# Accelerated adiabatic passage in cavity magnomechanics

Shi-fan Qi  Department of Physics, Zhejiang University, Hangzhou 310027,
Zhejiang, China  Jun Jing<EMAIL_ADDRESS>Department of Physics, Zhejiang
University, Hangzhou 310027, Zhejiang, China

###### Abstract

Cavity magnomechanics provides a readily-controllable hybrid system, that
consisted of cavity mode, magnon mode, and phonon mode, for quantum state
manipulation. To implement a fast-and-robust state transfer between the hybrid
photon-magnon mode and the phonon mode, we propose two accelerated adiabatic-
passage protocols individually based on the counterdiabatic Hamiltonian for
transitionless quantum driving and the Levis-Riesenfeld invariant for inverse
engineering. Both the counterdiabatic Hamiltonian and the Levis-Riesenfeld
invariant generally apply to the continuous-variable systems with arbitrary
target states. It is interesting to find that our counterdiabatic Hamiltonian
can be constructed in terms of the creation and annihilation operators rather
than the system-eigenstates and their time-derivatives. Our protocol can be
optimized with respect to the stability against the systematic errors of
coupling strength and frequency detuning. It contributes to a quantum memory
for photonic and magnonic quantum information. We also discuss the effects
from dissipation and the counter-rotating interactions.

##  I Introduction

Hybrid cavity-magnon systems Rameshti  _et al._ (2021); Lachance-Quirion  _et
al._ (2019); Li  _et al._ (2020) base on the expedient control of coherent
magnon-photon coupling have recently attracted intensive attention. They found
new avenues for quantum computing Ladd  _et al._ (2010), quantum communication
Reiserer and Rempe (2015), and quantum sensing Degen  _et al._ (2017). Analog
to the cavity quantum electrodynamics (QED) Blais  _et al._ (2021) and
optomechanics Aspelmeyer  _et al._ (2014), cavity magnomechanics Zhang  _et
al._ (2016) develops rapidly to become a mesoscopic platform for quantum
information processing in both theoretical Soykal and Flatté (2010a, b); Liu
_et al._ (2019); Li  _et al._ (2021) and experimental aspects Tabuchi  _et
al._ (2014); Zhang  _et al._ (2014); Harder  _et al._ (2016); Hyde  _et al._
(2017); Wang  _et al._ (2018a); Shen  _et al._ (2021); Tabuchi  _et al._
(2015); Lachance-Quirion  _et al._ (2020). Active investigations about magnon-
based quantum information transfer focus on the coupling between photons and
magnons and that between magnons and phonons in the ferrimagnetic material.
Typical applications of these couplings include the hybrid entanglement and
steering Li  _et al._ (2018); Li and Zhu (2019); Yu  _et al._ (2020); Chen
_et al._ (2021), the photon-phonon interface Qi and Jing (2021); Sarma  _et
al._ (2021), and the magnomechanical phonon laser Vashahri-Ghamsari  _et al._
(2021).

In particular, a cavity magnomechanical system Zhang  _et al._ (2016) consists
of a single-crystal yttrium iron garnet (YIG) sphere placed inside a microwave
cavity, where the magnon modes formed by the excitations the collective
angular momentum of the spins in such a magnetic-material sphere are coupled
with the deformation phonon modes via a magnetostrictive force, and also with
the electromagnetic cavity modes via a magnetic dipole interaction. The phonon
in the YIG sphere decays with a rate about $100$ Hz Zhang  _et al._ (2016); Li
_et al._ (2018), much smaller than its own frequency and those of the magnon
and the photon. That enables the storage-and-transfer of the microwave
photonic and magnonic states as long-lasting modes, constituting a key step
for the future quantum communication networks Chien  _et al._ (2001). Inspired
by the light-matter interface implemented within the cavity QED Cirac  _et
al._ (1997); Kimble (2008), the optomechanical systems Fiore  _et al._ (2011);
Genes  _et al._ (2008); Kumar and Bhattacharya (2019), and the optical
waveguides Ho and Tseng (2015), the stimulated Raman adiabatic passage between
photon and phonon Sarma  _et al._ (2021) and the magnon-assisted photon-phonon
conversion Qi and Jing (2021) have been proposed in the cavity magnomechanical
systems. These protocols however demand a long evolution time and then the
quantum system is prone to decoherence. Thus the shortcut-to-adiabatic (STA)
protocols are desired to realize a quick-and-faithful state transfer in the
cavity magnomechanical systems.

Various STA approaches Guéry-Odelin  _et al._ (2019); Torrontegui  _et al._
(2013), including the transitionless quantum driving (TQD) based on the
counterdiabatic Hamiltonian Berry (2009); Demirplak and Rice (2005); del Campo
(2013) and the inverse engineerings based on Lewis-Riesenfeld (LR) invariant
Lewis and Riesenfeld (1969), time-rescaling Bernardo (2020), or noise-induced
adiabaticity Wang  _et al._ (2018b); Jing and Wu (2015); Jing  _et al._
(2014), have been applied to several prototypes, such as, two- and three-level
atomic system Chen  _et al._ (2010); Wu  _et al._ (2021); Vepsäläinen  _et
al._ (2019), quantum harmonic oscillator Tobalina  _et al._ (2020),
optomechanical system Chen  _et al._ (2018); Zhang  _et al._ (2019a, b), and
coupled-waveguide device Ho and Tseng (2015). Comparing to the existing STA
methods, which are limited to the discrete systems or the continuous-variable
systems in a subspace with a fixed excitation number Wang  _et al._ (2021);
Zhang  _et al._ (2019a, b), our protocol in this work is independent of the
target state and can be applied to any coupled harmonic oscillators. The
stability of our STA protocol for state transfer will be examined with respect
to its robustness against the systematic errors Ruschhaupt  _et al._ (2012),
that result mainly from the intensity fluctuations or inaccurate realization
of the time-dependent driving laser. Then we can optimize the STA protocols in
the cavity magnomechanical system Ruschhaupt  _et al._ (2012); Wang  _et al._
(2021); Song  _et al._ (2021); Yu  _et al._ (2018).

The rest part of this work is structured as following. In Sec. II, we
introduce a hybrid quantum model for cavity magnomechanics and then provide
the effective Hamiltonian describing the interaction between the hybrid
photon-magnon mode and the phonon mode. The details of the derivation can be
found in Appendix A. Based on the effective Hamiltonian, we then propose two
STA protocols in Sec. III for fast quantum state transfer in the cavity
magnomechanical systems. In Sec. III.1, we construct the counterdiabatic
Hamiltonian for TQD in terms of the creation and annihilation operators of the
bosonic modes rather than the eigenstates of the effective Hamiltonian and
their time-derivatives. In Sec. III.2, we derive a general Levis-Riesenfeld
invariant for a two-coupled-bosonic-mode system, which applies to arbitrary
target state. In Sec. IV, we provide a general formalism for the systematic
error of the effective Hamiltonian. The error sensitivity or robustness of
various protocols, including the $\pi$-pulse, TQD and invariant-based STA, are
analyzed and optimized in Secs. IV.1, IV.2, and IV.3, respectively. In Sec. V,
we discuss the effects of decoherence and counter-rotating interaction in the
transport protocols by master equation and numerical simulation, respectively.
The whole work is summarized in Sec. VI.

##  II Model

Figure 1: Schematic diagram of a YIG sphere placed in a microwave cavity
nearby the maximum magnetic field of the cavity mode. The uniform bias
magnetic field exciting the Kittel mode in the YIG and establishing the
magnon-photon coupling is aligned along the $z$-axis. The photon mode is
driven by a microwave source along the $x$-axis (with a Rabi frequency
$\epsilon_{p}$). The insects shows how the dynamic magnetization of magnon
(vertical black arrows) causes the deformation (compression along the $y$
direction) of the YIG sphere (and vice versa), which rotates at the magnon
frequency.

Consider a hybrid system in cavity-magnonic setup shown in Fig. 1, where a YIG
sphere is inserted into a microwave cavity. The system is constituted by
microwave-mode photons, magnons and mechancical-mode phonons, that has been
experimentally realized in the dispersive regime. The magnons are coupled to
photons via the Zeeman interaction and simultaneously coupled to phonons by
the magnetization interaction. In particular, the temporally-varying
magnetization induced by the magnon excitation inside the YIG sphere leads to
the deformation of its geometrical structure, which forms the vibrational
modes (phonons) of the sphere. The Hamiltonian of the full system is given by
($\hbar=1$) Zhang  _et al._ (2016)

$\displaystyle H_{0}$
$\displaystyle=\omega_{a}a_{1}^{\dagger}a_{1}+\omega_{m}m_{1}^{\dagger}m_{1}+\omega_{b}b^{\dagger}b+g_{mb}m_{1}^{\dagger}m_{1}(b+b^{\dagger})$
(1)
$\displaystyle+g_{ma}(a_{1}m_{1}^{\dagger}+a_{1}^{\dagger}m_{1})+i(\epsilon_{p}a_{1}^{\dagger}e^{-i\omega_{p}t}-\epsilon^{*}_{p}a_{1}e^{i\omega_{p}t}),$

where $a_{1}(a_{1}^{\dagger})$, $m_{1}(m_{1}^{\dagger})$ and $b(b^{\dagger})$
are the annihilation (creation) operators of the microwave cavity mode, the
magnon of the ground Kittel mode, and the mechanical mode with transition
frequencies $\omega_{a}$, $\omega_{m}$ and $\omega_{b}$, respectively. The
frequency of the magnon mode $\omega_{m}=\gamma h$, where $\gamma$ is the
gyromagnetic ratio and $h$ is the external bias magnetic field. Thus the
frequency $\omega_{m}$ can be readily tuned by the external magnetic field.
$g_{ma}$ and $g_{mb}$ are respectively the single-excitation coupling strength
of the photon-magnon interaction and magnon-phonon interaction. The last term
in $H_{0}$ describes the external driving of the photon mode, where
$\omega_{p}$ is the frequency of the driven and $\epsilon_{p}$ is the Rabi
frequency of driving field.

Following the standard linearization approach Aspelmeyer  _et al._ (2014) and
under the proper driving condition, we can extract an effective Hamiltonian
describing the interaction between a hybridized photon-magnon mode and the
phonon mode (the derivation detail can be found in Appendix A):

$H=(\Delta-\omega_{b})m^{\dagger}m+(gm^{\dagger}b+g^{*}mb^{\dagger}).$ (2)

Here $m=\sin\phi a_{1}-\cos\phi m_{1}$ is the hybridized normal mode with
$\tan(2\phi)\equiv 2g_{ma}/{(\omega_{a}-\omega_{m})}$. $g$ is the driving-
enhanced coupling strength between the hybrid mode-$m$ and the mechanical
mode-$b$

$g=g_{mb}m_{s}\cos^{2}\phi-g_{mb}a_{s}\sin\phi\cos\phi,$ (3)

where

$m_{s}=\frac{\epsilon_{p}\sin\phi}{i\Delta+\kappa_{m}},\quad
a_{s}=\frac{\epsilon_{p}\cos\phi}{i\Delta^{\prime}+\kappa_{a}},$ (4)

with the effective frequencies of the hybridized modes

$\displaystyle\Delta=\frac{\omega_{a}+\omega_{m}}{2}-\omega_{p}-\sqrt{\left(\frac{\omega_{a}-\omega_{m}}{2}\right)^{2}+g^{2}_{ma}},$
(5)
$\displaystyle\Delta^{\prime}=\frac{\omega_{a}+\omega_{m}}{2}-\omega_{p}+\sqrt{\left(\frac{\omega_{a}-\omega_{m}}{2}\right)^{2}+g^{2}_{ma}},$

and the decay rates $\kappa_{m}$ and $\kappa_{a}$.

The coupling $g$ can then be varied by tuning the driving parameters
$\epsilon_{p}$ and $\omega_{p}$. The hybrid-mode frequency can be altered by
adjusting the strength of the external magnetic bias field Li  _et al._
(2018). Thus both terms in Eq. (2) can be modulated with time. The following
state transfer between the hybrid mode and the mechanical mode could be
formally started from the time-dependent Hamiltonian

$H(t)=\Delta(t)m^{\dagger}m+g(t)m^{\dagger}b+g^{*}(t)b^{\dagger}m.$ (6)

where $\Delta(t)\equiv\Delta-\omega_{b}$. In the framework of various STA
state-transfer protocols, it is instructive to transform the system
Hamiltonian $H(t)$ into the rotating frame with respect to
$U=\exp[i\int_{0}^{t}ds\Delta(s)/2(m^{\dagger}m-b^{\dagger}b)]$,

$H(t)=\frac{\Delta(t)}{2}\left(m^{\dagger}m-b^{\dagger}b\right)+g(t)m^{\dagger}b+g^{*}(t)b^{\dagger}m.$
(7)

##  III The state transfer protocols

A straightforward protocol to achieve the state transfer is using a $\pi$
pulse. In this case, one can hold the driving frequency to be resonant with
the hybrid mode $m$, i.e., $\Delta(t)=0$ for all of time in Hamiltonian (7).
Then a initial state
$|\psi(0)\rangle=(\sum_{k}C_{k}|k\rangle_{m})|0\rangle_{b}$ with arbitrary
normalized coefficients $C_{k}$’s could be converted to

$|\psi(T)\rangle=|0\rangle_{m}\left(\sum_{k}C_{k}e^{-i\frac{k\pi}{2}}|k\rangle_{b}\right)$
(8)

after a desired period $T$ as long as the coupling strength satisfies
$\int^{T}_{0}dt|g(t)|=\pi/2$. For example, for a flat $\pi$ pulse we can set
$g(t)=\pi/(2T)$. Note the final state for the mechanical mode in
$|\psi(T)\rangle$ is not exactly the same one for the hybrid mode $m$ in the
initial state $|\psi(0)\rangle$, regarding the dynamical phase $k\pi/2$.
However, the phase difference between the final and the initial states could
be compensated by the dynamical local phase $e^{-ik\omega_{b}\tau}$ after a
free evolution time $\tau$ by $H_{b}=\omega_{b}b^{\dagger}b$. The phase
difference vanished at the moments satisfying $\omega_{b}\tau+\pi/2=2n\pi$
with integer $n$. We therefore do not distinguish the distinction between the
states $\sum_{k}C_{k}e^{-i\frac{k\pi}{2}}|k\rangle$ and
$\sum_{k}C_{k}|k\rangle$ when calculating the state-transfer fidelity. The
state-transfer fidelity or efficiency is thus measured by the target-state
population

$P=\sum_{C_{k}\neq 0}|\langle 0k|\psi(t)\rangle|^{2},$ (9)

where $|\psi(t)\rangle$ is the dynamical state determined by the initial state
$|\psi(0)\rangle$ and the Hamiltonian (7). Note when the target state is a
Fock state $|N\rangle$, $P$ becomes the conventional fidelity.

The $\pi$-pulse protocol is straightforward but sensitive to both decoherence
and the systematic errors Ruschhaupt  _et al._ (2012) caused by the long-time
evolution and the fluctuations of the Hamiltonian, respectively. The state-
transfer protocol can by improved by the accelerated adiabatic passage or
shortcuts to adiabaticity that is robust to both decoherence and systematic
error. In the two subsequent subsections, we will introduce two STA protocols
about TQD and LR-invariant, to our hybrid magnomechanical model. we apply our
results to two extreme examples of the quantum state of mode-$m$ and mode-$b$.
One is based on the number state or Fock state and the other is on the cat
state as both superposed coherent state and Fock state.

###  III.1 Transitionless quantum driving for continuous-variable system

The TQD approach was proposed in the first decade of this century, depending
on the full knowledge about the instantaneous eigenstructure of the original
Hamiltonian. Conventionally, if the original time-dependent Hamiltonian $H(t)$
could be formally expressed in the spectral representation as
$H(t)=\sum_{n}E_{n}(t)|n(t)\rangle\langle n(t)|$, then assisted by an
ancillary Hamiltonian, or called the counterdiabatic (CD) Hamiltonian Berry
(2009); Guéry-Odelin  _et al._ (2019)

$\displaystyle H_{\rm CD}(t)=i\sum_{n}\left[1-|n(t)\rangle\langle
n(t)|\right]|\dot{n}(t)\rangle\langle n(t)|,$ (10)

the system could keep track of the instantaneous eigenstates of $H(t)$ in a
much faster speed. This approach is also believed to highly robust against the
control-parameter variations Guéry-Odelin  _et al._ (2019). One can understand
that it applies usually to the discrete systems, since it is explicitly
represented by the eigenstates and their time-derivatives.

In this work, we derive the CD term using operators in the continuous-variable
systems. Using the Bogolyubov transformation Tikochinsky (1978), the
Hamiltonian in Eq. (7) is rewritten as

$H(t)=\omega_{A}A^{\dagger}A+\omega_{B}B^{\dagger}B,$ (11)

where $\omega_{A,B}=\pm\sqrt{\Delta^{2}+4g^{2}}/2$ and

$\displaystyle A\equiv\cos\theta m+\sin\theta b,$ (12) $\displaystyle
B\equiv\sin\theta m-\cos\theta b$

with $\tan(2\theta)=2g(t)/\Delta(t)$. To simplify the formation of the TQD
protocol, the coupling strength $g(t)$ in this protocol is set to be real,
i.e., $g(t)=g^{*}(t)$.

In the subspace with a fixed and arbitrary excitation number $N$, the CD
Hamiltonian for the system Hamiltonian in Eq. (11) can be written as $H_{\rm
CD}=i\sum_{n=0}^{N}|\dot{\epsilon}_{n}\rangle\langle\epsilon_{n}|$, where the
orthonormal eigenstates read

$|\epsilon_{N-n}\rangle=\frac{1}{\sqrt{(N-n)!n!}}(A^{\dagger})^{N-n}(B^{\dagger})^{n}|0\rangle,$
(13)

with $|0\rangle$ the vacuum state for both modes. Due to the fact that

$\displaystyle\dot{A}=-\dot{\theta}\sin\theta m+\dot{\theta}\cos\theta
b=-\dot{\theta}B,$ (14) $\displaystyle\dot{B}=\dot{\theta}\cos\theta
m+\dot{\theta}\sin\theta b=\dot{\theta}A,$

we have

$\displaystyle|\dot{\epsilon}_{N-n}\rangle$
$\displaystyle=-\dot{\theta}\sqrt{(n+1)(N-n)}|\epsilon_{N-n-1}\rangle$ (15)
$\displaystyle+\dot{\theta}\sqrt{n(N-n+1)}|\epsilon_{N-n+1}\rangle.$

Then the CD Hamiltonian by Eq. (10) becomes

$\displaystyle H_{\rm CD}$
$\displaystyle=i\sum_{n=0}^{N}|\dot{\epsilon}_{n}\rangle\langle\epsilon_{n}|$
(16)
$\displaystyle=-\dot{\theta}\sum_{n=0}^{N-1}\sqrt{(N-n)(n+1)}|\epsilon_{N-n-1}\rangle\langle\epsilon_{N-n}|$
$\displaystyle+\dot{\theta}\sum_{n=1}^{N}\sqrt{n(N-n+1)}|\epsilon_{N-n+1}\rangle\langle\epsilon_{N-n}|.$

According to the definition in Eq. (13), the first term in Eq. (16) expands

$\displaystyle\sum_{n=0}^{N-1}\sqrt{(N-n)(n+1)}|\epsilon_{N-n-1}\rangle\langle\epsilon_{N-n}|$
(17) $\displaystyle=$
$\displaystyle\sum_{n=0}^{N-1}\sqrt{(N-n)(n+1)}\frac{(A^{\dagger})^{N-n-1}(B^{\dagger})^{n+1}A^{N-n}B^{n}}{(N-n-1)!n!\sqrt{(N-n)(n+1)}}$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{N-1}\frac{(A^{\dagger}A)^{N-n-1}(B^{\dagger}B)^{n}B^{\dagger}A}{(N-n-1)!n!},$
$\displaystyle=$
$\displaystyle\frac{(A^{\dagger}A+B^{\dagger}B)^{N-1}}{(N-1)!}B^{\dagger}A=IB^{\dagger}A=B^{\dagger}A.$

where $I$ is the identify operator in the subspace with $N-1$ excitations. In
the last line, we have applied the binomial theorem
$(x+y)^{N}=\sum^{N}_{n=0}C^{n}_{N}x^{n}y^{N-n},C_{N}^{n}=N!/[n!(N-n)!]$.
Similarly, the second term in Eq. (16) turns out to be

$\sum_{n=1}^{N}\sqrt{n(N-n+1)}|\epsilon_{N-n+1}\rangle\langle\epsilon_{N-n}|=A^{\dagger}B.$
(18)

Note $N$ is arbitrary, then in the whole Hilbert space, the CD Hamiltonian
could be expressed as

$H_{\rm
CD}=i\dot{\theta}(A^{\dagger}B-B^{\dagger}A)=i\dot{\theta}(b^{\dagger}m-m^{\dagger}b).$
(19)

And regarding the original Hamiltonian (7), then the total Hamiltonian for
transitionless quantum driving reads

$\displaystyle H_{\rm tot}=H(t)+H_{\rm CD}$ (20)
$\displaystyle=\frac{\Delta(t)}{2}(m^{\dagger}m-b^{\dagger}b)+[g(t)-i\dot{\theta}]m^{\dagger}b+[g(t)+i\dot{\theta})]mb^{\dagger},$

where the time-dependence and boundary-condition of

$\dot{\theta}=\frac{\dot{g}\Delta-\dot{\Delta}g}{\Delta^{2}+4g^{2}}$ (21)

determines the speed of the accelerated adiabatic passage.

By virtue of the definitions in Eqs. (12) and (13), the adiabatic path from
$|k\rangle_{m}|0\rangle_{b}$ to $|0\rangle_{m}|k\rangle_{b}$ under the total
Hamiltonian (20) is constructed by

$|k(t)\rangle=|\epsilon_{k}(t)\rangle=\frac{1}{\sqrt{k!}}(A^{\dagger})^{k}|0\rangle=|\epsilon_{k}\rangle$
(22)

under the boundary condition $\theta(t=0)=0$ and $\theta(t=T)=\pi/2$. And the
system wavefunction can be written as

$|\psi_{k}(t)\rangle=e^{-i\chi_{k}(t)}|k(t)\rangle,$ (23)

where the quantum phase is

$\displaystyle\chi_{k}(t)$
$\displaystyle=\int^{t}_{0}dt^{\prime}E_{k}(t^{\prime})-i\int^{t}_{0}dt^{\prime}\langle\epsilon_{k}(t^{\prime})|\partial_{t^{\prime}}\epsilon_{k}(t^{\prime})\rangle$
(24) $\displaystyle=\int^{t}_{0}dt^{\prime}E_{k}(t^{\prime})=k\chi(t).$

Note $E_{k}(t)$ is the instantaneous eigenvalue of $|\epsilon_{k}\rangle$,
$\langle\epsilon_{k}(t^{\prime})|\partial_{t^{\prime}}\epsilon_{k}(t^{\prime})\rangle=0$,
and $\chi(t)\equiv\int^{t}_{0}dt^{\prime}\omega_{A}(t^{\prime})$.
Alternatively, by setting $\theta(0)=\pi/2$ and $\theta(T)=0$, the adiabatic
path followed by the system can be constructed by
$|k(t)\rangle=|\epsilon_{0}(t)\rangle=\frac{1}{\sqrt{k!}}(B^{\dagger})^{k}|0\rangle$.
To be self-consistent, we stick to the boundary condition $\theta(0)=0$ and
$\theta(T)=\pi/2$ in this work.

In general situations, a superposed state
$|\psi(0)\rangle=\sum_{k}C_{k}|k0\rangle$ with normalized coefficients
$C_{k}$’s will adiabatically evolve to

$|\psi(T)\rangle=\sum_{k}C_{k}e^{-ik\chi(T)}|0k\rangle,$ (25)

at the desired moment $T$ and $C_{k}$ is invariant with time. The quantum
phase $\chi_{k}(T)$ for the Fock state with $k$ excitations is proportional to
$k$. By the preceding analyse, the phase difference can be periodically
cancelled by the bare Hamiltonian of the mechanical mode. The state transfer
thus has been indeed completed by Eq. (25).

Figure 2: (a): The time dependence of the driving-enhanced coupling strength
$g$, the effective frequency of the lower-frequency hybrid mode $\Delta$, and
the time-derivative of the control-parameter $\dot{\theta}$, in units of a
coupling strength $\Omega=\pi/T$. (b), (c), and (d): The target-state
population of the phonon mode $b$ for the initial state of the hybrid mode $m$
prepared as the Fock-state $|1\rangle$, the cat state with $\zeta=1$, and the
cat state with $\zeta=4$, respectively.

Now we can verify the TQD approach in the state transfer from mode-$m$ to
mode-$b$ by presenting the practical dynamics of the target-state population
$P$ given by Eq. (9). We choose the time-dependence of the effective frequency
$\Delta$ and the driving-enhanced coupling strength $g$ to be in a sinusoid
shape, and by Eq. (21),

$\Delta=2\Omega\cos(2\theta),\quad
g=\Omega\sin(2\theta),\quad\theta=\frac{\pi}{2}\frac{t}{T},$ (26)

where $\Omega$ is the coupling strength determined by the desired transfer
time $T$ as $\Omega T=\pi$. The shape functions of $\Delta$, $g$, and
$\dot{\theta}$ are plotted in Fig. 2(a). With various initial states of
mode-$m$, the blue solid lines and the red dashed lines in Figs. 2(b), (c),
and (d) describe the dynamics of the target-state population under $H_{\rm
tot}$ in Eq. (20) with the counterdiabatic term and that under $H(t)$ in Eq.
(7) without the counterdiabatic term, respectively.

One can find that practically the accelerated state-transfer could be
perfectly completed by the TQD approach for either Fock-state and superposed
states. The latter of the hybrid mode is prepared as an even cat-state
$(|\zeta\rangle+|-\zeta\rangle)/\sqrt{2+2e^{-2\zeta^{2}}}$, where
$|\zeta\rangle$ is the Glauber coherent state. For the Fock-state transfer
$|1\rangle_{m}|0\rangle_{b}\rightarrow|0\rangle_{m}|1\rangle_{b}$ in Fig.
2(b), $P$ approaches $0.97$ by the original Hamiltonian $H(t)$. In Figs. 2(c)
and (d) for the cat state with $\zeta=1$ and $4$, respectively, it is found
that the TQD approach manifests its power for larger cat-state by achieving
unit transfer population. In contrast, under the original Hamiltonian $H(t)$,
$P$ approaches respectively $0.98$ and $0.65$ for $\zeta=1$ and $4$.

###  III.2 Invariant-based inverse engineering

Another main-stream accelerated adiabatic-passage is the invariant-based
inverse engineering Guéry-Odelin  _et al._ (2019); Lewis and Riesenfeld
(1969), where the parametrical adiabatic-path of the system is designed
through a Hermitian operator $I(t)$ termed the Levis-Riesenfeld invariant. For
an arbitrary original Hamiltonian $H(t)$, the invariant satisfies

$\frac{\partial I(t)}{\partial t}=-i[H(t),I(t)].$ (27)

In the framework of invariant-based inverse engineering, the wavefunction of a
time-dependent Schrödinger equation
$i\partial_{t}|\psi(t)\rangle=H(t)|\psi(t)\rangle$ can be expressed as
$|\psi(t)\rangle=\sum_{n}C_{n}e^{-i\kappa_{n}(t)}|\epsilon_{n}(t)\rangle$,
where $C_{n}$ is a time-independent amplitude, $|\epsilon_{n}(t)\rangle$ is
the eigenstates of the invariant $I(t)$, and $\kappa_{n}$ is the Lewis-
Riesenfeld phase defined by

$\dot{\kappa}_{n}(t)=\langle\epsilon_{n}(t)|[-i\partial_{t}+H(t)]|\epsilon_{n}(t)\rangle.$
(28)

To ensure the desired state transfer rather than the full-time adiabatic
passage, $I(t)$ and $H(t)$ have to share the same eigenstates at both initial
and final moments.

To carry out the derivation of a general LR invariant for our two-coupled-
resonator system with arbitrary target state, we rewrite the system
Hamiltonian $H(t)$ (7) into

$\displaystyle H(t)$
$\displaystyle=\frac{\Delta(t)}{2}m^{\dagger}m+[g_{R}(t)-ig_{I}(t)]m^{\dagger}b$
(29)
$\displaystyle+[g_{R}(t)+ig_{I}(t)]mb^{\dagger}-\frac{\Delta(t)}{2}b^{\dagger}b.$

where $g_{R}(t)$ and $g_{I}(t)$ represent the real and imaginary parts of the
complex coupling strength $g(t)$.

Its corresponding Lewis-Riesenfeld invariant can be formulated by

$\displaystyle I(t)$
$\displaystyle=\cos\beta(m^{\dagger}m-b^{\dagger}b)+\sin\beta(e^{-i\alpha}m^{\dagger}b+e^{i\alpha}mb^{\dagger}),$
(30) $\displaystyle=A^{\dagger}A-B^{\dagger}B,$

where

$\displaystyle
A=\cos\left(\frac{\beta}{2}\right)e^{\frac{i\alpha}{2}}m+\sin\left(\frac{\beta}{2}\right)e^{-\frac{i\alpha}{2}}b,$
(31) $\displaystyle
B=\sin\left(\frac{\beta}{2}\right)e^{\frac{i\alpha}{2}}m-\cos\left(\frac{\beta}{2}\right)e^{-\frac{i\alpha}{2}}b,$

are the normalized annihilation operators for $I(t)$ and both
$\beta\equiv\beta(t)$ and $\alpha\equiv\alpha(t)$ are time-dependent functions
to be determined. Substituting Eq. (30) into Eq. (27), we have

$\displaystyle\dot{\beta}$ $\displaystyle=2g_{I}\cos\alpha-2g_{R}\sin\alpha,$
(32) $\displaystyle\dot{\alpha}$
$\displaystyle=\Delta-\cot\beta(2g_{R}\cos\alpha+2g_{I}\sin\alpha).$

With the annihilation and creation operators of $I(t)$, the original
Hamiltonian (29) can be rewritten as

$H=\omega(A^{\dagger}A-B^{\dagger}B)+g_{AB}A^{\dagger}B+g^{*}_{AB}B^{\dagger}A,$
(33)

where

$\displaystyle\omega$
$\displaystyle\equiv\frac{\Delta\cos\beta}{2}+g_{R}\sin\beta\cos\alpha+g_{I}\sin\beta\sin\alpha,$
(34) $\displaystyle g_{AB}$
$\displaystyle\equiv\frac{\Delta\sin\beta}{2}-g_{R}(\cos\beta\cos\alpha+i\sin\alpha)$
$\displaystyle-g_{I}(\cos\beta\sin\alpha-i\cos\alpha).$

And then in the subspace with a fixed excitation number $N$, the general
solution of the Schrödinger equation can be expressed as a superposition of
the eigenstates of the invariant,

$|\psi_{N}(t)\rangle=\sum^{N}_{n=0}p_{n}|\epsilon_{N-n}\rangle
e^{-i\kappa_{N-n}(t)},$ (35)

where $p_{n}$ is a normalized coefficient, $|\epsilon_{N-n}\rangle$ is the
normalized eigentstates of the LR invariant taking the same form as in Eq.
(13), and $\kappa_{N-n}(t)$ is the quantum phase defined in Eq. (28).

We consider the state transfer from $|N\rangle_{m}|0\rangle_{b}$ to
$|0\rangle_{m}|N\rangle_{b}$. With the definitions in Eqs. (31) and (13), one
can construct a particular solution via

$|\psi_{N}(t)\rangle=|\epsilon_{N}\rangle
e^{-i\kappa_{N}(t)}=\frac{e^{-i\kappa_{N}(t)}}{\sqrt{N!}}(A^{\dagger})^{N}|0\rangle.$
(36)

under the boundary conditions $\beta(0)=0$ and $\beta(T)=\pi$. As for the
Lewis-Riesenfeld phase, we have

$\displaystyle\dot{\kappa}_{N}$
$\displaystyle=-i\langle\epsilon_{N}|\partial_{t}|\epsilon_{N}\rangle+\langle\epsilon_{N}|H|\epsilon_{N}\rangle$
(37)
$\displaystyle=-i\left(-iN\frac{\dot{\alpha}\cos\beta}{2}\right)+N\frac{\Delta\cos\beta}{2}$
$\displaystyle+N(g_{R}\sin\beta\cos\alpha+g_{I}\sin\beta\cos\alpha)$
$\displaystyle=N\frac{g_{R}\cos\alpha+g_{I}\sin\alpha}{\sin\beta}=N\dot{\kappa},$

with $\dot{\kappa}\equiv(g_{R}\cos\alpha+g_{I}\sin\alpha)/\sin\beta$, in which
we have applied the time-derivative of the operators $A$ and $B$

$\displaystyle\frac{\partial}{\partial t}A^{\dagger}$
$\displaystyle=-\frac{\dot{\beta}}{2}B^{\dagger}-\frac{i\dot{\alpha}}{2}\left(\cos\beta
A^{\dagger}+\sin\beta B^{\dagger}\right),$ (38)
$\displaystyle\frac{\partial}{\partial t}B^{\dagger}$
$\displaystyle=\frac{\dot{\beta}}{2}B^{\dagger}-\frac{i\dot{\alpha}}{2}\left(\sin\beta
A^{\dagger}-\cos\beta B^{\dagger}\right),$

and Eqs. (32) and (33).

Given the time-dependent parameters $\beta(t)$, $\alpha(t)$, and $\kappa(t)$
in Eqs. (32) and (37), the coupling strengths and the effective frequency of
the hybrid mode that are directly relevant in quantum control can be expressed
by

$\displaystyle g_{R}$
$\displaystyle=\dot{\kappa}\cos\alpha\sin\beta-\frac{\dot{\beta}}{2}\sin\alpha,$
(39) $\displaystyle g_{I}$
$\displaystyle=\dot{\kappa}\sin\alpha\sin\beta+\frac{\dot{\beta}}{2}\cos\alpha,$
$\displaystyle\Delta$ $\displaystyle=\dot{\alpha}+2\dot{\kappa}\cos\beta.$

By virtue of their excitation-number-independence, the transfer of an
arbitrary superposed state from mode-$m$ to mode-$b$ can be achieved under the
boundary conditions $\beta(0)=0$ and $\beta(T)=\pi$. Thus in a general
situation, an initial state

$|\psi(0)\rangle=\sum_{k}C_{k}|k0\rangle=\sum_{k}C_{k}|\psi_{k}(0)\rangle,$
(40)

where $C_{k}$ is the time-independent normalized coefficients, will evolve to

$|\psi(T)\rangle=\sum_{k}C_{k}e^{-ik(\kappa+\alpha)}|0k\rangle,$ (41)

at the final time $T$, according to Eqs. (36) and (31).

Figure 3: (a) The shapes of the real and imaginary parts of the driving-
enhanced coupling strength $g_{R}$ and $g_{I}$ and the effective frequency of
the hybrid mode $\Delta$, in units of the coupling strength $\Omega$ used in
Eq. (26). (b) The dynamics of the state population of mode-$b$ under various
initial cat states of the hybrid mode-$m$. Here the control parameters are set
as $\beta=\pi t/T$, $\alpha=-4/3\sin^{3}\beta$, and
$\kappa=\beta-\sin(2\beta)/2$.

The state transfer assisted by the Lewis-Riesenfeld invariant can be verified
in Fig. 3 by the state population $P$ of the phonon mode-$b$ given by Eq. (9).
With the selected control functions of $\beta$, $\alpha$, and $\kappa$, one
can directly find the time-dependence of the real and imaginary parts of the
coupling strength $g_{R}$ and $g_{I}$ and the frequency of the hybrid mode
$\Delta$ through Eq. (39). They are plotted in Fig. 3(a). The initial states
in Fig. 3(b) are various cat states with $\zeta=1,2,4$. It is found that a
perfect transfer can always be achieved via the LR-invariant-based inverse
engineering.

##  IV State transfer under systematic errors

In practice the ideal trajectory of the control parameters is not implemented
exactly because of technical imperfections and constraints. These systematic
errors pose the need for studying the effect of perturbations on transport
protocols and optimizing protocols that are robust with respect to the
stochastic fluctuation in the Hamiltonian Ruschhaupt  _et al._ (2012). In this
section, the Hamiltonian implemented in experiments could be assumed to be

$H_{\rm exp}=H(t)+\gamma H_{g}+\eta H_{\Delta},$ (42)

where $H(t)=H_{g}+H_{\Delta}$ is the ideal or unperturbed Hamiltonian in Eq.
(7), $H_{g}$ and $H_{\Delta}$ are respectively the interaction Hamiltonian
between the hybrid mode and the phonon mode and the bare Hamiltonian for them,
i.e.,

$\displaystyle H_{g}$ $\displaystyle\equiv
g(t)m^{\dagger}b+g^{*}(t)mb^{\dagger},$ (43) $\displaystyle H_{\Delta}$
$\displaystyle\equiv\frac{\Delta(t)}{2}(m^{\dagger}m-b^{\dagger}b).$

And $\gamma$ and $\eta$ are the dimensionless perturbed coefficient for the
coupling strength and the frequency detuning, respectively.

With the practical Hamiltonian (42), the evolved state is obtained from the
Schrödinger equation

$i\frac{\partial}{\partial t}|\Psi(t)\rangle=[H(t)+\gamma H_{g}+\eta
H_{\Delta}]|\Psi(t)\rangle.$ (44)

Then the sensitivity to the systematic error is defined as

$\displaystyle q_{g}$
$\displaystyle=-\frac{\partial{P(T)}}{\partial(\gamma^{2})}\bigg{|}_{\gamma=0},$
(45) $\displaystyle q_{\Delta}$
$\displaystyle=-\frac{\partial{P(T)}}{\partial(\eta^{2})}\bigg{|}_{\eta=0}.$

where $P(T)$ is the state population evaluated by replacing the evolved state
in Eq. (9) with $|\Psi(T)\rangle$ at the final time $T$. We write $P(T)$ as
$P$ for simplicity in the subsequent subsections IV.1, IV.2, and IV.3.

###  IV.1 $\pi$-pulse

Before working on the two accelerated adiabatic passages, we first consider a
straightforward $\pi$-pulse for the state transfer. Then in Eq. (7), it is
found that $\Delta(t)=0$ and $\int^{T}_{0}dtg(t)=\pi/2$. Accordingly, Eq. (42)
becomes $H_{\rm exp}=(1+\gamma)H_{g}$, which can be diagonalized with the new
operators $A=(m+b)/\sqrt{2}$ and $B=(m-b)/\sqrt{2}$, similarly to the
transformation in Eqs. (11) and (12). The special initial state
$|\Psi(0)\rangle=|N\rangle_{m}|0\rangle_{b}=|N0\rangle$ could then expand by
the eigenstates in Eq. (13)

$|N0\rangle=\frac{1}{\sqrt{2^{N}}}\sum^{N}_{n=0}\sqrt{C^{n}_{N}}|\epsilon_{N-n}\rangle.$
(46)

The eigenvalue of $|\epsilon_{N-n}\rangle$ is now $(N-2n)(1+\gamma)g(t)$ with
respect to $H_{\rm exp}$. So that we have

$\displaystyle|\Psi(T)\rangle=\frac{1}{\sqrt{2^{N}}}\sum^{N}_{n=0}\sqrt{C^{n}_{N}}e^{-i(N-2n)(1+\gamma)\frac{\pi}{2}}|\epsilon_{N-n}\rangle.$
(47)

by the Schrödinger equation in Eq. (44). According to Eqs. (11) and (12), the
target state is

$|0N\rangle=\frac{1}{\sqrt{2^{N}}}\sum^{N}_{n=0}(-1)^{n}\sqrt{C^{n}_{N}}|\epsilon_{N-n}\rangle.$
(48)

Then the state-transfer fidelity measured by the target-state population $P$
in mode-$b$ is

$\displaystyle P=|\langle
0N|\Psi(T)\rangle|^{2}=\cos^{2N}\left(\frac{\pi}{2}\gamma\right).$ (49)

by virtue of Eq. (9). And by Eq. (45), the systematic-error sensitivity to
coupling strength for the initial Fock state $|N0\rangle$ is

$q_{g}=\frac{N\pi^{2}}{4}.$ (50)

The preceding derivation can be straightforwardly extended to arbitrary
superposed state $|\psi(0)\rangle=\sum_{k}C_{k}|k0\rangle$ by virtue of its
independence of the excitation number. In general situations, we have

$\displaystyle P=\sum_{C_{k}\neq
0}|C_{k}|^{2}\cos^{2k}\left(\frac{\pi}{2}\gamma\right),$ (51) $\displaystyle
q_{g}=\frac{\pi^{2}}{4}\sum_{C_{k}\neq
0}k|C_{k}|^{2}=\frac{\pi^{2}}{4}\bar{n}_{m}.$

It is interesting to find that the systematic-error sensitivity in the
$\pi$-pulse protocol is proportional to the average excitation number
$\bar{n}_{m}$ of the initial state.

###  IV.2 Transitionless quantum driving

Figure 4: State-transfer population $P(T)$ of the phonon mode-$b$ as a
function of the systematic errors associated to coupling-strength $\gamma$ and
frequency-detuning $\eta$, under various target states and shape functions of
TQD protocol. In (a) and (b), the initial state is the Fock state $|1\rangle$
and in (c) and (d), it is the cat state with $\zeta=1$. $T=\pi/\Omega$.

Now we analysis the stability for state transfer and error sensitivity of the
TQD protocol provided in Sec. III.1. Note the unperturbed Hamiltonian $H(t)$
in the total Hamiltonian in Eq. (20) is replaced with $H_{\rm exp}$ in Eq.
(42). In Figs. 4(a) and (c) $\eta=0$, and in Figs. 4(b) and (d) $\gamma=0$. In
Fig. 4, the TQD protocols are performed by the time-dependence of the
effective frequency $\Delta$ and the driving-enhanced coupling strength $g(t)$
(assumed to be real for TQD), that are determined by the shape functions of
$\theta=\pi/2(t/T)$ in Eq. (26) or $\theta=\pi/2(t/T)^{2}$. One can observe
that the protocol stability is not sensitive to the choice of the target
states. It is found that the impact of the coupling-strength fluctuation of
the interaction Hamiltonian $H_{g}$ is asymmetrical to the parameter $\gamma$
in the negative and positive axis. With the same magnitude, the decrement in
the state population $P$ induced by a positive $\gamma$ is clearly smaller
than that by a negative $\gamma$. In particular, $P=0.99$ for
$\gamma/\Omega=0.2$ and $P=0.96$ for $\gamma/\Omega=-0.2$. In contrast, the
state-transfer population is roughly symmetrical to the energy fluctuation
$\eta$ of the free Hamiltonian $H_{\Delta}$. Another difference between Figs.
4(a), (c) and Figs. 4(b), (d) manifests in the error sensitivity to the shape
of the control parameter $\theta(t)$. For a nonvanishing $\gamma$ ($\eta$),
the protocol is more robust with the linear function $\theta=\pi/2(t/T)$ [the
quadratic function $\theta=\pi/2(t/T)^{2}$] than that with the quadratic
function $\theta=\pi/2(t/T)^{2}$ [the linear function $\theta=\pi/2(t/T)$].

Figure 5: State-transfer population $P(T=\pi/\Omega)$ of the phonon mode-$b$
in the space of the systematic-error parameters $\gamma$ and $\eta$. The
target state is chosen as the cat state with $\zeta=1$.

In Fig. 5, we switch simultaneously on the systematic error in both
interaction Hamiltonian and free Hamiltonian and fix the target state as a cat
state with $\zeta=1$. Much to our anticipation, a nearly unit state-transfer
is found in a remarkable regime with $\gamma\approx\eta$. It could be readily
understood that when $\gamma\approx\eta$, the experimental Hamiltonian $H_{\rm
exp}$ in Eq. (42) is approximated by $(1+\gamma)H(t)$, equivalent to a
rescaling over the whole original Hamiltonian that renders the same counter-
diabatic Hamiltonian $H_{CD}$ in Eq. (19).

From both Figs. 4 and 5, with even $20\%$ fluctuations in system parameters,
the state-transfer fidelity could be maintained above $0.95$. One can
generally find that the TQD approach is robust against the systematic errors.

###  IV.3 Invariant-based inverse engineering

In this subsection, we construct an optimal protocol against the systematic
errors by using the Lewis-Riesenfeld invariant in Eq. (30). Now we employ the
ideal Hamiltonian $H(t)$ in Eq. (29) with a complex coupling strength $g(t)$.

From the Schrödinger equation in Eq. (44), one can obtain the time-evolved
state at the final time $T$ up to the second-order of $O(\gamma^{2})$ and
$O{(\eta^{2})}$:

$\displaystyle|\Psi(T)\rangle=|\psi(T)\rangle-i\gamma\int^{T}_{0}dtU_{0}(T,t)H_{g}|\psi(t)\rangle$
(52)
$\displaystyle-\gamma^{2}\int^{T}_{0}dt\int^{t}_{0}dt^{\prime}U_{0}(T,t)H_{g}(t)U_{0}(t,t^{\prime})H_{g}(t^{\prime})|\psi(t^{\prime})\rangle$
$\displaystyle-i\eta\int^{T}_{0}dtU_{0}(T,t)H_{\Delta}|\psi(t)\rangle$
$\displaystyle-\eta^{2}\int^{T}_{0}dt\int^{t}_{0}dt^{\prime}U_{0}(T,t)H_{\Delta}(t)U_{0}(t,t^{\prime})H_{\Delta}(t^{\prime})|\psi(t^{\prime})\rangle$
$\displaystyle+\cdots,$

where $|\psi(t)\rangle$ is the unperturbed solution determined by the ideal
evolution operator $U_{0}(t,0)$. Under the assumption that the initial state
is a Fock state $|N\rangle_{m}|0\rangle_{b}$ and by virtue of Eqs. (35) and
(36), the unperturbed solution reads

$|\psi(t)\rangle=|\epsilon_{N}(t)\rangle e^{-iN\kappa(t)},$ (53)

where $\kappa$ is defined in the last line of Eq. (37), and we have

$U_{0}(s,t)=\sum^{N}_{n=0}e^{-i(N-2n)[\kappa(s)-\kappa(t)]}|\epsilon_{N-n}(s)\rangle\langle\epsilon_{N-n}(t)|.$
(54)

Substituting Eqs. (53) and (54) to Eq. (52), the final state population is

$\displaystyle P$ $\displaystyle=|\langle
0N|\Psi(T)\rangle|^{2}=|\langle\psi(T)|\Psi(T)\rangle|^{2}$ (55)
$\displaystyle\approx
1-\gamma^{2}\sum^{N}_{n=1}\left|\int^{T}_{0}dte^{-2ni\kappa(t)}\langle\epsilon_{N-n}(t)|H_{g}|\epsilon_{N}(t)\rangle\right|^{2}$
$\displaystyle-\eta^{2}\sum^{N}_{n=1}\left|\int^{T}_{0}dte^{-2ni\kappa(t)}\langle\epsilon_{N-n}(t)|H_{\Delta}|\epsilon_{N}(t)\rangle\right|^{2}.$

By virtue of

$\displaystyle\langle\epsilon_{N-1}(t)|H_{g}|\epsilon_{N}(t)\rangle$
$\displaystyle=-\sqrt{N}\dot{\kappa}\sin\beta\cos\beta+\frac{i\sqrt{N}\dot{\beta}}{2},$
(56) $\displaystyle\langle\epsilon_{N-n}(t)|H_{g}|\epsilon_{N}(t)\rangle$
$\displaystyle=0,\quad n\neq 1$
$\displaystyle\langle\epsilon_{N-1}(t)|H_{\Delta}|\epsilon_{N}(t)\rangle$
$\displaystyle=\frac{\sqrt{N}\dot{\alpha}}{2}\sin\beta+\sqrt{N}\dot{\kappa}\sin\beta\cos\beta,$
$\displaystyle\langle\epsilon_{N-n}(t)|H_{\Delta}|\epsilon_{N}(t)\rangle$
$\displaystyle=0,\quad n\neq 1$

and the boundary condition $\beta(0)=0$ and $\beta(0)=\pi$, the systematic
error sensitivities for $|\psi(0)\rangle=|N0\rangle$ are found to be

$\displaystyle q_{g}$
$\displaystyle=N\left|\int^{T}_{0}dt\dot{\beta}\sin^{2}\beta
e^{-2i\kappa}\right|^{2},$ (57) $\displaystyle q_{\Delta}$
$\displaystyle=N\left|\int^{T}_{0}dt\sin\beta\left(\frac{\dot{\alpha}}{2}+\dot{\kappa}\cos\beta\right)e^{-2i\kappa}\right|^{2},$

according to their definitions in Eq. (45). Specially when $\kappa$ is
constant, we restore the result in the $\pi$-pulse case $q_{g}=N\pi^{2}/4$,
which is independent of $\beta(t)$.

It is interesting to find that $q_{g}=0$ can be attained when

$\kappa(t)=j\left[\beta-\frac{\sin(2\beta)}{2}\right]$ (58)

with a nonzero integer $j$. And $q_{\Delta}=0$ can be attained when

$\frac{\dot{\alpha}}{2}+\dot{\kappa}\cos\beta=0.$ (59)

Equations (58) and (59) render
$\dot{\alpha}=-4j\dot{\beta}\cos^{2}\beta\sin\beta$. An immediate choice is
$\alpha=-4j\cos^{3}\beta/3$. Therefore the optimized invariant-based inverse
engineering that is robust against the systematic errors are described by

$\displaystyle\beta(0)=0,\quad\beta(T)=\pi,$ (60)
$\displaystyle\kappa(t)=j\left[\beta-\frac{\sin(2\beta)}{2}\right],$
$\displaystyle\alpha(t)=-\frac{4j\cos^{3}\beta}{3}.$

Consequently, the control protocol in Eq. (39) turns out to be

$\displaystyle
g_{R}=2\dot{\beta}\sin^{3}\beta\cos\left(\frac{4}{3}\sin^{3}\beta\right)+\frac{\dot{\beta}}{2}\sin\left(\frac{4}{3}\sin^{3}\beta\right),$
(61) $\displaystyle
g_{I}=-2\dot{\beta}\sin^{3}\beta\sin\left(\frac{4}{3}\sin^{3}\beta\right)+\frac{\dot{\beta}}{2}\cos\left(\frac{4}{3}\sin^{3}\beta\right),$
$\displaystyle\Delta=0.$

where we have set $j=1$. It is more important to find that the optimized
protocol in Eq. (61) as well as the parametric setting in Eq. (60) is
independent of the excitation number. So that it applies to the general
superposed state $|\psi(0)\rangle=\sum_{k}C_{k}|k0\rangle$, where the
population is found to be

$P=1-\gamma^{2}\sum_{C_{k}\neq
0}|C_{k}|^{2}\left|\int^{T}_{0}dt\langle\epsilon_{k-1}(t)|H_{g}|\epsilon_{k}(t)\rangle\right|^{2}.$
(62)

Note in the optimized control by Eq. (61), $\Delta=0$ implies that $P$ is
strictly insensitive to $\eta$.

Figure 6: State-transfer population $P(T=\pi/\Omega)$ of the phonon mode-$b$
as a function of the systematic errors associated to coupling-strength
$\gamma$ under various protocols. (a) the initial state is the Fock state
$|1\rangle$, (b) it is the cat state with $\zeta=1$.

In Fig. 6, we compare the systematic-error sensitivity in various state-
transfer protocols, including the flat $\pi$ pulse (see the blue solid lines),
the transitionless quantum driving protocol described by the parametric
functions in Eq. (26) (see the red dashed lines), and the optimized protocol
based on the LR-invariant described by Eq. (61) (see the orange dotted lines).
It is found for both Fock-state and cat state, the optimized protocol assisted
by the LR-invariant demonstrates a much stronger robustness than the other two
protocols. The population is maintained as unit within a significant regime of
fluctuations of $10\%$ magnitude in terms of the coupling strength. The flat
$\pi$ pulse behaves as the most fragile protocol.

##  V Discussion

Alternatively, the robustness of the state-transfer protocols can be tested by
taking the effects of dissipative thermal baths into account. We now calculate
the transfer population with the standard Lindblad master equation under the
Born-Markovian approximation. The dynamical equation reads,

$\displaystyle\frac{\partial\rho(t)}{\partial t}$
$\displaystyle=-i[H(t),\rho(t)]$ (63)
$\displaystyle+\left[\kappa_{m}(\bar{n}_{m}+1)L(m)+\kappa_{m}\bar{n}_{m}L(m^{\dagger})\right]\rho$
$\displaystyle+\left[\kappa_{b}(\bar{n}_{b}+1)L(b)+\kappa_{b}\bar{n}_{b}L(b^{\dagger})\right]\rho,$

where the superoperation for any Lindblad operator $o$, $o=m,b$, is defined as

$L(o)\rho\equiv o\rho o^{\dagger}-\frac{1}{2}o^{\dagger}o\rho-\frac{1}{2}\rho
o^{\dagger}o.$ (64)

Here $\rho$ is the density operator of the two modes and $H(t)$ is the system
Hamiltonian in Eq. (20) for TQD protocol or that in Eq. (29) for invariant
protocol. $\kappa_{m}$ and $\kappa_{b}$ represent the decay rates of the
hybrid mode-$m$ and phonon mode-$b$, respectively. In the numerical
evaluation, we choose the mode frequencies as $\omega_{m}/2\pi=10$ GHz and
$\omega_{b}/2\pi=10$ MHz and the damping rates as $\kappa_{b}=100$ Hz and
$\kappa_{m}=10$ kHz Li  _et al._ (2018); Zhang  _et al._ (2016). The two
thermal baths are assumed to be at the temperature $T_{\rm th}$, then the
average excitation number for the mode-$o$ is
$\bar{n}_{o}=[\exp(\omega_{o}/k_{B}T_{\rm th})-1]^{-1}$. The evolution time is
fixed as $\Omega T=\pi$ with $\Omega/2\pi=1$ MHz.

Figure 7: The dynamics of target-state population in mode-$b$ in the presence
of the thermal baths with various temperatures. In (a) we use the TQD protocol
as described in Fig. 2, and in (b) we use the LR-invariant protocol as
described in Fig. 3. Here the initial state is set as the cat state with
$\zeta=1$.

The transfer population defined in Eq. (9) for a special cat state under
various temperatures is plotted in Fig. 7. Comparing Figs. 7(a) and 7(b), it
is found that for both protocols of accelerated adiabatic passage, i.e., TQD-
and invariant-STA, the effects of external thermal baths are almost the same.
They are found to be robust against the thermal baths surrounding the system
under low temperatures. $P$ could be maintained over $0.97$ when $T_{\rm
th}\leq 0.1$ K. Even under a high-temperature, e.g., $T_{\rm th}=1$ K, the
population is still above $0.88$.

Our proposals are based on the system Hamiltonian (7) under the rotating-wave
approximation. The ignorance of the counter-rotating terms means the coupling
strength $g$ should be kept much smaller than the eigenfrequency $\omega_{b}$.
However, $g\ll\omega_{b}$ might be invalid in any protocols employing
accelerated adiabatic passages. To consider the effect from the counter-
rotating terms, we should move back to the linearized Hamiltonian in Eq. (74)
having two hybrid modes and the phonon mode. After the unitary transformation
with respect to $U=\exp[i\int_{0}^{t}ds\Delta^{\prime}(s)a^{\dagger}a]$, we
have

$\displaystyle H$ $\displaystyle=\Delta
m^{\dagger}m+\omega_{b}b^{\dagger}b+g(m^{\dagger}b+m^{\dagger}b^{\dagger})+g^{*}(mb^{\dagger}+mb)$
(65)
$\displaystyle+g^{\prime}\left\\{a^{\dagger}be^{i[\int_{0}^{t}ds\Delta^{\prime}(s)]}+a^{\dagger}b^{\dagger}e^{i[\int_{0}^{t}ds\Delta^{\prime}(s)]}\right\\}$
$\displaystyle+g^{\prime*}\left\\{ab^{\dagger}e^{-i[\int_{0}^{t}ds\Delta^{\prime}(s)]}+abe^{-i[\int_{0}^{t}ds\Delta^{\prime}(s)]}\right\\}.$

Discarding the fast-oscillating terms and using the same conventions in Eq.
(6), the Hamiltonian turns out to be

$\displaystyle H$
$\displaystyle=[\omega_{b}+\Delta(t)]m^{\dagger}m+\omega_{b}b^{\dagger}b$ (66)
$\displaystyle+g(t)m^{\dagger}(b+b^{\dagger})+g^{*}(t)m(b+b^{\dagger}).$

Note it can be reduced to the Hamiltonian (7) under rotating-wave
approximation with a sufficiently large $\omega_{b}$.

Figure 8: The population dynamics in the presence of the counter-rotating
interaction under various state-transfer protocols. In (a) and (b), the
initial state is the Fock state $|1\rangle$ and in (c) and (d), it is the cat
state with $\zeta=1$. The phonon-mode frequency is set as
$\omega_{b}/\Omega=10$ in (a) and (c) and set as $\omega_{b}/\Omega=4$ in (b)
and (d). $\Omega T=\pi$.

Figure 8 demonstrates the effect of the counter-rotating terms under various
transfer protocols and target states. The TQD protocol and the invariant
protocol use the same groups of parameters as in Fig. 2(a) and Fig. 3(a),
respectively. And the transfer populations for the Fock state $|10\rangle$ and
the superposed state, i.e., the cat state of $|\zeta=1\rangle$, are presented
respectively in Figs. 8(a), (b) and Figs. 8(c), (d). It is found in Figs. 8(a)
and (c) that the counter-rotating Hamiltonian could be omitted with a
sufficiently large $\omega_{b}$. When $\omega_{b}/\Omega=10$, the population
under the $\pi$-pulse protocol would slightly deviate from the unit transfer.
It is interesting to find in Figs. 8(b) and (d) that with a smaller
$\omega_{b}/\Omega=4$, the invariant-based protocol [under the optimized
condition indicated by Eq. (61)] yields the most disappointing behavior among
the three protocols. Especially when the initial state is the Fock state
$|10\rangle$, $P$ becomes even smaller than $0.2$. And the performance of the
TQD protocol is still perfect, showing a great robustness to the presence of
the counter-rotating interactions.

##  VI Conclusion

In this work, we have presented two popular shortcut-to-adiabatiticy protocols
to realize a general state transfer in the cavity magnomechanical system,
where the magnon mode simultaneously coupled to a microwave cavity mode and
the mechanical-vibration mode in the same YIG sphere. Particularly, our work
demonstrates how to construct the counterdiabatic Hamiltonian in terms of the
creation and annihilation operators instead of the system-eigenstates and
their time-derivatives, which generally applies to the continuous-variable
systems. Also we derives in details the Levis-Riesenfeld invariant, based on
which an inverse engineering for arbitrary initial state could be carried on.
Through the analysis over the systematic error in various protocols, we obtain
an optimized regime for TQD protocol and an optimized parametric setting for
LR-invariant-based protocol. With the same systematic error in coupling
strength, the LR protocol is superior to the TQD protocol. While TQD protocol
is found to accommodate the counter-rotating interactions. Both of them are
robust against the external thermal noises.

Our work in pursuit of the quantum state transfer and protection provides an
important application of the cavity-magnomechanical system as a promising
hybrid platform for quantum information processing. Also it applies to
constructing the accelerated adiabatic passages in a general continuous-
variable system.

## Acknowledgments

We acknowledge grant support from the National Science Foundation of China
(Grants No. 11974311 and No. U1801661).

##  Appendix A Effective Hamiltonian for the hybrid magnomechanical system

This appendix contributes to deriving the effective Hamiltonian in Eq. (2).
With respect to the transformation
$U=\exp[i\omega_{p}t(a_{1}^{\dagger}a_{1}+m_{1}^{\dagger}m_{1})]$, the
original Hamiltonian in Eq. (1) turns out to be

$\displaystyle H_{0}$
$\displaystyle=\Delta_{a}a_{1}^{\dagger}a_{1}+\Delta_{m}m_{1}^{\dagger}m_{1}+\omega_{b}b_{1}^{\dagger}b_{1}+g_{ma}(a_{1}m_{1}^{\dagger}+a_{1}^{\dagger}m_{1})$
(67)
$\displaystyle+g_{mb}m_{1}^{\dagger}m_{1}(b+b^{\dagger})+i(\epsilon_{p}a_{1}^{\dagger}-\epsilon^{*}_{p}a_{1}),$

where $\Delta_{a}=\omega_{a}-\omega_{p}$ and
$\Delta_{m}=\omega_{m}-\omega_{p}$. In the strong coupling regime for the
magnon-photon interaction, we have $g_{ma}\gg\kappa_{1},\kappa_{2}$, where
$\kappa_{1}$ and $\kappa_{2}$ represent the decay rates of the photon and
magnon, respectively. The magnon mode, i.e., the collective spin-wave
excitations, can efficiently interface with microwave photons, thereby
consolidating the strength of dispersive interaction to produce well-separated
hybridized states. These dressed normal modes read,

$\displaystyle m=\sin\phi a_{1}-\cos\phi m_{1},$ (68) $\displaystyle
a=\cos\phi a_{1}+\sin\phi m_{1},$

where $\tan(2\phi)\equiv 2g_{ma}/{(\Delta_{a}-\Delta_{m})}$ and
$\phi\in[0,\pi/2]$. Then the Hamiltonian of the hybrid photon-magnon-phonon
system in Eq. (67) can be rewritten as

$\displaystyle H$ $\displaystyle=\Delta^{\prime}a^{\dagger}a+\Delta
m^{\dagger}m+\omega_{b}b^{\dagger}b$ (69)
$\displaystyle+i\epsilon_{p}(\cos\phi a^{\dagger}+\sin\phi
m^{\dagger})-i\epsilon_{p}^{*}(\cos\phi a+\sin\phi m)$
$\displaystyle+g_{mb}(b+b^{\dagger})(\sin^{2}\phi
a^{\dagger}a-\sin\phi\cos\phi a^{\dagger}m$ $\displaystyle-\sin\phi\cos\phi
am^{\dagger}+\cos^{2}\phi m^{\dagger}m),$

with

$\displaystyle\Delta=\frac{\omega_{a}+\omega_{m}}{2}-\omega_{p}-\sqrt{\left(\frac{\omega_{a}-\omega_{m}}{2}\right)^{2}+g^{2}_{ma}},$
(70)
$\displaystyle\Delta^{\prime}=\frac{\omega_{a}+\omega_{m}}{2}-\omega_{p}+\sqrt{\left(\frac{\omega_{a}-\omega_{m}}{2}\right)^{2}+g^{2}_{ma}}.$

Due to the input-output theory or the Heisenberg-Langevin equation, the time
evolution of the system operators satisfy

$\displaystyle\dot{a}$
$\displaystyle=-(i\Delta^{\prime}+\kappa_{a})a+\epsilon_{p}\cos\phi-
ig_{mb}\sin^{2}\phi a(b+b^{\dagger})$ (71) $\displaystyle-
ig_{mb}\sin\phi\cos\phi m(b+b^{\dagger})+\sqrt{2\kappa_{a}}a_{\rm in},$
$\displaystyle\dot{m}$
$\displaystyle=-(i\Delta+\kappa_{m})m+\epsilon_{p}\sin\phi-ig_{mb}\cos^{2}\phi
m(b+b^{\dagger})$ $\displaystyle-ig_{mb}\sin\phi\cos\phi
a(b+b^{\dagger})+\sqrt{2\kappa_{m}}m_{\rm in},$ $\displaystyle\dot{b}$
$\displaystyle=-(i\omega_{b}+\kappa_{b})b-ig_{mb}(\sin^{2}\phi
a^{\dagger}a-\sin\phi\cos\phi a^{\dagger}m$ $\displaystyle-\sin\phi\cos\phi
am^{\dagger}+\cos^{2}\phi m^{\dagger}m)+\sqrt{2\kappa_{b}}b_{\rm in},$

where $\kappa_{a}=\kappa_{1}\cos^{2}\phi+\kappa_{2}\sin^{2}\phi$,
$\kappa_{m}=\kappa_{1}\sin^{2}\phi+\kappa_{2}\cos^{2}\phi$ Zhang  _et al._
(2016), and $\kappa_{b}$ are the decay rates of the modes $a$, $m$, and $b$,
respectively.

The steady-state values $a_{s}\equiv\langle a\rangle$, $m_{s}\equiv\langle
m\rangle$, and $b_{s}\equiv\langle b\rangle$ are determined by letting
$\dot{a}=\dot{m}=\dot{b}=0$. We have

$\displaystyle-(i\Delta^{\prime}+\kappa_{a})a_{s}+\epsilon_{p}\cos\phi-
ig_{mb}a_{s}(b_{s}+b^{*}_{s})\sin^{2}\phi$ (72) $\displaystyle-
ig_{mb}m_{s}(b_{s}+b^{*}_{s})\sin\phi\cos\phi=0,$
$\displaystyle-(i\Delta+\kappa_{m})m_{s}+\epsilon_{p}\sin\phi-
ig_{mb}m_{s}(b_{s}+b^{*}_{s})\cos^{2}\phi$ $\displaystyle-
ig_{mb}a_{s}(b_{s}+b^{*}_{s})\sin\phi\cos\phi=0,$
$\displaystyle-(i\omega_{b}+\kappa_{b})b_{s}-ig_{mb}(|a_{s}|^{2}\sin^{2}\phi-a^{*}_{s}m_{s}\sin\phi\cos\phi$
$\displaystyle-a_{s}m^{*}_{s}\sin\phi\cos\phi+|m_{s}|^{2}\cos^{2}\phi)=0.$

Due to the fact that $g_{mb}\ll\omega_{b}$ Zhang  _et al._ (2016), the last
equation yields $b_{s}\approx 0$. And then we have

$m_{s}=\frac{\epsilon_{p}\sin\phi}{i\Delta+\kappa_{m}},\quad
a_{s}=\frac{\epsilon_{p}\cos\phi}{i\Delta^{\prime}+\kappa_{a}}.$ (73)

Following the standard linearization approach Aspelmeyer  _et al._ (2014), we
can rewrite the linearized Hamiltonian using the hybrid modes as

$\displaystyle H$ $\displaystyle=\Delta^{\prime}a^{\dagger}a+\Delta
m^{\dagger}m+\omega_{b}b^{\dagger}b+(g^{\prime}a^{\dagger}+g^{\prime*}a)(b+b^{\dagger})$
(74) $\displaystyle+(gm^{\dagger}+g^{*}m)(b+b^{\dagger}),$

by substituting the steady-state values in Eq. (73) to the Hamiltonian (69)
and ignoring all the high-order terms of fluctuations and operators, and the
coupling strengths are

$\displaystyle g=g_{mb}m_{s}\cos^{2}\phi-g_{mb}a_{s}\sin\phi\cos\phi,$ (75)
$\displaystyle g^{\prime}=g_{mb}a_{s}\sin^{2}\phi-
g_{mb}m_{s}\sin\phi\cos\phi,$

respectively.

According to Eq. (70), the two hybrid modes $a$ (with a higher frequency) and
$m$ (with a lower frequency) are well well-separated polaritonic states, or
normal modes, characterizing level repulsion, by the strong magnon-photon
coupling $\Delta^{\prime}-\Delta\geq 2g_{ma}\gg\kappa_{a},\kappa_{m}$. Then
under either the blue-detuning driving (with a higher $\omega_{p}$ yielding
negative $\Delta^{\prime}$ and $\Delta$) and the red-detuning driving (with a
lower $\omega_{p}$ yielding positive $\Delta^{\prime}$ and $\Delta$), one can
figure out $4$ situations for exacting a pair of nearly-resonant modes $m$ and
$b$ or $a$ and $b$. We should stress that our STA approach is applicable to
all of them. Now we choose the situation under the red-detuning driving and
the higher-frequency hybrid mode is far-off-resonant from $b$, i.e., we have
$\Delta^{\prime}-\omega_{b}\approx 2g_{ma}/\sin 2\phi\gg 0$.

Transforming the Hamiltonian (74) into the interaction picture with respect to
$U=\exp[i\int_{0}^{t}ds\Delta^{\prime}(s)a^{\dagger}a+i\omega_{b}(m^{\dagger}m+b^{\dagger}b)t]$,
we have

$\displaystyle H$ $\displaystyle=(\Delta-\omega_{b})m^{\dagger}m$ (76)
$\displaystyle+g^{\prime}\left\\{a^{\dagger}be^{i[\int_{0}^{t}ds\Delta^{\prime}(s)-\omega_{b}t]}+a^{\dagger}b^{\dagger}e^{i[\int_{0}^{t}ds\Delta^{\prime}(s)+\omega_{b}t]}\right\\}$
$\displaystyle+g^{\prime*}\left\\{ab^{\dagger}e^{-i[\int_{0}^{t}ds\Delta^{\prime}(s)-\omega_{b}t]}+abe^{-i[\int_{0}^{t}ds\Delta^{\prime}(s)+\omega_{b}t]}\right\\}$
$\displaystyle+g(m^{\dagger}b+m^{\dagger}b^{\dagger}e^{2i\omega_{b}t})+g^{*}(mb^{\dagger}+mbe^{-2i\omega_{b}t}).$

By discarding the fast-oscillating terms, the Hamiltonian turns out to be

$H=(\Delta-\omega_{b})m^{\dagger}m+(gm^{\dagger}b+g^{*}mb^{\dagger}).$ (77)

That is exactly the effective Hamiltonian in Eq. (2) describing the lower-
frequency hybrid mode coupled to the phonon mode.

## References

  * Rameshti  _et al._ (2021) B. Z. Rameshti, S. V. Kusminskiy, J. A. Haigh, K. Usami, D. Lachance-Quirion, Y. Nakamura, C.-M. Hu, H. X. Tang, G. E. Bauer, and Y. M. Blanter,  _Cavity magnonics,_ arxiv: 2106. 09312v1 (2021).
  * Lachance-Quirion  _et al._ (2019) D. Lachance-Quirion, Y. Tabuchi, A. Gloppe, K. Usami, and Y. Nakamura,  _Hybrid quantum systems based on magnonics,_Appl. Phys. Express 12, 070101 (2019).
  * Li  _et al._ (2020) Y. Li, W. Zhang, V. Tyberkevych, W. K. Kwok, and V. Novosad,  _Hybrid magnonics: physics, circuits, and applications for coherent information processing,_ J. Appl. Phys. 128, 130902 (2020).
  * Ladd  _et al._ (2010) T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien,  _Quantum computers,_Nature (London) 464, 45 (2010).
  * Reiserer and Rempe (2015) A. Reiserer and G. Rempe,  _Cavity-based quantum networks with single atoms and optical photons,_Rev. Mod. Phys. 87, 1379 (2015).
  * Degen  _et al._ (2017) C. L. Degen, F. Reinhard, and P. Cappellaro,  _Quantum sensing,_Rev. Mod. Phys. 89, 035002 (2017).
  * Blais  _et al._ (2021) A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff,  _Circuit quantum electrodynamics,_Rev. Mod. Phys. 93, 025005 (2021).
  * Aspelmeyer  _et al._ (2014) M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt,  _Cavity optomechanics,_Rev. Mod. Phys. 86, 1391 (2014).
  * Zhang  _et al._ (2016) X. Zhang, C.-L. Zou, L. Jiang, and H. Tang,  _Cavity magnonmechanics,_ Sci. Adv. 2, e1501286 (2016).
  * Soykal and Flatté (2010a) O. O. Soykal and M. E. Flatté,  _Strong field interactions between a nanomagnet and a photonic cavity,_Phys. Rev. Lett. 104, 077202 (2010a).
  * Soykal and Flatté (2010b) O. O. Soykal and M. E. Flatté,  _Size dependence of strong coupling between nanomagnets and photonic cavities,_Phys. Rev. B 82, 104413 (2010b).
  * Liu  _et al._ (2019) Z.-X. Liu, H. Xiong, and Y. Wu,  _Magnon blockade in a hybrid ferromagnet-superconductor quantum system,_Phys. Rev. B 100, 134421 (2019).
  * Li  _et al._ (2021) J. Li, Y.-P. Wang, W.-J. Wu, S.-Y. Zhu, and J. You,  _Quantum network with magnonic and mechanical nodes,_PRX Quantum 2, 040344 (2021).
  * Tabuchi  _et al._ (2014) Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura,  _Hybridizing ferromagnetic magnons and microwave photons in the quantum limit,_Phys. Rev. Lett. 113, 083603 (2014).
  * Zhang  _et al._ (2014) X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang,  _Strongly coupled magnons and cavity microwave photons,_Phys. Rev. Lett. 113, 156401 (2014).
  * Harder  _et al._ (2016) M. Harder, P. Hyde, L. Bai, C. Match, and C.-M. Hu,  _Spin dynamical phase and antiresonance in a strongly coupled magnon-photon system,_Phys. Rev. B 94, 054403 (2016).
  * Hyde  _et al._ (2017) P. Hyde, L. Bai, M. Harder, C. Dyck, and C.-M. Hu,  _Linking magnon-cavity strong coupling to magnon-polaritons through effective permeability,_Phys. Rev. B 95, 094416 (2017).
  * Wang  _et al._ (2018a) Y.-P. Wang, G.-Q. Zhang, D. Zhang, T.-F. Li, C.-M. Hu, and J. Q. You,  _Bistability of cavity magnon polaritons,_Phys. Rev. Lett. 120, 057202 (2018a).
  * Shen  _et al._ (2021) R.-C. Shen, Y.-P. Wang, J. Li, S.-Y. Zhu, G. S. Agarwal, and J. Q. You,  _Long-time memory and ternary logic gate using a multistable cavity magnonic system,_Phys. Rev. Lett. 127, 183202 (2021).
  * Tabuchi  _et al._ (2015) Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura,  _Coherent coupling between a ferromagnetic magnon and a superconducting qubit,_ Science 349, 405 (2015).
  * Lachance-Quirion  _et al._ (2020) D. Lachance-Quirion, S. Piotr Wolski, Y. Tabuchi, S. Kono, K. Usami, and Y. Nakamura,  _Entanglement-based single-shot detection of a single magnon with a superconducting qubit,_ Science 367, 425 (2020).
  * Li  _et al._ (2018) J. Li, S.-Y. Zhu, and G. S. Agarwal,  _Magnon-photon-phonon entanglement in cavity magnomechanics,_Phys. Rev. Lett. 121, 203601 (2018).
  * Li and Zhu (2019) J. Li and S.-Y. Zhu,  _Entangling two magnon modes via magnetostrictive interaction,_New J. Phys. 21, 085001 (2019).
  * Yu  _et al._ (2020) M. Yu, H. Shen, and J. Li,  _Magnetostrictively induced stationary entanglement between two microwave fields,_Phys. Rev. Lett. 124, 213604 (2020).
  * Chen  _et al._ (2021) Y.-T. Chen, L. Du, Y. Zhang, and J.-H. Wu,  _Perfect transfer of enhanced entanglement and asymmetric steering in a cavity-magnomechanical system,_Phys. Rev. A 103, 053712 (2021).
  * Qi and Jing (2021) S.-f. Qi and J. Jing,  _Magnon-assisted photon-phonon conversion in the presence of structured environments,_Phys. Rev. A 103, 043704 (2021).
  * Sarma  _et al._ (2021) B. Sarma, T. Busch, and J. Twamley,  _Cavity magnomechanical storage and retrieval of quantum states,_ New J. Phys. 23, 043041 (2021).
  * Vashahri-Ghamsari  _et al._ (2021) S. Vashahri-Ghamsari, Q. Lin, B. He, and M. Xiao,  _Magnomechanical phonon laser beyond the steady state,_Phys. Rev. A 104, 033511 (2021).
  * Chien  _et al._ (2001) L. Chien, Z. Dutton, C. H. Behroozi, and L. V. Hau,  _Observation of coherent optical information storage in an atomic medium using halted light pulses,_Nature 409, 490 (2001).
  * Cirac  _et al._ (1997) J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi,  _Quantum state transfer and entanglement distribution among distant nodes in a quantum network,_Phys. Rev. Lett. 78, 3221 (1997).
  * Kimble (2008) H. J. Kimble,  _The quantum internet,_ Nature 453, 1023 (2008).
  * Fiore  _et al._ (2011) V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. Wang,  _Storing optical information as a mechanical excitation in a silica optomechanical resonator,_Phys. Rev. Lett. 107, 133601 (2011).
  * Genes  _et al._ (2008) C. Genes, A. Mari, P. Tombesi, and D. Vitali,  _Robust entanglement of a micromechanical resonator with output optical fields,_Phys. Rev. A 78, 032316 (2008).
  * Kumar and Bhattacharya (2019) P. Kumar and M. Bhattacharya,  _Single-photon transfer using levitated cavityless optomechanics,_Phys. Rev. A 99, 023811 (2019).
  * Ho and Tseng (2015) C.-P. Ho and S.-Y. Tseng,  _Optimization of adiabaticity in coupled-waveguide devices using shortcuts to adiabaticity,_Opt. Lett. 40, 4831 (2015).
  * Guéry-Odelin  _et al._ (2019) D. Guéry-Odelin, A. Ruschhaupt, A. Kiely, E. Torrontegui, S. Martínez-Garaot, and J. G. Muga,  _Shortcuts to adiabaticity: Concepts, methods, and applications,_Rev. Mod. Phys. 91, 045001 (2019).
  * Torrontegui  _et al._ (2013) E. Torrontegui, S. Ibánez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guéry-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga,  _Shortcuts to adiabaticity,_Adv. At. Mol. Opt. Phys. 62, 117 (2013).
  * Berry (2009) M. V. Berry,  _Transitionless quantum driving,_J. Phys. A: Math. Theor. 42, 365303 (2009).
  * Demirplak and Rice (2005) M. Demirplak and S. A. Rice,  _Assisted adiabatic passage revisited,_J. Phys. Chem. B 109, 6838 (2005).
  * del Campo (2013) A. del Campo,  _Shortcuts to adiabaticity by counterdiabatic driving,_Phys. Rev. Lett. 111, 100502 (2013).
  * Lewis and Riesenfeld (1969) H. R. Lewis and W. B. Riesenfeld,  _An exact quantum theory of the timedependent harmonic oscillator and of a charged particle in a time‐dependent electromagnetic field,_J. Math. Phys. 10, 1458 (1969).
  * Bernardo (2020) B. d. L. Bernardo,  _Time-rescaled quantum dynamics as a shortcut to adiabaticity,_Phys. Rev. Research 2, 013133 (2020).
  * Wang  _et al._ (2018b) B.-X. Wang, T. Xin, X.-Y. Kong, S.-J. Wei, D. Ruan, and G.-L. Long,  _Experimental realization of noise-induced adiabaticity in nuclear magnetic resonance,_Phys. Rev. A 97, 042345 (2018b).
  * Jing and Wu (2015) J. Jing and L.-A. Wu,  _Overview of quantum memory protection and adiabaticity induction by fast signal control,_Sci. Bull. 60, 328 (2015).
  * Jing  _et al._ (2014) J. Jing, L.-A. Wu, T. Yu, J. Q. You, Z.-M. Wang, and L. Garcia,  _One-component dynamical equation and noise-induced adiabaticity,_Phys. Rev. A 89, 032110 (2014).
  * Chen  _et al._ (2010) X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga,  _Shortcut to adiabatic passage in two- and three-level atoms,_Phys. Rev. Lett. 105, 123003 (2010).
  * Wu  _et al._ (2021) S. Wu, W. Ma, X. Huang, and X. Yi,  _Shortcuts to adiabaticity for open quantum systems and a mixed-state inverse engineering scheme,_Phys. Rev. Applied 16, 044028 (2021).
  * Vepsäläinen  _et al._ (2019) A. Vepsäläinen, S. Danilin, and G. S. Paraoanu,  _Superadiabatic population transfer in a three-level superconducting circuit,_Sci. Adv. 5, eaau5999 (2019).
  * Tobalina  _et al._ (2020) A. Tobalina, E. Torrontegui, I. Lizuain, M. Palmero, and J. G. Muga,  _Invariant-based inverse engineering of time-dependent, coupled harmonic oscillators,_Phys. Rev. A 102, 063112 (2020).
  * Chen  _et al._ (2018) Y.-H. Chen, Z.-C. Shi, J. Song, and Y. Xia,  _Invariant-based inverse engineering for fluctuation transfer between membranes in an optomechanical cavity system,_Phys. Rev. A 97, 023841 (2018).
  * Zhang  _et al._ (2019a) H. Zhang, X.-K. Song, Q. Ai, H. Wang, G.-J. Yang, and F.-G. Deng,  _Fast and robust quantum control for multimode interactions using shortcuts to adiabaticity,_Opt. Express 27, 7384 (2019a).
  * Zhang  _et al._ (2019b) F.-Y. Zhang, W.-L. Li, W.-B. Yan, and Y. Xia,  _Speeding up adiabatic state conversion in optomechanical systems,_J. Phys. B 52, 115501 (2019b).
  * Wang  _et al._ (2021) Y. Wang, J.-L. Wu, Y.-K. Feng, J.-X. Han, Y. Xia, Y.-Y. Jiang, and J. Song,  _Optimal control for robust photon state transfer in optomechanical systems,_Ann. Phys. (Berlin) 533, 202000608 (2021).
  * Ruschhaupt  _et al._ (2012) A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga,  _Optimally robust shortcuts to population inversion in two-level quantum systems,_New J. Phys. 14, 093040 (2012).
  * Song  _et al._ (2021) X.-K. Song, F. Meng, B.-J. Liu, D. Wang, L. Ye, and M.-H. Yung,  _Robust stimulated raman shortcut-to-adiabatic passage with invariant-based optimal control,_Opt. Express 29, 7998 (2021).
  * Yu  _et al._ (2018) X.-T. Yu, Q. Zhang, Y. Ban, and X. Chen,  _Fast and robust control of two interacting spins,_Phys. Rev. A 97, 062317 (2018).
  * Tikochinsky (1978) Y. Tikochinsky,  _Transformation brackets for generalized bogolyubov-boson transformations,_J. Math. Phys. 19, 270 (1978).