Floquet Operators

The Floquet operator is the one-period time-evolution operator of a periodically driven closed quantum system. If

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

then, for a chosen reference time $t_0$,

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

This unitary operator is the central object in practical Floquet theory. Its eigenvectors define Floquet modes at the chosen drive phase, its eigenphases define quasienergies, and its powers generate stroboscopic dynamics. As a discrete-time unitary map, it is part of the broader language in Quantum Maps and Discrete-Time Evolution.

The broader theorem is explained in Floquet Theorem in Quantum Mechanics. This page focuses on the operator $U_F$ itself. Floquet Systems Preview explains why an exact matrix logarithm need not be a useful local many-body Hamiltonian and owns the heating and phase diagnostics.

Definition

The time-evolution operator satisfies

$$i\hbar\frac{\partial}{\partial t}U(t,t_0) = H(t)U(t,t_0), \qquad U(t_0,t_0)=I.$$

The Floquet operator is the evolution through one full period:

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

If $H(t)$ is self-adjoint and the closed-system evolution is well-defined, then $U_F(t_0)$ is unitary:

$$U_F^\dagger(t_0)U_F(t_0)=I.$$

This unitarity is why Floquet eigenvalues are phases rather than arbitrary complex numbers.

For noncommuting Hamiltonians at different times, the formal expression is time ordered:

$$U_F(t_0) = \mathcal T \exp\left[ -\frac{i}{\hbar} \int_{t_0}^{t_0+T} H(t)\,dt \right].$$

This is not usually an ordinary exponential of the averaged Hamiltonian.

Reference Phase

The reference time $t_0$ specifies where in the drive cycle the stroboscopic snapshot is taken. If $s$ is another phase offset, define

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

Then the one-period operators based at $t_0$ and $t_0+s$ are related by unitary conjugation:

$$U_F(t_0+s) = V(s)U_F(t_0)V^{-1}(s).$$

This follows from the composition law and periodicity of $H(t)$. Therefore the eigenvalues of $U_F(t_0)$ are independent of the reference phase, while the eigenvectors are transported around the drive cycle.

In applications this matters. Observing a driven system at the peak of a pulse and observing it halfway through the pulse can give different stroboscopic states, even though the quasienergy spectrum is the same.

Eigenvalues and Quasienergies

Because $U_F$ is unitary, its eigenvalues can be written

$$\lambda_\alpha = e^{-i\theta_\alpha}, \qquad \theta_\alpha\in\mathbb R \quad \text{mod }2\pi.$$

It is conventional to write

$$\theta_\alpha = \frac{\varepsilon_\alpha T}{\hbar},$$

so that

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

The quasienergy is therefore

$$\varepsilon_\alpha = \frac{\hbar\theta_\alpha}{T} \quad \text{mod }\hbar\Omega, \qquad \Omega=\frac{2\pi}{T}.$$

The eigenphase $\theta_\alpha$ is the invariant spectral datum. The quasienergy representative depends on a branch choice for $\theta_\alpha$.

Effective Hamiltonian Branches

One often defines a stroboscopic effective Hamiltonian $H_F$ by

$$U_F=e^{-iH_FT/\hbar}.$$

Formally,

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

The logarithm of a unitary is multi-valued. If

$$U_F = \sum_\alpha e^{-i\theta_\alpha} \lvert u_\alpha\rangle\langle u_\alpha\rvert,$$

then one branch gives

$$H_F = \sum_\alpha \frac{\hbar\theta_\alpha}{T} \lvert u_\alpha\rangle\langle u_\alpha\rvert.$$

Changing the phase by $2\pi m_\alpha$ changes the corresponding quasienergy representative:

$$\frac{\hbar\theta_\alpha}{T} \longrightarrow \frac{\hbar(\theta_\alpha+2\pi m_\alpha)}{T} = \varepsilon_\alpha+m_\alpha\hbar\Omega.$$

This branch freedom is harmless when one works directly with $U_F$, but it can become confusing when interpreting $H_F$ as an ordinary Hamiltonian. There is no globally preferred quasienergy zone in general.

Stroboscopic Evolution

The experimental and numerical sampling viewpoint, including micromotion loss and frequency aliasing, is developed in Stroboscopic Dynamics.

At integer multiples of the period,

$$U(t_0+nT,t_0) = U_F(t_0)^n.$$

For a state vector,

$$\lvert\psi_n\rangle \equiv \lvert\psi(t_0+nT)\rangle = U_F^n\lvert\psi_0\rangle.$$

For a density operator,

$$\rho_n = U_F^n\rho_0(U_F^\dagger)^n.$$

For an observable in the stroboscopic Heisenberg picture,

$$A_n = (U_F^\dagger)^n A_0 U_F^n.$$

The expectation value is therefore

$$\langle A\rangle_n = \operatorname{Tr}(\rho_n A_0) = \operatorname{Tr}(\rho_0 A_n).$$

This is a discrete-time dynamics generated by a unitary map. It describes snapshots taken at the same phase of the drive; it does not by itself describe intra-period micromotion.

Micromotion Versus Stroboscopic Dynamics

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

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

The operator $P(t,t_0)$ contains micromotion. It can strongly affect observables measured within a period. Two systems with the same $U_F$ can agree at stroboscopic times while differing between snapshots if their micromotion operators differ.

This distinction is especially important in driven experiments, where detection may occur at a specific phase of the drive rather than only after complete periods.

Two-Step Drive

A common exactly defined model is a two-step drive:

$$H(t)= \begin{cases} H_A, & 0\le t\lt\tau,\\ H_B, & \tau\le t\lt T. \end{cases}$$

The one-period unitary, with the period starting at $t=0$, is

$$U_F = e^{-iH_B(T-\tau)/\hbar} e^{-iH_A\tau/\hbar}.$$

The rightmost factor acts first. If $[H_A,H_B]=0$, then this can be combined into

$$U_F = \exp\left[ -\frac{i}{\hbar} \bigl(H_A\tau+H_B(T-\tau)\bigr) \right],$$

so that one may choose

$$H_F = \frac{H_A\tau+H_B(T-\tau)}{T}$$

up to quasienergy branch choices. If $[H_A,H_B]\ne0$, commutator corrections appear, and replacing the drive by the time-averaged Hamiltonian is generally wrong.

Numerical Computation

For a finite-dimensional driven system, the most direct numerical procedure is:

1. compute $U_F(t_0)$ by integrating the time-dependent Schrödinger equation or multiplying short-time propagators;
2. check that $U_F^\dagger U_F$ is close to $I$;
3. diagonalize the unitary $U_F$, not the instantaneous Hamiltonian $H(t)$;
4. extract eigenphases and choose a quasienergy branch;
5. use eigenvectors for Floquet modes and powers of $U_F$ for stroboscopic dynamics.

For a time grid

$$t_0\lt t_1\lt\cdots\lt t_N=t_0+T,$$

a first-order product approximation is

$$U_F \approx e^{-iH(t_{N-1})\Delta t/\hbar} \cdots e^{-iH(t_1)\Delta t/\hbar} e^{-iH(t_0)\Delta t/\hbar}.$$

Higher-order integrators are often preferable, especially for long-time simulations, but the ordering principle is the same: earliest time acts first and appears on the right.

Numerical pitfalls include loss of unitarity, inconsistent quasienergy branch choices, inadequate time resolution for sharp pulses, and interpreting a stroboscopic effective Hamiltonian as if it described every instant inside the period.

Common Mistakes

• Diagonalizing $H(t_0)$ and calling its eigenvalues quasienergies.
• Replacing $U_F$ by $\exp[-(i/\hbar)\int_0^T H(t)\,dt]$ when Hamiltonians at different times do not commute.
• Forgetting that the drive phase $t_0$ changes Floquet eigenvectors by unitary conjugation.
• Treating the logarithm of $U_F$ as single-valued.
• Assuming $H_F$ alone describes micromotion inside the period.
• Ordering piecewise drive factors in the order they are written rather than the order in which they act.

Cross-Links

• Floquet Theorem in Quantum Mechanics explains Floquet states and quasienergies.
• Quantum Maps and Discrete-Time Evolution places $U_F$ among repeated unitary maps, kicked systems, circuits, and product-formula steps.
• Quasienergies explains modular spectra, Floquet zones, and avoided crossings.
• Kicked Rotor Preview gives an exact factorized Floquet map with dynamical localization and quantum-resonance regimes.
• Quantum Chaos Preview explains symmetry-resolved quasienergy statistics and why unitary evolution needs diagnostics other than state-vector separation.
• Time-Evolution Operator gives the composition law and unitarity.
• Time Ordering explains the ordered exponential.
• Formula Sheet summarizes the core formulas.
• Floquet–Magnus Expansion gives the direct one-period Magnus construction of $H_F(t_0)$.
• High-Frequency Expansions develops van Vleck effective Hamiltonians, micromotion dressing, resonance tests, and prethermal validity.
• Rabi Oscillations: First Encounter gives a familiar driven two-level example.

References

• 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.
• N. Goldman and J. Dalibard, “Periodically driven quantum systems: effective Hamiltonians and engineered gauge fields,” Physical Review X 4, 031027, 2014.

Exercises

1. Prove that $U_F$ is unitary if the evolution operator is unitary.

Solution

For closed-system evolution,

$$U^\dagger(t,t_0)U(t,t_0)=I.$$

Set $t=t_0+T$. Then

$$U_F^\dagger U_F = U^\dagger(t_0+T,t_0)U(t_0+T,t_0) = I.$$

Thus the Floquet operator is unitary.

2. Show that shifting the reference phase conjugates the Floquet operator.

Solution

Let

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

Using composition,

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

Using periodicity,

$$U(t_0+s+T,t_0+T) = U(t_0+s,t_0)=V(s),$$

so another decomposition is

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

Equating the two expressions gives

$$U_F(t_0+s)V(s) = V(s)U_F(t_0).$$

Multiplying on the right by $V^{-1}(s)$ yields

$$U_F(t_0+s) = V(s)U_F(t_0)V^{-1}(s).$$

3. Write the Floquet operator for a two-step drive and explain the order of factors.

Solution

If $H_A$ acts first for time $\tau$ and $H_B$ acts second for time $T-\tau$, then

$$U_F = e^{-iH_B(T-\tau)/\hbar} e^{-iH_A\tau/\hbar}.$$

The factor generated by $H_A$ appears on the right because it acts on the initial state first. The later step appears on the left.

4. A Floquet eigenvalue is $\lambda=e^{-i\theta}$ with $-\pi\lt\theta\le\pi$. What quasienergy representative is selected by this branch?

Solution

The eigenvalue is written as

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

Thus the selected representative is

$$\varepsilon = \frac{\hbar\theta}{T}.$$

Other representatives are

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

5. Derive the stroboscopic density-operator evolution.

Solution

At one period,

$$\rho_1 = U_F\rho_0U_F^\dagger.$$

After two periods,

$$\rho_2 = U_F\rho_1U_F^\dagger = U_F^2\rho_0(U_F^\dagger)^2.$$

Continuing by induction gives

$$\rho_n = U_F^n\rho_0(U_F^\dagger)^n.$$