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A correspondence between the Rabi model and an Ising model with long-range interactions
by Bruno Scheihing-Hitschfeld and Néstor Sepúlveda
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Bruno Scheihing-Hitschfeld |
| Submission information | |
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| Preprint Link: | scipost_202505_00021v1 (pdf) |
| Date submitted: | May 10, 2025, 12:57 a.m. |
| Submitted by: | Bruno Scheihing-Hitschfeld |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
By means of Trotter’s formula, we show that transition amplitudes between coherent states in the Rabi model can be understood in terms of a certain Ising model featuring long-range interactions (i.e., beyond nearest neighbors) in its thermodynamic limit. Specifically, we relate the transition amplitudes in the Rabi Model to a sum over n binary variables of the form of a partition function of an Ising model with n spin sites, where n is also the number of steps in Trotter’s formula. From this, we show that a perturbative expansion in the energy splitting of the two-level subsystem in the Rabi model is equivalent to an expansion in the number of spin domains in the Ising model. We conclude by discussing how calculations in one model give nontrivial information about the other model, and vice-versa, as well as applications and generalizations this correspondence may find.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Report #3 by Emanuele Dalla Torre (Referee 3) on 2025-10-1 (Invited Report)
Report
Before being able to make a recommendation, I would need the authors to address the following four major comments.
Major comments:
1. While intellectually stimulating, this work has limited applicability: one is generically interested in the real time dynamics, rather than in the imaginary one. Imaginary times is usually introduced to obtain finite temperature or ground state expectation values, but in this case one does not compute overlaps between coherent states. Can the author propose at least one scenario in which the quantity that they compute has a physical meaning? (The text below Eq. (28) suggests that in principle one can use this approach to compute the partition function of the Rabi model at finite temperatures, but this is calculation is not carried over in practice and translated into physical observables).
2. The (imaginary) time evolution of the Rabi model can be computed by truncating the Hilbert space of the bosonic mode. Can the authors demonstrate explicitly that the present mode is more efficient in terms of classical computational resources?
3. The method allows one to compute the overlap with respect to the eigenstates of the operators b, which are not the operators of the original model (a). The authors state (page 4) that one can go from one to other through “appropriate linear combinations”. Can the author explain this point in detail?
4. Section 5 seems to me pretentious: The Ising model under present consideration is a fine tuned one with a very specific type of interactions. Why would anyone encounter such a model in real life, except through the present derivation? Alternatively, can one learn something generic about Ising models through this mapping?
5. The Rabi model is known to undergo a quantum phase transition (in the appropriate limit). Is the present method applicable to this limit? How does the phase transition affect the corresponding Ising model? Also, does the classical model at hand have a phase transition? (It probably depends on the scaling of the long range correlations)
Minor comments:
6. Fig. 1 – add labels on the subplots to help the reader understand the difference between the different plots without reading the caption.
7. Style: the text is at some times sloppy and can be improved using common LLM tools.
8. Typos: Page 6: haven’t --> have not
9. The mapping reminds me of the Coulomb gas formalism used to study imaginary time field theories. See for example https://en.wikipedia.org/wiki/Coulomb_gas . Perhaps there is a relation between these approaches (I’m not sure)?
Recommendation
Ask for major revision
Strengths
2) Full mathematical details which make following a complex set of arguments possible
Weaknesses
Report
Requested changes
1) The coherent states used in eqn 2 are not the usual ones, as is made clear in the appendix, they are a combination of the regular coherent states along with a particular state for the spin. This should be clarified in the main text.
2) It would be interesting to discuss in more detail exactly what form the coupling in the Ising model has. For example, how does K depend on distance for different coupling strengths in the Rabi model? Are there signatures there of the different dynamical regimes present at weak and strong coupling?
Recommendation
Publish (meets expectations and criteria for this Journal)
Report
The article under review establishes an interesting connection between the Rabi model (an important paradigm of quantum optics) and classical thermodynamics, by expressing the time-dependent transition amplitudes in terms of a partition function of a long-range classical Ising model.
I enjoyed reading the paper, which is well-written (except for a confusing point on the definition of coherent states - see below) and describes nontrivial and beautiful mathematical manipulations, through which the Authors work out in detail the relation between the two formulations. The main shortcoming of the article is, I would say, that the connection does not reveal new features of the two systems. After all, the Rabi model only involves a single atom in a cavity and its properties are relatively straightforward to extract in a direct manner.
This said, the Authors point out in Section 4 that the thermodynamic formulation of the transition amplitudes could be advantageous when the coherent states have large amplitudes, as it allows to avoid the use of a large Hilbert spaces (see After Eq. 22). A similar comment applies to the large-temperature limit of the partition function in Eq. 26 (see after Eq. 28). Therefore, the new formulation of the time evolution already offers some technical advantages. Furthermore, the method could have more general applicability. This is noted in the conclusion, where the Authors propose that the application of a similar approach to other quantum model could be an interesting direction for future studies.
I agree that the idea of this paper could have general applicability. In fact, it basically coincides with the well known quantum-to-classical correspondence in statistical mechanics, where a quantum model with d dimension is mapped to a classical model with d+1 dimensions. Here the Trotterization is performed in real time instead of imaginary time (as we are dealing with a transition amplitude, instead of a partition function). My impression is that this point should be emphasized more prominently in the article, in the introduction or even in the abstract. In the current version, I only noticed such a remark at the end of Section 2.
Finally, the Authors might be interested to consider the limit w/w0 --> 0, when a phase transition to a superradiant phase takes place place at gc=\sqrt{w w0}/2. This topic recently attracted considerable attention, especially following M.-J. Hwang, R. Puebla, and M. B. Plenio, Phys. Rev. Lett. 115, 180404 (2015). An earlier and quite transparent discussion can be also be found in L. Bakemeier, A. Alvermann, and H. Fehske, Phys. Rev. A 85, 043821 (2012). Perhaps the thermodynamic formulation in terms of the Ising model could allow to interpret this phase transition from a different angle. Upon inspection, I do not see this physics emerging in a very obvious way from the classical partition function, therefore I leave this as an optional suggestion.
Requested changes
1) When reading the article, I was confused by the definition of coherent states and amplitudes, in particular Eq. 2 and Eq. 3. For Eq. 2, it is clear that |alpha> is the regular coherent state, satisfying the usual condition a|alpha>= alpha|alpha> (as stated before Eq. 2). However, in this case A_\beta\alpha of Eq. 2 would be a spin operator. Instead, a few lines below A_\beta\alpha is called "amplitude" and amplitudes are usually c-numbers. Even more confusing was for me the following section 2, as |alpha> now indicates a different type of coherent state. The Authors actually note at the end of Section 1 that they use the coherent states of b (and not of a) but I just assumed that they take them in the simple form |alpha>|\pm>, where the spin states are eigenstates of sigma_x. This choice, I would say, is the most intuitive one since b=a sigma_x. However, also in Eq. 3 the spin degrees of freedom do not appear explicitly.
As it turns out, the form of the coherent states is different from my initial guess, and is only given in Appendices A and B. To avoid misunderstandings, I think it is better to define the coherent states of b explicitly in the main text, before introducing the amplitudes. Then, one can appreciate immediately that they are entangled states between spin and cavity with an additional subscript \pm (distinguishing |alpha>_\pm from the usual |alpha>), which is later omitted.
2) I was confused by the meaning of \delta_l,j+1 in Eq. 15 (perhaps because of the concomitant presence of \delta_n). Maybe it is not completely useless to remind the reader that \delta_l,j+1 is a Kronecker delta.
3) Equation A.1 does not coincide with Eq. 1, since \sigma^+\sigma^- is not equal to \sigma_z. The relation on line 284 is wrong and should be corrected. Perhaps it is better to directly use sigma_z in Eq. A.1, like in Eq. 1.
4) I noticed these minor misprints: - there is a space missing on line 283, before "hbar=1" - on line 325, evolution-->evolve? -|0> is missing in on the rightmost side of Eq. B.4
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
We appreciate the referee’s overall positive evaluation and the suggestion to emphasize more clearly that the Trotterization in our analysis is carried out in real time. In the revised version that we intend to resubmit, we plan to add sentences in the abstract and introduction highlighting this fact, as well as a further sentence in the introduction to contrast it with the Trotterization in imaginary time that we discuss later on in the manuscript. We also thank the referee for pointing out to us the limit $\omega/\omega_0 \to 0$ and the associated transition to a superradiant phase. While a full analysis of this regime is beyond the present scope of our work, we will mention this in the conclusion as a possible future direction.
Regarding the requested changes: 1) After re-reading our manuscript, we agree with the referee that we should have introduced the (generalized) coherent states in full detail in the introduction, to avoid confusion. We plan to add a couple of paragraphs discussing the differences of these states with the coherent states $|\alpha,\pm\rangle$, and also how one can write one kind of coherent states as a linear combination of the other kind. 2) We will add a sentence doing so. 3) We thank the referee for pointing out this mistake. Indeed, $\sigma^+\sigma^-$ is not equal to $\sigma_z$. We will correct Eq. (A1) by explicitly writing it in terms of $\sigma_z$, in agreement with Eq. (1). In addition to that, we will modify the corresponding sentence on line 284 to be consistent with this correction. 4) We thank the referee for carefully pointing out these misprints. We have corrected all three.
Author: Bruno Scheihing-Hitschfeld on 2025-09-24 [id 5857]
(in reply to Report 2 on 2025-09-03)We thank the referee for the positive assessment under “Strengths” and for pointing out that the analysis of the resulting Ising model was somewhat superficial. We will address this by expanding the discussion of the Ising couplings and their relation to different dynamical regimes of the Rabi model.
Regarding the requested changes: 1) We agree with the referee that we should have introduced the (generalized) coherent states in full detail in the introduction, to avoid confusion. We plan to add a couple of paragraphs discussing the differences of these states with the coherent states $|\alpha,\pm\rangle$, and also how one can write one kind of coherent states as linear combinations of the other kind. 2) We plan to expand the discussion in Section 4 (between Eqs. (23) and (24)), explaining in more detail the form of the Ising couplings $K_{j,\ell}^E$ and how they reflect the different dynamical regimes of the Rabi model. Concretely, we will explain that the kernel $K_{j,\ell}^E$ has two contributions: (i) a nearest-neighbor term, which plays the role of a strong coupling between consecutive time slices in the Trotter decomposition and depends on $\omega_0$, and (ii) a ferromagnetic, exponentially decaying tail whose amplitude grows with $g^2$ and whose range is set by $\omega$. In the continuum limit, this second contribution becomes a nonlocal interaction kernel proportional to $e^{-\omega\tau |u-v|}$. Taking the Fourier transform of this long-range part alone shows that its spectral weight has the form $\widetilde{K}(k) \propto [(\omega\tau)^2+k^2]^{-1}$, i.e. a Yukawa-type kernel. This Fourier representation is not used in the subsequent calculations, but it is useful to highlight the effective range and structure of the induced interactions on the Ising side. This structure reveals how different dynamical regimes of the Rabi model appear on the Ising side. For weak coupling $g \ll \omega,\omega_0$, the long-range piece is negligible and the dynamics is dominated by the short-range stiffness. Conversely, in the ultra-strong and deep-strong coupling regimes $g\sim \omega$ or $g \gg \omega$, the ferromagnetic tail becomes long-ranged and sizable. This behavior encodes the crossover between perturbative dynamics and the strongly correlated regime, providing an explicit Ising-side diagnostic of the dynamical phases of the Rabi model.