Theory of coherent Bragg spectroscopy of a trapped Bose-Einstein condensate

16 Pages·0.29 MB·English

by  P. B. Blakie

Theory ofcoherent Bragg spectroscopy ofa trapped Bose-Einstein condensate
P. B. Blakie∗ and R. J. Ballagh
Department of Physics, University of Otago, Dunedin, New Zealand
C. W. Gardiner
School of Chemical and Physical Sciences, Victoria University, Wellington, New Zealand
(Dated:February1,2008)
WepresentadetailedtheoreticalanalysisofBraggspectroscopyfromaBose-EinsteincondensateatT =0K.
We demonstrate that within the linear response regime, both a quantum field theory treatment and a mean-
fieldGross-Pitaevskii treatment leadtothe samevalue for the mean evolutionof thequasiparticle operators.
TheobservableforBraggspectroscopyexperiments,whichisthespectralresponsefunctionofthemomentum
transferredtothecondensate,canthereforebecalculatedinameanfieldformalism. Weanalysethebehaviour
2
ofthisobservable bycarryingoutnumericalsimulationsinaxiallysymmetricthree-dimensional casesandin
0
twodimensions. Anapproximateanalyticexpressionfortheobservableisobtainedandprovidesameansfor
0
identifyingthe relativeimportance of threebroadening and shiftmechanisms (meanfield, Doppler, and finite
2
pulseduration)indifferentregimes.Weshowthatthesuppressionofscatteringatsmallvaluesofqobservedby
n Stamper-Kurnetal.,[Phys.Rev.Lett.83,2876(1999)]isaccountedforbythemeanfieldtreatment,andcanbe
a interpretedintermsoftheinterferenceoftheuandvquasiparticleamplitudes. Wealsoshowthat,contraryto
J
theassumptionsofpreviousanalyses,thereisnoregimefortrappedcondensatesforwhichthespectralresponse

4 functionandthedynamicstructurefactorareequivalent.Ournumericalcalculationscanalsobeperformedout-
2 sidethelinearresponseregime,andshowthatatlargelaserintensitiesasignificantdecreaseintheshiftofthe
spectralresponsefunctioncanoccurduetodepletionoftheinitialcondensate.
2
v
PACSnumbers:03.75Fi
0
8
4 I. INTRODUCTION andthedynamicstructurefactor,andshowthatforatrapped
8
condensatethereisnoregimeinwhichonecansimplybeob-
0
tainedfromtheother.
1
0 In 1999 Ketterle’s group at MIT reported a set of experi- WebegininsectionIIwithintheframeworkofmanybody
/ ments in which condensate properties were measured using field theoryand calculate, in the Bogoliubovapproximation,
t
a the techniqueof Bragg spectroscopy[1, 2]. In those experi- the linear response of the condensate to an applied Bragg
m mentsalowintensityBraggpulsewasusedtoexciteasmall pulse. We obtain expressions for the temporal evolution of
- amount of condensate into a higher momentum state, and thequasiparticleoperators,andshowthattheyhaveanonzero
d theBraggspectrumofthecondensatewasfoundbymeasur- meanvalue,i.e. thatthequasiparticlesaregeneratedascoher-
n
ing the momentumtransferfora rangeof Bragg frequencies entstates. We demonstratethatwithinawelldefinedregime
o
c (ω) and momenta (¯hq). That work established Bragg spec- the meanvaluesof the Bogoliubovoperatorsare identicalto
: troscopy as a tool capable of measuring condensate proper- theamplitudesobtainedfromalinearizedmean-field(Gross-
v
tieswithspectroscopicprecision. Thetheoreticalanalysisof Pitaevskii)treatment. Themeanfieldtreatmentthereforepro-
i
X the measurementshowever,givesrise to a numberof issues. videsa valid descriptionof the experimentsin the regimeof
r Ketterleandhiscolleaguesassumedthatthespectragaveadi- smallexcitation.
a rectmeasurementofthedynamicstructurefactor,whichisthe A number of meanfield theoretical treatments of Bragg
Fouriertransformofthedensity-densitycorrelationfunction, scatteringfromcondensateshavebeengiven.BlakieandBal-
andisfamiliarastheobservableinneutronscatteringexperi- lagh have presented a quantitative meanfield description [7]
mentsinsuper-fluidhelium[3, 4, 5]. Theyalsoattributedthe whichconfirmedtheanalysisoftheBraggspectroscopyshift
suppression of imparted momentum they observed at low q givenbythe MITgroup,andprovidedanalyticestimatesfor
values to correlated pair excitations, and quantum depletion anumberofquantities,includingthemomentumwidthofthe
ofthecondensate,andspeculated[6]thatanaccuratedescrip- scatteredcondensate. Stringariandcolleagueshavealsoused
tionwouldrequireamorecompletequantumtreatment. The ameanfielddescriptiontoanalyzeBraggspectroscopy[8,9],
purposeof the currentpaperis to developa theoryof Bragg andinadditionhaveusedtheapproachtodeviseschemesfor
spectroscopy that is valid in the regime of the experiments, measuringquasiparticleamplitudes[10],andformakingspa-
andtouseittomakequantitativecalculationsandtoanalyze tiallyseparatecondensatesinterfere[11]. Inthecurrentpaper
the phenomena that occur in this regime. We also investi- weusetheGrossPitaevskiiformulationofBraggscatteringas
gate the relationship between the observable of Bragg spec- presented by Blakie and Ballagh [7] to analyze the behavior
troscopy (i.e. the momentum transferred to the condensate) observedintheBraggspectroscopyexperiments.
Theobservableintheexperimentsisthemomentumtrans-
ferredtothecondensate,andinsectionIIIwedefineanormal-
izedversionoftheexpectationvalueofthisquantitythatwe
∗Electronicaddress:[email protected] callthespectralresponsefunctionR(q,ω).Foratrappedcon-
2
densate the momentum transfer can arise from two sources, (a)
theBraggbeamsorthetrapitself,whichcomplicatestheanal-
ysis. The MIT group recognized this issue, and applied the
Bragglaserpulsesonlyforasmallfractionofthetrapperiod,
q
and thenreleased the trap. Howeverthe tradeoffinvolvedin BEC
minimizingmomentumtransferfromthetrapbyusingashort
Braggpulsesignificantlycompromisestheenergyselectivity k2,w2 k ,w
of the process. We examinethe influencethis has on the re- 1 1
lationshipbetweenR(q,ω)and thedynamicstructurefactor
S(q,ω),andweshowthatinthepresenceofatrap,theevalu- (b)
ationofS(q,ω)requiresR(q,ω)tobeknownforallpossible
pulselengths. t)
(
V
ThecentralquantityofBraggspectroscopyisthusthespec- yVp
tralresponsefunctionR(q,ω)andwederiveanapproximate it
s
analyticexpressionforthisquantity,incorporatingtheeffects n
e
t
of both the meanfield interaction and the finite duration of n
I 0
the Braggpulse, in sectionIV. In sectionV we use R(q,ω) 0 Tp
to characterize our numerical investigations of Bragg spec- Time
troscopy,andweconsiderawiderangeofthreedimensional
axially symmetric scenarios, for which we simulate the ex- FIG.1: BraggspectroscopyofaBose-Einsteincondensate(BEC).
perimentsusingthemeanfield(GrossPitaevskii)equationfor (a)Twolaserbeamswithwavevectorsk andk andfrequenciesω
1 2 1
Braggscattering [7]. We verifythe validityrangeof ourap- andω respectively,createamovingopticalpotentialwithwavevec-
2
proximateformforR(q,ω)bycomparingittothefullnumer- torq=k k andfrequencyω=ω ω (see[7]).(b)Temporal
1 2 1 2
− −
icalresults,andweidentifytheregimesinwhichoneorother behaviouroftheBraggpulseassumedinthispaper.
ofthemechanismsof: themeanfieldinteraction,theDoppler
effect,andthefinite pulseduration,dominatestheformation
of the Bragg spectrum. We also show that our approximate A. Many-bodyfieldtheoreticapproach
formforR(q,ω)willallowamoreaccurateestimationofthe
momentumwidthof a condensatethanobtainedbyprevious
The many-bodyHamiltonian for N identical bosons in a
0
analyses.
trapandsubjecttoatime-dependentBraggpulsecanbewrit-
Our numericalsimulationsallow us to calculate the effect ten
on Bragg spectroscopy of laser intensities sufficiently large
that linear response theory no longer holds. We investigate Hˆ =Hˆ0+HˆI(t), (1)
caseswherethe scatteredfractionofthe condensateisofor-
der20%,andshowthatthedepletionofthegroundstatecon- whereHˆ istheusualtrappedbosonHamiltonian
0
densateleadstoasignificantreductionofthefrequencyshift,
which has not been accounted for in previous analyses. We ¯h2 U
alsoconsiderthespectralresponsefunctionfromavortex,us- Hˆ0 = drΨˆ† 2+VT(r) Ψˆ + 0 drΨˆ†Ψˆ†ΨˆΨˆ.
−2m∇ 2
ing two dimensional simulations. Finally in section VI we Z (cid:20) (cid:21) Z
(2)
investigate the energy response of a condensate subject to a
Braggpulseofsufficientlylongdurationthatindividualquasi-
V (r) is the trapping potential, which we choose to be har-
particleexcitationscanberesolved. T
monic. The Bragg interaction Hˆ (t) arises from two over-
I
lappingplanewavelaserfields,whichhaveequalamplitudes
but frequency and wave vector differences ω and q respec-
tively(see Fig. 1(a)). Thelaser fieldsare treatedclassically,
andtheir interactionwith the internaltransitionof the atoms
II. LOW-INTENSITYBRAGGSCATTERINGTHEORY
is characterizedbya Rabi frequencyΩ(t) (forthe combined
fieldsattheintensitypeaks)andadetuning∆whichislarge
Inthissectionwecalculatetheresponseofthecondensate and essentially the same for both laser fields. In this regime
to a Bragg pulse within the linear regime, using two distinct the internal structure for the atoms can be eliminated (see
approaches.Inthefirstofthese(sectionIIA)weusethemany [7] for details) so that the field operatorΨˆ refersonly to the
bodyfieldtheoryformalismintheBogoliubovapproximation, groundinternalstate,andHˆ (t)takestheform
I
tocalculatethetemporalevolutionofthequasiparticleopera-
tors. Inthesecondapproach(sectionIIB)weuseameanfield Hˆ (t)= drΨˆ†[h¯V(t)cos(q r ωt)]Ψˆ, (3)
(Gross Pitaevskii) equation and obtain the amplitudesof the I · −
Z
linearized response. The two approaches are shown to give
identicalmeanresultsinsectionIIC. whereV(t)=Ω2(t)/2∆ (seeFig. 1(a)).
| |
3
1. Bogoliubovtransformation 2. BogoliubovHamiltonian
For a highly occupied stationary state (T 0K) the field ApplyingthetransformationinEq.(4),totheHamiltonians
≈
operatorcanbewrittenintheBogoliubovapproximationasa (2)and(3)weobtaintheirBogoliubovform,i.e.
sumofmeanfieldandoperatorparts(Ψˆ = Ψˆ +φˆ). Inthis
paperwemostlyconsideragroundstate,buthweialsoconsider Hˆ0 ≈ Hˆ0B
the case of a central vortex. Following standard treatments E + ¯hω ˆb†ˆb , (14)
(e.g. [12, 13]) we employ a Bogoliubov transformation for ≡ 0 i i i
i
theoperatorparttowrite X
Hˆ (t) HˆB(t)
I ≈ I
Ψˆ(r,t) = N ψ (r)e−iµt+ei(S0(r)−µt) (4)
0 0 × N dr ψ 2¯hV(t)cos(q r ωt) (15)
0 0
p ˆbi(t)e−iωitu˜i(r)+ˆb†i(t)eiωitv˜i∗(r) , ≡ Z | | · −
Xi (cid:16) (cid:17) + N0 ˆb†ieiωit dr(u˜∗i +v˜i∗)
where the condensateis representedby the first term, andˆbi p Xi h Z
and ˆb† are the quasiparticle destruction and creation opera- ¯hV(t)cos(q r ωt)ψ +h.c ,
i × · − | 0|
torsrespectively(inaninteractionpicturewithrespecttoHˆ ).
0 i
Thestateψ = ψ exp(iS ), isaneigenstatesolution,with whereE isthe energyofthe highlyoccupiedstate (see Eq.
0 0 0 0
eigenvalue¯hµof|the|timeindependentGross-Pitaevskiiequa- (75)). The quasiparticle transformation diagonalizes Hˆ to
0
tion quadraticorder,andwenotethattheorthogonalbasis u˜ ,v˜
i i
{ }
¯h2 is requiredforthisdiagonalizationto bevalid. Inevaluating
¯hµψ0 = −2m∇2+VT(r) ψ0+N0U0|ψ0|2ψ0, (5) HˆIB,termsinvolvingproductsofquasiparticleoperatorshave
(cid:20) (cid:21) beenignored.ThisamountstoneglectingBragginducedscat-
andthefunctions u˜ ,v˜ aretheorthogonalquasiparticleba- teringbetweenquasiparticlestates,whichisoforder1/√N
i i 0
{ }
sis states [14]. These basis states are orthogonalto the con- smallerthanthetermslinearinˆb† (orˆb ). Thoselinearterms
i i
densate mode and are related to the usual (non-orthogonal) areofprimaryinteresthere,astheydescribethescatteringbe-
quasiparticlebasisstates ui,vi (givenbelow)byprojection tweenthecondensateandquasiparticlestateswhichoccursas
{ }
intothesubspaceorthogonaltothecondensate,i.e. aresultoftheenergyandmomentumtransferfromtheoptical
potential.
u˜ = u a ψ , (6)
v˜∗i = v∗i−+ai∗|ψ0|, (7) Thetimedependentexponentials,exp(±iωit),whichmul-
i i i| 0| tiplythequasiparticleoperatorsinEq.(4)accountforthefree
where evolutionduetoHˆB,andsotheHeisenbergequation
0
a = dr ψ u = dr ψ v , (8)
i Z | 0| i −Z | 0| i i¯h∂∂t ˆbie−iωit = ˆbie−iωit ,Hˆ0B+HˆIB(t) , (16)
(see[14]).
(cid:16) (cid:17) h(cid:16) (cid:17) i
TheformofBogoliubovtransformationusedinEq.(4)ex- becomes
plicitlyincludesthephase(S )ofψ ,whichisconvenientin
0 0
∂
acansuemswbehreorefdψi0ffiesrennottsnigecnecsosanrvielyntaiognrsoaupnpdesatraitne.thWeelinteortaetuthraet, i¯h∂tˆbi = ˆbi,Hˆ0B+HˆIB(t) −¯hωiˆbi, (17)
h i
andoursdiffersfromthatinRef. [13]. Wediscussthediffer- = ˆb ,HˆB(t) . (18)
i I
entconventionsintheAppendix. TheBogoliubov-deGennes
h i
equationfor ui,vi are Thisiseasilysolvedtogive
{ }
u +N U ψ 2v = h¯ω u , (9)
i 0 0 0 i i i
L | |
L∗vi+N0U0|ψ0|2ui = −¯hωivi, (10) ˆbi(t)=ˆbi(0)−βi(t), (19)
where
whereβ (t)isac-number,
i
¯h2
= ( +i S )2+V (r) ¯hµ+2N U ψ 2 ,
0 T 0 0 0
L −2m ∇ ∇ − | |
(cid:20) ((cid:21)11) β (t) = i N tdt′V(t′)eiωit′ dr
i 0
−
withtheorthogonalityconditions (pu˜∗+v˜Z∗0)cos(q r ωt′Z)ψ . (20)
× i i · − | 0|
dr u u∗ v v∗ = δ , (12) We seethattheBraggexcitationcausesthequasiparticleop-
{ i j − i j} ij
Z erators to develop nonzero mean values. Note that this is a
dr u v v u = 0. (13) completesolutionofthephysicsinthelinearizedregime,from
i j i j
{ − } whichanyobservablequantitiescanbecomputed.
Z
4
3. Initialconditions wherea isdefinedinEq. (8). Sincethequasiparticlesform
j
acompleteset,wemayuseprojectiontoobtainasetofequa-
OurderivationsofarhasbeenbasedonaT = 0KBogoli- tions for quasiparticle amplitudes that are equivalent to Eq.
ubov treatment. For this case the initial state is the quasi- (23),namely
particle vacuum state (0 ). Had we started with this ini-
tial condition and cons|idiered evolution in the Schro¨dinger c˙ (t) = iV(t)eiωit dr u∗ei(µt−S0(r))cos(q r ωt)Ψ
j − j · −
picture we would have found that the system evolves as Z h
ˆ|b0i β→ ex=pβ(iθ)|β{βi}ain,dwθheisreso|{mβei}pihaisseafaccothoerr(esnetes[t1a5te]),.i.e. +vj∗e−i(µt−S0(r))cos(q·r−ωt)Ψ∗ . (26)
i|{ i}i i|{ i}i i
Equation(19)also providesinsightinto finite temperature Becausethequasiparticlesoccupationsareallsmallcompared
caseswheremostoftheatomsareinthecondensate.Tofully tothecondensatemode,wecansimplifyEq. (26)bysetting
treatthe finite temperaturecase, itis necessaryto generalize Ψ(r,t)=√N ψ (r) exp(iS (r) iµt),yielding
0 0 0
| | −
theBogoliubovtreatmenttoaccountforthethermaldepletion
(e.g. see[12]),requiringthefunctionsψ0,ui,vi andωi tobe c (t) = i N tdt′V(t′)eiωit′ dr(u∗+v∗)
solvedforinaself-consistentmanner(e.g. see[16, 17,18]). i − 0 i i
In that case, Eq. (19) for the evolutionofˆb will still apply, p Z0 ′ Z
i cos(q r ωt )ψ . (27)
andthusweseethattheinitialstatisticsofˆb (0)arepreserved × · − | 0|
i
andthemeanvalue ˆb isshiftedbyβ . This will of course only provide a good solution while the
i i
h i
quasiparticleoccupationsallremainsmall.
B. Gross-Pitaevskiiequationapproach
C. Comparisonofapproaches
TheGross-PitaevskiiequationforacondensateofN par-
0
ticlessubjecttoaBraggpulsewasderivedin[7],i.e. Direct comparison of the Gross-Pitaevskii and quantum
fieldtheoreticresultsiscomplicatedbythefactthattheyhave
∂ ¯h2 beenderivedfordifferentbasissets. Wenotethatalthoughthe
i¯h Ψ(r,t) = 2+V (r)+U Ψ2 Ψ(r,t)
∂t −2m∇ T 0| | Gross-Pitaevskii analysis was carried out using the u ,v
(cid:20) (cid:21) { i i}
+h¯V(t)cos(q r ωt)Ψ(r,t), (21) basis, it is equally straightforward to use the orthogonalba-
· − sis u˜ ,v˜ . Morganetal. [14]haveshownthat(inanalysis
i i
{ }
where Ψ(r,t) is the condensate meanfield wavefunction for oftheGross-Pitaevskiiequation)transformingbetweenthese
theatomsintheirinternalgroundstate,andisnormalizedac- bases affects only the groundstate populationand givesrise
cordingto dr Ψ2 =N . Thecondensatewavefunctioncan tonodifferenceinthequasiparticleoccupations.
0
| |
beexpandedintermsofaquasiparticlebasisintheform Connections between the field theory and simple Gross-
R
Pitaevskiiresults based on Eq. (21) can only be expectedto
Ψ(r,t) = N0ψ0e−iµt+ei(S0(r)−µt) (22) existatT = 0K, wherethermaleffectscan be ignored1. In
p× ci(t)uie−iωit+c∗i(t)vi∗eiωit , tpheicstaretigoinmoefwtheebqeugaisnipbayrtcicolnesoidpeerriantgorthferormeleEvqa.n(t1v9a)cuumex-
i
X(cid:0) (cid:1)
where c are the time dependent quasiparticle amplitudes.
i t
This expansion has been made using the non-orthogonal ˆb (t) = i N dt′V(t′)eiωit′ dr(u˜∗+v˜∗)
quasiparticle basis, and the ground state has been assumed h i i − 0 i i
Z0 Z
static. This latter assumption will only be valid while ex- cpos(q r ωt′)ψ . (28)
0
citation induced by the Bragg pulse remains small. The × · − | |
wavefunctiondecomposition(Eq.(22))transformstheGross- Thisexpressionemphasizesthecoherentnatureofthequasi-
PitaevskiiEq. (21)into particle states, and is the same as the Gross-Pitaevskii result
ofEq. (27),withtheidentificationc (t) ˆb (t) . We note
i i
i¯h c˙i(t)uie−iωit+c˙∗i(t)vi∗eiωit (23) thattheapparentdifferencebetweenequa↔tionhs(27i)and(28),
i where the former depends on the matrix elements involving
X(cid:0) (cid:1)
=e−i(S0(r)−µt)¯hV(t)cos(q r ωt)Ψ(r,t). (24) ui,vi andthelatteronmatrixelementsinvolving u˜i,v˜i ,
· − d{isappe}arswith the observationthat u +v = u˜ +{v˜ (se}e
i i i i
Morganetal. [14]haveshownhowto usethe orthogonality Eqs. (6) and (7)). Thus we have verified that the c (t) and
i
relationsof the quasiparticles, Eqs. (12)and (13), to project ˆb (t) areidentical.
i
outthequasiparticleamplitudesfromacondensatewavefunc- h i
tion,namely
cj(t) = eiωjt dr ei(µt−S0(r))u∗jΨ−vj∗e−i(µt−S0(r))Ψ∗ 1WeusethewordsimpleheretodistinguishourT =0KGross-Pitaevskii
2a∗Z, h (25i) ebqeuenatuiosnedfrtoominmveosrteigealtaebofirnaitteeftoemrmpseroaftuGrreoesfsf-ePcittsa.evFsokriiexthaemoprylewseheic[h1h9a].ve
− j
5
D. Quasiparticleoccupation where nˆ (0) is the initial occupation, e.g. thermal occu-
i
h i
pation at T = 0K , and is included for generality. In Eq.
6
For the physics we consider in this paper the mean occu- (29)wehavechosentousethe ui,vi quasiparticlebasisfor
{ }
pationofthei-thquasiparticlelevel(i.e. ˆb†ˆb or c (t)2 for easeofcomparisonwithpreviousresultsbyotherauthors(e.g.
h i ii | i | [8,20,21,22]).
themany-bodyorGross-Pitaevskiimethodsrespectively)isof
interest, asitis usedto make a quasi-homogeneousapproxi-
mation to the spectral response function in Eq. (58). Using
Eq. (19)wecalculate
t Toprogress,itisnecessarytospecifythetemporalbehavior
hnˆi(t)i = N0 dt′eiωit′V(t′) dr(u∗i +vi∗) oftheBraggpulse. Forsimplicitywetakethepulseshapeas
(cid:12)Z0 Z beingsquare, with V = V for 0 < t < T (see Fig. 1(b)),
c(cid:12)(cid:12)os(q r ωt′)ψ0 2+ nˆi(0) , (29) thenEq. (28)becomes(evapluatedatt=Tp)p
× · − | | h i
(cid:12)
(cid:12)
(cid:12)
sin((ω ω)T /2)
ˆb (T ) = i N V ei(ωi−ω)Tp/2 i− p dr(u∗+v∗)eiq·r ψ
h i p i − 0 p (ω ω) i i | 0|
p h(cid:18) i− (cid:19) Z
sin((ω +ω)T /2)
+eiωTp i p dr(u∗+v∗)e−iq·r ψ . (30)
(ω +ω) i i | 0|
(cid:18) i (cid:19)Z i
Inthisexpression,thesecondterm(withdenominatorω +ω) a measurement of the dynamic structure factor. We briefly
i
willtypicallybesignificantlysmallerthanthefirstterm(with reviewthedynamicstructurefactor,anddiscusswhyitisin-
denominatorω ω)sinceω >0,sotoagoodapproximation appropriatefortheseexperiments.
i
−
we can ignore the second term. Similarly, the quasiparticle
occupationresult(29)underthesameapproximationis
A. Spectralresponsefunction
πV2T N 2
nˆ (T ) = p p 0 dr(u∗+v∗)eiq·r ψ
h i p i 2 i i | 0| IntheMITexperiments[1,2]alowintensityBragggrating
(cid:12)Z (cid:12)
F(ω ω(cid:12),T )+ nˆ (0) , (cid:12) (31) wasusedtoexcitethecondensatein-situforlessthanaquar-
i (cid:12) p i (cid:12)
× − (cid:12) h i (cid:12) ter ofa trap period. Immediatelyfollowingthis thetrap was
inwhich turnedoff,thesystemallowedtoballisticallyexpand,andthe
momentumtransfertothesystemwasinferredbyimagingthe
2sin2(ωT/2) expandedspatial distribution. The experimentalsignal mea-
F(ω,T)= , (32)
πT ω2 suredis(see[1,2,6])
isafamiliartermintimedependentperturbationcalculations. R(q,ω)=γ P(Tp)−P(0), (34)
In particularF(ω ω,T ) in Eq. (31)is sharplypeakedin ¯hq
i p
−
frequency about ω = ω , encloses unit area and has a half
i where
wreimdtahinosfsσmωall∼)th2iπs/teTrpm. cIanntbheetlaimkeint Taspa→δ-fu∞nc(twiohnileexpVrpe2sTsp- γ−1 = π2N0Vp2Tp, (35)
ingpreciseenergyconservation,sothatonlyquasiparticlesof
andP isthemomentumexpectationofthesystem. Weshall
energy¯hωwillbeexcited,i.e.
refer to R(q,ω) as the spectral response function. For the
πV2T N 2 caseofaGross-Pitaevskiiwavefunction,thespectralresponse
lim nˆ (T ) = p p 0 dr(u∗+v∗)eiq·r ψ functioncanbewrittenas
Tp→∞h i p i δ(ω2 ωi)(cid:12)(cid:12)(cid:12)Z+ nˆi(0i) . i | 0(|(cid:12)(cid:12)(cid:12)33) R(q,ω)=γ drΨ∗(r,Tp)(−i¯h∇)Ψ(r,Tp), (36)
× − (cid:12) h i (cid:12) ¯hq
R
where it is assumed that the initial condensate has zero mo-
III. OBSERVABLEOFBRAGGSPECTROSCOPY mentumexpectationvalue.Thefactorsγand¯hqappearingin
Eqs. (34) and (36) effectively scale out effects of the Bragg
In this section we outline the experimental procedure of intensityandduration,themagnitudeofmomentumtransfer,
Braggspectroscopyoncondensates. We beginbydiscussing andcondensateoccupation,sothatR(q,ω)canbeinterpreted
themeasuredobservable,whichwerefertoasthespectralre- as a rate of excitation per atom (normalized with respect to
sponsefunction. In[1, 2]thisobservablewasassumedtobe V2)withinthecondensate.
p
6
B. Dynamicstructurefactor quantity
2
Thedynamicstructurefactorhasplayedan importantrole
S(q,ω) = dr(u∗+v∗)eiq·r ψ
in the analysis of inelastic neutron scattering in superfluid i i | 0|
4He. Ithasfacilitatedtheunderstandingofcollectivemodes, Xi (cid:12)(cid:12)Z (cid:12)(cid:12)
(((cid:12)nˆ +1)δ(ω ω )+ nˆ(cid:12) δ(ω+ω )).
and has enabled measurements of the pair distribution func- × (cid:12)h ii − i h (cid:12)ii i
tion and condensatefractionin thatsystem, as discussed ex- (40)
tensivelyin[5]. Inthoseexperimentsamonochromaticneu-
tronbeamofmomentum¯hk isdirectedontoasampleof4He Thisexpressiongeneralizesthedynamicstructurefactortothe
0
and the intensity of neutrons scattered to momentum ¯hk′ is caseoflightscatteringandappliesatfinitetemperaturesinthe
measured. van Hove [23] showed that the inelastic scatter- regime of linear response, where the Bogoliubov theory for
ing cross section of thermal neutrons, calculated in the first quasiparticlesis valid. AtT = 0K, wherethermaldepletion
Bornapproximation,canbedirectlyexpressedintermsofthe can be ignored, the dynamic structure factor (40) takes the
quantity form
1 2
S(q,ω) = e−βEm|hm|ρˆq|ni|2δ(h¯ω−Em+En), S0(q,ω) = dr(u∗i +vi∗)eiq·r|ψ0| δ(ω−ωi),
Z m,n i (cid:12)Z (cid:12)
X X(cid:12) (cid:12)
(37) (cid:12) (T =0K(cid:12)) (41)
(cid:12) (cid:12)
which is called the dynamic structure factor (see [3]). In for which a number of approximate forms have been calcu-
Eq. (37), m and Em are the eigenstates and energy lev- lated(e.g.see[8,20,22]).
| i
els of the unperturbed system, is the partition function,
Z
ρˆ = drΨˆ†(r)exp( iq r)Ψˆ(r)isthedensityfluctuation
q
− ·
operator,andwehavetaken
ThedynamicstructurefactorandBraggspectroscopy
R
¯h2k2 ¯h2k′2
¯hω = 0 , (38) Althoughthe dynamicstructurefactorandthe spectralre-
2m − 2m
sponsefunctionaredistinctlydifferentquantities,theydore-
′
q = k0 k . (39) sembleeachotherstrongly.Infact,thematrixelementsinthe
−
T = 0K dynamic structure factor in (41) resemble those in
Ourchoiceofnotationforthesequantitiesistofacilitatecom-
theexpression(33)fortheBragginducedquasiparticlepopu-
parison between the matrix elements which arise in the dy-
lation, nˆ inthelongtimelimit.Itiseasytoshow,beginning
i
namic structure factor and Bragg cases. We note that in the h i
from(41),that
Braggcontextqandωrefertothewavevectorandfrequency
respectively of the optical potential, whereas in a dynamic
structurefactormeasurement,¯hqand¯hω arethemomentum S (q,ω)= γ lim nˆ (T ) , (T =0K) (42)
0 i p
and energy respectively, transferred to the system from the ( Tp→∞h i)
i
X
scatteredprobe.
For the case of a trappedgasBose condensate, low inten- where nˆ isgivenbyEq. (33)andγ isdefinedinEq. (35).
i
h i
sityoffresonantinelasticlightscattering[24]providesaclose NotethatbecauseEq. (42)isevaluatedatT = 0K,wehave
analogyto neutron scattering from 4He. Csorda´s et al. [20] takentheinitialoccupation, nˆ (0) inEq. (33),tobezero.In
i
h i
haveshownthatthecrosssectionforsuchinelasticlightscat- practiceaverylongpulse(T 1/ω )ofsufficientlyweak
p T
tering, with energyand momentumtransfer to the photonof intensity(V2T 1)couldbe≫usedtoexcitethequasiparti-
p p ≪
¯hω and ¯hqrespectively,canbeexpressedintermsofthe clesintheregimenecessaryforEq. (42)tohold.
− −
In [1, 2] the spectral response function R(q,ω) was assumed to represent a measurement of the dynamic structure factor,
S(q,ω).TheargumentgiveninthosepaperswasbasedontheassumptionthattheBraggpulsewouldexcitequasiparticlesof
definitemomentum¯hqandthemomentumtransferishenceproportionaltotherateofquasiparticleexcitation.Thisisinfactnot
so,althoughwecanshowthat,inacertainsense,thedynamicstructurefactorandthespectralresponsefunctiondodetermine
eachother.WeusetheresultofBrunelloetal. [9],whohaveshownthatthemomentumimpartedcanberelatedtothedynamic
structurefactoraccordingto
dP (t) V 2 sin([ω ω′]t)
z = mω2Z+2N ¯hq dω′ [S(q,ω′) S( q, ω′)] − , (43)
dt − z 0 2 − − − ω ω′
(cid:18) (cid:19) Z −
wheretheBraggscatteringhasbeentakentobeinthez-directionandP isthez-componentofthemomentumexpectation.The
z
7
quantityZ = N z istheexpectationvalueofthezcenterofmasscoordinateandevolvesaccordingto
h j=1 ji
P
dZ P
= z. (44)
dt m
Takingtheinitialpositionandmomentumexpectationstobezero,Eqs.(43)and(44)canbesolvedusing(34)togivethespectral
responsefunctionintermsofthedynamicstructurefactoras
1 cos(ω T ) cos([ω ω′]T )
R(q,ω)= dω′[S(q,ω′) S( q, ω′)] z p − − p . (45)
πT − − − [ω ω′]2 ω2
p Z − − z
Thisformulacanbeinverted,buttheinversionformulaearedifferentdependingonwhetherω iszeroornot:
z
S(q,ω) S( q, ω) = lim R(q,ω,T ) forω =0, (46)
p z
− − − Tp→∞
∞
= ω2 R(q,ω,T )T dT forω =0. (47)
z p p p z 6
Z0
Inthelinearizedapproximationweareusing,wecanuse(40)toshowthat
S (q,ω) = S(q,ω) S( q, ω), ω 0, (48)
0
− − − ≥
so that the differencingcancels out finite temperature effects. Beyond the Bogoliubovapproximation, there will however be
residualfinite-temperatureeffects.
Thusweseeitispossibletodeterminethezerotemperature homogeneousquasiparticlesweightedbythecondensateden-
dynamic structure factor in a good degree of approximation sitydistribution. Thisapproximationplaysanimportantpart
fromexperimentsonatrappedcondensate,providedmeasure- in theanalysisof thenumericalresultswe presentin section
mentsareperformedforasufficientrangeofpulsetimesT . V.
p
It is clear that this could be a difficult experiment to imple-
ment,sincethespectralresponsefunctionwilldropofffaster
than1/T forlargeT ,sothemeasuredsignalcouldbecome A. Homogeneousspectralresponse
p p
very small. But in analogy with the results of Sect.VI, we
would expect that significant information could be obtained TheresultsdevelopedinsectionIIforthetrappedconden-
bymeasuringforpulsetimesTpuptoabout5trapperiods. satescanbeappliedtoahomogeneoussystemofnumberden-
Ontheotherhandthereisnodirectconnectionbetweenthe sitynbymakingthereplacements
spectral response function and the dynamic structure factor
foranysingletimeTp unlessωz = 0,inwhichcasewemust u (r) u (n)eik·r/√ , (49)
i k
take T . In fact, if we note that significant structure → V
inS(qp,ω→)is∞expectedonthefrequencyscaleofωz,thenthe vi∗(r) → vk(n)e−ik·r/√V, (50)
formula(45)showsthatasmearingoverthisfrequencyscale N ψ (r) N / =√n, (51)
0 0 0
isassuredbytheformoftheintegrandin(45),independently → V
p p
ofthevalueofT . where isthevolume,and
p
V
One therefore must conclude that the spectral response
function, notthe dynamic structure factor, is the appropriate u (n) = ωkB(n)+ωk , (52)
method of analysis for Bragg scattering experiments. How- k
2 ωB(n)ω
ever, it is in principlepossible to determinethe zero temper- k k
aturedynamicstructurefactorinacertaindegreeofapproxi- qωB(n) ω
v (n) = k − k , (53)
mationfromtheseexperimentsbyuseoftheinversionformula k
−2 ωB(n)ω
(47). k k
q
ωB(n) = ω2+2nU ω /¯h, (54)
k k 0 k
IV. QUASI-HOMOGENEOUSAPPROXIMATIONTO q¯hk2
R(q,ω) ω = . (55)
k
2m
Inthissectionwedevelopanapproximationforthespectral Wehaveexplicitlywrittentheseasfunctionsofdensitynfor
responsefunctionvalidfortimescales shorterthanaquarter laterconvenience.UsingEqs. (49)-(51)theexpectationofthe
trap period. Because trap effects are negligible on this time quasiparticle operators(30) (i.e. at T = 0K) resulting from
scale we employ a quasi-homogeneous approach, based on Braggexcitationisfoundtobe
8
n sin((ωB(n) ω)T /2)
hˆbki = −iVp dr(uk(n)e−ik·r+vk(n)e−ik·r)eiωkB(n)Tp/2 e−iωTp/2 (ωkB(n)− ω)p eiq·r
rV ZV h (cid:18) k − (cid:19)
sin((ωB +ω)T /2)
+eiωTp/2 k p e−iq·r . (56)
(ωB+ω)
(cid:18) k (cid:19) i
Since the homogeneousexcitationsare plane waves, evaluatingthe spatial integralin Eq. (56) selects out quasiparticleswith
wavevectork= q. Theoccupations ˆb†ˆb ofthesestatesare
± h k ki
πV2T N
nˆ = p p 0(u (n)+v (n))2F((ωB(n) ω),T ), (57)
h ±qi 2 q q q ∓ p
whereF(ω,T)isdefinedinEq. (32).
Sinceaquasiparticlecreatedbyˆb† carriesmomentum¯hk,the We shallrefer to R as the finite time quasi-homogeneous
k QH
total momentumtransferred to the homogeneouscondensate approximationtothespectralresponsefunction,orsimplythe
is quasi-homogeneousapproximation. Ignoring the finite time
broadening effects and assuming exact energy conservation
p =h¯q( nˆ nˆ ), (58)
h i h qi−h −qi inRQH withthereplacementF(ω,T) δ(ω), reducesEq.
→
and the spectral response function, as defined in Eq. (34), (64)tothesimplerline-shapeexpression
becomes
R (q,ω) I (ω), (65)
QH q
p →
R(q,ω)=γ |h i|, (59) 15¯h(ω2 ω2) ¯h(ω2 ω2)
¯hq = − q 1 − q , (66)
=γ ( nˆ nˆ ), (60) 8ωqN0U0 s − 2ωqN0U0
q −q
h i−h i
ω
= q F((ωB(n) ω),T ) F((ωB(n)+ω),T ) , wherewehaveadoptedthenotation,Iq,asusedintheoriginal
ωB(n) q − p − q p derivation[1]. Equation(66)is alsoknownasthelocalden-
q
h i
(61) sity approximationto the dynamic structure factor (see [8]),
andhasbeenusedtoanalyzeexperimentaldatain[1,2]. We
where we have used (uq(n)+vq(n))2 = ωq/ωqB(n) (from emphasizethatwithTp <(1/4)Ttraptheδ-functionreplace-
Eqs. (52)and(53))toarriveatthelastresult. mentisunjustifiable,asweverifywithournumericalresults
inthenextsection.
B. Quasi-homogeneousspectralresponsefunction
V. BRAGGSPECTROSCOPY
The density distribution of a Thomas-Fermicondensateis
givenby Braggspectroscopycanbroadlybedefinedasselectiveex-
15N citationofmomentumcomponentsinacondensate,byBragg
N(n)= 4n20n 1−n/np, (62) lightfields. Inthissectionwe considerthe spectralresponse
p q function as a Bragg spectroscopic measurement, and using
whereN(n)dnistheportionofcondensateatomsintheden- numerical simulations of the Gross-Pitaevskii equation (21)
sityrangen n+dn,andn =h¯µ/U ,isthepeakdensity and the analytic results of the previous section we identify
p 0
→
(see[1]). thedominantphysicalmechanismsgoverningthespectralre-
To approximate the spectral response function for the in- sponse behavior. We investigate the spectral response func-
homogeneouscasewemultiplytheportionofthecondensate tion for a vortex and identify parameter regimes in which a
at density n by the homogeneousspectral response function clearsignatureofavortexisapparent. Fromthefullnumeri-
(61) for a homogeneouscondensate (of density n) and inte- calsimulationswearealsoabletoassesstheeffectofhigher
grateoveralldensitiespresent,i.e. laserintensitiesonthespectroscopicmeasurements.
R (q,ω) dnN(n)R(q,ω), (63)
QH
≡ A. NumericalresultsforR(q,ω)
Z
15N np nω
0 q
= dn 1 n/n
4n2 ωB(n) − p 1. Procedure
(cid:18) p (cid:19)Z0 q q
F(ωB(n) ω,T ) F(ωB(n)+ω,T ) .
× q − p − q p The numerical results we present for R(q,ω), are found
h (64i) byevolvinganinitialstationarycondensatestate(typicallya
9
ground state) in the presence of the Bragg optical potential,
(a) q=2/r (b) q=16/r
using Eq. (21). At the conclusion of this pulse, the spectral 0 0
response is evaluated using Eq. (36). This differs slightly 0.1
fromthetypicalprocedureintheexperiments,wherethesys-
)
temisallowedtoexpandbeforedestructiveimagingisusedto ω
measurethe condensatemomentum. Howeverwe haveveri- (q,
R0.05
fied numerically (in cylindrically symmetric 3D cases) that
condensateexpansion(afterthepulse)doesnotalterthemo-
mentum expectation value. For each desired value of q and
ω, we repeatourprocedureofevolvingΨ accordingto (21),
0
and calculating R(q,ω) immediately after the optical pulse 5 15 25 240 260 280
terminates. ω/ω ω/ω
T T
For axially symmetric situations, the simulations are cal-
culatedinthreespatialdimensionswithqorientedalongthe (c) q=3.76/r (d) q=31.6/r
0 0
z-axis. Whentheinitialstateisavortex,theinterestingcase
ofscatteringinadirectionorthogonaltothevortexcore(lying 0.02
onthez-axis)wouldbreakthesymmetryrequirement,sofor )
ω
these cases 2D simulations with q directed along the y-axis q,
areused. Forconvenienceweusecomputationalunitsofdis- R(0.01
tance r = ¯h/2mω ; interaction strength w = h¯ω r3;
0 T 0 T 0
and time t = ω−1 ; where ω is the trappingfrequencyin
0 p T T
thedirectionofscattering.
0
10 30 50 70 1000 1100
ω/ω ω/ω
2. Parameterregimes T T
We use square pulses of intensity and duration such that FIG.2: SpectralresponsefunctionR(q,ω)ofa3Dcondensate. (a),
typicallylessthan1%ofthecondensateisexcited,exceptin (b) Spherical condensate, N0U0 = 104w0, µ = 14.2ωT. (c), (d)
Oblatecondensate withtrapasymmetry λ = √8, N U = 5.6
cseocntdioennsVatEe.wAhserleonwgeaisnvthesetiagmatoeutnhteonfoenxlicniteaatriorensipsosnmsealolf,twhee 105w0,µ = 70.9ωT. FullnumericalsolutionforRs0how0nassol×id
verifythatthespectralresponsefunctionR(q,ω)isindepen- line; the local density approximation (Iq) dash-dot; and the finite
timelocaldensityapproximation(RQH)dashed. Braggparameters
dent of V . However, the shape of R(q,ω) is dependenton
p areVp =0.2ωT andTp =0.4/ωT.
the pulse duration (in accordance with the frequency spread
about ω associated with the time limited pulse) and on the
magnitudeofq.
whereasFig. 2(d)isinthefreeparticleregime(q=31.6/r ).
Themomentum¯hq definedby 0
0 InallcaseswecomparetheGross-PitaevskiicalculationofR
with the lines-shape I and quasi-homogeneousresult R
¯hq = 2mn U , (67) q QH
0 p 0 fromEqs. (66)and(64)respectively.
p
(i.e. q = 1/ξ, where ξ is the condensate healing length)
0
characterizesthedivisionbetweenregimesofphononandfree
particle-likequasiparticlecharacter. We notethattheexperi- 1. Mechanisms
mentalresultsin[1]and[2]reportmeasurementsofR(q,ω)
inthefreeparticleandphononregime,respectively.
The two ground states used in calculating the results pre-
sentedinFig. 2arebothintheThomas-Fermilimit(i.e. they
satisfy the condition µ ω )2. We can see from the fig-
T
B. Underlyingbroadeningmechanisms ≫
ure that the local density approximation I Eq. (66) used
q
by previousauthorsdoesnotalways give a gooddescription
WepresentinFig. 2spectralresponsefunctionscalculated of R(q,ω), whereas our quasi-homogeneousapproximation
usingtheGross-Pitaevskiisimulationsinthreespatialdimen- R (q,ω) Eq. (64) is much more accurate. We have in-
QH
sions.ThespectralresponsefunctionsinFigs.2(a)and(b)are vestigatedtheaccuracyofR overawideparameterrange
QH
for a spherically symmetric ground state with q0 = 3.8/r0. whichhasallowedustoevaluatetherelativeimportanceofthe
Theqvalues(oftheBraggfields)werechosensothatthecase threeunderlyingbroadeningmechanismswhichcontributeto
in Fig. 2(a) is in the phononregime (q = 2/r ), while Fig.
0
2(b)isinthefreeparticlelimit(q =16/r ). Forcomparison,
0
we present in Figs. 2(c) and (d) spectral response functions
fora differentgroundstateofgreaternonlinearity,forwhich 2ForournumericalsimulationsweuseGross-Pitaevskiieigenstatescalcu-
q0 =8.4/r0. Fig.2(c)isinthephononregime(q=3.76/r0), latedfromEq.(5)
10
R(q,ω). Themechanismsandtheir contributionsareasfol- ThesefiguresareindicativeoftypicalBraggspectroscopyre-
lows: sults for a small to medium size condensate. The state used
in Figs. 2(c) and (d) corresponds to about 107 Na atoms in
i) Theshiftintheexcitationspectrumduetothemeanfield
a100Hztrap,andwaschosentomatchsomeofthefeatures
interaction(dependingonthelocaldensity). Therange
oftheexperimentsreportedin[1,2],e.g. thepeakdensityof
of densities present in the condensate cause a spread
this state is 4 1014 atoms/cm3 and the chemical poten-
in this shift. The frequencywidth associated with this ∼ ×
tial is 6.7kHz. The momentumvalueschosen correspond
spreadis proportionalto the chemicalpotentialµ, and ∼
tothoseusedtoprobethephononandfreeparticleregimesin
weshallrefertothisasthedensitywidth-see section thosepapers(Bragggratingformedby589nmbeamsat14◦
IV. and 180◦ respectively). Computationalconstraints mean we
cannotmatchtheexperimental(prolate)trapgeometry,butin
ii) TheDopplereffectduetothemomentumspreadofthe
a similar manner to the measurements made in [1] we scat-
condensate;theDopplerbroadenedfrequencywidthis
teralongatightlytrappeddirection,althoughfromaconden-
σ q/m,whereσ isthecondensatemomentumwidth.
p· p sate inanoblatetrapofaspectratio√8. We notethatin the
iii) The frequency spread in the Bragg grating due to the phonon probing experiments [2], scattering was performed
finite pulse time; the width arising from this effect is in the weakly trapped direction for imaging convenience. It
π/T . is worth emphasizing that the mechanisms accounted for in
p
∼
ourapproximateresponsefunctionR dependonlyonthe
QH
peakdensity(i.e. ¯hµ/U ),themagnitudeofq andtheBragg
0
2. Relativeimportanceofmechanisms pulseduration.ThereasonisthatcondensatesintheThomas-
Fermi regime with the same peak density will have identi-
Therelativeimportanceofthesemechanismsvariesaccord- cal density distributionsin either prolate or oblate traps (see
ingtotheparameterregime. InTableIwecomparetheesti- Eq. (62)). Thus the quasi-homogeneousapproximationpre-
mated values of these widths for the simulations in Fig. 2. dictsthesamespectralresponsefunctionforboth. Scattering
WeseethatforthecaseofFig. 2(a)bothfinitetimeandden- inatightlytrappeddirectionwillenhancemomentumeffects
notaccountedforinR ,sincespatiallysqueezingthecon-
QH
density finiteTp momentum densate causes the corresponding momentum distribution to
Fig. µ π/Tp σp q/m broaden. However, it is apparent from Table I, that for the
·
2(a) 14.2 7.9 0.86 case of large condensatesthis momentumeffect is relatively
2(b) 14.2 7.9 6.92 small, and thus we expectour resultin Figs. 2(c) and (d) to
2(c) 70.6 7.9 0.88 giveareasonablyaccuratedescriptionoftheMITexperiments
2(d) 70.6 7.9 7.4 [1,2].
TABLEI: Simpleestimateofwidthsofcomponentmechanismsof
C. Spectralresponsefunctionofavortex
thespectralresponsefunctionsinFig. 2. Allquantitiesexpressedin
unitsofωT.
In previous work [25] we showed how Bragg scattering
from a vortex can produce an asymmetric spatially selective
sity effects are important; hence the quasi-homogeneousre-
beam of scattered atoms which providesan in-situ signature
sult which includes both of these is in good agreement with
ofavortex. InFig. 3wecomparethespectralresponsefunc-
R(q,ω), whereas I (ω) which accounts for density effect
q
tionsfrom two dimensionalgroundand vortexstates, and in
aloneisquiteinaccurate. InFig. 2(b)thevalueofq ismuch
TableIIwesummarizethedensity,finitetimeandmomentum
larger, increasing the momentum width to a point where it
effects for the cases in Fig. 3. In the low momentumtrans-
is comparable to the other mechanisms. Since both I and
q
R fail to account for the condensate momentum, neither
QH
density finitetime momentum
approximationtothespectralresponsefunctionisinparticu-
larly good agreement with the numerically calculated R. In Fig. µ π/Tp σp q/m
·
bothFigs. 2(c)and(d)thedensitywidthisanorderofmagni- 3(a)ground 9.0 3.9 2.9
3(a)vortex 9.2 3.9 5.2
tudelargerthanboththe finite pulse andmomentumwidths,
3(b)ground 9.0 3.9 6.7
and so in this regime the simple line-shape expression I is
q 3(b)vortex 9.2 3.9 12.2
generallyadequate.
TABLEII: Simpleestimateofwidthsofcomponentmechanismsof
3. Experimentalcomparison thespectralresponsefunctionsinFig. 3. Allquantitiesexpressedin
unitsofωT.
Taking typical experimental parameters, the state used in
Figs. 2(a)and (b) correspondsto about2.8 105 Na atoms fer case Fig. 3(a), the density width is the most significant
×
ina50Hztrap,withtheBragggratingformedfrom589nm component of the response function width, although the in-
laserbeamsintersectingatanangleof5◦in(a)and39◦in(b). creasedmomentumdistributionofthevortexstaterelativeto