Lattice-Assisted Spectroscopy: A Generalized Scanning Tunneling Microscope for Ultracold Atoms

We propose a scheme to measure the frequency-resolved local particle and hole spectra of any optical lattice-confined system of correlated ultracold atoms that offers single-site addressing and imaging, which is now an experimental reality. Combining perturbation theory and time-dependent density matrix renormalization group simulations, we quantitatively test and validate this approach of lattice-assisted spectroscopy on several one-dimensional example systems, such as the superfluid and Mott insulator, with and without a parabolic trap, and finally on edge states of the bosonic Su-Schrieffer-Heeger model. We highlight extensions of our basic scheme to obtain an even wider variety of interesting and important frequency resolved spectra.

Figure 1

(a) Schematic overview of spectroscopic scheme: an extra outcoupling site (OS) (physical or internal state) is tunnel coupled with amplitude $J_{\perp }(t)$ to lattice site $x$. The time-evolving average $⟨{\widehat{n}}_{\mathrm{OS}}(t)⟩$ on the OS is obtained from repeated occupation-number imaging. (b) Local density for system in test case (A), 15 bosons in a 1D parabolic trap. The MI region is in the center, and SF regions occupy the wings. Red arrows indicate sites where the system is locally probed. (c) Symmetrized discrete absorption spectrum in center of the MI region, extracted from the full time-evolving average $⟨{\widehat{n}}_{\mathrm{OS}}(t){⟩}_{\omega }^{\text{init}=1}$ for $J_{\perp }/J=0.05$ (black diamond), 0.1 (red square), 0.2 (green circle), and compared with ${\mathcal{I}}_x^{\mathrm{ab}}(\omega )+{\mathcal{I}}_x^{\mathrm{ab}}(-\omega )$ obtained directly from tDMRG simulations (blue solid line). (d) $⟨{\widehat{n}}_{\mathrm{OS}}(t){⟩}_{\omega }^{\text{init}=1}$ over time at first resonance $\omega =6.62J$ for $J_{\perp }/J=0.05$ (black solid line), 0.1 (red dotted), 0.2 (green dashed). Rabi oscillations can be clearly observed at experimentally accessible time scales. (e) $⟨{\widehat{n}}_{\mathrm{OS}}(t){⟩}_{\omega }^{\text{init}=1}$ over time below the first resonance, at $\omega =6J$; values of $J_{\perp }/J$ match (d). Off resonance, oscillations lack all hallmarks of Rabi dynamics, exhibiting small amplitudes scaling with $J_{\perp }/J$, while frequency does not depend on it. (f) Equivalent figure to (c) for site 8 in the SF region.

Figure 2

Finite-resolution emission spectra in the thermodynamic limit for the MI and SF phases, reconstructed from $⟨{\widehat{n}}_{\mathrm{OS}}(t){⟩}_{\omega }^{\text{init}=0}$ for $J_{\perp }/J=0.05$ (black diamond), 0.1 (red square), 0.2 (green circle), and compared with ${\mathcal{I}}_x^{\mathrm{em}}(\omega )+{\mathcal{I}}_x^{\mathrm{em}}(-\omega )$ obtained directly from tDMRG simulations (blue $\times$; the blue solid line is a guide to the eye). (a) $U=10J$. (b) $U=2J$.

Figure 3

Bulk and edge spectra of the bosonic SSH model with $L=50$. Markers denote spectra reconstructed from $⟨{\widehat{n}}_{\mathrm{OS}}(t){⟩}_{\omega }^{\text{init}=0}$ for three different $J_{\perp }/J$: $J_{\perp }/J=0.05$ (black diamond), $J_{\perp }/J=0.1$ (red square), and $J_{\perp }/J=0.2$ (green circle). For comparison, we show ${\mathcal{I}}_x^{\mathrm{em}}(\omega )+{\mathcal{I}}_x^{\mathrm{em}}(-\omega )$ obtained directly from tDMRG simulations (dashed blue lines for bulk at $x=26$ and solid purple lines for edge spectra, at $x=1$ and $=50$, respectively). While in (a) the distinct edge mode persists for $U/(J+\delta J)=5$, in (b), (c) bulk and edge spectra start becoming indistinguishable for lower $U/(J+\delta J)$, with overall spectral weight on the edges collapsing.