Scalable spin squeezing from spontaneous breaking of a continuous symmetry

TL;DR

The paper shows that spontaneous breaking of a continuous symmetry in XXZ spin models enables scalable spin squeezing via adiabatic preparation: starting from a high-field coherent state and ramping the symmetry-breaking field down to $\Omega \sim 1/N$ yields a spin-squeezed state with $\xi_R^2 \sim N^{-1/2}$ and phase sensitivity $\delta\phi \sim N^{-3/4}$, with ramp times scaling as $\tau \sim N$. This mechanism leverages the Anderson tower of states and the persistence of a finite order parameter in the thermodynamic limit, while suppressing fluctuations of the symmetry generator $J^z$. The authors combine linear spin-wave theory, QMC, and TDVM/TD-LSW analyses to show universal low-field scaling $\mathrm{Var}(J^z)/N \sim \Omega^{1/2}$ (and $\Delta E/\mathcal{J} obreak\sim\nobreak\Omega^{1/2}$) across $d\ge 2$ and various interaction ranges, with beyond-LSW corrections enhancing squeezing in the 2D Heisenberg case. They demonstrate the approach's applicability to realistic platforms, including optical-lattice clocks with $^{87}$Rb or $^{174}$Yb, and discuss experimental routes to implement uniform or staggered fields, making adiabatic, strongly entangled spin states accessible in practical quantum sensors.

Abstract

Spontaneous symmetry breaking (SSB) is a property of Hamiltonian equilibrium states which, in the thermodynamic limit, retain a finite average value of an order parameter even after a field coupled to it is adiabatically turned off. In the case of quantum spin models with continuous symmetry, we show that this adiabatic process is also accompanied by the suppression of the fluctuations of the symmetry generator -- namely, the collective spin component along an axis of symmetry. In systems of $S=1/2$ spins or qubits, the combination of the suppression of fluctuations along one direction and of the persistence of transverse magnetization leads to spin squeezing -- a much sought-after property of quantum states, both for the purpose of entanglement detection as well as for metrological uses. Focusing on the case of XXZ models spontaneously breaking a U(1) (or even SU(2)) symmetry, we show that the adiabatically prepared states have nearly minimal spin uncertainty; that the minimum phase uncertainty that one can achieve with these states scales as $N^{-3/4}$ with the number of spins $N$; and that this scaling is attained after an adiabatic preparation time scaling linearly with $N$. Our findings open the door to the adiabatic preparation of strongly spin-squeezed states in a large variety of quantum many-body devices including e.g. optical lattice clocks.

Figures (11)

• Figure 1: Adiabatic squeezing from spontaneous symmetry breaking in the XXZ model. Starting from a coherent spin state at $\Omega = \infty$, an adiabatic reduction of the field $\Omega$ coupling to the order parameter leads to the appearance of scalable spin squeezing when $\Omega \sim 1/N$, due to the scaling of the uncertainty on the $J^z$ component, $\delta J^z = \sqrt{{\rm Var}(J^z)} \sim N^{1/4}$; and to the absence of scaling of the order parameter $m=\langle J^x \rangle/N$, as a consequence of spontaneous symmetry breaking (SSB). The red areas depict the uncertainty regions of the collective spin on a sphere of radius $\sqrt{\langle \bm J^2 \rangle} \sim N$. As a consequence the angular aperture of the uncertainty region along the $z$ axis is $\delta \phi \approx \delta J^z /\sqrt{\langle \bm J^2 \rangle} \sim N^{-3/4}$, defining the sensitivity of the state to rotations around the $y$ axis.
• Figure 2: Adiabatic squeezing from SSB in the 2$d$ Heisenberg model. (a) Field-induced magnetization $\langle J^x\rangle/N$ for various lattice sizes $N= L^2$; (b) Variance of the collective spin component $J^z$; (c) resulting spin-squeezing parameter $\xi_R^2$; (d) comparison between $\xi_R^{-2}$ and $4 {\rm Var}(J^y)/N$. In all panels the solid black line indicates the prediction of linear spin-wave (LSW) theory, and the dotted line in panels (b)-(c) shows the $\Omega^{1/2}$ scaling as a reference (multiplied by an arbitrary prefactor). Here and in the following figures, the error bars denote one standard deviation of the statistical fluctuations in QMC.
• Figure 3: Quasi-adiabatic preparation of the low-field state. (a) Comparison between the field dependence of the squeezing parameter $\xi_R^2$ for the ground state of the Heisenberg model in $d=$1, 2, and 3. For each value of $\Omega$, we use a system size $N$ such that $\Omega \geq {\cal J}/N$, at a temperature $T/{\cal J} = 1/N$ removing thermal effects. The dashed and solid lines show the prediction of LSW theory. (b-c) The two panels show tVMC results for the evolution of the spin-squeezing parameter in the 2$d$ Heisenberg model ($L=12$) along two field ramps starting from $\Omega_i/{\cal J} = 10$ and ending at (b) $\Omega_{\rm f}/{\cal J} = 10^{-1}$ and (c) $10^{-2}$ -- see text for the ramp protocol. Each panel shows three different ramps for different ramp durations $\tau$. The dashed lines show the ground-state spin squeezing parameters -- obtained by variational minimization of the energy with the spin-Jastrow Ansatz. (d) Spin squeezing parameter vs. applied field and entropy per spin in the 2$d$ Heisenberg model, $L=24$.
• Figure 4: Energy gap of the 2$d$ Heisenberg model as a function of the applied $\Omega$ field: the prediction of LSW theory for a large system ($L=100$) is compared to that of exact diagonalization (ED) for a small system ($L=4$), and to the expected low-field behavior $\Delta E \approx 2{\cal J} \sqrt{\Omega/{\cal J}}$, valid in the thermodynamic limit.
• Figure 5: LSW predictions for the 2d XXZ model with variable anisotropy and interaction range: (a) ${\rm Var}(J^z)/N$ and (b) excitation gap $\Delta E/{\cal J}$ for various anisotropies $\Delta$ in the 2d XXZ with nearest-neighbor interactions ($\alpha=\infty$); (c) ${\rm Var}(J^z)/N$ and (d) excitation gap $\Delta E/{\cal J}$ for the same model with dipolar interactions $\alpha=3$. In all the panels the dashed line indicates a $\sim\Omega^{1/2}$ dependence.