# Large deviations for continuous time random walks

Wanli Wang  Department of Physics, Institute of Nanotechnology and Advanced
Materials, Bar-Ilan University, Ramat-Gan 52900, Israel  Eli Barkai
Department of Physics, Institute of Nanotechnology and Advanced Materials,
Bar-Ilan University, Ramat-Gan 52900, Israel  Stanislav Burov  Department of
Physics, Bar-Ilan University, Ramat-Gan 52900, Israel

###### Abstract

Recently observation of random walks in complex environments like the cell and
other glassy systems revealed that the spreading of particles, at its tails,
follows a spatial exponential decay instead of the canonical Gaussian. We use
the widely applicable continuous time random walk model and obtain the large
deviation description of the propagator. Under mild conditions that the
microscopic jump lengths distribution is decaying exponentially or faster i.e.
Lévy like power law distributed jump lengths are excluded, and that the
distribution of the waiting times is analytical for short waiting times, the
spreading of particles follows an exponential decay at large distances, with a
logarithmic correction. Here we show how anti-bunching of jump events reduces
the effect, while bunching and intermittency enhances it. We employ exact
solutions of the continuous time random walk model to test the large deviation
theory.

##  I Introduction

Following the erratic motion of pollen under his microscope Robert Brown
discovered what is called today: Brownian motion. This phenomenon was modeled
by Einstein and others, with random walk theory while the mathematical
description of Brownian motion, i.e. the Wiener process Majumdar2005Brownian ,
was quickly established soon after. Two ingredients of this widely observed
dynamics are that the mean square displacement increases linearly with time
and that the spreading of the packet of particles all starting on a common
origin is Gaussian Montroll1965Random ; Haus1987Diffusion . The latter is the
physical manifestation of the central limit theorem. However, in Reference
Pinaki2007Universal  Chaudhuri, Berthier, and Kob, discovered what might turn
out to be a no less universal feature of random walks. Analysing experimental
and numerical data, e.g. the motion of particles in glass forming systems,
they revealed an exponential decay of the packet of spreading particles; see
related works in References Kege2000Direct ; Masolivera200dynamic ;
Weeks2000Three ; Pinaki2007Universal ; Wang2009Anomalous ; Hapca2009Anomalous
; Leptos2009Dynamics ; Eisenmann2010Shear ; Toyota2011Non ;
Skaug2013Intermittent ; Xue2016Probing ; Wang2017Three ;
Jeanneret2016Entrainment ; Chechkin2017Brownian ; Cherstvy2019Non ;
Witzel2019Heterogeneities ; Shin2019Anomalous ; Singh2020Non ; Mejia2020Tracer
; Xue2020Diffusion . Soon after this, similar behavior described by the
Laplace distribution, was found for many other systems, including for
molecules in the cell environment Weeks2000Three ; Wang2012Brownian ;
Munder2016transition . In Pinaki2007Universal  it was suggested that a
specific type of continuous time random walk (CTRW) model could explain the
physics of the observed behavior. Two of us have recently shown that the
phenomenon is indeed universal as it holds in general Barkai2020Packets  and
under very mild conditions (see below). The goal of this paper is to produce
further evidence for the phenomenon, present exact solutions of the model and
compare it with the new theory. We also present a more detailed analysis of
the model covering cases not discussed previously.

The observed behavior, is related to the way experimentalists record the
erratic paths. Following many trajectories one may find universal features of
the motion in at least two different ways. In some situations the measurement
time is very long such that many jumps occurred and consequently Gaussian
statistics will take place. However, in practice experiments are not conducted
for an infinite time. In fact in many experiments, the motion follows some
trapping and then released motion, i.e. a hop and then wait dynamics, as
modeled by the CTRW. In these experiments one follows many paths, e.g. many
single molecules, however typically each trajectory is recorded separately.
When the averaged number of jumps recorded under the microscope (or in
simulations) within the observation time $(0,t)$ is not large, one would
expect naively that this would imply the non-existence of universal
statistical laws. However a second limit is important, which is large
positions (and finite time). In this case, we will promote the idea that
exponential spreading is the rule. Thus the observation of either Gaussian or
exponential spreading depends on the time scale and length scale of the motion
and in statistical sense the temporal and spatial extent of the field of view.
The idea is that by observing the density of packet of particles in its tails
(large $x$) we are in fact considering trajectories where the number of jumps
is large, compared to the mean. We advocate that universal laws, that demand a
large number of steps, can be found for large displacements (and fixed time)
or for large time as usual. Theoretically since we have two parameters that
can be made large, i.e. $t/\langle\tau\rangle$ or $x/\sigma$ we may obtain
more than one limiting law. Here $\langle\tau\rangle$ is the average time
between jumps, and $\sigma$ is the variance of jumps lengths. The motion is
unbiased hence mean jump size is zero. We will consider, among other things
the case, when both $t/\langle\tau\rangle$ and $x/\sigma$ are large and their
ratio is finite.

The remainder of the manuscript is organized as follows. In Sects. II and III,
after presenting the CTRW model, we consider the far tail of the distribution
of the number of renewals and the position of the random walker, where the
waiting times are drawn from the exponential and the Erlang distributions,
respectively. The bunching effect is investigated in Sect. IV using the sum of
two exponential probability density functions (PDF). Finally, we conclude with
a discussion.

##  II Appetizer for exponential tails

We consider the well known CTRW model Montroll1965Random ;
Kindermann2017Nonergodic ; Kutner2017continuous . This model describes a wait
and then jump process. A particle in dimension one, starts on the origin at
time $t=0$. We draw a waiting time from the PDF $\psi(\tau)$ and then after
the wait, the particle will perform a jump whose length, $\chi$, is drawn from
$f(\chi)$. The process is then renewed. In our case the waiting times and the
jump lengths are not correlated. The position of the particle at time $t$ is
$x(t)=\sum_{i=1}^{n}\chi_{i}$, and here $n$ is the random number of jumps
within the time of observation $(0,t)$. The focus of this manuscript is the
PDF of finding the particle on $x$ at time $t$ which is denoted $P(x,t)$. In
particular the large $x$ limit of this distribution is of interest. We focus
on non-biased CTRWs, and then if the mean waiting times and the variance of
jump lengths are finite the density $P(x,t)$ will converge to a Gaussian as
expected from the central limit theorem. This limit theorem is valid for
$t\to\infty$ and $x\propto t^{1/2}$, while here we are interested in finite
time effects and the large $x$ limit, to be defined more precisely below.
Generally, the solution of the model is given by

$P(x,t)=\sum_{n=0}^{\infty}Q_{t}(n)P(x|n).$ (1)

Here the sum is over the possible outcomes of the number of jumps in the
process, $Q_{t}(n)$ is the probability of attaining $n$ jumps
Godreche2001Statistics , while $P(x|n)$ is the probability of finding the
particle on $x$ conditioned it made $n$ jumps. In Laplace space
Barkai2020Packets ,

$\hat{Q}_{s}(n)=\int_{0}^{\infty}Q_{t}(n)\exp(-st)dt=\frac{1-\hat{\psi}(s)}{s}\hat{\psi}^{n}(s),$
(2)

where $s$ is the Laplace pair of $t$. In particular, when $n=0$,
$Q_{t}(n=0)=\int_{t}^{\infty}\psi(\tau)d\tau$ is called the survival
probability.

Generally finding $Q_{t}(n)$ and $P(x|n)$ is non trivial. Here the goal is to
consider special choices of jump length distributions $f(\chi)$ and waiting
time PDFs $\psi(\tau)$ which allow us to express the solution as an infinite
sum over $n$ explicitly, i.e. without restoring to inverse Fourier and Laplace
transforms and/or numerical simulations. This allows us to compare between
exact solutions of the problem, and the large deviation theory we have
promoted previously Barkai2020Packets . Further, with the specific choices of
the input distributions $\psi(\tau)$ and $f(\chi)$ we can derive large
deviations theory in a straight forward way. The price that we pay is that we
do not consider here the theory in its full generality, since we stick to
exactly solvable models.

###  II.1 Displacement follows Gaussian distribution

We start the analysis with the simplest choice of waiting times and jump
length distributions. The jump length PDF is assumed to be Gaussian with zero
mean and variance $\sigma=1$, i.e., $f(\chi)=\exp(-\chi^{2}/2)/\sqrt{2\pi}$.
It then follows that $P(x|n)$ is Gaussian as well. The waiting times are
exponentially distributed $\psi(\tau)=\exp(-\tau)$ for $\tau>0$. Hence the
mean waiting time is $\langle\tau\rangle=1$ and $Q_{t}(n)$ obeys Poisson
statistics. The density of spreading particles all starting on the origin is
therefore

$\begin{split}P(x,t)=&\sum_{n=0}^{\infty}\frac{t^{n}\exp(-t)}{n!}\frac{\exp(-x^{2}/2n)}{\sqrt{2\pi
n}}\\\ \end{split}$ (3)

and in this sense we have an exactly solvable model valid for any time and
$x$. Here the term $n=0$ contains a delta function on the origin, which of
course is not plotted in the figures we present below. This describes
particles not moving at all.

To analyse this sum in the large $x$ limit we use Cramér-Daniels approach
Daniels1954Saddlepoint  to large deviations. Essentially this is a saddle
point method, exploiting the fact that one may relate the cumulant generating
function of $P(x,t)$ and the large deviations in the system. For that let us
take the Fourier transform of Equation (3) and then we have

$\widetilde{P}(k,t)=\sum_{n=0}^{\infty}{\frac{t^{n}\exp(-t)}{n!}}\widetilde{f}^{n}(k),$
(4)

where we used the Fourier transform of $f(\chi)$ denoted
$\widetilde{f}(k)=\exp(-k^{2}/2)$. We have exploited the fact that the jump
lengths are independent and identically (IID) hence from convolution theorem
of Fourier transforms we have the expression
$\widetilde{P}(k|n)=\widetilde{f}^{n}(k)$. The moment generating function is
$\langle\exp(ux)\rangle=\int_{-\infty}^{\infty}P(x,t)\exp(ux){\rm d}x$.
Replacing $k\to-iu$ and summing the series

$\langle\exp(ux)\rangle=\exp\left[-t(1-e^{u^{2}/2})\right].$ (5)

Then the cumulant generating function is simply the log of the above
expression, we denote it

$K(u)=\ln\left[\langle\exp(ux)\rangle\right]=-t\left(1-e^{u^{2}/2}\right).$
(6)

Now we are interested in the large $x$ limit of $P(x,t)$, which by definition
is the inverse Fourier transform $\widetilde{P}(k,t)$. Instead of integrating
over $k$ one may switch to integration over $u$ and using saddle point method
valid for the large $x$ one then finds a well known large deviation formula
Daniels1954Saddlepoint

$P(x,t)\sim{1\over\sqrt{2\pi
K^{\prime\prime}(\hat{u})}}\exp\left[K\left(\hat{u}\right)-\hat{u}x\right].$
(7)

Here $\hat{u}$ is given by the solution $K^{\prime}(u)=x$. While in this
section we are treating a special choice of jump length and waiting time, this
tool will serve us all along the paper. One should notice that a cumulant
generating function does not always exist. For example if $f(\chi)$ decays as
a power law for large $x$, the moments of the process will diverge. From here
we see the first condition of the theory to hold in generality: we assume that
cumulant generating function of $f(\chi)$ and $P(x,t)$ exist, so the decay of
$f(\chi)$ is faster than exponential Eli2020SRW .

For the Gaussian choice of $f(\chi)$, to find $\hat{u}$ we have

$K^{\prime}(\hat{u})=t\hat{u}\exp\left({\hat{u}^{2}\over 2}\right)=x.$ (8)

The solution of this equation is given by the Lambert $W_{0}$ function
Dence2013brief  where the subscript zero denotes the branch of this function.
The latter is the solution of the equation $y\exp(y)=z$ given by $y=W_{0}(z)$
if $z\geq 0$. Hence we get

$\hat{u}=\pm\sqrt{W_{0}\left[\left({x\over t}\right)^{2}\right]},$ (9)

where the sign of $\hat{u}$ is dictated by the sign of $x$. Using Equation (7)
we then find

$P(x,t)\sim{\exp\left(-t-|x|\left(\sqrt{W_{0}[\frac{x^{2}}{t^{2}}]}-{1\over\sqrt{W_{0}[\frac{x^{2}}{t^{2}}]}}\right)\right)\over\sqrt{2\pi
K^{\prime\prime}(\hat{u})}}$ (10)

with

$K^{\prime\prime}(\hat{u})=|x|\left(\sqrt{W_{0}[(x/t)^{2}]}+\frac{1}{\sqrt{W_{0}[(x/t)^{2}]}}\right).$

This is plotted in Figure 1. Using the large and small $z$ limits of the
Lambert function, i.e. $W_{0}(z^{2})\sim 2\ln(z)$ when $z>>1$ and
$W_{0}(z^{2})\sim z^{2}$ for $z<<1$, we find the following two limits:

$P(x,t)\simeq\exp\left[-x\sqrt{2\ln(x/t)}\right]$ (11)

valid when $x/t>>1$, and

$P(x,t)\simeq\exp[-x^{2}/2t]$ (12)

in the opposite limit when $|x|/t\to 0$. The first is the mentioned
exponential decay, and it includes what we call the $\ln(x)$ correction. As
far as we know, in the experimental literature this $\ln(x)$ is not reported.
However this fact is not surprising as the $\ln(x)$ is clearly a slowly
varying function and hence difficult to detect in reality. We note that the
$\log$ correction implies that the decay is in reality slightly faster than
exponential, implying the existence of the cumulant generating function. The
second case is the well known Gaussian behavior found in the centre of the
packet.

We may further formulate our results using two very much related approaches,
both consider the limits $x\to\infty$ and $t\to\infty$, and their ratio
$x/t=l$ is kept fixed. Using Equation (10) the first limiting law is

$\lim_{x\to\infty}-\ln(P(x,t))/x=\mathcal{I}_{x}(l)$ (13)

with $\mathcal{I}_{x}(l)=l^{-1}-1/\sqrt{W_{0}(l^{2}))}+\sqrt{W_{0}(l^{2})}$
and the second refers to

$\lim_{t\to\infty}-\ln(P(x,t))/t=\mathcal{I}_{t}(l)$ (14)

with $\mathcal{I}_{t}(l)=l\mathcal{I}_{x}(l)$. The functions
$\mathcal{I}_{t}(l)$ and $\mathcal{I}_{x}(l)$ are called the rate functions
Touchette2009large ; Daniel2018Anomalous ; Touchette2018Introduction . All
these approaches are essentially identical. Most experimentalists in the field
plot $P(x,t)$ versus $x$ on a semi-log scale, and for that aim Equation (10)
including the prefactor $1/\sqrt{2\pi K^{\prime\prime}(\hat{u})}$ is useful.
Mathematicians influenced by long time (large $x$) ideas, and the large
deviation literature, would possibly find it natural to use
$\mathcal{I}_{t}(l)$ ($\mathcal{I}_{x}(t)$) respectively. In demonstration
below we plot the exact solution for finite times, using both approaches; see
Figures 1, 2, and 3.

We now present a simple explanation for the exponential tail using a lower
bound. Clearly

$P(x,t)\geq{\exp(-\frac{x^{2}}{2n^{*}})\over\sqrt{2\pi
n^{*}}}{t^{n^{*}}\exp(-t)\over(n^{*})!}$

and here $n^{*}$ is the value of $n$ for which the summand in Equation (3) is
the maximum. To find $n^{*}$, we use Stirling’s approximation, i.e.,
$n!\sim\sqrt{2\pi n}(n/e)^{n}$, and

$\frac{d}{dn}\exp\left(-t+n\ln(t)-\frac{x^{2}}{2n}-\ln\left(\sqrt{2\pi
n}\left(\frac{n}{e}\right)^{n}\right)\right)=0.$ (15)

The solution of the above equation is
$n^{*}\sim|x|/\sqrt{W_{0}(x^{2}/t^{2})}$. As expected, in the limit
$|x|/t\to\infty$, we find the exponential decay again

$P(x,t)\geq\exp\left(-t-|x|\sqrt{\frac{\ln(\frac{|x|}{t})}{2}}+|x|\frac{\ln(\frac{te}{|x|}\sqrt{2\ln(\frac{|x|}{t}})}{\sqrt{2\ln(|x|/t)}}\right),$
(16)

where we used the asymptotic behavior of the Lambert function, i.e.,
$W_{0}(|x|)\sim\ln(|x|)$ with $|x|\to\infty$. The claim in Barkai2020Packets
is much more general as similar behavior is found for a large class of waiting
times and jump lengths PDFs, see details below.

Figure 1: The distribution $P(x,t)$ with exponentially distributed waiting
times and Gaussian displacements. The time is $t=2$ and for simulations we
used $5\times 10^{7}$ trajectories. Our theory Equation (10) performs
perfectly, while the Gaussian distribution Equation (12), black dash-dotted
line, completely fails for the far tails of the distribution of the position.
Note that the theoretical prediction Equation (10) works extremely well also
for the central part of the distribution. Here the exact result is obtained
from Equation (3).
Figure 2: The plot of the rate function $\mathcal{I}_{x}(l)$ with $l=|x|/t$.
The red solid line, predicted by Equation (13), describes the rate function
$\mathcal{I}_{x}(x/t)$ capturing the behavior of large deviations. Note that
for large $l$ we have $\mathcal{I}_{x}(l)\sim\sqrt{2\ln(l)}$ and this means
that $P(x,t)$ is decaying exponentially with the distance $x$ with a
correction. The dashed lines are the plot of $\ln(P(x,t))/(-|x|)$ calculated
from Equation (3).
Figure 3: Rate function $\mathcal{I}_{t}(l)$ versus $l=|x|/t$ for different
$t$. The red solid line is the (time) rate function Equation (14). As in
Figures 1 and 2 here $\psi(\tau)=\exp(-\tau)$ and
$f(\chi)=\exp(-\chi^{2}/2)/\sqrt{2\pi}$.

###  II.2 Displacement drawn from the discrete PDF

We now investigate a second example, which further demonstrates the
exponential far tail of $P(x,t)$. Consider a CTRW on a one dimensional
lattice, with exponential waiting times. Hence
$f(\chi)=[\delta(\chi-1)+\delta(\chi+1)]/2$ and $\psi(\tau)=\exp(-\tau)$. The
Fourier transform of the jump length distribution is
$\widetilde{f}(k)=\cos(k)$ and the moment generating function is
$\langle\exp(u\chi)\rangle=\cosh(u)$, where $\cosh(u)$ is hyperbolic cosine
function. Here $P(x|n)$ is the well known Binomial distribution

$P(x|n)={1\over
2^{n+1}}[1+(-1)^{n+x}]{n!\over\left(\frac{n+x}{2}\right)!\left(\frac{n-x}{2}\right)!},~{}~{}~{}~{}x\leq
n,$ (17)

where $x$ is an integer since the random walker is on a lattice with a unit
spacing. Notice that when $n$ is odd/even the same holds for $x$. The CTRW
probability is then

$P(x,t)=\sum_{n=0}^{\infty}{e^{-t}t^{n}\over n!}P(x|n).$ (18)

Switching to the Fourier space from convolution theorem, the Fourier transform
of $P(x|n)$ is $\widetilde{f}^{n}(k)$. Then after summation it is easy to find
the moment generating function
$M(u)=\langle\exp(ux)\rangle=\exp[-t(1-\cosh(u))]$. We then use, as before,
the cumulant generating function $K(u)=-t[1-\cosh(u)]$. We now need to find
the solution of $K^{{}^{\prime}}(u)=x$ which is given by
$\hat{u}=\sinh^{-1}(x/t)=\ln(x/t+\sqrt{1+x^{2}/t^{2}})$, then using Equation
(7) we get

$P(x,t)\sim\frac{\exp\left[-x\sinh^{-1}\left({x\over
t}\right)-t+t\sqrt{1+{x^{2}\over t^{2}}}\right]}{\sqrt{2\pi
K^{{}^{\prime\prime}}(\hat{u})}}$ (19)

with

$K^{{}^{\prime\prime}}(\hat{u})=\sqrt{x^{2}+t^{2}}.$

Equation (19) provides for small $x/t$ the standard Gaussian behavior,
$P(x,t)\simeq\exp[-x^{2}/2t]$. In the opposite limit of large $x$

$P(x,t)\simeq\exp\left[-x\ln\left({2x\over t}\right)\right],$ (20)

which demonstrates the exponential decay. We may also write this solution like

$P(x,t)\sim\frac{1}{\sqrt{2\pi
K^{{}^{\prime\prime}}(\hat{u})}}\exp\left[-t\mathcal{I}_{t}\left({x\over
t}\right)\right]$ (21)

with the rate function

$\mathcal{I}_{t}(l)=1+l\sinh^{-1}(l)-\sqrt{1+l^{2}}.$ (22)

Notice that in Equation (17) $P(x|n)=0$ for $x>n$ since the particle walking
on a lattice cannot reach a distance larger than the number of steps. Thus for
standard random walks, with a fixed number of jumps, the exponential tails are
not a generic feature. It implies that the fluctuation of the number of steps
is crucial for the observation of exponential tails, like those in Equation
(20).

Figure 4:  Plot of Erlang PDF (Equation (25)) for various $m$. Increasing $m$
leads to decreasing probability of obtaining small $\tau$ that leads to
transition from bunching to anti-bunching (see discussion preceding Equation
(54)).

##  III The exponential tails for Erlang waiting time PDF

So far we exposed the exponential tails of $P(x,t)$ assuming that the
statistics of the numbers of jumps is Poissonian. To advance the theory
further we now consider $Q_{t}(n)$ for the Erlang PDF of waiting times. For
Erlang distribution, the variance of the waiting time is finite. In the long
time limit, from the central limit theorem $n$ follows well known Gaussian
distribution Cox1977Theory ; Godreche2001Statistics . This indicates that the
spreading of the particles obeys Gaussian distribution for the central part of
the distribution of the position. For more details see Appendix. A. For the
tails, large number of jumps $n$ is the cause for large displacement from the
origin of the random walker. Hence the limit $n$ large and $t$ fixed, or both
these observables large while their ratio is finite is of interest.

###  III.1 The exponential tail of the number of events

In this regard, two of us have found a general solution of the problem
Barkai2020Packets . The large $n$ limit of $Q_{t}(n)$ is controlled by the
small $\tau$ limit of the waiting time PDF, since to have many jumps within a
finite fixed time $t$, the time intervals between jumps must be made small.
The main requirement we use here is that $\psi(\tau)$ is analytical in the
vicinity of $\tau\to 0$ and hence

$\psi(\tau)\sim C_{A}\tau^{A}+C_{A+1}\tau^{A+1}.$ (23)

Here $A$ is a non negative integer, and $C_{A}$ and $C_{A+1}$ are
coefficients. Then in the large $n$ limit, and $t$ fixed we have
Barkai2020Packets

$Q_{t}(n)\sim{\left\\{[C_{A}\Gamma(1+A)]^{1/(1+A)}t\right\\}^{n(1+A)}\over\Gamma\left[n(1+A)+1\right]}\exp\left(t{C_{A+1}\over
C_{A}}\right).$ (24)

Beyond $A$, $C_{A}$ and $C_{A+1}$ other properties of $\psi(\tau)$ are
irrelevant. Below, e.g. in Figure 11, we call Equation (24) the Qln formula
which denotes the limiting law of $Q_{t}(n)$ for $n\to\infty$.

We now consider an example, the Erlang distribution Forbes2011Statistical

$\psi(\tau)={\tau^{m-1}e^{-\tau}\over(m-1)!},$ (25)

where $m$ is an integer and the mean $\langle\tau\rangle=m$, see Figure 4.
Here exponential waiting times are recovered when $m=1$. The Laplace transform
of $\psi(\tau)$ is

$\hat{\psi}(s)={1\over(1+s)^{m}}.$ (26)

Thus, the Erlang PDF of order $m$ is the $m$ fold convolution of the
exponential PDF. As already mentioned, a well known formula Equation (2)
yields the Laplace transform $t\to s$ of $Q_{t}(n)$. Inserting Equation (26)
and using the Laplace pairs

${1\over s(1+s)^{nm}}\leftrightarrow
1-{\Gamma\left(nm,t\right)\over\Gamma(nm)},$ (27)

where $\Gamma(a,t)$ is the incomplete Gamma function Abramowitz1984Handbook ,
we find

$Q_{t}(n)={\Gamma\left(mn+m,t\right)\over\Gamma\left(nm+m\right)}-{\Gamma\left(nm,t\right)\over\Gamma(nm)},\
\ ~{}~{}~{}n\geq 1,$ (28)

and

$Q_{t}(0)=\Gamma(m,t)/\Gamma(m).$ (29)

Using the identity valid for a positive integer $m$

$\Gamma(m,y)=\left(m-1\right)!e^{-y}\sum_{j=0}^{m-1}{y^{j}\over j!},$ (30)

we get

$Q_{t}(n)=e^{-t}\sum_{j=nm}^{nm+m-1}{t^{j}\over j!},\ \ ~{}~{}~{}~{}~{}n\geq
0.$ (31)

We will soon use this expression to find an exact solution of the CTRW with
Erlang PDF of waiting times.

From the above theorem, and the definition of the Erlang PDF we have for this
example $A=m-1$, $C_{A}=1/(m-1)!$ and $C_{A+1}=-C_{A}$. Hence according to
Equation (24), we have

$Q_{t}(n)\sim e^{-t}{t^{nm}\over(nm)!}.$ (32)

As expected this is the same as the leading term in the expansion Equation
(31), when $t/nm\ll 1$. We now find the rate function of this example. Using
the Stirling approximation $n!\sim\sqrt{2\pi n}(n/e)^{n}$, Equation (31)
reduces to

$Q_{t}(n)\sim{H(t,l)\over(2\pi
mn)^{1/2}}\exp\left[-mn\mathcal{I}_{n}(l)\right],$ (33)

where $\mathcal{I}_{n}(l)=-\ln(l)+l-1$, $l=t/nm$ and

$\begin{split}H(t,l)&=1+\frac{1}{\frac{1}{l}+\frac{1}{t}}+\frac{1}{(\frac{1}{l}+\frac{1}{t})(\frac{1}{l}+\frac{2}{t})}+\cdots+\frac{1}{(\frac{1}{l}+\frac{1}{t})\cdots(\frac{1}{l}+\frac{j-1}{t})}.\end{split}$
(34)

For $l=t/nm\to 0$, i.e., $n\to\infty$ and $m$ is fixed, the asymptotic
behavior of Equation (34) follows

$\begin{split}H(t,l)&\sim 1+l+l^{2}+l^{j-1}=\frac{1-l^{j-1}}{1-l}.\end{split}$
(35)

This result is verified in Figure 5. From Equation (33), the rate function
becomes

$\lim_{n\to\infty}{-\ln Q_{t}(n)\over mn}=\mathcal{I}_{n}(l),$ (36)

where the limit is valid when $l$ is fixed so here $t$ is made large. See the
red solid line in Figure 6. Finally we consider also

$\lim_{t\to\infty}{-\ln Q_{t}(n)\over
t}=\overline{l}\ln(\overline{l})-\overline{l}+1$ (37)

and here the limit is taken such that $\overline{l}=mn/t$ remains fixed.

Remark As mentioned we, use in our plots two graphical representations of the
results, the distribution of observables of interest and the rate function. At
least to the naked eye we see that in Figures (1, 3, 5) the results converge
already for relatively short time, e.g. t=2, while in Figures 2 and 6, we see
that the convergence is achieved for much larger times, say $t=100$. This
discrepancy dissolves if all the prefactors like
$\ln\left(2K^{{}^{\prime\prime}}(\hat{u})\right)$ are included, and not only
the rate function. The rate function formalism is a limit where we take say
$x\to\infty$ or $n\to\infty$. Taking the $log$ of the distribution, be it
$P(x,t)$ or $Q_{t}(n)$ and dividing by a large number we get rid of the
prefactors. Hence the two representations are of course not identical.

Figure 5: Probability to observe $n$ renewals when the waiting time is
determined by Equation (25) for different $m$. The solid lines are theoretical
prediction Equation (33) with $H(t,l)$ obtained from Equation (35) and the
corresponding exact results, plotted by the symbols, are Equation (28). In our
setting, we fix $\bar{t}=t/\langle\tau\rangle=t/m=1$ and change the value of
$m$.
Figure 6: Rate function $\mathcal{I}_{n}(l)$ versus $l=t/(nm)$ for different
observation times $t$ with $m=2$. The dashed lines are the plot of the
function $\ln(Q_{t}(n))/(-nm)$ with $Q_{t}(n)$ calculated from Equation (31).
Clearly, with the increase of the observation time $t$, the function
$\ln(Q_{n}(t))/(-nm)$ versus $l$ approaches to the rate function
$\mathcal{I}_{n}(l)$.

###  III.2 The far tails of the positional PDF

For a Gaussian PDF of jump lengths and Erlang’s waiting times we obtain a
formal solution from Equations (1) and (31)

$\begin{split}P(x,t)=&\exp(-t)\sum_{n=1}^{\infty}\sum_{j=nm}^{nm+m-1}\frac{t^{j}}{j!}\frac{\exp(-\frac{x^{2}}{2n})}{\sqrt{2\pi
n}}+\frac{\Gamma(m,t)}{\Gamma(m)}\delta(x).\end{split}$ (38)

Similar to the previous examples, this function is easy to represent
graphically for any reasonable $x$ and $t$ with a program like MATHEMATICA.
Here our goal is to find the large $x$ behavior.

As briefly mentioned in the introduction, it was shown previously that
$P(x,t)$ decays exponentially with the position $x$, more correctly like $|x|$
multiplied by a slowly varying function. We briefly outline the main result in
Barkai2020Packets . Using Cramérs theorem Touchette2009large  it was shown
that under certain conditions, in particular that the PDF of jump length
decays faster than exponential decay, $P(x|n)\sim\exp(-n(x/\delta n)^{\beta})$
with $\beta>1$ and $|x|/n\to\infty$. Here we use Gaussian statistics for the
jump lengths so we have $\delta=\sqrt{2}$ and $\beta=2$. Then for large $x$
the main result in Barkai2020Packets  reads

$P(x,t)\simeq\exp\left(-t\left\\{\frac{|x|}{t}Z\left(\frac{|x|}{t}\right)-\frac{C_{A+1}}{C_{A}}\right\\}\right),$
(39)

where

$Z(y)=\frac{BW_{0}[g_{1}y^{\beta}]-g_{0}(A+1)}{W_{0}[g_{1}y^{\beta}]^{1/\beta}}$
(40)

with

$B=\frac{g_{0}(A+1)}{\beta}+\frac{(g_{0})^{1-\beta}}{\delta^{\beta}},$ (41)

$g_{0}=(\beta(\beta-1)/(A+1))^{1/\beta}/\delta$ and
$g_{1}=[g_{0}(A+1)/(C_{A}\Gamma(A+1))^{1/(1+A)}]^{\beta}$. Recall that the
Lambert function $W_{0}(y)$ is a monotonically increasing function of $y$.
Further $W_{0}(y)\sim\ln(y)$ for large $y$. Hence from Equation (39) we learn
that $P(x,t)$ decays exponentially (neglecting log terms) as mentioned. This
result establishes that the exponential decay of the PDF $P(x,t)$ is a
universal feature of CTRW, in the same spirit as experimental and numerical
evidence demonstrates in many examples.

To appreciate this result and further test it, let us derive it with our
example (in this case the proof is somewhat easier as compared to the general
case). When $x\to\infty$ and $t$ fixed, say $t$ of order of
$c\langle\tau\rangle$ and $c=1$ or $c=137$ etc, the terms contributing to the
infinite sum giving $P(x,t)$ are those with large $n$. Physically and
mathematically this is obvious, as to reach large $x$ in a finite time we need
many jumps since the average of displacement is finite. We therefore use large
$n$ approximation of $Q_{t}(n)$ Equation (31) and then

$P(x,t)\simeq\sum_{n=1}^{\infty}\frac{\exp(-t)t^{nm}}{(nm)!}\frac{\exp(-\frac{x^{2}}{2n})}{\sqrt{2\pi
n}}.$ (42)

Only large $n$ terms contribute, hence including small $n$ in the summation is
not a problem, since these terms are of order $\exp(-x^{2})$ while
$|x|\to\infty$. We now switch to Fourier space, and find

$\widetilde{P}(k,t)\simeq\exp(-t)S_{m}[t^{m}\exp(-k^{2}/2)]$ (43)

with the sum $S_{m}(y)=\sum_{n=0}^{\infty}y^{n}/(nm)!.$ The next step is to
find the asymptotic behavior of $S_{m}(y)$ for large $y$. In order to do so,
let us consider some special cases. Hence for $m=1$, $S_{1}(y)=\exp(y)$, and
similarly $S_{2}(y)=\cosh(\sqrt{y})\sim\exp(\sqrt{y})/2$,

$\begin{split}S_{3}(y)&=\frac{1}{3}e^{-\frac{\sqrt[3]{y}}{2}}\left(e^{\frac{3\sqrt[3]{y}}{2}}+2\cos\left(\frac{1}{2}\sqrt{3}\sqrt[3]{y}\right)\right)\\\
&\sim\frac{1}{3}\exp(-y^{1/3}),\end{split}$ (44)

etc. For a general $m$, we write the infinite sum as an integral

$\begin{split}S_{m}(y)&\to\int_{0}^{\infty}\frac{y^{n}}{(nm)!}dn\\\
&=\frac{1}{m}\int_{0}^{\infty}\frac{(y^{1/m})^{j}}{(j)!}dj\\\
&\sim\frac{1}{m}\exp(y^{1/m}).\end{split}$ (45)

Below, we will use the above equation to calculate the far tails of the
distribution of the position. As usual switching to $k=-iu$ we obtain the
moment generating function, and taking the log, we get the cumulant generating
function

$K(u)\simeq-t+t\exp(u^{2}/(2m)).$ (46)

This is valid for large $u$ since that limit is the relevant one for the
calculations of $P(x,t)$ for large $x$, similar to our previous examples. Here
we neglected a $\ln(m)$ term that is negligible in the limit under study.
Using the standard large deviation technique Equation (7), we find the
solution of $K^{{}^{\prime}}(u)=x$ denoted $\hat{u}$ and this solves the
equation

$\hat{u}\exp\left(\frac{(\hat{u})^{2}}{2m}\right)=\frac{mx}{t},$ (47)

and hence we find yet again the Lambert type of solution

$\hat{u}=\sqrt{mW_{0}\left[m\left(\frac{x^{2}}{t^{2}}\right)\right]}$ (48)

for $x>0$ otherwise the right hand side of Equation (48) has a negative sign.
If $m=1$, as expected, we relax to Equation (9). Using the large deviation
formula Equation (7), we find

$P(x,t)\sim\frac{\exp\left(-\sqrt{m}\left|x\right|\sqrt{W_{0}\left(\frac{mx^{2}}{t^{2}}\right)}+\frac{\sqrt{m}\left|x\right|}{\sqrt{W_{0}\left(\frac{mx^{2}}{t^{2}}\right)}}-t\right)}{\sqrt{\left|\left|x\right|2\pi
m^{-1/2}\left(\sqrt{W_{0}\left(\frac{mx^{2}}{t^{2}}\right)}+\frac{1}{\sqrt{W_{0}\left(\frac{mx^{2}}{t^{2}}\right)}}\right)\right|}}.$
(49)

This solution and the more general one Equation (39) are of course in
agreement. As shown in Figure 7, the far tails of the distribution of the
position follow exponential decay predicted by Equation (49) since as
mentioned $W_{0}(z)\sim\ln(z)$. But as shown in Figure 7 the rate of
convergence is slow, for example for $m=3$.

To check better the convergence issue, we consider the rate function. Based on
Equation (49), the rate function with respect to the position becomes

$\lim_{t\to\infty}-\frac{\ln(P(x,t))}{|x|}=\mathcal{I}_{x}(l)$ (50)

with $l=x/(t/\langle\tau\rangle)$ and

$\mathcal{I}_{x}(l)=\sqrt{m}\left(W_{0}(l^{2}/m)-\frac{1}{W_{0}(l^{2}/m)}\right)+\frac{m}{|l|}.$
(51)

It can be seen in Figure 8 that with the increase of observation time $t$, the
left-hand of Equation (50) tends to $\mathcal{I}_{x}(l)$ (Equation (51))
slowly. One contribution to this slow convergence effect, is that $t$ is not
very large, the prefactor of $\exp(-|x|\mathcal{I}_{x}(l))$, i.e.
$1/\sqrt{2\pi K^{{}^{\prime\prime}}(\hat{u})}$, is of importance; see Figure
8.

As expected, when $mx^{2}/t^{2}\to 0$, we get the Gaussian distribution

$P(x,t)\simeq\exp\left(-\frac{x^{2}m}{2t}\right)$ (52)

and since $\langle\tau\rangle=m$ from Equation (25) we have
$P(x,\bar{t})\simeq\exp(-x^{2}/2\bar{t})$ with $\bar{t}=t/\langle\tau\rangle$.
Recall that $W_{0}(y)\sim\ln(y)$ for $y\gg 1$, hence when $mx^{2}/t^{2}\gg 1$
we may approximate Equation (49) with

$P(x,t)\simeq\exp\left(-\sqrt{m}\left|x\right|\sqrt{\ln\left(\frac{mx^{2}}{t^{2}}\right)}-t\right).$
(53)

Thus using the rescaled time $\bar{t}=t/\langle\tau\rangle$ we have

$P(x,t)\simeq\exp\left(-\sqrt{m}|x|\ln\left[\frac{x^{2}}{m\bar{t}^{2}}\right]-t\right),$

implying that as we increase $m$ for fixed $\bar{t}$ we get a suppression of
the exponential tails.

As we increase $m$ the condition on $x\gg\sqrt{m}\bar{t}$ implies that we may
observe the nearly exponential decay of the packet but only for a very large
$x$, or only for very short times. This is because when we increase $m$, the
waiting time exhibits a strong anti-bunching effect. Namely, to find a large
$x$ we need to have a large fluctuations of $n$. For example if the average of
$\langle n(t)\rangle\sim 2$ we may still see realizations with many jumps,
leading to exponential decay in the far tails of $x$. However, this is less
likely for large $m$ if compared with small $m$; see Figure 4. This is because
when $m$ is large we have vanishing fluctuations of $n$. To quantify this we
may use a tool from quantum optic, namely the Mandel $Q$ parameter
Mandel1979Atomic  defined as

$Q=\frac{\langle n^{2}\rangle-\langle n\rangle^{2}}{\langle n\rangle}-1.$ (54)

The case $Q<0$ is called sub-Poissonian (anti-bunching ) and $Q>0$ sup-
Poissonian. If $Q=-1$, we have no fluctuations of $n$ at all. For the Erlang
PDF, and in the long time limit $Q=-1+1/m$. Namely, when $m\to\infty$ the
fluctuations of $n$ vanish (see Figure 9) , and hence with it the effect of
exponential tails of $P(x,t)$. This is easy to see since in the absence of
fluctuations of $n$, the number of jumps is fixed and since we have Gaussian
jump lengths, the total displacement will be Gaussian and not exponential.
Thus we see that as we increase $m$, namely make the process more anti-bunched
we see less exponential tails. Anti-bunching means the effective repulsion of
the dots on the time axis on which jump events takes place. And this is
controlled by the small $\tau$ behavior of $\psi(\tau)$. The anti-bunching $Q$
reduces to a negative value, and hence kills fluctuations of $n$ which are the
key to the observation of exponential tails of $P(x,t)$.

Figure 7:  Distribution of the position with Erlang distribution Equation
(25), where we chose the rescaled time $\bar{t}=1$. Here the solid lines are
the theory according to Equation (49) and the corresponding exact result is
obtained from Equation (1). See Figure 8 for the convergence of the theory.
Figure 8:  Comparison of analytical prediction Equation (50) (red line) for
$\mathcal{I}_{x}(l)$ with $\ln(P(x,t))/(-l)$. We show that $\ln(P(x,t))/(-l)$
obtained from Equation (38) converges to the rate function Equation (50) with
the growing of the observation time $t$. Here we choose $m=3$.

##  IV Bunching case of waiting time PDF

We further explore the behavior of probability to observe a large number of
renewals $n$. For some special distributions of the waiting times it is unwise
simply to take the $n\to\infty$ limit since the rate of convergence to
Equation (24) is very slow. Here we consider $\psi(\tau)$ as a sum of two
exponential waiting times PDFs

$\psi(\tau)=\frac{1}{2}\left(\frac{1}{a}\exp\left(-\frac{\tau}{a}\right)+\frac{1}{b}\exp\left(-\frac{\tau}{b}\right)\right).$
(55)

We choose $a<b$ while the opposite situation is merely relabeling. The average
waiting time is $\langle\tau\rangle=(a+b)/2$. In what follows we shall fix
$\langle\tau\rangle=1$, so $a+b=2$ and this means that we have one free
parameter say $a$ with $0<a<1$. We also consider the case where the jump size
distribution is Gaussian with a variance equal unity and zero mean. This
implies that in the long time limit, independently of the specific choice of
$a$, the mean square displacement $\langle x^{2}\rangle\sim t$ and the process
is Gaussian. However, for short times the dynamics of $P(x,t)$ is of course
$a$ sensitive. The question we wish to address: how does $a$ control the
exponential tails of $P(x,t)$?

This is related to effect of bunching. Here for short waiting times we have
$\psi(0)=(1/a+1/b)/2$. When $a=b=1$ we have a Poisson process, while when say
$a\to 0$, we get a large value of $\psi(0)$. This means that there is an
increased probability (compared to Poissonian case) to obtain a jump shortly
after a jump event. This is an effect of bunching where the jumps come in
groups, and then separated by the relatively large waiting times. Such
intermittent behavior is quantified with Mandel $Q$ parameter, which for large
time is

$Q=2(a-1)^{2},$ (56)

where we assumed that $a+b=2$. This is a super Poissonian behavior since
$Q>0$. Note that in the previous example of the Erlang PDF, we had the
opposite behavior, i.e. $Q<0$, since there $\lim_{\tau\to 0}\psi(\tau)=0$ for
any $m\neq 1$.

We can obtain $Q_{t}(n)$ in terms of an integral; see details in Appendix B.
Aside $Q_{t}(n)$ we also find $P(x,t)$, which is compared to the previous
theory when $x\to\infty$, i.e. Equation (39); see Figure 10. Here we see that
by making $a$ smaller, namely making the process more bunched, we get a large
exponential tail (see Figure 10). Thus bunching makes the observation of the
exponential tails effect more readily achievable in experiments. In principle
for any PDF of the waiting times (provided $\psi(\tau)$ is analytical in the
vicinity of $\tau=0$) an exponential decay of $P(x,t)$ is obtained, but this
decay can be sometimes achieved for extremely large values of $x$. We also
observe that simply including the asymptotic behavior of $Q_{t}(n)$ in
Equation (24) leads to slow convergence in the $a\to 0$ limit. To further
understand the effect of bunching we also plot $Q_{t}(n)$ versus $n$; see
Figure 11. Strong bunching in this model, implies a relatively high
probability for seeing a large $n$ (many short time intervals between jumps
since $a$ is small). As we see, the decay of $Q_{t}(n)$ with $n$ is relatively
slow when bunching is pronounced, and then the appearance of non negligible
probability for large $n$, implies that particles can travel large distances,
and then the tails posses more statistical weight. In Figure 11 the slow
convergence to asymptotic behavior of Equation (24) is observed in the $a\to
0$ limit and the delicate treatment of Appendix B is preferable.

Figure 9:  $Q_{t}(n)$ for the case of Erlang PDF of waiting times, i.e.
$\psi(\tau)$ is determined by Equation (25). The solid lines describe Equation
(28) the exact behavior of $Q_{t}(n)$ for various $m$. Here we choose $t=400$.
When $m$ increases the distribution narrows, indicating that fluctuations of
$n$ disappear. We further plot the central part of $Q_{t}(n)$ using the
symbols according to Gaussian approximation. Here for Gaussian approximation
we use $Q_{t}(n)\sim\exp(-(n-t/m)^{2}/(2t/m^{2}))/\sqrt{2\pi t/m^{2}}$ (see
Appendix A).
Figure 10:  $P(x,t)$ for the case where the jump lengths are Gaussian and
$\psi(\tau)$ is given by Equation (55). The solid line describe the exact
behavior for various $a$, as provided by Equations (1, 61), while the
corresponding large $|x|$ approximations (termed Plx) due to Equation (39)
described by the dashed lines. For small $a$, e.g., $a=0.1$, the convergence
of the large $|x|$ approximation is slow and achieved only for very large
values of $|x|$. See Appendix B for a proper discussion of the behavior in
this limit.
Figure 11: Probability $Q_{t}(n)$ of observing $n$ jump events versus $n$
while $\psi(\tau)$ is determined by Equation (55). The measurement time is
$t=2$. The solid lines describes the exact results for various $a$, i.e.
Equation (61), and the dashed lines are the corresponding Qln approximations,
i.e. Equation (24).

##  V conclusion

Following the work of Kob and co-workers we have formalized the problem of
nearly exponential decay of $P(x,t)$ using the CTRW framework. Exponential
decay is the rule, and it should be considered a natural consequence of large
deviation theory. In the long time limit the packet of particles is typically
Gaussian, hence the phenomenon can be found/measured for intermediate and
short times. To describe the dynamics we have considered a few examples where
we may find exact solutions to the problem. This allowed us to find the far
tail, obtain the rate function, and compare finite time solutions with
asymptotic expressions. We distinguished between bunching and anti-bunching
processes. These are determined by the behavior of $\psi(\tau)$ in the
vicinity of of $\tau$ close to zero. The short time behavior of $\psi(\tau)$
determines the statistical behavior of the number of jumps, when the latter is
large. And large number of jumps, leads the particle to non-typical large $x$,
where the phenomenon is found. Using the example of $\psi(\tau)$ expressed as
a sum of two exponential waiting times, we show that as we increase the
bunching effect, the exponential tail of $P(x,t)$ is more pronounced.
Similarly as we increase anti-bunching, by increasing the parameter $m$ in the
Erlang distribution, the exponential tails are suppressed. Indeed as
$m\to\infty$ we get the usual random walk where $n$ is not fluctuating and
then exponential tails cannot be found.

These effects are related to the rather universal behavior of $Q_{t}(n)$ found
for large $n$. As we have demonstrated here, for large $n$ $Q_{t}(n)$
universally attains exponential decay (with log corrections). This, as
mentioned, is valid for any $\psi(\tau)$ which is analytic for small $\tau$.
Thus the agreement with the experimental observation that finds exponential
tails, is a manifestation of the widely applicable CTRW model under study and
not merely consequence of a fitting procedure. This universal property of CTRW
is the crucial difference from the diffusing diffusivity model
Leptos2009Dynamics ; Chubynsky2014Diffusing ; Jain2016Diffusion ;
Thomas2017Cytoplasmic ; Cherstvy2019Non ; Grebenkov2019unifying ;
Metzler2020Superstatistics ; Hidalgo2020Hitchhiker .

As mentioned in Sect. III, in this manuscript we assume that the PDF of
waiting time $\psi(\tau)$ is analytic at the vicinity of $\tau=0$. Clearly not
all of the PDFs of waiting time satisfy this property, for example
$\psi(\tau)=\tau^{-3/2}\exp[-1/(4\tau)]/2\sqrt{\pi}$. Roughly speaking, for
this case the probability of performing many jumps at a finite time is much
smaller than the cases studied here since $\psi(\tau\to 0)=0$. More
specifically, for this example the analytical behavior Equation (23) is not
valid, and hence our main results do not apply. This implies that the far
tails of distribution of position decay faster than exponential tails. It
would be of interest to investigate this problem, which is left for future
work.

Finally, let us mention a few other open problems:

  * •

In the present case we assumed that the jump process starts at time $t=0$.
This is called an ordinary renewal process. If the processes started long
before the measurement, we will have a modification of the PDF of the first
waiting time Haus1987Diffusion ; Hou2018Biased . How does this effect the
large $x$ behavior of $P(x,t)$? This issue seems important since the
phenomenon can be found for relatively short times.

  * •

We focused on models that in the long time limit converge to Gaussian
statistics. What happens if $\psi(\tau)$ is fat tailed Bouchaud1990Anomalous ;
Metzler2000random  with diverging mean? Our results are certainly valid for
this case as well, however we did not explore this in detail.

  * •

What happens in dimension $d>1$?

  * •

What are ideal waiting time PDFs and jump length distributions, where
exponential tails are pronounce and if possible maintained for longer times.
We showed how this is related to bunching and anti-bunching, however more
refined work can help to clarify better the widely observed behaviour.

  * •

We used CTRW, instead one could use the noisy CTRW model Jeon2013Noisy . This
adds to the jumps also noise when the particle is waiting for its next jump.
Thus noisy CTRW is much more similar to real experiments.

  * •

Here we considered the decoupled CTRW, where jump lengths and waiting times
are uncorrelated. The general framework of CTRW, goes beyond this
simplification Klafter1994Levy ; Marcin2017Aging .

  * •

If the jump length PDF is sub-exponential, the far tail of $P(x,t)$ will
deviate from what we found here. Most likely the principle of the single big
jump Cistjakov1964theorem ; Alessandro2019Single ; Wang2019Transport  will
hold in some form, but the details of the theory are left unknown.

  * •

Recently Dechant et al. showed how the CTRW picture emerges from an under-
damped Langevin description of a particle in a periodic potential
Kindermann2017Nonergodic . And then showed how this model can be used to
analyse dynamics of Cesium atoms in optical lattices. Thus we expect to find
also here exponential tails of packets, however influence of the control
parameters of this phenomenon such as the depth of the optical potential, the
noise etc, are left unknown to us. Similarly, over damped Brownian motion in
corrugated channels, a model of biophysical transport, is likely related to
CTRW as a coarse grained description. In the former system exponential decay
was already explored in Ref. Li2019Non . Thus exponential tails are found both
via Langevin dynamics and within CTRW, the two approaches are related in some
limits.

## Acknowledgments

E.B. acknowledges the Israel Science Foundations grant 1898/17. S.B. was
supported by the Pazy foundation grant 61139927. W.W. was supported by Bar-
Ilan University together with the Planning and Budgeting Committee fellowship
program.

Figure A1: Comparison of exact result Equation (1) with integral approximation
Equation (58) and Gaussian approximation Equation (59). Here we choose $m=2$
and the observation time $t=10$ and $20$.

##  Appendix A PDFs of $n$ and $x$ in the long time limit

When the waiting time has a finite variance and the observation time $t$ is
large, $Q_{t}(n)$ obeys the limiting law described by Gaussian distribution
Cox1977Theory ; Godreche2001Statistics

$Q_{t}(n)\sim\frac{\exp\left(-\frac{\left(n-\langle
n(t)\rangle\right)^{2}}{2(\sigma_{n})^{2}}\right)}{\sqrt{2\pi(\sigma_{n})^{2}}}.$
(57)

In turn $\langle n(t)\rangle$ and the variance of $n$, i.e., $\langle
n^{2}(t)\rangle-\langle n(t)\rangle^{2}$ denoted as $(\sigma_{n})^{2}$, can be
obtained from

$\displaystyle\langle n(t)\rangle$ $\displaystyle=$
$\displaystyle\mathcal{L}^{-1}\left[\frac{\hat{\psi}(s)}{s(1-\hat{\psi}(s))}\right]$
$\displaystyle\langle n^{2}(t)\rangle$ $\displaystyle=$
$\displaystyle\mathcal{L}^{-1}\left[\frac{\hat{\psi}(s)(1+\hat{\psi}(s))}{s(1-\hat{\psi}(s))^{2}}\right],$

where $\mathcal{L}^{-1}[\hat{g}(s)]$ means the inverse Laplace transform of
$\hat{g}(s)$. In the limit of long $t$, we have $\langle n(t)\rangle\sim
t/\langle\tau\rangle$ and $(\sigma_{n})^{2}\sim
t(\langle\tau^{2}\rangle-\langle\tau\rangle^{2})/\langle\tau\rangle^{3}$.

We now consider the model with Erlang waiting time Equation (25) in the main
text and Gaussian jump length $f(\chi)=\exp(-\chi^{2}/2)/\sqrt{2\pi}$. In the
long time limit we have $\langle n(t)\rangle\sim t/m$ and
$(\sigma_{n})^{2}\sim t/m^{2}$. According to Equation (1), for large $t$,
$P(x,t)$ reads

$P(x,t)\simeq\int_{0}^{\infty}\frac{\exp(-x^{2}/(2n))}{\sqrt{2\pi
n}}\frac{\exp\left(-\frac{\left(n-\langle
n(t)\rangle\right)^{2}}{2(\sigma_{n})^{2}}\right)}{\sqrt{2\pi(\sigma_{n})^{2}}}dn.$
(58)

Figure A1 demonstrates that Equation (58), plotted by the solid lines, is an
excellent approximation. In the limit $t\to\infty$, Gaussian distribution is
found

$P(x,t)\sim\frac{1}{\sqrt{2\pi t/m}}\exp\left(-\frac{x^{2}}{2t/m}\right),$
(59)

where we used the relation $Q_{t}(n)\sim\delta(n-t/m)$. Here we stress that
Equation (59) is just valid for the central part of the distribution of the
position, while the large deviation theory developed in the main text is
applicable in the large x limit.

##  Appendix B Calculation of $Q_{t}(n)$ and $P(x,t)$ with waiting time
following Eq. (55)

Figure A2:  Probability $Q_{t}(n)$ of observing $n$ jump events versus $n$
while $\psi(\tau)$ is determined by Equation (55), for the case of $a=0.01$
while the measurement time is $t=2$. The solid line describes the exact
result, according to Equation (61). The approximation in the limit of $a\to 0$
(Equation (64)) is described by symbols and the dashed line is the asymptotic
behavior of this approximation (Equation (66)).

Here the aim is to find the exponential tails of the distribution of the
number of renewals and the position. As mentioned in the main text, we fix the
mean of waiting times, namely $\langle\tau\rangle=(a+b)/2=1$. From Equation
(2), we have

$\hat{Q}_{t}(s)=\frac{(\frac{1}{as+1}+\frac{1}{bs+1})^{n}}{s2^{n}}-\frac{(\frac{1}{as+1}+\frac{1}{bs+1})^{n+1}}{s2^{n+1}}.$
(60)

Thus, in real space, the formal solution is

$Q_{t}(n)=\sum_{j=0}^{n}G(t,n,j)+\sum_{j=0}^{n+1}G(t,n+1,j)$ (61)

with

$\begin{split}G(t,n,j)&=\frac{n!}{2^{n}j!(n-j)!}\int_{0}^{t}\left(1-\frac{\Gamma(j,\frac{t-y}{a})}{\Gamma(j)}\right)\frac{\exp(-\frac{y}{b})(\frac{y}{b})^{n-j-1}}{b\Gamma(n-j)}dy.\end{split}$
(62)

Let $a=0.01$. So $C_{A}=(1/a+1/b)/2\sim 1/(2a)$ and $C_{A+1}\sim-1/(2a^{2})$
according to Equation (23). This indicates that
$\exp(tC_{A+1}/C_{A})\propto\exp(-t/a)=\exp(-200)$. It means that the tails
are sensitive to value $a$ and it’s not measurable in real experiment for such
a small $a$. As shown in Figure 11, for small $a$, the limiting law Equation
(24) loses its role in real experiment since it is too small to observe this
phenomenon. Below, the aim is to find a new formula to predicte such behavior.
In the limit of $a\to 0$, Equation (60) reduces to

$Q_{t}(s)\sim\frac{(s+1)^{n}}{(2s+1)^{n+1}}.$ (63)

One can do the similar calculation, like Equation (60), but there is a simple
way. From Cauchy’s integral formula, the inverse Laplace transform of Equation
(63) yields

$Q_{t}(n)\sim\frac{1}{2^{n+1}n!}\frac{d^{n}}{ds^{n}}[\exp(st)(s+1)^{n}]|_{s=-\frac{1}{2}},$
(64)

where we used the fact that Equation (63) has poles at $s=-1/2$. The equation
(64), plotted by the symbols $(`+^{\prime})$ in Figure A2, is consistent with
the exact result for $a=0.01$. For this case, the law Equation (24) should be
modified since the rate of convergence is extremely slow (for example, see the
solid line for $a=0.1$ in Figure 11). Another interesting problem is how the
far tail of the distribution of $n$ decays with respect to $n$. Note that the
limit between $a$ and $n$ is a bit subtle. Let $a$ be small and $n$ be large
instead of infinity. Using the saddle point approximation, we get

$Q_{t}(n)\sim\frac{\exp\left(\frac{1}{4}\left(\sqrt{8nt+t^{2}+4t+4}-3t+2\right)-(n+1)\ln\left(\frac{\sqrt{8nt+t^{2}+4t+4}-t+2}{2t}\right)+n\ln\left(\frac{\sqrt{8nt+t^{2}+4t+4}+t+2}{4t}\right)\right)}{\sqrt{32\pi\left|t^{2}\left(\frac{n+1}{\left(-t+\sqrt{t^{2}+8nt+4t+4}+2\right)^{2}}-\frac{n}{\left(t+\sqrt{t^{2}+8nt+4t+4}+2\right)^{2}}\right)\right|}}.$
(65)

The corresponding asymptotic behavior of large $n$ follows

$Q_{t}(n)\sim\frac{\exp\left(-\ln\left(\frac{\sqrt{8nt}}{2t}\right)+\sqrt{2nt}-n\ln(2)+\frac{1}{2}-\frac{3t}{4}\right)}{\sqrt{4\pi
t^{2}\left|\frac{\sqrt{2}}{\sqrt{tn}}\right|}},$ (66)

which is plotted by the dash-dotted line in Figure A2. In this limit, we still
find the universal exponential decay of the far tail since the leading term of
Equation (66) is $n\ln(2)$.

Figure A3: $P(x,t)$ for the case where the jump lengths are Gaussian and
$\psi(\tau)$ is given by Equation (55), in the limit $a\to 0$. The exact
result (Equations (1, 61)) for the case of $a=0.01$ is described by the solid
line. The theoretical approximation in the limit $a\to 0$ (Equation (67)) is
presented by the dashed line and and the symbols present the asymptotic
behavior of the approximation (Equation (68)). The Gaussian distribution
(dashed-dotted line) is provided for comparison.

Utilizing Equation (66) and $P(x|n)=\exp(-x^{2}/(2n))/\sqrt{2n}$, and using
the saddle point method again, we find

$P(x,t)\sim\frac{\exp\left(\sqrt{t}\sqrt[4]{\frac{2}{\ln(2)}}\sqrt{\left|x\right|}-\ln\left(\frac{\sqrt[4]{\frac{2}{\ln(2)}}\sqrt{\left|x\right|}}{\sqrt{t}}\right)-\frac{1}{4}\ln\left(\frac{\left|x\right|}{\sqrt{2\ln(2)}}\right)-\sqrt{2\ln(2)}\left|x\right|-\frac{3t}{4}+\frac{1}{2}\right)}{\sqrt{4\pi\sqrt{2}t^{3/2}}\sqrt{\left|-\frac{2\sqrt{2}x^{2}\ln^{\frac{3}{2}}(2)}{\left|x\right|^{3}}-\frac{\sqrt{t}\ln^{\frac{3}{4}}(2)}{2^{3/4}\left|x\right|^{3/2}}+\frac{3\ln(2)}{2\left|x\right|^{2}}\right|}},$
(67)

For large $x$, we obtain

$P(x,t)\sim\frac{\exp\left(\sqrt[4]{\frac{2}{\ln(2)}}\sqrt{t\left|x\right|}+\ln\left(\frac{\sqrt{t\ln(2)}}{\sqrt[4]{\left|x\right|}}\right)-\sqrt{2\ln(2)}\left|x\right|-\frac{3t}{4}+\frac{1}{2}\right)}{\sqrt{24\pi
t^{3/2}\ln(2)}}.$ (68)

This is verified by the symbols in Figure A3 with $a=0.01$. Clearly, the
leading term $\sqrt{2\ln(2)}|x|$ in the numerator of the above equation is
responsible for exponential decay. While, note that with the growing of $x$
and $n$, Equation (67) fails since for large $x$ the parameter $a$ comes into
play, see the symbols in Figure A2. On the contrary, the law Equation (24)
will work for the extremely large $n$, which is difficult to see for small $a$
if compared with Equation (67). As mentioned in the main text this is related
to bunching effects.

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# On trapping geometry for cold atoms in a radio-frequency (rf) dressed
potential

Sourabh Sarkar  Laser Physics Applications Division, Raja Ramanna Centre for
Advanced Technology, Indore, 452013, India  Homi Bhabha National Institute,
Training School Complex, Anushakti Nagar, Mumbai, 400094, India  S. P. Ram
<EMAIL_ADDRESS>Laser Physics Applications Division, Raja Ramanna Centre
for Advanced Technology, Indore, 452013, India  Kavish Bhardwaj  Laser Physics
Applications Division, Raja Ramanna Centre for Advanced Technology, Indore,
452013, India  Gunjan Verma  Laser Physics Applications Division, Raja Ramanna
Centre for Advanced Technology, Indore, 452013, India  V. B. Tiwari  Laser
Physics Applications Division, Raja Ramanna Centre for Advanced Technology,
Indore, 452013, India  Homi Bhabha National Institute, Training School
Complex, Anushakti Nagar, Mumbai, 400094, India  S. R. Mishra  Laser Physics
Applications Division, Raja Ramanna Centre for Advanced Technology, Indore,
452013, India  Homi Bhabha National Institute, Training School Complex,
Anushakti Nagar, Mumbai, 400094, India

###### Abstract

We have investigated the atom trapping geometry for trapping of ${}^{87}{Rb}$
atoms in a radio-frequency (rf) dressed potential generated after superposing
a strong linearly polarized rf-field on a static magnetic trap. For this,
laser cooled atoms in a magneto-optical trap (MOT) in an ultra-high vacuum
(UHV) chamber (pressure $\sim$ 1.5 $\times$ $10^{-10}$ Torr) were trapped in a
quadrupole magnetic trap and evaporatively cooled before transferring them to
the rf-dressed potential. The experimentally observed hollow shell type atom
trapping geometry has been explained by theoretical modelling of the trapping
potential.

Rf-dressed potentials, Magnetic trap, Bose-Einstein condensation

###### pacs:

32.80.Pj, 33.55.Be, 67.85.-d.

##  I Introduction

Cooling and trapping of atoms give us a new and unique way to look into the
quantum world of atoms. Since the inception of laser cooling and trapping of
atoms, the physics of cold atoms spread its horizon to explore many
fundamental phenomena such as Bose-Einstein condensation (BEC) Davis et al.
(1995); Pethick and Smith (2008), complex phase transition Greiner et al.
(2002); Sunami et al. (2022), tunnelling Albiez et al. (2005), etc. Besides
that, now a days, many practical applications of cold atoms are emerging in
precision measurements of time Ludlow et al. (2015) , gravity Peters et al.
(2001); Poli et al. (2011); Abend et al. (2016), rotation Gustavson et al.
(1997); Wu et al. (2007); Ragole and Taylor (2016), electric Carter et al.
(2012) and magnetic fields Cohen et al. (2019). The cold atoms are also
considered important to develop qubits for quantum information Morgado and
Whitlock (2021) applications. All this is possible because of several new
developments have taken place in the past several years to control and
manipulate atoms using different techniques.

The technique of radio frequency (rf) dressing of atoms, first proposed by
Zobay and Garraway Zobay and Garraway (2001), is one such technique which
provides unique opportunity to tailor the trapping geometries for atoms using
rf-fields and static (i.e. DC) magnetic fields Perrin and Garraway (2017);
Chakraborty and Mishra (2014); Sarkar et al. (2021). A good control on atom
trapping geometries can be achieved in rf-dressed potential traps by changing
magnetic field and rf-field parameters. This has been earlier investigated by
several researchers Colombe, Y. et al. (2004); Morizot et al. (2006);
Chakraborty et al. (2016); Schumm et al. (2005); Sherlock et al. (2011);
Easwaran et al. (2010) which includes experimental demonstration of various
trapping geometries such as ring trap Morizot et al. (2006); Sherlock et al.
(2011); Heathcote et al. (2008), double-well trap geometry Schumm et al.
(2005); Harte et al. (2018); White et al. (2006); Gildemeister et al. (2010),
etc. The ring trap is an ideal candidate to study the superfluidity Ramanathan
et al. (2011); Heathcote et al. (2008); Gildemeister et al. (2012) and
rotation of cold atom cloud Gildemeister et al. (2012); Pandey et al. (2019,
2021). The double-well is useful for atom interferometry Schumm et al. (2005),
tunnelling Albiez et al. (2005), species selective confinement Bentine et al.
(2017), etc.

In this paper, we have investigated the trapping geometry of 87Rb atoms in a
rf-dressed potentials formed by a static quadrupole magnetic trap and a strong
rf-field. For this, the 87Rb atoms from a magneto-optical trap (MOT) in an
ultra-high vacuum (UHV) chamber (pressure $\sim$ 1.5 $\times$ $10^{-10}$ Torr)
were first trapped in a quadrupole magnetic trap. The atoms were then cooled
by rf-evaporative cooling method to bring them to a lower temperature suitable
for transferring them to the rf-dressed potential. With a linearly polarized
rf-field, the experimentally observed trapping geometry was a hollow shell-
like structure. The observed atom trapping geometry has been explained on the
basis of theoretical modelling of the trapping potential. This type of atom
trapping geometry may have potential applications in various studies such as
low dimensional Bose-Einstein condensation (BEC) Merloti et al. (2013),
superfluidity Ramanathan et al. (2011); Heathcote et al. (2008), atom
interferometry Navez et al. (2016), etc.

This article is organized as follows in section II we have briefly described
the theoretical formalism and understanding of rf-dressing. Section III has
been dedicated for the description of our experimental set-up, methods and
experimental techniques. In section IV we have discussed our experimental
results and finally we conclude in section V.

##  II Theoretical background

A static or DC magnetic field which is inhomogeneous in space provides
trapping potential for atoms via interaction with hyperfine spin. There are
varieties of magnetic traps Pethick and Smith (2008) that can be used for
magnetic trapping of atoms. Here we consider the simplest DC magnetic trap
which is known as quadrupole magnetic trap. The atoms are assumed to be in a
trappable Zeeman hyperfine state ($|F,m_{F}\rangle$). When a strong rf-field
at a particular frequency interacts with these atoms, the energy levels of
atoms are modified and the atoms experience the dressed state potential. This
potential is called RF-dressed potential Lesanovsky et al. (2006); Chakraborty
and Mishra (2014), can be tailored by changing the strength of trapping DC
magnetic field as well as by changing the magnitude, frequency and
polarization of rf-field,

Here we consider the DC magnetic trap as a quadrupole trap whose magnetic
field is described as,

$\mathbf{B_{S}(r)}=B_{q}(x\mathbf{\hat{x}}+y\mathbf{\hat{y}}-2z\mathbf{\hat{z}}),$
(1)

where $B_{q}$ is the radial ($x-y$) magnetic field gradient of quadrupole
magnetic trap.

The potential energy of an atom, in a Zeeman hyperfine state
$|F,m_{F}\rangle$, trapped in this trap at any point (x, y,z) can be written
as,

$V_{m_{F}}=g_{F}m_{F}\mu_{B}B_{q}\sqrt{x^{2}+y^{2}+4z^{2}},$ (2)

where $m_{F}$ is the Zeeman hyperfine quantum number corresponding to the
hyperfine state with angular momentum F, $g_{F}$ is the Lande’s g factor for
$|F,m_{F}\rangle$ state and $\mu_{B}$ is the Bohr magneton.

Now when a rf-field of frequency $\omega_{rf}$ and amplitude $B_{rf}$ is
applied on atoms trapped in the quadrupole magnetic trap, the modified
potential energy $V_{D}$ including the gravitational potential can be written
as Chakraborty et al. (2016),

$V_{D}=m_{F}\hbar\sqrt{{\Delta(r)}^{2}+\Omega_{R}(r)^{2}}+mgz,$ (3)

where $\Delta(r)$ is the detuning of rf-field with the local Larmor frequency
and $\Omega_{R}(r)$ is the inhomogeneous Rabi frequency, $m$ is the mass of
${}^{87}Rb$ atom, and g is the earth’s gravitational acceleration.

The expressions for detuning and Rabi frequency are given as Chakraborty and
Mishra (2014); Sarkar et al. (2021),

$\Delta(r)=\left|{\frac{g_{F}\mu_{B}B_{q}\sqrt{(x^{2}+y^{2}+4z^{2})}}{\hbar}-\omega_{rf}}\right|,$
(4)

and

${\mid\Omega_{R}(r)\mid}^{2}={\Big{(}\frac{{{g}_{F}{\mu}_{B}}}{2\hbar}\Big{)}}^{2}\Bigg{[}\frac{4{z}^{2}}{x^{2}+y^{2}+4{z}^{2}}\Bigg{(}\frac{{\Big{(}B_{rf}^{x}\Big{)}}^{2}{x}^{2}+{\Big{(}B_{rf}^{y}\Big{)}}^{2}{y}^{2}}{{x}^{2}+{y}^{2}}\Bigg{)}+\Bigg{(}\frac{{\Big{(}B_{rf}^{x}\Big{)}}^{2}{y}^{2}+{\Big{(}B_{rf}^{y}\Big{)}}^{2}{x}^{2}}{{x}^{2}+{y}^{2}}\Bigg{)}\\\
+{\Big{(}B_{rf}^{z}\Big{)}}^{2}\Bigg{(}\frac{{x}^{2}+{y}^{2}}{{x}^{2}+{y}^{2}+4{z}^{2}}\Bigg{)}-\frac{2{B}_{rf}^{x}{B}_{rf}^{y}xy\cos({\phi}_{rf}^{y})}{{x}^{2}+{y}^{2}+4{z}^{2}}+\frac{4{B}_{rf}^{x}{B}_{rf}^{y}z\sin({\phi}_{rf}^{y})}{\sqrt{{x}^{2}+{y}^{2}+4{z}^{2}}}+\frac{4{B}_{rf}^{y}{B}_{rf}^{z}yz\cos({\phi}_{rf}^{y}-{\phi}_{rf}^{z})}{{x}^{2}+{y}^{2}+4{z}^{2}}\\\
+\frac{2{B}_{rf}^{y}{B}_{rf}^{z}x\sin({\phi}_{rf}^{y}-{\phi}_{rf}^{z})}{\sqrt{{x}^{2}+{y}^{2}+4{z}^{2}}}+\frac{4{B}_{rf}^{z}{B}_{rf}^{x}xz\cos({\phi}_{rf}^{z})}{{x}^{2}+{y}^{2}+4{z}^{2}}+\frac{2{B}_{rf}^{z}{B}_{rf}^{x}y\sin({\phi}_{rf}^{z})}{\sqrt{{x}^{2}+{y}^{2}+4{z}^{2}}}\Bigg{]},$
(5)

where the symbols $B_{rf}^{x},B_{rf}^{y},B_{rf}^{z}$ denote the magnitudes of
the applied rf-field along $x$, $y$ and $z$-directions respectively. The
symbols $\phi_{rf}^{y}$ and $\phi_{rf}^{z}$ are the phase values of the $y$\-
and $z$-components of the rf-field relative to the $x$-component of rf-field.
From Eq. (3), it is obvious that the rf-dressed potential $V_{D}$ is position
dependent via detuning and Rabi frequency.

Now if we consider a special case where rf-field is linearly polarized along
$x$-direction, the expression for Rabi frequency $\Omega_{R}(r)$ can be given
by,

${\mid\Omega_{R}(r)\mid}^{2}={\Bigg{(}\frac{{{g}_{F}{\mu}_{B}B_{rf}^{x}}}{2\hbar}\Bigg{)}}^{2}\Bigg{[}\frac{4{x}^{2}{z}^{2}}{({x}^{2}+{y}^{2})(x^{2}+y^{2}+4{z}^{2})}+\frac{y^{2}}{{x}^{2}+{y}^{2}}\Bigg{]}.$
(6)

This expression can be used to understand the local variation in the rf-
dressed trapping potential $V_{D}$ when a linearly polarized ($x$-direction)
rf-field is used for rf-dressing.

##  III Experimental Set-Up

The experiments have been performed in an in-house developed setup, where a
single glass cell at ultra-high vacuum (UHV) pressure is used for performing
all the sequences in the experiment from loading of a magneto-optical trap
(MOT) to achieving a rf-dressed trap. The details of our setup has been given
elsewhere Bhardwaj et al. (2023). The schematic of the experimental set up has
been shown in Fig. 1. The UHV glass cell was evacuated to a pressure level of
$\sim$1.5 $\times$ $10^{-10}$ Torr. This UHV glass cell has its outer area of
cross-section as 30 mm $\times$ 30 mm, and 70 mm length usable for optical
beams. The glass cell wall thickness is 2.5 mm. The glass cell was attached to
a six-way cube and kept horizontal in orientation.

First of all, the magneto optical trap for ${}^{87}Rb$ atoms is prepared in
the glass cell. For this, six independent cooling laser beams (shown as UHV-
MOT beams in Fig. 1) are aligned on the glass cell. Each of these cooling beam
was derived from a single laser (TA-Pro, Toptica, Germany), and each beam had
a power of 20 mW at the glass cell entrance. A pair of repumping beam has been
derived from another laser (DL-100, Toptica, Germany) and these beams were
mixed with two cooling beams incident on the glass cell ($z$-directions
beams). Each repumping beam has a power of around 10 mW. Both the lasers were
respectively frequency locked to cooling and repumping transitions of
${}^{87}Rb$ and acousto-optic modulators (AOMs) were used to change the
frequency as well as amplitude of the cooling and repumping laser beams as per
requirement. For the MOT formation, the cooling beams are typically kept red-
detuned by 2.5$\Gamma$ from the cooling transition ($5^{2}S_{1/2}$, $F=2$
$\rightarrow$ $5^{2}P_{3/2}$, $F^{\prime}=3$), where $\Gamma=2\pi\times 6.06$
MHz is the natural linewidth of the ${}^{87}Rb$ atoms cooling transition. The
magnetic field gradient ($B_{q}$) of $\sim$5 G/cm is obtained for MOT from a
pair of quadrupole coils operating with DC current of 1 A in anti-Helmholtz
configuration. These coils, with higher currents, are also used for magnetic
trapping in a subsequent stage of experiment. The Rb-source (dispensers) is
kept inside a glass jacket which has a narrow opening near the centre of MOT
(Fig. 1). This jacket facilitates to load MOT without degradation of UHV in
the MOT region Bhardwaj et al. (2023). In the MOT, we can have $\sim$1
$\times$ $10^{8}$ atoms ${}^{87}Rb$ at temperature of 250 $\mu$K.

Figure 1: : The schematic of the experimental setup for rf-dressed atom trap.

After loading the MOT, the optical molasses was implemented for a few
milliseconds to reduce the temperature of atoms further. For molasses, the
magnetic field for MOT was switched-off and cooling beam detuning was
increased from -2.5$\Gamma$ to -8.0$\Gamma$. The temperature of the atom cloud
after molasses was reached to $\sim$70 $\mu$K. After molasses, the atom cloud
was prepared in a particular magnetic hyperfine state by implementing optical
pumping. Optical pumping was done by exposing the atoms (in $F=2$ state) to a
circularly polarized optical pulse ($\sim$ 500 $\mu$W of power and $\sim$ 500
$\mu$s of duration) in presence of a DC magnetic field of $\sim$ 2G. This
prepares atoms in $|F=2,m_{F}=2\rangle$ for magnetic trapping.

We used mechanical shutters in the path of the cooling beams which was
necessary to fully stop any light going to the magnetic trap region. Usually,
a leakage of laser beams through AOMs (when switched-off) is a primary source
of light which affects the number of atoms in magnetic trap and trap life-
time. For magnetic trapping, the optically pumped atom cloud was placed in
quadrupole magnetic trap by switching-on the quardupole magnetic field with a
lower gradient ($B_{q}=35$ G/cm) which was held at that value for $\sim$ 100
ms. This step provided an initial loading of magnetic trap. After this initial
loading, the gradient ($B_{q}$) was ramped up to a higher final value (e.g.
$B_{q}=126$ G/cm) slowly in 500 ms. At this final gradient, the atom cloud is
held for further experiments. We observed that when final gradient was 126
G/cm, the atom cloud was observable in magnetic trap up to 11 s duration with
a lifetime ($\frac{1}{e}$) of $\sim$ 4 s. In the magnetic trap, we have
measured the number and temperature of the atom cloud, because it was
important to know these parameters for subsequent rf-dressing experiment. Just
after reaching the final gradient of 126 G/cm in the magnetic trap, the number
of atoms in the magnetic trap was $\sim$ 1 $\times$ $10^{7}$ (initially) and
temperature was $\sim$ 240 $\mu$K.

Since the temperature of atom cloud in the magnetic trap was higher than the
trap depth of rf-dressed potential, the atom cloud in the magnetic trap was
evaporatively cooled to reduce the temperature. For this, a single loop
antenna (diameter of 25 mm) has been used for rf induced evaporation. The rf
source was connected to this antenna and the rf-field frequency was scanned
from 15 MHz to 2 MHz in 2 seconds, when atoms were placed in the magnetic
trap. The rf evaporation was initiated after 50 ms duration of reaching the
final value of gradient in magnetic trap. This time duration is good enough
for untrapped atoms to escape from the magnetic trap atom cloud. During the rf
evaporation process, the number of atoms and temperature, both are reduced in
the atom cloud. At the end of the evaporation sequence, we observed that atom
cloud in magnetic trap had $\sim$ 6$\times$ $10^{5}$ number of atoms at
temperature of $\sim$ 30 $\mu$K. With these parameters in the atom cloud, we
started loading atoms in rf-dressed trap. For doing this, the evaporatively
cooled atom cloud in the magnetic trap was exposed to a strong rf-field. A
different 20-turn coil of diameter $\sim$80 mm (shown in Fig. 1) was used to
generate the necessary rf-field for rf-dressing. The rf signal from the direct
digital synthesizer (DDS) (Tektronix, AFG3102C) has been fed to the rf-
amplifier (Hubert, model no: A 1020-15-200) to amplify the rf-signal. The rf-
amplifier output was finally connected to the 20-turn coil for generating the
dressing rf-field.

Finally, we detect the rf-dressed atom cloud in magnetic trap by a 2f-imaging
setup. For this, a weak absorption probe beam is passed though the atom cloud
and the transmitted probe was imaged on a CCD camera (PCO, pixelfly). All the
experimental sequences, starting from switching-on the MOT beams (through
AOMs) and MOT loading, molasses formation, optical pumping, magnetic trapping,
rf-evaporation, rf-dressing, and detection of atoms in rf-dressed trap were
controlled by a field programmable gate array (FPGA) based c-RIO (compact-
Reconfigurable Input Output Module) controller of National Instruments (NI
c-RIO 9035). This controller generates the analog and digital pulses to
control various instrument used in the controlled operation. The controlled
instruments include AOMs, power supplies of coils, shutters , DDS, CCD camera,
etc.

##  IV Results and discussion

Figure 2:  The measured variation (circles) in number of atoms in the
quadrupole magnetic trap with holding time in the trap. The magnetic trap was
operated with final gradient of $B_{q}=126$ G/cm. The continuous curve is an
exponential fit to the measured data. Figure 3:  Experimentally observed
absorption probe images (in $x-z$ plane) of ${}^{87}Rb$ atom cloud in rf-
dressed potential for different values of dressing rf-field amplitude
($B_{rf}^{x}$). The magnetic field gradient $B_{q}=126$ G/cm and rf-field
frequency $\omega_{rf}$ $=2\pi$ $\times$ $1.08$ MHz are common for all images
from (a) to (d).

We have first characterized our magnetic trap, which is a quadrupole magnetic
trap loaded in the single UHV (pressure $\sim$1.5 $\times$ $10^{-10}$ Torr)
chamber setup. The number of atoms in the trap and the trap life-time were
measured by analyzing the absorption probe images of the atom cloud captured
at different points of time after achieving the final gradient in the magnetic
trap. The variation in number atoms in the trap with time is shown in Fig. 2.
For each of these measurements in Fig. 2, once the image was captured for a
given data point, the cloud image for next data point was captured by running
a new cycle of MOT to magnetic trap loading with changed time for camera
trigger after magnetic trap loading. The error bar shows the deviation in the
measured values in repeated measurements for the same point. As shown in Fig.
2, the life-time ($\frac{1}{e}$) of the magnetic trap from the best fit was
$\sim 4$ s.

Since temperature of the atom cloud in the magnetic trap ($\sim$250 $\mu$K)
was higher as required for loading the rf-dressed trap, the magnetically
trapped atoms were cooled further by rf-evaporation. The rf-evaporation was
performed for 2 seconds with a frequency scan from 15 MHz to 2 MHz. After
evaporation, the temperature of the atom cloud was estimated to be $\sim$30
$\mu$K with $\sim$6 $\times$ $10^{5}$ number of atoms in the trap. In the
magnetic trap, without rf-evaporation, $\sim$ 1$\times$$10^{7}$ atoms survive
after 2 seconds duration in the magnetic trap.

After evaporative cooling and reduction in temperature, atoms in magnetic trap
were exposed to a strong rf-field for trapping them in rf-dressed potential.
For this we have used a 20-turn circular loop antenna placed at distance of
$\sim$6 cm from the atom cloud. The access of the coil was along $x$-axis
(Fig. 1). This coil was driven by the rf-source (DDS) at 1.08 MHz frequency.
The source signal was amplified and fed to the coil through impedance matched
circuits. After exposure to the dressing rf-field for $\sim$200 ms duration,
the cloud images were captured by absorption probe imaging technique. Fig. 3
shows the observed absorption images of atom cloud trapped in rf-dressed
potential for different values of rf-field amplitude. From these 2D images, we
note that the center of atom trap has low number density of atoms. More atoms
are distributed in the peripheral region of the 2D-image, which is an
indicative of trapping of atoms on a shell surface in 3D. In order to
understand these results, we have theoretically simulated the rf-dressed
potential for a linearly polarized rf-field which are discussed as follows.

Figure 4:  The 3D-plots ((a)-(c)) of rf-dressed potential $V_{D}$ for a
$x$-directional linearly polarized rf-field. Each plot shows $V_{D}$ values
from minimum ($\sim$1 $\mu$K) to a maximum $V_{D}$ value. The maximum value of
$V_{D}$ was set as 20 $\mu$K (a), 40 $\mu$K (b), and 60 $\mu$K (c) above the
minimum value. The rf-field magnitude is $B_{rf}^{x}=1.3G$ and frequency is
$\omega_{rf}=2\pi\times 1.08$ MHz. The value of magnetic field gradient is
$B_{q}=126$ G/cm.

Fig. 4 shows a 3D-plots of contours of rf-dressed potential given by Eq. (3)
for a $x$-directional linearly polarized rf-field configuration. Each plot has
contours of $V_{D}$ from a minimum value ($\sim$1 $\mu$K) of $V_{D}$ to a
maximum value of $V_{D}$. Here maximum $V_{D}$ value contour lies on the
surface of the 3D plot. In each plot ((a), (b) and (c)), the maximum value of
$V_{D}$ was respectively taken as 20 $\mu$K, 40 $\mu$K, and 60 $\mu$K above
the minimum value. From these plots, one can understand the 3D configuration
of trapping geometry, if kinetic energy (temperature) of atoms is known. For
example, from the plot (a) in Fig. 4, one can predict that for atoms with
kinetic energy smaller than 20 $\mu$K, the trapping geometry is expected to be
a double-well oriented along $x$-direction. This is as predicted earlier also
Lesanovsky et al. (2006); Chakraborty and Mishra (2014).

Figure 5:  Two dimensional ($x-z$) contour plots of rf-dressed potentials for
different maximum values of $V_{D}$. The parameters are : $B_{q}=126$ G/cm,
rf-field frequency $\omega_{rf}=2\pi\times$ 1.08 MHz, and amplitude of
$x$-polarized rf-field as $B_{rf}^{x}=1.3$ G. The contour (a) shows the range
of the trapping potential from minimum value to 20 $\mu$K above the minimum
value. Accordingly the contours in (b) and (c) show the maximum value of the
trapping potential to be 40 $\mu$K and 60 $\mu$K above the minimum value.

However, for atoms with kinetic energy higher than 20 $\mu$K, the atom
trapping geometry will correspond to potential contours shown in plots (b) and
(c) in Fig. 4, which will be a hollow shell type atom trap configuration and
not a double-well.

To understand it more clearly, we have also plotted the 2D contours ($x$-$z$
plane) of trapping potential $V_{D}$ with the color coding, and results are
shown in Fig. 5. Here ($x-z$) contours of potential $V_{D}$ have been plotted
for different maximum values of $V_{D}$. This figure shows that for the lower
values of $V_{D}$ (e.g. plot (a) in Fig. 5), the trapping potential is a
double-well. However, at higher values of $V_{D}$ (e.g. plot (c)), the
trapping potential takes a shape of ring structure in the $x-z$ plane, which
is corresponding to be a shell trap in 3D. In the experiments, we have
observed (Fig. 3) hollow ellipsoidal shell trap in which an atom cloud of 30
$\mu$K temperature has been used. The atom cloud of 30 $\mu$K temperature will
have significant fraction of atoms with kinetic energy more than 30 $\mu$K.
Such atoms are expected to follow the trap potential shapes as shown in plots
(b) and (c) in Fig. 5 and plots (b) and (c) in Fig. 4, which is corresponding
to a trap geometry of shell rather than a double-well. This explains our
experimental observation of a shell trap for ${}^{87}Rb$ atoms obtained after
rf-dressing with linearly polarized rf-field.

##  V Conclusion

We have investigated the atom trapping geometry formed by superposing a
linearly polarized radio-frequency (rf) field on a quadrupole magnetic trap.
The hollow ellipsoidal shell type atom trap has been experimentally observed,
which is explained by modelling theoretically the rf-dressed potentials. This
type of trap geometry may be useful in the study of low-dimensional Bose-
Einstein condensation (BEC), superfluidity, atom gyroscope, etc.

###### Acknowledgements.

We thank S. Singh and V. Singh for useful technical discussions. We are also
thankful to Shri D. Sharma and Shri Bhupinder Singh for the development of
switching circuit, Shri A. Pathak and Smt S. Tiwari for the development of
c-RIO based electronic controller, and P. S. Bagduwal for development of rf-
amplifier and rf-impedance matching circuits. S. Sarkar is grateful for
financial support from HBNI, RRCAT.

## Data Availability Statement

Data available on request from the authors.

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