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Finding a n0 such that Tr { ρD [i H n0, ρ] } ≠0 is required. As long as n0 is found, f n0 can be determined by f n0 =-Tr [ρD L (ρ)] /Tr { ρD [ρ,i H n0] }. Now we show that f n0 is real. By the definition of L (ρ), L (ρ) is Hermite, then Tr [ρD L (ρ)] is real. Identifying ρD with a time-independent density matrix, we have Tr { ρD [ρ,i H n0] } = t Tr (ρD ρ), where t ρ= [ρ,i H n0] is assumed. Then Tr { ρD [ρ,i H n0] } is also real. By the same procedure, we can show that all the control fields fn (t) are real.
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Finding a n0 such that Tr { ρD [i H n0, ρ] } ≠0 is required. As long as n0 is found, f n0 can be determined by f n0 =-Tr [ρD L (ρ)] /Tr { ρD [ρ,i H n0] }. Now we show that f n0 is real. By the definition of L (ρ), L (ρ) is Hermite, then Tr [ρD L (ρ)] is real. Identifying ρD with a time-independent density matrix, we have Tr { ρD [ρ,i H n0] } = t Tr (ρD ρ), where t ρ= [ρ,i H n0] is assumed. Then Tr { ρD [ρ,i H n0] } is also real. By the same procedure, we can show that all the control fields fn (t) are real.
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