Coherent fluctuation relations: from the abstract to the concrete
Zoë Holmes
1
, Sebastian Weidt
2
, David Jennings
1,3,4
, Janet Anders
5
, and Florian Mintert.
1
1
Controlled Quantum Dynamics Theory Group, Imperial College London, London, SW7 2BW, United Kingdom.
2
Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH, United Kingdom.
3
Department of Physics, University of Oxford, Oxford, OX1 3PU, United Kingdom.
4
School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, United Kingdom.
5
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.
1 Introduction
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
physics.
A quantum information theoretic approach has
proven a fruitful means of incorporating genuinely
quantum mechanical effects into thermal physics [1].
For example, the consequences of quantum entangle-
ment for Landauer erasure [2], the thermodynamic
arrow of time [3] and thermalisation [4] have been
investigated. In addition, for incoherent quantum
systems, general criteria for state conversion [5], gen-
eralisations of the second laws [6], and limits to work
extraction protocols [7] have been established. More
recently, these results have been extended to coherent
quantum states [811]. Much of this research [511]
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 [14].
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 [15]. They consider systems that are driven
out of equilibrium and establish exact relations be-
tween the resultant thermal fluctuations [16]. Ex-
perimental tests of fluctuation theorems, in partic-
ular the Jarzynski [17] and Crooks [18] equalities,
have been conducted in classical systems, including
stretched RNA molecules [19, 20], over-damped col-
loidal particles in harmonic potentials [21] and clas-
sical two-state systems [22], and quantum systems
such as trapped ions [23] and NMR systems [24]. An
experiment to test a quantum Jarzynski equality in
a weakly measured system using circuit QED has re-
cently been performed [25].
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 [2631].
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 [32] 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 [3336]. 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) [32], 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
harmonic oscillator battery that is prepared in op-
tical coherent states, squeezed states or cat states
and explore the physical content of these equalities.
In particular, we present the coherent state Crooks
equality, see Eq. (15), in which quantum corrections
to the classical Crooks equality can be attributed to
the presence of quantum vacuum fluctuations.
We propose an experiment to test the AQC
through which the role of coherence in thermal
physics can be probed. Using a trapped ion, one
can realise a two level system with a pair of the ion’s
internal energy levels and an oscillator battery with
the ion’s axial phonon mode. The motion of the ion
through an off-resonance laser beam induces a po-
sition dependent AC Stark shift on the internal en-
ergy levels. In this way an effectively time-dependent
Hamiltonian for the two level system can be realised.
2 The Classical and Quantum Crooks
Equalities
2.1 The classical Crooks equality
An initially thermal system, at temperature T , can
be driven from equilibrium with a time-dependent
Hamiltonian, changing from H
i
S
to H
f
S
. The classical
Crooks equality [18],
P
+
(W )
P
(W )
= exp
F
k
B
T
exp
W
k
B
T
, (1)
quantifies the ratio of the probability P
+
(W ) of the
work W done on a system in such a non-equilibrium
process, to the probability P
(W ) to extract the
work W in the time reversed process (where the
Hamiltonian is changed back from H
f
S
to H
i
S
). This
ratio is a function of the change in the system’s free
energy,
F
:
= F (H
f
S
) F (H
i
S
) with
F (H
S
)
:
= k
B
T ln
Tr
S
exp
H
S
k
B
T

,
(2)
and hence measurement of P (W ) and P (W ) for a
non-equilibrium process provides a means of inferring
free energy changes, an equilibrium property of a sys-
tem. More generally, Eq. (1) asserts that processes
that produce work are exponentially less likely than
processes that require work, irrespective of how the
change from H
i
S
to H
f
S
is realised.
2.2 The autonomous quantum Crooks equality
The realisation of a time-dependent Hamiltonian in
the classical Crooks equality, Eq. (1), necessarily im-
plies an interaction with a classical agent. The au-
tonomous framework [32] proposed by Johan Åberg
makes this control explicit by introducing a battery.
Specifically, the system evolves together with the bat-
tery according to the time-independent Hamiltonian,
H
SB
= H
S
B
+
S
H
B
+ V
SB
, (3)
comprised of the Hamiltonians H
S
and H
B
for sys-
tem and battery, and their interaction V
SB
.
For the initial and final Hamiltonians of the system
to be well defined, we need the system and battery
not to interact at these times. To ensure this, we
consider an interaction of the form
V
SB
= H
i
S
Π
i
B
+ H
f
S
Π
f
B
+ V
SB
, (4)
where Π
i
B
and Π
f
B
are projectors onto two orthog-
onal subspaces, R
i
and R
f
, of the battery’s Hilbert
space, and V
SB
has support only outside those two
subspaces, i.e. (X
S
Π
i
B
)V
SB
= (X
S
Π
f
B
)V
SB
= 0
for any system operator X
S
. The system Hamilto-
nian H
S
can always be absorbed into V
SB
, which we
will do in the following. Assuming the battery is ini-
tialised in a state in subspace R
i
only and evolves to a
final state in subspace R
f
only, the system Hamilto-
nian evolves from H
i
S
to H
f
S
, i.e. the system Hamilto-
nian is effectively time-dependent. Since the system
and battery are non-interacting in the regions R
i
and
R
f
, and since energy is globally conserved, the energy
required or produced by the system as its Hamilto-
nian evolves from H
i
S
to H
f
S
is necessarily provided
or absorbed by the battery.
The initially thermal system in the Crooks equal-
ity is incoherent in energy and so, to extend the rela-
tion to the quantum regime in this autonomous set-
ting, coherence
1
must begin in an additional non-
equilibrium quantum system. Here the coherence is
provided by the battery which we assume can be pre-
pared in superpositions of energy states. The evolu-
tion under H
SB
generates correlations between the
system and battery reducing the coherence that can
be attributed to the battery alone. Since the dynam-
ics conserves coherence globally, the coherence of the
final joint state of the system and battery is equal
to that initially provided by the battery. Thus this
framework allows us to describe the evolution of co-
herence under interactions with a thermal system.
1
We use the term ‘coherence’ in the sense of a ‘superposi-
tion of states belonging to different energy eigenspaces’. Such
energetic coherences are formally quantified in the thermal op-
erations framework [12, 13] and by measures such as the l
1
norm of coherence [39], C
l
1
(ρ)
:
=
P
j6=k
| hE
j
| ρ |E
k
i |, the sum
of the absolute value of the off diagonal elements of a state in
the energy eigenbasis.
Accepted in Quantum 2018-02-12, click title to verify 2
The difficulties surrounding how to define work in
the quantum regime [4045] are avoided by formu-
lating the AQC in terms of transition probabilities
between battery states. Considering a system ini-
tialised in the thermal state γ
i
and the battery ini-
tialised in the pure state |φ
i
i, the probability to ob-
serve the battery after a time-interval t in the state
|φ
f
i reads
P(φ
f
|φ
i
, γ
i
) = hφ
f
|Tr
S
[U(γ
i
|φ
i
ihφ
i
|)U
] |φ
f
i ,
(5)
with U = exp(iH
SB
t/~) the propagator of the sys-
tem and battery and the trace over the system de-
noted by Tr
S
. If we consider the transition probabil-
ity between energy eigenstates, the scheme consid-
ered here reduces to the usual two-point projective
energy measurement scheme [2628] and as such we
can recover the classical Crooks relation, Eq. (1). For
transition probabilities between more general quan-
tum states, such as between coherent states, squeezed
states and cat states [46] as we will consider, the post
measurement battery state has coherences in the en-
ergy eigenbasis and the measurement can induce a
non-classical updating of the state of the system.
The AQC relates the probabilities P(φ
f
|φ
i
, γ
i
) of a
forward process and P(ψ
i
|ψ
f
, γ
f
) of a reverse process
for two thermal system states
γ
x
exp
H
x
S
k
B
T
, with x = i, f (6)
and two pairs of pure initial and final battery states
satisfying
|ψ
f
i T exp
H
B
2k
B
T
|φ
f
i ,
|φ
i
i T exp
H
B
2k
B
T
|ψ
i
i .
(7)
The states |ψ
i
i and |ψ
f
i correspond to a time-
reversed process and are defined in terms of the time-
reversal operator
2
T [47]. This relationship between
the initial and final states of the forwards and re-
versed processes, Eq. (7), is forced by the deriva-
tion of the AQC, i.e. the structure arises natu-
rally when one reverses a general quantum process
in an autonomous setting. If the battery states
|φ
i
i and |φ
f
i are eigenstates of H
B
, the states |ψ
f
i
and |ψ
i
i of the reversed process are regular time-
reversed states. However, for any coherent super-
position of energy eigenstates the additional term
2
The time-reversal [47] operation T on a battery state |ψi
is defined as complex conjugation in the energy eigenbasis of
the battery, T |ψi = |ψ
i. We introduce T in more detail in
Appendix A.2.
exp(H
B
/2k
B
T ), which is linked with the Petz re-
covery map [48, 49], is essential to capture the influ-
ence of quantum coherence.
The AQC [32] reads
P(φ
f
|φ
i
, γ
i
)
P(ψ
i
|ψ
f
, γ
f
)
= exp
F
k
B
T
exp
˜
E
k
B
T
!
, (8)
where F is the change in the system’s equilibrium
free energy, Eq. (2), and thus the AQC, similarly
to the classical Crooks equality, provides a means of
inferring free energy changes. The term
˜
E(ψ
i
, φ
f
)
:
=
˜
E(ψ
i
)
˜
E(φ
f
) (9)
is a quantum mechanical generalisation of the energy
flow from the battery to the system. The function
˜
E(ψ)
:
= k
B
T ln
hψ|exp
H
B
k
B
T
|ψi
(10)
is an effective potential for the battery state |ψi that
specifies the relevant energy value within the fluctu-
ation theorem context. We provide a discussion of
its properties in Appendix A.7.
For the AQC to hold exactly we require |ψ
i
i and
|φ
f
i to have support only within subregions Π
i
B
and Π
f
B
respectively and for the battery Hamilto-
nian not to induce evolution between subregions, i.e.
(
B
Π
i
B
)H
B
Π
i
B
= 0 and (
B
Π
f
B
)H
B
Π
f
B
= 0. In
general, the second of these conditions does not hold
because a battery Hamiltonian that evolves states be-
tween subregions is required. However, despite this,
numerical simulations (discussed in Section 3.2) in-
dicate that any error can be made negligible as long
as |ψ
i
i and |φ
f
i are well localised in subregions Π
i
B
and Π
f
B
respectively. In Section 2.5 we discuss how
the accuracy of the equality can be quantified; how-
ever, the material is relatively technical and can be
skipped.
To make the predictions of the AQC, Eq. (8),
more concrete we now choose the battery to be
a quantum harmonic oscillator with Hamiltonian
H
B
=
P
n
~ω
n +
1
2
|nihn|. In order to derive
explicit expressions for the ratio of transition prob-
abilities, Eq. (8), we consider common states of
the harmonic oscillator, specifically coherent states,
squeezed states and cat states [46]. For each of these
three states we derive a specific AQC which is based
on the generalised energy flow, Eq. (9), and the re-
lation between the initial and final states of the for-
ward and reversed process, Eq. (7). While the bat-
tery Hamiltonian and states are fixed in the following
sections, the system Hamiltonians, H
i
S
and H
f
S
, and
the system and battery interaction, V
SB
, can still be
chosen at will.
Accepted in Quantum 2018-02-12, click title to verify 3
2.3 Coherent state Crooks equality
The physical content of the AQC is neatly illustrated
by considering transition probabilities between co-
herent states of the battery, |αi = exp(αa
α
a) |0i
with a
and a the creation and annihilation opera-
tors of the oscillator respectively, and α a complex
number. In the forwards process, we assume the
battery is initialised in |α
i
exp(χ)i with χ =
~ω
2k
B
T
and we are interested in the transition probability
P(α
f
|α
i
exp(χ), γ
i
) to the coherent state |α
f
i. Fol-
lowing Eq. (7), the initial and final states of the
reversed process are given by the coherent states
|α
f
exp(χ)i and |α
i
i. The symbol
denotes com-
plex conjugation in the Fock basis and arises from the
time reversal operation on a coherent state, T |αi =
|α
i.
The dimensionless parameter χ is the ratio of the
magnitude of quantum fluctuations,
~ω
2
, to the mag-
nitude of thermal fluctuations, k
B
T , and quantifies
the degree to which any regime is quantum mechan-
ical. In the limit in which thermal fluctuations dom-
inate, χ tends to 0, and the prefactor exp(χ) tends
to 1. Consequently, in this limit the reverse process
is the exact time-reversed process; however, in gen-
eral, the pairs of states considered in the forwards
and reverse process differ by an amount determined
by exp(χ).
As derived in more detail in B.1, the energy flow,
Eq. (9), takes the explicit form
˜
E(α
f
, α
i
) = k
B
T
|α
i
|
2
|α
f
|
2
)(1 exp(2χ)
.
(11)
In order to highlight similarities and differences to
the classical situation, it is instructive to explicitly
introduce the difference between the average energy
cost E
+
and gain E
of the forward and reverse
processes. We define the prefactor
q
:
=
˜
E
W
q
(12)
as the ratio between
˜
E and W
q
= (∆E
+
E
)/2.
As shown in B.1, we find that for coherent states the
prefactor takes the form
q (χ) =
1
χ
tanh(χ) (13)
and can be related to the average frequency, ω
T
, of
the oscillator battery in a thermal state, γ(H
B
)
exp
H
B
k
B
T
, at temperature T ,
~ω
T
:
= hH
B
i
γ(H
B
)
=
k
B
T
q(χ)
. (14)
0.5 1.0 1.5 2.0
χ
0.5
1.0
1.5
2.0
H
B
γ
B
ℏω
T
1
2
ℏω
h
2
mλ
th
2
k
B
T
Figure 1: The solid, dark blue line shows the average energy,
~ω
T
, of the oscillator in a thermal state at temperature T
as a function of χ. The red lines indicate the contribution
of thermal (dotted) and quantum (dashed) contributions to
~ω
T
as determined by Eq. (16). The grey line is the classical
limit in which the energy of the harmonic oscillator equals
k
B
T . Energies are given in units of k
B
T .
The coherent state AQC can thus be written as
P
α
f
α
i
exp(χ), γ
i
P
α
i
α
f
exp(χ), γ
f
= exp
F
k
B
T
exp
W
q
~ω
T
.
(15)
In this form, it is analogous to the classical Crooks
equality, Eq. (1), with the difference encoded in ω
T
.
The thermal frequency ω
T
can be understood in
terms of the thermal wavelength λ
th
=
h
p
[50] given
in terms of the momentum p of a particle. This is
a spatial scale that quantifies the regime in which
a thermal system should be treated as classical or
quantum. Broadly speaking, when λ
th
is small the
system is well localised and can be considered clas-
sical; however, quantum effects dominate when it is
larger.
The average energy of the oscillator is composed
of kinetic, potential, and vacuum fluctuation terms.
For an oscillator in a thermal state the kinetic and
potential energies are equal, and with the vacuum
energy E
vac
=
1
2
~ω, it follows that
~ω
T
=
h
2
2
th
+
1
2
~ω . (16)
This equality describes the splitting of ~ω
T
into its
thermal,
h
2
2
th
, and quantum,
1
2
~ω, contributions, as
is shown in Fig. 1.
In the classical limit the thermal wavelength λ
th
is
small and the dominant contribution to ~ω
T
is ther-
mal. In this limit ~ω
T
tends to k
B
T as predicted by
the equipartition theorem and in agreement with the
classical Crooks equality.
In the quantum limit of large λ
th
the thermal con-
tribution tends to zero and vacuum fluctuations dom-
inate. These vacuum fluctuation induce quantum
Accepted in Quantum 2018-02-12, click title to verify 4