Localized magnetic moments with tunable spin exchange in a gas of ultracold fermions
Abstract
We report on the experimental realization of a state-dependent lattice for a two-orbital fermionic quantum gas with strong interorbital spin exchange. In our state-dependent lattice, the ground and metastable excited electronic states of Yb take the roles of itinerant and localized magnetic moments, respectively, with a spin-exchange interaction analog to that in the well-known Kondo Hamiltonian. The exchange interaction arises from the difference in the on-site interaction of the nuclear spin singlet- and triplet configurations. In addition, we find that the resulting spin coupling can be tuned resonantly by varying the on-site confinement. We attribute this to a resonant coupling to center-of-mass excited bound states of the spin-singlet scattering channel.
pacs:
34.50.Cx, 37.10.Jk, 67.85.Lm, 75.20.HrIn many materials, such as transition metal oxides, electrons exhibit an orbital degree of freedom in addition to their spin Tokura and Nagaosa (2000). In these systems, electrons in different orbitals give rise to localized and mobile magnetic moments. Interorbital coupling of theses moments via spin-exchanging interactions leads to the appearance of the Kondo effect, heavy fermion physics or colossal magnetoresistance Hewson (1993); van den Brink et al. (2004); Coleman (2007); Löhneysen et al. (2007).
While ultracold fermionic atoms have been successfully used to realize a large variety of many-body phenomena Bloch et al. (2012), two-orbital models such as the Anderson or Kondo models Hewson (1993) have so far not been accessible. Alkaline-earth-like atoms (AEA), such as ytterbium and strontium, have been proposed to offer very direct and robust implementations of such systems Gorshkov et al. (2010); Foss-Feig et al. (2010a, b), compared to more complex approaches intended for alkali atoms Paredes et al. (2005); Nishida (2013); Bauer et al. (2013, 2015); Nishida (2016). The two orbitals can be represented by AEA in the two lowest-lying states of their electronic spin singlet and triplet manifold, (denoted ) and (). As the -state connects to only via an ultra-narrow clock transition, it is stable on typical experimental time scales Porsev et al. (2004).
Utilizing the different AC-polarizabilities of the two states Dzuba and Derevianko (2010) allows for the creation of state-dependent optical lattices (SDL) Daley et al. (2008); Gerbier and Dalibard (2010). In contrast to implementations with alkaline atoms using near-resonant light Mandel et al. (2003); Karski et al. (2009); Gadway et al. (2010) or field gradient modulation Jotzu et al. (2015), a SDL for AEA can be operated at large detuning and low scattering, and preserves the nuclear spin degree of freedom. The latter arises because the nuclear spin is strongly decoupled from the electronic states and Scazza et al. (2014); Zhang et al. (2014). For isotopes with nuclear spin , the strong decoupling even allows for generalization to SU()-symmetric forms of spin models Honerkamp and Hofstetter (2004); Hermele et al. (2009); Corboz et al. (2011); Cazalilla and Rey (2014).
Fermionic Yb stands out for the investigation of orbital magnetism, due to its strong interorbital spin-exchange coupling. This is a consequence of the unusually large difference in the scattering lengths and for an -pair in the nuclear spin singlet or triplet configuration Höfer et al. (2015). In the presence of a lattice potential, an on-site spin-exchanging interaction emerges in the corresponding single-band Hubbard model Gorshkov et al. (2010). For Yb, this fundamental exchange coupling was experimentally determined to be ferromagnetic and large (), both in free space and in deep state-independent potential wells Scazza et al. (2014); Cappellini et al. (2014).
In the present work, we implement a two-orbital system with state-dependent mobility, creating mobile and localized magnetic moments. Furthermore, we show that the spin-exchange coupling can be tuned via the external confinement.
In our experiment, the atoms are confined in an array of independent quasi-1D traps, created by two perpendicular state-independent lattices operated at the magic wavelength nm Barber et al. (2008) (see Fig. 1(a)). Along the remaining, longitudinal direction, we superimpose a state-dependent optical lattice. At our chosen wavelength of nm, the AC-polarizability of -atoms is times larger than for the -atoms SM . We operate the SDL at intermediate depths, where both states are in the tight-binding regime. However, the -atoms are effectively localized on experimentally relevant timescales while the -atoms retain their mobility. The trapping potential is independent of the nuclear spin component or ().
In analogy to the spin exchange in the well-known Kondo model, we consider a spin-exchanging collision, where a mobile -particle is scattered off a localized -state moment. In the tight-binding picture of the state-dependent Hubbard model, for two particles initially on separate sites, this requires tunneling of the -atom to the -atom and back, as depicted in Fig. 1(b). In this super-exchange-like process, the energy of the interacting intermediate state depends on the spin configuration of the two atoms. Neglecting bound states of the atom pair, the minimum on-site interaction energy is that of the nuclear spin triplet , whereas the energy of the nuclear spin singlet is much larger (see Fig. 1(c)). With both energies large compared to the hopping rate , the dominant spin-triplet state leads to an effective ferromagnetic spin-exchange coupling .
To probe this spin-exchange process experimentally, we prepare singly occupied lattice sites in a mixture of and -atoms. For this, a two-spin degenerate Fermi gas with is loaded into a deep 3D lattice ( and ) at a low atom density, such that double occupancy of sites is substantially suppressed. Here, the longitudinal (perpendicular) lattice depth is given in units of the longitudinal (perpendicular) lattice recoil energy () for -atoms. Applying a magnetic bias field of G allows us to then selectively transfer the atoms in state to with a fast -pulse on the clock transition.
In order to initiate the spin-exchange dynamics, the perpendicular magic-wavelength lattice and the longitudinal SDL are ramped to the values of interest, and . Simultaneously, the magnetic field is quenched to a small bias value of G. During a variable hold time , the -atoms are free to move longitudinally and interact with stationary -atoms. Finally, we image both excited-state spin components separately to measure the evolution of the spin populations SM .
Fig. 2(a) shows the resulting time dependence of the spin-exchange dynamics in a SDL of intermediate depth ( with tunneling rate for and for ). In the course of several tunneling times for the -atoms, the -component is repopulated through spin exchange and equilibrates with the -fraction. In Fig. 2(a), the black curve is attributed to the triplet-state dominated spin-exchange model described above, and in this particular case we observe a repopulation on a time scale .
In contrast to the behavior expected from this model, we find that the spin-exchange rate can be strongly modified when choosing certain values of the transverse confinement (blue curves in Fig. 2(a)) This is illustrated in Fig. 2(b), where a rate is determined with a linear fit to the initial dynamics of the -fraction. This exchange rate is resonantly enhanced with a maximum around and displays an asymmetric resonance profile. Fig. 2(a) shows that, together with the enhanced spin-exchange rate, also the loss rate of both exited state species is amplified (see SM for more details).
To probe the structure of this spin-exchange resonance, we map out the resonance position dependence on both perpendicular as well as longitudinal confinement. In Fig. 3(a), we show the repopulated -fraction after a short hold time . This scaling of is motivated by the super-exchange model for the spin-exchange process. Varying both confinement strengths, we identify two resonance branches in the experimentally accessible parameter regime. A cut along (see Fig. 3(b)) shows similar properties of the two observed branches.
A spin-exchange coupling which depends on the perpendicular confinement due to confinement-induced resonances (CIR) in a quasi-1D system has been proposed in Zhang et al. (2016). For the achievable lateral confinements in our system, we can rule out the observation of a 1D CIR. However, both available scattering channels are repulsive and therefore feature bound states of the relative motion (see Fig. 4 (a)). Together with the strong longitudinal confinement dependence of the exchange, this suggests a confinement-induced resonance effect in the combined transverse and longitudinal potential. One theory for this type of effect has been described in Cheng et al. , using a setting in which a harmonically localized impurity interacts with atoms in a quasi-1D potential.
Here, we concentrate on the tight-binding limit where both orbital states exhibit strong (but different) on-site confinements. In this picture, the very small binding energy of the spin-singlet bound state is in the range of a few longitudinal center-of-mass (c.m.) excitation energies. Therefore the lowest c.m. excited bound states can come into resonance with two non-interacting particles (see Fig. 1(c)). When this happens at certain combinations of transverse and longitudinal confinement, the spin singlet bound state becomes available as an additional intermediate state for the spin-exchange process. As the energy of this intermediate state depends on the tunable optical potentials, the overall process becomes controllable in strength and should even allow for an antiferromagnetic exchange coupling.
The qualitative behavior of the spin-exchange resonances can be explained by means of a simplified two-particle model for the strong interorbital interactions in the presence of a state-dependent confinement. In the limit of deep lattices, we treat the on-site physics using harmonic oscillator potentials. While the radial trapping frequency is the same for the two particles, the longitudinal frequencies and along the SDL depend on the orbital .
This setting of anisotropic and mixed confinement for an -pair can be approximately mapped to a sum of effective harmonic oscillator potentials in c.m. and relative (rel.) coordinates, and :
(1) | |||||
(2) |
Moreover, there is a coupling term between rel. and c.m. coordinate, with caused by the mixed confinement Deuretzbacher et al. (2008). Our polarizability ratio of for the SDL causes a considerable coupling of . Here, all energies are given in units of an effective longitudinal trapping frequency with , and all lengths are in units of the corresponding harmonic oscillator lengths. The ratio characterizes the anisotropy of an individual lattice site and is in the range . Interactions are modeled by a regularized contact potential in the rel. coordinate where the scattering length depends on the nuclear spin configuration, singlet or triplet , of the -pair. For more details on the model parameters, see SM .
In a first approximation, we set and neglect the trap anharmonicity, such that rel. and c.m. coordinate of the harmonic oscillator decouple. The problem can then be solved exactly for arbitrary Idziaszek and Calarco (2005). Fig. 4 depicts the relevant parts of the resulting spectrum, the interacting states in the rel. coordinate Hamiltonian (solid lines) as well as the lowest c.m. excitations (dashed lines) of the spin-singlet bound state. All energies are shown relative to the zero-point energy of the non-interacting harmonic oscillator which we use as the entrance energy of the scattering process starting with two particles on separate sites. The spin-singlet bound state is loosely bound and the lowest c.m. excitations can be resonant with , as shown in Fig. 4 (a). For a fixed longitudinal confinement , several resonances occur as a function of , varied by modifying the transverse confinement . Overlaying the zero-crossing positions (gray lines) with our experimental data in Fig. 3, we find that the resonances caused by the longitudinal c.m. excitations roughly match the functional dependence of the observed spin-exchange resonances. The validity of the harmonic oscillator model is improved with the SDL depth and for lower c.m. excitations. Indeed, for the branch with two c.m. excitations, also the position of the resonance is approximately reproduced.
An effect of the resonant singlet channel on the spin-exchange coupling requires a finite coupling for the process of a -atom tunneling to a neighboring site and forming a c.m.-excited bound state with an -atom. This coupling should be enhanced through the on-site c.m.-rel. motion coupling.
The total spin-exchange coupling for a given confinement results from the combined effect of all available exchange channels. In particular, for configurations in which the bound singlet channel provides the largest contribution to the exchange, the resulting effective can even switch sign between ferro- and antiferromagnetic exchange coupling.
To investigate the influence of the off-site bound state coupling, we consider a double-well model with two particles that incorporates the on-site interactions discussed above, as well as the mobility of the -atoms SM . Fig. 4(b) shows the expectation value of finding an -atom after letting an initial state on the two sites () of the double well coherently evolve for a time . If the c.m.-excited bound state is neglected (, solid line), tuning of the perpendicular confinement induces only weak variations in the dynamics, which are then dominated by the spin triplet intermediate state. In contrast, for finite couplings , a resonance in the spin exchange occurs, caused by the intermediate state . The width of the resonance becomes comparable to the one observed experimentally in Fig. 2 for on the order of . Also, the asymmetry with a steep flank on the side of low is reproduced. In the model, it is caused by the competition of ferromagnetic and antiferromagnetic coupling, leading to a zero crossing of the effective spin-exchange coupling, with an antiferromagnetic sign of the effective coupling term between the resonance and the minimum position in Fig. 4(b).
Despite the qualitative agreement, the two-particle model is not suited to fully describe the dynamics in our experiment for configurations close to the scattering resonances. Firstly, the description does not include the loss of atoms. The resonant structure of the loss feature implies that it is related to the coupling to c.m. excited singlet -pairs, which could de-excite. However, we observe that relative to the -population loss, -loss is significantly more pronounced than expected for an --symmetric process SM . Clearly, the loss feature is detrimental to the implementation of many-body physics very close to resonance, and an exact understanding of the process could help to minimize the effect.
Secondly, the nature of the coupling between c.m. and rel. motion is treated only on a simplified level. In particular, a more complete model of the coupling should lead to avoided crossings between the eigenenergy branches, affecting the resulting resonance positions. A precise numerical treatment of the optical lattice Hamiltonian, similar to Büchler (2010), would be required to determine more exact values for the effective Hubbard parameters.
In conclusion, we have implemented a two-orbital lattice system with both mobile and localized orbital. Interorbital spin exchange between the mobile and localized moments is observed in this configuration. The orbital lattice implementation is general for all AEA systems for an according choice of lattice wavelength. With appropriate filling factors of localized spins, it can be used for the realization of both Kondo- and Kondo lattice-type models Gorshkov et al. (2010). The spin-exchange coupling can be widely tuned in the vicinity of spin-singlet bound state resonances, in particular with the large singlet scattering length of Yb. Our novel tuning mechanism, as well as previously proposed schemes Gorshkov et al. (2010); Nakagawa and Kawakami (2015), rely on optical potentials for rapidly modifying the exchange coupling. This is advantageous for the investigation of non-equilibrium spin dynamics Hackl et al. (2009); Nuss et al. (2015). The SDL and the tuning implementation work independently of the nuclear spin, and both are expected to preserve the SU() symmetry of the interactions. Using more than two nuclear spin components of in the SDL could directly enable the realization of the Coqblin-Schrieffer model Coqblin and Schrieffer (1969); Hewson (1993); Gorshkov et al. (2010); Kuzmenko et al. (2016).
Acknowledgements.
We acknowledge the valuable and helpful discussions with Hui Zhai, Eugene Demler, Marton Kanász-Nagy, Peng Zhang, Meera M. Parish, Jesper Levinsen, Jan von Delft, Seung-Sup Lee and Frank Deuretzbacher. This work was supported by the ERC through the synergy grant UQUAM and by the European Union’s Horizon 2020 funding.References
- Tokura and Nagaosa (2000) Y. Tokura and N. Nagaosa, Science 288, 462 (2000).
- Hewson (1993) A. C. Hewson, The Kondo Problem to Heavy Fermions, Cambridge Studies in Magnetism (Cambridge University Press, 1993).
- van den Brink et al. (2004) J. van den Brink, G. Khaliullin, and D. Khomskii, “Orbital effects in manganites,” in Colossal Magnetoresistive Manganites (Springer Netherlands, Dordrecht, 2004) pp. 263–301.
- Coleman (2007) P. Coleman, “Heavy fermions: electrons at the edge of magnetism,” in Handbook of Magnetism and Advanced Magnetic Materials, Vol. 1 (Wiley, New York, 2007) pp. 95–148.
- Löhneysen et al. (2007) H. v. Löhneysen, A. Rosch, M. Vojta, and P. Wölfle, Rev. Mod. Phys. 79, 1015 (2007).
- Bloch et al. (2012) I. Bloch, J. Dalibard, and S. Nascimbène, Nat. Phys. 8, 267 (2012).
- Gorshkov et al. (2010) A. V. Gorshkov, M. Hermele, V. Gurarie, C. Xu, P. S. Julienne, J. Ye, P. Zoller, E. Demler, M. D. Lukin, and A. M. Rey, Nat. Phys. 6, 289 (2010).
- Foss-Feig et al. (2010a) M. Foss-Feig, M. Hermele, and A. M. Rey, Phys. Rev. A 81, 051603 (2010a).
- Foss-Feig et al. (2010b) M. Foss-Feig, M. Hermele, V. Gurarie, and A. M. Rey, Phys. Rev. A 82, 053624 (2010b).
- Paredes et al. (2005) B. Paredes, C. Tejedor, and J. I. Cirac, Phys. Rev. A 71, 063608 (2005).
- Nishida (2013) Y. Nishida, Phys. Rev. Lett. 111, 135301 (2013).
- Bauer et al. (2013) J. Bauer, C. Salomon, and E. Demler, Phys. Rev. Lett. 111, 215304 (2013).
- Bauer et al. (2015) J. Bauer, E. Demler, and C. Salomon, J. Phys.: Conf. Ser. 592, 012151 (2015).
- Nishida (2016) Y. Nishida, Phys. Rev. A 93, 011606 (2016).
- Porsev et al. (2004) S. G. Porsev, A. Derevianko, and E. N. Fortson, Phys. Rev. A 69, 021403 (2004).
- Dzuba and Derevianko (2010) V. A. Dzuba and A. Derevianko, J. Phys. B 43, 074011 (2010).
- Daley et al. (2008) A. J. Daley, M. M. Boyd, J. Ye, and P. Zoller, Phys. Rev. Lett. 101, 170504 (2008).
- Gerbier and Dalibard (2010) F. Gerbier and J. Dalibard, New J. Phys. 12, 033007 (2010).
- Mandel et al. (2003) O. Mandel, M. Greiner, A. Widera, T. Rom, T. W. Hänsch, and I. Bloch, Phys. Rev. Lett. 91, 010407 (2003).
- Karski et al. (2009) M. Karski, L. Förster, J.-M. Choi, A. Steffen, W. Alt, D. Meschede, and A. Widera, Science 325, 174 (2009).
- Gadway et al. (2010) B. Gadway, D. Pertot, R. Reimann, and D. Schneble, Phys. Rev. Lett. 105, 045303 (2010).
- Jotzu et al. (2015) G. Jotzu, M. Messer, F. Görg, D. Greif, R. Desbuquois, and T. Esslinger, Phys. Rev. Lett. 115, 073002 (2015).
- Scazza et al. (2014) F. Scazza, C. Hofrichter, M. Höfer, P. C. de Groot, I. Bloch, and S. Fölling, Nat. Phys. 10, 779 (2014).
- Zhang et al. (2014) X. Zhang, M. Bishof, S. L. Bromley, C. V. Kraus, M. S. Safronova, P. Zoller, A. M. Rey, and J. Ye, Science 345, 1467 (2014).
- Honerkamp and Hofstetter (2004) C. Honerkamp and W. Hofstetter, Phys. Rev. Lett. 92, 170403 (2004).
- Hermele et al. (2009) M. Hermele, V. Gurarie, and A. M. Rey, Phys. Rev. Lett. 103, 135301 (2009).
- Corboz et al. (2011) P. Corboz, A. M. Läuchli, K. Penc, M. Troyer, and F. Mila, Phys. Rev. Lett. 107, 215301 (2011).
- Cazalilla and Rey (2014) M. A. Cazalilla and A. M. Rey, Rep. Prog. Phys. 77, 124401 (2014).
- Höfer et al. (2015) M. Höfer, L. Riegger, F. Scazza, C. Hofrichter, D. R. Fernandes, M. M. Parish, J. Levinsen, I. Bloch, and S. Fölling, Phys. Rev. Lett. 115, 265302 (2015).
- Cappellini et al. (2014) G. Cappellini, M. Mancini, G. Pagano, P. Lombardi, L. Livi, M. Siciliani de Cumis, P. Cancio, M. Pizzocaro, D. Calonico, F. Levi, C. Sias, J. Catani, M. Inguscio, and L. Fallani, Phys. Rev. Lett. 113, 120402 (2014).
- Barber et al. (2008) Z. W. Barber, J. E. Stalnaker, N. D. Lemke, N. Poli, C. W. Oates, T. M. Fortier, S. A. Diddams, L. Hollberg, C. W. Hoyt, A. V. Taichenachev, and V. I. Yudin, Phys. Rev. Lett. 100, 103002 (2008).
- (32) “See supplemental material [url] for additional information about the experimental sequence and a detailed discussion of the theoretical models.” .
- Zhang et al. (2016) R. Zhang, D. Zhang, Y. Cheng, W. Chen, P. Zhang, and H. Zhai, Phys. Rev. A 93, 043601 (2016).
- (34) Y. Cheng, R. Zhang, P. Zhang, and H. Zhai, arXiv:1705.06878 .
- Deuretzbacher et al. (2008) F. Deuretzbacher, K. Plassmeier, D. Pfannkuche, F. Werner, C. Ospelkaus, S. Ospelkaus, K. Sengstock, and K. Bongs, Phys. Rev. A 77, 032726 (2008).
- Idziaszek and Calarco (2005) Z. Idziaszek and T. Calarco, Phys. Rev. A 71, 050701 (2005).
- Büchler (2010) H. P. Büchler, Phys. Rev. Lett. 104, 090402 (2010).
- Nakagawa and Kawakami (2015) M. Nakagawa and N. Kawakami, Phys. Rev. Lett. 115, 165303 (2015).
- Hackl et al. (2009) A. Hackl, D. Roosen, S. Kehrein, and W. Hofstetter, Phys. Rev. Lett. 102, 196601 (2009).
- Nuss et al. (2015) M. Nuss, M. Ganahl, E. Arrigoni, W. von der Linden, and H. G. Evertz, Phys. Rev. B 91, 085127 (2015).
- Coqblin and Schrieffer (1969) B. Coqblin and J. R. Schrieffer, Phys. Rev. 185, 847 (1969).
- Kuzmenko et al. (2016) I. Kuzmenko, T. Kuzmenko, Y. Avishai, and G.-B. Jo, Phys. Rev. B 93, 115143 (2016).
Supplemental Material
s.i Experiment sequence details
The ratio of the AC-polarizabilities of - and -atoms in the state-dependent lattice (SDL), and thereby of the lattice potential depths, has been determined by means of independent parametric heating of both species in deep lattice systems. At a wavelength of 670.0 nm, we obtain , in agreement with the theoretical prediction in Dzuba and Derevianko (2010). For the lattice depth range of used in the experiment, this results in a bandwidth range of (1.2 … 0.3) and (0.22 … 0.01) for and , respectively.
One of the two perpendicular, magic-wavelength lattice beams ensures Lamb-Dicke conditions for the coaligned 578.4 nm excitation beam for the clock transition.
We expect and have verified experimentally that, in the limit of strong perpendicular confinement, the position of the spin-exchange resonances only depends on the geometric mean of the individual perpendicular lattice depths and . Thus, the perpendicular lattice depth provided in the main text is obtained through . The measurements have been performed with a tiny anisotropy of 1.7% in the perpendicular trapping frequencies. This is due to a small mismatch in the experimental calibration of the perpendicular lattice depths and , determined in an independent measurement.
The initial, balanced mixture of - and -atoms is prepared by evaporative cooling in a crossed optical dipole trap and subsequent optical pumping into the nuclear spin components . In order to prepare singly occupied lattice sites with either or -atoms, we first load the two-spin -mixture with relatively low atom numbers of per spin state into a 3D lattice ( and ). We estimate an average filling of around 21 atoms per tube. The repulsively interacting fermions in the lattice suppress double occupancies. We verify this experimentally via clock-line spectroscopy on the initial state, showing no peaks attributable to interacting states.
A schematic overview of the experimental sequence driving the spin-exchange dynamics is shown in Fig. S1. The transfer to is done by a -pulse on the clock transition. A magnetic field of G provides enough Zeeman splitting to address the two spin states separately. It also suppresses undesired spin-exchange dynamics during the state preparation. Also, the mobility of the -atoms is reduced by the deep SDL, with a hopping rate of Hz. The -pulse is performed at high intensities (at Rabi frequencies of kHz) to compensate for spatially varying light shifts of the clock transition due to the harmonic confinement of the SDL beam.
To initiate the spin-exchange dynamics, the magnetic field is quenched to a small bias value of G immediately after the excitation pulse. The field is high enough to preserve the nuclear spin quantization axis while at the same time inducing only small Zeeman energies ( Hz) compared to the on-site interaction energies (e.g., kHz for and ). Simultaneously, the magic-wavelength lattice and the SDL are ramped, within 1 ms, to the values and , defining the confinement during a variable spin-exchange hold time .
Before separately imaging both excited-state spin components, we image both ground-state spin components by means of a high-intensity imaging pulse. This allows us to count the -atoms and also sets a well-defined end point for the spin-exchange dynamics. We also ramp the lattice depth up again (1 ms ramp time) in order to suppress further tunneling dynamics during the imaging of -atoms. To image the -atoms, we first ramp the magnetic field back up to 20 G (10 ms ramp time), spin-selectively deexcite to the ground-state with a -pulse and quench the field back to 1 G. By imaging the deexcited -atoms they are also removed from the lattice. This procedure is repeated analogously for the -atoms. Overall, we are able to count the , and atom numbers in a single experimental realization.
s.ii Lifetime in the SDL
The enhanced spin-exchange rate on the resonance branches observed in the main text is accompanied by amplified atom losses from the trap. A linear fit to the short-time dynamics, up to one tunneling time , yields a maximum initial loss rate of 52 Hz for the -atoms and 6 Hz for the -atoms. This indicates that the loss is not entirely caused by an --symmetric process.
In an independent measurement, the pure -atom losses are quantified. Therefore, a balanced --mixture is prepared at densities comparable to the spin-exchange measurements in the main text. The -atoms are held in the same lattice configuration as the -mixture in Fig. 2(a) (black curve) of the main text (, ). After a time of , only 6% of the -atoms are lost, compared to 35% in the -mixture in Fig. 2(a).
On longer timescales, the -lifetime in our experiment is limited by vacuum losses and repumping to by the magic-wavelength as well as state-dependent lattice light. By holding a spin-polarized sample in a 3D isotropic magic-wavelength lattice (30 depth), we measure a loss rate of 28 mHz induced by vacuum background collisions and an optical repumping rate of 175 mHz to the ground state, induced by the magic-wavelength light.
The longitudinal SDL at a lattice depth of adds an additional repumping rate of 147 mHz for the -atoms. This rate was obtained by fitting a rate equation that includes all single-particle loss processes to the and atom counts (see Fig. S2). By using 45 deep perpendicular magic-wavelength lattices in addition to the SDL, the total magic-wavelength light power was maintained compared to the previous measurement. The overall lifetime of a spin-polarized -sample is 2.8 s for the lattice configuration specified above, exceeding the experimentally relevant time scales in the main text.
s.iii Two particles in mixed anisotropic confinement
In order to estimate the on-site energies in the state-dependent lattice, we treat the lattice sites as independent harmonic oscillators. We consider one -atom and one -atom trapped in a mixed confinement, interacting via the regularized contact potential . While the radial trapping frequency (along the magic) is the same for the two particles, the longitudinal frequencies and (along the SDL) depend on the electronic orbital. The total Hamiltonian in cylindrical coordinates () is
(S.1) |
Let us now use center-of-mass (c.m.) and relative (rel.) coordinates and . One obtains , , and . The Hamiltonian becomes
with an effective longitudinal trapping frequency . The harmonic oscillators in the rel. and c.m. coordinate are coupled by a mixing term proportional to . In the main text, we use the energy unit , relative coordinate length unit (with the reduced mass) and c.m. coordinate length unit (with the combined mass for the c.m.):
where is the anisotropy parameter of the on-site confinement.
For the prediction of the spin-exchange resonance locations in Fig. 3 and Fig. 4 of the main text, we neglect all coupling terms between c.m. and rel. coordinates. Then, the c.m. problem is simply a non-interacting 3D harmonic oscillator with eigenenergies . Here, and are the number of longitudinal and perpendicular band excitations. Concerning the relative motion, the problem can be solved exactly, as in Idziaszek and Calarco (2005). States of the 3D harmonic oscillator with angular momentum do not see the contact interaction potential. For states with , the energy of the relative motion part is , with
The above integral only converges for . Yet, the following recursive formula can be used for negative values:
To fully treat the on-site interactions, one would need to take into account corrections from the anharmonic part of the lattice potential and the coupling term from the mixed confinement.
s.iv Model for spin-exchange tunability
As a model system for the two-particle spin-exchange dynamics, we consider two particles on two lattice sites (sites in a double well) of the state-dependent lattice. One particle is in the excited state , the other one is in the ground state . They can have different spins or . The -particle is localized on one of the two sites (site ). The -particle can hop between the two sites with tunnel coupling . Initially, the two particles are prepared on different lattice sites, with a spatial wave function . A spin-exchange process can then only be mediated by tunneling of -atoms. In contrast, the on-site spin-exchange coupling for doubly occupied sites is given as Scazza et al. (2014); Cappellini et al. (2014).
Considering only the typical lowest-band Hubbard parameters, the interorbital interaction strengths and for the spin-triplet and spin-singlet interaction channels are both positive and large compared to the hopping rate . For our experiment, this is true everywhere in the tight-binding regime for both interaction channels. In the sampled range of , we obtain at . Spin exchange then happens only via a virtual process and the effective coupling scales as . Mainly, the lower-energy spin triplet contributes to the exchange process, leading to a ferromagnetic coupling
(S.2) |
in the low-energy limit.
As explained in the main text and Sec. S.III, we expect a coupling of c.m. and rel. motion that can lead to two particles on separate sites coupling into a c.m.-exited on-site bound state. The on-site c.m. excitation is then labeled by . The tunnel coupling from neighboring sites into that state is defined as .
If two particles tunnel onto the same site () and go into an on-site wave function without c.m. excitation, the possible interaction channels are and , depending on whether they are in the spin triplet or spin singlet .
If two particles tunnel onto the same site while acquiring a band excitation into , they will dominantly interact via a c.m.-excited, spin singlet/triplet bound state close to the entrance energy, as described in he main text. The total on-site energy, considering also the c.m. excitation, is then for the states . The interaction energies and are from the on-site interaction model in Sec. S.III.
The corresponding Hamiltonian is:
In order to study the effect of bound-state coupling onto the spin-exchange dynamics, study the time evolution of the Hamiltonian. For computation we use a product state basis. With internal states and the spatial wave function , the relevant two-particle basis states are , , , , and (we omit the trivial fermionization). In this basis, the two-site model has the following form:
(S.3) |
Fig. S3(a) depicts the coherent spin-exchange dynamics happening in the double-well model for different confinements at a finite coupling strength into the c.m.-excited bound state. The data is centered around the energetic resonance of the second c.m. excitation of the bound spin-singlet state with the entrance energy . The -state is repopulated through spin-exchange with a rate that shows a resonance structure in the confinement similar to the one observed experimentally (see main text). In Fig. S3(b), the influence of the coupling strength on the width of the resonance is illustrated. On the low-confinement side of the resonance, the contributions of the intermediate states and to the effective spin-exchange coupling cancel and lead to a zero in the exchange rate.
This model is based on the decoupled harmonic-oscillator provided in Sec. S.III and therefore does not incorporate on-site coupling between c.m. and rel. coordinate, potentially leading to further coupling terms and energetic shifts of the Hubbard parameters.
References
- Dzuba and Derevianko (2010) V. A. Dzuba and A. Derevianko, J. Phys. B 43, 074011 (2010).
- Idziaszek and Calarco (2005) Z. Idziaszek and T. Calarco, Phys. Rev. A 71, 050701 (2005).
- Scazza et al. (2014) F. Scazza, C. Hofrichter, M. Höfer, P. C. de Groot, I. Bloch, and S. Fölling, Nat. Phys. 10, 779 (2014).
- Cappellini et al. (2014) G. Cappellini, M. Mancini, G. Pagano, P. Lombardi, L. Livi, M. Siciliani de Cumis, P. Cancio, M. Pizzocaro, D. Calonico, F. Levi, C. Sias, J. Catani, M. Inguscio, and L. Fallani, Phys. Rev. Lett. 113, 120402 (2014).