1.2.1 Optical lattice potential
The understanding of correlated many body phenomena in usual solid state physics are extremely difficult due to the complicated band structure, complexity in Coulomb interactions between the constituent particles, the presence of dis- order and impurities etc. Sometimes it is quite impossible to take care of all the under lying consequences in a single theoretical model and often hard to gauge out the relevance of a particular effect while synthesizing the outcome of an ex-
periment. An alternate way to bypass such restrictions is to use of an artificial, known as optical lattice potential [2] and thus provides a gateway to explore the phenomena at the interface of solid state physics and (ultracold) atomic physics.
An optical lattice is formed due to the interference of counter propagating laser beams from all directions that renders a periodic potential which replicates lattice structures as perceived by an electron in a real crystal. To create a one di- mensional optical lattice potential, consider two counter propagating laser beams each of amplitude E0, and linearly polarized in the x-direction with wave vector k=(±k,0,0)then the resultant electric field is written as,
E(x)=E0eˆz[eikx +e−ikx] 1.5
The field intensity is given by,
I(x)=|E(x)|2 =4|E0|cos2(kx) 1.6 Thus the optical lattice potential becomes,
Vop(x)=V0cos2(kx) 1.7 where V0 is the depth of the optical lattice and the lattice periodicity, a = λ/2.
Using further two pair of laser beams in y and z direction, a three dimensional optical lattice potential is formed as shown in [Fig.1.2] [2],
Vop(r)=V0xcos2(kx)+V0ycos2(ky)+V0zcos2(kz) 1.8
Figure 1.2: A 3-dimensional optical lattice potential is formed by superimposing three orthogonal standing waves. In the 3D case, the optical lattice can be approxi- mated by a 3D simple cubic array of tightly confining harmonic oscillator potentials at each lattice site. Figure courtesy of I. Bloch [57].
The height of the optical lattice potential, V0i is often expressed in terms of recoil energy via ER = ~2k2/2m. The superiority of the optical lattice potential
originates from the vast control of the lattice parameters as well as the potential height by smoothly calibrating only the laser intensity. More so, it is free from defects, impurities and lattice vibrations etc. Thus the tunability of different parameters that are agents of the phase transition yields an edge over experiments performed in the context of usual condensed matter physics.
1.2.2 Laser cooling
An optical lattice allows the neutral atoms to cool down at extremely low tem- perature using various cooling mechanism such as laser cooling [3–6], evaporative cooling [58] techniques etc. Thermal velocity of an atom is directly related with the temperature and hence slowing down their motion will help in decreasing the temperature. The central idea of laser cooling is to decelerate an atom by using a radiation (or scattering) pressure of a laser light near atomic resonance. In this process, an atom in the ground state absorbs a photon from an incident laser beam. After absorbing a photon of energy hν, the atom acquires a momentum impulse ofhν/c along the incoming direction. In order to absorb a photon again, the atom has to return to the ground state by emitting a photon. After emitting a photon, the atom recoils in the opposite direction which results in their slowing down for a span of few microseconds, a phenomenon responsible for the origin of
"optical molasses" [3] and subsequently producing low temperature. Apart from the laser cooling, few other techniques such as polarization gradient cooling [59], Raman cooling [59], Sisyphus cooling [4,6,60] etc have been developed to reach ultralow temperatures.
1.2.3 Atom-atom interaction: Feshbach resonance
In optical lattices, experimental navigation from a weak to a strong interac- tion region is possible by simply changing the optical lattice potential. However, without disturbing the lattice geometry so much, a strong interaction region can be achieved by controlling the atom-atom interaction using a Feshbach reso- nance [11,54] which in turn determine the on-site interaction potential. For ultracold gases at extremely low temperature, the two body interaction potential is essentially short range in nature and hence entirely depends upon thes- wave scattering length,as. The Feshbach resonance was first introduced in the context of nuclear physics which was later extended for ultracold atomic gases to tune the scattering lengths via magnetic field.
Apart from the Feshbach resonance, few other resonances such as a ’shape resonance’ which occurs in a potential barrier where the scattering cross section is a function of the angular momentum and a ’potential resonance’ that happens in a single channel where thes-wave phenomena is dominant [61]. Unlike shape and
potential resonances, a Feshbach resonance involves two particle collision process between multi channels and hence occurs when a bound state in a closed channel resonantly coupled with the scattering state of an entrance or open channel [11, 54]. Here the open and the closed channels correspond to the two different spin configurations of the molecule. Further by simply changing the magnetic field, the energy difference and hence the interaction potential can be controlled over a wide range of values. Instead of a magnetically induced resonance, a Feshbach resonance can be obtained by inducing an optical transition by detuning the light from the atomic resonance [11]. In a magnetically tuned Feshbach resonance, thes-wave atomic scattering length,as is now a function of the magnetic field, B as [62],
as(B)= abg(1−∆/B−B0) 1.9
-3 -2 -1 0 1 2 3
(B-B0)/∆ -4
-2 0 2 4
a s/a bg
Figure 1.3: The variation of the scattering length, as/abg in a magnetically tuned Feshbach resonance. Figure courtesy of Chin et al. [11].
Here abg is the background scattering length as a result of the background collision in the open channel represents an off-resonant value. B0is the resonance field at which scattering length diverges (a → ±∞) and∆is called resonance width over which the scattering length changes sign. Fig.1.3shows a schematic plot of the atomic scattering length as a function of magnetic field,B. Bothaandabgcan have positive as well as negative values which implies a repulsive and an attractive interactions respectively can be realized for the cold atoms. Such tunability of the interaction by using Feshbach resonances have resulted in successful observation of BEC [63], physics of the BEC-BCS crossover [64], and the existence of Efimov states [65] etc.