All-electrostatic valley qubit gates in tilted Dirac-Weyl semimetals

Abstract

Valley degrees of freedom in tilted Dirac materials offer a route toward fully electrical quantum control, but previous electrostatic barrier schemes have used the valley index only as a classical filtering resource. Here, we show that a smooth electrostatic barrier operated in a quantum point contact geometry at normal incidence instead realizes coherent valley phase control. In the single-mode regime, both valleys retain near-unit transmission while the tilt-induced valley-dependent traversal phase generates a controllable relative phase shift $Δd = δ_K - δ_{K'}$ between the $|K\rangle$ and $|K'\rangle$ components of the wavefunction. The resulting electrostatic element implements a tunable valley $Z$ rotation whose accessible phase range covers 99.5\% of the full $2π$ interval while maintaining a transmission-balance metric $B$ above 0.99 over a broad parameter window. Combined with a fixed valley-mixing element that supplies an $X$ rotation, this enables universal single-qubit control through a $Z$--$X$--$Z$ Euler decomposition. For realistic parameters, the ballistic gate time is $\sim$50\,fs, with particularly favourable operating windows in 8-$Pmmn$ borophene and WTe$_2$. These results establish tilted Dirac semimetals as a platform for coherent, all-electrical valley manipulation.

Figures (4)

• Figure 1: Smooth electrostatic barrier as a valley phase gate.a, Barrier potential profiles: sharp (dashed red), $\sigma = 5$ nm (gold), and $\sigma = 10$ nm (blue) for a barrier of width $d = 50$ nm and height $V_0 = 200$ meV. The Fermi energy $E_{\mathrm{F}} = 100$ meV is indicated. b, Valley-resolved transmission $T(\theta)$ comparing sharp (dashed) and smooth ($\sigma = 10$ nm, solid) barriers at tilt $t = 0.5$ and barrier rotation $\phi = 20^\circ$. The smooth barrier suppresses the Fabry-Pérot oscillations that produce valley-asymmetric amplitude modulation. c, Valley phase difference $\Delta\delta = \delta_K - \delta_{K'}$ at normal incidence ($\theta = 0$) as a function of $V_0$ for three barrier smoothness values. The smooth barrier ($\sigma = 10$ nm) produces a monotonic, continuously tunable phase spanning the full $[-\pi, \pi]$ range. d, Transmission-balance metric $\mathcal{B} = \min(T_K, T_{K'}) / \max(T_K, T_{K'})$ at $\theta = 0$. The green-shaded region indicates $\mathcal{B} > 0.99$ for $\sigma = 10$ nm, demonstrating near-equal transmission amplitudes over a broad voltage range. e, Comparison of the numerical phase with the analytical estimate based on the valley-dependent $k_{\perp}$ difference inside the barrier. f, Valley birefringent regime ($V_0 = 90$ meV $< E_{\mathrm{F}}$): the phase accumulates with barrier width $d$ at near-unit transmission balance (dashed green), enabling precision phase calibration.
• Figure 2: Phase gate parameter space.a, Valley phase difference $\Delta\delta(\sigma, V_0)$ at normal incidence for $t = 0.5$, $\phi = 20^\circ$, $d = 50$ nm. Smooth barriers ($\sigma \gtrsim 5$ nm) yield monotonic, broadband phase control; sharp barriers produce oscillatory structure. b, Transmission-balance metric $\mathcal{B}(\sigma, V_0)$. Dashed contours show $|\Delta\delta|$ levels. High transmission balance (dark green) occupies the entire smooth-barrier regime. c,$\Delta\delta$ versus barrier rotation angle $\phi$ for $\sigma = 10$ nm, $V_0 = 150$ meV at four tilt values. The phase increases with both $t$ and $\phi$. d, Transmission balance versus $\phi$: $\mathcal{B}$ remains near unity for $\phi \lesssim 30^\circ$ at all tilts shown. e,$\Delta\delta$ versus barrier width $d$ for three barrier heights, showing phase accumulation with propagation distance. f, Phase diagram $\Delta\delta(t, \phi)$ with transmission-balance contours (black curves: $\mathcal{B} = 0.90$ dashed, $\mathcal{B} = 0.99$ solid). The high-balance region covers most of the moderate-tilt, moderate-angle parameter space.
• Figure 3: Universal single-qubit gate via $Z$--$X$--$Z$ Euler decomposition.a, Bloch sphere trajectories under $Z$-gate operation: sweeping $V_0$ traces arcs on the Bloch sphere for two different input states $|\psi_{\mathrm{in}}\rangle$. Green and red dots mark the start and end points. b, Full Bloch-sphere coverage achieved by the $Z$--$X$--$Z$ protocol $U(\alpha, \beta, \gamma) = R_z(\alpha) \cdot R_x(\pi/2) \cdot R_z(\gamma)$ applied to $|K\rangle$ (black star). Coloured trajectories show the three-step paths to four target states; teal dots show the dense sampling of the reachable state space. c, Device architecture: a QPC-defined single-mode channel with two independently gated smooth barriers (tunable $V_1$, $V_2$) providing $R_z(\alpha)$ and $R_z(\gamma)$, separated by a fixed valley-mixing element providing $R_x(\beta)$.
• Figure 4: Material candidates and operating regimes.a, Valley phase shift $\Delta\delta$ versus tilt parameter $t$ at $V_0 = 150$ meV, $\sigma = 10$ nm, $d = 50$ nm, $\phi = 20^\circ$. Material candidates are marked: 8-$Pmmn$ borophene ($t = 0.46$), WTe$_2$ ($t = 0.35$), $\alpha$-(BEDT-TTF)$_2$I$_3$ ($t = 0.72$). The right axis shows the transmission-balance metric (dashed green), which remains near unity throughout. b, Operating regime map: valley conductance polarization $|\mathcal{P}|$ versus single-mode phase $|\Delta\delta|$, coloured by transmission balance $\mathcal{B}$. The quantum phase-gate regime (upper left, high $\mathcal{B}$) is distinct from the classical valley-filter regime (lower right, low $\mathcal{B}$). c, Gate time $\tau_{\mathrm{gate}} = d/v_{\mathrm{F}}$ compared with estimated valley decoherence times $T_2$ for different material platforms. d, Number of gate operations per coherence time $T_2/\tau_{\mathrm{gate}}$ as a function of $T_2$, with material estimates marked. The dotted line indicates the approximate fault-tolerance threshold ($\sim$100 operations).