# Quantumness and speedup limit of a qubit under transition frequency
modulation

Amin Rajabalinia  Department of Physics, University of Guilan, P. O. Box
41335–1914, Rasht, Iran  Mahshid Khazaei Shadfar  Dipartimento di Ingegneria,
Università di Palermo, Viale delle Scienze, 90128 Palermo, Italy  INRS-EMT,
1650 Boulevard Lionel-Boulet, Varennes, Québec J3X 1S2, Canada  Farzam Nosrati
Dipartimento di Ingegneria, Università di Palermo, Viale delle Scienze, 90128
Palermo, Italy  INRS-EMT, 1650 Boulevard Lionel-Boulet, Varennes, Québec J3X
1S2, Canada  Ali Mortezapour<EMAIL_ADDRESS>Department of Physics,
University of Guilan, P. O. Box 41335–1914, Rasht, Iran  Roberto Morandotti
INRS-EMT, 1650 Boulevard Lionel-Boulet, Varennes, Québec J3X 1S2, Canada
Rosario Lo Franco<EMAIL_ADDRESS>Dipartimento di Ingegneria,
Università di Palermo, Viale delle Scienze, 90128 Palermo, Italy

###### Abstract

Controlling and maintaining quantum properties of an open quantum system along
its evolution is essential for both fundamental and technological aims. We
assess the capability of a frequency-modulated qubit embedded in a leaky
cavity to exhibit enhancement of its dynamical quantum features. The qubit
transition frequency is sinusoidally modulated by an external driving field.
We show that a properly optimized quantum witness effectively identifies
quantum coherence protection due to frequency modulation while a standard
quantum witness fails. We also find an evolution speedup of the qubit through
proper manipulation of the modulation parameters of the driving field.
Importantly, by introducing a new figure of merit $\mathfrak{R_{g}}$, we
discover that the relation between Quantum Speed Limit Time (QSLT) and non-
Markovianity depends on the system initial state, which generalizes previous
connections between these two dynamical features. The frequency-modulated
qubit model thus manifests insightful dynamical properties with potential
utilization against decoherence.

##  I Introduction

The coherent superposition of quantum states is termed quantum coherence and
is regarded as the main distinguishing feature of the quantum world which has
no classical counterpart. Many quantum-enhanced technologies are founded on
this feature Streltsov et al. (2017); Walls and Milburn (2008); Sun et al.
(2022). Tomographic methods are usually employed to detect quantum coherence
by reconstructing the density matrix of quantum systems. Albeit the
implementation of such a strategy poses a technical challenge as to the
measurement settings of the experiment Nielsen and Chuang (2002); D’Ariano et
al. (2003). To overcome the complexity of detection in the experiment,
Leggett-Garg inequality Leggett and Garg (1985) and quantum witness Li et al.
(2012); Kofler and Brukner (2013) have been introduced as quantum indicators
to quantify the nonclassicality of a system. Quantum witness is based on the
classical no-signaling-in-time assumption, according to which a prior
experiment does not disturb the statistical outcome of the subsequent
experiment Li et al. (2012); Kofler and Brukner (2013); Clemente and Kofler
(2016). Hence, in recent years, finding a standard quantum witness has been
extensively studied in a wide variety of physical systems Schild and Emary
(2015); Wang et al. (2017); Friedenberger and Lutz (2017); Knee et al.
(2018a); Ban (2019); Bojer et al. (2019); Gholipour et al. (2020); Nosrati et
al. (2020a); Ku et al. (2020); Usui and Ban (2020); Gong et al. (2020);
Kurashvili et al. (2021); Karthik et al. (2021). Nonetheless, newly a pitfall
is emerged in the standard quantum witness which demonstrates this indicator
fails to detect the maximal violations of classicality of a moving qubit in a
dissipative cavity. Therefore, in order to remove the shortcoming, the
optimized quantum witness Nosrati et al. (2020a) with the suitable
intermediate blind measurement has been put forward which can reach the
maximum violation of its amplitude, coinciding with the coherence monotone.

Since a realistic quantum system inevitably interacts with the environment,
such interactions consequently give rise to coherence loss. Usually, the bath
is considered to play a destructive role in quantum coherence, with the well-
known phenomenon of decoherence Breuer and Francesco (2002). As a result of
decoherence, time constraint is imposed on quantum tasks. Therefore, various
strategies have been devised to control and maintain quantum resources (such
as coherence and entanglement) including dynamical decoupling Viola and Lloyd
(1998); Viola and Knill (2005); Lo Franco et al. (2014); D’Arrigo et al.
(2014); Orieux et al. (2015); Damodarakurup et al. (2009); Cuevas et al.
(2017); Facchi et al. (2005), decoherence-free subspaces Zanardi and Rasetti
(1997); Lidar et al. (1998), error correction Preskill (1998); Knill (2005);
Plenio et al. (1997); Shor (1995); Laflamme et al. (1996), topological
properties Kitaev (2003); Freedman et al. (2003), structured quantum and
classical environments Mazzola et al. (2009); Lo Franco et al. (2013); Lo
Franco (2016); Aaronson et al. (2013); Lo Franco et al. (2012); Xu et al.
(2013); Bellomo et al. (2008); Bellomo and Antezza (2013); Mortezapour et al.
(2017); Man et al. (2015a); Scala et al. (2008); Man et al. (2015b, c); Silva
et al. (2016); de Vega and Alonso (2017); Caruso et al. (2014); Aolita et al.
(2015); Tan et al. (2010); Tong et al. (2010); Scala et al. (2011), and
indistinguishability-based protection Nosrati et al. (2020b, c); Piccolini et
al. (2021); Perez-Leija et al. (2018); Piccolini et al. (2022). In fact,
optimizing such quantum techniques paves the way for the realization of
quantum information applications.

The minimal time during which a quantum system evolves between two
extinguishable states is recognized as the quantum speed limit time (QSLT).
This interval determines the maximal rate of evolution that a quantum system
can reach. Regarding the literature, Leonid Mandelstam and Igor Tammthe had a
pioneering role in formulating the QSLT concept by means of time-energy
uncertainty relation which bounded the speed of evolution reads as MT bound
Mandelstam and Tamm (1991). Later, Norman Margolus and Lev Levitin refined
this relation and derived a more rigorous relation (ML bound) according to
which the speed of evolution cannot exceed the mean energy Margolus and
Levitin (1998); Deffner and Lutz (2013). The MT and ML bounds had been
applicable for closed quantum systems, despite the fact that the realistic
quantum systems are not isolated. It was of crucial importance to generalize
the notion of QSLT to open quantum systems. Hence, Deffner and Lutz Deffner
and Lutz (2013) filled this gap by introducing a unified quantum speed limit
time. On the other hand, since in the open quantum systems the environmental
effects cannot be ignored, this question is posed that how non-Markovianity
can affect the QSLT. Although the early studies suggested that non-
Markovianity can speed up the quantum evolution Deffner and Lutz (2013), the
recent reports indicate the existence of a direct relationship between the
QSLT and memory effect is still open to debate for the most general dynamics
of open quantum systems. Thus, increasing the non-Markovianity, quantified by
the backflow of information Breuer et al. (2009), does not necessarily lead to
the speedup in the evolution Teittinen et al. (2019). This minimum bound also
determines the maximum rate of quantum information Bekenstein (1981),
computation Lloyd (2000), entropy production Deffner and Lutz (2010), the
ultimate precision in quantum sensing Giovannetti et al. (2011), and
scrambling the spectral form factor Del Campo et al. (2017). Moreover, the
QSLT serves as the inherent limitations of quantum optimal control algorithms
Caneva et al. (2009).

Recent studies show that modulating the transition frequency of a qubit
provides the foundation for the observation of a wide variety of novel
phenomena, such as the formation of sidebands transitions Strand et al.
(2013); Beaudoin et al. (2012) the coherent destruction of tunneling Grossmann
et al. (1991), dynamic Stark effect Alsing et al. (1992),
Landau–Zener–Stückelberg-interference Shevchenko et al. (2010), modifying
fluorescence spectrum Ficek et al. (2001), and the enhancement of the degree
of non-Markovianity Poggi et al. (2017). Moreover, this scheme allows
observing topological transitions Martin et al. (2017) and population trapping
Gray et al. (1978). On the other hand, one can employ the frequency modulation
method to decouple a qubit system from its environment Viola et al. (1999);
Agarwal (1999); Kofman and Kurizki (2001) and consequently protect quantum
resources from unwanted environmental noise Mortezapour and Lo Franco (2018);
Nourmandipour et al. (2021). In addition, frequency modulation of the qubit
can be harnessed to create fast two-qubit gates for quantum information
processing Beaudoin et al. (2012); Chu et al. (2020). It is noteworthy that
exertion of an external off-resonant field can be employed as a tactic to
perform a frequency modulation in an atomic qubit Noel et al. (1998); Silveri
et al. (2017). Moreover, regarding the most recent reports, the realization of
the frequency modulation in superconducting qubits is feasible with the aid of
state-of-the-art experimental techniques Tuorila et al. (2010); Nakamura et
al. (2001); Tuorila et al. (2013); Oliver et al. (2005); Trabesinger (2017).

In the present work, we consider a frequency-modulated qubit inside a leaky
cavity for pursuing two goals: (i) checking the consistency between the
standard quantum witness (SQW) and coherence monotone, motivated by the
results introduced in Ref. Nosrati et al. (2020a); (ii) studying the role of
frequency modulation in the speedup evolution, addressing the relationship
between QSLT and non-Markovianity as a highly controversial issue which
depends on the different initial states. Concerning the first goal, the
results suggest that SQW largely fails to indicate the non-classicality of the
system, certifying previous results Nosrati et al. (2020a). Moreover, it is
revealed that SQW either lags behind or exceeds the coherence monotone.
Differently, the optimized quantum witness (OQW) Nosrati et al. (2020a)
exhibits maximum values perfectly coinciding with the coherence monotone for
optimal parameters of modulation (frequency and amplitude): this feature
stresses the requirement of optimization to get a reliable quantum indicator.
Regarding the second goal, we demonstrate that the system can reach a high
evolution speedup for optimal modulation parameters. The achieved results
indicate that the former conception of the relation between QSLT and non-
Markovianity needs to be amended by introducing a new parameter, that we name
$\mathfrak{R_{g}}$, which is interestingly related to the ratio between non-
Markovianity and the population of the excited state.

The paper is organized as follows: In Sec. II, we present the Hamiltonian of a
single frequency-modulated qubit and discuss the state evolution of the
system. In Sec. III, we give a basis for assessing the quantum witness and its
optimization. This section is followed by providing results that demonstrate
the efficiency of OQW in measuring the quantumness of the frequency-modulated
qubit. Sec. IV is devoted to the discussion about the QSLT of the system and
its connection to non-Markovianity. Finally, we present an outline of the main
conclusions and prospects in Sec. V.

##  II Model and System

Figure 1: Sketch of the single driven qubit system. A qubit (two-level atom)
is embedded inside a structured leaky cavity. The qubit transition frequency
$\omega_{0}$ is sinusoidally modulated via an external applied field with a
modulation amplitude $\delta$ and a modulation frequency $\Omega$. The qubit
interacts with vacuum modes.

The system under consideration is a qubit (two-level system) coupled to a
zero-temperature reservoir which is comprised of the quantized modes of a
high-$Q$ cavity. It is assumed that the transition frequency of the qubit is
modulated sinusoidally by an external driving field, as depicted in Fig. 1. By
utilizing the electric-dipole and rotating-wave approximations, the
Hamiltonian of the system can be written as ($\hslash=1$):

$\displaystyle\hat{H}=\sum_{k}\omega_{k}\hat{b}_{k}^{\dagger}\hat{b}_{k}+\frac{1}{2}[\omega_{0}+\delta\cos(\Omega
t)]\hat{\sigma}_{z}$
$\displaystyle+\sum_{k}\\{g_{k}\hat{\sigma}_{+}\hat{b}_{k}+g_{k}^{*}\hat{\sigma}_{-}\hat{b}_{k}^{\dagger}\\}.$
(1)

where
$\hat{\sigma}_{z}=\left|e\right\rangle\left\langle{e}\right|-\left|g\right\rangle\left\langle{g}\right|$
is a Pauli operator for the qubit, $\hat{\sigma}_{\pm}$ represent the raising
and lowering operators of the qubit, $\hat{b}_{k}^{\dagger}$ and $\hat{b}_{k}$
are respectively the creation and the annihilation operators for a photon of
mode $k$ with the frequency $\omega_{k}$ and $g_{k}$ being the coupling
strength for this mode to the qubit. Furthermore, $\omega_{0}$ characterizes
the transition frequency of the qubit (see Fig. 1) in the absence of
modulation, while $\delta$ denotes the modulation amplitude with frequency
$\Omega$. After switching to a non-uniformly rotating frame (interaction
picture) by the unitary transformation

$\small\hat{U}=\exp\left[-i\left\\{\sum_{k}\omega_{k}\hat{b}_{k}^{\dagger}\hat{b}_{k}t+[\omega_{0}t+(\delta/\Omega)\sin(\Omega
t)]\hat{\sigma}_{z}/2\right\\}\right].$ (2)

One can obtain the effective hamiltonian
$\hat{H}_{\mathrm{eff}}=\hat{U}^{\dagger}\hat{H}\hat{U}+i(\partial\hat{U}^{\dagger}/\partial
t)\hat{U}$ as

$\displaystyle\hat{H}_{\mathrm{eff}}$ $\displaystyle=$
$\displaystyle\sum_{k}g_{k}\hat{\sigma}_{+}\hat{b}_{k}e^{-i(\omega_{k}-\omega_{0})t}e^{i(\delta/\Omega)\sin(\Omega
t)}$ (3) $\displaystyle+$
$\displaystyle\sum_{k}g_{k}^{*}\hat{b}_{k}^{\dagger}\hat{\sigma}_{-}e^{i(\omega_{k}-\omega_{0})t}e^{-i(\delta/\Omega)\sin(\Omega
t)}\ .$

Note that the exponential factors on the right-hand side of Eq. (3) are
periodic therefore, they can be rewritten as below with making use of the
Jacobi-Anger expansion.

$e^{\pm i(\delta/\Omega)\sin(\Omega
t)}=J_{0}\left(\frac{\delta}{\Omega}\right)+2\sum_{n=1}^{\infty}(\pm
i)^{n}J_{n}\left(\frac{\delta}{\Omega}\right)\cos(n\Omega t),$ (4)

where $J_{n}\left(\frac{\delta}{\Omega}\right)$ is the Bessel function of the
first kind with order $n$. Starting from the initial state
$\left|\Psi(0)\right\rangle={\cos(\frac{\theta}{2})\left|e\right\rangle+\sin(\frac{\theta}{2})e^{i\phi}\left|g\right\rangle}\left|0\right\rangle$,
where $\left|0\right\rangle$ is the vacuum state of the cavity, the
conservation of the number of excitations gives the following time evolution
of the whole system:

$\displaystyle{\left|\Psi(t)\right\rangle}=\cos(\theta/2)C(t){\left|e\right\rangle}{\left|0\right\rangle}+\sin(\theta/2)e^{i\phi}{\left|g\right\rangle}{\left|0\right\rangle}$
$\displaystyle+{\sum_{k}C_{k}(t){\left|g\right\rangle}{\left|1_{k}\right\rangle}},$
(5)

where ${\left|1_{k}\right\rangle}$ describes a single photon in mode $k$ of
the reservoir and $C_{k}(t)$ represents its probability amplitude. Solving the
Schrödinger equation for the total system results in the following
differential equations for the probability amplitudes $C(t)$ and $C_{k}(t)$

$\dot{C}(t)=-i(i/\alpha)\exp[i(\delta/\Omega)\sin(\Omega
t)]\sum_{k}g_{k}e^{-i(\omega_{k}-\omega_{0})t}C_{k}(t),$ (6)
$\dot{C}_{k}(t)=-i\alpha\exp[-i(\delta/\Omega)\sin(\Omega
t)]g_{k}^{*}e^{i(\omega_{k}-\omega_{0})t}C(t).$ (7)

Successively solving Eq. 7 formally and substituting the achieved result into
Eq. 6, yields

$\dot{C}(t)+\int_{0}^{t}dt^{\prime}F(t,t^{\prime})C(t^{\prime})=0,$ (8)

where the kernel $F(t,t^{\prime})$ is the correlation function defined in
terms of continuous limits of the environment frequency

$\displaystyle F(t,t^{\prime})=\exp[i(\delta/\Omega)\\{\sin(\Omega
t)-\sin(\Omega t^{\prime})\\}]$
$\displaystyle\times\int_{0}^{\infty}J(\omega_{k})e^{-i(\omega_{k}-\omega_{0})(t-t^{\prime})}d\omega_{k}.$
(9)

Here $J(\omega_{k})$ indicates the spectral density of reservoir modes. Since
we consider the case in which the qubit resonantly interacts with the cavity
modes, possessing Lorentzian spectral distribution, $J(\omega_{k})$ adopts the
following form Breuer and Petruccione (2002); Lo Franco et al. (2013):

$J(\omega)=\frac{1}{2\pi}\frac{\gamma\lambda^{2}}{[(\omega_{0}-\omega_{k})^{2}+\lambda^{2}]},$
(10)

with $\lambda$ the decay rate and $\gamma$ the spectral width of the coupling
Breuer and Petruccione (2002). Making use of the Lorentzian spectral density,
the kernel $F(t,t^{\prime})$ can be acquired as follows:

$F(t,t^{\prime})=\frac{\gamma\lambda}{2}e^{-\lambda(t-t^{\prime})}e^{[i(\delta/\Omega)\\{\sin(\Omega
t)-\sin(\Omega t^{\prime})\\}]}.$ (11)

By substituting it into Eq. (7), one gets

$\displaystyle\dot{C}(t)+\frac{\gamma\lambda}{2}e^{[i(\delta/\Omega)\sin(\Omega
t)]}\hskip 100.00015pt$
$\displaystyle\times\int_{0}^{t}dt^{\prime}e^{[-i(\delta/\Omega)\sin(\Omega
t^{\prime})]}e^{-\lambda(t-t^{\prime})}C(t^{\prime})=0.\hskip 20.00003pt$ (12)

the probability amplitude $C(t)$ can be obtained by numerically solving the
above equation. Then, the reduced density matrix of the qubit in the atomic
basis $\\{\left|e\right\rangle,\left|g\right\rangle\\}$ can be expressed as

$\rho^{q}(t)=\left(\begin{array}[]{cc}{\cos^{2}(\frac{\theta}{2})\left|C(t)\right|^{2}}&{\frac{1}{2}\sin(\theta)e^{-i\phi}C(t)}\\\
{\frac{1}{2}\sin(\theta)e^{i\phi}C^{*}(t)}&{1-\cos^{2}(\frac{\theta}{2})\left|C(t)\right|^{2}}\\\
\end{array}\right).$ (13)

With the evolved reduced density matrix above we can carry on our analysis, as
reported in the following sections.

##  III Standard and Optimized quantum witness

In this section, we apply the quantum witness as a criterion of
nonclassicality to quantify the quantumness of a frequency-modulated qubit
inside a leaky cavity. Our general aim is to demonstrate that SQW is no longer
a reliable quantum indicator for non-isolated systems and needs to be refined
as the Optimized Quantum Witness (OQW). Hence, the proposed scheme best
exemplifies the pitfall associated with SQW and highlights the importance of
such refinement one more time as it has already been reported in ref Nosrati
et al. (2020a). Let us mention that the attributed calculations are performed
considering the no-signaling-in-time condition and the results suggest that
SQW fails to detect the non-classicality of a frequency-modulated qubit in a
dissipative cavity.

###  III.1 Standard Quantum Witness

In the present subsection, we intend to analyze the nonclassicality of the
frequency-modulated qubit by employing the SQW which is defined as

$W_{q}(t)=\left|P_{m}(t)-P^{\prime}_{m}(t)\right|.$ (14)

where $P_{m}(t)$ denotes the quantum probability of finding the system in the
state $m$ at time $t$ without performing any prior measurement, while
$P^{\prime}_{m}(t)=\sum_{n=1}^{d}P(m,t|n,t_{0})P_{n}(t_{0})$ represents the
classical probability of finding the system in state $m$ at time $t$ before
which an intermediate nonselective measurement of the state $n$ has been
performed at time $t_{0}$. According to the classical no-signaling-in-time
assumption, the prior measurement at time $t_{0}$ is noninvasive and does not
perturb the statistical outcome of the subsequent measurement at time $t$,
thus $P_{m}(t)=P^{\prime}_{m}(t)$, $[W_{q}=0]$ and the system acts as a
classical one. However, inequality of these two probabilities namely nonzero
values of $W(t)$ violate this assumption and signify the nonclassicality of
the system state at time $t$. Also, the upper bound of the standard quantum
witness Schild and Emary (2015) is $W_{q}^{max}(\tau)=1-1/d$ where $d$ is the
system dimension and describes the number of possible outcomes of a blind
measurement. Before proceeding to calculate the quantum witness, it is
convenient to initially find a propagator for the reduced density matrix of
the qubit owing to the fact that the quantum and classical probabilities are
acquired by averaging projection operators on the system state at time $t$.
One can obtain the propagator $\aleph$ using Lindblad-type evolution for an
operator $\hat{X}$ in the Heisenberg picture $d\hat{X}/dt=L[\hat{X}]$ Breuer
and Petruccione (2002); Friedenberger and Lutz (2017). For a dissipative
system-environment model, the integro-differential equation reads as

$\hat{X}(t)+\int_{0}^{t}dt^{\prime}K_{t}[\hat{X}(t^{\prime})]=0,$ (15)

where

$\begin{split}K_{t}[X(t^{\prime})]&=F(t,t^{\prime})(\sigma_{+}\sigma_{-}X(t^{\prime})+X(t^{\prime})\sigma_{+}\sigma_{-}\\\
&-2\sigma_{+}\hat{X}(t^{\prime})\sigma_{-}).\end{split}$ (16)

Let us recall that the function $F(t,t^{\prime})$ has already defined in Eq.
(11). Considering the evolution of the basis of Pauli operators
$\\{I,\sigma_{x},\sigma_{y},\sigma_{z}\\}$,

$\left(\begin{array}[]{cc}{\sigma_{x}(t)}\\\ {\sigma_{y}(t)}\\\
{\sigma_{z}(t)}\\\ {I(t)}\\\
\end{array}\right)=\aleph(t,0)\left(\begin{array}[]{cc}{\sigma_{x}(0)}\\\
{\sigma_{y}(0)}\\\ {\sigma_{z}(0)}\\\ {I(0)}\end{array}\right),$ (17)

where

$\displaystyle\aleph(t,0)=\hskip 120.00018pt$
$\displaystyle\small\left(\begin{array}[]{cccc}{\frac{1}{2}(C(t)+C^{*}(t))}&{\frac{-i}{2}(C(t)-C^{*}(t))}&{0}&{0}\\\
{\frac{1}{2}(C(t)-C^{*}(t))}&{\frac{i}{2}(C(t)+C^{*}(t))}&{0}&{0}\\\
{0}&{0}&{{\left|C(t)\right|}^{2}}&{{\left|C(t)\right|}^{2}-1}\\\
{0}&{0}&{0}&{1}\\\ \end{array}\right).$ (22)

Then, finding the average values of the Pauli operators at a time $t$, such
that
$\left\langle\sigma_{i}(t)\right\rangle=\aleph(t,0)\left\langle\sigma_{i}(0)\right\rangle,(i=x,y,z)$,
the qubit density matrix at time $t$ is obtained as

$P(t)=\frac{1}{2}{\left(I+\left\langle\sigma_{x}(t)\right\rangle\sigma_{x}+\left\langle\sigma_{y}(t)\right\rangle\sigma_{y}+\left\langle\sigma_{z}(t)\right\rangle\sigma_{z}\right)}.$
(23)

The quantum probability $P_{\pm}$ of finding the state
$\left|{\pm}\right\rangle$ at time $\tau$ in the absence of the intermediate
nonselective measurement is given by

$P_{\pm}(\tau)={\left\langle\Pi_{\pm}(\tau)\right\rangle}=Tr\left(P(\tau)\Pi_{\pm}(\tau)\right).$
(24)

where $P(\tau)$ is the evolved reduced density matrix of the qubit. We let
${\Pi_{\pm}^{x}}=\left(I\pm\sigma_{x}\right)/2$ Friedenberger and Lutz (2017),
be the intermediate nonselective projections operators and assume the qubit is
initially in a coherent superposition
$\left|\Psi(0)\right\rangle={\cos(\frac{\theta}{2})\left|e\right\rangle+\sin(\frac{\theta}{2})e^{i\phi}\left|g\right\rangle}\left|0\right\rangle$.
Hence, the quantum and classical probabilities
$P_{\pm}(\tau),P^{\prime}_{\pm}(\tau)$ can be used to calculate the SQW
according to

$\displaystyle W_{q}(\tau)=\left|P_{+}(\tau)-P^{\prime}_{+}(\tau)\right|$
$\displaystyle=\frac{1}{4}{\left|\sin(\theta)\left(C(\tau)+C^{*}(\tau)-\frac{1}{2}{\left(C(\tau/2)+C^{*}(\tau/2)\right)}^{2}\right)\right|}.$
(25)

Figure 2: Standard Quantum witness $W(\tau),$ Optimized quantum witness
$W^{\prime}(\tau),$ and Coherence monotone $C(\tau)/2,$ as a function of the
dimensionless time interval $\gamma\tau$ for different values of modulation
frequency $\Omega$ and a fixed modulation amplitude $\delta=5~{}\gamma$. The
panel $(a)$ corresponds to the unmodulated condition $\delta=0$, $\Omega=0$,
$(b)$ $\Omega=0.1\gamma$, $(c)$ $\Omega=0.5\gamma$ and $(d)$ $\Omega=5\gamma$.
Other parameters are $\lambda=0.1\gamma$, $\theta=\pi/2$ and $\phi=0$. Figure
3: Standard Quantum witness $W(\tau),$ Optimized quantum witness
$W^{\prime}(\tau),$ and Coherence monotone $C(\tau)/2,$ as a function of the
dimensionless time interval $\gamma\tau$ with various values of the modulation
frequency and amplitude: $(a)$ $\delta=2.40483\Omega~{}[J_{0}(2.40483)=0]$,
$(b)$ $\delta=3.83170\Omega~{}[J_{1}(3.83170)=0]$, $(c)$
$\delta=5.13562\Omega~{}[J_{2}(5.13562)=0]$, $(d)$
$\delta=6.38016\Omega~{}[J_{3}(6.38016=0]$. Other parameters are
$\lambda=0.1\gamma$, $\theta=\pi/2$ and $\phi=0$.

###  III.2 Optimized Quantum Witness

It has been demonstrated that the SQW of an isolated system can reach its
maximum value equivalent to the coherence monotone
$W_{q}^{max}(\tau)={C(\tau)}/2$ in which ${C(\tau)}$ is the envelope of
quantum coherence. One can simply obtain ${C(\tau)}$ by employing the
$\ell_{1}-$ norm of coherence
$C_{\ell_{1}}(\rho)=\sum_{i\neq{j}}|\left<i|\rho|j\right>|$ Baumgratz et al.
(2014). The same maximal value of quantum witness has been also validated for
a damped qubit in a Markovian thermal reservoir. Namely the envelope of the
quantum witness, defined by means of the usual intermediate and final
measurements $\Pi_{\pm}^{x}$ of Eq. (24), perfectly matches the coherence
monotone Schild and Emary (2015). However, the most recent study concerning
quantum witness poses a challenging question whether for a generic nonisolated
quantum system in which the non-Markovian behavior appears it is guaranteed
that quantum witness coincides with the coherence monotone. Their results
suggest that intermediate blind measurement matters in determining the upper
bound of SQW and they optimize the quantum witness by substituting
$\Pi_{\pm}^{z}=\frac{1}{2}{\left(I\pm\sigma_{z}\right)}$ for the conventional
blind measurement i.e.,
$\Pi_{\pm}^{x}=\frac{1}{2}{\left(I\pm\sigma_{x}\right)}$ Nosrati et al.
(2020a). The proposed system once again testifies that such optimization is
crucially required. In this context, for the proposed dissipative system
following the $\ell_{1}-$ norm of coherence we simply obtain
$C_{\ell_{1}}=\left|\sin(\theta)C(t)\right|$ from the qubit reduced density
matrix of Eq. (13). Moreover, we apply the operator
${\Gamma(\rho(\tau/2))}=\sum_{i}{\left|i\right>\left<i\right|\rho(\tau/2)\left|i\right>\left<i\right|}$,
to classicalize the system which preserves the diagonal entries of the system
state and discard the off-diagonal ones Knee et al. (2018b). One can rewrite
the Eq. (24) by employing the optimal nonselective projections. The operations
effectively make the perturbed intermediate state classical, so that any
incoherent channel remains classical for the rest of the evolution. It is
analogous to maximizing the distance between the state of the system at
$t=\tau$ and its perturbed counterpart. Calculating these new blind
measurements and having the perturbed state ${\rho^{\prime}(\tau/2)}$ for the
subsequent evolution at the time leads to the qubit state at ${t=\tau}$

${\rho^{\prime}(\tau)}=\left(\begin{array}[]{cc}{\cos^{2}{(\frac{\theta}{2})}{\left|C(\tau)\right|}^{4}}&{0}\\\
{0}&{1-\cos^{2}{(\frac{\theta}{2})}{\left|C(\tau)\right|}^{4}}\\\
\end{array}\right),\\\ $ (26)

where ${\rho^{\prime}(\tau)}$ is the reduced density matrix of the qubit after
measuring the probability amplitude ${C(\tau)}$. Accordingly, the quantum and
classical probabilities are obtained with the usual final measurement set by
projector $\Pi_{x,+}$. The optimized quantum witness of the system under a
well-defined blind measurement is

$W^{\prime}(\tau)=\frac{1}{4}{\left|\sin(\theta)(C(\tau))+C^{*}(\tau))\right|=\frac{1}{2}\left|\sin(\theta)Re[C(\tau)]\right|}.$
(27)

The optimization procedure on the standard quantum witness ${W_{q}(\tau)}$ of
Eq. (III.1) coincides with the real part of the off-diagonal term of the
evolved qubit density matrix. In order to draw a comparison among the dynamic
behavior of the SQW (${W_{q}(\tau)}$), the OQW (${W^{\prime}_{q}(\tau)}$) and
the coherence monotone with respect to the amplitude and frequency modulation
parameters (respectively, $\delta$ and $\Omega$), we plot them simultaneously
in Fig. 2 as a function of the dimensionless time interval $\gamma\tau$. The
time evolution of the system is compared to the case where the external
driving is off ${(\delta=0,\Omega=0)}$ (see Fig. 2(a)), for which coherence
monotone monotonically decreases, and both ${W_{q}(\tau)}$ and
${W^{\prime}_{q}(\tau)}$ manifest this behavior well. Such behavior is also
observed in panel (d) when the modulation frequency is high. However, panels
(b) and (c) reveal the inconsistency between the pattern of the behavior of
SQW and the coherence monotone. So that, the SQW curve is unable to touch the
coherence monotone however, the OQW curve keeps up with the coherence
monotone. Such inconsistency is abundantly clear in panel (c), for the optimal
value of the frequency modulation, in which SQW approximately possesses the
constant zero value whereas OQW perfectly matches the coherence monotone.

In order to gain a far better understanding of the dynamic of the system, we
subsequently aim to compare SQW and OQW when the ratio of amplitude and
frequency of modulation parameter $(\delta/\Omega)$ is intentionally tuned to
the first zero of the n-th Bessel function ${J_{n}}$ in Fig. 3. By taking a
sweeping glance at all the panels, one would notice that SQW either lags
behind or exceeds (when the ratio $\delta/\Omega$ is adjusted to the first
zero of the third Bessel function) the coherence monotone. Whereas, the OQW
amplitude reaches its upper limit, being exactly equal to the coherence
monotone. Regarding such glaring inconsistency between the behavior of SQW and
coherence monotone, one can easily deduce that the SQW is truly incapable of
detecting the behavior of the generic quantum systems in which non-Markovian
environmental effect appears. Simultaneously, it is proved that the adaptive
blind measurement $\Pi_{\pm}^{z}$ is a faithful measure to classify quantum
from classical behavior in the experiment. Moreover, the proposed model not
only certifies the violation of SQW Knee et al. (2018b) but, in contrast to
the previous reports in which SQW could not reach the coherence monotone, it
also reveals the SQW can exceed the coherence monotone. The disclosure of such
serious shortcomings in SQW stresses the crucial requirement of optimization.

##  IV Interplay between QSLT and non-Markovianity of a frequency-modulated
qubit

The primary purpose of this section is to assess the impact of frequency
modulation on the speedup evolution of a qubit embedded in a leaky cavity.
Meanwhile, two questions immediately arise: (a) What is the connection between
QSLT and non-Markovianity in the modulated quantum system? (b) Would non-
Markovianity be the sole factor in speeding up the rate of evolution of the
frequency-modulated system or do other contributing factors participate? To
address these questions, quantum speed limit time (QSLT) can be employed to
analyze the maximal speed of the evolution of the frequency-modulated qubit.
Initially, it is required to provide preliminary explanations concerning QSLT.
The QSLT determines the minimal time required by the system to completely
evolve between two distinguishable states Mandelstam and Tamm (1991); Deffner
(2014). It consequently constrains the maximum speed of evolution that the
system can reach. For a generic driven open quantum system, this parameter is
defined as a unified expression based on the Schatten $p$ norm by Deffner and
Lutz Deffner and Lutz (2013):

$\begin{split}{\tau_{\mathrm{QSLT}}}=\max\left\\{\frac{1}{\Lambda^{\mathrm{op}}_{\tau}},\frac{1}{\Lambda^{\mathrm{tr}}_{\tau}},\frac{1}{\Lambda^{\mathrm{hs}}_{\tau}}\right\\}\sin^{2}(\mathcal{L}(\rho(0),\rho{(\tau)})),\end{split}$
(28)

$\mathcal{L}(\rho(0),\rho{(\tau)})=\arccos\sqrt{\langle\psi_{0}|\rho{(\tau)}|\psi_{0}\rangle}$
is the so-called Bures angle between initial pure state
$\rho(0)=|\psi_{0}\rangle\langle\psi_{0}|$ and its evolved state
$\rho{(\tau)}$, governed by the time-dependent master equation
$\dot{\rho}(t)=L_{t}\rho(0)$ ($L_{t}$ is a super-operator). Therefore, we have
$\sin^{2}(\arccos\sqrt{\langle\psi_{0}|\rho{(\tau)}|\psi_{0}\rangle}=1-\langle\psi_{0}|\rho{(\tau)}|\psi_{0}\rangle$.
It is worth mentioning that the Bures angle above is a measure of the distance
between a pure state and a mixed state. The denominator in Eq. (28) indicates
the average of $\dot{\rho}(t)$ over actual driving time duration $\tau$, i.e.,
$\Lambda^{\mathrm{n}}_{\tau}=\frac{1}{\tau}\int_{0}^{\tau}\mathrm{d}t\lVert\dot{\rho}(t)\rVert_{\mathrm{n}}$
(n $=$ $op$, $tr$, $hs$), with $op$, $tr$ and $hs$ denoting the operator,
trace and Hilbert-Schmidt norms, respectively. It has been shown that the
operator norm
$\lVert\dot{\rho}(t)\rVert_{\mathrm{op}}={\displaystyle{\underset{i}{max}}}\\{s_{i}\\}$
always maximizes Eq. (28), where $s_{i}$ is the singular value of evolved
density matrix $\dot{\rho}(t)$ Jozsa (1994); Bures (1969). Thus, the QSLT in
Eq. (28) can be simplified as

$\begin{split}\tau_{\mathrm{QSLT}}=\frac{1-\langle\psi_{0}|\rho{(\tau)}|\psi_{0}\rangle}{\frac{1}{\tau}\int_{0}^{\tau}\mathrm{d}t~{}{\underset{i}{max}}\\{s_{i}\\}}.\end{split}$
(29)

The QSLT has the merit of evaluating the speed of quantum evolution for two
regions: speedup and no speedup. The time-scaled notation of QSLT
$(\tau_{\mathrm{QSLT}}/\tau)$ divides the evolution into two regions, speedup
and no speedup respectively for $\tau_{\mathrm{QSLT}}/\tau<1$ and
$\tau_{\mathrm{QSLT}}/\tau=1$. For the speedup region, it is revealed that as
$\tau_{\mathrm{QSLT}}/\tau$ decreases, the process exhibits acceleration.
However, the growth of $\tau_{\mathrm{QSLT}}/\tau$ results in the deceleration
of the process. Regarding this significant result and with the aim of shedding
light on the relationship between QSLT and non-Markovianity, we study the QSLT
and non-Markovianity of a frequency-modulated qubit. Two scenarios have been
considered depending on the initial state, first, the initial state is assumed
to be an excited state, and second, it is a coherent superposition of ground
and excited states.

###  IV.1 First scenario: Initial excited state

We prepare the driven qubit initially in the excited state
$\rho_{1}(0)=|e\rangle\langle e|$ by taking parameters $\theta=\phi=0$ and
substitute in Eq. (29), thus the QSLT reduces to

$\begin{split}\frac{\tau_{\mathrm{QSLT}}}{\tau}=\frac{{1-{|{C}(\tau)|}^{2}}}{2\mathcal{N}{(\tau)}+{1-{|{C}(\tau)|}^{2}}},\end{split}$
(30)

where
$\mathcal{N}{(\tau)}=\frac{1}{2}\bigg{(}\int_{0}^{\tau}|\partial_{t}{|{C}(t)|}^{2}|\mathrm{d}t+{|{C}(\tau)|}^{2}-1\bigg{)}$
is the BLP non-Markovianity measure and ${|{C}(\tau)|}^{2}$ characterizes the
population of the excited state. As can be seen Eq. (30) indicates an
analytical relationship between the QSLT and backflow of information. For the
Markovian process $\mathcal{N}{(\tau)}=0$ and consequently
$\tau_{\mathrm{QSLT}}/\tau=1$, which means the controlled quantum evolution
has reached the highest possible speed and cannot speed the evolution further.
However, for the non-Markovian case $\mathcal{N}{(\tau)}>0$ then
$\tau_{\mathrm{QSLT}}/\tau<1$, which signifies the acceleration is possible so
that, the smaller $\tau_{\mathrm{QSLT}}/\tau$, the larger potential capacity
for speedup Xu et al. (2018).

Figure 4: Ratio of the QSLT to the actual driving time
$(\tau_{\mathrm{QSLT}}/\tau)$ and non-Markovianity $\mathcal{N}(\tau)$ as
function of actual driving time $\tau$ for different modulations parameters:
$\Omega=0$ and $\delta=0$ (blue solid line), $\Omega=5$ and $\delta=10$ (red
dash line), and $\Omega=5$ and $\delta=2.40483\Omega$ (green dash-dot line).
The excited state $\rho_{1}(0)=|e\rangle\langle e|$ is considered as the
initial state, and other parameters are $\gamma=0.1$ and $\lambda=1$. Figure
5: $\tau_{\mathrm{QSLT}}/\tau$ and non-Markovianity $\mathcal{N}(\tau)$ as a
function of $\gamma/\lambda$ for different values of actual driving times
$\tau=0.4$ (blue solid line), $\tau=0.6$ (red dash line), and $\tau=0.8$
(green dash-dot line) with $\lambda=1$. The excited state
$\rho_{1}(0)=|e\rangle\langle e|$ is considered as the initial state.

In Fig. 4, we plot the ratio of the QSLT to the actual driving time
$(\tau_{\mathrm{QSLT}}/\tau)$ and non-Markovianity $\mathcal{N}(\tau)$ in
terms of driving time for the coupling constant $\gamma=0.1$ and $\lambda=1$.
It can be seen that the system initially remains in no speedup region and
Markovian regime and then switches to speedup region and non-Markovian regime.
It is noteworthy that when $\delta/\Omega$ is tuned to the zero of the Bessel
function $J_{0}$, it not only appreciably accelerates the speedup process, but
it increases the non-Markovianity as well. The results indicate that the
frequency modulation parameters ($\delta$ and $\Omega$) have a key role in the
speedup process.

To gain more detailed information concerning the link between QSLT and non-
Markovianity, we plot $\tau_{\mathrm{QSLT}}/\tau$ and non-Markovianity as a
function of $\gamma/\lambda$ for different values of driving time $\tau$ under
different modulation frequencies $\Omega$ and amplitudes $\delta$ in Fig. 5.
As it is evidently displayed, there are specific values of the
$\gamma/\lambda$ for which the system makes the transition from no speedup to
speedup regions. Moreover, it is surprisingly disclosed that the switching
point from Markovian to non-Markovian regimes perfectly matches the transition
point, albeit such switching would occur regardless of the initial state.
Since the excited state population appears in the Eq. (30), we wonder how this
parameter affects the QSLT. It is speculated that the QSLT is affected by an
interplay between non-Markovianity and the population of the excited state
during the evolution. Hence, we introduce a new parameter $\mathfrak{R_{g}}$
which illustrates this interplay as

$\mathfrak{R_{g}}:=\frac{\mathcal{N}(\tau)}{1-\langle\psi_{0}|\rho_{\tau}|\psi_{0}\rangle}=\frac{\mathcal{N}(\tau)}{1-|{C}(\tau)|^{2}}.$
(31)

As this ratio suggests, it is far better to discuss the collective effect of
the non-Markovianity and population of the excited state on the QSLT to find
an explicit relationship. In order to support this claim, we plot derivatives
of $\mathfrak{R_{g}}$ and $\tau_{\mathrm{QSLT}}/\tau$ as a function of
$\gamma/\lambda$ for different values of actual driving time $\tau$ under
different values of modulation amplitudes $\delta$ and frequencies $\Omega$ in
Fig. 6. As the curves illustrate, there is approximately mirror symmetry
between the derivations of $\mathfrak{R_{g}}$ and $\tau_{\mathrm{QSLT}}/\tau$
conveying this message that an increase in the ratio $\mathfrak{R_{g}}$ can
always lead to the speedup in the quantum evolution.

Figure 6: Derivatives of $\tau_{\mathrm{QSLT}}/\tau$ (solid blue line) and
$\mathfrak{R_{g}}$ (dash-dotted red line) as a function of $\gamma/\lambda$
for different actual driving time $\tau$ and $\lambda=1$. Dashed gray line is
$y=0$. The excited state $\rho_{1}(0)=|e\rangle\langle e|$ is considered as
the initial state.

###  IV.2 Second scenario: Initial coherent superposition state

Figure 7: $\tau_{\mathrm{QSLT}}/\tau$ and non-Markovianity $\mathcal{N}(\tau)$
as a function driving time $\tau$ for different modulations parameters:
$\Omega=0$ and $\delta=0$ (blue solid line), $\Omega=5$ and $\delta=10$ (red
dash line), and $\Omega=5$ and $\delta=2.40483\Omega$ (green dash-dot line).
The excited state $\rho_{1}(0)=|e\rangle\langle e|$ is considered as the
initial state, and other parameters are $\gamma=1$ and $\lambda=3$. Figure 8:
$\tau_{\mathrm{QSLT}}/\tau$ and non-Markovianity $\mathcal{N}(\tau)$ as a
function of $\gamma/\lambda$ for different values of actual driving time
$\tau=0.4$ (blue solid line), $\tau=0.6$ (red dash line), and $\tau=0.8$
(green dash-dot line) with $\lambda=1$. The coherent superposition
$|\psi(0)\rangle=\frac{1}{\sqrt{2}}\left(|e\rangle+|g\rangle\right)$ is
considered as the initial state.

We resume the process considering the second scenario according to which the
system starts from a coherent superposition
$|\psi(0)\rangle=\frac{1}{\sqrt{2}}\left(|e\rangle+|g\rangle\right)$ by taking
parameters $\theta=\pi/2,\phi=0$. To carry out a comparative study, we proceed
with the parallel trend with the previous subsection. Thus, from Eq. (29), the
QSLT is given by

$\begin{split}\frac{\tau_{QSL}}{\tau}=\frac{1-\mathrm{Re}(C({\tau}))}{\int_{0}^{\tau}\mathrm{d}t\left|\sqrt{\left|\partial_{t}C(t)\right|^{2}+|\partial_{t}|C(t)|^{2}|^{2}}\right|}.\end{split}$
(32)

As is clear, no explicit expression is found for $\tau_{\mathrm{QSLT}}/\tau$
in terms of non-Markovianity. The dynamic behavior of
$\tau_{\mathrm{QSLT}}/\tau$ and non-Markovianity versus actual driving time is
respectively depicted in Fig. 7 (a) and (b). It is revealed that, by initially
preparing the system in a coherent superposition state, the evolution of
system is already laid in the speedup region whether the qubit is frequency
modulated or not. Nonetheless, for the case qubit is unmodulated,
$\tau_{\mathrm{QSLT}}/\tau$ plummets down initially and then gradually
increases which indicates deceleration. However, for the frequency-modulated
qubit, $\tau_{\mathrm{QSLT}}/\tau$ at first sharply decreases, and after a
short while with the commencement of non-Markovian behavior, falls one more
step and then incessantly fluctuates implying constant acceleration and
deceleration. Fig. 8 exhibits the behavior of $\tau_{\mathrm{QSLT}}/\tau$ and
non-Markovianity $\mathcal{N}(\tau)$ as a function of $\gamma/\lambda$ for
different values of actual driving time $\tau$ under different modulation
frequencies $\Omega$ and amplitudes $\delta$. As it is manifest, the dynamic
of the system for all values of $\gamma/\lambda$ is laid in the speedup region
regardless of the modulation process. However, modulation exerts an influence
on the non-Markovian behavior. Nevertheless, one can notice that as the system
switches from the Markovian to the non-Markovian regimes, the curves of
$\tau_{\mathrm{QSLT}}/\tau$ experience a declining trend signifying
acceleration. Moreover, as it is evident, the modulation process gives rise to
a marked shift in the initial value of $\tau_{\mathrm{QSLT}}/\tau$ for
$\tau=8$ the same as the case where the system is initially prepared in the
excited state. As a final investigation, we plot the derivatives of
$\mathfrak{R_{g}}$ and $\tau_{\mathrm{QSLT}}/\tau$ versus $\gamma/\lambda$ in
Fig. 9. As it is crystal clear, when the system is initially prepared in the
coherent superposition state, there is no significant relationship between
$\mathfrak{R_{g}}$ and speedup process.

Figure 9: Derivatives of $\tau_{\mathrm{QSLT}}/\tau$ (solid blue line) and
$\mathfrak{R_{g}}$ (dash-dotted red line) as a function of $\gamma/\lambda$
for actual driving time $\tau=1$ with $\delta=5$, $\Omega=5$, and $\lambda=1$.
Solid gray line is $y=0$. The superposition state
$\rho_{2}(0)=\frac{1}{2}(|e\rangle\langle e|+|e\rangle\langle
g|+|g\rangle\langle e|+|g\rangle\langle g|)$ is considered as the initial
state.

##  V Conclusion

In this work, we have carried out a thorough study of the quantumness and
speedup limit time of a frequency-modulated qubit embedded in a leaky cavity.
The validity of the standard quantum witness as a quantum indicator is checked
for the proposed system in comparison to the certain criterion of quantumness
namely, coherence monotone. The proposed model best exemplifies that SQW fails
to truly describe the non-classicality of a generic non-isolated system
(including non-Markovian environmental effect) certifying the result of ref
Nosrati et al. (2020a). Moreover, the results indicate that, in contrast to
the previous reports in which SQW could not reach the coherence monotone, the
SQW can exceed the coherence monotone as well. Such glaring inconsistency
between the behaviors of SQW and coherence monotone leads us to employ the
optimization method, introduced in Ref. Nosrati et al. (2020a), by choosing
the appropriate blind measurement and proceeding according to the optimal
quantum witness. We have demonstrated that OQW keeps up with the coherence
monotone and its maximum values perfectly coincide with the coherence monotone
for the different values of modulation parameters.

We have then assessed the impact of the frequency modulation on the speedup
evolution and shed light on the relationship between QSLT and non-
Markovianity, by considering two cases depending on the initial state.
Surprisingly, it has emerged that the non-Markovianity is not the sole
influential factor in speeding up the rate of evolution but the population of
the excited state also participates as a contributing factor. Hence, we have
introduced a new parameter $\mathfrak{R_{g}}$, defined as the ratio of non-
Markovianity to the population of the excited state, for exploring its
relationship with the usual ratio of QSLT to the actual driving time. The
findings indicate that the interplay of modulation parameters $\delta$ and
$\Omega$ has a key role in the speedup process.

For the case when the initial state is the excited state, we have observed
that the transition point from no speedup to speedup regions perfectly matches
to switching point from Markovian to non-Markovian regimes. We have also found
that the relationship between $\mathfrak{R_{g}}$ and
$\tau_{\mathrm{QSLT}}/\tau$ conforms to a meaningful pattern so that as the
$\mathfrak{R_{g}}$ increases, the ratio of $\tau_{\mathrm{QSLT}}/\tau$
decreases implying the acceleration of speedup process. However, when the
initial state is assumed to be a coherent superposition of the ground and
excited states, the evolution of system is already laid in the speedup region
whether the process is Markovian or non-Markovian and there is no significant
relationship between $\mathfrak{R_{g}}$ and $\tau_{\mathrm{QSLT}}/\tau$.
Therefore, all together the achieved results verify our intuition about the
effect of the excited state population on the QSLT. Also, as a common result
for both cases regardless of the initial state, non-Markovianity induces
fluctuation of the speedup process implying periodic acceleration and
deceleration.

As a prospect, the optimal quantum witness can be potentially applied to the
complex quantum systems such as the flux qubit Knee et al. (2016) and IBM
quantum computer Ku et al. (2020), for which the implementation of quantum
tomographic methods is problematic to indicate quantum coherence. Moreover,
frequency modulation can be employed as a simple but effective technique to
accelerate the process of performing quantum tasks. As a matter of fact,
experimentalists have recently utilized this technique to fabricate and
control quantum circuit devices in superconducting Josephson qubits
(artificial atoms) Silveri et al. (2017); Nakamura et al. (2001); Oliver et
al. (2005); Tuorila et al. (2010, 2013), enabling progress in the building
blocks of current quantum computer prototypes Trabesinger (2017). Hence,
qubits with optimal transition frequency modulation are good candidates for
quantum gates owing to the fact that they can perform fast quantum operations.

###### Acknowledgements.

A.M. acknowledges the support of the University of Guilan. R.M. acknowledges
support from NSERC, MEI and the CRC program in Canada. R.L.F. acknowledges
support from Unione europea – NextGenerationEU – fondi MUR D.M. 737/2021 –
progetto di ricerca “Indistinguibilità come risorsa controllabile per processi
di informazione quantistica e termodinamica quantistica”.

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# Laboratory rotational spectroscopy and astronomical search of ethynyl
substituted naphthalene

Carlos Cabezas,1 Isabel Peña2 and José Cernicharo1
1Departamento de Astrofísica Molecular, Instituto de Física Fundamental,
Consejo Superior de Investigaciones Científicas,
C/ Serrano 121, 28006 Madrid, Spain
2Departamento de Química Física y Química Inorgánica, Facultad de Ciencias,
Universidad de Valladolid, Paseo de Belén 7, Valladolid 47011, Spain E-mail:
<EMAIL_ADDRESS>(CC)E-mail<EMAIL_ADDRESS>(IP)

(Accepted 2022 December 13. Received 2022 December 11; in original form 2022
September 9)

###### Abstract

The recent interstellar detection of cyanonaphthalenes bring interest in
related aromatic molecular species that could be present in similar
astronomical environments. In this context, ethynyl derivatives of naphthalene
are promising candidates to be observed in the Taurus Molecular Cloud (TMC-1),
where cyanonaphthalenes together with cyano- and ethynyl- derivatives of
cyclopentadiene and benzene have been detected. To enable the interstellar
searches for ethynyl derivatives of naphthalene, their pure rotational spectra
need to be investigated in the laboratory. We have observed for the first time
the rotational spectra of 1- and 2-ethynylnaphthalene species using a
broadband Fourier-transform microwave spectrometer operating in the 2-8 GHz
frequency region. Accurate spectroscopic parameters are derived from the
analysis of the experimental spectra, allowing for reliable predictions for
astronomical searches. Our searches in TMC-1 for both isomers provide upper
limits for the abundances of these species.

###### keywords:

astrochemistry – ISM: molecules – methods: laboratory: molecular

††pubyear: 2022††pagerange: Laboratory rotational spectroscopy and
astronomical search of ethynyl substituted naphthalene–References

##  1 Introduction

Aromatic molecules are dominant in the chemistry occurring on Earth, many
playing important roles in biological systems, i.e., nucleobases and their
derivatives, nucleosides. They are ubiquitous also in the Universe, where
around 20% of all interstellar carbon, is thought to be incorporated into
polycyclic aromatic hydrocarbons (PAHs) (Tielens, 2008). PAHs are proposed as
carriers of the infrared emission features that dominate the spectra of most
galactic and extragalactic sources (Léger & Puget, 1984; Allamandola et al.,
1985; Tielens, 2008). However, the formation processes of these species have
been poorly understood due to the inability to detect individual molecules.
The first detection of an aromatic molecule in space was in 2001, when benzene
was identified in the protoplanetary carbon-rich star CRL618 (Cernicharo,
2001) through its infrared spectral signature. However, despite their expected
ubiquity, further astronomical identification of aromatic molecules had proven
elusive for almost two decades, until benzonitrile was detected in 2018 by
McGuire et al. (2018) in the dense molecular cloud TMC-1 through its
rotational spectrum. Very recently, four new aromatic molecules have been
discovered in TMC-1 by their rotational spectra, thanks to the high level of
sensitivity available with the Green Bank Telescope and the Yebes 40m
telescope and the observing projects GOTHAM111GBT Observations of TMC-1:
Hunting Aromatic Molecules (McGuire et al., 2018) and QUIJOTE222Q-band
Ultrasensitive Inspection Journey to the Obscure TMC-1 Environment (Cernicharo
et al., 2021a). These new molecular species are the two isomers of the cyano
derivatives of naphthalene, 1- and 2-cyanonaphthalene (McGuire et al., 2021)
and the first pure PAH discovered so far, indene (Cernicharo et al., 2021b;
Burkhardt et al., 2021) together with its cyano derivative (Sita et al.,
2022).

Indene, despite its large molecular size, is exceptionally abundant in TMC-1,
just five times less abundant than the ubiquitous cyclic hydrocarbon $c$-C3H2
(Cernicharo et al., 2021b). The detection of cyano derivatives of indene
serves to directly compare the ratio of a pure hydrocarbon PAH to its cyano
substituted counterpart in observations versus those predicted by
astrochemical models. Sita et al. (2022) posit that cyano derivatives of PAHs
indeed can be used as an excellent observational proxy for their hydrocarbon
counterparts for constraining models, at least within a factor of a few. Based
on this, benzene and naphthalene are also expected to be very abundant in
TMC-1, although they were supposed to be formed primarily in the circumstellar
envelopes of evolved stars (Tielens, 2008). In this scenario and due to the
extraordinary level of sensitivity of the current TMC-1 line surveys, it is
probable that further PAHs are awaiting detection. However, most of PAHs are
very high symmetry molecules, which implies that they lack of permanent
electric dipole moment preventing their radio and millimeter detection. As
such, two alternatives can be followed to improve our understanding on the
chemical processes leading to these molecular species. The first one is the
identification of the intermediate species that are probably behind the in-
situ bottom-up synthesis of PAHs in these environments (Agúndez et al., 2021),
and the second one, the direct detection of polar derivatives of the target
aromatic molecules, i.e. benzonitrile and cyanonaphthalenes (McGuire et al.,
2018, 2021). New results in any of this research lines will certainly help in
obtaining a complete chemical network to explain the molecular chemistry of
aromatics and PAHs in TMC-1.

As mentioned above, the detections of 1- and 2-cyanonaphthalene species in
TMC-1 (McGuire et al., 2021) suggest that naphthalene should be quite abundant
species in TMC-1. Cernicharo et al. (2021c) reported the abundances for -CN
and -CCH derivatives of cyclopentadiene and benzene and their results indicate
that the -CCH derivatives have larger abundances for both systems. McGuire et
al. (2021) discussed the formation routes for the cyanonaphthalenes and one of
the considered reactions is that between naphthalene molecule and CN radical.
If we consider an analogue reaction to form the ethynyl derivatives of
naphthalene, then these species could reach higher abundances than the cyano
derivatives, as it occurs for cyclopentadiene and benzene, due to the large
abundance of CCH radical compared to CN radical. It is then straightforward to
think that ethynylnaphthalene can be considered as a good candidate to be
detected through radio astronomy observations in TMC-1. In this work, we have
employed Fourier transform microwave (FTMW) spectroscopy to obtain and assign
the rotational spectra of the two isomers of ethynyl-substituted naphthalene,
1-ethynylnaphthalene (1-ETN) and 2-ethynylnaphthalene (2-ETN). The molecular
constants for both species are accurately determined, in good agreement with
previous theoretical predictions (Ye et al., 2022), and provide sufficient
data for interstellar searches in TMC-1.

Figure 1: Molecular structures and carbon atom labelling of the two isomers of
ethynylnaphthalene. The $a$ (red) and $b$ (green) principal inertial axes are
shown. The $c$ axis is perpendicular to the molecular plane.

##  2 Experimental

The rotational spectra of the 1-ETN and 2-ETN species were observed using a
chirped-pulsed Fourier-transform microwave spectrometer operating in the
frequency region of 2-8 GHz (Peña & Cabezas, 2020; Cabezas et al., 2021).
Samples of 1-ETN (stated purity 97%) and 2-ETN (stated purity >95%) were
purchased from Sigma Aldrich and were used without further purification. They
were vaporized in a heated nozzle (held near 370 K) attached to a solenoid-
driven pulsed valve (0.8 mm diameter, Parker General Valve, Series 9) with
neon and argon, respectively, at 3 bar backing pressure, used as carrier gases
creating a supersonic expansion into the vacuum chamber with a rotational
temperature of approximately 2 K. The use of neon or argon is just due to
logistic circumstances in our lab. For each gas pulse, the ensemble of
molecules was polarized with a series of 8 microwave chirps of 4 $\mu$s length
varying linearly in frequency from 2 to 8 GHz. The chirped pulse was generated
with an arbitrary waveform generator (Tektronix AWG 70002A), amplified to 200
W with a travelling wave tube amplifier and broadcast perpendicular to the
propagation of the jet expansion through a horn antenna. A molecular transient
emission spanning 40 $\mu$s is then detected through a second horn and
amplified by a low noise MW signal amplifier. A total of 4300k FIDs were co-
added and Fourier transformed with a Kaiser-Bessel window function to give the
broadband rotational spectrum in the frequency domain. The spectral resolution
is better than 10 kHz and given the signal to noise ratio observed, the
frequency measurements have an estimated accuracy of 5 kHz.

##  3 Results and Discussion

###  3.1 Rotational spectra analysis

Geometry optimization calculations for both 1-ETN and 2-ETN isomers (see Fig.
1) have been reported very recently before by Ye et al. (2022). In this work,
the authors report accurate molecular structures for both 1-ETN and 2-ETN
isomers (see Fig. 1) and provide theoretical values for the rotational and
centrifugal constants together with values for the electric dipole moment
components. The molecular constants for both isomers are shown in Table 1. We
have used these theoretical constants as starting point in the rotational
spectra analysis. Both species display no very large $a$ and $b$ dipole moment
components, $\mu_{a}$=0.58 D and $\mu_{b}$=0.33 D for 1-ETN and $\mu_{a}$=0.90
D and $\mu_{b}$=0.11 D for 2-ETN., and null $c$ dipole moment component due to
their $C_{s}$ symmetry.

Figure 2: Rotational spectrum of 1-ETN in the 2–8 GHz frequency range. The
black trace is the experimental spectrum, and the red trace is the simulated
spectrum of 1-ETN based on the fitted rotational constants at a rotational
temperature of 1 K.

1-ETN. In light of the dipole moment components for 1-ETN it is expected that
both $a$\- and $b$-type spectra for 1-ETN will be weak but observable. The
recorded broadband rotational spectrum in the 2–8 GHz frequency region for
1-ETN is shown in Fig. 2. On first inspection the broadband rotational
spectrum is dense with a strong characteristic $a$-type transitions pattern.
In addition, $b$-type lines were also observed, but with lower intensity.
Finally, a total of 160 rotational transitions, list included in the
Supporting Information, were assigned for 1-ETN. The best fit for all these
lines was obtained using the $S$-reduction of Watson’s Hamiltonian in the
$III^{r}$ representation (Watson, 1977). This fit was done using the Kisiel’s
ASFIT program (Kisiel & Bialkowska-Jaworska, 2019) which is best suited when
the $S$-reduction in the $III^{r}$ representation is required. The list of
determined constants includes the three rotational constants and the five
quartic distortion constants. They are shown in Table 1 together with the
predicted values from Ye et al. (2022). The calculated rotational constants
are in excellent agreement with the experimental values with relative errors
ranging from 0.02 to 0.05 %, indicating that the calculated geometry is very
accurate. The discrepancies between experiment and theory are even smaller
than those reported for 1-cyanonaphthalene (Ye et al., 2022). The
discrepancies between the experimental and predicted values for the quartic
distortion constants are larger, as expected, but the agreement can be
considered satisfactory as well.

Table 1: Spectroscopic constants for the two isomers of ethynylnaphthalene.

| 1-ETN | 2-ETN
---|---|---
Constants/Units | Exp. | Calc.a | Exp. | Calc.a
$A$/MHz | 1474.99376(15)b | 1475.332 | 2713.63397(37) | 2709.967
$B$/MHz | 953.707853(95) | 953.269 | 606.040212(41) | 606.269
$C$/MHz | 579.26886(10) | 579.129 | 495.478949(41) | 495.489
$D_{J}$/kHz | 0.0593(14) | 0.0327 | [0.00488]c | 0.00488
$D_{JK}$/kHz | -0.0915(15) | -0.0837 | [0.00142] | 0.00142
$D_{K}$/kHz | 0.0383(18) | 0.0553 | [0.465] | 0.465
$d_{1}$/kHz | -0.01309(71) | -0.0189 | [-0.00134] | -0.00134
$d_{2}$/kHz | -0.01989(30) | -0.0130 | [-0.00021] | -0.00021
$\sigma_{rms}$/kHz | 3.2 |  | 3.6 |
$N_{lines}$ | 167 |  | 78 |
$J_{min}$/$J_{max}$ | 0/11 |  | 0/13 |
$K_{a,min}$/$K_{a,max}$ | 0/7 |  | 0/5 |

  * •

$Notes$. a Ye et al. 2022 b Numbers in parentheses represent the derived
uncertainty ($1\,\sigma$) of the parameter in units of the last digit. c
Numbers in brackets have been fixed to the theoretical values.

A deeper inspection of the spectrum allowed us to identify the twelve 13C
isotopic species in natural abundance. The observed number of transitions for
each isotopic species vary between 21 and 40, including $a$\- and $b$-type
transitions. The analysis of the measured transitions was carried out in the
same manner than that for the 1-ETN parent species with the exception that the
quartic centrifugal distortion constant values were kept fixed to those
determined for the parent species. The results obtained from our analysis are
shown in Table 2 and a list of all the frequency transitions measured are
collected in the Supporting Information.

Figure 3: Rotational spectrum of 2-ETN in the 2–8 GHz frequency range. The
black trace is the experimental spectrum, and the red trace is the simulated
spectrum of 2-ETN based on the fitted rotational constants at a rotational
temperature of 1 K.

2-ETN. As it occurs for the 1-ETN isomer, the quantum chemical calculations
predict no null $a$\- and $b$-dipole moment components for 2-ETN. However, the
$\mu_{a}$ component (0.90 D) is much larger than the $\mu_{b}$ component, 0.11
D. The observed broadband rotational spectrum for 2-ETN in the 2–8 GHz
frequency region, shown in Fig. 3, is less populated than that for 1-ETN. The
main reason is the low value of the $\mu_{b}$ dipole moment component, which
considerably affect to the intensity of the $b$-type spectrum, compared to
that for 1-ETN. As mentioned above, the experiments for the 2-ETN were carried
out using Ar instead of Ne. However, no significant differences in terms of
spectral intensities were encountered between both experiments. In total, we
measured a total of 78 rotational transitions for 2-ETN, including 50 $a$-type
and 28 $b$-type transitions. The complete list of the assigned transitions is
available at the Supporting Information. The best fit for the observed
transitions was achieved utilising Watson’s $S$-reduced asymmetric rotor
Hamiltonian in the $I^{r}$ representation (Watson, 1977) using the SPFIT
program (Pickett, 1991). In contrast to the 1-ETN species, only the rotational
constants could be determined, while the quartic distortion constants were
kept fixed to the theoretical values, see Table 1. As for the 1-ETN isomer,
the experimental rotational constants agree very well with those predicted by
(Ye et al., 2022), except the $A$ constant which show larger discrepancies. In
this manner, $B$ and $C$ constants show relative errors of 0.04 and 0.002 %,
respectively, while $A$ constant differs from the predicted value by 0.14 %.

As for 1-ETN, all the 13C isotopic species of 2-ETN were observed in natural
abundance. Only $a$-type transition could be observed, due to weakness of the
$b$-type spectrum. The analysis of the measured transitions was carried out as
for the parent species and only the rotational constants were determined. As
for the 2-ETN parent isotopologue, the quartic centrifugal distortion constant
values were kept fixed to those obtained by quantum chemical calculations. The
results obtained from our analysis are shown in Table 3 and a list of all the
measured frequency is collected in the Supporting Information.

Table 2: Spectroscopic constants for the 13C isotopic species of 1-ETN.

Constants/Units |  13C1 |  13C2 |  13C3 |  13C4 |  13C5 |  13C6
---|---|---|---|---|---|---
$A$/MHz | 1475.00592(39) | 1466.71741(45) | 1449.91735(47) | 1455.56200(42) | 1472.33576(47) | 1474.09337(46)
$B$/MHz | 950.50826(19) | 948.26618(22) | 952.59009(29) | 953.12640(17) | 951.91083(26) | 943.57085(19)
$C$/MHz | 578.08900(10) | 575.98526(11) | 574.95303(15) | 576.034958(93) | 578.19622(14) | 575.37709(11)
$\sigma_{rms}$/kHz | 3.5 | 3.9 | 4.9 | 3.2 | 4.3 | 4.6
$N_{lines}$ | 37 | 39 | 39 | 35 | 38 | 42
$J_{min}$/$J_{max}$ | 1/6 | 1/6 | 1/6 | 1/6 | 1/6 | 1/7
$K_{a,min}$/$K_{a,max}$ | 0/3 | 0/4 | 0/4 | 0/3 | 0/4 | 0/3
Constants/Units |  13C7 |  13C8 |  13C9 |  13C10 |  13C11 |  13C12
$A$/MHz | 1472.10068(54) | 1460.14552(53) | 1464.07629(65) | 1474.7433(11) | 1471.19265(46) | 1461.50609(51)
$B$/MHz | 939.77460(18) | 947.52523(25) | 953.27127(27) | 953.73834(55) | 944.00278(19) | 935.73755(19)
$C$/MHz | 573.66045(10) | 574.69570(13) | 577.41815(14) | 579.24303(24) | 575.09433(11) | 570.542557(96)
$\sigma_{rms}$/kHz | 4.1 | 3.9 | 4.1 | 6.8 | 2.9 | 3.3
$N_{lines}$ | 38 | 34 | 37 | 23 | 34 | 35
$J_{min}$/$J_{max}$ | 1/6 | 1/6 | 1/6 | 1/6 | 1/6 | 1/6
$K_{a,min}$/$K_{a,max}$ | 0/3 | 0/3 | 0/3 | 0/3 | 0/3 | 0/3

  * •

$Notes$. Numbers in parentheses represent the derived uncertainty
($1\,\sigma$) of the parameter in units of the last digit. In all the fits the
quartic distortion constants have been kept fixed to the values determined for
the parent species.

Table 3: Spectroscopic constants for the 13C isotopic species of 2-ETN.

Constants/Units |  13C1 |  13C2 |  13C3 |  13C4 |  13C5 |  13C6
---|---|---|---|---|---|---
$A$/MHz | 2700.586(41) | 2713.729(45) | 2685.692(38) | 2669.111(37) | 2704.804(31) | 2694.9441(30)
$B$/MHz | 605.56726(32) | 603.69734(29) | 604.53161(30) | 606.03543(32) | 605.45232(23) | 602.288833(22)
$C$/MHz | 494.72788(30) | 493.91390(27) | 493.53457(28) | 493.97132(30) | 494.79014(22) | 492.348782(22)
$\sigma_{rms}$/kHz | 4.1 | 3.8 | 3.9 | 4.2 | 4.9 | 4.4
$N_{lines}$ | 22 | 23 | 24 | 22 | 20 | 21
$J_{min}$/$J_{max}$ | 2/7 | 2/7 | 2/7 | 2/7 | 2/7 | 2/7
$K_{a,min}$/$K_{a,max}$ | 0/3 | 0/3 | 0/3 | 0/3 | 0/3 | 0/3
Constants/Units |  13C7 |  13C8 |  13C9 |  13C10 |  13C11 |  13C12
$A$/MHz | 2713.247(37) | 2693.173(35) | 2678.321(42) | 2708.608(45) | 2711.924(37) | 2707.840(35)
$B$/MHz | 598.44889(25) | 599.99403(25) | 604.24577(33) | 605.83767(37) | 598.74852(28) | 592.54743(27)
$C$/MHz | 490.38143(24) | 490.75654(26) | 493.09641(31) | 495.17796(35) | 490.53742(26) | 486.23501(25)
$\sigma_{rms}$/kHz | 3.5 | 3.7 | 4.4 | 4.9 | 3.6 | 3.9
$N_{lines}$ | 21 | 22 | 25 | 25 | 23 | 23
$J_{min}/J_{max}$ | 2/7 | 2/7 | 2/7 | 2/7 | 2/7 | 2/7
$K_{a,min}$/$K_{a,max}$ | 0/3 | 0/3 | 0/3 | 0/3 | 0/3 | 0/3

  * •

$Notes$. Numbers in parentheses represent the derived uncertainty
($1\,\sigma$) of the parameter in units of the last digit. In all the fits the
quartic distortion constants have been kept fixed to the theoretical values
reported by Ye et al. 2022.

Figure 4: The top panel shows the computed rotational spectrum of 1-ETN in the
range 31.1-41.2 GHz. We have adopted a rotational temperature of 10 K, a line
width of 0.6 km s-1 and a column density of 1$\times$1013 cm-2. The other
panels show the QUIJOTE data at the frequency of selected rotational lines of
the 1-ETN isomer. The computed synthetic spectrum is shown in red. Abcisa is
frequency and ordinate is antenna temperature in milli Kelvins. Blanked
channels correspond to negative features produced in the folding of the
frequency switching observations.  Figure 5: Same as Figure 4 but for 2-ETN.

###  3.2 Astronomical search for ethynylnaphthalene

The observational data used in this article consist of spectra of TMC-1 taken
with the Yebes 40m telescope towards the cyanopolyyne peak of TMC-1,
$\alpha_{J2000}=4^{\rm h}41^{\rm m}41.9^{\rm s}$ and
$\delta_{J2000}=+25^{\circ}41^{\prime}27.0^{\prime\prime}$. The observations
are part of the ongoing QUIJOTE line survey (Cernicharo et al., 2021a) carried
out during different observing runs between November 2019 and May 2022.
QUIJOTE has reached now a level of sensitivity (0.12-0.25 mK across the
Q-band) that permits to detect new isotopologues and CN and CCH derivatives of
species with abundances of a few $\sim$10-10.

The observations were performed using the frequency-switching mode with a
frequency throw of 10 MHz in the very first observing runs, in November 2019
and February 2020, 8 MHz in the observations of January-November 2021, and 10
MHz again in the last observing run that took place between October 2021 and
May 2022. The total on-source telescope time is 546 h in each polarization
(twice this value after averaging the two polarizations), which can be split
into 293 and 253 hours for the 8 MHz and 10 MHz frequency throws. The QUIJOTE
line survey uses a 7 mm receiver covering the Q band (31.0-50.3 GHz) with
horizontal and vertical polarizations. Receiver temperatures in 2019 and 2020
varied from 22 K at 32 GHz to 42 K at 50 GHz. In 2021, some power adaptation
carried out in the down-conversion chains changed the receiver temperatures to
16 K at 32 GHz and 25 K at 50 GHz. The backends are 16 Fourier transform
spectrometers, which provide a bandwidth of 8 $\times$ 2.5 GHz in each
polarization, thus covering practically the whole Q band, with a spectral
resolution of 38.15 kHz. The system is described in detail by Tercero et al.
(2021).

The intensity scale used is antenna temperature, $T_{A}^{*}$, which is
calibrated using two absorbers at different temperatures and the ATM package
(Cernicharo, 1985; Pardo et al., 2001). Calibration uncertainties were assumed
to be 10% based on the observed repeatability of the line intensities between
different observing runs. All data were analysed using the software
GILDAS333http://www.iram.fr/IRAMFR/GILDAS/.

Table 4: Rotational partition function for 1-ETN and 2-ETN at different
temperatures.

Temperature/K | 1-ETN  | 2-ETN
---|---|---
300.000 | 971140.1 | 970842.2
225.000 | 630707.4 | 630564.7
150.000 | 343252.0 | 343199.5
75.000 | 121340.5 | 121330.2
37.500 | 42901.2 | 42898.4
18.750 | 15170.8 | 15169.6
9.375 | 5366.2 | 5365.4

The frequency predictions for both isomers 1-ETN and 2-ETN were implemented in
the MADEX code (Cernicharo, 2012), to compute the synthetic spectrum of both
isomers in the Q-band we have assumed a rotational temperature of 10 K, a line
width of 0.6 km s-1, and a vLSR of 5.83 km s-1 (Cernicharo et al., 2020a). We
used the dipole moment components reported by (Ye et al., 2022) and shown in
Section 3.1. The partition functions employed in the frequency predictions are
shown in Table 4. They were calculated using the SPCAT program (Pickett, 1991)
and at maximum value of $J$=300. We focused our search on the $a$\- and
$b$-type $R$-branch transitions with $K_{a}$ values 0, 1 and 2 which are
degenerated in frequency increasing the intensity of the expected lines. None
of these series of lines are clearly seen in our TMC-1 data. The sensitivity
of the QUIJOTE line survey varies between 0.12 and 0.25 mK in the 31-50 GHz
domain. Figures 4 and 5 show the predicted spectrum of both isomers across the
Q-band and the comparison of the strongest predicted lines with the data.
Frequency predictions for 1-ETN have uncertainties between 10-60 kHz in the
Q-band, while those of 2-ETN have larger uncertainties due to the
impossibility to get the distortion constants from the laboratory data.
Although in some cases the predicted lines agree with features in the QUIJOTE
data, most lines are missing. We tried to assign some of the spectral features
appearing close to the predicted lines, but we fail in getting a reasonable
fit when merging them with the laboratory data. Adopting the observed
3$\sigma$ limits to the intensity of strongest lines, we derive a 3$\sigma$
upper limit to the column density of 1-ETN in TMC-1 of 1.0$\times$1013 cm-2.
For the 2-ETN isomer, which have a larger $\mu_{a}$ dipole moment the derived
upper limit is smaller, 6.8$\times$1012 cm-2.

It is interesting to compare the abundance of the ethynyl and cyano
derivatives of naphtalene. McGuire et al. (2021) derived column densities for
1-cyano and 2-cyanonaphtalene following a stacking spectral method and fitting
four velocity components with different sizes. The total column density of
1-cyano and 2-cyano naphtalene is 7.35$\times$1011 cm-2 and 7.04$\times$1011
cm-2 respectively. Hence, the 3$\sigma$ upper limits to the abundance ratio of
ethynyl over cyano derivatives of naphtalene are 13.6 and 9.7 for the
positions 1 and 2 of the derivatives. The significant lower dipole moments of
the ethynyl derivatives compared to the cyano ones could explain that only
upper limits are obtained. By stacking the non-blended lines of 1- and 2-ETN,
these upper limits could be improved by a factor of 2 giving values for the
abundance ratios similar to those found for the ethynyl and cyano derivates of
cyclopentadiene, $\sim$7 (Cernicharo et al., 2021c). We note, however, that
for smaller hydrocarbons such as C2H4 the CH2CHCCH/CH2CHCN abundance ratio is
$\sim$2 (Cernicharo et al., 2021d). Future versions of the QUIJOTE line
survey, together with additional laboratory data at higher frequencies for the
ethynyl derivatives, could certainly improve the present results and, perhaps,
allow the detection of both isomers.

##  4 Conclusions

In this work, we have investigated the pure rotational spectra of the two
isomers of the ethynyl derivative of naphthalene. The measurements have been
carried out in the 2-8 GHz frequency region using a high-resolution broadband
spectrometer. From the analysis of the rotational spectra we obtained accurate
rotational parameters for both isomers which allowed us to obtain frequency
predictions in the Q-band frequency range and search for them using the
QUIJOTE line survey. Although they were not detected towards TMC-1 we obtained
upper limits for both species.

## Acknowledgements

C. C. and J. C. thank European Research Council for funding support under
Synergy Grant ERC-2013-SyG, G.A. 610256 (NANOCOSMOS) and Ministerio de Ciencia
e Innovación through the PID2019–107115GB–C21 project. I.P acknowledges
Ministerio de Ciencia e Innovación (Grant PID2020–117925GA–I00) and Junta de
Castilla y León (Grant INFRARED-FEDER IR2020-1-UVa02). We thank the referee
for a careful reading of the manuscript and for useful comments.

C.C. conceived the study, performed the analysis of the rotational
spectroscopy experiments, and wrote the original draft. I.P is responsible for
the resources and funding and carried out all the laboratory experiments. J.C.
performed the analysis of all TMC-1 surveys and carried out the astronomical
search of the observed molecules. He is also responsible for the resources and
funding for this investigation. All the authors reviewed and commented the
manuscript.

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding authors.

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