Dynamics of bosonic cold atoms in optical lattices.
Dynamics of bosonic cold atoms in optical lattices.
Dynamics of bosonic cold atoms in optical lattices.
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<strong>Dynamics</strong> <strong>of</strong> <strong>bosonic</strong> <strong>cold</strong><br />
<strong>atoms</strong> <strong>in</strong> <strong>optical</strong> <strong>lattices</strong>.<br />
K. Sengupta<br />
Indian Association for the Cultivation <strong>of</strong> Science, Kolkata<br />
borators: Anirban Dutta, Christian Trefzger, David Pekker,<br />
Shreyoshi Mondal, Brian Clark, and Michael Kobodurez
Overview<br />
1. Introduction to ultra<strong>cold</strong> bosons<br />
2. <strong>Dynamics</strong> <strong>of</strong> the Bose-Hubbard model<br />
3. <strong>Dynamics</strong> <strong>in</strong>duced freez<strong>in</strong>g<br />
4. Bosons <strong>in</strong> an electric field: Theory and<br />
experiments<br />
5. <strong>Dynamics</strong> <strong>of</strong> bosons <strong>in</strong> an electric field<br />
6. Conclusion
Introduction to ultra<strong>cold</strong> bosons
State <strong>of</strong> <strong>cold</strong> bosons <strong>in</strong> a lattice: experiment Bloch 2001
From BEC to the Mott state<br />
Apply counter<br />
propagat<strong>in</strong>g laser:<br />
stand<strong>in</strong>g wave <strong>of</strong> light.<br />
For a deep enough potential,<br />
the <strong>atoms</strong> are localized : Mott<br />
<strong>in</strong>sulator described by s<strong>in</strong>gle<br />
band Bose-Hubbard model.<br />
The <strong>atoms</strong> feel a potential V = -a |E|2<br />
Energy Scales<br />
δEn = 5Er ~ 20 U<br />
U ~10-300 t<br />
Model Hamiltonian<br />
20E r<br />
Ignore higher bands
Mott state with 1 boson per site<br />
Mott-Superfluid transition: prelim<strong>in</strong>ary analysis<br />
Stable ground state for 0 < m < U<br />
dd<strong>in</strong>g a particle to the Mott state<br />
Mott state is destabilized when<br />
the excitation energy touches 0.<br />
emov<strong>in</strong>g a particle from the Mott state<br />
ion <strong>of</strong> the Mott state via addition <strong>of</strong> particles/hole: onset <strong>of</strong> sup
Beyond this simple picture<br />
igher order energy calculation<br />
Freericks and Monien: Inclusion<br />
up to O(t3/U3) virtual processes.<br />
Mean-field theory (Fisher 89,<br />
Seshadri 93)<br />
Phase diagram for n=1 and d=3<br />
uantum Monte Carlo studies for<br />
D & 3D systems: Trivedi and Krauth,<br />
. Sansone-Capponegro<br />
MFT<br />
O(t2/U2) theories<br />
edicts a quantum phase<br />
nsition with z=2 (except at<br />
e tip <strong>of</strong> the Mott lobe where<br />
1).<br />
Mott<br />
Superfluid<br />
No method for study<strong>in</strong>g dynamics beyond mean-field theory
Projection Operator Method
Dist<strong>in</strong>guish<strong>in</strong>g between hopp<strong>in</strong>g processes<br />
Mott state<br />
Dist<strong>in</strong>guish between two types <strong>of</strong> hopp<strong>in</strong>g processes<br />
us<strong>in</strong>g a projection operator technique<br />
Def<strong>in</strong>e a projection operator<br />
Divide the hopp<strong>in</strong>g to classes (b) and (c)
Build<strong>in</strong>g fluctuations over MFT<br />
n a transformation which elim<strong>in</strong>ate hopp<strong>in</strong>g<br />
sses <strong>of</strong> class (b) perturbatively <strong>in</strong> J/U.<br />
ta<strong>in</strong> the effective Hamiltonian<br />
se the effective Hamiltonian<br />
compute the ground state<br />
nergy and hence the phase<br />
iagram
Equilibrium phase diagram<br />
Reproduction <strong>of</strong> the phase diagra<br />
with remarkable accuracy <strong>in</strong> d=3:<br />
much better than standard<br />
mean-field or strong coupl<strong>in</strong>g<br />
expansion <strong>in</strong> d=2 and 3.<br />
Accurate for large z as can be<br />
checked by compar<strong>in</strong>g with QMC<br />
data for 2D triangular (z=6)<br />
and 3D cubic lattice<br />
Allows for straightforward generalization for treatment <strong>of</strong> dynami
Non-equilibrium dynamics: L<strong>in</strong>ear ramp<br />
ider a l<strong>in</strong>ear ramp <strong>of</strong> J(t)=Ji +(Jf - Ji) t/t.<br />
ynamics, one needs to solve the Sch. Eq.<br />
a time dependent transformation<br />
dress the dynamics by project<strong>in</strong>g on<br />
nstantaneous low-energy sector.<br />
method provides an accurate description<br />
e ramp if J(t)/U
F<br />
Absence <strong>of</strong> critical scal<strong>in</strong>g: may<br />
be understood as the <strong>in</strong>ability <strong>of</strong><br />
the system to access the critical<br />
(k=0) modes.<br />
Fast quench from the Mott to the SF<br />
phase; study <strong>of</strong> superfluid dynamics.<br />
S<strong>in</strong>gle frequency pattern near the critica<br />
Po<strong>in</strong>t; more complicated deeper <strong>in</strong> the S<br />
phase.<br />
Strong quantum fluctuations near the Q<br />
justification <strong>of</strong> go<strong>in</strong>g beyond mft.
Experiments with ultra<strong>cold</strong> bosons on a lattice: f<strong>in</strong>ite rate dynamics<br />
EC conf<strong>in</strong>ed <strong>in</strong> a trap and <strong>in</strong> the presence<br />
<strong>optical</strong> lattice.<br />
e site imag<strong>in</strong>g done by light-assisted collision<br />
h can reliably detect even/odd occupation<br />
ite. In the present experiment they detect<br />
with n=1.<br />
from the SF side near the QCP to deep <strong>in</strong>side<br />
ott phase <strong>in</strong> a l<strong>in</strong>ear ramp with different<br />
rates.<br />
o. <strong>of</strong> sites with odd n displays plateau like<br />
vior and approaches the adiabatic limit<br />
the ramp time is <strong>in</strong>creased asymptotically.<br />
gnature <strong>of</strong> scal<strong>in</strong>g behavior. Interest<strong>in</strong>g<br />
al patterns.<br />
W. Bakr et al. Science 2010
Power law ramp<br />
Use a power-law ramp protocol<br />
Slope <strong>of</strong> both F and Q depends on<br />
a; however, the plateau-like behavior<br />
at large t is <strong>in</strong>dependent <strong>of</strong> a.<br />
Absence <strong>of</strong> Kibble-Zureck scal<strong>in</strong>g for<br />
any a due to lack <strong>of</strong> contribution <strong>of</strong><br />
small k (critical) modes <strong>in</strong> the dynamic
Periodic protocol: dynamics <strong>in</strong>duced freez<strong>in</strong>g
<strong>Dynamics</strong> <strong>in</strong>duced freez<strong>in</strong>g<br />
ne J from superfluid phase to Mott<br />
d back through the tip <strong>of</strong> the Mott lobe.<br />
Key result<br />
There are specific frequencies<br />
at which the wavefunction <strong>of</strong><br />
the system comes back to itself<br />
after a cycle <strong>of</strong> the drive<br />
lead<strong>in</strong>g to P=1-F -> 0.<br />
<strong>Dynamics</strong> <strong>in</strong>duced freez<strong>in</strong>g
Mean-field analysis: A qualitative picture<br />
Choose a gutzwiller wavefunction:<br />
The mean-field equations for fn<br />
En is the on-site energy <strong>of</strong> the<br />
state |n><br />
umerical solution <strong>of</strong> this equation<br />
icates that fn vanishes for n>2 for all<br />
ges <strong>of</strong> drive frequencies studied.<br />
nalysis <strong>of</strong> these equations leads to the<br />
elation <strong>in</strong>volv<strong>in</strong>g
us one can describe the system<br />
rms <strong>of</strong> three real variables : amplitude<br />
ate |1> and the sum and difference<br />
e relative phases.<br />
fs=f0+f2-2f1<br />
fd=f2-f0<br />
ne can construct a frequencypendent<br />
relation between r1 and fs<br />
Numerical solution <strong>of</strong> (6)<br />
There is a range <strong>of</strong> frequency for which<br />
r1 and fs rema<strong>in</strong> close to their orig<strong>in</strong>al<br />
values; dynamics <strong>in</strong>duced freez<strong>in</strong>g occur<br />
when fd/4p = n with<strong>in</strong> this range.
Robustness aga<strong>in</strong>st quantum fluctuations and presence <strong>of</strong> a trap<br />
Mean-field theory<br />
Projection operator formalism<br />
Density distribution <strong>of</strong><br />
the bosons <strong>in</strong>side a trap<br />
Robust freez<strong>in</strong>g<br />
phenomenon
Bosons <strong>in</strong> an electric field
Apply<strong>in</strong>g an electric field to the Mott<br />
state<br />
20E r<br />
Energy<br />
Scales:<br />
hων = 5Εr ≈ 20<br />
U<br />
U ≈10−300 J
Construction <strong>of</strong> an effective model:<br />
1D<br />
Parent Mott state
Charged excitations<br />
quasiparticle<br />
quasihole
Neutral dipoles<br />
Resonantly coupled to the<br />
parent Mott state when U=E.<br />
Neutral dipole state<br />
with energy U-E.<br />
Two dipoles which are not nearest<br />
neighbors with energy 2(U-E).
Effective dipole Hamiltonian: 1D
Weak Electric Field<br />
For weak electric field, the ground state is dipole vaccum and the low-energy<br />
excitations are s<strong>in</strong>gle dipole<br />
• The effective Hamiltonian for the dipoles for weak E:<br />
• Lowest energy excitations: S<strong>in</strong>gle band <strong>of</strong> dipole<br />
excitations.<br />
• These excitations s<strong>of</strong>ten as E approaches U. This is a<br />
precursor <strong>of</strong> the appearance <strong>of</strong> Is<strong>in</strong>g density wave with<br />
period 2.<br />
• Higher excited states consists <strong>of</strong> multiparticle<br />
cont<strong>in</strong>uum.
Strong Electric field<br />
The ground state is a state <strong>of</strong> maximum dipoles.<br />
Because <strong>of</strong> the constra<strong>in</strong>t <strong>of</strong> not hav<strong>in</strong>g two dipoles on<br />
consecutive sites, we have two degenerate ground states<br />
The ground state breaks Z2 symmetry.<br />
The first excited state consists <strong>of</strong> band <strong>of</strong> doma<strong>in</strong> walls<br />
between the two filled dipole states.<br />
Similar to the behavior <strong>of</strong> Is<strong>in</strong>g model <strong>in</strong> a transverse field.
Intermediate electric field: QPT<br />
Quantum phase transition<br />
at E-U=1.853w. Is<strong>in</strong>g<br />
universality.
Recent Experimental observation <strong>of</strong> Is<strong>in</strong>g order (Bakr et al Nature 20<br />
xperimental realization <strong>of</strong> effective Is<strong>in</strong>g model <strong>in</strong> ultra<strong>cold</strong> atom syst
Quench dynamics across the quantum critical po<strong>in</strong>t<br />
ne the electric field from<br />
i to Ef <strong>in</strong>stantaneously<br />
mpute the dipole order<br />
rameter as a function <strong>of</strong> time<br />
time averaged value <strong>of</strong> the order parameter is maximal near the QCP
Generic critical po<strong>in</strong>ts: A phase space argument<br />
system enters the impulse region when<br />
<strong>of</strong> change <strong>of</strong> the gap is the same order<br />
he square <strong>of</strong> the gap.<br />
slow dynamics, the impulse region is a<br />
ll region near the critical po<strong>in</strong>t where<br />
l<strong>in</strong>g works<br />
The system thus spends<br />
a time T <strong>in</strong> the impulse<br />
region which depends on<br />
the quench time<br />
his region, the energy gap scales as<br />
us the scal<strong>in</strong>g law for the defect<br />
nsity turns out to be<br />
neralization to f<strong>in</strong>ite system size<br />
ite-size scal<strong>in</strong>g
<strong>Dynamics</strong> with a f<strong>in</strong>ite rate: Kibble-Zureck scal<strong>in</strong>g<br />
ange the electric field l<strong>in</strong>early<br />
time with a f<strong>in</strong>ite rate v<br />
Theory<br />
uantities <strong>of</strong> <strong>in</strong>terest<br />
Experiment<br />
Scal<strong>in</strong>g laws for<br />
f<strong>in</strong>ite –size systems<br />
Q ~ v2 (v) for slow (<strong>in</strong>termediate) quench. These are<br />
termed as LZ(KZ) regimes for f<strong>in</strong>ite-size systems.
Kibble-Zureck scal<strong>in</strong>g for f<strong>in</strong>ite-sized system<br />
Observation <strong>of</strong> Kibble-Zurek law for<br />
Expected scal<strong>in</strong>g laws for Q and F<br />
<strong>in</strong>termediate v with Is<strong>in</strong>g exponents.<br />
Dipole dynamics<br />
Correlation function