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Dynamical phase diagram of the strongly
interacting Bose-Einstein Condensate in an
optical lattice
Bishwajyoti Dey
Department of Physics
University of Pune, Pune
with Galal Al- Akhaly
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
etc.
Transition from
superfluid (BEC) to
Mott insulator is
possible when BEC
is placed in a period
lattice (optical lattice).
Figure 2
(a) Two atoms (dark blue balls) occupy neighboring
potential wells. U is the energy cost for them to be in the
same well (pale blue balls). (b) Lowering one well relative to
the other allows atoms originally in separate wells (light
blue) to occupy the same well (dark blue). (Adapted from
ref. 1.)
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 effect (1970)
There exist bound states (Efimov states) of three bosons even if
the two-particle attraction is too weak to allow two bosons 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 2005
in ultra cold gas of cesium atoms.
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.
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
Mechanism not included in the GP theory are: three-body collisions which
become important when density of the system become large.
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 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.
We write 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:
We take gravitational force as external force (tilted wash board potential)
and obtain the quasimomentum as
, 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
High density BEC with deep optical lattice potential supports many
interesting phases. Consider the accelerating potential to be zero.
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 space trajectories 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.
Phase diagram of high density scalar BEC
(GA,BD 2011)
Parameters:
Inset:
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 in the
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.
Interest in BEC was lost since then, as it was believed that the conditions for BEC
can never be produced in a real system.
---Ideal gas
---Density and temperature relation
---Most of the gases will be solid at such low temperature
Tc = 34nK from the formula of Tc for rubidium vapor (Theory). Experimental value
170nK !
Super fluidity (1937) and superconductivity (1957- BCS theory) were believed to due
to Bose-Einstein condensation.
Super fluid Helium-4 (Boson) (1937) - transferring of fluid mass without transferring
energy.
Landau phenomenological theory (Noble Prize 1962):
-- super fluidity is destroyed above a critical velocity . Above this velocity, excitations
are created.
-- existence of roton (high momentum version of sound). Required to explain specific
heat data.
Microscopic theory of roton by Feynman.
-- existence of quantized vortex lines
--existence of second sound (application of heat to a spot in liquid helium results in a
heat wave conduction).
--super fluidity also observed in He-3 (fermions). Cooper pairing between the
atoms (rather than the electrons as in BCS theory of superconductivity).
Attractive interaction between the atoms is mediated by spin fluctuations rather than
phonons.
BEC is obtained with a two-step process of cooling and trapping.
The first stage uses laser light for the cooling and trapping.
The second stage uses magnetic fields for trapping and cooling by evaporation
(allowing high energy particles to escape).
Trapping provides a “thermos bottle” that keeps the very cold atoms from
coming into contact with hot wall only 1 cm away! No complex cryostats,
dilution refrigerators required! Only atom cloud is cooler than the room
temperature where the apparatus is!
Apparatus – a small glass cell
with some coils of wire around
it.
Now undergraduate lab. Expts!
Getting around the laws of thermodynamics in experiment.
Rubidium is a metallic solid in room temperature which is its true thermodynamic
equilibrium. Requirement is vapor state of rubidium.
The idea is to avoid reaching a true thermodynamic equilibrium.
Produce conditions (low temperature and low density) so that the gas remains
in its metastable supersaturated-vapor state for a long time. During this time
BEC is produced and studied. It takes long time to reach its true equilibrium
(solid) state. The condensate lives for 15-20 seconds.
This concept of needing to produce a sample with two very different time scales
for equilibration is a critical step for achieving BEC.
BEC (both bosonic and fermionic) is expected to show all properties exhibited
by superfluid helium.
Experiments on rotating BEC has already shown the existence of quantized
vortices. Experiments are on to determine the excitation spectra of BEC
(rotons), second sound etc.
BEC -BCS crossover: transition from the condensate formed by pair of
fermionic atoms forming pair through formation of molecules
(BEC condensate) to BCS condensate where atoms form pair through
“Cooper pairing” like mechanism.
The ability to realize ultra cold gases in periodic potential has enabled
detailed studies of many effects in solid state physics,
6. Study quark-gluon plasma (QGP)( state containing free quarks and gluons that
existed a fraction of second after Big Bang). When BEC is released from a cigar
shaped trap, it expands more rapidly in narrow direction than in the long direction.
Similar behavior seen in experiment built to produce quark-gluon plasma. Connection
between QGP and strongly interacting Fermi gas is their perfect hydrodynamics –
no damping and no viscosity.
7. Strongly interacting Fermi gas can be used as means of testing predictions in
other branches of physics where strong interaction dominate. String theory.
Theorists at Utrecht Univ., Netherlands proposed that superstrings could be
made in the laboratory by trapping ultracold cloud of fermionic atoms inside a
vortex in a BEC (A recipe for making strings in the lab).
8. Condensate to make quantum computers. Condensates have lot of potential
quantum bits.
Atom Laser : Laser that emits atoms rather than photons. Atoms that can be
collimated to travel long distance or brought to a tiny focus.
In BEC all atoms have same energy, same de Broglie wavelength, same phase
and is described by the same wave function. BEC therefore can be highly
monochromatic source of atoms when released from the condensate.
In conventional laser optical cavity is formed by mirror.
In BEC the inhomogeneous magnetic field provided a confining potential
around the atoms which plays the role of optical cavity.
To extract coherent beam from the cavity in conventional laser, partially transmitting
mirror is used (output coupling).
In atom laser the output coupling is achieved by changing the magnetic state of atom.
It is changed from the state that are confined to the state that are not.
This is done applying short radio frequency pulse to flip the spin of the atom and
therefore release from the trap. The extracted atoms are accelerated away from the
trap under gravity.
Application of atom laser
The discover of conventional laser was dubbed as “ a solution in search of a
problem”. Today laser is billions of dollars industry!
Atom laser is expected to spark similar revolution in the field of atomic optics.
Lens, mirrors, beam splitters have already been developed to control the
atomic beam.
Application in atom holography – the de-Broglie wavelength of atoms is much smaller
than the wavelength of the conventional lasers. This means atom laser can create
much higher resolution holographic images. It can be used to project complex
integrated-circuit patterns, just a few nanometer in scale, onto semiconductors.
Atom interferometry - atom beams can be split to create interference patterns –
more sensitive than optical interferometer. Useful in condensed matter physics and
quantum mechanics.
Problems with atom laser– getting continuous output. BEC has to be replenished
continuously. Steady state condensation formation required.
Also output energy is fixed as condensate is in ground state. BEC in excited state
requires to be created.
Experiment for showing superfluid (BEC) – Mott insulator quantum phase transition.
Experiment based on phase of the BEC and Mott insulator.
Number-phase uncertainty relation
in the condensate and its phase.
of the number of atoms
All atoms in BEC has same phase (same wave function, that of condensate). Two
BEC can show interference pattern. But number of particles in the lattice site is
not defined. Mott insulator on the other hand have number of particle on the lattice
sites fixed but no phase coherence between the particles.
This phase coherence and decoherence during the transition from the superfluid
(BEC) to Mott insulator phase is seen in experiment as presence or absence of
interference pattern.
Fig. Interference patterns in absorption images (gauged by scale on right) result when
a gas of cold atoms in a three-dimensional optical lattice is in its superfluid phase;
no interference is seen in a Mott insulating phase. The depth of the potential wells in
the lattice is systematically increased from 0 at (a) to 20 Er at (h), where Er is a
reference energy. The phase transition occurs somewhere between (f) and (g).
Dynamics of Bose-Einstein Condensate (Dey, Galal)
DST project sanctioned (July 2010) entitled
“Dynamics and phase diagram of scalar and spinor Bose-Einstein Condensates”
Spinor Condensate – In spinor BEC the atoms can be in an internal quantum state
that is a superposition of spin states.
The BEC of spin-1 particle, like the F=1 ground state, is a single condensate with
the atoms in a superposition of the three spin projection, mF =1, 0 and -1, or
the superposition of three coupled BECs with the same spatial wave function, one
in each of these spin states.
F-state of an atom
F=J+I. J=L+S (total electron angular momentum) and I= nuclear spin
87Rb has nuclear spin I=3/2. Electron configuration n=5, L=0, s=1/2, J=1/2
F=J+I =2 and F=J-I =1.
Spinor condensate is expected to possess a host of quantum phenomena
absent in scalar condensates. These include magnetization, spin wave mode,
spin domains, ferromagnetic, antiferromagnetic and cyclic phase of ground state
etc.
Cyclic phase has close analogy with d-wave BCS superfluids.
Problems to be studied
(a) Dynamics of the system
(b)Effect of two- and three body interactions on some model
properties of the system
(c) Examine nature of collapse of BEC
(d) Dynamic phase diagram of the system
(e) Phase transition
(f) Dynamics of coupled spinor condensates
(g) Magnetic states of the spinor condensates, in particular for
F=2 hyperfine states.
Bose-Einstein Condensation – Macroscopic occupation of a single particle state.
Peculiar to systems with integer spin (bosons). Total number of neutrons in
the atom decide whether the atom is fermion or boson.
S.N. Bose (1924) brought to the attention of Einstein a simple derivation of the
Planck radiation law that treated photons as particles that obeyed the
counting rule that are today called Bose-Einstein statistics.
Einstein extended the idea to atoms (bosons).
In a gas of atoms obeying Maxwell-Boltzmann statistics, states are
microscopically populated (number of particles occupying any one state is very
small fraction of total number of particles). Only at T=0, the ground state (E=0)
macroscopically populated.
Einstein showed that for a gas of identical bosons, macroscopic occupation of
the ground state, comparable to the total number of particles in the gas, can
occur at a finite nonzero temperature. This phase change is called Bose-Einstein
condensation (BEC).
Bose-Einstein statistics – thermal occupation of a state with energy
and temp. T
Since the occupation number should be +ve,
.
For a system held at constant T and V, when more particles are added
adjusts by increasing towards zero. For
the second term in N
can be evaluated as
and cannot increase further. (convert
to 3D integral over phase space volume).
At this point T represent critical value Tc.
Suppose we now add more particles to the system. How can
adjust further?
To see this expand
Thus as
becomes macroscopically occupied in a very controlled
way. As more particles are added below Tc, they all go into
For low density gas and high temperature, the de Broglie wavelength
, which
is a measure of delocalization of a particle, is small compared to interparticle separation.
For particles in the zero momentum state, the delocalization is infinite.
BEC occurs when the interparticle separation is of the order of delocalization.
Thermal de Broglie ( a measure of thermodynamics uncertainty in the localization of
particle) wavelength
From the expression of Tc, one can show that the interparticle spacing is
BEC does not occur in a 2-D gas (in absence of external potential). The second
term in N diverges faster than
, as
, and the occupation of
is
not favored.