Floquet Quasienergy Notebook

This notebook guide turns Floquet theory into a reproducible spectral calculation. Starting from a periodic two-level Hamiltonian, it constructs the one-period unitary, extracts its eigenphases with the correct sign and branch convention, tracks quasienergy branches through a resonance, and verifies that powers of the one-period map reproduce stroboscopic dynamics.

The formal theorem belongs to Floquet Theorem in Quantum Mechanics. The meanings of Floquet zones, gauge-equivalent representatives, and avoided crossings belong to Quasienergies. This page owns the numerical workflow and its validation tests. In particular, it treats an eigenphase as a point on the unit circle before assigning it a real-valued quasienergy representative.

The benchmark reuses the circularly driven spin from the Time-Dependent Hamiltonian Notebook. That page establishes the exact rotating-frame propagator and tests time-ordering algorithms. Here the same exact solution becomes an independent standard for Floquet spectral extraction.

Purpose

The notebook should demonstrate that:

• a Floquet operator is a time-ordered one-period propagator, not the exponential of a time-averaged Hamiltonian;
• a unitary eigenvalue determines a quasienergy only modulo the drive quantum;
• the sign relating a numerical complex argument to quasienergy must be handled explicitly;
• a reduced-zone plot can turn a small physical gap into a large vertical separation across the zone boundary;
• branch labels should be transported by eigenvector overlap and phase unwrapping rather than independently sorting every parameter point;
• individual Floquet eigenvectors are not well defined at an exact degeneracy;
• stroboscopic dynamics follows from powers of the one-period unitary, while micromotion requires additional intra-period evolution;
• small one-period operator errors can accumulate over many periods.

The central outputs are the numerical one-period unitary, its unitarity residual, principal and unwrapped eigenphases, reduced- and extended-zone quasienergies, the circular quasienergy gap, branch-overlap diagnostics, and stroboscopic transition probabilities.

Periodically Driven Two-Level Model

Use the laboratory-frame Hamiltonian

$$H(t) = \frac{\hbar}{2} \left[ A\cos(\omega_d t)\sigma_x + A\sin(\omega_d t)\sigma_y + \Delta\sigma_z \right].$$

The drive period and angular frequency are

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

Here $A$ is the transverse coupling, $\Delta$ is the longitudinal angular frequency, and $\omega_d$ is the drive angular frequency. All three have units of inverse time. The detuning and generalized rotating-frame frequency are

$$\delta=\Delta-\omega_d, \qquad q=\sqrt{A^2+\delta^2}.$$

The coupling $A$ will open an avoided crossing at $\delta=0$. Setting $A=0$ provides a control with an exact crossing.

Use the Pauli matrices in the $\sigma_z$ basis and record the basis convention beside the parameters. A silent interchange of basis vectors changes transition labels and Bloch-vector signs without changing the quasienergy spectrum.

Exact One-Period Benchmark

Define

$$R(t) = \exp\left( -\frac{i\omega_d t}{2}\sigma_z \right).$$

The exact rotating-frame derivation is given in the time-dependent notebook. Its result is

$$H_{\rm rot} = \frac{\hbar}{2} \left( A\sigma_x+\delta\sigma_z \right),$$

and

$$U(t,0) = R(t) \exp\left( -\frac{i}{\hbar}H_{\rm rot}t \right).$$

Because a spinor acquires a minus sign under a $2\pi$ rotation,

$$R(T)=-I.$$

The exact Floquet operator at reference phase $t_0=0$ is therefore

$$U_F^{\rm ex} = -\exp\left[ -\frac{iT}{2} \left( A\sigma_x+\delta\sigma_z \right) \right].$$

The eigenvalues of $H_{\rm rot}$ are $s\hbar q/2$, with $s=\pm1$. Hence

$$\lambda_s = \exp\left[ -i\left( \pi+s\frac{qT}{2} \right) \right].$$

One convenient extended-zone choice is

$$\varepsilon_s^{\rm ext} = \frac{\hbar\omega_d}{2} + s\frac{\hbar q}{2}.$$

The two extended branches have separation

$$\Delta\varepsilon^{\rm ext} = \hbar q = \hbar\sqrt{A^2+(\Delta-\omega_d)^2}.$$

At resonance, $\Delta=\omega_d$, the minimum gap is $\hbar\lvert A\rvert$. When $A=0$, the branches cross. This exact expression tests the entire numerical chain from time stepping to branch tracking.

Constructing the One-Period Unitary

Divide one period into $N_t$ equal steps,

$$\Delta t=\frac{T}{N_t}, \qquad t_j^*=\left(j+\frac12\right)\Delta t.$$

For midpoint sampling, define

$$S_j = \exp\left[ -\frac{i}{\hbar} H(t_j^*)\Delta t \right].$$

Column states evolve with the earliest factor on the right, so the numerical Floquet operator is

$$U_F^{(N_t)} = S_{N_t-1} S_{N_t-2} \cdots S_1S_0.$$

Each $S_j$ can be evaluated by the closed Pauli identity

$$\exp\left( -\frac{i\alpha}{2} \widehat{\mathbf n}\cdot\boldsymbol\sigma \right) = \cos\frac{\alpha}{2} I - i\sin\frac{\alpha}{2} \widehat{\mathbf n}\cdot\boldsymbol\sigma.$$

This makes every step unitary to roundoff and avoids hiding the benchmark behind a general matrix-exponential routine. A library matrix exponential should still be used once as an independent cross-check.

Midpoint freezing is globally second order for this smooth Hamiltonian. If

$$e_U(N_t) = \left\| U_F^{(N_t)}-U_F^{\rm ex} \right\|_2,$$

then doubling $N_t$ should reduce $e_U$ by approximately a factor of four before roundoff dominates. The observed order is

$$p_{\rm obs} = \log_2 \frac{e_U(N_t)}{e_U(2N_t)}.$$

Do not infer convergence from unitarity. Reversing the product order also yields a unitary matrix, but it approximates the wrong chronology.

Structural Checks Before Diagonalization

Before extracting eigenphases, report

$$r_{\rm unit} = \left\| U_F^\dagger U_F-I \right\|_2$$

and

$$r_{\det} = \left| \det U_F-1 \right|.$$

The Hamiltonian is traceless, so the exact propagator lies in $SU(2)$ and $\det U_F=1$. These checks diagnose implementation defects, but they do not establish spectral accuracy. Also compare the full operator with $U_F^{\rm ex}$ and verify the composition law across two subintervals.

If an ODE solver rather than exponential steps is used, diagonalize only after measuring its unitarity defect. Projecting an inaccurate matrix to the nearest unitary may improve structure while moving the eigenphases; preserve and report both the raw and projected results.

Extracting Eigenphases Without a Sign Error

Let a numerical eigensolver return

$$U_F\lvert u_a\rangle = \lambda_a\lvert u_a\rangle.$$

For a unitary matrix, $\lvert\lambda_a\rvert=1$ up to roundoff. The Floquet convention is

$$\lambda_a=e^{-i\theta_a}, \qquad \theta_a=\frac{\varepsilon_aT}{\hbar}.$$

Most numerical argument functions return the principal value $\operatorname{Arg}\lambda_a$ in $(-\pi,\pi]$. Therefore the principal Floquet phase is

$$\theta_a^{(0)} = -\operatorname{Arg}\lambda_a,$$

and the associated reduced-zone quasienergy is

$$\varepsilon_a^{(0)} = -\frac{\hbar}{T} \operatorname{Arg}\lambda_a.$$

Dropping the minus sign reverses the spectrum. A test with a static diagonal Hamiltonian catches this error immediately.

Normalize every eigenvector and verify both the eigenpair residual and orthogonality:

$$r_a = \left\| U_F\lvert u_a\rangle - \lambda_a\lvert u_a\rangle \right\|_2,$$ $$r_{\rm orth} = \left\| V^\dagger V-I \right\|_2.$$

A unitary matrix is normal, so an orthonormal eigenbasis exists. Close to degeneracy, however, individual numerical eigenvectors can rotate sharply inside the nearly degenerate subspace even when the invariant subspace is accurate.

Reduced Zones and Circular Gaps

Quasienergy representatives satisfy

$$\varepsilon_a \sim \varepsilon_a+m\hbar\omega_d, \qquad m\in\mathbb Z.$$

Choose and state a reduced zone, such as

$$-\frac{\hbar\omega_d}{2} \lt \varepsilon \le \frac{\hbar\omega_d}{2}.$$

For the baseline below, the two exact extended representatives are $0.375$ and $0.625$ in units with $\hbar=\omega_d=1$. Folding gives $0.375$ and $-0.375$. Their naive vertical separation is $0.75$, but their physical separation around the quasienergy circle is $0.25$.

Compute the phase separation on the unit circle:

$$d_{S^1}(\theta_a,\theta_b) = \left| \operatorname{Arg} e^{i(\theta_a-\theta_b)} \right|.$$

The branch-independent two-level gap is

$$\Delta\varepsilon_{S^1} = \frac{\hbar}{T} d_{S^1}(\theta_1,\theta_2).$$

This quantity returns $\hbar q$ throughout the scan as long as $q\le\omega_d$. A straight-line difference between reduced-zone representatives does not.

Scanning the Avoided Crossing

Fix $\omega_d$ and scan $\Delta$ through resonance for several couplings $A$. At each parameter point:

1. construct $U_F$ over one period;
2. check unitarity and convergence in $N_t$;
3. diagonalize $U_F$;
4. compute principal phases and reduced-zone quasienergies;
5. transport the branch labels from the preceding point;
6. unwrap each transported phase into an extended-zone branch;
7. compare the circular gap with $\hbar q$.

For $A=0$, the extended branches cross at $\Delta=\omega_d$. For $A\ne0$, the coupling opens an avoided crossing with minimum gap $\hbar\lvert A\rvert$. The Floquet eigenstates exchange their dominant $\sigma_z$ character across the avoided crossing. A color scale showing $\langle\sigma_z\rangle$ makes this exchange visible without relabeling the continuous adiabatic branches.

Plot reduced-zone and extended-zone spectra side by side. The extended plot displays the avoided crossing around $\hbar\omega_d/2$. In the symmetric reduced zone, the same two levels lie on opposite sides of the zone boundary. Neither representation is more physical; the eigenvalues on the unit circle are the invariant data.

Tracking Branches Responsibly

Independent sorting by quasienergy at every parameter point fails at zone boundaries and can swap labels near small gaps. Suppose $V_j$ and $V_{j+1}$ contain normalized Floquet eigenvectors at adjacent scan points. Form the overlap matrix

$$O_{ab} = \left| \langle u_a(p_j)|u_b(p_{j+1})\rangle \right|^2.$$

Choose the permutation at $p_{j+1}$ that maximizes the total overlap. For two states, compare the identity assignment with the swap. After assignment, fix each arbitrary eigenvector phase by multiplying the new vector so that

$$\langle u_a(p_j)|u_a(p_{j+1})\rangle$$

is real and nonnegative.

Then add an integer multiple of $2\pi$ to the new principal phase so that its change from the previous unwrapped phase has minimum magnitude. The unwrapped quasienergy is

$$\varepsilon_a^{\rm ext} = \frac{\hbar}{T} \left( \theta_a^{(0)}+2\pi m_a \right).$$

Record the smallest accepted overlap. A sudden decrease warns that the parameter mesh is too coarse or that a degeneracy has been approached.

At the exact crossing $A=0$ and $\Delta=\omega_d$, $U_F=-I$. Every vector is an eigenvector, so no algorithm can extract unique individual branches from that point. Skip the degenerate point when continuing one-dimensional labels, or track the projector onto the full degenerate subspace. Arbitrary eigensolver output at a degeneracy is not a physical state selection.

Drive Reference Phase

The one-period operator depends on the phase $t_0$ at which the drive is sampled:

$$U_F(t_0) = U(t_0+T,t_0).$$

For a shift $s$, define $V_s=U(t_0+s,t_0)$. Periodicity gives

$$U_F(t_0+s) = V_sU_F(t_0)V_s^\dagger.$$

Thus the eigenvalues and quasienergies are invariant, while the eigenvectors are transported by $V_s$. Repeat the one-period calculation at $t_0=0$, $T/5$, and $0.37T$. Compare unordered eigenvalue sets rather than raw eigensolver ordering. Then verify one transported eigenvector after fixing its phase.

This test catches a common implementation error: integrating from a shifted $t_0$ while resetting the Hamiltonian’s drive phase to zero.

Stroboscopic Dynamics

At integer periods,

$$\lvert\psi_n\rangle = \lvert\psi(nT)\rangle = U_F^n\lvert\psi_0\rangle.$$

If the Floquet eigenvectors form the columns of $V$ and

$$\lvert\psi_0\rangle = \sum_a c_a\lvert u_a\rangle,$$

then

$$\lvert\psi_n\rangle = \sum_a c_a e^{-in\theta_a} \lvert u_a\rangle.$$

Verify agreement among three routes:

• repeated multiplication by the numerical $U_F$;
• spectral reconstruction from its eigenpairs;
• direct time stepping over the entire interval $[0,nT]$ without restarting the drive phase.

For the initial state $\lvert0\rangle$, the exact rotating-frame result is

$$P_{0\to1}(nT) = \frac{A^2}{q^2} \sin^2\left( \frac{nqT}{2} \right).$$

On resonance, this reduces to

$$P_{0\to1}(nT) = \sin^2\left( \frac{nAT}{2} \right).$$

The global spinor factor $(-1)^n$ cancels from this probability but remains present in the operator.

Long-time agreement is stricter than a one-period test. For unitary matrices $U$ and $\widetilde U$,

$$\left\| \widetilde U^n-U^n \right\|_2 \le n\left\| \widetilde U-U \right\|_2.$$

The bound can be loose, but it explains why a harmless-looking phase error per period can become a large timing error after many cycles. Plot the stroboscopic error against both $n$ and $N_t$.

Micromotion Is Additional Data

For $t=nT+\tau$ with $0\le\tau\lt T$,

$$U(nT+\tau,0) = U(\tau,0)U_F^n.$$

Powers of $U_F$ determine only the sampled states at $\tau=0$. To reconstruct motion inside a period, store $U(\tau,0)$ on an intra-period mesh or compute the periodic micromotion operator. Compare a Bloch trajectory sampled only at $nT$ with the continuous trajectory over one period. Large loops between identical or slowly varying stroboscopic points are not numerical noise; they are micromotion.

The canonical interpretation and aliasing issues are developed in Stroboscopic Dynamics. The notebook should not infer a unique continuous Hamiltonian from $U_F$ alone.

Baseline Parameters

Set

$$\hbar=1, \qquad \omega_d=1, \qquad A=0.2.$$

For the convergence benchmark, use $\Delta=0.85$. Then

$$\delta=-0.15, \qquad q=0.25,$$

and the exact extended-zone representatives are

$$\varepsilon_-^{\rm ext}=0.375, \qquad \varepsilon_+^{\rm ext}=0.625.$$

In the symmetric reduced zone they become $0.375$ and $-0.375$, while the circular gap is $0.25$.

Use

$$N_t=16,32,64,128,256,512.$$

With midpoint exponential steps, the expected operator errors are approximately

$N_t$	$\lVert U_F^{(N_t)}-U_F^{\rm ex}\rVert_2$
$16$	$3.24\times10^{-3}$
$32$	$8.12\times10^{-4}$
$64$	$2.03\times10^{-4}$
$128$	$5.08\times10^{-5}$
$256$	$1.27\times10^{-5}$
$512$	$3.17\times10^{-6}$

The ratios exhibit second-order convergence. The unitarity residual should remain near floating-point roundoff rather than following the operator-error column.

For the avoided-crossing scan, use

$$0.6\le\frac{\Delta}{\omega_d}\le1.4$$

with enough points to resolve the minimum gap. At $\Delta/\omega_d=0.8,0.9,1.0,1.1,1.2$, the exact gaps for $A=0.2$ are approximately

$$0.282843, \quad 0.223607, \quad 0.200000, \quad 0.223607, \quad 0.282843.$$

Notebook Workflow

Organize the calculation into reproducible stages:

1. define Pauli matrices, units, basis order, and model parameters;
2. verify Hermiticity, tracelessness, and period $T$;
3. evaluate the exact rotating-frame Floquet operator;
4. construct midpoint one-period products for a refinement sequence;
5. measure operator error, unitarity, determinant, and observed order;
6. diagonalize the converged operator and test every eigenpair;
7. convert complex arguments to principal Floquet phases with the minus sign;
8. compare reduced-zone, extended-zone, and circular-gap descriptions;
9. scan detuning and track branches by overlap plus phase unwrapping;
10. repeat at shifted drive reference phases;
11. compare repeated-map, spectral, direct-integration, and analytic stroboscopic dynamics;
12. sample one intra-period trajectory to expose micromotion;
13. export parameters, tolerances, diagnostics, and package versions with any plot data.

Store raw eigenvalues as complex numbers or as sine-cosine pairs. A table containing only branch-cut-dependent quasienergy representatives cannot reconstruct the invariant spectrum reliably.

Validation Matrix

Check	Target
Hamiltonian period	$H(t+T)=H(t)$ at sampled times
One-period unitarity	$\lVert U_F^\dagger U_F-I\rVert_2$ near roundoff
Determinant	$\det U_F\simeq1$
Operator convergence	midpoint slope approaches two
Exact spectrum	numerical eigenvalues approach $\lambda_s$
Eigenpair residuals	small for both branches
Phase convention	static control confirms $\varepsilon=-\hbar\operatorname{Arg}\lambda/T$
Circular gap	approaches $\hbar q$
Resonant minimum	approaches $\hbar\lvert A\rvert$
Zero-coupling control	crossing restored for $A=0$
Branch transport	adjacent overlaps remain near one away from degeneracy
Reference phase	eigenvalue set independent of $t_0$
Stroboscopic dynamics	three numerical routes agree with the analytic probability
Long-time refinement	error decreases with $N_t$ at fixed period count
Micromotion	intra-period path differs from boundary-only samples

Common Numerical Failures

• Using $\operatorname{Arg}\lambda$ instead of $-\operatorname{Arg}\lambda$. This reverses every quasienergy representative.
• Sorting principal quasienergies independently. Labels jump at a Floquet-zone boundary.
• Measuring a gap by vertical distance in a reduced-zone plot. Levels separated across the boundary are close on the unit circle.
• Applying a generic matrix logarithm without stating its branch. The resulting effective Hamiltonian is branch dependent and can jump discontinuously.
• Declaring convergence because $U_F$ is unitary. Structure preservation and approximation accuracy are different tests.
• Restarting the drive phase every period during direct integration. This happens to preserve a strictly periodic drive at exact boundaries but fails for shifted starts and intra-period comparisons.
• Comparing raw eigenvector columns. Eigensolvers may permute columns and multiply each vector by an arbitrary phase.
• Forcing unique eigenvectors at a degeneracy. Only the degenerate invariant subspace is defined.
• Ignoring the spinor sign $R(T)=-I$. Dropping it shifts both quasienergies by $\hbar\omega_d/2$ and changes the operator.
• Raising a nonunitary approximation to a large power. Tiny radial eigenvalue errors can create artificial decay or growth.
• Reporting only one-period probabilities. A wrong relative eigenphase may be hard to see after one cycle and decisive after many.
• Calling $U_F^n$ the complete trajectory. It contains no information about motion between sampled phases.

Extensions

• Replace the circular drive by an elliptically polarized field and compare with the exact benchmark as ellipticity departs from one.
• Construct a branch-dependent effective Floquet Hamiltonian spectrally and verify $e^{-iH_FT/\hbar}=U_F$.
• Compare midpoint products with a commutator-corrected Magnus integrator.
• Add weak dephasing and distinguish unitary Floquet eigenphases from the spectrum of a one-period quantum channel.
• Scan both detuning and coupling to map the conical degeneracy at $A=\delta=0$ before Floquet-zone identification.
• Use adaptive parameter refinement when the minimum branch overlap falls below a declared threshold.

Cross-Links

• Periodic Hamiltonians
• Floquet Theorem in Quantum Mechanics
• Floquet Operators
• Quasienergies
• Rotating Frames
• Stroboscopic Dynamics
• Time-Dependent Hamiltonian Notebook
• Matrix Diagonalization
• Convergence Tests
• Validation Tests

References

• J. H. Shirley, “Solution of the Schrödinger Equation with a Hamiltonian Periodic in Time,” Physical Review 138, B979–B987 (1965), doi:10.1103/PhysRev.138.B979.
• H. Sambe, “Steady States and Quasienergies of a Quantum-Mechanical System in an Oscillating Field,” Physical Review A 7, 2203–2213 (1973), doi:10.1103/PhysRevA.7.2203.
• M. Grifoni and P. Hänggi, “Driven Quantum Tunneling,” Physics Reports 304, 229–354 (1998), doi:10.1016/S0370-1573(98)00022-2.
• A. Eckardt, “Colloquium: Atomic Quantum Gases in Periodically Driven Optical Lattices,” Reviews of Modern Physics 89, 011004 (2017), doi:10.1103/RevModPhys.89.011004.

Exercises

1. Derive the exact quasienergies

Starting from the rotating-frame propagator, derive $U_F^{\rm ex}$ and the extended-zone representatives $\varepsilon_s^{\rm ext}$.

Solution

At one period,

$$R(T) = e^{-i\pi\sigma_z} = -I.$$

The rotating-frame Hamiltonian has eigenvalues $s\hbar q/2$, so

$$U_F^{\rm ex} = -e^{-iH_{\rm rot}T/\hbar}$$

has eigenvalues

$$\lambda_s = -e^{-isqT/2} = e^{-i(\pi+sqT/2)}.$$

Since $\pi/T=\omega_d/2$, one representative satisfying $\lambda_s=e^{-i\varepsilon_sT/\hbar}$ is

$$\varepsilon_s^{\rm ext} = \frac{\hbar\omega_d}{2} + s\frac{\hbar q}{2}.$$

Adding $m\hbar\omega_d$ gives equivalent representatives.

2. Fix the argument sign

A numerical routine returns $\operatorname{Arg}\lambda=-0.6$ for a Floquet eigenvalue, using $\hbar=1$ and $T=2$. What is the principal quasienergy representative?

Solution

The convention is $\lambda=e^{-i\varepsilon T/\hbar}$, so

$$\varepsilon^{(0)} = -\frac{\hbar}{T} \operatorname{Arg}\lambda = -\frac{1}{2}(-0.6) = 0.3.$$

Using $+\operatorname{Arg}\lambda/T$ would produce the sign-reversed value.

3. Compute a circular gap

In units with $\hbar=\omega_d=1$, two reduced-zone quasienergies are $0.375$ and $-0.375$. Show that their circular gap is $0.25$, not $0.75$.

Solution

The period is $T=2\pi$. Their phase difference before reduction modulo $2\pi$ is

$$T(0.375-(-0.375)) = 1.5\pi.$$

The shortest distance on the unit circle is

$$2\pi-1.5\pi = 0.5\pi.$$

Therefore

$$\Delta\varepsilon_{S^1} = \frac{0.5\pi}{2\pi} = 0.25.$$

Equivalently, shift one representative by one drive quantum: $-0.375+1=0.625$, whose difference from $0.375$ is $0.25$.

4. Track two branches by overlap

At adjacent parameter values, the overlap matrix is

$$O = \begin{pmatrix} 0.08 & 0.92\\ 0.91 & 0.09 \end{pmatrix}.$$

Which assignment should be chosen, and what does the matrix say about independent energy sorting?

Solution

The identity assignment has total overlap

$$0.08+0.09=0.17,$$

whereas the swapped assignment has

$$0.92+0.91=1.83.$$

The columns at the new parameter point should therefore be swapped. Independent sorting has changed the column labels even though the physical eigenvectors continue smoothly. After swapping, each new vector should also be rephased so that its overlap with the preceding vector is real and nonnegative.

5. Derive the stroboscopic transition probability

For $\lvert\psi_0\rangle=\lvert0\rangle$, derive

$$P_{0\to1}(nT) = \frac{A^2}{q^2} \sin^2\left( \frac{nqT}{2} \right).$$

Solution

At integer periods, $R(nT)=(-1)^nI$, which contributes only a global phase. The remaining evolution is generated by

$$H_{\rm rot} = \frac{\hbar}{2} \left( A\sigma_x+\delta\sigma_z \right).$$

Using the Pauli exponential,

$$e^{-inH_{\rm rot}T/\hbar} = \cos\frac{nqT}{2}I - i\sin\frac{nqT}{2} \frac{A\sigma_x+\delta\sigma_z}{q}.$$

Only the $\sigma_x$ term connects $\lvert0\rangle$ to $\lvert1\rangle$. The transition amplitude has magnitude

$$\frac{\lvert A\rvert}{q} \left| \sin\frac{nqT}{2} \right|,$$

and squaring gives the stated result.

6. Shift the drive reference phase

Show that $U_F(t_0+s)$ and $U_F(t_0)$ have the same eigenvalues, and determine how a Floquet eigenvector is transported.

Solution

Let

$$V_s=U(t_0+s,t_0).$$

Periodicity and composition imply

$$U_F(t_0+s) = V_sU_F(t_0)V_s^\dagger.$$

If

$$U_F(t_0)\lvert u_a(t_0)\rangle = \lambda_a\lvert u_a(t_0)\rangle,$$

then

$$U_F(t_0+s) V_s\lvert u_a(t_0)\rangle = \lambda_a V_s\lvert u_a(t_0)\rangle.$$

The eigenvalue is unchanged, while the eigenvector at the shifted reference phase is $V_s\lvert u_a(t_0)\rangle$ up to an arbitrary phase and any freedom inside a degenerate eigenspace.