Spin qubit shuttling between coupled quantum dots with inhomogeneous Landé g-tensors

Abstract

By utilizing the site-dependent spin quantization axis in semiconductor quantum dot (QD) arrays, shuttling-based spin qubit gates have become an appealing approach to realize scalable quantum computing due to the circumvention of using high-frequency driving fields. The emergence of a spin deviation from the local quantization axis of one residing QD is the prerequisite to implement the qubit gates. In this work, we study the non-adiabatic dynamics of a spin qubit shuttling between coupled QDs with inhomogeneous Landé g-tensors and a small magnetic field. The spin dynamics is analyzed through solving the time-dependent Schrödinger equation of the qubit under the effects of spin-orbit interaction and rapid ramping inter-dot detuning. The precondition, imposed on the ramping time and the tunnel-coupling strength, to ensure a high-fidelity inter-dot transfer is estimated. We then calculate the change in the spin orientation of a transferred qubit, and study the dependences of the spin deviation on the difference in the quantization axes of the two QDs, the tunnel-coupling strength, and the ramping time. We also demonstrate that the effect of multiple rounds of inter-dot bidirectional shuttling can be captured by an operator matrix, and evaluate the idling times required for realizing the single-qubit Pauli-X and Pauli-Y gates. Intriguingly, it is confirmed that a generalized Hadamard gate can be achieved through tuning the idling times.

Figures (7)

• Figure 1: (a) Schematic of coupled quantum dots, i.e., left QD and right QD, occupied by a single spin qubit, in which $V^{}_{L/R}$ and $V^{}_{M}$ represent the left/right plunger-gate and middle barrier-gate potentials. (b) The spin quantization axes of the two QDs in a low magnetic field $B$ applied in the $y$-direction, i.e., $\mathbf{n}^{}_{l/r} = (\cos\theta^{}_{l/r}, \sin\theta^{}_{l/r},0 )$ with $\theta^{}_{l/r}$ being the direction angle of the left/right dot. (c) The "spin-flipped " tunneling $T^{}_{\downarrow \uparrow}$ of a qubit between the two QDs, with $|l/r,\uparrow\rangle$ and $|l/r,\downarrow\rangle$ indicating the local Zeeman-splitting states and $\epsilon$ sizing the inter-dot detuning. (d) A specific ramping up of $\epsilon(t)= V^{}_{0}\tanh(\pi t/w^{}_{0})$ to implement a leftward inter-dot transfer, with $w^{}_{0}$ being the ramping time.
• Figure 2: (a) Energy spectrum of the single qubit as a function of $\epsilon$ with $\theta^{}_{r}= 0.46\pi$, $\theta^{}_{l}= 1.11\pi$, $\varphi^{}_{\rm so} =0.12\pi$, $T^{}_{0}=0.8$, and $\Delta^{}_{0}$ being the unit of energy. (b) The spin orientation of a leftward transferred qubit defined in the Bloch sphere of the left QD with $\hat{n}^{}_{l }$ specifying the pole axis and the dashed/solid arrow indicating the spin direction after undergoing a adiabatic/non-adiabatic inter-dot transfer. (c) and (d) The time evolutions of $|C^{}_{j=1-4} (t)|^{2}_{}$ for a qubit preset in the ground-state ($E_1$) energy level with different $T^{}_{0}$ and $\epsilon = V^{}_{0}\tanh(\pi t/w^{}_{0})$, in which $V^{}_{0}= 10$, $w^{}_{0} = 6 t^{}_{0}$, and $t^{}_{0}= \pi\Delta^{}_{0}/\hbar$. In addition, the infidelity of the leftward inter-dot transfer ${\cal F}^{}_{c}$ is indicated in each case.
• Figure 3: (a) The infidelity of an inter-dot transfer ${\cal F}^{}_{ c}$ as a function of $\beta^{}_{ }$ for a qubit initialized in different spin states of the right QD, with $V^{}_{0}=10$, $T^{}_{0}=0.74$, and $w^{}_{0}=7.2 t^{}_{0}$. In addition to the numerical results (see lines), the discrete symbols specify the approximations from Eq. (\ref{['InF']}). (b) The magnitude of ${\cal F}^{}_{c}$ in terms of $T^{}_{0}$ and $w^{}_{0}$ for an initial spin-down qubit with $\beta^{}_{ }=0.2\pi$. The yellow- and cyan-colored regions correspond to the parameter ranges with ${\cal F}^{}_{c} >1\%$ and ${\cal F}^{}_{c}<1\%$, respectively, while the solid curve denotes the boundary between these two regions. Besides, the inset exhibits the possible LZTs. (c) For an initial spin-down qubit, the value of its spin-deviation angle $\vartheta^{}_{s}$ as a function of $\beta$, with $T^{}_{0}=1.5$ and $w^{}_{0}=7.2t^{}_{0}$. Here, the dashed curve represents the numerical results, while the discrete symbols specify the analytic results from Eq. (\ref{['lms']}). (d) The numerical distribution of $\vartheta^{}_{s}$ within the range of ${\cal F}^{}_{c}<1\%$ in (b), with the doubled-head arrow signifying the region of $\vartheta^{}_{s}>0.1\pi$. In addition, the inset exhibits the time dependence of $\Sigma^{}_{y,z}$ at the $P$ point with $T^{}_{0}= 1.8$ and $w^{}_{0}=6t_{0}$.
• Figure 4: (a) The time dependence of $\epsilon^{}_{\circlearrowleft } (t)$ to implement one round of inter-dot transfer, in which the up/down arrow indicates the leftward/rightward shuttling and $t^{}_{in}$ is the time of a qubit idling in the left QD. In addition, both the Larmor precession of qubit stay in the left QD and the detection of its spin flipping at the final moment are indicated. (b) The spin-flipping probability ${\cal F}^{}_{s}$ for an initial spin-down qubit as a function of $t^{}_{ in }$ and $V^{}_{c}$ with $\varphi^{}_{\rm so} =0.11\pi$, $\theta^{}_{r}=0.44\pi$, $\theta^{}_{l}=0.95\pi$, $V^{}_{e} =-10$, $T^{}_{0}=2.0$, $w^{}_{0}=2t^{}_{0}$, and $V^{\star}_{c}$ being the threshold of the plateau value ($V^{}_{c}$) for completing inter-dot transfers.
• Figure 5: (a) The modulation sequence of $\epsilon^{}_{2\circlearrowleft}(t)$ to realize the two rounds of inter-dot shuttling protocol, in which $t^{}_{in/out, n=1,2}$ indicates the time idling in the left/right QD and $\hat{S}^{}_{l/r,n}$ represents the connection matrix of the leftward/rightward shuttling in the $n$-th round. Therein, $k^{}_{1}=4w^{}_{0}+t^{}_{in,1}+ t^{}_{out,1}$, $k^{}_{2}=8w^{}_{0}+\sum^{2}_{n=1}( t^{}_{out,n} +t^{}_{in,n})$, and the Larmor precessions of a qubit idling in the QDs are also indicated. (b) Explicit equations for determining the requisite idling times $t^{}_{in, 1}$, $t_{in,2}$, $t^{}_{out,1}$ and $t^{}_{out,2}$ to implement the single-qubit X, Y, and generalized Hadamard (H) gates. (c) and (d) In the implementation of the shuttling-based X and H gates, the composition ratios $|\nu^{}_{r,\sigma}|^{2}_{}$ of the local basis state $|r,\sigma\rangle$ in the evolving state of a spin-polarized qubit at $t=0$ and $k^{}_{1,2}$. Besides, the specific values of $(t^{}_{ in,1}, t^{}_{out,1} ,t^{}_{in,2}, t^{}_{out,2} )$, in units of $t^{}_{0}$, are indicated in the panels and with the other parameters fixed as $\varphi^{}_{\rm so} =0.11\pi$, $\theta^{}_{r}=0.44\pi$, $\theta^{}_{l}=0.95\pi$, $T^{}_{1}=2.0$, $T^{}_{2}=2.2$, $w^{}_{0}= 5t^{}_{0}$, and $V^{}_{ c/e}=\pm10$.