Entanglement in S states of two-electron quantum dots with Coulomb impurities at the center.

Entanglement in states of two-electron quantum dots with Coulomb impurities at the center.


We study a system of two Coulombically interacting electrons in an external harmonic potential in the presence of an on-centre Coulomb impurity. Detailed results for the dependencies of the reduced von Neumann entropy on the control parameters of the system are provided for both the ground state and the triplet states with the lowest energy. Among other features, it is found that in the weak confinement regime the entanglement is strongly affected by the presence of an acceptor impurity.

I Introduction

In the last few years there has been an explosion of interest in the entanglement properties of systems of interacting particles, in view of their possible use in quantum information technology (1); (2). In particular, great efforts have been made to explore quantum entanglement in systems of two interacting particles, including model systems such as the Moshinsky atom (3); (4); (5); (6), the Crandall atom (7), or quantum dot systems with harmonically shaped traps (8); (9); (10); (11); (12). In the most recent years, research activity has expanded towards investigating the entanglement in real two-electron systems, i.e., the helium atoms and helium ions. For instance, Manzano et al. (7), Dehesa et al.(13) and, Benenti et al.(14) have addressed their investigations to the relation between entanglement and energy for the helium atom. Lin et al. (15) calculated the linear entropy for the helium ion states for up to , for a wide range of values of the nuclear charge (from to ). They found that in contrast to the ground state, the spatial entanglement of excited states increases with increasing nuclear charge. The entanglement in the spherical helium model, where the Coulombic interaction between the electrons is replaced by its spherical average, has also been discussed in the literature (16); (17). An overview of recent developments, both theoretical and experimental, in entanglement studies of quantum composite systems, including atoms and molecules, can be found in (18).

Few attempts have been made recently towards investigating the effect of a Coulomb impurity on the properties of quantum dots (19); (20); (21); (22). The systems of confined helium atoms and helium ions that are closely related to quantum dots with a Coulomb impurity have received much attention in recent years (23); (24); (25); (26); (27); (28). However, as far as we know, the entanglement properties of such systems have not been extensively studied in the literature. A simple candidate for studying the Coulomb impurity effect is the system of two interacting electrons in the dot modeled by a harmonic potential


where is the effective charge of the impurity. In particular, for , the system (1) describes the confined helium atoms and helium ions centred in a harmonic trap, respectively. The effects of the harmonic confinement on the ground state energy of a helium atom was estimated for a series of values of in (24).

The purpose of the present Letter is to gain insight into how both the effective charge and the confinement size influence the entanglement. To fully reveal the effect of the Coulomb impurity on the entanglement, we provide results for both the acceptor () and donor () impurities and compare them with the ones for impurity-free dots (). In particular, we study both and to study the confinement effect on their entanglement properties.

The scaling , turns the Schrödinger equation into




where . The limits as and correspond to situations in which the frequency of the trap tends to and , respectively. In the latter limit we recover a free space impurity system (atomic-like system), which has at least one bound state for (29); for , it consists of one electron bound to the impurity charge and one free outer electron.

Since the Hamiltonian (3) does not contain any terms that couple the spatial and the spin coordinates, its stationary states possess the form


where are the spin functions and the spatial wavefunctions are symmetric or antisymmetric under permutation of the electrons.

In this Letter, we restrict our investigation to the states, the spatial wavefunctions which depend explicitly only on the radial coordinates and , where is the inter-electronic angle coordinate, and the differential volume element is (30).

This Letter is organized as follows. In Section II, we discuss an effective procedure to analyze the entanglement of the singlet and triplet -symmetry states. Section III is devoted to the results and some concluding remarks are left for Section IV.

Ii Entanglement of -states

The real spatial wavefunction can be expanded in a Fourier-Legendre series of Legendre polynomials of the cosine of (31):




Being real and symmetric, the function has the Schmidt decomposition (32)


where the orbitals satisfy . On the other hand, the function is real and antisymmetric and so its Slater decomposition (32) is


where . With the help of the expansions (7) and (8), the addition theorem , and the identity , the Slater decompositions of the total singlet and triplet -symmetry wavefunctions (4) can be easily inferred. The number of non-zero expansion coefficients in the Slater decomposition is called the Slater rank (SR). An essential point is that a pure fermion state is non-entangled if, and only if, it can be expressed by a single Slater determinant (33), i.e., its SR is equal to one.

Quantum entanglement is characterized by the spectrum of the single-particle reduced density matrix (34), , and many ways of measuring its amount have been developed (32); (33); (35); (36); (37); (38). In this Letter, to quantify the entanglement we shall use the reduced von Neumann (vN) entropy


where is the ordinary vN entropy (32). The above measure vanishes for a two fermion pure state when its total wavefunction can be expressed as one single determinant (38). Since the total wavefunction factorizes into spatial and spin components, the same holds for the reduced density matrix and, in consequence, separates into , where the spin contribution depends only on (39), that is, . Therefore, the reduced vN entropy can be written as


where are the spatial reduced density matrices of the singlet and triplet states,


The eigenvalues (occupancies) and of the spatial reduced density matrices of the singlet and triplet states are related to the coefficients by . The former are -fold degenerate, whereas the latter are -fold degenerate, so that the normalization conditions give . As the vN entropies of the triplet states with and with differ from each other only by one, we will concentrate mainly on , without loss of generality. In terms of the occupancies, the vN entropies (10) of the singlet and the triplet with states are given by and , respectively.

The radial orbitals satisfy the following integral equations


with , where and for and , respectively. With the help of Eq. (6), we find that can be expressed by the following -dimensional integrals.


In particular, the orbitals and their eigenvalues can be determined by


One of the most efficient ways to determine is to solve the above integral equations through a discretization technique (see for example (9)). Since it is easier to obtain (6) than (13), we will deal throughout this Letter with (14) when the are needed.

Iii Numerical results and discussion

To compute the -state energies and wavefunctions of (3), we use a simple but effective variational wavefunction


where , and are the Hylleraas coordinates; , and is a non-linear variational parameter (30). In order to gain insight into the effectiveness of the method, we first determine the occupancies of the ground state of the free space impurity system with (helium) and assess their accuracy by comparing the linear entropy with the already available data in the literature, (13), 0.01606 (14), 0.015943 (15). We use an expansion given by a -term wavefunction (15) that includes all terms consistent with the condition , ( even), reproducing at the ground-state helium energy with to at least ten significant digits: (40). Discretizing the variables and with equal subintervals of length , we turn (14) into an algebraic eigenvalue problem


where . Diagonalization of the matrix yields thus a set of approximations to the coefficients . With the occupancies determined in such a way that , we can then obtain an approximation to the true value of , . For illustrative purposes, we present in Table 1 the values obtained for with different cut-offs and , where we obtain the value for the linear entropy, in excellent agreement with the results of Refs. (13); (14); (15) wherein they were obtained in different ways.

Table 1: computed with different and , for where the functions are mainly confined.
Figure 1: The von Neumann entropy of the ground-state as a function of for some different exemplary values of (). The horizontal lines mark the results for the free space impurity systems.

Now we come to the main goal of this Letter, which is to explore the entanglement in the ground state and the triplet states of lowest energy of (3). First we treat the systems in the presence of acceptor impurities . For all negative charge values considered in this Letter, one has that up to is typically sufficient to get a good estimate of the vN entropy, over the entire range of values of . Fig. 1 depicts the dependence of the ground-state vN entropy on for some exemplary values of smaller than . In particular, the results of this figure reveal the effects of the confinement on the entanglement in the ground states of the helium atom and the helium ion (the cases and , respectively).

Figure 2: (a) The von Neumann entropy of the ground state as a function of for some different negative values of (). The black-dashed line corresponds to the case . (b) The behaviour of the two largest occupancies determined as functions of for .

As far as we know, in none of the papers concerning confined helium ions have the results for their entanglement properties appeared. In the limit of , the vN entropy approaches strictly a value of the corresponding state of the free space impurity system, being smaller at smaller . A transition to the free space impurity system regime is manifested by the onset of the plateau in the behaviour of the vN entropy. The results of Fig. 1 indicate that the critical value of at which this occurs decreases with decreasing . This can be qualitatively understood by referring to the localization of the electrons, namely, for smaller , they are more localized around the center and, in consequence, a stronger confinement is needed (smaller ) to change their quantum state properties. The vN entropy deviates more and more from the free space impurity value as the confinement becomes stronger and stronger. The deviation is the largest in the limit of an infinitely strong confinement (), when the system behaves like non-interacting electrons in a harmonic potential well, which gives (SR=1).

We also calculated the ionization threshold value for the confined helium atom, which is defined as that for which , where and are the ground state energies of and confined in the same potential well(26). This means that for , the confined helium atom gets ionized, i.e., it consists of one electron bound to the nucleus and one unbound electron, but still confined within the harmonic potential well. Our highly numerical result is . There may be a general interest in noting that there is no characteristic change in the behaviour of the vN entropy near .

When becomes larger than , the entanglement exhibits a qualitatively different behaviour from that of Fig. 1; see Fig. 2 (a), namely the vN entropy has a visibly non-monotonic behaviour and in the limit of saturates at a constant value that is insensitive to . The last point can be understood by referring to Ref. (16) wherein it was found that the state of the free space impurity system, which corresponds to , has for only the two non-vanishing occupancies and , both equal to , giving (SR=2). We recall here that for , the state of the free space impurity system is no longer a bound state. For the sake of illustration, Fig. 2 displays in (b) the occupancies and determined as functions of for the example of . It is seen how they converge to an asymptotic doublet of the value when increases. Since both occupancies correspond to the purely radial natural orbitals, the bulk of the entanglement is manifested only in the radial variables as . To gain a deeper insight into the effect of the Coulomb impurity, the variation of for the impurity-free dots () is also shown in Fig. 2 (a) (the black dashed line). The vN entropy with grows monotonically with an increase in and goes to infinity as , reflecting the fact that as the dot size increases, more and more occupancies with different become substantial (41).

Figure 3: (a) The von Neumann entropy as a function of calculated for the same values of as in (a) of Fig. 2. The black-dashed line corresponds to the case . (b) Behaviour of the largest occupancy as a function of for .
Figure 4: Behaviour of the vN entropies (continuous lines) and (dashed lines), for as functions of .

Interestingly enough, one sees that the vN entropy starts to exhibit a clear local maximum after exceeding a value of about . The closer is to zero, the more pronounced is the maximum and the larger is the value of at which it occurs. At the same time, one observes that the range of values of around in which exhibits the behaviour of the impurity-free dots also becomes wider.

Fig. 3 (a) reveals the effects of changing both and on the entanglement in the lowest energy triplet state with , where for the sake of comparison the same values of as those for the ground state in Fig. 2 (a) are taken into account. Once again, as for the singlet ground state, the black dashed line represents the case with no impurity. Except for this case, for all remaining cases an analogous situation occurs, namely the vN entropy increases, attains a maximum value, and then diminishes until it vanishes as . The last point can be explained as follows: for the triplet state of the free space impurity system, which corresponds to , is an unbound state and has only one non-zero occupancy, that is (17). In the case , this gives (SR) and the corresponding states must therefore be regarded as non-entangled. The situation is different if one considers the triplet state with , namely: for its total wavefunction constitutes a sum of two Slater determinants (SR) and this state has to be regarded as an entangled state. Fig. 3 (b) demonstrates, for the example of , how the occupancy attains its asymptotic value with increasing . Except for the limits as and , where the dependence of the entanglement on disappears, the triplet state with is generally an entangled state. Nevertheless, as one can infer from Fig. 3 (a), the weakly entangled states with (SR) are realized for finite values of that are larger at bigger (negative) . For example, the vN entropy of the state of the system with starts to become vanishingly small already when exceeds the value . Comparing the results of Fig. 2 with those of Fig. 3, one can finally conclude that for a given , the ranges of values of in which the entropies of the ground and lowest triplet states make their most rapid variations are nearly the same.

We close our discussion with Fig. 4, which compares the behaviour of the vN entropies of systems with a donor impurity and without impurity charge, for both ground and triplet states. The vN entropy of the system with exhibits a monotonically increasing behaviour as increases and goes to infinity as , similarly to the case for the impurity-free dots. The presence of the donor impurity increases the entanglement for every except for , which is more pronounced for the triplet state than for the singlet one when is large (the weak confinement regime). It is well known that in the case of an impurity-free dot, the angular correlations carry the electrons at opposite sides of the centre of the trap as increases, i.e., the so-called linear Wigner molecule is formed (41). Being dependent only on the radial coordinate, the donor impurity has thus an effect mainly on the radial correlations when the confinement becomes weak. In this regime, the increase in the entanglement resulting from the impurity invasion comes thus mostly from an increase in radial correlations. As can be inferred from Fig. 4, the larger is the value of , the smaller is the change in the entanglement (in a relative sense) produced by the coming of a donor impurity. The impact of the donor impurity is therefore relatively small in the weak confinement regime in contrast to the acceptor impurity where the opposite behaviour occurs.

Iv Conclusions

In conclusion, we carried out a comprehensive study of the entanglement properties of two interacting electrons in the presence of Coulomb impurities in spherical harmonically shaped traps. Our results showed the dependencies of the vN entropy on the dot size and the effective charge for both the ground state and the triplet states with the lowest energy. From the results, it is apparent that the invasion of the charge impurity dramatically influences the entanglement. The effect is much more pronounced when the impurity is negatively charged. In such a case, when is large enough, the effect of the confining potential is negligible and the entanglement entropy approaches the value of the corresponding state of the free space impurity system. As a general trend, we found that the closer is to zero, the larger is the value of at which this occurs. On the other hand, when the confinement is strong (small ), the impact of the impurity is small and the entanglement approaches the value of the impurity-free dots. It turned out that the range of values of around in which this occurs tends to increase when . In the limit as , the vN entropy exhibits a discontinuity at the point , since it tends to a constant value as , while at , it tends to . In the case of a positively charged impurity, the vN entropy increases monotonically as the dot size increases, similarly to the case for the impurity-free dots. It was found that the impact of the donor impurity on the entanglement is relatively small in the weak confinement regime, i.e., where the Wigner molecule is formed. Except for the limit of infinitely strong confinement (), we found the lowest singlet state to have generally higher entanglement than the lowest triplet one with .

It would be desirable to gain a deeper insight into the properties of the system (1) by extending the current calculations to excited states. In particular, as the confined systems of atoms have important applications to modelling a variety of problems in physics, it would be interesting to explore the entanglement of the eigenstates of confined helium ions and check how the results deviate from those for their free counterparts (7); (13); (14); (15).



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