Abstract
Spin–orbit coupling in solids normally originates from the electron motion in the electric field of the crystal. It is key to understanding a variety of spintransport and topological phenomena, such as Majorana fermions and recently discovered topological insulators. Implementing and controlling spin–orbit coupling is thus highly desirable and could open untapped opportunities for the exploration of unique quantum physics. Here we show that dipole–dipole interactions can produce an effective spin–orbit coupling in twodimensional ultracold polar molecule gases. This spin–orbit coupling generates chiral excitations with a nontrivial Berry phase 2π. These excitations, which we call chirons, resemble lowenergy quasiparticles in bilayer graphene and emerge regardless of the quantum statistics and for arbitrary ratios of kinetic to interaction energies. Chirons manifest themselves in the dynamics of the spin density profile, spin currents and spin coherences, even for molecules pinned in a deep optical lattice and should be observable in current experiments.
Introduction
Polar molecules^{1,2,3,4,5,6} present a flexible platform for the exploration of quantum magnetism in manybody systems due to their strong and longrange dipole–dipole interactions and their rich internal structure of rotational levels. A few isolated rotational levels of a molecule represent an effective spin degree of freedom. Net spin–spin couplings can directly be generated by dipolar interactions even in frozen molecule arrays. Recent experiments^{7,8} with molecules pinned in a deep optical lattice have demonstrated dipolar spinexchange coupling. The anisotropic dipole–dipole interaction can also couple the spin degrees of freedom to the orbital motion of the molecules. Signatures of this type of coupling have been recently reported experimentally in bosonic magnetic atoms^{9,10}, and have been noted for their potential to generate topological phases^{11,12,13,14}. All previously predicted phenomena were limited to zerodimensional systems^{15} or weakly interacting bosonic systems at zero temperature or were tailored to particular experimental setups^{16,17,18,19,20}, requiring, for example, complicated dressing techniques^{12,13}.
Here we demonstrate that an effective spin–orbit coupling (SOC) is inherent in the excitations of any twodimensional (2D) system of polar molecules with a pair of degenerate N=1 rotational levels. These excitations, which we call chirons, are characterized by a nontrivial Berry phase 2π. The same Berry phase is responsible for, for example, an unconventional quantum Hall effect in bilayer graphene^{21,22}. Remarkably, in our system, SOC emerges due to interactions rather than being a singleparticle effect, which adds significant richness to the physics and removes the fundamental limitations imposed by spontaneous emission present when singleparticle SOC is artificially generated by light^{11,23,24,25,26}. We present ways to detect chirality and Berry phase, for instance, by exciting rotational degrees of freedom in a finite spatial region. Generically, this leads to two fronts of spin and density currents, corresponding to the two branches of the chiron spectrum. The Berry phase 2π manifests itself in the dwave symmetry ∝ cos(2φ) of the spin projection onto the plane, where φ is defined in Fig. 1a. In addition, the SOC leads to population transfer between the excited rotational levels together with the formation of a vortex structure in the spatial density profile. We discuss the experimental conditions necessary for the observation of the described phenomena and provide numerical examples germane to current polar molecule experiments in which molecules are pinned in a deep optical lattice with sparse filling.
Results
Manybody Hamiltonian for polar molecules in two dimensions
We consider an ensemble of polar molecules confined in a plane perpendicular to an external electromagnetic field that sets the quantization axis z (Fig. 1a). The rotational spectrum of each molecule can be indexed by the rotational angular momentum N and its projection M onto the z axis (Fig. 1b). Throughout the paper, we set ħ=k_{B}=1 and measure all lengths in units of the lattice constant a, unless specified otherwise.
We assume that most molecules are in the ground rotational state (N=0), and only the lowestenergy states, those with N=0 and N=1, participate in the dynamics. The N=1 states are separated from N=0 by a large gap 2B_{N} (~GHz), which significantly exceeds the characteristic interaction energy E_{d} (~kHz). In addition, the 1, 0› state is separated from the 1, ±1› states by an energy scale E_{1}, E_{d}≪E_{1}≪B_{N}, for example, due to the presence of external electric field E; more details on realising such level structure will be given below. Large lifetimes of the N=1 states (≳10 s) allow us to neglect relaxation between the N=1 and N=0 manifolds.
The Hamiltonian for the system of polar molecules can be written as
where the sums run over all particles in the system and is the singleparticle Hamiltonian. Here, m is the mass of a molecule, U(r) is the external periodic potential and is the Hamiltonian of the internal degrees of freedom giving the spectrum in Fig. 1b. The dipole–dipole interaction between two molecules i and j with dipole moments and and separated by vector R_{ij} (Fig. 1a) is given by
Introducing polar coordinates R_{ij}=(R_{ij}, φ_{ij}), the interaction Hamiltonian (2) can be decomposed as
where are spherical components of the dipole operator of molecule j.
The operator , equation (4), conserves both the total internal angular momentum of the interacting molecule and its zcomponent. As shown in Fig. 1c, exchanges the states 1, ±1› and 0, 0› of two molecules while preserving M_{i}+M_{j}. Such ‘spinexchange’ dipolar interactions have been observed in recent experiments on polar molecules^{7}, Rydberg atoms^{27,28} and magnetic atoms^{29}. In contrast, the operator transfers angular momentum between the internal and external orbital motion of the molecules, while preserving the total projection onto the z axis. Namely, the operator decreases the internal angular momentum by 2, while increases the orbital angular momentum of a molecule by 2, thus preserving the total angular momentum (see, Fig. 1d). As we shall show, this transfer of angular momentum is responsible for the generation of the effective SOC of the elementary excitations.
Phenomenological analysis
Let us assume that almost all the molecules are initially in their lowest rotational level 0, 0› and are in a spatially uniform, not necessarily equilibrium state, but with a relaxation time sufficiently long to be considered stationary. We then suppose that this state is slightly perturbed by a resonant microwave pulse, which excites a small number of molecules from 0, 0› to 1, ±1›. In what follows we show that the density and angular momentum dynamics of the 1, ±1› rotational levels after such excitation is equivalent to that of an ideal gas of spin1/2 chiral quasiparticles (chirons).
The emergence of the excitations can be phenomenologically understood as follows. Due to the translational invariance of the Hamiltonian of the system, the (quasi)momentum k (in the presence of an optical lattice), with polar coordinates (k, φ_{k}) (Fig. 2a) is a good quantum number. In the longwave limit k→0, there is a degeneracy between the excitations carrying molecules in the rotational states ↑›≡1, 1› and ↓›≡1, −1› due to the symmetry with respect to inverting the dipole moment. Note that the other excited rotational states are separated from ↑› and ↓› by large energy gaps and do not participate in the dynamics. This allows us to consider a reduced space {↑›, ↓›} of rotational states. Each of these states can be obtained from the other by acting with the operators . Hence, the most general form of the excitation Hamiltonian in the reduced space reads
where k^{±}=k_{x}±ik_{y}, and α(k) and ξ_{k} are some functions of k.
The Hamiltonian in equation (6) describes quasiparticles with a twobranch spectrum with energies
corresponding to the eigenstates
respectively, and a Berry phase of 2π. In a system with an inversionsymmetric Hamiltonian, ĥ(k)=ĥ(−k), the Berry phase can be defined^{30} modulo 4π as an integral along a contour C connecting two points k and −k in momentum space. Thus, the Berry phase 2π of chirons is nontrivial. As is necessary in a system with timereversal symmetry^{31,32}, Φ_{BP} is a multiple of π.
Let us notice that if α(k)=const, the Hamiltonian (6) coincides with that of the lowenergy excitations in bilayer graphene^{21,22} in the wavelength limit k→0. In the next section, we demonstrate that such Hamiltonian is indeed realised in the case of weak interactions between the molecules.
Microscopic calculation of the Hamiltonian
In the previous section, we have shown phenomenologically that the effective Hamiltonian of longwave excitations in a system of polar molecules has the form given by equation (6). In what immediately follows, we demonstrate that the Hamiltonian can be explicitly evaluated microscopically in the two opposite limits: when the kinetic energy is negligible compared with the characteristic interaction strength and when the interactions are small compared with the kinetic energy. While the first regime can be achieved by pinning the molecules in a deep optical lattice and is thus relevant for current experiments with reactive molecules^{7}, the second regime could be in principle realised in the future with molecules that are nonreactive and less susceptible to überresonant processes^{33}.
To address the first limit, we consider molecules in a deep, unitfilled square optical lattice. In this setting, the translational degrees of freedom are frozen and dynamics occurs only in the internal degrees of freedom. The dynamics can be mapped to that of a gas of bosons with spin σε{↑,↓} and longrange hopping. The vacuum corresponds to all molecules being in the 0, 0› state. The effective Hamiltonian describing the rotational excitations can be expressed in terms of the bosonic creation and annihilation operators of rotational excitations with σ character at lattice site i={i_{x}, i_{y}}, as
Here, the hopping constants J_{0} and J_{2} are determined by dipole matrix elements. We work in the hardcore limit, which restricts the occupation number on each site to 0 or 1. The hardcore constraint encapsulates that there is at most one molecule per lattice site and each molecule can harbour at most one N=1 rotational excitation. Physically, the hardcore constraint can stem either from strong elastic interactions or rapid inelastic loss rates, for example, twobody chemical losses, at short range^{7,34}.
The dispersions of a single rotational excitation are given by
and shown in Fig. 2. Here, F^{(n)}(k)=∑_{j≠0}exp(−i k·r_{j}+inφ_{j})r_{j}^{−3} with r_{j} a vector connecting sites in the square lattice and φ_{j} the polar angle of r_{j}. The phase ϕ_{k} of F^{(2)}, that is, , determines the polar angle of the Bloch vector (Fig. 2a). In the longwave limit, k→0, we obtain in accordance with equation (6) that ϕ_{k}≈2φ_{k}, ξ_{k}/J_{0}≈A+2π/k and α(k)=2πJ_{2}/(3k), with A≈9.03. For general ratios J_{2}/J_{0}, both branches have a conical dispersion for small k. For the case J_{2}=3J_{0}, as results from the geometry in Fig. 1a, there is a cancellation of the linear k component in the E_{+} branch. This leads to a locally flat dispersion E_{+}(k)/J_{0}≈A+(k^{2}) and a conical dispersion E_{−}(k)/J_{0}≈A+4πk+(k^{2}) (Fig. 2b).
In the case of a sufficiently shallow optical lattice or weak interactions, the kinetic energy of the molecules can dominate over the mean interaction energy. In this case, the dynamics can be analysed perturbatively in the interactions. By explicitly evaluating the Hamiltonian of the excitations, the details of which are provided in Supplementary Methods, we reproduce equation (6) with the offdiagonal entry
where the upper and the lower signs apply to bosonic and fermionic molecules, respectively; f_{00}(q) is the distribution function of the molecules in the 0, 0› state. It is assumed to be stationary and independent of the molecule position, but it is not restricted to be in thermal equilibrium; q^{+}=q_{x}+iq_{y}; and ∫_{q}…=∫(2π)^{−2}…d^{2}q. Due to the smallness of the interactions, the diagonal elements of the matrix in equation (6) are close to the kinetic energy of a single molecule and are only slightly modified by the interactions, ξ_{k}≈k^{2}(2m)^{−1} [1+(d^{2})]. This calculation is performed explicitly in Supplementary Methods.
The longtime dynamics is dominated by small momenta. For an isotropic distribution function f_{00}(q)=f_{00}(q), from equation (11) we find the value of the spin–orbital coupling in the limit k→0 to be
For a Fermi liquid of fermionic molecules at zero temperature [f_{00}(q)=θ(k_{F}−q)], equation (12) yields
where k_{F} is the Fermi momentum. At sufficiently high temperatures T≫n/m, where n is the density of the molecules (per nuclear spin), the distribution function is close to that of a Boltzmann gas, , and
For cold atoms in a quadratic trapping potential, the density of the molecules depends on temperature as n(T)∝T^{−1}, resulting in the temperature dependency of the SOC α(T)∝T^{−3/2}.
Chirality manifestations in spin and density dynamics
The chirality of the excitations can be observed in the dynamics of the spin1/2 operator, S={Ŝ_{x}, Ŝ_{y}, Ŝ_{z}}, in the reduced space of the rotational levels ↑› and ↓›. Let us assume that a short laser pulse excites a group of molecules in a small region of characteristic size Λ around r=0, 0, 0›→↑› (the results of this paper can be easily generalized to include more general excitation protocols, 0, 0›→A_{↑}↑›+A_{↓}↓›). The internal state ↑› corresponds to excitation by light with rightcircular polarization x+iy, and has a definite phase winding, as shown in Fig. 1b. Hence, for a spatially isotropic distribution of excitations, the laser polarization is what determines the spatial phase pattern emerging during the dynamics.
For sufficiently small Λ chirons leave the excited region quickly, reaching sufficiently low density, so that interactions between them can be neglected, and their dynamics is described by the kinetic equation for free particles, see Methods. In principle, chiron–chiron interactions may be important for hardcore particles in a deep optical lattice in the beginning of the dynamics, which, however, will not affect the results qualitatively. In the limit of pinned molecules, we additionally check whether chiron–chiron interactions can be neglected by numerically simulating the dynamics of two excitations, see Supplementary Methods and Supplementary Figs 2 and 3.
The chiral nature of the excitations is clearly visible in the density of Ŝ_{x}
where f_{↑↑}(r, q) is the distribution function of the molecules in the r, q› state at time t=0, v_{±}(q) are the velocities of the two chiron branches and φ_{q} is the polar angle of the vector q. In the longwave limit (see equation (7))
Equation (15) describes the spin distribution at sufficiently long times, when it is dominated by longwave chirons (q≪a^{−1}). In this limit, the phase factor cos(2φ_{q}) originates from the offdiagonal element of equation (6). To account for arbitrarymomenta excitations, 2φ_{q} in equation (15) has to be replaced by the polar angle ϕ_{q} of a chiron state on the Bloch sphere (Fig. 2a). For the small Λ under consideration, the distribution has a dwave symmetry at long times (Fig. 3a,b), which is a manifestation of the nontrivial Berry phase 2π of the excitations. The radial distribution of the spin component after applying a narrow laser pulse depends on the molecular statistics, interaction strength, optical lattice depth and so on, while its dwave symmetry is universal, being a consequence of the Berry phase 2π.
The spatial distribution of the spin coherences can be particularly easily understood in the case of a Fermi liquid (fermionic molecules at low temperatures) with weak interactions. In this case, the excitations propagate at the maximal speed υ_{+}(k_{F}), k_{F} being the Fermi momentum. At a distance r away from the initial excitation pulse, the spin distribution remains unaltered until time t=r/υ_{+}(k_{F}) when it is reached by the quickest branch of chirons with the angular distribution of spins . At a slightly later moment of time t=r/υ_{−}(k_{F}), the same point is reached by a wave of slower chirons with opposite spin, after which the spin density remains very small. The resulting distribution is shown in Fig. 3a.
In the case of a deep optical lattice, the branch of chirons with the dispersion E_{−}(q) is significantly faster than the other branch, leading to a quick spatial separation of the two branches (Fig. 3b) after applying the laser pulse. The outer circular density front corresponds to the faster chirons, which propagate at a nearly constant speed, while the more complex inner pattern comes from the slower branch of chirons. Despite different dispersions of the chirons, the angular distribution of the spin is again ∝cos(2φ).
Because of the SOC, spin is not conserved and the total numbers of molecules in the rotational states ↑› and ↓›,
are time dependent. At long times t→∞, both N_{↑}(t) and N_{↓}(t) saturate at a half of the number N_{ex} of the initially excited molecules (Fig. 4c) regardless of the details of the distribution function f_{↑↑}(r, q). The total number of molecules in excited rotational states is conserved, N_{↑}(t)+N_{↓}(t)=N_{ex}.
Thus, the SOC transfers the internal angular momentum of the molecules, all of which are in the ↑› state at t=0, to their orbital motion, leading to the formation of a vortex structure around r=0. This manifests itself, for instance, in a dip in the density of molecules in the ↓› rotational state, , around the centre of the vortex structure (Fig. 4b). In particular, if the chiron spectrum has a branch with quadratic dispersion E(k)~k^{2}(2M)^{−1}, which is realised, for example, for weak interactions in a shallow optical lattice or for strong interactions in a deep optical lattice, the density of the spindown molecules close to the centre of the vortex estimates n_{↓}(r, t)~(M^{2}r^{2}/t^{2})^{2} N_{ex}/Λ^{2} at sufficiently long times. Far from the centre of the vortex, the density profile is described by freely propagating chirons that do not interfere with each other:
The two contributions in the sum in equation (19) correspond to the two branches of chirons propagating with velocities υ_{+}(q) and υ_{−}(q), which leads to their spatial separation.
Experimental accessibility
In this section, we discuss some details related to the observation of dipolar SOC in present cold polar molecule experiments, taking as a representative example the KRb experiment at JILA^{7}. To prevent chemical reactions, KRb polar molecules are pinned in a deep 3D optical lattice. The relevant energy scales for equation (9) are and J_{2}=3J_{0}. Trapping in a deep optical lattice may also be required for molecular species that are chemically stable, as the presence of a very high density of resonances at ultracold energies has been proposed to lead to longlived collision complexes, which are highly susceptible to threebody loss^{33}.
The chirons’ spectra in the entire Brillouin zone can be measured by means of Rabi spectroscopy provided the probe beam can transfer the required quasimomentum k_{R}≃1/a to the molecules. Direct microwave transitions are insufficient since they have k_{R}a≪1, but k_{R}a≃1 can be achieved using optical Raman pulses. Here, k_{R}=k_{1}−k_{2}, with k_{i} the wavevector of the ith Raman beam. Raman transitions between internal states are already a key part of the production of groundstate molecules through STIRAP^{35}; our proposal requires only minor modifications of this wellestablished procedure.
Due to the inherent difficulty of directly cooling molecules, present experiments are not quantum degenerate, leading to a sparse latticefilling fraction near 10%^{7,8,34}. Figure 3c displays the density of Ŝ_{x} in a lattice with 10% filling, averaged over disorder realizations. The dwave symmetry of the distribution is still visible, albeit with reduced contrast compared with the unitfilled case (Fig. 3b). The dwave symmetry is a consequence of the Berry phase, a topological property, and so is robust against disorder. In contrast, disorder smears the vortex structure.
In the rotational structure of KRb, nuclear quadrupole interactions cause the states with predominant 1, −1› and 1, 1› character to be nondegenerate by about 70 hkHz at the 545 G magnetic fields used for magnetoassociation^{36}. These states can be made degenerate and out of resonance with the 1, 0› level by increasing the strength of the magnetic field B, even at zero electric field. In this scenario, the B field determines the quantization axis. For ^{40}K^{87}Rb, where the nuclear quadrupole moments are (eqQ)_{Rb}=−1.380 hMHz and (eqQ)_{K}=0.452 hMHz^{36}, the levels cross near B≃1,260 G, well within experimental feasibility. The energy difference between these levels close to the crossing is nearly linear, with a slope of roughly 40 Hz G^{−1}. Stabilization of magnetic fields at the 10 mG level, which is routine in ultracold gas experiments, would correspond to nondegeneracy on the order of 0.4 Hz, and will not significantly affect our results. Similar comments apply for the other alkali metal dimers. The level structure in Fig. 1b also results for ^{1}Σ molecules without hyperfine structure, for example, bosonic SrO, in the presence of a uniform electric field.
Finally, we note that several knobs can be used to manipulate the Hamiltonian, equation (9). For example, by changing the angle of the quantization axis with respect to the spacefixed z axis, one can tune the ratio J_{2}/J_{0} and remove the cancellation seen in the E_{+} branch. This can be used in turn to control the propagation velocity of the two branches of chirons. Chiron–chiron interactions that fall off as 1/r^{3} can also be controllably introduced by turning on an external static electric field.
Discussion
We have demonstrated that a 2D system of polar molecules behaves as a gas of chiral excitations with a Berry phase 2π. We have shown that signatures of those excitations, which resemble the lowenergy excitations exhibited by bilayer graphene, manifest in both the dynamics of the density and spin coherences.
The implementation of SOC in polar molecules presented here can open other exciting research avenues. In particular, by superimposing an effective magnetic field to the chirons via lightgenerated synthetic gauge fields^{23}, it might be possible to simulate the unconventional quantum Hall effect of bilayer graphene and to see its intriguing consequences^{21} in the lowest Landau levels in the limit of low chiron density and high synthetic magnetic fields. Such an effect would require fermionic statistics of the excitations, which can be realised with fermionic molecules in a shallow optical lattice.
Although so far the system in consideration is an ideal or nearly ideal gas of chirons and interactions between them can be neglected, chiron–chiron interactions tunable by the duration of the laser pulse, the size of the excited region or an external electric field may lead to very rich physics. For example, chiron–chiron interactions together with nonstationary background N=0 molecules or microwave dressing can give rise to interesting dynamic structures, new types of transport phenomena and even to fractional quantum Hall phases when combined with synthetic gauge fields^{13,37}.
Finally, we expect our predictions to be extendable to other dipole–dipole interacting systems such as Rydberg atoms, magnetic atoms and magnetic defects in solids.
Methods
Kinetic equation for polar molecules
To characterize the dynamics of a system of polar molecules, we introduce the nonequilibrium Green’s functions
where σ and σ′ label the internal states of the molecules (σ corresponds to (1, −1›, 0, 0›, 1, 1›)). The upper (lower) sign applies to bosonic (fermionic) particles.
The distribution functions of the molecules are defined as
where the upper and the lower signs apply to bosonic and fermionic particles, respectively, and is the result of the Wigner transformation^{38,39} of .
Using the functions (20) and (21), we define a 2 × 2 matrix in the Keldysh space^{38,39},
Each of the Green’s functions in equation (23) is a matrix in the space of the internal rotational levels of the molecules.
The function (23) satisfies the equation
(Dyson equation minus its conjugate), where ; 1=(t_{1}, r_{1}), 2=(t_{2}, r_{2}) and is the selfenergy part, determined by the dipole–dipole interactions.
In terms of the distribution functions , the kinetic equation reads
where U(r) is the external smooth (trapping) potential, E_{p} is the kinetic (quasi)energy, the summation over repeated indices is implied, and Stf is the collision integral, which accounts for the relaxation of the distribution function due to molecular collisions.
In this paper, we consider a model with a small relaxation rate, which can be neglected on the characteristic times of interest, so that the excitations propagate ballistically. Also, we assume that chirons reach sufficiently low density shortly after they are excited, so that chiron–chiron interactions can be neglected, and the problem becomes effectively single particle.
Introducing the chiron annihilation operator
where are the annihilation operators for the planewave states in the reduced space {↓›, ↑›} of the rotational levels of the molecules and the kinetic equation (25) is reduced, under the assumptions made above, to that for a singleparticle problem with the dispersion E_{±}(k):
In the case U=0, considered in this paper, the most general solution of equation (27) reads
G_{±}(r, p) being arbitrary functions of two arguments.
Additional information
How to cite this article: Syzranov, S. V. et al. Spin–orbital dynamics in a system of polar molecules. Nat. Commun. 5:5391 doi: 10.1038/ncomms6391 (2014).
References
Carr, L. D., Demille, D., Krems, R. V. & Ye, J. Cold and ultracold molecules: science, technology and applications. New J. Phys. 11, 055049 (2009).
Quéméner, G. & Julienne, P. S. Ultracold molecules under control!. Chem. Rev. 112, 4949–5011 (2012).
Barnett, R., Petrov, D., Lukin, M. & Demler, E. Quantum magnetism with multicomponent dipolar molecules in an optical lattice. Phys. Rev. Lett. 96, 190401 (2006).
Gorshkov, A. V. et al. Tunable superfluidity and quantum magnetism with ultracold polar molecules. Phys. Rev. Lett. 107, 115301 (2011).
Gorshkov, A. V. et al. Quantum magnetism with polar alkali dimers. Phys. Rev. A 84, 033619 (2011).
Wall, M. L. & Carr, L. D. Hyperfine molecular Hubbard Hamiltonian. Phys. Rev. A 82, 013611 (2010).
Yan, B. et al. Observation of dipolar spinexchange interactions with latticeconfined polar molecules. Nature 501, 521–525 (2013).
Hazzard, K. R. A. et al. Manybody dynamics of dipolar molecules in an optical lattice, Preprint at http://arxiv.org/abs/1402.2354 (2014).
Fattori, M. et al. Demagnetization cooling of a gas. Nat. Phys. 2, 765–768 (2006).
de Paz, A. et al. Resonant demagnetization of a dipolar BoseEinstein condensate in a threedimensional optical lattice. Phys. Rev. A 87, 051609(R) (2013).
Cooper, N. inManyBody Physics with Ultracold Gases: Lecture Notes of the Les Houches Summer School Vol. 94 (eds Salomon C., Shlyapnikov G. V., Cugliandolo L. F. )189–230Oxford Univ. Press (2013).
Manmana, S. R., Stoudenmire, E. M., Hazzard, K. R. A., Rey, A. M. & Gorshkov, A. V. Topological phases in ultracold polarmolecule quantum magnets. Phys. Rev. B 87, 081106 (2013).
Yao, N. Y. et al. Realizing fractional chern insulators in dipolar spin systems. Phys. Rev. Lett. 110, 185302 (2013).
Peter, D., Griesmaier, A., Pfau, T. & Büchler, H. P. Driving dipolar fermions into the quantum Hall regime by spinflip induced insertion of angular momentum. Phys. Rev. Lett. 110, 145303 (2013).
Pasquiou, B. et al. Spin relaxation and band excitation of a dipolar BoseEinstein condensate in 2D optical lattices. Phys. Rev. Lett. 106, 015301 (2011).
Sun, B. & You, L. Observing the Einsteinde Haas effect with atoms in an optical lattice. Phys. Rev. Lett. 99, 150402 (2007).
Santos, L. & Pfau, T. Spin3 chromium BoseEinstein condensates. Phys. Rev. Lett. 96, 190404 (2006).
Gawryluk, K., Brewczyk, M., Bongs, K. & Gajda, M. Resonant Einsteinde Haas effect in a rubidium condensate. Phys. Rev. Lett. 99, 130401 (2007).
Li, Y. & Wu, C. Spinorbit coupled Fermi liquid theory of ultracold magnetic dipolar fermions. Phys. Rev. B. 85, 205126 (2012).
Kawaguchi, Y., Saito, H. & Ueda, M. Einsteinde Haas effect in dipolar BoseEinstein condensates. Phys. Rev. Lett. 96, 080405 (2006).
McCann, E. & Fal'ko, V. I. Landaulevel degeneracy and quantum Hall effect in a graphite bilayer. Phys. Rev. Lett. 96, 086805 (2006).
Novoselov, K. S. et al. Unconventional quantum Hall effect and Berry's phase of 2π in bilayer graphene. Nat. Phys. 2, 177–180 (2006).
Galitski, V. & Spielman, I. B. Spinorbit coupling in quantum gases. Nature 494, 49–54 (2013).
Dalibard, J., Gerbier, F., Juzeliūnas, G. & Öhberg, P. Colloquium: Artificial gauge potentials for neutral atoms. Rev. Mod. Phys. 83, 1523–1543 (2011).
Miyake, H., Siviloglou, G. A., Kennedy, C. J., Burton, W. C. & Ketterle, W. Realizing the Harper Hamiltonian with laserassisted tunneling in optical lattices. Phys. Rev. Lett. 111, 185302 (2013).
Aidelsburger, M. et al. Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices. Phys. Rev. Lett. 111, 185301 (2013).
Günter, G. et al. Observing the dynamics of dipolemediated energy transport by interactionenhanced imaging. Science 342, 954–956 (2013).
Robicheaux, F., Hernández, J. V., Topçu, T. & Noordam, L. D. Simulation of coherent interactions between Rydberg atoms. Phys. Rev. A 70, 042703 (2004).
de Paz, A. et al. Nonequilibrium quantum magnetism in a dipolar lattice gas. Phys. Rev. Lett. 111, 185305 (2013).
Sun, K., Liu, W. V., Hemmerich, A. & Das Sarma, S. Topological semimetal in a fermionic optical lattice. Nat. Phys. 8, 67–70 (2012).
Blount, E. I. Solid State Physics Vol. 13,305–373Academic Press (1962).
Haldane, F. D. Berry curvature on the Fermi surface: anomalous Hall effect as a topological Fermiliquid property. Phys. Rev. Lett. 93, 206602 (2004).
Mayle, M., Quéméner, G., Ruzic, B. P. & Bohn, J. L. Scattering of ultracold molecules in the highly resonant regime. Phys. Rev. A 87, 012709 (2013).
Zhu, B. et al. Suppressing the loss of ultracold molecules via the continuous quantum Zeno effect. Phys. Rev. Lett. 112, 070404 (2014).
Ni, K.K. et al. A high phasespacedensity gas of polar molecules. Science 322, 231–235 (2008).
Neyenhuis, B. et al. Anisotropic polarizability of ultracold polar ^{40}K^{87}Rb molecules. Phys. Rev. Lett. 109, 230403 (2012).
Cooper, N. R. & Dalibard, J. Reaching fractional quantum Hall states with optical flux lattices. Phys. Rev. Lett. 110, 185301 (2013).
Rammer, J. & Smith, H. Quantum fieldtheoretical methods in transport theory of metals. Rev. Mod. Phys. 58, 323–359 (1986).
Kamenev, A. Field Theory of NonEquilibrium Systems Univ. Press (2011).
Acknowledgements
We appreciate useful discussions with M. Hermele, M. Lukin, N. Yao, K.R.A. Hazzard, T. Pfau and the KRb JILA experimental group. This work has been financially supported by NIST, JILANSF PFC1125844, NSFPIF1211914, NSFPHY1125915, ARO, ARODARPAOLE, AFOSR, AFOSRMURI and the NSF grants DMR1001240 and PHY1125844. M.L.W. thanks the NRC postdoctoral fellowship program for support. S.V.S. has been also partially supported by the Alexander von Humboldt Foundation through the Feodor Lynen Research Fellowship. A.M.R. and V.G. thank the Aspen Center for Physics and KITP.
Author information
Authors and Affiliations
Contributions
All authors contributed significantly to the work presented in this paper.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Supplementary information
Supplementary Information
Supplementary Figures 13, Supplementary Methods and Supplementary Reference (PDF 961 kb)
Rights and permissions
About this article
Cite this article
Syzranov, S., Wall, M., Gurarie, V. et al. Spin–orbital dynamics in a system of polar molecules. Nat Commun 5, 5391 (2014). https://doi.org/10.1038/ncomms6391
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/ncomms6391
This article is cited by

Tunable itinerant spin dynamics with polar molecules
Nature (2023)

Intrinsic topological magnons in arrays of magnetic dipoles
Scientific Reports (2022)

New frontiers for quantum gases of polar molecules
Nature Physics (2017)

Doublon dynamics and polar molecule production in an optical lattice
Nature Communications (2016)

Emergent Weyl excitations in systems of polar particles
Nature Communications (2016)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.