<s>
The	O
Bose	B-Algorithm
–	I-Algorithm
Hubbard	I-Algorithm
model	I-Algorithm
gives	O
a	O
description	O
of	O
the	O
physics	O
of	O
interacting	O
spinless	O
bosons	O
on	O
a	O
lattice	B-Algorithm
.	O
</s>
<s>
It	O
is	O
closely	O
related	O
to	O
the	O
Hubbard	B-Algorithm
model	I-Algorithm
that	O
originated	O
in	O
solid-state	O
physics	O
as	O
an	O
approximate	O
description	O
of	O
superconducting	O
systems	O
and	O
the	O
motion	O
of	O
electrons	O
between	O
the	O
atoms	O
of	O
a	O
crystalline	O
solid	O
.	O
</s>
<s>
The	O
Bose	B-Algorithm
–	I-Algorithm
Hubbard	I-Algorithm
model	I-Algorithm
can	O
be	O
used	O
to	O
describe	O
physical	O
systems	O
such	O
as	O
bosonic	O
atoms	O
in	O
an	O
optical	O
lattice	B-Algorithm
,	O
as	O
well	O
as	O
certain	O
magnetic	O
insulators	O
.	O
</s>
<s>
Here	O
,	O
denotes	O
summation	O
over	O
all	O
neighboring	O
lattice	B-Algorithm
sites	O
and	O
,	O
while	O
and	O
are	O
bosonic	O
creation	O
and	O
annihilation	O
operators	O
such	O
that	O
gives	O
the	O
number	O
of	O
particles	O
on	O
site	O
.	O
</s>
<s>
The	O
model	O
is	O
parametrized	O
by	O
the	O
hopping	O
amplitude	O
that	O
describes	O
boson	O
mobility	O
in	O
the	O
lattice	B-Algorithm
,	O
the	O
on-site	O
interaction	O
which	O
can	O
be	O
attractive	O
(	O
)	O
or	O
repulsive	O
(	O
)	O
,	O
and	O
the	O
chemical	O
potential	O
,	O
which	O
essentially	O
sets	O
the	O
number	O
of	O
particles	O
.	O
</s>
<s>
If	O
unspecified	O
,	O
typically	O
the	O
phrase	O
'	O
Bose	B-Algorithm
–	I-Algorithm
Hubbard	I-Algorithm
model	I-Algorithm
 '	O
refers	O
to	O
the	O
case	O
where	O
the	O
on-site	O
interaction	O
is	O
repulsive	O
.	O
</s>
<s>
The	O
dimension	O
of	O
the	O
Hilbert	O
space	O
of	O
the	O
Bose	B-Algorithm
–	I-Algorithm
Hubbard	I-Algorithm
model	I-Algorithm
is	O
given	O
by	O
,	O
where	O
is	O
the	O
total	O
number	O
of	O
particles	O
,	O
while	O
denotes	O
the	O
total	O
number	O
of	O
lattice	B-Algorithm
sites	O
.	O
</s>
<s>
Analogous	O
Hamiltonians	O
may	O
be	O
formulated	O
to	O
describe	O
spinless	O
fermions	O
(	O
the	O
Fermi-Hubbard	B-Algorithm
model	I-Algorithm
)	O
or	O
mixtures	O
of	O
different	O
atom	O
species	O
(	O
Bose	O
–	O
Fermi	O
mixtures	O
,	O
for	O
example	O
)	O
.	O
</s>
<s>
At	O
zero	O
temperature	O
,	O
the	O
Bose	B-Algorithm
–	I-Algorithm
Hubbard	I-Algorithm
model	I-Algorithm
(	O
in	O
the	O
absence	O
of	O
disorder	O
)	O
is	O
in	O
either	O
a	O
Mott	O
insulating	O
state	O
at	O
small	O
,	O
or	O
in	O
a	O
superfluid	O
state	O
at	O
large	O
.	O
</s>
<s>
The	O
phases	O
of	O
the	O
clean	O
Bose	B-Algorithm
–	I-Algorithm
Hubbard	I-Algorithm
model	I-Algorithm
can	O
be	O
described	O
using	O
a	O
mean-field	O
Hamiltonian:where	O
is	O
the	O
lattice	B-Algorithm
co-ordination	O
number	O
.	O
</s>
<s>
Ultracold	O
atoms	O
in	O
optical	O
lattices	O
are	O
considered	O
a	O
standard	O
realization	O
of	O
the	O
Bose	B-Algorithm
–	I-Algorithm
Hubbard	I-Algorithm
model	I-Algorithm
.	O
</s>
<s>
The	O
ability	O
to	O
tune	O
model	O
parameters	O
using	O
simple	O
experimental	O
techniques	O
and	O
the	O
lack	O
of	O
the	O
lattice	B-Algorithm
dynamics	O
that	O
are	O
present	O
in	O
solid-state	O
electronic	O
systems	O
mean	O
that	O
ultracold	O
atoms	O
offer	O
a	O
clean	O
,	O
controllable	O
realisation	O
of	O
the	O
Bose	B-Algorithm
–	I-Algorithm
Hubbard	I-Algorithm
model	I-Algorithm
.	O
</s>
<s>
The	O
biggest	O
downside	O
with	O
optical	O
lattice	B-Algorithm
technology	O
is	O
the	O
trap	O
lifetime	O
,	O
with	O
atoms	O
typically	O
trapped	O
for	O
only	O
a	O
few	O
tens	O
of	O
seconds	O
.	O
</s>
<s>
To	O
see	O
why	O
ultracold	O
atoms	O
offer	O
such	O
a	O
convenient	O
realization	O
of	O
Bose	O
–	O
Hubbard	O
physics	O
,	O
the	O
Bose	O
–	O
Hubbard	O
Hamiltonian	O
can	O
be	O
derived	O
starting	O
from	O
the	O
second	O
quantized	O
Hamiltonian	O
that	O
describes	O
a	O
gas	O
of	O
ultracold	O
atoms	O
in	O
the	O
optical	O
lattice	B-Algorithm
potential	O
.	O
</s>
<s>
where	O
is	O
the	O
optical	O
lattice	B-Algorithm
potential	O
,	O
is	O
the	O
(	O
contact	O
)	O
interaction	O
amplitude	O
,	O
and	O
is	O
the	O
chemical	O
potential	O
.	O
</s>
<s>
Here	O
,	O
is	O
a	O
Wannier	O
function	O
for	O
a	O
particle	O
in	O
an	O
optical	O
lattice	B-Algorithm
potential	O
localized	O
around	O
site	O
of	O
the	O
lattice	B-Algorithm
and	O
for	O
the	O
th	O
Bloch	O
band	O
.	O
</s>
<s>
When	O
considering	O
(	O
fast	O
)	O
lattice	B-Algorithm
dynamics	O
,	O
additional	O
terms	O
are	O
added	O
to	O
the	O
Hamiltonian	O
so	O
that	O
the	O
time-dependent	O
Schrödinger	O
equation	O
is	O
obeyed	O
in	O
the	O
(	O
time-dependent	O
)	O
Wannier	O
function	O
basis	O
.	O
</s>
<s>
Otherwise	O
,	O
the	O
lattice	B-Algorithm
dynamics	O
may	O
be	O
incorporated	O
by	O
making	O
the	O
key	O
parameters	O
of	O
the	O
model	O
time-dependent	O
,	O
varying	O
with	O
the	O
instantaneous	O
value	O
of	O
the	O
optical	O
potential	O
.	O
</s>
<s>
Quantum	O
phase	O
transitions	O
in	O
the	O
Bose	B-Algorithm
–	I-Algorithm
Hubbard	I-Algorithm
model	I-Algorithm
were	O
experimentally	O
observed	O
by	O
Greiner	O
et	O
al.	O
,	O
and	O
density	O
dependent	O
interaction	O
parameters	O
were	O
observed	O
by	O
Immanuel	O
Bloch	O
's	O
group	O
.	O
</s>
<s>
Single-atom	O
resolution	O
imaging	O
of	O
the	O
Bose	B-Algorithm
–	I-Algorithm
Hubbard	I-Algorithm
model	I-Algorithm
has	O
been	O
possible	O
since	O
2009	O
using	O
quantum	O
gas	O
microscopes	O
.	O
</s>
<s>
The	O
Bose	B-Algorithm
–	I-Algorithm
Hubbard	I-Algorithm
model	I-Algorithm
is	O
of	O
interest	O
in	O
the	O
field	O
of	O
quantum	O
computation	O
and	O
quantum	O
information	O
.	O
</s>
<s>
Then	O
the	O
local	O
Hilbert	O
space	O
dimension	O
is	O
The	O
dimension	O
of	O
the	O
full	O
Hilbert	O
space	O
grows	O
exponentially	O
with	O
the	O
number	O
of	O
lattice	B-Algorithm
sites	O
,	O
limiting	O
exact	O
computer	O
simulations	O
of	O
the	O
entire	O
Hilbert	O
space	O
to	O
systems	O
of	O
15-20	O
particles	O
in	O
15-20	O
lattice	B-Algorithm
sites	O
.	O
</s>
<s>
One-dimensional	O
lattices	O
may	O
be	O
studied	O
using	O
density	O
matrix	O
renormalization	O
group	O
(	O
DMRG	O
)	O
and	O
related	O
techniques	O
such	O
as	O
time-evolving	B-Algorithm
block	I-Algorithm
decimation	I-Algorithm
(	O
TEBD	B-Algorithm
)	O
.	O
</s>
<s>
This	O
includes	O
calculating	O
the	O
ground	O
state	O
of	O
the	O
Hamiltonian	O
for	O
systems	O
of	O
thousands	O
of	O
particles	O
on	O
thousands	O
of	O
lattice	B-Algorithm
sites	O
,	O
and	O
simulating	O
its	O
dynamics	O
governed	O
by	O
the	O
time-dependent	O
Schrödinger	O
equation	O
.	O
</s>
<s>
The	O
disorder	O
might	O
be	O
realised	O
by	O
a	O
speckle	O
pattern	O
,	O
or	O
using	O
a	O
second	O
,	O
incommensurate	O
,	O
weaker	O
,	O
optical	O
lattice	B-Algorithm
.	O
</s>
