Quantum Metrology via Adiabatic Control of Topological Edge States

TL;DR

This work links quantum metrology to topological phase transitions by showing that the sensitivity (as quantified by the Quantum Fisher Information) is governed by the order $p$ of band touching during a topological transition, with $\mathcal{F}_{Q} \sim L^{2p}$ in 1D. Extending to two dimensions, 2D HOTIs and Chern insulators with second-order band touchings likewise yield $\mathcal{F}_{Q} \sim L^{4}$, indicating universality across topological classes. Furthermore, encoding multipartite edge-mode entanglement via an adiabatic path to the transition yields $\mathcal{F}_{Q} \sim N^{2} L^{2p}$, combining Heisenberg scaling in particle number with critical scaling in system size. Overall, the paper proposes a topological phase-transition-based route to harness edge-state entanglement, large lattice sizes, and high-order band touchings for enhanced quantum metrology.

Abstract

Criticality-based quantum sensing exploits hypersensitive response to system parameters near phase transition points. This work uncovers two metrological advantages offered by topological phase transitions when the probe is prepared as topological edge states. Firstly, the order of topological band touching is found to determine how the metrology sensitivity scales with the system size. Engineering a topological phase transition with higher-order band touching is hence advocated, with the associated quantum Fisher information scaling as $ \mathcal{F}_Q \sim L^{2p}$, with $L$ the lattice size in one dimension, and $p$ the order of band touching. Secondly, with a topological lattice accommodating degenerate edge modes (such as multiple zero modes), preparing an $N$-particle entangled state at the edge and then adiabatically tuning the system to the phase transition point grows quantum entanglement to macroscopic sizes, yielding $\mathcal{F}_Q \sim N^2 L^{2p}$. This work hence paves a possible topological phase transition-based route to harness entanglement, large lattice size, and high-order band touching for quantum metrology.

Figures (4)

• Figure 1: (a) Phase diagram characterized by winding number $w$ of the eSSH model with intercell long-range coupling range up to $R=4$. The pink arrow shows the parameter-sweeping range used to evaluate the band structure, while fixing $\lambda_4=1$ in panel (b). The blue, black, and orange points denote the band-touching curves with regard to the corresponding colors. The rest parameters are set as $(\lambda_0,\lambda_1,\lambda_3)=(1,-4,-4)$. (b) Band structure of our model as a function of momenta $k$ and next-nearest-neighbor coupling strength $\lambda_2$, where the pink solid line highlights the band touching condition $E(\lambda_2,k)=0$. (c) By choosing $\lambda_2=-10,0,6$, various orders of band-touching near the critical momentum $k_c$ are depicted in blue, black, and orange colors, where the algebraic convergence law of energy gap fits $E\propto|k-k_c|^{p}$ with $p=2,1,4$ respectively.
• Figure 2: (a) The QFI and (b) the bulk-edge energy gap $\Delta E$ vs lattice size $L$ at different topological phase transition points with different orders of band touching, with the hopping range from $R=1$ at $(\lambda_{0},\lambda_{1})=(1,-1)$ to $R=4$ at $(\lambda_{0},\lambda_{1}, \lambda_{2},\lambda_{3},\lambda_{4})=(1,-4,6,-4,1)$. Symbols denote the $\mathcal{F}_{Q}$ and $\Delta E$, whereas solid lines indicate the power-law fitting curves, verifying the algebraic relation between scaling exponent and band-touching order as $\beta\simeq 2p$. (c) The QFI of the non-interacting many-body probe as a function of both system size $L$ and particle number $N$ with the range of intercell hopping $R=2$ at the phase transition point $(\lambda_{0}, \lambda_{1},\lambda_{2})=(1,2,1)$. Symbols are for computational results, whereas the surface plot denotes the power fitting function $\mathcal{F}_Q\propto N^2L^{2p}$ with $p=2$.
• Figure S1: (a) Phase diagram characterized by multipole chiral number $N_{xy}$ of the HOTI model with $\lambda_0=1$. (b) Band structure of the model as a function of momenta $k_x$ and $k_y$ with $\lambda_1=-2$ and $\lambda_2=1$. (c) QFI as a function of system size $L$. The fitting function exhibits a power-law scaling as $\mathcal{F}_{Q}\sim L^{4.09}$, where the blue dots represent numerical simulation results and the red line represents the polynomial fitting function.
• Figure S2: (a) Phase diagram characterized by Chern number $C$ of the CI model. (b) Band structure of the model as a function of momenta $k_x$ and $k_y$ with $m_0=1$ and $\lambda_0=-0.5$. (c) QFI as a function of system size $L$. The fitting function exhibits a power-law scaling as $\mathcal{F}_{Q}\sim L^{4.03}$, where the blue dots represent numerical simulation results and the red line represents the polynomial fitting function.