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.
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.
Spontaneous symmetry breaking (SSB) occurs when a many-body system governed by a symmetric Hamiltonian, and prepared in a symmetry-broken state by the application of a field coupling to its order parameter O, retains a finite O value even after the field is switched off. SSB is generally thought to occur only in the thermodynamic limit N→∞ (for N degrees of freedom). In this limit, the time to restore the symmetry once the field is turned off, either via thermal or quantum fluctuations, is expected to diverge. Here, we show that SSB can also be observed in finite-size quantum spin systems, provided that the spin is half-integer and three conditions are met: (1) the ground state of the system has long-range correlations, (2) the Hamiltonian conserves the (spin) parity of the order parameter, and (3) N is odd. Using a combination of analytical arguments and numerical results (based on time-dependent variational Monte Carlo and rotor + spin-wave theory), we show that SSB on finite-size systems can be achieved via a quasiadiabatic preparation of the ground state-which, in U(1)-symmetric systems, is shown to require a symmetry-breaking field vanishing over timescales τ∼O(N). In these systems, the symmetry-broken state exhibits spin squeezing with Heisenberg scaling.
Finite-size quantum spin systems can be magnetized by the application of a symmetry-breaking field, but in general their symmetry is expected to be restored once the field is turned off adiabatically. Recently (F. Caleca et al., arXiv:2412.15493) we have shown that systems of half-integer spins with an odd number of sites and a parity-preserving Hamiltonian can retain a finite magnetization, hence exhibiting spontaneous symmetry breaking (SSB) at finite size. Here we generalize this phenomenon to spin chains whose low-energy physics (in zero field) realizes a Luttinger-liquid phase. We observe that odd-sized chains can exhibit a phenomenon of finite-size quasi-SSB, in which a net sub-extensive magnetization, M∼N1−1/(4K) is retained, where N is the number of sites and K the Luttinger exponent. Interestingly, the states prepared by turning off the symmetry-breaking field quasi-adiabatically display scalable spin squeezing -- namely stronger the bigger the system -- regardless of the parity of N. The scaling of the squeezing parameter is dictated again by the Luttinger exponent, ξR2∼N−1+1/(2K). This result shows that scalable quantum correlations with metrological significance, associated typically with high-dimensional systems, can be found as well in gapless one-dimensional ones; and they are a direct consequence of the critical nature of Luttinger liquids.
Spin squeezed entanglement enables metrological precision beyond the classical limit. Understood through the lens of continuous symmetry breaking, dipolar spin systems exhibit the remarkable ability to generate spin squeezing via their intrinsic quench dynamics. To date, this understanding has primarily focused on lattice spin systems; in practice however, dipolar spin systems$\unicode{x2014}$ranging from ultracold molecules to nuclear spin ensembles and solid-state color centers$\unicode{x2014}$often exhibit significant amounts of positional disorder. Here, we develop a theory for scalable spin squeezing in a two-dimensional randomly diluted lattice of quantum dipoles, which naturally realize a dipolar XXZ model. Via extensive quantum Monte Carlo simulations, we map out the phase diagram for finite-temperature XY order, and by extension scalable spin squeezing, as a function of both disorder and Ising anisotropy. As the disorder increases, we find that scalable spin squeezing survives only near the Heisenberg point. We show that this behavior is due to the presence of rare tightly-coupled dimers, which effectively heat the system post-quench. In the case of strongly-interacting nitrogen-vacancy centers in diamond, we demonstrate that an experimentally feasible strategy to decouple the problematic dimers from the dynamics is sufficient to enable scalable spin squeezing.
Abstract We analyze the generation of spin-squeezed states via coupling of three-level atoms to an optical cavity and continuous quantum measurement of the transmitted cavity field in order to monitor the evolution of the atomic ensemble. Using analytical treatment and microscopic simulations of the dynamics, we show that one can achieve significant spin squeezing, favorably scaling with the number of atoms N. However, contrary to some previous literature, we clarify that it is not possible to obtain Heisenberg scaling without the continuous feedback that is proposed in optimal approaches. In fact, in the adiabatic cavity removal approximation and large N limit, we find the scaling behavior N − 2 / 3 for spin squeezing and N − 1 / 3 for the corresponding protocol duration. These results can be obtained only by considering the curvature of the Bloch sphere, since linearizing the collective spin operators tangentially to its equator yields inaccurate predictions. With full simulations, we characterize how spin-squeezing generation depends on the system parameters and departs from the bad cavity regime, by gradually mixing with cavity-filling dynamics until metrological advantage is lost. Finally, we discuss the relevance of this spin-squeezing protocol to state-of-the-art optical clocks.
Spin squeezing---a central resource for quantum metrology---can be generated via the nonlinear, entangling evolution of an initially factorized spin state. Here we show that robust (i.e., persistent) squeezing dynamics is generated by a very large class of S=1/2 spin Hamiltonians with axial symmetry, in relationship with the existence of a peculiar structure of the low-lying Hamiltonian eigenstates---the so-called Anderson's tower of states. Such states are fundamentally related to the appearance of spontaneous symmetry breaking in quantum systems; and, for models with sufficiently high connectivity, they are parametrically close to the eigenstates of a planar rotor (Dicke states), in that they feature an anomalously large value of the total angular momentum. Our central insight is that starting from a coherent spin state, a generic U(1)-symmetric Hamiltonian featuring the Anderson's tower of states generates the same squeezing evolution at short times as the one governed by the paradigmatic one-axis-twisting (or planar-rotor) model of squeezing dynamics. The full squeezing evolution of the planar-rotor model is seemingly reproduced for interactions decaying with distance r as ${r}^{\ensuremath{-}\ensuremath{\alpha}}$ when $\ensuremath{\alpha}<5d/3$ in d dimensions. Our results connect quantum simulation with quantum metrology by unveiling the squeezing power of a large variety of Hamiltonian dynamics that are currently implemented by different quantum simulation platforms.
Universal spin squeezing from the tower of states of U(1)-symmetric spin Hamiltonians
Spin squeezing -- a central resource for quantum metrology -- results from the non-linear, entangling evolution of an initially factorized spin state. Here we show that universal squeezing dynamics is generated by a very large class of S=1/2 spin Hamiltonians with axial symmetry, in relationship with the existence of a peculiar structure of the low-lying Hamiltonian eigenstates -- the so-called Anderson's tower of states. Such states are fundamentally related to the appearance of spontaneous symmetry breaking in quantum systems, and they are parametrically close to the eigenstates of a planar rotor (Dicke states), in that they feature an anomalously large value of the total angular momentum. We show that, starting from a coherent spin state, a generic U(1)-symmetric Hamiltonian featuring the Anderson's tower of states generates the same squeezing evolution at short times as the one governed by the paradigmatic one-axis-twisting (or planar rotor) model of squeezing dynamics. The full squeezing evolution is seemingly reproduced for interactions decaying with distance r as r−α when α<5d/3 in d dimensions. Our results connect quantum simulation with quantum metrology by unveiling the squeezing power of a large variety of Hamiltonian dynamics that are currently implemented by different quantum simulation platforms.