Generation of spin-squeezed states using dipole-coupled spins

TL;DR

The paper investigates generating spin-squeezed states in dipole-coupled spin systems to enhance quantum sensing. By simulating unitary dynamics under a rotating-frame dipole Hamiltonian and mapping 2D uncertainty distributions, it demonstrates $\sigma_{min} < 1/\sqrt{N}$ for several $N$ and coupling strengths, confirming squeezing beyond the standard quantum limit and linking squeezing to entanglement via the von Neumann entropy. The results show pronounced squeezing in small ($N=2-4$) and larger spin ensembles, with explicit analyses for triangle and linear 3-spin configurations that could be realized experimentally, including NV-center–based geometries. The work highlights a feasible route to spin-squeezed sensor networks and a diagnostic tool for entanglement in spin ensembles, while outlining future work on decoherence, optimized control, and hardware implementations.

Abstract

Spins in solids and molecules are promising for applications of quantum sensing technology. The sensitivity of the quantum sensing depends on how precisely spin observables can be determined in the measurement, and is intrinsically limited by the uncertainties of the observables. The use of a spin-squeezed state in a quantum sensor can reduce the uncertainty below the standard quantum limit when combined with an appropriate measurement procedure. Here, we discuss the simulation study of the generation of a squeezed state in an interacting spin system. We show that a spin system coupled by the magnetic dipole interaction can create a squeezed state. Model systems to realize the spin squeezing experimentally are also discussed. In addition, we find that a squeezed state is a type of entangled state. The present work paves the way to realize a squeezed state using a spin system to build a quantum sensor network with improved sensitivity, and to use it for the detection of quantum entanglement.

Figures (16)

• Figure 1: Models of interacting spin systems. (a) Dipole-coupled 2-spin system. (b) Dipole coupled 3-spin system.
• Figure 2: Calculation results of the expectation value, and uncertainty for $N = 3$ system. $d/(2\pi) = 1$ MHz. (a) Expectation values ($\langle J_x \rangle$, $\langle J_y \rangle$, and $\langle J_z \rangle$) as a function of the evolution time ($\tau$). The inset shows a pulse sequence to generate and measure the spin-squeezed state. (b) The uncertainty values as a function of angle $\theta$ at $\tau=89$ ns ($\Delta J_y^{\theta}$ and $\Delta J_z^{\theta}$). $\Delta J_b$ represents the minimum $\Delta J_y^{\theta}$. $\theta_{opt}$ is the corresponding angle for $\Delta J_b$. $\Delta J_a$ is the corresponding $\Delta J_z^{\theta}$ for $\theta_{opt}$. (c) $\Delta J_b^{\theta}$ and $\Delta J_a^{\theta}$ as a function of the evolution time. (d) $\theta_{opt}$ as a function of the evolution time. $\theta_{min}$ is the $\theta_{opt}$ value at $\tau_{min}$. The calculations were done with the step sizes of 1 ns and 1 degrees for $\tau$ and $\theta$, respectively.
• Figure 3: Emergence of spin-squeezed state. (a) Normalized expectation values ($\sigma_b$ and $\sigma_a$) as a function of the evolution time ($\tau$). Black vertical dashed line indicates the optimum evolution time $\tau_{min}$ to give the minimum $\sigma_b$ ($\sigma_{min}$). $\sigma_{min} = 0.440$ at $\tau_{min} = 89$ ns. (b) Distribution of $\sigma_y$ and $\sigma_z$ for $d/(2\pi) = 1$ MHz. $\sigma_a^{min}=0.961$. (c) Distribution of $\sigma_y$ and $\sigma_z$ for $d = 0$. (d) $S^i_{vN}$ as a function of the evolution time $\tau$. $i=1-3$. $S^1_{vN}=S^2_{vN}=S^3_{vN}$ in the present case. Black vertical dashed line indicates the optimum evolution time. Black horizontal dashed line represents the $S^i_{vN}$ value of a fully entangled state.
• Figure 4: Spin squeezing of $N=4$ system with $d/(2\pi) = 1$ MHz. (a) Expectation values as a function of the evolution time $\tau$. Vertical black dashed line represents the evolution time of $\tau_{min} = 74$ ns. (b) $\Delta J_b$ and $\Delta J_a$ vs $\tau$. The inset shows $\theta_{opt}$ vs $\tau$. (c) $\sigma_b$ and $\sigma_a$ as a function of $\tau$. Horizontal black solid line represents $\sigma_0$. The inset shows the 2D map of $\sigma_b$ and $\sigma_a$. $\sigma_{min} = 0.358$. $\sigma_a^{min} = 0.899$. $\tau_{min} = 73$ ns. $\theta_{min} = 54^{\circ}$.
• Figure 5: $\sigma_{min}$ as a function of the number of spin qubits ($N$) for cases with $d = 0$ and $d/(2\pi) = 1$ MHz. Blue dashed line indicates the standard quantum limit (SQL), i.e.$\sigma_0= 1/\sqrt{N}$. The inset shows the ratio of $\sigma_{min}/\sigma_0$ vs $N$.