Supremacy of the quantum many-body Szilard engine with attractive bosons

2.9 MB·

by  Jakob Bengtsson

Supremacyofthequantummany-bodySzilardenginewithattractivebosons
J.Bengtsson1,2,M.NilssonTengstrand1,2,A.Wacker1,2,P.Samuelsson1,2,M.Ueda4,5,H.Linke1,3 &S.M.Reimann1,2
1NanoLund, Lund University, P.O.Box 118, SE-22100 Lund, Sweden
2 Mathematical Physics, Lund University, Box 118, 22100 Lund, Sweden
3 Solid State Physics, Lund University, Box 118, 22100 Lund, Sweden
4DepartmentofPhysics, UniversityofTokyo, 7-3-1Hongo, Bunkyo-ku, Tokyo113-0033, Japan
5RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
(Dated:January30,2017)
Inaclassicthoughtexperiment,Szilard[1]suggestedaheatenginewhereasingleparticle,forexamplean
atomoramolecule,isconfinedinacontainercoupledtoasingleheatbath.Thecontainercanbeseparatedinto
twopartsbyamoveablewallactingasapiston. Inasinglecycleoftheengine,workcanbeextractedfrom
theinformationonwhichsideofthepistontheparticleresides. TheworkoutputisconsistentwithLandauer’s
principlethattheerasureofonebitofinformationcoststheentropyk ln2[2–4],exemplifyingthefundamental
B
7
relationbetweenwork,heatandinformation[5–11].HereweapplytheconceptoftheSzilardenginetoafully
1
interactingquantummany-bodysystem. WefindthataworkingmediumofanumberofN ≥ 2bosonswith
0
attractiveinteractionsisclearlysuperiortootherpreviouslydiscussedsetups[12–17]. Insharpcontrasttothe
2
classicalcase,wefindthattheaverageworkoutputincreaseswiththeparticlenumber. Thehighestovershoot

n occursforasmallbutfinitetemperature,showinganintricateinterplaybetweenthermalandquantumeffects.
a We anticipate that our finding will shed new light on the role of information in controlling thermodynamic
J fluctuationsinthedeepquantumregime,whicharestronglyinfluencedbyquantumcorrelationsininteracting

7 systems[18].
2

] I. INTRODUCTION tractiveinteractionsbetweentheparticles,astheycommonly
h
occur in, for example, ultra-cold atomic gases [30]. We
p
demonstrate quantum supremacy in the few-body limit for
-
The Szilard engine was originally designed as a thought
t N ≤ 5, whereasolutiontothefullmany-bodyproblemcan
n experiment with only a single classical particle [1] to illus-
a beobtainedwithveryhighnumericalaccuracy.Aperturbative
trate the role of information in thermodynamics (see, for ex-
u approach indicates that the supremacy further increases for
ample, [8] for a recent review). The apparent conflict with
q largerparticlenumbers.Surprisingly,thehighestovershootof
[ the second law could be resolved by properly accounting work compared to W = k T ln2 (i.e., the highest possible
1 for the work cost associated with the information process- classical work output)1occuBrs for a finite temperature, exem-
ing [2, 3, 19–22]. Although Szilard’s suggestion dates back
v plifyingtherelationbetweenthermodynamicfluctuationsand
to1929,onlymorerecentlytheconversionbetweeninforma-
8 themany-particleexcitationspectrum.
3 tionandenergywasshownexperimentallyusingaBrownian
1 particle [6]. A direct realisation of the classical Szilard cy-
8 clewasreportedbyRolda´netal.[23]foracolloidalparticle
0 in an optical double-well trap. In a different scenario, Koski
1. et al. [24, 25] measured kBT ln2 of work for one bit of in- II. MANY-BODYSZILARDCYCLEFORBOSONSWITH
0 formationusingasingleelectronmovingbetweentwosmall ATTRACTIVEINTERACTIONS.
7 metallicislands.Aquantumversionofthesingle-particleSzi-
1 lard engine was first discussed by Zurek [26]. In contrast to
Our claim is based on a fully ab initio simulation of the
:
v the classical case, insertion or removal of a wall in a quan- quantum many-particle Szilard cycle by exact numerical di-
i tum system shifts the energy levels, implying that the pro-
X agonalisation, i.e., the full configuration interaction method
cessmustbeassociatedwithnon-zerowork[27–29]. Kimet
(asfurtherdescribedinthesupplementarymaterial). Ahard-
r al.[12]showedthattheamountofworkthatcanbeextracted
a walled one-dimensional container of length L confines N
crucially depends on the underlying quantum statistics: two
bosons that constitute the working medium. We model the
non-interacting bosons were found superior to the classical
interactionsbytheusualtwo-bodypseudopotentialofcontact
equivalent,aswellastothecorrespondingfermioniccase.
type[30], gδ(x −x ), wherethestrengthoftheinteraction
1 2
ManydifferentfacetsofthequantumSzilardenginehavebeen g is given in units of g = (cid:126)2/(Lm). The single-particle
0
studied,includingoptimisationofthecycle[14,15]ortheef- ground state energy E = (cid:126)2π2/2mL2 sets the energy unit,
1
fect of spin [16] and parity [17], but all for non-interacting
where m is the mass of a single particle. The cycle of the
particles. Thecaseoftwoattractivebosonswasdiscussedin
Szilardenginegoesthroughfoursteps,assumedtobecarried
Ref. [13]; however, the authors assigned the increased work
outquasi-staticallyandinthermodynamicequilibriumwitha
output to a classical effect. The question thus remains how
single surrounding heat bath at temperature T: (i) insertion
the information-to-work conversion in many-body quantum
of a wall dividing the quantum many-body system at a posi-
systemsisaffectedbyinteractionsbetweentheparticles. tion(cid:96)ins,followedby(ii)ameasurementoftheactualparticle
Here, we present a full quantum many-body treatment of number n on the left side of the wall, (iii) reversible trans-
spin-0 bosonic particles in a Szilard engine with realistic at- lation of the wall to its final position (cid:96)rem depending on the
n
2
11..44
11..22
Bosons, N =4 b
BBoossoonnss, ,, NN=<= 544 Bosons
a 11..1144..44 BOSONS = b
10 11..22 interacting interacting
1100 11..11 non-interacting non-interacting
BBoossoonnss
11..1122..22 iwnetearkalyc tiinntgeracting
1 )] nnoonn--iinntteerraaccttiinngg
2 11
kT/ETemperature, B10011..11100 Aggt==tra0−ctgiv0e W/[kTln(Work output, BkT/Emperature, kT/Eperature, B1001B100001...0....118810246811..1110111011W/[kTln(2)]Work output, W/[kTln(2)]Work output, B 0B AggW/[kTln(2)]Work output, Agg1B1t001.==001.tt11==0t−...r...9915001.r118815a0a...0−011881c50−1155c.0t01g11tiNg11vg0i0v00e−ue−0−mc(55.n0l01ao55N..gbs1Nn0N0sgu-1euii011cunmg22mat0r−me00lb rbobaeg11ece−r−110f00tr ir− 00n o0g poog−.g0f1f )0fa pg 33−p1p0−ara10atr1110rrtig110tigct1155ciig0550c−c−55ll0−elle−ee22s2g0sc(ssc(0,nn0c(044llg,, aag,noo 0NlgNss 0Nannoss0N−--−s22iiiin22ccnn4s00a−−4at00t-0e22eli0lic1rngrg4aa0000a0tcc0getl550001ti0ignrn...0001ag4680g)2222...)c4685555tincgc22)55
em 1.05
1TTe.00050..11 11
1 1 BBosoosnons,s ,N N==33 0.2
10 0.2
g=g0 110 0.09.595 BBooCsslCaoolsannsssissc,ia,c la llNNi ml i=mi ti4=t 4
Repulsive
0.1 0.95
0.3 0.4 0.5 0.6 0.7 10 Bo00s.0.o000.1n101s, 00N..001.11= 3101.1 11001 11000010 110000000100
Insertion position, ℓins/L 1 0
0.951 TTeTememmppeperearratautturuerre,e ,, k kkBBBTTT///EEE111
g =Clga0ssical limit
g = g
FIG.1. Workoutputofthemany-bodySzilardengine. aFo0r1.N1 =Re4pbuolsso0invsetheoptimalworkoutputW (inunitsoftheclassicalsingle-
particleworkW1 = kBT ln2)isfoundforattractiveinteracti0o0n..s001g.13 R(uepppe0ur0l.ps1.a4inveel)at01a.fi5nitete1m0.0p6erature10a0.n70darou1n0d0a0symmetricinsertion
positionofthebarrier. (Theinteractionstrengthisinunitsofg0 =0g.3(cid:126)=2/(−ILnmsg0e0)r.4twioitnh pboox0s.ilt5eiongnt,hℓLin0sa./n6dLsingl0e-.p7ar1ticlemassm). Theoptimal
workoutputexceedsthecaseofnoninteracting(middlepanel)and rAeptturlasicvInetisbvTeoeesrtominosnp( lepoorwasetistutioprnaen,,e ℓlk)i.nsR/TeLpu/lEsivebosonsexhibitN peaksinthe
low-temperaturelimitfordifferentinsertionpositions. Thisb0eh.1aviorissimilartothecaseofnon-iBnteracting1fermionsandisasignatureof
1
thetransitionintoaTonks-Girardeaustate(seetext). bThEemaximalworkoutputW/W1increasessignificantlyw0.i8ththeparticlenumberfor
bosonswithattractiveinteractionswithg = −0.01g ,(so/lidredline). Itisalwayslargerthantheresultfornon-interactingbosons(dashed
0 T 10
orange line) and classical particles (dashed blue line). InBeach case, the temperature is chosen to maximise the relative work output. For
non-interactingbosons,thisoccursforT →0(weusedkkT/E =0.01),whiletheinteractingcaseisoptimisedatafinitetemperature(like
thewhiteregionina).ForclassicalparticlestheresultisiBne, depen1dentoftemperature.Optimalinsertionandremo0v.a6lpositionsofthewallare
r
usedtomaximiseW/W forallconsideredsystems. Theuinset1showsasketchofthemany-particleSzilardengineperformingworkatthe
1 at
expansionstepofthecycle.cTheworkoutputforN =3rasafunctionoftemperaturefordifferentstrengthsoftheattractiveinteraction.For
e
largeT,allcurvesconvergeintotheclassicalresult. p g = 0 0.4
m
e
T 0.1
outcomenofthemeasurement,andfinally(iv)removalofthe −(cid:80)N p ((cid:96)ins)lnp ((cid:96)ins).Goingbacktotheoriginalstate
n=0 n n 0.2
barrierat(cid:96)rem. 10 in the cycle, this information is lost, associated with an av-
n
The total average work output of a single cycle with pro- erage increase of entropy ∆S = kBI. This increase in en-
cesses(i)-(iv)hasbeendetermined[12]as tropyofthesystemallowsonetoextracttheaverageamount
0
ofworkW ≤ k TI whichcanbepositive. Here,theequal-
1 B
W =−kBT (cid:88)N pn((cid:96)ins)ln(cid:20)ppn(((cid:96)(cid:96)rienms))(cid:21) . (1)g =iogtfy0thoenlbyahrroieldrsisifreavllerpsnib(l(cid:96)ernefmor)e≡ach1.obInsetrhviesdcpaasretitchleenreummobvearl.
n=0 n n RepuTlhsiisvreeversibility had been associated with the conversion of
0.1
thefullinformationgainintowork[32],asexplicitlyassumed
Hleeftreo,fpthne((cid:96)w)adllelnooctaetsedthaetpproosbitaiobnili(cid:96)ty,atnodfiNnd−nnppaarrttiicclleessttoott0hh.ee3 Iisnnisn0eRg.4rleetfi-o.pn[a3 rp3tio0]c.s.l5eiWticoahnsi,eleℓw0pin.in6tsh/((cid:96)L(cid:96)rnreemm)0=.≡701ainsds(cid:96)trraemigh=tfoLrw,tahridsifsorhathrde
0 1
right, if the combined system is in thermal equilibrium. The torealiseforN ≥2[32].
N-particleeigenstatesΨ withenergyE ,obtainedbynumer-
i i
Forourcaseofamovingpiston,thefullworkcantypically
ical diagonalisation, can be classified by the particle number
notbeextracted. TooptimiseW,wechoosetheoptimal(cid:96)rem,
npni(i(cid:96)n)t=he(cid:80)lefitδsnui,bnsey−stEeim((cid:96))w/kiBthT0/Z<wxith<Z(cid:96).=T(cid:80)heinew−Eeifi((cid:96)n)/dkBthTa.t mpraoxciemduisriengispsinm((cid:96)ilrnaemrt)oftohreanlolns-yinstteemrasctcinognsciadseere[d15h]e).re. (nThe
Measuring the particle number on one side after insertion
of the wall, one gains the Shannon information [31] I = The highest relative work output is obtained for a many-
1.2
)] CLASSICAL BOSONS, T 0
2 →
n( 1 PARTICLES g=0
l g<0
T
B0.8 m
k
/[ mu
W0.6 xi
a 3
ut, M
p0.4
bodysystemofattractivebosonsatafinitetemperature. This 1.21ut.2
rcTo(abwbhiiwnse(cid:96)alcooaoliigtncsnstste→rihhsnuoormkWbeaornnW=escenss0ow/ct..-,uiiso/WifohtutLnmWFphni(rtt1t/oFuheepees12rtorea(cid:46)twafrr)rsWc(cid:46)aeecwsoovfgntl0erioaamo/igsn.k1rolr8rWtlggp.tanh8o0haeabis16u8bnesrriltgnie)o(psrit≈e,snneeoau≤tnososmshsntgenuete1v0wepisrlas.nW/[kTln(2)]tWork output, ta,et1sheetBeoontrsh2dlohapfcyil00001ewsooetcsfiuha.....prputSnom124682nsreeeapnruemsrnathfyosphaaoxeieg,W/[kTln(2)]output, nptefrliotrBiilsmho>ne(cid:96)nfeorym)et0001giemtunhfs.fe0ps....iotmelrde14682iFeWuounra(cid:54)=ndmiwlcrwgstelittCPaenie.ichLovoranAsLsle1yreolf/aW/[kTln(2)]tegk output, AkRsfsBr2Mdo((fsSTts(oa.loiitemi00001oSInohun)acuCeIw,.....IgnetotainprCL124682pedlwlteEeurAceCPadtpqsiph-hottSlLaAotLuptteaienahrlssAaapRr)arltiapt.nrweaatSclTittiaitcintioFSImauntnicflCehvynoeomeIelliCLeeesssrr)rltEABgCPSOL=ALSAR0OW/[kTln(2)]Work output, MaximumSTBNW/[kTln(2)]SIWork output, S0000CBI,C....L 012468Bg0000TEAWork o0....O=SL012468→ggS000O<>Maximum.g002N00>0S0,g00 0Bg<T.>CP→1O0=ALg→gS0.CPAR10O<TALSMaximumT0N0A RSI.,C01SSSIT0CLN,MaximumSI C.EATO2I00SCLLSP.E=A→gO2SrL0B<og.0b200a.bBg30iOl=i.BgtS3LSy00OOA=E,Maximum .NSCL3p00OCSIMSMaximum0.,IN ST4AT0SRAX.,AL 4→gITC0MP<.U4→g00M00<.50>00.g50.2468215.....00001BW/[kTln(2)]Work output,
qlouwanestut-mlyilnegvelle.veAlfitserinintsheertliaorng0eor0frork tehgei0ow.n2.a0lFl.,1otrhWor(cid:96)einesn0(cid:54)=e.r2gLegt/i>2caw0ll0ye.3 0.4 05..50 4P.0roPbroabba3ilib.t0yil,i typ,0 p20.0 1.0 00
knowbeforehandthelocationofthWepar0ticlges>an0Pdr0omb0eaasbuilriitnyg,0 p.10 0.2 0.3 0.4 0p ,0y.t5ilibaborP
the number of particles does not provide0any new0.1informa0-.2 FIGP.0r2o..3bWaobrkiliot0yu.,t4 pputper0cy.c5leforthetwo-particleSzilardengine.
tion, i.e., I = 0. Consequently, no work can be extractedPirnobabFiloirtya, spymmetric ins0ertion of the barrier the work output depends
0
thecycle. Attractiveinteractionsobviouslyenhancethisfea- solelyontheprobabilityp0tofindallparticlesontherightside.For
ture. However, this does not hold for repulsive interactions, classicalparticles,p0 =1/4holds,andthisresultisalsoreachedin
allothercasesinthelimitofhightemperatures.However,inthelimit
g > 0, as shown in the lowest panel of Fig. 1 (a). In this
oflowtemperatures,p differsforbosonswithdifferentinteractions.
case,theparticlesspreadoutondifferentsidesofthewallin 0
Theinsetsshowthetwo-particleconfigurationsintherepulsivecase
thegroundstate. Here,degeneraciesbetweendifferentmany-
(left)andtheattractivecase(right),aswellastheleveldegeneracy
particle states occur at particular values of (cid:96)ins, which allow
fornon-interactingbosons.
aninformationgaininthemeasurement. ThisexplainstheN
distinctpeaksasafunctionof(cid:96)ins forlowtemperatureinthe
lowestpanelofFig.1(a).
case. Onemightwonder,whethertheincreasedparticlenum-
The maximum of W/W for attractive bosons increases
1 bershouldnotimplyahigherpressureonthepistonandthus,
with particle number, as shown in Fig. 1 (b). The optimal
morework.This,however,isnotthecase,astheattractionbe-
relativeworkoutputishigherforattractivebosons(solidred
tweentheparticlesreducesthepressure. Also,wheninserting
line) than for non-interacting bosons (red dashed line) and
thebarrier, thedifferenceinworkduetotheinteractionshas
clearly beats the corresponding result for classical particles to be taken into account. With increasing temperature (i.e.,
(blue dashed line). Here, the data for N ≤ 5 were obtained k T ∼−3g(N−1)/L≈−0.6(N−1)E g/g ,forweakin-
B 1 0
by exact diagonalisation while a perturbative approach (see
teractionsasshowninthesupplementarymaterial)othermea-
supplementary material) was applied for N > 5. The peak surementoutcomesthann = 0orn = N becomeprobable.
workoutputforbosonswithattractiveinteractionsatafinite Sincep andp nowdecreasewithtemperatureweseeade-
0 N
temperatureisageneralfeature,whichholdsforawiderange
viationfromtheperformanceofthesingle-particleengine.
ofinteractionstrengths, seeFig.1(c)forthecaseofN = 3
bosons. Indeed,thetemperatureatwhichthepeakoccursin-
creaseswithlargerinteractionstrengths.
IV. THETWO-PARTICLEINTERACTINGENGINE.
Togetabetterunderstandingofthephysicsbehindtheen-
hancement of work output for bosons with attractive inter-
III. ONSETOFTHEPEAKATANINTERMEDIATE actions at finite temperatures, let us look at the two-particle
TEMPERATURE.
case in some more detail. For a central insertion of the bar-
rier, we find p (L/2) = p (L/2). For the same symmetry
0 2
For systems with attractive interactions, g < 0, the work reasons, p ((cid:96)) has a maximum at this barrier position. No
1
output equals kBT ln2 at low temperatures, independent of work can thus be extracted in cycles where the two particles
N. Due to the dominance of the attractive interaction, all are measured on different sides of the central barrier, since
N particles will be found on one side of the barrier. When p ((cid:96)ins)/p ((cid:96)rem) ≥ 1 in Eq. (1). Thus, the only contribu-
1 1 1
the barrier is inserted symmetrically, we have p0(L/2) = tionstotheworkoutputresultfromp0 andp2. Togetherwith
pN(L/2) = 1/2, whileallotherpn(L/2) = 0. Atthesame p0((cid:96)r0em =0)=p2((cid:96)r2em =L)=1weobtain
time, the removal position (cid:96)rem = 0 and (cid:96)rem = L provide
0 N
p ((cid:96)rem) = 1andp ((cid:96)rem) = 1, sothatEq.(1)providesthe W =−2k Tp (L/2)lnp (L/2) (2)
0 0 N N B 0 0
workoutputW =k T ln2fortheentirecycleasobservedin
B
Fig.1(c).Thiscase,withtwopossiblemeasurementoutcomes This function has its peak at p = 1/e with the peak value
0
and a full sweep of the piston, resembles the single-particle W ≈ 1.061k T ln2, see Fig. 2. This implies a finite value
B
4
p1 = 1−2/e. Even if no work can be extracted with one VI. REPULSIVEBOSONS
particleoneithersideofthebarrier,anon-zeroprobabilityp
1
ofsuchameasurementoutcomecanbepreferable. Finally, we consider the repulsive interactions between
bosons, see Figure 1(c). In the low-temperature limit, the
Two attractive bosons, initially at T → 0 and with p0 = relativeworkoutputisverysimilartothatofnon-interacting
1/2,willforincreasingT continuouslyapproachtheclassical
11
limitofp =1/4. Hence,atacertaintemperature,depending
0
ontheinteractionstrength, p passesthroughp = 1/epro-
0 0 0.08.8
dstscutiluadercneeiisdsno,gfttthhhateiehspnetepwobaropaokerpairretniaterrtttraihy,ncebgtoiorfvseueitlnmhabdteiiolvtesaoeonrgwngteosiotnhwrtehekirea.llsbPsayifhlnowytglhslalioeeycwisarplseaal:ynrttt,Adriocautlnceplteoicoowmannsa.tteyehT,meuhwnspehadecmirecyarhe--- pProbability, npProbability, n00..0046..46 BNgoBNg=so==os=−no4s−n04s.01.g10g0 nnnnn=====nnnnn43210=====43210
explains that W = kBT ln2 when p0 = 1/2. A less cor- 0.02.2
relatedsystem(obtainedwithincreasingT)providesalarger
expansionworkforcyclesinwhichbothparticlesareonone 00
side of the barrier. On the other hand, cycles with one par- 0.04.848 InI0sne.0s4re.9t4iro9tino np opso0its.i0o5it.ino5,n ℓ, inℓsin/0sL./05L.151 0.05.252
ticle on each side of the barrier, from which no work can be
extracted, become more frequent. For 1/e < p < 1/2, the
0 FIG. 3. Probability distributions. Work can be extracted in all
enhanced pressure is more important and the average work
cycleswithinsertionofthewallatthemidpoint,exceptforthecase
output increases with decreasing p . For lower values of p ,
0 0 withequallymanyparticlesoneitherside.Shownaretheprobability
i.e. p < 1/e, too few cycles contribute on average to the
0 distributionsassignedtothedifferentmeasurementoutcomesatthe
work production. The average work output decreases with temperatureofthemaximalrelativeworkoutput(k T/E ≈0.243)
B 1
decreasingp0 despitethecorrespondingincreaseinpressure. forN =4bosonswithweakattraction,g=−0.1g0.
Importantly, we note the absence of a similar maximum in
the non-interacting case, where W/W is found to decrease
1
spin-less fermions discussed in Refs. [13, 14]. This resem-
steadilytowardstheclassicallimitwithincreasingT.
blance becomes even more pronounced with increasing in-
teraction strength. This in fact is no coincidence, but rather
a property of one-dimensional bosons with strong, repulsive
interactions that have an impenetrable core: Indeed, in the
limitofinfiniterepulsion,bosonsactlikespin-polarisednon-
interactingfermions. Thisisthewell-knownTonks-Giradeau
regime [34]. Both for non-interacting fermions and strongly
repulsive bosons, the region where the quantum Szilard en-
V. SZILARDENGINESWITHN >2ATTRACTIVE
gine exceeds the classical single-particle maximum of work
BOSONS.
output,hasdisappeared.
The maximum of W/W1 tends to increase with the par- VII. CONCLUSIONS
ticle number N (as previously discussed in connection with
Fig. 1 b). The reason lies in the fact that work can be ex-
Wehavedemonstratedthattheworkoutputofthequantum
tractedfromalargernumberofmeasurementoutcomes. Sim-
Szilardenginecanbesignificantlyboostedbyshort-rangedat-
ilar to the two-particle engine, the combined contribution to
tractiveinteractionsforabosonicworkingmedium.Webased
theaverageworkoutputfromcyclesinwhichallparticlesare
ourclaimonthe(numerically)exactsolutionofthefullmany-
onthesamesideofabarrierinsertedat(cid:96)ins =L/2isgivenby
body Schro¨dinger equation for up to five bosons. It is likely
Eq.(2). However,alsocycleswithn = 1,2,...,N −1(ex-
thattheeffectisevenfurtherenhancedforlargerparticlenum-
ceptifn=N/2)ontheleftsideofthebarrierdocontributeto
bers; however, despitethesimpleone-dimensionalsetup, the
theaverageworkoutput,andworkoutputevenhigherthanin
numericaleffortgrowsverysignificantly(andbeyondourfea-
thetwo-particlecaseispossible. Themaximumofp1((cid:96))and sibility)forlargerN. Byincreasingthestrengthoftheinter-
that of p ((cid:96)) occurs for (cid:96) (cid:54)= L/2, as clearly indicated by
N−1 particleattraction,theengine’sworkoutputcanbeincreased
the probabilities for different measurement outcomes shown
significantlyalsoathighertemperatures,wheretheworkthat
forN = 4inFig.3. Thismeansthatp ((cid:96)ins)/p ((cid:96)rem) ≤ 1
n n n can be extracted generally is of larger magnitude. While we
ispossiblefor(cid:96)ins = L/2andthatworkmaybeextractedin
here restrict our analysis to idealised quasi-static processes,
agreementwithEq.(1). Forallsystemsconsideredhere,with
it would be of much interest to consider a finite speed in the
insertionofthebarrieratthemidpointtheoptimumisreached
ramping of the barrier, enabling transitions to excited states
for p = p ≈ 0.3 (see the example for N = 4 in Fig. 3),
0 N which by coupling to baths will lead to dissipation. Extend-
whichisclosetotheoptimalvalueof1/eforthecorrespond-
ing our approach to quantify irreversibility in real processes
ingtwo-particleengine. on the basis of a fully ab initio quantum description may in
thefutureallowtostudydissipativeaspectsinthekineticsof
theconversionbetweeninformationandwork.
5
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6
SupplementaryMaterial N − n in the right one. With this notation, Z (L) is thus
N
equivalent to the partition function of the initial system, be-
1. WorkoutputofthequantumSzilardengine foretheinsertionofthebarrier. Also,notethatpriortomea-
surement,thenumberofparticlesoneithersideofthebarrier
In contrast to other conventional heat engines that operate
isnotyetacharacteristicpropertyofthenewsystem.Weneed
by exploiting a temperature gradient, as discussed in many
thustosumoverallpossibleparticlenumbersinthenumera-
textbooks on thermodynamics, the Szilard engine [1] allows
tor ofEq. (5). Finally, wewant to stressthe fact that, unlike
forworktobeextractedalsowhenconnectedtoasingleheat
foraclassicaldescriptionoftheengine,theinsertionofabar-
bath at constant temperature. It is propelled by the informa-
rier costs energy in the form of work, due to the associated
tionobtainedabouttheworkingmediumanditsmicroscopical
changeinthepotentiallandscape.
properties. In the supplementary material, we briefly outline
(ii) Measurement. The number of particles located on the
the theoretical description of the quantum Szilard engine, in
differentsidesofthebarrierisnowmeasured.Here,following
close analogy to that of Refs. [12, 15]. An idealised version
[13],weassumethatthemeasurementprocessitselfcostsno
oftheSzilardenginecycleconsistsoffourwell-definedsteps:
work, i.e., we assume that W = 0 (see main text). The
(i)insertion,(ii)measurement,(iii)expansionandfinally(iv) (ii)
probabilitythatnparticlesaremeasuredtobeontheleftside
removal. First, an impenetrable barrier is introduced (i) that
ofthebarrier(andN −nontherightside)isgivenby
effectivelysplitstheworkingmediumintotwohalves. Then,
thenumberofparticlesoneachsideofthebarrierismeasured
Z ((cid:96)ins)
(ii). Dependingontheoutcomeofthismeasurement,thebar- p ((cid:96)ins)= n . (6)
rier moves (iii) to a new position and contraction-expansion n (cid:80)Nn(cid:48)=0Zn(cid:48)((cid:96)ins)
work can be extracted in the process. Finally, the barrier is
(iii) Expansion. The barrier introduced in (i) is assumed
removed(iv)whichcompletesasinglecycleoftheengine.
to move without friction. During this expansion/contraction
Allfoursteps,(i)-(iv),oftheSzilardengineareassumedto
process, the number of particles on either side of the barrier
becarriedoutquasi-staticallyandinthermodynamicequilib-
remains fixed. In other words, the barrier is assumed high
riumwithasurroundingheatbathattemperatureT. Now,the
enough such that tunnelling may be neglected. If the barrier
work associated with an isothermal process can be obtained
movesfrom(cid:96)ins to(cid:96)rem whennparticlesaremeasuredinthe
from n
leftsub-system, theaverageworkextractedfromthisstepof
W ≤−∆F =k T∆(lnZ), (3) thecyclereads
B
wHhelemrehokltBz,freFeenanerdgyZandartehetphaertiBtioolntzfmunacntnionconstant, the W(iii) =kBT (cid:88)N pn((cid:96)ins)ln(cid:20)ZZn(((cid:96)(cid:96)rnienms))(cid:21), (7)
n
(cid:88) n=0
Z = e−Ej/(kBT), (4)
wherep aretheprobabilitiesgivenbyEq.(6).
j n
(iv) Removal. The barrier at (cid:96)rem, that separates the left
n
where the sum runs over the energies Ej of, in principle, all sub-system with n particles from the right one with N −n
micro(orquantum)statesoftheconsideredsystem. Inprac- particles,isnowslowlyremoved. Astheheightofthebarrier
tice,however,weconstructanapproximatepartitionfunction shrinks, particles will eventually start to tunnel between the
from a finite number of energy states. Note that the work in twosub-systems. Thistransferofparticlesmakestheremoval
Eq.(3)ischosentobepositiveifdonebythesystem.Also,the of the barrier an irreversible process. Clearly, if we instead
equivalence between W and −∆F is reserved for reversible weretostartwithoutabarrierandintroduceoneat(cid:96)rem,then
n
processesalone. we can generally not be certain to end up with n particles
We now turn to the work associated with the individual to the left of the partitioner. Assuming that the particles are
stepsofthequantumSzilardengine. Forsimplicity, wecon- fullydelocalisedbetweenthetwosub-systemsalreadyinthe
sider an engine with N particles initially confined in a one- infiniteheightbarrierlimit,thentheaverageworkassociated
dimensional box of size L. All steps of the engine are, as withtheremovalprocessisgivenby
previouslymentioned,carriedoutquasi-staticallyandinther-
mT.alTeoqumilaibxrimiuimsewthitehwthoerskuorruotupnudt,inwgehfeuartthbearthaastsutemmeptehraattuarlel W(iv) =kBT (cid:88)N pn((cid:96)ins)ln(cid:34)(cid:80)N ZNZ(L()(cid:96)rem)(cid:35). (8)
involvedprocessesarereversible,unlessspecifiedotherwise. n=0 n(cid:48)=0 n(cid:48) n
(i) Insertion. A wall is slowly introduced at (cid:96)ins, where
Finally,theaveragedcombinedworkoutputofasingleSzi-
0≤(cid:96)ins ≤L. Intheend,theinitialsystemisdividedintoleft
lardcycle, W, isgivenbythesumofthepartialworksasso-
andrightsub-systemsofsizes(cid:96)ins andL−(cid:96)ins respectively.
ciated with the four steps (i)-(iv), i.e. W = W +W +
(i) (ii)
BasedonEq.(3),theworkofthisprocessisgivenby
W +W ,andsimplifiesinto
(iii) (iv)
(cid:20)(cid:80)N Z ((cid:96)ins)(cid:21)
W(i) =kBT ln n=Z0N(nL) , (5) W =−kBT (cid:88)N pn((cid:96)ins)ln(cid:20)ppn(((cid:96)(cid:96)rienms))(cid:21). (9)
where Z ((cid:96)ins) is the short-hand notation for the partition n=0 n n
n
function obtained with n particles in the left sub-system and whichisthecentralequation(1)inthemainarticle.
7
2. The interacting many-body Hamiltonian and exact di- numerical diagonalisation. Using (cid:96)ins = L/2 and determin-
agonalisation ing the optimal removal positions (cid:96)rem numerically, we get
n
the work output by Eq. (1). Again the optimal temperature
Tokeeptheschematicsetupofthemany-bodySzilardcycle
needstobechosentoobtaintheresultsplottedinFig.1(b).
assimpleaspossible,weconsideraquantumsystemofN in-
teractingparticles,initiallyconfinedinaone-dimensionalbox
ofsizeLthatisseparatedbyabarrierinsertedatacertainpo-
sition(cid:96).Wenotethatforcontactinteractionsbetweenthepar-
4. Estimateofthepeaktemperature
ticles,asdefinedinthemaintext,theexactenergiesE tothe
j
fullyinteractingmany-bodyHamiltonianHˆ arethosegivenin
termsoftwoindependentsystemswithnandN−nparticles. For the symmetric wall position, the ground state of the
In order to construct the partition functions and compute the system with attractive bosons has all particles on one side,
probabilitiesp ,theentireexactmany-bodyenergyspectrum say the left one. Using the perturbative approach discussed
n
isneeded. Forthesimplecaseofnon-interactingparticles(or above, the interaction energy is E(1)(L/2). If one boson
N
single-particlesystems)theseenergiesareknownanalytically. is transferred from the left side to the right side, the inter-
Forinteractingparticles,however,theymustbedeterminedby action energy changes to E(1) (L/2), while the level en-
solvingthefullmany-bodyproblem. Wehereapplythecon- N−1
ergies E(0)(L/2) are independent on n for the symmetric
figurationinteractionmethodwhereweuseabasisofthe5th n
wall position. Thus thermal excitations become likely for
orderB-splines[35],withalineardistributionofknot-points
withineachleft/rightsub-system,todeterminetheenergiesof kBT ∼ EN(1−)1(L/2)−EN(1)(L/2) = −3(N −1)g/L. For
each sub-system and parity at each stage. For N = 3, we these temperatures the particles do not cluster on the same
used62B-splines(orone-bodystates)toconstructthemany- sideofthewallanylongerandwehavep0 <1/2.
body basis for each sub-system. Since the dimension of the
many-body problemgrows drasticallywith N, weneeded to
5. Temperaturedependenceoftheworkoutputfordiffer-
decrease the number of B-splines to 32 for N = 4. Conse-
entinteractionstrengths
quently,inthiscasewecouldnotgotoequallyhightempera-
turesandinteractionstrengths.
AsacomplementtoFig.1(c)ofthemainarticle,weshow
3. Perturbative approach for weakly attractive bosons at
the case for N = 4 here in Fig. 4. For small to medium
lowtemperatures
couplings −g (cid:46) g < 0, the peaks have approximately the
0
Here we consider the case kBT (cid:28) E1, so that only the sameheightandtheyareshiftedproportionallytog.Thisshift
lowestquantumlevelsineachpartarethermallyoccupied. In follows the deviations from the low temperature limit W =
thecaseofvanishinginteraction,thestatewithnparticlesin W ,whichsetinatk T ≈−0.6(N−1)E g/g ,asshownby
1 B 1 0
thelowestleveloftheleftpartofthewallpositionedat(cid:96)(and theapproximativeapproachinthemainarticle. Asdiscussed
N −n particles in the lowest level of the right part) has the in the method section, for g ≈ −π2g /(N −1), correlation
0
energy effectsbecomeimportantandwefindareducedpeakatg =
−10g ,similartothecaseforN = 3inFig.1(c)ofthemain
E(0)((cid:96))=nE (cid:18)L(cid:19)2+(N −n)E (cid:18) L (cid:19)2 article0. Due to the high numerical demand on the numerical
n 1 (cid:96) 1 L−(cid:96) diagonalization,wedidnotobtainresultsforlarger|g|inthe
caseN = 4,whileforN = 3anincreaseofthepeakheight
(cid:112)
ApplyingthewavefunctionΨ0(x)=sin(πx/(cid:96)) 2/(cid:96)forthe forevenlarger|g|isobserved.
leftside,themutualinteractionenergybetweentwoparticles
inthislevelis
For all interaction strengths g < 0, the peak height is ac-
(cid:90) (cid:96) 3g
U((cid:96))=g dx|Ψ (x)|4 = (10) tuallylargerthanthepeakfortheattractivetwo-particlecase
0 0 2(cid:96) W2 ≈ 1.061kBT ln(2) depicted in Fig. 2 of the main arti-
cle. Thisisduetothefact, thatEq.(2)ofthemainarticleis
Nowweassumethatthisinteractionenergy(timesthenumber
a lower bound for the work output and p (L/2) necessarily
of interacting partners) is much smaller than the level spac- 0
ing, i.e. (n − 1)U((cid:96)) (cid:28) 3E (cid:0)L(cid:1)2, which is satisfied for movesfrom1/2atT →0totheclassicalresult1/2N atlarge
1 (cid:96) temperatures. Thus the maximum for p0 = 1/e is taken at
g (cid:28) π2g0. Thenwemaydeterminetheenergyofthemany- someintermediatetemperature.
N−1
particlestatebyfirst-orderperturbationtheory. Thisresultsin
theinteractionenergy
6. OperationoftheQuantumSzilardengineathightem-
E(1)((cid:96))= n(n−1)U((cid:96))+(N −n)(N −n−1)U(L−(cid:96)). peratures
n 2 2
(11)
Setting E ((cid:96)) ≈ E(0)((cid:96))+E(1)((cid:96)), we obtain an analytical For N = 4 particles, Fig. 5 shows that the symmetric in-
n n n
expression for the probabilities p ((cid:96)) without any need for sertionpoint(cid:96)ins =L/2isnotoptimalforhightemperatures.
n
8
(g ≤ 0)studied here, thesymmetricinsertionis favorablein
thelowtemperaturelimitasthoroughlydiscussedinthemain
article. On the other hand, for large temperatures the classi-
cal result needs to be recovered. This occurs via a pitchfork
bifurcation[36] at an intermediate temperature as shown in
Fig.5. FornoninteractingBosonsitoccursatk T ≈ 50E
B c 1
for N = 4, and at slightly larger values if attractive interac-
tionsareincluded.
Bosons, N=4
0.7
L Classical limit
/
insℓ 0.6
FIG.4. TemperaturedependenceoftheworkoutputforN = 4 on, 00..66
bosons. With increasing interaction strength, the peak in the rela- ositi 0.5 00..55
tiveworkoutputoccursatahighertemperature,similartowhatwas n p g=0
showninFig.1(c)forN =3. ertio 0.4 00.4.4 g=−g0
ns 5500 110000
I
Classical limit
0.3
Forclassicalparticles,theoptimalworkoutputisgivenby
)]
(cid:34)(cid:18)(cid:96)ins(cid:19)N (cid:18)(cid:96)ins(cid:19) n(2 1.1
Wtot =−NkBT L ln L TlB1.05
k
(cid:18) (cid:96)ins(cid:19)N (cid:18) (cid:96)ins(cid:19)(cid:35) W/[ 1
+ 1− ln 1− ut, 0.95
L L p
ut
o
−kBT N(cid:88)−1(cid:18)Nm(cid:19)(cid:18)(cid:96)Lins(cid:19)m(cid:18)1− (cid:96)Lins(cid:19)N−m Work 0.91 Classical limit 10 100 1000
m=1 Temperature, kBT/E1
(cid:16) (cid:17)m(cid:16) (cid:17)N−m
(cid:96)ins 1− (cid:96)ins
×ln L L . (12) FIG.5. Pitchforkbifurcationfortheoptimalinsertionposition
 (cid:0)m(cid:1)m(cid:0)1− m(cid:1)N−m  at a critical temperature for N = 4 bosons. For large temper-
N N
atures, theoptimalinsertionpositionbecomesasymmetricinorder
to recover the classical result (blue dashed line). The full red line
A numerical scan of different insertion positions shows that showsthecasewithoutinteractionsg = 0. Theinsetintheupper
an asymmetric insertion point (cid:96)ins (cid:54)= L/2 (as shown by the panelshowsthecorrespondingresultforg=−g (dark-redlines).
0
blue dashed line in Fig. 5) provides the highest work out-
put. Incontrast,thesymmetricpositionisoptimalforN ≤ 3
classicalparticles. Forthenon-repulsivelyinteractingbosons