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

TL;DR

This paper presents a novel protocol that uses inhomogeneities and interactions in a 3D optical lattice with spin-orbit-coupled fermions to generate robust, metrologically useful spin-squeezed states, enhancing quantum sensing.

Contribution

It introduces a spin-locking protocol leveraging inhomogeneities and interactions to produce strong, robust spin squeezing in 3D optical lattice clocks, compatible with current technologies.

Findings

Achieves 10-14 dB of spin squeezing in about 1 second.

Proposes a method compatible with existing 3D optical lattice clock schemes.

Demonstrates robustness of the protocol even with experimental imperfections.

Abstract

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 inter-particle 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 $\sim10$--$14$ dB of spin squeezing in $\sim1$ second with $\sim10^2$--$10^4$ atoms. This capability allows OLCs to enter a new era of quantum enhanced sensing using correlated quantum states of driven non-equilibrium systems.

Equations

$${\hat{H}}_{\text{FH}}^{(\varphi )}/\hslash$$

$$+U\sum _j{\hat{n}}_{j,\uparrow }{\hat{n}}_{j,\downarrow },$$

$${\hat{H}}_{\text{FH,single}}^{(\varphi )}/\hslash =-2J\sum _q\left[\cos (qa+\varphi ){\hat{n}}_{q,\uparrow }+\cos (qa){\hat{n}}_{q,\downarrow }\right].$$

$${\hat{H}}_{\text{spin}}/\hslash =-\frac{U}{L}\hat{\mathbf{S}}\cdot \hat{\mathbf{S}}-\sum _q{B}_q{\hat{s}}_q^{\text{z}},$$

$${\hat{H}}_{\text{eff}}/\hslash =-\frac{U}{L}\hat{\mathbf{S}}\cdot \hat{\mathbf{S}}-\overline{B}{\hat{S}}_{\text{z}}+\chi {\hat{S}}_{\text{z}}^2,$$

$${\xi }^2\equiv \underset{\theta }{\min }⟨\text{var}({\hat{S}}_{\theta }^{\perp })⟩\times N/\text{absolutevalue}\ast {⟨\hat{\mathbf{S}}⟩}^2,$$

$$H=H_0+V,$$

$$H_0=-\frac{U}{L}\mathbf{S}\cdot \mathbf{S},$$

$$\mathcal{O}V$$

$$\mathcal{L}V$$

$$H_{\text{eff}}^{(0)}={\mathcal{P}}_0{H}_0{\mathcal{P}}_0,$$

$$H_{\text{eff}}^{(2)}=-\frac{1}{2}{\mathcal{P}}_0{\left[\mathcal{O}V,\mathcal{L}V\right]}_{-}{\mathcal{P}}_0,$$

$$|m⟩\propto S_{+}^{N/2+m}{|\downarrow ⟩}^{\otimes n},$$

$$\overline{B}\equiv \frac{1}{N}\sum _n{B}_n,$$

$$V$$

$$=-\sum _n{b}_n{s}_{\text{z}}^{(n)}-\overline{B}{S}_{\text{z}}+\Omega S_{\text{x}},$$

$$H_{\text{eff}}^{(1)}$$

$$=-\sum _n{b}_n{\mathcal{P}}_0{s}_{\text{z}}^{(n)}{\mathcal{P}}_0-\overline{B}{S}_{\text{z}}+\Omega S_{\text{x}},$$

$$\begin{gathered} \sum _n{b}_n{\mathcal{P}}_0{s}_{\text{z}}^{(n)}{\mathcal{P}}_0=\sum _{n,m,m^{\prime }}{b}_n\text{outerproduct}mm{s}_{\text{z}}^{(n)}\text{outerproduct}{m}^{\prime }{m}^{\prime } \\ =\sum _n{b}_n\sum _{m,m^{\prime }}\text{outerproduct}mm{s}_{\text{z}}^{(1)}\text{outerproduct}{m}^{\prime }{m}^{\prime }=0, \end{gathered}$$

$$H_{\text{eff}}^{(1)}=-\overline{B}{S}_{\text{z}}+\Omega S_{\text{x}}.$$

$$\mathcal{I}\equiv \sum _{|\alpha ⟩\in {\mathcal{B}}_0\left({\mathcal{E}}_0\right)}\frac{\text{outerproduct}\alpha \alpha }{E_{\alpha }},$$

$$H_{\text{eff}}^{(2)}=-{\mathcal{P}}_{0}V\mathcal{I}V{\mathcal{P}}_0,$$

$$|mk⟩\propto \sum _{n=1}^N{e}^{2\pi ikn/N}{s}_{+}^{(n)}|m-1⟩,$$

$$\begin{gathered} H_{\text{eff}}^{(2)}=-\frac{1}{fU}\sum _{m,k,n,n^{\prime }}{b}_n{b}_{n^{\prime }}\text{outerproduct}mm{s}_{\text{z}}^{(n)}|mk⟩ \\ \times ⟨mk|s_{\text{z}}^{(n^{\prime })}\text{outerproduct}mm, \end{gathered}$$

$$⟨m⟩s_{\text{z}}^{(n)}mk=e^{2\pi ikn/N}\sqrt{\frac{(N/2{)}^2-m^2}{N^2(N-1)}},$$

$$\begin{gathered} H_{\text{eff}}^{(2)}=-\frac{1}{fU}\sum _m\frac{(N/2{)}^2-m^2}{N^2(N-1)}\text{outerproduct}mm \\ \times \sum _{k,n,n^{\prime }}{b}_n{b}_{n^{\prime }}{e}^{2\pi ik\left(n-n^{\prime }\right)/N}. \end{gathered}$$

$$\sum _{k,n,n^{\prime }}{b}_n{b}_{n^{\prime }}{e}^{2\pi ik\left(n-n^{\prime }\right)/N}$$

$$=\sum _{n,n^{\prime }}{b}_n{b}_{n^{\prime }}\sum _{k=0}^{N-1}{e}^{2\pi ik\left(n-n^{\prime }\right)/N},$$

$$\sum _{k,n,n^{\prime }}{b}_n{b}_{n^{\prime }}{e}^{2\pi ik\left(n-n^{\prime }\right)/N}=N^2{\tilde{B}}^2,$$

$${\tilde{B}}^2\equiv \frac{1}{N}\sum _n{b}_n^2=\frac{1}{N}\sum _n{\left(B_n-\overline{B}\right)}^2.$$

$$H_{\text{eff}}^{(2)}=-\sum _m\frac{{\left(N/2\right)}^2-m^2}{\left(N-1\right)fU}{\tilde{B}}^2\text{outerproduct}mm,$$

$$H_{\text{eff}}=-\frac{U}{L}\mathbf{S}\cdot \mathbf{S}-\overline{B}{S}_{\text{z}}+\Omega S_{\text{x}}+\chi S_{\text{z}}^2,$$

$$\chi \equiv \frac{{\tilde{B}}^2}{\left(N-1\right)fU}.$$

Full text

I Introduction

A major frontier of contemporary physics is the understanding of non-equilibrium behaviors of many-body quantum systems, and the application of these behaviors toward the development of novel quantum technologies with untapped capabilitiesEisert et al. (2015). To this end, ultracold atomic, molecular, and optical systems are ideal platforms for studying unexplored regimes of many-body physics due to their clean preparation and readout, high controllability, and long coherence timesBloch et al. (2008); Gross and Bloch (2017). The exquisite capabilities of these systems have pushed the frontiers of metrology, quantum simulation, and quantum information science.

Optical lattice clocks in particular have seen some of the most impressive developments in recent years, reaching record levels of precision ($\sim 3\times 10^{-19}$)Campbell et al. (2017); Marti et al. (2018) and accuracy ($\sim 1\times 10^{-18}$)Bloom et al. (2014); McGrew et al. (2018). These advancements required important breakthroughs, including the capability to cool and trap fermionic alkaline-earth atoms in spin-insensitive potentialsTakamoto and Katori (2003); Barber et al. (2006); Ye et al. (2008); the development of ultracoherent lasersKessler et al. (2012); Cole et al. (2013); Matei et al. (2017) to fully exploit an ultranarrow clock transitionLudlow et al. (2015); the detailed characterization of inter-atomic interactionsScazza et al. (2014); Cappellini et al. (2014); Zhang et al. (2014); and, more recently, the preparation of a quantum degenerate gas in a three-dimensional (3D) optical latticeCampbell et al. (2017); Marti et al. (2018); Goban et al. (2018). Nonetheless, all improvements in sensing capabilities to date have been based on single-particle control of internal atomic degrees of freedom. Such strategies will eventually have diminishing returns due to practical difficulties in (i) suppressing decoherence from external (motional) degrees of freedom, and (ii) interrogating more particles without additional systematic errors from interactionsMartin et al. (2013); Ludlow et al. (2015); Marti et al. (2018).

Pushing beyond the current independent-particle paradigm requires leveraging many-body quantum correlations. Entangled states such as spin-squeezed statesKitagawa and Ueda (1993); Wineland et al. (1992); Ma et al. (2011) can enhance measurement sensitivity, i.e. the uncertainty $\Delta\theta$ in the estimation of a parameter $\theta$, below the standard quantum limit $\Delta\theta\sim 1/\sqrt{N}$ for $N$ uncorrelated particlesItano et al. (1993); Degen et al. (2017). The major challenge for progress in this direction is that generating entanglement requires interactions, which are generally undesirable because they degrade atomic coherence, thereby limiting clock performanceSwallows et al. (2011); Martin et al. (2013); Rey et al. (2014); Ludlow et al. (2011); Lemke et al. (2011); Ludlow et al. (2015). In fact, the most precise and accurate optical lattice clocks were designed to operate with fermionic atoms in identical nuclear and electronic states to suppress collisional decoherenceCampbell et al. (2009); Swallows et al. (2011); Campbell et al. (2017), as identical fermions cannot interact via the otherwise dominant ($s$-wave) collisions at ultracold temperatures. However, an initially spin-polarized Fermi gas still exhibits interactions at later times due to spin-orbit coupling (SOC) that is induced by the laser that drives the clock transition (i.e. the “clock laser”)Wall et al. (2016); Kolkowitz et al. (2016); Livi et al. (2016); Bromley et al. (2018). Specifically, the momentum kick imparted by this laser imprints a position-dependent phase that induces inhomogeneous spin precession and generates spin dephasing, thereby making atoms distinguishable and vulnerable to collisions. While a deep lattice can suppress SOC, it also intensifies the light scattering which currently limits the coherence time of the clockDörscher et al. (2018); Goban et al. (2018); Hutson et al. (2019).

In this work, we describe a scheme that can lead to metrological advances in state-of-the-art optical lattice clocks through direct use of quantum entanglement by harnessing the interplay between nominally undesirable collisions and SOC. This scheme is made possible in the weak SOC regime by the formation of an interaction-energy gap that suppresses the SOC-induced population transfer from the exchange-symmetric Dicke manifold (spanned by spin-polarized, and thus non-interacting states) to the remainder of Hilbert space. Interactions thereby prolong inter-particle spin coherence through a spin-locking effect, which additionally transforms the dephasing effect of SOC into a collective spin squeezing process. To generate spin squeezing, our protocol only requires the capability to fix (i) the orientation of the clock laser and (ii) the optical lattice depth. These controls are straightforward to incorporate into current 3D clock interrogation sequences without sacrificing atom numbers or coherence times. Additionally, we show that by applying a modulated drive from the clock laser, one can further prepare states that saturate the Heisenberg limit $\Delta\theta\sim 1/N$ for phase sensitivityKitagawa and Ueda (1993); Ma et al. (2011); Degen et al. (2017). This capability mirrors efforts in other settings, such as nitrogen-vacancy centers in diamondBauch et al. (2018); Aiello et al. (2013) and trapped ionsBurd et al. (2019), to enhance quantum metrology through the use of driven non-equilibrium phenomena.

Despite an abundance of proof-of-principle experiments with entangled statesDegen et al. (2017); Pezzè et al. (2018), so far only the remarkable example of LIGOAasi et al. (2013); Abbott et al. (2016) has demonstrated a quantum advantage in a state-of-the-art quantum sensing or measurement system. The new generation of 3D optical lattice systems have fully quantized motional degrees of freedomCampbell et al. (2017), allowing for precise control of collisional interactions. We demonstrate how these interactions can naturally give rise to metrologically useful correlated many-body fermionic states, opening a path to not only generate entanglement, but also harness it to achieve a quantum advantage in a world-class sensor. Such an advance will ultimately deliver gains to real-world applications including timekeeping, navigation, telecommunication, and our understanding of the fundamental laws of natureSafronova et al. (2018).

II Spin squeezing with the Fermi-Hubbard model

We consider $N$ fermionic atoms with two spin states (labeled $\uparrow$ and $\downarrow$) trapped in a 3D optical lattice. In this discussion, these spin states are associated with the two electronic states of a nuclear-spin-polarized gas. At sufficiently low temperatures, atoms occupy the lowest Bloch band of the lattice and interact only through $s$-wave collisions. A schematic of this system is provided in Fig. 1(a), where tight confinement prevents motion along the vertical direction ($z$), effectively forming a stack of independent 2D lattices. For simplicity and without loss of generality, however, we first consider the case when tunneling can only occur along one direction, $x$, and thus model the system as living in one dimension.

An external laser with Rabi frequency $\Omega$ and wavenumber $k_{\text{L}}$ along the tunneling axis resonantly couples atoms’ internal states through the Hamiltonian $\hat{H}_{\text{laser}}/\hbar=\sum_{j}\Omega e^{ik_{\text{L}}x_{j}}\hat{c}_{j,\uparrow}^{\dagger}\hat{c}_{j,\downarrow}+\text{h.c.}$, where $\hat{c}_{j\alpha}$ is a fermionic annihilation operator for an atom on site $j$ with internal state $\alpha\in\left\{\uparrow,\downarrow\right\}$ and $x_{j}$ is the position of site $j$. This laser imprints a position-dependent phase that equates to a momentum kick $k_{\text{L}}$ when an atom changes internal states by absorbing or emitting a photon, thereby generating spin-orbit couplingWall et al. (2016); Livi et al. (2016). After absorbing the position dependence of the laser Hamiltonian into fermionic operators through the gauge transformation $\hat{c}_{j,\uparrow}\to e^{ik_{\text{L}}x_{j}}\hat{c}_{j,\uparrow}$, which makes $\hat{H}_{\text{laser}}$ spatially homogeneous, the atoms are well-described in the tight-binding limit by the Fermi-Hubbard HamiltonianEsslinger (2010)

[TABLE]

where $J$ is the nearest-neighbor tunneling rate; the SOC angle $\phi\equiv k_{\text{L}}a$ determines the phase gained by spin-up atoms upon tunneling from site $j$ to site $j+1$ (in the gauge-transformed frame) with lattice spacing $a=x_{j+1}-x_{j}$; $U$ is the on-site interaction energy of two atoms; and $\hat{n}_{j\alpha}\equiv\hat{c}_{j\alpha}^{\dagger}\hat{c}_{j\alpha}$ is a number operator.

The Fermi-Hubbard Hamiltonian can be re-written in the quasi-momentum basis with annihiliation operators $\hat{c}_{q\alpha}\equiv L^{-1/2}\sum_{j}e^{-iqx_{j}}\hat{c}_{j\alpha}$, where $q$ is a quasi-momentum and $L$ is the total number of lattice sites. In this basis, the single-particle Hamiltonian exhibits shifted dispersion relations that signify spin-orbit coupling [see Fig. 1(b)]:

[TABLE]

When $U\lesssim J$, interaction energies are too weak for collisions to change the occupancies of single-particle quasi-momentum modes. Atoms are then pinned to these modes, which form a lattice in quasi-momentum space [see Fig. 1(c)]Bromley et al. (2018). In this strong-tunneling limit, the Fermi-Hubbard Hamiltonian [Eqn. (2)] can be mapped to a spin-$1/2$ system with a collective ferromagnetic Heisenberg interaction and an inhomogeneous axial field, given byMartin et al. (2013); Rey et al. (2014); Bromley et al. (2018)

[TABLE]

where $\hat{\bm{S}}=\sum_{q}\hat{\bm{s}}_{q}$ is a collective spin operator; $\hat{\bm{s}}_{q}$ is a spin-1/2 operator for mode $q$ with components $\hat{s}_{q}^{j=\text{x},\text{y},\text{z}}\equiv\frac{1}{2}\sum_{\alpha,\beta}\hat{c}_{q\alpha}^{\dagger}\sigma^{j}_{\alpha\beta}\hat{c}_{q\beta}$ defined in terms of the Pauli matrices $\sigma^{j=\text{x},\text{y},\text{z}}$; the sums over $q$ run over all occupied quasi-momentum modes; and $B_{q}\equiv-4J\sin(qa+\phi/2)\sin(\phi/2)$ is the SOC-induced axial field.

On its own, the collective Heisenberg term ($\sim\hat{\bm{S}}\cdot\hat{\bm{S}}$) in Eqn. (3) opens an energy gap $fU$, with $f\equiv N/L$ the filling fraction of spatial modes, between the collective Dicke states $\Ket{S=N/2,M_{S}}$ and the remainder of Hilbert spaceRey et al. (2008); Martin et al. (2013); Norcia et al. (2018); Smale et al. (2019) with $S<N/2$. Here $S$ and $M_{S}$ respectively label the eigenvalues of the collective spin operators $\hat{\bm{S}}\cdot\hat{\bm{S}}$ and $\hat{S}_{\text{z}}$, with eigenvalues $S(S+1)$ for non-negative $S\in\left\{N/2,N/2-1,\cdots\right\}$ and $M_{S}\in\left\{-S,-S+1,\cdots,S\right\}$. The axial field $B_{q}$ generally couples states within the Dicke manifold to states outside it. In the weak SOC limit (i.e. $B_{q}\ll fU$), however, the interaction energy gap suppresses population transfer between states with different total spin $S$ [see Fig. 1(d)]. In this regime, the virtual occupation of states outside the Dicke manifold can be accounted for perturbatively. The symmetries of SOC as expressed in Eqn. (3) dictate that this treatment should yield powers of $\hat{S}_{\text{z}}$ when projected onto the collective Dicke manifold at higher orders in perturbation theory. At second order in perturbation theory (see Appendix A), we thus find that SOC effectively yields a one-axis twisting (OAT) model widely known to generate squeezing dynamicsKitagawa and Ueda (1993); Ma et al. (2011):

[TABLE]

where $\overline{B}\equiv\sum_{q}B_{q}/N$ is the mean and $\widetilde{B}^{2}\equiv\sum_{q}\left(B_{q}-\overline{B}\right)^{2}/N$ the variance of the axial field. The effect of the $\sim\hat{\bm{S}}\cdot\hat{\bm{S}}$ term is to generate a relative phase between states with different total spin $S$ and thus has no effect on dynamics restricted to a fixed $S$. Note also that the collective spin rotation from $\overline{B}\hat{S}_{\text{z}}$ can be eliminated by going into a rotating frame or by using a spin echo.

The entire protocol for preparing a squeezed state via OAT, sketched out in Fig. 1(e), reduces to a standard Ramsey protocol with a spin echo: after initially preparing a spin-down (i.e. $-\hat{\bm{z}}$) polarized sample of ultracold atoms populating the lowest Bloch band of a lattice, a fast $\pi/2$ pulse is applied with the clock laser to rotate all spin vectors into $+\hat{\bm{x}}$. The atoms then freely evolve for a variable time $t$ (possibly with spin-echo $\pi$-pulses), after which the amount of metrologically useful spin squeezing, measured by the Ramsey squeezing parameter

[TABLE]

can be determined experimentally from global spin measurements. Here $\braket{\hat{\bm{S}}}$ is the mean collective spin vector and $\braket{\text{var}(\hat{S}^{\perp}_{\theta})}$ is the variance of spin measurements along an axis orthogonal to $\braket{\hat{\bm{S}}}$, parameterized by the angle $\theta\in[0,2\pi)$.

The above protocol concerns only the preparation of a spin-squeezed state, which would then be used as an input state for a follow-up clock interrogation protocol without SOC. While increasing the lattice depth to turn off SOC during clock interrogation is the simplest approach, this will limit the interrogation time due to light scattering (see discussion below). Alternatively, it is possible to keep the same lattice depth used for the spin squeezing generation by adding a specific pulse sequence to suppress SOC. See details in Appendix E.

II.1 Model validity

The validity of the OAT model in Eqn. (4) relies on two key conditions concerning experimental parameter regimes. First, the on-site interaction energy $U$ should not be much larger in magnitude than the tunneling rate $J$ (clarified below); otherwise, one cannot assume frozen motional degrees of freedom (i.e. with atoms pinned to fixed quasi-momentum modes) and map the Fermi-Hubbard model to a spin model. Second, the SOC-induced fields $B_{q}\sim J\sin(\phi/2)$ should be considerably smaller in magnitude than the interaction energy gap $fU$, as otherwise one cannot perturbatively transform SOC into OAT. These two conditions can be satisfied by appropriate choices of $U/J$ and the SOC angle $\phi$, which are respectively controlled by tuning the lattice depth and changing the angle between the clock laser and the lattice axes [see Fig. 1(a)].

We demonstrate the importance of these conditions in Fig. 2, where we show numerical results from exact simulations of a 1D system with $L=12$ sites. Therein, optimal squeezing achievable under unitary dynamics is provided in dB, i.e. $-10\log_{10}(\xi_{\text{opt}}^{2})$, while the time at which this squeezing occurs is provided in units of the nearest-neighbor tunneling time $2\pi/J$. At $f=1$ atom per lattice site, i.e. half filling of all atomic states in the lowest Bloch band, the spin model [Eqn. (3)] agrees almost exactly with the Fermi-Hubbard (FH) model [Eqn. (2)] up through (and exceeding) $U/J=8$. The agreement at half filling ($f=1$) is assisted by Pauli blocking of mode-changing collisions. Below half filling ($f=5/6$), these two (FH and spin) models show good agreement at $U/J\lesssim 2$, while at $U/J\gtrsim 2$ mode-changing collisions start to become relevant and invalidate the frozen-mode assumption of the spin model. Note that we chose filling $f=5/6$ to demonstrate that our protocol should work, albeit sub-optimally, even in this highly hole-doped case; in practice, optimized experiments are capable of achieving fillings closer to the optimal $f=1$Brown et al. (2017). Interestingly, even with mode-changing collisions the Fermi-Hubbard model exhibits comparable amounts of squeezing to the spin model, and achieves this squeezing in less time. The spin and OAT models agree in the regime of weak SOC with $\widetilde{B}\sim J\sin(\phi/2)\ll fU$, and exhibit different squeezing behaviors outside this regime as single-particle spin dephasing can no longer be treated as a weak perturbation to the spin-locking interactions.

In realistic implementations, the Gaussian profile of the laser beams always introduces an additional effective harmonic potential that modifies the translational invariance assumed so far. We present a detailed discussion of the role of the harmonic trap in Appendix I, where we demonstrate that the addition of harmonic confinement barely modifies the achievable spin squeezing with currently accessible trapping frequencies. We find that the existence of single-particle localized modes in the lattice with harmonic confinementRey et al. (2005); Pupillo et al. (2006) helps to protect spin squeezing and shifts the optimal parameter window to $U/J\gtrsim 2$.

II.2 Two-axis twisting

The above scheme for OAT achieves optimal spin squeezing that scales as $\xi^{2}_{\text{opt}}\sim N^{-2/3}$ with minimal intervention, i.e. a standard Ramsey protocol. Further improvements upon this scheme can be made by introducing a time-dependent driving field that transforms the OAT Hamiltonian into a two-axis twisting (TAT) one. While the OAT model initially generates squeezing faster than the TAT model, the squeezing generation rate of OAT (measured in dB per second) falls off with time, while the squeezing generation rate for TAT remains approximately constant until reaching Heisenberg-limited amount of spin squeezing with $\xi^{2}_{\text{opt}}\sim N^{-1}$Kitagawa and Ueda (1993).

There are two general strategies for converting OAT into TAT: by use of either a pulsedLiu et al. (2011) or continuousHuang et al. (2015) drive protocol. For simplicity, we consider the latter in this work, although the pulsed protocol could provide additional advantages, as explained at the end of Appendix E. Following the prescription in Ref. [51], we use the clock laser to apply an amplitude-modulated drive $\hat{H}_{\text{drive}}(t)/\hbar=\Omega_{0}\cos(\omega t)\hat{S}_{\text{x}}$. If the modulation frequency $\omega$ satisfies $\omega\gg N\chi$ and $\mathcal{J}_{0}\left(2\Omega_{0}/\omega\right)=\pm 1/3$, where $\chi$ is the OAT squeezing strength in Eqn. (4) and $\mathcal{J}_{0}$ is the zero-order Bessel function of the first kind, then up to (i) an $\sim\hat{\bm{S}}\cdot\hat{\bm{S}}$ term that contributes only overall phase factors, and (ii) an $\sim\hat{S}_{\text{z}}$ term that can be eliminated with a simple dynamical decoupling pulse sequence (see Appendix D), the effective Hamiltonian becomes $\hat{H}_{\text{TAT}}^{(+)}/\hbar=(\chi/3)(\hat{S}_{\text{z}}^{2}-\hat{S}_{\text{x}}^{2})$ or $\hat{H}_{\text{TAT}}^{(-)}/\hbar=(\chi/3)(\hat{S}_{\text{y}}^{2}-\hat{S}_{\text{x}}^{2})$ (see Appendix C), which squeezes an initial state polarized along the $y$ or $z$ axis, respectively.

III Experimental implementation and practical considerations

Thus far, we have largely considered the general preparation of spin-squeezed states with the Fermi-Hubbard model. Here, we discuss the specific implementation of the above protocols in the state-of-the-art 3D 87Sr optical lattice clock (OLC). If successful, such an implementation would (to our knowledge) for the first time break through the proof-of-principle stage of spin squeezing efforts, and achieve a genuine metrological enhancement of a world-class quantum sensor.

As required for our protocol, 3D 87Sr OLC has demonstrated the capability to load a quantum degenerate gas into a 3D lattice at the “magic wavelength” ($\lambda_{\text{lattice}}=2a\approx 813$ nm) for which both the ground (${}^{1}S_{0},\downarrow$) and first excited (${}^{3}P_{0},\uparrow$) electronic states (i.e. the “clock states”) of the atoms experience the same optical potentialCampbell et al. (2017). Furthermore, the 3D 87Sr OLC currently operates at sufficiently low temperatures to ensure vanishing population above the lowest Bloch band, such that its dynamics are governed by the Fermi-Hubbard Hamiltonian [Eqn. (2)]Esslinger (2010).

An external clock laser with wavelength $\lambda_{\text{L}}\approx 698$ nm resonantly interrogates the ${}^{1}S_{0}$ and ${}^{3}P_{0}$ states of the atoms and generates spin-orbit coupling (SOC)Wall et al. (2016). While the relative wavelengths of the lattice and clock lasers do not allow for weak SOC along all three lattice axes, weak SOC along two axes can be implemented by, for example, (i) fixing a large lattice depth along the $z$ axis, effectively freezing atomic motion along $z$, and then (ii) making the clock laser nearly collinear with the $z$ axis, with only a small projection of its wavenumber $\bm{k}_{\text{L}}$ onto the $x$-$y$ plane [see Figure 1(a)]. The entire 3D OLC then factorizes into an array of independent 2D systems with $N=f\ell^{2}$ atoms each, where $\ell$ is the number of lattice sites along each axis of the lattice. As in the 1D case, atoms within the 2D system experience all-to-all interactions, as well as spin-orbit coupling along two directions characterized by SOC angles $\phi_{\text{x},\text{y}}=k_{\text{L}}^{\text{x},\text{y}}a$. Generally speaking, higher-dimensional systems (e.g. 2D vs. 1D) are more desirable because they allow packing more interacting atoms into a fixed system volume, thereby increasing the maximally attainable amount of spin squeezing.

Figure 3 shows, for both OAT and TAT protocols, the maximally attainable amount of spin squeezing and the shortest time at which it occurs as a function of the lattice depth $V_{0}$ and linear lattice size $\ell$ in a single half-filled 2D layer (i.e. $f=1,N=\ell^{2}$) of the 3D OLC. Atoms are confined along the direction transverse to the 2D layer by a lattice of depth 60 $E_{\text{R}}$, where $E_{\text{R}}$ is the atomic lattice recoil energy. The maximally attainable amount of spin squeezing by each protocol in Fig. 3 depends only on the atom number $N$, while the shortest attainable time is determined by choosing the largest SOC angles $\phi_{\text{x}}=\phi_{\text{y}}\equiv\phi$ which saturate $\widetilde{B}/U\approx 0.05$. We impose this constraint on $\widetilde{B}/U$ to ensure validity of the OAT Hamiltonian perturbatively derived in Appendix A (see also Appendix B).

Currently, light scattering from the lattice beams induces decoherence of the clock on a time scale of $\sim 10$ secondsGoban et al. (2018); Hutson et al. (2019), which is much shorter than the natural ${}^{3}P_{0}$ lifetime of $\sim 160$ seconds (see Appendix F). This limitation imposes significant constraints on achievable spin squeezing, as shown in Figure 4 where the maximal squeezing with spin decay in the OAT case was determined using exact expressions for spin correlators derived in Ref. [52], while in the TAT case these correlators were determined by solving Heisenberg equations of motion for collective spin operatorsPerlin and Rey (2019) (see Appendix H). Due to the fast growth of Heisenberg operators in systems with all-to-all interactions, the latter method is not always capable of simulating up to the optimal squeezing time, and thus only provides a lower bound on the maximal squeezing theoretically obtainable via TAT.

The results in Fig. 4 show that squeezing via OAT saturates with system size around $N\approx 10^{3}$ ($\ell\approx 30$), while TAT allows for continued squeezing gains through $N=10^{4}$ ($\ell=100$). Even with decoherence, our protocol may realistically generate $\sim 10$–$14$ dB of spin squeezing in $\sim 1$ second with $\sim 10^{2}$–$10^{4}$ atoms in a 2D section of the lattice, which is compatible with the atom numbers and interrogation times of state-of-the-art optical lattice clocksCampbell et al. (2017); Marti et al. (2018). This amount of spin squeezing exceeds those reported in the ground-state nuclear spin sublevels of a state-of-the-art 171Yb OLC ($\sim 6.5$ dB)Braverman et al. (2019). While the latter protocol might be used to transfer spin squeezing to the electronic clock state, to date there has been no demonstration of spin squeezing in an optical clock transition.

In addition to light scattering, $p$-wave losses from inelastic ${}^{3}P_{0}$ collisionsMartin et al. (2013); Zhang et al. (2014); Bishof et al. (2011) can also degrade the maximum achievable spin squeezing, which becomes more pronounced for shallower lattices. More details on $p$-wave losses are discussed in Appendix J, where we show that operating at lattice depths $V_{0}\gtrsim 7E_{\text{R}}$ may be necessary to suppress the impact of inelastic collisions on spin squeezing, at the cost of slightly increasing light scattering.

The sources of decoherence considered above are not fundamental, and can be avoided by, for instance, using two nuclear spin levels as spin-1/2 degrees of freedom that are interrogated by far-detuned Raman transitions instead of a direct optical transitionMancini et al. (2015). The strength of SOC for Raman transitions is tunable and, moreover, the lifetimes of ground nuclear spin levels are longer than 100 seconds in the latticeGoban et al. (2018). In this case, our protocol for preparing a squeezed state would additionally end with a coherent state transfer from nuclear to electronic degrees of freedom to retain metrological utility for the atomic clock. If, for example, the $-9/2$ and $-7/2$ nuclear spin states are used for the preparation of a squeezed state, then the collective-spin entanglement of atoms can be transferred to electronic states at the end of the spin squeezing protocol with a $\sigma^{-}$ polarized $\pi$ pulse. Such a pulse can transfer $\Ket{g,-7/2}$ to $\Ket{e,-9/2}$ without affecting $\Ket{g,-9/2}$, where $g$ and $e$ respectively denote the ground and excited (electronic) clock states.

IV Conclusions

We have proposed a new protocol to generate spin squeezing in a fermionic 3D optical lattice clock by combining nominally undesirable atomic collisions with spin-orbit coupling. To our knowledge, this is the first proposal to use quantum correlations in a many-body fermionic system to push state-of-the-art quantum sensors beyond the independent-particle regime, thereby achieving a genuine quantum advantage. Such capability could allow for major improvements in clock sensitivity and bandwidth, enhancing not only traditional timekeeping applications such as measurement standards, navigation (GPS), and telecommunications, but also geodesy and gravitational wave detection, precision tests of fundamental physics, and the search for new physics beyond the standard modelSafronova et al. (2018).

V Acknowledgments

We acknowledge helpful discussions with M. Norcia, C. Sanner, and M. Mamaev. This work is supported by the Air Force Office of Scientific Research (AFOSR) grant FA9550-18-1-0319; the AFOSR Multidisciplinary University Research Initiative (MURI) grant; the Defense Advanced Research Projects Agency (DARPA) and Army Research Office (ARO) grant W911NF-16-1-0576; the National Science Foundation (NSF) grant PHY-1820885; JILA-NSF grant PFC-1734006; and the National Institute of Standards and Technology (NIST).

Appendix A Derivation of the effective one-axis-twisting model

Suppose we have a Hamiltonian of the form ($\hbar=1$)

[TABLE]

with

[TABLE]

and we consider $N$-particle states initially in the ground-state manifold $\mathcal{G}_{0}$ of $H_{0}$, which have total spin $S=N/2$. If the largest eigenvalue of $V$ is smaller in magnitude than half of the collective spin gap $NU/L=fU$, i.e. the energy gap under $H_{0}$ between $\mathcal{G}_{0}$ and its orthogonal complement $\mathcal{E}_{0}$, then we can formally develop a perturbative treatment for the action of $V$ on $\mathcal{G}_{0}$. Such a treatment yields an effective Hamiltonian on $\mathcal{G}_{0}$ of the form $H_{\text{eff}}=\sum_{p}H_{\text{eff}}^{(p)}$, where $H_{\text{eff}}^{(p)}$ is order $p$ in $V$. Letting $\mathcal{P}_{0}$ ($\mathcal{Q}_{0}$) be a projector onto $\mathcal{G}_{0}$ ($\mathcal{E}_{0}$), we define the super-operators $\mathcal{O}$ and $\mathcal{L}$ by

[TABLE]

where $H_{0}=\sum_{\alpha}E_{\alpha}\outerproduct{\alpha}{\alpha}$. The first few terms in the expansion of the effective Hamiltonian $H_{\text{eff}}$ are then, as derived in Ref. [57],

[TABLE]

and

[TABLE]

with $\left[X,Y\right]_{-}\equiv XY-YX$. The zero-order effective Hamiltonian $H_{\text{eff}}^{(0)}=H_{0}$ within the ground-state manifold. To calculate $H_{\text{eff}}^{(1)}$, we note that the ground-state manifold $\mathcal{G}_{0}$ is spanned by the Dicke states

[TABLE]

in terms of which we can expand the collective spin-z operator as $S_{\text{z}}=\sum_{m}m\outerproduct{m}{m}$. We can likewise expand the collective spin-x operator $S_{\text{x}}$ in terms of $x$-oriented Dicke states $\Ket{m_{\text{x}}}$ as $S_{\text{x}}=\sum_{m}m\outerproduct{m_{\text{x}}}{m_{\text{x}}}$. The ground-state projector $\mathcal{P}_{0}$ onto $\mathcal{G}_{0}$ can be expanded in either basis as $\mathcal{P}_{0}=\sum_{m}\outerproduct{m}{m}=\sum_{m}\outerproduct{m_{\text{x}}}{m_{\text{x}}}$. Defining the mean and residual fields

[TABLE]

we can then write

[TABLE]

and in turn

[TABLE]

where we used the fact that $\mathcal{P}_{0}S_{j=\text{z},\text{x}}\mathcal{P}_{0}=S_{j}$ within the ground-state manifold. By construction, the residual fields are mean-zero, i.e. $\sum_{n}b_{n}=0$. Using the particle-exchange symmetry of the Dicke states, we can therefore expand

[TABLE]

which implies

[TABLE]

To calculate the second-order effective Hamiltonian $H_{\text{eff}}^{(2)}$, we let $\mathcal{B}_{0}\left(\mathcal{E}_{0}\right)$ denote an eigenbasis of $H_{0}$ for the excited subspace $\mathcal{E}_{0}$, and set the ground-state energy to 0. We then define the operator

[TABLE]

which sums over projections onto excited states with corresponding energetic suppression factors, in terms of which we can write

[TABLE]

which is simply an operator-level version of the textbook expression for second-order perturbation theory. The only part of $V$ which is off-diagonal with respect to the ground- and excited-state manifolds $\mathcal{G}_{0}$ and $\mathcal{E}_{0}$ is $-\sum_{n}b_{n}s_{\text{z}}^{(n)}$, and the individual spin operators in this remainder can only change the total spin $S$ by at most 1. It is therefore sufficient to expand $\mathcal{I}$ in a basis for states which span the image of $\mathcal{G}_{0}$ under all $s_{\text{z}}^{(n)}$ within the $S=N/2-1$ manifold. Such a basis is provided by the spin-wave states

[TABLE]

for $k=1,2,\cdots,N-1$Swallows et al. (2011). Using the fact that all spin-$z$ operators preserve the projection of total spin onto the $z$ axis, we then have that

[TABLE]

where the relevant matrix elements between the Dicke states and the spin-wave states areSwallows et al. (2011)

[TABLE]

which implies

[TABLE]

Using the fact that $\sum_{n}b_{n}=0$, we can expand

[TABLE]

where the sum over $k$ vanishes for $n\neq n^{\prime}$ and equals $N$ when $n=n^{\prime}$, so

[TABLE]

where

[TABLE]

We therefore have that

[TABLE]

where the $\left(N/2\right)^{2}$ term contributes a global energy shift which we can neglect, while the $m^{2}$ term is proportional to $m^{2}\outerproduct{m}{m}=S_{\text{z}}^{2}$. In total, the effective Hamiltonian through second order in perturbation theory is thus

[TABLE]

with

[TABLE]

We benchmark the validity of this effective Hamiltonian via exact simulations of the spin [Eqn. (3)] and OAT [Eqn. (4)] Hamiltonians in a system of 20 spins, finding that the relative error in maximal squeezing (in dB) of the OAT model is less than 3% when $\widetilde{B}/U<0.06$ (see Appendix B).

Appendix B Numerical benchmarking of the OAT model

Here we provide additional information about our benchmarking of the one-axis twisting model against the spin model. This benchmarking was performed via exact simulations of a 20-spin system. Fig. B1 shows the relative error in maximal squeezing of the OAT model (measured against the spin model) as a function of the reduced field variance $\widetilde{B}/U$. Here squeezing is measured in decibels (dB) by $-10\log_{10}\xi^{2}$ for the squeezing parameter $\xi^{2}$ define in Eqn. (5). The relative error in maximal squeezing (in dB) by the OAT model is less than 3% when $\widetilde{B}/U<0.06$.

In principle, spin-changing decoherence compromises the validity of the OAT model, as its perturbative derivation in Appendix A relies on spin population remaining primarily within the Dicke manifold. This assumption breaks down in the presence of, for example, spontaneous emission, which transfers population outside of the Dicke manifold. Nonetheless, we find decent agreement between the OAT and spin models when decoherence is sufficiently weak (see Fig. B2).

Appendix C Two-axis twisting, decoherence, and the residual axial field

The protocol we use to transform one-axis twisting (OAT) into two-axis twisting (TAT) is as previously proposed in Ref. [51]; we provide a summary of this protocol here, in addition to some brief discussion of its implications for decoherence and the residual $\sim\bm{S}\bm{\cdot}\bm{S}$ and $\sim S_{\text{z}}$ terms of our OAT protocol. The TAT protocol begins with the OAT Hamiltonian with a time-dependent transverse field,

[TABLE]

where $\beta$ is the modulation index of the driving field and the drive frequency $\omega\gg N\chi$, with $N$ the total number of spins. Moving into the rotating frame of $\Omega\left(t\right)S_{\text{x}}$ subtracts this term from the Hamiltonian, and transforms operators $\mathcal{O}$ as

[TABLE]

where

[TABLE]

In particular, the operators $\tilde{S}_{\pm}\equiv-S_{\text{z}}\pm iS_{\text{y}}$ (i.e. the raising and lowering operators in the $x$ basis) transform simply as

[TABLE]

For any operator $\mathcal{O}$ and drive frequency $\omega\gg\norm{\mathcal{O}}$, where $\norm{\mathcal{O}}\equiv\max_{\psi}\sqrt{\braket{\psi}{\mathcal{O}^{\dagger}\mathcal{O}}{\psi}}$ is the operator norm of $\mathcal{O}$ (i.e. the magnitude of the largest eigenvalue of $\mathcal{O}$), we can generally make a secular approximation to say

[TABLE]

where $\mathcal{J}_{n}$ is the $n$-th order Bessel function of the first kind. Expanding $S_{\text{z}}^{2}=\dfrac{1}{4}\left(\tilde{S}_{+}+\tilde{S}_{-}\right)^{2}$, one can thus work out that the effective Hamiltonian in the rotating frame of the drive is

[TABLE]

Driving with a modulation index $\beta$ for which $J_{0}\left(2\beta\right)=\pm 1/3$ then gives us the effective two-axis twisting Hamiltonians

[TABLE]

where $\simeq$ denotes equality up to the addition of a term proportional to $\bm{S}^{2}=S_{\text{z}}^{2}+S_{\text{x}}^{2}+S_{\text{y}}^{2}$, which is irrelevant in the absence of coherent coupling between states with different net spin. In a similar spirit, one can work out that single-spin operators transverse to the $x$-axis transform as

[TABLE]

which implies that shifting into the rotating frame of the time-dependent drive takes

[TABLE]

and

[TABLE]

As the TAT Hamiltonians $H_{\text{eff}}^{(\pm)}$ are realized in a rotating frame, to properly account for decoherence throughout the TAT protocol one must transform jump operators according to Eqns. (39)-(40).

In practice, our protocols realize the OAT Hamiltonian in Eqn. (30) with additional $\sim\bm{S}\cdot\bm{S}$ and $\sim S_{\text{z}}$ terms [see Eqn. (28)]. The effect of the $\sim\bm{S}\cdot\bm{S}$ term is to generate a relative phase between states with different total spin $S$ (with e.g. $S=N/2$ within the Dicke manifold). In the absence of coherent coupling between states with different total spin, therefore, the $\sim\bm{S}\cdot\bm{S}$ term has no effect on system dynamics. The $\sim S_{z}$ term, meanwhile, is important; the magnitude of this term (as measured by the operator norm) is generally comparable to that of the squeezing term $\chi S_{\text{z}}^{2}$. Unlike in the case of OAT, $S_{\text{z}}$ does not commute with the TAT Hamiltonians, so its effects cannot be eliminated by a single spin-echo $\pi$-pulse $\exp\left(-i\pi S_{\text{x}}\right)$ half way through the squeezing protocol. Nonetheless, we find that for $N=10^{2}$ ($10^{3}$) atoms, $\sim 5$ (10) $\pi$-pulses in a CPMG (Carr-Purcell-Meiboom-Gill) sequenceCarr and Purcell (1954); Meiboom and Gill (1958) suffice to mitigate the effects of the $S_{\text{z}}$ term in the TAT protocol (see Appendix D). Phase control over these pulses, specifically choices of whether to apply $\exp\left(\pm i\pi S_{\text{x}}\right)$ or $\exp\left(\pm i\pi S_{\text{y}}\right)$ in any given $\pi$-pulse, can be used to construct XY-$n$ pulse sequencesMaudsley (1986); Gullion et al. (1990) that are robust to pulse errors.

Appendix D Dynamical decoupling in the TAT protocol

The effective Hamiltonian resulting from a perturbative treatment of SOC is (see Appendix A)

[TABLE]

where $U$ is a two-atom on-site interaction strength; $L$ is the number of lattice sites; $\overline{B}\equiv\sum_{n}B_{n}/N$ is a residual axial field determined by the occupied quai-momentum modes $\left\{n\right\}$ (with $\absolutevalue{\left\{n\right\}}=N$ atoms total); $\Omega$ is the magnitude of a driving field; and $\chi$ is an effective OAT squeezing strength. The effect of the $\sim\bm{S}\cdot\bm{S}$ term is to generate a relative phase between states with different total spin $S$ (where $S=N/2$ within the Dicke manifold). In the absence of coherent coupling between states with different total spin, therefore, the $\sim\bm{S}\cdot\bm{S}$ term has no effect on system dynamics, and we are safe to neglect it entirely.

In the parameter regimes relevant to our discussions in the main text, the operator norms of $\overline{B}\hat{S}_{\text{z}}$ and $\chi\hat{S}_{\text{z}}^{2}$ in Eqn. (41) will typically be comparable in magnitude. The OAT protocol sets $\Omega=0$, and eliminates the effect of $\overline{B}\hat{S}_{\text{z}}$ with a spin-echo $\pi$-pulse $\exp\left(-i\pi\hat{S}_{\text{x}}\right)$ applied half way through the squeezing protocol. The TAT protocol, meanwhile, effectively takes $\chi\hat{S}_{\text{z}}^{2}+\Omega\hat{S}_{\text{x}}\to\hat{H}_{\text{TAT}}^{(\pm)}$ (as defined in Appendix C) and $\overline{B}\hat{S}_{\text{z}}\to\mathcal{J}_{0}\left(\beta_{\pm}\right)\overline{B}\hat{S}_{\text{z}}$, where $\mathcal{J}_{0}$ is the zero-order Bessel function of the first kind and $\beta_{\pm}$ is the modulation index of the amplitude-modulated driving field $\Omega$, satisfying $\mathcal{J}_{0}\left(2\beta_{\pm}\right)=\pm 1/3$. Unlike in the case of OAT, $\hat{S}_{\text{z}}$ does not commute with the TAT Hamiltonian, so its effect cannot be eliminated with a spin-echo. Nonetheless, this term can be eliminated with a dynamical decoupling pulse sequence that periodically inverts the sign of $\hat{S}_{\text{z}}$ while preserving $\hat{H}_{\text{TAT}}^{(\pm)}$.

Fig. D1 shows the maximal squeezing generated by $N=10^{2}$ and $10^{3}$ atoms via OAT, TAT, and TAT in the presence of the mean field $\mathcal{J}_{0}\left(\beta_{\pm}\right)\overline{B}\hat{S}_{\text{z}}$ as a function of the number of $\pi$-pulses performed prior to the optimal TAT squeezing time. These pulses are applied in a CPMG sequence $\left(\tau_{n}/2-\pi_{\text{x}}-\tau_{n}/2\right)^{n}$, where $\tau_{n}/2$ denotes Hamiltonian evolution for a time $\tau_{n}/2$, $\pi_{\text{x}}$ denotes the application of an instantaneous $\pi$-pulse $\exp\left(-i\pi\hat{S}_{\text{x}}\right)$, and $n$ is the number of pulses, such that the optimal TAT squeezing time is $t_{\text{opt}}^{\text{TAT}}=\left(\tau_{n}\right)^{n}$. The label TAT${}_{\pm,\text{z}}$ in Fig. D1 denotes squeezing through the Hamiltonian $\hat{H}_{\text{TAT}}^{(\pm,\text{z})}\equiv\hat{H}_{\text{TAT}}^{(\pm)}-\mathcal{J}_{0}\left(\beta_{\pm}\right)\Braket{\overline{B}}^{\text{rms}}_{f}\hat{S}_{\text{z}}$, where $\Braket{\overline{B}}^{\text{rms}}_{f}$ is the root-mean-square average of $\overline{B}$ over choices of occupied spacial modes $\left\{n\right\}$ at fixed filling $f$ of all spatial modes in the lowest Bloch band of a periodic 2D lattice. While the modulation index $\beta_{+}$ is uniquely defined by $\mathcal{J}_{0}\left(2\beta_{+}\right)=1/3$, there are two choices of $\beta_{-}$ for which $\mathcal{J}_{0}\left(2\beta_{-}\right)=-1/3$; we use that which minimizes $\absolutevalue{\mathcal{J}_{0}\left(\beta_{-}\right)}$. Fig. D1 assumes an SOC angle $\phi=\pi/50$ (although results are independent of $\phi$ for $\phi\ll 1$), a reduced field variance $\widetilde{B}/U=0.05$, and a filling $f=5/6$. Note that as the filling $f\to 1$, the residual axial field vanishes ($\overline{B}\to 0$), so TAT${}_{\pm,\text{z}}\to$ TAT.

Appendix E Clock interrogation after squeezing

The protocols in our work concern the preparation of spin-squeezed states in an optical lattice clock. Here, we discuss the use of these states in a follow-up clock interrogation protocol. For simplicity, we restrict our discussion to the case of squeezing in 1D, as in Section II of the main text, with the understanding that a generalization of this discussion to higher dimensions is straightforward.

A spin-squeezed state is generated by interactions and SOC that are generally undesirable during the clock interrogation protocol. In the parameter regimes considered in our work, interactions alone have no effect on clock interrogation: absent of coherence between states with different net spin $S$, collective $\bm{S}\cdot\bm{S}$ interactions only generate unobservable global phases within each fixed-$S$ sector of Hilbert space. Therefore, the remaining task to allow clock interrogation after spin squeezing is to turn off SOC, which inhomogeneously detunes atomic transition frequencies by an amount $B_{q}$ that depends on the quasi-momentum $q$ of an atom. The SOC-induced axial fields $B_{q}\sim J\sin(\phi/2)$ depend on two tunable parameters: the tunneling rate $J$ and the SOC angle $\phi$. The simplest way to turn off SOC is thus to increase the lattice depth prior to clock interrogation, taking $J\sim B_{q}\to 0$. Increasing the lattice depth to turn off SOC is compatible with the current clock interrogation sequence, but is incompatible with ongoing efforts to mitigate light scattering from the lattice beams (currently the primary source of decoherence in the clock; see Appendix F) by using shallower latticesHutson et al. (2019). We thus devote the rest of this section to discussing strategies for turning off SOC that are compatible with using the same lattice depth for clock interrogation as the spin squeezing generation.

If we cannot take the tunneling rate $J\to 0$, our remaining control parameter for turning off SOC is the SOC angle $\phi=k_{\text{L}}a$, where $k_{\text{L}}$ is the projection of the clock laser wavenumber onto the lattice axis and $a$ is the lattice spacing. The squeezing protocol needs a clock laser with a small but nonzero SOC angle $\phi\ll 1$, while the clock interrogation protocol requires a clock laser with $\phi=0$. Simply using one clock laser with $\phi\neq 0$ for the squeezing protocol and another clock laser with $\phi=0$ for the clock interrogation protocol, however, does not resolve this discrepancy, because a state that is squeezed with respect to spin operators that are homogeneous (i.e. of the form $S_{\text{x}},S_{\text{y}},S_{\text{z}}$) in a particular gauge is not necessarily squeezed with respect to spin operators that are homogeneous in a different gauge. In this appendix, we will work explicitly in the “lab gauge” of the clock interrogation protocol, in which the Fermi-Hubbard Hamiltonian is SOC-free and the $\phi=0$ clock laser is homogeneous. To resolve the fact that our squeezing protocol prepares a state that is squeezed in the “wrong gauge”, we will construct a simple pulse sequence that transforms the inhomogeneous spin operators accessible by the $\phi\neq 0$ laser into a homogeneous form in our lab gauge.

Starting with a spin-down-polarized initial state

[TABLE]

our OAT protocol prepares the state

[TABLE]

where $H_{\text{FH}}^{(0)}$ is the Fermi-Hubbard Hamiltonian in Eqn. (2) without SOC; $t$ is some free evolution time; and $S_{\text{x}}^{(\theta)}$ is the “rotated” spin-$x$-like generator induced by a clock laser with SOC angle $\phi=\theta$, namely

[TABLE]

Defining on-site spin operators (in the lab gauge)

[TABLE]

we can identify the rotated collective spin operators

[TABLE]

The state $\Ket{\xi_{\text{OAT}}^{(\theta)}}$ is squeezed with respect to components of the rotated collective spin vector $\bm{S}_{\theta}\equiv\left(S_{\text{x}}^{(\theta)},S_{\text{y}}^{(\theta)},S_{\text{z}}\right)$. Therefore, to take advantage of the squeezing in $\Ket{\xi_{\text{OAT}}^{(\theta)}}$, the clock interrogation protocol effectively needs to rotate this state by some unitary $\exp\left(-i\bm{\eta}\cdot\bm{S}_{\theta}\right)$, and then extract information about the rotation vector $\bm{\eta}$ from collective spin observables of the form

[TABLE]

where $\mathcal{O}_{\theta}$ is some product of the rotated collective spin operators in $\bm{S}_{\theta}$, e.g. $S_{\text{x}}^{(\theta)}$ or $S_{\text{x}}^{(\theta)}S_{\text{y}}^{(\theta)}$. In order to turn off SOC during clock interrogation, however, we are restricted to performing rotations of the form $\exp\left(-i\bm{\eta}\cdot\bm{S}_{0}\right)$ and measuring homogeneous operators $\mathcal{O}_{0}$. We thus seek a “gauge-switching” operation $G_{\theta}$ that maps homogeneous operators $\mathcal{O}_{0}$ onto rotated operators $\mathcal{O}_{\theta}$ via $G_{\theta}^{\dagger}\mathcal{O}_{0}G_{\theta}=\mathcal{O}_{\theta}$. Equipped with $G_{\theta}$, we could decompose

[TABLE]

for a transformed state

[TABLE]

that is now squeezed with respect to the homogeneous collective spin operators in $\bm{S}_{0}$, accessible with the SOC-free ($\phi=0$) clock laser used during clock interrogation.

Given the definitions of the rotated spin operators $S_{\text{x}}^{(\theta)},S_{\text{x}}^{(\theta)}$ in Eqns. (48), (49), a suitable candidate for the gauge-switching operator $G_{\theta}$ is the site-dependent rotation

[TABLE]

To implement $G_{\theta}$ with “global” (i.e. non-site-selective) experimental controls, we decompose each local rotation into a product of two reflections:

[TABLE]

where $\simeq$ denotes equality up to an overall phase, and $s_{\alpha}^{(j)}\equiv\cos\alpha s_{\text{x}}^{(j)}+\sin\alpha s_{\text{y}}^{(j)}$. This decomposition implies

[TABLE]

which can be implemented using one SOC-free ($\phi=0$) clock laser, and one clock laser with SOC angle $\phi=\theta/2$. Appending the two $\pi$-pulses given by Eqn. (55) to our squeezing protocol thus prepares a state $\Ket{\tilde{\xi}_{\text{OAT}}^{(\theta)}}$ that is squeezed with respect to the homogeneous collective spin operators $S_{\text{x}},S_{\text{y}},S_{\text{z}}$ accessible to the SOC-free ($\phi=0$) clock laser used during clock interrogation.

As presented, the combined squeezing and clock interrogation protocols require three clock lasers in total: one without SOC ($\phi=0$), and one each for SOC angles $\phi\in\left\{\theta/2,\theta\right\}$. We can use $G_{\theta}$, however, to decompose any pulse $\exp\left(-i\bm{\beta}\cdot\bm{S}_{\theta}\right)$ into composite pulses that use only $\bm{S}_{0}$ and $\bm{S}_{\theta/2}$:

[TABLE]

The state prepared by the OAT squeezing protocol, for example, can be equivalently prepared via

[TABLE]

Spin-echo pulses applied throughout OAT can likewise be decomposed according to (56), eliminating the need for a clock laser with SOC angle $\phi=\theta$. Applying a continuous drive during a squeezing protocol, however, still requires all three clock lasers. If carefully tuning the relative orientations of three clock lasers proves to be too difficult in practice, converting OAT into TAT would therefore need to be done with a pulsed drive protocol, as in Ref. [50].

Appendix F Decoherence in the 3D 87Sr optical lattice clock

Currently, light scattering from lattice beams in the 3D 87Sr optical lattice clock induces decoherence on a time scale of $\sim$10 secondsGoban et al. (2018); Hutson et al. (2019). This decoherence acts identically on all atoms in an uncorrelated manner, and can be understood by considering the density operator $\rho$ for a single atom, with effective spin states $\downarrow$ and $\uparrow$ respectively corresponding to the ${}^{1}\text{S}_{0}$ and ${}^{3}\text{P}_{0}$ electronic states. Empirically, the effect of decoherence after a time $t$ within the $\left\{\downarrow,\uparrow\right\}$ subspace of a single atom is to take $\rho\to\rho\left(t\right)$ with $\rho\left(0\right)\equiv\rho$ and

[TABLE]

where $\Gamma_{\uparrow\uparrow}\approx\Gamma_{\uparrow\downarrow}\approx\Gamma=0.1~{}\text{sec}^{-1}$ are respectively decay rates for $\rho_{\uparrow\uparrow}$ and $\rho_{\uparrow\downarrow}$. This form of decoherence can be effectively modeled by decay and dephasing of individual spins (respectively denoted $\Gamma_{\text{ud}}$ and $\Gamma_{\text{el}}$ in Ref. [52]) at rates $\Gamma$. In the language of the section that follows, we would say that this decoherence is captured by the sets of jump operators $\mathcal{J}_{-}\equiv\left\{s_{-}^{(j)}\right\}$ and $\mathcal{J}_{\text{z}}\equiv\left\{s_{\text{z}}^{(j)}\right\}$ with corresponding decoherence rates $\gamma_{-}=\gamma_{\text{z}}=\Gamma$.

Appendix G Time-series of squeezing via OAT and TAT

Figure G1 shows an example of squeezing over time via OAT and TAT, both with and without decoherence via decay and dephasing of individual spins. The OAT model initially generates squeezing faster than the TAT model, but the squeezing generation rate of OAT (measured in dB per second) falls off with time. The squeezing generation rate for TAT, meanwhile, remains approximately constant (in the absence of decoherence) until squeezing via TAT surpasses that of OAT. In the absence of decoherence, OAT achieves a maximal amount of squeezing that scales as $\xi^{2}\sim N^{-2/3}$, while TAT achieves Heisenberg-limited squeezing with $\xi^{2}\sim N^{-1}$. Note that our method for computing squeezing via TAT in the presence of decoherence (described in Appendix H) is not capable of computing squeezing for the full range of times shown in Fig. G1; the corresponding time-series data in this figure is therefore shown up to the point at which this method breaks down.

Appendix H Solving Heisenberg equations of motion for collective spin systems

In order to compute squeezing of a collective spin system, we need to compute expectation values of (homogeneous) collective spin operators. We compute these expectation values using a method recently developed in Ref. [53], and provide a short overview of the method here. Choosing the basis $\left\{\mathcal{S}_{\bm{m}}\right\}$ for all collective spin operators, where $\mathcal{S}_{\bm{m}}\equiv S_{+}^{m_{+}}S_{\text{z}}^{m_{\text{z}}}S_{-}^{m_{-}}$ with $\bm{m}\equiv\left(m_{+},m_{\text{z}},m_{-}\right)\in\mathbb{N}_{0}^{3}$, we can expand all collective spin Hamiltonians in the form

[TABLE]

The evolution of a general correlator $\Braket{\mathcal{S}_{\bm{n}}}$ under a Hamiltonian of the form in Eqn. (59) is then given by

[TABLE]

where $\left[X,Y\right]_{\pm}\equiv XY\pm YX$; $\mathcal{J}$ is a set of jump operators with corresponding decoherence rate $\gamma_{\mathcal{J}}$; the decoherence operator $\mathcal{D}$ is defined by

[TABLE]

and $T_{\bm{m}\bm{n}}$ is a matrix element of the time derivative operator $T\equiv d/dt$. These matrix elements can be calculated analytically using product and commutation rules for collective spin operators. We can then expand correlators in a Taylor series about $t=0$ to write

[TABLE]

where $T_{\bm{m}\bm{n};k}\equiv\left[T^{k}\right]_{\bm{m}\bm{n}}$ are matrix elements of the $k$-th time derivative. Expectation values of collective spin operators can thus be computed via the expansion in Eqn. (62), which at short times can be truncated at some finite order beyond which all terms have negligible contribution to $\Braket{\mathcal{S}_{\bm{n}}}$.

Appendix I Effect of a harmonic confining trap

Current 3D optical lattice implementations involve a harmonic confining potential, which significantly alters the underlying single-particle eigenstructure and can potentially degrade accessible squeezing within our protocol. In this appendix, we examine the effect of a harmonic trap on our protocol and discuss strategies to mitigate undesired effects. We model the trap by the addition of the term

[TABLE]

to our Fermi-Hubbard model (Eqn. (2)), where $j_{0}$ denotes the trap center and $\Omega=m(\omega a)^{2}/2\hbar$ characterizes the trap strength for atom mass $m$, trap frequency $\omega$, and lattice spacing $a$. In current state-of-the-art 3D 87Sr OLC implementations, values of $\omega\approx 56\times 2\pi~{}\text{sec}^{-1}$ can be achieved within each 2D layer of weak SOC by utilizing in-plane lattice depths of $5E_{R}$ and a lattice depth of $60E_{R}$ in the axial direction, resulting in a value of $\Omega/J\approx 0.01$. We restrict our discussion to 1D, although for a separable 3D lattice our arguments should extend in a straightforward manner.

We briefly review the structure of the single-particle eigenstates of the system, before discussing the effects on squeezing. In the quasi-momentum basis, the eigenfunctions $\psi_{n,\alpha}(q)=\braket{q}{n,\alpha}$ are given by the $\pi$-periodic Mathieu functions, with the corresponding energies described by the Mathieu characteristic valuesRey et al. (2005). In the presence of SOC, using the gauge transformation described in the main text, we obtain the relation

[TABLE]

In contrast to the case of a pure harmonic potential, which generically has spatially delocalized single-particle eigenstates, the addition of a tight-binding lattice causes eigenmodes with quantum number $n$ (index $n$ labels the eigenvalues in order of increasing energy) larger than $n_{c}\approx 2\sqrt{2J/\Omega}$ to become localized at corresponding lattice sites. Therefore the sites at a distance $n_{c}/2$ from the trap center with potential energy $2\hbar J$ define the boundary between the delocalized modes at the trap center and the high-energy localized trap edges. Tunneling in the region of localized modes is typically suppressed by large potential energy differences even in the presence of SOC. These modes are thus largely decoupled and do not contribute to the trap center dynamics. On the other hand the delocalized modes may be approximated by those of a quantum harmonic oscillator with effective mass $m^{*}=\hbar/(2Ja^{2})$ and frequency $\omega^{*}=\sqrt{4J\Omega}$.

As emphasized in the main text, the key requirements for our protocol are 1) the validity of the spin model, which depends on the pinning of particles in their initial single particle modes, and 2) the gap protection against SOC dephasing, which arises from collective spin interactions. Concerning the latter point, it is desirable to maintain a weak trap so as to enable a large number of delocalized modes in the trap center, which are the only type capable of contributing to the generation of squeezing. Though the interactions between these modes are not strictly all-to-all, they remain long-ranged, and can thus still lead to a spin-locking effect and a protective gapRey et al. (2014); Smale et al. (2019). For $\Omega/J=0.01$ we have $n_{c}=28$, enabling $\sim 10^{3}$ contributing modes in each 2D layer of our system. Concerning the validity of the spin model, from a single-particle perspective the eigenmodes of our $\uparrow$ states will be initially displaced in quasi-momentum space from equilibrium by $\phi/a$ as per Eqn. (64), and will generally undergo dipole oscillations and not remain strictly pinned to their initial modes. However, as long as we ensure the displacement is small enough to guarantee a constant density distribution across the trap center, the spin model will remain valid. The localized modes at the trap edges can actually help to satisfy this condition, since they can serve as a barrier against motion. This is demonstrated in Fig. I1 where we show that filling all delocalized modes guarantees that the trap center maintains a constant density; we characterize this by the time-averaged fluctuations of the $\uparrow$ density at each site $j$ about its initial value following the Ramsey pulse,

[TABLE]

choosing sufficiently large evolution times to ensure convergence.

In the presence of interactions, an additional point of concern is that the interplay between the trap and interactions may induce resonances that enable the formation of a significant doublon population,

[TABLE]

which in turn may alter the density distribution and invalidate the spin model. Since doublon formation in the localized edges will not have consequences for our squeezing protocol, we must only ensure that doublons are not formed in the trap center, which requires $U>\Omega(n_{c}/2)^{2}=2J$Pupillo et al. (2006). In Fig. I2, we perform exact simulations to assess the effect of the trap on our system. Though restricted to small system sizes, the results demonstrate that for $U/J\lesssim 2$, the trap will always lead to a decrease of squeezing due to the formation of doublons in the trap center, while for $U/J\gtrsim 2$, we are protected from this process even for trap strengths much stronger than the experimentally relevant ones.

Appendix J Accounting for $p$-wave inelastic collisions

Inelastic ${}^{3}\text{P}_{0}$ (electronic state $e$ or $\uparrow$) collisions are detrimental for optical lattice clocks. For the nuclear-spin-polarized gas discussed in this work, $ee$ losses are only possible via the $p$-wave scattering channel since $s$-wave collisions are suppressed by Fermi statistics. The big advantage here compared to prior experiments done in a 1D lattice at $\mu$K temperatureMartin et al. (2013) is that in a Fermi degenerate gas loaded in a 3D optical lattice, $p$-wave losses are further suppressed by the centrifugal barrier and Pauli blocking, and only happen through a wave-function overlap between atoms at different lattice sites. In this appendix, we quantify the effect of $p$-wave interactions on squeezing. To account for $p$-wave losses, we describe the dynamics using a master equation for the system’s density matrix $\hat{\rho}$:

[TABLE]

where $\hat{H}_{\text{eff}}=\chi\hat{S}_{z}^{2}$ is the effective one-axis twisting Hamiltonian obtained from the original Fermi-Hubbard Hamiltonian with spin-orbit coupling, and $\mathcal{L}$ is the Lindblad superoperator that accounts for $p$-wave $ee$ inelastic collisions. This latter term can be written using a pseudo-potential approximation asRey et al. (2014):

[TABLE]

where the jump operators are $\hat{A}_{{\bm{k}}{\bm{k}^{\prime}}}=\hat{c}_{{\bm{k}},\uparrow}\hat{c}_{{\bm{k}^{\prime}},\uparrow}$, and $\bm{k}$, $\bm{k}^{\prime}$ sum over all the populated quasi-momentum modes. The decay matrix elements $\Gamma_{{\bm{k}}{\bm{k}^{\prime}}}$ are given by:

[TABLE]

where $b_{\text{im}}=121a_{0}$Zhang et al. (2014); Goban et al. (2018) is the $p$-wave inelastic scattering length (with $a_{0}=5.29\times 10^{-11}$ m the Bohr radius), $\phi_{{\bm{k}}}({\bm{r}})$ is the Bloch function with quasi-momentum ${\bm{k}}$, and

[TABLE]

In Fig. J1, we show the averaged decay rate $\gamma\equiv\sum_{{\bm{k}}{\bm{k}^{\prime}}}\Gamma_{{\bm{k}}{\bm{k}^{\prime}}}/\ell^{2}$, where $\ell$ is the number of lattice sites along the $x$ and $y$ axes, as a function of the lattice depth $V_{0}$ along these axes. Here, we assume the same lattice depth in the $z$ direction used in the main text, $V=60E_{\text{R}}$. The decay rate $\gamma$ is suppressed exponentially with increasing lattice depth $V_{0}$. To quantify the effect of these losses on the spin squeezing generation process, we follow a similar methodology to the one described in detail in Ref. Rey et al. (2014). The basic idea is to take advantage of the so-called Truncated-Wigner Approximation (TWA)Polkovnikov (2010); Schachenmayer et al. (2015), which allows us to capture the development of spin squeezing using semi-classical phase-space methods. In the TWA the quantum dynamics are accounted for by solving mean field equations of motion supplemented by noise. The mean field equations are derived by assuming that the many-body density matrix of the system can be factorized as $\hat{\rho}=\bigotimes_{i}\hat{\rho}({i})$, where $\hat{\rho}(i)$ is the reduced density matrix of the particle in quasi-momentum mode ${\bm{q}}_{i}$ [see Eqn. (58)]. Under this assumption, the non-linear mean field equations are given by

[TABLE]

and

[TABLE]

where $\rho_{\sigma\sigma^{\prime}}\equiv\langle\hat{\rho}_{\sigma\sigma^{\prime}}\rangle$. Since we are interested in the collective behavior, one can define $\rho_{\sigma\sigma^{\prime}}^{T}=\sum_{j}\rho_{\sigma\sigma^{\prime}}(j)$. For these observables the equations of motion simplify to

[TABLE]

and

[TABLE]

where $f\equiv N/\ell^{2}$ is the filling fraction.

Under the TWA, one accounts for quantum fluctuations during the dynamics by averaging over different mean field trajectories generated by sampling over different initial conditions chosen to reconstruct the Wigner function of the initial coherent spin state at $t=0$Rey et al. (2014). This method has proven to be successful in simulating quantum spin dynamics. Using this approach, Fig. J2 shows numerical simulation results of squeezing over time in the presence of inelastic collisions. For shallow lattices ($V_{0}\lesssim 7E_{\text{R}}$), the effect of inelastic collisions can limit the spin squeezing to $\sim 10$ dB. Thus, in this regime, losses are as relevant as light scattering. The role of inelastic interactions could be mitigated by either operating at deeper lattices as shown in Fig. J2, or by using nuclear spin states to generate the squeezing instead of the clock states directly.