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Physical review. B./Physical review. B

Microscopic design of a synthetic spin-1 chain in an InAsP quantum dot array

Jacob Manalo, Daniel Miravet, Paweł Hawrylak

Published Feb 8, 2024• 7 citations
Publisher

Abstract

We present here the steps enabling the microscopic design of a synthetic spin-1 chain in an InAsP quantum dot (QD) array embedded in an InP nanowire. The chain is described by a two-leg multiorbital Hubbard Kanamori (HK) model with parameters obtained from the microscopic calculations of up to eight electrons in a single and double QD. In this HK model describing long arrays of QDs, using both exact diagonalization and matrix product state (MPS) tools, we demonstrate a fourfold quasidegenerate ground state separated from excited states by a finite energy gap similar to a Heisenberg spin-1 chain in the Haldane phase. We demonstrate characteristic behavior of spin-$\frac{1}{2}$ quasiparticles at the edges of the chain by observing the magnetic field dependence of the low-energy spectrum as a function of the applied magnetic field. The applied magnetic field isolates the singlet and ${S}^{z}=0$ triplet states from the other triplet components emulating a singlet-triplet qubit but with macroscopic quantum states. Most importantly, the regions in parameter space where the low-energy spectrum of the multiorbital Hubbard chain yields a Heisenberg spin-1 chain spectrum are mapped out.

DOI
10.1103/physrevb.109.085112

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arXiv:0907.1571•Physical Review B•Jul 30, 2009•22 citations
We present here the steps enabling the microscopic design of a synthetic spin-1 chain in an InAsP quantum dot (QD) array embedded in an InP nanowire. The chain is described by a two-leg multiorbital Hubbard Kanamori (HK) model with parameters obtained from the microscopic calculations of up to eight electrons in a single and double QD. In this HK model describing long arrays of QDs, using both exact diagonalization and matrix product state (MPS) tools, we demonstrate a fourfold quasidegenerate ground state separated from excited states by a finite energy gap similar to a Heisenberg spin-1 chain in the Haldane phase. We demonstrate characteristic behavior of spin-12\frac{1}{2}21​ quasiparticles at the edges of the chain by observing the magnetic field dependence of the low-energy spectrum as a function of the applied magnetic field. The applied magnetic field isolates the singlet and Sz=0{S}^{z}=0Sz=0 triplet states from the other triplet components emulating a singlet-triplet qubit but with macroscopic quantum states. Most importantly, the regions in parameter space where the low-energy spectrum of the multiorbital Hubbard chain yields a Heisenberg spin-1 chain spectrum are mapped out.
We present a theory of a two-qubit gate with macroscopic singlet-triplet (ST) qubits in synthetic spin-one chains in InAsP quantum dot nanowires. The macroscopic topologically protected singlet-triplet qubits are built with two spin-half Haldane quasiparticles. The Haldane quasiparticles are hosted by a synthetic spin-one chain realized in chains of InAsP quantum dots embedded in an InP nanowire, with four electrons each. The quantum dot nanowire is described by a Hubbard-Kanamori (HK) Hamiltonian derived from an interacting atomistic model. Using exact diagonalization and matrix product states tools, we demonstrate that the low-energy behavior of the HK Hamiltonian is effectively captured by an antiferromagnetic spin-one chain Hamiltonian. Next, we consider two macroscopic qubits and present a method for creating a tunable coupling between the two macroscopic qubits by inserting an intermediate control dot between the two chains. Finally, we propose and demonstrate two approaches for generating highly accurate two-ST qubit gates: (1) by controlling the length of each qubit and (2) by employing different background magnetic fields for the two qubits.
In this thesis I present a theory of a macroscopic singlet-triplet qubit in quantum dots embedded in nanowires, each containing 4 electrons and together simulating an artficial Haldane gap material. A Haldane gap material exhibits a 4-fold degenerate ground state separated by an energy gap from excitations. The ground state is equivalent to a degenerate spin-singlet and -triplet state. The 4 degenerate states exhibit the characteristics of spins-1/2 localized on either end of the chain. These states may be used as a coded qubit for quantum information processing. Using the effective mass approximation, I calculate single-particle energy levels of one and two quantum dots in a quantum wire. Using these energy levels I compute the Coulomb matrix elements of the interacting Hamiltonian. Using configuration interaction I demonstrate that the ground state of a quantum dot with 4 electrons is a spin-1 state. I then show that the two dot system behaves approximately like two spin-1 objects interacting via an antiferromagnetic Heisenberg Hamiltonian. While the Heisenberg model is approximate, the two dots have a spin-0 ground-state, indicating antiferromagnetic coupling. I then present a simpler spin model to illustrate the physical parameters which control this interaction. Finally, I present a brief solution to the Heisenberg Hamiltonian for finite spin-chains, and show how one can manipulate the singlet-triplet combined ground state of the spin-chain via localized magnetic field, realizing a singlet-triplet qubit in a macroscopic semiconductor device.
In this thesis, I present a microscopic theory of quantum circuits based on interacting electron spins in quantum dot molecules. We use the Linear Combination of Harmonic Orbitals-Configuration Interaction (LCHO-CI) formalism for microscopic calculations. We then derive effective Hubbard, t-J, and Heisenberg models. These models are used to predict the electronic, spin and transport properties of a triple quantum dot molecule (TQDM) as a function of topology, gate configuration, bias and magnetic field. With these theoretical tools and fully characterized TQDMs, we propose the following applications: 1. Voltage tunable qubit encoded in the chiral states of a half-filled TQDM. We show how to perform single qubit operations by pulsing voltages. We propose the "chirality-to-charge" conversion as the measurement scheme and demonstrate the robustness of the chirality-encoded qubit due to charge fluctuations. We derive an effective qubit-qubit Hamiltonian and demonstrate the two-qubit gate. This provides all the necessary operations for a quantum computer built with chirality-encoded qubits. 2. Berry's phase. We explore the prospect of geometric quantum computing with chirality-encoded qubit. We construct a Herzberg circuit in the voltage space and show the accumulation of Berry's phase. 3. Macroscopic quantum states on a semiconductor chip. We consider a linear chain of TQDMs, each with 4 electrons, obtained by nanostructuring a metallic gate in a field effect transistor. We theoretically show that the low energy spectrum of the chain maps onto that of a spin-1 chain. Hence, we show that macroscopic quantum states, protected by a Haldane gap from the continuum, emerge. In order to minimize decoherence of electron spin qubits, we consider using electron spins in the p orbitals of the valence band (valence holes) as qubits. We develop a theory of valence hole qubit within the 4-band k.p model. We show that static magnetic fields can be used to perform single qubit operations. We also show that the qubit-qubit interactions are sensitive to the geometry of a quantum dot network. For vertical qubit arrays, we predict that there exists an optimal qubit separation suitable for the voltage control of qubit-qubit interactions.
We theoretically explore the possibility of realizing the symmetry-protected topological Haldane phase of spin-1 chains in a tunable hybrid platform of superconducting islands (SIs) and quantum dots (QDs). Inspired by recent findings suggesting that an appropriately tuned QD-SI-QD block may behave as a robust spin-1 unit, we study the behavior of many such units tunnel-coupled into linear chains. Our efficient and fully microscopic modeling of long chains with several tens of units is enabled by the use of the surrogate model solver [V. V. Baran et al., Phys. Rev. B 108, L220506 (2023); Phys. Rev. B 109, 224501 (2024)]. Our numerical findings indicate that the QD-SI-QD chains exhibit emblematic features of the Haldane phase, such as fractional spin-1/2 edge states and nonvanishing string order parameters, and that these persist over a sizable region of parameter space.
Energy spectra, spin configurations, and entanglement characteristics of a system of four electrons in lateral double quantum dots are investigated using exact diagonalization (EXD), as a function of interdot separation, applied magnetic field, and strength of interelectron repulsion. A distinctly different quantum behavior is found compared to that of circular single quantum dots. As a function of the magnetic field, the energy spectra exhibit a low-energy band consisting of a group of six states, with the number six being a consequence of the conservation of the total spin of the four electrons and the ensuing spin degeneracies. These six states appear to cross at a single value of the magnetic field, with the crossing point becoming sharper for larger interdot distances. As the strength of the Coulomb repulsion increases, the six states tend to become degenerate and a well defined energy gap separates them from the higher-in-energy excited states. The appearance of the low-energy band is a consequence of the formation of a Wigner supermolecule, with the four electrons (two in each dot) being localized at the vertices of a rectangular parallelogram. Using the spin-resolved pair-correlation functions, one can map the EXD many-body wave functions onto the spin functions associated with the four localized electrons. Analogies with nanomagnets, such as finite Heisenberg clusters, are discussed. The ability to determine associated spin functions enables investigations concerning entanglement properties of the system of four electrons. In particular, the formation of Wigner supermolecules generates EXD solutions belonging to the class of strongly entangled states referred to as N-qubit Dicke states in the quantum-information literature.
Energy spectra and spin configurations of a system of N=4N=4N=4 electrons in lateral double quantum dots (quantum dot helium molecules) are investigated using exact diagonalization (EXD), as a function of interdot separation, applied magnetic field (B)(B)(B), and strength of interelectron repulsion. As a function of the magnetic field, the energy spectra exhibit a low-energy band consisting of a group of six states, with the number six being a consequence of the conservation of the total spin and the ensuing spin degeneracies for four electrons. The energies of the six states appear to cross at a single value of the magnetic field, and with increasing Coulomb repulsion they tend to become degenerate, with a well-defined energy gap separating them from the higher-in-energy excited states. The appearance of the low-energy band is a consequence of the formation of a Wigner supermolecule, with the four electrons (two in each dot) being localized at the vertices of a rectangle. Using spin-resolved pair-correlation distributions, a method for mapping the complicated EXD many-body wave functions onto simpler spin functions associated with a system of four localized spins is introduced. Detailed interpretation of the EXD spin functions and EXD spectra associated with the low-energy band via a four-site Heisenberg cluster (with BBB-dependent exchange integrals) is demonstrated. Aspects of spin entanglement, referring to the well-known NNN-qubit Dicke states, are also discussed.