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 Collaborators B A Malomed, L Salasnich

Contents Bose-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 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 BEC H: 1998 MIT Kleppner Rb: June 1995 JILA Cornell/Wieman Na: Sept 1995 MIT Ketterle Li: July 1995 Rice Hulet K: Oct 2001 Firenze Inguscio Cs: Oct 2002 Innsbruck Grimm He*: Feb-Mar 2001 Orsay/Paris Aspect & Cohen Tannoudji Cr: 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 gas Fermionic 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 Li- 23 Na – Ketterle K- 87 Rb – 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 conservation By manipulating a background magnetic field, a relative motion is generated between triplet free-particle threshold and Feshbach resonance As the Feshbach resonance moves past the triplet free- particle threshold the atomic scattering length changes from positive through infinity to negative values Interaction controlled by atomic scattering length By 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 units of ~ ms Space step 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: [-i  t -  zz +N BB |  | 2 +N BF |  | 2 ]  z,t  =0 [-i  t -  zz +N FB |  | 2 +N FF |  | 4/3 ]  z,t  =0 N BB = 4a BB N B /l, N BF = 8a BF N F /l, N FB = 8a BF N B /l, N FF = 9(6  N F ) 2/3 /5 l = harmonic oscillator length = 1  m

N B =1000, N F =10000 a BB = 0.5nm, a BF = -3.75nm, N BB = 20, N BF = -30, N FB = -300, N FF = 275 At t = 100 ms, the nonlinearities are jumped from: N BF = -30  -33 N FB = -300  -330

Formation of Soliton train upon jumping a BF 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)=-V 0 cos(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: [-i  t -  xx /2+n B |  | 2 +n BF |  | 2 -V 0 cos(2x)]  x,t  =0 [-i  t -  xx /2+n FB |  | 2 +n F |  | 4/3 -V 0 cos(2x)]  x,t  =0 n B =a BB   /  l , n BF = a BF  N F /  l , n FB = a BF  N B /  l , n F = 3(3N F  /4  l  ) 2/3 /10 l = harmonic oscillator length = h/(2  m  )=1  m V 0 =optical lattice strength = 5,  wavelength g B =n B /N B, g BF =n BF /N F, g F =n F /N F 2/3 G_G_

Profile of bosonic and fermionic fundamental 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 bosonic and fermionic nonfundamental gap 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 mixtures 40 K- 87 Rb 2002 Firenze Inguscio 6 Li- 23 Na 2002 MIT Ketterle 6 LI- 7 Li 2001 Rice Hulet & 2001 France Salomon Fermi-Fermi mixtures 40 K- 40 K* 1999 JILA Jin 6 Li- 6 Li* 2002 Duke Thomas

Collapse was observed in fermions in a 40 K- 87 Rb mixture. Inguscio (2002) Experiment was later refined: Bongs et al (2006). Controlled Collapse has been considered in: Zaccanti et al. Cond-mat/ Ospelkaus et al. Cond-mat/

Collapse/Bright Soliton observed in a BEC Collapse: 7 Li Jerton et al (Rice, 2001), 85 Rb 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 7 Li. 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 40 K- 87 Rb 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 m B =3m F =m( 87 Rb), 40 K- 87 Rb mixture a BB = 5 nm, l (harm-osc length) = 1  m, N B =4800, N F =1200 a BF = nm  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: N F = 1000, N B = 10000, a BB = 0.3 nm, a BF = -2.4 nm, Nonlinearities: N BB = 10, N BF = -19, N FB = -190, N FF = 275 Bright soliton/gap soliton in an optical-lattice potential V(z) = 100sin 2 (4y)