build based on 778cc50

docs/examples/bath-observables/index.html

@@ -51,7 +51,8 @@
 correlations_cdag = [
     cdagcdag_average[i, j, t] - cdag_average[i, 1, t] .* cdag_average[j, 1, t]
     for i in 1:size(cdagcdag_average, 1), j in 1:size(cdagcdag_average, 2), t in 1:size(cdagcdag_average,3)
-]</code></pre><p>It is possible to invert the thermofield transformation (details in <sup class="footnote-reference"><a id="citeref-riva_thermal_2023" href="#footnote-riva_thermal_2023">[riva_thermal_2023]</a></sup>). The expression of the mean value of the number operator for the physical modes can be expressed as a function of mean values in the extended bath, which we denote <span>$\langle \hat a_{2k}^\dagger \hat a_{2k} \rangle$</span>: $     \langle \hat b<em>k^\dagger \hat b</em>k \rangle = \cosh{\theta<em>k}\sinh{\theta</em>k} (\langle\hat a<em>{2k}\hat a</em>{1k}\rangle + \langle \hat a<em>{1k}^\dagger\hat a</em>{2k}^\dagger\rangle ) + \sinh^2{\theta<em>k} (1+ \langle \hat a</em>{2k}^\dagger \hat a<em>{2k} \rangle ) + \      + \cosh^2{\theta</em>k} \langle \hat a<em>{1k}^\dagger \hat a</em>{1k}, \rangle $ We remark that in the thermofield case, a negative frequency <span>$\omega_{2k}$</span> is associated to each positive frequency <span>$\omega_{1k}$</span>. The sampling is therefore symmetric around zero. This marks a difference with T-TEDOPA, where the sampling of frequencies was obtained through the thermalized measure <span>$d\mu(\beta) = \sqrt{J(\omega, \beta)}d\omega$</span>, and was not symmetric. To recover the results for the physical bath of frequencies starting from the results of our simulations, that were conducted using the T-TEDOPA chain mapping, we need to do an extrapolation for all of the mean values appearing in Eq. \ref{eq:physical<em>occupations}, in order to have their values for each <span>$\omega$</span> at <span>$-\omega$</span> as well. This is done in the code with the `physical</em>occup<code>function:</code>``julia bath<em>occup</em>phys = physical<em>occup(correlations</em>cdag[:,:,T], correlations<em>c[:,:,T], omeg, bath</em>occup[:,:,T], β, N)</p><pre><code class="language-none">
+]</code></pre><p>It is possible to invert the thermofield transformation (details in <sup class="footnote-reference"><a id="citeref-riva_thermal_2023" href="#footnote-riva_thermal_2023">[riva_thermal_2023]</a></sup>). The expression of the mean value of the number operator for the physical modes can be expressed as a function of mean values in the extended bath, which we denote <span>$\langle \hat a_{2k}^\dagger \hat a_{2k} \rangle$</span>:</p>$<pre><code class="language-none">\langle \hat b_k^\dagger \hat b_k \rangle = \cosh{\theta_k}\sinh{\theta_k} (\langle \hat a_{2k}\hat a_{1k}\rangle + \langle \hat a_{1k}^\dagger\hat a_{2k}^\dagger\rangle ) + \sinh^2{\theta_k} (1+ \langle \hat a_{2k}^\dagger \hat a_{2k} \rangle ) + \\ 
++ \cosh^2{\theta_k} \langle \hat a_{1k}^\dagger \hat a_{1k} \rangle</code></pre>$<p>We remark that in the thermofield case, a negative frequency <span>$\omega_{2k}$</span> is associated to each positive frequency <span>$\omega_{1k}$</span>. The sampling is therefore symmetric around zero. This marks a difference with T-TEDOPA, where the sampling of frequencies was obtained through the thermalized measure <span>$d\mu(\beta) = \sqrt{J(\omega, \beta)}d\omega$</span>, and was not symmetric. To recover the results for the physical bath of frequencies starting from the results of our simulations, that were conducted using the T-TEDOPA chain mapping, we need to do an extrapolation for all of the mean values appearing in Eq. \ref{eq:physical<em>occupations}, in order to have their values for each <span>$\omega$</span> at <span>$-\omega$</span> as well. This is done in the code with the `physical</em>occup<code>function:</code>``julia bath<em>occup</em>phys = physical<em>occup(correlations</em>cdag[:,:,T], correlations<em>c[:,:,T], omeg, bath</em>occup[:,:,T], β, N)</p><pre><code class="language-none">
 Finally, in the pure dephasing case, it is also possible to obtain the analytical prediction of the time evolution of the occupations of the bath&#39;s modes, so that we can compare our numerical results with the analytical ones, exploiting the Heisenberg time evolution relation:
 $$
 \frac{d \langle \hat b_\omega \rangle}{dt} = -i \langle[ \hat b_\omega, \hat H] \rangle = - i \omega \langle\hat b_\omega \rangle - i \frac{\langle \hat \sigma_x \rangle}{2} \sqrt{J(\omega, \beta)},     \\