|| 沙川 貴大 教授
|| Implementing noncommutative time-dependent Hamiltonians by a quantum clock
||When considering manipulating a quantum system with external control, there are two theoretical frameworks to treat the time dependence of the Hamiltonian. One is a nonautonomous framework in which the classical external field is considered and the Hamiltonian of the system is time-dependent, and the other is an autonomous framework in which the external quantum system interacting with the system is explicitly considered and the Hamiltonian of the whole system is time-independent. In this study, we investigate the relationship between these frameworks.
We focus on the fact that the time-dependent Hamiltonian can be implemented approximately to the system by using an ancilla system called a clock . In particular, we investigate how the error between the state evolution with a finite-dimensional clock as an ancilla system and the evolution of the time-dependent Hamiltonian scales with respect to the dimension of the clock. In a previous study, the scaling of the exponential decay at the time after one cycle of the clock has been theoretically shown, but the analysis of the implementation error of the state evolution was restricted to the case where the time-dependent Hamiltonian commutes with itself at different times . In our study, we perform numerical calculations for the case where the Hamiltonian does not commute with itself at different times and observe the scaling of the exponential decay of the implementation error at the time after one cycle as in the commutative case. The scaling of the error is also analyzed for times other than after one cycle, and in contrast, a power-law decay is observed. Furthermore, we have theoretically derived such a power-law decay. This result suggests that high accuracy implementation with the clock can only be achieved at specific times.
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 M. P. Woods, R. Silva and J. Oppenheim, Ann. Henri Poincare *20, *125-218(2019). Recently, a new criterion of the second law of thermodynamics is proposed, which is called concavity . An energy eigenstate that meets the criterion indeed obeys the principle of maximum work for adiabatic processes, which is an expression of the second law. The concavity also can be used to judge whether a mixed state obeys the second law or not and would be useful to study quantum heat engines.
In quantum heat engines , there are several examples that show higher efficiency than the Carnot efficiency . However, the reason why these quantum engines show high efficiency has not been clarified yet.
In this presentation, I will introduce the concept of concavity for a single energy eigenstate and a mixed state, and show a numerical estimate of the efficiency of a single qubit heat engine.
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