Unruh effect for interacting particles with ultracold atoms

Dear editor,

We resubmit to your attention the revised version of our manuscript. We have addressed all the comments and we have modified the manuscript accordingly to the suggestions of all the referees. Since we sent you our response to Referee 1 with the corresponding changes in the manuscript some time ago, we attach below only the response to Referee 2 and 3.

With best regards,
Arkadiusz Kosior
Maciej Lewenstein
Alessio Celi

Response to Referee 2

We thank the referee for the careful reading of our manuscript and the recognition of our work.

We understand the referee’s reservations and agree that a canonical observation of the Unruh effect requires a coupling between the field fluctuations and a De Witt detector. The transition rate of the De Witt detector is proportional to the response rate – the Wightman function in frequency domain – which is a key observable that we study in our paper. In the manuscript we argue, that in principle the Wightman function can be measured with ultracold atoms by using a one-particle excitation spectroscopy, and hence, we simulate a De Witt detector.

In order to stress this point and make it more apparent, we put more focus on this issue in the introduction to the new version of the manuscript:

<<In the experimental part, we discuss the feasibility of the model and argue that the Wightman response function of the De Witt detector can be, in principle, measured by using one-particle excitation spectroscopy (see also Ref. [107]).>>

Response to Referee 3

We thank the referee for the careful reading of the manuscript and the constructive criticism of our work. We believe that we have incorporated all the requested changes.

“In the introduction, saying the Unruh effect and Hawking radiation are the only phenomena experimentally accessible that can bring us closer to a true model of quantum gravity is a little misleading or strong for my liking, as for instance any kinematic effects of quantum field theory in curved space-time, could achieve something similar. “
We have rephrased the sentence:

<<Simulating the Unruh effect and the Hawking radiation (...) are among the experimentally accessible phenomena that can bring us closer to quantum gravity. >>

“1. Address the feasibility (for a realistic experiment) and improve the presentation for accessibility for the cold-atom community.”
“(…) To do this, the QFT calculations need to be presented in a more accessible way, and the discussion on what is capable and what would need to be done in an experiment could be extended. For example, on what scale can the interactions be tuned spatially? How would you measure the power-spectra? Is there some alternative to measuring the power spectra that gives you the same information? Comment on the feasibility to control the couplings at the boundary in an experiment: Is it feasible to assume the lattice couplings coincide at the boundary “

In the experimental part of our manuscript we focus on the feasibility of the interacting model and shortly review the results from our previous work (Ref. [107] in the new version) that deals with non-interacting case, where we discuss the details of the power spectrum measurement, quench protocol, the control of the tunneling amplitudes etc. At the same time, we believe that the theoretical part is needed in order to keep the article traceable and self-sufficient.

While here and in [107] we consider the detection of the Wightman function by one particle excitation spectroscopy as witness of the thermal behavior, one can in principle search for signatures of Unruh effect in other correlation functions, for instance density-density correlations.
This is an interesting research direction we plan to pursuit in the next future.

We have added the above paragraph to the Conclusions and Outlook section. More changes made in the experimental section of the new version of the manuscript :

<<The magnetic field (or in alternative the light shift) inducing the Feshbach resonance has to be then tuned spatially on the lattice spacing scale such to provide the desired $V$-shape interaction profile.>>

“(…) assume you can treat the interactions in a mean field approach, by taking the mean field averages of the interactions?”
This question was already answered in Sec. 4.

<<One may question the validity of the meanfield approach for values of the interactions of the order of the band width. While we can expect (small) quantitative deviations with respect to more precise approaches like diagrammatic quantum MonteCarlo, the qualitative behavior is known to be well captured by the mean field approach >>

“It seems quite odd that the thermal distribution statistics seen by an accelerated observer interchange for odd and even space-time dimensions. Why is this and why does this physically make sense? “

We discuss at length the mathematical origin and the physical implication of this phenomenon in Ref. 107. Here we have added few sentences to recall and summarize such discussion:

<<Let us stress that the later does not imply a violation of the canonical anticommutation (commutation) relations, but it is an apparent statistic interchange that comes from dimensional differences in wave propagation known as the Takagi inversion theorem [60]. In Ref. [107] we verified the interchange of statistics for noninteracting fermions with a dimensional crossover. >>

“2. Eqn 62: Remove extra spurious ‘+’ in the Wightman function.”
“3. Figure 1: Please fix the legend: This isn’t a linear function, it’s fit to a power function. Do you mean you use linear regression to find your best fit to this power function? “
We have fixed those minor errors. Again, we thank the referee for pointing them out.
“4. In Fig 3, One could argue that the Fermi-Dirac plateau is also not very obvious in the figure on the right and only features for |m|=40. Is this the cutoff |m| for which the plateau arises and what defines the critical number of Cooper pairs?”

The large distances to the effective horizon correspond to the least accelerating observers and the smallest Unruh temperature. The |m|=40 curve corresponds to the smallest Unruh temperature and therefore we observe an almost step-like sharp response at low energies. This behavior is smoother for other curves, still a clear inclination is visible. The number of Cooper pairs increase gradually with $\Delta$ (see Appendix B for the estimation of Cooper pair density), and the effect of statistics inversion can be observed when $\Delta$ becomes significant in comparison to other energy scales of Hamiltonian. In the new version of the manuscript we have expanded the discussion in a caption of Fig.3.