Gross Pitaevskii Equation with a Morse potential: bound states and evolution of wave packet

0.19 MB·English

by  Sukla Pal

Gross Pitaevskii Equation with a Morse potential: bound states and evolution of
wave packets
Sukla Pala, , Jayanta K. Bhattacharjeeb
∗
aDepartment of Theoretical Physics, S.N.Bose National Centre For Basic Sciences, JD-Block, Sector-III, Salt Lake City, Kolkata-700098,
India
bHarish-Chandra Research Institute,Chhatnag road, Jhunsi, Allahabad-211019, India
5Abstract
1
0WeconsidersystemsgovernedbytheGrossPitaevskiiequation(GPE)withtheMorsepotentialV(x)=D(e−2ax 2e−ax)
−
2as the trapping potential. For positive values of the coupling constant g of the cubic term in GPE, we find that the
critical value g beyond which there are no bound states scales as D3/4 (for large D). Studying the quantum evolution
n c
aof wave packets, we observe that for g <gc, the initial wave packet needs a critical momentum for the packet to escape
Jfrom the potential. For g >g , on the otherhand, all initial wave packets escape from the potential and the dynamics is
c
2like that of a quantum free particle. For g <0, we find that there can be initial conditions for which the escaping wave
1packet can propagate with very little change in width i,e., it remains almost shape invariant.

]Keywords: Critical coupling constant, Gaussian wave-packet,quantum evolution, Threshlod momentum, Quantum
h
fluctuation, Threshlod energy
p
-
t
n
a1. Introduction The primary importance of GPE lies in the fact that
u it describes the dynamics of the condensate in the pro-
q The wave packet dynamics of the usual Schrodinger cess of Bose Einstein Condensation (BEC)[10, 11]. The
[equation has been extensively studied for the free parti- emergence of different kinds of soliton (dark, bright, grey

1cle as well as for different kinds of external potential. For [12]-[14] is well known in case of GPE. Also the control-
vthe free particle, if a nonlinear term in the wave function lable soliton emission [15] has been investigated for GPE
9is added to the Schro¨dinger equation, then one has the withashallowtrapandwithnegativeinterspeciesinterac-
6
6non-linear Schro¨dinger equation (NLSE)[1, 2]. This too tion. In this article,we have dealtwith the attractiveand
2has been a subject of intense study and one has found repulsive interatomic interaction separately for the Morse
0variouskindsofexactsolutionslikesolitons,pulses,fronts potentialandthedynamicsinbothofthesetwocaseshave
1.etc.[3,4,5]. Ifoneaddsapotential,thenonehastheGross been explored both numerically and analytically.
0Pitaevskii equation (GPE)[6, 7] which has been very use- TimedependentGrossPitaevskiiequation(GPE)with
5ful for describing the process of Bose Einstein condensate one spatial dimension has the following form:
1(BEC)[8, 9] for a trapped gas (it should be noted that
v:experimentally BEC has only been observed in a trapped i~∂ ψ = ~2 2ψ+g ψ 2ψ+V (x)ψ (1)
igas). The trapping potential makes the problem far more t −2m∇ | | ext
X
difficult and the only case which has been reasonablywell
r
astudied is the simple harmonic potential. In an experi- 2. Bound state
ment, it is actually far more convenient to tune the trap-
ping potential and observe different behaviors of the con- ThestationarystatesofGrossPitaevskiiequation(GPE)
densate. With this in mind, we wanted to study the GPE havetheformψ(x,t)=eiµt/~u(x),sothatµu(x)= ~2 2u+
−2m∇
with a Morse potential so that one can achieve a much guu2 +V u and lowest value of µ is the ground state
ext
| |
greater flexibility in adjusting the potential. In fact, the energyE whichis to be obtainedbyminimizing E ofEq.
0
potential can have bound states or no bound state at all, (2).
depending upon certain parameters of the potential. The
~2 g
corresponding dynamics is treated both for g > 0 and E[ψ(x)]= ∞[ ψ 2+V ψ 2+ ψ 4]dx (2)
g < 0, where g is the coupling constant of the non linear Z 2m|∇ | ext| | 2| |
−∞
term.
where,V (x)=D(e 2ax 2e ax). Fora<<1,V xt(x)
ext − − e
− ∼
x2 indicatingtheoscillatorlimitoftheMorse. Ontheoth-
∗Correspondingauthor erhand, for a , Vext(0) = D and 0 else where for
Email address: [email protected] (SuklaPal) x>0 as is clea→rly∞depicted in Fi−g (1).
Preprint submittedto Elsevier January 13, 2015
and and writing λ = agm, we get as the minimization
smalla ~2a2
condition
largea
d Γ(4α)
2α+(1 2K)+λ =0 (8)
− dα24α[Γ(2α)]2
L
x
HV Sincethetightlyboundsituationcorrespondtolargevalue
ofK,weusetheaboveequationinthelimitwhenαisrea-
sonably largeandΓ(x) canbe replacedby the asymptotic
form
1
x Γ(x) e−xxxx−1/2√2π[1+ + ]
∼ 12x ···
We get from Eq. (8) to the leading order, α to be given
Figure 1: Schematic diagram of Morse potential depicting the two
by
extreme conditions. Blue curve shows approximately the oscillator
limit (a ≪ 1) and the violet one (for a → ∞) indicates the free α=K 1 λ 1
behavior forregionofx>0. − 2 − 2√2π(2K 1)1/2
−
This gives a ground state energy of
With the differential equation for u given as
~2a2 1 λ
~2 d2u E = [ (2K 1)2+ (2K 1)1/2] (9)
+D(e 2ax 2e ax)u+g uu =µu (3) 2m −4 − 2√π −
− −
− 2mdx2 − | |
anequationwhichisaccurateonly forK 1. InTable 1,
we can explore the asymptotic solution as x . The ≫
→−∞ weshowthe comparisonofgroundstateenergiesobtained
function u has to vanish as x being a bound state
→±∞ from the actual solution of the minimization condition of
function and hence the dominant part of Eq. (3) for x
→ Eq. (8) and the energy obtained from Eq. (9) for fixed
will be e 2ax and we have in that range
−∞ − values of λ of λ = 1.0 < λc and different values of K. As
~2 d2u expected the approximation of Eq. (9) gets closer to the
+De 2axu 0 (4)
− 2mdx2 − ∼ answerobtainedfromEq. (8)asK increases. Theincrease
of λ weakens the ground state and for large K, we have
Makingthesubstitutione ax =y,wefindtheexactasymp-
− the result that the critical λ for disappearanceof a bound
toticformofuforx (y )tobeofthefollowing
→−∞ →∞ state scales as K3/2 i.e., D3/4.
form
uasy =Ae−Ky (5) K E0(fromEq.(9)) E0(fromEq.(8))
2 -1.762 -2.243
where, K2 = 2mD and A is a constant to be determined
~2a2 3 -5.619 -6.25
from normalization in y-space. Since the ground state
4 -11.504 -12.25
wave function will have no zeroes at any finite nonzero
5 -19.404 -20.25
value of y (note that x corresponds to 0
−∞ ≤ ≤ ∞ ≤ 6 -29.315 -30.25
y ), we take u(y)=√Nyαe Ky, to be the trial wave
−
≤∞
function for the minimization of E given in Eq. (2) and
Table 1: Results compairing the ground state energies from varia-
we obtain the following expression for the ground state
tional calculation (third column) with the expression (Eq. (9) ob-
energy, tained considering the asymptotic series for different values of K
keepingλfixedat1.0
~2a2 agm Γ(4α)
E(α)= [α2+α(1 2K)+ ] (6)
2m − ~2a224α[Γ(2α)]2 Theinterestingthingwouldbetoexplorethedynamics
ofaninitialGaussianwavepacketintheregionλ>λ and
If g = 0 (i.e., usual Schro¨dinger equation with a Morse c
λ<λ for λ>0. In the next section we describethe wave
potential),wegetthecriticalαforminimizationofenergy c
packet dynamics both for λ>0 and λ<0.
is α=K 1 and the groundstate energy turns out to be
− 2
E = ~2a21(1 2K)2 (7) 3. Wave packet dynamics
− 2m 4 −
The effective Hamiltonian of GPE with a Morse trap
whichis the exactanswer.Itshouldbe notedthatfor K =
is given by
1, the binding energyreacheszero. i.e., forK < 1 (equiv-
2 2
p2
alentlyfora>q8m~2D,therecanbenoboundstateinthis H = 2m +D(e−2ax−2e−ax)+gZ dx|ψ(x)|4
potential. We now analyze the effect of non linear term.
Returning to Eq. (6), we consider the term involving ‘g’ We make H dimensionless by the following rescallings:
p = √mDp¯, ψ¯ = ψ 1, x = x¯, ∆ = ∆¯, t = t¯ a2D,
qa a a q m
2
V¯ = V and g = γD. Hence the relation between λ 3.1. Dynamics: (γ > 0 and γ < γ ); Corresponding
D √2π a c
and γ turns out to be λ = √2πγK2. Considering these figure: Fig 2
2
transformations the dimensionless Hamiltonian turns out
to be
H p¯2
H¯ = = +(e−2x¯ 2e−x¯)+√2πγ dx¯ψ¯(x¯)4(10)
D 2 − Z | |
We consider Gaussian wave packets of the form given be-
low
ψ(x,t)= 1 e−(x−2∆x(0t()t2))2eip(t)x/~ (11)
π1/4 ∆(t)
p
i.e., we assume an initially Gaussian form remains Gaus-
sian with a changing centre and width. The initial shape
corresponding to x¯ =0 and ∆=∆ has the energy
0 0
hED0i = 2K12∆¯ + p¯202 +[e∆¯20 −2e∆¯420]+ ∆¯γ (12)
0 0
The dynamics of the wave packet is governed by the fol-
lowingequations(wedropthebarswiththeunderstanding
that all quantities are dimensionless).
d x
h i = p
dt h i
d x2
h i = xp+px
dt h i
d p dV
h i = =2( e−2x e−x )
dt −hdxi h i−h i
d p2 dV dV d
h i = p + p √2πγ ψ 4dx (13)Figure2: Figure(a)showsthedynamicsofthepeak(x0)ofthewave
dt −h dx dx i− dtZ | | packetforγ(=0.5)<γc(=0.917)forfourdifferentvaluesofp0. The
d xp+px dV potential parameter K = 2 and the initial x0 = 0 while the width
h dt i =2hp2i−2aDhxdxi+√2πγZ |ψ|4dx ∆0 = 0.4. For low values of p0 the wave packet hits the potential
barrieratrightand reflects back andforthwithinthe trapshowing
=2 p2 +4aD( xe−2x xe−x ) oscillatory behavior. With increase of p0 the number of reflection
h i h i−h i reduces and finally at p0 = pth the wave packet comes out of the
+√2πγ ψ 4dx potential barrierwhichisrepresentedbythelinearincreaseofx0(t)
Z | | atlatertime(greencurve). Thefigureaboveshowsthatinthiscase
pth=0.45. Figure(b)showsthedynamicsofthewidth(∆). Allthe
With the help of the equations given in Eq. (13), we find parameters are same as are taken in figure (a). With lower values
that the dynamics of the wave packet will be governedby of p0 when the wave packet hits the potential barrier and reflects
back and forth within the trap, ∆(t) also shows nearly oscillatory
the two coupled equations given below
behavior. With increase of p0 the the wave packet finally escapes
ddt22x0 =2[e−(2x0−∆2)−e−(x0−∆42)] (14) withincreaseof∆(t)astincreases(greencurve).
Inthisparameterregionaftersolvingthecoupledequa-
tions given in Eq. (14) and Eq. (15) numerically, we have
d2 ∆2 = 2 1 + γ +4∆2[1e−(x0−∆42) foundthatwheninitialmomentum(p0)ofthewavepacket
dt2 K2∆2 ∆ 2 (15)
issmall,itreflectsbackfromthepotentialbarrieratright.
e−(2x0−∆2)] Thenitmovestotheleftandcollideswiththeinfinitebar-
−
rier or the potential at right. This back and forth oscil-
HerewehaveexploredthewavepacketdynamicsforK =2
lation within the trap continues with time. With the in-
which fixes γ =0.917.
c creaseof p , the oscillationof the peak ofthe wavepacket
0
within the trapgoesonwiththe higheramplitude up to a
certain threshold value ((p ) of p . At p =p it simply
th 0 0 th
comesoutofthepotentialbarrierandhencetheoscillatory
behavior of x (t) ceases. For p > p ) the graph for x (t)
0 0 t 0
vs. t shows sharp linear increase (not shown in given fig-
ures). ForK =2,weobservedp =0.45. Hence withthe
th
help of Eq. (12), we conclude that the initial wave packet
3
has to have the minimum average energy E = 0.751D to thatwavepacketin this parameterregionalwayshaveen-
come out of the potential barrier. We will call this mini- ergy E >E (=0.751D obtained for K =2 andγ =0.5).
th
mumaverageenergyrequiredforemittingthewavepacket Thus the wave packet always comes out of the potential
fromthepotentialasthethresholdenergy(E ). Itshould barrier and delocalises in space. In Fig 3, we clearly de-
th
be noted that in the classical case, for a partice to es- scribe this feature considering γ(= 1.2) > γ (= 0.917),
c
cape from the Morse potential, the averageE needs to be ∆ =0.4 and K =2. The behavior is alwaysthatof a free
0
greaterthanorequaltozero. Iftheparticleisattheorigin particle.
initially with a potentialenergy of-1,then the threshhold
momentum would be p = √2. In the quantum case the 3.3. Dynamics(γ <0); Corresponding figure: Fig 4
0
average initial momentum is 0.45 clearly showing the role and Fig5
of quantum fluctuations. The dynamics of the width also
follows the same qualitative feature as described above.
The dynamics in this parameter regionis clearly depicted
in Fig 2. For convenience we keep the initial width of the
wave packet fixed at ∆ =0.4 throughout the study.
0
3.2. Dynamics: (γ > 0 and γ > γ ); Corresponding
c
figure: Fig 3
Figure 4: Figure (a) shows the dynamics of the peak (x0) of the
wavepacketforγ=−0.5forthreedifferentvaluesofp0. K=2and
(∆0 = 0.4) are considered. Like Figure 2(a), with lower values of
p0 the wave packet hits the potential barrier and reflects back and
forth within the trap. x0(t) shows oscillatory behavior in this case
also. Withincreaseofp0thenumberofreflectionreduces,oscillatory
behavior ceases and finally when p0 =pth(=1.35) the wave packet
comesoutofthepotentialbarrierindicatedbythegradualincreaseof
x0(t). Infigure(b),thedynamicsofwidthshowsarandomoscillation
for p0 < pth(= 1.35) and as p0 → pth, the oscillation takes more
periodic manner with constant amplitude. But unlike the previous
casesherethewidthisboundedbyamaximumandminimumvalue
Figure3: Figure(a)showsthedynamicsofthepeak(x0)ofthewave (∆0).
packet for γ =1.2forthree differentvalues of p0. Thedynamics of
x0 alwaysshowslinearincreaseforallthesethreevaluesofp0unlike In this parameterregionthe dynamics ofx shows the
0
thepreviouscaseofγ<γc. Thisimpliesinthisparameterregionthe
same qualitative feature as is described in Figure 2. We
wave packet will always come out of the potential barrier whatever
betheinitialmomentum. Thewidthalsoincreasesandthedynamics set γ = 0.5, K = 2 and ∆0 = 0.4. But unlike the
−
islikethatofafreeparticle. previouscasesnow the dynamics ofthe width is bounded.
The random oscillationof width continues with time with
In this parameter region we have observed the linear moderate amplitude and as p p the oscillation takes
0 th
→
increase of x (t) irrespective of initial momentum (p ) of place in a more periodic manner with constant amplitude
0 0
thewavepacket. Thewidthalsoshowscontinuousincrease which implies that the wave packet will never delocalise
with time. With the help of Eq. (12), we now conclude in space after coming out of the potential barrier. This
4
case is depicted in Fig 4. In Fig 5 we have shown the K E (for γ =0.5) E (for γ = 0.5)
th th
−
dynamics of x and ∆ for a greater value of γ = 1.2.
0 2 0.751D -0.934D
−
The dynamics of x follows the same qualitative feature
0 3 0.904D -1.09D
as described earlier. But this time the dynamics of the
4 0.971D -1.14D
widthshowsaninterestingfeature. Thewidthofthewave
5 0.986D -1.17D
packethasatendencytoretainitsinitialshapewithinthe
6 1.004D -1.185D
trap as well as after coming out of the trap.
Table 2: Results showing how the threshold energy (Eth) behaves
with potential paramater K for two fixed values of γ (γ = 0.5 and
γ=−0.5)
4. Conclusion
Inconclusion,wehaveexploredthefeaturesoftheGPE
with the Morse potential. We have found that for a posi-
tive couplingconstantanda deeppotentialthere isa crit-
ical coupling g where the ground state disappears as a
c
bound state and g4/3 scales as the depth of the potential.
c
For the dynamics in this potential if g < g , the initial
c
wave packet needs to posses a threshold average momen-
tumforthepackettoescapefromthepotential. Forg >g
c
however,the wavepacketdynamics alwaysresemblesthat
for a quantum free particle. If the coupling g is negative,
we find that the packetdoes escape from the potential for
above threshold value of the average momentum and fur-
ther if g is more negative than a critical value, the width
of the packet remains constant in time.
Acknowledgments
Oneoftheauthors,SuklaPalwouldliketothankS.N.
Bose National Centre for Basic Sciences for the financial
support during the work. Sukla PalacknowledgesHarish-
Figure 5: Figure (a) shows the dynamics of the peak (x0) of the Chandra Research Institute for hospitality and support
wave packet for γ = −1.2 for three different values of p0. K = 2 during visit.
and (∆0 = 0.4) are considered. Like Fig. 4(a), with low values of
p0 the wave packet hits the potential barrier and reflects back and
forth within the trap, showing oscillatory behavior. With increase References
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6