∎

11institutetext: Varun Dubey 22institutetext: International Centre for
Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru
560089, India. 22email<EMAIL_ADDRESS>33institutetext: Cédric
Bernardin 44institutetext: Université Côte d’Azur, CNRS, LJAD Parc Valrose,
06108 NICE Cedex 02, France. 44email<EMAIL_ADDRESS>55institutetext: Abhishek Dhar 66institutetext: International Centre for
Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru
560089, India. 66email<EMAIL_ADDRESS>
#  Quantum dynamics under continuous projective measurements: non-Hermitian
description and the continuum space limit

Varun Dubey  Cédric Bernardin  Abhishek Dhar

(August 27, 2024)

###### Abstract

The problem of the time of arrival of a quantum system in a specified state is
considered in the framework of the repeated measurement protocol and in
particular the limit of continuous measurements is discussed. It is shown that
for a particular choice of system-detector coupling, the Zeno effect is
avoided and the system can be described effectively by a non-Hermitian
effective Hamiltonian. As a specific example we consider the evolution of a
quantum particle on a one-dimensional lattice that is subjected to position
measurements at a specific site. By solving the corresponding non-Hermitian
wavefunction evolution equation, we present analytic closed-form results on
the survival probability and the first arrival time distribution. Finally we
discuss the limit of vanishing lattice spacing and show that this leads to a
continuum description where the particle evolves via the free Schrodinger
equation with complex Robin boundary conditions at the detector site. Several
interesting physical results for this dynamics are presented.

###### Keywords:

Quantum time of arrival Projective measurements Non-Hermitian Hamiltonian
Survival probability

##  1 Introduction

The question of the time of arrival of a quantum particle at a detector is a
difficult problem since this requires one to make measurements on a system
which for quantum systems lead to a change in the system’s evolution dynamics.
The most dramatic manifestation of this fact is the so-called Zeno effect
where a system under constant observation is never detected misra1977 . In a
more general setting, consider a quantum system that is prepared in an initial
state $|\phi\rangle$ and then evolves under a unitary dynamics. One is
interested in the time of return of the system to the initial state or it’s
arrival to a state $|\chi\rangle$ that is orthogonal to $|\phi\rangle$. A
well-defined protocol to answer this question is to consider a sequence of
projective measurements done on the system at regular intervals of time, say
$\tau$, whose output is $1$ (a “yes”) if the system is in the desired final
state and is otherwise $0$ (a null measurement). The experiment stops when we
record a $1$ and, if this happens on the $n$-th measurement, we say that the
time of detection is $t=n\tau$. Till the time of detection the system’s
evolution thus consists of a unitary evolution that is conditioned by null
measurements (non-detection in the target state) at regular intervals of time
and hence repeated projections into the space complimentary to the target
space.

In the second of a series of three papers on the quantum time of arrival
problem allcock1969time , Allcock discussed the repeated projection protocol
in the context of the time of arrival of a particle which is released at some
point $x<0$ on the real line and eventually detected by a measuring device at
the origin $x=0$. Allcock considered a sequence of measurements at discrete
time intervals which project the particle to the domain $x<0$ till the time of
detection. It was argued that this process could be effectively modeled by
including an imaginary potential $-iV_{0}\theta(-x)$, where $\theta(x)$ is the
Heaviside function, in the free Schrodinger evolution of the particle. The
strength of the potential, $V_{0}$, is related to the intervals between
measurements, $\tau$, as $V_{0}\sim\tau^{-1}$. The limit $\tau\to 0$ gives the
Zeno effect. We are not aware of a formal derivation of the equivalence of the
repeated projection dynamics with the non-Hermitian Hamiltonian echanobe2008 ;
Halliwell2009 ; Halliwell2010 . In more recent work dhar2015a ; dhar2015b ;
lahiri2019  the first detection problem using the repeated measurement scheme
has been investigated in a lattice model for a quantum particle satisying the
discrete Schrodinger equation. It was numerically demonstrated that a
description of the repeated measurement dynamics by means of an effective non-
Hermitian Hamiltonian is in fact quite accurate for small values of $\tau$
(more precisely for $\tau<<\gamma^{-1}$ where $\gamma$ is the transition rate
between lattice sites). In an independent set of work bach2004one ;
grunbaum2013 , for systems with discrete Hilbert spaces, the arrival time
problem using the repeated measurements protocol has been treated using a
renewal approach similar to what one uses for the classical Polya first
passage problem. Using this approach, a number of interesting physical results
on first passage in the lattice Schrodinger problem have been obtained in
Refs. friedman2016 ; friedman2017 ; thiel2018 ; thiel2018spectral ;
thiel2020uncertainty . In particular it has been shown that the return
probability on an infinite one dimensional lattice is less than one and we do
not have the Polya recurrence of the 1-D random walk. The distribution of
first detection times has been shown to scale as $1/t^{3}$. Different aspects
of the equivalence to the non-Hermitian Hamiltonian have been explored in
Refs. lahiri2019 ; thiel2020 ; thiel2020quantization . Other interesting
results include discussions of the non-Hermitian description in many-body
systems elliott2016quantum ; kozlowski2016non  and in the context of
experiments barontini2013controlling ; ashida2016quantum .

The present work makes several contributions. First we show that, for systems
with a discrete Hilbert space, the equivalence between the repeated projection
dynamics and the non-Hermitian Hamiltonian can be established rigorously in a
particular limit where we let the measurement time interval $\tau\to 0$ while
taking a large value (of order $1/\tau^{1/2}$) for the strength of the
coupling strength between the system and detector. This choice is motivated by
similar ones used in the description of the trajectories of continuously
monitored quantum systems BauerBenoistBernard2012 . Secondly we study the
example of a quantum particle on different $1D$ lattices (finite, semi-
infinite and infinite) and, using a Fourier-Laplace transform of the effective
equations of motion, obtain closed-form expressions for the survival
probability and detection time distribution. We compute the spectrum of the
effective non-Hermitian Hamiltonian and show that the results from the
Fourier-Laplace solution also follow from a spectral analysis. Finally, we
consider the lattice-to-continuum space limit and show that the evolution is
described by a Schrodinger equation with complex Robin boundary condition at
the location of the detector. Apart from the survival probability and the full
spectrum, the form of the “surviving” wave-function is also discussed. Such
boundary conditions have earlier been discussed, somewhat formally, in the
context of the time of arrival problem werner1987 ; TumulkaTeufel2019 ;
Tumulka2019 .

The plan of the paper is as follows. In Sec. (2) we describe the limiting
procedure which gives the exact mapping between the repeated projective
measurements and the non-Hermitian Hamiltonian. In Sec. (3) we consider the
example of a quantum particle on three $1D$ lattices and write the form of the
effective non-Hermitian Hamiltonian for these cases. The exact results for
survival probability and detection time distribution using the Fourier-Laplace
transform approach and the Green’s function approach are presented in Sec.
(4). In Sec. (5) we discuss the continuum space limit and finally conclude
with a summary of the results in Sec. (6).

##  2 Description of dynamics under repeated projective measurements by an
effective non-Hermitian Hamiltonian: exact mapping

Consider a quantum system whose states belong to the Hilbert space
$\mathcal{H}$. We assume that $\mathcal{H}=\mathcal{S}\oplus\mathcal{D}$ can
be written as the sum of two orthogonal complementary subspaces, where
$\mathcal{S}$ is a “system” space and ${\mathcal{D}}$ a “detector” space. Let
$\left|i\right\rangle$ be an orthonormal basis of $\mathcal{S}$ and
$\left|\alpha\right\rangle$ an orthonormal basis of $\mathcal{D}$. The set of
states $\\{\left|i\right\rangle,\left|\alpha\right\rangle\\}$ together form a
complete orthonormal basis of $\mathcal{H}$. Hence we have

$\sum_{i}\left|i\right\rangle\left\langle
i\right|+\sum_{\alpha}\left|\alpha\right\rangle\left\langle\alpha\right|=I$
(1)

where $I$ is the identity operator. The most general Hamiltonian describing
such a system is given by

$\displaystyle H$
$\displaystyle=H^{(\mathcal{S})}+H^{(\mathcal{D})}+H^{(\mathcal{SD})},$ (2)
$\displaystyle H^{(\mathcal{S})}$
$\displaystyle=\underbrace{\sum_{i,j}H_{ij}^{(S)}\left|i\right\rangle\left\langle
j\right|}_{\text{System}},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
H^{(\mathcal{D})}=\underbrace{\sum_{\alpha,\beta}H_{\alpha\beta}^{(D)}\left|\alpha\right\rangle\left\langle\beta\right|}_{\text{Detector}},$
(3) $\displaystyle H^{(\mathcal{SD})}$
$\displaystyle=\underbrace{\sum_{i,\alpha}\left[H_{i\alpha}^{(\mathcal{SD})}\left|i\right\rangle\left\langle\alpha\right|+H_{\alpha
i}^{(\mathcal{SD})}\left|\alpha\right\rangle\left\langle
i\right|\right]}_{\text{System-Detector}}.$ (4)

Let $\left|\psi(t)\right\rangle\in{\mathcal{H}}$ denote the state of the total
system at time $t$. The unitary evolution of this state with the above
Hamiltonian $H$ is given by

$\left|\psi(t)\right\rangle=U_{t}\left|\psi(0)\right\rangle,\,\,\,\,U_{t}=\exp\left(-\imath
tH/\hbar\right).$

Consider instantaneous projective measurements made to find if the system is
in the detector subspace $\mathcal{D}$. This corresponds to the projection
operator

$P=\sum_{\alpha}\left|\alpha\right\rangle\left\langle\alpha\right|$ (5)

while the complementary operator corresponding to projection into the system
subspace $\mathcal{S}$ is given by

$Q=I-P=\sum_{i}\left|i\right\rangle\left\langle i\right|.$

Now imagine an experiment where initially the system starts in the space
$\mathcal{S}$ and begins to evolve unitarily. We keep making measurements, at
regular intervals of time $\tau$, to find if the system has arrived in the
detector subspace $\mathcal{D}$. If the result of the measurement is negative,
the system (now projected back into $\mathcal{S}$) continues its unitary
evolution, until the next measurement and the process is repeated. The
experiment stops when we get a positive result indicating arrival into
$\mathcal{D}$. Thus the evolution of the state vector consists of a sequence
of steps in $\mathcal{S}$, each consisting of a unitary evolution followed by
the projection $Q$. Looking at the state of the system conditioned on survival
(non-detection), let us denote by $\left|\psi(n\tau)\right\rangle$ the state
just after the $n$-th measurement. Then it can be shown dhar2015b  that

$\left|\psi(n\tau)\right\rangle=\widetilde{U}_{\tau}^{n}\left|\psi(0)\right\rangle,\quad\widetilde{U}_{\tau}=QU_{\tau}Q.$
(6)

The probability, $S(n\tau)$, that the system stays undetected after the $n$-th
measurement is given by the norm of the state

$\displaystyle S(n\tau)=\left\langle\psi(n\tau)|\psi(n\tau)\right\rangle.$ (7)

The normalized state after the $n$-th measurement is given by

$\left|\widetilde{\psi}(n\tau)\right\rangle=\frac{\left|\psi(n\tau)\right\rangle}{\sqrt{S(n\tau)}}.$
(8)

It was numerically demonstrated in Refs. dhar2015a ; dhar2015b ; lahiri2019
that for the case where the time between measurements $\tau$ is small
(compared to typical time scales in the unitary evolution) the above dynamics
is accurately described by a continuous time evolution with a non-Hermitian
effective Hamiltonian. However, to obtain a description with continuous time,
we need the limit $\tau\to 0$ and this is problematic because it leads to the
Zeno effect. Here we discuss a systematic approach where this problem is
overcome and the effective non-Hermitian description can be obtained in a
rigorous way. For this we let the system-detector couplings scale with the
detection interval $\tau$ as

$H_{i\alpha}^{({\mathcal{SD}})}=\sqrt{\frac{\gamma_{i\alpha}}{\tau}},\,\,\,H_{\alpha
i}^{({\mathcal{SD}})}=\sqrt{\frac{\gamma_{\alpha i}}{\tau}}.$

Further, denoting $H_{ij}^{(\mathcal{S})}=\gamma_{ij}$ and
$H_{\alpha\beta}^{(\mathcal{D})}=\gamma{}_{\alpha\beta}$ we note the form of
the Hamiltonian $H$

$H=\underbrace{\sum_{i,j}\gamma_{ij}\left|i\right\rangle\left\langle
j\right|}_{H^{\mathcal{S}}}+\underbrace{\sum_{\alpha,\beta}\gamma{}_{\alpha\beta}\left|\alpha\right\rangle\left\langle\beta\right|}_{H^{\mathcal{D}}}+\underbrace{\sum_{i,\alpha}\left[\sqrt{\frac{\gamma_{i\alpha}}{\tau}}\left|i\right\rangle\left\langle\alpha\right|+\sqrt{\frac{\gamma_{\alpha
i}}{\tau}}\left|\alpha\right\rangle\left\langle
i\right|\right]}_{H^{\mathcal{SD}}}.$ (9)

We assume from now on that $H$ is measured in units of $\hbar$ and so has
units of frequency, then the $\gamma$ coefficients (assumed to be positive) so
do too. Expanding $\widetilde{U}_{\tau}$ in a power series, one has

$\widetilde{U}_{\tau}=Q\exp\left(-\imath\tau
H\right)Q=QQ+\left(-\imath\tau\right)QHQ+\frac{\left(-\imath\tau\right)^{2}}{2}QHHQ+\dots$

Note that $\left|\psi(n\tau)\right\rangle$ belongs to $\mathcal{S}$, which
means that $Q\left|\psi(n\tau)\right\rangle=\left|\psi(n\tau)\right\rangle$.
The following identities follow from the orthonomality of the basis states Eq.
(1) and the definition of $Q$:

$\displaystyle QH^{(\mathcal{S})}Q$
$\displaystyle=H^{(\mathcal{S})},\,\,QH^{(\mathcal{D})}Q=0,\,\,QH^{(\mathcal{SD})}Q=0,$
(10) $\displaystyle Q\left[H^{(\mathcal{S})}\right]^{2}Q$
$\displaystyle=\left[H^{(\mathcal{S})}\right]^{2},\,\,Q\left[H^{(\mathcal{D})}\right]^{2}Q=0,$
(11) $\displaystyle Q\left[H^{(\mathcal{SD})}\right]^{2}Q$
$\displaystyle=\sum_{i,j,\alpha}H_{i\alpha}^{({\mathcal{SD}})}H_{\alpha
j}^{({\mathcal{SD}})}\left|i\right\rangle\left\langle
j\right|=\frac{1}{\tau}\sum_{i,j,\alpha}\left|i\right\rangle\left\langle
j\right|\sqrt{\gamma_{i\alpha}\gamma_{\alpha j}}.$ (12)

By use of these in the expansion of $\widetilde{U}_{\tau}$, we then get

$\widetilde{U}_{\tau}=I-\imath\tau H^{\mathcal{S}}-\tau
V^{\mathcal{S}}+\mathcal{O}(\tau^{2})$ (13)

where

$V_{ij}^{\mathcal{S}}=\frac{1}{2}\sum_{\alpha}\sqrt{\gamma_{i\alpha}\gamma_{\alpha
j}}$ (14)

We now take the continuum limit $\tau\to 0$, $n\to\infty$ while keeping
$t=n\tau$ finite. Let
$\left|\psi(t)\right\rangle=\left|\psi(n\tau)\right\rangle$. Then, using Eq.
(13), we write

$\imath\frac{\partial\left|\psi(t)\right\rangle}{\partial t}=\lim_{\tau\to
0}\imath\frac{\left|\psi(n\tau+\tau)\right\rangle-\left|\psi(n\tau)\right\rangle}{\tau}=\imath\frac{\widetilde{U}\left|\psi(n\tau)\right\rangle-\left|\psi(n\tau)\right\rangle}{\tau}$

and we get a new Shrodinger’s equation with an effective non-Hermitian
Hamiltonian $H^{\text{eff}}$

$\imath\frac{\partial\left|\psi(t)\right\rangle}{\partial t}=H^{{\rm
eff}}\left|\psi(t)\right\rangle,\quad H^{\text{eff}}=H^{\mathcal{S}}-\imath
V^{\mathcal{S}}.$ (15)

Thus we have shown that the repeated-measurement dynamics, in the special
limit where the time interval between measurements $\tau$ goes to zero while
the system-detector interaction diverges like $\tau^{-1/2}$, is given by the
non-Hermitian Hamiltonian in Eq. (15). Note that the dynamics is completely
evolving in the system subspace $\mathcal{S}$.

Note that in the limit $\tau\to 0$ the survival probability at time $t$ is
obtained from Eq. (7):

$S(t)=\left\langle\psi(t)|\psi(t)\right\rangle,$ (16)

with $|\psi(t)\rangle$ solution of the effective Shrodinger’s equation Eq.
(15). The distribution of first detection times is given by

$\displaystyle F(t)=\frac{dS}{dt}=-2\langle\psi|V^{\mathcal{S}}|\psi\rangle.$
(17)

##  3 Example of a quantum particle on different one dimensional lattice

We now consider some simple physical examples of the Hamiltonian Eq. (9)
corresponding to the case of a single particle hopping on a one-dimensional
lattice $\Lambda\subset\mathbb{Z}$ containing the site $0$ and subjected to
the action of a potential $v(n)$. A detector performs a position measurement
at the site $0$ after every time interval of length $\tau$. The basis vectors
$|i\rangle$ now correspond to the position eigenstates and we have
$P=|0\rangle\langle 0|$. We consider three different choices of lattices which
we specify now along with the corresponding Hamiltonian:

  1. 1.

Finite lattice $\Lambda=\\{0,\ldots,N\\}$ of size $N\geq 2$:

$\begin{split}H=&-\gamma_{0}\sum_{n=2}^{N}\left(\leavevmode\nobreak\
\left|n\right\rangle\left\langle n-1\right|+\left|n-1\right\rangle\left\langle
n\right|-2\left|n\right\rangle\left\langle n\right|\leavevmode\nobreak\
\right)+(2+\beta)\gamma_{0}\left|1\right\rangle\left\langle 1\right|\\\
&-\sqrt{\frac{\gamma}{\tau}}\left(\leavevmode\nobreak\
\left|0\right\rangle\left\langle 1\right|+\left|1\right\rangle\left\langle
0\right|\leavevmode\nobreak\ \right).\end{split}$ (18)

  2. 2.

Semi-infinite lattice $\Lambda={\mathbb{N}}$

$\begin{split}H=&-\gamma_{0}\sum_{n=2}^{\infty}\left(\leavevmode\nobreak\
\left|n\right\rangle\left\langle n-1\right|+\left|n-1\right\rangle\left\langle
n\right|-2\left|n\right\rangle\left\langle n\right|\leavevmode\nobreak\
\right)+(2+\beta)\gamma_{0}\left|1\right\rangle\left\langle 1\right|\\\
&-\sqrt{\frac{\gamma}{\tau}}\left(\leavevmode\nobreak\
\left|0\right\rangle\left\langle 1\right|+\left|1\right\rangle\left\langle
0\right|\leavevmode\nobreak\ \right).\end{split}$ (19)

  3. 3.

Infinite lattice $\Lambda=\mathbb{Z}$:

$\begin{split}H=&-\sum_{n\in\mathbb{Z}\setminus\\{0\\}}\left[\leavevmode\nobreak\
(1-\delta_{n,1})\left|n\right\rangle\left\langle
n-1\right|+(1-\delta_{n,-1})\left|n\right\rangle\left\langle
n+1\right|-2\left|n\right\rangle\left\langle n\right|\leavevmode\nobreak\
\right]\\\ &+(2+\beta)\gamma_{0}\left[\leavevmode\nobreak\
\left|1\right\rangle\left\langle
1\right|+\left|-1\right\rangle\left\langle-1\right|\leavevmode\nobreak\
\right]\\\ &-\sqrt{\frac{\gamma}{\tau}}\left[\leavevmode\nobreak\
\left|0\right\rangle\left\langle 1\right|+\left|1\right\rangle\left\langle
0\right|+\left|0\right\rangle\left\langle-1\right|+\left|-1\right\rangle\left\langle
0\right|\leavevmode\nobreak\ \right].\end{split}$ (20)

We introduce the dimensionless parameter $\alpha=\frac{\gamma}{2\gamma_{0}}$
which can be regarded as the strength of the measurement. We have assumed that
the potential $v(n)$ is constant and equal to $2\gamma_{0}$ except on the two
sites $-1,+1$ where it is equal to $(\beta+2)\gamma_{0}$. Hence $\beta$ is
also a dimensionless parameter measuring the impurity of the potential near
the detector. These parameters are capsulated in the complex number
$w=\alpha+\imath\beta.$ Denote then by

$H_{\Lambda}=\cfrac{H^{\text{eff}}}{\gamma_{0}}$

the corresponding scaled effective Hamiltonian. By using Eqs. (14,15) we get
the following effective Hamiltonians corresponding to the system-detector
Hamiltonians in Eqs. (18,19,20):

  1. 1.

Finite lattice $\Lambda_{N}=\\{1,\ldots,N\\}$ of size $N\geq 2$:

$H_{\Lambda_{N}}=-\sum_{n=2}^{N}\big{[}\left|n\right\rangle\left\langle
n-1\right|+\left|n-1\right\rangle\left\langle
n\right|-2\left|n\right\rangle\left\langle n\right|\big{]}+(2-\imath
w)\left|1\right\rangle\left\langle 1\right|.$ (21)

  2. 2.

Semi-infinite lattice $\mathbb{N}$:

$H_{\mathbb{N}}=-\sum_{n=2}^{\infty}\big{[}\left|n\right\rangle\left\langle
n-1\right|+\left|n-1\right\rangle\left\langle
n\right|-2\left|n\right\rangle\left\langle n\right|\big{]}+(2-\imath
w)\left|1\right\rangle\left\langle 1\right|.$ (22)

  3. 3.

Infinite lattice $\mathbb{Z}$:

$\begin{split}H_{\mathbb{Z}}=&-\sum_{n\in\mathbb{Z}\setminus\\{0\\}}\big{[}(1-\delta_{n,1})\left|n\right\rangle\left\langle
n-1\right|+(1-\delta_{n,-1})\left|n\right\rangle\left\langle
n+1\right|-2\left|n\right\rangle\left\langle n\right|\big{]}\\\ &-\imath
w\sum_{n,n^{\prime}\in\\{-1,1\\}}|n\rangle\,\langle n^{\prime}|.\end{split}$
(23)

Observe that these operators are not Hermitian (nor normal) and therefore a
priori not diagonalisable. In fact for e.g. $N=2$ we can explicitly show that
for specific values of $w$ the operator $H_{\Lambda_{2}}$ is not
diagonalisable. We discuss the spectral properties of the Hamiltonian
$H_{\mathbb{N}}$ and $H_{\mathbb{Z}}$ in App. (A).

##  4 Calculation of Survival Probability

Having mapped the measurement problem to that of non-unitary evolution given
by Eq. (15), the methods developed in Ref. krapivsky2014  can be employed to
calculate the survival probability. The three cases considered in the previous
section are discussed in the following sub-sections.

###  4.1 The lattice $\Lambda_{N}$

The Hamiltonian $H_{\Lambda_{N}}$ can be rewritten as

$H_{\Lambda_{N}}=-\Delta_{N}-\imath w|1\rangle\,\langle 1|$

where $\Delta_{N}$ is the discrete Laplacian on $\Lambda_{N}$. We first prove
that in this case the survival probabilty goes to $0$ with an exponential
decay at large times. We recall that the spectrum of the Laplacian
($\epsilon_{q}=-2(1-\cos(q))$, where $q=s\pi/(N+1),\leavevmode\nobreak\
s=1,2,\ldots,N$. Hence it is contained in
$\\{z\in{\mathbb{C}}\;;\;{\real}(z)<0,\;\imaginary(z)=0\\}$. Let $\lambda$ be
an eigenvalue of $-\imath H_{\Lambda_{N}}$ associated to some eigenvector
$|\psi\rangle$ then we have that

${\lambda}\langle\psi|\psi\rangle=\imath\,\langle\psi|\Delta_{N}|\psi\rangle-{w}\,|\langle
1|\psi\rangle|^{2}.$

If $\langle 1|\psi\rangle\neq 0$, $\real(\lambda)<0$ because $\real(w)>0$. If
$\langle 1|\psi\rangle=0$ then
$\imath\Delta_{N}|\psi\rangle=\lambda|\psi\rangle$, hence $|\psi\rangle$ is an
eigenvector of $\Delta_{N}$ associated to the eigenvalue $-\imath\lambda$.
Since the (explicit) eigenvectors of $\Delta_{N}$ are not such that $\langle
1|\psi\rangle=0$, the latter case never holds. We conclude that the spectrum
of $-\imath H_{\Lambda_{N}}$ is contained in
$\\{z\in{\mathbb{C}}\;;\;{\real}(z)<0\\}$. By using the Jordan-Chevalley
decomposition of $-\imath H_{\Lambda_{N}}$ we conclude that for any
$|\psi(0)\rangle$,

$S(t)=\langle\psi(t)|\psi(t)\rangle=O(e^{-\mu t})$

for some real $\mu>0$. Hence the survival probability $S(t)$ goes to $0$ as
$t\to\infty$ exponentially fast. We will see in the two next subsections that
when the lattice is infinite, it is no longer the case. For the case where
$-\imath H_{\Lambda_{N}}$ has a non-degenerate spectrum, let $\lambda_{m}$ be
the eigenvalue with the smallest negative real part. Then the decay constant
is given by $\mu=-\real[\lambda_{m}]$. The “surviving” wave function at large
times would be given by $|\psi(t)\rangle\sim e^{\lambda_{m}t}|\chi_{m}\rangle$
where $|\chi_{m}\rangle$ is the eigenvector of $-\imath H_{\Lambda_{N}}$
corresponding to the eigenvalue $\lambda_{m}$.

From Eq. (17) we see that the first passage time distribution is given by
$F(t)=-\tfrac{\partial S}{\partial t}(t)=2\alpha|\psi_{1}(t)|^{2}$, where we
use the notation $\psi_{i}=\langle i|\psi\rangle$. We now show that one can
write a formal solution for $\psi_{1}(t)$, by using the information on the
eigenspectrum of the Hermitian part of the effective Hamiltonian, which in
this case is the lattice Laplacian. We first write the equation of motion:

$\displaystyle\imath\frac{\partial|\psi\rangle}{\partial
t}=-\Delta_{N}|\psi\rangle-iw\langle 1|\psi\rangle|1\rangle.$ (24)

Taking a Laplace transform
$|\tilde{\psi}(s)\rangle=\int_{0}^{\infty}dte^{-st}|\psi(t)\rangle$, we get

$\displaystyle-|\psi(0)\rangle+s|\tilde{\psi}(s)\rangle=\imath\Delta_{N}|\tilde{\psi}(s)\rangle-w\langle
1|\tilde{\psi}(s)\rangle|1\rangle.$ (25)

Defining the Green’s function $G(s)=[s-i\Delta_{N}]^{-1}$, we get the
following formal solution

$\displaystyle|\tilde{\psi}(s)\rangle=G(s)|\psi(0)\rangle-w\langle
1|\tilde{\psi}(s)\rangle G(s)|1\rangle.$ (26)

Assuming an initial wavefunction, $|\psi(0)\rangle=|\ell\rangle$, localized at
a site $\ell$ we take a projection of the above equation on the state
$|1\rangle$ to get the Laplace transform ${\mathcal{L}}\psi_{1}$ of
$\psi_{1}$:

$\displaystyle[\mathcal{L}\psi_{1}](s):=\langle
1|\tilde{\psi}(s)\rangle=\frac{G_{1\ell}(s)}{1+wG_{11}(s)},$ (27)

where the matrix elements $G_{ij}(s)=\langle i|G(s)|j\rangle$ can be written
in terms of the eigenfunctions $\phi_{q}(j)=\sqrt{2/(N+1)}\sin(qj)$ of the
Laplacain $\Delta_{N}$, and corresponding eigenvalues $\epsilon_{q}=-2(1-\cos
q)$ with $q=s\pi/(N+1)$, $s=1,2,\ldots,N$. One gets

$\displaystyle
G_{ij}(s)=\sum_{q}\frac{\phi_{q}(i)\phi_{j}(q)}{s-i\epsilon_{q}}$ (28)

and hence an explicit expression for the Laplace transform
$({\mathcal{L}}\psi_{1})(s)$. By using an inverse Laplace transform formula we
can then get an explicit formula for the first passage time distribution
$F(t)$ which remains however difficult to exploit, even qualitatively. Similar
expressions have recently been discussed in Ref. thiel2020  who also point the
analogy with the renewal approach for the repeated measurement problem. In the
next section we will make use of the analogous formula as Eq. (27) to get
explicit results on first passage distribution on the infinite lattice. For
the finite lattice we now present some numerical results on the form of the
survival probability.

Numerical results: The numerical results are obtained by a direct solution of
the non-Hermitian Schrodinger equation. As shown above in this case we have
$\left[S(t)\right]_{t\rightarrow\infty}=0$ and we always eventually detect the
particle. In Fig. (1) (left panel) we plot the decay of survival probability
on a system for which $N=15$ for three different values of $w$. The survival
probability cascades to $0$ over time. We observe a non-monotonic dependence
with $S(t)$ decaying slowly for both very small $w=0.1$ as well large $w=5$.
The several plateaues in the survival probability can be understood as arising
from ballistic propagation of the particle with a a group velocity $\approx
2$, such that as the wave packet hits the detector, the surival probability
plunges considerably. The velocity $2$ corresponds to the maximum group
velocity $d\epsilon_{q}/dq$. In Fig. (1) (right panel) we show the decay of
survival probability for relatively larger systems. In all cases,
$\psi_{i}(0)=\delta_{i,20}$ and $w=2.0$. In this case the first short plateau
in $S(t)$ corresponds to the time taken for the wave-packet to reach the
detector while the second longer plateau corresponds to the wave-packet
traveling from the origin to the right end and back. Hence this second time-
scale can be seen to be proprtional to the lattice size $N$. The dashed line
indicates the infinite time survival probability $S_{\mathbb{N}}(w,20)$ on the
semi-infinite lattice ${\mathbb{N}}$ which is computed exactly in the next
section (see Eq. (40)). Similar finite size effects have been discussed
recently in Ref. lahiri2019 .

Figure 1:  In the graph on the left, the survival probability $S(t)$ is
plotted for $N=15$, $\psi_{i}(0)=\delta_{i,15}$ and for different values of
$w$. In the graph to the right, the survival probability $S(t)$ is plotted for
lattice sizes $N=100$ and $N=200$. In both cases
$\psi_{i}(0)=\delta_{i,20},\,w=2$. The dashed line is the value of survival
probability $S(\infty)$ obtained from Eq. (40) for the $\mathbb{N}$ Lattice.

###  4.2 $\mathbb{N}$ Lattice

The Schrodinger equation corresponding to the effective Hamiltonian
$H_{\mathbb{N}}$ given by Eq. (22) is

$\imath\cfrac{\partial\psi_{n}}{\partial t}=\begin{dcases}(2-\imath
w)\psi_{1}-\psi_{2},&n=1,\\\ 2\psi_{n}-\psi_{n-1}-\psi_{n+1},&n\geq
2.\end{dcases}$ (29)

By introducing the fictitious Dirichlet boundary condition $\psi_{0}(t)=0$ the
Eq. (29) can be written as

$\imath\cfrac{\partial\psi_{n}}{\partial
t}=2\psi_{n}-\psi_{n-1}-\psi_{n+1}-\imath w\delta_{1}(n)\psi_{1},\quad n\geq
1.$ (30)

Decompose the wave function on the following orthonormal sin-basis

$\displaystyle\psi_{n}(t)=\sqrt{\dfrac{2}{\pi}}\int_{0}^{\pi}dk\,\hat{\psi}(k,t)\sin(n\,k)\quad\text{with}\quad\hat{\psi}(k,t)=\sqrt{\dfrac{2}{\pi}}\sum_{n=1}^{\infty}\psi_{n}(t)\,\sin(n\,k).$

Observe that the Dirichlet boundary condition is automatically satisfied.
Assume $\psi_{n}(0)=\delta_{n,n_{0}}$ so that
$\hat{\psi}(k,0)=\sqrt{\dfrac{2}{\pi}}\sin\left(n_{0}k\right)$. Then it
follows from Eq. (30) that

$\imath\cfrac{\partial\hat{\psi}}{\partial
t}(k,t)-2(1-\cos\left(k\right))\hat{\psi}(k,t)=-\imath
w\,\sqrt{\dfrac{2}{\pi}}\,\psi_{1}(t)\,\sin\left(k\right).$ (31)

The Laplace transforms of $\hat{\psi}(k,t)$ and $\psi_{1}(t)$ are by
definition given by

$\displaystyle\tilde{\psi}(k,s)=[\mathcal{L}\hat{\psi}(k,\cdot)](s)=\int_{0}^{\infty}dt\,\exp(-st)\hat{\psi}(k,t),$
(32)
$\displaystyle[\mathcal{L}\psi_{1}](s)=\int_{0}^{\infty}dt\,\exp(-st)\,\psi_{1}(t)=\sqrt{\dfrac{2}{\pi}}\int_{0}^{\pi}dk\,\tilde{\psi}(k,s)\,\sin(k).$
(33)

Taking the Laplace Transform of Eq. (31) and by use of the above equations

$\tilde{\psi}(k,s)=\imath\sqrt{\dfrac{2}{\pi}}\,\frac{1}{\imath
s-2\left(1-\cos\left(k\right)\right)}\left\\{\sin\left(n_{0}k\right)-w\sin\left(k\right)\left[\mathcal{L}\psi_{1}\right](s)\right\\}.$
(34)

The two relations Eq. (33) and Eq. (34) between $\tilde{\psi}(k,s)$ and
$[\mathcal{L}\psi_{1}](s)$ above give

$\left[\mathcal{L}\psi_{1}\right](s)=\imath\dfrac{\displaystyle\dfrac{2}{\pi}\int_{0}^{\pi}dk\,\dfrac{\sin\left(k\right)\sin\left(n_{0}k\right)}{\imath
s-2(1-\cos(k))}}{1+\imath\dfrac{2w}{\pi}\displaystyle\int_{0}^{\pi}dk\,\dfrac{\sin^{2}k}{\imath
s-2(1-\cos(k))}}.$ (35)

Note that this also follows from Eq. (27) on taking the limit $N\to\infty$.
The integrals involved in Eq. (35) can be evaluated by means of contour
integration for any complex number $s$ such that ${\real}(s)>0$, see App.
(B.1.1). We get then that

$\left[\mathcal{L}\psi_{1}\right](s)==-\imath\frac{\imath^{n_{0}}\bigg{[}-\left(\frac{s}{2}+\imath\right)+\sqrt{\left(\frac{s}{2}+\imath\right)^{2}+1}\bigg{]}^{n_{0}}}{1+w\bigg{[}-\left(\frac{s}{2}+\imath\right)+\sqrt{\left(\frac{s}{2}+\imath\right)^{2}+1}\bigg{]}}$
(36)

where $\sqrt{z}$ denotes the principal square root of
$z\in\mathbb{C}\backslash{\mathbb{R}}^{*}_{-}$ (using the nonpositive real
axis as a branch cut). We have hence obtained an explicit expression of the
Fourier-Laplace transform $\tilde{\psi}$ of the wave function $\psi$ by
plugging Eq. (36) into Eq. (34).

We now turn to the computation of the survival probability. Recalling Eq.
(16), the survival probability after time $t$ is
$S(t)=\sum_{n=1}^{\infty}\left|\psi_{n}(t)\right|^{2}$. From Eq. (29), one has

$\imath\dfrac{\partial}{\partial
t}\left|\psi_{n}\right|^{2}=\begin{dcases}-2\imath\real(w)\left|\psi_{1}\right|^{2}-\psi_{1}^{*}\psi_{2}+\psi_{1}\psi_{2}^{*},&n=1,\\\
-\psi_{n}^{*}\psi_{n-1}+\psi_{n}\psi_{n-1}^{*}-\psi_{n}^{*}\psi_{n+1}+\psi_{n}\psi_{n+1}^{*},&n\geq
2.\end{dcases}$

Since $\psi_{n}$ goes to $0$ as $n\rightarrow\infty$, the equations can be
summed to obtain ${dS}/{dt}=-2\real(w)\left|\psi_{1}\right|^{2}.$ Integrating,
we get the survival probability

$S_{\infty}=\lim_{t\to\infty}S(t)=1-2\real(w)\int_{0}^{\infty}dt\,\left|\psi_{1}(t)\right|^{2}.$
(37)

For square integrable function $\psi_{1}$, we have Schiff , Chapter 4, that

$\int_{0}^{\infty}dt\,\left|\psi_{1}(t)\right|^{2}=\lim_{\epsilon\to
0^{+}}\dfrac{1}{2\pi\imath}\int_{\epsilon-\imath\infty}^{\epsilon+\imath\infty}ds\,\left|[\mathcal{L}\psi_{1}](s)\right|^{2}.$
(38)

For $s=\epsilon+2\imath(x-1)$ in the limit $\epsilon\rightarrow 0^{+}$, by Eq.
(36), we have that

$\left|[\mathcal{L}\psi_{1}](s)\right|^{2}=\begin{cases}\dfrac{\left(x+\sqrt{x^{2}-1}\right)^{2n_{0}}}{1+\left|w\right|^{2}\left(x+\sqrt{x^{2}-1}\right)^{2}+2\imaginary(w)\left(x+\sqrt{x^{2}-1}\right)},\quad
x<-1,\\\ \\\
\dfrac{1}{1+\left|w\right|^{2}+2\big{[}\real(w)\sqrt{1-x^{2}}+\imaginary(w)\,x\big{]}},\quad-1\leq
x\leq 1,\\\ \\\
\dfrac{\left(x-\sqrt{x^{2}-1}\right)^{2n_{0}}}{1+\left|w\right|^{2}\left(x-\sqrt{x^{2}-1}\right)^{2}+2\imaginary(w)\left(x-\sqrt{x^{2}-1}\right)},\quad
x>1.\end{cases}$ (39)

Denote by $S_{\mathbb{N}}(w,n_{0})$ the survival probability given by Eq. (37)
corresponding to the initial condition $|\psi(0)\rangle=\delta_{n_{0}}$, when
the strength measurement parameter is $\alpha={\real}(w)$ and the impurity
parameter is $\beta={\imaginary}(w)$. From Eq. (37) and Eqs. (38, 39), after
simplification, we obtain then that

$\begin{split}S_{\mathbb{N}}(w,n_{0})=1&-\frac{{\real}(w)}{\pi\left|w\right|}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{\cos\,\theta}{\frac{1}{2}\left(\left|w\right|+\frac{1}{\left|w\right|}\right)+\cos\left(\theta-\varphi\right)}d\theta\\\
&-\frac{2{\real}(w)}{\pi\,}\int_{0}^{1}\frac{u^{2n_{0}-2}(1-u^{2})(1+\left|w\right|^{2}u^{2})}{(1+\left|w\right|^{2}u^{2})^{2}-(2{\imaginary}(w)\,u)^{2}}du\end{split}$
(40)

where $\varphi={\rm{Arg}}(w)$. Only the second integral is depending on
$n_{0}$. Both can be explicitly computed but the formula obtained do not have
a simple form. For large $n_{0}$ a simple expansion can be done for the latter
at any order, see App.(B.1.2). Therefore for $n_{0}\gg 1$, we have e.g. that

$\begin{split}S_{\mathbb{N}}(w,n_{0})\approx
1&-\frac{{\real}(w)}{\pi\left|w\right|}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{\cos\,\theta}{\frac{1}{2}\left(\left|w\right|+\frac{1}{\left|w\right|}\right)+\cos\left(\theta-\varphi\right)}d\theta\\\
&-\cfrac{1}{n_{0}^{2}}\;\frac{{\real}(w)}{\pi}\dfrac{(1+|w|^{2})}{(1+|w|^{2})^{2}-4[{\imaginary}(w)]^{2}}.\end{split}$
(41)

Figure 2: Plots showing variation of $S_{\mathbb{N}}$ (Eq. (40)) with $w$ for
various starting positions $n_{0}$. The left plot is for $w=\alpha+\imath/2$
and the right plot for $w=1/2+\imath\beta$

In fact we can also get an explicit formula for the function $\psi_{1}$ and
therefore for the survival probability
$S(t)=1-2\real(w)\int_{0}^{t}dt^{\prime}\,\left|\psi_{1}(t^{\prime})\right|^{2}$
at time $t$. For that it is sufficient to recognize in Eq. (36) the Laplace
transform of some explicit function. In App. (B.1.3) we identify this function
by using properties of the Bessel functions of the first kind:

$J_{k}(t)=\cfrac{1}{2\pi}\int_{-\pi}^{\pi}dx\,e^{-\imath(kx-t\sin(x))}.$ (42)

We find that

$\psi_{1}(t)=-2\imath^{n_{0}+1}e^{-2\imath
t}\sum_{k=0}^{\infty}(k+n_{0})(-w)^{k}\dfrac{J_{k+n_{0}}(2t)}{2t},$ (43)

which immediately gives us the explicit form of the first detection
distribution $F(t)=2\real[w]|\psi_{1}(t)|^{2}$. At large times using the
asymptotic form of $J_{k}(2t)$ we get

$\Rightarrow\psi_{1}(t)\asymp\frac{(-\imath)^{n_{0}+1}e^{-\imath
2t}}{2\sqrt{\pi t^{3}}}\Bigg{[}e^{\imath\left(2t-\frac{\pi}{4}+\frac{\imath
n_{0}\pi}{2}\right)}f(n_{0},w)+e^{-\imath\left(2t-\frac{\pi}{4}+\frac{\imath
n_{0}\pi}{2}\right)}\left[f(n_{0},w^{*})\right]^{*}\Bigg{]}$ (44)

where

$f(n_{0},w)=\frac{n_{0}-\imath(n_{0}-1)w}{(1-\imath w)^{2}}.$

Hence we get the long time decay form $F(t)\sim 1/t^{3}$.

###  4.3 $\mathbb{Z}$ Lattice

The calculations here are along the same lines as for the $\mathbb{N}$ Lattice
case. For the effective Hamiltonian $H_{\mathbb{Z}}$ given by Eq. (23) the
Schrodinger equation is equivalent to

$\begin{split}&\imath\cfrac{\partial\psi_{n}}{\partial
t}=2\psi_{n}-\psi_{n+1}-\psi_{n-1}-\imath
w(\delta_{-1}(n)+\delta_{1}(n))(\psi_{-1}+\psi_{1}),\quad
n\in{\mathbb{Z}}\backslash\\{0\\},\\\ &\psi_{0}(t)=0.\end{split}$ (45)

Let us define

$\displaystyle\hat{\psi}^{\pm}(k,t)=\sqrt{\dfrac{2}{\pi}}\sum_{n=1}^{\infty}\psi_{\pm
n}(t)\,\sin(n\,k)\quad\Rightarrow\quad\psi_{\pm
n}(t)=\sqrt{\dfrac{2}{\pi}}\int_{0}^{\pi}dk\,\hat{\psi}^{\pm}(k,t)\sin(n\,k)$

which satisfy both the same equation

$\imath\cfrac{\partial\hat{\psi}^{\pm}}{\partial
t}(k,t)-2(1-\cos\left(k\right))\hat{\psi}^{\pm}(k,t)=-\imath
w\sqrt{\dfrac{2}{\pi}}\,[\psi_{1}(t)+\psi_{-1}(t)]\,\sin\left(k\right).$ (46)

We assume that $\psi_{n}(0)=\delta_{n,n_{0}}$ with $n_{0}\geq 1$ so that
${\hat{\psi}}^{+}(k,0)=\sqrt{\dfrac{2}{\pi}}\sin\left(n_{0}k\right)$ and
${\hat{\psi}}^{-}(k,0)=0$. The Laplace transforms of $\hat{\psi}^{\pm}(k,t)$
and $\psi_{\pm 1}(t)$ are by definition

$\displaystyle\tilde{\psi}^{\pm}(k,s)=[\mathcal{L}\hat{\psi}^{\pm}(k,\cdot)](s)=\int_{0}^{\infty}dt\,\exp(-st)\hat{\psi}^{\pm}(k,t),$
(47) $\displaystyle[\mathcal{L}\psi_{\pm
1}](s)=\int_{0}^{\infty}dt\,\exp(-st)\,\psi_{\pm
1}(t)=\sqrt{\dfrac{2}{\pi}}\int_{0}^{\pi}dk\,\tilde{\psi}^{\pm}(k,s)\,\sin(k).$
(48)

Taking the Laplace Transform of Eq. (46) and by use of the above equations

$\begin{split}\tilde{\psi}^{+}(k,s)&=\imath\sqrt{\dfrac{2}{\pi}}.\frac{1}{\imath
s-2\left(1-\cos\left(k\right)\right)}\left\\{\sin\left(n_{0}k\right)-w\sin\left(k\right)\mathcal{L}[\psi_{-1}+\psi_{1}](s)\right\\},\\\
\tilde{\psi}^{-}(k,s)&=-\imath\sqrt{\dfrac{2}{\pi}}.\frac{1}{\imath
s-2\left(1-\cos\left(k\right)\right)}w\sin\left(k\right)\mathcal{L}[\psi_{-1}+\psi_{1}](s).\end{split}$
(49)

Then we get

$\left[\mathcal{L}(\psi_{-1}+\psi_{1})\right](s)=\imath\,\dfrac{\displaystyle\dfrac{2}{\pi}\int_{0}^{\pi}dk\,\dfrac{\sin\left(k\right)\sin\left(n_{0}k\right)}{\imath
s-2(1-\cos(k))}}{1+\imath\dfrac{4w}{\pi}\displaystyle\int_{0}^{\pi}dk\,\dfrac{\sin^{2}k}{\imath
s-2(1-\cos(k))}}$ (50)

from which follows the explicit expression for the Fourier-Laplace transform
of the wave function. This is the same expression as Eq. (35) where $w$ has
been replaced by $2w$.

Proceeding as in derivation of Equation Eq. (37), in this case the expression
for the survival probability is

$S_{\infty}=\lim_{t\to\infty}S(t)=1-2\real(w)\int_{0}^{\infty}dt\,\left|\psi_{-1}(t)+\psi_{1}(t)\right|^{2}.$
(51)

If $S_{\mathbb{Z}}(w,n_{0})$ denotes the survival probability $S_{\infty}$
corresponding to this initial condition $\psi_{n}(0)=\delta_{n,n_{0}}$ we get
that

$\begin{split}S_{\mathbb{Z}}(w,n_{0})=S_{\mathbb{N}}(2w,n_{0}).\end{split}$
(52)

###  4.4 Green’s Function approach

It is of interest to recover the value Eq. (40) of the survival probability
and the value Eq. (43) of $\psi_{1}(t)$ by considering the spectral analysis
of $H_{\mathbb{N}}$ performed in App. (A.3). A similar analysis can be
performed on the lattice $\mathbb{Z}$ as in App. (A.4).

We recall that $w=\alpha+\imath\beta$ and we denote
$\zeta=\tfrac{\beta-\imath\alpha}{1+\beta-\imath\alpha}=\tfrac{w}{\imath+w}$.
It is proved in this appendix that the Hamiltonian $H_{\mathbb{N}}$ has a
complete “real orthonormal” basis of eigenfunctions. This basis is composed of
scattering states Eq. (87) $\\{\eta^{k}\;;\;k\in(0,\pi)\\}$ (associated with
the eigenvalue $E_{s}(k)=2(1-\cos(k))$, and, if
$|1-\xi^{-1}|<1\Leftrightarrow|w|>1$, supplemented by a bound state $\eta^{b}$
Eq. (88), associated to the eigenvalue $E_{b}(\zeta)=1+\zeta^{-1}$.

From this spectral analysis we have that the solution $\psi(t)$ of Eq. (29)
with initial condition $\psi(0)$ can be expressed as

$\psi(t)=G(t)\psi(0)$ (53)

where the entries of the infinite matrix $G(t)$ are given by

$G_{nm}(t)=\int_{0}^{\pi}dk\,\eta_{n}^{k}\eta_{m}^{k}\,e^{-\imath
E_{s}(k)t}\;+\;{\bf 1}_{|w|>1}\,\eta_{n}^{b}\eta_{m}^{b}\,e^{-\imath
E_{b}(\zeta)t}.$ (54)

In particular, starting with an initial condition
$|\psi(0)\rangle=\delta_{n_{0}}$ we have that $\psi_{1}(t)=G_{1,n_{0}}(t)$. By
using the explicit forms of $\\{\eta^{k}\;;\;k\in(0,\pi)\\}$, $E_{s}(k)$ in
Eq. (87) and using that $\zeta/(1-\zeta)=-\imath w$ we get

$\eta_{1}^{k}\eta_{n_{0}}^{k}\,e^{-\imath
E_{s}(k)t}=-\cfrac{\imath}{\pi}e^{-2\imath(1-\cos(k))t}\,\sin(k)\left[\cfrac{e^{\imath
n_{0}k}}{1-\imath we^{\imath k}}-\cfrac{e^{-\imath n_{0}k}}{1-\imath
we^{-\imath k}}\right].$

Assuming $|w|<1$ we can transform the previous expression in series since
$|\imath we^{\pm\imath k}|<1$, exchange the sum with the integrals, and by
using symmetries we get

$\int_{0}^{\pi}dk\,\eta_{n}^{k}\eta_{m}^{k}\,e^{-\imath
E_{s}(k)t}=-\cfrac{\imath}{\pi}\sum_{p=0}^{\infty}(\imath
w)^{p}\int_{-\pi}^{\pi}dk\,e^{-2\imath(1-\cos(k))t}\,\sin(k)e^{\imath(n_{0}+p)k}.$
(55)

By performing an integration by parts in the definition of the Bessel function
$J_{p}$ (see Eq. (42)) we get

$\int_{-\pi}^{\pi}dk\,\sin(k)e^{2\imath t\cos(k)}e^{\imath
pk}=\cfrac{\pi\imath^{p}}{t}pJ_{p}(2t)$

and plugging this in Eq. (55) we recover Eq. (43).

If $|w|>1$ we have to take into account the existence of a bound state but for
the rest we proceed in a similar way by rewriting for $|z|=1$

$\cfrac{1}{1-\imath
wz}=\cfrac{\imath}{wz}\,\cfrac{1}{1+\tfrac{\imath}{wz}}=\cfrac{\imath}{wz}\,\sum_{p=0}^{\infty}(-1)^{p}\,\left(\tfrac{\imath}{wz}\right)^{p}.$

Then we get the same expression as before Eq. (43) for $\psi_{1}(t)$. The
expression for $|w|=1$ is obtained from the previous expression by a
continuity extension.

##  5 Continuum limit

In this section we discuss the continuum limit of the infinite models defined
in Section (3) when the lattice spacing $\epsilon$ goes to $0$.

###  5.1 Half-line case

We consider the Hamiltonian $H_{\mathbb{N}}$ given by Eq. (22):

$H_{\mathbb{N}}=-\sum_{n=1}^{\infty}\big{[}\left|n\right\rangle\left\langle
n+1\right|+\left|n+1\right\rangle\left\langle
n\right|-2\left|n\right\rangle\left\langle n\right|\big{]}-\imath
w\left|1\right\rangle\left\langle 1\right|.$

We recall that Schrodinger equation take the form

$\imath\cfrac{\partial\psi_{n}}{\partial t}=\begin{dcases}(2-\imath
w)\psi_{1}-\psi_{2},&n=1,\\\ 2\psi_{n}-\psi_{n-1}-\psi_{n+1},&n\geq
2.\end{dcases}$

These are equivalent to the equations

$\imath\cfrac{\partial\psi_{n}}{\partial
t}=2\psi_{n}-\psi_{n+1}-\psi_{n-1},\leavevmode\nobreak\ \leavevmode\nobreak\
n\geq 1,$ (56)

with the boundary condition $\psi_{0}=\imath w\psi_{1}$ which can be rewritten
in the form

$\psi_{0}-\dfrac{\imath w}{1-\imath w}(\psi_{1}-\psi_{0})=0.$ (57)

We now define a small lattice spacing parameter $\epsilon$ and choose $w$ such
that

$\displaystyle\frac{\imath w}{\imath w-1}=\frac{\zeta}{\epsilon},$ (58)

where $\zeta$ is a finite complex number. We define the continuous wave
function $\Psi$ as 111We assume that the initiale wave fonction $\psi_{n}(0)$
satisfies this property.

$\Psi(x,t)=\lim_{\epsilon\to
0}\epsilon^{-1/2}\psi_{[x/\epsilon]}\big{(}t\epsilon^{-2}\big{)}.$

Then in the limit $\epsilon\to 0$ Eqs. (56,57) reduce to

$\begin{gathered}\imath\frac{\partial\Psi}{\partial
t}=-\frac{\partial^{2}\Psi}{\partial x^{2}},\quad\text{with complex Robin
b.c.}\quad\left[\Psi+\zeta\tfrac{\partial\Psi}{\partial
x}\right]_{x=0}=0.\end{gathered}$ (59)

Note that to obtain the desired limit in Eq. (58) we need to set
$\alpha=\real(w)=\mathcal{O}(\epsilon)$ and
$\beta=\Im{w}=-1+\mathcal{O}(\epsilon)$. Hence from the definition of $\zeta$
and the fact that $\alpha=\real(w)>0$ we observe that ${\imaginary}(\zeta)<0$
while $\real(\zeta)$ can have either sign. Robin boundary conditions for real
$\zeta$ have been studied and they arise in models of generic reflecting walls
allwright2016robin . Here we have $\zeta$ as a complex number.

We now proceed with the solution of the above boundary value problem with a
specified initial wave function $\Psi_{0}(x)=\Psi(x,0)$. The eigenspectrum of
the system in Eq. (59) is discussed in App. (A.1), where it is shown that for
$\real(\zeta)<0$, the spectrum only has scattering eigenfunctions while for
$\real(\zeta)>0$, one has in addition a bound state that is localized near the
origin. In this section we consider this latter case, the discussion would be
similar for the other case. The scattering functions $\\{\eta^{k}\;;\;k>0\\}$
and the bound state $\eta^{b}$ defined respectively by Eq. (75) and Eq. (76)
in App. (A.1) form a complete basis of eigenfunctions of the Laplacian
operator on the half line with the complex Robin boundary condition at the
origin, see Eq. (59). Moreover they satisfy the “real orthogonality relations”
Eq. (77). Then we can write the general time-dependent solution as

$\begin{split}\Psi(x,t)&=\int_{0}^{\infty}dk\leavevmode\nobreak\
c(k)\eta^{k}(x)e^{-\imath k^{2}t}+c_{b}\eta^{b}(x)e^{\imath
t/\zeta^{2}},\leavevmode\nobreak\ \\\ {\rm where}\quad
c(k)&=\int_{0}^{\infty}dx\leavevmode\nobreak\ \Psi_{0}(x)\eta^{k}(x),\quad
c_{b}=\int_{0}^{\infty}dx\leavevmode\nobreak\
\Psi_{0}(x)\eta^{b}(x).\end{split}$ (60)

The coefficients $c(k)$ as well as the oscillating integrals above are well
defined for any initial wave function $\Psi_{0}$ which is sufficiently smooth
since then the $c(k)$’s have a fast decay in $k$. If
$\Psi_{0}\in{\mathbb{L}}^{2}([0,\infty))$ is only square integrable the
previous formula has to be understood by using an approximation of the initial
condition by smooth initial conditions. The approximation scheme propagates in
time thanks to the decay Eq. (67) of the ${\mathbb{L}}^{2}$-norm.

From the explicit formula Eq. (60) and performing a standard saddle point
approximation (see App. (B.2) details) we find the non-universal asymptotic
form

$\lim_{t\to\infty}{\sqrt{2t}}\,\Psi(2ty,t)=-c(y)\;\dfrac{\left(1-\imath\zeta
y\right)}{\sqrt{1+\zeta^{2}y^{2}}}\exp\left[\imath\left(y^{2}t-\tfrac{\pi}{4}\right)\right].$
(61)

Equivalently we can say that in the limit $t\to\infty$, for any finite $x/t$,
the wavefunction tends to the limiting form $\Psi\to\Psi_{\infty}$, where

$\Psi_{\infty}(x,t)=-\frac{1}{\sqrt{2t}}c\big{(}{\tfrac{x}{2t}}\big{)}\;\dfrac{\left(1-\imath\zeta\tfrac{x}{2t}\right)}{\sqrt{1+\zeta^{2}\tfrac{x^{2}}{4t^{2}}}}\exp\left[\imath\left(\tfrac{x^{2}}{4t}-\tfrac{\pi}{4}\right)\right].$
(62)

For $x\lesssim\sqrt{t}$ we need the form of $c(k)$ at $k\to 0$. Using the
ecxplicit form of the wavefunctions in App. (A.1) we find immediately that

$\displaystyle c(k)\sim\frac{2\imath
k}{\sqrt{2\pi}}m_{\Psi_{0}},\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm
where}\leavevmode\nobreak\
m_{\Psi_{0}}=\int_{0}^{\infty}dx\,(x-\zeta)\Psi_{0}(x)$ (63)

Hence we have for $x\lesssim\sqrt{t}$

$\displaystyle\Psi_{\infty}(x,t)\approx m_{\Psi_{0}}\frac{x}{\sqrt{4\pi
t^{3}}}\exp\left[\imath\left(\tfrac{x^{2}}{4t}+\tfrac{3\pi}{4}\right)\right],$
(64)

which has a universal structure apart from the factor $m_{\Psi_{0}}$.

As an explicit numerical example, we now consider the evolution of an initial
wavefunction of the form

$\displaystyle\Psi_{0}(x)=1,\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm
for}\leavevmode\nobreak\ 1<x<2,$

and zero elsewhere. In this case the basis expansion coefficients are given:

$\begin{split}c(k)&=\frac{1}{\sqrt{2\pi(1+k^{2}\zeta^{2})}}\frac{\sin(k/2)}{k/2}\bigg{[}k\zeta\cos(3k/2)-\sin(3k/2)\bigg{]},\\\
c_{b}&=\sqrt{2\zeta}e^{-2/\zeta}(-1+e^{1/\zeta}).\end{split}$

We choose $\zeta=0.2-0.5\imath$. In Fig. (3) we how the evolution of the
wavefunction at early (a) and late times (inset of b). For the the scaled wave
function in Fig. (3)b and we see an excellent agreement with the analytic form
in Eq. (61).

Figure 3:  Plot showing $|\Psi(x,t)|^{2}$ at times $t=0,0.1,0.5$, obtained
from the solution in Eq. (60) with the square initial condition $\Psi(x,0)=1$
for $1<x<2$ and zero elsewhere. The parameter value $\zeta=0.2-0.5i$ was
taken.

The survival probability $S(t)$ is given by

$S(t)=\int_{0}^{\infty}\left|\Psi(x,t)\right|^{2}\,dx$ (65)

and after some straightforward manipulations one can show, using Eq. (59) that

$S(t)=1+2\frac{\imaginary(\zeta)}{\left|\zeta\right|^{2}}\int_{0}^{t}\left|\Psi(0,\tau)\right|^{2}\,d\tau$
(66)

which is the continuous counterpart of Eq. (37). For purely real $\zeta$, i.e.
$\alpha=0$ which corresponds to non-measurement, the system Eq. (59)
determines unitary evolution on half line. We note that the first passage time
distribution is given by

$F(t)=-\cfrac{dS(t)}{dt}=-2\frac{\imaginary(\zeta)}{\left|\zeta\right|^{2}}\left|\Psi(0,t)\right|^{2}\;>\;0.$
(67)

From the asymptotic scaling form in Eq. (61) we find that

$\begin{gathered}S_{\infty}=\lim_{t\to\infty}S(t)=\int_{0}^{\infty}\left|\frac{1-\imath\zeta
k}{1+\imath\zeta k}\right|\left|c({k})\right|^{2}dk.\end{gathered}$ (68)

We are not able to find more explicit forms for $F(t)$ or $S(t)$. However we
can obtain the asymtotic long time form of $F(t)$. We need the wavefunction at
the origin, $\Psi(0,t)$. This can be obtained from the scaling solution in Eq.
(64) by use of the boundary condition
$\Psi(0,t)=-\zeta\left[\tfrac{\partial\Psi}{\partial x}\right]_{x=0}$. This
gives us, for $t\to\infty$, $\Psi(0,t)=[m_{\Psi_{0}}{\zeta}/{\sqrt{4\pi
t^{3}}}]e^{-\imath{3\pi}/{4}}$. Hence we get

$\lim_{t\to\infty}t^{3}F(t)=-\frac{\imaginary(\zeta)}{2\pi}|m_{\Psi_{0}}|^{2}.$
(69)

In Fig. (4) we plot $F(t)$ for the same parameters and initial wavefunction
used in Fig. (3). At large times, we verify the above asymptotic form given in
Eq. (69).

Figure 4:  The first passage time distribution $F(t)$ for the initial
wavepacket and parameter values considered in Fig. (3). The inset shows the
decay $F(t)\sim t^{-3}$ at large times, with a pre-factor given by Eq. (69).

We observe that the righthand side of (69) may vanish for some special initial
condition $\Psi_{0}$. This means that the asymptotic decay of the first
passage time distribution is not universal. An exhaustive study can be
performed showing that it is always possible to start with a special initial
wave function $\Psi_{0}$ such that the asymptotic decay of the first passage
time distribution will be of order $t^{-(2s+1)}$ for some integer $s\geq 1$.
The detailed study is performed in App. (B.2).

###  5.2 Real-line case

The Schrodinger Eq. (45) can be rewritten in the form

$\begin{split}&\imath\cfrac{\partial\psi_{n}}{\partial
t}=2\psi_{n}-\psi_{n+1}-\psi_{n-1},\quad n\in{\mathbb{Z}}\\\ &\psi_{0}=\imath
w(\psi_{-1}+\psi_{1}).\end{split}$ (70)

Assuming that $\imath w=\tfrac{1}{2}+\tfrac{\epsilon}{\zeta}$ where $\zeta$ is
a complex number and defining the continuous wave function $\Psi$ as

$\Psi(x,t)=\lim_{t\to
0}\epsilon^{-1/2}\psi_{[x/\epsilon]}\big{(}t\epsilon^{-2}\big{)},$

we get in the limit $\epsilon\to 0$ the above equation reduces to the
Schrodinger equation with a complex Robin boundary condition at the origin

$\begin{split}&\imath\tfrac{\partial\Psi}{\partial
t}=-\tfrac{\partial^{2}\Psi}{\partial x^{2}},\\\
&2\Psi|_{{x=0}}+{\zeta}\left[\tfrac{\partial\Psi}{\partial
x}|_{x=0^{+}}-\tfrac{\partial\Psi}{\partial
x}|_{x=0^{-}}\right]=0.\end{split}$ (71)

The solution of Eq. (71) can be obtained in the following way. Let us define

$\Psi^{s}(x,t)=\cfrac{\Psi(x,t)+\Psi(-x,t)}{2},\quad\Psi^{a}(x,t)=\cfrac{\Psi(x,t)-\Psi(-x,t)}{2},$

the symmetric and antisymmetric part of the wave function $\Psi$. Both
functions are uniquely determined by their restrictions to the half line
$[0,\infty)$. Eq. (71) implies that on $[0,\infty)$, $\Psi^{s}$ is solution of
the Schrodinger Eq. (59) with a complex Robin boundary condition at the origin
and that on $[0,\infty)$, $\Psi^{a}$ is solution of the Schrodinger equation
with the Dirichlet boundary condition $\Psi^{a}(0,t)=0$. To solve the latter,
we observe that it is in fact sufficient to solve the free Schrodinger
equation on the real line with the initial antisymmetric wave function
$\Psi^{a}$ since this property will be preserved by the free propagator and in
particular the solution will vanish on $0$ at any time $t>0$. It follows that
the solution of Eq. (71) is given by

$\begin{split}\Psi(x,t)&=\int_{0}^{\infty}dk\,\big{[}{\hat{c}}({k}){\hat{\eta}}^{k}(x)+a({k})\sigma^{k}(x)\big{]}\,e^{-\imath
k^{2}t}+{\hat{c}}_{b}{\hat{\eta}}^{b}(x)\,e^{\imath\frac{t}{\zeta^{2}}},\\\
{\text{where}}\quad{\hat{c}}({k})&=\int_{-\infty}^{\infty}dx\,\Psi_{0}(x){\hat{\eta}}^{k}(x),\quad
a({k})=\int_{-\infty}^{\infty}dx\,\Psi_{0}(x)\sigma^{k}(x),\\\
{\hat{c}}_{b}&=\int_{-\infty}^{\infty}dx\,\Psi_{0}(x){\hat{\eta}}^{b}(x),\end{split}$
(72)

with the scattering states ${\hat{\eta}}^{k},\sigma^{k}$ and the bound state
${\hat{\eta}}^{b}$ defined by Eq. (80) and Eq. (81). The latter are nothing
but the eigenfunctions of the Laplacian operator on the real line with complex
Robin boundary condition at the origin as it appears in Eq. (71), see Appendix
(A.2).

The survival probability $S(t)$ is given by

$S(t)=\int_{-\infty}^{\infty}\left|\Psi(x,t)\right|^{2}\,dx$ (73)

and after some straightforward manipulations one can show, using Eq. (71) that

$S(t)=1+4\frac{\imaginary(\zeta)}{\left|\zeta\right|^{2}}\int_{0}^{t}\left|\Psi(0,t^{\prime})\right|^{2}\,dt^{\prime}.$
(74)

Since $\sigma^{k}(0)=0$ and ${\hat{\eta}}^{k}$, ${\hat{\eta}}^{b}$ are even,
we conclude that the first passage time distribution $F=-{\partial
S}/{\partial t}$ is up to a multiplicative constant the same as the one for
the half-line, but starting from the wave function $\Psi_{0}^{s}$ (instead of
$\Psi$) restricted to the half-line. We observe in particular that if we start
with an antisymmetric wave function $\Psi=\Psi^{a}$ then the particle is never
detected.

##  6 Conclusion

Subjecting a quantum particle to repeated projective measurements at regular
intervals of time ($\tau$) to ascertain its observation by a detector is one
of the standard methods to study the quantum time of arrival problem. Since
the work of Allcock allcock1969time , it is believed that an equivalent way to
study the first detection problem is to introduce an imaginary absorbing
potential. In the present work we discussed a precise limiting procedure of
taking $\tau\to 0$ and at the same time allowing the coupling strength between
system and detector to be large (as $\leavevmode\nobreak\ 1/\tau^{1/2}$). We
summarize here our main results:

  * •

Formulating the problem for general quantum systems with a discrete Hilbert
space, we rigorously showed the equivalence between the repeated measurement
protocol and the non-Hermitian description.

  * •

For a quantum particle on a 1D lattice with a detector at one site we then
solved the corresponding Schrodinger equation with a complex potential to
obtain closed form analytic results for the survival probability and the
distribution of first detection time of a particle starting from an arbitrary
initial lattice site. Various asymptotic cases were discussed.

  * •

We then studied the limit of lattice spacing going to $0$ to obtain a
formulation for the continuum case. For the semi-infinite lattice with a
detector at one end we find that the effective description is in terms of free
Schrodinger evolution with complex Robin boundary conditions at the detector
site. Again in this case we provide analytic results for several objects of
interest. The long time asymptotic form of the surviving wave-function was
obtained. We find that while the detection time probability density
generically decays as $1/t^{3}$, it is possible to construct special initial
states for which the decay is faster. A similar dependence of decay exponent
on initial states was observed in a lattice study thiel2018spectral  and it
will be interesting to relate these results.

We note that this non-Hermitian description on the ${\mathbb{N}}$-lattice
remains unchanged even if we consider the detector to be localized on all
sites on the negative axis. Taking the continuum space limit then gives us the
case considered by Allcock allcock1969time . However, we see that for with our
model of system-detector interactions, the non-Hermitian description is in
terms of one with complex Robin boundary conditions instead of one with an
imaginary potential Tumulka2019 . An interesting direction for future work
would be to consider other forms of the system-detector interaction that could
lead to different forms of effective non-Hermitian Hamiltonians. An extension
of our results to higher dimensions would be another interesting question to
explore.

###### Acknowledgements.

VD and AD acknowledge support of the Department of Atomic Energy, Government
of India, under project no.12-R$\&$D-TFR-5.10-1100. C.B. acknowledges support
of MITI-CNRS under the project StrongQU. This work has been supported by the
project RETENU ANR-20-CE40-0005-01 of the French National Research Agency
(ANR) and the French-Indian UCA project “Large deviations and scaling limits
theory for non-equilibrium systems.”

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##  Appendix A Spectrum Analysis

###  A.1 Spectrum on the half-line

The operator appearing in Eq. (59) has a complete set of basis functions
comprising of a continuum of scattering states given by

$\displaystyle\eta^{k}(x)=\cfrac{\imath}{\sqrt{2\pi(1+\zeta^{2}k^{2})}}\left[(1-\imath
k\zeta)e^{\imath kx}-(1+\imath k\zeta)e^{-\imath kx}\right],\quad k>0,$ (75)

and, for $\real[\zeta]>0$, a bound state given by

$\displaystyle\eta^{b}(x)=\sqrt{\frac{2}{\zeta}}e^{-x/\zeta}.$ (76)

Here $\sqrt{z}$ denotes the principal square root of
$z\in\mathbb{C}\backslash{\mathbb{R}}_{-}$ using the nonpositive real axis as
a branch cut. They satisfy the boundary condition at $x=0$ and the “real
orthonormality conditions”:

$\begin{split}&\int_{0}^{\infty}dx\leavevmode\nobreak\
\eta^{k}(x)\eta^{k^{\prime}}(x)=\delta(k-k^{\prime}),\\\
&\int_{0}^{\infty}dx\leavevmode\nobreak\ \eta^{k}(x)\eta^{b}(x)=0,\\\
&\int_{0}^{\infty}dx\leavevmode\nobreak\ [\eta^{b}(x)]^{2}=1.\end{split}$ (77)

This is easily proved using the identity
$\int_{0}^{\infty}dxe^{ikx}=iP(1/k)+\pi\delta(k)$, where $P$ denotes the
principal part. One also has the completeness relation:

$\displaystyle\begin{split}\int_{0}^{\infty}dk\leavevmode\nobreak\
\eta^{k}(x)\eta^{k}(x^{\prime})&=\delta(x-x^{\prime})\leavevmode\nobreak\
\leavevmode\nobreak\ {\rm for}\leavevmode\nobreak\ \real[\zeta]<0,\\\
\int_{0}^{\infty}dk\leavevmode\nobreak\
\eta^{k}(x)\eta^{k}(x^{\prime})+\eta^{b}(x)\eta^{b}(x^{\prime})&=\delta(x-x^{\prime})\leavevmode\nobreak\
\leavevmode\nobreak\ {\rm for}\leavevmode\nobreak\ \real[\zeta]>0.\end{split}$
(78)

To prove this, we note that

$\displaystyle\int_{0}^{\infty}dk\leavevmode\nobreak\
\eta^{k}(x)\eta^{k}(x^{\prime})$
$\displaystyle=-\frac{1}{2\pi}\int_{0}^{\infty}dk\frac{1}{{(1+\zeta^{2}k^{2})}}\left[(1-\imath
k\zeta)e^{\imath kx}-(1+\imath k\zeta)e^{-\imath
kx}\right]\leavevmode\nobreak\ \left[(1-\imath k\zeta)e^{\imath
kx^{\prime}}-(1+\imath k\zeta)e^{-\imath kx^{\prime}}\right]$
$\displaystyle=\delta(x-x^{\prime})-\frac{1}{2\pi}\int_{0}^{\infty}dk\left[\frac{(1-\imath
k\zeta)^{2}e^{\imath k(x+x^{\prime})}}{(1+\zeta^{2}k^{2})}+\frac{(1+\imath
k\zeta)^{2}e^{-\imath k(x+x^{\prime})}}{{(1+\zeta^{2}k^{2})}}\right]$
$\displaystyle=\delta(x-x^{\prime})-\frac{1}{2\pi}\int_{-\infty}^{\infty}dk\frac{(1-\imath
k\zeta)e^{\imath k(x+x^{\prime})}}{(1+ik\zeta)}$
$\displaystyle=\delta(x-x^{\prime})-\frac{1}{\pi}\int_{-\infty}^{\infty}dk\frac{e^{\imath
k(x+x^{\prime})}}{1+ik\zeta},$ (79)

where we used the fact that $\int_{-\infty}^{\infty}dke^{ik(x+x^{\prime})}=0$
since $x,x^{\prime}>0$.Performing the integral, we then get the completeness
relations in Eq. (78).

###  A.2 Spectrum on the real line

The operator appearing in Eq. (71) has a complete set of basis functions
comprising of a continuum of scattering states given by

$\begin{split}{\hat{\eta}}^{k}(x)&=\frac{\imath}{\sqrt{4\pi\left(1+\zeta^{2}k^{2}\right)}}\bigg{[}(1-\imath\zeta
k)e^{\imath k\left|x\right|}-(1+\imath\zeta k)e^{-\imath
k\left|x\right|}\bigg{]},\quad k>0,\\\
\sigma^{k}(x)&=\frac{1}{\sqrt{\pi}}\sin\left(k\,x\right),\quad
k>0.\end{split}$ (80)

and, for $\real[\zeta]>0$, a bound state given by

${\hat{\eta}}^{b}(x)=\frac{1}{\sqrt{\zeta}}\exp\left(-\frac{\left|x\right|}{\zeta}\right).$
(81)

Observe that, up to a multiplicative constant, ${\hat{\eta}}^{k}$ and
${\hat{\eta}}^{b}$ are respectively the symmetrization of ${\eta^{k}}$ and
${\eta^{b}}$. The eigenfunctions $\sigma^{k}$ are odd while
${\hat{\eta}}^{k},{\hat{\eta}}^{b}$ are even. All the functions
${\hat{\eta}}^{b},\,{\hat{\eta}}^{k}$ and $\sigma^{k}$ satisfy the boundary
condition in Eq. (71). Actually $\sigma^{k}$ satisfy the boundary condition
somewhat trivially since they are smooth and $\sigma^{k}(0)=0$. All these
eigenfunctions also satisfy the “real orthonormality conditions”:

$\begin{split}&\int_{-\infty}^{\infty}dx\,[{\hat{\eta}}^{b}]^{2}=1,\\\
&\int_{-\infty}^{\infty}dx\,{\hat{\eta}}^{k}{\hat{\eta}}^{k^{\prime}}=\int_{-\infty}^{\infty}dx\,\sigma^{k}\sigma^{k^{\prime}}=\delta(k-k^{\prime}),\quad
k,k^{\prime}>0,\\\
&\int_{-\infty}^{\infty}dx\,\sigma^{k}{\hat{\eta}}^{k^{\prime}}=\int_{-\infty}^{\infty}dx\,\sigma^{k}{\hat{\eta}}^{b}=\int_{-\infty}^{\infty}dx\,{\hat{\eta}}^{k}{\hat{\eta}}^{b}=0\end{split}$
(82)

and the completeness relation:

$\displaystyle\int_{0}^{\infty}dk\leavevmode\nobreak\
[{\hat{\eta}}^{k}(x){\hat{\eta}}^{k}(x^{\prime})+\sigma^{k}(x)\sigma^{k^{\prime}}(x)]=\delta(x-x^{\prime}),\leavevmode\nobreak\
\leavevmode\nobreak\ {\rm for}\leavevmode\nobreak\ \leavevmode\nobreak\
\real[\zeta]<0$ (83) $\displaystyle\int_{0}^{\infty}dk\leavevmode\nobreak\
[{\hat{\eta}}^{k}(x){\hat{\eta}}^{k}(x^{\prime})+\sigma^{k}(x)\sigma^{k^{\prime}}(x)]+{\hat{\eta}}^{b}(x){\hat{\eta}}^{b}(x^{\prime})=\delta(x-x^{\prime}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm for}\leavevmode\nobreak\
\real[\zeta]>0.$ (84)

The results of Eq. (82) are easily checked by using that
$\int_{-\infty}^{\infty}dxe^{\imath rx}=2\pi\delta(r)$. The completeness
relation Eq. (83) of these eigenfunctions is a consequence of the completeness
relation Eq. (78) of the $\\{\eta^{k}\;\;k>0\\},\eta^{b}$ when restricted to
the half line. Indeed, if $\Psi(x)$ is a square integrable wave function, we
can write it as $\Psi=\Psi^{s}+\Psi^{a}$ where $\Psi^{s}$ (resp. $\Psi^{a}$)
is the symmetric (resp. antisymmetric) part of $\Psi$. We can decompose
$\Psi^{s}$ on $[0,\infty)$ on the basis of the $\eta^{k},\eta^{b}$ thanks to
the completeness relation Eq. (78). Since $\Psi^{s}$, ${\hat{\eta}}^{k}$,
${\hat{\eta}}^{b}$ are symmetric, the decomposition also holds on
$(-\infty,\infty)$ with $\eta^{k}$ (resp. $\eta^{b}$) replaced by
${\hat{\eta}}^{k}$ (resp. ${\hat{\eta}}^{b}$), up to some multiplicative
constants. On the other hand we can decompose the function $\Psi^{a}$ on the
standard Fourier basis
$\Psi^{a}(x)=(2\pi)^{-1/2}\int_{-\infty}^{\infty}dk[{\mathfrak{F}}\Psi^{a}](k)e^{-\imath
kx}$ and by the oddness of $\Psi^{a}$ it follows that
$\Psi^{a}(x)=-\imath\sqrt{{2}}\int_{0}^{\infty}dk[{\mathfrak{F}}\Psi^{a}](k)\sigma^{k}(x)$.
This concludes the proof of the completeness.

###  A.3 Spectrum on the ${\mathbb{N}}$-lattice

We want to study the spectrum of the non-hermitian Hamiltonian
$H_{\mathbb{N}}$ defined by Eq. (22). We start to look for eigenfunctions
$(\psi_{n})_{n\geq 1}$ associated to the complex eigenvalue $E$:

$\begin{split}(2-\imath w)\psi_{1}-\psi_{2}&=E\psi_{1},\\\
2\psi_{n}-\psi_{n-1}-\psi_{n+1}&=E\psi_{n}.\leavevmode\nobreak\
\leavevmode\nobreak\ n>1.\end{split}$ (85)

This is equivalent to solving the second equation for $n\geq 1$ with the
boundary condition

$\psi_{0}+\xi(\psi_{1}-\psi_{0})=0,\quad\xi=\cfrac{-\imath w}{1-\imath w}.$
(86)

The most general scattering solutions are of the form $\psi_{n}=Ae^{\imath
kn}+Be^{-\imath kn}$. This satisfies the second equation of Eq. (85) with
$E_{s}(k)=2(1-\cos(k))=4\sin^{2}(k/2)$. Then the boundary condition Eq. (86)
above gives $B=-A\tfrac{1-\xi+\xi e^{\imath k}}{1-\xi+\xi e^{-\imath k}}$.

This gives us all the scattering solutions with and we define for each
$k\in(0,\pi)$ the eigenfunction $\eta^{k}$ associated to the eigenvalue
$2(1-\cos(k))$ by

$\eta_{n}^{k}=\cfrac{\imath}{\sqrt{2\pi(1-\xi+\xi e^{\imath k})(1-\xi+\xi
e^{-\imath k})}}\,\left[(1-\xi+\xi e^{-\imath k})e^{\imath kn}-(1-\xi+\xi
e^{\imath k})e^{-\imath kn}\right]$ (87)

for any $n\geq 1$.

We can then look for bound state solutions of the form $e^{-kn}$ with
$\real(k)>0$. In this case we find $2(1-\cosh(k))$ and using the boundary
condition it gives $1-\xi^{-1}=e^{-k}$, which has a solution with
${\real}(k)>0$ only for $\left|1-\xi^{-1}\right|<1$ or equivalently $|w|>1$.
In this case we define the bound state $\eta^{b}$ associated to the eigenvalue
$E_{b}=1+\xi^{-1}$ by

$\eta^{b}_{n}=\dfrac{(1-\xi^{-1})^{n}}{\sqrt{1-(1-\xi^{-1})^{2}}}$ (88)

for any $n\geq 1$.

“Real orthogonality relations” and completeness property similar to Eq. (77)
and Eq. (78) may be establish as in the continuum case.

###  A.4 Spectrum on the ${\mathbb{Z}}$-lattice

The spectrum of the non-hermitian Hamiltonian $H_{\mathbb{Z}}$ defined by Eq.
(23) can be deduced from the spectrum of $H_{\mathbb{N}}$ derived in the
previous section. We recall that the equations of motion are given by Eq.
(70). Let $(\psi_{n})_{n\neq 0}$ be an eigenstate associated to the eigenvalue
$E$. It satisfies

$\begin{split}(2-\imath w)\psi_{1}-\psi_{2}-\imath w\psi_{-1}&=E\psi_{1},\\\
(2-\imath w)\psi_{-1}-\psi_{-2}-\imath w\psi_{1}&=E\psi_{-1},\\\
2\psi_{n}-\psi_{n-1}-\psi_{n+1}&=E\psi_{n},\quad|n|\geq 2.\end{split}$ (89)

Introduce the symmetric part
$\psi^{s}=\Big{(}\tfrac{\psi_{n}+\psi_{-n}}{2}\Big{)}_{n\neq 0}$ and the
antisymmetric part
$\psi^{a}=\Big{(}\tfrac{\psi_{n}-\psi_{-n}}{2}\Big{)}_{n\neq 0}$ of $\psi$.
The functions $\psi^{s}$ satisfy Eq. (85) for $n\geq 1$ with $w$ replaced by
$2w$ and is therefore eigenfunctions of $H_{\mathbb{N}}$ associated to the
eigenvalue $E$. On the other hand, the function $\psi^{a}$ satisfies for
$n\geq 1$ the equations

$\begin{split}2\psi^{a}_{1}-\psi^{a}_{2}&=E\psi^{a}_{1},\\\
2\psi^{a}_{n}-\psi^{a}_{n-1}-\psi^{a}_{n+1}&=E\psi^{a}_{n},\quad n\geq
2,\end{split}$

which can be equivalently written as

$2\psi^{a}_{n}-\psi^{a}_{n-1}-\psi^{a}_{n+1}=E\psi^{a}_{n},\quad n\geq 1,$

with the boundary condition $\psi^{a}_{0}=0$. Since $\psi^{a}$ is
antisymmetric, this is equivalent to say that $\psi^{a}$ is an antisymmetric
eigenstate for the discrete Laplacian on $\mathbb{Z}$. Observe moreover that
$\psi^{s}$ and $\psi^{a}$ are “real orthogonal” since $\sum_{n\neq
0}\psi^{a}_{n}\psi_{n}^{s}=0$ due to the symmetry properties of the two
functions.

By introducing $\zeta=\tfrac{-2\imath w}{1-2\imath w}$ we deduce that the
spectrum of $H_{\mathbb{Z}}$ is composed of:

  * •

Symmetric scattering states ${\hat{\eta}}^{k}$ for each $k\in(0,\pi)$
associated to the eigenvalue $2(1-\cos(k))$ and defined for $|n|\geq 1$ by

${\hat{\eta}}_{n}^{k}=\cfrac{\imath}{\sqrt{4\pi(1-\zeta+\zeta e^{\imath
k})(1-\zeta+\zeta e^{-\imath k})}}\,\left[(1-\zeta+\zeta e^{-\imath
k})e^{\imath k|n|}-(1-\zeta+\zeta e^{\imath k})e^{-\imath k|n|}\right].$ (90)

  * •

Antisymmetric scattering states $\sigma^{k}$ for each $k\in(0,\pi)$ associated
to the eigenvalue $2(1-\cos(k))$ and defined for $|n|\geq 1$ by

$\sigma^{k}_{n}=\cfrac{1}{\sqrt{\pi}}\sin(kn).$ (91)

  * •

A symmetric bound state ${\hat{\eta}}^{b}$ associated to the eigenvalue
$E_{b}=1+\zeta^{-1}$ if $\left|1-\zeta^{-1}\right|<1$ or equivalently
$|2w|>1$. It is defined by

${\hat{\eta}}^{b}_{n}=\dfrac{(1-\zeta^{-1})^{|n|}}{\sqrt{2(1-(1-\zeta^{-1})^{2})}}.$
(92)

These eigenstates satisfy “real orthogonality relations” by construction and a
completeness property similar to Eq. (82) and Eq. (83) may be establish as in
the continuum case.

##  Appendix B Technical computations to estimate the survival probability

###  B.1 $\mathbb{N}$ Lattice

####  B.1.1 Computation of the expression given in Eq. (35)

We compute here the quantity

$Z(w,n_{0})=\imath\dfrac{\displaystyle\dfrac{2}{\pi}\int_{0}^{\pi}dk\,\dfrac{\sin\left(k\right)\sin\left(n_{0}k\right)}{\imath
s-2(1-\cos(k))}}{1+\imath\dfrac{2w}{\pi}\displaystyle\int_{0}^{\pi}dk\,\dfrac{\sin^{2}k}{\imath
s-2(1-\cos(k))}}$ (93)

where $w=\alpha+\imath\beta$. We will assume that $s\in\mathbb{C}$ satisfies
$\imaginary(s)\neq-2$ and $s\notin\imath{\mathbb{R}}$.

The integral above can be evaluated by means of contour integration. We first
note that

$\int_{0}^{\pi}dk\,\dfrac{\sin\left(k\right)\sin\left(n_{0}k\right)}{\imath
s-2(1-\cos(k))}=\dfrac{1}{8}\left[-I(s,n_{0}+1)+I(s,n_{0}-1)+I(s,-n_{0}+1)-I(s,-n_{0}-1)\right]$

where

$I(s,n)=\int_{-\pi}^{\pi}dk\,\dfrac{\exp(\imath nk)}{\imath
s-2(1-\cos(k))}=-\imath\int_{\mathcal{C}}dz\,\dfrac{z^{n}}{z^{2}-(2-\imath
s)z+1}.$

The integration contour ${\mathcal{C}}$ in the second integral above is the
unit circle. It is easily seen that $I(s,n)=I(s,-n)$ so that we need only to
compute $I(s,n)$ for $n\geq 0$. The poles of the integrand are

$z_{\pm}(s)=\imath\theta_{\pm}\left(\dfrac{s}{2}+\imath\right)$

where

$\theta_{\pm}(z)=-z\pm\sqrt{z^{2}+1}.$ (94)

We need first to know the positions of theses poles with respect to
${\mathcal{C}}$.

These two functions are well defined analytic functions in the subdomain
$\Omega$, defined as the complex plane $\mathbb{C}$ where the imaginary axis
has been removed apart from the open segment of the imaginary axis between
$-\imath$ and $\imath$. We claim that $\left|\theta_{+}(z)\right|<1$ and
$\left|\theta_{-}(z)\right|>1$ for
$z\in\Omega_{+}=\Omega\bigcap\\{z\in\mathbb{C}:\real(z)>0\\}$ and
$\left|\theta_{+}(z)\right|>1$ and $\left|\theta_{-}(z)\right|<1$ for
$z\in\Omega_{-}=\Omega\bigcap\\{z\in\mathbb{C}:\real(z)<0\\}$. Since the proof
of the two claims are similar we only prove the first one. Notice that
$\theta_{\pm}$ are analytic on the connected set $\Omega_{+}$ and satisfy

$\theta_{+}(z)\cdot\theta_{-}(z)=-1,\,\,\dfrac{\theta_{+}(z)+\theta_{-}(z)}{2}=-z.$

Consider $z\in\Omega_{+}$. If $\theta_{+}(z)$ or $\theta_{-}(z)$ was belonging
to $\mathcal{C}$ then both would be because of the first relation above. It
would then follow that $z=-\imath\sin\left[\arg(\theta_{+}(z))\right]$ which
is excluded since $z\in\Omega_{+}$. A similar ad absurdio argument shows that
$\theta_{-}(z)$ and $\theta_{+}(z)$ do not belong to the imaginary axis. Since
$\theta_{\pm}$ are analytic on the connected set $\Omega_{+}$, so is
$\theta_{\pm}(\Omega_{+})$. The imaginary axis being excluded from
$\theta_{\pm}(\Omega_{+})$, the domains $\theta_{\pm}(\Omega_{+})$ are
included into $\\{z\in\mathbb{C};\real(z)>0\\}$ or into
$\\{z\in\mathbb{C};\real(z)<0\\}$. Since $\theta_{+}(1)=\sqrt{2}-1>0$ and
$\theta_{-}(1)=-\sqrt{2}-1<0$ we get
$\theta_{+}(\Omega_{+})\subset\\{z\in\mathbb{C};\real(z)>0\\}$ and
$\theta_{-}(\Omega_{+})\subset\\{z\in\mathbb{C};\real(z)<0\\}$. Similarly,
since $\theta_{\pm}(\Omega_{+})\bigcap\mathcal{C}=\emptyset$ the domains
$\theta_{\pm}(\Omega_{+})$ are included into the interior of the unit disc or
into the exterior of the unit disc. The values above of $\theta_{\pm}(1)$
imply that $\theta_{+}(\Omega_{+})$ is included in the interior of unit disc
and $\theta_{-}(\Omega_{+})$ is included in the exterior of unit disc. Hence
we have that that if $\real(s)>0$, the pole inside the unit disc is
$z_{+}(s)$.

It follows by the Residues Theorem that

$I(s,n)=-\imath\pi\frac{[z_{+}(s)]^{n}}{\sqrt{\left(\frac{s}{2}+\imath\right)^{2}+1}},\quad\real(s)>0.$
(95)

This gives

$\int_{0}^{\pi}dk\,\dfrac{\sin\left(k\right)\sin\left(n_{0}k\right)}{\imath
s-2(1-\cos k)}=-\dfrac{\pi}{2}[z_{+}(s)]^{n_{0}},\quad{\real}(s)>0.$ (96)

From Eq. (93) and Eq. (96)

$Z(w,n_{0})=-\imath\dfrac{[z_{+}(s)]^{n_{0}}}{1-\imath
w[z_{+}(s)]},\quad{\real}(s)>0.$ (97)

####  B.1.2 Integrale approximation

Here we estimate the integral

$\begin{split}\int_{0}^{1}du\,\frac{u^{2n_{0}-2}(1-u^{2})(1+\left|w\right|^{2}u^{2})}{(1+\left|w\right|^{2}u^{2})^{2}-(2{\imaginary}(w)\,u)^{2}}&=\int_{0}^{\infty}dt\,e^{-(2n_{0}-2)t}g(t)\\\
&=\dfrac{1}{2(n_{0}-1)}\int_{0}^{\infty}dt\,e^{-t}g\left(\dfrac{t}{2(n_{0}-1)}\right)\end{split}$

where

$g(t)=e^{-t}\cfrac{(1-e^{-2t})(1+\left|w\right|^{2}e^{-2t})}{(1+\left|w\right|^{2}e^{-2t})^{2}-(2{\imaginary}(w)\,e^{-t})^{2}}.$

For $t$ near $0$ we have

$g(t)\sim\dfrac{2(1+|w|^{2})}{(1+|w|^{2})^{2}-4[{\imaginary}(w)]^{2}}\,t$

and therefore for large $n_{0}$ we get

$\int_{0}^{1}du\,\frac{u^{2n_{0}-2}(1-u^{2})(1+\left|w\right|^{2}u^{2})}{(1+\left|w\right|^{2}u^{2})^{2}-(2{\imaginary}(w)\,u)^{2}}\sim\cfrac{1}{n_{0}^{2}}\;\dfrac{(1+|w|^{2})}{2(1+|w|^{2})^{2}-8[{\imaginary}(w)]^{2}}.$
(98)

####  B.1.3 Expression of $\psi_{1}$ in terms of Bessel functions of first
kind

We obtain here the explicit formula Eq. (43) for $\psi_{1}$ whose Laplace
transform can be expressed as

$[{\mathcal{L}}\psi_{1}](s)=-\imath^{n_{0}+1}\chi\left(\dfrac{s}{2}+\imath\right)$

where

$\chi(z)=\dfrac{[\theta_{+}(z)]^{n_{0}}}{1+w\theta_{+}(z)}$

with the function $\theta_{\pm}$ defined in Eq. (94). We recall that this
function is analytic in the subdomain $\Omega$, defined as the complex plane
$\mathbb{C}$ where the imaginary axis has been removed apart from the open
segment of the imaginary axis between $-\imath$ and $\imath$. Moreover we have
seen that

$\begin{split}{\real}(z)>0\quad\text{implies}\quad|\theta_{+}(z)|<1,\quad|\theta_{-}(z)|>1.\end{split}$

Let us first identify $\chi(z)$ as the Laplace transform $[{\mathcal{L}}f](z)$
of some explicit function $f$. It is known that if $z\in\mathbb{C}$ satisfies
${\real}(z)>0$ then

$\int_{0}^{\infty}dte^{-zt}\dfrac{J_{k}(t)}{t}=\cfrac{(\theta_{+}(z))^{k}}{k}.$

Assuming that $z$ is such that $|w\theta_{+}(z)|<1$ we have then that

$\begin{split}\chi(z)&=\sum_{k=0}^{\infty}(-w)^{k}[\theta_{+}(z)]^{n_{0}+k}=\sum_{k=0}^{\infty}(k+n_{0})(-w)^{k}\int_{0}^{\infty}dte^{-zt}\dfrac{J_{k+n_{0}}(t)}{t}\\\
&=\int_{0}^{\infty}dte^{-zt}f(t)dt=[\mathcal{L}f](z)\end{split}$

with

$f(t)=\sum_{k=0}^{\infty}(k+n_{0})(-w)^{k}\dfrac{J_{k+n_{0}}(t)}{t}.$

The interversion of the sum and the integral is justified since
$J_{k}(2t)\sim(2\pi k)^{-1/2}(et/k)^{k}$ for large $k$. It follows that for
any time $t\geq 0$

$\psi_{1}(t)=-2\imath^{n_{0}+1}e^{-2\imath
t}\sum_{k=0}^{\infty}(k+n_{0})(-w)^{k}\dfrac{J_{k+n_{0}}(2t)}{2t}$

since the functions above have their Laplace transform coinciding on
$\\{s\in\mathbb{C}\;;\;\real(s)>\sigma\\}$ for some $\sigma>0$ sufficiently
large.

###  B.2 Estimate of the asymptotic forms of the wavefunction and of the first
passage time distribution on the half line

Saddle point approximation for the wavefunction: To seek the saddle point
approximation of

$\displaystyle\Psi(x,t)$
$\displaystyle=\int_{0}^{\infty}dk\,c(k)\,\eta^{k}(x)\,\exp(-\imath\,k^{2}t)$
(99)
$\displaystyle=\frac{\imath}{\sqrt{2\pi}}\Bigg{[}\int_{0}^{\infty}dk\,c(k)\frac{(1-\imath\zeta
k)e^{\imath kx-\imath
k^{2}t}}{\sqrt{1+\zeta^{2}k^{2}}}-\int_{0}^{\infty}dk\,c(k)\frac{(1+\imath\zeta
k)e^{-\imath kx-\imath k^{2}t}}{\sqrt{1+\zeta^{2}k^{2}}}\Bigg{]}$ (100)
$\displaystyle=\frac{\imath}{\sqrt{2\pi}}\int_{-\infty}^{\infty}dk\,c(k)\frac{(1-\imath\zeta
k)}{\sqrt{1+\zeta^{2}k^{2}}}e^{\imath t(k\frac{x}{t}-k^{2})},$ (101)

where we used the fact that $c(-k)=-c(k)$. We now use the following result
[29]: Let $g(k)$ be complex valued, $I(k)$ be real valued functions of the
real variable $k$. Let $t>0$ and define the integral

$F(t)=\int_{-\infty}^{\infty}dk\,g(k)\,e^{\imath t\,I(k)}.$ (102)

If $k_{0}$ is a stationary point of $I(k)$ such that
$\big{[}\frac{dI(k)}{dk}\big{]}_{k=k_{0}}=I^{\prime}(k_{0})=0$ and
$\big{[}\frac{d^{2}I(k)}{dk^{2}}\big{]}_{k=k_{0}}=I^{\prime\prime}(k_{0})\neq
0$ then for large $t$, the contribution to $F(t)$ from $k_{0}$ is

$\Bigg{[}\frac{2\pi}{t\left|I^{\prime\prime}(k_{0})\right|}\Bigg{]}^{1/2}g(k_{0})e^{\imath
t\,I(k_{0})+\imath\frac{\text{sign}(I^{\prime\prime}(k_{0}))\pi}{4}}+o(t^{-1/2}).$
(103)

In the current case

$g(k)=c(k)\frac{(1-\imath\zeta
k)}{\sqrt{1+\zeta^{2}k^{2}}}\,\,\,\,\&\,\,\,\,I(k)=k\frac{x}{t}-k^{2}.$ (104)

We get a unique saddle point at $k_{0}=x/(2t)$ and this then gives

$\left[\Psi(x,t)\right]_{t\rightarrow\infty}\asymp\frac{\imath}{\sqrt{2t}}c\left(\frac{x}{2t}\right)\sqrt{\frac{1-\imath\zeta\frac{x}{2t}}{1+\imath\zeta\frac{x}{2t}}}e^{\imath\left(\frac{x^{2}}{4t}-\frac{\pi}{4}\right)}.$
(105)

First detection distribution: Recall Eq. (60) giving the expression of the
solution of the Schrodinger equation on the half-line with complex Robin
boundary condition and initial condition $\Psi_{0}(x)=\Psi(x,0)$. To simplify
we assume that $\Psi_{0}$ has compact support and has a bounded derivative
defined almost everywhere. In particular all the moments of $\Psi_{0}$ and
$\tfrac{\partial\Psi_{0}}{\partial x}$ are well defined. After a change of
variables we can write $\Psi(0,t)$ as

$\Psi(0,t)=\sqrt{\frac{2}{\pi}}\frac{\zeta}{t}\int_{0}^{\infty}\frac{y}{\sqrt{1+\frac{\zeta^{2}}{t}y^{2}}}c\Big{(}{\tfrac{y}{\sqrt{t}}}\Big{)}\exp\left(-\imath
y^{2}\right)\,dy\;+\;\sqrt{\frac{2}{\zeta}}c_{b}\exp\left(\imath\frac{t}{\zeta^{2}}\right),$

where

$c({k})=-\sqrt{\frac{2}{\pi(1+\zeta^{2}k^{2})}}k\,H(k),$

$H(k)=\int_{0}^{\infty}\underbrace{\bigg{[}\Psi_{0}(x)+\zeta\frac{\partial\Psi_{0}}{\partial
x}(x)\bigg{]}}_{\Phi(x)}\frac{\sin\left(k\,x\right)}{k}dx.$

Up to factors, $H(k)$ is the sine transform of $\Phi(x)$. If $H(k)$ were to
vanish identically, then $\Phi(x)\equiv 0$ and $\Psi_{0}(x)$ would correspond
to the bound state. In this situation $\Psi(0,t)$ evolves solely by the second
term and the first passage time distribution $F(t)$ defined by (67) has
exponential decay for $\zeta$ lying in the fourth quadrant.

Assume $H(k)$ is not identically $0$. Substituting for $c({k})$ one has

$\Psi(0,t)\approx-\frac{2}{\pi}\frac{\zeta}{t^{\frac{3}{2}}}\int_{0}^{\infty}\frac{y^{2}}{1+\frac{\zeta^{2}}{t}y^{2}}H\left(\frac{y}{\sqrt{t}}\right)\exp\left(-\imath
y^{2}\right)\,dy$

where the exponentially decaying term is neglected. By expanding
$\sin\left(k\,x\right)$, and switching 222The switching is justified since
$\Psi_{0}$ has compact support so that the moments of $\Phi$ grows at most
geometrically. the summation and integration for $\Phi(x)$ one can obtain an
absolutely convergent series expansion for $H\left(\frac{y}{\sqrt{t}}\right)$.
We get that

$\begin{split}H\left(\frac{y}{\sqrt{t}}\right)&=\sum_{s=0}^{\infty}\frac{(-1)^{s}}{t^{s}}\frac{y^{2s}}{(2s+1)!}\int_{0}^{\infty}x^{2s+1}\Phi(x)\,dx\\\
&=\sum_{s=0}^{\infty}\frac{(-1)^{s}}{t^{s}}\frac{y^{2s}}{(2s+1)!}\left[M_{2s+1}-(2s+1)\zeta
M_{2s}\right]\end{split}$ (106)

where $M_{s}=\int_{0}^{\infty}x^{s}\Psi_{0}(x)\,dx$ the $s$th moment of
$\Psi_{0}$. In the second line we perform an integration by parts. This allows
us to finally write down the series

$\begin{split}\Psi(0,t)&\approx-\frac{2}{\pi}\frac{\zeta}{t^{\frac{3}{2}}}\bigg{[}\sum_{s=0}^{\infty}\frac{(-1)^{s}}{t^{s}}\frac{\left[M_{2s+1}-(2s+1)\zeta
M_{2s}\right]I_{s}(\frac{\zeta}{\sqrt{t}})}{(2s+1)!}\bigg{]}\\\
&\approx-\frac{2}{\pi}\frac{\zeta}{t^{\frac{3}{2}}}\left[\sum_{s=0}^{\infty}\frac{(-1)^{s}}{(2s+1)!\,t^{s}}{M_{2s+1}\left(1-\tfrac{\zeta}{\zeta_{s}}\right)I_{s}\Big{(}\tfrac{\zeta}{\sqrt{t}}\Big{)}}\right]\end{split}$
(107)

where

$I_{s}(z)=\int_{0}^{\infty}\frac{y^{2s+2}}{1+z^{2}y^{2}}\exp(-\imath
y^{2})\,dy$ (108)

and

$\zeta_{s}=\cfrac{1}{2s+1}\cfrac{M_{2s+1}}{M_{2s}}$ (109)

with the convention that $\zeta_{s}=\infty$ if $M_{2s}=0$. Noting that

$\lim_{z\rightarrow
0}I_{0}(z)=-\frac{\sqrt{\pi}}{4}e^{\frac{\imath\pi}{4}}\quad{\text{and}}\quad\lim_{z\rightarrow
0}I_{1}(z)=-\frac{3\sqrt{\pi}}{8}e^{-\frac{\imath\pi}{4}}$

one has then for large $t$

$\Psi(0,t)=\frac{1}{2\sqrt{\pi}}\frac{\zeta}{t^{\frac{3}{2}}}\bigg{[}\Big{(}1-\tfrac{\zeta}{\zeta_{0}}\Big{)}M_{1}e^{\frac{\imath\pi}{4}}\;-\;\Big{(}1-\tfrac{\zeta}{\zeta_{1}}\Big{)}\frac{M_{3}}{4t}e^{-\frac{\imath\pi}{4}}\bigg{]}\;+\;O(t^{-5/2}).$
(110)

This proves Eq. (69).

Consider the state

$\Psi_{0}(x)=\Theta(x-2)-\frac{\Theta(x-1)+\Theta(x-3)}{2}+\imath\frac{\Theta(x-1)-\Theta(x-3)}{2}$
(111)

where $\Theta(x)$ is the Heaviside step function. Then

$M_{1}=\frac{1}{2}+2\imath,\,\,\,\zeta_{0}=2-\frac{\imath}{2}$
$M_{3}=\frac{25}{4}+10\imath,\,\,\,\zeta_{1}=\frac{67}{82}-\frac{17i}{164}.$

The numerical evaluation of the integral in Eq. (60) can be performed to
obtain $\Psi(0,t)$. We choose the range $t\in(200,750)$ which is sufficient to
suppress the bound state contribution in Eq. (60). To begin with let
$\zeta=1-\imath$ . Then $\zeta$ is at a sufficient distance from $\zeta_{0}$
and the first term in the expansion Eq. (110) dominates. Therefore one has as
$t\to\infty$ that

$F_{3}(t)=-2\frac{\imaginary(\zeta)}{\left|\zeta\right|^{2}}\left|\Psi(0,t)\right|^{2}\sim\frac{5}{8\pi
t^{3}}.$

We use this analytic estimate to compare with the numerical evaluation via Eq.
(60) in Fig $\eqref{fig:FNumvsAsymp}$. This shows that the analytic estimate
is quite good and indeed $F(t)\sim\frac{1}{t^{3}}$.

Figure 5:  Plot comparing numerical estimates of $F(t)$ with $F_{3}(t)$ (resp.
$F_{5}(t)$) for $\zeta=1-\imath$ (resp. $\zeta(=\zeta_{0})=2-\imath/2$). The
discrete points correspond to numerical values evulated from Eq. (60)

.

Now choose $\zeta=\zeta_{0}$. Doing so causes the first term in the expansion
Eq. (110) to drop out. The estimate for $F(t)$ is obtained from Eq. (110)
which gives

$F_{5}(t)\asymp-\frac{\imaginary(\zeta_{0})}{32\pi
t^{5}}\left|M_{3}\Big{(}1-\frac{\zeta_{0}}{\zeta_{1}}\Big{)}\right|^{2}=\frac{5105}{1024}\frac{1}{\pi
t^{5}}.$

Figure $\eqref{fig:FNumvsAsymp}$ shows good agreement of the above estimate
with numerical values.

We could now claim that if $\Psi_{0}(x)$ was so constructed that

$\zeta_{0}=\zeta_{1}=\cdots=\zeta_{s-1}\neq\zeta_{s}(\neq 0)$

then one has

$F(t;\zeta=\zeta_{0})\sim\frac{1}{t^{3+2s}}$

while for other choices of $\zeta$ one has

$F(t,\zeta\neq\zeta_{0})\sim\frac{1}{t^{3}}.$

This is under the assumption that the moments
$\int_{0}^{\infty}x^{k}\Psi_{0}(x)\,dx$ do not vanish up until $k=2s+1$.
Lastly, we note that for the normalized state

$\Psi_{0}(x)=\sqrt{\frac{1}{\zeta_{0}}+\frac{1}{\zeta_{0}^{*}}}\exp\big(-\frac{x}{\zeta_{0}}\big{missing})$

with $\zeta_{0}$ in the fourth quadrant, $\zeta_{s}=\zeta_{0}$ for all $s$
follows from Eq. $\eqref{eq:Defzetas}$. For the measurement parameter
$\zeta=\zeta_{0}$, the estimate in Eq. $\eqref{eq:PsiApprox}$ is identically
$0$ and $F(t)$ falls exponentially, as has already been noted.