Unusual discrete soliton and breather modes
collective excitations in Bose-Einstein
condensates in optical lattice
Bishwajyoti Dey
Department of Physics
University 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 through
Each other without loss of
identity
7
Breathers in continuum systems
Breather solutions are known for PDEs. Nongeneric
• Well known sine-Gordon equation
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.
Nonlinear Lattices
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 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
For nonlinear lattice, onsite potential can be nonlinear, or W (intersite
interaction) can be nonlinear (anharmonic) or both can be nonlinear.
Linearize equation of motion around classical ground state
Dispersion relation for small amplitude plane wave
Page mode
(right),
Sievers-Takeno
Mode (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.
Dashed line
e5 vs time
e5
Solid line –Total energy
of 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.
Efimov effect
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 2006
in 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’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 an
external potential is given by
where
are boson field operators,
potential. The field operators can be written as
is the two-body interaction
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 twoand 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 of
atoms 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)
DNLS (discrete nonlinear Schrödinger) Equation
where
The DNLS equation is the equation of motion
and can be derived from the Hamiltonian
where
and
and the norm
are conjugate variable. Both the Hamiltonian
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
respectively of the density
associated momenta.
and
and
are center and width
and
are their
The dynamical evolution of the variational wave packet can be obtained
by a variational principle from the Lagrangian
Using Euler-Lagrange equation the variational equations of motion are
The pairs
Hamiltonian
and
are conjugate dynamical variables w.r.t. the
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
components
and
in terms of two
. 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
of condensate
oscillate as (exact solution)
Similarly, the width of the condensate density
No instability (decay) in linear regime.
, the center
oscillates as (exact solution)
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 is
fully 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
is obtained from the condition
For
This gives
and
for
.
.
which implies
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
(GA,BD 2011)
Self -trapping
Diffusion
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 numerical
integration 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
Phase diagram of the scalar BEC in deep optical lattice
(GA,BD 2011)
Parameters:
Inset:
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
trajectory shrink to zero.
, the area enclosed by
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 arises
at 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 the
interacting BEC around Dirac point. May be a correct choice of the Wannier function
is necessary?
Chen, Wu
PRL, 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.