Unusual discrete soliton and breather modes collective excitations in Bose-Einstein

condensates in optical lattice

Bishwajyoti DeyDepartment of PhysicsUniversity of Pune, Pune

with Galal Al- Akhaly

Nonlinear localised excitations – solitons, breathers, compactons.

These solutions are nonspreading – retain their shape in time.

Solitons, breathers form if the peaking due to nonlinearity is balanced by the spreading due to linear dispersion. For compactons balance with nonlinear dispersion is required.

For discrete systems the localization is due to the discreteness combined with the nonlinearity of the system.

For linear systems, the discrete translational invariance have to be broken (adding impurity) to obtain spatially localized mode (Anderson Localization).

For nonlinear systems one can retain discrete translational symmetry and still obtain localized excitations. Self localised solutions.

Bright solitons have been observed in BEC where the linear spreading due to dispersion is compensated by the attractive nonlinear interactions between the atoms.

Solitons and Breathers

Nonlinear energy localization in continuous media is well studied

• Small amplitude, long waves in a channel of slowly changing depths

• Kortweg de Vries (KdV) equation

Subscripts denote partial derivatives. q(x,t) wave shape.

Has spatially localized solutions called “ Soliton”

Two soliton pass throughEach other without loss of identity

7

Breathers in continuum systems

• Well known sine-Gordon equation

Breather solutions are known for PDEs. Nongeneric

has breather solution

Breather Excitations in continuous systems (described by PDE’s) are non generic

• Decade-long efforts to look for localized excitations like breathers in other partial differential equations (PDE’s) which describes evolution

of physical fields produced no result.

• Reason is simple. Resonances of the nonlinear localized excitation with spectrum of plane waves (linear) of the system.

• This resonance problem can be avoided if we consider a discrete system (LATTICE) instead of the continuum.

• In discrete lattice the plane wave frequency have always finite upper bound. We can excite the nonlinear localized excitation above the upper bound frequency to avoid resonances.

Fermi-Pasta-Ulam lattice, V=0 and W is anharmonic.

Klein-Gordon lattice, etc., V=nonlinear, W is harmonic.

Discrete Nonlinear Schrodinger (DNLS) Lattice.

Ablowitz-Ladik lattice Model.

Salerno Lattice Model.

More recently, dynamics of Bose-Einstein Condensate loaded on an deep optical lattice.

Nonlinear Lattices

Nonlinear Schrodinger equation with cubic nonlinearity (PDE)

Discrete nonlinear Schrodinger equation (nonlinear lattice model)

Ablowitz-Ladik equation (integrable lattice model)

Salerno equation

DNLS equation for the dynamics of Bose-Einstein condensate amplitude

The Lattice Problem : nonlinear lattice

• Spatial discreteness and Nonlinearity

Linearize equation of motion around classical ground state

For nonlinear lattice, onsite potential can be nonlinear, or W (intersite interaction) can be nonlinear (anharmonic) or both can be nonlinear.

Dispersion relation for small amplitude plane wave

Page mode(right),Sievers-TakenoMode (left)

Two-site DB(right)

ENERGY LOCALIZATION : Local energy density

If DBs are excited, initial local energy stay within the DB. The

function should not decay with time.

e5

Dashed linee5 vs time

Solid line –Total energyof the chain

First unambiguous observation of BEC was reported by Eric Cornell, Carl Wieman (1995) in Colorado (US).

BEC was observed cooling a gas of rubidium-87 to a temperature 170nK

Fig. Velocity distribution.

The axes are x and z velocities and third axis is number density of atoms.

Macroscopic fraction (~10%)of the atoms are in the ground state.

Optical lattice : an artificial crystal of light – a periodic intensity pattern that is formed by the interference of two or more laser beams. More lasers give 3D spatial structure.

Trapping atom in optical lattice – atoms can be trapped in the bright or dark regions of the optical lattice via Stark shift.

Strength of the optical potential confining can be increased by increasing laser intensity.

BEC mounted on a optical lattice is like electrons in a periodic potential of ions in conventional solid. Condensate atoms plays the role of electrons and optical lattice the role of ions.

Atoms trapped in an optical lattice move due to quantum tunneling even if the potential depth of the lattice point exceeds the kinetic energy. Strongly interacting limit.

However when the well depth is large then the interaction energy of the atoms become more than the hopping energy, then the atoms will be trapped in potential minima and cannot move freely. This phase is called Mott insulator.

Atoms in an optical lattice provide an ideal quantum system where all parameters can be controlled. This can be used to observe effects which are difficult to observe in real crystals. Examples:

Bloch oscillation,

Efimov effect,

Superfluid to Mott insulator transition Superconductivity etc.

Cold atom experiments tread into the land of two-dimensional superconductivity

M. Randeria Physics 5, 10 (2012).

Sommer et al. Phys. Rev. Lett 108, 045302 (2012).

A deeper understanding of strongly interacting two-dimensional superconductors and their normal states can give insights into high Tc superconductors.

2D attractive Fermi-gas has rich property even in normal state. Pair binding is enhanced in 2D. Pairing can occur without condensation over a large temperature gap leading to pseudogap effects.

Analytical meanfield calculations showed that in 2D the binding energy of many-body systems is same as the binding energy of just a pair. This is verified by experiment showing great prediction power of mean-field theory.

Sommer et al. [1] use radio-frequency (rf) spectroscopy to determine the binding energy of paired lithium atoms in a cold gas. This cartoon shows a gas with two species of fermions, denoted by red and blue, which are analogous to the spin-up and spin-down electrons in a metal. The presence of an optical lattice potential (blue curve) tunes the dimensionality and forces the gas into stacks of quasi-two-dimensional layers. The interaction between the two types of fermions is tuned using a Feshbach resonance. Absorption of the rf photon (red wavy line) converts a fermion in the red state to a different internal state shown as green, and measures the pair binding energy.

Bloch Oscillation

The quantum dynamics of accelerated particles in periodic potential leads to an oscillatory motion instead of a linear increase in velocity.This is termed as Bloch oscillation

The periodicity of the potential implies eigenfunctions obey relation

In presence of an accelerating force F, the quasimomentum evolves linearly in time

In combination with the periodicity of the band structure, this causes an oscillatory motion, the Bloch oscillation.

The oscillation period is .

Bloch Oscillation

In solid state systems scattering due to impurity of the crystals structure leads to damping of Bloch oscillations on time scales much shorter than the oscillation period itself. Hence difficult to observe experimentally.

Optical lattice on the other hand constitute a perfect optical crystal and BEC on optical lattice have enabled the first direct observation of Bloch oscillation.

Due to interactions between atoms, Bloch oscillation decays from dynamical instabilities.

Quantum Mechanics of three-body systems :

Efimov in 1970 predicted that there can exist bound states (Efimov states) of three particles even if the two-particle attraction is too weak to allow two particles to form a pair.

The sequence of three-body bound states have universal properties, it is insensitive to the details of two-body potential at short distances.

Efimov’s theoretical prediction could only be verified experimentally in 2006in ultra cold gas of cesium atoms 36 years after its predictions ( Kraemer et al, Nature 440, 315 (2006)).

Since then, Efimov effect have been observed in other BEC’s, Bose-Bose, Bose-Fermi, Fermi-Fermi and bosonic dipoles ( Ferlaino and Grimm, Physics 3, 9 (2010)).

Efimov effect

Efimov’s prediction that the binding of few particles is universal has also been confirmed experimentally (Berninger et al, Phys. Rev. Lett 107, 120401 (2011)).

It is this concept of universality which makes Efimov physics possible in various physical systems, such as, atomic physics, nuclear physics, strongly correlated system etc, where the relevant energy and length scales differ by many order of magnitude.

Dynamics of BEC in an optical lattice: order parameter and mean-field theory

The many-body Hamiltonian describing N-interacting Bosons confined by anexternal potential is given by

where are boson field operators, is the two-body interactionpotential. The field operators can be written as

where are the single-particle wave function and are the corresponding annihilation operators defined as, with commutation rules

Using the Heisenberg equation the time evolution of the field operator is given by

Bogoliubov first order theory for the excitations of interacting Bose-gas

where is a classical field, the order parameter or the wave function of the condensate. The condensate density .

Assuming that only binary collisions at low energy are relevant and these collisions are characterized by a single parameter, the s-wave scattering length, independent of the details of the two-body potential, we replace

The coupling constant where a is the scattering length.

This yields the equation for the order parameter, the Gross-Pitaevskii equation

The GP equation can be written as where

A system of atoms with attractive two-body interactions, is unstable against collapse above certain critical number of atoms Nc. An addition of a repulsive three-body interaction can overcome the collapse and region of stability for the condensate can be extended beyond Nc.

In presence of three-body interactions the Gross-Pitaevskii equation become

which depend on the Hamiltonian of a single trapped atom as well as two- and three-body coupling constants g2 and g3.

The three-body coupling constant g3 has been derived from a microscopic theory of three-body collisions in a BEC (Kohler, PRL (2002)).

The spatial coordinates are chosen as the vector from atom 1 to atom 2 ( r12) and the vector from the center of mass ofatoms 1 and 2 to atom 3

Dynamics of BEC: Gross- Pitaevskii (GP) equation – treating the condensate as classical field.

GP equation is a variant of the Nonlinear Schrodinger equation (NLS) incorporating an external potential used to confine the condensate.

Multicomponent GP equation for spinor condensate.

Dimensionality reduction possible in the presence of external periodic potential generated by the optical lattices and in the discrete limit.

Deep periodic optical potential limit – tight binding model

The linear Bloch waves exhibit strong localization in the deep potential limit. Condensate wave function is described with localized Wannier states associated with lowest band.

where is the condensate wave function localized in trap n

with the orthonormal conditions

Using in GP equation and integrating using the orthonormal

conditions above we get the dynamics of the condensate described by

the discrete nonlinear Schrodinger equation as (DNLS)

where

DNLS (discrete nonlinear Schrödinger) Equation

The DNLS equation is the equation of motion

and can be derived from the Hamiltonian

where and are conjugate variable. Both the Hamiltonian

and the norm are conserved quantity.

Variational Dynamics

To study the dynamical regime of a high density BEC in an array, we consider dynamical evolution of a Gaussian profile wave packet and introduce the variational wave function

where the variational parameters and are center and width respectively of the density and and are their associated momenta.

The dynamical evolution of the variational wave packet can be obtainedby a variational principle from the Lagrangian

Using Euler-Lagrange equation the variational equations of motion are

The pairs and are conjugate dynamical variables w.r.t. the Hamiltonian

The variational equations can be solved numerically to obtain the variational dynamics of the system.

The wave packet group velocity is given by

and the inverse effective mass is given by

where

Numerical solution of DNLS equation

DNLS equation is also solved numerically to compare with the variational dynamics results and also to check stability of the dynamics and phase diagrams over long period of time.

Write the amplitude of the order parameter in terms of two

components and . DNLS then can be written as

The coupled nonlinear equations are solved using Runge-Kutta method .

The variational wave function is used as initial condition

and

The Hamiltonian and the norm are checked at each steps of the integration to look for their constancy over time.

Dynamical instability of Bloch oscillation:

In presence of an accelerating force (tilted wash board potential) the quasimomentum is , where .

Linear regime: for zero condensate interactions , the center of condensate oscillate as (exact solution)

Similarly, the width of the condensate density oscillates as (exact solution)

No instability (decay) in linear regime.

Numerically, for Bloch oscillation, we calculate numerical average position

defined as

It is easy to show that , the average position of the center

of density.

Similarly,

gives the numerical width of the wavepacket.

Bloch Oscillation: no instability in absence of interactions

Nonlinear regime:

In this case the equation for the center of the density is given by

Note : even though there is a damping term in the equation, the dynamics isfully Hamiltonian.

The apparent damping is due to the divergence of the effective mass with time due to which the Bloch oscillation decays.

The Bloch oscillation decays as

Decay of Bloch oscillation: effect of nonlinear Interactions.(GA,BD 2011)

Anderson & Kasevich,Science (1998).

Phase diagram of the interacting BEC

BEC with deep optical lattice potential supports many interesting phases.

Phase diagrams can be obtained from the coupled variational equations and the corresponding Hamiltonian.

The trajectories in the plane can be obtained as

The condition implies for .

is obtained from the condition .

For and which implies

This gives and

The wave packet stops as the effective mass goes to infinity. This corresponds to the self-trapped regime in the phase diagram.

On the other hand, for ,

But and the effective mass

There is complete spreading of the wave packet giving rise to the diffusive regime.

The critical line separating these two regime (the self-trapped and the diffusive) is obtained from the condition as

Self -trapping

Diffusion

(GA,BD 2011)

SOLITON

For negative effective mass, i.e. for we get another interesting phase from the fixed point of the trajectory.

This gives a regime in the phase diagram where soliton solutions are allowed. The center of mass moves with constant velocity and the shape of the wavepacket do not change with time.

Soliton solutions are allowed for the parameter values

For there are no soliton solutions, as in this case the trajectory do not have fixed points.

Soliton solution from direct numericalintegration of the Gross-Pitaevskii equation. (GA,BD 2011)

DISCRETE BREATHER

Another interesting phase is the discrete breather which is a spatially localized and time-periodic solution. In this case oscillate with time.

The trajectories in the plane are closed. We have discrete breather solution with center of mass travelling with nearly constant velocity and with oscillating width. oscillate around constant value.

Phase space trajectories

Parameters: Inset:

(GA,BD 2011)

Phase diagram of the scalar BEC in deep optical lattice

Phase diagram in the Efimov region :

Numerical results shows that the soliton exist only for large value of .

For large value of , the soliton line approach the critical line .

Phase space trajectories in the Efimov region – no discrete breather.

In this region and when , the area enclosed by the trajectory shrink to zero.

However, for addition of a small two-body interaction in the Efimov region, the discrete breather solution reappears.

When the two- and three-body interactions have opposite sign, then solitons as well as discrete breathers are not allowed.

In this case, the soliton as well as the breather lines lies much below the critical line (deep inside the diffusion region) and it is not possible by increasing the width to get these lines approach the critical line.

Future work:

1.Dynamics of BEC in graphene optical lattice.

2.Nonlinear localized solutions in the gap region of the spectrum of BEC on optical lattice.

BEC in a Honeycomb optical lattice (Chen and Wu, PRL, August 2011).

Dirac point is changed completely by atomic interaction.

Dirac point is extended into a closed curve and an intersecting tube structure arisesat the original Dirac point.

The tube structure is caused by the superfluidity of the system.

This implies application of tight-binding model is not the correct one to describe theinteracting BEC around Dirac point. May be a correct choice of the Wannier function is necessary?

Chen, WuPRL, 2011

Tubed structure due to superfluidity of BEC.

2. Localized solutions in the spectrum gap region

Localized solutions can exist in the spectrum gaps forbidden for linear waves.

Such solutions are highly stable as they cannot decay by interacting with linear waves.

Gap Solitons in BEC in optical lattice was confirmed experimentally (PRL, 2004).

General problem of linearly coupled K-dV equations with nonlinear dispersion –localized excitations in the gap region of the spectrum.

GA, BD (PRE, 2011)

Spectrum can also occur in multi-component BEC – the spinor BEC, due to coupling between components.

Spectrum gap can also open due to interplay of lattice periodicity and nonlinearity.