I've been puzzled equation (2.2) of this paper,
I've looked into time-dependent perturbation theory and the adiabatic theorem and the closest I could come to deriving this was showing that
$$
i \hbar \dot{c_k} = (E_k – i \hbar < \psi_k | \dot{\psi}_n > ) c_k – i \hbar \sum_{n \neq k} \frac{\dot{H}_{kn}}{E_n-E_k} c_n,
$$
where $| \Psi_n(t)> = \sum_n c_n(t) | \psi_n (t) >$ and $H_{kn} = <\psi_k| H |\psi_n>$.
I was wondering if someone could direct me to a derivation of equation (2.2) as I am struggling to connect perturbation theory with this result.
Best Answer
The adiabatic expansion is not the same as that of time dependent perturbation theory. In tradition T-D perturbation we expand in a complete set of states of the original time-independent Hamiltonian. In the adiabtic perturbation series we expand in terms of the eigenstates of the current time-dependent Hamiltonian.
Start from the time dependent Schroedinger equation $$ i\partial_t|\psi_0(t)\rangle =\hat H(t)|\psi_0(t)\rangle $$ and expand $$ |\psi_0(t)\rangle=\sum_{n=0}^\infty a_n|n,t\rangle \exp\left\{-i\int_0^t E_0(t')dt'\right \}. $$ Choose the complete orthonormal set of states $|n,t\rangle$ to be eigenstates of the ``snapshot'' hamiltonian, $\hat H(t)$, $$ \hat H(t)|n,t\rangle=E_n(t)|n,t\rangle. $$ Insert the expansion into the Schroedinger equation, take components and assume that $|\psi_0(t)\rangle $ stays close to $|0,t\rangle$ so that $|a_0|$ is close to unity, and the other coeficients are small. This leads to $$ \dot a_0+a_0\langle 0,t|\partial_t|{0},t\rangle \approx 0\\ a_m\approx ia_0\langle{m,t}|{\partial_t}|{0,t}\rangle \frac1{(E_m-E_0)}. $$ Up to first order in time-derivatives of the states, we find $$ |\psi_0(t)\rangle =e^{-i\int_0^t E_0(t)dt+i\gamma_{Berry}}\left\{|{0,t}\rangle+ i\sum_{m\ne 0}\frac{|{m,t}\rangle\langle{m,t}|{\partial_t}|{0,t}\rangle}{E_m-E_0}+\ldots\right\}. $$ Berry's phase is the solution $a_0=\exp \{i\gamma_{Berry}\}$ to the first of the two equations above. It is a phase because a $|{0},t\rangle$ being normalized means that $\langle 0,t|\partial_t|{0},t\rangle$ is pure imaginary. This phase-factor is needed to take up the slack between the arbitrary phase choice made when defining $|0,t\rangle$, and the specific phase selected by the Schroedinger equation as it evolves the state.