Table 3 Laser fields needed for the implementation.
From: Local models of fractional quantum Hall states in lattices and physical implementation
Electric field |
|---|
Trap |
|
|
|
|
|
|
|
|
Hopping |
Er1(x, y, z, T)= −. |
Er2(x, y, z, T)= |
Eb3(x, y, z, T)=. |
- The eight laser fields listed below the word ‘Trap’ produce the checkerboard optical lattice in Fig. 1c, and the three standing wave fields listed below the word ‘Hopping’ bring about the hopping terms and ensure trapping in the z direction. T is time, E=E*, Er=
, E+ and E− determine the amplitudes, 
is the polarization vector of right circularly polarized light, 
is the polarization vector of left circularly polarized light, εz=(0,0,1) is the polarization vector of z polarized light, α and β are real, adjustable parameters, a is the lattice constant, c is the speed of light in vacuum, kx=ki(2+β2)−1/2, kz=kiβ(2+β2)−1/2, ki=ωi/c, the quantities ω1, ω2, ω3, ω4, ωr=cπ/a, and ωb are angular frequencies, and +c.c. stands for adding the complex conjugate of the preceding term. The frequencies ωi, i=1, 2 , 3, 4, are assumed to be slightly different such that coherent interference between fields with different ωi is avoided. This can be done while introducing only negligible differences in the lengths of the wavevectors, and we hence assume k1, k2, k3, and k4 to be equal. Note that β2≈2 is a particularly convenient choice because ωr≈ωi in this case, and therefore the same set of excited states can be used for implementing both the trap and the hopping terms. The value of α2 is chosen to get a suitable relative height of the red and blue potentials.
, E+ and E− determine the amplitudes, 
is the polarization vector of right circularly polarized light, 
is the polarization vector of left circularly polarized light, εz=(0,0,1) is the polarization vector of z polarized light, α and β are real, adjustable parameters, a is the lattice constant, c is the speed of light in vacuum, kx=ki(2+β2)−1/2, kz=kiβ(2+β2)−1/2, ki=ωi/c, the quantities ω1, ω2, ω3, ω4, ωr=cπ/a, and ωb are angular frequencies, and +c.c. stands for adding the complex conjugate of the preceding term. The frequencies ωi, i=1, 2 , 3, 4, are assumed to be slightly different such that coherent interference between fields with different ωi is avoided. This can be done while introducing only negligible differences in the lengths of the wavevectors, and we hence assume k1, k2, k3, and k4 to be equal. Note that β2≈2 is a particularly convenient choice because ωr≈ωi in this case, and therefore the same set of excited states can be used for implementing both the trap and the hopping terms. The value of α2 is chosen to get a suitable relative height of the red and blue potentials.