Floquet Synthetic Dimensions in Driven Systems

• Floquet synthetic dimensions are additional coordinates generated by periodic or quasiperiodic drives, effectively mapping a d-dimensional system to a higher-dimensional lattice.
• They enable the engineering of synthetic gauge fields and topological phases, observed in platforms like photonics, cold atoms, and magnonics.
• The framework uses Fourier harmonics and drive phases as lattice sites to control transport, energy pumping, and dynamical responses in advanced physical setups.

Floquet synthetic dimensions are additional coordinates generated by periodic or quasiperiodic driving, such that a periodically driven $d$-dimensional system may be treated as a $(d+1)$-dimensional system and, with $d_{\rm syn}$ incommensurate frequencies, as an effective $(d+d_{\rm syn})$-dimensional Floquet problem (Baum et al., 2017, Kim et al., 2022). In this enlarged description, Fourier harmonics, drive phases, frequency modes, time bins, parameter coordinates, or operator-Krylov indices can act as lattice sites, while periodic driving supplies hopping, Peierls phases, effective electric fields, and quasienergy structure. The resulting framework has been used to realize synthetic gauge fields, Chern bands, isolated topological surface theories, anomalous edge and corner responses, quantized energy pumping, and driven-dissipative transport in photonic, magnonic, cold-atom, and operator-space settings (Sriram et al., 2024, Pishehvar et al., 29 Jun 2026, Yeh et al., 2023).

1. Mathematical framework and synthetic coordinates

The canonical starting point is the Floquet eigenproblem

$$\bigl(H(t)-i\partial_t\bigr)|u_\alpha(t)\rangle=\varepsilon_\alpha |u_\alpha(t)\rangle,$$

with $H(t+T)=H(t)$ and $T=2\pi/\omega$. Expanding $H(t)$ and $|u_\alpha(t)\rangle$ in Fourier harmonics converts the time-dependent problem into a static Sambe-space lattice with matrix elements of the form

$$(\mathcal H_F)_{mn}=H_{m-n}+m\hbar\omega\,\delta_{mn},$$

so that the harmonic index $(d+1)$0 becomes a synthetic frequency coordinate and $(d+1)$1 acts as a linear potential along that direction (Wang et al., 2023). In a low-frequency regime this tilt can be weak enough that the synthetic direction is treated as approximately translationally invariant, whereas in a high-frequency regime it is more natural to integrate out off-resonant harmonics and work with an effective Floquet Hamiltonian (Cai et al., 2021).

Quasiperiodic driving sharpens this construction. For a $(d+1)$2-dimensional system with $(d+1)$3 incommensurate phases $(d+1)$4, the operators $(d+1)$5 and $(d+1)$6 identify the drive phases as synthetic quasimomenta and the Fourier indices as synthetic coordinates, yielding an exact mapping to an enlarged $(d+1)$7-dimensional Floquet problem (Kim et al., 2022). By contrast, a commensurate two-tone drive at $(d+1)$8 and $(d+1)$9 produces one Floquet ladder with multiple hopping ranges: the $d_{\rm syn}$0 tone connects $d_{\rm syn}$1, while the $d_{\rm syn}$2 tone connects $d_{\rm syn}$3 (Wang et al., 2023). This distinction is central: a commensurate multifrequency problem has a structured single synthetic frequency dimension, whereas incommensurate drives generate exact higher-dimensional synthetic lattices.

The literature also shows that the synthetic coordinate need not be the harmonic index alone. In non-Markovian Floquet theory, memory kernels generate $d_{\rm syn}$4-dependent onsite terms $d_{\rm syn}$5 in harmonic space and thereby create synthetic potentials and effective edges (Baum et al., 2017). In a periodically driven Raman lattice, discretized frequency components play the role of an additional synthetic dimension at low frequency, while at high frequency the same model is better described by a chiral effective Floquet Hamiltonian (Cai et al., 2021). This suggests that Floquet synthetic dimensions are best understood as a family of enlarged-space constructions rather than a single formalism.

2. Physical implementations and platform diversity

A major implementation route uses frequency modes as synthetic sites. In dynamically modulated photonics, resonator modes $d_{\rm syn}$6 are coupled by modulation at frequency $d_{\rm syn}$7, producing resonant mode conversion $d_{\rm syn}$8 and a synthetic tight-binding Hamiltonian

$d_{\rm syn}$9

In a non-Hermitian physical-synthetic photonic lattice, the physical coordinate is a resonator index $(d+d_{\rm syn})$0, the synthetic coordinate is the frequency index $(d+d_{\rm syn})$1, and each site $(d+d_{\rm syn})$2 carries two internal modes $(d+d_{\rm syn})$3 and $(d+d_{\rm syn})$4 (Ning et al., 5 Jun 2026). Closely related ring-resonator architectures modulated at the free spectral range realize a mixed $(d+d_{\rm syn})$5 lattice, where $(d+d_{\rm syn})$6 is the ring index and $(d+d_{\rm syn})$7 the longitudinal-mode index; spatially varying modulation phases map the device to a Harper–Hofstadter model and support a topological synthetic-space edge mode (Yang et al., 2021).

A distinct photonic route replaces explicit lattice-space construction by direct $(d+d_{\rm syn})$8-space engineering. In a driven-dissipative photonic molecule of two coupled rings, the effective two-level Hamiltonian

$(d+d_{\rm syn})$9

is programmed so that quasiperiodic phases $$\bigl(H(t)-i\partial_t\bigr)|u_\alpha(t)\rangle=\varepsilon_\alpha |u_\alpha(t)\rangle,$$0 reproduce the Bloch Hamiltonian of the Haldane or brick-wall Haldane model (Sriram et al., 2024). In that setting, non-square lattice geometry is transferred from physical fabrication to the engineering of complex Floquet drive signals.

Synthetic dimensions have also been realized in time-multiplexed lattices. In a two-loop fiber architecture, pulse arrival times define a 2D synthetic time lattice with amplitudes $$\bigl(H(t)-i\partial_t\bigr)|u_\alpha(t)\rangle=\varepsilon_\alpha |u_\alpha(t)\rangle,$$1, discrete-time step evolution, and complex quasienergies $$\bigl(H(t)-i\partial_t\bigr)|u_\alpha(t)\rangle=\varepsilon_\alpha |u_\alpha(t)\rangle,$$2 (Zheng et al., 2024). In magnonics, discrete standing-wave resonances of a single YIG device become synthetic sites, while periodic longitudinal-field modulation at $$\bigl(H(t)-i\partial_t\bigr)|u_\alpha(t)\rangle=\varepsilon_\alpha |u_\alpha(t)\rangle,$$3 couples modes whose spacing is close to $$\bigl(H(t)-i\partial_t\bigr)|u_\alpha(t)\rangle=\varepsilon_\alpha |u_\alpha(t)\rangle,$$4, producing an effective tight-binding Hamiltonian

$$\bigl(H(t)-i\partial_t\bigr)|u_\alpha(t)\rangle=\varepsilon_\alpha |u_\alpha(t)\rangle,$$5

in mode space (Pishehvar et al., 29 Jun 2026).

The platform notion has broadened further. A finite spin-$$\bigl(H(t)-i\partial_t\bigr)|u_\alpha(t)\rangle=\varepsilon_\alpha |u_\alpha(t)\rangle,$$6 manifold with basis $$\bigl(H(t)-i\partial_t\bigr)|u_\alpha(t)\rangle=\varepsilon_\alpha |u_\alpha(t)\rangle,$$7, $$\bigl(H(t)-i\partial_t\bigr)|u_\alpha(t)\rangle=\varepsilon_\alpha |u_\alpha(t)\rangle,$$8, forms a synthetic spin chain; periodic driving then adds a Floquet-Sambe harmonic coordinate $$\bigl(H(t)-i\partial_t\bigr)|u_\alpha(t)\rangle=\varepsilon_\alpha |u_\alpha(t)\rangle,$$9, giving a composite lattice $H(t+T)=H(t)$0 (Cao et al., 13 Jun 2026). Symmetry-breaking parameters such as polarization components $H(t+T)=H(t)$1 can define synthetic coordinates in Floquet symmetry analysis (Tzur et al., 2021). Even operator dynamics admits this language: the operator Krylov basis generated by repeated stroboscopic Floquet evolution defines a one-dimensional emergent synthetic lattice whose sites are orthonormal Krylov operators and whose couplings are set by Krylov angles $H(t+T)=H(t)$2 (Yeh et al., 2023).

3. Band topology, surface theories, and gapless phases

Floquet synthetic dimensions support a wide range of topological phases. In quasiperiodically driven quantum walks, a $H(t+T)=H(t)$3-dimensional protocol realizes a 3D Floquet metal in the same universality class as the surface of a 4D class A quantum Hall insulator, with winding number

$H(t+T)=H(t)$4

for the explicit model studied (Kim et al., 2022). The same work gives a $H(t+T)=H(t)$5-dimensional simulator for the surface of a 3D class AII topological insulator, with $H(t+T)=H(t)$6 invariant $H(t+T)=H(t)$7 (Kim et al., 2022). These constructions are notable because they engineer topological surface states in isolation, without a supporting higher-dimensional bulk in the actual device.

Commensurate two-tone driving provides a different route to topological band engineering. In a 2D lattice driven at $H(t+T)=H(t)$8 and $H(t+T)=H(t)$9, synthetic-frequency path interference between direct $T=2\pi/\omega$0 hops and composite $T=2\pi/\omega$1 hops produces competing $T=2\pi/\omega$2 and $T=2\pi/\omega$3 Chern phases, and transitions can be crossed by varying relative drive phases alone (Wang et al., 2023). In non-square Floquet synthetic dimensions, direct $T=2\pi/\omega$4-space engineering yields the Haldane and brick-wall Haldane models, with phase boundaries

$T=2\pi/\omega$5

for the two geometries, respectively (Sriram et al., 2024).

Gapless topology is more delicate. A one-dimensional Floquet system with a periodic control parameter $T=2\pi/\omega$6 can be treated as a 2D bandstructure in $T=2\pi/\omega$7 and realizes a Floquet semimetal with Floquet-band holonomy: after $T=2\pi/\omega$8, each band is carried into a different band rather than returning to itself (Zhou et al., 2016). A related mixed $T=2\pi/\omega$9-dimensional Weyl construction generated by two incommensurate drives realizes Weyl points in the mixed Floquet band structure and yields a momentum-resolved Chern number

$H(t)$0

but the total real-space response does not reproduce the static Weyl-semimetal phase diagram and instead follows an effective Rice–Mele-type pumping structure (Vinjamuri et al., 18 Jun 2026). This indicates that fully gapped topological insulators translate more straightforwardly to Floquet synthetic dimensions than gapless semimetals.

4. Boundary phenomena, anomalous responses, and non-Hermitian extensions

Boundary physics in Floquet synthetic dimensions is unusually rich. In the Floquet semimetal with synthetic parameter $H(t)$1, open boundaries support anomalous chiral edge modes localized only at one edge, winding around the entire quasienergy Brillouin zone, and yielding quantized or half-quantized pumping along the synthetic dimension according to

$H(t)$2

with $H(t)$3 for type-I modes and $H(t)$4 for one branch of type-II modes (Zhou et al., 2016). In the memory-engineered synthetic crystals of Baum and Refael, the memory kernel screens the usual linear $H(t)$5 tilt in harmonic space and creates effective topological-trivial interfaces at $H(t)$6, producing synthetic-dimension boundary states and chiral wave packets in the combined real-plus-synthetic lattice (Baum et al., 2017).

Non-Hermitian synthetic dimensions introduce further structure. In a non-Hermitian Floquet photonic lattice formed by a physical resonator coordinate and a synthetic frequency coordinate, a two-step modulation protocol realizes anomalous corner pairs at quasienergies $H(t)$7 and $H(t)$8 with three distinct layers of physics: topological existence, skin-effect-selected localization, and flux-controlled optical visibility (Ning et al., 5 Jun 2026). The higher-order corner diagnostic is

$H(t)$9

but this predicts existence only; the imaginary gauge fields $|u_\alpha(t)\rangle$0 determine where right eigenmodes accumulate, while the real flux $|u_\alpha(t)\rangle$1 controls the local interference matrix element $|u_\alpha(t)\rangle$2 that sets whether the doubled-period response is bright, skin-dark, or flux-dark (Ning et al., 5 Jun 2026). The same complex gauge also tunes an exceptional point of the two-period corner propagator, where the $|u_\alpha(t)\rangle$3 period-doubled sign alternation persists but the envelope becomes algebraic because of a Jordan block (Ning et al., 5 Jun 2026).

A complementary non-Hermitian route uses a time-multiplexed 2D synthetic time lattice. There, direction-dependent gain and loss generate complex-quasienergy winding, and gluing together four regions with opposite winding patterns produces a corner-localized non-Hermitian skin mode (Zheng et al., 2024). Because the intensity modulators are programmable at each step, one can realize spatial mode tapering, sequential non-Hermiticity on-off switching, dynamical corner-state relocation, and light steering, with robustness against intensity-modulation disorder up to a threshold $|u_\alpha(t)\rangle$4 (Zheng et al., 2024).

5. Transport, pumping, and dynamical diagnostics

Transport in Floquet synthetic dimensions is often measured not by spatial current but by energy flow between drives or by spreading in the physical coordinate. In the synthetic-surface simulators based on incommensurate quantum walks, the wave-packet spread

$|u_\alpha(t)\rangle$5

distinguishes topological metallicity from localization: the class A $|u_\alpha(t)\rangle$6 model exhibits $|u_\alpha(t)\rangle$7, while a trivial comparison model shows a metal-insulator transition and critical scaling $|u_\alpha(t)\rangle$8 (Kim et al., 2022).

In Haldane-type Floquet synthetic dimensions realized by a driven-dissipative photonic molecule, the work done by drive $|u_\alpha(t)\rangle$9,

$$(\mathcal H_F)_{mn}=H_{m-n}+m\hbar\omega\,\delta_{mn},$$0

reveals topological pumping. For the brick-wall and Haldane constructions, the net pumping is approximately twice the Chern number because next-nearest-neighbor pathways also contribute: $$(\mathcal H_F)_{mn}=H_{m-n}+m\hbar\omega\,\delta_{mn},$$1 in the topological regime (Sriram et al., 2024). In mixed Floquet Weyl lattices, the momentum-resolved energy transfer obeys

$$(\mathcal H_F)_{mn}=H_{m-n}+m\hbar\omega\,\delta_{mn},$$2

so drive-to-drive power conversion measures the $$(\mathcal H_F)_{mn}=H_{m-n}+m\hbar\omega\,\delta_{mn},$$3-resolved Chern number and therefore the separation of Weyl nodes, even though the total response collapses to a one-dimensional pump invariant (Vinjamuri et al., 18 Jun 2026).

Several works emphasize dynamical observables that are specific to Floquet settings. In the non-Hermitian higher-order photonic lattice, a coherent superposition of $$(\mathcal H_F)_{mn}=H_{m-n}+m\hbar\omega\,\delta_{mn},$$4 and $$(\mathcal H_F)_{mn}=H_{m-n}+m\hbar\omega\,\delta_{mn},$$5 corner modes yields a doubled-period local signal measured by a balanced interferometric observable $$(\mathcal H_F)_{mn}=H_{m-n}+m\hbar\omega\,\delta_{mn},$$6, with $$(\mathcal H_F)_{mn}=H_{m-n}+m\hbar\omega\,\delta_{mn},$$7 amplitude controlled by $$(\mathcal H_F)_{mn}=H_{m-n}+m\hbar\omega\,\delta_{mn},$$8 rather than by mode existence alone (Ning et al., 5 Jun 2026). In the periodically driven Raman lattice, Floquet DQPTs are diagnosed by zeros of the Loschmidt amplitude $$(\mathcal H_F)_{mn}=H_{m-n}+m\hbar\omega\,\delta_{mn},$$9, the rate function

$(d+1)$00

and a dynamic topological order parameter constructed from the Pancharatnam geometric phase (Cai et al., 2021). In the magnonic synthetic lattice, detuning-induced synthetic electric fields generate spectral Bloch oscillations, directly observed as oscillatory depletion and return of amplitude on a selected mode (Pishehvar et al., 29 Jun 2026).

Operator-space diagnostics supply yet another viewpoint. For any Hermitian operator under any Floquet unitary, the infinite-temperature autocorrelation

$(d+1)$01

is exactly the return amplitude at the left edge of a universal synthetic Krylov chain, $(d+1)$02, whose topology is that of an inhomogeneous Floquet transverse-field Ising model with possible $(d+1)$03- and $(d+1)$04-edge modes (Yeh et al., 2023). This suggests that Floquet synthetic dimensions can encode not only single-particle transport but also universal features of operator growth.

6. Conceptual distinctions, limitations, and scope

The literature makes clear that “Floquet synthetic dimensions” names a broader class of constructions than a single mode-frequency ladder. Harmonic indices in Sambe space, incommensurate drive Fourier coordinates, resonator frequency modes, time bins in long fiber loops, synthetic spin states, symmetry-breaking parameters, and operator-Krylov indices have all been used as synthetic coordinates (Zheng et al., 2024, Cao et al., 13 Jun 2026, Tzur et al., 2021, Yeh et al., 2023). A common misconception is therefore to equate the subject only with frequency-sideband lattices in modulated resonators.

A second distinction concerns exactness. The Sambe-space mapping itself is exact, but treating the synthetic direction as translationally invariant can be approximate because the term $(d+1)$05 acts as a uniform synthetic electric field. In the periodically driven Raman lattice, the effective 2D topological-insulator description in $(d+1)$06 requires $(d+1)$07; at larger $(d+1)$08, the high-frequency effective Floquet Hamiltonian becomes the appropriate description (Cai et al., 2021). Likewise, in the 3D Floquet Chern-vector insulator based on phase-delayed periodic hopping, the harmonic sectors appear in the extended Floquet matrix, but the final interpretation integrates them out and uses them to generate a 3D effective gauge field rather than retaining them as an explicit synthetic lattice direction (Ma et al., 2024).

A third distinction concerns observability. In non-Hermitian synthetic dimensions, topological existence, localization, and measurement visibility need not coincide. The non-Hermitian physical-synthetic photonic lattice separates these notions cleanly: non-Bloch invariants determine whether anomalous $(d+1)$09 corner pairs exist, imaginary gauge fields determine where right eigenmodes accumulate, and the real flux determines whether a local detector sees their doubled-period interference (Ning et al., 5 Jun 2026). This suggests that “topological mode existence” is not, by itself, a complete operational characterization of a driven synthetic system.

A plausible implication is that the field is best understood as a hierarchy of enlarged-space techniques for engineering topology, transport, and dynamical response in degrees of freedom that are easier to modulate than to fabricate. The strongest common thread across the literature is not a single geometry, but the systematic use of periodic or quasiperiodic driving to turn non-spatial labels into lattice coordinates with controllable hopping, gauge structure, and boundary phenomena.