Quantum sensing with discrete time crystals in the Lipkin-Meshkov-Glick Model

TL;DR

The paper addresses whether discrete time-crystal criticality in a periodically driven all-to-all interacting spin system can be leveraged for high-precision quantum sensing. It analyzes the Lipkin-Meshkov-Glick model under periodic modulation, computing the order parameter, quantum Fisher information, and time-averaged inverse participation ratio, and applies finite-size scaling to extract critical exponents and phase-diagram structure. The key findings show a DTC→non-DTC transition at $oldsymbol{ε_c \\approx 0.128}$ with exponents $oldsymbol{ν_m \\approx 2.369}$, $oldsymbol{ν_q \\approx 2.362}$, $oldsymbol{ζ_m \\approx -0.156}$, and $oldsymbol{ζ_q \\approx 3.534}$; the maximum QFI scales as $oldsymbol{F_Q^{max} \\propto N^{1.45}}$ and grows with drive cycles as $oldsymbol{n^{1.87}}$, indicating quantum-enhanced sensing beyond the SQL. TAIPR analyses and phase diagrams corroborate the critical behavior and show how drive parameters and transverse field shape the DTC region. The work suggests practical experimental routes for implementing DTC-based quantum sensors in platforms such as optical cavities, NV centers, and trapped ions.

Abstract

Quantum phase transitions have been shown to be highly beneficial for quantum sensing, owing to diverging quantum Fisher information close to criticality. In this work we consider a periodically modulated Lipkin-Meshkov-Glick model to show that discrete time crystal (DTC) phase transition in this setup can enable us to achieve quantum-enhanced high-precision sensing of field strength. We employ a detailed finite-size scaling analysis and a time-averaged Inverse Participation Ratio analysis to determine the critical properties of this second-order phase transition. Our studies provide a comprehensive understanding of how quantum criticality in DTCs involving long-range interactions can be harnessed for advanced quantum sensing applications.

Figures (6)

• Figure 1: Magnetization and susceptibility: (a) Stroboscopic magnetization $m_z(n, \epsilon)$ as a function of $\epsilon$ for $N = 1000$ over 400 periods. Inset: FFT of the magnetization signal. (b) Magnetization $m_z(n, \epsilon)$ shown for varying $n \in \{200, 203, \ldots, 221\}$ as a function of $\epsilon$. The inset is the plot of $m_z(n, \epsilon)$ and its second derivative (the susceptibility $\chi(n, \epsilon)$) versus $\epsilon$ for $n = 101$; the peak in susceptibility at $\epsilon = 0.128$ sets the critical imperfection threshold. We have fixed $h = 0.3$ and $\tau = 0.6$.
• Figure 2: Order parameter: (a) Order parameter $\overline{m}_z(\epsilon)$ [cf. Eq. \ref{['eqn order param']}] versus $\epsilon$ for $N = 60$ to $N = 1000$. (b) Finite-size scaling collapse: $\overline{m}_z(\epsilon) \cdot N^{-\zeta_m/\nu_m}$ versus $(\epsilon - \epsilon_c) \cdot N^{1/\nu_m}$. The value of the constants are $\nu_m = 2.369$, $\zeta_m = -0.156$ while the critical value is $\epsilon_c = 0.128$ (shown by the dashed black line). Here $h = 0.3$, $\tau = 0.6$ and $\mathcal{N} = 100$.
• Figure 3: Quantum Fisher Information: (a) QFI $\mathcal{F}_Q$ as a function of $\epsilon$ for various system sizes $N$ after time step $n=50$. (b) The maximum QFI $\mathcal{F}_Q^{\text{max}} = \mathcal{F}_Q (\epsilon^{\text{max}}_{\mathcal{F}})$ as a function of $N$. We fit the data to a function of the form $\mathcal{F}_Q = a \cdot N^b$ with $a=2493.07$, and $b=1.45$. (c) The value of $\epsilon^{\text{max}}_{\mathcal{F}}$ as a function of $N$, fitted to a Pareto-like function $\epsilon^{\text{max}}_{\mathcal{F}} = ( a + \frac{0.69}{N^{c}} )$ with $\tilde{a}=0.128$, $\tilde{b}=0.612$, and $\tilde{c} = 0.693$. (d) Finite-size scaling analysis that reveals the best data collapse using critical parameters $(\epsilon_c = 0.128, \nu_q = 2.362, \zeta_q = 3.534)$. (e) Dynamic growth of the QFI $\mathcal{F}_Q$ versus $\epsilon$ for different $n$ in a system of size $N=500$. (f) QFI $\mathcal{F}_Q$ as a function of time steps $n$ for fixed $\epsilon$ values. We fit the data to a function of the form $\mathcal{F}_Q (n, \epsilon) = \alpha \cdot n^{\beta}$ with $\alpha=13474.39$, and $\beta = 1.87$. We have set $h=0.3$, and $\tau = 0.6$.
• Figure 4: Time-averaged inverse participation ratio: (a) TAIPR $\overline{\mathcal{I}} (\epsilon)$ as a function of $\epsilon$ for various system sizes $N$. (b) $\epsilon^{\text{min}}_{\mathcal{I}}$ as a function of $N$, fitted to a Pareto-like function $\epsilon^{\text{min}}_{\mathcal{I}} (N) = ( 0.128 + \frac{0.334}{N^{0.62}} )$. Here $h=0.3$, $\tau = 0.6$, and $\mathcal{N} = 100$.
• Figure 5: Dynamical phase diagrams: (a) The second-order Rényi entropy, $\overline{\mathcal{S}}_2$, as a function of magnetic field $h$ and spin-flip imperfection $\epsilon$ with $\tau=0.6$. (b) The order parameter, $\overline{m}_z$, as a function of $h$ and $\epsilon$ with $\tau=0.6$. (c) The order parameter, $\overline{m}_z$, as a function of period of the drive $\tau$ and $\epsilon$ fixing $h=0.3$. The parameters are $N=400$, and $n=100$.