Abstract
In the laboratory, experiments with optical lattices are most often performed in the presence of an additional external potential for confining the atoms.
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- 1.
In the sense that the leading physics stems from the optical potential.
- 2.
Notice that the characteristic length of the trap \(l_{\text {trap}} \gg \lambda /2\), where \(\lambda \) is the wavelength of the laser.
- 3.
Since it is straightforward to write the matrix form of the ideal gas Hamiltonian in the 3D lattice, we only write explicit expressions of the 2D case here.
- 4.
Diagonalization is performed on the Hamiltonian of a truncated lattice.
- 5.
Notice, however, that because of the tunneling anisotropy, the local chemical potential is not the only factor determining local properties of the density.
- 6.
We remind that the coupling constants and tunneling coefficients are computed with lattice Wannier functions obtained from numerical solution of the Mathieu equation for the potential (4.18).
- 7.
In momentum space the term \(-t^\alpha _{\sigma }\partial ^2_\sigma \) corresponds to \(t^\alpha _{\sigma }k_{\alpha \sigma }^2\). Here however we use \(k^2_{\alpha \sigma }\rightarrow 2t^\alpha _{\sigma }(1 - \cos k_{\alpha \sigma })\) to account for the (inverted) shape of the p band and the discrete character of the system.
- 8.
It should be noticed, however, that the expression provided by \(S_{cont}\) is obtained in the limit of \(U_0 = 0\) and it does not approach 1 as \(V\rightarrow \infty \). On the other hand, since the kinetic term relative to interaction becomes negligible under these circumstances, any small \(U_0 > 0\) is sufficient to make \(S_{cont}\rightarrow 1\). For moderate values of the lattice depth V, \(S_{cont}\) increases monotonically with increasing values of V. This behavior is not predicted by the discrete model (Eq. (4.6)), and therefore we keep in mind that the two descriptions yield qualitatively different predictions in the limit of deep lattices.
- 9.
Cf. the analysis of Chap. 3.
- 10.
Recall here that \(J_{x,\varvec{j}}\) is always zero.
- 11.
The SU(3) structure of the Mott phase with a unit filling of the three-orbital system in the p band is discussed in details in Sect. 5.3.
- 12.
In the proper sense of changing the symmetry of the ground state.
- 13.
It is important to point out, however, that since here the densities of different orbital states are spatially different, adiabatic driving could lead to macroscopic flow of particles within the trap.
- 14.
Since the trap frequency also determines the susceptibility of the system to finite size effects, it can transform energy level crossings into avoided crossings, for example.
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Pinheiro, F. (2016). Confined p-Orbital Bosons. In: Multi-species Systems in Optical Lattices. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-43464-3_4
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DOI: https://doi.org/10.1007/978-3-319-43464-3_4
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