A quantum register using collective excitations in a Bose--Einstein condensate

Dear Editor,

Thank you very much for soliciting reports from the referees, and thank you to the referees for reviewing our manuscript. We have provided a short summary of the changes we made below, and have highlighted the changes in the manuscript in yellow. We hope that our work will now be accepted for publication.

Gratefully,
Elisha Haber, Zekai Chen, and Nicholas P. Bigelow

(1) We have removed the reference to Rydberg atoms from the abstract.

(2) The last sentence of the paragraph following Eq. (5) has been changed to explain that the energy shift due to the HO potential is contained in delta in Eq. (5) (and the following equations).

(3) Eq. (4) is no longer referred to as a Bose-Hubbard Hamiltonian.

(4) In the new manuscript the number of atoms in the various states are always denoted using lowercase n, while the total number is denoted using N.

(5) We thank the reviewer for pointing out this mistake in the last paragraph of section 3.2, and have changed omega_{01} to omega_{12}.

(6) We thank the reviewer for carefully considering our proposal to implement a sqrt{SWAP} gate. In this approach, a virtual transition is employed where an atom begins and ends in separate lattice sites, and in the intermediate state the atom is in the harmonic trap. The BEC is off-resonantly coupled to the harmonic trap states to modify their energy spectrum. The simple physical picture mentioned by the referee, where the atom is moved between lattice sites due to the momentum kick it receives from absorbing/emitting MW photons, is incorrect. In our model, this momentum is neglected because, as noted by the referee, it is small. Instead, a more accurate semiclassical picture is that of a gas expanding after being released from a box. If the atom were to absorb a photon and transition to the intermediate state it would go from being tightly confined in the lattice to being nearly free in the harmonic trap, and its wavefunction would rapidly expand, eventually overlapping with the lattice site we wish to transfer it to. It could then emit a photon into the field and end up in the second lattice site. In our case we consider a two-photon virtual process, but the semiclassical picture described above provides an intuition for our proposal. The speed of this process is set by the contributions from the many vibrational levels of the harmonic trap, the intensity of the field, and the locations of the lattice sites. We emphasize that the low-energy effective Hamiltonian we used to model this process was derived using Eq. (8), and follows the same steps shown in the CNOT gate section of the paper (but in a Hilbert space with different basis states). We also note that there are many papers (experimental and theoretical) that transport atoms between lattice sites through an intermediate virtual state that are modeled in exactly the same way (meaning no retardation due to the atom travelling between distant sites is added to the theory) [1,2]. Although we respectfully disagree with the reviewer about the validity of our sqrt{SWAP} gate, we have removed this section and all references to it from the new manuscript.

(7) The detunings in the equations for V_{HO} and V_{OL} in section 4.1 have been changed to delta_{1,2} and delta'_{1,2}.

(8) We have simplified the discussion on driving the atomic transitions using MW and RF fields by removing the discussion of optical Raman beams.

1. H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Burton, and W. Ketterle, Realizing the Harper Hamiltonian with Laser-Assisted Tunneling in Optical Lattices, Phys. Rev. Lett. 111, 185302 (2013), DOI: 10.1103/PhysRevLett.111.185302.

2. A. B. Deb, G. Smirne, R. M. Godun and C. J. Foot, A method of state-selective transfer of atoms between microtraps based on the Franck–Condon principle, Journal of Physics B: Atomic, Molecular and Optical Physics 40(21), 4131 (2007), DOI: 10.1088/0953-4075/40/21/001.