Quantum circuits based on topological pumping in optical lattices
Zijie Zhu Yann Kiefer Samuel Jele Marius Gächter Giacomo Bisson Konrad Viebahn Tilman Esslinger








[]viebahnk@phys.ethz.ch

Institute for Quantum Electronics & Quantum Center, ETH Zurich, 8093 Zurich, Switzerland

§ ABSTRACT 
Gate operations composed in quantum circuits form the basis of digital quantum simulation <cit.> and quantum processing <cit.>.
While two-qubit gates generally operate between nearest neighbours, many circuits require non-local connectivity, necessitating some form of quantum information transport, such as the repeated application of swap gates <cit.> or qubit shuttling <cit.>.
Preserving motional coherence during such transport remains a key challenge to improve gate fidelity <cit.> and qubit connectivity, as well as to connect local fermionic modes <cit.>.
Here we combine tuneable gate operations between fermionic potassium-40 atoms – based on superexchange interaction – with their bidirectional transport via topological Thouless pumping in an optical lattice <cit.>.
We demonstrate shuttling of atomic singlet pairs with a single-shift fidelity of 99.57(4)% over 50 lattice sites.
We spatially and coherently split a large number of randomly distributed fermionic spin singlet pairs and show (swap)^α-gate operations between atoms encountering each other during transport.
As a signature of entanglement between fermions separated over large distances and interwoven with each other, we observe multi-frequency singlet-triplet oscillations.
Topological pumping is generally applicable to long-lived atomic and molecular states, and specifically overcomes lifetime limitations inherent to transport using state-dependent optical lattices <cit.>.
Our work opens up new avenues for transport of quantum information and offers unprecedented possibilities for engineering connectivity in quantum circuits, including approaches based on fermionic modes <cit.>, as well as for atom interferometry <cit.>. 



Quantum circuits based on topological pumping in optical lattices 

    Tilman Esslinger
    Received May 17, 2024; accepted July 29, 2024
====================================================================



Optical lattices are a unique tool to prepare and position a large number of atoms in well-defined internal and motional states <cit.>, including the control over quantum tunnelling between lattice sites.
A crucial step towards employing optical lattices as a framework for quantum processing consists in transporting atoms between distant lattice sites <cit.>, leading to graph connectivity <cit.>.
However, lattice-based atom shuttling, either via state-dependent light shifts <cit.>, or via additional scanning tweezers <cit.>, has remained experimentally challenging, due to motional heating or atom loss.
Advances in the three areas of i) fast generation of degenerate quantum gases <cit.>, ii) dipolar and molecular interaction mechanisms <cit.>, and iii) novel concepts utilising fermionic modes for quantum computing <cit.>, have further increased the urgency to find robust methods to transport quantum information in optical lattices.

In this Letter we introduce bidirectional topological Thouless pumping <cit.> in combination with controlled superexchange interactions <cit.> as a coherent toolbox to transport, separate, interweave, and recombine atomic quantum states in an optical lattice.
The dynamic lattice potential creates an array of independent one-dimensional tubes along x, each realising a Thouless pump along its axis providing robust and state-independent adiabatic transport <cit.>, applicable to any type of polarisable quantum particles.
The lattice is characterized by tunnelling parameters t_x and t_x' on alternating bonds, and alternating lattice sites exhibit an offset energy ±Δ.
Thouless pumping is achieved via a periodic modulation of the lattice potential, cycling through the staggered (Δ=Δ_0, t_x=t_x') and the dimerised (Δ=0, t_x≠ t_x') configurations, which adiabatically shuttles the atoms within the lattice (Methods).
As a source of entanglement, we prepare atomic spin singlet pairs from a quantum degenerate two-component cloud of fermionic potassium-40 atoms inside the dynamical optical lattice.
With the singlet pairs in the lowest band of the topological pump, we unidirectionally transport the pairs across 50 lattice sites with a measured single-shift fidelity of 99.57(4)%.
Conversely, when ramping to strongly repulsive interactions between the atomic states, we adiabatically transfer each singlet pair into a correlated Bell state (|↓,↑⟩-|↑,↓⟩)/√(2) (`Heisenberg singlet'), where |↑,↓⟩ denotes a state where the spin-↑ atom occupies the left site and the spin-↓ atom occupies the right site of a double-well.
When considering the Bloch band basis instead of the Fock basis, exactly one atom occupies the lower and the other occupies the upper band of the topological pump, each experiencing opposite Chern numbers (C = ± 1, Fig. <ref>a).
Topological pumping therefore transports the two correlated components in opposite directions, separating them further with each pumping cycle.






We analyse the entangled two-particle state and its spatial separation in a measurement sequence.
To this end, the pump is stopped after several cycles and a magnetic field gradient is applied in the transport direction for a variable time.
The two spins that form the entangled state, m_F=-9/2 (↓) and -5/2 (↑) [F=9/2], have different magnetic moments.
Therefore, the energy splitting caused by the magnetic gradient increases with spatial separation of the two magnetic moments, leading to oscillations between the singlet and the triplet states (singlet-triplet oscillation <cit.>, STO).
By subsequently reverting the topological pumping directions the two components of the correlated state reconvene at their original position and we determine the singlet fraction of the state using a double occupancy measurement protocol (Methods).

The two counterpropagating modes in the topological pump connect atoms originating from  spatially separated lattice sites and encounters act as two-particle beam splitters with the coupling being adjustable via superexchange interaction between atoms of different spin.
In the language of quantum information, these beam splitters represent partial swap gates (swap)^α, acting on two qubits in the {↑,↓} basis, where α is fully tuneable.
For all values of α∉ℕ, (swap)^α gates are entangling and, when combined with single-qubit rotations <cit.>, would allow the construction of a universal gate set for quantum computation <cit.>.
For α = 1, the atoms pass effectively through each other and the entanglement between pairs can be rearranged, giving rise to spatially interwoven correlation patterns (Fig. <ref>b).
The reversal protocol described above demonstrates that the correlations remain intact through the circuit. 
For α = 2, atoms reflect off each other, giving rise to quantum states that exhibit multi-frequency STOs.
The combination of parallel layers of gate operations constitutes the assembly of quantum circuits using neutral atoms in optical lattices (Fig. <ref>b).

The indistinguishable nature of the fermionic constituents enables two complimentary (but equivalent) interpretations of the action of two-particle gates.
In the first interprepration, operations act on the spin-{↑,↓} degrees of freedom and a swap gate can be understood as two atoms swapping their spin states.
Strongly repulsive interactions prevent atoms from passing through each other, allowing individual atoms to be labelled by their order, which could be used to construct a quantum circuit representation based on effectively distinguishable qubits.
In the second, or `motional', interpretation a swap operation exchanges the atoms' positions and atoms can become delocalised in the lattice through actions such as √( swap), which is relevant for fermionic quantum computing <cit.>.

In a first experiment, we benchmark shuttle and gate operations. We assess the fidelity of shuttling by transporting paired atoms with opposite spins (Fig. <ref>).
Strong attractive interactions during the loading stage ensures that between 60% and 75% of the total number of 5.3(2)×10^4 atoms start off in doubly occupied unit cells.
In the weakly interacting regime (Hubbard U<2Δ_0) paired atoms can be shuttled together by one lattice site every half pump period, which we define as one operation cycle.
This is confirmed by in-situ measurements of the cloud displacement (Fig. <ref>b).
During the shuttling process, double occupancy—i.e., two atoms occupying the same site and orbital—varies in time since the atoms become alternatingly distributed over two sites and localised on one site (Fig. <ref>a).
In particular, the double occupancy reaches its maximum in the staggered lattice configuration (Δ=±Δ_0) and its minimum in the dimerised configuration (Δ=0), as shown in Fig. <ref>c.
Since the spectroscopic measurement of double occupancy is orbital-selective (Methods), this provides a microscopic observable to evaluate the fidelity of the shuttling operation.
A decrease in double occupancy indicates that an atom from a pair has been excited to higher bands, left behind, or has tunnelled in transverse directions.
The experimental sequence for Fig. <ref>c involves pumping forward and then returning to the original position by reversing the pump direction, ensuring consistent detection efficiency in the centre of the trap.
Exponential fits of the double occupancy in the staggered and dimerised lattice configurations yield an fidelity of F=0.9957(4) for a single shuttle operation by one lattice site (Methods).



Interparticle interactions allow the programmable application of gate operations when two atoms moving in opposite directions enter the same double well (Fig. <ref>d).
Specifically, the regime of strong repulsive Hubbard interactions between spins (Hubbard U ≫ t_x, t_x', Δ) gives rise to a `superexchange' process, characterised by J_ex = 4t_x^2/[U(1-(2Δ/U)^2)] <cit.> (Methods).
In our implementation, the superexchange coupling J_ex(τ) is negligible in the staggered configuration (-T/4 and T/4) due to small tunnelling t_x and finite Δ.
It increases to its maximum in the dimerised configuration (0T) and then decreases again (Fig. <ref>), resulting in discrete gate operations, which last half a pump period, equivalent to one operation cycle.
By changing the control parameters between cycles, we can build gate sequences over time.
The superexchange gate operations can be represented on a two-particle Bloch sphere, with the `Heisenberg' singlet state |s⟩=(|↓,↑⟩-|↑,↓⟩)/√(2) and the triplet state |t⟩=(|↓,↑⟩+|↑,↓⟩)/√(2) at the poles.
The equatorial states are the two product states (|↓,↑⟩, |↑,↓⟩), as well as |i_-⟩=(|↓,↑⟩-i|↑,↓⟩)/√(2) and |i_+⟩=(|↓,↑⟩+i|↑,↓⟩)/√(2) (Fig. <ref>e).
The fermionic superexchange Hamiltonian Ĥ_ex=J_exσ̂_z/2 induces a rotation around the z-axis by an angle φ=1/ħ∫_-T/4^T/4J_ex(τ) dτ, where σ̂_z is the third Pauli matrix.
Precise control of φ enables the realisation of the entire family of partial swap gates, such as √( swap) (φ=π/2), swap (φ=π), √( swap)^† (φ=3π/2), and ( swap)^2 (φ=2π) gates (see also refs. <cit.>).
Here we have simplified the gate action by considering only two opposite spins.
However, since all three triplet states are degenerate, the realisation of the partial swap gates remains valid in the full two-particle Hilbert space, including |↓,↓⟩ and |↑,↑⟩ (Methods).

In order to calibrate the gate operations and determine the rotation angle φ, we first prepare atom pairs in the product state |↑,↓⟩. Next, we pump for one half period, realising one gate operation as shown in Fig. <ref>d (-T/4 to T/4), during which the state evolves on the equator of the two-particle Bloch sphere. Finally, we determine φ by measuring the projection of the resulting state along the y-axis of the Bloch sphere (Methods).
We present two different methods to engineer the gate operations. In Fig. <ref>f, we vary the pump period T to control the interaction duration. 
In Fig. <ref>g, we realise different gates by adjusting the lattice depth V_X and thus the superexchange coupling J_ex. 
The insets indicate the parameters used for implementing the respective gates, determined from the fitted curves shown in solid red lines.




With full control of shuttle and gate operations, we are able to implement quantum circuits. The first circuit we realise consists solely of swap gates. It separates the initially prepared singlet pairs and swaps the spin state whenever two atoms meet.
This swap process can also be envisioned as two atoms exchanging positions, effectively passing through each other due to the indistinguishability of the fermions employed.
This is particularly useful for rearrangement purposes, especially given the strongly repulsive interactions between the fermions used here. 
As a result, the output state consists of interwoven singlet pairs, each separated by s=2N_cyc+1 lattice sites, where N_cyc is the number of operation cycles. In Fig. <ref>a, we show an exemplary circuit for N_cyc=3, outputting singlet pairs separated by 7 sites.
To measure the separation between two entangled atoms from an initial singlet pair, we apply a magnetic gradient Δ B for a specified time τ_STO.
This produces an energy offset Δ_↑↓∝Δ B × s between |↓,↑⟩ and |↑,↓⟩ states, corresponding to the Hamiltonian Ĥ_STO=Δ _↑↓σ̂_x/2 on the two-particle Bloch sphere (Fig. <ref>e), where σ̂_x is the first Pauli matrix. 
This leads to an oscillation between the singlet and triplet states at a frequency of Δ_↑↓/h, which is proportional to the separation s (Fig. <ref>b). 
After the STO, we reverse the pump to bring the entangled pairs back together for detection.
This sequence can be viewed as a many-particle interferometer.

In Fig. <ref>c, we present singlet-triplet oscillations for increasing numbers of operation cycles ranging from N_cyc=0 to N_cyc=5, corresponding to increasing atom separations s. We determine the oscillation frequency by fitting a sinusoidal curve to each time trace.
The results are shown in Fig. <ref>d, along with additional data up to s=19 sites, whose corresponding time traces are plotted in Fig. <ref>.
All measurements in Fig. <ref>c and Fig. <ref> are performed under the same gradient Δ B, whereas a reduction in contrast as function of s is attributed to fluctuating lattice depths and resulting deviations from perfect swap-operations.
A proportional fit of the data yields a slope of f_1=216.5(6)Hz, which gives the STO base frequency corresponding to entangled pairs on adjacent lattice sites. Pairs separated by s sites thus exhibit STOs at a frequency of s× f_1. We then map the time trace to the distribution of separations s by calculating the Fourier spectrum of the time trace.
Given that s must be an integer in lattice systems, we apply a multi-frequency sinusoidal fit F_singlet(τ)=Σ_sA_ssin(2π s f_1τ+θ_s), incorporating a global damping and offset.
The amplitude A_s is then proportional to the fraction of singlet pairs separated by a certain distance s.
In Fig. <ref>e, we show the amplitude A_s as a function of s from 1 to 12 lattice sites for N_cyc=0, 2, 3, 5. The dominant peaks at s=1, 5, 7, and 11 sites demonstrate the ability to coherently split fermionic singlet pairs by programmable distances.

We now turn to more complex quantum circuits that involve combinations of swap and ( swap)^2 gates. A ( swap)^2 gate does not alter the spin states when two atoms meet and can thus be considered a reflection. Incorporating ( swap)^2 processes into the circuit results in more intricate STO waveforms, indicating a non-trivial distribution of singlet pairs.
In Fig. <ref>, we present the measured STOs of three different circuits with five gate operation cycles each (N_cyc=5).
In the first example, we insert a single parallel ( swap)^2 operation at the second position of an otherwise purely swap sequence, leading to the emergence of multiple frequency components in the STO signal (Fig. <ref>a). We analyse the time trace analogously to Fig. <ref>e, using an independently calibrated base frequency of f_1'=218(1)Hz.
The resulting amplitudes A_s, representing the proportion of singlet pairs separated by s lattice sites, are shown in Fig. <ref>d.
We also apply a fast Fourier transform to the time trace (Fig. <ref>), which agrees with the fit.
We observe a significant contribution at s=4, in addition to s=2×5+1=11 lattice sites.
This can be explained by considering that during the initial operation cycle, two atoms originating from a singlet pair are separated by 2×1+1=3 lattice sites.
In the second operation cycle, one atom can be reflected while the other continues moving, resulting in a separation of 3+1=4 sites.
From then on, the two atoms shift in the same direction and their separation remains constant.
If both atoms originating from a singlet pair are reflected in the second operation cycle, they will be shuttled in opposite directions during the next three cycles, resulting in a separation of | 3-2×3 |=3 lattice sites.
This scenario is considerably less probable, in agreement with the low measured value at s=3. 
Both scenarios can be identified with trajectories in the circuit diagram shown in Fig. <ref>g.
The occurrence count of specific distances in the schematic (output) does not directly reflect the amplitudes A_s since the input state of the circuit is randomly initialised for every experimental realisation.
While STOs have been also observed on neighbouring lattice sites with SU(N) fermions <cit.>, in our experiment the occurrence of multiple frequencies is a direct consequence of the spatial distribution of singlet pairs.

Changing the gate composition of quantum circuits leads to additional contributions to the multi-frequency STOs.
For instance, Fig. <ref>b shows the time trace for a situation with two parallel ( swap)^2 gates, at the third and fourth position in the sequence.
The process involving a single reflection at N_cyc=3 or 4 results in two major contributions to the STO signal, corresponding to s=6 and s=8 lattice sites.
The process involving multiple reflections gives rise to signals at s=1, 3, 4, etc.
In addition, we implement a quantum circuit consisting solely of ( swap)^2 gates.
In this configuration, neighbouring singlets are constraining each other, generally resulting in small separations.
The time trace (Fig. <ref>c) and the fitted amplitudes A_s (Fig. <ref>f) confirm that the multi-frequency STO is dominated by low-frequency components, corresponding to small values of s.
The remaining fraction of singlets separated by s=11 lattice sites can be attributed to low-density regions near the edges of the system and imperfections in the gate operations.
The striking revival feature in the all-( swap)^2 time trace (Fig. <ref>c) shows the harmonicity of the frequencies and their phase coherence.
Considering that the signal is contributed by more than ten thousand individual entangled pairs, such phase coherence highlights the potential of topological pumping for scaling up quantum circuits.

In conclusion, we have experimentally achieved high-fidelity shuttle operations of indistinguishable fermionic atoms using topological pumping in an optical lattice.
Both motional ground-state coherence and the entanglement in the spin sector remain preserved during pumping, enabling us to transport entangled atoms together or separate them by tens of lattice sites. Utilising superexchange interactions, we further realise two-qubit (swap)^α gates, facilitating the construction of programmable quantum circuits. 
Compared to other platform targeting universal quantum computation, neutral atoms in lattices have a natural advantage in terms of atomic qubit density, scalability, and the ability to perform parallel operations <cit.>.
By improving the connectivity in optical lattices using topological pumping, our work opens up new possibilities for quantum information processing based on atoms and molecules.
This includes applications in fermionic quantum computing <cit.>, symmetry-protected operations <cit.>, exchange-only qubits <cit.>, and computing based on quantum walks <cit.>. 



§ ACKNOWLEDGEMENTS


We would like to thank Alex Baumgärtner and Peter Zoller for comments on a previous version of the  manuscript.
We thank Alexander Frank for assistance with electronics equipment.
We acknowledge funding by the Swiss National Science Foundation (Grant No. 200020_212168, Advanced grant TMAG-2_209376, as well as Holograph UeM019-5.1), as well as Quantera dynamite PCI2022 132919.











56
fxundefined [1]
 ifx#1

fnum [1]
 #1firstoftwo
 secondoftwo
 
fx [1]
 #1firstoftwo
 secondoftwo
 





noop [0]secondoftwo

ref[1]@startlink#1@href
href[1]#1@endlink
anitize@url [0]`
12`$12`&12`#12`1̂2`_12`%12
startlink[1]
endlink[0]

rl [1]href #1












@bib@innerbibempty

[Daley et al.(2022)Daley,
  Bloch, Kokail, Flannigan,
  Pearson, Troyer, and Zoller]daley_practical_2022
  author author A. J. Daley, author I. Bloch,
  author C. Kokail, author S. Flannigan, author
  N. Pearson, author M. Troyer, and author P. Zoller, title title
  Practical quantum advantage in quantum simulation, https://doi.org/10.1038/s41586-022-04940-6 journal journal Nature volume 607, pages
  667 (year 2022)NoStop
[Bharti et al.(2022)Bharti,
  Cervera-Lierta, Kyaw, Haug,
  Alperin-Lea, Anand, Degroote,
  Heimonen, Kottmann, Menke,
  Mok, Sim, Kwek, and  Aspuru-Guzik]bharti_noisy_2022
  author author K. Bharti, author A. Cervera-Lierta, author T. H. Kyaw, author T. Haug, author S. Alperin-Lea, author
  A. Anand, author M. Degroote, author H. Heimonen, author J. S.  Kottmann, author T. Menke, author W.-K. Mok,
  author S. Sim, author
  L.-C. Kwek, and author
  A. Aspuru-Guzik, title
  title Noisy intermediate-scale quantum algorithms, https://doi.org/10.1103/RevModPhys.94.015004 journal
  journal Reviews of Modern Physics volume
  94, pages 015004 (year 2022)NoStop
[Loss and DiVincenzo(1998)]loss_quantum_1998
  author author D. Loss and author D. P. DiVincenzo, title title Quantum computation
  with quantum dots, https://doi.org/10.1103/PhysRevA.57.120
  journal journal Physical Review A volume 57, pages 120 (year
  1998)NoStop
[Kandel et al.(2019)Kandel,
  Qiao, Fallahi, Gardner,
  Manfra, and Nichol]kandel_coherent_2019
  author author Y. P. Kandel, author H. Qiao,
  author S. Fallahi, author G. C. Gardner, author
  M. J. Manfra, and author
  J. M. Nichol, title
  title Coherent spin-state transfer via Heisenberg exchange,  https://doi.org/10.1038/s41586-019-1566-8 journal
  journal Nature volume 573, pages 553 (year 2019)NoStop
[Beugnon et al.(2007)Beugnon, Tuchendler, Marion, Gaëtan, Miroshnychenko, Sortais,
  Lance, Jones, Messin,
  Browaeys, and Grangier]beugnon_two-dimensional_2007
  author author J. Beugnon, author C. Tuchendler,
  author H. Marion, author A. Gaëtan, author
  Y. Miroshnychenko, author
  Y. R. P. Sortais, author
  A. M. Lance, author
  M. P. A. Jones, author
  G. Messin, author A. Browaeys, and author P. Grangier, title title
  Two-dimensional transport and transfer of a single atomic qubit in optical
  tweezers, https://doi.org/10.1038/nphys698 journal
  journal Nature Physics volume 3,  pages 696 (year 2007)NoStop
[Moses et al.(2023)Moses,
  Baldwin, Allman, Ancona,
  Ascarrunz, Barnes, Bartolotta, Bjork, Blanchard, Bohn, Bohnet, Brown, Burdick, Burton, Campbell, Campora, Carron, Chambers, Chan, Chen, Chernoguzov, Chertkov, Colina, Curtis, Daniel, DeCross, Deen, Delaney, Dreiling, Ertsgaard, Esposito, Estey, Fabrikant, Figgatt, Foltz, Foss-Feig, Francois, Gaebler, Gatterman, Gilbreth, Giles, Glynn, Hall, Hankin, Hansen, Hayes, Higashi, Hoffman, Horning, Hout, Jacobs, Johansen, Jones, Karcz, Klein, Lauria, Lee, Liefer,
  Lu, Lucchetti, Lytle,
  Malm, Matheny, Mathewson,
  Mayer, Miller, Mills,
  Neyenhuis, Nugent, Olson,
  Parks, Price, Price,
  Pugh, Ransford, Reed,
  Roman, Rowe, Ryan-Anderson,
  Sanders, Sedlacek, Shevchuk,
  Siegfried, Skripka, Spaun,
  Sprenkle, Stutz, Swallows,
  Tobey, Tran, Tran,
  Vogt, Volin, Walker,
  Zolot, and Pino]moses_race-track_2023
  author author S. Moses, author C. Baldwin,
  author M. Allman, author R. Ancona, author
  L. Ascarrunz, author
  C. Barnes, author J. Bartolotta, author B. Bjork, author P. Blanchard, author M. Bohn, author J. Bohnet, author N. Brown,
  author N. Burdick, author W. Burton, author
  S. Campbell, author
  J. Campora, author C. Carron, author J. Chambers, author J. Chan, author Y. Chen, author A. Chernoguzov,
  author E. Chertkov, author J. Colina, author
  J. Curtis, author R. Daniel, author M. DeCross, author D. Deen, author C. Delaney, author J. Dreiling,
  author C. Ertsgaard, author J. Esposito, author
  B. Estey, author M. Fabrikant, author C. Figgatt, author C. Foltz, author M. Foss-Feig, author D. Francois, author J. Gaebler, author T. Gatterman, author C. Gilbreth, author J. Giles, author E. Glynn, author A. Hall, author A. Hankin, author A. Hansen,
  author D. Hayes, author B. Higashi, author
  I. Hoffman, author B. Horning, author J. Hout, author R. Jacobs, author J. Johansen,
  author L. Jones, author J. Karcz, author
  T. Klein, author P. Lauria, author P. Lee, author D. Liefer, author S. Lu, author D. Lucchetti, author
  C. Lytle, author A. Malm, author M. Matheny, author B. Mathewson,
  author K. Mayer, author D. Miller, author
  M. Mills, author B. Neyenhuis, author L. Nugent, author S. Olson, author J. Parks, author G. Price, author Z. Price, author M. Pugh, author A. Ransford, author A. Reed,
  author C. Roman, author M. Rowe, author
  C. Ryan-Anderson, author
  S. Sanders, author J. Sedlacek, author P. Shevchuk, author P. Siegfried, author T. Skripka, author B. Spaun, author R. Sprenkle, author R. Stutz, author M. Swallows, author R. Tobey, author A. Tran, author T. Tran, author E. Vogt, author C. Volin, author
  J. Walker, author A. Zolot, and author J. Pino, title title A
  Race-Track Trapped-Ion Quantum Processor, https://doi.org/10.1103/PhysRevX.13.041052 journal journal Physical Review X volume 13, pages 041052 (year 2023)NoStop
[Bluvstein et al.(2023)Bluvstein, Evered, Geim, Li, Zhou, Manovitz, Ebadi,
  Cain, Kalinowski, Hangleiter,
  Bonilla Ataides, Maskara, Cong, Gao, Sales Rodriguez, Karolyshyn, Semeghini, Gullans,
  Greiner, Vuletić, and Lukin]bluvstein_logical_2023
  author author D. Bluvstein, author S. J. Evered, author A. A. Geim,
  author S. H. Li, author H. Zhou, author
  T. Manovitz, author
  S. Ebadi, author M. Cain, author M. Kalinowski, author D. Hangleiter, author J. P. Bonilla Ataides, author N. Maskara, author I. Cong,
  author X. Gao, author
  P. Sales Rodriguez, author
  T. Karolyshyn, author
  G. Semeghini, author
  M. J. Gullans, author
  M. Greiner, author V. Vuletić, and author M. D. Lukin, title title
  Logical quantum processor based on reconfigurable atom arrays, https://doi.org/10.1038/s41586-023-06927-3 journal journal Nature volume 626, pages
  58 (year 2023)NoStop
[Jenkins et al.(2022)Jenkins, Lis, Senoo, McGrew, and Kaufman]jenkins_ytterbium_2022
  author author A. Jenkins, author J. W. Lis,
  author A. Senoo, author W. F. McGrew, and author A. M. Kaufman, title title Ytterbium Nuclear-Spin Qubits in an
  Optical Tweezer Array, https://doi.org/10.1103/PhysRevX.12.021027 journal journal Physical Review X volume 12, pages 021027 (year 2022)NoStop
[Bravyi and Kitaev(2002)]bravyi_fermionic_2002
  author author S. B. Bravyi and author A. Y. Kitaev, title title Fermionic Quantum
  Computation, https://doi.org/10.1006/aphy.2002.6254 journal journal Annals of Physics volume 298, pages 210 (year
  2002)NoStop
[Oka and Kitamura(2019)]oka_floquet_2019
  author author T. Oka and author S. Kitamura,  title title Floquet Engineering of Quantum
  Materials, https://doi.org/10.1146/annurev-conmatphys-031218-013423 journal journal Annual Review of Condensed Matter Physics  volume 10, pages 387 (year
  2019)NoStop
[Citro and Aidelsburger(2023)]citro_thouless_2023
  author author R. Citro and author M. Aidelsburger, title title Thouless pumping and
  topology, https://doi.org/10.1038/s42254-022-00545-0 journal journal Nature Reviews Physics volume 5, pages 87 (year 2023)NoStop
[Nakajima et al.(2016)Nakajima, Tomita, Taie, Ichinose, Ozawa, Wang, Troyer, and Takahashi]nakajima_topological_2016
  author author S. Nakajima, author T. Tomita,
  author S. Taie, author
  T. Ichinose, author
  H. Ozawa, author L. Wang, author M. Troyer, and author Y. Takahashi, title title Topological Thouless
  pumping of ultracold fermions, https://doi.org/10.1038/nphys3622
  journal journal Nature Physics volume 12, pages 296 (year
  2016)NoStop
[Lohse et al.(2016)Lohse,
  Schweizer, Zilberberg, Aidelsburger, and Bloch]lohse_thouless_2016
  author author M. Lohse, author C. Schweizer,
  author O. Zilberberg, author M. Aidelsburger, and author I. Bloch, title
  title A Thouless quantum pump with ultracold bosonic atoms in
  an optical superlattice, https://doi.org/doi.org/10.1038/nphys3584
  journal journal Nature Physics volume 12, pages 350 (year
  2016)NoStop
[Koepsell et al.(2020)Koepsell, Hirthe, Bourgund, Sompet, Vijayan, Salomon, Gross, and Bloch]koepsell_robust_2020
  author author J. Koepsell, author S. Hirthe,
  author D. Bourgund, author P. Sompet, author
  J. Vijayan, author G. Salomon, author C. Gross, and author I. Bloch, title title Robust
  Bilayer Charge Pumping for Spin- and Density-Resolved Quantum
  Gas Microscopy, https://doi.org/10.1103/PhysRevLett.125.010403
  journal journal Physical Review Letters  volume 125, pages 010403 (year 2020)NoStop
[Minguzzi et al.(2022)Minguzzi, Zhu, Sandholzer, Walter, Viebahn, and Esslinger]minguzzi_topological_2022
  author author J. Minguzzi, author Z. Zhu,
  author K. Sandholzer, author A.-S. Walter, author
  K. Viebahn, and author
  T. Esslinger, title title Topological Pumping in a Floquet-Bloch Band, https://doi.org/10.1103/PhysRevLett.129.053201 journal
  journal Physical Review Letters volume
  129, pages 053201 (year 2022)NoStop
[Mandel et al.(2003)Mandel,
  Greiner, Widera, Rom,
  Hänsch, and Bloch]mandel_controlled_2003
  author author O. Mandel, author M. Greiner,
  author A. Widera, author T. Rom, author
  T. W. Hänsch, and author
  I. Bloch, title title Controlled collisions for multi-particle entanglement of optically
  trapped atoms, https://doi.org/10.1038/nature02008 journal journal Nature volume 425,  pages 937 (year 2003)NoStop
[Steffen et al.(2012)Steffen, Alberti, Alt, Belmechri, Hild, Karski, Widera, and Meschede]steffen_digital_2012
  author author A. Steffen, author A. Alberti,
  author W. Alt, author
  N. Belmechri, author
  S. Hild, author M. Karski, author A. Widera, and author D. Meschede, title title
  Digital atom interferometer with single particle control on a discretized
  space-time geometry, https://doi.org/10.1073/pnas.1204285109
  journal journal Proceedings of the National
  Academy of Sciences volume 109, pages
  9770 (year 2012)NoStop
[Robens et al.(2017)Robens,
  Zopes, Alt, Brakhane,
  Meschede, and Alberti]robens_low-entropy_2017
  author author C. Robens, author J. Zopes,
  author W. Alt, author
  S. Brakhane, author
  D. Meschede, and author
  A. Alberti, title title Low-Entropy States of Neutral Atoms in
  Polarization-Synthesized Optical Lattices, https://doi.org/10.1103/PhysRevLett.118.065302 journal
  journal Physical Review Letters volume
  118, pages 065302 (year 2017)NoStop
[Kumar et al.(2018)Kumar,
  Wu, Giraldo, and Weiss]kumar_sorting_2018
  author author A. Kumar, author T.-Y. Wu,
  author F. Giraldo, and author D. S. Weiss, title
  title Sorting ultracold atoms in a three-dimensional optical
  lattice in a realization of Maxwell’s demon, https://doi.org/10.1038/s41586-018-0458-7 journal journal Nature volume 561, pages
  83 (year 2018)NoStop
[González-Cuadra et al.(2023)González-Cuadra, Bluvstein, Kalinowski,
  Kaubruegger, Maskara, Naldesi, Zache, Kaufman, Lukin, Pichler, Vermersch, Ye, and Zoller]gonzalez-cuadra_fermionic_2023
  author author D. González-Cuadra, author D. Bluvstein, author M. Kalinowski, author R. Kaubruegger, author N. Maskara, author P. Naldesi,
  author T. V. Zache, author A. M. Kaufman, author
  M. D. Lukin, author
  H. Pichler, author B. Vermersch, author J. Ye, and author P. Zoller, title title
  Fermionic quantum processing with programmable neutral atom arrays, https://doi.org/10.1073/pnas.2304294120 journal journal Proceedings of the National Academy of Sciences volume 120, pages e2304294120 (year
  2023)NoStop
[Luo et al.(2024)Luo,
  Zhang, Koh, Wilson,
  Chu, Holland, Rey, and  Thompson]luo_momentum-exchange_2024
  author author C. Luo, author H. Zhang, author V. P. W. Koh, author
  J. D. Wilson, author
  A. Chu, author M. J. Holland, author A. M.  Rey, and author J. K.  Thompson, title title
  Momentum-exchange interactions in a Bragg atom interferometer suppress
  Doppler dephasing, https://doi.org/10.1126/science.adi1393
  journal journal Science volume 384, pages 551 (year
  2024)NoStop
[Panda et al.(2024)Panda,
  Tao, Egelhoff, Ceja,
  Xu, and Müller]panda_coherence_2024
  author author C. D. Panda, author M. Tao, author J. Egelhoff, author
  M. Ceja, author V. Xu, and author H. Müller, title title
  Coherence limits in lattice atom interferometry at the one-minute scale,  https://doi.org/10.1038/s41567-024-02518-9 journal
  journal Nature Physics volume 20,  pages 1234 (year 2024)NoStop
[Ludlow et al.(2015)Ludlow,
  Boyd, Ye, Peik, and  Schmidt]ludlow_optical_2015
  author author A. D. Ludlow, author M. M. Boyd,
  author J. Ye, author
  E. Peik, and author
  P. Schmidt, title title Optical atomic clocks, https://doi.org/10.1103/RevModPhys.87.637 journal journal Reviews of Modern Physics volume 87,  pages 637 (year 2015)NoStop
[Gross and Bloch(2017)]gross_quantum_2017
  author author C. Gross and author I. Bloch,  title title Quantum simulations with ultracold
  atoms in optical lattices, https://doi.org/10.1126/science.aal3837
  journal journal Science volume 357, pages 995 (year
  2017)NoStop
[Hartke et al.(2022)Hartke,
  Oreg, Jia, and Zwierlein]hartke_quantum_2022
  author author T. Hartke, author B. Oreg,
  author N. Jia, and author M. Zwierlein, title
  title Quantum register of fermion pairs, https://doi.org/10.1038/s41586-021-04205-8 journal journal Nature volume 601, pages
  537 (year 2022)NoStop
[Jaksch et al.(1999)Jaksch,
  Briegel, Cirac, Gardiner,  and Zoller]jaksch_entanglement_1999
  author author D. Jaksch, author H.-J. Briegel,
  author J. I. Cirac, author C. W. Gardiner, and author P. Zoller, title
  title Entanglement of Atoms via Cold Controlled
  Collisions, https://doi.org/10.1103/PhysRevLett.82.1975
  journal journal Physical Review Letters  volume 82, pages 1975 (year
  1999)NoStop
[Daley et al.(2008)Daley,
  Boyd, Ye, and Zoller]daley_quantum_2008
  author author A. J. Daley, author M. M. Boyd,
  author J. Ye, and author P. Zoller, title
  title Quantum Computing with Alkaline-Earth-Metal
  Atoms, https://doi.org/10.1103/PhysRevLett.101.170504 journal journal Physical Review Letters volume 101, pages 170504 (year
  2008)NoStop
[Young et al.(2024)Young,
  Geller, Eckner, Schine,
  Glancy, Knill, and Kaufman]young_atomic_2024
  author author A. W. Young, author S. Geller,
  author W. J. Eckner, author N. Schine, author
  S. Glancy, author E. Knill, and author A. M.  Kaufman, title title An atomic
  boson sampler, https://doi.org/10.1038/s41586-024-07304-4
  journal journal Nature volume 629, pages 311 (year
  2024)NoStop
[Scholl et al.(2023)Scholl,
  Shaw, Finkelstein, Tsai,
  Choi, and Endres]scholl_erasure-cooling_2023
  author author P. Scholl, author A. L. Shaw,
  author R. Finkelstein, author R. B.-S. Tsai, author
  J. Choi, and author
  M. Endres, http://arxiv.org/abs/2311.15580 title Erasure-cooling,
  control, and hyper-entanglement of motion in optical tweezers (year 2023), note arXiv:2311.15580 [cond-mat,
  physics:physics, physics:quant-ph]NoStop
[Norcia et al.(2024)Norcia,
  Kim, Cairncross, Stone,
  Ryou, Jaffe, Brown,
  Barnes, Battaglino, Bohdanowicz, Brown, Cassella, Chen, Coxe, Crow, Epstein,
  Griger, Halperin, Hummel,
  Jones, Kindem, King,
  Kotru, Lauigan, Li,
  Lu, Megidish, Marjanovic,
  McDonald, Mittiga, Muniz,
  Narayanaswami, Nishiguchi, Paule, Pawlak, Peng, Pudenz, Rodríguez Pérez, Smull,
  Stack, Urbanek, Van
  De Veerdonk, Vendeiro, Wadleigh,
  Wilkason, Wu, Xie,
  Zalys-Geller, Zhang, and Bloom]norcia_iterative_2024
  author author M. A. Norcia, author H. Kim,
  author W. B. Cairncross,
  author M. Stone, author A. Ryou, author
  M. Jaffe, author M. O. Brown, author K. Barnes, author P. Battaglino, author T. C.  Bohdanowicz, author A. Brown, author K. Cassella, author C.-A.  Chen, author R. Coxe, author D. Crow, author J. Epstein, author
  C. Griger, author E. Halperin, author F. Hummel, author A. M. W.  Jones, author J. M. Kindem, author J. King,
  author K. Kotru, author J. Lauigan, author
  M. Li, author M. Lu, author E. Megidish, author J. Marjanovic, author M. McDonald, author T. Mittiga,
  author J. A. Muniz, author S. Narayanaswami, author
  C. Nishiguchi, author
  T. Paule, author K. A. Pawlak, author L. S.  Peng, author K. L. Pudenz, author D. Rodríguez Pérez, author A. Smull, author D. Stack,
  author M. Urbanek, author R. J. M. Van De Veerdonk, author Z. Vendeiro, author
  L. Wadleigh, author
  T. Wilkason, author
  T.-Y. Wu, author X. Xie, author E. Zalys-Geller, author X. Zhang, and author B. J. Bloom, title title Iterative Assembly of 171
  Yb Atom Arrays with Cavity-Enhanced Optical Lattices, https://doi.org/10.1103/PRXQuantum.5.030316 journal journal PRX Quantum volume 5, pages
  030316 (year 2024)NoStop
[Gyger et al.(2024)Gyger,
  Ammenwerth, Tao, Timme,
  Snigirev, Bloch, and Zeiher]gyger_continuous_2024
  author author F. Gyger, author M. Ammenwerth,
  author R. Tao, author
  H. Timme, author S. Snigirev, author I. Bloch, and author J. Zeiher, title title
  Continuous operation of large-scale atom arrays in optical lattices,  https://doi.org/10.1103/PhysRevResearch.6.033104 journal journal Physical Review Research volume 6, pages 033104 (year
  2024)NoStop
[Urvoy et al.(2019)Urvoy,
  Vendeiro, Ramette, Adiyatullin, and Vuletić]urvoy_direct_2019
  author author A. Urvoy, author Z. Vendeiro,
  author J. Ramette, author A. Adiyatullin, and author V. Vuletić, title
  title Direct Laser Cooling to Bose-Einstein
  Condensation in a Dipole Trap, https://doi.org/10.1103/PhysRevLett.122.203202 journal
  journal Physical Review Letters volume
  122, pages 203202 (year 2019)NoStop
[Phelps et al.(2020)Phelps,
  Hébert, Krahn, Dickerson,
  Öztürk, Ebadi, Su, and  Greiner]phelps_sub-second_2020
  author author G. A. Phelps, author A. Hébert,
  author A. Krahn, author S. Dickerson, author
  F. Öztürk, author
  S. Ebadi, author L. Su, and author M. Greiner, http://arxiv.org/abs/2007.10807
  title Sub-second production of a quantum degenerate gas
  (year 2020), note arXiv:2007.10807 [cond-mat,
  physics:physics]NoStop
[Pür et al.(2023)Pür,
  Hetzel, Quensen, Hüper,
  Geng, Kruse, Ertmer, and  Klempt]pur_rapid_2023
  author author C. Pür, author M. Hetzel,
  author M. Quensen, author A. Hüper, author
  J. Geng, author J. Kruse, author W. Ertmer, and author C. Klempt, title title Rapid
  generation and number-resolved detection of spinor rubidium Bose-Einstein
  condensates, https://doi.org/10.1103/PhysRevA.107.033303
  journal journal Physical Review A volume 107, pages 033303 (year
  2023)NoStop
[Chomaz et al.(2023)Chomaz,
  Ferrier-Barbut, Ferlaino, Laburthe-Tolra, Lev, and Pfau]chomaz_dipolar_2023
  author author L. Chomaz, author I. Ferrier-Barbut, author F. Ferlaino, author B. Laburthe-Tolra, author B. L. Lev, and author T. Pfau,  title title Dipolar physics: a review of
  experiments with magnetic quantum gases, https://doi.org/10.1088/1361-6633/aca814 journal journal Reports on Progress in Physics volume
  86, pages 026401 (year 2023)NoStop
[Cornish et al.(2024)Cornish, Tarbutt, and Hazzard]cornish_quantum_2024
  author author S. L. Cornish, author M. R. Tarbutt, and author K. R. A. Hazzard, title title Quantum computation and
  quantum simulation with ultracold molecules, https://doi.org/10.1038/s41567-024-02453-9 journal journal Nature Physics volume 20, pages 730 (year 2024)NoStop
[Trotzky et al.(2008)Trotzky, Cheinet, Fölling, Feld, Schnorrberger, Rey, Polkovnikov, Demler, Lukin, and  Bloch]trotzky_time-resolved_2008
  author author S. Trotzky, author P. Cheinet,
  author S. Fölling, author M. Feld, author
  U. Schnorrberger, author
  A. M. Rey, author A. Polkovnikov, author E. A.  Demler, author M. D.  Lukin, and author I. Bloch, title title
  Time-resolved observation and control of superexchange interactions with
  ultracold atoms in optical lattices, @noop  journal
  journal Science volume 319, pages 295 (year 2008)NoStop
[Greif et al.(2013)Greif,
  Uehlinger, Jotzu, Tarruell,  and Esslinger]greif_short-range_2013
  author author D. Greif, author T. Uehlinger,
  author G. Jotzu, author L. Tarruell, and author T. Esslinger, title
  title Short-Range Quantum Magnetism of Ultracold
  Fermions in an Optical Lattice, https://doi.org/10.1126/science.1236362 journal journal Science volume 340, pages
  1307 (year 2013)NoStop
[Dai et al.(2016)Dai,
  Yang, Reingruber, Xu,
  Jiang, Chen, Yuan, and  Pan]dai_generation_2016
  author author H.-N. Dai, author B. Yang, author A. Reingruber, author
  X.-F. Xu, author X. Jiang, author Y.-A.  Chen, author Z.-S. Yuan, and author J.-W. Pan, title title Generation and detection of
  atomic spin entanglement in optical lattices, https://doi.org/10.1038/nphys3705 journal journal Nature Physics volume 12, pages 783 (year 2016)NoStop
[Barmettler et al.(2008)Barmettler, Rey, Demler, Lukin, Bloch, and Gritsev]barmettler_quantum_2008
  author author P. Barmettler, author A. M. Rey,
  author E. Demler, author M. D. Lukin, author
  I. Bloch, and author
  V. Gritsev, title title Quantum many-body dynamics of coupled double-well superlattices,  https://doi.org/10.1103/PhysRevA.78.012330 journal
  journal Physical Review A volume 78,  pages 012330 (year 2008)NoStop
[Vaucher et al.(2008)Vaucher, Nunnenkamp, and Jaksch]vaucher_creation_2008
  author author B. Vaucher, author A. Nunnenkamp, and author D. Jaksch, title title Creation of resilient
  entangled states and a resource for measurement-based quantum computation
  with optical superlattices, https://doi.org/10.1088/1367-2630/10/2/023005 journal
  journal New Journal of Physics volume
  10, pages 023005 (year 2008)NoStop
[Zhang et al.(2023)Zhang,
  He, Sun, Zheng, Liu, Luo, Wang, Zhu,
  Qiu, Shen, Wang,
  Lin, Yu, Li, Xiao, Li, Yang, Jiang,
  Dai, Zhou, Ma, Yuan, and Pan]zhang_scalable_2023
  author author W.-Y. Zhang, author M.-G. He,
  author H. Sun, author
  Y.-G. Zheng, author
  Y. Liu, author A. Luo, author H.-Y.  Wang, author Z.-H. Zhu, author P.-Y. Qiu,
  author Y.-C. Shen, author X.-K. Wang, author
  W. Lin, author S.-T. Yu, author B.-C.  Li, author B. Xiao, author M.-D. Li,
  author Y.-M. Yang, author X. Jiang, author
  H.-N. Dai, author Y. Zhou, author X. Ma, author Z.-S. Yuan, and  author J.-W. Pan, title title Scalable Multipartite Entanglement Created
  by Spin Exchange in an Optical Lattice, https://doi.org/10.1103/PhysRevLett.131.073401 journal
  journal Physical Review Letters volume
  131, pages 073401 (year 2023)NoStop
[Walter et al.(2023)Walter,
  Zhu, Gächter, Minguzzi,
  Roschinski, Sandholzer, Viebahn, and Esslinger]walter_quantization_2023
  author author A.-S. Walter, author Z. Zhu,
  author M. Gächter, author J. Minguzzi, author
  S. Roschinski, author
  K. Sandholzer, author
  K. Viebahn, and author
  T. Esslinger, title title Quantization and its breakdown in a Hubbard–Thouless pump,  https://doi.org/10.1038/s41567-023-02145-w journal
  journal Nature Physics volume 19,  pages 1471 (year 2023)NoStop
[Zhu et al.(2024)Zhu,
  Gächter, Walter, Viebahn,  and Esslinger]zhu_reversal_2024
  author author Z. Zhu, author M. Gächter,
  author A.-S. Walter, author K. Viebahn, and author
  T. Esslinger, title title Reversal of quantized Hall drifts at noninteracting and
  interacting topological boundaries, https://doi.org/10.1126/science.adg3848 journal journal Science volume 384, pages
  317 (year 2024)NoStop
[Viebahn et al.(2024)Viebahn, Walter, Bertok, Zhu, Gächter, Aligia, Heidrich-Meisner, and Esslinger]viebahn_interactions_2024
  author author K. Viebahn, author A.-S. Walter,
  author E. Bertok, author Z. Zhu, author
  M. Gächter, author
  A. A. Aligia, author
  F. Heidrich-Meisner, and author
  T. Esslinger, title title Interactions Enable Thouless Pumping in a Nonsliding
  Lattice, https://doi.org/10.1103/PhysRevX.14.021049 journal journal Physical Review X volume 14, pages 021049 (year
  2024)NoStop
[Trotzky et al.(2010)Trotzky, Chen, Schnorrberger, Cheinet, and Bloch]trotzky_controlling_2010
  author author S. Trotzky, author Y.-A. Chen,
  author U. Schnorrberger, author P. Cheinet, and author
  I. Bloch, title title Controlling and Detecting Spin Correlations of Ultracold
  Atoms in Optical Lattices, https://doi.org/10.1103/PhysRevLett.105.265303 journal
  journal Physical Review Letters volume
  105, pages 265303 (year 2010)NoStop
[Taie et al.(2022)Taie,
  Ibarra-García-Padilla, Nishizawa,
  Takasu, Kuno, Wei,
  Scalettar, Hazzard, and Takahashi]taie_observation_2022
  author author S. Taie, author E. Ibarra-García-Padilla, author N. Nishizawa, author Y. Takasu,
  author Y. Kuno, author
  H.-T. Wei, author R. T. Scalettar, author K. R. A. Hazzard, and author Y. Takahashi, title title
  Observation of antiferromagnetic correlations in an ultracold SU(N)
  Hubbard model, https://doi.org/10.1038/s41567-022-01725-6
  journal journal Nature Physics volume 18, pages 1356 (year
  2022)NoStop
[Weitenberg et al.(2011)Weitenberg, Endres, Sherson, Cheneau, Schauß, Fukuhara, Bloch, and Kuhr]weitenberg_single-spin_2011
  author author C. Weitenberg, author M. Endres,
  author J. F. Sherson, author M. Cheneau, author
  P. Schauß, author T. Fukuhara, author I. Bloch, and author S. Kuhr, title title
  Single-spin addressing in an atomic Mott insulator, https://doi.org/10.1038/nature09827 journal journal Nature volume 471, pages
  319 (year 2011)NoStop
[Duan et al.(2003)Duan,
  Demler, and Lukin]duan_controlling_2003
  author author L.-M. Duan, author E. Demler, and  author M. D. Lukin, title title Controlling Spin Exchange Interactions of
  Ultracold Atoms in Optical Lattices, https://doi.org/10.1103/PhysRevLett.91.090402 journal
  journal Physical Review Letters volume
  91, pages 090402 (year 2003)NoStop
[Anderlini et al.(2007)Anderlini, Lee, Brown, Sebby-Strabley, Phillips, and Porto]anderlini_controlled_2007
  author author M. Anderlini, author P. J. Lee,
  author B. L. Brown, author J. Sebby-Strabley, author W. D. Phillips, and author J. V. Porto, title
  title Controlled exchange interaction between pairs of neutral
  atoms in an optical lattice, https://doi.org/10.1038/nature06011
  journal journal Nature volume 448, pages 452 (year
  2007)NoStop
[Impertro et al.(2024)Impertro, Karch, Wienand, Huh, Schweizer, Bloch, and Aidelsburger]impertro_local_2024
  author author A. Impertro, author S. Karch,
  author J. F. Wienand, author S. Huh, author
  C. Schweizer, author
  I. Bloch, and author
  M. Aidelsburger, title
  title Local Readout and Control of Current and Kinetic
  Energy Operators in Optical Lattices, https://doi.org/10.1103/PhysRevLett.133.063401 journal
  journal Physical Review Letters volume
  133, pages 063401 (year 2024)NoStop
[Freedman et al.(2021)Freedman, Hastings, and Zini]freedman_symmetry_2021
  author author M. H. Freedman, author M. B. Hastings, and author M. S. Zini, title title Symmetry Protected
  Quantum Computation, https://doi.org/10.22331/q-2021-09-28-554
  journal journal Quantum volume 5, pages 554 (year 2021), note arXiv:2105.04649 [quant-ph]NoStop
[Rudolph and Virmani(2023)]rudolph_two-qubit_2023
  author author T. Rudolph and author S. S. Virmani, title title The two-qubit
  singlet/triplet measurement is universal for quantum computing given only
  maximally-mixed initial states, https://doi.org/10.1038/s41467-023-43481-y journal journal Nature Communications volume 14,  pages 7800 (year 2023)NoStop
[DiVincenzo et al.(2000)DiVincenzo, Bacon, Kempe, Burkard, and Whaley]divincenzo_universal_2000
  author author D. P. DiVincenzo, author D. Bacon,
  author J. Kempe, author G. Burkard, and author
  K. B. Whaley, title
  title Universal quantum computation with the exchange
  interaction, https://doi.org/10.1038/35042541 journal journal Nature volume 408,  pages 339 (year 2000)NoStop
[Childs et al.(2013)Childs,
  Gosset, and Webb]childs_universal_2013
  author author A. M. Childs, author D. Gosset,  and author Z. Webb, title title Universal Computation by Multiparticle
  Quantum Walk, https://doi.org/10.1126/science.1229957
  journal journal Science volume 339, pages 791 (year
  2013)NoStop
[Busch et al.(1998)Busch,
  Englert, Rzażewski, and Wilkens]busch_two_1998
  author author T. Busch, author B.-G. Englert,
  author K. Rzażewski, and  author M. Wilkens, title title Two Cold Atoms in a Harmonic Trap,  https://doi.org/10.1023/A:1018705520999 journal
  journal Foundations of Physics volume
  28, pages 549 (year 1998)NoStop






 



 
 




§ METHODS




 §.§ Superlattice potential


The time-dependent optical lattice potential is given by

    V(   x,y,z,τ) = 
       -V_Xcos^2(kx+θ/2)
       -V_Xintcos^2(kx)
       -V_Ycos^2(ky)
       -V_Zcos^2(kz)
       -√(V_XintV_Z)cos(kz)cos(kx+φ_SL(τ))
       -I_XZ√(V_XintV_Z)cos(kz)cos(kx-φ_SL(τ)),

where k=2π/λ, λ=1064 and the imbalance factor is given by I_XZ = 0.777(3). The lattice depths V_X, V_Xint, V_Y and V_Z are listed in Table <ref> in units of the recoil energy E_rec = h^2/2mλ^2, where m is the atomic mass.
In our experiments, the lattice depth along the y- and z-directions is sufficiently large to effectively freeze the dynamics in these two directions.
The resulting potential creates a superlattice structure along the x-direction, characterised by staggered tunnellings t_x(τ) and t_x'(τ), as well as a staggered site offset Δ(τ) (Fig. <ref>a).
A simple linear ramp of the superlattice phase φ_SL, imprinted via an acousto-optic modulator, leads to a periodic modulation of the model parameters with a period T (Fig. <ref>b).







 §.§ Topological pumping

The periodic modulation of the lattice potential can be considered as a static `short' lattice with a moving `long' lattice, describing a Thouless pump <cit.> (Fig. <ref>a).
Compared to its classical counterpart, the quantum nature of the Thouless pump is manifested in its directional dependence of the motional states.
In a band-structure picture, the pump in a bipartite one-dimensional lattice potential features two topologically distinct bands.
The topological properties of these bands are characterised by Chern numbers. These numbers are derived by mapping the space- and time-periodic Hamiltonian onto a time-independent 2D Harper-Hofstadter-Hatsugai (HHH) model, which incorporates both a real and a synthetic dimension.
The two lowest bands of the HHH model have Chern numbers C = ± 1 giving rise to quantised transport of two lattice sites per period in opposite directions <cit.>.

In contrast to bichromatic setups of topological pumps <cit.>, the depth of the `moving' lattice in our case is also periodically modulated. This modulation occurs automatically due to the time dependence of the two terms proportional to √(V_XintV_Z) in Eq. <ref> and requires no change in laser intensities.
The modulation ensures a smoother time evolution of the topological bandgap and it increases the duration of the superexchange interaction (width of the feature in Fig. <ref>c).











 §.§ Realisation of (swap)^α gates




We model two particles meeting in a unit cell by considering a double-well with sites labelled by L and R. With Hubbard interactions U, the corresponding Hamiltonian is given by

    Ĥ_DW = -t_x   ∑_σ=↑,↓[(ĉ^†_Lσĉ_Rσ+h.c.) + Δ(n̂_Lσ -n̂_Rσ)]
    
        +    ∑_α=L,R Un̂_α↑n̂_α↓,

where ĉ^(†)_α,σ is the fermionic annihilation (creation) on site α=L,R with spin σ=↑,↓ and n_ασ=c^†_ασc_ασ.  
We write |σ,σ'⟩ = ĉ^†_Lσĉ^†_Rσ'|0⟩.
Since |↑,↑⟩ and |↓,↓⟩ both have energy zero and do not couple to any other states, we can write the Hamiltonian in the reduced Hilbert space spanned by {|↑↓,0⟩, |↑,↓⟩, |↓,↑⟩,|0,↑↓⟩} as

    Ĥ_DW = [ U + 2Δ   -t_x    t_x      0;   -t_x      0      0   -t_x;    t_x      0      0    t_x;      0   -t_x    t_x U - 2Δ ]

In the limit U≫Δ,t_x, an effective low-energy Hamiltonian can be derived with a Schrieffer-Wolff transformation <cit.>. After projecting out the high-energy double occupancies (|↑↓,0⟩ and |0,↑↓⟩), the Hamiltonian reads

    Ĥ_eff = 1/2[ -J_ex  J_ex;  J_ex -J_ex ],

where the superexchange energy is given by

    J_ex = 4t_x^2/U(1-(2Δ/U)^2).

This Hamiltonian can also be written in `Heisenberg' form Ĥ_eff=J_ex(Ŝ_L·Ŝ_R - 1/4), where Ŝ_α = ∑_i,jĉ^†_α iσ_ijĉ_α j is the spin operator on site α and σ=(σ_x,σ_y,σ_z)^T is the vector of Pauli operators.
The time evolution operator in the Hilbert space without double occupancies but with the states |↑,↑⟩ and |↓,↓⟩ takes the form

    Û_α = [           1           0           0           0;           0 (1+e^iπα)/2 (1-e^iπα)/2           0;           0 (1-e^iπα)/2 (1+e^iπα)/2           0;           0           0           0           1 ],


realising a (swap)^α gate, where α = 1/ħπ∫_τ_start^τ_endJ_ex(τ) dτ  <cit.>. Note that in our realisation, t_x(τ) and Δ(τ) are periodically modulated (Fig. <ref>b), as is J_ex(τ).
In the experiment we are not fully in the limit U≫Δ,t_x. We therefore use the energy difference of the two lowest eigenstates of the full double-well Hamiltonian (Eq. <ref>) for the calculation of superexchange energy, which is used for the upper x-axis in Fig. <ref>g. Fig. <ref>c shows the resulting J_ex. During the staggered lattice configuration (± 0.25T), J_ex is negligible and increases (decreases) by several orders of magnitude as the balanced double-well configuration is entered (exited). Finally, Fig. <ref>d shows the Hubbard U, the maximum of Δ and t_x, and the resulting maximum of J_ex during a pump cycle as a function of lattice depth V_X.



 §.§ Singlet preparation

We load an evaporatively cooled, balanced spin mixture of fermionic potassium-40 atoms in the magnetic states F=9/2, m_F={-9/2, -7/2} into a crossed dipole trap and further evaporatively cool it, yielding 5.3(2) × 10^4 atoms at a temperature of 0.102(6) times the Fermi temperature.



The experimental sequence to achieve a high fraction of doubly occupied unit cells with two atoms in the spin-singlet configuration is shown in Fig. <ref>. First, we use the Feshbach resonance at 201.1G to tune the s-wave interactions between atoms in the -9/2 and -7/2 sublevels to be strongly attractive. The atoms are then loaded into a shallow chequerboard lattice over 200 ms and subsequently into a deep chequerboard lattice over 10 ms, resulting in a high fraction of paired atoms in the -9/2 and -7/2 sublevels <cit.>. To achieve strongly repulsive interactions, we transfer the -7/2 population to the -5/2 sublevel using a Landau-Zener sweep and adjust the magnetic field to reach the target scattering length. Finally, by ramping down V_X and ramping up V_Xint, the chequerboard lattice is split, resulting in between 60% and 75% of doubly occupied unit cells in the spin-singlet configuration.




 §.§ Product state preparation




The loading sequence for the product state |↓,↑⟩ is shown in Fig. <ref>. We start by loading singlets into a deep double-well configuration with the same sequence as in the singlet state preparation, but ramping to different lattice potentials during the split. In this lattice, both tunnellings t_x and t_x' are negligible. Then we apply a magnetic field gradient Δ B, which causes a rotation around the x-axis on the two-particle Bloch sphere with singlet state |s⟩=(|↓,↑⟩-|↑,↓⟩)/√(2) on the north pole and triplet state |t⟩=(|↓,↑⟩+|↑,↓⟩)/√(2) on the south pole (Fig. <ref>d). To see this, we write the Hamiltonian in Eq. <ref> in the {|s⟩,|t⟩} basis,
in which it takes the form

    Ĥ_eff = 1/2[  J_ex     0;     0 -J_ex ].

The magnetic field gradient couples the |s⟩ and the |t⟩ states (singlet-triplet oscillation, STO),

    Ĥ_STO = 1/2[  J_ex  Δ_↑↓;  Δ_↑↓ -J_ex ] ,

where Δ_↑↓ is the energy offset between |↑,↓⟩ and |↓,↑⟩ induced by the gradient.
This can be rewritten as Ĥ_STO = (J_exσ̂_z + Δ_↑↓σ̂_x)/2, where σ̂_x,z are Pauli matrices. Therefore, a time evolution under this Hamiltonian in the frozen lattice configuration, where J_ex=0, for a time τ=ħπ/2Δ_↑↓ rotates the state vector to the equatorial state |i_-⟩=(|↓,↑⟩-i|↑,↓⟩)/√(2). We calibrate this duration in a separate measurement. After this rotation, we turn off the magnetic-field gradient, unfreeze the lattice by ramping down V_X and pump for a quarter pump-cycle, such that the integrated J_ex corresponds to a π/2 rotation around the z-axis of the Bloch sphere, arriving at the target state |↑,↓⟩ in the staggered lattice configuration.



 §.§ Experimental sequence


In Fig. <ref> we show the experimental sequence used for the STO measurements shown in Fig. <ref>, Fig. <ref> and Fig. <ref>. After state preparation, we start the pump by linearly ramping φ_SL with a positive slope 2π/T. We tune the superexchange interaction J_ex by varying the lattice depth V_X between operation cycles, thereby implementing different gates.
The pump is then halted in the staggered lattice configuration to perform the STO. 
The lattice dynamics is frozen by ramping up V_X and by applying a magnetic-field gradient Δ B for a certain time τ_STO. After turning off the magnetic-field gradient and unfreezing the lattice, we reverse the pump by ramping φ_SL with a slope -2π/T for the same number of pump cycles to bring the atoms back to their initial position. We then measure the nearest-neighbour singlet fraction in the final state as a function of τ_STO. The last two steps (reversed pump and singlet detection) should be considered together as a detection scheme for long-distance correlation.
The entire sequence can be understood as a many-body atom interferometer.





 §.§ Detection of singlet fraction and double occupancy


To measure the fraction of spin-singlet in a doubly occupied unit cell, we first freeze the dynamics by quenching into a deep simple cubic lattice with half the periodicity of the double wells within 100. We clean the remaining double occupancies, which are suppressed by the strongly repulsive Hubbard interactions, by transferring all atoms in the -5/2 to the -3/2 state. When another atom in the -9/2 state is present, they will collide and leave the trap. After that, we transfer the remaining -3/2 population back to -7/2. Then, we merge adjacent sites and ramp the Hubbard U to the attractive regime. Singlets form double occupancies in the lowest band, while the Pauli exclusion principle forces triplets to convert to one atom in the lowest band and one in the first excited band.
This enables us to measure the fraction of singlets by detecting the double occupancies in the merged lattice. We sweep the magnetic field over the -7/2 and -9/2 Feshbach resonance then apply a Landau-Zener RF sweep. The interaction shift then causes only the -7/2 population on doubly-occupied sites to be transferred to the -5/2 state. The Zeeman sublevels are then separated by applying a magnetic-field gradient and 8 time of flight <cit.>. 
To measure ⟨σ̂_y⟩, we apply an additional STO corresponding to a π/2 and a 3π/2 rotation about the x-axis of the two-particle Bloch sphere before merging. This converts the projection on the y-axis to the projection on the z-axis. We then determine ⟨σ̂_y⟩ by calculating the difference in singlet fraction for the π/2 and 3π/2 rotations, normalised by the initial fraction of doubly occupied unit cells.

In Fig. <ref>c we directly measure the double occupancies in the simple cubic lattice without cleaning and merging adjacent sites. The double occupancy measurement relies on an energy shift and a spectroscopic transfer into a third hyperfine state, conditioned on the presence of a second atom in the same orbital <cit.>.
The double-occupancy detection is thus orbital-selective. To determine the shuttle operation fidelity, we calculate the average decay constant, β, from two exponential fits and convert it to a fidelity using the formula F=e^-1/β.








 §.§ Analysis of STO with multi-frequency fit and FFT


Since the energy difference under a homogeneous magnetic gradient between |↑,↓⟩ and |↓,↑⟩ states is linear in the separation of the atom pair, we expect the time-trace of the STO to be a superposition of sine-waves with frequencies sf_1, where s∈ℕ^+ is the atom pair separation in number of lattice sites and f_1 is the base STO frequency at a distance of one lattice site. We use the fit function 
    F_singlet(τ) =     e^-Γτ∑_s=1^12 A_ssin(2π sf_1τ+θ_s)
       + F_0.

The fit parameters are Γ,A_s,θ_s and F_0. We also calculate the Fast Fourier transform (FFT) of the time traces in Fig. <ref> to validate our fit function. The FFT for different gate sequences can be seen in Fig. <ref>. The dominant peaks are all at integer multiples of the base frequency f_1, which justifies the choice of fit function.

