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Schematic showing four regions characterizing spin and orbital degrees of freedom in several coupled quantum dots defined by the parameters in Eq. (1) at fixed confinement, ω0. The vertical axis depends on the inter-dot spacing, R, while the horizontal axis depends on the ratio between the cyclotron frequency, ωc, and ω0. The region of interest for quantum computing (spin Hamiltonian) yields a spin Hamiltonian dominated by Heisenberg exchange coupling, Jij Si ·Sj. Here, the higher orbital energy levels of the quantum dot have much higher energy than the exchange splitting, ∆ ≫ J. Above this region, R/a ≫ 1, the electrons in each dot do not interact strongly. At high magnetic fields and near R/a ∼ 1, the electrons capture vortices of the many-body wave function to form mixtures with ∆ J. Below the dotted line the small separation between dots allows single dot behavior, and therefore level crossing, ∆ J.
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Solid state quantum computing proposals rely on adiabatic operations of the exchange gate among localized spins in nanostructures. We study corrections to the Heisenberg interaction between lateral semiconductor quantum dots in an external magnetic field. Using exact diagonalization we obtain the regime of validity of the adiabatic approximation. W...
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... parameters in H define several regimes relevant for quantum computing architectures utilizing similarly con- fined single-electron quantum dots in an external mag- netic field. Fig. 1 depicts four separate pieces of the pa- rameter space with the confinement parameter, ω 0 , fixed. The solid line encloses an area in which the excited, or- bital states of the quantum dots have high energy, ∆, and the inter-dot coupling between two dots maps onto the Heisenberg Hamiltonian, J 12 S 1 · S 2 , as originally en- visaged ...
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... now discuss a set of variational states which model the low energy orbital states of Eq. (1) in an effort to go beyond the Hubbard limit discussed above. To obtain an accruate variational wavefunction we examine the form of the exact wavefunction in two limits: the upper left and lower right corners of Fig. 1. We then construct an ansatz which connects both regimes. We begin with the simplest system, N = 2. It is analytically soluble in two extreme regimes: Two well-separated one-electron "arti- ficial atoms" and a two-electron artificial atom in a high magnetic field. The first case is trivial and consists of two well separated quantum ...
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... Application of the ex- change gate to the two dot system will be an adiabatic process if the energy between the lowest, unwanted ex- cited state of the double quantum dot and the highest spin state storing quantum information is much larger than the exchange splitting, ∆ ≫ J ij . This condition is satisfied in the spin Hamiltonian regime in Fig. 1 and R, respectively. 9 In fact, experiments on coupled quantum dots, while pushing for shorter gate times (and hence larger exchange energies), may indeed leave the border defined by the solid line in Fig. 1. 9,28 It is therefore important to understand the low energy Hilbert space of the coupled dot system when J ij ∆. We will show, ...
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... quantum information is much larger than the exchange splitting, ∆ ≫ J ij . This condition is satisfied in the spin Hamiltonian regime in Fig. 1 and R, respectively. 9 In fact, experiments on coupled quantum dots, while pushing for shorter gate times (and hence larger exchange energies), may indeed leave the border defined by the solid line in Fig. 1. 9,28 It is therefore important to understand the low energy Hilbert space of the coupled dot system when J ij ∆. We will show, for N = 2, that the variational states discussed in Sec. IIB capture the magnetic field depen- dence of ∆. At large fields the variational states describe a bound state between electrons and vortices of the N ...
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... two body terms using the exchange operator: E ij = (4S i · S j + I ij )/2, where I is the identity operator. As apparent from Eq. (14), χ 123 yields an effective Zeeman splitting between the encoded basis states of the three spin qubit. In Sec. IVB we ver- ify numerically that the chiral term is actually sizable in the spin Hamiltonian regime of Fig. 1. We therefore arrive at a revealing inconsistency in seeking a decoher- ence free subsystem from a looped, three spin system. Part of our motivation for simultaneously coupling three spins was to remove the Zeeman term as a potential noise source. However, we have only enhanced the system's de- pendence on the external magnetic field ...
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... seek a quantitatively accurate description of the boundaries and underlying physics of all regions depicted in Fig. 1. While we find that the perturbative expansion in Sec. IIA is valid for R/a > 1 and ω c /ω 0 3, the remaining portions of the parameter space involve long range correlations. Using the N = 2 system we check the accuracy of the variational ansatz discussed in Sec. IIB in several limits. We expect that the variational states discussed ...
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Citations
... The successful manufacture of a QD system provides a practical value for theoretical research. [5][6][7][8] The symmetry and exchange interaction of the internal isolated system and quantum states can be controlled by tuning the gate voltage and Coulomb repulsion energy. [9][10][11] In recent years, as the triangle triple quantum dots (TTQDs) are the smallest elementary structures with topological phenomena, some unique internal characteristics such as chirality states, chiral currents, mixed valence bonds, and Kondo effect are being studied by theoretical researchers. ...
We utilize the calculation of hierarchical equations of motion to demonstrate that the spin-dependent properties between adjacent quantum dots (QDs) can be changed by breaking the internal symmetry configuration, corresponding to the inversion of dominant chiral states. In the linear triple quantum dots (LTQDs) connected to two electron reservoirs, we can observe that the blockage appears at the triangle triple quantum dots (TTQDs) by gradually increasing the coupling strength between next-nearest double QDs. When the initial coupling between LTQDs has altered, the internal chiral circulation also undergoes the corresponding transform, thus achieving qualitative regulation and detection of the blocking region. We also investigate the response of the chiral circulation to the dot–lead coupling strength, indicating the overall robust chiral circulation of the TTQDs frustration.
... where n is the average occupancy on each quantum dot. [25][26][27] J-term is the Heisenberg exchange interaction, where J = 4t 2 /U. Under the half-filling situation, n = 1, the first item disappears. ...
Based on the hierarchical equations of motion (HEOM) calculation, we theoretically investigate the corresponding control of a triangular triple-quantum-dots (TTQD) ring which is connected to two reservoirs. We initially demonstrate by adding bias voltage and further adjusting the coupling strength between quantum dots, the chiral current induced by bias will go through a transformation of clockwise to counterclockwise direction and an unprecedented effective Hall angle will be triggered. The transformation is very rapid, with a corresponding characteristic time of 80–200 ps. In addition, by adding a magnetic flux to compensate for the chiral current in the original system, we elucidate the relationship between the applied magnetic flux and the Berry phase, which can realize direct measurement of the chiral current and reveal the magnetoelectric coupling relationship.
... Under a perpendicular magnetic field, the magnetic flux φ threads the ring-like TTQD structure, we assume the magnetic flux act on the interdot tunneling t, that is, t = e ±i 2π 3 φ /φ 0 t. Then the t-J-χ effective model can be derived by treating t in H dots perturbatively, up to the third order [33,34] ...
We theoretically studied thermoelectric transport properties through the triangular triple quantum dots (TTQD) structure in the linear response regime using the hierarchical equations of motion approach. We demonstrated that large See-beck coefficient was obtained when properly matching the interdot tunneling strength and magnetic flux at the electronhole symmetry point, as a result of spin chiral interactions in TTQD system. We presented a systematic investigation of the thermopower (the Seebeck coefficient) dependence on the tunneling strength, magnetic flux, and on-site energy. The Seebeck coefficient showed a clear breakdown of electron-hole symmetry in the vicinity of the Kondo regime, accompanied by the deviation from the semiclassical Mott relation in the Kondo and mixed-valence regimes, which resulted from the many-body effects of the Kondo correlated induced resonance together with spin chiral interactions.
... When perpendicular magnetic field applied, magnetic flux threads the TTQD structure, the t-J-χ model can be derived by treating t in H dots perturbatively [32,49] ...
New characteristics of the Kondo effect, arising from spin chirality induced by the Berry phase in the equilibrium state, are investigated. The analysis is based on the hierarchical equations of motion (HEOM) approach in a triangular triple quantum-dot (TTQD) structure. In the absence of magnetic field, TTQD has four-fold degenerate chiral ground states with degenerate spin chirality. When a perpendicular magnetic field is applied, the chiral interaction is induced by the magnetic flux threading through TTQD and the four-fold degenerate states split into two chiral state pairs. The chiral excited states manifest as chiral splitting of the Kondo peak in the spectral function. The theoretical analysis is confirmed by the numerical computations. Furthermore, under a Zeeman magnetic field B , the chiral Kondo peak splits into four peaks, owing to the splitting of spin freedom. The influence of spin chirality on the Kondo effect signifies an important role of the phase factor. This work provides insight into the quantum transport of strongly correlated electronic systems.
... In the meantime, spin systems are also considered to be a promising platform to implement a quantum computer. In fact, most of the work on quantum computers is based on spin systems (Note: superconducting loops are actually artificial spins) [46][47][48][49][50][51][52][53][54][55] and Heisenberg model is the most popular model used in treating such systems [51][52][53][54][55] . If we take our pseudo-spin states |↓ and |↑ as qubits states, then the two methods coincide. ...
... In the meantime, spin systems are also considered to be a promising platform to implement a quantum computer. In fact, most of the work on quantum computers is based on spin systems (Note: superconducting loops are actually artificial spins) [46][47][48][49][50][51][52][53][54][55] and Heisenberg model is the most popular model used in treating such systems [51][52][53][54][55] . If we take our pseudo-spin states |↓ and |↑ as qubits states, then the two methods coincide. ...
We show that ultracold polar diatomic or linear molecules, oriented in an external electric field and mutually coupled by dipole-dipole interactions, can be used to realize the exact Heisenberg XYZ, XXZ and XY models without invoking any approximation. The two lowest lying excited pendular states coupled by microwave or radio-frequency fields are used to encode the pseudo-spin. We map out the general features of the models by evaluating the models' constants as functions of the molecular dipole moment, rotational constant, strength and direction of the external field as well as the distance between molecules. We calculate the phase diagram for a linear chain of polar molecules based on the Heisenberg models and discuss their drawbacks, advantages, and potential applications.
... When a perpendicular magnetic field is applied, a flux threads the ringlike TTQD structure. A t-J-χ Hamiltonian can be derived by treating t in H dots perturbatively [23,24]: ...
We theoretically investigate the quantum transport properties of a triangular triple quantum dot (TTQD) ring connected with two reservoirs by means of analytical derivation and accurate hierarchical-equations-of-motion calculation. A bias-induced chiral current in the absence of magnetic field is firstly demonstrated, which results from that the coupling between spin gauge field and spin current in the nonequilibrium TTQD induces a scalar spin chirality that lifts the chiral degeneracy and thus the time inversion symmetry. The chiral current is proved to oscillate with bias within the Coulomb blockade regime, which opens a possibility to control the chiral spin qubit by use of purely electrical manipulations. Then, a topological blockade of the transport current due to the localization of chiral states is elucidated by spectral function analysis. Finally, as a measurable character, the magnetoelectric susceptibility in our system is found about two orders of magnitude larger than that in a typical magnetoelectric material at low temperature.
... While a triangular shape provides some interesting new features, e.g., chirality [101][102][103][104][105][106][107] and faster qubit operations [26,108], these advantages currently do not seem to dominate the experimental drawbacks and difficulties. Therefore, and since almost all experiments and most theoretical studies use the linear geometry, we also mostly stick to the linear geometry in this thesis and implicitly assume that each TQD is linearly arranged, unless otherwise stated. ...
... A functioning threespin qubit device capable of quantum computation is demonstrated in a DQD [140,141,142,8] and in a linear TQD [131,4,136,143,137,9,144,145,146,147]. While a triangular shape provides some interesting new features, e.g., chirality [148,149,150,151,152,153,154] and faster qubit operations [2,12], the advantages currently do not seem to outweigh the experimental drawbacks and difficulties. Therefore and since almost all experiments and most theoretical studies use the linear geometry we also mostly stick in this review to the linear geometry and implicitly consider that each TQD is linearly arranged, unless otherwise mentioned. ...
... In the following, we briefly introduce triangularly arranged TQD systems (TQD molecules) where we mainly focus on the implementation of qubit rotations in such a system which differ from the linear case. For more details about the energy structure and properties we refer to the review by Chan-Yu Hsieh (see Ref. [126]) or the original works [148,149]. In addition to the exchange interaction J 13 between the first and the last dot an (equilateral) triangular shape adds another feature, the chirality, to the system. ...
... A unitary transformation connects them with the conventional eigenstates from Eq. (27). The low-energy subspace can also be approximated by a Heisenberg exchange Hamiltonian [149], however, the exchange couplings include additional terms arising from the circular structure and chirality [204,205,206,152]. Applying an electric field breaks the symmetry of the system and gives rise to terms ∝ σ y in the qubit space, corresponding to rotations around the yaxis on the Bloch sphere [151,152]. ...
The goal of this article is to review the progress of three-electron spin qubits from their inception to the state of the art. We direct the main focus towards the resonant exchange (RX) qubit and the exchange-only qubit, but we also discuss other qubit implementations using three electron spins. For each three-spin qubit we describe the qubit model, the physical realization, the implementations of single-qubit operations, as well as the read-out and initialization schemes. Two-qubit gates and decoherence properties are discussed for the RX qubit and the exchange-only qubit, thereby, completing the list of requirements for a viable candidate qubit implementation for quantum computation. We start with describing the full system of three electrons in a triple quantum dot, then discuss the charge-stability diagram and restrict ourselves to the relevant subsystem, introduce the qubit states, and discuss important transitions to other charge states. Introducing the various qubit implementations, we begin with the exchange-only qubit, followed by the spin-charge qubit, the hybrid qubit, and the RX qubit, discussing for each the single-qubit operations, read-out, and initialization methods, whereas the main focus will be on the RX qubit, whose single-qubit operations are realized by driving the qubit at its resonant frequency in the microwave range similar to electron spin resonance. Two different types of two-qubit operations are presented for the exchange-only and the RX qubit which can be divided into short-ranged and long-ranged interactions. Both of these interaction types can be expected to be necessary in a large-scale quantum computer. We also take into account the decoherence of the qubit through the influence of magnetic noise as well as dephasing due to charge noise.
... Each singlettriplet qubit consists of a pair of exchange-coupled electron spins localized in a double quantum dot, and the two qubits interact via an Ising-type capacitive coupling 11 . Compared with the exchange-coupled spin qubits 1 studied in a previous paper 15 , where two localized electron spins are coupled through the Heisenberg coupling 16,17 , the singlet-triplet system we consider here operates in a larger active Hilbert space due to the lack of spin conservation, and has more complicated dynamics and richer physics. This makes it harder to extract useful insights from analytical solutions 18 . ...
We study a pair of capacitively coupled singlet-triplet spin qubits. We characterize the two-qubit decoherence through two complementary measures, the decay time of coupled-qubit oscillations and the fidelity of entangled state preparation. We provide a quantitative map of their dependence on charge noise and field noise, and we highlight the magnetic field gradient across each singlet-triplet qubit as an effective tool to suppress decoherence due to charge noise.
... 19,20 This model is the appropriate description for the socalled exchange-gate architectures in semiconductor spin quantum computation, with the neighboring dots coupled through an exchange coupling arising from the combination of inter-dot Coulomb interaction and the singleparticle inter-dot wavefunction overlap. [21][22][23] The collective dynamics of the coupled spin qubits under a local magnetic field is described by the following Heisenberg Hamiltonian, 24 Here, the local magnetic field h k and the exchange coupling J k are independent random variables drawn from the normal distributions N (0, σ 2 h ) and N (J 0 , σ 2 J ), respectively. The standard deviations σ h and σ J describe the strengths of Overhauser and charge noise, 25 respectively, with the Overhauser noise being a measure of the background fluctuations in the local magnetic field at the qubits (which could arise, for example, from the very slow nuclear fluctuations whose dynamics is being ignored here). ...
We study the intrinsic, disorder-induced decoherence of an isolated quantum system under its own dynamics. Specifically, we investigate the characteristic timescale (i.e. the decoherence time) associated with an interacting many-body system losing the memory of its initial state. To characterize the erasure of the initial state memory, we define a new timescale, the intrinsic decoherence time, by thresholding the gradual decay of the disorder-averaged return probability. We demonstrate the system-size independence of the intrinsic decoherence time in different models, and we study its dependence on the disorder strength. We find that the intrinsic decoherence time increases monotonically as the disorder strength increases in accordance with the relaxation of locally measurable quantities. We investigate several interacting spin (e.g. Ising and Heisenberg) and fermion (e.g. Anderson and Aubry-Andr\'e) models to obtain the intrinsic decoherence time as a function of disorder and interaction strength.
