On leave of absence from]…

# The physics of boundary conditions in reaction–diffusion problems

Francesco Piazza  Dipartimento di Fisica & Astronomia, Università di Firenze,
Via G. Sansone 1, 50019 Sesto Fiorentino (FI), Italy
<EMAIL_ADDRESS>[

###### Abstract

The use of fully or partially absorbing boundary conditions for diffusion-
based problems has become paradigmatic in physical chemistry and biochemistry
to describe reactions occurring in solutions or in living media. However, as
chemical states may indeed disappear, particles cannot, unless such
degradation happens physically and should thus be accounted for explicitly.
Here, we introduce a simple, yet general idea that allows one to derive the
appropriate boundary conditions self-consistently from the chemical reaction
scheme and the geometry of the physical reaction boundaries. As an
illustration, we consider two paradigmatic examples, where the known results
are recovered by taking specific physical limits. More generally, we
demonstrate that our mathematical analysis delivers physical insight that
cannot be accessed through standard treatments.

Diffusion, Smoluchowski, boundary conditions

The work on diffusion-limited coagulation of colloidal particles of the Polish
physicist Marian von Smoluchowski [1] can be rightly considered as one of the
fundamental pillars of modern molecular physical chemistry, and, as a matter
of fact, of the entire biochemistry of living organisms. Smoluchowski’s
demonstration of how a non-equilibrium boundary value problem can be used to
describe chemical reactions in solution has been a scientific breakthrough
back then, and is to date a far-reaching achievement. Nonetheless, this
powerful idea rests on the somehow physically unsettling idea of using sinks
to model chemical reactions occurring upon diffusive encounters.
Let $A$ and $B$ be two particles of radius $R_{A}$ and $R_{B}$ and diffusion
coefficients $D_{A}$ and $D_{B}$, respectively, and let the goal be to compute
the typical rate of $A-B$ encounters. The reasoning goes as follows: If the
concentration of one of the species, say $c_{A}$, is much smaller then the
other, then the problem can be reduced to that of a single A surrounded by a
homogeneous fluid of $B$s. Moreover, if the latter are themselves diluted
enough to be considered as non-interacting, the whole problem reduces further
to a two-body problem. Smoluchowski’s brilliant intuition was that, if
diffusion is the rate-limiting step, i.e. the reaction that occurs upon
contact proceeds much faster than the typical time of diffusive approach, one
may consider a steady non-equilibrium diffusive flux of $B$ particles
disappearing instantly at the contact distance $R=R_{A}+R_{B}$. Then, the
total flux is the sought rate. Mathematically, one should solve the stationary
diffusion equation for the concentration field of $B$s around one $A$, $c(r)$,
with absorbing boundary conditions at $r=R$ and constant bulk concentration at
infinite $A-B$ separation, that is,

$\nabla^{2}c=0,\qquad\text{BC:}\quad c(R)=0,\lim_{r\to\infty}c(r)=c_{\infty}$
(1)

Integrating the current $J=-D\nabla c$ ($D=D_{A}+D_{B}$) over the reaction
sphere gives

$\left.k=4\pi DR^{2}\frac{dc}{dr}\right|_{R}=4\pi DRc_{\infty}$ (2)

which is known as the Smoluchowski rate 111Strictly speaking, as shown by
Szabo [7], this results is correct only in the limit $D_{A}\to 0$. This result
is simple and elegant and, indeed, it conveys correctly the essential physics
of diffusion-limited encounters in diluted solutions. A closer inspection,
however, reveals some unsettling troubles.
An absorbing boundary condition is certainly a clever mathematical trick, but
it implies nonetheless a somewhat mysterious mass annihilation. In fact, as
pointed out by Collins and Kimball (CK) in 1949, the result (2) cannot be
correct if not every $B$ particle that reaches the reaction radius eventually
reacts [3]. Based on this observation, CK surmised that a more physical
boundary condition should be of the Robin (mixed) type, namely

$\left.4\pi DR^{2}\frac{dc}{dr}\right|_{R}=k^{\ast}c(R)$ (3)

The rate constant $k^{\ast}$ (units of M-1$s^{-1}$) somehow gauges the finite
probability of a reaction indeed occurring and interpolates between reflecting
($k^{\ast}=0$) and absorbing ($k^{\ast}\to\infty$) BC. With the BC (3), the
stationary solution yields a total flux

$k=4\pi
DR^{2}\left.\frac{dc}{dr}\right|_{R}=k_{S}c_{\infty}\left(\frac{h}{1+h}\right)$
(4)

where $k_{S}=4\pi DR$ and $h=k^{\ast}/k_{S}$. This result has been famously
employed by Shoup and Szabo [4] to reinterpret the classical calculation by
Berg and Purcell [5] for the rate of irreversible binding of ligands to $N$
receptors on a cell’s membrane. In this case, each receptor acts as an
absorbing disk, so that $k^{\ast}\propto N$. The point, however, is that fully
or partially absorbing boundary conditions seem to be physically plausible
only if particles do really disappear. For example, in a process of intake
followed by degradation of some nutrient, say by a spherical colony of algae
[6]. By contrast, the use of fully or partially absorbing BCs to model binding
appears more elusive.
The unfathomable physical discomfort caused by the use of absorbing boundary
conditions to model chemical transformations resides in the fact that, by
doing this, particles cannot be properly distinguished from chemical states. A
clear illustration of this ambiguity are so-called annihilation reactions,
such as when a fluorescent molecule $B^{\ast}$ relaxes upon collision with a
quencher $A$, i.e. $B^{\ast}+A\to B+A$ [7]. In this case, the diffusing
particles can exist in either the excited or relaxed state, with a transition
in chemical space being promoted by a physical collision with the quencher.
However, while the excited state disappears, the particle is never gone. It
merely changes color, so to say.
The real problem seems that there is no general rigorous method at present to
derive self-consistently the appropriate boundary conditions for mass
diffusion coupled to chemical transformations among different states that may
occur in the bulk and on some reactive manifold. Of course, the boundary
conditions should ensure mass conservation if the chemistry at hand prescribes
so. In this paper, we introduce a simple, yet powerful idea that unveils a
self-consistent method to solve this problem in general. In the following, we
illustrate this method for a number of systems.
It is instructive to start from the very text-book case of annihilation
reactions, such as fluorescence quenching. Let $c^{\ast}$ and $c$ denote the
concentrations of excited and relaxed $B$ molecules, respectively. First of
all, the minimal physically transparent scheme should be laid out in chemical
space. This may take the following form:

$\displaystyle B\mathrel{\raisebox{-0.44481pt}{\hbox
to0.0pt{\raisebox{2.0pt}{$\ext@arrow
0359\arrowfill@\relbar\relbar\rightharpoonup{\hphantom{k_{-1}}}{k_{1}}$}\hss}\hbox{$\ext@arrow
3095\arrowfill@\leftharpoondown\relbar\relbar{k_{-1}}{\hphantom{k_{1}}}$}}}B^{\ast}\qquad\text{in
the bulk}$ $\displaystyle B+A\mathrel{\raisebox{-0.44481pt}{\hbox
to0.0pt{\raisebox{2.0pt}{$\ext@arrow
0359\arrowfill@\relbar\relbar\rightharpoonup{\hphantom{k^{\ast}_{-}}}{k^{\ast}}$}\hss}\hbox{$\ext@arrow
3095\arrowfill@\leftharpoondown\relbar\relbar{k^{\ast}_{-}}{\hphantom{k^{\ast}}}$}}}B^{\ast}+A\qquad\text{at
the reaction boundary}$ (5)

The reaction boundary is $r=R$, i.e. the contact distance between the
particles and the quencher, $A$. It can be seen that the reactions (The
physics of boundary conditions in reaction–diffusion problems) describe a two-
state system, where each molecule can switch between two states, either in the
bulk or at some prescribed boundary. Consequently, posing the problem properly
requires that mass be conserved. The reaction-diffusion equations
corresponding to the chemical scheme (The physics of boundary conditions in
reaction–diffusion problems) read

$\displaystyle\begin{split}\frac{\partial c}{\partial
t}=D\nabla^{2}c&+k_{-1}c^{\ast}-k_{1}c\\\ &+\frac{\delta(r-R)}{4\pi
r^{2}}(k^{\ast}c^{\ast}-k_{-}^{\ast}c)\end{split}$
$\displaystyle\begin{split}\frac{\partial c^{\ast}}{\partial
t}=D\nabla^{2}c^{\ast}&-k_{-1}c^{\ast}+k_{1}c\\\ &-\frac{\delta(r-R)}{4\pi
r^{2}}(k^{\ast}c^{\ast}-k_{-}^{\ast}c)\end{split}$ (6a)
$\displaystyle\lim_{r\to\infty}c=c_{\infty}\equiv\frac{k_{-1}}{k_{1}+k_{-1}}c_{0}$
$\displaystyle\lim_{r\to\infty}c^{\ast}=c^{\ast}_{\infty}\equiv\frac{k_{1}}{k_{1}+k_{-1}}c_{0}$
(6b)

The whole idea is that the boundary conditions for the system (6a) should
follow self-consistently from the chemical reactions and take into account
that no mass ought to disappear in this problem. This can be done by
integrating eqs. (6) from $r=0$ to $r=R+\delta R$ with $c(r)=c^{\ast}(r)=0$
for $r<R$. This exercise, followed by the limit $\delta R\to 0$, gives the
desired BCs,

$\displaystyle 4\pi DR^{2}\left.\frac{\partial c}{\partial
r}\right|_{R}+k^{\ast}c^{\ast}(R)-k_{-}^{\ast}c(R)=0$ (7) $\displaystyle 4\pi
DR^{2}\left.\frac{\partial c^{\ast}}{\partial
r}\right|_{R}-k^{\ast}c^{\ast}(R)+k_{-}^{\ast}c(R)=0$ (8)

Note that these correspond to reflecting BCs of the total density
$c+c^{\ast}$, which states correctly that real particles can exist in either
the relaxed or excited state, but should not vanish mysteriously.
In order to determine the encounter rate, we can restrict to the stationary
problem. The general solution of this takes the form

$\begin{pmatrix}c\\\ c^{\ast}\end{pmatrix}=\begin{pmatrix}c_{\infty}\\\
c^{\ast}_{\infty}\end{pmatrix}+\frac{a}{r}\begin{pmatrix}1\\\
\frac{k_{1}}{k_{-1}}\end{pmatrix}+\frac{b}{r}\begin{pmatrix}1\\\
-1\end{pmatrix}e^{-qr}$ (9)

where $q=\sqrt{(k_{1}+k_{-1})/D}$. The constants $a$ and $b$ are determined by
the BC (7) and (8), which gives the stationary profiles

$\displaystyle\frac{c(r)}{c_{\infty}}=1-\frac{R}{r}\left(1-e^{\beta\epsilon}\right)\frac{\displaystyle
k^{\ast}e^{-q(r-R)}}{\displaystyle k^{\ast}+k_{-}^{\ast}+k_{S}(1+qR)}$ (10)
$\displaystyle\frac{c^{\ast}(r)}{c^{\ast}_{\infty}}=1-\frac{R}{r}\left(1-e^{-\beta\epsilon}\right)\frac{\displaystyle
k_{-}^{\ast}e^{-q(r-R)}}{\displaystyle k^{\ast}+k_{-}^{\ast}+k_{S}(1+qR)}$
(11)

The energy $\epsilon=\beta^{-1}\log(k^{\ast}_{-}k_{1}/k^{\ast}k_{-1})$
quantifies the positive energy cost associated with increased non-equilibrium
depletion of the excited species $B^{\ast}$ at the reaction surface when the
boundary (i.e. contact with $A$ particles) acts as a quenching catalyst.
Indeed, it can be seen that the profiles are both flat at equilibrium, i.e.
when $k^{\ast}_{-}/k^{\ast}=k_{-1}/k_{1}$ or equivalently $\epsilon=0$. In
this case, they equal the two bulk molar fractions everywhere, as expected.
Note also that we get the apparently surprising result
$c(r)+c^{\ast}(r)=c_{\infty}$, i.e. point-wise mass conservation. Indeed, this
is the physically correct, albeit rather uninformative, solution of the
boundary problem for the total density with mass conservation.
As we are explicitly distinguishing particles from chemical states, the
correct way to compute the overall reaction rate at the quenching surface is
the following,

$\displaystyle k$ $\displaystyle=$ $\displaystyle J^{\ast}_{-}-J^{\ast}\equiv
k^{\ast}_{-}c^{\ast}(R)-k^{\ast}c(R)$ (12) $\displaystyle=$ $\displaystyle
k_{S}(1+qR)\frac{k^{\ast}_{-}c^{\ast}_{\infty}-k^{\ast}c_{\infty}}{k^{\ast}_{-}+k^{\ast}+k_{S}(1+qR)}$

The stationary profiles (10), (11) and the rate (12) convey a wealth of
physical insight. First and foremost, it is now clear that the rate should be
measured as an excess flow in chemical space relative to equilibrium. However,
the boundary condition (7) shows that this is nothing but the total flux of
excited molecules over the reactive boundary according to the definition (4).
Rather insightfully, our self-consistent theory reveals that the total
particle flux equals the net current at the catalytic surface in chemical
space. More generally, the sign of that current determines whether the
catalyst acts as an active quencher or adds up to the excitation in a non-
equilibrium steady state, i.e. when $k^{\ast}_{-}/k^{\ast}\neq k_{-1}/k_{1}$.
Indeed, direct inspection of the profiles (10) and (11) shows that one may
have enrichment of either species at the reaction surface, depending on the
relative magnitudes of the relaxation-to-excitation rates ratio, or
equivalently on the sign of $\epsilon$.
The familiar Smoluchowski setting is approached when the relaxation rate at
the surface is much greater than in the bulk. In the limit
$k^{\ast}_{-}\to\infty$ one has

$\displaystyle k\to k_{S}^{q}\equiv k_{S}c^{\ast}_{\infty}(1+qR)$ (13)

We discover that this is greater than the expected Smoluchowski limit,
$k_{S}c^{\ast}_{\infty}$. In the conventional picture of particles pouring
from infinity and vanishing into a sink, this may be interpreted as an extra
accumulation of particles in the bulk, where reactions are not infinitely slow
with respect to the typical diffusion time, $\tau_{D}=\sqrt{R^{2}/D}$. Indeed,
the Smoluchowski rate is recovered in the limit $qR\to 0$, i.e.
$k_{1}+k_{-1}\ll\tau_{D}^{-1}$. This may be considered as the limit of
infinitely slow equilibrium in the bulk, that is, $k_{1},k_{-1}\to 0$ with
$k_{1}/k_{-1}=const.$. The expression (12) also yields the physically sensible
result in the infinite-transport limit, i.e. $D\to\infty$. In this case one
has

$k=c_{0}(k^{\ast}_{-}k_{1}-k^{\ast}k_{-1})$ (14)

which correctly only accounts for relative flow in chemical space.
Interestingly, the present treatment of this chemical process allows one to
access quantities that are simply unavailable in the standard formulation. For
example, one may wish to quantify the efficiency $\eta$ of the surface
quenching reaction with respect to spontaneous relaxation occurring in the
bulk. This can be computed as

$\eta\equiv\frac{c(R)/c^{\ast}(R)}{c_{\infty}/c^{\ast}_{\infty}}=\frac{J^{\ast}_{-}}{J^{\ast}}e^{\beta\epsilon}=1+\frac{k^{\ast}(e^{\beta\epsilon}-1)}{k^{\ast}+\frac{k_{1}k_{S}(1+qR)}{k_{1}+k_{-1}}}$
(15)

We see that the efficiency of surface-catalyzed quenching is bounded from
above by $\eta_{\max}=e^{\beta\epsilon}$, which corresponds to the limit of
either fast excitation at the surface, $k^{\ast}/k_{S}\gg 1$, or fast
relaxation in the bulk, $k_{-1}/k_{1}\gg 1$, i.e. $c_{\infty}^{\ast}\ll
c_{0}$.
Let us now turn to another well-known problem, that of binding of a ligand
molecule to $\mathcal{N}$ receptors of size $b$ at the surface of a cell. The
celebrated Berg and Purcell (BP) formula [5] has the form of eq. (4) and thus
can be derived by solving a stationary diffusion problem subject to mixed BC,
provided $k^{\ast}=4DR\,b\mathcal{N}$ [4]. In other words, the surface
reaction rate should correspond to as many fully absorbing disks of radius $b$
as there are receptors on the cell’s surface. Unfortunately, despite the
correction worked out by Zwanzig through an elegant effective-medium argument
[8], the BP formula still suffers from the same ambiguity connected to the use
of sinks to describe binding reactions. Following our method, this problem can
be cast in a form that is more physically and biologically transparent and
that reduces to a BP-like formula in the appropriate physical limit.
Let a cell of radius $a$ be covered with $\mathcal{N}$ receptors that can
exist in their free form or in complex with some ligand. We consider the
following scheme of reactions, all occurring at the cell’s surface

$\displaystyle L+R\mathrel{\raisebox{-0.44481pt}{\hbox
to0.0pt{\raisebox{2.0pt}{$\ext@arrow
0359\arrowfill@\relbar\relbar\rightharpoonup{\hphantom{k_{-1}}}{k_{1}}$}\hss}\hbox{$\ext@arrow
3095\arrowfill@\leftharpoondown\relbar\relbar{k_{-1}}{\hphantom{k_{1}}}$}}}C$
$\displaystyle C\xrightarrow[]{k_{d}}R\qquad L\xrightarrow[]{k_{d}}\emptyset$
(16)

These correspond to complex formation and complex internalization with
concomitant unbinding, followed by receptor recycling and ligand degradation
within the cytoplasm. Taking into account the diffusion of ligand molecules in
the bulk, we can write the following set of equations,

$\frac{\partial\rho}{\partial t}=D\nabla^{2}\rho+\frac{\delta(r-a)}{4\pi
r^{2}}(k_{-1}N_{C}-k_{1}N_{L}N_{R})$ (17a)
$\frac{dN_{C}}{dt}=k_{1}N_{L}N_{R}-(k_{-1}+k_{d})N_{C}$ (17b)
$\lim_{r\to\infty}\rho(r)=\rho_{0}$ (17c)

where $\rho(r,t)$ is the ligand concentration field, $\rho_{0}$ its bulk
concentration and $N_{L},N_{R},N_{C}$ are the numbers of ligands, free
receptors and complexes at the surface of a single cell, respectively. The
number of ligands at the cell’s surface in eq. (17b) is
$N_{L}=\rho(a,t)/\rho_{c}$, where $\rho_{c}$ is the volume concentration of
cells. Combining eq. (17b) with the conservation law
$N_{R}+N_{C}=\mathcal{N}$, one easily obtains the number of complexes in the
stationary state, that is,

$N_{C}=\frac{\mathcal{N}\rho(a)}{\mathbb{K}+\rho(a)}$ (18)

where $\mathbb{K}=\rho_{c}(k_{-1}+k_{d})/k_{1}$ is the complex dissociation
constant. The appropriate boundary condition for the diffusion equation can be
obtained again by integrating eq. (17a) from $r=0$ to $r=a+\delta a$ and then
taking the limit $\delta a\to 0$. Taking into account eq. (18), this yields
the appropriate self-consistent BC, that is,

$4\pi
Da^{2}\left.\frac{d\rho}{dr}\right|_{a}-k_{d}\left(\frac{\mathcal{N}\rho(a)}{\mathbb{K}+\rho(a)}\right)=0$
(19)

The stationary ligand profile outside the cell has the form
$\rho(r)=\rho_{0}(1-\mu R/r)$, where $\mu$ is fixed by the BC (19),

$\mu=\frac{1}{2}\left(\frac{\mathbb{K}_{d}+\rho_{0}}{\rho_{0}}-\sqrt{\left(\frac{\mathbb{K}_{d}+\rho_{0}}{\rho_{0}}\right)^{2}-\frac{4k_{d}\mathcal{N}}{k_{S}\rho_{0}}}\right)$
(20)

with $\mathbb{K}_{d}=\mathbb{K}+k_{d}\mathcal{N}/k_{S}$. In this case, the
setting is intrinsically a non-equilibrium one as in the standard Smoluchowski
formulation, with the important difference that the present approach allows
one to build in ligand binding and degradation in a physically and
biologically transparent manner. The overall ligand intake rate follows
immediately as $k_{i}=4\pi Da^{2}(d\rho(a)/dr)=\mu k_{S}\rho_{0}$. This result
should be considered as the minimal extension of the Berg-Purcell formula in
the direction of physical transparency of the underlying biochemical process.
Once more, taking specific limits turns out to be insightful.
At first glance, one may expect that the Smoluchowski rate $k_{S}\rho_{0}$
should be recovered in the limit of infinite degradation, $k_{d}\to\infty$. In
fact, the maths reveal a slightly different physical picture, that is,

$\lim_{k_{d}\to\infty}k_{i}=k_{S}\rho_{0}\left(\frac{\kappa^{\ast}}{\kappa^{\ast}+k_{S}}\right)$
(21)

where $\kappa^{\ast}=\mathcal{N}k_{1}/\rho_{c}$. We thus learn that the BP
formula corresponds to the limit of infinite internal degradation of ligand
molecules when the kinetics of complex formation at the membrane is properly
accounted for. Moreover, the intuitive prescription
$\kappa^{\ast}\propto\mathcal{N}$ posited by Shoup and Szabo [4] is recovered
here self-consistently. Interestingly, our calculation unveils the correct
interpretation of the intrinsic reaction rate $\kappa^{\ast}$, beyond the
physical ambiguity of circular sinks. Not surprisingly, in this minimal model
it is the probability per unit time of complex formation, $k_{1}$, that
determines how effective a ligand absorber is a receptor-covered cell. Of
course, if the kinetics of ligand binding and the degradation chain were
described in more detail, the calculation would provide more insight into how
the inner workings of the whole biochemical pathway should enter the overall
intake rate. Perhaps a little more surprisingly, we also discover that
$\kappa^{\ast}\propto\rho_{c}^{-1}$. This simply means that $n$ cells with
similar numbers of receptors are $n$ times less effective individually at
ligand degradation 222This conclusion strictly holds provided
$\rho_{0}\gg\rho_{c}$, which allows one to describe ligand diffusion and
binding in the presence of a single cell in the first place., which is the
signature of mass-action law, encoded in the kinetics of ligand binding at the
cells’ membranes.
In summary, in this paper we have introduced a simple, yet powerful, idea that
allows one to address problems in molecular kinetics in solution on totally
new grounds. Instead of relying on the old idea of supplementing the diffusion
equation with empirical fully or partially absorbing boundary conditions, we
have shown that it is possible to derive the appropriate BCs self-consistently
from the geometry of the reaction boundary and the details of the chemical
reaction network. In our scheme, this can comprise reactions occurring in the
bulk and at the reaction surface alike. The extent of the proposed change of
perspective can be readily appreciated from the vastness of the potential
applications of our idea. For example, with reference to ligand binding at a
cell’s membrane, the biochemical kinetic network considered may be enriched by
including different receptors, other surface proteins that activate/inhibit
them, more complex steps associated with internalization and recycling and a
more detailed description of the degradation chain. In principle, our method
should provide a rigorous microscopic strategy to investigate how ligand
binding regulates the functioning of any signaling pathway to any desired
level of biochemical accuracy.
More generally, our idea may be readily employed to investigate time-dependent
reaction diffusion problems too, as well as complex non-spherical reaction
boundaries or even multiple disconnected boundaries arranged according to some
pattern [10]. The latter perspective appears especially intriguing, as our
method would allow one to explore how neighbouring reaction surfaces influence
each other with respect to a given microscopic kinetic model.

###### Acknowledgements.

The main idea this paper hinges upon occurred to the author as a result of a
question asked by Alberto Parola on the physical meaning of absorbing boundary
conditions in the Smoluchowski problem. The author is indebted to him for
demonstrating that questions are typically more insightful than answers. The
author is also grateful to Paolo De Los Rios and Giuseppe Foffi for their
precious comments.

## References

  * von Smoluchowski [1916] M. von Smoluchowski, Drei vortrage ubër diffusion brownsche molekular bewegung und koagulation von kolloidteichen, Physik Z 17, 557 (1916).
  * Note [1] Strictly speaking, as shown by Szabo [7], this results is correct only in the limit $D_{A}\to 0$.
  * Collins and Kimball [1949] F. C. Collins and G. E. Kimball, Diffusion-controlled reaction rates, Journal of Colloid Science 4, 425 (1949).
  * Shoup and Szabo [1982] D. Shoup and A. Szabo, Role of diffusion in ligand binding to macromolecules and cell-bound receptors,  _Biophysical Journal_ , Biophysical Journal 40, 33 (1982).
  * Berg and Purcell [1977] H. C. Berg and E. M. Purcell, Physics of chemoreception., Biophysical journal 20, 193 (1977).
  * Sozza  _et al._ [2018] A. Sozza, F. Piazza, M. Cencini, F. De Lillo, and G. Boffetta, Point-particle method to compute diffusion-limited cellular uptake, Physical Review E 97, 023301 (2018).
  * Szabo [1989] A. Szabo, Theory of diffusion-influenced fluorescence quenching, The Journal of Physical Chemistry 93, 6929 (1989).
  * Zwanzig [1990] R. Zwanzig, Diffusion-controlled ligand binding to spheres partially covered by receptors: an effective medium treatment., Proceedings of the National Academy of Sciences of the United States of America 87, 5856 (1990).
  * Note [2] This conclusion strictly holds provided $\rho_{0}\gg\rho_{c}$, which allows one to describe ligand diffusion and binding in the presence of a single cell in the first place.
  * Galanti  _et al._ [2016] M. Galanti, D. Fanelli, and F. Piazza, Macroscopic transport equations in many-body systems from microscopic exclusion processes in disordered media: A review, Frontiers in Physics 4, 33 (2016).


# Influence of initial states on memory effects: A study of early-time
superradiance

S. C. Hou<EMAIL_ADDRESS>School of Science, Dalian Maritime University,
Dalian 116026, China  G. Q. Shuai  School of Science, Dalian Maritime
University, Dalian 116026, China  X. Y. Zhang  School of Science, Dalian
Maritime University, Dalian 116026, China  J. Shen  Department of Criminal
Technology, Liaoning Police College, Dalian 116036, China  X. X. Yi
<EMAIL_ADDRESS>Center for Quantum Sciences and School of Physics, Northeast
Normal University, Changchun 130024, China

###### Abstract

The initial state of a quantum system can significantly influence its future
dynamics, especially in non-Markovain quantum processes due to the
environmental memory effects. Based on a previous work of ours, we propose a
method to quantify the memory effects of a non-Markovian quantum process
conditioned on a particular system initial state. We apply our method to study
the early-time dynamics of a superradiance model where $N$ atoms (the system)
interacting with a single-mode vacuum cavity (the environment) with several
types of initial states. We find that the value of the memory effects in the
early-time regime is half the environmental photon number for the (dephased)
Dicke states. Besides, the memory effects, the environmental photon number and
the degree of superradiance can be simultaneously enhanced by the coherence or
entanglement of some initial states. In our study, the transitions from non-
superradiant initial states to superradiant ones are always accompanied by the
enhancement of memory effects, showing the importance of memory effects in
superradiance.

##  I Introduction

The initial state of a quantum system can significantly influence its future
dynamics [1, 2], especially in non-Markovain quantum processes due to the
environmental memory effects. One trivial example is that if the system is
initially in a steady state in a non-Markovian quantum process, it can hardly
exhibit any non-Markovain features afterward, such as the revival of energy or
information [3, 4]. More intriguing phenomena may emerge when a system
consists of a collection of subsystems, such that the properties of its
initial state, such as entanglement, may significantly influence the future
dynamics. A well-known example is the concept of superradiance [5, 7, 6, 8, 9,
10, 11, 12] introduced by Dicke in 1954, where the emission intensity from an
ensemble of atoms interacting with a common electromagnetic field can be
enhanced compared with that from independent atoms. It is also well-known that
the superradiance behaviors are highly relevant to the initial states of the
system. In recent years, superradiance has received a large amount of
attention due to its theoretical significance and potential applications [14,
13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. Under certain
approximations, such as a coarse grained time scale, the superradiance process
could be regarded as Markovian [8, 9, 10, 11, 12], whereas, it is
intrinsically non-Markovian. With the advances in theories and technologies,
understanding the non-Markovian dynamics of superradiance becomes more
demanding [26, 27, 28, 29]. In a recent work [29], the author shows that non-
Markovian memory effects play an important role in superradiance beyond
retardation, featuring the quadratic dynamics in the early-time (Zeno regime).
In view of the significant influences of the system initial states on non-
Markovain quantum processes, especially, a superradiance process, some
interesting questions arise. For example, how to quantitatively evaluate the
memory effects of a quantum process conditioned on a particular system initial
state? How does the initial state of a system influence the memory effects?
What is the role of the memory effects in a superradiance process, especially
in its the early-time dynamics where the superradiance is created. Are there
quantitative relations between the memory effects and the superradiance
characteristics.

In this paper, we try to answer these questions by first proposing a method to
quantify the memory effects in a quantum process conditioned on a particular
system initial state. The method is based on a previous work [30] of ours. In
that work, we quantify the memory effects (past-future dependence) of a
quantum process through the inequality of completely positive dynamical maps
$T(t_{2},t_{0})\neq T(t_{2},t_{1})T(t_{1},t_{0})$. By acting the inequality on
a particular initial state $\rho_{S}(t_{0})$, the memory effects could be
understood by the difference between two final states $\rho_{S}(t_{2})$ and
$\rho_{S}^{\prime}(t_{2})$. The reason is that the evolutions corresponding to
$T(t_{2},t_{0})$ and $T(t_{2},t_{1})$ have the same system states at $t_{1}$
but different histories before $t_{1}$. Based on the above interpretation, we
suggest measuring the memory effects conditioned on the initial state
$\rho_{S}(t_{0})$ by the maximal difference of $\rho_{S}(t_{2})$ and
$\rho_{S}^{\prime}(t_{2})$ in a time interval of interest. Using this method,
we calculate the memory effects (as well as the radiation characteristics) of
a superradiance model in the early-time regime (Zeno regime). The model
describes $N$ two-level atoms interacting with a single-mode vacuum cavity.
Several types of initial states are studied, such as the (dephased) Dicke
states and factorized identical states. We find that the value of memory
effects for the (dephased) Dicke states is half the cavity photon number in
the early-time regime. Besides, it is observed that the single-atom coherence
in the factorized identical states and the coherence (entanglement) in the
dephased Dicke states can enhance the memory effects, the cavity photon number
and the degree of superradiance at the same time. In our studies, the
transitions from non-superradiant initial states to superradiant ones are
always accompanied by the enhancement of memory effects. The results
demonstrate that the memory effects is vital for the superradiance phenomenon
and the (change of) environment photon number is an important source of the
memory effects, especially for the Dicke states.

The paper is organized as follows. In Sec.II, we propose a method to evaluate
the initial-state-dependent memory effects in a quantum process. In Sec.III,
we obtain the early-time solution of $N$ two-level atoms interacting with a
vacuum cavity, after which we give the expressions for the memory effects, the
cavity photon number and the degree of superradiance for an arbitrary initial
state of the system. In Sec.IV, the influences of initial states on the memory
effects as well as the superradiance characteristics are calculated and
analyzed for several types of initial states. Conclusions and discussions are
presented in Sec.V.

##  II Initial-state-dependent memory effects

In this section, we first review the measure of non-Markovianity in [30] that
quantifies the memory effects (the past-future dependence) in a quantum
process. Using its physical interpretations, we then propose a method to
quantify the memory effects in a quantum process conditioned on a particular
initial state of the system. The object of study in [30] is a quantum process
of an open quantum system described by the total Hamiltonian

$\displaystyle H=H_{S}+H_{E}+H_{SE},$ (1)

an arbitrary system initial state $\rho_{S}(t_{I})$ and a fixed initial state
of the environment $\rho_{E}(t_{I})$ (independent of the system). Here $t_{I}$
is the initial time of an evolution which is also arbitrary. The initial
condition of an evolution is

$\displaystyle\rho_{SE}(t_{I})=\rho_{S}(t_{I})\otimes\rho_{E}(t_{I})$ (2)

which means that the system is initially uncorrelated with the environment
before the evolution. In general, $\rho_{E}(t_{I})$ is governed by [31]

$\displaystyle\rho_{E}(t_{I})=\mathcal{T}e^{-i\int_{0}^{t_{I}}H_{E}(\tau)d\tau}\rho_{E}(0).$
(3)

Typically, one deals with a time-independent environment Hamiltonian $H_{E}$
and assumes a steady state of $H_{E}$ as the environmental initial state
(e.g., a thermal state). Then, the initial condition of an evolution is

$\displaystyle\rho_{SE}(t_{I})=\rho_{S}(t_{I})\otimes\rho_{E}$ (4)

for any $t_{I}$. Since $\rho_{S}(t_{I})$ and $t_{I}$ is arbitrary, the measure
of non-Markovianity (memory effects) in [30] is determined by $H$ and
$\rho_{E}(t_{I})$ and the time interval of interest.

In a non-Markovian quantum process, the meaning of a dynamical map given by
$\rho_{S}(t_{2})=\varepsilon(t_{2},t_{1})\rho_{S}(t_{1})$ is ambiguous unless
$\rho_{SE}(t_{1})$, or from another perspective, the initial time $t_{I}$ is
specified ($t_{I}\leqslant t_{1}$). For clarity, we define $T(t_{2},t_{1})$ as
a memoryless dynamical map that transfers $\rho_{S}(t_{1})$ to
$\rho_{S}(t_{2})$ where $t_{1}$ is the initial time of the evolution
($t_{I}=t_{1}$) [30], i.e.,

$\displaystyle\rho_{S}(t_{2})$ $\displaystyle=$ $\displaystyle
T(t_{2},t_{1})\rho_{S}(t_{1})$ (5) $\displaystyle=$
$\displaystyle\text{Tr}_{E}[U(t_{2},t_{1})\rho_{S}(t_{1})\otimes\rho_{E}(t_{1})U(t_{2},t_{1})^{{\dagger}}]$

where $\rho_{E}(t_{1})$ is fixed, $\rho_{S}(t_{1})$ is arbitrary and
$U(t_{2},t_{1})=\mathcal{T}e^{-i\int_{t_{1}}^{t_{2}}H(\tau)d\tau}$ in general.
Particularly, when $H$ is time-independent and $\rho_{E}(t_{I})=\rho_{E}$ is a
steady state of $H_{E}$, Eq.(5) is simplified to

$\displaystyle\rho_{S}(t_{2})$ $\displaystyle=$ $\displaystyle
T(t_{2},t_{1})\rho_{S}(t_{1})$ (6) $\displaystyle=$
$\displaystyle\text{Tr}_{E}[e^{-iH(t_{2}-t_{1})}\rho_{S}(t_{1})\otimes\rho_{E}e^{iH(t_{2}-t_{1})}]$

such that $T(t_{2},t_{1})=T(t_{2}-t_{1},0)$ [32]. The dynamical map $T$ is
trace-preserving and completely positive and called a universal dynamical map
(UDM) that is independent of the state it acts upon [33].

Let $t_{0}\leqslant t_{1}\leqslant t_{2}$, a (time-dependent) quantum process
is Markovian if the dynamical maps (denoted by $\Lambda_{M}$) satisfy the
divisibility condition

$\displaystyle\Lambda_{M}(t_{2},t_{0})=\Lambda_{M}(t_{2},t_{1})\Lambda_{M}(t_{1},t_{0})$
(7)

where each dynamical map is uniquely defined and a UDM. Remark that the
dynamical map $\Lambda_{M}(t_{2},t_{1})$ is a UDM if and only if it is induced
by

$\displaystyle\rho_{S}(t_{2})$ $\displaystyle=$
$\displaystyle\Lambda_{M}(t_{2},t_{1})\rho_{S}(t_{1})$ (8) $\displaystyle=$
$\displaystyle\text{Tr}_{E}[U(t_{2},t_{1})\rho_{S}(t_{1})\otimes\rho_{E}(t_{1})U(t_{2},t_{1})^{{\dagger}}]$

where $\rho_{E}(t_{1})$ is fixed and $\rho_{S}(t_{1})$ is arbitrary [33]. In a
open quantum system, the condition
$\rho_{SE}(t)=\rho_{S}(t)\otimes\rho_{E}(t)$ [$\rho_{E}(t)$ does not dependent
on the system state] may not be satisfied exactly for $t>t_{I}$. Thus the
dynamics of an exact open quantum system is typically not Markovian [33].
However, $\rho_{SE}(t)\approx\rho_{S}(t)\otimes\rho_{E}(t)$ can be a good
approximation where the correlation between the system and the environment
does not affect the system’s dynamics so much [33]. It is observed that the
Markovian dynamical map $\Lambda_{M}(t_{2},t_{1})$ is unique and does not
depend on the initial time of a evolution, e.g., $t_{I}=t_{1}$ or
$t_{I}<t_{1}$. Therefore, $\Lambda_{M}(t_{2},t_{1})=T(t_{2},t_{1})$ according
to the definition of the memoryless dynamical map $T$. Then, the Markovian
divisibility condition Eq.(7) can also be expressed in terms of the memoryless
dynamical map $T$ by

$\displaystyle T(t_{2},t_{0})=T(t_{2},t_{1})T(t_{1},t_{0})$ (9)

whose violation is a sign of non-Markovianity. Unlike the dynamical maps used
in some non-Markovianity measures [34, 35], all the dynamical maps $T$ are
completely positive. The violation of Eq.(9) is manifested by the inequality

$\displaystyle T(t_{2},t_{0})\neq T(t_{2},t_{1})T(t_{1},t_{0}).$ (10)

The physical meaning of Eq.(10) can be explained with Fig.1. Let the left-hand
and the right-hand side of Eq.(10) act on a system initial state
$\rho_{S}(t_{0})$. On the left-hand side of Eq.(10), $\rho_{S}(t_{0})$ is
mapped to $\rho_{S}(t_{2})$ by $T(t_{2},t_{0})$ in evolution A. On the right-
hand side, $\rho_{S}(t_{0})$ is firstly mapped to $\rho_{S}(t_{1})$ by
$T(t_{1},t_{0})$ in evolution B. Then, as the initial state of evolution C,
$\rho_{S}(t_{1})$ is mapped to $\rho^{\prime}_{S}(t_{2})$ by $T(t_{2},t_{1})$,
which means that the initial condition of evolution C is
$\rho_{SE}(t_{1})=\rho_{S}(t_{1})\otimes\rho_{E}(t_{1})$ with fixed
$\rho_{E}(t_{1})$ defined by Eq.(3) or a steady one $\rho_{E}$. At the moment
$t_{1}$, evolution A and C have the same system state $\rho_{S}(t_{1})$ but
different histories: evolution A has a history in time interval
$[t_{0},t_{1}]$ that is encoded in
$\rho_{SE}(t_{1})=U(t_{1},t_{0})\rho_{S}(t_{0})\otimes\rho_{E}(t_{0})U(t_{1},t_{0})^{{\dagger}}$;
while evolution C (starting at $t_{1}$) have no history before $t_{1}$.
Therefore, $\rho_{S}(t_{2})\neq\rho^{\prime}_{S}(t_{2})$ is an evidence that
the future state (after $t_{1}$) of the system depends on its history (in
$[t_{0},t_{1}]$) in this quantum process. This is the physical meaning of the
memory effects in this paper and [30]. Otherwise, if the divisibility Eq.(9)
holds, the process is Markovain and $\rho_{S}(t_{2})=\rho^{\prime}_{S}(t_{2})$
for any $\rho_{S}(t_{0})$.

On the other hand, the inequality could be understood by focusing on the
change of the environment state between evolution B and C. At the end of
evolution B,
$\rho_{SE}(t_{1})=U(t_{1},t_{0})\rho_{S}(t_{0})\otimes\rho_{E}(t_{0})U(t_{1},t_{0})^{{\dagger}}$.
Then, at the beginning of evolution C, the environment is initialized by
$T(t_{2},t_{1})$ such that
$\rho_{SE}(t_{1})\rightarrow\rho_{S}(t_{1})\otimes\rho_{E}(t_{1})$ where
$\rho_{E}(t_{1})$ is independent of the system. The information of the
system’s history in $[t_{0},t_{1}]$ is erased by $T(t_{2},t_{1})$. In
contrast, such an initialization never happens in evolution A. Therefore,
$\rho_{S}(t_{2})\neq\rho^{\prime}_{S}(t_{2})$ signifies that the environment
(as well as the correlations between the system and environment [33])
remembers the system’s history and this memory can influence the future of the
system.

In [30], we define the maximal difference of $T(t_{2},t_{0})$ and
$T(t_{2},t_{1})T(t_{1},t_{0})$ as the non-Markovianity where $t_{1}$ and
$t_{2}$ are optimized. Based on the dynamical maps $T$, all the initial states
are potentially considered, thus the measure of non-Markovianity in [30] does
not depend on the initial state of the system. In this paper, our goal is to
evaluate the influence of the system initial state on the memory effects in a
quantum process. Using the physical interpretations discussed above, we define
the value of memory effects conditioned on $\rho_{S}(t_{0})$ as the maximal
trace distance between $\rho_{S}(t_{2})$ and $\rho^{\prime}_{S}(t_{2})$:

$\displaystyle
N_{M}[\rho_{S}(t_{0})]=\max_{t_{1},t_{2}}\frac{1}{2}\|\rho_{S}(t_{2})-\rho^{\prime}_{S}(t_{2})\|.$
(11)

Here $||A||=\text{Tr}(\sqrt{A^{\dagger}A})$ is the trace norm of an operator
$A$. Assuming a fixed $t_{0}$ for convenience, $N_{M}[\rho_{S}(t_{0})]$ could
be calculated by optimizing $t_{1}$ and $t_{2}$ in a time interval $[t_{0},t]$
of interest where $t_{0}\leqslant t_{1}\leqslant t_{2}\leqslant t$. For
example, the time interval might be $[t_{0},\infty]$ or $[t_{0},t_{0}+\tau]$
where $\tau$ is finite (as done later in this paper). Similar to the measure
in [30], $0\leqslant N_{M}[\rho_{S}(t_{0})]\leqslant 1$ is satisfied due to
the properties of the trace distance.

Using Eq.(11), the influence of initial states on the memory effects of an
quantum process could be quantitatively evaluated. Note that
$N_{M}[\rho_{S}(t_{0})]>0$ is a sufficient condition for the inequality
Eq.(10), but not a necessary one. The theoretical calculation and experimental
observation of Eq.(11) might be easier than those in [30] since the
determination of the dynamical maps $T$ is not compulsory.

Figure 1: Physical interpretation of Eq.(10) as memory effects. At the moment
$t_{1}$, the system states in evolution A and C are the same. Remind that
evolution A has a history from $t_{0}$ to $t_{1}$ while evolution C does not
have any history before $t_{1}$. Therefore,
$\rho_{S}(t_{2})\neq\rho^{\prime}_{S}(t_{2})$ is an evidence that the future
(after $t_{1}$) state of the system depends on its history (from $t_{0}$ to
$t_{1}$) in a quantum process.

##  III Early-time superradiance

###  III.1 Theoretical model

We consider a fundamental model that describes $N$ two-level atoms (the
system, denoted by “$S$”) interacting with a cavity (the environment, denoted
by “$E$” ) initially in a vacuum state. The Hamiltonian is given by

$\displaystyle H\\!$ $\displaystyle=$ $\displaystyle\\!H_{S}+H_{E}+H_{I}$ (12)
$\displaystyle=$
$\displaystyle\\!\omega_{A}\\!\sum_{n=1}^{N}\\!\sigma_{n}^{+}\sigma_{n}^{-}\\!+\\!\omega_{B}b^{{\dagger}}b\\!+\\!\sum_{n=1}^{N}g(\sigma_{n}^{+}b+\sigma_{n}^{-}b^{{\dagger}}).$

Here $H_{S}$ and $H_{E}$ represent $N$ noninteracting two-level atoms and a
single-mode electromagnetic field in the cavity, respectively.
$H_{I}=\sum_{n=1}^{N}g(\sigma_{n}^{+}b+\sigma_{n}^{-}b^{{\dagger}})$ describes
the interactions between the atoms and the cavity with the rotating wave
approximation, and the coupling strength $g$ is a real constant. The lowering
and raising operators for the $n$th atom is defined as
$\sigma_{n}^{-}=|g\rangle_{n}\langle e|_{n}$ and
$\sigma_{n}^{+}=|e\rangle_{n}\langle g|_{n}$.

The initial condition of the model is assumed to be
$\rho_{S\\!E}(t_{0})=\rho_{S}(t_{0})\otimes|0\rangle\langle 0|$ where
$\rho_{S}(t_{0})$ is arbitrary and $|0\rangle$ is the vacuum state of the
cavity. For simplicity, we use $t_{0}=0$ in the reminder of this paper without
loss of generality. The density matrix of the composite system $\rho_{S\\!E}$
is described by the master equation,

$\displaystyle\dot{\rho}_{S\\!E}$ $\displaystyle=$
$\displaystyle-i[H_{S}+H_{E}+H_{I},\rho_{S\\!E}]$ (13)
$\displaystyle+\gamma(b\rho_{S\\!E}b^{{\dagger}}-\frac{1}{2}b^{{\dagger}}b\rho_{S\\!E}-\frac{1}{2}\rho_{S\\!E}b^{{\dagger}}b)$

where $\gamma$ is the dissipation strength of the cavity. Assume the cavity is
in resonance with the atoms ($\omega_{A}=\omega_{B}$), then the density matrix
$\rho_{S\\!E}$ in the interaction picture is described by

$\displaystyle\dot{\rho}_{S\\!E}$ $\displaystyle=$
$\displaystyle-i[\sum_{n=1}^{N}g(\sigma_{n}^{+}b+\sigma_{n}^{-}b^{{\dagger}}),\rho_{S\\!E}]$
(14)
$\displaystyle+\gamma(b\rho_{S\\!E}b^{{\dagger}}-\frac{1}{2}b^{{\dagger}}b\rho_{S\\!E}-\frac{1}{2}\rho_{S\\!E}b^{{\dagger}}b).$

In this remainder of this paper, we focus on the early-time dynamics where
$gt\ll 1$ due to the following reasons. First, non-Markovian characters are
non-negligible on such a short time scale. Second, it stresses the influence
of a particular initial state since the state hardly changes in this time
interval. Furthermore, it is helpful for understanding the creation mechanism
of the superradiance. In the early-time limit $gt\rightarrow 0$, one might
ignore the influence of cavity dissipation $\gamma$, providing that $\gamma$
is not infinitely large. In this case, $\rho_{S\\!E}$ evolves unitarily via

$\displaystyle\rho_{S\\!E}(t)=U(t)\rho_{S}(0)\otimes\rho_{E}U^{{\dagger}}(t)$
(15)

where $U(t)=e^{-iH_{I}t}$ and $\rho_{E}=|0\rangle\langle 0|$. The influence of
ignoring $\gamma$ on the early-time dynamics will be discussed at the end of
this section.

###  III.2 Early-time solution

The reduced dynamics of the atoms could be determined by tracing out the
degrees of the environment:

$\displaystyle\rho_{S}(t)$ $\displaystyle=$
$\displaystyle\text{Tr}_{E}[U(t)\rho_{S}(0)\otimes\rho_{E}U^{{\dagger}}(t)]$
(16) $\displaystyle=$ $\displaystyle\sum_{k}\langle
k|_{E}U(t)\rho_{S}(0)\otimes\rho_{E}U^{{\dagger}}(t)|k\rangle_{E}$

where $|k\rangle_{E}$ are a set of basis in $\mathcal{H}_{E}$. With the help
of the Baker-Campbell-Hausdorff formula

$\displaystyle e^{\alpha A}Be^{-\alpha
A}=B+\alpha[A,B]+\frac{\alpha^{2}}{2!}[A,[A,B]]+...,$ (17)

after the replacement $A=H_{I}$, $B=\rho_{S}(0)\otimes\rho_{E}$ and
$\alpha=-it$, we have

$\displaystyle\rho_{S}(t)$ $\displaystyle\approx$
$\displaystyle\sum_{k}\langle
k|(\rho_{S}(0)\otimes\rho_{E}-it[H_{I},\rho_{S}(0)\otimes\rho_{E}]$ (18)
$\displaystyle-\frac{1}{2}t^{2}[H_{I},[H_{I},\rho_{S}(0)\otimes\rho_{E}]]|k\rangle$

where the terms with higher orders of $gt$ have been omitted due to the early-
time limit ($gt\rightarrow 0$). Using the number state basis of the cavity
$|k\rangle=|0\rangle,|1\rangle,...$ and the assumption
$\rho_{E}=|0\rangle\langle 0|$, it is straightforward to derive the solution
of the system evolution in the early-time limit $gt\rightarrow 0$,

$\displaystyle\rho_{S}(t)=\rho_{S}(0)+(gt)^{2}\mathcal{L}_{\sigma^{-}}[\rho_{S}(0)].$
(19)

Here $\sigma^{-}=\sum_{n}\sigma_{n}^{-}$ is the collective lowering operator
and $\mathcal{L}_{K}(\rho)=K\rho
K^{+}-\frac{1}{2}K^{\dagger}K\rho-\frac{1}{2}\rho K^{\dagger}K$ is the
Lindblad superoperator. $b^{{\dagger}}|0\rangle=|1\rangle$ and $\langle
0|b=\langle 1|$ have been used to obtain Eq.(19). Notice that the partial
trace over the second term of Eq.(18) is zero, thus the change of
$\rho_{S}(0)$ is quadratic in $t$ in the early-time limit.

###  III.3 Memory effects

With the above results, we now calculate the initial-state-dependent memory
effects defined in Sec.II. We consider the dynamics in a short time interval
$[0,t]$ where $t_{0}=0\leqslant t_{1}\leqslant t_{2}\leqslant t$. In
“evolution A” (with a history from $t_{0}$ to $t_{1}$) as mentioned before,
the initial state is $\rho_{S}(0)$ and the final state $\rho_{S}(t_{2})$ is
given by

$\displaystyle\rho_{S}(t_{2})$ $\displaystyle=$
$\displaystyle\rho_{S}(0)+[g(\tau_{10}+\tau_{21})]^{2}\mathcal{L}_{\sigma^{-}}[\rho_{S}(0)].$
(20)

Here $\tau_{10}=t_{1}-t_{0}$ and $\tau_{21}=t_{2}-t_{1}$ are used for
convenience. The initial state of “evolution C” at $t_{1}$ is

$\displaystyle\rho_{S}(t_{1})$ $\displaystyle=$
$\displaystyle\rho_{S}(0)+(g\tau_{10})^{2}\mathcal{L}_{\sigma^{-}}[\rho_{S}(0)].$
(21)

In “evolution C” (without any history before $t_{1}$), the final state
$\rho^{\prime}_{S}(t_{2})$ is given by

$\displaystyle\rho^{\prime}_{S}(t_{2})$ $\displaystyle=$
$\displaystyle\rho_{S}(t_{1})+(g\tau_{21})^{2}\mathcal{L}_{\sigma^{-}}[\rho_{S}(t_{1})].$
(22)

Substituting Eq.(21) into Eq.(22), we have

$\displaystyle\rho^{\prime}_{S}(t_{2})$ $\displaystyle=$
$\displaystyle\rho_{S}(0)+(g\tau_{10})^{2}\mathcal{L}_{\sigma^{-}}[\rho_{S}(0)]$
(23)
$\displaystyle+(g\tau_{21})^{2}\mathcal{L}_{\sigma^{-}}\\{\rho_{S}(0)+(g\tau_{10})^{2}\mathcal{L}_{\sigma^{-}}[\rho_{S}(0)]\\}$
$\displaystyle=$
$\displaystyle\rho_{S}(0)+(g\tau_{10})^{2}\mathcal{L}_{\sigma^{-}}[\rho_{S}(0)]$
$\displaystyle+(g\tau_{21})^{2}\mathcal{L}_{\sigma^{-}}[\rho_{S}(0)]$

where the high-order term
$g^{4}\tau_{10}^{2}\tau_{21}^{2}\mathcal{L}_{\sigma^{-}}\\{\mathcal{L}_{\sigma^{-}}[\rho_{S}(0)]\\}$
has been omitted due to the early-time limit. The memory effects are
manifested by the difference between Eq.(20) and Eq.(23):

$\displaystyle\rho_{S}(t_{2})\\!-\\!\rho^{\prime}_{S}(t_{2})\\!$
$\displaystyle=$
$\displaystyle\\!g^{2}[(\tau_{10}+\tau_{21})^{2}\\!-\\!\tau_{10}^{2}\\!-\\!\tau_{21}^{2}]\mathcal{L}_{\sigma^{-}}[\rho_{S}(0)]$
(24) $\displaystyle=$
$\displaystyle\\!2g^{2}\tau_{10}\tau_{21}\mathcal{L}_{\sigma^{-}}[\rho_{S}(0)].$

According to Eq.(11), the value of the memory effects for the initial state
$\rho_{S}(0)$ is

$\displaystyle N_{M}[\rho_{S}(0)]$ $\displaystyle=$
$\displaystyle\max_{\tau_{10},\tau_{21}}\frac{1}{2}\|\rho_{S}(t_{2})-\rho^{\prime}_{S}(t_{2})\|$
(25) $\displaystyle=$ $\displaystyle
g^{2}\|\mathcal{L}_{\sigma^{-}}[\rho_{S}(0)]\|\max_{\tau_{10},\tau_{21}}(\tau_{10}\tau_{21}).$

Since $g^{2}\|\mathcal{L}_{\sigma^{-}}[\rho_{S}(0)]\|$ is a constant for a
given $\rho_{S}(0)$, Eq.(25) can be calculated simply by maximizing
$\tau_{10}\tau_{21}$ with the constraint
$0\leqslant\tau_{10}+\tau_{21}=\tau_{20}\leqslant t$. Let
$\tau_{10}+\tau_{21}=t^{\prime}$, then
$\tau_{10}\tau_{21}=-(\tau_{10}-\frac{t^{\prime}}{2})^{2}+\frac{t^{\prime
2}}{4}\leqslant\frac{t^{\prime 2}}{4}$. Therefore, the maximum of
$\tau_{10}\tau_{21}$ is $\frac{t^{\prime 2}}{4}$ when
$\tau_{10}=\tau_{21}=\frac{t^{\prime}}{2}$. It is easy to see that the maximum
of $\tau_{10}\tau_{21}$ in the time interval $[0,t]$ happens when
$t^{\prime}=t$. Eventually, $N_{M}[\rho(t_{0})]$ in the early-time limit is

$\displaystyle N_{M}[\rho_{S}(0)]$ $\displaystyle=$ $\displaystyle
g^{2}\|\mathcal{L}_{\sigma^{-}}[\rho_{S}(0)]\|\max_{\tau_{10},\tau_{21}}(\tau_{10}\tau_{21})$
(26) $\displaystyle=$ $\displaystyle
g^{2}\|\mathcal{L}_{\sigma^{-}}[\rho_{S}(0)]\|(\frac{t}{2})^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{4}(gt)^{2}\|\mathcal{L}_{\sigma^{-}}[\rho_{S}(0)]\|.$

Eq.(26) demonstrates that in a short time interval $[0,t]$, $N_{M}[\rho(0)]$
grows quadratically with $t$. To focus on the influence of initial states
(rather than the time $t$ or the atom-cavity coupling $g$), it is convenient
to discuss the normalized value of memory effects

$\displaystyle\frac{N_{M}[\rho_{S}(0)]}{(gt)^{2}}=\frac{1}{4}\|\mathcal{L}_{\sigma^{-}}[\rho_{S}(0)]\|.$
(27)

It represents the strength of memory effects in the early-time limit as a
function of only $\rho_{S}(0)$. Notice that
$\frac{N_{M}[\rho_{S}(0)]}{(gt)^{2}}$ is not bounded as $N_{M}[\rho_{S}(0)]$.
In principle, there is $0\leqslant\frac{N_{M}[\rho_{S}(0)]}{(gt)^{2}}<\infty$.

###  III.4 Cavity photon number

In a non-Markovian proceess, the relation
$\rho_{SE}(t)\approx\rho_{S}(t)\otimes\rho_{E}$ does not hold in general.
Particularly, for the superradiance problem, the change of photo number in the
environment (the cavity) might be an important source of the memory effects so
that $\rho_{SE}(t)\neq\rho_{S}(t)\otimes\rho_{E}$. Thus it is desirable to
know the cavity photon number (denoted by $N_{P}$ in this paper) in the early-
time limit in order to understand the physics of the memory effects. The total
excitation number of our model represented by
$N_{ex}=b^{\dagger}b+\sum_{n}\sigma_{n}^{+}\sigma_{n}^{-}$ is conserved since
$[N_{ex},H]=0$. Besides, there are no photon in the cavity initially.
Therefore, the cavity photon number is equal to the loss of excitations of the
atoms that represents the emission intensity of superradiance (in the early-
time limit).

According to Eq.(19), the cavity photon number at $t$ for $\rho_{S}(0)$ is
given by,

$\displaystyle N_{P}[\rho_{S}(0)]$ $\displaystyle=$
$\displaystyle\text{Tr}[\sum_{n}\sigma_{n}^{+}\sigma_{n}^{-}\rho_{S}(0)]-\text{Tr}[\sum_{n}\sigma_{n}^{+}\sigma_{n}^{-}\rho_{S}(t)]$
(28) $\displaystyle=$
$\displaystyle\text{Tr}[\sum_{n}\sigma_{n}^{+}\sigma_{n}^{-}\rho_{S}(0)]-\text{Tr}[\sum_{n}\sigma_{n}^{+}\sigma_{n}^{-}\rho_{S}(0)]$
$\displaystyle-\text{Tr}\\{\sum_{n}\sigma_{n}^{+}\sigma_{n}^{-}(gt)^{2}\mathcal{L}_{\sigma^{-}}[\rho_{S}(0)]\\}$
$\displaystyle=$
$\displaystyle(gt)^{2}\text{Tr}\\{-\sum_{n}\sigma_{n}^{+}\sigma_{n}^{-}\mathcal{L}_{\sigma^{-}}[\rho_{S}(0)]\\}.$

Using $[\sigma^{+}\sigma^{-},\sum_{n}\sigma_{n}^{+}\sigma_{n}^{-}]=0$ and
$[\sum_{n}\sigma_{n}^{+}\sigma_{n}^{-},\sigma^{+}]=\sigma^{+}$, Eq.(28) can be
simplified to a more concise form

$\displaystyle
N_{P}[\rho_{S}(0)]=(gt)^{2}\text{Tr}[\sigma^{+}\sigma^{-}\rho_{S}(0)].$ (29)

It is observed that in the early-time limit, the cavity number increases
quadratically with time. As mentioned in the last subsection, it is convenient
to discuss the normalized cavity photon number

$\displaystyle\frac{N_{P}[\rho_{S}(0)]}{(gt)^{2}}=\text{Tr}[\sigma^{+}\sigma^{-}\rho_{S}(0)]$
(30)

such that it does not depend on the evolution time $t$ and atom-cavity
coupling $g$. Eq.(30) represents the emission intensity in the early-time
limit as a function of only the initial state. When each atom radiates
independently, e.g., each atom radiates in its own cavity, the normalized
cavity photon number for the $n$th atom is reduced to

$\displaystyle\frac{N_{P}^{(n)}[\rho_{S_{n}}(0)]}{(gt)^{2}}=\text{Tr}[\sigma_{n}^{+}\sigma_{n}^{-}\rho_{S_{n}}(0)]$
(31)

where $\rho_{S_{n}}(0)$ is the reduced density matrix of the $n$th atom. Then
the degree of superradiant might be measured by the ratio [7, 13]

$\displaystyle S[\rho_{S}(0)]$ $\displaystyle=$
$\displaystyle\frac{N_{P}[\rho_{S}(0)]/(gt)^{2}}{\sum_{n}N_{P}^{(n)}[\rho_{S}(0)]/(gt)^{2}}$
(32) $\displaystyle=$
$\displaystyle\frac{\text{Tr}[\sigma^{+}\sigma^{-}\rho_{S}(0)]}{\sum_{n}\text{Tr}[\sigma_{n}^{+}\sigma_{n}^{-}\rho_{S_{n}}(0)]}$

for nonzero denominator. If $S[\rho_{S}(0)]$ is greater (less) than 1, the
state $\rho_{S}(0)$ is superradiant (subradiant).

###  III.5 Early-time regime

Mathematically, Eq.(19) holds true as $gt\rightarrow 0$. One may wonder the
validity of the early-time solution Eq.(19) in a longer time interval and how
the cavity dissipation deteriorates the validity. In this subsection, we
discuss this problem by comparing the systems dynamics by Eq.(19) with that by
numerically solving the full dynamics in
$\mathcal{H}_{S}\otimes\mathcal{H}_{E}$ with Eq.(14) and tracing out the
environment. A dynamics could be represented by the dynamical maps $T(t,0)$
that turns all the possible initial state at $0$ to a final state at $t$.
Furthermore, the difference of two dynamical maps can be evaluated through
their Choi-Jamiółkowski matrices [36, 37]. Here we evaluate the error of
Eq.(19) at instant $t$ by the trace distance of $\rho_{T(t,0)}^{quad}$ and
$\rho_{T(t,0)}^{exact}$, i.e.,

$\displaystyle
Error=\frac{1}{2}\|\rho_{T(t,0)}^{quad}-\rho_{T(t,0)}^{exact}\|$ (33)

where $\rho_{T(t,0)}^{quad}$ and $\rho_{T(t,0)}^{exact}$ are the Choi-
Jamiółkowski matrices calculated by Eq.(19) and Eq.(14), respectively. The
Choi-Jamiółkowski matrix is defined as
$\rho_{T}(t,0)=T(t,0)\otimes\mathbb{I}(|\Psi\rangle\langle\Psi|)$ with
$|\Psi\rangle=\frac{1}{\sqrt{2^{N}}}\sum_{i=1}^{2^{N}}|i\rangle|i\rangle$ a
maximally entangled state of the $N$-atom system and an ancillary system of
the same dimension.

We numerically calculated the error of Eq.(19) for $\gamma/g=0,1,10$ and
$N=2,6$. The results are illustrated in Fig.2 in logarithmic scale. It is seen
that the order of magnitude of error decreases linearly with that of $t$.
Although the cavity dissipation and the increasing of atom number can increase
the error, there exists a time interval where Eq.(19) is a good approximation
as long as $\gamma$ and $N$ are finite. For example, in a time interval
$[0,0.01/g]$, the error is less than $10^{-5}$ even when $\gamma=10g$ for both
$N=2$ and $N=6$, implying that Eq.(19) is a good approximation in this
scenario. The above discussion implies that the early-time limit
$gt\rightarrow 0$ could be relaxed to an early-time regime, i.e., a time
interval $[0,\tau]$ where the decay is quadratic. Here $\tau$ is linked to the
Zeno time [38, 39, 40, 41].

Figure 2: The error of Eq.(19) as a function of $gt$ for (a) 2 atoms and (b) 6
atoms with different strengths of cavity dissipation. The error represents the
difference of the dynamical maps $T(t,0)$ corresponding to Eq.(19) (early-time
assumption) and Eq.(14) (exact). The error vanishes as $gt\rightarrow 0$ and
increases with the cavity dissipation $\gamma$ or the atom number $N$.

##  IV Influence of initial states

In this section, the results Eq.(27), Eq.(30) and Eq.(32) are applied to
several types of $N$-atom initial states. Our first aim is to study the
influence of initial states on the memory effects. Another aim is to reveal
the role of memory effect in creating superradiance. Conversely, it also helps
to understand the physical source of memory effects in our model.

###  IV.1 Dicke states

The Dicke states, written as $|JM\rangle$, are extensively studied in the
field of superradiance. It is defined as the common eigenstate of the
pseudospin operators $D^{2}=\frac{1}{2}(D^{+}D^{-}+D^{-}D^{+})+D_{z}^{2}$ and
$D_{z}$ with $J=N/2$. The eigenvalues are given by

$\displaystyle D^{2}|JM\rangle$ $\displaystyle=$ $\displaystyle
J(J+1)|JM\rangle$ (34) $\displaystyle D_{z}|JM\rangle$ $\displaystyle=$
$\displaystyle M|JM\rangle\qquad(M=-J,...,J)$ (35)

where the operators are defined by $D^{+}=\sum_{n}\sigma_{n}^{+}$,
$D^{-}=\sum_{n}\sigma_{n}^{-}$ and
$D_{z}=\frac{1}{2}\sum_{n}(|e\rangle_{n}\langle e|_{n}-|g\rangle_{n}\langle
g|_{n})$. A Dicke state is symmetrical and invariant by atom permutation with
$N_{e}=J+M$ excited atoms and $N_{g}=J-M$ ground-state atoms. It can be
constructed by the following formula [8]:

$\displaystyle|JM\rangle=\sqrt{\frac{(J+M)!}{N!(J-M)!}}(\sum_{n}\sigma_{n}^{-})^{J-M}|e,e,...,e\rangle.$
(36)

For example, for 3 two-level atoms, there is
$|3/2,-1/2\rangle=\frac{1}{\sqrt{3}}(|egg\rangle+|geg\rangle+|gge\rangle)$.

A well-known result of the free-space spontaneous emission rate for two-level
atoms is $W_{N}=\Gamma\langle\sigma^{+}\sigma^{-}\rangle$ where $\Gamma$ is
the natural linewidth of a single atom. For the Dicke states, there is

$\displaystyle W_{N}$ $\displaystyle=$ $\displaystyle\Gamma\langle
JM|\sigma^{+}\sigma^{-}|JM\rangle$ (37) $\displaystyle=$
$\displaystyle\Gamma(J+M)(J-M+1)$

When $M=J$ (fully excited), $W_{N}=N\Gamma$. Then the emission rate is
proportional to $N$, or say, the $N$-atom emission rate is equal to the
summation of those from independent atoms. Thus there is no superradiance.
When $M=0$ (half excited), $W_{N}=\Gamma\frac{N}{2}(\frac{N}{2}+1)$, the
emission rate increase with $N^{2}$, or say, the $N$-atom emission rate is
greater than the summation of those from independent atoms
($\frac{N}{2}\Gamma$). The superradiance with the most intensive emission
happens in this case. Note that the results holds under the Born-Markov
approximation and on a coarse-grained time scale [8, 9, 10, 11, 12].

Figure 3: (a) Normalized Memory effects $N_{M}$ and (b) normalized cavity
photon number $N_{p}$ in the early-time regime for different Dicke states
$|JM\rangle$. The number of atoms $N$ varies from 1 to 15, correspondingly,
$J=N/2=1/2,3/2,....15/2$. The results demonstrate that if the atoms are
initially in $|JM\rangle$, the normalized value of memory effects is half the
normalized cavity photon number in the early-time regime. The strongest memory
effects as well as photon emission happens when $M$ is 0 or next to 0.

In the non-Markovian early-time regime, we calculated the normalized value of
memory effects and cavity photon number with Eq.(27) and Eq.(30) for the Dicke
states of different $N$. The results are shown in Fig.3 where $N=1,2,...,15$
($J=N/2$) and $M=-J,-J+1,...,J$. It is observed that when the number of atoms
$N$ is fixed, the strongest memory effects as well as photon emission happens
when $M$ is 0 or next to 0, which is in agreement with Eq.(37). Interestingly,
we find that the value of memory effects for the Dicke states can be fully
determined by the cavity photon number by

$\displaystyle N_{M}(|JM\rangle)=\frac{1}{2}N_{P}(|JM\rangle)$ (38)

in the early-time regime. The relation is proved in the following. Using the
property $\sigma^{\pm}|JM\rangle=\sqrt{J(J+1)-M(M\pm 1)}|J(M\pm 1)\rangle$, it
is straightforward to see that

$\displaystyle\frac{N_{M}(|JM\rangle)}{(gt)^{2}}\\!$ $\displaystyle=$
$\displaystyle\\!\frac{1}{4}\|\mathcal{L}_{\sigma^{-}}(|JM\rangle)\|$ (39)
$\displaystyle=$ $\displaystyle\\!\frac{1}{4}\|\sigma^{-}|JM\rangle\langle
JM|\sigma^{+}\\!-\\!\frac{1}{2}\sigma^{+}\sigma^{-}|JM\rangle\langle JM|$
$\displaystyle\\!-\frac{1}{2}|JM\rangle\langle JM|\sigma^{+}\sigma^{-}\|$
$\displaystyle=$
$\displaystyle\\!\frac{1}{4}(J\\!+\\!M)(J\\!-\\!M\\!+\\!1)\big{\|}|JM^{\prime}\rangle\langle
JM^{\prime}|$ $\displaystyle-|JM\rangle\langle JM|\big{\|}$

where $M^{\prime}=M-1$. Remind that the trace norm could be calculated by
$\|A\|=\text{Tr}\sqrt{A^{\dagger}A}=\sum_{m}|\lambda_{m}|$ where $\lambda_{m}$
is the eigenvalue of operator $A$. Meanwhile, the eigenvalues of
$|JM^{\prime}\rangle\langle JM^{\prime}|-|JM\rangle\langle JM|$ is $1$ and
$-1$ whose absolute values sum up to 2. Therefore,

$\displaystyle\frac{N_{M}(|JM\rangle)}{(gt)^{2}}=\frac{1}{2}(J+M)(J-M+1).$
(40)

Considering that the normalized cavity photon number for Dicke states is

$\displaystyle\frac{N_{P}(|JM\rangle)}{(gt)^{2}}$ $\displaystyle=$
$\displaystyle\text{Tr}(\sigma^{+}\sigma^{-}|JM\rangle\langle JM|)$ (41)
$\displaystyle=$ $\displaystyle\text{Tr}[(J+M)(J-M+1)|JM\rangle\langle JM|]$
$\displaystyle=$ $\displaystyle(J+M)(J-M+1),$

Eq.(38) is proved. The relation Eq.(38) demonstrates that the (change of) the
cavity photon number is a fundamental reason of the memory effects in the
early-time regime for the Dicke states.

We now discuss the degree of superadiance for the Dicke states in the early-
time regime. The normalized cavity photon number for one independent atom in
its excited state is given by
$N_{P}(|e\rangle)/(gt)^{2}=N_{P}(|\frac{1}{2},\frac{1}{2}\rangle)/(gt)^{2}=1$
by Eq.(31). Using the results of Eq.(41), the degree of superradiance for the
Dicke states could be calculated by

$\displaystyle S(|JM\rangle)$ $\displaystyle=$
$\displaystyle\frac{N_{P}(|JM\rangle)/(gt)^{2}}{\sum_{n=1}^{J+M}N_{P}(|e\rangle)/(gt)^{2}}$
(42) $\displaystyle=$ $\displaystyle\frac{(J+M)(J-M+1)}{J+M}$ $\displaystyle=$
$\displaystyle J-M+1$

for $J+M>0$, where the denominator represents the contribution of the emission
from the $J+M$ independently excited atoms in $|JM\rangle$. Note that the
physical meaning of the denominator in Eq.(42) is different from that in
Eq.(32), but their values are the same.

In Fig.3(a) or (b), the left row represents $|J,-J\rangle$ (no atom excited),
thus $N_{M}$ and $N_{P}$ are both zero which is a trivial case. Another group
of Dicke states without superradiance are $|J,J\rangle$ (fully excited)
corresponding to the right row in Fig.3(a) or (b). Using Eq.(42), it is easy
to check that $S(|J,J\rangle)=1$. Except for the two rows, other Dicke states
in Fig.3 are superradiant which could be examined by $S(|JM\rangle)>1$ with
Eq.(42). For example, for the Dicke states $|J,-J+1\rangle$ (one atom excited)
represented by the second row from the left, there is $S(|J,-J+1\rangle)=N$.
The single-photon superradiance is called “the greatest radiation anomaly” by
Dicke. Meanwhile, the normalized value of memory effects for $|J,-J+1\rangle$
is $\frac{N}{2}$, which is also proportional to $N$. For the states in the
second row from the right, there is $S(|J,J-1\rangle)=2$ for $N>1$. The degree
of superradiance is a constant showing a weaker degree of superradiance. For
Dicke states, although the normalized value of memory effects is not
proportional to the degree of superradiance in general, it is observed that
the transition from non-supperradiant states to superradiant states (e.g.,
$|J,J\rangle\rightarrow|J,J-1\rangle$) is always accompanied by the increase
of memory effects. Thus the memory effects are important for the creation of
superradiance.

###  IV.2 Factorized identical states

In this subsection, we investigate $N$-atom initial states in a factorized
from

$\displaystyle\rho_{S}^{fact}(0)$ $\displaystyle=$
$\displaystyle\rho_{S_{1}}\otimes\rho_{S_{2}}\otimes...\otimes\rho_{S_{N}}$
(43)

with identical single-atom states

$\displaystyle\rho_{S_{1}}=\rho_{S_{2}}=...=\rho_{S_{N}}=\left(\begin{array}[]{cc}\rho_{ee}&\rho_{eg}\\\
\rho_{eg}^{*}&\rho_{gg}\\\ \end{array}\right).$ (46)

Such states are fully characterized by $\rho_{ee}$ and $\rho_{eg}$ and may
also be superradiant. For example, states of this form are experimentally
studied in [13] to realize the single-atom superradiance where the a serious
of atoms enter a cavity one by one. In [21], we show that such a model can be
used to achieve the Heisenberg Limit in a quantum metrology scheme for
measuring the atom-cavity coupling. The superradiance of two artificial atoms
with initial state
$\frac{1}{2}(|e\rangle+|g\rangle)\otimes(|e\rangle+|g\rangle)$ are
experimentally observed in [15]. In the above works, the single-atom coherence
($|\rho_{eg}|$) are crucial for creating superradiance. Thus it is desirable
to know the influence of such states on the memory effects in superradiance,
especially the role of the single-atom coherence.

In this subsection, the normalized value of memory effects and the normalized
cavity photon number for states described by Eq.(43) are calculated using
Eq.(27) and Eq.(30). When Eq.(46) is a pure state, i.e.,
$\rho_{Sn}=|\phi\rangle\langle\phi|$, $\rho_{S}^{fact}(0)$ can be decomposed
into the superpositions of different Dicke states [13]. Using the analytical
result of $\langle\phi|\sigma^{+}\sigma^{-}|\phi\rangle$ in [13], we obtain
the normalized cavity photon number for $\rho_{S}^{fact}(0)$:

$\displaystyle\frac{N_{P}[\rho_{S}^{fact}(0)]}{(gt)^{2}}=N(N-1)|\rho_{eg}|^{2}+N\rho_{ee}.$
(47)

According to Eq.(32), the degree of superradiance is given by

$\displaystyle S[\rho_{S}^{fact}(0)]$ $\displaystyle=$
$\displaystyle\frac{N_{P}[\rho_{S}^{fact}(0)]/(gt)^{2}}{\sum_{n=1}^{N}N_{P}^{(n)}(\rho_{S_{n}})/(gt)^{2}}$
$\displaystyle=$
$\displaystyle\frac{N(N-1)|\rho_{eg}|^{2}+N\rho_{ee}}{N\rho_{ee}}$
$\displaystyle=$ $\displaystyle 1+(N-1)|\rho_{eg}|^{2}/\rho_{ee}$

for $\rho_{ee}>0$. The result demonstrates that the state $\rho_{S}^{fact}(0)$
is superradiant when $|\rho_{eg}|$ is nonzero and $N>1$. Besides, the
normalized cavity photon number as well as the degree of superradiance grow
quadratically with the single-atom coherence $|\rho_{eg}|$ for $N>1$.

Figure 4: Normalized memory effects $N_{M}$ (left column) and normalized
cavity photon number $N_{p}$ (right column) for $\rho_{S}^{fact}(0)$ as a
function of $\rho_{ee}$ and $|\rho_{eg}|$ in the early-time regime. The atom
numbers are $N=1$, $2$ and $11$ from top to bottom. For any $\rho_{ee}$ and
$N$ ($N>1$), $N_{M}$ ($N_{p}$) can always be enhanced nonlinearly by the
single-atom coherence $|\rho_{eg}|$.

In contrast to Eq.(40), the analytical solution of Eq.(27) for the factorized
initial states is complicated in general. Therefore, Eq.(27) is solved
numerically in this section. The results of the normalized
$N_{M}[\rho_{S}^{fact}(0)]$ and $N_{P}[\rho_{S}^{fact}(0)]$ are depicted in
Fig.4 as a function of $\rho_{ee}$ and $|\rho_{eg}|$ with $N=1$, 2 and $11$.
The condition $|\rho_{eg}|^{2}\leqslant(1-\rho_{ee})\rho_{ee}$ are satisfied
in Fig.4 to guarantee the positivity of $\rho_{S_{n}}$. The single-atom
coherence $\rho_{eg}$ are chosen to be real in the simulations for simplicity.
Numerical results shows that the phase of $\rho_{eg}$ does not affect the
value of memory effects. Unlike the case for the Dicke states,
$N_{M}[\rho_{S}^{fact}(0)]/(gt)^{2}$ is not proportional to
$N_{P}[\rho_{S}^{fact}(0)]/(gt)^{2}$ in general. However, our simulation
results show that the normalized value of memory effects can always be
enhanced by $|\rho_{eg}|$ for any $N$. Similar situations happen for
$N_{P}[\rho_{S}^{fact}(0)]/(gt)^{2}$ and $S[\rho_{S}^{fact}(0)]$ when $N>1$ as
seen in Eq.(47) and (IV.2). Besides, when the atom number $N$ increases,
$N_{M}[\rho_{S}^{fact}(0)]/(gt)^{2}$ grows more and more fast with
$|\rho_{eg}|$, and so do $N_{P}[\rho_{S}^{fact}(0)]/(gt)^{2}$ and
$S[\rho_{S}^{fact}(0)]$ as seen in Eq.(47) and (IV.2). Thus we conclude that
the creation and enhancement of superradiance by $|\rho_{eg}|$ is accompanied
by the increase of memory effects for the factorized identical initial states.
In Fig.4(c) and (e), it is found that the states around $\rho_{eg}=0$ and
$\rho_{ee}=0.5$ tend to take lower values of memory effects compared with
their neighboring states, especially when $N$ is large. This might be
intuitively interpreted by the fact that $\rho_{eg}=0$ and $\rho_{ee}=0.5$
leads to a maximally mixed state of the $N$-atom system. Besides, Fig.4 shows
that the (change of) the cavity photon number is not the only factor to
determine the value of the memory effects. Other factors might include the
(change of) coherence of the cavity state between the basis $|n\rangle$.

###  IV.3 Other initial states

Entanglement and coherence of the initial states may play important roles in
superradiance [42, 22, 23, 24, 25], especially for the Dicke states. Next, we
examine the influence of the coherence (or the entanglement among the atoms)
in the dephased Dicke states $\rho^{deph}_{S}(0)$ on the memory effects as
well as the superradiance. The initial state we consider is given by

$\displaystyle\rho^{deph}_{S}(0)=\lambda|JM\rangle\langle
JM|+(1-\lambda)\mathcal{D}(|JM\rangle\langle JM|)$ (48)

where $0\leqslant\lambda\leqslant 1$. Here $\mathcal{D}$ turns
$|JM\rangle\langle JM|$ into a fully dephased state with vanishing off-
diagonal elements in the basis $|e(g),e(g),...,e(g)\rangle$. Meanwhile,
$\mathcal{D}(|JM\rangle\langle JM|)$ is a mixed state of separable states
whose entanglement is zero. Thus $\lambda$ reflects the strength of
entanglement or the value of coherence of $\rho^{deph}_{S}(0)$. With the help
of numerical solutions, we find by mathematical induction that
$\frac{N_{M}[\mathcal{D}(|JM\rangle\langle JM|)]}{(gt)^{2}}=\frac{1}{2}(J+M)$
and

$\displaystyle\frac{N_{M}[\rho^{deph}_{S}(0)]}{(gt)^{2}}$ $\displaystyle=$
$\displaystyle\lambda\frac{N_{M}(|JM\rangle\langle JM|)}{(gt)^{2}}$ (49)
$\displaystyle+(1-\lambda)\frac{N_{M}[\mathcal{D}(|JM\rangle\langle
JM|)]}{(gt)^{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(J+M)[(J-M)\lambda+1].$

It is seen that the normalized value of memory effects increases linearly with
$\lambda$ when $M\neq J,-J$. The normalized cavity photon number can be
calculated by Eq.(30). We obtain

$\displaystyle\frac{N_{P}[\rho^{deph}_{S}(0)]}{(gt)^{2}}=$ $\displaystyle=$
$\displaystyle\lambda\frac{N_{P}(|JM\rangle)}{(gt)^{2}}$ (50)
$\displaystyle+(1-\lambda)\frac{N_{P}[\mathcal{D}(|JM\rangle\langle
JM|)]}{(gt)^{2}}$ $\displaystyle=$ $\displaystyle(J+M)[(J-M)\lambda+1].$

Similarly, $N_{M}[\rho^{deph}_{S}(0)]=\frac{1}{2}N_{P}[\rho^{deph}_{S}(0)]$ is
satisfied for the dephased Dicke states in the early-time regime. The degree
of superradiance for the dephased Dicke states is given by

$\displaystyle S[\rho^{deph}_{S}(0)]$ $\displaystyle=$
$\displaystyle\frac{N_{P}[\rho^{deph}_{S}(0)]/(gt)^{2}}{\sum_{n=1}^{J+M}N_{P}(|e\rangle)/(gt)^{2}}$
(51) $\displaystyle=$ $\displaystyle\frac{(J+M)[(J-M)\lambda+1]}{J+M}$
$\displaystyle=$ $\displaystyle(J-M)\lambda+1$

for $J+M>0$. It is observed that the entanglement or coherence (represented by
$\lambda$) in the depahsed Dicke state is necessary for superradiance.
Besides, the degree of superradiance increases linearly with $\lambda$ for
superradiant Dicke states. Similarly, the creation and increase of
superradiance induced by $\lambda$ is accompanied by the increase of memory
effects.

At the end of this subsection, we calculate the value of the normalized value
of memory effects of a probabilistic mixture of Dicke states
$\rho_{S}^{mix}(0)=\sum_{M}p_{M}|JM\rangle\langle JM|$ where $J=N/2$,
$M=-J,-J+1,...,J$ and $\sum_{M}p_{M}=1$. Using similar treatment as in
Eq.(39), we find that

$\displaystyle\frac{N_{M}[\rho_{S}^{mix}(0)]}{(gt)^{2}}\\!$ $\displaystyle=$
$\displaystyle\\!\frac{1}{4}[\\!\sum_{M=-J}^{J-1}\\!|p_{M+1}f(M+1)-p_{M}f(M)|$
(52) $\displaystyle+p_{J}f(J)]$

where $f(X)=(J+X)(J-X+1)$. The result reduces to Eq.(40) when
$\rho_{S}^{mix}(0)$ is a pure state $|JM\rangle\langle JM|$.

##  V Conclusions and Discussions

In this paper, we propose a method to evaluate the memory effects in a quantum
process conditioned on a particular system initial state. The method is based
on the physical interpretations of the inequality $T(t_{2},t_{0})\neq
T(t_{2},t_{1})T(t_{1},t_{0})$ [30]. Some features of the non-Markovianity
measure in [30] are inherited. For example, nonzero memory effects can be
characterized even in regimes where monotonic behaviors do not occur, or have
not occurred yet. Besides, $0\leqslant N_{M}[\rho_{S}(t_{0})]\leqslant 1$ is
satisfied without additional normalizations. This allows us to compare the
memory effects of quantum systems with different dimensions (e.g., different
numbers of atoms in our model). Using our method, we calculate the influence
of several types of initial states on the early-time memory effects. For the
(dephased, mixed) Dicke states, we obtained simple analytic solutions for the
normalized memory effects in terms of $J$ and $M$, or say, $N_{e}$ and
$N_{g}$. The environmental photon number as well as the degree of
superradiance are also calculated. The aims of comparing the memory effects
with the superradiance characters is to reveal the role of memory effects in
generating superradiance, and conversely, understand the physical sources of
the memory effects.

The main conclusions are as follows: For the (dephased) Dicke states, the
value of memory effects is half the cavity photon number in the early-time
regime. For the factorized identical states, the single-atom coherence is
necessary for superradiance. Such coherence can always enhance the memory
effects for $N\geqslant 1$, meanwhile, it increases the cavity photon number
and the degree of superradiance for $N\geqslant 2$. For the dephased Dicke
states, the coherence (entanglement of atoms) represented by $\lambda$ is also
necessary for superrdiance. Meanwhile, the memory effects, the cavity photon
number and the degree of superradiance increase linearly with $\lambda$ for
superradiant states. In the above cases, the transitions from non-superradiant
states to superradiant ones are always accompanied by the enhancement of
memory effects, implying the memory effects are vital for creating
superradiance. On the other hand, we find that the (change of) environmental
photon number is an important (but may not be the only) source of the memory
effects in the superradiance process, especially for the Dicke states.

##  VI ACKNOWLEDGMENTS

The authors thank X. L. Huang, X. L. Zhao, J. Cheng and B. Cui for helpful
discussions. The work is supported by the National Natural Science Foundation
of China under Grant No. 11705026, the Fundamental Research Funds for the
Central Universities in China under Grant No. 3132020178 and the Natural
Science Foundation of Liaoning Province under Grant No. 2023-MS-333.

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