Topological pumping in Aharonov-Bohm rings

Topological Thouless pumping and Aharonov-Bohm effect are both fundamental effects enabled by the topological properties of the system. Here, we study both effects together: topological pumping of interacting particles through Aharonov-Bohm rings. This system can prepare highly entangled many-particle states, transport them with topological protection and interfere them, revealing a fractional flux quantum. The type of the generated state is revealed by non-trivial Aharonov-Bohm interference patterns that could be used for quantum sensing. The reflections induced by the interference result from transitions between topological bands. Other types of states can be robustly transported with a band gap scaling as the square-root of the particle number. Our system paves a new way for a combined system of state preparation and topological protected transport.

Introduction. Topological matter defines an important field in fundamental physics with far reaching implications for quantum technology: the correlations encoded in topological matter are a precious resource for quantum technology; at the same time, quantum technology can be exploited to study topological matter with unprecedented precision and control [1][2][3][4]. Among the several important contributions given by David Thouless in this field, the idea of topological pumping is particularly relevant for quantum technology: charge is transported through a one dimensional system using the topological band structure of an extended (many-body) system [5,6]. This is realized by driving the system periodically in time while protecting the band gaps. Topological pumping has been studied with the most recent approaches in experimental quantum technology, from cold atoms [7][8][9], photonic waveguides [10] and superconducting circuits [11].
Here, we couple topological bands with Atomtronics: atomic circuits of ultra-cold atoms manipulated in micromagnetic or laser-generated micro-optical guides [12][13][14]. Clearly, this design implies several advantages compared to electron-based networks, including a reduced decoherence rate due to charge neutrality of atomic currents, the ability to play with fermionic or bosonic carriers and change intrinsic parameters like the carrier-carrier interaction. By investigating quantum transport with such new twists [15][16][17][18], the scope of cold atoms quantum technology can be enlarged for quantum simulation, quantum devices and quantum sensors [19].
There has been much interest in elementary atomtronic circuits made of a bosonic condensate flowing in a ring-shaped guide and pierced by an effective magnetic field [20][21][22][23][24][25][26][27][28][29][30][31]. Recent progress in the integration of the know-how in cold atom quantum technology and micro-optics allows to construct more complex circuits with new functionalities: condensates can be loaded in basically arbitrary potentials with micron-scale resolution [32,33]. In particular, Digital Mi-cromirror Devices (DMD) can update the shape and intensity of optical potentials in time scales of tens to hundreds microseconds. These remarkable advances allow to modulate the local features of the circuit in time during experiments [34][35][36][37].
In this paper we study the local driving of atomtronic circuits to highlight these new perspectives in cold atoms quantum technology. In particular, we analyze the interplay between topological pumping and Aharonov-Bohm (AB) phase winding in interacting bosonic systems for which several new features have been evidenced recently [38,39] (for topological pumping in non-interacting rings see [40,41]). We note that since topological pumping is robust to disorder and particle loss [11,42], by our approach we can further enhance the flexibility of Atomtronics.
Summary of the results. The setup is depicted in Fig.1 see Eqs.1, 2, 3. The system is characterized by topologically distinct bands that can be addressed feasibly by changing the parameter of the driving (its phase φ 0 )-see Fig.2a,b. Interaction, AB oscillations and the topology of the bands couple together in a non-trivial way. As a result, a complex variety of flux quanta, types of entangled states and transmission coefficients are found. In particular, we understand that AB oscillations in our circuit result from transitions between the topological bands. Interestingly enough, the source-ring and the ring-drain interfaces act as non-linear matter-wave interferometers: NOON states between lower and upper arms of the ring can be generated-see Fig.2c. The speed of the state preparation and the robustness to disorder is limited by Landau-Zener transitions between bands; the curvature of the band implies different methods of transitions (resonant and off-resonant) -see Fig.2b. The lowest band is the best band for the transport of N atoms as the band gap scales ∝ √ N. The states are transported through the ring, and interfere in the drain. The transmission depends on the applied AB flux and different (entangled) states make it periodic with a characteristic flux quantum Φ 0 depending on the particle number (see Figs.3,4b): for the lowest topological band and even number of particles the density of the pumped particles does not depend on the AB flux; for odd number of particles, instead, the density of the pumped particles results in a nontrivial interference pattern -see Fig.4a. The central band of the system can generate different types of partial transmission, parity effects and (entangled) states depending on the initial phase shift of the potential and the length of the ring -see Fig.5. By analyzing the dynamics of the system, we evidence that the curvature of the band leads to a partial particle pumping with AB flux. Our results are schematically summarized in Table I.
Model and methods. A sketch of the ring-lead system is presented in Fig.1a whereâ j andâ † j are the annihilation and creation operator at site j in the ring, L R the number of ring sites,n a j =â † jâ j is the particle number operator of the ring, J is the inter-site hopping, U is the on-site interaction between particles and Φ is the total flux through the ring. Periodic boundary conditions are applied for the ring withâ † L R =â † 0 . In the following, we set J = 1, and all values of U, Ω are given in units of J. The lead Hamiltonian for source (similar for drain H D ) is given by where L S (L D ) is the length of the source (drain) andŝ j (d j ) the source (drain) annihilation operator. We set L S = L D = 1 in the following. The coupling Hamiltonian between leads and ring is . We modulate the potential landscape adiabatically to pump particles with the driving frequency Ω, the particle number operator n j at site j of the system and phase shift φ 0 . The potential has a period of three sites and its arrangement in the ring-lead system is shown in Fig.1a.In the case of zero interaction and no flux, the Hamiltonian is known to be topological nontrivial with three bands and non-zero Chern numbers [6] (top +1 and bottom -1 band has Chern number C = −1 and central band 0 ± has C = 2). The pumping is induced by breaking timetranslational symmetry via driving. The topological properties hold true within perturbation theory even in the interacting case [11]. Similar systems have been studied in [11,43]. After one period of the time evolution T = 2π/Ω, particles move by 3C sites (see Fig.1b). The protocol is as follows: Initially, a Fock state with N particles is prepared at a single site in the source lead. Here, we initialize the particle at the first source lead site that directly neighbors the ring and use φ 0 to select the band (see Fig.1a).
The equations of motions are solved with exact diagonalization, by evolving the Schrödinger equation In the following, we investigate positive interaction U > 0 without loosing generality. For U < 0, simply switch the results of band +1 with -1, and 0 + with 0 − . We operate in the limit of P 0 J, U, such that the eigenstates are strongly localized within single sites. Thus, tunneling between neighboring sites is suppressed (since they have in general a widely different local potential), as well as effective tunneling across three sites (to the nearest site with the same potential). To fulfill the adiabatic criterion Ω J, U, P 0 .We characterize the dynamics in terms of transmitted density into the source after a single pumping iteration (when particles reach the drain for the first time).
Topological pumping. For a single particle or U = 0 the flux dependence and smallest energy gap ∆E = 2J is the same for all bands. The speed of pumped particles solely depends on the Chern number of its band. We see the standard AB effect with full transmission through the ring for zero flux, and total reflection due to destructive interference at half-flux. Remarkably, reflections occur by transitions into bands with Chern number of opposite sign, such that the particle moves into the reverse direction. However, we believe they are distinct from Landau-Zener transitions as they occur for even very slow driving frequency.   Fig.1a), the two types of transitions will produce different end results: convex gives NOON states, while concave yields product states.
band ring length Chern φ 0 AB period parity transmission: N even transmission: N odd state in ring band gap 2J   TABLE I. Aharonov-Bohm (AB) period and number of particles transmitted after one topological pumping cycle through an AB ring with initially N particles. Results depend on pumped band, ring length, parity of particle number N and interaction (here U > 0). Bands are visualized in Fig.2a.For U < 0, exchange the band indices + ↔ − (e.g. for U < 0 band +1 behaves like band -1 for U > 0).
We find that for increasing |U| > Ω the standard AB effect disappears, and instead we observe a many-body AB effect. Then, the pumping dynamics substantially depends on the band and the sign of U. The energy gap becomes dependent on band, interaction and particle number as well.
Lower band -1. Here, we choose φ 0 = π -see Fig.2a. In this case, the local driving implies that the avoided crossing is approached from below (the concave band of Fig.2b). The particle transfer from one site to the next is found to happen via resonant transitions to intermediate many-body states, whenever these states become resonant. For U J, the energy gap is independent of U: ∆E = 2 √ N J (see supplementary materials). In this regime, we find that the pumping is parity dependent. The dynamics as particles are pumped through the ring for different particles numbers is plotted in Fig.3. For N = 2n all the particles reach the drain independently of the flux (Fig.3e). For N = 2n + 1, instead, the magnetic flux drives a fractional transmission: For zero flux, all particles are transported to the drain (Fig.3a). However, for half-flux, one particle is reflected and the rest is transmitted (Fig.3c). The back-reflection occurs at twice the speed in the central band with Chern number C = 2. The effect traces back to the different structure of the many-body states that are generated in the ring by the driving. We denote the wavefunction of a single particle in the upper half of the ring as |∩ , and |∪ the wavefunction in the lower half of the ring. For N = 2n, the ring state is |Ψ 2n = (|∩ ⊗ |∪ ) n causing no interference. For N = 2n + 1, the state in the ring has the form |Ψ 2n+1 = (|∩ ⊗ |∪ ) n ⊗ (|∩ + |∪ ); the last part of the wavefunction is the one causing interference. The resulting density pumped into the drain is shown in Fig.4a. Incidentally, we note that the circuit with two atom species can provide entangled Bell states (see the supplementary material); fractional statistics as introduced in Ref. [44] could not generate a different type of entangled states, but it would change the flux quantum. Upper band +1. Here, we choose φ 0 = 0 so that the avoided crossing is approached from above - Fig.2b. The density in the drain after pumping is shown in Fig.4b. In this case, we find that tunneling between neighboring sites occurs via effective tunneling between off-resonant states via nearly resonant states that are weakly occupied during the transition (e.g. for N = 3 and N j N j+1 denoting particles at neighboring sites j: when states |30 and |03 become resonant, they are off-resonantly coupled via the weakly occupied states |21 and |12 ). The effective coupling between the final states can by calculated with the Schrieffer-Wolff transformation [11]. Well defined AB oscillations are found: the AB minimum is at Φ = 1 2N . At the AB minimum, one particle is reflected, while the rest is transmitted for any number of particles or ring sites. The energy gap to the next band is ∆E ∝ J N /U N−1 ([45], see supplementary materials). The gap decreases sharply with increasing interaction U and the pumping is most efficient in the regime of small U. We observe NOON type superposition between upper and lower part of the junction. Central band 0 ± . The dynamics through the ring depends on the initial phase 0 + (φ 0 = π/2) or 0 − (φ 0 = −π/2) (see Fig.2a). The transmission depends on the ring length L R . For L R = 4n + 2 and U > 0, the initial phase φ 0 = π/2 has the same flux dependence as the upper band +1. For φ 0 = −π/2 it behaves as the lower band; however, because of the different Chern number C = 2, the particles move at twice the speed and in opposite direction (thus exchange Ω → −Ω). The anticrossing curvature alternates between convex and concave (see Fig.2a) and the smallest energy gap scales as in band +1.
For L R = 4n, the dynamics is quite different. For the band 0 + (Fig.5a) the AB flux quantum is Φ 0 = 1/N. The transmission is much lower and parity dependent: for even N, the transmission is zero for zero flux, and one particle for half a flux quantum. For odd N, we find the opposite behavior: one particle transmitted at zero flux, and zero transmitted at half flux-quantum. For band 0 − (Fig.5b), the flux quantum is Φ 0 = 1. For even N, the transmission is zero, while for odd N it changes from one to zero with flux. The dependence on L R comes from the switch between convex and concave transitions at every other site: For L R = 4n, the transitions at sourcering and ring-drain approach the avoided crossing from opposite ways; this feature implies the change in transmission behavior. For L R = 4n + 2, instead, the transmissions sourcering and ring-drain approach the avoided crossing from the same ways; this features implies that the transmission is similar to what found for the band ±1. In our numerical simulation, we see a finite probability of reaching the drain for even number of particles due to non-adiabatic transitions. The dependence on interaction is discussed in the supplementary materials.
Conclusions. We studied topological pumping in conjunction with an synthetic magnetic field in an interacting ringlead system. We find that the interplay between topological bands and Aharonov-Bohm phase produces a complex variety of flux quanta, types of entangled states and transmission coefficients. Adiabatic shortcuts could allow to reduce the preparation time for the entangled states [46]. Relying on the established time-dependent light-shaping techniques available in quantum technology, we believe that our results are of immediate interest for the experiments. Our results are also rel-evant for interacting photons in non-linear superconducting resonators. In these systems, topological pumping of interacting photons [11] and synthetic gauge fields [47] has been demonstrated. the next level at the crossing point for convex and concave band curvature. In the following, we derive the many-body energy gap for for concave crossings. We consider a reduce model, consisting of two neighboring sites only. Other sites can be neglected due to the large potential difference. We indicate the many-body states of this two-site model as |N 1 , N 2 , where N 1 (N 2 ) is the number of particles at the first (second) site. For concave, the many-body states are resonantly coupled. As a result, for e.g. N = 3, the following resonant transitions occur one after the other due to the adiabatic driving:

Creating NOON states
Here, we go in further detail how to create NOON states. The relevant part of our setup is the lead-ring junction, where one input site is connected to two output sites via tunneling. We remove all other sites of the setup. We initialize the particles at the input site, and evolve the system. Depending on φ 0 of the potential, the state after the pumping is either a NOON state or a product state. We can see this from the nature of the transitions: If the transition is resonant (concave case, φ 0 = π), the tunneling from one site to the next occurs via occupying the intermediate many-body states one after the other, meaning one after the other particle tunnels over when the state becomes resonant due to the driving. e.g. for N = 2 and same notation as in Fig.2c, |200 → |110 + |101 → |011 . The end result is a product state.
In the case of off-resonant transition (convex case, φ 0 = 0), the tunneling can only occur when the final states have the same energy, e.g. |200 and |020 +|002 . This happens offresonantly via weakly occupied intermediate states. In this case, the particles tunnel together as a whole.

Interaction dependence
In this section we study the transmission into the drain for varying interaction at different values of flux and particle number. For band +1 and -1, see Fig.7. For the central band 0, see . a) 3 particles: for U < 0 AB effect changes particle number in drain between 3 and 2, with minimum for Φ = 1/2. For U > 0 same change in density, however with minimum at Φ = 1/6. b) 4 particles: for U < 0 independent of flux, while for U > 0 minimum at Φ = 1/8. Close to U ≈ 0, minimal density for Φ = 1/2. The number of particle pumped into the drain decreases for larger U due to Landau-Zener transitions as the gap decreases, while for negative interactions it does not change with large negative interactions. The strong oscillations at small U result from nonadiabatic transitions when U ≈ Ω. Drain density taken at tJ = 1260 with L R = 8, Ω = 0.01J and P 0 = 60J.

Two species pumping
One can also consider a different setup: The same Hamiltonian as introduced in the main text with now two species of atoms (denoted as ↑, ↓). There is only one particle of each species. The two species interact with In Fig.9, we present the transmitted density for pumping of 2 species of atoms.