We present a method for the creation of closed-loop lattices for ultra-cold atoms using dressed potentials. We analytically describe the generation of trap lattices that are state-dependent with dynamically controlled lattice depths and positioning. In a design akin to a synchronous motor, the potentials arise from the combination of a static, ring-shaped quadrupole field and multipole radio-frequency fields. Our technique relies solely on static and radio-frequency (rf) magnetic fields, enabling the creation of robust atom traps with simple control via rf amplitudes and phases. Potential applications of our scheme span the range from quantum many-body simulations to guided Sagnac interferometers.
I. INTRODUCTION
Adiabatic radio frequency (rf) dressed potentials play an increasingly important role in recent developments with ultra-cold atomic physics, see Ref. 1 for a review. These potentials, produced by the combination of static and oscillating magnetic fields, enable the creation of a plethora of different trapping geometries that can support confinement as well as, under certain conditions, transport of atomic clouds.
The trapping potentials obtained with rf dressing techniques are inherently species-selective and state-dependent, because they can be controlled independently for atoms that have different Landè g-factors.2,3 For the case of often used alkali atoms, the different hyperfine levels of their electronic ground state have g-factors with near identical magnitude but opposite sign. Arbitrary superpositions of such internal states can be prepared, and the combination with dressed, state-dependent potentials allows for coherent beam splitting, using only fields oscillating in the rf and microwave (mw) regime. This results in a versatile workbench capable of generation, independent manipulation, and detection of internally labeled atomic superposition states, especially relevant for interferometric schemes.4,5 There are numerous examples of experimental implementations that demonstrate the generation and detection of dressed superposition states (see Ref. 1 and references therein), including also nondestructive detection methods.6 The transport of a single, dressed spin state has been demonstrated over macroscopic distances.7 By manipulating the amplitude, frequency, and polarization or relative phases of the contributing rf fields, versatile control over the resulting rf dressed potentials can be achieved.8–10 This ability to dynamically modify the potential landscape between different configurations within a single experimental run is significantly useful in the context of both fundamental as well as applied experiments with ultra-cold atomic clouds and Bose–Einstein condensates (BECs). The possibilities range from double wells11,12 to hollow-shell traps,13 ring-shaped matter-waveguides,14–17 and purely magnetic atom-trap lattices.18,19
In recent years, ultra-cold atoms in optical, magnetic, or hybrid trap lattices have constituted an important quantum simulation testbed for a variety of physical phenomena otherwise not straightforward to probe.20 These include studying the dynamics of strongly correlated particles,21 investigating new topological phases of matter,22,23 topological pumping and quantized transport24 as well as the thermalization of quantum systems and dynamics in the many-body regime.25 The use of dressed potentials may add to this, as magnetic atom-trap lattices with interesting topologies can be formed, including, e.g., ring structures, that are furthermore adjustable and dynamically controllable in a state dependent fashion. These ring-shaped atom-trap lattices have been proposed as analogues for superconducting flux qubits26 as well as platforms where artificial gauge fields can be studied27 and correlated many-body effects can be harnessed for the implementation of rotation sensors and gyroscopes with enhanced sensitivity.28 Here, we present a method that allows for the generation of such ring-shaped atom-trap lattices based on rf dressed potentials by combining a ring-shaped quadrupole potential with a multipole rf field. Our method is compatible with atom-chip technology29 and may enable robust, mechanically stable and compact quantum devices and sensors. The approach combines state-dependent dynamical control with periodic boundary conditions, furthering a range of possible quantum simulation studies.
II. DRESSED RING WAVEGUIDES
Rf dressed potentials arise from the combination of an inhomogeneous magnetic field and an oscillating magnetic field that drives atomic spin flips. The static field defines a two-dimensional manifold where Larmor precession can be resonantly excited, thus coupling low-field seeking states to high-field seeking states. This principle can be extended to any pair of such states, e.g., by coupling states from different hyperfine manifolds, where the nuclear spin changes orientation with respect to the electronic spin. A trap is formed in the regime where atoms traverse the region of resonance adiabatically. The trap topology of our scheme is based on an axially symmetric combination of a ring-shaped, static quadrupole field and an oscillating rf field with radial and axial components of different phases. Such an arrangement results in a dressed magnetic potential with toroidal geometry,8 which allows for the creation of ring-shaped and toroidal, i.e., hollow torus-shaped atom traps. The fields are specifically chosen such that connected potential minima without degenerate points are generated, which would otherwise cause atom loss. In the following, we first recapitulate the approach for generating hollow-torus and in particular ring-shaped atom traps before describing the method for partitioning these traps in order to form a lattice.
Illustration of the static ring quadrupole field (section with field lines and local polar coordinates in the inset). Such a field can be obtained by means of four counterpropagating current loops as indicated by the arrows. Global Cartesian and cylindrical coordinates are shown together with local polar coordinates, defining toroidal ( ) and poloidal ( ) angles.
Illustration of the static ring quadrupole field (section with field lines and local polar coordinates in the inset). Such a field can be obtained by means of four counterpropagating current loops as indicated by the arrows. Global Cartesian and cylindrical coordinates are shown together with local polar coordinates, defining toroidal ( ) and poloidal ( ) angles.
III. DRESSED RING LATTICES
We are in particular interested in the case of forming rings at the top and bottom of the torus ( ). This occurs when the radial rf amplitude is smaller in magnitude than the vertical rf amplitude and the corresponding real fields are out-of-phase, i.e., or . Since the local static field in those rings is parallel to the z-direction, the trapping potential can be conveniently modulated by interference of with a multipole rf field, polarized in the x, y-plane, that oscillates at the same frequency. As shown below, the resulting modulation leads to the creation of state-dependent ring lattices.
Top panel: Orthogonal, linearly polarized, interior quadrupole fields ( ) that can be driven with -phase difference to generate circular polarization. The shown field lines are the leading order approximation [Eq. (10)] to the fields generated by two sets of four blue (red) infinitely long wires. The wires cross the plane at the locations depicted by the symbols ( ) for currents going into (out of) the page. The rotation offset angles are ( ) for the blue (red) quadrupole. The bold arrows emphasize the local orthogonal field directions along a circular path, centered on the z-axis. Bottom panel: Following the circular path in anticlockwise direction, the local field directions (blue and red arrows) rotate times clockwise, while the direction of a radial field pointing outwards (green arrows), rotates once in the opposite direction. The interference between these fields leads to n minima and maxima. For example, when the radial field is in phase with the blue multipole component, destructive interference occurs at and . Shifting the multipole phases by such that the phase of the red component aligns with that of the radial field leads to destructive interference at and .
Top panel: Orthogonal, linearly polarized, interior quadrupole fields ( ) that can be driven with -phase difference to generate circular polarization. The shown field lines are the leading order approximation [Eq. (10)] to the fields generated by two sets of four blue (red) infinitely long wires. The wires cross the plane at the locations depicted by the symbols ( ) for currents going into (out of) the page. The rotation offset angles are ( ) for the blue (red) quadrupole. The bold arrows emphasize the local orthogonal field directions along a circular path, centered on the z-axis. Bottom panel: Following the circular path in anticlockwise direction, the local field directions (blue and red arrows) rotate times clockwise, while the direction of a radial field pointing outwards (green arrows), rotates once in the opposite direction. The interference between these fields leads to n minima and maxima. For example, when the radial field is in phase with the blue multipole component, destructive interference occurs at and . Shifting the multipole phases by such that the phase of the red component aligns with that of the radial field leads to destructive interference at and .
(a) Cuts through the approximated dressed potential (not to scale for realistic geometries). The potential landscape on the resonant torus surface is shown for traps. Potential minima are depicted as dark blue regions. On the right-hand side, a cut through the 3D potential is shown for the y, z-plane. The assumed parameters are , G, G, and G. For a single internal spin state, the angular orientations and depths of the two lattices at the bottom and top of the torus can be controlled independently. This is shown for the potential on the resonant surface for (b) and (c). For different species or spin states with different g-factors, independent potentials can be superimposed with some restrictions.
(a) Cuts through the approximated dressed potential (not to scale for realistic geometries). The potential landscape on the resonant torus surface is shown for traps. Potential minima are depicted as dark blue regions. On the right-hand side, a cut through the 3D potential is shown for the y, z-plane. The assumed parameters are , G, G, and G. For a single internal spin state, the angular orientations and depths of the two lattices at the bottom and top of the torus can be controlled independently. This is shown for the potential on the resonant surface for (b) and (c). For different species or spin states with different g-factors, independent potentials can be superimposed with some restrictions.
An important scenario is the trapping of atoms with different g-factors in the same rf dressed trap. Different atomic species may be controlled via different radio-frequencies,3 and a particular case is a mixture or superposition of the same species in different hyperfine states. Here, the magnitude of the g-factor may be approximately equal but of opposite sign. In our description, a negative g-factor leads to a negative resonant dressing frequency. However, the presence of any amplitudes with positive implies the presence of the corresponding amplitudes with negative and conjugated phase. The torus and ring forming potential in the poloidal direction is symmetric in this respect [see Eq. (9)]. A trap formed at frequency also leads to a trap formed at frequency . However, as it can be seen from Eq. (13), control of atoms via in the top ring and/or in the bottom ring imparts the same effects in the opposite ring on atoms with the opposite g-factor . It should be noted, however, that changing the sign of also inverts the orientation of trap misalignment, because the negative resonant frequency changes the sign of the anti-symmetric but not of the symmetric as and swap their values. Within one ring, the two types of atoms can be controlled independently. Atoms in different spin states can be transported in opposite directions, thus making the configuration a candidate for guided Sagnac interferometer gyroscopes and other atomtronic applications. A lattice filled with atoms in one spin state could be immersed in a homogeneous ring of atoms in another spin state to couple different sites via phonons or study quantum friction of impurities in a Bose–Einstein condensate.
We briefly exemplify a few experimental considerations. With micro-fabricated trapping structures29 high field gradients can be achieved, e.g., T/m = G/ m. For 87Rb atoms in their electronic ground state, with total spin , , and , a torus with m forms for a dressing frequency kHz. Assuming realistic amplitudes G, G, and G to form traps over a millimeter sized ring with mm leads to prolate traps with reasonable trap frequencies of Hz, Hz, and kHz. The coupling strength at the minima corresponds to G with a transition frequency to other dressed sub-levels, i.e., an rf Rabi frequency, of kHz. The Rabi frequency is sufficiently below the dressing frequency for the rotating wave approximation to be valid, but well above the highest trap frequency to avoid non-adiabatic atom loss. These parameters also allow for trap alignment via time-averaged, adiabatic potentials with a modulation of at tens of kHz that is sufficiently fast compared to atomic motion but slow enough for atomic spins to adiabatically follow.
The above consideration also sets a scale for the required alignment precision between field generating structures. The concentricity tolerance between static and rf fields is similar to . While small, dynamical alignments can be achieved using homogeneous, externally generated fields, the above parameters require micron-scale accuracy. While this is achievable with current technology and easily realized within a single microstructure, it also requires attention when joining and bonding different, micro-fabricated elements.
As an outlook, we can also consider higher-order multipoles to describe static fields for small by replacing in Eqs. (4)–(6) together with the radial dependence . In this case, rings occur, each with n lattice sites, shown for in Fig. 4. However, not all rings can be controlled independently. Tailored trap patterns could be generated by combining different rf dressing multipole fields of different orders. Such patterns in combination with the ability to control these lattice sites in a state-dependent and dynamic fashion will create novel platforms for quantum simulation of interesting new physics. In general, the underlying principles are not restricted to the described toroidal geometries but allow for other combinations of inhomogeneous static fields with inhomogeneous dressing fields and a plethora of possible designs.
Approximated dressed potential for multiple ring lattices, which form for static fields that are described by interior multipoles of order in the vicinity of a ring of zero field. The example shown uses the same dressing parameters as in Fig. 3, but for a ring-shaped hexapole static field with , which leads to ring lattices at locations where the static field is aligned with the z-axis.
Approximated dressed potential for multiple ring lattices, which form for static fields that are described by interior multipoles of order in the vicinity of a ring of zero field. The example shown uses the same dressing parameters as in Fig. 3, but for a ring-shaped hexapole static field with , which leads to ring lattices at locations where the static field is aligned with the z-axis.
IV. CONCLUSIONS
In summary, the scheme introduced in this work allows the generation of ring-shaped atom-trap lattices by only using static and rf magnetic fields. Through the modulation of rf-dressed, toroidal potentials by rf-multipole fields (of order n) atom-trap lattices (with n sites) can be created, that are state-dependent and also allow for dynamic, independent control over the potential landscapes for different atomic species or spin states. These features together with tight atom confinement that can be maintained also for large ring sizes make them a candidate for the realization of robust, guided Sagnac interferometer gyroscopes.
Since the components for the generation of the required fields for this scheme are also compatible with existing atom-chip technologies, integrated platforms can be designed and manufactured to produce such purely magnetic atom-trap lattices. This will further extend the possibilities for compact quantum sensors that can be employed in the field and may enable novel quantum devices that can put fundamental physics questions to the test.
ACKNOWLEDGMENTS
The authors gratefully acknowledge discussions with Anna Minguzzi and Beatriz Olmos. This work was funded by the Engineering and Physical Sciences Research Council (EPSRC), Grant Agreement No. EP/M013294/1—UK Quantum Technology Hub for Sensors and Metrology.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Fabio Gentile and Jamie Johnson contributed equally to this work.
Fabio Gentile: Conceptualization (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Jamie Johnson: Conceptualization (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Konstantinos Poulios: Conceptualization (supporting); Software (supporting); Visualization (supporting); Writing – original draft (supporting); Writing – review & editing (equal). Thomas Fernholz: Conceptualization (equal); Software (supporting); Supervision (lead); Visualization (supporting); Writing – original draft (supporting); Writing – review & editing (equal).
DATA AVAILABILITY
Data sharing is not applicable to this article as no new data were created or analyzed in this study.