# Floquet-Landau-Zener interferometry: Usefulness of the Floquet theory in
pulse-laser-driven systems

Tatsuhiko N. Ikeda  Institute for Solid State Physics, University of Tokyo,
Kashiwa, Chiba 277-8581, Japan  Satoshi Tanaka  Department of Physical
Science, Osaka Prefecture University, Sakai, Osaka 599-8531, Japan  Yosuke
Kayanuma  Department of Physical Science, Osaka Prefecture University, Sakai,
Osaka 599-8531, Japan  Laboratory for Materials and Structures, Institute of
Innovative Research, Tokyo Institute of Technology, 4259 Nagatsuta, Yokohama
226-8503, Japan

###### Abstract

We develop the Landau-Zener transfer matrix theory for the instantaneous
Floquet states (IFSs) for quantum systems driven by strong pulse laser.
Applying this theory to the pulse excitation probability in two-level quantum
systems, we show unexpectedly good quantitative agreements for few-cycle
pulses. This approach enables us to qualitatively understand the probability’s
peculiar behaviors as quantum path interference between IFSs. We also study
the pulse-width dependence, finding that this Floquet-state interpretation
remains useful for shorter pulses down to 2-cycle ones in the present model.
These results imply that the Floquet theory is meaningful in experimental few-
cycle lasers if applied appropriately in the sense of IFSs.

##  I Introduction

Intense few-cycle laser fields have opened opportunities for studying strong
light-matter interactions and for optically controlling material properties
[1]. Among various approaches, Floquet engineering is an emerging concept in
optical material control, in which the time-oscillating nature of fields is
utilized [2, 3, 4]. The guiding principle is Floquet theory [5, 6, 7], which
governs solutions of time-dependent Schrödinger equations (TDSE) under
perfectly periodic external fields, i.e., infinitely-long pulses.

However, it has not been fully clarified yet when and how Floquet theory is
justified under external pulse fields available in experiments. In this
direction, Holthaus and coworkers developed the instantaneous Floquet state
(IFS) formalism in their pioneering works [8, 9, 10, 11]. Rather than
approximating a pulse field crudely by a continuous wave, this formalism
utilizes the Floquet states as instantaneous basis states, on which the actual
quantum state evolves adiabatically or diabatically during the pulse. While
the theory was elegantly formulated and applied to some examples [11], its
actual implementation is generically challenging, and its advantages have not
been fully explored yet.

In this paper, we further develop the IFS formalism and find the situations
where this formalism appropriately describes quantum dynamics under strong
pulse fields. In particular, we introduce the Landau-Zener-type transfer
matrices in the Floquet extended Hilbert space, describing multiple
transitions between the IFSs quantitatively correctly. We apply our theory to
two-level quantum systems driven by strong pulse fields, showing its
applicability and usefulness. Recent studies showed that the pulse excitation
probability of two-level systems exhibits peculiar parameter dependence [12,
13, 14], but its mechanisms have remained uncovered yet. Our theory explains
even quantitatively that this peculiar behavior is due to quantum path
interference between IFSs. We also study the pulse-width dependence, finding
that those Floquet-state interpretations remain valid shorter pulses down to
2-cycle ones in the present model. These results imply that Floquet theory is
meaningful in experimental few-cycle lasers if applied appropriately in the
sense of IFSs.

The rest of this paper is organized as follows. In Sec. II, we introduce the
pulse excitation problem in a two-level quantum system and demonstrate that
the excitation probability exhibits complex behaviors when we vary the pulse
strength and the two levels’ energy difference. To uncover the underlying
mechanisms of these behaviors, we review the IFS formalism and develop the
Floquet-Landau-Zener (FLZ) theory using transfer matrices for the Floquet
Hamiltonian in Sec. III. In Secs. IV and V, we implement the FLZ numerically,
showing its quantitative success in calculating the pulse excitation
probability. We elucidate that the complex behaviors introduced in Sec. II
originate from quantum path interference among IFSs. We also show that these
Floquet-based interpretations remain valid for unexpectedly short pulses,
including 2-cycle pulse lasers. Finally in Sec. VI we summarize our results
and list some open problems for future study.

##  II Formulation of the problem

For concreteness, we consider an abstract two-level quantum system in strong
pulse fields. Being an effective model in various physical systems, the driven
two-level system may represent, e.g., lasing of N2 molecules [12, 13, 14],
two-band electrons in semiconductors [15], nitrogen-vacancy centers in
diamonds [16], to name a few. Throughout this work, we suppose that the
Hamiltonian is given by

$\displaystyle H_{\mathrm{pulse}}(t)=\frac{b}{2}\sigma_{z}+a(t)V(t).$ (1)

Here, $b$ $(>0)$ is the energy difference between the two levels
$\ket{\uparrow}$ and $\ket{\downarrow}$. The coupling to the external field
consists of the periodic part $V(t+T)=V(t)$ and the pulse envelope with peak
height $a_{0}$ $(>0)$

$\displaystyle a(t)=a_{0}f(t),$ (2)

where $T$ is the period and we define its corresponding angular frequency as
$\omega\equiv 2\pi/T$. We assume that the envelope is positive and normalized
so that $\max_{t}f(t)=1$ and $f(t)\to 0$ $(t\to\pm\infty)$. For concreteness,
we focus on the following prototypical coupling term

$\displaystyle V(t)=\cos(\omega t)\sigma_{x},$ (3)

which naturally arises in, e.g., two-level atoms coupled to linearly-polarized
lasers. We discuss, in Appendix C, modified problems corresponding to circular
and elliptic polarizations. We also specify, for concreteness, the envelope to
be Gaussian (generalization to other envelopes is straightforward)

$\displaystyle f(t)=\exp\left[-\left(\frac{t}{\nu T}\right)^{2}\right],$ (4)

where the dimensionless parameter $\nu$ $(>0)$ represents the pulse width in
units of $T$. Namely, we consider a $\nu$-cycle pulse with the envelope (4).

Our problem to address is the following. Suppose that our initial state at
$t=-\infty$ is the ground state $\ket{\Psi(-\infty)}=\ket{\downarrow}$, which
evolves in time according to the time-dependent Schrödinger equation (TDSE)

$\displaystyle i\frac{d}{dt}\ket{\Psi(t)}=H_{\mathrm{pulse}}(t)\ket{\Psi(t)}$
(5)

to become

$\displaystyle\ket{\Psi(+\infty)}=\exp_{+}\left(-i\int_{-\infty}^{\infty}H_{\mathrm{pulse}}(\tau)d\tau\right)\ket{\downarrow}$
(6)

after the pulse ($\hbar=1$ throughout this paper). Then, we ask the final
weight of the excited state

$\displaystyle P_{\uparrow}=|\braket{\uparrow}{\Psi(+\infty)}|^{2}.$ (7)

Figure 1: Excitation probability $P_{\uparrow}$ plotted against the pulse peak
height $a_{0}$ and the energy level difference $b$ with pulse width $\nu=6$.

Despite this simple setup, $P_{\uparrow}$ variously changes depending on the
energy level difference $b$, coupling energy $a$, driving frequency $\omega$,
and pulse width $s$. We show in Fig. 1 the results of $P_{\uparrow}$ in terms
of the numerical integration of TDSE (5) with frequency $\omega=1$ and pulse
width $\nu=6$. At weak coupling $a_{0}$, $P_{\uparrow}$ becomes significant
only near resonance $b/\omega\simeq 1$. This parameter regime is well
described by the rotating-wave approximation, and the oscillating behavior of
$P_{\uparrow}$ is understood in relation to the Rabi oscillation [17] (see
also Appendix C.1).

Furthermore, away from the resonance and at strong couplings, there is a
complex pattern of the region for nonvanishing $P_{\uparrow}$. For example, as
$a_{0}$ increases with $b=2.5$ held fixed, $P_{\uparrow}$ suddenly grows up at
$a_{0}\sim 1.0$, shows clear oscillations up to $a_{0}\sim 3.0$, and then
exhibits irregular behaviors $a_{0}\gtrsim 3.0$. The sudden grow-up was shown,
in the pioneering works by Holthaus and coworkers [11], to be due to avoided
crossing structures of Floquet quasienergies as we will review below in Sec.
III.1. However, the oscillations and irregular behaviors have not been studied
well. In the following, we extend their theory by combining the Landau-Zener
transfer matrix and elucidating those complex behaviors of $P_{\uparrow}$ in
the whole parameter region.

##  III Landau-Zener-Stückelberg theory for Floquet states

###  III.1 Instantaneous Floquet states (IFS)

The key to understanding the complex dynamics is using the basis of the
instantaneous Floquet states (ITS) [9, 11], which we briefly review here. We
note that this formulation was generalized, in Ref. [10], to the case where
$\omega$ also varies slowly.

The Floquet states are defined by the solutions to the time-dependent
Schrödinger equation for the continuous wave rather than the pulse. Namely,
according to Floquet theory [6, 7], the two independent solutions to

$\displaystyle i\frac{d}{dt}\ket{\psi(t)}=H_{\mathrm{cw}}(a,t)\ket{\psi(t)}$
(8)

with

$\displaystyle H_{\mathrm{cw}}(a,t)=\frac{b}{2}\sigma_{z}+aV(t)$ (9)

can be written in the following forms:

$\displaystyle\ket{\psi_{m}(a,t)}=e^{-i\epsilon_{m}(a)t}\ket{u_{m}(a,t)}\quad(m=1,2).$
(10)

Note that $H_{\mathrm{cw}}(a,t)$ is obtained by replacing the envelope $a(t)$
in $H_{\mathrm{pulse}}(t)$ by a constant $a$ (i.e., replacing the Gaussian
pulse by a continuous wave). In Eq. (10),
$\ket{u_{m}(a,t)}=\ket{u_{m}(a,t+T)}$ are periodic and called the Floquet
states, and the real numbers $\epsilon_{m}(a)$ are quasienergies. We
explicitly put the dependence on the coupling strength $a$ on the Floquet
states and quasienergies that will play crucial roles.

We remark the famous replicas of Floquet states. Note that Eq. (10) can also
be written as $\ket{\psi_{m}(a,t)}=e^{-i[\epsilon_{m}(a)+l\omega]t}e^{il\omega
t}\ket{u_{m}(a,t)}$ for an arbitrary integer $l\in\mathbb{Z}$. Being periodic,

$\displaystyle\ket{u_{m,l}(a,t)}\equiv e^{il\omega
t}\ket{u_{m}(a,t)}\quad(m=1,2)$ (11)

are all Floquet states, and their quasienergies are given by

$\displaystyle\epsilon_{m,l}(a)\equiv\epsilon_{m}(a)+l\omega.$ (12)

In Fig. 2, we plot the quasienergies with replicas numerically obtained for
$b=2.5$. They show avoided crossings near $a=1.0$ and $3.0$, where two
quasienergies repel each other. This is a manifestation of strong
hybridization between $\ket{\uparrow}$ and $\ket{\downarrow}$, and we will
discuss, in detail, how this hybridization leads to the complex pattern in
Fig. (1). We remark that the quasienergies of $m=1$ and $2$ do not repel but
cross near $a=2.2$, which is due to a selection rule prohibiting hybridization
(see Appendix C.2).

Figure 2: Quasienergies for $b=2.5$ plotted against coupling strength $a$.
Solid (dashed) lines show those for Floquet states of $m=2$ ($m=1$)
approaching $\ket{\downarrow}$ ($\ket{\uparrow}$) as $a\to 0$. Five Floquet
replicas are presented.

Although these replicas lead to the same solution to the Schrödinger equaion
(8), they are all necessary when one expands an arbitrary periodic function
$F(t)$ with Floquet states. In other words, the replicas satisfy the
completeness relation

$\displaystyle\sum_{m=1,2}\sum_{l=-\infty}^{\infty}\ket{u_{m,l}(a,t)}\bra{u_{m,l}(a,t^{\prime})}=T\delta_{T}(t-t^{\prime})I,$
(13)

where $\delta_{T}(t)\equiv\sum_{n=-\infty}^{\infty}\delta(t-nT)$ and $I$ is
the identity operator.

The IFS formalism is to expand the solution for the pulse problem
$\ket{\Psi(t)}$ in terms of the Floquet states:

$\displaystyle\ket{\Psi(t)}=\sum_{\alpha}c_{\alpha}(t)\ket{u_{\alpha}(a(t),t)},$
(14)

where $\alpha=(m,l)$ is a shorthand notation for the double indices.
Substituting Eq. (14) into Eq. (5), we have the time-evolution equation for
the expansion coefficients (see Appendix B),

$\displaystyle i\frac{dc_{\alpha}(t)}{dt}$
$\displaystyle=\sum_{\beta}\mathcal{H}_{\alpha\beta}(a(t))c_{\beta}(t),$ (15)

where $\mathcal{H}_{\alpha\beta}(a(t))$ is the infinite-dimensional
“Hamiltonian” defined by

$\displaystyle\mathcal{H}_{\alpha\beta}(a(t))$
$\displaystyle\equiv\delta_{\alpha\beta}\epsilon_{\alpha}(a(t))-i\frac{da}{dt}\mathcal{G}_{\alpha\beta}(a(t)),$
(16) $\displaystyle\mathcal{G}_{\alpha\beta}(a)$
$\displaystyle\equiv\int_{0}^{T}\frac{dt}{T}\braket{u_{\alpha}(a,t)}{\partial_{a}}{u_{\beta}(a,t)}.$
(17)

Here, $\partial_{a}\equiv\partial/\partial a$, and we have assumed that
$\ket{u_{\alpha}(a,t)}$ are differentiable for $a$ by requiring the gauge-
fixing condition $\braket{u_{\alpha}(a,t)}{\partial_{a}}{u_{\alpha}(a,t)}=0$.
Equation (16) means that the “Hamiltonian” in the extended (Sambe [7]) space
has the quasienegies in its diagonal elements, and $\mathcal{G}_{\alpha\beta}$
causes transitions between the Floquet states.

We remark an ambiguity in expanding the physical state $\ket{\Psi(t)}$ in
terms of the Floquet replica index $l$. To work in the IFS, we fix the initial
values of $c_{\alpha}(t)$ by
$\ket{\Psi(t_{\mathrm{ini}})}=\sum_{\alpha}c_{\alpha}(t_{\mathrm{ini}})\ket{u(a(t_{\mathrm{ini}}),t_{\mathrm{ini}})}$,
which has an infinite number of solutions due to the Floquet replicas.
However, when we calculate physical observables such as
$|\braket{\uparrow}{\Psi(t_{\mathrm{fin}})}|^{2}$, the results do not depend
on which initial condition is used [10]. Intuitively, this independence is
based on the fact that the ambiguity happens only between the physically-
equivalent states. Thus, in the following, we assume that only one $l$ is
weighted in the initial condition.

This formalism helps us to interpret physical results. Following Ref. [11],
let us interpret how $P_{\uparrow}$ suddenly grows up at $a_{0}\sim 1.0$ as
$a_{0}$ increases from zero at, e.g., $b=2.5$. Initially $(t\to-\infty)$, our
state is $\ket{\downarrow}$, and the Floquet states there coincide with the
energy eigenstates $\ket{\uparrow}$ and $\ket{\downarrow}$ since the coupling
vanishes $a(t=-\infty)=0$. Thus, we can set the initial state in the extended
space as $c_{m=2,l=0}(-\infty)=1$ and $c_{\alpha}(-\infty)=0$ for
$\alpha\neq(2,0)$. Then, this state evolves according to Eq. (15), where the
coupling envelope $a(t)$ slowly varies. Graphically, our initial state lies at
a single left-end of a solid line in Fig. 2, and it goes right as $a$
increases with time. This evolution is adiabatic and transitions between
$\alpha$’s are unlikely as long as
$-i\frac{da}{dt}\mathcal{G}_{\alpha\beta}(a(t))$ in Eq. (16) is negligible.
For slowly varying $a(t)$, this condition breaks down at the first avoided
crossing point $a=a_{\mathrm{AC},1}\sim 1.0$, where a part of the wave
function is transferred to another state represented by a dashed line.
Therefore, for $a_{0}<a_{\mathrm{AC},1}$, the whole dynamics is adiabatic, the
final state is almost the same as the initial state, and $P_{\uparrow}\sim 0$.
Meanwhile, for $a_{0}>a_{\mathrm{AC},1}$, the state experiences transitions
twice, the final state is a superposition of the states on the solid and
dashed lines in Fig. 2, and $P_{\uparrow}$ becomes nonvanishing. This is the
IFS interpretation for the sudden increase of $P_{\uparrow}$ along e.g.
$b=2.5$ in Fig. 1.

###  III.2 Transfer Matrices

One is naturally led to the question: Does the IFS viewpoint allow us to
understand the whole complex structure in Fig. 1? To the authors’ knowledge,
although the Landau-Zener-like transition probability at a single passage of
an avoided crossing was analyzed [10], the interference pattern has not been
well studied. Our aim is to introduce the Landau-Zener-Stückelberg transfer
matrix method in the extended space and to show that the IFS formalism is very
powerful even quantitatively.

As we discussed at the end of Sec. III.1, the “Hamiltonian”
$\mathcal{H}(a(t))$ depends on time through the envelope $a(t)$, and its
eigenvalues (quasienergies) form avoided crossings. In applying the transfer
matrix method to the IFS (see Appendix A for this method in the conventional
sense), we need two generalizations: (i) there are quasienergy replicas of
avoided crossings and (ii) the system passes avoided crossings multiple times
in $-\infty<t<\infty$ when the pulse peak $a_{0}$ is large.

Suppose that there are $N$ $(>0)$ avoided crossing points denoted by
$\\{a_{\mathrm{AC},n}\\}_{n=1}^{N}$ below the pulse peak height $a_{0}$ and
they are in ascending order:
$0<a_{\mathrm{AC},1}<a_{\mathrm{AC},2}<\dots<a_{\mathrm{AC},N}<a_{0}$. For
example, we have $N=2$ for $a_{0}=3.5$ in Fig. 2. Correspondingly, we define
the crossing times $t_{\mathrm{AC},n}$ $(>0)$ by

$\displaystyle
a(t_{\mathrm{AC},n})=a_{0}f(t_{\mathrm{AC},n})=a_{\mathrm{AC},n}\quad(n=1,2,\dots,N).$
(18)

For simplicity, we assume that the envelope is even, $f(-t)=f(t)$, and
monotonically decreasing in $t\geq 0$ as the Gaussian envelope is. Then, the
crossings happen also at $t=-t_{\mathrm{AC},n}$ $(n=1,2,\dots,N)$, and we have

$\displaystyle-
t_{\mathrm{AC},1}<\dots<-t_{\mathrm{AC},N}<0<t_{\mathrm{AC},N}<\dots<t_{\mathrm{AC},1}.$
(19)

At the $n$-th crossing point, the transfer matrix $\mathcal{T}_{n}$, which
will be defined in Eq. (21), connects the state vectors before and after the
crossing as

$\displaystyle\vec{c}\,(t=t_{\mathrm{AC},n}^{+})=\mathcal{T}_{n}\vec{c}\,(t=t_{\mathrm{AC},n}^{-}).$
(20)

Here $\vec{c}(t)$ is the vector notation for $c_{\alpha}(t)$’s,
$t_{\mathrm{AC},n}^{\pm}\equiv t_{\mathrm{AC},n}\pm 0$ (we will also use
$-t_{\mathrm{AC},n}^{\pm}\equiv-t_{\mathrm{AC},n}\pm 0$), and
$\mathcal{T}_{n}$ is an infinite-dimensional matrix given as follows. The
avoided crossing occurs between a pair of Floquet states, which we label as
$(m_{U},l_{U})$ and $(m_{L},l_{L})$. Here, the subscript $U$ ($L$) denotes the
upper (lower) levels at the crossing. For example, $(m_{U},m_{L})=(2,1)$ and
$(1,2)$ at the first and second crossings, respectively, in Fig. 2. In this
notation, the nonzero matrix elements of $\mathcal{T}_{n}$ are given as

$\displaystyle(\mathcal{T}_{n})_{\alpha\beta}$
$\displaystyle=\begin{pmatrix}\sqrt{1-P_{n}}e^{-i\varphi^{S}_{n}}&-\sqrt{P_{n}}\\\
\sqrt{P_{n}}&\sqrt{1-P_{n}}e^{i\varphi^{S}_{n}}\end{pmatrix}_{\alpha\beta},$
(21)

where $\alpha=(m_{U},l_{U})$ and $(m_{L},l_{L})$ correspond to the first and
second rows, respectively. The two parameters $P_{n}$ and $\varphi_{n}^{S}$
are the Landau-Zener transition probability and the Stokes phase for the
$n$-th avoided crossing (see Sec. A),

$\displaystyle P_{n}$ $\displaystyle=\exp(-2\pi\delta_{n}),$ (22)
$\displaystyle\delta_{n}$ $\displaystyle=\frac{\Delta_{n}^{2}}{4v_{n}},$ (23)
$\displaystyle\varphi_{n}^{S}$
$\displaystyle=-\frac{\pi}{4}+\delta_{n}\ln(\delta_{n}-1)+\arg\Gamma(1-i\delta_{n}).$
(24)

Here, $\Delta_{n}$ and $v_{n}$ are the quasienergy gap and the passing speed
at the $n$-th avoided crossing, respectively. These parameters are defined in
the approximate form of the pair quasienergies near $t=t_{\mathrm{AC},n}$

$\displaystyle\epsilon_{\alpha}(a(t))$
$\displaystyle\simeq\mathrm{const.}\pm\sqrt{\left(\frac{\Delta_{n}}{2}\right)^{2}+\left[\frac{v_{n}(t-t_{\mathrm{AC},n})}{2}\right]^{2}}$
(25)
$\displaystyle\simeq\mathrm{const.}\pm\left[\frac{\Delta_{n}}{2}+\frac{v_{n}^{2}(t-t_{\mathrm{AC},n})^{2}}{4\Delta_{n}}\right]$
(26)

for $\alpha=(m_{U},l_{U})$ and $(m_{L},l_{L})$. Note that $\Delta_{n}$ and
$v_{n}$ are well-defined in that they are the same for every Floquet replica.

To obtain $\Delta_{n}$ and $v_{n}$ in practice, we expand
$\epsilon_{\alpha}(a)$ around $a=a_{\mathrm{AC},n}$. Since
$d\epsilon_{\alpha}(a)/da$ vanishes at the avoided crossing, we have the
following second-order series expansion
$\epsilon_{\alpha}(a(t))\simeq\epsilon_{\alpha}(a_{\mathrm{AC},n})+\frac{1}{2}(d^{2}\epsilon_{\alpha}(a)/da^{2})(da/dt)^{2}(t-t_{\mathrm{AC},n})^{2}$,
where $d^{2}\epsilon_{\alpha}(a)/da^{2}$ and $da/dt$ are evaluated at
$a=a_{\mathrm{AC},n}$ and $t=t_{\mathrm{AC},n}$, respectively. Comparing this
with Eq. (26), we have

$\displaystyle\Delta_{n}$
$\displaystyle=\epsilon_{(m_{U},l_{U})}(a_{\mathrm{AC},n})-\epsilon_{(m_{L},l_{L})}(a_{\mathrm{AC},n}),$
(27) $\displaystyle v_{n}$
$\displaystyle=\sqrt{2\Delta_{n}\left|\frac{d^{2}\epsilon_{\alpha}(a)}{da^{2}}\right|}\left|\frac{da}{dt}\right|,$
(28)

where $\alpha$ is either $(m_{U},l_{U})$ or $(m_{L},l_{L})$ that give the same
$|d^{2}\epsilon_{\alpha}(a)/da^{2}|$. One can obtain these parameters by
numerical fitting as we will implement in Sec. IV or by analytical
calculations for some special cases as we will demonstrate in Appendix C.

Except for the crossing points, the evolution is assumed to be mere phase
acquisitions due to the first term on the right-hand side of Eq. (16). In the
vector notation, we have

$\displaystyle\vec{c}\,(t=t_{\mathrm{AC},n}^{-})=\mathcal{U}_{n,n+1}\vec{c}\,(t=t_{\mathrm{AC},n+1}^{+}),$
(29)

where $\mathcal{U}_{n,n+1}$ is diagonal and

$\displaystyle(\mathcal{U}_{n,n+1})_{\alpha\alpha}=\exp\left[-i\int_{t_{\mathrm{AC},n+1}}^{t_{\mathrm{AC},n}}ds\epsilon_{\alpha}(a(s))\right].$
(30)

For convenience, we define
$(\mathcal{U}_{N,N+1})_{\alpha\alpha}=\exp\left[-i\int_{0}^{t_{\mathrm{AC},N}}ds\epsilon_{\alpha}(a(s))\right]$.

Since we are considering a symmetric envelope $a(-t)=a(t)$, time evolution is
symmetric in $-\infty<t\leq 0$ and $0\leq t<\infty$. The transfer matrix
$\mathcal{T}_{n}$ describes the state transfer both at $t=\pm
t_{\mathrm{AC},n}$, and $\mathcal{U}_{n+1,n}$ represents the phase acquisition
not only from $t_{\mathrm{AC},n+1}$ to $t_{\mathrm{AC},n}$ but also from
$-t_{\mathrm{AC},n}$ to $-t_{\mathrm{AC},n+1}$. Thus, we obtain the state
transfer between the first and final avoided crossings,

$\displaystyle\vec{c}\,(t=t_{\mathrm{AC},1}^{+})=\left[\prod_{n=N}^{1}\mathcal{T}_{n}\mathcal{U}_{n,n+1}\right]\left[\prod_{n=1}^{N}\mathcal{U}_{n,n+1}\mathcal{T}_{n}\right]\vec{c}\,(t=-t_{\mathrm{AC},1}^{-}).$
(31)

The entire dynamics is obtained by the phase acquisitions before (after) the
first (final) avoided crossing:
$\vec{c}(t=-t_{\mathrm{AC},1}^{-})=\mathcal{U}_{<}\vec{c}(t=t_{\mathrm{ini}})$
and
$\vec{c}(t=t_{\mathrm{fin}})=\mathcal{U}_{>}\vec{c}(t=t_{\mathrm{AC},1}^{+})$,
where
$(\mathcal{U}_{<})_{\alpha\alpha}=\exp\left[-i\int_{t_{\mathrm{ini}}}^{-t_{\mathrm{AC},1}}ds\epsilon_{\alpha}(a(s))\right]$
and
$(\mathcal{U}_{>})_{\alpha\alpha}=\exp\left[-i\int_{t_{\mathrm{AC},1}}^{t_{\mathrm{fin}}}ds\epsilon_{\alpha}(a(s))\right]$
with $t_{\mathrm{ini}}=-\infty$ and $t_{\mathrm{fin}}=+\infty$.

Thus, we have obtained the whole evolution of wave vector $\vec{c}(t)$ in the
IFS based on the transfer matrix method. The physical interpretation is clear
in Eq. (31). The wave vector experiences adiabatic dynamics described by the
phase factors $\mathcal{U}_{n,n+1}$ and Landau-Zener-like diabatic dynamics
described by the transfer matrices $\mathcal{T}_{n}$. The phase factors due to
the Stokes phase in $\mathcal{T}_{n}$ and $\mathcal{U}_{n,n+1}$ amount to the
Stückelberg phase and cause interferences as we will see in the following.

Finally, we formulate how to calculate the physical observable $P_{\uparrow}$
of interest from $\vec{c}$. By using Eq. (14), we have

$\displaystyle P_{\uparrow}$
$\displaystyle=|\braket{\uparrow}{\Psi(t_{\mathrm{fin}})}|^{2}$ (32)
$\displaystyle=\left|\sum_{\alpha}c_{\alpha}(t_{\mathrm{fin}})\braket{\uparrow}{u_{\alpha}(a(t_{\mathrm{fin}}),t_{\mathrm{fin}}}\right|^{2}.$
(33)

We recall Eq. (11) and suppose that $t_{\mathrm{fin}}\to\infty$, in which
$\ket{u_{m}(a(t_{\mathrm{fin}})),t_{\mathrm{fin}}}\to\ket{u_{m}(0,\infty)}=\delta_{m1}\ket{\uparrow}+\delta_{m2}\ket{\downarrow}$.
Thus, the sum over $\alpha=(m,l)$ in Eq. (33) is trivially taken for $m$, and
we have

$\displaystyle P_{\uparrow}=\left|\sum_{l}c_{1,l}(t_{\mathrm{fin}})e^{il\omega
t_{\mathrm{fin}}}\right|^{2}.$ (34)

Note that $\vec{c}(t_{\mathrm{fin}})$ is connected to
$\vec{c}(t_{\mathrm{AC},1}^{+})$ by
$c_{m,l}(t_{\mathrm{fin}})=\exp[-i\int_{t_{\mathrm{AC},1}}^{t_{\mathrm{fin}}}\epsilon_{m,l}(a(s))ds]c_{m,l}(t_{\mathrm{AC},1}^{+})=\exp[-i\int_{t_{\mathrm{AC},1}}^{t_{\mathrm{fin}}}\epsilon_{m}(a(s))ds]e^{-il\omega(t_{\mathrm{fin}}-t_{\mathrm{AC},1})}c_{m,l}(t_{\mathrm{AC},1}^{+})$,
where we have used Eq. (12). Substituting this equation into Eq. (34), we
obtain

$\displaystyle
P_{\uparrow}=\left|\sum_{l}c_{1,l}(t_{\mathrm{AC},1}^{+})e^{il\omega
t_{\mathrm{AC},1}}\right|^{2}.$ (35)

Equation (35) is useful since we can compute the excitation probability
$P_{\uparrow}$ just after the final passage of the avoided crossing, and
$\vec{c}(t_{\mathrm{AC},1}^{+})$ is given in Eq. (31).

We remark that, in Eq. (31), we can set
$c_{m=2,l=0}(-t_{\mathrm{AC},1}^{-})=1$ and
$c_{\alpha}(-t_{\mathrm{AC},1}^{-})=0$ for $\alpha\neq(2,0)$. These conditions
are what we imposed for $t=t_{\mathrm{ini}}$ at the end of Sec. III.1.
Nonetheless, the evolution between $t=t_{\mathrm{ini}}$ and
$-t_{\mathrm{AC},1}$ merely gives an overall phase factor, which is irrelevant
for $P_{\uparrow}$.

To summarize the transfer matrix method for the IFS, our recipe for obtaining
$P_{\uparrow}$ consists of using $\vec{c}(-t_{\mathrm{AC},1}^{-})$ thus
specified, transferring the state by Eq. (31), and invoking Eq. (35). While
this method is an approximation, its physical interpretation is clear in that
the evolution is a close analog of the Landau-Zener-Stückelberg interferometry
on the Floquet states. In the following, we will implement this recipe and
show that it works well even quantitatively.

##  IV Numerical implementation

In this section, we apply the transfer matrix method to understand the complex
structure in Fig. 1. We will focus on 6-cycle pulses ($\nu=6$), which are so
long that the transfer matrix method works well. We will discuss how results
change with the pulse width $\nu$ later in Sec. V.

As shown in Sec. III.2, the necessary information to implement the method are
all obtained from the quasienergies plotted in Fig. 2. To be specific, we set
$b=2.5$, for which $a_{\mathrm{AC},1}=1.09$ ($\Delta_{1}=8.98\times 10^{-2}$)
and $a_{\mathrm{AC},2}=3.05$ ($\Delta_{2}=0.302$) are obtained numerically,
from which we can calculate the $a_{0}$-dependence of $P_{\uparrow}$ by the
Floquet-Landau-Zener (FLZ) theory (35). In Fig. 3, we compare $P_{\uparrow}$
for $b=2.5$ and $\nu=6$ obtained by the exact numerical simulation of the TDSE
(5) and by the FLZ method, where the results are shown for $0\leq a_{0}\leq
a_{\mathrm{AC},3}=4.75$.

Figure 3: Excitation probability $P_{\uparrow}$ for $b=2.5$ plotted against
the pulse peak height $a_{0}$ for the pulse width $\nu=6$. The solid and
dashed curves show $P_{\uparrow}$ obtained, respectively, by solving the TDSE
(5) numerically and by invoking the FLZ theory (35).

For $a_{0}<a_{\mathrm{AC},1}$, the transfer matrix approach tells us that
there is no state transfer between the Floquet states and hence
$P_{\uparrow}=0$ as shown in Fig. 3. This result agrees with $P_{\uparrow}$
obtained directly by the TDSE for $a_{0}$ well below $a_{\mathrm{AC},1}$. Near
$a_{0}=a_{\mathrm{AC},1}$, the transfer matrix deviates from the exact result.
This deviation originates from the adiabatic-impulse approximation in that the
state transfer occurs instantaneously right at the avoided crossing and is a
close analog of the deviation in the conventional Landau-Zener problem
explained in Appendix A. Except for $a_{0}\simeq a_{\mathrm{AC},n}$
$(n=1,2,\dots)$, we expect that the transfer matrix method works well.

For $a_{\mathrm{AC},1}<a_{0}<a_{\mathrm{AC},2}$, we see the essence of the
Landau-Zener-Stückelberg interferometry for the IFS. For this case, there are
two passages of the same avoided crossing point $a_{\mathrm{AC},1}$, and we
have

$\displaystyle\vec{c}\,(t=t_{\mathrm{AC},1}^{+})=\mathcal{T}_{1}(\mathcal{U}_{1,2})^{2}\mathcal{T}_{1}\vec{c}\,(t=-t_{\mathrm{AC},1}^{-}),$
(36)

which follows from Eq. (31). The physical interpretation of Eq. (36) is
schematically illustrated in Fig. 4. At the first crossing
$t=-t_{\mathrm{AC},1}$, a superposition of two Floquet states $(m,l)=(2,0)$
and $(1,-3)$ is created by $\mathcal{T}_{1}$, and these state acquire phase
factors due to $(\mathcal{U}_{1,2})^{2}$ until $t=t_{\mathrm{AC},1}$. At the
second crossing $t=t_{\mathrm{AC},1}$, the superposed states experience the
state mixing again by $\mathcal{T}_{1}$, and the final state has nonvanishing
weight on $(m,l)=(1,-3)$, which adiabatically approaches $\ket{\uparrow}$ as
$t\to+\infty$.

Figure 4: Schematic illustration of the Floquet-Landau-Zener interferometry
for $a_{\mathrm{AC},1}<a_{0}<a_{\mathrm{AC},2}$. The curved arrows show
quantum state trajectories along the IFS, and the two $\mathcal{U}_{1,2}$’s
produce a relative phase factor between the upper and lower trajectories
between the two LZ transitions denoted by the transfer matrices
$\mathcal{T}_{1}$.

Since there are only two Floquet states are involved in this case, we can
simplify Eq. (36) by ignoring irrelevant zero elements. Focusing on the
subspace for $(m,l)=(2,0)$ and $(1,-3)$, we set $\vec{c}={}^{t}(1,0)$, for
which Eqs. (21) and (30) give
$c_{1,-3}(t=t_{\mathrm{AC},1}^{+})=\sqrt{P_{1}(1-P_{1})}[e^{i(\varphi_{1}^{S}-2\Phi_{1,-3})}+e^{i(-\varphi_{1}^{S}-2\Phi_{2,0})}]$
with

$\displaystyle\Phi_{\alpha}\equiv\int_{0}^{t_{\mathrm{AC},1}}ds\,\epsilon_{\alpha}(a(s)).$
(37)

Thus, from Eq. (35), we obtain

$\displaystyle
P_{\uparrow}=4P_{1}(1-P_{1})\cos^{2}(\varphi_{1}^{S}+\Phi_{2,0}-\Phi_{1,-3}).$
(38)

Here, the quantum path interference effect in $P_{\uparrow}$ is evident, and
the phase $\varphi_{1}^{S}+\Phi_{2,0}-\Phi_{1,-3}$ is a Floquet generalization
of the Stückelberg phase [18].

The $a_{0}$ dependence of $P_{\uparrow}$ calculated from Eq. (38) well
describes that obtained numerically exactly for
$a_{\mathrm{AC},1}<a_{0}<a_{\mathrm{AC},2}=3.05$ as shown in Fig. 3. Let us
discuss two characteristic behaviors of $P_{\uparrow}$ in this region: (i)
$P_{\uparrow}$ oscillates, and (ii) the envelope of $P_{\uparrow}$ quickly
increases in $a_{\mathrm{AC},1}<a_{0}\lesssim 1.2$ and then slowly decreases
in $1.2\lesssim a_{0}<a_{\mathrm{AC},2}$. The first character (i) is mainly
due to $\Phi_{2,0}-\Phi_{1,-3}$, which is the integrated phase difference
between the Floquet states and corresponds to half the area of the shaded
region in Fig. 4. As $a_{0}$ increases, $t_{\mathrm{AC},1}$ increases, and so
does $\Phi_{2,0}-\Phi_{1,-3}$. Inside the cosine (see Eq. (38)), the increase
of $\Phi_{2,0}-\Phi_{1,-3}$ results in the oscillating behavior of
$P_{\uparrow}$. The second character (ii) is due to $P_{1}(1-P_{1})$ in Eq.
(38). We recall that $P_{1}$ depends on $a_{0}$ only through
$da/dt|_{t=t_{\mathrm{AC},1}}$ in the crossing speed $v_{1}$. Since our
envelope is Gaussian, as $a_{0}$ increases from $a_{\mathrm{AC},1}$,
$t_{\mathrm{AC},1}$ does from zero. During this,
$da/dt|_{t=t_{\mathrm{AC},1}}$ first increases and then decreases. This
nonmonotonic behavior results in the character (ii) through $\delta_{1}$ and
hence $P_{1}$.

While we have focused on $b=2.5$, the interpretation by two passages of
avoided crossings also apply to other points in Fig. 1. For example, roughly
in the region $(a_{0},b)\in[1,3]\times[1.5,3]$ as well as
$(a_{0},b)\in[3,4]\times[3,4]$, we see regular patterns of curves, in which
$P_{\uparrow}$’s behavior follows from similar mechanisms illustrated in Fig.
4.

Now, we come back to $b=2.5$ and consider
$a_{\mathrm{AC},2}<a_{0}<a_{\mathrm{AC},3}=4.75$ to elucidate the complex
pattern in $(a_{0},b)\in[3,4]\times[2,3]$ in Fig. 4. For this case, we have

$\displaystyle\vec{c}\,(t=t_{\mathrm{AC},1}^{+})$
$\displaystyle=\mathcal{T}_{1}\mathcal{U}_{1,2}\mathcal{T}_{2}(\mathcal{U}_{2,3})^{2}\mathcal{T}_{2}\mathcal{U}_{1,2}\mathcal{T}_{1}\vec{c}\,(t=-t_{\mathrm{AC},1}^{-}),$
(39)

with which Eq. (35) gives $P_{\uparrow}$. Even for this case, the FLZ theory
(35) well reproduces $P_{\uparrow}$ as shown in Fig. 3 away from the narrow
region near $a_{0}=a_{\mathrm{AC},2}=3.05$. The discrepancy in this narrow
region is due to the adiabatic-impulse approximation.

With the transfer matrix formulation, we can finally interpret the complex
pattern in $(a_{0},b)\in[3,4]\times[2,3]$ in Fig. 1. In this parameter region,
there are four passages of avoided crossings, under which the state flow of
Eq. (IV) is schematically illustrated in Fig. 5. After the strong pulse
irradiation, three Floquet states are superposed for each $m=1$ and $2$. Thus,
we have more quantum-path interference than other parameter regions like the
previous case (38). The complex pattern in $P_{\uparrow}$ is understood
qualitatively and quantitatively by the Landau-Zener-Stückelberg
interferometry in terms of the IFS.

Figure 5: Schematic illustration of the Floquet-Landau-Zener interferometry
for $a_{\mathrm{AC},2}<a_{0}<a_{\mathrm{AC},3}$. Solid (dashed) lines show
those for Floquet states of $m=2$ ($m=1$) approaching $\ket{\downarrow}$
($\ket{\uparrow}$) as $a\to 0$. The curved arrows show quantum state
trajectories along the IFS, and $\mathcal{T}_{1}$ ($\mathcal{T}_{2}$) denotes
the transfer matrix for the avoided crossing at $a_{\mathrm{AC},1}$
($a_{\mathrm{AC},2}$).

##  V Pulse-width dependence

In Sec. IV, we fixed the pulse width as $\nu=6$. Meanwhile, experimentally,
stronger peak amplitudes $a_{0}$ tend to be realized for shorter pulse widths
[1]. Thus, it is crucially important how small $\nu$ can be for the FLZ theory
remaining applicable. Naively speaking, the FLZ theory is expected to become
worse for shorter pulses because the envelope’s temporal change $da/dt$
increases, and the assumption of adiabaticity eventually breaks down (see also
Eq. (16)). Note that this tendency also holds for increasing amplitude $a_{0}$
as $da/dt$ increases with $a_{0}$ as well.

Figure 6: Excitation probability $P_{\uparrow}$ for $b=2.5$ plotted against
the pulse peak height $a_{0}$. The solid and dashed curves show $P_{\uparrow}$
obtained, respectively, by solving the TDSE (5) numerically and by invoking
the FLZ theory (35). The pulse width is $\nu=4,3,2,$ and $1$ from top to
bottom.

To address this issue of pulse-width dependence, we examine the FLZ theory’s
applicability with decreasing $\nu$. Figure 6 shows $P_{\uparrow}$ for $b=2.5$
obtained by the exact numerical simulation of the TDSE (5) and by the FLZ
method for different $\nu$’s. For $\nu=2,3$, and 4, we observe that the FLZ
theory captures quite well the exact results within the adiabatic-impulse
approximation except for the regions near the avoided crossing points
$a=a_{\mathrm{AC},1}=1.09$ and $a_{\mathrm{AC},2}=3.05$.

We note that these regions of disagreement become wider for smaller $\nu$,
which is consistent with the following intuitive argument on adiabaticity. Our
FLZ theory within the adiabatic-impulse approximation assumes that no
transition occurs between IFSs except $a=a_{\mathrm{AC},n}$ $(n=1,2,\dots)$.
However, this assumption is valid when the quasienergy difference is much
larger than the perturbation term ($\mathcal{G}$ in Eq. (16)) proportional to
$da/dt$. Thus, this assumption is not satisfied near the avoided crossings
where the quasienergy differences become small. Also, this tendency is
stronger for shorter pulses having larger $da/dt$. Transitions between IFSs
actually occur slightly away from the avoided crossings, while the adiabatic-
impulse approximation neglects them entirely. Nevertheless, it is remarkable
that the FLZ theory works quantitatively well and has wide-enough
applicability parameter regimes even if the pulse is as short as 2-cycle
($\nu=2$).

For the single-cycle pulse ($\nu=1$), the FLZ theory does not agree well with
the numerically exact results as seen in the range of
$a_{\mathrm{AC},1}<a_{0}<a_{\mathrm{AC},2}$, indicating that the FLZ theory
does not provide a good physical interpretation. There are two possible
reasons for the breakdown of the FLZ theory in $\nu<2$: (i) the adiabatic
impulse approximation becomes inaccurate, and (ii) transitions to distant
Floquet replicas become nonnegligible. As for (i), we notice that $\eta\equiv
1/\nu$ serves as the adiabatic parameter (see Appendix D.1 for details). Since
the adiabatic impulse approximation is justified in two-level systems by
power-series arguments for $\eta$ and is accurate for small $\eta$ [10, 19],
it is natural that the approximation here starts to fail at $\eta\sim 1$,
i.e., $\nu\sim 1$, although their precise values should depend on models. The
other possibility (ii) is unlikely because the transition matrix elements
$\mathcal{G}_{\alpha\beta}(a)$ between IFSs are actually small compared to the
quasienergy difference $\epsilon_{\alpha}(a)-\epsilon_{\beta}(a)$ (see
Appendix D.2 for details). Thus, the possibility (i) is likely to be the
reason for the FLZ’s failure at the ultrashort pulse as short as $\nu\sim 1$
in our model.

Despite this argument on the FLZ theory’s failure for ultrashort pulses, we
leave for future work to determine the precise value of $\nu$ for the
breakdown and to fully understand why this theory works even down to $\nu=2$.
It is worth noting that the FLZ theory seems to work for
$a_{0}>a_{\mathrm{AC},2}$ at $\nu=1$, even with the disagreement in
$a_{0}<a_{\mathrm{AC},2}$, for which the authors have not found its reason.
Systematically studying the FLZ in such an ultrashort-pulse regime is an open
problem.

##  VI Summary and Discussions

Considering the pulse excitation probability $P_{\uparrow}$ in a two-level
quantum system, we have studied the complex interference pattern in Fig. 1 in
the two-dimensional space spanned by the pulse peak height $a_{0}$ and the two
levels’ energy difference $b$. To understand these patterns, we have utilized
the instantaneous Floquet states (IFS), rather than the original energy
eigenstates, as a useful basis for understanding dynamics [9, 10]. The time
evolution driven by strong pulse fields can then be regarded as adiabatic
evolutions along the IFS and Landau-Zener-type (LZ-type) diabatic transitions
between them [11].

We have developed this idea quantitatively by applying the transfer matrix
method among Floquet states, formulated how to keep track of quantum states
under multiple LZ-type transitions, and termed this formulation as the
Floquet-Landau-Zener (FLZ) theory in Sec. III. Implementing this theory
numerically in Sec. IV (and analytically in Appendix. C), we have shown that
the FLZ theory well reproduces $P_{\uparrow}$ obtained by direct numerical
calculations. One advantage of the FLZ theory is that the physical
interpretation of dynamics is transparent; The complex interference patterns
in $P_{\uparrow}$ originate from quantum path interference between IFSs as
illustrated in Figs. 4 and 5.

We have demonstrated that the FLZ theory is valid for longer pulses (i.e.,
larger $\nu$). This is a natural tendency because the longer pulses mean
slower changes of pulse envelopes $a(t)$, validating the adiabatic
approximation. Rather surprisingly, however, the FLZ theory has worked in
appropriate parameter ranges if the pulse width is larger roughly than 2
cycles ($\nu\gtrsim 2$) as shown in Sec. V. This should be relevant for
experimental studies to address Floquet-related physics, which emerge ideally
under strong continuous external fields while strong laser fields
experimentally tend to be realized in short pulses. Our findings imply that
Floquet-related physics are present even in short-pulse experiments if we
interpret appropriately in the sense of IFSs. For the extension of the Floquet
formalism to the case not strictly time-periodic external fields, one of the
present authors and Mizumoto [20] studied the transition dynamics in the two-
level system under the level-crossing with a constant velocity plus time-
periodic modulation in the relative energy. An unexpected agreement has been
observed between the calculation by the Floquet-Landau-Zener transfer matrix
method and the numerical solutions of the TDSE for a wide range of parameter
values. A deep understanding of this success is also an open question.

As concluding remarks, we list some future directions. First, it is important
to validate the transfer matrix methods in the Floquet extended space and to
improve the adiabatic-impulse approximation systematically. For the
conventional Landau-Zener problem, the so-called Stokes phenomena are known to
underlie [21], and WKB theories [22] provide mathematical foundations. One
could generalize these insights to the Floquet extended space and validate the
FLZ theory mathematically. Second, it is intriguing to realize the FLZ
interferometry in an experiment. Our model should apply to any two-level
systems, but there are two experimental challenges: (i) a long-enough
coherence time and (ii) a strong-enough coupling to external fields with long-
enough pulses. In the present work, we neglected any decoherence/dissipation,
which, if strong, may destroy the clear interference pattern. It is intriguing
to investigate if we can overcome those effects experimentally together with
further theoretical investigations. Finally, it is interesting to generalize
the FLZ interferometry for other classes of systems with more than two levels
[23], including multiple two-level systems [24]. Generally speaking, the
denser the energy levels are, the worse the adiabatic approximation becomes.
Thus, we expect that the FLZ theory works in systems with not-so-many levels.
However, it could be possible to apply this theory for condensed-matter
systems with many energy levels but an energy gap above the ground state. We
leave the above problems open for future studies.

## Acknowledgements

T.N.I. thanks M. Holthaus for introducing the IFS formalism and M. Hongo and
H. Taya for fruitful discussions on related topics. Y.K. thanks Professors K.
Yamanouchi and E. Lötstedt for drawing his attention to the works of
population inversion in N2 molecules under irradiation of intense pulse laser
[12]. This work was supported by JSPS KAKENHI Grant No. JP21K13852 and
JP19K03696, and by Non-Equilibrium Working group (NEW) at RIKEN
Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS).

##  Appendix A Transfer matrix in the Landau-Zener problem

To supplement the discussions of transfer matrices for IFSs in the main text,
we briefly review the transfer matrix for the seminal Landau-Zener problem for
the linearly-time-dependent Hamiltonian
$H_{\mathrm{LZ}}(t)=-(vt/2)\sigma_{z}+(\Delta/2)\sigma_{x}=\sum_{n=1}^{2}E_{n}(t)\ket{E_{n}(t)}\bra{E_{n}(t)}$
with eigenenergies $E_{1}(t)\geq E_{2}(t)$. As shown in the level diagram in
Fig. 7(a), from $t=-\infty$ to $+\infty$, the system goes across, at $t=0$,
the avoided crossing of gap $\Delta=E_{1}(0)-E_{2}(0)$ at speed $v$.

Suppose that the initial state is given as $\ket{\Phi(t_{\mathrm{ini}})}$
($t_{\mathrm{ini}}<0$) and we are to solve its evolution $\ket{\Phi(t)}$ and
ask the population at the upper state $w(t)=|\braket{E_{1}(t)}{\Phi(t)}|^{2}$.
This population can be obtained analytically [19, 18] or numerically as
illustrated in Fig. 7(b), where $\Delta=5$, $v=10$, $t_{\mathrm{ini}}=-10$,
and $\ket{\Phi(t_{\mathrm{ini}})}=\ket{E_{2}(t_{\mathrm{ini}})}$.

Figure 7: (a) Eigenvalues of $H_{\mathrm{LZ}}(t)$ (see also text). (b) Upper-
state populations $w(t)$ calculated by the exact numerics (solid) and by the
transfer matrix method within the adiabatic-impulse approximation (dashed). In
both panels, we set $v=10$ and $\Delta=5$.

The transfer matrix method enables us to obtain an approximate solution with
clear physical interpretation. This method is based on the energy eigenbasis
$\ket{\Phi(t)}=\sum_{n=1}^{2}b_{n}(t_{\mathrm{ini}})\ket{E_{n}(t_{\mathrm{ini}})}$
and the assumption that no transition occurs away from the avoided crossing
point $t=0$. This assumption is known as the adiabatic-impulse approximation
[18]. With this method, we have, for $n=1$ and $2$,
$b_{n}(t<0)=\exp[-i\int_{t_{\mathrm{ini}}}^{t}dsE_{n}(s)]b_{n}(t_{\mathrm{ini}})$
and $b_{n}(t>0)=\exp[-i\int_{t_{\mathrm{ini}}}^{t}dsE_{n}(s)]b_{n}(0+)$. At
the avoided crossing, the state experiences transitions described by the
following transfer matrix,

$\displaystyle\bm{b}(0+)=\begin{pmatrix}\sqrt{1-P}e^{-i\varphi^{S}}&-\sqrt{P}\\\
\sqrt{P}&\sqrt{1-P}e^{i\varphi^{S}}\end{pmatrix}\bm{b}(0-),$ (40)

where

$\displaystyle P\equiv\exp(-2\pi\delta)$ (41)

is the prominent Landau-Zener transition probability with

$\displaystyle\delta$ $\displaystyle\equiv\frac{\Delta^{2}}{4v},$ (42)

and

$\displaystyle\varphi^{S}$
$\displaystyle\equiv-\frac{\pi}{4}+\delta\ln(\delta-1)+\arg\Gamma(1-i\delta),$
(43)

is the Stokes phase with $\Gamma(z)$ being the gamma function. As illustrated
in Fig. 7(b), the transfer matrix method well captures the exact solution
except in the vicinity of the avoided crossing. While the actual dynamics is
complicated within the avoided crossing region, it well describes, as a single
matrix, the integrated evolution from the input to the output.

The transfer matrix method applies not only to $H_{\mathrm{LZ}}(t)$ of linear
time dependence but also to other Hamiltonians of more generic dependence
[18]. For generic cases, we define $v$ and $\Delta$ by linearly approximating
the energy eigenvalues near the avoided crossing (see, e.g., Ref. [20]). This
generality enables us to apply this method for the IFS of our interest.

##  Appendix B Derivation of Eq. (15)

Here we derive Eq. (15) by substituting Eq. (14) into the Schrödinger equation
$id\ket{\Psi(t)}/dt=H_{\mathrm{pulse}}(t)\ket{\Psi(t)}$. Note that
$H_{\mathrm{pulse}}(t)=H_{\mathrm{cw}}(a(t),t)$ by definition, and the
Schrödinger equation reads

$\displaystyle
i\frac{d\ket{\Psi(t)}}{dt}=H_{\mathrm{cw}}(a(t),t)\ket{\Psi(t)}.$ (44)

The left-hand of Eq. (44) side becomes

$\displaystyle
i\frac{d\ket{\Psi(t)}}{dt}=i\sum_{\alpha}\left[\frac{dc_{\alpha}(t)}{dt}\ket{u_{\alpha}(a(t),t)}+c_{\alpha}(t)\frac{da}{dt}\frac{\partial\ket{u_{\alpha}(a(t),t)}}{\partial
a}+c_{\alpha}(t)\frac{\partial\ket{u_{\alpha}(a(t),t)}}{\partial t}\right].$
(45)

Here we recall the definition of Floquet states, $i\frac{\partial}{\partial
t}[e^{-i\epsilon_{\alpha}(a(t))t}\ket{u_{\alpha}(a(t),t)}]=H_{\mathrm{cw}}(a(t),t)[e^{-i\epsilon_{\alpha}(a(t))t}\ket{u_{\alpha}(a(t),t)}]$,
which implies

$\displaystyle i\frac{\partial\ket{u_{\alpha}(a(t),t)}}{\partial
t}=[-\epsilon_{\alpha}(a(t))+H_{\mathrm{cw}}(a(t),t)]\ket{u_{\alpha}(a(t),t)}.$
(46)

Then we substitute Eq. (45) together with Eq. (46) into Eq. (44), obtaining

$\displaystyle\sum_{\alpha}\left[i\frac{dc_{\alpha}(t)}{dt}-\epsilon_{\alpha}(a(t))\right]\ket{u_{\alpha}(a(t),t)}$
$\displaystyle+i\sum_{\alpha}c_{\alpha}(t)\frac{da}{dt}\frac{\partial\ket{u_{\alpha}(a(t),t)}}{\partial
a}=0,$ (47)

where the terms with $H_{\mathrm{cw}}$ canceled out between the left- and
right-hand sides.

Finally, to rewrite the second term of Eq. (47), we note the following
relations

$\displaystyle\frac{\partial\ket{u_{\alpha}(a(t),t)}}{\partial a}$
$\displaystyle=\int_{0}^{T}dt^{\prime}\delta_{T}(t-t^{\prime})\frac{\partial\ket{u_{\alpha}(a(t),t^{\prime})}}{\partial
a}$
$\displaystyle=\sum_{\beta}\int_{0}^{T}\frac{dt^{\prime}}{T}\ket{u_{\beta}(a(t),t)}\bra{u_{\beta}(a(t),t^{\prime})}\frac{\partial\ket{u_{\alpha}(a(t),t^{\prime})}}{\partial
a}$
$\displaystyle=\sum_{\beta}\ket{u_{\beta}(a(t),t)}\mathcal{G}_{\beta\alpha}(a(t)),$
(48)

where we used Eqs. (13) and (17) to obtain the third and fourth lines,
respectively. Substituting Eq. (48) into Eq. (47) and considering the
coefficients of each $\ket{u_{\alpha}(a(t),t)}$, we obtain

$\displaystyle
i\frac{dc_{\alpha}(t)}{dt}=\epsilon_{\alpha}(a(t))-i\frac{da}{dt}\sum_{\beta}\mathcal{G}_{\alpha\beta}(a(t))c_{\beta}(t),$
(49)

which is equivalent to Eq. (15) in the matrix representation.

##  Appendix C Analytical approach to circular and elliptic polarizations

In this appendix, we consider the following coupling term

$\displaystyle V(t)=\frac{1+\lambda}{2}\cos(\omega
t)\sigma_{x}+\frac{1-\lambda}{2}\sin(\omega t)\sigma_{y},$ (50)

which reduces to Eq. (3) for $\lambda=1$. For a single spin-1/2, this term
represents the Zeeman coupling $V(t)=\bm{B}(t)\cdot\bm{\sigma}$ to an
elliptically-polarized magnetic field
$\bm{B}(t)=(\frac{1+\lambda}{2}\cos(\omega t),\frac{1-\lambda}{2}\sin(\omega
t),0)$. The dimensionless parameter $\lambda$ quantifies the ellipticity, and
the special values $\lambda=0$ and $1$ correspond to the circular and linear
polarizations, respectively. Therefore, we call $\lambda=0$, 1, and the others
as the linear, circular, and elliptic polarizations, respectively, even if the
model does not necessarily suppose a single spin-1/2.

In the main text, we have shown that the FLZ theory works well for the linear
polarization $(\lambda=1)$. In those calculations, we implemented the transfer
matrices $\mathcal{T}_{n}$ and phase acquisition operators
$\mathcal{U}_{n+1,n}$ constructed from the quasienergies $\epsilon_{m}(a)$
obtained numerically. When $\lambda$ is zero or small, however, we can
analytically obtain the quasienergies approximately using the perturbation
theory for $\lambda$. In this appendix, using this analytical approach, we
extend analyses and gain deeper insights from the limit of $\lambda=0$ to
small $\lambda$.

###  C.1 Circular polarization

We begin by considering the circular polarization ($\lambda=0$), for which the
coupling term reads

$\displaystyle V_{0}(t)=e^{-i\omega t}\sigma_{+}+e^{i\omega t}\sigma_{-}$ (51)

with $\sigma_{\pm}=(\sigma_{x}\pm i\sigma_{y})/2$. Note that this case
corresponds to the rotating-wave approximation of Eq. (3). In this special
case, the continuous-wave problem (8) corresponds to the seminal Rabi model
[6]. We analytically obtain the two independent solutions as

$\displaystyle\ket{\psi_{A}(t)}$
$\displaystyle=\frac{e^{-i(\Omega+\frac{\omega}{2})t}}{\sqrt{2\Omega[\Omega+(\omega-b)/2]}}\begin{pmatrix}a/2\\\
\left(\frac{\omega-b}{2}+\Omega\right)e^{i\omega t}\end{pmatrix},$ (52)
$\displaystyle\ket{\psi_{B}(t)}$
$\displaystyle=\frac{e^{i(\Omega+\frac{\omega}{2})t}}{\sqrt{2\Omega[\Omega-(\omega-b)/2]}}\begin{pmatrix}\frac{a}{2}e^{-i\omega
t}\\\ \frac{\omega-b}{2}-\Omega\end{pmatrix},$ (53)

where

$\displaystyle\Omega=\frac{1}{2}\sqrt{a^{2}+(b-\omega)^{2}}$ (54)

is the Rabi frequency (energy).

We can read out the Floquet states and their quasienergies from Eqs. (52) and
(53). Recall that, in this work, we assign the Floquet states’ label $(m,l)$
so that $(m,0)$ approaches the undriven solutions $e^{-i(b/2)t}\ket{\uparrow}$
($e^{+i(b/2)t}\ket{\downarrow}$) for $m=1$ ($m=2$). To make this assignments,
it is convenient to consider the two cases, $b>\omega$ and $b<\omega$,
separately. For $b>\omega$,

$\displaystyle\ket{\psi_{A}(t)}\to e^{-i(b/2)t}\ket{\uparrow},\
\ket{\psi_{B}(t)}\to e^{+i(b/2)t}\ket{\downarrow}$ (55)

in the limit of $a\to 0$. Thus, in this case, we see that

$\displaystyle\ket{u_{1}(a;t)}$
$\displaystyle=\frac{1}{\sqrt{2\Omega[\Omega+(\omega-b)/2]}}\begin{pmatrix}a/2\\\
\left(\frac{\omega-b}{2}+\Omega\right)e^{i\omega t}\end{pmatrix},$ (56)
$\displaystyle\ket{u_{2}(a;t)}$
$\displaystyle=\frac{1}{\sqrt{2\Omega[\Omega-(\omega-b)/2]}}\begin{pmatrix}\frac{a}{2}e^{-i\omega
t}\\\ \frac{\omega-b}{2}-\Omega\end{pmatrix},$ (57)

and all the quasienergies (12) are

$\displaystyle\epsilon_{1,l}(a)$
$\displaystyle=+\Omega+\frac{\omega}{2}+l\omega,$ (58)
$\displaystyle\epsilon_{2,l}(a)$
$\displaystyle=-\Omega-\frac{\omega}{2}+l\omega.$ (59)

We plot the quasienergy for a representative off-resonant case $b=1.5$ in Fig.
8(a). As the analytical expressions imply, there is no avoided crossing.

On the other hand, for $b<\omega$,

$\displaystyle\ket{\psi_{A}(t)}\to e^{+i(b/2)t}\ket{\downarrow},\
\ket{\psi_{B}(t)}\to e^{i(\omega-b/2)t}\ket{\uparrow}$ (60)

as $a\to 0$. This means that

$\displaystyle\ket{u_{1}(a;t)}$ $\displaystyle=\frac{e^{i\omega
t}}{\sqrt{2\Omega[\Omega-(\omega-b)/2]}}\begin{pmatrix}\frac{a}{2}e^{-i\omega
t}\\\ \frac{\omega-b}{2}-\Omega\end{pmatrix},$ (61)
$\displaystyle\ket{u_{2}(a;t)}$
$\displaystyle=\frac{1}{\sqrt{2\Omega[\Omega+(\omega-b)/2]}}\begin{pmatrix}a/2\\\
\left(\frac{\omega-b}{2}+\Omega\right)e^{i\omega t}\end{pmatrix},$ (62)

and all the quasienergies (12) are

$\displaystyle\epsilon_{1,l}(a)$
$\displaystyle=-\Omega+\frac{\omega}{2}+l\omega,$ (63)
$\displaystyle\epsilon_{2,l}(a)$
$\displaystyle=+\Omega+\frac{\omega}{2}+l\omega.$ (64)

On the resonance $b=\omega$, $\ket{\psi_{A}(t)}$ and $\ket{\psi_{B}(t)}$ do
not converge to either $\ket{\uparrow}$ or $\ket{\downarrow}$ but to
superpositions of them in the limit of $a\to 0$. In fact, we have, for
$b=\omega$,

$\displaystyle\ket{\psi_{A}(t)}$ $\displaystyle\to
e^{-i(\omega/2)t}\frac{\ket{\uparrow}+\ket{\downarrow}}{\sqrt{2}},$ (65)
$\displaystyle\ket{\psi_{B}(t)}$ $\displaystyle\to
e^{+i(\omega/2)t}\frac{\ket{\uparrow}-\ket{\downarrow}}{\sqrt{2}}$ (66)

as $a\to 0$. We will see that this special property on the resonance
$b=\omega$ leads to nontrivial behaviors of excitation probabilities
$P_{\uparrow}$. We plot the quasienergy for the resonant case $b=1$ in Fig.
8(b), where they are degenerate at $a=0$.

Figure 8: Quasienergies for $\lambda=0$ in (a) $b=1.5$ and (b) $b=1.0$ plotted
against coupling strength $a$. (a) Solid (dashed) lines show those for Floquet
states of $m=2$ ($m=1$) approaching $\ket{\downarrow}$ ($\ket{\uparrow}$) as
$a\to 0$. (b) Solid and dashed lines show those for Floquet states
corresponding to $\ket{\psi_{B}(t)}$ and $\ket{\psi_{A}(t)}$ approaching
$(\ket{\uparrow}\mp\ket{\downarrow})/\sqrt{2}$ as Eqs. (65) and (66),
respectively.

With these Floquet states, let us now study the pulse excitation problem for
the circular polarization. Figure 9(a) illustrates the excitation probability
$P_{\uparrow}$ for $\nu=6$, showing (i) almost no excitation away from the
resonance $b\neq 1$ and (ii) an oscillating behavior on resonance $b=1$.

Figure 9: Excitation probability $P_{\uparrow}$ plotted against the pulse peak
height $a_{0}$ and the energy level difference $b$ for (a) the circular
($\lambda=0$) and (b) an elliptic ($\lambda=0.1$) polarizations with pulse
width $\nu=6$.

The property (i) is interpreted as follows. Away from resonance, we have a
quasienergy diagram like Fig. 8(a) without avoided crossings, and each Floquet
state approaches $\ket{\uparrow}$ or $\ket{\downarrow}$ as $a\to 0$. In such a
case, the IFS interpretation goes like at the end of Sec. III.1. Our initial
state for the pulse problem is $\ket{\downarrow}$, and it adiabatically moves
along $\ket{u_{2}(a(t);t)}$, which coincides with $\ket{\downarrow}$ at
$t=-\infty$. In the adiabatic move, there occurs no LZ-type transition to
other branches of Floquet states, and finally the state comes back to
$\ket{\downarrow}$, meaning that $P_{\uparrow}\approx 0$. Thus, the absence of
avoided crossings, a special property of $\lambda=0$, explains the suppressed
$P_{\uparrow}$ for $b\neq 1$ in Fig. 9(a).

The property (ii) is interpreted as follows. On resonance, the Floquet states
do not converge to either $\ket{\uparrow}$ or $\ket{\downarrow}$ but approach
superpositions of them in $a\to 0$. Thus, our initial state $\ket{\downarrow}$
is a superposition of $\ket{u_{1}(a(t);t)}$ and $\ket{u_{2}(a(t);t)}$ at
$t=-\infty$. While these Floquet states move adiabatically without LZ-type
transitions as there is no avoided crossing, they acquire relative phase
factors due to the quasienergies. Since the acquired phase from $t=-\infty$ to
$t=+\infty$ is an increasing function of the pulse peak height $a_{0}$, the
final excitation probability $P_{\uparrow}$ oscillates with $a_{0}$. This
mechanism is FLZ interferometry discussed in Sec. IV although the
superposition of Floquet states here is not created by LZ-type transitions but
by particular limiting behaviors (65) and (66).

###  C.2 Elliptic polarization

In Appendix C.1, we have shown that $\lambda=0$ is an ideal limit where the
quasienergies are obtained analytically and no avoided crossing occurs. For
$\lambda\neq 0$, $V(t)$ involves the counter-rotating component on top of Eq.
(51):

$\displaystyle V(t)$ $\displaystyle=V_{0}(t)+\lambda W(t),$ (67)
$\displaystyle W(t)$ $\displaystyle=e^{i\omega t}\sigma_{+}+e^{-i\omega
t}\sigma_{-}.$ (68)

This component hybridizes the independent solutions [Eqs. (52) and (53)] for
$\lambda=0$, giving rise to avoided crossings of quasienergies. To address
this scenario analytically, we here consider the case of small $\lambda$’s,
i.e., nearly-circular elliptic polarizations. In these cases, we can use the
Floquet states for $\lambda=0$ (see Appendix C.1) as the unperturbed solutions
and approximately obtain quasienergies with avoided crossings by perturbation
theory in terms of $\lambda a_{0}$ (see, e.g., Ref. [25] for technicalities).

We remark that this approach does not assume $a_{0}$ is small. In fact, the
unperturbed solutions (Floquet states for $\lambda=0$) can involve
nonperturbative effects of $a_{0}$. Thus, this approach is particularly useful
when

$\displaystyle\lambda a_{0}\ll 1\ll a_{0}$ (69)

since, for $a_{0}\ll 1$, we can apply the naive perturbation theory in terms
of $a_{0}$ for arbitrary $\lambda$. We cannot find such parameters (69) for
the linear polarization ($\lambda=1$) that we studied in the main text, where
we needed to calculate quasienergies numerically.

Figure 10: Quasienergies for $\lambda=0.1$ in (a) $b=1.5$ and (b) $b=0.5$
plotted against coupling strength $a$. Solid (dashed) lines show those for
Floquet states of $m=2$ ($m=1$) approaching $\ket{\downarrow}$
($\ket{\uparrow}$) as $a\to 0$. Thicker background curves show quasienergies
for $\lambda=0$ for reference.

Before developing detailed calculations, we qualitatively see how the FLZ
theory applies to elliptic polarizations. Figure 9(b) illustrates the
excitation probability $P_{\uparrow}$ for $\lambda=0.1$ and $\nu=6$. Unlike
the circular polarization case [see panel (a)], $P_{\uparrow}$ becomes
significant away from the resonance condition $b=1$. For, say, $b=1.5$ or
$0.5$, as $a_{0}$ increases, $P_{\uparrow}$ emerges at $a_{0}\approx 2$ and
then oscillates. This behavior is consistent with the quasienergy diagram in
Fig. 10, which shows that the first avoided crossing appears at
$a_{\mathrm{AC},1}\approx 2$. Thus, the FLZ interpretation discussed in Sec.
IV together with the quasienergy diagram qualitatively explains the
interference pattern in Fig. 9(b). We note the similarity between the
quasienergies for $\lambda=0.1$ and $0$ shown in Fig. 10; They are almost on
top of each other away from the avoided crossings whereas they slightly repel
each other at the avoided crossings. This suggests the validity of considering
$\lambda a_{0}$ as perturbation.

Let us now quantitatively apply the FLZ theory developed in Sec. III with its
inputs, such as $a_{\mathrm{AC},n}$ and $\epsilon_{m,l}(a)$, obtained
analytically. For concreteness, we first focus on $1<b<3$, for which the
quasienergy diagram looks like Fig. 10(a). As shown in the figure, although
the quasienegies $\epsilon_{1,l}(a)$ and $\epsilon_{2,l+2}(a)$ overlap at
$a\approx 1$, they do not repel. This is a selection rule due to the fact that
$W(t)$ does not have matrix elements between these Floquet states,

$\displaystyle\int_{0}^{T}\frac{dt}{T}\braket{u_{1,l}(a;t)}{W(t)}{u_{2,l+2}(a;t)}=0,$
(70)

where $\ket{u_{m,l}(t)}=e^{il\omega t}\ket{u_{m}(t)}$ with Eqs. (56) and (57).
More generally, similar selection rules follow from the fact that the matrix
elements vanish between $(1,l)$ and $(2,l+2k)$ ($k\in\mathbb{Z}$), which means
physically that $2k$-photon transitions are prohibited. Therefore, the first
avoided crossing occurs at the 3-photon resonance defined by
$\epsilon_{1,l}(a)=\epsilon_{2,l+3}(a)$, which gives

$\displaystyle
a_{\mathrm{AC},1}=\sqrt{(\omega+b)(3\omega-b)}\qquad(\text{for}\
\omega<b<3\omega).$ (71)

Note that we needed to calculate numerically $a_{\mathrm{AC},1}$ for the
linear polarization ($\lambda=1$) in Sec. IV.

The quasienergies in the presence of small $\lambda$ are obtained by
considering the coupling by $\lambda W(t)$ between the unperturbed Floquet
states $\alpha=(1,l)$ and $(2,l+3)$. The Floquet Hamiltonian within the
2-dimensional subspace reads

$\displaystyle\mathcal{H}_{\alpha\beta}=\begin{pmatrix}\epsilon_{1,l}(a)&\lambda
K\\\ \lambda K&\epsilon_{2,l+3}(a)\end{pmatrix}_{\alpha\beta},$ (72)

where $\alpha$ and $\beta$ denote either $(1,l)$ or $(2,l+3)$ and $\lambda
K\equiv\int_{0}^{T}\frac{dt}{T}\braket{u_{1,l}(a)}{\lambda
W(t)}{u_{2,l+3}(a)}$ yielding

$\displaystyle
K=\lambda\frac{a^{2}}{8\Omega}\sqrt{\frac{\Omega+\frac{\omega-b}{2}}{\Omega-\frac{\omega-b}{2}}}.$
(73)

The eigenvalues of Eq. (72) lead to the quasienergies with avoided crossing:

$\displaystyle\epsilon_{1,l}^{(\lambda)}(a;t)$
$\displaystyle\approx\frac{\omega}{2}-\sqrt{\left(\frac{\epsilon_{1,l}(a)-\epsilon_{2,l+3}(a)}{2}\right)^{2}+(\lambda
K)^{2}},$ $\displaystyle\epsilon_{2,l+3}^{(\lambda)}(a;t)$
$\displaystyle\approx\frac{\omega}{2}+\sqrt{\left(\frac{\epsilon_{1,l}(a)-\epsilon_{2,l+3}(a)}{2}\right)^{2}+(\lambda
K)^{2}}.$ (74)

In the approximation made here, we have ignored couplings outside the
2-dimensional subspace, and Eq. (74) involves higher-order terms in $\lambda$.
The quasienergy gap at the avoided crossing follows from Eq. (74) as

$\displaystyle\Delta_{1}$
$\displaystyle=\epsilon_{2,l+3}^{(\lambda)}(a_{\mathrm{AC},1})-\epsilon_{1,l}^{(\lambda)}(a_{\mathrm{AC},1})$
$\displaystyle=\lambda\frac{\sqrt{(\omega+b)(3\omega-b)^{3}}}{4\omega},$ (75)

where we used $\epsilon_{1,l}(a)=\epsilon_{2,l+3}(a)$ and hence
$\Omega=\omega$ at $a=a_{\mathrm{AC},1}$.

Figure 11: Excitation probability $P_{\uparrow}$ for (a) $b=1.5$ and (b)
$b=0.5$ plotted against the pulse peak height $a_{0}$ for an elliptic
polarization ($\lambda=0.1$) with pulse width $\nu=6$. The solid and dashed
curves show $P_{\uparrow}$ obtained, respectively, by solving the TDSE (5)
numerically and by invoking the FLZ theory (38).

Given that the avoided crossing point (71), quasienergies (74), and their gap
(75) are analytically obtained, we can implement the FLZ theory
quantitatively. Restricting ourselves to
$a_{\mathrm{AC},1}<a_{0}<a_{\mathrm{AC},2}$ for simplicity, we obtain
$P_{\uparrow}$ as in Eq. (38), where $P_{1}$ and $\varphi_{1}^{S}$ are
obtained using Eqs. (22), (23), and (24), and $\Phi_{m,l}$ are obtained
combining Eqs. (37) and (74) with numerical evaluation of the integral. The
excitation probability $P_{\uparrow}$ thus obtained is compared with the
direct numerical solution for $b=1.5$, $\lambda=0.1$, and $\nu=6$ in Fig.
11(a). Like in the linear polarization case studied in Sec. IV, the FLZ theory
well describes the numerical exact solution except for the vicinity of
$a=a_{\mathrm{AC},1}$, where the adiabatic-impulse approximation is not valid.
The oscillation of $P_{\uparrow}$ in $a_{0}>a_{\mathrm{AC},1}$ is due to the
FLZ interferometry, and thus we have quantitatively elucidated the stripe-
shaped pattern for $1<b<3$ in Fig. 9(b). Note that, unlike the linear
polarization case, we can implement the FLZ theory almost fully analytically
in nearly-circular elliptic polarizations except for the numerical integration
in obtaining $\Phi_{m,l}$ from analytically obtained quasienergies.

A similar analysis works for $0<b<1$ as well. For this case, the Floquet-state
hybridization occurs between $\alpha=(1,l+1)$ and $(2,l)$ as seen in Fig.
10(b). Namely, the first avoided crossing is caused by the 1-photon resonance.
Like in $1<b<3$ discussed above, we can perform perturbation-theory analyses
for these pairs of Floquet states, obtaining the avoided crossing point,
quasienergies and their gap. We plot the resulting $P_{\uparrow}$ calculated
by the FLZ theory in Fig. 11 together with the exact numerical results. Here
again, we obtain a quantitatively good agreement between these results even
though we resorted to perturbation-theory approximations.

##  Appendix D Justification of the FLZ theory

###  D.1 Relation to the semiclassical limit

Here we show that the limit of $\nu\to 0$ formally corresponds to the
semiclassical limit $\hbar\to 0$. For this purpose, we rewrite Eq. (15),
highlighting the approach to adiabaticity. Let us define
$\hat{a}(t)=a_{0}\exp[-(t/T)^{2}]$, which coincides with $a(t)$ at $\nu=1$ and
satisfies $a(t)=\hat{a}(t/\nu)=\hat{a}(\eta t)$. Here, $\eta\equiv 1/\nu$ is
the adiabaticity parameter [21], and the $\eta\to 0$ ($\nu\to\infty$) limit
corresponds to the infinitely slowly varying envelope function. Using
$\hat{a}$ and introducing the rescaled time $s\equiv\eta t$, we rewrite Eq.
(15) as

$\displaystyle
i\eta\frac{d\hat{c}_{\alpha}(s)}{ds}=\sum_{\beta}\mathcal{H}_{\alpha\beta}(\hat{a}(s))\hat{c}_{\beta}(s)$
(76)

with $\hat{c}_{\alpha}(s)\equiv c_{\alpha}(t/\eta)$. Equation (76) shows that
the adiabatic limit $\eta\to 0$ is formally equivalent to the semiclassical
limit $\hbar\to 0$. Considering that the Landau-Zener-type transition derives
from the series expansion for $\hbar$ and its resummation [10, 19], we
naturally expect that our FLZ theory works well when $\eta\ll 1$, i.e.,
$\nu\gg 1$.

The above argument is, at least, consistent with our numerics in Fig. 6, where
the FLZ theory works well for $\nu\geq 2$ but not for $\nu=1$. However,
characterizing the precise threshold, which should be model-dependent, and
showing why it is about 2 in the present model remain open for future work.

###  D.2 Transitions to distant IFSs

The transfer matrix $\mathcal{T}_{n}$ only mixes pairs of IFSs that have the
nearest quasienergies although the mixings can, in principle, happen between
other IFS pairs that are more distant in the quasienergy. These transitions
neglected in our FLZ theory could be nonnegligible when the pulse width $\nu$
becomes very small. However, we show here that these transitions are not
relevant in the present model.

To study this possibility quantitatively, we rewrite Eq. (15) so that the
$\nu$-dependence is evident:

$\displaystyle i\frac{d\tilde{c}_{\alpha}(\tau)}{d\tau}$
$\displaystyle=\sum_{\beta}\tilde{\mathcal{H}}_{\alpha\beta}(\tilde{a}(\tau))\tilde{c}_{\beta}(\tau),$
(77) $\displaystyle\tilde{\mathcal{H}}_{\alpha\beta}(\tilde{a}(\tau))$
$\displaystyle\equiv\nu
T\delta_{\alpha\beta}\epsilon_{\alpha}(\tilde{a}(\tau))-i\frac{d\tilde{a}}{d\tau}\mathcal{G}_{\alpha\beta}(\tilde{a}(\tau)).$
(78)

Here, $\tau\equiv\frac{t}{\nu T}$ is a dimensionless time,
$\tilde{c}_{\alpha}(\tau)=c_{\alpha}(\frac{t}{\nu T})$, and
$\tilde{a}(\tau)=a(\frac{t}{\nu T})=a_{0}\exp(-\tau^{2})$. In this
representation, $\nu$ effectively rescales the quasienergy as the first term
on the right-hand side of Eq. (78). For an IFS pair $(\alpha,\beta)$, their
transition is negligible if their effective quasienergy difference $\nu
T|\epsilon_{\alpha}-\epsilon_{\beta}|$ is much larger than their coupling
$|\frac{d\tilde{a}}{d\tau}\mathcal{G}_{\alpha\beta}(\tilde{a}(\tau))|$. Our
IFS theory neglects the pair transitions for
$|\epsilon_{\alpha}-\epsilon_{\beta}|>\omega$, and this treatment is justified
if

$\displaystyle\nu\gg\nu_{c}\equiv\frac{1}{T}\max_{\tau}\max_{\begin{subarray}{c}\alpha,\beta\\\
|\epsilon_{\alpha}-\epsilon_{\beta}|>\omega\end{subarray}}\left|\frac{d\tilde{a}}{d\tau}\frac{\mathcal{G}_{\alpha\beta}(\tilde{a}(\tau))}{\epsilon_{\alpha}(a(\tau))-\epsilon_{\beta}(a(\tau))}\right|.$
(79)

Figure 12: The ratio
$\mathcal{G}_{\alpha\beta}(a)/\\{T[\epsilon_{\alpha}(a)-\epsilon_{\beta}(a)]\\}$
for $\alpha=(2,0)$ and $\beta=(1,-3)$ (solid) and $(1,-1)$ (dashed). Here we
set $b=2.5$ as in Figs. 2 and 6.

Now we numerically confirm that $\nu_{c}$ is so small that the transitions to
distant IFS are negligible. Since $|d\tilde{a}/d\tau|<0.74$, we focus on the
ratio

$\displaystyle
R_{\alpha\beta}(a)\equiv\frac{\mathcal{G}_{\alpha\beta}(a)}{[\epsilon_{\alpha}(a)-\epsilon_{\beta}(a)]T}$
(80)

and verify $|R_{\alpha\beta}(a)|\ll 1$ for
$|\epsilon_{\alpha}(a)-\epsilon_{\beta}(a)|>\omega$. This ratio is shown in
Fig. 12 for $\alpha=(2,0)$ and $b=2.5$, which we mainly argued in the main
text. The ratio $R_{\alpha\beta}(a)$ shows large peaks at
$a=a_{\mathrm{AC},1}=1.09$ for $\beta=(1,-3)$ and at
$a=a_{\mathrm{AC},2}=3.05$ for $\beta=(1,-1)$, which correspond to
$|\epsilon_{\alpha}(a)-\epsilon_{\beta}(a)|<\omega$ and are responsible for
the transitions incorporated by the transfer matrices $\mathcal{T}_{1}$ and
$\mathcal{T}_{2}$, respectively (see Fig. 2). Except for these peaks,
$|R_{\alpha\beta}(a)|\lesssim 0.05$ for $0\leq a\leq 4$ as shown in the inset
($|R_{\alpha\beta}|$ is much smaller for other $\beta$’s not shown in the
figure). This means $\nu_{c}<0.05$ for $a_{0}\leq 4$ considered in the main
text.

Recall that the FLZ theory starts to fail when $\nu\sim 1$ as shown in Fig. 6.
However, as $\nu_{c}<0.05$, Eq. (79) still remains true. Thus, we conclude
that the transition to distant IFSs is not the main reason for the FLZ’s
failure at the ultrashort pulse as short as $\nu\sim 1$.

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