In the original paper1 we incorrectly claimed that the master eq 1 with the dissipator specified in eqs 3 and 8 is translationally invariant. However, the action of this dissipator can be made translationally invariant only in special cases identified below. The inverse Wigner transform of eq 8, mapping scalar functions on the phase space into linear operators, leads to Â = B+ + B- (1) B± = e±i◯/Lθ (∓p - ℏ/(2L)) √Lf(∓p) where θ(p) denotes the Heaviside step function. Comparing eq 2 with eq 8 below, we conclude that the dissipator (eqs 3 and 8 in ref 1) acts translationally invariant if f(p) vanishes for - ℏ/(2L) ≤ p ≤ ℏ/(2L) the density matrix ρ, underlying the Wigner function W, obeys 〈p|ρ|p'〉 = 0 for pp' < 0. Even for a state initially obeying this condition, the evolution governed by the master equation (eqs 3 and 8 in ref 1) may yield the state violating the constratint. The master equation with a translationally invariant isotropic (i.e., symmetric with respect to spatial inversion) dissipator valid for an arbitrary state reads d/dt W = -i/ℏ(H∗ W W ∗ H) + DA+ [W] + DA- [W] (3) DA [W.] = γ/ℏ(A∗W∗A∗ - 1/2W ∗A∗A 1/2 A∗ ∗A∗W) (4) A± = e±ix/L∗√Lf(∓p) (5) where f(p) is any real-valued function, H = p2/(2m) + U(x) denotes the Hamiltonian, and ∗ is the Moyal product defined in eq 5 in ref 1. As a result, the first moments of W satisfy the Ehrenfest relations d/dt〈x〉 = 1/m〈p〉 d/dt〈p〉 = -〈U'(x)〉 - γ〈f(p) f(-p)〉 (7) Note the difference between relation 7 and eq 7 in ref 1. The quantum thermodynamical properties of the quantum friction model (eqs 3-5) are studied in ref 2. Finally, we note that the inverse Wigner transform of eq 5 leads to Â± = e±i◯/L √Lf(∓p) Holevo3,4 and Vacchini5?7 have proven that an arbitrary translationally invariant dissipator can be recast in the form of eq 8. According to the reported error, the model in ref 1 must be replaced by the model in eqs 3-5 above (see also ref 2). This, however, does not affect the conclusions of the original paper. Additionally, the utilized formalism of operational dynamical modeling remains valid for open system dynamics. The erratum has resulted from the deployment of the formalism to the Ehrenfest relation 7 in ref 1 (rather than to eq 7 above) containing the nondifferentiable function sign. This required solving eq 14 in ref 1 for the cases of p > 0 and p < 0 separately, leading to the remarks on the limitations on the initial states. (Equation Presented).
All Science Journal Classification (ASJC) codes
- Materials Science(all)
- Physical and Theoretical Chemistry