The center-of-mass response of confined systems

4 Pages·0.09 MB·English

by  L. F. Lemmens

The center-of-mass response of confined systems
L. F. Lemmens
Dept. Natuurkunde, Universiteit Antwerpen RUCA, Groenenborgerlaan 171, B2020
Antwerpen, Belgium.
9
9 F. Brosens and J. T. Devreese
9 Dept. Natuurkunde, Universiteit Antwerpen UIA, Universiteitsplein 1, B2610
1 Antwerpen, Belgium.
n (September11, 1998, resubmitted December 22, 1998 to Phys. Rev. A)
a
Forconfinedsystemsofidenticalparticles,eitherbosonsorfermions,wearguethattheparabolic
J
nature of the confinement potential is a prerequisite for the non-dissipative character of the center
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of mass response to a uniform probe. For an excitation in a parabolic confining potential, the
1
halfwidthofthedensityresponsefunctiondependsneverthelessquantitativelyonpropertiesofthe

] internaldegreesoffreedom,asisillustratedhereforanidealconfinedgasofidenticalparticleswith
h harmonic interparticle interactions.
c
e 05.30.Jp, 03.75.Fi, 32.80.Pj
m
-
t
a
t I. INTRODUCTION
s
.
t
a Inthepresentpaperwestudytheresponseofaconfinedsystemofspin-polarizedidenticalparticlesifauniformbut
m
time-dependent force is applied. Our method is based on the Feynman–Kac functional representing the propagator
- of distinguishable particles, followed by the projection of this functional on the irreducible representations of the
d
permutationgroupasrequiredforidenticalparticles. Theprojectionintroducedin[1]is appliedonconfinedsystems.
n
Details have been given earlier [2] to derive the thermodynamical properties and the static response functions for
o
c bosons and for fermions. An application of the method to real systems can be found in Ref. [3] for rubidium.
[ The response function is calculated for a spatially homogeneous but time-dependent force. It is shown that if the

confinementpotentialisapolynomialofordertwo,themotionofthecenterofmassdoesnotinducetransitionsbetween
1
the modes representing the relative motion of the internal degrees of freedom. This follows from the independence of
v
2 these degrees of freedom, a property known [5] for harmonic systems. The analogue in extended systems is the Kohn
0 theorem[4],wellknownfor the cyclotronresonanceinaninteractingelectrongas. The connectionto the responseof
1 center-of-massexcitationbecomes especially relevantif the two-body interactiononly depends on the distance vector
1 between the positions. If one of these conditions is not satisfied, we shall show that the homogeneous external force
0
couplestotheinternaldegreesoffreedom. Evenifbothconditionsaresatisfiedtheshapeoftheresponsefunctionstill
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depends on the characteristics of the internal degrees of freedom, typically the frequency of the internal oscillation
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modes and the number of particles in the well. This suggests that these characteristics can be estimated from the
/
at shape of the response to a homogeneous time-dependent force. For instance, monitoring the density of a confined
m system whilst changing the direction of the gravitational force on the center of mass or changing slightly the form of
the confinement potential gives information on the internal degrees of freedom and the number of particles confined
-
d in the system.
n In view of the experimental progress made [6–9] on the Bose-Einstein condensed systems, the present response
o
propertyofaperfectharmonicconfinedsystemcouldbe relevanttochecktheparabolicityofthe trap. Furtherrecent
c
experiments[10]usestrongdeviationsfromtheparabolicconfinementpotentialtoprobethetimedependentbehavior
:
v of the confined atoms in a condensed phase. In this respect it seems important to know the implications of perfect
i
X parabolicity. Of course, the response of the system to such a local perturbation of the density would be of major
importance.
r
a The paper is organized as follows. In Sec. II the formal response theory is developed. In Sec. III we illustrate the
response properties for the exactly soluble model of a confined gas with harmonic interparticle interactions. In the
last section a discussion and the conclusions are given.
1
II. PARABOLICITY AND THE KOHN THEOREM
In a 3N dimensional configuration space with r (j =1,2,...,N) denoting the positions of the N particles, the
j
position ofthe center of mass is defined as R= 1 N r . For a quadratic confinement potential the center of mass
N j=1 j
can be introduced as follows:
P
N N
r2 =NR2+ (r R)2. (1)
j j −
j=1 j=1
X X
From mathematical statistics it is well known that for Gaussian distributions the average R can take any value
withoutaffectingthevalueof N (r R)2,i.e. thecenterofmasscouldbeindependentofthedeviationsfromthat
j=1 j −
center. This remains true in a physical system when the two-body potential depends only on the distance between
particles,because N V (rP r )= N V ((r R) (r R)).Intheharmonicmodelthefollowingtwo-body
j,l=1 j − l j,l=1 j − − l−
potential is used:
P P
N N
γ γN
(r r )2 = (r R)2. (2)
j l j
4 − 2 −
j,l=1 j=1
X X
Thistwo-bodypotentialallowsforexactsolutionsofthe propagatorandthe projectiononthe symmetricorantisym-
metric irreducible representation of the permutation group can be carried out.
Introducing a homogeneous time-dependent force:
N
f(τ) r =Nf(τ) R (3)
j
· ·
j=1
X
the potential confining N particles and the interaction of the particles with each other and the external field are
specifying the system for which the response will be calculated. Atomic units with ~ = m = 1 are used throughout
this paper.
If the particles were distinguishable, the Euclidean-time propagator K (¯r ,τ ¯r,0) for this interacting system can
D ′′ ′
be obtained from a straightforward generalization of our calculation in Ref. [2|]. Denoting by ¯r R3N the set of
∈
position vectors r ,...,r , the result turns out to be:
1 N
√Nf(τ)
K √NR ,τ √NR,0 N
′′ ′
K (¯r ,τ ¯r,0)= | Ω K r ,τ r ,0 , (4)
D ′′ | ′ (cid:16)K √NR ,τ √NR(cid:17),(cid:12)(cid:12)0 ′j′ | ′j w
′′ | ′ (cid:12) w jY=1 (cid:0) (cid:1)(cid:12)
(cid:16) (cid:17)(cid:12) (cid:12)
where K(r ,τ r ,0) denotes the well-knownpropagatorofa thr(cid:12)ee-dimensionalharmonic oscillatoroffrequency w:
τ | 0 |w (cid:12)
K(r ,τ r ,0) = w 3exp w r2τ +r20 coshwτ −2rτ ·r0 . (5)
τ | 0 |w r2πsinhwτ "−2 (cid:0) (cid:1)sinhwτ #
The quantity K(R ,τ R,0)[f(τ)] denotes the propagator of a three-dimensional harmonic oscillator of frequency
′′ | ′ Ω
Ω in the presence of a driving forcef(τ). If the driving force acts in the realtime t, the complex time τ =β+it has
to be introduced, with β =1/k T with k the Boltzmann constant and T the temperature. This propagatorcan be
B B
obtained in a direct way using the functional integration techniques illustrated in [2] leading to the following result:
K(R ,τ R,0)f(τ) = K(R ,τ R,0) (6)
′′ | ′ |Ω ′′ | ′ |Ω×
2 sinh 1Ωβ t s
exp 2 f(s) f(σ)sin(Ω(t s))sin(Ω(s σ))dsdσ
−Ωcosh(cid:0)21Ωβ(cid:1) Z0 Z0 · − − !×
t (cid:0) co(cid:1)s Ω s t+ 1iβ t cos Ω s t 1iβ
exp iR′·Z0 f(s) (cid:2)co(cid:0)sh−12Ωβ2 (cid:1)(cid:3)ds−iR′′·Z0 f(s) (cid:2)co(cid:0)sh−12Ω−β2 (cid:1)(cid:3)ds!.
¿Fromthe structureofthe propagatorK (¯r ,τ ¯r,0)(cid:0)fordi(cid:1)stinguishableparticles,itisclearth(cid:0)atpr(cid:1)ojectiononthe
D ′′ ′
|
representationsof the permutationgroupwill not affect the quotient of propagatorsthat contains the center of mass.
Therefore, the propagator K (¯r ,τ ¯r,0) for identical particles is given by
I ′′ ′
|
2
√Nf(τ)
K √NR ,τ √NR,0
′′ ′
K (¯r ,τ ¯r,0)= | Ω K (¯r ,τ ¯r,0), (7)
I ′′ | ′ (cid:16)K √NR ,τ √NR(cid:17),0(cid:12)(cid:12) I ′′ | ′
′′ | ′ (cid:12) w
where KI(¯r′′,τ ¯r′,0) accounts for the permutati(cid:16)ons of the particles,(cid:17)a(cid:12)(cid:12)nd is discussed extensively in [2]. Although the
| (cid:12)
present derivation has been given for the fully harmonic model (harmonic confinement and harmonic interaction),
the line of the derivation remains valid for more general two-body interactions. Indeed, the separation between the
propagationofthecenterofmassandoftheotherdegreesoffreedomisunaffectedbytheintroductionofthetwo-body
potential if it only depends on the difference vectors r r.
j l
−
III. RESPONSE TO AN UNIFORM FORCE
Knowing the propagator of the interacting many-particle system, the density response can be calculated in a
straightforwardway as the following expectation value:
1 N 1 3
nf(r,t) = δ(r r ) = nf (t)exp( iq r)dq,
N h − j i 2π q − ·
j=1 (cid:18) (cid:19) Z
X
N
1
nf (t) = exp iq r . (8)
q N · j
j=1
X(cid:10) (cid:0) (cid:1)(cid:11)
The averages A(¯r,t) are defined as:
h i
1
A(¯r,t) = d¯rA(¯r)K (¯r,β+it¯r,0), (9)
I
h i Z(β) |
Z
where Z(β) is the partition function for the system without externalforce at inverse temperature β. Performing this
averagesmeans the repetitive calculation of Gaussian integrals leading to the following result:
t sinΩ(t s)
nf (t) =nf (0)exp iq f(s) − ds
q q − · Ω
(cid:18) Z0 (cid:19)
q2 N 1cosh1wβ 1 cosh1Ωβ
nf (0) =exp − 2 + 2 (10)
q −4N w sinh1wβ Ω sinh1Ωβ
(cid:26) (cid:20) 2 2 (cid:21)(cid:27)
The response function nf(r,t) is then obtained by a Fourier transform leading to a Gaussian density distribution
function
2
r+ 1 tf(s)sin(Ω(t s))ds
nf(r,t)= 1 exp Ω 0 − , (11)
√2πS23 −h R 2S2 i 
in which the variance not only depends on the frequency Ω of the confining potential, but also on the frequency w of
the internal degrees of freedom:
1 N 1 wβ 1 Ωβ
S2 = − coth + coth . (12)
2 wN 2 ΩN 2
(cid:18) (cid:19)
In the absence of interparticle interactions, one readily obtains S2 = coth1Ωβ /(2Ω), which in the low-
|w=Ω 2
temperature limit β gives the variance of an harmonic oscillator with frequency Ω.
→∞ (cid:0) (cid:1)
In the average density response
1 t
r(t) = rnf(r,t)dr= f(s)sin(Ω(t s))ds (13)
h i −Ω −
Z Z0
one clearly sees the resonant structure in the presence of an oscillating driving force f(s) = f sinvs with a specific
v
frequency v:
f Ωsintv vsinΩt
r(t) = v − . (14)
h i|f(s)=fvsinvs −Ω Ω2 v2
−
This expectation value of the position does not depend on the parameters of the internal degrees of freedom, as
required by the Kohn theorem.
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IV. DISCUSSION
Fromthe derivationofthe density response to a uniform time-dependent force, it is clearthat the factorization(4)
is crucial to obtain a linear relation between the density and the applied force that does not depend on the internal
degreesoffreedom. Higher-ordertermsintheconfinementpotentialwillspoilthisrelation. Atermofthethirddegree
in the confinementpotential alreadyintroduces a contributionV ∼3R N (r )2,that mixes the center-of-mass
conf j=1 j
motionwith thatof the internaldegreesoffreedom,andmakes transitionspossible inthe internaldegreesof freedom
P
leading eventually to dissipative behavior of the center-of-mass excitation.
Furthermore the Gaussian nature of the density response may allow one to check the parabolic character of the
well by measuring the mean density and comparing it with the trap frequencies. The standard deviations of these
measurementscontaininformationabouttheexcitationfrequenciesoftheinternaldegreesoffreedomandthenumber
of particles contained in the well. The response of a center-of-mass mode behaving according to the Kohn theorem
contains information about the internal degrees of freedom. Its resonance structure reveals the parabolic quality of
the well.
Itshouldbenotedthatthe statisticsofthe internaldegreesoffreedomdoesnotenterintheexpressions. Therefore
the response of a parabolic well to a uniform external force does not distinguish between fermions or bosons merely
because it only applies tot the center of mass.
V. ACKNOWLEDGMENTS
ThisworkisperformedintheframeworkoftheFWOprojectsNo. 1.5.729.94,1.5.545.98,G.0287.95,G.0071.98and
WO.073.94N(WetenschappelijkeOnderzoeksgemeenschapover“Laagdimensionelesystemen”),the“Interuniversitaire
Attractiepolen – Belgische Staat, Diensten van de Eerste Minister – Wetenschappelijke, Technische en Culturele
Aangelegenheden”, and in the framework of the BOF NOI 1997 projects of the Universiteit Antwerpen. One of the
authors ( F.B.) acknowledges the FWO (Fonds voor Wetenschappelijk Onderzoek-Vlaanderen)for financial support.
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