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Roadmap on Atomtronics: State of the art and perspective

L. Amico, M. Boshier, G. Birkl, A. Minguzzi, C. Miniatura, L. -C. Kwek, D. Aghamalyan, V. Ahufinger, D. Anderson, N. Andrei, A. S. Arnold, M. Baker, T. A. Bell, T. Bland, J. P. Brantut, D. Cassettari, W. J. Chetcuti, F. Chevy, R. Citro, S. De Palo, R. Dumke, M. Edwards, R. Folman, J. Fortagh, S. A. Gardiner, B. M. Garraway, G. Gauthier, A. Günther, T. Haug, C. Hufnagel, M. Keil, W. von Klitzing, P. Ireland, M. Lebrat, W. Li, L. Longchambon, J. Mompart, O. Morsch, P. Naldesi, T. W. Neely, M. Olshanii, E. Orignac, S. Pandey, A. Pérez-Obiol, H. Perrin, L. Piroli, J. Polo, A. L. Pritchard, N. P. Proukakis, C. Rylands, H. Rubinsztein-Dunlop, F. Scazza, S. Stringari, F. Tosto, A. Trombettoni, N. Victorin, D. Wilkowski, K. Xhani, A. Yakimenko

arXiv:2008.04439v5cond-mat.quant-gasquant-ph

TL;DR

Stable matter-wave motion through atomtronic circuits remains challenging where junctions and bends can produce transverse instabilities. This review synthesizes recent atomtronics work and reports an experimental demonstration of current interference in a two-weak-link ring, revealed by flux-periodic critical-current modulation.

  • Problem

    Designing atomtronic circuits to reduce transverse instabilities remains a challenge for stable matter-wave motion through junctions and bends.

    Source: §VI · p. 29

  • Method

    The review synthesizes recent activities in atomtronics and discusses the field’s future.

    Source: §I · p. 4

  • Results

    Current interference was experimentally demonstrated in an atomtronic ring with two weak links, revealed by periodic critical-current modulation with applied flux.

    Source: F. First experimental demonstration of… · p. 65

  • Takeaways & Limitations

    Transport measurements may combine existing transport systems with single-atom-sensitive fluorescence counting or quantum-gas microscopy.

    Source: D. Concluding remarks and outlook · p. 72

    The presented Floquet treatment applies only to translationally invariant systems, so its results do not account for boundary conditions in merged-waveguide circuits.

    Source: B. Floquet Hamiltonian for the… · p. 33

Abstract

from arXiv · show

Atomtronics deals with matter-wave circuits of ultra-cold atoms manipulated through magnetic or laser-generated guides with different shapes and intensities. In this way, new types of quantum networks can be constructed, in which coherent fluids are controlled with the know-how developed in the atomic and molecular physics community. In particular, quantum devices with enhanced precision, control and flexibility of their operating conditions can be accessed. Concomitantly, new quantum simulators and emulators harnessing on the coherent current flows can also be developed. Here, we survey the landscape of atomtronics-enabled quantum technology and draw a roadmap for the field in the near future. We review some of the latest progresses achieved in matter-wave circuits design and atom-chips. Atomtronic networks are deployed as promising platforms for probing many-body physics with a new angle and a new twist. The latter can be done both at the level of equilibrium and non-equilibrium situations. Numerous relevant problems in mesoscopic physics, like persistent currents and quantum transport in circuits of fermionic or bosonic atoms, are studied through a new lens. We summarize some of the atomtronics quantum devices and sensors. Finally, we discuss alkali-earth and Rydberg atoms as potential platforms for the realization of atomtronic circuits with special features.

I. INTRODUCTION … A. General features of ring traps

This section frames atomtronics as coherently controlled neutral-atom matter-wave circuits, surveying methods for shaping their geometry and the role of coherence in their operation. It highlights ring traps as a setting for coherent flow and reconfigurable atomtronic experiments, alongside quantum-technology applications and control challenges.

I. INTRODUCTION

  • Atomtronics uses coherently controlled neutral-atom matter-wave circuits to enable tunable bosonic or fermionic currents, reconfigurable geometries, and device architectures beyond conventional electronics.The review highlights interferometric precision measurements and quantum information processing18,19, as well as mesoscopic studies of persistent currents and coherent transport20–22.Source: §I · pp. 3–4

A. Fast-scanning AODs

  • Fast-scanned AOD beams create time-averaged, spatially programmable traps; RF power controls local potential depth, while feed-forward and density-based iteration correct imperfections.An orthogonal light sheet can provide tight axial confinement and minimize excitations and phase fluctuations in that dimension.Source: A. Fast-scanning AODs · p. 5, §1 · p. 5

2. Phase evolution in time-averaged potentials

  • At slower scan rates, the time-varying potential imprints phase that can accumulate and produce residual condensate micromotion, a concern for phase-sensitive applications such as guided Sagnac interferometry.For sufficiently fast scans, the imprinted phase effect is negligible.Source: §2 · p. 5

3. Atomtronics with time-averaged optical traps

  • Time-averaged optical traps support flat-bottom lines, rings, lattices, and dumbbell reservoirs, while coherent propagation and phase splitting have been demonstrated in waveguides and beamsplitters.Repulsive barriers and stirring elements enable studies of persistent currents and superfluid transport.Source: §3 · p. 5

B. Optical potentials with liquid-crystal SLMs

  • Liquid-crystal SLM phase patterns shape output light fields; optimization algorithms can control phase and amplitude or remove vortices, while programmed phase structure enables near-diffraction-limit barriers.Flat programmed phase maintains intensity-profile shape for up to about 10 Rayleigh ranges, whereas random phase loses its shape sooner.Source: B. Optical potentials with… · p. 6

C. Direct imaged DMD optical potentials

  • DMDs combine rapidly refreshed mirror patterns with half-toning, density-based correction, and pulse-width modulation to improve projected-potential intensity control and smoothness.Their high-resolution masks and dynamic control support atomtronic geometries and time-dependent studies, including superfluid transport.Source: C. Direct imaged DMD optical potentials · p. 7, §1 · p. 7, §2 · p. 7

3. Turbulence with DMDs

  • DMD dynamics have been used to study Onsager vortices and their emergence, create tunable velocity solitons, and examine equilibration of chiral vortex clusters.Source: §3 · p. 7

D. Hybrid atomic-superconducting quantum systems

  • Optically heating superconducting films reshapes their current density and magnetic trapping potentials without changing the chip or applied electrical field, enabling split, crescent, ring, and lattice traps for compact on-chip devices.The review concludes that configurable optical and hybrid-chip technologies have substantially improved control and manipulation of ultracold neutral atoms.Source: D. Hybrid atomic-superconducting… · p. 8, E. Concluding remarks and outlook · p. 8

E. Concluding remarks and outlook

  • Dynamic optical manipulation has enabled studies of quantum turbulence, quenched condensate evolution, steerable-atom quantum simulation, and atomtronics, while further SLM improvements are expected to expand its impact.Dynamically modulated DMDs facilitated studies of two-dimensional quantum turbulence81,82,84 and condensate evolution under rapidly quenched potentials33,99; AOMs enabled steerable single-atom arrays100–102, and reconfigurable DMD traps enabled atomtronics30,35,83,103,104.Source: E. Concluding remarks and outlook · p. 8

III. IMPLEMENTING RING CONDENSATES

  • Ring systems are the simplest multiply connected geometry for coherent matter-wave guiding and potential atomtronic circuits, and can support persistent flow, quantum Hall states, and Sagnac interferometry.Magnetic, optical, and hybrid waveguides can form smooth, dynamically controllable loops for coherent quantum matter-wave flow.Source: §III · pp. 8–9

A. General features of ring traps

  • Approximately harmonic ring traps are parameterized by radius R and radial and vertical frequencies; connected geometry modifies Tc, while coherence requires a phase-correlation length exceeding πR.Elongated rings can exhibit thermally driven phase fluctuations below Tc; in the Thomas-Fermi regime, chemical potential depends on trapping parameters and the s-wave scattering length.Source: A. General features of ring traps · p. 9

B. Techniques based on magnetic traps … C. Trapping in rings with optical potentials

The section compares magnetic, RF-dressed, time-averaged, and optical approaches to ring trapping, describing their control capabilities, limitations, and experimental use. It highlights smooth TAAP guides and rapidly rotating dynamical rings, while noting challenges such as phase-coherent ring filling.

B. Techniques based on magnetic traps

  • Magnetic ring guides trade compactness and complex atom-chip geometries for surface corrugations and current-connection perturbations, while distant macroscopic conductors yield smoother, corrugation-free toroidal guides.Switching currents between wire sets can avoid connection-region bumps, but risks atom loss and heating; magnetic ring designs also face Maxwell-equation and symmetry constraints.Source: B. Techniques based on magnetic traps · pp. 9–10

1. RF dressing and bubbles

  • RF dressing uses adiabatic following of local eigenstates to create shell, ring, tube, and toroidal potentials, but a static quadrupole cannot trap a shell where RF coupling vanishes.Shell trapping instead requires a local extremum in the static-field magnitude and nonzero transverse RF coupling, as available with Ioffe-Pritchard configurations and variants.Source: §1 · pp. 10–11

2. Waveguides formed from Time-Averaged Adiabatic Potential (TAAP)

  • TAAPs time-average an adiabatic bubble potential when modulation is slower than the Larmor frequency but faster than the bubble-trap frequency, producing ring-like guides through greater potential increase at the poles than the equator.Tilting the ring or changing RF polarization provides azimuthal confinement; combined modulation can create two arbitrarily placed traps or longitudinal potentials containing harmonics up to 2φ.Source: §2 · p. 11
  • 40,000 ħ per atom was readily achieved in TAAP guides, which transported atoms tens of centimeters without propagation heating, while fully phase-coherent ring filling remained a challenge.The co-rotating potential was exceptionally flat, with density fluctuations corresponding to a few hundred picokelvin; reducing atom number and increasing ring radius can access the 1D regime.Source: §2 · p. 12

3. Dynamical ring in an rf-dressed adiabatic bubble potential

  • A dynamical ring reached a supersonic Mach number of 11 (v ≃ 6.9 mm/s; c ≃ 0.62 mm/s), rising above 18 after 45 s; its typical radius was ∼30 µm.The reported Mach number increased as the ring radius grew and the chemical potential decreased. The experiment is described in the passage with reference 136.Source: §3 · p. 13
  • A quadrupole-like mode in the dynamical ring rotated against the superfluid flow with an approximately 10 s period, a counterpropagating effect not predicted by mean-field theory.Resonant spectroscopy confirmed the effect; the discrepancy highlights the need for improved descriptions of fast-rotating superfluids in anharmonic traps.Source: §3 · p. 13

C. Trapping in rings with optical potentials

  • Optical dipole ring traps are insensitive to hyperfine state, support multicomponent and spinor condensates, permit flow imprinting or Raman angular-momentum transfer, and can form configurable geometries.Their trapping potential follows the intensity pattern of focused, far-detuned optical beams; spatial light modulators enable more complex configurations.Source: C. Trapping in rings with optical… · p. 13

1. Optical trapping … C. Cryogenic Atom Chips and Hybrid Quantum Systems

This section surveys optical, hybrid, and atom-chip approaches to ring traps, atomtronic circuits, sensing, and interfaces between ultracold atoms and solid-state quantum devices. It covers demonstrated currents, interferometry, and on-chip coherence, along with reported force sensitivity, advances toward atom–superconductor coupling, and proposed circuit capabilities.

1. Optical trapping

  • Optical ring traps use shaped, focused intensity profiles, while far-detuned light reduces scattering faster than it reduces trapping potential.For ring traps, attractive or repulsive ring-shaped light can be combined with a perpendicular light sheet for axial confinement.Source: §1 · p. 14

2. Optical ring traps

  • LG beams enable toroidal traps and circulation transfer; painted potentials enable local shaping, stirring, and atomtronic SQUIDs, while DMDs create dynamically controlled rings.LG modes carry orbital angular momentum and have been used for small rings and quantized flows [73, 150, 151]. Painting produced the first ring BEC [7] and demonstrated quantum interference in an atomtronic SQUID [16]; DMD stirring yielded a 21-quanta persistent current [157].Source: §2 · pp. 14–15
  • Conical refraction can produce configurable bright-ring patterns with near-unity conversion efficiency, while microfabricated optical elements offer flexible, scalable architectures.Blue-detuned atoms can be trapped between the bright rings, reducing scattering heating; BECs have been transferred into a Poggendorff ring, with sensors and SQUIDs proposed. Lithography can replicate systems, while direct laser writing supports rapid prototyping.Source: §2 · pp. 15–16

3. Imperfections in optical traps

  • Optical intensity variations should typically stay below 1% to avoid density perturbations, and condensate density can guide feedforward corrections to the potential.Because density closely follows the optical potential, it provides a sensitive probe of imperfections.Source: §3 · p. 16

D. Hybrid traps: RF bubble plus light sheet(s)

  • Combining an RF-dressed magnetic bubble with blue-detuned light sheets creates a toroidal gas; rotation above a critical frequency produces a hole indicating circulation.The setup trapped approximately 10^5 atoms; reduced optical imperfections could enable a 10^4-atom quasi-1D condensate. Future barriers could emulate Josephson junctions and support studies of 1D dynamics.Source: D. Hybrid traps: RF bubble plus light… · pp. 16–17

E. Concluding remarks and outlook

  • Optical and RF-dressed ring traps offer configurability and dynamic control; future challenges include atomtronic applications and larger or 3D structures, with atom-chip circuits possible where surface interactions are less limiting.Atom chips also offer guides and traps with varied architectures and control of atomic interactions and spin, and can support dynamic tunnelling barriers.Source: E. Concluding remarks and outlook · p. 17, §IV · p. 17

A. Progress towards on-chip interferometry

  • On-chip atomtronics seeks light-free circuits with sources and detection that work at any angle to gravity, but close atom–surface operation must address Johnson noise, wire-size effects, surface forces, fragmentation, and dimensionality.The envisioned Sagnac circuit will first test coherence over one and multiple turns, progressing from pulsed loading toward continuous-wave operation; high-Q micro-discs are studied for off-resonant detection.Source: A. Progress towards on-chip… · p. 18
  • Atoms maintained spatial coherence for at least 0.5 s at a 5 µm atom–surface distance in a proof-of-principle chip experiment.The work also reports light-free Stern–Gerlach interferometers observing spatial and spin-population fringes, T3 phase accumulation, clock interferometry, and geometric phase.Source: A. Progress towards on-chip… · pp. 18–19

B. Precision Sensing

  • CASPM uses ultracold atoms as scanning probe tips, with contact, dynamic, and overlap modes for sensing nanoscale interactions and imaging surfaces.207,208,209,210The modes monitor position-dependent losses, oscillation-frequency shifts, or time-dependent losses to probe near-surface potentials and van der Waals (Casimir-Polder) interactions.Source: B. Precision Sensing · p. 19
  • CASPM measurements of individual carbon nanotubes reached yN force sensitivity and working distances of several micrometers, extending conventional AFM force sensitivity by several orders of magnitude.208These capabilities enable investigation of fragile nano-objects.Source: B. Precision Sensing · p. 19
  • Single-atom detection enables real-time CASPM monitoring while losing few probe atoms and speeds probe-tip oscillation-frequency measurements by at least three orders of magnitude.211,212,213Continuous multiphoton-ionization subsampling and time-resolved ion detection underpin the scheme.Source: B. Precision Sensing · p. 19
  • A demonstrated quantum-galvanometer scheme transfers magnetic-field fluctuations through a BEC to an atom laser, whose statistics reveal microwave power spectra and noise correlations.214,215,216,217Second-order correlation analysis extracts these quantities within the BEC bandwidth, supporting proposed detection of nanoscale currents and current noise.Source: B. Precision Sensing · p. 19

C. Cryogenic Atom Chips and Hybrid Quantum Systems

  • Superconducting atom chips reduce surface magnetic noise and improve trapped-atom coherence; Tübingen experiments report coherence times of several seconds and coherent cavity coupling of hyperfine states.90,206,218,219,220,222Microwave dressing suppresses differential shifts, with the “double-magic point” identified as the optimum for quantum memories.221Source: C. Cryogenic Atom Chips and Hybrid… · pp. 19–20
  • Cryogenic-chip experiments implemented two-photon Rydberg excitation, optical detection of Rydberg populations and coherences, and measured increased Rydberg-state lifetimes.224,225Reference spectra and measurements near surfaces at room and cryogenic temperatures inform studies of surface-adsorbate fields and planned coherent manipulation.226,227,94,228,229,230,227,221,231Source: C. Cryogenic Atom Chips and Hybrid… · p. 20
  • Hybrid atom–superconductor experiments require millikelvin temperatures; 5x108 rubidium atoms were transported near a 70 mK stage, where trapped clouds lasted 13 minutes.232,233The group is developing atom interfaces with superconducting qubits, including 3D transmons and cavities designed for atomic access; first atom transport into a cavity is under way.233Source: C. Cryogenic Atom Chips and Hybrid… · pp. 20–21
  • YBCO vortices generate self-sufficient atom traps with low-noise potentials, and light-patterned heating shifts vortices to produce multiple trap geometries in situ.234,235,236,237Vortex-motion noise is expected to be an order of magnitude below Johnson noise in normal conductors; reconfiguring patterns does not require changing the chip or breaking vacuum.90Source: C. Cryogenic Atom Chips and Hybrid… · pp. 20–21

D. Concluding remarks and outlook … 1. The Lieb-Liniger model

This section presents atom chips as a platform for future sensing and fundamental research, and surveys integrable models for studying nonequilibrium dynamics. It focuses on quenches in the Lieb–Liniger model, including domain-wall steady states, fermionic correlations, work distributions, and long-time Loschmidt echoes, while noting that attractive interactions remain less understood because Bethe-state normalization is unknown in closed form.

D. Concluding remarks and outlook

  • Atom-chip research extends beyond bosonic rubidium to fermions, molecules and ions, including prospects for molecular microchips and scalable ion quantum-computing architectures.Molecular-chip trapping has been realized [243,244], while miniaturized surface-electrode traps are being developed to control and measure large numbers of ions [245].Source: D. Concluding remarks and outlook · p. 22
  • Atom chips are being used for matter-wave interferometry and quantum-gas microscopy, with applications in tests of quantum thermodynamics and many-body physics and in gravitational sensing.The outlook also identifies material research and tests of the Weak Equivalence Principle, antihydrogen interactions with matter and gravity, non-Newtonian gravity, and a fifth fundamental force as potential uses.Source: D. Concluding remarks and outlook · p. 22

V. QUENCH DYNAMICS OF INTEGRABLE MANY-BODY SYSTEMS

  • Integrable models offer a way to investigate nonequilibrium physics, for which no corresponding general framework to equilibrium statistical mechanics is established.The review discusses the Lieb-Liniger model and Heisenberg chain; dynamical fermionization also occurs in the nonintegrable Bose-Hubbard model [259].Source: §V · p. 22, B. Evolution under integrable… · p. 23

A. Quench Dynamics

  • Quench studies probe nonequilibrium dynamics by evolving an initial state under a different Hamiltonian and tracking local correlations, thermalization, entanglement, or global Loschmidt amplitudes and work distributions.For integrable post-quench Hamiltonians, conserved local charges constrain evolution and determine a generalized Gibbs ensemble from their initial values.Source: A. Quench Dynamics · pp. 22–23, B. Evolution under integrable… · p. 23

1. The Lieb-Liniger model

  • The Lieb-Liniger gas describes one-dimensional ultracold bosons with local interactions and is exactly solvable by the Bethe Ansatz, enabling quench calculations through eigenstates or the Yudson representation.The Yudson representation simplifies overlaps, particularly for coordinate-ordered initial states, which are typically difficult to calculate in the conventional approach.Source: §1 · pp. 23–24
  • A domain-wall quench in the repulsive Lieb-Liniger gas produces a long-time nonequilibrium steady state (NESS) with density independent of position and time inside the light cone.The density vanishes far ahead of the light cone, while a crossover separates that region from the NESS; the open-system limit L ≫ vt is essential to this behavior.Source: §1 · pp. 24–25
  • For repulsive interactions, long-time noise correlations develop a fermionic dip for any c > 0, whereas c = 0 yields a bosonic peak; this dynamical fermionization was observed experimentally in the Lieb-Liniger and Bose-Hubbard systems.The crossover occurs on the timescale t ∼ c^-2 as the effective coupling flows to strong coupling; the Bose-Hubbard model is nonintegrable.Source: §1 · p. 25
  • Repulsive interactions strongly reshape the low-work distribution, producing resonant peaks and a faster power-law decay near zero than in the free case.The average work is ⟨W⟩=Nω/4 in both cases, and the distributions agree at large W; the interaction-induced fluctuations are suppressed as particle number increases.Source: §1 · p. 26, Figure 24 · p. 27
  • The Loschmidt echo is interaction-independent at short times but decays as 1/t^N for free bosons and 1/t^N2 for interacting bosons at long times.The long-time exponent is independent of the initial repulsive coupling: any repulsive coupling flows to strong coupling, including the Tonks-Girardeau limit.Source: §1 · p. 26
  • The attractive model is less studied because its Bethe-state normalization is not known in closed form.For c<0, the ground state is a bound state of all N particles, and the Bethe Ansatz permits complex momenta for bound states.Source: §1 · p. 27
  • Attractive interactions split the work distribution into a free-particle contribution and a distinct bound-state contribution, with the former reflecting particle clustering.The free contribution is an analytic continuation of the repulsive expression to negative coupling, valid at arbitrary c<0 and reminiscent of super-Tonks-Girardeau behavior; attraction reduces the effective particle spacing.Source: §1 · p. 27
  • Bound-state formation is highly suppressed, yet it makes the work distribution nonzero at negative work, allowing work extraction while the average work remains positive.For a single two-particle bound state, the contribution is nonzero for −|c|^2/4m < W.Source: §1 · p. 27

2. The XXZ Heisenberg spin chain

The XXZ Heisenberg chain is an integrable model whose quench dynamics feature sharp light-cone propagation shaped by magnons and spinon bound states. Its anisotropy controls bound-state motion and spectral weight, while integrability gives the excitations infinite lifetimes.

2. The XXZ Heisenberg spin chain

  • The chain models anisotropic spin exchange; its isotropic ∆ = 1 case is SU(2)-invariant and was first solved by Bethe, with Orbach generalizing the model to anisotropy [283, 284].Eigenstates are characterized by Bethe momenta for down-spins in a background of up-spins.Source: §2 · p. 28
  • After a quench, spin excitations form a sharp light cone: free magnons set its boundary, while spinon bound states propagate within it and slow as anisotropy increases.Increasing anisotropy also shifts more spectral weight to the bound states.Source: §2 · p. 28
  • The time-evolving wavefunction of two adjacent flipped spins was compared with experiment without adjustable parameters.The calculation uses a Yudson representation to time-evolve the initial state.Source: §2 · p. 28
  • Integrability gives these excitations infinite lifetimes, preventing dispersion of the observed features; integrability-breaking terms may alter this picture through spinon decay [286].The magnetization evolution from three flipped spins is shown for different anisotropies.Source: §2 · p. 28

C. Concluding remarks and outlook … B. Floquet Hamiltonian for the Periodically Tilted Lieb-Liniger Model

The section surveys integrability-based approaches to nonequilibrium dynamics in one-dimensional atomtronic circuits, including Lieb–Liniger quenches and periodic driving. It also outlines Floquet descriptions for boundary-containing circuits and rotating rings, noting the conditions and symmetry considerations involved.

C. Concluding remarks and outlook

  • Microscopic integrability methods and the Yudson approach address nonequilibrium dynamics across Lieb-Liniger, Heisenberg, Gaudin-Yang, Kondo, and Anderson models.Generalized hydrodynamics offers a complementary long-scale, long-time description, but is limited to Euler-scale dynamics; comparing these approaches could clarify their limitations.Source: C. Concluding remarks and outlook · pp. 28–29

VI. NON-EQUILIBRIUM PROTOCOLS FOR ONE DIMENSIONAL BOSE GASES IN ATOMTRONIC CIRCUITS

  • One-dimensional waveguide circuits connect atomtronics with integrable systems: tight transverse confinement suppresses transverse instabilities, while Lieb-Liniger theory describes ultracold bosons in that regime.This connection supports applying integrability techniques to circuit dynamics and motivates further work on connected one-dimensional systems and mixtures.Source: §VI · p. 29

A. Quench dynamics in the Lieb-Liniger model

  • The Lieb-Liniger model provides a framework for homogeneous quench studies, where an abrupt change in interaction strength drives dynamics from an initial ground state.The chapter focuses on quench dynamics and Floquet engineering as protocols for controlling one-dimensional gases, enabled by experiments that vary interactions, trap geometry, and time-dependent potentials.Source: §VI · p. 29, A. Quench dynamics in the Lieb-Liniger… · p. 30

1. The quench problem

  • Integrability constrains post-quench relaxation: the generalized Gibbs ensemble includes local and quasi-local conservation laws, but computing dynamics and overlaps remains difficult.Earlier Lieb-Liniger studies focused on zero or infinitely repulsive post-quench interactions, leaving finite interactions an open problem before the Quench Action approach.Source: §1 · p. 30

2. The Quench Action

  • The Quench Action method predicts nonthermal steady states after repulsive interaction quenches and power-law relaxation of local observables.It replaces a sum over eigenstates with a functional integral over quasiparticle distributions, evaluated at a saddle point determined by the initial-state overlaps.Source: §2 · pp. 30–31

3. Quenches to the attractive regime

  • Attractive quenches from c0 = 0 form finite-density n-boson bound states, with the most populated bound-state size decreasing as γ = |c|/D increases.The resulting stationary state differs qualitatively from the super Tonks-Girardeau gas; its late-time g2 exceeds 2 and increases with γ.Source: §3 · p. 31

B. Floquet Hamiltonian for the Periodically Tilted Lieb-Liniger Model

  • Periodic tilting yields an integrable Floquet Hamiltonian for the Lieb-Liniger model, whose quasi-energies and eigenfunctions can be obtained with Bethe Ansatz techniques.A periodic tilt can control motion to and fro across atomtronic circuits; the derivation also applies to translationally invariant interacting systems in any dimension.Source: B. Floquet Hamiltonian for the… · pp. 31–33
  • The derivation applies only to translationally invariant systems; for merged waveguide circuits, boundary conditions make it only a first step toward the correct Floquet Hamiltonian.The needed circuit-specific derivation is identified as a worthwhile subject for atomtronic components and circuits.Source: B. Floquet Hamiltonian for the… · p. 33
  • For a periodically rotating ring, angle ϕ plays the role of coordinate x, and at stroboscopic times with f(t) = 0 the comoving-frame Floquet Hamiltonian has form (53), analogous to H − ΩLz [376, 129].An out-of-plane periodic rotation component is proposed as a ring-geometry counterpart to slightly deforming an isotropic potential to break translational invariance.Source: B. Floquet Hamiltonian for the… · p. 33

C. Concluding remarks and outlook … 1. How stirring a racetrack atom circuit produces flow

The outlook highlights generalized hydrodynamics and periodic tilting as directions for integrable-gas research, while the atomtronics contribution reviews persistent-current formation and vortex dynamics in closed circuits. GHD predictions have been experimentally verified with bosonic atoms on an atom chip, and racetrack stirring provides a setting for studying how vortex circulation becomes macroscopic flow.

C. Concluding remarks and outlook

  • Generalized hydrodynamics (GHD) extends exact predictions from homogeneous systems to inhomogeneous settings at hydrodynamic scales.The paper identifies GHD as a major recent advance in out-of-equilibrium integrable-systems theory.Source: C. Concluding remarks and outlook · p. 33
  • GHD predictions have been experimentally verified by monitoring bosonic cold atoms trapped on an atom chip.Recent work also applied GHD to repulsive 1D Bose gases with confining potentials, spatial inhomogeneities, and dephasing noise.Source: C. Concluding remarks and outlook · pp. 33–34
  • Future GHD studies could examine attractive inhomogeneous 1D Bose gases and more complex integrable mixtures of fermions and bosons.Hydrodynamic-scale phenomena have already been pointed out for mixtures in simple settings.Source: C. Concluding remarks and outlook · p. 34
  • Periodic tilting yields an integrable Floquet Hamiltonian in the repulsive Lieb–Liniger model and motivates studying boundary conditions and atomtronic circuits under tilting.The analysis uses quasi-energy spectra and stroboscopic dynamics; it can extend to systems such as the Yang–Gaudin model.Source: C. Concluding remarks and outlook · p. 34

VII. PERSISTENT CURRENTS AND VORTICES IN ATOMTRONIC CIRCUITS

  • A ring’s quantized circulation corresponds to a central “ghost” vortex, whose confinement by a potential barrier can stabilize even multicharged metastable states.Vortex dynamics govern persistent-current generation and decay, deterministically or randomly depending on the setting.Source: §VII · p. 34
  • Persistent currents in toroidal condensates can arise through stirring, angular-momentum transfer, or phase defects after a quench into the BEC state.The reviewed work also reports persistent currents in multicomponent ring condensates.Source: §VII · p. 34
  • Coupled atomic superflows offer a tunable setting for precision measurements and potential quantum gates, with phenomena ranging from Josephson effects to Kelvin–Helmholtz instability.In weakly interacting rings, superflow can decay by inducing phase slips.Source: §VII · p. 34

1. How stirring a racetrack atom circuit produces flow

  • The racetrack study uses a Gross–Pitaevskii model of a planar condensate stirred by a wide rectangular barrier to examine vortex formation, motion, and conversion into circuit-wide flow.The channel has two half-circles separated by straightaways of length L; L = 0 gives a ring, and the analysis considers conditions determining final flow.Source: §1 · p. 35

2. Creation of a single unit of flow: vortex swap … A. Critical transport and vortex dynamics in a thin atomic Josephson junction

This section examines the creation, control, and dynamics of superflow in atomic ring and junction geometries, including persistent flow, winding numbers, vortex dynamics, and dissipation through phase slips. It also identifies control of these flows as a route toward atomtronic technologies and sensors.

2. Creation of a single unit of flow: vortex swap

  • A critical vortex swap launches a vortex pair and compression wave in opposite directions, converting localized circulation into macroscopic racetrack flow.The disturbances travel at approximately the average local speed of sound; the backflow that precedes the swap is fastest at the channel edges, where the potential is largest.Source: §2 · p. 35

3. Final flow production: flow overtakes the barrier

  • Racetrack circulation tends toward the quantized flow speed nearest the stirring-barrier speed, and can oscillate around that value as flow overtakes the barrier.Recirculating disturbances can trigger an inverse swap that reduces circulation by one unit; above a critical barrier strength, transitions between curved and straight track sections can also change circulation by one unit.Source: §3 · pp. 35–36

1. Spontaneous Persistent Current Formation in a Ring Trap

  • Spontaneous ring-trap currents arise from randomly generated winding numbers centered on zero, with the possible winding set by geometry and quench rate.The effect is associated with local phase coherence and follows Kibble-Zurek expectations; phase fluctuations and nonzero winding can also affect dark-soliton dynamics.Source: §1 · pp. 36–37

2. Spontaneous Persistent Current Formation in Co-Planar Connected Ring Traps

  • In co-planar connected rings, locally forming phases in unconnected regions can yield distinct winding numbers, while each ring’s distribution matches that of an equivalent single ring.The distribution is centered on zero net winding in the studied geometry, and the conclusions are reported to extend to other planar-connected loops with one extended interface.Source: §2 · p. 37
  • Changing the connected potential can transfer winding between rings or enable oppositely oriented superflows to annihilate, suggesting controllable transfer across multi-loop architectures.Transfer was explicitly verified by connecting the two sides with a zero-potential region; deterministic transfer in architectures with multiple loops remains a prospective extension.Source: §2 · p. 37

C. Persistent Currents in Transversally coupled atomtronic circuits

  • Coupled rings with unequal topological charges form |m1 − m2| Josephson vortices and have zero net integral current through the junction.As the barrier is lowered and coupling increases, the Josephson vortices accumulate energy; merging then relaxes through three-dimensional dynamics involving Josephson vortices and persistent-current vortex lines.Source: C. Persistent Currents in… · p. 38
  • In merging rings, final vorticity depends on population imbalance and trap shape: in an oblate trap, the less populated ring determines it below Pcr ≈0.1755, and the stronger ring above.Counter-rotating quantized flows do not relax to the non-rotating ground state even at small imbalance; sufficiently elongated prolate traps instead produce robust hybrid vortex structures. Potential perturbations and initial imbalance can control the merger.Source: C. Persistent Currents in… · pp. 39–40

D. Concluding remarks and outlook

  • Controlled topological excitations in atomtronic circuits could support quantum computation and acceleration or rotation sensors, while transport, gate-like manipulation, and readout need further work.Sensor development also requires reduced decoherence and dissipation, rapid preparation of the desired initial state, and lower shot-to-shot atom-number fluctuations.Source: D. Concluding remarks and outlook · p. 40

A. Critical transport and vortex dynamics in a thin atomic Josephson junction

  • Dissipation combines transfer of incompressible flow energy to the vortex with acoustic emission; broader or higher barriers can instead support self-trapping [451,460].Thermal and quantum fluctuations and higher-order tunnelling progressively destabilize that self-trapped state [467–474].Source: A. Critical transport and vortex… · p. 42
  • Vortex-ring generation abruptly opposes particle transfer and can reverse background flow; the ring shrinks in the barrier, then travels outward at constant speed [451].The ring first appears as a ghost vortex in a low-density region and develops a roughly 2π relative-phase jump.Source: A. Critical transport and vortex… · p. 42, Figure 34 · p. 43
  • At larger imbalances, repeated ring nucleation produces sawtooth-like flow; overlapping rings can interact, reconnect, leapfrog, or generate turbulent-like dynamics [451].Rings may decay into rarefaction pulses or through interactions with transverse condensate boundaries.Source: A. Critical transport and vortex… · pp. 42–43
  • Above a critical initial imbalance, accelerated local flow nucleates a vortex ring that can enter the bulk, producing a 2π phase jump and dissipative flow [451].The Florence experiment observed bulk vortices rather than the expected self-trapping regime [442,450].Source: A. Critical transport and vortex… · pp. 41–42, Figure 34 · p. 43
  • Experiments observed individual vortices consistent with the mechanism; trap asymmetry, fluctuations, and barrier removal affect ring shape, defects, and lifetime [442,450,451].Asymmetric rings can exhibit Kelvin waves, while dynamical barrier removal significantly extends their lifetime.Source: A. Critical transport and vortex… · p. 43
  • Small thermal fluctuations scarcely change initial vortex generation, but high temperatures can induce extra rings and thermal effects destabilize and damp long-term motion [451].The damping arises through mutual friction with the thermal cloud.Source: A. Critical transport and vortex… · p. 44

B. Bose-Josephson junction among two one-dimensional atomic gases: a quantum impurity problem · C. Bose-Einstein condensate confined in a 1D ring stirred with a rotating delta link · D. Thermal and quantum phase slips in a one-dimensional Bose gas on a ring

These sections examine phase-slip and current dynamics in one-dimensional Bose systems, including a coupled-wire Josephson junction and ring gases subject to stirring or phase imprinting. They address distinct mechanisms and conditions for oscillation damping, metastability, and current decay, including thermal and coherent quantum phase slips.

B. Bose-Josephson junction among two one-dimensional atomic gases: a quantum impurity problem

  • In the Josephson regime, weak interactions yield undamped oscillations, while stronger interactions increase damping; in the Fock regime, small-imbalance Tonks-Girardeau oscillations are undamped, but larger quenches and finite temperature produce damping [453].For the Fock regime, the Josephson dynamics near excitation anticrossings correspond to Rabi oscillations with frequency EJ/ħ.Source: B. Bose-Josephson junction among two… · pp. 45–46
  • The coupled-wire Luttinger-liquid model maps to a quantum impurity coupled to an intrinsic oscillator bath; exact Tonks-Girardeau results validate its Josephson frequency and reveal additional damping modes [453].At finite temperature, low-energy excitations associated with the finite barrier contribute to damping beyond the linear-spectrum Luttinger-liquid description.Source: B. Bose-Josephson junction among two… · pp. 45–46

C. Bose-Einstein condensate confined in a 1D ring stirred with a rotating delta link

  • A rotating delta link splits ring-condensate soliton spectra into swallowtails whose critical velocities delimit where stationary trains exist; Bogoliubov analysis finds most such states stable and the upper branches unstable [497].The velocity ranges for dragging depend on weak-link strength; weaker interactions and stronger links narrow the range for dissipation-free soliton dragging.Source: C. Bose-Einstein condensate confined… · pp. 46–47
  • Adiabatic stirring paths can produce solitons or vortices with multiple angular-momentum quanta, while crossing critical lines yields hysteresis; time-dependent GPE simulations reproduce these mechanisms with narrow Gaussian links [497].The model qualitatively explains hysteresis cycles, whose widths decrease with weaker interactions or stronger links; Gaussian-link critical velocity decreases as its height increases.Source: C. Bose-Einstein condensate confined… · p. 47

D. Thermal and quantum phase slips in a one-dimensional Bose gas on a ring

  • For a weakly interacting ring gas at zero temperature, the imprinted current persists under weak-to-moderate barriers up to λGP ∼1, then rapidly decays and oscillates around zero as the dynamics shift from self-trapping to Josephson oscillations.The reported dynamics are for the weakly interacting regime at T = 0.Source: D. Thermal and quantum phase slips in… · p. 49
  • At finite temperature in the mean-field regime, weak barriers yield exponential current decay while stronger barriers produce damped oscillations and thermal phase slips at the barrier.Reflected slow solitons drive discrete current jumps; coupling between soliton dynamics and long-wavelength sound makes the average decay intrinsically stochastic [510].Source: D. Thermal and quantum phase slips in… · p. 49, Figure 39 · p. 49
  • At zero temperature in the Tonks-Girardeau regime, weak barriers produce Rabi-like current oscillations rather than self-trapping, corresponding to coherent quantum phase slips.Barrier-induced backscattering breaks rotational symmetry and couples angular-momentum states; the microscopic dynamics involves coherent multiple particle-hole excitations [418, 511].Source: D. Thermal and quantum phase slips in… · p. 49, Figure 39 · p. 49
  • In the Tonks-Girardeau regime, stronger barriers cause higher-mode population and dephasing, while the oscillation frequency crosses from Rabi-like ω = π2NλTG to Josephson-like ω ∝√λTG.At finite temperature, thermally occupied high-energy excitations damp the oscillations and suppress large-barrier revivals; incoherent phase slips contribute to exponential decay [171].Source: D. Thermal and quantum phase slips in… · pp. 49–50
  • In the mean-field regime, self-trapping prevents coherent phase slips, whereas finite-temperature incoherent slips arise through stochastic coupling of solitons to long-wavelength sound excitations.In the strongly interacting regime, coherent slips dominate, with finite-temperature high-energy excitations producing average current decay.Source: D. Thermal and quantum phase slips in… · pp. 49–50

E. Concluding remarks and outlook … C. Sagnac interferometry and rotation sensing

The concluding remarks connect phase-slip mechanisms to dissipation and coherence in atomtronic systems, while the device-and-sensor discussion surveys circuit components and rotation-sensing approaches. The sections highlight both the promise of atomtronic devices and the physical and technical challenges that remain.

E. Concluding remarks and outlook

  • Phase slips arise through distinct excitation mechanisms: vortex-ring generation in weakly interacting 3D gases and low-energy bulk modes or dark solitons in 1D systems.In the 3D bosonic analysis, increasing population imbalance can trigger sequential vortex-ring generation; its applicability to fermionic gases beyond the BEC regime remains open (442,450,512). In 1D rings, solitons couple to long-wavelength sound, while more realistic barriers, confinement, and beyond-Luttinger-liquid simulations are needed to extend the simplified models.Source: E. Concluding remarks and outlook · p. 50
  • Reducing dissipation and preserving coherence are central atomtronic challenges, with low-energy-excitation control, integrable dynamics, and topologically protected states identified as promising directions.Initial-state projections onto high-energy excitations can also produce complex damped dynamics; integrable or quasi-integrable systems have shown long-lived coherence and dissipation-free dynamics (258,302,310), while topologically protected states are proposed as robust device building blocks (513,514).Source: E. Concluding remarks and outlook · p. 50

IX. ATOMTRONICS ENABLED QUANTUM DEVICES AND SENSORS

  • Atomtronic devices are presented as novel components and applications, including rotation and magnetic sensing.The atomtronic SQUID is proposed as a sensor, circuit element for matter-wave interference, and possible store for quantum information.Source: §IX · p. 51, B. Atomtronic SQUIDs · pp. 52–53

A. Diodes, transistors, and other discrete components

  • Atomtronic circuit elements include reservoir-based batteries and an RLC-like circuit, with atom flow driven by reservoir chemical-potential differences.A cold-atom system demonstrated linear RLC dynamics, using two reservoirs linked by a channel whose finite resistance and apparent inductance discharge a capacitor-like reservoir (518). The battery experiment controls flow across a barrier and removes atoms from the second well with a terminator (516, 517).Source: A. Diodes, transistors, and other… · p. 51
  • Optical-lattice band gaps and double barriers enable proposals for atomtronic diodes, three-well transistors, and a five-site AND gate, with cascaded gates suggesting matter-wave quantum logic.The transistor’s transmission depends on the chemical potential between its barriers, and the proposed AND gate uses five lattice sites (12,31,181,521).Source: A. Diodes, transistors, and other… · pp. 51–52

B. Atomtronic SQUIDs

  • An atomtronic SQUID uses a ring waveguide and two barriers; rotation drives loop current, and oscillations in its AC-to-DC Josephson transition can reveal rotation, with hysteresis also reported (16,17).At excessive flow, the current can break up into vortex–antivortex pairs (73,170).Source: B. Atomtronic SQUIDs · p. 52

C. Sagnac interferometry and rotation sensing

  • In a ring Sagnac interferometer, the phase shift is ∆φ = 8πAΩ/λv, matching the optical form while cold atoms can have a much smaller λv; unguided-envelope-free states can accrue fringes at any time.Closed-path guided atom configurations have been demonstrated, but their sensitivity has yet to match free-space cold-atom gyros, despite recent experimental progress (554,555).Source: C. Sagnac interferometry and rotation… · p. 53
  • Typical repulsive interactions (positive g) make standing-wave fringes rapidly disperse, whereas a two-component system with very similar scattering lengths can support them; a π-swap-pulse protocol reads Ω from populations, most sensitively near 2ℓΩT ≈ π/2.The response is N1 = N[1+cos(2ℓΩT)]/2 and N2 = N[1−cos(2ℓΩT)]/2; the π swap pulse at T/2 counters relative-phase accumulation from internal-state energy and differences in g11 and g22 (557,558).Source: C. Sagnac interferometry and rotation… · pp. 53–54

D. Magnetometry … A. The Atomtronic quantum interference device:AQUID

The sections describe atomtronic magnetometry and ring-based quantum devices, showing how matter-wave coherence and interactions enable sensing and effective qubits. They also outline the operating conditions, experimental progress, and limitations of these approaches.

D. Magnetometry

  • An imbalanced superposition of counter-propagating l = ±1 modes creates a minimal-density line whose interaction-dependent rotation can encode scalar magnetic fields and rotations.The proposed readout uses direct imaging; the rotation frequency depends on nonlinear interactions and the population imbalance between the modes.Source: D. Magnetometry · p. 55
  • A few pT at 1 Hz is the projected magnetic-field-change sensitivity of a ring-BEC magnetometer that tracks rotation of a minimal-density line.The estimate assumes angular resolution of about 0.1 rad and a BEC lifetime around 1 s; Feshbach-resonance losses may limit measurement, though several atomic species are reported stable across resonances for seconds.Source: D. Magnetometry · pp. 55–56

E. Concluding remarks and outlook

  • Atomtronic devices use coherence and interactions in ultracold atoms to realize SQUID analogues, matter-wave interferometers, and rotation and magnetic sensors, with potential integration into larger circuits.Source: E. Concluding remarks and outlook · p. 56

X. TWO LEVEL QUANTUM DYNAMICS IN RING-SHAPED CONDENSATES AND MACROSCOPIC QUANTUM COHERENCE

  • A weak link in a ring of persistent-current states produces an AQUID; theory shows its two currents can form an effective two-level system, with inter-ring hopping enabling two-qubit gates [596–599, 608] [597, 609].Source: §X · p. 56
  • Experiments demonstrated flux-dependent critical-current oscillations in AQUIDs, but current entanglement was not demonstrated; ring lattices and time-of-flight readout methods have also been realized or described [16] [597] [610].Spatial-light-modulator ring lattices could in principle host stacked ring qubits, while interference patterns and image noise can reveal current and superposition state.Source: §X · p. 57

A. The Atomtronic quantum interference device:AQUID

  • At Ω = π, a single weak link creates a double-well potential for clockwise and anticlockwise currents; tunneling forms symmetric and antisymmetric qubit states, with a finite gap favored by a weak barrier and intermediate-to-strong interactions [597, 598].The two-level description requires other excitations to lie sufficiently far from the competing ground states; increasing ring size makes the qubit dynamics less addressable [597, 598].Source: Figure 44 · p. 57, A. The Atomtronic quantum interference… · p. 58
  • For a ring interrupted by three weak links, the two lowest levels near Ω = π are separated from higher excitations, supporting an effective qubit description [598, 599].The effective two-level result also holds at low filling, as shown by exact diagonalization of the Bose-Hubbard model [598, 599].Source: Figure 45 · p. 58, A. The Atomtronic quantum interference… · p. 59

B. Demonstration of the one qubit and two qubit unitary gates … 1. Interferometric detection of the current states

Optical ring lattices with impurities are proposed as atomtronic qubits for one- and two-qubit gates, while current states can be read out through interference and time-of-flight measurements. Density correlations and momentum noise reveal information that averaged density images can miss.

B. Demonstration of the one qubit and two qubit unitary gates

1. Single qubit gates

  • Near the symmetric double well, tuning the phase-slip term enables phase, NOT, and Hadamard gates through single-qubit rotations.Setting the gap energy close to zero by adjusting the impurity barrier gives NOT at α = π/2 and Hadamard at α = π/4.Source: §1 · pp. 59–60

2. Two-qubit coupling and gates

  • Coupled impurity rings support qubit-qubit interactions, with a CNOT obtainable using two √SWAP gates.The proposed tuning takes ε → 0 and Φ → π − δJ′′π/J′; one-qubit rotations and CNOT suffice for universal gates.Source: §2 · p. 60

C. Readout of atomtronic qubits

  • Atomtronic qubit currents can be read out by time-of-flight imaging or by interfering the ring condensate with a central condensate as a phase reference.In weakly interacting continuous rings, self-heterodyne in-situ measurements have detected both current orientation and intensity.Source: C. Readout of atomtronic qubits · p. 60

1. Interferometric detection of the current states

  • Averaging interferograms can wash out current-state spirals because each realization has a random relative phase, but density-density correlations recover the pattern.The ring condensate interferes with a central condensate at rest; the number of spirals indicates the total number of rotation quanta.Source: §1 · pp. 60–61
  • A spiral discontinuity in the density covariance reveals a superposition of winding numbers, distinguishing states whose ordinary interferometric images can look similar.The covariance’s discontinuity at the weak link is visible at intermediate expansion times; for U = 0 the current is a non-entangled superposition, while U = J gives a highly entangled NOON state.Source: §1 · p. 61, Figure 47 · p. 62
  • At the degeneracy point, momentum noise distinguishes one-particle superpositions with σGP = N/4 from NOON states with σNOON ∝ N/4, and rises by a factor N between minima and maxima.As interaction increases further, the system fermionizes and its ground state becomes a superposition of |K = 0⟩ and |K = N⟩.Source: §1 · p. 62
  • Momentum noise peaks at flux degeneracy when barrier and interaction are comparable, while weak-link-site filling marks entanglement onset and can be measured with site-resolved absorption.Moving flux away from degeneracy lowers noise; for weak interactions, the critical interaction for barrier-site filling scales linearly with barrier strength.Source: §1 · p. 62, Figure 48 · p. 62

D. Experimental realization of the ring-lattice potential with weak links … 1. Fermionic superfluidity and critical velocity

The section examines experimental ring-lattice platforms and their relevance to atomtronic devices and fermionic transport, including flux-dependent current interference in an atomic SQUID. It also considers fermionic critical velocity in light of competing excitations, counterflow, and experimental conditions.

D. Experimental realization of the ring-lattice potential with weak links

  • A programmable SLM produces scalable ring lattices with well depths of several Erec, 4.5% rms intensity variation, and dynamically adjustable barriers; camera-feedback iterations refine the target profile to below 2% discrepancy.The single-link design supports persistent flow-states, while the feedback method produces a scaled-down 5–10 µm-radius lattice with sufficient trapping power.Source: D. Experimental realization of the… · pp. 63–64

E. Setup for adjustable ring-ring coupling

  • Measured ring-lattice potentials remain almost undisturbed over axial shifts of ±2.2R, supporting a proposed vertical stack of more than 10 rings, while adjustable ring spacing could enable tunnel-coupled qubit stacks.The ring radius changes by ΔR/R = 0.0097 · z, with z normalized to the ring radius; potential quality gradually diminishes for larger shifts.Source: E. Setup for adjustable ring-ring… · pp. 64–65

F. First experimental demonstration of the interference of atomtronic currents

  • Experiments show critical-current modulation with applied flux in an atomic SQUID, revealing interference of currents, although the demonstrated dilute BEC is mean-field and does not demonstrate qubit entanglement.The experiment is presented as a first step toward qubits based on superpositions of currents.Source: F. First experimental demonstration of… · pp. 65–66

G. Concluding remarks and outlook

  • The proposed atomtronic qubit uses neutral-atom currents in weak-link ring lattices for effective two-level dynamics and single- and two-qubit gates; three weak links also enable a two-level flux-qubit architecture with time-of-flight readout.Readout can use spiral interference patterns, image noise, current magnitude and direction, or depletion at a weak link; entanglement remains un demonstrated in the cited interference experiment.Source: G. Concluding remarks and outlook · p. 67

XI. TRANSPORT AND DISSIPATION IN ULTRACOLD FERMI GASES

  • Cold-atom two-terminal systems have measured particle, spin, and heat conductances and off-diagonal transport coefficients in long channels, while short planar junctions realize the Josephson effect.Strongly interacting Fermi gases near a Feshbach resonance offer access to correlated-matter regimes across the BEC–BCS crossover.Source: §XI · p. 68

1. Fermionic superfluidity and critical velocity

  • Landau’s criterion predicts that an object moving below a superfluid’s critical velocity experiences no drag, based on energy- and momentum-conservation constraints.Source: §1 · p. 68
  • Landau’s threshold is min ω(k)/k; for a concave low-energy dispersion, it equals the sound velocity.The sound-velocity identification in the phonon sector assumes a convex dispersion; coupling to the broken-pair particle-hole continuum bends the fermionic dispersion downwards [660].Source: §1 · p. 69
  • The two excitation branches produce a Landau critical-velocity maximum near unitarity, observed by stirring an optical potential in a fermionic cloud [653, 654, 655].Phonons set the critical velocity in the BEC regime, whereas on the BCS side breaking Cooper pairs limits it to vc ≃ ∆/pF.Source: §1 · p. 69
  • For phonon-limited counterflow, the critical velocity is the sum of the two sound velocities, matching measurements in 6Li and 7Li mixtures [648].Relative motion can excite pairs of excitations in the two superfluids [647, 657–659].Source: §1 · p. 69
  • When dispersion is convex, Landau critical velocity in the phonon sector is identified with sound velocity; transverse trapping in experiments reduces critical velocity by inverting dispersion concavity at large momenta.Source: §1 · p. 69
  • Oscillatory disturbances suppress the Landau threshold, while quantized vortices cause dissipation for stronger disturbances [664, 665, 666, 512].Landau’s argument assumes constant disturbance velocity; by analogy with radiation from an accelerated charge, acceleration suppresses the critical velocity.Source: §1 · p. 69

2. Josephson currents … B. Ring-leads system

This section surveys atom-based Josephson transport across fermionic mesoscopic channels, spin drag, bosonic circuits and guides, and ring-lead systems. It covers observed transport phenomena, practical limits and emerging directions, including how ring-lead dynamics varies with coupling, flux and lead type.

2. Josephson currents

  • The Josephson critical current in strongly interacting Fermi gases peaks near unitarity, alongside opposing changes in chemical potential and pair-condensate fraction [35].Breakdown of coherent transport has also been accompanied by topological-defect nucleation; the dissipation mechanisms and excitation branches remain to be clarified [450,451,512,653–655].Source: §2 · p. 70

B. Fermionic transport in mesoscopic channels

  • Fermionic channels show conductance suppression at a lattice band gap that persists with superfluid reservoirs, while atomic point contacts realize an ideal spin filter [675,677].The persistence is attributed to a 1D Luther-Emery liquid; under weak interactions, near-resonant light scattering can be modeled as losses in a Landauer-Büttiker description [676,678].Source: B. Fermionic transport in mesoscopic… · pp. 70–71

C. Fast spin drag in normal Fermi gases

  • Fast spin drag in normal Fermi gases arises from spin-current interactions and vanishes only when Landau parameters F1 and G1 are equal [693].It is quadratic in k_Fa and beyond mean-field in character; Landau-theory applicability is expected at temperatures below T_F but above the superfluid transition, particularly in the weak-coupling BCS regime.Source: C. Fast spin drag in normal Fermi gases · p. 72

D. Concluding remarks and outlook

  • Weak fermionic currents make signal-to-noise a major measurement challenge; single-atom detection and cavity-based schemes could improve sensitivity toward the back-action limit [702–707].Sample-preparation noise scales unfavorably with particle number, while reaching the back-action limit would enable reconstruction of transport full counting statistics.Source: D. Concluding remarks and outlook · p. 72
  • Manipulating topological superfluids, including p-wave superfluids or Kitaev chains, is an overarching goal for studying protected quantum information and guiding topological quantum-computer development [708,709].Source: D. Concluding remarks and outlook · p. 73

XII. TRANSPORT IN BOSONIC CIRCUITS

  • Bosonic circuits can exhibit Andreev-like hole reflection, lack the electronic ring Aharonov-Bohm oscillation, and support nonreciprocal transport and topological pumping [711,712,716–719,721–726].Flux-driven ring transmission can be fractional in atom number and generate highly entangled states with potential sensing and information applications [713].Source: §XII · p. 73

A. Matterwave guides

  • TAAP rings reduce field-modulation effects by a factor of 10^-138 and have enabled Bose-Einstein condensates to propagate over 10 cm without additional heating [125].Artificially structured waveguides retain some roughness, whereas TAAP rings offer long-distance propagation; combining them with optical guides could enable tunable ring-lead couplings and velocity-selective routing [729–731].Source: A. Matterwave guides · pp. 73–74

B. Ring-leads system

  • The ring-leads system is modeled with a Bose-Hubbard lattice of source lead, ring, and drain chains coupled by tunneling, offering a controlled route to study bosonic transport [712,133].Although Aharonov-Bohm currents are established for fermionic rings, the effect is not well understood for bosonic systems.Source: B. Ring-leads system · p. 74
  • With single-site leads, weak coupling leaves the ring nearly empty, whereas strong coupling spreads atoms across ring and leads; near Φ = 0.5, oscillations slow [711,712,733].At weak coupling, increasing flux populates the ring and washes out density oscillations; strong coupling produces multiple oscillation frequencies and scattering also washes out oscillations.Source: B. Ring-leads system · p. 75, Figure 61 · p. 75
  • For extended leads, the source is initially filled and suddenly coupled to the ring; the dynamics is modeled with full-system DMRG or a Lindblad equation after tracing out bath leads [734,735].The Lindblad description uses the Born–Markov approximation.Source: B. Ring-leads system · p. 75
  • After flux-dependent initial dynamics, the interacting-boson steady state is nearly flux-independent, indicating the absence of a steady-state Aharonov–Bohm effect [737].The initial current retains a flux dependence, unlike the long-time steady state.Source: B. Ring-leads system · p. 75
  • Lindblad bath dynamics produces results similar to full ring-and-reservoir DMRG [734,735], but smooths its oscillations, supporting a memoryless Markovian-lead description.The comparison concerns the dynamics of the extended-lead system.Source: B. Ring-leads system · p. 75

C. Y-junctions … A. A boson ring ladder at weak interactions

Across atomtronic junctions, ring circuits, and bosonic ladders, the review describes tunable transport and reflection, pumping-generated entanglement, and phases shaped by flux and interactions.

C. Y-junctions

  • At K=1J, hard-core-boson Y-junctions show negative Andreev-like reflection (R≈−1/3) and T≈4/3; K≈0.5J nearly suppresses reflection, while weak K produces positive reflection.The strong-coupling transmission exceeds the incoming density wave because of the negative reflected contribution; similar dynamics are reported for finite interactions.Source: C. Y-junctions · pp. 76–77, Figure 63 · p. 77

D. Differences between fermions and hard-core boson

  • Hard-core bosons obey single-site exclusion while retaining a symmetric many-body wavefunction, and map to fermions in one dimension but not beyond it.The review connects these exchange-statistics differences to transport behavior.Source: D. Differences between fermions and… · p. 77
  • In ring-lead transport, fermionic density waves depend on flux and vanish at half-flux, whereas hard-core-boson transmission and reflection are flux-independent; bosons also show negative reflection in Y-junctions.The bosonic behavior is described as an absence of the Aharonov-Bohm effect, unlike the fermionic response.Source: Figure 64 · p. 77, D. Differences between fermions and… · pp. 77–78, Figure 65 · p. 78

E. Entangled state generation with topological pumping in ring circuits

  • Up to 6 particles can be pumped into NOON-like states with nearly unit fidelity; fidelity decreases for more particles or stronger interactions as the energy gap is exponentially suppressed.Periodic topological pumping moves atoms from a source through a ring, where the Bose-Hubbard interaction can create superpositions with all N atoms in either ring path.Source: Figure 66 · p. 78, E. Entangled state generation with… · p. 78

F. Concluding remarks and outlook

  • The review identifies low-heating transport, atom reservoirs, non-standard Hubbard models, and automated potential design as opportunities for atomtronic circuits.Time-dependent circuit potentials can enable topological pumping and NOON-state generation for prospective rotation sensing; DMDs or other light-based techniques can shape potentials, while machine learning may automate their design.Source: F. Concluding remarks and outlook · p. 79

XIII. ARTIFICIAL QUANTUM MATTER IN LADDER GEOMETRIES

  • Artificial-gauge-field ladders can host Meissner, vortex, density-wave, and quantum-Hall-analogue regimes, with phases depending on flux, interactions, hopping, and filling.The review also notes topological properties and quantum-magnetism simulation in a diamond-chain variant.Source: §XIII · p. 79

A. A boson ring ladder at weak interactions

  • Weakly interacting ring ladders support vortex, Meissner, and biased-ladder phases, with biased phases appearing only at weak interactions and disappearing as interactions increase.The Meissner phase has vanishing transverse current and homogeneous density; the vortex phase has transverse current and density modulations, and the chiral current jumps as vortices enter.Source: A. A boson ring ladder at weak… · p. 80, Figure 67 · p. 80
  • The Meissner phase has one Goldstone mode, while the vortex phase has two and exhibits supersolidity with a structure-factor peak and phase coherence.The second mode corresponds to broken discrete translational symmetry in the vortex phase.Source: A. A boson ring ladder at weak… · pp. 80–81, Figure 68 · p. 81

B. The boson ladder at strong interaction · C. Ultracold atoms carrying orbital angular momentum (OAM) in a diamond chain · D. Concluding remarks and outlook

The sections describe interaction-driven phases in bosonic ladders, topological and caged states in an OAM diamond chain, and prospective uses and challenges for ring-ladder atomtronics.

B. The boson ladder at strong interaction

  • For Ω ≠ 0 and U⊥ = 0, the ladder’s Meissner-to-Vortex transition is commensurate-incommensurate, marked by a split momentum peak and finite-wavevector rung-current correlations.In the Meissner state, which occurs for Ks > 1/4 in the stated limit, the chiral current rises linearly with small flux while the mean rung current vanishes and its correlations decay exponentially.Source: B. The boson ladder at strong… · p. 82, Figure 69 · p. 82
  • At U⊥ = 0, interactions enlarge the Meissner region and allow it to persist at sufficiently large Ω/J; stronger flux can produce a second incommensuration and localization tendencies near λ = π.The second incommensuration appears in rung-current correlations and the static structure factor, with extra rung-current correlation peaks at P = λ/2 + p(Ω²) and π ± P.Source: B. The boson ladder at strong… · p. 82, Figure 69 · p. 82
  • With U⊥ ≠ 0, the C-IC transition gives way to an Ising Meissner-to-ICDW transition, followed by vortex melting and a BKT transition into the Vortex phase.The reported signatures include a Lifshitz point before Lorentzian momentum peaks, zigzag charge-density-wave peaks, and—at larger flux—separated momentum and rung-current peaks indicating incommensuration.Source: B. The boson ladder at strong… · p. 82, Figure 70 · p. 83

C. Ultracold atoms carrying orbital angular momentum (OAM) in a diamond chain

  • The OAM diamond chain has a gap and four right-edge in-gap states that persist while the gap remains open; its topology requires mappings and has no transition across the gap-closing point.The spectrum has six bands in three degenerate pairs, becomes flat when J2 = J3, and maps through symmetric/antisymmetric subchains to modified SSH models; the original model is a square-root topological insulator.Source: C. Ultracold atoms carrying orbital… · p. 83, Figure 72 · p. 83
  • At J2 = J3, the diamond chain exhibits Aharonov–Bohm caging: central-site states oscillate to four neighboring sites without leaving a cage spanning two unit cells.The oscillation frequency is the absolute value of the top- or bottom-flat-band energy.Source: C. Ultracold atoms carrying orbital… · p. 83
  • Strongly interacting OAM bosons at unit filling can simulate spin-1/2 models, including XYZ Heisenberg models, with geometry and tunneling phases controlling the effective couplings and fields.For ring ladders with Θ_l = (2s+1)π/(2l), interaction-mediated single-spin flips vanish and the effective model is an XYZ Hamiltonian without external field; ring radius and separation further tune coupling strengths.Source: C. Ultracold atoms carrying orbital… · p. 84

D. Concluding remarks and outlook

  • Ring geometry enables accurate study of infinite-ladder phase diagrams and excitation spectra, finite-ring parity and commensurability effects, and phase probes including persistent currents, spiral interferometry, and Josephson modes.Source: D. Concluding remarks and outlook · p. 84
  • Outlook priorities include theories spanning weak to strong interactions, testing phase signatures against temperature and long-range interactions, and realizing ring ladders as atomtronic benchmarks.The paper describes experimental realization of ring ladders as close to reach.Source: D. Concluding remarks and outlook · p. 84

XIV. QUANTUM-ENHANCED ATOMTRONICS WITH BRIGHT SOLITONS … 3. Entangling solitons with different Lz

Attractive-boson solitons provide quantum resources for atomtronics, with behavior spanning barrier scattering, bound-state spectra and stability, rotation, and entanglement for enhanced rotation sensing.

XIV. QUANTUM-ENHANCED ATOMTRONICS WITH BRIGHT SOLITONS

  • The chapter presents attractive bosons’ quantum ground-state composition, barrier transmission, excitation spectrum, and rotational response as resources for atomtronics applications.Source: §XIV · p. 85

A. Scattering properties of attractive bosons against a barrier

  • Below one quarter of the soliton chemical-potential magnitude, mean-field scattering predicts a forbidden transmission window, but quantum fragmentation makes the transmission curve continuous and events random.The mean-field transmission jump occurs because splitting costs interaction energy that the incident kinetic energy may not supply [809].Source: A. Scattering properties of attractive… · p. 85
  • Even when only two orbitals are populated, scattering may impair macroscopic coherence through entanglement of center-of-mass motion with excitations and an orthogonality catastrophe; limited atom numbers may preserve it.The upper bound on a suitable atom number remains an open question requiring further numerical study.Source: A. Scattering properties of attractive… · p. 85

1. Quantum solitons in the Bose-Hubbard model

  • For two lattice bosons, the bound-state soliton band separates from scattering states at U/J ≥ 4; for larger samples, numerical results identify a low-energy bound-state band with exponentially decaying correlations.For larger particle numbers, the Bose–Hubbard model is not solvable by the coordinate Bethe ansatz because multi-particle occupancy prevents factorization of two-body scattering [832–834].Source: §1 · p. 86
  • Pinning and releasing a lattice soliton populates mostly bound states for U > Uc; increasing interaction stabilizes its density profile, and asymptotic expansion velocity shows peculiar scaling near Uc.The asymptotic expansion velocity acts like an order parameter, although the system has no criticality close to Uc.Source: §1 · p. 87

2. Solitons in rotation

  • Attractive bosons form many-body bound states whose persistent-current periodicity is 1/N in continuous rings, while lattice periodicity depends on interaction strength; fractionalization is a quantum many-body effect.The lattice behavior arises because center-of-mass and relative coordinates do not decouple at finite interaction; the observed features disappear in mean-field Gross–Pitaevskii theory.Source: §2 · pp. 87–88

3. Entangling solitons with different Lz

  • Breaking rotational invariance with a single-site barrier, then quenching Ω from 0 to Ω0/2, entangles Lz = 0 and Lz = N when the current reaches half its maximum.The resulting state is |ψ⟩NOON = 2^-1/2(|Lz = 0⟩ + |Lz = N⟩).Source: §3 · p. 88
  • The entangled state has quantum Fisher information FQ ∼ N^2, reaching the Heisenberg limit and enabling enhanced sensitivity to rotation.Thermal and technical fluctuations and particle losses are the main experimental limitations; losses are expected to play a minor role for the small particle numbers considered.Source: §3 · p. 88

C. Concluding remarks and outlook … C. Concluding remarks and outlook

These concluding remarks review atomtronic directions, including quantum-soliton interferometry, alkaline-earth-like atoms, SU(N) fermions, and Rydberg atoms. They cover persistent-current and driven-dissipative phenomena, implementation constraints and open directions, as well as Rydberg-based microwave-to-optical conversion and transport for atomtronic and hybrid circuits.

C. Concluding remarks and outlook

  • Attractive-boson bound states stabilize pinned solitons and give ring solitons a 1/N flux periodicity; a gauge-field quench can create angular-momentum superpositions with N-fold rotation sensitivity.A localized barrier can split and recombine the solitons; controlling decoherence and losses and finding optimal interferometer parameters remain challenges.Source: C. Concluding remarks and outlook · pp. 88–89

A. Why alkaline-earth-like metal atoms?

  • Alkaline-earth-like atoms offer efficient cooling and nuclear-spin SU(N) physics, but their weak magnetic sensitivity rules out magnetic trapping and Feshbach tuning; clock transitions also enable precision and correlated-physics applications.Experiments report ordered magnetic phases above the Néel temperature after Pomeranchuk entropy reduction; fermionic isotopes also support spin exchange between 1S0 and 3P0 orbitals and artificial gauge-field schemes.Source: A. Why alkaline-earth-like metal atoms? · pp. 89–90

B. Effective Abelian and non-Abelian gauge fields

  • A tripod scheme in 87Sr implements a homogeneous Abelian gauge field with zero Berry curvature and no Lorentz force; reversing a laser direction yields a non-Abelian field with spin-precession and momentum-space Josephson possibilities.The proposed circuit scales must exceed the gauge-field laser wavelength; tripod schemes can generate SU(2) fields and a double tripod can extend this to SU(3) [876].Source: B. Effective Abelian and non-Abelian… · pp. 90–91

C. Persistent Current of SU(N) Fermions

  • SU(N) fermions show spinon-driven fractional flux periodicity, an interaction-driven Mott transition for N > 2 at integer filling, and a parity-dependent diamagnetic or paramagnetic current.The flux fractionalization comes from coupling between spin and matter, unlike the bound-state mechanism in attractive bosons; a Mott gap suppresses ground-state spinon creation.Source: C. Persistent Current of SU(N) Fermions · pp. 91–92, Figure 80 · p. 91

D. Concluding remarks and outlook

  • Proposed SU(N) atomtronic circuits include non-Abelian ring interferometers and fermionic AQUIDs, while transport, spin-charge dynamics, disorder, and many-body localization remain research opportunities.The outlook also proposes SU(N) atomic SQUIDs and Datta-Das transistors, and current-based diagnostics of Hubbard, Heisenberg, and Kondo systems [881, 887–893].Source: D. Concluding remarks and outlook · p. 92

W. Li and O. Morsch

  • Rydberg atoms combine strong interactions, controllability, and long coherence times, but have not yet been used for atomtronics applications as defined in this review.Their excitations could support studies of transport in ordered or disordered arrays and add to the future atomtronics toolbox.Source: W. Li and O. Morsch · p. 93

A. Driven-dissipative Rydberg systems

  • In driven Rydberg gases, facilitation and spontaneous decay realize directed percolation, and experiments observed its absorbing-state transition in excitation number and variance.The experiment scanned the facilitation-to-decay rate ratio by varying off-resonant driving; future extensions include partially coherent driving, ordered arrays, and tailored dissipation.Source: A. Driven-dissipative Rydberg systems · pp. 93–94, Figure 82 · p. 93

B. Microwave-optical conversion using Rydberg atoms

  • Six-wave mixing in Rydberg atoms demonstrates coherent microwave-to-optical conversion of classical fields, a process relevant to coupling microwave superconducting qubits with photonic qubits.Such conversion is described as essential for connecting those qubit types in long-distance quantum communication [915].Source: B. Microwave-optical conversion using… · p. 94
  • Nonlinear frequency mixing of six waves in cold polarized 87Rb converts an input microwave field into light, with Rydberg EIT and induced ground-to-intermediate-state coherence enabling generation.The setup uses four auxiliary electromagnetic fields, the microwave field to be converted, and the generated optical field.Source: B. Microwave-optical conversion using… · p. 94
  • η = 0.051 photon conversion efficiency was measured, 17 times the value reported in Ref.913; theory predicts above 50% with a carefully selected low-absorption energy-level scheme.Conversion is most efficient near resonance, and generated-light power is approximately linear in microwave input intensity.Source: Figure 83 · p. 94, B. Microwave-optical conversion using… · p. 94
  • Stimulated Raman adiabatic passage or an effective two-photon transition are proposed routes toward near-unit single-photon state transfer; the conversion method may also enable sensitive microwave or THz detection.The two-photon approach could be realized by tuning fields such as C and A off resonance.Source: B. Microwave-optical conversion using… · p. 94

C. Concluding remarks and outlook

  • Future Rydberg-transport studies could inject excitations at defined positions and use patterned dipole-trap arrays [24] or microlens arrays [162] to explore source-drain configurations and ring or more complex topologies.Laser or microwave coupling could tailor interactions or add dissipation/dephasing, enabling study of the crossover from incoherent hopping to coherent transport.Source: C. Concluding remarks and outlook · pp. 94–95
  • Combining Rydberg-excitation transport with regular atomtronics could create hybrid systems transporting excitations and matter independently or in a coupled manner.The section notes that the experiments discussed do not yet directly contribute to atomtronics, while suggesting future transport studies could use their techniques.Source: C. Concluding remarks and outlook · pp. 94–95

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