Abstract
The transient quantum fluctuation theorems of Crooks and Jarzynski restrict and relate the statistics of work performed in forward and backward forcing protocols. So far, these theorems have been obtained under the assumption that the work is determined by two projective energy measurements, one at the end, and the other one at the beginning of each run of the protocol. We found that one can replace these two projective measurements only by special error-free generalized energy measurements with pairs of tailored, protocol-dependent post-measurement states that satisfy detailed balance-like relations. For other generalized measurements, the Crooks relation is typically not satisfied. For the validity of the Jarzynski equality, it is sufficient that the first energy measurements are error-free and the post-measurement states form a complete orthonormal set of elements in the Hilbert space of the considered system. Additionally, the effects of the second energy measurements must have unit trace. We illustrate our results by an example of a two-level system for different generalized measurements.
Highlights
During the past one-and-a-half decades, the transient fluctuation theorems [1], known as Jarzynski equality [2] and Crooks relation [3] have attracted wide interest
We found that generalized energy measurements at the beginning and the end of a force protocol lead to work pdfs which in general violate the transient fluctuation theorem of Crooks
The Jarzynski equality holds for universal energy measurements if the first one is error-free, as defined in (14), with a complete set of orthonormal post-measurement states, and the so-called effects [16] of the second measurement operator, given by Mm† (τ )Mm(τ ), have unit trace
Summary
During the past one-and-a-half decades, the transient fluctuation theorems [1], known as Jarzynski equality [2] and Crooks relation [3] have attracted wide interest. Huber et al [11] suggested an experimental proof by means of a cold atom sitting in a parabolic trap whose position can be varied The difficulty of this and of other experiments testing transient quantum fluctuation theorems lies in the measurement of the work performed on the system in a single realization. Two groups of researchers suggested alternative methods, in which the characteristic function of work is encoded in the reduced density matrix of an ancilla; measuring the state of the ancilla would allow us to verify transient fluctuation theorems in the quantum domain [17, 18] The realization of this idea requires a time-dependent system–ancilla coupling which follows the original force protocol amended by constant force periods of variable lengths [17, 18]. For the sake of the reader’s convenience, some technical arguments are presented in appendices A, B and C
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