# displacement_operator¶

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

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

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

• dimension (int) – The size of the truncated Fock space. By default, the Fock space is truncated as [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, 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

annihilation_operator()

Create an annihilation operator in the truncated Fock space.

coherent_state()

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

creation_operator()

Create a creation operator in the truncated Fock space.

fock_state()

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

number_operator()

Create a number operator in the truncated Fock space.

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

Examples

Create a displacement operator.

>>> graph.displacement_operator(1j, 2, name="displacement")
<Tensor: name="displacement", operation_name="displacement_operator", shape=(2, 2)>
>>> result = qctrl.functions.calculate_graph(graph=graph, output_node_names=["displacement"])
>>> result.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 = qctrl.functions.calculate_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]]])