In a time-periodically driven system like a qubit (e.g.,
H(t)=
H
1
+Cos[ωt]
H
2
), Floquet theory lets you express the state as a product of an exponential phase factor and a periodic function, whose Fourier expansion reveals how the drive
(
H
2
) creates sidebands in the dynamics. By truncating this expansion to include only the dominant harmonics—typically valid when the drive amplitude is small compared to the frequency,
||
H
2
||≪ω
—you reduce the problem to a finite-dimensional eigenvalue problem whose solution yields the quasi-energies and the corresponding Floquet states. These states provide an effective static description of the system over long times, allowing you to reconstruct the time evolution and compare it with direct numerical simulations, thereby confirming the validity of the approximation and giving insights into phenomena like dynamic stabilization and engineered effective Hamiltonians.
The reader can study the following references for more details on this topic:
◼
  • https://arxiv.org/pdf/1407.4803 : Bukov, M., D’Alessio, L., & Polkovnikov, A. (2015). Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to Floquet engineering. Advances in Physics, 64(2), 139–226. https://doi.org/10.1080/00018732.2015.1055918
    • ◼
  • https://arxiv.org/pdf/1804.03212 : Oka, T., & Kitamura, S. (2019). Floquet Engineering of Quantum Materials. Annual Review of Condensed Matter Physics, 10, 387-408. doi:10.1146/annurev-conmatphys-031218-013423. https://doi.org/10.1146/annurev-conmatphys-031218-013423
    • Consider time-dependent Schrodinger equation:
    
    ∂
    t
    |ψ(t)〉=H(t)|ψ(t)〉
    (
    1
    )
    with a periodic Hamiltonian
    H(t+T)=H(t)
    . Floquet’s theorem tells us that, due to the periodicity of
    H(t)
    , the solutions of the Schrödinger equation can be expressed as
    |ψ(t)〉=
    -ϵt
    
    |ϕ(t)〉
    (
    2
    )
    where
    -ϵt
    
    is an exponential phase factor with quasi-energy
    ϵ
    , and
    |ϕ(t)〉
    is a time-periodic function with the same period as the Hamiltonian
    |ϕ(t+T)〉=|ϕ(t)〉
    . Plugging this into Schrodinger equation yields:
    (H(t)-
    ∂
    t
    )|ϕ(t)〉=ϵ|ϕ(t)〉
    (
    3
    )
    This is the key eigenvalue equation in Floquet theory.
    Since
    |ϕ(t)〉
    is periodic, with
    T=2π/ω
    , one can write
    |ϕ(t)〉=
    ∞
    ∑
    n=-∞
    -nωt
    
    |
    ϕ
    n
    〉
    (
    4
    )
    with time-independent coefficients
    |
    ϕ
    n
    〉
    corresponds to a Fourier mode with a frequency shift of
    nω
    and they are vectors in the system’s Hilbert space. Note
    〈
    ϕ
    n
    |
    ϕ
    m
    〉=
    δ
    n,m
    p
    n
    with
    p
    n
    represents the weight (or probability) of the
    n
    -th harmonic in the Floquet state, i.e.
    ∑
    n
    p
    n
    =1
    .
    Eq.(4) allows one to map the original time-dependent problem to an effective time-independent problem in an extended Hilbert space (sometimes called Sambe space), where techniques from static quantum mechanics can be applied. Plugging Eq(4) into Eq.(3) yields
    (H(t)-
    ∂
    t
    )
    ∞
    ∑
    n=-∞
    -nωt
    
    |
    ϕ
    n
    〉=ϵ
    ∞
    ∑
    n=-∞
    -nωt
    
    |
    ϕ
    n
    〉
    (
    5
    )
    For the periodic Hamiltonian, let’s consider the following form, where
    H
    2
    is usually called as the drive term:
    H(t)=
    H
    1
    +Cos[ωt]
    H
    2
    =
    H
    1
    +
    1
    2
    (
    ωt
    
    +
    -ωt
    
    )
    H
    2
    (
    6
    )
    Putting Eq.(6) into Eq.(5), one gets:
    H(t)|ϕ(t)〉=
    ∞
    ∑
    n=-∞
    -nωt
    
    H
    1
    +
    1
    2
    -(n-1)ωt
    
    H
    2
    +
    1
    2
    -(n+1)ωt
    
    H
    2
    |
    ϕ
    n
    〉=
    ∞
    ∑
    n=-∞
    -nωt
    
    H
    1
    
    ϕ
    n
    +
    1
    2
    H
    2
    
    ϕ
    n-1
    +
    1
    2
    H
    2
    
    ϕ
    n+1
    
    (
    7
    )
    Therefore, for Floquet Hamiltonian one gets:
    (H(t)-
    ∂
    t
    )|ϕ(t)〉=
    ∞
    ∑
    n=-∞
    -nωt
    
    (
    H
    1
    -nω)
    ϕ
    n
    +
    1
    2
    H
    2
    (|
    ϕ
    n-1
    〉+|
    ϕ
    n+1
    〉)
    (
    8
    )
    ∞
    ∑
    n=-∞
    -nωt
    
    (
    H
    1
    -nω)
    ϕ
    n
    +
    1
    2
    H
    2
    (|
    ϕ
    n-1
    〉+|
    ϕ
    n+1
    〉)=ϵ
    ∞
    ∑
    n=-∞
    -nωt
    
    |
    ϕ
    n
    〉
    (
    9
    )
    Collecting terms that multiply
    -nωt
    
    gives us a set of coupled equations for the Fourier components:
    (
    H
    1
    -nω)|
    ϕ
    n
    〉+
    1
    2
    H
    2
    (|
    ϕ
    n-1
    〉+|
    ϕ
    n+1
    〉)=ϵ|
    ϕ
    n
    〉
    (
    10
    )
    Note the block matrix structure of
    ℋ
    F
    with diagonal blocks
    H
    1
    -nω
    and off-diagonals
    1
    2
    H
    2
    connecting
    n
    with
    n±1
    terms.
    The drive term of Hamiltonian,
    H
    2
    , couples
    |
    ϕ
    n
    〉
    with terms
    |
    ϕ
    n±1
    〉
    . If we assume
    H
    1
    is stronger than
    H
    2
    , then among all Fourier modes,
    n=0
    should be dominant one. This justifies truncating the Fourier series to the limited set of harmonics
    n=-1,0,+1
    only. Thus, one can write:
    Φ=
    ϕ
    -1
    ϕ
    0
    ϕ
    +1
    with
    ℋ
    F
    Φ=ϵΦ
    (
    11
    )
    where the Floquet Hamiltonian is
    ℋ
    F
    =
    H
    1
    +ω
    1
    2
    H
    2
    0
    1
    2
    H
    2
    H
    1
    1
    2
    H
    2
    0
    1
    2
    H
    2
    H
    1
    -ω
    (
    12
    )
    In a similar manner, if the truncation is limited to
    
    terms, one finds:
    Φ=
    ϕ
    -
    ⋮
    ϕ
    0
    ⋮
    ϕ
    +
    and
    ℋ
    F
    =
    H
    1
    +ω
    1
    2
    H
    2
    0
    0
    1
    2
    H
    2
    ⋱
    ⋱
    0
    0
    ⋱
    ⋱
    1
    2
    H
    2
    0
    0
    1
    2
    H
    2
    H
    1
    -ω
    (
    13
    )
    Let’s assume that
    H
    1,2
    are Hamiltonian operators of a qubit. Note the truncation is limited to
    ±,±(-1),…,±1,0
    (
    2+1
    terms). This means
    ℋ
    F
    is a
    (2+1)×(2+1)
    Hermitian matrix with
    2+1
    eigenvectors. Let’s label them with the index
    j
    . Thus we can write Eq.(4) as:
    
    (j)
    ϕ
    (t)=
    ωt
    
    
    (j)
    ϕ
    -
    +
    (-1)ωt
    
    
    (j)
    ϕ
    -(-1)
    +…+
    (j)
    ϕ
    0
    +…+
    -(-1)ωt
    
    
    (j)
    ϕ
    -1
    +
    -ωt
    
    
    (j)
    ϕ
    +
    
    (
    14
    )
    And likewise, Eq.(2) reads as follow
    
    (j)
    ψ
    (t)=
    -
    ϵ
    j
    t
    
    
    ωt
    
    
    (j)
    ϕ
    -
    +
    (-1)ωt
    
    
    (j)
    ϕ
    -(-1)
    +…+
    (j)
    ϕ
    0
    +…+
    -(-1)ωt
    
    
    (j)
    ϕ
    -1
    +
    -ωt
    
    
    (j)
    ϕ
    +
    
    (
    15
    )
    with
    ϵ
    j
    the corresponding
    j
    -th eigenvalue (i.e. quasi-energy). These states, which are also called as Floquet eigenstates, form a over-complete eigen-basis at any time:
    1
    2+1
    2+1
    ∑
    j=1
    
    (j)
    ψ
    (t)
    (j)
    ψ
    (t)=I
    (
    16
    )
    Install quantum framework:
    In[]:=
    PacletInstall[CloudObject["https://wolfr.am/DevWQCF"],ForceVersionInstall->True]​​<<Wolfram`QuantumFramework`
    Out[]=
    PacletObject
    Name: Wolfram/QuantumFramework
    Version: 1.5.0
    
    Consider a time-dependent Hamiltonian as
    H=Δ
    σ
    z
    +ACos[ωt]
    σ
    x
    which means
    H
    1
    =Δ
    σ
    z
    and
    H
    2
    =A
    σ
    x
    In[]:=
    ClearAll[h,Δ,ω,A,t]​​h[t_]=ΔQuantumOperator["Z"]+ACos[ωt]QuantumOperator["X"]
    Out[]=
    QuantumOperator
    Pure map
    ​
    Dimension: 2→2
    Order: {1}→{1}
    
    Check it is periodic:
    In[]:=
    $Assumptions={ω,t}∈Reals;​​h[t+2π/ω]==h[t]//FullSimplify
    Out[]=
    True
    Create the Floquet Hamiltonian:
    In[]:=
    ClearAll[hF]​​hF[{n_,ω_},{h1_,h2_}]:=ReplacePart[DiagonalMatrix[ConstantArray[0,2n+1]],{{i_,i_}->(n+1-i)ωIdentityMatrix[2]+h1,{i_,j_}/;Abs[i-j]==1->h2,{_,_}->DiagonalMatrix[{0,0}]}]
    For the case of
    =3
    , show the Floquet Hamiltonian
    Out[]=
    Δ+3ω
    0
    0
    A
    2
    0
    0
    0
    0
    0
    0
    0
    0
    0
    0
    0
    -Δ+3ω
    A
    2
    0
    0
    0
    0
    0
    0
    0
    0
    0
    0
    0
    0
    A
    2
    Δ+2ω
    0
    0
    A
    2
    0
    0
    0
    0
    0
    0
    0
    0
    A
    2
    0
    0
    -Δ+2ω
    A
    2
    0
    0
    0
    0
    0
    0
    0
    0
    0
    0
    0
    0
    A
    2
    Δ+ω
    0
    0
    A
    2
    0
    0
    0
    0
    0
    0
    0
    0
    A
    2
    0
    0
    -Δ+ω
    A
    2
    0
    0
    0
    0
    0
    0
    0
    0
    0
    0
    0
    0
    A
    2
    Δ
    0
    0
    A
    2
    0
    0
    0
    0
    0
    0
    0
    0
    A
    2
    0
    0
    -Δ
    A
    2
    0
    0
    0
    0
    0
    0
    0
    0
    0
    0
    0
    0
    A
    2
    Δ-ω
    0
    0
    A
    2
    0
    0
    0
    0
    0
    0
    0
    0
    A
    2
    0
    0
    -Δ-ω
    A
    2
    0
    0
    0
    0
    0
    0
    0
    0
    0
    0
    0
    0
    A
    2
    Δ-2ω
    0
    0
    A
    2
    0
    0
    0
    0
    0
    0
    0
    0
    A
    2
    0
    0
    -Δ-2ω
    A
    2
    0
    0
    0
    0
    0
    0
    0
    0
    0
    0
    0
    0
    A
    2
    Δ-3ω
    0
    0
    0
    0
    0
    0
    0
    0
    0
    0
    0
    A
    2
    0
    0
    -Δ-3ω
    Assign some numerical value to parameters of Hamiltonian:
    In[]:=
    Δ=2;ω=17;A=3;
    Set the initial state for simulation of the time evolution:
    Simulate the time evolution:
    Check the numerical deviation from norm-preserving for dynamical simulation:
    Find the final state from Floquet approach:
    Compare simulation vs Floquet approximation:
    A function to visualize the above plot for different truncation:
    Compare the inner product of the simulated states with the states obtained Floquet approach with different truncation:

    CITE THIS NOTEBOOK

    Floquet theory in quantum mechanics and Floquet states​
    by Mohammad Bahrami​
    Wolfram Community, STAFF PICKS, March 31, 2025
    ​https://community.wolfram.com/groups/-/m/t/3435831