Topological magnon insulator and quantized pumps from strongly-interacting bosons in optical superlattices

We propose a scheme realizing topological insulators and quantized pumps for magnon excitations, based on strongly-interacting two-component ultracold atoms trapped in optical superlattices. Specifically, we show how to engineer the Su-Schrieffer-Heeger model for magnons using state-independent superlattices, and the Rice-Mele model using state-dependent superlattices. We describe realistic experimental protocols to detect the topological signatures of magnon excitations in these two models. In particular, we show that the non-equilibrium dynamics of a single magnon can be exploited to directly detect topological winding numbers and phase transitions. We also describe how topological (quantized) pumps can be realized with magnons, and study how this phenomenon depends on the initial magnon state preparation. Our study opens a new avenue for exploring magnonic topological phases of matter and their potential applications in the context of topological magnon transport.


Introduction
The emblematic Su-Schrieffer-Heeger (SSH) model describes non-interacting fermions hopping on a onedimensional lattice with alternating hopping matrix elements [1]. This model constitutes a paradigmatic example of a non-trivial topological insulator in 1D, which results from its underlying chiral symmetry [2,3]. It exhibits a variety of topological features, including topological solitons [4], fractional charges [5], quantized Zak phases [6], degenerate zero-energy topological edge states [7], and a characteristic 'chiral' displacement [8]. Adding staggered on-site energies to this model yields the Rice-Mele (RM) model [9], which realizes a quantized (Thouless) pump upon varying its parameters in a cyclic and adiabatic manner [10,11]. The SSH and RM models have both been experimentally realized using non-interacting single-component ultracold bosonic atoms trapped in an optical superlattice [12], where quantized and fractional Zak phases were measured. More recently, cold-atom experiments further demonstrated geometric and topological pumping, both for fermionic [13] and bosonic gases [14,15], as well as the existence of a topological Anderson insulating phases in the presence of disorder [16]. Very recently, the SSH model was implemented for hard-core bosons, using Rydberg atoms [17].
On the theoretical side, it was predicted that the SSH model could spontaneously form in systems of interacting bosons described by an extended Hubbard model [18]. Besides, the interacting fermionic SSH model was shown to exhibit unusual correlation dynamics upon a quench [19], which can be attributed to a rich interplay between the presence of topological edge modes and spin-charge separation in one dimension. Finally, quantized pumps of interacting bosons [20] and fermions [21] were also recently investigated.
In this paper, we describe a scheme that realizes the SSH model for magnon excitations, in a system of strongly-interacting two-component ultracold atoms trapped in an optical superlattice. Our approach consists in starting from a spinful SSH model for non-interacting two-component ultracold bosonic atoms trapped in a state-independent optical superlattice; we then add (Hubbard-type) interactions between the atoms so as to reach the Mott-insulator regime at unit filling (where each lattice site is occupied by a single atom); by making use of a schrieffer-Wolf transformation, we demonstrate that this interacting system can be described by the SSH model for the underlying magnons (which are bosonic quasiparticle excitations around the ground state of an effective ferromagnet; see [22] and equation (3) below). In this way, we show that the single-particle topological properties of the underlying SSH model are transferred to the bosonic magnonic excitations. Such a transfer of topological band properties to interacting-particle settings is reminiscent of that discussed in [23,24], in the context of topological doublons, and in [25,26] in the context of topological polarons; see also [27][28][29][30][31][32] on interplays between topological properties and magnons.
In the single-magnon excitation subspace, the energy spectrum of the interacting system has two topological magnonic bands, which are characterized by topological winding numbers. Similarly to the more standard single-particle configuration, the system can be prepared in topologically trivial or nontrivial magnonic insulating phases by tuning the tunneling matrix elements of the underlying superlattice. We show that both the value of the topological winding number and the topological phase transition points can be unambiguously detected through nonequilibrium single-magnon quantum dynamics taking place in the optical superlattice. We note that the single-magnon excitation state and its dynamics can be precisely prepared and measured in experiments, based on single-site and time-resolved optical-lattice technologies [22]. Besides, we propose a method based on an array of optical superlattices and using parallel single-magnon state preparation and detection to efficiently measure the single-magnon quantum dynamics.
Furthermore, we show how to promote this system to a RM model for magnons, using a state-dependent optical superlattice. We build on the fact that this additional optical superlattice can be controlled dynamically and show that this time-dependent configuration can lead to quantized (topological) pumping of magnons, i.e. the band structure exhibits non-zero Chern numbers, when taking the dynamical feature of the model into account [11]. We describe how the shift of the magnon-density center reflects this non-trivial topological invariant, and how the latter depends on the initially prepared single-magnon Bell states. Finally, we discuss how to prepare such Bell states and how to efficiently detect the quantized topological magnon pumping with current optical-lattice technologies.

SSH model for magnons
We consider interacting two-component ultracold bosonic atoms trapped in a one-dimensional stateindependent optical superlattice. For instance, considering 87 Rb atoms, the two components could be represented by the hyperfine states F m 1, 1 , where F and m F label the hyperfine levels and Zeeman sublevels, respectively. The optical superlattice is generated by superimposing two standing optical waves that generate a state-independent lattice potential of the form where the state-independent lattice potential depths V l s x , and laser phase j can be varied by changing the laser power and the optical path difference. For sufficiently deep optical lattice potential and low temperature, this optical superlattice system can be described by the spinful Su-Schrieffer-Heeger-Bose-Hubbard model  = are the effective spin-exchange couplings. We assume the spinexchange couplings are ferromagnetic throughout this work.
In the present work, we restrict ourselves to system configurations that only feature a single spin-up excitation; in practice, this requires temperatures that are low compared to the effective spin-exchange couplings. In this case, the longitudinal spin coupling in equation (3) only gives rise to an energy offset and can therefore be neglected. Based on the Matsubara-Matsuda mapping [37], and restricting ourselves to the space of single spin-up excitations, the above effective spin model can be rewritten in terms of the following magnonic SSH model is the magnon creation operator associated with the lattice site a b x x ( ) in the xth unit cell, and Gñ =    ñ | | ·· can be seen as the magnon vacuum state. To study its topological features, we rewrite it in the momentum-space representation, H m hk m ( ), and x ŝ and y ŝ are the Pauli spin operators defined in the momentum space. We note that this 2 × 2 k-space Hamiltonian for magnons formally corresponds to that describing the standard SSH model for non-interacting fermions; in particular, it satisfies a chiral symmetry h k h k Besides, the energy spectrum of the system displays two magnonic bands, whose topological features are characterized by the topological winding number where the unit vector is defined as n n d d E n , , This indicates that the strongly-interacting bosonic system is in the topologically nontrivial (resp. trivial) magnonic insulating phase if the hopping matrix elements (or the related spin-exchange interactions) satisfy J 1 <J 2 (resp. J 1 >J 2 ).

Dynamical detection of topology via single-magnon quantum dynamics
In contrast to the experimental study performed in the Munich group [33], which investigated quantum dynamics of magnons in an optical lattice, we explore the topological features of single-magnon quantum dynamics in an optical superlattice. We suppose that the ultracold atoms that are trapped in the optical superlattice are initially prepared in the Mott-insulator regime at unit filling:each optical lattice site traps a single atom in the internal state . Based on single-site addressing technology, one can address a single atom, in the middle of the optical lattice, and flip its spin into F m 2, 2 Then the initial state of the optical superlattice system is prepared in a single-magnon state, which can be written as The quantum dynamics of such single-magnon state is governed by the single-magnon Hamiltonian in equation (4). After an evolution time t, the final state of the system becomes The time evolution of the density distribution of single magnons in the optical superlattice is shown in figure 1(a). The results shows that the single magnon spreads ballistically with two pronounced side peaks and low probability around the initial position. Current technologies allow for the time-and site-resolved observation of the dynamics associated with a single magnon state [22,33]. In the following, we will show the single-magnon quantum dynamics can be efficiently detected based on parallel state preparation and detection. At the initial time, as shown in figure 1(b), one can prepare a two-dimensional degenerate Bose gas of 87 Rb atoms in a two-dimensional optical lattice ). When V s y is tuned to be vary large, the hopping along the y direction can be neglected, and this lattice potential creates an array of independent onedimensional optical superlattices along the x direction. In the Mott-insulator regime, this optical lattice system can be seen as an array of parallel one-dimensional spin chains. Suppose all the atoms trapped in the lattice are prepared in the ground state F m 1, 1 Then, an addressing beam profile in form of a line generated by a spatial light modulator is used to flip the atoms in the middle lattice sites of all spin chains into the excited state F m 2, 2 F ñ = = = -ñ | | [22,33]. After that, the single-magnon excitation in each spin chain will propagate with time. Via rapidly increasing the lattice depth, the single-magnon dynamics can be frozen to a fixed time and its density distribution can be measured by single site-resolved fluorescence imaging. The measured magnon density is obtained by averaging data from the measurements in all spin chains. This parallel state preparation and detection strategy can greatly improve the experimental efficiency.

Dynamical detection of the topological winding number
A recent study, performed in the context of linear-optics topological quantum walk, has shown that the topological winding number can be connected with discrete-time quantum dynamics [38]. This method has been further generalized to systems with continuous-time dynamics [8]. Based on such a method, we show that the topological winding number also can be directly detected through the mean chiral displacement associated with the dynamics of a single-magnon quantum state. For the single-magnon quantum dynamics described in equation (8), the mean chiral displacement is expressed as where the mean chiral displacement operator is defined as X x z = Ĝˆ. In the long time limit, a relationship between the winding number and the time-averaged mean chiral displacement can be derived as [38] 2 , 10 where the time-averaged mean chiral displacement is As we can see, z  is the oscillation center of X z varying with time and the topological winding number is twice this center. Equation (10) also indicates that the quantum dynamics of a single-magnon state in a topological optical superlattice can give us the underling topological information.
To measure the time evolution of the mean chiral displacement based on single-magnon quantum dynamics in the optical superlattice, we need to transfer the mean chiral displacement into real space and rewrite it as is the magnon density operator. By substituting the above equation into equation (9), the time evolution of the mean chiral displacement associated with the dynamics of a singlemagnon state described in equation (8) is numerically calculated in figure 2. The numerical results in figure 2(a) show that, when the optical superlattice system is in the topological nontrivial magnon insulator phase, the mean chiral displacement oscillates around 0.5. While if the system is in the topological trivial magnon insulator phase, as shown in figure 2(b), it oscillates around 0. According to equation (10), the topological winding number is twice the oscillation center, which allows one to directly obtain the topological winding numbers as ν=1 and 0, respectively.
In addition to the optical superlattice potential, a weak external harmonic trap is also typically present in experiments. This harmonic trap introduces a site-dependent potential , m is the mass of an atom, w is the harmonic trap frequency, j is the coordinate of a lattice site, j 0 is the coordinate of the center of the harmonic trap and the lattice constant is chosen to be unity. In principle, the super-exchange couplings will become site-dependent due to the harmonic trap potential. Specifically, the super-exchange coupling between the jth and j+1th lattice site can be written as ].

Dynamical detection of the topological-phase-transition point
In addition to the topological winding number itself, the topological-phase-transition point can also be directly detected via single-magnon quantum dynamics. Indeed, a previous study showed that the topological phase transition of a one-dimensional topological (quantum walk) system can be observed via a second-order moment [39]. Building on this method, here we show that the second order moment associated with the single-magnon continuous-time quantum dynamics, as obtained in the long-time limit, can also be related to the topological phase transitions predicted in equation (6). The second order moment associated with the dynamics of a singlemagnon state is given by  This equation shows that the topological phase transition point J 1 =J 2 in the superlattice Bose-Hubbard model can be precisely measured through the second order moment in the long-time limit. In this way, the topological phase transition can also be observed based on single-magnon quantum dynamics. The second order moment is extracted from the quantum dynamics of a single-magnon state in real space. Thus, we rewrite the second order moment operator in the corresponding representation, After substituting the above equation into equation (14), we numerically calculate the change of the secondorder moment in the long-time limit, and study its behavior as a function of the ratio J J ; 2 1 the results are shown in figure 2 (c). We find that the numerical results for Y t agree well with the analytical result presented in equation (15). When J 1 is fixed and J 2 >J 1 , the second order moment is a constant and will not change with J 2 /J 1 , which is reflected by the plateaus in figure 2 (c). While if J 2 <J 1 , the second order moment is a quadratic function of J 2 /J 1 . In this way, the topological phase transition and also the transition point J 1 =J 2 can be measured. We also numerically calculate the influence of the external weak harmonic trap on the above topological phase transition signal. These results show that while the actual value of the plateau is modified by the presence of the harmonic trap, its position along the J 2 /J 1 axis remains essentially unchanged. One also find that the relationship between the second order moment and the topological phase transition breaks down when the trap energy is comparable to the spin-exchange couplings V

RM model for magnons
In this section, we analyze how the RM model and the related topological pump can be obtained by replacing the state-independent optical superlattice by a state-dependent one. We point out that such state-dependent optical superlattices have been realized in experiments; see [40] for details. The corresponding potential is taken in the where the potential depths V l s , s ( ) and the phases j s can be varied by changing the laser parameters [40]. The corresponding tight-binding Hamiltonian is described by the spinful RM-Bose-Hubbard model = + + . The integrand n n ń ¶ q · is simply the Jacobian of the aforementioned mapping. Its integration is a topological winding number giving the total area of the image of the Brillioun zone 2  on S 2 [2,3]. It means that, when k , x q ( ) wraps around the entire first Brillouin zone T 2 , this winding number is equal to the number of times the vector n wraps around the unit sphere S 2 , which is independent of the details of the band structure parameters.
For the sake of completeness, we plot the unit vector configurations (n x , n y ) and the contours of n z , for the lower and upper magnon bands, in figures 3(b)-(c), respectively. The results show that, for the lower (upper) magnon band, the unit vector n starts from the North (South) pole at the Brillouin zone center and ends at the South (North) pole at the Brillouin zone boundary after wrapping around the unit sphere once. Thus, the Chern number corresponding to the lower (upper) magnon band corresponds to C l =1 (C u =−1). Since θ is a periodic dynamical parameter introduced to construct the first Brillouin zone, the above nontrivial Chern numbers characterize a dynamical version of a topological magnon insulating phase.

Quantized topological magnon pumping
This section describes how the dynamical control over the topological magnonic bands allows for the implementation of a quantized pump for magnons. As described below, this requires a special preparation of the initial single-magnon state. The pump itself is then realized by adiabatically tuning the dynamical parameter t 0 q q = W + over one period, where Ω is the modulation frequency and 0 q is the initial phase.
At the initial time, we assume the whole optical superlattice system consists of series of independent double wells by tuning the optical superlattice potential to make J e2 =0. This is equivalent to require the initial periodic parameter t JJ 0 arcsin e p ). Suppose the initial system stays in the magnon vacuum state Gñ =    ñ  | | . Note that the Hamiltonian for the single-magnon excitation in each double well has two eigenstates which are two different Bell states. As shown in figure 4(a), suppose that one of middle double wells is prepared into the Bell states l c ñ | , while the other qubits stay in the ground state. The initial state of the system then can be described by l l y c ñ =    ñ   | | . Actually, one can find that l y ñ | ( u y ñ | ) is just the Wannier function corresponding to the lower (upper) magnon bands. The reason is that the coupling between nearest neighbor double wells is zero and so the Wannier function only localizes the middle double well.
After having prepared the initial single-magnon state, the parameter θ is adiabatically modulated from ) , the single-magnon wave packet will experience an adiabatic transfer. We employ a magnon density center to monitor this transfer. Specifically, the operator for such a center is defined as is the magnon density in the lattice site a x (b x ). Then, when the parameter θ is adiabatically modulated, the corresponding magnon density center can be written as , , . Therefore, the magnon density center depends on the gauge choice of the Bloch state. However, the change of the magnon density center is gauge invariant and thus can be well defined.
Suppose the periodic parameter θ is changed continuously from i q to f q . The resulting magnon density center shift is Hamiltonian in equation (18)), by switching on the parameter Δ; this step is realized by tuning the statedependent optical superlattice potential. This effective magnetic field modulates the phase of the entangled state, which allows one to generate the desired Bell state 2  ñ   ñ (|  |  ) . During the whole process, one assumes that the state of the spins in the other double wells remain unchanged. In this way, the single-magnon state l u l u , , | is prepared. The magnonic topological pumps proposed in this work could be efficiently measured in experiments through parallel state preparation and detection. As illustrated in figure 5(b), a two-dimensional degenerate Bose gas of 87 Rb atoms can be prepared in a two-dimensional optical lattice potential V(x, y) is the state-dependent superlattice potential in the x direction and V y V k y sin 2 ) is a state-independent potential in the y direction. When V y is tuned to be vary large, the hopping along the y direction can be ignored. Then V(x, y) creates an array of independent one-dimensional optical superlattices along the x direction. In the Mott-insulator regime, this system can be seen as an array of parallel onedimensional spin chains described by equation (18). One then proposes to simultaneously flip all the spin-down states located on the left part of the double wells, which form the central column [ figure 5(b)]; this could be realized using a single addressing beam, with a proper linear profile, as was used in a recent experiment on singlemagnon states [33]; in this way, extending the entanglement generation procedure described above, each superlattice system could display a single magnon state, l u , y ñ | . Finally, tuning θ over one period in each spin chain, one can realize different topological magnon pumps in parallel, defined in the two-dimensional optical lattice system. In this case, the magnon density can be efficiently measured by averaging the data extracted from the magnon density measurements in all spin chains [22,33].

Conclusion
In summary, we have proposed and analyzed schemes by which topological insulators and quantized pumps could be realized for magnons, which are excitations that can be addressed and detected in strongly-interacting bosonic systems. Our approach builds on the implementation of the emblematic SSH (resp. RM) models for magnons, using two-component bosonic atoms in state-independent (resp. dependent) optical superlattices.
We have studied single-magnon quantum dynamics and shown that such nonequilibrium dynamics can be directly exploited to detect topological winding numbers and topological phase transition points. We have also presented an experimentally realistic method to realize quantized topological pumps for magnonic excitations, and we have revealed the importance of the initial (single-magnon) state preparation in this context. We believe that our work will inspire future studies of magnonic topological phases of matter in optical lattice systems.
Importantly, our topological magnonic models are compatible with recent optical-lattice experiments on magnon physics [22,33]. In this sense, our proposal provides a promising platform for exploring bosonic excitations with topological properties [26]. Finally, we note that topological two-magnon bound states have been analyzed in the context of Chern-insulator models [30,31]; it would be interesting to study the properties of such bound states in symmetry-protected one-dimensional models, such as the magnonic SSH model explored in this work.