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Interacting Fermions in Optical Lattice Potentials

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From Atom Optics to Quantum Simulation

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Abstract

In this chapter I report on the experimental realization of the Fermi-Hubbard model. To this end, a repulsively interacting balanced spin mixture of ultracold \(^{40}\)K atoms is loaded into a three-dimensional optical lattice potential. The emerging quantum phases are probed by measuring the global compressibility of the quantum gas.

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Notes

  1. 1.

    In typical ultracold atom experiments it is notoriously difficult to vary the atom number from shot to shot in a controlled way. However, our setup offers exceptional control of the harmonic confinement. Therefore we actively vary the harmonic trapping potential, while the atom number is kept constant. The remaining shot-to-shot fluctuations of the atom number are largely suppressed in the experimental data, because all quantities are expressed in rescaled, atom number independent units, such as \(E_{\mathrm {t}} \) and \(R_\mathrm{{sc}}\) (see below).

  2. 2.

    However, the entropies are not low enough to reach magnetic ordering of the spins, which is discussed later in this chapter (see also Sect. 3.3.2).

  3. 3.

    Actually, it seems desirable to perform a preramp to a lattice depth above \(2.2\) \(E_\mathrm{rec }\) in order to ensure a real band gap. However, this would require a very slow compression beyond the time scale that is affordable due to technical heating. In this respect the preramp to \(1\) \(E_\mathrm{rec }\) is a already compromise.

  4. 4.

    For a Gaussian distribution this definition of the cloud radius yields the standard deviation.

  5. 5.

    At half-filling, the maximal entropy is accommodated in the case of vanishing interactions. In a system of \(N\) lattice sites and \(N/2\) atoms per spin state, each configuration \(j\) has the same probability \(p_j=\left[ \frac{N!}{(N/2)! (N/2)!}\right]^{-2}\). With \(S = - k_B \sum \nolimits _j p_j \ln (p_j)\) and Stirling’s formula \(\ln (N!) \approx N \ln (N)\) the entropy per particle \(S/N=k_B \ln (4) \) is readily obtained. For strong repulsion each lattice site is occupied by exactly one atom. Here, all configurations have equal probability \(p_j^\infty =\left[\frac{N!}{(N/2)! (N/2)!}\right]^{-1}\) leading to \(S/N=k_B \ln (2)\). In the case of long-range antiferromagnetic order, there are only two configurations for the global quantum state. In the limit of large \(N\) this results in vanishing entropy per particle \(S/N\gtrsim 0\).

  6. 6.

    Localized wavefunctions have a stronger curvature, which corresponds to higher momentum contributions and, consequently, higher kinetic energy.

  7. 7.

    In the limit of strong compression, this corresponds to the maximally packed state of an ideal band insulator.

  8. 8.

    To my knowledge, the first qualitative arguments pointing to these possibilities have been made by Nikolay Prokof’ev during a DARPA OLE team meeting in the summer of 2008 in Cambridge, MA.

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    Sebastian Will

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Will, S. (2013). Interacting Fermions in Optical Lattice Potentials. In: From Atom Optics to Quantum Simulation. Springer Theses. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33633-1_6

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