GHZ-like states in the Qubit-Qudit Rabi Model

Dear Prof. Babak Seradjeh, We thank the Referee for the comments on our manuscript and for stating that our work satisfies the SciPost criteria. In the revised version of the manuscript, we have addressed all the requests of the Referee. In particular, the discussion of the dynamics was substantially extended and the contents of the old Fig.5 is now split in two figures (Figs.5-6), with additional panels. Our reply to the reviewers and a summary of changes in the manuscript can be found below.

On behalf of all the authors, Yuan

Referee 1: Strengths 1) Timely topic with results relevant for quantum technology 2) Results appear to satisfy all the general criteria for SciPost Physics

Weaknesses 1) In the dynamics sections, an extended discussion of their results is necessary (requested changes). Report

The present manuscript performs a comprehensive study of a qubit-qudit coupled system within a Rabi-like minimal model. The authors use analytical approximations to study the weak coupling limit and ultra strong coupling regime. The validity of their approximations is demonstrated by comparing the approximate solution with exact numerical results. The main physical result is that the ground state in the strong regime corresponds to a maximally entangled Greenberg-Horne-Zeilinger state. This is demonstrated via negativity calculations as a measure of entanglement. Finally, the dynamics of the system is investigated in two limits: quench dynamics from the non-interacting limit, and in the adiabatic limit. I find this study very interesting and it satisfies the SciPost criteria in my view.

We thank the Referee for appreciating our work and for stating that it satisfies the SciPost criteria.

Requested changes and reply to Referee 1 1) I recommend to add perspectives for future work. We thank the referee for the suggestion. We foresee some further theoretical investigations. First, the system dynamics can be investigated in a dissipative situation, involving a description in terms of a modified Lindblad master equation. Second, our model can be embedded in a more general scheme where, for instance, the ultrastrong coupling is exploited to mediate the coherent state transfer between two given qubits. A similar idea has been put forward in a very recent preprint, which we cite in the new submission. In the revised version of the manuscript, we inserted a discussion of the future perspective of our work in the conclusions. The new text is highlighted in red.

2) A concrete proposal for a physical system where the interesting Greenberg-Horne-Zeilinger state could be realized would motivate further studies We thank the Referee very much for the suggestion. An option for GHZ implementation would be through cQED platforms. Alternatively, hybrid superconductor-semiconductor systems can be used. Indeed, both the ultrastrong and deep-strong regimes between the qubits and the resonators have been experimentally proven in cQED and, more recently, in semiconducting double quantum dots. The qudit in our model approximately describes a quantum system where the multi-level nature cannot be neglected. A standard example is the transmon qubit, which is characterized by weak anharmonicity. In the revised version of the manuscript, we inserted a paragraph in the conclusions to discuss the relevance of our model in current state-of-the-art experimental platforms.

3) In Fig. 5, the numeric and approximate solution depart from each other in roughly one period. An extended discussion on the reasons for this discrepancy is advised. Furthermore, in the text it is mentioned: "the approximate expressions capture the initial decrease in fidelity and the amplitude of its oscillations, as well as the main frequency components of the evolution.” It would be beneficial to compute the Fourier transform to show explicitly the frequency components are captured. We thank the referee for the comment, which prompted us to clarify some important points of our work. The approximation given in the main text for the ultrastrong coupling regime neglects the action of the “off-diagonal” terms in the displaced oscillator basis. The equations derived in this approach [Eqs. (17)-(18)] exactly describe the time evolution for $\Omega_1=\Omega_2=0$. Note that Eq.(18) has been added in the revised version of the manuscript; it generalizes the result presented in the first submission in which only the fundamental mode was retained. The terms depending on the frequencies of the qubit and the qudit have a sizeable impact on the dynamics. Hence, the (numeric) time evolution is only partially captured by our approximations. In the revised version of the manuscript, we give a more extended discussion of the dynamics. We have computed numerically the frequency spectrum (Fast Fourier Transform) of the various curves of Fig.5, comparing the analytical results to the numerical computations. To improve the visualization and help the discussion, we inserted a new figure (Fig.6 of the new submission). In the revised Fig.5, the fidelity plots are grouped with the corresponding frequency spectra. The time evolution of the expectation values and the associated Fourier transforms are reported in Fig.6.
The analysis shows that, in the qubit case, the analytical expression gives a fairly good approximation of the fundamental mode of the oscillation, with a small shift toward lower frequencies. For the qutrit and the ququart cases, in which the transitions can occur in a more complicated energy levels manifold, the approximation worsens.

List of changes 1- The discussion of the dynamics has been significantly extended, following the comment of the Referee. In particular, we inserted a new figure and included new plots, computing numerically the fast Fourier transform of the time evolved signals (both for the analytics and the numerics). We derived a new equation (Eq.18). The definition of $\tilde H$ is given in a new equation (Eq.10, previously defined as in-line equation). 2- We extended the discussion in the conclusions. More precisely, we discuss possible platforms for the experimental implementation of our model, and future perspectives, addressing the comments of the Referee. 3- We inserted new references: [27] Rev. Mod. Phys.91, 025005 (2019), [51] Phys. Rev. Lett. 105, 023601 (2010), [52] New J. Phys. 19, 023022 (2017), [53] J. Phys. A: Math. Theor. 50 294001 (2017), [54] Phys. Rev. Lett. 105, 237001 (2010), [55] Nat. Phys. 13, 39 (2017), [56] arXiv:2106.01669 [quant-ph], [57] arXiv:2104.03045 [cond-mat.mes-hall], [58] Nat. Commun.10, 3011 (2019), [59] arXiv:2104.14490 [quant-ph].

Attachment:

resubmission_version_Scipost1.pdf