# Simulating non-Hermitian quasicrystals with photonic quantum walks

Quan Lin  Beijing Computational Science Research Center, Beijing 100084, China
Tianyu Li  CAS Key Laboratory of Quantum Information, University of Science
and Technology of China, Hefei 230026, China  CAS Center For Excellence in
Quantum Information and Quantum Physics, Hefei 230026, China  Lei Xiao
Beijing Computational Science Research Center, Beijing 100084, China  Kunkun
Wang  Beijing Computational Science Research Center, Beijing 100084, China
Wei Yi<EMAIL_ADDRESS>CAS Key Laboratory of Quantum Information, University
of Science and Technology of China, Hefei 230026, China  CAS Center For
Excellence in Quantum Information and Quantum Physics, Hefei 230026, China
Peng Xue<EMAIL_ADDRESS>Beijing Computational Science Research Center,
Beijing 100084, China

###### Abstract

Non-Hermiticity significantly enriches the properties of topological models,
leading to exotic features such as the non-Hermitian skin effects and non-
Bloch bulk-boundary correspondence that have no counterparts in Hermitian
settings. Its impact is particularly illustrating in non-Hermitian
quasicrystals where the interplay between non-Hermiticity and quasiperiodicity
results in the concurrence of the delocalization-localization transition, the
parity-time (PT)-symmetry breaking, and the onset of the non-Hermitian skin
effects. Here we experimentally simulate non-Hermitian quasicrystals using
photonic quantum walks. Using dynamic observables, we demonstrate that the
system can transit from a delocalized, PT-symmetry broken phase that features
non-Hermitian skin effects, to a localized, PT-symmetry unbroken phase with no
non-Hermitian skin effects. The measured critical point is consistent with the
theoretical prediction through a spectral winding number, confirming the
topological origin of the phase transition. Our work opens the avenue of
investigating the interplay of non-Hermiticity, quasiperiodicity, and spectral
topology in open quantum systems.

Non-Hermiticity arises in open systems and can lead to intriguing properties
with no Hermitian counterparts QJ ; benderreview ; bender ; photonpt1 ;
review2 ; review3 ; nonHtopo1 ; nonHtopo2 ; nonHtopo3 ; WZ1 ; WZ2 ; Budich ;
mcdonald ; alvarez ; murakami ; ThomalePRB ; fangchenskin ; kawabataskin ;
Slager ; yzsgbz ; stefano ; tianshu ; lli ; luole ; yuce ; ngra ; nsa ; nta .
Exotic non-Hermitian phenomena such as the parity-time (PT) symmetry and
exceptional points bender ; luole ; yuce ; photonpt1 ; review3 ; nonHtopo1 ;
nonHtopo2 ; nonHtopo3 , the non-Hermitian skin effects WZ1 ; WZ2 ; Budich ;
mcdonald ; alvarez ; murakami ; ThomalePRB ; fangchenskin ; kawabataskin ;
Slager ; yzsgbz ; stefano ; tianshu ; lli , and the non-Bloch bulk-boundary
correspondence WZ1 ; WZ2  have attracted much attention. The interest is
further stimulated by their experimental observation in open systems teskin ;
teskin2d ; metaskin ; photonskin ; XDW+21 ; scienceskin ; nsm , with potential
applications in precision measurements, non-reciprocal quantum device, and
topological transport. So far, these experiments either focus on non-Hermitian
models with no spatial degrees of freedom, or on lattices with discrete
translational symmetry. The properties of non-Hermitian quasicrystals remain
largely unexplored experimentally.

In closed quantum systems, quasicrystals play host to a wealth of properties
harper ; AA ; sokoloff ; roati ; lahini ; verbin ; nsl ; naad . For instance,
in the one-dimensional Aubry-André-Harper (AAH) model, a finite strength of
quasiperiodicity drives the system from an extended metallic state into an
Anderson insulator where the bulk eigenstates become exponentially localized
harper ; AA ; sokoloff ; nsl ; naad . The AAH model can also be mapped to the
Hofstadter model, and hence to the integer quantum Hall system, revealing a
deeper connection with band topology hofstadter ; kraus ; chen1 . Variations
of the AAH model feature different forms of quasiperiodic disorder, as well as
interactions, which can further give rise to more exotic phases such as the
many-body localized mbl1 ; mbl2 ; mbl3  or the critically-localized states
crit1 ; crit2 ; crit3 . In a recent series of theoretical studies, it has been
shown that specific non-Hermitian variants of the AAH model can feature a
topological phase transition characterized by a spectral winding number
nonHtopo1 ; chen3 ; chen4 ; chen5 ; YX ; stefano2 . Remarkably, such a
critical point is a simultaneous demarcation for the delocalization-
localization transition, the PT-symmetry-breaking transition, as well as for
the onset of the non-Hermitian skin effects when an open boundary condition
(OBC) is imposed chen3 . Given the uniqueness of such a multi-critical point,
the experimental simulation of non-Hermitian quasicrystals and the underlying
rich critical physics is desirable.

In this work, we experimentally simulate a non-Hermitian, PT-symmetric AAH
model using photonic quantum-walk dynamics. Building upon our previous
experiments of quantum walks with non-Hermitian skin effects photonskin ;
XDW+21 , we further introduce position- and polarization-dependent phase
operators to implement quasiperiodicity. Through dynamic observables such as
the Lyapunov exponents stefano ; xueexp , the dynamic inverse participation
ratio (dIPR) crit3 , and the time-evolved photon counts, we confirm key
properties of the phases on either side of the multi-critical point, as
predicted by the spectral winding number. Interestingly, the measured dIPR
further provides evidence for the presence of a mobility edge in the system,
whereas mobility edges are known to be absent in the original AAH model.

Figure 1: Simulation of non-Hermitian quasicrystals. (a) A cartoon
illustration of quantum-walk dynamics. Operators $S$ and $R$ are defined in
the main text. Lattice sites are located at the crossing points of the laser
beams (red). The dashed grey curves illustrate the quasiperiodic phase (or a
quasiperiodic potential for the effective Hamiltonian) introduced by the phase
operator $P_{1,2}$. (b) A time-multiplexed implementation of the photonic
quantum walk illustrated in (a). ND: neutral density filter; HWP: half-wave
plate; AOM: optical switch acousto-optic modulator; EOM: electro-optic
modulator; (P)BS: (polarizing) beam splitter; APD: avalanche photo-diode. (c)
Phase diagram of the spectral winding number $\omega$. Insets are the typical
eigenspectra of the two phases on the complex plane with a fixed $\gamma=0.1$,
and $\theta=\pi/8$ for $\omega=1$ and $\theta=3\pi/8$ for $\omega=0$. The
phase boundary here not only marks the onset of the non-Hermitian skin effect,
but also the PT-symmetry breaking point and the delocalization-localization
transition. (d) Top panel: spatial distribution of all eigenstates of $U$
under the OBC, for the phase with $\omega=1$ (left) and $\omega=0$ (right),
respectively. Middle panel: The real components of the eigenergies $E$ of the
effective Hamiltonian $H$ (under PBC) with increasing $\theta$, colored
according to their respective inverse participation ratio IPR. Lower panel:
The imaginary components of the eigenenergies with increasing $\theta$ (under
PBC), colored according to the IPR. For the numerical simulations, we take the
lattice size $N=200$ and $\gamma=0.1$.  Figure 2: Experimental detection of
the non-Hermitian skin effect in a non-Hermitian quasicrystal. Measured photon
distributions up to $10$ time steps with a fixed $\gamma=0.1$. The initial
state is $\ket{x=0}\otimes\ket{H}$, and the coin parameters are
$\theta=0.18\pi$ in (a) and $\theta=0.33\pi$ in (b), respectively. The
corresponding numerical results are shown in the Supplemental Material. (c)(d)
Measured Lyapunov exponent $\lambda(v)$ versus the shift velocity $v$ with the
same parameters in (a) and (b). Red dots represent the experimental data, and
blue triangles are the corresponding numerical simulations. Grey dotted lines
denote the numerical results from $100$-step quantum walks. Orange dashed
lines indicate the threshold values below which experimental data are no
longer reliable due to photon loss. Error bars are due to the statistical
uncertainty in photon-number counting.

A time-multiplexed non-unitary quantum walk.— We simulate non-Hermitian
quasicrystals using a one-dimensional photonic quantum walk, governed by the
Floquet operator

$\displaystyle U=MSP_{2}R(\theta)MSP_{1}R(\theta),$ (1)

where the coin operator
$R(\theta)=\sum_{x}\ket{x}\bra{x}\otimes\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix}$, under the polarization basis
$\\{|H\rangle,|V\rangle\\}$, with $|H\rangle$ ($|V\rangle$) the horizontally
(vertically) polarized state and $x$ labeling the lattice sites. The shift
operator is given by
$S=\sum_{x}\ket{x-1}\bra{x}\otimes\ket{H}\bra{H}+\ket{x+1}\bra{x}\otimes\ket{V}\bra{V}$.
Non-Hermiticity is introduced through the partial measurement
$M=\sum_{x}\ket{x}\bra{x}\otimes\begin{pmatrix}e^{\gamma}&0\\\
0&e^{-\gamma}\end{pmatrix}$ with $\gamma$ the gain-loss parameter. The
position-dependent phase operators satisfy
$P_{1}=P_{2}^{-1}=\sum_{x}\ket{x}\bra{x}\otimes\begin{pmatrix}e^{\frac{i}{2}\cos(x\pi\phi)\pi}&0\\\
0&e^{-\frac{i}{2}\cos(x\pi\phi)\pi}\end{pmatrix}$ with $\phi=(\sqrt{5}-1)/2$,
the inverse of the golden Mean.

For the quantum-walk dynamics, the Floquet operator $U$ repeatedly acts on the
initial state $|\Psi(0)\rangle$, with the time-evolved state given by
$|\Psi(t)\rangle=U^{t}|\Psi(0)\rangle$, where $t$ labels the discrete time
steps (see Fig. 1(a) for a cartoon illustration of the state evolution). The
quantum walk is therefore a stroboscopic simulation of the time evolution
driven by an effective Hamiltonian $H$, defined as $U=e^{-iH}$ photonskin ;
XDW+21 .

In the absence of the phase operators $P_{i}$ ($i=1,2$), the quantum walk
governed by $U$ features non-Hermitian skin effect photonskin ; xueexp .
Therein, the interplay of the effective spin-orbit coupling (the coupling
between polarization and spatial modes) and the polarization-dependent loss,
leads to a finite, non-reciprocal probability current in the bulk that is
responsible for the accumulation of population at boundaries, the namesake
phenomenon of non-Hermitian skin effects. The quasiperiodicity is then
introduced through the phase operators $P_{1,2}$, which can be understood as a
polarization-dependent quasiperiodic potential for the effective Hamiltonian.
The resulting Floquet operator $U$ thus realizes a stroboscopic simulation of
the dynamics of a generalized non-Hermitian AAH model that features non-
reciprocal hopping and quasiperiodic modulation of the on-site potential supp
.

As illustrated in Fig. 1(b), we adopt a time-multiplexed scheme for the
experimental implementation of $U$ supp . Pulses from the photon source are
attenuated to the single-photon level using a neutral density filter, ensuring
a negligible probability of multi-photon events sch ; SGR+12 . Similar to the
setup in Ref. xueexp , the various operators are implemented using beam
splitters and half-wave plates, and are integrated into two optical fibre
loops through which each photon passes twice for each discrete time step. Here
different walker positions are encoded into the time domain, thanks to the
polarization-dependent time delay introduced through the two optical fibre
loops. With these, the position-dependent phase operators $P_{1,2}$ are
realized through an electro-optical modulator that imposes a time-dependent
phase within each discrete time step. Finally, we implement a polarization-
dependent loss operator $M_{\text{E}}=e^{-\gamma}M$, and the experimentally
realized time-evolved state $|\Psi_{\text{E}}(t)\rangle$ is related to
$|\Psi(t)\rangle$ through $|\Psi(t)\rangle=e^{2\gamma
t}|\Psi_{\text{E}}(t)\rangle$.

Figure 3: Observation of the Anderson localization in a non-Hermitian
quasicrystal. (a) Measured dIPR (blue) and dNPR (red) for non-Hermitian
quantum walks with the initial state $\ket{x=0}\otimes\ket{V}$. Solid and
hollow symbols represent the experimental data and their theoretical
predictions, respectively. (b) Numerical simulation of $\overline{\text{IPR}}$
(blue) and $\overline{\text{NPR}}$ (red) for a lattice size $N=1000$. See the
main text for the definitions of $\overline{\text{IPR}}$ and
$\overline{\text{NPR}}$. Here we take $\gamma=0.1$.

Winding number and phase diagram.— The quasicrystalline features of $U$ are
best illustrated using the phase diagram characterized by a spectral winding
number. To calculate the winding number, we consider a periodic boundary
condition (PBC) with $N$ lattices sites, and with an auxiliary magnetic flux
$\Phi$ threaded through the resulting ring. This is equivalent to making the
replacement $\gamma\rightarrow\gamma+i\Phi/(2N)$ in $U$. Subsequently, the
spectral winding number is defined as nonHtopo1 ; chen3

$\displaystyle\omega=\lim_{N\rightarrow\infty}\frac{1}{2\pi
i}\int^{2\pi}_{0}d\Phi\frac{\partial_{\Phi}\det\\{H(\Phi/N)-E\\}}{\det\\{H(\Phi/N)-E\\}},$
(2)

where $H$ is the effective Hamiltonian of $U$, and $E$ is a base point in the
complex energy plane. The quantum walk has a spectral topology, if there
exists an $E$ such that $\omega\neq 0$. In this case, the PBC eigenspectrum of
the effective Hamiltonian exhibits closed loops around the base point $E$,
implying a persist current in the bulk which is the origin of the non-
Hermitian skin effect under an OBC fangchenskin ; kawabataskin . Otherwise,
when $\omega=0$, the PBC eigenspectrum of $H$ does not have spectral topology,
and the quantum walk does not show the non-Hermitian skin effect under OBC. We
show in Fig. 1(c), the phase diagram of the system. In the region with
$\omega=1$, the emergence of non-Hermitian skin effect is confirmed in the top
panel of Fig. 1(d), where all eigen wave functions accumulate toward the
boundaries under OBC.

Under quasiperiodicity, eigenstates of the system undergo a delocalization-
localization transition with increasing $\theta$. The localization of the
$n$th eigenstate can be characterized by the inverse participation ratio (IPR)
crit2  defined as

$\displaystyle\text{IPR}_{n}=\sum_{x}\frac{|\psi_{n}(x)|^{4}}{\sum_{x}|\psi_{n}(x)|^{2}},$
(3)

where $\psi_{n}(x)$ is the support of the $n$th eigenstate of $U$ on site $x$.
The state is delocalized (localized) when $\text{IPR}_{n}$ is vanishingly
small (finite). As illustrated in the middle panel of Fig. 1(d), the critical
$\theta$ appears to be energy-dependent across the eigenspectrum, indicating
the existence of a mobility edge.

Interestingly, the aforementioned spectral topological phase boundary
simultaneously marks the transition point beyond which all eigenstates of the
system are localized. In Fig. 1(d), we indicate such a global delocalization-
location transition with a dashed vertical line in red. To the right of the
line, all $\text{IPR}_{n}$ are finite, the system is fully localized, and the
winding number $\omega=0$. To the left of the line, not all eigenstates are
localized (some states still feature finite $\text{IPR}_{n}$), and the winding
number $\omega=1$. One may further identify a transition point [indicated by
the black dashed line in Fig. 1(d)] where the first localized eigenstates (the
ground and the highest excited states) emerge in the spectrum. This transition
marks the onset of the mobility edge as $\theta$ increases, but does not
coincide with the spectral topological transition.

The spectral topological phase boundary is also the critical point at which
the PT-symmetry becomes broken under PBC. Due to the spectral topological
origin of the non-Hermitian skin effect, the region with $\omega=1$ is
necessarily PT-symmetry broken. Intriguingly, as shown in the lower panel of
Fig. 1(d), in the region with $\omega=0$, the eigensepctrum of $H$ is entirely
real, indicating the concurrence of the spectral topological transition, the
global delocalization-localization transition, and the PT-symmetry breaking
transition chen3 ; supp . In the following, we experimentally probe these
transitions one-by-one.

Non-Hermitian skin effects.— We experimentally probe the presence of the non-
Hermitian skin effects through bulk dynamics xueexp . While the direct
manifestation of the non-Hermitian skin effect is the accumulation of
population at open boundaries, it originates from a persistent, directional
bulk current that can be directly probed through bulk dynamics. For this
purpose, we initialize the walker near $x=0$, and record the photon
distribution at each time step up to $t=10$. For $\theta<\theta_{c}$, the
system possesses non-Hermitian skin effects. This is reflected as the
directional probability flow in the bulk, as shown in Fig. 2(a). By contrast,
for sufficiently large $\theta$, the system is in the Anderson-localized
state, and the probability would accumulate at the initial position, as shown
in Fig. 2(b).

The conclusion, while consistent with the phase diagram, is further confirmed
by measuring the Lyapunov exponent, defined as stefano ; xueexp

$\displaystyle\lambda(v)=\frac{1}{t}\log|\left(\langle x=vt|\otimes\langle
H|\right)|\Psi(t)\rangle|,$ (4)

where $v$ is the shift velocity. The Lyapunov exponent reveals the presence of
the bulk probability flow, since by definition, the location of its peak shows
how the wave function propagates along the lattice. Specifically, $\lambda(v)$
peaks at a finite shift velocity $v$ in the presence of non-Hermitian skin
effect; whereas it peaks at $v=0$ in the absence of non-Hermitian skin effect
stefano ; xueexp . Experimentally, we construct $\lambda(v)$ following a
$10$-step quantum walk, by measuring the photon distribution at each step. As
shown in Figs. 2(c)(d), the experimental data confirms the presence of non-
Hermitian skin effect in the region with $\omega=1$.

Figure 4: PT-symmetry transition in a non-Hermitian quasicrystal. (a)
Imaginary parts of the quasienegies of a non-Hermitian quantum walk with a
lattice size $N=200$, and $\gamma=0.1$. The vertical dashed line at
$\theta/\frac{\pi}{4}\approx 1.25$ separate the PT unbroken and broken phases.
(b) Measured overall corrected probabilities $P(t)$ of the quantum walk with
an initial state $\ket{x=0}\otimes\ket{H}$ for various $\theta$ corresponding
to the PT unbroken and broken phases. Symbols represent the experimental data
and curves for their theoretical predictions.

Localization and mobility edge.— To characterize the localization properties,
we measure the dIPR crit3 , defined as $\text{dIPR}=\sum_{x}P^{2}(x,t)$, where
$P(x,t)$ is the corrected probability of the walker on position $x$ at the end
of time step $t$. The probability is normalized by dividing the total photon
count after the $t$-th step, so that $P(x,t)=|\langle
x|\Psi_{\text{E}}(t)\rangle|^{2}/\sum_{x}|\langle
x|\Psi_{\text{E}}(t)\rangle|^{2}$.

A vanishingly small dIPR at long times suggests the delocalized, metallic
phase; while a finite dIPR indicates the onset of localization. As illustrated
in Fig. 3(a), the measured dIPR (blue) indicates a transition point near
$\theta/\frac{\pi}{4}\approx 1.05$, which is consistent with the black
vertical dashed line in Fig. 1(d). This is the location where some eigenstates
start to be localized, as a mobility edge emerges in the eigenspectrum.

To reveal the global delocalization-localization transition where all
eigenstates become Anderson localized, we construct the dynamic normalized
inverse participation ratio (dNPR), defined as
$\text{dNPR}=[2t\sum_{x}P^{2}(x,t)]^{-1}$. As shown in Fig. 3(a), the measured
dNPR (red) becomes vanishingly small near $\theta/\frac{\pi}{4}\approx 1.25$,
consistent with the global delocalization-localization transition in Fig.
1(d).

The measured dIPR and dNPR are respectively the dynamic probe to the averaged
IPR: $\overline{\text{IPR}}=(1/2N)\sum_{n}\text{IPR}_{n}$; and the average
normalized IPR: $\overline{\text{NPR}}=1/(4N^{2})\sum_{n}1/\text{IPR}_{n}$
huse ; chen4 . Key features in the numerical simulations of these quantities
agree well with our measurements [see Fig. 3(b)], indicating a mobility edge
in the range $\theta/\frac{\pi}{4}\in(1.05,1.25)$.

Breaking parity-time symmetry.— We confirm the breaking of PT symmetry, by
measuring the time evolution of the overall corrected probability of photons.
Here the overall corrected probability is defined as $P(t)=e^{4\gamma
t}\sum_{x}|\langle x|\Psi_{E}(t)\rangle|^{2}$, where the exponent $e^{4\gamma
t}$ recovers the probability evolution under $U$. When the system is in the PT
unbroken regime, the quasienergy is entirely real, and $P(t)$ would be on the
order of unity. By contrast, when the system is in the PT broken regime,
$P(t)$ should grow in time, as the eigenenergies can acquire positive
imaginary components. These understandings are confirmed in Fig. 4, where time
evolutions at different $\theta$ shows that the PT transition point is
consistent with the theoretical phase diagram.

Conclusion.— We experimentally simulate the dynamics of a one-dimensional non-
Hermitian quasicrystal using photonic quantum walks, and reveal an tricritical
point where the spectral topological transition, the global delocalization-
localization transition, and the PT-symmetry breaking transition
simultaneously occur. Our experiment thus unveils a highly non-trivial
phenomenon that is absent in Hermitian quasicrystals, and calls for further
study of the critical phenomena near these phase transitions.

Intriguingly, the coincidence of all three phase transitions can be model-
dependent. In a very recent theoretical study naad , it is shown that,
although the localization and the PT-symmetry breaking transitions coincide in
a general class of non-Hermitian quasicrystals, they are not necessarily
accompanied by a change of the spectral topology. While the study raises
important open questions regarding the conditions for the concurrence of phase
transitions, our experiment offers a highly tunable platform on which these
questions can be systematically addressed in the future.

##  A Supplemental Material for “Simulating non-Hermitian quasicrystals with
single-photon quantum walks”

###  A.1 Experimental scheme

We adopt a time-multiplexed scheme for the experimental simulations of non-
Hermitian quasicrystals with single-photon quantum walks sch ; SGR+12 ;
scienceskin . The photon source is provided by a pulsed laser with a central
wavelength of $808$nm, a pulse width of $88$ps, and a repetition rate of
$31.25$KHz. The pulses are attenuated to the single-photon level by using a
neutral density filter. An average photon number per pulse is less than
$2\times 10^{-4}$, which ensures a negligible probability of multi-photon
events. Photons pass through a polarizing beam splitter (PBS) and a half-wave
plate (HWP) for initializing the polarization states. In this process, the
coins of the quantum walk are encoded in the polarization state, and are
coupled in and out of a time-multiplexed setup through a beam splitter (BS)
with a reflectivity of $5\%$, corresponding to a low coupling rate of photons
into the network. Such a low-reflectivity BS also enables the out-coupling of
photons for measurement.

The setup consists of two optical fibre loops with different lengths, which
are coupled by PBSs. The shift operator $S$ is implemented by separating
photons corresponding to their two polarization components and routing them
through the two fibre loops, respectively. Polarization-dependent time delay
is then introduced and the walker’s position is encoded into the time domain.
The lengths of the two fiber loops are $167.034$m and $160.000$m,
respectively. A round-trip time is approximately given by $772$ns. The time
difference of photons traveling through two fiber loops is $33.046$ns, which
defines the temporal width of a time-bin. To implement the coin operator
$R(\theta)$, two HWPs provide a careful control over the parameter $\theta$.
The position-dependent phase operator is implemented by an electro-optical
modulator (EOM)

$P=\sum_{x}\ket{x}\bra{x}\otimes\begin{pmatrix}e^{i\frac{\phi^{\prime}}{2}}&0\\\
0&e^{-i\frac{\phi^{\prime}}{2}}\end{pmatrix}$ (S1)

with $\phi^{\prime}=\cos(x\pi\phi)\pi$. The rise/fall times of EOM ($4$ns) are
much less than the time difference of the adjacent position ($33.046$ns),
which enables us to control the parameter $\phi^{\prime}(x)$ precisely.

To realize a polarization-dependent loss operation
$M_{\text{E}}=e^{-\gamma}M$, two HWPs are introduced into each of the fiber
loops. For the short loop, after passing through the first PBS, horizontally
polarized photons are all transmitted by the second PBS and are subject to
further time evolution. Whereas for the long loop, by controlling the setting
angle of the HWP, part of the photons $(1-e^{4\gamma})$ with vertical
polarization are flipped into horizontal ones, transmitted by the second PBS
and subsequently leak out of the setup. Then we implement a half step of the
split-step quantum walk $U$. We therefore read out the time-evolved state
driven by $U$ by adding a time-dependent factor $e^{2\gamma t}$ to the
outcomes of our experimental measurement. For instance, a superposition of
multiple spatial positions at a given time step is translated into the
superposition of multiple well-resolved pulses within the same discrete time
step.

Three avalanche photo-diodes (APDs) are employed to record the temporal and
polarization properties of both the out-coupled photons and the lost photons,
yielding information regarding the number of time steps, as well as the
spatial and coin states of the walker.

The probability distribution of the walker $P(x,t)=\frac{|\langle
x|\Psi_{\text{E}}(t)\rangle|^{2}}{\sum_{x}|\langle
x|\Psi_{\text{E}}(t)\rangle|^{2}}$ at the position $x$ and step $t$ is
obtained by dividing the number of photons collected by APD at the position
$x$ using the total photons collected after the $t$th step, i.e.,

$P(x,t)=\frac{N(x,t)}{\sum_{x}N(x,t)}$ (S2)

with $N(x,t)=N_{H}(x,t)+N_{V}(x,t)$. With the probability distribution
$P(x,t)$, we can construct the dynamic IPR $\text{dIPR}=\sum_{x}P^{2}(x,t)$.
Note that, as defined in the main text, $|\Psi(t)\rangle=e^{2\gamma
t}|\Psi_{E}(t)\rangle$, where $|\Psi(t)\rangle$ and
$|\Psi_{\text{E}}(t)\rangle$ are respectively the time-evolved states under
$U$, and under the experimentally implemented dynamics.

The overall probability $P(t)=\sum_{x}|\langle x|\Psi(t)\rangle|^{2}$ can be
constructed as

$P(t)=e^{4\gamma
t}\prod_{t^{\prime}=1}^{t}\left[\frac{\frac{N^{\prime}(t^{\prime})}{r}}{\frac{N^{\prime}(t^{\prime})}{r}+N_{1L}(t^{\prime})}\times\frac{\frac{N(t^{\prime})}{r}}{\frac{N(t^{\prime})}{r}+N_{2L}(t^{\prime})}\right].$
(S3)

The photon loss caused by the partial measurement $M$ is also included. Here
$r=0.05$ is the reflectivity of the BS,
$N^{\prime}(t)=\sum_{x}N^{\prime}(x,t)$ is the total photon numbers reflected
by the BS after the first half step of the $t$th step, $N(t)=\sum_{x}N(x,t)$
is the total photon numbers reflected by the BS after the $t$th step, and
$N_{1L,2L}(t)=\sum_{x}N_{1L,2L}(x,t)$ are the total photon loss after the
first (second) half step of the $t$th step. Then, we can construct the
Lyapunov exponent

$\lambda(v)=\frac{1}{t}\log\sqrt{\frac{P(t)N_{H}(x,t)}{\sum_{x}N_{H}(x,t)+\sum_{x}N_{V}(x,t)}}$
(S4)

with the experimental results of $P(t)$, where $N_{H,V}(x,t)$ are the photon
numbers with horizontal (vertical) polarizations collected by APDs at the
position $x$ after the $t$th step.

###  A.2 Dynamics of the time-evolved state

The quantum-walk dynamics implemented in our experiment constitutes a
stroboscopic simulation of the time evolution driven by a generalized non-
Hermitian Aubry-André-Harper (AAH) model with both diagonal and off-diagonal
quasiperiodic disorder.

To see this, we consider the state at the $m$-th time step
$|\Psi\rangle=\sum_{x}\left(\begin{array}[]{l}\mu^{m}_{x}\\\
v^{m}_{x}\end{array}\right)\otimes|x\rangle$. Applying $U$ one on the state
once, we obtain the dynamic equation, connecting the state at the $m+1$-th
step with that at the $m$-th step

$\displaystyle\mu_{x}^{m+1}=$
$\displaystyle\left(\cos^{2}(\theta)\mu_{x+2}^{m}-\cos(\theta)\sin(\theta)v_{x+2}^{m}\right)\exp\left(2\gamma+\frac{i\phi^{\prime}(x+2)-i\phi^{\prime}(x+1)}{2}\right)$
$\displaystyle-\left(\sin^{2}(\theta)\mu_{x}^{m}+\cos(\theta)\sin(\theta)v_{x}^{m}\right)\exp\left(\frac{i\phi^{\prime}(x)+i\phi^{\prime}(x+1)}{2}\right),$
(S5) $\displaystyle v_{x}^{m+1}=$
$\displaystyle\left(\cos(\theta)\sin(\theta)\mu_{x-2}^{m}+\cos^{2}(\theta)v_{x-2}^{m}\right)\exp\left(-2\gamma-\frac{i\phi^{\prime}(x-2)-i\phi^{\prime}(x-1)}{2}\right)$
$\displaystyle+\left(\cos(\theta)\sin(\theta)\mu_{x}^{m}-\sin^{2}(\theta)v_{x}^{m}\right)\exp\left(\frac{i\phi^{\prime}(x)+i\phi^{\prime}(x-1)}{2}\right),$
(S6)

where $\phi^{\prime}(x)=\cos(x\pi\phi)\pi$.

From the above two equations, we have
$\sum_{x}\left(\begin{array}[]{c}\mu_{x}^{m+1}\\\
\bm{v}_{x}^{m+1}\end{array}\right)|x\rangle=U\sum_{x}\left(\begin{array}[]{c}\mu_{x}^{m}\\\
\bm{v}_{x}^{m}\end{array}\right)|x\rangle$, where

$\displaystyle U_{11}$
$\displaystyle=\sum_{x}-\sin^{2}(\theta)\exp\left(\frac{i\phi^{\prime}(x)+i\phi^{\prime}(x+1)}{2}\right)|x\rangle\langle
x|+\cos^{2}(\theta)\exp\left(2\gamma+\frac{i\phi^{\prime}(x+2)-i\phi^{\prime}(x+1)}{2}\right)|x\rangle\langle
x+2|,$ (S7) $\displaystyle U_{12}$
$\displaystyle=\sum_{x}-\sin(\theta)\cos(\theta)\exp\left(\frac{i\phi^{\prime}(x)+i\phi^{\prime}(x+1)}{2}\right)|x\rangle\langle
x|-\sin(\theta)\cos(\theta)\exp\left(2\gamma+\frac{i\phi^{\prime}(x+2)-i\phi^{\prime}(x+1)}{2}\right)|x\rangle\langle
x+2|,$ (S8) $\displaystyle U_{21}$
$\displaystyle=\sum_{x}\sin(\theta)\cos(\theta)\exp\left(\frac{i\phi^{\prime}(x-1)+i\phi^{\prime}(x)}{2}\right)|x\rangle\langle
x|+\sin(\theta)\cos(\theta)\exp\left(-2\gamma-\frac{i\phi^{\prime}(x-2)-i\phi^{\prime}(x-1)}{2}\right)|x\rangle\langle
x-2|,$ (S9) $\displaystyle U_{22}$
$\displaystyle=\sum_{x}-\sin^{2}(\theta)\exp\left(\frac{i\phi^{\prime}(x-1)+i\phi^{\prime}(x)}{2}\right)|x\rangle\langle
x|+\cos^{2}(\theta)\exp\left(-2\gamma-\frac{i\phi^{\prime}(x-2)-i\phi^{\prime}(x-1)}{2}\right)|x\rangle\langle
x-2|.$ (S10)

For comparison, we derive the corresponding dynamic equation for the original
AAH model below. We consider the Hamiltonian

$\displaystyle H=\sum_{x}J(|x\rangle\langle x+1|+|x+1\rangle\langle
x|)+\sum_{x}V(x)|x\rangle\langle x|,$ (S11)

where $J$ is the hopping rate, and $V(x)=V\cos(2\pi\phi x)$ is the on-site
quasiperiodic potential, with $\phi$ an irrational number.

We define the infinitesimal time-evolution operator by $U_{H}=e^{-iH\Delta
t}$, where the time interval $\Delta t$ is infinitesimal. Using the
Baker–Campbell–Hausdorff formula, we have

$\displaystyle U_{H}=$ $\displaystyle e^{-iH\Delta t}=e^{-i\Delta
tJ\sum_{x}(|x\rangle\langle x+1|+|x+1\rangle\langle x|)}e^{-i\Delta
t\sum_{x}V(x)|x\rangle\langle x|}+o(\Delta t^{2})$ $\displaystyle=$
$\displaystyle\sum_{x}e^{-i\Delta tV(x)}|x\rangle\langle x|-i\Delta
tJ\sum_{x}\left(e^{-i\Delta tV(x+1)}|x\rangle\langle x+1|+e^{-i\Delta
tV(x)}|x+1\rangle\langle x|\right)+o(\Delta t^{2}).$ (S12)

Now we consider the state $|\phi\rangle=\sum_{x}a^{m}_{x}|x\rangle$, where the
superscript indicates the $m$-th time step, meaning the operator $U_{H}$ has
been applied $m$ times on an initial state. Applying $U_{H}$ one more time,
and keeping terms of the order $\Delta t$, we have

$\displaystyle a^{m+1}_{x}=e^{-i\Delta tV(x)}a_{x}^{m}-i\Delta tJe^{-i\Delta
tV(x+1)}a_{x+1}^{m}-i\Delta tJe^{-i\Delta tV(x-1)}a_{x-1}^{m}.$ (S13)

Comparing Eqs. (S5)(S6) and Eq. (S13) [or equivalently Eqs. (S7)(S8)(S9)(S10)
and Eq. (S12)], it is straightforward to see that the implemented quantum-walk
dynamics corresponds to a discrete-time evolution governed by a lattice model
with quasiperiodic potential (given by $\phi^{\prime}(x)$). Further, different
signs of $\gamma$ in the exponentials of Eqs. (S5)(S6) indicate the lattice
model features a non-reciprocal hopping, which eventually gives rise to the
non-Hermitian skin effect.

###  A.3 Reality of eigenenergy and localization

From numerical simulations, it is straightforward to establish that for
eigenstates with real eigenenergies, the state is spatially localized. For
those with a finite imaginary components, the state is spatially extended.
This is illustrated in Fig. S1.

Figure S1: (a) Eigenspectrum of the effective Hamiltonian $H$ in the PT-broken
regime under a periodic boundary condition, where we take a lattice size
$N=200$, $\gamma=0.1$, and $\theta/(\frac{\pi}{4})=1.1$ for numerical
calculations. (b)(c) shows the spatial probability distribution of the
eigenstates with eigenenergies marked by red dot (b) and black square (c) in
the eigenspectrum (a), respectively.

In the parity-time (PT) symmetry broken regime, some of the eigenstates can
still have real eigenenergies; while in the PT symmetry unbroken regime, all
eigenstates have real eigenenergies. Thus, when the system crosses the PT
symmetry breaking point from the PT-broken side to the unbroken side, a global
delocalization-localization transition occurs, where all eigenstates are
localized on the unbroken side. This explains the concurrence of the PT-
symmetry breaking transition and the delocalization-localization transition at
$\theta\approx 1.25\frac{\pi}{4}$.

###  A.4 Dependence of the phase boundary on the gain-loss parameter

As shown in the phase diagram in Fig. 1 of the main text, the phase boundary
is dependent on the gain-loss parameter $\gamma$. We experimentally confirm
this by probing the PT-symmetry breaking and the delocalization-localization
transitions by adopting a different $\gamma$ compared to the main text.

In Fig. S2(a), we show the numerically calculated $\overline{\text{IPR}}$
(blue) and $\overline{\text{NPR}}$ (red) for $\gamma=0.02$ and $N=1000$. The
two transitions are now located at $\theta/\frac{\pi}{4}\approx 0.98$ and
$\theta/\frac{\pi}{4}\approx 1.2$, respectively. The latter should correspond
to the triple phase transition point. The experimentally measured dIPR and
dNPR in Fig. S2(b) are consistent with the numerical simulations, indicating a
global delocalization-localization transition near
$\theta/\frac{\pi}{4}\approx 1.2$.

In Fig. S2(c), we show the numerically evaluated $\text{Im}(E)$ for
$\gamma=0.02$ and $N=200$, where a PT-symmetry breaking transition is
identified at $\theta/\frac{\pi}{4}\approx 1.2$. The experimentally measured
overall corrected probability $P(t)$ for different $\theta$ in Fig. S2(d) are
mostly consistent with the PT-symmetry breaking transition. For the odd
$\theta/\frac{\pi}{4}=1.032$ (green), $P(t)$ should grow in time (see
numerical simulation in the inset). Due to the limitation of experimentally
accessible time steps and its closeness to the transition point, the measured
$P(t)$ for $\theta/\frac{\pi}{4}=1.032$ appears to be on the order of unity.
Nevertheless, together with the measured $P(t)$ for
$\theta/\frac{\pi}{4}=1.032$ in Fig. 4(b) of the main text, our data provide
evidence for the dependence of the PT transition on $\gamma$.

Figure S2: (a) Numerical simulation of $\overline{\text{IPR}}$ (blue) and
$\overline{\text{NPR}}$ (red) for $N=1000$ and $\gamma=0.02$. (b) Measured
dIPR (blue) and dNPR (red) for non-Hermitian quantum walks with the initial
state $\ket{x=0}\otimes\ket{V}$ and $\gamma=0.02$. Solid and hollow symbols
represent the experimental data and their theoretical predictions,
respectively. The black and red dashed lines in (a)(b) are at
$\theta/\frac{\pi}{4}\approx 0.98$ and $\theta/\frac{\pi}{4}\approx 1.2$,
respectively. (c) Imaginary parts of the quasienegies of a non-Hermitian
quantum walk with $N=200$ and $\gamma=0.02$. The vertical dashed line at
$\theta/\frac{\pi}{4}\approx 1.2$ separate the PT unbroken and broken phases.
(d) Measured overall corrected probabilities $P(t)$ of a non-Hermitian quantum
walk with the initial state $\ket{x=0}\otimes\ket{H}$ for various $\theta$
corresponding to the PT unbroken and broken phases. Symbols represent the
experimental data and curves for their theoretical predictions. Error bars are
due to the statistical uncertainty in photon-number counting. Inset:
numerically simulated $P(t)$ at longer time steps for
$\theta/\frac{\pi}{4}=1.32$ (blue) and $\theta/\frac{\pi}{4}=1.032$ (green),
respectively.

###  A.5 Additional numerical results

Additional numerical results corresponding to Figs. 2(a)(b) in the main text
are shown in Fig. S3.

Figure S3: The numerical simulations corresponding to experimental data in
Figs. 2(a) and (b), respectively.

###### Acknowledgements.

This work has been supported by the National Natural Science Foundation of
China (Grant Nos. 12025401, U1930402, 11974331 and 12088101). W. Y.
acknowledges support from the National Key Research and Development Program of
China (Grant Nos. 2016YFA0301700 and 2017YFA0304100).

## References

  * (1) H. J. Carmichael, Phys. Rev. Lett. 70, 2273 (1993).
  * (2) C. M. Bender, Rep. Prog. Phys. 70, 947 (2007).
  * (3) R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, Nat. Phys. 14, 11-19 (2018).
  * (4) C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).
  * (5) M.-A. Miri and A. Alù, Science 363 eaar7709 (2019).
  * (6) Y. Ashida, Z. Gong, and M. Ueda, Advances in Physics 69, 3 (2020).
  * (7) K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Phys. Rev. X 9, 041015 (2019).
  * (8) H. Zhou and J. Y. Lee, Phys. Rev. B 99, 235112 (2019).
  * (9) A. Ghatak and T. Das, J. Phys. Condens. Matter 31, 263001 (2019).
  * (10) J. Li, A. K. Harter, J. Liu, L. de Melo, Y. N. Joglekar, and L. Luo, Nat. Commun. 10, 1-7 (2019).
  * (11) C. Yuce, Phys. Lett. A. 378, 2024-2028 (2014).
  * (12) S. Yao and Z. Wang, Phys. Rev. Lett. 121, 086803 (2018).
  * (13) S. Yao, F. Song, and Z. Wang, Phys. Rev. Lett. 121, 136802 (2018).
  * (14) K. Yokomizo and S. Murakami, Phys. Rev. Lett. 123, 066404 (2019).
  * (15) C. H. Lee and R. Thomale, Phys. Rev. B 99, 201103(R) (2019).
  * (16) F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, Phys. Rev. Lett. 121, 026808 (2018).
  * (17) D. S. Borgnia, A. J. Kruchkov, and R.-J. Slager, Phys. Rev. Lett. 124, 056802 (2020)
  * (18) A. McDonald, T. Pereg-Barnea, and A. A. Clerk, Phys. Rev. X 8, 041031 (2018).
  * (19) V. M. Martinez Alvarez, J. E. Barrios Vargas, and L. E. F. Foa Torres, Phys. Rev. B 97, 121401(R) (2018).
  * (20) K. Zhang, Z. Yang, and C. Fang, Phys. Rev. Lett. 125, 126402 (2020).
  * (21) N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, Phys. Rev. Lett. 124, 086801 (2020).
  * (22) Z. Yang, K. Zhang, C. Fang, and J. Hu, Phys. Rev. Lett. 125, 226402 (2020).
  * (23) S. Longhi, Phys. Rev. Research 1, 023013 (2019).
  * (24) T.-S. Deng and W. Yi, Phys. Rev. B 100, 035102 (2019).
  * (25) L. Li, C. H. Lee, S. Mu, and J. Gong, Nat. Commun. 11, 5491 (2020).
  * (26) M. Gräfe, R. Heilmann, R. Keil, T. Eichelkraut, M. Heinrich, S. Nolte, and A. Szameit, New. J. Phys. 15, 033008 (2013).
  * (27) S. Scheel and A. Szameit, Europhysics Letters 122, 34001 (2018).
  * (28) T. Jiang, A. Fang, Z.-Q. Zhang, and C. T. Chan, Nanophotonics. 10, 443-452 (2021).
  * (29) T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. Molenkamp, C. Lee, A. Szameit, M. Greiter, and R. Thomale, Nat. Phys. 16, 747 (2020).
  * (30) T. Hofmann, T. Helbig, F. Schindler, N. Salgo, M. Brzezińska, M. Greiter, T. Kiessling, D. Wolf, A. Vollhardt, A. Kabas̆i, C. H. Lee, A. Bilušić, R. Thomale, and T. Neupert, Phys. Rev. Research 2, 023265 (2020).
  * (31) A. Ghatak, M. Brandenbourger, J. van Wezel, and C. Coulais, Proc. Natl. Ac. Sc. 117, 29561 (2020).
  * (32) L. Xiao, T. Deng, K. Wang, G. Zhu, Z. Wang, W. Yi, and P. Xue, Nat. Phys. 16, 761 (2020).
  * (33) L. Xiao, T. Deng, K. Wang, Z. Wang, W. Yi, and P. Xue, Phys. Rev. Lett. 126, 230402 (2021).
  * (34) S. Weidemann, M. Kremer, T. Helbig, T. Hofmann, A. Stegmaier, M. Greiter, R. Thomale, and A. Szameit, Science 368, 311 (2020).
  * (35) S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, Nat. Mater. 16, 433-438 (2017).
  * (36) P. G. Harper, Proc. Phys. Soc. London A 68, 874 (1955).
  * (37) S. Aubry and G. André, Ann. Israel Phys. Soc. 3, 133-140 (1980).
  * (38) J. B. Sokoloff, Phys. Rep. 126, 189-244 (1984).
  * (39) S. Longhi, Phys. Rev. B 103, 054203 (2021).
  * (40) A. P. Acharya, A. Chakrabarty, D. K. Sahu, and S. Datta, Phys. Rev. B 105, 014202 (2022).
  * (41) Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, and Y. Silberberg, Phys. Rev. Lett. 103, 013901 (2009).
  * (42) M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y. Silberberg, Phys. Rev. Lett. 110, 076403 (2013).
  * (43) G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature 453, 895-899 (2008).
  * (44) D. R. Hofstadter, Phys. Rev. B 14, 2239 (1976).
  * (45) Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, Phys. Rev. Lett. 109, 106402 (2012).
  * (46) L.-J. Lang, X. Cai, and S. Chen, Phys. Rev. Lett. 108, 220401 (2012).
  * (47) D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Rev. Mod. Phys. 91, 021001 (2019).
  * (48) M. Schreiber, S. S. Hodgman, P. Bordia, H. P. Lüschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I. Bloch, Science 349, 842 (2015).
  * (49) M. Rispoli, A. Lukin, R. Schittko, S. Kim, M. E. Tai, J. Léonard, and M. Greiner, Nature 573, 385 (2019).
  * (50) J. H. Han, D. J. Thouless, H. Hiramoto, and M. Kohmoto, Phys. Rev. B 50, 11365 (1994).
  * (51) Y. Wang, C. Cheng, X.-J. Liu, and D. Yu, Phys. Rev. Lett. 126, 080602 (2021).
  * (52) T. Xiao, D. Xie, Z. Dong, T. Chen, W. Yi, and B. Yan, Sci. Bull. 66, 2175 (2021).
  * (53) H. Jiang, L.-J. Lang, C. Yang, S. -L. Zhu, and S. Chen, Phys. Rev. B 100, 054301 (2019).
  * (54) Y. Liu, Q. Zhou, and S. Chen, Phys. Rev. B 104, 024201 (2021).
  * (55) Z. Xu, X. Xia, and S. Chen, Sci. China-Phys. Mech. Astron. 65, 227211 (2022).
  * (56) Q.-B. Zeng and Y. Xu, Phys. Rev. Research 2, 033052 (2020).
  * (57) S. Longhi, Phys. Rev. Lett. 122, 237601 (2019).
  * (58) Q. Lin, T. Li, L. Xiao, K. Wang, W. Yi, and P. Xue, arXiv:2108.01097.
  * (59) See Supplemental Material for details.
  * (60) A. Schreiber, K. N. Cassemiro, V. Potoc̆ek, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, Phys. Rev. Lett. 104, 050502 (2010).
  * (61) A. Schreiber, A Gábris, P. P. Rohde, K. Laiho, M. Štefaňák, V Potoček, C. Hamilton, I. Jex, and C. Silberhorn, Science 336, 55-58 (2012).
  * (62) S. Iyer, V. Oganesyan, G. Refael, and D. A. Huse, Phys. Rev. B 87, 134202 (2013).


# Effects of quantum geometric phase of a particle in an oscillating hard-wall
spherical trap

Reza Moazzemi<EMAIL_ADDRESS>Department of Physics, University of Qom,
Ghadir Blvd., Qom 371614-6611, I.R. Iran  Seyed Mahdi Fazeli  Department of
Physics, University of Qom, Ghadir Blvd., Qom 371614-6611, I.R. Iran

###### Abstract

We obtain the geometric phase for states of a particle in a spherical infinite
potential well with moving walls in two different cases; First, when the
radius of the well increases (or decreases) monotonically. Second, when the
radius changes oscillatory. In the latter case, we have solved the Schrödinger
equation and found its solutions approximately. We obtain the transition rate
for the possible real situation in an acousto-optic case which can reveal the
effect of the geometric phase. We show that the absorption or radiation peaks
will appear if the energy gap between the incident photon and the modified
energy difference of two levels by the geometric phase, is equal to the
integer multiple of oscillation frequency.

Berry phase, geometric phase, spherical well, adiabatic theorem

##  I Introduction

The Berry phase is a gauge invariant and geometrical phase which is firstly
defined by Berry in 1984 Berry1984 . As a matter of fact, many physical
situations are contained of objects whose behavior is specified up to a phase
by certain parameters. Berry phase appears when the transversal of a closed
path by the parameters, which at the end of the path they have returned to
their initial values, leads to a change of phase whose magnitude depends only
on the path (For a collection of papers on the subject, see wilczek ).

As a simple example consider a magnet precessing about a magnetic field. Here
the parameters are the three components of the field. The field is slowly
rotated and finally, returns to its original direction. In this situation, the
phase angle of the precession will be shifted from what it would be if the
field had been held constant Cina1986 . In addition, recently many works have
been published on the Berry phase and its applications in various fields. For
example, in Ref. marija  a system of strongly interacting one-dimensional (1D)
bosons on a ring pierced by a synthetic magnetic flux tube is studied. In Ref.
Antti , Employing the spectral interference law of electromagnetic waves, a
general expression for the Berry phase in Young’s two-pinhole setup is
established.

The most of applications of the Berry phase are in solid state physics,
especially in macroscopic phenomena which are slow in time and smooth in space
in comparison with the atomic scales (for review see Xiao2010 ). For example,
due to the Berry phase, when an electron completes a cycle around the Dirac
point (a particular location in graphene’s electronic structure), the phase of
its wave function changes by $\pi$. Scanning tunneling spectroscopy revealed
sudden jumps in conductivity as the external magnetic field was increased past
a threshold value science . As another important application of optical
manipulation of the Berry phase in a solid-state spin qubit, in Ref. nature
authors demonstrate an all-optical method to accumulate a geometric phase, in
an individual nitrogen–vacancy centre in diamond. Also, the measuring Berry
phase of graphene from wavefront dislocations in Friedel oscillations has been
done recently in Ref. nature2 . In Quantum field theory the subject of the
Berry phase is also considered baggio ; hsin .

The solutions of time-dependent Schrödinger equation for an infinite square-
well potential with a moving wall in one dimension are known Doescher1969 .
Here we derive its solutions in three dimensions for a spherical infinite
potential well with moving wall, which also has been reported previously in
Ref. musavi . Then we calculate the geometric phase for this case.

The other three-dimensional case which we consider in this paper is the
oscillatory moving wall, i.e. we assume the radius of potential well changes
oscillatory. In both cases, the changes in the radius of the well should occur
slow enough so that we can use the adiabatic theorem. In the latter case, the
solutions of Schrödinger equation are found approximately. We then calculate
the geometrical phase for this situation. Finally, we consider a more
realistic situation in which a system oscillates with an acoustic wave. For
this case, we derive the transition rate and discuss the absorption and
radiation modes. Note that, in this problem, we have only one time-dependent
parameter, the radius of the well. However, this does not mean that the
geometric phase is irrelevant in this problem. In fact, here, we present a way
to measure some manifestations of geometric phase for this problem.

##  II Moving with constant velocity

###  II.1 Solutions of Schrödinger Equation

In this section we first find the solutions of the time dependent Schrödinger
equation for an infinite spherical well in which the radius of the well move
with the constant velocity $v$. The time-dependent Schrödinger equation is

$H(\mathbf{r},t){\Phi_{nlm}}(\mathbf{r},t)=i\hbar\frac{{\partial{\Phi_{nlm}}(\mathbf{r},t)}}{{\partial
t}},$ (1)

where $H(\mathbf{r},t)=-\frac{\hbar^{2}}{2m}\nabla^{2}+V(\mathbf{r},t)$ is the
time-dependent Hamiltonian with

$V(\mathbf{r},t)=\left\\{\begin{array}[]{l}0{\qquad\ \,}r<a(t){\rm{}}\\\
\infty{\qquad}r\geq a(t).{\rm{}}\end{array}\right.$ (2)

where $a(t)=a_{0}+vt$. The general solution of the problem has the following
form:

$\Psi(\mathbf{r},t)=\sum_{nlm}c_{nlm}\Phi_{nlm}(\mathbf{r},t).$ (3)

Since the potential is spherically symmetric we can easily separate the
angular part of the solutions $\Phi_{nlm}(\mathbf{r},t)$, and write it down as

$\Phi_{nlm}(\mathbf{r},t)=\frac{u_{nl}(r,t)}{r}Y_{l}^{m}(\theta,\varphi)\,{e^{i\phi(r,t)}}$
(4)

where ${{\phi(r,t)}}$ is some phase, $Y_{l}^{m}(\theta,\varphi)$ are the
spherical harmonics and $u_{nl}(r,t)$ satisfy the Schrödinger radial equation

$-\frac{{{\hbar^{2}}}}{{2m}}\frac{{{\partial^{2}}{u_{nl}}(r,t)}}{{\partial{r^{2}}}}+\frac{{{\hbar^{2}}}}{{2m}}\frac{{l(l+1)}}{{{r^{2}}}}{u_{nl}}(r,t)=i\hbar\frac{{\partial{u_{nl}}(r,t)}}{{\partial
t}}.$ (5)

On the other hand, using adiabatic theorem we know that the solutions should
have the following form

$\displaystyle{\Phi_{nlm}}(\mathbf{r},t)=\frac{1}{{j_{l+1}({\beta_{nl}})}}\sqrt{\frac{2}{{{a(t)^{3}}}}}{j_{l}}[{\beta_{nl}}r/a(t)]$
$\displaystyle\times
Y_{l}^{m}(\theta,\varphi)\,{e^{i{\theta_{nl}}(t)}}{e^{if(r,t)}},$ (6)

with the energies ${E_{nl}}(t)={{{\hbar^{2}\beta_{nl}^{2}}}}/{{2m{a(t)^{2}}}}$
(similar to a well with radius of constant $a$). The requrement of the appling
adiabatic theorem is that the speed of the walls is much less than the
characteristic speed of the particle i.e. $v\ll\frac{\hbar}{ma}$. In (II.1),
$j_{l}(x)$ is the spherical Bessel function of order $l$, $\beta_{nl}$ is
$n$th zero of the $l$th spherical Bessel function. Now the dynamical phase is
defined by

$\displaystyle{\theta_{nl}}(t)$ $\displaystyle=$
$\displaystyle-\frac{1}{\hbar}\int_{0}^{t}{{E_{nl}}(t^{\prime})}dt^{\prime}$
(7) $\displaystyle=$
$\displaystyle-\frac{{\hbar\beta_{nl}^{2}}}{{2mv}}\left({\frac{1}{{{a_{\,0}}}}-\frac{1}{{{a_{\,0}}+vt}}}\right).$
(8)

To find $f(r,t)$ in (II.1), we try to solve the Eq. (1) by this solution. If
we solve the real and imaginary part separately, we find that

$f\left({r,t}\right)=\frac{{mv{r^{2}}}}{{2\hbar\left({{a_{0}}+vt}\right)}}.$
(9)

Therefore the solution of the Schrodinger equation for infinite spherical well
with moving wall is

$\displaystyle{\Phi_{nlm}}(\mathbf{r},t)$ $\displaystyle=$
$\displaystyle{\varphi_{nlm}}(\mathbf{r},t){e^{i{\theta_{nl}}(t)}}$
$\displaystyle=$
$\displaystyle\frac{1}{{j_{l+1}({\beta_{nl}})}}{\sqrt{\frac{2}{{{{({a_{0}}+vt)}^{3}}}}}}\,\,{j_{l}}\left[{\frac{{{\beta_{nl}}r}}{{{a_{0}}+vt}}}\right]$
$\displaystyle\times{\exp{\left[i{\frac{{mv{r^{2}}}}{{2\hbar\left({{a_{0}}+vt}\right)}}-i\frac{{\hbar\beta_{nl}^{2}}}{{2mv}}\left({\frac{1}{{{a_{0}}}}-\frac{1}{{{a_{0}}+vt}}}\right)}\right]}}Y_{l}^{m}(\theta,\varphi).$

These solutions are in agreement with the ones reported in Ref. musavi .

###  II.2 The geometric phase

Now, we are ready to obtain the geometric phase

${\gamma_{nl}}(t)=i\int_{0}^{t}{\left\langle{{\varphi_{nlm}}(\mathbf{r},t^{\prime})\left|{\frac{\partial}{{\partial
t^{\prime}}}{\varphi_{nlm}}(\mathbf{r},t^{\prime}}\right.)}\right\rangle\,}dt^{\prime}.$
(11)

The Berry phase is the above geometric phase when the system returns to its
original state (or to one of its degenerate state) in a cyclic path

${\gamma_{nl}}(C)=i\oint_{C}{\left\langle{{\varphi_{nlm}}(\mathbf{r},t^{\prime})\left|{\frac{\partial}{{\partial
t^{\prime}}}{\varphi_{nlm}}(\mathbf{r},t^{\prime}}\right.)}\right\rangle\,}dt^{\prime}.$
(12)

Computing the inner product of (11) we have

$\displaystyle\left\langle{{\varphi_{nlm}}\left|{\frac{{\partial{\varphi_{nlm}}}}{{\partial
t^{\prime}}}}\right.}\right\rangle=\int_{0}^{a}{\varphi_{nlm}^{*}(r,t)}\frac{\partial}{{\partial
t^{\prime}}}\varphi_{nlm}(r,t){r^{2}}drd\Omega$
$\displaystyle=-i\frac{{mv^{2}}}{{6\hbar\beta_{nl}^{2}}}{\left[{\frac{{j_{l-1}\left({{\beta_{nl}}}\right)}}{{j_{l+1}\left({{\beta_{nl}}}\right)}}}\right]^{2}}\left[{4l(l+1)-3+2\beta_{nl}^{2}}\right].$
(13)

Therefore,

$\displaystyle{\gamma_{nl}}(t)=\frac{{mv}}{{6\hbar\beta_{nl}^{2}}}{\left({\frac{{j_{l-1}\left({{\beta_{nl}}}\right)}}{{j_{l+1}\left({{\beta_{nl}}}\right)}}}\right)^{2}}\left[{4l(l+1)-3+2\beta_{nl}^{2}}\right]$
$\displaystyle\times\left[{a(t)-{a_{\,0}}}\right].$ (14)

The appearance of $v$ in the above relation shows how this geometric phase
depends on the path taken. We can write (II.2) as

$\displaystyle{\gamma_{nl}}(\mathbf{a})=f_{nl}a+\gamma_{0},$ (15)

where $f_{nl}$ and $\gamma_{0}$ are constant. It is clear that the above time
dependent geometric phase directly depends on the radius of the well which is
a geometric parameter.

##  III The oscillatory moving wall

###  III.1 Solutions of the Schrödinger equation

In this section we derive the Berry phase when the wall of the well is moving
oscillatory. Again the potential is the same as Eq. (2) but with
$a(t)=a_{0}+b\sin(\omega t)$. The dynamical phase is now

$\displaystyle\theta_{nl}^{\rm{osc}}(t)$ $\displaystyle=$
$\displaystyle-\frac{1}{\hbar}\int^{t}\frac{\hbar^{2}\beta_{nl}^{2}}{2m(a_{0}+b\sin\omega
t^{\prime})^{2}}\,dt^{\prime}$ (16) $\displaystyle=$
$\displaystyle-\frac{\hbar\beta_{nl}^{2}}{2m\omega}\Bigg{\\{}\frac{2a_{0}\arctan\left[\frac{b+a_{0}\tan(\omega
t/2)}{\sqrt{a_{0}^{2}-b^{2}}}\right]}{(a_{0}^{2}-b^{2})^{3/2}}$
$\displaystyle+\frac{b\cos\omega t}{(a_{0}^{2}-b^{2})(a_{0}+b\sin\omega
t)}\Bigg{\\}}+\phi_{0}$ $\displaystyle=$
$\displaystyle{-\frac{\bar{E}_{nl}}{\hbar}t+\zeta_{nl}(t)+\phi_{0}},$ (17)

where $\bar{E}_{nl}$ is the averaged energy in one period, $\zeta_{nl}(t)$
some periodic function of $t$ and $\phi_{0}$ a constant phase. We try the
following solution:

$\displaystyle{\Phi_{nlm}^{\rm{osc}}}(\mathbf{r},t)=\frac{1}{{j_{l+1}({\beta_{nl}})}}\sqrt{\frac{2}{{{a(t)^{3}}}}}{j_{l}}[{\beta_{nl}}r/a(t)]Y_{l}^{m}(\theta,\varphi)$
$\displaystyle\times e^{i{\theta_{nl}^{\rm{osc}}(t)}}{e^{ig(r,t)}},$ (18)

in the Schrödinger equation (1). We encounter two separate equations for
imaginary and real parts. To satisfy the imaginary part it is sufficient that
these two following relations hold:

$\displaystyle\left\\{\begin{array}[]{l}\hbar(a_{0}+b\sin\omega
t)\dfrac{\partial g(r,t)}{\partial r}-bm\omega r\cos\omega t=0\vspace{.2cm}\\\
\hbar(a_{0}+b\sin\omega t)\left(r\dfrac{\partial^{2}g(r,t)}{\partial
r^{2}}+2(l+1)\dfrac{\partial g(r,t)}{\partial r}\right)\\\ \hskip 71.13188pt-
bm\omega r(2l+3)\cos\omega t=0\end{array}\right.$ (22)

The solution of the first differential equation in (22) is

$g(r,t)=\frac{bm\omega r^{2}\cos\omega t}{2\hbar(a_{0}+b\sin\omega t)}+h(t),$
(23)

where $h(t)$ is some function of time. By the way, this result satisfies also
the second equation. But now, for the real part it is necessary that following
differential equation holds:

$\displaystyle\frac{\hbar}{2m}\left(\dfrac{\partial g(r,t)}{\partial
r}\right)^{2}+\dfrac{\partial g(r,t)}{\partial t}=0$ (24)

This equation leads us to take $h(t)=0$. Unfortunately, the resultant $g$ does
not satisfy the Eq. (24) exactly. However, it can be approximately hold if

$\frac{mb\omega^{2}r^{2}\sin\omega t}{2a{(t)}}\approx 0.$ (25)

When we compare the above expression with a particle energy level,
$\hbar^{2}\beta_{nl}^{2}/2ma^{2}$, we find

$\omega\ll{\frac{\hbar\beta_{nl}}{ma_{0}^{2}}\sqrt{\frac{a_{0}}{b}}\sim
10^{-2}\mbox{[m${}^{2}$/S]}a_{0}^{-2}}$ (26)

Note that this condition is weaker than adiabatic theorem requirement
$b\omega\ll\frac{\hbar}{ma}$. Satisfying the requirement (26), the solution of
the Scherödinger equation for an infinite oscillatory spherical well can be
written as

$\displaystyle{\Phi_{nlm}^{\rm{osc}}}(\mathbf{r},t)={\varphi_{nlm}^{\rm{osc}}}(\mathbf{r},t){e^{i{\theta_{nl}^{\rm{osc}}}(t)}}$
$\displaystyle\approx\frac{1}{{j_{l+1}({\beta_{nl}})}}{\left({\frac{2}{{{{({a_{\,0}}+b\sin\omega
t)}^{3}}}}}\right)^{1/2}}{j_{l}}\left({\frac{{{\beta_{nl}}r}}{{{a_{\,0}}+b\sin\omega
t}}}\right)$ $\displaystyle\times{\exp\left[i\frac{bm\omega r^{2}\cos\omega
t}{2\hbar(a_{0}+b\sin\omega
t)}+i\theta_{nl}^{\rm{osc}}(t)\right]}\,Y_{l}^{m}(\theta,\varphi).$ (27)

###  III.2 The geometric phase

To obtain the geometric phase, again we use the Eq. (11). In this case we need
to the following inner product

$\displaystyle\left\langle{{\varphi_{nlm}^{\rm{osc}}}\left|{\frac{{\partial{\varphi_{nlm}^{\rm{osc}}}}}{{\partial
t^{\prime}}}}\right.}\right\rangle=-i\frac{bm\omega^{2}}{{12\hbar\beta_{nl}^{2}}}\left(4l^{2}+4l-3+2\beta_{nl}^{2}\right)$
$\displaystyle\times j_{l-1}^{2}\left(\beta_{nl}\right)(a_{0}+b\sin\omega t).$
(28)

Note that the integral of the above expression is not elementary. We use the
following integral relation

$\displaystyle\int{{x^{4}}{j_{l}}{{\left(x\right)}^{2}}dx}=$ (29)
$\displaystyle\frac{{\pi{x^{2l+5}}{{}_{2}}{F_{3}}\left({l+1,l+\frac{5}{2};l+\frac{3}{2},l+\frac{7}{2},2l+2;-{x^{2}}}\right)}}{{{2^{2l+2}}\left({2l+5}\right)\Gamma{{\left({l+\frac{3}{2}}\right)}^{2}}}}+C$
$\displaystyle=\frac{1}{{12}}\left({-2l-3}\right){x^{2}}\left({4{l^{2}}+2{x^{2}}-1}\right){j_{l-1}}\left(x\right){j_{l}}\left(x\right)$
$\displaystyle\hskip
28.45274pt+\frac{1}{{12}}{x^{3}}\left({4{l^{2}}+8l+2{x^{2}}+3}\right){j_{l}}{\left(x\right)^{2}}$
$\displaystyle\hskip
28.45274pt+\frac{1}{{12}}{x^{3}}\left({4{l^{2}}+4l+2{x^{2}}-3}\right){j_{l-1}}{\left(x\right)^{2}}+C,$
(30)

where ${{}_{2}}{F_{3}}(a;b;x)$ is the generalized hypergeometric function and
$\Gamma(x)$ the gamma function. Then, for the geometric phase we have

$\displaystyle\gamma_{nl}^{\rm{osc}}(t)$ $\displaystyle=$
$\displaystyle\frac{mb\omega}{{12\hbar\beta_{nl}^{2}}}\left(4l^{2}+4l-3+2\beta_{nl}^{2}\right)$
(31) $\displaystyle\qquad{\times
j_{l-1}^{2}}\left(\beta_{nl}\right)\left[b\omega t+a_{0}(1-\cos\omega
t)\right]$ $\displaystyle=$
$\displaystyle{-\frac{\varepsilon_{nl}}{\hbar}t+\zeta^{\prime}_{nl}(t)},$ (32)

where

$\varepsilon_{nl}=\frac{-mb^{2}\omega^{2}}{{12\beta_{nl}^{2}}}j_{l-1}^{2}\left(\beta_{nl}\right)\left(4l^{2}+4l-3+2\beta_{nl}^{2}\right)$
$\zeta^{\prime}_{nl}(t)=\frac{mb\omega
a_{0}}{{12\hbar\beta_{nl}^{2}}}j_{l-1}^{2}\left(\beta_{nl}\right)\left(4l^{2}+4l-3+2\beta_{nl}^{2}\right)(1-\cos\omega
t).$

The second term in Eq. (32) is some periodic function and therefore the first
term is just the Berry phase if we put $t=T=\frac{2\pi}{\omega}$. In Fig. 1 we
have plotted the dynamical, geometrical and total phases separately for some
first quantum numbers. We have illustrated the ratio of geometric and dynamic
phases in Fig. 2.

Figure 1:  The dynamical, geometrical and total phases. Figure 2:  The ratio
of geometric and dynamic phases.

##  IV Acousto-optic example

Consider a metal sphere that fluctuates sinusoidally under an elastic acoustic
wave. Such waves can be easily created with frequencies much more than MHz, so
that for a well radius about 500 nm, the condition of Eq.(26) is satisfied.
For the approximation of the infinite potential well to be acceptable, the
metal must have a high work function. This structure can be considered as an
example of a soluble system. Now, we have

$H_{0}(t)|\phi_{nlm}^{\rm{osc}}\rangle=i\hbar\frac{\partial}{\partial
t}|\phi_{nlm}^{\rm{osc}}\rangle.$ (33)

Here,

$|\phi_{nlm}^{\rm{osc}}\rangle=|\phi^{0}_{nlm}\rangle e^{-i\eta_{nl}},$ (34)

where $\eta_{nl}$ is the total phase (dynamical plus geometrical, see Eq.
(32))

$\displaystyle\eta_{nl}$ $\displaystyle=$
$\displaystyle-\frac{\bar{E}_{nl}}{\hbar}t-\frac{\varepsilon_{nl}}{\hbar}t+\zeta_{nl}+\zeta^{\prime}_{nl}$
(35)

and $|\phi^{0}_{nlm}\rangle$ is the rest of wave function. We can generally
write Eq. (35) in the form

$\displaystyle\eta_{nl}=-\frac{\tilde{E}_{nl}}{\hbar}t+\tilde{\zeta}_{nl},$
(36)

where, $\tilde{E}_{nl}=\bar{E}_{nl}+\varepsilon_{nl}$ that we call modified
time-averaged energy in one period and
$\tilde{\zeta}_{nl}=\zeta_{nl}+\zeta^{\prime}_{nl}$ is some periodic function.
Now, if we expose this system to electromagnetic fields, we can consider the
Hamiltonian of the system as $H(t)=H_{0}+\lambda V(t)$. Considering

$|\psi\rangle=\sum_{nlm}c_{nlm}(t)|\phi_{nlm}^{\rm{osc}}\rangle$ (37)

we can solve the Schrödinger equation to have

$i\hbar\frac{\partial}{\partial
t}c_{nlm}(t)=\sum_{n^{\prime}m^{\prime}l^{\prime}}c_{n^{\prime}l^{\prime}m^{\prime}}(t)\langle\phi_{nlm}^{\rm{osc}}|\lambda
V(t)|\phi_{n^{\prime}l^{\prime}m^{\prime}}^{\rm{osc}}\rangle.$ (38)

Note that, one can easily show, that for a typical metal the penetration depth
of the electric field is much more than the dimensions of the metal sphere. On
the other hand, since $\lambda V=-e\bf E\cdot r$, we simply take the field in
the $z$ direction, so that $\lambda V=-eEr\cos\theta$. Therefore,

$\langle\phi_{nlm}|\lambda
V|\phi_{n^{\prime}l^{\prime}m^{\prime}}\rangle=\frac{a_{0}+b\sin\omega
t}{a_{0}}\langle\phi_{nlm}^{0}|\lambda
V(t)|\phi_{n^{\prime}l^{\prime}m^{\prime}}^{0}\rangle,$ (39)

where $|\phi_{n^{\prime}l^{\prime}m^{\prime}}^{0}\rangle$ is time independent
for constant sphere ($b=0$). Consequently we get

$\displaystyle i\hbar\frac{\partial}{\partial
t}c_{nlm}(t)=\sum_{n^{\prime}m^{\prime}l^{\prime}}c_{n^{\prime}l^{\prime}m^{\prime}}(t)\frac{a_{0}+b\sin\omega
t}{a_{0}}e^{-i(\eta_{n^{\prime}l^{\prime}}-\eta_{nl})}$ $\displaystyle\hskip
85.35826pt\times\langle\phi_{nlm}^{0}|\lambda
V(t)|\phi_{n^{\prime}l^{\prime}m^{\prime}}^{0}\rangle.$ (40)

Now, if at initial point, the electron is in $n_{0}l_{0}m_{0}$ level, then
$c_{nlm}(0)=\delta_{nn_{0}}\delta_{ll_{0}}\delta_{mm_{0}}$ and up to ${\cal
O}(\lambda)$ we have

$\displaystyle i\hbar\frac{\partial}{\partial
t}c_{nlm}(t)=\frac{a_{0}+b\sin\omega
t}{a_{0}}e^{-i(\eta_{n_{0}l_{0}}-\eta_{nl})}$ $\displaystyle\hskip
85.35826pt\times\langle\phi_{nlm}^{0}|\lambda
V(t)|\phi_{n_{0}l_{0}m_{0}}^{0}\rangle.$ (41)

and accordingly,

$\displaystyle c_{nlm}(t)=\frac{1}{i\hbar}\int_{0}^{t}\frac{a_{0}+b\sin\omega
t^{\prime}}{a_{0}}e^{-i\Delta\eta_{nl}}$ $\displaystyle\hskip
71.13188pt\times\langle\phi_{nlm}^{0}|\lambda
V(t^{\prime})|\phi_{n_{0}l_{0}m_{0}}^{0}\rangle dt^{\prime},$ (42)

where $\Delta\eta_{nl}=\eta_{n_{0}l_{0}}-\eta_{nl}$. Since the electric field
is periodic and Hermitian operator, we can write
$V(t)=V_{0}e^{-i\omega_{\rm{ph}}t}+V_{0}^{\dagger}e^{i\omega_{\rm{ph}}t}$,
where $\omega_{\rm{ph}}$ is the frequency of photon. On the other hand, since
$\tilde{\zeta}_{nl}(t)$ is a periodic function with angular frequency
$\omega$, we can write the oscillating terms in Fourier series form as
follows:

$\frac{a_{0}+b\sin\omega
t}{a_{0}}e^{-i\tilde{\zeta_{nl}}(t)}=\sum_{k=-\infty}^{\infty}f^{k}_{nl}e^{-ik\omega
t}.$ (43)

Therefore,

$\displaystyle c_{nlm}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{i\hbar}\sum_{k}f^{k}_{nl}\bigg{[}\langle\phi_{nlm}^{0}|\lambda
V_{0}|\phi_{n_{0}l_{0}m_{0}}^{0}\rangle\int_{0}^{t}e^{-i(\Delta\tilde{E}_{nl}+\hbar\omega_{\rm{ph}}+\hbar
k\omega)t/\hbar}dt^{\prime}$ (44) $\displaystyle\hskip
45.52458pt+\langle\phi_{nlm}^{0}|\lambda
V_{0}^{\dagger}|\phi_{n_{0}l_{0}m_{0}}^{0}\rangle\int_{0}^{t}e^{-i(\Delta\tilde{E}_{nl}-\hbar\omega_{\rm{ph}}+\hbar
k\omega)t/\hbar}dt^{\prime}\bigg{]}$ $\displaystyle=$
$\displaystyle\frac{1}{i\hbar}\sum_{k}f^{k}_{nl}\bigg{\\{}\langle\phi_{nlm}^{0}|\lambda
V_{0}|\phi_{n_{0}l_{0}m_{0}}^{0}\rangle
e^{-i(\Delta\tilde{E}_{nl}+\hbar\omega_{\rm{ph}}+\hbar
k\omega)t/2\hbar}\mbox{sinc}\left[(\Delta\tilde{E}_{nl}+\hbar\omega_{\rm{ph}}+\hbar
k\omega)t/2\hbar\right]t$ $\displaystyle\hskip
14.22636pt+\langle\phi_{nlm}^{0}|\lambda
V^{\dagger}_{0}|\phi_{n_{0}l_{0}m_{0}}^{0}\rangle
e^{-i(\Delta\tilde{E}_{nl}-\hbar\omega_{\rm{ph}}+\hbar
k\omega)t/2\hbar}\mbox{sinc}\left[(\Delta\tilde{E}_{nl}-\hbar\omega_{\rm{ph}}+\hbar
k\omega)t/2\hbar\right]t\bigg{\\}}.$

To have the probability, we should multiply Eq. (44) in its complex conjugate.
At large $t$ we have

$\left[t\ \mbox{sinc}(tx/2)\right]^{2}=\frac{\sin^{2}(tx/2)}{(x/2)^{2}}\cong
2\pi\delta(x)t.$

For the transition rate we have

$\displaystyle\Gamma_{n_{0}l_{0}m_{0}\to nlm}$ $\displaystyle=$
$\displaystyle\frac{2\pi}{\hbar^{2}}\sum_{k}|f^{k}_{nl}|^{2}\Bigg{\\{}|\langle\phi_{nlm}^{0}|\lambda
V_{0}|\phi_{n_{0}l_{0}m_{0}}^{0}\rangle|^{2}\delta\left[\frac{\Delta\tilde{E}_{nl}}{\hbar}+\omega_{ph}+k\omega\right]$
(45) $\displaystyle\hskip 56.9055pt+|\langle\phi_{nlm}^{0}|\lambda
V_{0}^{\dagger}|\phi_{n_{0}l_{0}m_{0}}^{0}\rangle|^{2}\delta\left[\frac{\Delta\tilde{E}_{nl}}{\hbar}-\omega_{ph}+k\omega\right]\Bigg{\\}}.$

This is just the transition rate for dielectric radiation modified with
respect to oscillation. From Eq. (45) we can understand that if the energy gap
between the incident photon $\omega_{\rm{ph}}$, and difference of the modified
energy of two levels $\Delta\tilde{E}_{nl}$, is equal to the integer multiple
of oscillation frequency $\omega$, we will also have absorption or radiation
peaks. Here, the effect of geometric phase is hidden in the
$\Delta\tilde{E}_{nl}$.

##  V Summary

The geometric phase for spherical infinite potential well with moving wall in
two different cases has been considered. First, when the radius of well
increases (or decreases) monotonically. The counterpart of this problem in one
dimension is standard and well-known. We have shown that there is a way to
observe some manifestations of geometric phase, though the parameter space of
this problem is one-dimensional. To solve this problem we have, at first,
found the wave functions which satisfy the Schrödinger equation. Second, when
the radius changes oscillatory. This problem is closer to physical situations.
For this case, we have also solved the Schrödinger equation and found its
solutions approximately. For both cases, we compute the geometrical phases. In
the other section, we have derived the transition rate for a possible real
situation which manifests the effect of geometric phase . We have shown that,
in an acousto-optic case, the absorption or radiation peaks will appear if the
energy gap between the incident photon and the difference of modified averaged
energy of two levels due to Berry phase, is equal to the integer multiple of
oscillation frequency.

## References

  * (1) M V Berry, P. Roy. Soc A-Math. Phy. 392, 45 (1984)
  * (2) A Shapere and F Wilczek, Geometric Phases in Physics (World Scientific, Singapore, 1989)
  * (3) J A Cina, Chem. Phys. Lett. 132, 393 (1986)
  * (4) D Xiao, M C Chang and Q Niu, Rev. Mod. Phys. 82, 1959 (2010)
  * (5) M Todorić, B Klajn, D Jukić and H Buljan Phys. Rev. A 102, 013322 (2020)
  * (6) A Hannonen, H Partanen, J Tervo, T Setälä and A.T. Friberg Phys. Rev. A 99, 053826 (2019)
  * (7) F Ghahari, et al. Science 356, 845 (2017)
  * (8) C G Yale, F J Heremans, B B Zhou, A Auer, G Burkard and D D Awschalom, Nat. photonics 10, 184 (2016)
  * (9) C Dutreix, H González-Herrero, I Brihuega, M I Katsnelson, C Chapelier and V T Renard,Nature 574, 219 (2019)
  * (10) M Baggio, V Niarchos and K Papadodimas. J. high Energy Phys. 04, 1 (2017)
  * (11) P S Hsin, A Kapustin and R Thorngren, Phys. Rev. B 102, 245113 (2020).
  * (12) S W Doescher and M H Rice, Am. J. Phys. 37, 1246 (1969)
  * (13) S V Mousavi, Europhys. Lett. 99, 30002 (2012)