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1、Articlehttps:/doi.org/10.1038/s41467-022-33667-1Dynamical control of quantum heat enginesusing exceptional pointsJ.-W.Zhang1,2,10,J.-Q.Zhang1,10,G.-Y.Ding1,3,10,J.-C.Li1,3,J.-T.Bu1,3,B.Wang1,3,L.-L.Yan4,S.-L.Su4,L.Chen1,2,F.Nori5,6,.K.zdemir7,F.Zhou1,2,H.Jing8,9&M.Feng1,2,4A quantum thermal machine
2、is an open quantum system coupled to hot andcold thermal baths.Thus,its dynamics can be well understood using theconcepts and tools from non-Hermitian quantum systems.A hallmark of non-Hermiticity is the existence of exceptional points where the eigenvalues of anon-HermitianHamiltonianoraLiouvillian
3、superoperatorandtheirassociatedeigenvectorscoalesce.Here,wereporttheexperimentalrealizationofasingle-ionheatengineanddemonstratetheeffectofLiouvillianexceptionalpointsonthe dynamics and the performance of a quantum heat engine.Our experi-ments have revealed that operating the engine in the exact-and
4、 broken-pha-ses,separated by a Liouvillian exceptional point,respectively during theisochoric heating and cooling strokes of an Otto cycle produces more workandoutputpowerandachieveshigherefficiencythanexecutingtheOttocyclecompletely in the exact phase where the system has an oscillatory dynamicsand
5、 higher coherence.This result opens interesting possibilities for the con-trolofquantumheatenginesandwillbeofinteresttootherresearchareasthatare concerned with the role of coherence and exceptional points in quantumprocesses and in work extraction by thermal machines.Quantum heat engines extract use
6、ful work from thermal reservoirsusing quantum matter as their working substance.Contrary to theirclassical counterparts,which do not include coherence in its micro-scopic degrees of freedom and suffer from irreversible loss during aclassical thermodynamic cycle,quantum heat engines are expected tobe
7、nefit from quantum features to surpass the output power and effi-ciency that can be attained by an equivalent classical heat engine14.The growing interest in quantum heat engines is also fueled by theinterest in understanding the quantum-classical transition in energy-informationandwork-heatconversi
8、on.Additionalmotivationsincludethe need to maximize the efficiency(the ratio of useful work to theinput heat)and the output power while keeping power fluctuationsminimal in micro-and nano-scale heat engines,in which quantumfluctuations and non-equilibrium dynamics play a crucial role59.Microscopic a
9、nd nanoscopic heat engines with and without theinvolvement of quantum coherences have been implemented withReceived:8 May 2022Accepted:27 September 2022Check for updates1State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,Wuhan Institute of Physics and Mathematics,Innovation
10、Academy ofPrecision Measurement Science and Technology,Chinese Academy of Sciences,Wuhan,China.2Research Center for Quantum Precision Measurement,Guangzhou Institute of Industry Technology,511458 Guangzhou,China.3School of Physics,University of the Chinese Academy of Sciences,100049Beijing,China.4Sc
11、hoolofPhysics,ZhengzhouUniversity,450001Zhengzhou,China.5TheoreticalQuantumPhysicsLaboratory,RIKEN,ClusterforPioneeringResearch,Wako-shi,Saitama351-0198,Japan.6PhysicsDepartment,TheUniversityofMichigan,AnnArbor,MI48109-1040,USA.7DepartmentofEngineeringScience and Mechanics,and Materials Research Ins
12、titute,Pennsylvania State University,State College,University Park,PA 16802,USA.8Key Laboratory ofLow-Dimensional Quantum Structures and Quantum Control of Ministry of Education,Department of Physics andSynergetic Innovation Center forQuantumEffects and Applications,Hunan Normal University,410081 Ch
13、angsha,China.9Synergetic Innovation Academy for Quantum Science and Technology,Zhengzhou University of Light Industry,450002 Zhengzhou,China.10These authors contributed equally:J.-W.Zhang,J.-Q.Zhang,G.-Y.Ding.e-mail:;Nature Communications|(2022)13:6225 11234567890():,;1234567890():,;single trapped i
14、ons810,ensembles of nitrogen-vacancy centres indiamond4,magneticresonance11,12,asingleelectronbox13,andimpurityelectron spins in a silicon tunnel field-effect transistor14.Many inter-esting proposals have also been put forward for their realization insuperconducting circuits15,16and optomechanics17,
15、18.Another field that has been attracting increasing interest is non-Hermiticity,including parity-time(PT)symmetry19in physical sys-tems.In particular,non-Hermitian spectral degeneracies known asexceptional points(EPs)have been shown to have tremendous effectson the dynamicsof physicalsystems,leadin
16、g to many counterintuitivefeatures which have led to the development of novel functionalitiesand classical devices2027.Effects of non-Hermiticity have generallybeen studied using an effective non-Hermitian Hamiltonian and itsspectral degeneracies.Recently,there is a growing interest to harnessnon-He
17、rmiticity and EPs for quantum applications2833.In quantumsystems,however,Hamiltonian EPs(known as HEPs)cannot capturethe whole dynamics because these exclude quantum jumps and theassociated noises.Instead,one should resort to the Liouvillian form-alism which takes into account both coherent non-unit
18、ary evolutionand quantum jumps3438.In this formalism,EPs are defined as thedegenerate eigenvalues of Liouvillian superoperators.Thus they arereferred as Liouvillian EPs(LEPs),whose properties and effects onquantum systems have remained largely unexplored except in somerecent experiments in supercond
19、ucting qubit systems38,39.As open quantum systems which exchange energy with thermalreservoirs,quantum heat engines naturally exhibit non-Hermitiandynamics,whichcanbecontrolledbyjudiciouslytuningparametersofthe heat engine to operate it in the exact-or broken-phases separatedby LEPs(i.e.,LEPs corres
20、pond to the transition points between theexact-and the broken-phases).Here we report the experimentalimplementation of a quantum Otto engine using a single40Ca+ionconfined in a linear Paul trap4042,and demonstrate the control of theengine efficiency and output power by harnessing LEPs and theirassoc
21、iateddynamics.Thisconstitutesaninterestingobservationofthesignatures of LEPs in a quantum heat engine.We note that previousexperiments studied non-Hermiticity in single-spin systems by con-sidering only HEPs28,31,32.In contrast,here we use LEPs and their rami-fications to control the performance of
22、a quantum heat engine.Thus,our study takes into account quantum jumps and the associateddynamics.As it will become clear below and discussed previously37,38,theLEP inthis systemcorresponds tothe critical damping pointwhichemerges in the parameter space as the system transits between theoscillatory a
23、nd non-oscillatory dynamics,in analogy with a dampedharmonic oscillator.ResultsSingle-ion quantum heat machineThe working substance of the quantum Otto engine we implementhere is a pseudo-spin 1/2 represented by the internal states of atrapped single40Ca+ion,i.e.,the ground state 42S1=2,mJ=+1=2ilabe
24、led as gi,and the metastable state 32D5=2,mJ=+5=2i labeled asei,with the magnetic quantum number mJ(see Fig.1A).In ourexperiment,we confine a single40Ca+ion in a linear Paul trap,whoseaxial and radial frequencies are z/2=1.1MHz and r/2=1.6MHz,respectively.We define a quantization axis along the axia
25、l directionby a magnetic field of 3.4 Gauss at the center of the trap.We thenperform Doppler and sideband cooling of the ion until an averagephonon number of?n4(weak coupling),both 3and 4are real with asplitting amount,corresponding to the broken phase characterizedby a non-oscillatory dynamics with
26、 purely exponential decay43,44.Foreff4(strong coupling),on the other hand,3and 4form acomplex conjugate pair which splits in their imaginary parts by,corresponding to the exact phase characterized by an oscillatorydynamics.Thus,the LEP divides the parameter space into a region ofoscillatory dynamics
27、(exact phase,eff4).As such,the LEP here issimilar to the critical damping point of a damped harmonic oscillatorandemergesinthe transitionbetweentheregionsofoscillatory(exactphase)and non-oscillatory(broken phase)dynamics.Quantum Otto cycles of a single ionThe question we address in this study is:How
28、 do the presence of LEPsand the associated transitions between the oscillatory and non-oscillatory dynamics affect the performance of an Otto engine?Atypical Otto cycle has four strokes:two adiabatic strokes,which resultin compression and expansion,and two isochoric strokes which con-nect the workin
29、g substance to cold and hot baths.Quantum Ottocycles differ from their classical counterparts in the varying andthe invariant thermodynamic quantities,and how these quantities aredefined,see Supplementary Note 3 and Supplementary Fig.2.Forexample,inaquantumisochoricstroke,thepopulationPnofeachleveln
30、ofthequbit,andhencetheentropySofthesystem,changesuntiltheworking substance reaches thermal equilibrium with the heat bath,while there is no change in the eigenenergies En14,45,46.In a classical isochoric process,the pressure P and the tempera-ture T change but the volume V remains unchanged,and the
31、workingsubstance reaches thermal equilibrium with the heat bath only at theend of this process.In a classical adiabatic stroke,all thermodynamicquantities P,T,and the volume V vary(i.e.,no invariant thermo-dynamic quantity)and there is no requirement that occupationprobabilities remain unchanged.The
32、refore,work is done only duringArticlehttps:/doi.org/10.1038/s41467-022-33667-1Nature Communications|(2022)13:6225 2the classical adiabatic strokes(no work is done during classical iso-choric strokes).Similarly,a quantum heat engine does work onlyduring the quantum adiabatic strokes(i.e.,no work is
33、done during thequantum isochoric strokes)but with a different underlying mechan-icsmthantheclassicaladiabaticstrokes:Inaquantumadiabaticstroke,Pnand S should remain unchanged during the process(thus no heatexchange)but Enmay shift.This change in Enleads to non-zero work.As demonstrated later,we find
34、 that the coherence-enabledimprovements in work and power output of a quantum heat enginearise if the coherence during the work strokes(i.e.,quantum adia-batic strokes)induces a hump in the thermal strokes(i.e.,quantumisochoric strokes).Therefore,to answer the question stated aboveandclarifytherelat
35、ionamongLEPs,thesurvivingcoherenceafterthethermal strokes,and the performance of a quantum heat engine,wehave designed experiments implementing Otto cycles with(i)bothisochoric strokes in the exact phase(eff4,non-oscillatory dynamics),and(iii)isochoric heating stroke in the exactbut isochoric coolin
36、g stroke in the broken phases.We note that afourth case would be the opposite of(iii),corresponding to a quan-tum refrigerator,totally reversing the process of the heat engineunder the setting(iii).This case is beyond the scope of the presentstudy and thus we have not performed experiments under thi
37、ssetting.In our system of a single trapped ion,we implement the ingre-dients of the Otto cycle as follows:Hot and cold heat baths are pre-pared by tuning/eff,which is the ratio of the driving laser beamstrengthtotheeffectivedecayeffofthequbit.Thisimpliesthatlaserirradiation together with the real en
38、vironment constitutes the baths,where the hot and cold baths correspond to strong and weak drives,respectively.The qubit absorbs or releases heat by coupling to the hotor the cold baths,respectively.Here,we adjust effby varying thepower of the laser with wavelength 854 nm,which is tuned to the P3/2-
39、D5/2transition and is adjusted by tuning the power of the 729nmlaser red-tuned to the S1/2-D5/2transition(Fig.1B).We evaluate theperformance of the heat engine,e.g.,the work output to the cold bathand heat absorbed from the hot bath,by monitoring the variation inthe populations of the two-level syst
40、em.Fig.1|Single-spin quantum heat engine in a trapped40Ca+ion exhibiting aLiouvillianexceptionalpoint(LEP).ASchematicoftheexperimentalsetup.AOM:acousto-optic modulator.PMT:photomultiplier tube.AWG:arbitrary waveformgenerator.B Energy levels of the40Ca+ion,where the straight red arrows representtrans
41、itions by laser irradiation with wavelengths labeled and the blue wavy arrowdenotesspontaneousemission.Suchathree-levelconfigurationequalsaneffectivetwo-levelsystemwithcontrollabledrivinganddecay,asplottedin(C).DSchematicdiagram forworkdoneintheexact-andbroken-phases separatedin the parameterspace b
42、y an LEP at/eff=1/4,where a bifurcation occurs due to coherence-inducedoscillationsintheexactphase.EFourstrokesofourquantumOttoengine,where strokes from Step 1 to Step 2 and from Step 3 to Step 4 are adiabaticprocesses;while strokes from Step 2 to Step 3 and from Step 4 to Step 1 representisochoricp
43、rocesses.ThegreendashedlinerepresentsanidealquantumOttocycle,andthesolidlinecorrespondstothecycleobtainedbysolvingthemasterequationusing experimentally available parameter values.F Experimental operationsequences for an Otto cycle,where the duration t2of the second stroke(isochoricheating)is varied
44、to quantify the quantumness involved in the cycle.Articlehttps:/doi.org/10.1038/s41467-022-33667-1Nature Communications|(2022)13:6225 3In our treatment under the rotating frame with respect to thedriving laser frequency,quantum adiabatic strokes are executed bytuningthefrequencyofthedrivinglaserwhic
45、hhelpsvarytheinternalenergy gap (but without population change)and the temperatureof the working substance.Similarly,quantum isochoric strokes areperformed by tuning/effwhich controls the heat exchangebetween the working substance and the thermal baths.Thus,whilethe power of the 854nm laser helps tu
46、ne the qubit decay,the powerand the frequency of the 729nm laser helps implement the fourstrokes of the Otto cycle as follows(Fig.1E,F and SupplementaryFig.2):Starting with the qubit at steady state with a small populationin the excited state,we first carry out an adiabatic compression byincreasing
47、the detuning linearly from minto max,where Peremains in a small and constant value(Step 1Step 2).Next,weperform isochoric heating by rapidly increasing/effto a largevalue,during which the detuning remains equal to max(Step2Step 3).Then,we carry out an adiabatic expansion by linearlydecreasing from m
48、axto min,with Pestaying unchanged(Step3Step 4).Finally,we perform isochoric cooling by rapidlydecreasing/effto a small value with the remaining unchanged asmin(Step 4Step 1).To accomplish a closed Otto cycle,we wait,afterfinishingthelaststroke,forthesystemreachingthesteadystateand returning to the i
49、nitial state.Performances of the single-ion quantum heat engineWe evaluate the role of coherence in the performance of the quantumheat engine by monitoring the oscillations in the populations of thequbit states in the isochoric stroke and then assess the net work,theoutputpower,andtheefficiencyoftheheatengineasafunctionoftheexecution time t2of the second stroke(isochoric heating)while theexecution times of other strokes are kept fixed(SupplementaryNotes 4 and 5).The strong coupling and the exact-phase regimes overlap for/eff1/4 and similarly the weak coupling and the broken-phaseregimes overl