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z. Since in the equilibrium state of the bath 〈X̂〉=0, the work done for the realization of this switching is zero.
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2k)〉 will be a sum of (2k-1)!! = (2k-1)(2k-3)...3 terms. Wick's theorem is related to the fact that the commutator of the quantum noise is a c-number; see Ref. [41].
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τ dttr[ρ̇(t)Ĥ(t)]. Note that the last integral is equal to zero due to the equation of motion (57).
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Note that in equilibrium thermodynamics, people frequently distinguish functions of a quasiequilibrium thermodynamical process (the one which can be viewed as a chain of equilibrium states) from functions of the state. In this context, the change in heat is written as dQ. Here we consider (possibly strongly) nonequilibrium situations, where almost any quantity (e.g., energy) is a function of the process. Therefore, we do not introduce the symbol d.
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79
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Here and after, we do not present the results of the full optimization of the work over the parameters of the involved pulses. The reason is that we do not want to make the pulsing setup too dependent on the details of the model. On the other hand, the results of this full optimization did not show any qualitative difference with the presented ones.
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80
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note
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The assumption that each spin has its bath is a natural one for cases when the spins are sufficiently well separated [37,38].
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81
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33646980440
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note
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The assumption on the Gaussian character of this distribution can be motivated by the central limit theorem, where the randomness of is viewed to be caused by many (nearly) independent small random factors.
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82
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note
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It is known in the literature that one can obtain decoupling of the spin from the means of strong pulses [22,34]. However, recently the restriction of sharp pulses was relaxed while still sucesfully achieving dynamical decoupling [61]. This prompts a hope that also for the present effect the strong and sharp pulse restriction is not necessary.
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