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Reply to “Comment on: ‘Multi-photon Rabi model: Generalized parity and its applications’ [Phys. Lett. A 377 (2013) 3205]” [Phys. Lett. A 378 (2014) 1969]
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Comment
Reply to “Comment on: ‘Multi-photon Rabi model: Generalized parity and its applications’ [Phys. Lett. A 377 (2013) 3205]” [Phys. Lett. A (2014)] Bartłomiej Gardas ∗ , Jerzy Dajka Institute of Physics, University of Silesia, PL-40-007 Katowice, Poland
a r t i c l e
i n f o
a b s t r a c t
Article history: Available online xxxx Communicated by P.R. Holland
We show that the claims stated in the comment made by Chi-Fai Lo on our paper [Phys. Lett. A 377 (2013) 3205] neither contradict nor falsify our results. © 2014 Elsevier B.V. All rights reserved.
Recently C.F. Lo has made a comment concerning mathematical aspects of k-photon Rabi model that we investigated in [1]. He claims that our reasoning is incorrect for k > 2. Below we show that the arguments of Lo and coworkers [2] motivating the comment, although formally correct and apposite, do not contradict ours. Validity of the analysis conducted in [1] for all k is independent from character of the model’s spectrum. Let us consider a relation between generalized parity Xk (Eq. (19) in [1]) and the spectrum of H i.e. σ (H). Since Xk is a bounded solution to the operator Riccati equation H can be block diagonalized:
U† HU =
H+ 0
0 , H−
1 with U := √ 2
1 Xk
−Xk 1
,
(1)
where
H± := ωa† a ± g ak + a†k ± α Xk .
(2)
The spectrum consists of two parts:
σ (H) = σ (H+ ) ∪ σ (H− ).
(3)
It follows from the work of Lo et al. [2] (which serves as a basis for the comment) that these parts necessarily consist of continuous eigenvalues when k > 2 (and for g > g c if k = 2). Therefore H± cannot possess only discrete energy levels. The same conclusion is reached by looking at the generalized eigenstates here denoted by |·) (and which can be discussed within
the rigged Hilbert space formalism [3]) of H. They are given by
|Ψ ) = and
|ψ) , Xk |ψ)
where H+ |ψ) = E|ψ),
−Xk |φ) |Φ) = , |φ)
where
H− |φ) = E|φ).
*
http://dx.doi.org/10.1016/j.physleta.2014.04.053 0375-9601/© 2014 Elsevier B.V. All rights reserved.
(5)
Indeed, such states are not spanned by the Fock basis as stated in the comment. Instead of generalized eigenstates one can use ‘usual’ eigenstates |· for either k = 1 or k = 2 (provided that g g c ). It seems to us that the source of Lo’s claim originates from the fact that we have used the Fock basis to establish Proposition 1 without discussing properties of the continuous spectrum. We have invoked a discrete formula to represent a continuous Hamiltonian. There is no contradiction in this because our representation is not a spectral decomposition of H. The latter would of course involve an integral due to the continuum of energy levels. Such a discrete representation is sufficient to establish an existence of the symmetry as we did in Proposition 1. However, it is clearly not sufficient to find the spectrum. We agree that to avoid potential confusion we should have explicitly mentioned of this fact. Acknowledgements This work was supported by the Polish Ministry of Science and Higher Education; NCN projects DEC-2011/01/N/ST3/02473 (B.G.) and DEC-2013/09/B/ST3/01659 (J.D.) References
DOI of original article: http://dx.doi.org/10.1016/j.physleta.2013.10.011. DOI of comment: http://dx.doi.org/10.1016/j.physleta.2014.04.044. Corresponding author. E-mail address: [email protected] (B. Gardas).
(4)
[1] B. Gardas, J. Dajka, Phys. Lett. A 377 (2013) 3205. [2] C.F. Lo, K.L. Liu, K.M. Ng, Europhys. Lett. 42 (1998) 1. [3] A. Böhm, Quantum Mechanics, Springer-Verlag, New York, 1979.
AID:22582 /DIS
[m5G; v 1.133; Prn:8/05/2014; 11:03] P.1 (1-1)
Physics Letters A ••• (••••) •••–•••
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Comment
Reply to “Comment on: ‘Multi-photon Rabi model: Generalized parity and its applications’ [Phys. Lett. A 377 (2013) 3205]” [Phys. Lett. A (2014)] Bartłomiej Gardas ∗ , Jerzy Dajka Institute of Physics, University of Silesia, PL-40-007 Katowice, Poland
a r t i c l e
i n f o
a b s t r a c t
Article history: Available online xxxx Communicated by P.R. Holland
We show that the claims stated in the comment made by Chi-Fai Lo on our paper [Phys. Lett. A 377 (2013) 3205] neither contradict nor falsify our results. © 2014 Elsevier B.V. All rights reserved.
Recently C.F. Lo has made a comment concerning mathematical aspects of k-photon Rabi model that we investigated in [1]. He claims that our reasoning is incorrect for k > 2. Below we show that the arguments of Lo and coworkers [2] motivating the comment, although formally correct and apposite, do not contradict ours. Validity of the analysis conducted in [1] for all k is independent from character of the model’s spectrum. Let us consider a relation between generalized parity Xk (Eq. (19) in [1]) and the spectrum of H i.e. σ (H). Since Xk is a bounded solution to the operator Riccati equation H can be block diagonalized:
U† HU =
H+ 0
0 , H−
1 with U := √ 2
1 Xk
−Xk 1
,
(1)
where
H± := ωa† a ± g ak + a†k ± α Xk .
(2)
The spectrum consists of two parts:
σ (H) = σ (H+ ) ∪ σ (H− ).
(3)
It follows from the work of Lo et al. [2] (which serves as a basis for the comment) that these parts necessarily consist of continuous eigenvalues when k > 2 (and for g > g c if k = 2). Therefore H± cannot possess only discrete energy levels. The same conclusion is reached by looking at the generalized eigenstates here denoted by |·) (and which can be discussed within
the rigged Hilbert space formalism [3]) of H. They are given by
|Ψ ) = and
|ψ) , Xk |ψ)
where H+ |ψ) = E|ψ),
−Xk |φ) |Φ) = , |φ)
where
H− |φ) = E|φ).
*
http://dx.doi.org/10.1016/j.physleta.2014.04.053 0375-9601/© 2014 Elsevier B.V. All rights reserved.
(5)
Indeed, such states are not spanned by the Fock basis as stated in the comment. Instead of generalized eigenstates one can use ‘usual’ eigenstates |· for either k = 1 or k = 2 (provided that g g c ). It seems to us that the source of Lo’s claim originates from the fact that we have used the Fock basis to establish Proposition 1 without discussing properties of the continuous spectrum. We have invoked a discrete formula to represent a continuous Hamiltonian. There is no contradiction in this because our representation is not a spectral decomposition of H. The latter would of course involve an integral due to the continuum of energy levels. Such a discrete representation is sufficient to establish an existence of the symmetry as we did in Proposition 1. However, it is clearly not sufficient to find the spectrum. We agree that to avoid potential confusion we should have explicitly mentioned of this fact. Acknowledgements This work was supported by the Polish Ministry of Science and Higher Education; NCN projects DEC-2011/01/N/ST3/02473 (B.G.) and DEC-2013/09/B/ST3/01659 (J.D.) References
DOI of original article: http://dx.doi.org/10.1016/j.physleta.2013.10.011. DOI of comment: http://dx.doi.org/10.1016/j.physleta.2014.04.044. Corresponding author. E-mail address: [email protected] (B. Gardas).
(4)
[1] B. Gardas, J. Dajka, Phys. Lett. A 377 (2013) 3205. [2] C.F. Lo, K.L. Liu, K.M. Ng, Europhys. Lett. 42 (1998) 1. [3] A. Böhm, Quantum Mechanics, Springer-Verlag, New York, 1979.