Floquet-Magnus Expansion

The Floquet–Magnus expansion applies the Magnus Expansion directly to one period of a periodically driven Hamiltonian. It constructs a stroboscopic generator tied to a chosen starting phase $t_0$. High-Frequency Expansions owns the broader inverse-frequency method, including the van Vleck effective Hamiltonian, Floquet-space block diagonalization, kick operators, resonant denominators, and prethermal validity.

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.

For periodic pulse control, Average Hamiltonian Theory uses the same one-cycle logarithm after transforming to the control toggling frame. It owns interval weights, pulse-order commutators, NMR selective averaging, and cycle symmetry; this page owns the direct Floquet–Magnus representative and its dependence on the chosen period origin.

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.

The full evolution can be written schematically as

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

where $P(t)$ is a periodic micromotion operator 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

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

with

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

and

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

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.

Heating and Validity Caveats

In a bounded few-level problem, a small ratio such as

$$\frac{\|H_m\|}{\hbar\Omega} \ll 1$$

for bounded few-level systems. In many-body systems, the situation is more subtle. Even if local terms are small compared with $\hbar\Omega$, rare high-order processes can eventually absorb energy from the drive.

Common regimes include:

• a rapidly convergent or asymptotic expansion for short and intermediate times,
• a long prethermal window governed by an approximate $H_F$,
• eventual heating in generic interacting systems,
• exact or approximate stability in special constrained, localized, or integrable settings.

Thus a Floquet–Magnus Hamiltonian can be extremely useful without being an eternal exact Hamiltonian. The local energy scale, resonant denominators, optimal truncation, and exponentially long prethermal window are developed on High-Frequency Expansions.

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.

Common Mistakes

• Identifying $H_F$ with the instantaneous Hamiltonian.
• Ignoring quasienergy branch choices.
• Forgetting micromotion when comparing with observables inside a period.
• Applying a high-frequency expansion near a strong resonance.
• Assuming a many-body Floquet-Magnus expansion guarantees no heating at arbitrarily late times.
• Dropping commutators when the Fourier components do not commute.
• Summing a positive-harmonic commutator formula over both signs and thereby double counting it.

Magnus Expansion Error supplies an exact two-step Floquet benchmark for truncation order, unitarity, the sufficient norm condition, repeated-period error, and equivalent logarithm branches.

Exercises

1. Show that quasienergies are defined modulo $\hbar\Omega$.

Solution

The Floquet eigenvalue is

$$e^{-i\epsilon T/\hbar}.$$

Replace $\epsilon$ by $\epsilon+m\hbar\Omega$. Since $\Omega T=2\pi$,

$$\begin{aligned} e^{-i(\epsilon+m\hbar\Omega)T/\hbar} &= e^{-i\epsilon T/\hbar} e^{-im\Omega T} \\ &= e^{-i\epsilon T/\hbar} e^{-i2\pi m} \\ &= e^{-i\epsilon T/\hbar}. \end{aligned}$$

Thus the same Floquet eigenvalue corresponds to quasienergies differing by integer multiples of $\hbar\Omega$.

2. For $H(t)=H_0+V e^{i\Omega t}+V^\dagger e^{-i\Omega t}$, compute the phase-independent order-$\Omega^{-1}$ correction and state the additional term in the direct Floquet–Magnus representative at phase $t_0$.

Solution

The only nonzero Fourier components with $m\ne0$ are

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

The phase-independent van Vleck correction is

$$H_{\mathrm{eff}}^{(1)} = \frac{[H_1,H_{-1}]}{\hbar\Omega} = \frac{[V,V^\dagger]}{\hbar\Omega}.$$

The direct Floquet–Magnus Hamiltonian also contains

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

This extra term changes with the period origin and represents micromotion dressing of the stroboscopic generator.

3. Why is micromotion relevant even when the stroboscopic Hamiltonian is known?

Solution

The stroboscopic Hamiltonian reproduces $U(nT,0)$ at integer periods. At intermediate times,

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

where $P(t)$ is periodic. Observables measured within the period depend on $P(t)$. Therefore $H_F$ alone is sufficient only for questions sampled at a fixed drive phase or for observables insensitive to intra-period motion.

References

• W. Magnus, “On the exponential solution of differential equations for a linear operator,” Communications on Pure and Applied Mathematics 7, 649-673, 1954.
• S. Blanes, F. Casas, J. A. Oteo, and J. Ros, “The Magnus expansion and some of its applications,” Physics Reports 470, 151-238, 2009.
• A. Eckardt, “Colloquium: Atomic quantum gases in periodically driven optical lattices,” Reviews of Modern Physics 89, 011004, 2017.
• 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.
• 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.