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arXiv:1712.09965 · Physical Review Letters

Conductivity Spectrum of Ultracold Atoms in an Optical Lattice

Rhys Anderson, Fudong Wang, Peihang Xu, Vijin Venu, Stefan Trotzky, Frédéric Chevy, Joseph H. Thywissen

Published Apr 18, 2019• 62 citations
arXivHTMLPDF

Abstract

We measure the conductivity of neutral fermions in a cubic optical lattice. Using in situ fluorescence microscopy, we observe the alternating current resultant from a single-frequency uniform force applied by displacement of a weak harmonic trapping potential. In the linear response regime, a neutral-particle analog of Ohm's law gives the conductivity as the ratio of total current to force. For various lattice depths, temperatures, interaction strengths, and fillings, we measure both real and imaginary conductivity, up to a frequency sufficient to capture the transport dynamics within the lowest band. The spectral width of the real conductivity reveals the current dissipation rate in the lattice, and the integrated spectral weight is related to thermodynamic properties of the system through a sum rule. The global conductivity decreases with increased band-averaged effective mass, which at high temperatures approaches a T-linear regime. Relaxation of current is observed to require a finite lattice depth, which breaks Galilean invariance and enables damping through collisions between fermions.

DOI
10.1103/physrevlett.122.153602

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We measure the conductivity of neutral fermions in a cubic optical lattice. Using in situ fluorescence microscopy, we observe the alternating current resultant from a single-frequency uniform force applied by displacement of a weak harmonic trapping potential. In the linear response regime, a neutral-particle analog of Ohm's law gives the conductivity as the ratio of total current to force. For various lattice depths, temperatures, interaction strengths, and fillings, we measure both real and imaginary conductivity, up to a frequency sufficient to capture the transport dynamics within the lowest band. The spectral width of the real conductivity reveals the current dissipation rate in the lattice, and the integrated spectral weight is related to thermodynamic properties of the system through a sum rule. The global conductivity decreases with increased band-averaged effective mass, which at high temperatures approaches a T-linear regime. Relaxation of current is observed to require a finite lattice depth, which breaks Galilean invariance and enables damping through collisions between fermions.
In this thesis, we observe and characterize the current response of neutral fermions to a periodic applied force, thereby obtaining their conductivity. The experiment takes place in a 3D optical lattice subject to weak harmonic confinement, where a quantum gas microscope enables high-resolution fluorescence imaging of atoms pinned to lattice sites. Using micron-scale periodic displacements of an underlying harmonic potential to provide the oscillating uniform force, we measure the global current response of the atoms up to frequencies sufficient to characterize intraband transport, but well below the bandgap. We observe that the current response scales linearly with the forcing, providing experimental verification that data is taken in the linear response regime. The resulting conductivity spectra are analyzed and fit using five different models, from which a broadening parameter and the spectral weight of the response may be obtained. Broadening of the current response spectrum provides a measure of the rate of dissipation, and increases with lattice depth, interaction strength, and density. This dissipation occurs purely due to fermion-fermion collisions, given the absence of phonons or impurities in our potentials. It is observed to require a finite lattice depth in order to break Galilean invariance, as well as to enable Umklapp scattering events, which play a significant role in the dynamics. Measured dissipation rates collapse onto the predictions of a kinetic theory under the wide range of conditions studied. The spectral weight of the conductivity spectrum predicts the impulse response of the global current. It satisfies a sum rule in the limit of small lattice depth, but diminishes as the depth increases, reflecting an increase in the band-averaged effective mass. Measured under varying temperatures, densities, interaction strengths, and lattice depths, the spectral weight is shown to collapse onto the predictions of a thermodynamic description of the system. Furthermore, it is shown to be unaffected by varying the strength of interactions between the fermions, illustrating a fundamental prediction for conductivity spectra. As our measurements approach a high-temperature regime, the inverse of the spectral weight is shown to approach T-linear behaviour.
We measure the conductivity of neutral fermions in a cubic lattice. A uniform force is applied at a single frequency by periodic displacement of the trapping potential, and the current response is observed though the global displacement of atoms. In the linear response regime, the steady-state ratio between current and force gives the ac conductivity. The tensor properties of the conductivity reveal temporal and spatial symmetries of the sample. Similarly to optical conductivity commonly measured in materials with reflectivity, exact sum rules relate dynamical measurements to thermodynamic properties. We explore the effect of lattice depth, temperature, interaction strength, and atom number on conductivity. Using a relaxation-time approximation, we extract the transport time of current. Our work enables a direct comparison between cold-atom experiments and conductivity studies of materials.
We investigate the interaction-induced resistivity of ultracold fermions in a three-dimensional optical lattice. In situ observations of transport dynamics enable the determination of real and imaginary resistivity. In the strongly interacting metallic regime, we observe a striking saturation of the current-dissipation rate toward a value that is independent of the interaction strength. This phenomenon is quantitatively captured by a dissipation model that uses a renormalized two-body scattering matrix. We further measure the temperature dependence of resistivity in the strongly interacting limit and discuss the predicted asymptotic high-temperature behavior. Our results provide a clear microscopic understanding of bounded resistivity of low-density metals, thus providing a useful benchmark for studies of strongly correlated atomic and electronic systems.
Quantum simulations are quickly becoming an indispensable tool for studying particle transport in correlated lattice models. One of the central topics in the study of transport is bad-metal behavior, characterized by the direct-current resistivity linear in temperature. In the fermionic Hubbard model, optical conductivity has been studied extensively, and a recent optical lattice experiment demonstrated bad-metal behavior in qualitative agreement with theory. Far less is known about transport in the bosonic Hubbard model. We investigate the conductivity in the Bose-Hubbard model and focus on the regime of strong interactions and high temperatures. We use numerically exact calculations for small lattice sizes. At weak tunneling, we find multiple peaks in the optical conductivity that stem from the Hubbard bands present in the many-body spectrum. This feature slowly washes out as the tunneling rate gets stronger. At high temperature, we identify a regime of TTT-linear resistivity, as expected. When the interactions are very strong, the leading inverse-temperature coefficient in conductivity is proportional to the tunneling amplitude. As the tunneling becomes stronger, this dependence takes quadratic form. At very strong coupling and half filling, we identify a separate linear resistivity regime at lower temperature, corresponding to the hard-core boson regime. Additionally, we unexpectedly observe that at half filling, in a big part of the phase diagram, conductivity is an increasing function of the coupling constant before it saturates at the hard-core-boson result. We explain this feature based on the analysis of the many-body energy spectrum and the contributions to conductivity of individual eigenstates of the system.
We explore the transport properties of an interacting Fermi gas in a three-dimensional optical lattice. The center of mass dynamics of the atoms after a sudden displacement of the trap minimum is monitored for different interaction strengths and lattice fillings. With increasingly strong attractive interactions the weakly damped oscillation, observed for the noninteracting case, turns into a slow relaxational drift. Tuning the interaction strength during the evolution allows us to dynamically control the transport behavior. Strong attraction between the atoms leads to the formation of local pairs with a reduced tunneling rate. The interpretation in terms of pair formation is supported by a measurement of the number of doubly occupied lattice sites. This quantity also allows us to determine the temperature of the noninteracting gas in the lattice to be as low as (27+/-2)% of the Fermi temperature.
We study a harmonically confined atomic gas which is subjected to an additional external potential such as an optical lattice. Using a linear response formulation, we determine the response of the gas to a small, time-dependent displacement of the harmonic trap and derive a simple exact relation showing that the centre-of-mass position of the atomic cloud is directly related to the global optical conductivity of the system. We demonstrate the usefulness of this approach by calculating the optical conductivity of bosonic atoms in an optical lattice. In the Mott insulating phase, there is clear evidence of an optical Mott gap, providing a proof-of-principle demonstration that the global optical conductivity gives high-quality information about the excitations of strongly correlated quantum gases.