Nonrelativistic Floquet Modes in Quantum Systems

• Nonrelativistic Floquet modes are quasi-stationary solutions of the Schrödinger equation with periodic Hamiltonians, defined by their modular quasi-energy spectrum.
• The methodology involves time- and frequency-domain representations and high-frequency expansions to diagonalize the Floquet operator and accurately compute quasienergies.
• Research reveals that these modes underpin phenomena such as Floquet topological phases and localized edge states, offering new avenues for coherent control in quantum systems.

Nonrelativistic Floquet modes are fundamental quasi-stationary solutions of the time-dependent Schrödinger equation with a strictly periodic Hamiltonian in nonrelativistic quantum systems. They appear universally in quantum systems subjected to time-periodic driving, underpinning diverse phenomena from coherent control in few-level systems to emergent topological phases in condensed matter platforms. Their mathematical structure, representation in real and frequency domains, physical classification, and application in strongly correlated and engineered quantum systems constitute major research directions.

1. Floquet Theorem and Fundamental Structure

Given a nonrelativistic quantum system described by the time-dependent Schrödinger equation with a periodic Hamiltonian,

$$i\hbar\,\frac{\partial}{\partial t}\,\psi(t) = H(t)\,\psi(t), \qquad H(t+T) = H(t),$$

Floquet's theorem asserts the existence of a complete set of solutions of the form

$$\psi_\alpha(t) = e^{-i \epsilon_\alpha t/\hbar} \phi_\alpha(t),\qquad \phi_\alpha(t+T) = \phi_\alpha(t),$$

where $\epsilon_\alpha$ are called quasi-energies, defined only modulo $\hbar\Omega$ ($\Omega = 2\pi/T$), and $\phi_\alpha(t)$ are the Floquet modes, periodic with the drive period.

Substituting this ansatz yields the eigenvalue equation in the so-called Sambe space,

$$\left[ H(t) - i\hbar\,\frac{\partial}{\partial t}\right]\phi_\alpha(t) = \epsilon_\alpha\,\phi_\alpha(t),$$

which represents an eigenproblem for a non-Hermitian operator acting on $T$-periodic functions. The spectrum's modular nature reflects the underlying discrete time-translation symmetry (quasienergy Brillouin zone).

A closely related representation involves the Floquet operator,

$$U(T, 0) = \mathcal{T}\exp\left[ -\frac{i}{\hbar} \int_0^T H(t')\,dt' \right],$$

whose eigenstates $|\phi_\alpha(0)\rangle$ satisfy

$$U(T,0) |\phi_\alpha(0)\rangle = e^{-i\epsilon_\alpha T/\hbar} |\phi_\alpha(0)\rangle.$$

Thus, the Floquet problem reduces either to diagonalization in the infinitely-extended Sambe (Fourier) space or to computation and diagonalization of the one-period evolution operator to obtain the quasienergy spectrum.

2. Representations and High-Frequency Expansions

Floquet modes may be constructed in either time or frequency domains. In the time domain, the $T$-periodic Floquet function $\phi_\alpha(t)$ is determined directly by integrating the Schrödinger equation and enforcing periodic boundary conditions. In frequency space, one Fourier expands both the Hamiltonian and Floquet mode: $$H(t) = \sum_j e^{-ij\Omega t} H_j, \qquad \phi_\alpha(t) = \sum_j e^{-ij\Omega t}\phi_\alpha^j,$$ yielding a static infinite-dimensional matrix eigenproblem

$$\sum_j (H_{j-k} - j\Omega \delta_{j,k}) \phi_\alpha^j = \epsilon_\alpha \phi_\alpha^k.$$

Truncation to $|j| \leq j_\text{max}$ produces a finite-dimensional “Floquet Hamiltonian” suitable for numerical computation.

In the high-frequency regime ($\Omega$ large compared to all bare energy scales), systematic expansions exist. The van Vleck and Magnus expansions provide corrections to the effective, stroboscopic Hamiltonian: $$H_\text{eff} = H_0 + \frac{1}{\hbar\Omega} \sum_{m\ne 0} \frac{[H_{-m}, H_{m}]}{m} + \cdots,$$ where $H_0$ is the period averaged Hamiltonian. This expansion controls the micromotion and long-term dynamics in the limit $\Omega \to \infty$ (Tsuji, 2023). Stroboscopic evolution is then governed by $H_\text{eff}$, while intra-period motion is described by the micromotion, or “kick,” operator $K(t)$.

3. Physical Classification and Manifestations

Floquet modes underpin both “conventional” and “anomalous” physical phenomena:

• Conventional Floquet edge or bound states: In one-dimensional driven superconducting wires, periodic modulation induces end modes with quasienergy $\varepsilon = 0$ or $\pi/T$ (so $e^{-i\varepsilon T} = \pm1$). These are Floquet-Majorana modes and directly generalize static Majorana edge states (Saha et al., 2016, Wu et al., 2023).
• Anomalous Floquet modes: Periodic driving can stabilize edge modes at quasienergies neither $0$ nor $\pi/T$, corresponding to complex-conjugate-pair eigenvalues $e^{\pm i\theta}$. These “anomalous” modes emerge for sufficiently strong driving (Saha et al., 2016) and underlie new topological phases that lack static counterparts, such as the $\pi$-mode in the Floquet-engineered SSH model (Cheng et al., 2018).
• Bulk–boundary correspondence: For small modulation strength, the quasienergies and momenta at which the Floquet operator in the bulk has extrema directly predict the corresponding localized end-mode properties, establishing a precise bulk-edge mapping (Saha et al., 2016).

In higher-dimensional systems, periodically driven lattices can support topological boundary states even when bulk Chern numbers vanish, as in the case of anomalous Floquet insulators with perfectly flat bands (Dag et al., 2022). Driven systems also host dynamical localization (i.e., coherent suppression of transport) and can realize prethermal or nonequilibrium steady states, depending on coupling to an environment (Tsuji, 2023).

4. Quantum Algorithms and Computational Approaches

The computation of Floquet modes and quasienergies is inherently demanding, especially in the low-frequency or strongly correlated regimes, as Hilbert space truncations rapidly lose accuracy. Quantum algorithms tailored to the Floquet problem are therefore of significant practical interest (Fauseweh et al., 2021). Two main strategies are prominent:

• Time-domain variational approach: A parametrized quantum circuit prepares an approximate Floquet eigenstate at $t=0$, variationally optimized so that its overlap with its one-period evolution is maximized. Quasienergy is then extracted via phase estimation on the evolution operator.
• Frequency-domain VQE (Variational Quantum Eigensolver): The truncated Floquet Hamiltonian is mapped onto an extended Hilbert space (physical $\otimes$ Fourier modes), and a variational ansatz targets the central part of the spectrum with an excited-state VQE for eigenpair extraction.

Benchmarks demonstrate that both algorithms can accurately capture Floquet spectra for driven spin and chain models on small quantum devices, with trade-offs in circuit depth, resource requirements, and error resilience. The time-domain algorithm is width-efficient but requires deep phase estimation circuits; the frequency-domain VQE is shallower but requires additional ancilla qubits (Fauseweh et al., 2021).

5. Analytical Examples and Applications

Floquet theory underpins a range of analytically tractable models with distinct physical implications:

• Driven two-level systems: For a sinusoidally driven spin-$\frac{1}{2}$, the exact energy splitting and resonant conditions are accessible. At high frequencies, the effective Hamiltonian recovers the static limit, with corrections appearing only at subleading order. At specific detuning, Rabi resonances occur (Tsuji, 2023).
• Tight-binding chain in an AC field: An electron in a periodically driven lattice experiences an effective hopping modulated by a Bessel function. At zeros of the Bessel function, coherent destruction of tunneling—dynamical localization—emerges (Tsuji, 2023).
• Driven SSH and superconducting Kitaev chains: Time-periodic modulations generate topological $\pi$-modes or large multiplicities of Floquet-Majorana edge states. The number and character of these states are classified by winding numbers associated with the corresponding quasienergy gaps; phase diagrams can feature infinite hierarchies of topological lobes and complex bulk-edge relations (Cheng et al., 2018, Wu et al., 2023).

A summary of representative examples is given below.

Model	Key Floquet Feature	Reference
Driven two-level system	Exact quasienergy splittings, Rabi resonances	(Tsuji, 2023)
AC-driven chain	Dynamical localization via effective hopping suppression	(Tsuji, 2023)
Kitaev chain	Floquet Majorana & anomalous edge modes	(Saha et al., 2016Wu et al., 2023)
Driven SSH model	$\pi$-modes, topological classification	(Cheng et al., 2018)
Flat-band systems	Chiral edge modes with zero Chern number	(Dag et al., 2022)

6. Topological, Geometric, and Thermodynamic Extensions

Recent research has extended the formalism of nonrelativistic Floquet modes in several important directions:

• Geometric Floquet Theory: The quasienergy folding structure arises from the reduction $U(1) \to \mathbb{Z}$ of the gauge group under time-periodicity (Schindler et al., 9 Oct 2024). A distinguished “parallel transport” gauge separates geometric (Berry phase/Wilson line) and dynamical (average energy/Kato Hamiltonian) contributions in the evolution operator, unambiguously sorting Floquet states and isolating nonequilibrium topological effects (e.g., $\pi$-mode splitting in time crystals).
• Floquet-Nambu-Goldstone Modes: In spontaneous Floquet states breaking both $U(1)$ and continuous time-translation symmetry, a quantum time operator emerges associated with a temporal Goldstone mode at zero quasienergy. The associated ballistic quantum fluctuations are confirmed by both theory and numerics (Nova et al., 16 Feb 2024).
• Universality in Nonrelativistic Field Theory: In nonrelativistic ($\epsilon/m \ll 1$) expansions about breathers, quasi-breathers, and oscillons in relativistic scalar theories, the Floquet spectrum consists of a universal continuum plus four discrete zero-modes tied to translational, boost, and amplitude moduli. The leading order structure is independent of the interaction potential (Evslin et al., 6 Nov 2025).

7. Challenges, Open Problems, and Research Frontiers

Several central issues remain in the theory and application of nonrelativistic Floquet modes:

• Heating and Thermalization: Isolated many-body systems generically absorb energy under periodic driving, heating to infinite temperature absent additional mechanisms. Exponentially long prethermal plateaux are possible under high-frequency driving, but rigorous control of approach to equilibrium in more general settings remains challenging (Tsuji, 2023, Schindler et al., 9 Oct 2024).
• Open Systems and Steady States: Coupling to a bath can stabilize well-defined nonequilibrium steady states with unique Floquet band structure; a full dynamical treatment can require the Floquet-Lindblad or Floquet-Keldysh formalisms (Tsuji, 2023).
• Computation in Strongly Correlated/Low-Frequency Regimes: Exact diagonalization of the Floquet Hamiltonian or operator rapidly becomes infeasible as system size or Fourier basis increases. Near-term quantum algorithms and hybrid methods offer promising strategies but require optimization of circuit architecture and noise mitigation (Fauseweh et al., 2021).
• Multi-frequency Driving and Beyond: Extensions to quasi-periodic or multi-frequency driving introduce effective higher-dimensional “Floquet lattices” in frequency space, supporting new classes of edge states and pumping phenomena (Park et al., 2022), but their classification and control are less developed than for single-frequency cases.

Nonrelativistic Floquet modes remain central to the theoretical and computational understanding of periodically driven quantum systems, enabling the design and interpretation of experiments probing coherent control, topological phenomena, and nonequilibrium quantum matter across a broad range of platforms.