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Topological quantum phase transition and
quantum magnetism simulation are the frontier researches in the field of
ultracold atomic physics. Changing a certain parameter of multi-particle
systems when under the absolute zero degree, such as the coupling strength, the
pressure or the external magnetic field strength, the system can be continuously
changed from one ordered state to another. Since the thermodynamic fluctuations
are as important as the quantum fluctuations when at the critical point, this
kind of phase transition is completely different from the thermodynamic phase
transition induced by temperature and is called as the quantum phase
transition. But when under the absolute zero degree, another kind of quantum
phase transition may also exist in a quantum system. This special kind of
quantum phase transition process does not exist of spontaneous symmetry
breaking of Landau theory and the behavior is determined by the bulk properties
of the wave function for the ground state of the system. The system is in
disordered phases at both sides of the phase transition point and there is no
localized order parameter to describe or distinguish the two phases. But a
non-localized order parameter may exist and this kind of quantum phase
transition is called as topological quantum phase transition^{ [1]}.
For example, the phase transition between the conductance quantum platform in
the fractional quantum Hall effect is a typical topological quantum phase
transition.

FIG.1 Quantum phase diagram under non zero temperature

For the two-component bosons or fermions in optical
lattices, tag them with the pseudospin variable ↑ and ↓ and interaction U≫J.
Using the two order perturbation theory of the kinetic energy term, we correspond
the Bose-Hubbard model to the Heisenberg model which has the equivalent spin-spin
interaction between adjacent lattice sites,
FIG.2 The principle diagram of the superexchange
interaction^{ [3]}

FIG.3 Ferromagnetic order FIG.4 Antiferromagnetic order

Ultracold quantum gases in optical lattices

An optical lattice is typically formed by two interfering laser beams, giving rise to a periodic intensity pattern which is seen as a periodic potential by the atoms. In the optical lattice potential well, the atoms can be trapped in the strongest or weakest position of the laser intensity due to the optical dipole forces arising from the AC Stark effect: If the light field is red-detuned (light frequency smaller than atomic transition frequency) the atoms will be drawn to the intensity maximum, provided they are in the ground state; A blue detuned light field repells the atoms to dark regions.

FIG.5 Schematic diagram of one dimensional, two
dimensional and

FIG.6 Band structure in optical lattice

FIG.7 Hexagonal optical lattices formed by three laser
beams with 120 degrees of each other

FIG.8 Optical lattices formed by two pairs of vertical laser beams, phase
difference between the two pairs of beams:0,π/6,π/3,π/2,2π/3,5π/6

Full dynamical control over the individual
laser-beam intensities as well as over the relative phase between the two
standing waves forming the lattice allows us to set all relevant parameters of
the atomic many-body system. We obtain 87Rb Bose-Einstein condensates with the temperature
of 30nK by magneto-optical trap and forced evaporative cooling in optical
dipole trap, then load the condensates into a three-dimensional optical lattice
formed by three standing wave laser beams. FIG.9 is the material wave
interference pattern of the quantum phase transition from the superfluid state
to the Mott insulating state we obtained in the three-dimensional optical
lattice. FIG.10 is the quantum Newton’s cradle formed by using one-dimensional
Bose gases.FIG.9 Quantum phase transition from superfluid to Mott insulating in a three dimensional optical lattice

FIG.10 Quantum Newton’s cradle

Summary of experimental system

The experimental vacuum system is composed of scientific cavity, two dimensional magnetic optical trap, atomic source and ion pump, as shown in Figure 11. The science chamber vacuum is 3×10-9 Pa and the two-dimensional magneto-optical trap cavity vacuum is 3×10-7 Pa. The science chamber is drum shaped, including eight 2.75-inch windows, sixteen 1.33-inch windows and two 8-inch windows in up and down direction. These two 8-inch windows are close to the center of the chamber, for future experimental realization of in situ detection with large numerical aperture.

FIG.11 Vacuum system

FIG.12 Rubidium 87 D2 transition hyperfine structure

FIG.13 Laser system

FIG.14 Schematic diagram of optical dipole trap

[1]S. Sachdev, Quantum
Phase Transition (Cambridge University Press, Cambridge, 1999).

[2]A. Auerbach,
Interacting Electrons and Quantum Magnetism (Springer, Berlin, 2006).

[3]S. Trotzky et
al., Time-Resolved Observation and Control of Superexchange Interactions with
Ultracold Atoms in Optical Lattices, Science **319**, 295 (2008).

[4]I. Bloch et al.,
Quantum simulations with ultracold quantum gases, Nature Phys. **8**, 267 (2012).