#  $\gamma-$rigid triaxial nuclei in the presence of a minimal length via a
quantum perturbation method

S. Ait Elkorchi  High Energy Physics and Astrophysics Laboratory, Faculty of
Sciences Semlalia, Cadi Ayyad University, P. O. B. 2390, Marrakech 40000,
Morocco

Preparation of Papers for Heron Press Science Series BooksS. Ait Elkorchi, M.
Chabab, A. El Batoul, A. Lahbas, M. Oulne

1, M. Chabab1, A. El Batoul1, A. Lahbas1, M. Oulne1

1

In this work, we derive a closed solution of the Shr$\ddot{o}$dinger equation
for Bohr Hamiltonien within the minimal length formalism. This formalism is
inspired by Heisenberg algebra and a generlized uncertainty principle (GUP),
applied to the geometrical collective Bohr- Mottelson model (BMM) of nuclei by
means of deformed canonical commutation relation and the Pauli-Podolsky
prescription. The problem is solved by means conjointly of asymptotic
iteration method (AIM) and a quantum perturbation method (QPM) for
transitional nuclei near the critical point symmetry Z(4) corresponding to
phase transition from prolate to $\gamma-rigid$ triaxial shape. A scaled
Davidson potentiel is used as a restoring potential in order to get physical
minimum. The agreement between the obtained theoretical results and the
experimental data is very satisfactory.

##  1 Introduction

Critical point symmetries in nuclear structure are recently receiving
considerable attention[1, 2] since they provide parameter-free solutions. The
pioneering ones amid them were E(5) [1, 3], X(5) [4] and Z(5)[5] corresponding
to shape phase transitions from U(5) to O(6), U(5) to SU(3) and from axial to
triaxial shapes respectively, with the recent addition of Y(5) [2] related to
the transition from prolate to oblate shape . Later, a $\gamma$-rigid (with
$\gamma=0$) version of X(5), called X(3), has been introduced [6]. In the same
way, other CPS have been developped like for example Z(4) (with
$\gamma=\pi/6$) the gamma rigid version of Z(5) corresponding to shape phase
transitions from prolate to triaxial symmetry [7, 8, 9, 10]. From a structural
point of view, the collective Bohr Mottelson represents a sound frame work to
describe many properties of the quadrupole collective dynamics in nuclei [3].
Its formulation has the ability to describe both rotational and vibrational
modes. On the other hand, recently, a great interest has been consecrated to
the quantum mechanical problems related to a generalized modifed commutation
relations involving a minimal length or generalized uncertainty principle [11,
12].
In the present work we focuse on the study of the quadrupole collective states
in $\gamma$-rigid case, by modifying Davydov-Chaban Hamiltonian in the
framework of the minimal length formalism [13] with Davidson potential [14]
for$\beta$-vibrations. The model is conventionally called Z(4)-D-ML. The
organization of this paper is as follows: in section 2, we present the
Z(4)-D-ML model with the quantum perturbation method which requires the study
of the model in the absence and in the presence of the minimal length,
presented in sections 3 and 4 respectively. Finally, section 5 is devoted to
the numerical results and brief discussion for energy spectrum of some
triaxial-rigid nuclei, while section 6 contains our conclusions.

##  2  Z(4)-D-ML with the quantum perturbation method

In the Z(4) model the $\gamma$ variable is “frozen” to $\gamma=\pi/6$ and only
four variables are involved; three Euler angles
$(\theta_{1},\theta_{2},\theta_{3})$, which obviously define the orientation
of the intrinsic principal axes in the laboratory frame and the deformation
parameter. So, in the frame of the Bohr-Mottelson model [3, 15], the
corresponding eigenvalue problem reduced to that of the Davydov-Chaban
hamiltonian [7]. Therefore, we aimed to study the minimal length effect on
energy spectrum in the context of $\gamma$-rigid nuclei Z[4].

$H_{DC}=-\dfrac{\hbar^{2}}{2B_{m}}\left[\dfrac{1}{\beta^{3}}\dfrac{\partial}{\partial\beta}\beta^{3}\dfrac{\partial}{\partial\beta}-\dfrac{1}{4\beta}\sum^{3}_{k=1}\dfrac{Q_{k}^{2}}{sin^{2}(\gamma-\frac{2\pi}{3}k)}\right]+U(\beta),$
(1)

Where $\beta$ and $\gamma$ are the usual collective coordinates [3],
$Q_{k}(k=1,2,3)$ are the components of angular momentum and $B_{m}$ is the
mass parameter. In this Hamiltonian $\gamma$ is treated as a parameter and not
as a variable.
By employing the mathematical formulation, including the minimal length
concept, presented in the original paper[13], the collective equation of
eigenstates, up to the first order of $\alpha$, is written as follows:

$\left(-\dfrac{\hbar^{2}}{2B_{m}}\Delta+\dfrac{\alpha\hbar^{4}}{B_{m}}\Delta^{2}+V(\beta)-E_{n,L}\right)\psi(\beta,\theta_{i})=0,$
(2)

where

$\Delta=\dfrac{1}{\beta^{3}}\dfrac{\partial}{\partial\beta}\beta^{3}\dfrac{\partial}{\partial\beta}-\dfrac{\Delta_{\theta}}{4\beta}$
(3)
$\Delta_{\theta}=\sum^{3}_{k=1}\dfrac{Q_{k}^{2}}{sin^{2}(\gamma-\frac{2\pi}{3}k)}$
(4)

and $\theta_{i}(i=1,2,3)$ are the Euler angles. This equation can be simplifed
by introducing an auxiliary wave function[13]:

$\psi(\beta,\theta_{i})=(1+2\alpha\hbar^{2}\Delta)\xi(\beta,\theta_{i}).$ (5)

Thus, we obtain the following differential equation satisfied by
$\xi(\beta,\theta_{i})$

$\left[(1+4B_{m}\alpha(E-V(\beta)))\Delta+\dfrac{2B_{m}}{\hbar^{2}}(E_{n,L}-V(\beta))\right]\xi(\beta,\theta_{i})=0.$
(6)

This equation can be simplified by using the usual following factorization

$\xi(\beta,\theta_{i})=\Phi(\beta)\chi(\theta_{i}).$ (7)

The separation of variables leads to two equations : one depending only on the
$\beta$ variable and the other depending on the $\gamma$ and the Euler angles
:

$\left[\dfrac{1}{\beta^{3}}\dfrac{\partial}{\partial\beta}\beta^{3}\dfrac{\partial}{\partial\beta}-\dfrac{\Lambda}{\beta^{2}}+\dfrac{2B_{m}}{\hbar^{2}}\bar{k}(E_{n,L},\beta)\right]\Phi(\beta)=0,$
(8)

where

$\bar{K}(E_{n,L},\beta)=\left(\dfrac{E_{n}-V(\beta)}{1+4B_{m}\alpha(E_{n,L}-V(\beta)}\right)$
(9)

$n$ is the radial quantum number.

$[\dfrac{1}{4}\sum^{3}_{k=1}\dfrac{Q_{k}^{2}}{sin^{2}(\gamma-\frac{2\pi}{3}k)}-\Lambda]\chi(\theta_{i})=0.$
(10)

In the case of $\gamma=\pi/6$, the last equation takes the form [16]:

$[\dfrac{1}{4}(Q_{1}^{2}+4Q_{2}^{2}+4Q_{3}^{2})-\Lambda]\chi(\theta_{i})=0.$
(11)

This equation has been solved by Meyer-ter-Vehn [16], the eigen functions
being:

$\chi(\theta_{i})=\chi(\theta_{i})_{\mu,\omega}^{L}=\sqrt{\dfrac{2L+1}{16\pi^{2}(1+\delta_{\omega},0)}}\times[D^{L}_{\mu,\omega}(\theta_{i})+(-1)^{L}D^{L}_{\mu,-\omega}(\theta_{i})].$
(12)

Here, $D^{L}_{\mu,\omega}(\theta_{i})$ represents the Weigner function of the
Euler angles, L are the eigenvalues of angular momentum, while $\omega$ and
$\mu$ are the eigenvalues of the projections of angular momentum on the body-
fixed ˆx-axis and the laboratory fixed ˆz-axis, respectively.[16] with

$\Lambda=\Lambda_{L,\omega}=L(L+1)-\frac{3}{4}\omega^{2}.$ (13)

Thanks to the smallness of the parameter $\alpha$, by expanding (9) in power
series of $\alpha$, one can obtain diferent order approximations of the
standard model Z(4)-ML. At the first order approximation, as it has been done
recently in [17] (9) becomes:

$\bar{K}(E_{n,L}-V(\beta))\approx(E_{n,L}-V(\beta))(1-4B_{m}\alpha(E_{n,L}-V(\beta)))$

$\quad=E_{n,L}-V(\beta)-4B_{m}\alpha(E_{n,L}-V(\beta))^{2}.$ (14)

In what concerns the $\beta$ degree of freedom, we will consider the Davidson
potential chosen to be of the following form:

$V(\beta)=a\beta^{2}+\dfrac{b}{\beta^{2}},\quad\beta_{0}=(\dfrac{b}{a})^{\frac{1}{4}}$
(15)

where a and b are two free scaling parameters, and $\beta_{0}$ represents the
position of the minimum of the potential. The special case of $b=0$
($\beta_{0}=0$) corresponds to the simple harmonic oscillator.
The diferential equation (8) was solved exactly, with an infinite square well
like potential, within the standard model [21] but it is not soluble
analytically for the Davidson-type potential. However, the quantum
perturbation theory one of its familiar forms, dubbed the quantum perturbation
method (QPM), is used to obtain approximate solutions for all values of
angular momentum L [17].

##  3 Z(4) model with Davidson potential Z(4)-D ($\alpha=0$)

It is preferable to write the equation in a Schr$\ddot{o}$dinger picture. This
is realized by changing the wave function as
$\Phi(\beta)=\beta^{-\frac{3}{2}}f(\beta)$. However one obtains an equation
which resembles the radial Schr$\ddot{o}$dinger equation for an isotropic
Harmonic Oscillator acting in four-dimensional space:

$\dfrac{d^{2}}{d\beta^{2}}f(\beta)+\left[\dfrac{2B_{m}E_{n,L}}{\hbar^{2}}-\dfrac{2B_{m}a}{\hbar^{2}}\beta^{2}-\dfrac{L(L+1)+\frac{3}{4}(1-\alpha^{2})+\dfrac{2bB_{m}}{\hbar^{2}}}{\beta^{2}}\right]f(\beta)=0.$
(16)

We define:

$\epsilon=\dfrac{2B_{m}E_{n,L}}{\hbar^{2}}$,
$\quad\omega=\dfrac{2B_{m}a}{\hbar^{2}},$

and

$\vartheta_{l,b}(\vartheta_{l,b}+1)=L(L+1)+\frac{3}{4}(1-\alpha^{2})+\dfrac{2bB_{m}}{\hbar^{2}}.$
(17)

However one obtains an equation which resembles to the Gol’dman and
Krivchenkov Hamiltonian [18]

$\dfrac{d^{2}}{d\beta^{2}}f(\beta)+\left[\epsilon-\omega\beta^{2}-\dfrac{\vartheta_{l,b}(\vartheta_{l,b}+1)}{\beta^{2}}\right]f(\beta)=0.$
(18)

To solve this diferential equation via the asymptotic iteration method
(AIM)[18], we propose the following ansatz[18]:

$f(\beta)=\beta^{1+\vartheta_{l,b}}+e^{-\frac{\sqrt{\omega}}{2}\beta^{2}}\Theta(\beta).$
(19)

Thus we obtain,

$\dfrac{d^{2}\Theta(\beta)}{d\beta^{2}}+(\frac{2p}{\beta}-4q\beta)\dfrac{d\Theta(\beta)}{d\beta}+[\epsilon-2q(1+2p)]\Theta(\beta)=0,$
(20)

where $q=\dfrac{\sqrt{\omega}}{2}$ and $p=1+\vartheta_{l,b}$.
After calculating $\lambda_{0}$ and $s_{0}$, by means of the recurrence
relations[18], we get the generalized formula of the reduced energy from the
roots of the quantization condition

$\epsilon=q[2+4p+8n];n=1,2,....$ (21)

from which, we obtain the energy spectrum :

$E_{n,L}=\dfrac{\hbar^{2}}{2B_{m}}\epsilon=\sqrt{\dfrac{\hbar^{2}}{2B_{m}}}a[3+4n_{\beta}+2\vartheta_{l,b}].$
(22)

From equation (17), we get $\vartheta_{l,b}$ as a function of the total
angular momentum L and the parameter b :

$\vartheta_{l,b}=-\dfrac{1}{2}+\dfrac{1}{2}\sqrt{4L(L+1)-3\alpha^{2}+8b+4}.$
(23)

The physical solutions to (8) are obtained as:

$\Phi(\beta)=N_{n\beta,L}\beta^{-\frac{3}{2}+p}e^{-q\beta^{2}}L^{p-\frac{1}{2}}_{n\beta}(2q\beta^{2}),$
(24)

where $L^{p-\frac{1}{2}}_{n\beta}(2q\beta^{2})$ denotes the associated
Laguerre polynomials and $N_{n\beta,L}$

is a normalization constant.

##  4 Z(4) model with Davidson potential via a minimal length (Z(4)-D-ML)

Here, we treat the additional term$\dfrac{\alpha\hbar^{4}}{B_{m}}\Delta^{2}$
shown in equation (2) as a perturbation and then estimate its effect on the
energy spectrum up to the first order of the perturbation theory. Hence, the
energy spectrum can be written as:

$E_{n,L}=E_{n,L}^{0}+\Delta E_{n,L},$ (25)

where $E_{n,L}^{0}$ is the unperturbed energy spectrum, and $\Delta E_{n,L}$
the correction induced by the minimal length, given by:

$\Delta
E_{n,L}=\dfrac{\alpha\hbar^{4}}{B_{m}}<\psi^{0}(\beta,\theta_{i})\mid\Delta^{2}\mid\psi^{0}(\beta,\theta_{i})>,$
(26)

where $\psi^{0}(\beta,\theta_{i})$ are the eigenfunctions, solutions to the
ordinary Schr$\ddot{o}$dinger equation ($\alpha=0$) .
So, the energy spectrum can be expressed as [17],

$\Delta E_{n,L}=4B_{m}\alpha\left[(E_{n,L}^{0})^{2}-2E_{n,L}^{0}<\psi^{0}\mid
V(\beta)\mid\psi^{0}>+<\psi^{0}\mid V(\beta)^{2}\mid\psi^{0}>\right].$ (27)

After substituting the Davidson potential (16), one obtains

$\Delta
E_{n,L}=4B_{m}\alpha\left[(E_{n,L}^{0})^{2}+2ab-2E_{n,L}^{0}(a\overline{\beta^{2}}+b\overline{\beta^{-2}})+(a^{2}\overline{\beta^{4}}+b^{2}\overline{\beta^{-4}})\right],$
(28)

Where $\overline{\beta^{i}}$ (i=2,- 2,4,- 4) are expressed as follows

$\left\\{\begin{aligned}
\overline{\beta^{2}}=\dfrac{4n+2\vartheta_{l,b}+3}{4q}\\\
\overline{\beta^{-2}}=\dfrac{4q}{2\vartheta_{l,b}+1}\\\
\overline{\beta^{4}}=\dfrac{4\vartheta_{l,b}^{2}+24n\vartheta_{l,b}+24n^{2}+16\vartheta_{l,b}+36n+15}{16q^{2}}\\\
\overline{\beta^{-4}}=\dfrac{16q^{2}(4n+2\vartheta_{l,b}+3)}{(2\vartheta_{l,b}+3)(4\vartheta_{l,b}^{2}-1)}\end{aligned}\right.$
(29)

Details of $\overline{\beta^{i}}$ calculations are given in [17].

##  5 Numerical examination

The model established in this work, called Z(4)-D-ML, is adequate for
description of $\gamma$-rigid nuclei for which the $\gamma$ parameter is fixed
to $\gamma=\pi/6$. Basically, the energy levels of the ground state band as
well as of the vibrational bands are characterized by the principal quantum
number $n_{\beta}$ and $n_{\omega}$, respectively. with $n_{\omega}$ is the
wobbling quantum number [16, 19] $n_{\omega}=L-\omega$. We briefly recall a
few interesting low-lying bands which are classified by the quantum numbers
$n_{\beta}$ and $n_{\omega}$

  * •

The ground state band (gsb) with $n_{\beta}=0$ and $n_{\omega}=0$

  * •

The $\beta$ band with $n_{\beta}=1$ and $n_{\omega}=0$

  * •

The$\gamma$ band composed by the even L levels with $n_{\beta}=0$ and
$n_{\omega}=2$ and the odd L levels with $n_{\beta}=0$ and $n_{\omega}=1$

  * •

For our subsequent calculations, we define the energy ratios as:
$R(n_{\beta},L,n_{\omega})=\dfrac{E_{n_{\beta},L,n_{\omega}}-E_{0,0,0}}{E_{0,2,0}-E_{0,0,0}}$

  * •

The mentioned results are thus found to have the smallest deviations from the
experimental data [20], evaluated by the quality measure
$\sigma=\sqrt{(\sum_{i=1}^{N}(E_{i}^{exp}-E_{i}^{th})^{2})/{N}},$

where N is the maximum number of levels. We have treated 32 nuclei among which
are depicted in Figure 1 those with good results

Figure 1: Comparison of the Z(4)-ML-D predictions for (normalized) energy
levels to experimental data for ${}^{124}Xe$ , ${}^{126}Xe$, ${}^{128}Xe$,
${}^{130}Xe$, ${}^{114}Pd$

The comparison between Z(4)-D-ML theoretical predictions and experimental
data[20] of selected candidates regarding energy levels is visualized
schematically in Fig.1. The agreement with experiment is very good for the
ground state band and $\beta$ band, despite the fact that there is not much
experimental data especially for the $\gamma$ band of these studied nuclei. As
a result, one concludes that the Z(4)- D-ML is more suitable for describing
the structural properties of nuclei having a structure in vicinity of the Z(4)
limit.

nuclei | ${\sigma_{D}}$ | ${\sigma_{D-ML}}$ | $\beta_{0}$
---|---|---|---
${}^{124}Xe$ | 0.588 | 0.44 | 0.82
${}^{126}Xe$ | 0.36 | 0.36 | 0.61
${}^{128}Xe$ | 0.44 | 0.40 | 0.69
${}^{130}Xe$ | 0.35 | 0.35 | 0.62
${}^{114}Pd$ | 0.84 | 0.63 | 0.78
Table 1: Standard deviation between experimental and theoretical results

##  6 Conclusion

The idea of Z(4)-ML is already used and presented with the square wel
potential [21], but this time it was used with the Davidson potential.
However, the Hamiltonian of the system is not soluble analytically for a
potential other than the square well. In order to overcome such a difficulty,
in the present work we used, a quantum perturbation method (QPM), to obtain
approximate solutions for all values of angular momentum L. Therefore, closed-
form analytical formula for the energy of the ground and vibrational bands was
derived for trixial $\gamma$-rigid nuclei within Davidson potential. Our
results indicate a better agreement with the experimental values, and
reproduced well the best Z(4) condidate nuclei already obtained in the Xe
region around A = 130 including the new one ${}^{114}Pd$.

## Acknowledgements

S. Ait Elkorchi would like to thank the organizing committee for the
hospitality and the wonderful scientific meet. Also, she acknowledges the
financial support (Type A) of Cadi Ayyad University.

## References

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# Enhanced quantum sensing mediated by a cavity in open systems

Quinn Langfitt1<EMAIL_ADDRESS>Zain H. Saleem1<EMAIL_ADDRESS>Tian Zhong2
<EMAIL_ADDRESS>Anil Shaji3<EMAIL_ADDRESS>Stephen K. Gray4
<EMAIL_ADDRESS>Mathematics and Computer Science Division, Argonne National
Laboratory, 9700 S Cass Ave, Lemont IL 60439 1 Pritzer School of Molecular
Engineering, University of Chicago, Chicago, Illinois 60637, USA 2 School of
Physics, IISER Thiruvananthapuram, Kerala, India 695551 3 Center for Nanoscale
Materials, Argonne National Laboratory, Lemont, Illinois 60439, USA 4

###### Abstract

We simulate the dynamics of systems with $N$ = 1-20 qubits coupled to a cavity
in order to assess their potential for quantum metrology of a parameter in the
open systems limit. The qubits and the cavity are both allowed to have losses
and the system is studied under various coupling strength regimes. The focus
is primarily on the coupling between the qubits using the quantum Fisher
information as the measured parameter. Some results on estimating the qubit-
cavity detuning parameter are also presented. We investigate the scaling of
the uncertainty in the estimate of the qubit-cavity coupling with the number
of qubits and for different initial states of the qubits that act as the
quantum probe. As initial probe states, we consider Dicke states with varying
excitation numbers, the GHZ state, and separable X-polarized states. It is
shown that in the strong coupling regime, i.e., when the coupling between the
qubits and the cavity is greater than the decay parameters of both the qubits
and the cavity, Dicke states with a large excitation number can achieve the
Heisenberg limit, with the precision scaling improving as the excitation
number increases. A particularly intriguing finding of our study is that in
the weak coupling regime, as well as in situations where either the qubit or
cavity decay parameters exceed the coupling, the separable $X$-polarized state
is the best in terms of scaling and is even able to achieve the Heisenberg
limit in these lossy regimes for the range of $N$ considered.

##  I Introduction

In quantum metrology, the goal is to estimate parameters more precisely using
quantum resources, such as entanglement and squeezing [1, 2, 3, 4, 5]. The
maximum achievable precision is set by the Heisenberg Limit (HL), which
corresponds to the precision scaling as $1/N$, where $N$ is the number of
elementary sub-units making up the quantum probe. HL can be achieved with
entangled states in a noiseless environment [6]. However, system noise can
hinder the probe’s capabilities, leading to a precision scaling of
$1/\sqrt{N}$, referred to as the Standard Quantum Limit (SQL) or worse. Our
research aims to surpass this limitation through the careful selection of
probe states. For instance, it has been demonstrated that using a Dicke state
with certain excitation numbers as the probe, under the Tavis-Cummings model
in the strong coupling regime, can exceed SQL and reach HL, when the duration
of the measurement is optimised, owing to the Dicke state’s resilience to
noise [7].

In this study, we focus on the $N$ qubits-cavity model, limiting ourselves to
permutationally symmetric states. Permutational symmetry reduces the
Hamiltonian’s dimension size to $O(N^{5})$, enabling us to extend simulations
to a larger number of qubits than otherwise possible [8, 9]. The $N$ qubits
are coupled to a cavity with a coupling strength of $g$, with decay rates
$\kappa$ and $\gamma$ for the resonator and spin states, respectively. We
normalize these decay rates with respect to $g$. Moreover, aligning with our
focus on quantum sensing in noisy environments, we primarily concentrate on
high-loss regimes — noisy-cavity, noisy-qubit, and weak coupling — where
either $\kappa/g$ or $\gamma/g$, or both, are significantly greater than one,
and compare these to the less noisy, strong coupling regime. A noticeable
difference in the selection of optimal probe states is observed when
transitioning from the strong-coupling to high-loss regimes. Fig. 1
illustrates these four regimes.

Strong CouplingNoisy-CavityNoisy-QubitWeak
Coupling$\kappa/g$$\gamma/g$$1$$3$$1$$3$$0$

Figure 1: Coupling regimes. The boxes colored in blue are the main focus of
this paper.

The quantum Fisher information (QFI) serves as our tool for measuring the
accuracy of our probe states [10, 11, 12, 13, 14, 15]. The QFI quantifies the
sensitivity of a quantum state to changes in the parameter being estimated,
which is inversely proportional to the variance of the estimated parameter
(see Appendix A for further detail). Due to the open evolution of our quantum
probe, the QFI has a non-trivial time dependence. For closed systems, the QFI
increases quadratically over time, but for open systems, there is a time for
which the QFI is maximized and the optimal time for measurement is achieved
[16]. In our study, we consider the maximum QFI over time, and then see how
the maximum QFI value scales with increasing system size $N$.

The remainder of the paper is organized as follows: Section II describes in
further detail the system under study. Section III examines a particularly
successful state for high-loss regimes, the polarized-$X$ state, and its max
QFI scaling across $N$. Section IV compares the results of the probe states
across all four regimes, as depicted in Fig. 1. Section V briefly discusses
our results when estimating detuning $\Delta$. Finally, Section VI provides a
conclusion and discussion of future work.

##  II N qubits-cavity system

In our study, we consider a system composed of $N$ two-level quantum systems
(qubits) interacting with a single electromagnetic cavity/resonator mode. We
include decays for both the resonator and the qubits, denoted as $\kappa$ and
$\gamma$, respectively. These decay rates appear in the equation that governs
the evolution of the open quantum system, which is a Gorini-Kossakowski-
Sudarshan-Lindblad (GKSL) type master equation [17, 18, 19, 20]:

$\dot{\rho}=-i[H,\rho]+\kappa\mathcal{D}[a](\rho)+\gamma\sum_{j=1}^{N}\mathcal{D}[\sigma_{-}^{(j)}](\rho),$
(1)

where $\mathcal{D}[O](\rho)=O\rho
O^{\dagger}-\frac{1}{2}\\{O^{\dagger}O,\rho\\}$ is the Lindblad dissipator for
an operator $O$. Eq. (1) encompasses both the unitary evolution driven by the
system Hamiltonian and the incoherent processes due to environmental
interactions. The system Hamiltonian, $H$ in Eq. (1) is given by,

$H=\frac{\omega_{q}}{2}\sum_{j=1}^{N}\sigma_{z}^{(j)}+\omega_{c}a^{\dagger}a+g\sum_{j=1}^{N}\big{(}a^{\dagger}\sigma_{-}^{(j)}+a\sigma_{+}^{(j)}\big{)}.$
(2)

where $\omega_{c}$ and $\omega_{q}$ are the frequencies of the cavity and the
qubits, $a^{\dagger}$ and $a$ are the photon creation and annihilation
operators for the cavity, $\sigma_{z}^{(j)}$ is the Pauli Z operator acting on
the $j^{\rm th}$ qubit, $g$ is the coupling strength between the qubits and
the cavity, and $\sigma_{-}^{(j)}$ and $\sigma_{+}^{(j)}$ represent the
lowering and raising operators acting on the $j^{\rm th}$ qubit.

We operate under resonance conditions and within a rotating frame, setting
$\omega_{q}=\omega_{c}=0$. Additionally, we consider the scenario of
estimating detuning, characterized by a nonzero difference between
$\omega_{q}$ and $\omega_{c}$, as an extra parameter of interest. In the case
of detuning estimation, we set this difference $\Delta=0.1g$, ensuring that
the system remains near-resonance and thereby avoiding significant alterations
to the model under investigation.

Within the $N$ qubits-cavity system, we examine the metrology performance of
the following probe states: the GHZ state, the Dicke states, and the
X-polarized state, each representing different levels of entanglement. The GHZ
state [21] is a maximally entangled state expressed as

$\ket{\text{GHZ}}=\frac{1}{\sqrt{2}}\big{(}\ket{0}^{\otimes
N}+\ket{1}^{\otimes N}\big{)}.$ (3)

The X-polarized state, so-named because it is an eigenstate of the sum of
$\sigma_{x}$ over the individual qubits, with eigenvalue $N$, is given by

$\ket{\text{X}}=\frac{1}{\sqrt{2^{N}}}(\ket{0}+\ket{1})^{\otimes N}.$ (4)

The Dicke states, containing intermediate levels of entanglement, are defined
as superpositions of all permutations of $N$ qubits with a fixed number of
excitations,

$\ket{\text{Dicke-n}}=\frac{1}{\sqrt{\binom{N}{n}}}\sum_{\text{permutations}}|\underbrace{0\ldots
0}_{N-n}\underbrace{1\ldots 1}_{n}\rangle.$ (5)

Figure 2: Scaling of maximum QFI with respect to number of qubits $N$

##  III Scaling of maximum QFI

Figure 2 compares the scaling of the maximum QFI as a function of the number
of qubits $N$ across Dicke states, GHZ state, and the $X$-state. We see that
though the $X$-state has smaller maximum QFI than the Dicke-$10$ state at
$N=20$, the $X$-state shows quadratic scaling, while the Dicke states appear
to have linear scaling with respect to $N$. The Dicke-$0$ state, which is
another seperable state, appears to perform quite poorly. The GHZ state also
seems to have scaling greater than linear with $N$.

Figure 3 illustrates the time-dependent behavior of QFI for the $X$-polarized
state, under various coupling regimes and sets of decay rates $\kappa/g$ and
$\gamma/g$, while keeping the number of qubits $N$ constant. The figure
illustrates a consistent pattern where the QFI first increases, peaks, and
then gradually diminishes. This trend is marked by oscillations, particularly
pronounced in regimes of poor cavity and qubit quality. Notably, the regions
exhibiting the highest QFI values and the corresponding maximal QFI both
escalate in size with an increase in $N$ for all the probe states we consider,
including the Dicke-$n$ states and the GHZ state [7].

Figure 3: Analysis of the $X$-state in various coupling regimes: top figure
shows time-dependent QFI, and bottom figure shows size scaling of maximum QFI.
The regimes are defined as follows: strong coupling ($\kappa/g=\gamma/g=0.8$),
noisy-cavity ($\kappa/g=3.0$, $\gamma/g=0.2$), noisy-qubit ($\kappa/g=0.2$,
$\gamma/g=3.0$), and weak coupling ($\kappa/g=\gamma/g=3.0$).

Our study focuses on the scaling of maximum QFI in high-loss regimes. By
fitting the function $y(N)=aN^{b}+c$ to the maximum QFI against the number of
qubits $N$, we focus on the scaling exponent $b$ to evaluate performance
against quantum metrology benchmarks. The standard quantum limit (SQL)
corresponds to the QFI scaling as $N$, i.e., $b$ = 1, and the Heisenberg limit
(HL) corresponds to the QFI scaling as $N^{2}$, i.e., $b$ = 2.

In Fig. 4, we show the scaling exponent $b$ for the $X$-state within the weak
coupling regime. Our results indicate that the $X$-state consistently
surpasses SQL, with the lowest observed exponent value being around $1.8$.
This plot highlights the $X$-state’s effectiveness in weak coupling
conditions, consistently approaching or even reaching Heisenberg limit
performance.

Figure 4: Map plot of the scaling exponent $b$ against different resonator and
qubit decay rates $\kappa/g$ and $\gamma/g$ for the $X$-state.

##  IV Comparison of Probe States

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 5: Bar plots illustrating comparative analysis of various initial probe
states for representative $N$ = 20 cases: The upper row corresponds to the
maximum QFI values, while the lower row corresponds to the scaling exponent
$b$. These comparisons span across four distinct regimes, providing a
comprehensive overview of performance variations in each scenario. (“Excited”
denotes the case that all $N$ qubits are initially excited.)

Fig. 5 presents a comparative analysis of the maximum QFIs and the scaling
exponents across different states for all four regimes. In the context of the
initially unentangled $X$-state, our bar plots in Fig. 5 show an intriguing
trend. For lower decay constant values, the $X$-state shows scaling closer to
SQL, aligning with expectations. However, as we explore higher decay constant
values, a notable shift towards scaling significantly better than SQL is
observed.

It is important to highlight that, while the $X$-state shows promising scaling
properties, its maximum QFI, as revealed in our bar plots, is not always
comparable to the magnitudes achievable with the Dicke states. In certain
high-loss scenarios, the $X$-state’s QFI approaches or matches that of the
Dicke states, but there are regimes where Dicke states maintain a clear
advantage in terms of QFI magnitude, particularly within the strong coupling
regime.

Saleem et al. showed that Dicke states can exceed SQL and even approach the
Heisenberg limit under specific conditions involving low but non-negligible
decay rates [7]. This finding is corroborated by the results shown in Fig.
LABEL:fig:strong_coupling_2. These results, however, are predominantly
observed in the strong coupling limit where $g\gg\kappa,\gamma$, a regime that
might be challenging to achieve in practical settings. In contrast, the
$X$-state demonstrates near Heisenberg limit scaling in a regime of weaker
coupling, characterized by $g\ll\kappa,\gamma$. This suggests that the
$X$-state could offer a more feasible route to high-performance quantum
metrology in environments where strong coupling is not achievable. One might
wonder why or how the $X$-state, which is a separable state, can lead to such
good performance. This state is only separable at $t$ = 0 and, through qubit-
cavity interactions the system becomes entangled [22]. It is also important to
note that the qubit-cavity coupling can be viewed as an interaction
“x-oriented” dipoles of the qubits ($\propto\sigma_{x}^{(j)}$ =
$\frac{1}{2}(\sigma_{-}^{(j)}+\sigma_{+}^{(j)})$) and cavity
($\propto(a+a^{\dagger})$). The $X$-polarized initial condition, with the
maximum possible projection of the qubit spins on the x-axis, might therefore
be expected to lead to strong qubit-cavity interactions.

Further analysis presented in Fig. 5 reveals that the $X$-state surpasses
other separable states in performance, particularly the Dicke-$0$ state (which
is the ground state, $\ket{0}^{\otimes N}$) and the fully excited state
$\ket{1}^{\otimes N}$, where $N$ denotes the total number of qubits comprising
our probe. This finding indicates that the $X$-state’s effectiveness in high-
loss regimes is not merely a result of its separability. Instead, it points to
unique characteristics of the $X$-state, likely tied to its initial
configuration in the quantum state space, which enhance its performance under
substantial losses. Additionally, the fully excited state shows comparable
performance to the $X$-state in terms of the scaling exponent within the
strong coupling and noisy-qubit regimes. However, our analysis indicates that,
at least for decay rates ranging from $0.2$ to $3.0$, the maximum scaling
exponent for the fully excited state does not surpass $1.5$. This contrasts
with the $X$-state’s ability to achieve the Heisenberg limit.

##  V Detuning estimation

In our simulations focusing on detuning estimation, we set the detuning
parameter $\Delta=\omega_{q}-\omega_{c}$ to $\Delta$ = 0.1 $g$, by assigning
$\omega_{q}$ a value of 0.1 $g$ and keeping $\omega_{c}$ at 0. This choice
ensures that $\Delta$ remains significantly smaller than $g$, thus maintaining
consistency in the system’s dynamics. Similar to coupling estimation, when
estimating detuning we are concerned with the QFI, except we take the partial
derivative of the state $\rho$ with respect to $\Delta$ instead of $g$ in Eq.
6. Across all four regimes, the $X$-state consistently outperformed other
states in detuning estimation tasks, although its performance did not
substantially exceed SQL.

Interestingly, the maximum QFI values observed for detuning estimation were
markedly lower than those for coupling estimation in all regimes. The highest
maximum QFI values were recorded in the strong coupling regime, yet these did
not surpass 10. For other states, including both GHZ and the Dicke states, the
scaling exponent $b$ hovered around 0. This result indicates that the GHZ and
Dicke states are relatively ineffective for estimating detuning, in stark
contrast to their performance in estimating the coupling strength. Example
results in the moderate coupling regime $\gamma/g=\kappa/g=1.0$ are shown in
Figure 6.

Figure 6: Scaling of maximum QFI against number of qubits $N$ when estimating
detuning $\Delta$.

##  VI Conclusions and future directions

We simulated a qubits-cavity system with up to $N=20$ qubits to evaluate the
metrology performance for various probe states, primarily focusing on high-
loss regimes, such as noisy-cavity, noisy-qubit, and weak coupling conditions.
Remarkably, we discovered that the fully separable $X$-polarized state not
only exceeds the standard quantum limit but often reaches the Heisenberg limit
in these scenarios. This highlights the $X$-state’s adaptability and noise
resilience, making it a promising candidate for practical quantum metrology
applications in environments characterized by significant losses.

One of the overarching themes of our research is the balance between
entanglement and noise resilience. Unlike highly entangled states, such as the
GHZ state which have diminished performance in noisy settings, the $X$-state
shows a robustness against noise, offering better performance in high-loss
conditions. This finding suggests that in certain practical quantum metrology
scenarios, especially those involving substantial losses, states with no
initial entanglement might be more advantageous. Future research will involve
understanding the unique properties of the $X$-state that contribute to its
good performance in noisy settings that not only set it apart from entangled
states like the GHZ and Dicke states but also other seperable states.

The practical nature of our study enhances the feasibility of applying these
theoretical models in experimental setups. Specifically, spin qubits in solids
are potential candidates for the proposed experiment: a small ensemble of
single electronic spins can be coupled to a superconducting microwave cavity
with realistic cQED parameters $\kappa/g$ = 1-10 and $\gamma/g$ =0.1-1, which
is conducive to the proposed configurations discussed above. By bridging the
gap between theory and practice, our research contributes to the advancement
of quantum technologies, particularly in noisy and loss-prone environments.

Acknowledgements: This material is based upon work supported by the U.S.
Department of Energy Office of Science National Quantum Information Science
Research Centers. Q.H.L., S.K.G. and Z.H.S. were supported by the Q-NEXT
Center. Work performed at the Center for Nanoscale Materials, a U.S.
Department of Energy Office of Science User Facility, was supported by thee
U.S. DOE Office of Basic Energy Sciences, under Contract No. DE-
AC02-06CH11357. A. S. was supported by QuEST grant No Q-113 of the Department
of Science and Technology, Government of India. The results in this work were
obtained from simulations using $\sim$1000 CPU-hours on the Bebop computing
cluster at Argonne National Laboratory.

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of Argonne National Laboratory (“Argonne”). Argonne, a U.S. Department of
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##  Appendix A Quantum Fisher Information

Consider a quantum state described by the density matrix $\rho(\theta)$, which
depends on the parameter $\theta$. The QFI, $F(\theta)$, is given by [23]:

$\displaystyle F(\theta)=\sum_{\\{a,b|\lambda_{a}+\lambda_{b}\neq
0\\}}\frac{2}{\lambda_{a}+\lambda_{b}}|\langle\lambda_{a}|\partial_{\theta}\rho|\lambda_{b}\rangle|^{2}~{}~{}~{}.$
(6)

In our work, $\theta$ is taken to be either the qubit-cavity coupling, $g$, or
the difference between qubit and cavity transition energies, $\Delta$.

In the context of quantum metrology, QFI is used to bound the precision of
parameter estimation. According to the Quantum Cramér-Rao Bound [24], the
variance of any unbiased estimator $\hat{\theta}$ of the parameter $\theta$ is
bounded by the inverse of the QFI:

$\text{Var}(\hat{\theta})\geq\frac{1}{MF(\theta)}~{}~{},$ (7)

where $M$ corresponds to the number of independent experiments carried out.
The square root of the variance corresponds to the uncertainty in the
parameter. This bound implies that higher QFI values allows for more precise
parameter estimation.