Ring-shaped atom-trap lattices using multipole dressing fields
Fabio Gentile, Jamie Johnson, Konstantinos Poulios, Thomas Fernholz
RRing-shaped atom-trap lattices using multipole dressing fields
Fabio Gentile, ∗ Jamie Johnson, ∗ Konstantinos Poulios, and Thomas Fernholz † School of Physics & Astronomy, University of Nottingham, University Park, Nottingham NG7 2RD, U.K. (Dated: September 4, 2019)We present a method for the creation of closed-loop lattices for ultra-cold atoms using dressedpotentials. We analytically describe the generation of trap lattices that are state-dependent, withdynamically 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 multipoleradio-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. Po-tential applications of our scheme span the range from quantum many-body simulations to guidedSagnac interferometers.
Adiabatic radio-frequency (rf) dressed potentials playan increasingly important role in recent developmentswith ultra-cold atomic, see physics[1] for a recent review.These potentials, produced by the combination of staticand oscillating magnetic fields, enable the creation of aplethora of different trapping geometries that can sup-port confinement as well as, under certain conditions,transport of atomic clouds.The trapping potentials obtained with rf dressingtechniques are inherently species-selective and state-dependent, because they can be controlled independentlyfor atoms that have different Land`e g-factors [2, 3]. Forthe case of often used alkali atoms, the different hyperfinelevels of their electronic ground state have g-factors withnear identical magnitude but opposite sign. Arbitrarysuperpositions of such internal states can be prepared,and the combination with dressed, state-dependent po-tentials allows for coherent beam splitting, using onlyfields oscillating in the rf and microwave (mw) regime.This results in a versatile workbench capable of gener-ation, independent manipulation, and detection of in-ternally labelled atomic superposition states, especiallyrelevant for interferometric schemes [4, 5]. There are nu-merous examples of experimental implementations thatdemonstrate the generation and detection of dressed su-perposition states [see [1] and references therein], includ-ing also non-destructive detection methods [6]. Recently,transport of a single, dressed spin state has been demon-strated over macroscopic distances [7]. By manipulat-ing the amplitude, frequency, and polarization or rela-tive phases of the contributing rf fields, versatile controlover the resulting rf dressed potentials can be achieved[8–10]. This ability to dynamically modify the potentiallandscape between different configurations within a sin-gle experimental run is significantly useful in the contextof both fundamental as well as applied experiments withultra-cold atomic clouds and Bose-Einstein condensates(BECs). The possibilities range from double wells [11, 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, magneticor hybrid trap lattices have constituted an important quantum simulation testbed for a variety of physicalphenomena otherwise not straightforward to probe [20].These include studying the dynamics of strongly cor-related particles [21] and investigating new topologicalphases of matter [22, 23] as well as the thermalization ofquantum systems and dynamics in the many-body regime[24]. The use of dressed potentials may add to this, asmagnetic atom-trap lattices with interesting topologiescan be formed, including, e.g., ring structures, that arefurthermore adjustable and dynamically controllable ina state dependent fashion. These ring-shaped atom-traplattices have been proposed as analogues for supercon-ducting flux qubits [25] as well as platforms where ar-tificial gauge fields can be studied [26] and correlatedmany-body effects can be harnessed for the implemen-tation of rotation sensors and gyroscopes with enhancedsensitivity [27]. Here we present a method that allowsfor the generation of such ring-shaped atom-trap lat-tices based on rf dressed potentials, by combining a ring-shaped quadrupole potential with a multipole rf field.Our method is compatible with atom-chip technology [28]and may enable robust, mechanically stable and compactquantum devices and sensors.Rf dressed potentials arise from the combination of aninhomogeneous magnetic field and an oscillating mag-netic field that drives atomic spin flips. The static fielddefines a two-dimensional manifold where Larmor pre-cession can be resonantly excited, thus coupling low-fieldseeking states to high-field seeking states. This principlecan be extended to any pair of such states, e.g., by cou-pling states from different hyperfine manifolds, where thenuclear spin changes orientation with respect to the elec-tronic spin. A trap is formed in the regime where atomstraverse the region of resonance adiabatically. The traptopology of our scheme is based on an axially symmet-ric combination of a ring-shaped, static quadrupole fieldand an oscillating rf field with radial and axial compo-nents of different phases. Such an arrangement results ina 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 specif-ically chosen such that connected potential minima with- a r X i v : . [ phy s i c s . a t o m - ph ] S e p out degenerate points are generated, which would other-wise cause atom loss. In the following, we first reca-pitulate the approach for generating hollow-torus and inparticular ring-shaped atom traps before describing themethod for partitioning these traps in order to form alattice.An atom interacting with a weak magnetic field, con-sisting of a static term B dc = B dc e and a time-dependent term B ac ( t ) = B rf e iω rf t / c.c., B rf ∈ C thatoscillates at frequency ω rf , is described by the Hamilto-nian ˆ H ( t ) = g F µ B ˆ F (cid:126) · ( B dc + B ac ( t )) , (1)where ˆ F is the atom’s total angular momentum, µ B is theBohr magneton, and g F is Land´e g-factor. In general, thestatic field direction depends on position, and the oscil-lating field can be expressed in a local spherical basis { e , e ± = ( − e ± i e ) / √ } as B rf = B + e + + B − e − + B e , with amplitudes B ± , of corresponding field polar-izations that drive σ ± - and π -polarised transitions withrespect to a quantization axis e . Vectors e , , form aright-handed system and we use corresponding, dimen-sionless spin operators ˆ F , , = e , , · ˆ F / (cid:126) . The Hamilto-nian can be transformed to a frame rotating at ω rf (the rfdressing frequency) about the static field direction e , i.e.ˆ H (cid:48) = ˆ U ˆ H ˆ U − + i (cid:126) ( ∂∂t ˆ U ) ˆ U − with the unitary operatorˆ U = e iω rf t ˆ F . Defining ˆ F ± = ˆ F ± i ˆ F , using the Baker-Haussdorff formula ˆ U ˆ F ± ˆ U − = e ± iω rf t ˆ F ± , and makingthe rotating wave approximation (RWA), leads to thetransformed Hamiltonianˆ H (cid:48) RWA = 12 g F µ B (cid:18) ( B dc − (cid:126) ω rf g F µ B ) ˆ F − B + √ F − (cid:19) + h.c. (2)The resulting spectrum of dressed state (quasi)energiesis given by E m F = m F g F µ B (cid:115)(cid:18) B dc − (cid:126) ω rf g F µ B (cid:19) + | B + | , (3)where m F is the magnetic quantum number, and the am-plitude of the remaining dressing field component is givenby B + = e + · B rf . The vanishing of the first term definesa resonance, which may occur for negative frequency ω rf and thus inverted rotational senses, depending on thesign of g F and the chosen sign of B dc .We assume a static, circular quadrupole field with zeromagnetic field along a ring of radius r , centered in thelaboratory’s x, y -plane at z = 0, see Fig. 1. Such a fieldcan be generated using four counter-propagating circularcurrents. We approximate the static field in the vicinityof this ring, and use the toroidal angle θ together withlocal polar coordinates ρ, φ to parameterize planes or-thogonal to the ring. The approximate field magnitudeis then given by B dc = qρ , where q is the quadrupole FIG. 1. Illustration of the static ring quadrupole field (sectionwith field lines and local polar coordinates in the inset). Sucha field can be obtained by means of four counterpropagatingcurrent loops as indicated by the arrows. Global Cartesianand cylindrical coordinates are shown together with local po-lar coordinates, defining toroidal ( θ ) and poloidal ( φ ) angles. gradient and ρ is the distance from the field zero. For m F g F >
0, the resonance condition qρ = (cid:126) ω rf /g F µ B minimizes the potential with respect to ρ , thus definingthe surface of a torus at ρ = ρ where atoms can betrapped [8]. The potential on this surface is given bythe locally varying amplitude B + , which is the focus ofthe remaining discussion. For its evaluation, we definea local coordinate system in the vicinity of the ring ofzero field with a basis given by the approximated staticfield direction e , the tangent to the ring e , and theright-handed completion e = e × e . Expressed in theCartesian laboratory representation this choice of localbasis is given by e = ( − cos θ cos φ, − sin θ cos φ, sin φ ) T , (4) e = ( − sin θ, cos θ, T , (5) e = ( − cos θ sin φ, − sin θ sin φ, − cos φ ) T . (6)In order to generate non-vanishing potential minima,the effective field amplitude B + must be non-zero. Onepossibility to achieve this with axial symmetry is to usean rf field tangential to the ring, i.e. parallel to e . Foran alternating current I along the setup’s central (c) z -axis, the field would be B (c)rf = e µ I/ r and lead to B (c)+ = e + · B (c)rf = − µ I/ √ r . More versatility can beachieved by using a dressing field that is elliptically po-larized in the ρ , φ planes. Such a toroidal (t) field can begenerated by combining a uniform field, linearly polar-ized along the z -direction, with a phase-shifted, axiallysymmetric quadrupole field that provides a radial compo-nent in the x, y -plane along r . For simplicity, we neglectradial dependence of this field and approximate it in thevicinity of the forming trap. In the Cartesian laboratorypresentation, its decomposition into orthogonal circularcomponents is given by B (t)rf = a + √ cos θ sin θi + a − √ cos θ sin θ − i , (7)with amplitudes a + and a − . In this case, the couplingfield component is given by the projection B (t)+ = e + · B (t)rf = (cid:0) − a + e − iφ + a − e iφ (cid:1) / . (8)By substituting this result in Eq. (3) it can be seen that avariation of the trapping potential over the poloidal angle φ can be controlled by the choice of the dressing field’spolarization. At ρ = ρ , the potential is determined by (cid:12)(cid:12)(cid:12) B (t)+ (cid:12)(cid:12)(cid:12) = | a + | + | a − | − | a + a − | φ − α ) , (9)where α = arg( a + ) − arg( a − ). For a single circular com-ponent, i.e. a + = 0 or a − = 0, the potential minimumis independent of both θ and φ , resulting in a flat po-tential over the toroidal surface. For an elliptical field,0 < | a + | (cid:54) = | a − | >
0, the trap splits into two poloidalminima, i.e. it forms two rings at φ , = α/ π ). Forthe extreme case of | a + | = | a − | , i.e. for linear polarizationof B (t)rf , the coupling at the minimum vanishes, leadingto degenerate potentials for different m F and thus atomloss.We are in particular interested in the case of form-ing rings at the top and bottom of the torus ( φ , = ± π/ a r =( a + + a − ) / √ a z = ( a + − a − ) / √ ◦ out-of-phase, i.e. arg( a z ) = arg( a r )or arg( a z ) = arg( a r ) ± π . Since the local static field inthose rings is parallel to the z -direction, the trapping po-tential can be conveniently modulated by interference of B (t)rf 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.A linearly polarized, interior cylindrical multipole (m)field of order n is described by B (m)rf = B (m) r n − sin (( n − θ − nθ )cos (( n − θ − nθ )0 , (10)where the offset angle θ describes a rotation about the z -axis and the moment B (m) sets the field strength. Theturning number of the field along a loop around the z axisis given by 1 − n , and the expression in Eq. 10 can beviewed as including the field B (c)rf as a special case for n =0. For n = 1, we obtain the homogeneous interior dipolefield, n = 2 describes a quadrupole field, etc. For n > θ = 0 and θ = FIG. 2. Top panel: Orthogonal, linearly polarized, interiorquadrupole fields ( n = 2) that can be driven with 90 ◦ -phasedifference to generate circular polarization. The shown fieldlines are the leading order approximation (Eq. 10) to the fieldsgenerated by two sets of four blue (red) infinitely long wires.The wires cross the plane at the locations depicted by thesymbols ⊗ ( (cid:12) ) for currents going into (out of) the page. Therotation offset angles are θ = 0 ( θ = π/
4) for the blue(red) quadrupole. The bold arrows emphasize the local or-thogonal field directions along a circular path, centered onthe z -axis. Bottom panel: Following the circular path in an-ticlockwise direction, the local field directions (blue and redarrows) rotate n − n minima and maxima. E.g., when the radial fieldis in phase with the blue multipole component, destructiveinterference occurs at θ = π/ θ = 3 π/
2. Shifting themultipole phases by 90 ◦ such that the phase of the red com-ponent aligns with that of the radial field leads to destructiveinterference at θ = π/ θ = 5 π/ π/ n , driven with individual rf amplitudes and phases,allows for the generation of a multipole field of arbitraryin-plane polarization. The example of two orthogonal,linearly polarized quadrupole fields is shown in Fig. 2.The generic (g) interior cylindrical multipole field canbe expressed in terms of orthogonal circular componentswith moments U ± , taking the form B (g)rf = U + √ re − iθ ) n − i + U − √ re iθ ) n − i − . (11)Again, we neglect the radial dependence of this field inthe vicinity of the traps ( r ≈ r ) and approximate thelocal field amplitudes as u ± = U ± r ( n − . The projection B (g)+ = e + · B (g)rf of this field onto the relevant sphericalcomponent of the static field basis then leads to B (g)+ = − u + φ e − inθ + u − − sin φ e inθ . (12)At the top of the torus, i.e. for φ = + π/
2, the static fielddirection coincides with the setup’s z -direction. Here, theamplitude u + leads to maximal coupling, while the con-tribution from u − vanishes. At the bottom of the torus,i.e. for φ = − π/
2, the static field direction is inverted,and we find the opposite situation.A ring lattice structure emerges from the combinationof the toroidal field with a multipole field, because theiramplitudes interfere. Assuming real amplitudes a ± forthe toroidal rf field and considering only the coupling atthe top and the bottom of the torus, leads to | B + | φ = ± π/ = | B (t)+ + B (g)+ | φ = ± π (13)= 12 a r + | u ± | ± √ a r | u ± | sin ( nθ ∓ ϕ ± ) , where we expressed u ± = | u ± | e iϕ ± . The result showsthat for | u ± | < | a r | / √ n , be- FIG. 3. (a) Cuts through the approximated dressed po-tential (not to scale for realistic geometries). The potentiallandscape on the resonant torus surface is shown for n = 10traps. Potential minima are depicted as dark blue regions. Onthe right-hand side a cut through the 3D potential is shownfor the y, z -plane. The assumed parameters are m F g F = 1, a r = 0 . a z = 1 . | u ± | = 0 . ϕ ± = 0 (b) and ϕ ± = π/ cause it is given by the relative number of rf field rota-tions along the closed path [see Fig. 2]. It also showsthat the two circularly polarized quadrupole componentscontrol lattices in the top and bottom ring independently.The modulation depth and positioning of the forming lat-tices can be dynamically controlled via amplitudes andphases of the multipole rf fields (relative to the phase ofthe toroidal rf field). The amplitudes enter with conju-gate phases, which means that a single, linearly polarizedmultipole field that contains both amplitudes u ± with afixed phase difference will lead to counter-propagatinglattices when this phase is ramped.Figure 3 provides a more comprehensive illustrationof the potential landscape on the resonant surface forthe case of two stacked rings, e.g., a + > a − < u ± present, generating n = 10traps in each ring. It can be seen that non-isotropictraps are formed that are not aligned with the toroidaland poloidal directions. In the harmonic approximation,this can be analysed by expressing the curvature ten-sor at the trap minima to determine aspect ratios andalignment. The squared trap frequencies are given by ω jk = ( ∂ /∂ j ∂ k ) E m F /m , where m is the atomic mass ofthe trapped species. For simplicity, we assume real a ± again and make the assumption of only the locally dom-inant field u + or u − being present, although potentialvariations in one ring influence the confinement in theother ring to some degree. To clarify radial, poloidal,and toroidal directions, and without loss of generality,we assume ϕ ± = π/ a r >
0) or maximum ( a r <
0) forming at θ = 0 and ex-press the corresponding tensor in Cartesian coordinates j, k ∈ { x, y, z } ω jk (cid:12)(cid:12) φ = ± π = γm δ + a r | u ± | ρ na z | u ± | r ρ na z | u ± | r ρ n a r | u ± | r
00 0 2 √ q , (14)using the definitions δ = ( a z − a r − | u ± | ) / √ γ = m F g F µ B / (2 √ B ), and the effective field strength at theextremum B = | a r −√ | u ± || . The comparison of diag-onal and off-diagonal elements shows that the rotationalsense of trap misalignment depends on the relative signbetween amplitudes a z and a r . In principle, the traps canbe aligned by modulating the sign of a z to form a timeaveraged dressed potential [9, 15, 16], which will make theaveraged curvature tensor diagonal in these coordinates.The Hamiltonian at the trap centres will remain unmod-ulated, where a z does not enter the coupling strength [seeEq. 13].An important scenario is the trapping of atoms withdifferent g-factors in the same rf dressed trap. Differ-ent atomic species may be controlled via different radio-frequencies [3], and a particular case is a mixture orsuperposition of the same species in different hyperfinestates. Here, the magnitude of the g-factor may be ap-proximately equal but of opposite sign. In our descrip-tion, a negative g-factor leads to a negative resonantdressing frequency. But the presence of any a ± , u ± am-plitudes with positive ω implies the presence of the cor-responding amplitudes a ∓ , u ∓ with negative ω and con-jugated phase. The torus and ring forming potential inthe poloidal direction is symmetric in this respect [seeEq. (9)]. A trap formed at frequency ω also leads toa trap formed at frequency − ω . But as it can be seenfrom Eq. (13), control of atoms via u + in the top ringand/or u − in the bottom ring imparts the same effectsin the opposite ring on atoms with the opposite g-factor g F . It should be noted, however, that changing the signof g F also inverts the orientation of trap misalignment,because the negative resonant frequency ω rf changes thesign of the anti-symmetric a z but not of the symmetric a r as a + and a − swap their values. Within one ring,the two types of atoms can be controlled independently.Atoms in different spin states can be transported in op-posite directions, thus making the configuration a can-didate for guided Sagnac interferometer gyroscopes andother atomtronic applications. A lattice filled with atomsin one spin state could be immersed in a homogeneousring of atoms in another spin state to couple differentsites via phonons or study quantum friction of impuritiesin a Bose-Einstein condensate.We briefly exemplify a few experimental consid-erations. With micro-fabricated trapping structures[28] high field gradients can be achieved, e.g., q =10 T/m=1 /
10 G/ µ m. For Rb atoms in their electronicground state, with total spin F = 2, g F = 1 /
2, and m F = 2, a torus with ρ ≈ µ m forms for a dress-ing frequency ω rf = 700 kHz. Assuming realistic am-plitudes a r = 0 . a z = 1 . | u + | = 0 . n = 10 traps over a millimeter sized ring with r = 0 . ω xx = 2 π ×
989 Hz, ω yy = 2 π ×
116 Hz, and ω zz = 2 π × . B = 0 .
31 G with a transition frequencyto other dressed sub-levels, i.e. an rf Rabi frequency, ofΩ = | g F | µ B B / (cid:126) = 2 π ×
227 kHz. The Rabi frequencyis sufficiently below the dressing frequency for the rotat-ing wave approximation to be valid, but well above thehighest trap frequency to avoid non-adiabatic atom loss.These parameters also allow for trap alignment via time-averaged, adiabatic potentials with a modulation of a z attens of kHz that is sufficiently fast compared to atomicmotion but slow enough for atomic spins to adiabaticallyfollow.Depending on the targeted geometry of the ring lat-tice, we have to revisit the approximations that weremade to describe the underlying field geometries. Weapproximated the static field to be described by a lo-cal quadrupole field and assumed the rf field amplitudes FIG. 4. Approximated dressed potential for multiple ring lat-tices, which form for static fields that are described by interiormultipoles of order l > l = 3,which leads to 2( l −
1) = 4 ring lattices at locations where thestatic field is aligned with the z -axis. to be constant near the forming traps. The static fieldapproximation is valid for ρ (cid:28) r , z where z is a min-imal distance from the field generating structures. Fora thin torus with ρ (cid:28) r , also the approximation for B (t)+ is valid, but B (g)+ requires more careful examination,as B (g)rf scales with r n − . Here, the relative amplitudevariation over the full resonant surface is given by η = ∆ B (g)rf / B (g)rf (cid:12)(cid:12)(cid:12) r = r = 2 ρ ( n − r , (15)which increases with the multipole order. For the numer-ical example above, we find η = 0 .
36, which will alreadyaffect the shape of the dressed potential. However, forsufficiently low temperatures of the atomic cloud, thefilled trap volumes will be confined to a much smallerrange of radii ∆ r < ρ , in particular when the traps arealigned by modulating the vertical rf field amplitude a z .As an outlook, we can also consider higher-order mul-tipoles to describe static fields for small ρ by replacing φ → ( l − φ in Eqs. (4-6) together with the radial de-pendence B dc → q l ρ l − . In this case, 2( l −
1) rings occur,each with n lattice sites, shown for l = 3 in Fig. 4. Butnot all rings can be controlled independently. Tailoredtrap patterns could be generated by combining differentrf dressing multipole fields of different orders. Such pat-terns in combination with the ability to control these lat-tice sites in a state-dependent and dynamic fashion willcreate novel platforms for quantum simulation of inter-esting new physics. In general, the underlying principlesare not restricted to the described toroidal geometriesbut allow for other combinations of inhomogeneous staticfields with inhomogeneous dressing fields and a plethoraof possible designs.In summary, the scheme introduced in this work allowsthe generation of ring-shaped atom-trap lattices by onlyusing static and rf magnetic fields. Through the mod-ulation of rf-dressed, toroidal potentials by rf-multipolefields (of order n ) atom-trap lattices (with n sites) can becreated, that are state-dependent and also allow for dy-namic, independent control over the potential landscapesfor different atomic species or spin states. These featurestogether with tight atom confinement that can be main-tained also for large ring sizes make them a candidate forthe realization of robust, guided Sagnac interferometergyroscopes.Since the components for the generation of therequired fields for this scheme are also compatible withexisting atom-chip technologies, integrated platformscan be designed and manufactured to produce suchpurely magnetic atom-trap lattices. 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