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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 clewasreportedbyRoldańetal.[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 barrier, 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 interaction 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. 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M. R. & Petrov, D. Universalfeaturesintheenergeticsofsymmetrybreaking. Nat Phys10,457–461(2014). 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 interactingparticles,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 article. 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 classical 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 output. Incontrast,thesymmetricpositionisoptimalforN ≤ 3 classicalparticles. Forthenon-repulsivelyinteractingbosons