# Functional acoustic metamaterial using shortcut to adiabatic passage in
acoustic waveguide couplers

Shuai Tang1 Jin-Lei Wu1<EMAIL_ADDRESS>Cheng Lü1 Jie Song1 Yongyuan
Jiang1,2,3,4<EMAIL_ADDRESS>1School of Physics, Harbin Institute of
Technology, Harbin 150001, China  2Collaborative Innovation Center of Extreme
Optics, Shanxi University, Taiyuan 030006, China  3Key Lab of Micro-Optics and
Photonic Technology of Heilongjiang Province, Harbin 150001, China  4Key Lab
of Micro-Nano Optoelectronic Information System, Ministry of Industry and
Information Technology, Harbin 150001, China

###### Abstract

Shortcut to adiabatic passage (STAP), initially proposed to accelerate
adiabatic quantum state transfer, has been widely explored and applied in
quantum optics and integrated optics. Here we bring STAP into the field of
acoustics to design compact couplers and functional metamaterial. The space-
varying coupling strengths of acoustic waveguides (WGs) are tailored by means
of dressed states in a three-level system, accounting for the desirable
acoustic energy transfer among three WGs with short length. We show that the
acoustic coupler has one-way feature when loss is introduced into the
intermediate WG. More uniquely, when the propagation of acoustic wave is
designed to mimic a Hadamard transformation, an acoustic metamaterial can be
constructed by arraying several couplers, possessing the beam-splitting
function and unidirectional, broadband performances. Our work bridges STAP and
the acoustic coupler as well as metamaterial, which may have profound impacts
on exploring quantum technologies for promoting advanced acoustic devices.

##  I Introduction

Coherent control of quantum states plays a significant role in extensive
practical applications including quantum information processing Nielsen and
Chuang (2010), high-precision measurement Hänsch (2006), and manipulation of
atoms and molecules Král  _et al._ (2007). To robustly obtain a target state
with high fidelity, adiabatic passage (AP) as well as its variants has been
demonstrated as an effective means through well-tuned radiative interactions
Vitanov  _et al._ (2017, 1999); Boozer (2008); Unanyan  _et al._ (2001); Dridi
_et al._ (2009), while suffering from lengthy evolution owing to the
limitation of adiabatic criterion. Recently, methods for speeding up AP,
called shortcut to adiabatic passage (STAP), provide new approaches for
rendering a system to reach the desired state quickly, accurately, and
robustly Demirplak and Rice (2008); Berry (2009); Chen  _et al._ (2010);
Guéry-Odelin  _et al._ (2019). The development of STAP makes it applicable
widely from quantum optics Zhou  _et al._ (2017); Vepsäläinen  _et al._
(2019); Yan  _et al._ (2019); Chu  _et al._ (2020); Chen  _et al._ (2021a) and
integrated optics Lin  _et al._ (2012); Martínez-Garaot  _et al._ (2014);
Huang  _et al._ (2018); Chung  _et al._ (2019); Taras  _et al._ (2021) to
mechanical engineering González-Resines  _et al._ (2017); Martínez-Cercós  _et
al._ (2020), physical chemistry Vitanov and Drewsen (2019); Wu  _et al._
(2019), and biology Weinreich (2021); Iram  _et al._ (2021). In this work, we
intend to apply STAP for designing compact couplers and functional
metamaterial in the field of acoustics.

An acoustic waveguide (WG) coupler always serves as a functional device to
achieve power transfer Cao  _et al._ (2021); Zhao  _et al._ (2018); Tang  _et
al._ (2020); Wu  _et al._ (2021); Xu (2022), mode conversion Tang  _et al._
(2021a); Zhu  _et al._ (2020); Qian  _et al._ (2020), or phase inversion Zhu
_et al._ (2016); Zhang  _et al._ (2020, 2021) by taking advantage of the
phononic crystals Menchon-Enrich  _et al._ (2014); Chen and Wu (2016); Li  _et
al._ (2011), Helmholtz resonators Li  _et al._ (2016); Liu  _et al._ (2018),
or coiling-up particles Ryoo and Jeon (2018); Li  _et al._ (2013); Esfahlani
_et al._ (2017); Li  _et al._ (2020). Various fascinating phenomena, including
focusing beam Xie and Hou (2021); Zhang  _et al._ (2009); Tang  _et al._
(2021b); Liu  _et al._ (2021); Tang  _et al._ (2022a), splitting beam Li  _et
al._ (2020); Tang  _et al._ (2020), self-bending beam Li  _et al._ (2021);
Tang  _et al._ (2021c) and vortex beam Jiang  _et al._ (2016); Ren  _et al._
(2021); Fan and Zhang (2019), can be observed by constructing acoustic devices
integrated by WG couplers. Nevertheless, complicated structural designs pose
challenges to modeling and analysis inevitably. Besides, the performance of
device is subject to the narrow band and single function, hindering
practicability and industrialization of acoustic coupler. In the past decade,
the combination of acoustic systems with other concepts (such as topological
state Zhang  _et al._ (2017); Lee and Iizuka (2019); Xue  _et al._ (2019); He
_et al._ (2020); Xia  _et al._ (2018); Wang  _et al._ (2021), parity-time
symmetry Yang  _et al._ (2019); Lan  _et al._ (2020); Fleury  _et al._ (2015);
Zhu  _et al._ (2014), quantum adiabaticity Shen  _et al._ (2020a); Zeng  _et
al._ (2021); Shen  _et al._ (2020b, 2019) and Landau-Zener transition Chen
_et al._ (2021b)) has received great interest and has led to surprising
advantages for designing advanced functional acoustic metadevices. As a
promising and pragmatic quantum technology, yet STAP in an acoustic system has
not been reported so far.

In this work, a technique of STAP is introduced to design compact acoustic WG
couplers. Due to the agreement in form between the Schrödinger equation in
quantum mechanics and the coupled-mode equation of classical waves Messiah
(1962), the coupled WGs for transporting energy of classical wave can be
mapped to quantum states with transitions being driven by external fields. The
propagation of acoustic wave along coupled WGs can mimic the quantum evolution
controlled with STAP by meticulously modulating discrete coupling strengths
between two adjacent WGs. The comparison between performances in couplers
designed by STAP and AP verifies the advantage of STAP with a shorter device
length. Furthermore, we show that the coupled-WG acoustic system exhibits a
one-way transmission feature when introducing an appropriate loss in the
intermediate WG. More uniquely, by designing a WG coupler where the
propagation of acoustic wave mimics a Hadamard transformation in quantum
computing, an asymmetric beam-shaping metamaterial with the beam-splitting
function is able to be constructed conveniently by arraying several WG
couplers. An incident plane wave can be splitted with a certain angle of
refraction in a relatively wide space for one side incidence, but can be
hardly transmitted for incidence from the other side. The present work bridges
acoustic metamaterials and quantum STAP technology, providing a new solution
to design advanced acoustic functional device with excellent performances.

Figure 1: (a) The three-level quantum system. The states $|1\rangle$,
$|2\rangle$, and $|3\rangle$ are coupled by pump field $\Omega_{12}(t)$ and
Stokes field $\Omega_{23}(t)$, respectively. (b) Acoustic three-WG coupler in
$y$-$z$ plane. The WGs 1 and 3 are coupled with WG 2 by space-dependent
strengths $C_{12}(x)$ and $C_{23}(x)$, respectively. The purple region
represents the rigid material and the cyan region represents the air
background. (c) The three dimensional schematic diagram of the acoustic three-
WG system.

##  II AP in an acoustic system of three coupled WGs

For a three-level quantum system, as illustrated in Fig. 1(a), ladder-type
transitions $|1\rangle\leftrightarrow|2\rangle$ and
$|2\rangle\leftrightarrow|3\rangle$ are driven resonantly by two external
fields (for example, visible laser fields for transitions of a valence
electron in an atom Dudin and Kuzmich (2012)), respectively, with time-
dependent Rabi frequencies $\Omega_{12}(t)$ and $\Omega_{23}(t)$. The
Hamiltonian in the interaction picture under the rotating-wave approximation
is written as (setting natural unit $\hbar=1$)

$\hat{H}(t)=\left(\begin{array}[]{ccc}0&~{}\Omega_{12}(t)&~{}0\\\
\Omega_{12}(t)&~{}0&~{}\Omega_{23}(t)\\\
0&~{}\Omega_{23}(t)&~{}0\end{array}\right).$ (1)

The two Rabi frequencies can be parametrized by a common amplitude $\Omega(t)$
and a mixed angle $\theta(t)$, as $\Omega_{12}(t)=\Omega(t)\sin\theta(t)$ and
$\Omega_{\mathrm{23}}(t)=-\Omega(t)\cos\theta(t)$, and then instantaneous
eigenstates of $\hat{H}(t)$ can be described by a unitary matrix with base
vectors $\\{|1\rangle,~{}|2\rangle,~{}|3\rangle\\}$

$U_{\mathrm{ad}}=\left(\begin{array}[]{ccc}\sin\theta(t)/\sqrt{2}&~{}1/\sqrt{2}&~{}-\cos\theta(t)/\sqrt{2}\\\
\cos\theta(t)&~{}0&~{}\sin\theta(t)\\\
\sin\theta(t)/\sqrt{2}&~{}-1/\sqrt{2}&~{}-\cos\theta(t)/\sqrt{2}\end{array}\right),$
(2)

where the first and third rows represent two bright states
$\left|\varphi_{\pm}(t)\right\rangle$ with eigenenergies
$E_{\pm}(t)=\pm\Omega(t)$, while the second row is called the dark state
$\left|\varphi_{0}(t)\right\rangle$ with eigenenergy $E_{0}(t)=0$ (the
intermediate state $|2\rangle$ does not get involved in the adiabatic passage
process). In the adiabatic basis, the Hamiltonian (1) becomes

$\hat{H}_{\mathrm{ad}}(t)=\Omega(t)\hat{M}_{z}+\dot{\theta}(t)\hat{M}_{y},$
(3)

for which three spin-1 operators are introduced

$\hat{M}_{x}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{rrr}0&-1&0\\\ -1&0&1\\\
0&1&0\end{array}\right),$ (4a)
$\hat{M}_{y}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{rrr}0&i&0\\\ -i&0&-i\\\
0&i&0\end{array}\right),$ (4b) $\hat{M}_{z}=\left(\begin{array}[]{rrr}1&0&0\\\
0&0&0\\\ 0&0&-1\end{array}\right).$ (4c)

The term $\dot{\theta}(t)\hat{M}_{y}$ corresponding to nonadiabatic couplings
can be ignored in the adiabatic limit
$|\dot{\theta}(t)|\ll\sqrt{2}|\Omega(t)|$ Král  _et al._ (2007); Bergmann  _et
al._ (1998); Vitanov  _et al._ (2017). Then according to the AP technique, a
perfect transfer from state $|1\rangle$ to $|3\rangle$ is capable of realizing
by utilizing the adiabatic medium state
$\left|\varphi_{0}(t)\right\rangle=\cos\theta(t)|1\rangle+\sin\theta(t)|3\rangle$,
where the pulse sequence is adopted for ${\theta}(t_{i})=0$ and
${\theta}(t_{f})=\pi/2$, $t_{i}$ and $t_{f}$ representing the start and end
instants, respectively.

We would like to mimic the quantum system of three discrete levels by
constructing an acoustic WG coupler and to achieve desirable energy transfer
from an input WG to the target WG. The mathematical similarity of the time-
dependent Schrödinger equation in quantum mechanics to the wave equation of
the classical wave makes it possible to apply the AP technique in the
classical-mechanics systems of coupled acoustic WGs Shen  _et al._ (2019,
2020b); Tang  _et al._ (2022b). Propagating properties in WGs of acoustic
waves make typical ultrafast phenomena in time intuitively visible in space,
and also WGs of acoustic waves provide an excellent platform to investigate
coherent dynamics accessible difficultly in quantum systems.

We therefore consider an analog of the three-level model in an acoustic
system. As shown in Fig. 1(b), three WGs 1, 2 and 3 for an acoustic coupler
can be viewed as the discrete states $|1\rangle$, $|2\rangle$, and
$|3\rangle$, respectively. The space-dependent coupling strength $C_{mn}$
$(m=1,2;~{}n=m+1)$ for transmitting sound through slits between WG $m$ and $n$
corresponds to the time-dependent Rabi frequency $\Omega_{mn}$ in the three-
level model. Figure 1(c) shows the three dimensional schematic of the acoustic
coupler. The width of air pipe (cyan regions) is $w_{1}$=10 mm. The thickness
of rigid material (purple regions) is $w_{2}$=8 mm. The space-dependent width
of slit is $d_{mn}(x)$ between WGs $m$ and $n$. The distance between two
spaced slits is $p=4$ mm. The width and height of whole waveguide are
$W_{0}=46$ mm and $h=10$ mm, respectively. By slotting the rigid wall between
two adjacent WGs, the WG 1 and WG 3 are coupled with WG 2 when propagating
acoustic waves. The coupling strength $C_{12}(x)$ and $C_{23}(x)$ can be
modulated along propagation direction ($x$-axis) by adopting different slit
widths with relation $C_{mn}(x)\propto d_{mn}(x)$, thus achieving desired
space dependence.

Based on the coupled-mode theory Messiah (1962), acoustic pressure fields,
$P_{1}$, $P_{2}$, and $P_{3}$ satisfying normalization
$\sum_{j=1}^{3}\left|P_{j}(x)\right|^{2}=1$, along three WGs in the coupler
are described as:

$\displaystyle i\frac{\partial P_{1}(x)}{\partial x}=C_{12}(x)P_{2}(x),$ (5)
$\displaystyle i\frac{\partial P_{2}(x)}{\partial
x}=C_{12}(x)P_{1}(x)+C_{23}(x)P_{3}(x),$ $\displaystyle i\frac{\partial
P_{3}(x)}{\partial x}=C_{23}(x)P_{2}(x),$

which can be rewritten as a similar form of Schrödinger equation in the
spatial dimension

$i\frac{\partial|\varphi(x)\rangle}{\partial x}=\hat{M}(x)|\varphi(x)\rangle,$
(6)

where $\hat{M}(x)$ is defined as the coupling matrix of the three-WG coupler,
corresponding to the Hamiltonian of the three-level system

$\hat{M}(x)=\left(\begin{array}[]{ccc}0&C_{12}(x)&0\\\
C_{12}(x)&0&C_{23}(x)\\\ 0&C_{23}(x)&0\end{array}\right).$ (7)

The acoustic pressure at each site of coupler,
$|\varphi(x)\rangle=P_{1}(x)|1\rangle+P_{2}(x)|2\rangle+P_{3}(x)|3\rangle$,
can be calculated by solving Eq. (5). Thereafter, the AP technique for
discrete quantum states in a three-level system is capable of mapping into the
acoustic three-WG coupler when a spatial adiabatic criterion
$|\partial\theta_{s}(x)/\partial x|\ll\sqrt{2}|C_{s}(x)|$ is satisfied, where
the parameters for the spatial AP Longhi  _et al._ (2007); Menchon-Enrich  _et
al._ (2016) are introduced by defining $C_{12}(x)=C_{s}(x)\sin\theta_{s}(x)$
and $C_{23}(x)=-C_{s}(x)\cos\theta_{s}(x)$, and the propagation of acoustic
wave could mimic evolution of AP in quantum optics Shen  _et al._ (2019). For
the acoustic energy transfer from WG 1 (input port) to WG 3 (output port), one
can render the sound distribution in the three WGs to follow the dark state
$|\Psi_{d}(x)=\cos\theta_{s}(x)|1\rangle+\sin\theta_{s}(x)|3\rangle$ and set
$\theta_{s}(x_{i})=0$ and $\theta_{s}(x_{f})=\pi/2$, where $x_{i}$ and $x_{f}$
label the start and end sites of inter-WG slits, respectively.

##  III STAP in the three-WG coupler

###  III.1 Modified coupling strengths

However, a perfect quantum state transfer needs a very long evolution time to
guarantee the temporal adiabatic criterion
$|\dot{\theta}(t)|\ll\sqrt{2}|\Omega(t)|$, analogous to which an adiabatic
acoustic energy transfer with criterion $|\partial\theta_{s}(x)/\partial
x|\ll\sqrt{2}|C_{s}(x)|$ requires a long spatial variation of device, costing
more space and resources. Therefore, it is necessary to apply a STAP technique
to design inter-WG slit widths for achieving acoustic energy transfer in an
compact coupler.

To shorten the length of coupled WGs, we map the time-dependent STAP technique
into the space-dependent acoustic system to engineer inter-WG coupling
strengths $C_{12}(x)$ and $C_{23}(x)$ as well as the coupling slits by
rendering the sound distribution in the three WGs to follow a STAP pathway
$|\Psi_{s}(x)\rangle$, a dressed state applying for achieving the desired
state transfer instead of the AP pathway (i.e., the dark state
$|\Psi_{d}(x)\rangle$):

$\displaystyle|\Psi_{s}(t)\rangle$ $\displaystyle=$
$\displaystyle\cos\mu_{s}(x)[\cos\theta_{s}(x)|1\rangle+\sin\theta_{s}(x)|3\rangle]$
(8) $\displaystyle+i\sin\mu_{s}(x)|2\rangle,$

which indicates that two ports of each WG has the same sound distribution as
that in the AP situation when the auxiliary parameter $\mu_{s}(x)$ is limited
by $\mu_{s}(x_{i})=\mu_{s}(x_{f})=0$. The STAP technique for a compact WG
coupler requires that there are an extra pair of inter-WG coupling strengths
$\tilde{C}_{12}(x)$ and $\tilde{C}_{23}(x)$ added into $C_{12}(x)$ and
$C_{23}(x)$, respectively, to eliminate the transitions from
$|\Psi_{s}(t)\rangle$ to other orthogonal dressed states, resulting in
modified inter-WG space-dependent coupling strengths
$C^{\prime}_{12}(x)=C_{12}(x)+\tilde{C}_{12}(x)$ and
$C^{\prime}_{23}(x)=C_{23}(x)+\tilde{C}_{23}(x)$. Such a pair of modified
coupling strengths can be obtained by several STAP methods, such as invariant-
based inverse engineering Lewis and Riesenfeld (1969); Chen and Muga (2012),
multiple Schrödinger dynamics Ibáñez  _et al._ (2013); Song  _et al._ (2016),
transitionless evolution engineering Kang  _et al._ (2016a); Chen  _et al._
(2016), three-mode parallel paths Wu  _et al._ (2017, 2019), and
superadiabatic dressed states Baksic  _et al._ (2016); Liu  _et al._ (2017),
which show different mathematical processes but similar underlying physics. In
Appendix A, we give a detailed derivation for obtaining a pair of modified
coupling strengths by using the STAP method of dressed states Baksic  _et al._
(2016) in the context of time-dependent Schrödinger equation. Accordingly, a
pair of modified inter-WG space-dependent coupling strengths
$C^{\prime}_{12}(x)$ and $C^{\prime}_{23}(x)$ for the acoustic coupler can be
obtained

$\displaystyle
C_{12}^{\prime}(x)=-\sin\theta_{s}(x)\cot\mu_{s}(x)\partial_{x}\theta_{s}(x)-\cos\theta_{s}(x)\partial_{x}\mu_{s}(x),$
$\displaystyle
C_{23}^{\prime}(x)=\cos\theta_{s}(x)\cot\mu_{s}(x)\partial_{x}\theta_{s}(x)-\sin\theta_{s}(x)\partial_{x}\mu_{s}(x).$

The related parameters are set as

$\displaystyle\theta_{s}(x)=\frac{\pi
x}{2x_{f}}-\frac{1}{3}\sin\left(\frac{2\pi
x}{x_{f}}\right)+\frac{1}{24}\sin\left(\frac{4\pi x}{x_{f}}\right),$ (10)
$\displaystyle\mu_{s}(x)=\frac{\beta_{m}}{2}\left[1-\cos\left(\frac{2\pi
x}{x_{f}}\right)\right],$

for which we have chosen $x_{0}=0$.

###  III.2 Energy transfer from WG 1 to WG 3

To obtain the relation between the inter-WG coupling strength $C$ and the slit
width $d$, we consider a two-WG acoustic system composed of WGs $A$ and $B$
with state function $|\Psi_{T}(x)\rangle=A(x)|A\rangle+B(x)|B\rangle$ and the
Hamiltonian $H_{T}(x)=C|A\rangle\langle B|+C|B\rangle\langle A|$, where $A(x)$
and $B(x)$ are the acoustic evolution waveforms along propagation direction in
WG $A$ and WG $B$, respectively Shen  _et al._ (2017, 2019). Through solving
the Schrödinger-like coupled-mode equation of the two-WG system
$i\frac{\partial}{\partial_{x}}\left|\Psi_{T}(x)\right\rangle=H_{T}\left|\Psi_{T}(x)\right\rangle$,
one can obtain
$\left|\Psi_{T}(x)\right\rangle=\exp\left(-iH_{T}x\right)\left|\Psi_{T}(0)\right\rangle$
with $\left|\Psi_{T}(0)\right\rangle=A(0)|A\rangle+B(0)|B\rangle$, which
yields

$A(x)=\cos(Cx)A(0)-i\sin(Cx)B(0),$ (11a) $B(x)=-i\sin(Cx)A(0)+\cos(Cx)B(0).$
(11b)

For a given initial condition of $A(0)=A_{0}$ and $B(0)=0$, the acoustic power
flow along the propagation direction in WG $A$ and WG $B$ can be obtained as
$P_{\mathrm{A}}(x)={|A(x)|^{2}}/{\left|A_{0}\right|^{2}}=\cos^{2}(Cx)$ and
$P_{B}(x)=|B(x)|^{2}/\left|A_{0}\right|^{2}=\sin^{2}(Cx)$.

Figure 2: (a) Acoustic intensity distributions in different acoustic couplers
at 8800 Hz, for which the slit widths are $d=0.8$ mm, $d=1.2$ mm and $d=1.6$
mm. Blue arrows represent the input direction and location. (b) The relation
between coupling length and incident frequency.

The propagation wave between WGs $A$ and $B$ will funnel back and forth,
leading to a periodic distribution of intensity field. The coupling strength
can be calculated by $C=\pi/L$ with $L$ being the coupling length defined by
the distance between two adjacent peaks of intensity field. In a three-WG
coupler, the WGs 1 and 3 are coupled with WG 2 by space-dependent strengths
$C_{12}^{\prime}(x)$ and $C_{23}^{\prime}(x)$ in Eqs. (III.1), respectively.
Here, $C_{mn}^{\prime}(x)(m=1,2;n=m+1)$ represents the space-dependent
coupling strength between WGs $m$ and $n$. To implement the space dependence
of coupling strength $C_{mn}^{\prime}(x)$, it is necessary to know how to vary
the slit widths between two WGs quantificationally and accurately to change
the coupling strengths in space. To this end, it is convenient to obtain the
$C$-$d$ correspondence from the results obtained by investigating a two-WG
coupler with space-independent slit widths. According to the relation of
$C=\pi/L$, we learn that the coupling strength in a two-WG system can be
modulated by adopting different values of coupling length. Next, we
numerically calculate the coupling length by employing a set of straight two-
WG couplers with different slit widths (from 0.2 mm to 2 mm) at 8800 Hz, where
the period of slit is set to be $p=4$ mm. Three cases of different slit
widths, $d=0.8$ mm, $d=1.2$ mm and $d=1.6$ mm, are shown in Fig. 2(a), while
the other cases are listed in Table 1. It can be seen that the coupling length
decreases as the slit width increases, and the relation between them can be
fitted by the function $L=\alpha/d$ with $\alpha=275\mathrm{~{}mm}^{2}$.
Therefore, the space-dependent coupling strengths of $C_{mn}^{\prime}(x)$ can
be achieved correspondingly by adopting different slit widths along the
propagating direction of the coupler. In addition, Fig. 2(b) shows the
relation between coupling length and incident frequency for the cases of
$d=0.8$ mm, $d=1.2$ mm and $d=1.6$ mm, from which the change of coupling
length is very slight within 8200-10600 Hz, providing the foundation for
broadband feature of the three-WG coupler.

Table 1: The coupling length ($L$) versus the slit width ($d$) $d$ (mm) |  $L$ (mm) |  $d$ (mm) |  $L$ (mm)
---|---|---|---
0.2 | 1235 | 0.4 | 627
0.6 | 422 | 0.8 | 322
1.0 | 260 | 1.2 | 219
1.4 | 191 | 1.6 | 169
1.8 | 151 | 2.0 | 142

The next step is to select an appropriate value of $\beta_{m}$ to determine
the shapes of coupling strengths, so as to determine the required widths of
coupling slit along propagation direction by mapping the time-dependent
external fields [$\Omega_{12}^{\prime}(t)$ and $\Omega_{23}^{\prime}(t)$] into
space-dependent coupling strength [$C_{12}^{\prime}(x)$ and
$C_{23}^{\prime}(x)$]. As we illustrated in Appendix A, the demand for maximum
coupling strength decreases when the value of $\beta_{m}$ increases, offering
a variety of options for the design of coupling strengths. Meanwhile, a higher
value of $\beta_{m}$ results in a shorter evolution time (or shorter device
length) with a fixed maximum coupling strength
[$C_{\max}=\pi/(142\mathrm{~{}mm})$ for $d_{\max}=2.0$ mm], indicating that
the value of $\beta_{m}$ is inversely related to the device length. Although a
compact acoustic device needs a high value of $\beta_{m}$ to reduce the length
in space, the discrete factor should be taken into account as well because the
discrete precision in shorter device is lower than that in a longer one
inevitably under a fixed slotting period, which may influence the efficiency
of the acoustic energy transfer. Here, we select $\beta_{m}=0.144\pi$ to
ensure sufficient accuracy of discrete slit arrays, from which the device
length can be calculated as $L_{1}=316$ mm. Note that we ignore the slits with
$d<0.2$ mm at both ends of the device to guarantee the feasibility of
fabrication craft in practice, and the actual length of the coupler is
$L_{1}=260$ mm in this case.

Figure 3: The acoustic fields of the coupler designed by STAP for (a) and (c),
and by AP with shortcut evolution for (e) and (g). The acoustic intensity
distributions along coupler designed by STAP for (b) and (d), and by AP with
shortcut evolution for (f) and (h). White arrows represent the input direction
and location. The working frequency is 8800 Hz.

To verify the energy transfer feature of the acoustic coupler designed by
STAP, the finite element method is adopted to simulate the intensity
distribution along each WG at 8800 Hz. The rigid materials are set as hard
boundary conditions, and the ports of each WG are set as planar wave radiation
conditions. As shown in Fig. 3(a), the acoustic wave input from the left side
of WG 1 can transfer to the right side of WG 3 eventually with $L_{1}=260$ mm.
The simulated intensity distributions along three WGs shown in Fig. 3(b) agree
with the analytical results obtained by solving Schrödinger-like equation, Eq.
(6), which confirms the feasibility of our design. It is notable that the
small discrepancies between simulated and analytical results are attributed to
the discrete precision, which can be diminished by lengthening the device. As
shown in Figs. 3(c) and (d), near-perfect energy transfer can be realized as
well in the acoustic coupler designed by STAP with actual length of
$L_{1}=360$ mm ($\beta_{m}=0.107\pi$), and the simulated intensity results fit
further well with the analytical ones. In addition, the maximum acoustic
intensity along WG 2 with $L_{1}=260$ mm is higher than that with $L_{1}=360$
mm, which verifies the conclusion obtained by Eq. (8) that a higher value of
$\beta_{m}$ leads to a large intensity distribution in WG 2.

Figure 4: (a) The relation between the value of $\beta_{m}$ and the intensity
of output wave in acoustic coupler designed by STAP. (b) Device length-
dependent intensity of the output wave in acoustic coupler designed by AP with
shortcut evolution. Figure 5: black(a) The relation between the intensity
distributions at the output ports of three WGs and the value of $\beta_{m}$.
The acoustic wave is incident from the left side of WG 3 at 8800 Hz. (b) The
acoustic intensity field of the lossless coupler with $L_{1}=260$ mm. (c) The
normalized intensity distributions along three WGs for incidence from WG 3.
Acoustic intensity field of the lossy coupler with $L_{1}=260$ mm for
incidences from (d) WG 1 and (f) WG 3. White arrows represent the input
direction and location. The dissipation factor is
$\gamma=0.7\max[C^{\prime}_{12}(x)]$ in WG 2. The normalized intensity
distributions along three WGs in lossy coupler for incidence from (e) WG 1 and
(g) WG 3.

On the other hand, we show the acoustic coupler designed by AP with shortcut
evolution to compare with the performance of the coupler designed by STAP. As
we discussed in Sec. II, a perfect adiabatic transfer from state $|1\rangle$
to $|3\rangle$ is capable of realizing by utilizing the medium state
$\left|\varphi_{0}(t)\right\rangle=\cos\theta(t)|1\rangle+\sin\theta(t)|3\rangle$
with ${\theta}(t_{i})=0$ and ${\theta}(t_{f})=\pi/2$, which can be mapped into
space dimension as well Shen  _et al._ (2019). As shown in Ref. Shen  _et al._
(2019), to achieve the asymptotic condition of $\lim_{x\rightarrow
0}\frac{C_{12}(x)}{C_{23}(x)}=1$ and $\lim_{x\rightarrow
L_{1}}\frac{C_{12}(x)}{C_{23}(x)}=0$, the space-dependent coupling slits
between two adjacent WGs are designed as
$d_{12}(x)=d_{0}e^{-(x-x_{i})^{2}/x_{c}^{2}}$ and
$d_{23}(x)=d_{0}e^{-\left(x-x_{i}-\chi\right)^{2}/x_{c}^{2}}$, where
$x_{c}=L_{1}/4.8$, $x_{i}=2x_{c}$, $\chi=0.86x_{c}$, and $d_{0}=343C_{\max}$.
When $L_{1}=260$ mm, as illustrated in Fig. 3(e), the acoustic wave inputs
from the left side of WG 1 can be transferred partially to the right side of
WG 3. Following the simulated and analytical results of normalized intensity
distribution shown in Fig. 3(f), the transfer efficiency in the coupler
designed by AP with shortcut evolution is merely around 0.6 due to the
imperfect satisfaction of adiabatic condition induced by short device length.
By increasing the length of coupler to $L_{1}=360$ mm, as shown in Figs. 3(g)
and (h), the efficiency of wave transfer is higher than that with $L_{1}=260$
mm but still lower than 0.9, indicating the drawback of the system designed by
AP with shortcut evolution.

Moreover, in Fig. 4(a) we show the analytic relation between the value of
$\beta_{m}$ and the intensity of output wave from acoustic coupler designed by
STAP, where the impact of discrete precision is not taken into account. It can
be observed that the normalized intensity of the output wave is independent of
the parameter $\beta_{m}$, and thus the high efficiency wave transfer from WG
1 to WG 3 can be obtained with arbitrary device length so long as the time-
dependent driving pulse can be fitted well in the spatial dimension. For
acoustic coupler designed by AP with shortcut evolution, however, the device
length-dependent intensity of the output wave shown in Fig. 4(b) indicates
that the transfer efficiency is closely related to the coupler length. Near-
perfect energy transfer from WG 1 to WG 2 can be realized only when
$L_{1}>400$ mm, which is much longer than that by STAP. Therefore, compared
with traditional AP system with shortcut evolution Shen  _et al._ (2019), the
STAP approach provides an effective way to design compact acoustic device.

###  III.3 One-way acoustic energy transfer

As another interesting feature, the acoustic coupler designed by STAP can
exhibit a one-way energy transfer. If acoustic wave is incident from the left
side of WG 3, the corresponding initial condition in STAP system is
$|\Psi(t_{i})\rangle=|3\rangle$, and the state function follows the evolution
of superposition states of $|\Psi_{\pm}(t)\rangle$ with
$a_{+}\left(t_{i}\right)=a_{-}\left(t_{i}\right)=1/\sqrt{2}$. Given by Eq.
(21), the state transfer from $|3\rangle$ to $|2\rangle$ is realized when
meeting the same conditions adopted in the state transfer from $|1\rangle$ to
$|3\rangle$:

$\displaystyle\mu\left(t_{i}\right)=\mu\left(t_{f}\right)=0,\quad\theta\left(t_{i}\right)=0,\quad\theta\left(t_{f}\right)=\pi/2,$
$\displaystyle\beta_{m}=0.144\pi.$ (12)

By mapping the results into spatial dimension, the acoustic wave input from
left side of WG 3 can be transferred to the right side of WG 2.

Indeed, the output port for incidence from WG 3 is closely related to
$\beta_{m}$. As numerical results shown in Fig. 5(a), the acoustic output
intensity of the propagation wave along WG 3 can be ignored while that along
the other two WGs will funnel back and forth with the increase of $\beta_{m}$.
Therefore, either WG 1 or WG 2 can play the role of output port by selecting
appropriate value of $\beta_{m}$. The point marked by P2 on red solid line
corresponds to $\beta_{m}=0.107\pi$ [the device shown in Fig. 3(c)]. The
exchange of acoustic energy can be realized in this case because the high-
intensity acoustic wave will output on WG 1 (WG 3) with incidence from WG 3
(WG 1). Meanwhile, the point marked by P1 on green dashed line corresponds to
$\beta_{m}=0.144\pi$ [the device shown in Fig. 3(a) and Fig. 5(b)], which
illustrates that a high-intensity acoustic wave will output on WG 2 (WG 3)
with incidence from WG 3 (WG 1), providing an inspiration for a one-way
acoustic energy transfer. The simulated and analytical results of acoustic
intensity distribution along three WGs agree well with each other, verifying a
port-selective feature of the coupler, i.e., $|1\rangle\rightarrow|3\rangle$
[Fig. 3(a)] and $|3\rangle\rightarrow|2\rangle$ [Fig. 5(b)] .

Taking the acoustic coupler with $L_{1}=260$ mm as an example, the acoustic
intensity distribution along WG 2 with incidence from WG 1 [Fig. 3(b)] is
quite lower than that from WG 3 [Fig. 5(c)]. Hence, when we introduce a strong
wave dissipation in WG 2 with a constant damping rate $\gamma$, the acoustic
energy attenuation for incidence from WG 3 will be faster than that from WG 1.
In this case, the coupling operator $\hat{M}(x)$ in Eq. (7) can be changed
into a non-Hermitian form

$\hat{M}_{\gamma}(x)=\left(\begin{array}[]{ccc}0&C_{12}(x)&0\\\
C_{12}(x)&-i\gamma&C_{23}(x)\\\ 0&C_{23}(x)&0\end{array}\right).$ (13)

The performance of lossy coupler with $L_{1}=260$ mm is illustrated in Figs.
5(d) and (f), from which we can observe that the normalized intensity of
output wave for incidence from WG 3 (lower than 0.1) is weaker than that from
WG 1 (higher than 0.5). The excellent agreement between simulated and
analytical results of the intensity field shown in Figs. 5(e) and (g) further
confirms that the output wave for incidence from WG 3 is more sensitive to
lossy WG 2 than that from WG 1, and thus the acoustic coupler designed in this
case has a one-way wave propagation feature. Note that the value of
$\beta_{m}$ should be as low as possible to reduce the intensity distribution
along WG 2 for incidence from WG 1 so as to guarantee a high contrast ratio
between WG 1 and WG 3 incidences, which is opposite to the condition for
shortening the device length mentioned before. Therefore, the acoustic coupler
designed in our work is a compromise, i.e., the discrete precision, device
length, and one-way feature need to be taken into account simultaneously.

##  IV Unidirectional acoustic metamaterial

Figure 6: The acoustic pressure field of the coupler with $L_{1}=200$ mm for
incidence from (a) WG 1 and (c) WG 3. White arrows represent the input
direction and location. The normalized intensity distributions along three WGs
for incidence from (c) WG 1 and (d) WG 3. Figure 7: Acoustic pressure fields
of the unidirectional mode converter for (a) left side incidence and (b) right
side incidence at 8800 Hz. Acoustic pressure fields of the unidirectional
metamaterial composed of mode converter array for (c) left side incidence and
(d) right side incidence at 8800 Hz. Figure 8: Acoustic pressure fields of the
mode converter and metamaterial at (a), (d) 8200 Hz, (b), (e) 9400 Hz, (c),
(f) 10600 Hz. The colored arrows represent the incident directions.

In this section, we intend to construct a unidirectional acoustic metamaterial
based on the coupler designed by STAP, through which the incident plane
acoustic wave can be split in a relatively wide space for one side incidence
but hardly transmitted for the other side incidence. A three-WG acoustic
coupler that renders a Hadamard-like transformation,
$|1\rangle\rightarrow\frac{1}{\sqrt{2}}(|1\rangle-|3\rangle)$ and
$|3\rangle\rightarrow\frac{1}{\sqrt{2}}(|1\rangle+|3\rangle)$, is adopted to
serve as a basic unit cell to generate desirable output phases for the design
of metamaterial. Here, we utilize the conditions

$\displaystyle\mu\left(x_{i}\right)=\mu\left(x_{f}\right)=0,\quad\theta\left(x_{i}\right)=0,\quad\theta\left(x_{f}\right)=\pi/4,$
$\displaystyle\beta_{m}=0.107\pi$ (14)

to realize the Hadamard-like wave transformation in acoustic coupler, and set
$\theta_{s}(x)=\frac{\pi x}{4x_{f}}-\frac{1}{3}\sin\left(\frac{\pi
x}{x_{f}}\right)+\frac{1}{24}\sin\left(\frac{2\pi x}{x_{f}}\right)$ and
$\mu_{s}(x)=\frac{\beta_{m}}{2}\left[1-\cos\left(\frac{2\pi
x}{x_{f}}\right)\right]$ for the coupling strengths in Eq. (III.1). The
performance of acoustic coupler is shown in Figs. 6(a) and (c), where acoustic
waves are output from WG 1 and WG 3 simultaneously with identical intensity
for a single port incidence from WG 1 or WG 3. The simulated and analytical
results of intensity distribution along three WGs shown in Figs. 6(b) and (d)
confirm the beam splitting feature of the coupler induced by Hadamard-like
wave transformation. Moreover, the phase responses of the output wave from WG
1 and WG 3 are opposite for incidence from WG 1, while those for incidence
from WG 3 are the same, providing a fascinating ability to construct
unidirectional acoustic device.

Before designing the acoustic metamaterial with beam splitting feature, as
shown in Figs. 7(a) and (b), we propose a unidirectional acoustic mode
converter constructed by three coupled WGs with the left sides of WG 2 and WG
3 being sealed. For left side incidence (LI), the 0-order plane acoustic wave
can only input from the left port of WG 1, and the 1-order wave is able to be
generated owing to the opposite phase responses of the output wave from WG 1
and WG 3 Qian  _et al._ (2020); Tang  _et al._ (2021a). For right side
incidence (RI), however, the reflection field is strong and the transmission
field is relatively weak compared with that for LI, which is attributed to
that most of propagation waves along WG 1 are coupled into WG 3 and reflected
by the hard boundary conditions, showing an excellent performance of
unidirectional transmission feature.

Next, we adopt a coding method Xie  _et al._ (2017) to construct the
unidirectional acoustic metamaterial with beam splitting property, where two
phase responses, 0 and $\pi$, are needed for structural design, which can be
satisfied by output waves from WGs 1 and 3 due to their phase difference of
$\pi$. By arraying the mode converter periodically along $y$-direction, as
shown in Fig. 7 (c), a beam splitting metamaterial is obtained, and the
pattern of transmission field is determined by the period length ($L_{p}$) and
the incident wavelength ($\lambda$). For LI, the diffraction of $\pm$1-order
wave are able to directly take one-pass propagation following the round-trip
process demonstrated in the work Fu  _et al._ (2019), which is preferential to
the 0-order wave, thus resulting in a splitting beam. Given the generalized
Snell’s law Yu  _et al._ (2011), the refraction angle can be calculated by
$\theta_{r}=\arcsin(\lambda/L_{p})$, which makes it necessary to guarantee the
relation of $L_{p}>\lambda$. Otherwise, surface acoustic wave will be produced
along $y$-direction Mei and Wu (2014) and the desirable high-intensity
splitting beam is not able to be obtained in this case. To avoid the
generation of evanescent wave, we set $L_{p}=0.064$ m, which is longer than
wavelength
$\lambda=(343~{}\mathrm{m/s})/(8800~{}\mathrm{Hz})=0.039~{}\mathrm{m}$. As
illustrated in Fig. 7 (c), the normally incident plane wave can radiate into
two directions with $\theta_{r}=\pm 37.5^{\circ}$, showing an excellent
performance of beam splitting. For RI, however, the intensity of transmission
field is weak owing to the strong reflection induced by mode converter array.
As illustrated in Fig. 7 (d), no apparent beam splitting can be observed,
which confirms the unidirectional feature of the acoustic metamaterial
proposed in our work.

Note that there are numerous achievements in the field of acoustic
metamaterial so far. From the perspective of structural design, however,
traditional approaches mainly utilize complicated unit design with resonance
feature to achieve desirable phase responses, which result in unavoidable
complexity of modeling and analysis. By taking advantage of the Hadamard-like
acoustic transformation, our work provides a new solution to construct
acoustic metamaterial with a simple means (slotting the rigid material
periodically). Furthermore, the dependence on resonance response makes the
bandwidth being a limitation for application of the traditional device, while
the acoustic device designed by STAP relies on the coupling effect rather than
resonance effect. Consequently, an excellent robustness to the incident
wavelength can be realized here. The broadband feature of the three-WG coupler
has been illustrated in Appendix B. Therefore, as shown in Figs. 8(a)-(f), the
unidirectional acoustic mode converter and metamaterial can work well at 8200
Hz, 9400 Hz and 10600 Hz, verifying the broadband feature of our design.
Meanwhile, the refraction angle marked in Figs. 8(d)-(f) for LI reduces with
the increase of working frequency following the relation of
$\theta_{r}=\arcsin(\lambda/L_{p})$, which indicates that the incident
wavelength can be regarded as a degree of freedom to modulate the angle of
splitting beam dynamically.

##  V Conclusion

An approach is exhibited in this work for efficient transfer of acoustic wave
energy by connecting shortcut to adiabatic passage (STAP) with the
construction of a compact acoustic coupler, expanding applications of STAP
into the field of acoustics. The acoustic wave propagating along the coupler
mimics the evolution of STAP for a quantum three-level system by modulating
the coupling strength between two adjacent waveguides (WGs), thus resulting in
a desirable behavior of energy exchange with one-way feature. Meanwhile, we
further illustrate that a unidirectional acoustic mode converter is able to be
designed via a three-WG coupler rendering a Hadamard-like wave transformation.
Thereafter, a broadband acoustic metamaterial with simultaneous unidirectional
transmission feature and beam splitting property can be designed by following
the generalized Snell’s law through periodically arraying the acoustic mode
converter, which improves the practicability of the device in engineering
applications including sound isolation, communication and stealth. Our work,
by introducing STAP into constructing acoustic coupler as well as acoustic
metamaterial, paves a new way to guide the design of advanced acoustic
functional device from the behind physical laws.

## Acknowledgements

The authors acknowledge the financial support by the National Natural Science
Foundation of China (Grant No. 62075048) and Natural Science Foundation of
Shandong Province of China (Grant No. ZR2020MF129).

##  Appendix A Time-dependent STAP method of dressed states in a quantum
three-level system

Figure 9: (a) Time-dependent coupling strength with different values of
$\beta_{m}$. The solid lines and dotted lines represent the driving pulses
$\Omega_{12}^{\prime}(t)$ and $\Omega_{23}^{\prime}(t)$, respectively. (b) The
relation between the values of $\beta_{m}$ and
$T$max[$\Omega_{12}^{\prime}(t)$]. Figure 10: Acoustic intensity distributions
along three WGs at 8200-10600 Hz for (a), (b) WG 1 incidence and (d), (e) WG 3
incidence. Energy exchange is realized in (a), (d), and Hadamard
transformation is realized in (b), (e). The transmission spectra of three WGs
within 8200-10600 Hz for (c) WG 1 incidence and (f) WG 3 incidence.

blackTo offset nonadiabatic errors when the adiabatic criterion is not
satisfied well, a correction Hamiltonian $\hat{H}_{\mathrm{c}}(t)$ is needed
to modify Hamiltonian (1) as
$\hat{H}_{\mathrm{mod}}(t)=\hat{H}(t)+\hat{H}_{\mathrm{c}}(t)$, and a unitary
$\hat{V}(t)$ selecting a basis of dressed states is required to guarantee the
agreement between dressed medium states and original medium states [i.e.,
eigenstates of Hamiltonian (1)] at instants $t_{i}$ and $t_{f}$. Here, the
construction of dressed states is parametrized as a rotation of the spin with
Euler angles $\xi(t)$, $\mu(t)$, and $\eta(t)$ Baksic  _et al._ (2016)

$\hat{V}_{g}=\exp\left[i\eta(t)\hat{M}_{z}\right]\exp\left[i\mu(t)\hat{M}_{x}\right]\exp\left[i\xi(t)\hat{M}_{z}\right].$
(15)

The correction Hamiltonian $\hat{H}_{\mathrm{c}}(t)$ is given by a general
form as

$\hat{H}_{c}(t)=\hat{U}_{\mathrm{ad}}^{\dagger}(t)\left[g_{x}(t)\hat{M}_{x}+g_{z}(t)\hat{M}_{z}\right]\hat{U}_{\mathrm{ad}}(t).$
(16)

which leads to a modified Hamiltonian with the same form as Hamiltonian (1)

$\hat{H}_{\mathrm{mod}}=\left(\begin{array}[]{ccc}0&\Omega^{m}_{12}(t)&0\\\
\Omega^{m}_{12}(t)&0&\Omega^{m}_{23}(t)\\\
0&\Omega^{m}_{23}(t)&0\end{array}\right),$ (17)

with
$\Omega^{m}_{12}(t)=\Omega(t)\sin\theta(t)-g_{x}(t)\cos\theta(t)+g_{z}(t)\sin\theta(t)$,
and
$\Omega^{m}_{23}(t)=-\Omega(t)\cos\theta(t)-g_{z}(t)\cos\theta(t)-g_{x}(t)\sin\theta(t)$.
Meanwhile, the Hamiltonian in the new frame defined by $\hat{V}(t)$ takes the
form of

$\displaystyle\hat{H}_{\text{new }}(t)$ $\displaystyle=$
$\displaystyle\hat{V}(t)\hat{H}_{\mathrm{ad}}(t)\hat{V}(t)^{\dagger}+\hat{V}(t)\hat{U}(t)\hat{H}_{c}(t)\hat{U}(t)^{\dagger}\hat{V}(t)^{\dagger}$
(18) $\displaystyle+i\frac{d\hat{V}(t)}{dt}\hat{V}(t)^{\dagger},$

in which the unwanted off-diagonal elements should be canceled to offset
nonadiabatic errors. Thus the parameters of $g_{x}(t)$ and $g_{z}(t)$ can be
determined by:

$g_{x}(t)=\frac{\dot{\mu}(t)}{\cos\xi(t)}-\dot{\theta}(t)\tan\xi(t),$ (19a)
$g_{z}(t)=-\Omega(t)+\dot{\xi}(t)+\frac{\dot{\mu}(t)\sin\xi(t)-\dot{\theta}(t)}{\tan\mu(t)\cos\xi(t)}.$
(19b)

For simplicity, $\xi(t)=0$ is adopted in this work. In this situation,
Hamiltonian Eq. (18) is diagonal in the representation of the following three
dressed states

$\displaystyle|\Psi_{0}(t)\rangle$ $\displaystyle=$
$\displaystyle\cos\mu(t)[\cos\theta(t)|1\rangle+\sin\theta(t)|3\rangle]+i\sin\mu(t)|2\rangle,$
$\displaystyle|\Psi_{\pm}(t)\rangle$ $\displaystyle=$
$\displaystyle\left\\{[\sin\theta(t)\mp
i\sin\mu(t)\cos\theta(t)]|1\rangle\mp\cos\mu(t)|2\rangle\right.$ (20)
$\displaystyle\left.-[\cos\theta(t)\pm
i\sin\mu(t)\sin\theta(t)]|3\rangle\right\\}/{\sqrt{2}}.$

blackThe state function of the system can be mapped into the dressed-state
space by

$\Psi(t)=\sum_{n=\pm,0}\tilde{a}_{n}(t)\Psi_{n}(t),$ (21)

where
$\tilde{a}_{n}(t)=a_{n}(t_{i})\exp\left[-i\int_{t_{i}}^{t}\varepsilon_{n}\left(t^{\prime}\right)dt^{\prime}\right]$
with $a_{n}(t_{i})=\langle\Psi_{n}(t_{i})|\Psi(t_{i})\rangle$ and
$\varepsilon_{n}(t)=\langle\Psi_{n}(t)|\hat{H}_{\mathrm{mod}}(t)|\Psi_{n}(t)\rangle$.
For a given condition of $|a_{0}\left(t_{i}\right)|=1$ and
$a_{+}\left(t_{i}\right)=a_{-}\left(t_{i}\right)=0$, the state function
follows the evolution of $|\Psi_{0}(t)\rangle$. Perfect state transfer from
$|1\rangle$ to $|3\rangle$ can be achieved when $\mu(t)$ and $\theta(t)$ meet
the condition of

$\mu\left(t_{i}\right)=\mu\left(t_{f}\right)=0,\quad\theta\left(t_{i}\right)=0,\quad\theta\left(t_{f}\right)=\pi/2.$
(22)

In this case, the adiabatic criterion can be avoided and the same result of
the final state as $\left|\psi_{0}\left(t_{f}\right)\right\rangle$ can be
achieved by designing $\mu(t)$ and $\theta(t)$ meticulously, resulting in a
shorter time of perfect evolution.

blackAccording to the analogy between two Hamiltonians (1) and (17), it can be
seen that the driving pulses in three-level system for STAP are changed to

$\displaystyle\Omega_{12}^{\prime}(t)=-\dot{\theta}(t)\sin\theta(t)\cot\mu(t)-\dot{\mu}(t)\cos\theta(t),$
$\displaystyle\Omega_{23}^{\prime}(t)=\dot{\theta}(t)\cos\theta(t)\cot\mu(t)-\dot{\mu}(t)\sin\theta(t).$
(23)

To meet the conditions in Eq. (22), we set the start and end instants
$t_{i}=0$, $t_{f}=T$ and pulse parameters Kang  _et al._ (2016b); Wu  _et al._
(2017):

$\displaystyle\theta(t)=\frac{\pi t}{2T}-\frac{1}{3}\sin\left(\frac{2\pi
t}{T}\right)+\frac{1}{24}\sin\left(\frac{4\pi t}{T}\right),$ (24)
$\displaystyle\mu(t)=\frac{\beta_{m}}{2}\left[1-\cos\left(\frac{2\pi
t}{T}\right)\right],$

where $\beta_{m}$ is a parameter related to evolutive populations of
$|1\rangle$. Two driving pulses $\Omega_{12}^{\prime}(t)$ and
$\Omega_{23}^{\prime}(t)$ for STAP are related to the value of $\beta_{m}$.
Figure 9(a) shows the time-dependent coupling strength with different values
of $\beta_{m}$, from which we can observe that the demand for maximum coupling
strength decreases when the value of $\beta_{m}$ increases, offering a variety
of options for the design of driving pulses. Meanwhile, from Fig. 9(b) we
learn that a higher value of $\beta_{m}$ results in a shorter evolution time
with a fixed maximum coupling strength, making it necessary to control the
value of $\beta_{m}$ as high as possible for designing compact acoustic
device. Nevertheless, as another constraint to determine the value of
$\beta_{m}$, as described in $|\Psi_{0}(0)\rangle$, a high value of
$\beta_{m}$ inevitably leads to a large population along mediate state
$|2\rangle$. Although the lossy caused by population along intermediate state
can be neglected in WG system compared with that in atomic system designed by
quantum adiabatic technology, it would have influence on the one-way feature
of the acoustic WG system (see Section III.3). Therefore, a compromise
consideration is needed to determine the value of $\beta_{m}$. For the initial
time $t_{i}=0$, Eq. (24) could agree well with the condition (22), providing a
practical approach to STAP, so as to design compact acoustic device.

##  Appendix B Broadband performance of acoustic three-WG couplers

In our work, we construct two types of three-WG coupler based on STAP, through
which energy exchange and Hadamard transformation can be realized
respectively. For energy exchange and Hadamard transformation, the acoustic
intensity distributions along three WGs at 8200-10600 Hz with WG 1 incidence
are shown in Figs. 10(a) and (b), respectively, while those with WG 3
incidence are shown in Figs. 10(d) and (e), respectively. The propagation
process within the operating band from 8200 Hz to 10600 Hz can almost remain
unchanged, verifying that the proposed couplers are robust to the incident
frequency. Meanwhile, the transmission spectra of three WGs within 8200
Hz-10600 Hz for WG 1 incidence and WG 3 incidence are shown in Figs. 10(c) and
(f), respectively. The normalized output intensities for three WGs (1, 2, 3)
in the cases of energy exchange and Hadamard transformation are about (0.03,
0.05, 0.92) and (0.49, 0.01, 0.50) in average for WG 1 incidence [Fig. 10(c)],
while those are about (0.11, 0.86, 0.03) and (0.49, 0.02,0.49) in average for
WG 3 incidence [Fig. 10(f)]. The output intensities of three WGs meet the
results of energy exchange and Hadamard transformation within 8200-10600 Hz,
which further confirm the broadband feature of the coupler. Therefore, as
illustrated in Fig. 8(d)-(f), the unidirectional acoustic beam splitter
composed of several couplers can work well in a continuous band within
8200-10600 Hz.

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