PHYSICAL REVIEW A 76, 043620 共2007兲
Collapse in boson-fermion mixtures with all-repulsive interactions
Vladyslav I. Prytula,* Vladimir V. Konotop,† Víctor M. Pérez-García,‡ and Vadym E. Vekslerchik§
ETS de Ingenieros Industriales, Departamento de Matematicas, and Instituto de Matemática Aplicada a la Ciencia y la Ingeniería,
Universidad de Castilla–la Mancha, Avenida Camilo Jose Cela, 3, 13071 Ciudad Real, Spain
共Received 11 May 2007; revised manuscript received 26 July 2007; published 18 October 2007兲
We describe the collapse of the bosonic component in a boson-fermion mixture due to the pressure exerted
on it by a large fermionic component, leading to collapse in a system with all-repulsive interactions. We
describe the phenomena of early collapse and superslow collapse of the mixture.
DOI: 10.1103/PhysRevA.76.043620
PACS number共s兲: 03.75.Lm, 03.75.Ss, 05.45.Yv
I. INTRODUCTION
II. THE MODEL
The achievement of Bose-Einstein condensation 共BEC兲
关1兴 has pushed the field of degenerate quantum gases to become one of the most active areas of physics. After the condensation of different bosonic species, degenerate mixtures
of bosonic and fermionic atoms were created 关2兴, providing a
highly controllable tool for study of systems of mixed quantum statistics.
Several interesting nonlinear phenomena have been studied in the context of boson-fermion mixtures. First, the fact
that interspecies interactions may result in attraction among
bosons 关3,4兴 implies that solitons, i.e., localized nondispersive waves sustained by nonlinear atom-atom interactions
关5兴, could also exist in quasi-one-dimensional boson-fermion
mixtures 关6–8兴. Also, collapse of the atomic cloud induced
by the interspecies attraction in boson-fermion mixtures was
observed experimentally 关9兴 and studied theoretically 关10兴.
The usual scenario of collapse in single-species quantum
gases corresponds to a BEC which collapses because of its
attractive self-interaction 关11兴. This phenomenon had been
long known in mathematical physics 关12兴 but the theoretical
description of its specifics for BECs is still motivating a lot
of theoretical research. Here we extend the previous studies
and consider the boson-fermion mixtures when all interatomic interactions are repulsive. While even in such a situation the interspecies interactions can lead to an effective
attraction between bosons 关3,7,10兴, leading us to expect such
phenomena as, for example, modulational instability of homogeneous bosonic distributions 共see Sec. IV below兲, in the
present paper we show that the induced attraction persists
also for strongly inhomogeneous distributions dominating
the boson-boson repulsion and driving the collapse of atomic
clouds. More specifically, we describe additional blowup
scenarios in degenerate quantum gases, including the phenomena of early collapse and of superslow collapse of the
mixtures.
We describe bosons in the mean-field approximation and
spin-polarized fermions at zero temperature in the hydrodynamic approximation, when the number of fermions is much
larger than the number of bosons. The corresponding dynamical system reads 关3,13兴
共6␲2兲2/3ប2
⳵ 2␳ 1
gbf 2
=
⵱
␳
⵱
兩␺兩
0
2 1/3 ␳1 +
2
⳵t
mf
3m f ␳0
共1b兲
冊册
.
Here ␺共r , t兲 is the macroscopic wave function of bosons. Due
to the Pauli exclusive principle only small numbers of fermions with energies near the Fermi surface are involved in the
dynamics. They are described by the density excitation
␳1共r , t兲, due to boson-fermion interactions, from the unperturbed density ␳0共r兲, so that the total fermionic density is
given by
␳共r,t兲 = ␳0共r兲 + ␳1共r,t兲.
共2兲
gbb and gbf are the coefficients of two-body boson-boson and
boson-fermion interactions, Vb共r兲 is the confining potential
for bosons, Vb = mb␻2br2 / 2. The trapping potential for fermions is accounted for by the unperturbed distribution, where it
appears explicitly when the Thomas-Fermi approximation
␳0共r兲 ⬇ ␳TF共r兲 =
冉 冊
2m f
ប2
3/2
关ប␻ f 共6N f 兲1/3 − m f ␻2f /2r2兴3/2
6␲2
共3兲
is used 关3,10兴, N f being the total number of fermions.
It is convenient to introduce the scaled variables x = r / a
and ␶ = ␻bt, where a = 冑ប / mb␻b is the linear oscillator length
for the bosons. We also define
†
1050-2947/2007/76共4兲/043620共5兲
共1a兲
冋 冉
*[email protected]
Permanent address: Centro de Física Teórica e Computacional,
Universidade de Lisboa, Av. Prof. Gama Pinto 2, Lisboa 1649-003,
Portugal and Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Ed. C8, Piso 6, Lisboa 1749016, Portugal. [email protected]
‡
[email protected]
§
[email protected]
⳵␺
ប2
=−
⌬␺ + Vb共r兲␺ + gbb兩␺兩2␺ + gbf ␳␺ ,
⳵t
2mb
iប
n0 = 共gbf /ប␻b兲␳0 ,
共4a兲
n = 共gbf /ប␻b兲␳1 ,
共4b兲
u = 冑gbf mb/ប␻bm f ␺ ,
共4c兲
and rewrite Eqs. 共1兲 in the form
043620-1
1
iu␶ = − ⵜ2u + V0u + nu + g兩u兩2u,
2
共5a兲
©2007 The American Physical Society
PHYSICAL REVIEW A 76, 043620 共2007兲
PRYTULA et al.
冋 冉
n␶␶ = ⵱ n0 ⵱
␣
n + 兩u兩2
n1/3
0
冊册
u p = A exp共iK · x − i⍀␶兲,
共5b兲
,
g = gbbm f / gbf mb,
␣
where
V0 ⬅ V0共x兲 = x / 2 + n0共x兲,
= 共6␲2兲2/3ប5/3mb / 3m2f 共gbf 兲2/3␻1/3
b , ⵱ is understood in terms of
the new coordinates, and x ⬅ 兩x兩 for x 苸 R3. We will refer to
this model as the nonlinear Zakharov system 共NZS兲 since it
can be seen as a generalization of the Zakharov system 关14兴
describing Langmuir waves in plasmas, the latter being given
by Eq. 共5兲 with gbb = 0 共i.e., in the absence of the two-body
interactions among bososns兲. We will consider Eqs. 共5兲 with
analytical initial data u共0 , x兲, n共0 , x兲, and n␶共0 , x兲.
2
共10兲
we get the corresponding excitation of the fermionic density
n p = −g0A2. Next we look for perturbed solutions of the form
u = u p共1 + u1兲,
共11a兲
n = n p共1 + n1兲,
共11b兲
where 兩u1 兩 Ⰶ 兩u p兩 and 兩n1 兩 Ⰶ 兩n p兩. After linearizing with respect to
共u1,n1兲 ⬀ exp共ik · x − ␻t兲,
共12兲
III. CONSERVED QUANTITIES
we find that the instability occurs for wave vectors satisfying
Equations 共5兲 can be rewritten in a Hamiltonian form. To
this end, following 关16兴, we introduce the vector function
k2 ⬍ 2A2共g0 − g兲 − 4K2 cos2 ␪ ,
冕 冉
␶
v=−
0
冊
␣
n0 ⵱ 1/3 n + 兩u兩2 d␶ ,
n0
共6兲
which describes the hydrodynamic velocity of excitations,
and rewrite the NZS in the form
n␶ = − ⵜ · v,
v␶ = − n0 ⵱
冋
冉
冊
␣
n + 兩u兩2 ,
n1/3
0
冉
n1/3 ␣
1
iu␶ = − ⵜ2u + V0 + 0
n + 兩u兩2
2
␣ n1/3
0
冊
2
+g−
共7a兲
册
n1/3
0
兩u兩2 u,
␣
共7b兲
which admits the Hamiltonian
H=
冕冋
R3
+
兩ⵜu兩2 +
冉
冉
冊
n1/3
1
g − 0 兩u兩4 + V0兩u兩2
2
␣
1 n1/3
␣
0
n + 兩u兩2
2 ␣ n1/3
0
冊
2
+
册
n0 2
兩v兩 dx.
2
储u储 p =
冉冕
R3
兩u兩 pdx
冊
1/p
,
where ␪ is the angle between K and k. Thus the condition of
modulational instability has the same functional form as that
for the purely bosonic condensate 关12兴 with the substitution
of −g by g0 − g, i.e., if g0 ⬎ g, then even when all two-body
interactions are repulsive, provided g ⬍ g0, modulational instability may set in. From the physical point of view, the
modulational instability develops for long-wavelength excitations. For such excitations the fermionic excitations are
smooth, and therefore the constant amplitude density n p enters the condition of the instability. This result means that
instabilities may develop in a system with repulsive interatomic interactions, which is very different from the case of
the purely bosonic condensate and is the first indication of
the phenomenona to be discussed in detail in this paper.
Moreover, as follows from Eq. 共12兲, both the bosonic and
fermionic components undergo the instability simultaneously
共below we will see that the same is true also for collapsing
solutions兲.
V. NONCOLLAPSING SOLUTIONS
共8兲
H is an integral of motion; thus H共␶兲 ⬅ H共0兲.
The total number of bosons NB = 储u储22, defined through the
standard notation for the L p norm,
共9兲
for integer p is another integral of motion. The density n
describes excitations of the Fermi sea and thus the number of
excited fermions is not conserved.
IV. MODULATIONAL INSTABILITY
We start with the simplest manifestation of instabilities in
nonlinear wave systems which is the appearance of modulational instability 共for the one-dimensional statement see also
关15兴兲. Neglecting the external trapping potential, i.e., supposing that n0, and hence g0 = n1/3
0 / ␣, are x-independent constants, and looking for plane-wave solutions of Eq. 共5a兲 of
the form
共13兲
First, we will prove that there are initial data which do not
blow up, i.e., the existence of global solutions of Eq. 共7兲.
More specifically, we want to establish uniform boundness in
time of U = 储⵱u储22 关16,17兴 for g ⬍ g0 共for g ⬎ g0 the proof is
trivial兲. Using the simplified version of the GagliardoNirenberg inequality 储w储44 艋 C1储⵱w储32储w储12 关18兴, with C1
= 0.449 27. . . being the best constant in the three-dimensional
case 关19兴, we get
H 艌 U2 − ␥U3 ,
共14兲
where ␥ = 共g0 − g兲C1NB1/2. Equation 共14兲 is valid for all ␶, and
thus, if there is at least one positive root of F共U兲 ⬅ ␥U3
− U2 + H = 0, to be denoted as u+: F共u+兲 = 0, and if initially
U共0兲 ⬍ u+, then Eq. 共14兲 guarantees that U共␶兲 ⬍ u+ for any
time, i.e., U is bounded uniformly in time. Hence, to establish the sufficient condition for the existence of the solution
we have to find the condition of existence of a positive root
u+. It is easy to see that F共U兲 has a local minimum at Umin
= 2 / 3␥ and a sufficient condition for the existence of a positive root is F共Umin兲 艋 0, what is equivalent to the constraint
043620-2
PHYSICAL REVIEW A 76, 043620 共2007兲
COLLAPSE IN BOSON-FERMION MIXTURES WITH ALL-…
(a)
10
logeNb
2000
N* =3.6x10 6
Nb
1000 No
blow-up
(b)
Ÿ共␶兲 =
5
No blow-up
0
0
N*
Nf
0
107 0
loge (Nf /N* )
1
FIG. 1. 共Color online兲 Ground-state solutions of Eq. 共1兲 for
abf = 10 nm,
abb = 5.25 nm,
m f = 0.66⫻ 10−25 kg,
mb = 1.44
⫻ 10−25 kg, ␻b = 188 Hz, and ␻ f = ␻b冑mb / m f = 269 Hz, corresponding to 87Rb- 40K mixtures such as those of Ref. 关9兴. 共a兲 Location on
the plane NB-N f of the ground states satisfying H = 4 / 27␥2; thus all
ground state solutions in the shaded region must be stable. 共b兲 Some
more data on a logarithmic scale.
H ⬍ H0 =
4 1
4
1
.
2 =
27 ␥
27 共g0 − g兲2C21NB
VI. COLLAPSING SOLUTIONS
It was shown numerically in 关20兴 that the Zakharov equations may undergo finite-time collapse. In our system, due to
the presence of repulsive boson-boson interactions, it may
seem counterintuitive that such a phenomenon could exist.
However, we will argue that the bosonic component in a
Bose-Fermi mixture, in which there are many more fermions
than bosons, can undergo collapse in finite time even when
both the self-interaction between bosons and the interspecies
interactions between bosons and fermions are repulsive.
To support this statement we will use “early collapse”
type arguments 关21兴, modified for the NZS since the standard
arguments based on the virial identities cannot be used.
Let us define the squared wave packet width as Y共␶兲
= 兰R3x2兩u兩2dx. By direct differentiation we obtain Ẏ共␶兲
= 4 Im 兰R3ū共x · ⵱u兲dx and
R3
共8兩⵱u兩2dx + 6g兩u兩4 − 4兩u兩2x · ⵱n兲dx
共16兲
共hereafter an overdot stands for the derivative with respect to
␶兲. In our case, and even in the case of the classical Zakharov
system 关22兴, the lack of control on the last term of Eq. 共16兲
makes it not posible to apply the usual virial-type arguments.
Nevertheless, by analyzing Eq. 共16兲 for small times, we can
obtain information about the early stage of the evolution.
First, we notice that Y共0兲 and Ẏ共0兲 depend only on the initial
distribution of the boson component u共0 , x兲. In particular, to
simplify the analysis, we can choose Ẏ共0兲 = 0 by choosing
u共0 , x兲 real. Considering the higher-order derivatives, one
can note that for given u共0 , x兲, Ÿ共0兲 can be made negative by
an appropriate choice of n共0 , x兲. For example, this occurs if
n共0 , x兲 is a cuplike profile. Moreover, from the expression
for the third derivative
共15兲
It can be seen that the condition U共0兲 ⬍ u+ is satisfied by
choosing U共0兲2 艋 H. This rigorous result, being a sufficient
condition, provides a lower bound for the largest energy allowing the existence of solutions, i.e., the physical region of
existence could be larger, in particular allowing H ⬎ H0.
Since Eq. 共15兲 involves both the Hamiltonian H and the
number of bosons NB, which are not independent, in Fig. 1
we plot the region of noncollapsing ground states by fixing
the number of fermions and computing numerically the
ground state for different NB until the limit of the inequality
is reached. This leads a domain in which we can guarantee
the existence of global solutions. It is interesting to note that
there exists a critical number of fermions N* below which all
solutions exist globally and no collapse can occur. The number of fermions separating the domain of globally existing
solutions decreases with increasing NB, which is explained
by the fact that the repulsion between bosons due to twobody interactions decreases with NB, and thus smaller fermionic densities dominate the interbosonic repulsion.
冕
ត 共0兲 = − 4
Y
冕
R3
兩u共0,x兲兩2x · ⵱n␶共0,x兲dx,
共17兲
ត 共0兲 can also be made negative by an approit follows that Y
priate choice of n␶共0 , x兲.
Summarizing the previous arguments, assuming that Y共␶兲
ត 共0兲 to be
is sufficiently smooth and requiring Ÿ共0兲 and Y
negative, which is always possible by means of the proper
choice of the initial conditions, we can expand Y共␶兲 in Taylor
series for sufficiently small ␶:
Y共␶兲 = 共␶2 − ␶2*兲
兩Ÿ共0兲兩 ␶3 ត
+ Y 共0兲 + o共␶3兲,
2
6
共18兲
and thus we can estimate
␶* ⯝ 冑− 2Y共0兲/Ÿ共0兲.
共19兲
The higher terms that are not written explicitly in Eq. 共18兲
depend on higher derivatives of u共␶ , x兲 and n共␶ , x兲 at ␶ = 0
关this can be verified by direct computation of the higher derivative of Y共␶兲, by analogy with Eqs. 共16兲 and 共17兲兴, and
thus can be chosen to satisfy a priori given properties 共which
in our case are reduced to existence of a nonzero radius of
convergence of the Taylor series兲. This means that one can
construct initial data such that Y共␶兲 crosses zero at ␶ 艋 ␶*.
This contradiction with the definition of Y共␶兲, which is nonnegative, implies a finite-time collapse for the chosen initial
data.
To make the previous discussion fully mathematically rigorous, we should prove that ␶* is smaller than the convergence radius of the series given by Eq. 共18兲. Although that
proof is not available, we can show that by proper choice of
the initial conditions ␶* can be made as small as necessary,
which supports our conjecture.
Let us consider a realistic situation, where initially the
fermions are unperturbed and the bosons have a Gaussian
distribution narrower than the size of the fermionic cloud. In
that situation, n0 can be taken as constant and we can compute ␶*, given below in physical units:
043620-3
PHYSICAL REVIEW A 76, 043620 共2007兲
PRYTULA et al.
T* =
冉
共2␲兲1/4a1/2
共6␳0兲1/3abf 2abb
mf
−
␻b共NBabf 兲1/2 mb + m f
␲1/3
abf
冊
Strauss lemma with R = R0 together with Eq. 共22兲 enables us
to estimate the right-hand side of Eq. 共24兲 as
−1/2
,
共20兲
where ␻b is the initial width of the bosonic cloud. From Eq.
共20兲 it can be clearly seen that controlling abf allows one to
make T* as small as necessary. Taking the same data as in
Fig. 1, but with abf = 10−7 m, we get a blowup time T*
⬇ 0.127 s.
− Ż 艌 −
3H C2 2 C3␤2共1 − g兲2 6
− 储u储2 −
储u储2 .
4␲ 2
16R40
Thus, if H is negative, choosing the constants C2 and R0 we
obtain −Ż共␶兲 ⬎ 0. Moreover, following 关22兴, we can show
that
冉 冕
VII. SUPERSLOW COLLAPSE
Finally, following Ref. 关22兴 where the possibility of collapse was studied for the conventional Zakharov system, we
outline the proof of the fact that the bosonic component for
g ⬍ g0 and H ⬍ 0 may undergo a completely different physical phenomenon: infinite-time blowup, which we will denote
as superslow blowup. We restrict our analysis to radially
symmetric solutions and for the sake of simplicity consider a
spatially homogeneous mixture with ␣ = 1 and n0 = 1 共i.e.,
g0 = 1兲, which can always be done by proper rescaling of the
independent variables. Recalling that x is a dimensionless
radial variable, we define
Z共␶兲 = Im
冕
⬁
共x2ūux + xnv · x兲f xdx,
共21兲
0
− Z共␶兲 艋 C4 1 +
共22a兲
f xx ⬍ 1,
共22b兲
兩⌬r2 f兩 艋 C2 ,
共22c兲
C3 艌 3 − ⌬r f 艌 0
3 − ⌬r f = 0
for x 艌 R0 ,
共22d兲
for x ⬍ R0 ,
共22e兲
where ⌬r ⬅ ⳵ / ⳵x + 共2 / x兲⳵ / ⳵x. By direct algebra we obtain
2
− Ż共␶兲 =
2
冕冋
⬁
0
冉
兩u兩2 2
2
⌬ f + 共3 − 2f xx兲兩ux兩2 + f x兩v兩2 + 共3 − ⌬r f兲
2 r
x
n2 + 兩v兩2
g
⫻ n兩u兩2 + 兩u兩4 +
2
2
冊册
x2dx −
3H
,
4␲
共23兲
and using the properties in Eqs. 共22兲 we estimate
− Ż共␶兲 艌
g−1
2
冕
⬁
关共3 − ⌬r f兲兩u兩4 + 兩ux兩2兴x2dx − C2 −
0
3H
.
4␲
共24兲
Radially symmetric functions u 苸 L 共R 兲, with 兩ux兩 苸 L 共R3兲,
satisfy the Strauss lemma 关24兴
2
兩u兩2 艋 ␤R−2储u储2储ux储2
3
for x ⬎ R,
2
共25兲
where R is an arbitrary positive constant and the constant ␤
is independent of R. Since g ⬍ 1, a simple application of the
⬁
冊
共兩ux兩2 + n2 + 兩v兩2兲x2dx ,
0
共27兲
where C4 is a positive constant depending only on 储u储2 and
兩f x兩, which leads to
lim
␶→⬁
冕
⬁
共兩ux兩2 + n2 + 兩v兩2兲x2dx = ⬁,
共28兲
0
which implies collapse for infinite times and physically gives
an indication that a large fermionic component would be able
to slowly compress a bosonic component even when the twobody interactions between bosons are repulsive, provided
they are not too strong. This compression leads to a superslow collapse.
where the weight function f共x兲 is chosen such that, for arbitrary positive constants R0, C2, and C3, the following conditions are satisfied 关23兴:
兩f x兩 艋 R0 ,
共26兲
VIII. DISCUSSION AND CONCLUSIONS
We have studied different collapse scenarios of the
bosonic component in boson-fermion mixtures at limited
positive boson-boson scattering lengths 共defined by g ⬍ g0兲
provided the fermionic component has a large enough density. They are 共i兲 a finite-time collapse, occurring at early
stages of the evolution, and 共ii兲 superslow collapse happening at infinite times. These phenomena occur even when all
interatomic interactions are repulsive. The physical reason
for the collapse in both cases are the attractive interactions
among bosons induced by boson-fermion interactions. The
physical mechanism of the induced attractive interactions
can be understood following 关3兴 as the tendency of bosons to
gather in the domain with smaller 共larger兲 density of fermions in the case of repulsive 共attractive兲 boson-fermion interactions. It turns out, as was discussed above, that such interactions, which are dominant at the initial stages of evolution
when the density has a smooth distribution, persist as relevant even near the collapse, when intuitive physical arguments should be verified by more detailed studies. This also
means that collapse occurs simultaneously in both components 共otherwise, if only one of the components collapsed
one should observe growing repulsive interactions, preventing the collapse兲.
In the real physical world 共unlike in pure mathematical
statements兲 a collapse does not end up with developing singularities, because other physical mechanisms like dissipation, dispersion, interactions with higher modes 共noncondensed atoms兲, etc., become relevant. In the case at hand,
there exists a different reason which prevents development of
the authentic singularity in the density distributions. This is
043620-4
PHYSICAL REVIEW A 76, 043620 共2007兲
COLLAPSE IN BOSON-FERMION MIXTURES WITH ALL-…
Our theoretical predictions can be tested with current experimental setups, and describe interesting singular quantum
phenomena with degenerate quantum gases.
the natural restriction on n, negative in the case of allrepulsive interactions, which must be less than n0: 兩n兩 ⬍ n0
共the total density n0 + n cannot be negative兲. Since the collapse occurs in the bosonic and fermionic components simultaneously, it is arrested when 兩n兩 becomes of the order of n0.
Physically, this means that the hydrodynamic approximation
for fermions is not valid any more and the problem becomes
essentially quantum. In other words, the described collapse is
arrested by quantum fluctuations, ruling out the mixture from
the mean-field regime.
This work has been partially supported by Grants No.
FIS2006-04190 and No. SAB2005-0195 共Ministerio de Educación y Ciencia, Spain兲, No. PAI-05-001 共Junta de Comunidades de Castilla–La Mancha, Spain兲, and No. POCI/FIS/
56237/2004 共FCT Portugal and European program FEDER兲.
关1兴 M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman,
and E. A. Cornell, Science 269, 198 共1995兲; K. B. Davis,
M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee,
D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75, 3969
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ACKNOWLEDGMENTS
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