Cusp bifurcation in the eigenvalue spectrum of PT-symmetric Bose-Einstein condensates

0.8 MB·English

by  Daniel Dizdarevic

Cusp bifurcation in the eigenvalue spectrum of -symmetric Bose-Einstein
PT
condensates
Daniel Dizdarevic, Dennis Dast,∗ Daniel Haag, Jörg Main, Holger Cartarius, and Günter Wunner
Institut für Theoretische Physik 1, Universität Stuttgart, 70550 Stuttgart, Germany
(Dated: January 16, 2015)
A Bose-Einstein condensate in a double-well potential features stationary solutions even for
attractivecontactinteractionaslongastheparticlenumberandthereforetheinteractionstrengthdo
notexceedacertainlimit. Introducingbalancedgainandlossintosuchasystemdrasticallychanges
the bifurcation scenario at which these states are created. Instead of two tangent bifurcations at
whichthesymmetricandantisymmetricstatesemerge,onetangentbifurcationbetweentwoformerly
independent branches arises [D. Haag et al., Phys. Rev. A 89, 023601 (2014)]. We study this
transition in detail using a bicomplex formulation of the time-dependent variational principle and
find that in fact there are three tangent bifurcations for very small gain-loss contributions which
5
coalesce in a cusp bifurcation.
1
0
2 PACSnumbers: 03.65.Ge,03.75.Hh,11.30.Er

n
a I. INTRODUCTION in a three-dimensional -symmetric double-well poten-
J PT
tial

5
Bose-Einstein condensates with attractive contact in-
1
teractions become unstable if the number of particles
1
] exceedsacertainlimit[1–3]. Inmean-fieldapproximation V (x)= [x2+ω2(y2+z2)]+v e−σx2+iγxe−ρx2, (2)
h wherethecondensateisdescribedbytheGross-Pitaevskii ext 4 0
p
equation this effect manifests itself in a vanishing ground
-
t state. Above this limit the mean interaction caused by
n
a the particles is strong enough to constrict and ultimately withω =2, v0 =4, σ =0.5andρ≈0.12. Thesymmetric
u collapse the condensate. Below this limit the stationary real part of the potential is a harmonic trap which is
q solutions of the Gross-Pitaevskii equation are observable separated into two wells by a Gaussian barrier. The
[ even though for attractive interactions the ground state antisymmetric imaginary part of the potential induces

is not the global minimum of the mean-field energy [4]. particle loss in the left well and particle gain in the right
1
well whose strength is given by the gain-loss parameter γ.
v Characteristic properties of Bose-Einstein condensates
5 with attractive interaction could be used for an atomic
It was found that four stationary states of the real
2 soliton laser [1]. However, this requires the realization
7 double-well potential are created in the two lowest-
of a particle flow into and out of the condensate. Model-
3 lying tangent bifurcations at strong attractive interac-
ing such effects in mean-field approximation is done via
0 tion strengths Na. If gain and loss is introduced into
imaginarypotentials,thusrenderingtheHamiltoniannon-
. the system, instead of two bifurcations there is only one
1
Hermitian [5, 6]. Such non-Hermitian systems have been
0 bifurcation between two previously independent states,
widely studied [6–12] and are supported by comparison
5 while the other two states vanish. However, the under-
with many-particle calculations [13, 14]. Both in- and
1 lying bifurcation mechanism remained unclear. It is the
v: outcouplingofparticleshavebeenexperimentallyrealized purposeofthispapertoclarifythismechanismandstudy
[15, 16].
i the bifurcation scenario in detail classifying it as a cusp
X Despite the non-Hermiticity real eigenvalues and thus
bifurcation [36].
r true stationary states can be found if the Hamiltonian
a is -symmetric [17–19] or pseudo-Hermitian [20–22]. Thenumberofsolutionsisnotconservedatbifurcations
PT
This unique property motivated a variety of theoretical since the Gross-Pitaevskii equation is non-analytic due
studies[23–31]andtheexperimentalrealizationinoptical to its nonlinear part ψ 2. This problem is addressed by
| |
wave-guide systems [23, 32–34]. applying an analytic continuation, where the complex
In [35] we studied the eigenvalue spectrum and dynam- wave functions are replaced by bicomplex ones. The
ical properties of a Bose-Einstein condensate, described numericalresultsarethenobtainedviathetime-dependent
by the dimensionless Gross-Pitaevskii equation variational principle (TDVP).
We start with a short introduction to bicomplex num-
( ∆+V +8πNaψ 2)ψ =µψ, (1)
− ext | | bers in Sec. II before discussing their application to the
Gross-Pitaevskii equation and the TDVP in Sec. III. A
detailed analysis of the eigenvalue spectrum and the bi-
furcation scenario is carried out in Sec. IV. Conclusions
∗ [email protected] are drawn in Sec. V.
2
II. BICOMPLEX NUMBERS into two coupled complex differential equations, which
evidently do not contain a non-analytical term anymore,
Consider a usual complex number z C, z = x +
iy with x,y R and the imaginary uni∈t i2 = 1. A (−∆+Vext+8πNaψ+ψ−∗)ψ+ =µ+ψ+, (9a)
bicomplex nu∈mber is constructed by replacing t−he real (−∆−Vext+8πNaψ−ψ+∗)ψ− =µ−ψ−. (9b)
and imaginary part of z by complex numbers with an
additional imaginary unit j which also fulfills j2 = 1, For complex wave functions the relation ψ+ =ψ− holds,
− as can be seen in Eqs. (5), and the Eqs. (9) are reduced
to the Gross-Pitaevskii equation (1). Thus, the coupled
z =x+iy =(x +jx)+i(y +jy) z +iz+jz +kz . (3)
1 j 1 j 1 i j k
≡ Eqs. (9) still support all solutions of the Gross-Pitaevskii
In the last step k=ij with k2 =1 was introduced. equation (1).
To solve the Eqs. (9) the TDVP is generalized for
Every bicomplex number can be uniquely written [37,
bicomplex differential equations. Our ansatz consists
38] as
of coupled Gaussian wave packets which yields accurate
results even for a small number of wave packets [39–41].
z =z e +z e , (4)
+ + − −
For the double-well potential studied in this work it was
shown that two coupled Gaussian wave packets suffice
with the components
andadditionalwavepacketsonlyleadtosmallcorrections
[35, 42]. For all calculations the following bicomplex
z =(z +z )+i(z z), (5a)
+ 1 k i j
− ansatz is used
z =(z z )+i(z +z), (5b)
− 1 k i j
−
2
(cid:88)
and the idempotent basis ψ(x,z(t))= exp( xTAnx+bn x+cn), (10)
− ·
n=1
1 k
e± = ±2 . (6) with the bicomplex three-dimensional diagonal matrix
An =diag(An,An,An),thebicomplexthree-dimensional
(cid:107) ⊥ ⊥
One can easily confirm that e± fulfill the relations vector bn = (bn,0,0)T, and the bicomplex number cn.
The vector z(t) contains all parameters An, bn, cn of
e++e− =1, e+−e− =k, e+e− =0, e2± =e±. (7) the wave function. The ansatz is chosen such that all
solutions are invariant under rotations around the x axis,
Notethatanequivalentrepresentationexistsinwhichthe i.e. we search only for wave functions that possess the
components z± are complex numbers with the imaginary symmetry of the potential. The complex components
unit j instead of the imaginary unit i. of the wave function can be written as ±
The introduction of the idempotent basis significantly
simplifies arithmetic operations of bicomplex numbers (cid:88)2
ψ (x,z (t))= exp( xTAnx+bn x+cn)
since for two bicomplex numbers z,w the relation ± ± − ± ±· ±
n=1
z◦w =(z+◦w+)e++(z−◦w−)e− (8) (cid:88)2 gn (11)
≡ ±
holds, where represents addition, subtraction, multi- n=1
◦
plication and division. Thus the bicomplex algebra is
using Eq. (8) since the exponential function is defined as
effectively reduced to a complex algebra within the coeffi-
a power series.
cients.
The TDVP makes use of the McLachlan variational
principle [43] which demands that the variation of the
functional
III. NUMERICAL METHOD
I = iϕ ψ 2 (12)
(cid:107) −H (cid:107)
As a next step we apply the analytic continuation
to the Gross-Pitaevskii equation (1) by allowing bicom- withrespecttoϕvanishes, thenafterthevariationϕ=ψ˙
plex values for the wave function ψ and the chemical isset. Usingtherepresentationwiththeidempotentbasis
potential µ. With the notation introduced in Eq. (4) (4) for ϕ, ψ and leads to
H
the modulus squared can be written as ψ 2 = ψψ∗ =
ψ ψ∗e +ψ ψ∗e . Hereψ∗ denotesaco|mp|lexconjuga- I = iϕ− −ψ− iϕ+ +ψ+ e+
+ − + − + − (cid:104) −H | −H (cid:105)
tion of ψ with respect to the complex unit i. Since k=ij + iϕ+ +ψ+ iϕ− −ψ− e−
(cid:104) −H | −H (cid:105)
this complex conjugation changes also the sign of k and I e +I e (13)
+ + − −
thus transforms e into e and vice versa. Due to the ≡
+ −
properties (7) and (8) the components are independent Thetwocoefficientsarecomplexconjugate,thus,thefunc-
and the bicomplex Gross-P±itaevskii equation can be split tional I˜=I =I∗ already contains the full information.
+ −
3
The variation of the functional I˜reads, 2.6
γ=0
γ=0.0004 A
δI˜= iδϕ− iϕ+ +ψ+ + iϕ− −ψ− iδϕ+ 2.4 γγ==00..00000068 | i
(cid:104) | −H (cid:105) (cid:104) −H | (cid:105)
γ=0.0010
= i∂ ψ iψ˙ ψ δz˙∗
(cid:104) z− −| +−H+ +(cid:105) − 2.2
+ iψ˙ ψ i∂ ψ δz˙ |Bi
(cid:104) −−H− −| z+ +(cid:105) + µ T1
=0. (14) 2 T2
Since the variations δz˙ and δz˙∗ are independent both 1.8 |Ci
+ − T3
coefficients have to vanish, which can be written in the
compact form 1.6 |Di
1.365 1.36 1.355 1.35 1.345 1.34 1.335 1.33
iψ˙ ψ ∂ ψ =0. (15) − − − − Na− − − −
(cid:104) ∓−H∓ ∓| z± ±(cid:105)
Using the idempotent basis we can reduce the bicomplex Figure 1. (Color online) The bifurcation scenario of the PT-
equation to a pair of coupled complex equations. symmetric stationary states of the three-dimensional double-
The next step is to insert the ansatz (11) of the wave well potential at strong attractive interactions. The chemical
functionintoEq.(15). Thecalculationsareinfullanalogy potential of the stationary solutions is shown as a function of
the interaction strength Na for different values of the gain-
with the TDVP for complex Gaussian wave packets [44]
loss parameter γ. For γ =0 (black dashed line) two tangent
and therefore we only list the results.
bifurcations T1 and T2 are present. The states |A(cid:105) and |D(cid:105)
The equations of motion for the parameters of the
emerge from the bifurcation T1 whereas T2 gives birth to the
Gaussian wave packets read
states |B(cid:105) and |C(cid:105). For small values of γ (solid lines) a third
bifurcation T3 appears at which the two states |B(cid:105) and |C(cid:105)
iA˙n =4(An)2 Vn , (16a)
± ± − 2± bifurcate which connects the so far independent branches of
ib˙n =4Anbn +vn , (16b) the states |C(cid:105) and |D(cid:105). For higher values of γ (dotted lines)
± ± ± 1±
the bifurcations T2 and T3 vanish, as does the intermediate
ic˙n =2trAn bn bn +vn . (16c)
± ±− ±· ± 0± state |C(cid:105), thus, the branches of the states |B(cid:105) and |D(cid:105) merge.
The coefficients of the effective potential Vn , vn and
2± 1±
v are obtained by numerically solving
0± briefly. In particular the bifurcation scenario was only
N shown for the gain-loss parameters γ =0 and γ =0.001,
(cid:88)
xkxlgm (xTVn x+vn x+vn )gn which, as we will see, does not suffice to understand the
(cid:104) α β ∓| 2± 1±· 0± ±(cid:105) bifurcation process in detail.
n=1
N Figure 1 shows the relevant part of the eigenvalue spec-
(cid:88)
= xkxlgm (V +8πNaψ∗ψ )gn , (17) trum as a function of the nonlinearity parameter Na
(cid:104) α β ∓| ext ∓ ± ±(cid:105) for different gain-loss parameters γ. All states are -
n=1
PT
symmetric, therefore, their chemical potentials µ are real.
with α,β = 1,...,d; m = 1,...,N and k +l = 0,1,2,
For a vanishing gain-loss contribution, γ = 0, the two
where d is the dimension and N the number of coupled
states A and B , which originate from the ground and
Gaussians. For the ansatz (11) used in this work we have | (cid:105) | (cid:105)
excited state of the double-well potential, emerge from
d=3 and N =2.
independent tangent bifurcations T1 and T2. The bifur-
Stationary solutions are obtained by varying the pa-
cation T1 gives birth to the states A and D whereas
rameters An, bn, (cn cN), such that they fulfill the | (cid:105) | (cid:105)
± ± ± − ± the states B and C emerge from the bifurcation T2.
equationsA˙n =0,b˙n =0,(c˙n c˙N)=0forn=1,...,N. At γ = 0.0|01(cid:105)the si|tu(cid:105)ation has changed fundamentally.
± ± ±− ±
Note that the differences (cn cN) are used since not all The two branches A and B are no longer independent
parameters cn are free due±t−o th±e norm constraint and but emerge from t|he(cid:105)same| b(cid:105)ifurcation T1. In addition
±
thechoiceofaglobalphase. Standardnumericalmethods the branches C and D have vanished.
| (cid:105) | (cid:105)
for complex numbers can be used to achieve this since This behavior can be understood by studying the sys-
all parameters in Eqs. (16) and (17) are complex. This temforverysmallparametersγ. Forγ =0.0004bothtan-
again shows the benefit of the idempotent basis. gent bifurcations T1 and T2, and thus, also the branches
C and D are still present. However, the two pairs
| (cid:105) | (cid:105)
emerging from T1 and T2 are now connected by a new
IV. RESULTS tangent bifurcation T3 at which the lower lying states
vanish. If the gain-loss parameter is slightly increased
Before discussing the analytic continuation the real the bifurcation T3 is shifted to lower values of Na and
spectrum is investigated. In [35] the real eigenvalue spec- approaches T2 (γ =0.0006). For γ =0.0008 both bifur-
trum of this system has already been studied, however, cations T2 and T3 have united and vanished.
the bifurcation scenario at strong attractive interactions, To discuss the propagation of the two tangent bifur-
onwhichweconcentrateinthispaper,wasonlydiscussed cations in the parameter space in more detail, a phase
4
−1.344 TT23 (a)2.4 γγ==00.0006
−1.346 |Bi 2.3 γγ==00..00000180
a 1.348 B, C, D 2.2
N− | i | i | i
1.35 2.1
−−1.352 |Di µ1 2
1.354 1.9
− 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008
1.8
γ
1.7
Figure 2. (Color online) Trajectories of the two tangent bi- (b)
furcations T2 and T3 in the γ-Na-parameter space. In the 0.3
shaded area all three states involved in the two bifurcations 0.2
are present while in the other areas only one of these states 0.1
exists. For increasing values of the gain-loss parameter γ, the
µj 0
bifurcation T2 is strongly shifted to lower nonlinearity param-
0.1
eters Na. At γc ≈0.000766 the two bifurcations coincide and −
the state |C(cid:105) vanishes while the states |B(cid:105) and |D(cid:105) merge. 0.2
−
0.3
−
1.37 1.365 1.36 1.355 1.35 1.345 1.34
diagram is shown in Fig. 2. The two bifurcations coalesce − − − − − − −
Na
and vanish at critical values of the gain-loss parameter
γc 0.000766 and interaction strength (Na)c 1.352. Figure 3. (Color online) Bicomplex bifurcation scenario show-
≈ ≈−
Their trajectories have the characteristic form of a cusp ing the µ and µ component of the chemical potential as a
1 j
in the phase diagram. In the area enclosed by the two function of the nonlinearity parameter Na for different values
trajectories all three states B , C , and D involved of the gain-loss parameter γ. The i and k components vanish
in this bifurcation scenario a|re(cid:105)pr|ese(cid:105)nt. Ou|ts(cid:105)ide of this for all states discussed, thus µ∗ =µ holds and all states are
area only one state remains. Below the trajectory of T2 PT symmetric. Thestateswithpurelyrealchemicalpotential,
i.e. µ = 0, are still present (solid lines). In addition a pair
only the state D exists and above the trajectory of T3 j
only the state|B(cid:105). Since the areas below T2 and above of bicomplex states with µ=µ1±jµj emerges at the tangent
T3 are connect|ed(cid:105)for γ >γ , the states B and D can bifurcations T1, T2, and T3 (dotted lines). For γ >γc, the
c bifurcations T2 and T3 have vanished leading to a bicomplex
| (cid:105) | (cid:105)
continuously be transferred into each other by following
branch that is completely independent from the remaining
a path in the parameter space around the cusp. scenario. Independent of the parameters γ and Na there are
We have seen that the transition from the γ =0 case, four stationary solutions.
where the states A and B are completely independent,
| (cid:105) | (cid:105)
to the γ = 0.001 case, where both states emerge from
a common bifurcation, occurs in two steps. The first analytic Eqs. (9) are obtained.
step is the formation of a new tangent bifurcation T3 The bicomplex chemical potential µ of the stationary
between the states C and D and the second step is solutions is shown in Fig. 3. For the discussion of the
| (cid:105) | (cid:105)
the coalescence of the two bifurcations. In the following resultsweagainusethecomponentsofbicomplexnumbers
we will show that introducing bicomplex numbers to ana- as introduced in Eq. (3) instead of the representation in
lytically continue the Gross-Pitaevskii equation allows a the idempotent basis since this renders the interpretation
more detailed discussion of these two steps. of the results much clearer. The first thing to note is that
Bifurcation scenarios are classified by comparison with allstateshavevanishingµ andµ components. Therefore
i k
normal forms [36]. The eigenvalue spectrum shows three µ∗ =µ holds and all states considered are symmetric.
PT
tangent bifurcations which are described by the normal Allstatesdiscussedsofararestillpresentinthespectrum
form x˙ = x2 σ. The stationary solutions, x˙ = 0, are and only have a non-vanishing µ component (solid lines).
1
−
obviously x = √σ. Two real solutions exist for σ > 0, However, additional states with µ =0 are found (dotted
j
± (cid:54)
they coalesce at σ = 0, where two complex conjugate lines). SincethepotentialintheGross-Pitaevskiiequation
solutions (σ < 0) emerge. Thus, the total number of is symmetric under a complex conjugation with respect
solutions stays constant if we allow x to be complex. to j, these states have to appear in pairs µ=µ +jµ and
1 j
Due to the non-analytic modulus squared in the Gross- µ=µ jµ [45].
1 j
−
Pitaevskii equation this conservation of solutions is not Firstwediscussthecaseγ <γ whereallthreetangent
c
guaranteed although the wave functions can be complex. bifurcations are present. At the bifurcation T1 the states
In analogy with the transition from real to complex A and D coalesce and vanish. On the other side of the
| (cid:105) | (cid:105)
numbersinthenormalformofthetangentbifurcation,we bifurcation two bicomplex states emerge whose chemical
expectatransitionfromcomplextobicomplexnumbersin potentials are complex conjugate with respect to j. Also
the eigenvalue spectrum of the Gross-Pitaevskii equation. at the bifurcations T2 and T3 two states of the real
By introducing bicomplex numbers the Gross-Pitaevskii spectrum vanish and on the other side of the bifurcations
equation is analytically continued and the two coupled two bicomplex states emerge. For γ > γ the situation
c
5
(a) (b) described by the normal form
1
1.95 0.5 x˙ =x3+ρx σ, (18)
µ1 1.9 xRe 0 −
whose stationary solutions x˙ =0 are found by Cardano’s
0.5
− method. The spectrum of the normal form has two
1.85 1
(c) (d)− tangent bifurcations which vanish at the critical values
0.1 γγγ<>=γγγccc 1 ρρρ=<>ρρρccc ρc =σc =0. In Fig. 4 the normal form is compared with
theeigenvaluespectrumofthedouble-wellpotential. The
0.05 x0.5
µj 0 Im 0 parameters ρ and σ play the role of the gain-loss param-
0.05 0.5 eter γ and the nonlinearity parameter Na, respectively.
− −
0.1 1 At the tangent bifurcations in the eigenvalue spectrum
− −
-1.354 -1.353 -1.352 -1.351 -1.350 -1.0 -0.5 0 0.5 1.0 real values of the chemical potential µ turn into bicom-
Na σ plex values with a j component, µ = µ +jµ, whereas
1 j
in the spectrum of the normal form real values become
Figure 4. (Color online) Comparison of the cusp bifurcation
ordinary complex numbers. Thus, we have to compare µ
occurring in the eigenvalue spectrum (left panels) and in 1
with Rex and µ with Imx. Again the real branches are
the normal form (18) (right panels). For parameter values j
drawn as solid lines whereas the complex and bicomplex
smaller than the critical value (γ < γ , ρ < ρ ) the tangent
c c
branches are drawn as dotted lines.
bifurcations T2 and T3 are separated. They coalesce at the
criticalparametervalue(γ =γc,ρ=ρc)andvanishforγ >γc, In the regime γ <γc, ρ<ρc the two tangent bifurca-
ρ>ρc. Realbranchesaredrawnassolidlinesand(bi)complex tions are still separated, at γ = γc, ρ = ρc the tangent
branches as dotted lines. bifurcations coalesce and finally for γ > γ , ρ > ρ the
c c
bifurcationshavevanished. Thenormalformcapturesthe
qualitative behavior of the bifurcation scenario found in
the eigenvalue spectrum, thus justifying its classification
as a cusp bifurcation.
at the bifurcation T1 has not changed, however, the
bifurcations T2 and T3 have vanished and the states B
| (cid:105)
and D havemerged. Thebicomplexstatesthatemerged
| (cid:105) V. CONCLUSION AND OUTLOOK
from T2 and T3 have also merged and form independent
branches that do not bifurcate with any real branch.
This answers the question raised in [35] regarding the
At the tangent bifurcations two eigenvectors and the bifurcation scenario at strong attractive interactions and,
corresponding eigenvalues become equal qualifying them thus, completes the discussion of the eigenvalue spectrum
as exceptional points of second order [46, 47]. The bicom- of the three-dimensional -symmetric double-well po-
PT
plex analysis reveals that at the cusp point, where the tential. Formulating the TDVP with bicomplex numbers
bifurcations T2 and T3 merge, a total of three states, one in the idempotent basis has proved to be useful to analyt-
with real eigenvalue and two with bicomplex eigenvalues, icallycontinuethenon-analyticGross-Pitaevskiiequation
coalesce. This is characteristic of an exceptional point of and analyze the bifurcation scenarios in the eigenvalue
third order which has already been observed in spectra spectrum. Using this formalism will help tackling prob-
with similar cusp-like behavior [48, 49]. lems with more complicated -symmetric potentials
PT
andadditionalinteractionssuchasthedipolarinteraction
The coalescence of two tangent bifurcations is the char- which might show an even richer variety of bifurcation
acteristic property of a cusp bifurcation [36] which is scenarios.
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