build based on e40d7f8

docs/examples/bath-observables/index.html

@@ -51,7 +51,7 @@
 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>:</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` function:</p><pre><code class="language-julia">bath_occup_phys = physical_occup(correlations_cdag[:,:,T], correlations_c[:,:,T], omeg, bath_occup[:,:,T], β, N)</code></pre><p>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<em>\omega \rangle}{dt} = -i \langle[ \hat b</em>\omega, \hat H] \rangle = - i \omega \langle\hat b<em>\omega \rangle - i \frac{\langle \hat \sigma</em>x \rangle}{2} \sqrt{J(\omega, \beta)},     \
+]</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, 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 <code>physical_occup</code> function:</p><pre><code class="language-julia">bath_occup_phys = physical_occup(correlations_cdag[:,:,T], correlations_c[:,:,T], omeg, bath_occup[:,:,T], β, N)</code></pre><p>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<em>\omega \rangle}{dt} = -i \langle[ \hat b</em>\omega, \hat H] \rangle = - i \omega \langle\hat b<em>\omega \rangle - i \frac{\langle \hat \sigma</em>x \rangle}{2} \sqrt{J(\omega, \beta)},     \
 \frac{d \langle \hat n<em>\omega \rangle}{dt} = -i \langle[\hat b</em>\omega^\dagger \hat b<em>\omega, \hat H] \rangle= 2 \frac{|J(\omega,\beta)|}{\omega} \sin(\omega t). $ To this end, it is convenient to choose one of the eigenstates of \hat \sigma</em>z$ as the initial state, so that <span>$\langle \hat \sigma_x \rangle = \pm 1$</span>. By solving these differential equations, one obtains the time evolved theoretical behavior of the bath. We define the function for the comparison with analytical predictions:</p><pre><code class="language-julia">Johmic(ω,s) = (2*α*ω^s)/(ωc^(s-1))
 
 time_analytical = LinRange(0.0, tfinal, Int(tfinal))