Probing pair correlations in Fermi gases with Ramsey-Bragg interferometry

The authors propose a new experimental method to extract information on the two-body density matrix in a two-component Fermi gas. The method is based on recording correlations between the numbers of spin-up and spin-down recoiling atoms after a Ramsey-Bragg sequence is applied to the system. In this way, information on both the presence of ODLRO associated with a pair-condensate and the internal spatial structure of pairs is obtained.

The description of the method is supplemented by a calculation of the proposed experimental signal based on a BCS mean-field approximation at T=0.

A direct experimental proof of superfluid ODLRO in an ultracold Fermi gas would be certainly a milestone. The method which is proposed here looks promising and the theoretical analysis is sound. I thus think that the paper should be published on SciPost Physics.

I have only just a few remarks that I would like to see addressed by the authors in a revised version of the manuscript.

1) Can the authors explain better how the duration of the pulse appear in inequality (1)?

2) In the second to the last sentence of the paragraph following eq (1), it is stated “fulfilling both inequalities”. However, only one inequality is introduced before this sentence (the second inequality is introduced in the last sentence of the paragraph).

3) I found the notation \Psi_{r,sigma}(r) for the recoil field operator rather confusing. It took me some time to grasp that the first “r” stand for recoil, the second for the position r. Could the authors introduce an alternative notation? For instance, using \psi (lowercase) for the recoil operator, or \Psi_{rec}?

4) The definition for the recoil operator in Eq. (4) is crucial. However, it comes with little explanation (a few words are given after Eq. (4)). Can the authors provide more details?

5) I found a bit confusing that Eq. (13) for S does not contain the term <\hat{N}_\up \hat{N}_down> - N_up N_down, but then this term pops up in Eq. (18). The reason is that <\hat{N}_\up \hat{N}_down> - N_up N_down = 0 if the number of particles is conserved, while this term is different from zero in Eq. (18) because the BCS wave-function does not conserve the number of particles. I think that the reader would be helped if a further line would be added in eq. (13) where the term <\hat{N}_\up \hat{N}_down> - N_up N_down is still present, and then in the next line this term is dropped (and the reason for dropping it would be explained immediately after).

6) The function S(x) in Fig. 2 approaches the ODLRO exponentially, over a length scale \xi_x which is related to the pair size. However, one would expect on general grounds that due to the presence of the Goldstone gapless mode in the broken symmetry phase, the asymptotic limit (yielding the condensate fraction) should be reached as a power law (see e.g. the book by D. Forster “Hydrodynamic Fluctuations, Broken Symmetry, and correlation functions (1975)”). The point is that to take into account this mode one should go beyond BCS mean field when calculating \rho_2 and include pairing fluctuations. This has been recently done by L. Pisani et al., PRB 105, 054505 (2022). The function h_2 shown in Fig. 9 of that paper corresponds to f_tr -1/2\rho_{1, up}\rho_{2,down} of the present manuscript. Within mean-field, this function would be constant, and equal to the condensate density. The inclusion of a Maki-Thompson-like diagram makes this function to depend on R and approach its asymptotic value as a power law. The various terms f_str of the present work, on the other hand, which are related to the stretching of a pair, should instead be described reasonably well by BCS mean-field. In summary, with the inclusion of fluctuations, the function S(x) shown in Fig. 2 should display first an exponential behavior over a scale \xi_x, followed by a power law decay when the contribution coming from f_str fades away, and only f_str survives. I suggest the authors to comment on this in their manuscript.

7) The term <\hat{N}_\up \hat{N}_down> - N_up N_down, which is different from zero within BCS, would become zero after projection of the BCS wave-function onto a number conserving wave-function. I thus wonder if it would not be better to drop altogether its contribution to S(x) (with appropriate explanation), in particular when presenting Fig. 2. We know that this constant term is spurious. In this way S(x) of Fig. 2 would become closer to the real physical S(x).

8) Eq. (22) for the length scale \xi_x coincides with the T-> 0 limit of Eq. (A11) for the length scale \xi_1 in the above work by Pisani et al. In that reference, \xi_1 is the scale over which the one-body density matrix decays. In addition, the same result is obtained by J.C. Obeso et al., New J. Phys 25, 113019 (2023) (see their eqs. (14) and (15)) for the scale \xi_alpha over which density-density correlators decay. Can the authors add a comment on this in their manuscript?