Suppression of the quantum-mechanical collapse by repulsive interactions in

bosonic gasesHidetsugu Sakaguchi

Department of Applied Science for Electronics and Materials, Interdisciplinary Graduate School of Engineering Sciences,

Kyushu University, Fukuoka, Japan Boris A. Malomed

Department of Physical Electronics, Faculty of Engineering, Tel Aviv University, Tel Aviv, Israel

A part of the results have been published in:

(1 )Introduction and the modelThe attractive potential ~1/r 2 plays the well-known critical role in the standard quantum mechanics, leading to the quantum collapse, alias “fall onto the center” in 3D

and 2D geometries. The scaled form of the Schrödinger equation with this potential (the harmonic trapping potential, Ω²r ²/2, is included too) is

002

2 2 21 1, .

2 2- 0U

i r Urt

The manifestation of the criticality:The 3D Schrödinger equation with potential -1/r 2 has no ground state for U0 > )U0(cr ≡

¼, and does have it at U0¼ > .The 2D Schrödinger equation has no

ground state for any U0 > 0 .This phenomenon of the quantum collapse is also called “fall onto the center” in the book Quantum Mechanics by Landau and Lifshitz.

The physical interpretation of both the 3D and 2D Schrödinger equations with this potential: the interaction between a particle carrying a permanent electric dipole moment d, and an electric charge Q placed at the origin. If the

orientation of the moment is locked to the local electric field (E), the effective interaction potential

is precisely -)1/2(U0/r2, with U0 = 2|Q|d :

20

sgn( ) ( / ),

( ) ( / 2) .

Q d r

U r U r

d r

d E

In particular, if Q = 1 is the electron’s charge, and the mass of the particle is taken as ~ 10 proton masses, the critical value of the attraction strength U0 = ¼, corresponds, in physical units, to a very small value of the dipole moment, d ~ 10-5 Debye, hence the case of U0 > ¼ is essential indeed.

In the 2D geometry (which corresponds to a pancake-shaped BEC configuration), another interpretation is possible too, in addition to the above-mentioned one: the interaction of a particle carrying a magnetic dipole moment with a current filament (e.g., an electron beam) which transversely intersects the 2D layer, assuming that the orientation of the magnetic moment is locked to the local magnetic field induced by the current.

A solution to the quantum anomaly problem in this quantum-mechanical setting (i.e., the nonexistence of the ground state) was proposed outside of the framework of quantum mechanics – in terms of the linear quantum-field theory. Essentially, that solution postulates that the ground state is created by the field-theory renormalization procedure. The so created ground state features an arbitrary spatial scale imposed by the renormalization:K. S. Gupta and S. G. Rajeev, Phys. Rev. D 48, 5940

(1993) ;H. E. Camblong, L. N. Epele, H. Fanchiotti, and C. A.Garcia Canal, Phys. Rev. Lett. 85, 1590 (2000).

Our objective is to propose a different solution of the quantum-collapse problem: taking into account, in the mean-field approximation, collisions between dipolar particles attracted to the center by the potential -1/r 2, i.e., replacing the Schrödinger equation by the corresponding Gross-Pitaevskii equation. This approach will create a ground state, with the spatial scale uniquely determined by physical parameters of the system.

The corresponding Gross-Pitaevskii equation, with the contact repulsive nonlinearity:

The realization of ultracold quantum gases formed by molecules carrying electric dipole moments may be possible, using (for instance) Li-Cs bosonic molecules:

J. Deiglmayr, A. Grochola, M. Repp, K. Mörtlbauer, C. Glück, J. Lange, O. Dulieu, R. Wester, and M. Weidemüller, Phys.Rev. Lett. 101, 133004 (2008).

2 2 2 202

1 1| | .

2 2

Ui rt r

Taking into account the dipole-dipole interactions in the Gross-Pitaevskii equation:

2

The density of the dipole moments, i.e., the

of the gas, considered as a

dielectric medium, is | | .

The additional electric field created by this distribution

of the polarizati

d

P d

E

local polarization

2

2 2

on is determined by the Poisson equation,

4 0, hence 4 4 | | .

The respective induced by

the dipole-dipole interactions is 4 | | ,

hence the c

U d

d d

dd d

additional mean field potent

E P E P d

al

d

i

E

2 2

2 2

orresponding extra term in the GP equation,

4 | | , implies a

: / .s s

U d

a a md

dd redifinition of the

effective scattering length

For m ~ 10 proton masses and

as ~ 10 nm, the renormalization

of the scattering length is essential

for d ~ 1 Debye.

(2 )Presentation of results

(i) The standard description of the quantum collapse will be recapitulated.

(ii) The central point: In the 3D case, it is demonstrated that the repulsive cubic nonlinearity suppresses the quantum collapse, i.e., it creates the ground state which is missing in the linear Schrödinger equation.

(iii) A quantum phase transition is found at U0

= 1 ≡ 4 )U0(cr . (iv) If the potential -)1/2(U0/r2 is not strong enough

to cause the quantum collapse in 3D, the action of the cubic nonlinearity in the combination with the harmonic trapping potential gives rise to a tri-stability of bound states.

(v) It will be shown that the cubic nonlinearity is insufficiently strong to suppress the quantum collapse in the 2D setting. However, the quintic nonlinearity does it (this results is a more formal one).

(vi) It will be demonstrated (in a brief form) that the nonlinearity also suppresses the quantum collapse and creates the missing ground state in the case when a strong external uniform polarizing field reduces the symmetry of the 3D setting from spherical to cylindrical.

(i )Recapitulation of the description of the quantum collapse in the linear Schrödinger equation

Solutions to the 3D and 2D linear Schrödinger equations (including the harmonic trap), with angular momentum ℓ, are sought for as, respectively,

3D 3D

2D 2D

Y , ( ),

( ).

for the radial wave function, ( ),can be found,

. In the case,

physically relevant solution

i tlm

i t il

e r

e e r

r

in the presence of the external harmonic potent

Exact sol

ial

s

3D

ution

0

3D

0 cr

0

s exist for 1 :

2 2 1 1 3/ 2( ) , , .2 4 2

The corresponds t

1

o (smaller

4

).

lU U l l

rr

U

r e Ul

+

±± ±

onl

ground state

y

20

In , the exact solution has a similar form, with

, , 1 , which is

relevant for (repulsion from the center).

The presence of the

0

in this setting is demonstrl

l lU U l

U

U

± ±

only

quantum collapse

2D

1/20

0

0

ated

by the , which can be found for

in , and for in , at 0 :

cos 1/ 4 ln / , in ,(

unnorm

)cos ln / , in

(these are formally exa a

1/ 4

any

ct liz

0

but

l

l

l

l

r

r U r rr

U r r

U

U

3D

formal asymptotic soluti

2

on

D

3D

2D

wave functions for 0).

The is seen in the fact that the

have an

at 0, while the ground state

able

r

form

nonexistence of the ground state

infinite number of nodes (zeral wave functions os)

must not have nodes.

(ii )The suppression of the quantum collapse by the repulsive cubic nonlinearity in 3D )no harmonic trap, Ω = 0(.

2 2 30

1

0

The ( ) solution to the

is looked for as ( , ) ( ), with

1.

2

At , the asymptotic value of solution is (0) / 2,

and at an

0

0

obviou

i t

l

r

r

r

r

t e r

U r r

U

isotropic Gross - Pitaevskii

equation

20

2(3D) 10 0

s asymptotic form of the solution is

( ) . The simplest analytical approximation, based

on the , yields:

/ 2 .

r

ri t

r e

U e r e

interpolation between the asymptotic forms

Using this approximation, one can calculate the norm of the solution, N, and thus derive an approximate μ)N( characteristic for the ground state:

2

0

2 2 2

0 0

1,

2

4 ( ) 4 ( ) .

U

N

N r r dr r dr

3D3D

3D

(Ω=0)

The family of the ground-state solutions was then found in a numerical form, as a solution of the radial equation for the 3D wave function. The solutions, both analytical and numerical, exist equally well for U0 < ¼ and U0 > ¼, i.e., in cases

when the linear Schrödinger equation does and does not have the ground state. Numerical simulations confirm that

all the ground states are stable .

Examples of the numerically found ground states of the 3D nonlinear equation without the external trap, Ω = 0 (continuous lines), and its comparison to the corresponding simplest analytical prediction (dashed lines):

Radial profiles μ)N( curves μ)N( curvesfor U0 = 0.8 > ¼ for U0 = 0.8 > ¼ for U0 = 0.1 < ¼

(here, N = 6.26)

5

For 1 elementary charge, ~ 1 Debye,

~ 10 proton masses, and ~ 10 , the

radius of the so predicted ground state

is .

Q d

m N

~ 2 μm

(iii )The quantum phase transition at U0

= 1 ≡ 4 )U0(cr

1

2 2 30

0

/2 201

0

Recall that the solution to the Gross-Pitaevskii

equation is looked for as ( , ) ( ), with

1.

2The expansion of the solution a t 0 and 1:

( ) 1 ,2 1

i t

s

r t e r r

U r r

r U

Ur r r

U

0

0

2 0

1 1 8 .

Precisely at 1, the expansion takes the form of

( ) 1/ 2 1 ln .2

s U

U

rr r

r

(2)

0 0 0cr cr

/

0

0

2

2

The to the constant value at small

is at / 2 2, i.e., at 1 4 , and

it is at 1.

Thus, the dependence of the power of in the dominating

correction on is

s

r

s U U U

U

r

r

U

r

dominant correction

0

(2)

0 0 0 cr

(2)

0 0 cr

, signaling the occurrence

of a at 1:

(1/ 2) 1 1 8 , at 1;

2, at 1.

U

U U U

U U

nonanalytical

quantum phase transiti

power

on

(iv )The effect of the additional harmonic trapping potential )Ω² > 0( in the 3D case

The harmonic trap (Ω² > 0) distorts the branch of the ground states which was created by the nonlinearity, and additionally supports extensions of the two bound modes that were found in the exact form as solutions to the linear Schrödinger equation with Ω² > 0. These additional modes exist precisely at U0 < ¼ (where the linear equation does not give rise to the quantum collapse). The two additional modes were shown (by dint of direct simulations) to be stable. Thus, the nonlinear equation gives

rise to the tri-stability at U0 .¼ >

The illustration of the tri-stability at Ω² = 0.1 and

U0 = 0.2. ¼ >

At N → ∞, the asymptotic form of all the three branches of μ)N( is given by the Thomas-Fermi approximation:

3

16 2.

15N

5/2TF 3D

(v )The suppression of the quantum collapse in the 2D setting by the quintic repulsive term

In the 2D situation, the cubic repulsive term formally gives rise to the wave function, which replaces the collapsing one, with the asymptotic form ~ 1/r at r → 0 (the same as in 3D) . However, unlike the 3D case, the norm of such a wave function diverges (logarithmically) at r → 0 in 2D, i.e., the cubic term is not strong enough to create the missing

physically acceptable 2D ground state .

The suppression of the 2D quantum collapse is provided by a quintic term, which may account for (elastic) triple collisions in the quantum gas:

F. K. Abdullaev, A. Gammal, L. Tomio, and T. Frederico, Phys. Rev. A 63, 043604 (2001).

2

2 2

1

Solutions for the and its counterparts with angular momentum ,

created by the self-r

21 1 1 2 2 40 | | .2 22 2

:

r

l

Ui rt r rr r

ground state

The 2D nonlinear equatioquintic n

2D

1

4

2

1

D 1

2

20

/

epulsive are looked for as

( , ) ( ).

The expansion of the

,

solution at 0:

1 1 1, 1 5 16 ,

2 4 2

.

i t il

sl l

l

r t e e r

r

U r s U

U U l

r

quintic nonlinearity

The for the family of

the , and the respective relation ( )

can be constructed as the between the

found at and (the

N

global analytical approximation

interpolat

asym

2D ground states

r

ion

pto 0tic f rorms same

approximation as used in ) :

0 0

11 1 24 ,2 4

21 ,4 2

where the is defined as2 22 ( ) 2 (

2

.

/

)

1 ri t ilU e el

Ul N

N r

r

rdr r dr

3D

(2D)Ω = 0

(2D)Ω = 0

2D2D norm

2D

Examples of the radial profiles of the 2D ground state created by the quintic nonlinear term, and respective μ)N( curves (solid and dashed curves depict numerical findings and the analytical approximation, respectively):

The radial profiles The μ)N( curves for U0 = +0.05 for U0 = +0.05 and U0 = -0.18

(vi )The anisotropic extension of the 3D setting :

If the local orientation of the electric dipoles is frozen into a strong external uniform field, the symmetry of the potential of the interaction of the dipoles with the central electric charge is

reduced from spherical to cylindrical ,U)r ,θ( = -(U0/2(r -²cosθ,where θ is the angular coordinate in 3D.

The cylindrically-symmetric 3D ground-state wave function is created by the nonlinearity in this case too:

H. Sakaguchi and B.A. Malomed, Phys. Rev. A 84, 033616 (2011).

(3 )CONCLUSIONS(i) The problem of the “fall onto the

center/quantum collapse/quantum anomaly”, induced by the attractive potential –U0/r ² in 3D and 2D settings, can be resolved by the self-repulsive cubic (in 3D) or quintic (in 2D) terms, which are added to the respective linear Schrödinger equation, transforming it into the

Gross-Pitaevskii equation .The nonlinear term creates the stable ground state, which was absent in the linear equation. The spatial scale of the newly created ground state is uniquely determined by physical parameters of the system.

These 3D and 2D models may be realized as BEC of molecules or atoms composed of dipolar molecules or atoms.

(ii) The quantum phase transition, signaled by the non-analyticity in the dependence of the wave-function structure at r → 0 on U0 ,

occurs at U0 ≡ 4 )U0(cr .

(iii) In the case when the linear equation does not yet give rise to the collapse, and the harmonic trapping potential is added, the corresponding nonlinear equation gives rise to the tri-stability of the bound states in the 3D case.

(iv) The ground state is created as well by the self-repulsive nonlinearity in the 3D model with the symmetry reduced from spherical to cylindrical by the strong external uniform field, which polarizes the dipole moments of the particles.

(v) Very recently, we have extended the analysis for the two-component system, including miscible and immiscible cases (γ < 1 and γ > 1, respectively):

2 2 2011 1 1 2 12

2 2 2022 2 2 1 12

1| | | | ,

2 21

| | | | .2 2

Uit r

Uit r

(4 )OULTLOOK

(i) In the 3D case, it is interesting to extend the analysis of the nonlinear equation for the vortical modes carrying the angular momentum.

(ii) A challenging issue is to extend the analysis to the quantum gas of fermionic particles.

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