Floquet engineering uses periodic driving to give a quantum system an effective Hamiltonian it does not naturally have. Examples: suppressing unwanted interactions, creating new couplings, or synthesizing gauge fields and topological bands. Pulse sequences in NMR, dynamical decoupling, parametric gates in superconducting circuits and shaken optical lattices are all instances of the same idea. Optimal control extends it by optimizing the drive waveform itself.

Floquet theory in brief

For a Hamiltonian with period \(T\), \(H(t+T) = H(t)\), the evolution operator can be written as

$$U(t, 0) = P(t)\,e^{-iH_F t/\hbar}, \qquad P(t + T) = P(t),$$

with a time-independent Floquet Hamiltonian \(H_F\) and a periodic micromotion \(P(t)\) [1]. Observed stroboscopically, at \(t = nT\), the system evolves as if under \(H_F\) alone. Floquet engineering means choosing the drive so that \(H_F\) is the Hamiltonian one wants.

The high-frequency (Magnus) expansion

When the drive is fast compared with the system’s own dynamics, \(H_F\) is computed order by order (with \(\hbar = 1\)):

$$H_F = \underbrace{\frac{1}{T}\int_0^T H(t)\,dt}_{H^{(0)}} \;+\; \underbrace{\frac{1}{2iT}\int_0^T\!dt_1\int_0^{t_1}\!dt_2\,\big[H(t_1), H(t_2)\big]}_{H^{(1)}} \;+\;\dots$$

The leading term is the time average. In magnetic resonance this is average Hamiltonian theory [2], which explains, for example, how the WAHUHA sequence averages away dipolar couplings to give high-resolution NMR in solids [3]. The higher-order terms matter. In several important classes of protocols — the Kapitza pendulum, artificial magnetic fields for neutral atoms, Floquet Chern insulators — the infinite-frequency Floquet Hamiltonian differs from the simple average [4, 5]. Decoupling sequences are the special case where the target average is zero; see Dynamical decoupling.

Where it is used

  • Ultracold atoms in shaken lattices. Periodic modulation renormalizes tunnelling, creates artificial gauge fields and topological bands [6]. A notable example is the realization of the topological Haldane model with fermions in a modulated honeycomb lattice [7]. See Ultracold atoms & BECs.
  • Interacting spins. Global periodic pulse sequences engineer target many-body Hamiltonians that are robust against disorder, unwanted interactions and pulse imperfections [8]. With Rydberg atoms, the type of spin Hamiltonian itself can be changed by driving [9]. See Spins — NMR, ESR & colour centres and Neutral atoms & Rydberg arrays.
  • Superconducting circuits. Modulating qubit frequencies activates selective parametric couplings, giving a universal gate set on fixed-coupling hardware [10]. See Superconducting circuits.
  • Quantum materials. Light-driven band structures and phases — “Floquet engineering of quantum materials” [11].
  • Counterdiabatic driving. Approximate counterdiabatic terms, built as nested commutators, can be realized with periodic driving using only the available controls [12]; see Shortcuts to adiabaticity.

Heating and its limits

Interacting many-body systems absorb energy from a periodic drive. For generic systems at finite drive period, the long-time state is featureless, equivalent to infinite temperature [13]. Fortunately, heating is exponentially slow in the drive frequency. A long-lived prethermal regime exists in which the engineered Floquet Hamiltonian governs the dynamics [14]. Choosing the drive frequency is therefore a trade-off: high enough to suppress heating and higher-order corrections, low enough to stay within the available bandwidth and avoid exciting other levels.

Floquet engineering and optimal control

Floquet engineering is usually analytic: choose a drive and compute \(H_F\). Optimal control turns it around. It searches for the drive waveform that produces a target Floquet Hamiltonian or a target quasi-energy band structure, for example to manipulate the Floquet bands of a two-dimensional material [15]. Robustness to pulse errors can be built into the same optimization; see Robust & ensemble optimal control.