Diagrammatic analysis of hidden- and open-charm tetraquark production in $B$
decays

Qin<EMAIL_ADDRESS>School of physics, Huazhong University of Science
and Technology, Wuhan 430074, China

Jing-Liang Qiu and Fu-Sheng<EMAIL_ADDRESS>corresponding author

School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000,
China
and Lanzhou Center for Theoretical Physics, and Frontiers Science Center for
Rare Isotopes, Lanzhou University, Lanzhou 730000, China
and Center for High Energy Physics, Peking University, Beijing 100871, China

###### Abstract

Discoveries of tetraquarks not only enrich the hadronic spectrum but also
provide more platforms to understand quantum chromodynamics. We study the
production processes of hidden-charm and open-charm tetraquarks in $B$ decays
by analyzing their topological amplitudes. Relations between different
channels are found, which confront tests by experiments to probe the nature of
the tetraquarks. Furthermore, promising channels to find more tetraquarks are
proposed.

##  1 Introduction

Since the discovery of $X(3872)$ [1], more and more exotic hadronic states
have been observed in experiments (see [2, 3, 4, 5, 6] for reviews) and also
are expected to found in future [7, 8, 9, 10]. Together with mesons and
baryons, they form a much more colorful and comprehensive ”periodic table of
hadrons”, which provide dramatic platforms for deciphering the quantum
chromodynamics. Many explanations for them such as multiquark states [11, 12,
13], molecular states [14, 15, 16], hybrid mesons [17, 18, 19] and missing
excited hadron states [20, 21] are proposed. Later, $Z^{\pm}(4430)$ was
observed [22] and confirmed [23] in the $B$ meson decay channel $B^{0}\to
K^{\mp}Z^{\pm}(4430),\ Z^{\pm}(4430)\to\pi^{\pm}\psi^{\prime}$, and its
partner particles $Z_{c}^{+}(4200)$ [24], $Z_{c}^{+}(4050)$ and
$Z_{c}^{+}(4250)$ [25] were also reported, which definitely have structures
containing four quarks, beyond the explanation of hybrid mesons or an excited
charmonia.333The discovery of $T_{cc}^{+}$ [26, 27, 28] also strengthens this
conclusion. More of such tetraquarks were discovered in $B$ meson decays, such
as $T^{\theta}_{\psi s1}(4000)^{+}$ and $T_{\psi s1}(4220)^{+}$ [29], and
$T_{cs0,1}[cs\bar{u}\bar{d}]$ [30]. Very recently, two new
$D_{s}^{+}\pi^{\pm}$ resonances
$T^{a}_{c\bar{s}0}(2900)^{++}[c\bar{s}{u}\bar{d}]$ and
$T^{a}_{c\bar{s}0}(2900)^{0}[c\bar{s}\bar{u}{d}]$444See [31] for a name
convention for exotic hadrons. were found in [32, 33]

$\displaystyle B^{+}\to D^{-}D_{s}^{+}\pi^{+}\;,\qquad
B^{0}\to\bar{D}^{0}D_{s}^{+}\pi^{-}\;,$ (1)

respectively. Such tetraquark candidates were studied by previous theoretical
works [34, 35, 36, 37, 38, 39, 40].

It can be found that the $B$ decays are a good place for the tetraquark
production[41, 42, 43]. In this work, we attempt to probe the mechanism of
such tetraquark production processes via $B$ meson decays, employing the
framework of the topological amplitude. We do not consider the states
containing a $q\bar{q}$ ($q=u,d,s$) pair, since they are not easy to be
identified from ordinary meson states, and also suffer large backgrounds from
ordinary states.

The dynamics of the tetraquark productions in $B$ decay are so complicated
that it is nearly impossible to perform the first principle calculation. In
such a case, the diagrammatic approach can nevertheless teach us essential
information about the physical processes. The diagrammatic approach has been
applied in $B$ decays [44, 45, 46, 47] and $D$ decays [48, 49, 50, 51, 53, 55,
54, 52]. It has also been proved to be able to recover the symmetry protected
relations between different decay amplitudes [56, 57]. We apply it to the $B$
decay channels into a tetraquark and a meson. Such an analysis can not give
explicit values of decay amplitudes, but can build relations between different
decay modes, which will be useful in at least two aspects. Firstly, with the
information of observed channels, we can predict which tetraquarks have great
potential to be discovered with the current database and which channels should
be used for the search. Secondly, the compact tetraquark structure is assumed
to obtain some relations between different channels, so the experimental tests
of these relations can examine validity of the assumption. The production of
heavy-flavor tetraquarks in $B$ decays and their decays have also been studied
based on the flavor SU(3) symmetry analysis [58, 59, 34, 60].

The hidden-charm and open-charm tetraquark production processes in $B$ decays
are studied. For the hidden-charm case, the $b\to c\bar{c}s$ processes are
considered, which have larger Cabbibo-Kobayashi-Maskawa (CKM) matrix elements
than other production processes. For the open-charm case, both the $b\to
c\bar{c}s$ and $b\to c\bar{u}d$ processes are considered, because both of them
are CKM favored. The topological amplitudes of all the processes are given,
and it is found that they can be classified into several groups, in which the
processes share the same topological amplitude. The decay processes in the
same group are expected to have similar branching ratios, and if one of them
have been discovered experimentally, the others should also have a large
potential to be observed.

The rest of the paper is organized as follows. In section 2, the hidden-charm
and open-charm tetraquark production processes in $B$ meson decays are
analyzed by using their topological amplitudes. We conclude by section 3.

##  2 Topological amplitude analysis

Figure 1: Topological diagrams that contribute to the $B$ meson decays into a
tetraquark and a meson. The wavy lines indicate the $W$ boson exchanged in the
four-quark vertices. All possible gluon exchanges between the quark lines are
in principle included but omitted in the diagrams.

The tetraquarks are produced in $B$ decays via weak interactions, so typically
a four-fermion weak-interaction vertex is involved in the effective theory of
$b$ decays. For simplicity, we only consider the most frequent processes
induced by tree-level quark transition with large CKM matrix elements, namely,
the $b\to c\bar{c}s$ and $b\to c\bar{u}d$ processes. The most promising
channels at experiments should be among them, since the involved CKM matrix
elements $V_{cb}$ is the largest $b$-decay one and $V_{ud,cs}\sim 1$. Based on
these weak transitions, five kinds of topological amplitudes contribute to the
processes of the $B$ meson (including $B^{-},B^{0},B_{s}^{0}$) decaying into a
tetraquark (including the hidden-charm tetraquarks $c\bar{c}q_{1}\bar{q}_{2}$
and the open-charm tetraquarks $cq_{1}\bar{q}_{2}\bar{q}_{3}$ with the light
quarks $q_{i}=u,d,s$) and a meson (including charmed mesons and light mesons),
as shown in Figure 1, labelled as $T_{1,2,3}$, $C$ and $E$. The quark lines
disconnected to the weak-interaction vertices are produced by pair from the
vacuum or transited from the spectator quark in the initial-state $B$ meson.
Note that the tetraquarks containing a $\bar{q}q$ pair are not considered, so
the diagrams with a vacuum $q\bar{q}$ pair entering a tetraquark are not
included.

If two decay channels share the identical topological amplitude, they are
expected to have analogous magnitudes of decay amplitudes, even if the final-
state particles have different spins or belong to different flavor SU(3)
representations, which was observed in mesonic $B$ meson decays [44, 45].
Therefore, these decay channels also have anomalous branching ratios. There
might be exceptions owing to some special symmetry rules such as isospin
cancellation and helicity suppression. Next, we will analyze the topological
amplitudes of the $B$ decays into a hidden-charm tetraquark or an open-charm
tetraquark.

###  2.1 Hidden-charm tetraquarks

Table 1: Topological amplitudes of the $B$ meson decaying into one hidden-charm open-flavor tetraquark and a meson. Modes | Topological amplitudes | Experimental processes | Experimental processes
---|---|---|---
${B^{-}\to T_{c\bar{c}s\bar{u}}\phi}$ |  $T_{2}$ $V_{cb}V_{cs}^{*}$ | $B^{-}\to D^{(*)0}D_{s}^{(*)-}\phi$ |  $B^{-}\to J/\psi K^{-}\phi$ [29]
$\overline{B}^{0}\to T_{c\bar{c}s\bar{d}}\phi$ |  $T_{2}$ $V_{cb}V_{cs}^{*}$ | $\overline{B}^{0}\to D^{(*)+}D_{s}^{(*)-}\phi$ | $\overline{B}^{0}\to J/\psi\overline{K}^{(*)0}\phi$
$\overline{B}^{0}\to T_{c\bar{c}u\bar{d}}K^{-}$ |  $T_{2}$ $V_{cb}V_{cs}^{*}$ | $\overline{B}^{0}\to D^{(*)+}\overline{D}^{(*)0}K^{-}$ |  $\overline{B}^{0}\to J/\psi\pi^{+}K^{-}$ [24]
$B^{-}\to T_{c\bar{c}d\bar{u}}\overline{K}^{(*)0}$ |  $T_{2}$ $V_{cb}V_{cs}^{*}$ | $B^{-}\to D^{(*)0}D^{(*)-}\overline{K}^{(*)0}$ | $B^{-}\to J/\psi\pi^{-}\overline{K}^{(*)0}$
$\overline{B}_{s}^{0}\to T_{c\bar{c}d\bar{s}}\overline{K}^{(*)0}$ |  $(T_{2}+E)$ $V_{cb}V_{cs}^{*}$ | $\overline{B}_{s}^{0}\to D_{s}^{(*)+}D^{(*)-}\overline{K}^{(*)0}$ | $\overline{B}_{s}^{0}\to J/\psi K^{0}\overline{K}^{(*)0}$
$\overline{B}_{s}^{0}\to T_{c\bar{c}u\bar{s}}K^{-}$ |  $(T_{2}+E)$ $V_{cb}V_{cs}^{*}$ | $\overline{B}_{s}^{0}\to D_{s}^{(*)+}\overline{D}^{(*)0}K^{-}$ | $\overline{B}_{s}^{0}\to J/\psi K^{+}K^{-}$
$\overline{B}_{s}^{0}\to T_{c\bar{c}u\bar{d}}\pi^{-}$ |  $E$ $V_{cb}V_{cs}^{*}$ | $\overline{B}_{s}^{0}\to D^{(*)+}\overline{D}^{(*)0}\pi^{-}$ | $\overline{B}_{s}^{0}\to J/\psi\pi^{+}\pi^{-}$
$\overline{B}_{s}^{0}\to T_{c\bar{c}d\bar{u}}\pi^{+}$ |  $E$ $V_{cb}V_{cs}^{*}$ | $\overline{B}_{s}^{0}\to D^{(*)0}D^{(*)-}\pi^{+}$ | $\overline{B}_{s}^{0}\to\par J/\psi\pi^{-}\pi^{+}$
$\overline{B}^{0}\to T_{c\bar{c}s\bar{u}}\pi^{+}$ |  $T_{1}$ $V_{cb}V_{cs}^{*}$ | $\overline{B}^{0}\to D^{(*)0}D_{s}^{(*)-}\pi^{+}$ | $\overline{B}^{0}\to\par J/\psi K^{-}\pi^{+}$
$B^{-}\to T_{c\bar{c}s\bar{d}}\pi^{-}$ |  $T_{1}$ $V_{cb}V_{cs}^{*}$ | $B^{-}\to D^{(*)+}D_{s}^{(*)-}\pi^{-}$ | $B^{-}\to J/\psi\overline{K}^{0}\pi^{-}$
$B^{-}\to T_{c\bar{c}s\bar{u}}\pi^{0}$ |  ${1\over\sqrt{2}}T_{1}$ $V_{cb}V_{cs}^{*}$ | $B^{-}\to D^{(*)0}D_{s}^{(*)-}\pi^{0}$ | $B^{-}\to J/\psi K^{-}\pi^{0}$
$\overline{B}^{0}\to T_{c\bar{c}s\bar{d}}\pi^{0}$ |  ${1\over\sqrt{2}}T_{1}$ $V_{cb}V_{cs}^{*}$ | $\overline{B}^{0}\to D^{(*)+}D_{s}^{(*)-}\pi^{0}$ | $\overline{B}^{0}\to J/\psi\overline{K}^{0}\pi^{0}$
$\overline{B}_{s}^{0}\to T_{c\bar{c}s\bar{d}}K^{(*)0}$ |  $(T_{1}+E)$ $V_{cb}V_{cs}^{*}$ | $\overline{B}_{s}^{0}\par\to D^{(*)+}D_{s}^{(*)-}K^{(*)0}$ | $\overline{B}_{s}^{0}\to\par J/\psi\overline{K}^{0}K^{(*)0}$
$\overline{B}_{s}^{0}\to T_{c\bar{c}s\bar{u}}K^{+}$ |  $(T_{1}+E)$ $V_{cb}V_{cs}^{*}$ | $\overline{B}_{s}^{0}\par\to D^{(*)0}D_{s}^{(*)-}K^{+}$ | $\overline{B}_{s}^{0}\to J/\psi K^{-}K^{+}$

In this section, we focus on the $B$ meson decays into a hidden-charm
tetraquark and a meson, $\bar{B}\to T_{c\bar{c}q_{1}\bar{q}_{2}}M$. The meson
$M$ must be a light meson because at most two $c(\bar{c})$ quarks are
generated in the weak-interaction vertices and the heavy $c\bar{c}$ pair
production from vacuum is highly suppressed and neglected here. All the decay
channels in this category are listed in Table 1, with also the corresponding
topological amplitudes and possible experimental processes to search for these
channels. The spins or parities of the tetraquarks are not specified, and they
may belong to the flavor SU(3) octet or decuplet. The tetraquarks
$T_{c\bar{c}q_{1}\bar{q}_{2}}$ typically decay strongly into two mesons
composed by $c\bar{c}$ and $q_{1}\bar{q}_{2}$, or $q_{1}\bar{c}$ and
$c\bar{q}_{2}$. The light meson could be a ground state pseudoscalar meson or
a vector meson. To avoid the troublesome quark flavor mixing in $\eta$ and
$\eta^{\prime}$, we do not consider these two mesons. For the $s\bar{s}$
component, we only consider the $\phi$ meson, and for $u\bar{u}$ and
$d\bar{d}$, we consider both $\pi^{0}$ and $\rho^{0}$, the latter having a
much higher detection efficiency at the LHC. For conciseness, we only display
$\pi^{0}$ while hiding $\rho^{0}$ in Table 1, but it should be understood in
the way that everywhere the $\pi^{0}$ appears the $\rho^{0}$ is an alternative
option.

It can be observed from Table 1 that all the decays channel can be classified
into 5 different categories according to their topological amplitudes $T_{2}$,
$T_{2}+E$, $E$, $T_{1}$ and $T_{1}+E$. The channels in each category have
similar branching ratios. Furthermore, because the weak-exchange amplitude $E$
is expected to be small compared to others, the branching ratios of the
$T_{2}$ and $T_{2}+E$ channels should be at the same order of magnitude, and
it also applies to the $T_{1}$ and $T_{1}+E$ channels. Besides the topological
amplitude argument, some channels are related by the flavor symmetries,
including

isospin symmetry: $\displaystyle A(B^{-}\to
T_{c\bar{c}s\bar{u}}\phi)=A(\overline{B}^{0}\to T_{c\bar{c}s\bar{d}}\phi)\;,$
(2) $\displaystyle A(\overline{B}^{0}\to
T_{c\bar{c}u\bar{d}}K^{(*)-})=A(B^{-}\to
T_{c\bar{c}d\bar{u}}\overline{K}^{(*)0})\;,$ (3) $\displaystyle
A(\overline{B}_{s}^{0}\to
T_{c\bar{c}d\bar{s}}\overline{K}^{(*)0})=A(\overline{B}_{s}^{0}\to
T_{c\bar{c}u\bar{s}}K^{(*)-})\;,$ (4) $\displaystyle A(\overline{B}_{s}^{0}\to
T_{c\bar{c}u\bar{d}}\pi^{-})=A(\overline{B}_{s}^{0}\to
T_{c\bar{c}d\bar{u}}\pi^{+})\;,$ (5) $\displaystyle A(\overline{B}^{0}\to
T_{c\bar{c}s\bar{u}}\pi^{+})=A(B^{-}\to T_{c\bar{c}s\bar{d}}\pi^{-})\;,$ (6)
$\displaystyle A(B^{-}\to T_{c\bar{c}s\bar{u}}\pi^{0})=A(\overline{B}^{0}\to
T_{c\bar{c}s\bar{d}}\pi^{0})\;,$ (7) $\displaystyle A(\overline{B}_{s}^{0}\to
T_{c\bar{c}s\bar{d}}K^{(*)0})=A(\overline{B}_{s}^{0}\to
T_{c\bar{c}s\bar{u}}K^{(*)+})\;.$ (8)

The above relations can be tested at future experiments. Because only the
$b\to c\bar{c}s$ processes are taken into account, there are no processes
related by the $U$-spin or $V$-spin symmetry.

Resonances of two types of four-quark constituents $T_{c\bar{c}u\bar{d}}$ and
$T_{c\bar{c}s\bar{u}}$ have been observed experimentally in the channels
$\overline{B}^{0}\to T_{c\bar{c}u\bar{d}}K^{-}\to J/\psi\pi^{+}K^{-}$ [24] and
${B^{-}\to T_{c\bar{c}s\bar{u}}\phi}\to J/\psi K^{-}\phi$ [29], respectively.
It should be noted that the topological amplitude of these two channels are
both $T_{2}$, so it is expected that $T_{2}$ is large and the $T_{2}$ and
$T_{2}+E$ channels should be searched with priority. We list the most
promising ones as follows,

$\displaystyle\overline{B}^{0}\to T_{c\bar{c}s\bar{d}}\phi\to
J/\psi\overline{K}^{(*)0}\phi\;,$ (9) $\displaystyle B^{-}\to
T_{c\bar{c}d\bar{u}}\overline{K}^{(*)0}\to
J/\psi\pi^{-}\overline{K}^{(*)0}\;,$ (10)
$\displaystyle\overline{B}_{s}^{0}\to
T_{c\bar{c}d\bar{s}}\overline{K}^{(*)0}\to J/\psi K^{0}\overline{K}^{(*)0}\;,$
(11) $\displaystyle\overline{B}_{s}^{0}\to T_{c\bar{c}u\bar{s}}K^{-}\to J/\psi
K^{+}K^{-}\;.$ (12)

###  2.2 Open-charm tetraquarks

In this section, we study the $B$ meson decays into a open-charm tetraquark
and a meson, $\bar{B}\to T_{\bar{c}\bar{q}_{1}q_{2}q_{3}}M$. The meson $M$ can
be a charmed meson $D^{(*)0,+}$ or $D_{s}^{(*)+}$ if the involved quark
transition is $b\to c\bar{c}s$, and it can also be a light meson as in the
hidden-charm tetraquark case if the quark transition is $b\to c\bar{u}d$. The
open-charm tetraquarks $T_{\bar{c}\bar{q}_{1}q_{2}q_{3}}$ are constituted by
one anti-charm quark, one light antiquark and two quarks. They are in the
following irreducible representations of the flavor SU(3) group,

${\bf 3}\otimes{\bf 3}\otimes{\bf\bar{3}}={\bf 15}\oplus{\bf\bar{6}}\oplus{\bf
3}\oplus{\bf 3}\;.$ (13)

Because the $\bar{q}q$ states are omitted, the 3 representations are excluded.
The 15 representation is partially symmetric containing e.g. the states
$uu\bar{s}$ and $\\{ud\\}\bar{s}$, and the ${\bf\bar{6}}$ representation is
partially antisymmetric containing e.g. the state $[ud]\bar{s}$. Therefore,
the tetraquarks with two identical light flavors such as
$T_{\bar{c}\bar{s}uu}$ must belong to the 15 representation, while the
tetraquarks with three different light flavors such as $T_{\bar{c}\bar{s}ud}$
can belong to either the 15 representation or the ${\bf\bar{6}}$
representation. All the CKM favored decays channels are listed in Table 2,
with the topological amplitudes and possible experimental processes to search
for them.

Table 2: Topological amplitudes of the $B$ meson decaying into one open-charm tetraquark and a meson. Modes | Topological amplitudes | Experimental processes | Experimental processes
---|---|---|---
$B^{-}\to T_{\bar{c}\bar{u}ds}D^{(*)+}$ |  $C$ $V_{cb}V_{cs}^{*}$ |  $B^{-}\to D_{s}^{(*)-}\pi^{-}D^{(*)+}$ [32, 33] | $B^{-}\to D^{(*)-}K^{-}D^{(*)+}$
$\overline{B}^{0}\to T_{\bar{c}\bar{d}us}D^{(*)0}$ |  $C$ $V_{cb}V_{cs}^{*}$ |  $\overline{B}^{0}\to D_{s}^{(*)-}\pi^{+}D^{(*)0}$ [32, 33] | $\overline{B}^{0}\to\overline{D}^{(*)0}\overline{K}^{0}D^{(*)0}$
$B^{-}\to T_{\bar{c}\bar{u}ss}D_{s}^{(*)+}$ |  $\sqrt{2}C$ $V_{cb}V_{cs}^{*}$ | $B^{-}\to D_{s}^{(*)-}K^{-}D_{s}^{(*)+}$ |
$\overline{B}^{0}\to T_{\bar{c}\bar{d}ss}D_{s}^{(*)+}$ |  $\sqrt{2}C$ $V_{cb}V_{cs}^{*}$ | $\overline{B}^{0}\to D_{s}^{(*)-}\overline{K}^{0}D_{s}^{(*)+}$ |
$B^{-}\to T_{cs\bar{u}\bar{d}}D^{(*)-}$ |  $T_{3}$ $V_{cb}V_{cs}^{*}$ |  $B^{-}\to D^{(*)+}K^{-}D^{(*)-}$ [30] | $B^{-}\to D^{(*)0}\overline{K}^{0}D^{(*)-}$
$\overline{B}^{0}\to T_{cs\bar{u}\bar{d}}\overline{D}^{(*)0}$ |  $T_{3}$ $V_{cb}V_{cs}^{*}$ | $\overline{B}^{0}\to D^{(*)+}K^{-}\overline{D}^{(*)0}$ | $\overline{B}^{0}\to D^{(*)0}\overline{K}^{0}\overline{D}^{(*)0}$
$B^{-}\to T_{cs\bar{u}\bar{u}}\overline{D}^{(*)0}$ |  $\sqrt{2}T_{3}$ $V_{cb}V_{cs}^{*}$ | $B^{-}\to D^{(*)0}K^{-}\overline{D}^{(*)0}$ |
$\overline{B}^{0}\to T_{cs\bar{d}\bar{d}}D^{(*)-}$ |  $\sqrt{2}T_{3}$ $V_{cb}V_{cs}^{*}$ | $\overline{B}^{0}\to D^{(*)+}\overline{K}^{0}D^{(*)-}$ |
$\overline{B}_{s}^{0}\to T_{cd\bar{u}\bar{s}}\pi^{0}$ |  ${1\over\sqrt{2}}(T_{3}-T_{2})$ $V_{cb}V_{ud}^{*}$ | $\overline{B}_{s}^{0}\to D_{s}^{(*)+}\pi^{-}\pi^{0}$ | $\overline{B}_{s}^{0}\to D^{(*)0}K^{0}\pi^{0}$
$\overline{B}_{s}^{0}\to T_{cd\bar{s}\bar{s}}K^{-}$ |  $\sqrt{2}T_{3}$ $V_{cb}V_{ud}^{*}$ | $\overline{B}_{s}^{0}\to D_{s}^{(*)+}K^{0}K^{-}$ |
$B^{-}\to T_{cd\bar{u}\bar{s}}K^{-}$ | ($T_{1}$ \+ $T_{3}$) $V_{cb}V_{ud}^{*}$ | $B^{-}\to D_{s}^{(*)+}\pi^{-}K^{-}$ | $B^{-}\to D^{(*)0}K^{0}K^{-}$
$B^{-}\to T_{cd\bar{u}\bar{u}}\pi^{0}$ | ($T_{1}$ \+ $T_{3}$ \- $T_{2}$) $V_{cb}V_{ud}^{*}$ | $B^{-}\to D^{(*)0}\pi^{-}\pi^{0}$ |
$\overline{B}_{s}^{0}\to T_{cd\bar{u}\bar{u}}K^{+}$ |  $\sqrt{2}T_{1}$ $V_{cb}V_{ud}^{*}$ | $\overline{B}_{s}^{0}\to D^{(*)0}\pi^{-}K^{+}$ |
$\overline{B}_{s}^{0}\to T_{cd\bar{u}\bar{s}}\phi$ |  $T_{1}$ $V_{cb}V_{ud}^{*}$ | $\overline{B}_{s}^{0}\to D_{s}^{(*)+}\pi^{-}\phi$ | $\overline{B}_{s}^{0}\to D^{(*)0}K^{0}\phi$
$\overline{B}^{0}\to T_{cd\bar{u}\bar{s}}\overline{K}^{(*)0}$ | ($E$ \+ $T_{1}$) $V_{cb}V_{ud}^{*}$ | $\overline{B}^{0}\to D_{s}^{(*)+}\pi^{-}\overline{K}^{(*)0}$ | $\overline{B}^{0}\to D^{(*)0}K^{0}\overline{K}^{(*)0}$
$\overline{B}^{0}\to T_{cd\bar{u}\bar{u}}\pi^{+}$ |  $\sqrt{2}$($E$ \+ $T_{1}$) $V_{cb}V_{ud}^{*}$ | $\overline{B}^{0}\to D^{(*)0}\pi^{-}\pi^{+}$ |
$B^{-}\to T_{cs\bar{u}\bar{u}}K^{(*)0}$ |  $\sqrt{2}T_{2}$ $V_{cb}V_{ud}^{*}$ | $B^{-}\to D^{(*)0}K^{-}K^{(*)0}$ |
$\overline{B}^{0}\to T_{cs\bar{u}\bar{d}}K^{(*)0}$ | ($E$ \+ $T_{2}$) $V_{cb}V_{ud}^{*}$ | $\overline{B}^{0}\to D^{(*)+}K^{-}K^{(*)0}$ | $\overline{B}^{0}\to D^{(*)0}\overline{K}^{0}K^{(*)0}$
$\overline{B}^{0}\to T_{cs\bar{u}\bar{u}}K^{+}$ |  $\sqrt{2}$ $E$ $V_{cb}V_{ud}^{*}$ | $\overline{B}^{0}\to D^{(*)0}K^{-}K^{+}$ |

It is observed from Table 2 that the $b\to c\bar{c}s$ channels can be
classified into two categories according to their topological amplitudes, the
$C$ one and the $T_{3}$ one. The decay channels sharing the same topological
amplitude should have similar decay widths. Some of them are also related by
the flavor symmetries, including

isospin symmetry: $\displaystyle A(B^{-}\to
T_{\bar{c}\bar{u}ds}D^{(*)+})=A(\overline{B}^{0}\to
T_{\bar{c}\bar{d}us}D^{(*)0})\;,$ (14) $\displaystyle A(B^{-}\to
T_{\bar{c}\bar{u}ss}D_{s}^{(*)+})=A(\overline{B}^{0}\to
T_{\bar{c}\bar{d}ss}D_{s}^{(*)+})\;,$ (15) $\displaystyle A(B^{-}\to
T_{cs\bar{u}\bar{d}}D^{(*)-})=A(\overline{B}^{0}\to
T_{cs\bar{u}\bar{d}}\overline{D}^{(*)0})\;,$ (16) $\displaystyle A(B^{-}\to
T_{cs\bar{u}\bar{u}}\overline{D}^{(*)0})=A(\overline{B}^{0}\to
T_{cs\bar{d}\bar{d}}D^{(*)-})\;.$ (17)

These relations can be tested at future experiments. For the $b\to c\bar{c}s$
processes, there are no processes related by the $U$-spin or $V$-spin symmetry
as in the hidden-charm case. There are no $b\to c\bar{u}d$ processes related
by the flavor SU(3) symmetry.

Three open-charm tetraquarks have been observed experimentally, including
$T_{cs\bar{u}\bar{d}}$ observed in the channel $B^{-}\to
T_{cs\bar{u}\bar{d}}D^{(*)-}$ [30], $T_{\bar{c}\bar{u}ds}$ observed in
$B^{-}\to T_{\bar{c}\bar{u}ds}D^{(*)+}$ [32, 33], and $T_{\bar{c}\bar{d}us}$
observed in $\overline{B}^{0}\to T_{\bar{c}\bar{d}us}D^{(*)0}$ [32, 33],
respectively. It indicates that the amplitudes $C$ and $T_{3}$ should be large
and the channels with amplitudes containing them should have a large chance to
be observed in the near future. Therefore, we suggest the experimental
searches for the following channels with priority,

$\displaystyle B^{-}\to T_{\bar{c}\bar{u}ss}D_{s}^{+}\to
D_{s}^{-}K^{-}D_{s}^{+}\;,$ (18) $\displaystyle\overline{B}^{0}\to
T_{\bar{c}\bar{d}ss}D_{s}^{+}\to D_{s}^{-}\overline{K}^{0}D_{s}^{+}\;,$ (19)
$\displaystyle B^{-}\to T_{cs\bar{u}\bar{u}}\overline{D}^{0}\to
D^{0}K^{-}\overline{D}^{0}\;,$ (20) $\displaystyle\overline{B}^{0}\to
T_{cs\bar{d}\bar{d}}D^{(*)-}\to D^{(*)+}\overline{K}^{0}D^{(*)-}\;,$ (21)
$\displaystyle\overline{B}_{s}^{0}\to T_{cd\bar{u}\bar{s}}\pi^{0}\to
D^{0}K^{0}\pi^{0}(D_{s}^{+}\pi^{-}\pi^{0})\;,$ (22)
$\displaystyle\overline{B}_{s}^{0}\to T_{cd\bar{s}\bar{s}}K^{-}\to
D_{s}^{+}K^{0}K^{-}\;,$ (23) $\displaystyle B^{-}\to
T_{cd\bar{u}\bar{s}}K^{-}\to D^{0}K^{0}K^{-}(D_{s}^{+}\pi^{-}K^{-})\;,$ (24)
$\displaystyle B^{-}\to T_{cd\bar{u}\bar{u}}\pi^{0}\to D^{0}\pi^{-}\pi^{0}\;.$
(25)

##  3 Conclusion

We have analyzed the $B$ decay processes into a meson and a tetraquark, which
can be hidden-charm or open-charm, by evaluating their topological amplitudes.
It is found that some of the channels share the same topological amplitudes,
which indicates that they have branching ratios of similar size and this can
be examined experimentally. Among the hidden-charm tetraquark production
processes, $\overline{B}^{0}\to T_{c\bar{c}u\bar{d}}K^{-}\to
J/\psi\pi^{+}K^{-}$ [24] and ${B^{-}\to T_{c\bar{c}s\bar{u}}\phi}\to J/\psi
K^{-}\phi$ have been observed experimentally, so their amplitude $T_{2}$
should be large. Therefore, we suggest $\overline{B}^{0}\to
T_{c\bar{c}s\bar{d}}\phi\to J/\psi\overline{K}^{(*)0}\phi$ and $B^{-}\to
T_{c\bar{c}d\bar{u}}\overline{K}^{(*)0}\to J/\psi\pi^{-}\overline{K}^{(*)0}$
should be searched experimentally with priority. Among the open-charm
tetraquark production processes, $B^{-}\to T_{cs\bar{u}\bar{d}}D^{(*)-}$,
$B^{-}\to T_{\bar{c}\bar{u}ds}D^{(*)+}$ and $\overline{B}^{0}\to
T_{\bar{c}\bar{d}us}D^{(*)0}$ have been observed experimentally, indicating
that $T_{3}$ and $C$ should be large. Therefore, we suggest the priority
channels $B^{-}\to T_{\bar{c}\bar{u}ss}D_{s}^{+}\to D_{s}^{-}K^{-}D_{s}^{+}$,
$\overline{B}^{0}\to T_{\bar{c}\bar{d}ss}D_{s}^{+}\to
D_{s}^{-}\overline{K}^{0}D_{s}^{+}$, $B^{-}\to
T_{cs\bar{u}\bar{u}}\overline{D}^{0}\to D^{0}K^{-}\overline{D}^{0}$ and
$\overline{B}^{0}\to T_{cs\bar{d}\bar{d}}D^{(*)-}\to
D^{(*)+}\overline{K}^{0}D^{(*)-}$ to be searched.

## Acknowledgement

This work is supported in part by the National Natural Science Foundation of
China under Grant Nos. 12005068, 11975112, and National Key Research and
Development Program of China under Contract No. 2020YFA0406400.

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# Advances in sequential measurement and control of open quantum systems

Stefano Gherardini<EMAIL_ADDRESS>Department of Physics and
Astronomy, University of Florence, via G. Sansone 1, I-50019 Sesto Fiorentino,
Italy.  LENS and QSTAR, via N. Carrara 1, I-50019 Sesto Fiorentino, Italy.
Andrea Smirne<EMAIL_ADDRESS>Institute of Theoretical Physics, and
IQST, Universität Ulm, Albert-Einstein-Allee 11, 89069 Ulm, Germany.  Matthias
M. Müller<EMAIL_ADDRESS>LENS and QSTAR, via N. Carrara 1, I-50019
Sesto Fiorentino, Italy.  Institute for Quantum Control, Peter Grünberg
Institut, Forschungszentrum Jülich, 52425 Jülich, Germany  Filippo Caruso
<EMAIL_ADDRESS>Department of Physics and Astronomy, University of
Florence, via G. Sansone 1, I-50019 Sesto Fiorentino, Italy.  LENS and QSTAR,
via N. Carrara 1, I-50019 Sesto Fiorentino, Italy.

###### Abstract

Novel concepts, perspectives and challenges in measuring and controlling an
open quantum system via sequential schemes are shown. We discuss how similar
protocols, relying both on repeated quantum measurements and dynamical
decoupling control pulses, can allow to: (i) Confine and protect quantum
dynamics from decoherence in accordance with the Zeno physics. (ii)
Analytically predict the probability that a quantum system is transferred into
a target quantum state by means of stochastic sequential measurements. (iii)
Optimally reconstruct the spectral density of environmental noise sources by
orthogonalizing in the frequency domain the filter functions driving the
designed quantum-sensor. The achievement of these tasks will enhance our
capability to observe and manipulate open quantum systems, thus bringing
advances to quantum science and technologies.

Keywords

open quantum systems, quantum Zeno physics, noise quantum sensing.

##  I Introduction

Let us consider a non-isolated quantum mechanical system $\mathbb{S}$, defined
within the finite-dimensional Hilbert space $\mathcal{H}_{\mathbb{S}}$. Its
dynamics is governed by a time-dependent Hamiltonian of the form
$H(t)=H_{0}+H_{\rm control}(t)$, where $H_{0}$ is the Hamiltonian of the
system but without taking into account any interaction with the external
environment $\mathbb{E}$. Instead, $H_{\rm
control}(t)=\lambda(t)\sigma_{\mathbb{S}}$ denotes a coherent control term
where $\sigma_{\mathbb{S}}$ is an arbitrary operator acting on $\mathbb{S}$.
$H_{\rm control}(t)$ depends on the function $\lambda(t)$ that is properly
modulated so as to fulfill the control tasks required by the user. In case
$\mathbb{S}$ is a closed system, the Hamiltonian $H_{0}$ is time-independent,
and the only ways to interact with the system are given by performing control
actions and measurements, usually on a portion of the system wave-function
modeling the system quantum state. Conversely, in case $\mathbb{S}$ is in
contact with other external systems PetruccioneBook , it has to be considered
as an open quantum system. The effects of such interactions affect only
$\mathbb{S}$ and they can be easily modeled by adding in the Hamiltonian
$H_{0}$ a non-deterministic term proportional to a stochastic field $E(t)$,
which can be effectively seen as an environmental noise contribution. Under
this hypothesis, the evolution of the system is described by a stochastic
quantum dynamics; in this regard, results stemming from the statistical field
theory ParisiBook  have been recently derived Gherardini2016NJP ;
GherardiniQST2017 ; MuellerANNALEN2017 . However, the noise can involve not
only $\mathbb{S}$ but also the control pulse and measurement apparata. Though
the effects of such noise sources lead to systematic errors that can be
selectively identified and attenuated, they need to be properly modeled so as
to avoid a substantial loss of efficiency and accuracy. Otherwise, one can
adopt the standard description of open quantum systems, whereby the system is
physically coupled to a structured non-equilibrium environment modeling its
surroundings. In this case, the global dynamics is governed by an Hamiltonian
$H(t)$ including also the term $H_{\rm int}$ that describes the interaction
between $\mathbb{S}$ and $\mathbb{E}$. In case $H_{\rm int}$ is fully known
and described by deterministic coupling terms, the dynamics of $\mathbb{S}$ is
deterministic as well; conversely, by including in $H_{\rm int}$ also the
action of stochastic fields, one can recover the stochastic dynamics like that
in MuellerANNALEN2017 ; MuellerSCIREP2016 .

Repeated quantum measurements. Let us assume to monitor the dynamics of
$\mathbb{S}$ within the time interval $[0,t_{\rm fin}]$, which is defined by
$m$ distinct instants $t_{\rm fin}=t_{m}>t_{m-1}>\ldots t_{1}>t_{0}=0$, not
necessarily equidistributed in time. Protocols allowing for this purpose are
given by a sequence of quantum measurements, locally performed on $\mathbb{S}$
and in correspondence of $t_{k}$, $k=1,\ldots,m$, according to the observables
$\mathcal{O}_{k}\equiv\mathcal{O}_{\mathbb{S},k}\otimes I_{\mathbb{E}}$, where
$\mathcal{O}_{\mathbb{S},k}\equiv F_{\theta_{k}}$ and $I$ denotes the identity
operator. Specifically, $\\{\theta_{k}\\}$ is the set of the possible
measurement outcomes, while $\\{F_{\theta}\\}$ denotes the set of positive
Hermitian semi-definite operators on $\mathcal{H}_{\mathbb{S}}$ satisfying the
relation $\sum_{\theta}F_{\theta}=I_{\mathbb{S}}$. Given the system density
matrix $\rho_{\mathbb{S},k}$ at time $t_{k}$, the probability that the outcome
$\theta$ associated with the measurement operator $F_{\theta}$ occurs is
returned by the trace ${\rm Tr}[\rho_{\mathbb{S},k}F_{\theta}]$, while the
post-measurement state of $\mathbb{S}$ equals to
$\widetilde{\rho}_{\mathbb{S},k}=(M_{\theta}\rho_{\mathbb{S},k}M_{\theta}^{\dagger})/{\rm
Tr}[M_{\theta}\rho_{\mathbb{S},k}M_{\theta}^{\dagger}]$, where $M_{\theta}$
fulfills the identity $F_{\theta}=M^{\dagger}_{\theta}M_{\theta}$ (notice that
for the same value of $\theta$ two different operators $F_{\theta}$ are not
allowed).

Coherent pulsed control couplings. Coherent (open-loop) control pulses are an
essential tool to efficiently perform quantum sensing Giovannetti_NatPhon_2011
; Degen_RevModPhys_2017 . Here, we briefly introduce the control strategy also
called Dynamical Decoupling (DD) Viola_PRL_1999 , which is given by applying a
sequence of $\pi-$pulses, i.e., short and strong control pulses that invert
the phase of the quantum system $\mathbb{S}$ \- usually a qubit - used as a
sensor. In the noise sensing context, the qubit-sensor is placed in
interaction with an external stochastic field $E(t)$ with the aim to infer its
fluctuation profile. For this purpose, the qubit is prepared in the ground
state $|0\rangle$ and a $\pi/2-$pulse is firstly performed so as to transfer
the system in the superposition state $(|0\rangle+|1\rangle)/\sqrt{2}$, where
$|1\rangle$ is the corresponding excited state. Only at this point the
sequence of $\pi-$pulse is applied to $\mathbb{S}$, which thus acquires a
phase $\phi(t)$, providing us (if measured) information about the fluctuating
field. In fact, at the end of the DD sequence at time $t_{\rm fin}$, the state
of the qubit-sensor is $[\mathrm{e}^{i\phi(t_{\rm
fin})}|0\rangle+\mathrm{e}^{-i\phi(t_{\rm fin})}|1\rangle]/\sqrt{2}$, where
$\phi(t_{\rm fin})\equiv\int_{0}^{t_{\rm
fin}}y(t^{\prime})E(t^{\prime})dt^{\prime}$ and $y(t)\in\\{-1,1\\}$ is the
control pulse modulation function that switches sign whenever a $\pi-$pulse
occurs. Finally, a second $\pi/2$-pulse brings the qubit-sensor into the final
state $[(\mathrm{e}^{i\phi(t_{\rm fin})}+\mathrm{e}^{-i\phi(t_{\rm
fin})})|0\rangle+(\mathrm{e}^{i\phi(t_{\rm fin})}-\mathrm{e}^{-i\phi(t_{\rm
fin})})|1\rangle]/2$ and the probability $p_{|0\rangle}(t_{\rm fin})$ that the
qubit-sensor is in the state $|0\rangle$ at time $t_{\rm fin}$ is measured.

##  II Stochastic Quantum Zeno physics

The main purpose to apply sequences of quantum measurements based on the
quantum Zeno physics Misra1977 ; ItanoPRA1990 ; FisherPRL2001 ; FacchiPRL2002
; FacchiJPA2008 ; SmerziPRL2012  is to force the dynamics of $\mathbb{S}$ to
be confined within the Hilbert subspace defined by the measurement observable.
Since their introduction, standard observation protocols, given by sequences
of repeated projective measurements, have been applied to closed quantum
systems by assuming that between each measurement the system freely evolves
with unitary dynamics for a constant small time interval $\tau$. More
formally, all the measurement observables $\mathcal{O}_{\mathbb{S},k}$ are set
equal for any $k$ to the projector (Hermitian, idempotent operator, in general
with dimension greater than 1) $\Pi$ that defines the confinement Hilbert
subspace $\mathcal{H}_{\mathbb{S}}^{\Pi}\equiv\Pi\mathcal{H}_{\mathbb{S}}$. In
$\mathcal{H}_{\mathbb{S}}^{\Pi}$ the dynamics of the system is described
exclusively by the projected (or Zeno) Hamiltonian $\Pi H\Pi$, as it has been
observed in Refs. SchaferZeno ; Signoles2014 .

Recently, the Probability Density Functions (PDF) $p(\tau)$ and $p(E(t))$
respectively of the time intervals $\tau$ between measurements or of the
stochastic field $E(t)$ have been taken into account and the acronym
Stochastic Quantum Zeno Dynamics (SQZD) has been introduced MuellerANNALEN2017
; MuellerSCIREP2016 . Here, the peculiarity is that the interaction model with
$\mathbb{E}$ is given by fluctuation profiles of one or more parameters
entering in the dynamics of $\mathbb{S}$. Such a noise involves a no longer
effective confinement of the system dynamics, but on the other side it
increases the capability of the system to explore a larger number of
configurations within $\mathcal{H}_{\mathbb{S}}^{\Pi}$. In a controlled setup
this viewpoint makes emerge the noise, i.e., the presence of an external
environment, as a resource GherardiniPhDThesis ; GherardiniPRE2018 . In this
regard, a first result is given by the analytical expression of the
probability distribution that $\mathbb{S}$ belongs to the confinement subspace
$\mathcal{H}_{\mathbb{S}}^{\Pi}$ after a large enough number of sequential
measurements. In particular, let us define the survival probability
$\mathcal{P}_{t_{\rm fin}}\equiv{\rm Prob}(\rho_{\mathbb{S},t_{\rm
fin}}\in\mathcal{H}_{\mathbb{S}}^{\Pi})$ to find the system in the confinement
subspace. In case the time intervals $\tau_{j}$ between measurements are
independent and identically distributed random variables, $\mathcal{P}_{t_{\rm
fin}}$ is equal to

$\mathcal{P}_{t_{\rm fin}}=\prod_{j=1}^{m}{\rm Tr}\left[\Pi
U_{j-1:j}\Pi\rho_{\mathbb{S},t_{j}}\Pi U_{j-1:j}^{\dagger}\right],$ (1)

where
$U_{j-1:j}\equiv\hat{T}\exp\left(-(i/\hbar)\int_{t_{j-1}}^{t_{j}}H(t)dt\right)$,
$\hat{T}$ is the time ordering operator, while $m$ denotes the total number of
projective measurements applied to $\mathbb{S}$. Being able to take values
from an ensemble of configurations, the density matrix of $\mathbb{S}$ at time
$t_{\rm fin}$ and the survival probability $\mathcal{P}_{t_{\rm fin}}$ are
random quantities. Thus, the prediction power of the method is constrained by
our ability to compute the most probable value $\mathcal{P}^{\ast}$ of the
survival probability after a single realization of the sequence of
measurements. In this regard, by using the Large Deviation (LD) theory, it has
been derived also the analytical expression of $\mathcal{P}^{\ast}$ for a
large enough value of $m$ MuellerANNALEN2017 , i.e.,

$\mathcal{P}^{\ast}=\exp\left(m\int_{\tau,\eta(t)}p(\tau)\,p(\eta(t))\ln(q(\tau,\eta(t)))\,d\tau\,d\eta(t)\right),$
(2)

where $q(\tau,\eta(t))$ is a functional identifying the probability that
$\mathbb{S}$ belongs to $\mathcal{H}_{\mathbb{S}}^{\Pi}$ at time $t$ after the
application of a couple of projective measurements interspersed by the time
interval $\tau$. In the limit of small $\tau$’s, $q(\tau,\eta(t))$ admits the
second-order expansion $q(\tau,\eta(t))\approx 1-\eta(t)^{2}\tau^{2}$. In
particular, $\eta(t)\equiv\Delta_{\rho_{\mathbb{S},t}^{\Pi}}H_{\Pi}(t)$ is the
standard deviation of $H_{\Pi}(t)\equiv H(t)-\Pi H(t)\Pi$ with respect to the
system density matrix within the confinement subspace, i.e.,
$\rho_{\mathbb{S},t}^{\Pi}\equiv
U_{t-\tau:t}^{\Pi}\rho_{\mathbb{S},t-\tau}^{\Pi}(U_{t-\tau:t}^{\Pi})^{\dagger}$
with $U_{t-\tau:t}^{\Pi}\equiv\hat{T}\exp\left(-(i/\hbar)\int_{t-\tau}^{t}\Pi
H(t^{\prime})\Pi dt^{\prime}\right)$. Moreover, in Eq. (2) $p(\eta(t))$
denotes an artificial PDF obeying the relation $\int
p(\eta(t))\eta(t)^{2}d\eta(t)=\frac{1}{t_{\rm fin}}\int_{0}^{t_{\rm
fin}}\Delta^{2}_{\rho_{\mathbb{S},t}^{\Pi}}H_{\Pi}(t)dt$, that fixes on
average the leakage dynamics of $\mathbb{S}$ outside the confinement subspace
$\mathcal{H}_{\mathbb{S}}^{\Pi}$.

SQZD is a special class of dynamics induced by protocols based on sequential
measurements. In the more general case, the measurement observables within the
sequence are no longer equal to a single projector, and also the presence of
coherent control terms in the Hamiltonian $H(t)$ can be taken into account.
Such protocols are expected to provide the proper tool to explore the whole
Hilbert space of a quantum system by engineering the occurrence of the
measurement operators in specific time instants, so as to move the system
population from one portion of the Hilbert space to another. This question is
still challenging, because it requires to properly modulate a control pulse
$\lambda(t)$ so that the probability distribution of $\mathcal{P}_{t_{\rm
fin}}$ is peaked in correspondence of a target value chosen by the user. As
further remark, let us also observe that coherent dynamical couplings with an
auxiliary system have been studied as tools playing the role of a measurement.
In other words the back-action induced by a quantum measurement has been
reproduced by using a coherent pulse, and an equivalence between sequences of
repeated measurements and pulsed control couplings have been established.
However, although such an approximation revealed to experimentally work quite
well in peculiar dynamical conditions, as shown for example in SchaferZeno ,
one has always to keep in mind that, even when the measurement outcomes are
not recorded, sequential measurements can lead to dissipative dynamics, thus
implying loss of quantum coherence. On the other side, pulsed control
couplings are not always able to reproduce the measurement back-action; e.g.,
the ideal confinement of quantum dynamics in ensured only by sequence of
projective measurements Gherardini2016NJP ; MuellerANNALEN2017 .

##  III Noise-Robust Quantum Sensing

The prediction power of the results shown in the previous section is ensured
by knowing (also partially) the fluctuation profiles of the parameters
entering in the dynamics of $\mathbb{S}$. In this regard, noise sensing (or
spectroscopy) aims to determine the spectral density of the noise originated
by the interaction between the quantum system $\mathbb{S}$ used as a probe and
its external environment $\mathbb{E}$ Degen_RevModPhys_2017 .

Let us simply take a qubit as sensing device to detect the presence of
stochastic (time-varying) magnetic fields $E(t)$. Thus, assuming that the
qubit-sensor is coherently manipulated, the application of different and
optimized sequences of control pulses allows to enhance the sensor sensitivity
in probing the target field Kofman_PRL_2001 ; Paz-SilvaPRL2014 ;
PoggialiPRX2018 ; Mueller_sensing_2018 . In this regard, the DD control
strategy has been successfully applied to noise sensing Alvarez_PRL_2011 . Its
main peculiarity is that the decay rate (or decoherence function) $\chi(t)$ of
the qubit-sensor due to the presence of $\mathbb{E}$ is related to the
probability $p_{|0\rangle}(t_{\rm fin})$ via the equation
$\chi(t)=-\ln(1-2p_{|0\rangle}(t_{\rm fin}))$ (see e.g., Mueller_sensing_2018
), and is simply given by the overlap in the frequency domain between the
environmental spectral density function $S(\omega)$ and the filter function
$F(\omega)$ of the DD sequence driving the qubit. More specifically

$\chi(t)=\frac{1}{2}\int_{0}^{t}\int_{0}^{t}y(t^{\prime})y(t^{\prime\prime})g(t^{\prime}-t^{\prime\prime})dt^{\prime}dt^{\prime\prime},$
(3)

where $y(t)$ is the control modulation function and

$\displaystyle g(t^{\prime}-t^{\prime\prime})$ $\displaystyle\equiv$
$\displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}p(E(0),E(t^{\prime}-t^{\prime\prime}))$
(4) $\displaystyle\times$ $\displaystyle
E(0)E(t^{\prime}-t^{\prime\prime})dE(0)dE(t^{\prime}-t^{\prime\prime})$

is the autocorrelation function
$\big{\langle}E(0)E(t^{\prime}-t^{\prime\prime})\big{\rangle}$ of the
fluctuating field $E(t)$, with $p(E(0),E(t^{\prime}-t^{\prime\prime}))$
denoting the joint PDF of $E(t)$ in the two time instants $t=0$ and
$t=t^{\prime}-t^{\prime\prime}$. The previous equation is valid if we assume
that the mean value of $E(t)$ is equal to zero, i.e.,
$\big{\langle}E(t)\big{\rangle}=0$, and $E(t)$ is a stationary process so that
$g(t^{\prime},t^{\prime\prime})=g(t^{\prime}-t^{\prime\prime})$. Then, being
$S(\omega)$ and $Y(\omega)$ defined respectively as the Fourier transform of
the autocorrelation function $g(t)$ and the pulse modulation function $y(t)$,
one has that

$\chi(t)=\int_{-\infty}^{\infty}S(\omega)F(\omega)d\omega,$ (5)

where $F(\omega)\equiv\frac{4}{\pi}|Y(\omega)|^{2}$ ($F(\omega)$ is usually
called filter function). As a result, from the measurement of $p_{|0\rangle}$
at the end of the protocol, one can obtain the corresponding value of the
decoherence function at time $t_{\rm fin}$, i.e., $\chi(t_{\rm fin})$.

As second step, different filter functions $F(\omega)$ can be designed by
engineering the pulse modulation function $y(t)$, with the aim to reconstruct
$S(\omega)$ in a range $\omega\in[0,\omega_{c}]$ for a given cut-off frequency
$\omega_{c}$. To this end, let us consider a set of $N$ filter functions
$F_{n}(\omega)$, $n=1,\dots,N$, generated by equidistant $\pi-$pulse sequences
with a different number of pulses placed in correspondence of the zeros of
$\cos[\omega_{max}\frac{n-1}{N}t^{\prime}]$. As given by the Filter
Orthogonalization (FO) protocol, introduced in Mueller_sensing_2018  for
noise-robust quantum sensing, we quantify the overlap between the $N$ filter
functions $F_{n}(\omega)$ in the frequency domain, by computing the following
$N\times N$ symmetric matrix $A$ with matrix elements

$A_{nl}\equiv\int_{0}^{\omega_{c}}F_{n}(\omega)F_{l}(\omega)\,d\omega.$ (6)

An accurate estimate of $S(\omega)$ is then obtained in case of no overlaps
between the $F_{n}(\omega)$’s, i.e., if the filter functions are orthogonal
and they span a $N-$dimensional space. Otherwise, we orthogonalize the matrix
$A$ by using the transformation
$VAV^{\dagger}=\mathrm{diag}(\lambda_{1},\dots,\lambda_{N})$, where $V$ is an
orthogonal matrix and $\lambda_{n}$ are the eigenvalues of $A$. In this way,
we will determine a transformed version of the filter functions
$F_{n}(\omega)$, i.e.,

$\widetilde{F}_{n}(\omega)=\frac{1}{\sqrt{\lambda_{n}}}\sum_{l=1}^{N}V_{nl}F_{l}(\omega),\quad
n=1,\dots,N,$ (7)

that are all orthogonal functions for any (integer) value of $n$. The
procedure is concluded by expanding $S(\omega)$ in the transformed orthogonal
basis, so that also the $\chi(t_{\rm fin})$’s are accordingly modified in the
transformed coefficients

$\widetilde{\chi}_{n}\equiv\int_{0}^{\infty}S(\omega)\widetilde{F}_{n}(\omega)=\frac{1}{\sqrt{\lambda_{n}}}\sum_{l=1}^{N}\chi_{l}(t_{\rm
fin})V_{nl},$ (8)

and the estimate of $S(\omega)$, i.e., $\widetilde{S}(\omega)$, is simply
given by the following expansion:

$\displaystyle\widetilde{S}(\omega)=\sum_{n=1}^{N}\widetilde{\chi}_{n}\widetilde{F}_{n}\,.$
(9)

To sum-up, the FO protocol is a reconstruction algorithm for the spectrum of a
signal that is based on the orthogonalization of the applied filter functions,
each of them corresponding to a properly engineered pulse modulation function
– usually $\pi-$pulses. The filter functions select specific frequency ranges
of the power spectral density $S(\omega)$, and, in order to correctly estimate
the functional behaviour of $S(\omega)$, a set of $K$ _orthogonal_ filter
functions $F_{k}(\omega)$ has to be employed. However, due to physical and/or
experimental constraints, a sufficiently large number of orthogonal filter
functions cannot be realized, and, mainly for this reason, the FO protocol
aims to solve this issue and thus speed-up the sensing procedure of different
forms of noise.

##  IV Conclusions and Outlook

In conclusion, we have shown two novel approaches of measurement and control
theory, respectively based on the physics of Zeno phenomena Gherardini2016NJP
; GherardiniQST2017 ; MuellerANNALEN2017  and noise-sensing spectroscopy
Mueller_sensing_2018 . The first relies on applying repeated quantum
measurements, while the second on performing DD sequences and using models
with fluctuating parameters entering in the dynamics of $\mathbb{S}$. Such
methods are believed to be a concrete step towards the realization of novel
quantum technologies, especially quantum-based sensing devices for future
biomedicine applications.

As main outlook, it is worth analyzing with the same formalism the non-
Markovianity (NM) Rivas_Review_2014 ; Breuer_Review_2016  of the open quantum
system $\mathbb{S}$ in reference to the multi-time statistics obtained by
locally measuring $\mathbb{S}$ Smirne2017Coherence ; Gherardini_Non-Markov .
Indeed, in case our knowledge of the system-environment interaction is
a-priori unknown (or partially known), our capability to evaluate the NM of
the system dynamics is simply given by the outcomes from a sequence of
measurements, i.e., by monitoring the change of the state of the system due to
the presence of $\mathbb{E}$. In the quantum mechanical context, this is still
a challenging issue, since it requires to understand which is the role and the
effects on the dynamics of the measurement back-action in probing the NM of
$\mathbb{S}$.

Acknowledgments: The authors thank Susana Huelga for useful discussions and
for the reading of the manuscript. This work was financially supported from
the Fondazione CR Firenze through the project Q-BIOSCAN.

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