Momentum interferences of a freely expanding Bose-Einstein condensate in 1D due to interatomic interaction change

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by  A. Ruschhaupt

Momentum interferences of a freely expanding Bose-Einstein condensate in 1D due to
interatomic interaction change
A. Ruschhaupt
∗
Institut fu¨r Mathematische Physik, TU Braunschweig,
Mendelssohnstrasse 3, 38106 Braunschweig, Germany
6
A. del Campo and J. G. Muga
0 † ‡
Departamento de Qu´ımica-F´ısica, Universidad del Pa´ıs Vasco, Apdo. 644, 48080 Bilbao, Spain
0
2
ABose-Einstein condensatemaybepreparedinahighlyelongated harmonictrapwithnegligible

n interatomicinteractionsusingaFeshbachresonance. Ifastrongrepulsiveinteratomicinteraction is
a switchedonandtheaxialtrapisremovedtoletthecondensateevolvefreelyintheaxialdirection,a
J
timedependentquantuminterferencepatterntakesplaceintheshorttime(Thomas-Fermi)regime,

9 in which the number of peaks of the momentum distribution increases one by one, whereas the
1 spatial density barely changes.

] PACSnumbers: 03.75.Kk,03.75.Be,39.20.+q
r
e
h
t The dynamics of Bose-Einstein condensates has been netically tunable Feshbachresonance[10] or an optically
o
much studied, both experimentally and theoretically, for induced Feshbach resonance [11].
.
t
a determining the properties of the condensate and its
In this paper we shall take advantage of these con-
m transport behavior in waveguides or free space with po-
trol possibilities and study a quantum interference ef-
- tential applications in nonlinear atomoptics, atom chips
d and interferometry. Fully free (three dimensional) or di- fectoriginatingfromachangeoftheinteratomicinterac-
n tionstrength. Thepreparationofthecondensatemaybe
mensionally constrained expansions have in particular
o carried out in a 1D harmonic trap with negligible inter-
been subjected to close scrutiny to obtain, from time
c atomic interaction strength such that the ground state
[ series of cloud images and the appropriate theoretical
is approximately Gaussian. Removing the trap under
models, information on the condensate and/or its con-
1 these conditions would preserve the momentum distri-
v fining potentials [1]. In reduced dimensions, the expan-
bution, but if the interatomic interaction is immediately
7 sions have also been examined to characterize and iden-
increased as the trap is removed, the momentum distri-
3 tify different dynamical regimes (the mean-field domi-
4 natedThomas-Fermilimit,quasi-condensatesandthedi- bution evolution changes dramatically and in a highly
1 non-classical way: a time dependent interference occurs
lute Tonks-Girardeaugas of impenetrable bosons [2, 3]),
0 consisting of an orderly,one-by-oneincrease of the num-
ortoinvestigatestatisticalbehaviorinapartialreleaseof
6
ber of peaks in the momentum distribution, see Fig. 1,
0 the condensateinto aboxoffinite size[4]. Aneffectively
during the early stage of the 1D expansion, which is de-
/ one dimensional (1D) Bose gas may be realized experi-
t scribedaccuratelyby the Thomas-Fermi(TF) model. In
a mentally by a tight confinement of the atomic cloud in
m the same period of time of the snapshots shown, the
two (radial) dimensions and a weak confinement in the
spatial profile has barely evolved from the initial pro-
- axial direction so that radial motion is constrained to
d file. Classical mechanics only explains the global broad-
the ground transversal state. Expansions are quite gen-
n ening of the momentum distribution due to the release
o erally described and observed in coordinate space but
of (mean field) interatomic potential energy, notice the
c very interesting phenomena occur in momentum space
: motion of the outer peaks in Fig. 1, but not the oscilla-
v [5, 6]. Also interferences, which show the wave nature
tory pattern, which will require a quantum interference
i of the condensate and may form the basis of metro-
X analysis. Awaytomakethemomentuminterferencevis-
logical applications, are usually apparent in coordinate
r ible in coordinate space is to switch off the interatomic
a space, between independent condensates [7] or as a self-
interactionagainso that the subsequent free flight maps
interference [8], but, as in our present case, they may
the momentum peaks into spatial ones.
be genuinely momentum space effects, possibly with an
indirect and less obvious spatial manifestation. (Other Let us now discuss the details. Assume that an effec-
striking example of quantum momentum-space interfer- tively 1D Bose-Einsteincondensate is prepared in a har-
ence phenomenon for an ordinary, non-condensate, one- monic trap. The condensate wavefunction is the ground
particle wavefunction colliding with a barrier has been state of the 1D (stationary) Gross-Pitaevskii equation.
discussed recently [9].) The expansions may be manipu- We shall assume first that the initial interatomic inter-
lated in different ways, e.g. by controlling the time de- action is zero (this will be relaxed later on). Then the
pendence and shape of the external trapping potentials Gross-Pitaevskii equation becomes a linear Schr¨odinger
or by varying the interatomic interaction using a mag- equationso thatthe groundstate condensate wavefunc-
2
(a) 50 0.3
40
m] 30 0.2
^ 2ψ|| [s/c 1200 m/s] 0 .01
0-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 v [c
v [cm/s] -0.1
(b) 12
-0.2
10
m] 8 -0.3
^ 2ψ|| [s/c 246 0 0.1 0.2 0.3 0.4 0.5 t [0m.6s] 0.7 0.8 0.9 1 1.1 1.2
0
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 FIG.2: Velocitiesofthepeaksofthewavefunction Ψˆ(t,v) 2:
v [cm/s]
(c) 8 g0 = 0 (solid lines), g0 = 0.002344cm/s (dashed lin(cid:12)es), g0(cid:12)=
6 0.02344cm/s (dotted lines); TF approximation ψˆ(cid:12)TF(t,v)(cid:12)2
m]
^ 2ψ|| [s/c 24 (o|ψcldi(rxcle)le|v2se)<l.sIu0n.c0ht0h1tihs∗aamtndtahxsexim′w|iψalav(rex′fif)ug|2nu.crteisonweisaalwssauymseadppt(cid:12)(cid:12)loybaetzherreosh(cid:12)(cid:12)i-f
0
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
v [cm/s]
(d) 8 mentum space distribution changes substantially: more
6 andmorepeaks arecreatedastime increases,see Fig. 1,
^ 2ψ|| [s/cm] 24 cwrietahtiψˆo(nt,cva)n=bepal2smπoh¯sReednxinψ(Ft,igx.)2ex(pso(cid:0)l−idilvih¯mnexs(cid:1)),.wThheerepetahke
velocities of the peaks versus time are plotted forming a
0
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 characteristic structure where bifurcations and creation
v [cm/s]
of a new central peak follow each other.
To understand this effect, we shall approximate Eq.
FIG. 1: Wave function in momentum space; exact result (2). If we neglect the kinetic energy (TF regime) we get
ψˆ(t,v) 2 (lines), TF approximation ψˆTF(t,v) 2 (crosses),
(cid:12)(cid:12)(ccla)stsi=ca(cid:12)(cid:12)l0r.4esmulst,P(d()t,tv=)(0d.a6smhesd. lines);(a(cid:12)(cid:12))t=0,(b(cid:12)(cid:12))t=0.2ms, i¯h∂∂tψTF(t,x)= ¯h2g1|ψTF(t,x)|2ψTF(t,x). (3)
tion ψ0(x) is a Gaussian, namely oTrh,einsomluotmioenntiusmψTsFpa(tc,ex,) = ψ0(x)exp(cid:16)−2itg1|ψ0(x)|2(cid:17)
mω mω
ψ0(x) = r4 π¯hx exp(cid:16)− 2h¯xx2(cid:17), (1) ψˆTF(t,v)=r2mπ¯hr4 mπω¯hx Z dx exp(cid:20)m2ωh¯xx2
where ωx is the axial (angular) frequency. For the rest i mωx mωx 2 vm
of the paper we choose m=mass(Na) and ωx =5/s. At −2g1tr π¯h exp(cid:16)− π¯h x (cid:17)−i ¯h x(cid:21). (4)
timet=0theconfiningtrapisswitchedoffandtheinter-
atomic interaction is switched on immediately. (A more Note that in the TF regime the spatial density remains
realisticswitchingprocedureneedingafinitetimewillbe unchanged. The results for Eq. (4) are also plotted in
discussedbelow.) Thetimeevolutionisnowdescribedby Figs. 1 and 2. There is a good agreement between the
the time-dependent Gross-Pitaevskiiequation, exactresultandTF,andbothshowthesameinterference
behavior. Itisclearthatthenonlinearityisplayingakey
∂ψ(t,x) ¯h2 ∂2ψ(t,x) ¯h 2 role in the effect.
i¯h ∂t =−2m ∂x2 + 2g1|ψ(t,x)| ψ(t,x), Wemaycomparethequantumdynamicswithasimilar
(2)
classicalone. Letusassumeanensembleofclassicalpar-
where g1 is the effective 1D coupling parameter. We
ticles where the probability density of initial positions
(atrheeexfaacmtoinrin1g/2s0h0oritstaimrbeistrtar<y,tcit:=sho1u/l(d20b0eωxt)ru=e 1thmast and velocities is given by ρ0(x,v) = ψ0(x)2 ψˆ0(v) 2,
| | (cid:12) (cid:12)
t 1/ωx) for which the absolute square of the Gaus- with ψ0(v) given by Eq. (1). The probabil(cid:12)ity de(cid:12)n-
≪ (cid:12) (cid:12)
sian in Eq. (1) under the free evolution (i.e. Eq. (2) sity is then evolved with the classical Liouville equation
with g1 =0) is nearly not changing in coordinate space. with the Hamiltonian H = h¯2g1 dv′ρt(x,v′), without
Even with g1 = 234.4cm/s and for times t < 1ms, kinetic term in analogy to the quRantum Hamiltonian of
2
ψ(t,x) is nearly not changing. In contrast, the mo- the TF regime, see Eq. (3). The solution is given by
| |
3
50 0.3
0.2
40
0.1
^ 2ψ|| [s/cm] 2300 v [cm/s] -0 .01
10
-0.2
0 -0.3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
t [ms] t [ms]
FIG. 3: Wave function for v = 0: exact result Ψˆ(t,0) 2 FIG. 4: Velocities of the peaks of the exact result ψˆ 2
(lines), TF approximation ΨˆTF(t,0) 2 (crosses), st(cid:12)(cid:12)ationary(cid:12)(cid:12)- (lines), thestationary-phaseapproximation ψˆs 2 (triang(cid:12)(cid:12)les(cid:12)(cid:12));
phaseapproximation Ψˆs(t,(cid:12)(cid:12)0) 2 (dash(cid:12)(cid:12)edline);thefilledcircle theouter thick-dottedlines mark thecritica(cid:12)(cid:12)l va(cid:12)(cid:12)lues ±vc(t).
marks t0. (cid:12) (cid:12)
(cid:12) (cid:12)
If 0 < κ/τ < exp( 1/2)/√2π there are two solu-
ρt(x,v) = ρ0(x,v + αt) with α = h¯2gm1∂∂x dvρ0(x,v). tions ζ0,|ζ1 w| ith 0 <−ζ0 < 1/√2 < ζ1 . Physically
| | | |
P(t,v) := dxρt(x,v) is also plotted in FRig. 1. The these are two positions in which the force exertedon the
classicalpicRtureprovides,approximately,theouterpeaks atom is equal because the Gaussian profile of the po-
moving outwards with time because of the release of po- tentialchangesfromconcave-downaroundthe center,to
tential energy,but not the centralones, which are there- concave-up in the tails. Thus these two positions con-
fore associatedwith quantum interference. Let us exam- tribute to the same momentum and the corresponding
ine this interference. amplitudeswillinterferequantummechanically. Thesta-
Defining dimensionless quantitiesζ = mωx/¯hx, τ = tionary phase approximation of Eq. (5) is
mωx/¯hg1t, and κ= m/¯hωxv, we wrpite Eq. (4) as
p ψˆTF(τ,κ)= 1 p 4 mωx ψˆs(τ,κ)= r4 m¯hωxeiπ4 √τ1ωx (6)
√2πω r π¯h
dζ exp ζ2x iτ 1 e ζ2 + κζ . (5) exp(cid:20)−iτ(e2−√ζπ02 + κτζ0)(cid:21) exp(cid:20)−iτ(e2−√ζπ12 + κτζ1)(cid:21)
× Z (cid:20)− 2 − (cid:18)2√π − τ (cid:19)(cid:21) × 1 2ζ02 −i 2ζ12 1 .
− −
Animportantcharacteristictimeisthevalueτ0 whenthe p p
secondmaximumof ψˆTF(τ0,κ 0) 2 inthe TFapprox-  2 
(cid:12) ≡ (cid:12) Fig. 4 shows also the peaks of ψˆ (τ,κ) resulting from
imationappears(the(cid:12)firstmaximum(cid:12)isatτ =0),seeFig. s
(cid:12) (cid:12) (cid:12) (cid:12)
3. This sets the scale of the oscillations and the value is using Eq. (6)in the range0< (cid:12)κ/τ <e(cid:12)xp( 1/2)/√2π.
|(cid:12) | (cid:12) −
found numerically, Except for the outer peaks and the κ = 0 line, Eq. (6)
givesthecorrectpeakbehavior. Theopeningofthecone
27.703 ¯h
τ0 27.703 t0 . of the effect can be approximated by the definition of a
≈ ⇒ ≈ g1 rmωx critical κc(τ) or vc(t) to make the interference possible,
Thistimet0shouldbemuchsmallerthanthecriticaltime
κ exp( 1/2) exp( 1/2)
tc when the kinetic energy starts to influence. Therefore c = − κc(τ)= − τ,
we get a necessary condition for g1, τ √2π ⇒ √2π
1 27.703 ¯h or vc(t)=exp( 1/2)ωxg1t/√2π.
tc = 200ωx >t0 ≈ g1 rmωx The interfere−nce pattern in momentum space can be
seenin coordinatespace if the interactionis switched off
¯hωx 2
⇒ g1 >27.703×200r m ≈65.12cm/s. ata giventime toff. Then ψˆ(v,t) is notchangingand
(cid:12) (cid:12)
thereforethe positionandn(cid:12)umber(cid:12)ofpeaks is notchang-
WewillsimplifyfurtherEq. (5)bymeansofthestation- (cid:12) (cid:12)
ing. On the other hand, the different peaks separate in
ary phase method. If κ/τ is constant and τ then
→ ∞ coordinatespace. Thereforefordifferenttimest adif-
the maincontributionto the integral(5) comes fromthe off
ferent peak pattern will appear in coordinate space (see
values ζ which fulfill
Fig. 5). Only the central peak cannot be seen.
∂ 1 e ζ2 + κζ =0 ζe−ζ2 + κ =0. Up to now we have considered an abrupt switching of
∂ζ (cid:20)2√π − τ (cid:21) ⇒− √π τ theinteraction,buttheeffectremainsforswitchingtimes
4
(a) 0.02 nated from the interatomic interaction change. The mo-
mentum distribution expands due to the release ofmean
m]
µ fieldenergyandthenumberofpeaksincreaseswithtime
2ψ| [1/ 0.01 becauseofthe interferenceoftwopositionsincoordinate
|
space contributing to the same momentum or velocity.
0
-200 -150 -100 -50 0 50 100 150 200 The effect is stable in a parameter range and and could
x [µm] be observed with current technology.
(b) 0.02
WearegratefultoC.Salomonforencouragementatan
m] earlystageofthe project,andto G. Garc´ıa-Calder´onfor
µ
2ψ| [1/ 0.01 commentingonthemanuscript. Thisworkhasbeensup-
| portedbyMinisteriodeEducaci´onyCiencia(BFM2003-
0 01003)andUPV-EHU(00039.310-15968/2004). A.C.ac-
-200 -150 -100 -50 0 50 100 150 200
x [µm] knowledgesfinancialsupportbytheBasqueGovernment
(c) 0.02 (BFI04.479).
m]
µ
2ψ| [1/ 0.01
|
0 ∗ Email address: [email protected]
-200 -150 -100 -50 0 50 100 150 200
x [µm] † Email address: [email protected]
‡ Email address: [email protected]
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≪
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a quantum interference effect in momentum space origi-