sf.ops.BSgate¶
-
class
BSgate(theta=pi/4, phi=0.)[source]¶ Bases:
strawberryfields.ops.GateBeamsplitter gate.
\[B(\theta,\phi) = \exp\left(\theta (e^{i \phi} a_1 a_2^\dagger -e^{-i \phi} a_1^\dagger a_2) \right)\]- Parameters
theta (float) – Transmittivity angle \(\theta\). The transmission amplitude of the beamsplitter is \(t = \cos(\theta)\). The value \(\theta=\pi/4\) gives the 50-50 beamsplitter (default).
phi (float) – Phase angle \(\phi\). The reflection amplitude of the beamsplitter is \(r = e^{i\phi}\sin(\theta)\). The value \(\phi = \pi/2\) gives the symmetric beamsplitter.
Details and Conventions
Definition
For the annihilation and creation operators of two modes, denoted \(\a_1\) and \(\a_2\), the beamsplitter is defined by
\[B(\theta,\phi) = \exp\left(\theta (e^{i \phi}\a_1 \ad_2 - e^{-i \phi} \ad_1 \a_2) \right)\]Action on the creation and annihilation operators
They will transform the operators according to
\[\begin{split}B^\dagger(\theta,\phi) \a_1 B(\theta,\phi) &= \a_1\cos \theta -\a_2 e^{-i \phi} \sin \theta = t \a_1 -r^* \a_2,\\ B^\dagger(\theta,\phi) \a_2 B(\theta,\phi) &= \a_2\cos \theta + \a_1 e^{i \phi} \sin \theta= t \a_2 +r \a_1.\end{split}\]where \(t = \cos \theta\) and \(r = e^{i\phi} \sin \theta\) are the transmittivity and reflectivity amplitudes of the beamsplitter respectively.
Therefore, the beamsplitter transforms two input coherent states to two output coherent states \(B(\theta, \phi) \ket{\alpha,\beta} = \ket{\alpha',\beta'}\), where
\[\begin{split}\alpha' &= \alpha\cos \theta-\beta e^{-i\phi}\sin\theta = t\alpha - r^*\beta\\ \beta' &= \beta\cos \theta+\alpha e^{i\phi}\sin\theta = t\beta + r\alpha\\\end{split}\]Action on the quadrature operators
By substituting in the definition of the creation and annihilation operators in terms of the position and momentum operators, it is possible to derive an expression for how the beamsplitter transforms the quadrature operators:
\[\begin{split}&\begin{cases} B^\dagger(\theta,\phi) \x_1 B(\theta,\phi) = \x_1 \cos(\theta)-\sin(\theta) [\x_2\cos(\phi)+\p_2\sin(\phi)]\\ B^\dagger(\theta,\phi) \p_1 B(\theta,\phi) = \p_1 \cos(\theta)-\sin(\theta) [\p_2\cos(\phi)-\x_2\sin(\phi)]\\ \end{cases}\\[12pt] &\begin{cases} B^\dagger(\theta,\phi) \x_2 B(\theta,\phi) = \x_2 \cos(\theta)+\sin(\theta) [\x_1\cos(\phi)-\p_1\sin(\phi)]\\ B^\dagger(\theta,\phi) \p_2 B(\theta,\phi) = \p_2 \cos(\theta)+\sin(\theta) [\p_1\cos(\phi)+\x_1\sin(\phi)] \end{cases}\end{split}\]Action on the position and momentum eigenstates
A 50% or 50-50 beamsplitter has \(\theta=\pi/4\) and \(\phi=0\) or \(\phi=\pi\); consequently \(|t|^2 = |r|^2 = \frac{1}{2}\), and it acts as follows:
\[\begin{split}& B(\pi/4,0)\xket{x_1}\xket{x_2} = \xket{\frac{1}{\sqrt{2}}(x_1-x_2)}\xket{\frac{1}{\sqrt{2}}(x_1+x_2)}\\ & B(\pi/4,0)\ket{p_1}_p\ket{p_2}_p = \xket{\frac{1}{\sqrt{2}}(p_1-p_2)}\xket{\frac{1}{\sqrt{2}}(p_1+p_2)}\end{split}\]and
\[\begin{split}& B(\pi/4,\pi)\xket{x_1}\xket{x_2} = \xket{\frac{1}{\sqrt{2}}(x_1+x_2)}\xket{\frac{1}{\sqrt{2}}(x_2-x_1)}\\ & B(\pi/4,\pi)\ket{p_1}_p\ket{p_2}_p = \xket{\frac{1}{\sqrt{2}}(p_1+p_2)}\xket{\frac{1}{\sqrt{2}}(p_2-p_1)}\end{split}\]Alternatively, symmetric beamsplitter (one that does not distinguish between \(\a_1\) and \(\a_2\)) is obtained by setting \(\phi=\pi/2\).
Attributes
Returns a copy of the gate with the self.dagger flag flipped.
Extra dependencies due to parameters that depend on measurements.
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H¶ Returns a copy of the gate with the self.dagger flag flipped.
H stands for hermitian conjugate.
- Returns
formal inverse of this gate
- Return type
Gate
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measurement_deps¶ Extra dependencies due to parameters that depend on measurements.
- Returns
dependencies
- Return type
set[RegRef]
-
ns= 2¶
Methods
apply(reg, backend, **kwargs)Ask a backend to execute the operation on the current register state right away.
decompose(reg, **kwargs)Decompose the operation into elementary operations supported by the backend API.
merge(other)Merge the operation with another (acting on the exact same set of subsystems).
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apply(reg, backend, **kwargs)¶ Ask a backend to execute the operation on the current register state right away.
Like
Operation.apply(), but takes into account the special nature of p[0] and applies self.dagger.- Returns
Gates do not return anything, return value is None
- Return type
None
-
decompose(reg, **kwargs)¶ Decompose the operation into elementary operations supported by the backend API.
Like
Operation.decompose(), but applies self.dagger.
-
merge(other)¶ Merge the operation with another (acting on the exact same set of subsystems).
Note
For subclass overrides: merge may return a newly created object, or self, or other, but it must never modify self or other because the same Operation objects may be also used elsewhere.
- Parameters
other (Operation) – operation to merge this one with
- Returns
other * self. The return value None represents the identity gate (doing nothing).
- Return type
Operation, None
- Raises
MergeFailure – if the two operations cannot be merged