Lattices

We can use Bloqade to simulate the quantum evolution of information stored in neutral atoms. Present-day neutral-atom hardware permits the arrangement of atoms in a regular lattice structure and even in nearly arbitrary geometries in 1D, 2D, and 3D. This makes neutral atom platform a natural playground for quantum simulation of statistical models and quantum matters. With Bloqade, we support several built-in lattice structures and also allow the users to specify atom positions by inputting coordinates. Please refer to the Rydberg Blockade page for recommendations on how to set the lattice constants for different lattices.

Types of Lattices

A crystal lattice is completely determined by a set of Bravais lattice vectors (in unit of μm) plus the locations of atoms within a unit cell. A Bravais lattice is an infinite array of discrete points generated by a set of discrete translation operations described by

\[\mathbf{R} = n_1 \mathbf{a}_1 + n_2 \mathbf{a}_2 + \ldots + n_d \mathbf{a}_d,\]

where $d$ is the dimension of space, $n_1, \ldots, n_d \in Z$ are integers. The unit cell of a Bravais lattice is defined by specifying its lattice vectors $(\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_d)$. To create a simple lattice, we first specify the locations of the atoms within a unit cell and then specify the lattice vectors of the Bravais lattice. For example, for a triangular lattice, we need just one site (atom) at the location (0.0, 0.0) in the unit cell and then lattice vectors (1.0, 0.0) and (0.5, 0.5*sqrt(3)):

julia> using Bloqade
julia> triangular = GeneralLattice([(1.0, 0.0), (0.5, 0.5*sqrt(3))], [(0.0, 0.0)])GeneralLattice{2, 1, Float64}(((1.0, 0.0), (0.5, 0.8660254037844386)), ((0.0, 0.0),))

For composite lattices, one should provide multiple sites as the second argument to specify their locations in a unit cell. For example, the honeycomb lattice can be defined by:

julia> honeycomb = GeneralLattice([(1.0, 0.0), (0.5, 0.5*sqrt(3))],
           [(0.0, 0.0), (0.5, 0.5/sqrt(3))])GeneralLattice{2, 2, Float64}(((1.0, 0.0), (0.5, 0.8660254037844386)), ((0.0, 0.0), (0.5, 0.2886751345948129)))

We provide a few shorthands for several useful lattices, including the ChainLattice, SquareLattice, HoneycombLattice, TriangularLattice, LiebLattice, and KagomeLattice shown below. One can use lattice_vectors and lattice_sites to access the lattice vectors and site locations in a unit cell as described in the above section.

`ChainLattice`

using Bloqade
chain = ChainLattice()

ChainLattice()

# to make the plot look good in both light and dark backgrounds.
BloqadeLattices.DEFAULT_BACKGROUND_COLOR[] = "#FFFFFF"

# to show the lattice vectors (rescaled a bit to shrink the head).
unitvectors(lattice::AbstractLattice{2}) = [((0.0, 0.0), v) for v in lattice_vectors(lattice)]

Bloqade.plot(generate_sites(chain, 10); vectors=[((0.0, 0.0), (0.9, 0.0))], bond_linewidth=0.015)

Note

You can see the above visualization in one of the following editors

a VSCode editor,
a Jupyter notebook,
or a Pluto notebook,

but not in a Julia REPL which does not have a graphical display.

lattice_vectors(chain)

((1.0,),)

lattice_sites(chain)

((0.0,),)

Once we have defined certain lattice shapes (which have fixed lattice vectors and site positions in the unit cell), we can generate the atom positions by specifying the number of atoms and the scale size of the lattice. This is done by using the function generate_sites , which will return a AtomList instance containing the coordinates of each atom, e.g.:

atoms = generate_sites(HoneycombLattice(), 3, 5; scale = 4.5)

where scale defines the unit distance in the unit μm of the lattice, and 3, 5 specifies the repetitions of unit cells in each lattice vector direction. The default scale is 1 μm.

Here are some examples of other lattices:

`SquareLattice`

square = SquareLattice()
Bloqade.plot(generate_sites(square, 10, 10); vectors=unitvectors(square), bond_linewidth=0.015)

Note that the indices showing on the sites are consistent with the indices of the qubits for performing computation. In other words, if we want to do measurement or apply operations on individual sites (qubits), we can refer to the numbering on the atoms for convenience. For more details on how to generate Hamiltonians by using the lattice as an argument, please see the section Hamiltonians.

lattice_vectors(square)

((1.0, 0.0), (0.0, 1.0))

lattice_sites(square)

((0.0, 0.0),)

`HoneycombLattice`

honeycomb = HoneycombLattice()
Bloqade.plot(generate_sites(honeycomb, 5, 5); vectors=unitvectors(honeycomb), bond_linewidth=0.015)

`TriangularLattice`

triangular = TriangularLattice()
Bloqade.plot(generate_sites(triangular, 8, 8); vectors=unitvectors(triangular), bond_linewidth=0.015)

`LiebLattice`

lieb = LiebLattice()
Bloqade.plot(generate_sites(lieb, 5, 5); vectors=unitvectors(lieb), bond_linewidth=0.015)

`KagomeLattice`

kagome = KagomeLattice()
Bloqade.plot(generate_sites(kagome, 5, 5); vectors=unitvectors(kagome), bond_linewidth=0.015)

Sorting Sites and Other Operations on Lattices

We also support different operations on the generated lattices. For instance, one can apply some predefined filters, e.g. rescale_axes, clip_axes, offset_axes, to manipulate atom locations:

atoms = generate_sites(HoneycombLattice(), 3, 5; scale = 4.5)
rescale_axes(atoms, 0.8)

where the above operation rescales the coordinates of the original sites by a factor of 0.8.

The code below restricts the atoms sitting in the window (0.0, 5.0), (0.0, 6.0) and throw away those outside this area:

clip_axes(atoms, (0.0, 5.0), (0.0, 6.0))

Furthermore, we can shift the origin of the atoms by some vector (5.0, 5.0) simply by typing the code:

offset_axes(atoms, 5.0, 5.0)

To sort the atoms by their x-coordinates, one can convert these locations to a MaskedGrid representation of the atoms:

atoms_in_grid = make_grid(atoms)

Then one can get the sorted atoms by typing:

sorted_atoms = collect_atoms(atoms_in_grid)

Note that the sorting will change the index numbering of the atoms.

You can also delete atoms given their index number:

deleteat!(atoms, 8, 9, 13, 14, 18, 20, 23)
atoms

Note that this permanently changes the contents of atoms and just like sorting atoms, will change the index numbering albeit to preserve the integer sequence without gaps.

User-Defined Arbitrary Geometries

One can also generate atoms located at arbitrary positions by directly inputting the coordinates of the atoms:

julia> atom_coordinate = AtomList([(0.0, 0.0), (0, 5), (0, 8), (5, 2), (6, 7), (9, 6)])6-element AtomList{2, Real}:
 (0.0, 0.0)
 (0, 5)
 (0, 8)
 (5, 2)
 (6, 7)
 (9, 6)

Query Neighbors

One can use make_kdtree to generate a k-d tree data type for the efficient querying of neighborhoods in a low-dimensional space.

tree = make_kdtree(sorted_atoms)

NearestNeighbors.KDTree{StaticArraysCore.SVector{2, Float64}, Distances.Euclidean, Float64}
  Number of points: 30
  Dimensions: 2
  Metric: Distances.Euclidean(0.0)
  Reordered: true

The return value is a KDTree instance, which is defined in the package NearestNeigbors. One can use it to query the neighbors of an atom: e.g. one can find the 20 nearest neighbors of the 5-th site by typing:

neighbors = grouped_nearest(tree, 5, 20)

DistanceGroup([5, 8, 2, 3, 10, 6, 11, 4, 7, 9, 14, 13, 1, 16, 12, 17, 15, 19, 20, 22], [1, 2, 5, 9, 11, 14, 16, 17, 18, 19, 20, 21])

The return value is a DistanceGroup instance, and the indices of the second nearest neighbors are:

neighbors[2]

4-element Vector{Int64}:
 10
  6
 11
  4

One can select and display these atoms with the correct labeling by typing:

Bloqade.plot(sorted_atoms[neighbors[2]]; texts=string.(neighbors[2]))

It shows the correct second nearest neighbors of the site 5. One can check the docstring of Bloqade.plot to know more about how to customize lattice visualization.

References

BloqadeLattices.AbstractLattice — Type

AbstractLattice{D}

Supertype for all D dimensional lattices.

Implementation

lattice_vectors and lattice_sites functions must be defined which should both return an indexable iterable containing the Bravais lattice vectors and lattice sites respectively. (e.g.: GeneralLattice returns a tuple of tuples containing the Bravais lattice vectors and lattice sites).