Hierarchic superradiant phases in anisotropic Dicke model

List of changes: 1. More references have been added. 2. A discussion on negative energy has been added. 3. The discussion on stability has been rephrased. 4. The discussions on the truncation of the Hilbert space have been added. 5. A discussion on the impact of the initial state on the phase boundaries has been added. The main revised text is marked in red in the manuscript.

Referee's comments: The authors consider a generalized (anisotropic) Dicke model (1), where counter-rotating-wave and and rotating-wave terms may have different amplitudes. By using a standard Holstein-Primakoff transform in the thermodynamic limit, they arrive at the usual quadratic harmonic oscillator system. Diverting from the standard treatment, they perform a rotation (5) and introduce non-Hermitian Nambu spinors with the aim to decouple the new modes. In the resulting Hamiltonian (6), the block matrices are non-hermitian and display a formal similarity to exceptional points where both eigenvalues and eigenvectors become degenerate. The exceptional points define additional phase boundaries in the superradiant phase, where sub-phases can be classified by the stability of the Hamiltonian. To highlight the physical relevance of the exceptional points, the authors consider the Loschmidt echo for a particular initial state and indeed find from extensive finite-size numerical simulations that this experimentally accessible quantity also shows the phase boundaries defined by the exceptional points with distinct dynamical features (Figs 2,3). They conclude that these hidden exceptional points signal significant changes in complete sets of eigenstates. In general, I found the paper interesting to read. I have a few objections though and would recommend the authors to revise their paper, taking into account the comments below.

Author's response: We sincerely appreciate the referee for the careful reading of our manuscript and the valuable and constructive comments. We have revised our manuscript accordingly based on the feedback provided. Below is our point-by-point response to the comments. All modifications have been highlighted in red in the revised version.

Referee's comments: (1) I am missing a discussion of the origin of negative energies e.g. in (16). It is perfectly acceptable to theoretically discuss Eq. (1) but the light-matter minimal coupling procedure would normally yield a quadratic term that prohibits the superradiant phase transition in standard setups, see e.g. Refs https://doi.org/10.1038/ncomms1069 and https://doi.org/10.1103/PhysRevA.86.053807 or similar, and always provides a lower bound to the spectrum of the Hamiltonian, unlike (15-16). Since the authors care about an experimental verification, I think some link to such discussions could be relevant.

Author's response: We thank the Referee for raising this important question. Regarding the origin of the negative energy in Eq. (16), we wish to clarify that it stems directly from the diagonalization of the Hamiltonian $H_2$ (Eq. (12)) under the specific parameter regime where $|\omega - g_1| > g_2$ and $\omega - g_1 < 0$. Within this parameter interval, the diagonalization process naturally yields the form presented in Eq. (16). Consequently, the emergence of negative energy is an inherent result of diagonalizing a system with a negative effective chemical potential. Concerning the second part of the comment, we fully acknowledge the Referee's key point that the $A^2$ term in the standard light-matter minimal coupling framework indeed suppresses the superradiant phase transition in conventional setups. However, we wish to note that recent research has demonstrated that the anisotropic Dicke model can circumvent this superradiant no-go theorem [Phys. Rev. A \textbf{110}, 063722 (2024)], thereby providing a theoretical basis for its potential experimental realization in platforms such as cavity QED. In our revised manuscript, we have refined the explanation for the negative energy terms and incorporated a discussion, along with the corresponding reference, concerning the no-go theorem for superradiance.

Referee's comments: (2) What freedom does one have when introducing the spinor form in (6)? As it is written, it just looks like a convenient way to identify the phase boundaries of the model (17,18). I suppose that these could have also been extracted from (4) by a two-mode Bogoliubov transform, and we could have inferred the phase boundaries also from the vanishing of the $\Omega _{i}$. What is the extra benefit of the exceptional point picture and can it be generalized to arbitrary bosonic/fermionic quadratic models and/or multiple modes?

Author's response: We thank the Referee for raising this insightful question. We acknowledge that there exists a certain degree of freedom in the choice of representation when decoupling the effective Hamiltonian. We fully agree that the phase boundaries of the model can indeed be obtained by performing a conventional two-mode Bogoliubov transformation on Eq. (4) and setting $\Omega_{i}=0$, which is consistent with the method we detailed in Appendix A. In general, there are two equivalent approaches to diagonalizing such quadratic models: (i) the Bogoliubov transformation, and (ii) diagonalizing the core matrix within the spinor form of the Hamiltonian. However, for bosonic systems like the one in this work, the core matrix is non-Hermitian. The key advantage of our chosen spinor representation (Eq. (6)) lies in its ability to directly and clearly reveal the exceptional point structure within the core matrix: the phase boundaries correspond precisely to the points where these non-Hermitian matrices exhibit exceptional points—characterized by the simultaneous degeneracy of eigenvalues and eigenvectors. This physical picture is not immediately intuitive within the standard Bogoliubov transformation framework. Furthermore, this spinor representation is generalizable and can be extended to other quadratic systems [New J. Phys. 22, 083004 (2020)]. In the revised manuscript, we have supplemented the discussion regarding the origin and advantages of choosing the spinor representation.

Referee's comments: (3) Below (17,18), I find the discussion of stability rather confusing. A Hamiltonian without a lower spectral bound like (16) would also be unstable in the sense that a thermal reservoir would induce more and more particles in the system. So please clarify stability here.

Author's response: We thank the Referee for raising this crucial point, which has helped us clarify the ambiguous statements in our manuscript. The Referee is absolutely correct that a Hamiltonian without a spectral lower bound would indeed lead to thermodynamic instability when coupled to a thermal reservoir. We wish to clarify that the term "stability" discussed in our original text specifically refers to the dynamical stability of the isolated system in the thermodynamic limit, i.e., whether the solutions of the equations of motion exhibit exponentially growing modes. Based on this clarification, we reiterate: (i) Harmonic oscillator (Eq.(14)): Possesses a spectral lower bound and is dynamically stable (exhibiting oscillatory solutions). (ii) Anti-harmonic oscillator (Eq.(16)): Features an inverted energy spectrum and lacks a thermodynamically stable ground state. However, within its defined equations of motion, the solutions are oscillatory, and it is therefore dynamically stable within our defined context. (iii) Inverted harmonic oscillator (Eq.(15)): Has no spectral lower bound, and its equations of motion support exponentially growing solutions, rendering it dynamically unstable. In the revised manuscript, we have rewritten the relevant paragraphs below Eqs.(17,18) to explicitly adopt the above stability definition based on dynamical behavior. We have also removed potentially confusing statements regarding "ground state stability" and similar expressions. We sincerely thank the Referee for prompting this important clarification.

Referee's comments: (4) How sensitive is the highlighting of phase boundaries to the chosen initial state in the Loschmidt echo? I could imagine that the current initial state has been chosen to highlight the importance of the counter-rotating terms. Another interesting choice could be to investigate a single reservoir version of Dicke superradiant decay (all atoms in their excited state, the cavity mode in its ground state).

Author's response: We thank the Referee for raising this highly insightful question. We agree that investigating the dynamics under different initial states is of significant interest. In our current work, since the effective Hamiltonian is derived under the condition that atomic excitations are much smaller than the total number of atoms, the phase boundaries revealed by the Loschmidt echo are insensitive to the specific choice of initial state, provided this condition is satisfied. The initial state proposed by the Referee---with all atoms in their excited state and the optical field in the vacuum state---indeed represents a particularly important special case. We have conducted a deeper analysis of this scenario and discovered a profound connection: through a global SU(2) spin rotation (specifically, mapping $J_z\rightarrow-J_{z},J_{\pm} \rightarrow J_{\mp}$), the dynamics under this initial state can be strictly mapped onto the problem we have already studied in detail, which starts from the atomic ground state. The direct physical consequence is that the effective phase diagram measured from this initial state would be a "mirror image" of the one presented in our paper---specifically, the $\mathrm{SP_1}$ and $\mathrm{SP_3}$ regions would be swapped, while the $\mathrm{SP_2}$ region remains unchanged. In the revised version, we will include a discussion of this point.

Referee's comments: (5) Finite-dimensional quantum systems will evolve (quasi-)periodically (e.g. https://doi.org/10.1103/PhysRevA.18.2379). I do not see that captured in (38). Please clarify the timescales over which the approximations can be expected to be valid.

Author's response: We fully concur with the Referee's perspective. The Referee correctly highlights the crucial point that finite-dimensional quantum systems exhibit quasi-periodic evolution. Consequently, the exponential decay behavior described in Eq. (38) for the $\mathrm{SP_2}$ and $\mathrm{SP_3}$ phases cannot persist indefinitely in a finite system. We would like to clarify the following: Our analytical expressions, including the exponential decay in Eq. (38), are derived in the thermodynamic limit ($N\rightarrow\infty$), where the energy spectrum of the system becomes continuous. For a system with finite $N$, our theoretical description remains valid over intermediate time scales before the onset of finite-size revivals, a duration that extends as the system size $N$ increases. In the revised manuscript, we have added clarifications in the text surrounding Eq. (38) and in the caption of Fig. 3. These explicitly state that the analytical results apply in the thermodynamic limit and emphasize that for the presented time scales and our chosen $N=100$, finite-size effects are negligible, and the numerical results are in excellent agreement with the theoretical predictions.

Referee's comments: (6) I also find the summary where EP-phase separations are proclaimed very different from quantum phase transitions a bit confusing. Also in paradigmatic models of quantum phase transitions like the 1d Ising model in a transverse field or Dicke model all eigenstates undergo sudden changes at the critical point, so I do not see a strong contrast here. It is true that mostly one focuses on the ground state but finite-temperature quantities may also be affected. Maybe the exceptional point picture rather highlights the role of the dynamics?

Author's response: We thank the Referee for this nuanced comment, which has helped us refine our statements. We fully concur with the Referee's observation that in standard quantum phase transitions, the structure of all eigenstates may change at the critical point (e.g., gap closure affects the entire spectrum). Our intended contrast is not with this fact, but rather pertains to a dynamical phase transition. The key point we wish to emphasize is that the phase separations induced by exceptional points (EPs) within the superradiant phase are distinct from the superradiant phase transition driven by spontaneous $Z_2$ symmetry breaking in the ground state. Instead, they manifest as singular changes in the excited-state structure and dynamics of the system, which can be experimentally measured using probes such as the Loschmidt echo. The EP framework provides a natural perspective for understanding these dynamical features, and we have further emphasized this point in the conclusion of our revised manuscript.

We sincerely thank you for the detailed list of minor comments. We have revised our manuscript accordingly based on your suggestions and hope that the revised version meets with your approval and recommendation.

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