# Search for rare decay $J/\psi\to\phi e^{+}e^{-}$

M. Ablikim1, M. N. Achasov10,d, S. Ahmed15, M. Albrecht4, M. Alekseev55A,55C,
A. Amoroso55A,55C, F. F. An1, Q. An52,42, Y. Bai41, O. Bakina27, R. Baldini
Ferroli23A, Y. Ban35, K. Begzsuren25, D. W. Bennett22, J. V. Bennett5, N.
Berger26, M. Bertani23A, D. Bettoni24A, F. Bianchi55A,55C, E. Boger27,b, I.
Boyko27, R. A. Briere5, H. Cai57, X. Cai1,42, A. Calcaterra23A, G. F. Cao1,46,
S. A. Cetin45B, J. Chai55C, J. F. Chang1,42, W. L. Chang1,46, G.
Chelkov27,b,c, G. Chen1, H. S. Chen1,46, J. C. Chen1, M. L. Chen1,42, P. L.
Chen53, S. J. Chen33, X. R. Chen30, Y. B. Chen1,42, W. Cheng55C, X. K. Chu35,
G. Cibinetto24A, F. Cossio55C, H. L. Dai1,42, J. P. Dai37,h, A. Dbeyssi15, D.
Dedovich27, Z. Y. Deng1, A. Denig26, I. Denysenko27, M. Destefanis55A,55C, F.
De Mori55A,55C, Y. Ding31, C. Dong34, J. Dong1,42, L. Y. Dong1,46, M. Y.
Dong1,42,46, Z. L. Dou33, S. X. Du60, P. F. Duan1, J. Fang1,42, S. S.
Fang1,46, Y. Fang1, R. Farinelli24A,24B, L. Fava55B,55C, S. Fegan26, F.
Feldbauer4, G. Felici23A, C. Q. Feng52,42, E. Fioravanti24A, M. Fritsch4, C.
D. Fu1, Q. Gao1, X. L. Gao52,42, Y. Gao44, Y. G. Gao6, Z. Gao52,42, B.
Garillon26, I. Garzia24A, A. Gilman49, K. Goetzen11, L. Gong34, W. X.
Gong1,42, W. Gradl26, M. Greco55A,55C, L. M. Gu33, M. H. Gu1,42, Y. T. Gu13,
A. Q. Guo1, L. B. Guo32, R. P. Guo1,46, Y. P. Guo26, A. Guskov27, Z.
Haddadi29, S. Han57, X. Q. Hao16, F. A. Harris47, K. L. He1,46, X. Q. He51, F.
H. Heinsius4, T. Held4, Y. K. Heng1,42,46, Z. L. Hou1, H. M. Hu1,46, J. F.
Hu37,h, T. Hu1,42,46, Y. Hu1, G. S. Huang52,42, J. S. Huang16, X. T. Huang36,
X. Z. Huang33, Z. L. Huang31, T. Hussain54, W. Ikegami Andersson56, M,
Irshad52,42, Q. Ji1, Q. P. Ji16, X. B. Ji1,46, X. L. Ji1,42, H. L. Jiang36, X.
S. Jiang1,42,46, X. Y. Jiang34, J. B. Jiao36, Z. Jiao18, D. P. Jin1,42,46, S.
Jin33, Y. Jin48, T. Johansson56, A. Julin49, N. Kalantar-Nayestanaki29, X. S.
Kang34, M. Kavatsyuk29, B. C. Ke1, I. K. Keshk4, T. Khan52,42, A. Khoukaz50,
P. Kiese26, R. Kiuchi1, R. Kliemt11, L. Koch28, O. B. Kolcu45B,f, B. Kopf4, M.
Kornicer47, M. Kuemmel4, M. Kuessner4, A. Kupsc56, M. Kurth1, W. Kühn28, J. S.
Lange28, P. Larin15, L. Lavezzi55C, S. Leiber4, H. Leithoff26, C. Li56, Cheng
Li52,42, D. M. Li60, F. Li1,42, F. Y. Li35, G. Li1, H. B. Li1,46, H. J.
Li1,46, J. C. Li1, J. W. Li40, K. J. Li43, Kang Li14, Ke Li1, Lei Li3, P. L.
Li52,42, P. R. Li46,7, Q. Y. Li36, T. Li36, W. D. Li1,46, W. G. Li1, X. L.
Li36, X. N. Li1,42, X. Q. Li34, Z. B. Li43, H. Liang52,42, Y. F. Liang39, Y.
T. Liang28, G. R. Liao12, L. Z. Liao1,46, J. Libby21, C. X. Lin43, D. X.
Lin15, B. Liu37,h, B. J. Liu1, C. X. Liu1, D. Liu52,42, D. Y. Liu37,h, F. H.
Liu38, Fang Liu1, Feng Liu6, H. B. Liu13, H. L Liu41, H. M. Liu1,46, Huanhuan
Liu1, Huihui Liu17, J. B. Liu52,42, J. Y. Liu1,46, K. Y. Liu31, Ke Liu6, L. D.
Liu35, Q. Liu46, S. B. Liu52,42, X. Liu30, Y. B. Liu34, Z. A. Liu1,42,46,
Zhiqing Liu26, Y. F. Long35, X. C. Lou1,42,46, H. J. Lu18, J. G. Lu1,42, Y.
Lu1, Y. P. Lu1,42, C. L. Luo32, M. X. Luo59, T. Luo9,j, X. L. Luo1,42, S.
Lusso55C, X. R. Lyu46, F. C. Ma31, H. L. Ma1, L. L. Ma36, M. M. Ma1,46, Q. M.
Ma1, X. N. Ma34, X. Y. Ma1,42, Y. M. Ma36, F. E. Maas15, M. Maggiora55A,55C,
S. Maldaner26, Q. A. Malik54, A. Mangoni23B, Y. J. Mao35, Z. P. Mao1, S.
Marcello55A,55C, Z. X. Meng48, J. G. Messchendorp29, G. Mezzadri24A, J.
Min1,42, T. J. Min33, R. E. Mitchell22, X. H. Mo1,42,46, Y. J. Mo6, C. Morales
Morales15, N. Yu. Muchnoi10,d, H. Muramatsu49, A. Mustafa4, S. Nakhoul11,g, Y.
Nefedov27, F. Nerling11,g, I. B. Nikolaev10,d, Z. Ning1,42, S. Nisar8, S. L.
Niu1,42, X. Y. Niu1,46, S. L. Olsen46, Q. Ouyang1,42,46, S. Pacetti23B, Y.
Pan52,42, M. Papenbrock56, P. Patteri23A, M. Pelizaeus4, J. Pellegrino55A,55C,
H. P. Peng52,42, Z. Y. Peng13, K. Peters11,g, J. Pettersson56, J. L. Ping32,
R. G. Ping1,46, A. Pitka4, R. Poling49, V. Prasad52,42, H. R. Qi2, M. Qi33, T.
Y. Qi2, S. Qian1,42, C. F. Qiao46, N. Qin57, X. S. Qin4, Z. H. Qin1,42, J. F.
Qiu1, S. Q. Qu34, K. H. Rashid54,i, C. F. Redmer26, M. Richter4, M. Ripka26,
A. Rivetti55C, M. Rolo55C, G. Rong1,46, Ch. Rosner15, A. Sarantsev27,e, M.
Savrié24B, K. Schoenning56, W. Shan19, X. Y. Shan52,42, M. Shao52,42, C. P.
Shen2, P. X. Shen34, X. Y. Shen1,46, H. Y. Sheng1, X. Shi1,42, J. J. Song36,
W. M. Song36, X. Y. Song1, S. Sosio55A,55C, C. Sowa4, S. Spataro55A,55C, F. F.
Sui36, G. X. Sun1, J. F. Sun16, L. Sun57, S. S. Sun1,46, X. H. Sun1, Y. J.
Sun52,42, Y. K Sun52,42, Y. Z. Sun1, Z. J. Sun1,42, Z. T. Sun1, Y. T Tan52,42,
C. J. Tang39, G. Y. Tang1, X. Tang1, M. Tiemens29, B. Tsednee25, I. Uman45D,
B. Wang1, B. L. Wang46, C. W. Wang33, D. Wang35, D. Y. Wang35, Dan Wang46, H.
H. Wang36, K. Wang1,42, L. L. Wang1, L. S. Wang1, M. Wang36, Meng Wang1,46, P.
Wang1, P. L. Wang1, W. P. Wang52,42, X. F. Wang1, Y. Wang52,42, Y. F.
Wang1,42,46, Z. Wang1,42, Z. G. Wang1,42, Z. Y. Wang1, Zongyuan Wang1,46, T.
Weber4, D. H. Wei12, P. Weidenkaff26, S. P. Wen1, U. Wiedner4, M. Wolke56, L.
H. Wu1, L. J. Wu1,46, Z. Wu1,42, L. Xia52,42, X. Xia36, Y. Xia20, D. Xiao1, Y.
J. Xiao1,46, Z. J. Xiao32, Y. G. Xie1,42, Y. H. Xie6, X. A. Xiong1,46, Q. L.
Xiu1,42, G. F. Xu1, J. J. Xu1,46, L. Xu1, Q. J. Xu14, X. P. Xu40, F. Yan53, L.
Yan55A,55C, W. B. Yan52,42, W. C. Yan2, Y. H. Yan20, H. J. Yang37,h, H. X.
Yang1, L. Yang57, R. X. Yang52,42, S. L. Yang1,46, Y. H. Yang33, Y. X. Yang12,
Yifan Yang1,46, Z. Q. Yang20, M. Ye1,42, M. H. Ye7, J. H. Yin1, Z. Y. You43,
B. X. Yu1,42,46, C. X. Yu34, J. S. Yu30, J. S. Yu20, C. Z. Yuan1,46, Y. Yuan1,
A. Yuncu45B,a, A. A. Zafar54, Y. Zeng20, B. X. Zhang1, B. Y. Zhang1,42, C. C.
Zhang1, D. H. Zhang1, H. H. Zhang43, H. Y. Zhang1,42, J. Zhang1,46, J. L.
Zhang58, J. Q. Zhang4, J. W. Zhang1,42,46, J. Y. Zhang1, J. Z. Zhang1,46, K.
Zhang1,46, L. Zhang44, S. F. Zhang33, T. J. Zhang37,h, X. Y. Zhang36, Y.
Zhang52,42, Y. H. Zhang1,42, Y. T. Zhang52,42, Yang Zhang1, Yao Zhang1, Yu
Zhang46, Z. H. Zhang6, Z. P. Zhang52, Z. Y. Zhang57, G. Zhao1, J. W. Zhao1,42,
J. Y. Zhao1,46, J. Z. Zhao1,42, Lei Zhao52,42, Ling Zhao1, M. G. Zhao34, Q.
Zhao1, S. J. Zhao60, T. C. Zhao1, Y. B. Zhao1,42, Z. G. Zhao52,42, A.
Zhemchugov27,b, B. Zheng53, J. P. Zheng1,42, W. J. Zheng36, Y. H. Zheng46, B.
Zhong32, L. Zhou1,42, Q. Zhou1,46, X. Zhou57, X. K. Zhou52,42, X. R.
Zhou52,42, X. Y. Zhou1, Xiaoyu Zhou20, Xu Zhou20, A. N. Zhu1,46, J. Zhu34, J.
Zhu43, K. Zhu1, K. J. Zhu1,42,46, S. Zhu1, S. H. Zhu51, X. L. Zhu44, Y. C.
Zhu52,42, Y. S. Zhu1,46, Z. A. Zhu1,46, J. Zhuang1,42, B. S. Zou1, J. H. Zou1
(BESIII Collaboration) 1 Institute of High Energy Physics, Beijing 100049,
People’s Republic of China
2 Beihang University, Beijing 100191, People’s Republic of China
3 Beijing Institute of Petrochemical Technology, Beijing 102617, People’s
Republic of China
4 Bochum Ruhr-University, D-44780 Bochum, Germany
5 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
6 Central China Normal University, Wuhan 430079, People’s Republic of China
7 China Center of Advanced Science and Technology, Beijing 100190, People’s
Republic of China
8 COMSATS Institute of Information Technology, Lahore, Defence Road, Off
Raiwind Road, 54000 Lahore, Pakistan
9 Fudan University, Shanghai 200443, People’s Republic of China
10 G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
11 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
12 Guangxi Normal University, Guilin 541004, People’s Republic of China
13 Guangxi University, Nanning 530004, People’s Republic of China
14 Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
15 Helmholtz Institute Mainz, Johann-Joachim-Becher-Weg 45, D-55099 Mainz,
Germany
16 Henan Normal University, Xinxiang 453007, People’s Republic of China
17 Henan University of Science and Technology, Luoyang 471003, People’s
Republic of China
18 Huangshan College, Huangshan 245000, People’s Republic of China
19 Hunan Normal University, Changsha 410081, People’s Republic of China
20 Hunan University, Changsha 410082, People’s Republic of China
21 Indian Institute of Technology Madras, Chennai 600036, India
22 Indiana University, Bloomington, Indiana 47405, USA
23 (A)INFN Laboratori Nazionali di Frascati, I-00044, Frascati, Italy; (B)INFN
and University of Perugia, I-06100, Perugia, Italy
24 (A)INFN Sezione di Ferrara, I-44122, Ferrara, Italy; (B)University of
Ferrara, I-44122, Ferrara, Italy
25 Institute of Physics and Technology, Peace Ave. 54B, Ulaanbaatar 13330,
Mongolia
26 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45,
D-55099 Mainz, Germany
27 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
28 Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-
Buff-Ring 16, D-35392 Giessen, Germany
29 KVI-CART, University of Groningen, NL-9747 AA Groningen, The Netherlands
30 Lanzhou University, Lanzhou 730000, People’s Republic of China
31 Liaoning University, Shenyang 110036, People’s Republic of China
32 Nanjing Normal University, Nanjing 210023, People’s Republic of China
33 Nanjing University, Nanjing 210093, People’s Republic of China
34 Nankai University, Tianjin 300071, People’s Republic of China
35 Peking University, Beijing 100871, People’s Republic of China
36 Shandong University, Jinan 250100, People’s Republic of China
37 Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
38 Shanxi University, Taiyuan 030006, People’s Republic of China
39 Sichuan University, Chengdu 610064, People’s Republic of China
40 Soochow University, Suzhou 215006, People’s Republic of China
41 Southeast University, Nanjing 211100, People’s Republic of China
42 State Key Laboratory of Particle Detection and Electronics, Beijing 100049,
Hefei 230026, People’s Republic of China
43 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
44 Tsinghua University, Beijing 100084, People’s Republic of China
45 (A)Ankara University, 06100 Tandogan, Ankara, Turkey; (B)Istanbul Bilgi
University, 34060 Eyup, Istanbul, Turkey; (C)Uludag University, 16059 Bursa,
Turkey; (D)Near East University, Nicosia, North Cyprus, Mersin 10, Turkey
46 University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China
47 University of Hawaii, Honolulu, Hawaii 96822, USA
48 University of Jinan, Jinan 250022, People’s Republic of China
49 University of Minnesota, Minneapolis, Minnesota 55455, USA
50 University of Muenster, Wilhelm-Klemm-Str. 9, 48149 Muenster, Germany
51 University of Science and Technology Liaoning, Anshan 114051, People’s
Republic of China
52 University of Science and Technology of China, Hefei 230026, People’s
Republic of China
53 University of South China, Hengyang 421001, People’s Republic of China
54 University of the Punjab, Lahore-54590, Pakistan
55 (A)University of Turin, I-10125, Turin, Italy; (B)University of Eastern
Piedmont, I-15121, Alessandria, Italy; (C)INFN, I-10125, Turin, Italy
56 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
57 Wuhan University, Wuhan 430072, People’s Republic of China
58 Xinyang Normal University, Xinyang 464000, People’s Republic of China
59 Zhejiang University, Hangzhou 310027, People’s Republic of China
60 Zhengzhou University, Zhengzhou 450001, People’s Republic of China
a Also at Bogazici University, 34342 Istanbul, Turkey
b Also at the Moscow Institute of Physics and Technology, Moscow 141700,
Russia
c Also at the Functional Electronics Laboratory, Tomsk State University,
Tomsk, 634050, Russia
d Also at the Novosibirsk State University, Novosibirsk, 630090, Russia
e Also at the NRC ”Kurchatov Institute”, PNPI, 188300, Gatchina, Russia
f Also at Istanbul Arel University, 34295 Istanbul, Turkey
g Also at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany
h Also at Key Laboratory for Particle Physics, Astrophysics and Cosmology,
Ministry of Education; Shanghai Key Laboratory for Particle Physics and
Cosmology; Institute of Nuclear and Particle Physics, Shanghai 200240,
People’s Republic of China
i Also at Government College Women University, Sialkot - 51310. Punjab,
Pakistan.
j Also at Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and
Institute of Modern Physics, Fudan University, Shanghai 200443, People’s
Republic of China

###### Abstract

Using a data sample of $448.1\times 10^{6}$ $\psi(3686)$ events collected at
$\sqrt{s}=$ 3.686 GeV with the BESIII detector at the BEPCII, we search for
the rare decay $J/\psi\to\phi e^{+}e^{-}$ via
$\psi(3686)\to\pi^{+}\pi^{-}J/\psi$. No signal events are observed and the
upper limit on the branching fraction is set to be $\mathcal{B}(J/\psi\to\phi
e^{+}e^{-})<1.2\times 10^{-7}$ at the 90% confidence level, which is still
about one order of magnitude higher than the Standard Model prediction.

###### pacs:

13.20.Gd, 13.40.Ks

##  I INTRODUCTION

The BESIII experiment has accumulated $4.48\times 10^{8}$ $\psi(3686)$ events
which is the largest $\psi(3686)$ data sample produced directly in
$e^{+}e^{-}$ annihilation in the world currently. By tagging the two soft
pions in the decay of $\psi(3686)\to\pi^{+}\pi^{-}J/\psi$, the final states
from $J/\psi$ decay can be well distinguished. This provides an almost
background free sample to investigate the rare $J/\psi$ decay, which may be
sensitive to new physics. The rare decay $J/\psi\to\phi e^{+}e^{-}$ is one
particulary interesting example guoxd . This decay channel occurs mainly
through the three dynamic processes shown in Figs. 1(a)-(c). These include:
(a) the leading-order electromagnetic (EM) process; (b) the EM and strong
mixed loop process; and (c) the EM process proceeding through three virtual
photons. In diagram (b), the non-perturbative strong loop can be treated as
proceeding through intermediate mesons, as discussed in Ref. guoxd . Within
the framework of the Standard Model (SM), the partial widths from the leading
EM and mixed loop processes are predicted to be at a level of $10^{-6}$ and
$10^{-9}$ keV, respectively, corresponding to branching fractions at the order
of $10^{-8}$ and $10^{-11}$ guoxd . However, if there is a new particle
involved in the intermediate process, such as a dark photon with a mass of
several MeV/$c^{2}$ or a glueball with certain quantum numbers, contribution
from Fig. 1(b) can be enhanced greatly. Thus, any deviations from the
predictions guoxd  would hint at the existence of new physics. Alternatively,
if a positive result were obtained with a branching fraction in the expected
range, this decay channel could be used to extract information of some
interesting mesons such as $f_{0}(980)$ or $a_{0}(980)$ since their form
factors are involved in the predictions.

Although BESIII has also available the currently world’s largest data sample
of directly produced $J/\psi$, this is not used in the present analysis due to
badly-controlled background contamination from QED processes. In this work, we
report on search for the rare decay of $J/\psi\to\phi e^{+}e^{-}$ via
$\psi(3686)\to\pi^{+}\pi^{-}J/\psi$.

Figure 1: Feynman diagrams contributing to the decay $J/\psi\to\phi
e^{+}e^{-}$: (a) the leading-order EM process, (b) the EM and strong mixed
loop process, and (c) the EM process proceeding through three virtual photons.

##  II BESIII AND BEPCII

The BESIII detector bes3  at the BEPCII $e^{+}e^{-}$ collider is a major
upgrade of the BESII experiment bes2  at the Beijing Electron-Positron
Collider (BEPC) and is optimized to study physics in the $\tau$-charm energy
region. The design peak luminosity of BEPCII, $1.0\times 10^{33}$ cm-2s-1 at a
center-of-mass energy of $3773$ MeV, was achieved in 2016. The BESIII
detector, with a geometrical acceptance of $93$% of the full solid angle,
consists of the following five main components. $(1)$ A small-celled multi-
layer drift chamber (MDC) with $43$ layers is used for charged track
reconstruction and measurement of ionization energy loss ($dE/dx$). The
average single-wire resolution is $135$ $\mu$m, and the momentum resolution
for $1.0$ GeV/$c$ charged particles in a $1$ T magnetic field is $0.5$%. The
specific $dE/dx$ resolution is 6% for electrons from Bhabha scattering. $(2)$
A time-of-flight (TOF) system surrounds the MDC. This system is composed of a
two-layer barrel, each layer consisting of $88$ pieces of $5$ cm thick and
$2.4$ m long plastic scintillators, as well as two end caps each with $96$
fan-shaped, $5$ cm thick, plastic scintillators. The time resolution is $80$
ps in the barrel and $110$ ps in the end caps, providing a $K/\pi$ separation
of more than $2\sigma$ for momenta up to $1.0$ GeV/$c$. $(3)$ An
electromagnetic calorimeter (EMC) is used to measure photon energies and
consists of $6240$ CsI(Tl) crystals in a cylindrically-shaped barrel and two
end caps. For $1.0$ GeV photons, the energy resolution is $2.5$% in the barrel
and $5$% in the end caps, and the position resolution is $6$ mm in the barrel
and $9$ mm in the end caps. $(4)$ A superconducting solenoid magnet
surrounding the EMC provides a 1 T magnetic field. $(5)$ The muon chamber
system is made of resistive plate chambers with $9$ layers in the barrel and
$8$ layers in the end caps and is incorporated into the return iron yoke of
the superconducting magnet. The global position resolution is about $2$ cm.

Interactions within the BESIII detector are simulated by the GEANT4-based
geant4  simulation software boost boost , which includes: geometric and
material descriptions of the BESIII detector; detector response and
digitization models; and a record of detector running conditions and
performances. The production of the $\psi(3686)$ resonance is simulated by the
Monte Carlo (MC) generator KKMC kkmc , which incorporates the effects of the
energy spread of the beam and initial-state radiation. Known decays are
generated by EVTGEN evtgen  using the branching fractions quoted by the
Particle Data Group (PDG) pdg18 , and the remaining unknown decays are
generated with the LUNDCHARM model lundcharm .

In this analysis, the process $J/\psi\to\phi e^{+}e^{-}$ is studied via
$\psi(3686)\to\pi^{+}\pi^{-}J/\psi$, and the $\phi$ meson is reconstructed
using its decay to $K^{+}K^{-}$. The transition
$\psi(3686)\to\pi^{+}\pi^{-}J/\psi$ is generated according to the results of
an amplitude analysis as described in Ref. bes2jpsipipi . The process
$J/\psi\to\phi e^{+}e^{-}$ is generated according to the amplitude given in
Ref. guoxd , in which the leading-order EM process is expected to be the
dominant contribution according to the SM prediction. The spin correlation of
$J/\psi$ produced in the previous decay is considered as in Ref. lifengyun .
The decay $\phi\to K^{+}K^{-}$ is generated using a $\sin^{2}\theta_{K}$
distribution, where $\theta_{K}$ is the helicity angle of the kaon in the
center-of-mass system of the $\phi$ meson.

##  III Event Selection

Charged tracks are reconstructed with MDC hits within the range
$|\cos\theta|<0.93$, where $\theta$ is the polar angle with respect to the
electron beam direction. They are required to originate from the interaction
region, defined as $R_{xy}<1$ cm and $R_{z}<10$ cm, where $R_{xy}$ and $R_{z}$
are the projections of the distances from the closest approach of the tracks
to the interaction point in the $xy$-plane and in the $z$-direction,
respectively.

Particle identification (PID) probabilities for good charged tracks are
calculated with the $dE/dx$ and TOF measurements under the hypothesis that the
track originated from a pion, kaon, proton or electron. For kaon candidates,
we require that the probability for the kaon hypothesis is larger than the
corresponding probability for the pion and proton hypotheses. For electron
candidates, the probability for electron hypothesis is required to be larger
than the probabilities for the pion and kaon hypotheses. To avoid
contamination from pions, electron candidates must satisfy the additional
requirement $E/p>0.8$, where $E$ and $p$ represent the energy deposited in the
EMC and the momentum of the electron, respectively.

For the two pions, no PID selection criterion is required. All pairs of
opposite charged tracks with momentum less than $0.45$ GeV/$c$ are assumed to
be pions, and their recoil masses $M_{\pi^{+}\pi^{-}}^{\rm rec}$ are
calculated and required to be within the range $(3.05,3.15)$ GeV/$c^{2}$.

To improve the mass resolution and suppress backgrounds, an energy-momentum
constrained kinematic fit ($4$C) to the initial beam four momentum is imposed
on the selected charged tracks. The resulting $\chi^{2}_{4\rm C}$ of the
kinematic fit is required to be less than $40$. If more than one combination
is found in an event, the combination with the least $\chi^{2}_{4\rm C}$ is
retained for further analysis.

##  IV Analysis

The process $J/\psi\to\phi e^{+}e^{-}$ is studied by examining the two-
dimensional distribution of the $M_{\pi^{+}\pi^{-}}^{\rm rec}$ versus the
invariant mass of the $K^{+}K^{-}$ pair, $M_{K^{+}K^{-}}$. Figure 2(a) shows
the distribution for the signal MC sample, where the signal region (shown as a
red solid box) is defined as $|M_{K^{+}K^{-}}-M_{\phi}|<0.010$ GeV/$c^{2}$ and
$|M^{\rm rec}_{\pi^{+}\pi^{-}}-M_{J/\psi}|<0.007$ GeV/$c^{2}$, where
$M_{\phi}$ and $M_{J/\psi}$ are the nominal masses of $\phi$ and $J/\psi$
mesons in taken from PDG pdg18 , respectively. The five boxes with equal area
around the signal region are selected as sideband regions, which are
categorized into three types. The first type is used to estimate the
background without a $J/\psi$ in the intermediate state; the second one is for
the estimation of the background without a $\phi$ in the intermediate state.
These first two are shown as the pink dashed boxes and the green dashed double
dotted box, respectively. The third type is for the estimation of the
background that includes neither a $J/\psi$ nor a $\phi$ in the intermediate
state, and is shown as blue dashed dotted boxes. Figure 2(b) shows the
corresponding plot for the $\psi(3686)$ data sample. No events are observed in
the signal region and two events are observed in the $\phi$ sideband. The non-
flat non-$\phi$ background, mainly due to the threshold effect, is estimated
by $\psi(3686)\to\pi^{+}\pi^{-}J/\psi$ with $J/\psi\to\phi\pi^{+}\pi^{-}$ and
$\phi\to K^{+}K^{-}$. The scale factor is determined to be $0.8$ for the
background in the $\phi$ signal region to that in the $\phi$ sideband region.
Therefore, the scaled background estimated by the sideband data is $1.6\pm
1.1$ events. The projections of Fig. 2(b) on $M^{\rm rec}_{\pi^{+}\pi^{-}}$
and $M_{K^{+}K^{-}}$ are shown in Figs. 3(a) and (b), respectively.

The backgrounds from $\psi(3686)$ decays are also studied with an inclusive MC
sample of 506 million $\psi(3686)$ events. No events survive in the signal
region, and only one event is found in the second type of sideband region.
This event is from $\psi(3686)\to\pi^{+}\pi^{-}J/\psi,J/\psi\to
K^{+}K^{-}\pi^{0},\pi^{0}\to\gamma e^{+}e^{-}$, which will not form a peak in
$\phi$ signal region. In addition, the potential peaking backgrounds from
$\psi(3686)\to\pi^{+}\pi^{-}J/\psi$ with $J/\psi\to\phi\eta/\pi^{0}$ and
$\eta/\pi^{0}\to\gamma e^{+}e^{-}$ are studied through exclusive MC events
generated with a size corresponding to more than 100 times of that of data.
The contribution from these channels is negligible.

Possible background sources from continuum processes are estimated with $44$
pb-1 of data collected at a center-of-mass energy $\sqrt{s}=3.65$ GeV condata
, which is about one fifteenth of the integrated luminosity of the
$\psi(3686)$ data. There are no events satisfying the above selection
criteria, therefore we neglect the continuum background.

Figure 2: Distribution of $M^{\rm rec}_{\pi^{+}\pi^{-}}$ versus
$M_{K^{+}K^{-}}$ from (a) signal MC sample and (b) $\psi(3686)$ data. The
signal region is defined as the solid box, and the three types of sideband
regions (described in the text) are represented by the dashed, dashed double
dotted, and dashed dotted boxes.  Figure 3: Distributions of (a) $M^{\rm
rec}_{\pi^{+}\pi^{-}}$ and (b) $M_{K^{+}K^{-}}$. The solid histograms
represent the $\psi(3686)$ data and the dashed histograms represent signal MC
with arbitrary scale. The region between the red arrows denotes the signal
region, and the one(s) between the blue arrows denotes the sideband region(s).

##  V Systematic Uncertainties

The systematic uncertainties originate mainly from the number of $\psi(3686)$
events, the tracking efficiency, the PID efficiency, the kinematic fit, the
selection of the $J/\psi$ and $\phi$ signal regions, background estimation, MC
statistics, and the branching fractions of intermediate decays. These are
discussed in detail in the following, and are summarized in Table 1.

The uncertainty from the total number of $\psi(3686)$ events is estimated to
be $0.7$% npsip .

The tracking efficiencies for $\pi^{\pm}$ mesons have been studied with the
process $\psi(3686)\to\pi^{+}\pi^{-}J/\psi$,
$J/\psi\to\ell^{+}\ell^{-}(\ell=e,\mu)$. The difference in the efficiencies
between data and MC simulation is $1.0$% per pion ktrkpid . The tracking
efficiencies for $K^{\pm}$ mesons as functions of transverse momentum have
been studied with the process $J/\psi\to K_{S}^{0}K^{\pm}\pi^{\mp}$,
$K_{S}^{0}\to\pi^{+}\pi^{-}$. The difference in the efficiencies between data
and MC simulation is $1.0$% per kaon ktrkpid . The tracking efficiencies for
$e^{\pm}$ are obtained with a control sample of radiative Bhabha scattering
$e^{+}e^{-}\to\gamma e^{+}e^{-}$ (including $J/\psi\to\gamma e^{+}e^{-}$ ) at
the $J/\psi$ resonance in Ref. etrk . The difference in tracking efficiencies
between data and MC simulation is calculated bin-by-bin over the distribution
of transverse momentum versus the polar angle of the lepton tracks. The
uncertainty is determined to be $1.0$% per electron/positron. The systematic
uncertainties arising from the different charged tracks are summed linearly to
be $6.0$%.

High purity control samples of $e^{+}e^{-}\to\gamma e^{+}e^{-}$ and $J/\psi\to
K^{+}K^{-}\pi^{0}$ have been selected to study the electron/positron and kaon
PID uncertainty. The difference of PID efficiency between data and MC
simulation is calculated in bins of momentum and cos$\theta$. Averaged
systematic uncertainties for electron/positron and kaon identification are
obtained by weighting the difference with the events in each bin of momentum
and cos$\theta$ from the signal MC sample, and determined to be $0.4$% per
electron/positron and $0.8$% per kaon. Adding these values linearly, the PID
systematic uncertainty is determined to be $2.4$%.

The systematic uncertainty of the $4$C kinematic fit is studied using a
control sample of $\psi(3686)\to\pi^{+}\pi^{-}J/\psi$ with
$J/\psi\to\phi\pi^{+}\pi^{-}$, $\phi\to K^{+}K^{-}$. The efficiency difference
between data and MC simulation with the $\chi^{2}_{\rm 4C}<40$ requirement is
$3.3$%, which is assigned as the systematic uncertainty.

The uncertainty from the signal regions of $M_{K^{+}K^{-}}$ and $M^{\rm
rec}_{\pi^{+}\pi^{-}}$, due to their resolution difference between data and MC
simulation, is studied by means of the control sample
$\psi(3686)\to\pi^{+}\pi^{-}J/\psi$, $J/\psi\to\phi\pi^{+}\pi^{-}$, $\phi\to
K^{+}K^{-}$. The efficiency differences in the $M_{K^{+}K^{-}}$ and $M^{\rm
rec}_{\pi^{+}\pi^{-}}$ signal regions between data and MC simulation are
$0.2$% and $1.8$%, respectively. Adding them in quadrature yields $1.8$%,
which is taken as the systematic uncertainty.

The uncertainty on the background estimation is studied by an alternative
estimation of the scale factor ($0.74$), estimated using the inclusive MC
sample instead of data. The difference between the resulting upper limits is
taken as the systematic uncertainty, which is $1.5$%.

The uncertainty of the detection efficiency attributed to the limited size of
the MC sample, $0.4$%, is taken as the systematic uncertainty from MC
statistics.

To estimate the uncertainty from the model used to simulate the $J/\psi\to\phi
e^{+}e^{-}$ decay, we generated a MC sample based on the phase-space
assumption. The difference between the efficiencies determined from the MC
sample described in Sec. II and the phase space MC sample is $10.7$%. We take
half of the difference ($5.4$%) as the systematic uncertainty from MC
modeling.

In the determination of the upper limit on the branching fraction of the
process of interest, we have accounted for the branching fractions of
$\psi(3686)\to\pi^{+}\pi^{-}J/\psi$ and $\phi\to K^{+}K^{-}$ by taking the
values given by the PDG pdg18 . The uncertainties of these cited values are
taken as a source of systematic uncertainty, which are $0.9$% and $1.0$%,
respectively.

The total systematic uncertainty, $\Delta_{\rm sys}$, is calculated by adding
the uncertainties from all sources in quadrature.

Table 1:  Summary of relative systematic uncertainties (in %).  Sources | Uncertainty
---|---
$N^{\rm tot}_{\psi(3686)}$ | 0.7
Tracking | 6.0
PID | 2.3
Kinematic fit | 3.3
Signal region | 1.8
Background estimation | 1.5
MC statistics | 0.4
MC modeling | 5.4
$\mathcal{B}(\psi(3686)\to\pi^{+}\pi^{-}J/\psi)$ | 0.9
$\mathcal{B}(\phi\to K^{+}K^{-})$ | 1.0
Total | 9.5

##  VI  Result

Since no candidate events are observed in the signal region and $1.6\pm 1.1$
background events are estimated, the upper limit on the number of
$\psi(3686)\to\pi^{+}\pi^{-}J/\psi$ events with $J/\psi\to\phi e^{+}e^{-}$ is
set to be $1.31$ at the $90$% confidence level (CL) using the Feldman-Cousins
feldman  method with the assumption of a Poisson process.

After taking into account the systematic uncertainty cousins , the upper limit
of the branching fraction of $J/\psi\to\phi e^{+}e^{-}$ is calculated with

$\mathcal{B}(J/\psi\to\phi e^{+}e^{-})<\frac{N^{\rm up}\times(1+N^{\rm
up}\times\Delta_{\rm sys}^{2}/2)}{N^{\rm
tot}_{\psi(3686)}\times\epsilon\times\prod\mathcal{B}_{i}},$ (1)

where $N^{\rm tot}_{\psi(3686)}$ is the number of $\psi(3686)$ events,
$\prod\mathcal{B}_{i}$ represents the branching fraction product
$\mathcal{B}(\psi(3686)\to\pi^{+}\pi^{-}J/\psi)\times\mathcal{B}(\phi\to
K^{+}K^{-})$ and $\epsilon$ is the detection efficiency, which is $(14.31\pm
0.05)$% determined by MC simulations as described in Sec. II. Table 2
summarizes the various values that were used as input to Eq. (1). We find an
upper limit on the branching fraction of the $J/\psi\to\phi e^{+}e^{-}$
process at the 90% CL of

$\mathcal{B}(J/\psi\to\phi e^{+}e^{-})<1.2\times 10^{-7}.$

Table 2:  Input values used to obtain the upper limit on the branching fraction of $J/\psi\to\phi e^{+}e^{-}$. $N^{\rm obs}$, $N^{\rm bkg}$ and $N^{\rm up}$ represent the number of observed events, background events, and the upper limit on the number of observed events, respectively.  Item | Value
---|---
$N^{\rm tot}_{\psi(3686)}$ | $(448.1\pm 2.9)\times 10^{6}$
$N^{\rm obs}$ | $0$
$N^{\rm bkg}$ | $1.6\pm 1.1$
$N^{\rm up}$(90% CL) | $1.31$
$\epsilon$ |  $(15.13\pm 0.05)$%
$\mathcal{B}(\psi(3686)\to\pi^{+}\pi^{-}J/\psi)$ |  $(34.49\pm 0.30)$%
$\mathcal{B}(\phi\to K^{+}K^{-})$ |  $(48.9\pm 0.5)$%
$\Delta_{\rm sys}$ | 9.5%
$\mathcal{B}(J/\psi\to\phi e^{+}e^{-})$ | $<1.2\times 10^{-7}$

##  VII  Summary

Using the $448.1\times 10^{6}$ $\psi(3686)$ events collected with the BESIII
detector, we report a search for the rare decay $J/\psi\to\phi e^{+}e^{-}$ via
$\psi(3686)\to\pi^{+}\pi^{-}J/\psi$. No signal events are observed and the
upper limit of the branching fraction for this decay is calculated to be
$\mathcal{B}(J/\psi\to\phi e^{+}e^{-})<1.2\times 10^{-7}$ by the Feldman-
Cousins method at the $90$% CL, which is one order of magnitude higher than
the prediction in Ref. guoxd . Our result shows that, even if there are new
particles involved in the Fig. 1(b) process, their contributions will not be
too large, and the masses of the possible existing new particles or their
coupling information to vector mesons ($J/\psi$ and $\phi$) can also be
constrained.

##  VIII Acknowledgements

The BESIII collaboration thanks the staff of BEPCII and the IHEP computing
center for their strong support. This work is supported in part by National
Key Basic Research Program of China under Contract No. 2015CB856700; National
Natural Science Foundation of China (NSFC) under Contracts Nos. 11575077,
11475090, 11235011, 11335008, 11425524, 11625523, 11635010, 11735014,
11835012; the Outstanding Youth project of Natural Science Foundation of Hunan
Province; the Chinese Academy of Sciences (CAS) Large-Scale Scientific
Facility Program; Joint Large-Scale Scientific Facility Funds of the NSFC and
CAS under Contracts Nos. U1532257, U1532258, U1732263, U1832207; CAS Key
Research Program of Frontier Sciences under Contracts Nos. QYZDJ-SSW-SLH003,
QYZDJ-SSW-SLH040; 100 Talents Program of CAS; INPAC and Shanghai Key
Laboratory for Particle Physics and Cosmology; German Research Foundation DFG
under Contract No. Collaborative Research Center CRC 1044; Istituto Nazionale
di Fisica Nucleare, Italy; Koninklijke Nederlandse Akademie van Wetenschappen
(KNAW) under Contract No. 530-4CDP03; Ministry of Development of Turkey under
Contract No. DPT2006K-120470; National Science and Technology fund; The Knut
and Alice Wallenberg Foundation (Sweden) under Contract No. 2016.0157; The
Swedish Research Council; U. S. Department of Energy under Contracts Nos. DE-
FG02-05ER41374, DE-SC-0010118, DE-SC-0012069; University of Groningen (RuG)
and the Helmholtzzentrum für Schwerionenforschung GmbH (GSI), Darmstadt.

## References

  * (1) X. D. Guo, S. R. Xue, H. W. Ke, X. Q. Li, Q. Zhao, Chin. Phys. C 40, 073104 (2016).
  * (2) M. Ablikim et al. [BESIII Collaboration], Nucl. Instrum. Meth. A 614, 345 (2010).
  * (3) J. Z. Bai et al. [BES Collaboration], Nucl. Instrum. Meth. A 344, 319 (1994); 458, 627 (2001).
  * (4) S. Agostinelli et al. [GEANT4 Collaboration], Nucl. Instrum. Meth. A 506, 250 (2003).
  * (5) Z. Y. Deng et al., High Energy Physics & Nuclear Physics 30, 371 (2006).
  * (6) S. Jadach, B. F. L. Ward, and Z. Was, Comput. Phys. Commun. 130, 260 (2000); Phys. Rev. D 63, 113009 (2001).
  * (7) R. G. Ping, Chin. Phys. C 32, 599 (2008); D. J. Lange, Nucl. Instrum. Meth. A 462, 152 (2001).
  * (8) M. Tanabashi et al. [Particle Data Group], Phys. Rev. D 98, 030001 (2018).
  * (9) J. C. Chen, G. S. Huang, X. R. Qi, D. H. Zhang, Y. S. Zhu, Phys. Rev. D 62, 034003 (2000); R. L. Yang, R. G. Rong and D. Chen, Chin. Phys. Lett. 31, 061301 (2014).
  * (10) M. Ablikim et al. [BES Collaboration], Phys. Rev. D 62, 032002 (2000).
  * (11) M. Ablikim et al. [BESIII Collaboration], arxiv:1809.00635.
  * (12) M. Ablikim, et al. [BESIII Collaboration], Chin. Phys. C 37, 123001 (2013).
  * (13) M. Ablikim et al. [BESIII Collaboration], Chin. Phys. C 42, 023001 (2018).
  * (14) M. Ablikim et al. [BESIII Collaboration], Phys. Rev. D 83, 112005 (2011).
  * (15) M. Ablikim et al. [BESIII Collaboration], Phys. Rev. D 89, 092008 (2014).
  * (16) G. J. Feldman and R. D. Cousins, Phys. Rev. D 57, 3873 (1998).
  * (17) R. D. Cousins and V. L. Highland, Nucl. Instrum. Meth. A 320, 331 (1992).


# Enhanced super-Heisenberg scaling precision by nonlinear coupling and
postselection

Lupei Qin<EMAIL_ADDRESS>Center for Joint Quantum Studies and Department
of Physics, School of Science,
Tianjin University, Tianjin 300072, China  Jialin Li  Center for Joint Quantum
Studies and Department of Physics, School of Science,
Tianjin University, Tianjin 300072, China  Yazhi Niu  Center for Joint Quantum
Studies and Department of Physics, School of Science,
Tianjin University, Tianjin 300072, China  Xin-Qi Li<EMAIL_ADDRESS>Center for Joint Quantum Studies and Department of Physics, School of Science,
Tianjin University, Tianjin 300072, China

###### Abstract

In quantum precision metrology, the famous result of Heisenberg limit scaling
as $1/N$ (with $N$ the number of probes) can be surpassed by considering
nonlinear coupling measurement. In this work, we consider the most practice-
relevant quadratic nonlinear coupling and show that the metrological precision
can be enhanced from the $1/N^{\frac{3}{2}}$ super-Heisenberg scaling to
$1/N^{2}$, by simply employing a pre- and post-selection (PPS) technique, but
not using any expensive quantum resources such as quantum entangled state of
probes.

Introduction

.— Quantum precision measurement (or, quantum metrology) is one of the main
quantum technology frontiers under explorations. It can reach unprecedented
metrological precision owing to using quantum resources such as entanglement
and squeezing Ld04 ; Ld06 ; Ld11 . However, the precision of quantum metrology
is also limited by measurement strategies and some fundamental quantum
principles. For instance, the quantum nature of the underlying state,
interacting evolution, and final measurement would make the output results of
measurement suffer strong statistical uncertainty. Fundamentally, the various
metrological precisions must be bounded by the Heisenbergy uncertainty
principle or, equivalently, the Heisenberg inequality Lui04 . For the
metrological schemes taking only classical strategies, i.e., not exploring any
quantum correlations between the probes, the precision can at most scale with
$N$ (the number of probes) as $1/\sqrt{N}$ Ld04 ; Ld06 ; Ld11 ; Lui04 . This
is actually the shot-noise-caused scaling limit (owing to the absence of
correlation), which is also, more often, referred to as standard quantum limit
(SQL), owing to taking standard classical measurement strategies. However, the
standard classical strategies are not optimal. If quantum correlation is
introduced, e.g., by quantum entanglement of the probes, the SQL can be
surpassed by achieving the better precision of $1/N$ scaling, which has a
$\sqrt{N}$ precision enhancement over the SQL Ld04 ; Ld06 ; Ld11 . This
scaling is usually referred to as Heisenberg limit (HL). For long time, HL was
believed as the ultimate limit, e.g., as the ultimate bound to the precision
of phase measurements in the case of interferometry.

However, the above conclusions of SQL and HL are achieved only for linear
coupling measurement. The so-called HL (scaling as $1/N$, but not really
limited by the Heisenberg principle) can be surpassed by considering nonlinear
coupling measurements Lui04 ; Lui06 ; Lui07 ; Cav07 ; Cav08a ; Cav08b ; Cav09
. In this work, following some references, we may dub the metrological
precision better than $1/N$ scaling as super-Heisenberg limit (or super-
Heisenberg scaling). For a general $k_{\rm th}$-order nonlinear coupling
measurement, the main conclusions are Lui04 ; Lui06 ; Lui07 ; Cav07 ; Cav08a ;
Cav08b ; Cav09 : (i) an optimal sensitivity that scales as $1/N^{k}$ can be
achieved, if using as well an entangled initial probe state; (ii) sensitivity
that scales as $1/N^{k-1/2}$ can be possible, if the probe is initially
prepared in a product state. In practice, nonlinear coupling can be realized
in such as quantum optical and condensed matter systems. For instance,
following the pioneering works, subsequent studies consider the specific
realizations by introducing nonlinear Kerr medium in the Mach-Zehnder
interferometer (MZI), and by exploring spin-based atom ensembles and condensed
matter systems Lui15 ; Zhao19 ; Shaj21 ; Mit10 . Moreover, experimental
demonstrations have also been carried out Mit11 ; Mit14 ; Peng18 ; Yuan20 .
For nonlinear coupling measurement, the most practice-relevant case is the
second-order nonlinearity with $k=2$. The both results scaling as $1/N^{2}$
(for entangled initial probe state) and $1/N^{\frac{3}{2}}$ (for product
initial probe state) are super-Heisenberg scaling, being thus of great
interest.

In this work, we consider to introduce the strategy of pre- and post-selection
(PPS), which was proposed by Aharonov, Albert, and Vaidman (AAV) in the
quantum weak value (WV) measurement AAV1 ; AAV2 . We will show that, without
need of additional quantum resources such as entangled state of the probes,
the PPS strategy can help to enhance the metrological precision scaling from
$1/N^{\frac{3}{2}}$ to $1/N^{2}$, for the quadratic nonlinear coupling
measurement. Actually, the weak-value-amplification (WVA) technique has been
successfully demonstrated in quantum precision measurement in various contexts
Kwi08 ; How09a ; How10a ; How10b ; How13 ; Sto12 ; Lun17 ; ZLJ20 . The basic
conclusion achieved by the WVA community is somehow twofold. In one aspect,
indeed, the WVA technique can amplify small signals beyond the resolution of
detector, can suppress the technical noise in some practical situations, and
can greatly outperform the conventional measurement (CM) in the presence of
power saturation of detectors Sim10 ; Ste11 ; Nish12 ; Bru15 ; Bru16 ; How17 ;
Jor17 ; Pan18 ; How15 ; Jor18 . In other aspect, owing to discarding data in
the postselection, the Fisher information encoded in the postselected data
(and thus the metrological precision) cannot surpass the result of
conventional measurement (without involving the postselection procedure)
Tana13 ; FC14 ; Kne14 ; Ked12 ; Jor14 ; Ren20 .

Very recently, by considering a two-state system (qubit) linearly coupled to
an optical coherent state (probe meter), it was found that the PPS strategy
employed in the WVA measurement can lead to some new results XQLi22 : (i) with
the increase of the measurement strength (to violate the AAV limit), the WVA
scheme can outperform the conventional approach; (ii) the WVA scheme can make
a mixture of coherent states work better than a pure coherent state with
identical average photon numbers, while the opposite conclusion was claimed in
context of conventional measurement (without using the PPS strategy) Lui10 .
In present work, we extend the study of Ref. XQLi22  from linear to quadratic
nonlinear coupling measurement, and obtain new results over the conventional
measurement, by means of the PPS strategy of WVA.

Formulation of the measurement scheme

.— Following Ref. XQLi22 , let us consider a quantum two-state (qubit) system,
with states denoted as $|1\rangle$ and $|2\rangle$, coupled to an optical
probe (meter) system prepared in the coherent state $|\alpha\rangle$. However,
in this work, rather than the linear coupling, we consider a quadratic
nonlinear coupling described by
$\hat{H}^{{}^{\prime}}=-\lambda\hat{\sigma}_{z}\hat{n}^{2}$, where the Pauli
operator $\hat{\sigma}_{z}=|2\rangle\langle 2|-|1\rangle\langle 1|$ describes
the qubit system, and the number operator $\hat{n}=\hat{a}^{\dagger}\hat{a}$
is for the optical meter, with $\hat{a}^{\dagger}$ ($\hat{a}$) the creation
(annihilation) operator of a photon. Accordingly, the evolution of the whole
qubit-plus-meter system is governed by the unitary operator
$U=\exp(i\chi\hat{\sigma}_{z}\hat{n}^{2})$, where
$\chi=\int_{0}^{\tau}dt\lambda=\lambda\tau$ is the integrated strength of
interaction over time $\tau$. For quantum precision measurement, we assume
$\chi$ the parameter to be estimated. Starting with a product state of the
qubit and meter, $|\Psi^{(i)}_{J}\rangle=|i\rangle|\alpha\rangle$, where
$|i\rangle=\cos\frac{\theta_{i}}{2}|1\rangle+\sin\frac{\theta_{i}}{2}e^{i\varphi_{i}}|2\rangle$,
the coupling interaction will evolve the entire system into an entangled state
as

$|\Psi_{J}\rangle=\cos\frac{\theta_{i}}{2}|1\rangle|\phi_{-}\rangle+\sin\frac{\theta_{i}}{2}e^{i\varphi_{i}}|2\rangle|\phi_{+}\rangle\,,$
(1)

where the meter’s state is affected differently by the qubit states
$|1\rangle$ and $|2\rangle$, being given by

$|\phi_{\mp}\rangle=e^{-|\alpha|^{2}/2}\sum_{n=0}^{\infty}\frac{\alpha^{n}}{\sqrt{n!}}e^{\mp
i\chi n^{2}}|n\rangle\,.$ (2)

Then, through the coupling interaction, the parameter $\chi$ under estimation
has been encoded into the meter states. One can perform various types of
measurement to extract the value of $\chi$. In this work, following the idea
of WVA, we consider applying the strategy of post-selection to amplify the
signal for $\chi$. To be specific, let us assume the post-selection
measurement of the qubit state with
$|f\rangle=\cos\frac{\theta_{f}}{2}|1\rangle+\sin\frac{\theta_{f}}{2}e^{i\varphi_{f}}|2\rangle$,
and keep the final measurement results (e.g., photon numbers) of the meter
only after successful post-selection of the qubit state. Mathematically, the
post-selection is simply described as $|\widetilde{\Phi}_{f}\rangle=\langle
f|\Psi_{J}\rangle$, resulting thus in a state for the meter as

$|\widetilde{\Phi}_{f}\rangle=\cos\frac{\theta_{i}}{2}\cos\frac{\theta_{f}}{2}|\phi_{-}\rangle+\sin\frac{\theta_{i}}{2}\sin\frac{\theta_{f}}{2}e^{i\varphi_{0}}|\phi_{+}\rangle\,.$
(3)

Here $\varphi_{0}$ is introduced to denote the phase difference of the PPS
states $|i\rangle$ and $|f\rangle$, i.e.,
$\varphi_{0}=\varphi_{i}-\varphi_{f}$. We further normalize the postselected
meter state as $|\Phi_{f}\rangle=|\widetilde{\Phi}_{f}\rangle/\sqrt{p_{f}}$,
with $p_{f}=\langle\widetilde{\Phi}_{f}|\widetilde{\Phi}_{f}\rangle$. We may
mention that $p_{f}$ is right the success probability of postselection, which
is explicitly given by

$p_{f}=A+Be^{-|\alpha|^{2}}\sum_{n=0}^{\infty}\frac{|\alpha|^{2n}}{n!}\cos(2\,\chi\,n^{2}+\varphi_{0})\,,$
(4)

where $A=\frac{1}{2}(1+\cos{\theta_{i}}\cos{\theta_{f}})$ and
$B=\frac{1}{2}\sin{\theta_{i}}\sin{\theta_{f}}$. Based on the state
$|\Phi_{f}\rangle$, one can consider certain specific measurement, e.g., the
photon number measurement to it. Straightforwardly, the probability of
obtaining $n$ photons is given by $P_{f}(n)=|\langle n|\Phi_{f}\rangle|^{2}$.
Explicitly, we find

$P_{f}(n)=\frac{e^{-|\alpha|^{2}}}{p_{f}}\left(\frac{|\alpha|^{2n}}{n!}\left[A+B\cos(2\,\chi\,n^{2}+\varphi_{0})\right]\right)\,.$
(5)

In order to carry out a characterization for the metrological precision of the
parameter $\chi$, we can calculate the Fisher information (FI) $F_{f}$ of
$\chi$ encoded in the distribution function $P_{f}(n)$, through

$F_{f}=\sum_{n}\frac{1}{P_{f}(n)}\left(\frac{\partial
P_{f}(n)}{\partial\chi}\right)^{2}\,.$ (6)

Then, the estimate precision for $\chi$ is bounded as $\delta(\chi)\geq
1/\sqrt{p_{f}F_{f}}$, following the Cramér-Rao bound (CRB) inequality. This is
the result associated with the specific scheme of photon-number measurement to
$|\Phi_{f}\rangle$. One can also consider the so-called quantum Fisher
information (QFI) encoded in the state $|\Phi_{f}\rangle$, which is given by

$Q_{f}=4\left(\frac{d\langle\Phi_{f}|}{d\chi}\frac{d|\Phi_{f}\rangle}{d\chi}-\left|\langle\Phi_{f}|\frac{d|\Phi_{f}\rangle}{d\chi}\right|^{2}\right)\,.$
(7)

Unlike the FI associated with certain specific measurement scheme, the QFI
represents the maximum amount of information for all possible measurement
schemes. Thus, we carry out the following result for the WVA scheme

$\displaystyle p_{f}Q_{f}$
$\displaystyle=4\,\left[A\langle\hat{n}^{4}\rangle-\frac{1}{p_{f}}\left(C\langle\hat{n}^{2}\rangle\right)^{2}\right.$
(8) $\displaystyle-
Be^{-|\alpha|^{2}}\sum_{n=0}^{\infty}\frac{|\alpha|^{2n}}{n!}\,n^{4}\,\cos(2\,\chi\,n^{2}+\varphi_{0})$
$\displaystyle-\left.\frac{1}{p_{f}}B^{2}\left(e^{-|\alpha|^{2}}\sum_{n=0}^{\infty}\frac{|\alpha|^{2n}}{n!}\,n^{2}\,\sin(2\,\chi\,n^{2}+\varphi_{0})\right)^{2}\right]\,.$

In this result, $\langle\bullet\rangle$ represents the expectation over the
optical coherent state $|\alpha\rangle$. Thus we have
$\langle\hat{n}^{4}\rangle=N^{4}+6N^{3}+4N^{2}+N$, and
$\langle\hat{n}^{2}\rangle=N^{2}+N$. Here we also introduced
$C=(\cos{\theta_{i}}+\cos{\theta_{f}})/2$.

In this work, we would like to term the FI (QFI) $p_{f}F_{f}$ ($p_{f}Q_{f}$)
as WVA-FI (WVA-QFI), and especially in numerical demonstrations choose the PPS
parameters as $\theta_{i}=\frac{\pi}{2}$, $\varphi_{0}=\pi$, and
$\theta_{f}=\frac{\pi}{2}$. This choice makes the WVA in the AAV regime, i.e.,
making the AAV-WV $\sigma_{z}^{w}=\langle f|\sigma_{z}|i\rangle/\langle
f|i\rangle$ divergent. Note that, for finite coupling strength, the
postselected average of readout results is not divergent, but maximally
amplified Ren20 . In the AAV limit, one can prove that the meter’s state after
postselection is $\widetilde{\Phi}_{f}\sim\langle f|i\rangle
e^{i\chi\sigma_{z}^{w}\hat{n}^{2}}|\alpha\rangle$, from which we find that the
coupling strength is amplified as $\widetilde{\chi}=\chi\sigma_{z}^{w}$.
Actually, this reveals the basic idea of WVA.

In order to carry out a comparison with the conventional measurement, let us
consider to encode the parameter $\chi$ into the probe state $|\alpha\rangle$
by coupling through a single state $|1\rangle$ or $|2\rangle$ of the qubit,
but not through their superposition. Actually, in the absence of
postselection, it can be proved that using the superposed state of $|1\rangle$
and $|2\rangle$ will result in the same QFI as using the single state
$|1\rangle$ or $|2\rangle$ XQLi22 . To be specific, let us consider coupling
the qubit through $|1\rangle$ to the probe field. In this case, after
interaction, the joint state for the entire qubit-plus-meter is
$|\Psi\rangle_{\rm{cm}}=|1\rangle|\phi_{-}\rangle$. Then, the QFI of $\chi$ in
the state $|\phi_{-}\rangle$ is simply obtained as

$Q_{\rm{cm}}=4\left(4\,N^{3}+6\,N^{2}+N\right)\,.$ (9)

In this result, we also use $N=|\alpha|^{2}$, to denote the mean photon number
of the probe field.

Figure 1:  The WVA-FI $p_{f}F_{f}$ (solid red lines) and WVA-QFI $p_{f}Q_{f}$
(dashed black lines) are compared with each other, and with the QFI $Q_{\rm
cm}$ of conventional measurement (dash-dotted blue lines, without using the
PPS technique). The results shown in (a)-(d) correspond to measurement
strengths $\chi=0.001$, 0.005, 0.01, and 0.1, respectively. Other parameters
used for the numerical calculations are: $\theta_{i}=\frac{\pi}{2}$,
$\varphi_{0}=\pi$, and $|\alpha|^{2}=N=8$.

Results and discussion

.— For linear coupling measurement, as studied in Ref. XQLi22 , the
interaction-strength-encoded meter states $|\phi_{\pm}\rangle$ have simple
analytic solutions. However, for the quadratic nonlinear coupling measurement
considered in this work, we are unable to find analytic solutions for
$|\phi_{\pm}\rangle$, but only present the formal expression in terms of power
series expansion, as shown by Eq. (2). Therefore, we would like to carry out
numerical results and make some discussions.

In Fig. 1 we plot the numerical results of $p_{f}F_{f}$ and $p_{f}Q_{f}$ for
the WVA measurement. Indeed, as expected, we find that $p_{f}F_{f}$ is always
bounded by $p_{f}Q_{f}$. Only at some special postselection angles, e.g., at
$\theta_{f}=3\pi/2$, $p_{f}F_{f}$ can reach the maximum value of $p_{f}Q_{f}$.
Since the WVA-FI and WVA-QFI are periodic functions of $\theta_{f}$ with
period of $2\pi$, here we only show the results of a single period. We notice
that the postselection dependence behavior varies drastically with the
measurement strength $\chi$. In Fig. 2 we show the particular $\chi$
dependence which, non-monotonically oscillating in the stronger coupling
regime, is quite different from the linear coupling measurement (see Fig. 2 in
Ref. XQLi22 ).

Importantly, we find that, with the increase of $\chi$, both $p_{f}Q_{f}$ and
$p_{f}F_{f}$ can exceed the QFI $Q_{\rm cm}$ of the conventional scheme, which
is free from $\chi$, as observed in Fig. 1 (c) and (d). This result is somehow
beyond the usual claim among the WVA community Nish12 ; Ked12 ; Jor14 ; Ren20
; Tana13 ; FC14 ; Kne14 . That is, it was generally believed that, from either
the perspective of Fisher information or the signal-to-noise ratio, the
intrinsic overall estimate precision cannot be enhanced by the WVA technique.
The basic reason is that, despite the enhanced signal after postselection, the
postselection will discard a large number of measurement data and thus result
in a larger uncertainty fluctuations. Therefore, the WVA technique was
conceived of having, at least, no theoretical advantages Tana13 ; FC14 ; Kne14
, but only having some technical advantages in practice such as not being
bounded by the saturation limit of photo-detectors Lun17 ; ZLJ20 .

We notice that the analysis of achieving such conclusion was largely based on
using the transverse spatial wavefunction of a light beam as probe meter, for
instance, in Ref. Jor14 , and in many other references. In Ref. XQLi22  and in
present work, the analysis is fully along the line in Ref. Jor14 . The only
difference is using an optical coherent state as probe meter. Therefore, the
claim that the WVA scheme cannot exceed the conventional measurement is not a
result imposed by fundamental physics. The result displayed in Fig. 1
indicates that the nonlinear coupling WVA measurement can reach a precision
better than the conventional approach in the absence of postselection. This is
a next example, subsequent to the previous investigation of linear coupling
measurement XQLi22 .

We also notice that the conventional nonlinear coupling measurement (without
postselection) can realize super-Heisenberg scaling metrological precision.
For instance, for quadratic nonlinear coupling measurement, it was found Lui04
; Lui06 ; Lui07 ; Cav07 ; Cav08a ; Cav08b ; Cav09  that the precision can
scale with the (average) photon numbers of probe as $1/N^{\frac{3}{2}}$.
Indeed, this is in agreement with the result of QFI shown in Eq. (9), which
scales with the photon numbers dominantly as $N^{3}$ in the large $N$ limit.
In this context, we may raise an interesting and practically important
question: Is it possible to surpass this super-Heisenberg scaling limit, under
the restriction of quadratic nonlinear coupling measurement and not using
expensive quantum resources such as entangled state of the probe photons? In
the following, we will show that the answer is yes. That is, by involving the
simple PPS strategy in WVA, we can boost the metrological precision to scale
with the probe photon numbers from $1/N^{\frac{3}{2}}$ to $1/N^{2}$. Actually,
preliminary insight from the result of the WVA-QFI given by Eq. (8) indicates
that this is possible, owing to the leading term proportional to $N^{4}$ in
large $N$ limit. The only unclear point is how fast this term becomes the
dominant one after accounting for the summed pre-coefficients, when compared
with other lower-order terms in the relatively complicated expression of Eq.
(8).

In Fig. 1, we have already shown that the WVA-FI can exceed the QFI of the
conventional scheme for a fixed average photon number $N$ of the probe field,
by properly (even slightly) increasing the measurement coupling strength
$\chi$. Now, we add to show in Fig. 3 that for a fixed coupling strength
$\chi$, the WVA-FI $p_{f}F_{f}$ can increase very fast with the average probe
photon numbers. Actually, the increasing is highly nonlinear, owing to the
nonlinear measurement coupling. This implies that the precision enhancement
boosted by the PPS technique of WVA will be very efficient for the nonlinear
coupling measurement, by increasing the photon numbers of the probe field. We
also notice that the WVA-FI enhancement with the increase of $N$ is more
efficient for stronger nonlinear coupling strength (but being restricted as
well in the weak coupling regime). This can be understood jointly with the
help of the result in Fig. 2.

Figure 2:  Measurement strength dependence of the WVA-FI $p_{f}F_{f}$ (solid
red line) and WVA-QFI $p_{f}Q_{f}$ (dashed blue line). The WVA related PPS
parameters are chosen as $\theta_{i}=\frac{\pi}{2}$, $\varphi_{0}=\pi$, and
$\theta_{f}=\frac{\pi}{2}$; and the average photon number is assumed as
$|\alpha|^{2}=N=8$.
Figure 3:  Average photon number dependence of the WVA-FI $p_{f}F_{f}$ and the
conventional QFI $Q_{{\rm cm}}$. Results shown are for measurement strengths
$\chi=0.001$, 0.01, and 0.1, while the result of $\chi=0.001$ in the plot has
been amplified by $10^{3}$ for better visual effect. Note also that the
conventional QFI $Q_{{\rm cm}}$ is independent of $\chi$. The WVA related PPS
parameters are the same as in Fig. 2.
Figure 4:  Super-Heisenberg scaling with the average photon number, re-plotted
using the result of measurement strength $\chi=0.01$ in Fig. 3 (but for larger
range of $N$). The numerical results are fitted by the solid straight lines
with slopes $\widetilde{k}=4$ and $3$, respectively. The slope fitted here
corresponds to the precision scaling $1/N^{k}$ with $k=\widetilde{k}/2$,
according to the CRB inequality. Then, it is evident that the WVA-FI
$p_{f}F_{f}$ has better precision scaling than the conventional QFI $Q_{{\rm
cm}}$, i.e., with a $\sqrt{N}$ precision enhancement purely owing to employing
the PPS strategy in WVA.

In Fig. 4, we re-plot the result of the measurement strength $\chi=0.01$ in
Fig. 3 to show super-Heisenberg scaling with the average photon numbers (for
larger range of $N$, but not much large, for instance, approximately larger
than ten). We explicitly demonstrate the dominant overall $N^{4}$-scaling
behavior for the WVA-FI $p_{f}F_{f}$, in comparison with the $N^{3}$-scaling
of the QFI $Q_{\rm cm}$ of the conventional scheme. The WVA-FI scaling
behavior shown in Fig. 4 simply means that the metrological precision can be
enhanced by the PPS strategy of WVA from $1/N^{\frac{3}{2}}$ to $1/N^{2}$
scaling limit, based on the CRB inequality. In this context, we may mention
the long standing issue of achieving the HL (beyond the SQL) in quantum
metrology, which is extremely demanding, e.g., needing expensive resources
such as quantum entanglement or squeezing. Noting that what has been gained
from SQL to HL is a $\sqrt{N}$ precision enhancement, it seems thus extremely
desirable to achieve this same precision enhancement by means of the simple
PPS strategy of WVA.

About the quadratic nonlinear coupling, it can be realized by various ways
based on available state-of-the-art experimental platforms. For instance, in
the Mach-Zehnder interferometer (MZI), inserting nonlinear optical kerr medium
into the interference arms can lead to quadratic nonlinear coupling with the
probe laser field, while the laser field is well described by a coherent
state. Since the interference paths can be described as the ‘system’ states
$|1\rangle$ and $|2\rangle$, the nonlinear MZI can fall into the theoretical
description of our present work. Recently, based on the nonlinear MZI,
interesting studies were carried out for parameter estimation of the
gravitational effect, such as estimating the characteristic parameters of
black-hole and wormholes Ral17 ; Sab17 ; Sab18 . In particular, the super-
Heisenberg scaling $\sim 1/N^{\frac{3}{2}}$ was highlighted Ral17 ; Sab17 ;
Sab18 , viewing that current techniques (e.g. the atomic interferometry) for
gravitational measurements are limited to the SQL. Even if using quantum
entanglement or squeezing resource, only the HL precision can be achieved. Our
present study, remarkably, indicates that, if further applying the PPS
strategy of WVA, the performance of the nonlinear MZI can be improved to
achieve the super-Heisenberg scaling $\sim 1/N^{2}$.

In addition to the realization in terms of the nonlinear optical MZI mentioned
above, it is also possible to realize the quadratic nonlinear coupling as
considered in this work by employing the optical cavity QED or solid-state
circuit QED systems. For instance, consider a three-level atom (or artificial
atom) coupled with the single-mode cavity field. For the three-level atom,
beside the ground state $|g\rangle$ and excited state $|e\rangle$, we assume
the existence of a third intermediate state $|e^{\prime}\rangle$ for the
purpose of mediating a two-photon virtual transition coupling between
$|g\rangle$ and $|e\rangle$, which can result thus in the nonlinear dispersive
coupling Hamiltonian $H^{\prime}=\chi\sigma_{z}(a^{\dagger}a)^{2}$, where
$\sigma_{z}=|e\rangle\langle e|-|g\rangle\langle g|$. The nonlinear coupling
strength $\chi$ reads as $\chi=|g_{1}g_{2}|^{2}/(\Delta_{1}^{2}\Delta_{2})$,
obtained by using the 4th-order perturbation theory. In this result, $g_{1}$
and $g_{2}$ are the single photon transition coupling strengths between the
states $|g\rangle$ and $|e^{\prime}\rangle$, and between $|e^{\prime}\rangle$
and $|e\rangle$, respectively. The energy detunings are
$\Delta_{1}=\omega-(E_{e^{\prime}}-E_{g})$ and
$\Delta_{2}=2\omega-(E_{e}-E_{g})$. We also mention that the interplay of
cavity driving and photon loss will result in a coherent state in steady
state. Then in the so-called ‘bad-cavity’ limit, the coupling of the cavity
field to the atom and the subsequent measurement of the field Kor11 ; Kor16 ;
Li15 ; Li16 , are completely captured by the treatment in present work.

Summary.— For the purpose of quantum metrology, we have shown that, without
need of expensive quantum resources such as entanglement of probes or
squeezing, applying the PPS strategy of WVA can help to enhance the
metrological precision scaling with the probe photon numbers from
$1/N^{\frac{3}{2}}$ to $1/N^{2}$, for quadratic nonlinear coupling
measurement. The basic reason of achieving this enhancement is that the PPS
technique of WVA encodes the parameter $\chi$ into the superposed state of Eq.
(3), resulting thus in the dominant $N$ scaling of QFI from $N^{3}$ to
$N^{4}$, which has a $\sqrt{N}$ precision enhancement over the conventional
approach. This better scaling means that the metrological precision can be
enhanced more efficiently by increasing the photon numbers of the probe field,
while the overall WVA-FI is also larger than the QFI of conventional approach.
The metrology scheme analyzed in this work also enjoys the general technical
advantages of WVA, such as breaking through the resolution limit of detector,
owing to the amplified ‘signal’ (from $\chi$ to $\widetilde{\chi}$).

The nonlinear metrology scheme is quite relevant to phase measurement based on
the the Mach-Zehnder interferometer (MZI), by considering to insert a piece of
nonlinear optical kerr medium into one of the interference arms, which also
resembles the laser interferometer for gravitational wave detection, such as
LIGO. Thus, LIGO may consider the use of nonlinear media and may even consider
to employ the PPS strategy of WVA. The PPS procedure of WVA in the MZI can be
implemented by simply introducing a controllable phase shift in one of the
arms, before collecting the probe light from one output port to perform photon
number (intensity) or quadrature measurement. We anticipate that the present
work can motivate further exploration along this research line.

Acknowledgements.

— This work was supported by the NNSF of China (Nos. 11675016, 11974011 &
61905174).

## References

  * (1) V. Giovannetti, S. Lloyd, and L. Maccone, Quantum-enhanced measurements: Beating the standard quantum limit, Science 306, 1330(2004).
  * (2) V. Giovannetti, S. Lloyd, and L. Maccone, Quantum metrology, Phys. Rev. Lett. 96, 010401 (2006).
  * (3) V. Giovannetti, S. Lloyd, and L, Maccone, Advances in quantum metrology, Nat. Photon. 5, 222 (2011).
  * (4) A. Luis, Nonlinear transformations and the Heisenberg limit, Phys. Lett. A 329, 8 (2004).
  * (5) J. Beltrán and A. Luis, Breaking the Heisenberg limit with inefficient detectors, Phys. Rev. A 72, 045801 (2005).
  * (6) A. Luis, Quantum limits, nonseparable transformations, and nonlinear optics, Phys. Rev. A 76, 035801 (2007).
  * (7) S. Boixo, S. T. Flammia, C. M. Caves, and J. M. Geremia, Generalized limits for single-parameter quantum estimation, Phys. Rev. Lett. 98, 090401 (2007).
  * (8) S. Boixo, A. Datta, M. J. Davis, S. T. Flammia, A. Shaji, and C. M. Caves, Quantum metrology: Dynamics versus entanglement, Phys. Rev. Lett. 101, 040403 (2008).
  * (9) S. Boixo, A. Datta, S. T. Flammia, A. Shaji, E. Bagan, and C. M. Caves, Quantum-limited metrology with product states, Phys. Rev. A 77, 012317 (2008).
  * (10) S. Boixo, A. Datta, M. J. Davis, A. Shaji, A. B. Tacla, and C. M. Caves, Quantum-limited metrology and Bose-Einstein condensates, Phys. Rev. A 80, 032103 (2009).
  * (11) A. Luis, and Á. Rivas, Nonlinear Michelson interferometer for improved quantum metrology, Phys. Rev. A 92, 022104 (2015).
  * (12) J.-D. Zhang, Z.-J. Zhang, L.-Z. Cen, J.-Y. Hu, and Y. Zhao, Nonlinear phase estimation: Parity measurement approaches the quantum Cramér-Rao bound for coherent states, Phys. Rev. A 99, 022106 (2019).
  * (13) B. J. Mathew and A. Shaji, Nonclassical states of light in a nonlinear Michelson interferometer, Phys. Rev. A 104, 062604 (2021).
  * (14) M. Napolitano and M. W. Mitchell, Nonlinear metrology with a quantum interface, New J. Phys. 12, 093016 (2010).
  * (15) M. Napolitano, M. Koschorreck, B. Dubost, N. Behbood, R. J. Sewell, and M. W. Mitchell, Interaction-based quantum metrology showing scaling beyond the Heisenberg limit, Nature (London) 471, 486 (2011).
  * (16) R. J. Sewell, M. Napolitano, N. Behbood, G. Colangelo, F. M. Ciurana, and M. W. Mitchell, Ultrasensitive atomic spin measurements with a nonlinear interferometer, Phys. Rev. X 4, 021045 (2014).
  * (17) X. Nie, J. Huang, Z. Li, W. Zheng, C. Lee, X. Peng, J. Du, Experimental demonstration of nonlinear quantum metrology with optimal quantum state, Sci. Bull. 63, 469 (2018).
  * (18) G.-F. Jiao, K. Zhang, L. Q. Chen, W. Zhang, and C.-H. Yuan, Nonlinear phase estimation enhanced by an actively correlated Mach-Zehnder interferometer, Phys. Rev. A 102, 033520 (2020).
  * (19) Y. Aharonov, D. Z. Albert, and L. Vaidman, How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100, Phys. Rev. Lett. 60, 1351 (1988).
  * (20) Y. Aharonov and L. Vaidman, Properties of a quantum system during the time interval between two measurements, Phys. Rev. A 41, 11 (1990).
  * (21) O. Hosten and P. G. Kwiat, Observation of the Spin Hall Effect of Light via weak measurements, Science 319, 787 (2008).
  * (22) P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, Ultrasensitive beam deflection measurement via interferometric weak value amplification, Phys. Rev. Lett. 102, 173601 (2009).
  * (23) D. J. Starling, P. B. Dixon, N. S.Williams, A. N. Jordan, and J. C. Howell, Continuous phase amplification with a sagnac interferometer, Phys. Rev. A 82, 011802 (2010).
  * (24) D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, Precision frequency measurements with interferometric weak values, Phys. Rev. A 82, 063822 (2010).
  * (25) G. I. Viza, J. Martínez-Rincón, G. A. Howland, H. Frostig, I. Shomroni, B. Dayan, and J. C. Howell, Weak-values technique for velocity measurements, Opt. Lett. 38, 2949 (2013).
  * (26) P. Egan and J. A. Stone, Weak-value thermostat with 0.2 mK precision, Opt. Lett. 37, 4991 (2012).
  * (27) J. Harris, R. W. Boyd,and J. S. Lundeen, Weak value amplification can outperform conventional measurement in the presence of detector saturation, Phys. Rev. Lett. 118 070802 (2017).
  * (28) L. Xu, Z. Liu, A. Datta, G. C. Knee, J. S. Lundeen, Y. Lu, and L. Zhang, Approaching quantum-limited metrology with imperfect detectors by using weak-value amplification, Phys. Rev. Lett. 125, 080501 (2020).
  * (29) N. Brunner and C. Simon, Measuring small longitudinal phase shifts: weak measurements or standard interferometry?, Phys. Rev. Lett. 105, 010405 (2010).
  * (30) A. Feizpour, X. Xingxing, and A. M. Steinberg, Amplifying single-photon nonlinearity using weak measurements, Phys. Rev. Lett. 107, 133603 (2011).
  * (31) A. Nishizawa, K. Nakamura, and M. K. Fujimoto, Weak value amplification in a shot-noise-limited interferometer, Phys. Rev. A 85, 062108 (2012).
  * (32) S. Pang and T. A. Brun, Improving the Precision of Weak Measurements by Postselection Measurement, Phys. Rev. Lett. 115, 120401 (2015).
  * (33) S. Pang, J. R. G. Alonso, T. A. Brun, and A. N. Jordan, Protecting weak measurements against systematic errors, Phys. Rev. A 94, 012329 (2016).
  * (34) J. Martínez-Rincón, C. A. Mullarkey, G. I. Viza, W. T. Liu, and J. C. Howell, Ultra sensitive inverse weak-value tilt meter, Opt. Lett. 42, 2479 (2017).
  * (35) J. Sinclair, M. Hallaji, A. M. Steinberg, J. Tollaksen, and A. N. Jordan, Weak-value amplification and optimal parameter estimation in the presence of correlated noise, Phys. Rev. A 96 052128 (2017).
  * (36) L. Li, Y. Li, Y.-L. Zhang, S. Yu, C.-Y. Lu, N.-L. Liu, J. Zhang, and J.-W. Pan, Phase amplification in optical interferometry with weak measurement, Phys. Rev. A 97 033851 (2018).
  * (37) G. I. Viza, J. Martínez-Rincón, G. B. Alves, A. N. Jordan, and J. C. Howell, Experimentally quantifying the advantages of weak-value-based metrology, Phys. Rev. A 92 032127 (2015).
  * (38) K. Lyons, J. C. Howell, and A. N. Jordan, Noise suppression in inverse weak value-based phase detection, Quantum Stud. Math. Found. 5 579 (2018).
  * (39) S. Tanaka and N. Yamamoto, Information amplification via postselection: a parameter-estimation perspective, Phys. Rev. A 88, 042116 (2013).
  * (40) C. Ferrie and J. Combes, Weak value amplification is suboptimal for estimation and detection, Phys. Rev. Lett. 112, 040406 (2014).
  * (41) G. C. Knee and E. M. Gauger, When amplification with weak values fails to suppress technical noise, Phys. Rev. X 4, 011032 (2014).
  * (42) Y. Kedem, Using technical noise to increase the signal-to-noise ratio of measurements via imaginary weak values, Phys. Rev. A 85, 060102(R) (2012).
  * (43) A. N. Jordan, J. Martinez-Rincon, and J. C. Howell, Technical advantages for weak-value amplification: when less is more, Phys. Rev. X 4, 011031 (2014).
  * (44) J. Ren, L. Qin, W. Feng, and X. Q. Li, Weak-value-amplification analysis beyond the Aharonov-Albert-Vaidman limit, Phys. Rev. A 102, 042601 (2020).
  * (45) Y. Liu, L. Qin, and X.-Q. Li, Fisher information analysis on weak-value-amplification metrology using optical coherent states, Phys. Rev. A 106, 022619 (2022).
  * (46) Á. Rivas and A. Luis, Precision quantum metrology and nonclassicality in linear and nonlinear detection schemes, Phys. Rev. Lett. 105, 010403 (2010).
  * (47) S. P. Kish and T. C. Ralph, Quantum limited measurement of space-time curvature with scaling beyond the conventional Heisenberg limit, Phys. Rev. A 96, 041801(R) (2017).
  * (48) C. Sabín, Quantum detection of wormholes, Sci. Rep. 7, 716 (2017).
  * (49) C. Sanchidrián-Vaca and C. Sabín, Parameter estimation of wormholes beyond the Heisenberg limit, arXiv:1811.02395v2.
  * (50) A. N. Korotkov, Quantum Bayesian approach to circuit QED measurement, arXiv:1111.4016; see also chapter 17 in Les Houches 2011 Session XCVI on Quantum Machines.
  * (51) A. N. Korotkov, Quantum Bayesian approach to circuit QED measurement with moderate bandwidth, Phys. Rev. A 94, 042326 (2016).
  * (52) P. Y. Wang, L. P. Qin, and X.-Q. Li, Quantum Bayesian rule for weak measurements of qubits in superconducting circuit QED, New J. Phys. 16, 123047 (2014); 17, 059501 (2015).
  * (53) W. Feng, P. F. Liang, L. P. Qin, and X.-Q. Li, Exact quantum Bayesian rule for qubit measurements in circuit QED, Sci. Rep. 6, 20492 (2016).