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Single-Particle State Mixing in Two-Electron Coupled Quantum Dots

Dmitriy V. Melnikov, Jean‐Pierre Leburton

Published Nov 4, 2021• 0 citations
Publisher

Abstract

The properties of an electron pair confined in a coupled quantum dot (QD) system are currently being intensively investigated both experimentally and theoretically. Indeed the manipulation of the two-electron spin states is paramount for the realization of controllable entanglement so as to produce the fundamental quantum gate operations in a solid state quantum computer. The works usually concentrate on the properties of a two-electron system without going into detailed analysis of the contributions from various single-particle states or the effects produced by the constituent single QDs. Zeeman splitting is neglected as it affects only the magnitude of the exchange energy but not the single-particle state mixing. In summary, after numerically diagonalizing the two-electron Schrodinger equation in the model confinement potential of both single and laterally coupled double quantum dots, we computed the contributions from the various single-particle states to the final two-particle wave function.

DOI
10.1201/9781003148494-15

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The properties of an electron pair confined in a coupled quantum dot (QD) system are currently being intensively investigated both experimentally and theoretically. Indeed the manipulation of the two-electron spin states is paramount for the realization of controllable entanglement so as to produce the fundamental quantum gate operations in a solid state quantum computer. The works usually concentrate on the properties of a two-electron system without going into detailed analysis of the contributions from various single-particle states or the effects produced by the constituent single QDs. Zeeman splitting is neglected as it affects only the magnitude of the exchange energy but not the single-particle state mixing. In summary, after numerically diagonalizing the two-electron Schrodinger equation in the model confinement potential of both single and laterally coupled double quantum dots, we computed the contributions from the various single-particle states to the final two-particle wave function.
In the last years, the electron spin in quantum dots has become a most promising candidate for the realization of qubits, the basic element for quantum computation. In quantum dot spin qubits the exchange interaction implements a fundamental two qubit gate, which originates in the Coulomb interaction and the Pauli principle. Compared to single qubit gates, the exchange based gates are much faster and easier to control locally, motivating the solely exchange energy based quantum computation. The control is based on the exponential sensitivity of the exchange energy on the interparticle distance. Manipulation then can proceed, for example, shifting the single particle states electrically or compressing them magnetically. The practical manipulation schemes require quantitative knowledge of the exchange energy. The spin-orbit interaction plays an important role in this nano-system. Indeed the usual exchange interaction changes its magnitude, while a new term, called anisotropic exchange, appears. In this thesis we propose an accurate effective spin Hamiltonian for modeling spin-orbit-induced spin dynamics in the presence of the magnetic field. We have also performed numerically exact calculations of the isotropic and anisotropic exchange coupling in realistic two-electron GaAs coupled quantum dots in the presence of both Dresselhaus and Bychkov-Rashba spin-orbit interactions. The numerics allow us to establish the goodness of our model and to see the limits of previous statements. Furthermore we find that in zero magnetic field the second-order spin-orbit effects are absent at all interdot couplings.
A two-level quantum system is the building block of a quantum computer. This pair of quantum states defines the computational unit of a quantum computer, and it is called a quantum bit or qubit in analogy to the bit that is the binary unit of a classical computer. The spin of a single electron naturally defines such a two-level quantum system. A quantum dot can be tuned to the single-electron regime, and an array of singly occupied quantum dots provides a multi-qubit register. This thesis describes the multielectron encoding of a spin qubit using a pair or a trio of quantum dots, and it analyzes the coherence properties and manipulation protocols of this system. The singlet-triplet qubit, coded using the singlet state and the spinless triplet state of a pair of singly occupied quantum dots, can be controlled all electrically when applying voltages to gates close to the quantum dot structure. Rapid, subnanosecond modifications of the double quantum dot's charge configuration have been realized successfully. A small magnetic field gradient across the double quantum dot enables universal qubit control. Two modifications of the normal singlet-triplet encoding are introduced. (1) The six-electron configuration of a double quantum dot encodes a singlet-triplet qubit in the same way as for the two-electron double quantum dot. Two electrons at each quantum dot are irrelevant for the qubit manipulations because they are paired in a singlet state; the remaining two electrons encode the qubit. The qubit's wave function is immune to charge noise at moderate out-of-plane magnetic fields. (2) A singlet-triplet qubit encoded using two quantum dots of different sizes has an orbital state degeneracy of the singlet state and the spinless triplet state at finite out-of-plane magnetic fields. Spin-orbit interactions lift this state degeneracy, while the magnitude of the state coupling is determined by the size difference of the QDs. This setup enables the manipulations of singlet-triplet qubits without the need for magnetic field gradients. Finally, two-qubit gates between singlet-triplet qubits are proposed that use mediated exchange interactions via one quantum state. These operations are well controlled and highly noise insensitive. Orbital interactions alone can control spin qubits coded in a three electron Hilbert space. For example, the exchange-only qubit is encoded using three singly occupied quantum dots. The exchange interactions of two quantum dot pairs need to be modified to manipulate this qubit. The noise sensitivity of the exchange-only qubit is discussed. Alternatively, a three-electron qubit at a double quantum dot can be operated when single electrons are transferred between the quantum dots. Fast, subnanosecond manipulations of the double quantum dot's charge configuration are required to realize single-qubit gates. A novel two-qubit pulse gate for the three-electron double quantum dot qubit is proposed.
We report on our recent progress in applying semiconductor quantum dots for spin-based quantum computation, as proposed by Loss and DiVincenzo (1998 Phys. Rev. A 57 120). For the purpose of single-electron spin resonance, we study different types of single quantum dot devices that are designed for the generation of a local ac magnetic field in the vicinity of the dot. We observe photon-assisted tunnelling as well as pumping due to the ac voltage induced by the ac current driven through a wire in the vicinity of the dot, but no evidence for ESR so far. Analogue concepts for a double quantum dot and the hydrogen molecule are discussed in detail. Our experimental results in laterally coupled vertical double quantum dot device show that the Heitler–London model forms a good approximation of the two-electron wavefunction. The exchange coupling constant J is estimated. The relevance of this system for two-qubit gates, in particular the SWAP operation, is discussed. Density functional calculations reveal the importance of the gate electrode geometry in lateral quantum dots for the tunability of J in realistic two-qubit gates.
Recent works indicate that coupled quantum dots (QD) are promising nanostructures for realizing quantum gates that are the basic elements of a quantum computer. In QDs, fundamental quantum logic operations can be achieved in a quantum gate by controlling the entanglement between two electron spins by means of external electric and magnetic fields. This chapter describes the computational approach used to solve the many-body Schrodinger equation. It provides results of the simulations for both the strongly coupled and weakly coupled QDs and analyzes the important differences between the two cases. The utilization of simple trial wavefunctions in VMC have lead to good agreement with the ED method, when electrons are decoupled, which validates two-level approaches such as the Heitler-London approximation in weakly coupled QDs.
An all electrical manipulation with singlet and triplet states of an electron pair in a quadruple quantum dot (QD) has been studied by a configuration interaction method. We have investigated two nanodevices that consist of four weakly coupled lateral QDs, in which the electron localization can be changed by tuning the external voltages: the gate voltage in the first nanodevice and the source-drain voltage in the second. The electrons that---in the initial state---singly occupy the two identical input QDs are transferred to the two asymmetric output QDs by the sufficiently high electric field. We have shown that---for the QDs with suitably chosen parameters---the localization of the electrons in the output QDs depends on the spin states of the electron pair and the final spin state can be uniquely determined by a single-shot measurement of the electric charge gathered in only one of the output QDs. This leads to a possibility of performing the controlled operations on the electron-spin qubits in the quadruple QDs. We have found that the classical exclusive OR (XOR) logic gate can be performed with the electric field-induced transitions and the output read out by the spin-to-charge conversion.
Recently, an implementation of a universal set of one- and two-quantum-bit gates for quantum computation using spin states of coupled single-electron quantum dots was proposed. It was demonstrated that it is possible to execute a coherent control of a quantum system based on two-electron spin states in a double quantum dot, allowing state preparation, coherent manipulation, and projective readout. This possibility is based on rapid electrical control of the spin exchange interaction. These results motivated us to develop a formal theoretical study of the corresponding model of two coupled spins placed in a magnetic field and subjected to a time-dependent mutual Heisenberg interaction. Using possible exact solutions of the corresponding quantum problem, we study entangling of different separable initial states in this model. It is demonstrated that the entanglement due to a time-dependent Heisenberg interaction is dominating in comparison with the entanglement due to the action of an external magnetic field.