1. Introduction
Quantum sensing exploits the peculiarities of the quantum world and has emerged as an important pillar of quantum technology [1–4]. It steers the precision measurement of physical observables to achieve a sensitivity beyond the classical limit mostly by exploiting quantum superposition [5–8] and quantum entanglement [9, 10], which offers unprecedented accuracy having diverse applications, including navigation [11, 12], atomic clocks [13, 14], gravimeters [15–17] and gradiometers [15, 18], electron microscopes [19], biomedical advancements [20] and applications in ultracold atoms [21, 22]. It has been customary to use the coherent states (CSs) of different physical systems to devise a novel mechanism of quantum sensing [23–27]. An optical CS maintains its minimum uncertainty and imposes the standard quantum limit (SQL) which can be visualized by the phase space area in Planck scale using the Wigner quasi-probability distribution [28, 29]. However, this limit can be further pushed down to a sub-Planck dimension for nonlocal quantum superposition of CSs, such as Schrödinger cat state, compass state etc [7, 23–25, 27]. These sub-Planck interference structures are physically meaningful and can set the limit of quantum sensitivity beyond the Heisenberg limit [25, 26, 30–34]. They have been extensively studied for systems modelled by the harmonic oscillator and also for other quantum systems with nonlinear energy spectra like a Kerr medium, where the time evolution manifests fractional revivals [35–44]. It has been observed that the sub-Planck structures mediated by fractional revivals play a vital role in quantum sensing [25, 27, 30, 45].
It is worth mentioning that the molecular system is one of the well-studied systems having a nonlinear energy spectrum [46, 47]. The sophisticated experiments involving ultra-short laser pulses and an ultrafast coherent control method have enabled observations and manipulations of tiny quantum interference structures in laboratories [48–51]. Though anharmonic systems are relatively nontrivial to analyse, significant progress has been observed towards characterising and improving the quantum precision limit for a diatomic molecular system [25, 30, 52, 53].
In this work, we explore the quantum sensitivity of the photon-added CS (PACS) and photon-subtracted CS (PSCS) of a diatomic molecule at fractional revival instances. Photon addition and subtraction actually change the average number of photons of the system. As a result, the sub-Planck structures formed by the mesoscopic superposition states get affected which modulates the limit of quantum sensitivity. The photon-added coherent state (PACS) was first introduced by G S Agarwal and K Tara [54], which was followed by a number of investigations by both theory and experiments of PACS and PSCS in both linear and nonlinear optical systems [34, 55–67]. However, insignificant emphasis has been given to such studies for a molecular system in comparison. Hence, it is interesting to explore the PACS or PSCS for a physical system like a diatomic molecule to further probe the quantum sensitivity limit and to report a reliable candidate for modern quantum technologies. We consider a vibrating diatomic molecular system for a photon-added/subtracted CS to analyze the quantum sensing limit. Though quantum sensing with a molecular wave packet is challenging, the system dynamics of this anharmonic system manifests diverse interesting phenomena.
In section 2 , we describe the particular system and construction of its SU(2) coherent state. The PACSs and PSCSs are evaluated in section 3 , which are followed by their temporal dynamics. Any quantum feature of a state is cohesive with its nonclassicality and becomes particularly vital for a molecular system with a nonlinear energy spectrum. For nonclassicality, we study the Mandel Q parameter in section 4 and emphasize the idea of sub-Poissonian photon statistics. An elaborate description of the corresponding phase space is provided in section 5 with the signature of nonclassicality through the Wigner negativity. Section 6 is devoted to evaluating the quantum sensitivity limit and a number of interesting artifacts. We conclude in section 7 with a brief summary of the work.
2. Potential and the Perelomov coherent state of the iodine molecule
The well-known Morse potential, which is a suitable model for a vibrating diatomic molecule, takes the form [68]
$\begin{eqnarray}V(x)=D({{\rm{e}}}^{-2\beta x}-2{{\rm{e}}}^{-\beta x}),\end{eqnarray}$
where $x=\frac{R}{{R}_{0}}-1$ gives the dimensionless displacement from the equilibrium position (R0) and R is the internuclear separation. The values of the dissociation energy (D), and the shape parameter (β), along with R0 identify the potential for a particular diatomic molecule. Here, we consider the vibrational levels of an I2 molecule with the values of R0, β and D in atomic units of 5.03, 4.954 and 0.057, respectively. The energy eigenfunction of this potential is usually written in terms of the parameters $\begin{eqnarray}\lambda =\sqrt{\frac{2\mu D{R}_{0}^{2}}{{\beta }^{2}{\hslash }^{2}}}\quad and\quad s=\sqrt{\frac{-8\mu {R}_{0}^{2}E}{{\beta }^{2}{\hslash }^{2}}},\end{eqnarray}$
where μ is the reduced mass and ℏ is the reduced Planck constant. $\begin{eqnarray}{\psi }_{m}^{\lambda }(\xi )=N{{\rm{e}}}^{-\frac{\xi }{2}}{\xi }^{\frac{s}{2}}{L}_{m}^{s}(\xi ),\end{eqnarray}$
where ξ = 2λe−βx (0 < ξ < ∞), and ${L}_{m}^{\lambda }$ is the associated Laguerre polynomial with $0\leqslant m\leqslant \left[\lambda -\frac{1}{2}\right]$. $\left[\lambda -\frac{1}{2}\right]$ is the largest integer smaller than $\lambda -\frac{1}{2}$, which includes $\left[\lambda -\frac{1}{2}\right]+1$ number of vibrational bound states, provided s + 2m = 2λ − 1 [25]. The parameters λ and s are directly related to the dissociation energy and the total energy of the molecule, respectively, where λ > 0, s > 0. The normalization constant N is given by $\begin{eqnarray}N={\left[\frac{\beta (2\lambda -2m-1){\rm{\Gamma }}(m+1)}{{\rm{\Gamma }}(2\lambda -m){R}_{0}}\right]}^{\frac{1}{2}}.\end{eqnarray}$
For studying the wave packet dynamics of an anharmonic potential like Morse, it is customary to consider a suitable coherent state to start with. The supersymmetric approach is adopted in the work of Nieto and Simmons for the coherent states of general potentials [69] and by Benedict and Molnar for the coherent states of the Morse potential [70]. Demonstrating the Schrödinger cat state of a NO molecule is also in place [71]. The Morse potential has a finite number of bound states and its dynamical group is SU(2) [72]. The corresponding group generators are ${\hat{K}}_{+}$, ${\hat{K}}_{-}$ and ${\hat{K}}_{0}$, which satisfy the commutation relation $\left[{\hat{K}}_{+},{\hat{K}}_{-}\right]\,=2{\hat{K}}_{0}$. The explicit expression of the raising operator is $\begin{eqnarray}{\hat{K}}_{+}=\left[\frac{{\rm{d}}}{{\rm{d}}\xi }(s-1)+\frac{1}{2\xi }s(s-1)-\lambda \right]\sqrt{1-\frac{2}{s}},\end{eqnarray}$
which performs the raising operation at the level of a wavefunction as $\begin{eqnarray}{\hat{K}}_{+}{\psi }_{m}^{\lambda }(\xi )=\sqrt{(m+1)(2\lambda -m-1)}{\psi }_{m+1}^{\lambda }(\xi ).\end{eqnarray}$
Similarly, the lowering operator $\begin{eqnarray}{\hat{K}}_{-}=-\left[\frac{{\rm{d}}}{{\rm{d}}\xi }(s+1)-\frac{1}{2\xi }s(s+1)+\lambda \right]\sqrt{1+\frac{2}{s}}\end{eqnarray}$
changes the state in the following manner: $\begin{eqnarray}{\hat{K}}_{-}{\psi }_{m}^{\lambda }(\xi )=\sqrt{m(2\lambda -m)}{\psi }_{m-1}^{\lambda }(\xi ).\end{eqnarray}$
Being the spectrum-generating operators, ${\hat{K}}_{-}$ annihilates the ground state, whereas ${\hat{K}}_{+}$ annihilates the uppermost bound state (${\hat{K}}_{+}{\psi }_{{m}_{{\rm{\max }}}}^{\lambda }=0$). The SU(2) Perelomov coherent state of a Morse system can be constructed by displacing the uppermost bound state, where the displacement operator is defined as ${{\rm{e}}}^{\alpha {\hat{K}}_{+}-{\alpha }^{* }{\hat{K}}_{-}}$ [25]. The CS is written in the form $\begin{eqnarray}\chi (\xi )=\displaystyle \sum _{m=0}^{{m}_{{\rm{\max }}}}{d}_{m}{\psi }_{m}^{\lambda },\end{eqnarray}$
for $\begin{eqnarray}{d}_{m}=\frac{{(-\alpha )}^{{m}_{{\rm{\max }}}-m}}{({m}_{{\rm{\max }}}-m)!}\sqrt{\left[\frac{{m}_{{\rm{\max }}}!{\rm{\Gamma }}(2\lambda -m)}{m!{\rm{\Gamma }}(2\lambda -{m}_{{\rm{\max }}})}\right]},\end{eqnarray}$
where α is the coherent state parameter, which decides the number of energy levels contributing to form the wave packet. As mentioned, we are considering the I2 molecule for which the potential is depicted in figure 1 [52, 73]. The total number of bound states is calculated as ${m}_{{\rm{\max }}}=116$. Choosing the exact initial wave packet of such an anharmonic system becomes crucial for studying the time evolution. We need to keep in mind that the weighting distribution (dm) will change upon photon additions and subtractions in the present context, and an efficient quantum sensing requires a well-localized CS during the whole course with minimal oscillatory tails. Based on that, we choose the contributing energy levels window for α = 2.2. This value of α is considered to ensure a well-localized wave packet, having the peak of its weighting distribution at n = 10.
Figure 1. Morse potential of an iodine molecule plotted against x, where the equilibrium position is represented by R = R0. The potential is in atomic units. |
3. Photon-added and -subtracted molecular wave packets
A time-dependent solution of the above CS can be written modulo normalization as
$\begin{eqnarray}\chi (\xi ,t)=\displaystyle \sum _{m=0}^{{m}_{{\rm{\max }}}}{d}_{m}{\psi }_{m}^{\lambda }{{\rm{e}}}^{-{\rm{i}}{E}_{m}t},\end{eqnarray}$
where the nonlinear energy spectrum is governed by $\begin{eqnarray}{E}_{m}=-\left(\frac{D}{{\lambda }^{2}}\right){\left(\lambda -m-\frac{1}{2}\right)}^{2}.\end{eqnarray}$
An r-PACS is obtained by operating ${\hat{K}}_{+}$ r-times and the resultant state can be written modulo normalization as $\begin{eqnarray}\begin{array}{rcl}{\hat{K}}_{+}^{r}\chi (\xi ,t) & = & \displaystyle \sum _{m=0}^{{m}_{{\rm{\max }}}}{d}_{m}{\hat{K}}_{+}^{r}{\psi }_{m}^{\lambda }{{\rm{e}}}^{-{\rm{i}}{E}_{m}t}\\ & = & \displaystyle \sum _{m=0}^{{m}_{{\rm{\max }}}}{d}_{m}\sqrt{\frac{(m+r)!(2\lambda -m-1)!}{m!(2\lambda -m-r-1)!}}\\ & & \times {\psi }_{m+r}^{\lambda }{{\rm{e}}}^{-i{E}_{m}t}\\ & = & \displaystyle \sum _{n=r}^{{m}_{{\rm{\max }}}}{d}_{n}^{r}{\psi }_{n}^{\lambda }{{\rm{e}}}^{-{\rm{i}}{E}_{n-r}t},\end{array}\end{eqnarray}$
where, (m + r) is replaced by n and ${d}_{n}^{r}\,={d}_{n-r}\sqrt{\frac{n!(2\lambda -n+r-1)!}{(n-r)!(2\lambda -n-1)!}}$. On the other hand, an r-PSCS is obtained by operating ${\hat{K}}_{-}$ r-times and the resultant state can be written as $\begin{eqnarray}\begin{array}{rcl}{\hat{K}}_{-}^{r}\chi (\xi ,t) & = & \displaystyle \sum _{m=0}^{{m}_{{\rm{\max }}}}{d}_{m}{\hat{K}}_{-}^{r}{\psi }_{m}^{\lambda }{{\rm{e}}}^{-{\rm{i}}{E}_{m}t}\\ & = & \displaystyle \sum _{m=0}^{{m}_{{\rm{\max }}}}{d}_{m}\sqrt{\frac{(m-1)!(2\lambda -m-r-1)!}{(m-r-1)!(2\lambda -m-1)!}}\\ & & \times {\psi }_{m-r}^{\lambda }{{\rm{e}}}^{-{\rm{i}}{E}_{m}t}\\ & = & \displaystyle \sum _{{n}^{{\prime} }=0}^{{m}_{{\rm{\max }}}-r}{d}_{{n}^{{\prime} }}^{r}{\psi }_{{n}^{{\prime} }}^{\lambda }{{\rm{e}}}^{-{\rm{i}}{E}_{{n}^{{\prime} }+r}t}.\end{array}\end{eqnarray}$
This state is also required to be numerically normalized prior to finding any physical quantity. Here, (m − r) is replaced by ${n}^{{\prime} }$ and ${d}_{{n}^{{\prime} }}^{r}={d}_{{n}^{{\prime} }+r}\sqrt{\frac{({n}^{{\prime} }+r-1)!(2\lambda -{n}^{{\prime} }-1)!}{({n}^{{\prime} }-1)!(2\lambda -{n}^{{\prime} }-r-1)!}}$. We observe that introducing additional photons to the system results in a noticeable shift of the peak of the weighting distribution as displayed in figure 2(a). This shift signifies an increase in the average photon number, causing the probability to shift from n = 10 to n = 13 and n = 16 for r = 2 and 4, respectively.
Figure 2. (a) $| {d}_{n}^{r}{| }^{2}$ plotted against vibrational energy levels, n, for different values of r. (b) Spatial variations of the corresponding probability densities. In both (a) and (b), the black solid line corresponds to r = 0, the blue dashed line to r = 2 and red dot-dashed line to r = 4. We use a CS parameter of α = 2.2. All the units are in atomic units. |
The main concern is the nature of the PACS for these parameters, depicted in figure 2(b). As we add photons, the peaks in the probability density begin shifting leftwards, indicating a shift towards the higher excited states. The changes in the height and the width of the probability density are artifacts of the conservation of total probability density (area under the curve) of the wave packets. One can also observe the ripples in the probability density which arise for r = 4 in figure 2(b). It is intuitive to understand that the photon subtraction will make the PSCS move towards the lower energy levels. It is worth noting that choosing a PACS for photon addition much beyond r = 4 may not be preferable, in order to avoid the said ripples and poorer performance in quantum sensing. Similarly, choosing a PSCS for larger r should be avoided to maintain the distinguishability for mesoscopic superpositions. The crucial factors which govern the experiments with coherent states are the intensity and the central frequency of the pump laser. The latter ensures the initial wave packet peaks around the 10th vibrational level in the present study. Pump-probe spectroscopy [74, 75] or stimulated Raman scattering [76, 77] use tailored photon pulses to manipulate the wave packet coherently. A pump can excite the molecule into superposed states and a probe pulse interacts with the evolving wave packet, either adding more photons or altering its structure by inducing further transitions. The parameter α indirectly affects the photon addition or subtraction to control the corresponding probability densities.
Time Evolution: It is well known that wave packet time evolution in the nonlinear energy spectrum manifests fractional revivals [35–40]. Linear energy makes it evolve with a classical time period (Tcl) like a harmonic oscillator. However, inclusion of a nonlinear term in energy will lead to spreading, splitting of the wave packet and, finally, it will regain its shape to revive at Trev. These two time scales are given by 11 ), provided only the linear term in the energy is considered. For $\frac{p}{q}=1/4$ and 1/8, one obtains l = q/2 = 2 and 4, respectively. The two-way split state (l = 2), called a Schrödinger cat-like state, is defined as
$\begin{eqnarray}{T}_{{\rm{cl}}}=\frac{{T}_{{\rm{rev}}}}{(2\lambda -1)},\quad {T}_{{\rm{rev}}}=\frac{2\pi {\lambda }^{2}}{D}.\end{eqnarray}$
Splitting into two fractions happens at time, $t=\frac{{T}_{{\rm{rev}}}}{4}$ and four fractions at time $t=\frac{{T}_{{\rm{rev}}}}{8}$ [25]. The general expression of a wave packet at time $t=\frac{p}{q}\times {T}_{{\rm{rev}}}$ can be written as the linear combinations of classical wave packets (χcl(ξ, t)): $\begin{eqnarray}{\chi }_{{\rm{gen}}}(\xi ,t\approx \frac{p}{q}{T}_{{\rm{rev}}})=\displaystyle \sum _{s=0}^{l-1}{a}_{s}\left[{\chi }_{{\rm{cl}}}(\xi ,t+\frac{s}{l}{T}_{{\rm{cl}}})\right],\end{eqnarray}$
where $\begin{eqnarray}{a}_{s}=\frac{1}{l}\displaystyle \sum _{m=0}^{l-1}{{\rm{e}}}^{2\pi {\rm{i}}(\frac{s}{l}m-\frac{p}{q}{m}^{2})},\end{eqnarray}$
which is obtained by the discrete Fourier transform [37–39, 78]. Here, χcl(ξ, t) represents the state is equation ( $\begin{eqnarray}\begin{array}{rcl}{\chi }_{{\rm{cat}}}(\xi ,t\approx \frac{1}{4}{T}_{{\rm{rev}}}) & = & \frac{1}{\sqrt{2}}\left[\Space{0ex}{0.25ex}{0ex}{{\rm{e}}}^{\frac{-{\rm{i}}\pi }{4}}{\chi }_{{\rm{cl}}}(\xi ,t)\right.\\ & & \left.+{{\rm{e}}}^{\frac{{\rm{i}}\pi }{4}}{\chi }_{{\rm{cl}}}(\xi ,t+\frac{{T}_{{\rm{cl}}}}{2})\right],\end{array}\end{eqnarray}$
whereas the four-way split (l = 4), known as the compass-like state, is expressed as $\begin{eqnarray}\begin{array}{rcl}{\chi }_{{\rm{com}}}\Space{0ex}{0.25ex}{0ex}(\xi ,t\approx \frac{1}{8}{T}_{{\rm{rev}}}\Space{0ex}{0.25ex}{0ex}) & = & \frac{1}{2}\left[{{\rm{e}}}^{\frac{-{\rm{i}}\pi }{4}}{\chi }_{{\rm{cl}}}(\xi ,t)+{\chi }_{{\rm{cl}}}\Space{0ex}{0.25ex}{0ex}(\xi ,t+\frac{{T}_{{\rm{cl}}}}{4}\Space{0ex}{0.25ex}{0ex})\right.\\ & & -{{\rm{e}}}^{\frac{-{\rm{i}}\pi }{4}}{\chi }_{{\rm{cl}}}\Space{0ex}{0.25ex}{0ex}(\xi ,t+\frac{{T}_{{\rm{cl}}}}{2}\Space{0ex}{0.25ex}{0ex})\\ & & \left.+{\chi }_{{\rm{cl}}}\Space{0ex}{0.25ex}{0ex}(\xi ,t+\frac{3{T}_{{\rm{cl}}}}{4}\Space{0ex}{0.25ex}{0ex})\right].\end{array}\end{eqnarray}$
As the classical wave packet completes one oscillation in time Tcl, a time deference of Tcl/2 means two wave packets at two opposite ends. The probability densities of PACS at these two times $t=\frac{{T}_{{\rm{rev}}}}{4}$ and $t=\frac{{T}_{{\rm{rev}}}}{8}$ are shown in figures 3(a)–(b), where the two-way split (though asymmetric) is clearly visible for cat-like state in figure 3(a) and the four-way split is present in figure 3(b), in which the two splits (among the four) fall in the same position around the middle by creating spatial interference ripples. We also observe that the distances between the fractional wave packets increases with the addition of photons for the Schrödinger cat-like state (figure 3(a)), but they remain localized. The fractions move away from the equilibrium position of the molecule. The fractions for the compass-like state in figure 3(b) also move apart due to photon addition with simultaneous changes in the middle spatial interference oscillations. Consequently, due to the asymmetry of the potential, the overlap region between the two spatial interference structures of the CS without and with photon addition also diminishes.
Figure 3. (a) Probability densities for the time-evolved states of the photon-added cat-like state at time t = 0.25 × Trev. r yields an increasing photon number as indicated in the plot legends. (b) Probability densities of the photon-added compass-like state at t = 0.125 × Trev. In both (a) and (b), the black solid line corresponds to r = 0, the blue dashed line to r = 2 and red dot-dashed line to r = 4. We use a CS parameter of α = 2.2. All the units are in atomic units. |
4. Mandel Q parameter and nonclassicality of the molecular PACS
Nonclassicality signifies the quantumness of a system. It may depend on the desired physical situations and also on the system which one is dealing with. Some prime causes of nonclassicality are often phenomena like quantum entanglement, squeezing, quantum superposition etc. In the present context, we are dealing with quantum superposition which is expected to impart greater quantum sensitivity. Hence, it would be an interesting aspect in a molecular system to examine nonclassicality and its commensurate quantum sensitivity.
The Mandel Q parameter is very useful for classifying the quantum states based on the photon number fluctuations. The mathematical expression of the parameter is given by [79, 80]
$\begin{eqnarray}Q=\frac{\langle {\rm{\Delta }}{n}^{2}\rangle -\langle n\rangle }{\langle n\rangle }.\end{eqnarray}$
The value of the Mandel Q parameter is important in characterising the nature of the photon distribution in a system. The distribution is Poissonian if Q = 0, sub-Poissonian if Q < 0 and super-Poissonian if Q > 0. The sub-Poissonian statistics is manifested by a quantum state. In this domain, a Fock state acquires the value, Q = − 1. Moreover, photon addition also alters the Mandel Q parameter.For a molecular system, these changes are not very well defined. Particularly, the understanding of nonclassicality becomes important when we perform quantum precision measurements. We calculate Mandel Q parameters for the PACS of an iodine molecule, as shown in figure 4(a). We observe that the value of the Mandel Q parameter with no photon addition is Q = − 0.13. This negative value signifies the sub-Poissonian statistics and nonclassicality of the considered SU(2) CS of the iodine molecule. This is very different from the CS of a harmonic oscillator, which is classical. The Mandel Q parameter in figure 4(a) becomes increasingly negative and nonclassical with the addition of photons (PACS). However, the variation is not linear and tends to saturate towards Q = − 1 for a large number of photon additions.
Figure 4. (a) Mandel Q parameter plotted as a function of photon addition to the system. Negative values signify the sub-Poissonian nature of the statistics. (b) Wigner negativity (η) plotted as a function of photon addition for cat-like and compass-like states as indicated in the plot legends. |
5. Quantum phase space of the molecular PACS
Earlier, we studied the probability distributions of cat-like and compass-like states in coordinate space. Now we will see the distributions in phase space to emphasize the quantum interference structures. The Wigner quasi-probability distribution function in phase space can clearly specify the quantum interference structures, where the negative part is the signature of nonclassicality. The Wigner function of a quantum state ψ(x) is expressed as
$\begin{eqnarray}W(x,p)=\frac{1}{\pi }\int \psi (x+\zeta ){\psi }^{* }(x-\zeta ){{\rm{e}}}^{-2{\rm{i}}p\zeta }{\rm{d}}\zeta ,\end{eqnarray}$
where p is the canonically conjugate momentum.The mini-wave packets and the interference patterns for cat-like states and compass-like states, resulting from fractional revivals, are not symmetric due to the asymmetric nature of the potential. Here, we have shown the Wigner functions, i.e., the phase space quasi-probability distribution of a CS wave packet, at two specific fractional revival times with the addition of photons to the system. From both figures 5 and 6, we see that adding photons pushes the fractional wave packets away from each other, thereby increasing the number of interference fringes for both cat-like and compass-like states. As the phase space area of each interference fringe is found much below the Planck constant, these structures are called sub-Planck scale structures. These structures consist of oscillations, and a higher number of oscillations will have a larger negative region in the Wigner function. Such Wigner negativity is defined by
$\begin{eqnarray}\eta =\int \int | W(x,p)| {\rm{d}}x{\rm{d}}p-1.\end{eqnarray}$
This can also work like an indicator for nonclassicality [81]. If the Wigner function is non-negative, η becomes zero for coherent and squeezed vacuum states. We calculate the Wigner negativity (η) for a molecular PACS with various numbers of photon additions and then plot it in figure 4(b) for both cat-like and compass-like states. The initial (r = 0) nonclassicality is greater for the compass-like state (η = 1.77) in comparison to the cat-like state (η = 0.75). Both of these increase with r for a molecular PACS. Though the Mandel Q parameter provides an idea about the nonclassicality variation, the Wigner negativity looks more useful for its typical variations at different evolution times. The slope of the increment of η is slightly higher for a compass-like molecular PACS. It is also intriguing to observe the linear nature of the curve for the cat-like state (dark circles in figure 4(b)), unlike the curve for the compass-like state. In addition, the numerical integration required to find the negativity of a Morse-PACS involves a highly oscillatory function, which causes numerical error. For a larger number of photon addition, this error is quite significant, which restricts us to physically analyze the curve beyond a certain number.
Figure 5. The Wigner distribution of the molecular PACS at t = 0.25 × Trev, when it becomes a cat-like state, is shown (a) without adding photons and with addition of (b) two photons and (c) four photons. The central interference regions are zoomed to observe the evolution of the area of the sub-Planck structures. |
Figure 6. The Wigner distribution of the molecular PACS at t = 0.125 × Trev, when it becomes a compass-like state, is shown (a) without adding photons and with the addition of (b) two photons and (c) four photons. The central interference regions are zoomed to observe the evolution of the area of the sub-Planck structures and to compare them with the cat-like state. |
6. Quantum sensitivity of the molecular PACS
Quantum sensitivity depends upon the smallest area in the phase space interference structures, which can be estimated from the area of the sub-Planck scale structures. The effective phase space area, or the classical action, is defined by A = ΔxΔp and the corresponding dimension of the smallest sub-Planck scale structure is $a\approx \frac{1}{A}$ which is known as the scaling law in atomic units [23]. The equality ($a=\frac{{\hslash }^{2}}{A}$) can be analytically proven for a harmonic oscillator compass state but is difficult to show for any system with some kind of anharmonicity. Hence, one verifies the scaling law numerically before applying it to estimate quantum sensitivity, as was shown in [53] for a vibrating diatomic molecule to calculate the slope of the a versus $\frac{1}{A}$ curve as 0.99 (≈ 1). This is also verified for the system under consideration.
The a values actually set the limits of quantum sensitivity, provided these become physically meaningful. An infinitesimal displacement of the quantum state by an amount equal to the dimension of these spotty structures should make the initial and final quantum states quasi-orthogonal and distinguishable. Otherwise, these structures will not contribute towards improving the sensitivity. Hence, the smaller the area of the smallest sub-Planck structure, the greater is the quantum sensitivity. Therefore, the quantum sensitivity is quantified by the inverse of the smallest sub-Planck area and is equal to the total phase space support. The sensitivity of a harmonic oscillator coherent state satisfies the SQL. However, this limit can be pushed beyond for quantum superpositions, squeezing, entanglement etc. In our molecular case, fractional revivals manifest quantum superposition structures in phase space. Hence, it is appropriate to express the quantum sensitivity by the inverse of the dimension of the interference structures, that is $\frac{1}{a}$. We have defined this term as quantum sensitivity, plotted in figure 7.
Figure 7. Quantum sensitivities corresponding to the states in figures 5 and 6 plotted as a function of photon addition. The nature of the curve for the cat-like state is linear, whereas the same for the compass-like state becomes parabolic. The dashed lines in both curves indicate the corresponding suitable curve fitting. |
As quantum interference is directly related to the average number of photons in the system, quantum sensitivity is also directly related to that. An increase in the number of photons in a mesoscopic superposition state has a direct consequence for the quantum sensitivity. The Wigner functions of the cat- and compass-like states are represented in figures 5 and 6, respectively. Two salient changes become responsible for engineering the quantum sensitivity. Firstly, the shapes of the fractional wave packets in both the figures manifest the anharmonicity of the potential. A steeper Morse surface towards negative-x supports a wider momentum width but a narrower position width of the wave packet. These become pronounced with increasing r, offering a larger area of overlap at the center of the phase space. Secondly, their phase space supports also increase with r due to the fractional mini-wave packets getting pushed away with photon addition. We have focused on the quantum superposition states with up to four photon additions. Firstly, the addition of photons to the coherent state wave packet makes the wave packet go towards the edge of the potential well, where the Morse potential has a steeper slope, which disturbs the localization of the initial wave packet. The ripples extend beyond r = 4 (figure 2(b)). Hence, one may not practically select beyond r = 4 when it comes to taking an appropriate initial wave packet for quantum sensing. Secondly, the sub-Planck structures in compass-like states, which are the main essence of the quantum sensitivity limit, are formed by the superposition of diagonal interferences between the odd and even Schrödinger cat-like states (figure 4). With the increase of r, the fractional mini-wave packets move apart from each other and the diagonal overlapped region is diminished; beyond r = 4, we almost do not obtain any checkerboard pattern, which compromises the quantum sensitivity limit.
The enlarged figures of the central interference regions (marked by dotted squares) in both figures 5 and 6 show the details of the sub-Planck structures. The first factor mentioned above suggests that the number of interference ripples, along with the area of the overlapped region, increases with a higher number of photon additions. Due to the asymmetry of the potential, not all the ripples completely contribute to the cross diagonal overlap which can be seen in the zoomed plots in figure 6. The second effect due to photon addition makes the interference ripples finer and helps us further to improve the precision. However, these observations are only qualitative; a precise analysis will be possible, provided we quantify the quantum sensitivity limit (a) of, for instance, a cat-like state and a compass-like state.
The SQL is given by $a\approx \frac{1}{A}=1$ (Planck scale) and we have used atomic unit throughout the present study. Quantum sensitivity for the fractional molecular state is always less than 1, implying quantum sensitivity in the sub-Planck domain. We have computed quantum sensitivity up to r = 25 and have shown in figure 7 both cat-like (blue circles) and compass-like (red squares) states. Both the curves in figure 7 suggest that photon addition increases the quantum sensitivity by reducing the area of the quantum interference structures. However, the nature of the curves are quite distinct, and thus, we have gone ahead with curve-fitting. Interestingly, the quantum sensitivity (Scat) varies linearly for the molecular cat-like PACS and fits with the straight line Scat = a1q + b1, for a1 = 1.07 and b1 = 5.4. In contrast, the sensitivity of the molecular compass-like PACS (Scompass) is nonlinear and fits well with a parabola, Scompass = a2q + b2q2 + c2 with a2 = 1.31, b2 = − 0.014 and c2 = 10.76.
We also observe that the quantum sensitivity for the compass-like PACS is always higher than that of the cat-like PACS. For a non-photon-added state, quantum sensitivities are obtained by putting q = 0 (or r = 0) in these equations to obtain 5.4 and 10.76, respectively. It is important to notice in figure 7 that the difference between the quantum sensitivities of cat- and compass-like PACSs is also not uniform with photon addition. The quantum sensitivity curves tend to merge for large r, which is attributed to the fact that, beyond a certain value of r for a compass-like PACS, the horizontal cat and vertical cat do not have a common interference region in phase space. Such a trend is also visible in the zoomed plots in figure 6. This makes them effectively distinct cat-like pairs, having no checkerboard structures in phase space and the quantum sensitivity becomes comparable to a cat-like PACS. Another interesting observation is regarding the maximum difference between the quantum sensitivities of cat-like and compass-like PACSs of the iodine molecule, which does not happen at r or q = 0 (but say at ${q}_{{\rm{\max }}}$). We calculate the value from the two fitted equations to obtain ${q}_{{\rm{\max }}}=8.57$. Hence, the maximum difference will occur for r = 9 and r = 8. It is very interesting to observe that the linear and parabolic natures of the nonclassicality, as shown in figure 4(b), are also reflected in the quantum sensitivity curve in figure 7.
Though we have studied the quantum sensitivity of a temporally evolved molecular PACS for cat- and compass-like situations, it may be worth addressing briefly the case of a molecular PSCS for comparison.
In figure 8, we have visualized a comparison between the quantum sensitivities for cat-like and compass-like states of both PACS and PSCS. Figure 8 clearly indicates that the quantum sensitivity increases with increasing number of photon additions for both cat-like and compass-like states, whereas it decreases for the photon-subtracted states. It is also interesting to observe the merging of the sensitivity curves of PACS (cat-like state) and PSCS (compass-like state) at r = 4 in figure 8.
Figure 8. Comparative analysis of the quantum sensitivities between molecular PACSs and molecular PSCSs with respect to the number of added or subtracted photons to the system for t = 0.25 × Trev (cat-like state) and t = 0.125 × Trev (compass-like state). |
7. Conclusions
We consider a suitably prepared PACS of an iodine molecule and study its the time evolution. The nature of the photon-added coherent states comprising 116 vibrational bound states of the molecule is quite distinct from that of a well-studied harmonic potential. We have chosen the coherent state parameters in such a way that the structures can be generated prominently by adding a subsequent number of photons, despite the asymmetric nature of the mini-wave packets. We have evaluated the Mandel Q parameter and Wigner negativity to quantify the sub-Poissonian nature and the nonclassicality of the photon-added molecular wave packet. Quantum sensitivities of Schrödinger cat-like and compass-like states are quantified and compared. The conditions for relative improvement in quantum sensitivity are revealed and corresponding physical reasons are clarified. Our study enhances the quantum sensitivity limits in quantum precision measurement, which can have significant implications across various anharmonic systems for improving technological performances, mediated by photon addition. It is fascinating to observe the reported physical analogy between the nonclassicality and quantum sensitivity. In particular, photon addition modulates the nonclassicality in a linear fashion for a cat-like state and parabolically for a compass-like state and the same variations are mimicked in their quantum sensitivities. Overall, the unveiled scientific artifacts are suitable for making an improved quantum sensor for vibrating molecule.