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Journal of Physics A Mathematical and Theoretical

Reply to Comment on ‘Initial states of qubit–environment models leading to conserved quantities’

Bartłomiej Gardas, Jerzy Dajka

Published Apr 2, 2014• 0 citations
Publisher

Abstract

We discuss a comment recently made by C F Lo concerning validity of one of the examples used in our paper (2013 J. Phys. A: Math. Theor. 46 235301). While we agree with Lo's statement regarding inapplicability of the example to the proposed schema, we disagree with the claim concerning mathematical correctness of that example. We are not convinced by Lo's arguments leading to the conclusion that the model is mathematically ill-defined.

DOI
10.1088/1751-8113/47/16/168002

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Comment on ‘Initial states of qubit–environment models leading to conserved quantities’

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Recently, Gardas and Dajka (2013 J. Phys. A: Math. Theor. 46 235301) proposed a theoretical scheme on how to prepare a proper initial state of a composite qubit–boson system which could assure no energy exchange between the two subsystems. For illustration, they applied the proposed scheme to both the Jaynes–Cummings model and the multi-photon Rabi model. However, it has already been rigorously proven that the k -photon Rabi model for k > 2 does not have eigenstates in the Hilbert space spanned by the eigenstates of the corresponding k -photon Jaynes–Cummings model and that the two-photon Rabi model has a discrete eigenenergy spectrum with normalizable eigenstates only if the coupling strength is smaller than a critical value. Hence, the proposed theoretical scheme is not applicable to the multi-photon Rabi model.

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Abstract In their comment, Angelini et al object to the conclusion of (2019 J. Phys. A: Math. Theor. 52 445002), where we show that in (2013 Phys. Rev. B 87 134201) the exponent ν has been obtained by applying a mathematical relation in a regime where this relation is not valid. We observe that the criticism above on the mathematical validity of such relation has not been addressed in the comment. Our criticism thus remains valid, and disproves the conclusions of the comment. This constitutes the main point of this reply. In addition, we provide a point-by-point response and discussion of Angelini et al ’s claims. First, Angelini et al claim that the prediction 2 1/ ν = 1 of (2019 J. Phys. A: Math. Theor . 52 445002) is incorrect, because it results from the relation λ max = 2 1/ ν between the largest eigenvalue of the linearized renormalization-group (RG) transformation and ν , which cannot be applied to the ensemble renormalization group (ERG) method, given that for the ERG λ max = 1. However, the feature λ max = 1 is specific to the ERG transformation, and it does not give any grounds for questioning the validity of the general relation λ max = 2 1/ ν specifically for the ERG transformation. Second, Angelini et al claim that ν should be extracted from an early RG regime (A), as opposed to the asymptotic regime (B) used to estimate ν in (2019 J. Phys. A: Math. Theor . 52 445002), and that (B) is dominated by finite-size effects. However, (A) is a small-wavelength, non-critical regime, which cannot characterize the critical exponent ν related to the divergence of the correlation length. Also, the fact that (B) involves finite-size effects is a feature specific to the ERG, and gives no rationale for extracting ν from (A). Finally, we refute the remaining claims made by Angelini et al As a result of our analysis, we stand by our assertion that the ERG method yields a prediction given by 2 1/ ν = 1.

arXiv:2102.06263•Journal of Physics A Mathematical and Theoretical•Apr 3, 2020•0 citations
We discuss a comment recently made by C F Lo concerning validity of one of the examples used in our paper (2013 J. Phys. A: Math. Theor. 46 235301). While we agree with Lo's statement regarding inapplicability of the example to the proposed schema, we disagree with the claim concerning mathematical correctness of that example. We are not convinced by Lo's arguments leading to the conclusion that the model is mathematically ill-defined.
Recently, Gardas and Dajka (2013 J. Phys. A: Math. Theor. 46 235301) proposed a theoretical scheme on how to prepare a proper initial state of a composite qubit–boson system which could assure no energy exchange between the two subsystems. For illustration, they applied the proposed scheme to both the Jaynes–Cummings model and the multi-photon Rabi model. However, it has already been rigorously proven that the k -photon Rabi model for k > 2 does not have eigenstates in the Hilbert space spanned by the eigenstates of the corresponding k -photon Jaynes–Cummings model and that the two-photon Rabi model has a discrete eigenenergy spectrum with normalizable eigenstates only if the coupling strength is smaller than a critical value. Hence, the proposed theoretical scheme is not applicable to the multi-photon Rabi model.
Abstract In their comment, Angelini et al object to the conclusion of (2019 J. Phys. A: Math. Theor. 52 445002), where we show that in (2013 Phys. Rev. B 87 134201) the exponent ν has been obtained by applying a mathematical relation in a regime where this relation is not valid. We observe that the criticism above on the mathematical validity of such relation has not been addressed in the comment. Our criticism thus remains valid, and disproves the conclusions of the comment. This constitutes the main point of this reply. In addition, we provide a point-by-point response and discussion of Angelini et al ’s claims. First, Angelini et al claim that the prediction 2 1/ ν = 1 of (2019 J. Phys. A: Math. Theor . 52 445002) is incorrect, because it results from the relation λ max = 2 1/ ν between the largest eigenvalue of the linearized renormalization-group (RG) transformation and ν , which cannot be applied to the ensemble renormalization group (ERG) method, given that for the ERG λ max = 1. However, the feature λ max = 1 is specific to the ERG transformation, and it does not give any grounds for questioning the validity of the general relation λ max = 2 1/ ν specifically for the ERG transformation. Second, Angelini et al claim that ν should be extracted from an early RG regime (A), as opposed to the asymptotic regime (B) used to estimate ν in (2019 J. Phys. A: Math. Theor . 52 445002), and that (B) is dominated by finite-size effects. However, (A) is a small-wavelength, non-critical regime, which cannot characterize the critical exponent ν related to the divergence of the correlation length. Also, the fact that (B) involves finite-size effects is a feature specific to the ERG, and gives no rationale for extracting ν from (A). Finally, we refute the remaining claims made by Angelini et al As a result of our analysis, we stand by our assertion that the ERG method yields a prediction given by 2 1/ ν = 1.