By Zagoskin A.M.
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Otherwise, the action of U (t) would reduce to producing phase factors multiplying the instant taneous basis states: U (t)|g(0) → exp[− 2ı 0 E0 (t ) dt ]|g(t) , U (t)|e(0) → t exp[− 2ı 0 E1 (t ) dt ]|e(t) , where E0,1 (t) = ∓ (t)/2. These formulas are, of course, wrong, but they are almost right in an important special case when H (t) changes adiabatically. ) A pure state of the qubit can always be written as |ψ(t) = C0 (t)|g(t) + C1 (t)|e(t) . , the asymptotes of the anticrossing levels) are asymmetrically inclined, would require a more general Hamiltonian, H =− 1 2 ˆ , σx + (λ)(ασz + β 1) but it would lead to trivial changes only.
58) where P means that when the expression is integrated around the singular point (in our case, x = 0), the integral must be taken in the sense of the principal value, we obtain ¯ = B− e−ıωB t π δ(ωE − ωB ) − ıP B(t) 1 ωE − ωB + B+ eıωB t π δ(ωE + ωB ) − ıP ≡ κ− B− e−ıωB t + κ+ B+ eıωB t . 59) In the general case, the spectra of both the system and the environment – especially the environment! – contain many frequencies, all of which will contribute to κ± (ω) ≡ Re κ± (ω) and κ± (ω) ≡ Im κ± (ω).
Therefore, from the mathematical point of view, the unitary evolutions constitute a group, while the non-unitary evolutions of Eq. 1 Bloch vector and Bloch sphere We are often – and rightly – awed by the power and beauty of universal mathematical methods. Nevertheless, all of them started as one-off tricks for solving some specific problem, and were later generalized. The approaches that defied generalization are not to be neglected, since they are often as powerful in their area and allow significant simplification.
Quantum Engineering by Zagoskin A.M.