’

U

S S

S S U

S S

S S u, M S S

S S u

S S nihilated) by the operators aj†s (ajs). Expanding the Coulomb interaction for |⃗rA − ⃗rB| j d, and assuming

2 that ℏÉ s k Cd03xls , with ls the harmonic oscillator lengths, we obtain the Hamiltonian:

H = ℏ X É saj†sajs + ℏ X

j,s s,s′ where

a†BsTss′ aAs′ + H.c. , ( 3 ) T (¹ ) =

C0 md3 1−3 cos2 θ 3 sin θ cos θ !

ω↑ √ω↑ω³ 3 sin θ cos θ 1−3 sin2 θ

√ω↑ω³ ω³ is the vibron-hopping matrix [46]. The system hence emulates the Hubbard-like dynamics of pseudo-spin-1/2 bosons hopping between the two micro-traps, with the spin encoded in the two oscillation directions of the micro-traps. Note that ( 4 ) ( 5 ) É ↑ − É ↓ k 3C0l↑l↓ sin 2¹

4d3 is necessary in order to guarantee that the local oscillator axes are only given by the microtrap potentials, independently of the Coulomb interaction with neighboring ions.

The hopping matrix is diagonal if one of the microtrap axes aligns with the line joining the micro-trap centers (¹ = 0 or Ã/ 2). In that case, the pseudo-spin vibron components decouple, a situation encountered in trapped-ion arrays employed for quantum computation [47]. An angle ¹ ̸= 0, Ã/ 2 is hence required to get off-diagonal hopping terms, a necessary requirement to establish non-Abelian dynamics. However, off-diagonal coupling is off-resonant, due to condition ( 5 ), and hence negligible. As a result, hopping would be effectively diagonal, precluding the study of non-Abelian dynamics.

Off-diagonal hopping may be however activated by means of Floquet techniques, assuming that the trap frequencies are periodically modulated with a driving frequency É d: É i,s=↑,↓(t) = É s + ¸ iÉ d cos(É dt + ϕ i).

( 6 ) We set equal driving phase ϕ i=A,B and strength ¸ i in the two axis (s =↑, ³ ) of the micro-traps. They may be however different opening further possibilities for Floquet engineering.

In order to resonantly couple the different vibron types we set É ↑ − É ↓ = nÉ d, with n = 1, 2, . . . . Assuming that the driving period 2Ã/É d is much shorter than the time scale md3xÉ s/C0 of vibron hopping, and employing Floquet formalism [44], the states ↑ and ³ acquire equal energy in the rotating frame, and are coupled by ( 7 ) ( 8 ) ( 9 ) the effective time-independent hopping matrix T˜ (see Fig. 1 (b)) [48]: with fss′ = inà sys′ , à y the Pauli matrix, Jm(x) the m-th Bessel function of the first kind, and

ZA2B = ¸ A2 + ¸ B2 − 2¸ A¸ B cos(ϕ B − ϕ A) tan µ AB =

¸ B sin(ϕ B − ϕ A) ¸ A − ¸ B cos(ϕ B − ϕ A)

Note that the elements of the hopping matrix have acquired an effective Peierls phase [49]. The effective Hamiltonian is then

H = ℏ X a†BsT˜ss′ (¹ )aAs′ + H.c. .

( 10 ) s,s′ The driving can be employed not only to bridge the energy difference between vibron states, but also to dynamically decouple two sites, using the nodes of Jm(x). We make use of this flexibility below in order to induce and monitor non-Abelian dynamics in the system.

Non-Abelian plaquette.– In the following, we employ the technique described above to engineer non-Abelian dynamics in a minimal instance, formed by four microtraps arranged in a square plaquette on the xy-plane, see Fig. 1 (c). However, the ideas can be extrapolated to larger two-dimensional lattices. We label the ions A, B, C and D in a counter-clockwise order as in Fig. 1 (c).

The tilting angle of all micro-traps is set to ¹ = Ã/ 4, which results in bare vibron hopping matrices along the two directions given by Tx = T (Ã/ 4) and Ty = T (−Ã/ 4), where T (¹ ) is given by Eq. ( 4 ). Ions diagonally opposite to each other are coupled with a weaker amplitude, Tx+y = T ( 0 )/2√2 and Tx−y = T (Ã/ 2)/2√2. We consider for simplicity the same driving strength ¸ in all micro-traps, and the same phase difference ¶ϕ = Ã between sites placed at opposite sites of a plaquette diagonal. Note that with the choice of ¹ = Ã/ 4, the hopping matrices Tx+y and Tx−y are diagonal.

Hence, ions at opposite sites of a diagonal are fully decoupled if ¸ is chosen such that J0(ZAC ) = 0 and J0(ZDB) = 0 (note that restricting the hopping to nearest neighbors simplifies the dynamics, but it is not fully necessary for establishing non-Abelian dynamics).

Due to our choice of ¶ϕ this requirement simplifies to ¸ = 0.5j01 with j01 the first root of J0(x) [48]. Setting ϕ A = 0, the only free phase parameter left is ϕ = ϕ D.

We can then write the effective hopping matrices of the driven system as:

C0 T˜x = − 2md3

C0 T˜y = − 2md3

J0(j01| cos(φ/2)|)

ω↑ −3 Jn(j01| cos(φ/2)|) einφ/2

√ω↑ω³ J0(j01| sin(φ/2)|)

ω↑ 3in Jn(j01| sin(φ/2)|) einφ/2 √ω↑ω³ −3 Jn(j01| cos(φ/2)|) e−inφ/2!

√ω↑ω³ J0(j01| cos(φ/2)|)

ω³ 3in Jn(j01| sin(φ/2)|) e−inφ/2!

√ω↑ω³ J0(j01| sin(φ/2)|) ω³ .

The dynamics acquires a non-Abelian character when [T˜x, T˜y] ̸= 0.

Path-dependent spin dynamics.– In order to probe the non-Abelian vibron dynamics, we initialize a single vibron in a fixed spin state in site A, and monitor its spin state when it reaches the diagonally opposite site C (see Fig. 1 (c)). Path-dependent spin dynamics is revealed if that spin state depends on whether the vibron travelled in a clockwise or counter-clockwise fashion around the plaquette. This may be probed by exploiting the flexibility of the proposed system to steer the vibron motion.

Setting the trap frequencies of trap B (D) off-resonant we may suppress the hopping to that site, forcing the vibron into the clockwise (counter-clockwise) path around the plaquette. This may be achieved by increasing the frequencies by (É ↑ − É ↓)/2, which places them within the energy gap of the other sites, and out of reach of any hopping process.

Figure 2 shows the results assuming a single vibron initially prepared at site A in a symmetric superposition of both spin states, √12 (aA,↑ + aA,↓)† |0ð. We consider É d/2Ã = 500 kHz, d = 40 µm and É ↑/2Ã = 4 MHz, within the range reported in recent experiments aiming at two-dimensional quantum simulations with ions [43].

We have compared our results using the effective timeindependent Hamiltonian ( 10 ), with those obtained using the time-dependent Hamiltonian finding a good agreement for the considered modulation frequency. Driving with n = 2, ϕ = Ã/ 2 and ¶ϕ = Ã , induces T˜x = T˜y, and, as it should, the vibron spin at C remains independent of the travelled path. When tuning the driving away from this point, the vibron spin at site C becomes pathdependent revealing the non-commutative dynamics. In order to quantify the non-Abelian character, we define:

Ds = |ns⟲(t) − ns⟳(t)|dt

( 13 ) where ns⟳(t) (ns⟲(t)) is the vibron occupation of ion C in the state s =↑, ³ with the clockwise (counter-clockwise) path open. Figure 3 depicts D↑ as a function of the driving phase difference ϕ and n, fixing ¶ϕ = Ã . We observe path-dependence for basically any choice of the driving, showing that non-commutative dynamics is a robust general feature of the set up that requires no precise fine-tuning. ( 11 ) ( 12 ) ( 14 ) ( 15 )

Spin-orbit coupling.– The spin-orbit coupling characteristic of the non-commutative nature of the vibron dynamics may be alternatively probed by monitoring how the preparation of a specific initial spin state can affect the vibron motion. We consider different initial spin states at site A: |È ð =

eiβ sin ³ a †A,↑ + cos ³ a †A,↓ |0ð , and analyze the time-averaged vibron angular momentum

L = T 0

Pj nj R⃗ j is the vibron position, and ⃗v = Pj ddntj R⃗ j is the vibron velocity, with nj = ïPs aj†,sajsð the local vibron number. Non-Abelian vibron dynamics results in a spin dependent angular momentum, as shown in Fig. 4, where we depict L as a function of ³ and ´ for the same case as in Fig. 2 (b), i.e. n = 2, ¶ϕ = Ã , and ϕ = Ã/ 3 (the orientation of the rotation is inverted when ϕ → −ϕ ). Note that this way of probing the dynamics d FIG. 3. D↑ (see text) as a function of the phase difference φ of the driving of micro-traps A and D (see Fig. 1 (c)), for n = 1 (red), n = 2 (blue), n = 3 (green) and n = 4 (orange).

The initial state and the tilting angle θ are the same as in Fig. 2. The integration time is T = 10mω↑d3/C0.

0 0.2 0.8

1 0.4

0.6 α/π FIG. 4. Average angular momentum L as a function of the initial state given by Eq. ( 14 ). The driving has been set to n = 2, φ = π/3 and δφ = π. The integration time T is the same as in Fig. 3. The value of L is normalized by C0/mω↑d. merely involves free evolution in the plaquette, without the need of engineering any given path, as above.

The results for the time average of the angular momentum as well as the spin occupation difference presented in the section above depend on the integration time and the driving parameters. In particular choosing larger T reduces the observable signal for L since in any case the angular momentum of the vibron is oscillating around 0.

Note in this sense, that unlike a charged particle subjected to a magnetic field, vibrons do not perform a directed cyclotron motion.

Non-Abelian dynamics in a square lattice.– Finally, we consider a square lattice of microtraps consisting of copies of the square plaquette discussed in the section above. In this system, vibrons may hop between sites beyond nearest neighbors, with a hopping rate decaying as 1/|⃗ri − ⃗rj |3. This decay is partially countered by −6 −4 −2 2 4

6 0 x/d FIG. 5. Expectation values ïσizð in a square lattice created from copies of the square plaquette shown in Fig. 1(c). The initial vibron excitation with spin state given by α = π/4, β = 3π/5 has an equal projection to the central four sites. The evolution time was 2mω↑d3/C0. The driving phase has been chosen to φ = π/3 (top) and φ = π/2 (bottom). resonances between the driving phases, e.g. the microtrap frequencies of sites with distance 2d are oscillating in phase and hence the Floquet drive enhances the respective hopping rates. As a consequence we cannot neglect the long-range hopping matrices for the calculation of the vibron dynamics. In Fig. 5 we investigate the evolution of a vibron excitation initially delocalized equally over the sites of one square plaquette with a fixed spin state.

The non-Abelian hopping matrices lead to a different diffusion of the two spin components and hence the buildup of spin state separation. This mechanism is expressed in a breaking of the Z4 symmetry of the initial state, i.e. the spin state distribution is not invariant under the rotation of the system by multiples of Ã/ 2.

Interestingly, long-range hops, which constitute a relevant feature of the dynamics of driven vibrons in ion lattices, result in non-Abelian dynamics for all possible parameters. In this sense, as seen in Fig. 5 the choice of n = 2, ϕ = Ã/ 2, leading to T˜x = T˜y (and hence to Abelian dynamics in the single-plaquette case), results in non-Abelian dynamics as well, caused by hops over a distance of √5d. This is yet another sign of the robust non-Abelian character of the vibron dynamics in the considered driven ion arrays.

Conclusions and outlook.– The exquisite control possibilities offered by ions in micro-trap arrays open interesting perspectives for the quantum simulation of nonAbelian lattice models. Vibrational excitations along the two axes of the micro-traps can be understood as effective spin-1/2 particles (vibrons) hopping amongst the microtraps. Vibron hopping may be accompanied by a spin flip, resulting in a 2 × 2 hopping matrix with off-diagonal terms. Crucially, the hopping matrices in different directions may be engineered as non-commuting, resulting in non-Abelian vibron transport.

We have shown that non-commutative vibron transport may be robustly induced, without the need of a precise fine-tuning of the parameters, by combining two main ingredients. First, the micro-trap axes are arranged sustaining an angle with the links joining the micro-trap centers. Second, Floquet techniques based on the periodic modulation of the trap frequencies allow for the activation of originally off-diagonal couplings. Moreover, we have discussed how these state-of-the-art techniques may be employed to probe in various ways the non-Abelian character by monitoring the induced coupling between the spin and motional degrees of freedom.

The possibilities for a local individual control of the micro-traps open intriguing possibilities for the engineering of more complex non-Abelian lattice models.

In particular, note that we have only discussed singleparticle dynamics as the excitations considered here are non-interacting oscillator modes. However, the motional states that encode the vibron spin can be coupled to internal electronic states of the ions by means of a side-band laser, resulting in vibron-vibron interactions [50, 51]. The study of non-commutative dynamics in interacting many-body systems constitute an exciting future direction open by the present work.

Vibron Hamiltonian. – In the following we sketch the 4 derivation of the effective Hamiltonian (Eq. ( 9 ) in the 2 main text) and explicitly discuss the approximations ap0 plied to obtain it. In our work we deal with the quantum 2 vibrations of the ions around their classical equilibrium n positions, the vibrons. If we consider the case of two ions, Ja A and B, with mass m confined in microtraps with trap 92 schenifttesrtshR⃗e jc=laAs,sBica=l e±qud2⃗eilxibtrhiae amwuatyufarlomCotuhleomtrbaprecpeunltseirosn.

However, for large d and stiff local confinement the ] classical equilibrium positions are in good approximation hp given by R⃗ j . The Hamiltonian for the displacement vec- tors of the ions ⃗ri is then given by Eq. ( 1 ) of the main m text. We expand the Coulomb term up to second order o in |⃗rA − ⃗rB|/d and neglect the first order term since it ta vanishes for the equilibrium positions as argued above. .s The remaining term of the expansion can be written in c the form i s 1 yh Hhop = 2d3 3[⃗ex · (⃗rA − ⃗rB)]2 − [⃗rA − ⃗rB]2 . ( 1 ) [p Introducing the creation and annihilation operators

ajs and ajs we write the displacement operators as v1 √2⃗ri = Ps ls⃗es ais + ais . The expansion yields num2 ber non-conserving terms which are proportional to 02 rCo0tdlas3ltse′ awisitahjsf′r+eqHu.ecn..cyI nÉ sa+nÉ inst′esroacthtiaotnwpeicctaunrenethgleescetttehremms 61 if . 10 mÉ s2d2 k Cd0 . ( 2 ) 4 2 After this rotating wave approximation the vibron :v Hamiltonian takes the form i X r a

H = ℏ X (É s + dÉ s) ajsaj,s

j,s + ℏg X

j + ℏ X s,s′

aj↑aj³ + aj³ aj↑ aB,sTss′ aAs′ + H.c. , Supplementary Material for ”Non-Abelian vibron dynamics in trapped-ion arrays”

L. Timm,1, ∗ H. Weimer,1, 2 and L. Santos1 1Institut fu¨r Theoretische Physik, Leibniz Universita¨t Hannover, Appelstr. 2, 30167 Hannover, Germany 2Institut fu¨r Theoretische Physik, Technische Universita¨t Berlin, Hardenbergstr. 36, 10623 Berlin, Germany (Dated: January 30, 2024)

In this Supplementary Material we provide a detailed derivation of the vibron Hubbard Hamiltonian and the Floquet driving discussed in the main text. with dÉ s = 2Cℏ0dls23 3(⃗es · ⃗ex)2 − 1 , g = 3C4ℏ0ld³ 3l↑ sin 2¹ and with Tss′ the vibron hopping matrix given in Eq. ( 4 ) of the main text. We can neglect the energy shifts of the vibrons dÉ s since É s k dÉ s due to condition ( 2 ).

The second term of Eq. ( 3 ) couples the two local vibron directions with coupling strength g, leading to a rotation of the local vibron eigenstates. Employing again rotating wave arguments we can drop these terms for

3C0 É ↑ − É ³ k 4m√É ↑É ³ d3 sin 2¹, so that we arrive at the vibron Hubbard Hamiltonian of Eq. ( 3 ) of the main text. Note that the last condition is qualitatively different from ( 2 ) as it does not deals with the absolute values of the trap frequencies but with their difference, which needs to be sufficient such that the local vibron eigenstates are imposed by the microtraps.

Floquet driving. – In the following we derive the time-independent model that applies if the trap frequencies are modulated in time with a sufficiently large frequency É d. After inserting the periodically driven frequencies É is = É s + ¸ iÉ d cos(É dt + ϕ i) into Eq. ( 3 ), we transform the Hamiltonian into the interaction picture with respect to its first term ℏ Pj,s É sajsajs. The transformation yields a time-dependent hopping matrix with elements that can be written via a Jacobi-Anger expansion in the form =

X Jm(¸ B)Jm′ (¸ A)ei(ωs−ωs′ +(m−m′)ωd)t Tss′ (¹, t )

When using the shorthand notation É s − É s′ = fss′ É d, where fss′ = inà sys′ is defined as in the main text, we find that terms with m′ = m + fss′ are resonant as their timedependence cancels. All other terms in the sum oscillate with a frequency that is given by a multiple of É d. These off-resonant terms can be neglected if and the timescale of the rotation separates from the timescale of the vibron hopping given by the amplitude of Tss′ (¹ ).

The sum of the resonant terms is then given by = F (¸ B, ¸ A, fss′ , 2¶ϕ BA)e−ifss′ φ¯BA , ( 4 ) ( 5 ) ( 6 ) 1, 0.8 0 j/a 0.7 1.2 1.1

1 0.9 0.6 0.5 0.4 0.3 φ/π with 2¶ϕ ij = ϕ i − ϕ j and 2ϕ ¯ij = ϕ i + ϕ j and with

F (a, b, f, ϕ ) =

Jm(a)Jm+f (b)ei(m+f/2)φ ∞

X m=−∞ = Jf (Z)ei(γ+φ/2)f , ( 8 ) ( 9 ) where Jm(x) m-th Bessel function of the first kind and we used the Neumann-Graf formular in the second line [1]. This introduces the parameters

Z2 = a2 + b2 − 2ab cos ϕ, tan µ =

a sin ϕ b − a cos ϕ For a = b these parameters simplify to Z = 2|a sin φ2 | and tan µ = cot φ2 , which we can solve to 2µ = Ã − ϕ so that the hopping matrix terms are given by

Dynamical decoupling. – In the main text we use the modulation in order to decouple the ions which are located in sites diagonally opposite in a square. This requires to tune the corresponding driving amplitudes and the phase differences such that |F (a, b, 0, ϕ )| vanishes. As can be seen from Eq. ( 9 ), this demands Z2 = j0,k where

2 jm,k is the k-th positive root of Jm(x). For arbitrary val( 11 ) ( 12 ) ues a and b we introduce c = a − b and find the solutions

These solutions are shown in Fig. 1 as a function of the angle ϕ with different values of c, choosing k = 1. A special case relevant for this work is c = 0 which simplifies the solution to a =

j0,k 2| sin φ2 | , which has its minimal value of all solutions for ϕ = Ã .

Hopping matrices in the square plaquette.– When the derived model is applied to the square plaquette discussed in the main text, there are are in general 9 parameters (4 driving amplitudes, 4 driving phases and n), however they cannot be chosen arbitrarily in order to engineer non-Abelian dynamics. Let us assume that the driving strengths of sites diagonally opposite are equal such that we have ¸ A = ¸ C = ¸ 1 and ¸ B = ¸ D = ¸ 2 for the two diagonals. In order to ensure that the hopping matrices in x direction do not depend on the y positions of the coupled ions (and vice versa) we need to impose that the respective matrix elements given by Eq. ( 7 ) are equal. For equal amplitudes we need ZAB = ZDC and ZAD = ZBC which demands cos(ϕ B − ϕ A) = cos(ϕ C − ϕ D), cos(ϕ D − ϕ A) = cos(ϕ C − ϕ B).

( 13 ) An obvious solution to these two equations is 2¶ϕ CA = 2¶ϕ BD = Ã which also ensures that the respective phases of the matrix elements are equal.

If we in addition decouple the ions diagonally opposite the driving strengths are given by Eq. ( 12 ), i.e. 2¸ 1 = j0k and 2¸ 2 = j0f . This includes the case of the main text with k = f = 1 for which all driving strengths are equal.

With these restrictions to the driving parameters the general form of the hopping matrices is given by

T˜sxs′ (¹ )