Enhancing Gaussian quantum metrology with position-momentum correlations

Our quantum estimation protocol schematized in figure 1 is represented in three parts: initialization, interaction, and estimation. The initialization step is characterized by the generation of fullerene Gaussian wave packets with momentum $\overrightarrow{k}$ and an initial coherent length ℓ₀. In the sequence, the parameterization process of unknown parameters is separated into two parts. The first one, here denoted as focalization, describes the unitary parameterization associated with the encoding of the PM correlation parameter γ employing a real-noisy source (details about this process are discussed in appendix B). The second one, denoted as interaction, represents a non-unitary parameterization for which the probe interacts with a Markovian bath producing an encoding of the effective environment coupling parameter Λ and in turn its temperature T in the correlated Gaussian state. Finally, the estimation procedure of the unknown parameters is carried out during the estimation stage. This stage includes (I) PM correlations and (II) thermometry estimations.

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We consider as the initial state the following correlated Gaussian state of transverse width σ₀

$$\begin{eqnarray}&&{\psi }_{0}({x}_{i})=\frac{1}{\sqrt{{\sigma }_{0}\sqrt{\pi }}}\exp \left[-\frac{{x}_{i}^{2}}{2{\sigma }_{0}^{2}}+\frac{i\gamma {x}_{i}^{2}}{2{\sigma }_{0}^{2}}\right],\end{eqnarray} \tag{ 1 }$$

that represents a PM-correlated Gaussian state [52]. The real parameter γ ensures that the initial state is correlated [52, 53]. Considering the initial state, equation (1), the variance in position becomes ${\sigma }_{xx}={\sigma }_{0}^{2}/2$, whereas the variance in momentum is ${\sigma }_{pp}=(1+{\gamma }^{2}){\hslash }^{2}/2{\sigma }_{0}^{2}$, and the initial correlations PM is ${\sigma }_{xp}=\langle (\hat{x}\hat{p}+\hat{p}\hat{x})/2\rangle -\langle \hat{x}\rangle \langle \hat{p}\rangle =\hslash \gamma /2$. Notice that, for γ = 0 we have a simple uncorrelated Gaussian wave packet, i.e. σ_xp = 0. Exploring the Pearson correlation coefficient between $\hat{x}$ and $\hat{p}$, i.e. $r={\sigma }_{xp}/\sqrt{{\sigma }_{xx}{\sigma }_{pp}}\,(-1\leqslant r\leqslant 1)$, the γ parameter turns out to be $\gamma =r/\sqrt{1-{r}^{2}}\,(-\infty \leqslant \gamma \leqslant \infty )$, illustrating the physical meaning of the γ as a parameter that encoded the initial correlations between position and momentum for the initial state.

In the standard quantum phase estimation protocols, it is usual to estimate the physical parameter that implements a relative phase shift [54, 55]. Here, our main goal will be to explore whether initial PM correlations can work as a resource to improve quantum metrology under noise. To estimate the initial PM correlations, we consider the production of beam particles from a non-ideal or partially incoherent source (see figure 1), where each particle has momentum p_x = ℏk_x in the x-direction within a certain range ${\sigma }_{{k}_{x}}$ of wavenumber k_x and can be modeled by the following initial density matrix [37, 56, 57]

$$\begin{eqnarray}&&{\rho }_{\gamma }(x,x^{\prime} )=\int d{k}_{x}p({k}_{x}){\psi }_{0{k}_{x}}({x}_{0}){\psi }_{0{k}_{x}}^{* }(x^{\prime} ),\end{eqnarray} \tag{ 2 }$$

with ${\psi }_{0{k}_{x}}({x}_{0})={\psi }_{0}({x}_{0}){e}^{i{k}_{x}{x}_{0}}$. In this model, we consider wave effects only in the x-direction since the energy associated with the momentum of the particles in the z-direction is high enough such that the momentum component p_z is sharply defined, i.e. Δp_z ≪ p_z. Then, the behavior in the z-direction can be considered classical [37]. The geometry of the collimation device is such that it selects only particles with a transverse wave number k_x inside a specific range defined by the width σ_kx, being described by a classical Maxwell–Boltzmann distribution, $p({k}_{x})={e}^{-({k}_{x}^{2}/2{\sigma }_{kx}^{2})}/\sqrt{2\pi }{\sigma }_{kx}$. After some algebraic modifications, the initial density matrix of the coherent correlated Gaussian wave packet becomes

$$\begin{eqnarray}&&{\rho }_{\gamma }({x}_{0},x^{\prime} )={{ \mathcal N }}_{0}\exp \left\{-{{ \mathcal A }}_{0}{x}_{0}^{2}-{{ \mathcal B }}_{0}{x}_{0}^{{\prime} 2}+{{ \mathcal C }}_{0}{x}_{0}{x}_{0}^{{\prime} }\right\},\end{eqnarray} \tag{ 3 }$$

where ${{ \mathcal N }}_{0}=1/{\sigma }_{0}\sqrt{\pi }$, ${{ \mathcal A }}_{0}=1/2{\ell }_{0}^{2}+(1-i\gamma )/2{\sigma }_{0}^{2}$, ${{ \mathcal B }}_{0}=1/2{\ell }_{0}^{2}+(1+i\gamma )/2{\sigma }_{0}^{2}$, ${{ \mathcal C }}_{0}=1/{\ell }_{0}^{2}$, and ${\ell }_{0}={\sigma }_{kx}^{-1}$ is related with the coherence level of the initial state. It is important to note that, in the limit σ_kx → 0 (ideal collimation), one has ℓ₀ → ∞ (${{ \mathcal C }}_{0}\to 0$), meaning that the initial state (3) is completely coherent or pure. On the other hand, if σ_kx → ∞, then one has ℓ₀ → 0 (${{ \mathcal C }}_{0}\to \infty$), indicating a completely incoherent or mixed state.

According to the protocol depicted in figure 1, after the focalization process, the system interacts with an environment that is considered to be a Markovian bath [58], and its evolution is given by

$$\begin{eqnarray}&&{\rho }_{{\rm{\Lambda }}}(x,{x}^{{\prime} },t)=\int \int d{x}_{0}d{x}_{0}^{{\prime} }{K}_{{\rm{\Lambda }}}(x,{x}^{{\prime} },t;{x}_{0},{x}_{0}^{{\prime} },0){\rho }_{\gamma }({x}_{0},{x}_{0}^{{\prime} }),\end{eqnarray} \tag{ 4 }$$

where

$$\begin{eqnarray}\begin{array}{rcl}{K}_{{\rm{\Lambda }}}(x,{x}^{{\prime} },t;{x}_{0},{x}_{0}^{{\prime} },0) & = & \frac{m}{2\pi \hslash t}\,\exp \left\{\right.\frac{im}{2\hslash t}\left[{(x-{x}_{0})}^{2}+{({x}^{{\prime} }-{x}_{0})}^{2}\right]\\ & & -\frac{{\rm{\Lambda }}t}{3}\left[{(x-{x}^{{\prime} })}^{2}+{({x}_{0}-{x}_{0}^{{\prime} })}^{2}+(x-{x}^{{\prime} })({x}_{0}-{x}_{0}^{{\prime} })\right]\left\}\right.,\end{array}\end{eqnarray} \tag{ 5 }$$

is the propagator which includes the environmental effect [37, 59]. We consider the propagation of fullerene molecules and the decoherence effect produced by air molecules scattering [37]. In this case, the effective scattering constant is given by ${\rm{\Lambda }}=(8/3{\hslash }^{2})\sqrt{2\pi {m}_{\,\rm{air}\,}}{({k}_{B}T)}^{3/2}N{w}^{2}$, where N is the total number density of the air, m_air is the mass of the air molecule, w is the size of the molecule of the quantum system, k_B is the Boltzmann constant, and T is the bath temperature [59]. In the propagator described by equation (5), it is assumed that the center-of-mass state of the system remains undisturbed by the scattering events, a valid condition for fullerene molecules (m ≫ m_air) [37]. The environmental scattering decoherence is a ubiquitous effect constantly monitoring the position of quantum systems in the cosmos, that can be caused by air molecules, light (optical photons), solar neutrinos, cosmic muons, background radioactivity, and even the Universe's 3 K cosmic background radiation [59]. It is also essential to emphasize that in the derivation of this model, it was also assumed the following conditions [59, 60]: the system and the surroundings are not initially correlated; when the composite object–environment system is translated, the scattering interaction remains unchanged; the rate of scattering is much faster than the characteristic rate of change in the state of the system; and the distribution of momenta of the particles in the environment is isotropic and obeys a Maxwell–Boltzmann distribution. In physical terms, the Λ parameter quantifies the rate at which spatial coherence over a given distance Δx is suppressed, motivating the introduction of a decoherence timescale given by [59]

$$\begin{eqnarray}&&{\tau }_{\,\rm{dec}\,}=\frac{1}{{\rm{\Lambda }}{({\rm{\Delta }}x)}^{2}}.\end{eqnarray} \tag{ 6 }$$

This equation will be useful for interpreting how PM correlation is related to loss of purity and gain of Fisher information. After integration and manipulating equation (4), we get

$$\begin{eqnarray}&&{\rho }_{{\rm{\Lambda }}}(x,{x}^{{\prime} },t)={{ \mathcal N }}_{t}\exp \left\{-{{ \mathcal A }}_{t}{x}^{2}-{{ \mathcal B }}_{t}{x}^{{\prime} 2}+{{ \mathcal C }}_{t}x{x}^{{\prime} }\right\},\end{eqnarray} \tag{ 7 }$$

where ${{ \mathcal A }}_{t}$, ${{ \mathcal B }}_{t}$, ${{ \mathcal C }}_{t}$ and ${{ \mathcal N }}_{t}$ are parameters that include the interaction with the Markovian bath (see appendix C).

Employing equation (7), we can cast the purity of the state $\mu (\gamma ,{\rm{\Lambda }},t)=\,\rm{tr}\,\left[{\rho }_{{\rm{\Lambda }}}^{2}(x,{x}^{{\prime} },t)\right]$ in the concise form

$$\begin{eqnarray}&&\mu (\gamma ,{\rm{\Lambda }},t)=\sqrt{\frac{{{ \mathcal A }}_{t}+{{ \mathcal B }}_{t}-{{ \mathcal C }}_{t}}{4{{ \mathcal A }}_{R}{{ \mathcal B }}_{R}+{{ \mathcal C }}_{R}^{2}}},\end{eqnarray} \tag{ 8 }$$

where ${{ \mathcal A }}_{R}=\,\rm{Re}\,[{{ \mathcal A }}_{t}]$, ${{ \mathcal B }}_{R}=\,\rm{Re}\,[{{ \mathcal A }}_{t}]$ and ${{ \mathcal C }}_{R}=\,\rm{Re}\,[{{ \mathcal A }}_{t}]$. This result produces μ = 1 when ℓ₀ → ∞ (${{ \mathcal C }}_{t}\to 0$), i.e. for a completely coherent source and no environment effect Λ = 0. Outside this regime, we always have μ(γ, Λ, t) < 1, typical of a noisy scenario represented by mixed states (details in appendix C).

In the following subsections, we present the dynamics of the QFI and the purity in each part of the parameterization process illustrated in our estimating protocol in figure 1. We begin estimating PM correlations in a scenery where the quantum system unavoidably interacts with its surrounding environments and then we explore how these correlations can enhance noisy quantum metrology.

From the mixed Gaussian state described in equation (7), it is easy to check that the first moments $\overrightarrow{d}=(\langle \hat{x}\rangle ,\langle \hat{p}\rangle )$ are null and the dimensionless second moments are given by

$$\begin{eqnarray}&&{\sigma }_{xx}=\frac{{t}^{2}}{{\tau }_{0}^{2}}\left[\frac{1}{2}+2{\left(\frac{{\tau }_{0}}{2t}+\frac{\gamma }{2}\right)}^{2}+\frac{{\sigma }_{0}^{2}}{{\ell }_{0}^{2}}+\frac{2{\rm{\Lambda }}t{\sigma }_{0}^{2}}{3}\right];\,\,{\sigma }_{pp}=\frac{(1+{\gamma }^{2})}{2}+\frac{{\sigma }_{0}^{2}}{{\ell }_{0}^{2}}+2t{\rm{\Lambda }}{\sigma }_{0}^{2};\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{\sigma }_{xp}=\frac{\gamma }{2}+\frac{t}{2{\tau }_{0}}(1+{\gamma }^{2})+\frac{t{\sigma }_{0}^{2}}{{\tau }_{0}{\ell }_{0}^{2}}+\frac{{t}^{2}{\rm{\Lambda }}{\sigma }_{0}^{2}}{{\tau }_{0}}.\end{eqnarray} \tag{ 10 }$$

Here, ${\tau }_{0}=(m{\sigma }_{0}^{2})/\hslash$ is the time at which the distance of the order of the wave packet extension is traversed with speed corresponding to the dispersion in velocity [61]. Then, the covariance matrix σ and its inverse σ⁻¹ can be written as

$$\begin{eqnarray}{\boldsymbol{\sigma }}(\gamma ,{\ell }_{0},{\rm{\Lambda }},t)=\left(\begin{array}{cc}{\sigma }_{xx} & {\sigma }_{xp}\\ {\sigma }_{px} & {\sigma }_{pp}\end{array}\right);\,{{\boldsymbol{\sigma }}}^{-1}(\gamma ,{\ell }_{0},{\rm{\Lambda }},t)={\mu }^{2}\,\rm{adj}\,({\boldsymbol{\sigma }}),\end{eqnarray} \tag{ 11 }$$

where σ_px = σ_xp, adj(σ) is the adjugate matrix of σ and we have used that, for Gaussian states, det(σ) = μ⁻² [62]. The mean number of excitations, which is proportional to the energy of the system, can then be expressed for the single-mode system as follows [63]:

$$\begin{eqnarray}&&n=({\sigma }_{xx}+{\sigma }_{pp}+{\overrightarrow{d}}^{2}-1)/2=\frac{1}{2}\left[\left(\right.\frac{1}{2}+\frac{{\sigma }_{0}^{2}}{{\ell }_{0}^{2}}+\frac{2{\rm{\Lambda }}{\sigma }_{0}^{2}t}{3}\left)\right.\frac{{t}^{2}}{{\tau }_{0}^{2}}+\frac{{\sigma }_{0}^{2}}{{\ell }_{0}^{2}}+2{\rm{\Lambda }}{\sigma }_{0}^{2}t\right]+\left(\right.\frac{t}{{\tau }_{0}}\gamma +\frac{3{\gamma }^{2}}{4}\left)\right..\end{eqnarray} \tag{ 12 }$$

In optics, the quantity n represents the mean photon number. This quantity will be important to analyze how the QFI scales with n, allowing a comparison with the shot noise and the Heisenberg limit (HL), more details in figure.

Therefore, we can write the QFI for single-mode Gaussian states (see equation (A5) in appendix A) as following [64, 65]

$$\begin{eqnarray}&&{{ \mathcal F }}_{{\rm{\Theta }}}^{\,\rm{Q}\,}({\rm{\Theta }})=\frac{{\mu }^{4}}{2(1+{\mu }^{2})}\,\rm{Tr}\,\{{[\,\rm{adj}\,({\boldsymbol{\sigma }}){\partial }_{{\rm{\Theta }}}{\boldsymbol{\sigma }}]}^{2}\}+2\frac{{({\partial }_{{\rm{\Theta }}}\mu )}^{2}}{1-{\mu }^{4}}.\end{eqnarray} \tag{ 13 }$$

The above expression will be useful for interpreting our results, where we can explicitly understand the role of purity and its derivative as a resource for estimating unknown parameters in the presence of noise. Furthermore, we also investigate the dynamics of Classical Fisher Information (CFI) obtained from the density distribution in position space as

$$\begin{eqnarray}&&{{ \mathcal F }}_{{\rm{\Theta }}}^{\,\rm{C}\,}({\rm{\Theta }})=\int \frac{1}{P({x}_{i}| {\rm{\Theta }})}\left(\right.\frac{\partial P({x}_{i}| {\rm{\Theta }})}{\partial {\rm{\Theta }}}{\left)\right.}^{2}d{x}_{i},\end{eqnarray} \tag{ 14 }$$

where $P({x}_{i}| {\rm{\Theta }})={\rho }_{{\rm{\Lambda }}}(x,{x}^{{\prime} }=x,t)$. In what follows, the estimated parameters will be the initial position-momentum correlation Θ = γ and the effective environmental coupling constant Θ = Λ.

We employ the QFI and CFI for the estimation of the initial correlations. They were calculated from equations (13) and (14) by making Θ = γ. After some simplifications, we obtain

$$\begin{eqnarray}&&{{ \mathcal F }}_{\gamma }^{\,\rm{Q}\,}(\gamma ,t,{\rm{\Lambda }})=\frac{{\mu }^{4}}{2(1+{\mu }^{2})}\,{{\rm{\Phi }}}_{\gamma }(\gamma ,t,{\rm{\Lambda }})+2\frac{{({\partial }_{\gamma }\mu )}^{2}}{1-{\mu }^{4}},\end{eqnarray} \tag{ 15 }$$

and

$$\begin{eqnarray}&&{{ \mathcal F }}_{\gamma }^{\,\rm{C}\,}(\gamma ,t,{\rm{\Lambda }})=\frac{1}{8{\sigma }_{0}^{4}{B}^{4}}{\left(\frac{m}{\hslash t}+\frac{\gamma }{{\sigma }_{0}^{2}}\right)}^{2},\end{eqnarray} \tag{ 16 }$$

where the parameter B is explicitly defined in appendix C and Φ_γ(γ, t, Λ) is a monotonic increasing function of the parameters γ, t and Λ (see appendix D). Since this function is multiplied by the fourth power of μ, the purity behavior determines the first term in QFI. This can be illustrated using the following parameters: fullerene mass m = 1.2 × 10⁻²⁴ kg, molecular size w = 7 Å, the width of the initial wave packet σ₀ = 7.8 nm, the mass of the air molecule m_air = 5.0 × 10⁻²⁶ kg and density of the air molecules N = 10¹² molecules/m³[37, 38, 56]. It is assumed that the collimator apparatus selects wave numbers in the Ox direction, with a transverse wavenumber dispersion of ${\sigma }_{{k}_{x}}\approx 1{0}^{7}\,{{\rm{m}}}^{-1}$. This corresponds to an initial coherence length of ${\ell }_{0}={\sigma }_{{k}_{x}}^{-1}\approx 50\,{\rm{nm}}$. Parameters of this order of magnitude were previously used in experiments with fullerene molecules by Zeilinger [66, 67]. With the other parameters fixed, the variation in Λ corresponds to the variation in the environmental temperature such that, Λ ∝ T^3/2. For instance, Λ of the order of 10¹⁹ m⁻²s⁻¹, corresponds to an effective temperature around 205 K. The value 3.2 × 10¹⁵ m⁻²s⁻¹ for the scattering by air molecules estimated in [37] for the experiment with fullerene molecules corresponds to a temperature of 300 K and a density of the air molecules of N = 1.8 × 10⁸ molecules/m³, smaller than the densities we are considering. In other words, the air molecule density and temperature range we are exploring here are experimentally feasible with the current technology.

To analyze the role of each term in the QFI (13) and the influence of the environmental effect, we present in figure 2 the behavior of the QFI (${{ \mathcal F }}_{\gamma }^{\,\rm{Q}\,}$), CFI (${{ \mathcal F }}_{\gamma }^{\,\rm{C}\,}$), purity μ, and its absolute relative derivative $\frac{1}{\mu }| {\partial }_{\gamma }\mu |$ as a function of the initial correlation γ for a characteristic interaction time t = 1.0μs and different environmental effect coupling constants. Panels (a) and (b) demonstrate that the QFI behaves similarly to the purity when the environment has a relatively small effect (low-temperature regime). In contrast, panels (c) and (d) demonstrate that the QFI behaves according to the absolute purity variation when the environment effect is stronger. Also, we see that only in the regime of strong environmental effect (quantum to classical transition in the high-temperature regime) does the CFI become comparable with the QFI.

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One of the most intriguing features here is that besides the suppression of the QFI when the noise increases, we can find PM correlations that enhance the corresponding QFI compared to the usual Gaussian state. It is also important to note that figure 2(d) shows that even a maximal purity value (corresponding to a null PM correlation) does not provide the maximum value for the QFI. This happens because, in the regime of strong environmental effect, the change in purity (second term in equation (15)) is dominant over the purity value itself is close to zero, and will be elevated to the fourth power (first term in equation (15)). To comprehend in more detail how the QFI behavior is governed by either the purity or by the purity variation, and how this transition occurs, we present in appendix D these quantities where we fixed Λ and plotted them as a function of the PM correlations and the propagation time.

Here, we analyze how the PM correlations affect the estimation of the effective environmental coupling constant, showing a specific regime in which the correlated Gaussian state provides a quantum advantage over the standard one in estimating the effective parameter related to an environmental interaction. Using equations (13) and (14) and setting Θ = Λ, the QFI and CFI reads

$$\begin{eqnarray}&&{{ \mathcal F }}_{{\rm{\Lambda }}}^{\,\rm{Q}\,}(\gamma ,t,{\rm{\Lambda }})=\frac{{\mu }^{4}}{2(1+{\mu }^{2})}\,{{\rm{\Phi }}}_{{\rm{\Lambda }}}(\gamma ,t,{\rm{\Lambda }})+2\frac{{({\partial }_{{\rm{\Lambda }}}\mu )}^{2}}{1-{\mu }^{4}},\end{eqnarray} \tag{ 17 }$$

and

$$\begin{eqnarray}&&{{ \mathcal F }}_{{\rm{\Lambda }}}^{\,\rm{C}\,}(\gamma ,{\rm{\Lambda }},t)=\frac{{t}^{2}}{12{\sigma }_{0}^{4}{B}^{4}},\end{eqnarray} \tag{ 18 }$$

where Φ_Λ(γ, t, Λ) is another monotonic increasing function of the parameters γ, t and Λ (see appendix E). Similarly, this function does not determine the profile of QFI in the first term.

In figure 3, we exhibit the curves for QFI (${{\rm{\Lambda }}}^{2}{{ \mathcal F }}_{{\rm{\Lambda }}}^{\,\rm{Q}\,}$), CFI (${{\rm{\Lambda }}}^{2}{{ \mathcal F }}_{{\rm{\Lambda }}}^{\,\rm{C}\,}$), purity μ, and its absolute relative variation $\frac{1}{\mu }| {\partial }_{\gamma }\mu |$ as a function of initial correlation γ for t = 50.0μs and considering (a) weak environmental effect (Λ = 1.0 × 10¹⁵ m⁻²s⁻¹ or T = 442 mK) and (b) strong environmental effect (Λ = 1.0 × 10²¹ m⁻²s⁻¹ or T = 4.4 × 10³ K). Again, we observe two characteristic regimes, the gain in QFI is related either to purity or relative variation in purity when the environmental effect is strong (high-temperature regime) or weak (low-temperature regime), respectively. Such regimes are associated with the first and second term in equation (17), respectively. One of the more interesting results of this work, rarely explored in the literature, is investigating the relation between loss of purity and gain in Fisher information. Also, we can observe that in the low-temperature regime, the PM can improve the QFI, and this behavior paves the way for exploring the enhancement of noisy quantum metrology by PM correlations which will be discussed in the following. On the other hand, these PM correlations quantum advantage ceases to be valid in the limit of strong environmental effect (Λ = 1.0 × 10²¹ m⁻²s⁻¹ or T = 4.4 × 10³ K), with the standard uncorrelated Gaussian state γ = 0 providing the best sensitivity for thermometry in this regime [see figure 3 (b)].

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Also, it is important to mention that the effect of increasing the interaction time t or changing the environmental effect Λ is almost equivalent from a theoretical point of view, they both decrease the purity and coherence of the quantum state. However, from an experimental point of view, the variation of these two parameters is very different. The first is varied just by moving the position of the detector to regions further away from the source. In contrast, the second can be varied by changing, for example, the environmental temperature or the air pressure.

In figure 4 we investigate the time dependence of (a) QFI (${{\rm{\Lambda }}}^{2}{{ \mathcal F }}_{{\rm{\Lambda }}}^{\,\rm{Q}\,}$) and (b) purity μ for different values of the initial correlation γ and considering Λ = 1.0 × 10¹⁵ m⁻²s⁻¹ (T = 442 mK). We observe a temporal quantum enhancement due to PM correlation γ. Therefore, employing an initial correlated (position-momentum) Gaussian state can reduce the time required to obtain the maximum amount of information until it is saturated. Moreover, note the relationship [inset of panel (b)] between the maximum temporal rate of purity variations and the time of information saturation. We interpret this gain in QFI as being associated with the rate of change of purity. This relationship can be seen directly through the inset in figure 4 (b), showing that for large times and regardless of the value of γ the second term in equation (17) ceases to contribute to an increase in QFI since purity goes to zero at this limit. In other words, the second term contributes to an enhancement in the QFI only up to approximately 300 microseconds, during which the presence of PM-correlations can further improve this gain. In this context, a similar effect was recently observed in [68], where it was noted that the rate of change of coherence concerning the channel parameter—rather than the amount of coherence—can produce a parameter estimation gain.

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Besides this, to quantify such an enhancement or acceleration in the time required to saturate the QFI due to PM correlation γ, we introduce the Temporal Gain of Information (TGI), defining it on the decibel scale as follows

$$\begin{eqnarray}&&\,\rm{TGI}\,(\gamma )=-(10\,\,\rm{dB}\,)\mathrm{log}\left[\frac{{\tau }_{\,\rm{max}\,}({\rm{\Lambda }},\gamma )}{{\tau }_{\,\rm{max}\,}({\rm{\Lambda }},\gamma =0)}\right],\end{eqnarray} \tag{ 19 }$$

where the time τ_max corresponds to the maximization point of the relative purity variation, i.e. the maximum of $\frac{1}{\mu }| {\partial }_{t}\mu |$ [see inset of figure 4(b)]. In this definition, the acceleration in the time to saturate the QFI is compared with the correspondent time needed for the standard uncorrelated Gaussian state (γ = 0), such that the uncorrelated state is always associated with a null temporal gain.

To obtain a simplified and closed expression for the TGI as a function of PM correlations γ, we consider interaction times in the order of some microsecond and since ${({\sigma }_{0}/{\ell }_{0})}^{2}\approx 0.024$ the cubic term in equation (C5) becomes dominant, and therefore the purity can be approximately expressed as

$$\begin{eqnarray}&&{\mu }_{\,\rm{approx}\,}(\gamma ,{\rm{\Lambda }},t)\approx {\left[1+\frac{4\hslash {\rm{\Lambda }}({\gamma }^{2}+1)}{3{\tau }_{0}m}{t}^{3}\right]}^{-\frac{1}{2}}.\end{eqnarray} \tag{ 20 }$$

Then, the time τ_max that maximizes the relative purity variation, i.e. the maximization point of $\frac{1}{{\mu }_{\rm{approx}\,}}{\partial }_{t}{\mu }_{\,\rm{approx}}$ is given by

$$\begin{eqnarray}&&{\tau }_{\,\rm{max}\,}({\rm{\Lambda }},\gamma )=\left[\right.\frac{3{\tau }_{0}^{2}}{2(1+{\gamma }^{2}){\rm{\Lambda }}{\sigma }_{0}^{2}}{\left]\right.}^{1/3}.\end{eqnarray} \tag{ 21 }$$

Note that this time can be reduced by increasing the initial correlation γ. In this way, using τ_max(Λ, γ) from equation (21), the approximate TGI(γ) can be written explicitly as

$$\begin{eqnarray}&&{\rm{TGI}\,}_{\,\rm{approx}}(\gamma )=(10\,\,\rm{dB}\,)\mathrm{log}{(1+{\gamma }^{2})}^{1/3}.\end{eqnarray} \tag{ 22 }$$

In figure 5, we examine the enhancement in TGI as a function of PM correlation γ. The solid red line corresponds to the approximate gain calculated from equation (22). The full circles correspond to particular TGI values determined by a precise number that optimizes the function $\frac{1}{\mu }| {\partial }_{t}\mu |$ without any approximation in the purity expression. These points are obtained directly by probe inspection in the inset of figure 4(b) and are summarized in table 1. Moreover, this procedure was applied to obtain an approximate and simple expression for the TGI and its dependence on the PM correlations γ. Note that for PM correlation on the order of 150, the gain achievable is almost 15 dB over the standard uncorrelated Gaussian state. In experimental terms, this parameter can be controlled, for example, by changing the laser frequency or intensity in the initialization step (see appendix B).

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Table 1. Values with the time improvement in the estimation environmental effect for Λ = 10¹⁵ m⁻²s⁻¹.

γ	τmax (μs)	μ(τmax)	$\frac{1}{\mu }| \frac{d\mu }{dt}| ({\tau }_{\,\rm{max}\,})$(s−1)	${{\rm{\Lambda }}}^{2}{{ \mathcal F }}_{{\rm{\Lambda }}}^{\,\rm{Q}\,}({\tau }_{\,\rm{max}\,})$	TGI (dB)
−50.0	17.1	0.563	58 488	0.246	11.28
−25.0	27.2	0.563	36 861	0.247	9.24
−1.0	183.2	0.563	5 472	0.247	0.97
0.0	228.4	0.563	4 377	0.247	0
35.0	21.7	0.563	46 117	0.247	10.22
70.0	13.7	0.563	73 191	0.248	12.23
150.0	8.2	0.563	121 646	0.247	14.45

The quantum Cramér-Rao bound often provides a generalized uncertainty relation, even if no Hermitian operator can be simply associated with a given observable [69], as is the case, for phase estimation. In this context, the vacuum fluctuation (shot noise) imposes that one can estimate the phase values with a statistical error ΔΘ proportional to $1/\sqrt{n}$, using a classical beam with n average photons or n separate single-photon beams. However, it has been shown that the injection of a squeezed vacuum into the typically unutilized port of an interferometer enables the attainment of a sensitivity of 1/n^3/4 [70]. Conversely, sensitivity can be further enhanced by employing entangled states, resulting in a resolution that scales inversely with n, thereby achieving the Heisenberg limit (HL). To investigate the scaling of the QFI with n and compare the quantum advantage provided by PM correlations against shot noise, squeezing, and the Heisenberg limit, we exhibit in figure 6 the QFI (${{\rm{\Lambda }}}^{2}{{ \mathcal F }}_{{\rm{\Lambda }}}^{\,\rm{Q}\,}$) dependence with the mean number of excitations n in log scale for different values of the initial correlation γ and considering Λ = 1.0 × 10¹⁵ m⁻²s⁻¹ (T = 442 mK). The solid straight lines correspond to a power law fitting of the form ${{\rm{\Lambda }}}^{2}{{ \mathcal F }}_{{\rm{\Lambda }}}^{\,\rm{Q}\,}={n}^{s}+b$, where s adjusts how the Fisher scales with n. This plot was generated in part from figure 4, in terms of temporal parameterization t of the QFI behavior and the mean number of excitations n [see equation (12)]. In the context of statistical error and using the Cramér-Rao inequality (A2), it can be observed that the sensitivity in estimating the effective environmental coupling constant Λ scales as 1/n^(1+s)/2. The values s = 0 (s = 1) correspond to the shot noise limit (HL), respectively, while s = 1/2 to the sensitivity of a squeezed vacuum state in a closed quantum system. Thus, the value of s ≈ 0.2 for γ = 70 indicates a dependence of 1/n^0.6, demonstrating that PM correlations can be applied to beat the shot noise limit and is almost close to the squeezing dependence of 1/n^0.75 (see also appendix F where we explore the connection of PM correlation with squeezing), but is still relatively far from reaching HL. Since the generation of entanglement is typically a resource-intensive process, this finding is particularly relevant, as it suggests the potential for metrological enhancement solely through the correlation between two degrees of freedom associated with conjugate observables. This paves the way for advancing quantum metrology without requiring resources like entanglement.

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The Fisher information is a measure of information that an observable random variable carries about an unknown parameter [79]. It expresses the level of uncertainty of a measured physical quantity. Let P(x_i∣Θ) be the conditioned probability of measuring data x_i given a specific value of Θ. The Classical Fisher Information (CFI) for estimation of the parameter Θ is defined by

$$\begin{eqnarray}&&{{ \mathcal F }}_{{\rm{\Theta }}}^{\,\rm{C}\,}({\rm{\Theta }})=\displaystyle \sum _{i}P({x}_{i}| {\rm{\Theta }})\left(\right.\frac{\partial \mathrm{ln}[P({x}_{i}| {\rm{\Theta }})]}{\partial {\rm{\Theta }}}{\left)\right.}^{2}=\displaystyle \sum _{i}\frac{1}{P({x}_{i}| {\rm{\Theta }})}\left(\right.\frac{\partial P({x}_{i}| {\rm{\Theta }})}{\partial {\rm{\Theta }}}{\left)\right.}^{2}.\end{eqnarray} \tag{ A1 }$$

If P(x_i∣Θ) exhibits a peak as a response to variations in Θ, then, the data x_i provides the information needed to estimate the parameter Θ. On the other hand, when P(x_i∣Θ) is flat, numerous samples of x_i would be required to estimate the value of Θ, which would be determined only by using the complete sample population. In this context, the Cramér-Rao inequality [80] establishes the lower bound on standard deviation ${\rm{\Delta }}{\rm{\Theta }}=\sqrt{\langle {{\rm{\Theta }}}^{2}\rangle -{\langle {\rm{\Theta }}\rangle }^{2}}$ of the estimated parameter

$$\begin{eqnarray}&&{\rm{\Delta }}{\rm{\Theta }}\geqslant \frac{1}{\sqrt{n{{ \mathcal F }}_{{\rm{\Theta }}}^{\,\rm{C}\,}({\rm{\Theta }})}},\end{eqnarray} \tag{ A2 }$$

where n is the number of times that the experiment is repeated. Therefore, the Fisher information determines the reachable accuracy of the estimated quantity and represents the figure of merit in parameter estimation problems. Import to mention, that this inequality is valid only for unbiased estimators, i.e. estimators for which 〈Θ_est〉 = Θ_real.

In quantum theory, we use a set of positive operator-values measure (POVM) $\{\hat{A}(\kappa )\}$, parameterized by κ, to describe the measurement procedure. These operators are positive and satisfy $\displaystyle \sum _{\kappa }\hat{A}(\kappa )=\hat{I}$ to ensure normalization. The probability can be expressed as $P(\kappa | {\rm{\Theta }})=\,\rm{Tr}\,[\hat{\rho }\{\hat{A}(\kappa )\}]$. Using P(κ∣Θ) as the probability distribution, the CFI is defined as

$$\begin{eqnarray}&&{{ \mathcal F }}_{{\rm{\Theta }}}^{\,\rm{C}\,}({\rm{\Theta }};\hat{A})=\displaystyle \sum _{\kappa }\frac{1}{P(\kappa | {\rm{\Theta }})}\left(\right.\frac{\partial P(\kappa | {\rm{\Theta }})}{\partial {\rm{\Theta }}}{\left)\right.}^{2},\end{eqnarray} \tag{ A3 }$$

where $\hat{\rho }$ is the density operator describing the system. Naturally, optimal POVMs are those characterized by a statistical distribution of measurement results that is maximally sensitive to changes in the parameter Θ [81].

The Quantum Fisher information (QFI), here denoted by ${{ \mathcal F }}_{{\rm{\Theta }}}^{\,\rm{Q}\,}$, is defined by maximizing equation (A3) over all possible POVMs $\{\hat{A}(\kappa )\}$ as follows [82–85]

$$\begin{eqnarray}&&{{ \mathcal F }}_{{\rm{\Theta }}}^{\,\rm{Q}\,}({\rm{\Theta }})=\mathop{\max }\limits_{\hat{A}}{{ \mathcal F }}_{{\rm{\Theta }}}^{\,\rm{C}\,}({\rm{\Theta }};\hat{A}).\end{eqnarray} \tag{ A4 }$$

The quantum estimation is then related to the optimal possible measurement precision. Note that this maximization procedure turns the QFI into an upper bound for the classical one, i.e. ${{ \mathcal F }}_{{\rm{\Theta }}}^{\,\rm{Q}\,}({\rm{\Theta }})\geqslant {{ \mathcal F }}_{{\rm{\Theta }}}^{\,\rm{C}\,}({\rm{\Theta }})$ [84]. Quantum metrology is not the only application of quantum Fisher information, alternatives include quantum cloning [86], entanglement detection [87, 88], and quantum phase transition [89]. Feng and Wei [90] have highlighted the relationship between quantum coherence and Quantum Fisher Information (QFI), demonstrating that QFI is useful for quantifying quantum coherence. This relationship is established because QFI satisfies monotonicity under typical incoherent operations and convexity when quantum states are mixed. In [91], it was demonstrated how QFI can be used to identify quantum correlations such as steering, showing that such correlations can be useful for quantum-enhanced measurement protocols.

For a single-mode Gaussian state, with covariance matrice σ and first moments d, the QFI is given by [62, 92]

$$\begin{eqnarray}&&{{ \mathcal F }}_{{\rm{\Theta }}}^{\,\rm{Q}\,}({\rm{\Theta }})=\frac{\,\rm{Tr}\,[{({{\boldsymbol{\sigma }}}^{-{\bf{1}}}{\partial }_{{\rm{\Theta }}}{\boldsymbol{\sigma }})}^{2}]}{2(1+{\mu }^{2})}+2\frac{{({\partial }_{{\rm{\Theta }}}\mu )}^{2}}{1-{\mu }^{4}}+2{({\partial }_{{\rm{\Theta }}}{\boldsymbol{d}})}^{\,\rm{T}\,}({{\boldsymbol{\sigma }}}^{-{\bf{1}}})({\partial }_{{\rm{\Theta }}}{\boldsymbol{d}}),\end{eqnarray} \tag{ A5 }$$

where $\mu =1/\sqrt{\,\rm{det}\,({\boldsymbol{\sigma }})}$ is the purity of the quantum state and ∂_Θ denotes differentiation with respect to the parameter Θ. The first term is associated with the dynamical dependence of the covariance matrix with the Θ parameter. The second one is the dynamic of purity under Θ variation, and the third is the contribution of the moments dynamics of the Gaussian state for the estimated parameter. For example, this result has been utilized in metrological protocols that take advantage of the superradiant phase transition in the Rabi model [93]. For our system under investigation, we will apply the result (A5) throughout the work to quantify the QFI.