\section{Introduction}

The interference between quantum transition amplitudes is the underlying principle of coherent control \cite{doi:10.1146/annurev.pc.43.100192.001353,doi:10.1146/annurev.physchem.48.1.601,doi:10.1146/annurev.physchem.040808.090427}. Given some independent excitation pathways connecting an initial state to a final target state of a quantum system, the total transition probability depends on the modulus squared of the sum of the corresponding complex transition amplitudes. Therefore, upon changing the relative phase of these amplitudes, it is possible to modify the final yield. This concept has found several applications in atomic, molecular and semiconductor physics, among other areas \cite{Petta2180}. In addition, it has contributed towards the development of quantum optimal control theory (QOCT), which seeks to find external controls to drive a given transition, maximizing some performance criteria \cite{rabitz2000,brif2010,peirce1988}.

Coherent control has been applied in the context of ultracold atomic gases \cite{koch2012,PhysRevLett.96.173002,PhysRevA.75.051401,PhysRevA.95.013411} and in particular to Bose-Einstein condensates (BEC) \cite{Thomasen_2016,Bohi2009,Lozada_Vera_2013}. For instance, the control of the onset of self-trapping of a condensate in a periodically modulated double well has been demonstrated \cite{PhysRevA.64.011601}. Also, SU(2) rotations on spinor condensates has been coherently controlled by stimulated Raman adiabatic passage \cite{PhysRevA.94.053636}. In addition, there have been several experimental and theoretical successful implementations of quantum optimal control algorithms for BEC \cite{Nat1,Nat2,hohenester2014,hohenester2007,Mennemann_2015,0953-4075-46-10-104012,PhysRevA.90.033628,PhysRevA.98.022119}.

A challenging control problem is the generation of non-ground-state condensates, which can exhibit several interesting features such as mode locking, critical dynamics, interference patterns and atomic squeezing \cite{yukalov69}. In the mean-field picture, non-ground-states can be represented as stationary solutions of the Gross-Pitaevskii equation (GPE), and are termed nonlinear topological modes \cite{yukalov66}. It has been shown that a given nonlinear mode can be generated by a time-dependent modulation of the trapping potential tuned to the resonance between the ground state and the desired excited nonlinear mode \cite{yukalov56}. Alternatively, it is also possible to generate a nonlinear mode by means of a resonant time-dependent modulation of the scattering length \cite{yukalov78,PhysRevA.81.053627}, which can be produced by an alternating magnetic field close to a Feshbach resonance \cite{PhysRevA.67.013605,PhysRevA.81.053627,clark2015,arunkumar2019}. Since quantum transition amplitudes can be attributed to these different modulations, their relative phase can be used to control the overall transition. Nevertheless, due to the nonlinear nature of the GPE, the transition amplitudes of the trapping and scattering length modulations are not completely independent, and new aspects of the coherent control can be expected in comparison with the linear Schr\"odinger equation. It is worth noting that QOCT has been applied to maximize the transition between nonlinear modes utilizing temporal and spatial modulation of both the trapping and the scattering length \cite{,PhysRevA.93.053612}. However, the control fields obtained from QOCT are often complex and difficult to be implemented in laboratory. Additionally, the detailed role of the interference between both modulations has not been addressed.

In the present work, we explore the possibility of using both modulations to resonantly generate non-ground-state condensate. In particular, we consider the role of the relative phase of the modulations on the transitions between nonlinear modes. In sec.~\ref{sec2}, we introduce the theoretical framework for the production of the nonlinear coherent modes. In sec.~\ref{twolevels}, a time average technique is applied to the GPE allowing the description of the dynamics by only two modes. A perturbative treatment is used to obtain an analytic expression for the transition probability in sec.~\ref{sec4}. Numerical results confirming the predictions are presented in sec.~\ref{sec5}. Finally, conclusions are drawn in sec.~\ref{sec6}.

\section{Excitation of nonlinear modes}\label{sec2}

We consider the dynamics of a Bose gas wavefunction $\Psi({\bf r},t)$ described by the Gross-Pitaevskii equation \cite{RevModPhys.71.463,pethick},
\begin{equation}
{\rm i}\hbar\frac{\partial}{\partial t}\Psi({\bf r},t)=\left(-\frac{\hbar^2}{2m}\nabla^2+V({\bf r},t)+g({\bf r},t)N|\Psi({\bf r},t)|^2\right)\Psi({\bf r},t),
\label{eq:TDGPE}
\end{equation}
where $m$ is the atomic mass and $N$ is the number of atoms in the condensate. The trapping potential $V({\bf r},t)$ is composed by two parts,
\begin{equation}
V({\bf r},t)=V_{\rm trap}({\bf r})+V_{\rm mod}({\bf r},t),
\label{eq:trappingwm}
\end{equation}
where $V_{\rm trap}({\bf r})$ is a fixed trapping potential and $V_{\rm mod}({\bf r},t)$ a time-dependent modulating potential. The nonlinear interaction amplitude $g({\bf r},t)$ is also composed by two parts,
\begin{equation}
g({\bf r},t)=g_{0}+g_{\rm mod}({\bf r},t),
\label{eq:nonlinearitygwm}
\end{equation}
where $g_0=4\pi\hbar^2a_0/m$ is a fixed nonlinearity and $g_{\rm mod}({\bf r},t)=4\pi\hbar^2a({\bf r},t)/m$ is a modulating nonlinearity, with the s-wave scattering length $a_s$ near a Feshbach resonance being written as $a_s=a_0+a({\bf r},t)$. The normalization condition of the wavefunction is $\int d{\rm \bf r}|\Psi({\bf r},t)|^2=1$.

With both modulations turned off, i.e., $V_{\rm mod}({\bf r},t)=0$ and $g_{\rm mod}({\bf r},t)=0$, the system can be described by the nonlinear Hamiltonian $H_0$,

\begin{equation}
    H_0[\phi({\bf r})]=-\frac{\hbar^2}{2m}\nabla^2+V_{\rm trap}({\bf r})+{g}_{0}N|\phi({\bf r})|^2.
\end{equation}

We consider the nonlinear topological modes of $H_0$, which are solutions of the eigenvalue problem \cite{yukalov56},
\begin{equation}
    H_0[\phi_n({\bf r})]\phi_n({\bf r})=\mu_n\phi_n({\bf r}),
    \label{multiindex0}
\end{equation}
with $n$ generally being a multi-index label for the quantum states and $\mu_n$ the corresponding chemical potential. Here, we are concerned with inducing transitions between stationary solutions $\phi_n({\bf r})$. This task can be accomplished by means of modulating the trapping potential with an oscillatory field with frequency $\omega_t$ \cite{yukalov56,courteille}. Alternatively, one may also modulate the atomic scattering length with frequency $\omega_g$ \cite{yukalov78}. We assume that both modulations are present and that they possess a phase difference given by $\theta$,

\begin{equation}
    V_{\rm mod}({\bf r},t)=V({\bf r})\cos(\omega_t t+\theta),
\label{Vmod}
\end{equation}
and
\begin{equation}
   g_{\rm mod}({\bf r},t)=g({\bf r})\cos(\omega_g t).
   \label{gmod}
\end{equation}

For definiteness, we consider a transition between an initial state $\phi_1({\bf r})$ to a final state $\phi_2({\bf r})$, with $\mu_1<\mu_2$, and we associate the resonance frequency $\omega_{21}=(\mu_2-\mu_1)/\hbar$ with this transition. As we have already pointed out, the transition can be induced by resonant modulations and in this case the system can be approximately described solely by the topological modes involved in the transition \cite{yukalov56,courteille}. Thus, we assume that the modulating frequencies $\omega_t$ and $\omega_g$ are close to $\omega_{21}$. More specifically, we assume that $|\Delta\omega_{t}/\omega_t|\ll1$ and $|\Delta\omega_{g}/\omega_g|\ll1$, with the detunings defined by $\Delta\omega_{t}=\omega_{t}-\omega_{21}$ and $\Delta\omega_{g}=\omega_{g}-\omega_{21}$.

\section{Two-Level approximation}\label{twolevels}

In order to simplify the dynamical equations, we consider that the wave function $\Psi({\bf r},t)$ can be written as an expansion in terms of nonlinear modes \cite{yukalov56},
\begin{equation}
\Psi( {\bf r},t) =\sum_{m}c_{m}(t)\phi_{m}({\bf r})\exp\left(-i\mu_m t/\hbar\right),
\label{expan0}
\end{equation}
and that the following condition for mode separation is valid,
\begin{equation}
    \frac{\hbar}{\mu_m} \left|\frac{dc_{m}}{dt}\right|\ll1,
    \label{condvary0}
\end{equation}
meaning that the $c_m(t)$ are slow function of time in comparison with $\exp(-i\mu_m t/\hbar)$.

Substituting expansion (\ref{expan0}) in the GPE and performing a time-averaging procedure, with the coefficients $c_m(t)$ treated as quasi-invariants, will result in a set of coupled nonlinear differential equations for the coefficients $c_m(t)$ (see Ref. \cite{yukalov66} for details). As a consequence, if at the initial time only  the levels $n=1,2$ are populated and the frequencies of the modulations are close to $\omega_{21}$, the only relevant coefficients are $c_1(t)$ and $c_2(t)$ and we obtain the set of equations,

\begin{subequations}\label{pop2}
\begin{alignat}{2}
i\frac{d{c}_{1}}{dt} = &  {\alpha}_{12}{\left| c_{2} \right|}^{2}{c}_{1} +\frac{1}{2}{\beta}_{12}c_{2}{\rm{e}}^{i\left(\Delta \omega_{t} t +\theta \right)}+ \frac{1}{2}{\rm{e}}^{-i\Delta\omega_{g} t}c_{2}^{*}{c_{1}}^{2}\gamma_{21}  \nonumber
 \\ & +\frac{1}{2}{\rm{e}}^{i\Delta\omega_{g} t}\left( {\left|c_{2} \right|}^{2}c_{2}\gamma_{12} + 2{\left|c_{1} \right|}^{2}c_{2}{\gamma}_{21}^{*} \right)   \ ,
 \\
i\frac{d{c}_{2}}{dt}=  &  {\alpha}_{21}{\left| c_{1} \right|}^{2}{c}_{2}+\frac{1}{2}{\beta}_{12}^{*}c_{1}{\rm{e}}^{-i\left(\Delta \omega_{t} t +\theta\right)} +
\frac{1}{2}{\rm{e}}^{i\Delta\omega_{g} t}c_{1}^{*}{c_{2}}^{2}\gamma_{12} \nonumber
 \\ & + \frac{1}{2}{\rm{e}}^{-i\Delta\omega_{g} t}\left( {\left|c_{1} \right|}^{2}c_{1}\gamma_{21} + 2{\left|c_{2} \right|}^{2}c_{1}{\gamma}_{12}^{*} \right),
  \end{alignat}
 \end{subequations}
with the coupling constants $\alpha_{mn}$, $\beta_{mn}$ and $\gamma_{mn}$ given by
\begin{equation}
    \alpha_{mn} \equiv g_{0}\frac{N}{\hbar}\int{d{\bf r}{|\phi_{m}({\bf r})|}^{2}\left[ 2{|\phi_{n}({\bf r})|}^{2}-{|\phi_{m}({\bf r})|}^{2}\right]},
    \label{alpha0}
\end{equation}
\begin{equation}
    \beta_{mn}\equiv \frac{1}{\hbar}\int{d{\bf r}{\phi}_{m}^{*}({\bf r})V({\bf r}){\phi}_{n}({\bf r})} \ ,
    \label{beta0}
\end{equation}
and
\begin{equation}
    \gamma_{mn}\equiv \frac{N}{\hbar}\int{d{\bf r}{\phi}_{m}^{*}({\bf r})g({\bf r}){|\phi_{n}({\bf r})|}^{2}}\phi_{n}({\bf r}) \ .
    \label{I0}
\end{equation}
In order to fulfill condition (\ref{condvary0}), the couplings are assumed to be much smaller than the transition frequency, i.e., $ |\alpha_{mn}/\omega_{21}|\ll1 $,  $            |\beta_{mn}/\omega_{21} |\ll1 $ and $ |\gamma_{mn}/\omega_{21} |\ll1 $ \cite{yukalov66}.

From the dynamical equations (\ref{pop2}), we note that the modulation of the trap couples the modes by means of the linear term containing $\beta_{mn}$ and the nonlinear term with $\alpha_{mn}$, while the modulation of the scattering length couples the modes by means of distinct nonlinear terms containing $\gamma_{mn}$. Therefore, there exists interference between the linear and nonlinear terms and by varying the phase $\theta$, this interference can be controlled. We also note that when the modulation of the scattering length is absent, $g_{\rm mod}({\bf r},t)=0$, an approximate analytic solution to (\ref{pop2}) has been derived, which shows that the population oscillates between the two states with a Rabi-like chirped frequency \cite{yukalov56}. This chirped frequency depends on the populations $|c_m(t)|^2$. Unfortunately, such approximate solutions for $g_{\rm mod}({\bf r},t)\neq0$ is not possible due to the presence of terms with $c_m(t)^2$. Thus, we resort to perturbation theory to gain more insight into the role of $\theta$ in the transition.

\section{Perturbative Approximation}\label{sec4}

We assume that modulating fields (\ref{Vmod}) and (\ref{gmod}) can be considered as small perturbations in order to apply canonical perturbation theory \cite{cohen}. To this end, we introduce a perturbation parameter $\lambda\ll1$ such that the Hamiltonian can be written as

\begin{equation}
    H[\Psi]=H_0[\Psi]+\lambda \left[V_{\rm mod}({\bf r},t)+g_{\rm mod}({\bf r},t)N|\Psi|^2\right] \ .
    \label{pertH}
\end{equation}
We are interested in the transition probability from state $\phi_1$ to state $\phi_2$, often defined as $P_{1\rightarrow 2}(t)=\left|\bra{\phi_2}\left.\Psi({\bf r},t)\right>\right|^2$ and we assume the initial conditions $c_1(0)=1$ and $c_2(0)=0$. However, from the approximations of the last section, one can deduce the normalization condition for the coefficients, $\sum_{m}\left|c_m(t)\right|^2=1$. Thus, despite of the fact that the set of nonlinear modes is not orthogonal, we can define the transition probability simply as $P_{1\rightarrow 2}(t)=\left|c_2(t)\right|^2$.

As usual, we write the coefficients $c_{j}(t)$ as a power series in $\lambda$,
\begin{equation}
    c_{j}(t)={c}_{j}^{(0)}(t)+\lambda{c}_{j}^{(1)}(t)+{\lambda}^{2}{c}_{j}^{(2)}(t)+\cdots,
    \label{cserie}
\end{equation}
and substitute this series into dynamic equations (\ref{pop2}) equating the like powers of $\lambda$. To zeroth order, this yields,
\begin{subequations}\label{zerotho}
\begin{alignat}{2}
i\frac{d{c}_{1}^{(0)}}{dt}=\alpha_{12}{|{c}_{2}^{(0)} |}^{2}{c}_{1}^{(0)} ,
\label{c1lambda0} \\
i\frac{d{c}_{2}^{(0)}}{dt}=\alpha_{21}{|{c}_{1}^{(0)}|}^{2}{c}_{2}^{(0)} \ .
\label{c2lambda0}
  \end{alignat}
 \end{subequations}
And we obtain the zeroth order solution as being ${c}_{1}^{(0)}(t)=1$, ${c}_{2}^{(0)}(t)=0$.

To first order in $\lambda$, the equations are
\begin{subequations}\label{firsto}
\begin{alignat}{2}
\label{c1lambda1} i\frac{d{c}_{1}^{(1)}}{dt}=\alpha_{12}\left[\left({c}_{2}^{*(0)}{c}_{2}^{(1)}+{c}_{2}^{*(1)}{c}_{2}^{(0)} \right){c}_{1}^{(0)} +{|{c}_{2}^{(0)} |}^{2}{c}_{1}^{(1)} \right]+\frac{1}{2}{\beta}_{12}{c}_{2}^{(0)}{\rm{e}}^{i\left(\Delta\omega_{t} t+\theta \right)} \nonumber \\
+\frac{1}{2}{\rm{e}}^{i\Delta\omega_{g} t}\left[ {| {c}_{2}^{(0)}|}^{2}{c}_{2}^{(0)}\gamma_{12}+2{|{c}_{1}^{(0)} |}^{2}{c}_{2}^{(0)}{\gamma}_{21}^{*} \right]+\frac{1}{2}{\rm{e}}^{-i\Delta\omega_{g} t} {c}_{2}^{*(0)}{{c}_{1}^{(0)}}^{2}\gamma_{21} \ , \\
 i\frac{d{c}_{2}^{(1)}}{dt}=\alpha_{21}\left[\left({c}_{1}^{*(0)}{c}_{1}^{(1)}+{c}_{1}^{*(1)}{c}_{1}^{(0)} \right){c}_{2}^{(0)} +{|{c}_{1}^{(0)} |}^{2}{c}_{2}^{(1)} \right]+\frac{1}{2}{\beta}_{12}^{*}{c}_{1}^{(0)}{\rm{e}}^{-i\left(\Delta\omega_{t} t+\theta \right)} \nonumber \\
+\frac{1}{2}{\rm{e}}^{-i\Delta\omega_{g} t}\left[ {| {c}_{1}^{(0)}|}^{2}{c}_{1}^{(0)}\gamma_{21}+2{|{c}_{2}^{(0)} |}^{2}{c}_{1}^{(0)}{\gamma}_{12}^{*} \right]+\frac{1}{2}{\rm{e}}^{i\Delta\omega_{g} t} {c}_{1}^{*(0)}{{c}_{2}^{(0)}}^{2}\gamma_{12} \ .
\label{c2lambda1}
  \end{alignat}
 \end{subequations}

Substituting the zeroth order solutions into (\ref{c1lambda1}) and (\ref{c2lambda1}), these equations simplify to

\begin{subequations}\label{firsto2}
\begin{alignat}{2}
i\frac{d{c}_{1}^{(1)}}{dt} & =0 \ ,
\label{c11lambda1} \\
i\frac{d{c}_{2}^{(1)}}{dt} & =\alpha_{21}{|{c}_{1}^{(0)}|}^{2}{c}_{2}^{(1)}+\frac{1}{2}{\beta}_{12}^{*}{\rm{e}}^{-i\left(\Delta\omega_{t} t +\theta \right)}+\frac{1}{2}\gamma_{21}{\rm{e}}^{-i\Delta\omega_{g} t} \ .
\label{c22lambda1}
  \end{alignat}
 \end{subequations}
Thus, within first order, ${c}_{1}^{(1)}(t)=0$ and
\begin{equation}
   {c}_{2}^{(1)}(t)=-\frac{1}{2}\frac{{\beta}_{12}^{*}}{(\alpha_{21}-\Delta\omega_{t})}{\rm{e}}^{-i\theta}\left( {\rm{e}}^{-i\Delta\omega_{t} t}-1\right)-\frac{1}{2}\frac{\gamma_{21}}{(\alpha_{21}-\Delta\omega_{g})}\left( {\rm{e}}^{-i\Delta\omega_{g}t}-1\right) \ .
    \label{sc22lambda1}
\end{equation}
Thus, we can write the transition probability within first order as
\begin{eqnarray}
P_{1\rightarrow 2}(t)\approx \frac{{|\beta_{12}|}^{2}}{2{|\alpha_{21}-\Delta\omega_{t}|}^{2}}\left[1- \cos(\Delta\omega_{t} t)\right]+\frac{{|\gamma_{21}|}^{2}}{2{|\alpha_{21}-\Delta\omega_{g}|}^{2}}\left[1- \cos(\Delta\omega_{g} t)\right] \nonumber \\
+ \frac{{\beta}_{12}\gamma_{21}}{4{{(\alpha_{21}-\Delta\omega}_{t})}^{*}(\alpha_{21}\Delta\omega_{g})}{\rm{e}}^{i\theta}\left[1+{\rm{e}}^{i(\Delta\omega_{t}-\Delta\omega_{g})t}-{\rm{e}}^{i\Delta\omega_{t}t}-{\rm{e}}^{-i\Delta\omega_{g}t}\right]  \nonumber \\
+\frac{{\gamma}_{21}^{*}{\beta}_{12}^{*}}{4{(\alpha_{21}-\Delta{\omega}_{g})}^{*}(\alpha_{21}-\Delta\omega_{t})}{\rm{e}}^{-i\theta}\left[1+{\rm{e}}^{-i(\Delta\omega_{t}-\Delta\omega_{g})t}-{\rm{e}}^{-i\Delta\omega_{t}t}-{\rm{e}}^{i\Delta\omega_{g}t}\right]    \ .
\label{P12t}
\end{eqnarray}
When the frequencies of the modulations are equal, $\Delta \omega_{t}=\Delta\omega_{g}=\Delta\omega$, the expression for the transition probability simplifies to

\begin{equation}
P_{1\rightarrow 2}\approx \left[  \frac{{|\beta_{12}|}^{2}+{|\gamma_{21}|}^{2}+2\Re\{{\beta}_{12}{\gamma}_{21}\exp(i\theta)\}}{{|\alpha_{21}-\Delta\omega|}^{2}}\right]{\sin}^{2}\left(\frac{\Delta\omega t}{2}\right) \ ,
    \label{P12twr}
\end{equation}
where $\Re\{\cdot\}$ stands for the real part.

Although the above expression is only valid for very short times, for which the population of the state $\phi_1$ is still very close to $1$, expression (\ref{P12twr}) evidences the role of the relative phase $\theta$ on the transition. For instance, if $({\beta}_{12}\gamma_{21})$ is real and positive, then for $\theta=\pi$ the modulations will act destructively decreasing the transition probability, whereas for $\theta=0$ they will act constructively. The extent of the interference will be dictated by the magnitude of the couplings parameters $\beta_{12}$ and $\gamma_{12}$. Additionally, according to (\ref{P12twr}), if the modulation of the scattering length is absent, then variation of $\theta$ plays no role in the dynamics.

\section{numerical results}\label{sec5}

We have carried out direct numerical calculations of the GPE solving Eq.~(\ref{eq:TDGPE}) in its 1D version,

\begin{equation}
{\rm i}\frac{\partial}{\partial t}\Psi(x,t)=H[\Psi]\Psi(x,t),
    \label{eq:TDGPE_1D}
\end{equation}
with the nonlinear Hamiltonian given by
\begin{equation}
 H[\Psi]= -\frac{\partial^2}{\partial x^2}+V(x,t)+g(x,t)|\Psi(x,t)|^2,
\end{equation}
and considering arbitrary units such that $\hbar=m=N=g_0=1$. The nonlinear Hamiltonian operator has been written as a matrix over a grid of points according to the Chebyshev spectral method \cite{trefethen,mason}.

In order to solve the time-dependent equation (\ref{eq:TDGPE_1D}), we express the corresponding time evolution operator, which connects the initial time $t=0$ to the final time $t=t_f$, in $N$ small time-step $\Delta t$ evolution operators,
\begin{equation}
U(t_f,0)=\prod_{k=1}^{N}U\left(k\Delta t,(k-1)\Delta t\right).
    \label{evolEP}
\end{equation}
Each one of the small time-step evolution operators is calculated as an expansion in Chebyshev polynomials \cite{kosloff1984,LEFORESTIER199159,formanek2010},

\begin{equation}
   U\left(k\Delta t,(k-1)\Delta t\right)\approx\sum_{n=0}^{N_p}a_n\chi_n(-iH[\Psi((k-1)\Delta t)]\Delta t),
\end{equation}
where $a_n$ are the expansion coefficients, $\chi_n$ are the complex Chebyshev polynomials and $N_p$ sets the number of terms in the expansion. The propagation of the wavefunction in the $k$-th time step is obtained by applying $U\left(k\Delta t,(k-1)\Delta t\right)$ to the wavefunction calculated in the previous step $\Psi((k-1)\Delta t)$. The relaxation method, which in essence consists in performing propagation with imaginary time $t\rightarrow {\rm i}t$, has been applied to obtain the ground state \cite{kosloff1986}. The excited modes of the condensate have been determined by the spectrum-adapted scheme described in Ref. \cite{PhysRevA.93.053612}. We have found very good agreement comparing our results with  those from Refs. \cite{MURUGANANDAM20091888,PhysRevA.93.053612}.

For harmonic trapping potentials and modulating fields with linear behavior with distance, no transition to excited modes is possible through modulation of the trap \cite{yukalov69}. Thus, we have fixed the trapping potential to $V_{trap}(x,t)=x^4/4$, allowing for a simple form of the spatial dependence of $V(x)$. For this trap, we have obtained the chemical potentials $\mu_{0}= 0.808$,  $\mu_{1}=1.857$, and $\mu_{2}= 3.279$, for the ground, first and second nonlinear modes, respectively.

We have considered transitions from the ground state to the first and to the second excited modes. In the first case, we have set $g(x)=A_g x$ and $V(x)=A_t x$, while in the second case, $g(x)=A_g x^2$ and $V(x)=A_t x^2$. The frequencies of the modulations are set to be equal $\omega_t=\omega_g=\omega$ and are chosen to satisfy the resonance condition for each target.

Figure~\ref{figure1} compares single modulation with double modulation for $\theta=0$ by showing the corresponding target population dynamics, denoted by $n_j\equiv\left|\bra{\phi_j}\left.\Psi(x,t)\right>\right|^2$. In panel (a), the target is the first excited state, while in panel (b) the target is the second excited state. In both cases, we observe the double modulation performing a faster transition than the individual modulations. Additionally, the double modulation enhances the target population beyond that of the sum of the individual modulations, which is an evidence of quantum interference.

Panels (a) and (c) of Fig.~\ref{figure2} show the population of the target modes, the first and second modes, respectively, as a function of the relative phase of the modulations for some fixed times. For $\theta=\pi$ the transition is essentially inhibited, whereas for $\theta=0$ the target population is enhanced, in agreement with the perturbative analysis. Panels (b) and (d) show the corresponding population dynamics for some fixed phases. We observe that as the phase varies from $0$ to $\pi$, the transitions become slower while transferring fewer atoms. This behavior has not been captured by the perturbative expression and may be attributed to the nonlinear character of the GPE.

Figure~\ref{figure3} illustrates the comparison of the two-level approximation obtained by solving Eq.~(\ref{pop2}) using a fourth-order Runge-Kutta method with the numerical solution of the GPE. We have obtained for the coupling parameters: $\alpha_{21}\approx0.124$, $\beta_{12}\approx7.7\times{10}^{-2}$ and $\gamma_{21}\approx4.6\times{10}^{-2}$ for the transition to the first excited mode. Panel (a) shows the target population dynamics for $\theta=0$, while panel (b) shows the target population at $t=31$ as a function of the relative phase. Although the two-level approximation departs from the expected solution as the evolution takes place, these two panels illustrate the good qualitative agreement between the approaches that we have generally found in our calculations, corroborating our analysis.

Figure \ref{figure4} considers the impact of the phase when only a single modulating field is present. In panels (a) and (b), the modulation of the nonlinearity is turned off $g_{\rm mod}(x,t)=0$, whereas in panels (c) and (d) the modulation of the trap is turned off $V_{\rm trap}(x,t)=0$ and Eq.(\ref{gmod}) reads $g_{\rm mod}=g(x)\cos(\omega_g t+\theta)$. The upper panels show the population dynamics of the first mode, while the lower panels show the population of the first mode as a function of $\theta$ for some fixed times. In contrast with the prediction of the perturbative approach, in both cases the phase does have an impact on the population dynamics. However, this impact is indeed very small compared to the case when the two modulating fields are present.

\section{Conclusion}\label{sec6}

Non-ground-state BEC can be created from the ground state by resonantly modulating either the trapping potential or the atomic interactions. We have explored the simultaneous use of the modulations on the population dynamic and the impact of their relative phase on the formation of the excited modes in the framework of the GPE. Numerical as well as approximated analytical methods have been applied. We have shown that the relative phase can be used to coherently control the transition to the excited modes by enhancing or suppressing the transition. We have also shown that the double modulation can affect the speed of the transitions. This behavior, which is not often found in ordinary quantum dynamics, can be attributed to the nonlinear character of the GPE. This work should stimulate the search for experimental evidences of coherent control of transitions between nonlinear modes induced by double modulation. It should also motivate the study of different control problems using double modulation.

\section*{Acknowledgments}\label{sec7}

This study was financed in part by the Coordination for the Improvement of Higher Education Personnel (CAPES) - Finance Code 001. EFL acknowledges support from Sao Paulo Research Foundation, FAPESP (grant 2014/23648-9) and from National Council for Scientific and Technological Development, CNPq (grant 423982/2018-4).

\begin{figure}[ht]
		\begin{center}
					\includegraphics[width=\textwidth]{archive1_new_mod.pdf}
\end{center}
\caption{\label{figure1} a) Population of first excited state versus time for a system driven by trap, scattering length and double modulation ($\theta=0$), with $A_{t}=0.1$ and $A_{g}=0.3$ amplitudes. b) Population of second excited state by trap, scattering length and double modulations, with $A_{t}=0.08$ and $A_{g}=0.4$ amplitudes.}
\end{figure}

\pagebreak

\begin{figure}[ht]
\begin{center}
	\includegraphics[width=\textwidth]{archive2_new.pdf}
\end{center}
\caption{\label{figure2} a) Population of the first excited mode versus relative phase of the modulations for some fixed times and parameters of panel a) of Fig. \ref{figure1}. b) Population of the first excited state versus time for some fixed phases (same parameters of a)). c) Population of the second excited state for some fixed times and parameters of panel b) of Fig. \ref{figure1}. d)  of the second excited mode versus time for some fixed phases (same parameters of c)).}
\end{figure}

\pagebreak

\begin{figure}[ht]
	\begin{center}
		\includegraphics[width=\textwidth]{archive3_new.pdf}
\end{center}
\caption{\label{figure3} a) Population of the first excited mode versus time under comparison of the direct numerical calculations and the two levels model for the same system and parameters as we introduced in Fig.  \ref{figure1} with $\theta=0$ phase . b) Population versus relative phase of the modulations at $t=31$.}
\end{figure}

\pagebreak

\begin{figure}[ht]
		\begin{center}
	\includegraphics[width=\textwidth]{archive4_new.pdf}
\end{center}
\caption{\label{figure4} a) Population of first excited mode versus time for a system driven by the trap modulation only, with $\theta=0$ and $A_{t}=0.1$ amplitude. b) Population versus relative phase for some fixed times (same parameters as a)). c) Population of first excited state versus time for a system driven by the scattering length modulation only with $\theta=0$ and $A_{g}=0.3$ amplitude. d) Population of the first excited mode versus relative phase for some fixed times (same parameters as c).}
\end{figure}

\pagebreak


\section{Introduction}

A chemical reactor could be any vessel containing chemical
reactions. In general, a reactor is designed such as to maximize the
yield of some particular products while requiring the least amount
of money to purchase and operate. Normal operating expenses include
energy input, energy removal, raw material costs, labor, etc. Energy
changes can occur in the form of heating or cooling, or agitation.
The latter is quite important because an appropriate mixing has a
large influence on the yield. Therefore, the design and operation of
mixing devices often determines the profitability of the whole
plant.Theoretically, the effect of stirring in reactant media have
also attracted considerable attention \cite{theory}

In particular, in the widely developed continuous stirred tank
reactors (CSTR) one or more fluid reagents are introduced into a
tank equipped with an impeller while the reactor effluent is removed
\cite{Epstein}. The impeller stirs the reagents to ensure proper
mixing. Classical CSTR dynamical models, based on coupled
deterministic ordinary differential equations (ODEs), are the usual
approach to chemical systems at the macroscopic scale. It has been
demonstrated to have considerable usefulness. However, their
validity relies on many assumptions that limit the situations in
which they can be applied. One of the most important is that
chemical systems are discrete at the molecular level and statistical
fluctuations in concentration and temperature occur at the local
scale. The elaboration of models considering this discreteness is
important. Seyborg \cite{Seybold 1997} and Neuforth \cite{Neuforth
2000} have shown that stochastic cellular automata models can be
successfully applied in simulating first order chemical reactions.
In their papers, they worked on a squared arrangement of cells, each
of them having a chemical reactive. The reactions are performed by
considering a probability of change, from reactive A to reactive B,
proportional to the kinetics constant that defines the chemical
equation. However, this type of calculation does not apply directly
to the CSTR case, where a chemical feed flow is present.

 In this paper we extend the stochastic CA model to CSTRs by
simulating the feed flow flux by means of a random selection of a
subset of cells to which the flux conditions with respect to
chemical concentration and temperature are imposed. We would like to
remark that mixing in a stirred tank is complicated and not well
described despite the extensive usage of dimensionless numbers and
models based on ODEs \cite{Chakraborty 2003}. Therefore, more
accurate models are essential for developing and testing control
strategies or even to explore new reactor geometries. The
organization of the paper is the following. Section \ref{S2}
presents shortly the standard ODE-based CSTR model. Section \ref{S3}
describes the CA method that we implemented for jacketed CSTRs. The
simulations are displayed and briefly discussed in Section \ref{S4}.
The paper ends up with several concluding remarks.

\section{The Deterministic Dynamical Systems Model}\label{S2}

As already mentioned, we consider an ideal jacketed CSTR where the
following exothermic and irreversible first-order reaction is taking
place:
\begin{eqnarray}
    A \longrightarrow B \nonumber
\end{eqnarray}

The CSTR modeling equations in dimensionless form are the following
\cite{Silva 1999}
\begin{eqnarray}
    \frac{d\,X_1}{d\,\tau} &=& -\phi\,X_1\,k(X_2)+q\,(X_{1_{f}}-X_1)\\
    \frac{d\,X_2}{d\,\tau} &=& \beta\,\phi\,X_1\,k(X_2)-(q+\delta)\,X_{2}+\delta\,X_3+q\,X_{2_{f}}\\
    \frac{d\,X_3}{d\,\tau} &=&
    \frac{q_c}{\delta_1}\,\left(X_{3_{f}}-X_3\right)+\frac{\delta}{\delta_1\,\delta_2}\,(X_2-X_3)~,
\end{eqnarray}
where $X_1$, $X_2$, and $X_3$ are the dimensionless concentration,
temperature, and cooling jacket temperature, respectively. We note
that it is possible to use the dimensionless coolant flow rate,
$q_c$, to manipulate $X_2$.

\begin{figure}[htb]
        \begin{center}
            \includegraphics[height=5cm]{cstr2.eps}
                \put(-109,-10){$A+B$}
                \put(-160,122){\line(-1,0){15}} \put(-160,122){\vector(1,0){2}}
                \put(-172,126){$A$}
                \put(-155,0){\tiny{Coolant}}
                \put(-159,-7){\tiny{inlet-stream}}
                \put(-30,100){\tiny{Coolant}}
                \put(-34,95){\tiny{outlet-stream}}
                \put(-108,70){CSTR}
        \end{center}
\caption{Schematic representation of the jacketed CSTR.}
\end{figure}
The relationships between the dimensionless parameters and variables
and the physical variables are the following:
\begin{eqnarray}
    k(X_2)=\exp\left(\frac{X_2}{1+X_2\,\gamma^{-1}}\right),\ \gamma=\frac{E}{R\,T_{f_{0}}},\ X_3=\frac{T_c-T_{f_{0}}}{T_{f_{0}}}\,\gamma,\ X_2=\frac{T-T_{f_{0}}}{T_{f_{0}}}\,\gamma\nonumber\\
    \beta=\frac{(-\Delta H)\ C_f}{\rho C_p t_{f_0}},\
    \delta=\frac{U\ A}{\rho \ C_p\ Q_0},\ \phi=\frac{V}{Q_0}k_0\
    e^{-r}, \ X_1=\frac{C}{C_{f_0}},\ X_{1_f}=\frac{C_f}{C_{f_0}},\
    \nonumber\\
    X_{2_f}=\frac{T_f-T_{f_0}}{T_{f_0}}\gamma,\
    \delta_1=\frac{V_c}{V},\ \tau=\frac{Q_0}{V}t,\
    X_{3_f}=\frac{T_{c_f}-T_{f_0}}{T_{f_0}}\gamma,\
    \delta_2=\frac{\rho_c\ C_{p_c}}{\rho\ C_p} \nonumber
\end{eqnarray}
where the meaning of the symbols is given in Table 1. In the rest of
the paper we shall use the solution of this ODE model as the
theoretical case with which the CA simulations will be compared.

\begin{table}[htb]
\centering \caption{Parameters of the model} \label{tab:1}
\begin{tabular}{lll}
\hline\noalign{\smallskip}
Symbol & \ \ \ \ \ \ \ \ \ \ \ Meaning & \ \ \ \ Value   \\
            & \ \ \ \ \ \ \ \ \ \             & (arb. units) \\
\noalign{\smallskip}\hline\noalign{\smallskip}
$C$ & Reactor composition&\ \ \ \ \ $0.001$   \\
$C_f$ & Feed composition&\ \ \ \ \ $1.0$  \\
$q$ & Dimensionless reactor feed flow rate&\ \ \ \ \  $1.0$  \\
$q_c$ & Dimensionless coolant flow rate&\ \ \ \ \ $1.65$  \\
$q_{cs}$ & Steady-state value of $Q$ &\ \ \ \ \ $1.0$  \\
$T$ & Reactor temperature &\  \ \ \ \ $1.0$  \\
$T_c$ & Coolant temperature &\ \ \ \ \ $1.0$  \\
$UA$ & Heat transfer coefficient times the heat transfer area &\ \ \ \ \ $1.0$ \\
$V$ & Reactor volume &\ \ \ \ \ $1.0$  \\
$V_c$ & Cooling jacket volume &\ \ \ \ \ $1.0$  \\
$X_{1f}$ & Dimensionless feed concentration &\ \ \ \ \ $1.0$  \\
$X_{2f}$ & Dimensionless feed temperature &\ \ \ \ \ $0.0$  \\
$X_{3f}$ & Dimensionless coolant feed temperature&\ \ \ \ \ $1.0$  \\
$\beta$ & Dimensionless heat of reaction &\ \ \ \ \ $8.0$  \\
$\delta _1$ & Dimensionless volume ratio &\ \ \ \ \ $0.1$  \\
$\delta _2$ & Dimensionless density multiplied by the heat capacity of coolant &\ \ \ \ \ $1.0$  \\
$\phi$ & Hill's threshold parameter &\ \ \ \ \ $0.072$  \\
$\gamma$ & Dimensionless activation energy &\ \ \ \ \ $20.0$  \\
$\rho _c C_{p_c}$ & Density multiplied by the heat
capacity of coolant&\ \ \ \ \ $1.0$  \\
$\tau$ & Dimensionless time &\ \ \ \ \ $--$  \\
\\\noalign{\smallskip}\hline
\end{tabular}
\end{table}

The nominal parameter values used here are given by:
\begin{eqnarray}
\beta = 8.0,\ \delta = 0.3,\ X_{1_f} = 1.0,\ X_{3_f} = 1.0,\ q =
1.0,\ q_{c_s} = 1.65\nonumber\\ \gamma = 20.0,\ X_{2_f} = 0.0,\ \phi
= 0.072,\ \delta_1 = 0.1,\ \delta_2 = 0.5~.\nonumber
\end{eqnarray}

\section{Stochastic CA Model for Jacketed CSTR}\label{S3}

The process simulated in this work is the exothermic reaction that
converts a chemical A into a product B in a jacketed CSTR. Our model
is composed of three squared arrangements of cells, all of the same
size. The first lattice is for chemicals A and B and represents the
chemical distribution in the tank reactor. In each cell there is
only one unit of reactive A or one unit of product B (not necessary
representing a single molecule), under the condition that all cells
are occupied. The second arrangement is for the tank temperature. It
contains the temperatures $t_{ij}$ in real values, with each cell in
this arrangement corresponding to the respective cell in the first
arrangement. The third arrangement represents the coolant system. We
have used a squared lattice of the same size as the temperature
array in such a way that each temperature tank cell is in
``contact'' with a coolant jacket cell.

In our model the first process in each time step is the irreversible
conversion of chemical A into product B. The conversion rate is
determined by $\phi\,k(X_{2})$  as in \cite{Silva 1999}, where
$X_{2}$ is the average temperature of the tank temperature
arrangement. This first order kinetics
 ``constant'' is multiplied by the time step in order to get the proportion
of the reactive A that is expected to be converted into product B in
each evolution step. This number could also be considered as the
probability that a molecule of chemical A would be converted in
product B if the time step is small enough. Such proportion is
compared with a randomly generated number, one different random
number for each cell in the arrangement containing reactive A. If
the random number is less than the proportion, the reactive A is
changed for product B in the cell. Since the reaction is exothermic,
the temperature value in the
temperature array is increased by $\beta$ (according to Eq.~(2)) in the corresponding cell.\\

The second simulated process in our model is the tank temperature
diffusion that can be simultaneously considered as an energy
diffusion. It can be performed by means of finite differences, but
in order to obtain a model almost ODE independent we have
implemented a moving average method, where the value of the
temperature in a cell at the next time step is the average
temperature of its neighborhood. This procedure gives similar
results to those of finite differences, as shown by Weimar for
reaction-diffusion systems simulated by cellular automata
\cite{Weimar 1997}. We used a square neighborhood formed by
$(2R+1)^{2}$ cells, where $R$ is the number of steps that we have to
walk from the center of the cell in order to reach the most far
horizontal (vertical) cell in the neighborhood.

The third simulated process is the tank feed flow. We have simulated
the feed flow rate $q$ in a stochastic way. In order to get an
approximation to the proportion of the tank that must be replaced by
the incoming flow, $q$ is multiplied by the time step and by the
total number of cells in the arrangement. This give us a real number
$x$. Then, following Weimar \cite{Weimar 1997}, we used a
probabilistic minimal noise rule, \textit{i.e.} it is defined the
probability $p = x- \lfloor x \rfloor$ in order to decide if
$\lfloor x \rfloor$ or $\lfloor x \rfloor +1$ cells will be replaced
by the flow. We choose $\lfloor x \rfloor$ with probability $1-p$
and $\lfloor x \rfloor +1$ with probability $p$. This method
conserves the proportion $x$ in a statistical way. Subsequently, a
cell is selected in a random way, by means of two random numbers
which are used to select a row and a column, in such a way that all
cells has the same probability of being selected. If the cell has
been selected in the same time step, a new selection is made. This
is repeated until we have reached the number of cells that must be
replaced. Finally, the selected cells are changed in the temperature
arrangement by the feed flow temperature $X_{2f}$, and in the
reactive arrangement it is put a unit of reactive A with probability
$X_{1f}$, that represent the concentration of chemical A in the feed
flow. In the simulations presented in this work we used $X_{1f}=1$.
This method of flow simulation could be improved in several ways, in
order to simulate different tank geometries or for showing the flow
direction. However, in this work we want only to show that the CA
method could fit the CSTR behavior in a very good approximation,
with the advantage of spatial analysis.

The fourth simulated process is the energy interchange between the
tank and the jacket. This has been done by directly calculating the
energy/temperature interchange between each tank temperature cell
and its corresponding jacket temperature cell. This interchange is
dictated by the difference between the two temperatures and it is
weighted by $\delta$, $\delta_{1}$, and $\delta_{2}$ as in Eqs.~(1)
and (2). The fifth and the sixth simulated processes are the coolant
flow and the coolant temperature diffusion, respectively. Both of
them are performed in a similar way as for the concentration and
temperature tank.

\section{Simulations}\label{S4}

In this section we first present the comparison between the curves
obtained by differential equations and those obtained from the
implemented cellular automata model (see Fig.~\ref{comparativo}). It
is clear that cellular automata simulations resemble with excellent
agreement the values for the concentration of chemical A, the tank
temperature and the jacket temperature at all times. Using a time
step of 0.001 it is shown that the curves coincide at the initial
time, transient time and for stable state. We have found that we can
maintain this remarkable fitness by properly adjusting the time step
to a sufficiently small value.

\begin{figure}[htb]
\centering
\includegraphics[scale=0.5]{comparativo2.ps}
\caption[]{Comparison between the curves from the differential
equations and the curves obtained from simulations with the cellular
automata approach. Initial values are: $X_{1}=0.1$, $X_{2}=0.1$,
$X_{3}=0.1$. For the six curves: 100 points separated by a
dimensionless time of 0.2 were taken from 20000-point simulations
with a dimensionless time step of 0.001.}

 \label{comparativo}
\end{figure}

When a kinetic constant based on the average temperature is used, it
is implicity assumed that the mixing in the CSTR is perfect,
vanishing any temperature inhomogeneities. One could ask what could
be the change in the tank behavior if the mixing is almost perfect.
It could be studied by using a model that consider the spatial
distribution of temperature. We studied this effect by calculating
the kinetic constant for each cell based in the its correspondent
temperature. The effect for a $1000 \times 1000$ cells array and a
time step of 0.001 is shown in Fig~\ref{local}. It could be observed
that the tank temperature curve for the perfect mixing and the one
for the locally calculated kinetic constant are the same for almost
all times. However, they separate during the transient period,
leading to a reduction in the magnitude of the peak and a little
delay in its appearance. The curve was calculated with a tank
temperature diffusion process per time step with a $R=1$
neighborhood. If more diffusion steps are used per time step the
curves obtained tend to the theoretical one as is expected. Besides,
it could be also interpreted as the effect of a perfect mixed tank,
but with material
where each component tends to maintain its energy.\\

\begin{figure}[htb]
\centering
\includegraphics[scale=0.55]{local.ps}
\caption[]{Comparison between the tank temperature evolution curves
for a kinetic constant based on average temperature and for kinetics
constant calculated on the base of the local temperature. Initial
values are: $X_{1}=0.1$, $X_{2}=0.1$, $X_{3}=0.1$. The array is of
$1000 \times 1000$ cells and the dimensionless time step is of
0.001; one temperature diffusion step per time step.} \label{local}
\end{figure}

The ODEs generally represent the characteristics of the global
system, where it has enough number of elements such that the
statistical fluctuations are of small amplitude.
However, when the system size and the quantity of elements are
diminished, the statistical fluctuations could be of increasing
importance. In this way, another advantages of the cellular automata
method proposed here is its flexibility with respect to the reactor
size, and its stochastic nature,
that allows to study how much the system could be affected by the initial conditions and by the
stochastic features of the process.\\

We have performed several simulations applying the parameters
presented above in arrangements of small size, where the model is no
longer representative for a 3-dimensional CSTR at such scales.
However, we can think of it as a representation of a catalytic
surface dividing two regions, one carrying the chemical A and the
other as a temperature reservoir. Therefore, this model is a simple
approach, useful as a first approximation, in the analysis and study
of microreactors or even nanoreactors. We recall that the usage of
microreactors for {\em in situ} and on-demand chemical production is
gaining increasing importance as the field of microreaction
engineering has already demonstrated potential to impact a wide
spectrum of chemical, biological, and process system applications
\cite{Jensen}. There are already many successfully developed
microreactors for chemical applications such as partial
oxidation reactions \cite{partial},
phosgene synthesis \cite{phosgene}, multiphase processing
\cite{multiphase}, and (bio)chemical detection \cite{detection}.

Figure \ref{smallconc} displays the variability that could be found
in CSTR systems at small scales. It is clear evidence that the
statistical fluctuations are a primordial issue at this scale. In
addition, one could notice that the dynamical behavior could be
totally different to the expected behavior of a larger system e.g.,
$1000 \times 1000$ cells arrangement.

\begin{figure}[htb]
\centering
\includegraphics[scale=0.55]{smallconc.ps}
\caption[]{Behaviors that could be found in systems with small
number of elements (cells for the CAs and clusters of molecules in
the real case). These behaviors are different from that expected in
systems with a large number of elements. The initial values are
$X_{1}=0.1$, $X_{2}=0.1$, $X_{3}=0.1$. The employed time step is
0.001. One hundred points separated by a time lag of 0.2 were taken
from 20000-point simulations with a time step of 0.001;
one temperature diffusion step per time step.} \label{smallconc}
\end{figure}

Finally, the study of small systems by direct simulation using
stochastic simulations could give us insight in how an open CSTR
system could behave when the statistical fluctuations and the
initial configuration are important. In Fig.~\ref{rep} one can see
that the possible behaviors of a system of size $20 \times 20$ have
big deviations from the average (theoretical) value. This kind of
variability is not provided by pure ODEs (without a stochastic
term). We think that this stochastic CA approach could be an
important tool for testing control strategies since the CA approach
could be seen as a step between the ODEs models and the specific
experimental situation.

\begin{figure}[htb]
\centering
\includegraphics[scale=0.55]{rep.ps}
\caption[]{Different concentration behaviors of chemical A for
systems of the same size ($20 \times 20$) that are treated by the
same method and could be the underlying dynamic characteristic of
microreactors. Five simulations are displayed together with the
average curve (thick line) of 20 individual simulations. The initial
configuration and the stochasticity introduced in the model lead to
a time-distributed behavior. The initial values are: $X_{1}=0.1$,
$X_{2}=0.1$, $X_{3}=0.1$. The time step is 0.001; one temperature
diffusion step per time step.} \label{rep}
\end{figure}

\section{Concluding Remarks}\label{S5}

A cellular automata approach for the CSTR with cooling jacket has
been presented in this paper. It is able to reproduce the CSTR
dynamical behavior calculated by ODE's with a good approximation and
in an easy way. The stochastic model presented allow us to study
what could be the behavior of the variables of the tank when the
reaction probability depends on the local temperature. It also give
us an approach to study systems with few elements, as could be micro
and nanoreactors, as could be catalytic membranes separating two
phases. The main advantages of the CA approach presented here are
its stochastic nature and the direct involvement of a spatial
structure. This also represents a tool for studying the role of
initial configuration and stochastic fluctuations in systems with
few elements. Additionally, the CA approach is a clear improvement
of the CSTR modeling and moreover can be applied to different
reactor and jacket geometries, as well as for considering in more
detail the real mass flow in the tank reactor-geometry.