Interacting electrons in silicon quantum interconnects

TL;DR

Interacting electrons in silicon quantum interconnects study gate-defined 1D channels in a Si/SiGe heterostructure, revealing a density-driven Wigner regime ($4k_F$ correlations) transitioning to a Friedel regime ($2k_F$ correlations) as density increases. The work combines bosonization and large-scale DMRG to map ground-state density, incompressibility, and capacitive coupling, and proposes transport and charge-sensing signatures to identify the crossover while testing robustness to short-range disorder and valley disorder. It demonstrates that the Wigner regime enables long-range capacitive coupling between quantum dots across the interconnect, offering a route to nonlocal entanglement and scalable interconnects. These findings position silicon interconnects as a platform for exploring Luttinger liquid physics and for architectures enabling nonlocal quantum error correction and quantum simulation.

Abstract

Coherent interconnects between gate-defined silicon quantum processing units are essential for scalable quantum computation and long-range entanglement. We argue that one-dimensional electron channels formed in the silicon quantum well of a Si/SiGe heterostructure exhibit strong Coulomb interactions and realize strongly interacting Luttinger liquid physics. At low electron densities, the system enters a Wigner regime characterized by dominant 4kF correlations; increasing the electron density leads to a crossover from the Wigner regime to a Friedel regime with dominant 2kF correlations. We support these results through large-scale density matrix renormalization group (DMRG) simulations of the interacting ground state under both screened and unscreened Coulomb potentials. We propose experimental signatures of the Wigner-Friedel crossover via charge transport and charge sensing in both zero- and high-magnetic field limits. We also analyze the impact of short-range correlated disorder - including random alloy fluctuations and valley splitting variations - and identify that the Wigner-Friedel crossover remains robust until disorder levels of about 400 micro eV. Finally, we show that the Wigner regime enables long-range capacitive coupling between quantum dots across the interconnect, suggesting a route to create long-range entanglement between solid-state qubits. Our results position silicon interconnects as a platform for studying Luttinger liquid physics and for enabling architectures supporting nonlocal quantum error correction and quantum simulation.

Figures (10)

• Figure 1: (a) Cross-section of the resistive interconnect, consisting of a Si/SiGe heterostructure. A positively biased wire, or "topgate", creates a one-dimensional electron channel (1DEC) in the silicon well. (b) Top-view of the interconnect where the topgate creates a 1DEC with a lateral width $d$ under it in the Si well and assists in momentum-incoherent shuttling of electrons between quantum dots. (c) Ground state charge density profile of the 1DEC obtained from density matrix renormalization group calculations in the Wigner (magenta, $n=0.01\text{ nm}^{-1}$) and Friedel (blue, $n=0.083$ nm$^{-1}$) regimes when the external magnetic field is set to zero. The Wigner regime charge density has been offset by 0.3 nm$^{-1}$ for better visualization. (d) The absolute value of the Fourier transform of the ground state charge density within the 1DEC displays a crossover from a potential energy dominated Wigner regime with $4k_F$ correlations to a kinetic energy dominated Friedel regime with $2k_F$ correlations with increasing density, $n \geq 0.05 \text{ nm}^{-1}$ in the interconnect.
• Figure 2: Wigner crystal regime as the ground state of electrons within the one-dimensional Si-well. (a) Total charge density of the ground state in a $L=$600 nm interconnect with different numbers of electrons interacting via the screened-Coulomb interaction with screening length $D=75$ nm (charge density for every particle number is offset by an arbitrary constant). As the number of electrons in the channel increases, we observe a crossover from a potential energy dominated Wigner regime with $4k_F$ correlations to kinetic energy dominated Friedel regime with $2k_F$ correlations. In the Wigner regime, $N$ electrons in the channel display $N$ distinct peaks, whereas the Friedel regime only displays $N/2$ such peaks. (b) Charge density at the midpoint of the interconnect for different screening lengths. For low densities, we observe strong oscillations in the charge density with the parity of the electron number, which vanishes in the higher-density Friedel regime; this observation informs our later charge-sensing protocol. (c-f) Absolute value of the Fourier transform of charge density plotted as a function of the wavevector and electron density for different screening lengths. Red (blue) lines denote the $4k_F(2k_F)$ wavevectors $(k_F=n\pi/2)$ for each electron density. As the density of the electrons increases beyond 0.05 nm$^{-1}$ or around $(na_B)^{-1} \approx 6$ in the channel, there is a crossover in the peak of the Fourier transform of the ground state wavefunction, which is observed across different effective screening lengths.
• Figure 3: Wigner crystal regime as the ground state of electrons within the one-dimensional Si-well at high-magnetic fields. (a) Charge density of the ground state in a 600nm interconnect with different number of electrons interacting via the screened-Coulomb interaction with screening length $d=75$ nm (charge density for every particle number is offset by a constant). As the number of electrons in the channel increases, we do not observe a crossover from a Wigner-dominated $4k_F$ regime to a Friedel-dominated $2k_F$ oscillation regime. (b) Charge density at the midpoint of the interconnect for different screening lengths. In contrast to Fig. \ref{['fig:spinful-screening']}, we observe strong oscillations in the charge-density with odd-even number of electrons up to the highest densities we consider. (c-f) Absolute value of the Fourier transform of charge density plotted as a function of the wavevector and electron density for different screening lengths. (Blue, red, black) lines denote $(2k_F, k_{max}, 4k_F)$ for each electron density. As the density of the electrons increases, the peak of the Fourier transform of the ground state wavefunction remains at $4k_F$ consistently across all screening lengths.
• Figure 4: (a) Electronic incompressibility $\kappa^{-1}$ of the interconnect exhibits a strong decrease with decreasing density for both zero and high magnetic field B in the interconnect. $\kappa^{-1}$ in the high magnetic field limit (diamond markers) shows a monotonic increase with both density and screening lengths. Results from spinless bosonization theory are plotted as solid lines and show excellent agreement with results from DMRG at high densities, while exhibiting similar qualitative trends at lower densities. In the zero B limit (triangular markers), $\kappa^{-1}$ exhibits two distinct regimes---at low densities, there is a monotonic increase and at higher densities, strong oscillations occur with particle number. This is absent in the high B limit where electronic spins are polarized and $\kappa^{-1}$ increases with additional particles. (b) Phenomenological conductance $G$ of the interconnect with the number of electrons in the channel determined from the incompressibility for $D=75$ nm. Elliptical patches indicate two nearby peaks of opposite spin. At low densities, the spacing between peaks, given by $\kappa^{-1}$ is small, while at high densities, $\kappa^{-1}$ and the gap between peaks is found to alternate as electrons of opposite spin either pair-up or remain unpaired in the Luttinger liquid. (c) In the high $B$ limit, the spacing between conductance peaks continues to monotonically increase with the number of electrons. (d,e) Conductance for the Coulomb interaction without screening $(D=\infty)$ shows alternations in the peak-spacing in the low-B limit and a monotonic increase in the high-B limit.
• Figure 5: Charge sensing setup to probe the Wigner-Friedel crossover. (a) Setup of a charge sensing dot near the center of the interconnect that helps detect the Wigner-Friedel crossover with increasing density. $D_L(D_R)$ are quantum dots to the left (right) of the interconnect. (b) Energy of the ground state charge density measured by a single charge-sensor placed near the middle of the interconnect ($E_s^m$) for different screening lengths in the Si-channel. The sensor measurements depict small oscillations at very low densities and then the energy measured follows a linear trend with the number of electrons. The black vertical line at $n \approx 0.05 \text{ nm}^{-1}$ is the approximate density after which the system displays a Friedel-like $2k_F$ oscillation. (b) The first order difference $\Delta E_s^m(N)= E_s^m(N+1) - E_s^m(N)$ displays strong oscillations before the crossover density and lowers down to zero after the crossover. (c) The second order difference in the sensor energy $\Delta^2 E_s^m(N)= E_s^m(N+1) - 2E_s^m(N) + 2E_s^m(N-1)$ also displays oscillations at low density, which reduces to zero after the crossover density.