# Measurement of inclusive jet spectra in pp, p–Pb, and Pb–Pb collisions with
the ALICE detector

Rüdiger Haake for the ALICE Collaboration  Institut für Kernphysik,
Westfälische Wilhelms-Universität Münster, 48149 Münster, Germany
<EMAIL_ADDRESS>
###### Abstract

Highly energetic jets are sensitive probes for the kinematic properties and
the topology of high energy hadron collisions. Jets are collimated sprays of
charged and neutral particles, which are produced in fragmentation of hard
scattered partons from an early stage of the collision. In ALICE, jets have
been measured in pp, p–Pb, and Pb–Pb collisions at several collision energies.
While analyses of Pb–Pb events unveil properties of the hot and dense medium
formed in heavy-ion collisions, pp and p–Pb collisions can shed light on
hadronization and cold nuclear matter effects in jet production. Additionally,
pp and p–Pb serve as a baseline for disentangling hot and cold nuclear matter
effects. A possible modification of the initial state is tested in p–Pb
analyses. For the extraction of a jet signal, the exact evaluation of the
background from the underlying event is an especially important ingredient.
Due to the different nature of underlying events, each collision system
requires a different analysis technique for removing the effect of the
background on the jet sample. The focus of this publication is on the ALICE
measurements of nuclear modification factors connecting p–Pb and Pb–Pb events
to pp collisions. Furthermore, the radial jet structure is explored by
comparing jet spectra reconstructed with different resolution parameters.

##  1 Introduction

Jets can conceptually be described as the final state produced in a hard
scattering. Therefore, jets are an excellent tool to access a very early stage
of the collision. The jet constituents represent the final state remnants of
the fragmented and hadronized partons that were scattered in the reaction.
While all the detected particles have been created in a non-perturbative
process (i.e. by hadronization), ideally, jets represent the kinematic
properties of the originating partons. Thus, jets are mainly determined by
perturbative processes due to the high momentum transfer and the cross
sections can be calculated with pQCD. This conceptual definition is
descriptive and very simple, the technical analysis of those objects is quite
complicated though.

In pp collisions, jets can serve as a probe for undisturbed pQCD physics. In
Pb–Pb collisions, the medium effects like those from the quark-gluon plasma
(QGP) can be probed. For p–Pb collisions, formation of a hot, dense, and
extended medium is not expected and therefore cold nuclear matter effects,
e.g. those described by nuclear parton distribution functions (nPDFs), should
be dominant.

Here, the focus is on the inclusive jet results currently available from the
ALICE experiment. Results and conclusions of recent ALICE measurements in pp,
p–Pb, and Pb–Pb are presented in Sec. 4. Beforehand, a short description of
the ALICE detector with regard to jet measurements is given in Sec. 2.
Additionally, important details of the analyses are shown in Sec. 3.

##  2 The ALICE detector

Figure 1: A sketch of the ALICE detector

The ALICE detector (see Fig. 1) is the dedicated heavy-ion experiment at the
LHC studying properties of the quark-gluon plasma and the QCD phase diagram in
general. The detector is designed as a general-purpose heavy-ion detector [1]
to measure and identify hadrons, leptons, and also photons down to very low
transverse momenta.

One of ALICE’s strengths is the excellent charged particle tracking
capability. The tracking is performed in the central barrel, which mainly
consists of the Inner Tracking System (ITS) and the Time Projection Chamber
(TPC). The ITS [2] is a cylindrical six-layered device consisting of three
different semiconductor subdetectors: silicon pixel, drift, and strip
detectors (SPD, SDD, and SSD). It directly surrounds the beam pipe. Around the
ITS, the Time Projection Chamber is placed. The ALICE TPC [3] is mainly filled
with neon gas at atmospheric pressure, the gas detector has a radius of
$250\mathrm{~{}cm}$ and a length of $500\mathrm{~{}cm}$. Combining the ITS and
the TPC, a straight track can be reconstructed within a pseudorapidity
interval of $\left|\eta\right|<0.9$.

To measure fully reconstructed jets, also neutral particles have to be
detected. For this purpose, a lead-scintillator sampling electromagnetic
calorimeter – the EMCal [4] – is used. The calorimeter covers the
pseudorapidity interval $\left|\eta\right|<0.7$ and roughly $107\degree$ in
azimuth. While the EMCal is very sensitive to photons, e.g. from neutral pion
decays, the neutron reconstruction capability is less precise.

The VZERO [5] scintillation counter is used for event characterization and
centrality estimation.

##  3 Jet reconstruction and correction techniques

Jet reconstruction is usually a multistep procedure. Next to the conceptual
definition of jets as hadronized partons, a technical jet definition is
necessary. There is no unambiguous jet definition and no common way how to
measure a jet. The currently most prevalent jet definition at the LHC is given
by the anti-$k_{\mathrm{T}}$ algorithm [6] implemented in the FastJet [7]
package. For all presented analyses, the anti-$k_{\mathrm{T}}$ algorithm is
used to measure signal jets. Additionally, the $k_{\mathrm{T}}$ algorithm is
used for some of the background correction techniques.

The input passed to the jet finding algorithm is given by charged particles
and calorimeter clusters. The track cuts are chosen to obtain a uniform
charged track distribution in the full $\eta-\phi$ plane. Additionally, only
tracks from $\left|\eta\right|<0.9$ and with $p_{\mathrm{T}}>150\mbox{ MeV/}c$
are used in the jet finding procedure.

While charged jets are reconstructed using only the charged part of a jet,
full jets also include neutral jet energy reconstructed from calorimeter
clusters. Since the EMCal is also sensitive to charged tracks, the energy of
each cluster has to be corrected for charged energy to avoid double-counting.
This is done by extrapolating tracks to the EMCal and matching them to the
deposited energy. The corrected cluster energy is given by

$E^{\mathrm{corr}}_{\mathrm{clus}}=E^{\mathrm{orig}}_{\mathrm{clus}}-\sum{p^{\mathrm{matched}}}.$
(1)

Eventually, only clusters with $E^{\mathrm{corr}}_{\mathrm{clus}}>300$ MeV are
passed to FastJet. After constituent selection, the tracks and calorimeter
clusters are clustered by FastJet using the so-called
$p_{\mathrm{T}}$-recombination scheme, assuming massless jets.

In ALICE, jets have been measured with several jet resolution parameters
(often called jet radii) for different collision systems. To avoid edge
effects, only jets fully contained within the acceptance are used for further
analysis.

Due to the very high multiplicities in heavy-ion collisions, there is a non-
negligible contribution of jets at lower $p_{\mathrm{T}}$ made of so-called
combinatorial jets mostly containing background. To suppress those jets, a
bias on the leading particle’s transverse momentum can be imposed. On the
other hand, this biases the jets to show a harder fragmentation but,
eventually, the systematic uncertainty from the fragmentation bias is smaller
than that from combinatorial jets.

###  3.1 Background correction

In principle, every particle that does not originate from the hard parton
scattering in question can be considered as background. Like the definition of
a jet, also the definition of the background is not unambiguous. All presented
analyses use background correction techniques but the methods differ depending
on the considered collision system: While the background subtraction is a
large correction to the jet momentum in Pb–Pb collisions, the background
density in pp and p–Pb collisions is roughly smaller by a factor of 100
[8][9].

A typical approach to correct for background is to subtract the background on
a jet-by-jet basis. The average background energy density $\rho$ is evaluated
using $k_{\mathrm{T}}$ jets (p–Pb and Pb–Pb) or by probing the background in a
cone perpendicular to the jet (pp) on an event-by-event basis. Then, the
transverse momentum of every jet is corrected for the background momentum
density $\rho$ depending on the jet area $A$ by

$p^{\mathrm{corr}}_{\mathrm{T,\;jet}}=p^{\mathrm{orig}}_{\mathrm{T,\;jet}}-\rho
A.$ (2)

In p–Pb and Pb–Pb collisions, region-to-region fluctuations of the background
are also taken into account. These are quantified by probing the transverse
momentum in randomly distributed cones and comparing it to the average
background. Quantitatively, this distribution is given by

$\delta p_{\mathrm{T}}=\sum_{\mathrm{i}}{p_{\mathrm{T,\,i}}-\rho
A},~{}~{}~{}A=\pi R^{2}.$ (3)

###  3.2 Unfolding

Spectra in p–Pb and Pb–Pb collisions are corrected for background fluctuations
and detector effects – e.g. from the limited single-particle tracking
efficiency – in an unfolding procedure. In pp collisions, background
fluctuations are neglected and only detector effects are corrected for.
Depending on the analysis, Bayesian, SVD, or $\chi^{2}$-unfolding or also a
conceptually simpler bin-by-bin correction is used. The detector response
matrix is created with a full detector simulation using PYTHIA6 to generate
jets and GEANT3 for the particle transport through the detector.

##  4 Results

In the following, the results on jet production measurements are presented in
terms of the nuclear modification factors and jet cross section ratios.

The nuclear modification factor shows the effect of the nuclear environment on
jet production. It is defined by

$R_{\mathrm{AA}}=\frac{\left.\frac{\mathrm{d}N}{\mathrm{d}p_{\mathrm{T}}\mathrm{d}\eta}\right|_{\mathrm{AA}}}{\left<T_{\mathrm{AA}}\right>\cdot\left.\frac{\mathrm{d}\sigma}{\mathrm{d}p_{\mathrm{T}}\mathrm{d}\eta}\right|_{\mathrm{pp}}}$
(4)

where $T_{\mathrm{AA}}$ is the nuclear overlap function [10], which connects
the heavy-ion collisions in Pb–Pb or p–Pb to a reference measurement in pp.

Figure 2: Nuclear modification factor for $R=0.2$ (Pb–Pb, full jets,
$\sqrt{s_{\mathrm{NN}}}=$ 2.76 TeV)

Figure 3: Nuclear modification factor for
$R=0.4$ (p–Pb, charged jets, $\sqrt{s_{\mathrm{NN}}}=$ 5.02 TeV)

Preliminary ALICE measurements for full jets in Pb–Pb [11] and charged jets in
p–Pb [12] are presented here, see Figs. 3 and 3, respectively. The data in
Pb–Pb clearly shows jet quenching, i.e. a strong jet suppression, consistent
with the charged hadron result [13]. In addition, the measurement shows that
the suppression has a centrality dependence and a small transverse momentum
dependence.

For p–Pb collisions (Fig. 3), no nuclear modification is observed. This is a
strong hint that the jet quenching originates mainly from medium effects due
to the formation of a quark-gluon plasma and not from cold nuclear effects due
to the presence of the Pb-nucleus.

A conceptually simpler quantity is $R_{\mathrm{CP}}$, the ratio of central
over peripheral jet spectra scaled by the respective number of binary
collisions. Like the nuclear modification factor, it is a measure for the
nuclear modification but it is simpler because no pp reference is needed. It
is shown for charged jets in Pb–Pb collisions [14] in Fig. 5. Like the full
jet $R_{\mathrm{AA}}$, the charged jet $R_{\mathrm{CP}}$ shows a centrality-
dependent suppression that only has a small transverse momentum dependence.

Figure 4: Central to peripheral ratio of jets for $R=0.2$ (Pb–Pb, charged
jets, $\sqrt{s_{\mathrm{NN}}}=$ 2.76 TeV)

Figure 5: Jet cross section ratio compared to PYTHIA calculations (Pb–Pb,
charged jets, $\sqrt{s_{\mathrm{NN}}}=$ 2.76 TeV)

A basic observable to study the internal radial structure of jets, i.e. jet
broadening and narrowing, is the ratio of jet cross sections reconstructed
with different resolution parameters. For pp collisions, the ratio has been
measured for full (Fig. 7) and charged jets (Fig. 7) at different energies.
The full jet analysis in pp [15] shows a good theory-data agreement when
comparing to an NLO calculation with hadronization effects. The charged jet
analysis in pp [16][17] shows a good agreement with PYTHIA and HERWIG
simulations.

For charged jets in Pb–Pb (Fig. 5), no significant medium modification of the
radial jet structure is observed comparing to pp collisions. It is compatible
with PYTHIA simulations. The same holds for p–Pb collisions: The radial
structure in both, full (cf. [18]) and charged jets in p–Pb, agree well with
proton-proton results (Figs. 9 and 9).

Thus, the radial jet structure is essentially unmodified by the nuclear
environment and the medium in p–Pb and Pb–Pb collisions.

Figure 6: Jet cross section ratio (pp, full jets, $\sqrt{s}=$ 2.76 TeV)

Figure 7: Jet cross section ratio (pp, charged jets, $\sqrt{s}=$ 7 TeV)

Figure 8: Jet cross section ratio in p–Pb (full jets,
$\sqrt{s_{\mathrm{NN}}}=$ 5.02 TeV) compared to that in pp at $\sqrt{s}=$ 2.76
TeV

Figure 9: Jet cross section ratio in p–Pb (charged jets,
$\sqrt{s_{\mathrm{NN}}}=$ 5.02 TeV) compared to that in pp at $\sqrt{s}=$ 7
TeV

##  5 Summary

ALICE has measured charged and full jets in pp, p–Pb, and Pb–Pb collisions at
several energies. In pp, the spectra and radial structure ratios for different
resolution parameters show a good agreement with (next-to-)leading-order
calculations. The same also holds for p–Pb collisions: In p–Pb collisions,
there is no sign for a nuclear modification within the uncertainties. In Pb–Pb
collisions, jet quenching and its centrality dependence has been measured for
full and charged jets. Neither in p–Pb nor in Pb–Pb, the jet cross section
ratio shows a significant modification.

## References

  * [1] Aamodt K et al. (ALICE Collaboration) 2008 JINST 3 8002
  * [2] Aamodt K et al. (ALICE Collaboration) 2010 JINST 5 3003 (Preprint 1001.0502)
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  * [11] Aiola S (ALICE Collaboration) 2014 QM2014 proceedings, to be published (Preprint 1408.0479)
  * [12] Haake R (ALICE Collaboration) 2013 PoS EPS-HEP2013 (proceedings) 176 (Preprint 1310.3612)
  * [13] Aamodt K et al. (ALICE Collaboration) 2011 Phys. Lett. B 696 30–39 (Preprint 1012.1004)
  * [14] Abelev B et al. (ALICE Collaboration) 2014 JHEP 03 013 (Preprint 1311.0633)
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Learning mappings between infinite dimensional function spaces has achieved empirical success in many disciplines of machine learning, including generative modeling, machine learning solving Partial Difference Equations , functional data analysis, causal inference, and multi-agent reinforcement learning. In this paper, we study the statistical limit of learning a Hilbert-Schmidt operator between two infinite-dimensional Sobolev reproducing kernel Hilbert spaces. We establish the information-theoretic lower bound in terms of the Sobolev Hilbert-Schmidt norm and show that a regularization that learns the spectral components below the bias contour and ignores the ones that above the variance contour can achieve the optimal learning rate. At the same time, the spectral components between the bias and variance contours give us the flexibility in designing computationally feasible machine learning algorithms. Based on this observation, we develop a multilevel kernel operator learning algorithm that is optimal when learning linear operators between infinite-dimensional function spaces.

§ INTRODUCTION

§ PROBLEM FORMULATION

§.§ Examples

§ INFORMATION THEORETIC LOWER BOUND

§ ON THE  SHAPE OF REGULARIZATION

§ MULTILEVEL KERNEL OPERATOR LEARNING

§ CONCLUSION AND DISCUSSION

We considered the sample complexity of learning an operator between two infinite-dimensional Sobolev kernel Hilbert spaces. We provided an information theoretical lower bound for this problem along with a multi-level machine learning algorithm. Our lower bound is determined by the harder rate of two polynomial rates: one is fully determined by the hardness of input space while the other is fully controlled by the hardness of the output space. The second rate is new to the literature. We explained the our bound from the viewpoint of variance and bias counters in <Ref> and <Ref>. The optimal estimator should learn all the spectral element under the bias contour but learn no information above the variance contour. To meet this requirement, we combined the idea of multilevel Monte Carlo with kernel operator learning, using successive levels to fit higher frequency information while keeping the variance at the same scale in order to reduce the bias. Our paper is the first paper on the non-parametric statistical optimality for multilevel algorithms. We leave estimation from discretely observe functional covariates with noise as future work  <cit.>.

Jikai Jin is partially supported by the elite undergraduate training program of School of
Mathematical Sciences in Peking University. Yiping Lu is supported by the Stanford Interdisciplinary Graduate
Fellowship (SIGF). Jose Blanchet is supported in part by the Air Force Office of Scientific Research under award number FA9550-20-1-0397.  Lexing Ying is supported is supported by National Science Foundation under award DMS-2208163.

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

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# Elements of random analysis about the gamma generalized hyperbolic
distribution Levy stochastic process

Nafy Ngom ,  Aladji Babacar Niang ,  Soumaila Dembelé and  Gane Samb Lo

###### Abstract.

In this paper, we study some aspects on random analysis on the Lévy stochastic
processes with margins following generalized hyperbolic distributions
generated by gamma laws. In particular we study the boundedness of its total
variations and the quadratic variations. Next we give an empirical
construction that enables the graphical representation of the paths of such
stochastic processes. Comparisons with the Brownian motions are considered.

Nafy Ngom
LERSTAD, Gaston Berger University, Saint-Louis, Sénégal.

Dr Aladji Babacar Niang
LERSTAD, Gaston Berger University, Saint-Louis, Senegal
HEC Faculty, UNIL, University of Lausanne, Suisse
Emails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Suisse address: 1027, Chemin de la Fauvette 2, Lonay
Senegal address: 088, HLM Grand Yoff, Dakar

Dr Soumaila Dembelé
Université des Sciences Sociale et de Gestion de Bamako (USSGB)
Faculté des Sciences Économiques et de Gestion (FSEG)
Email<EMAIL_ADDRESS><EMAIL_ADDRESS>

† Gane Samb Lo.
LERSTAD, Gaston Berger University, Saint-Louis, Sénégal (main affiliation).
LSTA, Pierre and Marie Curie University, Paris VI, France.
AUST - African University of Sciences and Technology, Abuja, Nigeria
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Permanent address : 1178 Evanston Dr NW T3P 0J9,Calgary, Alberta, Canada.

Keywords. generalized hyperbolic distribution (GHD); Lévy processes wth GHD
margins; Asymptotic boundednessn of variation; quadratic variation; infinitely
decomposable laws
AMS 2010 Mathematics Subject Classification: 60G51; 60F15; 60E07

##  1\. Introduction

In the recent years, the generalized hyperbolic distribution (GHD) is
attracting much attention in various branches of Statistics and Probability
theory, especially in statistical estimation and in finance modeling (see
McNeil et al. (2019)). This class of laws is seen as a credible alternative to
the Gaussian assumption of data both in skewness and in shape. Also, as
infinitely divisible laws, the Lévy processes they generate began to be
studied (see Applebaum  (2004)).

To be able to expose the aim of this paper and to describe its motivation, we
need to develop a little the three aspects aforementioned in the next line.

###  1.1. Infinitely divisibility laws or random variables

The probability law of a real-valued random variable $X$ associated to the
characteristic function $\psi_{X}$ is infinitely divisible, denoted as
[idecomp], if and only if it satisfies one of the criteria below.

(Criterion 1) A random variable $X$ is infinitely divisible or infinitely
decomposable (idecomp) if and only if for any $n\geq 1$, it can be equal in
law to a sum of $n$ independent and identically distributed random variables.

(Criterion 2) A random variable $X$ of characteristic function $\psi_{X}$ (or
of third moment generating function $\phi_{X}$ if $X\geq 0$) is idecomp if and
only if: for any $n\geq 1$, there exists a characteristic function $\psi$ (or
a third moment generating function $\phi$ of a non-negative random variable)
such that

$\psi_{X}=\psi^{n}\ \ (or\ \ \phi_{X}=\phi^{n}).$

(Criterion 3) A random variable $X$ of characteristic function $\psi_{X}$ (or
of third moment generating function $\phi_{X}$ if $X\geq 0$) is idecomp if and
only if: for any $n\geq 1$, $\psi_{X}^{1/n}$ is a characteristic function (or
$\phi_{X}^{1/n}$ is a third moment generating function).

Examples. Let us consider three classical idecomp probability laws: normal law
$U_{1}=\mathcal{N}(m,\sigma^{2})$, translation of a Poisson law
$\mathcal{P}(\lambda)$ by $a\geq 0$:
$U_{2}=a+\mathcal{P}(\lambda)=:\mathcal{P}(a,\lambda)$, and translation of a
gamma law $\gamma(a,\beta)$ by $a\geq 0$:
$U_{3}=a+\gamma(a,\beta)=:\gamma(a,a,\beta)$ with characteristic functions,
for $t\in\mathbb{R}$,

$\displaystyle\psi_{U_{1}}(t)=\exp(imt-\sigma^{2}t^{2}/2),\ \ \
\psi_{U_{2}}(t)=\exp\left(iat+\lambda(e^{it}-1)\right)$
$\displaystyle\psi_{U_{3}}(t)=e^{iat}\left(1-it/\beta\right)^{-a}.$

By taking the powers at $1/n$, $n\geq 1$, we get for any $t\in\mathbb{R}$,

$\psi_{U_{1}}(t)^{1/n}=\exp\left(i\\{m/n\\}t-(\sigma/\sqrt{n})^{2}t^{2}/2\right),\
\
\psi_{U_{2}}(t)^{1/n}=\exp\left(i\\{a/n\\}t+\\{\lambda/n\\}(e^{it}-1)\right),$

$\ \ \ \ \
\psi_{U_{3}}(t)^{1/n}=e^{i\\{a/n\\}t}\left(1-it/\beta\right)^{\\{-a/n\\}}.$

By identifying the characteristic functions, we can see that the
$\psi_{U_{i}}(t)^{1/n}$, $1\leq i\leq 3$, are cha.f’s of random variables of
law $\mathcal{N}(m/n,\sigma^{2}/n)$, $\mathcal{P}(a/n,\lambda/n)$ and
$\gamma(a/n,a/n,\beta)$ respectively. So, these probability laws are idecomp.

Full exposure of idecomp laws are treated in Loève  (1997), Feller  (1968b),
Zolotarev (1986), etc., and the readers are directed to them for getting
deeper properties.

###  1.2. Generalized Hyperbolic Distributions (GHD)

A real valued-random variable $X$ is said to follow a normal variance mixture
(NVM) law if it is of the form

(1.1) $X=\mu+\sigma Z\sqrt{W},$

where $\mu\in\mathbb{R}$, $\sigma>0$, $Z\sim\mathcal{N}(0,1)$, $Z$ independent
of $W$, $W$ is some positive random variable.

As in McNeil et al. (2019), we consider a few number of distributions for $W$:

a) The gamma distribution $\gamma(a,\beta)$ of parameters $a>0$ and $\beta>0$
of probability density function (pdf)

$f_{1}\left(x\right)=\frac{\beta^{a}}{\Gamma\left(a\right)}x^{a-1}e^{-\beta
x},\ x>0;$

b) Inverse gamma distribution $Ig(a,\beta)$ of $a>0$ and $\beta>0$ of pdf

$f_{2}\left(x\right)=\frac{\beta^{a}}{\Gamma\left(a\right)}x^{-(a+1)}e^{-\beta/x},\
\ x>0;$

c) The generalized inverse gamma $\textit{GIG}(a,b,c)$ distribution of
parameters $a\geq 0$, $b\geq 0$, $c\geq 0$ of pdf

$f_{3}(x)=\frac{1}{C\left(a,b,c\right)}x^{a-1}e^{-\left(bx+c/x\right)/2},\
x>0,$

where

$C\left(a,b,c\right)=\int_{0}^{+\infty}x^{a-1}e^{-\left(\left(bx+c/x\right)/2\right)}dx.$

Some authors use the Bessel functions $K_{a}$ to express the constant of the
distribution, and we have

$\frac{1}{C\left(a,b,c\right)}=\frac{b^{-a}\left(bc\right)^{\frac{a}{2}}}{2K_{a}\left(\left(bc\right)^{\frac{1}{2}}\right)},$

with

$K_{a}\left(\left(bc\right)^{\frac{1}{2}}\right)=\frac{c^{-a}\left(bc\right)^{\frac{a}{b}}}{2}\int_{0}^{+\infty}x^{a-1}e^{-\left(\left(bx+c/x\right)/2\right)}dx.$

The parameters $a$, $b$ and $c$ satisfy the conditions

$\displaystyle a$ $\displaystyle=$ $\displaystyle 0:b>0\ \ and\ c>0,$
$\displaystyle a$ $\displaystyle>$ $\displaystyle 0:b>0\ \ and\ c\geq 0,$
$\displaystyle a$ $\displaystyle<$ $\displaystyle 0:b\geq 0\ and\ c>0.$

We study the cases ($a>0,b>0,c=0$) and ($a<0,b=0,c>0$) apart. The first case
reduces to a gamma law $\gamma\left(a,b/2\right)$ of pdf

$f_{4}\left(x\right)=\frac{(b/2)^{a}}{\Gamma\left(a\right)}x^{a-1}e^{-bx/2},\
\ x\geq 0$

and the second to the following inverse gamma $Ig(a,c/2)$ distribution of pdf

$f_{2,1}\left(x\right)=\frac{1}{D\left(a,c\right)}x^{-\left(a+1\right)}e^{-c/2x},\
\ x>0,$

with $a>0$, $c>0$, and

$D\left(a,c\right)=\int_{0}^{+\infty}x^{-\left(a+1\right)}e^{-c/2x}\ dx.$

We also have

$D\left(a,c\right)=\left(\frac{c}{2}\right)^{-a}\Gamma^{-}\left(a\right),\ \
\Gamma^{-}\left(a\right)=\int_{0}^{+\infty}x^{-\left(a+1\right)}e^{-c/2x}\
dx,\ \ a>0,c>0.$

If we take as $W$ one of this sample of distributions, the NVM law becomes the
Generalized Hyperbolic Distribution (GHD). The GHD model is full if we have
$a\neq 0$, $b>0$ and $c>0$.

For any random variable $W>0$ with the density $f_{W}$, the pdf of a NVM
random variable can be obtained by the classical diffeomorphical change of
variables (see Lo  (2018), Chapter 2, Section 7, Part II) as follows. Let us
consider the following diffeomorphism

$\displaystyle h$ $\displaystyle:$
$\displaystyle\mathbb{R}\times\mathbb{R}_{+}^{\ast}\rightarrow\mathbb{R}\times\mathbb{R}_{+}^{\ast}$
$\displaystyle\left(z,w\right)$ $\displaystyle\longmapsto$
$\displaystyle\left(zw^{1/2},w\right)=\left(u,v\right).$

The new pdf of $\left(U,V\right)=h\left(Z,W\right)$ is given by

$f_{U,V}\left(u,v\right)=f_{Z}\left(uv^{-1/2}\right)f_{W}\left(v\right)v^{-1/2}1_{\mathbb{R}\times\mathbb{R}_{+}^{\ast}}\left(u,v\right).$

The pdf of $ZW^{1/2}$ is then

$f_{ZW^{1/2}}\left(u\right)=\frac{1}{\sqrt{2\pi}}\int_{0}^{+\infty}v^{-1/2}\
\exp(-u^{2}/(2v))\ f_{W}\left(v\right)dv,\ \ u\in\mathbb{R}.$

We are going to apply that formula for the distributions given above.

The distributions of $ZW^{1/2}$ corresponding to the pdf $f_{i}$,
$i\in\\{1,2,3\\}$ of $W$ are given as follows. Here we do not give the
computations related to the application of the diffeomorphic transform, in
particular their jacobian:

$\displaystyle
f_{g,ZW^{1/2}}(u)=\frac{\beta^{a}}{\sqrt{2\pi}\Gamma(a)}C\left(a-1/2,2\beta,u^{2}\right),\
u\in\mathbb{R},$

$\displaystyle
f_{ig,ZW^{1/2}}(u)=\frac{2^{a+1/2}\beta^{a}\Gamma(a+1/2)}{\Gamma\left(a\right)\sqrt{2\pi}}\left(u^{-2}+2\beta\right)^{-(a+1/2)},\
u\in\mathbb{R},$

$\displaystyle f_{gig,ZW^{1/2}}(u)=\frac{1}{\sigma
C\left(a,b,c\right)\sqrt{2\pi}}C\left(a-1/2,b,c+u^{2}\right),\ \
u\in\mathbb{R}.$

Hence we deduce the GHD associated to each case given by:

1. For $W\sim\gamma(a,\beta)$, $a>0$, $b>0$, we have

$f_{g,X}\left(u\right)=\frac{\beta^{a}}{\sigma\sqrt{2\pi}\Gamma(a)}C\left(a-1/2,2\beta,\sigma^{-2}(u-\mu)^{2}\right),\
u\in\mathbb{R};$

2. For $W\sim Ig(a,\beta)$, $a>0$, $\beta>0$, we have

$f_{ig,X}\left(u\right)=\frac{2^{a+1/2}\beta^{a}\Gamma(a+1/2)}{\sigma\Gamma\left(a\right)\sqrt{2\pi}}\left(\sigma^{-2}(u-\mu)^{2}+2\beta\right)^{-(a+1/2)},\
u\in\mathbb{R};$

3. For $W\sim Gig(a,b,c)$, $a>0$, $b>0$, $c>0$,

$f_{gig,X}\left(u\right)=\frac{1}{\sigma
C\left(a,b,c\right)\sqrt{2\pi}}C\left(a-1/2,b,c+\sigma^{-2}(u-\mu)^{2}\right),\
u\in\mathbb{R}.$

To find the characteristic function $\psi_{X}$ of $X$, we denote by
$H_{W}(t)=\mathbb{E}\left(e^{-tW}\right)$, $t\geq 0$, the moment function of
$W$.

In general, by using the mathematical expectation tool, we get

$\displaystyle\psi_{X}(t)$ $\displaystyle=$
$\displaystyle\mathbb{E}\left(\exp(i\mu t+i\sigma tW^{1/2}Z)\right)$
$\displaystyle=$ $\displaystyle\exp(i\mu t)\
\mathbb{E}\biggr{(}\mathbb{E}\biggr{(}\exp\left(i\sigma
tW^{1/2}Z\right)\biggr{|}W\biggr{)}\biggr{)}$ $\displaystyle=$
$\displaystyle\exp(i\mu t)\
\mathbb{E}\biggr{(}\exp(-\sigma^{2}t^{2}W/2)\biggr{)}$ $\displaystyle=$
$\displaystyle\exp(i\mu t)\int_{0}^{+\infty}\exp(-\sigma^{2}t^{2}w/2)f_{W}(w)\
dw$ $\displaystyle=$ $\displaystyle\exp(i\mu t)\ H_{W}(\sigma^{2}t^{2}/2).$

But for our three choices of $W$, i.e.
$W\in\left\\{\gamma(a,\beta),Ig(a,\beta),\ gig(a,b,c)\right\\}$, we have,
respectively

$H_{g}(t)=(1+t/\beta)^{-a},\
H_{Ig}(t)=\frac{\beta^{a}}{\Gamma(a)}C(-a,2t,2\beta)\ and\
H_{gig}(t)=\frac{C(a,b+2t,c)}{C(a,b,c)},\ t>0.$

So for $H_{W}\in\\{H_{g},H_{Ig},H_{gig}\\}$, we have

$\displaystyle\psi_{X}(t)$ $\displaystyle=$ $\displaystyle\exp(i\mu t)\
H_{W}(\sigma^{2}t^{2}/2),\ t>0.$

This may help showing that $X$ is infinitely decomposable (idecomp). Indeed,
for any $k\geq 1$, we have

$\displaystyle\psi_{X}(t)^{(1/k)}$ $\displaystyle=$
$\displaystyle\exp(i(\mu/k)t)\ H_{W}^{(1/k)}(\sigma^{2}t^{2}/2),\ t>0,$

and $X$ is idecomp whenever $W$ is idecomp. But it is possible, as in
Grosswald  (1976), to show that any GHD is decomp, but this task is out of the
scope of the paper. Since we focus on the GHD associated to the gamma law, the
characteristic function of $X$ is

(1.2) $\forall t\in\mathbb{R},\ \ \psi_{X}(t)=\exp(i\mu
t)\biggr{(}1+\frac{\sigma^{2}t^{2}}{2\beta}\bigg{)}^{-a},$

and the application of Criterion 3, in page 1.1, ensures that $Z$ is idecomp.

Generally, for $W$ following the $\gamma(a,b)$ law, the $Ig(a,c)$ law or the
$gig(a,b,c)$ law, $a>0$, $b>0,c>0$, we denote the GHD respectively as the
gamma-gh$(a,b,0,\mu,\sigma)$ law, or the ig-gh$(a,0,c,\mu,\sigma)$ law or the
gig-gh$(a,b,c,\mu,\sigma)$ law.

In this paper, we only treat with, gamma-gh$(a,b,0,\mu,\sigma)$, the gamma DH
distribution of parameters $a>0$, $b>0$, $c=0$, $\sigma>0$ and
$\mu\in\mathbb{R}$.

Our main source on GHD distribution is McNeil et al. (2019).

###  1.3. Lévy processes

By definition, a Lévy process
$\biggr{(}\Omega,\mathcal{A},\mathbb{P},(K_{t})_{(t\geq 0)}\biggr{)}$ is
defined as a stochastic process with marginal laws $\\{K(t),\ t\geq 0\\}$ such
that:

(ML) $\forall t\geq 0$, $L(t)\sim K(t)$, with $L(0)=0$ a.s;

(II) $\forall k\geq 0$, $\forall\ 0=t_{0}<t_{1}<\cdots<t_{k}$, the r.v’s
$L(t_{j})-L(t_{j-1})$ are independent;

(IS) $\forall 0\leq s<t$, $L(t)-L(s)\sim K(t-s)$;

(CS) The process $\\{L(t),\ t\geq 0\\}$ is stochastically continuous at zero.

NB. Each marginal law $K(t)$ is an infinitely decomposable (idecomp) law.

Examples. Here are three examples. (1) The Brownian process $(B_{t})_{t\geq
0}$ with $K(t)=\mathcal{N}(0,t)$, $t\geq 0$; (2) The simple Poisson process
$(N_{t})_{t\geq 0}$ with $K(t)=\mathcal{P}(t)$, $t\geq 0$; (3) The
$\gamma(a,b)$-process $(G_{a,b,t})_{t\geq 0}$ with $K(t)=\gamma(at,b)$.

For Lévy processes, we mainly refer to Applebaum  (2004).

###  1.4. Motivation and aim

After this round-up on the key objects of that paper, we precise that we aim
at doing some random analysis of the Lévy process $\\{Y^{\ast}(t),t\geq 0\\}$
whose margins are defined by

$Y(t)\sim K(t)=\textit{gamma-gh}(at,b,0,\mu t,\sigma).$

The existence of such Lévy processes is justified in Niang et Lo  (2022).
Because of the more involvement of $GHD$ laws in Finance, we see it
appropriate to have a detailed analysis of Lévy processes using GHD margins
should be done at the level of the details of Brownian motions for example.
There are more studies on these stochastic processes in general. The specific
studies we are going to do will help as example and counter-examples in those
investigations.

We recall that we already have applied the Kolmogorov Existence Theorem (KET)
to prove the existence of Lévy processes.

Hereafter, we begin with the gamma-gh$(a,b,0,\mu,\sigma)$ Lévy process because
of the explicit form of the characteristic function.

(a) The non-where Totally Bounded variation of $T$.

(a) The non convergence of the Quadratic variation on compact sets in $L^{2}$.

(c) The empirical construction of $Y$ providing graphical illustrations.

##  2\. Study of the Variation of gamma-gh Lev́y processes

Let $\mathcal{P}ar[a,b]$ be the collection of all finite partitions of
$[0,T]$, $T>0$, of the form $\pi=(0=t_{0}<t_{1}<\cdots<t_{k}=T)$, $k\geq 1$.
We define the variation of $Y^{\ast}$ on $\pi$ by

$V_{T}(\pi)=\sum_{j=0}^{k-1}\left|Y^{\ast}(t_{j+1})-Y^{\ast}(t_{j})\right|.$

The total variation of $Y^{\ast}$ on $[0,T]$ is defined as

$V_{T}=\sup_{\pi\in\mathcal{P}ar[a,b]}V_{T}(\pi).$

The stochastic process $Y$ is of bounded variation on some measurable subset
$\Omega_{0}\subset\Omega$ if for any $\omega\in\Omega_{0}$
$V_{T}(\omega)<+\infty$. If $\mathbb{P}(\Omega_{0})>0$, a path-wise Riemann-
Stieltjes integration theory of continuous stochastic processes $f(t,\omega)$,
i.e. treating integrals of the form,

$\Omega_{0}\in\omega\rightarrow\int_{0}^{T}f(t,\omega)\ dY(t,\omega),$

is meaningful and interesting. Otherwise, i.e. if $Y$ is non-where of bounded
variation, that is

$\mathbb{P}(V_{T}<\infty)=0,$

then a Riemann-Stieltjes integration theory leads to a dead end. Methods of
stochastic integration should be tried instead. The last described situation
actually happens.

This is the commonly formulation of the problem. However, a firm look of the
proof of that result (as given for example in mestuto-ang) allows to see that
we need less restrictive condition than the boundedness of total variation. We
need only what we call the asymptotic boundedness of total variation we denote
below. The full details related to that notion is to be found in Lo  (2022).

Let $k\geq 1$ and
$\pi_{k}=(0=t_{0}^{(k)}<t_{1}^{(k)}<\cdots<t_{\ell(k)}^{(k)}=T)$, be a any
finite partition of $[0,T]$, $T>0$, with

$\ell(k)\rightarrow+\infty,\ \ m(\pi_{k})=\max_{0\leq
j\leq\ell(k)-1}\left(t_{j+1}^{(k)}-t_{j}^{(k)}\right)\rightarrow 0\ as\
k\rightarrow+\infty.$

Let $F:[a,\ b]\rightarrow\mathbb{R}$ be a mapping. The variation of $F$ on
$\pi_{k}$ is

$V_{k}=\sum_{j=0}^{\ell(k)-1}\left|F(t_{j+1}^{(k)})-F(t_{j}^{(k)})\right|.$

We say that $F$ has an if and only if there exists a finite real number $M>0$
such that

$\limsup_{n\rightarrow+\infty}V_{k}<M$

whatever be the sequence of partitions $(V_{k})_{k\geq 1}$ as
$ell(k)\rightarrow+\infty$ and $m(\pi_{k})asymptoticboundedvariation$. It is
shown in Lo  (2022) the asymptotic boundedness of variation is enough for
making integrable all real-valued continuous functions on $[a,\ ]$ in the
sense of the Riemann-Stieltjes approach.

We have

######  Theorem 1.

The gamma-gh$(a,b,0,\mu,\sigma)$ Lévy process is almost-surely of asymptotic
bounded variation.

Instead of directly analyzing $Y^{\ast}$, we choose to study the centered
process

$\\{Y(t),\ \ t\geq 0\\}=\\{Y^{\ast}(t)-\mu t,\ \ t\geq 0\\}.$

Proof of Theorem 1. It is clear that $Y$ and $Y^{\ast}$ are of bounded
variation at the same time or none of them is. From (1.1), we have that:

$\displaystyle\forall\ 0<s<t,\ \ Z(s,t)=Y(t)-Y(s)$ $\displaystyle\sim$
$\displaystyle\textit{gamma-gh}(a(t-s),b,0,0,\sigma)$ $\displaystyle\sim$
$\displaystyle\sigma Z_{(s,t)}\ L_{(s,t)},$

where $Z_{(s,t)}\sim\mathcal{N}(0,1)$ is independent of
$L_{(s,t)}\sim\gamma(a(t-s),b)^{1/2}$. Let $k\geq 1$ and
$\pi_{k}=(0=t_{0}^{(k)}<t_{1}^{(k)}<\cdots<t_{\ell(k)}^{(k)}=T)$, be a
partition of $[0,T]$, $T>0$, with

$\ell(k)\rightarrow+\infty,\ \ m(\pi_{k})=\max_{0\leq
j\leq\ell(k)-1}\left(t_{j+1}^{(k)}-t_{j}^{(k)}\right)\rightarrow 0\ as\
k\rightarrow+\infty.$

The variation of $Y$ on $\pi_{k}$ is

$V_{k}=\sum_{j=0}^{\ell(k)-1}\left|Y(t_{j+1}^{(k)})-Y(t_{j}^{(k)})\right|.$

Let us show that $V_{k}$ has a finite variance. Let, for $0\leq
j\leq\ell(k)-1$,

$\Delta_{j}=|Y(t_{j+1}^{(k)})-Y(t_{j}^{(k)})|\ \ and\ \
\delta_{j}=t_{j+1}^{(k)}-t_{j}^{(k)}.$

Each $\Delta_{j}$, $j\in\\{0,\cdots,\ell(k)-1\\}$ is of the form
$\Delta_{j}=\sigma|Z_{j}|L_{j}$, $Z_{j}\sim\mathcal{N}(0,1)$ independent of
$L_{j}\sim\gamma(a\delta_{j},\beta)^{1/2}$. We have

$\displaystyle\mathbb{E}\Delta_{j}$ $\displaystyle=$
$\displaystyle\sigma\sqrt{\frac{2}{\pi}}\int_{0}^{+\infty}x^{1/2}f_{\gamma(a\delta_{j},\beta)}(x)\
dx$ $\displaystyle=$
$\displaystyle\sigma\sqrt{\frac{2}{\pi}}\times\frac{\beta^{a\delta_{j}}}{\Gamma(a\delta_{j})}\int_{0}^{+\infty}x^{a\delta_{j}+1/2-1}e^{-\beta
x}\ dx$ $\displaystyle=$
$\displaystyle\sigma\sqrt{\frac{2}{\pi}}\times\frac{\beta^{a\delta_{j}}}{\Gamma(a\delta_{j})}\frac{\Gamma(a\delta_{j}+1/2)}{\beta^{a\delta_{j}+1/2}}$
$\displaystyle=$
$\displaystyle\sigma\sqrt{\frac{2}{\pi\beta}}\times\frac{\Gamma(a\delta_{j}+1/2)}{\Gamma(a\delta_{j})}.$

Let us denote

$h_{j}=\frac{\Gamma(a\delta_{j}+1/2)}{\Gamma(a\delta_{j})},\ 0\leq
j\leq\ell(k)-1$

and

$\Gamma(a\delta_{j}+1/2)=\int_{0}^{1}x^{a\delta_{j}-1/2}\ e^{-x}\
dx+\int_{1}^{+\infty}x^{a\delta_{j}-1/2}\ e^{-x}\ dx=:b_{1,j}+b_{2,j}.$

Then, accordingly to: $e^{-1}\leq e^{-x}\leq 1$, we have

$\frac{2/e}{1+2a\delta_{j}}\leq b_{1,j}\leq\frac{2}{1+2a\delta_{j}},$

i.e.,

$(2/e)(1+\overline{o}(1))\leq b_{1,j}\leq 2(1+\overline{o}(1)),$

where $\overline{o}(1)$ is a small o that is uniform in $j$ over
$a\delta_{j}\leq am(\pi_{k})$. As well,

$b_{2,j}=\int_{1}^{+\infty}e^{a\delta_{j}\log x}x^{-1/2}\ e^{-x}\ dx$

and

$I_{1}=\int_{1}^{+\infty}x^{-1/2}\ e^{-x}\ dx\leq
b_{2,j}\leq\int_{1}^{+\infty}e^{am(\pi_{k})\log x}x^{-1/2}e^{-x}\ dx.$

The function $e^{am(\pi_{k})\log x}x^{-1/2}e^{-x}$ is bounded by $e^{\log
x}x^{-1/2}e^{-x}=x^{1/2}e^{-x}\in L^{1}([1,+\infty])$ and converges to
$x^{-1/2}e^{-x}$ as $k\rightarrow+\infty$ (with $m(\pi_{k})\rightarrow 0$).
Then

$b_{2,j}\rightarrow I_{1},$

uniformly in $j$. We then have

$((2/e)+I_{1})(1+\overline{o}(1))\leq\Gamma(a\delta_{j}+1/2)\leq(2+I_{1})(1+\overline{o}(1)).$

Similarly, we have

$I_{2}=\int_{1}^{\infty}x^{-1}\ e^{-x}\ dx,$

and for each $j$,

$\frac{1/e}{a\delta_{j}}+I_{2}(1+\overline{o}(1))\leq\Gamma(a\delta_{j})\leq\frac{1}{a\delta_{j}}+I_{2}(1+\overline{o}(1))$

i.e.,

$\frac{1/e+I_{2}a\delta_{j}(1+\overline{o}(1))}{a\delta_{j}}\leq\Gamma(a\delta_{j})\leq\frac{1+I_{2}a\delta_{j}(1+\overline{o}(1))}{a\delta_{j}}.$

Finally, we have

$\frac{\left(2/e+I_{1}\right)a\delta_{j}\left(1+\overline{o}\left(1\right)\right)}{\left(1+I_{2}a\delta_{j}\right)\left(1+\overline{o}\left(1\right)\right)}\leq
h_{j}\leq\frac{\left(2+I_{1}\right)a\delta_{j}\left(1+\overline{o}\left(1\right)\right)}{\left(1/e+I_{2}a\delta_{j}\right)\left(1+\overline{o}\left(1\right)\right)},$

which implies

$\frac{\left(2/e+I_{1}\right)a\delta_{j}\left(1+\overline{o}\left(1\right)\right)}{\left(1+I_{2}a\delta_{j}\right)}\leq
h_{j}\leq\frac{\left(2+I_{1}\right)a\delta_{j}\left(1+\overline{o}\left(1\right)\right)}{\left(1/e+I_{2}a\delta_{j}\right)}.$

Then we have

$\sigma\sqrt{\frac{2}{\pi\beta}}\frac{\left(2/e+I_{1}\right)a\delta_{j}\left(1+\overline{o}\left(1\right)\right)}{\left(1+I_{2}a\delta_{j}\right)}\leq\mathbb{E}\Delta_{j}\leq\sigma\sqrt{\frac{2}{\pi\beta}}\frac{\left(2+I_{1}\right)a\delta_{j}\left(1+\overline{o}\left(1\right)\right)}{\left(1/e+I_{2}a\delta_{j}\right)}$

Let $\underline{m}\left(\pi_{k}\right)=\min_{1\leq
j\leq\ell(k)-1}\delta_{j}\leq m(\pi_{k}$, then

$\sigma\sqrt{\frac{2}{\pi\beta}}\frac{\left(2/e+I_{1}\right)a\delta_{j}\left(1+\overline{o}\left(1\right)\right)}{\left(1+I_{2}am\left(\pi_{k}\right)\right)}\leq\mathbb{E}\Delta_{j}\leq\sigma\sqrt{\frac{2}{\pi\beta}}\frac{\left(2+I_{1}\right)a\delta_{j}\left(1+\overline{o}\left(1\right)\right)}{\left(1/e+I_{2}a\underline{m}\left(\pi_{k}\right)\right)}$

and

$\sigma\sqrt{\frac{2}{\pi\beta}}\frac{\left(2/e+I_{1}\right)aT\left(1+\overline{o}\left(1\right)\right)}{\left(1+I_{2}am\left(\pi_{k}\right)\right)}\leq\mathbb{E}V_{k}\leq\sigma\sqrt{\frac{2}{\pi\beta}}\frac{\left(2+I_{1}\right)aT\left(1+\overline{o}\left(1\right)\right)}{\left(1/e+I_{2}a\underline{m}\left(\pi_{k}\right)\right)}.$

Put $E_{1}=2/e+I_{1}$ and $E_{2}=\frac{2+I_{1}}{1/e}$. So, as
$m\left(\pi_{k}\right)\rightarrow 0$, we get

$\sigma\sqrt{\frac{2}{\pi\beta}}E_{1}aT\leq\mathbb{E}V_{k}\leq\sigma\sqrt{\frac{2}{\pi\beta}}E_{2}aT$

Now, let us compute $\mathbb{V}ar(V_{k})$. Since the $\Delta_{j}$’s are
independent, we have

$\mathbb{V}ar(V_{k})=\sum_{j=0}^{\ell(k)-1}\mathbb{V}ar(\Delta_{j}).$

But we have for each $j$,

$\mathbb{E}\Delta_{j}^{2}=\mathbb{E}(\sigma^{2}Z_{j}^{2}L_{j}^{2})=\sigma^{2}\frac{a\delta_{j}}{\beta}.$

Hence

$\displaystyle\sigma^{2}\frac{a\delta_{j}}{\beta}-E_{2}^{2}\frac{2\sigma^{2}}{\pi\beta}(a\delta_{j})^{2}\
(1+\overline{o}(1))$ $\displaystyle\leq\mathbb{V}ar(\Delta_{j})$
$\displaystyle\leq\sigma^{2}\frac{a\delta_{j}}{\beta}-E_{1}^{2}\frac{2\sigma^{2}}{\pi\beta}(a\delta_{j})^{2}\
(1+\overline{o}(1)).$

We get, as $k\rightarrow+\infty$,

$\sum_{j=0}^{\ell(k)-1}\delta_{j}^{2}\leq m(\pi_{k})T\rightarrow 0.$

By summing over $j$, we arrive at

$\displaystyle\sigma^{2}\frac{aT}{\beta}-\overline{o}(1)\leq\mathbb{V}ar(V_{k})\leq\sigma^{2}\frac{aT}{\beta}-\overline{o}(1).$

Then

$\mathbb{V}ar(V_{k})=\sigma^{2}\frac{aT}{\beta}+\overline{o}(1).$

We conclude as follows. Let $K_{1}=\sigma\sqrt{\frac{2}{\pi\beta}}E_{2}aT$ and
for $K_{2}=\sigma^{2}\frac{aT}{\beta}$. Then for any $p\geq 1$

(2.1) $(\sup_{k\geq 1}V_{k}=+\infty)=\bigcap_{n\geq 1}\biggr{(}\sup_{k\geq
1}V_{k}\geq n\biggr{)}\subset\biggr{(}\sup_{k\geq 1}V_{k}\geq
p\biggr{)}=A_{p}.$

Next we have

$A_{p,q}=\biggr{(}\sup_{1\leq k\leq q}V_{k}\geq p\biggr{)}\nearrow A_{p}\ as\
q\nearrow+\infty.$

Now, for the we may superpose the partitions $(\pi_{k})_{1\leq k\leq q}$ into
the partition

$\pi^{\ast}_{q}=(0=c^{\ast}_{1,q}<c^{\ast}_{2,q}<\cdots<c^{\ast}_{1,\ell^{\ast}(q)})$

Let us denote $V^{\ast}_{q}$ as the variation of $Y$ over the partition
$\pi^{\ast}_{q}$. We can directly see that:

(a) For any $k\in[1,q]$, $m(\pi^{\ast}_{q})\leq m(\pi_{k})$;

(b) For any $k\in[1,q]$, $V_{k}\leq V^{\ast}_{q}$.

Then we have

$\mathbb{P}(A_{p,q})\leq\mathbb{P}(V^{\ast}_{q}\geq
p)\leq\frac{\mathbb{E}V^{\ast}_{q}}{p}\leq\frac{K_{1}}{p}.$

We let $q\nearrow+\infty$ to get

$\mathbb{P}(A_{p})\leq\mathbb{P}(V^{\ast}_{q}\geq p)\leq\frac{K_{1}}{p}.$

Next, we combine this with Inequality (2.1) and let $p\nearrow+\infty$ to have

Then

$\mathbb{P}(\sup_{k\geq 1}V_{k}=+\infty)=0.$

We conclude that

$\mathbb{P}(\limsup_{n\rightarrow+\infty}V_{k})=0.\ \ \ \ \blacksquare$

Now, we move the quadratic variation.

##  3\. Quadratic Variation

The Quadratic Variation of $Y$ over $[0,\ T]$, by definition, is the limit (in
some sense), as $k\rightarrow+\infty$, of

$V_{Q}(T,\pi_{k})=\sum_{j=0}^{\ell(k)-1}(Y(t_{j+1}^{(k)})-Y(t_{j}^{(k)}))^{2}.$

Here is the result.

######  Theorem 2.

The Quadratic Variation $V_{Q}(T,\pi_{k})$ does not converge to a constant in
$L^{2}$ as $m(\pi_{k})\rightarrow 0$.

Proof of Theorem 2. We have

$V_{Q}(T,\pi_{k})=\sum_{j=0}^{\ell(k)-1}\Delta_{j}^{2}$

and

$\mathbb{E}V_{Q}(T,\pi_{k})=\sum_{j=0}^{\ell(k)-1}\mathbb{E}\Delta_{j}^{2}=\sum_{j=0}^{\ell(k)-1}\sigma^{2}\mathbb{E}Z_{j}^{2}\mathbb{E}\left(\gamma(a\delta_{j},\beta)\right).$

Hence

$\mathbb{E}V_{Q}(T,\pi_{k})=\sum_{j=0}^{\ell(k)-1}\sigma^{2}a\delta_{j}/\beta=a\sigma^{2}T/\beta.$

By independent of increments,

$\mathbb{V}ar\left(V_{Q}(T,\pi_{k})\right)=\sum_{j=0}^{\ell(k)-1}\mathbb{V}ar(\Delta_{j}^{2})$

and we already know that

$\mathbb{E}\Delta_{j}^{2}=\sigma^{2}a\delta_{j}/\beta.$

Now, we have

$\mathbb{E}\Delta_{j}^{4}=\mathbb{E}\biggr{(}\sigma
Z_{j}\gamma(a\delta_{j},\beta)^{1/2}\biggr{)}^{4}=3\sigma^{4}\frac{(a\delta_{j})(a\delta_{j}+1)}{\beta^{2}},$

i.e.,

$\mathbb{E}\Delta_{j}^{4}=3\sigma^{4}\frac{(a\delta_{j})^{2}}{\beta^{2}}+3\sigma^{4}\frac{(a\delta_{j})}{\beta^{2}}.$

This leads to

$\mathbb{V}ar(\Delta_{j}^{2})=2\sigma^{4}\frac{(a\delta_{j})^{2}}{\beta^{2}}+3\sigma^{4}\frac{(a\delta_{j})}{\beta^{2}}.$

We conclude that

$\mathbb{V}ar(V_{Q}(T,\pi_{k}))=2\frac{\sigma^{4}a^{2}}{\beta^{2}}\sum_{j=0}^{\ell(k)-1}\delta_{j}^{2}+3\frac{a\sigma^{4}T}{\beta^{2}}.$

Since

$0\leq\sum_{j=0}^{\ell(k)-1}\delta_{j}^{2}\leq m(\pi_{k})T\rightarrow 0,$

we get

$\mathbb{V}ar(V_{Q}(T,\pi_{k}))\rightarrow C=3\frac{a\sigma^{4}T}{\beta^{2}}.$

By means of Huygens Formula, for any $c\in\mathbb{R}$, for
$\|\circ\|^{2}=\mathbb{E}(\circ^{2})$, we obtained

$\|V_{Q}(T,\pi_{k})-c\|^{2}=\|V_{Q}(T,\pi_{k})-\mathbb{E}V_{Q}(T,\pi_{k})\|^{2}+(c-\mathbb{E}V_{Q}(T,\pi_{k}))^{2}\geq\biggr{(}3\frac{a\sigma^{4}T}{\beta^{2}}+o(1)\biggr{)}.$

Hence $V_{Q}(T,\pi_{k})$ does not converge in $L^{2}$. $\blacksquare$

A quick comparison with the Brownian process. The gamma-gh stochastic process
from with the Brownian motion and diverges regarding the quadratic variation
and the asymptotic boundedness of variation.

Now, let us move to the empirical construction that will allow us to get
graphical representations.

##  4\. Empirical construction

Let $T>0$. For each $n\geq 1$, let $X_{j,n}$, $1\leq j\leq n$, be independent
random variables following the same distribution gamma$(aT/n,b,0,\mu
T/n,\sigma)$. By the infinite divisibility process, we easily get that

$\sum_{1\leq j\leq n}X_{j,n}=:Z_{n}\sim\textit{gamma}(aT,b,0,\mu
T,\sigma)\textit{ for }n\geq 1.$

Now define, for $n\geq 1$, for any $T>0$

(4.1) $Y^{\ast}_{n}(t)=\biggr{(}\ \sum_{1\leq
j\leq[nt/T]}X_{j,n}\biggr{)}1_{([nt/T]\geq 1)},\ \ t\in[0,T].$

In the frame of weak convergence of stochastic processes in
$\ell^{\infty}(0,T)$, we have to check that the finite distribution
convergence of $Y^{\ast}_{n}$. For $k\geq 1$, for
$0=t_{0}<t_{1}<\ldots<t_{k}=T$, we have that the vectors, $1\leq j\leq k$,

$\Delta_{j}=Y^{\ast}_{n}(t_{j})-(t_{j-1})=\sum_{[nt_{j-1}/T]+1}^{[nt_{j}/T]}X_{j,n},$

are independent. For each $j\in\\{1,\ldots,k\\}$, by using (1.2), we clearly
have

$\displaystyle\psi_{\Delta_{j}}(u)$ $\displaystyle=$
$\displaystyle\exp\biggr{(}iu\mu
T([nt_{j}/T]-[nt_{j-1}/T])/n\biggr{)}\biggr{(}1+\frac{\sigma^{2}u^{2}}{2b}\biggr{)}^{-aT([nt_{j}/T]-[nt_{j-1}/T])/n}$
$\displaystyle\rightarrow$
$\displaystyle\exp\biggr{(}i\mu(t_{j}-t_{j-1})u\biggr{)}\biggr{(}1+\frac{\sigma^{2}u^{2}}{2b}\biggr{)}^{-a(t_{j}-t_{j-1})},$

as $n\rightarrow+\infty$. So

$\Delta_{j}\rightsquigarrow\textit{gamma}(a(t_{j}-t_{j-1}),b,0,\mu(t_{j}-t_{j-1}),\sigma),\textit{
as }n\rightarrow+\infty.$

By the Slutsky’s theorem, we conclude that the finite distribution of
$Y^{\ast}_{n}$ weakly converges those of gamma$(at,b,0,\mu t,\sigma)$-Lévy
process. To formally have the full weak convergence, we have to show the
asymptotic, which is to be done following the methods in Niang et al. (2022).
We will not do it here. Rather, we focus one approximation graphical
representations using those of $Y^{\ast}_{n}$.

We propose approximations of the centered gamma$gh(at,b,c,\mu t,\sigma)$-Lévy
processes by $Y_{n}$ in $[0,\ T]$. We take $n=500$, $b=\mu=1$, $\sigma=0.5$.
We give four different values to $a$, i.e. $a\in\\{0.5,1,3,10\\}$. Finally, we
represent the Brownian motion on $[0,\ T]$ as the weak limit of the sequence
of stochastic processes

$Z^{n}(t)=\frac{1}{\sqrt{n}}\sum_{j=1}^{[nt/T]}Z_{j},$

where the $Z_{i}$ are independent $\mathcal{N}(0,T)$ random variables. In
Figures 1 and 2 (pages 1 and 2), for small values $a\in\\{0.5,1\\}$, the
gamma$gh(at,b,0,\mu t,\sigma)$-Lévy processes have shapes very different that
of the Brownian motion in Figure 5 (page 5). But as $a$ gets bigger, like
$a\in\\{3,10\\}$, the shapes of the gamma$gh(at,b,c,\mu t,\sigma)$-Lévy
processes in Figures 3 and 4 (pages 3 and 4) and the Brownian motion seems to
be alike. It was important for us to see what might be like paths of the
studied process.

##  5\. Conclusion

This study is the first on a series of random analysis studies on
$gh(at,b,c,\mu t,\sigma)$-Lévy processes. Different aspects of stochastic
analysis will be investigated later.

Acknowledgment.

## References

  * Applebaum (2004) Applebaum D..(2004) Levy processes and stochastic calculus. Cambridge University Press. Cambridge.
  * Barndorff-Nielsen and Halgreen (1977) Barndorff-Nielsen O. and C. Halgreen (1977). Infinite Divisibility of the Hyperbolic and Generalized Inverse Gaussian Distribution. Z. Wahrs. verw. Gebiete. Vol. 38, 309-311
  * Billingsley (1995) Patrick Billingsley (1995). Probability and Measure. Wiley. Third Edition.

  * Feller (1968b) Feller W.(1968b) An introduction to Probability Theory and its Applications. Volume II. Third Editions. John Wiley & Sons Inc., New-York.
  * Grosswald (1976) Grosswald E.(1976) The Student t-distribution on any degree is infinitely divisible. Z. Wahrs. verw. Gebiete. Vol. 36, 103-109
  * Lo et al. (2016) Lo, G.S.(2016). Weak Convergence (IA). Sequences of random vectors. SPAS Books Series. Saint-Louis, Senegal - Calgary, Canada. Doi : 10.16929/sbs/2016.0001. Arxiv : 1610.05415. ISBN : 978-2-9559183-1-9
  * Lo and Niang (XXXX) Lo, G.S. and Niang A.B. (20XX).
  * Lo (2022) Lo, G.S. (2022). A note on Asymptotic total bounded variation versus uniform total bounded variation. Journal of Mathematical Facts and Short papers. to be publish
  * Lo (2018) Lo, G.S.(2018). Mathematical Foundation of Probability Theory. SPAS Books Series. Saint-Louis, Senegal - Calgary, Canada.
  * Loève (1997) Michel Loève (1997). Probability Theory I. Springer Verlag. Fourth Edition.
  * McNeil et al. (2019) McNeil A.J, Frey R. and Embrechts P.(2005) Quantitative Risk Management. Princeton University Press.
  * Niang et al. (2022) A. B. Niang, G. S. Lo, C. M. M. Traoré , A. Ball(2022). $\ell^{\infty}$-Poisson invariance principles from two classical Poisson limit theorems and extension to non-stationary independent sequences. Afrik. Statist., 17 (1), pp.
  * Niang et al. (2021) Niang A.B., Lo G.S. and Diallo M.(2004) Asymptotic laws of summands I: square integrable independent random variables. Imhotep International Mathematical Center. ArXiv:2108.03394. Doi: 10.16929/hs/imhotep.2021.x.002
dx.doi.org/10.16929/hs/imhotep.2021.x.002

  * Niang et Lo (2022) Niang A.B. and Lo G.S.Explicit constructions of some scaled Lévy Stochastic processes by means of weak convergences in $\ell^{\infty}$. To be posted in Arxiv
  * Zolotarev (1986) Zolotarev V.M (1986).One dimensional stable laws distributions. Translation of mathematical Monographs, n°65, Americal mathmatical society.

Figure 1. Construction of the gamma-gh Lévy process for
$a=0.5;b=1;\mu=1;\sigma=0.5$ Figure 2. Construction of the gamma-gh Lévy
process for
$a=1;b=1;\mu=1;\sigma=0.5$ Figure 3. Construction of the gamma-gh Lévy process
for
$a=3;b=1;\mu=1;\sigma=0.5$ Figure 4. Construction of the gamma-gh Lévy process
for
$a=10;b=1;\mu=1;\sigma=0.5$ Figure 5. Construction of the Brownian motion


# Gatemon qubit based on a thin InAs-Al hybrid nanowire

Jierong Huo  equal contribution State Key Laboratory of Low Dimensional
Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084,
China  Zezhou Xia  equal contribution State Key Laboratory of Low Dimensional
Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084,
China  Zonglin Li  equal contribution State Key Laboratory of Low Dimensional
Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084,
China  Shan Zhang  equal contribution State Key Laboratory of Low Dimensional
Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084,
China  Yuqing Wang  Beijing Academy of Quantum Information Sciences, Beijing
100193, China  Dong Pan  State Key Laboratory of Superlattices and
Microstructures, Institute of Semiconductors, Chinese Academy of Sciences, P.
O. Box 912, Beijing 100083, China  Qichun Liu  Beijing Academy of Quantum
Information Sciences, Beijing 100193, China  Yulong Liu  Beijing Academy of
Quantum Information Sciences, Beijing 100193, China  Zhichuan Wang  Beijing
National Laboratory for Condensed Matter Physics, Institute of Physics,
Chinese Academy of Sciences, Beijing 100190, China  Yichun Gao  State Key
Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua
University, Beijing 100084, China  Jianhua Zhao  State Key Laboratory of
Superlattices and Microstructures, Institute of Semiconductors, Chinese
Academy of Sciences, P. O. Box 912, Beijing 100083, China  Tiefu Li  School of
Integrated Circuits and Frontier Science Center for Quantum Information,
Tsinghua University, Beijing 100084, China  Beijing Academy of Quantum
Information Sciences, Beijing 100193, China  Jianghua Ying
<EMAIL_ADDRESS>Yangtze Delta Region Industrial Innovation Center of
Quantum and Information, Suzhou 215133, China  Runan Shang  Beijing Academy of
Quantum Information Sciences, Beijing 100193, China  Hao Zhang
<EMAIL_ADDRESS>State Key Laboratory of Low Dimensional Quantum
Physics, Department of Physics, Tsinghua University, Beijing 100084, China
Beijing Academy of Quantum Information Sciences, Beijing 100193, China
Frontier Science Center for Quantum Information, Beijing 100084, China

###### Abstract

We study a gate-tunable superconducting qubit (gatemon) based on a thin InAs-
Al hybrid nanowire. Using a gate voltage to control its Josephson energy, the
gatemon can reach the strong coupling regime to a microwave cavity. In the
dispersive regime, we extract the energy relaxation time $T_{1}\sim$0.56
$\upmu$s and the dephasing time $T_{2}^{*}\sim$0.38 $\mathrm{\upmu}$s. Since
thin InAs-Al nanowires can have fewer or single sub-band occupation and recent
transport experiment shows the existence of nearly quantized zero-bias
conductance peaks, our result holds relevancy for detecting Majorana zero
modes in thin InAs-Al nanowires using circuit quantum electrodynamics.

Topological quantum computation [1, 2] aims to solve the decoherence problem
at the device level by encoding information into Majorana zero modes [3, 4]. A
promising material candidate is the semiconductor-superconductor hybrid
nanowires [5, 6]. Tremendous efforts have been put into searching for possible
Majorana signatures in InAs and InSb nanowires [7, 8, 9, 10, 11, 12, 13, 14].
Meanwhile, proposals on topological qubits have been theoretically explored
with great enthusiasm [15, 16, 17, 18, 19, 20, 21]. A major technique in those
proposals is the circuit quantum electrodynamics (cQED), similar to that in
the superconducting transmon qubit [22, 23]. Moreover, cQED could also be used
to probe Majorana signatures if incorporating the nanowire into a transmon-
like device [24, 25, 26, 27, 28]. Motivated by this, transmon qubits based on
InAs-Al nanowires have been realized and studied in recent years [29, 30, 31,
32, 33, 34, 35, 36, 37]. The InAs wire diameters in those gate-tunable
transmons (gatemons) are typically large, $\sim$75-160 nm. Though the junction
region can be easily gate-tuned, the proximitized InAs region is heavily
screened by the covered Al film and is still in the multi-subband regime.
Thick wire and multi-band bring challenges into the Majorana detection [38,
39, 40, 41, 42, 43]. To overcome this issue, thin InAs-Al nanowires have been
explored and nearly quantized zero bias conductance peaks have been reported
[44, 11, 12]. Here, we report the realization of gatemon qubit based on these
thin wires. The InAs diameter is $\sim$35 nm, significantly smaller than those
in previous gatemons. Our result paves the way for future Majorana cQED
experiments.

Figure 1: (a) Optical image of a gatemon chip and measurement set-up. The
grounding plane is in orange. The common feed line is in blue (false colored).
The $\lambda/4$ cavity (for device A) is in green. The T-shape qubit capacitor
(for device A) is in red. The gate line is in pink. All these elements above
are NbTiN (thickness 100 nm) and were fabricated in one lithography step using
reactive ion etching. (b) An enlargement of (a) (the dashed box). (c) SEM
(false colored) of the InAs-Al region of the gatemon (the dashed box in (b)).
The top contact connects to the T-shape capacitor. The bottom contact is
grounded. A side gate (pink) connects to the gate line. The gate and contacts
were fabricated in another lithography step by sputtering Ti/NbTiN (1/100 nm).
(d) SEM of the Josephson junction region of the qubit (the blue box in (c)).
(e) Schematic of the measurement circuit. (f) Feed line transmission as a
function of the readout power. $V_{\text{G}}$ = -4.554 V. The cavity shift
indicates the coupling of the cavity to a nonlinear circuit (the qubit).

Qubit device and measurement circuit. Figure 1(a) shows the optical image
(false colored) of the device chip. One gatemon qubit (device A, the dashed
box) was measured with results shown in Figs. 2-4. The other five qubits on
this chip were not working. We have characterized three working qubits, see
the supplementary material (SM) for the other two. For the device fabrication,
a 100-nm-thick NbTiN superconducting film (the orange region) was first
sputtered onto a sapphire substrate. Reactive ion etching was then performed
to etch away part of the film (the dark regions). This lithography step
defines the co-planar wave-guide feed line (blue), the resonator/cavity
(green) and the shunt capacitor of the gatemon (red). Figure 1(b) is a zoomed-
in image of the qubit. The feed line capacitively couples to a $\lambda/4$
cavity for the qubit readout. The cavity internal quality factor, $Q_{i}$, is
$\sim$14000 and the bare resonance frequency $f_{\text{C}}$ is $\sim$4.6 GHz.
The cavity further couples to the T-shape capacitor whose capacitance is
estimated to be $\sim$100 fF. This capacitor connects to the InAs-Al nanowire
Josephson junction (Figs. 1(c) and 1(d)) and together, they form the gatemon
qubit. The large capacitance determines the charging energy to
$E_{\text{C}}\sim$$e^{2}/2C\sim$190 MHz. The other side of the Josephson
junction connects to the ground. A side gate (pink) tunes the junction
transparency and therefore controls the Josephson energy, $E_{\text{J}}$. The
junction was defined by removing (etching) a small Al segment on the InAs wire
(diameter $\sim$35 nm). Transports on these wires show a gate-tunable
supercurrent ($I_{\text{c}}$) on the order of 100 nA [45]. The corresponding
$E_{\text{J}}=\hbar I_{\text{c}}/2e\sim$50 GHz is much larger than
$E_{\text{C}}$. This ensures that the qubit can be operated in the transmon
regime. For details of the device fabrication, circuit set-up and cavity
calibration, see SM (the method session, Figs. S1 and S2).

Figure 2: (a) Cavity transmission in the single-tone measurement as a function
of the cavity drive frequency and $V_{\text{G}}$. The anti-crossings are the
vacuum Rabi splittings. (b) Fine scan of an anti-crossing. The gate voltage
has a shift compared to the same feature in (a) due to hysteresis or charge
jumps. The two dashed lines are the bare cavity frequency (pink,
$f_{\text{C}}$) and the extracted qubit frequency (yellow, $f_{\text{Q}}$).
(c) A line cut from (b) at the crossing point of the two dashed lines. The
peak spacing $\delta=f_{+}-f_{-}$ is indicated by the black arrow in (b). (d)
$\delta$ as a function of the qubit frequency. The red line is the theoretical
fit. Inset, energy schematic of the cavity-qubit hybridization and the
parameters. (e) Gatemon qubit spectroscopy (two-tone) as a function of
$V_{\text{G}}$ at three different ranges. (f) A line cut of (e) (at the black
arrow) with a Lorentzian fit (blue dashed line).

Figure 1(e) draws the equivalent circuit diagram. For the cavity and qubit
readout, a microwave tone of frequency $f_{\text{r}}$ (near the cavity
resonant frequency $f_{\text{C}}$) is applied to the feed line (see Fig.
1(a)). The transmission of this microwave tone, $S_{21}$, is measured by a
vector network analyzer (VNA). Figure 1(f) is such a “single-tone” measurement
by sweeping $f_{\text{r}}$ and its power while monitoring the transmission
amplitude $|S_{21}|$. At high power, the qubit is “overwhelmed” and the dip in
$|S_{21}|$ corresponds to the bare resonant frequency of the cavity
$f_{\text{C}}$ [46]. At low power, the resonant frequency is qubit-state
dependent due to the cavity-qubit dispersive interaction. The repulsion of the
qubit and cavity causes the shift of the resonance frequency as shown in Fig.
1(f). To excite and control the qubit, a second microwave tone of frequency
$f_{\text{d}}$ can be applied in the standard “two-tone” spectroscopy.

Vacuum Rabi splitting and qubit spectroscopy. In Fig. 2(a), we keep the
readout power low and scan the gate voltage ($V_{\text{G}}$). The resonant
frequency of the cavity is gate tunable, indicating the presence of the
gatemon. For better visibility, a signal background, contributed by standing
waves in the circuit, was subtracted from $|S_{21}|$ (see Fig. S3 in SM for
details). The jumps in the spectrum are due to charge instabilities in the
InAs-Al devices which are also commonly observed in the transport
characterizations. The qubit frequency ($f_{\text{Q}}$) is given by the energy
difference between the ground state and the first excited state,
$hf_{\text{Q}}=E_{01}\sim$$\sqrt{8E_{\text{C}}E_{\text{J}}}-E_{\text{C}}$.
$V_{\text{G}}$ tunes $E_{\text{J}}$ therefore controls $f_{\text{Q}}$. When
$f_{\text{Q}}$ is tuned close to the cavity frequency ($f_{\text{C}}$), the
strong qubit-cavity hybridization leads to the anti-crossings. This anti-
crossing is observed in the single-tone spectrum shown in Fig. 2(a). See Fig.
2(b) for a fine scan (an enlargement) of an anti-crossing. Figure 2(c) is the
line cut at $V_{\text{G}}$ = -3.310 V where the peak spacing is the smallest.
In this strong coupling regime, the frequencies of the two peaks are
$f_{\pm}=\left[f_{\text{Q}}+f_{\text{C}}\pm\sqrt{(f_{\text{Q}}-f_{\text{C}})^{2}+4(g/2\pi)^{2}}\right]/2$.
The peak spacing,
$\delta=f_{+}-f_{-}=\sqrt{(f_{\text{Q}}-f_{\text{C}})^{2}+4(g/2\pi)^{2}}$, is
a function of $f_{\text{Q}}$. The qubit frequency can be obtained by
$f_{\text{Q}}=f_{+}+f_{-}-f_{\text{C}}$. Figure 2(d) plots the extracted
$\delta$ and $f_{\text{Q}}$. The red line is the theory fit based on the
formula above. We extract the qubit-cavity coupling strength $g/2\pi\sim$74
MHz.

Next we tune $V_{\text{G}}$ to bias $f_{\text{Q}}$ away from $f_{\text{C}}$
and reach the dispersive regime. The large detuning,
$|\Delta/2\pi|=|f_{\text{Q}}-f_{\text{C}}|\gg g$, could effectively suppress
the energy relaxation due to the Purcell effect [47]. Figure 2(e) shows the
two-tone spectroscopy, $|S_{21}|$ as a function of the qubit drive
$f_{\text{d}}$ (the second tone) at three different $V_{\text{G}}$ ranges. The
readout frequency $f_{\text{r}}$ (the first tone) was fixed near the cavity
resonance ($f_{\text{C}}$). When $f_{\text{d}}$ is scanned on resonance with
$f_{\text{Q}}$, the qubit can be excited and the resonator frequency is
shifted. A signal in the readout tone can be observed due to the cavity shift.
The spectroscopy in Fig. 2(e) reveals the gate tunable nature of
$f_{\text{Q}}$. The non-monotonic fluctuations are associated with the non-
ballistic property of the InAs-Al junction, indicating the presence of
disorder.

Figure 3: (a) Rabi oscillations as a function of the qubit drive frequency and
the drive pulse duration. (b) A line cut at the qubit resonant frequency
$f_{\text{Q}}$ = 4.426 GHz (see the arrow in (a)). The dashed line is a
theoretical fit. (c) Rabi oscillations as a function of the drive power at a
fixed drive frequency $f_{\text{d}}=4.446$ GHz. (d) The Rabi oscillation
frequency ($\omega/2\pi$) extracted from (c) versus the driving amplitude
(over the range where the oscillations are visible). $10^{P/20}$ has a linear
relation with the driving amplitude. The dashed line is a linear fit.
$V_{\text{G}}=-2.890$ V for all panels. Figure 4: Energy relaxation time
$T_{1}$ and dephasing time $T_{2}^{*}$. (a) Pulse sequence schematic for the
$T_{1}$ measurement. (b)-(c) $T_{1}$ measurements at $V_{\text{G}}$ of -2.890
V and -4.442 V, respectively. The black dashed lines are the exponential decay
fits. The increasing vs decreasing trend between (b) and (c) are due to
different selected working points, resulting in reversed readout signal
strength. (d) Pulse sequence schematic for the $T_{2}^{*}$ measurement.
(e)-(f) Ramsey oscillations at different $V_{\text{G}}$ values. The dashed
lines are fits with an exponential decaying envelope.

The multiple peaks in Fig. 2(e) are likely caused by the photon-number-
dependent frequency shift of the qubit [48]. This is obvious when the cavity
readout tone was continuously applied (instead of pulsed) on the feed line,
see Fig. S4 in SM for detailed analysis. The extracted peak spacing ($\sim$12
MHz) roughly matches our estimation of $2|\chi|/2\pi=g^{2}/|\Delta|\pi\sim$10
MHz. Figure 2(f) shows a line cut of the qubit excitation peak from Fig. 2(e).
From the Lorentzian fit we extract the full width at half maximum (FWHM
$\sim$3.8 MHz). This width corresponds to a coherence time $\sim$0.26
$\upmu$s, consistent with the time-domain measurement in Figs. 3 and 4.

Rabi oscillations. We now manipulate the gatemon qubit in time domain. A qubit
drive pulse ($f_{\text{d}}$) was first applied for a duration time of $t$ and
then followed by a readout pulse of the cavity. Figure 3(a) shows the typical
Rabi oscillations as a function of $f_{\text{d}}$ and $t$. The Rabi
oscillation frequency
$\omega\propto\sqrt{(f_{\text{d}}-f_{\text{Q}})^{2}+{\text{const.}}}$. The
term, const., is proportional to the square of the driving amplitude. The
oscillation pattern shows a Chevron feature. We note that the pattern gets
disrupted near 4.40 GHz, probably due to the presence of another cavity on the
chip whose resonant frequency is around 4.40 GHz. From Fig. 3(a), we can
estimate the qubit frequency $f_{\text{Q}}\sim$4.426 GHz. Note that this is
not in the dispersive regime yet since the detuning is not large enough.
Figure 3(b) shows the line cut near $f_{\text{Q}}$. Fitting the decaying
oscillation using an empirical formula
$y=Ae^{-t/T_{\text{R}}}\text{cos}(\omega t+\phi)+C$ (the dashed line), we
extract the Rabi coherence time $T_{\text{R}}\sim$0.23 $\upmu$s.

In Fig. 3(c), the qubit drive power is varied. Higher power drives the qubit
faster, resulting in a shorter oscillation period. We extract the oscillation
frequency (the inverse of the period) and plot it as a function of its driving
amplitude (converted from the power) in Fig. 3(d). The dashed line is a linear
fit, confirming its Rabi oscillation nature.

Gatemon quantum coherence. To extract the gatemon energy relaxation time
$T_{1}$, a $\pi$ pulse was first applied to excite the qubit to the $\ket{1}$
state. The readout was performed after the waiting time $t_{1}$ (Fig. 4(a)).
Figures 4(b) and 4(c) show the exponential fit at two different gate voltages.
A relaxation time $T_{1}\sim$0.56 $\upmu$s can be extracted. The dephasing
time $T_{2}^{*}$ was determined by the Ramsey experiment: inserting a waiting
time $t_{2}$ between two slightly detuned $\pi$/2 pulses before the readout
(Fig. 4(d)). Figures 4(e) and 4(f) show two Ramsey oscillations. $T_{2}^{*}$
can reach $\sim$0.38 $\upmu$s. The fitting assumes an exponential decaying
envelope: $A\text{cos}(\omega t+\phi)\text{exp}(-(t_{2}/T_{2}^{*})^{2})+C$
[49]. Note that the gate voltage of Figs. 4(c) and 4(f) corresponds to the
dispersive regime ($f_{\text{Q}}$ here is $\sim$3.446 GHz), therefore a longer
coherence time is expected. As a comparison, the coherence times for Figs.
4(b) and 4(e) are shorter due to the Purcell effect. $T_{2}^{*}<2T_{1}$
indicates that the coherence of our qubit is not entirely limited by energy
relaxation. For the Rabi oscillations at $V_{\text{G}}$ = -4.442 V (same with
Figs. 4(c) and 4(f)), see Fig. S5. In Fig. S6, we show the measurement of two
more gatemon qubits.

In summary, we have studied the gatemon qubit based on a thin InAs-Al hybrid
nanowire. The gatemon can reach strong coupling to a cavity. Coherent Rabi
oscillations can be observed. The qubit relaxation time $T_{1}$ and dephasing
time $T_{2}^{*}$ can reach 0.56 $\upmu$s and 0.38 $\upmu$s, respectively.
Future work on these thin-wire-based gatemons could aim for possible Majorana
signatures in a finite magnetic field.

Acknowledgment We thank Chunqing Deng and Luyan Sun for valuable discussions.
We also thank the Teaching Center for Experimental Physics of Tsinghua
University for using their equipment. This work is supported by Tsinghua
University Initiative Scientific Research Program, Alibaba Innovative Research
Program, National Natural Science Foundation of China (Grant Nos. 12204047,
92065106, 61974138). D.P. also acknowledges the support from Youth Innovation
Promotion Association, Chinese Academy of Sciences (Nos. 2017156 and
Y2021043). Raw data and processing codes within this paper are available at
https://doi.org/10.5281/zenodo.7620737.

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