Partial Floquet Transformation

• Partial Floquet transformation is a method that isolates dominant dynamics by separating fast micromotion from slowly varying effective evolution in time-dependent systems.
• It employs invariant-subspace restrictions, finite-order truncations, and gauge transformations to achieve reduced-order models and controlled approximations.
• These techniques enhance computational efficiency and provide theoretical insights for analyzing modulated periodic systems in quantum and classical settings.

Searching arXiv for the cited papers on partial Floquet transformation and related Floquet methods. A partial Floquet transformation is a family of reductions that isolate a tractable part of the dynamics of a time-dependent system without constructing the full Floquet representation. In the works considered here, the term refers to several related operations: factorizing the propagator of a modulated periodically driven open quantum system into micromotion and slowly varying effective evolution; exposing only selected dominant Floquet modes of a large linear time-periodic system; truncating the Floquet gauge or micromotion transformation at finite order in a high-frequency expansion; or absorbing only part of the time dependence by a gauge-type change of variables in a more general non-autonomous system (Dai et al., 2017, Bender et al., 31 Jul 2025, Xu et al., 2024, Gaeta et al., 17 Jun 2025, Wang et al., 2024).

1. Classical Floquet structure and the meaning of “partial”

For a linear time-periodic system

$x'(t)=A(t)x(t)+B(t)u(t), \qquad y(t)=C(t)^*x(t),$

with $A(t+T)=A(t)$, $B(t+T)=B(t)$, and $C(t+T)=C(t)$, the state-transition operator $\Phi(t,\tau)$ satisfies $\Phi(\tau,\tau)=I$ and $\partial_t\Phi(t,\tau)=A(t)\Phi(t,\tau)$. Classical Floquet theory asserts that there exists an invertible, $T$-periodic matrix $P(t)$ and a constant matrix $F$ such that the change of variables $A(t+T)=A(t)$0 yields

$A(t+T)=A(t)$1

with $A(t+T)=A(t)$2. Equivalently, if $A(t+T)=A(t)$3 is chosen so that $A(t+T)=A(t)$4, the Floquet multipliers, then $A(t+T)=A(t)$5 (Bender et al., 31 Jul 2025).

In periodically driven open quantum systems, an analogous factorization separates a periodic micromotion operator from a time-independent effective generator: $A(t+T)=A(t)$6 The partial constructions discussed in the present literature preserve this separation principle while relaxing one or more ingredients of the full theorem. The relaxation may take the form of slow modulation that breaks exact periodicity, restriction to a selected invariant subspace, truncation of the micromotion series at finite order, or a gauge transformation that leaves a residual time-dependent term rather than eliminating the full time dependence (Dai et al., 2017).

2. Generalized factorization for modulated periodic open systems

For an open quantum system governed by the time-local master equation

$A(t+T)=A(t)$7

the generalized Floquet theorem of Dai, Platero, and Hänggi assumes a generator of the form

$A(t+T)=A(t)$8

where $A(t+T)=A(t)$9, the phase $B(t+T)=B(t)$0 is fast, and the parameters $B(t+T)=B(t)$1 and possibly $B(t+T)=B(t)$2 vary slowly. The two-parameter propagator

$B(t+T)=B(t)$3

admits the factorization

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

With $B(t+T)=B(t)$5, this becomes

$B(t+T)=B(t)$6

where $B(t+T)=B(t)$7 carries the fast $B(t+T)=B(t)$8 micromotion and $B(t+T)=B(t)$9 governs the long-time evolution (Dai et al., 2017).

The construction proceeds by embedding the problem in an extended space with angle coordinate $C(t+T)=C(t)$0 and defining

$C(t+T)=C(t)$1

After Fourier expansion in $C(t+T)=C(t)$2, one obtains

$C(t+T)=C(t)$3

where $C(t+T)=C(t)$4 shifts the Fourier index and $C(t+T)=C(t)$5 is the number operator. One then seeks a time-dependent similarity transform $C(t+T)=C(t)$6 such that

$C(t+T)=C(t)$7

is block-diagonal in Fourier space: $C(t+T)=C(t)$8 Projection onto the zeroth Fourier block yields $C(t+T)=C(t)$9, while the residual nonzero-$\Phi(t,\tau)$0 blocks are absorbed into $\Phi(t,\tau)$1 (Dai et al., 2017).

The high-frequency expansion is organized as

$\Phi(t,\tau)$2

with $\Phi(t,\tau)$3. At zeroth order,

$\Phi(t,\tau)$4

The first and second orders generate commutator corrections involving the Fourier blocks $\Phi(t,\tau)$5, their time derivatives, and $\Phi(t,\tau)$6. In the laboratory frame,

$\Phi(t,\tau)$7

The expansion converges when the off-diagonal Fourier couplings $\Phi(t,\tau)$8 and their time-derivatives are $\Phi(t,\tau)$9, so the method rests on a separation between the fast drive period $\Phi(\tau,\tau)=I$0 and the slow evolution of $\Phi(\tau,\tau)=I$1 and $\Phi(\tau,\tau)=I$2. In the strictly periodic case with constant $\Phi(\tau,\tau)=I$3 and $\Phi(\tau,\tau)=I$4, the standard Floquet–Magnus series is recovered, with time-independent $\Phi(\tau,\tau)=I$5 and strictly periodic $\Phi(\tau,\tau)=I$6 (Dai et al., 2017).

The spin-$\Phi(\tau,\tau)=I$7 example in the paper illustrates the interpretation. For a qubit in a fast rotating magnetic field with dissipation,

$\Phi(\tau,\tau)=I$8

the effective generator describes slow precession plus dissipation, whereas the micromotion

$\Phi(\tau,\tau)=I$9

reproduces the fast spin nutation. The instantaneous steady state of $\partial_t\Phi(t,\tau)=A(t)\Phi(t,\tau)$0 tracks the slow ramp of $\partial_t\Phi(t,\tau)=A(t)\Phi(t,\tau)$1 almost adiabatically, while $\partial_t\Phi(t,\tau)=A(t)\Phi(t,\tau)$2 imprints small rapid oscillations (Dai et al., 2017).

3. Invariant-subspace partial transforms for large linear time-periodic systems

In large-order linear time-periodic systems, the full Floquet transform may be computationally prohibitive because forming $\partial_t\Phi(t,\tau)=A(t)\Phi(t,\tau)$3 and its logarithm is typically intractable, with complexity $\partial_t\Phi(t,\tau)=A(t)\Phi(t,\tau)$4 or worse. The construction of Beattie, Gugercin, and Mehrmann therefore targets only a low-dimensional dominant Floquet subspace of dimension $\partial_t\Phi(t,\tau)=A(t)\Phi(t,\tau)$5 rather than the entire $\partial_t\Phi(t,\tau)=A(t)\Phi(t,\tau)$6-dimensional Floquet basis (Bender et al., 31 Jul 2025).

Their formulation uses the unbounded operator

$\partial_t\Phi(t,\tau)=A(t)\Phi(t,\tau)$7

on the Hilbert space $\partial_t\Phi(t,\tau)=A(t)\Phi(t,\tau)$8 of $\partial_t\Phi(t,\tau)=A(t)\Phi(t,\tau)$9-periodic $T$0-functions. Classical Floquet theory implies that $T$1 has discrete spectrum

$T$2

with eigenfunctions $T$3, where $T$4 are the columns of $T$5. Selecting an index set $T$6 of size $T$7, one forms $T$8 from the corresponding $T$9-periodic eigenfunctions. Then

$P(t)$0

where $P(t)$1 is constant if a single $P(t)$2-branch is chosen. Splitting the state as

$P(t)$3

and projecting onto the $P(t)$4-columns yields the reduced partial-Floquet coordinate

$P(t)$5

The time-periodic structure is thereby transferred from the state dynamics into the input map associated with the selected modes (Bender et al., 31 Jul 2025).

The dominant subspace is identified without computing the full Floquet transform. The paper builds a frequency-domain proxy for the LTP transfer operators, namely the Harmonic Transfer Function or Principal Harmonics Vector, whose poles occur at Floquet exponents shifted by integer multiples of the base frequency. A scalar dominance measure is defined as

$P(t)$6

where $P(t)$7 are left-eigenfunctions of $P(t)$8. A Newton-type Dominant Pole Algorithm is then applied directly in the LTP setting,

$P(t)$9

and a subspace-accelerated SADPA variant accumulates Krylov-type trial functions $F$0 and $F$1, orthonormalizes them into bases $F$2 and $F$3, and extracts Ritz pairs after projection. Previously found poles and all of their integer-frequency shifts are deflated by removing their residue contributions from the proxy function. The resulting procedure costs $F$4 solves of periodic-coefficient linear ODEs, each solve costing $F$5 where $F$6 is the number of time-collocation points, rather than $F$7 (Bender et al., 31 Jul 2025).

Once $F$8 and a dual basis $F$9 satisfying $A(t+T)=A(t)$00 are available, a Petrov–Galerkin projection yields the reduced-order LTP model

$A(t+T)=A(t)$01

where

$A(t+T)=A(t)$02

is constant or block-triangular. The periodic time dependence then appears only in the input and output maps. In the lifted LTI representation, the $A(t+T)=A(t)$03-norm of the transfer-function error bounds the $A(t+T)=A(t)$04 output error: $A(t+T)=A(t)$05 The synthetic example in the paper uses a $A(t+T)=A(t)$06-dimensional LTP system with $A(t+T)=A(t)$07 slowly decaying eigenvalues in $A(t+T)=A(t)$08 and $A(t+T)=A(t)$09 rapidly decaying ones in $A(t+T)=A(t)$10. The LTP-SADPA algorithm computes the $A(t+T)=A(t)$11 most dominant $A(t+T)=A(t)$12 in $A(t+T)=A(t)$13 Newton iterations, producing a $A(t+T)=A(t)$14-dimensional reduced model with better than $A(t+T)=A(t)$15 relative $A(t+T)=A(t)$16 error and pointwise $A(t+T)=A(t)$17 error of order $A(t+T)=A(t)$18 (Bender et al., 31 Jul 2025).

4. Truncated micromotion gauges and order-by-order cancellation

A different use of partial Floquet transformation appears in perturbative Floquet engineering. For a time-periodic Hamiltonian $A(t+T)=A(t)$19, the one-period propagator

$A(t+T)=A(t)$20

defines a Floquet Hamiltonian $A(t+T)=A(t)$21 through

$A(t+T)=A(t)$22

Equivalently, a $A(t+T)=A(t)$23-periodic micromotion operator $A(t+T)=A(t)$24 with $A(t+T)=A(t)$25 may be introduced so that

$A(t+T)=A(t)$26

where $A(t+T)=A(t)$27 is time independent. The Floquet–Magnus expansion writes

$A(t+T)=A(t)$28

with $A(t+T)=A(t)$29. The corresponding gauge expansion is

$A(t+T)=A(t)$30

The $A(t+T)=A(t)$31 are chosen recursively so that the transformed Hamiltonian

$A(t+T)=A(t)$32

is time independent up to the required order. In practice,

$A(t+T)=A(t)$33

and periodicity $A(t+T)=A(t)$34 is enforced by integrating over one period. A full Floquet gauge would require summing all orders, whereas a partial Floquet transformation truncates both $A(t+T)=A(t)$35 and $A(t+T)=A(t)$36 at order $A(t+T)=A(t)$37. The resulting error in $A(t+T)=A(t)$38 is $A(t+T)=A(t)$39, so the stroboscopic quasienergies and eigenstates are correct to that order (Xu et al., 2024).

Xu and Guo use this truncation as an order-by-order cancellation scheme. Rather than constructing $A(t+T)=A(t)$40 explicitly, they add a small correction drive $A(t+T)=A(t)$41 of order $A(t+T)=A(t)$42 whose rotating-wave contribution cancels the undesired Floquet–Magnus term $A(t+T)=A(t)$43. The construction is expressed through the Non-commutative Fourier Transform: $A(t+T)=A(t)$44 or, in polar form,

$A(t+T)=A(t)$45

If $A(t+T)=A(t)$46 denotes the NcFT weight of the correction to be removed, the compensating drive is chosen as

$A(t+T)=A(t)$47

For the example $A(t+T)=A(t)$48, the first-order correction $A(t+T)=A(t)$49 cancels all $A(t+T)=A(t)$50 errors, yielding $A(t+T)=A(t)$51. In the $A(t+T)=A(t)$52-fold-symmetric multi-component cat-state example, the corrected quasienergies and states agree with the target up to $A(t+T)=A(t)$53 (Xu et al., 2024).

5. Gauge-theoretic extension beyond strict periodicity

A broader formulation replaces periodic Floquet structure by a time-dependent gauge transformation on a general smooth linear system

$A(t+T)=A(t)$54

Writing

$A(t+T)=A(t)$55

with $A(t+T)=A(t)$56 smooth and invertible, one obtains

$A(t+T)=A(t)$57

where

$A(t+T)=A(t)$58

If one wants an autonomous linear part $A(t+T)=A(t)$59 with constant $A(t+T)=A(t)$60, then $A(t+T)=A(t)$61 must satisfy the gauge-determining equation

$A(t+T)=A(t)$62

Classical Floquet theory is recovered when $A(t+T)=A(t)$63 is periodic and one chooses $A(t+T)=A(t)$64 in the factorization $A(t+T)=A(t)$65 (Gaeta et al., 17 Jun 2025).

In this setting, “partial Floquet” means that the chosen constant matrix $A(t+T)=A(t)$66 absorbs only part of the original time dependence. If $A(t+T)=A(t)$67 and one sets $A(t+T)=A(t)$68, then the transformed equation becomes

$A(t+T)=A(t)$69

with

$A(t+T)=A(t)$70

The residual term $A(t+T)=A(t)$71 measures the part of $A(t+T)=A(t)$72 not absorbed by the gauge transformation. This formulation is explicitly non-periodic and geometric: $A(t+T)=A(t)$73 is interpreted as a covariant derivative on the trivial bundle over time, and the gauge change $A(t+T)=A(t)$74 corresponds to a moving frame or vielbein (Gaeta et al., 17 Jun 2025).

The paper gives several concrete examples. In the $A(t+T)=A(t)$75 example,

$A(t+T)=A(t)$76

choosing $A(t+T)=A(t)$77 and $A(t+T)=A(t)$78 leads to $A(t+T)=A(t)$79, so the transformed system has constant linear part. In the related partial-removal example, the same rotation removes $A(t+T)=A(t)$80 from the linear dynamics, but a nonlinear radial term retains time dependence through $A(t+T)=A(t)$81. Further examples use explicit $A(t+T)=A(t)$82 gauges in $A(t+T)=A(t)$83 and $A(t+T)=A(t)$84 gauges for lifted Riccati equations, again converting non-autonomous systems into autonomous or partially autonomous ones (Gaeta et al., 17 Jun 2025).

6. Floquet–Schrieffer–Wolff block-diagonalization

Another closely related construction is the Floquet Schrieffer–Wolff transform, presented as a time-domain block-diagonalization of a periodically driven many-body Hamiltonian

$A(t+T)=A(t)$85

The method seeks a time-periodic unitary

$A(t+T)=A(t)$86

such that the transformed Hamiltonian

$A(t+T)=A(t)$87

has its oscillatory Fourier components removed up to a given perturbative order. Expanding

$A(t+T)=A(t)$88

the first-order cancellation condition leads, for each $A(t+T)=A(t)$89, to the operator-valued Sylvester equation

$A(t+T)=A(t)$90

Its formal solution can be written as an integral, and in the large-$A(t+T)=A(t)$91 regime it has the expansion

$A(t+T)=A(t)$92

Higher orders generate analogous Sylvester equations with sources built from commutators of lower-order terms (Wang et al., 2024).

The resulting effective Hamiltonian reproduces Floquet–Magnus in the high-frequency limit. Up to order $A(t+T)=A(t)$93,

$A(t+T)=A(t)$94

while the micromotion operator is encoded by

$A(t+T)=A(t)$95

so that

$A(t+T)=A(t)$96

The paper characterizes this as what one may call a “partial Floquet” block-diagonalization, because the oscillatory pieces are perturbatively eliminated and the remaining periodic structure is pushed into micromotion rather than removed exactly at all orders (Wang et al., 2024).

The driven single-band Fermi–Hubbard model provides the worked example: $A(t+T)=A(t)$97

$A(t+T)=A(t)$98

At second order, the effective Hamiltonian contains a renormalized nearest-neighbour hopping,

$A(t+T)=A(t)$99

and a correlated-hopping interaction proportional to

$B(t+T)=B(t)$00

The paper emphasizes that this correlated hopping appears already at $B(t+T)=B(t)$01 in FSWT, while a naive high-frequency expansion places it at much higher order. In the zero-doublon sector for $B(t+T)=B(t)$02, one recovers a superexchange form with

$B(t+T)=B(t)$03

The regime of validity is explicitly non-resonant: denominators such as $B(t+T)=B(t)$04 must remain far from zero, so one requires $B(t+T)=B(t)$05 (Wang et al., 2024).

7. Regimes of validity, conceptual distinctions, and scope

The cited literature does not present a single canonical definition of partial Floquet transformation. Instead, the phrase names a class of reductions that preserve the Floquet idea of separating micromotion from effective dynamics while relaxing exact periodicity, dimensional completeness, or all-order elimination of time dependence. In one case the construction is an exact generalized theorem for modulated open systems; in another it is an invariant-subspace restriction for large linear time-periodic systems; in another it is a finite-order truncation of a Floquet gauge; in another it is a gauge transform with residual $B(t+T)=B(t)$06; and in the Floquet–Schrieffer–Wolff setting it is a perturbative block-diagonalization in the time domain (Dai et al., 2017, Bender et al., 31 Jul 2025, Xu et al., 2024, Gaeta et al., 17 Jun 2025, Wang et al., 2024).

A common misconception is that “partial” necessarily means uncontrolled. The sources show several distinct control mechanisms. In the generalized open-system theorem, the factorization is exact under the stated modulated-periodic structure, while the associated high-frequency series is controlled by $B(t+T)=B(t)$07, their derivatives, and the separation between slow and fast scales (Dai et al., 2017). In the invariant-subspace approach, the reduced model inherits an $B(t+T)=B(t)$08-based output-error bound through the lifted LTI embedding (Bender et al., 31 Jul 2025). In the truncated Floquet–Magnus gauge, the error after an $B(t+T)=B(t)$09th-order partial transformation is $B(t+T)=B(t)$10 (Xu et al., 2024). In FSWT, validity is tied to non-resonant denominators and the perturbative organization in drive amplitude and, in the high-frequency regime, in $B(t+T)=B(t)$11 (Wang et al., 2024).

Another misconception is that Floquet reduction is inherently periodic. The gauge-theoretic formulation makes explicit that the classical Floquet factorization $B(t+T)=B(t)$12 is a special periodic solution of the more general matrix ODE

$B(t+T)=B(t)$13

so non-periodic gauge choices can still produce autonomous or partially autonomous transformed systems (Gaeta et al., 17 Jun 2025).

Taken together, these constructions place partial Floquet transformation at the intersection of micromotion separation, invariant-subspace extraction, perturbative effective dynamics, and gauge reduction. The recurring objective is not the universal elimination of time dependence, but the isolation of the part of the dynamics that is structurally dominant, computationally accessible, or asymptotically controlled.