Dynamics of bosonic cold atoms in optical
lattices.
K. Sengupta
Indian Association for the Cultivation of
Science, Kolkata
Collaborators: Anirban Dutta, Brian Clark, Uma Divakaran, Michael Kobodurez,
Shreyoshi Mondal, David Pekker, Rajdeep Sensarma & Christian Trefzger
Overview
1. Introduction to ultracold bosons
2. Dynamics of the Bose-Hubbard model
3. Dynamics induced freezing
4. Bosons in an electric field: Theory and experiments
5. Dynamics of bosons in an electric field
6. Extension to trapped system [work in progress]
7. Conclusion
Introduction to ultracold bosons
Cooling atoms: how cold is ultracold
Perspective about how cold
these atoms really are: coldest
place in the universe
How do you measure
such temperatures?
Form a BEC and measure
the width of the central
peak in its momentum
distribution.
Hot
Cold
Cooling techniques: two methods
Doppler cooling of atoms
Creation of optical molasses
leading to microkelvin temperatures
Further evaporative cooling
of the atoms leading to
temperature ~ 10 nK
This is well below critical
Temperature of a BEC
Emulating lattices for bosons with light
Apply counter propagating laser:
standing wave of light.
The atoms feel a potential V = -a |E|2
For positive a, the atoms sits at the bottom
of the potential generated by the lasers:
Bloch theory for bosons
State of cold bosons in a lattice: experiment
Bloch 2001
From BEC to the Mott state
Apply counter propagating laser:
standing wave of light.
The atoms feel a potential V = -a |E|2
Energy Scales
For a deep enough potential, the atoms
are localized : Mott insulator described
by single band Bose-Hubbard model.
dEn = 5Er ~ 20 U
U ~10-300 t
Model Hamiltonian
20Er
Ignore higher bands
Mott-Superfluid transition: preliminary analysis
Mott state with 1 boson per site
Stable ground state for 0 < m < U
Adding a particle to the Mott state
Mott state is destabilized when
the excitation energy touches 0.
Removing a particle from the Mott state
Destabilization of the Mott state via addition of particles/hole: onset of superfluidity
Beyond this simple picture
Higher order energy calculation
by Freericks and Monien: Inclusion
of up to O(t3/U3) virtual processes.
Mean-field theory (Fisher 89,
Seshadri 93)
Quantum Monte Carlo studies for
2D & 3D systems: Trivedi and Krauth,
B. Sansone-Capponegro
Phase diagram for n=1 and d=3
O(t2/U2) theories
MFT
Superfluid
Predicts a quantum phase
transition with z=2 (except at
the tip of the Mott lobe where
z=1).
Mott
No method for studying dynamics beyond mean-field theory
Projection Operator Method
Distinguishing between hopping processes
Mott state
Distinguish between two types of hopping processes
using a projection operator technique
Define a projection operator
Divide the hopping to classes (b) and (c)
Building fluctuations over MFT
Design a transformation which eliminate hopping
processes of class (b) perturbatively in J/U.
Obtain the effective Hamiltonian
Use the effective Hamiltonian
to compute the ground state
energy and hence the phase
diagram
Equilibrium phase diagram
Reproduction of the phase diagram
with remarkable accuracy in d=3:
much better than standard
mean-field or strong coupling
expansion in d=2 and 3.
Accurate for large z as can be
checked by comparing with QMC
data for 2D triangular (z=6)
and 3D cubic lattice
Allows for straightforward generalization for treatment of dynamics
Non-equilibrium dynamics: Linear ramp
Consider a linear ramp of J(t)=Ji +(Jf - Ji) t/t.
For dynamics, one needs to solve the Sch. Eq.
Make a time dependent transformation
to address the dynamics by projecting on
the instantaneous low-energy sector.
The method provides an accurate description
of the ramp if J(t)/U <<1 and hence can
treat slow and fast ramps at equal footing.
Takes care of particle/hole production
due to finite ramp rate
F
Absence of critical scaling: may
be understood as the inability of
the system to access the critical
(k=0) modes.
Fast quench from the Mott to the SF
phase; study of superfluid dynamics.
Single frequency pattern near the critical
Point; more complicated deeper in the SF
phase.
Strong quantum fluctuations near the QCP;
justification of going beyond mft.
Experiments with ultracold bosons on a lattice: finite rate dynamics
2D BEC confined in a trap and in the presence
of an optical lattice.
Single site imaging done by light-assisted collision
which can reliably detect even/odd occupation
of a site. In the present experiment they detect
sites with n=1.
Ramp from the SF side near the QCP to deep inside
the Mott phase in a linear ramp with different
ramp rates.
The no. of sites with odd n displays plateau like
behavior and approaches the adiabatic limit
when the ramp time is increased asymptotically.
No signature of scaling behavior. Interesting
spatial patterns.
W. Bakr et al. Science 2010
Power law ramp
Use a power-law ramp protocol
Slope of both F and Q depends on
a; however, the plateau-like behavior
at large t is independent of a.
Absence of Kibble-Zureck scaling for
any a due to lack of contribution of
small k (critical) modes in the dynamics.
Periodic protocol: dynamics induced freezing
Dynamics induced freezing
Tune J from superfluid phase to Mott
and back through the tip of the Mott lobe.
Key result
There are specific frequencies
at which the wavefunction of
the system comes back to itself
after a cycle of the drive
leading to P=1-F -> 0.
Dynamics induced freezing
Mean-field analysis: A qualitative picture
1. Choose a gutzwiller wavefunction:
2. The mean-field equations for fn
En is the on-site energy of the
state |n>
3. Numerical solution of this equation
indicates that fn vanishes for n>2 for all
ranges of drive frequencies studied.
4. Analysis of these equations leads to the
relation involving
5. Thus one can describe the system
in terms of three real variables : amplitude
of state |1> and the sum and difference
of the relative phases.
fs=f0+f2-2f1 fd=f2-f0
6. One can construct a frequencyindependent relation between r1 and fs
Numerical solution of (6)
There is a range of frequency for which
r1 and fs remain close to their original
values; dynamics induced freezing occurs
when fd/4p = n within this range.
Robustness against quantum fluctuations and presence of a trap
Mean-field theory
Projection operator formalism
Density distribution of
the bosons inside a trap
Robust freezing
phenomenon
Bosons in an electric field
Applying an electric field to the Mott state
20Er
Energy Scales:
hwn = 5Er  20 U
U -3 J
Construction of an effective model: 1D
Parent Mott state
Charged excitations
quasiparticle
quasihole
Neutral dipoles
Resonantly coupled to the parent Mott
state when U=E.
Neutral dipole state with
energy U-E.
Two dipoles which are not nearest neighbors
with energy 2(U-E).
Effective dipole Hamiltonian: 1D
Weak Electric Field
For weak electric field, the ground state is dipole vaccum and the low-energy
excitations are single dipole
• The effective Hamiltonian for the dipoles for weak E:
• Lowest energy excitations: Single band of dipole excitations.
• These excitations soften as E approaches U. This is a precursor of the
appearance of Ising density wave with period 2.
• Higher excited states consists of multiparticle continuum.
Strong Electric field
• The ground state is a state of maximum dipoles.
• Because of the constraint of not having two dipoles on
consecutive sites, we have two degenerate ground states
• The ground state breaks Z2 symmetry.
• The first excited state consists of band of domain walls
between the two filled dipole states.
• Similar to the behavior of Ising model in a transverse field.
Intermediate electric field: QPT
Quantum phase transition at EU=1.853w. Ising universality.
Recent Experimental observation of Ising order (Bakr et al Nature 2010)
First experimental realization of effective Ising model in ultracold atom system
Quench dynamics across the quantum critical point
Tune the electric field from
Ei to Ef instantaneously
Compute the dipole order
Parameter as a function of time
The time averaged value of the order parameter is maximal near the QCP
Generic critical points: A phase space argument
The system enters the impulse region when
rate of change of the gap is the same order
as the square of the gap.
For slow dynamics, the impulse region is a
small region near the critical point where
scaling works
The system thus spends a time T
in the impulse region which
depends on the quench time
In this region, the energy gap scales as
Thus the scaling law for the defect
density turns out to be
Generalization to finite system size
finite-size scaling
Dynamics with a finite rate: Kibble-Zureck scaling
Change the electric field linearly
in time with a finite rate v
Quantities of interest
Scaling laws for
finite –size systems
Q ~ v2 (v) for slow (intermediate) quench. These are
termed as LZ(KZ) regimes for finite-size systems.
Kibble-Zureck scaling for finite-sized system
Expected scaling laws for Q and F
Dipole dynamics
Observation of Kibble-Zurek law for
intermediate v with Ising exponents.
Correlation function
Periodic dynamics: obtaining zero defect density
Protocol
Choose dE such that one starts from
the dipole vacuum at t=0, goes to the
maximally ordered state at t=p/w so tha
E(t) vanishes at t= (n+1/2)p/w
The dipole excitation density for an intermediate frequency range vanishes at
special points where the instantaneous electric field vanishes E(t)=U i.e. at
t=
(n+1/2)p/
Extension to trapped boson systems
Presence of a trap: spatially varying chemical potential
The chemical potential increases
as we move from the center to the
edges of the trap:
One gets multiple MI and SF phases
as one moves from the center to the
edges of the trap.
Clearly the projection operator method will
fail since at some sites one may have a value
of the chemical potential which equally favors
both n=1 and n=2 (or 0) occupation.
One can not project to a fixed number and hence
can not determine the projection operator
uniquely.
Hopping expansion technique for bosons
1. Consider two neighboring sites in the trap with boson occupations
can take positive or negative integer values
2. The hopping between these two
sites costs an energy
i=1(2) for right(left) hopping: these cost
different energy due to the trap.
3. One can rewrite the hopping term
Right hopping
Left hopping
4. We identify the value of
for which
can be both a positive or negative
Integer and may not even exist for a given link
5. Next, one designs a canonical transformation operator S which eliminates
all hoppings to linear order which has
6. One then finds the effective low-energy Hamiltonian using
Generalization of the projection
operator formalism
Quench and Ramp dynamics
Protocol: Linear ramp or quench of
the hopping amplitude J
Schrodinger equation to be solved
One can convert this into an Schrodinger
equation for the effective low-energy
Hamiltonian
One can obtain a set of equations for
The Gutzwiller coefficient which can
be solved numerically
Advantage: General method for treatment of non-equilibrium
trapped/disordered bosons beyond mean-field theory
Weakness: Can not take long-wavelength quantum fluctuations
into account.
Quench from superfluid to MI phase: preliminary results
The order parmeter oscillations
die after reaching a point for which
dm r= J or r = Int[J/K-0.5]
The time period dimishes linearly
With increasing K