The 2014 observation of two-orbital spin-exchange interactions in an SU(6)-symmetric 173Yb^{173}\mathrm{Yb} optical-lattice Fermi gas (Scazza et al.) characterized the microscopic interaction set of the 1S0^{1}S_0-3P0^{3}P_0 two-orbital system, but left its exploitation for many-body physics open: dynamics were modeled with classical two-body rate equations, coherent exchange oscillations were inaccessible in that setup, and the target correlated phases (Kondo lattice, orbital magnetism, SU(N) states) were not realized. Later work closed much of the two-body agenda: coherent inter-orbital exchange oscillations were observed (Cappellini et al. 2014), the interaction was made magnetically and confinement-tunably adjustable through the orbital Feshbach resonance (Höfer et al. and Pagano et al. 2015; Riegger et al. 2018), and the near-resonant symmetric-channel scattering length was extracted (aeg+=1878(37) a0a_{eg}^{+}=1878(37)\,a_0 from resonance fits, still in method-dependent tension with 3300(300) a03300(300)\,a_0 from coherent-oscillation data). What remains open is the many-body core: a quantitative beyond-rate-equation account of the coherent, correlated dynamics of the two-orbital SU(N) gas, and the identification plus experimental realization of stable parameter regimes for Kondo-lattice, orbital-magnetic, and SU(N)-correlated phases under the actual loss landscape, supported by a consolidated, regularization-controlled determination of aeg+a_{eg}^{+}.

Background

Fermionic alkaline-earth-like atoms such as 173Yb^{173}\mathrm{Yb} provide two long-lived electronic states that act as independent orbitals in a quantum gas: the ground state ∣g⟩=1S0|g\rangle={}^{1}S_0 and the metastable clock state ∣e⟩=3P0|e\rangle={}^{3}P_0, both with zero electronic angular momentum, so the nuclear spin (I=5/2I=5/2) is decoupled from the electronic degrees of freedom. Because the scattering properties are, to high precision, independent of the nuclear-spin projection, an NN-component spin mixture realizes an SU(NN)-symmetric Fermi gas with NN up to 6. Contact interactions between a gg and an ee atom split into two channels, ∣eg+⟩|eg^{+}\rangle and ∣eg−⟩|eg^{-}\rangle, the symmetric and antisymmetric combinations of the electronic orbital labels, which coincide with the nuclear-spin singlet and triplet two-particle states. They are characterized by s-wave scattering lengths aeg±a_{eg}^{\pm} or, in an optical lattice, by on-site Hubbard energies Ueg±U_{eg}^{\pm}. The splitting defines the inter-orbital spin-exchange coupling Vex=(Ueg+−Ueg−)/2V_{\mathrm{ex}}=(U_{eg}^{+}-U_{eg}^{-})/2, which swaps the nuclear spins of an ∣eg ↑↓⟩|eg\,\uparrow\downarrow\rangle pair at frequency 2∣Vex∣/h2|V_{\mathrm{ex}}|/h and plays the role of the impurity-conduction-electron coupling in proposals to simulate the Kondo lattice model — the paradigmatic Hamiltonian of heavy-fermion condensed matter, in which exchange-coupled local moments and itinerant fermions form a screened, strongly correlated state. In a lattice the relation UX∝aXU_X\propto a_X (with the overlap integral of the lowest-band Wannier functions, i.e. the lowest-band site wavefunctions) is valid only when aXa_X is much smaller than the harmonic-oscillator length ahoa_{\mathrm{ho}} of a lattice site; for interactions approaching the lattice band gap the pair wavefunction is strongly modified (up to 'fermionization', where the interaction integral vanishes), and extracting scattering lengths then requires multi-band modelling with proper short-range regularization of the contact potential, the complete electronic and nuclear internal structure, and the lattice anisotropy. Scazza et al. (Nature Physics 10, 779 (2014)) measured this system for 173Yb^{173}\mathrm{Yb} by clock-shift spectroscopy — measuring the interaction-induced shift of the ultranarrow g→eg\to e 'clock' transition — in a 3D lattice: SU(6) symmetry of the inter-orbital channels at the about 1% level; agg=199.4 a0a_{gg}=199.4\,a_0 and aeg−=219.5 a0a_{eg}^{-}=219.5\,a_0 (consistent across spin mixtures and with the single-band relation); and a very large exchange coupling, VexV_{\mathrm{ex}} up to about h×22 kHzh\times22\,\mathrm{kHz} with Ueg+U_{eg}^{+} at least h×44 kHzh\times44\,\mathrm{kHz}, exceeding the roughly h×25 kHzh\times25\,\mathrm{kHz} band gap, so that only loose bounds on the symmetric-channel scattering length were possible (Δaeg+=aeg+−agg\Delta a_{eg}^{+}=a_{eg}^{+}-a_{gg} of order 22-4×103 a04\times10^{3}\,a_0). Directly observed spin-exchange equilibration between gg and ee ensembles was fitted with a classical two-body rate-equation model including inelastic losses; the specific ∣eg−⟩|eg^{-}\rangle pair loss was shown to be tiny (loss coefficient below 3×10−15 cm3/s3\times10^{-15}\,\mathrm{cm^3/s}, imaginary scattering length ∣ηeg−∣<0.006 a0|\eta_{eg}^{-}|<0.006\,a_0), the aggregate gg-ee loss being dominated by the eg+eg^{+} nuclear-singlet channel, while fast ee-ee collisions (βee=2.2(0.6)×10−11 cm3/s\beta_{ee}=2.2(0.6)\times10^{-11}\,\mathrm{cm^3/s}) suppress pair coherence. A corrigendum (Nature Physics 11, 514 (2015)) inverted the ∣eg+⟩|eg^{+}\rangle/∣eg−⟩|eg^{-}\rangle assignment of the original paper; the corrected record is used here. A companion experiment with 87Sr^{87}\mathrm{Sr} (Zhang et al., Science 345, 1467 (2014)) measured the analogous channels (aeg+=169(8) a0a_{eg}^{+}=169(8)\,a_0, aeg−=68(22) a0a_{eg}^{-}=68(22)\,a_0). Subsequent work added the orbital Feshbach resonance (OFR): a magnetically tunable resonance between an open two-orbital channel and a near-threshold bound state of the strongly interacting eg+eg^{+} channel, enabled by the different nuclear gg-factors of the two orbitals, proposed by Zhang, Cheng, Zhai and Zhang (2015) and observed for 173Yb^{173}\mathrm{Yb} by Höfer et al. and Pagano et al. (2015), with quantum-defect theory supplying the regularization-correct two-body description (Cheng, Zhang and Zhang 2017).

Problem Statement

Given the now-measured microscopic interaction set of the two-orbital SU(6)-symmetric 173Yb^{173}\mathrm{Yb} Fermi gas (agg=199.4 a0a_{gg}=199.4\,a_0, aeg−=219.5(2.0) a0a_{eg}^{-}=219.5(2.0)\,a_0, a near-resonant symmetric channel aeg+a_{eg}^{+} of order 2×103 a02\times10^{3}\,a_0, on-site exchange coupling 2Vex2V_{\mathrm{ex}} between about h×14 kHzh\times14\,\mathrm{kHz} and h×44 kHzh\times44\,\mathrm{kHz} depending on lattice depth, inelastic loss dominated by the eg+eg^{+} nuclear-singlet channel with the ∣eg−⟩|eg^{-}\rangle channel essentially lossless, and fast e-e losses), establish the coherent many-body regime of this system. Specifically: (1) develop and experimentally validate a quantitative description of the many-body dynamics of the two-orbital gas that goes beyond the two-body rate-equation treatment used in the source experiment - i.e. the coherent, correlation-driven evolution, quantitatively treating at least one of many-body exchange dynamics, quasiparticle formation and decay, and Kondo-type screening dynamics, in regimes where e-e losses are eliminated and residual eg+eg^{+} losses compete with elastic exchange; (2) identify at least one stable parameter regime - giving the sign and magnitude of the effective Kondo coupling relative to tunnelling, temperature and loss rates together with the lattice depth, density, temperature and spin polarization of the regime - in which at least one of the targeted extended phases (Kondo-lattice/heavy-fermion behaviour, orbital magnetic order, SU(N)-correlated states) can be realized, and realize and detect at least one such phase in a two-orbital alkaline-earth gas through a stated discriminating observable (spectroscopic, thermodynamic or transport-like) that distinguishes the correlated phase from loss-driven or thermal artifacts; and (3) as the supporting microscopic input, consolidate the determination of the symmetric-channel scattering length aeg+a_{eg}^{+} of 173Yb^{173}\mathrm{Yb}, whose current extractions (1878(37) a01878(37)\,a_0 from orbital-Feshbach resonance fits versus 3300(300) a03300(300)\,a_0 from coherent-oscillation data) disagree beyond their uncertainties. These three parts together address one objective: converting the experimentally established two-body interaction set of the two-orbital SU(N) gas into controlled, coherent, correlated many-body physics. The parts of the original 2014 question that are already settled - the observation of coherent inter-orbital exchange oscillations, magnetic and confinement-based tunability of the exchange, and first quantitative extractions of aeg+a_{eg}^{+} - are to be treated as completed background, not as targets.

References

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