Conductivity Spectrum of Ultracold Atoms in an Optical Lattice

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

Phys. Rev. Lett. 122, 153602 – Published 18 April 2019

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.

Received 27 May 2018

DOI:https://doi.org/10.1103/PhysRevLett.122.153602

Physics Subject Headings (PhySH)

Cold atoms & matter waves Cold gases in optical lattices Electrical conductivity Optical conductivity

Fermi gases

Hubbard model

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Rhys Anderson¹, Fudong Wang¹, Peihang Xu¹, Vijin Venu¹, Stefan Trotzky¹, Frédéric Chevy², and Joseph H. Thywissen^1,3

¹Department of Physics, University of Toronto, Ontario M5S 1A7 Canada
²Laboratoire Kastler Brossel, ENS-PSL Research University, CNRS, UPMC-Sorbonne Université, Collège de France, 24 Rue Lhomond, 75005 Paris, France
³Canadian Institute for Advanced Research, Toronto, Ontario M5G 1M1 Canada

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Vol. 122, Iss. 15 — 19 April 2019

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Images

Figure 1
Measurement. (a) Atoms in a 3D optical lattice are driven by periodic displacement of one or both trapping beams (XDT). (b),(c) In situ images are taken after various drive times $t_{d}$ and the center of mass is extracted. (d) The displacement of the center of mass is fit to a single-frequency response (solid line) typically across two periods in trap displacement (dashed line). Data is for a 40 Hz drive and $V = 2 E_{R}$ lattice. (e) The response amplitude $A_{x}$ is shown versus drive $d_{x}$ in typical conditions and at several $ω$ . The linear response limit is found for $A_{x} ≲ 1 μ m$ , as seen by comparison to the lines fit at low $A_{x}$ .
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Figure 2
Conductivity spectra. (a) Real and imaginary on-diagonal conductivity, and (b) difference in real off-diagonal conductivities, versus drive frequency. Here, $V = 2.5 E_{R}$ , with background $a_{s}$ . Lines show fits to Eq. (2). (c) For $V = 0$ , a Fourier-limited response is observed, with $m S_{x x} / N = 1.01 (8)$ and $Γ = 18 (1) s^{- 1}$ . (d),(e),(f) For increasing $V$ , the response broadens and spectral weight (shaded) decreases. By $V = 4 E_{R}$ , $m S_{x x} / N = 0.37 (2)$ , and $Γ = 370 (140) s^{- 1}$ . The upwards shift in frequency is due to increased confinement from the lattice beams.
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Figure 3
Transport time. (a) The best-fit broadening $Γ$ found from $σ_{x x} (ω)$ spectra in a variety of conditions. Varying initial $N$ from $5 \times 10^{3}$ to $50 \times 10^{3}$ creates variable density-weighted filling per spin state $n_{↑}$ , which is measured in situ, and ranges from 0.09(1) to 0.19(2). Varying scattering length $a_{s}$ from $- 240 a_{0}$ to $470 a_{0}$ results in $U / t$ that ranges from $- 0.9$ to 1.8. Depositing additional heat energy $Q$ with a nonadiabatic lattice pulse before loading results in a range of $T / t$ from 1.5 to 3.0. The spectra in Figs. 2 correspond to the circled points in the variable- $V$ series. The red band shows the current damping rate $τ^{- 1}$ calculated with a kinetic theory over the range of measured temperatures and densities; $Γ_{Pol}$ (dashed) is measured for a noninteracting gas. (b) The same data and theory plotted with axes nondimensionalized to account for temperature scaling of the current dissipation rate. Some data for smallest $Γ$ is omitted for clarity.
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Figure 4
Spectral weight of data sets with various $N$ , $a_{s}$ , $Q$ , and $V$ , as described in the caption of Fig. 3. (a) The partial $f$ sum $S_{x x} / N$ , scaled by the TB effective mass $m_{0}^{*}$ , is shown versus measured $T / t$ . Data are compared to three treatments of the uniform-lattice HM: a single band populated using MB statistics (red band), $S_{x x}^{TB}$ (purple solid line), and the asymptotic $m_{0}^{*} S_{x x} / N = t / T$ (black dashed line). The width of the red band is due to the variation of $S_{x x}$ with $V$ , for fixed $T / t$ . The red line is for $V = 2.5 E_{R}$ . (b) Measured $S_{x x}$ robustly agrees with the noninteracting single-band calculation, even up to $a_{s} \sim 1.2 \times 10^{3} a_{0}$ .
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