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Unruh effect for interacting particles with ultracold atoms

by Arkadiusz Kosior, Maciej Lewenstein, Alessio Celi

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Submission summary

As Contributors: Alessio Celi · Arkadiusz Kosior
Arxiv Link: http://arxiv.org/abs/1804.11323v2 (pdf)
Date submitted: 2018-05-08 02:00
Submitted by: Kosior, Arkadiusz
Submitted to: SciPost Physics
Academic field: Physics
Specialties:
  • High-Energy Physics - Theory
  • Quantum Physics
Approach: Theoretical

Abstract

The Unruh effect is a quantum relativistic effect where the accelerated observer perceives the vacuum as a thermal state. Here we propose the experimental realization of the Unruh effect for interacting ultracold fermions in optical lattices by a sudden quench resulting in vacuum acceleration with varying interactions strengths in the real temperature background. We observe the inversion of statistics for the low lying excitations in the Wightman function as a result of competition between the spacetime and BCS Bogoliubov transformations. This paper opens up new perspectives for simulators of quantum gravity.

Ontology / Topics

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Ultracold atoms
Current status:
Has been resubmitted



Reports on this Submission

Anonymous Report 3 on 2018-10-24 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1804.11323v2, delivered 2018-10-24, doi: 10.21468/SciPost.Report.625

Strengths

1. This is an interesting paper proposing a potentially new way to realise the Unruh effect in a table-top experiment, in particular with a lattice of ultra-cold fermions, extending the previous theory from this group and adding to the existing literature on artificial gravity.

Weaknesses

1. If the aim of the paper is to suggest a potential cold atom experiment, some care needs to be taken to make it accessible to the cold-atom community. How the paper is presented is as a quite heavy theoretical quantum field theory / gravity manuscript, and it is not always clear what the observation in a cold-atom experiment should be. There is very little discussion on if the suggested scheme is within current capabilities, or the limitations and potential challenges of the proposed scheme.
2. In presentation of the research, grammar and English could be polished a little, and some further clarification in some areas could be helpful.

Report

I found this to be an interesting paper, proposing a potential new way to realise a simulation of the Unruh effect in a table-top experiment, with ultra-cold fermions in optical lattices. I cannot really comment on the ease of realisation in an experiment, but imagine it would be quite challenging. I think this is an interesting proposal, worthy of publication, and suggest some minor comments for the authors consideration.

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.

As mentioned earlier, I think this would be a stronger paper if it was made more more accessible 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, and also assume you can treat the interactions in a mean field approach, by taking the mean field averages of the interactions?

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?

Going further, the potential for extension to trans-Plankian physics and Lorentz-violation toy models is interesting although this dicussion lacks citations to related toy-models looking at trans-Plankian physics in the analogue gravity field, see for example papers by Liberati and Visser for example. It would be certainly worthwhile to see what ultra-cold fermions in lattice simulators bring to the table in comparison, and consequently the limitations they put on potential models of quantum gravity.

Requested changes

1. Address the feasibility (for a realistic experiment) and improve the presentation for accessibility for the cold-atom community.

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?

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?

  • validity: good
  • significance: good
  • originality: good
  • clarity: low
  • formatting: reasonable
  • grammar: reasonable

Author:  Arkadiusz Kosior  on 2018-11-14  [id 343]

(in reply to Report 3 on 2018-10-24)

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.

Anonymous Report 2 on 2018-10-23 (Contributed Report)

  • Cite as: Anonymous, Report on arXiv:1804.11323v2, delivered 2018-10-22, doi: 10.21468/SciPost.Report.622

Strengths

See report

Weaknesses

See report

Report

*** This report was sent to the editors via email, rather than through the online refereeing system. The referee has not responded to any follow up email. The criteria have been selected by the editor without regard to the paper. ***

I would like to directly comment on the questions raised by first referee: “cannot judge whether the core theoretical part is novel or even correct”. The theoretical part is correct and also novel. I on the other hand cannot judge the feasibility of the experimental implementation.

My only reservation is that I don’t think an experimental implementation of the proposal can be regarded as measurement of the analogue Unruh effect. The system is prepared (through a quench of a lattice with varying interaction strength) in a certain state, and only afterwards the measurement is being made. In my opinion, the Unruh effect requires a coupling between an accelerated detector (e.g. in its simplest case a two-level system) and the fluctuations of the analogue spacetime geometry. Through this interaction the detector will heat up.

However, I find the paper interesting and novel, and hence I recommend it’s publication.

Requested changes

None specific

  • validity: top
  • significance: top
  • originality: top
  • clarity: top
  • formatting: perfect
  • grammar: perfect

Author:  Arkadiusz Kosior  on 2018-11-14  [id 342]

(in reply to Report 2 on 2018-10-23)

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]).>>

Anonymous Report 1 on 2018-5-21 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1804.11323v2, delivered 2018-05-21, doi: 10.21468/SciPost.Report.461

Strengths

1- Originality: the paper explores a rather uncharted territory, namely the possibility to simulate aspects of quantum gravity with cold atoms in optical lattices
2- Timeliness: the proposal makes use of the state of the art experimental techniques and suggests novel ways to use the existing capabilites (or some that may be developped in reasonable future)

Weaknesses

1- Technicality: the paper is targeted at the cold atoms community, which is not used to the concepts from quantum gravity. Even though the authors make efforts to make the paper readable, this remains quite a large obstacle.
2- Experimental relevance: the experimental proposal described is extremely complicated. Even though this may stimulate further thinking or trigger extra theoretical steps, it is unlikely that in the present form it may be implemented by an experimental team.

Report

In this paper, Kosior and coworkers discuss the possibility to explore the combination of quantum mechanics with gravity using a quantum simulation approach, by which neutral atoms in a tailored optical lattice are used to mimic the Unruh effect. The possibility to use cold atoms to simulate some aspects of high-energy physics is very attractive. It represents a new line of research, where a lot remains to be done to bridge the theory-experiment gap. The paper is timely, since the main experimental tools proposed by the authors, namely local control of interactions in optical lattices and gauge field engineering are the topic of intensive experimental investigations.

I was asked for my opinion as experimentalist. The work is clearly of theoretical nature, and quite technical. Efforts have been done by the authors to describe the concepts in simple words, but the underling complexity of the problem makes it hard for the typical cold atoms physicist to follow, which I guess cannot be avoided.

The part describing the possible experimental realizations is quite short, but lists the most salient points. The authors seem to be well aware of the current technical limitations of the experiments, and their proposal combines the state-of-the-art techniques for lattice control, with a few novel schemes which are in principle reachable but still represents a huge challenge. Overall, the proposal is quite frightening in terms of complexity, even though each individual aspect is not in principle out of range. Most of the technical descriptions are already present in the previous paper of the same team (ref 54), with the addition of locally controlled interactions via a Feshbach resonance. This has been demonstrated by the groups of C. Chin (PRL 115, 155301 (2015)) and J. Thomas (PRL 116, 075301 (2016)), to cite only the most recent developments. These references might be relevant and could be included in the paper.

One naive question (from the uneducated experimentalist), is the crucial role played by the linear dispersion relation. In existing experiments, no Dirac Fermions have actually been directly prepared, only the linear dispersion of the band close to the Dirac point was probed. Creating actual Dirac Fermions by tuning the density exactly to fill the band up to the Dirac point is a very difficult task due to the small density of states, and all experimental imperfections, such as inhomogeneities come into play. How robust would the scheme be to non perfectly linear dispersion ?

I cannot judge whether the core theoretical part is novel or even correct, but from the point of view of experiments the paper is welcome, even though still quite ambitious in terms of realizability. I see no obstacle to publication provided the theoretical aspects are reviewed by a proper expert.

Requested changes

1- Please comment on the robustness to not-fully-linear dispersion for the Fermions (see report above).

  • validity: ok
  • significance: good
  • originality: good
  • clarity: ok
  • formatting: good
  • grammar: excellent

Author:  Arkadiusz Kosior  on 2018-07-11  [id 290]

(in reply to Report 1 on 2018-05-21)
Category:
answer to question

We thank the referee for the appreciation of our work, for the careful reading of the manuscript, and for the constructive criticism.

The question is raised by the referee is very interesting. We expect that having a moderate non-zero chemical potential it is not critical for the observation of the Unruh effect in optical lattices. A non-zero chemical means that what we are preparing is not the vacuum but a state that it is slightly populated by particles or antiparticles, depending on the sign of the chemical potential. We expect that the Planckian signature in the Wightman response function to be robust against chemical potentials up to values of the order of the Unruh temperature. Such situation is indeed not so different than starting with a gas of atoms at non-zero temperature. As we show for both interacting and non-interacting Dirac fermions, distinctive features of the Planckian spectrum survive up to temperature of the order or even slightly higher that the Unruh temperature itself. We also expect that the finite temperature of the sample will be in general larger in typical ultracold atom experiments than the error in the calibration of the atom filling, or that at worst they are of the same order. Thus, we conclude that the essential requirement for the experiment is that they are at most of the order of the maximal Unruh temperature, with is a fraction of the band width.

We back the arguments raised above with the numerical study of a non-zero chemical potential for non-interacting fermions (as argued above we expect a similar behavior also in presence of interactions. We prepare two figures (in attachment) that will be included in the revised version of the manuscript together with the following captions:

1)"The influence of non-zero chemical potential and the non-zero temperate of atoms on the power spectrum of Wightman response function. In the plot (top panel) the distance to the horizon is fixed $|m|=30$ which corresponds to the Unruh temperature $T_U\approx 0.015$ (see Fig. 1). As expected, $\mu=0.1$ and $\mu=0.15$ data deviate from $T=0$ curve only for small positive frequencies. One can see that $\mu=0.1$ curve is very close to $T=0$ result. The $\mu=0.15 $ curve is more thermal, still it is better then $T=0.2$ curve." 2) "The power spectrum of Wightman response function for a fixed nonzero chemical potential $\mu=0.15$ (bottom panel). Although the chemical potential is about an order of magnitude larger then the Unruh temperature, we can still distinguish the characteristic thermal properties of power spectra (see Eq. 60)."

Attachment:

wightman_mu.pdf

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