Orbital Many-Body Dynamics of Bosons in the Second Bloch Band of an Optical Lattice

J. Vargas, M. Nuske, R. Eichberger, C. Hippler, L. Mathey, and A. Hemmerich

Phys. Rev. Lett. 126, 200402 – Published 17 May 2021

Abstract

We explore Josephson-like dynamics of a Bose-Einstein condensate of rubidium atoms in the second Bloch band of an optical square lattice providing a double well structure with two inequivalent, degenerate energy minima. This oscillation is a direct signature of the orbital changing collisions predicted to arise in this system in addition to the conventional on-site collisions. The observed oscillation frequency scales with the relative strength of these collisional interactions, which can be readily tuned via a distortion of the unit cell. The observations are compared to a quantum model of two single-particle modes and to a semiclassical multiband tight-binding simulation of $12 \times 12$ tubular sites of the lattice. Both models reproduce the observed oscillatory dynamics and show the correct dependence of the oscillation frequency on the ratio between the strengths of the on-site and orbital changing collision processes.

Received 12 August 2020
Accepted 27 April 2021

DOI:https://doi.org/10.1103/PhysRevLett.126.200402

Physics Subject Headings (PhySH)

Bose gases Bose-Einstein condensates Cold atoms & matter waves Cold gases in optical lattices Decoherence in quantum gases Optical lattices & traps Ultracold collisions

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

J. Vargas¹, M. Nuske^1,2,3, R. Eichberger^1,2, C. Hippler¹, L. Mathey^1,2,3, and A. Hemmerich^1,2,3

¹Institut für Laserphysik, Universität Hamburg, 22761 Hamburg, Germany
²Zentrum für Optische Quantentechnologien, Universität Hamburg, 22761 Hamburg, Germany
³The Hamburg Center for Ultrafast Imaging, Luruper Chaussee 149, Hamburg 22761, Germany

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Vol. 126, Iss. 20 — 21 May 2021

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Images

Figure 1
(a) The bipartite lattice geometry with deep $A$ sites and shallow $B$ sites. The unit cell is shown by the gray rectangle. (b) The second Bloch band of the lattice in (a) is plotted across the first Brillouin zone with the two inequivalent energy minima at $X_{\pm}$ and the energy maximum at $Γ$ highlighted. Blue denotes low and white denotes high energy. (c) and (d) show contour plots of the Bloch functions $ψ_{\pm}$ , corresponding to the $X_{\pm}$ points of the second band.
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Figure 2
(a) The temporal evolution of the relative population difference $(n_{-} - n_{+}) / (n_{-} + n_{+})$ is shown for fixed $Δ V_{f} = 0.725 \times V_{0, f}$ after initially a BEC is formed at the $X_{+}$ point. The red solid line is a fit by an exponentially damped harmonic oscillation. The error bars show the standard deviations of the mean for a set of ten measurements. (b) The observed oscillation frequencies obtained from fitting data as in (a) are plotted versus $Δ V_{f}$ . The errors show the standard deviations found in the fits. The red disks and the blue squares represent two measurement series evaluated via momentum spectra and band mapping, respectively. The solid black line shows a calculation using a two-mode model.
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Figure 3
(a) Relative population difference of $X$ points $(n_{-} - n_{+}) / (n_{-} + n_{+})$ according to our classical-field-theory simulations at $Δ V_{f} = 0.69 V_{0}$ . The panel shows the oscillation for a single random initialization. (b) Fourier spectrum of the oscillation shown in (a) averaged over 500 random initializations, see black circles. The peak of the Fourier transform determines the main oscillation frequency at the given value of $Δ V_{f}$ . The red line shows a Gaussian fit to the Fourier spectrum. For details on the fitting routine see Ref. [30]. For both panels the temperature is $T = 76.8 nK \approx 0.8 E_{rec} / k_{B}$ and hence similar to the experimental temperature.
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Figure 4
Dominant oscillation frequencies obtained from $c$ -field simulations, see black diamonds, are compared to experimental data, shown by blue squares and red disks, which are repeated from Fig. 2. For each $Δ V_{f}$ we determine the averaged Fourier spectrum, as exemplarily shown in Fig. 3 and plot the dominant frequency obtained from a Gaussian fit. The errors in the determination of the positions of the maxima of the fitted Gaussians are mostly smaller than the data symbols.
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Figure 5
Phase-space diagram of the relative population in the two lowest-energy many-body states $m_{r}$ for a single initialization of our simulations. Here, $m_{r} = m_{+} (t) - m_{-} (t)$ and $m_{\pm} (t) = | ⟨ Ψ_{\pm} | ψ (t) ⟩ |^{2} / m_{0}$ . Furthermore $ψ (t)$ is the wave function obtained within our simulations and $m_{0} = | ⟨ Ψ_{+} | ψ (t) ⟩ |^{2} + | ⟨ Ψ_{-} | ψ (t) ⟩ |^{2}$ is the total number of condensed atoms in the second band. We consider an idealized case at lower temperature $T = 0.5 nK$ and 4 times stronger interactions as compared to Fig. 4. We also use a different quench protocol that keeps $V_{0} = 7 E_{rec}$ throughout the quench and changes $Δ V$ from $Δ V_{i} = - 0.6 V_{0}$ to $Δ V_{f} = 0.35 V_{0}$ .
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