neurophox.components package

neurophox.components.mzi module

class neurophox.components.mzi.BlochMZI(theta, phi, hadamard, epsilon=0.0, dtype=<class 'numpy.complex128'>)

Bases: neurophox.components.mzi.MZI

Mach-Zehnder Interferometer (Bloch basis, named after the Bloch sphere qubit formula)

Class simulating an ideal phase-shifting Mach-Zehnder interferometer. As usual in our simulation environment, we have \(\theta \in [0, \pi]\) and \(\phi \in [0, 2\pi)\).

In Hadamard convention, the corresponding transfer matrix is:

\[\begin{split}U_2(\theta, \phi) = H D(\theta) H L(\phi) = \begin{bmatrix} e^{i \phi}\cos \frac{\theta}{2} & i\sin \frac{\theta}{2} \\ ie^{i \phi}\sin \frac{\theta}{2} & \cos \frac{\theta}{2} \end{bmatrix}\end{split}\]

In beamsplitter convention, the corresponding transfer matrix is:

\[\begin{split}U_2(\theta, \phi) = B D(\theta) B L(\phi) = i \begin{bmatrix} e^{i \phi}\sin \frac{\theta}{2} & \cos \frac{\theta}{2} \\ e^{i \phi}\cos \frac{\theta}{2} & -\sin \frac{\theta}{2} \end{bmatrix}\end{split}\]

Parameters

• theta (float) – Amplitude-modulating phase shift

• phi (float) – External phase shift,

• hadamard (bool) – Whether to use Hadamard convention

• epsilon (Union[float, Tuple[float, float]]) – Beamsplitter error

• dtype – Type-casting to use for the matrix elements

class neurophox.components.mzi.MZI(internal_upper, internal_lower, external_upper, external_lower, hadamard, epsilon=0.0, dtype=<class 'numpy.complex128'>)

Bases: neurophox.components.transfermatrix.PairwiseUnitary

Mach-Zehnder Interferometer

Class simulating the scattering matrix formulation of an ideal phase-shifting Mach-Zehnder interferometer. This can implement any \(2 \times 2\) unitary operator \(U_2 \in \mathrm{U}(2)\).

The internal phase shifts, \(\theta_1, \theta_2\), and the external phase shifts \(\phi_1, \phi_2\), are used to define the final unitary operator as follows (where we define \(\theta := \theta_1- \theta_2\) for convenience):

In Hadamard convention, the corresponding transfer matrix is:

\[\begin{split}U_2(\theta, \phi) = H L(\theta_1) R(\theta_2) H L(\phi_1) R(\phi_2) = e^{i\frac{\theta_1 + \theta_2}{2}} \begin{bmatrix} e^{i \phi_1}\cos \frac{\theta}{2} & ie^{i \phi_2}\sin \frac{\theta}{2} \\ ie^{i \phi_1}\sin \frac{\theta}{2} & e^{i \phi_2}\cos \frac{\theta}{2} \end{bmatrix}\end{split}\]

In beamsplitter convention, the corresponding transfer matrix is:

\[\begin{split}U_2(\theta, \phi) = B L(\theta_1) R(\theta_2) B L(\phi_1) R(\phi_2) = i e^{i\frac{\theta_1 + \theta_2}{2}} \begin{bmatrix} e^{i \phi_1}\sin \frac{\theta}{2} & e^{i \phi_2}\cos \frac{\theta}{2} \\ e^{i \phi_1}\cos \frac{\theta}{2} & -e^{i \phi_2}\sin \frac{\theta}{2} \end{bmatrix}\end{split}\]

Parameters

• internal_upper (float) – Upper internal phase shift

• internal_lower (float) – Lower internal phase shift

• external_upper (float) – Upper external phase shift

• external_lower (float) – Lower external phase shift

• hadamard (bool) – Whether to use Hadamard convention

• epsilon (Union[float, Tuple[float, float]]) – Beamsplitter error

• dtype – Type-casting to use for the matrix elements

property matrix

Returns: \(U_2\), a \(2 \times 2\) unitary matrix implemented by this component

property reflectivity

Returns: Reflectivity, \(r\), for the matrix \(U\)

property transmissivity

Returns: Transmissivity, \(t\), for the matrix \(U\)

class neurophox.components.mzi.SMMZI(theta, phi, hadamard, lower_theta=False, lower_phi=False, epsilon=0.0, dtype=<class 'numpy.complex128'>)

Bases: neurophox.components.mzi.MZI

Mach-Zehnder Interferometer (single-mode basis)

Class simulating an ideal phase-shifting Mach-Zehnder interferometer. As usual in our simulation environment, we have \(\theta \in [0, \pi]\) and \(\phi \in [0, 2\pi)\).

In Hadamard convention, the corresponding transfer matrix is:

\[\begin{split}U_2(\theta, \phi) = H L(\theta) H L(\phi) = e^{-i \theta / 2} \begin{bmatrix} e^{i \phi}\cos \frac{\theta}{2} & i\sin \frac{\theta}{2} \\ ie^{i \phi}\sin \frac{\theta}{2} & \cos \frac{\theta}{2} \end{bmatrix}\end{split}\]

In beamsplitter convention, the corresponding transfer matrix is:

\[\begin{split}U_2(\theta, \phi) = B L(\theta) B L(\phi) = ie^{-i \theta / 2} \begin{bmatrix} e^{i \phi}\sin \frac{\theta}{2} & \cos \frac{\theta}{2} \\ e^{i \phi}\cos \frac{\theta}{2} & -\sin \frac{\theta}{2} \end{bmatrix}\end{split}\]

Parameters

• theta (float) – Amplitude-modulating phase shift

• phi (float) – External phase shift,

• hadamard (bool) – Whether to use Hadamard convention

• epsilon (Union[float, Tuple[float, float]]) – Beamsplitter error

• dtype – Type-casting to use for the matrix elements

classmethod nullify(vector, idx, lower_theta=False, lower_phi=False)

neurophox.components.mzi.get_mzi_transfer_matrix(internal_upper, internal_lower, external_upper, external_lower, hadamard, epsilon, dtype)

Mach-Zehnder interferometer

Parameters

• internal_upper (float) – Upper internal phase shift

• internal_lower (float) – Lower internal phase shift

• external_upper (float) – Upper external phase shift

• external_lower (float) – Lower external phase shift

• hadamard (float) – Whether to use Hadamard convention

• epsilon (Tuple[float, float]) – Beamsplitter error

• dtype – Type-casting to use for the matrix elements

Return type

ndarray

Returns

MZI transfer matrix

neurophox.components.mzi.get_tdc_transfer_matrix(kappa, delta, external_upper, external_lower, dtype)

Tunable directional coupler

Parameters

• kappa (float) – Phase-matched phase shift (from coupled mode theory)

• delta (float) – Phase-mismatched phase shift (from coupled-mode theory)

• external_upper (float) – Upper external phase shift

• external_lower (float) – Lower external phase shift

• dtype – Type-casting to use for the matrix elements

Return type

ndarray

Returns

MZI transfer matrix

neurophox.components.transfermatrix module

class neurophox.components.transfermatrix.Beamsplitter(hadamard, epsilon=0, dtype=<class 'numpy.complex128'>)

Bases: neurophox.components.transfermatrix.PairwiseUnitary

Ideal 50/50 beamsplitter

Implements the Hadamard or beamsplitter operator.

Hadamard transformation, \(H\):

\[\begin{split}H = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1\\ 1 & -1 \end{bmatrix}\end{split}\]

Beamsplitter transformation \(B\):

\[\begin{split}B = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 & i\\ i & 1 \end{bmatrix}\end{split}\]

Hadamard transformation, \(H_\epsilon\) (with error \(\epsilon\)):

\[\begin{split}H_\epsilon = \frac{1}{\sqrt{2}}\begin{bmatrix} \sqrt{1 + \epsilon} & \sqrt{1 - \epsilon}\\ \sqrt{1 - \epsilon} & -\sqrt{1 + \epsilon} \end{bmatrix}\end{split}\]

Beamsplitter transformation \(B_\epsilon\) (with error \(\epsilon\)):

\[\begin{split}B_\epsilon = \frac{1}{\sqrt{2}}\begin{bmatrix} \sqrt{1 + \epsilon} & i\sqrt{1 - \epsilon}\\ i\sqrt{1 - \epsilon} & \sqrt{1 + \epsilon} \end{bmatrix}\end{split}\]

Parameters

• hadamard (bool) – Amplitude-modulating phase shift

• epsilon (float) – Errors for beamsplitter operator

• dtype – Cast values as dtype for this pairwise unitary operator

property matrix

Returns: \(U_2\), a \(2 \times 2\) unitary matrix implemented by this component

Return type

ndarray

property reflectivity

Returns: Reflectivity, \(r\), for the matrix \(U\)

Return type

float

property transmissivity

Returns: Transmissivity, \(t\), for the matrix \(U\)

Return type

float

class neurophox.components.transfermatrix.PairwiseUnitary(dtype=<class 'numpy.complex128'>)

Bases: object

Pairwise unitary

This can be considered also a two-port unitary linear optical component (e.g. unitary scattering matrix for waveguide modal interaction).

This unitary matrix \(U_2\) has the following form:

\[\begin{split}U_2 = \begin{bmatrix} U_{11} & U_{12} \\ U_{21} & U_{22} \end{bmatrix},\end{split}\]

where we define the reflectivity \(r = |U_{11}|^2 = |U_{22}|^2\) and transmissivity \(t = |U_{12}|^2 = |U_{21}|^2\). We also require that the determinant \(|\det U| = 1\), from which we ultimately get the property \(U_2^\dagger U_2 = I_2\), where \(I_2\) is the \(2 \times 2\) identity matrix. All children of this class have this property.

Parameters

dtype – Cast values as dtype for this pairwise unitary operator

givens_rotation(units, m, n=None)

Givens rotation matrix \(T_{m, n}\) which rotates any vector about axes \(m, n\).

Parameters

• units (int) – Input dimension to rotate

• m (int) – Lower index to rotate

• n (Optional[int]) – Upper index to rotate, requires \(n > m\) (defaults to None, which implies \(n = m + 1\))

Return type

ndarray

Returns

The Givens rotation matrix

property inverse_matrix

Return type

ndarray

property matrix

Returns: \(U_2\), a \(2 \times 2\) unitary matrix implemented by this component

Return type

ndarray

property reflectivity

Returns: Reflectivity, \(r\), for the matrix \(U\)

Return type

float

property transmissivity

Returns: Transmissivity, \(t\), for the matrix \(U\)

Return type

float

class neurophox.components.transfermatrix.PhaseShiftCommonMode(phase_shift, dtype=<class 'numpy.complex128'>)

Bases: neurophox.components.transfermatrix.PairwiseUnitary

Common mode phase shift operator

Implements the common-mode phase shift operator \(C(\theta)\) given phase \(\theta\):

\[\begin{split}C(\theta) = \begin{bmatrix} e^{i\theta} & 0\\ 0 & e^{i\theta} \end{bmatrix}\end{split}\]

Parameters

• phase_shift (float) – Phase shift \(\theta\)

• dtype – Cast values as dtype for this pairwise unitary operator

property matrix

Returns: \(U_2\), a \(2 \times 2\) unitary matrix implemented by this component

Return type

ndarray

class neurophox.components.transfermatrix.PhaseShiftDifferentialMode(phase_shift, dtype=<class 'numpy.complex128'>)

Bases: neurophox.components.transfermatrix.PairwiseUnitary

Differential phase shift operator

Implements the differential-mode phase shift operator \(D(\theta)\) given phase \(\theta\):

\[\begin{split}D(\theta) = \begin{bmatrix} e^{i\theta / 2} & 0\\ 0 & e^{-i\theta/2} \end{bmatrix}\end{split}\]

Parameters

• phase_shift (float) – Phase shift \(\theta\)

• dtype – Cast values as dtype for this pairwise unitary operator

property matrix

Returns: \(U_2\), a \(2 \times 2\) unitary matrix implemented by this component

Return type

ndarray

class neurophox.components.transfermatrix.PhaseShiftLower(phase_shift, dtype=<class 'numpy.complex128'>)

Bases: neurophox.components.transfermatrix.PairwiseUnitary

Lower phase shift operator

Implements the lower phase shift operator \(R(\theta)\) given phase \(\theta\):

\[\begin{split}R(\theta) = \begin{bmatrix} 1 & 0\\ 0 & e^{i\theta} \end{bmatrix}\end{split}\]

Parameters

• phase_shift (float) – Phase shift \(\theta\)

• dtype – Cast values as dtype for this pairwise unitary operator

property matrix

Returns: \(U_2\), a \(2 \times 2\) unitary matrix implemented by this component

Return type

ndarray

class neurophox.components.transfermatrix.PhaseShiftUpper(phase_shift, dtype=<class 'numpy.complex128'>)

Bases: neurophox.components.transfermatrix.PairwiseUnitary

Upper phase shift operator

Implements the upper phase shift operator \(L(\theta)\) given phase \(\theta\):

\[\begin{split}L(\theta) = \begin{bmatrix} e^{i\theta} & 0\\ 0 & 1 \end{bmatrix}\end{split}\]

Parameters

• phase_shift (float) – Phase shift \(\theta\)

• dtype – Cast values as dtype for this pairwise unitary operator

property matrix

Returns: \(U_2\), a \(2 \times 2\) unitary matrix implemented by this component

Return type

ndarray