I am planing to describe time evolution of two-component BEC. I was thinking about non-equilibrium Green's functions, but I don't if the method can be applied to the problem describe below. I know nothing about Green's functions that is why I need an advice (is it worth spending my time learning the theory?).

Hamiltonian in second quantization reads: $$\mathcal{H} = \int d^{3}x \left\{ \sum\limits_{\epsilon = a,b}\Psi^{\dagger}_{\epsilon}\left( -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{x})\right)\Psi_{\epsilon} + \frac{g_{aa}}{2}\Psi^{\dagger}_{a}\Psi^{\dagger}_{a}\Psi_{a}\Psi_{a} + \frac{g_{bb}}{2}\Psi^{\dagger}_{b}\Psi^{\dagger}_{b}\Psi_{b}\Psi_{b} + g_{ab}\Psi^{\dagger}_{b}\Psi^{\dagger}_{a}\Psi_{a}\Psi_{b}\right\}$$ where field operators $\Psi_{a,b}(\mathbf{x})$ creates particle of type a,b at position $\mathbf{x}$.

Quantities I would like to calculate during evolution are spin operators and their variances: $$S_{x}(t) = \frac{1}{2}\int d^3 x \left[\Psi^{\dagger}_{a}(\mathbf{x},t) \Psi_{b}(\mathbf{x},t) + \Psi^{\dagger}_{b}(\mathbf{x},t)\Psi_{a}(\mathbf{x},t)\right]$$
$$S_{y}(t) = \frac{1}{2i}\int d^3 x \left[\Psi^{\dagger}_{a}(\mathbf{x},t) \Psi_{b}(\mathbf{x},t) - \Psi^{\dagger}_{b}(\mathbf{x},t)\Psi_{a}(\mathbf{x},t)\right]$$ $$S_{z}(t) = \frac{1}{2}\int d^3 x \left[\Psi^{\dagger}_{a}(\mathbf{x},t) \Psi_{a}(\mathbf{x},t) - \Psi^{\dagger}_{b}(\mathbf{x},t)\Psi_{b}(\mathbf{x},t)\right]$$ Quantities I need: $$\langle N_a', N_b'| S_{k}(t)|N_a, N_b\rangle$$ $$\Delta_{ij}(t) = \langle N_a', N_b'| S_{i}(t) S_{j}(t)|N_a, N_b\rangle - \langle N_a', N_b'| S_{i}(t)| N_a, N_b\rangle \langle N_a', N_b'| S_{j}(t)|N_a, N_b\rangle$$ Initial states are the following: $$|N_a, N_b\rangle = \frac{(a^{\dagger})^{N_a} (b^{\dagger})^{N_b}}{\sqrt{N_a !}\sqrt{N_b !}}|0\rangle$$ where $a^{\dagger}$ creates particle of type $a$ in mode $\phi_{0}(\mathbf{x})$, and $b^{\dagger}$ creates particle of type $b$ also in mode $\phi_{0}(\mathbf{x})$. Using field operators we can write: $$a^{\dagger} = \int d^3 x\ \phi_{0}(\mathbf{x}) \Psi^{\dagger}_{a}(\mathbf{x})$$ $$b^{\dagger} = \int d^3 x\ \phi_{0}(\mathbf{x}) \Psi^{\dagger}_{b}(\mathbf{x})$$ So we can also write for the initial state: $$|N_a, N_b\rangle = \frac{1}{\sqrt{N_a!}\sqrt{N_b!}}\prod\limits_{i=1}^{N_a}\int d^{3}x_i\ \phi_{0}(\mathbf{x}_i)\Psi^{\dagger}_{a}(\mathbf{x}_i)\ \prod\limits_{j=1}^{N_b}\int d^{3}x_j\ \phi_{0}(\mathbf{x}_j)\Psi^{\dagger}_{b}(\mathbf{x}_j)|0\rangle$$

Do you think I can somehow perturbatively find time-dependence of such quantities?

  • $\begingroup$ Are you sure you are not better off looking at something a bit simpler, like the Gross-Pitaevskii equation? Your choice of initial state would seem ideally suited to such a treatment. $\endgroup$ – Mark Mitchison Feb 26 '16 at 1:29
  • $\begingroup$ @MarkMitchison How would you proceed with Gross-Pitaevskii? This is a Hartree-Fock approximation so it should also be derived using Green's function method. I would like to derive equation of motion for the condensate cloud then also for non-condensed atoms. Apart from wavefunction there should be some sign of quantum nature - some creation and annihilation operators. $\endgroup$ – WoofDoggy Feb 26 '16 at 11:03

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