Understanding correlation effects continues to be one of the most central problems in theoretical quantum chemistry and condensed matter physics. As a matter of fact, this has been the main goal of the majority of theories and models introduced in this research area [1,2]. In particular, the behavior of electrons in low dimensions such as the two-dimensional (2D) electron gas [3] or 2D semiconductor systems [4][5][6][7][8] has been intensively studied during the past years. The interplay between confinement, quantum spin, delocalization effects and Coulomb repulsion between electrons under the dictate of quantum rules leads to many interesting physical phenomena and unanticipated patterns [9][10][11]. A system where considerable effort has been devoted in the past decade is the study of GaAs/AlGaAs concentric double quantum rings [12][13][14][15][16]. A double quantum ring system can be used to study the electronic transport and optical properties of various semiconducting materials [12][13][14][15][16]. For instance, the electronic transport in the system can be studied and controlled by means of external metallic electrodes that can be attached. Similarly, one can manipulate the electronic and optical properties of a double quantum ring by applying a laser field. By tuning the laser field one can create new degenerate energy levels and, thus, affect the optical properties of the system. Overall, a double quantum ring system has many possible technological applications with research topics ranging from opto-electronic properties to spin transport and/or quantum computation. The fabrication of such quantum rings was made possible by the invention of precise droplet-epitaxial techniques [17]. These nanosystems are extremely useful in order to gauge subtle quantum phenomena such as the Aharonov-Bohm effect [18] and the influence of the Coulomb interaction on the magnetic properties of small quantum systems of electrons [19].
In this study, we consider a concise model consisting of mass of the electrons, k is Coulomb's electric constant, e is the charge of the electron and d(φ 1 , φ 2 ) ≡ d(φ 1 , φ 2 ; R 1 , R 2 , α, H) is the separation distance between the pair of electrons. The square of such distance is given by:
No other subtler effects coming from electron's effective mass in a semiconductor host and/or from presence of a magnetic field are taken into account [26,27].
The scenario with added tilting allows one to tune the quantum properties of the system of electrons by incorporating drastically different limiting cases that range from coplanar concentric double rings to vertically separated double rings with an arbitrary tilting. Therefore, the mere change of the geometric parameters may lead to fundamentally different quantum behavior of the system in a controllable fashion. Any effort in simplifying the quantity in Eq.(2) will not lead to a useful expression for our endeavor. Therefore, our main objective becomes the numerical solution of the resulting stationary Schrödinger's equation. Atomic units are used in which distances are measured in units of the Bohr radius, a B while the energy is measured in the atomic unit of k e 2 /a B (a Hartree), which is a commonly used in quantum atomic physics. At a formal level, one simply sets = m e = k = e = 1. For this choice of units, the stationary Schrödinger's equation reads:
where Ψ(φ 1 , φ 2 ) is the wave function for the system of two electrons, E is the energy eigenvalue and 0 ≤ φ i < 0 for i = 1 and 2. Note that the wave function for noninteracting particles is that of a particle confined in a circular ring with infinite potential wells outside the ring. The solution of the quantum problem is obviously periodic, Ψ(0,
We use the following normalization condition:
The case α = 0 is quasi-exactly solvable by using the distance between particles as a new variable. Although this analytic approach does not apply here, we can use such results to gauge the accuracy of our numerical computations.
The model of three electrons is a variation of the geometry of Fig. 1. For the setup of three electrons, we will have three vertically separated, parallel and concentric rings with the same radius, R 1 = R 2 = R 3 = R = 1 situated on three different equidistant parallel planes. Each ring will contain one single electron. For example, one ring is situated below the xy plane by a vertical distance, H (ring 1), a second ring is above such plane at a distance, H (ring 2) while the third ring is on the xy plane (ring 3). We believe that the reader can easily visualize the geometry of such a ring setup without need to draw a schematic presentation. The quantum Hamiltonian for the system of three electrons can be easily generalized from Eq.( 1). The extension of the stationary Schrödinger's equation for such a case is straightforward (in atomic units):
where Ψ(φ 1 , φ 2 , φ 3 ) is the wave function for three electrons, E is the corresponding energy eigenvalue of the system and, in this scenario, d(φ i , φ j ) ≡ d(φ i , φ j ; R, R, α = 0, H) are the separation distances between pairs of electrons written in short-hand notation.
Obviously, the same periodicity properties are imposed on the wave function.
A natural question that emerges is what happens to the quantum system under consideration when the geometry of the Hamiltonian changes. One quantum informationbased measure of the spatial entanglement (mixedness) in quantum states is the so-called linear entropy defined as:
where Tr is the trace operation and ρ r is the singleparticle reduced density matrix [28] of the state:
One can check that:
Note that the calculation of the linear entropy is a rather involved mathematical process. The linear entropy defined in Eq.( 6) is quite popular for the analysis of entanglement in two-particle systems and has been pursued at length in the literature [29][30][31][32][33][34].
Our objective in this work is to obtain the entanglement measure, E defined in Eq.( 6) for various combinations of the geometry of the system. To be more specific, we will consider the following four cases:
• α = 0, H = 0 which reduces to the case of two coplanar concentric rings studied in [35]. The special case, R 1 = R 2 implies two electrons in the same ring.
• α = 0 and H = 0 reduces to the case of two electrons in vertically coupled parallel rings studied in [36].
• α = 0, H = 0 and R 1 ≈ R 2 where both radii increase, but they differ slightly from each other.
• α = 0 and R 1 ≈ R 2 which corresponds to two rings that almost share a point of contact since α is tilted and R 1 = R 2 (since they differ slightly).
All these cases are studied numerically by solving the the stationary Schrödinger's equation via the exact diagonalization method.
Let us illustrate the exact numerical diagonalization approach for the system of two electrons. One easy way of preserving the periodicity of the solution is to span the (unknown) wave function, Ψ(φ 1 , φ 2 ) in the basis of the eigenstates for two non-interacting electrons each one in its respective ring and then truncate the expansion to N + 1 terms where N even. That is:
(9) By substituting Eq.( 9) into Eq.( 3), multiplying by
√ R1R2 e -ikφ1 e -ilφ2 and integrating over the angles, φ 1 and φ 2 we obtain:
for indices, m, n = -N 2 , .., N 2 . Let us denote by H klmn the first line in Eq. (10). Solving Eq.( 10) for c k,l is tantamount to providing an approximate solution to Eq.( 3) for both ground and excited states. One can increase the accuracy of the calculations by augmenting the number of terms in the expansion.
The Coulomb interaction matrix element shown in Eq.( 10) reads explicitly as:
The set of equations in Eq.( 10) for c k,l does not read yet as a standard eigenvalue problem. In order to do so, one must transform H klmn -→ A ij and c k,l -→ g j , i, j = 1, .., (N +1)
With this transformation, we have the usual eigenvalue and eigenvector problem:
where i = 1, 2, .., (N + 1) 2 . Finding the corresponding energy eigenvalues will give as the energy spectrum of the system. In order to find the eigenvectors, the inverse transformation g j -→ c k,l can be proved to be unique. In other words, given j and N , we find a sole couple (k, l).
Once the coefficients c k,l are obtained, it is straightforward to obtain the wave function and, thus, calculate the linear entropy as defined in Eq.( 6). To this aim, we use the fact that the ultimate integration turns out to be on separate arguments provided that we use Eq.( 9) together with the orthogonality condition. The value of linear entropy:
is obtained numerically once the desired numerical accuracy is met.
In order to validate the accuracy of the numerical results, we compare them to the analytic case of two electrons in concentric rings [35]. The results for the expansion coefficients of the ground state wave function are shown in Table . I. The matching is perfect.
The corresponding ground state wave function is shown in Fig. 2 where Ψ(φ 1 , φ 2 ) is plotted. Note that the wave function is expressed as a linear combination of terms for negative, zero and positive indices. It happens that the expansion coefficients, c k,l have the same value for opposite indices. Therefore, the outcome is a purely real ground state wave function meaning that Ψ(φ 1 , φ 2 ) = |Ψ(φ 1 , φ 2 )| where the latter would represent the (real) modulus of the wave function. It is worth mentioning that, as in Table . I, it will be enough to have a In this Appendix we present the explicit values of the expansion coefficients for the ground state wave function of three electrons. The values of c k1,k2,k3 for R 1 = R 2 = R 3 = 1 and H = 1 are shown in Table . III. Those for H = 0.001 are shown in Table . IV.
-1 4 -3 5.40595983×10 -6 0 -4 4 -5.26775736×10 -6 0 -3 3 -6.4193788×10 -5 0 -2 2 -0.0011496827 0 -1 1 -0.00899787456 0 0 0 0.976219937
TABLE III. Expansion coefficients, c k 1 ,k 2 ,k 3 for the ground state wave function solution for the system of three electrons when R1 = R2 = R3 = 1 and H = 1. The rest of the coefficients not shown are obtained by changing the sign of the set of integers {ki}. They are exactly equal to the c k 1 ,k 2 ,k 3
shown here. We have used a basis set with N = 8.
k1 k2 k3 c k 1 ,k 2 ,k 3 -4 0 4 -0.00300389246 -4 1 3 0.00264379189 -4 3 1 -0.00264379189 -4 4 0 0.00300389246 -3 -1 4 0.00264382094 -3 0 3 -0.00815690883 -3 1 2 0.00950729392 -3 2 1 -0.00950729392 -3 3 0 0.00815690883 -3 4 -1 -0.00264382094 -2 -1 3 0.00950731754 -2 0 2 -0.0377186212 -2 2 0 0.0377186212 -2 3 -1 -0.00950731754 -1 -3 4 -0.00264370077 -1 -2 3 -0.00950718227 -1 0 1 0.406169353 -1 1 0 -0.406169353 -1 3 -2 0.00950718227 -1 4 -3 0.00264370077 0 -4 4 0.00300376913 0 -3 3 0.0081566846 0 -2 2 0.0377181052 0 -1 1 -0.406169831 TABLE IV. Expansion coefficients, c k 1 ,k 2 ,k 3 for the ground state wave function solution for the system of three electrons when R1 = R2 = R3 = 1 and H = 0.001. This is the scenario for which the three electrons are practically located in the same ring. The rest of the coefficients not shown are obtained by changing the sign of the set of integers {ki} and are exactly the opposite of the c k 1 ,k 2 ,k 3 coefficients shown here. Notice the change of structure with respect to the H = 1 case. We have used a basis set with N = 8.