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Here we outline a number of physics problems which we solved using ANISE. In all cases the equations were integrated with a requested tolerance of 1 x 10-12. Since the equations are highly nonlinear exact solutions are not known for individual trajectories. However, exact solutions are known for some average quantities. We therefore compare the exact average quantity to a numerically calculated averaged quantity obtained using ANISE for 100000 realizations.

1.   Nonlinear Absorber  

The first example we consider is the Gisin-Percival stochastic wave equation for the nonlinear absorber

\begin{eqnarray}\html{eqn0} d\vert\Psi(t,W_t)\rangle &=&.1(a^{\dag }-a)\vert\Psi... ...{2}(a^2-\langle a^2\rangle) \vert\Psi(t,W_t)\rangle dW_t\nonumber \end{eqnarray}

where a denotes the usual harmonic oscillator lowering operator. We chose an initial state $\vert\Psi(0,0)\rangle=\vert\rangle$ where $\vert\rangle$ is the lowest eigenstate of $a^{\dag }a$. We choose $W_0=0$ in this and later examples and the Wiener process $W_t$ is real. The notation $\langle Y\rangle$ indicates the quantum expectation $\langle \Psi\vert Y\vert\Psi\rangle$. The ensemble average over statistical realisations of the Wiener processes is denoted via $M[Y]$ for any $Y$. The quantity of interest for this example is the average density

\begin{eqnarray}\html{eqn0} \rho(t)&=&M[\vert\Psi(t,W_t)\rangle\langle \Psi(t,W_t)\vert]\nonumber \end{eqnarray}


which obeys the deterministic master equation

\begin{eqnarray}\html{eqn0} \frac{d\rho(t)}{dt}&=&.1[a^{\dag }-a,\rho(t)]+2a^2\rho(t)a^{\dag 2}-a^{\dag 2}a^2\rho(t)-\rho(t)a^{\dag 2}a^2.\nonumber \end{eqnarray}


To implement our approach we need to find the derivatives of $\vert\Psi(t,W_t)\rangle$ with respect to $t$ and $W_t$. These are

\begin{eqnarray}\html{eqn0} \frac{\partial \vert\Psi(t,W_t)\rangle}{\partial W_t... ...sqrt{2}(a^2-\langle a^2\rangle ) \vert\Psi(t,W_t)\rangle\nonumber \end{eqnarray}


and

\begin{eqnarray}\html{eqn0} \frac{\partial \vert\Psi(t,W_t)\rangle}{\partial t}&... ...\vert\langle a^2\rangle \vert^2)\vert\Psi(t,W_t)\rangle.\nonumber \end{eqnarray}


The dynamics was solved in a basis consisting of the lowest 11 eigenstates $\vert n\rangle $ of $a^{\dag }a$ with n = 0,..., 10. Thus, including real and imaginary parts of $\langle n\vert\Psi(t,W_t)\rangle$ for n = 0,..., 10 our equations consist of a total of 22 real nonlinear coupled stochastic equations with one real Wiener process.
The exact and approximate mean occupation number $~n_t=~Tr[a^{\dag }a\rho(t)]$ are plotted against time in the figure.

2.   Quantum Cascade  

The second example is the Gisin-Percival stochastic wave equation for a quantum cascade with absorption and stimulated emission

\begin{eqnarray}\html{eqn0} d\vert\Psi(t)\rangle &=&-.1i(a^{\dag }+a)\vert\Psi(t... ...sqrt{2}(a-\langle a\rangle ) \vert\Psi(t)\rangle dW_t^2.\nonumber \end{eqnarray}


Here again we chose the initial state $\vert\Psi(0)\rangle =\vert\rangle$. There are now 2 real statistically independent Wiener processes (i.e. $M[dW_t^1dW_t^2]=0$). The quantity of interest is again the density which in this case obeys the master equation

\begin{eqnarray}\html{eqn0} \frac{d\rho(t)}{dt}&=&-.1i[a^{\dag }+a,\rho(t)]+2a^{... ...o(t)a^{\dag }-.01a^{\dag }a\rho(t)-.01\rho(t)a^{\dag }a.\nonumber \end{eqnarray}


The derivatives of $\vert\Psi(t)\rangle$ with respect to $t$, $W_t^1$ and $W_t^2$ are given by

\begin{eqnarray}\html{eqn0} \frac{\partial \vert\Psi(t)\rangle}{\partial W_t^1}&... ...ngle a^2\rangle -\langle a\rangle^2)\vert\Psi(t)\rangle.\nonumber \end{eqnarray}


We used the same basis set as in the Nonlinear Absorber.
The exact and approximate mean occupation number $n_t=Tr[a^{\dag }a\rho(t)]$ are plotted against time in the figure.

3.   Coupled Oscillators  

Consider the set of stochastic wave equations

\begin{eqnarray}\html{eqn0} d\vert\psi_1\rangle &=&-i a_1^{\dag }a_1\vert\psi_1\... ..._2)^2\rangle_2-(\langle a_2^{\dag }+a_2\rangle_2)^2}dW_3\nonumber \end{eqnarray}


which decompose the exact solution $\vert\Phi(t)\rangle$ of the Schrödinger equation
\begin{eqnarray}\html{eqn0} d\vert\Phi(t)\rangle&=&-i[a_1^{\dag }a_1+a_2^{\dag }... ...a_1^{\dag }+a_1)(a_2^{\dag }+a_2)]\vert\Phi(t)\rangle dt\nonumber \end{eqnarray}


in a sort of stochastic Hartree product

\begin{eqnarray}\html{eqn0} \vert\Phi(t)\rangle&=&M[\vert\psi_1(t)\rangle \vert\psi_2(t)\rangle e^{\sqrt{\lambda}\theta(t)}].\nonumber \end{eqnarray}


Here the derivatives are

\begin{eqnarray}\html{eqn0} \frac{\partial \vert\psi_1\rangle}{\partial W_1}&=&\... ...-(\langle a_2^{\dag }+a_2\rangle_2)^2)\vert\psi_2\rangle\nonumber \end{eqnarray}


We set $\lambda=.1$ and calculated $\langle \Phi(0)\vert\Phi(t)\rangle$ for initial conditions $\vert\psi_1(0)\rangle=\vert 1\rangle$, $\vert\psi_2(0)\rangle=\vert 1\rangle$ and $\theta(0)=0$. We chose to represent the wavefunctions in the eigenbasis of $a^{\dag }a$. Specifically, we chose a basis consisting of the first 10 eigenvectors. Thus, there are in total 41 equations and three stochastic processes.

The exact and numerically computed real part of $\langle \Phi(0)\vert\Phi(t)\rangle$ are shown in the figure.

4.   Helium  

The fourth example consists of stochastic wave equations for a stochastic decomposition of the Schrödinger equation for Helium.

Neglecting nuclear motion about the center of mass, the Helium wavefunction $\Phi({\bf r}_1,{\bf r}_2,t)$ obeys the deterministic Schrödinger equation (in atomic units $\hbar=1$, $m_e=1$, and $e=1$)

\begin{eqnarray}\html{eqn0} \frac{\partial \Phi({\bf r}_1,{\bf r}_2,t)}{\partial... ...t{\bf r}_1-{\bf r}_2\vert}\}\Phi({\bf r}_1,{\bf r}_2,t),\nonumber \end{eqnarray}


for any specified antisymmetric initial state $\Phi({\bf r}_1,{\bf r}_2,0)$. Here ${\bf r}_1$ and ${\bf r}_2$ denote positions of electrons 1 and 2 with respect to the nucleus. For our example calculation we choose an initial wavefunction of the form

\begin{eqnarray}\html{eqn0} \Phi({\bf r}_1,{\bf r}_2,0)& =& \beta \left( \Psi_{1... ...2,0) - \Psi_{2}({\bf r}_1,0)\Psi_{1}({\bf r}_2,0)\right)\nonumber \end{eqnarray}


(where $\beta=1/\sqrt{2(1-\vert\langle \Psi_{1}(0)\vert\Psi_{2}(0)\rangle\vert^{2})}$ is a normalization factor) which is obviously antisymmetric in ${\bf r}_1$ and ${\bf r}_2$. Note that we are implicitly incorporating the two-component electron spins into the definitions of $\Psi_1$ and $\Psi_2$. For our purposes it is important that $\langle \Psi_{1}(0)\vert\Psi_{2}(0)\rangle\neq 0$. The actual initial conditions for this example calculation were chosen randomly as a mixture of 1s and 2s He+ states for each electron.

It can be shown that the exact deterministic wavefunction $\Phi({\bf r}_1,{\bf r}_2,t)$ can be decomposed into stochastic waves via an average of the form

\begin{eqnarray}\html{eqn0} \Phi({\bf r}_1,{\bf r}_2,t)&=&\beta M[\Psi_{1}({\bf ... ...\bf r}_2,t)- \Psi_{2}({\bf r}_1,t)\Psi_{1}({\bf r}_2,t)]\nonumber \end{eqnarray}


where $\Psi_{1}$ and $\Psi_{2}$ satisfy stochastic wave equations

\begin{eqnarray}\html{eqn0} d\Psi_{1}({\bf r},t) &=&  [-i (-\frac{1}{2}\nabla^2-... ...i_{1} \vert\Psi_{2}\rangle \right\}}\Psi_1({\bf r},t)dt.\nonumber \end{eqnarray}


We have used a notation $\langle F \rangle_j=\langle \Psi_j\vert F\vert\Psi_j\rangle$ in the above equations. Here the $\omega_s$ and operators $O_s$ arise through the one-body expansion of the Coulomb interaction

\begin{eqnarray}\html{eqn0} \frac{1}{\vert{\bf r}_1-{\bf r}_2\vert}=\sum_{s=1}^p \omega_s O_s(1)O_s(2)\nonumber \end{eqnarray}


which we performed numerically in a basis of He+ eigenstates. In the calculation reported here p = 8 which means that there are eight real Wiener processes. Since the initial states are of s type we included only the basis functions of s type with a principle He+ quantum number of 4 or less. This means that the total number of equations was 32.

The derivatives of $\Psi_j({\bf r},t)$ are given by

\begin{eqnarray}\html{eqn0} \frac{\partial \Psi_{j}({\bf r},t)}{\partial W_t^s}&... ...j-\vert\langle O_s \rangle_j\vert^2)\Psi_{j} ({\bf r},t)\nonumber \end{eqnarray}


where $k\neq j$ and j, k = 1, 2.

The exact and numerically computed real part of $\langle \Phi(0)\vert\Phi(t)\rangle$ are shown in the figure.

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