Engineering spin squeezing in a 3D optical lattice with interacting spin-orbit-coupled fermions

One of the most important tasks in modern quantum science is to coherently control and entangle many-body systems, and to subsequently use these systems to realize powerful quantum technologies such as quantum-enhanced sensors. However, many-body entangled states are difficult to prepare and preserve since internal dynamics and external noise rapidly degrade any useful entanglement. Here, we introduce a protocol that counterintuitively exploits inhomogeneities, a typical source of dephasing in a many-body system, in combination with interactions to generate metrologically useful and robust many-body entangled states. Motivated by current limitations in state-of-the-art three-dimensional (3D) optical lattice clocks (OLCs) operating at quantum degeneracy, we use local interactions in a Hubbard model with spin-orbit coupling to achieve a spin-locking effect. In addition to prolonging interparticle spin coherence, spin locking transforms the dephasing effect of spin-orbit coupling into a collective spin-squeezing process that can be further enhanced by applying a modulated drive. Our protocol is fully compatible with state-of-the-art 3D OLC interrogation schemes and may be used to improve their sensitivity, which is currently limited by the intrinsic quantum noise of independent atoms. We demonstrate that even with realistic experimental imperfections, our protocol may generate $\sim 10-14\mathrm{dB}$ of spin squeezing in $\sim 1$ second with $\sim {10}^2-{10}^4$ atoms. This capability allows OLCs to enter a new era of quantum-enhanced sensing using correlated quantum states of driven nonequilibrium systems.

Figure 1

Schematic of the setup for spin squeezing. (a) We consider $N$ fermionic atoms with two (pseudo)spin components, represented by red and blue spheres, trapped in the ground band of an optical lattice (shown in 2D for the sake of presentation). Atoms tunnel to neighboring sites at a rate $J$ and experience on-site interactions with strength $U$. An external laser carrying a position-dependent phase $e^{i{\mathit{k}}_{\text{L}}·\mathit{r}}$ couples the spin states of the atoms. (b) After a gauge transformation, different spin states exhibit different dispersion relations with a relative phase $\varphi =k_{\text{L}}a$, where $a$ is the lattice spacing. The external laser couples spin states with identical quasimomenta $q$ in the gauge-transformed frame. (c) If interactions are sufficiently weak, all motional degrees of freedom become frozen in momentum space, with atoms effectively pinned to fixed quasimomentum modes $\mathit{q}$. The dynamics on the frozen $\mathit{q}$-space lattice can then be mapped to a spin model in which collisional interactions correspond to a uniform, all-to-all ferromagnetic Heisenberg Hamiltonian with strength $U/L$, where $L$ is the total number of lattice sites. (d) The spin dependence of the dispersion relation is captured by a mode-dependent axial field $B_q$ that generates inhomogeneous spin precession. This axial field couples exchange-symmetric many-body Dicke states with total spin $S=N/2$ to spin-wave states with $S=N/2-1$. The all-to-all interaction opens an energy gap $fU$ (with $f=N/L$ the filling fraction of spatial modes) between the Dicke states and the spin-wave states, which forbids population transfer between them in the weak-field limit. (e) To generate spin squeezing via one-axis twisting, we initialize a product state with all spins polarized in $-\widehat{\mathit{z}}$ (i.e., in $|\downarrow \rangle$), and apply a fast external laser pulse to rotate all spins into $+\widehat{\mathit{x}}$. We then let atoms freely evolve for a variable time $t$ (with a spin-echo pulse), after which the amount of spin squeezing can be determined experimentally from global spin measurements. The spin-squeezed state can be used for a follow-up clock interrogation protocol (see Appendix pp1).

Figure 2

Benchmarking the spin and one-axis twisting models. Comparisons of maximum squeezing [top panels, (a.i) and (b.i)] and optimal squeezing time [lower panels, (a.ii) and (b.ii)] between the Fermi-Hubbard (FH), spin, and one-axis twisting (OAT) models; obtained numerically via the protocol depicted in Fig. 1 in a 1D lattice with $L=12$ sites. Results are shown for half filling with $N=12,f\equiv N/L=1$ [left panels, (a.i.) and (a.ii)] and filling $f=5/6$ [right panels, (b.i.) and (b.ii)] as a function of $U/J$ and the SOC angle $\varphi$. In both cases, the system is initialized in the corresponding ground state. Insets for both $f=1$ and $f=5/6$ show (in green) regions of the $U-\varphi$ plane in which both the optimal squeezing (in dB) and the corresponding squeezing time of all three models agree to within 20%. At half filling [(a.i) and (a.ii)], mode-changing collisions are suppressed by Pauli blocking, resulting in almost exact agreement between the FH and spin models; both of these models converge onto the OAT model in the gap-protected, weak SOC regime of large $U/J$, and small $\varphi$. The spin and OAT models show similar behavior away from half filling [(b.i) and (b.ii)], but the presence of mode-changing collisions results in their disagreement with the FH model as interactions begin to dominate at larger $U/J$. Even below half filling, however, the FH exhibits comparable amounts of squeezing to the spin model across a broad range of $U/J$ and $\varphi$, albeit at earlier times when $U/J\gtrsim 2$.

Figure 3

Optimal squeezing with one- and two-axis twisting in a 2D section of the 3D ${}^{87}\mathrm{Sr}$ optical lattice clock. (a) The maximum amount of squeezing depends only on the atom number $N={\ell }^2$, where $\ell$ is the number of lattice sites along each axis of the lattice. While the timescales for squeezing generally depend on several experimental parameters, the time at which maximal squeezing occurs can be minimized at any given lattice depth $V_0$ by choosing SOC angles $\varphi$ that saturate $\tilde{B}/U\approx 0.05$, where $\tilde{B}$ is the variance of the SOC-induced axial field and $U$ is the two-atom on-site interaction energy. Panels (b) and (c) show these minimal squeezing times as a function of the depth $V_0$ and linear size $\ell$ of the lattice. Lattice depths $V_0$ are normalized to the atomic lattice recoil energy $E_{\text{R}}$, and the upper axis on panels (b) and (c) marks values of $U/J$ at fixed lattice depths. In general, TAT achieves more squeezing than OAT for any system size, and achieves optimal squeezing faster for $N\gtrsim 400$ atoms, as denoted by a dotted line in panels (b) and (c).

Figure 4

Optimal squeezing with decoherence via one- and two-axis twisting in a 2D section of the 3D ${}^{87}\mathrm{Sr}$ optical lattice clock (OLC). In practice, decoherence due to light scattering limits the amount of squeezing that is attainable in the the 3D ${}^{87}\mathrm{Sr}$ OLC. Due to growing squeezing times with increasing system size, the maximal squeezing obtainable via OAT saturates past $\ell \approx 30$ sites along each axis of the lattice, with $N\approx {10}^3$ atoms total. The more favorable size dependence of TAT timescales, however, allow for continued squeezing gains through $\ell =100$ ($N={10}^4$). While the OAT results in (a) are exact, the TAT results in (b) reflect only a lower bound on the maximum squeezing obtainable, albeit one that is likely close (within a few dB) to the actual value. Optimal squeezing times in the presence of decoherence are generally smaller than the corresponding times shown in Fig. 3, as decoherence typically degrades squeezing before it reaches the decoherence-free maximum. The decoherence considered in this paper also limits maximally achievable squeezing to $\sim 20\mathrm{dB}$ less than the decoherence-free maxima shown in Fig. 3. Sample plots of squeezing over time for particular choices of lattice size ($\ell$) and depth ($V_0/E_{\text{R}}$) are provided in Appendix pp10.

Figure 5

Dynamics of noninteracting spin-orbit coupled fermions in a 1D lattice with SOC angle $\varphi =\pi /50$, plus a harmonic trap with $\mathrm{\Omega }/J=0.01$. Starting with a spin-polarized cloud in $\downarrow$ ground state, an initial clock laser pulse is applied to rotate spins into $x$, and the atoms are allowed to evolve during the dark time. We track the dynamics of the $\uparrow$ particle density for the cases of (a) $N=20$ and (b) $N=60$ atoms. Panel (c) shows the time-averaged fluctuations of the $\uparrow$ particle density for each site index $j$ from its initial value following the Ramsey pulse; see Eq. (C3). For $N=60$, we have filled all delocalized modes as well as several localized modes, resulting in a large region of no density fluctuations at the trap center. Panel (d) contains the eigenspectrum for a single internal state in the presence of the trap (with the index $n$ labeling the eigenvalues in order of increasing energy), where the critical mode $n_c$ dividing the spatially delocalized and localized modes is indicated by a black dash-dotted line. The highest occupied mode in the $\downarrow$ ground state for $N=20$ and $N=60$ is indicated by the green and red solid lines, respectively.

Figure 6

Dynamics of interacting spin-orbit coupled fermions in a 1D lattice plus a harmonic trap for $U/J=1$ (a), 2 (b), and 4 (c). For a 1D lattice with 10 sites and an SOC angle $\varphi =\pi /50$, we apply a $\pi /2$ clock laser pulse to the $\downarrow$ ground state and let the system evolve during the dark time. In (a.i)–(c.i) we show the squeezing dynamics of the system for both $N=10$ (solid lines) and $N=9$ (dashed lines) for a variety of trapping strengths. In (a.ii)–(c.ii), we plot the time-averaged fluctuations in total particle density, $\overline{\delta n_j}$ [as in Eq. (C3) but with ${\widehat{n}}_{j,\uparrow }$ replaced by ${\sum }_{\alpha }{\widehat{n}}_{j,\alpha }$]. In (a.iii)–(c.iii), we plot the growth of the doublon population $N_d\left(t\right)$ [see Eq. (C4)] as a function of time, noting the absence of squeezing in the presence of a large doublon population. For the chosen trap strengths, the corresponding values of $n_c$ are 28 ($\mathrm{\Omega }/J=0.01$), 14 ($\mathrm{\Omega }/J=0.04$), and 6 ($\mathrm{\Omega }/J=0.2$). In panels where the results for the homogeneous case (orange curves) are not visible, they are nearly identical to the results for $\mathrm{\Omega }/J=0.01$ (green curves). Here, we utilize periodic boundary conditions to minimize finite size effects.

Figure 7

Optimal squeezing as a function of $\pi$ pulses applied prior to the optimal TAT squeezing time in a CPMG sequence with (a) $N=100$ and (b) $N=1000$ atoms. Results are shown for OAT, TAT, and ${\mathrm{TAT}}_{\pm ,\text{z}}$, where ${\mathrm{TAT}}_{\pm ,\text{z}}$ denotes squeezing via the Hamiltonian ${\widehat{H}}_{\text{TAT}}^{(\pm ,\text{z})}\equiv {\widehat{H}}_{\text{TAT}}^{(\pm )}-{\mathcal{J}}_0\left({\beta }_{\pm }\right){〈\overline{B}〉}_f^{\text{rms}}{\widehat{S}}_{\text{z}}$. Details about experimental parameters for these simulations are provided in the text.

Figure 8

Relative error between maximal squeezing (measured in dB) obtained by the OAT [Eq. (4)] and spin [Eq. (3)] models of the main text in a system of 20 particles. The OAT model correctly captures the maximal squeezing (in dB) of the spin model to within 3% (marked by the horizontal reference line) within the gap-protected regime $\tilde{B}/U<0.06$.

Figure 9

Comparison between the OAT and the spin model in the presence of decoherence. (a) The difference between the maximal squeezing (measured in dB) obtained by the OAT [Eq. (4)] and spin [Eq. (3)] models increases with the particle number $N$ and the single-particle spontaneous emission rate $\gamma$. This disagreement is attributed in part to the fact that spontaneous emission transfers population of the collective spin state outside of the Dicke manifold, violating an assumption of the OAT model; see panel (b). The rate of population transfer outside of the Dicke manifold increases with both particle number and spontaneous emission rate. (Parameters for simulations in this figure: $U=1000\mathrm{Hz}, J=200\mathrm{Hz}$, and $\varphi =\pi /20$).

Figure 10

$p$-wave loss rates. Both the averaged $p$-wave inelastic collision rate $\gamma$ (orange) and the ratio of this collision rate to the optimal squeezing rate ${\chi }_{\text{opt}}$ (blue) are suppressed as the lattice depth increases. ${\chi }_{\text{opt}}$ is obtained by choosing SOC angles $\varphi$ that saturate $\tilde{B}/U\approx 0.05$, where $\tilde{B}$ is the variance of the SOC-induced axial field and $U$ is the two-atom on-site interaction energy.

Figure 11

Squeezing via OAT in the presence of inelastic collisions. (a) For fixed particle number $N=100$, the optimal squeezing decreases as the inelastic collision rate increases. Panel (b) shows squeezing over time for $\gamma /{\chi }_{\text{opt}}=0.04$ (solid lines), which corresponds to $U/J=6$, and compares it with $\gamma =0$ (dashed lines) for different particle numbers. Inelastic collisions prevent the growth of optimal squeezing with particle number. For $N=1000$, the maximum squeezing saturates to $\sim 10\mathrm{dB}$.

Figure 12

Squeezing via OAT and TAT in a 2D section of the 3D ${}^{87}\mathrm{Sr}$ optical lattice clock, shown for (a) $\ell =40$ and (b) $\ell =100$ sites per axis (with $N={\ell }^2$ atoms total), and a lattice depth of $V_0=4E_{\text{R}}$, where $E_{\text{R}}$ is the atomic lattice recoil energy. Atoms are confined along the direction transverse to the 2D layer by a lattice of depth 60 $E_{\text{R}}$. Squeezing over time is shown for OAT (blue) and TAT (green), both with (solid lines) and without (dashed lines) decoherence via uncorrelated decay and dephasing of individual spins at rates of $0.1{\text{sec}}^{-1}$ (see Appendix pp7).