Computation time for adiabatic evolution

[mupsi]

Hi,

my problem: following the adiabatic theorem we get an equation for the coefficients:

$\dot{a}_{m}=-a_{m} \langle{\psi_{m}} | \dot{\psi}_{m}\rangle - \sum_{n \neq m} \frac{\langle \psi_{m} | \dot{H}| \psi_{m}\rangle}{E_{n}-E_{m}} exp(\int_ 0^t E_{n}(t')-E_{m}(t') \, dt')$

we start in an eigenstate m. When the gap is large and the time derivative of H is small. We can neglect the sum and get for the coefficent m:

$a_m^0= a_m(0) exp(-i \gamma_m (t))$

so far so good. Now we want to determine the 1st order coefficients and make the ansatz:

$a_m= a_m^0 + a_m^1$

and we get:

$a_m^1= \sum_{n \neq m} \frac{\langle \psi_{m} | \dot{H}| \psi_{m}\rangle}{(E_{n}-E_{m})^2} a_n^0 exp(\int_ 0^t E_{n}(t')-E_{m}(t') \, dt')$

the last step is problematic. It's been a while since I used perturbation theory and I don't know how you get this result. The computation time should be:

$t<<\frac{\langle \psi_{m} | \dot{H}| \psi_{m}\rangle}{(E_{n}-E_{m})^2}$

which is clear once you obtain the result above. Can anyone help?

 

[mupsi]

Hi,

my problem: following the adiabatic theorem we get an equation for the coefficients:

$\dot{a}_{m}=-a_{m} \langle{\psi_{m}} | \dot{\psi}_{m}\rangle - \sum_{n \neq m} \frac{\langle \psi_{m} | \dot{H}| \psi_{m}\rangle}{E_{n}-E_{m}} exp(\int_ 0^t E_{n}(t')-E_{m}(t') \, dt')$

we start in an eigenstate m. When the gap is large and the time derivative of H is small. We can neglect the sum and get for the coefficent m:

$a_m^0= a_m(0) exp(-i \gamma_m (t))$

so far so good. Now we want to determine the 1st order coefficients and make the ansatz:

$a_m= a_m^0 + a_m^1$

and we get:

$a_m^1= \sum_{n \neq m} \frac{\langle \psi_{m} | \dot{H}| \psi_{m}\rangle}{(E_{n}-E_{m})^2} a_n^0 exp(\int_ 0^t E_{n}(t')-E_{m}(t') \, dt')$

the last step is problematic. It's been a while since I used perturbation theory and I don't know how you get this result. The computation time should be:

$t<<\frac{\langle \psi_{m} | \dot{H}| \psi_{m}\rangle}{(E_{n}-E_{m})^2}$

which is clear once you obtain the result above. Can anyone help?

not sure how to edit posts... a few remarks: it should be t>>... and gamma is the geometric phase.