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Quantum limits to center-of-mass measurements Timothy Vaughan∗ and Peter Drummond Australian Centre of Quantum-Atom Optics, University of Queensland, St Lucia, Queensland, Australia. Gerd Leuchs Institut fr Optik, Information und Photonik, Max-Planck Forschungsgruppe, Universitt Erlangen-Nrnberg, Gnther-Scharowsky-Strae 1/Bau 24, 91058 Erlangen, Germany 7 (Dated: February 6, 2008) 0 0 We discuss the issue of measuring the mean position (center-of-mass) of a group of bosonic or 2 fermionicquantumparticles,includingparticlenumberfluctuations. Weintroduceastandardquan- n tum limit for these measurements at ultra-low temperatures, and discuss this limit in the context a of both photons and ultra-cold atoms. In the case of fermions, we present evidence that the Pauli J exclusion principle has a strongly beneficial effect, giving rise to a 1/N scaling in the position 2 standard-deviation – as opposed to a 1/√N scaling for bosons. The difference between the ac- tualmean-position fluctuation and thislimit is evidencefor quantumwave-packet spreading in the ] center-of-mass. This macroscopic quantum effect cannot be readily observed for non-interacting r e particles,duetoclassical pulsebroadening. Forthisreason,wealsostudytheevolutionofphotonic h and matter-wave solitons, where classical dispersion is suppressed. In the photonic case, we show t thattheintrinsicquantumdiffusionofthemeanpositioncancontributesignificantlytouncertainties o insolitonpulsearrivaltimes. Wealsodiscusswaysinwhichtherelativelylonglifetimesofattractive . t bosonsin matter-wavesolitons may beusedtodemonstrate quantuminterference between massive a objects composed of thousands of particles. m - PACSnumbers: 03.75.Pp,42.50.Dv d n o I. INTRODUCTION Peierls[2], free particle momentum is a QND observable. c Infact,themomentumofaquantumsolitoncanbe non- [ destructivelymeasuredviathepositionofaprobesoliton 2 The topic of mesoscopic quantum effects is of great aftertheircollision[3]. Thequantumpositionfluctuation v current interest, especially in photonic or in ultra-cold increaseswithpropagationdistancedue toaninitialmo- 1 atomic systems where a large decoupling from the envi- mentumuncertainty. This placesafundamentallimiton 0 ronment is possible. An important degree of freedom is 2 momentum QND measurements. the average or center-of-mass position of a large num- 6 ber of particles. It is now possible to observe quantum We generalize the concept of the standard quantum 0 6 diffractionofthecenter-of-massinmoleculeslikeC60 and limit of a mean position measurement [4, 5] to any kind 0 related fullerenes[1]. Larger physical systems would pro- ofquantumfield. Thestandardquantumlimitis defined / vide an even a stronger test of quantum theoretical pre- hereas the positionuncertaintyofamany-bodyground- t a dictions. Aswellastestingquantumtheoryinnew,meso- state for non-interacting particles with a given density m scopic regimes, these types of experiment have potential distributionandtotalmomentum. Formassiveparticles, - applications to novel sensors. For example, ultra-precise thiscorrespondstotheHeisenberg-limiteduncertaintyof d measurements of position may be useful in measuring thecenter-of-massofagroupofnon-interactingparticles n gravitational interactions, or other forces that couple to in an external potential, at zero temperature. Intrigu- o c conserved quantities. ingly, we find completely different scaling laws for dif- : ferent particle statistics: while bosons have a position v In this paper, we wish to examine the quantum limits uncertainty (standard deviation) that scales as 1/√N, i to center-of-mass position fluctuations. We further dis- X fermionic center-of-mass uncertainties scale as 1/N for cuss these limits in the context of mesoscopic systems of N particles. This is caused by the particle correlations r interacting particles, atoms or molecules. This is an im- a induced by Fermi statistics. The scaling indicates that portant issue in quantum-optical or ultra-cold atom en- ultra-cold fermions are the preferred system for ultra- vironments, where center-of-mass motion places a limit sensitive measurements of center-of mass. on coherence properties of photonic or atom lasers. A localized system of particles features a dual particle and Anecessaryrequirementforquantum-limitedmeasure- wave nature, that is, it has conjugate observables of mo- mentsisanextremelylowleveloffluctuationsandnoise, mentum and position, as well as conjugate observables which makes laser pulses and ultra-cold atoms the most of number and phase. As pointed out by Landau and reasonable choices at present. As an example, the localized solitons of the attractive one-dimensional Bose gas can now be observed both in quantum-optical and in atom-optical environments. As well as technologi- ∗Electronicaddress: [email protected] cal applications in communications, these types of soli- 2 ton show intrinsic quantum effects. That is, the ef- to the conserved quantities fectsofquantumphasediffusion(squeezing)[4,6,7]have been observed in experiments[8, 9, 10]. Quantum cor- J = ˆj (x)dDx . (1) µ µ relations established by soliton collision (quantum non- Z demolitionmeasurement)[3]havealsobeendemonstrated Inaquantumstatebthatisaneigenstateoftheconserved in experiment[11]. Even though they involve up to 109 charge with J > 0, defined on a D- dimensional space, particles, these effects simply do not occur in classical 0 the definition of the centroid is: soliton theory. Phasesqueezingorphase QNDmeasurementdoesnot 1 xˆ = xˆj (x)dDx . (2) lead to center-of-mass changes, which are especially in- J J 0 0 Z teresting for massive particles, as this degree of freedom is coupled directly to the gravity field. Due to prob- After partial integration, the conservation law, lems in quantizing gravity, there have been suggestions that gravitational effects may lead to wave-packet col- ∂j0 + ˆj=0 , (3) lapse and/or dissipation[12, 13, 14, 16, 17]. While this ∂t ∇· remainsspeculative,itis clearlyanareaofquantumme- b then implies that the centroid position translates with a chanicswheretherearenoexistingtests. Toexcludesuch velocity given by the spatial part of the conserved cur- theories, one would need to violate a classical inequality rent: involvingmesoscopicnumbersofmassiveparticles,whose center-of-mass was in some type of quantum superposi- tion state[18]. In real experiments, weak localization[19] ∂xˆ Jˆ J cantake place due to interactionswith the environment. = . (4) ∂t J 0 Itisatleastinterestingtoinvestigatehowthismayoccur, asafirststeptowardstestingmorefundamentaltypesof While the concept of measuring the center of mass localization. We note that steps in this direction have position is well-known, there are a number of issues in- been taken recently for non-interacting, massless parti- volved. The most obvious is that there is a difference cles, via positionobservationswith photons, atthe stan- between the center-of-mass and the average position of dard quantum limit or better[20, 21]. a group of non-identical particles. This difference van- Quantum wave-packetspreading of the center-of-mass ishes for non-relativistic particles all of the same mass: is an intrinsic quantum prediction for untrapped parti- for example, isotopically pure, ultra-cold atoms. How- cles at all particle numbers. However, this is masked ever, there is a real difference in the case of particles of by the single-particle effects analogous to classical pulse different masses, and for extremely relativistic particles broadening for an optical pulse in a linear dispersive and photons, where the effective mass depends on the medium. Asolitonorquantumbound-statecansuppress momentum. Here the average position differs from the this single-particle or classical pulse broadening, and al- center-of-mass. Anothersubtletywhichthispapertreats lows the intrinsic center-of-mass quantum spreading to in some detail, occurs when the system is in a quantum be observed [5], in a similar way to the known quantum mixture of different particle numbers: the effects of this noiseeffectsfoundinamplifiedsolitons[23]. Wefindthat are treated in the remainder of this section. thisisanexactlysolubleproblemforany initialquantum state. No linearization or decorrelation approximation is needed, which allows us to examine quantum wave- A. Massive fields packet spreading over arbitrary distance scales. In order to distinguish classical pulse broadening from quan- We first consider non-relativistic massive quantum tum wave-packet spreading, we calculate the standard fields Ψˆ (x), where each component has an equal i quantumlimitofthismeasurementforaninitiallycoher- mass. The fundamental particle statistics can be either ent soliton pulse. This requires knowledge of the time- fermionic or bosonic. As usual for quantum fields, the evolution of the density distribution for an interacting commutators are: many-body system, which is nontrivial, and requires numerical solutions. Ψˆ (t,x),Ψˆ†(t,x′) =δ δ3(x x′). (5) i j ± ij − h i Introducing the particle density II. CENTER OF MASS OPERATORS nˆ(x)= nˆ (x)= Ψ†(x)Ψˆ (x), (6) i i i We startby notingthatwhile a centroidexistsforany Xi Xi localized, measurable quantity, it is most interesting for we can define the total particle number a conserved quantity. For any physical system with a conservedcurrentfour-densityj =(j j),itis possibleto 0, Nˆ = dDxnˆ(x), (7) define a center-of-charge (or mass, energy, etc) relative Z b 3 which is exactly conserved in the absence of dissipative that: processes like absorptionor amplification. In this paper, we will consider cases where the system has an uncer- ˆ = Eˆ +iBˆ E tainty in Nˆ, but we will exclude states with zero parti- = ˆ ∇×A cle number. In terms of operational measurements with = ˆ + ˆ . + − small particle numbers, this may require post-selection E E toexclude statesofthis type -since ofcoursethe center- ItiseasilycheckedfromthecomplexformofMaxwell’s of-mass is undefined in such cases. equationsthati∂ˆ /∂t= ˆ andi∂ ˆ /∂t= ˆ . σ σ σ σ In order to define center-of-mass or mean position for The photon densiEty is then∇:×E A ∇×A systems with particle number fluctuations, several defi- σ nitions are possible, depending on measurement proce- nˆ(x)= nˆ (x)= : ˆ† ˆ + ˆ ˆ† : , (9) dures. There is more than one measurement procedure σ 4 Eσ·Aσ Eσ·Aσ possible, since the quantum state need not be an eigen- Xσ (cid:16) (cid:17) state of the number operator. Hence, different number with a corresponding conserved current of : states could have an arbitrary relative weighting. In the absence of externalpotentials, systems of inter- σ acting particles also have an invariant quantity due to J = 4i : Eˆσ† ×Aˆσ +Aˆ†σ×Eˆσ . (10) translational invariance, which is the total momentum Xσ (cid:16) (cid:17) (in D dimensions): b We note that with these definitions, the photon density is not positive definite in small volumes and time- ~ Pˆ = Ψˆ†(x) Ψˆ (x) Ψˆ†(x)Ψˆ (x) dDx (8) intervals. However, the total photon number N is well- 2i i ∇ i −∇ i i Z h i definedandpositivedefinite,sothatapositioncentroidis well-defined for this conservedquantity. For thbe remain- Here we use the Einstein summation convention for re- derofthispaperwewillfocusonnon-relativisticmassive peatedindices. We useacapitallettertoemphasizethat particles for definiteness, while noting that many results this is anextensive quantity,proportionaltothe number thatonly dependonthe existence ofa conservedcurrent of particles. will hold in the general case of a photon field. B. Massless fields C. Intensive position Masslessfields-photons-arethemostcommonlyused For quantum states that are eigenstates of number particles where measurements are made at the quantum with N >0, the obvious definition is: level. In this case, we can distinguish two different important cases. xnˆ(x)dDx xˆ(N) = (11) N R Narrowbandfieldsinone-dimensionalwave-guides • Thisisundefinedforthevacuumstate,butiswell-defined or fibers experience dispersion, which causes them for all other number eigenstates. For the non-relativistic tobehaveasmassiveparticles. Inasimilarway,the massive particle case, this obeys the expected commuta- paraxialapproximationforbeamsallowsdiffractive tion relation with the total momentum operator: behaviorto be treatedin aneffective massapprox- imation. These cases can be handled in the same way as the treatment given above. xˆ(iN),Pˆj =i~δij. ± h i More fundamental issues arise in free-space, where Thepresenceofthevacuuminstatescomposedofmix- • there is a long-standing fundamental problem of tures or superpositions of number eigenstates introduces howtodefinephotonposition. Inaddition,asthere therequirementofdecidingwhichrelativeweightstouse is no photon mass, even the idea of center-of-mass for different particle numbers. The important fact to requirescare. Instead,onemusteitherusethecon- note is that the mean position of any vacuum compo- ceptofa center-of-energy,or ofa center ofphoton- nent is undefined and, as such, its presence does not number. We shallfocus onthe latter concept here. contribute to one’s knowledge of the mean position. Ac- cordingly,as mentionedpreviously,we chooseto workin To treat the free-space average position, we can use a arestrictedHilbertspacewhichdoesnotincludethevac- conservedlocaldensityobtainedfromthedualsymmetry uum state. Practically, this is equivalent to performing properties of Maxwell equations[22]. We introduce ˆ , heralded particle number measurements where any null σ E where σ = 1, which are the helicity components of the results are discarded. From a mathematical perspective, complex Ma±xwell fields in units where c = ε = µ = 1, we work with a projected density operator ρ′ = ˆρˆ, 0 0 andthe correspondingcomplex vectorpotentials ˆ , so where ˆ = (Iˆ 0 0). In order to extract posPitioPn σ A P − | ih | 4 information from linear superpositions or mixtures of E. Quasi-intensive position operator number states, we introduce the center-of-mass position operator[40]whichiswelldefinedinourprojectedHilbert We will finally define a quasi-intensive position opera- space: tor,whichcorrespondstothetypicaldirectmeasurement xnˆ(x)dDx procedures in optics. With this operator, the position is xˆ = (12) first measured with techniques that are extensive - but Nˆ R thefinalresultisnormalizedbytheaverageparticlenum- This intensive quantity also obeys the expected commu- ber. Thishasthepropertythatinanensemblewithvari- tation relation with Pˆ: able numbers of particles, more weight is given to those measurements that involve the most particles. Here: xˆ ,Pˆ =i~δ (13) i j ij ± h i Xˆ Thus, provided any measurement of the vacuum is dis- Xˆ = (21) carded,xˆ and Pˆ form a pair of canonicalconjugate vari- Nˆ h i ables with the dimensionless uncertainty relation whichcanstillbeconsideredausefulmeasureofthemean ~2 position of a particle distribution, especially in the limit h∆xˆ2iih∆Pî2i≥ 4 (14) of large Nˆ where total number fluctuations are sup- h i pressed. Indeed, this is the variable that corresponds for states projected into our restricted Hilbert space. most closely to some earlier proposed operational mea- suresofpulse position,measuredusinglocaloscillatoror homodyne techniques. It is interesting to note that the D. Extensive position conjugate variable, the mean momentum, is the same as for an extensive position. This is analogous to the Although xˆ has the usual definition of the mean po- mean frequency of a laser pulse in a dispersive medium, sition, it’s operational denominator makes it is difficult which also corresponds closely to standard operational to measure using many standard techniques, or even to measurements used in laser physics. represent in a normally ordered form. We can also in- In order to measure fluctuations about Xˆ or Xˆ , troduce an extensive position operator, which produces h i h i however,caremustbetakenduetothefactthatXˆ trans- a result proportional to the number of particles as well forms in the following way under translation: as their position. We define: Xˆ =xˆNˆ, (15) Nˆ Xˆ′ =Xˆ +∆x . (22) Nˆ which is well-defined without projection in the Hilbert h i space. However,Xˆ andPˆ donotformapairofcanonical By considering fluctuations about x′ =0 in a new refer- conjugate variables, since ence frame where x′ =x X , we define [Xˆ ,Pˆ ] =i~δ Nˆ. (16) −h i i j ± ij Nˆ Through the uncertainty relation, we see that Xˆ is sen- ∆Xˆ =Xˆ Xˆ (23) −h i Nˆ sitive not only to positional fluctuations but also fluctu- h i ations in the total particle number or beam intensity: and thus minimize the number fluctuation contribution. ~2 Nˆ2 FollowingfromEq.17,thisobeystheuncertaintyrelation ∆Xˆ2 ∆Pˆ2 h i. (17) h iih i i≥ 4 ~2 Nˆ2 Furthermore, as an extensive operator, Xˆ transforms in h∆Xˆ2iih∆Pîi≥ 4hNˆ 2i the following way. In a new reference frame S′ where h i x′ =x+∆x, we find that asthe totalnumberandmomentumoperatorscommute. Inordertoavoidcomplicationsarisingfromthisissue, Xˆ′ =Xˆ +∆xNˆ (18) we will only consider isolated systems where it is always rather than just being translated by a c-number. possibletofindaninertialreferenceframeinwhich Xˆ = h i Wenotethattheextensivepositiondoes haveacanon- Pˆ =0. h i icalconjugate partner,in terms ofthe intensive or mean momentum operator: III. QUANTUM LIMITS ON POSITION Pˆ UNCERTAINTIES pˆ = (19) Nˆ with a corresponding commutator of: Intrinsic quantum uncertainty in the results of projec- tivemeasurementsisperhapsthemoststrikingdifference [Xî,pˆj]± =i~δij (20) between quantum mechanical observables and classical 5 variables. Here we seek to characterize this uncertainty 1. Number states in the center-of-mass (COM) measurement by introducing a standard quantum limit to the variance in the dis- Zerotemperaturebosonicnumberstatescanexpressed tribution of results. This is not a lower bound - there is in the following way: none - but rather a natural limit that one can expect to achieveusingstandardcoolingand/orstabilizationmeth- 1 ods. The question of exactly which state to use to cal- N = (aˆ†)N 0 . (28) culate such a characteristic COM is difficult to answer | i √N | i uniformly, especially as the states accessible to bosonic Evaluating the COM variance of this state using the systemsaredifferenttothoseaccessibletofermionicsys- above decomposition is straightforward as the expecta- tems. tion value of the normally ordered term : ∆xˆ 2 : van- For a system with known density distribution and ex- | | ishes, leaving change statistics, we define the standard quantum limit to the COM uncertainty to be the variance remaining σ2 when a non-interacting system of the same particles is ∆xˆ 2 = (29) h| | i N reducedtozerotemperatureinanexternalpotentialthat reproducesthegivendensitydistribution. Itisimportant where σ2 = σˆ2 . h i to note that this definition makes no restriction on the statistics of the characteristic system. However, we will find that the statistics do have a large effect on the way 2. Coherent states that the quantum limit scales for a given density. In the following analysis, we consider the variance in For bosons, the first term also vanishes for any co- the intensive mean position operator, in the form: herent state of the field operators, given our co-moving reference frame. Here we define a coherent state using 1 ∆xˆ 2 = dDxdDy ∆x ∆ynˆ(x)nˆ(y) . (24) a projection, as explained earlier,which projects out the | | Nˆ2 { · } vacuum state in order to correspond to a heralded or Z post-selected measurement. Such states can be formally Re-arranging this expression by using normal-ordering defined as: and commutators gives the result that: 1 α = e|α|2 1 −1/2 ∞ αa†χ n 0 . (30) ∆xˆ 2 = Nˆσˆ2+: ∆Xˆ 2 : (25) | iP − n! | i | | Nˆ2 | | h i nX=1(cid:2) b (cid:3) (cid:16) (cid:17) This state can be viewed as a suitable reference state where we define the wavepacketvariance operator forquantumnoise,inwhichtheuncertaintyinaposition 1 measurement is governed entirely by the spread in the σˆ2 = dDx ∆x2nˆ(x) (26) particle distribution, χ(x)2. After normalizing by the Nˆ | | | | Z n o particle number, we find and the normally ordered COM variance 1 ∆xˆ 2 =σ2 . (31) h| | i Nˆ : ∆Xˆ : = dDxdDy∆x ∆y (cid:28) (cid:29) | | · Z This is different from the following result, obtained by Ψˆ†(y)Ψˆ†(x)Ψˆ (x)Ψˆ (y). (27) considering the variance in the quasi mean position op- × ij i j j i erator Xˆ which for the coherent state above is given by X Note that the exactoperatororderingin the last expres- ∆Xˆ 2 = σ2 . (32) sionisimportantifthedecompositionistobecorrectfor h| | i Nˆ both Fermi and Bose field operators. h i From the graphical comparison shown as Fig. 1 it is clear that the standard quantum noise limit for the extensive(quasimean)positionmeasurementislowerthan A. Bosonic fields that for the intensive position measurement. This can be understood as a consequence of the fact that an ex- The simplest configuration for a system of degenerate tensive position measurement for a coherent state effec- bosons has all particles described by a single normalized tively weights all individual particle position measure- mode function χ(x). Thus, the characteristicstates con- ments equally. This is advantageous for a coherent state sidered here are formed using functions of the creation - as in a laser pulse - where all particles arrive indepen- operator a†χ = dDx χ(x)Ψˆ†(x) . dently,andcarrythesameinformation. However,notall R n o b 6 where Ψ(N) = N−1 aˆ† (1−δjk) 0 . Taking the −j k=0 k | i inner p(cid:12)(cid:12)roducEt of th(cid:18)iQs result(cid:16)wi(cid:17)th it’s c(cid:19)onjugate leads to (cid:12) N−1 1 σ2 = dDx ∆x2 χ (x) . (36) j N | | | | Xj=0 Z n o Toevaluatethesecondtermintheexpectationvalueof Eq.25weconsidertheactionoftwoFermifieldoperators on the system state vector: N Ψˆ(y)Ψˆ(x) Ψ(N) = ( 1)j+k−θ(k−j)(1 δ ) jk − − (cid:12) E jXk=1 (cid:12) Fvaigriuarnec1e:,σD2e)voiafttihoneqofutahsieivnatreinasnicvee(CreOlaMtivceootordtihneatwea(vdeapsahcekde)t (cid:12) ×χj(x)χk(y) Ψ(N) (37) fromthetrueintensiveCOMcoordinate(solid)foraheralded (cid:12) E Again taking the norm of the result, w(cid:12)e find: coherent state. (cid:12) N : X2 : = dDxdDy x y[χ (x)2 χ (y)2 pulses have the same number of independent particles. h | | i { · | j | | k | An intensive measurement weights all collective parti- jXk=1Z cle position measurements equally, regardless of particle −χ∗j(x)χk(x)χ∗k(y)χj(y)]} (38) number. Thisgiveslessweightperparticlemeasurement Up until now, the single particle basis χ (x) has been whenthe particlenumber is large,andhence is lessthan j arbitrary. Fordefiniteness,wewillassumethatthe func- optimal. tions χ (x) = xj are energy eigenstates of a simple j h | i harmonicoscillatorof mass m andfrequency ω. Thatis, B. Fermionic fields N−1 1 σ2 = dDx x2 j x xj N {| | h | ih | i} IdentifyingacharacteristicCOMuncertaintyforacold j=0 Z X Fermi gas is a natural extensionto the above discussion. N−1 1 We thereforeconsidera spin-polarizedgasofN identical = j qˆ2 j , N h | | i fermions at zero temperature. In other words, we con- j=0 X siderasysteminwhichallmodesbelowthecharacteristic where we have also defined the first quantized position Fermi energy contain a single fermion each, whilst those operatorqˆ =(qˆ ,qˆ ,qˆ )suchthatqˆ x =x x . Recall- abovethisenergyarevacant. UsingtheHartree-Fockas- x y z µ µ | i | i ing that this operator can also be expressed in terms of sumptionofwavefunctionfactorization,thisstatecanbe raising (αˆ†) and lowering (αˆ ) operators,we find written µ µ N−1 |Ψ(N)i=j=0 aˆ†j|0i (33) σ2 = N1 2m~ω N−1 D (hj|αˆµαˆ†µ|ji+hj|αˆ†µαˆµ|ji Y (cid:18) (cid:19) j=0 µ=1   X X where the fermion creation operator for the j-th free- and hence the variance due to the wavepacket extent is particle energy level is defined as in this case D~ aˆ† = dDxχ (x)Ψˆ†(x) (34) σ2 = N. (39) j j 2mω Z ThenormallyorderedcomponentoftheCOMvariance and χ (x) is an orthonormal set of mode functions (en- j canbe evaluated using a similar approach. The first line ergy eigenstate basis). inEq.38vanishesduetotheoddparityoftheintegrand. We now calculate the expectation value of the mean The remaining term gives position variance operator given by Eq. 25. For the first term in this equation, we employ the commutation rela- N−1 D tion Ψˆ(x),aˆ†j + =χj(x) to show that h:|Xˆ|2 :i = − hj|qˆµ|kihk|qˆµ|ji jk=0µ=1 h i X X N−1 D~ N−1 Ψˆ(x) Ψ(N) = ( 1)jχ (x) Ψ(N) (35) = ((j+1)δ +jδ ) − j −j −2mω j+1,k j−1,k (cid:12) E Xj=0 (cid:12) E jXk=0 (cid:12) (cid:12) (cid:12) (cid:12) 7 which after combining the two terms in brackets leaves wheremcanbeeitherthemassofanatomortheeffective mass of a polariton, as discussed in section IIA above. D~ : Xˆ 2 : = N(N 1). (40) For extreme relativistic or massless photon propagation, h | | i −2mω − the lack of a longitudinal dispersion mechanism implies that wave-packet spreading is analogous to diffraction, Taking into account spin degeneracy of an S-component and occurs transversely to the propagation direction. Fermi gas, which gives rise to S independent measure- Thismustbetreatedusingtheconservednumberdensity ments each with N/S particles, the COM variance for current operator of Eq. (10). an N-particle Fermi gas at zero temperature held in an Thisresultcanbegeneralizedtootherstates,provided harmonic trap is that the total number is conservedduring the evolution. Sσ2 Underthiscondition,theHeisenbergpictureevolutionof ∆xˆ 2 = . (41) h| | i N2 theintensiveandquasi-intensivemeanpositionoperators are respectively Thecontributionfromthenormallyorderedtermvan- ishesforN =1andislessthanzeroforallN >1. Asthis Pˆ term always vanishes for a bosonic number state, while xˆ(t)=xˆ(0)+t , and (43) mNˆ thewavepacketvarianceisidenticalforabosonicgascon- strained to the same density profile as the fermionic gas we have just considered (i.e. nˆ(x) = N χ (x)2), Pˆ h i j=1| j | Xˆ(t)=Xˆ(0)+t . (44) it is clear that the standard quantum limit for COM m Nˆ measurements is intrinsically lower for ferPmions than for h i bosons. In the case of a harmonic trap, the variance is Inthepulseframe,takingthesquaresgivestheevolution N times smaller for fermions than for bosons. ofthe(trueintensiveandquasiintensive)meanposition: t ∆xˆ(t)2 = xˆ(0)2 + (xˆ(0) Pˆ +Pˆ xˆ(0))/Nˆ IV. QUANTUM WAVE PACKET SPREADING h| | i h| | i mh · · i t2 + Pˆ 2/Nˆ2 , and (45) Inpurestateevolution,anyincreaseintheuncertainty m2h| | i above thestandardquantumlimitisregardedasoriginat- ing in quantum wave-packet spreading of the center-of- t ∆Xˆ(t)2 = Xˆ(0)2 + Xˆ(0) Pˆ +Pˆ Xˆ(0) / Nˆ mass. We thus define h| | i h| | i mh · · i h i t2 σ = ∆X2 σ2 (42) + Pˆ 2 / Nˆ 2. (46) QM h i− SQL m2h| | i h i q to measure this phenomenon. We note that states for whichσQM 0 arethe “strangequantumstates”which A. Bosonic coherent state evolution ≫ are mentioned in the photonic case by some previous workers[5],andalsoplayaroleinthe theoryofquantum When the system is initially prepared in a coherent non-demolition position measurements and gravity-wave state(projectedtoremovethevacuumstate,asinEq.30) detection. This must be carefullydistinguishedfromun- the mean position uncertainties evolve as certaintiesinmixedstates,whicharenotcausedbyquan- tum superpositions. 1 ~t ~t 2 A similar property to quantum wave-packetspreading ∆Xˆ(t)2 = α+β +γ (47) is already known, and observed experimentally. In the h| | iP hNîP " m (cid:18)m(cid:19) # Gordon-Haus effect[23] in amplifying optical fibers, the where α,β,γ are functions of the initial wave-packet soliton time-of-arrival is perturbed by the effect of laser shape only. These are given by: amplification. Soliton timing jitter of this type is an ex- trinsic property of the amplifying system. Instead, we willtreatthe intrinsicquantumeffects whicharepresent α = x2 χ(x)2dDx | | | | even in the absence of the spontaneous emission noise of Z a laser amplifier. β = i x [( χ∗(x))χ(x) χ∗(x)( χ(x))]dDx TosolveforthetimeevolutionoftheCOMuncertainty · ∇ − ∇ Z ofarbitrarystates,wefirstnotethatEq.4canbedirectly γ = χ(x)2dDx integratedtogivethefollowingresult,forany(effectively |∇ | Z non-relativistic) system prepared in an eigenstate of the where χ(x) is again the normalized spatial mode in the total particle number operator: co-moving frame. Pˆ This result agrees with previous approximate lin- xˆN(t)=xˆN(0)+t . earized results obtained for initial coherent solitons[4], mN 8 but is much more general. It is valid for any initial co- A. Effective field theory herentstate,atlargephotonnumber,independentofthe nonlinearinteraction,duetothefactthatthecoefficients Our starting point is the standard Hamiltonian oper- depend only on the initial coherent state. If we recall ator describing a 1D ensemble of spinless bosons which that the Hamiltonian corresponds to coupled propaga- interact through a simple delta-function potential. In tion of massive bosons, then the reason for this exact terms of a dimensionless particle density amplitude, φˆ, independence of the coupling is transparent. The defini- the Hamiltonian can be written: tion of Xˆ corresponds to a center-of-mass measurement, 1 which never depends on the two-body coupling. How- Hˆ = dζ φˆ†( ∂2)φˆ φˆ†2φˆ2 (49) ever, the standard quantum limit does depend on the 2 ζ − ζ ζ ± ζ ζ Z h i nonlinearity. This is because the coupling changes the where the field operators have the commutation relation pulse-shape Ψˆ†(x)Ψˆ(x) , as an initially coherent pulse h i with uncorrelatedbosons evolves into a correlated state. 1 [φˆ (τ),φˆ† (τ)]= δ(ζ ζ′). (50) Thus, for example, if a 1D coherent source produces ζ ζ′ n¯ − a sech input pulse, so that χ(x) = sech(x/2)/2, then Heisenberg’s equations of motion for the field operators α= π2/3 m2;β =0; γ =1/12 m−2. This is not a mini- are then mum uncertainty state, as: i∂ φˆ =n¯[φˆ ,Hˆ]. (51) τ ζ ζ ∆Xˆ(0)2 P ∆Pˆ2 c =π2/36=0.27415..>0.25. (48) These equations simultaneously describe at least two h i h i physically distinct many-body systems, the details of whichareimplicitinthecharacteristictimet =t/τ and CoherentGaussianinputpulses,ontheotherhand,are a 0 length x = x/ζ which link the physical and dimension- 0 minimum uncertainty state in momentum and position, lesscoordinatesystems. Togetherwiththemeanparticle as one might expect. However, these have a large con- number n¯ and the sign of the nonlinearity, these scaling tinuum radiationwhen used to form solitons. This leads factors completely specify a particular system. to a paradox: the sech-type coherentsoliton has a larger Loss (or gain) mechanisms can be formally incorpo- uncertainty product than a coherentGaussianpulse, yet rated into this model by including terms in the Hamil- experimentally appears more localized! tonian which couple the system to external degrees of More complex input states than coherent states can freedom. After tracing over these reservoir states, this be considered, and they also can be treated exactly. For procedure leads to a master equation for the reduced example,Haus andLai[4]haveconsideredaninput state density operatorin generalizedLindblad form, which in- ofspatiallycorrelatedphotons,consistingofasuperposi- cludes c-number damping coefficients. Although these tionofeigenstatesoftheinteracting Hamiltonian. Inthis are frequently added in an ad hoc fashion, it is impor- case,themeanpulse-shapeisasechpulse. However,due tantto note that indoing so one implicitly assumes that toboson-bosoncorrelations,theresultingstateisamini- the reservoirisatverylowtemperature andhasno ther- mumuncertaintystateinpositionandmomentum[27]. It mal contribution to the system modes.[31] isunlikelythatanyexistinglasersourcecanproducethe required correlations. Nevertheless, this example shows that, when dealing with correlated states, it is possi- B. Numerical results ble to reach the minimum uncertainty limit with a non- Gaussian pulse-shape. Typical results for the mean position variances dur- ing the propagation of a 104 particle coherent state soliton with initial shape φˆ = 1sech(1ζ) are shown in h ζi 2 2 Fig. 2(a). Here the dashed line represents the analytic solution for the total variance ∆Xˆ2 /x2, while the solid V. SOLITON PROPAGATION h i 0 lines are numerical results generated using a stochastic computer simulation equivalent to the quantum nonlin- We next consider weakly interacting particle distribu- ear Schrdinger equation[6]. These results include the to- tions confined to a single transverse mode of a waveg- tal variance (in complete agreement with the analytical uide, which can therefore be treated by an effective one result), the standard quantum limit σS2QL and the mea- dimensional field theory. Examples of such systems in- sure of quantum wave packet spreading σ2 . For com- QM cludephotonicwavepacketsinsingle-modeopticalfibers parison,Figs.2(b)and2(c)displaytheevolutionofthese andatomicBECcloudsmovingin1Dwaveguides. Such variancesforwiderandnarrowerpulsesthantheclassical physical systems have the property that they can form soliton solution. solitons, in which the classical wave-packet spreading is Thestandardquantumlimitforthesolitonisconstant, minimized, thus forming excellent candidates for obser- which indicates each soliton envelope remains nearly in- vation of these center-of-mass uncertainties. variant; as one might expect from a classical analysis. 9 This is a remarkable macroscopic quantum effect. An optical coherent state soliton consists of linear superpositions of continuum photons and different number- momentum eigenstates. This is clearly different to a classicalsolitonwithfrequencyjitter, yetitbehavesvery similarly as far as the center position is concerned. On theotherhand,thestandardquantumlimitforapulsein alinear mediumincreasesinthesamewayasthe soliton quantum wave-packet spreading, due to the dispersive spreading in the average intensity. In the linear case, an initial coherent state remains coherent, so the increased quantum positional uncertainty can be attributed to the shot-noiseerrorintrinsictothedetectionofapulsewhose envelope is not sharply localized. This indicates that the intrinsic quantum wave-packet spreading can be observed only for the soliton, where it is easily distinguishable from coherent shot-noise effects. However, this does not mean that a quantum soliton is unstable. Each soliton preserves its soliton pulse shape. In this sense, the soliton quantum diffusion is different from the classical pulse broadening due to linear dispersion. In fact, in the case of the correlated initial state, the second-order and the fourth-order correlation functions are invariant[29]. VI. PRACTICAL EXAMPLES In this section we consider some practical examples and numerical estimates with currently available experi- mental technologies. A. Photonic systems In the optical case, the soliton propagation model de- scribes a distribution of interacting photons (or strictly, polaritons) propagating in a single-mode fiber with a third-order Kerr nonlinearity, χ(3). Here the sign of the interaction term is positive for normal dispersion and negativeforanomalousdispersion. Forχ(3) solitonprop- agation,werequirethe carrierfrequencyto be inside the anomalous dispersion regime. Typical orders of magnitude of soliton parameters are n¯ = 109, t = 10−12s, x = 1m. With these parameters, 0 0 weobtainastandardquantumlimitof10−8mforasingle coherent pulse. In 1km of propagation, the wave packet Figure 2: Mean position variances for 1D coherent pulses of wouldthereforespreadto 10−5m, muchgreaterthan the n¯ = 104 particles propagating in a dispersive medium. The standardquantum limit. Although still less than the ac- threefigurescorrespondtothreedifferentwavepacketshapes: tual soliton extent of 10−4m, this is a remarkably large (a)thesolitonsolutiontothe1DnonlinearSchrödingerequa- quantum wave-packet spreading in a composite object tion hφˆ(ζ)i = 12sech(21ζ), (b) a wider pulse of the form of 109 particles. An experiment on quantum-mechanical φˆ(ζ) sech(1ζ) and (c) a narrower pulse φˆ(ζ) sech(ζ). wave-packetspreadingina composite objectofthis type h i∝ 4 h i∝ The solid lines represents the analytically determined total (such as that described in [5]) would test quantum me- variance ∆Xˆ2 /x20, while the dashed and dash-dotted lines chanics in a region of much larger particle number than represenththe niumerical results for the classical (σS2QL/x20) previously achieved. and quantum (σQ2M/x20) contributions respectively. Unfortunately, however, even fiber attenuations lower than the usual 0.1dB/km translate to losses of over 107 10 photons from such a pulse over this distance. Since each For example, using numbers from the experiment of particle lost can be used to obtain information about Khaykovichetal.[32]andassumingacondensatetemper- the mean position of a pulse, these losses would destroy ature of 0.5T 80nK we find that the standard devia- c ≃ the coherence properties of a wave packet. Even so, the tion in the mean position due to thermal noise increases remaining statistical uncertainty is a practical concern – linearly at a rate of 1.3 10−4m/s. This is quite fast, × it is, for example, likely to have a severe effect on the whencomparedtotheeffectofquantumspreadingofthe error-rate in pulse-position logic[28]. The effect may be soliton wave packet, which increases the standard devia- enhanced by using dispersion-engineered fibers[5]. tionatthe muchslowerrateof4.2 10−5m/s. As this is × an order of magnitude smaller than the thermal uncertainty, whose growth rate goes as the square root of the B. Degenerate Bose gasses cloud temperature, this implies that temperatures less than1nKarerequiredto directly observequantumwave The same effective field theory can also describe a di- packet spreading. Although this is well below what is lute, single-species gas of massive bosonic particles, in- achievedincurrent7Liexperiments,Leanhardtet al.[35] teracting through low-energy S-wave (l = 0) scattering have achieved temperatures of less than 0.5nK in con- events and held in a potential which confines the parti- densates of spin-polarized 23Na, indicating that it may cles to a single transverse mode but allows longitudinal not be impossible to achieve such low temperatures in propagation. In this case, one finds that: matter-wavesolitonexperiments. Wenotethatadiabatic changesintheinteractionstrengthcouldbeusedtocom- ~2 x = press the soliton, thereby increasing the quantum posi- 0 mn¯g 1D tion uncertainty relative to the standard quantum limit. ~3 t = (52) 0 mn¯2g2 1D C. Degenerate Fermi gasses where m is the single-particle mass, and g = 2~a ω 1D s ⊥ is the effective 1D interaction strength[30]. For temperatures well below the BEC transition temperature, such Ultra-cold degenerate Fermi gases have recently been gassesareusuallyconsideredto be the matter-waveana- obtained in several laboratories[36, 37]. The prospects log of coherent laser output, and are thus sometimes re- for fermionic solitons have not yet been established ferred to as atom-lasers. clearly,althoughtunable Feshbachresonanceswith both Although there are various paths one cantake to real- attractive and repulsive interactions are known to exist. izing matter-wave solitons, we consider here only bright The species observed experimentally include 40K , 6Li solitons in the absence of any periodic potential – i.e., and 3He∗. Atom correlations[38] have already been ex- solitons for which the required nonlinearity is provided perimentallyobservedinopticalmeasurementswith40K. byanattractiveinteractionbetweenatoms. Suchsolitons Oneofthemostinterestingcasesismetastablefermionic have been observed[32, 33] in BECs of 7Li atoms, where Helium, in which atoms can be counted directly using the presence of a Feshbachresonance is used to tune the multi-channel plate detectors[37]. This affords an in- effective scattering length from repulsive to attractive. teresting possible avenue to testing the prediction that Theseexperimentspresentthe veryrealpossibilityofdi- center-of-mass fluctuations are reduced for fermions rel- rectly observing the quantum dispersionof a mesoscopic ative to bosons. object composed of massive particles, thanks to the extremely slow rate of atom loss from these systems . For instance,thetotalpredictedlossrateforatrappedcloud D. Double-slit interference of soliton ‘particles’ of attractive 7Li atoms of roughly the same number as that in [32] is around 50/s.[34] (This is analogous to an Providedonecouldovercomethisdifficulty,itwouldbe optical pulse traveling along a fiber with an attenuation extremely interesting to conduct double slit interference of less than 10−6dB/km!) experiments with such large quantum objects. One pos- Drawbacks in dealing with atomic clouds include the sible implementation of this idea – essentially an atomic effect of finite temperatures. This places a classical un- implementation of the fiber experiment proposal in [5] – certainty in the center of mass momentum and therefore isshowninschematicasFig.3. Hereamatter-wavesoli- leadsto aclassicalspreadin centerofmass positionover ton is allowedto propagatewithout disturbance until its time,obscuringthequantumdiffusion. Fora1DgasofN quantum wave packetis large comparedwith the soliton bosons of mass m and temperature T, the equipartition width. A tightly focused laser pulse is then used to re- theorem gives move the central and outer elements of the distribution, effectivelycreatingapairofaperturesthroughwhichthe k T σ˙ = B (53) remaining components of the distribution pass. At some th rmN later time, the distribution is imaged and the mean po- where σ˙ is the linear rate of increase in the thermal sition of the soliton measured. Another approach - pro- th mean position uncertainty. vided the separation was large compared to the soliton