General results of reducibility for first-order differential equations with quasi-periodic coefficients are not to be expected [39], [40], but the situation is better understood if the driving amplitude is small as compared to the norm |**ω**| of the frequency vector. Using unitless variables *τ*=|**ω**|*t*|, which are common in the mathematical literature, the corresponding differential equation reads

$$i{\partial}_{\tau}U\mathrm{(}\tau \mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{H\mathrm{(}\tau \mathrm{)}}{\mathrm{|}\omega \mathrm{|}}U\mathrm{(}\tau \mathrm{}\mathrm{)}\mathrm{.}$$(31)

The regime of the new rescaled Hamiltonian, which is quasi-periodic with frequencies **ω**/|**ω**|, is referred to as close-to-constant, whereas the term fast driving is more common in the physics literature. In this regime and under suitable hypothesis of regularity, non-degeneracy, and strong nonresonance of the frequencies, reducible and non-reducible systems are mixed like Diophantine and Liouvillean numbers; most systems are reducible, but non-reducible ones are dense [37], [41], [42], [54]. Moreover, the generalised Floquet decomposition of the time-evolution operator in (2) can be found with any given accuracy, for |**ω**|^{−1} that is sufficiently small, provided it exists [55], [56], [57]. In practice, this is often done through an expansion in terms of powers of |**ω**|^{−1}.

In this section, we will derive a generalisation of the Floquet-Magnus expansion [52], [58], [59] to quasi-periodic systems and provide a perturbative exponential expansion of the time-evolution operator with the desired Floquet representation. This will allow us to identify *H*_{Q} as the effective Hamiltonian that captures the dynamics of the system in a suitable fast-driving regime.

We start the derivation by reproducing the steps of the regular Floquet-Magnus expansion [52] and introducing the desired decomposition of the time-evolution operator

$$U\mathrm{(}t\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{U}_{Q}^{\u2020}\mathrm{(}t\mathrm{)}{e}^{-i{H}_{Q}t}$$(32)

into the Schrödinger equation *i*∂_{t}*U*(*t*)=*H*(*t*)*U*(*t*), which yields the differential equation

$$i{\partial}_{t}{U}_{Q}^{\u2020}\mathrm{(}t\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}H\mathrm{(}t\mathrm{)}{U}_{Q}^{\u2020}\mathrm{(}t\mathrm{)}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{U}_{Q}^{\u2020}\mathrm{(}t\mathrm{)}{H}_{Q}\mathrm{.}$$(33)

Then we define the quasi-periodic Hermitian operator *Q*(*t*) as the generator of the quasi-periodic unitary *U*_{Q}(*t*) via the relation

$${U}_{Q}\mathrm{(}t\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{e}^{iQ\mathrm{(}t\mathrm{)}}\mathrm{.}$$(34)

Introducing the expression in (34) into (33) and using a power series expansion for the differential of the exponential [52], [59], [60], one obtains the non-linear differential equation [52]

$${\partial}_{t}Q\mathrm{(}t\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\displaystyle \sum _{k\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0}^{\infty}}\frac{{B}_{k}}{k\mathrm{!}}{\mathrm{(}-i\mathrm{)}}^{k}{\text{ad}}_{Q\mathrm{(}t\mathrm{)}}^{k}\mathrm{(}H\mathrm{(}t\mathrm{)}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\mathrm{(}-1\mathrm{)}\mathrm{}}^{k\text{\hspace{0.17em}}+\text{\hspace{0.17em}}1}{H}_{Q}\mathrm{)}\mathrm{,}$$(35)

where *B*_{k} denotes the Bernoulli numbers and ad is the adjoint action defined via ${\text{ad}}_{A}^{k}B\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\mathrm{[}A\mathrm{,}{\text{\hspace{0.17em}ad}}_{A}^{k\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}B\mathrm{]}$ for *k* ≥ 1 and ${\text{ad}}_{A}^{0}B\text{\hspace{0.17em}}=\text{\hspace{0.17em}}B.$

The next step in the derivation is to consider a series expansion for the operators *H*_{Q} and *Q*(*t*) of the form

$${H}_{Q}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\displaystyle \sum _{n\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1}^{\infty}}{H}_{Q}^{\mathrm{(}n\mathrm{)}}$$(36)

$$Q\mathrm{(}t\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\displaystyle \sum _{n\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1}^{\infty}}{Q}^{\mathrm{(}n\mathrm{)}}\mathrm{(}t\mathrm{}\mathrm{)}\mathrm{,}$$(37)

with *Q*^{(n)}(0)=0 and where the superscript indicates the order of the expansion. After introducing the series in (36) and (37) into (35) and equating the terms with the same order, one obtains the differential equation

$${\partial}_{t}{Q}^{\mathrm{(}n\mathrm{)}}\mathrm{(}t\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{A}^{\mathrm{(}n\mathrm{)}}\mathrm{(}t\mathrm{)}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{H}_{Q}^{\mathrm{(}n\mathrm{)}}\mathrm{,}$$(38)

with *A*^{(1)}(*t*)=*H*(*t*) and

$${A}^{\mathrm{(}n\mathrm{)}}\mathrm{(}t\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\displaystyle \sum _{k\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1}^{n\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}}\frac{{B}_{k}}{k\mathrm{!}}\mathrm{(}{X}_{k}^{\mathrm{(}n\mathrm{)}}\mathrm{(}t\mathrm{)}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\mathrm{(}-1\mathrm{)}\mathrm{}}^{k\text{\hspace{0.17em}}+\text{\hspace{0.17em}}1}{Y}_{k}^{\mathrm{(}n\mathrm{)}}\mathrm{)}$$(39)

for *n* ≥ 2. The operators ${X}_{k}^{\mathrm{(}n\mathrm{)}}\mathrm{(}t\mathrm{)}$ and ${Y}_{k}^{\mathrm{(}n\mathrm{)}}\mathrm{(}t\mathrm{)}$ in (39) are given recursively by

$${X}_{k}^{\mathrm{(}n\mathrm{)}}\mathrm{(}t\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\displaystyle \sum _{m\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1}^{n\text{\hspace{0.17em}}-\text{\hspace{0.17em}}k}}\mathrm{[}{Q}^{\mathrm{(}m\mathrm{)}}\mathrm{(}t\mathrm{}\mathrm{)}\mathrm{,}\text{\hspace{0.17em}}{X}_{k\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}^{\mathrm{(}n\text{\hspace{0.17em}}-\text{\hspace{0.17em}}m\mathrm{)}}\mathrm{(}t\mathrm{)}]$$(40)

$${Y}_{k}^{\mathrm{(}n\mathrm{)}}\mathrm{(}t\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\displaystyle \sum _{m\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1}^{n\text{\hspace{0.17em}}-\text{\hspace{0.17em}}k}}\mathrm{[}{Q}^{\mathrm{(}m\mathrm{)}}\mathrm{(}t\mathrm{}\mathrm{)}\mathrm{,}\text{\hspace{0.17em}}{Y}_{k\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}^{\mathrm{(}n\text{\hspace{0.17em}}-\text{\hspace{0.17em}}m\mathrm{)}}\mathrm{(}t\mathrm{)}]$$(41)

for 1 ≤ *k* ≤ *n* − 1, with ${X}_{0}^{\left(1\right)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-iH\mathrm{(}t\mathrm{)},$ ${X}_{0}^{\mathrm{(}n\mathrm{)}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0$ for *n* ≥ 2, and ${Y}_{0}^{\mathrm{(}n\mathrm{)}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-i{H}_{Q}^{\mathrm{(}n\mathrm{)}}$ for all *n*.

An important feature of the differential equation in (38) is the structure of the operator *A*^{(n)}(*t*), which only contains terms involving the Hamiltonian *H*(*t*) or operators *Q*^{(m)}(*t*) and ${H}_{Q}^{\mathrm{(}m\mathrm{)}}$ of a lower order, i.e. with *m*<*n*. This allows to solve (38) by just integrating the right hand side of the equation, which leads to

$${Q}^{\mathrm{(}n\mathrm{)}}\mathrm{(}t\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\displaystyle {\int}_{0}^{t}}\mathrm{(}{A}^{\mathrm{(}n\mathrm{)}}\mathrm{(}t\mathrm{)}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{H}_{Q}^{\mathrm{(}n\mathrm{)}}\mathrm{)}\text{d}t\mathrm{.}$$(42)

Moreover, even though (38) describes a differential equation for *Q*^{(n)}(*t*), the solutions for both *Q*^{(n)}(*t*) and ${H}_{Q}^{\mathrm{(}n\mathrm{)}}$ can be inferred from it by imposing suitable conditions on the time dependence of *Q*^{(n)}(*t*). In the periodic case, for example, the operators ${H}_{Q}^{\mathrm{(}n\mathrm{)}}$ are fixed by the requirement that *Q*^{(n)}(*t*) is a periodic operator [52].

In the quasi-periodic case, we can determine ${H}_{Q}^{\mathrm{(}n\mathrm{)}}$ by exploiting the quasi-periodicity of *Q*^{(n)}(*t*) and *A*^{(n)}(*t*). This essentially results from the fact that, in order for the integral of a quasi-periodic operator $O\mathrm{(}t\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\displaystyle {\sum}_{n}{O}_{n}{e}^{in\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\omega t}}$ to be quasi-periodic, it must satisfy that ${O}_{0}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\mathrm{lim}}_{T\text{\hspace{0.17em}}\to \text{\hspace{0.17em}}\infty}\frac{1}{T}{\displaystyle {\int}_{0}^{T}}O\mathrm{(}t\mathrm{)}\text{d}t\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0.$ As a consequence, in order for *Q*^{(n)}(*t*) in (42) to be quasi-periodic, ${H}_{Q}^{\mathrm{(}n\mathrm{)}}$ must read

$${H}_{Q}^{\mathrm{(}n\mathrm{)}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\underset{T\text{\hspace{0.17em}}\to \text{\hspace{0.17em}}\infty}{\mathrm{lim}}\frac{1}{T}{\displaystyle {\int}_{0}^{T}}{A}^{\mathrm{(}n\mathrm{)}}\mathrm{(}t\mathrm{)}\text{d}t\mathrm{.}$$(43)

Equations (42) and (43) can be solved for any *n*>1 provided that the solutions for *m*<*n* are known. As they can be readily solved for *n*=1, (42) and (43) thus contain the necessary information to recursively derive all the terms in the expansions of *Q*(*t*) and *H*_{Q} in (36) and (37), respectively.

After performing the integrations in (42) and (43), the first two terms of the series for *H*_{Q} become

$${H}_{Q}^{\left(1\right)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{H}_{0}$$(44)

$${H}_{Q}^{\left(2\right)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{1}{2}{\displaystyle \sum _{n\text{\hspace{0.17em}}\ne \text{\hspace{0.17em}}0}}\frac{\mathrm{[}{H}_{n}\mathrm{,}\text{\hspace{0.17em}}{H}_{-n}\mathrm{]}}{\omega \text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}n}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\displaystyle \sum _{n\text{\hspace{0.17em}}\ne \text{\hspace{0.17em}}0}}\frac{\mathrm{[}{H}_{0}\mathrm{,}\text{\hspace{0.17em}}{H}_{n}\mathrm{]}}{\omega \text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}n}\mathrm{,}$$(45)

where *H***n** are the Fourier coefficients of the quasi-periodic Hamiltonian, as defined in (1). Similarly, the first two terms of *Q*(*t*) read

$${Q}^{\left(1\right)}\left(t\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-i{\displaystyle \sum _{n\text{\hspace{0.17em}}\ne \text{\hspace{0.17em}}0}}\frac{{H}_{n}}{n\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\omega}({e}^{in\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\omega t}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1)\mathrm{}$$(46)

$$\begin{array}{c}{Q}^{\left(2\right)}\left(t\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{i}{2}{\displaystyle \sum _{n\text{\hspace{0.17em}}\ne \text{\hspace{0.17em}}0}}\frac{\mathrm{[}{H}_{0}\mathrm{,}\text{\hspace{0.17em}}{H}_{n}\mathrm{]}}{{(n\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\omega )}^{2}}({e}^{in\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\omega t}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1)\mathrm{}\\ +\frac{i}{2}{\displaystyle \sum _{n\text{\hspace{0.17em}}\ne \text{\hspace{0.17em}}0;\text{\hspace{0.17em}}m\text{\hspace{0.17em}}\ne \text{\hspace{0.17em}}-n}}\frac{\mathrm{[}{H}_{n}\mathrm{,}\text{\hspace{0.17em}}{H}_{m}\mathrm{]}}{n\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\omega \text{\hspace{0.17em}}(n\text{\hspace{0.17em}}+\text{\hspace{0.17em}}m)\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\omega}({e}^{i(n\text{\hspace{0.17em}}+\text{\hspace{0.17em}}m)\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\omega t}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1)\mathrm{}\\ +\frac{i}{2}{\displaystyle \sum _{n\text{\hspace{0.17em}}\ne \text{\hspace{0.17em}}0;\text{\hspace{0.17em}}m\text{\hspace{0.17em}}\ne \text{\hspace{0.17em}}0}}\frac{\mathrm{[}{H}_{n}\mathrm{,}\text{\hspace{0.17em}}{H}_{m}\mathrm{]}}{n\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\omega m\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\omega}({e}^{im\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\omega t}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1)\mathrm{.}\end{array}$$(47)

Consistently with the periodic case, the results obtained here reduce the regular Floquet-Magnus expansion when the frequency vector **ω** contains only one element. Moreover, by using the Baker-Campbell-Hausdorff formula [61], one can verify that the expressions in (44)–(47) coincide with the first terms of the regular Magnus expansion [59], which applies for general time-dependent systems. This formal expansion can also be linked [62] to the method of averaging for quasi-periodic systems [55] to obtain exponentially small error estimates in the quasi-periodic case.

The expansion of the operators *H*_{Q} and *Q*(*t*) introduced in (36) and (37) can be interpreted as a series expansion in powers of |**ω**|^{−1} such that, in a suitable fast-driving regime, the lowest order terms of the series are the most relevant to describe the dynamics of the system [63]. Even though the convergence of the quasi-periodic Floquet-Magnus expansion is in general not guaranteed and requires further investigations, this permits us to identify effective Hamiltonian analogously as done for periodic systems.

In fast-driving regimes where the fundamental driving frequencies are the largest energy scales of the system, the two unitaries *U*_{Q}(*t*) and ${e}^{-i{H}_{Q}t}$ of the time-evolution operator in (32) capture two distinct behaviours of the system’s dynamics. On the one hand, the unitary *U*_{Q}(*t*) describes fast quasi-periodic fluctuations dictated by the fast frequencies **ω**. On the other hand, the operator ${e}^{-i{H}_{Q}t}$ captures the slower dynamics of the system characterised by the internal energy scales of *H*_{Q}, which can be thus identified as the effective Hamiltonian of the system.

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.