displacement_operator

Graph.displacement_operator(alpha, dimension, offset=0, *, name=None)

Create a displacement operator (or a batch of them).

Parameters:

  • alpha (number or np.ndarray) – A number \(\alpha\) that characterizes the displacement operator. You can also pass a 1D array to generate a batch of displacement operators.

  • dimension (int) – The size of the truncated Fock space. By default, the Fock space is truncated at [0, dimension). If non-zero offset is passed, the space is then truncated at [offset, dimension + offset).

  • offset (int, optional) – The lowest Fock state. Defaults to 0.

  • name (str or None, optional) – The name of the node.

Returns:

A tensor representing the displacement operator. The shape is 2D if there is no batch in alpha, otherwise 3D where the first axis is the batch dimension.

Return type:

Tensor

See also

Graph.annihilation_operator

Create an annihilation operator in the truncated Fock space.

Graph.coherent_state

Create a coherent state (or a batch of them).

Graph.creation_operator

Create a creation operator in the truncated Fock space.

Graph.fock_state

Create a Fock state (or a batch of them).

Graph.number_operator

Create a number operator in the truncated Fock space.

Graph.squeeze_operator

Create a squeeze operator (or a batch of them) in the truncated Fock space.

Notes

The displacement operator is defined as:

\[D(\alpha) = \exp(\alpha \hat{a}^\dagger - \alpha^\ast \hat{a})\]

where \(\hat{a}\) and \(\hat{a}^\dagger\) are the annihilation and creation operators respectively.

This operation uses truncated annihilation/creation operators to calculate the displacement operator. As such, for nonzero offset, generating a coherent state by applying the displacement operator to a Fock state will not generate the same result as using coherent_state, which uses a full operator and then truncates the resulting state. As the energy levels increase, however, the two coherent states will increasingly agree.

For more information about the displacement operator, see Displacement operator on Wikipedia.

Examples

Create a displacement operator.

>>> graph.displacement_operator(1j, 2, name="displacement")
<Tensor: name="displacement", operation_name="displacement_operator", shape=(2, 2)>
>>> result = bo.execute_graph(graph=graph, output_node_names="displacement")
>>> result["output"]output["displacement"]["value"]
array([[ 5.40302306e-01+0.00000000e+00j, -1.03050475e-16+8.41470985e-01j],
    [ 5.55111512e-17+8.41470985e-01j,  5.40302306e-01+6.61679489e-17j]])

Create a batch of displacement operators.

>>> graph.displacement_operator([1j, 3], 2, name="displacement_batch")
<Tensor: name="displacement_batch", operation_name="displacement_operator", shape=(2, 2, 2)>
>>> result = bo.execute_graph(graph=graph, output_node_names="displacement_batch")
>>> result["output"]["displacement_batch"]["value"]
array([[[ 5.40302306e-01+0.00000000e+00j, -1.11022302e-16+8.41470985e-01j],
        [ 5.55111512e-17+8.41470985e-01j, 5.40302306e-01+1.11022302e-16j]],
    [[-9.89992497e-01+0.00000000e+00j, -1.41120008e-01+0.00000000e+00j],
        [ 1.41120008e-01+0.00000000e+00j, -9.89992497e-01+0.00000000e+00j]]])