Coherent ﬂuctuation relations: from the abstract to the concrete
, Sebastian Weidt
, David Jennings
, Janet Anders
, and Florian Mintert.
Controlled Quantum Dynamics Theory Group, Imperial College London, London, SW7 2BW, United Kingdom.
Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH, United Kingdom.
Department of Physics, University of Oxford, Oxford, OX1 3PU, United Kingdom.
School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, United Kingdom.
CEMPS, Physics and Astronomy, University of Exeter, Exeter, EX4 4QL, United Kingdom.
February 20, 2019
Recent studies using the quantum informa-
tion theoretic approach to thermodynamics
show that the presence of coherence in quan-
tum systems generates corrections to classi-
cal ﬂuctuation theorems. To explicate the
physical origins and implications of such cor-
rections, we here convert an abstract frame-
work of an autonomous quantum Crooks rela-
tion into quantum Crooks equalities for well-
known coherent, squeezed and cat states. We
further provide a proposal for a concrete ex-
perimental scenario to test these equalities.
Our scheme consists of the autonomous evo-
lution of a trapped ion and uses a position
dependent AC Stark shift.
The emergent ﬁeld of quantum thermodynamics
seeks to extend the laws of thermodynamics and non-
equilibrium statistical mechanics to quantum sys-
tems. A central question is whether quantum me-
chanical phenomena, such as coherence and entan-
glement, generate corrections to classical thermal
A quantum information theoretic approach has
proven a fruitful means of incorporating genuinely
quantum mechanical eﬀects into thermal physics .
For example, the consequences of quantum entangle-
ment for Landauer erasure , the thermodynamic
arrow of time  and thermalisation  have been
investigated. In addition, for incoherent quantum
systems, general criteria for state conversion , gen-
eralisations of the second laws , and limits to work
extraction protocols  have been established. More
recently, these results have been extended to coherent
quantum states [8–11]. Much of this research [5–11]
utilised the resource theory framework and in par-
ticular the concept of thermal operations [12, 13].
However, despite signiﬁcant theoretical progress, the
results have been seen as rather abstract and una-
mendable to experimental implementation .
A separate line of enquiry has explored ﬂuctuation
theorems, which can be seen as generalisations of the
second law of thermodynamics to non-equilibrium
processes . They consider systems that are driven
out of equilibrium and establish exact relations be-
tween the resultant thermal ﬂuctuations . Ex-
perimental tests of ﬂuctuation theorems, in partic-
ular the Jarzynski  and Crooks  equalities,
have been conducted in classical systems, including
stretched RNA molecules [19, 20], over-damped col-
loidal particles in harmonic potentials  and clas-
sical two-state systems , and quantum systems
such as trapped ions  and NMR systems . An
experiment to test a quantum Jarzynski equality in
a weakly measured system using circuit QED has re-
cently been performed .
However, the use of two point energy measure-
ments or continual weak measurements to obtain a
work probability distribution reduces a system’s co-
herence with respect to the energy eigenbasis [26–31].
This limits the extent to which these previous exper-
iments probe quantum mechanical phenomena that
arise from coherences.
A recent theoretical proposal of a new quantum
ﬂuctuation relation  that connects naturally with
quantum information theory, models the work sys-
tem explicitly as a quantum battery. Rather than
implicitly appealing to an additional classical system
to drive the system out of equilibrium, the quan-
tum system and battery here evolve autonomously
under a time independent Hamiltonian [33–36]. This
proposal does not require projective measurements
onto energy eigenstates and so does not destroy
coherences. As a result coherence actively con-
tributes to the Autonomous Quantum Crooks equal-
ity (AQC) , which will be deﬁned in Eq. (8).
In this paper, we exploit the combined strengths
of the quantum information and ﬂuctuation theorem
approaches to quantum thermodynamics [37, 38]. We
develop quantum Crooks equalities for a quantum
Accepted in Quantum 2018-02-12, click title to verify 1
arXiv:1806.11256v4 [quant-ph] 19 Feb 2019