Spatial effects in superradiant Rayleigh scattering from Bose-Einstein condensates

0.55 MB·English

by  O. Zobay

Spatial effects in superradiant Rayleigh scattering from Bose-Einstein condensates
O. Zobay and Georgios M. Nikolopoulos
Institut fu¨r Angewandte Physik, Technische Universit¨at Darmstadt, 64289 Darmstadt, Germany
(Dated: February 6, 2008)
We present adetailed theoretical analysis of superradiant Rayleigh scattering from atomic Bose-
Einstein condensates. A thorough investigation of the spatially resolved time-evolution of optical
6
and matter-wave fields is performed in the framework of the semiclassical Maxwell-Schr¨odinger
0
equations. Ourtheoryisnotonlyabletoexplainmanyoftheknownexperimentalobservations,e.g.,
0
the behavior of the atomic side-mode distributions, but also provides further detailed insights into
2
thecoupleddynamicsofopticalandmatter-wavefields. Toworkoutthesignificanceofpropagation

n effects, we compare our results to other theoretical models in which these effects are neglected.
a
J PACSnumbers: 03.75.Kk,32.80.Lg,42.50.Ct

1
3
I. INTRODUCTION wards with respect to the direction of the applied laser

] field. Since the latter process requires to overcome an
r
e The beautiful recent experiments of Refs. [1, 2] have energy mismatch, it only occurs for sufficiently intense
h laser pulses. In this way, one is led to distinguish be-
opened up the possibility of studying superradiance
ot within the context of Bose-Einstein condensates of ul- tween the weak- and the strong-pulse regimes of super-
t. tracold atomic gases. In a typical experimental setup, radiance. The behavior of the system strongly differs in
a these two regimes. In particular, one important differ-
a cigar-shaped condensate is exposed to a far-off reso-
m ence concerns the atomic side-mode population patterns
nant laser pulse travelling in a direction perpendicular
- to the long condensate axis [see Fig. 1(a)]. Condensate which can be investigated through time-of-flight imag-
d ing. In the weak-pulse regime, where forward scattering
atoms can then undergo Rayleigh scattering thereby ex-
n
isprevalent,distributionstakeacharacteristicfanshape,
o periencingarecoilkick. Themovingatomstogetherwith
whereas in the strong-pulse regime, an X-like pattern is
c the condensate at rest form matter-wave gratings from
[ which further laser photons are scattered. This in turn observed.

produces additional recoiling atoms, causing the ampli- Basic features of the atomic momentum distributions
1
v tudesofthematter-wavegratingsandthescatteredlight weretheoreticallydiscussedbyanumberofauthors(see,
4 fields to grow rapidly in a self-amplifying process. Due e.g., [6, 7, 8, 9, 10]) without, however, giving a com-
1 to phase-matching effects induced by the elongated con- prehensive description of the experimental observations.
7 densateshape,thefastestgrowthisexperiencedbythose Reference [10], for example, examined a model in which
1 gratings for which the scattered photons travel parallel tworegimesofeitherpureforwardscatteringorcombined
0
to the condensate axis in the two so-called optical “end- backward and forward scattering could be distinguished
6
fire modes.” After an initial start-upandmode competi- depending on the external parameters. However, this
0
/ tion process [3], the ensuing superradiant light emission study only considered a strictly one-dimensional system
at is concentrated into the endfire modes, and the recoil- so that a direct comparison to the experiments was not
m ing atoms have well-defined momenta thus forming two possible.
- first-order atomic “side modes” [Fig. 1(a)]. Amorecomprehensiveinvestigationoftheatomicside-
d Comparing this process to “conventional” superradi- mode distributions was recently presented in our work
n
ance in an inverted atomic medium [4, 5], we see that [11]. In Ref. [11], the typical X- and fan-shape patterns
o
the role of the electronically excited atoms is now taken alongwithsomeoftheircharacteristicpropertieswerere-
c
: by the combination of the condensate and the applied produced and explained in terms of the underlying cou-
v
laser pulse, spontaneous photon emission is replaced by pled dynamics of optical and matter-wave fields. Our
i
X Rayleigh scattering, and de-excited atoms correspond to approach is built upon two main concepts: First of all,
r atoms inthe momentum side modes [2]. Another impor- we use a semiclassicaldescription for the field dynamics.
a tant difference to the conventional process, which stops Following earlier studies of conventional superradiance
after each atom has undergone a single transition, is the [5,12,13],wecanexpectthatthesemiclassicalapproach
possibilityofrepeatedormodifiedscatteringcycleswhich is valid as soon as the numbers of atoms and photons
lead to the population of multiple atomic side modes. in the atomic side and optical endfire modes, respec-
Theseprocessesoccurnaturallyiftheappliedlaserpulse tively, become large compared to one. It is this macro-
is of sufficient duration or strength. On the one hand, scopic regime of superradiance that we are focussing on
anatomin aside mode canagainscattera laserphoton, throughout this work. A fully quantized model, on the
thereby transferring to a higher-order side mode. Alter- other hand, such as the one used by Meystre and co-
natively, however, atoms may also interact with endfire workers [3, 7], allows to investigate the initial startup
mode photons and scatter them back into the laser field of the superradiant process, but is not easily extended
[2]. This leads to the production of atoms moving back- to the study of long-time dynamics. The second main
2
feature of our approachis the inclusion of spatial propa- (a) ~k + ez
E
gation effects. This aspect has been neglected in almost
(0,0) ex
all previous quantal as well as semiclassical treatments
of Bose-Einstein condensate (BEC) superradiance (see, El ~kl
e.g., [3, 6, 7, 8, 9, 10]). References [14, 15] examine ~k
l
some spatial effects in the BEC-light interaction, but do ~k
not provide a detailed comparison to the experimental −
oevbesre,rvsahtoiwontshoaftRtehfes.in[1c,lu2s].ioTnhoefrsepsautltiasloefffReecft.s[i1s1]c,rhuociwa-l −~k E− (1,−1)
(b)
for a full understanding of BEC superradiance. In fact,
our results also allowedus to resolve the controversybe-
(n 1,m+1) (n+1,m+1)
tween Refs. [2] and [7] regarding the origin of the spa- −
tial asymmetry between forward- and backward-moving
atomicsidemodesobservedinthestrong-pulseregimeof
superradiance.
(n,m)
Thissuccessfulexplanationofseveralkeyobservations
of the superradianceexperiments [1, 2] suggeststhat the
model of Ref. [11] provides an adequate description of
the system dynamics in the semiclassical regime. The (n 1,m 1) (n+1,m 1)
− − −
purpose of the present paper is therefore to extend our
previous work by providing an in-depth examination of FIG.1: SchematicrepresentationofBECsuperradiance. (a)
this model and, in this way, to obtain further detailed Acigar-shapedBECoflengthL(filledellipsoid)isexposedto
insights into the system dynamics. We also continue to alinearlypolarizedlaserpulsewithamplitudeEl(t)andwave
stress the significance of spatial propagation effects by vector kl. Stimulated Rayleigh scattering of laser photons
explicitly comparing our results to a related mean-field off the condensate produces recoiling atoms in side modes
with well-defined momenta. The side mode (0,0) refers to
modelinwhichtheseeffectsareneglected(see,e.g.,[10]).
the BEC at rest whereas only one of the two first-order side
Thepaperisorganizedasfollows. InSec.II,wepresent
modes, i.e., (1,−1), is shown. The scattered photons propa-
our theoretical framework which is based on the semi-
gate mainly along the condensate axis in the endfire modes
classical Maxwell-Schr¨odinger equations and the use of
E±. (b)Populationtransfertoandfromthesidemode(n,m)
the slowly-varying-envelope approximation. We also in- duetointeractionwiththelaserpulseasimpliedbyEq.(7).
dicate how one may include photon scattering within or
between the optical endfire modes. Section III analyti-
cally discusses the short-time gain regime. In particular,
faroffresonantfromthe atomictransitiontothe excited
itisshownthatduetothepropagationeffects,thegrowth
electronic state e .
of the atomic side-mode populations and opticalendfire- | i
As discussed in the Introduction, in this system the
mode intensities is slower than exponential. In Sec. IV,
coherent nature of the BEC leads to strong correlations
we discuss the strong-pulse regime and explain in de-
betweensuccessiveRayleighscatteringeventsandtocol-
tail how the characteristic observations of X-shape side-
lective superradiant behaviour [1, 2, 3]. Moreover, as a
mode patterns and the spatial asymmetry come about.
result of the cigar shape of the condensate, the gain is
Wealsoexhibitexplicitlythefailureofspatiallyindepen-
largest when the scattered photons leave the condensate
dent models to account for these effects. In Sec. V, the
travellingupand downits long axis in the so calledend-
weak-pulseregimeisinvestigated. Besidesdiscussingthe fire modes (x,t)e e−i(ωt∓kz), where (x,t) are the
± y ±
atomic side-mode distribution patterns, we also use our E E
envelope functions of the modes. As a consequence, the
model to study characteristic spatial effects in the cou-
recoiling atoms have well-defined momenta and appear
pled dynamics of optical and matter-wave fields. It will
in distinct atomic side modes. In the side mode (n,m),
turnoutthattheseeffectsaresignificantlymoreinvolved atoms possess momentum ~(nk e +mke ), while their
l x z
than in the strong-pulse regime. The paper concludes kineticenergyisgivenby~ω =~2(n2k2+m2k2)/2M.
with a short summary and outlook in Sec. VI. n,m l
Inthisnotation,the“sidemode”(0,0)describesthecon-
densate at rest. The wave vectors ke of the scattered
z
±
photons are fixed by energy conservation for the tran-
II. MAXWELL-SCHRO¨DINGER EQUATIONS sitions between the side modes (0,0) and (1, 1) which
initiate the process, i.e., ~ck = ~ck+~ω . G±iven that
l 11
Our model involves an elongated condensate of length ωn,m ωl, we approximately have k kl and kl k
≪ ≈ − ≪
L oriented along the z axis. The BEC consists of k,kl. Thus, the side-mode frequency is approximately
N two-level atoms which are coupled via the electric- givenbyωn,m ≈(n2+m2)ωr,whereωr =~kl2/2M isthe
dipoleinteractiontoalinearlypolarizedpumplaserpulse recoil frequency.
l(t)ey(ei(klx−ωlt)+c.c.)/2withωl =ckl,travelinginthe Maxwell-Schr¨odinger equations of motion for the cou-
E
x direction [see Fig. 1(a)]. The laser is considered to be pled matter-wave and electromagnetic fields offer a very
3
generaltheoreticalframeworkforthedescriptionofprob- E(−)(x,t)=E(+)∗(x,t), with ω =kc and A the average
lemspertainingtotheinteractionofultracoldatomswith condensatecrosssectionperpendiculartothezaxis. The
light[16,17,18]. Fortheproblemathand,afteradiabati- electricfieldinEq.(5)containstheimpinginglaserpulse
callyeliminatingtheexcitedatomicstate e ,thecoupled togetherwiththetwoopticalendfiremodesthatarepro-
| i
Maxwell-Schr¨odingerequationsforthemean-fieldmacro- duced by collective Rayleigh scattering. The atoms in
scopicatomicwavefunctionψ(x,t)andthepositive-and the side mode (n,m) are characterizedby a slowly vary-
negative-frequencycomponentsE(±)(x,t)oftheclassical ingspatialenvelopeψ (z,t). ThesummationinEq.(4)
nm
electric field read [5, 16, 17] canberestrictedtoterms(n,m)withm+neven,asEq.
(7) below will show.
∂ ~2 (d E(−))(d E(+))
i~ ψ = ∆ψ+ · · ψ, (1)
∂t −2M ~δ
Note that in the ansatz (4) and (5) we disregard the
∂2E(±) 1 ∂2P(±)
= c2∆E(±) (2) dependenceoftheenvelopefunctionsψnm and ± onthe
∂t2 − ε0 ∂t2 transversedirectionsxandy. Forthe matterwEaves,this
iscertainlyagoodapproximationsincetheradialdegrees
withδ the detuning ofthe laserfromthe electronictran-
of freedom are tightly confined by the trap. Moreover,
sition, d the atomic dipole moment and M the atomic
theuseofthesameapproximationfortheopticalfieldsis
mass. The polarization is given by
justified by the fact that in the experiments the Fresnel
number of the system is close to 1 [1, 5].
d E(+)(x,t)
P(+)(x,t)= dψ(x,t)2 · , (3)
− | | ~δ
Inordertointroduceaconcisenotation,werescalethe
with P(−) = P(+)∗. Note that in Eq. (1) we neglect optical and matter waves as
the externaltrapping potentialandinteractionsbetween
atoms, since they do not play a significant role on the
time scales of the process under consideration. However,
itwouldbestraightforwardtoincludetheminthemodel.
In order to solve Eqs. (1) and (2) under the slowly- ~ωk ψ √k
l nm l
e ; ψ . (6)
varying-envelope approximation (SVEA), we decompose E±,l → ±,lr2ε0A nm → √A
the fields as
ψ (z,t)
ψ(x,t) = nm e−i(ωn,mt−nklx−mkz), (4)
√A
X
(n,m)
Then, using the ansatz (4) and (5) and introducing the
E(+)(x,t) = e e−i(ωlt−klx)/2+ (z,t)e e−i(ωt−kz)
El y E+ y dimensionless time τ =2ωrt and length ξ =klz, Eq. (1)
+ (z,t)e e−i(ωt+kz), (5) in the SVEA reads
− y
E
∂ψ (ξ,τ) 1∂2ψ (ξ,τ) ∂ψ (ξ,τ)
nm nm nm
i = im
∂τ −2 ∂ξ2 − ∂ξ
+κ e∗(ξ,τ)ψ (ξ,τ)ei(n−m−2)τ +e∗(ξ,τ)ψ (ξ,τ)ei(n+m−2)τ
+ n−1,m+1 − n−1,m−1
h
+e (ξ,τ)ψ (ξ,τ)e−i(n−m)τ +e (ξ,τ)ψ (ξ,τ)e−i(n+m)τ
+ n+1,m−1 − n+1,m+1
i
+λ e∗(ξ,τ)e (ξ,τ)ψ (ξ,τ)e2i(m−1)τ +e∗(ξ,τ)e (ξ,τ)ψ (ξ,τ)e−2i(m+1)τ
− + n,m−2 + − n,m+2
h i
+λ(e (ξ,τ)2+ e (ξ,τ)2)ψ (ξ,τ), (7)
+ − nm
| | | |
with the coupling constants where
d2 ~ω
l l
g = | | E . (9)
2~2δ r2ε AL
0
g
κ = k L, (8a)
2ω l Thefirsttermontheright-handsideofEq. (7)describes
rp
κ thequantum-mechanicaldispersionoftheenvelopefunc-
λ = , (8b)
e tion, while the second one leads to a spatial translation
0
4
with velocity v = m~k/M. The other terms describe backwardscattering [2].
m
theinteractionbetweenthematter-waveandelectromag-
Anotherclassofphysicalprocessesinvolvesthephoton
neticfields. Thisinteractionleadstospatiallydependent
exchangebetweenthetwosidemodesandisdescribedby
shifts and couplings of the momentum side mode (n,m)
the terms with e e∗. In this case the accompanying re-
to other modes. Let us discuss the underlying physical ± ∓
coil transfers the atom from (n,m) to the side modes
processes in more detail.
(n,m 2). Finally, the terms containing e (ξ,τ)2 de-
The terms involving the coupling constant κ refer to ± | ± |
scribe the absorption of a photon from an endfire mode
photon exchange between one of the endfire modes and
and its subsequent emissioninto the same mode. Hence,
the laser beam [see Fig. 1(b)]. In particular, through
such processes give only rise to shifts which depend on
stimulatedscattering,anatominasidemode(n,m)can
both time and space, but do not introduce couplings to
absorba laser photonand deposit it into one of the end-
other side modes. The corresponding term for the laser
fire modes. The accompanying recoil transfers the atom
beam is related to a constant ac Stark shift and as such
into one of the side modes (n+1,m 1). Alternatively,
± is not included in Eq. (7).
theatommayabsorbanendfire-modephotonandemitit
into the laser beam, thereby ending up in the side mode In the SVEA, the envelope functions e of the endfire
±
(n 1,m 1). Thislatterprocessisresponsibleforatomic modes obey the equations
− ±
∂e ∂e
+ +χ + = i κei(n−m)τψ (ξ,τ)ψ∗ (ξ,τ)
∂τ ∂ξ − X h nm n+1,m−1
(n,m)
+λe (ξ,τ)e−2i(m−1)τψ (ξ,τ)ψ∗ (ξ,τ)+λe (ξ,τ)ψ (ξ,τ)2 , (10)
− nm n,m−2 + | nm | i
∂e ∂e
− χ − = i κei(n+m)τψ (ξ,τ)ψ∗ (ξ,τ)
∂τ − ∂ξ − X h nm n+1,m+1
(n,m)
+λe (ξ,τ)e2i(m+1)τψ (ξ,τ)ψ∗ (ξ,τ)+λe (ξ,τ)ψ (ξ,τ)2 (11)
+ nm n,m+2 − | nm | i
where our model (at least for the time-scales of interest) [1, 2].
Hence, Eq. (7) may be simplified to
ck
l
χ = . (12)
2ωr i∂ψnm(ξ,τ) = 1∂2ψnm im∂ψnm (15a)
∂τ −2 ∂ξ2 − ∂ξ
Introducing the atomic natural linewidth Γ =
a
d2ω3/(3πε0~c3), the coupling constant g can be ex- +κ e∗+ψn−1,m+1ei(n−m−2)τ +e∗−ψn−1,m−1ei(n+m−2)τ
pressed in terms of the Rayleigh scattering rate R = +e(cid:2)ψ e−i(n−m)τ +e ψ e−i(n+m)τ .
d2 2Γ /(4~δ2) as + n+1,m−1 − n+1,m+1
El a (cid:3)
Sincetheinteractionbetweentheopticalandthematter-
3πc3R wave fields is restricted to the condensate volume, any
g = . (13)
r2ω2AL relevant retardations are expected to be of the order of
L/c 10−12s and can be neglected. Thus, formal inte-
Moreover,the collective superradiant gain is given by grati≃on of Eqs. (10)-(11) for λ = 0 yields the following
equations for the envelope functions e :
g2N ±
G= , (14)
γ κ ξ
e (ξ,τ) = i dξ′ ei(n−m)τ
+
− χZ
with the photon damping rate γ =2c/L (note slight dif- −∞ (Xn,m)
ferencesfromRef.[10]regardingthenumericalprefactors ψ (ξ′,τ)ψ∗ (ξ′,τ), (15b)
× nm n+1,m−1
inthedefinitionsofGandγ). Experimentalobservations κ ∞
clearly distinguish between two different regimes of pa- e (ξ,τ) = i dξ′ ei(n+m)τ
−
− χZ
rameters, the so-called strong- and weak-pulse regimes ξ X
(n,m)
[1,2,3]. Theformerregimeischaracterizedbyascatter- ψ (ξ′,τ)ψ∗ (ξ′,τ). (15c)
ing ratecomparableto the recoilfrequencyandG ω , × nm n+1,m+1
r
≫
while for the latter R ω and G ω . From these equations one may clearly see how the
r r
≪ ≤
For both of these regimes, one can verify that photon buildup of the endfire-mode fields at (ξ,t) is driven by
exchange between endfire modes can be disregarded in the coherences ψ ψ∗ .
nm n+1,m±1
5
AsmentionedintheIntroduction,previoustheoretical the macroscopic dynamics that are observed in individ-
studies of BEC superradiance have neglected the spa- ual experimental realizations, such as those reported in
tial dependence of optical and matter-wave fields. It is Refs. [1, 2].
one of the main purposes of this paper to compare the Tothisend,wehavenumericallysolvedEqs.(15)with
model derived above to this simpler approach in order fixed externalparameters for a large set of different seed
to assess the significance of propagation effects. For- functions, and we find the characteristic qualitative fea-
mally, we can obtain the equations of motion for the tures, e.g., the side-mode distribution patterns, to show
spatially independent model from Eqs. (15) by dropping up independent of the choice of the specific initial con-
any spatial dependence and setting ψ C /√k L dition. This shows that these features are caused by the
nm nm l
→
and e b /√k L. The amplitudes C (τ) and b (τ) macroscopic dynamics and are not related to quantum
± ± l nm ±
→
canbe interpretedasamplitudesofthe matter-waveand fluctuation effects. In Sec. III we will explain how this
optical fields, respectively, and they obey the equations insensitivity to the initial conditions comes about.
of motion
dC (τ) g
nm = i b∗C ei(n−m−2)τ III. EARLY STAGE OF SUPERRADIANT
dτ − 2ωrh + n−1,m+1 SCATTERING
+b∗C ei(n+m−2)τ +b C e−i(n−m)τ
− n−1,m−1 + n+1,m−1
+b C e−i(n+m)τ , (16a) In this section, we investigate the early stages of the
− n+1,m+1
i superradiant process in detail. More precisely, we wish
g
b (τ)= i ei(n−m)τC (τ)C∗ (τ), (16b) toexaminetheregimewherethepopulationsofthefirst-
+ − γ X nm n+1,m−1 order side modes still remain far below the number of
(n,m)
g atoms in the condensate, but are large compared to
b−(τ)=−iγ ei(n+m)τCnm(τ)Cn∗+1,m+1(τ). (16c) 1, so that the semiclassical model is applicable. The
(Xn,m) undepleted-pump approximation for the condensate can
thenbeinvoked,whichallowstoderiveanalyticalresults
A similar model, which distinguished only the momen-
for the spatial distributions of the side-modes and their
tum side modes in the x diretion, was presented in [10].
totalpopulations. Withtheseresults,wecan(i)compare
ItshouldbeemphasizedthatonecannotconsiderC (τ)
nm the growth of the side modes in the weak- and strong-
and b (τ) as spatial averages of ψ (ξ,τ) and e (ξ,τ).
± nm ± pulse regimes, (ii) work out differences to the spatially-
Infact,wewillseethatthepredictionsofthetwomodels
independent model introduced at the end of Sec. II, and
show significant differences. It should also be noted that
(iii) explain why the time-evolution of the side modes is
the damping coefficient γ, although of the order of c/L,
rather insensitive to the details of the initial seed func-
cannot unambiguously be ascribed a precise value.
tion. Themostinterestingresultofthesestudiesisprob-
In general, superradiant scattering is initiated by
ably the observationthat the propagationeffects lead to
quantum-mechanical noise, i.e., spontaneous Rayleigh
a subexponential growth of the side-mode populations
scattering from individual condensate atoms [3]. Sub-
which is in contrast to the spatially independent model
sequent stimulated scattering and bosonic enhancement
where an exponential growth is predicted.
lead to rapid growth of the side-mode populations. The
In the startup regime described above, only the first-
semiclassical model derived in this section describes the
order side modes for forward- and backward-scattering
macroscopic stage of the superradiant process where the
become populated significantly, so that we can restrict
populationsofthesidemodesarealreadylargecompared
Eqs. (15) to these modes and the condensate. Fur-
to one.
thermore, the depletion of the condensate is assumed
Adapting the discussion of Refs. [5, 12, 13] regarding
to be negligible so that we set ψ (ξ,τ) ψ (ξ,0)
“conventional” superradiance, we can take the effects of 00 ≈ 00
(undepleted-pump approximation). The equations for
theinitialquantum-mechanicalfluctuationsintoaccount
the sets of modes ( 1, 1) and ( 1, 1) then decouple.
by solving the semiclassical equations of motion with ± ± ± ∓
Neglecting the effects of free propagation, which is well
stochastic initial conditions (seeds) for the side modes
justified for short times, the time evolution of the side
ψ (ξ,τ = 0). Since the noise is practically relevant
1,±1 modes ( 1, 1), for example, is governed by
forthe initialpopulationofthosemodes only,wecanset ± ∓
ψ (ξ,τ =0)=0 for all other side modes. The conden-
nm ∂ψ (ξ,τ) κ2 ξ
saattte=ψ000.iFsocrhaonsyenoftothbeeseinstothcheamstaiccrionsitcioapliccognrdoiutinodnss,ttahtee i 1,−∂1τ = i χ ψ00(ξ)Z−∞dξ′(cid:2)e2iτψ−∗1,1(ξ′,τ)
solution of the semiclassical equations of motion corre- ψ (ξ′)+ψ∗ (ξ′)ψ (ξ′,τ) , (17a)
× 00 00 1,−1
sponds to one possible realizationof the experiment. By ∂ψ (ξ,τ) κ2 ξ (cid:3)
studyingalargesetofsimulationswithvaryingseedfunc- i −1,1 = i ψ (ξ) dξ′ ψ (ξ′,τ)
00 −1,1
tions drawn from an appropriate distribution, one could ∂τ − χ Z−∞ (cid:2)
obtain information on, e.g., averages and fluctuations of ψ∗ (ξ′)+e2iτψ (ξ′)ψ∗ (ξ′,(τ1)7b.)
× 00 00 1,−1
relevant experimental observables. In the present paper, (cid:3)
however, we intend to focus on characteristic features of Beforediscussingthe solutionsofEqs.(17),letus briefly
6
consider the corresponding startup regime in the spa- space, but again display different behaviors in the weak-
tially independent model. Setting C (τ) = √N and and strong-pulse limits.
00
C (τ =0)=0, one obtains from Eqs. (16) For a spatially varying condensate wavefunction, e.g.,
−1,1
a BEC ground state in a trap, it becomes more difficult
C (τ) = C1,−1(0) eλ1τ (2λ Γ) eλ2τ(2λ Γ) , to derive analytical results for the side modes. However,
1,−1 2 1
4(i λ ) − − − the numerical solution of Eqs. (17) with the condensate
− 1 (cid:2) (cid:3)
(18a) having the Thomas-Fermi shape yields side-mode popu-
lationsthatareveryclosetothe onesforahomogeneous
C (0)
C (τ) = 1,−1 Γ eλ1τ eλ2τ (18b) condensatewiththesameatomnumber. Wethusexpect
−1,1
4(i λ ) −
− 1 (cid:0) (cid:1) laws similar to Eqs. (20) and (22) to apply in this case
e
as well.
withC (τ)=e2iτC∗ (τ),Γ=G/ω ,andthegrowth
−1,1 −1,1 r The study of Eqs. (17) also allows us to understand
ratesλe1 =i−√−1−iΓ,λ2 =i+√−1−iΓ. Intheweak- why, after an initial transient, the spatial shape of the
pulseregimeofΓ 1,onehasC1,−1(τ) C1,−1(0)eΓτ/2 side-mode wave functions becomes rather insensitive to
and C−1,1/C1,−1≪2 Γ2/16. For stro≈ng pulses with the details of the initial seed function. To this end, we
| | ≈
Γ 1, on the other hand, one finds C1,−1(τ) considertheFourierdecompositionofanarbitraryinitial
≫ ≈
C (0) Γ/16 e(1−i)√Γ/2τ+iπ/4 and C /C 1 seed
1,−1 −1,1 1,−1
| | ≈
(see alsop[10]). We thus see that the spatially indepen- ∞
dent model always predicts an exponential increase of ψ (ξ,0)= c exp(i2πnξ/Λ). (23)
1,−1 n
the side-mode populations, but with growth rates vary- n=X−∞
ing between Γ and √Γ.
Tofacilitatethecomparisonwiththesepredictions,we Since Eqs. (17) are linear, ψ1,−1(ξ,τ) is obtained
examine the spatially dependent equations (17) for con- as the linear superposition c ψ(n) (ξ,τ), where
n n 1,−1
stant initial conditions ψ1,−1(ξ,0) = ψ0, ψ−1,1(ξ,0) = 0 ψ(n) (ξ,0)=exp(i2πnξ/Λ). CoPnsidering the weak-pulse
and a homogeneous condensate ψ (ξ) = N/Λ, 0 1,−1
00 regime for concreteness, the Laplace transform method
≤
ξ ≤ Λ. Here, Λ = klL denotes the scaledpcondensate shows that, after some time, the wave functions ψ(n)
length. As shown in the Appendix, Eqs. (17) can then 1,−1
behave like
be solved approximately by Laplace transform and sub-
sequent inversion through the saddle-point method. In
exp 2 Γτξ/Λ
the weak-pulse limit, one obtains ψ1(n,−)1(ξ,τ)≈ (cid:16) p 2πn (cid:17) .
ψ √4π 4 Γτξ/Λ i ξ/4 Γτξ/Λ
ψ (ξ,τ) 0 exp 2 Γτξ/Λ , (19) (cid:18) − Λ (cid:19)
1,−1 p p
≈ √4π4 Γτξ/Λ (cid:16) p (cid:17) (24)
p From this result, we can draw two conclusions. (i) All
so that the side-mode population grows like wavefunctionsψ(n) acquiremoreorlessthesameshape,
1,−1
Λ N (0) i.e.,theirmoduligrowsubexponentiallyinspace,whereas
N (τ)= dξ ψ (ξ,τ)2 1,−1 exp 4√Γτ their phases vary only slowly. We can thus expect their
1,−1 1,−1
Z0 | | ≈ 8πΓτ (cid:16) (cid:17) superposition to qualitatively show the same behavior.
(20)
(ii)Thecontributionsofhigher-orderFouriercomponents
with the initial population N (0)= ψ 2Λ.
1,−1 | 0| are suppressedby a factor of n, approximately. In fact,
In the strong-pulse regime, we find | |
forgrowing n,wefindnumericallythatittakesincreas-
| |
ψ e5πi/12 Γξ 1/6 1 ingly more time to reach the form (24), and before this
ψ (ξ,τ) 0 time is reached, the suppression is even stronger. These
1,−1 ≈ 22/3√6π (cid:18) Λ (cid:19) τ2/3 observations explain the insensitivity of the side-mode
exp iτ +3e−iπ/6(Γτ2ξ/2Λ)1/3 .(21) wave functions to the details of the initial seed.
× (cid:16) (cid:17)
The total side-mode population is approximately given
IV. STRONG-PULSE REGIME
by
N (0) 1 33/2 In Secs. IV and V, we will use the theoretical model
N (τ) 1,−1 exp Γ1/3τ2/3 . (22)
1,−1 ≈ 12π√3 τ2 (cid:18) √38 (cid:19) developed in Sec. II to present a thorough discussion
of the coupled dynamics of optical and matter waves
Incontrastto Eqs. (18), equations(20) and(22)show in the strong- and weak-pulse regimes of superradiance.
thatpropagationeffects leadtoasubexponentialgrowth For the numerical calculations, we focus on the exper-
in the side-mode populations: for weak pulses, N in- imental data of Refs. [1, 2]. In particular, we con-
1,−1
creasesasexp(√τ)/τ,whereasforstrongpulses,itgrows sider a 87Rb BEC of N = 2 106 atoms, with length
like exp(τ2/3)/τ2. Moreover, from Eqs. (19) and (21), L = 200µm and cross-sectio×n diameter 15µm. The
weseethatthesidemodesalsogrowsubexponentiallyin condensate is in the Thomas-Fermi regime, so that we
7
can model its wave function as ψ (z) = n(z) with (0,0)
n(z)=C[(L/2)2 z2]Θ(L/2 z ),0C0=3N/p4L3. Asdis- +0.25 (a)
− −| |
cussedinSecs.IIandIII,theresultsarenotsignificantly (1,−1)
(1,1)
influenced by the shape of the seed function. Hence, for
the sake of simplicity, throughout our simulations the
first-order atomic side modes are seeded according to
ψ (z,0) = ψ (z)/√N, which corresponds to one de-
1,±1 00 er
localized atom in each of the modes. The applied laser as
L
pulse is modeled as rectangular lasting from t=0 up to 1.5
t=t . Equations(15)arethenpropagatedusingasplit- (−1,−1)
f
0.75
step algorithm [19]. The grid (n,m) for the side-mode
(−1,1)
ordersis chosensufficiently large,so that the population −4 0 m (cid:1)
of the highest-order side modes remains negligible at all −2 4
−
tstihumochseesa.ussTepdhuelisnechtshotreseeennxgptvhearluiamnesdenfdtosurr[t1ah,tei2o]ne.x,taerrenaclompapraarmabelteertso, (cid:0)z10−4m0 2 4 −1.5−0.75x10(cid:0)
Let us first consider the regime of strong laser pulses (cid:1)
(b)
which is characterized by Rayleigh scattering rates R (0,0)
comparable to ω . In this regime of parameters, ex- +0.25
paserwimelelnbtsacwkiwthardshr-roerctoillainsegraptuolmsess, hfoarvmeinshgoawnvefroyrwcharadr-- units) E− E+ units)
aacbtleerissptaictiaXl-sahsaypmempeattrtyerbne.twMeeonreofovrewr,artdhearendisbaacnkowtiacred- 2(arb.m| (1,1) (1,-1) 2(arb.|±
peaks. According to Ref. [2], these observations suggest n E
ψ (-1,-1) (-1,1) |
a purely optical picture of superradiance in which atoms |
arediffractedfromtheopticalgratingformedbytheend-
fireandpumpmodes. Theintensityoftheendfiremodes, -1.0 -0.5 0 0.5 1.0
andthustheopticalgrating,areassumedtobepeakedat z 10−4m
(cid:0) (cid:1)
the edges of the condensate. Our model has enabled us
toreproducethe experimentalobservationsandtoverify FIG.2: Spatiallydependentmodel. (a)Spatialdistributionof
thefirst-orderforward(1,±1)andbackward(−1,±1)atomic
this conjecture.
side modes, after applying a laser pulse of duration t = 8.5
f
µs and strength g = 3×106 s−1 to the condensate followed
by a free propagation for a time tp = 25ms. (b) Spatial
A. Side-mode patterns distributionsoftheatomicsidemodesandtheopticalendfire
modes E± at time tf. For the sake of illustration, in both
casesthepopulationoftheBEC(0,0)hasbeenmultipliedby
InFig.2(a),wedisplayasnapshotoftheatomicspatial
0.25.
distribution after applying a strong laser pulse to the
condensatefollowedbyafreepropagationforatimet
p
≫
t . Since we work with a one-dimensional model, we
f
calculate the displacement ∆x between the condensate cordingtoEq.(15a),thescatteringprocessisstrongestin
andthefirst-ordersidemodesinthexdirectionas∆x= theseareasoflargeelectricfields,andtherecoilingatoms
v t withtherecoilvelocityv =~k/M =5.9 10−3m/s. mainly originate from the edges of the condensate. Dur-
r p r
Ourresultclearlyreproducestheasymmetry×observedin ing the free time evolution following the strong pulse,
Fig. 2 of Ref. [2]. the forward-scattered atoms in the side modes (1, 1)
±
The physical mechanism behind this asymmetry can initially travel towards the center of the BEC, whereas
be understood from the spatially resolved dynamics of the atoms in the backwards side modes ( 1, 1) imme-
− ±
the optical and matter-wave fields as described by Eqs. diately move away from the center [dashed lines in Fig.
(15). For the sakeof illustration,the spatialdistribution 2(a)]. The net effect is the observed asymmetry in the
of the condensate, the first-order atomic side-modes and spatial distribution of forward and backward peaks. So,
the optical-field modes at the end of the strong pulse in agreement with the experiments, our model predicts
(i.e., at time t ) are plotted in Fig. 2(b). Clearly, the enhanced photon scattering near the edges of the con-
f
atomic side-modes and the optical-field modes are well densate in the strong-pulse regime. This effect strongly
localized near the condensate edges. This preferential supports an interpretation of the strong-pulse regime in
growth of the optical fields can be explained in the con- the framework of atomic scattering from optical fields as
text of Eqs. (15b) and (15c) which imply that (at least was first conjectured by Ketterle and co-workers[2].
forshorttimes)theelectricfields (z,t)growmonoton-
±
E
ically in the z and z directions, respectively, and are Our model also explains the experimentally observed
−
strongest at the ends of the condensate. As a result, ac- X-shape patterns [see Fig. 1(A) of Ref. [2]]. In Fig.
8
and this is possible only if the atomic side modes and
p the optical endfire modes are located at the condensate
nm
edges. To make this point clear, let us consider, for ex-
0 0.5 1
ample, the off-diagonal side mode (2,0). Figure 4 shows
5 (a)
the possible population transfers among the low-order
4
atomic side-modes and the optical fields. According to
3
Eq.(15a), althoughthe side mode (2,0)is resonantwith
2
themodes(1, 1),itcanonlybepopulatedifthe(1, 1)
1 ± ±
modes overlap with the endfire modes , respectively.
m0 As depicted in Fig. 2(b), however, thisE±overlap is very
-1
small since the modes are localized near the edges of
-2
the condensate, and the growth of the side mode (2,0)
-3
is therefore suppressed. On the other hand, population
-4
transfer to the side modes (2, 2) is easily accomplished
-5 duetothestrongoverlapbetw±eenthesidemodes(1, 1)
-5-4-3-2-1 0 1 2 3 4 5 ±
n and the fields ∓, respectively.
E
FIG. 3: Spatially dependent model. Atomic side-mode dis-
tributions in the strong-pulse regime for g = 3.0×106s−1 B. Discussion
and t = 12µs. The gray level of each square represents the
f
“relative” probability pnm as defined in the text. The arrow
Our studies show that the experimentally observed
indicatesthedirection oftheincoming laserpulseandpoints
asymmetryandX-shapepatternsarebasicallyduetothe
towards the BEC “side mode” (0,0).
spatial inhomogeneity of the optical fields. In an earlier
attempt to theoreticallydescribe these observations,Pu,
(2,2)
Zhang, and Meystre [7] adopted a model which does not
includeanyspatialpropagationeffects. Intheframework
ofthismodel,theyconcludedthattheobservedasymme-
E−
(−1,1) (1,1) tryisdue toangulareffects,thusquestioningthepicture
ofatomicscatteringfromopticalfieldssuggestedbyKet-
E+ E+ terle and co-workers [2]. In particular, they found that
E− forward recoiling atoms should have a symmetric dis-
(0,0) (2,0) tribution around 45o, whereas backward recoiling atoms
should favor larger angles due to a reduced energy mis-
+
E
match.
(−1,−1) E− (1,−1) E− Unfortunately, the analysis of [7] is restricted to short
timesforwhichthecondensateremainspracticallyunde-
E+ pleted and only first-order side modes are slightly popu-
lated. Asaresultthereisnoevidencethattheirtheoryis
(2,−2). able to explain the characteristic X-shape pattern which
involves higher-order atomic side modes. On the con-
FIG. 4: (Color online) Origin of the X-pattern: schematic trary, there are several different arguments which show
representation. Arrowsshowpopulationtransferbetweendif- that the explanation of [7] cannot account for the ob-
ferentsidemodes(n,m)viacouplingstotheopticalfieldsE±. served asymmetry and some of them have already been
The side mode (2,0) does not become populated due to the discussed elsewhere [11]. Here we would like to give a
missing spatial overlap between the side modes (1,±1) and
few morearguments whichshow the failure ofthe model
theendfiremodes E±, respectively. employedin[7]. Tothisend,byanalyzingFig.1(A)of[2]
which depicts a typical experimental outcome, we have
obtained a rather precise picture of the distribution of
3, we show the atomic side-mode distribution forming
the atomic side modes which, for the sake of compari-
an X-shape pattern at the end of a strong pulse with
son, is presented in Fig. 5. Clearly, the atoms recoil at
g = 3 106s−1. The gray level of each square repre- angles much smaller than 45o in the forward direction
×
sents the “relative” probability pnm =Pnm/Pn(mmax) with when measured with respect to the BEC center and the
P = dz ψ (z,t)2 and P(max) =max P . As direction of the applied laser pulse (dashed lines in Fig.
nm nm nm {n,m} nm
| |
amatteRroffact,theenhancedphotonscatteringnearthe 5). Thecorrespondingangleinthe backwarddirectionis
edgesofthecondensateisalsoresponsibleforthecharac- larger than 45o.
teristic X-shape distribution of the side modes. Indeed, These observations are incompatible with Fig. 3(b) of
the appearance of an X-shape pattern requires the sup- [7]. More precisely, this figure shows a very broad an-
pression of all off-diagonal side modes with n = m gular distribution for the forward and backward recoil-
| | 6 | |
9
FIG.5: (Color online) Observedspatial asymmetryin anac-
curateschematicrepresentationbasedonFig.1(A)of[2]. The
forward (backward) peaks clearly appear at angles smaller
(larger) than 45o. The dashed lines indicate the angles at
45o.
ing atoms. However, the angular width of the individual
backwards and forwards travelling atomic wave packets
observed in the experiments is actually much smaller.
This means that the curves of Fig. 3(b) should be inter-
preted as probability distributions for the directions of
recoiling atoms averaged over many realizations. In this
case, however, Fig. 3(b) of [7] shows that atoms should
predominantly recoil at 45o in both forward and back-
warddirections. Moreover,oneshouldalsoobservemany
experimentalruns where the forwardrecoiling atomsare
emitted under a larger angle than the backwards recoil-
ing ones. None of the experiments, however, reports on
such types of observations. Furthermore, as depicted in
our Fig. 5, experiments show the forward peaks to have
a much smaller distance from the BEC center than the
backwardpeaks. In an explicationbased on angular dis-
tributions,however,these distanceshavetobethe same.
These contradictions between theory and experiment
FIG. 6: Spatially independentmodel. Atomic side-modedis-
clearly indicate that angular effects cannot account for
tributions (compare with Fig. 3). (a) Strong-pulse regime:
the experimentally observed asymmetry and X-shape t = 10.4µs and g = 3.0×106s−1; (b) weak-pulse regime:
f
pattern. The main reason for the failure of the model t = 151.5µs and g = 3.5×105s−1; (c) weak-pulse regime:
f
employedin[7]isthatitdoesnottakespatialeffectsinto t =182µs and g=3.5×105s−1.
f
accountwhich,as weshowhere,arecrucialinproducing
both the asymmetry and the X-shape pattern. Indeed,
keeping only the two optical endfire modes, one can eas- become strongly populated. It is therefore obvious that
ilyverifythatinthesemiclassicalregimethemodelof[7] spatial effects play a significant role in the strong-pulse
turnsinto aspatially-independentmean-fieldmodelsim- regime.
ilar to the ones employed e.g., in [9, 10] and introduced
at the end of Sec. II.
As depicted in Fig. 6(a), this simpler model cannot V. WEAK-PULSE REGIME
explain the characteristic X-shape pattern: There is no
mechanismpresentto preventthe growthin off-diagonal In this section, we will discuss some characteristic dy-
side modes such as ( 2,0) and (0, 2), and they soon namic effects arising in the weak-pulse regime of BEC
± ±
10
(a) (b) nits] (a) (b)
u
b.
ar
nits] 2[0|
u ψ0
b. |
[ar (c) (d) -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1
2m| z 10−4m z 10−4m
ψn (cid:2) (cid:3) (cid:2) (cid:3)
|
FIG. 8: Depletion of condensate center in the weak-pulse
regime (g = 6.0×105s−1). Spatial distribution of the con-
densate at times t =215 (a) and t =350µs (b).
f f
-1 -0.5 0 0.5 1-1 -0.5 0 0.5 1.0
z 10−4m z 10−4m
(cid:2) (cid:3) (cid:2) (cid:3)
lution of the matter-wave fields. Behavior of this kind is
FIG. 7: Spatially dependent model and weak-pulse regime
(g=3.0×105s−1). Spatialdistributionofthecondensate(full observedforalargerangeofvaluesoftheexternalparam-
curves),sidemodes(1,1)(dotted),(1,−1)(dash-dotted),and eters. Forveryshorttimes[Fig.7(a)],wehaveasituation
(2,0) (dashed) at times t =200 (a); t =270 (b); t =350 similartothestrong-pulseregime,i.e.,thefirst-orderside
f f f
(c); tf =420µs (d). modes (1, 1) start to grow at the edges of the conden-
±
satewheretheopticalfieldsarestrongest. Thesidemode
(2,0), although resonantwith the modes (1, 1), cannot
±
superradiance. Together with Secs. III and IV, these yet be populated because the necessary overlap between
results illustrate how our model allows to obtain new matter-waveandopticalfieldsismissing. However,since
insights into the system behavior. First of all, we de- the appliedlaserfield isweak,higher-orderdiagonalside
scribe typical stages of the spatially resolvedtime evolu- modes are now populated less efficiently because of the
tion of the matter-wave fields. In particular, we address detuning barrier.
the depletion of the condensate center reported in Ref. Instead, a sort of Rabi oscillation sets in between the
[2]. Secondly, we discuss the time development of the first-orderside modesandthe condensate: afterthe con-
opticalfieldsandshowthataminimuminthesuperradi- densate population at some point z has been pumped
ant light emission does not imply that, at that time, the completelytothefirst-ordersidemode,itissubsequently
opticalfieldsarealsosmallinsidetheatomicsample. Fi- transferredback. This leads to the appearanceof a min-
nally, we show that our model is capable of reproducing imum in the condensate density (marking the point of
the characteristic fan patterns for the side-mode distri- complete transfer) and a concomitant regrowth of the
butions that have been observed experimentally. Addi- populationfromtheedges[Fig.7(b)]. Astobeexpected,
tionally,wepointoutsimilaritiesanddiscrepanciesinthe the minima stay near the density maxima of the first-
predictions of the spatially dependent and independent order side modes. Once these maxima get close the cen-
models. ter of the sample, the overlap between the (1, 1) side
±
modes and the endfire mode e becomes large, and the
±
(2,0) side mode starts to grow rapidly. Simultaneously,
A. Time evolution of matter-wave fields the minima in the condensate density merge at the cen-
ter [Figs. 7(c) and 7(d)]. This scenario thus shows that
In this subsection, we examine characteristic features there is a clear correlation between the onset of popula-
of the spatially resolved matter-wave dynamics in the tion growth in the (2,0) side mode and the depletion of
weak-pulse regime. We describe the early stages in the the condensate center. We expect that this effect should
time evolution of the (1, 1) and (2,0) side modes – be observable experimentally. One should also note the
±
which are resonantly coupled to the condensate – and localization of the (2,0) side mode around the center of
we discuss the depletion of the condensate center [2]. the system.
A main difference between the weak- and strong-pulse In Ref. [2], depletion of the condensate center was re-
regimes is the fact that in the latter only diagonal side ported after the system had already undergone a large
modes with n = m are populated significantly. As ex- number of superradiant emission cycles and high-order
| | | |
plainedinSec.IV,thisisduetothemissingspatialover- side modes were populated. As evidenced in Figs. 8, our
lap between, e.g., the side modes (1, 1) and the endfire simulations are able to reproduce this effect (note that
±
modee whichprohibitsthepopulationofthe(2,0)and g is increased by a factor of 2 compared to Fig. 7). At
±
other non-diagonalside modes. Instead, higher-orderdi- the times shown in these pictures, side modes of order
agonal side modes are populated, since this is not im- n = 3 and 4 are populated, respectively. We find that
peded by spatial effects and the strong fields allow to the depletion of the center is clearly maintained over a
overcome the concomitant detuning barriers. long period of time. It should be noted, however, that
A different situation arises in the weak-pulse regime. the central population increases again briefly after the
In Figs. 7, we show a typical scenario for the time evo- initial depletion described above. The simulations also