Scalable Spin Squeezing from Spontaneous Breaking of a Continuous Symmetry

Spontaneous symmetry breaking 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.

Figure 1

Adiabatic squeezing from spontaneous symmetry breaking (SSB) in the $XXZ$ model. Starting from a coherent spin state at $\mathrm{\Omega }=\infty$, an adiabatic reduction of the field $\mathrm{\Omega }$ coupling to the order parameter leads to the appearance of scalable spin squeezing when $\mathrm{\Omega }\sim 1/N$, due to the scaling of the uncertainty on the $J^z$ component, $\delta J^z=\sqrt{\text{Var}(J^z)}\sim N^{1/4}$, and to the absence of scaling of the order parameter $m=⟨J^x⟩/N$, as a consequence of SSB. The red areas depict the uncertainty regions of the collective spin on a sphere of radius $\sqrt{⟨{\mathit{J}}^2⟩}\sim N$. As a consequence, the angular aperture of the uncertainty region along the $z$ axis is $\delta \varphi \approx \delta J^z/\sqrt{⟨{\mathit{J}}^2⟩}\sim N^{-3/4}$, defining the sensitivity of the state to rotations around the $y$ axis.

Figure 2

Adiabatic squeezing from SSB in the $2d$ Heisenberg model. (a) Field-induced magnetization $⟨J^x⟩/N$ 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 $4\text{Var}(J^y)/N$. In all panels the solid black line indicates the prediction of linear spin-wave (LSW) theory, and the dotted line in (b) and (c) shows the ${\mathrm{\Omega }}^{1/2}$ scaling as a reference (multiplied by an arbitrary prefactor). Here and in Fig. 3, the error bars denote one standard deviation of the statistical fluctuations in QMC.

Figure 3

Quasiadiabatic 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 $\mathrm{\Omega }$, we use a system size $N$ such that $\mathrm{\Omega }\ge \mathcal{J}/N$, at a temperature $T/\mathcal{J}=1/N$ removing thermal effects. The dashed and solid lines show the prediction of LSW theory. (b),(c) tVMC results for the evolution of the spin-squeezing parameter in the $2d$ Heisenberg model ($L=12$) along two field ramps starting from ${\mathrm{\Omega }}_i/\mathcal{J}=10$ and ending at (b) ${\mathrm{\Omega }}_f/\mathcal{J}={10}^{-1}$ 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 versus applied field and entropy per spin in the $2d$ Heisenberg model, $L=24$.