Optical lattice with spin-dependent sub-wavelength barriers

We would like to thank the Referees for carefully reading our manuscript and providing helpful comments and suggestions. We have revised the manuscript taking into account the Referee comments. In the revised manuscript we have added four new figures and two extra references. We refer to the new ordering of figures and references in the list of changes and remarks for the referees.

MAIN CHANGES:

1. Two new references [33,34] on cooling atoms and ions using the tripod scheme have been added in the Introduction.

2. A new figure in Sec. 3.2.2 shows the functions characterizing the effective potentials (Figure 3). We have also added references to this figure in the appropriate places of Sec. 3.2.2.

3. In the numerical code, we use the wavenumber 2pi/a, while in the paper we use the wavenumber pi/a to define the recoil energy. As such, some of the system parameters written under the Figures were originally missing a factor of 4. We have corrected the parameters under the figure captions and have redrawn the figures with the correct parameters where necessary to address this issue. As a consequence, Figs. 4 through 12 have been changed in the way just described.

4. A paragraph has been added before Section 4 to discuss the limit for alpha -> pi/2.

5. A new figure before Sec. 4.2 (Figure 5) shows the energy dispersions for zero detuning, obtained by the 3 different methods presented in the paper. Additionally, the paragraph from Section 4.3 has been moved next to Figure 5. This paragraph has also been rewritten to represent more accurately our results, namely we emphasize that the 4x4 scattering approach recreates very accurately the exact odd band structure for large values of the phase parameter. This is emphasized again in Sec. 4.3 where the tight binding tunnelling parameters are discussed.

6. A new figure in Sec 4.3 (Figure 9) shows the tight binding tunnelling parameters of the first two energy bands as a function of phase, demonstrating the free particle-like behaviour of the odd bands, and the even bands approaching Lambda-like bands with only non-zero nearest neighbor (NN) tunnelling for alpha -> pi/2. Two paragraphs were added in the same Section to discuss these results.

7. The last paragraph in Section 4.3 about the detuning has been extended to explain its effects on the Wannier functions and the tunnelling parameters. We have also made a connection there with the Lambda-like setup.

8. The “concluding remarks” section has been extended to discuss the adiabatic loading of ultracold atoms and future prospects.

9. At the end of the Concluding Remarks section a note has been added on a recent preprint by Kubala et al.[44].

10. Appendix A has been altered considerably. First, a state vector orthogonal to that given by Eq. 40 has been introduced in Eq. 43. The corresponding two components of the Wannier function have been plotted for the first energy band in Fig. 13 for various of alpha, demonstrating the effects of non-zero alpha on the Wannier functions, and helping understanding the connection between the Wannier functions and the tunnelling parameters. Secondly, we have further explained how the Wannier functions from the adiabatic dark state Hamiltonian are optimized.

MINOR CHANGES:
1. A missing factor of 1/2 was added in the second part of Eq. 1.
2. There were notational inconsistencies between Eqs. 17 and 38. Namely, Eq. 38 was previously marked with a ket, while Eq. 17 was not. This issue has been fixed.
3. A typo in “the expanded 1BZ covering the range $2\pi/a<q\le2\pi/a$” was fixed, which now reads “the expanded 1BZ covering the range $-2\pi/a<q\le2\pi/a$”
4. Figure 6 now contains dotted lines that represent the Wannier functions obtained from the adiabatic dark state Hamiltonian. We have also added grid lines for better readability.
5. Figure 4 now contains dotted lines that represent solutions of the adiabatic dark state Hamiltonian for non-zero detuning.
6. Epsilon has been replaced with epsilon=Omega_p / Omega_c in many places in the paper to help the reader recollect the definition.
7. Equation 20 was rewritten in terms of newly defined row and column vectors constituting the geometric scalar potential.
8. Figure 7 caption was shortened and the discussion contained in this caption was moved to the main text in Section 4.3.
9. An acknowledgement to Ian B. Spielman was added.