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Journal of Physics A Mathematical and Theoretical

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

Chi‐Fai Lo

Published Apr 2, 2014• 1 citation
Publisher

Abstract

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.

DOI
10.1088/1751-8113/47/16/168001

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Initial states of qubit–environment models leading to conserved quantities

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It is possible to prepare a composite qubit–environment system so that its time evolution will guarantee the conservation of a preselected qubit's observable. In general, this observable is not associated with a symmetry. The latter may not even be present in the subsystem. The initial states which lead to such a quantity conserved dynamics form a subspace of the qubit–environment space of states. General construction of this subspace is presented and illustrated by two examples. The first one is the exactly solvable Jaynes–Cummings model and the second is the multi-photon Rabi model.

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We regret that such a misleading comment [Opt. Express (2013)] has been made to our paper. First Lo states in his abstract that "However, the nonlinear Rabi model has already been rigorously proven to be undefined" to later recoil and use the contradictory statement "(. . . ) regarding the BS model with the counter-rotating terms (. . . ) Lo and his co-authors have proven that the model is well defined only if the coupling stregth g is smaller than a critical value gc = ω/4". While Lo focuses on the validity of the quantum optics Hamiltonians and gives a misleading assesment of our manuscript, the focus of our paper is the method to map such a set of Hamiltonians from quantum optics to photonic lattices. Our method is valid for the given class of Hamiltonians and, indeed, precaution must be exerted on the paramater ranges where those Hamiltonians are valid and where their classical simulation is feasible. These parameter ranges have to be specified in for each particular case studied. Furthermore, we gave as example the Buck-Sukumar model including counter-rotating terms which is a valid Hamiltonian for some coupling parameters.

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

Bartłomiej Gardas, Jerzy Dajka

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.

Journal of Physics A Mathematical and Theoretical•Apr 2, 2014•0 citations

A classical simulation of nonlinear Jaynes-Cummings and Rabi models in photonic lattices: comment

Chi‐Fai Lo

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.

Optics Express•Jan 17, 2014•0 citations
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.
It is possible to prepare a composite qubit–environment system so that its time evolution will guarantee the conservation of a preselected qubit's observable. In general, this observable is not associated with a symmetry. The latter may not even be present in the subsystem. The initial states which lead to such a quantity conserved dynamics form a subspace of the qubit–environment space of states. General construction of this subspace is presented and illustrated by two examples. The first one is the exactly solvable Jaynes–Cummings model and the second is the multi-photon Rabi model.
We regret that such a misleading comment [Opt. Express (2013)] has been made to our paper. First Lo states in his abstract that "However, the nonlinear Rabi model has already been rigorously proven to be undefined" to later recoil and use the contradictory statement "(. . . ) regarding the BS model with the counter-rotating terms (. . . ) Lo and his co-authors have proven that the model is well defined only if the coupling stregth g is smaller than a critical value gc = ω/4". While Lo focuses on the validity of the quantum optics Hamiltonians and gives a misleading assesment of our manuscript, the focus of our paper is the method to map such a set of Hamiltonians from quantum optics to photonic lattices. Our method is valid for the given class of Hamiltonians and, indeed, precaution must be exerted on the paramater ranges where those Hamiltonians are valid and where their classical simulation is feasible. These parameter ranges have to be specified in for each particular case studied. Furthermore, we gave as example the Buck-Sukumar model including counter-rotating terms which is a valid Hamiltonian for some coupling parameters.
The present work focuses on the strong/weak interaction of many-body spin-systems with a cavity mode. It introduces the necessity of redefining the physical conditions determining the strong/weak coupling regime in those systems. In more complex systems, the effective coupling emerging from the collective dynamics may differ indeed from the actual coupling of each individual subsystem with the bosonic field. This is shown by highlighting some counter-intuitive dynamical effects properly related to the coupling regime: a Jaynes-Cummings dynamics emerging although a strong interaction is present. The universality of this result is demonstrated through the analysis of three distinct systems: a two-qubit, a two-qutrit, and an NNN-qubit chain quantum Rabi models.