Direct Measurement of Topological Numbers with Spins in Diamond

Topological numbers can characterize the transition between different topological phases, which are not described by Landau’s paradigm of symmetry breaking. Since the discovery of the quantum Hall effect, more topological phases have been theoretically predicted and experimentally verified. However, it is still an experimental challenge to directly measure the topological numbers of various predicted topological phases. In this Letter, we demonstrate quantum simulation of topological phase transition of a quantum wire (QW), by precisely modulating the Hamiltonian of a single nitrogen-vacancy (NV) center in diamond. Deploying a quantum algorithm of finding eigenvalues, we reliably extract both the dispersion relations and topological numbers. This method can be further generalized to simulate more complicated topological systems.

Figure 1

Phase diagram and geometric illustration of the topologically distinct phases. (a) Phase diagram of quantum wire system (calculated at $B_x=1.3$). The green line gives the boundary between the SC phase and TP phase. The energy dispersion relations of a SC point and a TP point are plotted in the insets. (b) Geometric illustration of the topological difference between the SC and the TP phases. In order to clearly visualize the two different trajectories associated with the 1st and 2nd energy eigenstates, the Bloch spheres are turned along the $Z$ axis with $p$ changes from 0 to $\infty$. This rotational transformation does not change the topological properties of the bitrajectories.

Figure 2

NV system and its correlation with the QW system. (a) Structure and energy levels of the NV centers. (b) Hyperfine structure of the coupling system with NV electron spin and ${}^{14}\mathrm{N}$ nuclear spin. The 9 energy levels are labeled as $|1⟩$ to $|9⟩$. The quantum simulation is carried out in the subspace spanned by $\{|4⟩,|5⟩,|7⟩,|8⟩\}$. Two MW pulses (purple arrows) and two RF pulses (orange arrows) are applied simultaneously to selectively drive the corresponding electron and nuclear spin transitions. (c) Four basis states of the QW system corresponding to the four NV states inside the square box.

Figure 3

Simulation of the QW Hamiltonian and detection of its eigenvalues. (a) The pulse scheme. The superscript $\alpha$($\beta$) of the RF pulses indicates the nuclear spin operation between $|4⟩$ and $|5⟩$($|5⟩$ and $|6⟩$). For the initialization and readout parts there are two pulse sequences, which correspond to the two initial state cases $|\mathrm{\Psi }⟩=|5⟩$ (the upper pulse sequence) and $|4⟩$ (the lower bracketed pulse sequence), respectively. (b) Photoluminescence (PL) changes versus different the RF $\pi /2$ pulse phase $\theta$ for fixed evolution time $m\tau$. The points are the experimental data and the curve is the sine function fit. Error bars indicate $\pm 1$ standard deviation induced by the photon shot noise. (c) Measurement of $a_{\psi ,m}$ with different evolution time $m\tau$. Black and red lines are the numerical calculation results. (d) The energy spectrum of the simulated Hamiltonian yielded from the Fourier transform of the time-domain data in (c). A Gaussian fit (the curve) is performed to get the exact eigenenergy value.

Figure 4

Energy dispersion relations and topological phase transition. (a) Energy dispersion relations with different chemical potential $\mu$. The points, light cyan lines, and red lines represent the experimental, analytical, and numerical results, respectively. Error bars given by fit error are smaller than the symbols. As the energy bonds are symmetrical about $p=0$, only the right half points (i.e., $p\ge 0$) are actually measured in the experiment. (b) The measured topological number $\nu$ versus the chemical potential $\mu$, which shows a topological phase transition happened near $\mu \approx -1.3$. The cyan line is the theoretical prediction.