Photon-Assisted Tunneling in a Biased Strongly Correlated Bose Gas

We study the impact of coherently generated lattice photons on an atomic Mott insulator subjected to a uniform force. Analogous to an array of tunnel-coupled and biased quantum dots, we observe sharp, interaction-shifted photon-assisted tunneling resonances corresponding to tunneling one and two lattice sites either with or against the force and resolve multiorbital shifts of these resonances. By driving a Landau-Zener sweep across such a resonance, we realize a quantum phase transition between a paramagnet and an antiferromagnet and observe quench dynamics when the system is tuned to the critical point. Direct extensions will produce gauge fields and site-resolved spin flips, for topological physics and quantum computing.

Figure 1

Photon-assisted tunneling. (a) For a single atom in a biased double well, tunneling is induced by lattice depth modulation at a frequency given by the energy gap $E$. (b) When there is one atom in each well, the energy gaps become $U+E$ for the right atom to tunnel onto the left atom (i) and $U-E$ for tunneling in the other direction (ii). The effective tunneling under modulation is Bose-enhanced by a factor of $\sqrt{2}$. (c) The double well physics may be extended to a one-dimensional lattice at unity filling (e.g., $n=1$ Mott insulator). There is then an interaction blockade preventing photon-assisted tunneling on adjacent sites due to the resulting energy mismatch. Arrows denote photon-assisted tunneling, with open (closed) circles denoting the initial (final) location of the atom.

Figure 2

Modulation spectroscopy. (a) Schematic of the observed tunneling processes. (b) The occupation probability $p_{\mathrm{odd}}$ versus the modulation frequency. $p_{\mathrm{odd}}$ drops at each of the tunneling resonances, as many doublon-hole pairs are produced. Each resonance is labeled in accordance with the corresponding process described in (a). The modulation of the $9E_r$ $x$ lattice by $\pm 15\%$ corresponds to a Bose-enhanced photon-assisted tunneling rate of $2\pi \times 8\mathrm{Hz}$. $p_{\mathrm{odd}}$ is averaged over a region of 25 lattice sites for 8 realizations. The solid curve is a four-Lorentzian fit to the $E\pm U$, $2E\pm U$ features. In this Letter, all errors in $p_{\mathrm{odd}}$ reflect $1\sigma$ statistical uncertainties in the region averages.

Figure 3

Number-sensitive photon-assisted tunneling using multiorbital shifts. (a) Modulation spectra of the $U-E$ resonant peaks in $n=1,2,3$ Mott insulator shells. Fitted to Gaussian profiles (solid curves), the peaks are located at 306(2), 280(3), and 260(3) Hz, respectively (dashed lines). (b) In situ images of the $n=1,2,3$ Mott shells [(i)–(iii)] without modulation, with the 49-site region studied in (a) enclosed by the red solid line.

Figure 4

Quantum magnetism by lattice modulation. For an $n=1$ Mott shell tilted by $E=2\pi \times 710(25)\mathrm{Hz}$, the modulation frequency is chirped linearly from 150 to 450 Hz and back in a total of 1 s, thus tuning through the $|U-E|$ resonance to produce doublon-hole pairs and then back to restore the double-hole pairs to singly occupied sites (illustrated above). The magnetic model [31] maps this to a quantum phase transition from the paramagnet to an antiferromagnet and back. The lattice depths are $18E_r$ and $45E_r$ in the $x$ and $y$ directions, respectively, corresponding to an on-site interaction energy of $U=2\pi \times 416(15)\mathrm{Hz}$. The $x$-lattice depth is modulated by $\pm 70\%$, producing a Bose-enhanced photon-assisted tunneling rate of $2\pi \times 7.4\mathrm{Hz}$. Inset: Dynamics of a many-body quench, produced by driving at ${\omega }_{\mathrm{mod}}=E-U$, for $E=2\pi \times 890(10)\mathrm{Hz}$ and $U=2\pi \times 317(10)\mathrm{Hz}$, for a reduced $x$-lattice depth of $9E_r$. The photon-assisted tunneling rate (with $\pm 17\%$ lattice depth modulation) is $2\pi \times 8.1\mathrm{Hz}$.