Chaoticity and Coherence in Bose-Einstein Condensation and Correlations

0.8 MB·English

by  Cheuk-Yin Wong

5
1
0
2

n
a
J

3
2

Chaoticity and Coherence in
]
h
p Bose-Einstein Condensation and Correlations∗
-
p
e
Cheuk-Yin Wong1† ,Wei-Ning Zhang2, Jie Liu2, Peng Ru2,
h
[
1Physics Division,Oak Ridge National Laboratory,Oak Ridge,TN,USA

3 2School of Physics and Optoelectric Technology,
v
Dalian University of Technology, Dalian, Liaoning, China
0
3
5 January 26, 2015
4
0
.
1
0
Abstract
5
1 WereviewthepropertiesofchaoticityandcoherenceinBose-Einsteincon-
: densation and correlations, for a dense boson system in its mean-field repre-
v
sentedapproximatelybyaharmonicoscillatorpotential. Theorderparameter
i
X and the nature of the phase transition from the chaotic to the condensate
r states are studied for different fixed numbers of bosons. The two-particle
a correlation function in momentum space is calculated to investigate how the
Bose-Einstein correlation depends on the degree of condensation and other
momentumvariables. WegeneralizetheBose-Einsteincorrelationanalysisto
three-particle correlations to show its dependence on the degree of conden-
sation.
∗Invited talk presented at the Xth Workshop on Particle Correlations and Femtoscopy
(WPCF14)atGy¨ongyo¨s,HungaryonAugust25to29,2014
†Speaker
1
1 Introduction
As is well known, a fundamental assumption for the occurrence of Bose-Einstein
correlation(BEC)isthepresenceofachaoticsourceofidenticalbosons[1,2]. The
Bose-Einstein correlation occurs in a chaotic source but not in a coherent source
[3, 4, 5, 6, 7, 8].
The properties of chaoticity and coherence are complementary attributes. Both
chaoticityandcoherenceshouldbeexaminedonequalfootingsinasingletheoretical
framework with the description of both the BE condensation and BE correlations.
In such a unified framework, it is then possible to investigate not only the states of
chaoticity and coherence, but also the transition from a chaotic state to a coherent
state. How can the degrees of chaoticity or coherence be quantified? Is the transi-
tion from a chaotic state to a coherent state a first-order with a sudden onset, or is
it a gradual transition that is closer to a second-order? What is the relevant order
parameterthatbestdescribesthetransition? HowdoesBose-Einsteincondensation
quantitativelyaffectsthetwo-particleandthree-particleBose-Einsteincorrelations?
QuestionsofBose-Einsteincorrelationsandcondensationarisenotonlyinatomic
physics [9, 10, 11] but also in high-energy heavy-ion collisions [5, 6, 8] where pions
are the most copiously produced particles. The use of two-pion Bose-Einstein cor-
relations to probe the source coherence was proposed at the end of 1970s [12, 3].
The introduction of the “chaoticity” parameter λ of BEC in pions is only a tool to
represent experimental data. However, the experimental measurement of λ is beset
by the presence of many other effects such as particle misidentification, long-live
resonance decay, final state Coulomb interaction, non-Gaussian source distribution,
etc. [5,7]. Theexplanationoftheexperimentalλresultsremainsanopenquestion.
In 1993, S. Pratt proposed a pion laser model in high energy collisions and studied
the influence of pion laser on two-pion Bose-Einstein correlation function and the
chaoticityparameter[13]. In1998,T.Cs¨org˝oandJ.Zim´anyiinvestigatedtheeffect
ofBose-Einsteincondensationontwo-pionBose-Einsteincorrelations[14]. Theyuti-
lizedGaussianformulasdescribingthespaceandmomentumdistributionsofastatic
non-relativisticbosonsystem,andinvestigatedtheinfluenceofthecondensationon
pionmultiplicitydistribution. In2007,C.Y.WongandW.N.Zhangstudiedhowλ
in Bose-Einstein correlations depends on the degree of Bose-Einstein condensation
orchaoticity,forstaticnon-relativisticandrelativisticbosongaseswithinaspherical
mean-field harmonic oscillator potential [15]. The model can be analytically solved
in the non-relativistic case and be used in atomic physics [9, 10, 11]. The limiting
conditions and circumstances under which the parameter λ can be approximately
related to the degrees of chaoticity were clarified [15]. A similar study for cylindri-
cal static boson gas sources was completed [16] and the chaoticity parameter λ in
two-pion Bose-Einstein correlations in an expanding boson gas model was recently
examined [17]. The investigation of chaoticity and coherence was also carried out
usingamodelofq-deformedoscillatoralgebraiccommutativerelations[18]andthe
model of partial indistinguishability and coherence of closely located emitters [19].
Inanotherrelatedtopic,initialconditionssuchasthecolor-glasscondensate(CGC)
with the coherent production of partons [20] in heavy-ion collisions may also lead
to condensate formation [21].
2
Recently, experimental investigation of the source coherence in Pb-Pb collisions
√
at s = 2.76 TeV at the Large Hadron Collider (LHC) was carried out by the
NN
ALICE collaboration [22]. A substantial degree of source coherence was measured
[22] using a new three-pion Bose-Einstein correlations technique with an improve-
ment over past efforts [23, 24, 25, 26]. Earlier work on three-particle correlations
were carried out in [6, 13, 27, 28, 29, 30, 31, 32, 33, 34].
A proper theoretical framework to study the above topics is the theory of the
Bose-Einstein condensation and correlations in their own mean field potential [15].
Wewouldliketoreviewtheessentialelementshereandexaminefurthertherelated
question of three-body correlations.
2 Bose-Einstein Condensation for attractively In-
teracting Bosons
WeseekadescriptionofchaoticityinBose-Einsteincorrelationsthroughtheconsid-
eration of Bose-Einstein condensation. Why is Bose-Einstein condensation relevant
to Bose-Einstein correlations (BEC)? Glauber in many private communications and
in his talk in QM2005 suggested that the consistent experimental observations of
λ<1maybeduepartlytothecoherenceofthepionsinBoseEinsteincorrelations
[35]. Furthermore,therehavebeenmajoradvancesinBose-Einsteincondensationin
atomicphysics[9,10,11]. Inparticular,theworksofPolitzer[9],andNaraschewski
&Glauber[10]revealsthatBEcondensationandtheBEcorrelationsareintimately
related.
We envisage the possibility of the occurrence of a Bose-Einstein condensation
in dense boson media of identical bosons with the following reasoning [15, 16, 17]
1. Identical bosons with mutual attractive interaction generate a mean field po-
tential, which depends on the boson density ρ(r) as [36]
1 (cid:16)r(cid:17)2
V(r)=−4πf(0)ρ(r)∼ ¯hω , (1)
2 a
where f(0) is the forward scattering amplitude, and a is the lengthscale that
defines the spatial region of boson occupation.
2. Therefore, for a given length scale a, the ¯hω of the underlying mean-field
potential increases with increasing density ρ of the produced bosons.
3. TheorderparameterthatdeterminesthedegreeofBEcoherenceorchaoticity
is T/¯hω. Thus the order parameter T/¯hω decreases with increasing boson
density.
4. ForagiventemperatureT atfreezeout,ahighdensityofproducedbosonswill
leadtoalowervalueoftheorderparameterT/¯hω,whichinturnwillleadtoa
greatercondensatefractionf =N /N,whereN isthetotalnumberofbosons
0 0
and N is the number of bosons in the lowest state. A greater condensate
0
fractionf bringsaboutagreatercoherenceinBose-Einsteincorrelationsand
0
a reduction in the degree of chaoticity.
3
In high energy heavy-ion collisions when bosons (gluons or pions) are copiously
produced within a small region in a short time interval, the density of the bosons
increases as the collision energy increases. Following the above reasoning, general-
izedtosystemswithdifferentialtransverseandlongitudinalspatialdistributions,we
expect the occurrence of boson condensation in high energy heavy-ion collisions at
somehighcollisionenergies. ItisusefultoexaminetheBose-Einsteincondensation
for bosons in an exactly solvable model.
3 Bose-Einstein Condensation for Bosons in a
Spherical Harmonic Oscillator Potential
We consider a system of N bosons in a spherical harmonic oscillator potential,
which arises either externally or from the bosons’ own mean-fields. We study how
theoccupationnumbersofdifferentstateschangeasafunctionofthetemperature
T, in relation to the oscillator frequency ¯hω. Bose-Einstein condensation occurs
when the occupation number N for the lowest state (the condensate state) is a
0
substantial fraction of the total particle number N. The degree of coherence or
chaoticity is quantified by the condensate fraction f = N /N, which varies as a
0 0
function of the order parameter T/¯hω.
In such a study, it is important to use the proper statistical ensemble [9]. In
a grand canonical ensemble, we fix the chemical potential µ and the temperature
T, and we allow the number of particles N in the n-th single-particle state to
n
vary. We obtain the average occupation number for the single-particle state n to
be N =(cid:104)a+a (cid:105). The square fluctuation of N is then given by
n n n n
(cid:104)(a+a −(cid:104)a+a (cid:105))2(cid:105)≈N (N +1). (2)
n n n n n n
As the fluctuation of N in a grand canonical ensemble is of the same order as
n
the occupation number itself, we cannot treat the lowest n=0 state in the grand
canonical ensemble. The lowest n = 0 state needs to be treated in the canonical
ensemble with a fixed total number of bosons.
It was shown however that while the lowest n = 0 state needs to be treated
in the canonical ensemble, the n > 0 state can be treated in the grand canonical
ensemble without incurring large errors [9]. We shall follow such a description for
the ensemble of N identical bosons in a spherical harmonic oscillator potential. In
suchacanonicalensembleforthelowestn=0statebutagrandcanonicalensemble
for the n>0 states, the total number of bosons is fixed and yields the condensate
number condition
N =N + (cid:88) N = z + (cid:88) gnze−((cid:15)n−(cid:15)0)/T , (3)
0 n 1−z 1−ze−((cid:15)n−(cid:15)0)/T
n=1,2,3,... n=1,2,3,...
where z = eµ/T is the fugacity of the system, g is the degeneracy number
n
g =(n+1)(n+2)/2 for the n-th single-particle level, and (cid:15) is the single-particle
n n
energy in the spherical harmonic oscillator potential
(cid:15) =(n+3/2)¯hω. (4)
n
4
ForagivenN,equation(3)containsonlyasingleunknown,z,whichcanbesolved
as a function of the order parameter T/¯hω. The solutions of z for N=25, 500,
1000 and 2000 are given in Fig. 1, and the corresponding condensate fractions
f =N /N are shown in Fig. 2.
0 0
1
N=2000
0.998 N=1000
N= 500
0.996 N= 250
z
0.994
0.992 (b)
0.990
1.0
0.8
0.6
z
0.4
(a)
0.2
0
0 5 10 15 20
T / h_ω
Figure1: (Coloronline)(a)Thefugacityparameterz satisfyingthecondensate
number condition Eq. (3) for different boson numbers N in a spherical har-
monic oscillator potential, as a function of the order parameter T/¯hω and (b)
an expanded view in the z ∼1 region.
We observe in Fig. 1 that the fugacity parameter z is close to unity in the
strongly coherent region at low temperatures. In fact, the fugacity parameter z
at T=0 assumes the value z(T=0) = N/(N +1). For a given boson number N,
the fugacity z decreases very slowly in the form of a plateau, as the temperature
increasesfromT =0. Theplateauregionpersistsuntilthecondensatetemperature
T is reached, and z then decreases rapidly thereafter. The greater the number of
c
bosons N, the greater is the plateau region, as shown in Fig. 1(b). For example,
for N =2000 the value of z is close to unity for 0<T/¯hω <11 in the plateau,.
We note in Fig. 2 that for a given value of the total number of bosons N in
the spherical harmonic oscillator potential, the condensate fraction f is close to
0
unity when the order parameter T/¯hω is below a limit, and this limit depends on
N. We can plot the condensate fraction f as a function of the order parameter
0
T/¯hω. The functional form of f (T) can be approximated by
0
(cid:26)
1−[(T/¯hω)/(T /¯hω)]3 for (T/¯hω)≤(T /¯hω),
f (T)= c c (5)
0 O(1/N)→0 for (T/¯hω)≥(T /¯hω).
c
5
1 0.0
N = 2000
0.8 0.2Tf
1000
=
0.6 500 0.4
(
f0 250 1
0.4 0.6
-

f
0.2 0.80
)
0 1.0
0 2 4 6 8 10 12 14
_ ω
T / h
Figure 2: (Color online) Solid curves represent the condensate fractions f (T),
0
calculatedwiththecondensatenumberconditionEq.(3),asafunctionofT/¯hω
for different boson numbers N in a spherical harmonic oscillator potential. The
abscissa labels for the corresponding chaotic fraction f (T)=[1 − f (T)] are
T 0
indicated on the right. The dashed curves are the fits to the solid curve results
of f (T) with the function 1−[(T/¯hω)/(T /¯hω)]3 of Eq. (5) where the values of
0 c
T /¯hω for different N values are listed in Table I.
c
The results from the above one-parameter fit to f (T) are shown as the dashed
0
curves in Fig. 2, to be compared with the f (T) calculated with the condensate
0
configuration condition Eq. (3) shown as the solid curves. The values of T /¯hω
c
that give the best fit to f (T) for different N values are listed in Table I.
0
The above results provide a comprehensive description for the transition from
a chaotic state to a coherent state. Fig. 2 indicates that the transition from the
completelychaoticstatewithf =0tothestateofcoherencewithf →1isasecond-
0 0
order-type transition under a gradual decrease of the order parameter T/¯hω. It is
not a first-order phase transition.
Table I. Critical order parameter T /¯hω obtained from (i) fitting f as a function
c 0
of T/¯hω with Eq. (5), and from (ii) the analytical formula of Eq. (6).
Number of Bosons T /¯hω obtained T /¯hω obtained
c c
N from fitting f with Eq. (5) with Eq.(6)
0
2000 10.97 11.00
1000 8.56 8.53
500 6.63 6.62
250 5.12 5.13
It is remarkable that the critical order parameter T /¯hω and the boson number
c
N obeys the following simple relationship
T /¯hω = 0.6777N0.36666, (6)
c
6
as shown by the third column in Table I. Thus, the knowledge of N suffices to
determinethecriticalorderparameterT /¯hω bytheabovesimpleequationandthe
c
knowledge of T /¯hω subsequently yields the approximate condensate fraction at all
c
other temperatures by Eq. (5).
4 Single-particle and Two-Particle Density
Matrices in Momentum Space
The determination of the fugacity z from the condensate number condition (3)
allows the calculation of various physical quantities. Specifically, the one-body
density matrix in momentum space is given by
∞
(cid:88)
G(1)(p ,p(cid:48))= u∗(p(cid:48))u (p )(cid:104)aˆ†aˆ (cid:105), (7)
1 1 n 1 n 1 n n
n=0
where u (p) is the single-particle wave function and the occupation number
n
N =(cid:104)aˆ†aˆ (cid:105) can be inferred from the terms in the summation in Eq. (3). The
n n n
two-particle density matrix in momentum space
(cid:88)
G(2)(p ,p ;p(cid:48),p(cid:48))= u∗(p(cid:48))u∗(p(cid:48))u (p )u (p )(cid:104)aˆ†aˆ†aˆ aˆ (cid:105) (8)
1 2 1 2 k 1 l 2 m 2 n 1 k l m n
klmn
can be written in terms of one-body density matrices as [10, 15]
G(2)(p ,p ;p(cid:48),p(cid:48))=G(1)(p ,p(cid:48))G(1)(p ,p(cid:48))+G(1)(p ,p(cid:48))G(1)(p ,p(cid:48))
1 2 1 2 1 1 2 2 1 2 2 1
∞ (cid:26) (cid:27)
(cid:88)
+ u∗(p(cid:48))u∗(p(cid:48))u (p )u (p ) (cid:104)aˆ†aˆ aˆ (cid:105)−2(cid:104)aˆ†aˆ (cid:105)(cid:104)aˆ†aˆ (cid:105) . (9)
n 1 n 2 n 2 n 1 n n n n n n n
n=0
1 1′ 1 1′
2 2′ 2 2′
(a) (b)
Figure 3: Two-particle distribution function expanded in terms of products of
one-particle distribution functions in uncorrelated mean-field approximation.
Theuncorrelatedpartinthefirsttwotermsoftheabovetwo-particledensityma-
trix, G(1)(p ,p(cid:48))G(1)(p ,p(cid:48))+G(1)(p ,p(cid:48))G(1)(p ,p(cid:48)), is represented schemati-
1 1 2 2 1 2 2 1
callybythediagraminFig.3. Ourtaskistoobtainthecorrelatedpartarisingfrom
Bose-Einstein condensation represented by the last term in Eq. (9).
In the limit of a large number of bosons N in a grand canonical ensemble for
the non-condensed states, the contributions from the set of {n > 0} states in the
7
summationinEq.(9)canbeneglected. Weareleftwithonlythen=0condensate
state contribution for this summation.
To describe the contribution from the n = 0 condensate state, we shall follow
Ref. [9, 10] and use the canonical ensemble which gives the canonical fluctuation
(cid:104)(aˆ†aˆ −(cid:104)aˆ†aˆ (cid:105))2(cid:105)=(cid:104)aˆ†aˆ†aˆ aˆ (cid:105)−(cid:104)aˆ†aˆ (cid:105)(cid:104)aˆ†aˆ (cid:105)=O(N ). (10)
0 0 0 0 0 0 0 0 0 0 0 0 0
Thus, we have
(cid:104)aˆ†aˆ†aˆ aˆ (cid:105)−2(cid:104)aˆ†aˆ (cid:105)(cid:104)aˆ†aˆ (cid:105)=−(cid:104)aˆ†aˆ (cid:105)(cid:104)aˆ†aˆ (cid:105)+O(N ). (11)
0 0 0 0 0 0 0 0 0 0 0 0 0
In the limit of a large number of particles, we can neglect the last term O(N ) in
0
the above equation which is small in comparison with the first term of order N2.
0
The two-particle distribution of Eq. (9) is therefore
G(2)(p ,p ;p ,p )=G(1)(p ,p )G(1)(p ,p )+|G(1)(p ,p )|2
1 2 1 2 1 1 2 2 1 2
−N2|u (p )|2|u (p )|2, (12)
0 0 1 0 2
which gives the conditional probability for the occurrence of a pion of momentum
p in coincidence with another identical pion of momentum p .
1 2
5 Two-Particle Momentum Correlation Function
InBEcorrelationmeasurements, wenormalizetheprobabilityrelativetotheproba-
bilityofdetectingparticlep andp ,anddefinethemomentumcorrelationfunction
1 2
C(p ,p ) as
1 2
G(2)(p ,p ;p ,p )
C(p ,p )= 1 2 1 2 . (13)
1 2 G(1)(p ,p )G(1)(p ,p )
1 1 2 2
It is convenient to introduce the average and the relative momenta of the pair
p=(p +p )/2, q =p −p . (14)
1 2 1 2
Themomentumcorrelationfunctioncanbeexpressedasafunctionofthekinematic
variablesp andp oralternativelyofpandq. FromEq. (12),wehavethegeneral
1 2
expression for the correlation function
|G(1)(p ,p )|2−N2|u (p )|2|u (p )|2
C(p,q)=C(p ,p )=1+ 1 2 0 0 1 0 2 . (15)
1 2 G(1)(p ,p )G(1)(p ,p )
1 1 2 2
This is the general Bose-Einstein correlation function for all situations: coherent,
chaotic, and the transition between coherent and chaotic systems.
TheevaluationofthecorrelationfunctionC(p,q)inEq.(15)requirestheknowl-
edge of G(1)(p ,p ) and the ground state wave function u (p ). For a system of
1 2 0 1
bosonsinasphericalharmonicoscillator, thewavefunctionsareallknown, andthe
correlation function can be written out analytically. Specifically, we have
8
∞
G(1)(p ,p )=(cid:88)zkG˜ (p ,p ;kβ¯hω), (16)
1 2 0 1 2
k=1
(cid:18) a2 (cid:19)3/2 (cid:18) a2 (p2+p2)(coshτ−1)+(p −p )2(cid:19)
G˜ (p ,p ;τ)= exp − 1 2 1 2 ,(17)
0 1 2 π¯h2(1−e−2τ) ¯h2 2sinhτ
and the ground state wave function is
(cid:18) a2 (cid:19)3/4 (cid:26) a2 p2(cid:27)
u (p)= exp − . (18)
0 π¯h2 ¯h2 2
The knowledge of the single-particle G(1)(p ,p ) and u (p) will then allow the
1 2 0
determination of the two-particle correlation function C(p,q).
The correlation function C(p,q) in Eq. (15) possesses the proper coherent and
chaotic limits. For a nearly completely coherent source with almost all particles
populating the ground condensate state, N →N, the two terms in the numerator
0
canceleachotherandwehaveC(p,q)=1,withtheabsenceoftheBEcorrelation.
FortheotherextremeofacompletelychaoticsourcewithN (cid:28)N,thesecondterm
0
in the numerator proportional to N2 in Eq. (15) gives negligible contribution and
0
can be neglected. The correlation function C(p,q) then becomes the usual BE
correlation for a completely chaotic source.
6 Evaluation of the Two-Particle Momentum Cor-
relation Function
For a given number of bosons N in a spherical harmonic oscillator, the solution
of fugacity z obtained as a function of the order parameter T/¯hω allows us to
evaluate the momentum correlation function C(p,q) with Eqs. (15)-(18). In Fig.
4, we show C(p,q) for example for the case of N=2000 for which the critical
order parameter is T /¯hω=10.97, as tabulated in Table I. We observe that the
c
correlation function is a complicated function of the average pair momentum p
and the order parameter T/¯hω. For p=¯h/a in Fig. 4(a), the correlation function
C(p,q) at q=0 is close to unity for temperatures below and up to T/¯hω=9 (below
T /¯hω), but increases to 2 rather abruptly at T/¯hω=12, (above T /¯hω). For
c c
p=2¯h/a in Fig. 3(b), the correlation function C(p,q) at q = 0 is substantially
above unity and increases gradually as temperature increases. For p=3h¯/a in Fig.
4(c),thecorrelationfunctionC(p,q)atq=0isabout2forallcasesoftemperatures
examined. Ifonefollowsthestandardphenomenologicalanalysisandintroducesthe
“chaoticity” parameter λ to represent the intercept of the correlation function at
zero relative momentum, then this parameter λ is a function of the average pair
momentum p and temperature T
λ(p,T)=[C(p,q =0;T)−1]. (19)
9
q) 1.82 N=2000 (a) TT//hh__ωω==192
p,1.6 _ T/h_ω=6
C(1.4 p=h /a T/h_ω=3
1.2

1
2
1.8 (b)
)
,q1.6 p=2h_ /a
p
(1.4
C
1.2
1
2
1.8 (c)
q)1.6 p=3h_ /a
,
p
(1.4
C
1.2
1
0 0.2 0.4 0.6 0.8 1
_
q in (h /a)
Figure 4: (Color online) The correlation function C(p,q) at different values of
the pair average momentum pa/¯h and temperatures. Figures (a), (b), and (c)
are for p=1, 2, and 3h¯/a, respectively.
We display explicitly the dependence λ(p,T) as a function of p in Fig. 5(a) for
differentorderparametersT/¯hω,forthecaseofN =2000. AtT/¯hω=12,whichis
above the critical condensate order parameter of T /¯hω=10.97, the λ parameter is
c
1 for all p values. At T/¯hω=9, as p increases the λ parameter rises gradually from
∼0.1atp=h¯/aandreachestheconstantvalueof1atp=2.4¯h/a. AtT/¯hω=6and
3, for which the systems are significantly coherent with large condensate fractions,
the λ parameter starts close to zero at p=¯h/a, but as p increases the λ parameter
increases gradually to unity at p=2.9 and 3.1¯h/a for T/¯hω=6 and 3 respectively.
The location where the λ parameter attains unity changes with temperature. The
lowerthetemperature,thegreateristhevalueofpatwhichtheλparameterattains
unity.
We conclude from our results that the parameter λ(p,T) is a sensitive function
of both p and T, and λ(p,T)=1 is not a consistent measure of the absence of the
condensate fraction, as it attains the value of unity in some kinematic regions for
significantlycoherentsystemswithlargecondensatefractionsattemperaturesmuch
below T . Only for the region of small p will the parameter λ(p,T) be correlated
c
with the chaotic fraction f (T) of the system.
T
It is interesting to note that experimentally measured values of λ from different
collaborations and different method of analysis [37, 38] exhibit an increase of λ as
p of the average momentum of the pair increase as shown in Fig. 5(b). There
T
is a similar trend of increasing λ as a function of p . This may be an indication
T
of the dependence of the correlation function on the average momentum of the
10