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IOP Institute of Physics

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17 October 2014

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4February2015

Realizing unconventional quantum magnetism with symmetric top molecules

M L Wall12, K Maeda2 and Lincoln D Carr2

1 JILA, NIST, and Department of Physics, University of Colorado, Boulder, CO 80309, USA

2 Department of Physics, Colorado School of Mines, Golden, CO 80401, USA

E-mail: mwall.physics@gmail.com

Keywords: ultracold molecules, symmetric top, XYZ magnetism, quantum spin model, methyl fluoride

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Abstract

We demonstrate that ultracold symmetric top molecules loaded into an optical lattice can realize highly tunable and unconventional models of quantum magnetism, such as an XYZ Heisenberg spin model. We show that anisotropic dipole-dipole interactions between molecules can lead to effective spin-spin interactions which exchange spin and orbital angular momentum. This exchange produces effective spin models which do not conserve magnetization and feature tunable degrees of spatial and spin-coupling anisotropy. In addition to deriving pure spin models when molecules are pinned in a deep optical lattice, we show that models of itinerant magnetism are possible when molecules can tunnel through the lattice. Additionally, we demonstrate rich tunability of effective model parameters using only a single microwave frequency, in contrast to proposals with diatomic molecules, which often require many microwave frequencies. Our results are germane not only for experiments with polyatomic symmetric top molecules, such as methyl fluoride (CH3F), but also diatomic molecules with an effective symmetric top structure, such as the hydroxyl radical OH.

1. Introduction

Lattice models of exchange-coupled quantum mechanical spins such as the Heisenberg model have long served as paradigmatic examples of strongly correlated many-body systems [1,2]. The exquisite tunability and precise microscopic characterization of ultracold gases makes them promising candidates for exploring quantum magnetism. However, the most prominent platform for ultracold gas quantum simulation, neutral atoms loaded into optical lattices, has difficulty reaching the regime where quantum magnetism is manifest [3-5]. The reason for the difficulty is that the short-range interactions experienced by neutral atoms require two atoms to occupy the same lattice site in order to significantly interact. For two-component (effective spin-1/2) atoms, effective models of quantum magnetism emerge when on-site interactions U are significantly larger than the tunneling amplitude tbetween neighboring lattice sites, pinning the atoms in a Mott insulator phase with one atom in each lattice site. Effective spin interactions are then mediated by a superexchange process [ 1 ] which requires virtual tunneling to doubly occupied sites. Because the resulting effective spin couplings scale as t 2/U with t ^ U, the temperature scales required to see the onset of magnetism are extraordinarily small.

Systems which feature long-range interactions can generate effective spin-spin interactions which are not mediated by tunneling, and so can display coherent internal state many-body dynamics even without quantum degeneracy in the motional degrees of freedom. Such long-range effective spin couplings have been realized using trapped ions [6-8],Rydberg atoms [9], and magnetic atoms [10], and have been proposed for other platforms, such as atoms in optical cavities [11]. In this work, we focus on the realization oflong-range effective spin interactions with polar molecules, as has been recently demonstrated experimentally [12,13]. A unique feature ofdipolar realizations ofquantum magnetism, as polar molecules in optical lattices provide, compared to non-dipolar systems (e.g., trapped ions) is that dipolar interactions are anisotropic. Anisotropic interactions do not conserve the internal (e.g., rotational) or the spatial angular momentum separately, but only their sum. By mapping the internal angular momentum of a molecule, in particular its rotational angular momentum, to

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(a) (b)

Figure 1. Resonant dipolar processes which exchange internal and external angular momentum. (a) An example of a Forster resonance which involves four different internal states satisfying the resonance condition En + Em ^ En + Em. Single-particle dipole-allowed transitions | n) ^ |n') and|m) ^ |m') drive the interaction-induced two-particle transition |nm) ^ |n'm'). (b) The resonances utilized in this work involve single-particle levels | n) and | n') which are nearly degenerate and posses a single-particle dipole-allowed transition | n) ^ |n') that mayvanish upon time averaging. Interactions cause a two-particle transition |nn) ^ |n'n'), changing the net rotational projection of the molecules.

Figure 2. Symmetric top molecules (STMs) in optical lattices. (a) Rotational angular momentum geometry of the STM CH3F. (b) Schematic of field and lattice geometry. Purple and green denote two internal states.

an effective spin, the dipole-dipole interaction hence generates the possibility of unconventional models of quantum magnetism which do not conserve the total magnetization. As we will show in this paper, such models feature tunable degrees of both spin and spatial anisotropy.

The exchange of internal and external angular momentum projection by dipole-dipole interactions requires two pairs of internal states which are nearly degenerate in energy (on the scale of dipole-dipole interactions) and also have dipole-allowed transitions between them. We show two such scenarios in figure 1. The first scenario is that we have two pairs ofinternal states, call them (n, m) and (n ', m') with energies En I Em _ En' I

Em nearly

degenerate. Further, we assume that at least one of the latter states is not a member of the former pair of states3, see figure 1(a). Such a two-particle near-degeneracy with non-radiative dipole coupling is generally called a Förster resonance, and such resonances have been fruitfully applied to control the interactions in Rydberg atoms [14,15]. Additionally, such resonances may occur at isolated points in the spectra of polar molecules, those with no orbital or spin angular momentum [16,17].

In this work, we instead exploit a resonant process such as is shown in figure 1(b). Here, two particles in the same internal state n are transferred to a different internal state n' which is dipole-coupled to the first and brought into resonance by external fields. In contrast to the Förster resonance, this latter type of resonance involves only two single-particle states, and so naturally leads to a description in terms of a spin-1/2 system. Such resonances are a generic feature ofmagnetic dipoles, and lead to phenomena such as spontaneous demagnetization of spinor Bose gases [18]. In contrast, electric dipoles have parity and time-reversal selection rules which would appear to preclude a dipole-coupled resonance such as is shown in figure 1 (b). A key finding of this work is that resonances like figure 1 (b) with an electric dipole transition are also a generic feature of symmetric top molecules (STMs) with microwave and static field dressing, even in the presence of hyperfine or other detailed molecular structure. Polyatomic STMs, such as the methyl fluoride molecule, CH3F, shown in figure 2(a), have a high degree of symmetry in their rotational structure which makes them behave as 'electric analogs' of pure magnetic dipoles [19]. Further details on STMs and their interactions with external fields is provided in section 3.

Processes which exchange two internal states through the transition | nm) ^ | mn) do not change the total angular momentum projection by definition, and so are not of the type we are discussing here. However, these terms do contribute to the total Hamiltonian, see, e.g., equation (7).

In the present work, we show that two isolated internal states of an STM tuned near a resonance of the form shown in figure 1 (b) form an effective spin-1/2 which is governed by a model with tunable anisotropy in both the spatial and spin-component dependence of the effective spin couplings. In contrast to related proposals, such as the realization of spin-component anisotropic XYZ Heisenberg models using bosonic atoms in excited p-orbitals of an optical lattice [20] or in a synthetic gauge field [21-24], our spin couplings are non-perturbative in the particle-particle interaction strength and are not mediated by tunneling through the lattice. Hence, magnetic phenomena in our proposal can be realized even in the absence of motional quantum degeneracy for the molecules. The ability to observe coherent many-body dynamics in a non-degenerate sample ofultracold molecules is important, as cooling of molecules is difficult and the fully quantum degenerate regime has not yet been reached [3,25]. Additionally, our microwave dressing proposal applies for present ~500 nm optical lattice spacings, in distinction to proposals for 2Z molecules, where appreciable couplings require trapping at quite small lattice spacings [26]. Finally, our proposal requires only a single microwave frequency, in contrast to many proposals for lZ molecules where multiple frequencies are required [17,27-29].

This paper is organized as follows. Section 2 provides a phenomenological analysis of how the resonances shown in figure 1 (b) lead to effective quantum spin systems, e.g. XYZ Heisenberg spin models, which do not conserve magnetization. In section 3 we present an overview of STMs and their coupling to external fields. In particular, section 3.2 discusses the interaction of STMs with microwave radiation that is near-resonant with a rotational transition. We then focus on how to engineer the external fields to obtain tunable resonances like those in figure 1(b), and analyze the dipole-dipole interactions between microwave-dressed states. Section 4 derives effective many-body models of quantum magnetism which are applicable near the field-induced level crossings. Finally, in section 5, we conclude. Some more technical details ofthe dipole-dipole interactions in the microwave-dressed basis states are given in the appendix.

2. Phenomenological realization of an XYZ spin model

In this section, we provide a phenomenological analysis of how a resonance such as that shown in figure 1 (b) leads to an effective XYZ spin model for a collection of STMs pinned in a quasi-2D lattice geometry, as in figure 2(b). A more detailed analysis will be provided in section 4, which also considers the general case in which molecules are not pinned.

We will label the two resonant internal states ofthe molecule as |0)and|T) ,and assume that all other states are far-detuned on the scale of interactions and so maybe neglected. Further, let us assume that the states 10) and |T) have well-defined internal angular momentum projections M = 0 and 1, respectively, for simplicity, though we stress that the assumption of well-defined angular momentum is not essential. Two molecules, call them molecules i and j, interact through the dipole-dipole interaction

dW- d(j) - ( dW)(ey- d(j)) Hddi =-—-, (l)

where R ij is the vector connecting two molecules, ej is a unit vector along Rj, and d is the molecular dipole operator for molecule i. For our purposes, it is useful to recast the dipole-dipole interaction as

HHddi =—^C(2)(Rij) " [d(i) ® d(j)f, (2)

Rij L J

where C® (Rj) = Y® (Rj) is an un-normalized spherical harmonic,

I d(i) ^ dj I = 2L=-l (l> m, l, q — m |2, q) d{id(lm is a component of the rank-two decomposition of the L q

tensor product of dipole operators with (j, mb j2, m21/, M) a Clebsch-Gordan coefficient and

d±l = + (dX ± idY

)/V2, d0 = dZ spherical components of the dipole operator, and the dot denotes tensor contraction. Expanding out the tensor contraction equation (2), we have

Hddi = Hq=0 + Hq=±i + Hq=±2, (j)

The bars indicate that these are microwave-dressed levels, as discussed in section 3.2.

HHq=0 =

1 — 3 cos2 Qij

Hq=±1 =

3 sin Qij cos Qij

5 (i) 5 (j) 5 (i) 5 (j)

d1 d—1 + d—1 d1 a (i) a (j)

2 + d 0 d0

[ (d^^(i)dA0(j) + d0(i)dA1(j) ) e—i* + h.c.

3 sin2 Q,

y| j(i)j(j) — 2i*..

d ii1 e 2ip,j

+ h.c.].

Here, Qij is the polar angle between Ry and the quantization axis and * is the azimuthal angle in the XY plane.

For the 2D geometry of figure 2(b), Qij = nl 2, and so the interactions equation (5) vanish identically. The term equation (4) conserves the internal and orbital angular momenta separately. Hence, when projected into our basis {10), 11)} of states with well-defined internal angular momentum, the most generic spin-spin coupling that can result is

HHq=0 1 ■

1 — 3 cos2 Qij R3

3 cos2 Qij

Ji (¿5+° s—j)+h.c.)+^ e

J /c(i)c(j) + S(i) S(J)

Jl ( ¿x ^x + sy sy

) + Jz Sfsf

where S±, Sx, Sy, and Sz are pseudospin-1/2 operators in the effective spin space {|f), |4)} = {10), |I)} and we have ignored constant terms and single-spin terms [3]. The coupling constants Jl and Jz are set by dipole matrix elements, and in general are affected by external confinement. The terms proportional to Jl are responsible for the 'spin exchange' or 'state swapping' [30] dipolar interactions which were observed recently in the KRb experiment at JILA [12], and the Jz terms account for the fact that interactions between molecules in the 10) state maybe different from interactions between molecules in the |I) state.

In contrast to the q = 0 term, equation (6) does not conserve internal and external angular momentum separately, but transfers two units of angular momentum from the molecular rotation to the orbital motion or vice versa. Projected into our two-state basis, these read

Hq=±2 = — J

3 sin2 Qj [ JA sfs+V + h.c.],

= —3-

sin2 Qij

! ( 2*ij) (Sx Sx Sy Sy \

Hence, the complete dipole-dipole interaction, equation (3), projected into these two states allows for vast control over the X, Y, and Z components of the Heisenberg spin couplings via geometry and dipole matrix elements. Models with unequal X and Ycoupling strengths do not conserve the total magnetization. Quantum spin models which do not conserve magnetization are of interest because they can generate quantum phases with no counterpart in magnetization-conserving systems, and also for their connection to Majorana fermions and other topological phenomena, see section 4. We stress that the q = ±2 components, equation (6), which are responsible for the terms which do not conserve magnetization, equation (9), only contribute near a resonance such as in figure 1(b). In the remainder of this work, we will show how to engineer such resonances for symmetric top molecules, and also how to tune the effective spin-spin couplings Jl, Jz, and (see equations (20) and (21) for the final spin model results). Also, we relax many of the simplifying assumptions made in this section, such as the restriction that the molecules are pinned and that the molecular states have well-defined internal angular momentum projection.

3. STMs and their interaction with external fields

In this section, we review the basic properties and energy scales of STMs and their interactions with both static and dynamic external fields. A key result of this section is that STMs display a linear Stark effect, which is to say the energy varies linearly with the applied electric field strength at moderate fields. A linear Stark effect has the consequence that a large portion of the dipole moment of an STM can be accessed with very modest electric fields. We also show that the linear Stark effect can be used together with microwave dressing of low-lying rotational states to engineer level crossings in the single-molecule energy spectrum. Such level crossings enable

the realization of the resonances shown in figure 1 (b) that are key for the unconventional magnetism described in this work.

3.1. Rotational structure and interaction with static electric fields

Polyatomic STMs, of which methyl fluoride, CH3F, is a canonical example, are defined by a doubly degenerate eigenvalue of the inertia tensor. Such a doubly degenerate eigenvalue corresponds to a cylindrical symmetry of the molecule, see figure2(a), and has key consequences for the rotational level structure ofSTMs. The rotational degrees of freedom of a STM in the lowest electronic and vibrational state maybe characterized by the rigidrotorbasis (a>m |/, K, M) = 2 +l Df*K (rnm), where J is the rotational quantum number, M is the projection of

rotation J on a space-fixed quantization axis, K is the projection of J on the symmetry axis of the molecule, and

D MK (®m) are the matrix elements of the Wigner D-matrix rotating the space-fixed frame to the molecule-fixed

frame by the Euler angles mm [31], see figure 2(a). The corresponding rotational eigenenergies are

EJKM = B0 J (J + l) + (A0 — B0) K2, where the rotational constants B0 « 25 GHz, A0 « l55 GHzforCH3F.

Diatomic l£ molecules, such as the alkali dimers, cannot have a projection of J on the body axis, and so K =0

identically.

Just as the isotropy of space requires that the states with differing projections M of J onto a space-fixed axis are degenerate in the absence of external fields, the cylindrical symmetry of STMs requires that states with opposite projection ±K of J onto a molecule-fixed axis are degenerate. Corrections to the rigid rotor approximation in the vibration-rotation Hamiltonian, such as the well-known inversion of ammonia, can cause mixing of the K levels and result in a splitting of this degeneracy. For simplicity of discussion we will focus on molecules such as CH3F which do not have an inversion splitting in the body of this paper, though we will revisit this issue at the end of the next subsection.

The presence ofa nonzero molecule-frame projection ofrotational angular momentum Kin STMs means that STMs can display a linear response to an externally applied static electric field. This is in stark contrast to the quadratic response exhibited by £-state molecules such as the alkali metal dimers [19]. In particular, in a static electric field of strength Ejc ^ B0/d defining the quantization axis, with d the permanent dipole moment, the matrix elements of the dipole operator along space-fixed spherical direction p, d p, take the form of a spherical

tensor with reduced matrix element (J, K '||d ||J, K) = dK^j $ K ,K' [19]. Hence, STMs in this field regime

display a linear Stark effect with Stark shifts Ejkm = —dKMEic/[J (J + l)].

The strong coupling of STMs to external fields enables them to be effectively decelerated by electric fields [32], and is the basis of opto-electrical cooling, a novel route to bring generic STMs to quantum degeneracy [33,34]. Furthermore, the nonzero reduced matrix element of the dipole operator within a rotational state manifold enables STMs to simulate the physics ofmagnetic dipoles and quantum magnetism with greatly enhanced dipolar interaction energies [19]. In [19], we showed how this correspondence between STMs with rotational quantum number J and an elemental quantum magnet with spin J gives rise to long-range and anisotropic spin models. In what follows, we introduce microwave dressing of rotational states as an additional handle with which to modify the forms and relative strengths of interactions that appear in such effective spin models.

3.2. Microwave dressing of STMs

Microwave radiation couples together neighboring rotational states of a molecule when the frequency of the radiation is near-resonant with the rotational energy level difference. For simplicity, we first consider applying a microwave field Eac with linear polarization along the space-fixed quantization axis, eac = eZ, see figure 2 (b), which is red-detuned an amount A ^ B05 from resonance with the |J, K, 0) —^ | J + l, K, 0) transition, as shown in figure 3(a). While the frequency of this transition in CH3F is larger than the corresponding rotational transition in the alkali dimers, the wavenumber of the transition k « 2(J + l)B0/(hc) is much less than l/a,with a the average separation between molecules, of order a few 100 nm for typical optical lattices. Hence, we neglect the spatial dependence of the microwave field. Applying the rotating wave approximation and transforming to the Floquet picture [35], the quasienergies are obtained by solving the Schrodinger equation for fixed M with the 2x2 Hamiltonians6:

5 We adopt units in which h = l unless specified otherwise.

6 States in the (J + l, K)-manifold which do not couple to any states in the (J, K)-manifold have trivial 1x1 Hamiltonians and we ignore them.

dEdc/A

Figure3. Microwave-dressing of symmetric top molecules with linear polarization. (a) The manifolds (J, K) = (1, 1) and(2, 1) have different linear Stark effects. Coupling by a z-polarized microwave of frequency w and detuning A generates dressed state 10) (|1) )in green (purple). (b) Combined dc and ac Stark shift cause a level crossing of 10) and|I) (Q/A = 1). Inset: enlargement of crossing.

-A -2 1

— —A o

i 2 3 irfSdc - —f 6

KM/2 = -2

-=lo/ J = 21

4.0 6.0 dEdc/A

Figure 4. Microwave-dressing of symmetric top molecules with circular polarization. (a) The manifolds (J, K) = (2, 2) and (3, 2) display different linear Stark effects. Coupling by a right-circularly polarized microwave of frequency w and detuning A generates dressed state 10) (|I)) in green (purple). (b) A level crossing of| 0) and|I) occurs when the differential dc Stark shift becomes on the order of the detuning (Q/A = 1).

dKMEdc 'J (J + 1)

dKMEdc

(J + 1)(J + 2)

Q JKM = Q<

[ (J + 1)2 — K 2][ (J + 1)2 — M2]

(J + 1)2 (2J + 1) (2J + 3)

with the Rabi frequency Q = dEac. Single-particle eigenstates of equation (11) in the rotating frame will be denoted by an overbar, e.g., |0).

In the perturbative regime where Q, dEdc ^ A, the quasienergies are split into manifolds EjKM;± separated by roughly A, see figure 3 (b). The M dependence of the off-diagonal components Q jkm introduces an effective tensor shift between states of different M which is proportional to Q2, similar to the microwave-induced quadratic Zeeman effect in spinor Bose gases [36,37]. Including the static field Edc can cause two such quasienergy levels with different M to cross as the static field energy dE dc becomes of the order of the effective tensor shift, as shown for the case of the (J, K) = (1, 1) ^ (2, 1) transitionin figure 3(a). The abilityto engineer generic quasienergy level crossings by tuning the static electric field strength is a consequence of the linear Stark effect exhibited by STMs. The Stark effect in 1Z molecules is quadratic, and so shifts all levels with identical J and | M | in the same fashion.

Level crossings can also be engineered outside of the perturbative regime, as well as for arbitrary polarization and rotational quantum number J. As an example, we consider the transition (J, K) = (2, 2) ^ (3, 2) with right-circularly polarized light in figure 4. Panel (b) of figure 4 shows two levels which cross outside of the perturbative regime. Here, the linear Stark energy must overcome not only the effective tensor shift, but also the detuning A of the microwave field from resonance. In what follows, we will denote the parametric relationship of the Rabi frequency Q and the electric field at a quasienergy level crossing as Q (Edc).

In our analysis of the field dressing of STMs we have neglected hyperfine structure. Though the hyperfine structure of STMs is complicated [19], a single hyperfine component maybe selected via a strong magnetic field, similarly to the alkali dimers [38]. Alternatively, working at microwave detuning large compared to the typical hyperfine splittings, A » Ehfs « 10 kHz for CH3F, one can address all hyperfine states equally with a readily achievable microwave power on the order of tens of W cm-2. While we have focused on polyatomic STMs in which all states with a given J and K are degenerate in zero dc field, we expect similar level crossings in other systems with a linear Stark effect but no zero-field degeneracy, such as the Lambda doublet of OH [39], its fermionic analog OD [40], or other species with non-zero projection of orbital angular momentum along the symmetry axis of the molecule | A | > 0. Generally, one can take the detuning A much larger than any fine energy scale which is not to be resolved and simply rescale the static field energy dEdc and the Rabi frequency Q accordingly.

3.3. Dipole-dipole interactions in microwave-dressed states

We now turn to the effective dipole-dipole interactions (equation (3)) in the microwave-dressed states. The components of the dressed states 10) and |T) in the | J + l, K, M) manifold oscillate in time with frequency m. Hence, the dipole moments of the dressed states contain both static and time-oscillating pieces. While the oscillating terms time-average to zero for a single molecule, the dipole-allowed exchange of rotational quanta for two molecules can be resonant due to the two dipoles oscillating in phase [12,41]. In more detail, let us consider the Floquet-picture eigenstates in the presence of a microwave with spherical polarization p = 0, which couples states|JKM) to| (J + l) KM ),see figure 3(a). Consider two such eigenstates

|M) = a \JKM) + be-1mt |(J + l)KM, (l3)

\M') = c \JKM' ) + de—imt |(J + l)KM'>. (l4)

Because we are interested in levels which cross, we assume that M ^ M'. From equation (11), two levels with the same M do not cross except in trivial cases. The matrix elements of the dipole operators are

(M \dp\M = a 2Jp + b 2DJ+MJ;V + ab (MK e—imt + D+MK eimt), (l5)

(m'| dp\M') = c ^ftf* + d 2DjMiM+;?;k + cd (dMj+MK e—imt + DJ eimt), (l6)

(m |dp| M= acDMJ,M';p + bdDMMjiplK + ade~imtDjMMljp + bce^Dj+MI, (l7)

J s (—l)M—kV(j+i)(2j+l) x (—M' p m)(—K 0 K), (l8)

are the dipole matrix elements in pure rotational STM states. Here, we recall that J is the rotational principal quantum number, M is the projection of rotation on a space-fixed quantization axis, and K is the projection of the rotational angular momentum on the symmetry axis of the molecule.

In order to find the effective dipole-dipole interactions, we take the matrix elements of the dipole operators inequations (4)-(6) using the matrix elements of equations (15)-(17) and then perform the long-time average. Here, 'long' time refers to a time which is long compared to the period of the microwave field. The long-time average is justified by the fact that the characteristic timescales of the translational motion of molecules are orders of magnitude longer than the period of the dressing field. The resulting time-averaged interactions for our two example polarization schemes are discussed in appendix, and specific numerical examples of interactions for these two polarizations are given in the next section. The only assumption we use in this work is that the dipole moments of two states near a level crossing only have static components along a single space-fixed spherical direction. Practically, the microwave field can contain either p = ± l components orp = 0 components, but not both. This is equivalent to the statement that terms which transfer only a single molecule between dressed state components are all proportional to sin d cos d, and so vanish in the geometry of figure 2. The requirement of only a single microwave frequency is in contrast to proposals with l£ molecules, which often require precise frequency and polarization control of multiple microwaves [ 17,27-29].

4. Unconventional Hubbard and spin models with STMs

In this section we incorporate the single-particle physics discussed in the previous section into an interacting many-body description in second quantization. Following a translation of the many-body problem to a Hubbard-type lattice model for the lowest lattice band, we then show how limiting cases of this description, for

example when the molecules are pinned to lattice sites, leads to spin models with unconventional magnetic couplings. Our main results are equation (19), the most complete Hubbard-type description of the physics of STMs near a quasienergy level crossing, and equations (21) and (23), which are the reductions to the Heisenberg XYZ and XY models, respectively.

4.1. Second-quantized description of physics near a quasienergy level crossing

In order to derive an effective model for the microwave-dressed STMs trapped in an optical lattice, we use the standard prescription [42] of expanding the field operator in the second quantized representation of the Hamiltonian in a basis ofWannier functions and keep only the terms corresponding to the lowest band of the lattice. Additionally, in what follows we will assume hard-core particles, which can be either bosons or fermions. By hardcore we specifically mean no more than one molecule can simultaneously occupy a given lattice site irrespective ofinternal state considerations. Such a constraint can arise either from a large positive elastic interaction energy, or from rapid inelastic losses via the quantum Zeno effect. The quantum Zeno effect has been shown to enforce a hard-core constraint for KRb, where two-body losses are due to chemical reactions, and gives rise to lifetimes which are long compared to the typical time scales of interactions [12,43]. Because our scheme populates multiple dressed states consisting of different rotational levels, molecules undergo possibly rapid rotationally inelastic processes at short range which will cause a loss of molecules from the trap even if the molecules themselves are chemically stable. The numerical examples given in this work have sufficiently large elastic on-site interactions that we do not need to worry about the nature of short-range inelastic collisions, and we can attribute the hard-core constraint to elastic interactions alone.

For two dressed states a £ {0, I} which are separated from all others by an energy large compared to the characteristic dipole-dipole energy scale, an expansion of the full many-body description in terms of the lowest band Wannier functions reads

H = — £ ta a I a ja + 5 £n ti + 2 £ [ Ehj S+S — + WtjS+S j + h.c.]

(i,j) ,a i i,j,i ^j

+ 1 £ Uj n w j. (19)

2 frd....

a,a ,i,j,i.j

Here, aia destroys a STM in Wannier state wia (r) centered at site i, hiia = aaia, and i>i+ = «0 a.i, S. = (i>i+)t are spin-1/2 operators. In order, the terms in equation (19) are state-dependent tunneling t a of molecules between neighboring lattice sites (i, j)7;a single-particle energy offset 5 of state |1 ) with respect to state|0 ); state-exchanging collisions El. of molecules at sites i and j; state-transferring collisions W,i. which transform two molecules in state 10 ) at sites i and j into the state |1 ) and vice versa; and state-preserving collisions U. between molecules in states a and a ' at lattice sites i and j, respectively. A schematic view of the processes in equation (19) is given in figures 5(a)-(c). Note that equation (19) applies to any 2D lattice geometry.

The Hamiltonian equation (19) bears a strong resemblance to the molecular Hubbard Hamiltonian (MHH) which has been derived for 1Z molecules in optical lattices [45-47], and many of the terms here have the same meaning as in the MHH. In particular, the interaction terms U.a correspond to the direct terms (aa' | HDDI |aa' ), and the interaction terms Ei>j correspond to the exchange terms (aa' | HDDI |a'a), where HDDI is given in equation (1). Expressed in the language of quantum magnetism for a spin-1/2 encoded in the states {|0 ), |i)}, the U terms correspond to SzSz or Ising-type interactions and the E terms correspond to S+S or spin-exchange-type interactions [27]. The new terms here, which have no counterpart in the MHH description of 1Z molecules, are the W,,. terms, which correspond to (aa | HDDI | a' a' ). These terms correspond to S+S+-type interactions in the spin language, and are absent from typical Heisenberg XXZ-type models of quantum magnetism, including those realized with 1Z polar molecules [27].

All of the interaction coefficients U, E, and W maybe tuned by adjusting the static and microwave field dressing strengths, ensuring that the two quasienergy levels remain near resonance. The magnitudes of the Hubbard parameters for the specific level crossings in figures 3 and 4 are displayed in figures 5(d)-(e) as a function of the Edc -dependent Rabi frequency at the level crossing, i2(Edc ), with analytic expressions given in the appendix. For these dressing schemes, the Hubbard parameters U and E are overlaps of the q = 0 component of the dipole-dipole potential, equation (4), in the basis ofWannier functions [48], while the Wterms involve overlaps of the q = ±2 components of the dipole-dipole potential, equation (6). All dipolar parameters U, E,

As has been discussed in previous work considering long-range interacting systems [47], truncation of tunneling to the nearest neighbor distance may not represent a consistent order of approximation when the interactions have infinite range. Instead, one can keep the tunneling between sites to a range rt and the interactions out to a range rU such that all discarded terms in the Hamiltonian have a bounded magnitude.

0.4 % 03

cc Ö o

"XXXMXKKXXX^....

Real space (lattice index i)

nmüin

g O^öBflosafleBaE^.

0.5 1.5 2.5 3.5

Ö(£dc)/A

\w\ m a

□ \E\ * U11 IN 3- 0.3

xU01 ce Ö 0.2

+ U°0 += CJ

<a !H 0.1

1.5 2.5 3.5

tl*(Edc)/A

Figure 5. Interaction processes in the molecules: effective lattice Hamiltonian. The two internal states |0} and| 1} maybe viewed as a discrete spatial degree of freedom, e.g. a ladder. (a) Tunneling rates ta depend on the internal state due to polarizability anisotropy [19,44]. (b) E and W interactions change the internal state of the molecules. E processes preserve the number in each internal state, W processes change it by ±2. (c) U interactions preserve the internal state of the molecules. (d) Nearest-neighbor Hubbard parameters (W = Wij+1 etc) of the many-body model equation (19) with (E^c-dependent) Rabi frequency ¿2 at the level crossing in figure 3. (e) Same as (d) for the level crossing in figure 4. Symbols correspond to processes marked in panels (a)-(c). The unit of interaction energy, d2/a3~ 4 kHz for CH3F in a lattice with a 500 nm lattice spacing.

and W have an approximately 1/li — j|3 decay between lattice sites, and the W terms additionally feature a dependence on the azimuthal angle $. Other dressing schemes, for example those involving both p = ± 1 polarizations, divide this angular dependence between U, E, and W. The order of magnitude of the interactions is given by d2/a3, which is ~4 kHz for CH3F in a lattice with a lattice spacing of500 nm.

The Hamiltonian equation (19) has a U(1) symmetry generated by the total number operator N = N0 + N with Na = 2i n-ia. The W term breaks number conservation within each internal state, but preserves the parities definedby Pa = exp(—inNa ). Due to the U(1) symmetry, the two parities are redundant, both being proportional to P = exp[—y (N0 — NN1)], which is the parity ofthe number difference between internal states. Hence, the internal symmetry of the model equation (19)is U(1) X Z2 [49]. We can interpret the W term as being a hopping ofpairs between two quantum wires or layers, where the wire indices correspond to the dressed states of the molecule, see figure 5. Due to the fact that exchange of rotational quanta only occurs when the dipoles oscillate in phase and the particular geometry, dipolar excitation of a single molecule is forbidden. Single excitation processes which break the Z2 symmetry can be included systematically by other choices of geometry or field polarization, see section 3.3 and the appendix.

4.2. Mapping to a pure spin model

In ultracold gases it is often easier to achieve low temperatures for the internal degrees of freedom even when the motional degrees of freedom remain hot. As an example, a collection of molecules all prepared in the same quantum state has zero effective spin temperature. Provided that the motional temperature of the molecules is lower than the rotational excitation energy (typically on the order of a few 100 mK), the spin and motional temperatures are effectively decoupled, and only the former is important for the dynamics. Hence, a natural first step for many-body physics is to freeze the motional degrees of freedom by loading into a deep optical lattice and consider the dynamics of only the internal degrees of freedom [12]. In the limit in which the quasi-2D confinement is so deep that the tunneling is negligible, equation (19) becomes a long-range and anisotropic spin model

H = 2 [( + Wïj)Sx + ( - Wj¡)/ - Wï(s'sf + siysJx) i,j,i+j

+ (Ui0o + - 2U#))] + £hiSiZ, (20)

where h = 5 + 12 j (Uj - Uj)) "j<x is the effective magnetic field at site i, we have ignored a constant

term, and W^ (Wt^) is the real (imaginary) part of Wij, the S*Sj+ coupling. Again, we would like to stress that equation (20) is defined on any 2D lattice geometry. The Hamiltonian equation (20) does not conserve magnetization due to the non-zero WK and W1 terms easily accessible in our scheme, in contrast to the XXZ models realized with alkali dimer molecules [ 12,27]. In deep optical lattices, where the Wannier functions become well-localized [48], the dipolar coupling constants can be approximated as

uaa ' E e—2i<P i,j Uj «-, E, j «-, Wt, « W-,

ij I • • |3 i,j | • • |3 ' i,j | • -13'

Ii - j \ \i -j \ \i - j \

where ^ j is the angle between the vector connecting sites i and j and the x axis, and U, E, and W are related to geometrical factors and expected dipole moments. With these approximations, we can rewrite equation (20) as

h = 2 ytj [ (e + W cos 2^i,j + (e - W cos )$/

i,j, i+j

+ W sin 2$ij (stxs? + S,ySjX) + (u00 + U11 - 2U] + , (21)

which makes the spatial anisotropy of the model more explicit.

Some simplification of equation (20) occurs in one spatial dimension (1D), which corresponds, e.g., to taking a single row of the 2D square lattice shown in figure 2(b). Such a reduction can be performed in experiment by applying electric field gradients to select a single row of a 2D optical lattice. In a 1D geometry, we can always choose coordinates such that the x axis lies along the lattice direction, and so vanishes and W^ is a monotonically decreasing function of | i - j |. Here, equation (20) reduces to a spin-1/2 XYZHeisenberg model in a longitudinal field

hxyz = 2 [J%$>X$> + + JZjsZs*] + 2ihi$i , (22)

i,j,i^j i

where JX = (E + W)/|i - j|3, ]'Yj = (E - W)/|i - j|3, and J* = (U00 + U11 - 2U01)/|i - j|3. Hence, the degree of spin anisotropy is tunable by changing the ratio between E and W, see figure 5. The phase diagram of the nearest-neighbor version of this model has been investigated recently in [20], displaying Berezinsky-Kosterlitz-Thouless, Ising, first-order, and commensurate-incommensurate phase transitions. Further, considering that the coefficient of S* S * vanishes8, equation (20) becomes a long-ranged version of the XY model in a longitudinal field

hxy = 2 [ Ji*$ix$j + JYiS^sy ] - h* 2$iz. (23)

i,j,i i

The nearest-neighbor XY model is equivalent to the Kitaev wire Hamiltonian [50], which has connections to topological phases and Majorana fermions9. It was also pointed out that long-range interactions may not qualitatively change the nature of topological phases [28]. Finally, we note that in the limit of motionally quenched molecules, equations (20), (22), and (23), the quantum statistics of the underlying molecules are unimportant; one can also realize equation (20) with bosonic or fermionic STMs, in contrast to XYZ models realized by ultracold atoms in the p-band of an optical lattice, where only the bosonic case can be realized [20].

5. Conclusion

We have identified a general mechanism for generating level crossings between internal states with a finite transition dipole matrix element in symmetric top molecules (STMs) by a combination of microwave dressing

8 The coefficient of S^S j can be made to vanish for a continuous range of Rabi frequencies with a single microwave frequency; this does not require fine-tuning to a specific point in parameter space.

The short-range XY model becomes the Kitaev wire Hamiltonian only following a Jordan-Wigner transformation, which is non-local, and

so the topological properties of states in the two models are not generally the same. It is the Z2 fermionic parity symmetry which protects the topological phase in the Kitaev wire. In contrast, the Z2 symmetry in our model is associated with relative number parity (either bosonic or fermionic), and this symmetry can spontaneously break. Hence, topological states in our setup only exist with fine tuning.

and the linear Stark effect. Such a pair of near-degenerate dressed states form an effective spin-1/2. The dipoledipole interaction generates resonant pair transitions between such nearly degenerate levels. By appropriate choices of geometry and field polarization, transfer of a single molecule between internal states can be forbidden, and the resulting many-body system features tunable degrees of spatial and spin-component anisotropy. Using only a single microwave frequency, we show rich tunability of the effective model parameters over a wide range. As special cases of our general many-body description, we show that Heisenberg XYZ and XYspin models arise when molecules are confined to a 1D line in a deep optical lattice. Our results provide a newroute towards the study of unconventional quantum magnetic phenomena by harnessing the rich internal structure of molecules.

Acknowledgments

We acknowledge useful conversations with Christina Kraus and Ryan Mishmash during initial development and exploration of the ideas in this work, and thank Kaden Hazzard and Ana Maria Rey for their comments on the manuscript. This work was supported by the AFOSR under grants FA9550-11-1-0224 and FA9550-13-1-0086, ARO grant number 61841PH, ARO-DARPA-OLE, and the National Science Foundation under grants PHY-1207881, PHY-1067973, PHY-0903457, PHY-1211914, PHY-1125844, andPHY-1125915. We also acknowledge the Golden Energy Computing Organization at the Colorado School of Mines for the use of resources acquired with financial assistance from the National Science Foundation and the National Renewable Energy Laboratories. We thank the KITP for hospitality.

Note added: after this work was submitted, we learned of a related proposal by Glaetzle etal for generating XYZ Heisenberg models using Rydberg atoms (arxiv: 1410.3388). We believe Rydberg atoms offer an exciting alternate scenario for the realization of anisotropic XYZ models, complementary to those described in this paper.

Appendix. Dipole-dipole interactions in microwave-dressed states

Here, we consider the long-time averaged matrix elements of the dipole-dipole interaction equation (3) in the basis of microwave-dressed states, see section 3.3. For two states

\M) = a \JKM) + be—iajt |(J + 1)KM), \M') = c IJKM' > + de~imt | (J + 1)KM'), with M ^ M ' in the polarization scheme of figure 3(a), we find (see equations (15)-(17))

(A.1) (A.2)

(MM |HDDI| MM = {MM'|HDDI| MM' )

(1 — 3 cos2 d)

[ (M M

1 — 3 cos2 d)

[ (r + b JJ+V)2 + 2(abJ0)2], (A.3)

JJ ,J;K

b2DJ+MJ +01;K) (c(KM';0 + d2Djk) (A.4)

(MM'IHDDI| M'M') =

I 1„l„.JnM,M;^ M' ,M' ;0 ]

+ 2abcdDjJM,KDjJ±iK ^ (1 — 3 cos2 9)

DJM;0 + d )2 + 2 (cdDJJ+1K0 )2)

(A. 5) (A.6)

{MM'|HDDI| MM)

3 cos2 9 — 1

2 [ ((M'* + bdDJx)) + (adDMMK) + (b(K)2], (A.7)

{MM IHddiI M'M') = — {e—2i$ [ (MM;1 + bdDf+M}i\.K ) + abcd (^ji)2 ], (A.8)

acDMf;—1 + bdD+Ml—K)) + abcd (Mf1)2]| }. (A.9)

In addition, terms of the form {MM | iiDDI |MM '}, which cause a transition |M} ^ |M '} for one molecule while the other molecule's state is unchanged are also present. All such terms are proportional to sin 9 cos 9 in the present dressing scheme, and so vanish for the 2D geometry of figure 2(b) in which the dc electric field is perpendicular to the plane. Note that this geometry only refers to the orientation of the dc electric field with respect to the plane, and makes no assumptions about the lattice structure in the plane. In equations (A.3)-(A.8) we explicitly show the 9-dependent factors in to provide clarity about the origins of each term. In what follows, we assume 9 = nl2 as in figure 2(b).

To see how these matrix elements can be modified by polarization, let us consider that we have polarization p =1 and consider M' = M + 1, as shown in figure 4. Here, again neglecting terms which vanish in the 2D geometry of figure 2(b) (Q = w/2),wefind

(MM |HDD| mm = R[(C + bDMtl+MT)2 - (Mr1)2)

(MM' \Hdd \ MM'> = ^[ (Dmj'm;0 + b2DM++}'Mi+k'°)M^M+V'0 + dJ2^0)

— abcdDjM Mili-1D}M +1M+2;-1],

d 2Dj++>2 M+2'0)2 — (cdDjMM +1M+2;—1)2 ], ;0)2 — ±("DMjM+1;—1 + bdDM+t} MiK2i—1)

(M' M'\HDD\ M'M') = — 1

^n M + 1,M+1;0 , c DJ J ;K +

(MM'HdD M ' M = -¡3

bcDM + 1,M+1;0\2 1 bcDJ+1,J;K

(MM \Hdd\M'M'> = e-2i^(acDjMiiM +1;-1 + MDjJ+K2-1)2.

(A.10)

(A.11)

(A.12)

Hence, maintaining a single frequency for the microwave field but allowing for different polarizations and intensities realizes extraordinary tunability over the various interaction processes through the coefficients a, b, c, d, the dipole matrix elements involved, and the components of the dipole-dipole interaction equations (4)-(6) which contribute to each process.

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