Electrons in Surface Acoustic Waves as Spin Qubits

Electrons in Surface Acoustic Waves as Spin Qubits Mikel Olano EHU/UPV Donostia International Physics Center A project submitted for the degree of PhD in Physics of Advanced Nanostructures 2025 REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 (cc) 2025 Mikel Olano Aranburu (cc by 4.0) REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 ii eta aurrera joanen gaitun guztiak batak bestearen inbidirik sentituko ez duen momenturarte Mikel Arregi REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 Abstract Solid state coupled quantum dots (QDs) have been long proposed as a platform to control spin qubits and perform quantum computation [ 54 , 14 ]. One of the most popular methods to create these arrays consists in creating a two-dimensional electron gas (2DEG) in the junction between two semiconductors with a similar band gap, being able to create electron depleted zones by applying electrical voltages with controlled gates. As a possible solution to the short range interactions that can usually be obtained in these scenarios “flying” qubits [ 25 , 5 ] have been proposed to transport electrons while performing the one and two-particle interactions. Surface acoustic waves (SAWs), which can be created by interdigital transducers (IDTs) in the surface of a piezoelectric material [ 21 , 82 ], have been proposed and demonstrated as the carriers of single electrons [ 7 , 33 ]. This Thesis covers the main interactions in the transport of single electrons from static to moving dots, including the spin-orbit interaction and the possible spin-flip processes that may happen due to the hyperfine interaction between the nuclear spin bath present in GaAs and the electron’s spin. The numerical analysis of the transfer process shows the possibility of treating the evolution of the system with few low-energy states, which reduces the number of terms to be taken into account to describe and optimize it. At the end, the Coulomb interaction between two particles in different dots is briefly discussed, mainly through the theoretical expressions that enter in its definition and a numerical approximation for a particular impact parameter. REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 Contents 1 Introduction 1 1.1 Quantum Simulation &Computation ................. 1 1.2 Quantum Dots ............................. 8 1.3 Surface Acoustic Waves ......................... 11 1.4 Proposal and Hypothesis ........................ 12 2 Effective model and numerical methods 15 2.1 Numeric definition of the Hamiltonian ................ 17 2.2 Numerical time evolution ........................ 20 3 The transfer process 25 3.1 Potentials ................................ 26 3.2 Varying the impact parameter ..................... 29 3.3 Few-level approach ........................... 33 3.3.1 Simplified version and parameter variations ......... 35 4 The spin orbit interaction 43 4.1 Well-separated minima ......................... 45 4.2 Effects on transfer probability ..................... 47 4.3 Entanglement .............................. 51 5 The hyperfine interaction 55 5.1 Approximated expressions ....................... 56 5.2 Spin-flip process time-evolution .................... 65 6 Two-particle interaction 71 6.1 The coulomb interaction ........................ 72 6.2 Further corrections ........................... 75 6.2.1 Real space expressions ..................... 76 6.2.2 Momentum space expressions ................. 78 6.3 Numerical considerations and results ................. 81 7 Conclusions 83 v REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 vi Contents Appendices A Coulomb term calculation 87 Bibliography 91 REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 1 Introduction Contents 1.1 Quantum Simulation &Computation . . . . . . . . . . . 1 1.2 Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Surface Acoustic Waves . . . . . . . . . . . . . . . . . . . 11 1.4 Proposal and Hypothesis . . . . . . . . . . . . . . . . . . 12 1.1 Quantum Simulation & Computation Quantum computation is a field that has drawn the attention of a great amount of scientists for the last half century. As electronic computers are reaching atomic sizes for their transistors [ 22 ], their quantum nature starts to become increasingly difficult to manage. The incapability of scaling both algorithms and computers that were powerful enough to simulate quantum phenomena properly, physicists and mathematicians started thinking in an alternate manner of studying such systems, focusing on the use of quantum states for their benefit to understand better the many-atom reality. One of the great fields that was opened following this path is quantum simulation, which allows to study Hamiltonians that are too complex to be computed in a classical computer [ 31 ]. One can encode a Hamiltonian in a programmable fashion in a quantum system such that the time evolution is 1 REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 8 1.2. Quantum Dots in order to have a decent fidelity. This is not necessary in measurement-based computations, though, that uses massively entangled states and sequences of adapted single-qubit measurements to perform quantum computation. The remaining platform that has shown great potential for performing quantum computation are spin qubits in quantum dots, which are this thesis’ choice of study. 1.2 Quantum Dots For quantum computational purposes, quantum dots (QDs) are small volumes in semiconducting materials where a potential creates a confining energy that single or multiple electrons can fill. Due to the three-dimensional confinement that gives rise to quantized levels which can be filled, they are also referred as “artificial atoms” [ 4 ]. One of the most popular techniques to create a potential that can be controlled starts by creating a two-dimensional electron gas (2DEG). This is done by creating insulating top and bottom layers on a semiconductor. Metallic gates are afterwards placed on top of the insulating material, such that applied voltages let a definite amount of electrons travel to the dot, controlled by the barrier created with the Coulomb interaction between the QD and the 2DEG [ 13 ]. The spatial zone in which electrodes allow electrons are called “source” and “drain”, which can be varied allowing electrons in their conduction band to enter the dot. Varying the strength of the gate potential, the equilibrium population on the QD changes. If there is no intermediate state that allows for “source” electrons to reach the “drain”, this population can be an integer with minimum fluctuations, which can be interpreted as a finite number of electrons in the dot [ 67 ]. Just like in regular atoms, the first electron to enter the QD will fill one of the states that can be defined by the Schrödinger equation for a free particle in a three dimensional “box” defined by the potential with an effective mass m∗ that depends on the curvature of the material’s lowest energy part of the conduction band [ 57 ]. Once the first electron enters the QD, an additional barrier is created in order to allow a second electron in the same dot, which is precisely due to the REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 1. Introduction 9 Coulomb interaction between them. This energy is usually characterized by the QD’s capacitance C , being its value U = e2/C [ 38 ], and it’s the main cause of what is called the Coulomb blockade. If the energy difference between either the source or the drain and the quantum’s dot next energy level is not big enough, then no other electron will enter the QD. Lowering the gate voltage and maintaining the same value for both the drain and the source can lead to an additional electron to enter the dot. If either one of the lateral structure’s potential is higher than the second electron energy but the other one remains below, then a steady current of single electrons will flow between them. Depending on how many states the QD allows inside, one can see a current from source to drain which is proportional to the number of states in the transport window created by the bias. This quantization of electron motion due to Coulomb blockade can also be seen in systems of coupled quantum dots [ 53 ], where currents through both dots can be either allowed or stopped by the number of electrons in both dots. One may block the stream of single particles by having a state occupied in one of the dots, independently to the accessible states for the other one. A common way to visualize this effect is to plot ∂I/∂Vsd for different values of gate and drain-source voltage [ 67 , 46 ]. The images show zones in which the value of the derivative is close to zero, where the number of electrons in the double-dot system is constant. Due to its dependence of this value on the independent parameters, these zones have a rhomboid shape and are called “Coulomb diamonds”, which are surrounded by positive and negative values indicating exchange of electrons between the two QDs. Being able to vary the tunneling barrier between the two dots one can also access states that belong to both of them, which are often called “molecular” states, due to their creation from two coupled artificial “atoms”. Apart from Coulomb blockade there is another interesting effect called “Pauli blockade” [ 38 , 63 ] that can occur in double quantum dots for which source and drain are connected by sequential tunneling through the two dots. This is based on the effect that the spin state of an electron in the double QD has on the states that are available for tunneling [ 48 ]. If one occupies one of the dots with two electrons REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 10 1.2. Quantum Dots in its lowest state, the total spin configuration that they acquire is the singlet state. Once one of the electrons leaves the double dot, the newly entering electron can only be in the singlet state to go “through” the already occupied dot, since the energy of the triplet states will be energetically unavailable. This can allow controlled spin loads in quantum computers, since one can load electrons on one side making sure that the first populated state belongs to the correct total spin state. Both the Pauli spin blockade and external magnetic fields can be used to initialize spin states in semiconductor quantum dots [ 13 ]. Once the electrons are inserted in the QDs from the surrounding Fermi sea, they can be transported between different dots [ 59 ] or made interact with each other if the potentials are either detuned or the potential barrier between them is decreased. The effect that a change of the potential’s shape can have on the interaction allows for a controlled spin interaction between the two electrons via the exchange coupling Hex = −J12  S1· S2 and can be further controlled by an external magnetic field. This was suggested to be a way of performing quantum computation with quantum dots [ 54 , 14 ] and still remains as a possible and developed candidate. Figure 1.1: Sketch showing the principle of Pauli blockade in a double quantum dot. a) The relative position of source and drain allows electron passages through singlets, b) blocking the singlet triplet transition. One of the biggest advantages that quantum-dot platforms have, is the already developed silicon-based industry, that has improved the reproducibility of quantum REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 1. Introduction 11 dots with the same characteristics [ 86 ]. The closeness with which these can be packed is greatly impacted by these type of techniques, which allows for many qubits that are interconnected one to each other. However, long-range interactions are still an issue since the exchange interaction is short-ranged, and most quantum dot interactions are just possible between nearest neighbors. Furthermore, their possible interactions with nuclear spin baths (strong in the case of GaAs and more diluted for purified 28 Si) pose challenges that need to be targeted for realizing quantum computation, as is decoherence arising from charge noise. 1.3 Surface Acoustic Waves Acoustic waves that advance along the surface of a material can be created by inducing both a vertical and a longitudinal force on the outer layers of the material [ 65 , 37 , 81 ]. This creates stress on the surrounding atoms at the surface, which propagates in the longitudinal direction in which the initial force has been applied. Due to the lack of bulk material above the surface, the stress created by the force has to be zero in it, while it penetrates into the material with exponentially decreasing strength defined by the strength of the atomic or molecular bonds and the applied force. The strength of the SAW decreases with distance from the surface with a typical penetration depth on the order of the waves’ wavelength λ [ 80 ]. Since SAWs have lower phase velocity than bulk waves they are not very well coupled, thus enabling low-loss propagation over long distances. The piezoelectric effect refers to the induction of electric polarization under mechanical deformation. Depending on the arrangement of the lattice (which must not have an inversion center to show this effect) and the direction in which the atoms are displaced, each material has a different piezoelectric response to mechanical stress. The effect is reversible, which means that applying an electric field to the material the atoms will be displaced under it. The relation between the mechanical stress and the electrical field induced by it creates a coupled mechanicalelectromagnetic equation that allows for waves in which both stress and field evolve at the same time. The coupling between the electric fields that are created by REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 12 1.4. Proposal and Hypothesis the mechanical stress and its effect on further compression in neighbouring atoms allows for creating diverse shapes for the surface acoustic waves that can be created through the voltages applied by deposited metallic interdigital transducers (IDTs) on the surface of the piezoelectric material. These gates are positioned in a way such that applying time-varying pulses corresponds to summing several plane waves that form a wave packet with the desired shape [ 84 , 82 ]. Several uses for surface acoustic waves produced by IDTs have been thought of since their discovery, and currently there are several scientific and industrial techniques that take advantage of this phenomenon [ 49 , 52 ]. Furthermore, their usefulness in quantum computation has long been predicted [ 5 ], where trapped single electrons can perform singleand two-qubit interactions between static and “flying” spin qubits [ 50 ] and their prospects vary in many fields surrounding quantum computation and metrology [ 21 ]. 1.4 Proposal and Hypothesis The system this Thesis studies is based on an idea that combines quantum dots and surface acoustic waves in order to address the connectivity issues that a grid of quantum dots may have. The limitation that this platform poses is the interaction between electrons that are not nearest neighbours. Usually, if one wants an interaction between non-nearest neighbour quantum dots, it has to be mediated by the intermediate qubit(s), which greatly slows the computational process and leaves little room for scaling or error correcting. Some suggestions propose shuttling processes as a mean to distribute entanglement or couple distant qubits [ 34 ], using microwave resonators for long-distance coupling [ 56 ] and using “floating” gates [ 78 ]. Based on the proposal by CHW Barnes [ 5 ], Christopher Bäuerle’s group has been specially prolific in experimentally demonstrating the use of SAWs to initialize and operate with single spin qubits [ 7 , 16 ], as well as achieving the initialization, transport and readout of two-electron spin states[ 7 ]. Their improvements in SAW definition and characterization have also been prominent [ 82 ]. The method that these experiments use to inject electrons in the moving minima is based on sending REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 1. Introduction 13 a pulse at the same time that the targeted minimum of the SAW passes the static quantum dot. As precise as it is, this does not ensure the electron to be in any particular state once it is transferred. Their theoretical treatment of said process is also scarce. Transfers to distant quantum dots that are not aligned are also limited by the time the electron takes to tunnel from side to side of the double channel scheme they propose in [ 75 ]. Moreover, the single-electron transport and the two-electron interaction that they have shown until now have not yet produced a necessary entangling gates between electrons. Figure 1.2: Simplified scheme of the proposed platform. In order to solve the issues that these groups have shown in the development of this platform, the proposal that this Thesis studies consists in electronic spins as qubits of the system that are transported (as in Barnes’ proposal) but moving them from and to static dots to perform single-qubit operations. Controlling this process and creating arrays of static quantum dots at a particular distance from the SAW, electrons that have been injected may interact with incoming electrons in a minimum created by the SAW. Although this proposal only includes static dots in a row near the channel where the SAW propagates, more flexible setups may be possible REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 14 1.4. Proposal and Hypothesis with varying architectures or mediums. The advantage of having static dots in which single-qubit operations can be more controlled adds to the benefits of letting several static electrons interact with the same moving one, thus enabling long-range tunable interactions between electrons that are not necessarily nearest neighbors. The purpose of this Thesis is to present a toy model that covers the main interactions in the transfer process of a single electron in a static QD into a SAW minimum and vice versa. Being GaAs heterostructures of high interest for their piezoelectric properties, the main interactions that affect the spin state of the moving quantum dot, i.e., the Rashba and Dresselhaus spin-orbit interactions [ 40 ] and the hyperfine interaction with the Ga and As magnetic nuclei [ 27 ] are taken into account. Looking at the current situation in which this proposal lies, this Thesis proposes three questions and hypotheses: • Assuming that a single minimum potential created by a SAW can interact with a laterally placed quantum dot, which are the optimal characteristics to have a high-fidelity transfer? • If the mentioned transfer can happen, Can the transfer process be described by considering only a low-dimensional subspace and how small can it be chosen? • What is the effect of the spin-orbit and the hyperfine interactions in this transfer process? • What are the necessary conditions to create an entangling gate between the moving and the static qubit? Can this be faster than between neighboring QDs? REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 2 Effective model and numerical methods Contents 2.1 Numeric definition of the Hamiltonian .......... 17 2.2 Numerical time evolution ................. 20 The electron whose quantum state’s time evolution is described in this thesis lives in a three-dimensional space. There are two initial considerations for describing its state in this first part. First, a complete description of its spatial representation includes both the envelope function that describes its probability distribution in space and the shape that the electron wavefunction has around the nuclei of the material depending on the band to which it belongs. Secondly, following the architectures that propose a tight potential in one of the spatial directions ( ωz>> ωx, ωy ) to trap the electron in a two-dimensional space, the envelope function can be described as a product of a function in the z-direction and another that includes x and y. One can consider the state to be in a particular energetic state in the z-direction (usually the groundstate) that will not be able to couple to other eigenstates because there is no change in the Hamiltonian for this direction. For a general quantum state of the electron | Φ ⟩ , its real space representation will be ⟨r|Φ⟩=ϕ(r)u(r)(2.1) 15 REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 16 2. Effective model and numerical methods where u (r)represents the shape of the electron around the nuclei and ϕ (r)is the envelope function. As mentioned above, the factorization ϕ(r) = ϕ2D(x, y)·ϕz(z)(2.2) can be applied because of the energy separation between the eigenstates in the z-direction and its invariability over time. The state of a single electron in a quantum well can be approximated as a free particle with an effective mass that depends on the conduction band curvature of the material. In our case, the potential’s shape in the z-direction will be considered to be well represented by a harmonic potential with a curvature of m∗ω2 z/ 2, giving a Gaussian lateral confinement for the electron: ϕz(z) = s1 2π∆−1 ze−z2/4∆2 z.(2.3) where ∆ z refers to the width of the state in the z-direction. In the other two directions, one can solve the Schrödinger equation for the Hamiltonian that includes the kinetic and electric potential contributions and has the eigenstates at each point in time. From now on, any time the position operator or a state’s position representation is mentioned, it refers to its expression in the XY plane unless explicitly stated. Let us first work on the transfer process between the static and moving QDs described in the proposal. The initial quantum state of the electron will evolve over time with the changes that occur in the potential landscape to which it is subjected. Here, we consider a potential that is the sum of two potentials with a single minimum each, one remaining static, representing a quantum dot, while the other is created by a propagating SAW and thus forms a moving potential minimum that passes near the quantum dot. The initial expression that is going to be considered for the two-dimensional envelope function that describes the electron’s quantum state will just take into account the effect of the potentials in its evolution. Let us consider two different potentials with a single global minimum each, one of them representing a static quantum dot and the other one the particular minimum created by the SAW to REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 2. Effective model and numerical methods 17 which the electron must be transferred. Assuming that the harmonic approximations around their minima in r r rs (for static) and r r rm ( t ) = r r rm (0) + v v vSAW ·t (for the moving) are similar, then the two-dimensional Hamiltonian H0(t) = p p p2 2m∗+Vs(r r r−r r rs) + Vm(r r r−r r rm(t)) (2.4) has for large separation of potentials almost doubly degenerate low-energy eigenstates and, in particular, an almost degenerate two-dimensional groundstate subspace, the splitting of which depends on the spatial separation |r r rm ( t ) −r r rs| between the potentials. Our aim is to find a deterministic way to transfer an initial state located in one of the minima to the other in a short period of time. This suggests that the evolution must include a non-adiabatic process. Since there is interest in knowing the intermediate states that are populated during its evolution, the description of the state’s evolution on the eigenstate basis of the system will be done at some point. Therefore, independently of the method that is used to evolve the system, one needs to know the solutions to the eigenvalue problem at different points in time during the retrieval/injection of the electron. 2.1 Numeric definition of the Hamiltonian The description of the system and its evolution has been translated into a numerical problem by discretizing the necessary quantities, namely both position and momentum, and time. Each of the spatial directions has N points, whereas the time differential must meet some requirements mentioned later. When performing the discretization, we map our operators (the potentials and the kinetic energy) to the discrete space. If the spatial points are defined such that r ≡rij = [ xi, yj ] where both subscripts i, j ∈ { 1 , N} , then the electric potential is a diagonal matrix with Vij,ij = V ( xi, yj ). Spatial derivatives (for the momentum and kinectic energy operators) include off-diagonal terms Tij,i′j′ = 0 for i = i′, j  = j′ that are defined by their center difference expressions with periodic boundary conditions. Since the spatial grid is a N2 -sized object, both operators are defined as a N2×N2 size REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 24 2.2. Numerical time evolution potential. This is why, every time a transfer probability is calculated, one must use the “moving” version of the state to calculate the value. 10-1 10-2 10-3 10-4 -12 -10 -8 -6 -4 -2 0 Figure 2.2: Probability to find the analytic evolved state after evolving it for t = 167 ps . The time differentials are relative to the time unit ut= 4.17 ps. Once we have the case well defined, it’s time to see which of the evolutions takes the smallest distance from the solution. Knowing what the final state would look like after moving the state in the minimum of the moving potential for vSAWT = 500 nm, one can evolve the system with different methods and see how the absolute value of the overlap changes with ∆ t . Trying the evolution formulas given in equations 2.10 and 2.13, figure 2.2 shows how well the probability to find the right state is maintained at the end of the process for different timesteps. The condition previously defined to accept a ∆ t as valid already happens for the value ∆ t = 4 . 17 · 10 −2ps . Nevertheless, to be certain that more complex evolutions will be covered as well, an order of magnitude lower differential will be chosen, this is, ∆ t = 4 . 17 · 10 −3ps . REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 3 The transfer process Contents 3.1 Potentials ........................... 26 3.2 Varying the impact parameter ............... 29 3.3 Few-level approach ..................... 33 3.3.1 Simplified version and parameter variations ....... 35 Looking at our proposal, one of the main questions that arise is whether it is possible or not to populate the moving potential while the initial state is completely localized in the static dot and vice versa. Since we are interested in an architecture with a single SAW propagating channel connected to various QDs along the way, they will have to be located with a lateral distance to the channel. Otherwise, any electron that would come with an incoming SAW pulse would strongly interact with electrons located in the static dots, which we may want to avoid if two-particle gates are to be avoided. This is a different approach compared to the one seen in the experiments described in most of the cited papers [ 33 , 16 ], that use a potential pulse timed such that the static electron enters in the wanted minimum of the SAW. This is also one of the reasons why the two-particle interaction is done while the electrons are in the channel and not in the static dots. This chapter will first describe the potentials that have been used for the 25 REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 26 3.1. Potentials numerical calculations, then the evolution that they provoke to the system where the electron starts located in the static QD. Given the simplicity of the approach, one can expect to see at least something similar to the process that is described in the introduction: a one-time transfer between the static and the moving dot, involving the groundstate and the first excited state of the static Hamiltonian at each point in time. If this is not the case, and there is a need of introducing more states for the process to happen, an explanation is due: one needs to have clear the effect that different parameters have in the wanted evolution. 3.1 Potentials As previously states, the potentials that are going to be considered will have a single global minimum, which recent advances in the creation of potentials with IDTs allow [ 82 ]. They will also share the value of the second derivative around it, such that ∂2 iV|min(V)=1 2m∗ω2 i(3.1) where the subindex refers to the spatial direction i∈ {x, y} , m∗ = 0 . 067 me is the effective mass of the electron in GaAs and ωi = 3 meV is the chosen energy gap for both potentials, which is consistent with GaAs QD models [ 50 ]. Due to the possibility of changing the shape of the static dot via metallic gates, it will be considered as exactly equal to the moving minimum except for an energy gap of 10 −4 , just so the states are not completely degenerate. The main particular function that has been chosen is the Gaussian function Vgauss =V0·exp−kx,gauss(x−x0)2−ky,gauss(y−y0)2;(3.2) with kx/y,gauss =−m∗ωx/y 2V0 .(3.3) The second proposed function for the electric potential is the squared cosine function truncated after half a period: REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 3. The transfer process 27 Vcosqr =V0·cos2kx,cosqr(x−x0)·cos2ky,cosqr(y−y0)· ·Hx(π/2−kx,cosqr ·|x−x0|)·Hy(π/2−ky,cosqr ·|y−y0|); (3.4) where now kx/y,cosqr =qkx/y,gauss.(3.5) and the H x/y are heaviside functions to ensure the truncation. -500 -400 -300 -200 -100 0 100 200 300 400 500 -40 -30 -20 -10 0 Figure 3.1: Gaussian (’gauss’) and squared cosine (’cosqr’) electric potential functions y dependence for a constant x position. Figure 3.1 shows a cut of the two dimensional function for a constant value of x while both potentials are centered around the center of the grid. For reference, there is a dotted line indicating the harmonic function with the same minimum value and second derivative around it. As can be seen, they overlap to the point where they become indistinguishable. Notice that the Gaussian potentials wings spread longer than the squared cosine’s, whose value increases faster. REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 28 3.1. Potentials The last proposal for a potential function is inspired in the function that defines the potential created by the SAW in [ 50 ]. It has a parameter by which one can tune the number of minima that the incoming potential may have. Vtr[∆tr] = V0·exp−kx,gauss(x−x0)2·cos2(y−y0)· ·tanh(y+π∆tr/ky,cosqr)−tanh(y−π∆tr/ky,cosqr)/2(3.6) -500 -400 -300 -200 -100 0 100 200 300 400 500 -40 -30 -20 -10 0 Figure 3.2: Cut for a constant x value of the function defined in (3.6) using different values for the ∆tr parameter. The difference between the hyperbolic tangents can be understood as a soft heaviside function that allows a different number of minima along the movement of the SAW. Figure 3.2 shows the same cut as the one done for the other two potentials. The value of the parameter ∆ tr goes from 0 . 5, where the minimum in the center is much bigger than the ones in the wings, to 1 . 5, where there are three equally strong minima. The values used in this chapter range from 0 . 5to 0 . 8, where the minima on the side start to be noticeable by the electron in the static dot. REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 3. The transfer process 29 Out of the three proposed potentials the gaussian has been chosen as the reference potential for its continuity (including its derivatives) and simplicity. 3.2 Varying the impact parameter Looking for the transfer process to happen, one must set some boundaries in the parameters. Let us define the impact parameter as the lateral distance between the center of the static dot and the moving one when they are far apart from each other, gp = ( r r rm (0) −r r rs ) ·ˆ i (where ˆ i is the unit vector in the x direction). If it is possible to have the transition between the two states without involving any other, then it must be when the two potentials are separated and there is a double well structure during the whole evolution. In this scenario, the eigenstates maintain a certain resemblance to the even and odd combinations of the local states around each potential. In the case of the Gaussian potential, the closest distance at which there are still two different minima is 145 . 1 nm for the parameters chosen. The following state evolutions will be done for several impact parameters, starting from this minimum to bigger ones. Starting from a point in space in which the individual potential eigenstates have a negligible overlap, the electron located in the static dot is evolved to a final point at the same distance between the two dots as the initial by the potential, which iteratively changes at a velocity of 3 . 0 µm·ns−1 . There, one solves the eigenvalue problem for the moving minimum and multiplies the solution with the momentum displacement operator with the value p0 = m∗vSAW to obtain the target state as in 2.14. If the groundstate of the moving potential at the end of the process is | Ψ 0,m ( T ) ⟩ , then the target state would be: |Ψtar⟩= exp im∗vSAW ℏyΨ0,m(T)⟩(3.7) Figure 3.3 shows the fidelity to obtain the target state after the evolution with the three different potential shapes. Several maxima are close to one in the cases of the Gaussian and the minima train function with the single minimum, while a single maxima can be found in the case of the cosine square before it merges with the REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 30 3.2. Varying the impact parameter 150 155 160 165 170 175 0 0.2 0.4 0.6 0.8 1 Figure 3.3: Probability to find the target state 3.7 after evolving the initial electron located in the static QD for the three aforementioned potentials in 3.2,3.4 and 3.6. For the last potential, the parameter’s value is ∆tr = 0.5. static potential at the closest approach. It is clear then that the potential’s lateral extension is key to have the possibility of transferring between these two states at different impact parameters. Moreover, one can also notice that the maxima are closer between them the shorter gp’s values are, which makes them narrower. The rightmost peak (the widest of them) occurs for the same value of gp for the three relevant potentials, suggesting some common effect. As expected, for gp values bigger than certain limit there is no transfer probability between the two states. It can also be interesting to see how multiple minima in the incoming potential can affect this transfer probability. Figure 3.4 shows this by varying the parameter in the definition of the last proposed potential Vtr [∆ tr ]. The case where the parameter value is ∆ tr = 0 . 5has a global minimum much bigger than the lateral ones and it essentially acts as the Gaussian potential as it can be seen in figure 3.3. Even REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 3. The transfer process 31 145 150 155 160 165 170 175 0 0.2 0.4 0.6 0.8 1 Figure 3.4: Probability to find the target state after evolving the initial electron located in the static QD with the potential defined in 3.6 for different values of the parameter ∆tr = 0.5. changing its value to ∆ tr = 0 . 6has a sizeable effect in the transfer porbability, lowering the rightmost maximum to about 0.8. Whereas the increase to ∆tr = 0.8 does not decrease this maximum as much, the lower the value of the impact parameter the more the peaks’ values decrease compared to the previous case. As it could have been predicted, the existence of several minima in the incoming decreases the transfer probability of the process. Independently of where the rest of the wavefunction has ended, it is clear that a strong single minimum is necessary to have a successful transfer process in this architecture. Having a 2D two-minima potential means that, looking at the nearest excited states, four eigenstates are close to degeneracy when the two dots are apart. Once the potentials are closer, two of them get closer to the two-fold groundstate basis, which leads to possible excitations during the evolution of the system. Knowing already REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 32 3.2. Varying the impact parameter the gp values for various high-fidelity points, one can ask whether the excitation occurs only in the two-fold groundstate or if some part of the evolved state excites further. The probability of finding the state in the two first instantaneous eigenstates of the system can be calculated at various points along its path. The remaining probability belongs to the proportion of the state that has been excited. 0 20 40 60 80 100 120 140 160 0 0.5 1 1.5 10-3 Figure 3.5: Total probability of finding the time-evolved state out of the twofold instantaneous groundstate in time for two different values of the grazing parameter. Figure 3.5 shows this for the leftmost peak in fidelity that has been previously pictured, gp = 145 . 9 nm and the rightmost one at gp = 160 . 5 nm . Similar behaviors have been found for the rest of the peaks in figure 3.3. There are two times at which this probability increases, decreasing shortly after to leave a partial of this peak as a reminiscent probability to find higher-energy states. Notice that since these projections have been done in the instantaneous basis and not the true eigenstates of the system (taking into account the momentum given by the moving potential) REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 3. The transfer process 33 some of the final probability of finding the state in an excited state belongs to the actual eigenstate of the moving potential, previously called target state. In either way, the probability of finding the state in the two-fold ground state along the path is always higher than 0 . 999 in the case of the farthest maximum, suggesting that a few-level description can be used to describe the effect and better understand the process. 3.3 Few-level approach The evolution of the state gives us some guidance on the effect that can be expected from the system, but it does not show what the mechanisms are and why this occurs. Trying to express the evolution with a known basis can give the missing information on what are the optimal conditions for our purpose. The need of the eigenstates of the system to evaluate our results motivates their use as a time-dependent base in this next section. Using a basis defined by a Hamiltonian that changes at each point in time, its evolving eigenstates can be inserted into the time-dependent Schrödinger equation (TDSE) to obtain an expression for the coupling between them at each instance, avoiding the need to write the evolved state at time t in the basis at t + ∆ t . The single-particle Hamiltonian at some particular time in the instantaneous eigenstate basis can be written as ˆ H=X i|Ψi⟩εi⟨Ψi|(3.8) being | Ψ i⟩ the eigenstates at time t , with energies εi . The TDSE applied to a general state looks like iℏ∂ ∂t|Ψ⟩=ˆ H|Ψ⟩ ⇒ iℏ∂ ∂tX n cn|Ψn⟩=X i |Ψi⟩εi⟨Ψi|X n cn|Ψn⟩ ⇒ ⇒iℏX n˙cn|Ψn⟩+cn˙ |Ψn⟩=X n cnεn|Ψn⟩, (3.9) which projected to a particular outgoing eigenstate | Ψ m⟩ REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 40 3.3. Few-level approach Looking at figure 3.3, one can already see that the rightmost maximum, being the widest, seems to be the best point to optimize the electron retrieval process with respect to the grazing parameter. At shorter distances, the values for ∂2 gpF are higher at the extreme points, making them less reliable for any noise that the system may have. Even in this best case, a change in ∆ gp∼ 3 nm already decreases the fidelity to zero. 150 160 170 180 190 200 0 0.2 0.4 0.6 0.8 Figure 3.9: Values of γ depending on gp for different values of the potential’s strength. The analysis of the change in the potential’s strength is a more complicated issue. One must take into account that both terms that enter in the description of the groundstate’s evolution change with it. Moreover, the strength with which the twofold groundstate is coupled to the excited state during the evolution is also changed. In general, one can ensure that bigger strengths (tighter potentials) will decrease the possibility of exciting any state living in the twofold groundstate. On the other hand, one would need to get the dots closer to have the possibility of fulfilling the condition in (3.22) , as can be seen in figure 3.9. At shorter distances, the change REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 3. The transfer process 41 in energy occurs faster, implying that the condition in (3.21) is met faster between fidelity maxima, increasing the error in transfer probability for smaller errors in gp . Generally speaking, if one would like to avoid errors resulting from having too big of a change in fidelity varying the grazing parameter, lower velocities (which most of the time requires a change of material) and lower potential strengths are the solution. Obviously, this decreases the number of particles that can be retrieved/injected per time unit, which may not be useful for quantum computation purposes. REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 42 REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 4 The spin orbit interaction Contents 4.1 Well-separated minima ................... 45 4.2 Effects on transfer probability ............... 47 4.3 Entanglement ......................... 51 Since the electron is moving in a space with a broken symmetry, its spin will evolve under the influence of a spin-orbit coupling that can be described by the Rashba [ 15 ] and Dresselhaus [ 26 ] spin-orbit coupling HSO =αR(ˆpxσy−ˆpyσx) + βD(−ˆpxσx+ ˆpyσy). (4.1) This interaction couples the spin and orbital degrees of freedom of the particle in a generally non-trivial way. Although there have been some studies describing the evolution of such systems for moving potentials, [ 40 , 23 ] as well as using them to perform single-qubit gates, [ 32 , 30 ] the loading process to a static dot, which includes also movement in the transverse direction of the electron carrying potential has not been yet tackled. Note that, even if the strength of the spinorbit interaction is smaller than the ones responsible for the transfer of the state (and therefore can be considered a perturbative effect on energy), the qualitative difference between the states can potentially change the time derivatives of the 43 REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 44 4. The spin orbit interaction eigenstates of the system in a significant manner. Reordering the terms according to their momentum direction proportionality ˆ HSO =−(βDσx−αRσy)·ˆpx−(αRσx−βDσy)·ˆpy=−σa·ˆpx−σb·ˆpy,(4.2) that can be seen as a momentum shift on the original Hamiltonian, since (ˆpx−m∗σa)2+ (ˆpy−m∗σb)2 2m∗=p 2m∗−σa·ˆpx−σb·ˆpy+1 2m∗(σ2 a+σ2 b)(4.3) where the last term is just a global shift in energy. Knowing this, one can be tempted to use the displacement operator to obtain the solutions of the total Hamiltonian, but as σa = σb are usually different, the two-momenta displacement operator does not give us the wanted Hamiltonian ˆ H(t) = ˆ H0(t)+ ˆ HSO =ˆ Dp(m∗σa, m∗σb)ˆ H0(t)ˆ D† p(m∗σa, m∗σb)−1 2m∗(σ2 a+σ2 b)(4.4) where the shift in momentum would be obtained by applying ˆ Dp , having the form ˆ Dp(m∗σa, m∗σb) = eim∗(σax+σby)/ℏ.(4.5) The Zassenhaus formula gives an infinite series that describes the total displacement operator as a multiplication of separate displacements. From the second order on, one finds that the commutator between σa and σb appears in all individual displacements. This expression looks like [σa, σb]=2i(α2 R−β2 D)σz. (4.6) which is precisely the reason for (4.4) . Since σa and σb do not commute in general, the sum in the exponent is not the same as doing each displacement separately and the exponentiation of such a matrix is a numerically expensive task. The condition for which the displacements can be applied separately is |αR| = |βD| , which is really specific but has been treated as possible [ 40 ] and experimentally REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 4. The spin orbit interaction 45 achievable [ 23 ]. Taking into account these limitations, this work will deal with the particular case where the commutator is zero. Choosing x and y along GaAs’s [110] and [ ¯ 1 10] orientations, the experimental strength for both the Rashba and Dresselhaus interaction can be tuned to αR = 300 nm ·ns−1 [ 40 , 23 ]. Different parameter choices can be made, but as long as this same order of magnitude is chosen, the physical meaning and effects will be, broadly speaking, the same as those described here. 4.1 Well-separated minima The previous section has parametrized the evolution of the spinless electron such that it can be described by the two-fold groundstate. For sufficiently separated dots, this is spanned by the groundstates of the free Hamiltonian with a single potential. With the inclusion of the spin-orbit interaction, the subspace of interest is doubled, each of the free states having the spin degree of freedom tied to their orbital state. In the initial and final configurations of the complete Hamiltonian the electrons are located around each particular potential and are not coupled between them, so they can be separately described. Let us examine the effect of the spin-orbit interaction in this configuration. The static spinless state | Ψ s⟩ is the eigenstate of the Hamiltonian ˆ H0,s = p / 2 m∗ + Vs . Considering the case of αR = βD makes σa = σb≡σ and the displacement operator simplifies to ˆ Dp(σ) = eim∗σ(ˆx+ˆy)/ℏ, (4.7) which changes the partial Hamiltonian to ˆ Dp(σ)ˆ H0,s ˆ Dp(σ)†=(ˆpx+m∗σ)2+ (ˆpy+m∗σ)2 2m∗+Vs−m∗σ2(4.8) and the corresponding eigenstate to (ˆ H0,s +ˆ HSO)|Ψs, ms⟩=εs,SO|Ψs, ms⟩,(4.9) REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 46 4.1. Well-separated minima where ms represents the coupled spin state. The case of the moving dot’s eigenstate is a bit different. Since the instantaneous eigenstate of the moving potential shifts in time in the y direction, it can be understood as a time-dependent basis state in our description, |Ψm(t)⟩=ei(vSAW·t)·ˆpy/ℏ|Ψm⟩(4.10) similar to the target state in the previous chapter. The way to obtain the eigenstate of the moving potential with the spin-orbit interaction is arriving to an expression that includes both the displacement in momentum and time. Applying the momentum operator to the time displacement, one obtains ˆ Dp(σ)ei(vSAW·t)·ˆpy/ℏ=ei(m∗σ(ˆx+ˆy)+(vSAW·t)·ˆpy−m∗(vSAW·t)σ/2)/ℏ(4.11) where the Baker-Campbell-Hausdorff formula has been used, along with the usual commutation relation [ ˆy, ˆpy ] = iℏ . The final displacement operator that we are looking for includes the two first addends in the right hand side exponential, which leaves us with the relation ˆ Dvt,σ =eim∗σ(ˆx+ˆy)/ℏei(vSAW·t)·ˆpy/ℏeim∗(vSAW·t)σ/(2ℏ)(4.12) that applied to the instantaneous eigenstate of the moving potential as in equation 4.9 (ˆ H0,m(t) + ˆ HSO)|Ψm, ms(t)⟩=εm,SO|Ψm, ms(t)⟩,(4.13) obtains the eigenstate of the moving electron under spin-orbit interaction at time t . This allows us to compute Ω mm from 3.11 to describe a non-adiabatic process related to the electron’s spin precession under these circumstances: Ωmm =⟨Ψm, ms(t)|˙ Ψm, ms(t)⟩(4.14) =im∗v 2ℏ⟨Ψm, ms(t)|σ|Ψm, ms(t)⟩(4.15) REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 4. The spin orbit interaction 47 which has the effect of an effective magnetic field in the direction of σ/√2αR and strength Bσ = m∗vαR/√2µBℏ , which has been observed experimentally. This means that if one chooses the eigenstates of the total Hamiltonian with a spin in the σ = αR ( σx + σy )direction, there will be two uncoupled subspaces by the spin value and two states in each subspace, shifted in energy depending on whether they move or not. 4.2 Effects on transfer probability Following the same principles that have been described in section 2the energies and eigenstates of the Hamiltonian ˆ H ( t ) = ˆ H0 ( t ) + ˆ HSO have been obtained at several points in time for the trajectory of the moving potential. The coupling terms between different eigenstates can be calculated with the exact same expression as before, but now one needs to also include the energy shift that comes by having moving eigenstates with a spin-orbit interaction. The new interaction is not strong enough to significantly increase the transfer to excited states in the vicinity of the transfer probability maximum located at 160 . 5 nm , which is where we are interested in operating our system. As previously reasoned, in the (now doubled) two-fold groundstate there are two uncoupled subspaces if the eigenstates’ spins are directed in the σ direction, such that ˆ H(t) = H+(t) 0 0H−(t)!(4.16) where the subscripts in H± refer to the spin values and both objects are 2 × 2 matrices that will evolve the initial state of the electron. To evaluate how these new expressions will affect the evolution of the electron depending on its spin, let us compare them with H0 . Taking into account that the new terms are smaller in strength and they come from the Hamiltonian’s time-derivative part, we can write H±(t) = H0(t)±ℏ δΩs(t)δΩsm(t) δΩsm(t)δΩm(t)!(4.17) REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 48 4.2. Effects on transfer probability where the second matrix encapsulates the changes to the evolution added by the SOI. The subscripts have been decided depending on whether the eigenstates is initially static ( s ) or moving ( m ). Its diagonal elements are related such that: δΩm=−δΩs+ Ω0 mm.(4.18) Writing it as in equation 3.15, changes appear as an addition to the energy gap and a new perpendicular term: H±= [fz(t)±δfz(t)]σz+fy(t)σy±δfx(t)σx(4.19) where δfz(t) = ℏδΩs−Ω0 mm/2(4.20) δfx(t) = ℏδΩsm.(4.21) When the potentials get closer to each other, the orbital parts can be more related to the even and odd combinations of the localized states, depending on the tilt between the two potentials. Moreover, the diagonal parts of the non-adiabatic contribution are expected to be strictly real since all states’ spins are required to point in the direction set by the spin-orbit interaction. As figure 4.1 shows, the initial and final diagonal elements of the non-adiabatic part are constant, coinciding with the strength previously mentioned as an effective magnetic field acting in the moving particle. The middle part, where one expects to have even and odd combinations of orbital parts in the eigenstates, shows half of the strength found in the beginning and the end of the process. This indicates precisely that there is a part of the eigenstates that keeps moving in the same direction as the potential. Since the electron is not only moving forward but also laterally, one would also expect to see imaginary components in the non-diagonal elements of the nonadiabatic contribution. This is shown in the lower part, where both the real and the imaginary parts of the non-diagonal elements in the non-adiabatic REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 4. The spin orbit interaction 49 0 20 40 60 80 100 120 140 160 -4 -3 -2 -1 0 1 Figure 4.1: All elements that are included in the Hamiltonian due to spin-orbit interaction 4.17. contribution are depicted. The real part partially corresponds to the nonadiabatic transition of the orbital part, which can also be found in the free-electron evolution process. It will also have a contribution to the spin-orbit interaction coming from the lateral movement of the electron. On the other hand, the imaginary part may exclusively come from the spin-orbit interaction. This suggests that an effective time-dependent magnetic field can be tracked during the evolution of the system, which implies a possibility of disturbing the process of transferring the electron from one dot to the other, given that the orbital and spin parts are entangled. Once the evolution matrix is defined for a particular path of the moving potential, one can evolve the system with different initial spin states and see the change in fidelity for the transfer process to analyze its dependence on the effective timedependent magnetic field created by the spin-orbit interaction. Given that during most of the interaction the effective field is directed in the σ direction, it can be REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 56 5.1. Approximated expressions Once this is done and taking into account the scale difference in interaction strength between the hyperfine and the terms governing the evolution of the electron, one can evolve the spin dynamic between the electron and the nucleus’s spin by evolving the state defined in that subspace in the interaction picture as a perturvative effect. Having the upper bound of the spin-flip rate for a single nucleus, one can obtain an approximate value for the overall probability of flipping the electron’s spin once it goes in and out of the static dot. 5.1 Approximated expressions The hyperfine interaction describes the energy that exists between two magnetic moments, which can be described as distributions or point-like. It is commonly used to describe the electron’s spin interaction with the nucleus’ [ 9 ], which only depends on their relative position r=r e− r n is given by ˆ Hhf = 2ℏµBγnhr−3I·(L−S)+3r−5(I·r)(S·r) + (8π/3)δ(r)I·Si(5.1) where µB is Bohr’s magneton, γn is the nuclear gyromagnetic ratio. The Srefers to the electron’s spin, whereas Irepresents the nucleus’. The first hamiltonian term refers to the interaction of an electron that is orbiting a nucleus, the second one takes into account the static case at a distance and the last one, the contact interaction, solves the indeterminate case for two magnetic moments at the same point in space. In semiconductors, the interaction of the electron spin with the (fully or partially) randomly oriented lattice nuclei leads to fast decoherence of its spin state [ 70 ] (on a time scale of ns, depending on the material). The effect is reduced if the wave function is spread out over more nuclei (coupling more weakly to each one) or if it is moving [ 27 , 40 ] (thereby effectively coupling more weakly to a larger number of nuclei). Nuclear spins, however, have longer decoherence times due to distances between them and nuclear magnetic moment strengths. The same way the nuclear spin bath can be an inconvenient feature of the system as it affects the coherence of the electron’s spin state, this same effect can be used favorably. Due to the hyperfine REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 5. The hyperfine interaction 57 interaction that exists between the electron and the nuclei in its vicinity, there have been proposals to create quantum memories by polarizing the nuclear spins through shuttling of electrons [ 77 , 85 , 47 , 28 ] an recently demonstrated experimentally [ 2 ]. Taking advantage of the knowledge that the model presented in this article gives on the evolution of the electron’s wave function for both spin directions, it is interesting to analyze the possible use that it may have as nucleus spin polarizer and evaluating how good the approximation of instantaneous loading done in [ 28 ] is. The intention of this section is to give an approximate number on the polarization rate of nuclei in a quantum dot by choosing a representative case to estimate the overall behavior. In many semiconductors, including GaAs, the most relevant effects are well described assuming the electron lies in a s-type conduction band, which means that there is no orbital momentum contribution in the interaction [ 70 ]. Therefore, the Hamiltonian term for the electron’s interaction with the surrounding nuclei can be simplified as: ˆ Hhf =X α ˆ SαX n Anδ(r−rn)ˆ Iα n.(5.2) where An is the interaction strength of the n -th nucleus with the electron. The operator that couples a single nucleus with the electron is therefore of the form ˆ Sˆ In=X α ˆ Sαˆ Iα n=ˆ Szˆ Iz n+1 2(ˆ S−ˆ I+ n+ˆ S+ˆ I− n).(5.3) where the plus and minus signs are the usual ladder operators that increase or decrease the spin’s value in the σ direction. 5.2 includes all magnetic nuclei in general, which can be reduced to an effective number Neff using the inverse participation ratio ( IPR ). Since our interest lies in the polarization of nuclei that are in the vicinity of a static quantum dot’s center, let’s use the wavefunction of the eigenstate in the static dot at time t = 0 ps , previously named | Ψ s ( r, 0) ⟩ . Let us define the total number of magnetic nuclei on our three-dimensional grid as Nn . Using the definition of the inverse participation ratio (which estimates 1 /Neff where Neff is the number of particles REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 58 5.1. Approximated expressions interacting with a state defined by a spread probability function), we can set the number of interacting nuclei as Neff =Nn X i=1 |ϕs(ri)|4v2 0−1(5.4) where the index i refers to the points in the three-dimensional grid that defines the state’s probability density in real space, whereas v0 refers to the volume unit around a single atom. The normalization of the state is ensured by the condition PNn i=1 |ϕs ( ri ) |2v0 = 1. Neff has a value between 7 . 9 · 10 5 and 3 . 9 · 10 6 for ∆ z varying from 2 nm to 10 nm . During the next calculations, the strongest potential strength will be considered, putting an upper limit on the interaction’s effect. The overall Hamiltonian this far has considered the kinetic energy of the electron, the electric potential created by the SAW that allows its transport and the spinorbit term that defines a preferential direction for the spin state of the electron. Including the hyperfine interaction one can write ˆ HT(t) = ˆ T+ˆ Ve(t) + ˆ HSO +ˆ Hhf =ˆ H0(t) + ˆ Hhf (5.5) as the total Hamiltonian. Since the z component of the hyperfine interaction is diagonal in the basis defined by ˆ H0 ( t )it can be separated form the rest of the hyperfine terms and added to this original Hamiltonian ˆ H(t) = ˆ H0(t) + ˆ Sz Neff X i=1 Aiδ(r−ri)ˆ Iz i(5.6) leaving the spin-flipping terms as the only interaction Hamiltonian ˆ Hsp =ˆ S+ Neff X i=1 Aiδ(r−ri)ˆ I− i+ˆ S− Neff X i=1 Aiδ(r−ri)ˆ I+ i.(5.7) The unitary that evolves the system from an initial time to t in the Schrödinger picture without the hyperfine interaction is defined as ˆ U(t, 0) = Te−iRt 0ˆ H(t′)dt′/ℏ,(5.8) REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 5. The hyperfine interaction 59 not to be confused with the total unitary that evolves the state to time t , since this would have to take into account the hyperfine interaction. The matrix representation of the Hamiltonian in our reduced basis has been defined up until now in the basis of | Ψ s, ms⟩ and | Ψ m, m′ s⟩ , the instantaneous eigenstates of the Hamiltonian without the hyperfine interaction. Instead of this, we could change the basis such that it follows the evolved state. To do this, one needs to find a matrix that, applied to the previous coefficients, always projects the state to the first component. Taking into account that for the aforementioned conditions on the spin-orbit parameters there is no possible spin-exchange in the z direction, let us call the time-evolved state |Φj(t)⟩±≡ |Ψj(t),±⟩ =c(j) s,±|Ψs,±⟩+c(j) m,±|Ψm,±⟩,(5.9) where the time dependence comes from the time-dependent basis states | Ψ i⟩ and the corresponding components ci . The j underscript makes reference to the initial state of the electron, which determines the values of the components during the evolution of the electron. An orthogonal state that lives in the same Hilbert space and is not populated during the evolution is |¯ Φ(t)⟩±=c(j)∗ m,±|Ψs,±⟩−c(j)∗ s,±|Ψm,±⟩ (5.10) which will be the second state of a new basis that is useful to us. The matrix that would meet the needed requirements is the basis change matrix from our previous basis to the new one: MΦ¯ Φ sm j ± = c(j)∗ s,±c(j)∗ m,± c(j) m,±−c(j) s,±!.(5.11) Since the evolution of the orbital state is independent for each ms value, this basis change matrix has to be applied for both directions, which leaves us with the complete basis change matrix MΦ¯ Φj sm =     MΦ¯ Φ sm j + 0 0MΦ¯ Φ sm j −      .(5.12) REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 60 5.1. Approximated expressions After it is applied, the time-evolution unitary will just be a diagonal matrix giving each populated state its corresponding phase. For simplicity, the notation for states or operators will not change, but all elements are now calculated with the populated orbital states as a time-evolving basis, instead of the instantaneous eigenstates of the total Hamiltonian without the hyperfine interaction. In the interaction picture, one can define the evolution of the system with a simplified operator if the effect of a part of the total on the system is known. Since all our previous considerations make the unitary ˆ U ( t, 0) manageable, it is convenient to describe the Hamiltonian and the time evolved state in the interaction picture as ˆ ˜ Hsp(t) = ˆ U†(t, 0) ˆ Hsp ˆ U(t, 0) (5.13) leaving the simplified interaction as time dependent, while the states evolve to take into account the phase they acquire with ¯ H ( t ): |˜ Ψ(t)⟩=ˆ U†(t, 0)|Ψ(t)⟩.(5.14) Since we are interested in the evolution of the orbital state of the injected electron, its spin and those of the surrounding nuclei, the general state of the system will include those degrees of freedom. A general state can be written as a superposition of states described by |Φj, ms,m(t)⟩=|Φj(r, t)⟩ms⊗|m1. . . mNeff ⟩(5.15) whose first term is the electron’s state that has evolved over time and the second part is a particular configuration of Neff nuclei around the static dot at time t . The extreme magnetization of the system would be obtained for all values mi having maximal values all up or down (in our case, +3 ℏ/ 2or − 3 ℏ/ 2). There are only two extremal states that can change one out of the magnetic moment of Neff states to change its particular configuration. In the opposite case, at minimum magnetization, one can find the highest amount of possible states with same total Iz . REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 5. The hyperfine interaction 61 Following 5.14, one can write an expanded version of it, which can be helpful to understand both the terms that have already come as well as the ones that will come forward. The state in the interaction picture is |˜ Ψ(t)⟩=X j,ms eiRt 0εj,ms,m(t′)dt′/ℏ˜cj,ms,m(t)|Φj, ms,m(t)⟩(5.16) where the “pseudoenergy” that evolves the phase of each of the states has components corresponding to both the orbital and hyperfine (Overhauser in this case, since it only involves the z component) interaction. Its separation based on the parameters over which it depends can be written as εj,ms,m(t) = ε0,SO(j, ms, t) + εov(ms,m, t)(5.17) where the first term is the diagonal term of the unitary time evolution obtained after changing the basis with the matrix in 5.11 corresponding to the evolving state | Φ( t ) ⟩ . The second term, though, involves applying the Overhauser term of the hyperfine interaction to the spin state of the basis elements as: εov =⟨Φj, ms,m(t)|ˆ Sz  Neff X n=1 Anδ(r−rn)ˆ Iz n |Φj, ms,m(t)⟩= =ms·  Neff X n=1 φn(t)·mn , (5.18) where the term that weights the sum inside the parenthesis is φj,n(t) = An⟨Φj(t)|δ(r−rn)|Φj(t)⟩(5.19) Following [ 70 ], one can calculate the value of the two components that define the hyperfine interaction’s strength between the electron and a particular nucleus: An=4µ0µB 3 µI,n In (5.20) ⟨Φj(t)|δ(r−rn)|Φj(t)⟩=ZΦ∗ j(r, t)δ(r−rn)Φj(r, t)d3r(5.21) REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 62 5.1. Approximated expressions where n refers to the position and species of the nucleus. Following 2.1, the amplitude of the electron wave function at the position of the particular nucleus determines the strength of the hyperfine interaction, which can be parametrized with a (in general species-dependent) constant ηn such that ⟨Φj(t)|δ(r−rn)|Φj(t)⟩=ηnZϕ∗ j(r, t)δ(r−rn)ϕj(r, t)d3r=ηn|ϕ(rn)|2.(5.22) With this definition of the weight inside the sum in 5.18, let us define a magnetization number as ⟨M(t)⟩= Neff X n=1 φn(t)·mn.(5.23) 0 50 100 150 200 250 300 -2 -1 0 1 2 3 Figure 5.1: Weights inside the magnetization factor measured in the center of the static dot. The two lines indicate each an electronic evolution: the solid line for the initially moving electron and the dashed for the initially static one. The dash and dotted plot is a scaled down pseudo-energy gap between the moving and the static state, which is relevant for the evolution of the electron-nucleus spin system evolution. REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 5. The hyperfine interaction 63 Taking into account the values for abundance, nuclear magnetic moment and confinement constants found in Table 1 in [ 70 ], one can calculate a weighted average for the value of An with a value of Aav = 1 . 73 · 10 −8meV . Figure 5.1 uses this averaged value to show the strength of the weights inside the mean magnetization’s sum for different electronic evolutions. The duration of the interaction is limited in this case such that the second moving minimum is in the initial position of the first one at time t = T . As a reference, this calculation has been done for a nucleus found in the middle of the static dot, which can be considered to be an upper limit for these values. Regarding the magnetic moment distribution without any external magnetic field, one can find an expected value of ⟨PNeff n=1 ·mn⟩ = 0, but a non-zero distribution width of ⟨PNeff n=1 m2 n⟩ ∝ qNeff . This gives a good measure for the limit to look at while varying the mean magnetization when evolving the states. One has to take into account that this interaction couples different orbital states inside each electronic spin subpace too. To measure the probability for this effect to happen, one has to compare φjj′,n(t) = 4µ0µB 3 µI,n In ηnϕ∗ j(rn, t)ϕj′(rn, t)(5.24) to the energy gap between these two orbital states εm−εs . Taking into account an approximate number of qNeff ∼ 10 3 nuclei that can create a magnetic moment difference, one can compare |φjj′,n ( t ) |· 10 3 to the energy gap that it must overcome to populate the other state. From figure 5.2 one sees that the maximum value of the coupling term coincides with the energy gap’s maximum value and is 10 −6 times smaller. This gives an approximated 10 3 factor by which the energy gap is bigger to our worst expectation. Therefore, this transfer probability is hugely reduced and will be ignored. As a last note, the non-zero value of this coupling term before the middle-point of the evolution comes from the remaining probability of the outgoing electron to remain in the static dot, which is truncated at time T coming from a charge measurement to see where the electron is at this point. REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 64 5.1. Approximated expressions 0 50 100 150 200 250 300 0 0.5 1 Figure 5.2: Absolute value of the Overhauser term coupling different orbital terms measured at the center of the static dot and the scaled-down energy gap. For the expanded expression of 5.13, the identity matrix composed by the basis elements can be inserted between the unitaries and the Hamiltonian, after which the phases arrive as in 5.16: ˆ ˜ Hsp(t) = X j,ms,m j′,m′ s,m′ eiRt 0(ε−ε′)dt′/ℏ|Φj, ms,m(t)⟩Hj′,m′ s,m′ j,ms,m⟨Φj′, m′ s,m′(t)|(5.25) where both “pseudoenergies” and the matrix elements Hj′,m′ s,m′ j,ms,m depend both on all the parameters over which this sum is done. Looking at the values found in figure 5.1, the spin-flip can be considered to be perturbative compared to the terms that evolve the electronic state. Therefore, the subspace defined by the four possible time-evolved initial states and their orthogonal counterparts is sufficient to describe the necessary Hilbert space. Since the electron’s orbital evolution is defined by the terms calculated previously, evolving the initial states | Φ j, ms⟩ is enough to REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 5. The hyperfine interaction 65 have all the necessary information about the strength of the hyperfine interaction between the electron and any nucleus along its path. 5.2 Spin-flip process time-evolution Now that the interaction’s strength between a single nucleus and the electron has been calculated, it can be useful to model the evolution of the whole system by generalizing with what would happen to a single nucleus with its environment’s effect. The new basis elements can be defined as |ms,m⟩=|ms, mk⟩⊗|m¯ k⟩(5.26) where the interacting nucleus’ spin has been separated from the rest of the nuclei. The notation of the state describing the N−1nuclei that remain indicates which nucleus is missing and the original configuration from which it comes. Introducing this notation to the expression in 5.25 one gets ˆ ˜ Hsp(t) = X χ,χ′ eiRt 0(εχ−ε′ χ)dt′/ℏ|ms, mk⟩⊗|m¯ k⟩Hχ,χ′⟨m′ ¯ k|⊗⟨m′ s, m′ k|(5.27) where the parameters on the sum have changed accordingly to the new notation and summarized in χ . The term inside the sum can be summarized as Hχ,χ′=φj,kδm¯ k,m′ ¯ k(I−S+δms,m′ s+1δmk,m′ k−1+I+S−δms,m′ s−1δmk,m′ k+1)(5.28) where I± ( mk ) = qI(I+ 1) −mk(mk±1) and a similar expression for the electron spin S± ( ms ) = qS(S+ 1) −ms(ms±1) . The first delta function simplifies the sum in 5.27 since | m ¯ k⟩ = | m ′ ¯ k⟩ is a necessary condition for the Hamiltonian element. Also, following the expression of S± , its values are always 1as long as they appear. Taking into account these simplifications, one can rewrite 5.27 as ˆ ˜ Hsp(t) = X mk,m˜φ↑↓ kI−| ↑, mk⟩⊗|m¯ k⟩⟨m¯ k|⊗⟨↓, mk+ 1|+ + ˜φ↓↑ kI+| ↓, mk⟩⊗|m¯ k⟩⟨m¯ k|⊗⟨↑, mk−1| (5.29) REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 72 6.1. The coulomb interaction 6.1 The coulomb interaction The energy gap between the two total spin subspaces comes from the selectivity that coulomb interaction imposes on spins added to the fact that electrons are indistinguishable fermions. All the involved calculations to determine the strength of this interaction can be separated in terms that do not necessarily imply any of those considerations. For two distinguishable charge distributions with an total electric charge of e , the Coulomb interaction reads Cijlm =kee2Zdr2 1dr2 2Ψ∗ i(r1)Ψ∗ j(r2)1 |r1−r2|Ψl(r1)Ψm(r2),(6.1) in real space, where the subindices refer to the label that each distinguishable particle has. In the case of two interacting particles on two different states those indices can take two different values, each referencing one particle in a particular state. Conveniently enough, these indices will be labeled s and m . There are Nr = 2 4 combinations for which these two states can fill four positions (in this case, indices). The value of the integral, though, has a smaller number of possible values due to the symmetry on particle exchange. If one exchanges r1 and r2 the expression remains constant, which means that values can be categorized for the number of repeated indices. In the case of two particles in two states •Cmmmm •Csmmm =Cmsmm =Cmmsm =Csmms •Cssmm =Csmsm =Csmms =Cmssm =Cmsms =Cmmss •Csssm =Cssms =Csmss =Cmsss •Cssss which means that there are only five values of the integral that need to be evaluated in order to have all possible information in this subspace. In the case of using the same potential strengths, the first and last term are also the same. Once REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 6. Two-particle interaction 73 the terms are calculated, one can create a first approach to the Hamiltonian created by these particles as Hdis =X ij |i0⟩εij⟨j0|+X ijlm |i0j0⟩Cijlm⟨l0m0|.(6.2) The vectors denote |i0j0⟩=|i0(r1)⟩⊗|j0(r2)⟩ ⟨l0m0|=⟨l0(r1)|⊗⟨m0(r2)|. (6.3) The first sum corresponds to the single-particle energies whose values are calculated in chapter 4. The two-particle states are, in this case, bare multiplications between single-particle states, since they are distinguishable. Taking into account now that fermions have antisymmetric wavefunctions that carry spin information, one can identify the symmetric spatial combinations between exchanged pairs as belonging to the antisymmetric spin configuration, i.e., the singlet. Following the same logic, the odd combination of spatial states corresponds to the triplet that is shared by states having both spins up and both down. On the other hand, since the doubly occupied states are coupled to the singlet state defined by the two eigenstates in the two different dots, they must have an antisymmetric spin configuration. All these states can be represented as •Singlet: |s↑s↓⟩ ≡ |s0s0⟩⊗ 1 √2(| ↑↓⟩−| ↓↑⟩)(6.4) |m↑m↓⟩ ≡ |m0m0⟩⊗ 1 √2(| ↑↓⟩−| ↓↑⟩)(6.5) 1 √2|m↑s↓⟩−|m↓s↑⟩≡Nsm|s0m0⟩+|m0s0⟩⊗1 √2(| ↑↓⟩−| ↓↑⟩)(6.6) •Triplet: 1 √2|m↑s↓⟩+|m↓s↑⟩≡Nsm|s0m0⟩−|m0s0⟩⊗1 √2(| ↑↓⟩+| ↓↑⟩)(6.7) REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 74 6.1. The coulomb interaction The total spin of the system is zero for the singlet and one for the triplet, and since there is no spin changing interaction in the Hamiltonian, the populations in the singlet and triplet subspaces do not change. It is the energy difference between the two last states that motivates our research. One can initialize a state |m↑s↓⟩ by picking up an electron with a spin in a particular direction with a surface acoustic wave and move it nearby a static dot in which there is an electron with the opposite spin direction. Their interaction will create a rotation in the subspace created by the triplet and the singlet states of Sz = 0 which will depend on the energy gap between these two states. If the evolution of the system can be kept between the two lowest energy states, the final state will look like |Ψ(T)⟩=1 √2       0 0 1 e−iRT 0J(t′)dt′       (6.8) up to a global phase, where J ( t ) = ( ET ( t ) −ES ( t )) /ℏ and therefore the gap between the two subspaces along the trajectory defines the evolution if there is no transfer to excited states. The rotation may leave the state with the spins flipped if the integral of the energy gap is π , which is commonly known as a SWAP gate. If instead, the solution to the integral is half that, π/ 2, the gate is refered to as √SWAP and the final state is partially entangled. The √SWAP gate (together with the single-qubit gates) is universal, and it can be used to implement the standard two-qubit quantum gates necessary for quantum computation [ 8 ]. Burkard et al. [ 14 ] proposed a model of this problem to calculate the singletriplet energy gap depending on external fields using the Heither-London and Hund-Mulliken techniques. The two-particle interaction is a particularly tricky integral to do for generic distributions, mainly due to the difficulty related to the computation of the value at |r2−r1| = 0. The gaussian functions come in handy in this situation, allowing a transformation of variables such that REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 6. Two-particle interaction 75 r+= (r1+r2)/2 r−= (r1−r2)/2(6.9) which allows the separation of the integral in two parts, first integrating over r+ considering r− constant, which reads Cijlm =K+2π qdet(A+)Zdr−exp1 2B+⊤A+−1B++C+1 qx−2+y−2= =K−Zdr−exp−x−y−A− x− y−!+B−⊤ x− y−!1 qx−2+y−2 (6.10) For more information about the constant and variables in the integral, check Appendix A. Changing to cylindrical coordinates, the Jacobian function that enters in the integral is |r−| , which takes care of the dividing distance. The remaining integral over the distance can be done by hand, therefore leaving the integral over the azimutal angle as a numerically solvable problem, which can be done really fast Cijlm =K−Zdrdθexp−Aθr2+Bθr=K−√π 2Z2π 0dθexpBθ2/4Aθ √Aθ (6.11) Once the calculation of both the singleand two-particle interactions are done, one can look for the energy gap between the two lowest energy eigenstates, which is the interaction strength. If this is done for the whole duration of the process, the integral over that gives the angle at which the final state has evolved. 6.2 Further corrections Although using Gaussian functions as a representation of the in-plane wavefunctions can be a useful approach for modelling some dependencies, in our case the nonadiabatic time-evolution and the vicinity of the potentials at some points require further steps to be able to properly describe the system’s evolution. In order to do this, we would like to know the eigenstates of the system including the Coulomb REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 76 6.2. Further corrections interaction so that we can measure the strength of both the energies and nonadiabatic terms that may cause the system to be excited. Therefore, we want to have a description (as analytic as possible) of the Coulomb interaction in 2D taking into account that our states live in a 3D space. There are approaches that have used less points in k-space to define the eigenstates of the system [ 50 ], so the description of the interaction should be given also in this space. 6.2.1 Real space expressions The real space integral expression of the two-particle interaction between the two-particle states | Ψ i⟩ and | Ψ j⟩ is defined as Cij ≡ ⟨Ψi|C|Ψj⟩=Z∞ −∞ Ψ∗ i(r1, r2)Ψj(r1, r2) q(x2−x1)2+ (y2−y1)2+ (z2−z1)2d3r1d3r2.(6.12) In our particular case, due to the energy levels in the z-direction being much further apart from each other than the ones in x and y, we can write any relevant state for the evolution of the system as: ⟨r|Ψi⟩= Ψxy i(x1, x2, y1, y2)Ψz i(z1, z2)(6.13) as it was done for one-particle states in 2.2, where the z-direction probability distribution function can be modeled as a gaussian function with a ∆ z width for both particles: Ψz i=s1 2π∆−1 ze−(z2 1+z2 2)/4∆2 z.(6.14) so we can introduce it in the expression that we want to evaluate (minus the normalizing constants written in the previous expression): Cij =Z∞ −∞ Ψxy i ∗Ψxy je−(z2 1+z2 2)/2∆2 z q(x2−x1)2+ (y2−y1)2+ (z2−z1)2d3r1d3r2=(6.15) =Z∞ −∞ Ψxy i ∗Ψxy jdx1dx2dy1dy2Z∞ −∞ e−(z12+z22)/2∆2 z qf2+ (z2−z1)2dz1dz2(6.16) REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 6. Two-particle interaction 77 with f2≡f(x1, x2, y1, y2)2= (x2−x1)2+ (y2−y1)2.(6.17) To simplify the integral over the z-coordinates further, let us define gxy ≡g(x1, x2, y1, y2) = Z∞ −∞ e−(z12+z22)/2∆2 z qf2+ (z2−z1)2dz1dz2.(6.18) To proceed with the integral it is convenient to change variables to zs = z1 + z2 and zd = z1−z2 which leaves it as: gxy = 2Z∞ −∞ e−(zs2+zd2)/4∆2 z qf2+z2 d dzsdzd= 4∆z√πZ∞ −∞ e−zd2/4∆2 z qf2+z2 d dzd= = 4∆z√πZ∞ −∞ e−z′ d 2 qf′2+z′ d 2dz′ d (6.19) and here f′ = f/ 2∆ z . Now we can use the fact that the function inside the integral is even and integrated over an even range around 0: gxy = 8∆z√πZ∞ 0 e−z′ d 2 qf′2+z′ d 2dz′ d.(6.20) This integral can be found in [ 35 ], page 367, integral number 3.462 (25). The solution that is given comes from a sequence of variable changes. First, let’s call zf = z′ d/f′ such that gxy = 8∆z√πZ∞ 0 e−f′2zf2 q1 + zf2dzf.(6.21) There are two ways of proceeding here. If the variable change zf = sinh ( t )is considered: gxy = 8∆z√πZ∞ 0e−f′2sinh2tdt (6.22) which is a pretty compact form of the integral and, most importantly, has a monotonic tendency, which makes the estimation of the error of the approximated REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 78 6.2. Further corrections integral easy evaluating the value of the function. The other possible change of variable is t = z2 f , which leaves equation 6.21 as: gxy = 4∆z√πZ∞ 0 e−f′2t √1 + t2√tdt. (6.23) This also appears in [ 35 ], page 1023, integral number 9.211 (4). It is also similar to the expression in [ 50 ], Appendix B, equation B4. The main difference with that expression is that they appear to have it in the denominator of the Coulomb term and the constants in the function are different. Notice, for example, that their α = − 1 / 2value is negative while in [ 35 ] the expression is given only for α > 0. This time, the expression that the book gives can be related to Bessel functions such that: gxy = Ψ(1 2,1, f′2) = Γ(0)Γ(1) Γ(1/2) ef′2/2J0(if′2/2) (6.24) combining equations 9.210 (2) and 9.215 (2) from the same section. 6.2.2 Momentum space expressions Another way of approaching this integral is to change the space in which it is defined such that the matrix elements that need to be computed to solve the eigenvalue problem are easier to compute. Let us define our new two-particle basis as {|k1k2⟩} , where ⟨r|k1k2⟩=e−i(k1·r1+k2·r2)(6.25) describe the three-dimensional form for particle 1 having momentum k1 and k2 in the case of particle 2. Inserting the unitary defined by this basis in the Coulomb interaction operator ˆ Ck=X k1k2k′ 1k′ 2|k′ 1k′ 2⟩Ck1k2k′ 1k′ 2⟨k1k2|(6.26) asks for the Fourier transform of the Coulomb interaction for a particular combination of momenta: REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 6. Two-particle interaction 79 Ck1k2k′ 1k′ 2=Ze−i(k1−k′ 1)r1−i(k2−k′ 2)r2 |r2−r1|d3r1d3r2.(6.27) This expression gets simplified by performing a change of variable such that r2−r1=rd r2+r1=rs (6.28) leaving (6.27) as Ck1k2k′ 1k′ 2=1 2Ze−i[(k1−k′ 1)+(k2−k′ 2)]rs/2d3rsZe−i[(k1−k′ 1)−(k2−k′ 2)]rd/2 rd d3rd= =δh(k1−k′ 1)+(k2−k′ 2)i 2Ze−i[(k1−k′ 1)−(k2−k′ 2)]rd/2 rd d3rd (6.29) where the 1 / 2factor comes as the Jacobian for the variable change and the delta function in the last expression ensures the momentum conservation in all directions. Defining q= k1−k′ 1 , one can write the remaining integral as the limit of the Yukawa potential with the scaling parameter going to zero: Ze−iq·rd rd d3rd= lim λ→0Ze−λrde−iq·rd rd d3rd.(6.30) Changing to spherical coordinates and doing the usual variable changes, one can obtain an expression depending only on the distance rd that reads lim λ→0Ze−λrde−iq·rd rd d3rd= lim λ→0 2π qi Z∞ 0e(iq−λ)rd−e−(iq+λ)rddrd(6.31) which can be integrated and evaluated leaving a final expression lim λ→0Ze−λrde−iq·rd rd d3rd= lim λ→0 2π qi 2qi q2+λ2=4π q2(6.32) which depends only on the modulus of the momentum exchanged between the involved plane waves. Going back to what we intend to obtain, one can write (6.26) as ˆ Ck=X k1k2q|k1−q k2+q⟩2π q2⟨k1k2|.(6.33) REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 80 6.2. Further corrections Similarly to what was done in section 6.2.1, in order to obtain an expression that can be used for a two-dimensional probability distribution one needs to insert the information of the third direction in this last equation. Bracketing it with the z component that is common for the two particles ˆ C2D,k =⟨Ψz|ˆ Ck|Ψz⟩(6.34) one can obtain such expression. First, one must find the Fourier transformed form of the z component of the states, since it is required by the variable change. Following the convention that matlab’s ’fft’ function gives us, let us define our normalized z component of the wavefunction as Ψkz=s2 π∆ze−∆2 z(k2 z1+k2 z2),(6.35) which can be inserted in the previous expression giving ˆ C2D,k = 4∆2 zZe−∆2 z[(k1z+qz)2+(k2z−qz)2]e−∆2 z(k2 1z+k2 2z) q2 x+q2 y+q2 z dk1zdk2zdqz= = 4∆2 zZe−2∆2 z[k2 1z+k2 2z+qz(k1z−k2z)+q2 z] q2 x+q2 y+q2 z dk1zdk2zdqz= = 4∆zrπ 2Ze−2∆2 z[k2 2z−k2zqz+q2 z−q2 z/4] q2 x+q2 y+q2 z dk2zdqz= = 2πZ+∞ −∞ e−∆2 zq2 z q2 x+q2 y+q2 z dqz (6.36) where the simplifications have been made using the integral number 3.323 (2) in page 339 on [ 35 ]. The last term that has been written in the simplification appears as part of an alternative expression for the error function (’erf’), integral number 8.252 (4) in page 898 of the same reference: ˆ C2D,k = 2π2e∆2 zq2 xy qxy [1 −erf(∆zqxy)] (6.37) where qxy = qq2 x+q2 y is the norm of the in-plane exchanged momentum between the corresponding plane waves. This expression is ill-defined for qxy = 0, where the limit goes to + ∞ . REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 6. Two-particle interaction 81 6.3 Numerical considerations and results Having the explicit expressions for the two-dimensional Coulomb energy that the particles will have defines the numerical Hamiltonian’s entrances for the grid points that one is considering. Being a two-particle state, the total amount of points that define a real-space representation of the wavefunction is N4 , which in our case has a value over 4 million. The opeartors, moreover, would have to be N4×N4 objects. It is clear that some dimensionality reduction has to be done for this otherwise untractable problem. 0 20 40 60 80 100 120 140 160 0 0.02 0.04 0.06 0.08 0.1 Figure 6.1: Singlet-triplet energy gap for the two-particle eigenstates of the Coulomb interacting Hamiltonian with two potentials. Here is where the description in the momentum space can simplify these issues. Since the real-space translation corresponds to the multiplication of a plan wave in k-space one can define the moving potential by the necessary gridpoints around | k | = 0 and then translate it as needed, allowing the use of a lower number of grid REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 88 A. Coulomb term calculation where K+=AiAjAlAmeγx+γy+γxy A+=−2 αxαxy αxy αy! B+= 2 βxx−+βxyy−−˜αx βxyx−+βyy−−˜αy! C+=αxx−2+αyy−2+ 2αxyx−y−−2˜ βxx−−2˜ βyy− (A.5) and Ai refers to the normalizing factor of state i . Note that both B+ and C+ have terms involving x− and y− , which are being integrated afterwards. The constants αuv,..., ˜ βu are given below in Eqs. (A.10-A.15). For the moment, one can integrate over r+ : Cijlm =K+2π qdet(A+)Zdr−exp1 2B+⊤A+−1B++C+1 qx−2+y−2 =K−Zdr−exp−x−y−A− x− y−!+B−⊤ x− y−!1 qx−2+y−2, (A.6) where the new parameters appearing on the right hand side a defined as K−=K+2π qdet(A+)expαy˜α2 x+αy˜α2 y−2αxy ˜αx˜αy α2 xy −αxαy A−=1 αxαy−αxy2 α′ xα′ xy α′ xy α′ y!+ αxαxy αxy αy!= α′′ xα′′ xy α′′ xy α′′ y! α′ x=αyβx2+αxβxy2−2αxyβxβxy α′ y=αxβy2+αyβxy2−2αxyβyβxy α′ xy =βxy(αyβx+αxβy)−αxy(βxβy+βxy2) B−=−2 αxαy−αxy2 β′ x β′ y!−2 ˜ βx ˜ βy! β′ x=αxy(βx˜αy+βxy ˜αx)−αyβx˜αx−αxβxy ˜αy β′ y=αxy(βxy ˜αy+βy˜αx)−αyβxy ˜αx−αxβy˜αy (A.7) which are all independent of ( x−, y− ). Rewriting the remaining 2d integral in terms of cylindrical coordinates ( r, θ ), the expression for the Jacobian coincides with the denominator and enables the integration over r : REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 A. Coulomb term calculation 89 Cijlm =K−Zdrdθ exp−(α′′ xcos2θ+α′′ xy2 sin θcosθ+α′′ ysin2θ)r2+ B−(1)cos θ+B−(2)sinθr=K−Zdrdθ exp−Aθr2+Bθr(A.8) where Aθ and Bθ just depend on the angle variable introduced by cylindrical coordinates. A last possible simplification can be made, leaving an integral that is numerically easy to solve: Cijlm =K−√π 2Z2π 0dθexpBθ2/4Aθ √Aθ(A.9) REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 90 A. Coulomb term calculation These are the expressions of previously used constants, all of them being sums over indices u∈ {i, j, l, m} αx=X u kux αy=X u kuy αxy =X u kuxy (A.10) αk x=X u kuxxu αk y=X u kuyyu αkx xy =X u kuxyxu αky xy =X u kuxyyu (A.11) γx=X u kuxxu2 γy=X u kuyyu2 γxy =X u kuxyxuyu (A.12) ˜αx=αk x+αky xy ˜αy=αk y+αkx xy (A.13) βx=kix +kmx −kjx −klx βy=kiy +kmy −kjy −kly βxy =kixy +kmxy −kjxy −klxy (A.14) βk x=kixxi+kmxxm−kjxxj−klxxl βk y=kiyyi+kmyym−kjyyj−klyyl βkx xy =kixyxi+kmxyxm−kjxyxj−klxyxk βky xy =kixyyi+kmxyym−kjxyyj−klxyyk ˜ βx=βk x+βky xy ˜ βy=βk y+βkx xy (A.15) REGISTRO TELEMÁTICO Sarreren Erregistro Orokorra / Registro General de Entradas 02/09/2025 13:23 EHU2025E042506 Bibliography [1] Rajeev Acharya, Dmitry A. Abanin, Laleh Aghababaie-Beni, Igor Aleiner, Trond I. Andersen, Markus Ansmann, Frank Arute, Kunal Arya, Abraham Asfaw, Nikita Astrakhantsev, Juan Atalaya, Ryan Babbush, Dave Bacon, Brian Ballard, Joseph C. 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