Correlated topological pumping of interacting bosons assisted by Bloch oscillations

Thouless pumping [1, 2], a quantized transport in periodically modulated systems, can be viewed as a dynamical version of integer quantum Hall effect. In a Thouless pump, when a uniform band occupation and adiabatic cyclic modulation are satisfied, the Chern number of the occupied band can be identified by the displacement per driven period. Due to potential applications such as current standard, quantum state transfer and entanglement generation, Thouless pumping has been theorized [3, 4, 5, 6, 7, 8], and observed in ultracold atom [9, 10, 11, 12], photon [13, 14, 15] and spin [16]systems. However, if initial state is restricted to a single momentum state, a nonquantized geometric pumping has been observed in a Bose-Einstein Condensate [17]. Recently, a new topological pumping assisted by Bloch osillations is proposed, in which adding a tilting potential recovers a quantized transport of a single momentum state [18]. Owing to the tilt potential, two main obstacles in Thouless pumping, initial-state preparation with a uniform band occupation and wave-packet dispersion, are solved.

Since particle-particle interaction is ubiquitous and inevitable, Thouless pumping has been naturally generalized to interacting systems [19, 20, 21, 22, 23]. Relying on co-translational symmetry, Thouless pumping of a multiparticle Wannier state is related to the Chern number of a given multiparticle Bloch band. For few particles, the interactions support the Thouless pumping of bound states in which particles are transport as a whole [19, 22], and topologically resonant tunneling in which particles are transported one by one [20]. For many particles, the quantized topological pumping may occur in the Mott-insulating regime with one boson per unit cell [21] and break down due to the vanishing many-body energy gap. On the other hand, the onsite interactions can be treated as Kerr nonlinearity in the mean field approximation. Moderate nonlinearity supports quantized transport of a solition while strong nonlinearity makes the soliton localized, which has been observed in curved waveguide arrays [23, 24]. Notice that a tilting potential has some advantages in topological pumping of a single particle. It is of great interest how the new topological pumping assisted by Bloch oscillations is affected by particle-particle interaction.

In this paper, we study topological transport of interacting bosons in a periodically driven tilted optical superlattice, as depicted in Fig. 1. In a rotating-wave framework, the tilting potential is transferred to the role of linearly varying phase in the tunneling rate, and co-translational symmetry is recovered. With the help of multi-particle Bloch theorem, the energy bands consist of scattering-state bands and bound-state bands in the strong interaction region. For an initial bound state with an arbitrary center-of-mass momentum, we find that the center-of-mass displacement in a overall period is nearly quantized and independent of the initial quasi-momentum, in contrast to non-quantized displacement in the absence of tilting potential. The quantization is related to a reduced Chern number defined as one-dimensional time integral of Berry curvature, which can be viewed a perfect approximation of two-body Chern number. We derive an effective single-particle model for the bound states by the many-body degenerate perturbation theory, and find the reduced Chern number of the single-particle model is consistent with the one mentioned above. The reduced Chern number can be used to precisely identify the boundary of topological phase transitions. For the scattering states, when the two particles are far away from each other and occupy the topologically-nontrivial scattering-state bands, the two particles as a Fock state can be transferred as a whole. However, when the two particles occupy the topologically-trivial scattering-state bands, the two particles can be independently shifted toward each other and come back to the initial state after crossing each other. When on-site potential difference between two neighboring sites matches the interaction, we find topologically resonant tunnelings occur where two particles move one by one. Surprisingly, the center-of-mass momentum is linearly and periodically varied as a function of time.

This paper is organized as follows. In Sec. II, we introduce an interacting Rice-Mele model in a tilted optical lattice. In Sec. III, under the guidance of effective single-particle model, we clarify the topological pumping of bound states assisted by Bloch oscillations. In Sec. IV, we show the scattering states can be topologically transported as a whole or maintain the same after certain cycles, depending on the initial positions of the two particles. In Sec. V, we discuss the topologically resonant tunnelings assisted by Bloch oscillations when on-site potential difference between neighboring sites matches the interaction. In Sec. VI, we give a brief summary and discussion.

We consider an interacting Rice-Mele model with a tilted potential, described by the following Hamiltonian

Here, $\hat{a}_{j}^{\dagger}$($\hat{a}_{j}$) creates (annihilates) a boson at site $j$ with the atom number operator $\hat{n}_{j}=\hat{a}_{j}^{\dagger}\hat{a}_{j}$. For simplicity, we set the Planck constant $\hbar=1$. $J$ is the tunneling constant, $\delta_{0}$ and $\Delta_{0}$ respectively represent the modulation amplitudes of tunneling strength and on-site potential. $U$ denotes the interaction which can be tuned by Feshbach resonance [25, 26]. A double-well optical superlattice can be formed by a superposition of a short-wavelength optical lattice $V_{s}(t)=-V_{s}\cos^{2}(\pi x/d)$ with a period $d$ and a long-wavelength optical lattice $V_{l}(t)=-V_{l}\cos^{2}[\pi x/(2d)-\phi(t)/2]$ with a period $2d$. The relative phase $\phi(t)=\phi_{0}+\omega t$ has been experimentally realized [9, 10].

$\omega_{F}$ is a tilt formed by applying a gradient magnetic field or by placing the lattice along the gravitational field direction. After applying additional tilts along one [27, 28] or two [29] directions in two-dimensional systems, the shifts of wave-packet centroid are also proposed to dynamically detect band topology. The tilt $\omega_{F}$ forms an on-site potential difference between two neighboring sites, which breaks the lattice translational invariance. To obtain the energy band under periodic boundary conditions, we apply a time-dependent unitary transformation $\hat{U}=\exp({i\sum_{j}j\omega_{F}t\hat{n}_{j}})$ to the Hamiltonian (1), the lattice translational invariance is recovered in the rotating frame given by

Obviously, the tilted potential in the Hamiltonian (1) amounts to a time-dependent phase factor in the tunneling term in the rotating frame. The Hamiltonian (2) involves in two frequencies where one is the modulation frequency $\omega$ depending on the parameter modulation period $T_{m}=2\pi/\omega$ and the other is the tilt frequency $\omega_{F}$ depending on the tilt period $T_{F}=2\pi/\omega_{F}$. $p$ and $q$ are chosen as coprime numbers, and a rational number is defined as $\omega_{F}/\omega=p/q$. In the following sections, the evolution time $T_{\mathrm{tot}}$ is chosen as the common multiple of periods $T_{m}$ and $T_{F}$ with $T_{\mathrm{tot}}=nqT_{m}$, $n=1,2,3,...$.

We consider $N$ interacting bosons in an optical superlattice consisting of $L$ cells with $d$ sites per cell. For Hamiltonian (2), particles as a whole satisfy a cotranslational invariance, that is, the system remains invariant as long as all particles as a whole shift integer cells [30, 20]. The quasimomentum of the center of mass, as a good quantum number, contributes to the multiparticle Bloch bands by solving the eigenequation

Here, $\left|\psi_{m}(k)\right\rangle$ represents the multiparticle Bloch state with quasimomentum $k$ in the $m$th multiparticle Bloch bands whose corresponding eigenvalue is $E_{m}(k)$. Besides the cotranslational invariance in the optical superlattice, the Hamiltonian (2) possesses a periodicity in time domain with $\hat{H}_{\mathrm{rot}}\left(t+T_{\mathrm{tot}}\right)=\hat{H}_{\mathrm{rot}}(t)$ with a overall period $T_{\mathrm{tot}}$. Combining the periodic parameters $t$ and $k$, we can construct a closed surface where $t$ serves as the quasimomentum in the second dimension [31, 32, 18]. Referring to real two-dimensional periodic systems [33], we define the Chern number in two-dimensional closed surface as

with the Berry curvature of the $m$th band $\mathcal{F}_{m}=i\left(\left\langle\partial_{t}\psi_{m}\mid\partial_{k}\psi_{m}\right\rangle-\left\langle\partial_{k}\psi_{m}\mid\partial_{t}\psi_{m}\right\rangle\right)$.

Given $\left[\hat{H}_{\text{rot}},\sum_{j}\hat{n}_{j}\right]=0$, subspaces with different particle numbers are decoupled and the particle number is conserved. We mainly analyze the topological properties of two interacting bosons in the Rice-Mele model where the system is confined in two-boson basis $\left\{\left|l_{1}l_{2}\right\rangle=\left(1+\delta_{l_{1}l_{2}}\right)^{-1/2}\hat{a}_{l_{1}}^{\dagger}\hat{a}_{l_{2}}^{\dagger}|\mathbf{0}\rangle\right\}$ with $1\leq l_{1}\leq l_{2}\leq L_{t}$ and $L_{t}=dL$ being the system size. After introducing $C_{l_{1}l_{2}}=\left\langle\mathbf{0}\left|\hat{a}_{l_{2}}\hat{a}_{l_{1}}\right|\bm{\psi}\right\rangle$, an arbitrary two-boson state can be expanded as $|\bm{\psi}\rangle=\sum_{l_{1}\leq l_{2}}\psi_{l_{1}l_{2}}\left|l_{1}l_{2}\right\rangle$ where $\psi_{l_{1}l_{2}}=C_{l_{1}l_{2}}(1+\delta_{l_{1}l_{2}})^{-1/2}$. Further, the eigenequation $\hat{H}_{\mathrm{rot}}|\bm{\psi}\rangle=E|\bm{\psi}\rangle$ turns to

In this section we discuss the quantized topological pumping of two-boson bound states assisted by Bloch oscillations, where bosons tend to stay at the same site. By solving the two-particle eigenequation (3), we obtain two-particle Bloch bands; see Fig. 2(a). The parameters are chosen as $J=-1$, $\delta_{0}=0.8$, $\Delta_{0}=2$, $U=30$, $\omega=0.005$, $\phi_{0}=0$, $\omega_{F}/\omega=10/3$, $L_{t}=26$. The two-particle energy spectrum consists of five isolated bands ordered for decreasing values of the energy marked with bands (i-v), of which the bands (i-ii) correspond to bound states and the rest bands belong to scattering states. According to the definition (4), the Chern numbers of multiparticle Bloch bands are calculated: $C_{\rm{i}}=-3$, $C_{\rm{ii}}=3$, $C_{\rm{iii}}=-3$, $C_{\rm{iv}}=0$ and $C_{\rm{v}}=3$. The corresponding bulk-boundary correspondence under open boundary condition is presented in Appendix A. Fig. 2(b) shows the change of energies for Bloch states with $k=0$, where the band gaps remain open as time evolves.

The adiabatic transport theorem indicates the velocity for a state with momentum $k$ in the $m$th band is written to first order as [34]

After inputting a Bloch state, the pumping distance at the moment $\tau$ is determined by a semiclassical formula

Without loss of generality, we pay attention on the band (ii) of the two-boson Bloch bands. Fig. 3 displays the center-of-mass displacements of multiparticle Bloch states with different quasimomenta in the band (ii) after an evolution time $T_{\mathrm{tot}}=3T_{m}$. When $\omega_{F}=0$, similar to a geometric pumping [17], the center-of-mass displacements of Bloch states are associated with the quasimomentum $k$, whose amplitudes are significantly reduced by interaction. In the presence of $\omega_{F}$, $\Delta X\left(3T_{m}\right)/d$ for each $k$ are quite close to the band Chern number ($C_{\rm{ii}}=3$), independent of the values of interaction. It implies a quantized transport can be realized even for initial Bloch states.

The multiparticle Bloch state is written in the two-particle basis as $|\bm{\psi}\rangle=\sum_{l_{1}\leq l_{2}}\psi_{l_{1}l_{2}}\left(k_{0},t_{0}\right)\left|l_{1}l_{2}\right\rangle$ with the probability amplitude $\psi_{l_{1}l_{2}}\left(k_{0},t_{0}\right)$. Since multiparticle Bloch states independently from $k$ can realize quantized transports, it is reasonable to predict that a quantized transport can be reached with an arbitrary initial wave packet prepared on a focused band. In order to verify this, two typical initial states are considered. One is a strongly localized initial state in real space, e.g. $\left|\bm{\psi}\left(t_{0}\right)\right\rangle=\left|L_{t}/2,L_{t}/2\right\rangle$. The multiparticle Bloch state with quasimomentum $k_{0}=0$ in the band (ii) is constructed into Gaussian wave packet as another type of initial state, e.g.

Here $\sigma$ and $l_{0}$ are the width and center-of-mass position of the initial wave packet, respectively. Such Gaussian wave packet (8) can be prepared by adding an additional harmonic potential. The evolved state $|\bm{\psi}(t)\rangle=\sum_{l_{1}\leq l_{2}}\psi_{l_{1}l_{2}}(t)\left|l_{1}l_{2}\right\rangle$ is determined by $|\bm{\psi}(t)\rangle=\mathcal{T}\exp\left\{-i\int_{t_{0}}^{t}\hat{H}(t)dt\right\}\left|\bm{\psi}\left(t_{0}\right)\right\rangle$ with the time-ordering operator $\mathcal{T}$. We extract the center-of-mass displacement

from the particle density distribution

with $X(t)=\sum_{l_{1}\leq l_{2}}\frac{l_{1}+l_{2}}{2}\left|\psi_{l_{1}l_{2}}(t)\right|^{2}$.

By numerically calculating the wave-packet dynamics of different initial states and interaction strengths, we evaluate the effect of tilt in Fig. 4. Relying on the probability of initial states projecting onto band (ii), the motion of real-space wave packet undergoes distinct dynamics. In the weak interaction regime, the tightly localized initial states actually spread in bands (i) and (ii) where fidelity between localized initial states and band (ii) increases as interaction increases. For $U=10$, the fidelity between initial Fock state and band (ii) is $0.858$, accompanied by the nonquantized centroid displacements $2.644d$ ($\omega_{F}=0$) and $2.607d$ ($\omega_{F}=10\omega/3$). Despite the lack of tilt, the centroid displacements of initial Gaussian wave packet gradually reach to band topology $C_{\rm{ii}}=3$ as interaction increases. In agreement with semiclassical results in Fig. 3, this behavior is directly attributed to the reduced group velocity by the interaction. We find that, the role of a tilt ensures the dispersionless quantized centroid displacements once the initial states are prepared at a specific band following adiabatic evolutions, while for $\omega_{F}=0$ the dispersion quantized centroid displacements are obtained with the initial states uniformly filling at a specific band. Our results confirm that the initial states that lack a uniform band occupation can support quantized transport when applied to a tilt. At the same time, the wave packet oscillates in real space with a slight width due to the reduced tunneling strength by strong interaction, whose effect of tilt is obviously exhibited in quasimomentum space in Appendix B.

The continuum band of scattering states is divided into three isolated cluster bands (iii)-(v). At the initial moment ($t/T_{m}=0$), in band (v) two independent bosons mainly occupy the different odd sublattices; in band (iv) one (the other) boson mainly occupies the odd (even) sublattice; and in band (iii) two independent bosons mainly occupy the different even sublattices. Under the appropriate parameters ($J=-1$, $\delta_{0}=0.8$, $\Delta_{0}=7$, $U=30$, $\omega=0.05$, $\phi_{0}=0$, $\omega_{F}/\omega=31/3$, $L_{t}=58$), initial state $\left|\bm{\psi}\left(t_{0}\right)\right\rangle=\left|21,35\right\rangle$ ($\left|\bm{\psi}\left(t_{0}\right)\right\rangle=\left|23,36\right\rangle$) supports fidelity $0.9805$ ($0.9806$) with band (v) (band (iv)). Fig. 6 show the dynamic evolution from the initial state $\left|\bm{\psi}\left(t_{0}\right)\right\rangle=\left|21,35\right\rangle$ with an evolved time $T_{\mathrm{tot}}=3T_{m}$. Individual transport is revealed via the particle density distribution $n_{j}=\left\langle\bm{\psi}(t)\left|\hat{n}_{j}\right|\bm{\psi}(t)\right\rangle$ in Fig. 6(a) and two-boson correlation $R_{ij}=\langle\bm{\psi}(t)|\hat{a}_{i}^{\dagger}\hat{a}_{j}^{\dagger}\hat{a}_{j}\hat{a}_{i}|\bm{\psi}(t)\rangle$ in Fig. 6(b). The time evolution of centroid position is added in Fig. 6(a) with red line, which reflects the centroid shifts 2.9414 cells during the evolution process. Similar to the behavior in Fig. 6, initial two bosons mainly occupying in band (iii) will freely propagate in another direction due to $C=-3$.

Fig. 7 manifests the case of initial state $\left|\bm{\psi}\left(t_{0}\right)\right\rangle=\left|23,36\right\rangle$ with $T_{\mathrm{tot}}=6T_{m}$. Two bosons propagate toward each other until to occupy neighboring sites, then are forbidden to stay at the same site due to the strong interaction strength (named as interaction blockade), and finally yields a counter-propagating, see Fig. 7(a). After an evolved time $T_{\mathrm{tot}}=6T_{m}$, the centroid remains almost constant with a zero centroid shifting in Fig. 7(b). Figs. 7(c) and (d) respectively show the two-boson correlations at moments $t/T_{m}=0$ and $3$. During the interaction blockade, two bosons behave similarly with Fig. 7(d). At last, two bosons nearly return to the initial position as Fig. 7(c).

Except for the topological transport of bound and scattering states, when $U\gg\left(J,\delta_{0},\omega_{F}\right)$ and $\Delta_{0}>U/2$ there will occur Topological resonant tunnelings as one of typical interaction effects. The on-site potential difference between two neighboring sites will match the interaction strength four times over a period $T_{m}$, where the two bosons travel one by one rather than as a whole. The effective single-particle model (19) will be invalid owing to the important role of coupling between the subspaces $\mathcal{U}$ and $\mathcal{V}$. Taking the Fock state as an example, once on-site potential difference between neighboring sites matches interaction, resonant tunneling occurs between state $|2\rangle_{j}$ and state $|1\rangle_{j}|1\rangle_{j+1}$ (or state $|1\rangle_{j-1}|1\rangle_{j}$) [20].

When the parameters remain the same as Fig. 2 except for $\Delta_{0}=20$, the multiparticle Bloch bands are significantly changed, as shown in Fig. 8(a) with the lowest three energy bands. Fig. 8(b) shows the change of energies for Bloch states with $k=0$, where a band marked with red dashed line is isolated from other bands to allow resonant tunnelings with band Chern number $C=3$. The energy difference between the states $|1\rangle_{j}|1\rangle_{j+1}$ and $|2\rangle_{j}$ (or $|2\rangle_{j+1}$) almost vanishes at the energy avoided-crossings. Within a modulation period $T_{m}$, four resonant tunnelings occur at four energy avoided-crossings.

The multiparticle Bloch state with quasimomentum $k=0$ is constructed into Gaussian wave packet as the initial state. The parameters are chosen as $J=-1$, $\delta_{0}=0.8$, $\Delta_{0}=20$, $U=30$, $\omega=0.005$, $\phi_{0}=0$, $\omega_{F}/\omega=10/3$ and $\sigma=5$. The system size is $L_{t}=74$ and the evolution time is $T_{\mathrm{tot}}=qT_{m}=3T_{m}$. As shown in Fig. 9(a), the initial Gaussian wave packet propagates locally in a certain range. The wave-packet centroid moves up 2.99 cells in Fig. 9(b), that is, $\Delta X(3T_{m})/d=2.99\approx 3$. There is a slight deviation between the centroid displacement extracted from the wave-packet dynamics and the band Chern number, which is caused by the nonadiabatic effect in the time evolution due to the slight band gap at the energy avoided-crossings. Based on the semiclassical formula (7), we numerically calculate the reduced Chern number of multiparticle Bloch states with quasimomentum $k_{0}=0$, and obtain $C_{\mathrm{red}}\left(3T_{m}\right)=3$. Both $\Delta X(3T_{m})/d$ and $C_{\mathrm{red}}\left(3T_{m}\right)$ are nearly the band Chern number. Despite a very obvious difference in the real-space density distribution between topological resonant tunnelings and topological pumping of bound states, they both behave as linear scanning in quasimomentum space with a tilt $\omega_{F}$, see Fig. 9(c) and Appendix B. The reason is that, most of the time two bosons stay at the same lattice site, except near the avoided-crossing points.

We explore the correlated topological pumping by applying a tilted potential in an interacting Rice-Mele model. Owing the role of tilt, the time variable not only provides an artificial dimension, but also carries out a uniform sampling in quasimomentum space. Our scheme reduces the difficulty of initial-state preparation as well as the wave-packet dispersion, which any states prepared on the focused band can realize a dispersionless quantized transport. The topological pumping of bound states assisted by Bloch oscillations is analytically explained via an effective single-particle model. For scattering states, the fruitful topological transports can be engineered by choosing the two-particle initial positions. The topological resonant tunnelings of two particles moving one by one are clarified when on-site potential difference between neighboring sites matches the interaction.

Our results are of great significance to topological pumping and can be generalized to spinor systems such as spin-1/2 bosons [37, 38], and other types of Thouless pumping such as nonlinear Thouless pumping [23, 24], Non-Abelian Thouless pumping [39, 40, 41], fractional Thouless pumping [42] and higher-order topological pumping [43]. Take the spin component into account, it is possible to obtain a dispersionless topological spin transport with accessible initial states, which has potential application prospect for designing robust and flexible topological quantum devices, such as topological beam splitters.

Note added: During the preparation of the manuscript, we became aware that Bloch oscillations contribute to the engineering of Floquet bands by applying a tilted linear potential [44].