Floquet operator ($U_F=U(T,0)$)

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.

For an instantaneous rotation about $x$ followed by free precession about $z$, the just-before-kick one-period operator is $U=e^{-i\Omega T{S}^z}{e}^{-i\Phi S^x}$. Sampling just after kicks reverses the displayed order. The spin generators are dimensionless and the pulse area $\Phi$ is an angle.

Multiplying the two matrix exponentials with the Pauli matrix multiplication law givesOne has $C^2+|\mathbf{v}{|}^2=1$. If $|\mathbf{v}|\ne 0$, put $\theta =2\arccos C$ and $\mathbf{n}=\mathbf{v}/|\mathbf{v}|$. The exponential of a Pauli vector gives $U_F^N=I\cos (N\theta /2)-i\mathbf{n}\cdot \mathbf{\sigma }\sin (N\theta /2)$. Its action on a Bloch vector is a rotation through $N\theta$ about $\mathbf{n}$, and its quasienergies are $\pm \hslash \theta /(2T)$ modulo $2\pi \hslash /T$. When $U_F=\pm I$, its axis is arbitrary and the Bloch vector is unchanged.

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.