Bright and Gap Solitons and Vortex Formation in a Superfluid

Boson-Fermion Mixture

Sadhan K. Adhikari

Institute of Theoretical Physics

UNESP – São Paulo State University,

São Paulo, Brazil

Adhikari44@yahoo.com

Collaborators B A Malomed, L Salasnich

ContentsBose-Einstein Condensation of Trapped Atoms

Mean-field Gross-Pitaevskii Equation

Boson-Fermion Superfluid: Pauli Repulsion among Fermions

Mean-field Equation for Boson-Fermion Superfluid

Bright Solitons for Boson-Fermion attraction

Gap Solitons in periodic Optical-Lattice potential

Bosonic Vortex in a Boson-Fermion mixture

Conclusion

Some characteristics of BEC

Pure coherent state of atoms: more than just all atoms having the same energy as in atomic orbitals.

Macroscopic quantum state observed, manipulated and measured in laboratory: wave-particle duality, interference, vortices, super-fluidity, atom laser etc.

Quantum statistics necessary

Experimental condition of low density and weak interaction allows mean-field models

Experimental difficulties

Solid versus Ideal gas at low temp

Artificial condition created for controlled BEC in Laboratory: Density 1012-13 atoms/cc

Magneto-optic trapping of Alkali metal atoms

Laser cooling to m K & evaporative cooling to ~ 100 n K

h-> few n K, T c -> hundred n K

Evaporative cooling

At the end of laser cooling one has a distribution of atoms with mean temperature μK

Magnetic trap is lowered so that the hottest atoms escape (90%)

Remaining 10% atoms have much reduced velocity (cm/sec) & temperature (n K) & undergoes BEC

History of BECH: 1998 MIT KleppnerRb: June 1995 JILA Cornell/WiemanNa: Sept 1995 MIT KetterleLi: July 1995 Rice HuletK: Oct 2001 Firenze InguscioCs: Oct 2002 Innsbruck GrimmHe*: Feb-Mar 2001 Orsay/Paris Aspect & Cohen TannoudjiCr: Mar 2005 Stuttgart Pfau

Mean-field Gross-Pitaevskii Equation

Application in nonlinear optics

This potential gives scattering length a in the Born approximation

The time-dependent equations yield the same result as the time-independent equations for stationary states.

However, the time-dependent equations can also be used for the study of non-stationary states that appear in collapse dynamics and generation of soliton trains: nonequilibrium transition between two stationary states.

Degenerate Fermi gasFermionic atoms are strongly repulsive at short distances due to Pauli blocking and are impossible to condense by evaporative cooling. Due to Pauli principle two identical fermions avoid each other.This is responsible for the stability of a neutron star against gravitational attraction.Condensation possible in Boson-Fermion and Fermion-Fermion mixtures by sympathetic cooling.

Degenerate Boson-Fermion Mixture

6,7 Li – Hulet 2001

6Li-23Na – Ketterle 2002

40K-87Rb – Inguscio 2002

Feshbach resonance: Ketterle, Nature 392, 151 (1998)

Feshbach resonance is a bound or quasi-bound singlet molecular state (zero magnetic energy) coupled resonantly to the triplet state by total angular momentum conservationBy manipulating a background magnetic field, a relative motion is generated between triplet free-particle threshold and Feshbach resonanceAs the Feshbach resonance moves past the triplet free-particle threshold the atomic scattering length changes from positive through infinity to negative valuesInteraction controlled by atomic scattering lengthBy changing a background magnetic field near a Feshbach resonance the effective interaction may change from repulsive to attractive through infinite values

BCS Superfluid and BEC of Molecules of Fermion pairs

• Using a Feshbach resonance it is possible to introduce a weak effective attraction among the fermionic atoms appropriate for the formation of a BCS (Bardeen-Cooper-Schrieffer) condensate.

• For a strong attraction one has the BEC of fermion pairs.

The eqs. are solved by real (and imaginary time) propagation after discretization by the Crank-Nicholson rule. Real time procedure preserves the norm of the wave function and is useful in the study of nonequilibrium dynamics. In the imaginary time method norm is not conserved and one needs to normalize at each step.

Typical time step 0.0005 units of ~ ms

Space step 0.025 units of m

Bright Soliton

• Moving normalizable solution of nonlinear equation formed due to attractive nonlinear interaction

• One and three dimensions• It is possible to have bright soliton in binary

mixture with repulsive intra-species interaction and attractive inter-species interaction

• This is how one can have bright solitons in binary boson-fermion mixtures

Bright soliton in a quasi-one-dimensional (attractive) Li BEC (Hulet, Rice)

Fermionic bright soliton

Usually, there cannot be bright soliton (moving integrable state) in fermions due to Pauli repulsion.

In a many-body study of a boson-fermion mixture bright solitons were formed for attraction between components (Feshbach resonance)

Such a many-body study is complicated and only a small number of fermions could be handled.

We find that the time-dependent mean-field boson-fermion equations can be used to study the essentials of the formation of bright soliton and soliton trains using a Feshbach Resonance.

.

Quase-one-dimensional dimensionless equations in cigar-shaped trap for boson-fermion bright soliton:

[-it-zz+NBB||2+NBF||2]z,t=0

[-it-zz+NFB||2+NFF||4/3]z,t=0

NBB = 4aBBNB/l , NBF = 8aBFNF/l ,

NFB = 8aBFNB/l , NFF= 9(6NF)2/3/5

l = harmonic oscillator length = 1 m

NB=1000, NF=10000 aBB = 0.5nm, aBF= -3.75nm, NBB = 20, NBF = -30, NFB = -300, NFF = 275

At t = 100 ms, the nonlinearities are jumped from:

NBF = -30 -33 NFB = -300 -330

Formation of Soliton train upon jumping aBF to a large attractive value at t = 0.

Larger jump creates more solitons.

Gap Soliton

• Normalizable solution of repulsive nonlinear equation formed due to periodic optical-lattice potential, provided the chemical potential falls in the bandgap of the linear equation.

• The system possess a negative effective mass, which together with a repulsive nonlinearity allows the formation of gap solitons.

• This is how gap solitons are formed in a repulsive BEC, and in a BCS fermionic superfluid.

Gap soliton in a quasi-one-dimensional (repulsive) Rb BEC in a periodic optical-lattice potential (M. Oberthaler, Heidelberg)

Band Structure in V(x)=-V0cos(2x)

One has localized solution in the gap and plane-wave-type (Bloch wave) solution in the conduction band. In the repulsive nonlinear Schrödinger equation one could have a gap soliton.

Quasi-one-dimensional dimensionless equations in cigar-shaped trap for boson-fermion gap soliton:

[-it-xx/2+nB||2+nBF||2-V0cos(2x)]x,t=0

[-it-xx/2+nFB||2+nF||4/3-V0cos(2x)]x,t=0

nB=aBB / l , nBF= aBFNF/ l

,

nFB = aBFNB / l , nF= 3(3NF /4 l )2/3/10

l = harmonic oscillator length = h/(2m)=1 m

V0=optical lattice strength = 5, wavelength

gB=nB/NB, gBF=nBF/NF, gF=nF/NF2/3G

_

Profile of bosonicand fermionicfundamental gap solitons for all repulsive interactions.Numerical vs variational results.

Chemical potential of bosonic and fermionic fundamental gap solitons for all repulsive interactions.

Numerical vs variational results.

Profile of bosonicand fermionic nonfundamentalgap solitons for all repulsive interactions.

Numerical results.

Stability of Gap solitons

At t = 10 the wave form is modified as i(x,t)1.1 X i(x,t), i=B,F

Bosonic vortex in a superfluid boson-fermion mixture

• Usually fermions do not form quantized vortex in the BCS limit. However, they can form such vortex in the molecular BEC limit.

• As we consider only the BCS limit we consider only a quantized vortex in the bosonic component.

S

Boson-fermion profiles in the trapped mixture for different values of boson-fermion scattering lengths mixing and demixing

Profiles of bosonic vortex in trapped boson-fermion mixture for different values of boson-fermion scattering lengths mixing and demixing

Stability and Collapse of a bosonic vortex in a trapped boson-fermion mixture for attractive

boson-fermion interaction

Conclusion

• Mean-field model is effective in the study of a trapped superfluid boson-fermion mixture for

• Bright and gap solitons, vortex formation, and mixing and demixing

• We have applied similar considerations to a trapped superfluid fermion-fermion mixture

There cannot be a collapse/bright soliton in a degenerate fermi gas due to Pauli repulsion.

Pauli repulsion stabilizes a neutron star.

In a boson-fermion mixture with interspecies attraction [using Feshbach resonance, Ketterle (2004), Jin (2004)], one can have an effective attraction between fermions which can lead to collapse or form bright solitons.

,

For stationary states Eq. (1) yields identical results as the GP Eqn.

Three-body recombination loss of atoms from a BEC

Collapse and Stability of Matter

Matter is stable due to short-range repulsion among its constituents: nucleons atoms.

In absence of short-range repulsion matter collapses.

Matter in a star like sun due to expanding force of nuclear explosion, in absence of which it may turn to a cold object like neutron star after a supernova explosion.

BEC in trapped atoms

In a free gas BEC is not possible in one or two dimensions

In a trapped gas BEC is possible in one and two dimensions, as the density of states changes and the BEC integral does not diverge resulting in a finite critical Temp

By putting a strong trap in transverse directions BEC has been realized in quasi one and two dimensions

Bose-Fermi and Fermi-Fermi degenerate gas

Boson-Fermi mixtures40K-87Rb 2002 Firenze Inguscio6Li-23Na 2002 MIT Ketterle6LI-7Li 2001 Rice Hulet & 2001 France Salomon

Fermi-Fermi mixtures40K-40K* 1999 JILA Jin6Li-6Li* 2002 Duke Thomas

Collapse was observed in fermions in a 40K-87Rb mixture. Inguscio (2002)

Experiment was later refined: Bongs et al (2006).

Controlled Collapse has been considered in:

Zaccanti et al. Cond-mat/0606757 Ospelkaus et al. Cond-mat/0607091

Collapse/Bright Soliton observed in a BEC

Collapse: 7Li Jerton et al (Rice, 2001), 85Rb Donley et al (JILA, 2001, Feshbach control)

Bright Soliton: Strecker/Khaykovich et al (2002)

Strecker et al (2002) produced a soliton train using a Feshbach resonance in 7Li. By suddenly producing a strong attraction in a cigar shaped trap the BEC collapses and forms a soliton train.

Collapse/Bright Soliton in boson-fermion mixture?

To study equilibrium phenomena, time-independent version of present equations were used by

(1)P. Capuzzi et al, Phys. Rev. A (2003) to study different properties of a boson-fermion mixture in agreement with many-body theory.

(2)Many of the expt. results on the stability of 40K-87Rb mixture, critical atom number for collapse etc., have been explained by Modugno et al, Phys. Rev. A (2003) [Thomas-Fermi version].

The formation of a BCS state is avoided by rapidly turning a repulsive boson-fermion mixture into a strongly attractive one, which would naturally decay due to three-body recombination and not form a BCS state.

In our study of collapse we consider

mB=3mF=m(87Rb), 40K-87Rb mixture

aBB = 5 nm, l (harm-osc length) = 1 m,

NB=4800, NF=1200

aBF = -12.5 nm -37.5 nm

B=200 Hz, F=345 Hz

Choice of atoms

Alkali metal atoms with two distinct levels are ideal

H, Li, Na, K, Rb, Cs used

Have permanent electronic magnetic moments and can be manipulated by external magnets

Other BOSONIC atoms: Excited He* and Cr atoms

A dynamical mean-field-hydrodynamic model seems to be very useful for the study of collpse in a boson-fermion mixture.

Observation of revival of collapse is in agreement with theory.

First observation of collapse was initiated by imbalance in population.

New experiment with Feshbach resonance will produce challenging results for theory.

Parameters:

NF = 1000, NB = 10000, aBB

= 0.3 nm, aBF = -2.4 nm,

Nonlinearities:

NBB = 10, NBF

= -19, NFB = -190, NFF = 275

Bright soliton/gap soliton in an optical-lattice potential

V(z) = 100sin2(4y)