# Detecting Topological phase transitions in a double kicked quantum rotor

Nikolai Bolik  Institute of Theoretical Physics, Heidelberg University,
Philosophenweg 16, 69120 Heidelberg, Germany  Caspar Groiseau  Departamento de
Física Teórica de la Materia Condensada and Condensed Matter Physics Center
(IFIMAC), Universidad Autónoma de Madrid, 28049 Madrid, Spain  Jerry H. Clark
Department of Physics, Oklahoma State University, Stillwater, Oklahoma
74078-3072, USA  Gil S. Summy<EMAIL_ADDRESS>Department of Physics,
Oklahoma State University, Stillwater, Oklahoma 74078-3072, USA  Airy3D, 5445
Avenue de Gaspé Suite 230, Montréal, Québec H2T 3B2, Canada  Yingmei Liu
<EMAIL_ADDRESS>Department of Physics, Oklahoma State University,
Stillwater, Oklahoma 74078-3072, USA  Sandro Wimberger
<EMAIL_ADDRESS>Dipartimento di Scienze Matematiche, Fisiche e
Informatiche, Università di Parma, Parco Area delle Scienze 7/A, 43124 Parma,
Italy  INFN, Sezione di Milano Bicocca, Gruppo Collegato di Parma, Parco Area
delle Scienze 7/A, 43124 Parma, Italy

###### Abstract

We present a concrete theoretical proposal for detecting topological phase
transitions in double kicked atom-optics kicked rotors with internal spin-1/2
degree of freedom. The implementation utilizes a kicked Bose-Einstein
condensate evolving in one-dimensional momentum space. To reduce influence of
atom loss and phase decoherence we aim to keep experimental durations short
while maintaining a resonant experimental protocol. Experimental limitations
induced by phase noise, quasimomentum distributions, symmetries, and the AC-
Stark shift are considered. Our results thus suggest a feasible and optimized
procedure for observing topological phase transitions in quantum kicked
rotors.

Discrete-time quantum walk, Bose-Einstein condensates, Atom-optics kicked
rotor, Topological phases, Floquet-Bloch engineering

##  I Introduction

Studies of topological phases have found many applications, including
revealing topologically protected edge states in topological insulators [1, 2,
3, 4, 5]. Stabilities of topological invariants are translated upon these edge
states, enabling them to be stable against a great variety of perturbations.
This robustness to decoherence makes topological phenomena intriguing for many
potential applications, e.g., in quantum computing [6] and quantum walks [7].

Topological effects can be simulated by periodically driven systems in a well-
controlled manner [8, 7, 9]. A double kicked quantum rotor (DKQR) with
internal spin-1/2 degree of freedom is an example of such systems and
therefore a potential candidate to experimentally realize topological phase
transitions [10, 11]. In the DKQR system a quantity known as a winding number
$\nu$ is topologically invariant under a wide range of transformations [10]. A
similar phenomenon occurs in solid state systems where holes are preserved
under certain transformations due to geometrical topology [12]. Preservation
of topological winding numbers requires preservation of chiral symmetry and
the band gap [10, 11]. A consequence of permanently containing chiral symmetry
is that the topological invariant can only change when the system is changed
to a configuration in which the band gap closes. This closure makes the phase
undetermined but also allows for its direct experimental control by scanning a
system parameter through it. The same is true for periodically driven systems
with Floquet spectra, as in our case of the DKQR [10]. Here, the quasienergy
spectrum itself is periodic and gaps can be controlled by the driving
parameters. Floquet topological states were observed in different experimental
settings, including ultracold atom [13, 14], photonic [15, 16, 17], phononic
and acoustic systems [18, 19, 20]. To understand which topological phases the
system has, one examines its spectral symmetries and the related "protected
gaps" in the quasieneregy spectrum of the Floquet Hamiltonian, see e.g. [21,
22, 7]. In our DKQR systems, the gaps and the phases can be tuned by the
kicking strengths, as described in detail in [10].

The experimental setup under consideration as described in [23] consists of a
Bose-Einstein condensate (BEC) with two Zeeman hyperfine states $|1\rangle$
and $|2\rangle$ participating in the dynamics, effectively forming a
spin-$1/2$ system. DKQR is based on a singly quantum kicked rotor (QKR) [24,
25] and is described by the Hamiltonian

$\begin{split}\hat{\mathcal{H}}&=\frac{\hat{p}^{2}\otimes\mathbf{1}}{2}+k_{1}\leavevmode\nobreak\
\mathrm{cos}(\hat{\theta})\otimes\hat{\sigma}_{x}\sum_{n=0}^{\infty}\delta(t-2n\tau)\\\
\ &+k_{2}\leavevmode\nobreak\
\mathrm{sin}(\hat{\theta})\otimes\hat{\sigma}_{y}\sum_{n=0}^{\infty}\delta(t-(2n+1)\tau).\end{split}$
(1)

Here, $\hat{p}$ and $\hat{\theta}$ are respectively the momentum and angular
position operators, $\tau$ describes the duration between two kicks of
different kicking strengths $k_{1}$ and $k_{2}$, and the Pauli-matrices
$\hat{\sigma}_{x}$ and $\hat{\sigma}_{y}$ act on internal spin-$1/2$ degree of
freedom. Under the on-resonance condition of $\tau=4\pi$ [26, 27, 28, 29, 30,
31, 32], corresponding to a full revival (at the Talbot time) of the free
evolution of the momentum degree of freedom, the quasiperiodicity of the
system can lead to a Hofstadter butterfly-like quasienergy spectrum [33, 34,
35], resolving a band structure rich in displaying topological properties
[10].

The previously proposed experimental sequences for achieving topological phase
transitions in DKQRs include a sequence of resonant microwave (MW) and
standing-wave kicking laser pulses [10, 11]. Similar sequences have been
successfully conducted on QKR systems, including ours consisting of 87Rb BECs
[36, 23, 37]. For observing the predicted topological effects, the
experimental procedure needs to be carefully designed to overcome a number of
experimental challenges. In particular, phase noise arising from random phase
fluctuations in the MW pulses and a finite quasimomentum distribution of the
BEC must be considered. In this paper we demonstrate how to transform the
proposals for detecting topological phases in DKQRs [10, 11] into a feasible
experimental procedure by presenting solutions to some experimental
limitations/challenges that appear in the BEC-based quantum walk and DKQR
setups [36, 23, 37].

##  II Previous Theoretical Proposals

We briefly review previous proposals for the measurement of the topological
phases in the DKQR setup [10, 11]. In these proposals, the MW operations
correspond to a Rabi coupling between the two internal states
$\ket{F=1,m_{F}=0}$ and $\ket{F=2,m_{F}=0}$ of the atoms and are expressed as
a unitary rotation on the Bloch sphere:

$\hat{M}(\alpha,\chi)=\begin{pmatrix}\cos(\frac{\alpha}{2})&e^{-i\chi}\sin(\frac{\alpha}{2})\\\
-e^{i\chi}\sin(\frac{\alpha}{2})&\cos(\frac{\alpha}{2})\end{pmatrix},$ (2)

The kicking laser is detuned between these internal states in such a way where
the potential is equal in strength but opposite in sign, as expressed by a
$\hat{\sigma}_{z}$ matrix. The kick operators, $\hat{K}_{1}$ and $\hat{K}_{2}$
differing only by a shift of $\theta=\pi/2$ in position space, are defined as

$\displaystyle\hat{K}_{1}$
$\displaystyle=e^{-ik_{1}\mathrm{cos}(\hat{\theta})\hat{\sigma}_{z}}$ (3)
$\displaystyle\hat{K}_{2}$
$\displaystyle=e^{-ik_{2}\mathrm{sin}(\hat{\theta})\hat{\sigma}_{z}}.$ (4)

As discussed in detail in [10], the system consists of two chirally symmetric
time frames expressed with two Floquet operators, $\hat{U}_{1}$ and
$\hat{U}_{2}$, possessing chiral symmetry. These operators are best realised
as a sequence of MW and kick operators on the atoms’ initial wave function,
e.g., $\ket{\psi_{\rm in}}=|n=0\rangle\otimes|2\rangle$, and take the
following form:

$\displaystyle\begin{split}\hat{U}_{1}=&\hat{M}\left(-\frac{\pi}{2},0\right)\hat{K}^{\frac{1}{2}}_{1}\hat{M}\left(\frac{\pi}{2},0\right)\hat{M}\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\hat{K}_{2}\hat{M}\left(\frac{\pi}{2},\frac{\pi}{2}\right)\\\
&\cdot\hat{M}\left(-\frac{\pi}{2},0\right)\hat{K}^{\frac{1}{2}}_{1}\hat{M}\left(\frac{\pi}{2},0\right)\end{split}$
(5)
$\displaystyle\begin{split}\hat{U}_{2}=&\hat{M}\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\hat{K}^{\frac{1}{2}}_{2}\hat{M}\left(\frac{\pi}{2},\frac{\pi}{2}\right)\hat{M}\left(-\frac{\pi}{2},0\right)\hat{K}_{1}\hat{M}\left(\frac{\pi}{2},0\right)\\\
&\cdot\hat{M}\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\hat{K}^{\frac{1}{2}}_{2}\hat{M}\left(\frac{\pi}{2},\frac{\pi}{2}\right)\end{split}$
(6)

with

$\hat{K}_{1,2}^{1/2}\equiv\hat{K}_{1,2}\left(\frac{k}{2},\theta\right).$ (7)

Due to the structural similarity of both operators we will further focus on
the discussion of $\hat{U}_{2}$, referred to hereafter as $\hat{\mathcal{U}}$
in this paper. Analogous reasoning can be found for $\hat{U}_{1}$, but is not
explicitly shown here.

As an abstract quantity, the topological winding number is not often directly
measurable. Instead a quantity, the mean chiral displacement (MCD), is
introduced in this context [10]. In DKQRs the MCD, describing the difference
between momentum distributions of the two internal states that evolve under
$\hat{\mathcal{U}}$, is defined as:

$\begin{split}C(t)&=\langle\psi_{t}|\hat{n}\otimes-\hat{\sigma}_{z}|\psi_{t}\rangle\\\
\
&\equiv\langle\psi_{0}|\hat{\mathcal{U}}^{-t}(\hat{n}\otimes-\hat{\sigma}_{z})\hat{\mathcal{U}}^{t}|\psi_{0}\rangle.\end{split}$
(8)

The average of the MCD over several discrete evolution steps $t$ converges to
half of the topological winding number $\nu$ [10]:

$\bar{C}(t)=\frac{1}{t}\sum_{t_{i}=1}^{t}C(t_{i})\xrightarrow{t\gg
1}\frac{\nu}{2}.$ (9)

To observe the topological phase transitions it is necessary to repeat
application of Eq. (6) for a series of configurations of $k_{1}$ and $k_{2}$
where the empirical results can be compared to the ideal phase diagram
computed in [10].

##  III Optimized experimental Sequence

The first and last MW rotations within Eq. (6) are inverses of each other. If
$\hat{\mathcal{U}}$ is applied subsequently for a larger number of evolution
steps $t\in\mathbb{N}$, this sequence of MW rotations and kicks can be
simplified as demonstrated in full detail in App. A. Considering the full
evolution of the system an alternative expression of $\hat{\mathcal{U}}^{t}$
is therefore found as:

$\begin{split}\hat{\mathcal{U}}^{t}=&\hat{M}(-\frac{\pi}{2},\frac{\pi}{2})\hat{K}_{2}^{\frac{1}{2}}\hat{M}(\frac{\pi}{2},\frac{\pi}{2})\\\
\
&\cdot\left[\hat{M}(-\frac{\pi}{2},0)\hat{K}_{1}\hat{M}(\frac{\pi}{2},0)\right.\left.\hat{M}(-\frac{\pi}{2},\frac{\pi}{2})\hat{K}_{2}\hat{M}(\frac{\pi}{2},\frac{\pi}{2})\right]^{t-1}\\\
\
&\cdot\hat{M}(-\frac{\pi}{2},0)\hat{K}_{1}\hat{M}(\frac{\pi}{2},0)\hat{M}(-\frac{\pi}{2},\frac{\pi}{2})\hat{K}_{2}^{\frac{1}{2}}\hat{M}(\frac{\pi}{2},\frac{\pi}{2}).\end{split}$
(10)

Although this new expression looks more complex, rewriting the sequence in
this form significantly reduces the number of operations necessary to realize
the complete evolution $\hat{\mathcal{U}}^{t}$. This is indeed of great
interest because it shortens experimental durations, reducing the impact of
atom loss and other sources of decoherence (See, e.g., Ref. [23] for a short
summary of these effects).

So far each application of the MW operator is assumed to be infinitely short
in time corresponding to a fully quantum resonant atom-optics kicked rotor
[38, 25]. However, this assumption does not apply to typical experiments since
every MW rotation has a finite duration $\mathcal{T}$. A free evolution
operator thus needs to be added after each MW operator as follows:

$\hat{M}(\alpha,\chi)\rightarrow
e^{-i\frac{\hat{p}^{2}}{2}\mathcal{T}}\hat{M}(\alpha,\chi).$ (11)

Experimental durations can be further reduced by keeping $\mathcal{T}$ as
short as possible. When $\mathcal{T}=4\pi$ the free evolution operator reduces
to unity and thus reflects a full Talbot period. This in turn corresponds to
an on-resonance atom-optics kicked rotor [25, 38]. In a specific case of Eq.
(10), it is possible to produce a resonant configuration with
$\mathcal{T}=\pi$, corresponding to a good MW signal with a quarter Talbot
period. In this paper, we consider $\mathcal{T}=\pi$ and free evolution of the
form ${e^{-i\frac{\hat{p}^{2}}{2}\pi}\equiv\hat{P}_{\pi}}$. In App. B the free
evolution operator is identified with the shift operator
$\hat{T}(\hat{\theta})=e^{i\hat{n}\hat{\theta}}$ where $\theta=\pi$ in
position space, thus $\hat{P}_{\pi}^{2}=\hat{T}(\pi)$. As a consequence of
introducing the free evolution $\hat{P}_{\pi}$, every second kick operator
experiences an effective inversion:

$\begin{split}\hat{\mathcal{U}}^{t}&\rightarrow\hat{P}_{\pi}\hat{M}(-\frac{\pi}{2},\frac{\pi}{2})\hat{K}_{2}^{\frac{1}{2}}\hat{M}(\frac{\pi}{2},\frac{\pi}{2})\\\
\
&\left[\hat{M}(-\frac{\pi}{2},0)\hat{K}_{1}^{-1}\hat{M}(\frac{\pi}{2},0)\right.\left.\cdot\hat{M}(-\frac{\pi}{2},\frac{\pi}{2})\hat{K}_{2}\hat{M}(\frac{\pi}{2},\frac{\pi}{2})\right]^{t-1}\\\
\
&\cdot\hat{M}(-\frac{\pi}{2},0)\hat{K}_{1}^{-1}\hat{M}(\frac{\pi}{2},0)\hat{M}(-\frac{\pi}{2},\frac{\pi}{2})\hat{K}_{2}^{\frac{1}{2}}\hat{P}_{\pi}\hat{M}(-\frac{\pi}{2},\frac{\pi}{2}).\end{split}$
(12)

More calculation details can be found in App. C. Although this free evolution
effect is undesired, it fortunately does not affect the measurement of
topological phase transitions as discussed in Sec. III.1. Note that this
induced inversion symmetry (see Eq. (12)) simplifies the observation of
topological phase transitions in our 87Rb system where the required $\pi$
pulse has a duration of $\approx 26\mu s$ achievable via a high-power MW
amplifier [39, 36, 23, 37]. If $\hat{\mathcal{U}}^{t}$ is computed for a MW
rotation of $\mathcal{T}=\pi$ without the simplification using Eq. (10), the
resultant operator sequence is antiresonant inducing periodic oscillations in
momentum space. This is understood analytically and verified with numerical
calculations. Therefore, a combination of our proposal (see Eqs. (10) and
(12)) and setting $\mathcal{T}=\pi$ is needed for an optimized experimental
sequence for realizing topological phase transitions in DKQRs.

###  III.1 Inversion symmetry

A finite duration $\mathcal{T}=\pi$ of each MW rotation results in every
second kick operator to be effectively inverted (see App. C), i.e.,

$\displaystyle\hat{K}_{1}$
$\displaystyle=e^{-ik_{1}\mathrm{cos}(\hat{\theta})\hat{\sigma}_{z}}\rightarrow
e^{+ik_{1}\mathrm{cos}(\hat{\theta})\hat{\sigma}_{z}}$ (13)
$\displaystyle\hat{K}_{2}$
$\displaystyle=e^{-ik_{2}\mathrm{sin}(\hat{\theta})\hat{\sigma}_{z}}\rightarrow
e^{-ik_{2}\mathrm{sin}(\hat{\theta})\hat{\sigma}_{z}}.$ (14)

Instead of a relative shift of $\theta=\pi/2$ in position space in between the
two kicks, see Eqs. (3) and (4), Eqs. (13) and (14) indicate that the relative
shift is $\theta=-\pi/2$. This is equivalent to an inversion in position/angle
space around the zero angle. As a result, momentum distributions of the
internal states evolve along opposite directions. This causes the expectation
value of momentum for each internal state respectively to change sign, as
illustrated in Fig. 1.

Figure 1: Momentum distributions of the internal states after $t=5$
applications of $\hat{\mathcal{U}}$ for $k_{1}=\frac{\pi}{2}$ and
${k_{2}=2.5\pi}$. The initial state is $|\psi_{\rm
in}\rangle=|n=0\rangle\otimes|2\rangle$. $\mathcal{T}=0$ corresponds to fully
resonant conditions, while $\mathcal{T}=\pi$ specifies MW pulses with the
finite duration. If $\mathcal{T}=\pi$ is chosen, the momentum distributions of
the internal states are mirrored respectively to the ideal case. Note that the
initial state $\ket{2}$ evolves symmetrically in momentum space thus not
contributing to the MCD.

Eqs. (8) and Eq. (9) also indicate that the topological curves computed from
the average MCD change their signs due to introducing the finite MW duration
of $\mathcal{T}=\pi$. This does not change the underlying physics and thus the
expected phase transitions are still visible although with a change in sign.
This undesired effect can be compensated by changing several phase angles of
the MW operators. For instance, we find that

$\begin{split}\hat{M}(-\frac{\pi}{2},0)e^{-ik_{1}\mathrm{cos}(\hat{\theta})\hat{\sigma}_{z}}\hat{M}(\frac{\pi}{2},0)\\\
=\hat{M}(\frac{\pi}{2},0)e^{ik_{1}\mathrm{cos}(\hat{\theta})\hat{\sigma}_{z}}\hat{M}(-\frac{\pi}{2},0).\end{split}$
(15)

This change in the phases within the MW rotations causes a change in sign in
the exponent and thus compensates for the aforementioned change of the two
internal states.

###  III.2 Initial state dependence

Momentum distributions are good observables in QKR and quantum walk
experiments, as shown in Refs. [36, 23, 37] where the complete momentum
distribution ${P(t)=P(t)_{|1\rangle}+P(t)_{|2\rangle}}$ is measured over one
experimental cycle. The momentum distributions for both internal states,
$P(t)_{|1\rangle}$ and $P(t)_{|2\rangle}$, however, need to be measured
separately to compute the MCD using Eq. (8). This may double the experimental
effort, because one MCD measurement requires two consecutive experimental
cycles with one cycle addressing one individual state in systems similar to
those shown in [25, 24]. The presence of additional symmetries can ease this
requirement because a state symmetrically evolving in momentum space has a
zero mean momentum and no contribution to the MCD. Therefore, if one of the
internal states is symmetric in momentum space, the information of the
topological winding number can be conveniently extracted from the momentum
distributions of the other internal state. As a result this effectively
reduces the number of experimental measurements in practice. A good example is
illustrated in Fig. 1.

##  IV Stability

Figure 2: The predicted MCD and topological phase diagrams as a function of
the noise strength $\phi$ when $\beta=0$ (a) and $\Delta_{\beta}=0.025$ (b)
derived for our proposed sequence Eq. (12) at $\mathcal{T}=\pi$. The change in
sign is compensated using Eq. (15) and the MCD is computed exploiting the
symmetry of the initial state $|\psi_{\rm
in}\rangle=|n=0\rangle\otimes|2\rangle$ as discussed in Sec. III.2. The
red/gray line corresponds to $\phi=\pi/3$, the maximum phase noise acceptable
for observing topological phase transitions. (c) The black dotted and black
dashed lines represent cuts at $\phi=\pi/9$ respectively from panel (a) and
(b), and the red/gray solid line corresponds to a cut at $\phi=\pi/3$ from
panel (a). While the steps are visible at small noise strength $\phi$ (see the
black lines), they get washed out as $\phi$ becomes larger than $\pi/3$.  For
each $k_{2}$, the topological number is averaged over 1000 noise trajectories
in all cases (see text).

Some experimental imperfections, for example fluctuations in kicking strengths
and pulse durations, have little effects on the MCD measurements [10] because
the kicks can be timed with good precision and resonant atom-optics kicked
rotors are intrinsically stable with respect to such imperfections [25]. In
this section we focus on three unavoidable experimental limitations impacting
phase evolutions of the system. First, we will investigate the most crucial
problem of uncontrolled phases of the MW pulses. There are many MW pulses
during a single experimental cycle (see Eq. (12)), thus these phase errors
would accumulate quickly. Second, different quasimomenta result in coherent
but unwanted deviations from quantum resonance in atom-optics kicked rotors.
It is necessary to verify whether the typical spread in momentum reported
previously [23, 36, 37] has some impact on the measurement of topological
phases. The third problem arises from a relative energy shift between the two
internal states. This leads to a relative dynamical phase in the evolution
that must be corrected.

###  IV.1 Phase noise

One major source of perturbation originates from fluctuations on the precise
timing of the internal MW rotations across the ensemble [23]. This effectively
leads to fluctuations on the phase parameter $\chi$ in the MW rotation
operator $\hat{M}(\alpha,\chi)$ (see Eq. (2)) as follows

$\hat{M}(\alpha,\chi)\rightarrow\hat{M}(\alpha,\chi+\Delta_{\chi}),$ (16)

modeled numerically as a time-dependent random walk within an additional
dynamical phase $\Delta_{\chi}=\sum_{i=1}^{l}\delta_{\chi,i}$. Here
${\delta_{\chi,i}\in\mathrm{uniform}[-\phi,\phi]}$ is drawn individually for
each MW application, and $l$ counts the number of subsequent MW applications.

DKQRs are much more complicated than a single kicked atom-optics kicked rotor
(see [25, 24]), however, our experimental system is capable of generating
DKQRs with $t\approx 5$ steps [23, 36, 37]. Figure 2 shows our numerical
results of the MCD using Eq. (8) with $1000$ noise trajectories, and
topological diagrams using Eq. (9) with a time average over up to $t=5$
applications of $\hat{\mathcal{U}}$. Similar to [11], our simulations keep the
kick strength $k_{1}$ at $\pi/2$ while scanning $k_{2}$ within a range of
$k_{2}\in[0,\ 2.5\pi]$. Here both kick strengths are expressed in
dimensionless units. Scanning $k_{2}$ reveals topological phase transitions
for each configuration of the kick strengths where the band gap in the Floquet
spectrum closes [10]. A typical example is shown in Fig. 2, which indicates
that the signature of the topological phase transitions decays continuously as
$\phi$ increases but remains visible for $\phi\leq\pi/3$ (see the red/gray
curves in Fig. 2). Therefore, the MW phase noise should be kept within
$\phi<\pi/3$ in experiments.

###  IV.2 Finite quasimomentum distributions

The periodicity of the kicking potential enables the use of Bloch’s theorem
with the momentum expressed as $p=n+\beta$. Here $n$ is an integer and $\beta$
is the dimensionless quasimomentum. Up to this point, we have exclusively
discussed the case of $\mathcal{T}=\pi$ and $\beta=0$, however, experiments
with BECs usually start at a momentum close to $n=0$ with a finite width
$\Delta_{\beta}$ in the Brillouin zone. For example, $\Delta_{\beta}$ is found
to be $0.025$ in Refs. [23, 36, 37]. To model this finite quasimomentum
distribution, a Gaussian distribution with zero mean value and FWHM
$\Delta_{\beta}=0.025$ is used in our calculations. Figure 2 shows topological
phase transitions as a function of the phase noise $\phi$ when different
quasimomentum distributions are considered. Comparisons between Fig. 2(a) and
Fig. 2(b) indicate that the quasimomenta present in experiments have little
effect on the topological diagrams.

###  IV.3 Light-shift compensation

Our system has two internal hyperfine states, the kicking potential created by
a virtual transition to a third state thus creates an additional energy gap
between the two states. In our system, an atom-optics kicked rotor with two
internal states, this light (or AC-Stark) shift effectively changes the
kicking potential to [40, 41, 42]

$\displaystyle\hat{\mathcal{K}}\equiv
2k\cdot\mathrm{cos}^{2}(\frac{\hat{\theta}}{2})$
$\displaystyle=k(\mathrm{cos}(\hat{\theta})+1),$ (17)

where the constant term independent of the angle $\theta$ represents the shift
of the internal states. The total energy shift to be corrected for is thus
$2k$ for a single kick of the kicking strength $k$ in our atom-optics kicked
rotor systems and the quantum walk experiments [23, 36, 37], because the
second internal state effectively sees a kick with an opposite sign.

To observe topological phases, it is important to have a relative shift of
$\Delta\theta=\pi/2$ in position space between the two kicks $\hat{K}_{1}$ and
$\hat{K}_{2}$ (see Eqs. (3) and (4)). The effective kick operators are then

$\displaystyle\hat{K}_{1,\rm eff}$
$\displaystyle=e^{-ik_{1}\hat{\sigma}_{z}(\mathrm{cos}(\hat{\theta})+1)}\equiv
e^{-i\hat{\sigma}_{z}\hat{\mathcal{K}}_{1}}$ (18) $\displaystyle\hat{K}_{2,\rm
eff}$
$\displaystyle=e^{-ik_{2}\hat{\sigma}_{z}(\mathrm{sin}(\hat{\theta})+1)}\equiv
e^{-i\hat{\sigma}_{z}\hat{\mathcal{K}}_{2}}.$ (19)

Figure 3: (a) The winding number and (b) values of the function
$\mathrm{cos}(\mathcal{K}_{1})\mathrm{cos}(\mathcal{K}_{2})$ respectively
derived from Eqs. (18) and (19) when light-shift effects are NOT compensated.
(c) The winding number and (d) values of the function
$\mathrm{cos}(\mathcal{K}_{1})\mathrm{cos}(\mathcal{K}_{2})$ respectively
derived from Eqs. (20) and (21) when light-shift effects are effectively
compensated. In panel (c), the averaged MCD converges to the predicted phase
transitions (black solid line) as the evolution step $t$ increases. If the
$\mathrm{cos}(\mathcal{K}_{1})\mathrm{cos}(\mathcal{K}_{2})$ function changes
sign, the topological phase changes thus implying instabilities. This is
satisfied everywhere in panel (b) while only at specific points in panel (d),
see the main text.

The effective kick potentials are described by $\hat{\mathcal{K}}_{1}$ and
$\hat{\mathcal{K}}_{2}$. No topological phase transitions are observed if
these operators are implemented without compensating light shifts as shown in
Fig. 3(a). This is because a topological winding number can only change if
$\mathrm{cos}(\mathcal{K}_{1})\mathrm{cos}(\mathcal{K}_{2})=\pm 1$
corresponding to a closing band gap (see Ref. [10]). Figure 3(b) shows that
this condition is always fulfilled within the simulated range of $k_{2}$, thus
it becomes impossible to distinguish between topological phase transitions and
possible instability, if light shifts are not compensated. This suggests that
compensation of light shifts is necessary for observation of topological phase
transitions in experiments.

In walk experiments as reported in Refs. [23, 36, 37], compensation of light
shifts was done by adjusting the phase of MW pulses. Such a procedure is
necessary for each kick in the DKQR system where two kicks have different
kicking strengths. Our simulations, however, indicate that compensating one of
the two kicking pulses is sufficient. For example, a simpler method of
compensation can be conducted by correcting light shift originating from
$k_{1}$, therefore the kicking operators are

$\displaystyle\hat{K}_{1,\rm eff}$ $\displaystyle\rightarrow
e^{-ik_{1}\hat{\sigma}_{z}\mathrm{cos}(\hat{\theta})}\equiv
e^{-i\hat{\sigma}_{z}\hat{\mathcal{K^{\prime}}}_{1}}$ (20)
$\displaystyle\hat{K}_{2,\rm eff}$ $\displaystyle\rightarrow
e^{-ik_{2}\hat{\sigma}_{z}\mathrm{sin}(\hat{\theta})}e^{-i(k_{2}-k_{1})\hat{\sigma}_{z}}\equiv
e^{-i\hat{\sigma}_{z}\hat{\mathcal{K^{\prime}}}_{2}},$ (21)

where the effective kick dynamics described by
$\hat{\mathcal{K}^{\prime}}_{1}$ and $\hat{\mathcal{K}^{\prime}}_{2}$ are
partially compensated. This compensation may not remove light-shift effects
completely, however, it is adequate for observing topological phase
transitions as shown in Figs. 3(c) and 3(d). Our calculations show that the
number of topological phase transitions doubles in the same range of $k_{2}$
when the light shift is compensated. This implies that it may be possible to
observe topological phase transitions by scanning $k_{2}$ even in a smaller
range after light shifts are effectively compensated.

##  V Conclusion

We have investigated the feasibility of measuring topological phase
transitions with the DKQR platform and demonstrated how impacts of decoherence
and experimental durations can be minimized by setting the duration of each MW
rotation at $\mathcal{T}=\pi$ corresponding to a quarter of Talbot period. A
proper choice of initial states possessing an inversion symmetry is found to
further simplify experimental realizations.

Our results have suggested that an successful protocol for observing
topological phase transitions must minimize the phase noise to $\phi\leq\pi/3$
and at least partially compensate the light-shift effects. Compensating light-
shift effects can also facilitate experiments as it doubles the amount of
expected topological steps within a same $k_{2}$ scanning range. Our findings
have thus confirmed that the DKQR is a promising platform to realize and
measure topological phases in a time-dependent Floquet setup.

###### Acknowledgements.

J. H. C. and Y. L. thank the Noble Foundation for financial support.

##  Appendix A Simplification of original proposed sequence

For the evolution operator $\hat{\mathcal{U}}$ the first and the last MW-
rotation are inverse to each other:

$\displaystyle\hat{M}(\frac{\pi}{2},0)=\hat{M}(-\frac{\pi}{2},0)^{-1}\qquad\qquad\qquad\qquad\hat{M}(-\frac{\pi}{2},\frac{\pi}{2})=\hat{M}(\frac{\pi}{2},\frac{\pi}{2})^{-1}.$
(22)

When $\hat{\mathcal{U}}$ is applied subsequently, this can be exploited to
reduce the amount of necessary MWs and kicks in $\hat{\mathcal{U}}$. It is
sufficient to study this effect for $\hat{\mathcal{U}}^{2}$.

$\displaystyle\hat{\mathcal{U}}^{2}=$
$\displaystyle\left(\hat{M}(-\frac{\pi}{2},\frac{\pi}{2})\hat{K}^{\frac{1}{2}}_{2}\hat{M}(\frac{\pi}{2},\frac{\pi}{2})\hat{M}(-\frac{\pi}{2},0)\hat{K}_{1}\hat{M}(\frac{\pi}{2},0)\hat{M}(-\frac{\pi}{2},\frac{\pi}{2})\hat{K}^{\frac{1}{2}}_{2}\hat{M}(\frac{\pi}{2},\frac{\pi}{2})\right)^{2}$
(23) $\displaystyle\ =$
$\displaystyle\hat{M}(-\frac{\pi}{2},\frac{\pi}{2})\hat{K}^{\frac{1}{2}}_{2}\hat{M}(\frac{\pi}{2},\frac{\pi}{2})\hat{M}(-\frac{\pi}{2},0)\hat{K}_{1}\hat{M}(\frac{\pi}{2},0)\hat{M}(-\frac{\pi}{2},\frac{\pi}{2})K^{\frac{1}{2}}_{2}$
(24)
$\displaystyle\cdot\hat{K}^{\frac{1}{2}}_{2}\hat{M}(\frac{\pi}{2},\frac{\pi}{2})\hat{M}(-\frac{\pi}{2},0)\hat{K}_{1}\hat{M}(\frac{\pi}{2},0)\hat{M}(-\frac{\pi}{2},\frac{\pi}{2})\hat{K}^{\frac{1}{2}}_{2}\hat{M}(\frac{\pi}{2},\frac{\pi}{2})$
(25) $\displaystyle\ =$
$\displaystyle\hat{M}(-\frac{\pi}{2},\frac{\pi}{2})\hat{K}^{\frac{1}{2}}_{2}\hat{M}(\frac{\pi}{2},\frac{\pi}{2})\hat{M}(-\frac{\pi}{2},0)\hat{K}_{1}\hat{M}(\frac{\pi}{2},0)\hat{M}(-\frac{\pi}{2},\frac{\pi}{2})\hat{K}_{2}\hat{M}(\frac{\pi}{2},\frac{\pi}{2})$
(26)
$\displaystyle\cdot\hat{M}(-\frac{\pi}{2},0)\hat{K}_{1}\hat{M}(\frac{\pi}{2},0)\hat{M}(-\frac{\pi}{2},\frac{\pi}{2})\hat{K}^{\frac{1}{2}}_{2}\hat{M}(\frac{\pi}{2},\frac{\pi}{2})$
(27)

##  Appendix B Conditions necessary for sequence resonance considerations

We consider the free evolution operator
$\hat{P}=e^{-i\frac{\hat{p}^{2}}{2}\mathcal{T}}$ for a quarter Talbot time
$\mathcal{T}=\pi$. Using Bloch’s theorem we can decompose momentum $\hat{p}$
into an integer and a quasimomentum $\beta$: $\hat{p}=\hat{n}+\beta$, with
$\beta\in[0,1)$. In the following we choose resonant values for $\beta$, for
instance $\beta=0$ [25], to simplify the discussion. We can identify similarly
to [43, 38] the free evolution operator with the shift operator in momentum
space:

$\hat{P}_{\pi}^{2}=(e^{-i\frac{\hat{n}^{2}}{2}\pi})^{2}=e^{-i\hat{n}^{2}\pi}=\left\\{\begin{array}[]{ll}1&n=2j\\\
-1&n=2j+1\\\ \end{array}\right.\equiv e^{-i\hat{n}\pi}\equiv\hat{T}(\pi).$
(28)

$\hat{T}(\pi)$ can thus denote the shift operator in angular momentum space
for $\theta=\pi$. The free evolution operator for the two-state system
commutes with the microwave operators since the entries of the MW-matrix are
scalar values.

$\displaystyle\hat{P}=e^{-i\frac{(\hat{n}+\beta)^{2}}{2}\tau\otimes\mathbf{1}}$
$\displaystyle=e^{-i\frac{(\hat{n}+\beta)^{2}}{2}\tau}\cdot\mathbf{1}$ (29)
$\displaystyle\Rightarrow\hat{P}\hat{M}(\alpha_{1},\chi_{1})\hat{P}\hat{M}(\alpha_{2},\chi_{2})$
$\displaystyle=\hat{M}(\alpha_{1},\chi_{1})\hat{M}(\alpha_{2},\chi_{2})\hat{P}^{2}$
(30)
$\displaystyle=\hat{M}(\alpha_{1},\chi_{1})\hat{M}(\alpha_{2},\chi_{2})\hat{T}(\pi).$
(31)

The translation operator affects the kick operators $\hat{K}_{1}$ and
$\hat{K}_{2}$ as follows:

$\displaystyle\hat{T}(\pi)\hat{K}_{1,2}(k,\theta)$
$\displaystyle=\hat{K}_{1,2}(k,\theta+\pi)=\hat{K}_{1,2}(-k,\theta)\equiv\hat{K}_{1,2}^{-1}$
(32) $\displaystyle\hat{T}(2\pi)\hat{K}_{1,2}(k,\theta)$
$\displaystyle=\hat{K}_{1,2}(k,\theta+2\pi)=\hat{K}_{1,2}(k,\theta).$ (33)

##  Appendix C Introduction of free evolution

The free evolution operator $\hat{P}_{\pi}$ for $\mathcal{T}=\pi$ after
neglecting quasimomentum is not unity. When each MW-rotation is considered to
have a finite duration, this affects the evolution of the system described by
$\hat{U}^{t}$ in the following way:

$\displaystyle\begin{split}\hat{\mathcal{U}}^{t}&=\hat{P}_{\pi}\hat{M}(-\frac{\pi}{2},\frac{\pi}{2})\hat{K}_{2}^{\frac{1}{2}}\hat{P}_{\pi}\hat{M}(\frac{\pi}{2},\frac{\pi}{2})\left[\hat{P}_{\pi}\hat{M}(-\frac{\pi}{2},0)\hat{K}_{1}\hat{P}_{\pi}\hat{M}(\frac{\pi}{2},0)\hat{P}_{\pi}\hat{M}(-\frac{\pi}{2},\frac{\pi}{2})\hat{K}_{2}\hat{P}_{\pi}\hat{M}(\frac{\pi}{2},\frac{\pi}{2})\right]^{t-1}\\\
\
&\qquad\qquad\qquad\qquad\qquad\qquad\cdot\hat{P}_{\pi}\hat{M}(-\frac{\pi}{2},0)\hat{K}_{1}\hat{P}_{\pi}\hat{M}(\frac{\pi}{2},0)\hat{P}_{\pi}\hat{M}(-\frac{\pi}{2},\frac{\pi}{2})\hat{K}_{2}^{\frac{1}{2}}\hat{P}_{\pi}\hat{M}(\frac{\pi}{2},\frac{\pi}{2})\end{split}$
(34)
$\displaystyle\begin{split}&=\hat{P}_{\pi}\hat{M}(-\frac{\pi}{2},\frac{\pi}{2})\hat{K}_{2}^{\frac{1}{2}}\hat{M}(\frac{\pi}{2},\frac{\pi}{2})\left[\hat{M}(-\frac{\pi}{2},0)\hat{T}(\pi)\hat{K}_{1}\hat{M}(\frac{\pi}{2},0)\hat{M}(-\frac{\pi}{2},\frac{\pi}{2})\hat{T}(\pi)\hat{K}_{2}\hat{M}(\frac{\pi}{2},\frac{\pi}{2})\right]^{t-1}\\\
\
&\qquad\qquad\qquad\qquad\qquad\qquad\cdot\hat{M}(-\frac{\pi}{2},0)\hat{T}(\pi)\hat{K}_{1}\hat{M}(\frac{\pi}{2},0)\hat{M}(-\frac{\pi}{2},\frac{\pi}{2})\hat{T}(\pi)\hat{K}_{2}^{\frac{1}{2}}\hat{P}_{\pi}\hat{M}(\frac{\pi}{2},\frac{\pi}{2})\end{split}$
(35)
$\displaystyle\begin{split}&=\hat{P}_{\pi}\hat{M}(-\frac{\pi}{2},\frac{\pi}{2})\hat{K}_{2}^{\frac{1}{2}}\hat{M}(\frac{\pi}{2},\frac{\pi}{2})\left[\hat{M}(-\frac{\pi}{2},0)\hat{K}_{1}^{-1}\hat{M}(\frac{\pi}{2},0)\hat{M}(-\frac{\pi}{2},\frac{\pi}{2})\hat{T}(2\pi)\hat{K}_{2}\hat{M}(\frac{\pi}{2},\frac{\pi}{2})\right]^{t-1}\\\
\
&\qquad\qquad\qquad\qquad\qquad\qquad\cdot\hat{M}(-\frac{\pi}{2},0)\hat{K}_{1}^{-1}\hat{M}(\frac{\pi}{2},0)\hat{M}(-\frac{\pi}{2},\frac{\pi}{2})\hat{T}(2\pi)\hat{K}_{2}^{\frac{1}{2}}\hat{P}_{\pi}\hat{M}(\frac{\pi}{2},\frac{\pi}{2})\end{split}$
(36)
$\displaystyle\begin{split}&=\hat{P}_{\pi}\hat{M}(-\frac{\pi}{2},\frac{\pi}{2})\hat{K}_{2}^{\frac{1}{2}}\hat{M}(\frac{\pi}{2},\frac{\pi}{2})\left[\hat{M}(-\frac{\pi}{2},0)\hat{K}_{1}^{-1}\hat{M}(\frac{\pi}{2},0)\hat{M}(-\frac{\pi}{2},\frac{\pi}{2})\hat{K}_{2}\hat{M}(\frac{\pi}{2},\frac{\pi}{2})\right]^{t-1}\\\
\
&\qquad\qquad\qquad\qquad\qquad\qquad\cdot\hat{M}(-\frac{\pi}{2},0)\hat{K}_{1}^{-1}\hat{M}(\frac{\pi}{2},0)\hat{M}(-\frac{\pi}{2},\frac{\pi}{2})\hat{K}_{2}^{\frac{1}{2}}\hat{P}_{\pi}\hat{M}(\frac{\pi}{2},\frac{\pi}{2}).\end{split}$
(37)

Therefore, every second kick operator (here $\hat{K}_{1}$) effectively
experiences an inversion.

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