I'm trying to recreate some results from a paper:(adsbygoogle = window.adsbygoogle || []).push({});

https://arxiv.org/pdf/1406.1711.pdf

Basically they take the Hamiltonian of graphene near the Dirac point (upon irradiation by a time periodic external field) and use Floquet formalism to rewrite it in an extended Hilbert space incorporating time periodic functions; in doing this they effectively remove the time dependence of the periodically modulated Hamiltonian. Skipping a few steps, they rewrite this Floquet Hamiltonian in a subspace of two potentially degenerate eigenvector branches near the Dirac point (eigenvalues of this matrix are equation (15)). The matrix is (for only the m=0 and m=1 modes)

[itex]

H_f =

\left[

\begin{array}{cc}

\hbar\Omega -\hbar v_fk_p& \frac{v_fe}{2c}A_0e^{i\theta}\\

\frac{v_fe}{2c}A_0e^{-i\theta}& \hbar v_f k_p

\end{array}

\right]

[/itex]

The eigenvectors of this matrix are quasienergy states [itex]|\phi^\alpha\rangle[/itex] from the equation [itex]H_F |\phi^\alpha\rangle = \epsilon_\alpha|\phi^\alpha\rangle[/itex]. My problem is from going from equation (9)

[itex] |\phi^\alpha\rangle = (|u^\alpha_1\rangle, |u^\alpha_0\rangle)^T [/itex]to (8) [itex]|\phi^\alpha(t)\rangle = |u^\alpha_0\rangle + |u^\alpha_1\rangle e^{i\Omega t}[/itex] in which the coefficients [itex] |u^\alpha_m\rangle[/itex] are 1x2 kets. Expanding [itex]|\phi^\alpha\rangle [/itex]:

[itex]

|\phi^\alpha\rangle =

\left[

\begin{array}{cc}

u^\alpha_{1a}&u^\alpha_{1b}\\

u^\alpha_{0a}&u^\alpha_{0b}

\end{array}

\right]

[/itex]

I need to apply this to the 2x2 Hamiltonian and solve for the [itex] u^\alpha_m[/itex]coefficients. Then take those coefficients and put them into

[itex]

|\phi^\alpha(t)\rangle =

\left[

\begin{array}{c}

u^\alpha_{0a}\\

u^\alpha_{0b}

\end{array}

\right]+

\left[

\begin{array}{c}

u^\alpha_{1a}\\

u^\alpha_{1b}

\end{array}

\right]e^{i\Omega t}

[/itex]

to get the final answer they have in equation (17). In doing so I get a system of 2 equations and 4 unknowns. When I work this out my answer doesn't come out like theirs. Is this the right way to go about going from the form in the extended Hilbert space

[itex] |\phi^\alpha\rangle = (|u^\alpha_1\rangle, |u^\alpha_0\rangle)^T [/itex]to the form in the standard Hilbert space (8) [itex]|\phi^\alpha(t)\rangle = |u^\alpha_0\rangle + |u^\alpha_1\rangle e^{i\Omega t}[/itex]?

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# A Eigenvectors of a Floquet Hamiltonian

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