\section{Introduction}

Rydberg atom based electrometry is a field that has gained considerable attention in the past decade~\cite{first_ryd_elet_shaff, gor1, holl1, holl2, Fan_2015}. This is primarily due to the large polarizabilty offered by highly excited atoms~\cite{gallagher_book} and their ability to act as a calibrated standard for various measurements~\cite{holl1, si_trace,pow_stand}. For this reason, Rydberg states are used for greatly enhanced field sensing in communications~\cite{Song:19,9054945,8778739,dual_species,waveguide_SA,rydberg_array,comm_sense_fancher,AM_FM_anderson}, power measurements~\cite{pow_stand}, temperature sensing~\cite{Norrgard_2021,Fan_2015,9054945}, and imaging~\cite{terehertz_imaging}. In addition to the enhanced sensing offered by Rydberg atoms, the electric field measurements are traceable to the international system of standards (SI)~\cite{SI_standard} through Planks constant~\cite{gor1, holl1, pow_stand}.

\begin{figure}[h!]
    \centering
\scalebox{.22}{\includegraphics*{repump_effect.pdf}}
\caption{(a) Level diagram of EIT coupling 5S state to the 50D Rydberg state through 5P$_{3/2}$ intermediate state: repumping from unused 5S$_{1/2}$,F=2 state to the 5P$_{1/2}$,F=3 through 5P$_{1/2}$ state in $^{85}$RB.
(b) 780 laser transmission plotted against the coupling laser detuning.}
    \label{fig:repump_effect_sample}
\end{figure}

Many Rydberg electrometry techniques rely on electromagnetically induced transparency (EIT) to probe the interaction between the atoms and the fields of interest ranging from DC-THz. EIT is a nonlinear process that couples two states that have a dipole-forbidden transition with a two-photon resonance utilizing an intermediate state, as shown in Fig.~\ref{fig:repump_effect_sample} (a). EIT results in a narrow resonance feature that is used to probe highly excited states with increased accuracy and precision. With this narrow feature, we can probe the energy level shifts of Rydberg states due to various field atom interactions like the DC Stark shift~\cite{ma2021measurement,plate_cell_sandia}, AC Stark shift~\cite{r12, r13, davecon,PhysRevLett.98.203005}, and Autler-Townes (AT) splitting~\cite{first_ryd_elet_shaff, gor1, holl1, pow_stand,si_trace}, thereby characterizing the field of interest. While this EIT resonance is narrow, it is weak and Doppler limited. Experimental techniques have yet to achieve the expected natural decay width~\cite{si_trace,freq_det,dual_species,pow_stand,9054945,Fan_2015}, ultimately limiting the sensitivity of these Rydberg-based techniques. However, through the use of the AT splitting and a RF local oscillator field, the frequency splitting measurement can be converted into a an modulated amplitude measurement. With a lock-in amplifier, this method results in three orders of magnitude improvement and a field sensitivity of 5.5 $\mu$Vm$^{-1}$Hz$^{-1/2}$ ~\cite{gor3,Jing2020}.

There are several aspects which limit us to this sensitivity in the atom-based mixer approach. The primary effect comes down to the doppler effect and the residual doppler shifts due to the frequency mismatch of the probe (780 nm) and coupling (480 nm) lasers. Because of this, some of the key parameters used to tune the sensitivity of Rydberg sensors are coupled. In order to enhance the amplitude of the EIT peak, the power of the lasers must be increased. However this lends itself to power broadening that ultimately limits the fundamental sensitivity. Alternatively, we can increase the number of atoms in the interaction by increasing the temperature or optical power. However, this increases the doppler broadening and increases the residual doppler mismatch of the probe and coupling fields. In addition to this, the competing effects of increased number of atoms and their collision with the Rydberg atoms cause decoherence and can weaken and broaden the EIT line further~\cite{Fan_2015}.

In this manuscript, we use a ground state repumping to increase the number of Rydberg atoms in the EIT interaction without doppler-, power-, or collisional-broadening from other methods.
Such a method has been used to increase the frequency conversion efficiency for optical beams~\cite{Prajapati:18,Akulshin_2011}, showing an over a factor of 5 increase in conversion efficiency. With a ground state repumping laser, shown by orange ray in Fig.~\ref{fig:repump_effect_sample} (a), we excited atoms from the 5S$_{1/2}$, F=2 state to the 5P$_{1/2}$ state which then decay to the 5S$_{1/2}$, F=3 state. This effectively doubles the number of atoms in the EIT interaction without changing the velocity group of the atoms. This is possible since the rubidium atoms in the interaction region are naturally split between the two hyperfine ground states (5S$_{1/2}$, F=2 and F=3) of the alkali atom. With the repump we are simply incorporating these atoms into the interaction.
Additionally, when optical fields are used to control the populations of the excited states, decays to the non-interacting ground state become more common. These atoms are then trapped in this state due to the dipole selection rules and are naturally reintroduced very slowly through collisional effects. However, by reintroducing these atoms into the interaction with a repump laser, we can add atoms into the interaction and increase the strength of the interaction, shown by red trace in Fig.~\ref{fig:repump_effect_sample} (b). Additionally, these "new" atoms exist in the same doppler profile and do not contribute to a broadening of the EIT width.

In this demonstration, we use Rydberg atoms to measure the field strengths of weak radio frequency (RF) fields. When such an alternating field is resonant with atomic transitions, the energy levels experience a splitting proportional to the field strength and the dipole moment of the transition. The AT splitting becomes more sensitive for higher lying Rydberg states, but is limited for a fixed state by the amplitude and width of the EIT peak. These factors are determined by the power of the lasers, density of the atoms, and the doppler mismatch of the lasers being used. By using a repump field, we can enhance the EIT amplitude line beyond what is possible by only optimizing these parameters.

\begin{figure}[h!]
    \centering
    \scalebox{.22}{\includegraphics*{setup_eps.pdf}}
    \caption{
    Experimental setup: the 795~nm laser is  used to
    pump atoms into the hyperfine state of interest.
The acronyms are polarizing beam displacer (PBD) and polarizing beam  splitter (PBS).}

    \label{fig:setup}
\end{figure}

To observe the effects of the repump field, we look at the alterations to the EIT line due to the repump laser. The experimental apparatus is shown in Fig.~\ref{fig:setup}.
We generate EIT with a 780 nm external cavity diode laser (ECDL) as our probe and a 480 nm laser derived from second harmonic generation of a 960 nm ECDL as our coupling beam. Both lasers are collimated and pass through the rubidium vapor cell in counter-propagating directions to account for the Doppler shift in the ladder configuration. The 795 nm repump optical field is also generated by an ECDL and combined to be co-propagating with the blue coupling laser. While the 795 nm optical field does not introduce wave vector-dependent Doppler effects, we want to avoid the light leakage into the photo-detector (PD) that would add noise. To assure that laser noise is limited, we use balanced detection to cancel common mode noise (discussed further later). The beam widths for the probe and coupling optical fields at the location of the cell are $\approx$2~mm. The large beams were required to counter the power broadening effects while maximizing the number of atoms in the interaction. This was done to maximize the EIT amplitude while maintaining a small width. While even larger beams might show improved results, we were limited by the laser power available in the coupling laser. This limitation arises from the small dipole transition element for the energy levels connected by the coupling laser, thereby increasing the required laser power to achieve the same Rabi frequency. The repump field was diverging entering the cell with a 2.5 $\mu$m width.

For this demonstration, we tune the 780 nm probe laser to the $5S_{1/2},F=3\rightarrow5P_{3/2},F'=4$ transition and the 480 nm coulping laser to the $5P_{3/2},F'=4\rightarrow50D_{5/2}$ transition. Since this is a two-photon interaction, the coupling laser scan will probe the Rydberg states of interest. When resonant with a Rydberg state, the probe will experience a transmission to readout the Rydberg state and any effects on it, as shown in Fig.~\ref{fig:repump_effect_sample} (b). Additionally, we calibrate our coupling laser scan by taking advantage of the hyperfine structure present the traces in Fig.~\ref{fig:repump_effect_sample} (b). This also allows us to accurately determine the width of the EIT peak and the separation of the peak when measuring the AT splitting. Therefore, we obtain the EIT amplitude and width through a fit of the traces in Fig.~\ref{fig:repump_effect_sample} to gauge the effects of the repump.

The ultimate sensitivity is limited by how well we can determine the AT splitting due to the RF field. The factors determining sensitivity are the EIT amplitude, EIT width, and noise on the measurement. Common methods to increasing EIT signal strength include tuning the beam size, powers, and atomic density, but these have their limitations. For instance, an increase in the probe or coupling optical power will saturate and show only modest improvement.

Figs.~\ref{fig:repump_probe_dependance} (a) and (b) show the effects of the probe power and coupling power, respectively on the EIT amplitude (solid) and widths (dashed) for cases where the repump is present (red) and is not present (black). It can be seen that increasing the probe field power has immediate power broadening accompanied with the amplitude increase, shown by dashed black trace in Fig.~\ref{fig:repump_probe_dependance} (a). This broadening is detrimental when trying to identify the AT splitting. Here, the costs outweigh the benefits. However, with the introduction of the repump field, we can immediately see the EIT amplitude and saturation level increase, shown by solid red trace in Fig.~\ref{fig:repump_probe_dependance}. At 300 $\mathrm{\mu}$W of probe power, we nearly a double the EIT amplitude due to the repump field, but the width of the peak does not change much, shown by dashed traces in Fig.~\ref{fig:repump_probe_dependance}(a). In addition to this, no increase in probe power will reach the same EIT amplitude attained by introducing the repump field due to saturation, solid traces in Fig.~\ref{fig:repump_probe_dependance}(a).

While the effects of the repump scale nonlinearly with the probe power, the effects are linear with the coupling field power, solid traces in Fig.~\ref{fig:repump_probe_dependance}(b). Also, unlike the case of the probe power, the saturation occurs at the same coupling field power.  This effect is likely due to limited population change from the coupling field, whereas the probe field has direct consequences on the hyperfine split ground states. The effects of the repump field on the EIT amplitude provide over a factor of 2 increase in the amplitude while showing no increase in the width of the EIT resonance for any given probe and coupling field powers. Additionally, repumping requires only a modest amount of power. Even though the repump beam has a larger width than the probe and coupling beams, it saturates the pumping transition near the same power as the probe.
For the remainder of this manuscript, we use a probe field optical power of 550$~\mu\mathrm{W}$, a coupling power of 72$~\mathrm{mW}$, and a repump power of 10$~\mathrm{mW}$ measured before entering the cell. These powers offer roughly a factor of 1.5 increase on the EIT amplitude while not substantially increasing the peak width. While the large repump power is not necessary, the strong saturation helps avoid repump laser intensity fluctuations from being transferred to the probe.

\begin{figure}[h!]
    \centering
    \scalebox{.28}{\includegraphics*{power_dependance.pdf}}
    \caption{a) EIT amplitude (solid) and EIT half-width-half-max (HWHM) (dashed) for different probe powers with (red) and without (black) the repump field, coupling fixed to 40 mW. (b) Same as (a), but plotted against coupling field power with probe power fixed to 400 $\mu$W.}
    \label{fig:repump_probe_dependance}
\end{figure}

The most sensitive RF field detection techniques rely on the concept of the atom-based field mixer~\cite{doi:10.1063/1.5088821,8878963,gor3, Jing2020}. An RF field (say a local oscillator--LO) causes AT splitting, shown by black trace in Fig.~\ref{fig:beat_sample}(a). Then a second RF field interferes with the LO resulting in interference, shown by red trace in Fig.~\ref{fig:beat_sample}(a). While the fields sum in free space, atoms only perceive the field amplitude, thereby mixing the signals and observing the beat-note as a modulation on the AT splitting. For an optimum LO field, the modulation of the AT splitting translates to an amplitude modulation of the EIT peak, shown by Figs.~\ref{fig:beat_sample}(a) and (b). By locking to the EIT peak and measuring the amplitude of this beat-note through a lock-in amplifier, weak signal fields can be detected. Field sensitivity down to $5\mathrm{\mu V(m\sqrt{Hz})^{-1}}$ has been shown with this technique\cite{gor3,Jing2020}.

\begin{figure}[h!]
    \centering
        \scalebox{.28}{\includegraphics*{mixer_detection.PNG}}
    \caption{(a) AT splitting of EIT by LO horn (black trace) and AT splitting amplitude modulated by Sig and LO field (red trace)

    (b) EIT signal with probe and coupling laser locked to resonance showing beat-note from Sig and LO RF fields separated by 10 kHz with (red) and without (black) repump.}

    \label{fig:beat_sample}
\end{figure}

By introducing a repump field, we can effectively double the strength of the beat-note, shown by red trace in Fig.~\ref{fig:beat_sample} (b). Ideally, this enhancement should increase our sensitivity. We use a lock-in amplifier to detect the strength of the beat-note signal. For this demonstration, we used a LO resonant with the 50$D_{5/2}\rightarrow51P_{3/2}$ Rydberg states that corresponds to a frequency of 17.04340 GHz. The LO was generated using a signal generator (SG) set to an output RF power of -14 dBm.

The AT splitting provides an immutable  measurement of the E-field traceable to the International System of standards (SI) through Plank's constant through
\begin{equation}
    |E| = \frac{h\cdot\Delta_{AT}}{\wp},
\label{eq:at_effect}
\end{equation}
where $\Delta_{AT}$ is the AT splitting, $\wp$ is the transition dipole moment, $h$ is Plank's constant, and E is the electric field strength.  Using Eq.~(\ref{eq:at_effect}), we determined that this corresponds to the LO field strength of 330 mV/m. We generated our signal field with a separate SG set to 17.04341 GHz and controlled the RF output power to adjust the field strength at the cell. The LO and signal were summed using a power divider and fed into a horn antenna with a gain of 17.5 dB for this band.

However, to avoid misrepresenting the propagation loss and effects from the glass, we calibrated the SG's using the AT splitting and Eq.~(\ref{eq:at_effect}).  These measurements cannot determine the actual field down to the field strengths of interest. However, we can determine the field at the cell location for larger RF powers and extrapolate to determine the expected field strength for the lower SG output powers, see \cite{gor3} for details. As a precaution, we also verify the linearity of the SG's down to the lower powers with a calibrated power meter and spectrum analyzer.

With this calibration, we take measurements of low output power of the SG and map the sensitivity of our sensor. We input the signal from the photodetector (Fig.~\ref{fig:beat_sample} (b)) into a lock-in amplifier which outputs the beat-note amplitude as a voltage. We use a 1 s time constant and 10 mV/A gain on the lock-in amplifier when collecting data. We took five sets of data while varying the output RF power of the signal SG (ie. signal RF field strength) for the cases with (red trace) and without (black trace) the repump field present, shown in Fig.~\ref{fig:cal} (a) and (b). The data is the average of five data sets, and the error bars are the standard deviation. The effect of the repump field is immediately apparent for large signal field strengths. The lock-in signal is 50\% larger for the trace with the repump optical field present. While we have shown that we can double the EIT peak, we limit ourselves to a probe power of 550 $\mu$W to limit the broadening effect that outweigh the effects of the gain in amplitude. The coupling laser power is 70 mW since increasing its power does not cause significant broadening.

\begin{figure}[h!]
    \centering
    \includegraphics[width=.9\columnwidth]{field_strength.pdf}
    \caption{(a) beat-note amplitude plotted against field strength for with (red) and without (black) repump field. (b) Same plot, but zoom depicted by dashed box. The dashed lines show the respective detection limits.}
    \label{fig:cal}
\end{figure}

As we apply smaller fields, we reach the detection noise floor and see that the minimum field detectable without the repump field present is $\approx$ 5 $\mathrm{\mu Vm^{-1}}$, shown by black trace in Fig.~\ref{fig:cal} (b). For the case of the data with the repump field present, the minimum field detectable is $\approx$ 3 $\mu$Vm$^{-1}$, shown by black trace in Fig.~\ref{fig:cal} (b). Since the collection time for this data is 1 s, the sensitivity for these two measurements is the same as the minimum detectable field strength. This increase in sensitivity is close to the level of gain attained on the EIT amplitude from the repump field. However, the noise floor seems to be higher when the repump is present. This noise is likely from the laser noise projection through repumping onto the probe optical field. Since the repump optical field directly alters the probe field absorption through population transfer, the fluctuations are also strongly coupled. Therefore any frequency shifts of the repump field play strongly play a role in the amplitude of the EIT. However, we are free from intensity fluctuations of the repump since the transition is saturated.

We utilized balanced differential detection to remove any classical noise as effectively as possible. This includes the noise introduced by the repump field. For this purpose, the repump field was split into two arms that overlapped with the signal probe and the reference probe. After analysis of the noise floor, we find that the limit in the detection is likely the probe laser noise composed of 1/f noise and photon shot noise. We compare different sources of noise measured by a spectrum analyzer in Fig.~\ref{fig:noise_comp}. The yellow trace shows the noise form the differential detection of the probe laser, in other words the shot noise limit. The detector noise (green) and SA noise (red) are several decades in power lower than the shot noise and will have no effect to the measurement noise floor. Additionally, in the presence of the blue laser and the mixer detection method (blue trace), we observe no apparent shift in the noise floor. Therefore, we can be certain the detection floor is the laser noise of the probe field. Also present in the blue trace is the expected beat-note peak at 10 kHz detection frequency.

For a more quantitative analysis, we calculate the expected noise power for the optical probe power. The photo-current variance for shot noise limited light is given in ~\cite{Fox:1001868} by
\begin{equation}
    (\Delta i)^2 = \frac{2\cdot \eta_{qe} \cdot P\cdot \lambda\cdot f_{BW}}{h\cdot c},
\end{equation}
where $h$ is Plank's constant, $c$ is speed of light, $\lambda$ is wavelength, $P$ is optical power, $f_{BW}$ is bandwidth, and $\eta_{qe}$ is the quantum efficiency of the detector. Since we are performing balanced detection with 70\% quantum efficient photo-diodes, being shot noise limited is a good assumption. The expected noise power ($P_{noise}$) on the SA in dBm is then
\begin{equation}
    P_{noise} = 10\cdot log_{10}(\frac{(G^2\cdot (\Delta i)^2}{\Omega}\cdot1000),
\end{equation}
where $G$ is PD gain, and $\Omega$ is load resistance of the spectrum analyzer. Accounting for resolution bandwidth (100 Hz), gain ($626\cdot10^3$ V/A), and resistance (100$~\Omega$), we can estimate the noise power on the spectrum analyzer. The probe power before the cell was $560~\mu W$, but actual power reaching the detector was 220 $\mu$W. The expected noise power from this is then -76 dBm. The yellow and blue traces in Fig.~\ref{fig:noise_comp} fall to this level at detection frequencies above 300 kHz. This establishes shot noise as a clear detection limit in the near future. As for the lower frequencies, we believe the dominating noise is 1/f laser noise since it is present in probe laser noise. While outside the scope of this study, there have been several studies that have demonstrated cancellation of 1/f noise down to frequencies less than 200 Hz and reach the shot noise floor~\cite{Prajapati:19,8426633}. The shot noise marks a stark quantum limit for improvement of this type of sensor. However, non-classical sources of light may provide further improvement. In ~\cite{Prajapati:21}, twin-beams were used to probe an excited atomic state. While this study explored an off-resonant effect, there are forms of squeezing that incorporated resonant enhancement.

\begin{figure}[h!]
    \centering
    \includegraphics[width=.7\columnwidth]{final_relative_noise2.jpg}
    \caption{Noise power for atom-mixer method (blue), differential probe detection (yellow), PD dark noise (green), and SA dark noise (red). SA settings: RBW = 100 Hz, VBW = 100 Hz, and 100 averages.

    }

    \label{fig:noise_comp}
\end{figure}

In this manuscript we have demonstrated the enhancement of Rydberg sensing through the use of a repumping field. We showed that the repump increases the saturation threshold, and amplifies without increasing EIT linewidth. Additionally, this is done with modest repump power. By increasing the EIT amplitude, we managed to double the signal response to a signal RF field. The current limitations in this technology are from the width and height of the amplitude of the EIT. This broadening free enhancement marks progress towards better Rydberg-based technology. While we demonstrated increased sensitivity, we also identified another fundamental bound that must be surpassed. We identified the noises in the system and determined the noise floor to be the 1/f laser noise for lower frequencies and laser shot noise for higher frequencies. Future applications of quantum enhanced light will play an important role in the development of these sensors.

\section*{Acknowledgements}
We thank Dr. Svenja A. Knappe of Fieldline Inc in Boulder, CO for the use of some equipment used here. We also Dr. Eric Norrgard of NIST and Dr. Eugeniy Mikhailov of William and Mary for their useful technical discussions. This work was partially funded by the DARPA SAVaNT program and the NIST-on-a-Chip Program.

\section*{Data Availability Statement}
Data is available upon request.

\newpage
\bibliographystyle{aipnum4-1.bst}


\section{\label{Introduction}Introduction}
\textit{Introduction:}
{The entangled states in hybrid quantum systems offer a promising platform to study various quantum mechanical phenomena ranging from microscopic to macroscopic scales. A few of the leading fields are high precision measurements in quantum metrology \cite{Quantum_metrology_sensitivity_entanglement}, quantum information processing \cite{entanglement_quantum_cryptography, entanglement_quantum_communication, entanglement_swapping_two_mechanical_Vitali, Quantum_entanglement, Quantum_Computation_Trapped_Ions}, noise suppression and advancement in gravitational-wave (GW) detection \cite{LIGO_sensitivity_squeezing, study4roadmap, Danilishin_GW_2019,  Polzik_GW_Spin_PRD, study4calculation} and others. One of the important features of such systems is that they also provide a testbed to study a transition from classical to quantum behavior \cite{classical_quantum_boundary}, along with significant technology applications. One of those relates to GW problematic addressing the sensitivity enhancement of advanced GW detectors. An important problem herein is the inability to suppress at the same time the two main noise sources, namely radiation pressure noise (RPN) and shot noise. One can be suppressed only at the cost of another by, for example, using squeezed light  \cite{Danilishin_GW_2019}. The hybrid nature of the above-mentioned system provides a playground to study novel methods for suppressing RPN. In this aspect, a promising scheme has recently been proposed by introducing an auxiliary atomic spin system with a negative effective mass, which was supposed to offer a back-action evading (BAE) measurement, improving the GW detector sensitivity in relevant spectral ranges \cite{Danilishin_GW_2019, study4roadmap, Polzik_GW_Spin_PRD, study4calculation}. In this BAE measurement scheme,  for transferring the input squeezed state to the output beam, the entanglement between mechanical and spin systems plays a crucial role. However, we demonstrate in this letter that the entanglement criteria like Duan bound and logarithmic negativity limit the choice of parameters for a realistic detection scenario.
The investigation of the transfer of squeezing may be divided into two parts namely, the transfer of squeezing through the entanglement between subsystems of the hybrid system, followed by a transfer of squeezing to the output beams. Here we are focusing on the first part i.e., investigating the entanglement between all possible subsystems quantitatively under two mentioned criteria namely, the negativity and the Duan bound to find out the operational parameter space for the proposed scheme. Recently, an entangled state has been generated experimentally between a macroscopic mechanical oscillator and a collective atomic spin oscillator \cite{Entanglement_distant_macroscopic_spin},  where the output of the spin system is directly used as the input of the optomechanical system. Though this can be seen as a step for exploring entanglement between subsystems of the hybrid system, it differs from the originally suggested setup in the BAE scheme for GW detection. In a realistic scenario, it is important to find out the operational parameter space required for entanglement. However, the situation is more complex. In this article, we, therefore, study the entanglement between subsystems of the hybrid system which has been proposed in \cite{Danilishin_GW_2019, study4roadmap, Polzik_GW_Spin_PRD}. }

In particular, even though it is witnessed that the collective EPR-variance of the spin and mechanical modes are obtainable below the separability limit for the system mentioned in \cite{Entanglement_distant_macroscopic_spin}, however, importantly, we could not find any possibility of non-separability in the case of the system mentioned in refs. \cite{Danilishin_GW_2019, study4roadmap, Polzik_GW_Spin_PRD}, and which we are interested in.

{It has been supposed that satisfying Duan bound criterion automatically means that the spin system and the mechanical oscillator are entangled \cite{Duan_Inseparability_entanglement}. According to this criterion, for all Gaussian two-party continuous variable (CV) states there exist collective EPR-type of quadratures $X_\Sigma = c_1 X_1 + c_2 X_2$ and $Y_\Delta = c_1 Y_1 - c_2 Y_2$ whose variances violates the Duan inequality. Here, $X_{n}$ and $Y_{n}$ are the quadrature operators
that fulfil the canonical commutation relation $[X_n, Y_m] = i \delta_{n,m}$, where the index $n,m \in (1,2)$ for the two systems. The Duan inequality for such a system is \cite{Duan_Inseparability_entanglement},}

\begin{equation}\label{duan_ineq_initial}
\langle (\Delta X_\Sigma)^2 \rangle + \langle \Delta (Y_\Delta)^2 \rangle \geq 1.
\end{equation}

$\langle \Delta (X_\Sigma)^2 \rangle = \frac{1}{2}\int \frac{\mathrm{d}\omega}{2\pi} \langle \{ X_{\Sigma,\omega}, X_{\Sigma,-\omega} \} \rangle$ and analogously $\langle \Delta (Y_\Delta)^2 \rangle$, $c_1$ and $c_2$ are system determined nonzero real number.

In this letter, we start with
a brief description of the hybrid system, followed by the method for calculating the entanglement between the cavity and spin modes by transferring the input TMSV into the system including an analysis of the role played by light-matter interactions. Hereafter, we provide a detailed study of the spin-cavity entanglement with the variation of the parameters of the hybrid system. The investigation ends up with the search of the possibility of entanglement between mechanical and spin modes, along with justification on this aspect.

\begin{figure}[t]
\includegraphics[width=8 cm]{block_GW_diagram.jpg
\caption{\label{block_GW_diagram}Schematic block diagram of the BAE measurement setup }
\end{figure}

\begin{figure}[b]
\includegraphics[width=8 cm]{cav_spin1.jpg
\caption{\label{cav_spin} Logarithmic negativity $(E_N)$ or entanglement between the atomic spin and optomechanical cavity vs input squeezing $r$ in the presence and absence of the interaction to mechanical motion for a fixed parameter  $\Delta_0 = 60 \kappa_0, \omega_{m0}= 60 \kappa_0, \omega_{s0}= -60 \kappa_0, \gamma_{m0}= \kappa_0, \gamma_{s0}= \kappa_0, \Gamma_{S0} = 25.14\kappa_0, n_m=0.8, n_s=0.5,  \eta_I=\eta_S=1$. The dashed lines represent corresponding Duan quantity and the dashed black line indicates the limit below which the inequality violets and makes systems non-separable.   Inset: Entanglement between modes of input TMSV. }
\end{figure}

\begin{figure*}[ht]
\includegraphics[width=15cm,trim=2cm 0cm 2cm 0cm   ]{ENvsALL.jpg
\caption{\label{ENvsALL}Logarithmic negativity $(E_N)$ or entanglement between the atomic spin and optomechanical cavity vs input squeezing $r$ with the variation of parameters  ( I(a) cavity frequency $\Delta$ and I(b) cavity linewidth $\kappa$, II(a) spin Larmor frequency $\omega_s $ and II(b) linewidth $\gamma_s $, III(a) mechanical oscillation frequency $\omega_m $ and III(b) linewidth $\gamma_m $,  IV(a) optomechanical coupling strength $G$ and  IV(b) spin readout rate $\Gamma_S $, thermal excitation of the V(a) mechanical $(n_m)$ and V(b) spin $(n_s)$ reservoir, and input channel efficiency of the VI(a) cavity $(\eta_I)$ and VI(b) spin $(\eta_s)$ ). All other parameters remain same with Fig. \ref{cav_spin}. }
\end{figure*}

\begin{figure}[ht]
\includegraphics[width=8cm]{duanvsGGamma.jpg
\caption{\label{duanvsGGamma} Plot of the Duan quantity between the atomic spin and mechanics with the increment of the squeezing factor at temperature $T=0$ for both the reservoirs spin and mechanics. Inset: Entanglement between cavity and mechanics. All other parameters remain the same with Fig. \ref{cav_spin}.}
\end{figure}

\textit{Model: }The hybrid scheme of GW detector consists of an optomechanical system located at a distance from its counterpart which is an auxiliary atomic spin ensemble \cite{Danilishin_GW_2019, study4roadmap, Polzik_GW_Spin_PRD}, as demonstrated in Fig. \ref{block_GW_diagram}.
Both the mechanics and optical cavity continuously interact to their corresponding bosonic thermal bath which is Markovian in nature, typically approximated to have a flat and infinite spectral density \cite{Cavity_optomechanics_Aspelmeyer}.
The  Hamiltonian of the optomechanical system  is \cite{study4calculation}

\begin{align}\label{optomechanical_Hamiltonian}
\hat{H}_M/\hbar = \omega_c a^\dagger a +\frac{\omega_m}{2} (q_m^2+p_m^2) - g_0 a^\dagger a  q_m q_{zpf} \nonumber \\
  + i \left(a^\dagger E e^{-i\omega_l t} -h.c. \right) ,
\end{align}

where $|E|$ is the strength of the external drive with frequency $\omega_l$,
 $a(a^\dagger)$ are the annihilation(creation) operators of the cavity modes, and $q_m$ and $p_m$ are dimensionless position and momentum operators of the mechanical mode, follows the canonical commutation relation $[q_m,p_m]=i$. $\omega_c$ and $\omega_m$ are the resonance frequencies of the cavity and the mechanics, respectively; determining radiation pressure coupling energy $(g_0=\frac{\omega_c}{L})$ and the zero-point fluctuation  $(q_{zpf} = \sqrt{\frac{\hbar}{m\omega_m} })$, respectively. Here, $L, m$ and $\hbar$ are length of the cavity, mass the mechanical resonator and Plank's constant, respectively.
The rates of dissipation due to continuous interaction with the corresponding reservoir are $\kappa$ and $\gamma_m$ for the cavity and mechanics, respectively, bringing stability to the system
.
The external drive and the cavity parameters altogether determines steady state cavity field amplitude $(\alpha_s)$ which in turn tunes the optomechanical coupling strength $(G=  \sqrt{2} g_0 |\alpha_s| q_{zpf})$ in linearised regime. In order to make the Hamiltonian time-independent, the effective cavity detuning frequency expressed in the rotating frame of the laser is $\Delta = \omega_l -\omega_c + g_0 q_s $ where $q_s = \frac{g_0}{\omega_m} |\alpha_s|^2 q_{zpf}^2  $ is the steady-state displacement of the mechanical oscillator.

In addition to this, the spins of the auxiliary atomic ensemble are polarised initially under the influence of a magnetic field oriented along a direction (say z), and therefore, the collective spin exhibits an average projection $|\langle J_z \rangle|>> \hbar$.
Anticipating that, considering the atomic spin mainly reside in their excited state and using the Holstein-Primakoff (HP)
transformation \cite{HP_transform, nandi_pra},
we map the spin operators in terms of bosonic modes: $X_s = J_x/\sqrt{\hbar \langle J_z \rangle }, Y_s = J_y/\sqrt{\hbar \langle J_z \rangle}$, which in turn gives the Hamiltonian

\begin{equation}\label{spin_Hamiltonian}
\hat{H}_S/\hbar =  \omega_s \langle J_z \rangle - \frac{ \omega_s}{2}  (X_s^2 + Y_s^2) ,
\end{equation}
where $\omega_s$ is the Larmor frequency with linewidth broadening $\gamma_s$ causes due to continuous interaction to its reservoir. the first term is contributed by the mean spin polarization, therefore it turns out to be an irrelevant energy offset. The second term is equivalent to the mechanical oscillator expressed in Eq. \eqref{optomechanical_Hamiltonian}, with a negative mass. In the limit of large detuning and low saturation, the interaction Hamiltonian between the light polarization and the atomic spin is
$\hat{H}_{LS}/\hbar = \alpha S_x J_x$, where  $S_x$ is the Stokes component of light, measures the degree of circular polarization and $\alpha$ is the coupling strength that depends on laser detuning frequency and the area of interaction.  Using HP transformation, this can be mapped to $\hat{H}_{LS}/\hbar = \sqrt{\Gamma_S} X_s X_{in}^S$   where $\Gamma_S = \frac{\alpha^2}{2} \Phi|\langle J_z \rangle|$ is the spin readout rate for a given  photon flux $\Phi$ \cite{spin_readout}. The negative mass spin oscillator ensures that each quantum of excitation corresponds to a deexcitation from the highest energy levels of the inverted spin population. Therefore, the anti-Stokes interaction with the incoming laser becomes predominant over the Stokes interaction.

The distant optomechanical and spin systems are feed by a TMSV as an input signal which is generated by non-linear optical crystal through the parametric down-conversion process (PDC), satisfying

\begin{subequations}
\begin{align}\label{TMSV_field}
	\begin{bmatrix}
		a^I_{in}\\
		{a^S_{in} }^\dagger
	\end{bmatrix} &= \begin{bmatrix}
		\cosh r & \sinh r\\
		\sinh r & \cosh r
	\end{bmatrix} \begin{bmatrix}
	a^I_{z}\\
	{a^S_{z} }^\dagger
\end{bmatrix} \\
	\begin{bmatrix}
	a^S_{in}\\
	{a^I_{in} }^\dagger
\end{bmatrix} &= \begin{bmatrix}
	\cosh r & \sinh r\\
	\sinh r & \cosh r
\end{bmatrix} \begin{bmatrix}
	a^S_{z}\\
	{a^I_{z} }^\dagger
\end{bmatrix}
\end{align}
\end{subequations}

where $r$ is the squeezing factor, $a^I_{z} ({a^I_{z} }^\dagger)$ and $a^S_{z} ({a^S_{z} }^\dagger)$ are the annihilation (creation) operators of the two independent vacuum fields, and $a^I_{in}, a^S_{in}$ are the input fields to the optomechanical and spin systems respectively.

\begin{comment}
\begin{align*}
	\begin{bmatrix}
		X^I_{in}\\
		X^S_{in}
	\end{bmatrix} &= \begin{bmatrix}
	\cosh r & \sinh r\\
	\sinh r & \cosh r
\end{bmatrix} \begin{bmatrix}
		X^I_{z}\\
		X^S_{z}
	\end{bmatrix} \\
	\begin{bmatrix}
	Y^I_{in}\\
	Y^S_{in}
\end{bmatrix} &= \begin{bmatrix}
	\cosh r & -\sinh r\\
	-\sinh r & \cosh r
\end{bmatrix} \begin{bmatrix}
	Y^I_{z}\\
	Y^S_{z}
\end{bmatrix}
\end{align*}
\end{comment}

\textit{Spin-Cavity Entanglement: }
The proposed scheme for the BAE measurement imposes a necessary condition which fixes both the susceptibility equal $(\chi_m = \chi_s)$, where  $\chi_m (\omega) \approx \frac{\omega_m}{\omega_m^2 -\omega^2 - 2i \omega \gamma_m}$ is the optomechanical susceptibility and $\chi_s (\omega) \approx \frac{\omega_s}{\omega_s^2 -\omega^2 - 2i \omega \gamma_s}$ for spin \cite{Danilishin_GW_2019, study4roadmap, Polzik_GW_Spin_PRD, study4calculation}. This implies the condition $\omega_{m0} = \omega_{s0}$ and $\gamma_{m0} = \gamma_{s0}$. Accepting the condition, we plotted the logarithmic negativity between the cavity and spin in Fig \ref{cav_spin},  for both the presence and absence of the interaction to the mechanics.
Even though the entanglement between the atomic spin system and the optical cavity is created by the transfer of the input TMSV to the system, unlike the entanglement of the input beam (inset of Fig \ref{cav_spin}), we see that the spin-cavity entanglement saturates to a finite value for highly squeezed input beams when no mechanics are involved.
The involvement of mechanics interrupts the coherent transfer of input TMSV into the system, causing a reduction of the entanglement. Further, the entanglement drops down to zero for a highly squeezed input signal due to more injection of photons into the cavity generates more quanta in the mechanical oscillator which in turn causes more disturbance in this process. The phenomenon can also be anticipated by looking at the Duan quantity plotted with the dashed line in  Fig \ref{cav_spin}, where we see the collective uncertainty of the EPR-like Duan operators obeys the inequality (Eq. \eqref{duan_ineq_initial})
 beyond the regime of entanglement
, making the systems separable \cite{Duan_Inseparability_entanglement}.

The entanglement, present only in a finite interval of $r$, has shown explicit dependency on the system parameters.
 Fig.  \ref{ENvsALL}-I(a) shows that, in a particular range of cavity detuning parameters at blue sideband regime $(\frac{\Delta_{0}}{\omega_{m0}}=1)$, the entanglement rapidly decreases when $\Delta$ moves away from $(\Delta_{0})$. Besides, it is also notable that the peak moves towards higher $r$ when $\Delta$ approaches to $(\Delta_{0})$. In the case of cavity linewidth, a similar pattern is observed where the entanglement becomes robust at $\kappa = \kappa_0$ and the pick moves to the higher value of $r$ (\ref{ENvsALL}-I(b)).  Following cavity parameters, the changes of entanglement exhibit a similar pattern while detuning the spin Larmor frequency (Fig. \ref{ENvsALL}-II(a)) and the spin bandwidth (Fig. \ref{ENvsALL}-II(b)). All these phenomenons are governed by the occurrence of the balanced transfer of input TMSV to the cavity and the spin system when the condition $\Delta = -\omega_{s}$ and $\kappa = \gamma_{s}$ satisfies, as highlighted in Fig 3 (I and II).  The entanglement moreover exhibits robustness with the variation of the frequency of the mechanical mode (Fig. \ref{ENvsALL}-III(a)), whereas it increases significantly with the increment mechanical bandwidth  (Fig. \ref{ENvsALL}-III(b))
 .
 A more interesting situation is depicted in Fig. \ref{ENvsALL}-IV where we study the robustness of entanglement while changing optomechanical and spin-laser coupling strength. We observe an overall increment of entanglement with the reduction of optomechanical coupling, especially in the case of the higher value of $r$ (Fig. \ref{ENvsALL}-IV(a)). Whereas, an opposite phenomenon is observed in the case of spin readout rate, shown in Fig. \ref{ENvsALL}-IV(b)
 where the higher readout rate enhances the entanglement transfer from the input state to the system. Fig. \ref{ENvsALL}-V ((a) and (b)) shows how the entanglement reduces with the temperature of both the reservoirs of mechanics and the spin system. Finally,  Fig. \ref{ENvsALL}-VI ((a) and (b)) shows the change of entanglement against the input quantum efficiency of the input path. Both figures exhibit a reduction of entanglement with the reduction of efficiency due to less delivery of the initial entanglement of the input TMSV into the system.

Note, importantly,  that the range of parameters, especially $G$ and $\Gamma_S$, used here are far different from the range of parameters used in articles \cite{Danilishin_GW_2019, study4roadmap, Polzik_GW_Spin_PRD}. The possibility of obtaining entanglement is almost negligible with the parameters used in those articles aiming to obtain squeezing in output modes

\textit{Spin-Mechanics Entanglement: }
Furthermore, we study the possibility of achieving entanglement between the spin and the mechanical mode in  Fig. \ref{duanvsGGamma}. The study reveals that, even in a situation where the entanglement is present in between the spin and optical the cavity, and the cavity and the mechanics (inset of Fig. \ref{duanvsGGamma}(I)
; the entanglement is still impossible to obtain between spin and the mechanics.
Fig. \ref{duanvsGGamma}(I) shows that the Duan quantity increases while increasing the strength of optomechanical coupling and always remains above the unit value satisfying the separability condition. On the other hand, a similar phenomenon is also observed while increasing the spin readout rate in Fig. \ref{duanvsGGamma}(II). In both cases the Duan quantity is lower bounded by unit value when no coupling is present, making the systems completely separable.

Note that we fix input quantum efficiency to be maximum and the temperature to be zero for the reservoirs of both mechanical oscillator and the spin system to obtain maximum entanglement, which therefore justifies the impossibility of obtaining entanglement in any range of those parameters.

\textit{Conclusion:}
We demonstrated under what condition entanglement can be generated between the distant spin system and the optical cavity introduced in \cite{Danilishin_GW_2019, study4roadmap, Polzik_GW_Spin_PRD}, by transferring input TMSV states which are generated through the PDC process. We see the entanglement becomes robust for weaker optomechanical interaction and stronger spin-optical interaction. The study shows that the entanglement is highly dependent on the system parameters, especially on the cavity resonance and the spin Larmor frequency and their linewidths. The mismatch of both frequencies and their linewidths causes a rapid reduction of entanglement. However, the entanglement shows robustness against the variation of the frequency and the linewidth broadening of the mechanical oscillator.
Even though the entanglement between the spin-cavity and the cavity-mechanics is prominent, it is still impossible to obtain entanglement between the spin and the mechanical motion, which has been justified by the Duan separability limitation. The scheme, proposed in  \cite{Danilishin_GW_2019,study4roadmap, Polzik_GW_Spin_PRD}, though looks quite promising, while considering from the perspective of practical implementation, our investigation raises some concern and one need to choose the parameter space wisely, to obtain effective BAE measurement.
Apart from sensitivity increment of GW detectors \cite{ Danilishin_GW_2019, study4roadmap, Polzik_GW_Spin_PRD, study4calculation}, this scheme provides a promising platform for technology applications, such as quantum teleportation \cite{entanglement_swapping_two_mechanical_Vitali}, cryptography \cite{entanglement_quantum_cryptography}, CV dense coding \cite{entanglement_quantum_communication} and quantum computation \cite{Quantum_entanglement, Quantum_Computation_Trapped_Ions}. In addition, the hybrid system provides a platform for studying the effect of decoherence induced quantum to classical transitions\cite{classical_quantum_boundary}. Such a study requires proper modeling of environment-induced decoherence and its effect on the dynamics of the entangled state of systems along with the existing parameter space. Therefore, we believe our findings of operational parameter space are of due importance in studying classical to quantum transition.

\nocite{*}

\section{}
\subsection{}
\subsubsection{}

\section{\label{Introduction}Introduction}
\textit{Introduction:}
{The entangled states in hybrid quantum systems offer a promising platform to study various quantum mechanical phenomena ranging from microscopic to macroscopic scales. A few of the leading fields are high precision measurements in quantum metrology \cite{Quantum_metrology_sensitivity_entanglement}, quantum information processing \cite{entanglement_quantum_cryptography, entanglement_quantum_communication, entanglement_swapping_two_mechanical_Vitali, Quantum_entanglement, Quantum_Computation_Trapped_Ions}, noise suppression and advancement in gravitational-wave (GW) detection \cite{LIGO_sensitivity_squeezing, study4roadmap, Danilishin_GW_2019,  Polzik_GW_Spin_PRD, study4calculation} and others. One of the important features of such systems is that they also provide a testbed to study a transition from classical to quantum behavior \cite{classical_quantum_boundary}, along with significant technology applications. One of those relates to GW problematic addressing the sensitivity enhancement of advanced GW detectors. An important problem herein is the inability to suppress at the same time the two main noise sources, namely radiation pressure noise (RPN) and shot noise. One can be suppressed only at the cost of another by, for example, using squeezed light  \cite{Danilishin_GW_2019}. The hybrid nature of the above-mentioned system provides a playground to study novel methods for suppressing RPN. In this aspect, a promising scheme has recently been proposed by introducing an auxiliary atomic spin system with a negative effective mass, which was supposed to offer a back-action evading (BAE) measurement, improving the GW detector sensitivity in relevant spectral ranges \cite{Danilishin_GW_2019, study4roadmap, Polzik_GW_Spin_PRD, study4calculation}. In this BAE measurement scheme,  for transferring the input squeezed state to the output beam, the entanglement between mechanical and spin systems plays a crucial role. However, we demonstrate in this letter that the entanglement criteria like Duan bound and logarithmic negativity limit the choice of parameters for a realistic detection scenario.
The investigation of the transfer of squeezing may be divided into two parts namely, the transfer of squeezing through the entanglement between subsystems of the hybrid system, followed by a transfer of squeezing to the output beams. Here we are focusing on the first part i.e., investigating the entanglement between all possible subsystems quantitatively under two mentioned criteria namely, the negativity and the Duan bound to find out the operational parameter space for the proposed scheme. Recently, an entangled state has been generated experimentally between a macroscopic mechanical oscillator and a collective atomic spin oscillator \cite{Entanglement_distant_macroscopic_spin},  where the output of the spin system is directly used as the input of the optomechanical system. Though this can be seen as a step for exploring entanglement between subsystems of the hybrid system, it differs from the originally suggested setup in the BAE scheme for GW detection. In a realistic scenario, it is important to find out the operational parameter space required for entanglement. However, the situation is more complex. In this article, we, therefore, study the entanglement between subsystems of the hybrid system which has been proposed in \cite{Danilishin_GW_2019, study4roadmap, Polzik_GW_Spin_PRD}. }

In particular, even though it is witnessed that the collective EPR-variance of the spin and mechanical modes are obtainable below the separability limit for the system mentioned in \cite{Entanglement_distant_macroscopic_spin}, however, importantly, we could not find any possibility of non-separability in the case of the system mentioned in refs. \cite{Danilishin_GW_2019, study4roadmap, Polzik_GW_Spin_PRD}, and which we are interested in.

{It has been supposed that satisfying Duan bound criterion automatically means that the spin system and the mechanical oscillator are entangled \cite{Duan_Inseparability_entanglement}. According to this criterion, for all Gaussian two-party continuous variable (CV) states there exist collective EPR-type of quadratures $X_\Sigma = c_1 X_1 + c_2 X_2$ and $Y_\Delta = c_1 Y_1 - c_2 Y_2$ whose variances violates the Duan inequality. Here, $X_{n}$ and $Y_{n}$ are the quadrature operators
that fulfil the canonical commutation relation $[X_n, Y_m] = i \delta_{n,m}$, where the index $n,m \in (1,2)$ for the two systems. The Duan inequality for such a system is \cite{Duan_Inseparability_entanglement},}

\begin{equation}\label{duan_ineq_initial}
\langle (\Delta X_\Sigma)^2 \rangle + \langle \Delta (Y_\Delta)^2 \rangle \geq 1.
\end{equation}

$\langle \Delta (X_\Sigma)^2 \rangle = \frac{1}{2}\int \frac{\mathrm{d}\omega}{2\pi} \langle \{ X_{\Sigma,\omega}, X_{\Sigma,-\omega} \} \rangle$ and analogously $\langle \Delta (Y_\Delta)^2 \rangle$, $c_1$ and $c_2$ are system determined nonzero real number.

In this letter, we start with
a brief description of the hybrid system, followed by the method for calculating the entanglement between the cavity and spin modes by transferring the input TMSV into the system including an analysis of the role played by light-matter interactions. Hereafter, we provide a detailed study of the spin-cavity entanglement with the variation of the parameters of the hybrid system. The investigation ends up with the search of the possibility of entanglement between mechanical and spin modes, along with justification on this aspect.

\begin{figure}[t]
\includegraphics[width=8 cm]{block_GW_diagram.jpg
\caption{\label{block_GW_diagram}Schematic block diagram of the BAE measurement setup }
\end{figure}

\begin{figure}[b]
\includegraphics[width=8 cm]{cav_spin1.jpg
\caption{\label{cav_spin} Logarithmic negativity $(E_N)$ or entanglement between the atomic spin and optomechanical cavity vs input squeezing $r$ in the presence and absence of the interaction to mechanical motion for a fixed parameter  $\Delta_0 = 60 \kappa_0, \omega_{m0}= 60 \kappa_0, \omega_{s0}= -60 \kappa_0, \gamma_{m0}= \kappa_0, \gamma_{s0}= \kappa_0, \Gamma_{S0} = 25.14\kappa_0, n_m=0.8, n_s=0.5,  \eta_I=\eta_S=1$. The dashed lines represent corresponding Duan quantity and the dashed black line indicates the limit below which the inequality violets and makes systems non-separable.   Inset: Entanglement between modes of input TMSV. }
\end{figure}

\begin{figure*}[ht]
\includegraphics[width=15cm,trim=2cm 0cm 2cm 0cm   ]{ENvsALL.jpg
\caption{\label{ENvsALL}Logarithmic negativity $(E_N)$ or entanglement between the atomic spin and optomechanical cavity vs input squeezing $r$ with the variation of parameters  ( I(a) cavity frequency $\Delta$ and I(b) cavity linewidth $\kappa$, II(a) spin Larmor frequency $\omega_s $ and II(b) linewidth $\gamma_s $, III(a) mechanical oscillation frequency $\omega_m $ and III(b) linewidth $\gamma_m $,  IV(a) optomechanical coupling strength $G$ and  IV(b) spin readout rate $\Gamma_S $, thermal excitation of the V(a) mechanical $(n_m)$ and V(b) spin $(n_s)$ reservoir, and input channel efficiency of the VI(a) cavity $(\eta_I)$ and VI(b) spin $(\eta_s)$ ). All other parameters remain same with Fig. \ref{cav_spin}. }
\end{figure*}

\begin{figure}[ht]
\includegraphics[width=8cm]{duanvsGGamma.jpg
\caption{\label{duanvsGGamma} Plot of the Duan quantity between the atomic spin and mechanics with the increment of the squeezing factor at temperature $T=0$ for both the reservoirs spin and mechanics. Inset: Entanglement between cavity and mechanics. All other parameters remain the same with Fig. \ref{cav_spin}.}
\end{figure}

\textit{Model: }The hybrid scheme of GW detector consists of an optomechanical system located at a distance from its counterpart which is an auxiliary atomic spin ensemble \cite{Danilishin_GW_2019, study4roadmap, Polzik_GW_Spin_PRD}, as demonstrated in Fig. \ref{block_GW_diagram}.
Both the mechanics and optical cavity continuously interact to their corresponding bosonic thermal bath which is Markovian in nature, typically approximated to have a flat and infinite spectral density \cite{Cavity_optomechanics_Aspelmeyer}.
The  Hamiltonian of the optomechanical system  is \cite{study4calculation}

\begin{align}\label{optomechanical_Hamiltonian}
\hat{H}_M/\hbar = \omega_c a^\dagger a +\frac{\omega_m}{2} (q_m^2+p_m^2) - g_0 a^\dagger a  q_m q_{zpf} \nonumber \\
  + i \left(a^\dagger E e^{-i\omega_l t} -h.c. \right) ,
\end{align}

where $|E|$ is the strength of the external drive with frequency $\omega_l$,
 $a(a^\dagger)$ are the annihilation(creation) operators of the cavity modes, and $q_m$ and $p_m$ are dimensionless position and momentum operators of the mechanical mode, follows the canonical commutation relation $[q_m,p_m]=i$. $\omega_c$ and $\omega_m$ are the resonance frequencies of the cavity and the mechanics, respectively; determining radiation pressure coupling energy $(g_0=\frac{\omega_c}{L})$ and the zero-point fluctuation  $(q_{zpf} = \sqrt{\frac{\hbar}{m\omega_m} })$, respectively. Here, $L, m$ and $\hbar$ are length of the cavity, mass the mechanical resonator and Plank's constant, respectively.
The rates of dissipation due to continuous interaction with the corresponding reservoir are $\kappa$ and $\gamma_m$ for the cavity and mechanics, respectively, bringing stability to the system
.
The external drive and the cavity parameters altogether determines steady state cavity field amplitude $(\alpha_s)$ which in turn tunes the optomechanical coupling strength $(G=  \sqrt{2} g_0 |\alpha_s| q_{zpf})$ in linearised regime. In order to make the Hamiltonian time-independent, the effective cavity detuning frequency expressed in the rotating frame of the laser is $\Delta = \omega_l -\omega_c + g_0 q_s $ where $q_s = \frac{g_0}{\omega_m} |\alpha_s|^2 q_{zpf}^2  $ is the steady-state displacement of the mechanical oscillator.

In addition to this, the spins of the auxiliary atomic ensemble are polarised initially under the influence of a magnetic field oriented along a direction (say z), and therefore, the collective spin exhibits an average projection $|\langle J_z \rangle|>> \hbar$.
Anticipating that, considering the atomic spin mainly reside in their excited state and using the Holstein-Primakoff (HP)
transformation \cite{HP_transform, nandi_pra},
we map the spin operators in terms of bosonic modes: $X_s = J_x/\sqrt{\hbar \langle J_z \rangle }, Y_s = J_y/\sqrt{\hbar \langle J_z \rangle}$, which in turn gives the Hamiltonian

\begin{equation}\label{spin_Hamiltonian}
\hat{H}_S/\hbar =  \omega_s \langle J_z \rangle - \frac{ \omega_s}{2}  (X_s^2 + Y_s^2) ,
\end{equation}
where $\omega_s$ is the Larmor frequency with linewidth broadening $\gamma_s$ causes due to continuous interaction to its reservoir. the first term is contributed by the mean spin polarization, therefore it turns out to be an irrelevant energy offset. The second term is equivalent to the mechanical oscillator expressed in Eq. \eqref{optomechanical_Hamiltonian}, with a negative mass. In the limit of large detuning and low saturation, the interaction Hamiltonian between the light polarization and the atomic spin is
$\hat{H}_{LS}/\hbar = \alpha S_x J_x$, where  $S_x$ is the Stokes component of light, measures the degree of circular polarization and $\alpha$ is the coupling strength that depends on laser detuning frequency and the area of interaction.  Using HP transformation, this can be mapped to $\hat{H}_{LS}/\hbar = \sqrt{\Gamma_S} X_s X_{in}^S$   where $\Gamma_S = \frac{\alpha^2}{2} \Phi|\langle J_z \rangle|$ is the spin readout rate for a given  photon flux $\Phi$ \cite{spin_readout}. The negative mass spin oscillator ensures that each quantum of excitation corresponds to a deexcitation from the highest energy levels of the inverted spin population. Therefore, the anti-Stokes interaction with the incoming laser becomes predominant over the Stokes interaction.

The distant optomechanical and spin systems are feed by a TMSV as an input signal which is generated by non-linear optical crystal through the parametric down-conversion process (PDC), satisfying

\begin{subequations}
\begin{align}\label{TMSV_field}
	\begin{bmatrix}
		a^I_{in}\\
		{a^S_{in} }^\dagger
	\end{bmatrix} &= \begin{bmatrix}
		\cosh r & \sinh r\\
		\sinh r & \cosh r
	\end{bmatrix} \begin{bmatrix}
	a^I_{z}\\
	{a^S_{z} }^\dagger
\end{bmatrix} \\
	\begin{bmatrix}
	a^S_{in}\\
	{a^I_{in} }^\dagger
\end{bmatrix} &= \begin{bmatrix}
	\cosh r & \sinh r\\
	\sinh r & \cosh r
\end{bmatrix} \begin{bmatrix}
	a^S_{z}\\
	{a^I_{z} }^\dagger
\end{bmatrix}
\end{align}
\end{subequations}

where $r$ is the squeezing factor, $a^I_{z} ({a^I_{z} }^\dagger)$ and $a^S_{z} ({a^S_{z} }^\dagger)$ are the annihilation (creation) operators of the two independent vacuum fields, and $a^I_{in}, a^S_{in}$ are the input fields to the optomechanical and spin systems respectively.

\begin{comment}
\begin{align*}
	\begin{bmatrix}
		X^I_{in}\\
		X^S_{in}
	\end{bmatrix} &= \begin{bmatrix}
	\cosh r & \sinh r\\
	\sinh r & \cosh r
\end{bmatrix} \begin{bmatrix}
		X^I_{z}\\
		X^S_{z}
	\end{bmatrix} \\
	\begin{bmatrix}
	Y^I_{in}\\
	Y^S_{in}
\end{bmatrix} &= \begin{bmatrix}
	\cosh r & -\sinh r\\
	-\sinh r & \cosh r
\end{bmatrix} \begin{bmatrix}
	Y^I_{z}\\
	Y^S_{z}
\end{bmatrix}
\end{align*}
\end{comment}

\textit{Spin-Cavity Entanglement: }
The proposed scheme for the BAE measurement imposes a necessary condition which fixes both the susceptibility equal $(\chi_m = \chi_s)$, where  $\chi_m (\omega) \approx \frac{\omega_m}{\omega_m^2 -\omega^2 - 2i \omega \gamma_m}$ is the optomechanical susceptibility and $\chi_s (\omega) \approx \frac{\omega_s}{\omega_s^2 -\omega^2 - 2i \omega \gamma_s}$ for spin \cite{Danilishin_GW_2019, study4roadmap, Polzik_GW_Spin_PRD, study4calculation}. This implies the condition $\omega_{m0} = \omega_{s0}$ and $\gamma_{m0} = \gamma_{s0}$. Accepting the condition, we plotted the logarithmic negativity between the cavity and spin in Fig \ref{cav_spin},  for both the presence and absence of the interaction to the mechanics.
Even though the entanglement between the atomic spin system and the optical cavity is created by the transfer of the input TMSV to the system, unlike the entanglement of the input beam (inset of Fig \ref{cav_spin}), we see that the spin-cavity entanglement saturates to a finite value for highly squeezed input beams when no mechanics are involved.
The involvement of mechanics interrupts the coherent transfer of input TMSV into the system, causing a reduction of the entanglement. Further, the entanglement drops down to zero for a highly squeezed input signal due to more injection of photons into the cavity generates more quanta in the mechanical oscillator which in turn causes more disturbance in this process. The phenomenon can also be anticipated by looking at the Duan quantity plotted with the dashed line in  Fig \ref{cav_spin}, where we see the collective uncertainty of the EPR-like Duan operators obeys the inequality (Eq. \eqref{duan_ineq_initial})
 beyond the regime of entanglement
, making the systems separable \cite{Duan_Inseparability_entanglement}.

The entanglement, present only in a finite interval of $r$, has shown explicit dependency on the system parameters.
 Fig.  \ref{ENvsALL}-I(a) shows that, in a particular range of cavity detuning parameters at blue sideband regime $(\frac{\Delta_{0}}{\omega_{m0}}=1)$, the entanglement rapidly decreases when $\Delta$ moves away from $(\Delta_{0})$. Besides, it is also notable that the peak moves towards higher $r$ when $\Delta$ approaches to $(\Delta_{0})$. In the case of cavity linewidth, a similar pattern is observed where the entanglement becomes robust at $\kappa = \kappa_0$ and the pick moves to the higher value of $r$ (\ref{ENvsALL}-I(b)).  Following cavity parameters, the changes of entanglement exhibit a similar pattern while detuning the spin Larmor frequency (Fig. \ref{ENvsALL}-II(a)) and the spin bandwidth (Fig. \ref{ENvsALL}-II(b)). All these phenomenons are governed by the occurrence of the balanced transfer of input TMSV to the cavity and the spin system when the condition $\Delta = -\omega_{s}$ and $\kappa = \gamma_{s}$ satisfies, as highlighted in Fig 3 (I and II).  The entanglement moreover exhibits robustness with the variation of the frequency of the mechanical mode (Fig. \ref{ENvsALL}-III(a)), whereas it increases significantly with the increment mechanical bandwidth  (Fig. \ref{ENvsALL}-III(b))
 .
 A more interesting situation is depicted in Fig. \ref{ENvsALL}-IV where we study the robustness of entanglement while changing optomechanical and spin-laser coupling strength. We observe an overall increment of entanglement with the reduction of optomechanical coupling, especially in the case of the higher value of $r$ (Fig. \ref{ENvsALL}-IV(a)). Whereas, an opposite phenomenon is observed in the case of spin readout rate, shown in Fig. \ref{ENvsALL}-IV(b)
 where the higher readout rate enhances the entanglement transfer from the input state to the system. Fig. \ref{ENvsALL}-V ((a) and (b)) shows how the entanglement reduces with the temperature of both the reservoirs of mechanics and the spin system. Finally,  Fig. \ref{ENvsALL}-VI ((a) and (b)) shows the change of entanglement against the input quantum efficiency of the input path. Both figures exhibit a reduction of entanglement with the reduction of efficiency due to less delivery of the initial entanglement of the input TMSV into the system.

Note, importantly,  that the range of parameters, especially $G$ and $\Gamma_S$, used here are far different from the range of parameters used in articles \cite{Danilishin_GW_2019, study4roadmap, Polzik_GW_Spin_PRD}. The possibility of obtaining entanglement is almost negligible with the parameters used in those articles aiming to obtain squeezing in output modes

\textit{Spin-Mechanics Entanglement: }
Furthermore, we study the possibility of achieving entanglement between the spin and the mechanical mode in  Fig. \ref{duanvsGGamma}. The study reveals that, even in a situation where the entanglement is present in between the spin and optical the cavity, and the cavity and the mechanics (inset of Fig. \ref{duanvsGGamma}(I)
; the entanglement is still impossible to obtain between spin and the mechanics.
Fig. \ref{duanvsGGamma}(I) shows that the Duan quantity increases while increasing the strength of optomechanical coupling and always remains above the unit value satisfying the separability condition. On the other hand, a similar phenomenon is also observed while increasing the spin readout rate in Fig. \ref{duanvsGGamma}(II). In both cases the Duan quantity is lower bounded by unit value when no coupling is present, making the systems completely separable.

Note that we fix input quantum efficiency to be maximum and the temperature to be zero for the reservoirs of both mechanical oscillator and the spin system to obtain maximum entanglement, which therefore justifies the impossibility of obtaining entanglement in any range of those parameters.

\textit{Conclusion:}
We demonstrated under what condition entanglement can be generated between the distant spin system and the optical cavity introduced in \cite{Danilishin_GW_2019, study4roadmap, Polzik_GW_Spin_PRD}, by transferring input TMSV states which are generated through the PDC process. We see the entanglement becomes robust for weaker optomechanical interaction and stronger spin-optical interaction. The study shows that the entanglement is highly dependent on the system parameters, especially on the cavity resonance and the spin Larmor frequency and their linewidths. The mismatch of both frequencies and their linewidths causes a rapid reduction of entanglement. However, the entanglement shows robustness against the variation of the frequency and the linewidth broadening of the mechanical oscillator.
Even though the entanglement between the spin-cavity and the cavity-mechanics is prominent, it is still impossible to obtain entanglement between the spin and the mechanical motion, which has been justified by the Duan separability limitation. The scheme, proposed in  \cite{Danilishin_GW_2019,study4roadmap, Polzik_GW_Spin_PRD}, though looks quite promising, while considering from the perspective of practical implementation, our investigation raises some concern and one need to choose the parameter space wisely, to obtain effective BAE measurement.
Apart from sensitivity increment of GW detectors \cite{ Danilishin_GW_2019, study4roadmap, Polzik_GW_Spin_PRD, study4calculation}, this scheme provides a promising platform for technology applications, such as quantum teleportation \cite{entanglement_swapping_two_mechanical_Vitali}, cryptography \cite{entanglement_quantum_cryptography}, CV dense coding \cite{entanglement_quantum_communication} and quantum computation \cite{Quantum_entanglement, Quantum_Computation_Trapped_Ions}. In addition, the hybrid system provides a platform for studying the effect of decoherence induced quantum to classical transitions\cite{classical_quantum_boundary}. Such a study requires proper modeling of environment-induced decoherence and its effect on the dynamics of the entangled state of systems along with the existing parameter space. Therefore, we believe our findings of operational parameter space are of due importance in studying classical to quantum transition.

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