Observation of Magnon-Polarons in the Fermi-Hubbard Model

Abstract

The interplay of magnetic excitations and itinerant charge carriers is a ubiquitous phenomenon in strongly correlated electron systems. In the vicinity of magnetically ordered phases, strong interactions between itinerant quasiparticles and magnetic excitations can result in the dramatic renormalization of both. A key theoretical question is understanding the renormalization of the magnon quasiparticle, a collective spin excitation, upon doping a magnetic insulator. Here, we report the observation of a new type of quasiparticle arising from the dressing of a magnon with the doped holes of a cold atom Fermi-Hubbard system, i.e. a magnon-Fermi-polaron. Utilizing Raman excitation with controlled momentum in a doped, spin-polarized band insulator, we address the spectroscopic properties of the magnon-polaron. In an undoped system with strong interactions, photoexcitation produces magnons, whose properties are accurately described by spin wave theory. We measure the evolution of the photoexcitation spectra as we move away from this limit to produce magnon-polarons due to dressing of the magnons by charge excitations. We observe a shift in the polaron energy with doping that is strongly dependent on the injected momentum, accompanied by a reduction of spectral weight in the probed energy window. We anticipate that the technique introduced here, which is the analog of inelastic neutron scattering, will provide atomic quantum simulators access to the dynamics of a wide variety of excitations in strongly correlated phases.

Figures (10)

• Figure 1: Raman spectroscopy of a Fermi-Hubbard system. (a) Left: schematic spectral function corresponding to a spin-flip injected into a band insulator, showing a well-defined magnon quasiparticle state (green line) and a doublon-hole scattering continuum at higher energies (grey). Right: in the doped system, the spin-flip is dressed by scattered holes, producing a polaron, so the injected magnon has a finite lifetime (green band). (b) The Raman beams are sent through the objective at a controlled angle and illuminate the 2D Fermi-Hubbard system. (c) Injected momentum calibration: application of a Raman pulse followed by a global RF rotation converts the spatially varying Raman phase to a population pattern. (d) Measured two-point spin correlations after the momentum calibration pulse sequence for $\Delta \mathbf{k}\approx (0,1) \, \pi/a$ (left) and $(1,1)\, \pi/a$ (right). The measured momentum is reported from a 2-D sinusoidal fit to the correlations resulting in measured values of $(0.000(1),1.011(1))\,\pi/a$ and $(1.0093(4),0.9941(4)) \, \pi/a$, respectively. (e) Raman spectroscopy of an itinerant, non-interacting Fermi gas at the high symmetry points $\Delta \mathbf{k} \approx (0,\pi)\, a^{-1}$ and $\Delta k \approx (\pi,\pi)\,a^{-1}$. For these momenta, the experimental photoexcitation spectrum approximates the ideal densities of states of a one-dimensional (1D) and two-dimensional (2D) lattice, respectively (lightly shaded curves). Solid lines are numerical simulations taking into account measured temperature, doping and harmonic confinement.
• Figure 2: Interaction dependence of the quasiparticle energy. (a) Raman spectroscopy with injected momentum $\Delta \mathbf{k}=(0,\pi)a^{-1}$. Green: Injection spectrum for isolated atoms in a deep lattice. Blue: Injection spectrum for $U/t=6.0(3)$ and low hole doping ($7(2)\%$). Points are transferred atom counts normalized to maximum and solid lines are Lorentzian fits to the data. The frequency axis zero is calibrated to the fitted isolated atom resonance and the dashed line is the energy of a magnon in the Heisenberg model with coupling $J=4t^2/U$. (b) Measured photoexcitation spectra versus inverse interaction $t/U$ at average doping 7(2)%. The upper (lower) panel corresponds to magnons injected at the $X$ ($M$) point of the BZ. Colorbar displays the density of transferred atoms in the lattice. (c) Magnon energy versus $t/U$. Blue (red) data corresponds to magnon excitation at the $X$ ($M$) point, respectively. Dashed lines are the energy of a single magnon in the Heisenberg model. Solid lines are based on numerics using a non-Gaussian variational wavefunction and include the effects of finite temperature and hole doping.
• Figure 3: Doping dependence of the magnon-polaron energy. (a) Photoexcitation spectra versus doping taken at the $X$ point (top) and the $M$ point (bottom). For all dopings where the signal is strong enough to fit a peak, we overlay the peak transfer resonance as a guide to the eye. (b) Polaron binding energy at the $X$ point (blue) and $M$ point (red). Experimental data (circles) is compared to three variational calculations: molecular Chevy ansatz (dotted lines), non-Gaussian ansatz ground state (dashed lines) and non-equilibrium non-Gaussian ansatz (solid lines). All theories converge to the magnon energies at zero doping (diamonds). (c) Calculated $g_2$ density correlation function around the impurity in the polaron frame as a function of doping $\delta$ and transferred momentum. (d) Integrated atom transfer within the probed detuning window $\Delta\in[-2.1,1.3]t/h$ as a function of doping at the $X$ point. Experimental transfer (circles) is compared to the theoretical transfer integrated over the experimental window (solid lines) and all calculated frequencies (dashed line) with an overall scaling factor as a free parameter. (e) Theoretical spectra versus energy at the $X$ point, showing spectral weight leaving the experimentally probed window (vertical lines) as a result of dressing with high-energy excitations.
• Figure 4: Magnon-polaron dispersion and effective mass. (a) Tuning the injected magnon momentum. Adjusting the position of the Raman beams in the Fourier plane of the objective tunes the transferred momentum $\Delta \mathbf{k}$. (b) First Brillouin zone of the square lattice. Momentum scans are taken along a line extending from the $M$ point $(\pi,\pi)$ to the $\Gamma$ point (red line) to characterize the dispersion of the polaron along this line. (c) Measured dispersion of the polaron for $U/t=11.7(5)$ (green), 8.3(5) (red) and 7.0(2) (blue). Dashed lines are fits to a single magnon dispersion with the bandwidth as a fit parameter. Inset: the effective mass extracted from the fitted dispersion (circles) together with the calculated effective mass of the polaron (solid line) in units of $m^*=\hbar^2/{2a^2 t}$, the effective mass of a single atom in an empty square lattice.
• Figure S1: Experimental Schematic. Linearly polarized Raman beams (solid red lines) are reflected by a polarizing beam splitter (PBS) followed by half- and quarter-waveplates (not shown) for full polarization control to optimize the two-photon Rabi rate $\Omega_R$. Florescence imaging light (shaded area) is largely unpolarized and the $\sim 50\%$ that is transmitted through the PBS is collected on a CMOS camera used for single-atom imaging.