# Superresolution imaging of two incoherent optical sources with unequal
brightnesses

Jian-Dong Zhang<EMAIL_ADDRESS>School of Mathematics and Physics,
Jiangsu University of Technology, Changzhou 213001, China  Yiwen Fu  School of
Mathematics and Physics, Jiangsu University of Technology, Changzhou 213001,
China  Lili Hou  School of Mathematics and Physics, Jiangsu University of
Technology, Changzhou 213001, China  Shuai Wang  School of Mathematics and
Physics, Jiangsu University of Technology, Changzhou 213001, China

###### Abstract

Resolving the separation between two incoherent optical sources with high
precision is of great significance for fluorescence imaging and astronomical
observations. In this paper, we focus on a more general scenario where two
sources have unequal brightnesses. We give the ultimate precision limit with
respect to separation by using the quantum Fisher information. Through the
calculation of the classical Fisher information, we analyze and compare
several specific measurement schemes including direct measurement, Gaussian
mode measurement and zero-photon measurement. The results indicate that
Gaussian mode measurement is the nearly optimal for a small separation. Our
work provides a positive complement to the aspect of superresolution imaging
of incoherent sources.

##  I Introduction

For a realistic imaging system, diffraction effects are unavoidable due to
various spatial restrictions. In this circumstance, an ideal optical point
passing through the imaging system evolves into an optical spot with a certain
spatial size. When there are two closely incoherent optical sources on the
object plane, their images on the image plane will overlap extensively. At
this time, the estimation with respect to the separation between two images is
crucial, for the results can provide the position information regarding
optical sources. The estimation precision is directly proportional to the
resolution of an imaging system. A high-resolution imaging system is
beneficial for many applications ranging from microscopic fluorescence imaging
to macroscopic astronomical imaging.

In 2016, Tsang _et al._ analyzed the ultimate precision limit with respect to
the separation between two incoherent sources through the use of the quantum
Fisher information Tsang  _et al._ (2016). An interesting result is that the
precision limit remains constant irrespective of the separation. In stark
contrast, the precision given by classical direct measurement becomes worse as
the separation tends to zero. This indicates that, for a small separation,
there exist measurement schemes which are superior to direct measurement in
precision. Since then, incoherent imaging has received a lot of attention and
many super-resolved measurement schemes have been discussed. Tsang proposed a
spatial-mode demultiplexing (SPADE) measurement scheme Tsang (2018, 2017).
Nair _et al._ showed a measurement scheme based on super localization by image
inversion interferometry (SLIVER) Nair and Tsang (2016). Tham _et al._
analyzed the measurement scheme using super-resolved position localisation by
inversion of coherence along an edge (SPLICE) Tham  _et al._ (2017); Bonsma-
Fisher  _et al._ (2019). The corresponding proof-of-principle experiments have
been also presented Tang  _et al._ (2016); Paúr  _et al._ (2016); Larson  _et
al._ (2019); Zhou  _et al._ (2019); Boucher  _et al._ (2020); Tan  _et al._
(2023).

The above studies are a beneficial exploration for improving the resolution of
imaging systems. However, these studies focused on two equal-brightness
sources. Related to this, some works paid attention to a more general scenario
where two sources have unequal brightnesses Řehaček  _et al._ (2017); Řeháček
_et al._ (2018); Prasad (2020). The precision limit was analyzed through the
quantum Fisher information, and the optimal decomposition modes were
calculated. More recently, Xin _et al._ showed a specific super-resolved
measurement scheme by utilizing fractional Hilbert transform Xin  _et al._
(2021). In this paper, we provide a positive complement to the aspect of
separation estimation of two unequal-brightness sources. We report the
precision of Gaussian mode measurement and that of zero-photon measurement,
which are two specific super-resolved measurement schemes.

The remainder of this paper is organized as follows. Section II introduces the
fundamental model and the corresponding quantum Fisher information. In Sec.
III, we analyze direct measurement, Gaussian mode measurement and zero-photon
measurement with the method of the classical Fisher information. The
comparison between three measurement schemes is also shown. Finally, we
summarize our work in Sec. IV.

##  II Model and corresponding precision limit of incoherent sources

In Fig. 1, we give the model of two incoherent sources with unequal
brightnesses. Two sources passing through an optical system are imaged on the
image plane. The precision limit of a symmetrically distributed model has been
discussed in Refs. Řehaček  _et al._ (2017) and Řeháček  _et al._ (2018).
Throughout this paper we consider an asymmetrically distributed model in Refs.
Zanforlin  _et al._ (2022) and Huang and Lupo (2021). Specifically, the images
of two sources are located at positions of 0 and $d$, respectively. The
separation $d$ is the estimated parameter.

Figure 1: (a) Schematic diagram of superresolution imaging of two incoherent
optical sources with unequal brightnesses. (b) The images on the image plane.
The centers of two images are located at 0 and $d$.

We first direct our attention to the precision limit of our model. Without
loss of generality, we assume that the point spread function is Gaussian and
the number of photons on the image plane satisfies $\epsilon\ll 1$. At this
time, the quantum state on the image plane can be well approximated as

$\displaystyle\rho\approx\left({1-\epsilon}\right){\rho_{0}}+\epsilon{\rho_{1}}$
(1)

where
${\rho_{0}}=\left|{{\rm{vac}}}\right\rangle\left\langle{{\rm{vac}}}\right|$ is
the zero-photon state, and the one-photon state can be written as

$\displaystyle{\rho_{1}}=\left({1-q}\right)\left|0\right\rangle\left\langle
0\right|+q\left|d\right\rangle\left\langle d\right|,$ (2)

where $q$ is a parameter that describes the relative brightness of two
sources.

In the coordinate representation, the amplitude profiles of two sources are
found to be

$\displaystyle\left\langle{x}{\left|{\vphantom{x{{0}}}}\right.\kern-1.2pt}{{{0}}}\right\rangle={\left({\frac{1}{{2\pi}}}\right)^{{1\mathord{\left/{\vphantom{14}}\right.\kern-1.2pt}4}}}\exp\left({-\frac{{{x^{2}}}}{{4}}}\right),$
(3)
$\displaystyle\left\langle{x}{\left|{\vphantom{x{{d}}}}\right.\kern-1.2pt}{{{d}}}\right\rangle={\left({\frac{1}{{2\pi}}}\right)^{{1\mathord{\left/{\vphantom{14}}\right.\kern-1.2pt}4}}}\exp\left[{-\frac{{{{\left({x-d}\right)}^{2}}}}{4}}\right],$
(4)

where we used ${\sigma=1}$. It is not difficult to find that the states of two
sources on the image plane are not orthogonal, i.e.,

$\displaystyle\left\langle{0}{\left|{\vphantom{0d}}\right.\kern-1.2pt}{d}\right\rangle=\exp\left({-\frac{{{d^{2}}}}{{8}}}\right)\equiv\delta\neq
0.$ (5)

In order to calculate the quantum Fisher information, we need to find a set of
orthogonal eigenstates of $\rho_{1}$. For our model, we note that $\rho_{1}$
only contains two states $\left|0\right\rangle$ and $\left|d\right\rangle$.
This feature enables us to expand $\rho_{1}$ in terms of two eigenstates
$\left|{{\lambda_{1}}}\right\rangle$ and $\left|{{\lambda_{2}}}\right\rangle$
with nonzero eigenvalues $\lambda_{1}$ and $\lambda_{2}$. Namely, the one-
photon state can be expressed in the form of spectral decomposition

$\displaystyle{\rho_{1}}={\lambda_{1}}\left|{{\lambda_{1}}}\right\rangle\left\langle{{\lambda_{1}}}\right|+{\lambda_{2}}\left|{{\lambda_{2}}}\right\rangle\left\langle{{\lambda_{2}}}\right|,$
(6)

where
$\left\langle{\lambda_{1}}{\left|{\vphantom{{\lambda_{1}}{\lambda_{2}}}}\right.\kern-1.2pt}{\lambda_{2}}\right\rangle=0$.
The specific forms of eigenvalues and eigenstates can be taken as

$\displaystyle\lambda_{1}$ $\displaystyle={\frac{1}{2}}\left(1+\Delta\right)$
(7) $\displaystyle\lambda_{2}$
$\displaystyle={\frac{1}{2}}\left(1-\Delta\right)$ (8)
$\displaystyle\left|{{\lambda_{1}}}\right\rangle$
$\displaystyle={A_{1}}\left|0\right\rangle+{B_{1}}\left|d\right\rangle,$ (9)
$\displaystyle\left|{{\lambda_{2}}}\right\rangle$
$\displaystyle={A_{2}}\left|0\right\rangle+{B_{2}}\left|d\right\rangle,$ (10)

where

$\displaystyle\Delta=\sqrt{{{\left({1-2q}\right)}^{2}}+{\delta^{2}}\left[{1-{{\left({1-2q}\right)}^{2}}}\right]}$
(11)

and the parameters are given by

$\displaystyle{A_{1}}$
$\displaystyle=\sqrt{\frac{{\left({1-q}\right)\left[{\Delta+\left({1-2q}\right)}\right]}}{{\Delta\left({1+\Delta}\right)}}}$
(12) $\displaystyle{B_{1}}$
$\displaystyle=\sqrt{\frac{{q\left[{\Delta-\left({1-2q}\right)}\right]}}{{\Delta\left({1+\Delta}\right)}}}$
(13) $\displaystyle{A_{2}}$
$\displaystyle=\sqrt{\frac{{\left({1-q}\right)\left[{\Delta-\left({1-2q}\right)}\right]}}{{\Delta\left({1-\Delta}\right)}}}$
(14) $\displaystyle{B_{2}}$
$\displaystyle=-\sqrt{\frac{{q\left[{\Delta+\left({1-2q}\right)}\right]}}{{\Delta\left({1-\Delta}\right)}}}.$
(15)

Based on the above results, the quantum Fisher information can be calculated
in terms of Liu  _et al._ (2019)

$\displaystyle{\cal Q}=$
$\displaystyle\sum\limits_{i=1}^{2}{\frac{1}{{{\lambda_{i}}}}{{\left({\frac{{\partial{\lambda_{i}}}}{{\partial
d}}}\right)}^{2}}}+\sum\limits_{i=1}^{2}{4{\lambda_{i}}\left({\frac{{\partial\left\langle{{\lambda_{i}}}\right|}}{{\partial
d}}}\right)\left({\frac{{\partial\left|{{\lambda_{i}}}\right\rangle}}{{\partial
d}}}\right)}$
$\displaystyle-\sum\limits_{i,j=1}^{2}{\frac{{8{\lambda_{i}}{\lambda_{j}}}}{{{\lambda_{i}}+{\lambda_{j}}}}}\left({\frac{{\partial\left\langle{{\lambda_{i}}}\right|}}{{\partial
d}}\left|{{\lambda_{j}}}\right\rangle}\right)\left({\left\langle{{\lambda_{j}}}\right|\frac{{\partial\left|{{\lambda_{i}}}\right\rangle}}{{\partial
d}}}\right).$ (16)

The ultimate precision limit is the reciprocal of square root of the quantum
Fisher information.

In order to intuitively observe the quantum Fisher information against the
separation, we provide the result in Fig. 2. It turns out that the quantum
Fisher information does not remain constant when the separation changes. The
minimum value occurs at approximately $d=2$ for any value of $q$. However, the
fluctuation degree of the quantum Fisher information is different for
different values of $q$. In addition, the quantum Fisher information increases
with the increase of $q$. This is due to the fact that the second source
becomes brighter as the value of $q$ increases, and this source carries
information about the separation.

Figure 2: The quantum Fisher information as a function of the separation with
different values of $q$.

Figure 3: The classical Fisher information of three measurement schemes as a
function of the separation, (a) $q=0.1$, (b) $q=0.3$, (c) $q=0.7$, (d)
$q=0.9$.

##  III Calculation and comparison of specific measurement schemes in
precision

In this section, we turn our attention to analyze two specific measurement
schemes including Gaussian mode measurement and zero-photon measurement. As a
benchmark, we first analyze the classical direct measurement scheme. On the
image plane, the intensity profile reads

$\displaystyle\Lambda$ $\displaystyle=\left\langle
x\right|{\rho_{1}}\left|x\right\rangle$
$\displaystyle=\frac{{1-q}}{{\sqrt{2\pi}}}\exp\left({-\frac{{{x^{2}}}}{2}}\right)+\frac{q}{{\sqrt{2\pi}}}\exp\left[{-\frac{{{{\left({x-d}\right)}^{2}}}}{2}}\right].$
(17)

Since the intensity profile is continuous in the coordinate representation,
the corresponding classical Fisher information can be calculated according to

$\displaystyle{{\cal F}_{\rm
D}}=\int_{-\infty}^{\infty}{dx\frac{1}{\Lambda}{{\left({\frac{{\partial\Lambda}}{{\partial
d}}}\right)}^{2}}}.$ (18)

Gaussian mode measurement is a binary scheme belonging to the SPADE
measurement scheme. Specifically, a single-mode optical fiber is aligned with
the position of 0 and resolves the spatial mode of photons on the image plane.
Correspondingly, only photons in Gaussian mode can be coupled into the fiber.
The probabilities of identifying Gaussian and non-Gaussian modes are denoted
as ${P_{\rm G}}$ and $1-{P_{\rm G}}$, respectively. The probability ${P_{\rm
G}}$ is given by

$\displaystyle{P_{\rm G}}=\left\langle
0\right|{\rho_{1}}\left|0\right\rangle=1+q\left[{\exp\left({-\frac{{{d^{2}}}}{4}}\right)-1}\right],$
(19)

and the corresponding classical Fisher information turns out to be

$\displaystyle{{\cal F}_{\rm G}}=\frac{1}{{{P_{\rm
G}}}}{\left({\frac{{\partial{P_{\rm G}}}}{{\partial
d}}}\right)^{2}}+\frac{1}{{1-{P_{\rm{G}}}}}{\left[{\frac{{\partial\left({1-{P_{\rm{G}}}}\right)}}{{\partial
d}}}\right]^{2}}.$ (20)

Zero-photon measurement is also a binary scheme, which belongs to the SLIVER
measurement scheme. In this scheme, we need to spatially separate the photons
on the image plane according to symmetric and antisymmetric parts. This can be
achieved through the use of a Mach-Zehnder interferometer embedded with a Dove
prism Nair and Tsang (2016). Then we record the photons originating from the
antisymmetric part. The probabilities of identifying zero and non-zero photons
are denoted as ${P_{\rm Z}}$ and $1-{P_{\rm Z}}$, respectively. This
measurement can be achieved by using a Gm-APD that can only distinguish the
presence or absence of photons. By using semiclassical method (see Appendix
for details), the probability ${P_{\rm Z}}$ is found to be

$\displaystyle{P_{\rm{Z}}}=\frac{2}{{2+q\left[{1-\exp\left({-{{{d^{2}}}\mathord{\left/{\vphantom{{{d^{2}}}2}}\right.\kern-1.2pt}2}}\right)}\right]}}.$
(21)

Accordingly, the classical Fisher information is given by

$\displaystyle{{\cal F}_{\rm
Z}}=\frac{1}{{{P_{{\rm{Z}}}}}}{\left({\frac{{\partial{P_{{\rm{Z}}}}}}{{\partial
d}}}\right)^{2}}+\frac{1}{{1-{P_{\rm{Z}}}}}{\left[{\frac{{\partial\left({1-{P_{\rm{Z}}}}\right)}}{{\partial
d}}}\right]^{2}}.$ (22)

The specific precision of each measurement scheme is the reciprocal of square
root of the corresponding classical Fisher information.

As a comparison among the above measurement schemes, we give the dependence of
the classical Fisher information on the separation, as shown in Fig. 3. The
results indicate that, for a small separation, there is a significant
difference between the classical Fisher information of the direct measurement
and the quantum Fisher information. On the other hand, the classical Fisher
information of Gaussian mode measurement and that of zero-photon measurement
can saturate with the quantum Fisher information when the separation
approaches 0. By comparison, the classical Fisher information of Gaussian mode
measurement is superior to that of zero-photon measurement. This suggests that
Gaussian mode measurement is the approximately optimal scheme for a small
separation.

To quantify the optimality, in Fig. 4 we show the ratio of classical Fisher
information of Gaussian mode measurement to the quantum Fisher information.
This ratio does not change significantly with the value of $q$, especially
within the interval of $0\leq d\leq 0.5$. In particular, when the separation
satisfies $d<0.283$, the classical Fisher information of Gaussian mode
measurement can exceed 99% of the quantum Fisher information. That is,
Gaussian mode measurement can maintain approximately optimal precision over a
considerable range of the separation.

It should be noted that the use of Gaussian mode measurement in this paper is
for the sake of simplicity in experimental implementation. In theory, one can
further improve the classical Fisher information if the measured modes are
increased. A direct result is that the measurement can provide approximately
optimal precision over a larger separation range.

Figure 4: The ratio of classical Fisher information of Gaussian mode
measurement to the quantum Fisher information as a function of the separation
with different values of $q$.

##  IV Conclusion

In summary, the separation estimation problem in the scenario of two unequal-
brightness incoherent sources was addressed. We calculated the quantum Fisher
information of our model. Three measurement schemes, direct measurement,
Gaussian mode measurement and zero-photon measurement, were analyzed through
the use of the classical Fisher information. Gaussian mode measurement was
found to be the approximately optimal scheme, and the corresponding classical
Fisher information can exceed 99% of the quantum Fisher information as long as
$d\leq 0.283$. These results may be beneficial for many applications such as
fluorescence imaging and astronomical observations.

## Acknowledgment

This work was supported by the Program of Zhongwu Young Innovative Talents of
Jiangsu University of Technology (20230013).

## Appendix

In this section, we provide the calculation of the probability $P_{\rm Z}$.
According to the semiclassical theory, the complex amplitude of the states on
the image plane can be written as

$\displaystyle
E\left(x\right)={C_{1}}\psi\left(x\right)+{C_{2}}\psi\left({x-d}\right).$ (23)

The amplitudes are circular-complex Gaussian random variables satisfying

$\displaystyle{\mathbf{E}}\left[{{C_{1}}}\right]={\mathbf{E}}\left[{{C_{2}}}\right]=0,$
(24)
$\displaystyle{\mathbf{E}}\left[{{C_{1}}{C_{2}}}\right]={\mathbf{E}}\left[{C_{1}^{*}{C_{2}}}\right]={\mathbf{E}}\left[{{C_{1}}C_{2}^{*}}\right]=0,$
(25) $\displaystyle{\mathbf{E}}\left[{C_{1}^{*}{C_{1}}}\right]=1-q,$ (26)
$\displaystyle{\mathbf{E}}\left[{C_{2}^{*}{C_{2}}}\right]=q.$ (27)

For SLIVER measurement, the complex amplitude of antisymmetric part can be
calculated from

$\displaystyle{E_{\rm
a}}\left(x\right)=\frac{{E\left(x\right)-E\left({-x}\right)}}{2}=\frac{{{C_{2}}}}{2}\left[{\psi\left({x-d}\right)-\psi\left({x+d}\right)}\right],$
(28)

where we used the symmetry of the point spread function, i.e.,

$\displaystyle{\psi\left(x\right)}={\psi\left(-x\right)}.$ (29)

Further, the mean photon number in the antisymmetric part is given by

$\displaystyle{N_{\rm
a}}=\int_{-\infty}^{\infty}{{{\left|{{E_{a}}\left(x\right)}\right|}^{2}}dx}=\frac{{q}}{2}\left[{1-\exp\left({-\frac{{{d^{2}}}}{2}}\right)}\right].$
(30)

Since the photon number satisfies Bose-Einstein distribution, the probability
of zero-photon events is found to be

$\displaystyle{P_{0}}=\frac{1}{{1+{N_{\rm a}}}}.$ (31)

Finally, the probability $P_{\rm Z}$ in Eq. (21) is obtained by combining Eqs.
(30) and (31).

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# Effect of decay of the final states on the probabilities of the Landau-Zener
transitions in multistate non-integrable models

Rajesh K. Malla  Center for Nonlinear Studies and Theoretical Division, Los
Alamos National Laboratory, Los Alamos, New Mexico 87545, USA  M. E. Raikh
<EMAIL_ADDRESS>Department of Physics and Astronomy, University of
Utah, Salt Lake City, UT 84112

###### Abstract

For a Landau-Zener transition in a two-level system, the probability for a
particle, initially in the first level, to survive the transition and to
remain in the first level, does not depend on whether or not the second level
is broadened [V. M. Akulin and W. P. Schleicht, Phys. Rev. A 46, 4110 (1992)].
In other words, the seminal Landau-Zener result applies regardless of the
broadening of the second level. The same question for the multistate Landau-
Zener transition is addressed in the present paper. While for integrable
multistate models, where the transition does not involve interference of the
virtual paths, it can be argued that the independence of the broadening
persists, we focus on non-integrable models involving interference. For a
simple four-state model, which allows an analytical treatment, we demonstrate
that the decay of the excited states affects the survival probability provided
that the widths of the final states are different.

##  I Introduction

In Ref. Akulin1992,  (see also Ref. Vitanov1997, ) a problem of time-dependent
crossing between a sharp level and a broadened level was considered. It was
assumed that, initially, at times $t\rightarrow-\infty$, the sharp level is
the ground state and is occupied. The authors demonstrated that the
probability for a sharp level to stay occupied after the transition, at
$t\rightarrow\infty$, does not depend on the width of the broadened level. The
authors of Ref. Akulin1992,  had arrived at their conclusion by a direct
calculation although it could be argued on general grounds (see e.g. Ref.
Shytov, ) that the level width drops out from the $t\rightarrow\infty$ limit
of the survival probability since this probability emerges as a result of
integration along the contour of a large radius in the complex time plane.

A natural question to ask is whether the decay of the excited states affects
the survival probability in the case of multilevel transitions. These
transitions are characterized by $N\times N$ scattering matrix with $N\geq 3$.
Finding the elements of this matrix analytically is possible only for certain
mutual arrangements of the levels, Demkov1968 ; Brunhobler ; Ostrovsky1997 ;
Ostrovsky2000 ; Ostrovsky2001 ; Sinitsyn2004 ; Sinitsyn1 ; Volkov .
RecentlySinitsyn2015 ; Sinitsyn2017 ; Yuzbashyan2017 ; Sinitsyn2018 ;
Sinitsyn2020  a new class of exactly solvable multistate models was uncovered
and the conditions under which the problem is solvable were formulated. A
common feature of the solvable models is that the elements of the scattering
matrix are expressed via partial Landau-Zener probabilities and do not depend
on the time intervals between the subsequent level crossings. This fact is
crucial for the effect of the level decay on the survival probability. For
exactly solvable (integrable) models the levels’ decay should drop out from
the survival probabilities. On the contrary, for non-integrable multistate
models, it might be expected that the probabilities should depend on the ratio
of the time intervals between the level crossing and the lifetimes of the
crossing levels. In the present paper, we consider a non-integrable four-state
model for which the consideration can be carried out analytically. By
including the broadening of the two possible final states of the transition,
we trace how this broadening modifies the survival probability.

##  II Model

For illustration purposes, we consider two tunnel-coupled quantum dots in
which the levels are swept past each other by the gate voltage, see Fig. 1.
Denote with $v$ the relative velocity of the levels and with $J$ the tunnel
splitting of the levels at the crossing point. In order to mimic the decay, we
assume that the electron can tunnel out of the dot. This tunneling causes an
imaginary correction, $\frac{i}{\tau}$, to the energy level position,
$\frac{vt}{2}$, in the left dot, where $\tau$ is the tunnel lifetime. Then the
amplitudes $a$ and $b$ to find a particle in the left and right dot,
respectively, are related via the Schrödinger equation

$\displaystyle i\left({\dot{a}}+\frac{a}{\tau}\right)=-\frac{vt}{2}a+Jb,$ (1)
$\displaystyle i{\dot{b}}=\frac{vt}{2}b+Ja.$ (2)

The solutions of the system Eqs. (1), (2) is governed by two dimensionless
parameters: $\frac{J^{2}}{v}$ and $\frac{J}{v\tau}$. Then the result of Refs.
Akulin1992, , Vitanov1997,  reduces to the fact that the celebrated Landau-
Zener formula

$\frac{|b({\infty})|^{2}}{|b(-\infty)|^{2}}=\exp\left(-\frac{2\pi
J^{2}}{v}\right).$ (3)

applies independently of the parameter $\frac{J}{v\tau}$.

A hint on how the decay in the left dot affects the survival probability can
be inferred from the semiclassical solution of the system Eqs. 1 and 2. Upon
substituting $a(t)=a_{0}e^{-i\lambda t},~{}~{}b(t)=b_{0}e^{-i\lambda t}$, we
arrive at the system of algebraic equations

$\displaystyle\left(\lambda+\frac{i}{\tau}+\frac{vt}{2}\right)a_{0}=Jb_{0}$
$\displaystyle\left(\lambda-\frac{vt}{2}\right)b_{0}=Ja_{0}.$ (4)

This system yields the following equation for $\lambda$

$\left(\lambda+\frac{i}{2\tau}\right)^{2}=\left(\frac{vt}{2}+\frac{i}{2\tau}\right)^{2}+J^{2}.$
(5)

At large times the term $\frac{i}{2\tau}$ is negligible compared to
$\frac{vt}{2}$. Since, according to the contour integral approach Demkov1968 ;
Shytov ; Sinitsyn2004 , the survival probability is fully determined by the
behavior of $a(t)$ and $b(t)$ at large times, it can be expected that the
decay does not affect this probability. In the absence of decay, the
quantitative criterion of large times is $t\gg\frac{J}{v}$. In the presence of
the decay, there is an additional criterion of large times is
$t\gg\frac{1}{v\tau}$. The physical meaning of the latter criterion is that
the change, $vt$, of the level positions during the time, $t$, must exceed the
level width. The above consideration, however, does not capture the behavior
of the population of a narrow level, $|b(t)|^{2}$, at finite times. We address
this question in the next section.

Figure 1: (Color online) Microscopic illustration of the Landau-Zener
transition with decay. Shown are two quantum dots coupled by tunneling with a
matrix element, $J$. Due to the temporal change of the gate voltage, the
levels in the dots are swept past each other with relative velocity, $v$. The
left dot is coupled to the continuum; the possibility of tunneling out leads
to the broadening, $\frac{1}{\tau}$, of the level in the left dot. At time
$t\rightarrow-\infty$ an electron resides in the right dot. Survival
probability is the probability for the electron to stay in the right dot at
$t\rightarrow\infty$.

##  III Effect of decay at finite times

We start by reproducing more rigorously the $t\rightarrow\infty$ result of
Refs. Akulin1992, , Vitanov1997,  and then turn to finite $t$. To this end, we
express $b$ from Eq. 1 and substitute it into Eq. 2. Searching for $b(t)$ in
the form $b_{0}(t)\exp(-\frac{t}{2\tau})$, we find that $b_{0}(t)$ satisfies
the second-order differential equation

${\ddot{b}_{0}}+\Biggl{[}\frac{1}{4}\left(vt+\frac{i}{\tau}\right)^{2}+\frac{iv}{2}+J^{2}\Biggr{]}b_{0}=0,$
(6)

which is the equation for the parabolic cylinder functions. Disappearance of
$\tau$ from the probability, $|b(t)|^{2}$, in the long-time limit follows from
the fact that in this limit the semiclassical approach applies. The
semiclassical phase is given by $\Phi(t)=\pm
i\int\limits_{0}^{t}dt\biggl{\\{}\frac{1}{4}\left(vt+\frac{i}{\tau}\right)^{2}+\frac{iv}{2}+J^{2}\biggr{\\}}^{1/2}$.
Keeping only two leading terms, in the long-time limit, we have
$\Phi(t)\approx\pm i\left(\frac{vt^{2}}{4}+\frac{it}{2\tau}\right)$. The sign
”-” leads to a growing exponent $\propto\exp\left(\frac{t}{2\tau}\right)$ in
$b_{0}(t)$. This term is canceled by the prefactor
$\exp\left(-\frac{t}{2\tau}\right)$ in $b(t)$. This cancellation illustrates
the message of Refs. Akulin1992, , Vitanov1997,  that the broadening drops out
from the survival probability at $t\rightarrow\infty$. However, as the
numerics in Ref. Vitanov1997,  shows, at finite times, the oscillations in
$|b(t)|^{2}$ are suppressed due to the leakage from the left dot into
continuum. To trace this suppression analytically, we examine the solution of
Eq. (6) at complex argument $t+\frac{i}{v\tau}$.

At $\tau\rightarrow\infty$, i.e. without the decay, the solution of Eq. (6),
which satisfies the right “boundary” condition
$\bigl{[}\exp(\frac{ivt^{2}}{4})$ behavior at $t\rightarrow\infty\bigr{]}$, is
given by the function $D_{\nu}(z)$ with integral representation

$D_{\nu}(z)=\Bigl{(}\frac{2}{\pi}\Bigr{)}^{1/2}e^{\frac{z^{2}}{4}}\int\limits_{0}^{\infty}du~{}e^{-\frac{u^{2}}{2}}u^{\nu}\cos\left(zu-\frac{\pi\nu}{2}\right).$
(7)

Argument $z$ and parameter $\nu$ in Eq. 7 are defined as

$z=v^{1/2}e^{\frac{\pi i}{4}}t,~{}~{}~{}\nu=-\frac{iJ^{2}}{v}.$ (8)

Following Ref. Zhuxi, , it is convenient to divide the integral Eq. (7) into
two contributions

$D_{\nu}(z)=I_{+}(z)e^{\frac{i\pi\nu}{2}}+I_{-}(z)e^{-\frac{i\pi\nu}{2}},$ (9)

where the functions $I_{+}(z)$ and $I_{-}(z)$ are defined as

$I_{\pm}(z)=D_{\nu}(z)=\Bigl{(}\frac{1}{2\pi}\Bigr{)}^{1/2}e^{\frac{z^{2}}{4}}\int\limits_{0}^{\infty}du~{}u^{\nu}e^{-\frac{u^{2}}{2}\pm
iuz}.$ (10)

Both integrals are evaluated with the help of steepest descent approach. The
corresponding saddle points are given by

$u_{\pm}=\frac{iz}{2}\pm\frac{(4\nu-z^{2})^{1/2}}{2}.$ (11)

Performing the Gaussian integration around the saddle points, we arrive to the
following asymptotic expressions for $I_{+}$ and $I_{-}$

$\displaystyle
I_{+}(z)\approx(4\nu-z^{2})^{-1/4}\exp\Biggl{[}\frac{1}{4}iz(4\nu-z^{2})^{1/2}-\frac{1}{2}\nu\Biggr{]}$
$\displaystyle\times\Biggl{[}\frac{1}{2}iz+\frac{1}{2}(4\nu-z^{2})^{1/2}\Biggr{]}^{\nu+\frac{1}{2}},$
(12)

$\displaystyle
I_{-}(z)\approx(4\nu-z^{2})^{-1/4}\exp\Biggl{[}-\frac{1}{4}iz(4\nu-z^{2})^{1/2}-\frac{1}{2}\nu\Biggr{]}$
$\displaystyle\times\Biggl{[}-\frac{1}{2}iz+\frac{1}{2}(4\nu-z^{2})^{1/2}\Biggr{]}^{\nu+\frac{1}{2}}.$
(13)

The condition of applicability of the steepest descent method is that the
typical value of $\left(u-u_{+}\right)$ contributing to the integral Eq. 10 is
much smaller than $u_{+}$. With characteristic time of the transition being
$\frac{J}{v}$, we conclude that characteristic $z$ is
$\sim\frac{J}{v^{1/2}}\sim|\nu|^{1/2}$. Thus, for $|\nu|\gg 1$, when the
position of the saddle point is $u_{+}\sim|\nu|^{1/2}\gg 1$ the saddle-point
result applies not only for large, but for arbitrary $z$.withEugene

It is instructive to rewrite $D_{\nu}(z)$ in terms of time

$\displaystyle D_{\nu}(z)=e^{\frac{\pi
i}{8}+\frac{iJ^{2}}{2v}}\biggl{(}\frac{v}{W^{2}(t)}\biggr{)}^{1/4}\Biggl{(}\frac{e^{-\frac{\pi
i}{4}}}{v^{1/2}}\Biggr{)}^{\frac{1}{2}-\frac{iJ^{2}}{v}}$
$\displaystyle\times\Bigg{\\{}e^{{\frac{it}{4}}W(t)}\Bigg{(}\frac{-vt+W(t)}{2}\Bigg{)}^{\frac{1}{2}-\frac{iJ^{2}}{v}}e^{-\frac{\pi
J^{2}}{2v}}$
$\displaystyle+e^{{-\frac{it}{4}}W(t)}\Bigg{(}\frac{vt+W(t)}{2}\Bigg{)}^{\frac{1}{2}-\frac{iJ^{2}}{v}}e^{\frac{\pi
J^{2}}{2v}}\Bigg{\\}},$ (14)

where $W(t)=\left(4J^{2}+v^{2}t^{2}\right)^{1/2}$. Interference of two
oscillating exponents in the curly brackets is responsible for the beatings in
$|b_{0}(t)|^{2}$. The magnitudes of these terms are related as
$\exp(-\frac{\pi J^{2}}{v})$, in agreement with classical result Eq. 3. In the
limit $t\rightarrow\pm\infty$ we have $W(t)\rightarrow\pm vt$, so either
$\bigl{(}W(t)-vt\bigr{)}$ or $\bigl{(}W(t)+vt\bigr{)}$ approaches zero. In
other words, only one of the two terms in the curly brackets survives in both
limits. Moreover, the product of the two terms in the curly brackets does not
depend on time.

Upon substituting $t+\frac{i}{v\tau}$ into
$\exp\Big{(}\frac{itW(t)}{4}\Big{)}$ and
$\exp\Big{(}-\frac{itW(t)}{4}\Big{)}$, we realize that the $\tau$-dependence
is canceled by the prefactor $\exp\left(-\frac{t}{2\tau}\right)$ both at large
positive and large negative $t$. Still, at finite $t$, the occupation of the
right dot, in which the particle initially (at $t\rightarrow-\infty$) resides,
depends on $\tau$. Namely, it is multiplied by $\exp\Big{(}-2{\cal
F}(t)\Big{)}$, where the function ${\cal F}(t)$ is defined as

${\cal F}(t)=\frac{\tau_{\scriptscriptstyle
LZ}}{\tau}\Biggl{\\{}\frac{8}{(4+{\tilde{t}}^{2})^{1/2}\left[|{\tilde{t}}|+(4+{\tilde{t}}^{2})^{1/2}\right]^{2}}\Biggr{\\}},$
(15)

where ${\tilde{t}}=\frac{t}{\tau_{\scriptscriptstyle LZ}}$ is the
dimensionless time measured in the units of $\tau_{\scriptscriptstyle
LZ}=\frac{J}{v}$, which is the characteristic time of the Landau-Zener
transition.

We now turn to the interference effects in the level occupation. This
occupation is proportional to
$\exp\left(-\frac{t}{\tau}\right)|b_{0}(t)|^{2}$. The essential time
dependence of $b_{0}$ is contained in the curly brackets in Eq. III. Upon
taking the absolute value square of this curly brackets, we obtain

$\displaystyle\Big{|}\Big{\\{}...\Big{\\}}\Big{|}^{2}=\exp\Big{(}-\frac{\pi
J^{2}}{v}\Big{)}F+\exp\Big{(}\frac{\pi J^{2}}{v}\Big{)}G$
$\displaystyle+2\left(FG\right)^{1/2}\cos\Big{(}\frac{t}{2}W(t)-\frac{J^{2}}{v}\ln\frac{F}{G}\Big{)},$
(16)

where the functions $F(t)$ and $G(t)$ are defined as

$F(t)=\frac{W(t)-vt}{2},~{}~{}~{}G(t)=\frac{W(t)+vt}{2},$ (17)

so that the product $F(t)G(t)$ is equal to $J^{2}$, i.e. it is time-
independent. We then conclude that the time decay of the interference term is
$\exp\big{(}-\frac{t}{\tau}\big{)}$. This dependence comes from the prefactor.

Figure 2: (Color online) (a) Illustration of $2\times 2$ model with decay.
Both left and right dots contain two levels. We assume that the level
separation in both dots is the same. It is also assumed that the coupling
constants between the levels $1$ and $2$ in the left dot and the levels
$1^{\prime}$ and $2^{\prime}$ in the right dot are the same and are equal to
$J$. Similar to Fig. 1, the levels $1^{\prime},2^{\prime}$ are swept by the
levels $1,2$ with relative velocity, $v$. Levels in the left dot are coupled
to the continuum. Tunnel lifetimes into continuum, $\tau_{1}$ and $\tau_{2}$,
of the levels $1$ and $2$ are different. As a result of this difference, the
survival probability in the right dot depends on the decay from the right dot.
(b) Time evolution of the levels in the left and right dot. Unlike the
conventional Landau-Zener transition, illustrated in Fig. 1, there are two
pathways (red and blue) from the states in the right dot into the states in
the left dot, which interfere with each other.

##  IV $2\times 2$ non-integrable model with decay

We now turn to the simplest non-integrable model illustrated in Fig. 2. In
this model, there are two sharp levels in the right dot, which are spaced in
energy by $2\Delta$, and two broadened levels in the left dot, which are
coupled to the continuum. For simplicity, we assume that the level separation
in the left dot is also $2\Delta$. Also, to keep the analysis as simple as
possible, we assume that the tunnel couplings between all the levels are the
same and are equal to $J$. Similar to the situation considered in the previous
sections, we assume that the levels in the left and right dots are swept past
each other with velocity $v$. We denote the amplitudes to find the particle in
the level $1$ and $2$ of the left dot with $a_{1}$ and $a_{2}$, respectively.
Correspondingly, $b_{1}$ and $b_{2}$ are the amplitudes to find the particle
in the first level (level 1’ in Fig. 2 and the second level (level 2’ in the
right dot. In the semiclassical limit, four amplitudes are related by the
following system of equations similar to the system Eq. II

$\displaystyle\left(\lambda+\frac{i}{\tau_{\scriptscriptstyle
1}}\right)a_{1}=-\left(\Delta+\frac{vt}{2}\right)a_{1}+J\left(b_{1}+b_{2}\right),$
(18) $\displaystyle\left(\lambda+\frac{i}{\tau_{\scriptscriptstyle
2}}\right)a_{2}=\left(\Delta-\frac{vt}{2}\right)a_{2}+J\left(b_{1}+b_{2}\right),$
(19) $\displaystyle\lambda
b_{1}=\left(-\Delta+\frac{vt}{2}\right)b_{1}+J\left(a_{1}+a_{2}\right),$ (20)
$\displaystyle\lambda
b_{2}=\left(\Delta+\frac{vt}{2}\right)b_{2}+J\left(a_{1}+a_{2}\right).$ (21)

Here $\tau_{1}$ and $\tau_{2}$ are the tunnel lifetimes of the levels in the
left dot, and $\lambda$ is the time-dependent eigenvalue. Introducing the sums
and differences

$\displaystyle A_{1}=a_{1}+a_{2},~{}~{}~{}~{}~{}~{}~{}B_{1}=b_{1}+b_{2},$ (22)
$\displaystyle A_{2}=a_{1}-a_{2},~{}~{}~{}~{}~{}~{}~{}B_{2}=b_{1}-b_{2},$ (23)

The system of four equations for the new variables is obtained by adding Eqs.
(18) and (19) and also by adding Eqs. (20) and (21). This yields

$\displaystyle\left(\lambda+\frac{vt}{2}+\frac{i}{2\tau_{\scriptscriptstyle
1}}+\frac{i}{2\tau_{\scriptscriptstyle 2}}\right)A_{1}-2JB_{1}$
$\displaystyle=\left(-\Delta-\frac{i}{2\tau_{\scriptscriptstyle
1}}+\frac{i}{2\tau_{\scriptscriptstyle 2}}\right)A_{2},$ (24)
$\displaystyle\left(\lambda-\frac{vt}{2}\right)B_{1}-2JA_{1}=-\Delta B_{2}.$
(25)

The other two equations emerge upon subtracting Eq. (19) from Eq. (18) and Eq.
(21) from Eq. (20)

$\displaystyle\left(\lambda+\frac{vt}{2}+\frac{i}{2\tau_{\scriptscriptstyle
1}}+\frac{i}{2\tau_{\scriptscriptstyle
2}}\right)A_{2}=-\left(\Delta+\frac{i}{2\tau_{\scriptscriptstyle
1}}-\frac{i}{2\tau_{\scriptscriptstyle 2}}\right)A_{1},$ (26)
$\displaystyle\left(\lambda-\frac{vt}{2}\right)B_{2}=-\Delta B_{1}.$ (27)

In the limit of large $J$ we can neglect the first terms in Eqs. (IV) and (25)
and express $B_{1}$ via $A_{2}$ and $A_{1}$ via $B_{2}$. This yields

$\displaystyle B_{1}=\frac{\left(\Delta+\frac{i}{2\tau_{\scriptscriptstyle
1}}-\frac{i}{2\tau_{\scriptscriptstyle 2}}\right)}{2J}A_{2},$ $\displaystyle
A_{1}=\frac{\Delta}{2J}B_{2}.$ (28)

As a last step, we substitute $B_{1}$ and $A_{1}$ into the right-hand sides of
Eqs. (26) and (27), respectively, and arrive to the final system

$\displaystyle\left(\lambda-\frac{vt}{2}\right)B_{2}=-\frac{\Delta\left(\Delta+\frac{i}{2\tau_{\scriptscriptstyle
1}}-\frac{i}{2\tau_{\scriptscriptstyle 2}}\right)}{2J}A_{2},$ (29)
$\displaystyle\left(\lambda+\frac{vt}{2}+\frac{i}{2\tau_{\scriptscriptstyle
1}}+\frac{i}{2\tau_{\scriptscriptstyle
2}}\right)A_{2}=-\frac{\Delta\left(\Delta+\frac{i}{2\tau_{\scriptscriptstyle
1}}-\frac{i}{2\tau_{\scriptscriptstyle 2}}\right)}{2J}B_{2}.$ (30)

Multiplying Eqs. (29), (30), we arrive to the closed equation for $\lambda(t)$

$\left(\lambda(t)+\frac{i}{4\tau_{\scriptscriptstyle
0}}\right)^{2}=\left(\frac{vt}{2}+\frac{i}{4\tau_{\scriptscriptstyle
0}}\right)^{2}+\frac{\Delta^{2}\left(\Delta+\frac{i}{2\tau_{\scriptscriptstyle
c}}\right)^{2}}{4J^{2}},$ (31)

where $\tau_{0}$ and $\tau_{c}$ are defined as

$\frac{1}{\tau_{\scriptscriptstyle 0}}=\frac{1}{\tau_{\scriptscriptstyle
1}}+\frac{1}{\tau_{\scriptscriptstyle
2}},~{}~{}~{}~{}~{}~{}\frac{1}{\tau_{\scriptscriptstyle
c}}=\frac{1}{\tau_{\scriptscriptstyle 1}}-\frac{1}{\tau_{\scriptscriptstyle
2}}.$ (32)

Comparing Eq. 5 to Eq. 31, we see that the semiclassical description of the
Landau-Zener transition in $2\times 2$ model reduces to the conventional
Landau-Zener theory in which the tunnel splitting $J$ is replaced by

${\tilde{J}}=\frac{\Delta\left(\Delta+\frac{i}{2\tau_{c}}\right)}{2J}={\tilde{J}}_{1}+i{\tilde{J}}_{2}.$
(33)

Naturally, in the absence of decay $\tau_{c}\rightarrow\infty$, Eq. 33
reproduces the earlier result obtained in Ref. Rajesh,  together with a
nontrivial message that the survival probability increases upon the increasing
of the tunnel matrix element, $J$. Later this message was confirmed by a
semiclassical calculation in Ref. Rajesh1, .

Qualitative novelty of the system Eqs. 29, 30 is that the four-level $2\times
2$ model is reduced to the conventional Landau-Zener system Eq. II with an
effective tunnel matrix element, ${\tilde{J}}$, being complex. This complexity
reflects the effect of decay in generic non-integrable models.

A straightforward consequence of the effective tunneling matrix element being
complex is that the dimensionless parameter defined in Eq. 8 becomes complex

$\nu=\nu_{1}+i\nu_{2}=-\frac{i{\tilde{J}}^{2}}{v}=-\frac{\Delta^{3}}{2J^{2}\tau_{c}v}-\frac{i\Delta^{2}\left(\Delta^{2}-\frac{1}{4\tau_{c}^{2}}\right)}{4J^{2}v}.$
(34)

In the absence of decay, the survival probability emerges as an absolute value
square of the ratio of the long-time asymptotes of the two terms in Eq. 9.
With parameter $\nu$ being complex, upon taking the absolute value square, the
real part of $\nu$ drops out. This leads us to the main result: the survival
probability in a non-integrable $2\times 2$ model illustrated in Fig. 2 is
given by

$\exp{\Big{(}-2\pi|\nu_{2}|\Big{)}}=\exp{\Bigg{[}-\frac{2\pi\Delta^{2}\Big{|}\Delta^{2}-\frac{1}{4\tau_{c}^{2}}\Big{|}}{J^{2}v}\Bigg{]}}.$
(35)

At first glance, this result seems counterintuitive. Indeed, Eq. 35 suggests a
nontrivial dependence of the survival probability on the decay, which is
peaked at $\frac{1}{|\tau_{c}|}=2\Delta$. On the other hand, the decay enters
in combination with a half of inter-level energy separation, which is
intuitively correct. As explained in Ref. Rajesh, , the inverse dependence of
the survival probability on the tunnel splitting is the result of the
interference of the multiple tunnel paths and is the consequence of the
$2\times 2$ model being non-integrable.

##  V Relation to Ref. [20]

In general, the transition probabilities in multi-state Landau-Zener models
cannot be evaluated analytically. This is due to the virtual paths
interference that emerges when multiple paths lead up to a particular
transition. This interference is illustrated in Fig. 2b. The systems of
general type with finite path interference are not fully solvable. In Ref.
Rajesh,  such not-fully solvable models were referred to as non-integrable.
However, in Ref. Rajesh1,  integrability was defined in a different sense.
Namely, the integrability condition for a time-dependent Hamiltonian
$H(t,t^{\prime})$ was defined as a possibility to find analytical form of a
nontrivial operator $H^{\prime}(t,t^{\prime})$ such that commute

$\displaystyle\frac{\partial H}{\partial t^{\prime}}-\frac{\partial
H^{\prime}}{\partial t}$ $\displaystyle=$ $\displaystyle 0,$ (36)
$\displaystyle[H,H^{\prime}]$ $\displaystyle=$ $\displaystyle 0.$ (37)

In particular, in Ref. Rajesh1,  it was demonstrated that the Landu-Zener
grids, with two sets of parallel levels, similar to Fig. 1a, are formally
integrable. These grids have non-trivial commuting partners having the form of
time-quadratic-polynomial operators. While these models are still not fully
solvable, integrability in the sense of Ref. Rajesh1,  allows to establish
exact non-trivial relations between the partial transition probabilities. It
also allows finding asymptotically exact expressions for the leading exponents
describing the transition probabilities in the nearly adiabatic limit.

In Ref. Rajesh1,  general theoretical predictions were put to numerical test
for a linearly driven double quantum dot system, described by a Landau-Zener
grid with two sets of parallel levels crossing each other, see Fig. 2a.

Within the approach of Ref. Rajesh1, , the time-dependent Hamiltonian is
described by a $4\times 4$ matrix. Integrability reduces the system to an
effective $2\times 2$ matrix Hamiltonian, and the transition probabilities are
obtained by a modified Dykhne formula. Most importantly, the strong coupling
limit of Ref. Rajesh1, , similar to the limit $\Delta\ll J$ in Section IV, the
numerical test reproduced the result of Ref. Rajesh, .

In the presence of decay, the Hamiltonian representing the system of equations
(18), (19), (20), (21) is non-Hermitian. Nevertheless, this Hamiltonian
satisfies the integrability conditions (36) and (37), where the commuting
partner operator is also non-Hermitian. The integrability allows for rescaling
of the parameters by changing $t^{\prime}$ as outlined in Ref. Rajesh1,  and
could lead to a semiclassical solution even in the presence of decaying terms.
This is however beyond the scope of the present paper.

##  VI Concluding remarks

In the present paper, we demonstrated for the first time that, unlike the case
of integrable multistate models, where the decay into the continuum does not
affect the survival probability, in non-integrable models, this decay enters
explicitly into the survival probability, see Eq. 35. The model considered in
the present paper is the simplest example of the Landau-Zener grid.Rajesh1 ;
Suzuki2022  Note that, for decay to enter the result of the transition, the
tunnel widths of the two final states should be different, as it follows from
Eq. 32. The underlying reason is that the imbalance of the tunnel widths
results in the mixing of the symmetric and antisymmetric modes in the two
dots. A crucial step that allowed us to solve the system 18 is the large-$J$
limit. Quantitatively, this large-$J$ limit corresponds to the condition
${\tilde{J}}\ll J$, where ${\tilde{J}}$ is defined by Eq. 33. As follows from
Eq. 33, this condition amounts to $|\Delta|\ll J$ and
$\frac{|\Delta|}{\tau_{c}}\ll J^{2}$. It also follows from Eq. IV that the
smallness of the ratio $\frac{\Delta}{J}$ insures that $B_{1}\ll A_{2}$ and
$A_{1}\ll B_{2}$. This, in turn, suggests that anomalously high survival
probability is promoted by antisymmetric combination, $a_{1}-a_{2}$ (see Eq.
22). Symmetric combination, on the other hand, yields a conventional survival
probability with the exponent $\sim\frac{J^{2}}{v}$. As a final remark, our
treatment of the $2\times 2$ model was semiclassical. This approach is
justified when the velocity $v$ is small enough, namely, the probability Eq.
35 should be much smaller than one. More rigorous derivation of Eq. 35 based
on the Schrödinger equation in the absence of decay can be found in Ref.
Rajesh, .

## Acknowledgements

We would like to thank Nikolai Sinitsyn for useful discussions. R.K.M. would
like to thank the U.S. Department of Energy, Office of Science, Basic Energy
Sciences, Materials Sciences and Engineering Division, Condensed Matter Theory
Program. R.K.M. was also supported by the Center for Nonlinear Studies.

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