Download Analytical study of resonant transport of Bose

PHYSICAL REVIEW A 73, 033608 共2006兲
Analytical study of resonant transport of Bose-Einstein condensates
K. Rapedius, D. Witthaut, and H. J. Korsch*
Technische Universität Kaiserslautern, FB Physik, D-67653 Kaiserslautern, Germany
共Received 19 December 2005; published 13 March 2006兲
We study the stationary nonlinear Schrödinger equation, or Gross-Pitaevskii equation, for a one-dimensional
finite square-well potential. By neglecting the mean-field interaction outside the potential well it is possible to
discuss the transport properties of the system analytically in terms of ingoing and outgoing waves. Resonances
and bound states are obtained analytically. The transmitted flux shows a bistable behavior. Novel crossing
scenarios of eigenstates similar to beak-to-beak structures are observed for a repulsive mean-field interaction.
It is proven that resonances transform to bound states due to an attractive nonlinearity and vice versa for a
repulsive nonlinearity, and the critical nonlinearity for the transformation is calculated analytically. The boundstate wave functions of the system satisfy an oscillation theorem as in the case of linear quantum mechanics.
Furthermore, the implications of the eigenstates on the dymamics of the system are discussed.
DOI: 10.1103/PhysRevA.73.033608
PACS number共s兲: 03.75.Dg, 03.75.Kk, 42.65.Pc
I. INTRODUCTION
Due to the experimental progress in the field of BoseEinstein condensates 共BECs兲, e.g., atom-chip experiments
achieved in the past few years 关1–3兴, it is now possible to
study the influence of an interatomic interaction on transport
关4,5兴. At low temperatures, BECs can be described by the
nonlinear Schrödinger equation 共NLSE兲 or Gross-Pitaevskii
equation 共GPE兲
iប
冉
冊
⳵␺共r,t兲
ប2 2
= −
ⵜ + V共r兲 + g兩␺共r,t兲兩2 ␺共r,t兲
⳵t
2m
共1兲
in a mean-field approach 关6–9兴. Another important application of the NLSE is the propagation of electromagnetic
waves in nonlinear media 共see, e.g., Ref. 关10兴, Chap. 8兲. In
the case of vanishing interaction 共g = 0兲 it reduces to the linear Schrödinger equation of single particle quantum mechanics.
The treatment of transport within this mean-field theory
reveals new interesting phenomena which arise from the
nonlinearity of the equation. For example a bistable behavior
of the flux transmitted through a double barrier potential has
been found 关5兴. For the same system transitions from resonances to bound states can occur due to the mean-field interaction 关11,12兴. However, previous studies of nonlinear transport and nonlinear resonances usually relied strongly on
numerical solutions, such as complex scaling. There are only
few model systems of the NLSE which have been studied
analytically and most of these treatments concentrate on
bound states. Among these few examples are the infinite
square-well 关13–15兴, the finite square-well 关16–18兴, the potential step 关19兴, ␦ potentials 关20,21兴 and the ␦ comb 关22,23兴
as an example of a periodic potential. Therefore analytical
solutions of the NLSE for other simple model potentials are
of fundamental interest.
In the present paper we solve the time-independent NLSE
for a simple one-dimensional square-well potential in order
*Electronic address: [email protected]
1050-2947/2006/73共3兲/033608共12兲/$23.00
to analytically indicate and discuss the nonlinear phenomena
described above. To unambiguously define ingoing and outgoing waves and thus a transmission coefficient 兩T兩2, we neglect the mean-field-interaction g兩␺共x兲兩2 outside the potential
well. Since the one-dimensional mean-field coupling constant g is inversely proportional to the square of the system’s
radial extension a⬜, this can be achieved in an atom-chip
experiment by a weaker radial confinement of the condensate
outside the potential well 关6,17,24兴. Especially for bound
states our approximation is well justified since in this case
the condensate density 兩␺共x兲兩2 is much higher inside the potential well than outside.
The paper is organized as follows. In Sec. II we analyze
stationary scattering states of the NLSE with a finite squarewell potential. Then we investigate the transition from resonances to bound states for an attractive 共Sec. III兲 and a repulsive mean-field interaction 共Sec. IV兲. In addition to our
discussion in terms of stationary states we have a brief look
at the dynamics of the system in Sec. V, where we solve the
time-dependent NLSE numerically. In Appendix A we
briefly review some basic results for the linear Schrödinger
equation with a square-well potential which has been treated
in various textbooks on quantum mechanics 关25–27兴,
whereas the Appendixes B and C contain details of our calculations of resonances and bound states.
II. SCATTERING STATES
We consider the one-dimensional NLSE with a squarewell potential, where we set the mean-field term g兩␺共x兲兩2 to
zero outside the potential well. This means that for 兩x兩 ⬎ a the
wave function must satisfy the free linear Schrödinger equation
冉
−
冊
ប2 d2
− ␮ ␺共x兲 = 0
2m dx2
共2兲
and for 兩x兩 艋 a the time-independent NLSE
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©2006 The American Physical Society
PHYSICAL REVIEW A 73, 033608 共2006兲
RAPEDIUS, WITTHAUT, AND KORSCH
冉
−
冊
ប2 d2
+ g兩␺共x兲兩2 + V0 − ␮ ␺共x兲 = 0
2m dx2
with a constant potential V0 ⬍ 0, where ␮ ⬎ 0 is the chemical
potential.
For an incoming flux from x = −⬁ we make the ansatz
冦
Aeikx + Be−ikx , x ⬍ − a,
兩x兩 艋 a,
␺共x兲 = f共x兲,
Ceikx ,
x⬎a
冧
f共x兲 = 冑S共x兲ei⌽共x兲 ,
␣
and S共x兲 = ␧ + ␸dn2共x + ␦兩p兲
S共x兲
ប2 2
,
gm
册
S⬘共− a兲 i⌽共−a兲
e
.
2k
关S共− a兲 − S共a兲兴2 + S⬘共− a兲2/共4k2兲
关S共− a兲 + S共a兲兴2 + S⬘共− a兲2/共4k2兲
共15兲
and the transmission probability
兩T兩2 =
4S共a兲S共− a兲
S共a兲
.
2
2
2 =
关S共− a兲 + S共a兲兴 + S⬘共− a兲 /共4k 兲 兩A兩2
共16兲
It can easily be verified that 兩R兩2 + 兩T兩2 = 1. The scattering
phase ␪共␮兲, defined by T共␮兲 = 兩T共␮兲兩exp关i␪共␮兲 − 2ika兴, can be
expressed as
␪共␮兲 = arg共C兲 − arg共A兲 + 2ka
冕
+a
−a
S⬘共− a兲
kS共a兲
dx + arctan
. 共17兲
S共x兲
2k关S共− a兲 + S共a兲兴
The condition 共10兲 can be further exploited by rewriting it as
−2␸p dn共u 兩 p兲sn共u 兩 p兲cn共u 兩 p兲 = 0 with the abbreviation u
= a + ␦. As the dn function is always nonzero, the solutions
of this equation are given by the zeros of either of the two
functions cn共u 兩 p兲 or sn共u 兩 p兲.
Case 1.
sn共a + ␦兩p兲 = 0 Û a + ␦ = 2jK共p兲 Þ ␦ = 2jK共p兲 − a.
3
ប2
共2 − p兲2 ,
␮ = V0 + g␧ −
2
2m
−
兩R兩2 =
=
共7兲
S共− a兲 − S共a兲 + i
As in the linear case 共see Appendix A兲, we define the amplitudes of reflection and transmission as R ª B / A and T
ª C / A. The above equations then lead to the reflection probability
共6兲
in terms of the Jacobi elliptic function dn with the real period
2K共p兲, where K共p兲 is the complete elliptic integral of the first
kind and p is the elliptic parameter 关28–30兴. The real parameters ␣, p, ␧, ␸, , and ␦ have to satisfy the relations
冋
共14兲
共5兲
where S共x兲 and ⌽共x兲 are real functions. As already shown by
Carr, Clark, and Reinhard 关13,14兴 a complex solution of the
NLSE with a constant potential is given by the equations
␸=−
2
共4兲
with the wave number k = 冑2m␮ / ប. Inside the potential we
assume an intrinsically complex solution
⌽⬘共x兲 =
冑S共− a兲Be+ika = 1
共3兲
共8兲
ប2 2
关 ␸␧共p − 1兲 − ␣2兴 + g␧3 − 共␮ − V0兲␧2 = 0.
2m
共9兲
The variable j denotes an integer number. This means
dn2共a + ␦ 兩 p兲 = dn2共2jK共p兲 兩 p兲 = 1 or
S共a兲 = ␧ + ␸ .
共18兲
Case 2. cn共a + ␦ 兩 p兲 = 0. In analogy to case 1 we get ␦
= 共2j + 1兲K共p兲 − a and
The wave function and its derivative must be continuous at
x = ± a. At x = + a we obtain the equations f共a兲 = Ceika and
f ⬘共a兲 = ikCeika. By inserting Eq. 共5兲 we arrive at 21 S⬘共a兲 + i␣
= ikS共a兲. Since we are considering positive chemical potentials ␮ ⬎ 0, the wave number k is real so that we obtain the
conditions
For given values of the chemical potential ␮ and the
incoming amplitude 兩A兩, the parameters p, ␧, and ␸
= −ប22 / gm are completely determined by the equations 共8兲
and 共9兲 and the absolute square of Eq. 共13兲,
S⬘共a兲 = 0 and ␣ = kS共a兲.
共10兲
16k2兩A兩2S共− a兲 = 4k2关S共− a兲 + S共a兲兴2 + S⬘共− a兲2 .
Ae−ika + Beika = f共− a兲,
共11兲
ik共Ae−ika − Beika兲 = f ⬘共− a兲.
共12兲
The positions of the resonances are defined by 兩T兩2 = 1 or
兩R兩 = 0. With Eq. 共15兲 this is equivalent to the conditions
S⬘共−a兲 = 0 and S共−a兲 = S共a兲. Now we consider these conditions for the two cases mentioned above.
Case 1. sn共a + ␦ 兩 p兲 = 0 or ␦ = 2jK共p兲 − a. The condition
S⬘共−a兲 = 0 now reads
At x = −a we obtain
S共a兲 = ␧ + ␸共1 − p兲.
共19兲
共20兲
2
S⬘共− a兲 = − 2␸p cn共u兩p兲sn共u兩p兲dn共u兩p兲 = 0
With Eqs. 共5兲 and 共10兲 these two conditions can be written as
冑S共− a兲Ae−ika = 1
2
冋
S共− a兲 + S共a兲 − i
册
S⬘共− a兲 i⌽共−a兲
e
,
2k
共13兲
with u = −a + ␦. The solutions of this equation are again
given by the zeros of either the sn function or the cn function. One can show easily that only the zeros of the sn function are compatible with the condition S共−a兲 = S共a兲. Thus we
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obtain −2a + 2jK共p兲 = 2lK共p兲 where l and j are integer numbers. The resonance positions are then given by
=
K共p兲
n,
a
共21兲
where the integer number n is defined as n = j − l.
Case 2. cn共a + ␦ 兩 p兲 = 0 or ␦ = 共2j + 1兲K共p兲 − a. In analogy to case 1 one can show that the resonances are also
determined by Eq. 共21兲.
In the case of a resonance there is no reflected wave 共B
= 0兲 so that Eqs. 共11兲, 共18兲, and 共19兲 lead to
兩A兩2 = S共− a兲 = S共a兲 =
or
␧共p兲 =
再
再
␧ + ␸,
case 1,
␧ + ␸共1 − p兲, case 2
兩A兩2 − ␸ ,
case 1,
兩A兩 − ␸共1 − p兲, case 2.
2
冎
冎
共22兲
共23兲
With the help of Eqs. 共23兲, 共7兲, and 共8兲 we obtain a formula
for the chemical potential of a resonance
再
3
ប2K2共p兲 2 共1 + p兲, case 1,
␮R共p兲 = V0 + g兩A兩2 +
n
2
共1 − 2p兲, case 2.
2ma2
冎
共24兲
The chemical potential depends on the parameter p and the
integer number n which must be sufficiently large to make
␮R 艌 0. As in the linear case 关see Eq. 共A5兲兴 there is a term
which grows quadratically in n. This term can be viewed as
an effective increase of the chemical potential in comparison
to the energy of the linear system in case 1 and, respectively,
as a decrease in case 2. However, it turns out 共see below兲 that
the energy shift due to nonlinear interaction is mainly determined by the term 23 g兩A兩2, which is a direct consequence of
the effective increase or decrease of the potential well by the
mean-field term g兩␺共x兲兩2. As we will show below, we have
p → 0 if g → 0 so that K共p兲 → ␲ / 2 and Eq. 共24兲 reduces to the
equivalent equation 共A5兲 for the resonance energies of the
linear system. The Jacobi elliptic parameter p can be calculated analytically in terms of n and g兩A兩2 共Appendix B兲.
Case 2 is valid for 0 艋 g 艋 兩V0兩 / 兩A兩2 and case 1 is valid for
any other value of g 共see Appendix B兲. Since the parameter p
and the chemical potential ␮R only depend on the product
g兩A兩2 we henceforth assume 兩A兩2 = 1 in all figures and numerical calculations. Using Eq. 共23兲 in Eqs. 共5兲 and 共6兲 we obtain
the squared modulus of the resonance wave functions inside
the potential well 兩x兩 艋 a. In dependence of p and n it is given
by
兩␺R共x兲兩2 = 兩A兩2 −
ប 2 2 2
关dn „x − nK共p兲兩p… − 1兴
gm
共25兲
in case 1 and
兩␺R共x兲兩2 = 兩A兩2 −
ប 2 2 2
关dn „x + 共1 − n兲K共p兲兩p… − 1 + p兴
gm
共26兲
in case 2 with = nK共p兲 / a.
FIG. 1. 共Color online兲 Squared modulus of the wave functions
of the two most stable resonances of the potential V0 = −10, a = 2 for
the attractive nonlinearity g = −2. The normalization 兩A兩 = 1 has been
used.
Figure 1 shows the squared modulus of resonance wave
functions for attractive nonlinearities in the region of the
potential well 兩x兩 艋 a where scaled units with ប = m = 1 are
used, as for all figures and numerical calculations in this
paper. The functions are symmetric with respect to x = 0 and
have no nodes. The quantum number n indicates the number
of the minima. The smaller the chemical potential ␮R共n兲 is,
the smaller is the value of 兩␺共x兲兩2 at its minima. In the case of
repulsive nonlinearity the wave functions show exactly the
same behavior.
Figure 2 shows the transmission coefficient 兩T共␮兲兩2 in
dependence of the chemical potential ␮ for the potential V0
= −10 and a = 2 for different attractive nonlinearities g ⬍ 0.
For small values of 兩g兩, the behavior of the transmission coefficient is very similar to the linear system. With increasing
attractive interaction, the curves bend more and more to the
left so that the resonances are shifted towards smaller values
of ␮. This shift is mainly determined by the term 23 g兩A兩2 in
Eq. 共24兲. For g = −2, the lowest resonance so far has disappeared into the bound state regime ␮ ⬍ 0. Between g = −6
and g = −8 the next resonance disappears. It can be shown
that these resonances become bound states of the system 共see
Sec. III兲. For higher values of 兩g兩 the bending of the curve in
the vicinity of the resonances leads to a bistable behavior.
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RAPEDIUS, WITTHAUT, AND KORSCH
FIG. 2. 共Color online兲 Transmission coefficients for different
attractive nonlinearities for the square-well potential V0 = −10, a
= 2. The positions of the resonances, calculated with the formula
共24兲, are marked by asterisks.
For a fixed value of ␮, there are now different wave functions which lead to different transmission coefficients
兩T共␮兲兩2. Due to the nonlinear mean field interaction the state
adopted by the system for an incoming flux from x = −⬁ is
not only determined by the chemical potential ␮ and the
amplitude A but also by the density distribution 兩␺共x , t兲兩2 of
the condensate inside the potential well so that the system
has a memory 共hysteresis兲. This will be further investigated
in Sec. V.
Now we consider repulsive nonlinearities g ⬎ 0. Figure 3
shows the transmission coefficient 兩T兩2 in dependence of ␮
for a potential V0 = −12, a = 4 for different repulsive nonlinearities. Here we observe a shift of the resonance positions
towards higher values of ␮, since the term 23 g兩A兩2 is now
positive. New resonances appear for increasing 兩g兩 which
originate from the bound states of the linear system 共see Sec.
IV and Appendix C兲. Now the curves bend to the right and,
as in the attractive case, the transmission coefficient shows a
bistable behavior. Between g = 4.5 and g = 5, we observe a
bifurcation phenomenon at ␮ ⬇ 1 which resembles an
avoided crossing. This turns out to be a beak-to-beak scenario which also occurs in two-level systems. These phenomena will be discussed in detail in a subsequent paper
关31兴. In all transmission curves we observe that, for a fixed
value of g兩A兩2, the effect of the nonlinearity on the shape of
FIG. 3. 共Color online兲 Transmission coefficients for different
repulsive nonlinearities for the square-well potential V0 = −12, a
= 4. The positions of the resonances, calculated with the formula
共24兲, are marked by asterisks.
the resonances is stronger for small values of ␮ than for large
ones, because the kinetic energy is higher for larger ␮ and
the mean-field energy has a comparatively smaller influence.
This also manifests itself in formula 共24兲 for ␮R, where the
term proportional to n2 counteracts the dominant term
3
2
2 g兩A兩 .
Figures 4 and 5 show scattering phases in dependence of
the chemical potential ␮ which have been calculated by numerical integration of Eq. 共18兲 for the parameters of Figs. 2
and 3. For weak nonlinearities the scattering phase behaves
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FIG. 4. 共Color online兲 Scattering phases for different attractive
nonlinearities for the potential V0 = −10, a = 2. The positions of the
resonances, calculated with the formula 共24兲, are marked by dotted
vertical lines.
similar to the scattering phase of the linear system which is
zero at the positions ␮ = ␮R of the resonances. For stronger
nonlinearities the resonance scattering phase increasingly deviates from zero. Since the squared modulus of the wave
function enters Eq. 共18兲, the scattering phase ␪共␮兲 inherits
the bistable behavior of the transmission coefficient 兩T共␮兲兩2.
Wherever beak-to-beak structures occur in the transmission
coefficient the scattering phase shows loops.
III. BOUND STATES IN THE CASE OF ATTRACTIVE
NONLINEARITY
In this section, we consider bound states of the nonlinear
system described by Eqs. 共3兲 and 共2兲 for attractive nonlinearities g ⬍ 0. In analogy to the linear case, bound states
occur only for chemical potentials ␮ ⬍ 0. As in the linear
case, we look for solutions of even and odd parity. We make
the ansatz
冦
␥ ±e ␬x ,
x ⬍ − a,
␺共x兲± = I±cn共±x + K±兩p±兲, 兩x兩 艋 a,
± ␥ ±e −␬x ,
x⬎a
冧
FIG. 5. 共Color online兲 Scattering phases for different repulsive
nonlinearities for the potential V0 = −12, a = 4. The positions of the
resonances, calculated with the formula 共24兲, are marked by dotted
vertical lines.
The solutions inside the potential well 兩x兩 艋 a are given by
Jacobi elliptic functions cn with amplitudes
I± =
冑
ប2±2 p±
gm
共28兲
2m共␮ − V兲
.
ប2共1 − 2p±兲
共29兲
−
and wave numbers
共27兲
for even 共⫹ sign, K+ = 0兲 and odd parity solutions 关⫺ sign,
K− = K共p−兲兴 with ␬ = 冑−2m␮ / ប.
± =
冑
共1兲 Symmetric solutions. The continuity of the wave function and its derivative at x = ± a yield
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RAPEDIUS, WITTHAUT, AND KORSCH
␥+e−␬a = I+cn共u兩p+兲
共30兲
and ␬␥+e−␬a = I++sn共u 兩 p+兲dn共u 兩 p+兲, with the abbreviation
u = +a. Thus we arrive at the condition
␬cn共u兩p+兲 − +sn共u兩p+兲dn共u兩p+兲 = 0
共31兲
for symmetric bound states. In the limit p → 0, the Jacobi
elliptic functions converge to trigonometric functions and we
regain the condition 共A9兲 of the linear system.
共2兲 Antisymmetric solutions. In analogy to the case of
even parity we obtain the condition
␬cn共u兩p−兲 − −sn共u兩p−兲dn共u兩p−兲 = 0
共32兲
for antisymmetric bound states with u = −a + K共p−兲, which
also converges to Eq. 共A10兲 in the limit p → 0.
Now we show explicitly that the resonances discussed in
Sec. II become bound states for a sufficiently strong attractive mean-field interaction. Due to the behavior of the wave
function outside the potential well, transitions from resonances to bound states occur at ␮ = 0. For ␮ = 0 the condition
共31兲 for symmetric bound states becomes sn共+a 兩 p+兲 = 0
which is solved by
+ = 2j+K共p+兲/a
共33兲
FIG. 6. 共Color online兲 Squared modulus of the wave function of
the most stable resonance of the potential V0 = −10, a = 2 exactly at
the critical point for the transition from resonances to bound states
共see Table I兲. Here, the minima are also zeros of the wave function
共cf. Fig. 1兲. The normalization 兩A兩 = 1 has been used.
共1 − 2pT−兲K2共pT−兲 =
共1 − 2p+兲K 共p+兲 =
2
2ប2 j+2
pT− = −
共34兲
.
24
+
23
冑冉 冊
24
23
冉
+ ␨+ + O共p+3兲
共35兲
冊 冉
冊
2m兩V0兩a2
32
32
共R0/␲兲2
.
1− 2 2 2 =
1−
23
23
ប ␲ j+
j+2
共36兲
The quantity R0 / ␲ in ␨+ determines the number of bound
states N+ = 关R0 / ␲兴⬎ for g = 0 共see Appendix A兲. Since p+
must be positive, ␨+ must be positive as well and we obtain
the condition j+ 艌 N+. One can prove 共see Appendix C兲 that
for a critical nonlinearity gT+ the wave function of a symmetric bound state continuously merges into a resonance wave
function with even quantum number n.
In a similar way one can show that resonances of odd
quantum number n become antisymmetric bound states for a
sufficiently strong attractive nonlinearity gT− ⬍ 0. We obtain
关see Eq. 共33兲兴 − = 共2j− + 1兲K共p−兲 / a with integer j− and n
= 2j− + 1. If we choose the normalization 兩␥−兩 = 兩A兩, the transition occurs at
ប n K共pT−兲 pT−
.
ma2兩A兩2
2 2
gT− = −
␨− =
2
as an approximate solution for the parameter p+ where the
transition occurs. Here we have introduced the abbreviation
␨+ =
2
The parameter pT− ª p− = pR is determined by
24
+
23
冑冉 冊
24
23
2
+ ␨− + O共p−3兲,
共39兲
where we have introduced the abbreviation
For small values of p+ we can make a Taylor approximation
and obtain
p+ = −
共38兲
or
with integer j+. Inserting Eq. 共33兲 into 共29兲 yields
m兩V0兩a2
2m兩V0兩a2
ប2共2j− + 1兲2
共37兲
冉
冊
8m兩V0兩a2
32
1− 2 2
.
23
ប ␲ 共2j− + 1兲2
共40兲
The condition pT− ⬎ 0 implies ␨− ⬎ 0 and thus j− 艌 N− where
N− = 关R0 / ␲ − 1 / 2兴⬎ is the number of antisymmetric bound
states for g = 0 as defined in Eq. 共A11兲.
Figure 6 shows the squared modulus of wave functions at
the transition from resonances to bound states. As in the case
of resonances the functions are symmetric and have n
minima 共cf. Fig. 1兲. The quantum number n counts all the
resonances and bound states defined above and is equal to
the number of minima of the squared modulus of the wave
function. For bound states, these minima are also nodes of
the wave function. Thus, the famous oscillation theorem of
linear quantum mechanics 共see Ref. 关32兴兲 is also valid in the
nonlinear case considered here.
TABLE I. Critical parameters for the transition of the two most
stable resonances of the potential V0 = −10, a = 2 to bound states.
The approximations 共35兲 for p+ and 共39兲 for p− have been used.
pT
gT
gTeff
033608-6
symmetric 共n = 6兲
antisymmetric 共n = 7兲
numerically exact approx.
numerically exact approx.
0.0644
−1.4772
−0.7325
0.0643
−1.4749
−0.7331
0.2044
−6.9159
−3.3592
0.205
−6.969
−3.346
PHYSICAL REVIEW A 73, 033608 共2006兲
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FIG. 7. Dependence of the effective nonlinear interaction
strength g±eff on the parameter p± for the symmetric bound state n
= 0 共solid line兲 and the antisymmetric bound state n = 1 共dashed
line兲.
In order to study the shift of the bound state energies due
to mean-field interaction it is useful to introduce an effective
nonlinear interaction strength
g±eff =
g
2a
冕
a
兩␺±共x兲兩2dx
共41兲
−a
共see Ref. 关20兴兲 in the interaction region 兩x兩 艋 a. The integration can be carried out analytically. We obtain
FIG. 8. 共Color online兲 Wave functions of the two lowest bound
states of the potential V0 = −10, a = 2 for geff = −0.15 共solid lines兲 in
comparison with the respective wavefunctions of the linear system
a
共dashed lines兲. The normalization 兰−a
兩␺共x兲兩2dx = 1 has been used.
from its linear counterpart than the wave function of the state
n = 1.
Figure 9 shows the shift of bound states in dependence of
兩geff兩. The appearance of new bound states can be observed.
For large values of 兩geff兩, which correspond to elliptic parameters p ⬎ 0.5, the states all lie below the potential well depth
V0 = −10.
For small values of p± the chemical potential ␮共g±eff兲 can
be approximated linearly in terms of g±eff,
g−eff = −
ប −
关E„−a + K共p−兲兩p−… − E共p−兲 − 共1 − p−兲−a兴
−
ma
2
g−eff =
ប 2 +
关E共+a兩p+兲 − 共1 − p+兲+a兴
g+eff = −
ma
冋
gT±兩A兩2 E共pT±兲
p T±
K共pT±兲
册
− 共1 − pT±兲 .
a
−a
cn2„−x + K共p−兲…dx
冉
冊
sin共2qa兲
+ O共p−2兲,
2qa
共45兲
where
共43兲
for symmetric bound states where E共p兲 and E共u 兩 p兲 are the
complete and incomplete elliptic integral of the second kind,
respectively. The critical effective nonlinear interaction
strength for the transition from resonances to bound states is
given by
gTeff =
±
冕
= − p−共E0 − V0兲 1 −
共42兲
for antisymmetric and
ប2−2 p−
2ma
E0 = V0 + ប2q2/共2m兲
共46兲
is the energy of the respective bound state of the linear system. For symmetric states we get
共44兲
As seen from Eqs. 共42兲 and 共43兲, 兩geff共p±兲兩 increases faster
for bigger wave numbers ± 共respectively, bigger chemical
potentials ␮, see Fig. 7兲. This means that for a given value of
geff, states which are strongly bound are more affected by the
nonlinear interaction than weakly bound states. As in the
case of resonances this is due to the fact that for weakly
bound states the energy of the mean-field interaction is much
smaller than the kinetic energy and thus has a comparatively
weak influence. This can also be observed in Fig. 8 which
compares the wave functions of the two lowest bound states
of the potential V0 = −10, a = 2 for geff = −0.15 with the respective wave functions of the linear system 共geff = 0兲. The
wave function of the state n = 0 differs much more clearly
FIG. 9. 共Color online兲 Shift of the bound states towards lower
chemical potentials in dependence of 兩geff兩 for p± 艋 0.99 for the
potential V0 = −10, a = 2, for symmetric 共⫻兲 and antisymmetric 共䊊兲
bound states 共attractive nonlinearity兲.
033608-7
PHYSICAL REVIEW A 73, 033608 共2006兲
RAPEDIUS, WITTHAUT, AND KORSCH
冉
g+eff = − p+共E0 − V0兲 1 +
冊
sin共2qa兲
+ O共p+2兲.
2qa
Then, for small values of p±, we have
p± ⬇ −
冋
g±eff
冉
sin共2qa兲
共E0 − V0兲 1 ±
2qa
冊册
共47兲
␬sn共−a兩p−兲 + −cn共−a兩p−兲dn共−a兩p−兲 = 0
共48兲
␮± = V0 + ប2±2共1 − 2p±兲/2m,
共49兲
where ± is the wave number of a Jacobi elliptic function cn
with quarter period K共p±兲. In the limit p± → 0 the cn function
becomes a trigonometric function with quarter period ␲ / 2
and wave number q. This motivates the approximation ±
⬇ 2qK共p±兲 / ␲ which leads to
ប 2q 2 4
共1 − p±兲K2共p±兲.
2m ␲2
␮± ⬇ E0 +
冉
3
sin共2qa兲
1±
2
2qa
冊
n=
3
␮± ⬇ E0 + g±eff
2
共52兲
As in the previous section we look for solutions of odd or
even parity. We make the ansatz
␥ ±e ␬x ,
x ⬍ − a,
␺共x兲± = I±sn共±x + K±兩p±兲, 兩x兩 艋 a,
␥ ±e −␬x ,
x⬎a
冧
共53兲
for even 关⫹ sign, K+ = K共p+兲兴 and odd parity solutions
共- sign, K− = 0兲, where ␬ = 冑−2m␮ / ប.
For 兩x兩 艋 a the solutions are given by Jacobi elliptic sn
functions with amplitude
I± =
冑
冎
共58兲
ប2n2K共pT±兲2 pT±
,
ml2兩A兩2
共59兲
where the transition parameters pT± are given by
共1 + pT±兲K2共pT±兲 =
2m兩V0兩2a2
ប 2n 2
24
24
3
兲 − ␨共n兲 + O共pT±
+ 冑共 25
兲 with
or pT± = − 25
共60兲
2
␨共n兲 =
IV. BOUND STATES IN THE CASE OF REPULSIVE
NONLINEARITY
even parity,
2j− + 1, j− 苸 N0 , odd parity.
gT± =
for an infinitely deep square-well potential 关13–15兴.
ប2±2 p±
gm
2j+, j+ 苸 N0 ,
The transition occurs at
共51兲
A similar relation was derived for a delta-shell potential 关20兴.
For small values of p± this approximation is in good agreement with the numerically exact calculations. In the limit
V0 → −⬁ and thus q → ⬁ we obtain the formula
冦
再
−1
g±eff .
共57兲
for odd parity, which determine the bound states of the system. In the limit p → 0 the conditions 共56兲 and 共57兲 converge
to Eqs. 共A9兲 and 共A10兲 of the linear system.
For the case of an attractive nonlinearity, we have shown
that resonances with even/odd quantum number n become
bound states of even/odd parity due to the attractive meanfield interaction. In a similar way one can show for the case
of repulsive nonlinearity that even/odd bound states become
resonances with even/odd quantum number n.
As before, we use the normalization 兩␥±兩 = 兩A兩. Thus we
obtain the condition ±a = nK共p±兲, where
共50兲
Inserting Eqs. 共46兲 and 共48兲 and expanding K2共p兲共1 − 2p兲
= 共␲ / 2兲2共1 − 23 p兲 + O共p2兲 we obtain
共56兲
for even parity, with u = +a + K共p+兲, and
−1
and the chemical potential is given by
␮± ⬇
␬sn共u兩p+兲 + +cn共u兩p+兲dn共u兩p+兲 = 0
冉
冊
8m兩V0兩a2
32
1− 2 2 2 .
25
ប␲n
共61兲
Since pT± has to be positive this implies the conditions j±
艋 N±, where N+ and N− are the numbers of even and odd
bound states of the linear system. This expresses the obvious
fact that the number of newly created resonances cannot be
greater than the number of bound states of the linear system.
As in the case of an attractive nonlinearity, the quantum
number n numbers the the resonances as well as the symmetric and antisymmetric bound states and is equal to the number of the minima of the squared modulus of the wavefunction. In the case of bound states, these minima are also zeros
of the wavefunction, so that the oscillation theorem is valid.
In terms of the effective nonlinearity defined by Eq. 共41兲,
we obtain
g−eff =
共54兲
ប 2 −
关−a − E共−a兩p−兲兴
ma
共62兲
for antisymmetric bound states and
and wave number
± =
冑
2m共␮ − V兲
.
ប2共1 + p±兲
g+eff =
共55兲
Again the wave functions and their derivatives must be
continuous at x = ± a. Thus we obtain the equations
ប 2 +
关+a + E共p+兲 − E„+a + K共p+兲兩p+…兴
ma
共63兲
for symmetric bound states. For the transition we get in both
cases
033608-8
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ANALYTICAL STUDY OF RESONANT TRANSPORT OF¼
FIG. 10. 共Color online兲 Shift of the bound states towards higher
chemical potentials in dependence of geff for p± 艋 0.995 for the
potential V0 = −10, a = 2. ⫹: symmetric bound states, 䊊: antisymmetric bound states. The sudden disappearance of the curves at
geff共p = 0.995兲 is due to the fact that in this figure the Jacobi elliptic
parameter had to be “artificially” limited to p 艋 0.995 for numerical
reasons.
gTeff =
±
gT±兩A兩2
p T±
冋
1−
E共pT±兲
K共pT±兲
册
共64兲
.
The magnitude of geff has the same qualitative dependence
on p± as in the attractive case so that the nonlinearity has a
stronger effect on lower bound states.
Figure 10 shows how the bound states of the system are
shifted towards higher chemical potentials if geff is increased,
until they reach ␮ = 0 and develop into resonance states. Approximating the chemical potential linearly in terms of geff
results in
␮± ⬇ E0 +
冉
3
sin共2qa兲
1±
2
2qa
冊
−1
geff ,
共65兲
which is identical with the respective Eq. 共51兲 for attractive
nonlinearities.
FIG. 11. Transmission of the square-well potential V0 = −12, a
= 4 with a repulsive nonlinearity g兩A兩2 = 4. Shown is the transmission coefficient of the stationary states calculated as described in the
previous sections 共solid lines兲 in comparison to the transmission
coefficients obtained from a numerical solution of the timedependent NLSE as described in the text 共crosses兲.
initial state is an empty waveguide ␺共x , t = 0兲 = 0. The source
strength S共t兲 is then ramped up adiabatically, so that finally a
steady transmitting solution ␺共x , tend兲 is found. To determine
the incident current and the transmission coefficient, a superposition of plane waves with momenta k = ± 冑2␮ is fitted to
the wave function ␺共x , tend兲 for x0 ⬍ x ⬍ −a 共upstream region兲
and x ⬎ a 共downstream region兲. The NLSE is numerically
integrated in real space using a predictor-corrector step 关33兴.
To avoid reflections at the edges of the computational interval, one-way boundary conditions are used as described in
Ref. 关34兴. An example of the wave function calculated in this
way is shown in Fig. 12.
The results of these simulations are shown in Fig. 11,
where the transmission coefficient is plotted in dependence
of the chemical potential ␮ for a fixed repulsive nonlinearity
g兩A兩2 = 4. First, we notice that the final state ␺共x , tend兲 is indeed a stationary solution as discussed in the previous sections. The transmission coefficient 兩T共␮兲兩2 follows closely
one of the steady state results which are computed as described in the previous sections. The small deviations are
caused by numerical errors, in fact mainly by the 共still not
V. DYNAMICS
In this section we briefly discuss the implications of the
results for the nonlinear eigenstates presented above on the
dynamics. To this end we assume that a source outside of the
square-well potential emits condensed atoms with a fixed
chemical potential ␮. In fact, we numerically solve the timedependent NLSE with an additional delta-localized source
term 共see Ref. 关5兴兲
i
冉
冊
⳵␺共x,t兲
1 ⳵2
= −
+ V共x兲 + g共x兲兩␺共x,t兲兩2 ␺共x,t兲
2 ⳵x2
⳵t
+ S共t兲exp共− i␮t兲␦共x − x0兲,
共66兲
where units with ប = m = 1 are used. As above, V共x兲 = V0H共a
− 兩x兩兲 is a square-well potential with depth V0 = −12 and halfwidth a = 4. The nonlinearity is nonzero only within the
square-well g共x兲 = gH共a − 兩x兩兲. Here, H共x兲 denotes the Heaviside step function. The source is located at x0 = −15.5. The
FIG. 12. Instability due to a beak-to-beak crossing scenario in
the transmission through a square-well potential with V0 = −12 and.
Shown is the squared modulus of the wave function 兩␺共x , t兲兩2 for
␮ = 2 in a grayscale plot 共lower panel兲 and the source strength S共t兲
共upper panel兲 for comparison 共repulsive nonlinearity兲.
033608-9
PHYSICAL REVIEW A 73, 033608 共2006兲
RAPEDIUS, WITTHAUT, AND KORSCH
perfectly absorbing兲 boundary conditions. Secondly, the final
state ␺共x , tend兲 is the one with the smallest transmission. This
behavior may suppress resonant transport, because a resonance state is not populated if different stationary states exist
with a smaller transmission. This can be seen in Fig. 11
around ␮ ⬇ 5 and ␮ ⱗ 2. However, it is possible to access
states with a higher transmission by a suitable preparation
procedure as it has been shown in Ref. 关5兴. If there is no
bistability in the vicinity of a resonance 共e.g., around ␮
⬇ 7.5 in Fig. 11兲, resonant transport is found.
A more complicated situation arises when a beak-to-beak
bifurcation scenario occurs 共see Fig. 3兲. As an example, we
consider the solution of the time-dependent NLSE with a
source term as in Ref. 共66兲 for the same square-well potential
as above 共V0 = −12, a = 4兲 and a fixed chemical potential ␮
= 2. Again the source strength is increased adiabatically 关35兴
to a level S共t兲 = S1 so that the system assumes a stationary
transporting state, where g兩A兩2 = 3.5 is well below the critical
value for the beak-to-beak bifurcation. Then the source
strength is again slowly increased up to S共t兲 = S2, so that g兩A兩2
becomes larger than the critical value of the beak-to-beak
bifurcation. The lowly transmitting stationary state ceases to
exist and the time evolution cannot follow the stationary solutions adiabatically any longer. This is shown in Fig. 12.
Indeed one observes that the system becomes unstable and
finally does not assume a stationary state any more. However, the transmission through the square-well increases. The
beak-to-beak crossing scenario and its implications on the
dynamics will be discussed in detail in a subsequent paper
关31兴.
VI. CONCLUSION
The scattering of a BEC by a finite square-well potential
has been discussed in terms of stationary states of the nonlinear Schrödinger equation. Neglecting the mean-field interaction outside the potential, ingoing and outgoing waves and
thus amplitudes of reflection and transmission can be defined. The transmitted flux shows a bistable behavior in the
vicinity of the resonances. For repulsive nonlinearities, the
transmission coefficient also shows beak-to-beak bifurcations. Wherever these beak-to-beak scenarios occur, the scattering phase shows loops.
An analytical expression for the position of the resonances is derived. The transitions from resonances to bound
states and vice versa due to attractive or repulsive mean-field
interaction are proven analytically and an explicit formula
for the transition interaction strength gT is found. Furthermore, the positions of the bound states of the system are
calculated and compared with the case of vanishing interaction g = 0. It is found that the oscillation theorem also holds
for interaction parameters g ⫽ 0.
Finally, we solve the time-dependent NLSE numerically
in order to discuss the dynamics of the system for g ⬎ 0. If
the amplitude 兩A兩 of the incoming flux is slowly ramped up
for a fixed value of the chemical potential ␮, the system
populates the lowest branch of the stationary transmission
coefficient, as it was first shown in Ref. 关5兴 for a double
barrier potential, so that resonant transport is usually be sup-
pressed. In the vicinity of a beak-to-beak bifurcation, the
system can become unstable if g兩A兩2 becomes larger than the
critical value of the beak-to-beak bifurcation so that the lowest branch of the stationary transmission coefficient suddenly
ceases to exist. These beak-to-beak crossing scenarios will
be further investigated in a subsequent paper 关31兴.
ACKNOWLEDGMENTS
Support from the Deutsche Forschungsgemeinschaft via
the Graduiertenkolleg “Nichtlineare Optik und Ultrakurzzeitphysik” is gratefully acknowledged. We thank Peter
Schlagheck and Tobias Paul for inspiring discussions.
APPENDIX A: REVIEW OF THE LINEAR SCHRÖDINGER
EQUATION WITH A FINITE SQUARE-WELL
POTENTIAL
The scattering states of the linear time-independent
Schrödinger equation
冉
−
冊
ប2 d2
+ V共x兲 ␺共x兲 = E␺共x兲
2m dx2
共A1兲
for a particle with mass m where the potential is given by
V共x兲 = V0H共a − 兩x兩兲, where V0 ⬍ 0 and H共x兲 denotes the Heaviside step function, are obtained for positive particle energies
E ⬎ 0. We assume an incoming flux from x = −⬁ and make
the ansatz
冦
x ⬍ − a,
Aeikx + Be−ikx ,
␺共x兲 = Fe + Ge
Ceikx ,
iqx
−iqx
, 兩x兩 艋 a,
x ⬎ a,
冧
共A2兲
where the wave numbers inside 共兩x兩 艋 a兲 and outside 共兩x兩
⬎ a兲 the potential well are given by q = 冑2m共E − V0兲 / ប and
k = 冑2mE / ប, respectively. The amplitudes of transmission
and reflexion are then determined by T ª C / A and R ª B / A.
They satisfy the relation 兩R兩2 + 兩T兩2 = 1 as the Schrödinger
equation is flux preserving. We have to make the wave function and its derivative continuous at x = ± a. Together with the
normalisation condition A = 1 these boundary conditions determine the wavefunction of the system. By eliminating F
and G we obtain the amplitude of transmission
T共E兲 =
exp共− 2ika兲
1
共A3兲
cos共2qa兲 1 − 共i/2兲关共q/k兲 + 共k/q兲兴tan2共qa兲
and thus the transmission probability
冋 冉 冊
兩T共E兲兩2 = 1 +
1 q k
−
4 k q
2
sin2共2qa兲
册
−1
.
共A4兲
We define a resonance as a maximum of the transmission
coefficient where the potential well is fully transparent, that
is 兩T共E兲兩2 = 1. Then Eq. 共A4兲 implies the resonance condition
2qa = n␲. The resonance energies are then given by
En = V0 +
ប 2q 2
ប 2␲ 2 2
= V0 +
n ,
2m
8ma2
共A5兲
where n is an integer number and sufficiently large to make
En ⬎ 0. A Taylor expansion of Eq. 共A3兲 in terms of E − En
yields
033608-10
PHYSICAL REVIEW A 73, 033608 共2006兲
ANALYTICAL STUDY OF RESONANT TRANSPORT OF¼
兩T共E兲兩2 ⬇
共⌫n/2兲2
,
共E − En兲2 + 共⌫n/2兲2
共A6兲
where ⌫n is given by
2冑2ប 冑En共En − V0兲
⌫n =
冑ma 2En − V0 .
negative this means either g 艋 0 or g兩A兩2 艌 兩V0兩. For small
values of p the left-hand side of Eq. 共B1兲 can be approxi4
mated by pK4共p兲 = 共 ␲2 兲 共p + p2兲 + O共p3兲 and we obtain
p⬇−
共A7兲
This means that the transmission coefficient can be approximated by a Lorentzian with width ⌫n in the vicinity of the
resonances. We define the scattering phase ␪共E兲 by T共E兲
and
obtain
tan ␪共E兲 = 共q / k
= 兩T共E兲兩exp关i␪共E兲 − 2ika兴
+ k / q兲tan共2qa兲 / 2. In the vicinity of a resonance this reduces
to ␪共E兲 ⬇ arctan关2共E − En兲 / ⌫n兴. Note that the scattering phase
is zero for E = En.
Bound states are found for energies V0 ⬍ E ⬍ 0. The area
兩x兩 ⬎ a outside the potential well is classically forbidden and
the wavefunction decays exponentially with the coefficient
␬ = 冑−2mE / ប. Inside the well it still oscillates with the wavenumber q = 冑2m共E − V0兲 / ប. Due to the symmetry V共x兲
= V共−x兲 of the potential we can choose eigenfunctions which
are either of odd or even parity. We make the ansatz
冦
␥ ±e ␬x ,
x ⬍ − a,
␺±共x兲 = ␩± cos共qx + ␩±兲, 兩x兩 艋 a,
␥ ±e −␬x ,
x⬎a
冧
共A8兲
for even 共⫹ sign, ␩+ = 0兲 and odd parity solutions 共⫺ sign,
␩− = −␲ / 2兲. As in the case of scattering states the wave function and its derivative must be continuous at x = ± a. For even
parity we get the condition
q tan共qa兲 = ␬ ,
共A9兲
which determines the energy of the bound states. For odd
parity we obtain
q cot共qa兲 = − ␬ .
共A10兲
冋 册
R0
␲
⬎
and N− =
冋 册
R0 1
−
␲ 2
,
⬎
共A11兲
冑
1 2g兩A兩2共V0 + g兩A兩2兲m2a4
+
.
4
共␲/2兲4ប4n4
共B2兲
2m2a4
.
ប 4n 4
共B3兲
Case 2.
p共1 − p兲K4共p兲 = − g兩A兩2共V0 + g兩A兩2兲
For g = 0, this equation is solved by p = 0. The solution p = 1
is incompatible with Eq. 共24兲 because it leads to a negative
chemical potential ␮R. From p共1 − p兲K4共p兲 艋 0 we obtain the
condition 0 艋 g兩A兩2 艋 兩V0兩. For small values of p, the lefthand side of Eq. 共B3兲 can be approximated by p共1
4
− p兲K4共p兲 = 共 ␲2 兲 p + O共p3兲 and we get
p⬇−
2g兩A兩2共V0 + g兩A兩2兲m2a4
.
共␲/2兲4ប4n4
共B4兲
APPENDIX C: PROOF THAT THE BOUND-STATE WAVE
FUNCTIONS CONTINUOUSLY MERGE INTO
RESONANCE WAVE FUNCTIONS
Next we want to demonstrate that the wave function of a
symmetric bound state continuously merges into a resonance
wavefunction with an even quantum number n at ␮ = 0 if we
use an appropriate normalization. To avoid confusion, all the
parameters of the resonance wave function now carry an additional index R. For ␮ = 0 we have k = ␬ = 0 so that the
squares of the wave functions outside the potential well
agree if we use the normalization 兩␥+兩 = 兩A兩. Equations 共28兲,
共30兲, and 共33兲 then imply
The transcendental equations 共A9兲 and 共A10兲 can be
solved numerically. One can show 关27兴 that the number of
symmetric or antisymmetric solutions is given by
N+ =
1
+
2
兩A兩2 = 兩I+兩2 = −
4ប2 j+2K共p+兲2 p+
.
gma2
共C1兲
Inside the potential well the squared modulus of the resonance wavefunction is given by
respectively, where 关y兴⬎ denotes the smallest integer greater
than y and R0 is given by R0 = 冑2ma2V0 / ប.
兩␺R共x兲兩2 = 兩A兩2 −
冋 冉
冊 册
nK共pR兲
ប2K2共pR兲n2
x兩pR − 1
dn2
2
gma
a
共C2兲
APPENDIX B: DETERMINATION OF THE JACOBI
ELLIPTIC PARAMETER
The elliptic parameter p can be determined by inserting
the expressions 共24兲 and 共23兲 into Eq. 共9兲. We obtain the
following.
Case 1.
2m2a4
pK 共p兲 = g兩A兩 共V0 + g兩A兩 兲 4 4 .
បn
4
2
2
关see Eq. 共25兲兴 as g ⬍ 0 corresponds to case 1 in Sec. II. Because of k = 0, the phase of the resonance wave function is
constant and the phases of ␺R and ␺+ can be chosen identical. Using Eq. 共C1兲 and the relation between the squares of
the Jacobi elliptic functions 关28–30兴 we arrive at
共B1兲
For g = 0 this equation can only be solved by p = 0. The inequality pK4共p兲 艌 0 implies g兩A兩2共V0 + g兩A兩2兲 艌 0. Since V0 is
033608-11
兩␺R共x兲兩2 = −
ប24j+2K共p+兲2 p+ ប2K2共pR兲n2
−
pR
gma2
gma2
冋 冉
⫻ cn2
冊 册
nK共pR兲
x兩pR − 1 .
a
共C3兲
PHYSICAL REVIEW A 73, 033608 共2006兲
RAPEDIUS, WITTHAUT, AND KORSCH
By comparing 兩␺R共x兲兩2 with the squared modulus of the
bound-state wave function
冉
冊
gT+ = −
ប2n2K共pT+兲2 pT+
.
ma2兩A兩2
共C5兲
we see that 兩␺R共x兲兩2 is equal to 兩␺+共x兲兩2 if pR = p+¬pT+ and
n = 2j+, where the index T denotes transition. Equation 共C1兲
implies that the transition occurs for the nonlinear interaction
strength
Inserting n = 2j+ and Eqs. 共C5兲 and 共34兲 into Eq. 共24兲 for ␮R
and Eq. 共B1兲, which connects g and pR, we see that ␮R共p
= pT+兲 = 0 and that Eq. 共B1兲 holds. Thus we have proved that
pR = p+ = pT+ is valid so that resonances of even quantum
number n indeed become symmetric bound states. In a similar way one can show that resonances of odd quantum number n become antisymmetric bound states.
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兩␺+共x兲兩2 = −
ប24j+2K共p+兲2 p+ 2 2jK共p+兲
cn
x兩p+ ,
gma2
a
共C4兲
033608-12