1 Introduction

In recent years, there has been a growing interest in studying the properties of systems consisting of particles confined in external potentials [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]. This interest has been fuelled by significant experimental advances that enabled the creation of such systems in laboratories. Entanglement in these systems has received considerable attention because of its relevance to quantum information technology and the challenge of quantifying the degree of correlation [17,18,19,20,21]. Significant scientific effort has been dedicated to understanding the correlation properties of harmonically trapped systems. These systems include those with harmonics [22], delta contact [23, 24], finite-range soft-core [25], inverse power law [26,27,28,29,30], and \(\wp \)-wave interactions [15, 16, 23]. Several studies have explored entanglement in natural systems such as helium and helium-like atoms [31,32,33,34]. A review of entanglement in composite systems, including atoms and molecules, can be found in [35]. In addition, recent advances in machine learning and deep learning are opening up new opportunities to study entanglement properties in various quantum systems [36, 37], including few-body systems [38,39,40]

The goal of our research is to better understand the correlation properties of a two-dimensional system consisting of two bosonic atoms in a central harmonic trap that interact through a finite-range soft-core potential.

The Hamiltonian that describes this system is given by

$$\begin{aligned} {{\mathcal {H}}} = \sum _{i=1}^2\left[ -\frac{\hbar ^2\nabla ^{2}_{{\textbf{r}}_{i}}}{2\textrm{m}} + V({\textbf{r}}_i)\right] +U(|{\textbf{r}}_1-{\textbf{r}}_2|), \end{aligned}$$
(1)

where \(V({\textbf{r}})=m\omega ^2 {\textbf{r}}^2/2\) and

$$\begin{aligned} { U}(\varvec{r}) = \left\{ \begin{array}{lll} \kappa , &{} 0\le &{}{\varvec{r}}\le \sigma \\ 0, &{} {\varvec{r}}>\sigma , \end{array} \right. \ \end{aligned}$$
(2)

where \(\kappa \) and \(\sigma \) are the strength and range of interactions, respectively. Interaction (2) can be applied to simulate various atomic systems [41,42,43,44,45], including those with Rydberg-dressed atoms [46,47,48,49]. The analysis of this system can be simplified because the trapping potential is in harmonic form, i.e., the Hamiltonian (1) can be decomposed into relative motion and center of mass motion [50]. An additional advantage of this system is that its solutions can be expressed by special functions [51, 52]. Exact closed-form solutions are available for certain values of the control parameters [52]. Some properties of this system, including the energy levels and the radial and angular distributions, are studied in [52].

Here, we present our findings in a structured manner. First, in Sect. 2, we analyze particle correlations in the ground state across a broad range of control parameters. In Sect. 3, we examine the correlations between the subsystems associated with radial and angular variables. A notable observation in this section is that in the hardcore limit \(\kappa \rightarrow \infty \), the correlations between the atoms can be accurately decomposed into radial and angular correlations. A detailed analysis of these correlations is provided. Finally, the conclusions in Sect. 4 highlight the main findings of this study.

2 Particle correlations

In this study, we focus on exploring the ground states of two bosonic atoms. This state has a total orbital angular momentum equal to zero. It depends only on the radial coordinates \(r_{1}\) and \(r_{2}\) as well as the angular coordinate between the particles, \(\theta _{12}=\varphi _{1}-\varphi _{2}\). \(\Psi ({\textbf{r}}_{1},{\textbf{r}}_{2})={\Psi }(r_{1}, r_{2},\textrm{cos}(\theta _{12}))\), where the normalization condition gives \(2\pi \int _{0}^{2\pi }\int _{0}^{\infty }\int _{0}^{\infty } d\theta _{12}dr_{1}dr_{2}r_{1}r_{2}|\Psi |^2=1\). Schmidt decomposition is a tool for analyzing the correlations between two bosons [53]. To obtain the Schmidt formula for the wave function \(\Psi \), we first decompose it into a Fourier-Lagrange series represented as follows:

$$\begin{aligned} \sqrt{r_{1}r_{2}}\Psi ({\textbf{r}}_{1},{\textbf{r}}_{2})={g_{0}(r_{1}, r_{2})\over {2\pi }}+\sum _{l=1}{ g_{l}(r_{1},r_{2}) \textrm{cos}[l(\varphi _{1}-\varphi _{2})]\over {\pi }}, \end{aligned}$$
(3)

where the term \(\sqrt{r_{1}r_{2}}\) provides the correct normalization in the radial directions and the component \(g_{l}(r_{1}, r_{2})\) is calculated using the integral

$$\begin{aligned} g_{l}(r_{1}, r_{2})=\sqrt{r_{1}r_{2}}\int _{0}^{2\pi }d\theta _{12} \Psi ({\textbf{r}}_{1},{\textbf{r}}_{2}) \textrm{cos}(l\theta _{12}). \end{aligned}$$
(4)

The function \(g_l(r_{1}, r_{2})\) is both real and symmetric, which means we can express it using the Schmidt decomposition as

$$\begin{aligned} g_l(r_{1}, r_{2}) = \sum _{n=1} k_{nl} \chi _{nl}(r_1) \chi _{nl}(r_2),\end{aligned}$$
(5)

where \(\langle \chi _{nl} | \chi _{n'l} \rangle = \delta _{nn'}\). Using the above expression and the identity for the cosine function in terms of exponential functions \(\textrm{cos}(l\theta ) = (e^{{\mathfrak {i}} l\theta } + e^{-{\mathfrak {i}}l\theta })/2\) (where \({\mathfrak {i}}\) is an imaginary unit), we can represent the wave function \(\Psi \) as

$$\begin{aligned} \Psi ({\textbf{r}}_{1},{\textbf{r}}_{2})=\sum _{\begin{array}{c} n=1\\ l=-\infty \ldots \infty \end{array}}k_{nl}u^{*}_{nl}({\textbf{r}}_{1})u_{nl}({\textbf{r}}_{2}), \end{aligned}$$
(6)

where the single-particle orbitals are given by

$$\begin{aligned} u^{}_{nl}({\textbf{r}})={\chi _{nl}(r)\over \sqrt{r }}{e^{{\mathfrak {i}} l\varphi }\over \sqrt{2\pi }},\end{aligned}$$
(7)

\(k_{nl}=k_{n|l|}\) and \(\chi _{nl}(r)=\chi _{n|l|}(r)\). These orbitals form an orthonormal basis set; that is, \(\int _{0}^{2\pi } \int _{0}^{\infty }d\varphi dr [r u^{*}_{nl}({\textbf{r}})u^{}_{n^{'}l^{'}}({\textbf{r}})]=\delta _{nn^{'}}\delta _{ll^{'}}\). Therefore, we conclude that Eq. (6) represents the Schmidt decomposition of the wave function \(\Psi \). Note that both the orbital \(u^{}_{nl}({\textbf{r}})\) and its complex conjugate are eigenfunctions of the angular momentum operator \((-{\mathfrak {i}}\hbar \partial _{\varphi })\) and the spatial reduced density matrix (RDM),

$$\begin{aligned} \rho ({\textbf{r}},{\textbf{r}}^{'})=\int \Psi ^{*}({\textbf{r}},{\textbf{r}}_{2})\Psi ({\textbf{r}}^{'},{\textbf{r}}_{2})d{\textbf{r}}_{2}, \end{aligned}$$
(8)

that is,

$$\begin{aligned} \rho ({\textbf{r}},{\textbf{r}}^{'})=\sum _{\begin{array}{c} n=1\\ l=-\infty \ldots \infty \end{array}}\lambda _{nl}u_{nl}({\textbf{r}})u^{*}_{nl}({\textbf{r}}^{'}), \end{aligned}$$
(9)

where the eigenvalues \(\lambda _{nl}\) (occupancies) are related to the Schmidt coefficients \(k_{nl}\) by \(\lambda _{nl}=k_{nl}^2\), all occupancies except those with \(l=0\) are doubly degenerate and the conservation of probability yields \(\sum _{nl=0}\lambda _{nl}+2\sum _{nl=1}\lambda _{nl}=1\). The state of two bosonic atoms is nonentangled if and only if it can be represented by a single permanent [53]. This occurs when there exists a Schmidt coefficient equal to either \(k_{n0}=1\) or \(k_{nl}=1/\sqrt{2}\). Note that the remaining Schmidt coefficients disappear in each of these cases owing to the conservation of probability.

Fig. 1
figure 1

The graphs labeled a and b depict the participation ratio K and the corresponding collective occupancies \(f_{l}\) in the hardcore limit \(\kappa \rightarrow \infty \) as functions of \(\sigma \), respectively. Graph c illustrates the participation ratio K for the three different values of \(\sigma \) as a function of \(\kappa \). Plot d shows the collective occupancies \(f_{l}\) for \(\sigma \) equal to 1 (solid lines) and 1.5 (dashed lines) as functions of \(\kappa \). The interaction range \(\sigma \) and strength \(\kappa \) are measured in \(\sqrt{\hbar /\textrm{m}\omega }\) and \(\hbar \omega \), respectively

Full size image

To quantify the degree of correlation in the ground state \(\Psi \), we use the participation ratio \({K}={{\mathcal {P}}}^{-1}\) [54], where \({{\mathcal {P}}}=\textrm{Tr}{\hat{\rho }}^{2}\) is the purity of the RDM, which is given by

$$\begin{aligned} {{\mathcal {P}}}=\sum _{{\begin{array}{c} n=1\\ l=-\infty \ldots \infty \end{array}}}\lambda _{nl}^2.\end{aligned}$$
(10)

The participation K counts approximately the number of one-particle orbitals actively involved in Schmidt decomposition (6). A helpful tool for analyzing particle correlations is the collective occupancy,

$$\begin{aligned} f_{l}=(2-{\delta }_{l0})\sum _{n=1}\lambda _{nl},\end{aligned}$$
(11)

which calculates the probability of finding a pair of particles, where one has an angular momentum of \(\hbar l\) and the other has an angular momentum of \(-\hbar l\), \(\sum _{l=0}f_{l}=1\). Note that the values of \({{\mathcal {P}}}\) and \(f_{l}\) can be determined without explicitly calculating occupancies. Instead, they can be expressed in terms of integrals \({{\mathcal {P}}}=\sum _{l}\int [\rho _{l}(r,r^{'})]^2dr dr^{'}\) and \(f_{l}=(2-{\delta }_{l0})\int \rho _{l}(r, r)dr\), where \(\rho _{l}(r,r^{'})=\int g_{l}(r, r_{2})g_{l}(r^{'},r_{2})dr_{2}\).

Figure 1 summarizes our results for particle correlations in the ground state. The left panel shows the results obtained for the hardcore limit \(\kappa \rightarrow \infty \) as a function of \(\sigma \). The plots labeled (a) and (b) depict the results of the participation ratio and corresponding collective occupancies, respectively. The participation ratio increases almost linearly with increasing \(\sigma \), indicating that the number of significant Schmidt orbitals in Eq. (6) increases in a similar manner. The fraction of particle pairs with only radial correlations \((f_{0})\) decreases steadily with \(\sigma \). In contrast, the collective occupancies for higher values of l exhibit more complicated behavior. Initially, the value of collective occupancy \(f_{1}\) increases with increasing \(\sigma \) and reaches parity with \(f_{0}\) around \(\sigma = 2\). Beyond this point, the contribution of \(f_{1}\) decreases, and the components with higher l become more prominent, indicating more complex correlation effects. The right panel of Fig. 1 shows the results obtained for finite interaction strengths \(\kappa \). We observe that the effect of varying \(\sigma \) on the correlations in the attraction regime is the opposite of that in the repulsion regime. Notably, it is worth noting that increasing \(\sigma \) leads to a significant expansion in the range of negative \(\kappa \) values, where the entanglement is weak. The considered state becomes nearly unentangled (\(K\approx 1\)) over a wide range of attractive forces starting from \(\sigma =2\).

3 Radial and angular correlations

Schmidt’s formula for analyzing the correlations between subsystems associated with the radial variable \(\textbf{r}=(r_{1},r_{2})\) and angular variable \(\mathbf {\varphi }=(\varphi _{1},\varphi _{2})\) is as follows:

$$\begin{aligned} \sqrt{r_{1}r_{2}}\Psi ({\textbf{r}}_{1},{\textbf{r}}_{2})=\sum _{n=1}{q}_{n}{{\mathcal {V}}}_{n}(\textbf{r}){\Phi }_{n}(\mathbf {\varphi }), \end{aligned}$$
(12)

\(\langle {{\mathcal {V}}}_{n}|{{\mathcal {V}}}_{n^{'}}\rangle =\langle {\Phi }_{n}|{\Phi }_{n^{'}}\rangle =\delta _{nn^{'}}\). To obtain form (12), we proceed as follows. First, we note that the wave function \(\Psi \) can be decomposed as

$$\begin{aligned} \sqrt{r_{1}r_{2}}\Psi ({\textbf{r}}_{1},{\textbf{r}}_{2})=\sum _{lm}W_{lm}{\Theta }_{l}(\mathbf {\varphi }){ v}_{m}(\textbf{r}), \end{aligned}$$
(13)

where

$$\begin{aligned} \Theta _{1}(\mathbf {\varphi })={1\over 2\pi },\Theta _{l}(\mathbf {\varphi })={\textrm{cos}[(l-1) (\varphi _{1}-\varphi _{2})]\over \sqrt{2}\pi },\end{aligned}$$
(14)

\(\langle \Theta _{l}|\Theta _{l^{'}}\rangle =\delta _{ll^{'}}\) and \(\{{v}_{m}(\textbf{r})\}\) is some set of orthonormal basis functions, \(\langle v_{m}|v_{m^{'}} \rangle =\delta _{mm^{'}}\). The expansion coefficients are given by

$$\begin{aligned} W_{lm}=\sqrt{2-\delta _{l1}}\int g_{l}(r_{1},r_{2}){v}_{m}(\textbf{r})dr_{1}dr_{2}. \end{aligned}$$
(15)

Next, using the singular value decomposition theorem \({\textbf {W}}={\textbf {U}}{\textbf{Q}}{} {\textbf {V}}^T\), \({\textbf {W}}=[W_{lm}]\), we obtain the Schmidt coefficients \({q}_{n}={\textbf{Q}}_{nn}\) and the Schmidt orbitals

$$\begin{aligned} {{\mathcal {V}}}_{n}(\textbf{r})=\sum _{j=1}({\textbf{V}})_{jn}{v}_{j}(\textbf{r}),\end{aligned}$$
(16)

and

$$\begin{aligned} \Phi _{n}(\mathbf {\varphi })=\sum _{j=1} ({\textbf {U}})_{jn}\Theta _{j}(\mathbf {\varphi }).\end{aligned}$$
(17)

In our calculations we used a basis set \(\{v_{m}\}\) with permanent \({v}_{m}(\textbf{r})=\textrm{per}[{\tilde{v}}_{m_{1}}(r_{1}),{\tilde{v}}_{m_{2}}(r_{2})]\) formed by single-particle orbitals. We have chosen a simple basis of trigonometric functions \({\tilde{v}}_{s}(r)=(\sqrt{2/L})\textrm{sin}(s \pi r/L)\) for a sufficiently large box size L, which gives good convergence. The eigenvalues of the RDMs for the radial and angular subsystems, denoted by \({\gamma }_{n}\), are identical and given by \({\gamma }_{n}={q}_{n}^2\) (\(\sum \gamma _{n}=1\)). To assess the coupling between the radial and angular correlations, we rely on the largest eigenvalue \(\gamma _{1}\). A given state is the ideal product of the radial and angular components when \({\gamma }_{1}=1\), which is the case for the ground state without interactions. Alternatively, \({\gamma }_{0}\) can be interpreted as the fraction of a pair of particles with independent radial and angular correlations.

Fig. 2
figure 2

Results obtained in the hardcore limit \(\kappa \rightarrow \infty \) as a function of \(\sigma \). Graph a shows the behavior of the largest eigenvalue of the RDM for the radial/angular subsystem \({\gamma }_{1}={q}_{1}^2\). Plots b and c show the behavior of the participation ratios \(K_{\textbf{r}}\) and \(K_{\mathbf {\varphi }}\), respectively. The potential range \(\sigma \) is measured in \(\sqrt{\hbar /\textrm{m}\omega }\)

Full size image

First, we focus on the limit \(\kappa \rightarrow \infty \). Our analysis indicates that \({\gamma }_{1}\) is almost equal to one regardless of the interaction range \(\sigma \). This is illustrated in Fig. 2a. Based on this finding, we can conclude that the following approximation is valid in the strong repulsion regime:

$$\begin{aligned} \sqrt{r_{1}r_{2}}\Psi ({\textbf{r}}_{1},{\textbf{r}}_{2})\approx {{\mathcal {V}}}_{1}(r_{1},r_{2}){\Phi }_{1}(\varphi _{1},\varphi _{2}).\end{aligned}$$
(18)

This means that the correlations between the radial and angular coordinates are nearly negligible. However, correlations still exist within both radial and angular variables. As discussed in [52], for a sufficiently large value of \(\sigma \), crystallization is allowed in the strong repulsion regime, i.e., the angular correlations carry the particles on opposite sides of the trap center. The minimum in the behavior of \(\gamma _{1}\), which appears at about \(\sigma = 0.8\), indicates an increase in the correlation between radial and angular coordinates and can be identified with the transition to the crystallization regime.

To obtain the Schmidt form of \({{\mathcal {V}}}_{1}(r_{1},r_{2})\), diagonalization is necessary, whereas using the identity for the cosine function in terms of exponential functions, the orbital \({\Phi }_{1}(\varphi _{1},\varphi _{2})\) can be written in Schmidt form as follows:

$$\begin{aligned} \Phi _{1}({\varphi }_{1},\varphi _{2})=\sum _{l=-\infty \ldots \infty }w_{l} \phi ^{*}_{l}(\varphi _{1})\phi _{l}(\varphi _{2}),\end{aligned}$$
(19)

where \(\phi _{l}(\varphi )={e^{{\mathfrak {i}} l\varphi }/\sqrt{2\pi }} \), \( w_{l}=({\textbf {U}})_{|l|+1,1}/\sqrt{2-\delta _{l0}}\). Figure 2b, c shows the results of the participation ratios calculated for the angular and radial components \({{\mathcal {V}}}_{1}(r_{1},r_{2})\) and \({\Phi }_{1}(\varphi _{1},\varphi _{2})\) denoted by \(K_{\textbf{r}}\) and \(K_{\mathbf {\varphi }}\), respectively. As can be seen, the radial correlations are generally weaker than the angular correlations, and the difference becomes more pronounced as \(\sigma \) increases.

Fig. 3
figure 3

The plot shows the behavior of the eigenvalue \(\gamma _{1}\) in a soft-core interaction scenario. The range \(\sigma \) and strength \(\kappa \) are measured in \(\sqrt{\hbar /\textrm{m}\omega }\) and \(\hbar \omega \), respectively

Full size image

For completeness, we also computed the value of \(\gamma _{1}\) for finite values of \(\kappa \). Our results are shown in Fig. 3, where it can be seen that when \(\sigma \) is sufficiently large, \(\gamma _{1}\) exhibits a local minimum. As \(\sigma \) increases, the position of this minimum shifts toward larger values of \(\kappa \), and its value decreases. The parameter value at which the minimum occurs indicates the transition point to the regime of strong repulsion. Regarding the attractive forces, we observe that for fixed \(\sigma \), \(\gamma _{1}\) decreases monotonically as \(\kappa \) decreases.

4 Conclusions

We studied the correlation between two atoms interacting through a finite-range soft-core potential and confined in a harmonic trap. By employing Schmidt decomposition, we determined the correlations between the particles and between the subsystems associated with their radial and angular variables. When the interaction range is fixed, the entanglement between the particles is a monotonic function in both the repulsive and attractive regimes. Our results showed that the coupling between the radial and angular correlations has a complicated dependence on the system parameters. However, in the hardcore limit, it becomes almost insignificant and independent of the interaction range. Consequently, in this limit, the particle correlations are almost exactly decomposed into radial and angular correlations, which can be analyzed separately. Entanglement in both radial and angular components increases monotonically as the interaction range increases.

Our results enhance the understanding of quantum correlations and suggest that studying the correlations between the radial and angular subsystems can yield further insights into particle correlations. To gain a deeper understanding, it is crucial to examine how factors such as the type of interaction, particle number, and confinement potential affect these correlations.