Periodic and Driven Dynamics

Periodic and driven dynamics studies quantum systems whose Hamiltonians are deliberately or externally varied in time. A generic drive requires the full time-ordered propagator. A periodic drive adds a discrete time-translation symmetry, allowing one-period evolution, quasienergies, and micromotion to organize the problem.

The chapter moves through three levels:

$$\begin{gathered} H(t) \quad \text{continuous driven dynamics}, \\ H(t+T)=H(t) \quad \text{periodic structure}, \\ U_F=U(t_0+T,t_0) \quad \text{one-period quantum map}. \end{gathered}$$

These levels are related but not interchangeable. A Floquet operator determines same-phase snapshots, while the full Hamiltonian also determines motion inside each period. A rotating frame changes representation continuously; it is not itself a stroboscopic approximation. An effective Hamiltonian reproduces a selected map only after branch and validity choices are declared.

What This Chapter Owns

This chapter is the canonical home for

• unitary dynamics under externally prescribed time-dependent drives;
• the additional structure supplied by exact temporal periodicity;
• the quantum Floquet theorem and periodic Floquet modes;
• one-period operators, eigenphases, and quasienergies;
• exact time-dependent rotating-frame transformations;
• same-phase stroboscopic evolution and micromotion reconstruction;
• the kicked rotor as a concrete Floquet map with classical and quantum transport;
• the bridge from driven propagators to coherent quantum-control design.

The chapter does not own every approximation or application involving a drive. The rotating-wave approximation, direct Floquet–Magnus construction, and general High-Frequency Expansions belong to Effective Hamiltonians and Scale Separation. Detailed pulse protocols, optimal control, feedback, decoherence, and noise limits belong to Quantum Control and Feedback. Detailed quantum-chaos diagnostics and many-body driven phases belong to later volumes.

Core Equations

The propagator of a driven closed system satisfies

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

When Hamiltonians at different times do not commute,

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

For an exactly periodic Hamiltonian,

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

The Floquet theorem permits solutions of the form

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

with

$$\lvert u_\alpha(t+T)\rangle = \lvert u_\alpha(t)\rangle.$$

The one-period unitary is

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

and its eigenvalue equation is

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

Because an eigenphase is defined modulo $2\pi$,

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

At same-phase stroboscopic times,

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

At an intra-period offset $0\leq\tau\lt T$, the missing micromotion is restored by

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

This last formula is the cleanest statement of what the one-period map preserves and what it omits.

Three Simplifications, Three Meanings

Rotating frame

With

$$\lvert\psi_R(t)\rangle = R(t)^\dagger\lvert\psi(t)\rangle,$$

the exact transformed Hamiltonian is

$$H_R(t) = R^\dagger HR - i\hbar R^\dagger\dot R.$$

No information is discarded when states, observables, and endpoints are transformed consistently. A later rotating-wave approximation may discard rapidly oscillating terms, but that is a separate step.

Stroboscopic map

Replacing continuous time by $t_0+nT$ keeps one fixed drive phase and produces the discrete map $U_F^n$. It removes direct access to intra-period motion and aliases frequencies that differ by integer multiples of $\Omega$.

Effective Hamiltonian

One may write

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

but $H_F$ is not unique because the logarithm of a unitary is multivalued. High-Frequency Expansions supplies controlled inverse-frequency approximations in suitable regimes; it does not eliminate logarithm branches or guarantee locality, convergence, or absence of heating in every driven system.

Page Map

Question	Canonical page	Main object
What changes when a closed system is externally driven?	Driven Closed Quantum Systems	$H(t)$ and $U(t,t_0)$
What extra structure follows from exact periodicity?	Periodic Hamiltonians	$H(t+T)=H(t)$
Why can solutions be separated into periodic modes and phases?	Floquet Theorem in Quantum Mechanics	$e^{-i\varepsilon t/\hbar}u(t)$
How is one cycle represented and diagonalized?	Floquet Operators	$U_F(t_0)$
Why are driven spectra modular?	Quasienergies	$\varepsilon\bmod\hbar\Omega$
How can rapid known motion be removed exactly?	Rotating Frames	$R^\dagger HR-i\hbar R^\dagger\dot R$
What do same-phase measurements preserve or lose?	Stroboscopic Dynamics	$U_F^n$ and micromotion
How do Floquet maps connect chaos, diffusion, and localization?	Kicked Rotor Preview	kick–free standard map
How do driven propagators become pulse-design variables?	Links to Quantum Control	$H_0+\sum_a u_aH_a$

Together these nine articles form the planned chapter. Each owns one layer of the subject so that Floquet proofs, quasienergy conventions, sampling theory, and control design are not repeated in several places.

Suggested Reading Routes

First route through driven dynamics

Begin with Driven Closed Quantum Systems and Periodic Hamiltonians. Then read the Floquet Theorem, Floquet Operators, and Quasienergies in that order.

Experimental and control route

Read Rotating Frames before Links to Quantum Control. Continue to Rabi and Ramsey Control, Pulse Sequences, and Optimal Control according to the protocol.

Floquet simulation route

Read Floquet Operators and Stroboscopic Dynamics, then pair them with Quantum Maps and Discrete-Time Evolution. Continue to Floquet–Magnus Expansion for the direct one-period series or High-Frequency Expansions for van Vleck, Floquet-space, resonance, and prethermal analysis. Use either only after identifying a small parameter and an error strategy.

Quantum-chaos entry route

Read Stroboscopic Dynamics and then Kicked Rotor Preview. The rotor page develops the exact map, dynamical localization, and resonance exceptions while reserving general quantum-chaos diagnostics for their later canonical home.

Modeling Checklist

Before applying Floquet or control language, state

1. whether the retained system is closed on the timescale of interest;
2. whether periodicity is exact, approximate, or only a feature of the chosen protocol;
3. the period $T$, angular frequency $\Omega$, and reference phase $t_0$;
4. the ordering convention for piecewise or kicked evolution;
5. the rotating-frame convention and any approximation made after changing frames;
6. the quasienergy zone or logarithm branch used for plots;
7. whether the observable is sampled stroboscopically or resolved within each cycle;
8. the truncation, convergence, and noise checks used in numerics or experiment.

Most disagreements between otherwise correct Floquet calculations can be traced to one of these undeclared conventions.

Common Mistakes

• Replacing a time-ordered exponential by the exponential of the period average without checking commutators.
• Treating every time-dependent Hamiltonian as periodic.
• Calling instantaneous energy eigenvalues quasienergies.
• Forgetting that quasienergies are defined modulo $\hbar\Omega$.
• Assuming a Floquet effective Hamiltonian uniquely determines micromotion.
• Omitting the inertial term in a rotating frame.
• Confusing an exact frame change with the rotating-wave approximation.
• Comparing Floquet eigenvectors at different reference phases without transporting them.
• Reading products of segment unitaries from left to right in time.
• Claiming dynamical localization without checking quantum-resonance conditions or finite-grid saturation.
• Treating ideal closed-system controllability as evidence of robust control in a noisy device.

References

• J. H. Shirley, “Solution of the Schrödinger equation with a Hamiltonian periodic in time,” Physical Review 138, B979–B987, 1965.
• H. Sambe, “Steady states and quasienergies of a quantum-mechanical system in an oscillating field,” Physical Review A 7, 2203–2213, 1973.
• M. Grifoni and P. Hänggi, “Driven quantum tunneling,” Physics Reports 304, 229–354, 1998.
• 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, “Colloquium: atomic quantum gases in periodically driven optical lattices,” Reviews of Modern Physics 89, 011004, 2017.
• F. Haake, Quantum Signatures of Chaos, 3rd ed., Springer, 2010.
• D. D’Alessandro, Introduction to Quantum Control and Dynamics, Chapman and Hall/CRC, 2007.

Exercises

1. Choose the canonical page for each task: proving the Floquet decomposition, selecting a quasienergy zone, reconstructing motion at $t_0+nT+\tau$, and designing a robust pulse.

Solution

Use Floquet Theorem in Quantum Mechanics for the decomposition, Quasienergies for zone conventions, Stroboscopic Dynamics for intra-period reconstruction, and Optimal Control for constrained robust pulse design. Links to Quantum Control is the bridge into the last topic.

2. Explain why two periodic drives can have the same Floquet operator but different continuous trajectories.

Solution

The Floquet operator records only the net unitary over one complete period. Different time orderings or pulse shapes can multiply to the same endpoint unitary while producing different intermediate propagators. For $0\lt\tau\lt T$, the factor

$$U(t_0+\tau,t_0)$$

depends on the detailed drive. Therefore

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

can differ even when $U_F$ is identical. The missing factor is micromotion.

3. A calculation transforms to a rotating frame, drops rapidly oscillating terms, and samples once per period. Identify the three logically distinct operations.

Solution

The rotating-frame transformation is an exact change of representation when the frame term and transformed observables are retained. Dropping rapidly oscillating terms is a rotating-wave or related approximation requiring a scale and error argument. Sampling once per period is a stroboscopic restriction of the observations, which discards direct access to intra-period motion. Performing all three in one calculation does not make them the same operation.