# Intermittent random walks under stochastic resetting

Rosa Flaquer-Galmés  Grup de Física Estadística. Departament de Física,
Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain  Daniel Campos
Grup de Física Estadística. Departament de Física, Universitat Autònoma de
Barcelona, 08193 Bellaterra, Spain  Vicenç Méndez  Grup de Física Estadística.
Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra,
Spain

###### Abstract

We analyze a one-dimensional intermittent random walk on an unbounded domain
in the presence of stochastic resetting. In this process, the walker
alternates between local intensive search, diffusion, and rapid ballistic
relocations in which it does not react to the target. We demonstrate that
Poissonian resetting leads to the existence of a non-equilibrium steady state.
We calculate the distribution of the first arrival time to a target along with
its mean and show the existence of an optimal reset rate. In particular, we
prove that the initial condition of the walker, i.e., either starting
diffusely or relocating, can significantly affect the long-time properties of
the search process. Moreover, we demonstrate the presence of distinct
parameter regimes for the global optimization of the mean first arrival time
when ballistic and diffusive movements are in direct competition.

††preprint: APS/123-QED

##  I Introduction

In recent years, significant efforts have been dedicated to studying optimal
search strategies [1] in the field of behavioral ecology, both theoretically
and experimentally [2, 3, 4, 5]. A paradigmatic kind of model often invoked to
capture the characteristics of animal trajectories is that of intermittent
movement, where different motion patterns or mechanisms are alternated in time
by the organism [6, 7, 8]. One such of these intermittent search strategies is
the Intermittent Random Walk (IRW) which combines two distinct phases. The
first one (known as ’extensive’ search) consists of rapid ballistic
relocations that are often unresponsive to the presence of the target, as it
is considered that these movements are used by the animals just for choosing a
new region to explore, and cognitive abilities are partially or completely
suppressed during this transit [9]. This relocation is then followed by a
slower diffusive phase (often denoted as ’intensive local’ search) during
which the target can be in general detected if approached [9].

IRWs represent then a natural and significant extension of Brownian
trajectories in many contexts [10]. In biological systems, they have been
tested as plausible mechanisms for seed dispersal [11] or copepod dispersal
[12]. Their use in physical systems, however, is also widespread. They have
been proved, for instance, to offer a plausible framework to describe particle
trajectories at liquid-solid interfaces [13], supercooled liquids [14],
colloidal suspensions [15], or in different types of disordered media [16, 17,
18], among many other [19]. Also, their convenience as a mechanism to optimize
energy minimization algorithms have also been explored [20]. Finally, their
formal connection to self-trapping and/or self-avoiding interactions has also
been suggested as a possible underlying mechanism to explain such trajectories
at microscopic scales [21].

Theoretical studies have demonstrated that IRWs are also an effective search
strategy in a wide range of scenarios [22, 23, 24, 25, 26, 27, 28]. For the
case of Poissonian switching between the relocation and diffusion modes, an
optimal relationship between the two switching rates can minimize the mean
search time for detecting a randomly localized target in bounded domains [22].
Despite the potential interest of this model, a handicap of IRWs when regarded
as search trajectories is that their mean search time is still infinite in
unbounded domains, as happens with Brownian walkers and other strategies
totally or partially controlled by a homogeneous Gaussian noise. As a remedy
to this, the mechanism of stochastic resetting (in which the walker returns to
its initial position after a random sojourn time, and so arbitrary departures
from the target are avoided) has been advocated. In the particular case of
biological foraging, stochastic resetting can additionally offer a new layer
of realism to the mechanism of search. Indeed, many organisms repeatedly
return to a central nest, or to previously visited sites, as a part of their
regular foraging strategies [29]. This revisiting process can then be
effectively described by introducing a stochastic resetting of the
individual/animal position to those preferential points. In such cases,
stochastic resetting is capable of turning the mean first passage finite.
Furthermore, there exists an optimal value for the rate at which stochastic
resets are executed such that the corresponding mean first passage time
attains its minimum value.[30].

In this work, we explore the main transport properties of an IRW model
combining diffusive and relocation modes and study its dynamics under
stochastic Poissonian resetting. This will allow us to study whether the
existence of an unresponsive mode (with relocations during which the target
cannot be detected) modifies significantly the arrival statistics to a target.
In Section II we review the properties of the mean square displacement of the
walker, and the first arrival time to a hidden target in the absence of
resetting. After that, in Section III we compute the non-equilibrium
stationary state reached by the system when resetting is introduced, how the
statistics of the first arrival probability get modified accordingly, and its
dependence on the initial condition chosen. An exhaustive analysis of the
optimal mean first arrival time in terms of the characteristic parameters is
performed. It reveals an interesting trade-off of the switching rates between
the relocation and diffusive states. Finally, we summarize in Section IV the
main conclusions of our work.

##  II IRW model

Let $\rho_{+}(x,t)$ and $\rho_{-}(x,t)$ be the probability density of a walker
at position $x$ at time $t$ with velocities $+v$ and $-v$, respectively.
States $+$ and $-$ correspond to a relocation mode in which the walker moves
ballistically during a random (and exponentially distributed) time before
switching to the diffusive mode. After that, the walker moves diffusively for
a random time (in which it is also able to detect the target), also
exponentially distributed, before switching back to the relocation mode. If
the probability density for the walker to be at position $x$ at time $t$
moving diffusively is $\rho_{0}(x,t)$, then

$\displaystyle\frac{\partial\rho_{+}}{\partial t}$
$\displaystyle=-v\frac{\partial\rho_{+}}{\partial
x}-\frac{1}{\tau_{v}}\rho_{+}+\frac{1}{2\tau_{D}}\rho_{0},$ (1)
$\displaystyle\frac{\partial\rho_{-}}{\partial t}$
$\displaystyle=v\frac{\partial\rho_{-}}{\partial
x}-\frac{1}{\tau_{v}}\rho_{-}+\frac{1}{2\tau_{D}}\rho_{0},$ (2)
$\displaystyle\frac{\partial\rho_{0}}{\partial t}$
$\displaystyle=D\frac{\partial^{2}\rho_{0}}{\partial
x^{2}}+\frac{1}{\tau_{v}}\big{(}\rho_{+}+\rho_{-}\big{)}-\frac{1}{\tau_{D}}\rho_{0},$
(3)

where $\tau_{D}^{-1}$ and $\tau_{v}^{-1}$ are the switching rates from
diffusive to relocation mode, and from relocation to diffusive mode,
respectively. Note that the factor $1/2$ in the last term of the right-hand
side of Eqs. (1) and (2) implies that when switching from diffusion to
relocation, right and left moving states are equally probable as there is no
predilect direction. Nevertheless, for systems where the left-right symmetry
is broken in the relocation phase, one can generalize Eqs. (1) to (3) by
introducing the corresponding probability, $p$ to relocate to the right and
$1-p$ to relocate to the left. Let us introduce now the Fourier transform
(denoted with a bar symbol) and the Fourier-Laplace transform (denoted with a
hat symbol) as follows,

$\displaystyle\bar{\rho_{i}}(k,t)=\int_{-\infty}^{\infty}e^{-ikx}\rho_{i}(x,t)dx,\;\;$
$\displaystyle\hat{\rho_{i}}(k,s)=\int_{-\infty}^{\infty}e^{-ikx}\int_{0}^{\infty}e^{-st}\rho_{i}(x,t)dxdt$

with $i=0,+,-$. Transforming Eqs. (1-3) by Fourier-Laplace we can write a
single equation for $\hat{\rho}_{0}(k,s)$:

$\displaystyle s\hat{\rho}_{0}$
$\displaystyle(k,s)-\bar{\rho}_{0}(k,0)=-Dk^{2}\hat{\rho}_{0}(k,s)$
$\displaystyle+\frac{1}{\tau_{D}}\left[\frac{(s\tau_{v}+1)-(s\tau_{v}+1)^{2}-(kv\tau_{v})^{2}}{(s\tau_{v}+1)^{2}+(kv\tau_{v})^{2}}\right]\hat{\rho}_{0}(k,s)$
$\displaystyle+\frac{(s\tau_{v}+1)\left(\bar{\rho}_{+}(k,0)+\bar{\rho}_{-}(k,0)\right)}{(s\tau_{v}+1)^{2}+(kv\tau_{v})^{2}}$
$\displaystyle+ikv\tau_{v}\frac{\left(\bar{\rho}_{+}(k,0)-\bar{\rho}_{-}(k,0)\right)}{(s\tau_{v}+1)^{2}+(kv\tau_{v})^{2}},$
(4)

where $\bar{\rho_{i}}(k,0)$ is the Fourier transform of the initial condition
at state $i$. We study two different initial conditions for a walker located
at $x=x_{0}$ at time $t=0$. For the first case (case I in the following) we
consider the walker being at the diffusive state $i=0$, so that
$\bar{\rho}_{0}(k,0)=e^{-ikx_{0}}$ and
$\bar{\rho}_{+}(k,0)=\bar{\rho}_{-}(k,0)=0$. For the second one (case II), we
consider a walker in the relocation state with the same probability of being
at states $i=+,-$, so that $\bar{\rho}_{0}(k,0)=0$ and
$\bar{\rho}_{+}(k,0)=\bar{\rho}_{-}(k,0)=e^{-ikx_{0}}/2$. To facilitate
reading of the manuscript, we will use superscripts D and v (referring to the
initial state, diffusive or ballistic, in each case) to identify our results
for cases I and II. One could also consider that after the resetting the
system is in a random phase, that is, it could be diffusing or relocating with
the same probability after resetting. This third case is included in Appendix
A as the results obtained for it can be understood via the first two cases
here presented. The distinction between these initial conditions is relevant
as in Section III we show that in the presence of resetting the effect of the
initial condition cannot be neglected. For case I the propagator
$\hat{\rho}(k,s)=\hat{\rho}_{0}(k,s)+\hat{\rho}_{+}(k,s)+\hat{\rho}_{-}(k,s)$
reads:

$\displaystyle\hat{\rho}^{D}(k,s)=\hat{\rho}_{0}^{D}(k,s)\left[1+\frac{\tau_{v}}{\tau_{D}}\frac{s\tau_{v}+1}{(s\tau_{v}+1)^{2}+(v\tau_{v}k)^{2}}\right],$
(5)

where $\hat{\rho}^{D}_{0}(k,s)$ is defined as

$\displaystyle\hat{\rho}_{0}^{D}(k,s)=\frac{e^{-ikx_{0}}}{s+Dk^{2}+\frac{1}{\tau_{D}}\big{[}1-\frac{s\tau_{v}+1}{(s\tau_{v}+1)^{2}+(vk\tau_{v})^{2}}\big{]}}.$
(6)

Analogously, for case II these expressions are

$\displaystyle\hat{\rho}^{v}(k,s)=$
$\displaystyle\hat{\rho}_{0}^{v}(k,s)\left[1+\frac{\tau_{v}}{\tau_{D}}\frac{s\tau_{v}+1}{(s\tau_{v}+1)^{2}+(v\tau_{v}k)^{2}}\right]$
$\displaystyle+\tau_{v}e^{-ikx_{0}}\frac{s\tau_{v}+1}{(s\tau_{v}+1)^{2}+(v\tau_{v}k)^{2}},$
(7)

$\displaystyle\hat{\rho}_{0}^{v}(k,s)=\hat{\rho}_{0}^{D}(k,s)\frac{(s\tau_{v}+1)}{(s\tau_{v}+1)^{2}+(vk\tau_{v})^{2}}.$
(8)

Figure 1: Mean squared displacement for case I (green circles) and case II
(blue triangles). In solid lines we show the analytic result, Eq. (9), and in
dots data obtained from numerical simulations. The parameters used are
$D=v=1$, $\tau_{D}=3.8$, $\tau_{v}=3.3$ and $x_{0}=0$.

To study the transport properties of this IRW, we start by computing the Mean
Squared Displacement (MSD) of the whole process. We also note that, due to the
symmetry of the initial conditions, the mean distance traveled is
$\left\langle
x(s)\right\rangle=\frac{1}{i}\left[\frac{\partial\hat{\rho}(k,s)}{\partial
k}\right]_{k=0}=0,$ where $\hat{\rho}(k,s)$ is the propagator of the whole
process (Eq. (5) or (7) depending on the initial condition). Then, the MSD can
be computed from $\left\langle
x^{2}(s)\right\rangle=-\left[\frac{\partial^{2}\hat{\rho}(k,s)}{\partial
k^{2}}\right]_{k=0}.$ Assuming $x_{0}=0$ for simplicity, when transforming
back from Laplace space the MSD reads

$\displaystyle\left\langle x^{2}(t)\right\rangle$
$\displaystyle=\frac{2(D+\frac{(v\tau_{v})^{2}}{\tau_{D}})}{(1+\frac{\tau_{v}}{\tau_{D}})}t+A_{1}$
$\displaystyle+A_{2}e^{-\frac{t}{\tau_{v}}}+A_{3}e^{-(\frac{1}{\tau_{v}}+\frac{1}{\tau_{D}})t},$
(9)

where the coefficients $A_{i}$ with $i=1,2,3$ take different values depending
on whether the walker is initially in the diffusive or relocation state;
specific values of $A_{i}$ are included in Appendix B. Note that in the large
time limit, regardless of the initial condition, the MSD is essentially
diffusive and behaves according to

$\displaystyle\left\langle
x^{2}(t)\right\rangle=2\frac{D\tau_{D}+(\tau_{v}v)^{2}}{\tau_{D}+\tau_{v}}t\quad\textrm{as$\quad
t\rightarrow\infty$}$ (10)

On the other hand, for short times one recovers a diffusive MSD for case I,
while for case II is ballistic, as expected given the initial conditions:

$\displaystyle\left\langle x^{2}(t)\right\rangle^{D}$ $\displaystyle\approx
2Dt\quad$ $\displaystyle\mathrm{as}\quad t\to 0$ (11)
$\displaystyle\left\langle x^{2}(t)\right\rangle^{v}$
$\displaystyle\approx\left(v^{2}+\frac{D}{\tau_{v}}\right)t^{2}\quad$
$\displaystyle\mathrm{as}\quad t\to 0$ (12)

In Fig. 1 we have compared Eq. (9) for cases I and II with numerical
simulations. We conclude then that the switching processes between diffusive
and relocation states on affects the transient temporal behaviour of the MSD.

###  II.1 First arrival time

A widely used measure of the efficiency of a search process is the study of
the statistics of a walker arriving for the first time at a given position
where a hidden target is located, that is, the First Arrival Time (FAT) when
the target position is considered as an absorbing boundary. In particular, we
consider the First Arrival Time Density (FATD) and its Mean (MFAT). As stated
above, we work under the assumption that the target can only be detected when
the walker is in the diffusive phase, as is assumed that in the relocation
phase the high velocity prevents the walker from detecting the target [9].
Considering searchers starting from an initial position $x=x_{0}$ and a target
located at $x=0$ one can rewrite Eq. (3) to account for the target

$\displaystyle\frac{\partial\rho_{0}}{\partial t}$
$\displaystyle=D\frac{\partial^{2}\rho_{0}}{\partial
x^{2}}+\frac{\left(\rho_{+}+\rho_{-}\right)}{\tau_{v}}-\frac{\rho_{0}}{\tau_{D}}-f(x_{0},t)\delta(x),$
(13)

while Eq. (1) and Eq. (2) remain unchanged. Here, $f(x_{0},t)$ is the first
arrival probability density to a target at the origin. The role of the last
term in (13) is to remove the particle when it arrives at a target placed at
$x=0$. To compute the FATD we transform Eqs. (1), (2) and (13) by Fourier-
Laplace and combine them in a single equation for $\rho_{0}$. Integrating this
equation over $k$ and using that
$\int_{-\infty}^{\infty}dk\hat{\rho}_{0}(k,s)=\bar{\rho}_{0}(x=0,s)=0$, we
obtain the FATD for case I, denoted with the superscript $D$, which reads

$\displaystyle\tilde{f}^{D}(x_{0},s)=\frac{I(x_{0},s)}{I(x_{0}=0,s)},$ (14)

where $\tilde{f}(x_{0},s)$ is the Laplace transform of $f(x_{0},t)$ and

$\displaystyle I(x_{0},s)$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}\hat{\rho}_{0}^{D}(k,s)dk$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{A(k,s)e^{-ikx_{0}}}{k^{4}+B(s)k^{2}+C(s)}dk.$
(15)

The coefficients $A(k,s)$, $B(s)$ and $C(s)$ are defined as:

$\displaystyle
A(k,s)=\frac{\left(1+s\tau_{v}\right)^{2}}{D(v\tau_{v})^{2}}+\frac{k^{2}}{D},$
(16)

$\displaystyle
B(s)=\frac{1+s\tau_{D}}{D\tau_{D}}+\frac{\left(1+s\tau_{v}\right)^{2}}{(v\tau_{v})^{2}},$
(17)

$\displaystyle
C(s)=\frac{\left(1+s\tau_{v}\right)\left[s\tau_{v}+s\tau_{D}\left(1+s\tau_{v}\right)\right]}{D\tau_{D}(v\tau_{v})^{2}}.$
(18)

The integral (15) can be rewritten in the more suitable form

$I(x_{0},s)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{A(k,s)e^{-ikx_{0}}}{\left[k^{2}+\lambda_{+}^{2}\right]\left[k^{2}+\lambda_{-}^{2}\right]}dk,$

with

$\lambda_{\pm}=\left[\frac{B(s)\pm\sqrt{B(s)^{2}-4C(s)}}{2}\right]^{1/2}.$
(19)

It is not difficult to show that $B^{2}(s)>4C(s)$ so that $\lambda_{\pm}$ are
always real values. From the residue theorem, the integral can be computed

$\displaystyle I(x_{0},s)$
$\displaystyle=\frac{A(k=i\lambda_{-},s)e^{-|x_{0}|\lambda_{-}}}{2\lambda_{-}(\lambda_{+}^{2}-\lambda_{-}^{2})}-\frac{A(k=i\lambda_{+},s)e^{-|x_{0}|\lambda_{+}}}{2\lambda_{+}(\lambda_{+}^{2}-\lambda_{-}^{2})}$
$\displaystyle=\frac{1}{2(\lambda_{+}^{2}-\lambda_{-}^{2})}\left[\frac{\phi_{-}e^{-|x_{0}|\lambda_{-}}}{D}-\frac{\phi_{+}e^{-|x_{0}|\lambda_{+}}}{D}\right],$
(20)

where we have defined $\phi_{\pm}(s)$ as

$\phi_{\pm}(s)=\frac{\left(1+s\tau_{v}\right)^{2}}{(v\tau_{v})^{2}\lambda_{\pm}}-\lambda_{\pm}.$
(21)

From this, the FATD finally reads

$\tilde{f}^{D}(x_{0},s)=\frac{\phi_{-}e^{-|x_{0}|\lambda_{-}}-\phi_{+}e^{-|x_{0}|\lambda_{+}}}{\phi_{-}-\phi_{+}}$
(22)

Now, for case II, denoted with the superscript $v$, the FATD in the Laplace
space follows straightforwardly,

$\tilde{f}^{v}(x_{0},s)=\frac{J(x_{0},s)}{I(x_{0}=0,s)},$ (23)

where $J(x_{0},s)$ is defined as

$\displaystyle J(x_{0},s)$
$\displaystyle=\frac{1}{2\pi}\frac{1+s\tau_{v}}{D(v\tau_{v})^{2}}\int_{-\infty}^{\infty}\frac{e^{-ikx_{0}}}{k^{4}+B(s)k^{2}+C(s)}dk$
$\displaystyle=\frac{1+s\tau_{v}}{D(v\tau_{v})^{2}}\frac{\lambda_{+}e^{-|x_{0}|\lambda_{-}}-\lambda_{-}e^{-|x_{0}|\lambda_{+}}}{2\left(\lambda_{+}^{2}-\lambda_{-}^{2}\right)\lambda_{-}\lambda_{+}}.$
(24)

Following the same reasoning as for case I, the final expression for the first
arrival distribution reads

$\tilde{f}^{v}(x_{0},s)=\frac{1+s\tau_{v}}{(v\tau_{v})^{2}}\frac{\lambda_{+}e^{-|x_{0}|\lambda_{-}}-\lambda_{-}e^{-|x_{0}|\lambda_{+}}}{\lambda_{+}\lambda_{-}(\phi_{-}-\phi_{+})}.$
(25)

If one tries to compute the MFAT as $\langle
T(x_{0})\rangle=\int_{0}^{\infty}tf(x_{0},t)dt$ it goes to infinity as walkers
are not confined. We note that this is different from the results reported
previously for the IRW, e.g. in [22], [25] as in there the walker was
considered to be in a finite domain or interval. Thus, it is clear that in
unbounded domains the walker needs another mechanism apart from the IRW in
order to optimize the search of a target.

##  III Resetting

Figure 2: Example of two trajectories the green one for case I and the blue
one for case II. The dashed lines represent a reset to the initial position.
The parameters used are $D=v=1$, $\tau_{D}=3.8$, $\tau_{v}=3.3$, $x_{0}=0$ and
$r=0.1$. In the inset, other two trajectories for the case without reset, for
case I (blue) and case II (green) respectively.

As we have shown, in unbounded domains the MFAT to a hidden target becomes
infinity. We expect, and later show, that the presence of stochastic resetting
to the initial position yields a finite MFAT to the target, as in numerous
other works (see Refs. [31, 32] for example). The process with resetting is as
follows: the walker initially starts at $x_{0}$ either diffusing (case I) or
relocating (case II). After some random time the walker is returned
instantaneously to its initial position and the movement starts again. It is
worth noting that a reset implies that the process is started all over again.
This implies that if initially the walker started diffusing, after a reset the
walker will be in the diffusing state, or vice versa if the walker is
initially relocating. In Fig. 2 we present an example of two sampled
trajectories of the IRW with stochastic resetting, where the dashed lines
reveal that a reset has taken place (example trajectories without reset are
included in the inset of the figure for better comparison).

###  III.1 Stationary state

It is known [33] that the propagator in the presence of resets, namely
$\rho_{r}(x,t)$, satisfies the renewal equation

$\displaystyle\rho_{r}(x,t)=\varphi^{*}(t)\rho(x,t)+\int_{0}^{t}\varphi(t^{\prime})\rho_{r}(x,t-t^{\prime})dt^{\prime}$
(26)

where $\rho(x,t)$ is the propagator in absence of resets, $\varphi(t)$ is the
probability of a resetting at time $t$, and $\varphi^{*}(t)$ is the
complementary probability of not having a reset up to time $t$, that is
$\varphi^{*}(t)=\int_{t}^{\infty}\varphi(t^{\prime})dt^{\prime}$. Considering
a Poissonian resetting process, i.e., exponentially distributed resets
$\varphi(t)=\tau_{r}^{-1}e^{-t/\tau_{r}}$, the propagator in the presence of
resets in the Fourier-Laplace space is related to the one without resets by
means of:

$\displaystyle\hat{\rho}_{r}(k,s)=\frac{1+s\tau_{r}}{s\tau_{r}}\hat{\rho}\left(k,s+\frac{1}{\tau_{r}}\right).$
(27)

It is known that the presence of resets induces a Non-Equilibrium Stationary
State (NESS) [31, 32]. Hence, we compute the long time limit of Eq. (27) in
order to study the existence and properties of such NESS.

$\displaystyle\bar{\rho}_{r}^{(s)}(k)=\lim_{s\to
0}s\hat{\rho}_{r}(k,s)=r\hat{\rho}(k,s=r).$ (28)

where $r=1/\tau_{r}$ is the constant reset rate and the superscript $(s)$
denotes that it is the stationary propagator. It is interesting to note that
since the propagator depends on the initial state of the walker, Eqs. (5) and
(7), the stationary state will also be different if the walker was initially
either at the relocation or diffusive states. In case I,

$\displaystyle\rho_{r}^{(s),D}(x)$
$\displaystyle=r\left[\frac{1}{2\pi}\int_{-\infty}^{\infty}\hat{\rho}^{D}(k,s)e^{ikx}dk\right]_{s=r}$
$\displaystyle=r\left[I(|x-x_{0}|,r)+\frac{\tau_{v}}{\tau_{D}}J(|x-x_{0}|,r)\right],$
(29)

where $I(x,s)$ is defined in Eq. (15) and $J(x,s)$ in Eq. (24). We note that
in the absence of relocations ($\tau_{D}\to\infty$) one recovers the result in
Ref. [30]. On the other hand, for case II the propagator at the stationary
state is given by

$\displaystyle\rho_{r}^{(s),v}(x)=r\left[\frac{1}{2\pi}\int_{-\infty}^{\infty}\hat{\rho}^{v}(k,s)e^{ikx}dk\right]_{s=r}$
$\displaystyle=r\bigg{[}J(|x-x_{0}|,r)+\frac{e^{-|x-x_{0}|\alpha}}{2v}$
$\displaystyle+\frac{\tau_{v}(r\tau_{v}+1)^{2}}{(2D\tau_{D}(v\tau_{v})^{4}}\bigg{(}\frac{e^{-|x-x_{0}|\lambda_{-}}}{\lambda_{-}(\lambda_{+}^{2}-\lambda_{-}^{2})(\alpha^{2}-\lambda_{-}^{2})}$
$\displaystyle-\frac{e^{-|x-x_{0}|\lambda_{+}}}{\lambda_{+}(\lambda_{+}^{2}-\lambda_{-}^{2})(\alpha^{2}-\lambda_{+}^{2})}+\frac{e^{-|x-x_{0}|\alpha}}{\alpha(\lambda_{+}^{2}-\alpha^{2})(\lambda_{-}^{2}-\alpha^{2})}\bigg{)}\bigg{]},$
(30)

where $\alpha\equiv(r\tau_{v}+1)/(v\tau_{v})$. In Fig. 3 we show the
comparison between the NESS (computed through Eqs. (29) and (30)) and
numerical simulations, confirming the validity of our results. We can see
that, as expected, the NESS is different depending on the initial condition of
the walker. This is a result that we are going to recover for all other
magnitudes computed in this work. This is due to the fact that resetting makes
the walker return to the initial condition and so, its effect is not smoothed
out at long times or long distances as is expected in IRW processes without
resetting.

Figure 3: The propagator at the stationary state for both initial conditions,
case I (green circles) and case II (blue triangles) in the system with
resetting. In solid lines we present the analytic solution, Eq. (29) and Eq.
(30), and in dots data obtained from numerical simulations. The parameters
used are $D=v=1$, $\tau_{D}=3.8$, $\tau_{v}=3.3$, $x_{0}=0$ and $r=0.1$.

###  III.2 Mean squared displacement

Next, we compute the MSD when resetting is present in the model. Multiplying
(26) by $x^{2}$ and integrating over $x$ one obtains

$\displaystyle\left\langle
x^{2}(t)\right\rangle_{r}=\int_{0}^{t}Q(t-t^{\prime})\varphi^{*}(t^{\prime})\left\langle
x^{2}(t^{\prime})\right\rangle dt^{\prime};$ (31)

where $\left\langle x^{2}(t)\right\rangle$ is the MSD in absence of resetting
and $Q(t)$ is the resetting rate function whose Laplace transform is

$Q(s)=\frac{1}{1-\varphi(s)}=\frac{1+s\tau_{r}}{s\tau_{r}}.$

Inserting (9) into (31) we finally obtain that the MSD in the presence of
resetting is given by

$\displaystyle\left\langle x^{2}(t)\right\rangle_{r}$
$\displaystyle=2\frac{D+\frac{\tau_{v}^{2}v^{2}}{\tau_{D}}}{(1+\frac{\tau_{v}}{\tau_{D}})}\tau_{r}\left(1-e^{-t/\tau_{r}}\right)$
$\displaystyle+A_{1}+\frac{A_{2}}{1+\frac{\tau_{r}}{\tau_{v}}}\left(1+\frac{\tau_{r}}{\tau_{v}}e^{-t\frac{(\tau_{r}+\tau_{v})}{\tau_{r}\tau_{v}}}\right)$
$\displaystyle+\frac{A_{3}\left[1+\frac{\tau_{r}}{\tau_{v}}\left(1+\frac{\tau_{v}}{\tau_{D}}\right)e^{-t\frac{(\tau_{r}+\tau_{v}+\tau_{D})}{\tau_{D}\tau_{r}\tau_{v}}}\right]}{1+\frac{\tau_{r}}{\tau_{v}}\left(1+\frac{\tau_{v}}{\tau_{D}}\right)}$
(32)

where again the constants $A_{1}$, $A_{2}$ and $A_{3}$ are as in (54) - (56)
for case I and (57) - (59) for case II. From Eq. (32) it can be easily checked
that

$\displaystyle\left\langle x^{2}(t)\right\rangle_{r}$ $\displaystyle=$
$\displaystyle
2\frac{D+\frac{\tau_{v}^{2}v^{2}}{\tau_{D}}}{(1+\frac{\tau_{v}}{\tau_{D}})}\tau_{r}+A_{1}+\frac{A_{2}}{1+\frac{\tau_{r}}{\tau_{v}}}$
$\displaystyle+$
$\displaystyle\frac{A_{3}}{1+\frac{\tau_{r}}{\tau_{v}}\left(1+\frac{\tau_{v}}{\tau_{D}}\right)},\;\textrm{as}\;t\to\infty.$

so $\left\langle x^{2}(t)\right\rangle_{r}$ tends to a constant value as time
goes to infinity, which is expected since it is the second moment of the NESS.
In Fig. 4 we compare our analytical result (32) with numerical simulations and
see that the analytical expressions are in good agreement with the numerical
data.

Figure 4: MSD in the presence of resetting for case I (green circles) and case
II (blue triangles) for the system with resetting. In solid lines we present
the analytic solution, Eq. (32), and in dots data obtained from numerical
simulations. The parameters used being $D=v=1$, $\tau_{D}=3.8$,
$\tau_{v}=3.3$, $x_{0}=0$ and $r=0.1$.

###  III.3 FAT under resetting

For the case without resetting, we have seen that the MFAT goes to infinity
for a target located in an unbounded space. But as we have mentioned above
many systems can optimize the MFAT in the presence of stochastic resetting.
Following this reasoning we proceed to compute the FATD for the IRW under
resetting. We do so in Laplace space in terms of the FATD of the system
without resetting. Following [33], let $G_{0}(x_{0},t)$ be the survival
probability for the system without resetting and $G_{r}(x_{0},t)$ the one for
the system with resetting; both magnitudes are related via a last renewal
equation

$\displaystyle G_{r}(x_{0},t)$ $\displaystyle=e^{-rt}G_{0}(x_{0},t)$
$\displaystyle+r\int_{0}^{t}e^{-r\tau}G_{0}(x_{0},\tau)G_{r}(x_{0},t-\tau)d\tau,$
(33)

Eq. (33) in Laplace space becomes

$\displaystyle\tilde{G}_{r}(x_{0},s)=\frac{\tilde{G}_{0}(x_{0},s+r)}{1-r\tilde{G}_{0}(x_{0},s+r)}.$
(34)

In order to find the FATD, $f(t)$, we note that it is related to the survival
probability via $f(t)=-\frac{\partial G(x_{0},t)}{\partial t}$, so, from Eq.
(34) the FATD with resetting in Laplace space reads

$\displaystyle\tilde{f}_{r}(s)=\frac{(s+r)\tilde{f}_{0}(s+r)}{s+r\tilde{f}_{0}(s+r)},$
(35)

where $\tilde{f}_{0}(s)$ is the first passage probability density without
resetting. In principle (35) can be transformed back to real space, via the
inverse Laplace transform. In Fig. 5 we present a particular case for the FATD
for case I and II in real space from simulation data. We have checked that the
FATD decays exponentially, though there is not a simple relation between the
decaying constant and the model’s parameters.

Figure 5: FATD in the presence of resetting for both case I (green circles)
and case II (blue triangles) obtained from numerical simulations. The dotted
lines are added as a visual guide. The parameters used being $D=v=1$,
$\tau_{D}=3.8$, $\tau_{v}=3.3$, $x_{0}=10$ and $r=0.1$.

We can further compute the MFAT $\left\langle T(x_{0})\right\rangle_{r}$ from
the first passage probability density

$\displaystyle\left\langle T(x_{0})\right\rangle_{r}$
$\displaystyle=\int_{0}^{\infty}tf_{r}(x_{0},t)dt=\tilde{G}_{r}(x_{0},s=0)$
$\displaystyle=\frac{\tilde{G}_{0}(x_{0},r)}{1-r\tilde{G}_{0}(x_{0},r)}=\frac{1-\tilde{f}_{0}(s=r)}{r\tilde{f}_{0}(s=r)}.$
(36)

Combining Eqs. (22) and (36) the expression of the MFAT for the case I is
given by

$\displaystyle\left\langle
T^{D}(x_{0})\right\rangle_{r}=\frac{1}{r}\left[\frac{\phi_{-}-\phi_{+}}{\phi_{-}e^{-|x_{0}|\lambda_{-}}-\phi_{+}e^{-|x_{0}|\lambda_{+}}}-1\right]$
(37)

where $\phi_{+}$, $\phi_{-}$, $\lambda_{+}$ and $\lambda_{-}$ are as in Eqs.
(19) and (21) but setting $s=r$. It can be seen that if the searcher does not
relocate ($\tau_{D}\rightarrow\infty$), i.e., only diffuses then
$\lambda_{+}\simeq\sqrt{r/D}$ and $\phi_{-}\simeq 0$. Therefore, (37) reduces
to the well-known case of Brownian walkers with stochastic resetting [30].
Analogously, for case II the MFAT reads

$\displaystyle\langle
T^{v}(x_{0})\rangle_{r}=\frac{1}{r}\bigg{[}\frac{(v\tau_{v})^{2}(\phi_{-}-\phi_{+})/(r\tau_{v}+1)}{\lambda_{-}^{-1}e^{-|x_{0}|\lambda_{-}}-\lambda_{+}^{-1}e^{-|x_{0}|\lambda_{+}}}-1\bigg{]}.$
(38)

We note that this result is different from that obtained in Eq. (37) which
reveals the explicit dependence of the MFAT on the initial condition, i.e., if
the walker is initially either relocating or diffusing. In this case, in the
limit of no relocation, $\tau_{D}\to\infty$, the MFAT goes to

$\left\langle
T^{v}(x_{0})\right\rangle\simeq\frac{1}{r}\left[\frac{\lambda_{+}^{2}-\lambda_{-}^{2}}{\lambda_{+}e^{-|x_{0}|\lambda_{-}}+\lambda_{-}e^{-|x_{0}|\lambda_{+}}}-1\right].$
(39)

This shows that the MFAT is dependent on the initial condition also in this
limit. It can easily be understood if noticing that $\tau_{D}\to\infty$ means
that the searcher only diffuses, so for case I the walker does not relocate.
It is not the case for case II, where the walkers relocate at rate
$\tau_{v}^{-1}$ before switching to diffusion state. In the opposite limit,
when $D=0$ the searcher only relocates i.e., it never detects the target so
that the MFAT (37) and (38) diverge. Effectively, if $D=0$ then
$\lambda_{\pm}\rightarrow\infty$ and $\left\langle
T(x_{0})\right\rangle_{r}\rightarrow\infty$. One may notice that while
$\left\langle T^{D}(x_{0})\right\rangle_{r}\to 0$ as $x_{0}\to 0$, $\langle
T^{v}(x_{0})\rangle_{r}$ tends to a constant value given by

$\langle
T^{v}(x_{0})\rangle_{r}\simeq\frac{1}{r}\bigg{[}\frac{(v\tau_{v})^{2}}{r\tau_{v}+1}\frac{(\phi_{-}-\phi_{+})\lambda_{+}\lambda_{-}}{\lambda_{+}-\lambda_{-}}-1\bigg{]}$
(40)

in the same limit. This difference arises because for case II the walkers do a
relocation first (during which target detection is not possible), so in
general this is detrimental in order to find the target for $x_{0}\to 0$. More
interesting is the case for $x_{0}\to\infty$, which satisfies

$\frac{\langle T^{D}(x_{0})\rangle_{r}}{\langle
T^{v}(x_{0})\rangle_{r}}\simeq\frac{r\tau_{v}+1}{(v\tau_{v})^{2}}\frac{1}{\phi_{-}\lambda_{-}}.$
(41)

In general, $\langle T^{D}(x_{0})\rangle_{r}\neq\langle
T^{v}(x_{0})\rangle_{r}$ showing that the effect of the initial condition is
not smoothed out by a long time evolution. As we have mentioned this behavior
can be understood because, in the presence of resetting, the walker is
consistently returned to its initial condition during the time evolution.

Figure 6: MFAT in the presence of resetting for both case I (green circles)
and case II (blue triangles). In solid lines we show Eq. (37) and Eq. (38) and
in dots data obtained from numerical simulations. The parameters used being
$D=v=1$, $\tau_{D}=3.8$, $\tau_{v}=3.3$, $x_{0}=10$ and $r=0.1$.

###  III.4 Existence of an optimal reset rate

[h!]

Figure 7: MFAT vs $r$ in the presence of resetting for different parameter
values. In panels a), b) and c) case I is represented, the MFAT computed with
Eq. (37); Panels d), e) and f) are for case II, its MFAT computed with Eq.
(38). The parameters used are $D=v=1$, $\tau_{D}=3.8$, $\tau_{v}=3.3$ and
$x_{0}=1$ unless stated the contrary. In panels a) and d) the purple curve is
for $\tau_{v}=0.3$, the blue one for $\tau_{v}=3$ and the yellow one for
$\tau_{v}=10$. For panels b) and e) the purple curve is for $\tau_{v}=0.3$,
the blue one for $\tau_{v}=3$ and the yellow one for $\tau_{v}=10$. In panels
c) and f) the purple curve is for $x_{0}=0.5$, the blue one for $x_{0}=1$ and
the yellow one for $x_{0}=2$. All axes are on a logarithmic scale.

In Fig. 6 we compare Eq. (37) and Eq. (38) to numerical simulations, so we can
observe the emergence of an optimal value of $r$ minimizing the MFAT of the
IRW, as happens for the case of Brownian walkers too [30]. In the absence of
resetting the walker takes an infinite time to reach the target due to the
non-recurrent nature of Brownian particles. On the other hand, for large reset
rates the walker’s motion is localized near the reset point, also resulting in
a very large MFAT. To explore analytically the existence of this optimal rate,
we analyze Eq. (37) both for small and large $r$. Let us first expand the MFAT
for large $r$. From Eqs. (17) and (18) we have $B(r)\simeq r^{2}/v^{2}$ and
$C(r)\simeq r^{3}/Dv^{2}$ so that

$\displaystyle\lambda_{+}\simeq
r/v,\;\lambda_{-}\simeq\sqrt{r/D}\quad\text{as}\quad r\rightarrow\infty$ (42)

Finally,

$\displaystyle\left\langle
T^{D}(x_{0})\right\rangle_{r}\simeq\frac{1}{r}e^{|x_{0}|\sqrt{r/D}}\quad\text{as}\quad
r\rightarrow\infty$ (43)

so that the MFAT tends to infinity as $r$ does. This is the same behaviour as
in the absence of resetting. To prove that the optimal reset rate exists we
just need to verify if the MFAT decreases with $r$ for small values of $r$,
this is, if the condition

$\displaystyle\lim_{r\rightarrow 0}\left(\frac{\partial\left\langle
T^{D}(x_{0})\right\rangle}{\partial r}\right)<0$ (44)

is met. Expanding $\lambda_{\pm}$ for small $r$ we find

$\displaystyle\lambda_{+}\simeq\lambda_{0}+O(r),\quad\lambda_{-}\simeq\lambda_{1}\sqrt{r}+O(r),$
(45)

where

$\lambda_{0}=\sqrt{\frac{1}{D\tau_{D}}+\frac{1}{(v\tau_{v})^{2}}},\quad\lambda_{1}=\frac{\tau_{D}+\tau_{v}}{(v\tau_{v})^{2}+D\tau_{D}}.$

After some calculations, the dominant term in the expansion of the MFAT for
small $r$ reads

$\displaystyle\left\langle
T^{D}(x_{0})\right\rangle\simeq\frac{T_{0}}{\sqrt{r}}\quad\text{as}\quad
r\rightarrow 0,$ (46)

where

$T_{0}=\lambda_{1}(v\tau_{v})^{2}\left[\left(e^{-|x_{0}|\lambda_{0}}-1\right)\left(\frac{1}{\lambda_{0}(v\tau_{v})^{2}}-\lambda_{0}\right)+\frac{|x_{0}|}{(v\tau_{v})^{2}}\right].$

Finally, the condition (44) applied to (46) is equivalent to $T_{0}>0$, or
simply

$\frac{|x_{0}|\lambda_{0}}{1-e^{-|x_{0}|\lambda_{0}}}>-\frac{(v\tau_{v})^{2}}{D\tau_{D}},$

which is always fulfilled. This confirms the existence of the optimal reset
rate.

One can also prove that (38) has an optimal reset rate following an analogous
procedure. In the limit of large reset rate $r$, using (42) we find that

$\displaystyle\left\langle
T^{v}(x_{0})\right\rangle\simeq\tau_{r}e^{|x_{0}|\sqrt{r/D}}\quad\text{as}\quad
r\rightarrow\infty.$ (47)

Analogously, using (45) we find that when $r\rightarrow 0$ the MFAT is given
by (38) and behaves as in (46) but with

$T_{0}=\frac{\lambda_{1}}{\lambda_{0}}\left[\frac{(v\tau_{v})^{2}}{D\tau_{D}}+\lambda_{0}|x_{0}|+e^{-|x_{0}|\lambda_{0}}\right].$

Since $T_{0}$ is positive, the optimal reset rate also exists in this case. In
Fig. 7 we present some illustrative cases of the dependence of the MFAT with
$r$ computed from Eq. (37) and Eq. (38) for case I and case II, respectively,
to check the existence of the optimal value of $r$. Though we are not able to
find a compact relationship between the model parameters and the minimal MFAT,
in the next Section we identify numerically the existence of different regime
behaviours of the MFAT.

###  III.5 Phase space parameter exploration

Figure 8: Panel a) Phase diagram of the behavior of MFAT in $(x_{0},r)$ space
for the diffusive initial condition (case I). Panels b), c), and d) are
representative examples of the MFAT in each phase and the interphase in
$(\tau_{D},\tau_{v})$ space for a given $x_{0}$ and $r$. The Roman number
shows their corresponding position in the phase diagram. In panel b)
$x_{0}=50$, $r=0.05$; In panel c) $x_{0}=7.9$, $r=0.05$; And in panel d)
$x_{0}=0.05$, $r=0.05$. For all plots $D=1$ and $v=1$. Figure 9: Panel a)
Phase diagram of the behavior of MFAT in $(x_{0},r)$ space for the relocation
initial condition (case I). Panels b), c), and d) are representative examples
of the MFAT in each phase in $(\tau_{D},\tau_{v})$ space for a given $x_{0}$
and $r$. The Roman number shows their corresponding position in the phase
diagram. In panel b) $x_{0}=50$, $r=0.002$; In panel c) $x_{0}=50$, $r=0.3$;
And in panel d) $x_{0}=0.02$, $r=0.03$. For all plots $D=1$ and $v=1$.

We have numerically studied the MFAT behavior for distinct parameter values
$\tau_{D}$, $\tau_{v}$, $x_{0}$, and $r$. We have kept $v=1$ and $D=1$ (note
that one may change $x$ and $t$ to nondimensional variables $x\to
x\frac{v}{D}$ and $t\to t\frac{v^{2}}{D}$). This allows us to focus the study
on the most interesting regime in which ballistic and diffusive movements are
in direct competition.

We have centered our study on finding the combination of parameter values that
globally minimize the MFAT. The analysis was conducted in the following
manner: for a set of $(\tau_{D},\tau_{v})$ in the range from $10^{-4}$ to
$10^{7}$ we used Eq. (37) and Eq. (38) to compute the MFAT for different
combinations within the ranges $r\in[10^{-3},10^{0}]$ and
$x_{0}\in[10^{-2},10^{2}]$.

For the diffusive initial condition, two distinct phases can be observed
(distinguished by the colors red and cyan in Fig. 8a). For the phase in cyan,
which we name the localized phase, the MFAT presents a well-localized minimum
in the $(\tau_{D},\tau_{v})$ phase space. This can be better identified in
Fig. 8b, where we directly show for a specific choice of $(x_{0},r)$ values
that the minimum value lies in a confined region of the space
($\tau_{D},\tau_{v}$). On the other hand, for $x_{0}$ small we obtain the red
(non-localized) phase, for which the MFAT does not have a well-defined minimum
but rather there exists a plateau region of $(\tau_{D},\tau_{v})$ values for
which the MFAT becomes minimum. In particular, any combination of
$(\tau_{D},\tau_{v})$ out of the region $\tau_{v}\gg\tau_{D}$ will suffice to
optimize the MFAT, as can be seen for the case depicted in Fig.8d. For the
sake of completeness, in panel 8c we show the behavior of the MFAT for the
interface between both (localized and non-localized) phases, where we observe
the transition from one behavior to the other.

In Fig. 9 we present the same analysis for the relocation initial condition,
case II. In this case, we observe three different phases. Apart from the
localized and non-localized phases above, we observe an intermediate situation
(which we name ballistic localization, in light yellow in Fig. 9a). For this
region, the minimum of the MFAT is confined to a fine line of
$(\tau_{D},\tau_{v})$ values where $\tau_{v}$ is (regardless of the value of
$\tau_{D}$) at least of the order of the timescale $x_{0}/v$, which
corresponds to the case where the target can be reached on average in a single
relocation. Similarly to case I, in panels b-d of Fig. 9 we show the behavior
of the MFAT for the three different phases. Remarkably, the behavior for the
non-localized phase is slightly different in that now the region where the
minimum MFAT occurs does not include the region where $\tau_{v}$ is large.
This is because for $x_{0}$ small a very large value of $\tau_{v}$ now implies
that the walker will depart on average much from the target position before
switching to diffusive motion when the target can be detected.

In parallel, we have studied how the values $\tau_{D},\tau_{v}$ which minimize
the MFAT depend on the model parameters $x_{0},r$ in the localized phase for
both initial conditions. The corresponding numerical results are presented in
Figs. 10 and 11 as a function of $x_{0}$ and $r$. While we have tried, it has
not been possible to detect any straightforward scaling relation that is
satisfied in the region of interest apart from the conditions stated in the
discussion above. This is due to the several timescales (resetting,
relocation, diffusive), all of them of a similar order, directly competing in
the process.

Figure 10: Plots of $\tau_{D}$ and $\tau_{v}$ that minimize the MFAT for the
localized phase of case I. The MFAT has been computed with Eq. (37). Panels a)
and b) represent $\tau_{D}$ and $\tau_{v}$ respectively as a function of $r$,
the different curves for different values of
$x_{0}=9.1,$$11.4,$$14.4,$$18.2,$$23.0,$$29.0,$$36.6,$$46.1,$$58.2,$$73.4,$
and $92.5$ from purple to yellow. Panels c) and d) represent $\tau_{D}$ and
$\tau_{v}$ respectively as a function of $x_{0}$, the different curves being
for different values of $r$, again, the purple is for smaller values of $r$
and yellow for larger ones. The $r$ represented are
$r=0.001,$$0.0018,$$0.0035,$$0.0068,$$0.012,$$0.024,$$0.046,$$0.088,$$0.16,$$0.31,$
and $0.6$. The dotted line on all panels signals the transition to the other
phase. All axes are on a logarithmic scale. Figure 11: Plots of $\tau_{D}$ and
$\tau_{v}$ that minimize the MFAT for the localized phase of case II. The MFAT
has been computed with Eq. (38). Panels a) and b) represent $\tau_{D}$ and
$\tau_{v}$ respectively as a function of $r$, the different curves for
different values of $x_{0}=9.1,$$11.4,$$14.4,$$18.2,$$23.0,$
$29.0,$$36.6,$$46.1,$$58.2,$$73.4,$ and $92.5$ from purple to yellow. Panels
c) and d) represent $\tau_{D}$ and $\tau_{v}$ respectively as a function of
$x_{0}$, the different curves being for different values of $r$, again, the
purple is for smaller values of $r$ and yellow for larger ones. The $r$
represented are $0.001,$$0.0014,$$0.002,$$0.003,$
$0.0043,$$0.0063,$$0.0091,$$0.013,$$0.019,$$0.027,$ and $0.04$. The dotted
line on all panels signals the transition to the other phase. All axes are on
a logarithmic scale.

##  IV Conclusions

We analyzed a two-state model for a one-dimensional intermittent random walk
subject to stochastic resetting at a constant rate. One state corresponds to
diffusive motion, while the other represents a relocating phase where the
walker moves ballistically with a constant velocity in either the right or
left direction. We consider the possibility that the walker may be initially
diffusing (case I) or relocating (case II). We studied the MSD and the FAT for
this process first without resetting and observed that, in the unbounded
domain, the MFAT diverges.

We then studied the system with resetting and demonstrated the existence of a
NESS. In addition, we computed the FATD and we proved that its mean is finite
in the system with resetting and that there exists a value of the reset rate
that, given all the other parameters of the model, minimizes the MFAT. Our
study of the NESS, FATD, and MFAT for both cases I and II shows that they
differ depending on the initial condition of the walker.

We argue that this behavior occurs because, in the presence of resetting, the
initial condition is consistently revisited, which prevents the smoothing out
of its effect even in the long-term limit. Thus, our results demonstrate the
importance of the initial condition (whether the individual starts diffusing
or relocating) for the search dynamics.

Next, we conducted a numerical analysis of the MFAT, focusing on determining
the parameter combination that maximizes its global optimization in the regime
where ballistic and diffusive movements directly compete. For case I, we
observed two distinct behaviors. There is a localized phase in which the MFAT
exhibits a well-localized minimum in $(\tau_{D},\tau_{v})$ phase space. On the
other hand, for small $x_{0}$ or large $r$ there is a non-localized phase in
which a plateau region of values $(\tau_{D},\tau_{v})$ optimizes the MFAT. In
this phase, the specific combination of $(\tau_{D},\tau_{v})$ is not relevant
for optimizing the search. In the localized phase, however, it is. For case II
a third intermediate phase arises, ballistic localization. Here the minimum of
the MFAT is restricted to a fine line region of $(\tau_{D},\tau_{v})$ values,
corresponding to the situation where the target can be reached on average in a
single relocation. The existence of the localized phase for a non-trivial
combination of $(\tau_{D},\tau_{v})$ reveals that having an unresponsive
relocation phase is beneficial for the optimization of the MFAT of the search
process for 1D IRW in unbounded domains in the presence of resetting in
agreement with [22, 24, 25, 28].

The analysis presented in this work contributes to the existing literature on
search strategies and provides a foundation for future research. Recently, it
has been observed that some kinds of birds combine the relocating mode with a
subdiffusive detection mode [34]. Therefore, extending our IRW model in this
direction in the future would be of great value. Other recent studies have
focussed on the energetic cost of resetting [35, 36, 37, 38], in these lines a
worthwhile study of our model is to consider not only the cost of resetting
but also the cost of switching phases or considering a phase more energetic
than the other. In this case, the optimization of the MFAT would not suffice
to characterize the efficiency of the search strategy, but what has to be
optimized is the total cost of the exploration. Moreover, we note that we have
presented a model with instantaneous resetting, while, for many physical and
biological systems, it would be more realistic to consider it non-
instantaneous such as in [39, 40, 41, 42, 43, 44]. In the present work we
wanted to highlight the role of the trade-off between the timescales diffusion
and relocation and its interplay with resetting. Thus, we chose the
instantaneous resetting framework, which is more understood in the literature
to put forward these results as we considered that adding a non-instantaneous
reset would also introduce an extra timescale which could obscure the role of
the other timescales. For future works, we aim to consider it non-
instantaneous and see how these results may change, we expect a similar
qualitative behavior for walkers resetting at constant speed. In a similar
line, in our study we considered both the diffusive phase and the ballistic
phase as part of the search process, following the approach of O. Bénichou et
al in [9]. Hence, resetting can occur in either phase, restarting the search
process as a whole. However, for some applications, it may be useful to modify
this model so that resetting only occurs during the diffusive state and see
how the optimization of the MFAT changes.

## Acnowledgements

The Authors acknowledge the financial support of the Spanish government under
grant PID2021-122893NB-C22.

##  Appendix A Random initial condition

If one considers that the walker can start in either state, diffusion or
relocation, randomly then: $\bar{\rho}_{0}(k,0)=e^{-ikx_{0}}/2$ and
$\bar{\rho}_{+}(k,0)=\bar{\rho}_{-}(k,0)=e^{-ikx_{0}}/4$. From Eq. (4) the
propagator for the diffusing phase $\hat{\rho}^{R}_{0}(k,s)$, reads:

$\hat{\rho}^{R}_{0}(k,s)=\frac{\hat{\rho}^{D}_{0}(k,s)}{2}+\frac{\hat{\rho}^{v}_{0}(k,s)}{2}$
(48)

where the superscript $R$ means random initial condition. similarly the
propagator for the complete process:

$\hat{\rho}^{R}(k,s)=\frac{\hat{\rho}^{D}(k,s)}{2}+\frac{\hat{\rho}^{v}(k,s)}{2}$
(49)

Therefore, the MSD, FAT PDF with and without resets, and steady state are
simply linear combinations of cases I and II:

$\left\langle x^{2}(t)\right\rangle^{R}=\frac{\left\langle
x^{2}(t)\right\rangle^{D}}{2}+\frac{\left\langle
x^{2}(t)\right\rangle^{v}}{2}$ (50)

$\tilde{f}^{R}(x_{0},s)=\frac{\tilde{f}^{D}(x_{0},s)}{2}+\frac{\tilde{f}^{v}(x_{0},s)}{2}$
(51)

$\rho_{r}^{(s),R}(x)=\frac{\rho_{r}^{(s),D}(x)}{2}+\frac{\rho_{r}^{(s),v}(x)}{2}$
(52)

On the other hand, From Eq. (36) and Eq. (51) the MFAT

$\displaystyle\left\langle
T^{R}(x_{0})\right\rangle_{r}=\frac{1}{r}\left[\frac{2(\phi_{-}-\phi_{+})}{\sigma_{-}e^{-|x_{0}|\lambda_{-}}-\sigma_{+}e^{-|x_{0}|\lambda_{+}}}-1\right]$
(53)

where
$\sigma_{\pm}=\frac{\left(1+r\tau_{v}\right)\left(2+r\tau_{v}\right)}{(v\tau_{v})^{2}\lambda_{\pm}}-\lambda_{\pm}$.
Then, we can see that the MFAT cannot be written as a linear combination of
cases I and II. For this case, we have performed a similar parameter
exploration to the one done in the section III.5 and we have found the same
qualitative behavior of the MFAT to the one of case II.

##  Appendix B Coefficients MSD

If the walker starts at the diffusive state (case I), the coefficients of the
Mean Squared Displacement, Eq. (9)

$\displaystyle A_{1}$
$\displaystyle=2\frac{\tau_{v}^{2}}{\tau_{D}}\frac{D-2\tau_{v}v^{2}-\frac{(\tau_{v}v)^{2}}{\tau_{D}}}{(1+\frac{\tau_{v}}{\tau_{D}})^{2}},$
(54) $\displaystyle A_{2}$ $\displaystyle=2(\tau_{v}v)^{2},$ (55)
$\displaystyle A_{3}$
$\displaystyle=-2\frac{\tau_{v}^{2}}{\tau_{D}}\frac{(D+\tau_{D}v^{2})}{(1+\frac{\tau_{v}}{\tau_{D}})^{2}}.$
(56)

and if it is initially at the relocation state (case II),

$\displaystyle A_{1}$
$\displaystyle=2\frac{{\tau_{v}}^{2}(v^{2}-\frac{D}{\tau_{v}})-\frac{{\tau_{v}}^{3}v^{2}}{\tau_{D}}(1+\frac{\tau_{v}}{\tau_{D}})}{(1+\frac{\tau_{v}}{\tau_{D}})^{2}},$
(57) $\displaystyle A_{2}$ $\displaystyle=2\tau_{v}v^{2}(\tau_{v}-\tau_{D}),$
(58) $\displaystyle A_{3}$
$\displaystyle=2\tau_{v}\frac{(D+\tau_{D}v^{2})}{(1+\frac{\tau_{v}}{\tau_{D}})^{2}}.$
(59)

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∗Correspondence<EMAIL_ADDRESS>Tel.: +52-222-266-3100

# Quasiprobability distribution functions from fractional Fourier transforms

Jorge A. Anaya-Contreras 1,A. Zúñiga-Segundo1,and Héctor M. Moya-
Cessa${}^{4}*$ 1 Instituto Politécnico Nacional, ESFM, Departamento de Física.
Edificio 9, Unidad Profesional “Adolfo López Mateos, CP 07738 Cd. de México,
Mexico
2 Instituto Nacional de Astrofísica, Óptica y Electrónica, Calle Luis Enrique
Erro 1, Santa María Tonantzintla, Pue., 72840 Mexico

###### Abstract

We show, in a formal way, how a class of complex quasiprobability distribution
functions may be introduced by using the fractional Fourier transform. This
leads to the Fresnel transform of a characteristic function instead of the
usual Fourier transform. We end the manuscript by showing a way in which the
distribution we are introducing may be reconstructed by using atom-field
interactions.

####

Keywords: Quasiprobability distribution functions; Fractional Fourier
transform; Reconstruction of the wave function.

##  1 Introduction

It has been already shown that quasiprobability distribution functions may be
reconstructed by the measurement of atomic properties in ion-laser
interactions [1] and two-level atoms interacting with quantized fields [2, 3].
Such measurements of the wave function are realized usually by measuring
atomic observables, namely, the atomic inversion and polarization.

Although the first quasiprobability distribution functions were introduced in
the quantum realm [4, 5, 6, 8, 7], they may be also used to analyze classical
signals [9, 10].

Ideal interactions, i.e., without taking into account an environment, have
shown to lead to the reconstruction of the Wigner function [3] by taking
advantage of its expression in terms of the parity operator. However, the
interaction of a system with its environment [11] leads to $s$-prametrized
quasiprobability distribution functions [12, 13, 14]

$F(\alpha,s)=\frac{2}{\pi(1-s)}\sum_{k=0}^{\infty}\left(\frac{s+1}{s-1}\right)^{k}\langle
k|\hat{D}^{\dagger}(\alpha)\hat{\rho}\hat{D}(\alpha)|k\rangle$ (1)

where $\hat{D}(\alpha)=\exp(\alpha\hat{a}^{\dagger}-\alpha^{*}\hat{a})$, with
$\hat{a}$ and $\hat{a}^{\dagger}$ the annihilation and creation operators of
the harmonic oscillator, respectively, is the Glauber displacement operator
[15]. The state $\hat{D}(\alpha)|k\rangle=|\alpha,k\rangle$ is a so-called
displaced number state [16]. Note that, in order to reconstruct a given
quasiprobability function it is needed to do displace the system by an
amplitude $\alpha$ and then measure the diagonal elements of the displaced
density matrix.

The parameter $s$ defines different orderings and therefore different
quasiprobability distribution functions (QDF). The Glauber-Sudarshan
$P$-function [15, 17] is given for $s=1$, and is used to obtain averages of
functions of normal ordered creation and annihilation operators; $s=-1$ gives
the Husimi $Q$-function, used to obtain averages of functions of anti-normal
ordered creation and annihilation operators, while $s=0$ is used for the
symmetric ordering and gives the Wigner function.

Equation (1) may be rewritten as

$F(\alpha,s)=\frac{2}{\pi(1-s)}Tr\left\\{\left(\frac{s+1}{s-1}\right)^{\hat{a}^{\dagger}\hat{a}}\hat{D}^{\dagger}(\alpha)\hat{\rho}\hat{D}(\alpha)\right\\},$
(2)

that, by using the commutation properties under the symbol of trace, and the
system in a pure state $|\psi\rangle$, may be casted into

$\displaystyle F(\alpha,s)$ $\displaystyle=$
$\displaystyle\frac{2}{\pi(1-s)}Tr\left\\{\hat{D}(\alpha)\left(\frac{s+1}{s-1}\right)^{\hat{a}^{\dagger}\hat{a}}\hat{D}^{\dagger}(\alpha)\hat{\rho}\right\\}$
(3) $\displaystyle=$
$\displaystyle\frac{2}{\pi(1-s)}\langle\psi|\hat{D}(\alpha)\left(\frac{s+1}{s-1}\right)^{\hat{a}^{\dagger}\hat{a}}\hat{D}^{\dagger}(\alpha)|\psi\rangle.$

Recent studies have openned the possibility of measuring, instead of
observables, non-Hermitian operators [18]. It would be plausible then that
such measurements could be related to complex quasiprobability distributions
like the McCoy-Kirkwood-Rihaczek-Dirac distribution functions [5, 6, 8, 19].

In this contribution we would like to introduce other kind of complex
quasiprobabilities that, although they could be introduced simply by taking
$s$ as a complex number, we introduce them in a formal way by considering the
fractional Fourier transform (FrFT) [20, 21, 22] of a signal. Then, by writing
the Dirac-delta function in terms of its FrFT, we are able to write a general
expression for complex quasiprobability distributions in terms of the Fresnel
transform. Indeed, the representation of these complex quasiprobability
distributions in terms of a Fresnel transform implies that they are solutions
of a paraxial wave equation [3]. Finally, by using an effective Hamiltonian
for the atom-field interaction, we show how this quasiprobability distribution
function may be reconstructed.

##  2 Fractional Fourier Transform

Up to a phase, the fractional Fourier Transform of a signal $\psi(x)$ can be
written by the following expression [20, 21, 22]

$\mathcal{F}_{\omega}\left[\psi(x)\right]=\exp\left(-i\omega\hat{a}^{\dagger}\hat{a}\right)\psi(x)\;,$
(4)

that may be expressed in terms of an integral transform as

$\mathcal{F}_{\omega}\left[\psi(x)\right]=\int_{-\infty}^{+\infty}dx^{\prime}K(x,x^{\prime};\omega)\psi(x^{\prime}),$
(5)

where

$K(x,x^{\prime};\omega)=\frac{1}{\sqrt{2\pi
i}}\sqrt{\frac{e^{i\omega}}{\sin\omega}}\exp\left[i\frac{x^{2}}{2}\cot\omega+i\frac{x^{\prime
2}}{2}\cot\omega-ixx^{\prime}\csc\omega\right]\,.$ (6)

Then, if we consider equation (6) as a propagator, Dirac’s delta distribution
function takes the form

$\displaystyle\delta(x-x^{\prime})$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}dx^{\prime\prime}K(x,x^{\prime\prime};-\omega)K(x^{\prime\prime},x^{\prime};\omega)$
(7) $\displaystyle=$ $\displaystyle\frac{1}{2\pi\sin\omega}e^{i\frac{x^{\prime
2}}{2}\cot\omega-i\frac{x^{2}}{2}\cot\omega}\int_{-\infty}^{+\infty}dx^{\prime\prime}e^{ix^{\prime\prime}(x-x^{\prime})\csc\omega}$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi}\exp\left[i\frac{x^{\prime
2}}{2}\cot\omega-i\frac{x^{2}}{2}\cot\omega\right]\int_{-\infty}^{+\infty}dx^{\prime\prime}\exp\left[ix^{\prime\prime}(x-x^{\prime})\right]\,.$

Now, if we apply the fractional Fourier transform to the Dirac delta function
we obtain

$\mathcal{F}_{\omega}\left[\delta(x-y)\right]=\int_{-\infty}^{+\infty}dx^{\prime}K(x,x^{\prime};\omega)\delta(x^{\prime}-y)=K(x,y;\omega).$
(8)

Then, applying the inverse fractional Fourier transform to equation (8) we
obtain an alternative representation of the Dirac delta distribution function

$\displaystyle\delta(x-y)$ $\displaystyle=$
$\displaystyle\mathcal{F}_{-\omega}\left[\mathcal{F}_{\omega}\left[\delta(x-y)\right]\right]=\int_{-\infty}^{+\infty}dx^{\prime\prime}K(x,x^{\prime\prime};-\omega)K(x^{\prime\prime},y;\omega)$
(9) $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\exp\left[i\frac{y^{2}}{2}\cot\omega-i\frac{x^{2}}{2}\cot\omega\right]\int_{-\infty}^{+\infty}dx^{\prime}\exp\left[ix^{\prime}(x-y)\right]\,.$

From the above equation it may be seen that there is a phase multiplying the
usual integral representation of the Dirac delta function, that although could
be omitted by using properties of the delta function, we keep in order to
obtain a quasiprobability distribution function as a fractional Fourier
(Fresnel) transform of the characteristic function.

##  3 Probability distribution in the phase space

We define $J(q,p)$ a probability distribution in the phase space as

$J(q,p)=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}dq^{\prime}dp^{\prime}\mathcal{P}(q^{\prime},p^{\prime})\delta(q^{\prime}-q)\delta(p-p^{\prime})\,,$
(10)

then, using equation (9), the distribution $J(q,p)$ may be rewritten as

$\displaystyle J(q,p)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}}e^{i\frac{q^{2}}{2}\cot\alpha-i\frac{p^{2}}{2}\cot\beta}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}dudv\,e^{iup-
ivq}\,\times$ (11) $\displaystyle\times$
$\displaystyle{Tr}\left\\{\hat{\rho}e^{iv\hat{q}-iu\hat{p}+i\frac{\hat{p}^{2}}{2}\cot\beta-i\frac{\hat{q}^{2}}{2}\cot\alpha}\right\\}\,,$

and because

$\displaystyle
e^{iv\hat{q}-iu\hat{p}+i\frac{\hat{p}^{2}}{2}\cot\beta-i\frac{\hat{q}^{2}}{2}\cot\alpha}$
$\displaystyle=$ $\displaystyle
e^{-i\frac{u^{2}}{2}\tan\beta}e^{i\frac{v^{2}}{2}\tan\alpha}e^{iu\tan\beta\hat{q}}e^{-iv\tan\alpha\hat{p}}\times$
(12) $\displaystyle\times$ $\displaystyle
e^{i\frac{\hat{p}^{2}}{2}\cot\beta-i\frac{\hat{q}^{2}}{2}\cot\alpha}e^{iv\tan\alpha\hat{p}}e^{-iu\tan\beta\hat{q}}\,,$

equation (11) takes the form

$\displaystyle J(q,p)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}}e^{i\frac{q^{2}}{2}\cot\alpha-i\frac{p^{2}}{2}\cot\beta}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}dudv\,e^{iup-
ivq}e^{-i\frac{u^{2}}{2}\tan\beta}e^{i\frac{v^{2}}{2}\tan\alpha}\,\times$ (13)
$\displaystyle\times$ $\displaystyle
Tr\left\\{\hat{\rho}e^{iu\tan\beta\hat{q}}e^{-iv\tan\alpha\hat{p}}e^{i\frac{\hat{p}^{2}}{2}\cot\beta-i\frac{\hat{q}^{2}}{2}\cot\alpha}e^{iv\tan\alpha\hat{p}}e^{-iu\tan\beta\hat{q}}\right\\}\,,$

that by using the equivalence

$e^{iu\tan\beta\hat{q}}e^{-iv\tan\alpha\hat{p}}=e^{\frac{i}{2}uv\tan\alpha\tan\beta}e^{iu\tan\beta\hat{q}-iv\tan\alpha\hat{p}}\,,$
(14)

may be casted into the expression

$\displaystyle J(q,p)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}}\,e^{i\frac{q^{2}}{2}\cot\alpha-i\frac{p^{2}}{2}\cot\beta}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}dudv\,e^{iup-
ivq}e^{-i\frac{u^{2}}{2}\tan\beta}e^{i\frac{v^{2}}{2}\tan\alpha}\,\times$ (15)
$\displaystyle\times$ $\displaystyle
Tr\left\\{\hat{\rho}e^{iu\tan\beta\hat{q}-iv\tan\alpha\hat{p}}e^{i\frac{\hat{p}^{2}}{2}\cot\beta-i\frac{\hat{q}^{2}}{2}\cot\alpha}e^{iv\tan\alpha\hat{p}-iu\tan\beta\hat{q}}\right\\}\,.$

###  3.1 Case $\cot\alpha=-\cot\beta=\pi$

The above quasiprobability distribution function is defined for a range of
parameters $\alpha$ and $\beta$, however, for the sake of simplicity, we will
consider the case $\cot\alpha=-\cot\beta=\pi$.

We may relate the quasiprobability distribution function $J(q,p)$ to the
Wigner function, by noting that, for $\cot\alpha=-\cot\beta=\pi$, equation
(15) has the form

$\displaystyle J(q,p)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}i}e^{i\pi\left(\frac{p^{2}}{2}+\frac{q^{2}}{2}\right)}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}dudv\,e^{iup-
ivq}e^{i\frac{u^{2}}{2\pi}+i\frac{v^{2}}{2\pi}}\times$ (16)
$\displaystyle\times$ $\displaystyle
Tr\left\\{\hat{\rho}e^{-i\frac{u\hat{q}}{\pi}-i\frac{v\hat{p}}{\pi}}(-1)^{\hat{n}}e^{i\frac{u\hat{q}}{\pi}+i\frac{v\hat{p}}{\pi}}\right\\}.$

According to trace representation of Wigner function [14]

$W\left(\frac{v}{\pi},-\frac{u}{\pi}\right)=Tr\left\\{\hat{\rho}\,\frac{1}{\pi}\,e^{-i\frac{u\hat{q}}{\pi}-i\frac{v\hat{p}}{\pi}}(-1)^{\hat{n}}e^{i\frac{u\hat{q}}{\pi}+i\frac{v\hat{p}}{\pi}}\right\\},$
(17)

we write the distribution $J(q,p)$ as the Fresnel transform of the Wigner
function

$J(q,p)=\frac{1}{4\pi
i}e^{i\pi\left(\frac{p^{2}}{2}+\frac{q^{2}}{2}\right)}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}dudv\,e^{iup-
ivq}e^{i\frac{u^{2}}{2\pi}+i\frac{v^{2}}{2\pi}}W\left(\frac{v}{\pi},-\frac{u}{\pi}\right)\,.$
(18)

It is easy to show that the quasiprobability distribution (18) can be
normalized

$\displaystyle\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}dqdp\,J(q,p)$
$\displaystyle=\frac{\pi}{2}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}dxdy\,\left[\frac{1}{2\pi}\int_{-\infty}^{+\infty}dpe^{ixp}\right]\left[\frac{1}{2\pi}\int_{-\infty}^{+\infty}dqe^{-iyq}\right]e^{-i\frac{x^{2}}{2\pi}-i\frac{y^{2}}{2\pi}}Tr\left\\{\hat{\rho}\,e^{iy\hat{q}-ix\hat{p}}\right\\}$
$\displaystyle=\frac{\pi}{2}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}dxdy\,\delta(x)\delta(y)e^{-i\frac{x^{2}}{2\pi}-i\frac{y^{2}}{2\pi}}Tr\left\\{\hat{\rho}\,e^{iy\hat{q}-ix\hat{p}}\right\\}=\frac{\pi}{2}Tr\left\\{\hat{\rho}\right\\}=\frac{\pi}{2}\,.$
(19)

Therefore, for normalization reasons, the quasiprobability distribution is
finally given in the form

$J(q,p)=\frac{1}{4\pi^{2}}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}du\,dv\,e^{iup-
ivq}e^{-i\frac{u^{2}}{2\pi}-i\frac{v^{2}}{2\pi}}Tr\left\\{\hat{\rho}\,e^{iv\hat{q}-iu\hat{p}}\right\\}\,,$
(20)

that, by applying the change of variables $\beta=u/\sqrt{2}+iv/\sqrt{2}$ takes
the form

$J(\alpha)=\frac{1}{2\pi^{2}}\int
d^{2}\beta\,e^{\alpha\beta^{*}-\alpha^{*}\beta}e^{-\frac{i}{\pi}|\beta|^{2}}Tr\left\\{\hat{\rho}\,\hat{D}(\beta)\right\\}\,,$
(21)

with $\alpha=q/\sqrt{2}+ip/\sqrt{2}$.

From the above expression it is direct to show that the Wigner function

$W(\alpha)=\int d^{2}\beta
e^{\alpha\beta^{*}-\alpha^{*}\beta}\\{\hat{\rho}\hat{D}(\beta)\\},$ (22)

and the function $J(\alpha)$ may be easily related by the differential
relation

$J(\alpha)=\exp\left\\{\frac{i}{\pi}\frac{\partial^{2}}{\partial\alpha\partial\alpha^{*}}\right\\}W(\alpha).$
(23)

The above quasiprobability function may be written as a trace by noting that

$\frac{1}{2\pi^{2}}\int
d^{2}\beta\,\exp\left(-\frac{i}{\pi}|\beta|^{2}\right)\hat{D}(\beta)=\frac{1}{2i+\pi}\left(\frac{2i-\pi}{2i+\pi}\right)^{\hat{n}}$
(24)

that leads to the trace representation of $J(q,p)$

$J(q,p)=\frac{1}{2i+\pi}Tr\left\\{\hat{\rho}\,\hat{D}(\alpha)\,\left(\frac{2i-\pi}{2i+\pi}\right)^{\hat{n}}\,\hat{D}^{\dagger}(\alpha)\right\\}\,.$
(25)

Last equation allows us to show that $J(q,p)$ is correctly normalized, for
this we do the double integration

$\displaystyle\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}J(q,p)dqdp$
$\displaystyle=$ $\displaystyle
Tr\left\\{\hat{\rho}\frac{2}{\pi+2i}\int\,d^{2}\alpha\hat{D}(\alpha)\hat{D}^{\dagger}\left(\alpha
e^{i\theta}\right)e^{i\theta\hat{n}}\right\\}$ (26) $\displaystyle=$
$\displaystyle Tr\left\\{\hat{\rho}\,\hat{A}\,e^{i\theta\hat{n}}\right\\}\,,$

where we have defined

$e^{i\theta}=\frac{2i-\pi}{2i+\pi}\,,$ (27)

and

$\displaystyle\hat{A}$ $\displaystyle=$ $\displaystyle\frac{2}{\pi+2i}\int
d^{2}\alpha\,e^{i\sin\theta|\alpha|^{2}}\hat{D}\left(\alpha\left(1-e^{i\theta}\right)\right)$
(28) $\displaystyle=$ $\displaystyle\frac{1}{\pi^{2}}\int\int
d^{2}z_{1}d^{2}z_{2}\left|z_{1}\right\rangle\left\langle
z_{2}\right|B(z_{1},z_{2},z^{*}_{1},z^{*}_{2})\,,$

with

$\displaystyle B(z_{1},z_{2},z^{*}_{1},z^{*}_{2})=\frac{2}{\pi+2i}\int
d^{2}\alpha\,e^{i\sin\theta|\alpha|^{2}}\langle
z_{1}|\hat{D}\left(\alpha\left(1-e^{i\theta}\right)\right)|z_{2}\rangle$
$\displaystyle=\frac{2\langle
z_{1}|z_{2}\rangle}{\pi+2i}\int_{-\infty}^{+\infty}\,d\alpha_{x}\exp\left(-\left(1-e^{i\theta}\right)\alpha_{x}^{2}+\alpha_{x}\left(\left(1-e^{i\theta}\right)z^{*}_{1}-\left(1-e^{-i\theta}\right)z_{2}\right)\right)\times$
$\displaystyle\times\int_{-\infty}^{+\infty}\,d\alpha_{y}\exp\left(-\left(1-e^{i\theta}\right)\alpha_{x}^{2}+i\alpha_{y}\left(\left(1-e^{i\theta}\right)z^{*}_{1}+\left(1-e^{-i\theta}\right)z_{2}\right)\right)$
$\displaystyle=\frac{2\langle
z_{1}|z_{2}\rangle}{\pi+2i}\frac{\pi}{1-e^{i\theta}}\left[-\left(1-e^{-i\theta}\right)z^{*}_{1}z_{2}\right]=\left(-\frac{|z_{1}|^{2}}{2}-\frac{\left|z_{2}e^{-i\theta}\right|^{2}}{2}+z^{*}_{1}\left(e^{-i\theta}z_{2}\right)\right)$
$\displaystyle=\langle z_{1}|e^{-i\theta\hat{n}}|z_{2}\rangle\,.$ (29)

By replacing equation (3.1) into equation (28) we obtain

$\hat{A}=e^{-i\theta\hat{n}}\,,$ (30)

that shows that equation (26) is correctly normalized

$\displaystyle\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}J(q,p)dqdp=Tr\left\\{\hat{\rho}\,e^{-i\theta\hat{n}}\,e^{i\theta\hat{n}}\right\\}=Tr\left\\{\hat{\rho}\right\\}=1\,.$
(31)

##  4 Kirkwood distribution and $J(q,p)$ distribution

Being the QDF $J(q,p)$ and Kirkwood distributions complex functions we show
now some differences between them.

The Kirkwood distribution is defined as [8, 19, 23, 24]

$K(q,p)=\frac{1}{4\pi^{2}}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}du\,dv\,e^{iup-
ivq}e^{i\frac{uv}{2}}Tr\left\\{\hat{\rho}e^{iv\hat{q}-iu\hat{p}}\right\\}\,,$
(32)

or an alternative way to write it as an expectation value [25] is

$K(q,p)=\frac{1}{\sqrt{2}\pi}e^{\frac{q^{2}}{2}+\frac{p^{2}}{2}+iqp}\langle-i\sqrt{2}p|e^{\frac{\hat{a}^{2}}{2}}\hat{\rho}e^{-\frac{\hat{a}^{\dagger\,2}}{2}}|\sqrt{2}q\rangle.$
(33)

###  4.1 Number state

The Kirkwood $K(q,p)$ and $J(q,p)$ distributions for number state
$\left|n\right\rangle$, are represented by the following equations

$K_{n}(q,p)=\frac{i^{n}}{2^{n}n!\pi\sqrt{2}}e^{-\frac{q^{2}}{2}-\frac{p^{2}}{2}+iqp}H_{n}(q)H_{n}(p)$
(34)

and

$\displaystyle
J_{n}(q,p)=\frac{1}{2i+\pi}\left(\frac{2i-\pi}{2i+\pi}\right)^{n}\exp\left(-\frac{\pi(q^{2}+p^{2})}{2i+\pi}\right)L_{n}\left(\frac{2\pi^{2}(q^{2}+p^{2})}{4+\pi^{2}}\right)\,,$
(35)

where, $H_{n}(x)$ and $L_{n}(x)$ are Hermite and Laguerre polynomials,
respectively.

###  4.2 Superposition of two coherent states

Now, we consider a superposition of two coherent states as:

$\left|\psi_{\pm}\right\rangle=\frac{1}{\sqrt{2\pm
2\mathrm{Re}\langle\alpha_{1}|\alpha_{2}\rangle}}\left(\left|\alpha_{1}\right\rangle\pm\left|\alpha_{2}\right\rangle\right)\,,$
(36)

$\left|\psi_{\pm}\right\rangle=\frac{1}{\sqrt{2\pm
2\mathrm{Re}\langle\alpha_{1}|\alpha_{2}\rangle}}\left(\left|\alpha_{1}\right\rangle\pm\left|\alpha_{2}\right\rangle\right)\,,$
(37)

where $\alpha_{k}=q_{k}/\sqrt{2}+ip_{k}/\sqrt{2}$, such that the Kirkwood
$K(q,p)$ and the $J(q,p)$ distributions for the superposition of two coherent
states, $\left|\psi_{\pm}\right\rangle$, is given by

$\displaystyle J_{\pm}(q,p)=\frac{1}{2i+\pi}\frac{1}{2\pm
2\mathrm{Re}\langle\alpha_{1}|\alpha_{2}\rangle}\left(\exp\left(-\frac{\pi}{2i+\pi}\left((q-q_{1})^{2}+(p-p_{1})^{2}\right)\right)\right)$
$\displaystyle\frac{1}{2i+\pi}\frac{1}{2\pm
2\mathrm{Re}\langle\alpha_{1}|\alpha_{2}\rangle}\left(\exp\left(-\frac{\pi}{2i+\pi}\left((q-q_{2})^{2}+(p-p_{2})^{2}\right)\right)\right)$
$\displaystyle\pm\frac{1}{2i+\pi}\frac{1}{2\pm
2\mathrm{Re}\langle\alpha_{1}|\alpha_{2}\rangle}\left(\exp\left(\frac{i}{2}\left(q\left(p_{1}-p_{2}\right)-p\left(q_{1}-q_{2}\right)\right)\right)\langle\alpha_{2}-\alpha|e^{i\theta}(\alpha_{1}-\alpha)\rangle\right)$
$\displaystyle\pm\frac{1}{2i+\pi}\frac{1}{2\pm
2\mathrm{Re}\langle\alpha_{1}|\alpha_{2}\rangle}\left(\exp\left(-\frac{i}{2}\left(q\left(p_{1}-p_{2}\right)-p\left(q_{1}-q_{2}\right)\right)\right)\langle\alpha_{1}-\alpha|e^{i\theta}(\alpha_{2}-\alpha)\rangle\right)$

and

$\displaystyle
K_{\pm}(q,p)=\frac{1}{\sqrt{2}\pi}\frac{\exp\left(-\frac{q^{2}}{2}-\frac{p^{2}}{2}+iqp\right)}{2\pm
2\mathrm{Re}\langle\alpha_{1}|\alpha_{2}\rangle}\exp\left(-\frac{1}{2}(q^{2}_{1}+p^{2}_{1})+(qq_{1}-pp_{1})\right)\times$
$\displaystyle\times\exp\left(\frac{i}{2}q_{1}(p_{1}+2p)+\frac{i}{2}p_{1}(q_{1}-2q)\right)$
$\displaystyle+\frac{1}{\sqrt{2}\pi}\frac{\exp\left(-\frac{q^{2}}{2}-\frac{p^{2}}{2}+iqp\right)}{2\pm
2\mathrm{Re}\langle\alpha_{1}|\alpha_{2}\rangle}\exp\left(-\frac{1}{2}(q^{2}_{2}+p^{2}_{2})+(qq_{2}-pp_{2})\right)\times$
$\displaystyle\times\exp\left(\frac{i}{2}q_{2}(p_{2}+2p)+\frac{i}{2}p_{2}(q_{2}-2q)\right)$
$\displaystyle\pm\frac{1}{\sqrt{2}\pi}\frac{\exp\left(-\frac{q^{2}}{2}-\frac{p^{2}}{2}+iqp\right)}{2\pm
2\mathrm{Re}\langle\alpha_{1}|\alpha_{2}\rangle}\exp\left(-\frac{1}{2}(q^{2}_{1}+p^{2}_{2})+(qq_{1}-pp_{2})\right)\times$
$\displaystyle\times\exp\left(\frac{i}{2}q_{2}(p_{2}+2p)+\frac{i}{2}p_{1}(q_{1}-2q)\right)$
$\displaystyle\pm\frac{1}{\sqrt{2}\pi}\frac{\exp\left(-\frac{q^{2}}{2}-\frac{p^{2}}{2}+iqp\right)}{2\pm
2\mathrm{Re}\langle\alpha_{1}|\alpha_{2}\rangle}\exp\left(-\frac{1}{2}(q^{2}_{2}+p^{2}_{1})+(qq_{2}-pp_{1})\right)\times$
$\displaystyle\times\exp\left(\frac{i}{2}q_{1}(p_{1}+2p)+\frac{i}{2}p_{2}(q_{2}-2q)\right)\,,$
(39)

respectively.

Figure 1: In figures a) and c) we can see the phase space distribution of the
real and imaginary parts of the Kirkwood function for a number state
$\left|n=3\right\rangle$. In figures b) and d) we see the distribution
$J(q,p)$, for the same number state, again, the real and imaginary parts,
respectively.

We plot both distribution in Figures 1 and 2. In both figures a more uniform
behaviour may be seen in the QDF $J_{\pm}(q,p)$ than in the Kirkwood function.
In fact, the real and imaginary parts of the distribution we have introduced
here, look like Wigner function for number states (Fig. 1) and Scrhödinger cat
states (Fig. 2).

Figure 2: In figures a) and c) we can see the phase space distribution of the
real and imaginary parts of the Kirkwood function for two superposition of
coherent states $\left|\psi_{+}\right\rangle$ wiht $q_{1}=-q_{2}=4$ and
$p_{1}=p_{2}=0$. In figures b) and d) we see the distribution $J(q,p)$, again,
the real and imaginary parts, respectively.

##  5 Reconstruction of distribution $J(\alpha)$

It is not difficult to show that the real part of QDF $J(\alpha)$ may be
measured. This can be achieved by measuring the atomic polarization in the
dispersive interaction between an atom and a quantized field [3], whose
Hamiltonian reads

$\hat{H}=-\chi\hat{a}^{\dagger}\hat{a}\hat{\sigma}_{z},$ (40)

with $\hat{\sigma}_{z}=|e\rangle\langle e|-|g\rangle\langle g|$, the Pauli
matrix corresponding to the atomic inversion operator, where $|g\rangle$ and
$|e\rangle$ represent the ground and excited states of the two-level atom. The
parameter $\chi$ is the dispersive coupling constant. The above Hamiltonian
yields the evolution operator

$\hat{U}(t)=\exp\\{-i\chi t\hat{a}^{\dagger}\hat{a}\hat{\sigma}_{z}\\},$ (41)

from where we can obtain the evolved wavefunction
$|\psi(t)\rangle=\hat{U}(t)|\psi(0)\rangle$, that allows the calculation of
averages of different observables.

The average of observable $\hat{\sigma}_{x}=|e\rangle\langle
g|+|g\rangle\langle e|$ then can be obtained for an arbitrary initial field,
which we conveniently write as
$|\psi_{F}(0)\rangle=\hat{D}^{\dagger}(\alpha)|\phi(0)\rangle$ and the atom is
initially in a superposition of atomic states,
$|\psi_{A}(0)\rangle=\frac{1}{\sqrt{2}}(|g\rangle+|e\rangle)$. Then we write

$\langle\hat{\sigma}_{x}(t)\rangle=\frac{1}{2}\left(\langle\phi(0)|\hat{D}(\alpha)\exp\\{2i\chi
t\hat{a}^{\dagger}\hat{a}\\}\hat{D}^{\dagger}(\alpha)|\phi(0)\rangle+c.c.\right).$
(42)

It is also easy to show that the imaginary part of the QDF may be associated
to the observable $\hat{\sigma}_{y}=i(|e\rangle\langle g|-|g\rangle\langle
e|)$

$\langle\hat{\sigma}_{y}(t)\rangle=\frac{i}{2}\left(\langle\phi(0)|\hat{D}(\alpha)\exp\\{2i\chi
ta^{\dagger}a\\}\hat{D}^{\dagger}(\alpha)|\phi(0)\rangle-c.c.\right).$ (43)

If we set the interaction time
$t=\frac{\arctan\frac{4\pi}{\pi^{2}-4}}{2\chi}$, we obtain that

$Re\\{J(\alpha)\\}\propto\langle\hat{\sigma}_{x}\rangle,\qquad
Im\\{J(\alpha)\\}\propto\langle\hat{\sigma}_{y}\rangle.$ (44)

Therefore, by measuring the polarizations $\hat{\sigma}_{x}$ and
$\hat{\sigma}_{y}$ we are able to measure the QDF $J(\alpha)$.

##  6 Conclusions

We have introduced a set of parametrized (in terms of $\alpha$ and $\beta$)
quasiprobability distribution functions, equation (15), by using the
fractional Fourier transform. This has lead us to generalize QDF to Fresnel
transforms of the characteristic function instead of their usual Fourier
transforms. We have also shown how such QDF may be recosntructed in the
dispersive atom-field interaction. We have also given a (differential)
relation that allows the calculation of the newly introduced QDF from the
Wigner function.

Acknowledgments: We thank CONACYT for support.

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