Floquet Exceptional Points (FEPs)

• Floquet Exceptional Points (FEPs) are non-Hermitian degeneracies in periodically driven systems where two or more Floquet quasi-energies and eigenstates coalesce under conditions like multiphoton resonance.
• FEPs can be engineered via periodic modulation, explicit EP relocation, and dissipative coupling, providing a tunable platform to control chiral dynamics and symmetry transitions.
• Research on FEPs has advanced diagnostics and revealed novel topological band phenomena, enhanced quantum responses, and potential applications across photonic, atomic, and electronic systems.

Floquet exceptional points (FEPs) are non-Hermitian degeneracies in periodically driven systems at which two or more Floquet quasi-energies and the corresponding Floquet eigenstates coalesce. They are the time-periodic counterparts of static exceptional points, but their location and order are controlled by the drive itself, so they can arise under multiphoton resonance, periodic modulation of coupling, gain, loss, or dissipation, and in settings ranging from Hamiltonians and Liouvillians to scattering matrices and quantum channels. This makes FEPs central to chiral transport, adiabatic state-flips, anti-$\mathcal{PT}$ and $\mathcal{PT}$-symmetry transitions, skin effects, topological band singularities, and enhanced response in driven open systems (Longhi, 2017, Zhang et al., 18 Apr 2025).

1. Formal definition and mathematical setting

For a time-periodic non-Hermitian Hamiltonian $\mathcal{H}(t+T)=\mathcal{H}(t)$, the evolution equation

$$i \frac{d\mathbf{a}}{dt} = \mathcal{H}(t)\mathbf{a}(t)$$

admits the Floquet form

$$\mathbf{a}(t)=\Phi(t)\exp(-i\mathcal{R} t)\mathbf{a}(0),$$

where $\Phi(t+T)=\Phi(t)$ and the eigenvalues $\{\mu_n\}$ of $\mathcal{R}$ are the Floquet quasi-energies. The associated Floquet eigenstates are

$$\mathbf{f}^{(n)}(t)=\Phi(t)\mathbf{q}_n \exp(-i\mu_n t).$$

A Floquet exceptional point of order $M$ occurs when

$\mathcal{PT}$0

and the corresponding Floquet eigenstates coalesce, so the Floquet generator becomes defective and generalized eigenstates are required (Longhi, 2017).

Equivalent formulations use the one-period evolution operator. In periodically driven open systems described by a time-dependent Liouvillian $\mathcal{PT}$1, one defines

$\mathcal{PT}$2

In this setting, a Floquet EP is a point where two or more eigenvalues and eigenmatrices of $\mathcal{PT}$3 merge, so $\mathcal{PT}$4 becomes non-diagonalizable. This extends the notion of FEP from wavefunction dynamics to Lindblad evolution and other superoperator problems (Gunderson et al., 2020).

The spectral object that becomes singular depends on the platform: a Floquet Hamiltonian, a monodromy matrix, a Floquet Liouvillian, or a Floquet scattering matrix. The shared feature is defectiveness of the stroboscopic evolution over one drive period.

2. Mechanisms for generating and relocating FEPs

A basic mechanism is multiphoton resonance. If the eigenvalues $\mathcal{PT}$5 of the static part of the Hamiltonian are real, distinct, and their differences are not integer multiples of the drive frequency $\mathcal{PT}$6, then the quasi-energies are non-degenerate. If instead a subset satisfies

$\mathcal{PT}$7

then a FEP appears at that frequency. In this sense, FEPs “generally arise when the oscillation frequency satisfies a multiphoton resonance condition” (Longhi, 2017).

A second mechanism is explicit Floquet relocation of an EP in parameter space. In the two-timescale model

$\mathcal{PT}$8

the instantaneous Floquet Hamiltonian is

$\mathcal{PT}$9

with quasi-energies

$\mathcal{H}(t+T)=\mathcal{H}(t)$0

The FEP condition is

$\mathcal{H}(t+T)=\mathcal{H}(t)$1

Varying the fast Floquet frequency $\mathcal{H}(t+T)=\mathcal{H}(t)$2 therefore re-allocates the exceptional point without changing the slow adiabatic loop (Halpern et al., 2017).

A third mechanism is Floquet dissipative coupling between sidebands. In thermal atoms, the static effective Hamiltonian is

$\mathcal{H}(t+T)=\mathcal{H}(t)$3

with static EP condition $\mathcal{H}(t+T)=\mathcal{H}(t)$4. Under periodic Zeeman modulation, the sideband Hamiltonian becomes

$\mathcal{H}(t+T)=\mathcal{H}(t)$5

with eigenvalues

$\mathcal{H}(t+T)=\mathcal{H}(t)$6

and FEP criterion

$\mathcal{H}(t+T)=\mathcal{H}(t)$7

This makes the FEP location tunable by detuning, drive frequency, and sideband index, even when the static system is far from the EP threshold (Zhang et al., 18 Apr 2025).

3. Physical platforms and realizations

Laser-driven molecular dynamics provided an early Floquet setting in which all field-free vibrational states become resonances with complex quasi-energies. In $\mathcal{H}(t+T)=\mathcal{H}(t)$8, exceptional points appear in clusters in the wavelength-intensity plane, and encircling a single EP yields $\mathcal{H}(t+T)=\mathcal{H}(t)$9 transfer while encircling a cluster yields $$i \frac{d\mathbf{a}}{dt} = \mathcal{H}(t)\mathbf{a}(t)$$0 transfer. For the example $$i \frac{d\mathbf{a}}{dt} = \mathcal{H}(t)\mathbf{a}(t)$$1, four successive $$i \frac{d\mathbf{a}}{dt} = \mathcal{H}(t)\mathbf{a}(t)$$2 transfers left only $$i \frac{d\mathbf{a}}{dt} = \mathcal{H}(t)\mathbf{a}(t)$$3 molecules undissociated, whereas a single multi-EP cascade pulse yielded $$i \frac{d\mathbf{a}}{dt} = \mathcal{H}(t)\mathbf{a}(t)$$4 survival (Lefebvre et al., 2011).

In thermal atomic ensembles, two spatially separated optical channels in a paraffin-coated $$i \frac{d\mathbf{a}}{dt} = \mathcal{H}(t)\mathbf{a}(t)$$5 vapor cell are coupled by atomic motion, which transfers spin coherence between the channels and mediates a nonlocal dissipative coupling. A static bias magnetic field $$i \frac{d\mathbf{a}}{dt} = \mathcal{H}(t)\mathbf{a}(t)$$6 produces a Zeeman shift $$i \frac{d\mathbf{a}}{dt} = \mathcal{H}(t)\mathbf{a}(t)$$7, while an oscillating field $$i \frac{d\mathbf{a}}{dt} = \mathcal{H}(t)\mathbf{a}(t)$$8 generates Floquet sidebands. This platform demonstrated Floquet dissipative coupling in a quantum system and observed an anomalous anti-$$i \frac{d\mathbf{a}}{dt} = \mathcal{H}(t)\mathbf{a}(t)$$9 symmetry phase transition at an exceptional point far from the phase-transition threshold of the static counterpart (Zhang et al., 18 Apr 2025).

Coupled optical parametric oscillators realize a non-dissipative non-Hermitian setting in which phase-sensitive amplification and de-amplification replace conventional gain and loss. With periodic pump modulation,

$$\mathbf{a}(t)=\Phi(t)\exp(-i\mathcal{R} t)\mathbf{a}(0),$$0

the Floquet exponents of the monodromy matrix exhibit tunable FEPs. Increasing modulation depth $$\mathbf{a}(t)=\Phi(t)\exp(-i\mathcal{R} t)\mathbf{a}(0),$$1 shifts the FEP location, dynamical encirclement produces chiral asymmetric evolution, and the system supports squeezed vacuum around exceptional points (Roy et al., 2020).

In $$\mathbf{a}(t)=\Phi(t)\exp(-i\mathcal{R} t)\mathbf{a}(0),$$2-symmetric magnonics, two coupled magnetic waveguides sandwich a Pt spacer whose AC current produces balanced, time-periodic gain and loss. Unlike the static case, where there is a single EP at $$\mathbf{a}(t)=\Phi(t)\exp(-i\mathcal{R} t)\mathbf{a}(0),$$3, AC driving creates multiple EPs and multiple islands of broken $$\mathbf{a}(t)=\Phi(t)\exp(-i\mathcal{R} t)\mathbf{a}(0),$$4-symmetry whose number and extension are tunable by the drive frequency and amplitude. The first FEP appears at much reduced $$\mathbf{a}(t)=\Phi(t)\exp(-i\mathcal{R} t)\mathbf{a}(0),$$5, magnetization auto-oscillations occur at low current, and high-frequency spin waves can be generated with low-frequency current (Wang et al., 2023).

Linear chains of inductively coupled RLC oscillators provide an additional route to high-order FEPs. In these systems, periodic square-wave modulation of inductances realizes Floquet exceptional contours of order $$\mathbf{a}(t)=\Phi(t)\exp(-i\mathcal{R} t)\mathbf{a}(0),$$6 in the $$\mathbf{a}(t)=\Phi(t)\exp(-i\mathcal{R} t)\mathbf{a}(0),$$7 plane, showing that arbitrary-order EPs can be engineered in a linear time-modulated circuit platform (Huerta-Morales et al., 2023).

4. Topological structure and long-time dynamics

FEPs have distinctive long-time consequences. In slowly cycled non-Hermitian systems, they generate chiral dynamics over many periods: the final dominant state depends on whether the cycle is traversed clockwise or counterclockwise. When a FEP is absent, adiabatic following persists even over many cycles; when a FEP is present, generalized Floquet eigenstates generate secular terms and adiabaticity eventually breaks down (Longhi, 2017).

Floquet relocation of an EP turns this into a control mechanism. In the two-timescale protocol, a fixed slow parameter loop can produce either a state-flip or return to the original eigenstate depending only on the fast Floquet frequency. In the notation of that model, $$\mathbf{a}(t)=\Phi(t)\exp(-i\mathcal{R} t)\mathbf{a}(0),$$8 gives a state-flip, while $$\mathbf{a}(t)=\Phi(t)\exp(-i\mathcal{R} t)\mathbf{a}(0),$$9 gives return to the original state. The topological effect of the slow cycle is therefore reconfigured by the fast drive rather than by changing the cycle itself (Halpern et al., 2017).

In periodically quenched non-Hermitian lattices, FEPs act as topological band-touching defects. Hermitian Dirac points can split into FEPs, and additional type-II EPs unique to the non-Hermitian case can appear. These Floquet bulk singularities are accompanied by the Floquet non-Hermitian skin effect, which breaks conventional bulk-edge correspondence. Under open boundary conditions, zero and $\Phi(t+T)=\Phi(t)$0 edge modes can individually coalesce and localize at two different boundaries, and their accumulation boundaries can switch as parameters are tuned (Zhang et al., 2019).

A distinct subclass is the Floquet $\Phi(t+T)=\Phi(t)$1 exceptional point, defined by a quasienergy at $\Phi(t+T)=\Phi(t)$2 and by an eigenvector that rotates on the Bloch sphere and accumulates a geometric phase of exactly $\Phi(t+T)=\Phi(t)$3 over one period. Two order-$\Phi(t+T)=\Phi(t)$4 Floquet $\Phi(t+T)=\Phi(t)$5 EPs with the same dynamical structure can merge to one order-1 EP, whereas those with opposite dynamical structure cannot merge. In Floquet bipartite lattices, the order-1 Floquet $\Phi(t+T)=\Phi(t)$6 EP appears at the transition point between quasimomentum gap phases and quasienergy gap phases, and its scattering response is “perfect transparency but detectable in reflection for one of two sides” (Zhu, 2024).

Periodically quenched two-dimensional non-Hermitian models extend this topological structure to multiple FEPs in the Floquet Brillouin zone. Zero and $\Phi(t+T)=\Phi(t)$7 FEPs carry robust integer quantized winding numbers, pair creation and annihilation of FEPs can be driven by the time period, and the same models exhibit a higher-order skin effect with skin modes localized at both edges and corners (Dash et al., 2 Sep 2025).

5. Diagnostics and observables

FEPs are identified through a combination of spectral, dynamical, and response-based diagnostics. Different platforms use different observables, but all exploit the same defectiveness of the one-period evolution.

Setting	Diagnostic	Signature
Floquet-Lindblad qubit (Gunderson et al., 2020)	$\Phi(t+T)=\Phi(t)$8	$\Phi(t+T)=\Phi(t)$9 at EP contours
Driven NH bulk bands (Dash et al., 2 Sep 2025)	Bi-orthogonal Floquet fidelity susceptibility $\{\mu_n\}$0	Peaks at FEP momenta; momentum-summed susceptibility diverges when the number of FEPs changes
Thermal atoms (Zhang et al., 18 Apr 2025)	EIT resonance positions and amplitudes	Floquet sidebands and peak coalescence at the FEP
Coupled electronic oscillators (Huerta-Morales et al., 2023)	$\{\mu_n\}$1 and $\{\mu_n\}$2	Bounded, amplifying, or algebraic growth; at EP$\{\mu_n\}$3, $\{\mu_n\}$4
Floquet scattering systems (Globosits et al., 3 Oct 2025)	Floquet scattering-matrix eigenvalues	Eigenvalues on the unit circle meet at EPs; at parametric resonance one eigenvalue vanishes and its partner diverges

In periodically driven Lindblad equations, adiabatic diagonalization yields instantaneous Liouvillian eigenvalues

$\{\mu_n\}$5

so EPs occur when

$\{\mu_n\}$6

At these points the evolution shows rapid dephasing and a staircase-like loss of coherence, which can be observed through population inversion or purity. In the Floquet picture, weak decay gives a Wannier-Stark ladder spectrum with Stark-localized eigenstates, whereas larger decay dissolves the ladders and produces new, less localized states whose eigenvalues are exponentially sensitive to perturbations (Larson et al., 2023).

In time-modulated photonic scattering, the decisive object is the multispectral Floquet scattering matrix. For real, dispersionless, and time-periodic permittivities it is pseudounitary, and increasing modulation strength drives its eigenvalues from the unit circle to inverse complex-conjugate pairs through EPs. At parametric resonance, one eigenvalue vanishes while its partner diverges, signifying simultaneous coherent perfect absorption and lasing (Globosits et al., 3 Oct 2025).

6. Conceptual boundaries and terminological distinctions

A common overextension is to treat every Floquet transition with anomalously slow decay as an EP transition. Dissipative Floquet synthetic circuits provide a counterexample: slowly decaying eigenmodes can emerge and disappear without exceptional points, and the relevant transitions are driven by avoided level crossings rather than eigenvalue coalescence. In that system, slowly decaying modes emerge at vanishingly small dissipation strength in the weak-coupling limit, and multiple transitions appear under Floquet modulation, but the paper explicitly distinguishes these from EP-driven behavior (León-Montiel et al., 2018).

Another boundary concerns the operator whose spectrum hosts the degeneracy. In some works the FEP belongs to a Hamiltonian or effective Floquet Hamiltonian; in others it belongs to a Liouvillian, a monodromy matrix, a scattering matrix, or a reset-driven quantum channel. In reset-driven Floquet quantum channels, increasing a chaos-control parameter causes real eigenvalues to drift, coalesce at exceptional points, and bifurcate into complex-conjugate pairs, producing an exceptional-point-induced transition from a symmetry-constrained ergodic regime to a fully chaotic regime (Feng et al., 12 May 2026).

The acronym “FEP” is also not unique across the non-Hermitian literature. “Fragmented exceptional points” are partially defective degeneracies characterized by $\{\mu_n\}$7, where only a subset of eigenvectors coalesce. That usage refers to a different class of non-Hermitian degeneracies than Floquet exceptional points, even though the same abbreviation is used (Bid et al., 29 Jul 2025).

7. Directions opened by recent work

Recent studies position FEPs as a route to non-Hermitian phases and responses with no static analogue. In thermal atoms, Floquet dissipative coupling “sets the stage for Floquet engineering of non-Hermitian topological spectra, and for engineering new quantum phases that cannot exist in static systems” (Zhang et al., 18 Apr 2025). In periodically quenched two-dimensional models, zero and $\{\mu_n\}$8 FEPs, bulk topological charges, edge states, Dirac-like dispersion around $\{\mu_n\}$9, and higher-order skin modes point toward a broader Floquet engineering program for driven dissipative lattices (Dash et al., 2 Sep 2025).

Quantum-enhanced response is another major direction. In coupled OPOs, FEPs coexist with phase-sensitive amplification, squeezed vacuum, and dynamically reconfigurable non-Hermitian control, suggesting low-noise sensing and quantum information applications (Roy et al., 2020). In multi-qubit systems, Floquet driving is proposed to engineer effective multi-body Ising-type interactions, with the maximal energy-response order bounded by $\mathcal{R}$0 for $\mathcal{R}$1 qubits and realizable in trapped ions or superconducting qubits (Shi et al., 23 Feb 2025).

Driven many-body and wave-scattering settings further widen the scope. Modulated Floquet parametric driving in plasmonic media produces two lines of exceptional points connected by dispersionless states, enhances plasmon quality below threshold, and above threshold yields a crystal-like state with soft, Goldstone-like phononic excitations (Kiselev et al., 2023). In periodically time-modulated photonic structures, Floquet EPs of the scattering matrix organize symmetry-breaking transitions and coherent-perfect-absorption/lasing points in slabs, spheres, and metasurfaces, including quasi-bound-state-in-the-continuum resonances where the required modulation strength can be minimal (Globosits et al., 3 Oct 2025).

Taken together, these results indicate that FEPs are no longer a narrow variant of static exceptional-point physics. They constitute a general framework for drive-enabled defectiveness, tunable quasi-energy topology, and non-Hermitian control across atomic, molecular, optical, magnonic, circuit, many-body, and open-quantum platforms.