a r X i v :c o n d -m a t /0702281v 1 [c o n d -m a t .o t h e r ] 12 F e
b 2007
EPJ manuscript No.
(will be inrted by the editor)
Finite-well potential in the 3D nonlinear Schr¨o dinger equation:Application to Bo-Einstein condensation
Sadhan K.Adhikari a
Instituto de F ´ısica Te´o rica,UNESP −S˜a o Paulo State University,01.405-900S˜a o Paulo,S˜a o Paulo,Brazil
Received:date /Revid version:date
Abstract.Using variational and numerical solutions we show that stationary negative-energy localized
(normalizable)bound states can appear in the three-dimensional nonlinear Schr¨o dinger equation with
a finite square-well potential for a range of nonlinearity parameters.Below a critical attractive nonlinearity,the system becomes unstable and experiences collap.Above a limiting repulsive nonlinearity,the system becomes highly repulsive and cannot be bound.The system also allows nonnormalizable states of infinite norm at positive energies in the continuum.The normalizable negative-energy bound states could be created in BECs and studied in the laboratory with prent knowhow.
PACS.45.05.+x General theory of classical mechanics of discrete systems –05.45.-a Nonlinear dynam-ics and chaos –03.75.Hh Static properties of condensates;thermodynamical,statistical,and structural properties
1Introduction
The nonlinear Schr¨o dinger (NLS)equation with cubic or Kerr nonlinearity appears in many areas of physics and mathematics [1].Of the,two areas have drawn much attention in recent time.They are pul propagation in nonlinear medium [2,3,4]and Bo-Einstein condensation (BEC)in confining traps [5].A quantum-mechanical mean-field description of BEC is done using the nonlinear Gross-Pitaevskii (GP)equation which is esntially the NLS equation with cubic nonlinearity.Though the NL
S equa-tion in the two areas have similar mathematical struc-ture,the interpretation of the different terms in it is quite distinct.In BEC it is an extension of the Schr¨o dinger equa-tion to include a nonlinear interaction among bosons.In optics it describes electromagnetic pul propagation in a nonlinear medium.Also,usually there is no external po-tential in the NLS equation in optics [1],whereas in BEC a trapping potential is to be included in it [5].In most stud-ies in BEC an infinite parabolic harmonic potential has been included in the NLS equation which simulates the infinite or nearly infinite experimental parabolic magnetic trap.
In this paper we consider a finite square-well trapping potential in the NLS equation with cubic nonlinearity.Al-though,we consider a square-well potential for obvious analytical knowledge about this potential,most of our re-sults should be valid for any finite potential and the ex-periments are really carried out on finite traps.This po-tential is piecewi constant and leads to analytic solution
2S.K.Adhikari:Finite-well potential in the3D nonlinear Schr¨o dinger equation We show that it is possible to have normalizable sta-
tionary BEC bound states in localizedfinite3D square-
well potentials for a range of nonlinearity parameters.A
too strongly attractive nonlinearity parameter is found to
lead to collap,whereas a very strong nonlinear repulsion
does not bind the system.In addition to the normalizable
stationary bound states,the repulsive NLS equation with
square-well interaction is also found to yield nonnormal-
izable states where the probability density has a central
peaking on a constant background extending to infinity.
Obviously,the nonnormalizable states do not satisfy the
boundary condition that the wave functionψ(r)at a ra-
dial distance r should tend to0as r→∞.The forma-
1375年
tion and the study of the normalizable states could be of
utmost interest in veral ,optics[1],nonlin-
ear physics[1]and BEC[5],whereas the nonnormalizable
states will draw the attention of rearchers in mathemat-
ical and nonlinear physics.We u both variational as well
as numerical solutions of the NLS equation in our study.
In this connection we mention that in an exponentially
decayingfinite potential well one could have the interest-
ing possibility of quantum tunneling and the appearance
of quasi-bound states,which has been studied in detail in
Ref.[14].In the prent study with square-well potential
this possibility is not of concern.
In Sec.II we prent the theoretical model which we
u in our investigation.In Sec.III we explain how to ob-
tain numerically the usual normalizable solutions of the
NLS equation with thefinite and infinite square-well po-
tentials.We also explain the origin of the nonnormaliz-
able solutions and how to obtain them numerically.Then
we develop a time-dependent variational method for the
study of this problem.The nonlinear problem is reduced
to an effective potential well.The possibility of the ap-
pearance of stable bound states in this effective potential
for a wide range of the parameters is discusd.In Sec.IV
we consider the complete numerical solution of the NLS
equation for afinite and infinite square-well potentials.
We obtain the condition of stability of the bound states
numerically andfind their wave functions.We also ob-
tain the nonnormalizable solutions of the NLS equation
numerically.Finally,in Sec.V a brief summary is given.
2The Nonlinear Schr¨o dinger Equation
We begin with the radially-symmetric time-dependent qu-
antum-mechanical GP equation ud to describe a BEC
at0K[5].As we shall not be concerned with a particular
experimental system,we write the GP equation in dimen-
sionless variables.The radial part of the3D spherically-
symmetric GP equation for the Bo-Einstein condensate
wave functionΦ(R;τ)at position R and timeτcan be
written as[15]
−i¯h∂2m ∂2
R ∂
¯h/(mω)and whereωis an external reference
angular frequency.In terms of the new variables the GP
equation becomes
−i∂∂r2−2∂r+V(r)+g|Ψ(r;t)|2 Ψ(r;t)=0,(2)
where g=8πaN/l.The square well potential is taken as
V(r)=−γ2,r≤Λ;=0,r>Λ,(3)
withγ2the depth andΛthe range.Equation(2)with
cubic nonlinearity is the usual NLS equation often ud
in problems of optics and nonlinear physics and will be
子的词语
referred to as the NLS equation in the following.If we t
g=0in Eq.(2),this equation becomes the usual linear
Schr¨o dinger equation.In BEC t denotes time and r the
space variable.In nonlinear optics,t denotes the direction
of propagation of pul,r denotes the transver direc-
tions,andΨrefers to components of electromagneticfield.
In nonlinear optics the3D NLS equation is spatiotemporal
in nature where for anomalos dispersion the time variable
can be combined with the two space variables in transver
directions to define the3D vector r with r=|r|.There
have been many numerical studies of the3D NLS equa-
tion in optical pul propagation[4,16].In BEC a scaled
nonlinearity n is often defined by n=g/(8π)=Na/l.
The normalization condition in Eq.(2)is
d3r|Ψ(r;t)|2=1.(4)
3Analytic Consideration
3.1Normalizable Solution
The localized normalizable solutions to nonlinear equation
(2)with potential(3)are allowed only at negative ener-
gies.We solve numerically Eq.(2)starting from a time
iteration of the linear problem obtained by tting g=0
in this equation.Hence we prent a brief summary of the
linear problem in the following[17].The stationary solu-
tion of the nonlinear equation(2),which we look for,has
the formΨ(r;t)=ψ(r)exp(−iµt)withµthe chemical
potential,so that
−∂2r∂
(γ2−|µ|),for r≤Λ;and
S.K.Adhikari:Finite-well potential in the3D nonlinear Schr¨o dinger equation3
β=
αr
,r≤Λ,(6)
=−
B
√µ−V(r).(9)As V(r)is piecewi constant,in the TF approximation ψTF(r)will also be piecewi constant.Usually,to imple-ment the TF approximation we need the parameter|µ|. But in this ca we determineµby requiring that in the TF approximationψTF(r)→c asymptotically.The actual numerical solution also tends to this asymptotic limit.
If the solution of Eq.(5)is generated from the initial constant solutionψ(r)=c,the TF approximation to the wave function becomes
ψTF(r)≡C= µ/g,r>Λ,(11) with
C= g.(12) 3.3Variational Result
To understand how the normalizable bound states are formed in the square-well potential we employ a varia-tional method with the following Gaussian wave function for the solution of Eq.(2)[16]
ψ(r,t)=A(t)exp −r22β(t)r2+iα(t) ,(13)
2022元宵节
where A(t)≡[π3/4R3/2(t)]−1,R(t),β(t),andα(t)are the normalization,width,chirp,and pha ofψ,respectively. The Lagrangian density for generating Eq.(2)is[16,18]
L(ψ)=i∂r 2
−V(r)|ψ|2−1
dt
∂L eff
∂q(t)
,(15)
where q(t)stands for the generalized displacements R(t),β(t),A(t)orα(t).
公务员职级
The following expression for the effective Lagrangian can be calculated after some straightforward algebra[18]
L eff=
π3/2A2(t)R3(t)
2
˙β(t)R2(t)−1
2
gA2(t)−2˙α(t)−3
√2R e−Λ2/R2−
√
4
Erf Λ
4S.K.Adhikari:Finite-well potential in the3D nonlinear Schr¨o dinger equation with the error function Erf(x)defined by
Erf(x)=2
π x0e−t2dt.(18)
The Euler-Lagrange equations(15)forα(t),A(t),β(t), and R(t)are given,respectively,by
π3/2A2R3=constant=1,(19)
3˙β+4˙α
R4
+
2
√R2
+
2
R2FΛγ(R)+10β2 +
16
π
γ2Λ3
√R4−4β2+
gA2
2R2
−16πγ2Λ3
dt2≡−
dU(R)
R3+
8n
2π
1
3
√R4e−Λ2/R2,(24)
where n=g/(8π)and U(R)is the effective potential of motion given by
U(R)=2
3
√R3+2
S.K.Adhikari:Finite-well potential in the 3D nonlinear Schr¨o dinger equation 5
-
1.1
-1-0.9-0.8-0.7-0.60
0.2
0.40.6
0.8
1
C r i t i c a l N o n l i n e a r i t y
γ−2 (Ε0−1
)Λ = 2 µm (a)numerical variational
-4-3-2-100
2
4
6810
C r i t i c a l N o n l i n e a r i t y
Λ (µm)
γ2 = 4E 02
(b)numerical variational
Fig.2.Critical nonlinearity n crit (a)vs.γ−2for Λ=2µm and (b)vs.Λfor γ2=4E 0≡4¯h ω:full line -numerical;dotted line -variational.
nonlinearity n =0.Then during time iteration the non-linearity n is switched on slowly until its desired value is attained.The change in the parameter should be such that
it does not greatly alter the eigenvalue of the Hamiltonian (after time propagation).We also calculated the nonnor-malizable bound states in the continuum for a positive n .To obtain this solutio
n,the time iteration is started with a constant wave function for a finite positive n with γ=0.Then in the cour of time iteration the strength γof the square-well potential is switched on slowly until its de-sired value is attained.If stabilization upon time iteration could be achieved for the chon parameters one already obtains the required nonlinear bound state in the square-well potential.In previous studies we compared critically the prent numerical scheme for the time-dependent NLS equation with other approaches [20,21]including the time-independent schemes [22]and assured that the prent approach leads to results with high precision not only for the NLS equation with one space variable but also for NLS equations in two and three space variables [23].The agree-ment between the results obtained with real and imagi-nary time propagation also assures the correctness of our results.Although we calculate our results in dimensionless units,typical parameters for an experimental realization can be easily obtained therefrom for a particular atom.In the following we prent results for the Rb atom.For Rb let 30
60901201501800
5
10152025
L i m i t i n g N o n l i n e a r i t y
γ2
(E 0)
Λ = 2 µm
variational Fig.3.Limiting nonlinearity n lim vs.γ2for Λ=2µm.
the length l be 1µm;for this to be possible the reference frequency is ω≈2π×38Hz.Conquently,the unit of energy is E 0≡¯h ω≈1.57×10−13eV.4.1Normalizable States
Stable normalizable bound states are indeed found in all cas for various ranges of parameters.Some plausible
properties of the bound state are found in agreement with the above variational study.For a given nonlinear-ity,the bound states are only formed for a sufficiently strong square-well potential determined by Λand γ.For
weaker potentials,from the wisdom obtained in varia-tional calculation,the effective potential U (R )does not have a minimum and there cannot be any bound state.
电脑怎么共享屏幕
怀孕可以补牙吗For a given square-well potential,bound states are found for n lim >n >n crit .
It is difficult to obtain the limiting nonlinearity n =n lim numerically as this corresponds to a state with zero energy which extends to a very large r .However,the crit-ical value n =n crit can be obtained numerically in a controlled fashion as the wave function is highly localized in this limit.In Figs.2we plot the critical nonlinearity for collap n crit for different parameters of the square-well potential.In Fig.2(a)we plot n crit vs.γ−2for Λ=2µm whereas in Fig.2(b)we plot n crit vs.Λfor γ2=4E 0.In the figures we show results of variational and full numerical calculations.The variational calculation always leads to a larger |n crit |.In ca of the infinite harmonic potential also,the variational estimate of |n crit |is larger than the result of full numerical calculation [5].For the infinite square-well potential with γ2=∞and Λ=2µm,n crit =−0.62;for the infinite parabolic potential n crit =−0.575[5,24].Becau of the different shapes of the two infinite potentials n crit is different in the two cas.Although it is difficult to obtain n lim from a numeri-cal solution of the NLS equation,it is possible to obtain it from the variational calculation.The limiting nonlinear-ity n lim corresponds to the largest value of n for which U (R )has a minimum.In Fig.3we plot n lim vs.γ2for Λ=2µm.We e that n lim increas linearl
y with the
6S.K.Adhikari:Finite-well potential in the 3D nonlinear Schr¨o dinger equation
numerical
variational Thomas-Fermi 300.10.20.30.40.50
1
2
456
ψ(r )
r (µm)
γ2 = 4E 0Λ = 2 µm
n = -0.75
n = 0n = 10
(a)4γn = -0.65-0.100.10.20.302
68ψ(r )
r (µm)
2 = 4E 0Λ = 4 µm
n = 6
(b)
4γ-0.100.10.20.3
0.40
26810
ψ(r )
r (µm)
2 = 4E 0Λ = 4 µm
涛组词n = -0.4
n = 6
(c)
Fig.4.(a)Ground-state wave function ψ(r )of the NLS equa-tion (5)with the square-well potential for γ2=4E 0,Λ=2µm and scaled nonlinearity n −0.75,0and 10(upper to lower curves).In all cas the numerical and variational results are shown,in addition,for n =10the TF approximation is also shown.(b)Same for the first excited soliton (numerical calcu-lation only)with γ2=4E 0,Λ=4µm and n =−0.65and 6.(c)Same for the cond excited soliton (numerical calculation only)with γ2=4E 0,Λ=4µm and n =−0.4and 6.
strength of the square well γ2.The variational calcula-tion underbinds the system.The repulsive nonlinearity destroys binding and a smaller repulsive nonlinearity can destroy the weaker binding of the variational model.Con-quently,the variational limiting nonlinearity is smaller than the actual limi
不知心恨谁
ting nonlinearity,which we verified in our numerical calculation.The numerical calculation re-lies on the existence of a localized wave function and it is difficult to calculate the limit when this wave function extends to infinity and a localized wave function ceas to
00.2
0.40.60.8100.5
1 1.52
ψ(r )
r (µm)
Λ = 2 µm n = -0.62
n = 0n = 10
(a)
1n = -0.3-0.400.40.81.2
00.5
1.52
ψ(r )
r (µm)
Λ = 2 µm n = 0n = 10
(b)
Fig.5.(a)Ground-state wave function ψ(r )of the NLS equa-tion (5)for the infinite square-well potential of range Λ=2µm and scaled nonlinearity n =−0.62,0and 10(upper to lower curves);(b)same for the first excited soliton for n =−0.3,0and 10.
exist.The TF wave function (9)is always fully localized within r <Λand is inadequate for calculating the limit-ing nonlinearity for µ→0.The TF wave function leads to a much smaller value n lim =Λ3γ2/6,bad on imposing µ=0within the TF regime.
In Fig.4(a)we plot the wave function for the bound state of the NLS equation for a square-well potential with γ2=4E 0,Λ=2µm and for nonlinearities n =−0.75,0and 10.The system with n =−0.75is attractive,n =0noninteracting,and n =10repulsive.In the cas results for both the full numerical calculation and variational ap-proximation are shown.In addition,for n =10the TF approximation (10)with c =µ=0is also shown.In Fig.4(b)we show the wave functions for the first excited soli-ton with a single node for γ2=4E 0and Λ=4µm and for n =−0.65and 6.In Fig.4(c)we show the wave functions for the cond excited soliton with two nodes for γ2=4E 0and Λ=4µm and for n =−0.4and 6.In Figs.4we find that the bound state for the attractive system with nega-tive n values is more centrally peaked than the bound state for the repulsive system with positive n values.This is true for both ground and excited solitonic states in the finite square-well potential as well as for states in the infinite square-well potential studied below.The central peaking of the wave function for the attractive system correspond-
