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: coldestplace in the universe

How do you measure such temperatures?

Form a BEC and measure the width of the centralpeak in its momentum distribution.

Hot Cold

Doppler cooling of atoms

Creation of optical molassesleading to microkelvin temperatures

Cooling techniques: two methods

Further evaporative coolingof 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

Energy ScalesdEn = 5Er ~ 20 U

U ~10-300 t

rE20

For a deep enough potential, the atoms are localized : Mott insulator described by single band Bose-Hubbard model.

From BEC to the Mott state

Ignore higher bands

The atoms feel a potential V = -a |E|2

Model Hamiltonian

Apply counter propagating laser: standing wave of light.

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 stateMott 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 calculationby Freericks and Monien: Inclusionof up to O(t3/U3) virtual processes.

Mean-field theory (Fisher 89, Seshadri 93)

Phase diagram for n=1 and d=3

MFT

O(t2/U2) theories

Predicts a quantum phase transition with z=2 (except atthe tip of the Mott lobe where z=1).

Mott

Superfluid

Quantum Monte Carlo studies for 2D & 3D systems: Trivedi and Krauth,B. Sansone-Capponegro

No method for studying dynamics beyond mean-field theory

Projection Operator Method

Distinguishing between hopping processes

Distinguish between two types of hopping processes using a projection operator technique

Define a projection operator

Divide the hopping to classes (b) and (c)

Mott state

Building fluctuations over MFT

Design a transformation which eliminate hoppingprocesses 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.

Allows for straightforward generalization for treatment of dynamics

Accurate for large z as can be checked by comparing with QMCdata for 2D triangular (z=6) and 3D cubic lattice

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 onthe instantaneous low-energy sector.

The method provides an accurate descriptionof 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

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 SFphase.

Strong quantum fluctuations near the QCP;justification of going beyond mft.

F

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 collisionwhich can reliably detect even/odd occupation of a site. In the present experiment they detectsites with n=1.

Ramp from the SF side near the QCP to deep insidethe 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

Slope of both F and Q depends on a; however, the plateau-like behaviorat large t is independent of a.

Use a power-law ramp protocol

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 Mottand back through the tip of the Mott lobe.

Key result

There are specific frequenciesat which the wavefunction ofthe system comes back to itselfafter 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 thestate |n>

3. Numerical solution of this equation indicates that fn vanishes for n>2 for allranges 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.

6. One can construct a frequency-independent 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.

fs=f0+f2-2f1 fd=f2-f0

Robustness against quantum fluctuations and presence of a trap

Mean-field theory Projection operator formalism

Density distribution ofthe bosons inside a trap

Robust freezing phenomenon

Bosons in an electric field

Applying an electric field to the Mott state

Energy Scales:

hwn = 5Er » 20 U

U »10-300 J

rE20

Construction of an effective model: 1D

Parent Mott state

Charged excitationsquasiparticle quasihole

Neutral dipoles

Neutral dipole state with energy U-E.

Resonantly coupled to the parent Mott state when U=E.

Two dipoles which are not nearest neighbors with energy 2(U-E).

Effective dipole Hamiltonian: 1D

Weak Electric Field

• 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.

For weak electric field, the ground state is dipole vaccum and the low-energyexcitations are single dipole

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 E-U=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 whenrate of change of the gap is the same orderas the square of the gap.

For slow dynamics, the impulse region is a small region near the critical point wherescaling 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 sizefinite-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

Theory

Experiment

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 Correlation function

Observation of Kibble-Zurek law for intermediate v with Ising exponents.

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 thatE(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/w

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 willfail since at some sites one may have a valueof the chemical potential which equally favors both n=1 and n=2 (or 0) occupation.

One can not project to a fixed number and hencecan not determine the projection operator uniquely.

Hopping expansion technique for bosons

1. Consider two neighboring sites in the trap with boson occupations

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.

can take positive or negative integer values

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 projectionoperator formalism

Quench and Ramp dynamicsProtocol: Linear ramp or quench ofthe hopping amplitude J

Schrodinger equation to be solved

One can convert this into an Schrodingerequation 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