I'm familiar with the adiabatic theorem and have basic knowledge of quantum mechanics. While doing some independent learning, I stumbled on the concept of Berry's phase.
I would like to understand the practical significance of Berry's phase.
From what I learned, I got the following:
According to the adiabatic theorem, if we start in some eigenstate $|\psi_n(t_0)\rangle$, then if we evolve slowly enough, the state of a system $|\psi(t)\rangle$ will tend to be close to $e^{i\theta_n(t)}e^{i \gamma_n(t)} |\psi_n(t)\rangle$ where $t_0 \leq t \leq T$. Specifically, we have $$ \tag{1} \left|\left| \ |\psi(t)\rangle - e^{i\theta_n(t)}e^{i \gamma_n(t)} |\psi_n(t)\rangle \ \right|\right| = O(1/T).$$
In the above equation, Berry's phase is represented by $\gamma_n(t)$, and interestingly, it is time-independent.
Unfortunately, I fail to understand the significance of why we should care about this phase in Eq (1). Does it somehow help solve the Schrodinger equation or re-express eigenstates? Or maybe it is important for adiabatic quantum computation?
I have no physics background whatsoever so I would appreciate answers in simpler terms. I'm aware of this stack question, but I'm not able to extract any intuition from that.
