Ultracold atomic gases in optical lattices provide a flexible and controllable platform well isolated from the environment for quantum simulation of complex and controversial problems in condensed matter physics.^[1–3] Neutral atoms allow practically perfect realizations of a variety of Hubbard models and rich quantum phases.^[4–9] They are also good candidates for emulating diverse spin models,^[10–12] where spin–spin interactions between atoms or molecules can be implemented and manipulated by controlled collisions on a single site,^[13] on-site exchange interaction,^[14,15] dipolar spin-exchange interaction,^[16] or superexchange interaction in optical superlattices.^[17–19] In particular, the interplay between tunneling and superexchange interaction has been experimentally observed by recording magnetization dynamics of atomic spin configurations in a two-dimensional (2D) optical lattice.^[20]

In parallel with the aforementioned achievements with alkali atoms, increasing theoretical and experimental efforts have evidenced the potential of alkaline-earth atoms (AEAs) as quantum simulators of extraordinary strongly correlated systems,^[21,22] owing to the emergent SU(N)-symmetric spin-exchanging collisions between the atoms in ¹S₀ ( ) and ³P₀ ( ) electronic clock states, respectively.^[23–25] The SU(N)-symmetric feature of the two-orbital spin-exchange interactions offer experimental access to explore the Kondo lattice model and heavy-fermion physics,^[26–30] where g and e atoms represent mobile electron and localized magnetic spin, respectively.

Now a question naturally arises: how do the superexchange-mediated magnetization dynamics of the g atoms behave with doped impurities (the e atoms). It is well-known that the superexchange mechanism can work for g atoms in different spins on neighboring sites, while the inter-orbital spin-exchange interaction generally happens locally when single g and e atoms coexist in a single lattice well. In this situation, the local spin-exchange interaction gives rise to spin-flip events that actually change the spin configuration of the g atoms and interrupt the superexchange process. Intuitively, one may come to the conclusion that spin-exchange interaction with the impurity atoms is negative to maintain the magnetization dynamics governed by superexchange interactions. Ultracold fermionic AEAs provide unprecedented opportunities to address the issue, and we will demonstrate the invalidity of the above statement and show an interaction-modulation feature of the magnetization dynamics. Furthermore, we will also demonstrate that the filling of the e atoms has remarkable influence in the modulated magnetization dynamics.

The paper is organized as follows. In Section 2, we introduce a Hamiltonian to describe the system, taking account of tunneling and interatomic interactions. In Section 3, we calculate the time-resolved spin dynamics of the system and present the analytic results based on perturbation method to explain the modulation feature even with the dominant superexchange interaction. Section 4 discusses the effect of the filling of e atoms on the superexchange mediated magnetization. In Section 5, we discuss the possibility of experimental observation in our setup. Finally, we present a summary of our results in Section 6. For completeness, mathematical details on the perturbation method at half and unit filling are presented in Appendices A, B, and C.

Let us consider an array of isolated double-well potentials created by optical superlattice as depicted in Fig. 1(a). The g and e atoms are independently trapped in state-dependent lattices. For simplicity, they are assumed to be located in the lowest band of the respective lattices, as long as the vibrational level splitting is sufficiently larger than the other relevant energy scales. In each double-well, there are only two g atoms and the number of the e atoms can be one (half-filling) or two (unit-filling) by changing the atomic population in . The lattice depth of the e atoms are very deep so that the tunneling is completely forbidden. We first study the half-filling case of the doped e atoms, that is to say, a single e atom is assumed to be fixed in the left subwell. We are interested in the spin configuration of the g atoms, and label the relevant states: the antiferromagnetic (AFM) states , non-magnetic state , and doubly-occupied states . For the situations presented here, the system is well characterized by the onsite energy U_gg, the tunneling rate J_g and the spin-exchange interaction strength V_ex, representing the transitions between the states marked by the arrows in Fig. 1(a).

Fig. 1.

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Fig. 1. (color online) (a) Schematic diagram of the superexchange and inter-orbital spin-exchange interactions. The atoms are independently trapped in the shallow (deep) superlattice, whose nuclear spin states are marked by and . The superexchange process (green arrow) happens between AFM states ( ) via virtual tunneling (black dashed arrows), while two-orbital spin-exchange interaction (red arrows) couples with the magnetic state . (b) Time evolution of the spin imbalance Ngz of the g atoms (black) and the population of the state (blue) for a small magnetic field (the resulting differential Zeeman shift can be ignored) and the initial state . The envelope of Ngz is given by (red). The collapse and revival happen at and , respectively. (c) Energy levels of the perturbation model (left) and the eigenstates (right). . are the coupling matrix elements. δ is the energy shift. The red solid and blue dashed arrows indicate the near-resonant coupling and far-off-resonant coupling, respectively. (d) Time evolution of the spin-imbalance of g atoms with (black line). The envelope (red solid) corresponds to . The parameters are , , in panels (b) and (d).

We focus on the strong-interacting regime , and assume that the superexchange-dominated antiferromagnetic strength predominates over the spin-exchange interaction, . The limit assures that doubly-occupied states of the g atoms are allowed only via virtual tunneling and the system is antiferromagnetic ordered with the strength . It is well understood that, without the magnetic impurity (e atom), the system can be mapped onto a Heisenberg model with the antiferromagnetic magnetic strength .^[31] When the e atoms are introduced as magnetic impurities, Néel order of the g atoms may be destroyed due to the inter-orbital spin-exchange mechanism. Under these conditions the system is described by a two-site SU(N)-symmetric Fermi–Hubbard Hamiltonian as follows:

We begin to study how the spin dynamics governed by the superexchange interaction is perturbed by the spin-exchange interaction in the presence of the localized e atom. In fact, the superexchange process of the two g atoms in a double-well potential can be visualized by tracing the evolution of g atom spin imbalance (Néel order parameter). We numerically calculate with the full Hamiltonian Eq. (1) and show the results in Fig. 1(b). In a clean system without the e atoms, ideally, exhibits a simple oscillation at the frequency , which has been experimentally used to extract the superexchange interaction strength.^[18] It is obvious here that the superexchange-dominating magnetic dynamics features an oscillation with a modulated amplitude. The envelope of the curve is characterized by the collapse (revival) time ( ) and the depth . One can immediately realize the fact that the spin imbalance is zero for the nonmagnetic state . results in evolving the system out of the AFM states, and consequently degrades the AFM superexchange. It should be kept in mind, however, that the spin-exchange interaction process is coherent and cannot be regarded as a pure damping mechanism for the superexchange process. It is therefore instructive to monitor the population of the non-magnetic state . As shown in Fig. 1(b), manifests itself as an oscillating sine-like curve. In particular, the envelope of this curve is out of phase with that of . In other words, the envelope of N_gz approaches the minimum at when the envelope of P₃ reaches the maximum simultaneously and vice versa. These results are obviously inherit to the role of inter-orbital spin-exchange interactions in the dynamics of the system.

To quantitatively understand the emergence of the modulating oscillation in the magnetization dynamics, we propose an improved perturbation strategy for revealing the behavior of the system analytically and extracting the modulation parameters. At first, similar to the superexchange process solely, the states with double occupancy ( and ) are regarded as virtual intermediate states and can be adiabatically eliminated. As a result, equation (1) is reduced to the following low-energy effective Hamiltonian^[31]

In the aforementioned discussions, we have shown that the appearance of a single impurity atom indeed leads to significant modulation of the amplitude of the g atom spin dynamics in the double-well potential. Next we will explore the effect of the filling of the e atoms on the superexchange-mediated quantum magnetism. As shown in Fig. 2(a), the spin-polarized e atoms are deeply located in both wells. Similarly, we first numerically solve the full Hamiltonian to obtain N_gz as shown in Fig. 2(b). For comparison, the envelopes of the time-resolved spin dynamics at unit- and half-filling cases are depicted by the red solid curve and the blue dashed curve, respectively. They have a distinct feature that the amplitude of the antiferromagnetic exchange is periodically varied. Moreover, in comparison with the doping impurity at half-filling case, the modulated spin dynamics seems to be accelerated. In fact, the periodicity of the amplitude envelop decreases, which is consistent with smaller and . To illustrate the effect of the impurity filling, we adopt the same steps in the analysis of Fig. 1. Starting with the schematic diagram Fig. 2(a), after adiabatically eliminating the double occupancy and transforming the system into the basis composed of the spin singlet/spin triplet states , we have a symmetric level diagram as shown in Fig. 2(c). It is not surprising since the spin-exchange interaction can happen in both subwells and there are two coupling paths between the antiferromagnetic and non-magnetic states with the strength . These couplings in the real process contribute all the transitions labeled by . Following the routine analysis on the half-filling situation, we need to perform the perturbation calculations in the low-energy subspace including the relevant states. In Fig. 2(c), the two magnetic states and the state form a degenerate manifold that are resonantly coupled and separated with the state by . Since the coupling strengths are the same in magnitude, the lowest state is just slightly shifted by off-resonant coupling from states and . We can then obtain the eigenstates , , , and . As a direct consequence, the state is orthogonal to the initial state , which means should not be populated during the evolution. Finally, the dynamical evolution of the spin imbalance still contains two frequencies , i.e.,

Fig. 2.

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Fig. 2. (color online) (a) Schematic diagram of the superexchange and inter-orbital spin-exchange interactions at unit-filling of the e atoms. (b) Time evolution of spin imbalance Ngz (black solid). The envelopes of at unit-filling (red solid) and half-filling (blue dashed) have been shown. and are the collapse and revival periods, respectively. The parameters are , , . The envelope (red solid) is . The collapse and revival happen at and , respectively. (c) Energy levels in the low-energy effective Hamiltonian (left) and the eigenstates (right). . and are the coupling matrix elements. is the energy shift. The red solid and blue dashed arrows indicate the near-resonant and far-off-resonant couplings, respectively.

Now we turn to discuss experimental requirements for observing the amplitude-varying superexchange-dominating magnetization dynamics. It is requisite to prepare the initial state with the AFM order of the g atoms and one e atom. First, we have to prepare a spin mixture with two atoms and one in the long lattice, which can be realized by carefully tuning the state transfer from to in a spin-polarized Fermi gas. Then we load a band insulator with atoms in the ground band and one in the first excited band per site on the deep long lattice. Subsequently we ramp up the short lattice with a bias to build a double-well array that it is energetically favorable to have two atoms with different spins in the left well and one atom in the right well. Finally we perform a π-polarized clock excitation solely resonant with the transition that is off-resonant for when the differential Zeeman shift is larger than the linewidth of the clock excitation. In addition, the single-occupied site can be spectroscopically resolved due to large U_gg. To measure N_gz, we release the e atoms in the lattice by transferring them to the state, and subsequently band mapping technique is used to extract the information of the g atoms. These can be reached within the-state-of-the-art experimental efforts. The initial state in Fig. 2 can also be prepared with the similar strategy. Moreover, the spin-exchange interaction is expected to be tuned by the recently proposed orbital Feshbach resonance technique,^[33] which has been experimentally demonstrated with ultracold ¹⁷³Yb atoms.^[34,35]

In conclusion, we have theoretically studied the superexchange-mediated magnetization dynamics with ultracold AEAs in an optical superlattice. We found that, the doped ³P₀ atoms and the associated spin-exchange interaction between the clock states give rise to the modulation superexchange dynamics, especially when the superexchange interaction predominates in the system. As a particular example, the appearance of the modulating oscillation in the magnetization dynamics has shown the doped magnetic impurities indeed play an important role in the superexchange interaction by coherent spin-exchange mechanism. We have also found that, compared to the half-filling case of the e atom, the modulated spin dynamics of the g atoms can indeed be accelerated at unit-filling. Our findings may provide new insights in coherent control of quantum magnetism with hidden symmetric intrinsic interactions between ultracold atoms.