Understanding and manipulating work fluctuations in microscale and nanoscale systems are of both fundamental and practical interest. For example, aspects of work fluctuations will be an important factor in designing nanoscale heat engines. In this work, an optimal control approach directly exploiting Jarzynski's equality is proposed to effectively suppress the fluctuations in the work statistics, for systems (initially at thermal equilibrium) subject to a work protocol but isolated from a bath during the protocol. The control strategy is to minimize the deviations of individual values of e −βW from their ensemble average given by e −β∆F , where W is the work, β is the inverse temperature, and ∆F is the free energy difference between two equilibrium states. It is further shown that even when the system Hamiltonian is not fully known, it is still possible to suppress work fluctuations through a feedback loop, by refining the control target function on the fly through Jarzynski's equality itself. Numerical experiments are based on linear and nonlinear parametric oscillators. Optimal control results for linear parametric oscillators are also benchmarked with early results based on accelerated adiabatic processes.
We investigate Landau-Zener processes modeled by a two-level quantum system, with its finite bias energy varied in time and in the presence of a single broadened cavity mode at zero temperature. By applying the hierarchy equation method to the Landau-Zener problem, we computationally study the survival fidelity of adiabatic states without Born, Markov, rotating-wave or other perturbative approximations. With this treatment it also becomes possible to investigate cases with very strong system-bath coupling. Different from a previous study of infinite-time Landau-Zener processes, the fidelity of the time-evolving state as compared with instantaneous adiabatic states shows non-monotonic dependence on the system-bath coupling and on the sweep rate of the bias. We then consider the effect of applying a counter-diabatic driving field, which is found to be useful in improving the fidelity only for sufficiently short Landau-Zener processes. Numerically exact results show that different counter-diabatic driving fields can have much different robustness against environment effects. Lastly, using a case study we discuss the possibility of introducing a dynamical decoupling field in order to eliminate the decoherence effect of the environment and at the same time to retain the positive role of a counter-diabatic field. Our work indicates that finite-time Landau-Zener processes with counter-diabatic driving offer a fruitful test bed to understand controlled adiabatic processes in open systems.
Understanding and manipulating work fluctuations in microscale and nanoscale systems are of both fundamental and practical interest. For example, in considering the Jarzynski equality e −βW = e −β∆F , a change in the fluctuations of e −βW may impact on how fast the statistical average of e −βW converges towards the theoretical value e −β∆F , where W is the work, β is the inverse temperature, and ∆F is free energy difference between two equilibrium states. Motivated by our previous study aiming at the suppression of work fluctuations, here we obtain a principle of minimal work fluctuations. In brief, adiabatic processes as treated in quantum and classical adiabatic theorems yield the minimal fluctuations in e −βW . In the quantum domain, if a system initially prepared at thermal equilibrium is subject to a work protocol but isolated from a bath during the time evolution, then a quantum adiabatic process without energy level crossing (or an assisted adiabatic process reaching the same final states as in a conventional adiabatic process) yields the minimal fluctuations in e −βW , where W is the quantum work defined by two energy measurements in the beginning and at the end of the process. In the classical domain where the classical work protocol is realizable by an adiabatic process, then the classical adiabatic process also yields the minimal fluctuations in e −βW . Numerical experiments based on a Landau-Zener process confirm our theory in the quantum domain, and our theory in the classical domain explains our previous numerical findings regarding the suppression of classical work fluctuations [G. Y. Xiao and J. B. Gong, Phys. Rev. E 90, 052132 (2014)].
The quantum analog of Carnot cycles in few-particle systems consists of two quantum adiabatic steps and two isothermal steps. This construction is formally justified by use of a minimum work principle. It is then shown, without relying on any microscopic interpretations of work or heat, that the heat-to-work efficiency of the quantum Carnot cycle thus constructed may be further optimized, provided that two conditions regarding the expectation value of some generalized force operators evaluated at equilibrium states are satisfied. In general the optimized efficiency is systemspecific, lower than the Carnot efficiency, and dependent upon both temperatures of the cold and hot reservoirs. Simple computational examples are used to illustrate our theory. The results should be an important guide towards the design of favorable working conditions of a realistic quantum heat engine.Introduction -The big energy challenge of this century calls for diversified energy research, including a bottomup approach towards energy efficiency. Apart from two stimulating implementations of microscale heat engines [1,2], some theoretical aspects as well as possible realizations of nanoscale heat engines [3][4][5][6][7][8][9][10][11][12][13][14][15][16] have been studied. For purely quantum heat engines at the nanoscale where the working medium may consist of few particles only (e.g., few trapped ions [6]), both quantum fluctuations and thermal fluctuations become significant. General understanding of the design of such energy devices are also of fundamental interest to nanoscale thermodynamics [17][18][19][20]. In particular, as the size of the working medium shrinks to a quantum level, one must reexamine the implications of the second law of thermodynamics for the efficiency of quantum heat engines. To that end, we construct and look into the quantum analog of Carnot cycles [21,22].The construction of the quantum analog of a Carnot cycle is not as straightforward as it sounds. Consider first the two quasi-static isothermal steps during which the working medium is in thermal equilibrium with a reservoir. Regardless of the size of the quantum medium, its thermodynamic properties can therefore be well defined in the standard sense. As such isothermal steps can be directly carried over to the quantum case. However, translating the two adiabatic steps of a Carnot cycle into a quantum analog is by no means obvious. One intuition [3,9,10] is to replace quasi-static adiabatic steps in thermodynamics (without heat exchange) by quantum adiabatic processes (as defined in the celebrated quantum adiabatic theorem [23]). The starting point of this work is to formally justify such an intuitive construction by revealing a fundamental reason related to energy efficiency.Below we simply call the quantum anolog of a classical Carnot cycle (two isothermal steps and two quantum adiabatic processes) as a quantum Carnot cycle. It is yet fundamentally different from a conventional Carnot cycle. During the two adiabatic steps, the working medium
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