1. The problem statement, all variables and given/known
data
Consider a time-dependent harmonic oscillator with
Hamiltonian
$$\hat{H}(t)=\hat{H}_0+\hat{V}(t)$$
$$\hat{H}_0=\hbar \omega \left( \hat{a}^{\dagger}\hat{a}+\frac{1}{2} \right)$$
$$\hat{V}(t)=\lambda \left( e^{i\Omega t}\hat{a}^{\dagger}+e^{-i\Omega t}\hat{a} \right)$$
*(i) Compute $\hat{U}_S(t,0)$ using the interaction
representation formula (Equation 1 in next section) to
second order perturbation theory.
(ii) Compute $\hat{U}_S(t,0)$ using (Equation 2 in next
section) to second order perturbation theory.
2. Relevant equations
EQUATION 1:
$$U_I(t,0)=1-\frac{i}{\hbar}\int_0^t dt' V_I(t')+\left( \frac{-i}{\hbar} \right)^2 \int_0^t dt' \int_0^{t'} V_I(t')V_I(t'') + \dots$$
EQUATION 2:
$$U(t,0)=1+\sum_{n=1}^{∞}\left( \frac{-i}{\hbar} \right)^n\int_0^t dt_1 \int_0^{t_1} dt_2 \dots \int_0^{t_{n-1}}dt_n H(t_1)H(t_2)\dots H(t_n)$$
3. The attempt at a solution
So I know that for the interaction picture the transformation of the operator $\hat{V}_I$ is..
$$\hat{V}_I=e^{\frac{i}{\hbar}\hat{H}_0 t} \hat{V} e^{\frac{-i}{\hbar}\hat{H}_0 t}$$
I also know that both operators and kets evolve in time. So I use the interaction picture equation of motion on the ladder operators so I can obtain an expression for them as a function of time.
$$\frac{d\hat{a}}{dt}=\frac{1}{i\hbar}\left[ \hat{a},\hbar \omega \left(\hat{a}^{\dagger}\hat{a} + \frac{1}{2} \right) \right]$$
$$\frac{d\hat{a}^{\dagger}}{dt}=\frac{1}{i\hbar}\left[ \hat{a}^{\dagger},\hbar \omega \left( \hat{a}^{\dagger}\hat{a} + \frac{1}{2} \right) \right]$$
I then got..
$$\hat{a}(t)=\hat{a}(0)e^{-i\omega t}$$
$$\hat{a}^{\dagger}(t)=\hat{a}^{\dagger}(0)e^{i\omega t}$$
I plugged these into the expression for V to get,
$$\hat{V}=\lambda \left[ \hat{a}^{\dagger}(0)e^{i(\Omega + \omega)t} + \hat{a}(0)e^{-i(\Omega + \omega)t} \right]$$
So now what needs to be done, is to transform this into the interaction picture and then plug it into Equation 1 from above and integrate. But this seems very messy and I am having doubts if this is the correct way to I also know that both operators and kets evolve in time.
So I use the interaction picture equation of motion on the ladder operators so I can obtain an expression for them as a function of time. go about this problem. If anyone can shed some light onto this I would really appreciate it!
Best Answer
Let's start from your EQ1:
$\begin{eqnarray}U_I(t,0) = \mathbf{Id} + \frac{1}{i\hbar}\int_0^t dt_1V_H(t_1) +\cdots+\left(\frac{1}{i\hbar}\right)^k\int_0^tdt_1...\int_0^{t_{k-1} }V_H(t_1)\cdots V_H(t_k) \end{eqnarray}$
where $V_H$ means $V$ evolved by heisenberg. The operator is totally symmetric so we can adjust the integral extrema to write the well know path-order exponentail:
$\begin{eqnarray} U_I(t,0)=\mathbf{Id}+\sum_{k=1}^{+\infty}\frac{1}{k!}\int_0^tdt_1..\int_0^tdt_{k-1}V_H(t_1)...V(t_k) = \text{Texp}\left[\frac{1}{i\hbar}\int_0^tdt'V_H(t')\right] \end{eqnarray}$
So now you can use the form of potential that you fine in the path-order exponential, and with GellMann and Low theorem find the ground state of your hamiltonian.