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Erratum: Optimal control theory for a unitary operation under dissipative evolution (New Journal of Physics (2014) 16 (055012) DOI: 10.1088/1367-2630/16/5/055012)

Goerz MH, Reich DM, Koch CP (2021)

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Publication Type: Journal article, Erratum

Publication year: 2021

Journal

Book Volume: 23

Article Number: 039501

Journal Issue: 3

DOI: 10.1088/1367-2630/abe970

Abstract

We correct a typographical error in equation (9b) of our previous paper New J. Phys. 16, 055012 (2014), where the dissipator LD is missing a dagger. The numerical results shown in the paper use the correct equation. Thus, the error does not affect our conclusions. Equation (9b) of reference [1] should read ^dd^σtⁱ = −i[Ĥ, σi] − L^†D(σi), (C1 where L^†D(ρ) = ∑ k ( Â^†kρÂk − ¹2Â^†kÂkρ − ¹2ρÂ^†kÂk ). (C2 The only difference is the use of L^†D instead of LD. The adjoint dissipator L^†D differs from the original dissipator LD only by switching Âk and Â^†k in the first term. We can see that L^†D = LD if and only if the set of Lindblad operators {Âk} is identical to the set of adjoint operators {Â^†k}, for example if one were to consider pure dephasing (a single Hermitian Lindblad operator) as the only dissipative process. In general, for an optimization with Krotov's method where the forward propagation takes the form ^dd^ρt = L(ρ), (C3 cf equation (8) in reference [1], the equation of motion for the backward-propagation is [2-4] ^dd^σt = −L^†(σ). (C4 For numerical efficiency, equation (C3) is often evaluated directly, by vectorizing the density matrices ρ and σ, and applying the super-operator L as a (sparse) matrix, using matrix-vector multiplication. The matrix representation of L^† is then simply the adjoint of the matrix representation of L. This is in fact the approach we took for the numerical results shown in reference [1]. A complication arises only when further writing out equation (C3) in Lindblad form, d^dtρ(t) = −i [ Ĥ, ρ(t) ] + LD(ρ(t)) (C5a = −i [ Ĥ, ρ(t) ] + ∑ k (ÂkρÂ^†k − ¹2Â^†kÂkρ − ¹2ρÂ^†kÂk ) , (C5b with the Hamiltonian Ĥ, the dissipator LD, and the Lindblad operators {Âk}, prompting us to write equation (C4) in the corresponding form, equations (C1) and (C2), replacing the incorrect equation (9b). Since our numerical simulations avoid this extra step, the error in equation (9b) does not affect the remainder of the paper. For an in-depth discussion of how equations (C1) and (C2) can be derived, see appendix B of reference [5].

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How to cite

APA:

Goerz, M.H., Reich, D.M., & Koch, C.P. (2021). Erratum: Optimal control theory for a unitary operation under dissipative evolution (New Journal of Physics (2014) 16 (055012) DOI: 10.1088/1367-2630/16/5/055012). New Journal of Physics, 23(3). https://dx.doi.org/10.1088/1367-2630/abe970

MLA:

Goerz, Michael H., Daniel M. Reich, and Christiane P. Koch. "Erratum: Optimal control theory for a unitary operation under dissipative evolution (New Journal of Physics (2014) 16 (055012) DOI: 10.1088/1367-2630/16/5/055012)." New Journal of Physics 23.3 (2021).

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