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A Floquet Hamiltonian generates the evolution over one period of a time-periodic Quantum Hamiltonian. For period $T$, choose $H_F$ such that $U(T,0)=e^{-i{H}_{F}T/\hslash }$. Its eigenvalues are quasienergies, defined modulo $\hslash \omega$ with $\omega =2\pi /T$.

Taking a logarithm of the same unitary permits different branches, so the quasienergy convention must be specified. A complete description within the period also needs the periodic micromotion; the stroboscopic generator alone does not determine every intermediate observable. At sufficiently high drive frequency, an inverse-frequency expansion can approximate $H_F$, but resonances or long-time heating can limit its validity. The construction is useful for separating repeated-cycle dynamics from motion within a cycle.

Put $\alpha =\Delta \tau /\hslash$ and $\beta =E{x}_0\tau /\hslash$. The Pauli matrix multiplication law gives the pulse-and-precession spin one-half mapBecause $C^2+|\mathbf{v}{|}^2=1$, a suitable angle and unit vector for the rotation areIf the denominator vanishes, $U=\pm I$ and the axis is arbitrary. The Bloch vector $\mathbf{r}=⟨\mathbf{\sigma }⟩$ rotates through $\theta$ about $\mathbf{n}$ per period; after $N$ periods the angle is $N\theta$. The eigenphases equivalently give quasienergies $\pm \hslash \theta /(2\tau )$ modulo $2\pi \hslash /\tau$.

In the two-orbital subspace the position operator is $X=x_0{\sigma }_x$. The Heisenberg equation of motion between pulses gives $P/m=(i/\hslash )[H_0,X]=(2\Delta x_0/\hslash ){\sigma }_y$, henceThus two components of the Bloch vector directly describe the mean position and momentum; its third gives the orbital population imbalance. The electric kick rotates about the position axis, leaving $⟨X⟩$ unchanged at the kick, while free evolution mixes the position and momentum components. Starting in $|1s⟩$, one complete period giveswhich exhibits the induced motion explicitly. These are projected two-orbital observables.

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.