Observing Topological Invariants Using Quantum Walks in Superconducting Circuits

The direct measurement of topological invariants in both engineered and naturally occurring quantum materials is a key step in classifying quantum phases of matter. Here, we motivate a toolbox based on time-dependent quantum walks as a method to digitally simulate single-particle topological band structures. Using a superconducting qubit dispersively coupled to a microwave cavity, we implement two classes of split-step quantum walks and directly measure the topological invariant (winding number) associated with each. The measurement relies upon interference between two components of a cavity Schrödinger cat state and highlights a novel refocusing technique, which allows for the direct implementation of a digital version of Bloch oscillations. As the walk is performed in phase space, our scheme can be extended to higher synthetic dimensions by adding additional microwave cavities, whereby superconducting circuit-based simulations can probe topological phases ranging from the quantum-spin Hall effect to the Hopf insulator.

Popular Summary

The modern classification of phases of matter (such as solids and liquids) is based on what are known as topological properties—characteristics which, like the number of holes in a surface or the number of twists in looped ribbon, are unchanged by small deformations like stretching or bending. Despite 30 years of study, a direct experimental measurement of these so-called topological invariants remains elusive. Recent theoretical work has shown that quantum walks—the quantum cousin of the familiar random walk—exhibit topological features exactly like the ones found in naturally occurring phases of matter, making them a powerful tool for quantum simulation. By performing such a simulation in a new experimental platform, we have been able to measure, for the first time, a topological invariant.

Quantum walks comprise a repeatedly tossed quantum coin that controls the motion of a particle known as the walker. In our experiment, a superconducting circuit is the coin, while an electromagnetic field trapped in a resonator plays the role of the walker. We show that measuring the topological invariant of a quantum walk can be accomplished by using a so-called Schrödinger cat state, a quantum superposition of two classical states of the electromagnetic field. Allowing one of these states to undergo the quantum walk and observing the way it interferes with the stationary state enables us to directly extract the topological invariant.

Our quantum simulation opens the door to understanding more complex topological phases of matter. Future work might extend our protocol to walks in multiple dimensions, which could simulate exotic topological insulators.

Figure 1

Quantum-walk implementation in cavity phase space. (a) Schematic representation of a split-step quantum walk on a line, with rotations $\widehat{R}({\theta }_1)$ and $\widehat{R}({\theta }_2)$ and spin-dependent translation $T_{\uparrow \downarrow }$. Red (blue) lines show spin-up (-down) components moving left (right). The opacity of each circle indicates the population on the corresponding lattice site. (b) Set of ten cavity coherent states on which the walk takes place, in the phase space of the ${\mathrm{TE}}_{210}$ cavity mode. (c) Cavity resonator and qubit. The fundamental (${\mathrm{TE}}_{110}$, orange) mode at ${\omega }_R=2\pi \times 6.77\mathrm{GHz}$ is used to measure the qubit state. This mode couples strongly [$\kappa =2\pi \times 600\mathrm{kHz}=1/(260\mathrm{ns})$] to a 50-ohm transmission line via the readout port at the center of the cavity. The ${\mathrm{TE}}_{210}$ cavity mode (green) at ${\omega }_c=2\pi \times 7.41\mathrm{GHz}$ is long lived with an inverse lifetime, $\kappa =2\pi \times 4\mathrm{kHz}=1/(40\mu \mathrm{s})$. The transmon qubit (coin) has transition frequency ${\omega }_q=2\pi \times 5.2\mathrm{GHz}$, relaxation times $T_1=40\mu \mathrm{s}$ and $T_2^{*}=5.19\mu \mathrm{s}$, and is dispersively coupled to both cavity modes, with the dispersive shift of the walker mode, ${\chi }_{qc}=2\pi \times 1.61\mathrm{MHz}$.

Figure 2

Quantum-walk protocol and resulting populations. (a) Protocol used to perform the quantum walk, showing cavity state preparation (blue), quantum walk (green), qubit state measurement (blue), and Q function measurement (pink). The dashed boxes with ${\sigma }_z$ gates are performed to implement the Bloch oscillation. (b) Cavity Q functions after each step of the quantum walk without Bloch oscillations, ${\widehat{U}}_0$ (top strip) and ${\widehat{U}}_1$ (bottom strip). Spin-up (red) and spin-down (blue) Q functions are superimposed. The average fidelities of the populations compared to theoretical predictions are 0.97 and 0.96 for ${\widehat{U}}_0$ and ${\widehat{U}}_1$, respectively. (c) Cavity Q functions after each step of the refocusing quantum walk with Bloch oscillations. The state refocuses after ten steps, as shown in the final frame for both ${\widehat{U}}_0$ and ${\widehat{U}}_1$. Refocusing fidelities (to the initial state) for ${\widehat{U}}_0$ and ${\widehat{U}}_1$ are 0.83 and 0.87, respectively.

Figure 3

Topological classes of split-step quantum walks. Calculated band structures, quasienergy $\epsilon$ versus quasimomentum $k$, corresponding to the two walks we perform in the experiment, ${\widehat{U}}_0={\widehat{T}}_{\uparrow \downarrow }\widehat{R}(\pi /4){\widehat{T}}_{\uparrow \downarrow }\widehat{R}(3\pi /4)$ (a) and ${\widehat{U}}_1={\widehat{T}}_{\uparrow \downarrow }\widehat{R}(3\pi /4){\widehat{T}}_{\uparrow \downarrow }\widehat{R}(\pi /4)$ (b). Though the energy bands of the two walks are identical, they are topologically distinct, with the topology given by the winding of $\overrightarrow{n}(k)$ as $k$ varies through the Brillouin zone, shown in diagrams (c) and (d). In diagram (c), the trivial case ${\widehat{U}}_0$, $\overrightarrow{n}(k)$ does not complete a full revolution around the Bloch sphere, while in the topological case ${\widehat{U}}_1$ diagram (d), it does perform a full revolution. This also provides a direct connection to the Berry phase, as for a spin-$1/2$ system the Berry phase is simply half the subtended solid angle of the Bloch sphere path. A schematic representation of the variation of $\overrightarrow{n}(k)$ is shown by the ribbons below the Bloch spheres. The arrows on these strips point in the direction of $\overrightarrow{n}(k)$. Analogous to the number of twists in closed ribbons, winding numbers are quantized and robust to local perturbations.

Figure 4

Winding number measurement via direct Wigner tomography of refocused Schrödinger cat states. (a) Protocol for measuring topology via a time-dependent walk (Bloch oscillations). The Schrödinger cat state is first prepared (blue), after which the ten-step refocusing quantum walk is performed (green). The qubit and cavity state are then disentangled, the qubit state is purified (blue), and direct Wigner tomography on the cavity state is performed (pink). Wigner tomography of (b) the cat undergoing no quantum walk, (c) the cat after undergoing the trivial ${\widehat{U}}_0$ walk, and (d) the cat after undergoing the topological ${\widehat{U}}_1$ walk. Fidelities of these resulting cat states compared to pure cat states are 0.68, 0.69, and 0.67, respectively. (e) A cut of the Wigner function, showing the fringes that encode the relative phase between the two cat components for no walk (black), a trivial walk (red), and a topological walk (blue). The relative phase corresponds to the phase of the measured interference fringes following the relation $A\exp [-2|\mathrm{Im}(\alpha ){|}^2]\cos [2\sqrt{\overline{n}}\mathrm{Im}(\alpha )+\varphi ]$, where $A$, $\varphi$ are the amplitude and phase of the fringes. The Berry phase—captured by the phase difference between the topological and the trivial walks—is ${\varphi }_B=1.05\pi \pm 0.06\pi$ in experiment, consistent with the theoretical expectations of $\pi$.