Floquet–Magnus Expansion

The Floquet–Magnus expansion applies the Magnus series directly to one period of a drive. It constructs a stroboscopic generator tied to the starting phase $t_0$; changing that phase changes the effective matrix by micromotion conjugation. High-frequency expansions develops the complementary phase-independent van Vleck Hamiltonian and its kick operator.

Required background. The Magnus expansion gives the commutator series; time-dependent Hamiltonians and the evolution operator establish the propagator and its composition law.

Consider a Hamiltonian satisfying

$$H(t+T)=H(t), \qquad \Omega=\frac{2\pi}{T}.$$

The goal is to approximate the one-period evolution by a stroboscopic effective Hamiltonian:

$$U(T,0) = \exp\left[ - \frac{i}{\hbar}H_FT \right].$$

The operator $H_F$ is called a Floquet Hamiltonian. It governs evolution observed at integer multiples of the drive period, but it does not by itself describe all motion within a period.

Floquet Operator

The one-period evolution operator

$$U_F \equiv U(T,0)$$

is the Floquet operator. Its eigenvalues have the form

$$e^{-i\epsilon_\alpha T/\hbar},$$

where $\epsilon_\alpha$ are quasienergies. Because the phase is unchanged under

$$\epsilon_\alpha \longrightarrow \epsilon_\alpha + m\hbar\Omega, \qquad m\in\mathbb Z,$$

quasienergies are defined modulo $\hbar\Omega$.

This branch structure is one reason effective Floquet Hamiltonians must be interpreted with care.

Stroboscopic Effective Hamiltonian

Choose a branch of the logarithm and define

$$H_F = \frac{i\hbar}{T} \log U_F.$$

Then

$$U(nT,0) = e^{-iH_FnT/\hbar}$$

for integer $n$ if the same period origin is used. This is stroboscopic evolution: it compares the system only at the same phase of the drive.

For a finite-dimensional periodic Hamiltonian, the full evolution has the exact factorization

$$U(t,0) = P(t) e^{-iH_Ft/\hbar},$$

where $P(t)=U(t,0)e^{iH_Ft/\hbar}$ is a unitary micromotion operator with $P(0)=I$, satisfying

$$P(t+T)=P(t).$$

The effective Hamiltonian and micromotion together describe the driven system.

Floquet–Magnus Terms

Expand the periodic Hamiltonian in Fourier modes:

$$H(t) = \sum_{m\in\mathbb Z} H_m e^{im\Omega t}, \qquad H_{-m}=H_m^\dagger$$

for Hermitian $H(t)$. The time average is

$$H_0 = \frac1T \int_0^T H(t)\,dt.$$

The leading Floquet–Magnus Hamiltonian is the time average:

$$H_{\mathrm{FM}}^{(0)}=H_0.$$

At a period origin $t_0$, the result through order $\Omega^{-1}$ is

$$H_{\mathrm{FM}}(t_0) ={} H_0 + H_{\mathrm{vV}}^{(1)} + H_K^{(1)}(t_0) + O(\Omega^{-2}).$$

with

$$H_{\mathrm{vV}}^{(1)} = \sum_{m=1}^{\infty} \frac{[H_m,H_{-m}]} {m\hbar\Omega}$$

and

$$H_K^{(1)}(t_0) = \sum_{m\ne0} \frac{e^{im\Omega t_0}} {m\hbar\Omega} [H_0,H_m].$$

The positive-harmonic commutator is the first phase-independent van Vleck correction. The second line dresses that effective Hamiltonian into the stroboscopic frame selected by $t_0$. Exact Floquet Hamiltonians at different starting phases are unitarily conjugate and have the same quasienergies. At finite truncation order, the phase-dependent term must be handled consistently with micromotion; otherwise approximate spectra can acquire artifacts beyond the controlled order.

In the van Vleck organization, one instead writes

$$U(t,t_0) = e^{-iK(t)} e^{-iH_{\mathrm{eff}}(t-t_0)/\hbar} e^{iK(t_0)}.$$

Then

$$H_F(t_0) = e^{-iK(t_0)} H_{\mathrm{eff}} e^{iK(t_0)}$$

on compatible logarithm branches. This factorization makes the difference between phase-independent effective evolution and phase-dependent stroboscopic dressing explicit.

Relation to Rotating-Wave Approximation

The rotating-wave approximation keeps terms that are slow in a rotating frame and drops terms that oscillate rapidly. Floquet–Magnus instead treats periodic driving through the one-period evolution operator and expands in inverse powers of the drive frequency.

The two methods often agree in overlapping weak-drive, near-resonant regimes after choosing an appropriate rotating frame. They answer slightly different questions:

• RWA isolates near-resonant slow dynamics.
• Floquet–Magnus constructs a stroboscopic generator for a periodic Hamiltonian.

For strong resonant driving, one should not blindly use a high-frequency expansion in the laboratory frame.

Micromotion

The micromotion operator $P(t)$ describes intra-period motion. Even when $H_F$ is simple, observables measured within a drive cycle can depend strongly on $P(t)$.

For this reason, matching only $H_F$ is not enough when:

• the observable is sampled inside a period,
• the drive phase matters,
• the initial state is prepared at a specific phase,
• kicks or pulses produce large intra-period rotations.

The effective Hamiltonian is a stroboscopic object. It is not the whole driven dynamics.

Convergence and the Observation Time

For finite matrices, a sufficient one-period Magnus condition is $\int_{t_0}^{t_0+T}\|H(t)\|dt<\pi\hbar$, after any exactly treated frame change. The size of each Fourier harmonic alone is insufficient: the static splitting, the sum of harmonics, and resonant couplings also matter. Unitarity of a truncated exponential proves no accuracy statement.

For unitary exact and approximate cycle maps, an error $\varepsilon$ per period implies the conservative bound $\min(2,n\varepsilon)$ after $n$ periods. An expansion useful for one period can therefore fail on the time scale of a precision phase measurement. For local interacting many-body systems, a global norm bound generally grows with size; high-frequency methods explains the additional locality assumptions behind a long but finite prethermal interval.

Simple Commutator Example

Let

$$H(t) = H_0 + V e^{i\Omega t} + V^\dagger e^{-i\Omega t}.$$

Then

$$H_1=V, \qquad H_{-1}=V^\dagger.$$

The phase-independent first correction is

$$H_{\mathrm{eff}}^{(1)} = \frac{[V,V^\dagger]}{\hbar\Omega}.$$

The direct Floquet–Magnus representative also contains

$$\frac{ e^{i\Omega t_0}[H_0,V] - e^{-i\Omega t_0}[H_0,V^\dagger] }{ \hbar\Omega }.$$

If $V$ commutes with $V^\dagger$, the phase-independent correction vanishes. If not, the rapidly oscillating drive can generate a static effective term at order $\Omega^{-1}$. The extra $t_0$-dependent term is the leading micromotion conjugation of that effective Hamiltonian.

Exercises

1. A phase-dependent generator with unchanged quasienergies

Take $H(t)=\Delta\sigma_z/2+A\sigma_x\cos\Omega t$, where $\Delta,A$ are energies. Compute the first correction to the direct Floquet–Magnus Hamiltonian at $t_0$. Why does its phase dependence not imply an order-$\Omega^{-1}$ change in quasienergies?

Solution

Here $H_1=H_{-1}=A\sigma_x/2$, so the van Vleck commutator vanishes. Since $[H_0,H_1]=i\Delta A\sigma_y/2$, the direct representative is

$$H_F(t_0)=\frac{\Delta}{2}\sigma_z -\frac{\Delta A}{\hbar\Omega}\sin(\Omega t_0)\sigma_y +O(\Omega^{-2}).$$

The correction rotates the eigenvectors. It changes the eigenvalues of this truncated matrix only at order $\Omega^{-2}$, where other terms have been omitted. Exact conjugate representatives have identical quasienergies; comparing inconsistent truncations beyond their accuracy creates a spurious phase dependence.

2. Nontrivial motion with a zero cycle Hamiltonian

For $H(t)=A\cos\Omega t\,\sigma_z$, choose the branch $H_F=0$ and an initial $+1$ eigenstate of $\sigma_x$. Find $\langle\sigma_x\rangle$ at $T/4$ and $T$.

Solution

All instantaneous Hamiltonians commute, giving

$$U(t,0)=\exp\!\left[-i\frac{A}{\hbar\Omega}\sin(\Omega t)\sigma_z\right].$$

Thus $\langle\sigma_x(T/4)\rangle=\cos[2A/(\hbar\Omega)]$, whereas $\langle\sigma_x(T)\rangle=1$. The whole motion is micromotion on this branch. Knowing the identity cycle map alone does not recover it.

References

• D. A. Abanin, W. De Roeck, W. W. Ho, and F. Huveneers, “Effective Hamiltonians, prethermalization, and slow energy absorption in periodically driven many-body systems,” Physical Review B 95, 014112, 2017.
• S. Blanes, F. Casas, J. A. Oteo, and J. Ros, “The Magnus expansion and some of its applications,” Physics Reports 470, 151-238, 2009.
• M. Bukov, L. D’Alessio, and A. Polkovnikov, “Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to Floquet engineering,” Advances in Physics 64, 139-226, 2015.
• A. Eckardt and E. Anisimovas, “High-frequency approximation for periodically driven quantum systems from a Floquet-space perspective”, New Journal of Physics 17, 093039 (2015), doi:10.1088/1367-2630/17/9/093039.
• A. Eckardt, “Colloquium: Atomic quantum gases in periodically driven optical lattices,” Reviews of Modern Physics 89, 011004, 2017.
• W. Magnus, “On the exponential solution of differential equations for a linear operator,” Communications on Pure and Applied Mathematics 7, 649-673, 1954.