# Dynamic quantum-enhanced sensing without entanglement in central spin
systems

Wenkui Ding<EMAIL_ADDRESS>Beijing National Laboratory for Condensed
Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing
100190, China  Yanxia Liu  Beijing National Laboratory for Condensed Matter
Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190,
China  Department of Physics, Yunnan University, Kunming 650500, China  Zhenyu
Zheng  Beijing National Laboratory for Condensed Matter Physics, Institute of
Physics, Chinese Academy of Sciences, Beijing 100190, China  Shu Chen
<EMAIL_ADDRESS>Beijing National Laboratory for Condensed Matter Physics,
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing, 100049, China  Yangtze River Delta Physics Research Center, Liyang,
Jiangsu 213300, China

(August 27, 2024)

###### Abstract

We propose a dynamic quantum sensing scheme by using a quantum many-spin
system composed of a central spin interacting with many surrounding spins.
Starting from a generalized Ising ring model, we investigate the error
propagation formula of the central spin and it indicates that Heisenberg
scaling can be reached while the probe state only needs to be a product state.
Particularly, we derive an analytical form of the dynamic quantum Fisher
information in a limit case, which explicitly exhibits the Heisenberg scaling.
By comparing with numerical results, we demonstrate that the general case can
be well approximated by the analytical result when the coupling strength among
the surrounding spins is much weaker than the coupling strength between the
central and surrounding spins. This analytic result guides us to find the
appropriate probe state and the proper measurement time, to achieve the
Heisenberg scaling in realistic situations. Furthermore, we investigate
various effects which are important in practical quantum systems, including
the central spin Zeeman term, the anisotropy of the hyperfine interaction and
the inhomogeneity of the hyperfine coupling strength. Our result indicates
that the dynamic quantum-enhanced sensing scheme seems feasible in realistic
quantum central spin systems, like semiconductor quantum dots.

##  I Introduction

Quantum sensing [1, 2] and quantum metrology [3, 4, 5] are becoming the
frontiers of quantum technologies nowadays. Specifically, theoretical research
on quantum parameter estimation utilizing quantum properties has attracted
great attentions recent years. On one hand, most of the research on quantum
metrology has been the employment of entangled probe state to achieve the sub-
shot-noise limit [6, 7, 8, 9, 10]. On the other hand, since the entangled
probe state is extremely difficult to generate and very prone to decoherence
[11, 12, 13, 14], many techniques to realize the quantum-enhanced metrology
without entanglement have also been proposed [15, 16, 17]. Particularly, the
quantum effects of quantum many-body systems are employed to realize metrology
schemes without entanglement. For example, the quantum phase transition has
been proposed to realize quantum-enhanced parameter estimation [18, 19, 20,
21]. Besides, the non-linear many-body Hamiltonian has also been proposed to
enhance the measurement precision, even reaching beyond the conventional
Heisenberg limit [22, 23, 24, 25, 26, 27].

The central spin model, which consists of a central spin interacting with many
surrounding spins, can be used to describe realistic quantum many-spin
systems, such as the semiconductor quantum dots [28, 29], the Nitrogen-Vacancy
center in the diamond [30, 31], etc. Recently, these solid-state central spin
systems have been widely investigated in quantum information and quantum
computation [32, 29], as well as quantum metrology and quantum sensing [33,
34, 35]. Particularly, in Ref. [33, 34], the central spin model has been
proposed to achieve the quantum-enhanced sensing. However, this sensing scheme
is still based on the generation of the entangled probe state. The non-linear
interaction between the central spin and the surrounding spins implies that it
is possible to realize the quantum-enhanced sensing of the magnetic field
without entanglement. Recently a dynamic framework for criticality-enhanced
quantum sensing in the quantum Rabi model has been proposed [20]. This work
inspires us to look for a dynamic scheme for the realization quantum-enhanced
sensing without resorting to the entanglement. Particularly, we shall explore
a dynamic routine to sense the magnetic field in a central spin model. We
discover that the Heisenberg scaling can be achieved in our sensing scheme,
while neither the entangled probe state nor the quantum criticality is
required. A similar model known as ZZXX-model was studied in Ref. [36] via the
numerical calculation of the quantum Fisher information, in which the authors
concluded that the Heisenberg scaling cannot be reached when only the central
spin was measured [36]. However, in this work we demonstrate that the
Heisenberg scaling can still be reached dynamically by only measuring the
central spin, when appropriate probe state and measurement time is applied.
Especially, we can obtain an explicit analytic form of the dynamic quantum
Fisher information for a limit case of our studied central spin model, which
correctly predicts the dynamics of the ZZXX-model when the number of
surrounding spins becomes large enough. Besides, we investigate more realistic
central spin systems and our result indicates that quantum-enhanced
magnetometry using our dynamic sensing framework seems feasible. In
particular, our scheme has the great advantage that neither entangled probe
state nor quantum criticality is required.

The rest of paper is organized as follows. In Sec.II, we first describe the
formalism for the calculation of quantum metrology and introduce our model
system. In Sec.III, we study both the local and global quantum Fisher
information for the central spin systems and derive analytical expression of
the dynamic quantum Fisher information in a limit case. In Sec.IV, we discuss
effects related to more realistic central spin system. A conclusion summary is
given in the last section.

##  II Formalism and model

Before the study of concrete models, firstly we give a brief review of the
basics of quantum metrology for the convenience of following calculations. The
quantum fidelity between two quantum states $\hat{\rho}_{1}$ and
$\hat{\rho}_{2}$ is defined as,

$\mathcal{F}(\hat{\rho}_{1},\hat{\rho}_{2})=\text{Tr}(\sqrt{\sqrt{\hat{\rho}_{1}}\hat{\rho}_{2}\sqrt{\hat{\rho}_{1}}}).$
(1)

In particular, when the quantum state is continuously dependent on parameter
$\lambda$, we can define the fidelity susceptibility [37], which is equivalent
to the quantum Fisher information [38],

$F_{\lambda}=-4\frac{\partial^{2}\mathcal{F}(\rho(\lambda),\rho(\lambda+\delta_{\lambda}))}{\partial\delta_{\lambda}^{2}}|_{\delta_{\lambda}=0}.$
(2)

Specifically, when the quantum state is a pure state, namely
$\rho(\lambda)=|\Psi(\lambda)\rangle\langle\Psi(\lambda)|$, we can calculate
the quantum Fisher information as follows,

$F_{\lambda}=4(\langle\Psi(\lambda)|\overleftarrow{\frac{\partial}{\partial\lambda}}\overrightarrow{\frac{\partial}{\partial\lambda}}|\Psi(\lambda)\rangle-|\langle\Psi(\lambda)|\overrightarrow{\frac{\partial}{\partial\lambda}}|\Psi(\lambda)\rangle|^{2}).$
(3)

Furthermore, for a unitary parameter imprint process [3, 39],

$|\Psi(\lambda)\rangle=e^{-iH_{\lambda}t}|\Psi_{0}\rangle,$ (4)

where $H_{\lambda}=H_{0}+\lambda H_{1}$ is a general parameter dependent
Hamiltonian, the quantum Fisher information is given by

$F_{\lambda}=4(\langle\Psi_{0}|G_{\lambda}^{2}|\Psi_{0}\rangle-|\langle\Psi_{0}|G_{\lambda}|\Psi_{0}\rangle|^{2}).$
(5)

Here, the transformed local generator

$G_{\lambda}\equiv
ie^{iH_{\lambda}t}\frac{\partial}{\partial\lambda}e^{-iH_{\lambda}t}$

can be calculated as follows [40],

$G_{\lambda}=\int_{0}^{t}e^{iH_{\lambda}t}H_{1}e^{-iH_{\lambda}t}ds=-i\sum_{n=0}^{\infty}\frac{(it)^{n+1}}{(n+1)!}[H_{\lambda},H_{1}]_{n},$
(6)

where the commutation relation is defined as
$[H_{\lambda},H_{1}]_{n+1}=[H_{\lambda},[H_{\lambda},H_{1}]_{n}]$, with
$[H_{\lambda},H_{1}]_{0}=H_{1}$.

The quantum Fisher information sets a bound to the estimation sensitivity,
which is called the Cramér-Rao bound [38],

$\Delta_{\delta_{\lambda}}(\hat{A},\lambda)\geq F_{\lambda}^{-\frac{1}{2}}.$
(7)

Here, the standard derivation of the measurement value is calculated via the
error propagation formula,

$\Delta_{\delta_{\lambda}}(\hat{A},\lambda)=\frac{\sqrt{\langle\hat{A}^{2}\rangle_{\rho(\lambda)}-\langle\hat{A}\rangle^{2}_{\rho(\lambda)}}}{|\frac{\partial\langle\hat{A}\rangle_{\rho(\lambda+\delta\lambda)}}{\partial\delta_{\lambda}}|_{\delta_{\lambda}=0}|},$
(8)

where $\hat{A}$ is the observable to be measured in the experiment.

We begin with a generalized cental spin model [41] which is described by the
Hamiltonian of the Ising ring in a transverse magnetic field interacting with
a central spin 111The spins considered in this article is always
spin-$\frac{1}{2}$, so here the Pauli matrix is defined as
$\sigma_{z}=\frac{1}{2}\begin{bmatrix}1&0\\\ 0&-1\end{bmatrix}$, and likewise
for $\sigma_{x}$ and $\sigma_{y}$.,

$H=-J\sum_{i=1}^{N}\sigma_{i}^{x}\sigma_{i+1}^{x}-h\sum_{i=1}^{N}\sigma_{i}^{y}+A\sigma_{0}^{z}\sum_{i=1}^{N}\sigma_{i}^{z}.$
(9)

Here, $\sigma_{0}^{\alpha=x,y,z}$ are the operators of the central spin,
$\sigma_{i=1,...,N}^{\alpha=x,y,z}$ are the operators of spins in the Ising
ring, $J$ is the strength of the Ising coupling, and $A$ is the coupling
strength between the central spin and surrounding spins in the Ising ring.
Different from the usual central spin model studied in Ref.[41], here we apply
the transverse magnetic field along the y-axis, instead of the z-axis coupled
with the central spin. In this work, the strength of transverse magnetic field
$h$ is the parameter to be estimated.

##  III Local and global quantum Fisher information

###  III.1 Local quantum Fisher information

We shall first consider the case with only the central spin being measured. In
order to evaluate the performance of utilizing this parameter-dependent
Hamiltonian as the parameterization generator, we will first calculate the
error propagation formula when the expectation value of central spin is
measured. Specifically, the initial state (or the probe state) is chosen to be
a product state

$|\Psi_{0}\rangle=\frac{1}{\sqrt{2}}(\left|\uparrow\right\rangle+\left|\downarrow\right\rangle)\otimes|\Phi_{n}\rangle,$
(10)

where $\left|\uparrow(\downarrow)\right\rangle$ is the central spin state and
$|\Phi_{n}\rangle$ is the collective state of the spins in the Ising ring. The
time evolution of the system is governed by
$|\Psi(t)\rangle=e^{-iHt}|\Psi_{0}\rangle$ and the central spin expectation
value can be expressed as

$\langle\sigma_{0}^{x}(t)\rangle=\frac{1}{2}\text{Re}(\langle\Phi_{n}|e^{iH_{+}t}e^{-iH_{-}t}|\Phi_{n}\rangle),$
(11)

where

$H_{\pm}=-J\sum_{i=1}^{N}\sigma_{i}^{x}\sigma_{i+1}^{x}-h\sum_{i=1}^{N}\sigma_{i}^{y}\pm\frac{A}{2}\sum_{i=1}^{N}\sigma_{i}^{z}.$
(12)

Furthermore, we have the relation

$e^{-iH_{\pm}t}|\Phi_{n}\rangle=e^{-i\theta\sum_{i=1}^{N}\sigma_{i}^{x}}e^{-iH_{\text{eff}}^{(\pm)}t}e^{i\theta\sum_{i=1}^{N}\sigma_{i}^{x}}|\Phi_{n}\rangle,$
(13)

where $\theta=\arctan(2h/A)$ and

$H_{\text{eff}}^{(\pm)}=-J\sum_{i=1}^{N}\sigma_{i}^{x}\sigma_{i+1}^{x}\pm\sqrt{h^{2}+\frac{A^{2}}{4}}\sum_{i=1}^{N}\sigma_{i}^{z}.$
(14)

We now consider a specific product probe state given by
$|\Phi_{n}\rangle=|+,+,...+\rangle\equiv|N/2,M_{x}=N/2\rangle$ with
$|+\rangle=1/\sqrt{2}(\left|\uparrow\right\rangle_{n}+\left|\downarrow\right\rangle_{n})$,
namely all spins in the Ising ring are polarized along the x-axis. Now, we
have

$\displaystyle\langle$
$\displaystyle\Phi_{n}|e^{iH_{+}t}e^{-iH_{-}t}|\Phi_{n}\rangle=e^{-i\theta
N}\times$ (15)
$\displaystyle\langle\frac{N}{2},M_{x}=\frac{N}{2}|e^{iH_{\text{eff}}^{(+)}t}e^{2i\theta\sum_{i=1}^{N}\sigma_{i}^{x}}e^{-iH_{\text{eff}}^{(-)}t}|\frac{N}{2},M_{x}=\frac{N}{2}\rangle.$

Next, we discuss two important limiting situations. If
$J\gg\sqrt{h^{2}+{A^{2}}/{4}}$, then
$H_{\text{eff}}^{(+)}=H_{\text{eff}}^{(-)}\approx-J\sum_{i=1}^{N}\sigma_{i}^{x}\sigma_{i+1}^{x}$,
and approximately, $|\Phi_{n}\rangle=|N/2,M_{x}=N/2\rangle$ is the eigenstate
of $H_{\text{eff}}^{(+)}$ and $H_{\text{eff}}^{(-)}$ with eigenenergy
$\epsilon_{0}$. This leads to
$\langle\Phi_{n}|e^{iH_{+}t}e^{-iH_{-}t}|\Phi_{n}\rangle\approx 1$, which
indicates that no information on the parameter can be retrieved by monitoring
the central spin expectation value. Meanwhile, for the opposite limiting
situation with $J\ll\sqrt{h^{2}+{A^{2}}/{4}}$, approximately,
$H_{\text{eff}}^{(+)}\approx\sqrt{h^{2}+{A^{2}}/{4}}\sum_{i=1}^{N}\sigma_{i}^{z}$
and
$H_{\text{eff}}^{(-)}\approx-\sqrt{h^{2}+{A^{2}}/{4}}\sum_{i=1}^{N}\sigma_{i}^{z}$.
If we choose the evolution time $t=t_{0}=\pi/\sqrt{h^{2}+{A^{2}}/{4}}$, then
we have

$\displaystyle\langle$
$\displaystyle\Phi_{n}|e^{iH_{+}t}e^{-iH_{-}t}|\Phi_{n}\rangle$ (16)
$\displaystyle=e^{-i\theta
N}\langle\Phi_{n}|e^{i\pi\sum_{i=1}^{N}\sigma_{i}^{z}}e^{2i\theta\sum_{i=1}^{N}\sigma_{i}^{x}}e^{i\pi\sum_{i=1}^{N}\sigma_{i}^{z}}|\Phi_{n}\rangle$
$\displaystyle=e^{-i\theta
N}\langle\frac{N}{2},M_{x}=-\frac{N}{2}|e^{2i\theta\sum_{i=1}^{N}\sigma_{i}^{x}}|\frac{N}{2},M_{x}=-\frac{N}{2}\rangle$
$\displaystyle=e^{-2i\theta N}.$

Using Eq.(11), the expectation value of the central spin is given by

$\displaystyle\langle\sigma_{0}^{x}(t_{0})\rangle$
$\displaystyle=\frac{1}{2}\cos(2\theta N)$ (17)
$\displaystyle=\frac{1}{2}\cos[2\arctan(\frac{2h}{A})N].$

By substituting this result into the error propagation formula in Eq.(8), we
obtain

$\displaystyle\mathcal{E}_{h}^{-1}=$
$\displaystyle\frac{\langle[\sigma_{0}^{x}(t_{0})]^{2}\rangle-\langle\sigma_{0}^{x}(t_{0})\rangle^{2}}{|\frac{\partial\langle\sigma_{0}^{x}(t_{0})\rangle}{\partial
h}|^{2}}$ (18) $\displaystyle=$
$\displaystyle\frac{(A^{2}+4h^{2})^{2}}{16A^{2}N^{2}},$

which indicates the Heisenberg scaling with respect to $N$, i.e. the number of
spins in the Ising ring.

Since here the measurement is only done on the central spin, now in order to
estimate the precision bound of this local measurement, we need to calculate
the local quantum Fisher information corresponding to the reduced density
matrix of the central spin. We can calculate this quantity by using the
formula for the quantum Fisher information of one qubit [43]:

$F_{h}^{0}=\left\\{\begin{array}[]{lr}|\partial_{h}\mathbf{V}|^{2}+\frac{(\mathbf{V}\cdot\partial_{h}\mathbf{V})^{2}}{1-|\mathbf{V}|^{2}},&\text{
if }|\mathbf{V}|<1,\\\ |\partial_{h}\mathbf{V}|^{2},&\text{ if
}|\mathbf{V}|=1,\end{array}\right.$ (19)

where
$\mathbf{V}=(2\langle\sigma_{0}^{x}(t)\rangle,2\langle\sigma_{0}^{y}(t)\rangle,2\langle\sigma_{0}^{z}(t)\rangle)$
is the spin vector on the Bloch sphere. Following the same procedure for
deriving Eq.(17), we have

$\langle\sigma_{0}^{y}(t_{0})\rangle=\frac{1}{2}\sin[2\arctan(\frac{2h}{A})N],$
(20)

and $\langle\sigma_{0}^{z}(t_{0})\rangle=0$. Then, by substituting these
expectation values and their derivatives into Eq. 19, we get

$F_{h}^{0}=\mathcal{E}_{h}=\frac{16A^{2}N^{2}}{(A^{2}+4h^{2})^{2}},$ (21)

which happens to be the same form as the reciprocal of the error propagation
formula in Eq. 8. This indicates that $\sigma_{0}^{x}$ is indeed the optimal
observable to saturate the sensitivity bound for this specific probe state and
local measurement.

In fact, employing the standard Ising Hamiltonian with transverse field (with
$A=0$ in Eq. 9) to estimate parameters has been investigated in Ref. [44].
Particularly, the result in that paper demonstrated that quantum-enhanced
metrology cannot be realized if the probe state is restricted to be the
product state. Therefore, the interaction of the central spin with the Ising
ring introduced in our model is crucial to realize the quantum-enhanced
sensing without entanglement.

In order to investigate the effect of finite Ising coupling strength, we
numerically calculate the local quantum Fisher information as function of $N$
for different values of $J$. The result is plotted in Fig. 1(a) and it
indicates that the analytical result for the case of $J=0$ can give good
approximation for the general case when the coupling strength among the
surrounding spins ($J$) is much weaker than the coupling strength between the
central and surrounding spins ($A$). As the Ising coupling strength $J$
increases, the Heisenberg scaling deteriorates. This implies that it is better
to measure a magnetic field with large amplitude (which can be realized by
adding a large reference magnetic field) to satisfy the condition,
$J\ll\sqrt{h^{2}+{A^{2}}/{4}}$, for the optimal sensitivity.

Figure 1:  (a) The scaling of the local quantum Fisher information with
respect to $N$ for different Ising coupling strength $J$. The dashed line
corresponds to the analytic result for $J=0$ in Eq. 21, while the squares
correspond to $J=0.1$ and the stars correspond to $J=0.2$, respectively. (b)
The scaling of the global quantum Fisher information with respect to $N$ for
different Ising coupling strength $J$. The dashed line corresponds to the
analytic result for $J=0$ in Eq. 35, while the squares correspond to $J=0.1$
and the stars correspond to $J=0.2$, respectively. The other parameters in
both figures are set to be $A=1$, $h=1$.

###  III.2 Global quantum Fisher information

As discussed above, the calculation of local quantum Fisher information
reveals that the case of $J=0$ can realize the quantum-enhanced sensing
without entanglement. Now we will focus on the case of $J=0$ and use Eq. 5 and
Eq. 6 to deduce the analytic form of the global quantum Fisher information. In
comparison with the local quantum Fisher information, the global one gives the
ultimate sensitivity bound for all possible measurements, namely, not
restricted to measurements only on the central spin.

Before we calculate the dynamic quantum Fisher information for the specific
case with $J=0$, we have to calculate the transformed local generator using
Eq. 6. The Hamiltonian of the system now becomes,

$\displaystyle H$
$\displaystyle=-h\sum_{i=1}^{N}\sigma_{i}^{y}+A\sigma_{0}^{z}\sum_{i=1}^{N}\sigma_{i}^{z}$
(22) $\displaystyle\equiv-hI_{y}+AS_{z}I_{z}.$

Here, since the Hamiltonian does not contain the Ising terms any longer, we
have used the collective spin operator
$I_{\alpha=x,y,z}\equiv\sum_{i=1}^{N}\sigma_{i}^{\alpha=x,y,z}$, and redefined
the central spin operator $S_{z}\equiv\sigma_{0}^{z}$. For convenience, we may
call the central spin the electron spin and the surrounding spin the nuclear
spin in the following sections.

We now use Eq. 6 to calculate the transformed local generator. For the even
commutation terms, we have

$[H,H_{1}]_{2n}=-A(A^{2}S_{z}^{2}+h^{2})^{n-1}(hS_{z}I_{z}+AS_{z}^{2}I_{y}),$
(23)

while for the odd terms, we have the commutation relation as follows,

$[H,H_{1}]_{2n+1}=iA(A^{2}S_{z}^{2}+h^{2})^{n}S_{z}I_{y}.$ (24)

Then, the transformed local generator to estimate $h$ is calculated as
follows,

$\displaystyle G_{h}=$ $\displaystyle ie^{iHt}\frac{\partial}{\partial
h}e^{-iHt}$ (25) $\displaystyle=$ $\displaystyle-tI_{y}-A\frac{\sin(\Omega
t)-\Omega t}{\Omega^{3}}(hS_{z}I_{z}+AS_{z}^{2}I_{y})$
$\displaystyle+A\frac{\cos(\Omega t)-1}{\Omega^{2}}S_{z}I_{x},$

where since the central spin $S=1/2$, we have the oscillation frequency
$\Omega=\sqrt{A^{2}S_{z}^{2}+h^{2}}=\sqrt{A^{2}/4+h^{2}}$.

Using Eq. 5, it is easy to verify that the maximized quantum Fisher
information is [40]

$F_{max}=(E_{max}-E_{min})^{2},$ (26)

where $E_{max}$ and $E_{min}$ are the maximal and minimal eigenvalues of
$G_{h}$, respectively. Here for convenience, we denote the transformed local
generator in Eq. 25 as

$G_{h}\equiv\alpha I_{y}+\beta S_{z}I_{z}+\gamma S_{z}I_{x}.$ (27)

Since $[G_{h},S_{z}]=0$, we can separately find the eigenvalues in the
$S_{z}=1/2$ subspace and the $S_{z}=-1/2$ subspace. The calculated result of
the maximized quantum Fisher information is

$F_{max}=(4\alpha^{2}+\beta^{2}+\gamma^{2})I^{2},$ (28)

where $I=N/2$ and it clearly shows the Heisenberg scaling. Generally, we have
the relation

$\mathcal{E}_{h}\leq F_{h}^{0}\leq F_{max}.$ (29)

The first relation is simply due to the Cramér-Rao bound. The second relation
results from the fact that the local quantum Fisher information is obtained by
tracing out the degree of freedom of the nuclear spins, which unavoidably
leading to the loss of information on the parameter.

To achieve this maximized quantum Fisher information, the probe state in Eq. 5
should be 222due to the degeneracy of the spectrum of $G_{h}$, the choice of
the optimal probe state is actually not unique.,

$|\Psi_{0}\rangle=\frac{1}{\sqrt{2}}(|E_{max}\rangle+|E_{min}\rangle).$ (30)

The eigenstate corresponding to the maximal eigenvalue is

$|E_{max}\rangle=|\uparrow\rangle\otimes(e^{-iI_{z}\phi}e^{-iI_{y}\theta}|I,M_{z}=I\rangle),$
(31)

and the eigenstate corresponding to the minimal eigenvalue is

$|E_{min}\rangle=|\uparrow\rangle\otimes(e^{-iI_{z}\phi}e^{-iI_{y}\theta}|I,M_{z}=-I\rangle),$
(32)

with $\theta=\arctan(\frac{\sqrt{4\alpha^{2}+\gamma^{2}}}{\beta})$ and
$\phi=\arctan(\frac{2\alpha}{\gamma})$, where $|I,M_{z}\rangle$ is the
eigenstate of $I_{z}$. This probe state $|\Psi_{0}\rangle$ is generally an
entangled state, however, what interests us most is whether the Heisenberg
scaling still maintains when the probe state is restricted to be a product
state [46].

Similar to the dynamical sensing scheme in Ref. [20], we emphasize on the
discussion of the dynamic quantum Fisher information at a specific sensing
time $t=t_{0}=2\pi/\Omega$. Now the transformed local generator in Eq. 25
becomes

$\displaystyle G_{h}$
$\displaystyle=(\frac{\pi}{2}\frac{A^{2}}{\Omega^{3}}-\frac{2\pi}{\Omega})I_{y}+\frac{2\pi
Ah}{\Omega^{3}}S_{z}I_{z}$ (33)
$\displaystyle\equiv\alpha_{0}I_{y}+\beta_{0}S_{z}I_{z}.$

Here, we consider a specific product probe state,

$|\Psi_{0}\rangle=\frac{1}{\sqrt{2}}(\left|\uparrow\right\rangle+\left|\downarrow\right\rangle)\otimes|I,M_{z}=I\rangle,$
(34)

where
$|I,M_{z}=I\rangle=\left|\uparrow_{n},\uparrow_{n},\cdots,\uparrow_{n}\right\rangle$.
It is readily to verify $\langle\Psi_{0}|G_{h}|\Psi_{0}\rangle=0$, and using
Eq. 5 we obtain

$F_{h}=4(\langle\Psi_{0}|G_{h}^{2}|\Psi_{0}\rangle)=2\alpha_{0}^{2}I+\beta_{0}^{2}I^{2}.$
(35)

This again obviously manifests the Heisenberg scaling with respect to the
number of nuclear spins ($I=N/2$).

It needs to be mentioned that the probe state (Eq. 34) used to achieve the
Heisenberg scaling in $F_{h}$ is different from the probe state (Eq. 10) used
to calculate the error propagation formula in the previous subsection. This is
due to the fact that the global quantum Fisher information discussed in this
subsection corresponds to the ultimate bound that the measurement on the
observable can be done globally, not limited to the central spin only.

Similar to the discussions for the local quantum Fisher information, here we
also numerically investigate the effect of finite Ising coupling strength
($J\neq 0$) to the global quantum Fisher information. In Fig. 1(b), we
illustrate the scaling of the global quantum Fisher information with respect
to $N$ for different Ising coupling strength $J$. Compared to the local
quantum Fisher information in Fig. 1(a), the scaling of the global quantum
Fisher information is less sensitive to the strength of Ising coupling, and
the analytic result for $J=0$ in Eq. 35 approximates very well with the
numerical result when the coupling strength between the nuclear spins is much
weaker than the coupling strength among the surrounding spins. This is
reasonable, since the local quantum Fisher information corresponds to the
reduced state of central spin, while the finite Ising coupling will accelerate
the decoherence of the central spin, the decoherence will unavoidably lead to
loss of information on the parameter. Meanwhile, since the global quantum
Fisher information corresponds to the quantum state of the composite system
(central spin and surrounding spins), the Ising coupling between the nuclear
spins may not lead to loss of the parameter information encoded in the total
quantum state.

##  IV Realization of the protocol in realistic quantum systems

###  IV.1 Effect of the electronic Zeeman term

In the above section, the coupling between the central spin and the
surrounding spins may be implemented by the interaction of the light with two-
level atoms, where the interaction of the central spin with the field is
neglected. However, it becomes necessary to take into account the electronic
Zeeman term if we want to implement our dynamic sensing scheme using solid-
state spin systems, like semiconductor quantum dots. By taking into account
the electronic Zeeman term, the Hamiltonian of the system now becomes

$\displaystyle H$
$\displaystyle=-h(\sigma_{0}^{y}+\sum_{i=1}^{N}\sigma_{i}^{y})+A\sigma_{0}^{z}\sum_{i=1}^{N}\sigma_{i}^{z}$
(36) $\displaystyle\equiv-h(S_{y}+I_{y})+AS_{z}I_{z},$

which corresponds to the ZZXX-model numerically investigated in Ref. [36]. In
this subsection we will show that, for the dynamic sensing scheme proposed in
this paper, the dynamics and the dynamic quantum Fisher information
corresponding to this Hamiltonian, can actually be well approximated by the
Hamiltonian in Eq. 22, where the electronic Zeeman term is neglected.

As shown in Fig. 2, the performance of this approximation (by neglecting the
electronic Zeeman term) is actually dependent on $N$, the number of nuclear
spins. This is reasonable, since for our dynamic sensing scheme, the central
spin is initialized along the x-axis and then it will precess about the
effective magnetic field, from the view of semi-classical picture. This
effective magnetic field consists of the magnetic field $h$ along the y-axis
and the nuclear field, $B_{nuc}\propto A\langle I_{z}\rangle$, along the
z-axis. As the nuclear spin number $N$ increases, the nuclear field $B_{nuc}$
becomes significantly larger than the magnetic field $h$ and the approximated
analytic result becomes even closer to the exact numerical result. Thus, in
realistic solid state central spin system, which contains a large number of
nuclear spins, the analytic result obtained by neglecting the electronic
Zeeman term can be safely utilized in our dynamic sensing scheme. In other
words, when the dynamic sensing scheme by employing the Hamiltonian in Eq. 36
is applied, our analytic result can be used to determine the appropriate probe
state, to devise the proper measurement strategy or to estimate the overall
sensitivity.

Figure 2:  The effect of the electronic Zeeman term on the performance of the
dynamic sensing scheme. In all figures, the solid lines correspond to the
Hamiltonian that taking into account the electronic Zeeman term, while the
dashed lines corresponds to the analytic result that neglecting this term. The
left column [(a),(c),(e)] corresponds to $N=8$ and the right column
[(b),(d),(f)] corresponds to $N=40$. The time evolution of the central spin
expectation value $\langle S_{x}(t)\rangle$ is compared in (a) and (b), with
$A=1$ and $h=1$. The time evolution of the global quantum Fisher information
is compared in (c) and (d), with $A=1$ and $h=1$, where the vertical dashed
line in these figures corresponds to the measurement time $t_{0}$ deduced from
the analytic result. In (e) and (f), we plot the local quantum Fisher
information for the central spin as a function of the external field $h$,
while the coupling strength $A=1$ and the measurement time $t=t_{0}$.

###  IV.2 Effect of the anisotropy of the hyperfine interaction

In many realistic quantum central spin systems, the XX-term of the hyperfine
interaction may not be neglected, which is described by the so-called XXZ
central spin model [47]. The Hamiltonian for such a general central spin
system with homogeneous coupling is as follows,

$\displaystyle H$
$\displaystyle=\mathbf{h}\cdot(\mathbf{\sigma}_{0}+\sum_{k=1}^{N}\mathbf{\sigma}_{k})+\sum_{k=1}^{N}[\frac{\Delta}{2}(\sigma_{0}^{+}\sigma_{k}^{-}+\sigma_{0}^{-}\sigma_{k}^{+})+A\sigma_{0}^{z}\sigma_{k}^{z}]$
(37)
$\displaystyle=\mathbf{h}\cdot(\mathbf{S}+\mathbf{I})+\frac{\Delta}{2}(S^{+}I^{-}+S^{-}I^{+})+AS_{z}I_{z},$

where $\Delta$ is the hyperfine coupling strength of the XX-term and
$I^{\pm}=I_{x}\pm iI_{y}$, etc. When $\Delta=A$, namely no anisotropy in the
hyperfine coupling, we can check that
$[\mathbf{h}\cdot(\mathbf{S}+\mathbf{I}),H]=0$. It is easy to verify that
$G_{h}=\mathbf{n}\cdot(\mathbf{S}+\mathbf{I})$ with $\mathbf{n}=\mathbf{h}/h$,
which indicates that no quantum enhancement can be obtained when the probe
state is restricted to be only a product state. Therefore, the anisotropy in
the hyperfine interaction is necessary to achieve the quantum-enhanced sensing
without entanglement for our dynamic sensing scheme. Furthermore, when
$\Delta\neq A$ and the magnetic field is applied along the z-axis, it is
readily to verify that $G_{h}=S_{z}+I_{z}$ and again no quantum enhancement is
possible when the initial state is restricted to be a product state. Thus, in
order to obtain the quantum-enhanced sensing without entanglement, first,
there should be anisotropy in the hyperfine coupling; second, the magnetic
field to be estimated cannot be applied along the z-axis.

We numerically calculate the scaling of the dynamic quantum Fisher information
with respect to the nuclear spin number $N$ for various values of $\Delta$ in
Fig. 3. The result indicates that, increasing the anisotropy in the hyperfine
coupling will lead to a better performance of our dynamic sensing scheme.

Figure 3:  Investigate the effect of the anisotropy of the hyperfine
interaction on the performance of the dynamic sensing scheme. The local
quantum Fisher information for the central spin is plotted in (a), while the
global quantum Fisher information is plotted in (b). In both figures, the
solid line corresponds to the analytic result when the electronic Zeeman term
is neglected and the hyperfine anisotropy $\Delta=0$. The dashed lines in both
figures correspond to the standard quantum limit, which is the ultimate limit
when $\Delta=A=0$ in Eq. 37 and the probe state is restricted to be the
product state. In both figures, the circles correspond to $\Delta=0$, $A=1$ in
Eq. 37, while the squares correspond to $\Delta=0.1$, $A=1$ and the stars
correspond to $\Delta=0.2$, $A=1$, respectively.

###  IV.3 Effect of the inhomogeneity of the hyperfine coupling strength

For realistic solid-state quantum central spin systems, like semiconductor
quantum dots, the hyperfine coupling strength between the central electron
spin and surrounding nuclear spins is usually inhomogeneous. The Hamiltonian
describing such a system is

$H=-h(\sigma_{0}^{y}+\sum_{k=1}^{N}\sigma_{k}^{y})+\sum_{k=1}^{N}A_{k}\sigma_{0}^{z}\sigma_{k}^{z},$
(38)

where $A_{k}$ is the coupling strength between the central electron spin and
the $k$-th nuclear spin. Due to the inhomogeneity of the coupling strength
($A_{k}\neq A$), the collective nuclear spin operator cannot be used any
longer and we have to resort to numerics to research the performance of the
dynamic sensing scheme.

Here, we employ the Chebyshev method (by expanding the time evolution operator
in terms of Chebyshev polynomials [48]) to calculate the dynamics of this
inhomogeneous central spin system and retrieve the scaling of the dynamic
quantum Fisher information with respect to $N$. The result is shown in Fig. 4
and it indicates that the performance of such an inhomogeneous sensor can be
well approximated by the homogeneous one, as long as the coupling strength in
the homogeneous case is set to be $A=\sum_{k}{A_{k}}/N$. Crucially, the
Heisenberg scaling still exists for the inhomogeneously coupled central spin
system.

Figure 4:  Investigate the performance of the inhomogeneity of the hyperfine
coupling strength on the performance of the dynamic quantum sensing. (a). The
solid line corresponds to the time evolution of the global quantum Fisher
information for $N=16$ nuclear spins with inhomogeneous hyperfine coupling,
while the dashed line corresponds to the homogeneous case. The vertical dashed
line corresponds to the measurement time. (b). The global quantum Fisher
information as a function of the nuclear spin number $N$. The dashed line
corresponds to the analytic result of the homogeneous case, while the squares
correspond to the numerical result of the inhomogeneous case.

###  IV.4 The realistic semiconductor quantum dot system

We now elaborately consider a practical solid-state quantum central spin
system, namely the semiconductor quantum dots, which has been widely
researched in the area of quantum computation [32, 29] and quantum sensing
[33, 34, 35] recent years. The detailed Hamiltonian describing such a quantum
many-spin system is as follows [49, 29],

$H=H_{\text{D}}+H_{\text{Z}}+H_{\text{HF}},$ (39)

with,

$\displaystyle H_{\text{D}}$
$\displaystyle=\frac{\gamma_{n}^{2}}{2}\sum_{j=1}^{N}\sum_{k=1}^{N}[\frac{\mathbf{I}_{j}\cdot\mathbf{I}_{k}}{r_{jk}^{3}}-\frac{3(\mathbf{I}_{j}\cdot\mathbf{r}_{jk})(\mathbf{I}_{k}\cdot\mathbf{r}_{jk})}{r_{jk}^{5}}],$
(40) $\displaystyle H_{\text{Z}}$
$\displaystyle=\mathbf{h}\cdot(\gamma_{e}\mathbf{S}+\gamma_{n}\sum_{k=1}^{N}\mathbf{I}_{k}),$
$\displaystyle H_{\text{HF}}$
$\displaystyle=\sum_{k=1}^{N}A_{k}[\frac{\delta}{2}(S_{+}I_{k-}+S_{-}I_{k+})+S_{z}I_{kz}],$

where $\mathbf{S}$ is the electronic spin operator and $\mathbf{I}_{k}$ is the
spin operator of the $k-$th nuclei in the quantum dot. $H_{\text{D}}$
describes the dipole-dipole interaction between nuclear spins, where
$\mathbf{r}_{jk}$ is the position vector from the $j$-th nuclei to the $k-$th
nuclei and $\gamma_{n}$ is the gyromagnetic ratio of the nuclear spin.
$H_{\text{Z}}$ corresponds to the electronic Zeeman term and nuclear Zeeman
terms, where $\gamma_{e}$ is the gyromagnetic ratio of the electron spin.
$H_{\text{HF}}$ describes the hyperfine interaction between the electron spin
and nuclear spins, where $\delta$ stands for the anisotropy in the hyperfine
interaction. The hyperfine coupling strength
$A_{k}\propto|\phi(\mathbf{x}_{k})|^{2}$, where $|\phi(\mathbf{x}_{k})|^{2}$
is the electron density at the site $\mathbf{x}_{k}$ of the $k$-th nuclear
spin.

The dipole-dipole interaction between nuclear spins in the quantum dot
resembles the role of the Ising coupling in Eq. 9. Typically, in semiconductor
quantum dots, the hyperfine coupling strength between the electron spin and
nuclear spins is several orders of magnitude stronger than the strength of the
nuclear dipole-dipole coupling [50, 49]. Similar to the discussion on the
Ising coupling in the previous section, this indicates that the effect from
the nuclear dipolar coupling to the overall sensitivity of the dynamic sensing
scheme can be negligible. Another point that needs to be mentioned is the
difference in the gyromagnetic ratio between the electron spin and the nuclear
spins, namely $\gamma_{e}\neq\gamma_{n}$. However, the discussions in Sec.
IV.1 still applies despite the difference in the gyromagnetic ratio. This is
because for typical semiconductor quantum dots, the number of nuclear spins
$N\sim 10^{4}-10^{6}$ in a single quantum dot, and the effective nuclear filed
experienced by the electron spin suppresses the magnetic field. Thus, in
realistic semiconductor quantum dots, the electronic Zeeman term can be safely
neglected to analyze the performance of our dynamic sensing scheme.

##  V Conclusion

In summary, by studying the Ising ring model interacting with a central spin,
we show that the Heisenberg scaling in the error propagation formula can be
reached by only measuring the central spin dynamics. While the the dynamic
quantum Fisher information for the general case can be numerically calculated,
we can obtain an analytical form of the dynamic quantum Fisher information to
estimate the magnetic field in a limit case, which gives good approximation
for the general case as long as $J\ll\sqrt{h^{2}+{A^{2}}/{4}}$ is fulfilled.
This analytic result explicitly manifests that the Heisenberg scaling can be
reached with an appropriate product probe state and the proper measurement
time. Furthermore, we gradually investigate more realistic central spin models
analytically and numerically and our results indicate that the Heisenberg
scaling can still be achieved in practical quantum central spin systems, like
semiconductor quantum dots. Our dynamic sensing scheme processes the great
advantage that neither entangled probe state nor quantum phase transition is
required to obtain the quantum enhancement. Our theoretical result paves the
way to the implementation of quantum-enhanced magnetometry without
entanglement in practical quantum many-spin systems.

###### Acknowledgements.

The work is supported by National Key National Key Research and Development
Program of China (Grant No. 2021YFA1402104), the NSFC under Grants No.12174436
and No.T2121001 and the Strategic Priority Research Program of Chinese Academy
of Sciences under Grant No. XDB33000000.

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