Exploring 2D Synthetic Quantum Hall Physics with a Quasiperiodically Driven Qubit

Eric Boyers, Philip J. D. Crowley, Anushya Chandran, and Alexander O. Sushkov

Phys. Rev. Lett. 125, 160505 – Published 16 October 2020

Abstract

Quasiperiodically driven quantum systems are predicted to exhibit quantized topological properties, in analogy with the quantized transport properties of topological insulators. We use a single nitrogen-vacancy center in diamond to experimentally study a synthetic quantum Hall effect with a two-tone drive. We measure the evolution of trajectories of two quantum states, initially prepared at nearby points in synthetic phase space. We detect the synthetic Hall effect through the predicted overlap oscillations at a quantized fundamental frequency proportional to the Chern number, which characterizes the topological phases of the system. We further observe half-quantization of the Chern number at the transition between the synthetic Hall regime and the trivial regime, and the associated concentration of local Berry curvature in synthetic phase space. Our Letter opens up the possibility of using driven qubits to design and study higher-dimensional topological insulators and semimetals in synthetic dimensions.

Received 18 May 2020
Accepted 13 September 2020

DOI:https://doi.org/10.1103/PhysRevLett.125.160505

Physics Subject Headings (PhySH)

Chern insulators Geometric & topological phases Topological phase transition Topological phases of matter

Optically detected magnetic resonance

Condensed Matter, Materials & Applied PhysicsQuantum Information, Science & Technology

Authors & Affiliations

Eric Boyers¹, Philip J. D. Crowley ¹, Anushya Chandran¹, and Alexander O. Sushkov ^1,2,3,*

¹Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA
²Department of Electrical and Computer Engineering, Boston University, Boston, Massachusetts 02215, USA
³Photonics Center, Boston University, Boston, Massachusetts 02215, USA

^*asu@bu.edu

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Vol. 125, Iss. 16 — 16 October 2020

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Images

Figure 1
(a) Synthetic Brillouin zone of drive phases $(θ_{1}, θ_{2})$ conjugate to the synthetic photon lattice. The phases evolve linearly in time corresponding to dynamics under an effective electric field $\vec{Ω}$ . Two states, marked with black circles, are initialized with initial drive phases ${\vec{θ}}_{0}$ and ${\vec{θ}}_{0} + \vec{δ θ}$ . The states acquire a relative phase $Φ$ due to the Berry curvature in the region bounded by their trajectories. Incrementing evolution time by $δ t$ changes this phase by $δ Φ$ . (b) Synthetic lattice of photon occupation numbers of each drive tone. (c) Phase diagram and band structures of the half-BHZ model in the topological $(0 < m < 2, C = 1)$ and trivial $(m > 2, C = 0)$ regimes and at the critical point $(m = 2, C = 1 / 2)$ . (d) Experimental setup schematic. The amplitude of a carrier tone at frequency $ω_{0}$ , created by a signal generator (SG), is modulated by an arbitrary waveform generator (AWG), so that the rf fields oscillate at frequencies $Ω_{1}$ , $Ω_{2}$ in the rotating frame. Signals are injected into a waveguide generating a rf magnetic field that drives the electronic spin of a single NV center in diamond. The spin Bloch sphere trajectories of the two states with initial drive phases ${\vec{θ}}_{0}$ and ${\vec{θ}}_{0} + \vec{δ θ}$ are shown as red and blue lines, diverging as they evolve and acquire a relative Berry phase.
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Figure 2
Measurements of spin projection evolution for trajectories starting with different initial drive phases and the same initial spin state $| ψ (0) ⟩ = [| ϕ^{1} ({\vec{θ}}_{0}) ⟩ + | ϕ^{2} ({\vec{θ}}_{0}) ⟩] / \sqrt{2}$ . All plots show on $y$ axis the difference in expectation values, $⟨ σ_{i} ⟩ - ⟨ σ_{i} ⟩^{'}$ ( $i = x$ , $y$ , $z$ ). Dashed lines are envelopes to the data determined by NV fluorescence shot noise and the accumulated Berry phase, which bounds the differences in the expectation values of projections. Envelopes are given by $\pm [2 | \sin [(| \vec{δ θ} \times \vec{Ω} | C / 2 π) t] | + 2 σ]$ , where $σ$ is the shot noise standard deviation [37]. Experimental parameters’ values: $γ B_{0} = 2 π \times 0.25 MHz$ , $\vec{Ω} = 2 π (0.5, 0.5 φ) MHz$ , ${\vec{θ}}_{0} = (0, π / 2)$ , ${\vec{δ θ}}_{0} = (φ / 30, - 1 / 30)$ , with $φ$ equal to the golden ratio so that the drive frequencies are incommensurate. (a) Trivial phase ( $m = 3$ ): trajectories of the two states stay close to each other. (b) Topological phase ( $m = 1$ ): trajectories of the two states diverge as the states acquire increasing geometric phase. (c) Topological phase ( $m = 1$ ) with larger initial distance ${\vec{δ θ}}_{0} = (φ / 8, - 1 / 8)$ and longer evolution time: the states periodically diverge and rephase.
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Figure 3
(a) Evolution of quantum state overlap $F (t)$ . In the topological phase ( $m = 1$ , green circles, solid line fit), the overlap oscillates with frequency proportional to the Chern number. In the trivial phase ( $m = 3$ , orange diamonds, dashed line fit), the overlap slowly decays due to qubit decoherence. At the critical point ( $m = 2$ , purple triangles, dot-dashed line fit), the overlap oscillates at half the frequency of the topological regime. Fits are performed using the model $F (t) = 1 / 2 + (1 / 2 - A \sin^{2} ω_{fit} t) e^{- t / τ} + δ$ , where $ω_{fit} = (| \vec{Ω} \times {\vec{δ θ}}_{0} | / 2 π) | C_{\exp} |$ . Data are binned for clarity, error bars represent 1 standard error. Experimental parameters are the same as in Fig. 2. (b) The overlap near the critical point, $m = 1.95$ , for the same parameters as in (a). Sudden changes in overlap occur when either of the two states pass near the point $\vec{θ} = (0, 0)$ around which the Berry curvature is concentrated. (c) Phase diagram of the synthetic half-BHZ model. Red data points show Chern number measurements. Parameters are the same as in (a), error bars are of the order of data marker size and may be obscured.
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Figure 4
Local Berry curvature in the synthetic Brillouin zone for three values of $m$ approaching the critical point at $m = 2$ from below. Color scale represents the magnitude of the local Berry curvature, which concentrates near the point $\vec{θ} = (0, 0)$ . Experimental measurements (left) are consistent with the exact Berry curvatures (right). Parameters are the same as in Fig. 3, except for values of $m$ as noted.
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