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arXiv:quant-ph/0403122 · Journal of Applied Physics

Effect of electron-nuclear spin interactions for electron-spin qubits localized in InGaAs self-assembled quantum dots

Seungwon Lee, Paul von Allmen, Fabiano Oyafuso, Gerhard Klimeck, K. Birgitta Whaley

Published Jan 24, 2005• 25 citations
arXivHTMLPDF

Abstract

The effect of electron-nuclear spin interactions on qubit operations is investigated for a qubit represented by the spin of an electron localized in an InGaAs self-assembled quantum dot. The localized electron wave function is evaluated within the atomistic tight-binding model. The electron Zeeman splitting induced by the electron-nuclear spin interaction is estimated in the presence of an inhomogeneous environment characterized by a random nuclear spin configuration, by the dot-size distribution, alloy disorder, and interface disorder. Due to these inhomogeneities, the electron Zeeman splitting varies from one qubit to another by the order of 10−6, 10−6, 10−7, and 10−9eV, respectively. Such fluctuations cause errors in exchange operations due to the inequality of the Zeeman splitting between two qubits. However, the error can be made lower than the quantum error threshold if an exchange energy larger than 10−4eV is used for the operation. This result shows that the electron-nuclear spin interaction does not hinder quantum-dot based quantum computer architectures from being scalable even in the presence of inhomogeneous environments.

DOI
10.1063/1.1850605

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The effect of electron-nuclear spin interactions on qubit operations is investigated for a qubit represented by the spin of an electron localized in an InGaAs self-assembled quantum dot. The localized electron wave function is evaluated within the atomistic tight-binding model. The electron Zeeman splitting induced by the electron-nuclear spin interaction is estimated in the presence of an inhomogeneous environment characterized by a random nuclear spin configuration, by the dot-size distribution, alloy disorder, and interface disorder. Due to these inhomogeneities, the electron Zeeman splitting varies from one qubit to another by the order of 10−6, 10−6, 10−7, and 10−9eV, respectively. Such fluctuations cause errors in exchange operations due to the inequality of the Zeeman splitting between two qubits. However, the error can be made lower than the quantum error threshold if an exchange energy larger than 10−4eV is used for the operation. This result shows that the electron-nuclear spin interaction does not hinder quantum-dot based quantum computer architectures from being scalable even in the presence of inhomogeneous environments.
A pair of self-assembled InGaAs quantum dots filled with two electrons can act as a singlet-triplet spin qubit that is robust against nuclear spin fluctuations as well as charge noise. This results in a T2* coherence time two orders of magnitude longer than that of a single electron, provided the qubit is operated at a particular "sweet spot" in gate voltage. However, at this fixed operating point the ground-state splitting can no longer be tuned into resonance with e.g. another qubit, limiting the options for coupling multiple qubits. Here, we propose using a four-electron coupled quantum dot to implement a singlet-triplet qubit that features a magnetically tunable level splitting. As a first step towards full experimental realization of this qubit design, we use optical spectroscopy to demonstrate the tunability of the four-electron singlet-triplet splitting in a moderate magnetic field.
The time evolution of spin states of two electrons interacting with a nuclear spin bath in a quantum dot system is studied. The hyperfine interaction between the electrons and the nuclear spins is modeled by an isotropic Heisenberg interaction, and the interaction between the electron spins by Heisenberg exchange. Depending on the extent of the overlap between the spatial wave functions of the electrons, there are two physically different cases, namely, the two qubits either interact with the same set of nuclear spins or they see different nuclear spin environments. In the two cases, the decoherence of the two-qubit state is studied analytically. We have identified a class of two-qubit states that have a rich dynamics when the exchange interaction between the qubits becomes large in comparison to the hyperfine interaction strengths. The decoherence time scale is determined as a function of the bath-spin distribution and the polarizations of the initial two-qubit state. States with large decoherence times are identified by performing a minimization over all the two-qubit pure states.
The decoherence and fidelity of spin states in a localized single-electron quantum dot in the presence of a dc magnetic field, arising either from the nuclear hyperfine interaction within the dot or due to its coupling with another localized quantum dot, are examined in detail. A general framework for determining the time evolution of the reduced density matrix. for a single dot is presented, which is exact up to the second order in interaction with any reservoir. In particular, it is applied to the problem of nuclear hyperfine coupling, and approximate estimates of coherence decay time are made when the nuclear spins are either polarized or unpolarized and the internal dynamics of nuclear spins is determined mainly by the nuclear magnetic dipole-dipole interaction. The fidelity of a pure unperturbed electronic one-qubit spin state is obtained as a function of time, which is exact even on a very short timescale of logic gate operations. The time variation of the fidelity of the same one-qubit state on the localized dot as a part of the direct product with another one-qubit state on another localized dot arising because of coupling between these quantum dots is also calculated in this paper. In this case, we include both the single-particle tunnelling between the dots as well as the direct and exchange Coulomb interactions, including on-site Coulomb repulsion. This allows for the double occupation of a single dot. It is found that the loss of fidelity of such two-qubit states due to double occupancy and additional phase errors in the presence of appreciable dot-dot coupling can become a more severe limiting factor than that due to the hyperfine interaction in individual dots.
We determine the spin-exchange coupling J between two electrons located in two vertically tunnel-coupled quantum dots, and its variation when magnetic (B) and electric (E) fields (both in-plane and perpendicular) are applied. We predict a strong decrease of J as the in-plane B field is increased, mainly due to orbital compression. Combined with the Zeeman splitting, this leads to a singlet-triplet crossing, which can be observed as a pronounced jump in the magnetization at in-plane fields of a few T, and perpendicular fields of the order of 10 T for typical self-assembled dots. We use harmonic potentials to model the confining of electrons, and calculate the exchange J using the Heitler-London and Hund-Mulliken techniques, including the long-range Coulomb interaction. With our results we provide experimental criteria for the distinction of singlet and triplet states, and therefore for microscopic spin measurements. In the case where dots of different sizes are coupled, we present a simple method to switch the spin coupling on and off with exponential sensitivity using an in-plane electric field. Switching the spin coupling is essential for quantum computation using electronic spins as qubits.
Spins localized in semiconductor nanostructures have been intensively investigated for quantum spintronics. These include the spin of single electron localized by quantum dots or impurities, and spins of the lattice nuclei. These localized spins can be exploited as carriers of quantum information, while in some circumstances they also play the role of deleterious noise sources for other quantum objects through their couplings. Quantum control of the spins in semiconductor nanostructures is therefore of central interest for quantum applications. In this thesis, we address several problems related to the quantum control of electron or hole spin and nuclear spins in semiconductor quantum dots and impurity centers. The first problem studied is the control of nuclear spin bath for a hole spin qubit in III-V semiconductor quantum dot. In quantum dots formed on III-V compounds, the direct band gap of the host material allows ultrafast optical addressability of a single electron or hole spin qubit. However, nonzero nuclear spins of group III and group V elements result in a large statistical fluctuation in the Zeeman splitting of the spin qubit which then dephases in nanosecond time scale. We present a novel feedback scheme to suppress the statistical fluctuation of the nuclear spin field for enhancing the coherence time of the hole spin qubit. We also find positive feedback control which can amplify the magnitude of the nuclear field, so that a bimodal distribution can develop, realizing a quantum environment that can not be described by a single temperature. The second problem addressed here is the control of donor spin qubits in silicon architecture which have ultra-long quantum coherence time. We developed the quantum control scheme to realize the quantum metrology of magnetic field gradient, based on the celebrated Kane’s architecture for quantum computation. The scheme can also be generalized to calibrate the locations of the donors. In the third part of the thesis, we investigate a novel type of quantum dot formed in a new class of two-dimensional semiconductors, monolayer transition metal dichalcogenides (TMDs), which exhibit interesting spin and pseudospin physics. This novel quantum dot system may offer new opportunity for quantum spintronics in the ultimate 2D limit, and we investigate here the valley pseudospin as a possible quantum bit carrier. A main finding is that, contrary to the intuition, the lateral confinement by the quantum dot potential does not lead to noticeable valley hybridization, and therefore the valley pseudospin in monolayer TMDs QD can well inherit the valley physics such as the valley optical selection rules from the 2D bulk which implies a variety of quantum control possibilities.
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.