounds.— Given a system of $N$ particles of
mass $m$, its superfluid fraction is a tensor defined through the response
along the direction $\alpha$ to a perturbation moving with velocity
$v_{\beta}$ along $\beta$:

$f_{\alpha\beta}=1-\lim_{v\rightarrow
0}\langle\hat{P}_{\alpha}\rangle/(Nmv_{\beta}),$ (1)

where $\langle\hat{P}_{\alpha}\rangle$ is the mean total momentum along
$\alpha$. In the following, we will only consider the parallel response to a
drag along $x$, and denote for simplicity $f_{xx}$ by $f$. The Leggett’s lower
and upper bounds to $f$ read Leggett (1970, 1998)

$f_{\downarrow}=\frac{1}{\bar{n}}\biggl{\langle}\frac{1}{\langle\frac{1}{n}\rangle_{x}}\biggl{\rangle}_{{\mathbf{r}}_{\perp}}\;\;\;\textrm{
and
}\;\;\;f_{\uparrow}=\frac{1}{\bar{n}}\frac{1}{\bigl{\langle}\frac{1}{\langle
n\rangle_{{\mathbf{r}}_{\perp}}}\bigl{\rangle}}_{x}.$ (2)

The upper bound is derived variationally and therefore rigorous (always
exact), while the lower one is heuristic and expected to hold only for dilute
bosonic gases. Here $\bar{n}=N/L$ stands for the mean density,
${\mathbf{r}}_{\perp}$ represents all coordinates in the hyperplane
perpendicular to $x$, and $\langle\cdot\rangle_{\alpha}$ denotes a spatial
average over coordinate $\alpha=\\{x,{\mathbf{r}}_{\perp}\\}$. Whenever the
particle density factorizes as
$n({\mathbf{r}})=n_{x}(x)n_{\perp}({\mathbf{r}}_{\perp})$, the two bounds
coincide:

$f_{\downarrow}=f_{\uparrow}=\frac{1}{\langle
n_{x}\rangle_{x}\langle\frac{1}{n_{x}}\rangle}_{x}\quad\textrm{(factorized
}n),$ (3)

and thus this expression gives the exact result for the superfluid density.
This factorized form for $n({\mathbf{r}})$ arises for example for Bose gases
in 1D optical lattices, the situation studied in the recent Refs. (Chauveau
_et al._ , 2023; Tao  _et al._ , 2023).

A physical intuition about these bounds may be obtained by examining the two
sketches presented in Fig. 1. The lower bound is the easiest to understand.
Consider slicing the system into 1D tubes along the axis parallel to the drag.
To get $f_{\downarrow}$, one computes Eq. (3) along each tube (finding $f<1$
whenever a barrier obstructs it), and averages the results over all tubes. By
ignoring the cross-talk between the tubes, one obtains naturally a lower bound
to $f$. The upper bound is slightly more subtle to grasp. Here one slices the
system perpendicularly to the force, then extracts the superfluid fraction
traversing each plane, and finally uses this quantity to compute Eq. (3) along
the direction of the force. By first computing the transverse average, one
erases all information about the particular path the superfluid took to
traverse each plane. For example, the path shown in the right panel contains
sharp turns, and the superfluid is not likely to follow it because of its high
kinetic cost. As a consequence, this procedure produces an upper bound
$f_{\uparrow}$. Importantly, whenever the system presents transverse nodal
surfaces [i.e., regions where $n(x_{0},{\bf r}_{\perp})=0$ for all values of
${\bf r}_{\perp}$, see the red box in Fig. 1(b)], then the upper bound
immediately vanishes, and therefore the system must be fully normal.

Superfluid fraction in 2D Bose gases—In this work, we study the superfluid
response of a two-dimensional bosonic system by comparing a mean-field
approximation, specifically the 2D Gross-Pitaevskii model, with numerically
exact Diffusion Monte Carlo (DMC) simulations of 3D Bose gases under strong
transverse confinement. Our investigation covers systems in the presence of
potentials which are periodic, disordered, and hybrid (ordered along one
direction, and disordered along the other).

Our primary objective is to quantify the superfluid behavior of the system. In
the reference frame of the moving potential, the Gross-Pitaevskii equation
(GPE) reads

$\left[-\frac{\hbar^{2}}{2m}\left(\nabla+\frac{mv}{\hbar}\hat{\mathbf{x}}\right)^{2}+V(\mathbf{r})+g|\psi(\mathbf{r})|^{2}\right]\psi(\mathbf{r})=\mu\psi(\mathbf{r}),$
(4)

where $V(\mathbf{r})$ is the external potential, $g$ the 2D coupling constant,
and $\mu$ the chemical potential.

The key point is that only the superfluid component reacts to the drag. In
particular, the perturbed and unperturbed ground state energies are related by

$E(v)=E(0)+f\frac{mv^{2}}{2}\hskip 10.0pt(v\xrightarrow{}0),$ (5)

an expression directly yielding the superfluid fraction $f$. This approach is
particularly suitable for a numerical study because very robust numerical
techniques have been developed to study the ground state properties of the
Gross-Pitaevskii equation (see, e.g., Refs. McLachlan (1993); Suzuki (1991)).

To independently assess the validity of the 2D GPE description of this system,
we perform numerically exact Diffusion Monte Carlo simulations in three
dimensions, adding to the potential $V$ in the $xy$ plane a tight harmonic
oscillator in the $z$ direction. The interaction potential between the
particles is modeled by a soft-sphere potential with the diameter equal to the
transverse oscillator length $a_{\rm ho}$ and scattering length $a_{3D}$ small
compared to it, $a_{3D}/a_{\rm ho}=0.1$. Under these conditions, to a good
approximation, the mean-field relation between the three-dimensional
parameters and two-dimensional coupling constant can be used,
$g=4\pi\hbar^{2}(a_{3D}/\sqrt{2\pi}a_{\rm ho})/m$. We consider $N=100$
particles in a box of size $L\times L$ with periodic boundary conditions, and
we keep the 2D density $n=N/L^{2}$ small, so that $na_{\rm ho}^{2}=0.1$.
Within the MF relation for the 2D scattering length, $\xi=3.16a_{\rm ho}$. The
guiding wave function is taken in a similar form as in Ref. Astrakharchik  _et
al._ (2013), i.e. it is constructed as a product of one-body terms (Gaussian
along $z$, corresponding to the ground-state of the transverse harmonic
oscillator, and GPE solution in $(x,y)$ plane, taking into account the effect
of the potential $V$) and two-body Jastrow terms (constructed by matching the
solution to the two-body scattering problem at short distances and an
exponential decay at large distances). The superfluid fraction in DMC
calculation is obtained by using the winding number technique Pollock and
Ceperley (1987).

Figure 2:  Superfluid fraction in a 2D optical lattice and in a regular grid
of repulsive square columns (shown in the inset) plotted as a function of the
mean value $\langle V\rangle$ of the potential divided by
$E_{0}=\hbar^{2}/(2md^{2})$. Solid symbols correspond to an optical lattice
potential with lattice constant $d$ and two different values of the coupling
strength. Squares show results for the column potential with $\xi/d=1$. Dotted
lines show the weak-perturbation Bogoliubov approximation, Eq. (6).

Regular and hybrid potentials— In this paper we investigate the behavior of
the Leggett’s bounds in 2D potentials. It is important to highlight the
significant conceptual difference with the case of 1D optical lattices
recently investigated in Refs. Chauveau  _et al._ (2023); Tao  _et al._
(2023). In the latter studies the density remained separable, and therefore by
Eq. (3) the bounds coincide. However, the density of an interacting gas in a
2D potential is non-separable, and therefore the accuracy of the bounds is an
open question.

We start by considering two kinds of regular potentials: a symmetric 2D
optical lattice with equal spatial period $d$ and intensity along both
directions, and a 2D grid of square columns whose centers are separated by a
distance $d$111Along each direction, a column of this potential is modeled as
$V_{\text{col}}(x)\propto\tanh{\left[16\left(x+l/2\right)/d\right]}-\tanh{\left[16\left(x-l/2\right)/d\right]}$,
with $l\approx d/2$ determining the linear size of the column.. We present in
Fig. 2 our results for the superfluid fraction $f$, along with its upper and
lower bounds $f_{\downarrow}$ and $f_{\uparrow}$ given by Eq. (2), and with
the analytic Bogoliubov result valid for weak perturbations Singh and Rokhsar
(1994); Lugan and Sanchez-Palencia (2011); Astrakharchik  _et al._ (2013)

$f_{s}=1-\frac{2}{L^{2}}\int\limits_{{\bf k}\neq 0}d^{2}k\frac{|V({\bf
k})|^{2}}{\left[\hbar^{2}k^{2}/(2m)-2gn\right]^{2}}$ (6)

with $V({\bf k})=L^{-2}\int d^{2}r\,e^{-i{\bf k}\cdot{\bf r}}\,V({\bf r})$,
for various values of the repulsive interactions. We consider several values
of the healing length $\xi=\sqrt{\hbar^{2}/(2m\mu)}$. We find that as long as
$\xi$ does not exceed substantially the spatial period $d$ the bracketing
provided by the bounds is so tight that, for all practical purposes, both
bounds coincide with the exact $f$. The reason underlying the closeness of
$f_{\downarrow}$ and $f_{\uparrow}$ is that when $\xi<d$ the density is well
approximated by a Thomas-Fermi profile $n_{\rm TF}({\bf r})=[\mu-V({\bf
r})]/g$, which is naturally factorized, and therefore Eq. (3) approximately
holds. The bracketing is tighter for a 2D optical potential than for a grid of
columns, because the latter presents sharp edges. Near the edges the ground
state density varies rapidly, making it harder to approximate by a factorized
form, and this results in a looser bracketing.

The superfluid-to-normal transition takes place when the mean potential height
becomes comparable to the chemical potential $\mu$. A notable feature visible
in Fig. 2 is that $f$ for the column potential approaches a constant value for
large $\langle V\rangle$. This is easy to understand. Our grid of columns is
aligned parallel to both the $x$ and $y$ axes, so that there are long
unobstructed channels between them along the direction of the drag. In Fig. 2
we chose $\xi=d$, so that the gas can easily penetrate in between the columns,
no matter how high they are. When they become effectively impenetrable, the
lower bound saturates to a constant, and the gas will keep flowing around the
columns without being further affected by the changes in the potential.

Figure 3:  Superfluid fraction in a hybrid potential. Stars denote disordered
potential (speckles) along the direction of the drag and ordered (optical
lattice) along it, while filled circles denote the opposite situation. The
results are for $\xi/L_{c}=1$, and averaged over five different speckle
realizations.

We continue by studying the superfluid fraction in a hybrid potential,
obtained by summing 1D disordered speckles along one direction and a 1D
optical lattice along the other. This asymmetric potential is illustrative
because rotating it by 90∘ allows us to probe directly the tensor nature of
the superfluid fraction 222Since here we consider time-reversal symmetric
Hamiltonians, the off-diagonal elements of the superfluid fraction vanish, and
in this case strictly-speaking the latter is more simply a vector. Speckle
patterns are obtained by diffracting light off a rough surface, and constitute
the simplest form of disorder since they are characterized by a single
dimensionless parameter, the ratio of their mean amplitude $\langle V\rangle$
to the characteristic energy $E_{0}=\hbar^{2}/(2mL^{2}_{c})$ obtained from the
correlation length $L_{c}$ of the speckles Clément  _et al._ (2006). We
generate these patterns by computing the Fourier transform of aperture
functions whose entries are random complex numbers with unit modulus, thereby
mimicking the random and uncorrelated scattering events on the rough surface.
We consider the simplest situation where the spatial period $d$ of the optical
lattice coincides with the correlation length $L_{c}$ of the speckles, and the
mean amplitude equals $\langle V\rangle$ in both directions. As shown in Fig.
3, radically different behaviors are observed when the drag is applied
parallel to the disordered direction as opposed to the periodic one. The first
case closely resembles the idealization used to discuss the lower bound at the
beginning of this work. The optical lattice “slices” the system into 1D tubes
along the axis of the drag, with limited cross-talk between them, and all
tubes contain the same speckle potential. The dynamics is then basically 1D,
and as soon as the disorder contains a speckle which is sensibly higher than
the chemical potential, all tubes are suddenly blocked (i.e., there appears a
transverse nodal surface), and the superfluid density vanishes rapidly through
a rather abrupt transition. The second case is very different: now we have
tubes with very variable cross-talk, and along each tube the fluid sees a
periodic potential. In this situation, it is much harder to form transverse
nodal surfaces, and the resulting superfluid-normal transition is smoother and
takes place at much larger values of $\langle V\rangle$.

Figure 4:  Superfluid fraction in 2D speckle disorder, plotted against the
mean height $\langle V\rangle$ of the speckles. Left/right curves with filled
symbols represent the GPE results for a gas with weaker/stronger repulsive
interactions, the open stars denote DMC data, the dotted lines are the weak-
perturbation Bogoliubov approximation (6), and the shaded regions show the
windows left open by Leggett’s lower and upper bounds $f_{\downarrow}$ and
$f_{\uparrow}$. Results averaged over five speckle realizations. The
statistical errors due to the disorder averaging are smaller than the symbols’
size. No significant difference between GPE and DMC was found for the bounds.

2D speckle disorder—We conclude our analysis by studying the superfluidity in
2D speckle potentials. Our results for the superfluid fraction $f$ and its
bounds $f_{\downarrow}$ and $f_{\uparrow}$ are presented in Fig. 4. Finite-
size effects are important for 2D disorder, and we account for them by doing
calculations for increasing system sizes $L$, and extrapolating to the
infinite size limit 333In agreement with Ref. Astrakharchik  _et al._ (2013),
we find that the superfluid fraction scales linearly with the system size:
$f(\infty)=f(L)+c/L+\ldots$, with $c<0$. . In analogy with the cases studied
before, the superfluid fraction starts to decrease when the mean disorder
amplitude exceeds the chemical potential $\mu$, and it becomes significantly
suppressed when $\langle V\rangle\gg\mu$. The GPE compares very favorably with
exact DMC when the gas is sufficiently dilute, a condition which is well-
verified for $\xi/L_{c}=3$.

The Leggett’s bounds are seen to bracket correctly the superfluid density, and
the window between them is particularly narrow for weak repulsive
interactions, such that $\xi>L_{c}$. Under these conditions, the superfluid-
insulator transition happens for very weak potentials, so that the ground
state wavefunction is approximately separable, and Eq. (3) shows that the two
bounds should be very close to the actual value of $f$. On the other hand, for
strong disorder (large $\langle V\rangle)$ the superfluid fraction and its
bounds all tend rapidly to zero, signaling the appearance of transverse nodal
surfaces in the ground state density.

Discussion and conclusions — We have shown that the bounds derived by Leggett
bracket with surprising accuracy the superfluid fraction of dilute Bose gases,
which can be described by the Gross-Pitaevskii equation, under many
experimentally-relevant situations. Our findings complement previous studies
where the bounds were used to study Josephson junctions Zapata  _et al._
(1998); Smith  _et al._ (2023); Pezzè  _et al._ (2023).

When Leggett first derived the upper bound for the superfluid fraction, he was
discussing the viability of a supersolid phase in crystalline systems Leggett
(1970). Using this bound Leggett estimated that the superfluid fraction in
solid helium should be very small (“probably $\leq 10^{-4}$”).

On the other hand, supersolidity has been recently observed in spin-orbit
coupled BECs, dipolar quantum gases and dilute Bose-Bose mixtures Li  _et al._
(2017); Tanzi  _et al._ (2019); Böttcher  _et al._ (2019); Chomaz  _et al._
(2019); Recati and Stringari (2023); Ripley  _et al._ (2023); Šindik  _et al._
(2023); Hirthe  _et al._ (2024). There, density modulations appear
spontaneously in the so-called “stripe-phase”, and the mean-field description
remains accurate. As such, we expect that the lower Leggett’s bound will
apply, and that the two bounds will narrowly bracket the actual value of $f$.

Nonetheless, there are various situations where the lower bound (which is
heuristic, i.e., not rigorous) completely fails. First, this happens in
systems possessing gapless excitations at finite momentum. In such cases, the
Landau criterion predicts that superfluidity is strongly reduced Baym and
Pethick (2012). At the same time the energy might remain quadratic in the
phase twist and the system could have a uniform density, so that both the
lower Leggett’s bound $f_{\downarrow}$ (2) and the definition of $f$ in Eq.
(5) would predict a fully superfluid system. This is the case for example in
an ideal Bose gas (which is pathologic as it has a vanishing sound speed), but
also in ideal Fermi gases in any dimensionality or in 1D Luttinger liquids,
which possess a non-zero sound speed, but have gapless Umklapp processes at
twice the Fermi momentum Cazalilla (2004); Astrakharchik and Pitaevskii
(2004); Carusotto and Castin (2004).

Similarly, $f_{\downarrow}$ fails for systems which are unstable to
infinitesimal perturbations, such as the ideal Fermi gas (which in this case
displays an Anderson orthogonality catastrophe) or Luttinger liquids with
Luttinger parameter $K<3/2$ in presence of disorder Giamarchi and Schulz
(1988); Giamarchi (2004); Cazalilla  _et al._ (2011). These systems retain an
almost uniform density (so that $f_{\downarrow}\approx 1$) even if their
energy is unaffected by the drag (and thereby one must have $f=0$).

The upper bound $f_{\uparrow}$ is instead rigorous, and therefore applies to
generic many-body systems. For example, Orso and Stringari proved by means of
sum rules that $f_{\uparrow}$ accurately approximates $f$ in a unitary two-
component Fermi gas, provided that the external perturbation probes the
phononic regime of small momentum, i.e., the paired nature of the system Orso
and Stringari (2023). At higher momenta, however, the spectrum is no-longer
simply phononic, and $f_{\uparrow}$ overestimates $f$.

Interesting open directions for future work include studying the accuracy of
Leggett’s bounds for strongly-interacting bosons Yao  _et al._ (2023), and
their extension to non-zero temperatures.

###### Acknowledgements.

We wish to thank David Clément, Jean Dalibard, Laurent Sanchez-Palencia and
Sandro Stringari for insightful discussions. We acknowledge support by the
Spanish Ministry of Science and Innovation (MCIN/AEI/10.13039/501100011033,
grant PID2020-113565GB-C21), by the Spanish Ministry of University (grant FPU
No. FPU22/03376 funded by MICIU/AEI/10.13039/501100011033), and by the
Generalitat de Catalunya (grant 2021 SGR 01411). P.M. further acknowledges the
ICREA Academia program, the Institut Henri Poincaré (UAR 839 CNRS-Sorbonne
Université) and the LabEx CARMIN (ANR-10-LABX-59-01).

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XX XX 8

# Serverless Predictions: 2021-2030

Pedro Garcia Lopez  Universitat Rovira i Virgili/IBM Watson Research,
Spain/USA  Aleksander Slominski  IBM Watson Research, USA  Michael Behrendt
IBM Deutschland Research & Development GmbH, DE  Bernard Metzler  IBM Research
Zurich, CH

###### Abstract

Within the next 10 years, advances on resource disaggregation will enable full
transparency for most Cloud applications: to run unmodified single-machine
applications over effectively unlimited remote computing resources. In this
article, we present five serverless predictions for the next decade that will
realize this vision of transparency – equivalent to Tim Wagner’s Serverless
SuperComputer or AnyScale’s Infinite Laptop proposals.

Serverless gained popularity in industry and in academia in last few years
[1]. It attracted companies and developers with a simple Function-as-a-Service
(FaaS) programming model that realized original promise of the Cloud:
elasticity and fine grained pay-as-you-go for actual usage. When a code is
running in FaaS developers do not have control over where the code is running
and do not need to worry about how to do scaling - serverless cloud providers
create transparency by removing servers, or at least making them more
transparent.

Transparency is an archetypal challenge in distributed systems that has not
yet been adequately solved. Transparency implies the concealment from the user
and the application programmer of the complexities of distributed systems.
According to Colouris [2], access transparency enables local and remote
resources to be accessed using identical operations.

Nevertheless, Waldo et al. [3] explain that the goal of merging the
programming and computational models of local and remote computing is not new.
They state that around every ten years “a furious bout of language and
protocol design takes place and a new distributed computing paradigm is
announced“. In every iteration, a new wave of software modernization is
generated, and applications are ported to the newest and hot paradigm.

We believe the Serverless Compute paradigm, as emerging today [1, 9], will
converge at the needed level of resource abstraction to enable transparency.
What we call The Serverless End Game is the process of mapping this principle
on emerging disaggregated computing resources (compute, storage, memory),
eventually enabling unlimited flexible scaling.

The major hypothesis of this paper is that full transparency will become
possible in the next years thanks to predicted advances on latency reduction
in distributed systems [4, 6, 16]. This will put an end to the aforementioned
cycles of software modernization. The consequences for the field will be
enormous, by considerably simplifying development and maintenance of software
systems for the majority of users.

## BACKGROUND

Latency improvements [4, 6] are boosting resource disaggregation in the Cloud,
which is the definitive catalyst to achieve transparency. As we can see in
table 1, current data center networks already enable disk storage
disaggregation, where reads from local disk are comparable (10ms) to reads
over the network. In contrast, creating a thread in Linux takes about 10µs,
still far better than 15ms/100ms (warm/cold) as achieved today in Function-as-
a-Service (FaaS) settings. The level of resource disaggregation as possible
today is specifically utilized by Serverless Platforms, and focus of research
in Disaggregated Data Centers (DDC) in general [8].

Table 1: Latencies for remote resource access Resource |  Local |  Remote today |  Remote soon
---|---|---|---
Storage |  10ms |  10ms |  0.05ms (NVM)
Compute |  0.01ms (Thread) |  15-100ms (FaaS) |  0.001-0.1ms (RPC)
Memory |  0.0001ms |  0.25ms (Redis) |  0.002-0.01ms (PMEM)

Providing access transparency over DDC resources is the aim of LegoOS: A
disseminated, distributed OS for hardware resource disaggregation [13]. LegoOS
exposes a distributed set of virtual nodes (vNode) to users. Each vNode is
like a virtual machine managing its own disaggregated processing, memory and
storage resources. LegoOS achieves transparency and backwards compatibility by
supporting the Linux system call interface, so that unmodified Linux
applications can run on top of it. For example, LegoOS executes two unmodified
applications: Phoenix (a single-node multi-threaded implementation of
MapReduce) and TensorFlow.

A good example of providing access transparency over serverless resources is
Lithops [14]. Lithops intercepts Python language libraries (multiprocessing)
in order to access remote serverless resources in a transparent way. Lithops
is however limited to running Python aplications using that library.

Another example of transparency in a serverless context is Faasm [19]. Faasm
exposes a specialized system interface which includes some POSIX syscalls,
serverless-specific tasks, and frameworks such as OpenMP and MPI. Faasm
transparently intercepts calls to this interface to automatically distribute
unmodified applications, and execute existing HPC applications over serverless
compute resources.

Faasm allows colocated functions to share pages of memory and synchronizes
these pages across hosts to provide distributed state. However, this is done
through a custom API where the user must have knowledge of the underlying
system, hence breaking full transparency. Furthermore, when functions are
widely distributed, this approach exhibits performance similar to traditional
distributed shared memory (DSM), which has proven to be poor without hardware
support.

Nevertheless, resource disaggregation is still in its infancy, and there is no
current solution to provide flexible scaling and access transparency over
remote shared memory. Container instantiation is slow comparing to local
threads, and even fast NVMs[16] are an order of magnitude slower than local
memory accesses which are in the nanosecond range [6].

Besides of the aforementioned constraints of current resource disaggregation,
serverless computing has a number of well known limitations[9] like: focus on
stateless computations, lack of efficient communication between executed tasks
or functions, maximum code runtime limitations, and deficiencies in
transparent integration of hardware accelerators.

## PREDICTIONS

The major hypothesis of this paper is that transparency will be achieved in
the next ten years thanks to novel advances in networking, disaggregation, and
middleware services. The huge consequence is the unification of local and
remote paradigms, which will democratize distributed programming for a
majority of users. This will realize the old and ultimate goal of hiding the
complexity of distributed systems.

The projected developments to reach the ultimate goal (Serverless End Game)
include the following:

  * •

Prediction 1: Serverless Clusters (Multi-tenant Kubernetes) will overcome the
current limitations of direct communication among functions, hardware
acceleration, and time limits.

  * •

Prediction 2: Serverless Granular computing will offer 1-10 µs microsecond
latencies for remote functions thanks to lightweight virtualization and fast
RPCs.

  * •

Prediction 3: Serverless memory disaggregation will offer shared mutable state
and coordination at 2-10 µs microsecond latencies over persistent memory.

  * •

Prediction 4: Serverless Edge Computing platforms leveraging 6G’s ms latencies
and AI optimizations will facilitate a Cloud Continuum for remote
applications.

  * •

Prediction 5: Transparency will become the dominant software paradigm for most
applications, when computing resources become standardized utilities.

We will discuss the basis of these predictions as well as technical challenges
and risks. As we can see in Figure 1, the predictions are mapped to phases to
reach the final goal. Let’s review the proposed forecasts.

Figure 1: Serverless End Game

### Prediction 1

In less than two years, a second generation of Serverless platforms will
overcome major limitations of previous FaaS offerings.

This is already starting to happen around the so called serverless containers
platforms like Google Cloud Run, IBM Code Engine, or Amazon Fargate. Multi-
tenant Kubernetes (K8s) clusters may become the necessary glue to build multi-
Cloud API-agnostic applications.

As a consequence, a number of Cluster toolkits will accelerate the transition
to K8s like Web frameworks (Flask, Rails, Node.js), Microservices (Quarcus,
Spring, Micronaut), Analytics (Spark, Ray, Dask), Actors (Akka), and even High
Performance Computing (MPI).

There will be intense efforts in cluster scheduling and workload analysis
using machine learning techniques to optimize the deployment of different
applications and frameworks.

Nevertheless, container instantiation times will still be slow (tens to 100s
ms) and similar to current FaaS technologies. And the lack of memory
disaggregation will preclude the transition for many applications.

### Prediction 2

Serverless Granular Computing will drastically reduce instantiation and
execution times thanks to more lightweight virtualization and execution
technologies[7] and fast RPCs[5]. Microsecond latencies will also propagate to
other middleware services such as messaging and collective communication.

Accessing disaggregated resources in a transparent manner requires a form of
lightweight, flexible virtualization that does not currently exist. This new
virtualization must intercept computation and memory management to provide
access to disaggregated resources, and must do so with native-like performance
and no input from the programmer. Several alternatives for lightweight
virtualization will emerge around micro-VMs, software-based virtualization
(Krustlet and WebAssembly) and platform-independent runtimes like GraalVM.

In summary, serverless granular computing will represent a performance boost
for most applications in the Cloud, provoking a massive redesign of many
toolkits (Web, games, Desktop).

### Prediction 3

In the next years, public Cloud providers will start offering Serverless
Memory Disaggregation at microsecond speeds. We leverage here previous work on
memory disaggregation like [20, 11, 10]. Services like Serverless Redis will
arrive soon with limited consistency requirements.

Nevertheless, disaggregated memory at microsecond latencies will never reach
the nanosecond latencies of local memory. As a consequence, the combination of
local memory with remote consistent shared memory will then create truly
Serverless Mutable State and Coordination services. This will open
transparency avenues for most stateful applications like: Serverless In-Memory
Databases, Analytics (TensorFlow, PyTorch, Phoenix, OpenMP), Game engines and
Multi-user Worlds (Unity, Minecraft), Desktop applications (Image editing,
Video Editing), and Collaborative applications (games, conferencing).

When Serverless Granular Computing and Disaggregated Memory become available,
the three types of resources (compute, storage, memory) may be offered through
virtual interfaces against disaggregated Serverless services.

This will imply parallel work in different layers of the software stack. The
first approach, as followed by Lithops[14] will consist in intercepting
language libraries and toolkits to transparently interact and combine local
and remote resources.

The second approach will adapt polyglot execution runtimes like GraalVM, WASM,
or .NET Common Language Runtime among others. This second phase is more
ambitious, and it will consistently arrive to all languages and applications
built on top of them.

Finally, the more advanced but also more complex approach is to redesign the
Kernel Space to accomodate Serverless Remote Resources in a seamless way. This
is more in line with initial DDC works like LegoOS[13] or Arrakis[18], but
they will certainly require hard engineering efforts to cover the entire
kernel APIs, and probably novel co-designs and OS improvements at all levels.
We outline here the virtual file system and network device drivers, virtual
memory management, and also process/thread management layers.

In this line, Arrakis[18] comes from previous efforts aimed at optimizing the
kernel code paths to improve data transfer and latency in the OS. In Arrakis,
applications have direct access to virtualized I/O devices, which allows most
I/O operations to bypass the kernel entirely without compromising process
isolation. Arrakis virtualized control plane approach allows storage solutions
to be integrated with applications (co-design), even allowing the development
of higher level abstractions like persistent data structures. Arrakis shows us
how its control plane may become a first step towards integration with a
serverless data center resource allocator.

Redesign of OS kernels will then enable the final goal of full transparency
for any native applications using OS public Call Interfaces. This will require
the collaboration of DDC and OS researchers, but also the research work of the
broader Systems community.

### Prediction 4

The computing power at the edge, along 1-2 millisecond latencies facilitated
by 6G will produce Serverless Edge platforms that will alter the operating
systems of mobile phones and terminals at the edge. The smooth integration
with Cloud resources will create a Cloud Continuum where transparent
applications directly access data in remote resources.

Micro-second latencies will also produce a flourishing of remote use
interfaces. In line with remote displays (X-Windows, VNC), many applications
could just focus edge devices in user interface interaction, moving all
computing logic to remote execution. This could provoke a revival of dumb
terminals that delegate their execution entirely to the cloud.

Nevertheless, dumb terminals cannot address more sophisticated use cases where
edge computing resources are relevant. In cases where you cannot neglect the
computing resources in the edge, local and remote resources should be
intelligently combined in the Edge/Cloud Continuum.

This will certainly become the last frontier for software modernization, which
will then define how to write local applications that use remote resources in
a transparent way. A plethora of novel applications (tele-presence, games,
collaborative work, augmented/mixed reality, virtual reality) will emerge
leveraging the novel programming abstractions, and existing ones will be re-
engineered or improved considerably.

### Prediction 5

In ten years, no application domain will remain unaltered for transparency as
a consequence of the Serverless End Game. The convergence of Granular
Computing, Disaggregated memory, and the Edge-Cloud continuum will definitely
unify the local and remote paradigms. Computing will finally become a utility
thanks to standards accepted by all providers.

## IMPACT

Achieving transparency will produce a profound impact in the computing field
and it will change drastically how applications are built in the future.

On the technical side, transparency has been the ultimate goal of distributed
systems for many decades now. Developers will still care about parallel
programming, object oriented programming, user interfaces, event-based
systems, or functional programming. However, most developers will no longer
worry about middleware, transport layers and marshaling, web-based protocols,
or Cloud-specific APIs among others.

On the technical side, the disaggregation (and standardization) of remote
computing resources will also have a profound impact on the edge: in the
redesign of Operating Systems for computers, mobile phones, and edge devices.
Novel development environments will emerge to cover the entire life cycle of
applications in the Cloud/Edge continuum.

On the economic side, transparency will also imply a revolution that will
shake the entire industry. The major economic impact will be in developer
costs and productivity for most applications. If Serverless technologies
transparently handle the distributed systems complexity, programming remote
applications will become as simple as programming local ones with infinite
resources.

Transparency will boost the consumerization of software, facilitating the
creation of many applications in different domains. More users will be able to
customize and develop software, benefiting from novel visual and no-code
tools.

The economic impact will also affect all Cloud providers on a first wave, in a
fierce competition to be the most efficient disaggregated backend for
applications. It will also affect Operating Systems in the edge (like Apple,
Windows, Linux, Android, or iOS), which will seamlessly integrate remote
resources.

In this integration of Cloud and Edge, some providers with presence in both
sides (Google, Microsoft, Apple) will have a certain advantage over the rest
of the industry. Nevertheless, open source giants like IBM/RedHat could also
offer integrated solutions in the Cloud and Edge based on Linux technologies.

In any case, microsecond latencies enabled in the data center, and 1-5 ms
latencies of edge resources will profoundly change how we interact with
technology. Analogously to the unification of the local and remote paradigms
in programming, we will see a blurring and unification of real and virtual
environments using Augmented and Virtual Reality.

## TRADEOFFS OF DISAGGREGATION AND TRANSPARENCY

Blurring the borders between local and remote resources may have important
advantages, but it also implies some tradeoffs and limitations that we must be
aware.

Waldo et al.[3] already considered 25 years ago that future hardware
improvements could make the difference in latency irrelevant, and that
differences between local and remote memory could be masked. But they still
claimed that concurrency and partial failures preclude the unification of
local and remote computing. Let´s study the major challenges and tradeoffs:

  1. 1.

Partial Failure: Existing applications are a blackbox for the Cloud, but the
transition will imply a “compile to the Cloud“ process. In this case, the
Cloud will have access over applications’ life cycle and it will be able to
offer failure transparency, and to optimize their execution performance and
cost. This means that they can perform static analysis to predict resource
requirements, failures, dependencies and potential for hardware acceleration.
In the past, extensive optimizations have been performed to declarative
dataflows, but there are also opportunities to improve imperative programs
with static analysis [15].

Transparency efforts for different types of applications will also require
customizable control planes. In particular, customized co-designs [17] may be
necessary to optimize resource usage. Such customization will be based on
advanced observability and fast orchestration mechanisms relying on standard
services and protocols. Monitoring and interception of the different resources
(compute, storage, memory, network) should be available and even integrated
into the data center, enabling coordinated actuators at different levels.

  2. 2.

Concurrency: Another important limitation when developing concurrent
applications is scaling transparency, which means that applications can expand
in scale without changes to the system structure or the application
algorithms. If the local programming model was designed to use a fixed amount
of resources, there is no magic way of transparently achieving scalability,
not to mention elasticity.

Some workloads that do not need elasticity, such as enterprise batch jobs or
scientific simulations, can just port existing code to the Cloud. If the
application is configured to run on a fixed number of resources (threads or
processes), it may run unmodified over the same fixed amount of disaggregated
resources. However, for more user driven and interactive services, such as
internal enterprise web applications, simple porting of the executables
(sometimes referred as “lift-and-shift“) is rarely enough. The unchanged code
is not able to take advantage of the elasticity of disaggregated resources.

We will then need elastic programming models that can be used without change
when running over Cloud resources. Such elastic models should take care of
providing the different transparency types (scaling, failure, replication,
location, access) and other aspects of application behavior when it is moved
between local and distributed environments. The local executable APIs may need
to be expanded to include elastic programming abstractions for processes,
memory, and storage.

  3. 3.

Locality: Advances in datacenter networking and NVMs have reduced access to
networked storage to a few 10ths of µs, however this is still an order of
magnitude slower than local memory accesses which are in the nanosecond range
[6] (100ns), and local cache accesses in the 4ns-30ns range. Existing efforts
in memory disaggregation [11, 10] strive to play in the µs range, which can be
a limiting factor for some applications. This means that local memory cannot
be neglected, since it will always offer an order of magnitude faster access
than remote memory. Future approaches should smartly leverage local memory and
combine it with remote memory.

Locality still plays a key role in stateful distributed applications. For
example: (i) where huge data movements represent a penalty and memory-locality
can be useful; (ii) where specialized hardware like GPUs must be used; in
(iii) some iterative machine-learning algorithms; in (iv) simulators,
interactive agents or actors.

Future serverless middleware addressing transparency will have to provide
affinity and grouping requirements for stateful entities. Serverless Stateful
services will support very different requirements of coordination,
consistency, scalability and fault tolerance.

  4. 4.

Cost: Another limiting factor today for current serverless disaggregated
technologies is cost. For example, Amazon Lambda compute time is 2x more
expensive than on-demand VMs, and around 6x more expensive than Spot
Instances. For problems like batch analytics that have 100% utilization, VMs
are now a better solution for intensive computing applications.

We foresee that cost limitations will be overcome in the future, when Cloud
providers improve scheduling of lightweight micro-VMs, reducing both start
time and overall costs thanks to locality. Furthermore, sophisticated cost
planes should be available to users in order to have full control of costs
derived by remote resources.

## CONCLUSIONS

We argue that full transparency will be possible soon thanks to low latency
and resource disaggregation. The Serverless End Game will unify local and
remote programming paradigms, changing completely the way we currently create
distributed applications. This is the ultimate goal of distributed systems, to
become invisible using transparent middleware, and to simplify how users
access remote resources.

## ACKNOWLEDGMENT

This work has been partially supported by the EU Horizon2020 programme under
grant agreement No 825184.

## References

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Pedro Garcia Lopez is full professor at Universitat Rovira i Virgili (Spain)
and visiting scientist at IBM Watson Research. He leads the “Cloud and
Distributed Systems Lab” research group and coordinates the european research
project ”CloudButton: Serverless Data Analytics”. Contact him at
<EMAIL_ADDRESS>
Aleksander Slominski is research staff member in the Serverless Group in Cloud
Platform, Cognitive Systems and Services Department at IBM T.J. Watson
Research Center in Yorktown Heights, NY, USA. Contact him at https://aslom.net

Michael Behrendt is a Distinguished Engineer and technical executive at IBM
Deutschland Research. He leads key Serverless services like IBM Cloud
Functions and IBM Code Engine. Contact him at<EMAIL_ADDRESS>
Bernard Metzler is a Principal Research Staff Member and Technical Leader at
IBM Zurich Research Laboratory. His main research interests are in enhancing
network and storage IO of distributed systems, and the integration of modern
high performance IO hardware with distributed applications. Contact him at
<EMAIL_ADDRESS>