Strengthening the atom-field coupling through the deep-strong regime via pseudo-Hermitian Hamiltonians

To the Editor-in-charge of our submitted manuscript, entitled "Strengthening
the atom-field coupling through the deep-strong regime via pseudo-Hermitian
Hamiltonians", authored by M.A. de Ponte, F. de Oliveira Neto, and M.H.Y.
Moussa.

Dear Editor,

Thank you for your message of May 21, regarding our submitted manuscript. We
also thank Dr. Yao for his constructive comments that helped us eliminate an
incorrect statement that was in the first version of the manuscript. Below
we present our answers to Dr. Yao.

The first comment made by Dr. Yao is the following:

\textbf{1.The author discussed construction of pseudo-hermitian Hamiltonian,
following ref.7. These are surely correct, however, if I understand
correctly, a easier way to conclude Eq.(3)-Eq.(4) and discussion related to
those, is just saying you construct your Hamiltonian }$H_{eff}$\textbf{,
such that it equal to an hermitian Hamiltonian }$h$\textbf{\ via a
similarity transformation }$\eta $\textbf{. And the inner product defined in
Eq.(4), is just }$\left\langle \Psi (t)\left\vert O\right\vert \Psi
(t)\right\rangle $\textbf{.}

\textbf{(And the right eigenstate for }$H_{eff}$\textbf{, are related to
eigenstate of }$h$\textbf{\ via }$\left\vert \Psi _{n,R}\right\rangle =\eta
^{-1}\left\vert \psi _{n}\right\rangle $\textbf{, while left eigenvector is }%
$\left\langle \Psi _{n,R}\right\vert =\left\langle \psi _{n}\right\vert \eta
$\textbf{. Actually these are more widely used way of describing
non-hermitian systems, based on development of recent years.)}


We fully agree with this observation. We can indeed construct the effective
Hamiltonian $H_{eff}$, such that it equal to a Hermitian Hamiltonian $h$ via
a similarity transformation $\eta $, where the non-unitary operator $\eta $
defines a new metric $\Theta =\eta ^{\dag }\eta $, through which we redefine
the inner product. In our manuscript, however, we chose to closely follow
the usual form presented by Mostafazadeh in Ref. [7].


The second comment made by Dr. Yao is:


\textbf{2. I would like the authors to clarify Eq.(12a,12b), where the
operator has a large coefficient }$A$\textbf{\ and }$B$\textbf{, in the
untransformed basis. I understand this does not necessarily cause a problem,
because that coefficients are not necessarily connects to physical
quantities, but can the author comments more on how experimentally feasible
for measuring this operator? Does measuring this operator cost a high energy?%
}\bigskip

The simultaneous measurements of canonically conjugated variables seem to be
perfectly feasible in conventional quantum optics. As noted by U. Leonhardt
and H Paul, in Ref. [14], these measurements are supported by "well-known
schemes (...) based on beam-splitting, amplification and heterodyning."
These measurements are really a sensitive issue in our scheme, and in the
manuscript we limit ourselves to drawing attention to the fact. Equally
difficult as engineering a Hamiltonian with a significantly small degree of
Hermiticity is to carry out these measurements of canonically conjugated
variables. We hope that these difficulties have been properly highlighted in
the manuscript.


Regarding the other question very appropriately raised by the referee: "Does
measuring this operator cost a high energy?", we note that we have
completely modified the discussion on this topic presented in the
manuscript. In fact, we cannot make the claim found in the manuscript: "The
higher the Rabi frequencies, the higher the energies required for measuring
properties of the radiation field, as higher as the lower the error
tolerances." For an in-depth discussion of this topic, we refer to the
well-known works on the relationship between energy-time uncertainty and
quantum measurements:


V. Fock and N. Krylov, J. Phys., USSR, 11, 112 (1947);

Y. Aharonov and D. Bohm, Phys. Rev. 122, 1649 (1961);

V.A. Fock, Sov. Phys. - JETP 15,784 (1962);

Y. Aharonov and D. Bohm, Phys. Rev. 134B, 1417 (1964);

Y. Aharonov and J. L. Sajko, Ann. Phys. (N.Y.) 91, 279 (1975);

Yu. 1. Vorontsov, Sov. Phys. - Uspekhi 24, 150 (1981);

M. Moshinsky, Am. J.Phys. 44, 1037 (1976);

J. Rayski and J. M. Rayski, Jr., "On the meaning of the time-energy
uncertainty relation", in The Uncertainty Relation and Foundations of

Quantum Mechanic, edited by W. C. Price and S. S. Chissick (John Wiley, New
York, 1971).\bigskip

In the new version of the manuscript we have modified the sentence correctly
questioned by the referee, replacing it with the paragraph:


\textquotedblleft Regarding achieving faster than Hermitian quantum
mechanics, we note that the effective coupling strength $G$ defines a
typical time $1/G$ to carry out an elementary logical operation. The minimum
energy required for this operation, over a given error tolerance $%
\varepsilon $, is estimated to be $E_{min}\approx \hbar G/\varepsilon $
[13]. The higher the Rabi frequencies, the higher the energies required for
this fast than Hermitian quantum operation, as higher as the lower the error
tolerances.\textquotedblright

\textquotedblleft We mention here a recently presented result [14], where it
is demonstrated that the construction of coherent many-body Rabi
oscillations, through the coherent interaction of an atomic sample with a
field mode, allows increasing the Rabi frequency $g$ by the factor $\sqrt{N}$%
, where $N$ is the number of atoms in the sample. In this case, the typical
time to carry out an elementary logical operation decreases from $1/g$ to $1/%
\sqrt{N}g$. Therefore, in addition to the gain in computational time that
results from the quantum nature of the operation, i.e., from the use of
qubits as information carriers [15], we have here the gain that results from
the collective nature of the radiation-matter interaction. In the present
proposal, the gain in computational time comes from strengthening the Rabi
frequency through pseudo-Hermiticity instead of taking advantage of
collective effects in the coherent interaction between atomic samples and
cavity fields.\textquotedblright


With the changes made to the new version of the manuscript, resulting from
the observations by Dr. Yao, we made the manuscript clearer and now more
complete.


Best regards,

Flavio de Oliveira Neto, Mickel de Ponte and Miled Moussa.

In the new version of the manuscript we have modified the sentence

"The higher the Rabi frequencies, the higher the energies required for measuring properties of the radiation field, as higher as the lower the error tolerances."

by

"Regarding achieving faster than Hermitian quantum
mechanics, we note that the effective coupling strength $G$ defines a
typical time $1/G$ to carry out an elementary logical operation. The minimum
energy required for this operation, over a given error tolerance $%
\varepsilon $, is estimated to be $E_{min}\approx \hbar G/\varepsilon $
[13]. The higher the Rabi frequencies, the higher the energies required for
this fast than Hermitian quantum operation, as higher as the lower the error
tolerances."

"We mention here a recently presented result [14], where it
is demonstrated that the construction of coherent many-body Rabi
oscillations, through the coherent interaction of an atomic sample with a
field mode, allows increasing the Rabi frequency $g$ by the factor $\sqrt{N}$%
, where $N$ is the number of atoms in the sample. In this case, the typical
time to carry out an elementary logical operation decreases from $1/g$ to $1/%
\sqrt{N}g$. Therefore, in addition to the gain in computational time that
results from the quantum nature of the operation, i.e., from the use of
qubits as information carriers [15], we have here the gain that results from
the collective nature of the radiation-matter interaction. In the present
proposal, the gain in computational time comes from strengthening the Rabi
frequency through pseudo-Hermiticity instead of taking advantage of
collective effects in the coherent interaction between atomic samples and
cavity fields."