Floquet Engineering in Quantum Matter

Periodic driving can reshape tunnelling, interactions, and band geometry, but the engineered Hamiltonian is only one part of the dynamics. A controlled Floquet description specifies the stroboscopic effective Hamiltonian, the micromotion within each period, the preparation protocol, resonances, and the time window before heating or loss changes the state.

Required background. Prethermalization and Floquet ensembles supplies the general high-frequency and heating framework; driven steady states distinguishes isolated from bath-stabilized dynamics; and real-time observable extraction supplies numerical checks. Helpful background. Band topology and symmetry protection is useful when the drive is intended to engineer a topological band.

Floquet operator and gauge

For $H(t+T)=H(t)$ with angular frequency $\Omega=2\pi/T$, define

$$U(t_0+T,t_0) =\mathcal T\exp\!\left[-i\int_{t_0}^{t_0+T}\!\mathrm dt\,H(t)\right] =e^{-iH_F(t_0)T}.$$

The quasienergies of $H_F$ are defined modulo $\Omega$, because multiplying a Floquet mode by $e^{im\Omega t}$ shifts its quasienergy by $m\Omega$. The choice of initial phase $t_0$ changes $H_F(t_0)$ by a unitary transformation. Physical predictions are unchanged only when the states and observables are transformed consistently.

A useful factorization is

$$U(t,t_0)=P(t)e^{-iH_{\mathrm{eff}}(t-t_0)}P^\dagger(t_0), \qquad P(t+T)=P(t),$$

where $P(t)$ is micromotion. Measurements made at arbitrary drive phase contain $P(t)$; a stroboscopic calculation using only $H_{\mathrm{eff}}$ cannot predict them by itself.

Bukov, D’Alessio, and Polkovnikov 2015 give a convention-conscious review of quasienergy gauge, effective Hamiltonians, and micromotion.

Expand $H(t)=\sum_m H_m e^{im\Omega t}$. In the van Vleck convention,

$$H_{\mathrm{eff}} =H_0+\sum_{m=1}^{\infty}\frac{[H_m,H_{-m}]}{m\Omega} +O(\Omega^{-2}).$$

Different high-frequency expansions distribute terms differently between $H_{\mathrm{eff}}$ and micromotion, but agree on observables to the retained order. A truncation is credible only if successive orders decrease for the chosen parameters and if exact one-period evolution agrees with the truncated result on held-out observables.

The van Vleck organization and its use for engineered gauge fields are developed by Goldman and Dalibard 2014.

Shaken-lattice tunnelling

Consider a one-dimensional tight-binding model in a sinusoidal force,

$$H(t)=-J\sum_j(c_{j+1}^\dagger c_j+\mathrm{h.c.}) +K\cos(\Omega t)\sum_j j n_j.$$

Transforming to the accelerated frame removes the force and gives the hopping a Peierls phase $e^{i\zeta\sin(\Omega t)}$, with $\zeta=K/\Omega$ in units where the lattice spacing and $\hbar$ are one. Since

$$e^{i\zeta\sin(\Omega t)} =\sum_{m=-\infty}^{\infty}\mathcal J_m(\zeta)e^{im\Omega t},$$

the period average yields

$$H_{\mathrm{eff}}^{(0)} =-J_{\mathrm{eff}}\sum_j(c_{j+1}^\dagger c_j+\mathrm{h.c.}), \qquad J_{\mathrm{eff}}=J\mathcal J_0(\zeta).$$

The sign of the effective hopping can be reversed, and its leading value vanishes at a zero of $\mathcal J_0$. “Dynamical localization” at such a zero is exact only in special noninteracting idealizations. Longer-range hopping, interactions, trap inhomogeneity, finite-frequency corrections, and drive noise set the residual bandwidth in a platform.

For optical-lattice realizations and their calibration limits, see Eckardt 2017.

This example also exposes why calibration must state what $K$ means. If the laboratory input is a displacement, voltage, or magnetic-field modulation, converting it to the energy gradient $K$ requires a separate transfer function with uncertainty.

Resonances and the prethermal window

For a bounded local lattice Hamiltonian with local scale $g$, sufficiently large $\Omega/g$ can produce a heating time that is exponentially long in $\Omega/g$ up to model-dependent constants. Within

$$g^{-1}\ll t\ll t_*, \qquad t_*\sim g^{-1}e^{c\Omega/g},$$

the system can evolve under a quasi-conserved effective Hamiltonian. This is a prethermal statement, not an assertion of eternal stability. The rigorous bounded-local assumptions do not directly cover an untruncated bosonic site or a continuum with arbitrarily high energies.

The exponentially slow absorption theorem and its quasi-conserved Hamiltonian are given by Abanin et al. 2017.

Resonant processes occur when $m\Omega$ matches a many-body energy difference with a non-negligible matrix element. Avoiding all single-particle band gaps is insufficient: interactions open multiparticle channels, and a many-body spectrum becomes dense with size. Useful checks therefore scan absorbed energy versus frequency and amplitude, vary the observation time, and compare the exact Floquet spectrum or short-time propagator with the truncated expansion.

Preparation matters as much as the plateau. A sudden turn-on populates multiple Floquet branches and adds micromotion-induced excitations. A smooth envelope can suppress them, but quasienergy avoided crossings make a globally adiabatic Floquet ramp generally impossible in the thermodynamic limit. The experimental target is instead a controlled loading path into the desired prethermal sector.

From engineered Hamiltonian to phase claim

An effective coupling is established when calibrated dynamics agree with $H_{\mathrm{eff}}$ and its micromotion corrections over a stated window. A Floquet phase requires additional evidence: an invariant or order parameter, robustness to symmetry-preserving perturbations, size and lifetime trends, and exclusion of ordinary synchronization or finite-size recurrence. Floquet phases and time crystals develops those stronger criteria.

Exercises

1. Bessel-renormalized hopping. Perform the accelerated-frame transformation for the shaken chain and show that the period-averaged hopping is $J\mathcal J_0(K/\Omega)$.

Solution

Choose $R(t)=\exp[-i\zeta\sin(\Omega t)\sum_j jn_j]$. The term $-iR^\dagger\dot R$ cancels the sinusoidal gradient for $\zeta=K/\Omega$. Because $R^\dagger c_{j+1}^\dagger c_jR=e^{i\zeta\sin(\Omega t)}c_{j+1}^\dagger c_j$, averaging the Jacobi–Anger expansion over one period keeps only its $m=0$ component, $\mathcal J_0(\zeta)$.

2. A commuting drive. Suppose $H(t)=H_0+V\cos\Omega t$ and $[H_0,V]=0$. Find $U(T,0)$ exactly and compare it with the first-order van Vleck expression.

Solution

All Hamiltonians commute at different times, so time ordering is irrelevant. The cosine integrates to zero over one period and $U(T,0)=e^{-iH_0T}$. Here $H_{\pm1}=V/2$, so $[H_{-1},H_1]=0$ and the first correction also vanishes. There can still be within-period micromotion generated by the time integral of $V\cos\Omega t$.

References

• Abanin, Dmitry A., Wojciech De Roeck, Wen Wei Ho, and François Huveneers. “Effective Hamiltonians, Prethermalization, and Slow Energy Absorption in Periodically Driven Many-Body Systems.” Physical Review B 95, 014112 (2017). DOI.
• Bukov, Marin, Luca D’Alessio, and Anatoli Polkovnikov. “Universal High-Frequency Behavior of Periodically Driven Systems: From Dynamical Stabilization to Floquet Engineering.” Advances in Physics 64, 139–226 (2015). DOI.
• Eckardt, André. “Colloquium: Atomic Quantum Gases in Periodically Driven Optical Lattices.” Reviews of Modern Physics 89, 011004 (2017). DOI.
• Goldman, Nathan, and Jean Dalibard. “Periodically Driven Quantum Systems: Effective Hamiltonians and Engineered Gauge Fields.” Physical Review X 4, 031027 (2014). DOI.