Spin Squeezing with Itinerant Dipoles: A Case for Shallow Lattices

Entangled spin squeezed states generated via dipolar interactions in lattice models provide unique opportunities for quantum enhanced sensing and are now within reach of current experiments. A critical question in this context is which parameter regimes offer the best prospects under realistic conditions. Light scattering in deep lattices can induce significant decoherence and strong Stark shifts, while shallow lattices face motional decoherence as a fundamental obstacle. Here we analyze the interplay between motion and spin squeezing in itinerant fermionic dipoles in one dimensional chains using exact matrix product state simulations. We demonstrate that shallow lattices can achieve more than 5dB of squeezing, outperforming deep lattices by up to more than 3dB, even in the presence of low filling, loss and decoherence. We relate this finding to SU(2)-symmetric superexchange interactions, which keep spins aligned and protect collective correlations. We show that the optimal regime is achieved for small repulsive off-site interactions, with a trade-off between maximal squeezing and optimal squeezing time.

Here, we study the exact quantum dynamics of fermionic itinerant dipoles trapped in a 1D chain achiev-able for example by imposing additional lattices along the 2D pancakes.Using matrix product states (MPS), and starting from a spin-coherent initial state, we find that in all cases considered spin squeezing and coherence time are increased by reducing the lattice depth.For shallow lattices, particles remain itinerant [62,63] reducing positional disorder at non-unit filling fractions, 0 < f < 1, while undesirable lossy on-site collisions can be suppressed by the quantum Zeno effect [30,64,65].Fig. 1(c) summarizes these main results.We further find that squeezing is enhanced when the signs of nearest neighbor dipole-dipole interactions and on-site interactions match, such that superexchange and dipole-dipole interactions add up.Smaller dipolar interactions give rise to larger squeezing, albeit at the cost of slower dynamics.We qualitatively explain these effects in a spin model valid for unit filling and sufficiently small tunneling.We also find that dephasing noise e.g.due to differential lattice polarizability can be echoed away, as observed in recent experiments [53], without affecting squeezing dynamics.Even though we focus the analysis on polar molecules, our predictions apply to generic itinerant fermionic systems featuring both contact and short-range off-site interactions.
Model -We consider a 1D chain of fermionic dipoles trapped in an optical lattice with a spin-1/2 degree of freedom, which can for example be realized in the rotational states of molecules as |↑ = |N = 1, N Z = 0 and |↓ = |N = 0, N Z = 0 , where a weak external field (electric field E or magnetic field B) defines a preferred polarization axis [see Fig. 1(a)].The system is modeled as an extended Hubbard model with Hamiltonian [66,67] (see Supplemental [61] for a discussion of approximations) Ĥ = ĤFH + Ĥdip .
(1) Here, ĤFH is the single-band Fermi-Hubbard Hamiltonian describing tunneling and on-site dipolar and contact interactions.It reads with fermionic annihilation operators on site j with spin σ, bj,σ , and number operators njσ = b † j,σ bj,σ .The tunneling rates J σ , and the contact interaction U contact are controlled by the optical lattice depth, while the on-site dipolar interactions U dd can be tuned via lattice depth, lattice anisotropy, and electric field (U = U contact + U dd ).In general, a differential polarizability of the rotational states leads to spin-dependent tunneling rates, which can be tuned by the lattice polarization axis and are equal at a magic angle [68].Close to zero field, the states |↑ and |↓ are spherically symmetric and have no induced dipole moments.Then, interactions between dipoles on different lattice sites are given by the dipolar exchange Hamiltonian Ĥdip = i>j (3) Here, the spin-operators ŝα j = σα j /2 with Pauli matrices σx,y,z are defined by σ− j = b † j,↓ bj,↑ .In 1D, the 1/r 3 tail of the interactions neglected in Eq. (3) speeds up the squeezing dynamics, but the maximum attainable squeezing remains unchanged within numerical precision [61].The interaction strength V ⊥ ∝ [1 − 3 cos 2 (θ)] is controlled by the angle θ between the field and the orientation of the 1D chain [Fig.1(a)].
We assume that particles are prepared in their ground state |↓ and uniformly distributed along the lattice such that each lattice site is occupied with probability 0 < f ≤ 1. Subsequently, a π/2-pulse prepares the dipoles in an x-polarized product state [see Fig. 1(b)] with density matrix ρ(t = 0) = j ρj with Here, |0 is an empty lattice site and |→ = (|↑ +|↓ )/ √ 2. The system's dynamics is described by the Lindblad master equation On-site two-body losses e.g.due to chemical reactions are described by Lindblad operators of the form Lj = Γ/2 bj,↓ bj,↑ , where the loss rate Γ increases with lattice depth [61,64].We numerically simulate the dynamics of Eq. ( 5) by representing the vectorized density matrix as an infinite MPS directly in the thermodynamic limit, which we time-evolve with an infinite time evolving block-decimation algorithm [61, [69][70][71].We measure squeezing by the Wineland squeezing parameter, which quantifies the precision gain in a Ramsey spectroscopy experiment [25][26][27] Here, S = ( Ŝα ) α=x,y,z = ( j ŝα j ) α=x,y,z is the Bloch vector, 4 S 2 /N 2 = 4 Ŝx 2 /N 2 is the square of the contrast (since Ŝy = Ŝz = 0) , and (∆S ⊥ ) 2 min is the minimal spin variance perpendicular to the Bloch vector [see illustration in Fig. 1(b)].We use a novel method to compute squeezing from infinite MPS directly in the thermodynamic limit, details of which are given in the Supplemental [61].In all Figures, we use parameter values for fermionic KRb molecules, but we expect our findings to be relevant for arbitrary fermionic dipoles.
Optimal parameter regimes -We start by discussing the numerical findings for J ↑ = J ↓ = J, and give an analytical understanding in the following section.The maximal squeezing achieved within the first 10ms is shown in Fig. 1(c) as a function of initial filling fraction and lattice depth.For all filling fractions, decreasing the lattice depth increases the squeezing.This is the main result of our paper and will be discussed in the remainder by considering time traces for parameters along the indicated arrows.We can see that while for deep lattices with frozen molecules (J/U < 10 −3 ) even for unit filling f = 1 squeezing is limited to around 3dB, which constitutes a global maximum [see Fig. 3(a) below], shallow lattices can match and even out-perform these results for f 0.4.As will be discussed below in Fig. 4(b), for such small filling fractions squeezing is limited by the evolution time and does not constitute a global maximum, as indicated by the grey striped area.This implies that if times longer than 10ms were considered, the results would shift even more in favor of shallow lattices at low filling since the apparent saturation with lattice depth for small f is limited only by the short-time growth, which we will show to be independent of lattice depth [Fig.2(a) and Fig. 3(a)].
Figs. 2(a),(b) show the dynamics for different lattice depths at fixed filling f = 0.8.Changing the lattice depth modifies both tunneling rate and on-site interaction, such that shallower lattice lead to a larger value of J/U .At short times, squeezing is generated at a rate independent of the lattice depth.For deep lattices, squeezing peaks at ξ 2 ≈ 2dB.In contrast, for shallower lattices when molecules are itinerant, the growth persists longer, leading to larger maximal squeezing at later times.This is mirrored in the contrast decay in panel (b).While for deep lattices the contrast decays quickly, it remains much larger for shallower lattices.One might expect that Fig. 2(c),(d) show the squeezing dynamics for a range of dipolar interaction strengths V ⊥ .First, focusing on the results for V ⊥ /h = 40Hz and V ⊥ /h = −40Hz, it is clear that positive values of V ⊥ are preferable: The growth of squeezing persists longer and the coherence is maintained for longer.In order to observe the dependence on |V ⊥ |, consider the curves for V ⊥ /h = (20, 40, 80)Hz.We find that increasing the interaction strength leads to a speed up of the dynamics, however at the cost of reducing the maximal squeezing.In order to achieve maximum squeezing, one thus wants to work with shallow lattices, and repulsive interactions.The optimal value of |V ⊥ | is then determined by any dephasing mechanisms, which set a time scale limiting how slow the dynamics can be made.
Analytical explanation: Spin model -In order to provide insight into the underlying physics and qualitatively understand the results discussed above, we now consider the limit J U , f = 1, and Γ = 0, where we can define an effective spin model [see Fig. 1(c)].Due to a combination of Pauli and interaction blockade mechanisms at a small tunneling rate, molecules are essentially frozen in space.Each molecule can then be described as a localized spin [72].The spins' interactions are governed by Eq. ( 3) and additional super-exchange interactions from virtual hopping processes.The resulting Hamiltonian is an XXZ model given by: The XXZ model generates spin squeezing, which is largest for small negative values of V z,eff /V sym,eff [51].The term proportional to V sym,eff in Eq. ( 8) is SU(2) symmetric and thus cannot generate squeezing by itself, but favors spin alignment.It is largest for shallow lattices where one can reach larger values of J/U , and for sgn(V ⊥ ) = sgn(U ) = +1.For these parameters the contrast is enhanced and the squeezing remains large (see Fig. 2).Additionally decreasing |V ⊥ | and thus |V z,eff /V sym,eff | further increases the maximal attainable squeezing.Finally, choosing V ⊥ > 0 maximizes squeezing by ensuring V z,eff /V sym,eff < 0. Since V sym,eff is SU(2) symmetric, the initial squeezing speed is determined solely by |V z,eff | = |V ⊥ |, independent of the lattice depth or the sign of V ⊥ .It can be estimated by restricting dynamics to the fully symmetric manifold | S | = N/2, where the model reduces to the analytically solvable one axis twisting (OAT) model Ĥ = −ξ Ŝ2 z with ξ = V ⊥ /N .In Fig. 3 we analyze the validity of the spin model for different lattice depths in absence of losses (Γ = 0).We find that, except for the shallowest lattice, the squeezing dynamics is well reproduced by the spin model [panel (a)].For that case, while the initial growth rate is consistent with the OAT model, it overestimates squeezing at later times.A direct indicator of beyond spin model physics is the doublon population nj↑ nj↓ [panel (b)] which we find remains small nj↑ nj↓ < 1% at the time of maximal squeezing except for the shallowest lattice.For the latter, however, the doublon population becomes significantly larger nj↑ nj↓ ≈ 5% at the peak time, and the spin model breaks down.In the presence of losses, the small doublon population at moderate lattice depths also translates into losses, but they are small and typically less than 10% of the initial molecules at the time of maximal squeezing [61].
Experimental considerations -Finally, we consider the impact of experimental imperfections on the generation of squeezing in Fig. 4. Panel (a) shows the effect of spin-dependent tunneling rates, and panel (b) shows different filling fractions.Spin dephasing naturally arises due to the distinct polarizabilities of the spin states and the resulting state-dependent trapping potentials and tunneling rates J ↑ = J ↓ .Typical values for KRb at a lattice depth of 3E R for the |↑ state, are J ↑ /h = 153Hz, J ↓ /h = 131Hz [61].In Fig. 4(a) we find that this leads to a reduction of spin squeezing from ∼ 4dB to ∼ 2dB.The tunnelling anisotropy can in principle be removed by a dynamical decoupling sequence, which effectively averages the tunneling rates of both states [23,24,53].Here, we consider a sequence of (infinitely fast) X pulses exp(iπ Ŝx ) spaced by a time τ and find that pulses with a pulse spacing of τ = 500µs are sufficient to almost fully recover the peak squeezing.
Panel (b) shows the squeezing dynamics at different filling fractions, corresponding to a horizontal cut through the diagram in Fig. 1(c), versus time scaled by the initial filling fraction.In experiments, the filling fraction is limited by the temperature of the gas before loading it into a lattice.We can compute the maximal achievable filling fraction by matching the entropy of free space gases to the entropy in the respective optical lattice.While in 2015 experiments in optical lattices achieved filling fractions up to f = 0.25 [73], for T /T F = 0.3 reported in Ref. [74], theoretically filling fractions up to f = 0.9 should be reachable [75].
We observe a collapse of all curves when plotted as a function of the rescaled time t × f .The slowdown of the dynamics is due to the reduction of average interactions ∝ f .At later times, systems with lower filling fractions have reduced squeezing compared to the f = 1 case, indicating that small filling fractions lead to a reduction of maximal attainable squeezing.Since the contrast is barely affected, the reduction in squeezing is due to an increase in the variance in Eq. ( 7), which may be e.g.due to enhanced motion at lower filling or disorder in the initial state.
Nevertheless, the most important reduction of the maximally reported spin squeezing for smaller filling fractions in Fig. 1(c) is imposed by the runtime of the dynamics, which here we set to 10ms, but will ultimately be limited by additional sources of spin dephasing in an experiment.Previous experiments in pancakes had coherence times limited by collisions [53] which are already included in our analysis.In a lattice, interaction-limited spin coherence times can be larger than 400ms [22], leading to negligible coherence loss on the 10ms time scales considered here [61], thus supporting the possibility to generate several dB squeezing in current experiments.

Conclusion -
In this paper, we have shown that spin squeezing is maximal in shallow lattices.While our study is focused on 1D due to the availability of exact numerical methods, we expect these results to extend to higher dimensions.In fact, the spin model arguments generalize directly to higher dimensions, and the better lattice connectivity in higher dimensions was shown to be beneficial in the case of deep lattices [51].Larger E-fields and Floquet engineering provide additional tuning knobs, which can turn the XY into an XXZ model [67,76], and thus further control V z,eff .Additional density-spin interaction terms [66] for large E-fields may constitute an additional source of dephasing, which can however be removed by the pulse sequence discussed in Fig. 4(a).
Furthermore, we emphasize that although in this paper we focused on KRb, we expect our results to generalize to other short-range interacting systems such as magnetic or Rydberg atoms.Indeed, the only necessary ingredients for our results are short-range interactions, the trapping in a tight-binding lattice, and the fermionic nature of the particles.The observed increase of squeezing with decreased interaction strength is particularly interesting for emerging experiments with magnetic atoms, which typically have much weaker interactions than molecules but longer coherence times [4].It is an interesting prospect to consider if our results can be further extended to bosons, and how they ultimately translate when pushing to even shallower lattices when corrections to the Fermi-Hubbard model become important [77,78].
We acknowledge careful review of this manuscript and useful comments from Jun-Ru Li and Jacob Higgins.We thank Johannes Schachenmayer for valuable contributions to the MPS code, which makes use of the intelligent tensor library (ITensor) [79].The work is supported by the AFOSR MURI, the ARO single investigator award W911NF-19-1-0210, the NSF JILA-PFC PHY-1734006 grants, NSF QLCI-2016244 grants, by the DOE Quantum Systems Accelerator (QSA) grant and by NIST.In order to represent the full density matrix as an MPS directly in the thermodynamic limit, we use the translation invariance of the state [69].In particular, we write the density matrix as in,αn,αn+1 λ [n]   αn n êin , where the i n run over the physical dimension, and the α n run over the virtual bond dimension and are truncated such that 1 ≤ α n ≤ χ.The êin form a basis set of the local density matrices (i.e. 1 ≤ i n ≤ 16 for four possible states on each site), for which we here choose the generalized Gell-Mann matrices [84].For a translation invariant state, also the Γ and λ tensors are translation invariant, i.e.Γ [n] = Γ [n+2] and λ [n] = λ [n+2] , and we only keep two Γ and λ tensors.In principle, Γ and λ are translation invariant by one site, but during the time evolution this translation invariance is intermittently broken and then restored, making all four tensors necessary.
To time evolve the state, we use an infinite time-evolving block decimation algorithm, i.e. a Trotter decomposition of the Hamiltonian into two site gates.However, a naive implementation of this algorithm for density matrices would destroy the orthogonality of the MPS.This is fixed by re-orthogonalizing the MPS after every gate application [69].For all simulations except the spin model simulation, we use a fourth order Trotter decomposition [85] (1)(1) T (1)(−2) T (1)(1)(1)(1)(1) T (1)(1) T (1) T (1) T (1) T (−2)(1) T (1)(1) T . (C2) Here, the number in brackets indicates the length of the timestep, negative numbers for evolution backwards, and the superscript T indicates that we apply the gates in transposed order.For the spin model simulation, we use a smaller time step for which the second order trotterization (1)(1) T suffices.

Convergence
In order to ensure convergence of our simulation, we successively decrease the timestep ∆t and increase the bond dimension until convergence is reached.In practice, we halve the timestep and increase the bond dimension in steps of χ = 512, 750/800, 1024, 1400, 2048, until the curves overlap.For the time step, we have confirmed for all results with f = 0.1 and for the shallowest lattice with f = 0.8 that a timestep ∆t = 1ms [time for a full Trotter sequence In Fig. S4, we consider the effect of dephasing on the system dynamics.Here, we model dephasing as white noise on the two different spin states, described by Lindblad operators Lj = Γ deph ŝz j .For spin coherence times of τ deph = 1/Γ deph = 100ms, we find a significant reduction in squeezing.In contrast, for spin coherence times of 500ms, squeezing is only slightly reduced, and for τ deph = 2s, changes induced by dephasing are barely visible.Since recent experiments have reported interaction limited coherence times > 400ms, we can assume that the real coherence times are even longer, such that additional dephasing does not need to affect the spin squeezing.

FIG. 1 .
FIG.1.(a) Schematic of the system: Fermionic dipoles encoding a spin-1/2 degree of freedom in two internal levels (for molecules two rotational states |N , NZ ) are loaded in a 1D chain.The rotation axis is set by an external electromagnetic field at an angle θ to the lattice axis ( E for electric dipoles).Our model includes tunneling J, dipole interactions V ⊥ , on-site interactions U , and two-body losses Γ.(b) A π/2 pulse prepares all dipoles in a superposition of both spin states, followed by free evolution with Liouvillian L. Bottom: Schematic illustration of time evolution in the S y -S z -plane.With time, the interaction V ⊥ generates spin squeezing by shearing the quantum noise distribution.(c) Maximal squeezing ξ 2 for t < 10ms versus filling fraction f and lattice depth.In x and y/z directions (V latt,x , V latt,⊥ )/ER = (3, 3),(3,40),(5,40),(40,40) (top to bottom, see Supplemental for detailed parameters [61]).The black line indicates where the system can be approximated by a spin model.The striped area indicates where squeezing is growing past 10ms.

TABLE II .
Bond dimension χ used for different filling fractions f and lattice depths indicated by J/U (see Tab. I).Entries marked by (X) are not conclusively converged until t = 10ms.This does not affect any results shown in the paper.