# Optimal multiple-phase estimation with multi-mode NOON states against photon
loss

Min Namkung  Center for Quantum Information, Korea Institute of Science and
Technology (KIST), Seoul, 02792, Korea  Dong-Hyun Kim  Center for Quantum
Information, Korea Institute of Science and Technology (KIST), Seoul, 02792,
Korea  Department of Physics, Yonsei University, Seoul 03722, Korea  Seongjin
Hong  Department of Physics, Chung-Ang University, Seoul 06974, Korea  Yong-Su
Kim  Center for Quantum Information, Korea Institute of Science and Technology
(KIST), Seoul, 02792, Korea  Division of Nanoscience and Technology, KIST
School, Korea University of Science and Technology, Seoul 02792, Korea
Changhyoup Lee<EMAIL_ADDRESS>Korea Research Institute of Standards
and Science, Daejeon 34113, Korea  Hyang-Tag Lim<EMAIL_ADDRESS>Center for Quantum Information, Korea Institute of Science and Technology
(KIST), Seoul, 02792, Korea  Division of Nanoscience and Technology, KIST
School, Korea University of Science and Technology, Seoul 02792, Korea

###### Abstract

Multi-mode NOON states can quantum-enhance multiple-phase estimation in the
absence of photon loss. However, a multi-mode NOON state is known to be
vulnerable to photon loss, and its quantum-enhancement can be dissipated by
lossy environment. In this work, we demonstrate that a quantum advantage in
estimate precision can still be achieved in the presence of photon loss. This
is accomplished by optimizing the weights of the multi-mode NOON states
according to photon loss rates in the multiple modes, including the reference
mode which defines the other phases. For practical relevance, we also show
that photon-number counting via a multi-mode beam-splitter achieves the
useful, albeit sub-optimal, quantum advantage. We expect this work to provide
valuable guidance for developing quantum-enhanced multiple-phase estimation
techniques in lossy environments.

##  I Introduction

Quantum resources can enhance performance of precisely estimating multiple
parameters over classical counterparts. For this reason, most studies have
focused on finding optimal quantum resources achieving the ultimate quantum
limit in multiple-parameter estimation e.polino ; m.szczy ; j.f.haase ,
including the use of the Greenberger-Horne-Zeilinger states l.z.liu ;
s.-r.zhao , single-photon Fock states e.polino2 , squeezed states y.xia ;
r.schnabel ; b.j.lawrie ; x.guo ; m.gessner ; c.oh ; s.-i.park , entangled
coherent states j.joo ; j.liu ; s.-y.lee , Holland-Burnett states z.su ;
m.a.ciampini ; g.y.xiang , and multi-mode NOON states p.c.humphreys ; l.zhang
; l.zhang2 ; s.hong ; s.hong2 ; j.urrehman . Among them, the multi-mode NOON
states are particularly known to achieve the enhanced estimate precision
outperforming the other probe states in the mean photon number for multiple-
phase estimation, consequently beating the standard quantum limit (SQL)
p.c.humphreys . From this fact, it is conceivable that multi-mode NOON states
could serve as a useful resource for developing novel quantum sensing
technology.

However, when a multi-mode NOON state is exposed to noisy or lossy
environment, it does not always provide the quantum advantage in precisely
estimating multiple-phases r.demkowicz ; u.doner ; r.demkowicz2 ; j.kolodynski
; m.kacprowicz . Disadvantage caused by photon loss when using multi-mode NOON
states in multiple-phase estimation becomes serious as the photon number $N$
increases u.doner . Fortunately, the number of modes in a multi-mode NOON
state is known to be a quantified resource for improving the multiple-phase
estimation in lossless case j.-d.yue . Thus, it is utmost importance to verify
whether a quantum advantage is also improved by increasing the number of modes
in a multi-mode NOON state against photon loss, even in the case that a
reference mode is more severely exposed to lossy environments than the other
modes.

In this work, we propose an optimal multiple-phase estimation scheme using the
weighted multi-mode NOON states in the presence of photon loss. Analytically
deriving the estimation precision for the proposed scheme, we investigate the
robustness of the scheme against photon loss and compare it with the scheme
using a particular type of the multi-mode NOON state proposed in Ref.
p.c.humphreys , which is known to be optimal in the lossless case. A quantum
advantage is also studied in comparison with the SQL set by a scheme using the
weighted multi-mode coherent states. As a result, we find that our scheme is
not only more robust against photon loss than the scheme in Ref. p.c.humphreys
, but also exhibits a quantum advantage over the SQL even in the presence of
loss. These findings are elaborated in more details for the cases of three-
and four-mode two-photon NOON states that have been experimentally
demonstrated s.hong ; s.hong2 , followed by the generalization of the scheme
with increasing the numbers of modes and photons. The proposed schemes are
also investigated with a measurement scheme with general structure consisting
of a multi-mode beam-splitter and multiple photon-number-resolving detectors
(PNRDs), offering a practical means to achieve the quantum advantage in
multiple-phase estimation under a lossy environment. We expect that our
results and methodology can be used for various studies in robust quantum
metrology, given that loss and decoherence are inevitable.

##  II Multiple-phase estimation scheme under lossy environment

Figure 1: Multiple-phase estimation using a $d+1$-mode probe state for
estimating $d$ unknown phases $\\{\phi_{1},\cdots,\phi_{d}\\}$ with a
measurement. Without loss of generality, the mode 0 is considered as a
reference. It is assumed that photon loss rate in the mode $j$ is
$\gamma_{j}\in[0,1]$.

###  II.1 Scheme

Let us consider a scheme for simultaneously estimating $d$ unknown phases
$\bm{\phi}=\\{\phi_{1},\cdots,\phi_{d}\\}$ in the presence of photon loss as
shown in Fig. 1. We assume that the mode 0 is used as a reference upon which
the phases of all the other modes are defined. For the multiple-phase
estimation, we employ a ($d+1$)-mode NOON state probe written by

$\displaystyle|\psi_{N}\rangle=\sum_{j=0}^{d}\sqrt{p_{j}}\frac{\hat{a}_{j}^{\dagger
N}}{\sqrt{N!}}|0\rangle^{\otimes d+1},$ (1)

where $\hat{a}_{j}^{\dagger}$ is the photon creation operator in the mode $j$
and $p_{j}$ with $\sum_{j=0}^{d}p_{j}=1$ and $p_{j}\in[0,1]$ is the weight of
the mode $j$. This probe state undergoes $d$ individual phase shifts described
by a unitary operator
$\hat{U}_{\bm{\phi}}=\bigotimes_{j=1}^{d}\exp(i\phi_{j}\hat{a}_{j}^{\dagger}\hat{a}_{j})$.
For practical relevance, we take into account photon loss that could occur in
the transmission channels or inefficient detectors. The photon loss can be
modeled by a fictitious beam-splitter represented by the mode transformation
$\hat{a}_{j}^{\dagger}\rightarrow\sqrt{1-\gamma_{j}}\hat{a}_{j}^{\dagger}+\sqrt{\gamma_{j}}\hat{e}_{j}^{\dagger}$
p.kok  with photon loss rate $\gamma_{j}\in[0,1]$, where
$\hat{e}_{j}^{\dagger}$ is a virtual vacuum input mode for describing
environment. It can be shown, by the Kraus representation of photon loss, that
photon losses occurring before and after the phase shifters are equivalent.
This allows to consider only the photon loss taking place onto the phase-
encoded probe state
$|\psi_{N,\bm{\phi}}\rangle=\hat{U}_{\bm{\phi}}|\psi_{N}\rangle$ before
measurement. The phase-encoded ($d+1$)-mode NOON state is thus transformed to
the output state

$\displaystyle\hat{\rho}_{N,\bm{\phi}}=\Lambda_{\gamma_{0}}\otimes\cdots\otimes\Lambda_{\gamma_{d}}(|\psi_{N,\bm{\phi}}\rangle\langle\psi_{N,{\bm{\phi}}}|),$
(2)

where $\Lambda_{\gamma_{j}}$ is a photon loss channel in the mode $j$ m.grassl
. If $\gamma_{j}$ is zero, then the quantum channel $\Lambda_{\gamma_{j}}$
becomes an identity channel. Finally, the estimation of $d$ unknown phases is
made based on the measurement results obtained by a measurement performed onto
the output state of Eq. (2). We also consider an estimation scheme with
classical light to examine a quantum advantage of the proposed scheme through
quantum parameter estimation theory that is introduced below. As a classical
benchmark, we consider the scheme that employs a ($d+1$)-mode coherent state
described by $\otimes_{j=0}^{d}|\alpha_{j}\rangle$ with
$\alpha_{j}=\sqrt{q_{j}N}$ ($q_{0}+\cdots+q_{j}=1$, $q_{j}\in[0,1]$) under the
same environment, leading to the output state
$|\sqrt{1-\gamma_{0}}\alpha_{0}\rangle\otimes|e^{i\phi_{1}}\sqrt{1-\gamma_{1}}\alpha_{1}\rangle\otimes\cdots\otimes|e^{i\phi_{d}}\sqrt{1-\gamma_{d}}\alpha_{d}\rangle$
followed by measurement and estimation.

###  II.2 Estimation precision

Denoting
$|\Delta\phi_{j}|^{2}=\langle(\phi_{\mathrm{est},j}-\phi_{j})^{2}\rangle$ as
an estimation uncertainty of the unknown phase $\phi_{j}$, the total
uncertainty of simultaneous estimation of $d$ phases can be defined as the sum
of individual uncertainties, i.e., $\sum_{j=1}^{d}|\Delta\phi_{j}|^{2}$. When
an unbiased estimator is used, the total uncertainty is lower-bounded by
p.c.humphreys ; j.urrehman ; j.liu2

$\displaystyle\sum_{j=1}^{d}|\Delta\phi_{j}|^{2}\geq\frac{\mathrm{Tr}\big{[}{F}_{\rm
C}^{-1}(\bm{\phi})\big{]}}{\mu}\geq\frac{\mathrm{Tr}\big{[}{F}_{\rm
Q}^{-1}(\bm{\phi})\big{]}}{\mu},$ (3)

where $\mu$ is the number of measurement being repeated, $F_{\rm
C}(\bm{\phi})$ is the Fisher information matrix (FIM), and $F_{\rm
Q}(\bm{\phi})$ is the quantum Fisher information matrix (QFIM). Here, an FIM
element is defined as

$\displaystyle{F}_{{\rm
C}(j,k)}=\sum_{\bm{l}}\frac{1}{P_{\bm{l}}}\left(\frac{\partial
P_{\bm{l}}}{\partial\phi_{j}}\right)\left(\frac{\partial
P_{\bm{l}}}{\partial\phi_{k}}\right),$ (4)

with the conditional probability $P_{\bm{l}}$ to obtain a measurement outcome
$\bm{l}$ conditioned on $\bm{\phi}$. QFIM is defined as j.liu2

$\displaystyle{F}_{{\rm
Q}(j,k)}=\frac{1}{2}\mathrm{Tr}\Big{[}\hat{\rho}_{N,\bm{\phi}}\left\\{\hat{L}_{j},\hat{L}_{k}\right\\}\Big{]},$
(5)

where $\\{\cdot,\cdot\\}$ denotes the anti-commutation and $\hat{L}_{j}$ is a
symmetric logarithmic derivative operator with respect to an unknown phase
$\phi_{j}$. In Eq. (3), ${\rm Tr}\left[F_{\rm C}^{-1}(\bm{\phi})\right]$ and
${\rm Tr}\left[{F}_{\rm Q}^{-1}(\bm{\phi})\right]$ are called the Cramer-Rao
bound (CRB) and the quantum Cramer-Rao bound (QCRB) (per a single measurement
try), respectively. Note that the CRB is a lower bound of the total
uncertainty determined for a given probe state and a chosen measurement
scheme, whereas the QCRB is a further lower bound that would be obtained when
an optimal measurement is employed for a given probe state. That is, the CRB
becomes the QCRB if a chosen measurement scheme is optimal. However, we need
an entanglement-based measurement to attain the QCRB in general, which is
experimentally challenging to implement s.hong ; j.urrehman . Thus, it is
necessary to consider the CRB provided by a measurement experimentally
configurable by linear optics. We also note that the CRB is asymptotically
attained by using maximum likelihood estimator when $\mu\gg 1$ l.pezze ;
c.w.helstrom .

##  III Quantum advantage in terms of the QCRB

###  III.1 Minimum QCRB of multi-mode NOON states

To evaluate the QCRB, we first derive the output state
$\hat{\rho}_{N,\bm{\phi}}$ defined in Eq. (2) for an arbitrary photon number
$N$ and phases $d$, as

$\displaystyle\hat{\rho}_{N,\bm{\phi}}=\left(\sum_{j=0}^{d}p_{j}(1-\gamma_{j})^{N}\right)|\widetilde{\psi}_{N,\bm{\phi}}\rangle\langle\widetilde{\psi}_{N,\bm{\phi}}|+\hat{\sigma}_{N}.$
(6)

Here, $\hat{\sigma}_{N}$ is a non-negative density matrix (see Appendix A)
diagonalized in the Fock state basis up to $(N-1)$, and
$|\widetilde{\psi}_{N,\bm{\phi}}\rangle$ is a lossy multi-mode NOON state in
the $N$-photon manifold written as

$\displaystyle|\widetilde{\psi}_{N,\bm{\phi}}\rangle$
$\displaystyle=\mathcal{N}\Bigg{(}\sqrt{p_{0}(1-\gamma_{0})^{N}}\frac{\hat{a}_{0}^{\dagger
N}}{\sqrt{N!}}$
$\displaystyle+\sum_{j=1}^{d}\sqrt{p_{j}(1-\gamma_{j})^{N}}\frac{\hat{a}_{j}^{\dagger
N}e^{iN\phi_{j}}}{\sqrt{N!}}\Bigg{)}|0\rangle^{\otimes d+1},$ (7)

with a normalization constant $\mathcal{N}$. One can analytically find the
minimum QCRB over all the weighted multi-mode NOON states taking the form of
Eq. (1) as

$\displaystyle\min_{\bm{p}}\mathrm{Tr}\left[{F}_{\rm
Q}^{-1}(\bm{\phi})\right]$
$\displaystyle=\frac{1}{4N^{2}}\left\\{\frac{\sqrt{d}}{(1-\gamma_{0})^{N/2}}+\sum_{j=1}^{d}\frac{1}{(1-\gamma_{j})^{N/2}}\right\\}^{2},$
(8)

and the optimal weights as

$\displaystyle p_{0}$
$\displaystyle=\frac{\sqrt{d}\lambda_{0}^{N}}{\sqrt{d}\lambda_{0}^{N}+\lambda_{1}^{N}+\cdots+\lambda_{d}^{N}},$
$\displaystyle p_{j}$
$\displaystyle=\frac{\lambda_{j}^{N}}{\sqrt{d}\lambda_{0}^{N}+\lambda_{1}^{N}+\cdots+\lambda_{d}^{N}},\
\ \mathrm{for}\ \ j\not=0$ (9)

with coefficients $\lambda_{j}=\prod_{l\not=j}(1-\gamma_{l})^{1/2}$ (see
Appendix B for the details). The minimum QCRB written in Eq. (III.1)
successfully encapsulates the previous results in Refs. s.hong ; s.hong2 ;
j.urrehman  as lossless case. The difference between $p_{0}$ and $p_{j}$
($j\not=0$) arises because the reference phase is assumed to be known
beforehand but the other phases are unknown. We prove that the QCRB of the
multi-mode NOON states is attainable (see Appendix C). Thus, the QCRB of the
multi-mode NOON states is understood as the minimum estimation uncertainty
over all possible measurements.

For the SQL, the minimum QCRB over all multi-mode coherent states can also be
found (see Appendix D for the details) and written as

$\displaystyle\min_{\bm{q}}\mathrm{Tr}\left[{F}_{\rm
Q,coh}^{-1}(\bm{\phi})\right]=\frac{1}{4N}\left\\{\sqrt{\frac{d}{1-\gamma_{0}}}+\sum_{j=1}^{d}\frac{1}{\sqrt{1-\gamma_{j}}}\right\\}^{2},$
(10)

and the optimal weights are

$\displaystyle q_{0}$
$\displaystyle=\frac{\sqrt{d}\lambda_{0}}{\sqrt{d}\lambda_{0}+\lambda_{1}+\cdots+\lambda_{d}},$
$\displaystyle q_{j}$
$\displaystyle=\frac{\lambda_{j}}{\sqrt{d}\lambda_{0}+\lambda_{1}+\cdots+\lambda_{d}},\
\ \mathrm{for}\ \ j\not=0.$ (11)

When the photon loss rates $\gamma_{j}$ $\forall j\not=0$ are all equal to
$\gamma$ except the reference mode $\gamma_{\rm ref}=\gamma_{0}$, the QCRB of
Eq. (III.1) is rewritten as

$\displaystyle\min_{\bm{p}}\mathrm{Tr}\left[{F}_{\rm
Q}^{-1}(\bm{\phi})\right]$
$\displaystyle=\frac{1}{4N^{2}}\left\\{\frac{\sqrt{d}}{(1-\gamma_{\rm
ref})^{N/2}}+\frac{d}{(1-\gamma)^{N/2}}\right\\}^{2},$ (12)

which reduces to

$\displaystyle\min_{\bm{p}}\mathrm{Tr}\left[{F}_{\rm
Q}^{-1}(\bm{\phi})\right]=\frac{1}{4N^{2}}\frac{(\sqrt{d}+d)^{2}}{(1-\gamma)^{N}}$
(13)

when $\gamma=\gamma_{\rm ref}$. For the latter case, the optimal weights are
given as

$\displaystyle p_{0}=\frac{\sqrt{d}}{\sqrt{d}+d},\ \
p_{j}=\frac{1}{\sqrt{d}+d}\ \ \mathrm{for}\ \ j\not=0,$ (14)

and interestingly these recover the particular multi-mode NOON state proposed
by Humphreys’ et al. in Ref. p.c.humphreys . This implies that Humphreys’ NOON
states are optimal only when $\gamma=\gamma_{\rm ref}$, including the lossless
case. On the other hand, the SQL of Eq. (10) is rewritten as

$\displaystyle\min_{\bm{q}}\mathrm{Tr}\left[{F}_{\rm
Q,coh}^{-1}(\bm{\phi})\right]=\frac{1}{4N}\left\\{\sqrt{\frac{d}{1-\gamma_{\rm
ref}}}+\frac{d}{\sqrt{1-\gamma}}\right\\}^{2},$ (15)

when $\gamma=\gamma_{j}$ $\forall j\in[1,\cdots,d]$, and

$\displaystyle\min_{\bm{q}}\mathrm{Tr}\left[{F}_{\rm
Q,coh}^{-1}(\bm{\phi})\right]=\frac{1}{4N}\frac{(\sqrt{d}+d)^{2}}{1-\gamma},$
(16)

when all $\gamma=\gamma_{\rm ref}$ for which the optimal weights $\bm{q}$ take
the same form as Eq. (14). A more detailed discussion about these findings is
provided below.

###  III.2 Quantum advantage from two-photon NOON states

Let us here investigate a quantum advantage of the proposed scheme using the
multi-mode NOON states in comparison with the SQL under the same lossy
environment. The environment that can be characterized in terms of the
reference mode $\gamma_{\rm ref}$ and the other modes $\gamma=\gamma_{j}$
$\forall j\in[1,\cdots,d]$, which is inspired by the distinct roles that
appear through Eq. (III.1) to Eq. (10). To quantify a quantum advantage, we
define the ratio of the QCRB for the considered scheme under study to that for
the optimized multi-mode coherent states, written as

$r_{\rm QA}=1-\frac{\mathrm{Tr}\left[{F}_{\rm
Q}^{-1}(\bm{\phi})\right]}{\mathrm{Tr}\left[{F}_{\rm
Q,coh}^{-1}(\bm{\phi})\right]}.$ (17)

A positive value of $r_{\rm QA}$ means that a methological performance of the
considered scheme surpasses the SQL. In a specific case that
$\gamma=\gamma_{\rm ref}$, $r_{\rm QA}$ can reduce to

$r_{\rm QA}=1-\frac{1}{N}(1-\gamma)^{1-N}.$ (18)

Here, note that a quantum advantage is independent of the mode number $d$ and
diminishes as $N$ becomes larger. On the other hand, the role of the mode
number $d$ is significant when $\gamma\not=\gamma_{\rm ref}$. It can be easily
shown by using Eqs. (III.1) and (15) that, in the case of $\gamma<\gamma_{\rm
ref}$, the quantum advantage quantified by Eq. (17) is improved by increasing
the mode number $d$. Moreover, when a multi-mode NOON state has sufficiently
large $d$, the quantum advantage in Eq. (17) converges to
$\lim_{d\rightarrow\infty}r_{\rm QA}=1-\frac{1}{N}(1-\gamma)^{1-N}$ as written
in Eq. (18), which is independent of the photon loss rate of the reference
mode. It is interesting that, for $\gamma<\gamma_{\rm ref}$, the quantum
advantage of the proposed scheme is irrelevant to the photon loss rate in a
reference mode.

Figure 2: Quantum advantage ($r_{\rm QA}$) in terms of the photon loss rate
$(\gamma_{\rm ref})$ in the reference mode and that ($\gamma$) in the other
modes for (a) the scheme proposed in Ref. p.c.humphreys  and (b) our scheme in
comparison with the SQL when $N=2$ and $d=2$.

Figure 2 illustrates the regions of photon loss rates where a quantum
advantage is achieved for the scheme proposed in Ref. p.c.humphreys  (see Fig.
2(a)) and for the scheme we propose with optimal multi-mode NOON states (see
Fig. 2(b)) when $N=d=2$. It is evident that the colored area of Fig. 2(b) is
larger than that of Fig. 2(a), which implies that a multiple-phase estimation
scheme proposed in this work is more robust to the photon loss than the scheme
proposed in Ref. p.c.humphreys . Also note that the proposed scheme is
slightly more robust against the photon loss in the reference mode than that
of the other modes.

To look into more detailed behaviors, we take two specific values of
$\gamma_{\rm ref}$ from Fig. 2(b), and compare different cases including the
case using balanced NOON states (i.e., $p_{j}=\frac{1}{d}$ $\forall j$). In
Fig. 3(a), we show that when $\gamma=\gamma_{\rm ref}$, our scheme is
equivalent to the scheme proposed in Ref. p.c.humphreys , already pointed out
in Eq. (14), and the balanced NOON states are also nearly optimal due to equal
losses. It can be shown that the equivalence holds for any $N$ and $d$, but
the near-optimality of the balanced NOON states is degraded with increasing
$\gamma$. Interestingly, on the other hand, our scheme becomes significantly
advantageous when $\gamma\not=\gamma_{\rm ref}$, which is clearly shown in
Fig. 3(b) for the case of $\gamma_{\rm ref}=0.5$ that precision difference
between these two schemes is well-visible. It reveals that our scheme is
better than the scheme in Ref. p.c.humphreys  and the balanced NOON states are
far from being optimal due to unequal losses. Also note that our scheme can
outperform the SQL in the lossy environment, which can be observed in Fig. 3.

Figure 3: QCRB in terms of the photon loss rate $\gamma$ when (a) $\gamma_{\rm
ref}=\gamma$ and (b) $\gamma_{\rm ref}=0.5$ for the schemes using the optimal
NOON states (blue), the NOON states proposed in Ref. p.c.humphreys  (green),
the balanced NOON states (orange), and the coherent states (red) setting the
SQL. Here $N=2$ and $d=2$ are considered for all the schemes.

Figure 4: Critical photon loss rate $\gamma_{\rm crit}$ such that a quantum
advantage remains until $\gamma\leq\gamma_{\rm crit}$ exhibits the robustness
of the considered scheme for $N,d\geq 2$. Blue area shows the region that
$r_{\rm QA}$ is lower than zero, wherein a multi-mode NOON state does not
exhibit a quantum advantage.

###  III.3 Quantum advantage from $d$-mode $N$-photon NOON states

Here, we examine if a quantum advantage is still achievable by $d$-mode
$N$-photon NOON states with $N,d>2$ and how large it is. For the purpose, we
define the critical photon loss rate $\gamma_{\rm crit}$ such that a quantum
advantage is observed when $\gamma_{j}\equiv\gamma\leq\gamma_{\rm crit}$ for
$j\in\\{1,\cdots,d\\}$ where $\gamma_{\rm ref}=0.5$. That is, $\gamma_{\rm
crit}$ indicates a robustness of the scheme under study. Figures 4(a) and (b)
present $\gamma_{\rm crit}$ for the scheme proposed in Ref. p.c.humphreys  and
$\gamma_{\rm crit}$ for our scheme, respectively, when $\gamma<\gamma_{\rm
ref}=0.5$ as an example that the photon loss of the reference mode is larger
than that of the other modes. It is clearly shown that the scheme proposed in
this work is more robust against photon loss than Humphreys et al.’s scheme
proposed in Ref. p.c.humphreys . Interestingly, the proposed scheme becomes
more robust as the number of phases or modes increase. This means that our
scheme is useful when the number of phases to be estimated is many, and offers
robust multiple-phase estimation schemes even when $N<d$.

##  IV Quantum advantage in terms of the CRB

Until the previous section, we have demonstrated that the QCRB of an optimal
multi-mode NOON state can surpass the SQL. Exploiting the argument written in
Appendix A together with Ref. l.pezze , it is easy to show that a measurement
achieving the QCRB has an entanglement-based projection
$|\widetilde{\psi}_{N,\bm{\phi}}\rangle\langle\widetilde{\psi}_{N,\bm{\phi}}|$.
However, it is known that such the measurement is experimentally challenging
to implement in general s.hong2 ; j.urrehman . Thus, for experimental
feasibility, we need to verify whether an estimation scheme using a
measurement with a linear optical structure can provides the SQL.

We consider a measurement scheme illustrated as Fig. 5 that employs PNRDs with
a multi-mode beam-splitter being optimized in the multiple-phase estimation
scheme using multi-mode NOON states, and investigate whether the associated
CRB evaluated in Appendix E can surpass the SQL and how close it is to be the
QCRB of Eq. (III.1). For the optimization over all multi-mode beam-splitters,
we exploit Clement’s configuration w.r.clements  to represent a general multi-
mode beam-splitter as a $(d+1)$-mode transformation

$\mathrm{U}=\mathrm{D}\prod_{j}\mathrm{V}_{j}(\theta_{j},\chi_{j}).$ (19)

Here, $\mathrm{D}$ is a diagonal unitary matrix and
$\mathrm{V}_{j}(\theta_{j},\chi_{j})$ is a block unitary matrix composed of a
$(d-1)$-dimensional identity matrix and 2 dimensional unitary matrix with real
parameters $\theta_{j}$ and $\chi_{j}$, which is implemented by a beam-
splitter and a phase shifter. Note that $\mathrm{D}$ consists of redundant
phases disappeared from the CRB. In other words, the optimization over all
possible multi-mode beam-splitters is equivalent to that over all the real
parameters in Eq. (19).

Figure 5: Multiple-phase estimation using a $d+1$-mode probe state for
estimating $d$ unknown phases $\\{\phi_{1},\cdots,\phi_{d}\\}$ with a
measurement consisting of a multi-mode beam-splitter and PNRDs.

Figure 6: CRB of our schemes with PNRDs performed through the multi-mode beam-
splitter are compared with the QCRB of Eq. (6) (orange) and the SQL of Eq.
(III.1) (red), in terms of $\gamma$ when $\gamma_{\rm ref}=0.5$ for (a) $N=2$,
$d=2$ and (b) $N=2$, $d=3$. The PNRDs are applied to not only the multi-mode
NOON states having been optimized with respect to Eq. (6) (green), but also
the case that both the multi-mode NOON state and multi-mode beam-splitter are
simultaneously optimized to maximize the CRB (blue).

Now, we use the configuration introduced in Eq. (19) for the optimal multi-
mode beam-splitter and evaluate the two cases: (i) we optimize the multi-mode
beam-splitter for the multi-mode NOON states having minimized the QCRB of Eq.
(III.1), and (ii) we optimize simultaneously the multi-mode beam-splitter and
the NOON state to minimize the CRB of Eq. (4). The results are shown in Fig. 6
for $N=2$ and $d=2,3$ when $\gamma_{\rm ref}=0.5$. It is clear that the
considered measurement scheme provides mostly inferior performance when
applied to the NOON states that have been optimized with respect to the QCRB.
However, when both the multi-mode beam-splitter and the NOON states are
simultaneously optimized to minimize the CRB, their precision becomes
significantly improved and even nearly optimal with respect to the minimum
QCRB. Experimental structure of optimal multi-mode beam-splitter depends on
the number of modes $d$. The minimum CRB of Fig. 6(a) is attained by a multi-
mode beam-splitter with unbalanced ratio, but that of Fig. 6(b) is exhibited
from a balanced one.

##  V Conclusion

We have proposed and studied an optimal scheme of multiple-phase estimation
using the weighted multi-mode NOON states under a lossy environment. We have
shown that the proposed scheme surpasses the SQL even in the presence of loss,
and is more robust to loss than the scheme proposed in Ref. p.c.humphreys
that employs only a particular type of multi-mode NOON state. These behaviors
persist when the numbers of modes and photons increase, and especially a
quantum advantage even increases with the number of modes when the reference
mode is lossier than the signal modes carrying the multi-mode NOON states.
When the proposed scheme is equipped with a measurement setting configurable
by using a multi-mode beam-splitter and PNRDs, a quantum advantage is still
achieved although sub-optimal, which provides a practical means in multiple-
phase estimation under a lossy environment. We believe the results obtained
and the methodology used in this work will motivate various relevant studies
in robust quantum metrology under lossy and noisy environments.

An interesting future study would be to apply our methodology to mitigate
other types of noises, e.g., the phase-diffusion m.t.dimario , or to estimate
other types of multiple parameters of interest, e.g., absorption parameters
a.karsa . One may also investigate if similar advantage and robustness can be
obtained by employing other strategies such as reference configurations
a.z.goldberg  or post-selection drm . Another application of this work can be
made to estimate a linear function of multiple parameters such as distributed
quantum sensing s.-r.zhao ; x.guo ; j.liu3 ; d.-h.kim .

## Appendix A. Phase-encoded multi-mode NOON states under photon loss

Here we derive the explicit form of a phase-encoded multi-mode NOON state in
Eq. (6) exposed to photon loss. We first note that a quantum channel
$\Lambda_{\gamma_{j}}(\cdot)$ is described by

$\displaystyle\Lambda_{\gamma_{j}}(\cdot)=\mathrm{Tr}_{e_{j}}\left\\{\hat{B}_{j}(\cdot)_{a_{j}}\otimes|0\rangle\langle
0|_{e_{j}}\hat{B}_{j}^{\dagger}\right\\}$ (20)

with the vacuum state $|0\rangle$ in the environment mode $j$ and unitary
operator $\hat{B}_{j}$ with respect to a mode transformation
$\hat{B}_{j}\hat{a}_{j}^{\dagger}\hat{B}_{j}^{\dagger}=\sqrt{1-\gamma_{j}}\hat{a}_{j}^{\dagger}+\sqrt{\gamma_{j}}\hat{e}_{j}^{\dagger}$
s.m.barnett , where $\mathrm{Tr}_{e_{j}}\left\\{\cdot\right\\}$ is the partial
trace over the $j$-th environment system. From Eq. (20), It is straightforward
to derive three terms $\Lambda_{\gamma_{j}}(|N\rangle\langle N|)$,
$\Lambda_{\gamma_{j}}(|N\rangle\langle 0|)$, and
$\Lambda_{\gamma_{j}}(|0\rangle\langle 0|)$ as

$\displaystyle\Lambda_{\gamma_{j}}(|N\rangle\langle N|)$
$\displaystyle=\sum_{k=0}^{N}\frac{N!}{(N-k)!k!}(1-\gamma_{j})^{N-k}\gamma_{j}^{k}|N-k\rangle\langle
N-k|,$ $\displaystyle\Lambda_{\gamma_{j}}(|N\rangle\langle 0|)$
$\displaystyle=(1-\gamma_{j})^{N/2}|N\rangle\langle 0|,\ \
\Lambda_{\gamma_{j}}(|0\rangle\langle 0|)=|0\rangle\langle 0|.$ (21)

Substituting the above terms in Eq. (2), the output state is derived as

$\displaystyle\hat{\rho}_{N,\bm{\phi}}=\left\\{\sum_{j=0}^{d}p_{j}(1-\gamma_{j})^{N}\right\\}|\widetilde{\psi}_{N,\bm{\phi}}\rangle\langle\widetilde{\psi}_{N,\bm{\phi}}|+\underbrace{\sum_{j=0}^{d}p_{j}\left\\{\sum_{r=1}^{N}\frac{N!}{r!(N-r)!}(1-\gamma_{j})^{N-r}\gamma_{j}^{r}\frac{\hat{a}_{j}^{\dagger
N-r}|0\rangle\langle 0|^{\otimes
d}\hat{a}_{j}^{N-r}}{(N-r)!}\right\\}}_{\hat{\sigma}_{N}}.$ (22)

## Appendix B. Evaluation of the minimum QCRB

Here we analytically evaluate the minimum QCRB of $d+1$-mode NOON states. The
non-negative operator $\hat{\sigma}_{N}$ in Eq. (22) is diagonalized by
vectors of the Fock basis whose photon number is strictly less than $N$.
Meanwhile, $|\widetilde{\psi}_{N,\bm{\phi}}\rangle$ in Eq. (22) is a linear
combination of the Fock basis vectors with photon number equal to $N$. Thus,
the support of $\hat{\sigma}_{N}$ in Eq. (22) is orthogonal to
$|\widetilde{\psi}_{N,\bm{\phi}}\rangle$, and $\hat{\rho}_{N,\bm{\phi}}$ is in
the spectral decomposition with the eigenvectors
$|0\rangle,\cdots,|N-1\rangle$ together with
$|\widetilde{\psi}_{N,\bm{\phi}}\rangle$. We can employ the theoretical
framework written in Ref. j.liu2  to derive components of the QFIM

$\displaystyle{F}_{\mathrm{Q}(a,b)}(\bm{\phi})=4N^{2}\left\\{\sqrt{p_{a}p_{b}(1-\gamma_{a})^{N}(1-\gamma_{b})^{N}}\delta_{ab}-\frac{p_{a}p_{b}(1-\gamma_{a})^{N}(1-\gamma_{b})^{N}}{p_{0}(1-\gamma_{0})^{N}+\cdots+p_{d}(1-\gamma_{d})^{N}}\right\\}.$
(23)

By using the Sherman-Morrison formula sherman-morrison , the QCRB is evaluated
as

$\displaystyle\mathrm{Tr}\left\\{{F}_{\rm
Q}^{-1}(\bm{\phi})\right\\}=\frac{1}{4N^{2}}\left\\{\frac{d}{p_{0}(1-\gamma_{0})^{N}}+\sum_{j=1}^{d}\frac{1}{p_{j}(1-\gamma_{j})^{N}}\right\\},$
(24)

which is the convex function of $\bm{p}=(p_{0},\cdots,p_{d})$. Since a
feasible set of $\bm{p}$ is convex, a global minimum is at the point where
gradient of the QCRB in Eq. (24) is vanished s.boyd . Hence, the minimum QCRB
is evaluated as Eq. (10) by the optimal weights in Eq. (III.1).

## Appendix C. Attainability of the QCRB

It was shown that the necessary and sufficient condition that the QCRB of a
quantum state $\hat{\rho}$ is attained by a measurement is j.liu2

$\displaystyle\mathrm{Tr}\left(\hat{\rho}[\hat{L}_{a},\hat{L}_{b}]\right)=0,$
(25)

where $\hat{L}_{a}$ and $\hat{L}_{b}$ are symmetric logarithm derivative
operators. By using the above proposition, we show that there is a measurement
attaining the QCRB in Eq. (24). From the output state described by Eq. (22),
the trace of the left hand side in Eq. (25) is written as linear combination
of
$\langle\widetilde{\psi}_{N,\bm{\phi}}|[\hat{L}_{a},\hat{L}_{b}]|\widetilde{\psi}_{N,\bm{\phi}}\rangle$
and $\langle 0|^{\otimes
d+1}\hat{a}_{j}^{N-r}[\hat{L}_{a},\hat{L}_{b}]\hat{a}_{j}^{\dagger
N-r}|0\rangle^{\otimes d+1}$ with $r\not=0$. First, since
$|\widetilde{\psi}_{N,\bm{\phi}}\rangle$ in Eq. (22) is a multi-mode NOON
state, there are symmetric logarithm derivative operators $\hat{L}_{a}$ and
$\hat{L}_{b}$ such that
$\langle\widetilde{\psi}_{N,\bm{\phi}}|[\hat{L}_{a},\hat{L}_{b}]|\widetilde{\psi}_{N,\bm{\phi}}\rangle=0$
p.c.humphreys . Second, as we consider $\hat{L}_{a}$ and $\hat{L}_{b}$ of the
multi-mode NOON state $|\widetilde{\psi}_{N,{\bm{\phi}}}\rangle$, their
supports are orthogonal to $\hat{a}_{j}^{\dagger N-r}|0\rangle^{\otimes d+1}$
as $r\not=0$. Thus, $\langle 0|^{\otimes
d+1}\hat{a}_{j}^{N-r}[\hat{L}_{a},\hat{L}_{b}]\hat{a}_{j}^{\dagger
N-r}|0\rangle^{\otimes d+1}$ also become zero. Finally, the symmetric
logarithm derivative operators satisfy the equality in Eq. (25), which leads
us to understand the attainability of the QCRB.

## Appendix D. Evaluation of the SQL

Here, we evaluate the SQL quantified by the QCRB of multi-mode coherent
states. If the mean photon number of an initial coherent states
$\bigotimes_{j=0}^{d}|\alpha_{j}\rangle$ with $\alpha_{j}=\sqrt{q_{j}N}$ is
not sufficiently large, then the coherent state in the reference mode is not
understood as strong. Thus, it is reasonable to describe a probe state as a
phase-averaged multi-mode coherent state a.z.goldberg ; m.jarzyna :

$\displaystyle\hat{\rho}_{\alpha,\bm{\phi}}=\frac{1}{2\pi}\int_{0}^{2\pi}d\theta\bigotimes_{j=0}^{d}|\beta_{j}e^{i\theta}\rangle\langle\beta_{j}e^{i\theta}|,$
(26)

with $\beta_{j}=\alpha_{j}e^{i\phi_{j}}$. Of course, the above probe state
encapsulates the case of the strong reference beam when $|\alpha|^{2}$ is
sufficiently large and $|\beta_{0}|^{2}$ is much larger than any
$|\beta_{j}|^{2}$ with $j\not=0$ m.jarzyna . Photon loss channels transform
the above phase-averaged multi-mode coherent state to

$\displaystyle\Lambda_{\gamma_{0}}\otimes\cdots\otimes\Lambda_{\gamma_{d}}\left(\hat{\rho}_{\alpha,\bm{\phi}}\right)=\frac{1}{2\pi}\int_{0}^{2\pi}d\theta\bigotimes_{j=0}^{d}|\beta_{j}\sqrt{1-\gamma_{j}}e^{i\theta}\rangle\langle\beta_{j}\sqrt{1-\gamma_{j}}e^{i\theta}|.$
(27)

Finally, the QCRB of the above noisy state is analytically evaluated as
a.z.goldberg

$\displaystyle\mathrm{Tr}\left\\{{F}_{\rm
Q,coh}^{-1}(\bm{\phi})\right\\}=\frac{d}{4(1-\gamma_{0})|\beta_{0}|^{2}}+\frac{1}{4}\sum_{j=1}^{d}\frac{1}{(1-\gamma_{j})|\beta_{j}|^{2}}=\frac{1}{4N}\left\\{\frac{d}{(1-\gamma_{0})q_{0}}+\sum_{j=1}^{d}\frac{1}{(1-\gamma_{j})q_{j}}\right\\}.$
(28)

Thus, the minimum value of the QCRB over all $q_{j}$ satisfying the relation
$\sum_{j=0}^{d}q_{j}=1$ is straightforwardly evaluated as Eq. (10).

## Appendix E. Measurement probabilities for the CRB

For evaluating the CRB, we first derive the FIM in Eq. (4) by deriving the
measurement probabilities $P_{\bm{l}}$. Let us start from expressing the
measurement probability as

$\displaystyle P_{\bm{l}}=\left|\langle\bm{l}|\hat{U}_{\rm
BS}\hat{\rho}_{N,{\bm{\phi}}}\hat{U}_{\rm BS}|\bm{l}\rangle\right|^{2},$ (29)

where
$|\bm{l}\rangle=|l_{0}\rangle\otimes|l_{1}\rangle\otimes\cdots\otimes|l_{d}\rangle$
is a multi-mode Fock basis vector with respect to measurement outcomes
$\bm{l}=(l_{0},\cdots,l_{d})$, and $\hat{U}_{\rm BS}$ is a unitary operator of
a multi-mode beam-splitter. By representing the unitary transformation with
$\hat{U}_{\rm BS}$ as a scattering matrix $\mathrm{U}$, the above measurement
probability is derived as

$\displaystyle
P_{\bm{l}}=\left|\sqrt{\frac{N!}{\prod_{j=0}^{d}l_{j}!}}\left\\{\sum_{j=0}^{d}p_{j}^{1/2}(1-\gamma_{j})^{N/2}e^{iN\phi_{j}}\mathrm{U}_{j0}^{*l_{0}}\mathrm{U}_{j1}^{*l_{1}}\cdots\mathrm{U}_{jd}^{*l_{d}}\right\\}\right|^{2}+\langle\bm{l}|\hat{U}_{\rm
BS}\hat{\sigma}_{N}\hat{U}_{\rm BS}|\bm{l}\rangle.$ (30)

We note that the second term of $P_{\bm{l}}$ is independent of the phases
$\bm{\phi}$. Thus, the CRB is only determined by the first term of Eq. (30) by
the definition in Eq. (4).

## Acknowledgement

This work was partly supported by National Research Foundation of Korea (NRF)
grant funded by the Korea government (MSIT) (2022M3K4A1097123), Institute for
Information & communications Technology Planning & Evaluation (IITP) grant
funded by the Korea government (MSIT) (2020-0-00947, RS-2023-00222863), and
the KIST research program (2E32941).

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# Hyperfine structure of the $2\,^{3}\\!P$ state in 9Be and the nuclear
quadrupole moment

Mariusz Puchalski  Faculty of Chemistry, Adam Mickiewicz University,
Uniwersytetu Poznańskiego 8, 61-614 Poznań, Poland  Jacek Komasa  Faculty of
Chemistry, Adam Mickiewicz University, Uniwersytetu Poznańskiego 8, 61-614
Poznań, Poland  Krzysztof Pachucki  Faculty of Physics, University of Warsaw,
Pasteura 5, 02-093 Warsaw, Poland

###### Abstract

We have performed accurate calculations of the hyperfine structure of the
$2\,^{3}\\!P$ state in the 9Be atom with the help of highly optimized,
explicitly correlated functions, accounting for the leading finite nuclear
mass, radiative, nuclear structure and relativistic effects. By comparison
with measurements, we have determined the 9Be nuclear quadrupole moment to be
$Q_{\rm N}=0.05350(14)$ barns, which is not only the most accurate result, but
also disagrees with previous determinations.

††preprint: Version 1.2

The electric quadrupole moment $Q_{\rm N}$ of nuclei is a measure of the
deformation of the nuclear charge distribution with respect to the spherical
symmetry. This deformation comes from a strong dependence of nuclear forces on
the orientation of the nucleon spin and is present in atomic nuclei with a
spin equal to or greater than one, as it is for $\,{}^{9}$Be with $I=3/2$. In
principle, $Q_{\rm N}$ could be determined from the nuclear structure theory,
but such calculations with controlled accuracy have not yet been feasible.
Only very recently and only for such a simple nucleus as the deuteron,
theoretical predictions based on the chiral effective field theory have
reached the uncertainty of 1% [1], and this result agreed with the more
precise value from the rotational spectroscopy of the deuterium hydride (HD)
molecule [Puchalski:20].

Originally, $Q_{\rm N}$ for many nuclei was determined by the electron
scattering of the nuclei, for example for 7Li [Voelk:91]. However, for 9Be the
available experimental data are not only limited in accuracy but also differ
noticeably from each other, indicating that the scattering results are not
always conclusive [Vinciguerra:69, Bernheim:69]. Nowadays, reliable $Q_{\rm
N}$ values can be derived from a combination of theoretical and atomic or
molecular spectroscopic data. This has been realized for LiH, LiF, and LiCl
molecules [Urban:90], leading to a $Q_{\mathrm{N}}(^{7}\mathrm{Li})$
consistent with the nuclear scattering value [Voelk:91]. Although appropriate
theoretical calculations were performed for BeH+ [Diercksen:89, Borin:92] as
well as for the 7,9Be- anion [Nemouchi:03] years ago, there have been no
corresponding measurements reported so far. Therefore, the method of choice
for finding $Q_{\mathrm{N}}(^{9}\mathrm{Be})$ is atomic spectroscopy.

In the accurate determination of $Q_{\rm N}$ from atomic spectroscopy, it is
important to understand the electron-nucleus interaction at the fundamental
level. Recent advances in measurements of electronic [CODATA:18] and muonic
atoms [Krauth:21], together with progress in the theoretical description of
atomic spectra [Pachucki:18], indicate the importance of the nuclear structure
effects. Therefore, accurate knowledge of the nuclear quadrupole moment will
give access to details of the electron-nucleus interactions which, so far,
have not been visible, like for example the nuclear quadrupole polarizability
[Friar:05].

The ground state $2\,^{1}\\!S_{0}$ of Be is fully symmetric; thus, the nuclear
quadrupole moment does not lead here to any observable effect. In contrast, in
the lowest excited $2\,^{3}\\!P$ level, which is metastable, the hyperfine
interactions cause level splitting, and this splitting can be accurately
measured [Blachman:67]. So, it is atomic structure theory that must interpret
the hyperfine structure in terms of the magnetic dipole and the electric
quadrupole moments of the nucleus. Still, the same hyperfine interactions that
lead to the hyperfine splitting also cause the hyperfine mixing of different
$2\,^{3}\\!P_{J}$ levels. This mixing had been accounted for in Ref.
Blachman:67 but in a very simplified way. Although the accuracy of these
approximations has been questioned [Diercksen:89], consecutive atomic
structure calculations, using modern and sophisticated approaches, have relied
on these original inaccurate calculations of the hyperfine mixing. An
exception is the work by Beloy et al. [Beloy:08], in which this mixing was
calculated in the second order of perturbation theory using the relativistic
CI+MBPT approach, but its numerical precision turned out to be insufficient to
obtain $Q_{\rm N}$ with a competitive accuracy. This hyperfine mixing is not
specific just to Be atoms, it is certainly present in all the other atomic
systems. Therefore, the determinations of quadrupole moments for many other
nuclei might have lower accuracy than anticipated, what should be verified, as
we have done here for the 9Be nucleus.

In this work, we employ well-optimized explicitly correlated Gaussian wave
functions to have good control over the numerical accuracy. With these wave
functions we perform calculations of the 9Be hyperfine structure, with a
complete treatment of the hyperfine mixing of different $2\,^{3}\\!P_{J}$
levels. Moreover, we fully account for the leading radiative (QED), finite
nuclear mass, and nuclear structure effects, while dealing with relativistic
and higher order corrections approximately. Our result for the electric
quadrupole moment $Q_{\rm N}(^{9}{\rm Be})$ show that all of its previous
determinations were not as accurate as claimed, due to a very approximate
treatment of the hyperfine mixing and neglect of the radiative (QED)
corrections.

## Fine and hyperfine structure Hamiltonian.—

The most accurate approach for light atomic systems solves at first the
nonrelativistic Hamiltonian

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{\vec{p}^{\,2}_{\mathrm{N}}}{2\,m_{\mathrm{N}}}+\sum_{a}\frac{\vec{p}^{\,2}_{a}}{2}-\sum_{a}\frac{Z}{r_{a}}+\sum_{a<b}\frac{1}{r_{ab}},$
(1)

and next includes relativistic effects as a perturbation. Relevant
relativistic effects are split into the fine structure Hamiltonian

$\displaystyle H_{\rm fs}=$ $\displaystyle\ Q_{1}+Q_{2}$ (2) $\displaystyle
Q_{1}=$ $\displaystyle\
\sum_{a}\,\frac{Z\,\alpha}{4}\,\vec{\sigma}_{a}\cdot\Big{[}\frac{(g-1)}{m^{2}}\frac{\vec{r}_{a}}{r_{a}^{3}}\times\vec{p}_{a}-\frac{g}{m\,m_{\rm
N}}\,\frac{\vec{r}_{a}}{r_{a}^{3}}\times\vec{p}_{\rm N}\Big{]}$
$\displaystyle+\sum_{a\neq
b}\,\frac{\alpha}{4\,m^{2}}\,\vec{\sigma}_{a}\cdot\Big{[}g\,\frac{\vec{r}_{ab}}{r_{ab}^{3}}\times\vec{p}_{b}-(g-1)\,\frac{\vec{r}_{ab}}{r_{ab}^{3}}\times\vec{p}_{a}\Big{]}$
(3) $\displaystyle Q_{2}=$ $\displaystyle\
-\sum_{a<b}\frac{3\,\alpha\,g^{2}}{16\,m^{2}}\,\sigma_{a}^{i}\,\sigma_{b}^{j}\,\biggl{(}\frac{r_{ab}^{i}\,r_{ab}^{j}}{r_{ab}^{5}}-\frac{\delta^{ij}}{3\,r_{ab}^{3}}\biggr{)}$
(4)

and the hyperfine structure Hamiltonian

$\displaystyle H_{\rm hfs}$
$\displaystyle=\vec{I}\cdot\vec{Q}+\frac{3\,I^{i}\,I^{j}}{I\,(2\,I-1)}\,\frac{Q^{ij}}{6}$
(5) $\displaystyle\vec{Q}$ $\displaystyle=\vec{Q}_{1}+\vec{Q}_{2}+\vec{Q}_{3}$
(6) $\displaystyle\vec{Q}_{1}$
$\displaystyle=\sum_{a}\frac{1}{3}\,\frac{Z\,\alpha\,g\,g_{\rm N}}{m\,m_{\rm
N}}\,\vec{\sigma}_{a}\,\pi\,\delta^{3}(r_{a})$ (7) $\displaystyle\vec{Q}_{2}$
$\displaystyle=\sum_{a}\frac{Z\,\alpha\,g_{\mathrm{N}}}{2\,m\,m_{\mathrm{N}}}\,\frac{\vec{r}_{a}}{r_{a}^{3}}\times\vec{p}_{a}-\frac{Z\,\alpha\,(g_{\mathrm{N}}-1)}{2\,m_{\mathrm{N}}^{2}}\,\frac{\vec{r}_{a}}{r_{a}^{3}}\times\vec{p}_{\rm
N}$ (8) $\displaystyle Q_{3}^{j}$
$\displaystyle=\sum_{a}\frac{Z\,\alpha\,g\,g_{\mathrm{N}}}{8\,m\,m_{\mathrm{N}}}\,\sigma_{a}^{i}\,\biggl{(}\frac{3\,r_{a}^{i}\,r_{a}^{j}}{r_{a}^{5}}-\frac{\delta^{ij}}{r_{a}^{3}}\biggr{)}$
(9) $\displaystyle Q^{ij}$
$\displaystyle=-\sum_{a}\alpha\,Q_{\mathrm{N}}\,\biggl{(}\frac{3\,r_{a}^{i}\,r_{a}^{j}}{r_{a}^{5}}-\frac{\delta^{ij}}{r_{a}^{3}}\biggr{)}\,,$
(10)

where $\vec{p}_{\rm N}=-\sum_{a}\vec{p}_{a}$ is the nuclear momentum,
$\vec{I}$ is the nuclear spin, $g$ is the free-electron g-factor,
$Q_{\mathrm{N}}$ is the electric quadrupole moment of the nucleus, and $g_{\rm
N}$ is the nuclear g-factor

$g_{\rm N}=\frac{m_{\rm N}}{Z\,m_{\rm p}}\,\frac{\mu}{\mu_{\rm
N}}\,\frac{1}{I}.$ (11)

Leading QED effects are included in the free electron g-factor, while higher
order corrections are accounted for as described later on.

Because the hyperfine splittings are only about 100 times smaller than the
fine splitting, one cannot assume that hyperfine levels have a definite
angular momentum $\vec{J}=\vec{L}+\vec{S}$, but only definite $\vec{L}$ and
$\vec{S}$. It is convenient therefore to extend the original formulation of
the hyperfine splitting by Hibbert in [Hibbert:75] and represent the fine and
the hyperfine structure of the $2\,^{3}\\!P$ state in terms of an effective
Hamiltonian, instead of expectation values only

$\displaystyle H_{\mathrm{eff}}=$ $\displaystyle\
c_{1}\,\vec{L}\cdot\vec{S}+c_{2}\,(L^{i}L^{j})^{(2)}(S^{i}S^{j})^{(2)}$
$\displaystyle\
+a_{1}\,\vec{I}\cdot\vec{S}+a_{2}\,\vec{I}\cdot\vec{L}+a_{3}\,(L^{i}L^{j})^{(2)}S^{i}I^{j}$
$\displaystyle\
+\frac{b}{6}\,\frac{3\,(I^{i}\,I^{j})^{(2)}}{I\,(2\,I-1)}\,\frac{3\,(L^{i}L^{j})^{(2)}}{L\,(2\,L-1)},$
(12)

where the coefficients $a_{1},a_{2},a_{3},b,c_{1},c_{2}$ are independent of
$J$, but are specific to the particular state. These coefficients can be
obtained, for example, from the matrix elements of $H_{\rm fs}$ and $H_{\rm
hfs}$ in the decoupled $|M_{L},M_{S}\rangle$ or $|J,M_{J}\rangle$ basis. Once
these coefficients are known, the above Hamiltonian can be diagonalized
yielding the hyperfine levels. Alternatively, the hyperfine structure can be
represented in terms of $J$-dependent $A_{J}$ and $B_{J}$ coefficients

$\displaystyle H_{\rm
hfs,\,eff}=A_{J}\vec{I}\cdot\vec{J}+\frac{B_{J}}{6}\,\frac{3\,(I^{i}\,I^{j})^{(2)}}{I(2\,I-1)}\,\frac{3\,(J^{i}\,J^{j})^{(2)}}{J(2\,J-1)},$
(13)

which are conventionally used to represent the measured values, while the fine
structure is given in terms of differences in the centroid energies.

If the atomic levels had a definite value $J$, the fine structure would be
given by the expectation value of $H_{\rm eff}$, namely

$\displaystyle E_{\rm fs}(J)=$ $\displaystyle\
\left\\{\begin{array}[]{lr}c_{1}+c_{2}/6&\mbox{\rm for}\;J=2\\\
-c_{1}-5\,c_{2}/6&\mbox{\rm for}\;J=1\\\ -2\,c_{1}+5\,c_{2}/3&\mbox{\rm
for}\;J=0\end{array}\right.,$ (17)

and the hyperfine structure by

$\displaystyle A_{J}=$ $\displaystyle\
\left\\{\begin{array}[]{lr}a_{1}/2+a_{2}/2+a_{3}/6&\mbox{\rm for}\;J=2\\\
a_{1}/2+a_{2}/2-5\,a_{3}/6&\mbox{\rm for}\;J=1\end{array}\right.\,,$ (20)
$\displaystyle B_{J}=$ $\displaystyle\ \left\\{\begin{array}[]{lr}b&\mbox{\rm
for}\;J=2\\\ -b/2&\mbox{\rm for}\;J=1\end{array}\right.\,.$ (23)

Because, in general, one cannot assume that the levels have a definite value
of $J$, the effective hyperfine Hamiltonian in Eq. (Fine and hyperfine
structure Hamiltonian.—) has to be diagonalized numerically. Nonetheless, the
hyperfine structure can still be represented in terms of $A_{J}$ and $B_{J}$
coefficients, and we use them for comparison of theoretical predictions with
experimental results and for the determination of the nuclear quadrupole
moment $Q_{\rm N}$.

## Relativistic, radiative, and finite nuclear size corrections.—

The hyperfine Hamiltonian in Eq. (5) represents the leading hyperfine
interactions, but there are also many small corrections which are often
overlooked in literature. These corrections contain terms with higher powers
of the fine structure constant $\alpha$. Because most of them are proportional
to the Fermi contact interaction, i.e. to $\vec{I}\cdot\vec{Q}_{1}$ in Eq.
(5), we account for them in terms of the following factor

$\displaystyle\tilde{a}_{1}=a_{1}(1+\epsilon).$ (24)

Below, we briefly describe contributions included in the $\epsilon$ term.

The $\mathcal{O}(\alpha)$ correction is analogous to that in hydrogenic
systems [Eides:01] and consists of two parts. The first part is due to the
nuclear recoil

$\displaystyle H^{(5)}_{\rm rec}=$ $\displaystyle\
\biggl{[}-\frac{3\,Z\,\alpha}{\pi}\,\frac{m}{m_{\rm
N}}\,\ln\Bigl{(}\frac{m_{\rm
N}}{m}\Bigr{)}\biggr{]}\vec{I}\cdot\sum_{a}\frac{2}{3}\,\frac{Z\,\alpha\,g_{\rm
N}}{m\,m_{\rm N}}\,\vec{\sigma}_{a}\,\pi\,\delta^{3}(r_{a})$ (25)

and numerically is very small, almost negligible. The recoil contribution to
$\epsilon$ amounts to $\epsilon_{\rm rec}=-0.000\,011$. The second part of the
$\mathcal{O}(\alpha)$ correction is due to the finite nuclear size and the
nuclear polarizability, and is given by [Eides:01, Puchalski:09]

$\displaystyle H^{(5)}_{\rm fs}=$ $\displaystyle\
\bigl{[}-2\,Z\,\alpha\,m\,r_{Z}\bigr{]}\vec{I}\cdot\sum_{a}\frac{2}{3}\,\frac{Z\,\alpha\,g_{\rm
N}}{m\,m_{\rm N}}\,\vec{\sigma}_{a}\,\pi\,\delta^{3}(r_{a}),$ (26)

where $r_{Z}$ is a kind of effective nuclear radius called the Zemach radius.
Disregarding the inelastic effects, this radius can be written down in terms
of the electric charge $\rho_{E}$ and magnetic-moment $\rho_{M}$ densities as

$\displaystyle r_{Z}=\int
d^{3}r\,d^{3}r^{\prime}\,\rho_{E}(r)\,\rho_{M}(r^{\prime})\,|\vec{r}-\vec{r}^{\prime}|.$
(27)

Nevertheless, the inelastic, i.e. polarizability, corrections can be
significant, but because they are very difficult to calculate, they are
usually neglected. In this work we account for possible inelastic effects by
employing $r_{Z}$ from a comparison of very accurate calculations of hfs in
9Be+ with the experimental value, namely $r_{Z}=4.07(5)\;{\rm fm}$
[Puchalski:09]. Because this correction is also proportional to the contact
Fermi interaction, we represent it in terms of $\epsilon_{\rm
fs}=-0.000\,615$.

Next, there are radiative and relativistic corrections of the relative order
$\mathcal{O}(\alpha^{2})$. The radiative correction, beyond that included by
the free electron g-factor, is [Eides:01]

$\displaystyle H^{(6)}_{\rm rad}=Z\,\alpha^{2}\biggl{(}\ln
2-\frac{5}{2}\biggr{)}\vec{I}\cdot\sum_{a}\frac{2}{3}\frac{Z\,\alpha\,g_{\rm
N}}{m\,m_{\rm N}}\,\vec{\sigma}_{a}\,\pi\,\delta^{3}(r_{a})$ (28)

and the corresponding $\epsilon$ factor is $\epsilon_{\rm rad}=-0.000\,384$.
The $\mathcal{O}(\alpha^{2})$ relativistic and higher order corrections are
much more complicated. They have been calculated for the ground state of
$\,{}^{9}$Be+ [Puchalski:09]. Here we take this result and assume that it is
proportional to the Fermi contact interaction, and obtain $\epsilon_{\rm
rel}=0.001\,664$. This is the only approximation we assume in this work, and
as a consequence we neglect the mixing of the $2\,^{3}\\!P_{1}$ state with the
nearby lying $2\,^{1}\\!P_{1}$. Exactly for this reason we will use only the
hyperfine splitting of the $2\,^{3}\\!P_{2}$ state, which does not mix with
$2\,^{1}\\!P_{1}$ for the determination of $Q_{\rm N}$. The resulting total
correction is

$\displaystyle\epsilon=\epsilon_{\rm rec}+\epsilon_{\rm fs}+\epsilon_{\rm
rad}+\epsilon_{\rm rel}=0.654\cdot 10^{-3}.$ (29)

Some previous works present these multiplicative correction for all individual
hyperfine contributions, but in our opinion this cannot be fully correct
because higher order relativistic corrections may include additional terms,
beyond that in $H_{\rm eff}$ in Eq. (Fine and hyperfine structure
Hamiltonian.—). These corrections are expected to be much smaller than the
experimental uncertainty for the $B_{J}$ coefficient, and therefore are
neglected.

## ECG wave function and expectation values.—

Let us now move to the calculations of the $a_{i},b$, and $c_{i}$
coefficients. To obtain sufficiently high accuracy for these parameters, we
express the four-electron atomic wave function as a linear combination of
properly symmetrized explicitly correlated Gaussian (ECG) functions,

$\Psi\left(\\{\vec{r}_{a}\\}\right)=\sum_{n=1}^{K}t_{n}\,{\cal
A}\left[\,\phi_{n}\left(\\{\vec{r}_{a}\\}\right)\,\chi_{\\{a\\}}\right]\,,$
(30)

where $t_{n}$ are linear coefficients, $\cal A$ is the antisymmetrization
operator over electronic indices, and $\\{a\\}$ and $\\{\vec{r}_{a}\\}$ denote
the sequence of electron indices and coordinates, respectively. The electronic
$P$ symmetry of the states was enforced using the following spatial functions

$\displaystyle\phi^{i}(\\{\vec{r}_{a}\\})$ $\displaystyle=$ $\displaystyle
r^{i}_{a}\,\exp\big{[}-\sum_{b}w_{b}\,r^{2}_{b}-\sum_{c<d}u_{cd}\,r^{2}_{cd}\big{]}$
(31)

with $w_{b}$ and $u_{cd}$ being nonlinear variational parameters, while the
spin functions corresponding to different spin $S=1$ projections were, for
$\\{a\\}=(1,2,3,4)$:

$\displaystyle\chi^{1-1}_{\\{a\\}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\alpha_{1}\,\beta_{2}-\beta_{1}\,\alpha_{2})\,\beta_{3}\,\beta_{4}\,,$
(32) $\displaystyle\chi^{10}_{\\{a\\}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\alpha_{1}\,\beta_{2}-\beta_{1}\,\alpha_{2})(\alpha_{3}\,\beta_{4}-\beta_{4}\,\alpha_{3}),$
(33) $\displaystyle\chi^{11}_{\\{a\\}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\alpha_{1}\,\beta_{2}-\beta_{1}\,\alpha_{2})\,\alpha_{3}\,\alpha_{4}\,,$
(34)

where $\alpha_{i}$ and $\beta_{i}$ are the one-electron spin-up and spin-down
functions. To control the uncertainty of our results we performed the
calculations with several basis sets successively increasing their size. The
nonlinear parameters were optimized variationally with respect to the
nonrelativistic energy $E$ until the energy reached stability in a desired
number of digits. From the analysis of convergence we obtained the
extrapolated nonrelativistic energies and mean values used in Table I. The
largest wave functions optimized variationally were composed of 6144 terms,
leading to the nonrelativistic energy in agreement with the result obtained by
Kedziorski et al. [Kedziorski:20].

Table 1: Convergence of the nonrelativistic energy and theoretical fine and hyperfine structure parameters for the $2\,^{3}\\!P$ state of 9Be (in MHz). Mass $m_{\mathrm{N}}=9.012\,183\,07(8)$ u [Wang:17] and magnetic moment $\mu/\mu_{\rm N}=-1.177\,432(3)$ [Stone:15] were used for the 9Be nucleus. $K$ | $E/\mathrm{a.u.}$ | $c_{1}$ | $c_{2}$ | $a_{1}$ | $a_{2}$ | $a_{3}$ | $b/Q_{\mathrm{N}}\,{\rm(MHz/barn)}$
---|---|---|---|---|---|---|---
1536 | $-14.566\,340\,608\,5$ | $32\,986.385$ | $5\,399.032$ | $-231.116\,70$ | $-22.698\,781$ | $14.788\,494$ | $27.150\,476$
2048 | $-14.566\,341\,144\,7$ | $32\,987.849$ | $5\,399.186$ | $-231.126\,66$ | $-22.698\,923$ | $14.788\,389$ | $27.149\,896$
3072 | $-14.566\,341\,380\,0$ | $32\,989.401$ | $5\,399.240$ | $-231.130\,99$ | $-22.698\,986$ | $14.788\,467$ | $27.149\,112$
4096 | $-14.566\,341\,441\,5$ | $32\,989.854$ | $5\,399.190$ | $-231.129\,61$ | $-22.699\,124$ | $14.788\,579$ | $27.148\,954$
6144 | $-14.566\,341\,466\,0$ | $32\,990.027$ | $5\,399.172$ | $-231.128\,99$ | $-22.699\,186$ | $14.788\,630$ | $27.148\,898$
$\infty$ | $-14.566\,341\,474(8)$ | $32\,990.2(2)$ | $5\,399.15(2)$ | $-231.128\,4(6)$ | $-22.699\,24(5)$ | $14.788\,68(5)$ | $27.148\,87(3)$

The optimized nonrelativistic wave functions were subsequently employed to
evaluate the fine and hyperfine parameters from the expectation values of the
corresponding quantum-mechanical operators (2)-(10). For this purpose we chose
$J=2$, $L=1$, and for an arbitrary $M$

$\displaystyle c_{1}$ $\displaystyle=\langle J,M|Q_{1}|J,M\rangle,$ (35)
$\displaystyle c_{2}$ $\displaystyle=6\,\langle J,M|Q_{2}|J,M\rangle,$ (36)
$\displaystyle a_{1}$ $\displaystyle=\langle
J,M|\vec{Q}_{1}\cdot\vec{J}|J,M\rangle/3,$ (37) $\displaystyle a_{2}$
$\displaystyle=\langle J,M|\vec{Q}_{2}\cdot\vec{J}|J,M\rangle/3,$ (38)
$\displaystyle a_{3}$ $\displaystyle=\langle
J,M|\vec{Q}_{3}\cdot\vec{J}|J,M\rangle,$ (39) $\displaystyle b$
$\displaystyle=\langle L,M|Q^{ij}\,L^{i}\,L^{j}|L,M\rangle/5.$ (40)

The numerical values of these parameters and their convergence with increasing
size of the ECG basis are shown in Table 1. Their relative accuracy can be
estimated collectively as better than $6\cdot 10^{-6}$.

## Fine and hyperfine structure.—

The hyperfine transition frequencies $\nu_{\\!J}(F;F+1)$ can be expressed in
terms of the $A_{J}$ and $B_{J}$ parameters and vice versa. For this, one
writes

$\displaystyle\langle H_{\rm hfs,eff}\rangle_{F}=$ $\displaystyle\
A_{J}\,A_{IJF}+B_{J}\,B_{IJF}+C_{J}\,C_{IJF}$ (41)

where

$\displaystyle A_{IJF}$ $\displaystyle=\frac{1}{2}K$ (42) $\displaystyle
B_{IJF}$ $\displaystyle=\frac{3/4\,K(K+1)-I(I+1)J(J+1)}{2I(2I-1)J(2J-1)}$ (43)

with $K=F(F+1)-I(I+1)-J(J+1)$ and the total angular momentum
$\vec{F}=\vec{S}+\vec{L}+\vec{I}$. The octupole term $C_{IJF}$ is given e.g.
by Schwartz [Schwartz:55] and Jaccarino [Jaccarino:54].

For $J=1$ we arrive at

$\displaystyle A_{1}$
$\displaystyle=-\frac{1}{6}\,\nu_{1}\\!\left(\frac{1}{2};\frac{3}{2}\right)-\frac{3}{10}\,\nu_{1}\\!\left(\frac{3}{2};\frac{5}{2}\right)$
(44) $\displaystyle B_{1}$
$\displaystyle=\frac{1}{3}\,\nu_{1}\\!\left(\frac{1}{2};\frac{3}{2}\right)-\frac{1}{5}\,\nu_{1}\\!\left(\frac{3}{2};\frac{5}{2}\right)\,,$
(45)

and for $J=2$

$\displaystyle A_{2}$
$\displaystyle=-\frac{3}{50}\,\nu_{2}\\!\left(\frac{1}{2};\frac{3}{2}\right)-\frac{7}{50}\,\nu_{2}\\!\left(\frac{3}{2};\frac{5}{2}\right)-\frac{4}{25}\,\nu_{2}\\!\left(\frac{5}{2};\frac{7}{2}\right)$
(46) $\displaystyle B_{2}$
$\displaystyle=\frac{2}{5}\,\nu_{2}\\!\left(\frac{1}{2};\frac{3}{2}\right)+\frac{2}{5}\,\nu_{2}\\!\left(\frac{3}{2};\frac{5}{2}\right)-\frac{16}{35}\,\nu_{2}\\!\left(\frac{5}{2};\frac{7}{2}\right)$
(47) $\displaystyle C_{2}$
$\displaystyle=-\frac{1}{50}\,\nu_{2}\\!\left(\frac{1}{2};\frac{3}{2}\right)+\frac{1}{50}\,\nu_{2}\\!\left(\frac{3}{2};\frac{5}{2}\right)-\frac{1}{175}\,\nu_{2}\\!\left(\frac{5}{2};\frac{7}{2}\right).$
(48)

Table 2: $A_{J}$, $B_{J}$, and $C_{J}$ parameters determined using Eqs. (44)-(48) from experimental [Blachman:67] and theoretical hfs frequencies $\nu_{\\!J}$ (in MHz). The theoretical $B_{2}$ is by definition equal to the experimental one, which serves for determination of $b$ in Eq. (49). | Experimental | Theoretical | Difference
---|---|---|---
$A_{1}$ | $-140.1564(8)$ | $-140.1091$ | $-0.0473(8)$
$B_{1}$ | $-3.2363(7)$ | $-3.2337$ | $-0.0026(7)$
$A_{2}$ | $-124.6165(5)$ | $-124.6073$ | $-0.0092(5)$
$B_{2}$ | $0.7800(21)$ | $0.7800$ | $0.0$
$C_{2}$ | $-0.00030(9)$ | $-0.00032$ | $0.00002(9)$

The above listed parameters (44)-(48) were evaluated using both experimental
and theoretical transition frequencies $\nu_{\\!J}(F;F+1)$, and their
numerical values are presented in Table 2. The experimental values of
$\nu_{\\!J}$ (in MHz) were taken from Ref. Blachman:67, assuming their
uncertainties are statistical and uncorrelated. The resulting experimental
values for $A_{1}$ and $B_{1}$ are the same as in Ref. Blachman:67, while for
$A_{2}$ and $B_{2}$ are twice more accurate due to inclusion of the $C_{2}$
coefficient. On the other hand, theoretical values were found by
diagonalization of the effective Hamiltonian of Eq. (Fine and hyperfine
structure Hamiltonian.—). Most importantly, the parameter $b$ was fixed by
matching theoretical and experimental $B_{2}$, which holds for

$\displaystyle b=1.4525(40)\;{\rm MHz},$ (49)

where the uncertainty originates from that of the experimental one in $B_{2}$.

The differences between experimental and theoretical values, shown in Tab. 2,
are not discrepancies but result from neglected higher order relativistic and
QED effects, which are beyond those proportional to the Fermi contact
interaction. For the $A_{2}$ coefficient these effects are of relative order
$0.07\cdot 10^{-3}$, while for $A_{1}$ they are $0.33\cdot 10^{-3}$. They are
larger for $A_{1}$ due to hyperfine mixing of $2\,^{3}\\!P_{1}$ with the
nearby $2\,^{1}\\!P_{1}$ state, which we have not taken into account in our
calculation. Moreover, the relative difference for the $B_{1}$ coefficient is
$\delta B_{1}/B_{1}=0.80\cdot 10^{-3}$, which most probably is also due to
this mixing. For this reason, we have chosen $B_{2}$ and not $B_{1}$ for the
determination of the $b$ coefficient.

The centroids of the calculated hyperfine energy levels are shown in Tab. 3 in
the column ‘fs levels’ for all $J$ values of the $2\,^{3}\\!P$ state. This
table contains also experimental [Blachman:67] and calculated fine-structure
transition frequencies as well as their difference.

Table 3: Theoretical fine structure (fs) energy levels of the $2\,^{3}\\!P$ state of 9Be, and the comparison of experimental [Blachman:67] and theoretical fs transition frequencies (in GHz). $J$ | fs levels | $\nu^{\mathrm{exp}}(J;J+1)$ | $\nu^{\mathrm{the}}(J;J+1)$ | Diff.
---|---|---|---|---
2 | $33.890$ |  |  |
1 | $-37.489$ | $71.86(24)$ | $71.379$ | $0.48(24)$
0 | $-56.987$ | $19.41(19)$ | $19.498$ | $-0.09(19)$

This difference is much larger than the estimate of relativistic corrections,
which is $\mathcal{O}((Z\,\alpha)^{2})$. Therefore, we presume, that these
differences are due to inaccurate experimental results, because these fine
splittings have not been directly measured but determined from the magnetic
field dependence of the hfs splittings [Blachman:67].

## Determination of $Q_{N}(^{9}{\rm Be})$.—

Having calculated the value of $b/Q_{\mathrm{N}}=27.148\,87(3)$ MHz/barn, see
Tab. 1, and fixed $b$ by matching theoretical and experimental values of
$B_{2}$, we obtain the electric nuclear quadrupole moment $Q_{\mathrm{N}}$ of
$\,{}^{9}$Be

$\displaystyle Q_{\rm N}$ $\displaystyle=0.053\,50(14)\text{ barn}\,.$ (50)

The uncertainty in $Q_{\mathrm{N}}$ comes from the experimental error in the
$\nu_{2}$ hyperfine frequencies. In this context, the uncertainties from the
neglected higher order relativistic and QED effects in the $b$ parameter, as
well as the numerical uncertainties, are much smaller and are not shown.

Among the literature results listed in Tab. 4 and depicted in Fig. 1, only the
$Q_{\mathrm{N}}$ reported by Beloy et al. [Beloy:08], thanks to its large
uncertainty, agrees with our quadrupole moment. The lack of agreement with the
other previous studies indicates that it is very difficult to estimate
theoretical uncertainties using well-established, atomic structure methods
such as MCHF or CI+MBPT.

Table 4: Comparison of the electric quadrupole moment (in barns) of the 9Be nucleus obtained using atomic structure calculations. Source | $Q_{\mathrm{N}}(^{9}{\rm Be})$
---|---
Blachman and Lurio, 1967 [Blachman:67] | $0.049(3)$
Ray et al. , 1973 [Ray:73] | $0.052\,5(3)$
Sinanoǧlu and Beck, 1973 [Sinanoglu:73] | $0.054\,94$
Beck and Nicolaides, 1984 [Beck:84] | $0.055\,45$
Sundholm and Olsen, 1991 [Sundholm:91] | $0.052\,88(38)$
Jönsson and Fisher, 1993 [Jonsson:93] | $0.052\,56$
Nemouchi et al., 2003 [Nemouchi:03] | $0.052\,77$
Beloy et al. , 2008 [Beloy:08] | $0.053(3)$
This work | $0.053\,50(14)$
Figure 1: Comparison of the electric quadrupole moment (in barns) of the 9Be
nucleus obtained using atomic structure calculations.

## Conclusions.—

We have determined the electric quadrupole moment $Q_{\rm N}(^{9}{\rm Be})$
with significantly higher accuracy and in disagreement with previous values.
The improvement achieved in this work has several independent sources. The
first one is the recalculation of the $A_{2}$, $B_{2}$, and $C_{2}$
coefficients from experimental hyperfine splitting. More precisely, including
$C_{2}$ in the effective Hamiltonian $H_{\rm hfs,eff}$ in Eq. (13) enabled us
to decrease the experimental uncertainties by a factor of 2. The second
improvement is due to the inclusion of the hyperfine mixing by exact
diagonalization of the effective fine- and hyperfine-structure Hamiltonian of
Eq. (Fine and hyperfine structure Hamiltonian.—). The third improvement comes
from the very accurate calculation of the expectation values of the fine- and
hyperfine-structure operators using explicitly correlated functions allowing
for the complete electron correlations. The fourth one is due to the exact
inclusion of the finite nuclear mass in the fine and hyperfine interaction in
Eqs. (2) and (5). Finally, the fifth source of improvement is due to
accounting for the relativistic, radiative and nuclear structure corrections
by appropriate rescaling of the $a_{1}$ parameter.

Regarding our theoretical uncertainties, they come exclusively from neglected
higher order relativistic and QED contributions, in particular those due to
hyperfine mixing of the $2\,^{3}\\!P_{1}$ and $2\,^{1}\\!P_{1}$ states. The
calculation of the complete $\mathcal{O}(\alpha^{2})$ correction is possible
but technically difficult. It has been performed for Be+ in [Puchalski:14] due
to the availability of the exponentially correlated basis functions there.
Nevertheless, if a complete $\mathcal{O}(\alpha^{2})$ correction is known, one
can use both $B_{1}$ and $B_{2}$ parameters to obtain an even more accurate
nuclear quadrupole moment of 9Be.

## Acknowledgments.—

This research was supported by National Science Center (Poland) Grant No.
2014/15/B/ST4/05022 as well as by a computing grant from Poznań Supercomputing
and Networking Center and by PL-Grid Infrastructure.

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