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For a periodic Hamiltonian $H(t+T)=H(t)$, the Floquet operator is the unitary operator evolving a quantum state through one complete period, at a fixed starting phase of the drive. The composition law gives $U(NT,0)=U_F^N$. Choosing a different starting phase conjugates $U_F$ by a unitary operator, so its eigenvalues are unchanged.

Use the ordered orthonormal basis $(|1s⟩,|2p_x⟩)$. The projected Hamiltonian between pulses is $H_0=-\Delta {\sigma }_z$, so its free time-evolution operator through a period isAcross one instantaneous pulse, integrating the Schrödinger equation givesusing the exponential of a Pauli vector. The finite free Hamiltonian contributes vanishing evolution during the zero-duration kick. A period measured immediately before successive pulses therefore consists of the kick followed by free evolution. Composition givesThis is the Floquet operator at the specified phase of the pulse train. Actual hydrogenic energy levels are not opposite numbers: the displayed convention subtracts their midpoint, with $\Delta =(E_{2p}-E_{1s})/2$. Restoring that midpoint multiplies the result by an unobservable global phase. Likewise $E$ in the given Hamiltonian denotes the electric coupling per unit length; if it is instead a physical electric-field strength, its charge factor must be included in $E{x}_0$.

A quasienergy is defined by an eigenvalue of the Floquet operator. It is determined modulo $2\pi \hslash /T$, because adding an integer multiple of that quantity leaves the complex exponential unchanged. A time-independent effective Hamiltonian can be defined by a branch of the matrix logarithm; its eigenvalues are the corresponding chosen quasienergies.