# Kerr nonlinearity induced strong spin-magnon coupling

Feng-Zhou Ji* — Key Laboratory of Quantum Theory and Applications of MoE,
Lanzhou Center for Theoretical Physics, and Key Laboratory of Theoretical
Physics of Gansu Province, Lanzhou University, Lanzhou 730000, China  Jun-Hong
An* —<EMAIL_ADDRESS>Key Laboratory of Quantum Theory and Applications of
MoE, Lanzhou Center for Theoretical Physics, and Key Laboratory of Theoretical
Physics of Gansu Province, Lanzhou University, Lanzhou 730000, China

###### Abstract

One pillar of quantum magnonics is the exploration of the utilization of the
mediation role of magnons in different platforms to develop quantum
technologies. The efficient coupling between magnons and various quantum
entities is a prerequisite. Here, we propose a scheme to enhance the spin-
magnon coupling by the magnonic Kerr nonlinearity in a YIG sphere. We find
that the Kerr-enhanced spin-magnon coupling invalidates the widely used
single-Kittel-mode approximation to magnons. It is revealed that the spin
decoherence induced by the multimode magnons in the strong-coupling regime
becomes not severe, but suppressed, manifesting as either population trapping
or persistent Rabi-like oscillation. This anomalous effect is because the spin
changes to be so hybridized with the magnons that one or two bound states are
formed between them. Enriching the spin-magnon coupling physics, the result
supplies a guideline to control the spin-magnon interface.

##  I Introduction

Magnons are the elementary excitation of a collective spin wave in magnetic
materials. The quantized interactions of magnons with different quantum
platforms have inspired many novel applications in quantum technologies [1, 2,
3, 4, 5, 6, 7, 8, 9, 10, 11]. Besides quantum transduction [12], memory [13],
sensing [14, 15], and unidirectional invisibility [16], using the coupling
between magnons and photons or phonons, the efficient couplings of magnons to
spins have attracted much attention due to their potential realization of
quantum networks [17] and quantum sensing [18, 19, 20]. The efficient spin-
magnon couplings via either direct interactions [21, 22, 23, 24, 25, 26, 27,
28] or the indirect way by exchanging photons or phonons [29, 30, 31, 32, 31]
have been proposed. How to enhance the spin-magnon coupling strength is a
prerequisite to explore their applications.

The Kerr nonlinearity of magnons in magnetic materials supplies a useful
mechanism in quantum-state engineering [33, 34]. Based on it, magnon-polariton
bistability [35, 36, 37] and tristability [38, 39], which are useful in the
microwave nonreciprocal transmission [40], high-order sideband [41, 42, 43]
and entanglement generations [44], and quantum phase transition [45], have
been reported. A scheme to enhance the spin-magnon coupling using the Kerr
nonlinearity was proposed in Ref. [46]. These works on quantum magnonics were
generally based on an approximation in which the magnons are effectively
treated as a single first-order Kittel mode [47, 48, 49, 50]. It has been
revealed that the higher-order magnonic modes are non-negligible in the
presence of the Kerr nonlinearity [51, 52, 53]. References [23, 25, 24]
studied the interactions between spins and multimode magnons. However, they
are under the Markovian approximation, which is only valid in the weak-
coupling condition.

Figure 1:  Schematic illustration of the system. A magnetic emitter interacts
with the magnons in a YIG sphere with radius $R$ in a static magnetic field
$\textbf{H}_{e}$, and an induced magnetic field $\textbf{M}_{s}$. A driving
field with frequency $\omega_{d}$ is applied.

Here, we investigate the non-Markovian dynamics of a spin defect coupled to
magnons in a YIG sphere. A scheme to enhance the spin-magnon coupling by the
magnonic Kerr nonlinearity is proposed. We find that the increasing of the
coupling invalidates the widely used single-mode approximation of magnons to
describe the matter-magnon coupling. The strong coupling also causes the spin
to exhibit features with a suppressed decoherence, i.e., from the conventional
oscillating damping to either the population trapping or the persistent Rabi-
like oscillation. Our analysis reveals that such an anomalous decoherence is
due to the formation of different numbers of spin-magnon bound states.
Indicating that the Kerr nonlinearity endows the spin-magnon interface with a
good controllability, our result paves the way to design quantum magnon
devices.

##  II System and spectral density

We consider a spin defect as a magnetic emitter coupled to the magnons
attached to a YIG sphere in the presence of the Kerr nonlinearity (see Fig.
1). Its Hamiltonian reads [22, 26]

$\displaystyle\begin{aligned}
\hat{\mathcal{H}}_{\text{Kerr}}=&\hbar\omega_{0}\hat{\sigma}^{\dagger}\hat{\sigma}+\sum_{k}\big{[}\hbar\omega_{k}\hat{b}_{k}^{\dagger}\hat{b}_{k}-(\hbar
K/2)\hat{b}_{k}^{\dagger 2}\hat{b}_{k}^{2}\\\
&-(g_{k}\hat{b}_{k}^{\dagger}\hat{\sigma}-\Omega_{d}e^{i\phi_{d}}\hat{b}_{k}^{\dagger}e^{-i\omega_{d}t}+\text{H.c.})\big{]}.\end{aligned}$
(1)

Here, $\hat{\sigma}=|g\rangle\langle e|$ is the transition operator of the
spin defect with frequency $\omega_{0}$ from the excited state $|e\rangle$ to
the ground state $|g\rangle$, $\hat{b}_{k}$ is the annihilation operator of
the $k$th magnon mode with frequency $\omega_{k}$, and
$g_{k}=\mu_{0}\textbf{m}\cdot\tilde{\textbf{H}}_{k}^{\ast}(\textbf{r})$, with
m being the spin magnetic moment and $\tilde{\textbf{H}}_{k}(\textbf{r})$
being the vacuum amplitude of the $k$th magnon mode concentrated around the
YIG sphere, is their coupling strength. An inhomogeneous rf magnetic excited
field is needed to trigger the multimode magnons [54]. The magnons are further
driven by a microwave field with frequency $\omega_{d}$, amplitude
$\Omega_{d}$, and phase $\phi_{d}$. The Kerr nonlinearity quantified by $K$ is
caused by the magnetocrystalline anisotropy of the YIG sphere. It has been
used to generate the magnon squeezing [34], which is attractive to the
application of the magnons [1, 55]. Rewriting the magnon operators as the sum
of their steady-state mean value and fluctuations, i.e.,
$\hat{b}_{k}=\langle\hat{b}_{k}\rangle+\delta\hat{b}_{k}$, and neglecting the
high-order fluctuation terms in the strong driving condition, Eq. (1) in the
rotating frame with
$\hat{\mathcal{H}}_{0}=\omega_{d}(\hat{\sigma}^{\dagger}\hat{\sigma}+\sum_{k}\hat{b}_{k}^{\dagger}\hat{b}_{k})$
becomes (see Supplemental Material [56])

$\displaystyle\hat{\mathcal{H}}_{\text{Linear}}=$
$\displaystyle\hbar\Delta_{0}\hat{\sigma}^{\dagger}\hat{\sigma}+\sum_{k}\big{\\{}\hbar(\omega_{k}-\Pi_{k})\hat{b}_{k}^{\dagger}\hat{b}_{k}$
(2)
$\displaystyle-[g_{k}\hat{b}_{k}^{\dagger}\hat{\sigma}+(\hbar\mathcal{K}_{k}/2)\hat{b}_{k}^{2}+\text{H.c.}]\big{\\}},$

where $\Delta_{0}=\omega_{0}-\omega_{d}$,
$\Pi_{k}=\omega_{d}+2\mathcal{K}_{k}$, and
$\mathcal{K}_{k}=K\langle\hat{b}_{k}\rangle^{2}$. We have rewritten
$\delta\hat{b}_{k}$ as $\hat{b}_{k}$ for brevity. Making the Bogoliubov
transformation
$\hat{\mathcal{S}}=\exp[\sum_{k}r_{k}(\hat{b}_{k}^{2}-\hat{b}_{k}^{{\dagger}2})/2]$,
with
$r_{k}=\frac{1}{4}\ln(\frac{\omega_{k}-\Pi_{k}+\mathcal{K}_{k}}{\omega_{k}-\Pi_{k}-\mathcal{K}_{k}})$,
to Eq. (2) and neglecting the counter-rotating terms [46], we obtain

$\hat{\mathcal{H}}=\hbar\Delta_{0}\hat{\sigma}^{\dagger}\hat{\sigma}+\sum_{k}[\hbar\zeta_{k}\hat{b}_{k}^{\dagger}\hat{b}_{k}-(\mathcal{G}_{k}\hat{b}_{k}^{\dagger}\hat{\sigma}+\text{H.c.})],$
(3)

where $\zeta_{k}=(\omega_{k}-\Pi_{k})/\cosh{(2r_{k})}$ and
$\mathcal{G}_{k}=e^{r_{k}}g_{k}/2$. We find that the spin-magnon coupling is
exponentially enhanced and the magnon frequencies are suppressed by the Kerr
nonlinearity assisted by the microwave driving. This is one of our main
results. To simplify our discussion, we approximate $\Pi_{k}\simeq\omega_{d}$
due to $\omega_{d}\gg\mathcal{K}_{k}$ and $r_{k}\equiv r$ by neglecting their
$k$-dependence, which is valid by properly choosing $\Omega_{d}$ and
$\omega_{d}$ in our finite magnonic bandwidth case [54].

Consider that the YIG sphere is at low temperature such that the magnons are
initially in the vacuum state [44, 57]. After tracing over the magnonic
degrees of freedom from the dynamics of the spin-magnon system, we derive an
exact master equation of the spin as (see Supplemental Material [56])

$\dot{\rho}(t)=i\Omega(t)[\rho(t),\hat{\sigma}^{\dagger}\hat{\sigma}]+\Gamma(t)[2\hat{\sigma}\rho(t)\hat{\sigma}^{\dagger}-\\{\hat{\sigma}^{\dagger}\hat{\sigma},\rho(t)\\}].$
(4)

The renormalized frequency is $\Omega(t)=-\text{Im}[\dot{c}(t)/c(t)]$ and the
decay rate is $\Gamma(t)=-\text{Re}[\dot{c}(t)/c(t)]$, where $c(t)$ satisfies

$\dot{c}(t)+i\Delta_{0}c(t)+\int_{0}^{t}d\tau c(\tau)f(t-\tau)=0,$ (5)

under $c(0)=1$. The convolution in Eq. (5) makes the dynamics non-Markovian
with all the memory effects incorporated in the time-dependent coefficients in
Eq. (4). The magnonic correlation function is
$f(t-\tau)=\int_{\zeta_{\text{min}}}^{\zeta_{\text{max}}}d\zeta
J(\zeta)e^{-i\zeta(t-\tau)}$ and the spectral density is [22]

$J(\zeta)=\frac{\eta\mu_{0}}{4\hbar\pi}\text{Im}[\textbf{m}^{\ast}\cdot
k^{2}\textbf{G}(\textbf{r},\textbf{r},\zeta\cosh(2r)+\Pi)\cdot\textbf{m}].$
(6)

where $k=[\zeta\cosh(2r)+\Pi]/c$ and $\eta=e^{2r}\cosh{(2r)}$. The Green’s
tensor reads
$\bar{k}^{2}\textbf{G}(\textbf{r},\textbf{a},\bar{\omega})=\sum_{\alpha,\beta\in\\{r,\theta,\varphi\\}}[H_{0,\alpha}^{\beta}+H_{\alpha}^{\beta}]\textbf{e}_{\alpha}\textbf{e}_{\beta}$,
where ${\bf H}=-{\boldsymbol{\nabla}}\phi$ and $\bf{H}_{0}$ takes the similar
form as ${\bf H}$ but in the absence of the YIG sphere. The potential $\phi$
caused by the YIG satisfying $\phi={1\over
4\pi}{\boldsymbol{\nabla}}_{a}{1\over|{\bf r}-{\bf a}|}$ is subject to the
boundary condition of $(1+\chi)(\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial
y^{2}})\phi+\frac{\partial^{2}\phi}{\partial z^{2}}=0$ in $r\leq R$ and
$\nabla^{2}\phi=0$ in $r>R$ [58]. $\boldsymbol{\chi}$ is the magnetic
susceptibility tensor and determined by the Landau-Lifshitz-Gilbert equation
as [59, 60]

$\displaystyle\chi_{xx}$
$\displaystyle=\chi_{yy}=\frac{\gamma^{2}h_{0}M_{s}}{\gamma^{2}h_{0}^{2}-\omega^{2}-i\Gamma_{0}\omega}\equiv\chi,$
(7) $\displaystyle\chi_{xy}$
$\displaystyle=\chi^{\ast}_{yx}=i\frac{\gamma\omega
M_{s}}{\gamma^{2}h_{0}^{2}-\omega^{2}-i\Gamma_{0}\omega}\equiv i\kappa,$

where $\gamma$ is the gyromagnetic ratio, $\Gamma_{0}=2\gamma h_{0}\alpha$,
with $\alpha$ being the Gilbert parameter, is the damping parameter,
$\textbf{h}_{0}=h_{0}\textbf{e}_{z}=\textbf{H}_{e}+\textbf{H}_{d}$, with
$\textbf{H}_{e}$ being the external static field and
$\textbf{H}_{d}=-\textbf{M}_{s}/3$ being the demagnetization field, and ${\bf
M}_{s}$ is the saturation magnetization. Putting the spin on the equatorial
plane of the YIG sphere, i.e., $\theta=\pi/2$, and choosing
$\textbf{m}=-\mu_{B}({\bf e}_{x}+i{\bf e}_{y})=-\mu_{B}e^{i\varphi}({\bf
e}_{r}+i{\bf e}_{\varphi})$, we have the nonzero components of the Green’s
tensor as
$\text{Im}[\textbf{m}^{\ast}\cdot\textbf{G}\cdot\textbf{m}]=\mu_{B}^{2}[\text{Im}(\textbf{G}_{rr}+\textbf{G}_{\varphi\varphi})+\text{Re}(\textbf{G}_{r\varphi}-\textbf{G}_{\varphi
r})]$ [48, 22, 24]. The analytic form of
$\textbf{G}(\textbf{r},\textbf{a},\bar{\omega})$ is given in (see Supplemental
Material [56]). The Green’s tensor and the spectral density show resonance
peaks determined by [59, 60]

$(n+1-m\kappa)P_{n}^{m}(\xi_{0})+\xi_{0}P_{n}^{m{\prime}}(\xi_{0})=0,$ (8)

where $P_{n}^{m}$ is the associated Legendre polynomial and
$\xi_{0}=(1+1/\chi)^{1/2}$. The magnon modes corresponding to $n=-m=1$, $2$,
and $3$ in the absence of the Kerr nonlinearity are the dipole or Kittel mode
$\omega_{\text{K}}=\gamma(h_{0}+M_{s}/3)$, the quadrupolar mode
$\omega_{\text{Q}}=\gamma(h_{0}+2M_{s}/5)$, and the octupolar mode
$\omega_{\text{O}}=\gamma(h_{0}+3M_{s}/7)$, respectively. In the presence of
the Kerr nonlinearity, they become
$\zeta_{\text{K,Q,O}}=(\omega_{\text{K,Q,O}}-\Pi)/\cosh(2r)$. Ranging all $m$
and $n$, it was found that the frequency range of $J(\zeta)$ is from
$\zeta_{\text{min}}=(\gamma h_{0}-\Pi)/\cosh(2r)$ to
$\zeta_{\text{max}}=[\gamma(h_{0}+M_{s}/2)-\Pi]/\cosh(2r)$ [54].

##  III Spin dynamics

A widely used approximation in studying matter-magnon coupling is the
Markovian approximation [61, 23, 47, 50, 24, 62, 63, 25]. It is valid when
their coupling is weak and the time scale of $f(t-\tau)$ is much smaller than
the one of the matter. After replacing $c(\tau)$ by $c(t)$ and extending the
upper bound of the time integral to infinity, the Markovian approximate
solution of Eq. (5) is
$c_{\text{MA}}(t)=e^{-[\Lambda+i\Upsilon(\Delta_{0})]t}$, where
$\Upsilon(\Delta_{0})=\mathcal{P}\int d\zeta J(\zeta)/(\Delta_{0}-\zeta)$ is
the Lamb shift and $\Lambda=\pi J(\Delta_{0})$ is the spontaneous emission
rate. The exponential-decay feature of $|c_{\text{MA}}(t)|^{2}$ characterizes
a unidirectional energy flow from the spin to the magnons and the destructive
effect of the magnons on the spin. This approximation cannot reflect the
energy back flow induced by the strong spin-magnon coupling [22, 26].

A pseudocavity method was proposed to study the strong light-matter coupling
in an absorptive medium [64, 65, 66, 67, 68]. Keeping only the Kittel mode
$\zeta_{\text{K}}$, we approximate $J(\zeta)$ as a Lorentzian form
$\mathcal{J}(\omega)=\frac{J(\zeta_{\text{K}})(\gamma_{p}/2)^{2}}{(\omega-\zeta_{\text{K}})^{2}+(\gamma_{p}/2)^{2}}$.
The system is effectively seen as a spin coherently interacting with a
pseudocavity mode $\hat{a}$ with frequency $\zeta_{\text{K}}$ and damping rate
$\gamma_{p}$ in a coupling strength $g^{2}=\pi
J(\zeta_{\text{K}})\gamma_{p}/2$. Here, $\gamma_{p}$ is relevant to the
damping parameter $\Gamma_{0}$. Thus, the spin dynamics is phenomenologically
described by

$\displaystyle\dot{\rho}(t)=$ $\displaystyle
i[\rho(t),\Delta_{0}\hat{\sigma}^{\dagger}\hat{\sigma}+\zeta_{\text{K}}\hat{a}^{\dagger}\hat{a}+g(\hat{a}\hat{\sigma}^{\dagger}+\text{H.c.})]$
(9)
$\displaystyle+\frac{\gamma_{p}}{2}[2\hat{a}\rho(t)\hat{a}^{\dagger}-\\{\hat{a}^{\dagger}\hat{a},\rho(t)\\}].$

Although partially reflecting the energy back flow from the magnons to the
spin, this method misses important physics from the magnonic higher-order
resonant modes.

To fully capture the physics of the strong spin-magnon coupling enhanced by
the Kerr nonlinearity and uncover the condition under which the pseudocavity
method is applicable, we investigate the exact spin dynamics by choosing
$\Delta_{0}=\zeta_{\text{K}}$. The steady-state solution of Eq. (5) is
computable by a Laplace transform. It converts Eq. (5) into
$\tilde{c}(s)=[s+i\Delta_{0}+\int_{\zeta_{\text{min}}}^{\zeta_{\text{max}}}d\zeta\frac{J(\zeta)}{s+i\zeta}]^{-1}$.
$c(t)$ is obtained by making an inverse Laplace transform to $\tilde{c}(s)$,
which requires finding its poles via (see Supplemental Material [56])

${E\over\hbar}=\Delta_{0}+\int_{\zeta_{\text{min}}}^{\zeta_{\text{max}}}\frac{J(\zeta)}{E/\hbar-\zeta}d\zeta\equiv
Y(E),$ (10)

where $E=i\hbar s$. First, the roots $E$ of Eq. (10) are exactly the
eigenenergies of the total spin-magnon system. To prove this, we expand the
eigenstate as
$|\phi_{E}\rangle=x|e,\\{0_{k}\\}\rangle+\sum_{k}y_{k}|g,1_{k}\rangle$.
Substituting $|\phi_{E}\rangle$ into
$\hat{\mathcal{H}}|\phi_{E}\rangle=E|\phi_{E}\rangle$, we readily obtain Eq.
(10). Second, because $Y(E)$ is a decreasing function in the regimes
$E\in(-\infty,\hbar\zeta_{\text{min}}]$ and
$[\hbar\zeta_{\text{max}},+\infty)$, Eq. (10) has one isolated root
$E^{\text{b}}$ in $(-\infty,\hbar\zeta_{\text{min}}]$ or
$[\zeta_{\text{max}},+\infty)$ provided
$Y(\hbar\zeta_{\text{min}})<\hbar\zeta_{\text{min}}$ or
$Y(\hbar\zeta_{\text{max}})>\hbar\zeta_{\text{max}}$. The eigenstate
corresponding to $E^{\text{b}}$ is called the bound state. On the other hand,
$Y(E)$ is non-analytical in the regime
$E\in[\hbar\zeta_{\text{min}},\hbar\zeta_{\text{max}}]$ due to the singularity
in its integration. Therefore, Eq. (10) has an infinite number of roots in
this regime, which form an energy band. Using the residue theorem, we have
$c(t)=\sum_{j=1}^{M}Z_{j}e^{{-i\over\hbar}E^{b}_{j}t}+\int_{\zeta_{\text{min}}}^{\zeta_{\text{max}}}\Theta(E)e^{-iEt}dE$,
where $\Theta(E)={J(E)\over[E-\Delta_{0}-\Upsilon(E/\hbar)]^{2}+[\pi
J(E)]^{2}}$, $M$ being the number of the bound states, and
$Z_{j}=[1+\int_{\zeta_{\text{min}}}^{\zeta_{\text{max}}}\frac{J(\zeta)d\zeta}{(E^{\text{b}}_{j}/\hbar-\zeta)^{2}}]^{-1}$
the residue contributed by the $j$th bound state. Oscillating with time in
continuously changing frequencies ${E/\hbar}$ of the band energies, the
integrand tends to zero in the long-time limit due to the out-of-phase
interference. Thus, the steady-state solution of Eq. (10) is [69]

$\lim_{t\rightarrow\infty}c(t)=\begin{cases}0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{no
bound state}\\\
\sum_{j=1}^{M}Z_{j}e^{{-i\over\hbar}E^{\text{b}}_{j}t},~{}M~{}\text{bound
states}\end{cases}.$ (11)

Assuming the spin is initially in $|e\rangle$ and solving Eq. (4), we obtained
that the excited-state population is just $|c(t)|^{2}$ (see Supplemental
Material [56]). Thus, Eq. (11) reveals that thanks to the Kerr-nonlinearity-
enhanced spin-magnon coupling, the formation of the bound states would prevent
the spin from relaxing to its ground state. Since it is not obtained from both
the Markovian approximation and the pseudocavity method, such an anomalous
decoherence manifests the distinguished role played by the non-Markovian
effect and the feature of the energy spectrum of the total spin-magnon system
in the decoherence of the spin. This is another main result of our work.

Figure 2: (a) Spectral density $J(\zeta)$ in different spin-YIG distance $a$.
(b) Kittel-mode frequency $\zeta_{\text{K}}$ and enhancement coefficient
$\eta$ in different $r$. (c) Evolution of the excited-state population
$|c(t)|^{2}$ from the Markovian approximation (red dotted line), the
pseudocavity method (cyan dashed line), and the non-Markovian dynamics (blue
line). The inset is $J(\zeta)$ and the fitted Lorentzian form
$\mathcal{J}(\zeta)$. We use $\omega_{d}=1$ GHz,
$\gamma=28~{}\text{GHz}\cdot\text{T}^{-1}$, $\Gamma_{0}=8\times 10^{-3}$ GHz,
$M_{s}=0.178$ T, $h_{0}=0.5$ T, $R=30$ nm, $r=2$ in (a) and (c), and $a=26$ nm
in (c).

##  IV Numerical results

We plot in Fig. 2(a) the spectral density $J(\zeta)$ in different spin-YIG
distance $a$ for $r=2$. The driving-field frequency $\omega_{d}$ is chosen as
$\omega_{d}=1$ GHz and the Kerr coefficient $K$ relates to the volume $V$ of
the three-dimensional YIG sphere, i.e., $K\varpropto V^{-1}$ [34] and is about
kHz [46], which makes the validity of $\Pi\simeq\omega_{d}$. We really see
that $J(\zeta)$ exhibits obvious peaks at $\zeta=537$, $549$, and $554$ MHz
irrespective of the value of $a$, which match well with our analytical
frequencies $\zeta_{\text{K,Q,O}}$ of the Kittel, quadrupolar, and octupolar
modes evaluated from Eq. (8). With decreasing $a$, $J(\zeta)$ shows an
increase due to the near-field enhancement [26]. It signifies a strong spin-
magnon coupling in the small-$a$ regime. Figure 2(b) shows the effect of the
Kerr nonlinearity on enhancing the spin-magnon coupling. It reveals that, with
increasing $r$ from zero to 2, $\zeta_{\text{K}}$ decreases from $15$ GHz to
$537$ MHz, while the prefactor $\eta$ of $J(\zeta)$ increases from $1$ to
$1200$. An efficient increase of about four orders of magnitude of
$\eta/\zeta_{\text{K}}$ manifests a dramatic boost of the spin-magnon coupling
strength. It confirms that the Kerr nonlinearity can be used to enhance the
spin-magnon coupling [46, 37, 38].

Figure 3:  (a) Evolution of $|c(t)|^{2}$ in different $a$. The black dashed
lines are the corresponding steady-state values from Eq. (11). The time for
the cases of $a=9$ and $13$ nm is magnified by a factor of 0.03. Energy
spectrum of the whole system in different (b) $a$ and (d) $r$ obtained by
solving Eq. (10). $|c(\infty)|^{2}$ from solving Eq. (5) denoted by the dots
and from Eq. (11) denoted by the solid lines in different (c) $a$ and (e) $r$.
The red region covers the values during its persistent oscillation. (f)
$|c(\infty)|^{2}$ when $\Delta_{0}=\zeta_{\text{K}}$ (brown dot),
$\zeta_{\text{Q}}$ (purple square), and $\zeta_{\text{O}}$ (cyan rhombus).
$r=2.0$ in (b) and (c), $a=4$ nm in (d) and (e), and others are the same as
Fig. 2(c).

Figure 2(c) shows the comparison of $|c(t)|^{2}$ obtained by three methods
when $a=26$ nm. The exponential decay in the Markovian result entirely fails
to describe the rapid spin-magnon energy exchanges obtained via numerically
solving Eq. (5), which is fully captured by the pseudocavity method. It is the
signature of the non-Markovian memory effect owned by the strong-coupling
dynamics [70]. In this case, $J(\zeta)$ is dominated by the Kittel mode such
that a Lorentzian fitting centered at $\zeta_{\text{K}}$ is sufficient and the
pseudocavity method works well. However, with further decreasing $a$ [see Fig.
2(a)] or increasing $r$, the high-order magnon modes become dominated, where
the pseudocavity method no longer work.

Figure 3(a) shows the exact $|c(t)|^{2}$ in different $a$ when $r=2$. The
strong spin-magnon coupling favored by both the near-field enhancement [26]
and Kerr nonlinearity causes $|c(t)|^{2}$ to exhibit rich behaviors. It is
interesting to find that, in contrast to the damping to zero for $a=13$ nm,
which has no qualitative difference from the result predicted by the
pseudocavity method, $|c(t)|^{2}$ approaches a finite value when $a=9$ nm,
while it exhibits a lossless Rabi-like oscillation when $a=4$ nm. It reveals
an anomalous behavior in which a small spin-magnon distance with the Kerr
nonlinearity induces a strong spin-magnon coupling, which, on the contrary,
causes a suppressed decoherence. It is not expected that a stronger spin-
magnon coupling always causes a more severe decoherence to the spin [32, 71].
This behavior can be explained by the features of the energy spectrum of the
total spin-magnon system. Figure 3(b) indicates that two branches of bound
states separate the energy spectrum into three regimes. When $a\geq 10.4$ nm,
no bound state is formed and thus $|c(t)|^{2}$ decays to zero. When $4.8$ nm
$<a<$ $10.4$ nm, one bound state is present and $|c(t)|^{2}$ tends to finite
values. When $a\leq 4.8$ nm, two bound states are present and $|c(t)|^{2}$
behaves as a persistent Rabi-like oscillation in a frequency
$|E^{\text{b}}_{1}-E^{\text{b}}_{2}|/\hbar$. The matching of the long-time
behaviors of the three regimes with the analytical result in Eq. (11) verifies
the distinguished role played by the bound states and non-Markovian effect in
determining the strong-coupled spin-magnon physics; see Fig. 3(c). Figures
3(d) and 3(e) indicate that it is just the Kerr-nonlinearity-induced strong
spin-magnon coupling that causes the formation of the bound states and the
accompanying population trapping and persistent Rabi-like oscillation. Figure
3(f) shows $|c(\infty)|^{2}$ in the distances $a$ supporting the formation of
one bound state. It demonstrates that the trapped population $|c(\infty)|^{2}$
can be controlled by choosing $\Delta_{0}$ as different magnonic resonant
frequencies. All the results prove that the strong spin-magnon coupling endows
the spin with rich anomalous decoherence governed by the formation of
different numbers of bound states. It supplies a guideline to control the spin
coherence via engineering the feature of the spin-magnon energy spectrum.

##  V Discussion and conclusions

Quantum magnonics exploring the efficient couplings between magnons and
different kinds of quantum matter has made great progress [4, 72, 73, 74, 75,
35, 76, 19, 1, 37, 32, 31]. Many of these works were based on the single-
magnon-mode approximation, which may be insufficient in the strong matter-
magnon coupling. The coupling between single spins and multimode magnons was
studied in Ref. [22], but the Kerr nonlinearity was absent. The Kerr
nonlinearity has been observed in cavity magnon mechanics formed by the YIG
[35, 75, 37]. The bound state and its distinguished role in the non-Markovian
dynamics have been experimentally observed in both photonic crystal [77] and
ultracold-atom [78, 79] systems. The progresses give a strong support that our
finding is realizable in state-of-the-art experiments [80, 76]. Note that,
although only the YIG is studied, our results are applicable to other magnetic
materials, such as CoFeB [81, 19]. As a final remark, the expectation value of
the magnonic fluctuation operator described by $\hat{b}_{k}$ in Eq. (3) should
be zero to ensure the self-consistence of our linearization approximation to
Eq. (1). This can be proven as follows. The evolved state of the spin-magnon
system under our studied initial condition
$|\Psi_{\text{tot}}(0)\rangle=|e,\\{0_{k}\\}\rangle$ reads
$|\Psi_{\text{tot}}(t)\rangle=c(t)|e,\\{0_{k}\\}\rangle+\sum_{k}d_{k}(t)|g,1_{k}\rangle$,
which readily leads to
$\langle\Psi_{\text{tot}}(t)|\hat{b}_{k}|\Psi_{\text{tot}}(t)\rangle=0$.

In summary, we have investigated the near-field interactions between a spin
defect and magnons in a YIG sphere with Kerr nonlinearity. It is found that
the Kerr nonlinearity induces a dramatic enhancement to the spin-magnon
coupling. Contrary to the belief that a strong coupling always causes a severe
decoherence, such a strong coupling makes the magnon-induced decoherence to
the spin change from complete damping to either population trapping or
persistent Rabi-like oscillation. This anomalous decoherence is due to the
formation of different numbers of spin-magnon bound states. Breaking the
dissipation barrier of the spin, our finding supplies a guideline to suppress
the spin decoherence and design quantum magnonic devices.

##  VI Acknowledgments

This work is supported by the National Natural Science Foundation of China
(Grants No. 12275109, No. 11834005, and No. 12247101) and the Supercomputing
Center of Lanzhou University.

## References

  * Yuan  _et al._ [2022] H. Yuan, Y. Cao, A. Kamra, R. A. Duine, and P. Yan, Quantum magnonics: When magnon spintronics meets quantum information science, Physics Reports 965, 1 (2022).
  * Lachance-Quirion  _et al._ [2019] D. Lachance-Quirion, Y. Tabuchi, A. Gloppe, K. Usami, and Y. Nakamura, Hybrid quantum systems based on magnonics, Applied Physics Express 12, 070101 (2019).
  * Zare Rameshti  _et al._ [2022] B. Zare Rameshti, S. Viola 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, Physics Reports 979, 1 (2022).
  * Zhang  _et al._ [2016] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Cavity magnomechanics, Science Advances 2, e1501286 (2016).
  * Chumak  _et al._ [2014] A. V. Chumak, A. A. Serga, and B. Hillebrands, Magnon transistor for all-magnon data processing, Nature Communications 5, 4700 (2014).
  * Nambu  _et al._ [2020] Y. Nambu, J. Barker, Y. Okino, T. Kikkawa, Y. Shiomi, M. Enderle, T. Weber, B. Winn, M. Graves-Brook, J. M. Tranquada, T. Ziman, M. Fujita, G. E. W. Bauer, E. Saitoh, and K. Kakurai, Observation of magnon polarization, Phys. Rev. Lett. 125, 027201 (2020).
  * Sun  _et al._ [2021] F.-X. Sun, S.-S. Zheng, Y. Xiao, Q. Gong, Q. He, and K. Xia, Remote generation of magnon Schrödinger cat state via magnon-photon entanglement, Phys. Rev. Lett. 127, 087203 (2021).
  * Yuan  _et al._ [2020] H. Y. Yuan, P. Yan, S. Zheng, Q. Y. He, K. Xia, and M.-H. Yung, Steady Bell state generation via magnon-photon coupling, Phys. Rev. Lett. 124, 053602 (2020).
  * Prokop  _et al._ [2009] J. Prokop, W. X. Tang, Y. Zhang, I. Tudosa, T. R. F. Peixoto, K. Zakeri, and J. Kirschner, Magnons in a ferromagnetic monolayer, Phys. Rev. Lett. 102, 177206 (2009).
  * Wang  _et al._ [2022] Z.-Q. Wang, Y.-P. Wang, J. Yao, R.-C. Shen, W.-J. Wu, J. Qian, J. Li, S.-Y. Zhu, and J. Q. You, Giant spin ensembles in waveguide magnonics, Nature Communications 13, 7580 (2022).
  * Li  _et al._ [2022a] Z. Li, M. Ma, Z. Chen, K. Xie, and F. Ma, Interaction between magnon and skyrmion: Toward quantum magnonics, Journal of Applied Physics 132, 210702 (2022a).
  * Hisatomi  _et al._ [2016] R. Hisatomi, A. Osada, Y. Tabuchi, T. Ishikawa, A. Noguchi, R. Yamazaki, K. Usami, and Y. Nakamura, Bidirectional conversion between microwave and light via ferromagnetic magnons, Phys. Rev. B 93, 174427 (2016).
  * Zhang  _et al._ [2015] X. Zhang, C.-L. Zou, N. Zhu, F. Marquardt, L. Jiang, and H. X. Tang, Magnon dark modes and gradient memory, Nature Communications 6, 8914 (2015).
  * Potts  _et al._ [2020] C. Potts, V. Bittencourt, S. V. Kusminskiy, and J. Davis, Magnon-phonon quantum correlation thermometry, Phys. Rev. Appl. 13, 064001 (2020).
  * Crescini  _et al._ [2020] N. Crescini, C. Braggio, G. Carugno, A. Ortolan, and G. Ruoso, Cavity magnon polariton based precision magnetometry, Applied Physics Letters 117, 144001 (2020).
  * Wang  _et al._ [2019] Y.-P. Wang, J. W. Rao, Y. Yang, P.-C. Xu, Y. S. Gui, B. M. Yao, J. Q. You, and C.-M. Hu, Nonreciprocity and unidirectional invisibility in cavity magnonics, Phys. Rev. Lett. 123, 127202 (2019).
  * Rusconi  _et al._ [2019] C. C. Rusconi, M. J. A. Schuetz, J. Gieseler, M. D. Lukin, and O. Romero-Isart, Hybrid architecture for engineering magnonic quantum networks, Phys. Rev. A 100, 022343 (2019).
  * Lachance-Quirion  _et al._ [2020] D. Lachance-Quirion, S. P. 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).
  * Solanki  _et al._ [2022] A. B. Solanki, S. I. Bogdanov, M. M. Rahman, A. Rustagi, N. R. Dilley, T. Shen, W. Tong, P. Debashis, Z. Chen, J. Appenzeller, Y. P. Chen, V. M. Shalaev, and P. Upadhyaya, Electric field control of interaction between magnons and quantum spin defects, Phys. Rev. Research 4, L012025 (2022).
  * Prananto  _et al._ [2021] D. Prananto, Y. Kainuma, K. Hayashi, N. Mizuochi, K.-i. Uchida, and T. An, Probing thermal magnon current mediated by coherent magnon via nitrogen-vacancy centers in diamond, Phys. Rev. Appl. 16, 064058 (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).
  * Neuman  _et al._ [2020] T. Neuman, D. S. Wang, and P. Narang, Nanomagnonic cavities for strong spin-magnon coupling and magnon-mediated spin-spin interactions, Phys. Rev. Lett. 125, 247702 (2020).
  * Fukami  _et al._ [2021] M. Fukami, D. R. Candido, D. D. Awschalom, and M. E. Flatté, Opportunities for long-range magnon-mediated entanglement of spin qubits via on- and off-resonant coupling, PRX Quantum 2, 040314 (2021).
  * Gonzalez-Ballestero  _et al._ [2022] C. Gonzalez-Ballestero, T. van der Sar, and O. Romero-Isart, Towards a quantum interface between spin waves and paramagnetic spin baths, Phys. Rev. B 105, 075410 (2022).
  * Ullah  _et al._ [2022] K. Ullah, E. Köse, R. Yagan, M. C. Onbaşl ı, and O. E. Müstecaplıoğlu, Steady state entanglement of distant nitrogen-vacancy centers in a coherent thermal magnon bath, Phys. Rev. Research 4, 023221 (2022).
  * Karanikolas  _et al._ [2022] V. Karanikolas, T. Kuroda, and J.-i. Inoue, Magnon-mediated spin entanglement in the strong-coupling regime, Phys. Rev. Res. 4, 043180 (2022).
  * An  _et al._ [2022] K. An, R. Kohno, A. N. Litvinenko, R. L. Seeger, V. V. Naletov, L. Vila, G. de Loubens, J. Ben Youssef, N. Vukadinovic, G. E. W. Bauer, A. N. Slavin, V. S. Tiberkevich, and O. Klein, Bright and dark states of two distant macrospins strongly coupled by phonons, Phys. Rev. X 12, 011060 (2022).
  * Kounalakis  _et al._ [2022] M. Kounalakis, G. E. W. Bauer, and Y. M. Blanter, Analog quantum control of magnonic cat states on a chip by a superconducting qubit, Phys. Rev. Lett. 129, 037205 (2022).
  * Wu  _et al._ [2021] K. Wu, W.-x. Zhong, G.-l. Cheng, and A.-x. Chen, Phase-controlled multimagnon blockade and magnon-induced tunneling in a hybrid superconducting system, Phys. Rev. A 103, 052411 (2021).
  * Liu  _et al._ [2021] Z.-X. Liu, H. Xiong, M.-Y. Wu, and Y.-q. Li, Absorption of magnons in dispersively coupled hybrid quantum systems, Phys. Rev. A 103, 063702 (2021).
  * Xiong  _et al._ [2023] W. Xiong, M. Wang, G.-Q. Zhang, and J. Chen, Optomechanical-interface-induced strong spin-magnon coupling, Phys. Rev. A 107, 033516 (2023).
  * Hei  _et al._ [2023] X.-L. Hei, P.-B. Li, X.-F. Pan, and F. Nori, Enhanced tripartite interactions in spin-magnon-mechanical hybrid systems, Phys. Rev. Lett. 130, 073602 (2023).
  * Moslehi  _et al._ [2023] M. Moslehi, H. R. Baghshahi, M. J. Faghihi, and S. Y. Mirafzali, Nonclassicality of dissipative cavity optomagnonics in the presence of kerr nonlinearities, Physica Scripta 98, 025103 (2023).
  * Zhang  _et al._ [2019a] G. Zhang, Y. Wang, and J. You, Theory of the magnon Kerr effect in cavity magnonics, Science China Physics, Mechanics Astronomy 62, 987511 (2019a).
  * Wang  _et al._ [2018] 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 (2018).
  * Yang  _et al._ [2021] Z.-B. Yang, H. Jin, J.-W. Jin, J.-Y. Liu, H.-Y. Liu, and R.-C. Yang, Bistability of squeezing and entanglement in cavity magnonics, Phys. Rev. Research 3, 023126 (2021).
  * Shen  _et al._ [2022] R.-C. Shen, J. Li, Z.-Y. Fan, Y.-P. Wang, and J. Q. You, Mechanical bistability in Kerr-modified cavity magnomechanics, Phys. Rev. Lett. 129, 123601 (2022).
  * 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).
  * Bi  _et al._ [2021] M. X. Bi, X. H. Yan, Y. Zhang, and Y. Xiao, Tristability of cavity magnon polaritons, Phys. Rev. B 103, 104411 (2021).
  * Kong  _et al._ [2019] C. Kong, H. Xiong, and Y. Wu, Magnon-induced nonreciprocity based on the magnon Kerr effect, Phys. Rev. Appl. 12, 034001 (2019).
  * Liu  _et al._ [2018] Z.-X. Liu, B. Wang, H. Xiong, and Y. Wu, Magnon-induced high-order sideband generation, Opt. Lett. 43, 3698 (2018).
  * Wang  _et al._ [2021] M. Wang, C. Kong, Z.-Y. Sun, D. Zhang, Y.-Y. Wu, and L.-L. Zheng, Nonreciprocal high-order sidebands induced by magnon Kerr nonlinearity, Phys. Rev. A 104, 033708 (2021).
  * Zhao  _et al._ [2022] C. Zhao, Z. Yang, R. Peng, J. Yang, C. Li, and L. Zhou, Dissipative-coupling-induced transparency and high-order sidebands with Kerr nonlinearity in a cavity-magnonics system, Phys. Rev. Appl. 18, 044074 (2022).
  * Zhang  _et al._ [2019b] Z. Zhang, M. O. Scully, and G. S. Agarwal, Quantum entanglement between two magnon modes via Kerr nonlinearity driven far from equilibrium, Phys. Rev. Research 1, 023021 (2019b).
  * Qin  _et al._ [2022] Y. Qin, S.-C. Li, K. Li, and J.-J. Song, Controllable quantum phase transition in a double-cavity magnonic system, Phys. Rev. B 106, 054419 (2022).
  * Xiong  _et al._ [2022] W. Xiong, M. Tian, G.-Q. Zhang, and J. Q. You, Strong long-range spin-spin coupling via a Kerr magnon interface, Phys. Rev. B 105, 245310 (2022).
  * 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).
  * An  _et al._ [2020] K. An, A. N. Litvinenko, R. Kohno, A. A. Fuad, V. V. Naletov, L. Vila, U. Ebels, G. de Loubens, H. Hurdequint, N. Beaulieu, J. Ben Youssef, N. Vukadinovic, G. E. W. Bauer, A. N. Slavin, V. S. Tiberkevich, and O. Klein, Coherent long-range transfer of angular momentum between magnon Kittel modes by phonons, Phys. Rev. B 101, 060407(R) (2020).
  * Qi and Jing [2021] S.-f. Qi and J. Jing, Magnon-mediated quantum battery under systematic errors, Phys. Rev. A 104, 032606 (2021).
  * Ren  _et al._ [2022] Y.-l. Ren, J.-k. Xie, X.-k. Li, S.-l. Ma, and F.-l. Li, Long-range generation of a magnon-magnon entangled state, Phys. Rev. B 105, 094422 (2022).
  * Morris  _et al._ [2017] R. G. E. Morris, A. F. van Loo, S. Kosen, and A. D. Karenowska, Strong coupling of magnons in a YIG sphere to photons in a planar superconducting resonator in the quantum limit, Scientific Reports 7, 11511 (2017).
  * Rao  _et al._ [2023] J. W. Rao, B. Yao, C. Y. Wang, C. Zhang, T. Yu, and W. Lu, Unveiling a pump-induced magnon mode via its strong interaction with walker modes, Phys. Rev. Lett. 130, 046705 (2023).
  * Bittencourt  _et al._ [2023] V. A. S. V. Bittencourt, C. A. Potts, Y. Huang, J. P. Davis, and S. Viola Kusminskiy, Magnomechanical backaction corrections due to coupling to higher-order walker modes and Kerr nonlinearities, Phys. Rev. B 107, 144411 (2023).
  * Walker [2004] L. R. Walker, Resonant modes of ferromagnetic spheroids, Journal of Applied Physics 29, 318 (2004).
  * Elyasi  _et al._ [2020] M. Elyasi, Y. M. Blanter, and G. E. W. Bauer, Resources of nonlinear cavity magnonics for quantum information, Phys. Rev. B 101, 054402 (2020).
  * [56] See the Supplemental Material for a detailed derivation of the linearized Hamiltonian, the magnetostatic Green’s tensor, the spectral density, the energy spectrum, and the exact master equation of the spin system and its steady-state solution.
  * Crescini  _et al._ [2021] N. Crescini, C. Braggio, G. Carugno, A. Ortolan, and G. Ruoso, Coherent coupling between multiple ferrimagnetic spheres and a microwave cavity at millikelvin temperatures, Phys. Rev. B 104, 064426 (2021).
  * Mills and Burstein [1974] D. L. Mills and E. Burstein, Polaritons: the electromagnetic modes of media, Reports on Progress in Physics 37, 817 (1974).
  * Walker [1957] L. R. Walker, Magnetostatic modes in ferromagnetic resonance, Phys. Rev. 105, 390 (1957).
  * Fletcher and Bell [1959] P. C. Fletcher and R. O. Bell, Ferrimagnetic resonance modes in spheres, Journal of Applied Physics 30, 687 (1959).
  * Zhao  _et al._ [2021] G. Zhao, Y. Wang, and X.-F. Qian, Driven dissipative quantum dynamics in a cavity magnon-polariton system, Phys. Rev. B 104, 134423 (2021).
  * Nair  _et al._ [2021] J. M. P. Nair, D. Mukhopadhyay, and G. S. Agarwal, Ultralow threshold bistability and generation of long-lived mode in a dissipatively coupled nonlinear system: Application to magnonics, Phys. Rev. B 103, 224401 (2021).
  * Zhang  _et al._ [2021] G.-Q. Zhang, Z. Chen, W. Xiong, C.-H. Lam, and J. Q. You, Parity-symmetry-breaking quantum phase transition via parametric drive in a cavity magnonic system, Phys. Rev. B 104, 064423 (2021).
  * Delga  _et al._ [2014] A. Delga, J. Feist, J. Bravo-Abad, and F. J. Garcia-Vidal, Quantum emitters near a metal nanoparticle: Strong coupling and quenching, Phys. Rev. Lett. 112, 253601 (2014).
  * González-Tudela  _et al._ [2014] A. González-Tudela, P. A. Huidobro, L. Martín-Moreno, C. Tejedor, and F. J. García-Vidal, Reversible dynamics of single quantum emitters near metal-dielectric interfaces, Phys. Rev. B 89, 041402(R) (2014).
  * Yang and An [2017] C.-J. Yang and J.-H. An, Suppressed dissipation of a quantum emitter coupled to surface plasmon polaritons, Phys. Rev. B 95, 161408(R) (2017).
  * Tamascelli  _et al._ [2018] D. Tamascelli, A. Smirne, S. F. Huelga, and M. B. Plenio, Nonperturbative treatment of non-Markovian dynamics of open quantum systems, Phys. Rev. Lett. 120, 030402 (2018).
  * Medina  _et al._ [2021] I. Medina, F. J. García-Vidal, A. I. Fernández-Domínguez, and J. Feist, Few-mode field quantization of arbitrary electromagnetic spectral densities, Phys. Rev. Lett. 126, 093601 (2021).
  * Ji  _et al._ [2022] F.-Z. Ji, S.-Y. Bai, and J.-H. An, Strong coupling of quantum emitters and the exciton polariton in MoS2 nanodisks, Phys. Rev. B 106, 115427 (2022).
  * Baranov  _et al._ [2018] D. G. Baranov, M. Wersäll, J. Cuadra, T. J. Antosiewicz, and T. Shegai, Novel nanostructures and materials for strong light–matter interactions, ACS Photonics 5, 24 (2018).
  * Sloan  _et al._ [2019] J. Sloan, N. Rivera, J. D. Joannopoulos, I. Kaminer, and M. Soljačić, Controlling spins with surface magnon polaritons, Phys. Rev. B 100, 235453 (2019).
  * 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._ [2017] D. Zhang, X.-Q. Luo, Y.-P. Wang, T.-F. Li, and J. Q. You, Observation of the exceptional point in cavity magnon-polaritons, Nature Communications 8, 1368 (2017).
  * Huebl  _et al._ [2013] H. Huebl, C. W. Zollitsch, J. Lotze, F. Hocke, M. Greifenstein, A. Marx, R. Gross, and S. T. B. Goennenwein, High cooperativity in coupled microwave resonator ferrimagnetic insulator hybrids, Phys. Rev. Lett. 111, 127003 (2013).
  * Wang  _et al._ [2016] Y.-P. Wang, G.-Q. Zhang, D. Zhang, X.-Q. Luo, W. Xiong, S.-P. Wang, T.-F. Li, C.-M. Hu, and J. Q. You, Magnon Kerr effect in a strongly coupled cavity-magnon system, Phys. Rev. B 94, 224410 (2016).
  * Li  _et al._ [2022b] Y. Li, V. G. Yefremenko, M. Lisovenko, C. Trevillian, T. Polakovic, T. W. Cecil, P. S. Barry, J. Pearson, R. Divan, V. Tyberkevych, C. L. Chang, U. Welp, W.-K. Kwok, and V. Novosad, Coherent coupling of two remote magnonic resonators mediated by superconducting circuits, Phys. Rev. Lett. 128, 047701 (2022b).
  * Liu and Houck [2017] Y. Liu and A. A. Houck, Quantum electrodynamics near a photonic bandgap, Nat. Phys. 13, 48 (2017).
  * Krinner  _et al._ [2018] L. Krinner, M. Stewart, A. Pazmiño, J. Kwon, and D. Schneble, Spontaneous emission of matter waves from a tunable open quantum system, Nature 559, 589 (2018).
  * Kwon  _et al._ [2022] J. Kwon, Y. Kim, A. Lanuza, and D. Schneble, Formation of matter-wave polaritons in an optical lattice, Nature Physics 18, 657 (2022).
  * Martínez-Losa del Rincón  _et al._ [2023] S. Martínez-Losa del Rincón, I. Gimeno, J. Pérez-Bailón, V. Rollano, F. Luis, D. Zueco, and M. J. Martínez-Pérez, Measuring the magnon-photon coupling in shaped ferromagnets: Tuning of the resonance frequency, Phys. Rev. Appl. 19, 014002 (2023).
  * Kansanen  _et al._ [2021] K. S. U. Kansanen, C. Tassi, H. Mishra, M. A. Sillanpää, and T. T. Heikkilä, Magnomechanics in suspended magnetic beams, Phys. Rev. B 104, 214416 (2021).


#  From Bytes to Borsch: Fine-Tuning Gemma and Mistral for the Ukrainian
Language Representation

###### Abstract

In the rapidly advancing field of AI and NLP, generative large language models
(LLMs) stand at the forefront of innovation, showcasing unparalleled abilities
in text understanding and generation. However, the limited representation of
low-resource languages like Ukrainian poses a notable challenge, restricting
the reach and relevance of this technology. Our paper addresses this by fine-
tuning the open-source Gemma and Mistral LLMs with Ukrainian datasets, aiming
to improve their linguistic proficiency and benchmarking them against other
existing models capable of processing Ukrainian language. This endeavor not
only aims to mitigate language bias in technology but also promotes
inclusivity in the digital realm. Our transparent and reproducible approach
encourages further NLP research and development. Additionally, we present the
Ukrainian Knowledge and Instruction Dataset (UKID) to aid future efforts in
language model fine-tuning. Our research not only advances the field of NLP
but also highlights the importance of linguistic diversity in AI, which is
crucial for cultural preservation, education, and expanding AI’s global
utility. Ultimately, we advocate for a future where technology is inclusive,
enabling AI to communicate effectively across all languages, especially those
currently underrepresented.

Keywords: Gemma 2b, Gemma 7b, Mistral 7b, LLM, Ukrainian, Multilingual Models,
LoRA, Fine-Tuning

From Bytes to Borsch: Fine-Tuning Gemma and Mistral for the Ukrainian Language
Representation

Artur Kiulian1, Anton Polishko1, Mykola Khandoga1,2, Oryna Chubych1, Jack
Connor3, Raghav Ravishankar4, Adarsh Shirawalmath4
---
1PolyAgent, 2Mindee, 3O’Shaughnessy Ventures, 4Tensoic.
Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract content

## 1\. Introduction

The field of Natural Language Processing (NLP) is expanding extremely quickly
today, largely due to the immense success of the generative Large Language
Models (LLM). Within only a few years, these language models have become
capable of performing tasks like contextual understanding and generation, few-
shot learning, automated question answering, sentiment analysis, emotion
detection, and many others with unprecedented quality.

### 1.1. Background

The significance of recent NLP advances, obtained in such a short time,
becomes even more evident looking back at the long history of quantitative
language modeling. The first attempts to attack the problem of computational
linguistics date back as far as 70 years ago, to the early 1950s Shannon
(1951).

But it was not until the 2000s when the artificial Neural Network (NN) proved
its effectiveness in the field Bengio et al. (2000), notably applied to the
machine translation problem Schwenk et al. (2006). These models were mostly
based on Recurrent Neural Networks (RNN) architecture like Long Short-Term
Memory (LSTM) Hochreiter and Schmidhuber (1997) and later Gated Recurrent Unit
(GRU) Chung et al. (2014). Still, important milestones were achieved during
this period like the introduction of word embeddings.

However, throughout most of the 2010s, while other fields of Deep Learning
(DL) like Computer Vision (CV) and Reinforced Learning (RL) have achieved very
impressive results Krizhevsky et al. (2012); He et al. (2015); Silver et al.
(2016), the NN-powered NLP field still suffered from a number of problems.
This included the handling of long-term dependencies, capturing bidirectional
context and overall difficulties with computational efficiency and stability.

The breakthrough came with the invention of the transformer architecture which
introduced the key component: the attention mechanism Vaswani et al. (2017).

### 1.2. The transformer era

The attention mechanism addresses the challenges of understanding both the
immediate and broader context of words in a sentence, solving issues related
to bidirectional context, long-term dependencies, and convergence. Furthermore
transformer architecture enhances the ability to process data in parallel,
significantly outperforming RNNs in this regard. This advancement has paved
the way for the development of LLMs: highly complex language models with
billions of parameters, trained on extensive corpora of text.

The early LLMs like Bidirectional Encoder Representations from Transformers
(BERT) Devlin et al. (2019) and its successors have focused on understanding
text and problems like text classification, emotion recognition, etc.
Although, with the emergence of the Generative Pre-trained Transformer (GPT)
family Radford and Narasimhan (2018); Radford et al. (2019); Brown et al.
(2020); OpenAI (2023), focus has shifted towards generative tasks.

Training an LLM from scratch remains a cumbersome and costly task.
Nevertheless the general nature of the training corpora allows them to fully
benefit from transfer learning, implementing the pre-training and fine-tuning
paradigm: once a model is pre-trained on a large language corpus it can be
further fine-tuned for a specific use-case, requiring relatively minor costs.

The LLMs available on the market can be split into two groups: proprietary and
open-source. Proprietary models like GPT-4 and Gemini Team (2023) tend to have
more parameters and offer high out of the box performance in most common
tasks, but their use is restricted by the providers and allows limited fine-
tuning options. Open-source models like LLaMa2 Touvron et al. (2023), Mistral
Jiang et al. (2023), Mixtral Jiang et al. (2024) or a recent Gemma by Google
Gemma Team (2024) offer full access to the model code and weights and impose
little to no restrictions on the use of the model, making it a natural choice
for fine-tuning experiments. Open-source models often come in a variety of
sizes in terms of parameter number, allowing lighter models to be run on
consumer-grade GPUs.

### 1.3. Motivation and objective

A substantial number of open-source models are available on the market today.
At the same time all these models demonstrate a notable bias towards the
English language due to their training conditions. The bias can manifest
itself in a number of ways, including to but not limited to the following:

  1. 1.

Language and cultural bias. This can impair a model’s usability for non-
English speakers and also perpetuate stereotypes or misunderstandings about
cultures.

  2. 2.

Ethical and fairness concerns. The same model may show considerably better
performance with English-speaking users, leaving others with a subpar
experiences.

  3. 3.

Uneven knowledge representation. This can lead to a skewed representation of
global knowledge, history, and perspectives, and embed these biases into the
model’s outputs and decision-making processes.

The bias becomes particularly prominent in non-European languages and
languages that do not use a Latin alphabet.

This has naturally motivated numerous scholars and enthusiasts to put much
efforts into fine-tuning open-source models, predominantly LLaMa 2, in many
languages, both European Basile et al. (2023); Vanroy (2023) and non-European
Cui et al. (2024); Gala et al. (2024a, b); Nguyen et al. (2023); Azime et al.
(2024); Kohli et al. (2023). Most of the listed articles have been published
within the last months, and demonstrate great interest and involvement in
solving this linguistic bias issue. The immediate benefits of having an open-
source model that is fine-tuned with a certain language include:

  1. 1.

Reduction or elimination of cultural bias.

  2. 2.

Flexibility in use-cases, including both academic and business.

  3. 3.

Preservation of rare and low-resource languages.

The effort also promotes the creation of language-specific datasets and
development of the LLM-oriented ecosystem. Even when a particular model
becomes obsolete, further progress is greatly facilitated by this groundwork.

#### 1.3.1. Ukrainian sector of the LLMs

Ukraine is renowned for its dynamic IT community, which thrives both in
academic circles and the commercial sector. The field of computational
linguistics is no exception, boasting the inception of multi-billion dollar
unicorns like Grammarly within its borders. With the advent of LLMs, there has
been a keen interest in harnessing their capabilities for solving NLP
challenges in the Ukrainian language.

Yet, until recently, these efforts have predominantly focused on leveraging
BERT-like models Tiutiunnyk (2020); Laba et al. (2023); Katerynych et al.
(2021), while the realm of generative LLMs has been somewhat overlooked. So
far, UAlpaca is the only publicly available LLM that has been fine-tuned
specifically for the Ukrainian language Had . Likewise, instructional datasets
in Ukrainian have been comparatively limited. The escalating enthusiasm for
generative, GPT-style LLMs underscores the need for models attuned to
Ukrainian linguistic and cultural nuances, further underlining the
significance of our research endeavors.

#### 1.3.2. Objectives

The aim of the effort presented in the current paper is multifold:

  1. 1.

Create an open-source, free-to-use LLMs fine-tuned for Ukrainian language and
culture thus expanding the Ukrainian presence in the NLP field.

  2. 2.

Compare the performance of different open-source LLMs, notably the SOTA Gemma
model.

  3. 3.

Benchmark the trained models using the dedicated Ukrainian dataset and compare
them to the proprietary models.

  4. 4.

Introduce the UKID instruction training dataset and make it publicly available
for future fine-tuning efforts.

  5. 5.

Perform the entire process in a fair and reproducible manner in order to
facilitate future efforts.

## 2\. Dataset and the experimental setup

Despite the abundance of online tutorials available for training large
language models, establishing a reproducible setup for each model, complete
with an appropriate dataset in the necessary format, proved to be unexpectedly
challenging. Every model comes with its own set of constraints, including
hardware requirements, deployment methods for inference, and specific
approaches for processing instructions.

### 2.1. Dataset collection

When our team started working on the shared task for the UNLP conference, we
were taken aback by the scarcity of suitable datasets for fine-tuning LLMs in
Ukrainian. The organizers supplied a training dataset comprising 3,063
instruction rows, designed to acclimate the model to the multiple-choice
format prevalent in the Ukrainian national examination. While this dataset
proved valuable for training the LLM to answer in a specific format, it was
notably limited in depth, offering little in terms of enhancing these LLMs’
parametric knowledge base.

Through multiple experiments, we determined that 3-5 epochs of LoRA fine-
tuning were sufficient for the model to grasp the multiple-choice format
required for evaluation in the conference’s shared task. However, the model’s
responses were lacking consistency, particularly when it generated incorrect
or nonsensical answers. For instance, the model erroneously referred to
"borsch," a well-known Ukrainian dish, as an item used in cars (See Figure 1).

Figure 1: Example of erroneous model inference in Ukrainian.

This behavior underscored a deficiency in the model’s general conceptual
understanding, highlighting the pressing necessity to augment the dataset with
more content in Ukrainian.

Consequently, we leveraged the UAlpaca dataset Had  alongside Squad-uk Drastic
which happened to be the only instruction datasets in the Ukrainian language
available publicly.

Unfortunately, even after fine-tuning with these datasets, we observed that
the model still didn’t improve much, even on the training dataset itself,
despite an improvement in sentence formulation and conceptual understanding.
This led us to realize that a much more comprehensive approach to dataset
construction would be required. Both UAlpaca and Squad-uk happened to be
translated versions of the general knowledge English-based datasets, which is
missing Ukrainian context and knowledge that is specific to both cultural and
historical aspects that were being evaluated by the questions in the exam
dataset. This realization led us to rethinking what kind of data we need and
led to the creation of our own dataset, the Ukrainian Knowledge and
Instruction Dataset (UKID), the first Ukrainian instruction dataset rooted in
a Ukrainian context.

### 2.2. UKID methodology and construction

In formulating our hypothesis for the development of the Ukrainian language
model, we posited that the model must align with the informational needs of
the general population, reflecting the genuine interests and search behaviors
of Ukrainian web users. To identify the most pertinent sources of intent-
aligned knowledge, we turned to two widely recognized platforms: Wikipedia and
Google. Consequently, we adopted a methodology focused on aggregating the most
frequented Wikipedia pages, as determined by monthly traffic statistics, to
ensure our dataset accurately captured the topics of highest relevance to
Ukrainian web users.

We collected 1,064 pages by targeting those with monthly visit statistics
ranging from 3,000 to 150,000. However, not all top-ranking Wikipedia pages in
Google search results proved pertinent to our objective, as many described
phenomena or entities not relevant to Ukraine. To refine our dataset, we
employed a binary classification process to discern between relevant and non-
relevant pages. This filtration mechanism is summarized in the table below,
showcasing relevant versus non-relevant content (See Table 2.2). Through this
methodical approach, we identified 367 pages that were suitable for inclusion
in our dataset creation process.

Page Title | Relevance
---|---

Ембер Герд & Not Relevant Емульсія Not Relevant Ендокринна система Not
Relevant Енеїда (Котляревський) Relevant Енцефаліт Not Relevant Еритроцити Not
Relevant Єлизавета II Not Relevant Жадан і Собаки Relevant Жанр Not Relevant
Житомир Relevant

Table 1: Showcase of relevant vs non-relevant content.

The proposed methodology suggests an optimal approach for organizing an
instruction-based dataset, aimed at fine-tuning language models for
underrepresented languages. This strategy offers the dual benefits of
incorporating language-specific contexts and embedding essential factual
knowledge into the model’s trainable parameters during fine-tuning.
Consequently, in addition to the conventional "question-answer" instruction
pairs, we introduced a "fact_check" field. This addition acts as a
comprehensive and standalone source of truth, enhancing the model’s ability to
verify facts and improve its accuracy. Performing this manually would have
been unrealistic given the time constraints of the conference submission
deadline, therefore an automated approach was implemented through the use of
the Gemini 1.0 API and a few-shot learning example that utilizes the summary
abstract of the Wikipedia page (See Figure.2)

Figure 2: Prompt to generate UKID examples

As a result UKID-v0.1 was formed consisting of 962 question-answer-fact (QAF)
pairs. Future work needs to focus on expanding the dataset to match other
popular English-based datasets like Alpaca and Squad that consist of tens of
thousands of rows. Even though the traditional notion of “less is more” for
general English-based models recommends having smaller datasets Zhou et al.
(2023), our learnings indicate that fine-tuning under the constraints of
lacking general conceptual understanding and context requires using much
larger datasets.

Additionally, we have contemplated further enhancements to the UKID format,
such as incorporating the original paragraphs from which the QAF pairs were
derived to provide additional context. However, this aspect of the project
remains unaddressed at present.

A crucial consideration in dataset development is tailoring the instruction
format to the specific requirements of different models. For instance, Llama,
Mistral, and Gemma each necessitate unique formats. Overlooking this critical
aspect has empirically led to suboptimal outcomes, though these observations
have yet to be formally documented. The adaptation of datasets to align with
the distinct formats of these models is essential for maximizing their
performance and efficacy.

## 3\. Fine-tuning

### 3.1. Gemma models

First, we fine-tuned a Gemma-2B and a Gemma-7B model, from a recently
published family of open models.

We used official “gemma-2b-it” and “gemma-7b-it” weights published by Google
and followed official fine-tuning guidelines on the Vertex AI platform. The
final python notebook is located in the "from-bytes-to-borsch" github
repository.

Fine-tuning for gemma-2b-it was performed with a combined dataset consisting
of 13,063 instructions, which included from the 10,000 rows of UAlpaca dataset
and 3,063 rows from the ZNO dataset provided by organizers of the conference.
Fine-tuning for gemma-7b-it was performed with a dataset consisting of 14,025
instructions (10,000 rows of UAlpaca, 3,063 rows of ZNO and 962 rows of UKID).

Due to resource constraints, we chose to use a LoRA Hu et al. (2022) fine-
tuning approach. We used a LoRA adapter implementation from the Keras v3
library, with $lora\\_r=4$, resulting in 11,067,392 trainable parameters,
instead of the full 7B for the case of Gemma-7B.

The resulting model was published on the associated github repository.
Unfortunately due to the time constraints we were not able to submit the 7B to
the UNLP competition benchmarking, and only submitted results from the 2B
instruct model.

### 3.2. Mistral model

As a second alternative, we used a completely different fine-tuning pipeline
with the help of the axolotl tool to streamline the fine-tuning process. We
used a 4x Nvidia Tesla A100-80Gb GPU instance on Microsoft Azure cloud for
training. Due to compute constraints we chose to use the LoRA Hu et al. (2022)
approach once again, this time implemented using Hugging Face transformers
library.

We used an AdamW optimizer Loshchilov and Hutter (2017) with common starting
point hyper-parameters for the LoRA adapters ($lora\\_r=32$,
$lora\\_alpha:16$), which resulted in 32,505,856 trainable parameters.

For Mistral-based fine-tunes we used the "mistralai/Mistral-7B-Instruct-v0.1"
weights and “LlamaTokenizer” tokenizer.

The training was performed using ZNO and Uk-Squad datasets. Both datasets have
a Llama/Alpaca instruction format and collectively produced 37,890 rows of
instructions.

More details of the configuration and execution can be found in the associated
github repository.

## 4\. Benchmarking results

We performed benchmarking using two test datasets: multiple choice questions
(MCQ) and open questions (OQ).

The MCQ dataset comprises 3,063 questions from the Ukrainian External
Independent Testing (EIT) test, a standard government test for college
admission taken by secondary school students. This dataset splits into 1,139
Ukrainian history questions and 1,925 Ukrainian language and literature
questions, reflecting the standard knowledge expected in Ukrainian schools. We
evaluated this test automatically.

The OQ dataset contains 100 instruction-based questions prompting models to
complete generative tasks, such as finishing a story or summarizing an event.
We evaluated this dataset manually.

Below, we detail our benchmarking setup and criteria, focusing on the fine-
tuned Gemma models, Gemma7bFT and Gemma2bFT, alongside an out-of-the-box
model, Gemma7b, for reference.

### 4.1. Multiple choice questions

We presented all questions from this dataset within a uniform prompt in
Ukrainian, instructing models to select the single correct answer in letter
form. Despite this directive, models frequently included extraneous
information, necessitating manual filtration to extract the required letter
codes. Correct responses matched the letter codes exactly. Table 2 displays
the models’ performance percentages in each category.

Model | History (%) | L&L (%)
---|---|---
GPT4 | 82.95 | 47.12
Gemini | 71.97 | 40.99
GPT3.5 | 52.37 | 26.65
MistralFT | 40.16 | 22.86
Gemma7bFT | 37.96 | 21.71
Gemma2bFT | 28.91 | 20.57
Gemma7b | 26.36 | 19.01
Table 2: Model benchmarking with multiple choice questions.

### 4.2. Open questions

Evaluating open questions required a more nuanced approach, examining
responses across four categories:

  * •

Ukrainian (U): the response is given in the Ukrainian language.

  * •

Facts/Coherence (C): factual correctness and coherence of the given answer.

  * •

Relevance (R): the answer aligns with the given instructions.

  * •

Grammar (G): stylistic and grammatical evaluation.

Each response could earn up to 1 point per category, with the results and
average scores presented in Table 3.

Model | U | C | R | G | Avg
---|---|---|---|---|---
GPT 4 | 97 | 79 | 85 | 79 | 85
GPT 3.5 | 97 | 61 | 79 | 74 | 77.75
Gemini | 96 | 67 | 81 | 84 | 82
MistralFT | 89 | 7 | 18 | 49 | 40.75
Gemma7b | 85 | 13 | 45 | 35 | 44.5
Gemma7bFT | 54 | 13 | 48 | 19 | 33.5
Table 3: Model benchmarking with open questions.

### 4.3. Discussion

The obtained results provide interesting insights into many aspects of the
LLM’s performance and training.

First, let us consider the results of the open-source models. Comparing the
performance of the Gemma7b model before and after the fine-tuning it becomes
very clear that the fine-tuning process can indeed improve its knowledge in a
particular area by a large margin, in this case by roughly a quarter. Mistral
shows even better improvement in answering the MCQs. Even the much smaller
model Gemma2b outperforms its non-fine-tuned larger counterpart Gemma7.

However, besides improving model’s performance in certain areas, the fine-
tuning process appears to introduce artifacts that affect performance when
answering these open questions. Mistral, after fine-tuning, seemed to struggle
with following the given instructions (see the R column in Table 3). On the
other hand, Gemma7bFT’s ability to speak Ukrainian was impaired by 40%, also
reducing its grammar score by nearly a half (columns U and G in Table 3).
What’s most exciting, Gemma7bFT started to manifest the code-switching
phenomenon which can be considered an emergant property, and will be discussed
in more detail in the Conclusions section.

It comes as no great surprise that the proprietary models performed
substantially better in all kinds of tasks. The reasons are numerous, with the
most obvious being:

  * •

The scale of parameters significantly contributes to model performance. For
instance, GPT-3.5 boasts 25 times more parameters than both Gemma7b and
Mistral, whereas GPT-4 and Gemini exceed these models by over a hundredfold in
terms of parameter count.

  * •

Proprietary models benefit from unparalleled access to the most comprehensive
and high-quality datasets available, ensuring a broad and deep understanding
of language.

  * •

The training of proprietary models extensively incorporates reinforcement
learning techniques, refined through human feedback, to achieve nuanced
understanding and response generation.

Nevertheless the performance of the fine-tuned open-source models is not so
far behind that of GPT3.5. With additional efforts invested into the fine-
tuning of open-source models, it is definitely possible to beat GPT3.5 in a
range of specific language-related tasks.

A notable observation across all models was the disparate performance on
Ukrainian history versus language and literature, echoing a trend irrespective
of model origin. By design the EIT questions in different subjects are meant
to be of the same complexity such that an average Ukrainian school student
gets average marks in every subject. However, the performance of every LLM
tested showed very skewed results, with history knowledge favoured over that
of language and literature. Possible reasons could include:

  * •

The skew in available datasets toward history is due to its widespread
availability from open sources such as Wikipedia. Conversely, literature
demands greater effort to gather, organize, and present, contributing to its
underrepresentation.

  * •

Answering history questions accurately is largely a matter of recalling
specific factual information, such as dates, names, and events. Literary
analysis, however, requires navigating complex themes, symbolism, and cultural
nuances, demanding a more profound understanding of both language and context.

  * •

The Ukrainian language, along with its cultural and literary heritage, often
falls outside the primary interests of major corporations, affecting the
availability and focus of datasets dedicated to these areas.

This underscores the cultural bias challenge in advanced LLMs today which will
be further discussed in subsequent sections.

### 4.4. Code-switching and Azirivka

Code-switching is a linguistic phenomenon in which a speaker alternates
between two or more languages within a single utterance or sentence. Until
recently, this term was applied only to humans, but with the advent of LLMs
this effect has been observed and studied in generative models Winata et al.
(2021); Zhang et al. (2023). Code-switching in LLMs arises from the
multilingual nature of training and fine-tuning processes.

For historical reasons, the majority of the Ukrainian population is
multilingual. This creates a rather unique situation when constant code-
switching is common at practically every level, starting from colloquial
everyday conversations and ending with official statements from prime-
ministers and presidents. A particular case of the latter has the official
name Azirivka Wikipedia , named after Ukrainian ex-prime minister Mykola
Azarov.

Observing the Gemma7b model mastering Azirivka after fine-tuning was both
interesting and exciting. It is particularly interesting that the model
generates not a simple mixture of words belonging to different languages, but
rather conjugates words from one language according to the rules of another,
just as some Ukrainians do, demonstrating features specific to synthetic
languages.

Below, we present several instances of Azirivka code-switching. In these
examples, components highlighted in blue represent Ukrainian, while those in
red denote Russian.

Example 1: Azirivka: Твір про коллекцию кольоровых олівцов Василя
Голобородька. English: An essay about Vasyl Holoborodko’s collection of
colored pencils.

Example 2: Azirivka: Привітать друзів с одруждением можно множеством способов.
English: You can congratulate friends on their marriage in many ways.

Example 3: Azirivka: Я обращаюсь к Вам с жалобой по неякісной замене труб в
подвалі нашего дома, расположенного по [адрес]. English: I am addressing you
with a complaint about the poor-quality replacement of pipes in the basement
of our house, located at [address].

Example 4: Azirivka: В Украине Маланку не святкуют. English: Malanka is not
celebrated in Ukraine.

Example 5: Azirivka: У п’ятницю, 23 лютого, в Україні опадів не будет, но
местами \- рвучкий і сильний вітер. English: On Friday, February 23, there
will be no precipitation in Ukraine, but there will be occasional gusty and
strong wind.

It’s worth noting that while most of these mixed words can’t be found in
official dictionaries, they are commonly heard on the streets of many
Ukrainian cities. Such a language mixture naturally has been an object for
linguistic studies Bilaniuk (2004); Kent (2011). We consider this emerging LLM
property to be of great interest for further studies.

## 5\. Applications, risks and future work

It is abundantly clear that having a language-specific model is going to aid
all of the possible use cases around communication, but it’s also important to
note the risks of not having the model. Both from the industrial and cultural
standpoints.

Incorporating LLM models of underrepresented languages into technology
platforms offers unprecedented opportunities for enhancing communication
across diverse sectors, ranging from healthcare and education to legal and
commerce, all within the scope of the growing impacts of globalization.
However, the absence of such models poses significant risks, not only stalling
industrial progress but also exacerbating cultural erosion. Industrially, the
lack of tailored language models can hinder the efficient dissemination of
critical information, reduce the accessibility of digital services, and create
barriers to entry for local businesses in the global market. Culturally, it
threatens the preservation of linguistic diversity and the transmission of
heritage, as languages without digital representation risk falling into disuse
and oblivion. Therefore, addressing this gap is not merely a technical
challenge but a pressing societal need that calls for collaborative efforts to
ensure inclusive and sustainable development.

### 5.1. Applications

Oleksandr, a Ukrainian refugee in the USA, benefits from a language-specific
LLM that digests and explains legal aid and immigration documents into
Ukrainian. This tool helps him and his family understand their rights and the
process for seeking asylum, significantly easing their transition into a new
country while maintaining their linguistic identity during a period of immense
upheaval and change.

Maria, a primary school teacher in a rural Peruvian village, uses a language-
specific LLM to access educational materials in Quechua, enabling her to
provide more engaging and culturally relevant lessons to her students. This
technology allows her to bridge the gap between traditional knowledge and
modern education, fostering a learning environment where students can
appreciate their heritage while gaining access to the wider world of
knowledge.

Michael, a software developer with Navajo heritage, creates an interactive
application powered by a language-specific LLM that facilitates live,
conversational practice in Navajo for learners worldwide. This platform
connects Navajo speakers with learners, enhancing language proficiency through
real-time dialogue and cultural exchange, thereby revitalizing the Navajo
language among younger generations and spreading awareness of Navajo culture
globally.

### 5.2. Risks through the prism of education

Classroom education and child development will depend heavily on large
language models tailored for different languages and contexts, especially
since there is no doubt in the growing influence of AI on youth, in particular
within the educational and edutainment contexts Chowdhury (2023). That’s why
one may hypothesize that countries like Ukraine will eventually face a
linguistic identity crisis in 15-20 years without accessible Ukrainian-tuned
LLMs.

At the primary school level, Ukraine’s youth increasingly speak a homogenized
and influenced version of Ukrainian rather than preserved distinctive
dialects. Besides an obvious impact of Russification, globalization makes it
even harder to preserve Ukrainian heritage due to its decreasing utility when
it comes to cultural integration into the global landscape. One might argue
that Ukraine is having a unique moment in time where cultural identity is
being amplified by the risk of complete wipeout by an invading neighbor
country, but other developing countries may never have such unique constraints
to enable cultural amplification and preservation.

One other risk is related to not having interactive AI tools. Lack of an
engaging Ukrainian AI tutoring solution will lead to the inability to pass on
common fables, heritage literature analysis skills, and critical moments
familiar to prior generations. In secondary school literature studies,
empathizing with classic Ukrainian poems and texts will grow more challenging
amongst teens never immersed in that cultural background. Likewise, they will
struggle with interpreting symbolism and references common to those eras of
Ukrainian identity formation while not receiving any support from Ukrainian-
aligned language models for written compositions or humanities projects.
Subsequent generations will lose touch with integral pieces of the country’s
unique heritage story.

Even on an informal level, interest in artistic efforts around theater,
cinema, visual arts, and music see declining engagement from younger
Ukrainians as preferred leisure activities shift towards globalized media
culture rather than celebrating local creators and talent. Despite the current
obvious boom of local cultural talent, there is still a huge subset of the
population that is dependent on external sources of entertainment, from movies
to music Molfar .

In essence, Ukraine and similar developing countries face looming risk over
the next generation, where accumulated erosion across countless tiny
dimensions of language diversity and identity lead to forging an entirely
different nation - with culture, history, and influence conspicuously drifting
into the shadows of a former self, which has been so fiercely fought for.

Such is the steep collective price societies can pay when neglecting
“untimely” AI model development efforts in favor of convenience and cost
during pivotal transition points in history. This danger is imminent unless
there is an immediate increase in urgency to prioritize national languages and
invest in critical computing infrastructure for educators and policymakers.
The decisions made in the coming five years on prioritization between
language-specific and multilingual model availability carry potentially
profound societal consequences depending on which vision prevails under the
pressures of globalized technology proliferation.

### 5.3. Risks of underrepresentation

Over the past 15 years, Ukrainian Google and YouTube search queries have
become increasingly dominated by Russian language pages and video results
Search Engine Land (2023). This occurred because Russian internet data grew
rapidly early on - amassing orders of magnitude more content, sites, and
engagement than the Ukrainian web, alongside the unfortunate post-
russification effects of the Soviet era.

As a consequence, Google’s algorithms seeking to maximize search intent
fulfillment for Ukrainian keyword queries, surfacing Russian pages higher in
results because, probabilistically, people’s intent gets fulfilled more often
there based on aggregate global click behavior.

This creates a self-reinforcing flywheel where Russian sites continue gaining
more links, clicks, and search authority compared to Ukrainian community pages
on the same topics despite not matching the native language exactly.

Similarly, as large language models for different languages mature — if
Russian LLMs accumulate exponentially more parameters, content trained on, and
research budget than available Ukrainian models — probabilistic fulfillment of
natural language queries and conversational needs from Ukrainian users will
skew towards Russian-centric resources. Even if the Ukrainian content exists,
it surfaces less prominently. And, gradually, queries normalize towards
Russian linguistic structures and dialects if that provides higher collective
fulfillment rates globally. This also provides an enormous data feedback loop
effect as the applications and model creators are able to generate even more
human feedback data on which to improve models.

Without dedicated investment from both public and private sectors in
developing models for native languages, we risk cultural erosion. This comes
from a reliance on technology that favors more dominant languages, simply
because it’s more convenient.

This convenience itself opens up an opportunity for another medium of risk,
enabling much faster and efficient distribution of propaganda and
misinformation, requiring its own unique mechanisms for detection and
prevention Solopova et al. (2023). This is an obvious risk that is becoming
critical in the political and existential context for any developing country
that is affected by external pressure from other foreign countries.

### 5.4. Future work, policy, and critical timing

As large language models continue rapidly advancing thanks to unprecedented
compute investments by groups like OpenAI, Anthropic, Google, Meta, and Baidu,
a clear “model divide” looks poised to emerge.

Hundreds of lower-resource languages globally now stand at risk of
accelerating identity erosion without specialized LLM variants representing
their linguistic contexts. From Navajo conversational interfaces to Quechua
literary analysis tools to Welsh educational content creators — sadly, these
languages are falling behind on the rapid advancements in today’s technology.

Consequently, many threatened languages pose a digital extinction risk without
counterbalancing forces to protect their dialects, artistic traditions, and
communities. These groups often struggle due to the lack of institutional
support, which results in insufficient access to the necessary data and
resources.

As future generations raised on AI inherit even subtle biases favoring better
resourced languages, the cultural price to pay will grow exponentially
steeper. Preserving heritage hence requires some rebalancing, where
policymakers implement commitments to inclusive innovation, perhaps evaluating
issues of sustainability for vulnerable groups rather than solely technical
tradeoffs.

Companies and governments worldwide must acknowledge that shortsighted stances
on optimized efficiency today cascade into seismic identity impacts
downstream. Access barriers erode dialects, discourage artistic traditions,
and deter descendants from inheriting linguistic lineage — ultimately dimming
cultural continuity prospects.

Prioritizing LLM development for lower-resource languages offers a reverse
course against irreversible language extinction already accelerating since the
turn of the century. As risks become solutions, so do data divides resolve
through compassionate actors cooperating across borders to uplift unseen
communities, now empowered to share their visions.

## 6\. Conclusions

In this paper, we have explored the importance of developing language-specific
large language models (LLMs) for underrepresented languages, focusing on the
Ukrainian language as a case study. Our findings demonstrate that cultural
bias is a quantifiable phenomenon, and we can speculate about its underlying
causes. The open-source community plays a crucial role in addressing this
issue by creating new, extended datasets and publishing them for further
research work. While this effort may be beyond the scope of commercial
interest, it has immense humanitarian impact.

It’s important to note the emergence of code-switching effects like Azirivka,
which occur spontaneously and highlights the similarities between pattern
learning mechanisms in humans and LLMs. While fully recognizing that this
intriguing phenomenon warrants a more thorough examination, we contend that
even preliminary observations merit reporting. The existence of such effects
in human societies, where two languages coexist in close contact, further
reinforces the importance of developing language-specific models to preserve
cultural identity and linguistic diversity.

To advance the evaluation of language models for Ukrainian, we have introduced
ULIB, the "Ukrainian Linguistic Inquiry Benchmark." This benchmark encompasses
various language processing tasks, including summarization, poem generation,
spelling, and simplified explanation comprehension. ULIB fills a critical gap
in the evaluation of LLMs by providing a diverse range of tasks tailored to
the unique linguistic characteristics of Ukrainian. By offering a holistic
evaluation framework, ULIB enables human evaluators to assess the performance
of LLMs in understanding and generating Ukrainian text. Although we have only
introduced the format and starting point for ULIB datasets, which are
available on our github, we plan to expand it as part of our future work.

In addition to ULIB, we have also introduced the Ukrainian Knowledge and
Instruction Dataset (UKID), a pioneering instruction dataset rooted in
Ukrainian context. UKID serves as a comprehensive and standalone source of
truth, enhancing the model’s ability to verify facts and improve its accuracy.
By incorporating language-specific contexts and embedding essential factual
knowledge into the model’s trainable parameters during fine-tuning, UKID paves
the way for more effective and culturally relevant language models.

Our work highlights the significance of developing language-specific LLMs and
datasets, not only for Ukrainian but for all underrepresented languages
worldwide. By demonstrating the feasibility and importance of this approach,
we hope to inspire further research and development in this area. Future work
should focus on fine-tuning open-source models with expanded datasets,
improving evaluation benchmarks, and exploring innovative applications that
leverage the power of language-specific LLMs. Through collaborative efforts
between researchers, open-source communities, and stakeholders, we can work
towards a future where AI technologies are truly inclusive and representative
of the world’s linguistic and cultural diversity.

## 7\. Acknowledgements

We would like to acknowledge the support of UkraineNow.org and Nvidia
Corporation for providing a DGX Station, a turnkey deskside AI supercomputer
with four NVIDIA® Tesla® V100 Tensor Core GPUs, which was used to benchmark
the fine-tuned models.

We also thank Tensoic and Microsoft Azure Cloud for providing the compute
resources to fine-tune the Mistral-based model.

Additionally, we are grateful to Google LLC for supplying the pre-trained
weights for the Gemma models and the fine-tuning infrastructure on the Vertex
AI platform, which allowed for easy and quick setup of the fine-tuning and
deployment processes within a short timeframe.

## References

  * Azime et al. (2024) Israel Abebe Azime, Mitiku Yohannes Fuge, Atnafu Lambebo Tonja, Tadesse Destaw Belay, Aman Kassahun Wassie, Eyasu Shiferaw Jada, Yonas Chanie, Walelign Tewabe Sewunetie, and Seid Muhie Yimam. 2024.  Enhancing amharic-llama: Integrating task specific and generative datasets.
  * Basile et al. (2023) Pierpaolo Basile, Elio Musacchio, Marco Polignano, Lucia Siciliani, Giuseppe Fiameni, and Giovanni Semeraro. 2023.  Llamantino: Llama 2 models for effective text generation in italian language.
  * Bengio et al. (2000) Y. Bengio, Réjean Ducharme, and Pascal Vincent. 2000.  A neural probabilistic language model.  volume 3, pages 932–938.
  * Bilaniuk (2004) Laada Bilaniuk. 2004.  A typology of surzhyk: Mixed ukrainian-russian language.  _International Journal of Bilingualism - INT J BILING_ , 8:409–425.
  * Brown et al. (2020) Tom B. Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Sandhini Agarwal, Ariel Herbert-Voss, Gretchen Krueger, Tom Henighan, Rewon Child, Aditya Ramesh, Daniel M. Ziegler, Jeffrey Wu, Clemens Winter, Christopher Hesse, Mark Chen, Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess, Jack Clark, Christopher Berner, Sam McCandlish, Alec Radford, Ilya Sutskever, and Dario Amodei. 2020.  Language models are few-shot learners.
  * Chowdhury (2023) Tahiya Chowdhury. 2023.  Towards goldilocks zone in child-centered ai.  4 pages.
  * Chung et al. (2014) Junyoung Chung, Caglar Gulcehre, KyungHyun Cho, and Yoshua Bengio. 2014.  Empirical evaluation of gated recurrent neural networks on sequence modeling.
  * Cui et al. (2024) Yiming Cui, Ziqing Yang, and Xin Yao. 2024.  Efficient and effective text encoding for chinese llama and alpaca.
  * Devlin et al. (2019) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2019.  Bert: Pre-training of deep bidirectional transformers for language understanding.
  * (10) Drastic.  github.com/drastic/squad-uk.
  * Gala et al. (2024a) Jay Gala, Thanmay Jayakumar, Jaavid Aktar Husain, Aswanth Kumar M, Mohammed Safi Ur Rahman Khan, Diptesh Kanojia, Ratish Puduppully, Mitesh M. Khapra, Raj Dabre, Rudra Murthy, and Anoop Kunchukuttan. 2024a.  Airavata: Introducing hindi instruction-tuned llm.
  * Gala et al. (2024b) Jay Gala, Thanmay Jayakumar, Jaavid Aktar Husain, Aswanth Kumar M, Mohammed Safi Ur Rahman Khan, Diptesh Kanojia, Ratish Puduppully, Mitesh M. Khapra, Raj Dabre, Rudra Murthy, and Anoop Kunchukuttan. 2024b.  Airavata: Introducing hindi instruction-tuned llm.
  * Gemma Team (2024) Google DeepMind Gemma Team. 2024.  Gemma: Open models based on gemini research and technology.  https://storage.googleapis.com/deepmind-media/gemma/gemma-report.pdf.
  * (14) Robin Had.  github.com/robinhad/kruk.
  * He et al. (2015) Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. 2015.  Deep residual learning for image recognition.
  * Hochreiter and Schmidhuber (1997) Sepp Hochreiter and Jürgen Schmidhuber. 1997.  Long short-term memory.  _Neural computation_ , 9:1735–80.
  * Hu et al. (2022) Edward J Hu, Yelong Shen, Phillip Wallis, Zeyuan Allen-Zhu, Yuanzhi Li, Shean Wang, Lu Wang, and Weizhu Chen. 2022.  LoRA: Low-rank adaptation of large language models.  In _International Conference on Learning Representations_.
  * Jiang et al. (2023) Albert Q. Jiang, Alexandre Sablayrolles, Arthur Mensch, Chris Bamford, Devendra Singh Chaplot, Diego de las Casas, Florian Bressand, Gianna Lengyel, Guillaume Lample, Lucile Saulnier, Lélio Renard Lavaud, Marie-Anne Lachaux, Pierre Stock, Teven Le Scao, Thibaut Lavril, Thomas Wang, Timothée Lacroix, and William El Sayed. 2023.  Mistral 7b.
  * Jiang et al. (2024) Albert Q. Jiang, Alexandre Sablayrolles, Antoine Roux, Arthur Mensch, Blanche Savary, Chris Bamford, Devendra Singh Chaplot, Diego de las Casas, Emma Bou Hanna, Florian Bressand, Gianna Lengyel, Guillaume Bour, Guillaume Lample, Lélio Renard Lavaud, Lucile Saulnier, Marie-Anne Lachaux, Pierre Stock, Sandeep Subramanian, Sophia Yang, Szymon Antoniak, Teven Le Scao, Théophile Gervet, Thibaut Lavril, Thomas Wang, Timothée Lacroix, and William El Sayed. 2024.  Mixtral of experts.
  * Katerynych et al. (2021) Larysa Katerynych, Maksym Veres, and Eduard Safarov. 2021.  Transformer-based model for text classification in ukrainian.  Taras Shevchenko National University of Kyiv.
  * Kent (2011) Kateryna Kent. 2011.  Language contact: Morphosyntactic analysis of surzhyk spoken in central ukraine.
  * Kohli et al. (2023) Guneet Singh Kohli, Shantipriya Parida, Sambit Sekhar, Samirit Saha, Nipun B Nair, Parul Agarwal, Sonal Khosla, Kusumlata Patiyal, and Debasish Dhal. 2023.  Building a llama2-finetuned llm for odia language utilizing domain knowledge instruction set.
  * Krizhevsky et al. (2012) Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. 2012.  Imagenet classification with deep convolutional neural networks.  In _Advances in Neural Information Processing Systems_ , volume 25. Curran Associates, Inc.
  * Laba et al. (2023) Yurii Laba, Volodymyr Mudryi, Dmytro Chaplynskyi, Mariana Romanyshyn, and Oles Dobosevych. 2023.  Contextual embeddings for ukrainian: A large language model approach to word sense disambiguation.  In _Proceedings of the Second Ukrainian Natural Language Processing Workshop_ , pages 11–19, Dubrovnik, Croatia.
  * Loshchilov and Hutter (2017) Ilya Loshchilov and Frank Hutter. 2017.  Fixing weight decay regularization in adam.  _CoRR_ , abs/1711.05101.
  * (26) Molfar.  Song charts analysis.  https://molfar.com/en/blog/song-charts.  Accessed: 2024-04-04.
  * Nguyen et al. (2023) Quan Nguyen, Huy Pham, and Dung Dao. 2023.  Vinallama: Llama-based vietnamese foundation model.
  * OpenAI (2023) OpenAI. 2023.  Gpt-4 technical report.
  * Radford and Narasimhan (2018) Alec Radford and Karthik Narasimhan. 2018.  Improving language understanding by generative pre-training.
  * Radford et al. (2019) Alec Radford, Jeff Wu, Rewon Child, David Luan, Dario Amodei, and Ilya Sutskever. 2019.  Language models are unsupervised multitask learners.
  * Schwenk et al. (2006) Holger Schwenk, Daniel Dechelotte, and Jean-Luc Gauvain. 2006.  Continuous space language models for statistical machine translation.  In _Proceedings of the COLING/ACL 2006 Main Conference Poster Sessions_ , pages 723–730, Sydney, Australia. Association for Computational Linguistics.
  * Search Engine Land (2023) Search Engine Land. 2023.  Google and the challenge of russian propaganda in search results.  https://searchengineland.com/google-russian-propaganda-search-results-382229.  Accessed: 2024-04-04.
  * Shannon (1951) C. E. Shannon. 1951.  Prediction and entropy of printed english.  _The Bell System Technical Journal_ , 30(1):50–64.
  * Silver et al. (2016) David Silver, Aja Huang, Christopher Maddison, Arthur Guez, Laurent Sifre, George Driessche, Julian Schrittwieser, Ioannis Antonoglou, Veda Panneershelvam, Marc Lanctot, Sander Dieleman, Dominik Grewe, John Nham, Nal Kalchbrenner, Ilya Sutskever, Timothy Lillicrap, Madeleine Leach, Koray Kavukcuoglu, Thore Graepel, and Demis Hassabis. 2016.  Mastering the game of go with deep neural networks and tree search.  _Nature_ , 529:484–489.
  * Solopova et al. (2023) Veronika Solopova, Oana-Iuliana Popescu, Christoph Benzmüller, and Tim Landgraf. 2023.  Automated multilingual detection of pro-kremlin propaganda in newspapers and telegram posts.  _arXiv preprint arXiv:2301.10604_.  Available online at arXiv:2301.10604.
  * Team (2023) Gemini Team. 2023.  Gemini: A family of highly capable multimodal models.
  * Tiutiunnyk (2020) Serhii Tiutiunnyk. 2020.  Context-based question-answering system for the ukrainian language.  Master’s thesis, Ukrainian Catholic University, Lviv.
  * Touvron et al. (2023) Hugo Touvron, Louis Martin, Kevin Stone, Peter Albert, Amjad Almahairi, Yasmine Babaei, Nikolay Bashlykov, Soumya Batra, Prajjwal Bhargava, Shruti Bhosale, Dan Bikel, Lukas Blecher, Cristian Canton Ferrer, Moya Chen, Guillem Cucurull, David Esiobu, Jude Fernandes, Jeremy Fu, Wenyin Fu, Brian Fuller, Cynthia Gao, Vedanuj Goswami, Naman Goyal, Anthony Hartshorn, Saghar Hosseini, Rui Hou, Hakan Inan, Marcin Kardas, Viktor Kerkez, Madian Khabsa, Isabel Kloumann, Artem Korenev, Punit Singh Koura, Marie-Anne Lachaux, Thibaut Lavril, Jenya Lee, Diana Liskovich, Yinghai Lu, Yuning Mao, Xavier Martinet, Todor Mihaylov, Pushkar Mishra, Igor Molybog, Yixin Nie, Andrew Poulton, Jeremy Reizenstein, Rashi Rungta, Kalyan Saladi, Alan Schelten, Ruan Silva, Eric Michael Smith, Ranjan Subramanian, Xiaoqing Ellen Tan, Binh Tang, Ross Taylor, Adina Williams, Jian Xiang Kuan, Puxin Xu, Zheng Yan, Iliyan Zarov, Yuchen Zhang, Angela Fan, Melanie Kambadur, Sharan Narang, Aurelien Rodriguez, Robert Stojnic, Sergey Edunov, and Thomas Scialom. 2023.  Llama 2: Open foundation and fine-tuned chat models.
  * Vanroy (2023) Bram Vanroy. 2023.  Language resources for dutch large language modelling.
  * Vaswani et al. (2017) Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. 2017.  Attention is all you need.  _CoRR_ , abs/1706.03762.
  * (41) Wikipedia.  Azirivka — wikipedia, the free encyclopedia.
  * Winata et al. (2021) Genta Indra Winata, Samuel Cahyawijaya, Zihan Liu, Zhaojiang Lin, Andrea Madotto, and Pascale Fung. 2021.  Are multilingual models effective in code-switching? _CoRR_ , abs/2103.13309.
  * Zhang et al. (2023) Ruochen Zhang, Samuel Cahyawijaya, Jan Christian Blaise Cruz, Genta Indra Winata, and Alham Fikri Aji. 2023.  Multilingual large language models are not (yet) code-switchers.
  * Zhou et al. (2023) Chunting Zhou, Pengfei Liu, Puxin Xu, Srini Iyer, Jiao Sun, Yuning Mao, Xuezhe Ma, Avia Efrat, Ping Yu, Lili Yu, Susan Zhang, Gargi Ghosh, Mike Lewis, Luke Zettlemoyer, and Omer Levy. 2023.  Lima: Less is more for alignment.