Dressed basis sets for the modeling of exchange interactions in double quantum dots

TL;DR

This work tackles the challenge of modeling exchange interactions in double quantum dots for spin-qubit applications, where full configuration interaction is prohibitively expensive for dynamics. The authors construct a reduced, dressed basis from a reference CI calculation that accurately describes the lowest singlet and triplet states across realistic gate voltage ranges using about $\sim$100 basis functions. They demonstrate that exchange results from a complex interplay of inter-dot tunneling, Coulomb exchange, and correlations, and that magnetic-field effects with spin-orbit coupling introduce anisotropies and $ST$ mixings that are captured by the dressed basis. Importantly, time-dependent simulations of singlet-triplet qubits using the dressed basis achieve results in excellent agreement with full CI but at vastly reduced computational cost, enabling efficient design and optimization of qubit operations in Ge-based hole-dot devices.

Abstract

We discuss the microscopic modeling of exchange interactions between double semiconductor quantum dots used as spin qubits. Starting from a reference full configuration interaction (CI) calculation for the two-particle wave functions, we build a reduced basis set of dressed states that can describe the ground-state singlets and triplets over the whole operational range with as few as one hundred basis functions (as compared to a few thousands for the full CI). This enables fast explorations of the exchange interactions landscape as well as efficient time-dependent simulations. We apply this methodology to a double hole quantum dot in germanium, and discuss the physics of exchange interactions in this system. We show that the net exchange splitting results from a complex interplay between inter-dot tunneling, Coulomb exchange and correlations. We analyze, moreover, the effects of confinement, strains and Rashba interactions on the anisotropic exchange and singlet-triplet mixings at finite magnetic field. We finally illustrate the relevance of this methodology for time-dependent calculations on a singlet-triplet qubit.

Figures (13)

• Figure 1: A double unit cell of the 2D array of spin qubit devices. The heterostructure is a 16-nm-thick Ge quantum well (red) grown on a thick Ge$_{0.8}$Si$_{0.2}$ buffer capped with a 50-nm-thick Ge$_{0.8}$Si$_{0.2}$ barrier (blue). The dots are shaped by plunger (L, R) and barrier (J, B$_i$) gates (gray). The yellow contour is the isodensity surface that encloses 90% of the hole charge in the ground $(1,1)$ state at bias $V_\mathrm{L}=V_\mathrm{R}=-40$ mV and $V_\mathrm{J}=-15$ mV (B gates grounded). The orientation of the magnetic field $\boldsymbol{B}$ is characterized by the angles $\theta$ and $\varphi$ in the crystallographic axes set $x=[100]$, $y=[010]$ and $z=[001]$.
• Figure 2: Outline of the methodology. The $N$ lowest eigenstates of the finite-differences Luttinger-Kohn Hamiltonian are used as a basis set for fast one-particle calculations and for CI calculations. Then the lowest eigenstates of the CI Hamiltonian, sampled at different gate voltages $\{V_\mathrm{G}^k\}$ and orthogonalized against each other, are used as a "dressed" basis set for two-particle calculations over the whole operational gate voltages range.
• Figure 3: (a) The tunnel coupling $|t|$ as a function of $V_\mathrm{J}$, calculated on the FD mesh and in finite basis sets of $N=8$ to $96$ lowest-lying eigenstates $\lvert\psi_n(V_\mathrm{J}^0=-15\,\mathrm{mV})\rangle$. (b) The error in the finite basis sets (with respect to the FD solution). The B gates are grounded and $V_\mathrm{L}=V_\mathrm{R}=-40$ mV.
• Figure 4: (a) Exchange energy $J$ as a function of $V_\mathrm{J}$, for different sizes of the single-particle basis set $N$ and CI basis set $M=N(N-1)/2$. The $N$ single-particle wave functions are either computed at $V_\mathrm{J}^0=V_\mathrm{J}$ (dots), or borrowed from a unique $V_\mathrm{J}^0=-15$ mV (lines). (b) Exchange energy $J$ at $V_\mathrm{J}=0$, $-5$, $-10$, $-15$, $-20$, $-25$ mV as a function of the number of single-particle states $N$ calculated at $V_\mathrm{J}^0=-15$ mV.
• Figure 5: The two-hole spectrum as a function of the detuning voltage $\delta V_\mathrm{d}=2\delta V_\mathrm{L}=-2\delta V_\mathrm{R}$ with respect to the bias point $V_\mathrm{L}=V_\mathrm{R}=-40$ mV, $V_\mathrm{J}=-15$ mV, computed in the original CI basis set ($M=4560$) and in the dressed basis set ($M^\prime=112)$.