Quantum
Phase Transition from a Superfluid to a Mott Insulator in a Gas of
Ultracold Atoms
(Experiments by
M. Greiner, O. Mandel, T. Esslinger, T.W.
Hänsch, and I. Bloch in Munich)
One
of the most intriguing aspects of quantum mechanics is that even at
absolute zero temperature quantum fluctuations prevail in a system,
whereas all thermal fluctuations are frozen out. Although these quantum
fluctuations are microscopic in nature, they are nevertheless able to
induce a macroscopic phase transition in the ground state of a many
body system, when the strength of two competing terms in the underlying
Hamiltonian is varied across a critical value.
When
atoms with repulsive interactions are transferred from a magnetically
trapped BoseEinstein condensate into a threedimensional optical
lattice potential, they will undergo such a quantum phase transition
from a Superfluid to a Mott Insulator as the potential depth of the
lattice is increased. For low potential depths, the atoms form a
superfluid phase, where each atom is spread out over the entire
lattice, with longrange phase coherence across the lattice. For high
potential depths the repulsive interactions between the atoms cause a
transition to a Mott insulator phase. In this phase the atoms are
localized at the lattice sites with an exactly defined atom number, but
no phase coherence throughout the lattice. The Mott insulator phase is
characterized by a gap in the excitation spectrum, which is detected in
the experiment. We also demonstrate that it is possible to reversibly
change between the two ground states of the system.

Fig. 1:
Superfluid state with coherence, Mott Insulator state without
coherence, and superfluid state after restoring the coherence 
Entering
the Mott Insulator Phase
In
the weakly interacting regime a BoseEinstein condensate in a 3D
optical lattice can be well described by a macroscopic wave function.
In the ground state only one bloch state is occupied, which means that
the phase of the macroscopic wave function is the same at each lattice
site. This state can be clearly identified by detecting sharp peaks in
the multiple matter wave interference pattern after a time of flight
expansion (see
also the section about the 3D lattice).
If
we now increase the lattice depth to deeper and deeper values, we
observe that the interference peaks dim out and a non coherent
background gains strength. For a deep lattice the phase coherence is
totally lost.
One
might think that the system in the deep lattice is still superfluid but
that the coherence is lost because the lattice sites are dephased,
which means that one has a statistical mixture of Bloch states. But we
were able to show that this is not the case. Instead a totally
different regime is entered, the regime of a Mott Insulator.
Restoring
Coherence
The
phase coherence can be restored very rapidly when the lattice potential
is lowered again. A significant amount of coherence is restored already
after about one tunneling time. This is shown in the graph below
(filled circles), where the width of the central interference peak,
which is a measure for coherence in the system, is plotted against the
ramp down time.
The open circles show the same measurement for an ensemble, which was
intentionally statistically dephased before reaching the Mott insulator
phase. The coherence is not restored at all in this case, even for of
times up to 400 ms. This is one of the evidences that the state in the
deep lattice is not just a statistically dephased superfluid state, but
really the Mott insulator
The
BoseHubbard Hamiltonian
The system of
interacting bosonic particles in a lattice potential is very well
described by a BoseHubbard Hamiltonian in second quantisation. It can
be derived by expanding the field operator in the wannier basis of
localized wave functions at each lattice site, and consists of three
parts:

The first term is the kinetic energy term
with the tunneling matrix element J. J
is basically determined by the overlap between adjacent localized wave
functions and decreases exponentially with the lattice depth.

The second term describes an energy offset in
each lattice site for example due to an external confinement.

The third term is the potential energy term
characterized by the onsite atomatom interaction energy U.
In our case U is very large, about h*1.6 kHz, due to the strong
confinement at each lattice site in a 3D lattice. U tells you
how much energy it costs to put a second atom into a lattice with
already one atom present at this lattice site.
For a shallow lattice J is large compared
to U
and the Hamiltonian is dominated by the kinetic energy term with a
superfluid ground state (left side of diagram below). For a deep
lattice J becomes small, the Hamiltonian is dominated by the
interaction energy term and the ground state is the state of a Mott
insulator for commensurate filling (right side).
In
the superfluid case, each atom is delocalized over the entire lattice.
Each lattice site is populated with a superposition of different number
states which form a coherent state with a well defined phase for each
lattice site. In the Mott Insulator state the atoms are localized to
lattice sites with a defined number of atoms at each site. Now only one
number state is occupied, for example with exactly one atom per site.
But with a minimized number uncertainty, the phase uncertainty is
maximized and the interference pattern vanishes. The phase coherence is
lost but replaced by atom number correlations.
Measuring
the Gap in the Excitation Spectrum
The gas of atoms can move freely through the lattice in the superfluid
regime. But when the Mott insulator regime is entered, the mobility of
the atoms is blocked due to the repulsive interaction between the
atoms. With an exact number of atoms at each lattice site, it costs the
onsite interaction energy U for an atom to tunnel to an adjacent site.
This energy U must be provided to create a particlehole excitation.
Since this excitation is the lowest possible excitation in the Mott
insulator state, a gap in the excitation spectrum is formed.
This
gap, which is an important characteristic of a Mott insulator, was
measured by applying a potential gradient to the system. The tunneling
of atoms to adjacent sites is blocked until the gradient becomes large
enough to overcome the gap: A difference of the potential energy
between neighbouring lattice sites of U can provide the energy needed
for tunneling (see right picture above).
The
following graphs show the response to an applied potential gradient,
plotted against the strength of the gradient, for different lattice
potential depths. The response is the perturbation of the system. It is
measured as the width of the interference peaks after the lattice depth
is reduced again and the coherence is restored.

For
a shallow lattice with a depth of 10 recoil energies the system can be
easily perturbed, even for small gradients. But at a depth of about 13
recoil energies two resonances start to appear, and for a lattice depth
of 20 recoil energies the situation has dramatically changed: We
observe two narrow resonances ontop of a otherwise flat perturbation
probability. For small gradients, the system cannot be perturbed at
all, but for an energy difference between neighbouring lattice sites of
the onsite interaction energy U, the system is perturbed resonantly.
This gap in the excitation spectrum directly proves that we have indeed
entered the Mott insulator regime. 
The position of
the first resonance in the threedimensional lattice
(closed circles) is in good agreement with the calculated value for the
onsite interaction energy U. 


M.
Greiner, O. Mandel, T. Esslinger, T.W.
Hänsch, and I. Bloch
Quantum Phase Transition from a
Superfluid to a
Mott Insulator in a Gas of Ultracold Atoms
Nature
415, 3944 (2002) 

