Table Of ContentMatter Wave Interference Pattern in the collision of bright solitons (Bose Einstein
condensates) in a time dependent trap
V. Ramesh Kumar1, R. Radha1,∗ and Prasanta K. Panigrahi2†
1 Centre for Nonlinear Science, Department of Physics,
Government College for Women, Kumbakonam 612001, India
2 Indian Institute of Science Education and Research, Kolkatta-700106, India
Weshowthat itispossible toobservematterwaveinterferencepatternsin thecollision of bright
solitons(BoseEinsteincondensates)withoutfreeballisticexpansionforsuitablechoicesofscattering
9 length and time dependenttrap.
0
0 PACSnumbers: 03.75.Lm,05.45.Yv
2
n I. INTRODUCTION ference just as it occurs with two interesting laser radia-
a
tions. In other words, when BECs were made to collide
J
upon release from the trap, de Broglie wave interference
0 Whenagasofmassivebosonsarecooledtoatempera-
pattern containing stripes of high and low density were
1 tureveryclosetoabsolutezeroinanexternalpotential,a
clearly observed. These experiments which underlined
largefractionofthe atomscollapseintothe lowestquan-
] the high degree of spatial coherence of BECs led to the
tum state ofthe externalpotential forminga condensate
r
creation of atom laser [14]. Can one observe the same
e known as a Bose Einstein condensate (BEC) [1-4]. Bose
h matter wave interference pattern by allowing the bright
Einsteincondensationisanexoticquantumphenomenon
t solitons which are condensates themselves to collide in a
o observed in dilute atomic gases and has made a huge
trap? Motivatedbythisconsideration,weinvestigatethe
. turnaround in the fields of atom optics and condensed
at matter physics. The recent experimental realization of collisional dynamics of condensates in a time dependent
m trapping potential.
BECs in rubidium [5] has really kickstarted the upsurge
- in this area of research leading to flurry of activities in
d
thisdirectionwhiletheobservationofdark[6]andbright
n II. GROSS-PITAEVSKII EQUATION AND
o solitons [7], periodic waves [8], vortices and necklaces [9] LAX-PAIR
c hasgivenanimpetus tothe investigationofthis singular
[ state of matter.
At the mean field level, the time evolution of macro-
1 The dynamics of BECs is in general governed by an scopic wavefunction of BECs is governed by the Gross-
v inhomogeneous nonlinear Schrodinger (NLS) equation
Pitaevskii (GP) equation,
0 called the Gross-Pitaevskii (GP) equation and the be-
6 haviour of the condensates depends on the scattering ∂ψ(~r,t) ~2
3 i~ = − 2+g ψ(~r,t)2+V ψ(~r,t) (1)
length (binary interatomic interaction) and the trapping ∂t 2m ∇ | |
1 (cid:18) (cid:19)
potential. EventhoughtheGPequationisingeneralnon-
1. integrable, it has been recently investigated for specific where g = 4π~2as(t)/m, V = V0 + V1, V0(x,y) =
0 choicesofscatteringlengths andtrapping potentials [10- mω⊥2(x2+y2), V1 =mω02(t)z2/2. In the above equation,
9 12]. V0 andV1 representatomsin a cylindricaltrapand time
0
dependent trap along z-direction respectively. The time
: It is known that a BEC comprises of coherent matter
v dependent trap could be either confining or expulsive
waves analogous to coherent laser pulses and the wave
i while the time dependent atomic scattering length a (t)
X nature of each atom is precisely in phase with that of s
can be either attractive (a < 0)or repulsive (a > 0).
r every other atom. In other words, the atoms occupy the s s
a same volume of space, move at identical speeds; scatter As a result, the condensatesconfrontwith both time de-
pendent scattering length and time dependent trapping
light of the same color and so on. Hence, it looked fun-
potential.
damentally impossible to distinguish them by any mea-
ConsideringBECsasanassemblyofweaklyinteracting
surement. This quantum degeneracy arising out of high
atomic gases, eq. (1) can be effectively reduced to a one
degreeofcoherencehasbeen recentlyexploitedinanex-
dimensional GP equation taking the following form (in
periment by Andrews et al. and Ketterle’s group [13]
dimensionless units),
showing that when two seperate coluds of BECs over-
lap under free ballistic expansion, the result is a fringe ∂ψ 1∂2ψ M(t)
patternofalternatingconstructiveanddestructiveinter- i + +γ(t)ψ 2ψ z2ψ =0 (2)
∂t 2 ∂z2 | | − 2
where γ(t) = 2a (t)/a , M(t) = ω2(t)/ω2, a is the
− s B 0 ⊥ B
Bohr radius, M(t) describes time dependent harmonic
∗Electronicaddress: radha˙[email protected] trap which can be either attractive (M(t) > 0) or ex-
†Electronicaddress: [email protected] pulsive (M(t) < 0). Now, to generate bright solitons
2
of eq. (2) for both regular and expulsive potentials, we ΓHtL
introduce the following modified lens transformation
3
2
ψ(z,t)= A(t)Q(z,t)exp(iΦ(z,t)) (3)
1
where the phase haspthe following simple quadratic form t
2 4 6 8 10
1 -1
Φ(z,t)= c(t)z2. (4)
−2 -2
-3
Substitutingthemodifiedlenstransformationgivenby
eq. (3) in eq. (2), we obtain the modified NLS equation
FIG. 1: The variation of scattering length γ[t]=0.2sec[√2t]
1 corresponding to case (i).
iQ + Q ic(t)zQ ic(t)Q+γ(t)A(t)Q2Q=0, (5)
t zz z
2 − − | |
with
where
M(t)=c′(t) c(t)2, (6)
− A = 2β [(α α )2 (β2 β2)]
and 1 {− 2 2− 1 − 1 − 2
4iβ β (α α ) e(θ1+iξ2)
1 2 2 1
d − − }
c(t)= lnA(t). (7)
−dt
A = 2β [(α α )2+(β2+β2)]e(−θ1+iξ2)
Equation (5) admits the following linear eigen value 2 − 2 2− 1 1 2
problem
A = 2β [(α α )2+(β2 β2)]
iζ(t) Q 3 {− 1 2− 1 1 − 2
φz = Uφ, U = Q∗ iζ(t) (8) + 4iβ β (α α ) e(iξ1+θ2)
1 2 2 1
(cid:18)− − (cid:19) − }
φ = Vφ,
t
iζ(t)2+ic(t)zζ(t) (c(t)z ζ(t))Q A = 4iβ β [(α α ) i(β β )]e(iξ1−θ2)
−+iγ(t)A(t)Q2 +−iQ 4 − 1 2 2− 1 − 1− 2
V = 2 | | 2 z (9)
−(c(t)+z2−iQζ∗z(t))Q∗ iζ−(t2i)γ2(−t)Aic((tt))|zQζ|(2t) BB12 == −24cβo1shβ(2θ[s1i)nhc(oθs1h)(sθi2n)h[((θα22)−+αco1s)(2ξ+1−(βξ122)+] β22)]
In the above linear eigenvalue problem, the spectral and
parameterwhichiscomplexisnonisospectralobeyingthe
t
following equation ′
θ = 2β z 4 (α β )dt +2δ
1 1 1 1 1
−
ζ′(t)=c(t)ζ(t) (10) Z0
t
ξ = 2α z 2 (α2 β2)dt′ 2φ
with γ(t)=1/A(t). It is obvious that the compatability 1 1 − 1− 1 − 1
condition (φ ) =(φ ) generates eq. (5). Z0
z t t z t
′
Substituting eq. (7) with γ(t)=1/A(t) in eq. (6), we θ = 2β z 4 (α β )dt +2δ
2 2 2 2 2
get − Z0
t
γ′′(t)γ(t) 2γ′(t)2 M(t)γ(t)2 =0 (11) ξ = 2α z 2 (α2 β2)dt′ 2φ
− − 2 2 − 2− 2 − 2
Z0
Thus,thesolvabilityoftheGPeqn.(2)depends onthe
suitable choices of scattering length γ(t) and the trap
frequency M(t) consitent with eq. (11). Recently eq. α1 = α10eR0tc(t′)dt′, β1 =β10eR0tc(t′)dt′,
(g2e)nehraastebdeeemnpinlovyeisntgiggaateudgeatnrdanbsrfoigrhmtastoiolintoanpsprhoaavcehb[1e1en]. α2 = α20eR0tc(t′)dt′, β2 =β20eR0tc(t′)dt′,
γ(t) = γ0eR0tc(t′)dt′.
III. BRIGHT SOLITON INTERACTION AND Togeneratetheinterferencepattern,wenowallowthe
MATTER WAVE INTERFERENCE bright solitons to collide with each other in the presence
of trap for suitable choices of scattering length γ(t) and
To investigate the collisional dynamics of condensates trap frequency M(t) (or c(t)) consistent with eq. (11).
inthepresenceofatimedependenttrap,wenowconsider Case (i): When c(t)=√2tan√2t, the variation of scat-
the two bright soliton of the following form, tering length γ(t)=0.2sec√2t is shown in fig.1. Ac-
cordingly, the trap frequency M(t) becomes a constant
ψ2(z,t)= 1 A1+A2+A3+A4 e−2ic(t)z2 (12) (M(t)=2). Under this condition, the collisional dynam-
sγ(t) B1+B2 ics of bright solitons in the harmonic trap is shown in
3
HaL ΓHtL
0.2
0.1
t
5 10 15 20
1
ÈΨ0È.75 -0.1
0.5 6
0.25
-0.2
0
4
t
--2200 FIG. 4: The variation of scattering length γ[t]=0.2sin[√2t]
2
00 corresponding to case (ii).
zz
2200
0
HbL HaL
10
8
6 0.2
0.15
t ÈΨÈ 8
0.1
4 0.05 6
0
2 --2200 4 t
00
0 zz 2
2200
-20 -10 0 10 20
HbL
z
10
FIG.2: Interactionoftwobrightsolitonsforminginterference
pattern for the choice c[t]=√2tan[√2t], γ[t]= 0.2sec[√2t], 8
α10=.01, α20=0.8, β10=0.06, β20=0.012, γ0=0.2, φ1 =0.01,
φ2=0.1, δ1=0.1, δ2=0.01 and (b) the corresponding contour
6
plot depicting matter wavedensity evolution
t
4
6
5 2
-30 -20 -10 0 10 20 30
4
z
t
3 FIG. 5: Two soliton interaction forming interference pattern
for the choice c[t]=√2cot[√2t], γ[t]=0.2 sin[√2t] with the
other parameters as in fig (1a) and (b) the corresponding
2
contourplot depicting matter wave density evolution.
1
-100 -50 0 50 100
varies sinusoidally with time as in the case of Fesh-
z
bach resonance (shown in fig.4) while the trap becomes
M(t)= 2cot2√2t 2cosec2√2t. The evolution of the
FIG. 3: The phase change evolutions of fig.(2a) − −
condensates in the above time dependent trap and the
correspondingdensityevolutionarenowshowninfigs5a
and5b. Fromtheinterferencepattern,oneobservesthat
fig.2aandthe correspondingdensity evolutioninfig. 2b. the matter wave density is maximum at the origin and
The density evolution consists of alternating bright and it decreases gradually on either sides. Again, the phase
dark fringes of high and low density respectively while change between the condensates oscillates with time as
the phase difference between the condensates shown in shown in fig.6.
fig. 3 continuously changes with time. Thus, our results reinforce the fact that the matter
Case(ii): When c(t)=√2cot√2t, the scattering length waves originating from the condensates (bright solitons)
4
6 types of temporal variation of the trap frequency M(t)
whilethescatteringlengthγ(t)canbecontrolledbothby
5 Feshbachresonanceaswellasthroughthetrapfrequency
M(t). Thephasechangeevolutionwhichgivesameasure
ofthe coherenceofthe condensatesshowninfigs.3and6
4
shows that it oscillates with time and this is reminiscent
t
of Josephseneffect wherein the phase difference between
3
the superconducting wave functions oscillates with time.
2
IV. CONCLUSION
1
-100 -50 0 50 100
In conclusion, we have shown that it is possible to ob-
z
serve matter wave interference pattern by studying the
collisional dynamics of condensates (bright solitons) for
FIG. 6: The phase change evolution of fig. 5a
suitable choices of scattering length γ(t) and trap fre-
quencyM(t)andthiscouldhappenwithoutfreeballistic
expansion. We alsoobservethatthe phasedifference be-
do interfere and produce a fringe pattern analogous to
tween the condensates oscillates with time clearly show-
the coherent laser beams and the interference pattern is
ing the manifestation of Josephson effect. It would be
a clear signature of the long range spatial coherence of
interesting to interpret the matter wave destructive in-
the condensates. It should be mentioned that the inter-
terference andwhether this wouldmeanthat atoms plus
ference patterns were obtained earlier by Andrews et al.
atoms add up to vaccuum.
[13]undertheconditionoffreeballisticexpansion(atoms
V. ACKNOWLEDGEMENTS
are released from the trap) while we selectively tune the
frequency of the trap M(t) in accordance with the scat-
tering length γ(t) (consistent with eq.(11)) to observe VRwishestothankDSTforofferingaSeniorResearch
matter wave interference in the collisional dynamics of Fellowship. RR acknowledges the financial assistance
bright solitons. It should be mentioned that the optical from DST and UGC in the form of major and minor
trapshaveopenedupthepossibilityofrealizingdifferent researchprojects.
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