Robust spin squeezing from the tower of states of U(1)-symmetric spin Hamiltonians

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 $r^{-\alpha }$ when $\alpha <5d/3$ 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.

Figure 1

Spin squeezing generated by the $\alpha -\mathit{XX}$ Hamiltonian, with $d=1$ and $\alpha =0$. For each system size $N$, the result of tVMC with the pair-product ansatz (solid lines) perfectly matches with the exact solution for the OAT Hamiltonian [2].

Figure 2

Low-energy spectrum of the $d=1\alpha -\mathit{XX}$ model for (a)–(d) $\alpha =1$, (b)–(e) $\alpha =3$, and (c)–(f) $\alpha =\infty$. The spectrum is shown as a function of the two quantum numbers $E=\langle \widehat{\mathcal{H}}\rangle$ and ${\left(J^z\right)}^2$ in (a)–(c), and as a function of $E$ and the average total spin $\langle {\widehat{\mathit{J}}}^2\rangle$ in (d)–(f). Black dots correspond to states of the ToS, while orange (gray) dots are outside the ToS; straight lines are linear fits of the ToS states.

Figure 3

Dynamics of the squeezing parameter for the $1d\alpha -\mathit{XX}$ Hamiltonian with (a) $\alpha =1$, (b) $\alpha =3$, and (c) $\alpha =\infty$, for system size from $N=16$ (bottom line) to $N=128$ (top line). Time is rescaled by the Kac prefactor $K_N^{\left(\alpha \right)}$; see text. The dashed line is the exact result for $N=16$, while the other data are obtained via tVMC (solid lines).

Figure 4

Finite-size scaling (a) of the optimal time $t_{\min }$ and (b) of the minimal squeezing ${\xi }_{R,\min }^2$, for several values of $\alpha$. Dots are tVMC results; straight lines are power-law fits with $t_{\min }\mathcal{J}{K}_N^{\left(\alpha \right)}\sim N^{\mu }$ and ${\xi }_{R,\min }^2\sim N^{-\nu }$. The fit range is $32\le N\le 128$. (c) Exponent $\mu$ for the scaling of the optimal time. (d) Exponent $\nu$ for the scaling of the minimum squeezing parameter. In (c) and (d), the horizontal black line shows the OAT exponents, and the vertical dotted line marks $\alpha =5d/3$ (see text).

Figure 5

Universal short-time squeezing dynamics. Squeezing dynamics for the $\alpha -\mathit{XX}$ model in (a) $d=1$ ($N=16$) and (b) $d=2$ ($N=4\times 4$), for various values of $\alpha$ from $\alpha =0$ (darkest line) to $\alpha =\infty$ (lightest line). Collapse of the curves at short times is obtained by rescaling time with the $I^{\left(\alpha \right)}$ parameter; (c),(d) Comparison between the three estimates of the moment of inertia: from the Hamiltonian projection onto Dicke states ($I^{\left(\alpha \right)}$), from the Fourier transform of the coupling constant ($I_{\gamma }^{\left(\alpha \right)}$), and from the ToS spectrum ($I_{\mathrm{ToS}}^{\left(\alpha \right)}$). System Hamiltonian and sizes as in (a)  and (b).

Figure 6

Overlap of the low-energy eigenstates of the $\alpha -\mathit{XX}$ model (with $d=1$ and $N=16$) with $|{\mathrm{CSS}}_x\rangle$, as a function of either ${\left(J^z\right)}^2$ (top panels) or the total angular momentum (lower panels). Black points are states in the ToS, while orange (gray) dots are for the rest of the spectrum.

Figure 7

Time evolution of spin squeezing generated by the $\alpha -\mathit{XX}$ Hamiltonian, with $d=1$ and $\alpha \in \{0,1/2,1,3/2,2,5/2,3,\infty \}$. Results are obtained via tVMC with the pair-product ansatz (solid lines) for system sizes from $N=16$ (bottom line) to $N=128$ (top line). For each $\alpha$, we also show the exact result for $N=16$ (dashed black line). Time is rescaled by the Kac prefactor. Statistical error bars on the tVMC curves are of the order of the line width or smaller.

Figure 8

Diagonal matrix elements of the $\alpha -\mathit{XX}$ Hamiltonian $\widehat{\mathcal{H}}$ (for $d=1$ and $N=16$) on the $|J_{\max },J^z\rangle$ states, as a function of ${\left(J^z\right)}^2$. For each $\alpha$, straight lines are linear fits. The exponent $\alpha$ ranges from $\alpha =\infty$ (top) to $\alpha =0$ (bottom).

Figure 9

Dynamics of the total spin $\langle {\widehat{\mathit{J}}}^2\rangle$ in the one-dimensional $\alpha -\mathit{XX}$ model, for several values of $\alpha$. For each $\alpha$, tVMC results (solid lines) are shown for system sizes from $N=16$ (top line) to $N=128$ (bottom line), and compared with the exact curve for $N=16$ (dashed black line).

Figure 10

Upper row: $J^z$-resolved spectra of $\alpha -\mathit{XX}$ chains ($N=16$) with long-range interactions ($\alpha =1$), with short-range interactions ($\alpha =\infty$), and with dimerized short-range interactions ($r={\mathcal{J}}^{\prime }/\mathcal{J}<1$). The blue (empty) circles highlight the ground states in each $J^z$ sector, while the green (filled, dark gray) dots indicate the spectrum of the Dicke-projected Hamiltonian, $E_{\mathrm{Dicke}}\left(J^z\right)=\langle J_{\max },J^z|\widehat{\mathcal{H}}|J_{\max },J^z\rangle$. Lower row: $J^z$-resolved overlap of the Hamiltonian eigenstates with the CSS. Symbols are the same as in the upper row. For $\alpha =\infty$, slight asymmetries are visible between the positive and negative $J^z$ values: they are an artifact of exact diagonalization mixing degenerate eigenstates.

Figure 11

Spin-squeezing dynamics of an $N=16 \mathit{XX}$ chain with dimerization. The time axis is rescaled with the moment of inertia $I_r^{\left(\infty \right)}$ of the Dicke-projected spectrum, so that the squeezing dynamics becomes identical to that of the OAT model at very short times. The $r$ parameter ranges from $r=0.2$ (lightest line) to $r=1$ (darkest line).