## Abstract

Recently Rodriguez-Lara *et al*. [Opt. Express **21**(10), 12888 (2013)] proposed a classical simulation of the dynamics of the nonlinear Rabi model by propagating classical light fields in a set of two photonic lattices. However, the nonlinear Rabi model has already been rigorously proven to be undefined by Lo [Quantum Semiclass. Opt. **10**, L57 (1998)]. Hence, the proposed classical simulation is actually not applicable to the nonlinear Rabi model and the simulation results are completely invalid.

© 2013 Optical Society of America

In recent years much attention has been attracted to the classical simulation of quantum and relativistic systems due to the technological advancement in manufacturing arrays of optical waveguides. For instance, Crespi *et al*. [1] reported a successful realization of a classical simulator of the quantum Rabi model, which is also known as the Jaynes-Cummings (JC) model with the counter-rotating term, via light transport in femtosecond-laser-written waveguide supperlattices. The classical simulator is able to provide an experimentally accessible test bed to explore the physics of light-matter interaction in the deep strong coupling regime. Similarly, Rodriguez-Lara *et al*. [2] proposed to simulate the dynamics of the nonlinear Rabi model (or equivalently, the nonlinear JC model with the counter-rotating term) by propagating classical light fields in a set of two photonic lattices. For illustration, they performed the classical simulation to the Buck-Sukumar (BS) model, *i.e.* the intensity-dependent JC model, with the counter-rotating term, which is a special case of the nonlinear Rabi model, and presented the simulation results of the model.

The aim of this comment is to point out that the nonlinear Rabi model considered in Ref. [2] has already been rigorously proven to be undefined by Lo [3]. In other words, the system does not have eigenstates in the Hilbert space spanned by the eigenstates of the corresponding nonlinear JC model. The nonlinear Rabi model is thus qualitatively different from the nonlinear JC model for the counter-rotating term does drastically alter the nature of the nonlinear JC model. The same conclusion has also been drawn for some other generalised JC models with the counter-rotating terms, namely the *k*-photon JC model with the counter-rotating term for *k* > 2 [4], the *k*-photon intensity-dependent JC model with the counter-rotating term for *k* > 1 [5], the nonlinear k-photon JC model with the counter-rotating term for *k* ≥ 1 [3], and the k-photon q-deformed JC model with the counter-rotating term for *k* ≥ 1 [6]. Accordingly, the classical simulation analysis discussed in Ref. [2] is actually not applicable to the nonlinear Rabi model at all.

Moreover, regarding the BS model with the counter-rotating term:

*g*is smaller than a critical value

*g*=

_{c}*ω*/4. Thus, the dynamics of the BS model with the counter-rotating term is significantly different from the BS model. Besides, previous studies have also shown that in the presence of the counter-rotating terms other generalised JC models, like the two-photon JC model [7], the intensity-dependent JC model [8] and the two-mode two-photon JC model [9], become undefined if the values of the coupling parameters are increased beyond some critical values. As a consequence, the illustrative simulation analysis of the BS model with the counter-rotating term presented in Section 4 of Ref. [2] is completely invalid because the adopted coupling strength is much larger than the critical value,

*i.e. g*= 2

*ω*.

In conclusion, the classical simulation analysis proposed by Rodriguez-Lara *et al*. is applicable to the nonlinear JC model only because the nonlinear Rabi model is undefined.

## References and links

**1. **A. Crespi, S. Longhi, and R. Osellame, “Photonic Realization of the Quantum Rabi Model,” Phys. Rev. Lett. **108**, 163601 (2012). [CrossRef] [PubMed]

**2. **R. M. Rodriguez-Lara, F. Soto-Eguibar, A. Z. Cardenas, and H. M. Moya-Cessa, “A classical simulation of nonlinear Jaynes-Cummings and Rabi models in photonic lattices,” Opt. Express **21**(10), 12888–12898 (2013). [CrossRef] [PubMed]

**3. **C. F. Lo, “Nonlinear multiquantum Jaynes-Cummings model with counter-rotating terms,” Quantum Semiclass. Opt. **10**L57–L61 (1998). [CrossRef]

**4. **C. F. Lo, K. L. Liu, and K. M. Ng, “The multiquantum Jaynes-Cummings model with the counter-rotating terms,” Europhys. Lett. **42**(1), 1–6 (1998). [CrossRef]

**5. **C. F. Lo, K. L. Liu, and K. M. Ng, “The multiquantum intensity-dependent Jaynes-Cummings model with the counterrotating terms,” Physica A **265**557–564 (1999). [CrossRef]

**6. **C. F. Lo and K. L. Liu, “The multiquantum q-deformed Jaynes-Cummings model with the counter-rotating terms,” Physica A **271**, 405–410 (1999). [CrossRef]

**7. **K. M. Ng, C. F. Lo, and K. L. Liu, “Exact eigenstates of the two-photon Jaynes-Cummings model with the counter-rotating term,” Eur. Phys. J. D **6**, 119–126 (1999).

**8. **K. M. Ng, C. F. Lo, and K. L. Liu, “Exact eigenstates of the intensity-dependent Jaynes-Cummings model with the counter-rotating term,” Physica A **275**, 463–474 (2000). [CrossRef]

**9. **K.M. Ng, C.F. Lo, and K.L. Liu, “Exact dynamics of the two-mode two-photon Jaynes-Cummings
model without the rotating-wave approximation,” *Proceedings of the
International Conference on Frontiers in Quantum Physics (July 9–11, 1997)*,
S.C. Lim, R. Abd-Shukor, and K.H. Kwek, eds. (Springer-Verlag,
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