Otto engine beyond its standard quantum limit

1.3 MB·

by  Bruno Leggio

Otto engine beyond its standard quantum limit
Bruno Leggio1 and Mauro Antezza1,2
1Laboratoire Charles Coulomb (L2C), UMR 5221 CNRS-Universit´e de Montpellier, F- 34095 Montpellier, France
2Institut Universitaire de France, 1 rue Descartes, F-75231 Paris Cedex 05, France
(Dated: February 17, 2016)
We propose a quantum Otto cycle based on the properties of a two-level system in a realistic
out-of-thermal-equilibrium electromagnetic field acting as its sole reservoir. This steady configura-
tion is produced without the need of active control over the state of the environment, which is a
non-coherent thermal radiation, sustained only by external heat supplied to macroscopic objects.
Remarkably, even for non-ideal finite-time transformations, it largely over-performs the standard
ideal Otto cycle, and asymptotically achieves unit efficiency at finite power.
6 PACSnumbers: 05.70.-d,07.20.Pe,42.50.Ct
1
0
2 I. INTRODUCTION ture of such non-equilibrium reservoir for the quantum
workingfluid,bothcycleefficiencyandpoweroutputcan
b
largelyovercometheirstandardequilibriumvalues. This
e Motivated by recent advancement in experimental
F work is structured as follows: in Section II we briefly
techniques for the manipulation of single or few-body
review the definition and the physical properties of a
6 quantum systems [1–3], a thermodynamic description of
standard equilibrium QOC for a two-level system. Sec-
1 microscaleandnanoscalephenomenahasbeenattracting
tion III is devoted to the description of the interaction
a huge deal of attention [4]. Among its many different
] of quantum emitters with a particular and realistic out-
h topics, one can notably list the quantum formulation of
of-thermal equilibrium (OTE) electromagnetic field pro-
p thelawsofthermodynamics[5–7],thephysicsofstrongly
duced by a macroscopic object embedded in a thermal
- non-equilibrium quantum dynamics [8–10], the charac-
t blackbody radiation; this will be employed in Section IV
n terizationofquantumthermalmachines[11–14],andthe
to give the main result of this paper, namely, a non-
a study of energy transport phenomena [15–17]. All these
u equilibrium quantum Otto cycle with remarkably high
research lines imply the descritpion of the interaction of
q performances. Finally,remarksaregivenandconclusions
quantum systems with large, usually classical environ-
[ are drawn in Section V.
ments. In particular, the interaction of quantum emit-

2 terswithelectromagneticradiationhasbeenlargelystud-
v
ied both in equilibrium and non-equilibrium thermody-
7 II. STANDARD QUANTUM OTTO CYCLE
namic contexts: out-of-thermal equilibrium electromag-
3
netic fields have been, for instance, shown to provide an
1 As any standard thermodynamic cycle, the Otto cy-
8 ideal playground to induce and exploit stationary quan-
cle happens between two temperatures imposed by ideal
0 tum properties in a many-emitters system.
thermal reservoirs. Classically it consists of four stages:
.
1 One of the most promising outcome of quantum ther- two isochoric processes during which the working sub-
0 modynamics is the characterization of quantum-scale
stanceexchangesheatwithoneofthetwothermalreser-
6 heat engines. These are quantum systems, referred to as
voirs, and two adiabatic processes through which work
1
working fluid, undergoing well-established cycles during
is exchanged with the external world. Its quantum ver-
:
v whichtheyinteractwithclassicalreservoirsandexchange sion for a quantum two-level system (TLS) as working
i workwithanexternaldevice. Inparticular,theso-called
X fluid consists of four stages between two different tem-
Ottocycleisoneofthemainthermodynamiccycles,both
peratures T > T [20, 29] as schematically depicted in
r 1 2
in classical [18] and quantum contexts [4]. Thanks to its
a Fig. 1. The standard quantum Otto cycle (s-QOC) is
theoreticalsimplicity,itallowstoexploreprofoundphys-
realized by directly putting the working fluid in contact
ical ideas, while still representing nowadays one of the
with the thermal reservoirs in the equivalent of isochoric
mostemployedcycles,notablyatthecoreofthefunction-
stages. The internal energy U of the TLS depends on
ing of many four-stroke engines. In quantum contexts, two parameters only: its frequency ω (such that (cid:126)ω is
alongsidetheCarnotcycle,ithasbeenamilestoneofthe
the energy separation of its two levels), and the excited
development of a quantum thermodynamics formalism state population p . In particular, U =(cid:126)ωp . Heat flow-
e e
[4, 19–22]. Furthermore, many micro- and nanoscopic
ing into/out of the TLS will change U by affecting p ,
e
realizations of thermodynamic cycles have recently been
whereas work contributions will change the value of ω.
proposed and achieved [23–28].
The working fluid is initially prepared, at frequency
In this paper, we employ non-equilibrium electromag- ω , in a thermal state at temperature T with excited
a 1
neticradiationtoenhancetheperformancesofthequan- state population p (ω ,T ), having introduced the ex-
e a 1
tum Otto cycle (QOC) of a two-level light emitter. We cited state population of a two-level system of frequency
show that, thanks to the realistic and non-trivial struc- ω and in equilibrium at temperature T as p (ω,T) =
e
2
On the other hand, the heat absorbed by the fluid (stage
D) reads
(cid:16) (cid:17)
Q =(cid:126)ω p (ω ,T )−p (ω ,T ) . (2)
abs a e a 1 e b 2
A
D B Note that not just any value ωb can be chosen. Indeed,
for the cycle to be thermodynamically convenient one
C
must require that W < 0 (i.e., one is extracting net
wf
workfromthesystem). Thisrequirementleadstotheso-
calledpositive-workconditionPWCwhich,directlyfrom
Eq. (1), reads ω /ω ≥ T /T ; moreover, the efficiency
b a 2 1
of work extraction defined as η = −W /Q is readily
wf abs
FIG. 1. Standard quantum Otto cycle. During adiabatic
evaluated as
stages A and C the two-level working fluid exchanges work
with the external world, while during stages B and D the η =1− ωb ≤1− T2 =η , (3)
TLS is put in contact with reservoirs at, respectively, T and ω T C
2 a 1
T >T .
1 2 η beingtheCarnotefficiencybetweenthesametwotem-
C
peratures T and T . As such, the natural requirement
1 2
(cid:0)1+exp[(cid:126)ω/(k T)](cid:1)−1. The TLS then undergoes four that work extraction vanishes at the Carnot limit, i.e.,
B η = η ⇒ W = 0, is obeyed, as one immediately veri-
transformations: C wf
fies by using the condition ω =ω T /T in Eq. (1).
b a 1 2
Recently, however, it has been shown that the intro-
• A: “expansion” ω →ω <ω . Since the energy of
a b a duction of non-equilibrium features in the two reservoirs
theTLSdecreases,workisdonebythefluidonthe
the working fluid interacts with in stages B and D can
external world. Adiabaticity is given by the fact
allowtogobeyondthesefundamentalbounds[30–32]. In
that p =p (ω ,T ) is constant;
e e a 1 particular, squeezing or, in general, coherence into elec-
tromagnetic reservoirs has been shown to provide higher
• B: thermalization of the system at frequency ω
b cycle performances. Coherence requires however a de-
with the reservoir at low temperature T . No work
2 tailed and steady control over the state of the baths,
isdonebyoronthesystem,whichreleasesheatinto
which can be cumbersome and usually implies the need
thereservoir,changingitspopulationtop (ω ,T );
e b 2 of external work to be supplied. One would thus like to
have an equivalent enhancement of cycle performances
• C:“compression”ω →ω . TheenergyoftheTLS
b a without the need of active control over the state of the
is now increased, such that work is exerted on the
reservoirs and, possibly, without the need of any work
working fluid; as in A, the adiabatic assumption
supply. In this work we propose ascheme toachieve this
means that p =p (ω ,T ) stays constant;
e e b 2 idea,byexploitingtheout-of-thermal-equilibrium(OTE)
propertiesofarealisticelectromagneticfieldproducedby
• D: thermalization of the system at frequency ωa a body at a fixed temperature embedded in a blackbody
with a reservoir at high temperature T1, such that radiation not in thermal equilibrium with it [33–37].
the initial cycle condition is restored. No work is
donebyoronthesystem,whichabsorbsheatfrom
the reservoir until pe =pe(ωa,T1). III. OUT-OF-THERMAL-EQUILIBRIUM FIELD
AND ITS INTERACTION WITH QUANTUM
The adiabaticity of stages A and C can be achieved by EMITTERS
changingthefrequencyoveratimeintervalmuchshorter
thantheoneneededfortheworkingfluidtointeractwith Let us then assume to have at disposal the same two
athermalbath. Inwhatfollows,whenthinkingaboutthe thermal reservoirs at T and T <T . Instead of directly
1 2 1
standardquantumdescriptionofthecycle,wewillalways couplingthemtotheworkingfluid,wesuggesttoemploy
have in mind the standard ideal (i.e., infinite frequency- them to produce an out-of-thermal equilibrium (OTE)
tuningspeed)quantumOttocycle,referredtoassi-QOC electromagnetic field, whose features can be exploited to
or simply QOC. In this configuration, the efficiency and enhancethecycleperformances. Imaginethustoconnect
the power delivered depend only on fundamental quan- thereservoirat T toamacroscopic objectofsome kind,
1
tities, independently on the practical realization of the forinstanceaslabofdielectricmaterialoffinitethickness
cycle [20, 29]. δ, and to embedded it in a thermal blackbody radiation
At the end of a si-QOC the net work made by the atT ,asdepictedinFig.2. Thisconfigurationgenerates
2
working fluid (wf) on the external world is given by the in the whole space around the slab a steady OTE field,
internal energy change during stages A and C: whose properties depend on the dielectric and geometric
properties of the slab through its reflection and trans-
W =(cid:126)(ω −ω )p (ω ,T )+(cid:126)(ω −ω )p (ω ,T ). (1) mission matrices. As such, the characterization of such
wf b a e a 1 a b e b 2
3
each transition frequency, dipole magnitude and orien-
tation, on the geometric and dielectric properties of the
slab and on the atom-atom and atom-slab distances.
The self-correlation functions of components l ∈
{x,y,z} and m ∈ {x,y,z} of the field at, respectively,
point R and R in space are defined as
z r i j
T T
2 2 1
cij (ω)= (cid:104)E (R ,ω)E† (R ,ω)(cid:105), (7)
lm (cid:126)2 l i m j
T1 cij (−ω)= 1 (cid:104)E†(R ,ω)E (R ,ω)(cid:105). (8)
lm (cid:126)2 l i m j
In what follows, it is more convenient to separate in R
i
the position r in the x−y plane (parallel to the slab
i
surface) from the z position(z =0being thecoordinate
FIG. 2. Out-of-thermal equilibrium configuration. A slab of i
of the slab surface), thus writing R = {r ,z }. Func-
dielectricmaterialiskeptatafixedtemperaturebymeansof i i i
athermalreservoiratT ,andembeddedinablackbodyfield tions (7)-(8) can be given an expression in terms of the
1
atT2 <T1. Quantumemittersplacedatadistancezfromthe slab and blackbody temperatures T1 and T2, and of the
slabsurfaceinteractwithanon-trivialsteadyelectromagnetic transmission and reflection scattering operators of the
field. slab, which in turn depend on the thickness and dielec-
tricpermittivityoftheslabmaterial[33,35]. Introducing
theaveragephotonnumberatfrequencyω andtempera-
a field is fully realistic when one employs real dielectric tureT asn(ω,T)=(cid:104)exp(cid:0)(cid:126)ω/k T(cid:1)−1(cid:105)−1,theirexplicit
functions for the particular material of the slab. B
When quantum emitters are placed in this field, they expressions read
ccoouupplleinwgihtahsitth.eIfnortmheHdIip=ola−r(cid:80)apipdrio·xEim(Rait)io,nwhliemreiti,rtuhniss (cid:104)El(Ri,ω)Em† (Rj,ω)(cid:105)= 3π(cid:126)εω03c3(cid:110)(cid:2)1+n(ω,T1)(cid:3)α1i,j(ω)(cid:12)(cid:12)lm
oinvearbaslelnpcoessoifblpeetrrmanansietniotnastoofmtihcedqiupaonletsu,mdiemisitttheersfiaenldd-, +(cid:2)1+n(ω,T2)(cid:3)α2i,j(ω)(cid:12)(cid:12)lm(cid:111), (9)
itnodaucqeudandtiupmoleemmiotmteernltocoafttehdeait-tRhit.ransition, belonging (cid:104)El†(Ri,ω)Em(Rj,ω)(cid:105)= 3π(cid:126)εω03c3(cid:110)n(ω,T1)α1i,j(ω)(cid:12)(cid:12)lm
solIenatthome iwcepaakrtccoaunplbinegdelismcriitb[e3d8]b,ytaheMdayrnkoavmiaicnsmofasttheer +n(ω,T2)α2i,j(ω)(cid:12)(cid:12)lm(cid:111), (10)
equation [35]. Be σi−(+) the lowering (raising) operator where the 3×3 matrices αi,j(ω) are given by
of transition i: the emitters master equation reads 1,2
dρ i(cid:2) (cid:3) αi,j(ω)=3πc(cid:88)(cid:90) d2k d2k(cid:48) ei(k·ri−k(cid:48)·rj)
dt =−(cid:126) Heff,ρ +Dloc(ρ)+Dnl(ρ), (4) 1 2ω p,p(cid:48) (2π)2(2π)2
(cid:26)
wemhietrteerHseHffa=m(cid:80)iltoinωiiaσni+,σini−+wh(cid:80)icrih,ejstΛhiejσfri−eeσj+pairstthisemeffoedcitfiievde ×(cid:104)p,k| ei(kzzi−kz(cid:48)∗zj)Xp+,p+(cid:48)(k,k(cid:48),ω) (11)
(cid:16)
by a field-induced dipole-dipole coupling of strength Λij × Ppw−RPpwR†−PewR†−TPpwT†
betweentworesonant transitionsiandj. Thesum(cid:80)res −1 −1 −1 −1
runs over any possible pair of resonant transitions i,j.i,j (cid:17)(cid:27)
+RPew |p(cid:48),k(cid:48)(cid:105),
The dissipative effects induced by the field are de- −1
scribed by the dissipators D and D , each given in
loc nl
terms of σ± as
D i=(cid:88)(cid:0)γ+L(σ−)+γ−L(σ+)(cid:1), (5) α2i,j(ω)=32πωc(cid:88)(cid:90) (d2π2k)2(d22πk)(cid:48)2ei(k·ri−k(cid:48)·rj)
loc i i i i p,p(cid:48)
i (cid:26)
D =(cid:88)res (cid:0)γ+R(σ−,σ−)+γ−R(σ+,σ+)(cid:1), (6) ×(cid:104)p,k| ei(kzzi−kz(cid:48)∗zj)Xp+,p+(cid:48)(k,k(cid:48),ω)
nl ij i j ij i j
(cid:16) (cid:17)
i,j × TPpwT†+RPpwR† (12)
−1 −1
having introduced the non-diagonal and diagonal lind- +ei(kzzi+kz(cid:48)∗zj)X+−(k,k(cid:48),ω)RPpw
blad dissipators as, respectively, R(K ,K ) = K ρK†− p,p(cid:48) −1
1/2(cid:8)K2†K1,ρ(cid:9) and L(K) = R(K,K).1All2the ra1tes γ2i±, +e−i(kzzi+kz(cid:48)∗zj)Xp−,p+(cid:48)(k,k(cid:48),ω)P−pw1R†
γ± andΛ aredirectlyobtainedfromtheself-correlation (cid:27)
ij ij +e−i(kzzi−kz(cid:48)∗zj)X−−(k,k(cid:48),ω)Ppw |p(cid:48),k(cid:48)(cid:105),
functionsoftheelectromagneticfield[35]anddependon p,p(cid:48) −1
4
(cid:113)
being k = ω2 −k2, and where the operator Ppw(ew)
z c2 −1
TLS
is the projector on the propagative (evanescent) sector
divided by k . We have introduced the 3×3 matrices
z
Xµν (k,k(cid:48),ω)(cid:12)(cid:12) =(cid:15)ˆµ(k,ω)(cid:12)(cid:12) ×(cid:15)ˆν (k(cid:48),ω)(cid:12)(cid:12) , (cid:15)ˆµ(k,ω) be-
p,p(cid:48) lm p l p(cid:48) m p
ing the polarization unit vector of the electromagnetic M
field,correspondingtopolarizationp∈{TE,TM}andz-
component of the propagation direction µ ∈ [+,−] [33].
The operators R and T describe, respectively, reflection
andtransmissionofelectromagneticradiationbytheslab
and, as such, depend on the slab dielectric permittivity
ε(ω) and slab thickness δ as
FIG.3. SchematicrepresentationoftheeffectsofOTEfieldin
(cid:104)p,k|R|p(cid:48),k(cid:48)(cid:105)=(2π)2δ(k−k(cid:48))δ ρ (k,ω), (13) the steady-state of a three level atom M, resonantly coupled
pp(cid:48) p
to a TLS. The transition |0(cid:105) ↔ |2(cid:105) has the same frequency
(cid:104)p,k|T|p(cid:48),k(cid:48)(cid:105)=(2π)2δ(k−k(cid:48))δ τ (k,ω), (14)
pp(cid:48) p ω astheelectronicresonanceoftheslabmaterial,whilethe
S
transition |1(cid:105) ↔ |2(cid:105) is resonant with the TLS at ω . Here
with a
T (ω )>T (ω −ω ) due to a transition-slab resonance
env S env S s
1−e2ikzmδ effect. Inthissituation,aredistributionofsteadypopulation
ρ (k,ω)=r (k,ω) , (15)
p p 1−rp2(k,ω)e2ikzmδ of M (schematically represented by yellow circles) brings the
transitionatω toamuchmoreenergeticstate,abletoinduce
a
τ (k,ω)=(1−r2(k,ω)) ei(kzm−kz)δ , (16) a steady very high or even negative temperature θwf to the
p p 1−rp2(k,ω)e2ikzmδ TLS.
where r and r are the standard vacuum-medium
TE TM
(cid:113)
Fresnel reflection coefficients and kzm = ε(ω)ωc22 −k2. of the OTE field with each atomic transition by means
ItisworthstressingatthispointthatEqs.(11)and(12) on an effective field temperature. Note however that
give the total field correlators as a result of four contri- such temperature fundamentally depends on the tran-
butions: the blackbody radiation at tempertaure T , the sition frequency: two different transitions exchange pho-
2
blackbody radiation reflected by the slab, the blackbody tons with the same field at different rates and, as such,
radiation transmitted by the slab and finally the radia- perceive the same field as having two different tempera-
tion directly emitted by the slab at T . Note that all tures. In particular, thanks to the strong dependence of
1
but the first contribution depend on the slab properties γ± on the slab dielectric properties, as previously com-
i
through the operators R and T. In particular, in corre- mented, this effective field temperature will be more or
spondencewitharesonanceinthedielectricpermittivity less close to the real temperature T of the slab, depend-
1
ε(ω) for a value ω = ω (i.e., in correspondence with a ing on the relative importance of α and α in Eqs. (9)-
S 1 2
peak in the spectrum of ε(ω)), the slab-dependent con- (10). Therefore, when ω = ω , i.e., the electronic reso-
S
tributions to Eqs. (11) and (12) become dominant for a nancefrequencyoftheslabmaterial, suchthatbothreal
broad range of atom-slab distances. andimaginarypartofitsthedielectricpermittivityε(ω)
Equations(9)and(10)canbeusedtocharacterizethe showasharphighpeakintheirspectrum,theslabcontri-
influence of atom-field coupling on the atomic dynamics, bution to the field correlation functions (7)-(8) becomes
through the dissipation rates in Eq. (4). Indeed, for real dominant, and the rates (17) are profoundly affected by
dipolesd ofcartesiancomponentsdx,y,z,theratesγ±(ω) it: transitions at ω feel a temperature much closer to
i i ij S
(including γ±(ω)=γ±(ω)) are [35] the one of the slab than to the background blackbody
i ii
radiation.
(cid:88)
γ±(ω)= cij (±ω)dldm. (17) The dipole-dipole coupling strength Λij has also a sim-
ij lm i j
ilar expression, partly depending on the slab properties
l,m=x,y,z
andpartlyoriginatingfromaT =0vacuumcontribution
Therateofabsorptionandemissionofphotonsfrom/into [35], which we do not report here for the sake of brevity.
the field is the standard way of characterising the field The term (cid:80)resΛ σ−σ+ allows two resonant transitions
i,j ij i j
temperature, at least the one perceived by the transi- intwodifferentatomstoexchangeenergyundertheform
tion involved in the photons exchange. Introducing the of heat.
vacuum emission rate γ (ω) = ω3(3π(cid:126)c3ε )−1, one can
0 0 Consider now the case of two quantum emitters only,
write
aTLSQandathree-levelsystemM,placedinthisOTE
2γ±(ω)=γ (ω)(cid:0)1±1+2n (ω)(cid:1). (18) field. M has three non-degenerate transitions 1,2 and 3,
i 0 env one of which (the one at lowest frequency, labeled as 2)
where nenv(ω)=(cid:104)exp(cid:0)(cid:126)ω/kBTenv(ω)(cid:1)−1(cid:105)−1 is the av- isstrruecstounraenotfwMithsucQhathtaftretqhueetnrcaynsωitai.onB2ecnonownectthselelevveelsl
eragethermalphotonnumbercorrespondingtoatemper- |1(cid:105) and |2(cid:105), whereas the high frequency transition be the
atureT (ω). Thisallowstocharacterizetheinteraction one between levels |0(cid:105) and |2(cid:105), and suppose this latter to
env
5
be resonant with the slab at ω . Due to the non-trivial
S
dependence of γ± on the transition frequency, the three-
i
level system with three different transitions exchanges
photonswiththefieldatdifferentratesand,assuch,per-
ceives three different temperatures. In particular, since A
T >T , the transition |0(cid:105)↔|2(cid:105) at ω perceives a much
1 2 S D B
highereffectivetemperaturethantherestofatomictran-
sitions. Thesituationisthereforesomewhatanalogousto C
athree-levelsystem, havingeachtransitionconnectedto
a different thermal reservoir. As explained in [13] and
schematically shown in Fig. 3, the net effect is a redistri-
bution of population in each level of M (represented in
Fig.3throughyellowcircles), leadingtoaveryenergetic
transition|1(cid:105)↔|2(cid:105). Asshownin[13,14],duetothefact
that this transition of M is resonant with Q, M can de-
FIG. 4. Schematic OTE Otto cycle. During stages A and
liver into the TLS a large amount of energy through the
C the two-level working fluid exchanges work with the exter-
dipole-dipole interaction H . This energy redistributes
eff nal world and, possibly, heat with the baths due to the non-
the populations in the two energy levels of Q, inducing
perfectadiabaticityoftheprocess. DuringstagesBandDthe
in it a Gibbs-form steady state ρQ ∝ e−(cid:126)ωa/kBθwf, cor- TLSisputincontactwithreservoirsat,respectively,TL and
responding to an atomic temperature θwf far outside the TH, corresponding respectively to Tenv(ωb) and θwf induced
range [T ,T ] and even up to negative values. As such, bytheOTEfieldwithoutandwiththehelpoftheadditional
2 1
theneteffectoftheOTEstructureofthefieldistoallow three-level system. The stages A and C are supposed to last
the temperature of a TLS to be brought to values which for a time α−1, such that an ideal cycle is achieved when
would not be accessible just by direct thermal contact α→∞
of the atom with the real reservoirs at T and T . Note
1 2
thatthisispossibleonlywhenMandQareinresonance.
the presence of M (via atom-atom quantum coherence
Thus,ifthefrequencyofQwerechangedtoanothervalue
[13]), when the TLS has frequency ω its steady temper-
ω , Q would not interact at all with M and would thus a
b ature will be θ >T . Therefore, before stage A begins,
thermalize to the effective temperature T (ω ). wf 1
env b the working fluid feels the presence of a much more en-
ergeticeffectiveenvironmentthansimplythebathatT ,
1
sincenowT =θ . TheinteractionbetweenMandthe
H wf
IV. OTE QUANTUM OTTO CYCLE TLS is however only possible when the transition of the
workingfluidisresonantwithoneofM[35](incidentally,
We suggest then to exploit this effect to enhance the note that a heat engine based on two resonant emitters
performances of an Otto cycle using the TLS Q as work- in equilibrium baths has been studied in [41]). Chang-
ingfluid. Duetothefundamentalroleplayedherebythe ing ω from ω to ω < ω puts the TLS and M out of
a b a
OTE field, we refer to this modified cycle as OTE quan- resonance and switches off their interaction. The TLS
tum Otto cycle. As commented, this OTE field configu- thus only interacts with the non-equilibrium field, which
ration can be produced by the same two thermal baths induces a temperature T ∈ [T ,T ] with a non-trivial
env 2 1
considered for the s-QOC, one fixing the temperature of dependence on the TLS frequency ω. As a consequence,
the slab and the other producing the blackbody radia- thetwonewtemperaturesofthecyclearenowT =θ
H wf
tion. Theslabisconnectedtothethermalbathathigher (felt at ω ) and T =T (ω ).
a L env b
temperature T , while the one at lower temperature T Consider for instance a slab of SiC (ω = 1.495 ×
1 2 S
is used to produce a thermal blackbody radiation im- 1014rad/s) of δ = 1µm, and be ω = 0.1 × ω . The
a S
pinging on the slab itself. To maintain the steady OTE three-level atom and the working fluid be at a distance
configuration one only needs heat inputs from reservoirs z = 26µm from the slab surface and at a distance
T and T . Such an input will in the following be con- r = 1µm from each other. Solving the long-time limit
1 2
sidered a structural feature of our setup in the form of of Eq. (4), using Eqs. (7)-(17) and employing a Drude-
housekeeping heat [39, 40], and thus not taken into ac- Lorentz model for the dielectric permittivity ε(ω), one
count in the evaluation of efficiency, as commonly done can (numerically) find the two temperatures T and T .
H L
in non-equilibrium scenarios [31]. WhenthetwoexternaltemperaturesareT =700Kand
1
Due to the OTE properties of the field, as commented T = 200K, the interaction with M brings the TLS to a
2
in the previous section, the TLS will not interact any- temperature θ =−537K, i.e., to population inversion.
wf
more with reservoirs at T and T but will rather per- On the other hand, T = 313K for ω = ω /2. Note
1 2 env b a
ceive effective environments depending on its frequency. that, strictly speaking, here T < T since T is nega-
H L H
To stress this difference, we call here T and T the ef- tive. However, what matters is clearly the fact that the
H L
fective temperatures perceived by the working fluid at, (effective) bath at T be more energetic than the one at
H
respectively, ω and ω , as shown in Fig. 4. Thanks to T , which is the case here.
a b L
6
With this in mind, let us revisit all the four stages 10
of the Otto cycle in light of this new structure of the
8
(effective) thermal baths of the working fluid: si-QOC
6
• A: ω → ω happens now at constant p =
a b e
p (ω ,θ ), higher than the standard value;
e a wf 4
• B: the thermalization changes p (ω ,θ ) →
e a wf 2
pe(ωb,Tenv(ωb)) at constant frequency ωb; (a)
0
• C: ωb →ωa is at constant pe =pe(ωb,Tenv(ωb)); 0 0.2 0.4 0.6 0.8 1
1
• D:p (ω ,T (ω ))→p (ω ,θ )isatconstantω .
e b env b e a wf a
0.8
The work done by the working fluid becomes now
0.6 si-QOC
W =(cid:126)(ω −ω )p (ω ,θ )+(cid:126)(ω −ω )p (ω ,T ). (19)
wf b a e a wf a b e b L
0.4
Note that, thanks to the much broader gap between
TH = θwf and TL = Tenv(ωb), the PWC in this OTE 0.2
(b)
configuration can become much less restrictive. In par-
ticular, when θ < 0 and T > 0, work can be ex- 0
wf env 0 0.2 0.4 0.6 0.8 1
tractedfromtheTLSforeach valueofω ≤0. Moreover,
b k
due to the interaction with M, p (ω ,θ ) (cid:29) p (ω ,T )
e a wf e a 1
suchthatmuchmoreenergyisgainedwhenreducingthe
TLS frequency. On the other hand, T (ω ) ∈ [T ,T ] FIG. 5. Work extracted W [panel (a)] and cycle efficiency η
env b 2 1
[panel(b)]fortheOTEOttocycleversustheratiok=ω /ω .
byconstruction,keepingp (ω ,T (ω ))relativelycloser b a
e b env b The different curves correspond to different tuning times in
to p (ω ,T ). The net effect is thus to gain an enormous
e b 2 stagesAandC.Perfectadiabaticityisachievedwhenα=∞
quantityofworkcomparedtothesi-QOCcase. Theheat
(fullredline). Thefigureshowsalsothesamequantitiesforan
absorbed by the TLS is
infinitespeedstandardquantumOttocycle(si-QOC,dashed
Qabs =(cid:126)ωa(cid:16)pe(ωa,θpw)−pe(ωb,Tenv)(cid:17), (20) bTl2a.cTkhlienes)tahnadvainrdgCthaernsaomteeffitwcioenecxyteηrCnailstaelmsopreerpatourrteesdTo1natnhde
left vertical scale and by the horizontal full line in panel (b).
such that the efficiency is again given by Eq. (3). How- AlltheplotsareobtainedforaSiCslab,withδ=1µm,ω =
a
ever, when compared to the si-QOC, η can now become 0.1×ωS, z=26µm, r=1µm, T1 =700K and T2 =200K.
much higher thanks to the new allowed values for ω .
b
where U (t)=−iω (t)(cid:2)σ+σ−,ρ(cid:3).
n n
A. Non-ideal OTE Otto cycle Solving Eq. (21) with the linear time-dependence of
the frequency, one obtains a nontrivial dependence of
Up to now, to preserve adiabaticity we considered an the excited state population on time. The state of the
idealcyclewheretheTLSfrequencyischangedsuddenly. TLS after stage A or C will thus depend on α and
Inrealisticmodels,afinite-timechangeoffrequencycor- will be referred to as ρA(C)(α−1), with excited state
responds to a non-adiabatic process during which the population pA(C)(α−1), as depicted in Fig. 4. Focus-
e
working fluid exchanges work with the external world ing now only on stage A of the cycle (stage C can be
and heat with the field reservoir. We account for this by treated analogously), the total change in internal energy
allowingdissipationoftheTLSduringstagesAandCof is ∆E(A) = (cid:82)1/αdt tr(cid:0)ρ˙ H +ρ H˙(cid:1). This can be split
the cycle. In particular, due to the fact that the dynam- U 0 A A
into a work and a heat part as
ics begins when the working fluid is set out-of-resonance
with M (stage A) and ends when the two emitters are (cid:90) 1 (cid:90) 1
brought back in resonance (stage C), the dissipative ef- W (α)= α dttr(cid:0)ρ H˙(cid:1)= α dtp(A)(t)ω˙ (t),(22)
A A e A
fects are only induced by the electromagnetic field. 0 0
staWgeesaAssaunmdeCainlintehaerfotirmmeω-tu(nti)n=g oωf(it)h+e(Tω(Lf)S−foωr(i)b)oαtth, QA(α)=(cid:90) α1 dttr(cid:0)ρ˙AH(cid:1)=(cid:90) α1 dtp˙(eA)(t)ωA(t).(23)
n n n n 0 0
n = A,C, and ω(i) = ω , ω(f) = ω . Here α is
A(C) a(b) A(C) b(a) The total work done by the system during the cycle
theadiabaticparameterwhichcharacterizesthespeedof
is W (α) = W (α) + W (α), whereas the absorbed
the stage, in the sense that both stages A and C last for wf A C (cid:0) (cid:1)
heat now reads Q (α) = Q + Θ Q (α) Q (α) +
α−1seconds, and become fully adiabatic when α → ∞. abs D A A
(cid:0) (cid:1)
Θ Q (α) Q (α),whereΘ(x)istheHeavisidestepfunc-
In the time interval [0,1/α], Eq. (4) thus reduces to C C
tion of x. Fig. 5(a) shows, for an exemplary configu-
ρ˙ =γ+(cid:0)ω (t)(cid:1)L(σ−)+γ−(cid:0)ω (t)(cid:1)L(σ+)+U (t), (21) ration, the work extracted −W (α) from the TLS at
n n n n wf
7
0.8 at asymptotically unitary efficiency is thus the most pe-
(a) culiar characteristics of our OTE cycles, impossible to
0.6
achieve in standard equilibrium contexts. The main re-
0.4 10
si-QOC
0.2 8
0.0
10-3 10-2 0.1 1 10 100 6
8
(b)
4
6
4 2 si-QOC
2 0
0 0.2 0.4 0.6 0.8 1
si-QOC
0
10-3 10-2 0.1 1 10 100
FIG. 7. Same configuration as in Fig. 5: Work extracted
FIG. 6. Same configuration as in Fig. 5: (green dashed line) versusthecorrespondingvalueofefficiencyfordifferentvalues
Efficiency at maximum power η (panel (a)) and work at of the cycle adiabatic parameter α.
MW
maximum efficiency W (panel (b)) for an OTE quantum
Mη
Ottocycleversustheadiabaticparameterα. Bothquantities
are also shown for a si-QOC between the same two external sults are evident in Fig. 7, showing for different values
temperatures. Note that W is identically zero for the si- of α the curves of W versus the efficiency η. Remark-
Mη
QOC (as expected), but it is always positive for OTE cycles ably, contrarily to s-QOC, for α (cid:54)= 0 the OTE cycle has
with α(cid:54)=0. non-zero W , which is a behavior opposite to standard
Mη
thermodynamic expectations .
different α, together with the same quantity for a si-
V. CONCLUSIONS
QOC between the same two temperatures, as a func-
tion of the ratio k = ω /ω . For a wide range of val-
b a InthisworkweintroduceaquantumOttocyclescheme
ues of the adiabatic parameter α, the work extracted is
which is realized by using a simple non-equilibrium real-
much higher than for an infinite-speed standard quan-
istic configuration of the electromagnetic field. A two-
tumOttocycle,andisalwayspositiveinthewholerange
level system undergoes 4 transformations with the help
0 ≤ ω ≤ ω . In particular, the maximum of work ex-
b a of a resonant 3-level system. We show that its perfor-
tracted in the si-QOC is WQOC = 2.6×10−23J, which
max mances are drastically enhanced, overcoming standard
becomes WOTE = 9.6 × 10−23J for the α = ∞ OTE
max equilibrium thermodynamic bounds. This scheme allows
cycle, nearly 4 times bigger. In addition, as shown in
to considerably increase both work extraction and its
Fig. 5(b), the OTE efficiency of work extraction asymp-
efficiency. In particular, finite (and almost maximal)
totically approaches 1 as ω → 0, situation forbidden in
b work can be extracted at asymptotically unitary effi-
the si-QOC due to the value of the PWC.
ciency, largely overperforming any standard ideal Otto
Figure 6 shows, in panel (a) and (b) respectively, the cycle working between the same two temperatures. The
efficiencyatmaximumpowerη andtheworkatmax- cycle is obtained using a single non-coherent reservoir
MW
imum efficiency W for the standard ideal cycle (short- producedbyheatfluxesprovidedtomacroscopicobjects,
Mη
dashed black line), and for both the infinite speed limit withouttheneedofanyactivecontrolonitsstate. Itex-
(solid red line) and the finite speed (long-dashed green ploitsquantumatomiccoherencewhichallowsthesystem
line)OTEcycle. Notethatη canbecomegreaterthan workingbetweeneffectivethermostatsatlargelydifferent
MW
its correspondent value for the standard ideal Otto cycle temperatures. Thisprovidesaninnovativeframeworkfor
already at finite speed. The infinite-speed limit of the highly efficient energy management and quantum ther-
OTE cycle greatly overperforms the infinite-speed stan- mal engines at the quantum scale.
dard cycle as ηOTE > 2ηQOC. Furthermore, the work
MW MW
at maximal efficiency W is of no interest in standard
Mη
thermodynamic cycles, since it corresponds to the work ACKNOWLEDGMENTS
performedbythecycleworkingatitsCarnotlimit,which
is known to vanish. In the case of OTE cycles, how- We acknowledge insightful discussions with P. Doyeux
ever, W is positive for any non-zero value of α, and and R. Messina, and financial support from the Julian
Mη
is very close, for ideal cycles, to WOTE. Finite work Schwinger Foundation.
max
8
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