How does the macroscopic wavefunction build up from a zero value to nonzero value?

I can understand how does spontaneous magnetization (the order parameter in paramagnetic to ferromagnetic transition) gradually build up as the temperature is lowered below the critical temperature. Microscopically, the atomic magnetic moments gradually align in some direction so that a macroscopic magnetization appears. As the temperature is lowered more and more moments are aligned in the same direction so that the Magnetization grows in magnitude until it saturates.

Similarly, the order parameter in case normal to superfluid transition is the macroscopic wavefunction. I have the following questions.

1. Is there a similar way to understand how does the macroscopic wavefunction (the order parameter) build up from zero value to a non-zero value during the normal to the superfluid transition of Helium?

2. Like the magnetization, does this wavefunction grow in magnitude further and further as the temperature is lowered below the transition temperature? If yes, how?

I'll describe here a general principle of the formation of a condensate by means of cooling. This principle should be essentially valid for all cases such as BEC, BCS, superfluidity etc.

The description given here is on the conceptual, toy model level only. The details of the theoretical and experimental implementation in a realistic case are much more involved. The actual implementation can differ significantly between the various cases and between the experimental methods.

For system with spontaneous symmetry breaking, the macroscopic condensate ground state is a coherent state; please see for example the reasoning given in section 2 of the following work by Yulakov, in the case of Bose-Einstein condensation. In this description, the macroscopic wave function is the eigenvalue of the field operator $\hat{\Psi}(\mathbf{r})$:

$$\hat{\Psi}(\mathbf{r}) |\Phi \rangle = \psi(\mathbf{r}) |\Phi \rangle$$

$|\Phi \rangle$ is the macroscopic ground state, and $\psi(\mathbf{r})$ is the macroscopic wave function.

Thus, formally the macroscopic state can be written as: $$|\Phi \rangle = e^{\int \psi(\mathbf{r}) a^{\dagger}(\mathbf{r})) dr} |0 \rangle$$

Where $|0 \rangle$ is the unbroken vacuum and $a^{\dagger}(\mathbf{r})$ are the creation operators at the location $\mathbf{r}$ (of course this equation is a field theoretical generalization of the single mode coherent state: $e^{\alpha a^{\dagger}}|0 \rangle$.

The cooling process should transform the state of the system from a thermal state, to the coherent state described above. Of course the dynamics of the cooling process cannot be Hamiltonian, since Hamiltonian dynamics cannot convert a mixed state into a pure state. The dynamics of the cooling process can however be approximated by Lindblad dynamis, which generalize the Schrödinger dynamics in the case of open or stochastically driven systems. The Lindblad dynamis can describe dissipation and it can also describe the above process of purification. The full details in our case can depend on the cooling method and the specific system and can be quite involved. Therefore, in the following I'll describe this process in principle in the case of a single mode: How a thermal state evolves into a coherent state:

In Lindblad dynamics, the evolution of the density operator is given by:

$$\dot{\rho} = \frac{i}{\hbar} [H, \rho] + \mu L \rho L^{\dagger} - \frac{1}{2} \{ L^{\dagger} L, \rho \}$$ This equation conserves the trace of the density matrix at 1. The first term is the usual Hamiltonian term. The operator $L$ is the Lindblad operator. The choice of the Lindblad operator controls the dissipation or the purification in our case. In the case when the Hamiltonian is Linear $$H = - i (\bar{\lambda} a - \lambda a^{\dagger})$$

and the Lindblad operator is chosen to be the annihilation operator $L=a$, the Lindblad system evolves into the coherent state $e^{\alpha a^{\dagger}}$, with $\alpha = \frac{2 \lambda}{\mu}$ from every initial state it starts in.

The following work by Barnett explains the above point in detail. He explains also the stability of the coherent state compared to the number states.

The property of certain Lindblad operators to make the dynamics to converge into a coherent state is general and not confined to the above single mode case.

Now, during the evolution process, the density matrix interpolates between the initial thermal state and the final coherent state. Barnett chooses the fidelity:

$$F = \langle \Phi_{\infty} | \rho(t) |\Phi_{\infty} \rangle$$

as a measure of this purification process ( $|\Phi_{\infty} \rangle$ is the steady state.). In our case the fidelity starts at a very small number for the thermal stste and reaches one when the system becomes a full condensate.

As I mentioned earlier, there are many more details in the actual implementation of the cooling process, please see the following presentation for some additional details.