Floquet Numerics

Floquet numerics solves a Hermitian Hamiltonian periodic in time by computing its one-period evolution or an enlarged Fourier-space eigenproblem. Quasienergies are defined modulo the drive quantum, so branch wrapping is part of the representation rather than a physical discontinuity. The Floquet theorem explains the general decomposition; here we develop the numerical choices.

Helpful background. Matrix exponentials and Magnus integrators supply period propagation.

Diagonalizing the period propagator

Let $H(t+T)=H(t)$ and $\omega=2\pi/T$. Compute $U(T,0)$ and solve

$$U(T,0)v_\alpha=e^{-i\varepsilon_\alpha T/\hbar}v_\alpha.$$

A principal representative is $\varepsilon_\alpha=-(\hbar/T)\operatorname{Arg}\lambda_\alpha$. Adding $m\hbar\omega$ gives the same eigenvalue, so all quoted quasienergies require a zone convention.

Propagating the identity obtains the full period matrix for a modest Hilbert space. For a large space, supply its action through time propagation to an iterative eigensolver, taking care that repeated actions have consistent accuracy. The period matrix is unitary, not Hermitian. A Hermitian eigensolver is inappropriate even though the instantaneous Hamiltonian is Hermitian.

Check $\|U^\dagger U-I\|$, eigenpair residuals, and convergence under time-step refinement. Tiny nonunitarity can shift eigenvalues away from the unit circle. Projecting onto a nearby unitary matrix may be a controlled diagnostic, but should not conceal propagation error.

Micromotion and time-origin checks

From an eigenvector at time zero, reconstruct the periodic mode

$$u_\alpha(t)=e^{i\varepsilon_\alpha t/\hbar}U(t,0)v_\alpha.$$

It should satisfy $u_\alpha(T)=u_\alpha(0)$ within error. The physical solution is $e^{-i\varepsilon_\alpha t/\hbar}u_\alpha(t)$.

Changing the time origin conjugates the period operator by a unitary evolution, leaving its eigenvalues unchanged while rotating its eigenvectors. Compare spectra at two time origins as a strong check of propagation order and periodicity. A quasienergy branch jump by $\hbar\omega$ can be removed for plotting by consistent unwrapping, but gaps and crossings must still be interpreted modulo the zone.

Near degeneracy, track the entire Floquet eigenspace. Individual eigenvectors and a chosen matrix logarithm become poorly conditioned at branch cuts or eigenvalue collisions.

Harmonic-space construction

Expand $H(t)=\sum_\ell H_\ell e^{i\ell\omega t}$ and $u(t)=\sum_m u_m e^{im\omega t}$. The equation $(H-i\hbar\partial_t)u=\varepsilon u$ becomes

$$\sum_m\left[H_{n-m}+n\hbar\omega\,\delta_{nm}\right]u_m =\varepsilon u_n.$$

Truncate $m$ to $-M,\ldots,M$. The enlarged Hermitian problem has many zone-shifted copies of physical modes. Do not count each copy as a new physical state. Identify modes with appreciable weight away from the harmonic boundary and compare representatives modulo $\hbar\omega$ as $M$ increases.

A stronger drive or sharp time waveform generally needs more harmonics. Fourier truncation and Hilbert-space truncation are independent. Agreement between the harmonic method and period propagation is especially useful because their dominant numerical errors differ.

A commuting driven example

For

$$H(t)=\frac{\hbar}{2}[\Omega_0+A\cos(\omega t)]\sigma_z,$$

Hamiltonians commute at all times. The oscillatory term integrates to zero over one period, so

$$U(T)=e^{-i\Omega_0T\sigma_z/2}.$$

Quasienergies are $\pm\hbar\Omega_0/2$ modulo $\hbar\omega$, independent of $A$, although micromotion depends on $A$. This checks the distinction between stroboscopic spectrum and within-period dynamics.

Exercises

If a constant Hamiltonian has energies $0$ and $1.2\hbar\omega$, what are equivalent representatives in the zone centered at zero?

Solution

The representatives are $0$ and $0.2\hbar\omega$. The second state has not changed its physical energy; its Floquet phase is unchanged by subtracting one drive quantum. Folding a static spectrum is a representation operation.

References

• Blanes, S., F. Casas, J. A. Oteo, and J. Ros, “The Magnus expansion and some of its applications,” Physics Reports 470, 151–238 (2009), doi:10.1016/j.physrep.2008.11.001 — time-ordered propagation and effective exponents.
• Shirley, J. H., “Solution of the Schrödinger equation with a Hamiltonian periodic in time,” Physical Review 138, B979–B987 (1965), doi:10.1103/PhysRev.138.B979 — harmonic-space Floquet construction.