Floquet Theorem in Quantum Mechanics

The periodic setup, propagator covariance, and motivating examples are developed in Periodic Hamiltonians. Floquet theory is the spectral theory of quantum systems whose Hamiltonian is periodic in time:

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

For a closed system, the state satisfies

$$i\hbar\frac{d}{dt}\lvert\psi(t)\rangle = H(t)\lvert\psi(t)\rangle,$$

and the full time evolution is generated by the time-evolution operator $U(t,t_0)$. The Floquet theorem says that solutions may be organized into a phase factor with a constant quasienergy and a periodic state vector:

$$\lvert\psi_\alpha(t)\rangle = e^{-i\varepsilon_\alpha(t-t_0)/\hbar} \lvert u_\alpha(t)\rangle, \qquad \lvert u_\alpha(t+T)\rangle = \lvert u_\alpha(t)\rangle.$$

The periodic factor $\lvert u_\alpha(t)\rangle$ is the micromotion. The number $\varepsilon_\alpha$ is a quasienergy, not an ordinary conserved energy. It is defined only modulo $\hbar\Omega$.

The theorem is exact for few-body and many-body systems alike. Floquet Systems Preview owns the additional thermodynamic questions: exponentially dense quasienergies, generic heating, prethermal control, and interacting Floquet phases.

Analogy with Bloch Theorem

Floquet theory is often described as the time-domain analogue of Bloch theory. For a spatially periodic crystal potential, a Bloch wave has the form

$$\psi_{n k}(x) = e^{ikx}u_{n k}(x), \qquad u_{n k}(x+a)=u_{n k}(x).$$

The wave is a plane-wave phase factor times a spatially periodic function. For a time-periodic Hamiltonian, a Floquet solution has the parallel form

$$\lvert\psi_\alpha(t)\rangle = e^{-i\varepsilon_\alpha t/\hbar} \lvert u_\alpha(t)\rangle, \qquad \lvert u_\alpha(t+T)\rangle = \lvert u_\alpha(t)\rangle.$$

The analogy is useful, but not perfect. Bloch momentum is defined modulo a reciprocal lattice vector; quasienergy is defined modulo $\hbar\Omega$. Bloch theory concerns spatial translations; Floquet theory concerns time translation by one drive period. See Bloch Theorem for the spatial version.

One-Period Evolution

The central object is the one-period evolution operator

$$U_F(t_0) \equiv U(t_0+T,t_0).$$

For a closed system, $U_F(t_0)$ is unitary. Its eigenvalues lie on the unit circle, so they can be written as phases:

$$U_F(t_0)\lvert u_\alpha(t_0)\rangle = e^{-i\varepsilon_\alpha T/\hbar} \lvert u_\alpha(t_0)\rangle.$$

This eigenvalue equation defines the Floquet states at the reference phase $t_0$ and the corresponding quasienergies. The reference phase matters for the representative eigenvectors, but the eigenphases of $U_F(t_0)$ are invariant under shifting $t_0$ when the evolution is related by unitary conjugation.

Once an eigenvector at $t_0$ is chosen, define

$$\lvert u_\alpha(t)\rangle = e^{i\varepsilon_\alpha(t-t_0)/\hbar} U(t,t_0)\lvert u_\alpha(t_0)\rangle.$$

Then the associated physical solution is

$$\lvert\psi_\alpha(t)\rangle = U(t,t_0)\lvert u_\alpha(t_0)\rangle = e^{-i\varepsilon_\alpha(t-t_0)/\hbar} \lvert u_\alpha(t)\rangle.$$

This is the closed-system quantum Floquet theorem in its most useful form.

Why the Micromotion Is Periodic

The periodicity of $H(t)$ implies

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

Using the composition law,

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

Acting on a Floquet eigenvector gives

$$U(t+T,t_0)\lvert u_\alpha(t_0)\rangle = e^{-i\varepsilon_\alpha T/\hbar} U(t,t_0)\lvert u_\alpha(t_0)\rangle.$$

Therefore

$$\begin{aligned} \lvert u_\alpha(t+T)\rangle &= e^{i\varepsilon_\alpha(t+T-t_0)/\hbar} U(t+T,t_0)\lvert u_\alpha(t_0)\rangle \\ &= e^{i\varepsilon_\alpha(t-t_0)/\hbar} U(t,t_0)\lvert u_\alpha(t_0)\rangle \\ &= \lvert u_\alpha(t)\rangle. \end{aligned}$$

The theorem is therefore not a guess about the solution form. It follows from the spectral decomposition of one-period unitary evolution.

Floquet Eigenvalue Equation

Substituting the Floquet form

$$\lvert\psi_\alpha(t)\rangle = e^{-i\varepsilon_\alpha t/\hbar} \lvert u_\alpha(t)\rangle$$

into the Schrödinger equation gives

$$\bigl(H(t)-i\hbar\partial_t\bigr) \lvert u_\alpha(t)\rangle = \varepsilon_\alpha \lvert u_\alpha(t)\rangle.$$

This is sometimes called the Floquet eigenvalue equation. It acts on $T$-periodic Hilbert-space-valued functions. A useful inner product on this extended space is

$$\langle\!\langle u|v\rangle\!\rangle = \frac{1}{T} \int_0^T \langle u(t)|v(t)\rangle\,dt.$$

This formulation makes the analogy with time-independent spectral theory vivid, but it must be handled with care. The operator $H(t)-i\hbar\partial_t$ acts on periodic functions, and its eigenvalues inherit the modular quasienergy structure.

Quasienergies Modulo Drive Frequency

The eigenvalue of $U_F$ is a phase. Therefore

$$e^{-i\varepsilon_\alpha T/\hbar} = e^{-i(\varepsilon_\alpha+m\hbar\Omega)T/\hbar}$$

for any integer $m$, because $\Omega T=2\pi$. Thus

$$\varepsilon_\alpha \sim \varepsilon_\alpha+m\hbar\Omega, \qquad m\in\mathbb Z.$$

Changing branches can be absorbed into the periodic factor:

$$\varepsilon_\alpha' = \varepsilon_\alpha+m\hbar\Omega, \qquad \lvert u_\alpha'(t)\rangle = e^{im\Omega(t-t_0)} \lvert u_\alpha(t)\rangle.$$

Then

$$e^{-i\varepsilon_\alpha'(t-t_0)/\hbar} \lvert u_\alpha'(t)\rangle = e^{-i\varepsilon_\alpha(t-t_0)/\hbar} \lvert u_\alpha(t)\rangle.$$

The physical state is unchanged. A choice of quasienergy “zone” is therefore like a branch choice for the logarithm of a unitary phase.

Effective Hamiltonian and Stroboscopic Motion

Choose a branch of the logarithm and define a stroboscopic Floquet Hamiltonian by

$$H_F(t_0) = \frac{i\hbar}{T}\log U_F(t_0).$$

Then

$$U_F(t_0) = e^{-iH_F(t_0)T/\hbar}.$$

At integer numbers of periods,

$$U(t_0+nT,t_0) = U_F(t_0)^n = e^{-iH_F(t_0)nT/\hbar}.$$

This is stroboscopic dynamics: it describes the state only at the same phase of the drive. It does not describe what happens inside a period unless the micromotion is also included.

The full evolution can be decomposed as

$$U(t,t_0) = P(t,t_0) e^{-iH_F(t_0)(t-t_0)/\hbar},$$

where the micromotion operator is periodic:

$$P(t+T,t_0)=P(t,t_0).$$

The logarithm is branch-dependent, so $H_F$ is not unique. The direct one-period series belongs to Floquet–Magnus Expansion; the general inverse-frequency method, including van Vleck and prethermal constructions, belongs to High-Frequency Expansions.

General State Expansion

In a finite-dimensional closed system, the unitary $U_F(t_0)$ has an orthonormal eigenbasis if degeneracies are resolved by choosing an orthonormal basis within each degenerate subspace. A general initial state can be expanded as

$$\lvert\psi(t_0)\rangle = \sum_\alpha c_\alpha \lvert u_\alpha(t_0)\rangle.$$

The time-evolved state is then

$$\lvert\psi(t)\rangle = \sum_\alpha c_\alpha e^{-i\varepsilon_\alpha(t-t_0)/\hbar} \lvert u_\alpha(t)\rangle.$$

This resembles expansion in energy eigenstates, but the analogy has limits:

• $\varepsilon_\alpha$ is modular;
• $\lvert u_\alpha(t)\rangle$ is time periodic;
• the drive can exchange energy with the system;
• avoided crossings and resonances often organize the quasienergy spectrum.

In infinite-dimensional systems, domain questions and spectral subtleties can become important. The finite-dimensional statement is the clean reference case.

Simple Examples

Time-Independent Hamiltonian

If $H(t)=H_0$, then the system is periodic for any chosen $T$. Energy eigenstates

$$H_0\lvert E_n\rangle = E_n\lvert E_n\rangle$$

give

$$U_F\lvert E_n\rangle = e^{-iE_nT/\hbar} \lvert E_n\rangle.$$

Thus $E_n$ is a quasienergy representative, but only modulo $\hbar\Omega$. Floquet theory reduces to ordinary stationary dynamics plus an artificial branch structure introduced by the chosen period.

Commuting Periodic Hamiltonian

Suppose

$$H(t)=f(t)A, \qquad [H(t),H(t')]=0$$

for all times. Then

$$U(T,0) = \exp\left[ -\frac{i}{\hbar} A\int_0^T f(t)\,dt \right].$$

If $A\lvert a\rangle=a\lvert a\rangle$, then

$$U_F\lvert a\rangle = \exp\left[ -\frac{ia}{\hbar} \int_0^T f(t)\,dt \right] \lvert a\rangle.$$

The quasienergy representative is the period average of the instantaneous eigenvalue:

$$\varepsilon_a = \frac{a}{T} \int_0^T f(t)\,dt \quad \text{mod }\hbar\Omega.$$

This is a special case. Most driven systems have noncommuting Hamiltonians at different times, so one must use time ordering.

Driven Two-Level System

A near-resonantly driven two-level system is a standard Floquet problem:

$$H(t) = \frac{\hbar\omega_0}{2}\sigma_z + \hbar\Omega_R\cos(\omega t)\sigma_x.$$

The rotating-wave approximation often converts the near-resonant dynamics into an effective time-independent problem in a rotating frame. Floquet theory instead treats the exact periodic Hamiltonian through $U_F$. The two descriptions overlap in weak-drive, near-resonant regimes, but they are not the same approximation. See Rabi Oscillations: First Encounter and Rotating-Wave Approximation.

Limitations and Cautions

Floquet theory is powerful, but several cautions matter:

• Exact periodicity is essential. Slowly drifting, noisy, or pulsed drives need additional approximations.
• Closed-system Floquet theory assumes unitary evolution. Open systems require a Floquet analysis of a master equation or channel, not just a Hamiltonian.
• Quasienergies are modular. There is no absolute quasienergy ground state without extra structure.
• The effective Hamiltonian depends on a logarithm branch and on the chosen drive phase $t_0$.
• Stroboscopic dynamics can look simple while intra-period micromotion is large.
• High-frequency expansions are approximations, not part of the theorem itself.
• In many-body systems, long-time heating and prethermal regimes require physics beyond the finite-dimensional theorem.

The common quasienergy pitfall is summarized in Common Pitfalls.

Cross-Links

• Time-Dependent Hamiltonians sets up driven closed systems.
• Time-Evolution Operator defines $U(t,t_0)$ and its composition law.
• Time Ordering explains the ordered exponential needed for noncommuting drives.
• Floquet Operators focuses on the one-period unitary, branch choices, stroboscopic observables, and numerical construction.
• Quasienergies focuses on modular spectra, Floquet zones, resonances, and avoided crossings.
• Formula Sheet lists the core Floquet formulas.
• Floquet–Magnus Expansion develops the direct one-period Magnus representative.
• High-Frequency Expansions develops inverse-frequency effective Hamiltonians, micromotion, resonance tests, and prethermal validity.
• Rabi Oscillations: First Encounter gives the canonical driven two-level example.
• Dressed States separates a classical Floquet harmonic index from physical photon number and connects the near-resonant Floquet block to quantized atom–mode doublets.

References

• G. Floquet, “Sur les équations différentielles linéaires à coefficients périodiques,” Annales scientifiques de l’École Normale Supérieure 12, 47, 1883.
• J. H. Shirley, “Solution of the Schrödinger equation with a Hamiltonian periodic in time,” Physical Review 138, B979, 1965.
• H. Sambe, “Steady states and quasienergies of a quantum-mechanical system in an oscillating field,” Physical Review A 7, 2203, 1973.
• M. Grifoni and P. Hänggi, “Driven quantum tunneling,” Physics Reports 304, 229, 1998.
• 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, 2015.

Exercises

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

Solution

The eigenvalue of $U_F$ is

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

If $\varepsilon'=\varepsilon+m\hbar\Omega$, then

$$e^{-i\varepsilon'T/\hbar} = e^{-i\varepsilon T/\hbar} e^{-im\Omega T} = e^{-i\varepsilon T/\hbar} e^{-i2\pi m} = e^{-i\varepsilon T/\hbar}.$$

Thus the same unitary eigenvalue corresponds to all representatives $\varepsilon+m\hbar\Omega$.

2. Starting from an eigenvector of $U_F(t_0)$, show that the associated micromotion is periodic.

Solution

Let

$$U_F(t_0)\lvert u(t_0)\rangle = e^{-i\varepsilon T/\hbar}\lvert u(t_0)\rangle$$

and define

$$\lvert u(t)\rangle = e^{i\varepsilon(t-t_0)/\hbar} U(t,t_0)\lvert u(t_0)\rangle.$$

Using periodicity and composition,

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

Therefore

$$\begin{aligned} \lvert u(t+T)\rangle &= e^{i\varepsilon(t+T-t_0)/\hbar} U(t+T,t_0)\lvert u(t_0)\rangle \\ &= e^{i\varepsilon(t+T-t_0)/\hbar} U(t,t_0)e^{-i\varepsilon T/\hbar} \lvert u(t_0)\rangle \\ &= \lvert u(t)\rangle. \end{aligned}$$

3. Treat a time-independent Hamiltonian as a Floquet problem. What are the quasienergies?

Solution

For $H(t)=H_0$, choose any period $T$. If

$$H_0\lvert E_n\rangle=E_n\lvert E_n\rangle,$$

then

$$U_F\lvert E_n\rangle = e^{-iE_nT/\hbar}\lvert E_n\rangle.$$

Thus $E_n$ is a quasienergy representative. Because only the phase is fixed, the full equivalence class is

$$E_n+m\hbar\Omega, \qquad m\in\mathbb Z.$$

4. Why does a stroboscopic Hamiltonian not determine all intra-period motion by itself?

Solution

The stroboscopic Hamiltonian is defined from

$$U_F=U(T,0) = e^{-iH_FT/\hbar}.$$

It fixes

$$U(nT,0)=U_F^n,$$

so it describes motion observed once per period. But for times between $0$ and $T$, the evolution also depends on the micromotion operator $P(t)$:

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

Different periodic micromotion can give the same one-period operator while producing different intra-period observables.

5. Explain why Floquet theory does not imply energy conservation.

Solution

Energy conservation follows from time-translation symmetry under arbitrary continuous time shifts. A periodic Hamiltonian has only discrete time-translation symmetry:

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

The conserved spectral object associated with one-period evolution is the eigenphase of $U_F$, usually written as a quasienergy modulo $\hbar\Omega$. The system can still exchange energy with the external drive, so ordinary energy need not be conserved.