Editing Beam splitter (section)

== Quantum mechanical description ==
In quantum mechanics, the electric fields are operators as explained by [[second quantization]] and [[Fock state|Fock states]].  Each electrical field operator can further be expressed in terms of [[Mode (electromagnetism)|modes]] representing the wave behavior and amplitude operators, which are typically represented by the dimensionless [[creation and annihilation operators]]. In this theory, the four ports of the beam splitter are represented by a photon number state <math>|n\rangle</math> and the action of a creation operation is <math>\hat{a}^\dagger|n\rangle=\sqrt{n+1}|n+1\rangle</math>. The following is a simplified version of Ref.<ref name="teich" /> The relation between the classical field amplitudes <math>{E}_{a},{E}_{b}, {E}_{c}</math>, and <math>{E}_{d}</math> produced by the beam splitter is translated into the same relation of the corresponding quantum creation (or annihilation) operators <math>\hat{a}_a^\dagger,\hat{a}_b^\dagger, \hat{a}_c^\dagger</math>, and <math>\hat{a}_d^\dagger</math>, so that

:<math>
\left(\begin{matrix}
\hat{a}_c^\dagger\\
\hat{a}_d^\dagger
\end{matrix}\right)= \tau
\left(\begin{matrix}
\hat{a}_a^\dagger\\
\hat{a}_b^\dagger
\end{matrix}\right)
</math>
where the transfer matrix is given in  [[#Classical lossless beam splitter|classical lossless beam splitter]] section above:

:<math>
\tau=\left(\begin{matrix}
r_{ac} & t_{bc}\\
t_{ad} & r_{bd}
\end{matrix}\right)
=e^{i\phi_0}\left(\begin{matrix}  
\sin\theta e^{i\phi_R} & \cos\theta e^{-i\phi_T} \\   
\cos\theta e^{i\phi_T} & -\sin\theta e^{-i\phi_R} \end{matrix}\right).
</math>

Since <math>\tau</math> is unitary, <math>\tau^{-1}=\tau^\dagger</math>, i.e.
:<math>
\left(\begin{matrix}
\hat{a}_a^\dagger\\
\hat{a}_b^\dagger
\end{matrix}\right)= 
\left(\begin{matrix}
r_{ac}^\ast & t_{ad}^\ast\\
t_{bc}^\ast & r_{bd}^\ast
\end{matrix}\right)
\left(\begin{matrix}
\hat{a}_c^\dagger\\
\hat{a}_d^\dagger
\end{matrix}\right).
</math>

This is equivalent to saying that if we start from the vacuum state <math>|00\rangle_{ab}</math> and add a photon in port ''a'' to produce 
:<math>|\psi_\text{in}\rangle=\hat{a}_a^\dagger|00\rangle_{ab}=|10\rangle_{ab},</math> 
then the beam splitter creates a superposition on the outputs of 
:<math>|\psi_\text{out}\rangle=\left(r_{ac}^\ast\hat{a}_c^\dagger+t_{ad}^\ast\hat{a}_d^\dagger\right)|00\rangle_{cd}=r_{ac}^\ast|10\rangle_{cd}+t_{ad}^\ast|01\rangle_{cd}.</math>

The probabilities for the photon to exit at ports ''c'' and ''d'' are therefore  <math>|r_{ac}|^2</math> and <math>|t_{ad}|^2</math>, as might be expected.


Likewise, for any input state <math>|nm\rangle_{ab}</math> 

:<math>
|\psi_\text{in}\rangle=|nm\rangle_{ab}
=\frac{1}{\sqrt{n!}}\left(\hat{a}_a^\dagger\right)^n\frac{1}{\sqrt{m!}}\left(\hat{a}_b^\dagger\right)^m|00\rangle_{ab}
</math>
and the output is
:<math>
|\psi_\text{out}\rangle
=\frac{1}{\sqrt{n!}}
\left(r_{ac}^\ast\hat{a}_c^\dagger+t_{ad}^\ast\hat{a}_d^\dagger\right)^n
\frac{1}{\sqrt{m!}}
\left(t_{bc}^\ast\hat{a}_c^\dagger+r_{bd}^\ast\hat{a}_d^\dagger\right)^m
|00\rangle_{cd}.
</math>

Using the [[Binomial theorem#Multi-binomial theorem|multi-binomial theorem]], this can be written
:<math>
\begin{align}
|\psi_\text{out}\rangle
&=\frac{1}{\sqrt{n!m!}} \sum_{j=0}^n \sum_{k=0}^m 
\binom{n}{j}
\left(  r_{ac}^\ast \hat{a}_c^\dagger \right)^j \left( t_{ad}^\ast \hat{a}_d^\dagger \right) ^{(n-j)} 
\binom{m}{k}
\left(  t_{bc}^\ast \hat{a}_c^\dagger \right)^k \left( r_{bd}^\ast \hat{a}_d^\dagger \right) ^{(m-k)} 
|00\rangle_{cd}
 \\
&=\frac{1}{\sqrt{n!m!}} \sum_{N=0}^{n+m}  \sum_{j=0}^N 
\binom{n}{j}
r_{ac}^{\ast j}  t_{ad}^{\ast (n-j)} 
\binom{m}{N-j} 
t_{bc}^{\ast (N-j)} r_{bd}^{\ast (m-N+j)} 
\left(\hat{a}_c^\dagger\right)^N
\left( \hat{a}_d^\dagger\right)^{M}|00\rangle_{cd}, \\
&=\frac{1}{\sqrt{n!m!}} \sum_{N=0}^{n+m}  \sum_{j=0}^N 
\binom{n}{j}
\binom{m}{N-j}
r_{ac}^{\ast j}  t_{ad}^{\ast (n-j)} t_{bc}^{\ast (N-j)} r_{bd}^{\ast (m-N+j)} 
\sqrt{N!M!} \quad |N,M\rangle_{cd},\end{align}
</math>
where <math>M=n+m-N</math> and the <math>\tbinom{n}{j}</math> is a binomial coefficient and it is to be understood that the coefficient is zero if <math>j\notin\{ 0,n \}</math> etc. 

The transmission/reflection coefficient factor in the last equation may be written in terms of the reduced parameters that ensure unitarity:

:<math>
r_{ac}^{\ast j}  t_{ad}^{\ast (n-j)} t_{bc}^{\ast (N-j)} r_{bd}^{\ast (m-N+j)}
=(-1)^j\tan^{2j}\theta(-\tan\theta)^{m-N}\cos^{n+m}\theta\exp-i\left[(n+m)(\phi_0+\phi_T)-m(\phi_R+\phi_T)+N(\phi_R-\phi_T)\right].
</math>
where it can be seen that if the beam splitter is 50:50 then <math>\tan\theta=1</math> and the only factor that depends on ''j'' is the <math>(-1)^j</math> term. This factor causes interesting interference cancellations. For example, if <math>n=m</math>  and the beam splitter is 50:50, then
:<math>
\begin{align}
\left(\hat{a}_a^\dagger\right)^n\left(\hat{a}_b^\dagger\right)^m
&\to \left[\hat{a}_a^\dagger\hat{a}_b^\dagger\right]^n \\
&=    \left[\left(r_{ac}^\ast\hat{a}_c^\dagger+t_{ad}^\ast\hat{a}_d^\dagger\right)
            \left(t_{bc}^\ast\hat{a}_c^\dagger+r_{bd}^\ast\hat{a}_d^\dagger\right) \right]^n \\
&=    \left[\frac{e^{-i\phi_0}}{\sqrt{2}}\right]^{2n}
      \left[\left(e^{-i\phi_R}\hat{a}_c^\dagger+e^{-i\phi_T}\hat{a}_d^\dagger\right)
            \left(e^{i\phi_T}\hat{a}_c^\dagger-e^{i\phi_R}\hat{a}_d^\dagger\right) \right]^n \\
&=    \frac{e^{-2in\phi_0}}{2^n}\left[e^{i(\phi_T-\phi_R)} \left(\hat{a}_c^\dagger\right)^2  
            +e^{-i(\phi_T-\phi_R)}\left(\hat{a}_d^\dagger\right)^2 \right]^n
\end{align}
</math>  
where the <math> \hat{a}_c^\dagger \hat{a}_d^\dagger </math> term has cancelled. Therefore the output states always have even numbers of photons in each arm. A famous example of this is the [[Hong–Ou–Mandel effect]], in which the input has <math>n=m=1</math>,  the output is always <math>|20\rangle_{cd}</math> or <math>|02\rangle_{cd}</math>, i.e. the probability of output with a photon in each mode (a coincidence event) is zero. Note that this is true for all types of 50:50 beam splitter irrespective of the details of the phases, and the photons need only be indistinguishable. This contrasts with the classical result, in which equal output in both arms for equal inputs on a 50:50 beam splitter does appear for specific beam splitter phases (e.g. a symmetric beam splitter <math>\phi_0=\phi_T=0,\phi_R=\pi/2</math>), and for other phases where the output goes to one arm (e.g. the dielectric beam splitter <math>\phi_0=\phi_T=\phi_R=0</math>) the output is always in the same arm, not random in either arm as is the case here. From the  [[correspondence principle]] we might expect the  quantum results to tend to the classical one in the limits of large ''n'', but the appearance of large numbers of indistinguishable photons at the input is a non-classical state that does not correspond to a classical field pattern, which instead produces a statistical mixture of different <math>|n,m\rangle</math> known as [[Photon statistics#Poissonian light|Poissonian light]].

Rigorous derivation is given in the Fearn–Loudon 1987 paper<ref>{{cite journal|last1=Fearn|first1=H.|last2=Loudon|first2=R.|date=1987|title=Quantum theory of the lossless beam splitter|journal=Optics Communications|volume=64|pages=485–490|doi=10.1016/0030-4018(87)90275-6|number=6|bibcode=1987OptCo..64..485F }}</ref> and extended in Ref <ref name="teich" /> to include statistical mixtures with the [[density matrix]].

=== Non-symmetric beam-splitter ===
In general, for a non-symmetric beam-splitter, namely a beam-splitter for which the transmission and reflection coefficients are not equal, one can define an angle <math>\theta</math> such that

<math>\begin{cases}
|R| = \sin(\theta)\\
|T| = \cos(\theta)
\end{cases}</math>

where <math>R</math> and <math>T</math> are the reflection and transmission coefficients. Then the unitary operation associated with the beam-splitter is then

<math>
\hat{U}=e^{i\theta\left(\hat{a}_{a}^{\dagger}\hat{a}_{b}+\hat{a}_{a}\hat{a}_{b}^{\dagger}\right)}.
</math>

=== Application for quantum computing ===
In 2000 Knill, Laflamme and Milburn ([[KLM protocol]]) proved that it is possible to create a universal [[quantum computer]] solely with beam splitters, phase shifters, photodetectors and single photon sources. The states that form a qubit in this protocol are the one-photon states of two modes, i.e. the states |01⟩ and |10⟩ in the occupation number representation ([[Fock state]]) of two modes. Using these resources it is possible to implement any single qubit gate and 2-qubit probabilistic gates. The beam splitter is an essential component in this scheme since it is the only one that creates [[Quantum entanglement|entanglement]] between the [[Fock states]].

Similar settings exist for [[continuous-variable quantum information|continuous-variable quantum information processing]]. In fact, it is possible to simulate arbitrary [[Bogoliubov transformation|Gaussian (Bogoliubov) transformations]] of a quantum state of light by means of beam splitters, phase shifters and photodetectors, given [[Squeezed coherent state|two-mode squeezed vacuum states]] are available as a prior resource only (this setting hence shares certain similarities with a Gaussian counterpart of the [[KLM protocol]]).<ref>{{cite journal|last1=Chakhmakhchyan|first1=Levon|last2=Cerf|first2=Nicolas|title= Simulating arbitrary Gaussian circuits with linear optics|journal=Physical Review A|date=2018|volume=98|issue=6 |page=062314|doi=10.1103/PhysRevA.98.062314|arxiv=1803.11534|bibcode=2018PhRvA..98f2314C }}</ref> The building block of this simulation procedure is the fact that a beam splitter is equivalent to a [[Squeezed coherent state#Operator representation|squeezing transformation]] under ''partial'' [[T-symmetry|time reversal]].