Phase diagram for interacting Bose gases

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Bose-Einstein condensation in a one-dimensional interacting system due to power-law trappin

a r X i v :c o n d -m a t /9810197v 1 [c o n d -m a t .s t a t -m e c h ] 16 O c t 1998Accepted to PHYSICAL REVIEW A for publicationBose-Einstein condensation in a one-dimensional interacting system due to power-lawtrapping potentialsM.Bayindir,B.Tanatar,and Z.GedikDepartment of Physics,Bilkent University,Bilkent,06533Ankara,TurkeyWe examine the possibility of Bose-Einstein condensation in one-dimensional interacting Bose gas subjected to conﬁning potentials of the form V ext (x )=V 0(|x |/a )γ,in which γ<2,by solving the Gross-Pitaevskii equation within the semi-classical two-ﬂuid model.The condensate fraction,chemical potential,ground state energy,and speciﬁc heat of the system are calculated for various values of interaction strengths.Our results show that a signiﬁcant fraction of the particles is in the lowest energy state for ﬁnite number of particles at low temperature indicating a phase transition for weakly interacting systems.PACS numbers:03.75.Fi,05.30.Jp,67.40.Kh,64.60.-i,32.80.PjI.INTRODUCTIONThe recent observations of Bose-Einstein condensation (BEC)in trapped atomic gases [1–5]have renewed inter-est in bosonic systems [6,7].BEC is characterized by a macroscopic occupation of the ground state for T <T 0,where T 0depends on the system parameters.The success of experimental manipulation of externally applied trap potentials bring about the possibility of examining two or even one-dimensional Bose-Einstein condensates.Since the transition temperature T 0increases with decreasing system dimension,it was suggested that BEC may be achieved more favorably in low-dimensional systems [8].The possibility of BEC in one -(1D)and two-dimensional (2D)homogeneous Bose gases is ruled out by the Hohen-berg theorem [9].However,due to spatially varying po-tentials which break the translational invariance,BEC can occur in low-dimensional inhomogeneous systems.The existence of BEC is shown in a 1D noninteracting Bose gas in the presence of a gravitational ﬁeld [10],an attractive-δimpurity [11],and power-law trapping po-tentials [12].Recently,many authors have discussed the possibility of BEC in 1D trapped Bose gases relevant to the magnetically trapped ultracold alkali-metal atoms [13–18].Pearson and his co-workers [19]studied the in-teracting Bose gas in 1D power-law potentials employing the path-integral Monte Carlo (PIMC)method.They have found that a macroscopically large number of atoms occupy the lowest single-particle state in a ﬁnite system of hard-core bosons at some critical temperature.It is important to note that the recent BEC experiments are carried out with ﬁnite number of atoms (ranging from several thousands to several millions),therefore the ther-modynamic limit argument in some theoretical studies [15]does not apply here [8].The aim of this paper is to study the two-body interac-tion eﬀects on the BEC in 1D systems under power-law trap potentials.For ideal bosons in harmonic oscillator traps transition to a condensed state is prohibited.It is anticipated that the external potentials more conﬁn-ing than the harmonic oscillator type would be possible experimentally.It was also argued [15]that in the ther-modynamic limit there can be no BEC phase transition for nonideal bosons in 1D.Since the realistic systems are weakly interacting and contain ﬁnite number of particles,we employ the mean-ﬁeld theory [20,21]as applied to a two-ﬂuid model.Such an approach has been shown to capture the essential physics in 3D systems [21].The 2D version [22]is also in qualitative agreement with the results of PIMC simulations on hard-core bosons [23].In the remaining sections we outline the two-ﬂuid model and present our results for an interacting 1D Bose gas in power-law potentials.II.THEORYIn this paper we shall investigate the Bose-Einstein condensation phenomenon for 1D interacting Bose gas conﬁned in a power-law potential:V ext (x )=V 0|x |κF (γ)G (γ)2γ/(2+γ),(2)andN 0/N =1−TF (γ)=1x 1/γ−1dx1−x,(4)and G (γ)=∞x 1/γ−1/2dxNk B T 0=Γ(1/γ+3/2)ζ(1/γ+3/2)T 01/γ+3/2.(6)Figure 1shows the variation of the critical temperature T 0as a function of the exponent γin the trapping po-tential.It should be noted that T 0vanishes for harmonic potential due to the divergence of the function G (γ=2).It appears that the maximum T 0is attained for γ≈0.5,and for a constant trap potential (i.e.V ext (x )=V 0)the BEC disappears consistent with the Hohenberg theorem.0.00.5 1.0 1.5 2.0γ0.00.20.40.6k B T 0 (A r . U n .)FIG.1.The variation of the critical temperature T 0withthe external potential exponent γ.We are interested in how the short-range interactioneﬀects modify the picture presented above.To this end,we employ the mean-ﬁeld formalism and describe the col-lective dynamics of a Bose condensate by its macroscopictime-dependent wave function Υ(x,t )=Ψ(x )exp (−iµt ),where µis the chemical potential.The condensate wavefunction Ψ(x )satisﬁes the Gross-Pitaevskii (GP)equa-tion [24,25]−¯h 2dx 2+V ext (x )+2gn 1(x )+g Ψ2(x )Ψ(x )=µΨ(x ),(7)where g is the repulsive,short-range interaction strength,and n 1(x )is the average noncondensed particle distribu-tion function.We treat the interaction strength g as a phenomenological parameter without going into the de-tails of actually relating it to any microscopic descrip-tion [26].In the semi-classical two-ﬂuid model [27,28]the noncondensed particles can be treated as bosons in an eﬀective potential [21,29]V eﬀ(x )=V ext (x )+2gn 1(x )+2g Ψ2(x ).(8)The density distribution function is given byn 1(x )=dpexp {[p 2/2m +V eﬀ(x )−µ]/k B T }−1,(9)and the total number of particles N ﬁxes the chemical potential through the relationN =N 0+ρ(E )dE2mgθ[µ−V ext (x )−2gn 1(x )],(12)where θ[x ]is the unit step function.More precisely,the Thomas-Fermi approximation [7,20,30]would be valid when the interaction energy ∼gN 0/Λ,far exceeds the kinetic energy ¯h 2/2m Λ2,where Λis the spatial extent of the condensate cloud.For a linear trap potential (i.e.γ=1),a variational estimate for Λis given by Λ= ¯h 2/2m (π/2)1/22a/V 0 1/3.We note that the Thomas-Fermi approximation would breakdown for tem-peratures close to T 0where N 0is expected to become very small.The above set of equations [Eqs.(9)-(12)]need to be solved self-consistently to obtain the various physical quantities such as the chemical potential µ(N,T ),the condensate fraction N 0/N ,and the eﬀective potential V eﬀ.In a 3D system,Minguzzi et al .[21]solved a simi-lar system of equations numerically and also introduced an approximate semi-analytical solution by treating the interaction eﬀects perturbatively.Motivated by the suc-cess [21,22]of the perturbative approach we consider aweakly interacting system in1D.To zero-order in gn1(r), the eﬀective potential becomesV eﬀ(x)= V ext(x)ifµ<V ext(x)2µ−V ext(x)ifµ>V ext(x).(13) Figure2displays the typical form of the eﬀective po-tential within our semi-analytic approximation scheme. The most noteworthy aspect is that the eﬀective poten-tial as seen by the bosons acquire a double-well shape because of the interactions.We can explain this result by a simple argument.Let the number of particles in the left and right wells be N L and N R,respectively,so that N=N L+N R.The nonlinear or interaction term in the GP equation may be approximately regarded as V=N2L+N2R.Therefore,the problem reduces to the minimization of the interaction potential V,which is achieved for N L=N R.FIG.2.Eﬀective potential V eﬀ(x)in the presence of in-teraction(x0=(µ/V0)1/γa).Thick dotted line represents external potential V ext(x).The number of condensed atoms is calculated to beN0=2γa√ze x−1+ 2µ/k B Tµ/k B TH(γ,µ,xk B T)(2µ/k B T−x)1/γ−1/2dxexp[(E−µ)/k B T]−1=κ(k B T)1/γ+1/2J(γ,µ,T),(18) whereJ(γ,µ,T)= ∞2µ/k B T x1/γ+1/2dxze x−1.and Ecis the energy of the particles in the condensateE c=g(1+γ)(2γ+1)gV1/γ.(19)The kinetic energy of the condensed particles is neglected within our Thomas-Fermi approximation to the GP equa-tion.III.RESULTS AND DISCUSSIONUp to now we have based our formulation for arbitrary γ,but in the rest of this work we shall present our re-sults forγ=1.Our calculations show that the results for other values ofγare qualitatively similar.In Figs. 3and4we calculate the condensate fraction as a func-tion of temperature for various values of the interaction strengthη=g/V0a(at constant N=105)and diﬀerent number of particles(at constantη=0.001),respectively. We observe that as the interaction strengthηis increased, the depletion of the condensate becomes more apprecia-ble(Fig.3).As shown in the correspondingﬁgures,a signiﬁcant fraction of the particles occupies the ground state of the system for T<T0.The temperature depen-dence of the chemical potential is plotted in Figs.5and 6for various interaction strengths(constant N=105) and diﬀerent number of particles(constantη=0.001) respectively.0.00.20.40.60.8 1.0T/T 00.00.20.40.60.81.0N 0/NN 0/N=1−(T/T 0)3/2η=10−5η=10−3η=10−1η=10FIG.3.The condensate fraction N 0/N versus temperature T /T 0for N =105and for various interaction strengths η.Eﬀects of interactions on µ(N,T )are seen as large de-viations from the noninteracting behavior for T <T 0.In Fig.7we show the ground state energy of an interacting 1D system of bosons as a function of temperature for dif-ferent interaction strengths.For small η,and T <T 0, E is similar to that in a noninteracting system.As ηincreases,some diﬀerences start to become noticeable,and for η≈1we observe a small bump developing in E .This may indicate the breakdown of our approxi-mate scheme for large enough interaction strengths,as we can ﬁnd no fundamental reason for such behavior.It is also possible that the Thomas-Fermi approximation em-ployed is violated as the transition to a condensed state is approached.0.00.20.40.60.8 1.0T/T 00.00.20.40.60.81.0N 0/NN 0/N=1−(T/T 0)3/2N=108N=105N=103N=101FIG.4.The condensed fraction N 0/N versus temperature T /T 0for η=0.001and for diﬀerent number of particles N .0.00.20.40.60.8 1.0 1.2T/T 0−100100200300400µ/V 0η=1η=0.1η=0.001η=0.00001FIG.5.The temperature dependence of the chemical potential µ(N,T )for various interaction strength and for N =105particles.Although it is conceivable to imagine the full solution of the mean-ﬁeld equations [Eq.(9)-(12)]may remedy the situation for larger values of η,the PIMC simulations [19]also seem to indicate that the condensation is inhibited for strongly interacting systems.The results for the spe-ciﬁc heat calculated from the total energy curves,i.e.C V =d E /dT ,are depicted in Fig.8.The sharp peak at T =T 0tends to be smoothed out with increasing in-teraction strength.It is known that the eﬀects of ﬁnite number of particles are also responsible for such a be-havior [20].In our treatment these two eﬀects are not disentangled.It was pointed out by Ingold and Lam-brecht [14]that the identiﬁcation of the BEC should also be based on the behavior of C V around T ≈T 0.0.00.20.40.60.8 1.0 1.2T/T 0−5050100µ/V 0N=107N=105N=103N=101FIG.6.The temperature dependence of the chemical po-tential µ(N,T )for diﬀerent number of particles N and for η=0.001.0.00.20.40.60.8 1.0 1.2T/T 00.00.20.40.60.8<E >/N k B T 0η=0η=0.001η=0.1η=1Maxwell−BoltzmannFIG.7.The temperature dependence of the total energy of 1D Bose gas for various interaction strengths ηand N =105particles.Our calculations indicate that the peak structure of C V remains even in the presence of weak interactions,thus we are led to conclude that a true transition to a Bose-Einstein condensed state is predicted within the present approach.0.00.20.40.60.81.01.2T/T 00.00.20.40.60.81.0C V /N k Bη=0η=0.001η=0.1Maxwell−BoltzmannFIG.8.The temperature dependence of the speciﬁc heat C V for various interaction strengths ηand N =105particles.IV.CONCLUDING REMARKSIn this work we have applied the mean-ﬁeld,semi-classical two-ﬂuid model to interacting bosons in 1D power-law trap potentials.We have found that for a range of interaction strengths the behavior of the thermo-dynamic quantities resembles to that of non-interactingbosons.Thus,BEC in the sense of macroscopic occu-pation of the ground state,occurs when the short-range interparticle interactions are not too strong.Our results are in qualitative agreement with the recent PIMC sim-ulations [19]of similar systems.Both 2D and 1D sim-ulation results [19,23]indicate a phase transition for a ﬁnite number system,in contrast to the situation in the thermodynamic limit.Since systems of much larger size can be studied within the present approach,our work complements the PIMC calculations.The possibility of studying the tunneling phenomenon of condensed bosons in spatially diﬀerent regions sepa-rated by a barrier has recently attracted some attention [31–34].In particular,Dalfovo et al .[32]have shown that a Josephson-type tunneling current may exist for bosons under the inﬂuence of a double-well trap potential.Za-pata et al .[34]have estimated the Josephson coupling energy in terms of the condensate density.It is inter-esting to speculate on such a possibility in the present case,since the eﬀective potential in our description is of the form of a double-well potential (cf.Fig.2).In our treatment,the interaction eﬀects modify the single-well trap potential into one which exhibits two minima.Thus if we think of this eﬀective potential as the one seen by the condensed bosons and according to the general ar-guments [31–34]based on two weakly connected systems we should have an oscillating ﬂux of particles when the chemical potential in the two wells is diﬀerent.Any con-ﬁguration with N L =N R which is always the case for odd number of bosons will result in an oscillatory mo-tion.It would be interesting to explore these ideas in future work.ACKNOWLEDGMENTSThis work was supported by the Scientiﬁc and Techni-cal Research Council of Turkey (TUBITAK)under Grant No.TBAG-1736and TBAG-1662.We gratefully ac-knowledge useful discussions with Prof.C.Yalabık and E.Demirel.[5]D.J.Han,R.H.Wynar,Ph.Courteille,and D.J.Heinzen,Phys.Rev.A57,R4114(1998).[6]I.F.Silvera,in Bose-Einstein Condensation,Ed.by A.Griﬃn,D.W.Snoke,and S.Stringari(Cambridge Uni-versity Press,Cambridge,1995).[7]F.Dalfovo,S.Giorgini,L.P.Pitaevskii,and S.Stringari,preprint,cond-mat/9806038(to be published in Reviews of Modern Physics);A.S.Parkins and D.F.Walls,Phys.Rep.303,1(1998).[8]W.Ketterle and N.J.van Druten,Phys.Rev.A54,656(1996).[9]P.C.Hohenberg,Phys.Rev.158,383(1967).[10]A.Widom,Phys.Rev.176,254(1968).[11]L.C.Ioriatti,Jr.,S.G.Rosa,Jr.,and O.Hipolito,Am.J.Phys.44,744(1976).[12]V.Bagnato and D.Kleppner,Phys.Rev.A44,7439(1991).[13]T.Haugset and H.Haugerud,Phys.Rev.A57,3809(1998).[14]G.-L.Ingold and mbrecht,Eur.Phys.J.D1,29(1998).[15]W.J.Mullin,J.Low Temp.Phys.110,167(1998)[16]W.Deng and P.M.Hui,Solid State Commun.104,729(1997).[17]H.Monien,M.Linn,and N.Elstner,preprint,cond-mat/9711178.[18]F.Brosens,J.T.Devreese,and L.F.Lemmens,SolidState Commun.100,123(1996).[19]S.Pearson,T.Pang,and C.Chen,Phys.Rev.A58,1485(1998).[20]S.Giorgini,L.P.Pitaevskii,and S.Stringari,Phys.Rev.A,54,4633(1996);J.Low Temp.Phys.109,309(1997).[21]A.Minguzzi,S.Conti,and M.P.Tosi,J.Phys.Cond.Matter9,L33(1997).[22]M.Bayindir and B.Tanatar,Phys.Rev.A58,3134(1998).[23]S.Heinrichs and W.J.Mullin,preprint,cond-mat/9807331.[24]E.P.Gross,Nuovo Cimento20,454(1961).[25]L.P.Pitaevskii,Zh.Eksp.Teor.Fiz.40,646(1961)[Sov.Phys.JETP13,451(1961)].[26]M.Olshanii,Phys.Rev.Lett.81,938(1998).[27]R.J.Dodd,K.Burnett,M.Edwards,and C.W.Clark,Acta Phys.Pol.A93,45(1998).[28]M.Naraschewski and D.M.Stamper-Kurn,Phys.Rev.A58,2423(1998).[29]V.Bagnato,D.E.Pritchard,and D.Kleppner,Phys.Rev.A35,4354(1987).[30]G.Baym and C.J.Pethick,Phys.Rev.Lett.76,6(1996).[31]S.Grossmann and M.Holthaus,Z.Naturforsch.50a,323(1995).[32]F.Dalfovo,L.P.Pitaevskii,and S.Stringari,Phys.Rev.A54,4213(1996).[33]A.Smerzi,S.Fantoni,S.Giovanazzi,and S.R.Shenoy,Phys.Rev.Lett.79,4950(1997).[34]I.Zapata,F.Sols,and A.J.Leggett,Phys.Rev.A57,28(1998).。

凯镭思互调仪操作手册(中英文)

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升级标题页添加Summitek/Triasx 商标和联系方式
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Updates. New state editor GUI. (IR7402)
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升级，新版本的测试设置文件编辑器用户界面
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Authorisation 批准
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Changes to section 1.5 by adding report number to the state
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升级1.5在测试设置文件编辑器中添加报告编号
Changes to AC specifications in section 1.2 and section
WA R N I N G- RF HAZARD 警告—射频危害
This equipment is designed for use in association with radio frequency (RF) radiating systems and is capable of producing up to 50W of RF power in the 800 to 2200 MHz region. Users are reminded that proper precautions must be taken to minimise exposure to these RF fields to the recommended limits. Please pay particular care to the following areas: 此设备设计用于无线电射频（RF）发射系统，能够在800至2200兆赫的射频区域内产生高达50瓦的射频功率（RF）。用户应注意，必须采取适当的预防措施，尽量减少暴露在射频区域里，保持在建议的范围内。请特别注意以下几个方面：

Bose FreeSpace E4 Series II 商务音乐系统电子部件说明书

P R O F E S S I O N A L S Y S T E M S D I V I S I ONProduct OverviewThe Bose ®FreeSpace ®E4 Series II system is an integrated four-zone system providing signal processing, routing and amplification for business music applications.System setup and configuration are accom-plished using the included FreeSpace Installer ™software.Bose Standard or Auto Volume or Paging interfaces allow the system owner to control system operation.ApplicationsThe FreeSpace E4 Series II system is ideal for business music applications requiring up to four system zones such as:•Restaurants •Supermarkets •Conference centers •Shopping centersProduct InformationAs a single component, the FreeSpace E4Series II system provides all of the necessary processing and control functions for most business music applications requiring one to four zones with up to 400W of system power.Signal processing and routing for up to four sources, three audio and a page source, is accomplished using digital signal processing.Each source may be assigned to one of four output zones.Each zone output supports Auto Volume,Dynamic Equalization, Room EQ, Speaker EQ and Output Gain functions.A proprietary Power Sharing amplifier distributes up to 400W of system power across the four output zones.Each output zone may draw as little as 1W of power, to as much as 400W.The amount of power delivered to an output zone is dependent on the quantity and tap of loudspeakers that are connected.The amplifier can be configured for either 70V or 100V constant voltage systems.FreeSpace Installer software is included with the FreeSpace E4 Series II ed for the setup and configuration of the system, the FreeSpace Installer software requires a PC for operation.Key Features•Proprietary Power Sharing amplifier dynamically distributes 400W of system power across the four output zones.•Auto Volume offers automatic control of individual zone volume.Volume levels are adjusted to compensate for changes in background noise according to the desired settings.•Each output zone may be independently paged.•Opti-voice ®paging provides a smooth transition between the music and page signals.•Opti-source ®level management monitors the input level of up to four sources.Source levels arecontinually adjusted to maintain a consistent volume level among different sources.•Scheduling allows system ON/OFF , zone volume,mute, Auto Volume and source change events to be programmed.Up to 64 events may be programmed to occur on specific days and times.Detailed Product SpecificationsFront- and Rear-Panel Controls and IndicatorsInput Signal Level Indicators for each of the four inputs.Each LED indicates:■YELLOW – Low Input Signal Level > -13 dBV,signal,< -2 dBV■GREEN – Good Input Signal Level > -2 dBV,signal,< +14 dBV ■RED – Input Signal Clipping signal > +14 dBV Output Status Indicators display the output status for each of the four output zones.Each LED indicates:■GREEN – Good Input Signal Level,zone operating normally■RED – Input Signal Clipping or Amplifier Fault System Status Indicator displays the current state of the system:■OFF – System is not operating■GREEN – System is operating normally ■RED – System faultStandby Indicator displays the operating state of the system:■OFF – System is operating■YELLOW – System is in standbyStandby Button is used to place the system into and out of standby.USB for future connection.Direct Input Indicator displays the current state of the Direct Input:■OFF – Direct Input is not active ■YELLOW – Direct Input is activeRS-232is used for the connection to the PC for setup and configuration with the FreeSpace ®Installer ™software.AC Mains Power is used to turn the AC Mains power ON and OFF .125768934Rack-mount ears used for mounting the E4 sys-tem within a standard 19-inch equipment rack.Standard User Interface provides control of sources, volume and mute for a single zone.Standard UI, North America/Japan PC029856Standard UI, Europe/AustraliaPC029857Multi-Zone Paging Interface provides selection of individual zone, all page and page initiation.Page UI, North America/Japan PC030103Page UI, Europe/AustraliaPC030104Auto Volume User Interface provides control of sources, volume and Auto Volume operation for a single zone.Auto Volume UI, North America/Japan PC030101Auto Volume UI, Europe/Australia PC030102Auto Volume Sense Microphone, placed within asystem zone to measure background noise.Auto Volume Sense Mic, North America/Japan PC029859Auto Volume Sense Mic, Europe/AustraliaPC029860FreeSpace E4 Series II System Setup FreeSpace E4 Series II Event Schedule FreeSpace E4 Series II System Error LogSafety and Regulatory ComplianceThe FreeSpace E4 Series II system complies withUL6500 2nd edition, EN60065 and IEC60065:1998 (6th).Architects’and Engineers’SpecificationsThe unit shall be an integrated signal processingand amplification system.The system shall use adigital signal processing architecture running at44.1 kHz sample rate.The frequency responseshall be from 20 Hz to 20 kHz, +-1 dB.Thesignal-to-noise ratio shall be 90 dB or greaterA-weighted.The power amplification section shall deliver amaximum of 400W with less than 1.0% THD.Channel separation shall be >70 dB at 1 kHz.The system shall consume AC power of 60W orless at idle, 300W at maximum continuous ratedpower.The system shall perform the followingprocessing functions:•Input gain•Input leveling•Source routing•Paging with adjustable ducking depth, holdand release time•Automatic Volume control for each output zone•Music on hold•Three-band graphic equalization per zone•Loudspeaker EQ for Bose®loudspeakers•Output gain with Mute•Loudspeaker protection limiting•System diagnosticsThe system shall be the Bose FreeSpace®E4Series II.Limited WarrantyThe FreeSpace E4 Series II system is covered bya five-year transferable warranty.Details of thewarranty and its coverage are included in theFreeSpace E4 Series II Owner’s Guide.LiteratureFreeSpace E4 Series II T echnical Data PC036723FreeSpace E4 Series II Brochure PC036724How Our Products Are Measured1.Amplifier PowerEIA Power –With the amplifier operating in 70Vor 100V mode, a single channel is driven tofull power with the minimum load impedance.Output power is measured using a 1 kHz sinewave with 1% THD, as measured at theamplifier output.FTC Continuous –With the amplifier operatingin 70V mode, any combination of channelsare driven to full power with the minimumload impedance.Output power is measuredusing test signals between 20 Hz and 12 kHzwith 1% THD, as measured at the amplifieroutput.2.Signal-to-Noise RatioThe output of the amplifier is connected to the ratedload impedance with a unity gain of a frequencyof 1 kHz.A dB-calibrated voltmeter is connectedto the amplifier’s output through an A-weightingfilter (in accordance with IEC 60651).A 1 kHzsignal is connected to one of the line inputs andthe level is adjusted to achieve the amplifier’s ratedoutput power.The signal source is removed, and theline input is shorted.The dB-calibrated voltmeternow reads the A-weighted output noise level.For more information, visit U LC USLISTED 917DAUDIOEQUIPMENTN123BosePSE。

基于ads仿真的梳状谱发生器的设计与实现

基于ads仿真的梳状谱发生器的设计与实现梳状谱发生器是一种广泛应用于通信、雷达、光谱分析等领域的信号发生器，它能够提供稳定、高分辨率、频率连续可调的信号输出。

本文将针对梳状谱发生器的设计与实现，基于ads仿真进行详细的介绍。

梳状谱发生器的基本原理是利用频率合成技术，通过将多个相位可调的载波信号进行混频合成，形成一个频率分辨率非常高的梳状谱，从而实现高精度的信号产生。

梳状谱发生器通常包括振荡器、相位调节器、混频器、滤波器和放大器等基本组成部分。

首先，我们需要确定梳状谱发生器的工作频率范围和分辨率要求。

在设计时，我们可以选择使用基于DDS（直接数字合成）技术的数字振荡器作为基频信号源，以实现频率连续可调的要求。

结合相位调节器可以实现每个频率分量的相位调节，从而实现梳状谱的产生。

其次，我们需要设计混频器和滤波器模块，用于将多路频率合成的信号进行混频和滤波处理，以获得稳定、纯净的输出信号。

在ads仿真中，我们可以通过建立适当的混频器和滤波器模型，进行电路的仿真分析和参数优化，以满足梳状谱发生器的输出性能要求。

另外，对于梳状谱发生器的放大器设计也非常关键。

放大器需要具备高线性度、宽带和低噪声等特性，以保证输出信号的幅度和谱纯度。

在ads仿真中，我们可以利用模拟器进行放大器电路的设计优化，以提高整个梳状谱发生器的整体性能。

在实际的工程应用中，还需要根据具体的系统需求，考虑梳状谱发生器的集成度、尺寸和功耗等方面的问题。

通过优化电路结构和选择合适的器件，可以实现梳状谱发生器的小型化和低功耗化。

总的来说，基于ads仿真的梳状谱发生器设计与实现涉及到频率合成技术、混频和滤波技术、放大器设计和电路优化等多个方面。

通过逐步分析和优化各个模块，可以实现高性能、高稳定性的梳状谱发生器设计，满足不同领域的应用需求。

物理学百篇经典文献目录(125篇)

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NORMA 4000 5000 Power Analyzer 用户说明手册说明书

Since some countries or states do not allow limitation of the term of an implied warranty, or exclusion or limitation of incidental or consequential damages, the limitations and exclusions of this warranty may not apply to every buyer. If any provision of this Warranty is held invalid or unenforceable by a court or other decision-maker of competent jurisdiction, such holding will not affect the validity or enforceability of any other provision.
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往复压缩机气阀振动信号的二阶循环谱特征提取

雷娜，，９０年６月生，士研究生。黑龙江省大庆市，６３８女１８博１３１。
第３９卷
第４期
化
工
机
械
４９３
稳信号来说，统计量是时变的，其且呈周期或多周期变化。
谱对比，行了仿真分析，以信号（）例进进现ｔ为
随着旋转机械故障诊断方法的成熟和完善，故障诊断领域逐渐向往复机械设备延伸，统的传特征提取方法面临挑战。往复压缩机是石油、化工等部门广泛应用的重要机械设备，气阀是往而复压缩机中最为重要的部件，是最容易损坏的也
行说明。
（）：［ｃｓ２￡］ｏ（ｔｔ１＋ｏ（）ｃｓ２）（０１）
循环平稳随机信号（）ｔ的统计量随时间呈
现周期性变化，即：
ｍ（，一，）ｔｒｔｒ＝ｍ（＋，一，）ｔｒｒ
ｃｔ丁一，１（，ｒ一）＝ｃｔ，一，）（＋下ｒ
核心问题之一。往复压缩机结构复杂，励源激
多，塞的往复运动是一种典型的周期运动，活产生
的振动信号中含有特定的周期 ’ 。特别是发生故障时，号的统计量（信如均值、自相关函数及高阶统计量等）随时间呈现出周期或者多周期变会化，是一类特殊的非平稳信号—— 循环平稳信这号。针对如此复杂的振动信号，统的信号处理传方法难以提取故障的有用信息，分析和处理这而类特殊信号最有力的工具是循环统计量理论。在传统的信号处理中，稳的高斯信号模型平占据主导地位。很多情况下（如旋转机械转子的振动信号）信号和噪声的平稳高斯分布假设可，以通过中心极限定理得到证明是合理的，对于但往复压缩机具有的规律性变化的循环平稳脉冲信号则不再适合。循环平稳过程理论的起源可以追溯到２０世纪５０年代，其迅速发展是从８但０年代

Bose-Einstein condensation in dense quark matter

arXiv:0809.2892v1 [hep-ph] 17 Sep 2008
Bose-Einstein condensation in dense quark matter
Jens O. Andersen
Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway
Preprint submitted to Elsevier 17 September 2008
In this talk, I would like to present some results for Bose condensation of charged pions using the linear sigma model as a low-energy eﬀective for two-ﬂavor QCD. The talk is based on Refs. [8,9] and part of the work is done in collaboration with Tomas Brauner. 2. Interacting Bose gas and 2PI 1/N -expansion The Euclidean Lagrangian for a Bose gas with N species of massive charged scalars is
2 2 2 ωn + m2 4 + ΠLO . The term m4 + ΠLO is thus a local mass term, which we denote by M . The exact propagator D satisﬁes the equation δ Γ/δD = 0. In the present case, it reduces to a local gap equation for M 2 :

人大考研-理学院物理系研究生导师简介-张威

爱考机构-人大考研-理学院物理系研究生导师简介-张威原子与分子物理、量子信息与计算(点击次数：14358)张威我的教学：电磁学超冷原子气体物理基本信息职称：副教授办公地点：理工楼712电子邮箱:wzhangl@电话:0086-10-62511881传真:0086-10-62517887教育经历1997年9月至2001年7月北京大学物理系学士2001年8月至2006年12月美国佐治亚理工学院物理系博士2003年8月至2006年8月美国佐治亚理工学院数学系硕士工作经历2006年11月至2008年10月美国密歇根大学物理系博士后2008年10月至今中国人民大学物理系副教授研究兴趣1.强相互作用的超冷原子气体2.玻色爱因斯坦凝聚3.有机超导体和高温超导体主要论著1.Xiang-FaZhou,Guang-CanGuo,WeiZhang,andWeiYi,arXiv:1302.1303ExoticpairingstatesinaFermigaswiththree-dimensionalspin-orbitcoupling2.FanWu,Guang-CanGuo,WeiZhang,andWeiYi,Phys.Rev.Lett.110,110401(2013).Unconventionalsuperfluidinatwo-dimensionalFermigaswithanisotropicspin-orbitco uplingandZeemanfields3.R.Zhang,F.Wu,J.-R.T ang,G.-C.Guo,W.Yi,andWeiZhang,Phys.Rev.A87,033629(2013). Significanceofdressedmoleculesinaquasi-two-dimensionalpolarizedFermigas4.P.Zhang,L.Zhang,andWeiZhang,Phys.Rev.A86,042707(2012).Interatomiccollisionsintwo-dimensionalandquasi-two-dimensionalconfinementswi thspin-orbitcoupling5.W.YiandWeiZhang,Phys.Rev.Lett.109,140402(2012). Moleculeandpolaroninahighlypolarizedtwo-dimensionalFermigaswithspin-orbitco upling.6.J.Zhou,WeiZhang,W.Yi,Phys.Rev.A84,063603(2011).Topologicalsuperfluidinatrappedtwo-dimensionalpolarizedFermigaswithspin-orbit coupling.7.T.Yin,P.Zhang,WeiZhang,Phys.Rev.A84,052727(2011).Stableheteronuclearfew-atomboundstatesinmixeddimensions.8.X.Liu,X.Zhou,WeiZhang,T.Vogt,B.Lu,X.Yue,andX.Chen,Phys.Rev.A83,063604(2011).Exploringmulti-bandexcitationsofinteractingBosegasesina1Dopticallatticebycoherentscattering.9.WeiZhangandP.Zhang,Phys.Rev.A83,053615(2011).Confinement-inducedresonancesinquasi-one-dimensio naltrapswithtransverseanisotropy.。

综述BEC(Bose-Einstein Condensation) 【英文版】

Theory of Bose-Einstein condensation in trapped gasesFranco Dalfovo and Stefano GiorginiDipartimento di Fisica,Universita`di Trento and Istituto Nazionale per la Fisica dellaMateria,I-38050Povo,ItalyLev P.PitaevskiiDipartimento di Fisica,Universita`di Trento and Istituto Nazionale per la Fisica dellaMateria,I-38050Povo,Italy;Department of Physics,TECHNION,Haifa32000,Israel;and Kapitza Institute for Physical Problems,ul.Kosygina2,117334MoscowSandro StringariDipartimento di Fisica,Universita`di Trento and Istituto Nazionale per la Fisica dellaMateria,I-38050Povo,ItalyThe phenomenon of Bose-Einstein condensation of dilute gases in traps is reviewed from a theoretical perspective.Mean-ﬁeld theory provides a framework to understand the main features of the condensation and the role of interactions between particles.Various properties of these systems are discussed,including the density proﬁles and the energy of the ground-state conﬁgurations,the collective oscillations and the dynamics of the expansion,the condensate fraction and the thermodynamic functions.The thermodynamic limit exhibits a scaling behavior in the relevant length and energy scales.Despite the dilute nature of the gases,interactions profoundly modify the static as well as the dynamic properties of the system;the predictions of mean-ﬁeld theory are in excellent agreement with available experimental results.Effects of superﬂuidity including the existence of quantized vortices and the reduction of the moment of inertia are discussed,as well as the consequences of coherence such as the Josephson effect and interference phenomena.The review also assesses the accuracy and limitations of the mean-ﬁeld approach.[S0034-6861(99)00103-8]CONTENTSI.Introduction463II.The Ideal Bose Gas in a Harmonic Trap466A.The condensate of noninteracting bosons466B.Trapped bosons atﬁnite temperature:thermodynamic limit468C.Finite-size effects470D.Role of dimensionality471E.Nonharmonic traps and adiabatic transformations472 III.Effects of Interactions:Ground State472A.Order parameter and mean-ﬁeld theory472B.Ground state474C.Collapse for attractive forces477rge-N limit for repulsive forces478E.Beyond mean-ﬁeld theory479 IV.Effects of Interactions:Dynamics480A.Excitations of the condensate and time-dependent Gross-Pitaevskii equation480rge Na/a ho limit and collisionlesshydrodynamics482C.Sum rules and collective excitations485D.Expansion and large-amplitude oscillations486E.Density of states:collective vs single-particleexcitations489 V.Effects of Interactions:Thermodynamics491A.Relevant energy scales491B.Critical temperature492C.Below T c493D.Thermodynamic limit and scaling495E.Results for the thermodynamic functions496F.Collective modes atﬁnite temperature498 VI.Superﬂuidity and Coherence Phenomena500A.Rotational properties:vortices and moment ofinertia501B.Interference and Josephson effect504C.Collapse and revival of collective oscillations506 VII.Conclusions and Outlook507 Acknowledgments509 References509 I.INTRODUCTIONBose-Einstein condensation(BEC)(Bose,1924;Ein-stein,1924,1925)was observed in1995in a remarkable series of experiments on vapors of rubidium(Anderson et al.,1995)and sodium(Davis et al.,1995)in which the atoms were conﬁned in magnetic traps and cooled down to extremely low temperatures,of the order of fractions of microkelvins.Theﬁrst evidence for condensation emerged from time-of-ﬂight measurements.The atoms were left to expand by switching off the conﬁning trap and then imaged with optical methods.A sharp peak in the velocity distribution was then observed below a cer-tain critical temperature,providing a clear signature for BEC.In Fig.1,we show one of theﬁrst pictures of the atomic clouds of rubidium.In the same year,ﬁrst signa-tures of the occurrence of BEC in vapors of lithium were also reported(Bradley et al.,1995).Though the experiments of1995on the alkalis should be considered a milestone in the history of BEC,the experimental and theoretical research on this unique phenomenon predicted by quantum statistical mechanics is much older and has involved different areas of physics (for an interdisciplinary review of BEC see Grifﬁn, Snoke,and Stringari,1995).In particular,from the very beginning,superﬂuidity in helium was considered by463Reviews of Modern Physics,Vol.71,No.3,April19990034-6861/99/71(3)/463(50)/$25.00©1999The American Physical SocietyLondon (1938)as a possible manifestation of BEC.Evi-dence for BEC in helium later emerged from the analy-sis of the momentum distribution of the atoms measured in neutron-scattering experiments (Sokol,1995).In re-cent years,BEC has been also investigated in the gas of paraexcitons in semiconductors (see Wolfe,Lin,and Snoke,1995,and references therein),but an unambigu-ous signature for BEC in this system has proven difﬁcult to ﬁnd.Efforts to Bose condense atomic gases began with hy-drogen more than 15years ago.In a series of experi-ments hydrogen atoms were ﬁrst cooled in adilutionFIG.2.(Color)Collective excitations of a Bose-Einstein condensate.Shown are in situ repeated phase-contrast images taken of a ‘‘pure’’condensate.The excitations were produced by modulating the magnetic ﬁelds which conﬁne the condensate,and then letting the condensate evolve freely.Both the center-of-mass and the shape oscillations are visible,and the ratio of their oscillation frequencies can be accurately measured.The ﬁeld of view in the vertical direction is about 620␮m,corresponding to a condensate width of the order of 200–300␮m.The time step is 5ms per frame.From Stamper-Kurn and Ketterle(1998).FIG.1.(Color)Images of the velocity distribution of rubidium atoms in the experiment by Anderson et al.(1995),taken by means of the expansion method.The left frame corresponds to a gas at a temperature just above condensation;the center frame,just after the appearance of the condensate;the right frame,after further evaporation leaves a sample of nearly pure condensate.The ﬁeld of view is 200␮m ϫ270␮m,and corresponds to the distance the atoms have moved in about 1/20s.The color corresponds to the number of atoms at each velocity,with red being the fewest and white being the most.From Cornell (1996).464Dalfovo et al.:Bose-Einstein condensation in trapped gasesRev.Mod.Phys.,Vol.71,No.3,April 1999refrigerator,then trapped by a magneticﬁeld and fur-ther cooled by evaporation.This approach has come very close to observing BEC,but is still limited by re-combination of individual atoms to form molecules(Sil-vera and Walraven,1980and1986;Greytak and Klepp-ner,1984;Greytak,1995;Silvera,1995).At the time of this review,ﬁrst observations of BEC in spin-polarized hydrogen have been reported(Fried et al.,1998).In the 1980s laser-based techniques,such as laser cooling and magneto-optical trapping,were developed to cool and trap neutral atoms[for recent reviews,see Chu(1998), Cohen-Tannoudji(1998),and Phillips(1998)].Alkali at-oms are well suited to laser-based methods because their optical transitions can be excited by available lasers and because they have a favorable internal energy-level structure for cooling to very low temperatures.Once they are trapped,their temperature can be lowered fur-ther by evaporative cooling[this technique has been re-cently reviewed by Ketterle and van Druten(1996a)and by Walraven(1996)].By combining laser and evapora-tive cooling for alkali atoms,experimentalists eventually succeeded in reaching the temperatures and densities re-quired to observe BEC.It is worth noticing that,in these conditions,the equilibrium conﬁguration of the system would be the solid phase.Thus,in order to observe BEC,one has to preserve the system in a metastable gas phase for a sufﬁciently long time.This is possible be-cause three-body collisions are rare events in dilute and cold gases,whose lifetime is hence long enough to carryout experiments.So far BEC has been realized in87Rb(Anderson et al.,1995;Han et al.,1998;Kasevich,1997;Ernst,Marte et al.,1998;Esslinger et al.,1998;So¨dinget al.,1999),in23Na(Davis et al.,1995;Hau,1997and 1998;Lutwak et al.,1998),and in7Li(Bradley et al.,1995and1997).The number of experiments on BEC invapors of rubidium and sodium is now growing fast.Inthe meanwhile,intense experimental research is cur-rently carried out also on vapors of caesium,potassium,and metastable helium.One of the most relevant features of these trappedBose gases is that they are inhomogeneous andﬁnite-sized systems,the number of atoms ranging typicallyfrom a few thousands to several millions.In most cases,the conﬁning traps are well approximated by harmonicpotentials.The trapping frequency␻ho also provides a characteristic length scale for the system,a ho ϭ͓ប/(m␻ho)͔1/2,of the order of a few microns in the available samples.Density variations occur on this scale.This is a major difference with respect to other systems,like,for instance,superﬂuid helium,where the effects ofinhomogeneity take place on a microscopic scaleﬁxedby the interatomic distance.In the case of87Rb and 23Na,the size of the system is enlarged as an effect of repulsive two-body forces and the trapped gases can be-come almost macroscopic objects,directly measurable with optical methods.As an example,we show in Fig.2 a sequence of in situ images of an oscillating condensate of sodium atoms taken at the Massachusetts Institute of Technology(MIT),where the mean axial extent is of the order of0.3mm.The fact that these gases are highly inhomogeneous has several important consequences.First BEC shows up not only in momentum space,as happens in super-ﬂuid helium,but also in coordinate space.This double possibility of investigating the effects of condensation is very interesting from both the theoretical and experi-mental viewpoints and provides novel methods of inves-tigation for relevant quantities,like the temperature de-pendence of the condensate,energy and density distributions,interference phenomena,frequencies of collective excitations,and so on.Another important consequence of the inhomogene-ity of these systems is the role played by two-body inter-actions.This aspect will be extensively discussed in the present review.The main point is that,despite the very dilute nature of these gases(typically the average dis-tance between atoms is more than ten times the range of interatomic forces),the combination of BEC and har-monic trapping greatly enhances the effects of the atom-atom interactions on important measurable quantities. For instance,the central density of the interacting gas at very low temperature can be easily one or two orders of magnitude smaller than the density predicted for an ideal gas in the same trap,as shown in Fig.3.Despite the inhomogeneity of these systems,which makes the solution of the many-body problem nontrivial,the dilute nature of the gas allows one to describe the effects of the interaction in a rather fundamental way.In practice a single physical parameter,the s-wave scattering length, is sufﬁcient to obtain an accuratedescription.FIG.3.Density distribution of80000sodium atoms in the trap of Hau et al.(1998)as a function of the axial coordinate.The experimental points correspond to the measured optical den-sity,which is proportional to the column density of the atom cloud along the path of the light beam.The data agree well with the prediction of mean-ﬁeld theory for interacting atoms (solid line)discussed in Sec.III.Conversely,a noninteracting gas in the same trap would have a much sharper Gaussian distribution(dashed line).The same normalization is used for the three density proﬁles.The central peak of the Gaussian is found at about5500␮mϪ2.Theﬁgure points out the role of atom-atom interaction in reducing the central density and en-larging the size of the cloud.465Dalfovo et al.:Bose-Einstein condensation in trapped gases Rev.Mod.Phys.,Vol.71,No.3,April1999The recent experimental achievements of BEC in al-kali vapors have renewed a great interest in the theoret-ical studies of Bose gases.A rather massive amount of work has been done in the last couple of years,both to interpret the initial observations and to predict new phe-nomena.In the presence of harmonic conﬁnement,the many-body theory of interacting Bose gases gives rise to several unexpected features.This opens new theoretical perspectives in this interdisciplinaryﬁeld,where useful concepts coming from different areas of physics(atomic physics,quantum optics,statistical mechanics,and condensed-matter physics)are now merging together. The natural starting point for studying the behavior of these systems is the theory of weakly interacting bosons which,for inhomogeneous systems,takes the form of the Gross-Pitaevskii theory.This is a mean-ﬁeld ap-proach for the order parameter associated with the con-densate.It provides closed and relatively simple equa-tions for describing the relevant phenomena associated with BEC.In particular,it reproduces typical properties exhibited by superﬂuid systems,like the propagation of collective excitations and the interference effects origi-nating from the phase of the order parameter.The theory is well suited to describing most of the effects of two-body interactions in these dilute gases at zero tem-perature and can be naturally generalized to also ex-plore thermal effects.An extensive discussion of the application of mean-ﬁeld theory to these systems is the main basis of the present review article.We also give,whenever possible, simple arguments based on scales of length,energy,and density,in order to point out the relevant parameters for the description of the various phenomena.There are several topics which are only marginally discussed in our paper.These include,among others, collisional and thermalization processes,phase diffusion phenomena,light scattering from the condensate,and analogies with systems of coherent photons.In this sense our work is complementary to other recent review ar-ticles(Burnett,1996;Parkins and Walls,1998).Further-more,in our paper we do not discuss the physics of ul-tracold collisions and the determination of the scattering length which have been recently the object of important experimental and theoretical studies in the alkalis (Heinzen,1997;Weiner et al.,1999).The plan of the paper is the following:In Sec.II we summarize the basic features of the non-interacting Bose gas in harmonic traps and we introduce theﬁrst relevant length and energy scales,like the oscil-lator length and the critical temperature.We also com-ment onﬁnite-size effects,on the role of dimensionality and on the possible relevance of anharmonic traps.In Sec.III we discuss the effects of the interaction on the ground state.We develop the formalism of mean-ﬁeld theory,based on the Gross-Pitaevskii equation.We consider the case of gases interacting with both repulsive and attractive forces.We then discuss in detail the large-N limit for systems interacting with repulsive forces,leading to the so-called Thomas-Fermi approxi-mation,where the ground-state properties can be calcu-lated in analytic form.In the last part,we discuss the validity of the mean-ﬁeld approach and give explicit re-sults for theﬁrst corrections,beyond meanﬁeld,to the ground-state properties,including the quantum deple-tion of the condensate,i.e.,the decrease in the conden-sate fraction produced by the interaction.In Sec.IV we investigate the dynamic behavior of the condensate using the time-dependent Gross-Pitaevskii equation.The equations of motion for the density and the velocityﬁeld of the condensate in the large-N limit, where the Thomas-Fermi approximation is valid,are shown to have the form of the hydrodynamic equations of superﬂuids.We also discuss the dynamic behavior in the nonlinear regime(large amplitude oscillations and free expansion),the collective modes in the case of at-tractive forces,and the transition from collective to single-particle states in the spectrum of excitations.In Sec.V we discuss thermal effects.We show how one can deﬁne the thermodynamic limit in these inho-mogeneous systems and how interactions modify the be-havior compared to the noninteracting case.We exten-sively discuss the occurrence of scaling properties in the thermodynamic limit.We review several results for the shift of the critical temperature and for the temperature dependence of thermodynamic functions,like the con-densate fraction,the chemical potential,and the release energy.We also discuss the behavior of the excitations atﬁnite temperature.In Sec.VI we illustrate some features of these trapped Bose gases in connection with superﬂuidity and phase coherence.We discuss,in particular,the structure of quantized vortices and the behavior of the moment of inertia,as well as interference phenomena and quantum effects beyond mean-ﬁeld theory,like the collapse re-vival of collective oscillations.In Sec.VII we draw our conclusions and we discuss some further future perspectives in theﬁeld.The overlap between current theoretical and experi-mental investigations of BEC in trapped alkalis is al-ready wide and rich.Various theoretical predictions, concerning the ground state,dynamics,and thermody-namics are found to agree very well with observations; others are stimulating new experiments.The comparison between theory and experiments then represents an ex-citing feature of these novel systems,which will be fre-quently emphasized in the present review.II.THE IDEAL BOSE GAS IN A HARMONIC TRAPA.The condensate of noninteracting bosonsAn important feature characterizing the available magnetic traps for alkali atoms is that the conﬁning po-tential can be safely approximated with the quadratic formV ext͑r͒ϭm2͑␻x2x2ϩ␻y2y2ϩ␻z2z2͒.(1) Thus the investigation of these systems starts as a text-466Dalfovo et al.:Bose-Einstein condensation in trapped gases Rev.Mod.Phys.,Vol.71,No.3,April1999book application of nonrelativistic quantum mechanics for identical pointlike particles in a harmonic potential. Theﬁrst step consists in neglecting the atom-atom in-teraction.In this case,almost all predictions are analyti-cal and relatively simple.The many-body Hamiltonian is the sum of single-particle Hamiltonians whose eigenval-ues have the form␧nx n y n zϭͩn xϩ12ͪប␻xϩͩn yϩ12ͪប␻yϩͩn zϩ12ͪប␻z,(2)where͕n x,n y,n z͖are non-negative integers.The ground state␾(r1,...,r N)of N noninteracting bosons conﬁned by the potential(1)is obtained by putting all the particles in the lowest single-particle state(n xϭn y ϭn zϭ0),namely␾(r1,...,r N)ϭ⌸i␸0(r i),where␸0(r) is given by␸0͑r͒ϭͩm␻ho␲បͪ3/4expͫϪm2ប͑␻x x2ϩ␻y y2ϩ␻z z2͒ͬ,(3)and we have introduced the geometric average of the oscillator frequencies:␻hoϭ͑␻x␻y␻z͒1/3.(4) The density distribution then becomes n(r)ϭN͉␸0(r)͉2 and its value grows with N.The size of the cloud is in-stead independent of N and isﬁxed by the harmonic oscillator length:a hoϭͩបm hoͪ1/2(5)which corresponds to the average width of the Gaussian in Eq.(3).This is theﬁrst important length scale of the system.In the available experiments,it is typically of the order of a hoϷ1␮m.Atﬁnite temperature only part of the atoms occupy the lowest state,the others being ther-mally distributed in the excited states at higher energy. The radius of the thermal cloud is larger than a ho.A rough estimate can be obtained by assuming k B T ӷប␻ho and approximating the density of the thermal cloud with a classical Boltzmann distribution n cl(r)ϰexp͓ϪV ext(r)/k B T͔.If V ext(r)ϭ(1/2)m␻ho2r2,the width of the Gaussian is R Tϭa ho(k B T/ប␻ho)1/2,and hence larger than a ho.The use of a Bose distribution function does not change signiﬁcantly this estimate. The above discussion reveals that Bose-Einstein con-densation in harmonic traps shows up with the appear-ance of a sharp peak in the central region of the density distribution.An example is shown in Fig.4,where we plot the prediction for the condensate and thermal den-sities of5000noninteracting particles in a spherical trap at a temperature Tϭ0.9T c0,where T c0is the temperature at which condensation occurs(see discussion in the next section).The curves correspond to the column density, namely the particle density integrated along one direc-tion,n(z)ϭ͐dx n(x,0,z);this is a typical measured quantity,the x direction being the direction of the light beam used to image the atomic cloud.By plotting di-rectly the density n(r),the ratio of the condensed and noncondensed densities at the center would be even larger.By taking the Fourier transform of the ground-state wave function,one can also calculate the momentum distribution of the atoms in the condensate.For the ideal gas,it is given by a Gaussian centered at zero mo-mentum and having a width proportional to a hoϪ1.The distribution of the thermal cloud is,also in momentum space,ing a classical distribution function oneﬁnds that the width is proportional to(k B T)1/2.Ac-tually,the momentum distributions of the condensed and noncondensed particles of an ideal gas in harmonic traps have exactly the same form as the density distribu-tions n0and n T shown in Fig.4.The appearance of the condensate as a narrow peak in both coordinate and momentum space is a peculiar fea-ture of trapped Bose gases having important conse-quences in both the experimental and theoretical analy-sis.This is different from the case of a uniform gas where the particles condense into a state of zero mo-mentum,but BEC cannot be revealed in coordinate space,since the condensed and noncondensed particles ﬁll the same volume.Indeed,the condensate has been detected experimen-tally as the occurrence of a sharp peak over a broader distribution,in both the velocity and spatial distribu-tions.In theﬁrst case,one lets the condensate expand freely,by switching off the trap,and measures the den-sity of the expanded cloud with light absorption(Ander-son et al.,1995).If the particles do not interact,the ex-pansion is ballistic and the imaged spatial distribution of the expanding cloud can be directly related to the initial momentum distribution.In the second case,one mea-sures directly the density of the atoms in the trapby FIG.4.Column density for5000noninteracting bosons in a spherical trap at temperature Tϭ0.9T c0.The central peak is the condensate,superimposed on the broader thermal distri-bution.Distance and density are in units of a ho and a hoϪ2,re-spectively.The density is normalized to the number of atoms. The same curves can be identiﬁed with the momentum distri-bution of the condensed and noncondensed particles,provided the abscissa and the ordinate are replaced with p z,in units of a hoϪ1,and the momentum distribution,in units of a ho2,respec-tively.467Dalfovo et al.:Bose-Einstein condensation in trapped gases Rev.Mod.Phys.,Vol.71,No.3,April1999means of dispersive light scattering(Andrews et al., 1996).In both cases,the appearance of a sharp peak is the main signature of Bose-Einstein condensation.An important theoretical task consists of predicting how the shape of these peaks is modiﬁed by the inclusion of two-body interactions.As anticipated in Fig.3,the interac-tions can change the picture drastically.This effect will be deeply discussed in Sec.III.The shape of the conﬁningﬁeld alsoﬁxes the symme-try of the problem.One can use spherical or axially sym-metric traps,for instance.Theﬁrst experiments on ru-bidium and sodium were carried out with axial symmetry.In this case,one can deﬁne an axial coordi-nate z and a radial coordinate rЌϭ(x2ϩy2)1/2and the corresponding frequencies,␻z and␻Ќϭ␻xϭ␻y.The ra-tio between the axial and radial frequencies,␭ϭ␻z/␻Ќ,ﬁxes the asymmetry of the trap.For␭Ͻ1the trap is cigar shaped while for␭Ͼ1is disk shaped.In terms of␭the ground state Eq.(3)for noninteracting bosons can be rewritten as␸0͑r͒ϭ␭1/43/4aЌ3/2expͫϪ12aЌ2͑rЌ2ϩ␭z2͒ͬ.(6)Here aЌϭ(ប/m␻Ќ)1/2is the harmonic-oscillator length in the xy plane and,since␻Ќϭ␭Ϫ1/3␻ho,one has also aЌϭ␭1/6a ho.The choice of an axially symmetric trap has proven useful for providing further evidence of Bose-Einstein condensation from the analysis of the momentum distri-bution.To understand this point,let us take the Four-ier transform of the wave function Eq.(6):␸˜0(p)ϰexp͓ϪaЌ2(pЌ2ϩ␭Ϫ1p z2)/2ប2͔.From this one can calculate the average axial and radial widths.Their ratio,ͱ͗z2͗͘Ќ2͘ϭͱ␭,(7) isﬁxed by the asymmetry parameter of the trap.Thus the shape of the expanded cloud in the xz plane is an ellipse,the ratio between the two axis(aspect ratio)be-ing equal toͱ␭.If the particles,instead of being in the lowest state(condensate),were thermally distributed among many eigenstates at higher energy,their distribu-tion function would be isotropic in momentum space, according to the equipartition principle,and the aspect ratio would be equal to1.Indeed,the occurrence of anisotropy in the condensate peak has been interpreted from the very beginning as an important signature of BEC(Anderson et al.,1995;Davis et al.,1995;Mewes et al.,1996a).In the case of the experiment at the Joint Institute for Laboratory Astrophysics(JILA)in Boul-der,the trap is disk-shaped with␭ϭͱ8.Theﬁrst mea-sured value of the aspect ratio was about50%larger than the prediction,ͱ␭,of the noninteracting model (Anderson et al.,1995).Of course,a quantitative com-parison can be obtained only including the atom-atom interaction,which affects the dynamics of the expansion (Holland and Cooper,1996;Dalfovo and Stringari,1996; Holland et al.,1997;Dalfovo,Minniti,Stringari,and Pi-taevskii,1997).However,the noninteracting model al-ready points out this interesting effect due to anisotropy.B.Trapped bosons atﬁnite temperature:thermodynamic limitAt temperature T the total number of particles is given,in the grand-canonical ensemble,by the sumNϭ͚n x,n y,n z͕exp͓␤͑␧n x n y n zϪ␮͔͒Ϫ1͖Ϫ1,(8) while the total energy is given byEϭ͚n x,n y,n z␧n x n y n z͕exp͓␤͑␧n x n y n zϪ␮͔͒Ϫ1͖Ϫ1,(9) where␮is the chemical potential and␤ϭ(k B T)Ϫ1.Be-low a given temperature the population of the lowest state becomes macroscopic and this corresponds to the onset of Bose-Einstein condensation.The calculation of the critical temperature,the fraction of particles in the lowest state(condensate fraction),and the other ther-modynamic quantities,starts from Eqs.(8)and(9)with the appropriate spectrum␧nxn y n z(de Groot,Hooman, and Ten Seldam,1950;Bagnato,Pritchard,and Klepp-ner,1987).Indeed the statistical mechanics of these trapped gases is less trivial than expected atﬁrst sight. Several interesting problems arise from the fact that these systems have aﬁnite size and are inhomogeneous. For example,the usual deﬁnition of thermodynamic limit(increasing N and volume with the average density kept constant)is not appropriate for trapped gases. Moreover,the traps can be made very anisotropic, reaching the limit of quasi-two-dimensional and quasi-one-dimensional systems,so that interesting effects of reduced dimensionality can be also investigated.As in the case of a uniform Bose gas,it is convenient to separate out the lowest eigenvalue␧000from the sum (8)and call N0the number of particles in this state.This number can be macroscopic,i.e.,of the order of N,when the chemical potential becomes equal to the energy of the lowest state,␮→␮cϭ32ប␻¯,(10)where␻¯ϭ(␻xϩ␻yϩ␻z)/3is the arithmetic average of the trapping frequencies.Inserting this value in the rest of the sum,one can writeNϪN0ϭ͚n x,n y,n z 01exp͓␤ប͑␻x n xϩ␻y n yϩ␻z n z͔͒Ϫ1.(11) In order to evaluate this sum explicitly,one usually as-sumes that the level spacing becomes smaller and smaller when N→ϱ,so that the sum can be replaced by an integral:NϪN0ϭ͵0ϱdn x dn y dn zexp͓␤ប͑␻x n xϩ␻y n yϩ␻z n z͔͒Ϫ1.(12) This assumption corresponds to a semiclassical descrip-tion of the excited states.Its validity implies that the relevant excitation energies,contributing to the sum (11),are much larger than the level spacingﬁxed by the468Dalfovo et al.:Bose-Einstein condensation in trapped gases Rev.Mod.Phys.,Vol.71,No.3,April1999。

Critical Point of an Interacting Two-Dimensional Atomic Bose Gas

a r X i v :c o n d -m a t /0703200v 2 [c o n d -m a t .o t h e r ] 11 D e c 2007Critical Point of an Interacting Two-Dimensional Atomic Bose GasPeter Kr¨u ger,Zoran Hadzibabic,and Jean DalibardLaboratoire Kastler Brossel and CNRS,Ecole Normale Sup´e rieure,24rue Lhomond,75005Paris,France(Dated:February 4,2008)We have measured the critical atom number in an array of harmonically trapped two-dimensional (2D)Bose gases of rubidium atoms at diﬀerent temperatures.We found this number to be about ﬁve times higher than predicted by the semi-classical theory of Bose-Einstein condensation (BEC)in the ideal gas.This demonstrates that the conventional BEC picture is inapplicable in an interacting 2D atomic gas,in sharp contrast to the three-dimensional case.A simple heuristic model based on the Berezinskii-Kosterlitz-Thouless theory of 2D superﬂuidity and the local density approximation accounts well for our experimental results.PACS numbers:03.75.Lm,32.80.Pj,67.40.-wBose-Einstein condensation (BEC)at a ﬁnite ature is not possible in a homogeneous (2D)system,but an interacting Bose ﬂuid can less become superﬂuid at a ﬁnite critical temperature This unconventional phase transition is described by Berezinskii-Kosterlitz-Thouless (BKT)theory [2,3],does not involve any spontaneous symmetry breaking emergence of a uniform order parameter.It is instead sociated with a topological order embodied in the of vortices with opposite circulations;true long-range der is destroyed by long wavelength phase even in the superﬂuid state [4,5].Recent advances in producing harmonically trapped,weakly interacting (quasi-)2D atomic gases [6,7,8,9,10,11,12,13,14]have opened the possibility for de-tailed studies of BKT physics in a controllable envi-ronment.There has been some theoretical debate on the nature of the superﬂuid transition in these sys-tems [15,16,17,18,19]because the harmonic con-ﬁnement modiﬁes the density of states compared to the homogenous case.This allows for “conventional”ﬁnite temperature Bose-Einstein condensation in the ideal 2D gas [20].Early experiments have been equally consistent with the BEC and the BKT picture of the phase transi-tion.For example,the density proﬁles at very low tem-peratures [6]are expected to be the same in both cases.However,recent studies of matter wave interference of in-dependent 2D atomic clouds close to the transition have revealed both thermally activated vortices [12,13]and quasi-long-range coherence properties [13]in agreement with the BKT theory [21,22].In this Letter,we study the critical atom number in an array of 2D gases of rubidium atoms,and observe stark disagreement with the predictions of the ideal gas BEC theory.We detect the critical point by measuring (i)the onset of bimodality in the atomic density distribution and (ii)the onset of interference between independent 2D clouds.These two measurements agree with each other,and for the investigated range of temperatures T ≈50–110nK give critical atom numbers N c which are ∼5times higher than the ideal gas prediction forz yxtial V (x,z )in the y =0plane for the lattice phase such that the two central planes are symmetric with respect to the trap center.conventional Bose-Einstein condensation in our trap [20].For comparison,in three-dimensional (3D)atomic gases,where conventional BEC occurs,the increase of the criti-cal atom number due to repulsive interactions is typically on the order of ten percent [23,24].A simple heuris-tic model based on the BKT theory of 2D superﬂuidity and the local density approximation gives good agree-ment with our measurements.In [13]we studied quasi-long-range coherence of a trapped 2D gas,which is directly related to the super-ﬂuid density ρs [22].In that case,signatures of the BKT transition emerge only once a signiﬁcant part of the cloud becomes superﬂuid.Since the atomic density in the trap is not uniform,this happens slightly below the true criti-cal temperature for the onset of superﬂuidity in the trap center,and the observed transition is rounded oﬀ.The present study concentrates on the exact critical point and relates to the total density at criticality ρc ,which has been of long standing theoretical interest [25,26].Our experimental procedure for the preparation of cold 2D Bose gases has been described in [13].We start with a 87Rb 3D condensate in a cylindrically symmetric mag-netic trap with trapping frequencies ωx =2π×10.6Hz and ωy =ωz =2π×125Hz.To split the sample into 2D clouds we add a blue detuned one-dimensional opti-2lattice and the density distribution is recorded by absorptionimaging along y after t=22ms of time ofﬂight.The mea-sured line densities¯ρ(x)(•)for an atom number just below(left)and just above(right)the critical number are displayedtogether with bimodalﬁts(solid lines).The dashed line in theleft panel shows the expected distribution of the2D ideal gasat the threshold of conventional BEC in our potential at thesame temperature(T=92nK).The dotted line in the rightpanel indicates the Gaussian part of the bimodal distribution.cal lattice with a period of d=3µm along the verticaldirection z(see Fig.1).The lattice is formed by twolaser beams with a532nm wavelength and focussed towaists of about120µm,which propagate in the yz planeand intersect at a small angle.The depth of the latticepotential around x=0is h×35kHz,corresponding toa vertical conﬁnement ofωz=2π×3.0kHz.The tun-neling rate between adjacent sites at the center of thetrap(x=0)is negligible on the time scale of the exper-iment.Theﬁnite waists of the lattice beams result in aslow variation ofωz along x,and the variation of the zeropoint energy ωz(x)/2modiﬁesωx to2π×9.4Hz at thetrap center.Fig.1shows contour lines for the full trapping po-tential.The number of signiﬁcantly populated latticeplanes is∼2−4in the investigated temperature range(50-110nK).The vast majority of atoms is trapped inthe central x region where the2D criterion kT< ωz(x)is fulﬁlled.However,the exchange of particles betweenlattice sites is still possible via the far wings of the en-ergy distribution(at energies above460nK).This en-sures thermal equilibrium between the planes[28]on thetime scale of∼100collision times[29],which in our casecorresponds to a fraction of a second.The2D interac-tion strength is g=( 2/m)˜g,where the dimensionlesscoupling constant˜g=a s6 kTFigure 3:Measurement of the critical point.The number of atoms in the Thomas-Fermi part of the bimodal distribution N 0(⋄)is plotted as a function of the total atom number N .The solid line shows the linear ﬁt we use to determine N c ,and the dashed line is its extrapolation.For comparison,the inter-ference amplitude |ˆρ(0,k 0)|(•)is also displayed as a function of N .It shows the same threshold N c within our experimen-tal precision.The inset shows that the temperature deduced from the Gaussian part of the ﬁt is to a good approximation constant for all data points.Horizontal and vertical dashed lines indicate the average temperature and the critical atom number,respectively.The solid line marks the region used to determine the average temperature T =92(6)nK close to the transition.Each data point is based on 5–10images,all error bars represent standard deviations.ical atom number N c (•)is measured at four diﬀerent tem-peratures T .Displayed error bars are statistical.The solidline shows the ideal 2D gas BEC prediction N multic ,id .The dashed line is the best empirical ﬁt to the data,which givesN c =αN multic ,idwith α=5.3(5).N c =αN multi c ,id (T ),with the scaling factor αas the only free parameter,ﬁts the data well and gives α=5.3(5),where the quoted error is statistical.We also study the coherent fraction of the 2D gas and compare its behavior with the bimodal density proﬁles.We investigate the interference patterns that form af-ter releasing the independent planar gases from the trap (Fig.5)[13].Fourier transforming the density proﬁle ρ(x,z )→F [ρ(x,z )]≡ˆρ(x,k z )allows us to quantify the size of the coherent,i.e.interfering part of the gas asxk z30µmk z =0k z =k 0k z =2k 0part of the cloud with the part following the Thomas-Fermi density distribution.Left:Interference patterns in the xz plane (see example in inset)are Fourier transformed along the expansion axis z and averaged over ten images taken un-der identical conditions,to obtain |ˆρ(x,k z )|.Right:Within experimental precision,ﬁts to the total density proﬁle |ˆρ(x,0)|and the interference amplitude proﬁle |ˆρ(x,k 0)|give the same Thomas-Fermi diameter 2R TF ,indicated by the dashed lines.The weak second harmonic peak at k z =2k 0reveals small occupation of the outer lattice planes.a function of N .The spatial frequency corresponding to the fringe period for the interference of neighboring planes is k 0=md/ t .We ﬁnd that ˆρ(x,k 0)is well ﬁtted by a pure Thomas-Fermi proﬁle.Within our experimen-tal accuracy,the radii R TF (k 0)of these proﬁles are equal to those obtained from a bimodal ﬁt to the density.In particular,the onsets of interference and bimodality co-incide (circles and diamonds in Fig.3,respectively).We now turn to the interpretation of our measurements in the framework of the BKT theory of 2D superﬂuidity.The theory predicts a universal jump of the superﬂuid density at the transition,from ρs =0to ρs λ2=4,where λ=h/√ ¯ω 2=ρc λ264 peak density in the most populated plane toρc,and sumthe contributions of all planes j using the correspondingpotentials V j(here we neglect the small non-harmoniceﬀects due toﬁnite laser waists).The total population inthe lattice is then N multic,BKT =p′N c,BKT,where the eﬀectivenumber of planes p′varies from2.4at50nK to3.5at110nK.We thus obtain N multic,BKT /N multic,id≃4.7,which isclose to the experimental ratioα=5.3.One could try to reproduce our observations within the self-consistent Hartree-Fock(HF)theory[19](see also [32,33,34]),by replacing V(r)with the eﬀective mean ﬁeld potential V(r)+2gρ(r)and again setting the peak density to the BKT thresholdρc.For very weak interac-tions,log(1/˜g)≫1,analytical HF calculation gives criti-cal numbers which are only slightly larger than N c,id[19]. This approach could in principle be implemented numer-ically for our value of˜g and our lattice geometry.How-ever,it has been suggested[35]that treating interactions at the meanﬁeld level is insuﬃcient for˜g∼10−1,be-cause the interactions are strong enough for the critical region to be a signiﬁcant fraction of the sample.In fu-ture experiments with atomic gases˜g could be varied between1and10−4using Feshbach resonances,allowing for detailed tests of the microscopic BKT theory and the possible breakdown of the meanﬁeld approximation.In conclusion,we have shown that the ideal gas the-ory of Bose-Einstein condensation,which is extremely successful in3D,cannot be used to predict the critical point in interacting2D atomic gases,where interactions play a profound role even in the normal state.A much better prediction of the critical point is provided by the BKT theory of2D superﬂuidity.We have also shown that,despite the absence of true long-range order,the low temperature state displays density proﬁles and local coherence largely analogous to3D BECs.We thank B.Battelier,M.Cheneau,P.Rath, D. Stamper-Kurn,N.Cooper,and M.Holzmann for useful discussions.P.K.and Z.H.acknowledge support from the EU(contracts MEIF-CT-2006-025047and MIF1-CT-2005-007932).This work is supported by R´e gion Ile de France(IFRAF),CNRS,the French Ministry of Research,and boratoire Kastler Brossel is a research unit of Ecole Normale Sup´e rieure,Universit´e Pierre and Marie Curie and CNRS.[1]D.J.Bishop and J.D.Reppy,Phys.Rev.Lett.40,1727(1978).[2]V.L.Berezinskii,Sov.Phys.JETP34,610(1972).[3]J.M.Kosterlitz and D.J.Thouless,J.Phys.C:SolidState Physics6,1181(1973).[4]N.D.Mermin and H.Wagner,Phys.Rev.Lett.17,1133(1966).[5]P.C.Hohenberg,Phys.Rev.158,383(1967).[6]A.G¨o rlitz et al.,Phys.Rev.Lett.87,130402(2001).[7]S.Burger,F.S.Cataliotti,C.Fort,P.Maddaloni,F.Mi-nardi,and M.Inguscio,Europhys.Lett.57,1(2002). [8]D.Rychtarik,B.Engeser,H.-C.N¨a gerl,and R.Grimm,Phys.Rev.Lett.92,173003(2004).[9]Z.Hadzibabic,S.Stock, B.Battelier,V.Bretin,andJ.Dalibard,Phys.Rev.Lett.93,180403(2004). [10]N.L.Smith,W.H.Heathcote,G.Hechenblaikner,E.Nu-gent,and C.J.Foot,Journal of Physics B38,223(2005).[11]M.K¨o hl,H.Moritz,T.St¨o ferle, C.Schori,andT.Esslinger,J.Low Temp.Phys.138,635(2005). [12]S.Stock,Z.Hadzibabic,B.Battelier,M.Cheneau,andJ.Dalibard,Phys.Rev.Lett.95,190403(2005). [13]Z.Hadzibabic,P.Kr¨u ger,M.Cheneau,B.Battelier,andJ.Dalibard,Nature441,1118(2006).[14]I.B.Spielman,W.D.Phillips,and J.V.Porto,Phys.Rev.Lett.98,080404(2007).[15]D.S.Petrov,M.Holzmann,and G.V.Shlyapnikov,Phys.Rev.Lett.84,2551(2000).[16]D.S.Petrov and G.V.Shlyapnikov,Phys.Rev.A64,012706(2001).[17]J.O.Andersen,U.Al Khawaja,and H.T.C.Stoof,Phys.Rev.Lett.88,070407(2002).[18]T.P.Simula and P.B.Blakie,Phys.Rev.Lett.96,020404(2006).[19]M.Holzmann,G.Baym,J.-P.Blaizot,and lo¨e,A104,1476(2007).[20]V.Bagnato and D.Kleppner,Phys.Rev.A44,7439(1991).[21]D.R.Nelson and J.M.Kosterlitz,Phys.Rev.Lett.39,1201(1977).[22]A.Polkovnikov,E.Altman,and E.Demler,Proc.Natl.A103,6125(2006).[23]F.S.Dalfovo,L.P.Pitaevkii,S.Stringari,andS.Giorgini,Rev.Mod.Phys.71,463(1999).[24]F.Gerbier et al.,Phys.Rev.Lett.92,030405(2004).[25]D.S.Fisher and P.C.Hohenberg,Phys.Rev.B37,4936(1988).[26]N.Prokof’ev,O.Ruebenacker,and B.Svistunov,Phys.Rev.Lett.87,270402(2001).[27]Y.Kagan,B.V.Svistunov,and G.V.Shlyapnikov,Sov.Phys.JETP66,314(1987).[28]Y.Shin et al.,Phys.Rev.Lett.92,150401(2004).[29]O.J.Luiten,M.W.Reynolds,and J.T.M.Walraven,Phys.Rev.A53,381(1996).[30]By imaging along x,we veriﬁed that the proﬁles along yare alsoﬁtted well by a Gaussian.[31]Note that the slope dN0/dN∼0.7for N>N c is lessthan unity,contrary to what is expected for a saturated gas.This lack of saturation will be discussed elsewhere.[32]R.K.Bhaduri et al.,J.Phys.B:At.Mol.Opt.Phys.33,3895(2000).[33]J.P.Fern´a ndez and W.J.Mullin,J.Low Temp.Phys.128,233(2002).[34]C.Gies and D.A.W.Hutchinson,Phys.Rev.A70,043606(2004).[35]N.Prokof’ev and B.Svistunov,Phys.Rev.A66,043608(2002).。

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Installation ....................................................................................................... 17 System Mounting .............................................................................................. 17 Unpacking and Inspection .................................................................... 17 Wall Mount Preparation and Procedure ............................................... 18 Free Standing Mount Instructions ........................................................ 20 Electrical Installation......................................................................................... 21 Gas & Air Supply Installation ........................................................................... 22

超流、涡流、有限温度效应

1
Let us consider the circulation of velocity in a super ow. It follows from the relation between the velocity and the phase, Eq. (6), that the circulation around any contour C obeys I h ; = v dr = m 2 l (8) C with some integer l. We see that The circulation is quantized in multiples of h=m The ows in multi-connected geometries, such as a ring-shape tube, are discrete Quantum leaps are required to change a ow. The fact that a super ow, due to the dicreteness of circulation, cannot be dissipated gradually, but only in dicrete steps, is the origin of super uidity. The only way to eliminate a super ow is to produce excitations with discrete vorticity and then remove them (along with the vorticity) from the system. Also mention the Landau criterion for super uidity: The quasiparticle energy (k) = (k) ; v k, Doppler-shifted due to the ow, should be positive, to prevent massive production of quasiparticles. This criterion de nes a critical velocity (9) vc = min (k)=jkj k

211126683_SPME-GC-MS技术结合rOAV分析不同加工工艺紫娟白茶的关键香气物质

李沅达，吴婷，黄刚骅，等. SPME-GC-MS 技术结合rOAV 分析不同加工工艺紫娟白茶的关键香气物质[J]. 食品工业科技，2023，44（9）：324−332. doi: 10.13386/j.issn1002-0306.2022060240LI Yuanda, WU Ting, HUANG Ganghua, et al. SPME-GC-MS Technique Combined with rOAV for the Analysis of Key Aroma Substances of Zijuan White Tea with Different Processing Processes[J]. Science and Technology of Food Industry, 2023, 44(9):324−332. (in Chinese with English abstract). doi: 10.13386/j.issn1002-0306.2022060240· 分析检测 ·SPME-GC-MS 技术结合rOAV 分析不同加工工艺紫娟白茶的关键香气物质李沅达，吴　婷，黄刚骅，任　玲，马晨阳，周小慧，李亚莉，周红杰*（云南农业大学茶学院，云南昆明 650000）摘　要：为研究不同加工工艺紫娟白茶的关键香气化合物，采用顶空固相微萃取-气相色谱-质谱联用技术（Head-space solid-phase micro extraction and gas chromatography mass spectrometry ，HS-SPME-GC-MS ），结合感官审评、PCA 验证、OPLS-DA 分析以及相对香气活度值（Relative odor activity value ，rOAV ），分析筛选紫娟白茶的关键香气化合物。

结果表明：共检测出82种主要香气组分，以醇类、酯类、杂环及芳香族化合物为主，芳樟醇、水杨酸甲酯、苯甲醇、β-紫罗兰酮等相对含量较高，rOAV 法分析结果显示：1-辛烯-3-醇、α-紫罗兰酮、苯乙醛、水杨酸甲酯等9种挥发性有机化合物对自然萎凋的紫娟白茶（TZW ）花果香馥郁且带毫香、辛香、药香的香气形成具有较大贡献，1-辛烯-3-醇、α-紫罗兰酮、β-紫罗兰酮、苯甲醛等9种挥发性有机化合物对复式萎凋紫娟白茶（CZW ）果香浓郁且带花香、辛香的香气形成具有较大贡献。

伯斯电力力学5说明书

The Challenge:Using a Customized Waveform to Mimic theHuman Cartilage Mechanical EnvironmentBackgroundApproximately 40 million Americans suffer from localized damage to the cartilage and subchondral bone. This leads to pain, loss of joint function and osteoarthritis. There is a pervasive need for effective clinical treatments to repair cartilage injuries.Regenerative medicine approaches are currently investigated through the replacement of the damaged cartilage with tissue-engineered cartilage constructs. Porous scaffolds not only provide a boundary for retention of cells, but also act as a substrate to which the anchorage-dependent chondrocytes can adhere.It is known that mechanical modulation has a significant impact on cell differentiation and proliferation. Thus,applying accurate and efficient mechanical stimuli is crucial in quality control of the tissue product. This may also in turn guide diagnosis and future therapy improvement. In this study, the Bose ® ElectroForce ® 5500 test instrument (Figure 1) was used to impose a customized waveform on a hydrogel, and the changes in sample properties were monitored over time.Sinusoidal cyclic waveforms are typically used when studying relationships between cell growth and mechanical stimulation; however, there is limitedinformation on using customized waveforms. It would be beneficial to use a waveform that mimics the mechanical environment of a human knee joint while walking during the in vitro tissue-engineered cartilage development.Polyethylene glycol (PEG) hydrogel sheets (4” x 4”) were purchased from MedlineIndustries. Hydrogel specimens, punched from the hydrogel sheet, were 12 mm in diameter and 6 mm in height (Figure 2).The ElectroForce 5500 systemhas a maximum force capacityof 200 N and a maximumdisplacement of 13 mm. The system was equipped with a 200 N load cell and a pair of 25 mm diameter platens. A preloading force of 0.1 N was used to ensure that the entire scaffold surface was in contact with the compression platens prior to testing (Figure 3).External waveform is a feature of the WinTest software that offers users with the ability to run custom waveforms whenmore complex mechanicalanalysis is required. Externalwaveform allows the importation of point by point files whichdefine evenly spaced data points as a function of time.Meeting the ChallengeThe ElectroForce 5500 test instrument, in combination with WinTest ® software, is ideal for mechanical studies in biomedical research. It provides precise force and displacement control throughout the experiment. Customized waveforms can be realized by externallyimporting them into the WinTest software at which point they can be reproduced by the patented Bose linearactuator that features a frictionless moving-magnet design.Materials and MethodsFigure 2 - Hydrogel Specimen Figure 1 - Bose ® ElectroForce ®5500 Test InstrumentFigure 3 - Specimen Loaded Between Compression PlatensBose Corporation – ElectroForce Systems Group10250 Valley View Road, Suite 113, Eden Prairie, Minnesota 55344 USA Email:*********************–Website: Phone: 952-278-3070 – Fax: 952-278-3071©2014 Bose Corporation. Patent rights issued and/or pending in the United States and other countries. Bose, the Bose logo, ElectroForce and WinT est are registered trademarks of Bose Corporation. 063014In order to create a WinTest ® software readable point by point file, the following steps were used:The waveform was extracted and replotted in Excel.According to normal human walking speed of 5 km/h, a one gait cycle time of 1.1 sec was obtained and used as a new X axis (Figure 5).An ASCII file was constructed by the text editor. The Y axis strain points were scaled according to the specimenthickness and used for the ASCII file. The above waveform contained 1100 points, so the time interval between points was set to 0.001 sec to match with the gait cycle time.Each pass through the waveform = 1100 x 0.001 = 1.1 sec. The ASCII file was imported into WinTest software. A point by point file was exported and used for the test setup.The same external waveform was successfully applied to all the specimens (Figure 6). Similar testing results of three specimens were achieved and reliable repeatability of this testing method was demonstrated (Displacement difference between samples: <2.5%; Load difference between samples:<13.9%). Compared to the original extracted waveform in the ASCII file, the majority of the waveform details were retained accurately.The Bose ® ElectroForce ® 5500 test instrument is a powerful tool, which is not only capable of generating sinusoidal, triangle, square, ramp and block waveforms, but also excels in precise waveform customization. Combined with easy to use WinTest software, the ElectroForce 5500 testinstrument is able to deliver waveform profiles that fit the needs of a particular experiment and gives researchers the ability to implement their ideas.SummaryFigure 4 - Strain at the Contact of Joint Cartilage during the Gait Cycle (Halonen et al., 2013)A waveform model (Figure 4, pink line) of strain vs. gait cycle based on simulation of human walking was used inthis study (Halonen, et al., 2013).Figure 5 - Extracted WaveformResultsThree specimens were tested with the externalwaveform, and displacement and load data were tracked during the test.(1) Halonen, K.S., M. E. Mononen, J. S. Jurvelin, J. Töyräs, and R.K. Korhonen. “Importance of depth-wise distibution of collagen and proteoglycans in articular cartilage - a 3D finite elment study of stresses and strains in the human knee joint.” Journal of Biomechanics (2013).ReferenceFigure 6 - Average Result of Three Specimens。

耦合的凝聚态Bose-Einstein方程的双周期解(英文)

耦合的凝聚态Bose-Einstein方程的双周期解(英文)
王军霞;刘安平;郭艳凤
【期刊名称】《应用数学》
【年(卷),期】2011(24)3
【摘要】本文通过引入参数假设,利用雅可比椭圆函数展开法,得到了自散焦的耦合非线性Schrdinger(NLS)方程的四种双周期解(雅可比椭圆函数).
【总页数】7页(P493-499)
【关键词】Bose—Einstein凝聚;双周期;雅可比椭圆函数
【作者】王军霞;刘安平;郭艳凤
【作者单位】中国地质大学教理学院,湖北武汉430074;中国地质大学环境学院,湖北武汉430074;广西工学院,广西柳州545006
【正文语种】中文
【中图分类】O175.29
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1.Kupershmidt方程与耦合KdV方程组的椭圆周期解 [J], 田贵辰;曹志军
2.一类耦合KdV方程的孤波解和周期波解及其相互关系 [J], 张卫国;徐伟;李想
3.非线性脉冲Fredholm积分方程的耦合拟解及解(英文) [J], 翟成波;杨晨
4.关于带有双侧共振的非线性微分方程周期解的一点注记(英文) [J], 董玉君;姜广军
5.Bose-Einstein凝聚态中的一维Gross-Pitaevskii方程的包络周期解和孤立波解[J], 高斌;刘式适;付遵涛;刘式达
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一种利用混沌测量低频频率的新方法

一种利用混沌测量低频频率的新方法
赵向阳;刘崇新;刘君华;李卫平
【期刊名称】《数据采集与处理》
【年(卷),期】2002(017)002
【摘要】非线性Duffing方程在从混沌状态到大周期转变时,表现出良好的等Q共振锁频特性.本文研究了在不同谐振频率下,系统从混沌状态到大周期状态时驱动力信号的幅值及频率条件,指出不同的驱动力幅值,存在着相应的锁频范围.当驱动力幅值大于阈值的余度越大,则相应的锁频范围越宽.进而通过虚拟仪器的实现方式,将被测信号的幅值做不同程度的归一化,利用混沌阵列进行了精密测量频率.实验测试表明,该方法可达到较高的精度,不足之处在于由于计算机要完成不同程度的归一化,同时需要人工判别相图是处于混沌还是大周期状态,所以需要测试时间较长.由此可知该方法特别适合于低频、单次信号的高精度测频.
【总页数】4页(P183-186)
【作者】赵向阳;刘崇新;刘君华;李卫平
【作者单位】西安交通大学电气工程学院,西安,710049;西安交通大学电气工程学院,西安,710049;西安交通大学电气工程学院,西安,710049;西安交通大学电气工程学院,西安,710049
【正文语种】中文
【中图分类】TN911.7
【相关文献】
1.一种低频下测量电滞回线的新方法 [J], 张孝林;行朝至
2.基于DTFT的一种低频振动信号相位差测量新方法 [J], 张海涛;涂亚庆
3.强噪声下利用混沌系统测量频率的新方法 [J], 聂春燕;石要武;衣文索;王有维
4.一种直接测量压电换能器串联谐振频率和并联谐振频率的新方法 [J], 鲍善惠
5.一种测量超低频绝对振动的新方法 [J], 余水宝
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高触发信号和功率合成的高峰值功率皮秒脉冲源

高触发信号和功率合成的高峰值功率皮秒脉冲源
张雅茹;陈袭;李杨;杨宏春;魏召唤
【期刊名称】《强激光与粒子束》
【年(卷),期】2022(34)6
【摘要】对于目标的攻击、干扰和探测,超宽带时域脉冲源的幅值直接影响其攻击、干扰和探测的强度和效果。

基于雪崩晶体管的Marx电路被广泛应用在产生此类信号源上,传统的Marx电路可以一定程度上提高输出电压的幅值,但由于雪崩晶体管
功率容量较低等原因,雪崩晶体管的Marx电路输出电压幅度会随级数增加而达到
饱和。

针对此类问题,为了产生更高幅值的脉冲信号,综合采用提高触发信号和使用
宽带功率合成器的手段。

最终利用26级Marx电路作为触发信号,4路40级
Marx电路进行功率合成的方法,实现了输出电压幅值为8.7 kV、上升沿约为180 ps的技术指标,并通过机理分析了高触发信号对雪崩晶体管Marx电路的影响,通过实验得到了印证。

【总页数】8页(P118-125)
【作者】张雅茹;陈袭;李杨;杨宏春;魏召唤
【作者单位】电子科技大学物理学院
【正文语种】中文
【中图分类】TN782
【相关文献】
1.皮秒脉冲泵浦的高功率超连续谱光源
2.DSRD高功率超宽谱脉冲源及其功率合成初探
3.由内腔倍频直接获取高功率皮秒准单脉冲
4.紧凑型高峰值功率高光束质量百皮秒激光器技术
5.高功率线偏振皮秒脉冲簇Yb光纤激光器
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