Simulation and Manipulation of Tunable Weyl-Semimetal Bands Using Superconducting Quantum Circuits

We simulated highly tunable Weyl-semimetal bands using superconducting quantum circuits. Driving the superconducting quantum circuits with microwave fields, we mapped the momentum space of a lattice to the parameter space, realizing the Hamiltonian of a Weyl semimetal. By measuring the energy spectrum, we directly imaged the Weyl points, whose topological winding numbers were further determined from the Berry curvature measurement. In addition, we manipulated the band structure with an additional pump microwave field, producing a momentum-dependent Weyl-point energy together with an artificial magnetic field, which are indispensable for generating chiral magnetic topological currents in some special Weyl semimetals and may have significant impact on topological physics.

Figure 1

(a) The energy diagram of a transmon without (left) and with (right) microwave driving. The construct (pump) microwave field is applied to $|1⟩$ and $|2⟩$ ($|0⟩$ and $|1⟩$). The system transforms to the dressed states, as shown in the right-hand panel. (b) Left: Microwaves with various frequencies, amplitudes, and phases are applied to construct the Hamiltonian. Right: Schematic for the effect of an extra pump microwave field, which is to generate a parameter-dependent energy offset.

Figure 2

(a) Illustration of Weyl points in the first Brillouin zone. Weyl points with positive (negative) charges are denoted by red (green) color. (b) Schematic of the formation of winding numbers at a positive (negative) Weyl point. Measured winding number from $⟨{\sigma }_y⟩$ are close to $+1$ ($-1$), corresponding to Weyl points with positive (negative) charges. (c)–(e) Measured three-dimensional energy spectra of the Weyl semimetal in the first Brillion zone for $\lambda =0$, $-0.5$, $-1$. The brightest points are Weyl points with the charges labeled. The locations of gapless Weyl points are found to shift with $\lambda$.

Figure 3

Effects of an offset $A_z$ and momentum-dependent $u_0$ are illustrated in (a) and (b). (c) Spectrum along the $k_z$ direction for Hamiltonian in Eq. (1) for $u_0(\mathbf{k})=f(k_z)\cos (k_x)\cos (k_y)$ (blue line) with $k_x=k_y=0$, where $f(k_z)$ is given in Eq. (3) with $\alpha =1$. The vector potential $A_z$, which causes the shift of the spectrum, is 0 (green) and $\pi /8$ (red), respectively. (d) Plot of $A_z$ as a function of $Y$. Error bars are derived from the width of resonant peaks in the spectrum.

Figure 4

(a) Energy spectrum with various $b_0$ obtained by changing $\lambda$ and $u_0$. The upper panel shows the spectrum for different $\alpha$ of $u_0$ with $\lambda =0$: $\alpha =1$ (green) and $\alpha =3$ (red). The lower panel shows the spectrum for $\lambda =0.5$ (red) and $-0.5$ (green) with $\alpha =1$. Solid dots and lines are experimental data and theoretical calculations, respectively. (b) The total topological current derived from Eq. (2), which is contributed to by all four pairs of Weyl points, is plotted as a function of $b_0$ for various $B_x$. From bottom to top, $B_x$ is 0.09, 0.13, 0.19, and 0.38, respectively. Blue and red circles are experimental data obtained from the shifts of $\lambda$ and $\alpha$, respectively, while dashed lines are theoretical results. The inset of (b) shows the normalized topological current $J/b_0$ as a function of $B_x$ for the two methods.