# Heisenberg-limited spin squeezing in a hybrid system with Silicon-Vacancy
centers

Zhen-Qiang Ren  School of Physics, Sun Yat-sen University, Guangzhou 510275,
China  Xian-Liang Lu  School of Physics, Sun Yat-sen University, Guangzhou
510275, China  Ze-Liang Xiang<EMAIL_ADDRESS>School of Physics,
Sun Yat-sen University, Guangzhou 510275, China

###### Abstract

In this paper, we investigate spin squeezing in a hybrid quantum system
consisting of a Silicon-Vacancy (SiV) center ensemble coupled to a diamond
acoustic waveguide via the strain interaction. Two sets of non-overlapping
driving fields, each contains two time-dependent microwave fields, are applied
to this hybrid system. By modulating these fields, the one-axis twist (OAT)
interaction and two-axis two-spin (TATS) interaction can be independently
realized. In the latter case the squeezing parameter scales to spin number as
$\xi_{R}^{2}\sim 1.61N^{-0.64}$ with the consideration of dissipation, which
is very close to the Heisenberg limit. Furthermore, this hybrid system allows
for the study of spin squeezing generated by the simultaneous presence of OAT
and TATS interactions, which reveals sensitivity to the parity of the number
of spins $N_{tot}$, whether it is even or odd. Our scheme enriches the
approach for generating Heisenberg-limited spin squeezing in spin-phonon
hybrid systems and offers the possibility for future applications in quantum
information processing.

##  I Introduction

The spin-squeezed state [1, 2, 3], one of the typical quantum many-body
entangled states [4, 5], is a critical resource in quantum information
processing [6, 7, 8, 9, 10] and quantum metrology [11, 12, 13]. Such a state
can be used to overcome the shot-noise limit [14, 15, 16, 17] and to study
many-body entanglement [18, 19, 20, 21, 22]. The ability to efficiently
generate spin-squeezed states is the first step in unlocking its potential
applications. There are various approaches to produce spin squeezing, such as
transferring the squeezing from squeezed light to spin ensembles [23, 24, 25,
26], quantum nondemolition (QND) measurements of collective spins [27, 28, 29,
30], and utilizing the OAT and two-axis twist (TAT) squeezing interactions [1,
3, 31, 32, 33, 34, 35, 36, 37]. While the OAT interaction is able to
deterministically generate spin squeezing and has been effectively implemented
in different experimental systems, including the atomic systems [38, 39, 40]
and trapped ion systems [41], it cannot reach the Heisenberg limit in
principle [3]. Conversely, although the TAT interaction allows for Heisenberg-
limited spin squeezing, it presents various implementation challenges. Thus,
many efforts have been devoted to engineering the TAT interaction and
generating Heisenberg-limited spin squeezing in current state-of-the-art
systems [3, 33, 31, 42, 43, 44, 45, 34, 46].

The quantum hybrid systems based on color centers have been extensively
developed in quantum information processing benefiting from their longer
coherence time and better controllability [47, 48, 49, 50, 51, 52, 53, 54, 55,
56, 57, 58]. The SiV center, a type of color centers, has the fourfold-
degenerate ground state, which enables various interactions between
corresponding electronic levels [59, 60, 61, 62, 63]. Moreover, due to the
strong coupling between SiV centers and acoustic modes, the entanglement of
SiV centers can be generated mediated by phonons [63, 64, 65]. Such a strong
coupling can also be employed to achieve better spin squeezing in spin-phonon
hybrid systems based on SiV centers [66, 67, 63, 64, 68, 52].

In this work, we propose a scheme for generating spin-squeezed states in a
hybrid system consisiting of an ensemble of SiV centers coupled to the
acoustic mode of a diamond waveguide via the strain interaction. This SiV
ensemble is partitioned into two different segments resulting from two sets of
non-overlapping microwave fields. The strain-induced coupling enables
effective spin-spin interactions mediated by virtual phonons, then the OAT and
TATS interactions can be induced independently, where the latter one can
realize Heisenberg-limited spin squeezing [1, 2, 3]. Furthermore, we
investigate the spin-squeezed states generated by the mixed Hamiltonian of OAT
and TATS interactions and show the sensitivity of these states to the even-odd
spin particles, which holds potential for sensing applications. Considering
practical dissipations in the system, the squeezing parameter $\xi_{R}^{2}$
has a trend as $\xi_{R}^{2}\sim 1.61N^{-0.64}$, which can be used to achieve a
measurement precision close to the Heisenberg limit. Compared to other schemes
that necessitate the use of squeezed field injection, complex pulse drive or
parametric drive to generate better spin-squeezed states, our scheme requires
only the appropriate modulation of microwave fields and allows better spin-
squeezed states based on this spin-phonon hybrid system.

Our paper is organized as follows. In Sec. II, we introduce the theoretical
model of a hybrid quantum system consisting of two SiV-center segments
embedded in a quasi 1D acoustic waveguide. Section III shows the time
evolution of squeezing parameters $\xi_{S}^{2}$ and $\xi_{R}^{2}$ in the case
of the OAT, TATS and mixed OAT-TATS Hamiltonians. In Sec. IV, we discuss the
experimental feasibility of this scheme and analyze the influence caused by
the experimental dissipation of this hybrid system. Finally, we make a summary
in Sec. V.

##  II Model

Figure 1: (a) Sketch of an array of SiV centers embedded in a 1D diamond
waveguide. The length, width, and thickness of the waveguide are L, w, d,
respectively. Molecular structure of the SiV center is shown as the inset. In
this system, there are two different segments $S_{1}$ and $S_{2}$ in the SiV-
center ensemble, which contains $N_{1}$ and $N_{2}$ SiV centers, respectively,
resulted from the different set of driving fields. (b) The level structure of
the electronic ground state of the SiV center. The time-dependent microwave
driving fields induce the transitions between levels
$|1\rangle\leftrightarrow|4\rangle$ and $|2\rangle\leftrightarrow|3\rangle$,
while the transitions between levels $|1\rangle\leftrightarrow|3\rangle$ and
$|2\rangle\leftrightarrow|4\rangle$ are caused by the strain-induced coupling.

As depicted in Fig. 1 (a), we consider a spin-phonon hybrid system, where two
sets of $N_{1}$ and $N_{2}$ SiV centers in segment $S_{1}$ and segment
$S_{2}$, respectively, are coupled to an acoustic mode of a 1D diamond
waveguide via the strain-induced interaction. This interaction arises from the
change of Coulomb energy of the electronic states due to the displacement of
atoms forming the defect. First, we consider the SiV centers in segment
$S_{1}$ which are driven by two time-dependent microwave fields
$\Omega_{1}(t)$ and $\Omega_{2}(t)$, and this system can be described by the
Hamiltonian [63, 65]

$H_{S_{1}}=H_{\rm SiV_{S_{1}}}+H_{\rm ph}+H_{\rm strain_{S_{1}}},$ (1)

where $H_{\rm SiV_{S_{1}}}$ and $H_{\rm ph}$ are the Hamiltonians of SiV
centers in segment $S_{1}$ and the acoustic mode, respectively, and $H_{\rm
strain_{S_{1}}}$ denotes the strain-induced coupling between the orbital
degree of the SiV center in segment $S_{1}$ and the common acoustic mode of
the waveguide, as shown in Fig. 1(b).

The SiV center is an interstitial point defect in which a silicon atom is
positioned midway between two adjacent missing carbon atoms in the diamond
lattice, as depicted in the inset of Fig. 1(a). Its ground state is four-fold
degenerate, with the corresponding energy splitting
$\Delta=[\lambda_{g}^{2}+\left(\Upsilon_{x}^{2}+\Upsilon_{y}^{2}\right)]^{1/2}\approx
2\pi\times 46$ GHz, where $\lambda_{g}=2\pi\times 45$ GHz is the spin-orbit
coupling strength, and $\Upsilon_{x(y)}$ describes the strength of the Jahn-
Teller (JT) effect along $\vec{x}(\vec{y})$ direction [59, 63]. Two time-
dependent microwave fields $\Omega_{1,2}(t)$ with frequencies $\omega_{1,2}$
will induce transitions between states $|1\rangle\leftrightarrow|4\rangle$ and
$|2\rangle\leftrightarrow|3\rangle$, as shown in Fig. 1(b). Consequently, the
dynamics of SiV centers can be described by the Hamiltonian$(\hbar=1)$ [63,
65]

$\begin{split}H_{\rm
SiV_{S_{1}}}=&\sum_{j}^{N_{1}}[\omega_{B}|2\rangle_{j}\langle
2|+\Delta|3\rangle_{j}\langle 3|+(\omega_{B}+\Delta)|4\rangle_{j}\langle 4|\\\
&+\frac{\Omega_{1}(t)}{2}|1\rangle_{j}\langle
4|e^{i\omega_{1}t}+\frac{\Omega_{2}(t)}{2}|2\rangle_{j}\langle
3|e^{i\omega_{2}t}\left.+{\rm H.c.}\right],\end{split}$ (2)

where $\omega_{B}=\gamma_{s}B_{0}$ denotes the energy-level splitting induced
by the Zeeman effect, and we set $\omega_{B}\approx 2\pi\times 5$ GHz here.
$\gamma_{s}$ is the spin gyromagnetic ratio, and $j$ labels the $j$-th SiV
center in segment $S_{1}$.

Now we consider acoustic modes in the quasi 1D diamond waveguide. The length,
width, and thickness of the waveguide are $L$, $w$, $d$, respectively, as
shown in Fig. 1(a), satisfying $L\gg w,d$ meanwhile. The quantized Hamiltonian
of acoustic modes can be written as

$H_{\rm ph}=\sum_{n,k}\omega_{n,k}a_{n,k}^{{\dagger}}a_{n,k},$ (3)

where $a_{n,k}$ is the annihilation operator of one acoustic mode.

Within the framework of linear elasticity theory, the strain-induced coupling
Hamiltonian between SiV centers in segment $S_{1}$ and acoustic modes in the
waveguide is given by [63, 59]

$H_{\rm strain_{S_{1}}}\approx\sum_{n,k}[g_{n,k}j_{+}a_{n,k}+\rm{H.c.}],$ (4)

where $j_{+}=j_{-}^{\dagger}=|3\rangle_{j}\langle 1|+|4\rangle_{j}\langle 2|$
is the spin-conserving raising operator, and $g_{n,k}$ describes the coupling
strength with the form

$g_{n,k}=d\sqrt{\dfrac{\hbar k^{2}}{2\rho V\omega_{n,k}}}\xi_{n,k}(y,z),$ (5)

where $d/2\pi\approx 1$ PHz is the strain sensitivity and the dimensionless
function $\xi_{n,k}(y,z)$ denotes specific strain distribution [63, 62]. For a
small quasi 1D diamond acoustic waveguide, the length, width, and thickness
can be chosen as $L\sim 10-100~{}\mu$m, $w,d\lesssim 200$ nm2, respectively,
and it has the group velocity $\upsilon\sim 1\times 10^{4}~{}\rm{m/s}$, thus
the strain-induced coupling strength is $g=2\pi\times(4\sim 14)$ MHz [63, 62,
64].

Considering that the acoustic modes are well separated from frequency
($\Delta\omega_{n}\geq 2\pi\times 50$ MHz) in the waveguide with small size,
we could treat the mechanical mode as a single standing wave with
$\omega_{n}\approx 2\pi\times 46$ GHz for simplicity. Then, the Hamiltonian
Eq. (1) can be written as

$\displaystyle H_{{S_{1}}}=$
$\displaystyle\sum_{j}^{N_{1}}[\omega_{B}|2\rangle_{j}\langle
2|+\Delta|3\rangle_{j}\langle 3|+(\omega_{B}+\Delta)|4\rangle_{j}\langle 4|$
(6) $\displaystyle+\frac{\Omega_{1}(t)}{2}|1\rangle_{j}\langle
4|e^{i\omega_{1}t}+\frac{\Omega_{2}(t)}{2}|2\rangle_{j}\langle
3|e^{i\omega_{2}t}+\mathrm{H.c.}]$
$\displaystyle+\omega_{n}a^{\dagger}a+\sum_{j}^{N_{1}}[g_{n}^{j}(|3\rangle_{j}\langle
1|+|4\rangle_{j}\langle 2|)a+\rm{H.c.}].$

Performing a unitary transformation with respect to $U=e^{-iH_{0}t}$, where
$H_{0}=\sum_{j}[\left({{\omega_{n}}-{\omega_{2}}}\right)|2_{j}\rangle\langle
2|+\omega_{n}|3_{j}\rangle\langle 3|+{\omega_{1}}|4_{j}\rangle\langle
4|+{\omega_{n}}{a^{\dagger}}a]$, the Hamiltonian in the interaction picture
reads

$\begin{split}H_{I_{S_{1}}}=&\sum_{j}^{{N_{1}}}[(\nu-\delta_{2})|2\rangle_{j}\langle
2|+{\nu}|3\rangle_{j}\langle 3|+\delta_{1}|4\rangle_{j}\langle 4|\\\
&+\dfrac{\Omega_{1}(t)}{2}|1\rangle_{j}\langle
4|+\dfrac{\Omega_{2}(t)}{2}|2\rangle_{j}\langle 3|\\\
&+g_{n}(|3\rangle_{j}\langle 1|+|4\rangle_{j}\langle
2|)ae^{iw_{1}t}+\rm{H.c.}],\end{split}$ (7)

where $\nu,\delta_{1},\delta_{2}$ are the corresponding detunings between the
frequencies $\omega_{n,k}$, $\omega_{1}$, $\omega_{2}$ and eigenfrequencies of
states $|3\rangle,|4\rangle$, as shown in Fig. 1(b), and
$w_{1}=\omega_{1}+\omega_{2}-\omega_{n}$. Under the condition,
$\delta_{1,2}\gg\Omega_{1,2}$, we may further eliminate the higher energy
levels $|3\rangle$ and $|4\rangle$ via Froehlich-Nakajima transformation [69,
70, 71]. Finally, we obtain an equivalent two-level Hamiltonian

$H_{\rm eq_{S_{1}}}=\sum_{j}^{N_{1}}[\varepsilon_{1}|2\rangle_{j}\langle
2|+\lambda_{1}a|2\rangle_{j}\langle 1|+\Lambda_{1}a|1\rangle_{j}\langle
2|e^{iw_{1}t}+\rm{H.c.}],$ (8)

where the parameters $\varepsilon_{1},\lambda_{1},\Lambda_{1}$in Eq. (8) have
the forms as following

$\begin{split}&\varepsilon_{1}=\nu-\delta_{2}-\dfrac{\Omega_{2}^{2}}{4\delta_{2}}+\dfrac{\Omega_{1}^{2}}{4\delta_{1}},\\\
&\lambda_{1}=-\left(\dfrac{1}{\nu}+\dfrac{1}{\delta_{2}}\right)\cdot\dfrac{\Omega_{2}g_{n}}{4},\\\
&\Lambda_{1}=-\left(\dfrac{1}{\delta_{1}}+\dfrac{1}{\delta_{1}+\delta_{2}-\nu}\right)\cdot\dfrac{\Omega_{1}g_{n}}{4}.\end{split}$
(9)

Next we consider the total hybrid system with two segments $S_{1,2}$, which
are connected by a common acoustic mode. The effective Hamiltonian of this
whole hybrid system can be written as

$\begin{split}H=&\dfrac{\varepsilon_{1}}{2}J_{1}^{z}+\lambda_{1}aJ_{1}^{+}+\Lambda_{1}aJ_{1}^{-}e^{iw_{1}t}\\\
+&\dfrac{\varepsilon_{2}}{2}J_{2}^{z}+\lambda_{2}aJ_{2}^{+}+\Lambda_{2}aJ_{2}^{-}e^{iw_{2}t}+\rm{H.c.},\end{split}$
(10)

where the operators $J_{1,2}^{z}=\sum_{j=1}^{N_{1,2}}(|2\rangle_{j}\langle
2|-|1\rangle_{j}\langle
1|),J_{1,2}^{+}=\sum_{j=1}^{N_{1,2}}|2\rangle_{j}\langle
1|,J_{1,2}^{-}=\sum_{j=1}^{N_{1,2}}|1\rangle_{j}\langle 2|$ are the collective
spin operators of the SiV centers, subscripts 1 and 2 denote the two parts and
$N_{1},N_{2}$ are the total spin numbers of corresponding SiV-center segments.
Moreover, the parameters $\varepsilon_{2},\lambda_{2},\Lambda_{2}$ in Eq. (10)
have the forms as following

$\begin{split}&\varepsilon_{2}=\nu-\delta_{4}-\dfrac{\Omega_{4}^{2}}{4\delta_{4}}+\dfrac{\Omega_{3}^{2}}{4\delta_{3}},\\\
&\lambda_{2}=-\left(\dfrac{1}{\nu}+\dfrac{1}{\delta_{4}}\right)\cdot\dfrac{\Omega_{4}g_{n}}{4},\\\
&\Lambda_{2}=-\left(\dfrac{1}{\delta_{3}}+\dfrac{1}{\delta_{3}+\delta_{4}-\nu}\right)\cdot\dfrac{\Omega_{3}g_{n}}{4}.\end{split}$
(11)

As shown in Eq. (9), Eq. (11), by properly adjusting the microwave fields, the
effective detunings can be set as
$\varepsilon_{1}=-\varepsilon_{2}=\Delta_{s}$, which implies that the two
parts of the SiV centers are physically different. In addition, we set
$w_{1}=w_{2}=0$.

Assuming that the setup works at 100mK temperature, thus the phonon number of
acoustic mode is close to 0, i.e. $\left\langle a^{\dagger}a\right\rangle\sim
0$. Moreover, with the condition
$\Delta_{s}\gg\lambda_{1,2}g_{n},\Lambda_{1,2}g_{n}$ is satisfied, applying
the canonical transformation $H\rightarrow e^{-S}H_{eff}e^{S}$ with [71, 72]

$\begin{split}S=&\dfrac{1}{\Delta_{s}}\cdot
g_{n}[\lambda_{1}(a^{\dagger}J_{1}^{-}-aJ_{1}^{+})+\Lambda_{1}(aJ_{1}^{-}-a^{\dagger}J_{1}^{+})\\\
&+\lambda_{2}(aJ_{2}^{+}-a^{\dagger}J_{2}^{-})+\Lambda_{2}(a^{\dagger}J_{2}^{+}-aJ_{2}^{-})].\end{split}$
(12)

Finally, we could obtain an effective projected Hamiltonian in the spin-
ensemble subspace [72, 73] as following

$\begin{split}H_{eff}=\dfrac{\Delta_{s}}{2}&J_{1}^{z}-\dfrac{\Delta_{s}}{2}J_{2}^{z}\\\
+\dfrac{1}{\Delta_{s}}&\cdot\left\\{\lambda_{1}^{2}J_{1}^{+}J_{1}^{-}-\lambda_{2}^{2}J_{2}^{+}J_{2}^{-}\right.\\\
&-\Lambda_{1}^{2}J_{1}^{-}J_{1}^{+}+\Lambda_{2}^{2}J_{2}^{-}J_{2}^{+}\\\
&\left.+(\lambda_{1}\Lambda_{2}-\lambda_{2}\Lambda_{1})(J_{1}^{+}J_{2}^{+}+J_{1}^{-}J_{2}^{-})\right\\}.\end{split}$
(13)

Here the terms in lines 2 and 3 represent the OAT interaction, while the term
in line 4 indicates the TATS interaction [23, 2, 3]. Thus, by tunning the
driving fields, one could realize an OAT Hamiltonian along the z axis, a TATS
Hamilatonian, and a mixed Hamiltonian containing the OAT and TATS
interactions, respectively. In addition, the dynamical evolution of the system
can be described by the quantum master equation

$\begin{split}\dfrac{d\rho(t)}{dt}=&-i\left[H_{eff},\rho\right]+(n_{th}+1)\varGamma_{eff}D\left[J_{-}\right]\rho(t)\\\
&+n_{th}\varGamma_{eff}D\left[J_{+}\right]\rho(t),\end{split}$ (14)

where $n_{th}$ is the average thermal phonon number, and
$D\left[\hat{o}\right]\hat{\rho}=\hat{o}\hat{\rho}\hat{o}^{\dagger}-\hat{o}^{\dagger}\hat{o}\hat{\rho}/2-\hat{\rho}\hat{o}^{\dagger}\hat{o}/2$
is the standard Linblad superoperator.
$\varGamma_{eff}=\varGamma_{m}\cdot(\lambda_{1,2}/\Delta_{s})^{2}$ indicates
the collective spin relaxation induced by mechanical dissipation
$\varGamma_{m}$ of the corresponding acoustic mode.

##  III Spin squeezing

In this section, we quantify the degree of spin-squeezed states by calculating
two most frequently used squeezing parameters, $\xi_{S}^{2}$ and
$\xi_{R}^{2}$. $\xi_{S}^{2}$ is closely linked to quantum correlations
(entanglement), whereas $\xi_{R}^{2}$ finds wide applications in quantum
metrology. These two parameters are defined as follows [23, 2, 74, 3]

$\begin{split}&\xi_{S}^{2}=\dfrac{4(\Delta
J_{\overrightarrow{n}_{\bot}})^{2}_{min}}{N_{tot}}\\\
&\xi_{R}^{2}=\dfrac{N_{tot}(\Delta
J_{\overrightarrow{n}_{\bot}})^{2}_{min}}{\left|\overrightarrow{\langle
J\rangle}\right|^{2}},\end{split}$ (15)

where $(\Delta J_{\overrightarrow{n}_{\bot}})^{2}_{min}$ is the minimum
variance in a direction which perpendicular to the mean spin direction, and
$\left|\overrightarrow{J}\right|=\sqrt{\left\langle
J_{x}^{2}\right\rangle+\left\langle J_{y}^{2}\right\rangle+\left\langle
J_{z}^{2}\right\rangle}$ denotes the magnitude of the mean spin.
$N_{tot}=N_{1}+N_{2}$ is the total number of SiV centers in the waveguide, and
for the sake of simplicity, we assume that $N_{1}\simeq N_{2}$.

###  III.1 OAT interaction Hamiltonian

When the terms in Eq. (13) are set as
$\lambda_{1}\Lambda_{2}-\lambda_{2}\Lambda_{1}=0$ and
$\lambda_{1}^{2}\neq\Lambda_{1}^{2},\lambda_{2}^{2}\neq\Lambda_{2}^{2}$
through tunning the amplitudes and frequencies of the driving fields, we can
obtain the following OAT Hamiltonian along the z axis,

$\begin{split}H_{OAT}=\dfrac{\Delta_{s1}}{2}J_{1}^{z}-\dfrac{\Delta_{s2}}{2}J_{2}^{z}+G_{OAT1}J_{1}^{+}J_{1}^{-}+G_{OAT2}J_{2}^{+}J_{2}^{-},\end{split}$
(16)

where $G_{OAT~{}1,2}=(\lambda_{1,2}^{2}-\Lambda_{1,2}^{2})/\Delta_{s}$
describe the OAT interaction strengths of corresponding spin ensembles, and
$\Delta_{s1,s2}=\Delta_{s}+2\rm
min\left\\{\lambda_{1,2}^{2},\Lambda_{1,2}^{2}\right\\}/\Delta_{s}$.
Considering that $J_{+}J_{-}=J^{2}-J_{z}^{2}+J_{z}$, the last two terms in Eq.
(13) indicate a standard OAT interaction Hamiltonian [3].

Figure 2: The effects of spin number to the spin squeezing generated by the
OAT interaction. (a) Time evolution of the squeezing parameter $\xi_{S}^{2}$
with $N_{1}=N_{2}=N=20,30,50$. (b) Time evolution of the squeezing parameter
$\xi_{R}^{2}$ with $N_{1}=N_{2}=N=20,30,50$.

Figure 2 shows the time evolution of the squeezing parameters $\xi_{S}^{2}$
and $\xi_{R}^{2}$. The red line, black line, and green dotted line represent
the squeezing parameters of $N_{1}=N_{2}=N=20,30,50$, respectively. The hybrid
system evolves from a spin coherent state distributed on the x-axis, in which
state the values of both parameters $\xi_{S}^{2}$ and $\xi_{R}^{2}$ are 1, as
depicted in the figure 2 at time 0. As the system begins to evolve, these two
parameters become smaller than 1, indicating that the spin squeezing has been
generated in this hybrid system. As shown in figure 2, the generated spin-
squeezed states reaches their optimal squeezing at time $t\sim 20us$, with
minimum values are $\xi_{S}^{2}\approx 0.18,0.11,0.08$ and $\xi_{R}^{2}\approx
0.22,0.15,0.13$ with the case of $N=20,30,50$, respectively. In addition, we
find that $\xi_{S}^{2}<\xi_{R}^{2}$ for the same spin numbers, which is
consistent with the results as mentioned in Ref. [3].

###  III.2 TATS interaction Hamiltonian

Similar to the case of OAT interaction, we can also set
$\lambda_{1}\Lambda_{2}-\lambda_{2}\Lambda_{1}\neq 0$ and
$\lambda_{1}^{2}=\Lambda_{1}^{2},\lambda_{2}^{2}=\Lambda_{2}^{2}$ by tunning
the amplitudes and frequencies of the driving fields. Then, the Hamiltonian
with a TATS interaction could be obtained from Eq. (13) as follows

$\begin{split}H_{TATS}&=\dfrac{\Delta_{s1}}{2}J_{1}^{z}-\dfrac{\Delta_{s2}}{2}J_{2}^{z}+G_{TATS}(J_{1}^{+}J_{2}^{+}+J_{1}^{-}J_{2}^{-}),\end{split}$
(17)

where $G_{TATS}=(\lambda_{1}\Lambda_{2}-\lambda_{2}\Lambda_{1})/\Delta_{s}$
indicates the TATS interaction strength, and
$\Delta_{s1,s2}=\Delta_{s}+2\lambda_{1,2}^{2}/\Delta_{s}$.

Figure 3: The effects of spin number to the spin squeezing generated by the
TATS interaction. (a) Time evolution of the squeezing parameter $\xi_{S}^{2}$
with $N_{1}=N_{2}=N=20,30,50$. (b) Time evolution of the squeezing parameter
$\xi_{R}^{2}$ with $N_{1}=N_{2}=N=20,30,50$.

Figure 3 depicts the time evolution of squeezing parameters $\xi_{S}^{2}$ and
$\xi_{R}^{2}$ with different spin numbers in the case of TATS interaction
Hamiltonian. The red, blue, green lines in this figure represent the squeezing
parameters $\xi_{S}^{2}$ and $\xi_{R}^{2}$ of $N_{1}=N_{2}=N=20,30,50$,
respectively. We can see that the minimum values of $\xi_{S}^{2}$ and
$\xi_{R}^{2}$ have decreased significantly compared to the OAT interaction
case in fig. 3 with the same spin number, specifically, $\xi_{S}^{2}\approx
0.03,0.021,0.013$ and $\xi_{R}^{2}\approx 0.113,0.079,0.049$ with the case of
$N=20,30,50$, respectively. Similarly, $\xi_{S}^{2}<\xi_{R}^{2}$ for the same
spin numbers. Figure 3 shows that both squeezing parameters would reach their
minimum values more quickly with the increment of the spin numbers. Moreover,
we can see that the spin-squeezed state generated by this TATS interaction
Hamiltonian could approach the Heisenberg limit $1/N$ for large spin numbers,
which is not possible in the OAT case [3].

###  III.3 Mixed Hamiltonian of OAT and TATS interaction

With appropriate tuning of the microwave driving fields, it is also possible
to obtain a mixed Hamiltonian that comprises both OAT and TATS interactions
from Eq. (13),

$\begin{split}H_{mix}=&\dfrac{\Delta_{s1}}{2}J_{1}^{z}-\dfrac{\Delta_{s2}}{2}J_{2}^{z}\\\
&+G_{mix}(J_{1}^{-}J_{1}^{+}+J_{2}^{+}J_{2}^{-}+J_{1}^{+}J_{2}^{+}+J_{1}^{-}J_{2}^{-}),\end{split}$
(18)

where $G_{mix}$ represents the mixed interaction strength, and
$\Delta_{s1,s2}=\Delta_{s}+2\rm
min\left\\{\lambda_{1,2}^{2},\Lambda_{1,2}^{2}\right\\}/\Delta_{s}$.

Figure 4: The effects of spin number to the spin squeezing generated by the
mixed Hamiltonian. (a) Time evolution of the squeezing parameter $\xi_{S}^{2}$
with $N_{1}=N_{2}=N=20,30,50$. (b) Time evolution of the squeezing parameter
$\xi_{R}^{2}$ with $N_{1}=N_{2}=N=20,30,50$.

We also plotted the time evolution of the squeezing parameters $\xi_{S}^{2}$
and $\xi_{R}^{2}$ for different spin numbers in Fig. 4, and the black, red,
green lines in this figure represent the squeezing parameters $\xi_{S}^{2}$
and $\xi_{R}^{2}$ of $N_{1}=N_{2}=N=20,30,50$, respectively. In the case of
mixed Hamiltonian of the OAT and TATS interaction, the minimum values of
corresponding squeezing parameters are $\xi_{S}^{2}\approx 0.08,0.063,0.047$
and $\xi_{R}^{2}\approx 0.146,0.109,0.075$ with the case of $N=20,30,50$,
respectively, which are smaller than the case of OAT interaction induced spin
squeezing, but also slightly larger than the ideal TATS case. From the Fig. 4,
we can also see that the time for the system to reach the optimal squeezing is
significantly smaller than the OAT (Fig. 2) and TATS (Fig. 3) cases.

In particular, in the mixed OAT-TATS interaction case, we find that the spin
squeezing effect differs significantly depending on whether the total number
of spins is odd or even. This property may be utilized to detect changes of
the number, $N_{tot}$, of coupled spins at the single-particle level. Figure 5
shows the time evolution of squeezing parameters $\xi_{S}^{2}$ with
$N_{tot}=40$ and $N_{tot}=39$. Notably, during the first instance of spin
squeezing, the parameters $\xi_{S}^{2}$ of $N_{tot}=40$ and $N_{tot}=39$ are
almost identical. However, as the hybrid evolves from the spin coherent state
to spin-squeezed state for the second time, the spin squeezing in the
$N_{tot}=39$ case is significantly poorer compared to the the $N_{tot}=40$
case, as shown in Fig. 5(a). When the number of total spins is odd, there will
be a difference in the parity of spin numbers between the two segments,
resulting in the overall dynamics of spin squeezing, the combination of two
parts with different periods and parities. Therefore, like destructive and
constructive interference, the squeezing parameters $\xi_{S}^{2}$ with odd
total spins will display the maximum squeezed value that alternates between
large and small in odd and even periods. In contrast, in the case with even
total spins, such alternations in the maximum value of spin squeezing are
absent. Figure 5(b) illustrates that how this odd-even sensitivity could be
used for sensing. We plot the value of $\left\langle J_{X}^{2}\right\rangle$
with different total spin numbers $N_{tot}=40,39,38,37,36$. When the spins
leaving or decoupling from the waveguide one by one, the corresponding values
of $\left\langle J_{X}^{2}\right\rangle$ would become smaller and smaller, as
shown in Fig. 5(b).

Figure 5: The effects of even and odd total spin numbers to the spin squeezing
generated by the mixed Hamiltonian. (a) Time evolution of the squeezing
parameter $\xi_{S}^{2}$ with $N_{tot}=40$ and $N_{tot}=39$. (b) An example of
how the even-odd sensitivity of the spin squeezing could be used for sensing.
Time evolution of the value of $\left\langle J_{X}^{2}\right\rangle$ with
$N_{tot}=40,39,38,37,36$.

##  IV Experimental feasibility

Figure 6: The optimal squeezing parameter $(1/\xi_{R}^{2})_{max}$ versus the
total spin numbers $N_{tot}$ with the consideration of experimental disspation
in TATS interaction case. The black dots is the value of
$(1/\xi_{R}^{2})_{max}$ with corresponding spin numbers, and the red line
represents the curve fitting of numerical results.

In this section, we discuss the relevant parameters used in numerical
simulations to assess the practical feasibility of this schemement. First, we
take the value of the strain-induced coupling strength between the acoustic
mode in the diamond waveguide and SiV centers to $g=2\pi\times 5\rm{MHz}$. To
embed the SiV centers into the 1D diamond waveguide, we can utilize ion
implantation techniques based on state-of-the-art nanofabrication techniques
[75]. The ground state splitting of SiV centers is $\Delta\approx$ 46GHz, and
the transitions between states
$\left|1\right\rangle\leftrightarrow\left|4\right\rangle$ and
$\left|2\right\rangle\leftrightarrow\left|3\right\rangle$ can be induced by
the microwave driving fields or via an equivalent optical Raman process, which
has already been experimentally realized [63, 76, 77]. At 100 mK, the spin
dephasing rate of a single SiV center is about $\gamma_{d}\sim 100$Hz,
corresponding to a coherent time is $T_{s}\sim$10ms [47, 60, 78]. The driving
fields adopted here are $\Omega_{1},\Omega_{2},\Omega_{3},\Omega_{4}\sim
2\pi\times(30\sim 50)\rm{MHz}$ with
$\delta_{1},\delta_{2},\delta_{3},\delta_{4}\sim 2\pi\times(300\sim
500)\rm{MHz}$, respectively. It should be noted that we have not taken into
account the effect of dissipation in numerical simulations of the squeezing in
the previous section. A quality factor of $Q\approx\times 10^{5}$ for the
mechanical phonon modes of the small-sized diamond waveguide has been
demonstrated [79, 80], which leads to a mechanical dissipation value of
$\varGamma\sim 2\pi\times 500~{}\rm kHz$. Consequently, the effective
collective decay rate induced by mechanical dissipation in Eq. (14) has a
value as $\varGamma_{eff}\sim 2\pi\times 50\rm{Hz}$ in the TATS interaction
case, which is of the same order of magnitude as the spin dephasing rate.
Here, we modify the master equation Eq. (14) by including the spin dephasing
term as follows

$\begin{split}\dfrac{d\rho(t)}{dt}=&-i\left[H_{eff},\rho\right]+(n_{th}+1)\varGamma_{eff}D\left[J_{-}\right]\rho(t)\\\
&+n_{th}\varGamma_{eff}D\left[J_{+}\right]\rho(t)+\gamma_{d}\textstyle\sum_{j}D\left[\sigma_{z}^{j}\right]\rho(t).\end{split}$
(19)

In Fig.6, we plot the optimal squeezing parameter $(1/\xi_{R}^{2})_{max}$
versus the total number of spins $N_{tot}$ in TATS interaction case, taking
into account the collective decoherence induced by mechanical dissipation of
the acoustic mode and the dephasing rate of SiV centers. Using the numriacl
results obtained from Eq. (19), we fit a curve and obtain the trend of the
optimal squeezing parameters $(\xi_{R}^{2})_{max}$ with respect to the number
of spins as $\xi_{R}^{2}\sim 1.61N^{-0.64}$. As such, our scheme can generate
highly squeezed-spin states under currently available experimental conditions
in the hybrid system base on SiV centers.

##  V Conclusion

In summary, we have designed a hybrid quantum system, consisting of an
ensemble of SiV centers coupled to the acoustic mode of a diamond waveguide
via the strain-induced coupling. The system is partitioned into two segments
with different sets of microwave driving fields, and by ajusting the
frequencies and amplitudes of fields, we can achieve the OAT interaction, the
TATS interaction and mixed Hamiltonian with both OAT and TATS interactions.
The scheme can still work when the numbers of SiV centers in the two segments
differ, despite a reduction in the squeezing effect. In the ideal TATS
scenario with large numbers of spins, the two spin-squeezing parameters
$\xi_{R}^{2}$ and $\xi_{S}^{2}$ scale with total spin numbers as
$\xi_{S}^{2},\xi_{R}^{2}\sim N^{-1}$, reaching the Heisenberg limit. In the
mixed interaction case, our hybrid system can generate the optimal spin
squeezing more rapidly, and these spin-squeezed states is sensitive to the
parity of the total number of spins. Moreover, we have provided a possible
method for measuring the change of spin numbers at the single particle level.
Considering the realistic dissipation, $\xi_{R}^{2}$ scales with the total
number of spins as $\xi_{R}^{2}\sim 1.61N^{-0.64}$, demonstrating its
potential for application in quantum metrology. Consequently, our scheme can
work well under experimental conditions and extend the applications of the
SiV-based hybrid quantum systems in quantum information processing and quantum
metrology.

##  VI Acknowledgments

This work was supported by the National Key R&D Program of China (Grant No.
2019YFA0308200) and the National Natural Science Foundation of China (Grant
No. 11874432). We thank Yuan Zhou for valuable discussions.

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