Engineering and Revealing Dirac Strings in Spinor Condensates

Artificial monopoles have been engineered in various systems, yet there has been no systematic study of the singular vector potentials associated with the monopole field. We show that the Dirac string, the line singularity of the vector potential, can be engineered, manipulated, and made manifest in a spinor atomic condensate. We elucidate the connection among spin, orbital degrees of freedom, and the artificial gauge, and show that there exists a mapping between the vortex filament and the Dirac string. We also devise a proposal where preparing initial spin states with relevant symmetries can result in different vortex patterns, revealing an underlying correspondence between the internal spin states and the spherical vortex structures. Such a mapping also leads to a new way of constructing spherical Landau levels, and monopole harmonics. Our observation provides insights into the behavior of quantum matter possessing internal symmetries in curved spaces.

Introduction -Despite of the lack of unambiguous experimental evidence for their existence, magnetic monopoles have a central place in our understanding of quantum matter and modern cosmology [1][2][3][4].Remarkably, recent theories and experiments have provided ample evidences for the emergence of artificial monopoles in various physical systems [5][6][7][8][9][10][11][12][13][14][15][16].It is well known that the vector potential associated with the monopole field contains line singularities (known as Dirac strings) that terminate at the monopole, even though the monopole magnetic field itself is smooth everywhere (except at the position of the monopole) [17].This does not pose any problem as the vector potential, unlike the field, is not "real" in the sense that it cannot be directly measured.This is also reflected in the fact that the positions of these Dirac strings are gauge dependent.
This conventional wisdom, however, is not necessarily true in systems with artificial gauge fields.In such systems, one often directly realizes and controls the artificial gauge potential, rather than the associated 'magnetic' field, rendering the former directly measurable.Indeed, the physical effects of artificial gauge potentials on time-of-flight images of cold atoms have been reported in several experiments [18][19][20][21][22].
One widely used platform to realize artificial gauge field is spinor Bose-Einstein Condensates (BECs).When the atomic spin [23] adiabatically follows an external magnetic field, the system accumulates a geometric phase [24], which induces an artificial gauge potential governing the spatial wave function; a manifestation of the spin-orbit coupling.If the atoms occupy a spatial region that contains a degenerate point with vanishing magnetic field, the artificial gauge potential can develop a line singularity, which can be regarded as an analog of the Dirac string and a consequence of the local spin-gauge symmetry [9,25].The purpose of this work is to elucidate the relationship among spin, orbit and the artificial gauge field.As conceptually represented in Fig. 1, through the spin-orbit coupling and the spin-gauge symmetry, there exists a mapping between the vortex filament and the Dirac string.This directly leads to a novel adiabatic scheme for preparing vortex configurations on a sphere where some initial spin state is prepared for the spinor condensate, and then the artificial magnetic field strength is turned on adiabatically.As a result of conservation of the total angular momentum, some of the initial spin angular momentum is transferred to the orbital degrees of freedom, arXiv:2402.14705v2[cond-mat.quant-gas]9 Apr 2024 resulting in the formation of vortices.Preparing different initial spin states can therefore result in different vortex patterns.From the point of view of the artificial gauge field, this amounts to different gauge choices that lead to different Dirac strings.
Spin, vortex, and Dirac strings -Let us consider a spin-F atom of mass M confined in an isotropic harmonic trapping potential with frequency ω, subjected to a hedgehog magnetic field B(r) ∝ r.The realization of the hedgehog field has been proposed in our earlier work [26].Working in units where ℏ = M = ω = 1, the single-particle Hamiltonian of the system reads Here α characterizes the strength of the hedgehog field, which we assume can be dialled from zero to large values.In the limit of large field strength αr, the lowest spin state follows the local magnetic field and satisfies (r • F)|F r ⟩ = F |F r ⟩, where |F r ⟩ can be obtained from |F ẑ ⟩ (the spin state polarized along the z-axis) via rotations: |F r ⟩ = e −iφF ẑ e −iθF ŷ |F ẑ ⟩.Here θ and φ are the polar and azimuthal angles respectively.The total wave function, Ψ(r), can be written as Ψ(r) = ψ(r)|F r ⟩, with the effective Hamiltonian of the scalar wave function ψ(r) being: Here the effective vector potential A = i⟨F r |∇F r ⟩ = êφ F cos θ/(r sin θ) reflects a 'magnetic' monopole (of magnetic charge F ) at the origin.Furthermore, the last term in Eq. (2) dictates that this monopole is also 'electrically polarized', with an electric dipole moment F/2.The vector potential A is singular for θ = 0 and π, which corresponds to two antipodal Dirac strings.It can be seen that the effective system realises a Haldane's sphere [27]: atoms of unit 'electric charge', are confined within a thin spherical shell (of order unity width in units of ∼ ℏ/M ω) centered at r 0 ≈ αF (in units of ∼ ℏ/M ω), in the presence of an 'electrically polarized magnetic monopole' at the origin.
If we rewrite the scalar function above as ψ(r) = n(r)e iϕ (r) , where n(r) represents the local atomic number density, then the total wavefunction can be written as Ψ(r) = n(r)| Fr ⟩, where | Fr ⟩ = e iϕ(r) |F r ⟩.In other words, we have absorbed the phase factor of the scalar function into a redefinition of the radial spin state.The corresponding vector potential associated with | Fr ⟩ is given by Ã = i⟨ Fr |∇ Fr ⟩ = A + ∇ϕ, which is related to A by a gauge transformation.This is just a manifestation of the spin-gauge symmetry in spinor gases [25].On the other hand the velocity field associated with the total wavefunction Ψ(r) is given by v mass = −∇ϕ − A = − Ã, which allows us to clearly see the connection between FIG. 2. The single-particle energy spectrum EñjF (shifted by α 2 F 2 /2) as a function of α, for the F = 1 case.Upon increasing α, different oscillator levels approach each other, and a crossover from the 3D isotropic oscillator levels to Landau levels is seen.The inset shows the lowest energy E011 in the adiabatic regime, which is obtained from GP equations (circle), numerical diagonalization (star) and the analysis in Eq. ( 5) (solid).
vortices (line singularities of v mass ) and line singularities of Ã.As we will show in the following, by preparing different initial spin configurations in the absence of the monopole field followed by its adiabatic turn on, we may result in final states with different phase structure ϕ(r), and hence different vortex or Dirac string orientations.In a sense, changing the initial spin configuration amounts to choosing a different gauge for the vector potential Ã.
Engineering Dirac Strings -To substantiate the argument we laid out above, here we provide a more quantitative description.
Single-particle spectrum : Let us first consider the single-particle spectrum for the Hamiltonian (1).It can be seen that while for α = 0 both L and F are conserved, for α ̸ = 0 only the total angular momentum (TAM) J = L + F is conserved.This means that the energy eigenstates are also eigenstates of {J 2 , J z }.These are the well-known spinor harmonics , where, c's are the Clebsch-Gordan coefficients, Y 's are the usual spherical harmonics, and |m F ẑ ⟩ are the spin multiplicity states (in the z basis).Furthermore, while F 2 is conserved, L 2 is not.Summing over the ℓ quantum number then (and with ñ as the radial quantum number), the eigenstates take the form With (j, m) being good quantum numbers, and since α = 0 corresponds to a simple 3D harmonic oscillator (with an intrinsic hyperfine spin), we can find the energy spectrum for any α by projecting the Hamiltonian H 0 onto the (n, ℓ) subspace.Here, n and ℓ = j + m F are the radial and orbital quantum numbers for the oscillator.This gives a tri-diagonal matrix, which can be numerically diagonalized by incorporating enough 3D oscillator levels [29].See [30] for details.In Fig. 2 we provide different energy curves as a function of α, for different n and j values.More importantly, from the point of view of the adiabatic flow of local spin, at large α the radial part of the scalar function ψ approaches the 1D harmonic oscillator centered at radius r 0 (c.f.Eq. ( 2)), while the angular structure is dictated by the Hamiltonian , where h ñ are the 1D harmonic oscillator states and g m F,j are the eigenstates of the angular Hamiltonian H Ω (known as the monopole harmonics [31]).Then with the ansatz f ℓ ñjF (r) ≡ r −1 h ñ(r − r 0 ) β ℓ F,j in Eq. ( 3), the β coefficients must be such that the following holds This is the radial spin flow correspondence (which holds for any j ≥ F ). Using this, the energy spectrum comes out to be [30] where the last term is just the spherical Landau levels (LLs) [27], plus a shift F/(2α 2 F 2 ) owing to the 'electric dipole moment'.We note that the radial spin flow correspondence (4), without any reference to H Ω , fetches both the g functions and the β coefficients, giving us all the LLs on the sphere [30].Our construction therefore reveals an alternative approach of constructing spherical LLs.It is clear from the energy spectrum that all the different j levels approach one another as α increases, because the energies of different states get increasingly dominated by the Zeeman term.As an explicit example, consider the spin-1 case.For the j = 1 level, we get , and the following three degenerate states The inset in Fig. 2 compares the energy with the Gross-Pitaevskii (GP) equation, and numerical diagonalization.It is evident that the radial spin flow correspondence holds well within ∼ 0.1% for α ≳ 4.         Creating vortices from different spin states: Conservation of J can be exploited to create vortex patterns/Dirac strings, as follows.Starting at α = 0, we can prepare the system in its ground state, carrying zero orbital angular momentum (OAM) ⟨L⟩ ini = 0 and any desired spin configuration |ζ x ⟩ carrying spin angular momentum (SAM) ⟨F⟩ ini .Then α is increased adiabatically, and in the process some of the SAM gets transferred to the OAM while keeping the total ⟨J⟩ fixed.At sufficiently large α owing to adiabatic spin flow, we converge to a vortex pattern in the final state carrying final OAM ⟨L⟩ fin = ⟨F⟩ ini /(1 + F ), and SAM ⟨F⟩ fin = F ⟨F⟩ ini /(1 + F ) [30].
We can also predict the orientation of these Dirac strings (which are lines of singularities) by considering the geometric/Bloch sphere representation for spin-F .We note that at points where the strings/vortices intersect the sphere, the 'wavefunction' must vanish.Then, owing to the transfer of initial SAM to final TAM during the adiabatic spin flow, these intersection points should be where the initial spin configuration |ζ x ⟩ was orthogonal to the final spin configuration |F r ⟩.With e −iθFy |F ẑ ⟩, this means those points (θ, φ) on the sphere where ⟨ζ x |F r ⟩ = 0: Note that these points are nothing but the so called Majorana stars, and our engineering of the Dirac strings reveals the connection between spin and real space.This connection is embodied in the SO(3) symmetry in both the spinor Boson gas and the simulated monopole system.More explicitly, the symmetries of |ζ x ⟩ correspond to the operations under which the set of vortex locations {(θ i , φ i )} on the Haldane sphere are invariants.
Using the integrator i-SPin 2 [33], in Fig. 4 we show the real time implementation of our idea, for the mixed state |ζ η=1/4 ⟩.Starting with the 3D harmonic oscillator ground state dressed with the spin texture |ζ η=1/4 ⟩, we adiabatically increase α from 0 to 6 [30].As α increases, the initial mass density at the origin is pushed outwards, with the spin density aligning radially outwards.At large enough α, two vortices intersecting the atomic cloud at (θ, φ) = ( 2 To summarize, we have established a one-to-one mapping between the spinor state and the vortex state on a sphere.

Effects of Interaction and Experimental Feasibility -
Now we briefly discuss the effects of the mean-field interaction.The associated energy functional is , where F (r) is the local spin density, and the first and the second terms correspond to the spin-independent anddependent interaction, respectively.Typically the two interaction strengths satisfy |c 0 | ≫ |c 2 |. Figure 5 illustrates the final density profiles for the mixed state (with η = 1/4), for attractive, non-interacting and repulsive interaction cases, when starting from the respective ground states in the absence of the spin-dependent interaction we ramp up α from 0 to 6.While some of the qualitative physics remains the same, there are some important distinctions worth pointing out: (1) Without interactions, it can be seen that all single-particle states Y m F,F (or linear combinations thereof), are left invariant under SO(2) Lz+ϕ (here ϕ corresponds to gauge transformations).Including the interaction breaks this invariance and the states get rotated along the z axis, as reflected by the rotation of vortex pairs.(2) In the final state, the angular momenta are not evenly distributed in the spin and the orbital sector.This is because the spin tex- < l a t e x i t s h a 1 _ b a s e 6 4 = " C u q 9 s u 9 u K 1 a A 6 5 I s 5 q T e g u 6 t r N E B e T o j M r v w 7 H A S U S 4 x g w p 5 b l O r P 0 U S U 0  ture becomes more (an-)isotropic under the (attractive) repulsive interaction [34].
For N = 10 4 87 Rb atoms (F = 1) in a trap with frequency ω = 2π × 100 Hz, and Zeeman field B 1 r with strength B 1 = 1 G cm −1 , α = r 0 ≃ 6 is large enough where the spin flow correspondence holds well.Furthermore, the energy scale of the contact interaction (per particle) is E c0 ≃ c 0 /(8πα 2 ) ≈ 0.7, and is only comparable with the corresponding LL gap ∆E LL ≈ 1. Therefore in such an experimental setup, the analytical results presented for non-interacting system remain qualitatively correct [30].
Conclusions -The simulation of singular monopole potentials and the concomitant correspondence between the spin and the real space is a new feature of the spinor system under hedgehog magnetic field.The key ingredient is the rotational invariance of the Zeeman term r • F. These will persist even when the S 2 manifold is deformed, as long as J z remains conserved.Due to this rationale, we can investigate such features in other SO(3) systems, such as the isotropic spin-orbital-coupling term p • F, where bent vortex lines in solitons have been discovered [35].Remarkably, the correspondence presented in this paper allows us to reveal symmetries within the internal degrees of freedom, as manifesting in coordinate space.
In this work, we have found the spin real correspondence in the LLLs.Since stronger interactions may make the atoms occupy higher LLLs [36], correspondence in FIG. 5. Snapshots of the mixed state atomic cloud for η = 1/4 at α = 6, for the three cases of attractive interaction (c0 = −6 and c2 = −0.1,left column), no interaction (c0 = c2 = 0, middle column), and repulsive interaction (c0 = 6 and c2 = 0.1, right column), respectively.Other parameters are the same as in Fig. 4. The upper panel shows a 3D view, whereas the bottom panel shows the slice in the plane where the vortices intersect the atomic cloud (at polar angle θ = 2 tan −1 (1/3) 1/4 ≈ 74.46 degrees).Our results also confirmed that for the values of c2 used here, the effect of the spindependent interaction is negligible.
these states may be found.The investigation in Haldane's spherical geometry was originally proposed for the study of the fractional quantum Hall effect.Thus, fertile vortex configurations may enrich the exploration of many-body quantum matter in curved spatial geometry [37][38][39][40][41][42][43][44].

Ratio of orbital to spin angular momentum
In our adiabatic evolution, we are confined within the lowest energy states, i.e. the j = F lowest Landau levels.With arbitrary initial spin dressing, the final state takes the general form which is the same as (B2) with g m F,j=F (θ, φ) replaced by a sum over monopole harmonics with different magnetic quantum number m.With this and ⟨F r |F|F r ⟩ = F r, we have for the full expectation where With the general form we have for r± ≡ r • x + i r • ŷ = sin θ e ±iφ and rz = r • ẑ = cos θ, the following The second equalities in each of the two expressions above is simply due to Y m F,F (r) being eigenstates of {J 2 , J z }, and can be obtained straightforwardly using angular momentum operator algebra.Therefore, from (B7), we have that In our adiabatic evolution, the total angular momentum J is conserved throughout, with all of it initially being in the spin sector, ⟨J⟩ = ⟨F⟩ ini .Eq. (B11) then dictates that F/(F + 1) of the total initial spin remains in the spin sector, with the remaining 1/(F + 1) is transferred to the orbital sector.
with [S µ , L ν ] = 0. Using the fact that r • L = 0 and J = F + L, the full Hamiltonian can then be written as H = S 0 − α r • F = S 0 − α r • J, where we further decompose the Zeeman term as Here, r = (a + a † )/ √ 2, J ± = J x ± iJ ŷ and J 0 = J ẑ .With α = 0, we simple have a 3D oscillator with an intrinsic (hyperfine-)spin, the Hamiltonian for which is just H = S 0 .The energy eigenstates are common eigenstates of the operators S 0 , the Casimir S 2 ≡ S 2 0 − (S + S − + S − S + )/2, L 2 , and L z .These can be labelled by the quantum numbers (n, ℓ, m ℓ ), with eigenvalues 2n + ℓ + 3/2.We shall denote them by |n, ℓ, m ℓ ⟩, and can be obtained by repeated actions of S n Here |n, ℓ, m − m F ⟩ are the 3D oscillator states as stated above.With this, and noting that J is conserved in the system, we wish to find the matrix elements of r • J for a fixed (j, m).These should be independent of m since J is conserved.To find these matrix elements, we first recapitulate the action of a † µ on a state |n, ℓ, m ℓ ⟩ [29]: . For our purposes, m ℓ = m − m F in the above.The above can be obtained using the various commutation relations between a µ and S µ , and a µ and L µ (which can be obtained straightforwardly using the commutation relations (C4)).The action of a µ on |n, ℓ, m ℓ ⟩ can be obtained in a similar fashion.Also for convenience/better illustration, µ = ± → ±1 in the expressions for d ± µ .Next, the action of J on a state |n, j, m, F, ℓ⟩ is J ± |n, j, m, F, ℓ⟩ = j(j + 1) − m(m ± 1) |n, j, m ± 1, F, ℓ⟩ and J 0 |n, j, m, F, ℓ⟩ = m|n, j, m, F, ℓ⟩.With the above, we can work out the matrix elements of A and A † , in the subspace of fixed (j, m):  Eq. (D2) include the total particle number N , and the total angular momentum The ground state Ψ g of the system can then be obtained by minimizing the energy functional subjected to the two constraints of number and angular momentum conservation.This can be simply done by introducing Lagrange multipliers µ and λ for the 4 conserved numbers, meaning one minimizes the following using the continuous normalized gradient flow method (imaginary time evolution), as described in Ref. [32].Setting both c 0 = c 1 = 0, in Fig. 3 of the main text, we show the single particle ground states (for F = 1) for the three different spin configurations considered there.

Full, real time simulations
We have performed real time adiabatic flow simulations of our rescaled system (D2), to confirm and validate our results in this paper.We used the integrator i-SPin 2 [33] developed by some of us, in order to perform these real time simulations.In general, the pseudo-spectral algorithm in i-SPin 2 is time-reversible, along with norm and spin preserving to machine precision.It can also handle self-interactions as well as couplings to time-dependent external fields.For the interested reader, the details of the algorithm and numerical implementation can be found in that paper.
For our present purpose, we begin by constructing the ground state of the initial system (with α = 0 and c 2 = 0).For c 0 = 0 this is simply a 3D oscillator ground state (with an overall desired spin structure), whereas for c 0 ̸ = 0 it is not.In general, we get to this state by using imaginary time evolution of the system (keeping the total particle number fixed, say N = 1).Taking this initial state, we then perform real time evolution of the system wherein α is increased from 0 to some large number adiabatically.For this, we used a hyperbolic tangent function: where the parameters p i , t 0 , and τ are chosen such that α(0) = 0 and the final value approaches some desired number.
For the simulations presented in the paper, we set α(t f ) = 6.The time step used was ∆t ≈ 0.04 with t f ≈ 107.This meant t 0 ≈ 53.5 and τ ≈ 23 in the above parameterization.Finally, the total box was a 71 3 grid, with the length of the box in each direction being 25.
Starting with different ground states (with c 0 and different spin textures) with c 2 = 0, we have performed real time simulations for both the cases when c 2 was kept to zero, and was turned on to some small but finite value.In the main text, we show simulation results for the mixed state.Fig. 4 shows the time evolution for the "single particle" case (c 0 = 0).Fig. 5 shows the same for c 0 ̸ = 0, with c 1 turned on during real time evolution.

Appendix E: Experimental Feasibility
To estimate the energy and the observability of vortices, we take 87 Rb atoms as an example (atomic mass 87u).For a harmonic trapping potential with typical frequency of ω, 87 Rb atoms have typical length scales l s = ℏ/M ω ≃ 1.1 (2π × 100 Hz/ω) 1/2 µm.With ω B = 2g F µ B B 1 l s /ℏ ≃ 2 × 10 3 g F (B 1 /G cm −1 )(2π × 100 Hz/ω) 1/2 Hz as the Larmor frequency, strength of the Zeeman coupling is 2π × 100 Hz ω where N is the total number of atoms in the condensate.The energy scale of the contact interaction per particle on the sphere (of radius r 0 = l s αF and width l s ), is E c0 ≃ 0.5 c0 (4πr 2 0 )l s (N/(4πr 2 0 l s )) 2 /N : (E3) On the other hand, the energy gap per particle, between the lowest (ñ = 0) and next (ñ = 1) Landau levels is ∆E LL ≃ ℏω.We see that for the chosen parameters N = 10 4 and ω = 2π × 10 Hz for 87 Rb atoms (F = 1), having α > 5 renders E c0 < ∆E LL .In this case, the effect of interactions can be neglected, and the simpler "single particle" analysis becomes valid.

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T h j a c g g 3 B W 3 x 5 m T T O y 9 5 l u X J / U a r e Z H H k 4 Q i O 4 R Q 8 u I I q 3 E E N 6 s A g g G d 4 h T d n 5 L w 4 7 8 7 H v D X n Z D O H 8 A f O 5 w / t M o z 6 < / l a t e x i t > l a t e x i t s h a 1 _ b a s e 6 4 = " K M J 6 k V E v O u n A Z 4 f d E 5 R h w d R b s K g = " > A A A B 6 H i c b V D L T g J B E O z F F + I L 9 e h l I j H x R H a N q E e i F 4 + Q y C O B D Z k d e m F k d n Y z M 2 t C C F / g x Y P G e P W T v P k 3 D r A H B S v p p F L V n e 6 u I B F c G 9 f 9 d n J r 6 x u b W / n t w s 7 u 3 v 5 B 8 f C o q e N U M W y w W M S q H

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5 H g e 2 M q B n q Z W 8 m / u d 1 U h P e + B M u k 9 S g Z I t F Y S q I i c n s a 9 L n C p k R Y 0 s o U 9 z e S t i Q K s q M z a Z g Q / C W X 1 4 l z Y u y d 1 W u 1 C 9 L 1 d s s j j y c w C m c g w f X U I V 7 q E E D G C A 8 w y u 8 O Y / O i / P u f C x a c 0 4 2 c w x / 4 H z + A I P t j M M = < / l a t e x i t > l a t e x i t s h a 1 _ b a s e 6 4 = " G Q u M Y Z h M m w P A n d Q 0 t T F N 4 4 p Z F Y g = " > A A A B 6 H i c b V D L S g N B E O z 1 G e M r 6 t H L Y B A 8 h V 3 x d Q x 6 8 Z i A e U C y h N l J b z J m d n a Z m R X C k i / w 4 k E R r 3 6 S N / / G S b I H T S x o K K q 6 6 e 4 K E s G 1 c d 1 H p 0 X 5 9 3 5 m L e u O P n M E f y B 8 / k D f F m M v g = = < / l a t e x i t > 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " K z I B W + 5 9 w u / x d X 7 u 0 b E C 8 w z H 6 + 0 = " > A A A B 6 X i c b V D L S g N B E O y N r x h f U Y 9 e B o P g K e y K r 2 P Q i 8 c o 5 g H J E m Y n v c m Q 2 d l l Z l Y I S / 7 A i w d F v P p H 3 v w b J 8 k e N L G g o a j q p r s r S A T X x n W / n c L K 6 t r 6 R n G z t L W 9 s 7 t X 3 j 9 o 6 j h V D B s s F r F q B 1 S j 4 B I b h h u B 7 U Q h j Q K B r W B 0 O / V b T 6 g 0 j + W j G S f o R 3 Q g e c g Z N V Z 6 8 N x e u e J W 3 R n I M v F y U o E c 9 V 7 5 q 9 u P W R q h N E x Q r T u e m x g / o 8 p w J n B S 6 q Y a E 8 p G d I A d S y W N U P v Z 7 N I J O b F K n 4 S x s i U N m a m / J z I a a T 2 O A t s Z U T P U i 9 5 U / M / r p C a 8 9 j M u k 9 S g Z P N F Y S q I i c n 0 b d L n C p k R Y 0 s o U 9 z e S t i Q K s q M D a d k Q / A W X 1 4 m z b O q d 1 m 9 u D + v 1 G 7 y O I p w B M d w C h 5 c Q Q 3 u o A 4 N Y B D C M 7 z C m z N y X p x 3 5 2 P e W n D y m U P 4 A + f z B + u y j P k = < / l a t e x i t > 10 < l a t e x i t s h a 1 _ b a s e 6 4 = " K z I B W + 5 9 w u / x d X 7 u 0 b E C 8 w z H 6 + 0 = " > A A A B 6 X i c b V D L S g N B E O y N r x h f U Y 9 e B o P g K e y K r 2 P Q i 8 c o 5 g H J E m Y n v c m Q 2 d l l Z l Y I S / 7 A i w d F v P p H 3 v w b J 8 k e N L G g o a j q p r s r S A T X x n W / n c L K 6 t r 6 R n G z t L W 9 s 7 t X 3 j 9 o 6 j h V D B s s F r F q B 1 S j 4 B I b h h u B 7 U Q h j Q K B r W B 0 O / V b T 6 g 0 j + W j G S f o R 3 Q g e c g Z N V Z 6 8 N x e u e J W 3 R n I M v F y U o E c 9 V 7 5 q 9 u P W R q h N E x Q r T u e m x g / o 8 p w J n B S 6 q Y a E 8 p G d I A d S y W N U P v Z 7 N I J O b F K n 4 S x s i U N m a m / J z I a a T 2 O A t s Z U T P U i 9 5 U / M / r p C a 8 9 j M u k 9 S g Z P N F Y S q I i c n 0 b d L n C p k R Y 0 s o U 9 z e S t i Q K s q M D a d k Q / A W X 1 4 m z b O q d 1 m 9 u D + v 1 G 7 y O I p w B M d w C h 5 c Q Q 3 u o A 4 N Y B D C M 7 z C m z N y X p x 3 5 2 P e W n D y m U P 4 A + f z B + u y j P k = < / l a t e x i t > 10 < l a t e x i t s h a 1 _ b a s e 6 4 = " K z I B W + 5 9 w u / x d X 7 u 0 b E C 8 w z H 6 + 0 = " > A A A B 6 X i c b V D L S g N B E O y N r x h f U Y 9 e B o P g K e y K r 2 P Q i 8 c o 5 g H J E m Y n v c m Q 2 d l l Z l Y I S / 7 A i w d F v P p H 3 v w b J 8 k e N L G g o a j q p r s r S A T X x n W / n c L K 6 t r 6 R n G z t L W 9 s 7 t X 3 j 9 o 6 j h V D B s s F r F q B 1 S j 4 B I b h h u B 7 U Q h j Q K B r W B 0 O / V b T 6 g 0 j + W j G S f o R 3 Q g e c g Z N V Z 6 8 N x e u e J W 3 R n I M v F y U o E c 9 V 7 5 q 9 u P W R q h N E x Q r T u e m x g / o 8 p w J n B S 6 q Y a E 8 p G d I A d S y W N U P v Z 7 N I J O b F K n 4 S x s i U N m a m / J z I a a T 2 O A t s Z U T P U i 9 5 U / M / r p C a 8 9 j M u k 9 S g Z P N F Y S q I i c n 0 b d L n C p k R Y 0 s o U 9 z e S t i Q K s q M D a d k Q / A W X 1 4 m z b O q d 1 m 9 u D + v 1 G 7 y O I p w B M d w C h 5 c Q Q 3 u o A 4 N Y B D C M 7 z C m z N y X p x 3 5 2 P e W n D y m U P 4 A + f z B + u y j P k = < / l a t e x i t > 10 < l a t e x i t s h a 1 _ b a s e 6 4 = " 4 X 3 N v 5 0 r S P b U s l j V D 7 2 f T U M T m 2 S o + E s b I l D Z m q v y c y G m k 9 i g L b G V E z 0 P P e R P z P a 6 c m v P Y z L p P U o G S z R W E q i I n J 5 G / S 4 w q Z E S N L K F P c 3 k r Y g C r K j E 2 n a E P w 5 l 9 e J I 2 z i n d Z u b g / L 1 d v 8 j g K c A h H c A I e X E E V 7 q A G d W D Q h 2 d 4 h T d H O C / O u / M x a 1 1 y 8 p k D + A P n 8 w d V N Y 0 w < / l a t e x i t > 10 < l a t e x i t s h a 1 _ b a s e 6 4 = " 4 X 3 N v 5 0 r S P b U s l j V D 7 2 f T U M T m 2 S o + E s b I l D Z m q v y c y G m k 9 i g L b G V E z 0 P P e R P z P a 6 c m v P Y z L p P U o G S z R W E q i I n J 5 G / S 4 w q Z E S N L K F P c 3 k r Y g C r K j E 2 n a E P w 5 l 9 e J I 2 z i n d Z u b g / L 1 d v 8 j g K c A h H c A I e X E E V 7 q A G d W D Q h 2 d 4 h T d H O C / O u / M x a 1 1 y 8 p k D + A P n 8 w d V N Y 0 w < / l a t e x i t > 10 < l a t e x i t s h a 1 _ b a s e 6 4 = " 4 X 3 N v 5 0

4 < 8 <
H p 0 X 5 9 3 5 m L e u O P n M E f y B 8 / k D f F m M v g = = < / l a t e x i t > 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " V F u K p c P 2 s z K C 8 d 4 w a t p j z 6 o E 7 O E = " > A A A B 5 H i c b V B N S 8 N A E J 3 U r x q / q l c v i 0 X w V B L x 6 1 j 0 4 r G C / Y A 2 l M 1 2 0 q 7d b M L u R i i h v 8 C L B 8 W r v 8 m b / 8 Z t m 4 O 2 P h h 4 v D f D z L w w F V w b z / t 2 S m v r G 5 t b 5 W 1 3 Z 3 d v / 6 D i H r Z 0 k i m G T Z a I R H V C q l F w i U 3 D j c B O q p D G o c B 2 O L 6 b + e 1 n V J o n 8 t F M U g x i O p Q 8 4 o w a K z 1 c 9 C t V r + b N Q V a J X 5 A q F G j 0 K 1 + 9 Q c K y G K V h g mr d 9 b 3 U B D l V h j O B U 7 e X a U w p G 9 M h d i 2 V N E Y d 5 P N D p + T U K g M S J c q W N G S u / p 7 I a a z 1 J A 5 t Z 0 z N S C 9 7 M / E / r 5 u Z 6 C b I u U w z g 5 I t F k W Z I C Y h s 6 / J g C t k R k w s o U x x e y t h I 6 o o M z Y b 1 4 b g L 7 + 8 S l r n N f + q d l m t 3 x Z h l O E Y T u A M f L i G O t x D A 5 r A A O E F 3 u D d e X J e n Y 9 F Y 8 k p J o 7 g D 5 z P H x f r i 5 g = < / l a t e x i t > l a t e x i t s h a 1 _ b a s e 6 4 = " Y w 0 M j 6 t n 6 C c L M e 6 0 3 j C D i i d D B j c = " > A A A B 6 H i c b V D L T g J B E O z F F + I L 9 e h l I j H x R H a N D 4 5 E L x 4 h k U c C G z I 7 9 M L I 7 O x m Z t a E E L 7 A i w e N 8 e o n e f N v H G A P C l b S S a W q O 9 1 d Q S K 4 N q 7 7 7 e T W 1 j c 2 t / L b h Z 3 d v f 2 D 4 u F R U 8 e p Y t h g s Y h V O 6 A a B Z f Y M N w I b C c K a R Q I b A W j u 5 n f e k K l e S w f z D h B P 6 I D y U P O q L F S v d I r l t y y O w d Z J V 5 G S p C h 1 i t + d f s x S y O U h g m q d c d z E + N P q D K c C Z w W u q n G h L I R H W D H U k k j 1 P 5 k f u i U n F m l T 8 J Y 2 Z K G z N X f E x M a a T 2 O A t s Z U T P U y 9 5 M / M / r p C a s + B M u k 9 S g Z I t F Y S q I i c n s a 9 L n C p k R Y 0 s o U 9 z e S t i Q K s q M z a Z g Q / C W X 1 4 l z Y u y d 1 2 + q l + W q r d Z H H k 4 g V M 4 B w 9 u o A r 3 U I M G M E B 4 h l d 4 c x 6 d F + f d + V i 0 5 p x s 5 h j + w P n 8 A Y h 5 j M Y = < / l a t e x i t > l a t e x i t s h a 1 _ b a s e 6 4 = " m 3 Y c 9 P t J i 9 W H y h k g m u h f Q e A w V j 4 = " > A A A B 6 X i c b V D L S g N B E O z 1 G e M r 6 t H L Y B A 8 h d 3 g 6 x j 0 4 j G K e U C y h N n J b D J k d n a Z 6 R X C k j / w 4 k E R r / 6 R N / / G S b I H T S x o K K q 6 6 e 4 K E i k M u u 6 3 s 7 K 6 t r 6 x W d g q b u / s 7 u 2 X D g 6 b J k 4 1 4 w 0 W y 1 i 3 A 2 q 4 F I o 3 U K D k 7 U R z G g W S t 4 L R 7 d R v P X F t R K w e c Z x w P 6 I D J U L B K F r p w a v 2 S m W 3 4 s 5 A l o m X k z L k q P d K X 9 1 + z N K I K 2 S S G t P x 3 A T 9 j G o U T P J J s Z s a n l A 2 o g P e s V T R i B s / m 1 0 6 I a d W 6 Z M w 1 r Y U k p n 6 e y K j k T H j K L C d E c W h W f S m 4 n 9 e J 8 X w 2 s + E S l L k i s 0 X h a k k G J P p 2 6 Q v N G c o x 5 Z Q p o W 9 l b A h 1 Z S h D a d o Q / A W X 1 4 m z W r F u 6 x c 3 J + X a z d 5 H A U 4 h h M 4 A w + u o A Z 3 U I c G M A j h G V 7 h z R k 5 L 8 6 7 8 z F v X X H y m S P 4 A + f z B + 6 6 j P s = < / l a t e x i t > 12 < l a t e x i t s h a 1 _ b a s e 6 4 = " k Q k H g 5 M E 6 G O F + e 2 V q l m 4 K u f L 2 Q U = " > A A A B + H i c b V D J S g N B E K 2 J W 4 x L R j 1 6 a Q y C F 8 O M x O U Y 9 O I x g l k g i a G n 0 0 m a 9 P Q M 3 T V C H P I l X j w o 4 t V P 8 e b f 2 F k O m v i g 4 P F e F V X 1 g l g K g 5 7 3 7 W R W V t f W N 7 K b u a 3 t n d 2 8 u 7 d f M 1 G i G a + y S E a 6 E V D D p V C 8 i g I l b 8 S a 0 z C Q v B 4 M b y Z + / Z F r I y J 1 j 6 O Y t 0 P a V 6 I n G E U r d d y 8 a q E I u S G + 9 5 C e l s Y d t + A V v S n I M v H n p A B z V D r u V 6 s b s S T k C p m k x j R 9 L 8 Z 2 S j U K J v k 4 1 0 o M j y k b 0 j 5 v W q q o X d Z O p 4 e P y b F
< l a t e x i t s h a 1 _ b a s e 6 4 = " d t Z n 9 D 7 x E R S x 2 A u C 3 v n I A / 3 W 5 J c = " > A A A B 7 n i c b V D L S g N B E O z 1 G e M r 6 t H L Y B C 8 G H b F 1 z H o x W M E 8 4 B k D b O T 3 m T I 7 O w y M yu E J R / h x Y M i X v 0 e b / 6 N k 2 Q P m l j Q U F R 1 0 9 0 V J I J r 4 7 r f z t L y y u r a e m G j u L m 1 v b N b 2 t t v 6 D h V D O s s F r F q B V S j 4 B L r h h u B r U Q h j Q K B z W B 4 O / G b T 6 g 0 j + W D G S X o R 7 Q v e c g Z N V Z q e u 5 j d u q N u 6 W y W 3 G n I I v E y 0 k Z c t S 6 p a 9 O L 2 Z p h N I w Q b V u e 2 5 i / I w q w 5 n A c b G T a k w o G 9 I + t i 2 V N E L t Z 9 N z x + T Y K j 0 S x s q W N G S q / p 7 I a K T 1 K A p s Z 0 T N Q M 9 7 E / E /r 5 2 a 8 N r P u E x S g 5 L N F o W p I C Y m k 9 9 J j y t k R o w s o U x x e y t h A 6 o o M z a h o g 3 B m 3 9 5 k T T O K t 5 l 5 e L + v F y 9 y e M o w C E c w Q l 4 c A V V u I M a 1 I H B E J 7 h F d 6 c x H l x 3 p 2 P W e u S k 8 8 c w B 8 4 n z 8 + q Y 7 f < / l a t e x i t > 10 < l a t e x i t s h a 1 _ b a s e 6 4 = " j s a j e 7 B T h z S K h i q e S S C L r d F 5 x m o = " > A A A B 7 n i c b V D L S g N B E O y N r x h f U Y 9 e B o P g x b A b f B 2 D X j x G M A 9 I 1 j A 7 m S R D Z m e X m V 4 h L P k I L x 4 U 8 e r 3 e P N v n C R 7 0 M S C h qK q m + 6 u I J b C o O t + O 7 m V 1 b X 1 j f x m Y W t 7 Z 3 e v u H / Q M F G i G a + z S E a 6 F V D D p V C 8 j g I l b 8 W a 0 z C Q v B m M b q d + 8 4 l r I y L 1 g O O Y + y E d K N E X j K K V m p 7 7 m J 5 V J t 1 i y S 2 7 M 5 B l 4 m W k B B l q 3 e J X p x e x J O Q K m a T G t D 0 3 R j + l G g W T f F L o J I b H l I 3 o g L c t V T T k x k9 n 5 0 7 I i V V 6 p B 9 p W w r J T P 0 9 k d L Q m H E Y 2 M 6 Q 4 t A s e l P x P 6 + d Y P / a T 4 W K E + S K z R f 1 E 0 k w I t P f S U 9 o z l C O L a F M C 3 s r Y U O q K U O b U M G G 4 C 2 + v E w a l b J 3 W b 6 4 P y 9 V b 7 I 4 8 n A E x 3 A K H l x B F e 6 g B n V g M I J n e I U 3 J 3 Z e n H f n Y 9 6 a c 7 K Z Q / g D 5 / M H Q C 6 O 4 A = = < / l a t e x i t > 10 < l a t e x i t s h a 1 _ b a s e 6 4 = " s W z m s O S h s I v H Y x x v g u 9 W K X 5 j w a E = " > A A A B 7 n i c b V D J S g N B E K 2 J W 4 x b 1 K O X x i B 4 M c y 4 H 4 N e P E Y w C y R j 6 O n U J E 1 6 e o b u H r + e 9 s f i f 1 0 p N e O 1 n X C a p Q c m m i 8 J U E B O T 8 e + k y x U y I 4 a W U K a 4 v Z W w P l W U G Z t Q w Y b g z b + 8 S O q n Z e + y f H F / X q r c z O L I w w E c w j F 4 c A U V u I M q 1 I D B A J 7 h F d 6 c x H l x 3 p 2 P a W v O m c 3 s w x 8 4 n z 9 B s 4 7 h < / l a t e x i t > 10 < l a t e x i t s h a 1 _ b a s e 6 4 = " D u r D J 2 L e 8 b T / H c 1 n v V L B 3 s y 0 / O o = " > A A A B 7 n i c b V D L S g N B E O y N r x h f U Y 9 e B o P g x b A r 8 X E M e v E Y w T w g W c P s p J 9 p 6 t F 6 s 1 1 V r w c p n j s E P W G + f 9 U S Q a w = = < / l a t e x i t > ↵ = 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " p X K j y x j 8 1 b l C v 0 S x 4 A S G b b y j b q g = " > A A A B 8 n i c d V D L S s N A F J 3 4 r P V V d e l m s A i u Q l L 6 X A h F N y 4 r 2 A e k o U y m k 3 b o Z C b M T I Q S + h l u X C j i 1 q 9 x 5 9 8 4 a S O o 6 I E L h 3 P u 5 d 5 7 g p h R p R 3 n w 1 p b 3 9 j c 2 i 7 s F H f 3 9 g 8 O S 0 f H P S U S i U k X C y b k I E C K M M p J V 1 P N y C C W B E U B I / 1 g d p 3 5 / X s i F R X 8 T s 9 j 4 k d o w m l I M d J G 8 o a I x V M E L 2 G l O C q V H b v l u K 1 a A 6 5 I s 5 q T e g u 6 t r N E G e T o j E r v w 7 H A S U S 4 x g w p 5 b l O r P 0 U S U 0 x I 4 FIG. 4. Evolution of the mixed state ζ η=1/4 when the field strength α is increased adiabatically from 0 to 6.The density profile (background color) and local spin expectation vector ⟨F(r)⟩ (arrows) in the x = 0 plane are plotted at four different times with the instataneous values of α indicated in the plots.For each plot, the length of the box along each direction is 25, and the calculation is done with a grid size N 3 = 71 3 .A simulation animation is available here.
+ and L ℓ−m ℓ − , and (a † + ) ℓ [29].Including the α dependent Zeeman term, couples the different 3D oscillator states.The quantum numbers associated with the full Hamiltonian are (ñ, j, m, F ), and the eigenstates are some n and ℓ superpositions (with |j −F | ≤ ℓ ≤ j +F as required by the triangle inequality) of the basis set |n, j, m, F, ℓ⟩ ≡