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.

Helpful background. Time-dependent Hamiltonians and time ordering explain how to construct driven evolution.

Evolution over one driving period

Begin with a finite-dimensional closed system and a piecewise-continuous Hermitian $H(t)$. Infinite-dimensional versions require additional existence and spectral assumptions. The propagator 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,

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

In the finite-dimensional setting the Floquet theorem gives a complete set of 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	Start here
What changes when a closed system is externally driven?	Driven Closed Quantum Systems
What extra structure follows from exact periodicity?	Periodic Hamiltonians
Why can solutions be separated into periodic modes and phases?	Floquet Theorem in Quantum Mechanics
How is one cycle represented and diagonalized?	Floquet Operators
Why are driven spectra modular?	Quasienergies
How can rapid known motion be removed exactly?	Rotating Frames
What do same-phase measurements preserve or lose?	Stroboscopic Dynamics
How do Floquet maps connect chaos, diffusion, and localization?	Kicked Rotor Preview
How do driven propagators become pulse-design variables?	Links to Quantum Control

A drive can return the state while moving it between snapshots

Consider $H(t)=\hbar a\cos(\Omega t)\sigma_z/2$, where $a$ and $\Omega$ have units of angular frequency. Since all Hamiltonians commute,

$$U(t,0)=\exp\left[-\frac{i a}{2\Omega}\sin(\Omega t)\sigma_z\right].$$

At $T=2\pi/\Omega$, $U_F=I$. This has the same one-period map as $H=0$, although the spin generally moves during the cycle. An initial $|+x\rangle$ has

$$\langle\sigma_x\rangle(t) =\cos\left[\frac{a}{\Omega}\sin(\Omega t)\right],$$

whereas it remains $1$ under $H=0$. A one-period effective Hamiltonian therefore cannot determine all intermediate observables without micromotion.

For practical pulse design, continue from Rotating Frames to Rabi and Ramsey Control. For approximate Floquet Hamiltonians, compare the Floquet–Magnus expansion with other high-frequency expansions; a chosen quasienergy branch is distinct from an approximation’s error bound.

Exercises

For the sinusoidal drive above, take $a/\Omega=\pi$. What are $\langle\sigma_x\rangle$ at $t=T/4$ and $t=T$? Can sampling only at integer periods distinguish it from $H=0$?

Solution

At $T/4$, $\sin(\Omega t)=1$ and $\langle\sigma_x\rangle=-1$. At $T$, it is $1$. All integer-period states agree with their initial states because $U_F=I$, so these snapshots alone cannot distinguish the two drives. Measurements inside a period can.

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.

References

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