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<section id="pygpe-an-introduction">
<h1>PyGPE: An introduction<a class="headerlink" href="#pygpe-an-introduction" title="Link to this heading">#</a></h1>
<section id="welcome-to-pygpe">
<h2>Welcome to PyGPE!<a class="headerlink" href="#welcome-to-pygpe" title="Link to this heading">#</a></h2>
<p>PyGPE is an open source Python library for use in simulating the dynamics
of Bose-Einstein condensate (BEC) systems.
It offers an easy-to-use API that makes simulating such dynamics painless.
PyGPE solves the Gross-Pitaevskii equations using CuPy meaning above all else,
PyGPE is <strong>fast</strong>.</p>
</section>
<section id="installing-pygpe">
<h2>Installing PyGPE<a class="headerlink" href="#installing-pygpe" title="Link to this heading">#</a></h2>
<p>Installation is through pip:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">pip</span> <span class="n">install</span> <span class="n">pygpe</span>
</pre></div>
</div>
<p>Requirements are installed automatically <strong>except</strong> CUDA Toolkit.
Ensure you have the required version of CUDA Toolkit (>=11.2) installed before attempting to install PyGPE.</p>
</section>
<section id="how-to-import-pygpe">
<h2>How to import PyGPE<a class="headerlink" href="#how-to-import-pygpe" title="Link to this heading">#</a></h2>
<p>Since PyGPE supports multiple types of BEC systems, we need to first select
the system we want to work with.
PyGPE currently supports scalar, spin-1, and spin-2 systems.
To access the correct functions & classes for a given system type, we
import the relevant module into Python like:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">pygpe.scalar</span> <span class="k">as</span> <span class="nn">gpe</span>
</pre></div>
</div>
<p>We shorten the import name to <cite>gpe</cite> for better readability.
The table below lists the system types and their respective import statements</p>
<table class="table">
<colgroup>
<col style="width: 50.0%" />
<col style="width: 50.0%" />
</colgroup>
<thead>
<tr class="row-odd"><th class="head"><p>System type</p></th>
<th class="head"><p>Import statement</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p>Scalar BEC</p></td>
<td><p><code class="code docutils literal notranslate"><span class="pre">import</span> <span class="pre">pygpe.scalar</span></code></p></td>
</tr>
<tr class="row-odd"><td><p>Two-component BEC</p></td>
<td><p><code class="code docutils literal notranslate"><span class="pre">import</span> <span class="pre">pygpe.spinhalf</span></code></p></td>
</tr>
<tr class="row-even"><td><p>Spin-1 BEC</p></td>
<td><p><code class="code docutils literal notranslate"><span class="pre">import</span> <span class="pre">pygpe.spinone</span></code></p></td>
</tr>
<tr class="row-odd"><td><p>Spin-2 BEC</p></td>
<td><p><code class="code docutils literal notranslate"><span class="pre">import</span> <span class="pre">pygpe.spintwo</span></code></p></td>
</tr>
</tbody>
</table>
<div class="admonition warning">
<p class="admonition-title">Warning</p>
<p>Importing multiple different systems into the same project can have
disastrous side effects.
Ensure you are only importing <strong>one</strong> of the above modules in your project.</p>
</div>
</section>
<section id="using-pygpe">
<h2>Using PyGPE<a class="headerlink" href="#using-pygpe" title="Link to this heading">#</a></h2>
<dl class="simple">
<dt>The use of PyGPE can be broken down into a few simple steps:</dt><dd><ul class="simple">
<li><p>Set up numerical grid.</p></li>
<li><p>Define condensate and time parameters.</p></li>
<li><p>Set up wavefunction & set initial state.</p></li>
<li><p>Set up DataManager (if using).</p></li>
<li><p>Evolve the system.</p></li>
</ul>
</dd>
</dl>
<section id="setting-up-the-grid">
<h3>Setting up the grid<a class="headerlink" href="#setting-up-the-grid" title="Link to this heading">#</a></h3>
<p>PyGPE offers a Grid class that handles all the details of the numerical grid.
It supports 1D, 2D and 3D grids.
To create a grid we first define the number of grid points per
dimension and their respective grid spacings, then generate a Grid object:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">pygpe.spinone</span> <span class="k">as</span> <span class="nn">gpe</span>
<span class="n">grid_points</span> <span class="o">=</span> <span class="p">(</span><span class="mi">64</span><span class="p">,</span> <span class="mi">64</span><span class="p">,</span> <span class="mi">64</span><span class="p">)</span>
<span class="n">grid_spacings</span> <span class="o">=</span> <span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
<span class="n">grid</span> <span class="o">=</span> <span class="n">gpe</span><span class="o">.</span><span class="n">Grid</span><span class="p">(</span><span class="n">grid_points</span><span class="p">,</span> <span class="n">grid_spacings</span><span class="p">)</span> <span class="c1"># Creates our grid object</span>
</pre></div>
</div>
<p>The above code generates a 3D grid with 64 points and a grid spacing of 0.5 in
each dimension.
To create grids of different dimensionality you only need to change the grid_points
and grid_spacings to match the desired dimensionality.
For example, to create a 2D grid we would instead have
<code class="code docutils literal notranslate"><span class="pre">grid_points</span> <span class="pre">=</span> <span class="pre">(64,</span> <span class="pre">64)</span></code> and <code class="code docutils literal notranslate"><span class="pre">grid_spacings</span> <span class="pre">=</span> <span class="pre">(0.5,</span> <span class="pre">0.5)</span></code>.
Similarly, for 1D we would simply have <code class="code docutils literal notranslate"><span class="pre">grid_points</span> <span class="pre">=</span> <span class="pre">64</span></code> and
<code class="code docutils literal notranslate"><span class="pre">grid_spacing</span> <span class="pre">=</span> <span class="pre">0.5</span></code>.</p>
<p>The grid object is stored in the parameter you specified, in our case <cite>grid</cite>.
We can access useful attributes about our grid:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">grid</span><span class="o">.</span><span class="n">ndim</span><span class="p">)</span> <span class="c1"># 3</span>
<span class="nb">print</span><span class="p">(</span><span class="n">grid</span><span class="o">.</span><span class="n">total_num_points</span><span class="p">)</span> <span class="c1"># 262144</span>
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The grid class is shared between all system types, so what works here
for the spin-1 system will work for all other systems.</p>
</div>
</section>
<section id="defining-condensate-and-time-parameters">
<h3>Defining condensate and time parameters<a class="headerlink" href="#defining-condensate-and-time-parameters" title="Link to this heading">#</a></h3>
<p>PyGPE uses a simple dictionary to keep track of condensate and time parameters.
For a spin-1 system we can define it as:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">params</span> <span class="o">=</span> <span class="p">{</span>
<span class="s2">"c0"</span><span class="p">:</span> <span class="mi">10</span><span class="p">,</span> <span class="c1"># Spin-independent interaction</span>
<span class="s2">"c2"</span><span class="p">:</span> <span class="mf">0.5</span><span class="p">,</span> <span class="c1"># Spin-independent interaction</span>
<span class="s2">"p"</span><span class="p">:</span> <span class="mf">0.</span><span class="p">,</span> <span class="c1"># Linear Zeeman shift</span>
<span class="s2">"q"</span><span class="p">:</span> <span class="mf">0.</span><span class="p">,</span> <span class="c1"># Quadratic Zeeman shift</span>
<span class="s2">"trap"</span><span class="p">:</span> <span class="mf">0.</span><span class="p">,</span> <span class="c1"># Trapping potential</span>
<span class="s2">"n0"</span><span class="p">:</span> <span class="mi">1</span><span class="p">,</span> <span class="c1"># Background density</span>
<span class="c1"># Time params</span>
<span class="s2">"dt"</span><span class="p">:</span> <span class="mf">1e-2</span><span class="p">,</span> <span class="c1"># Numerical time step</span>
<span class="s2">"nt"</span><span class="p">:</span> <span class="mi">1000</span><span class="p">,</span> <span class="c1"># Number of time steps</span>
<span class="s2">"t"</span><span class="p">:</span> <span class="mi">0</span> <span class="c1"># Current time</span>
<span class="p">}</span>
</pre></div>
</div>
<p>Each system requires specific parameters to be defined in order for the evolution functions to work correctly.
See <a class="reference internal" href="../reference/parameters.html"><span class="doc">Condensate and time parameters</span></a> for more details on parameters and their definitions.</p>
</section>
<section id="setting-up-the-wavefunction">
<h3>Setting up the wavefunction<a class="headerlink" href="#setting-up-the-wavefunction" title="Link to this heading">#</a></h3>
<p>Now that we have a grid class, we can use this to set up our wavefunction.
Setting up the initial wavefunction class is easy, we just need to pass in the
grid we have constructed.
The name of the wavefunction class in the respective system is given in the
table below:</p>
<table class="table">
<colgroup>
<col style="width: 50.0%" />
<col style="width: 50.0%" />
</colgroup>
<thead>
<tr class="row-odd"><th class="head"><p>System type</p></th>
<th class="head"><p>Wavefunction Class</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p>Scalar BEC</p></td>
<td><p><code class="code docutils literal notranslate"><span class="pre">ScalarWavefunction(grid)</span></code></p></td>
</tr>
<tr class="row-odd"><td><p>Two-component BEC</p></td>
<td><p><code class="code docutils literal notranslate"><span class="pre">SpinHalfWavefunction(grid)</span></code></p></td>
</tr>
<tr class="row-even"><td><p>Spin-1 BEC</p></td>
<td><p><code class="code docutils literal notranslate"><span class="pre">SpinOneWavefunction(grid)</span></code></p></td>
</tr>
<tr class="row-odd"><td><p>Spin-2 BEC</p></td>
<td><p><code class="code docutils literal notranslate"><span class="pre">SpinTwoWavefunction(grid)</span></code></p></td>
</tr>
</tbody>
</table>
<p>Then we can use the class methods to manipulate the wavefunction into the
desired initial state:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">wavefunction</span> <span class="o">=</span> <span class="n">gpe</span><span class="o">.</span><span class="n">SpinOneWavefunction</span><span class="p">(</span><span class="n">grid</span><span class="p">)</span>
<span class="n">wavefunction</span><span class="o">.</span><span class="n">set_ground_state</span><span class="p">(</span><span class="s2">"polar"</span><span class="p">)</span>
<span class="n">wavefunction</span><span class="o">.</span><span class="n">add_noise</span><span class="p">(</span><span class="n">components</span><span class="o">=</span><span class="s2">"outer"</span><span class="p">,</span> <span class="n">mean</span><span class="o">=</span><span class="mf">0.</span><span class="p">,</span> <span class="n">std</span><span class="o">=</span><span class="mf">1e-2</span><span class="p">)</span>
</pre></div>
</div>
<p>This first creates a wavefunction in a polar state <span class="math notranslate nohighlight">\(\psi=(0,1,0)^T\)</span> then
subsequently adds numerical noise drawn from a normal distribution with mean
<span class="math notranslate nohighlight">\(\mu=0\)</span> and variance <span class="math notranslate nohighlight">\(\sigma=10^{-2}\)</span> to the outer
(<span class="math notranslate nohighlight">\(\psi_\pm\)</span>) components.</p>
</section>
<section id="setting-up-the-data-manager">
<h3>Setting up the data manager<a class="headerlink" href="#setting-up-the-data-manager" title="Link to this heading">#</a></h3>
<p>PyGPE provides an easy way to save data throughout the simulation.
Once the initial grid, wavefunction and condensate parameters have been defined,
we instantiate a DataManager class,
which saves all the initial details of the system.
To do this, we write:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">data</span> <span class="o">=</span> <span class="n">gpe</span><span class="o">.</span><span class="n">DataManager</span><span class="p">(</span><span class="n">filename</span><span class="o">=</span><span class="s1">'data.hdf5'</span><span class="p">,</span> <span class="n">data_path</span><span class="o">=</span><span class="s1">'../../data/'</span><span class="p">,</span> <span class="n">wfn</span><span class="o">=</span><span class="n">wavefunction</span><span class="p">,</span> <span class="n">params</span><span class="o">=</span><span class="n">params</span><span class="p">)</span>
</pre></div>
</div>
<p>The constructor takes four parameters: <cite>filename</cite>, the path where we want to
save the data, <cite>data_path</cite>, the wavefunction object, <cite>wfn</cite>, and the
parameters’ dictionary, <cite>params</cite>.
Calling the constructor automatically generates the file and populates it with
the parameters of our system.</p>
<p>To save the current wavefunction object to the dataset (i.e., whilst a simulation
is running), we simply write:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">data</span><span class="o">.</span><span class="n">save_wavefunction</span><span class="p">(</span><span class="n">wavefunction</span><span class="p">)</span>
</pre></div>
</div>
<p>For more detail on how the DataManager class works see <a class="reference internal" href="../reference/datamanager.html"><span class="doc">The DataManager class</span></a>.</p>
</section>
<section id="evolving-the-wavefunction">
<h3>Evolving the wavefunction<a class="headerlink" href="#evolving-the-wavefunction" title="Link to this heading">#</a></h3>
<p>Now that everything is set up, we get to the important part: evolving the wavefunction.
PyGPE provides a simple function for evolving stepping the wavefunction forward one time step.
To step the wavefunction forward for a set number of time steps we include a for loop:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">params</span><span class="p">[</span><span class="s2">"nt"</span><span class="p">]):</span>
<span class="n">gpe</span><span class="o">.</span><span class="n">step_wavefunction</span><span class="p">(</span><span class="n">wavefunction</span><span class="p">)</span>
</pre></div>
</div>
<p>That’s it! All the evolution happens behind the <cite>step_wavefunction</cite> method.</p>
</section>
</section>
<section id="imaginary-complex-time-evolution">
<h2>Imaginary/complex time evolution<a class="headerlink" href="#imaginary-complex-time-evolution" title="Link to this heading">#</a></h2>
<p>Imaginary time evolution is an extremely useful way of computing ground states
of Bose-Einstein condensate systems and PyGPE readily supports it.
To use imaginary time evolution we simply have to define an imaginary time step
in our parameters dictionary:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">params</span> <span class="o">=</span> <span class="p">{</span><span class="s2">"dt"</span><span class="p">:</span> <span class="o">-</span><span class="mi">1</span><span class="n">j</span> <span class="o">*</span> <span class="mf">1e-2</span><span class="p">}</span>
</pre></div>
</div>
<p>PyGPE handles re-normalizing the wavefunction automatically.
To switch back to real time, re-define the time step as a float.</p>
</section>
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