Physical Review A

Reply to “Comment on ‘Solvability of the two-photon Rabi Hamiltonian’ ”

Igor Travěnec

Published Mar 10, 2015• 26 citations

Abstract

In the Comment, it is shown that functions used to find the spectrum of two-photon Rabi Hamiltonian vanish. This is true; nevertheless, our method evidently works. The trick is based on a substitution making the coefficients of following truncated Taylor series explicitly dependent on energies, and then the roots of the above-mentioned functions approach the eigenenergies. Thus it is the combination of entire exponential functions and truncated Taylor series that gives nontrivial results.

DOI: 10.1103/physreva.91.037802

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Physical Review A

Reply to “Comment on ‘Solvability of the two-photon Rabi Hamiltonian’ ”

Igor Travěnec

Published Mar 10, 2015• 26 citations

Publisher

Abstract

DOI: 10.1103/physreva.91.037802

Related papers

Comment on “Solvability of the two-photon Rabi Hamiltonian”

Andrzej J. Maciejewski, Maria Przybylska, Tomasz Stachowiak

An implicit formula for the spectrum of the two-photon Rabi model was presented by Trav\ifmmode \check{e}\else \v{e}\fi{}nec in Phys. Rev. A 85, 043805 (2012), in analogy to the method of Braak [Phys. Rev. Lett. 107, 100401 (2011)]. The spectrum is given by common zeros of four functions ${G}_{c}(z,E)$ taken with a fixed value of $z\ensuremath{\in}\mathbb{C}$. We point out that all the ${G}_{c}(z,E)$ functions defined by Trav\ifmmode \check{e}\else \v{e}\fi{}nec are identically zero, thus the spectrum is not in fact given by their roots.An implicit formula for the spectrum of the two-photon Rabi model was presented by Trav\ifmmode \check{e}\else \v{e}\fi{}nec in Phys. Rev. A 85, 043805 (2012), in analogy to the method of Braak [Phys. Rev. Lett. 107, 100401 (2011)]. The spectrum is given by common zeros of four functions ${G}_{c}(z,E)$ taken with a fixed value of $z\ensuremath{\in}\mathbb{C}$. We point out that all the ${G}_{c}(z,E)$ functions defined by Trav\ifmmode \check{e}\else \v{e}\fi{}nec are identically zero, thus the spectrum is not in fact given by their roots.

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{G}_{c}(z,E)

taken with a fixed value of $z\ensuremath{\in}\mathbb{C}$. We point out that all the

{G}_{c}(z,E)

functions defined by Trav\ifmmode \check{e}\else \v{e}\fi{}nec are identically zero, thus the spectrum is not in fact given by their roots.

c

are in fact the eigenvalues of the respective parity operators. Four analytic functions are introduced by a recurrence scheme for the coefficients of their series expansion. All their roots yield the complete spectrum of the Hamiltonian. Eigenstates in Bargmann space can also be found.

As established in the seminal work by Berry et al.[1], expanding the time evolution operator using truncated Taylor series (up to some order

K

) makes a good candidate for simulating Hamiltonian dynamics. Here, we adapt the method but present an alternative quantum circuit that maintains an equivalent asymptotic elementary gate cost but has an exponentially reduced number of ancilla qubits. This is realized by utilizing mid-circuit measurements (with early abort-and-restart of circuit execution), and transforming a series of multi-controlled

(H^k)

to a series of singly-controlled

(H^{k'})

, where

H

is a linear combination of unitaries and

k, k'

are integers. The proposed circuit utilizes a total of

\lceil \log(K) \rceil + \lceil \log(L) \rceil +n

qubits, where

L

is the number of terms in the Hamiltonian and

n

is the system qubit size. Our shorter width circuit with mid-measurements protocol is implemented and evaluated using the programming language Guppy[2,3].