Twistor theory

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https://en.wikipedia.org/wiki/Twistor_theory 

In theoretical physics, twistor theory was proposed by Roger Penrose in 1967 as a possible path to quantum gravity and has evolved into a widely studied branch of theoretical and mathematical physics. Penrose's idea was that twistor space should be the basic arena for physics from which space-time itself should emerge. It has led to powerful mathematical tools that have applications to differential and integral geometry, nonlinear differential equations and representation theory, and in physics to general relativity, quantum field theory, and the theory of scattering amplitudes. Twistor theory arose in the context of the rapidly expanding mathematical developments in Einstein's theory of general relativity in the late 1950s and in the 1960s and carries a number of influences from that period. In particular, Roger Penrose has credited Ivor Robinson as an important early influence in the development of twistor theory, through his construction of so-called Robinson congruences.

Overview

Projective twistor space ${\displaystyle \mathbb{P}\mathbb{T}}$ is projective 3-space ${\displaystyle {\mathbb{C}\mathbb{P}}^3}$, the simplest 3-dimensional compact algebraic variety. It has a physical interpretation as the space of massless particles with spin. It is the projectivisation of a 4-dimensional complex vector space, non-projective twistor space ${\displaystyle \mathbb{T}}$, with a Hermitian form of signature (2, 2) and a holomorphic volume form. This can be most naturally understood as the space of chiral (Weyl) spinors for the conformal group ${\displaystyle SO(4,2)/{\mathbb{Z}}_2}$ of Minkowski space; it is the fundamental representation of the spin group ${\displaystyle SU(2,2)}$ of the conformal group. This definition can be extended to arbitrary dimensions except that beyond dimension four, one defines projective twistor space to be the space of projective pure spinors for the conformal group.

In its original form, twistor theory encodes physical fields on Minkowski space in terms of complex analytic objects on twistor space via the Penrose transform. This is especially natural for massless fields of arbitrary spin. In the first instance these are obtained via contour integral formulae in terms of free holomorphic functions on regions in twistor space. The holomorphic twistor functions that give rise to solutions to the massless field equations can be more deeply understood as Čech representatives of analytic cohomology classes on regions in ${\displaystyle \mathbb{P}\mathbb{T}}$. These correspondences have been extended to certain nonlinear fields, including self-dual gravity in Penrose's nonlinear graviton construction and self-dual Yang–Mills fields in the so-called Ward construction; the former gives rise to deformations of the underlying complex structure of regions in ${\displaystyle \mathbb{P}\mathbb{T}}$, and the latter to certain holomorphic vector bundles over regions in ${\displaystyle \mathbb{P}\mathbb{T}}$. These constructions have had wide applications, including inter alia the theory of integrable systems.

The self-duality condition is a major limitation for incorporating the full nonlinearities of physical theories, although it does suffice for Yang–Mills–Higgs monopoles and instantons (see ADHM construction). An early attempt to overcome this restriction was the introduction of ambitwistors by Isenberg, Yasskin and Green, and their superspace extension, super-ambitwistors, by Edward Witten. Ambitwistor space is the space of complexified light rays or massless particles and can be regarded as a complexification or cotangent bundle of the original twistor description. By extending the ambitwistor correspondence to suitably defined formal neighborhoods, Isenberg, Yasskin and Green showed the equivalence between the vanishing of the curvature along such extended null lines and the full Yang–Mills field equations. Witten showed that a further extension, within the framework of super Yang–Mills theory, including fermionic and scalar fields, gave rise, in the case of N = 1 or 2 supersymmetry, to the constraint equations, while for N = 3 (or 4), the vanishing condition for supercurvature along super null lines (super ambitwistors) implied the full set of field equations, including those for the fermionic fields. This was subsequently shown to give a 1-1 equivalence between the null curvature constraint equations and the supersymmetric Yang-Mills field equations. Through dimensional reduction, it may also be deduced from the analogous super-ambitwistor correspondence for 10-dimensional, N = 1 super-Yang–Mills theory.

Twistorial formulae for interactions beyond the self-dual sector also arose in Witten's twistor string theory, which is a quantum theory of holomorphic maps of a Riemann surface into twistor space. This gave rise to the remarkably compact RSV (Roiban, Spradlin and Volovich) formulae for tree-level S-matrices of Yang–Mills theories, but its gravity degrees of freedom gave rise to a version of conformal supergravity limiting its applicability; conformal gravity is an unphysical theory containing ghosts, but its interactions are combined with those of Yang–Mills theory in loop amplitudes calculated via twistor string theory.

Despite its shortcomings, twistor string theory led to rapid developments in the study of scattering amplitudes. One was the so-called MHV formalism loosely based on disconnected strings, but was given a more basic foundation in terms of a twistor action for full Yang–Mills theory in twistor space. Another key development was the introduction of BCFW recursion. This has a natural formulation in twistor space that in turn led to remarkable formulations of scattering amplitudes in terms of Grassmann integral formulae and polytopes. These ideas have evolved more recently into the positive Grassmannian and amplituhedron.

Twistor string theory was extended first by generalising the RSV Yang–Mills amplitude formula, and then by finding the underlying string theory. The extension to gravity was given by Cachazo & Skinner, and formulated as a twistor string theory for maximal supergravity by David Skinner. Analogous formulae were then found in all dimensions by Cachazo, He and Yuan for Yang–Mills theory and gravity and subsequently for a variety of other theories. They were then understood as string theories in ambitwistor space by Mason and Skinner in a general framework that includes the original twistor string and extends to give a number of new models and formulae. As string theories they have the same critical dimensions as conventional string theory; for example the type II supersymmetric versions are critical in ten dimensions and are equivalent to the full field theory of type II supergravities in ten dimensions (this is distinct from conventional string theories that also have a further infinite hierarchy of massive higher spin states that provide an ultraviolet completion). They extend to give formulae for loop amplitudes and can be defined on curved backgrounds.

The twistor correspondence

Denote Minkowski space by ${\displaystyle M}$, with coordinates ${\displaystyle x^a=(t,x,y,z)}$ and Lorentzian metric ${\displaystyle {\eta }_{ab}}$ signature ${\displaystyle (1,3)}$. Introduce 2-component spinor indices ${\displaystyle A=0,1;A^{\prime }=0^{\prime },1^{\prime },}$ and set

${\displaystyle x^{A{A}^{\prime }}=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}t-z& x+iy\\ x-iy& t+z\end{array}\right).}$

Non-projective twistor space ${\displaystyle \mathbb{T}}$ is a four-dimensional complex vector space with coordinates denoted by ${\displaystyle Z^{\alpha }=\left({\omega }^A,{\pi }_{A^{\prime }}\right)}$ where ${\displaystyle {\omega }^A}$ and ${\displaystyle {\pi }_{A^{\prime }}}$ are two constant Weyl spinors. The hermitian form can be expressed by defining a complex conjugation from ${\displaystyle \mathbb{T}}$ to its dual ${\displaystyle {\mathbb{T}}^{\ast }}$ by ${\displaystyle {\overline{Z}}_{\alpha }=\left({\overline{\pi }}_A,{\overline{\omega }}^{A^{\prime }}\right)}$ so that the Hermitian form can be expressed as

${\displaystyle Z^{\alpha }{\overline{Z}}_{\alpha }={\omega }^A{\overline{\pi }}_A+{\overline{\omega }}^{A^{\prime }}{\pi }_{A^{\prime }}.}$

This together with the holomorphic volume form, ${\displaystyle {\epsilon }_{\alpha \beta \gamma \delta }{Z}^{\alpha }d{Z}^{\beta }\wedge d{Z}^{\gamma }\wedge d{Z}^{\delta }}$ is invariant under the group SU(2,2), a quadruple cover of the conformal group C(1,3) of compactified Minkowski spacetime.

Points in Minkowski space are related to subspaces of twistor space through the incidence relation

${\displaystyle {\omega }^A=i{x}^{A{A}^{\prime }}{\pi }_{A^{\prime }}.}$

The incidence relation is preserved under an overall re-scaling of the twistor, so usually one works in projective twistor space ${\displaystyle \mathbb{P}\mathbb{T},}$ which is isomorphic as a complex manifold to ${\displaystyle {\mathbb{C}\mathbb{P}}^3}$. A point ${\displaystyle x\in M}$ thereby determines a line ${\displaystyle {\mathbb{C}\mathbb{P}}^1}$ in ${\displaystyle \mathbb{P}\mathbb{T}}$ parametrised by ${\displaystyle {\pi }_{A^{\prime }}.}$ A twistor ${\displaystyle Z^{\alpha }}$ is easiest understood in space-time for complex values of the coordinates where it defines a totally null two-plane that is self-dual. Take ${\displaystyle x}$ to be real, then if ${\displaystyle Z^{\alpha }{\overline{Z}}_{\alpha }}$ vanishes, then ${\displaystyle x}$ lies on a light ray, whereas if ${\displaystyle Z^{\alpha }{\overline{Z}}_{\alpha }}$ is non-vanishing, there are no solutions, and indeed then ${\displaystyle Z^{\alpha }}$ corresponds to a massless particle with spin that are not localised in real space-time.

Variations

Supertwistors

Supertwistors are a supersymmetric extension of twistors introduced by Alan Ferber in 1978. Non-projective twistor space is extended by fermionic coordinates where ${\displaystyle \mathcal{N}}$ is the number of supersymmetries so that a twistor is now given by ${\displaystyle \left({\omega }^A,{\pi }_{A^{\prime }},{\eta }^i\right),i=1,\dots ,\mathcal{N}}$ with ${\displaystyle {\eta }^i}$ anticommuting. The super conformal group ${\displaystyle SU(2,2|\mathcal{N})}$ naturally acts on this space and a supersymmetric version of the Penrose transform takes cohomology classes on supertwistor space to massless supersymmetric multiplets on super Minkowski space. The ${\displaystyle \mathcal{N}=4}$ case provides the target for Penrose's original twistor string and the ${\displaystyle \mathcal{N}=8}$ case is that for Skinner's supergravity generalisation.

Higher dimensional generalization of the Klein correspondence

A higher dimensional generalization of the Klein correspondence underlying twistor theory, applicable to isotropic subspaces of conformally compactified (complexified) Minkowski space and its super-space extensions, was developed by J. Harnad and S. Shnider.

Hyperkähler manifolds

Hyperkähler manifolds of dimension ${\displaystyle 4k}$ also admit a twistor correspondence with a twistor space of complex dimension ${\displaystyle 2k+1}$.

Palatial twistor theory

The nonlinear graviton construction encodes only anti-self-dual, i.e., left-handed fields. A first step towards the problem of modifying twistor space so as to encode a general gravitational field is the encoding of right-handed fields. Infinitesimally, these are encoded in twistor functions or cohomology classes of homogeneity −6. The task of using such twistor functions in a fully nonlinear way so as to obtain a right-handed nonlinear graviton has been referred to as the (gravitational) googly problem. (The word "googly" is a term used in the game of cricket for a ball bowled with right-handed helicity using the apparent action that would normally give rise to left-handed helicity.) The most recent proposal in this direction by Penrose in 2015 was based on noncommutative geometry on twistor space and referred to as palatial twistor theory. The theory is named after Buckingham Palace, where Michael Atiyah suggested to Penrose the use of a type of "noncommutative algebra", an important component of the theory. (The underlying twistor structure in palatial twistor theory was modeled not on the twistor space but on the non-commutative holomorphic twistor quantum algebra.)

Beam splitter

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Beam_splitter

Schematic illustration of a beam splitter cube.
1 - Incident light
2 - 50% transmitted light
3 - 50% reflected light
In practice, the reflective layer absorbs some light.

Beam splitters

A beam splitter or beamsplitter is an optical device that splits a beam of light into a transmitted and a reflected beam. It is a crucial part of many optical experimental and measurement systems, such as interferometers, also finding widespread application in fibre optic telecommunications.

Designs

In its most common form, a cube, a beam splitter is made from two triangular glass prisms which are glued together at their base using polyester, epoxy, or urethane-based adhesives. (Before these synthetic resins, natural ones were used, e.g. Canada balsam.) The thickness of the resin layer is adjusted such that (for a certain wavelength) half of the light incident through one "port" (i.e., face of the cube) is reflected and the other half is transmitted due to FTIR (frustrated total internal reflection). Polarizing beam splitters, such as the Wollaston prism, use birefringent materials to split light into two beams of orthogonal polarization states.

Aluminium-coated beam splitter.

Another design is the use of a half-silvered mirror. This is composed of an optical substrate, which is often a sheet of glass or plastic, with a partially transparent thin coating of metal. The thin coating can be aluminium deposited from aluminium vapor using a physical vapor deposition method. The thickness of the deposit is controlled so that part (typically half) of the light, which is incident at a 45-degree angle and not absorbed by the coating or substrate material, is transmitted and the remainder is reflected. A very thin half-silvered mirror used in photography is often called a pellicle mirror. To reduce loss of light due to absorption by the reflective coating, so-called "Swiss-cheese" beam-splitter mirrors have been used. Originally, these were sheets of highly polished metal perforated with holes to obtain the desired ratio of reflection to transmission. Later, metal was sputtered onto glass so as to form a discontinuous coating, or small areas of a continuous coating were removed by chemical or mechanical action to produce a very literally "half-silvered" surface.

Instead of a metallic coating, a dichroic optical coating may be used. Depending on its characteristics, the ratio of reflection to transmission will vary as a function of the wavelength of the incident light. Dichroic mirrors are used in some ellipsoidal reflector spotlights to split off unwanted infrared (heat) radiation, and as output couplers in laser construction.

A third version of the beam splitter is a dichroic mirrored prism assembly which uses dichroic optical coatings to divide an incoming light beam into a number of spectrally distinct output beams. Such a device was used in three-pickup-tube color television cameras and the three-strip Technicolor movie camera. It is currently used in modern three-CCD cameras. An optically similar system is used in reverse as a beam-combiner in three-LCD projectors, in which light from three separate monochrome LCD displays is combined into a single full-color image for projection.

Beam splitters with single-mode^{[clarification needed]} fiber for PON networks use the single-mode behavior to split the beam.^{[citation needed]} The splitter is done by physically splicing two fibers "together" as an X.

Arrangements of mirrors or prisms used as camera attachments to photograph stereoscopic image pairs with one lens and one exposure are sometimes called "beam splitters", but that is a misnomer, as they are effectively a pair of periscopes redirecting rays of light which are already non-coincident. In some very uncommon attachments for stereoscopic photography, mirrors or prism blocks similar to beam splitters perform the opposite function, superimposing views of the subject from two different perspectives through color filters to allow the direct production of an anaglyph 3D image, or through rapidly alternating shutters to record sequential field 3D video.

Phase shift

Phase shift through a beam splitter with a dielectric coating.

Beam splitters are sometimes used to recombine beams of light, as in a Mach–Zehnder interferometer. In this case there are two incoming beams, and potentially two outgoing beams. But the amplitudes of the two outgoing beams are the sums of the (complex) amplitudes calculated from each of the incoming beams, and it may result that one of the two outgoing beams has amplitude zero. In order for energy to be conserved (see next section), there must be a phase shift in at least one of the outgoing beams. For example (see red arrows in picture on the right), if a polarized light wave in air hits a dielectric surface such as glass, and the electric field of the light wave is in the plane of the surface, then the reflected wave will have a phase shift of π, while the transmitted wave will not have a phase shift; the blue arrow does not pick up a phase-shift, because it is reflected from a medium with a lower refractive index. The behavior is dictated by the Fresnel equations. This does not apply to partial reflection by conductive (metallic) coatings, where other phase shifts occur in all paths (reflected and transmitted). In any case, the details of the phase shifts depend on the type and geometry of the beam splitter.

Classical lossless beam splitter

For beam splitters with two incoming beams, using a classical, lossless beam splitter with electric fields E_a and E_b each incident at one of the inputs, the two output fields E_c and E_d are linearly related to the inputs through

${\displaystyle {\mathbf{E}}_{\text{out}}=\left[\begin{array}{c}{E}_c\\ E_d\end{array}\right]=\left[\begin{array}{cc}{r}_{ac}& t_{bc}\\ t_{ad}& r_{bd}\end{array}\right]\left[\begin{array}{c}{E}_a\\ E_b\end{array}\right]=\tau {\mathbf{E}}_{\text{in}},}$

where the 2×2 element ${\displaystyle \tau }$ is the beam-splitter transfer matrix and r and t are the reflectance and transmittance along a particular path through the beam splitter, that path being indicated by the subscripts. (The values depend on the polarization of the light.)

If the beam splitter removes no energy from the light beams, the total output energy can be equated with the total input energy, reading

${\displaystyle |E_c{|}^2+|E_d{|}^2=|E_a{|}^2+|E_b{|}^2.}$

Inserting the results from the transfer equation above with ${\displaystyle E_b=0}$ produces

${\displaystyle |r_{ac}{|}^2+|t_{ad}{|}^2=1,}$

and similarly for then ${\displaystyle E_a=0}$

${\displaystyle |r_{bd}{|}^2+|t_{bc}{|}^2=1.}$

When both ${\displaystyle E_a}$ and ${\displaystyle E_b}$ are non-zero, and using these two results we obtain

${\displaystyle r_{ac}{t}_{bc}^{\ast }+t_{ad}{r}_{bd}^{\ast }=0,}$

where "${\displaystyle {}^{\ast }}$" indicates the complex conjugate. It is now easy to show that ${\displaystyle {\tau }^{†}\tau =\mathbf{I}}$ where ${\displaystyle \mathbf{I}}$ is the identity, i.e. the beam-splitter transfer matrix is a unitary matrix.

Expanding, it can be written each r and t as a complex number having an amplitude and phase factor; for instance, ${\displaystyle r_{ac}=|r_{ac}|e^{i{\varphi }_{ac}}}$. The phase factor accounts for possible shifts in phase of a beam as it reflects or transmits at that surface. Then is obtained

${\displaystyle |r_{ac}||t_{bc}|e^{i({\varphi }_{ac}-{\varphi }_{bc})}+|t_{ad}||r_{bd}|e^{i({\varphi }_{ad}-{\varphi }_{bd})}=0.}$

Further simplifying, the relationship becomes

${\displaystyle \frac{|r_{ac}|}{|t_{ad}|}=-\frac{|r_{bd}|}{|t_{bc}|}{e}^{i({\varphi }_{ad}-{\varphi }_{bd}+{\varphi }_{bc}-{\varphi }_{ac})}}$

which is true when ${\displaystyle {\varphi }_{ad}-{\varphi }_{bd}+{\varphi }_{bc}-{\varphi }_{ac}=\pi }$ and the exponential term reduces to -1. Applying this new condition and squaring both sides, it becomes

${\displaystyle \frac{1-|t_{ad}{|}^2}{|t_{ad}{|}^2}=\frac{1-|t_{bc}{|}^2}{|t_{bc}{|}^2},}$

where substitutions of the form ${\displaystyle |r_{ac}{|}^2=1-|t_{ad}{|}^2}$ were made. This leads to the result

${\displaystyle |t_{ad}|=|t_{bc}|\equiv T,}$

and similarly,

${\displaystyle |r_{ac}|=|r_{bd}|\equiv R.}$

It follows that ${\displaystyle R^2+T^2=1}$.

Having determined the constraints describing a lossless beam splitter, the initial expression can be rewritten as

${\displaystyle \left[\begin{array}{c}{E}_c\\ E_d\end{array}\right]=\left[\begin{array}{cc}R{e}^{i{\varphi }_{ac}}& T{e}^{i{\varphi }_{bc}}\\ T{e}^{i{\varphi }_{ad}}& R{e}^{i{\varphi }_{bd}}\end{array}\right]\left[\begin{array}{c}{E}_a\\ E_b\end{array}\right].}$^[2]

Applying different values for the amplitudes and phases can account for many different forms of the beam splitter that can be seen widely used.

The transfer matrix appears to have 6 amplitude and phase parameters, but it also has 2 constraints: ${\displaystyle R^2+T^2=1}$ and ${\displaystyle {\varphi }_{ad}-{\varphi }_{bd}+{\varphi }_{bc}-{\varphi }_{ac}=\pi }$. To include the constraints and simplify to 4 independent parameters, we may write ${\displaystyle {\varphi }_{ad}={\varphi }_0+{\varphi }_T,{\varphi }_{bc}={\varphi }_0-{\varphi }_T,{\varphi }_{ac}={\varphi }_0+{\varphi }_R}$ (and from the constraint ${\displaystyle {\varphi }_{bd}={\varphi }_0-{\varphi }_R-\pi }$), so that

${\displaystyle \begin{array}{rl}{\varphi }_T& =\frac{1}{2}\left({\varphi }_{ad}-{\varphi }_{bc}\right)\\ {\varphi }_R& =\frac{1}{2}\left({\varphi }_{ac}-{\varphi }_{bd}+\pi \right)\\ {\varphi }_0& =\frac{1}{2}\left({\varphi }_{ad}+{\varphi }_{bc}\right)\end{array}}$

where ${\displaystyle 2{\varphi }_T}$ is the phase difference between the transmitted beams and similarly for ${\displaystyle 2{\varphi }_R}$, and ${\displaystyle {\varphi }_0}$ is a global phase. Lastly using the other constraint that ${\displaystyle R^2+T^2=1}$ we define ${\displaystyle \theta =\arctan (R/T)}$ so that ${\displaystyle T=\cos \theta ,R=\sin \theta }$, hence

${\displaystyle \tau =e^{i{\varphi }_0}\left[\begin{array}{cc}\sin \theta e^{i{\varphi }_R}& \cos \theta e^{-i{\varphi }_T}\\ \cos \theta e^{i{\varphi }_T}& -\sin \theta e^{-i{\varphi }_R}\end{array}\right].}$

A 50:50 beam splitter is produced when ${\displaystyle \theta =\pi /4}$. The dielectric beam splitter above, for example, has

${\displaystyle \tau =\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right],}$

i.e. ${\displaystyle {\varphi }_T={\varphi }_R={\varphi }_0=0}$, while the "symmetric" beam splitter of Loudon has

${\displaystyle \tau =\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& i\\ i& 1\end{array}\right],}$

i.e. ${\displaystyle {\varphi }_T=0,{\varphi }_R=-\pi /2,{\varphi }_0=\pi /2}$.

Use in experiments

Beam splitters have been used in both thought experiments and real-world experiments in the area of quantum theory and relativity theory and other fields of physics. These include:

• The Fizeau experiment of 1851 to measure the speeds of light in water
• The Michelson–Morley experiment of 1887 to measure the effect of the (hypothetical) luminiferous aether on the speed of light
• The Hammar experiment of 1935 to refute Dayton Miller's claim of a positive result from repetitions of the Michelson-Morley experiment
• The Kennedy–Thorndike experiment of 1932 to test the independence of the speed of light and the velocity of the measuring apparatus
• Bell test experiments (from ca. 1972) to demonstrate consequences of quantum entanglement and exclude local hidden-variable theories
• Wheeler's delayed choice experiment of 1978, 1984 etc., to test what makes a photon behave as a wave or a particle and when it happens
• The FELIX experiment (proposed in 2000) to test the Penrose interpretation that quantum superposition depends on spacetime curvature
• The Mach–Zehnder interferometer, used in various experiments, including the Elitzur–Vaidman bomb tester involving interaction-free measurement; and in others in the area of quantum computation

Quantum mechanical description

In quantum mechanics, the electric fields are operators as explained by second quantization and Fock states. Each electrical field operator can further be expressed in terms of 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 ${\displaystyle |n⟩}$ and the action of a creation operation is ${\displaystyle {\hat{a}}^{†}|n⟩=\sqrt{n+1}|n+1⟩}$. The following is a simplified version of Ref. The relation between the classical field amplitudes ${\displaystyle E_a,E_b,E_c}$, and ${\displaystyle E_d}$ produced by the beam splitter is translated into the same relation of the corresponding quantum creation (or annihilation) operators ${\displaystyle {\hat{a}}_a^{†},{\hat{a}}_b^{†},{\hat{a}}_c^{†}}$, and ${\displaystyle {\hat{a}}_d^{†}}$, so that

${\displaystyle \left(\begin{array}{c}{\hat{a}}_c^{†}\\ {\hat{a}}_d^{†}\end{array}\right)=\tau \left(\begin{array}{c}{\hat{a}}_a^{†}\\ {\hat{a}}_b^{†}\end{array}\right)}$

where the transfer matrix is given in classical lossless beam splitter section above:

${\displaystyle \tau =\left(\begin{array}{cc}{r}_{ac}& t_{bc}\\ t_{ad}& r_{bd}\end{array}\right)=e^{i{\varphi }_0}\left(\begin{array}{cc}\sin \theta e^{i{\varphi }_R}& \cos \theta e^{-i{\varphi }_T}\\ \cos \theta e^{i{\varphi }_T}& -\sin \theta e^{-i{\varphi }_R}\end{array}\right).}$

Since ${\displaystyle \tau }$ is unitary, ${\displaystyle {\tau }^{-1}={\tau }^{†}}$, i.e.

${\displaystyle \left(\begin{array}{c}{\hat{a}}_a^{†}\\ {\hat{a}}_b^{†}\end{array}\right)=\left(\begin{array}{cc}{r}_{ac}^{\ast }& t_{ad}^{\ast }\\ t_{bc}^{\ast }& r_{bd}^{\ast }\end{array}\right)\left(\begin{array}{c}{\hat{a}}_c^{†}\\ {\hat{a}}_d^{†}\end{array}\right).}$

This is equivalent to saying that if we start from the vacuum state ${\displaystyle |00{⟩}_{ab}}$ and add a photon in port a to produce

${\displaystyle |{\psi }_{\text{in}}⟩={\hat{a}}_a^{†}|00{⟩}_{ab}=|10{⟩}_{ab},}$

then the beam splitter creates a superposition on the outputs of

${\displaystyle |{\psi }_{\text{out}}⟩=\left(r_{ac}^{\ast }{\hat{a}}_c^{†}+t_{ad}^{\ast }{\hat{a}}_d^{†}\right)|00{⟩}_{cd}=r_{ac}^{\ast }|10{⟩}_{cd}+t_{ad}^{\ast }|01{⟩}_{cd}.}$

The probabilities for the photon to exit at ports c and d are therefore ${\displaystyle |r_{ac}{|}^2}$ and ${\displaystyle |t_{ad}{|}^2}$, as might be expected.

Likewise, for any input state ${\displaystyle |nm{⟩}_{ab}}$

${\displaystyle |{\psi }_{\text{in}}⟩=|nm{⟩}_{ab}=\frac{1}{\sqrt{n!}}{\left({\hat{a}}_a^{†}\right)}^n\frac{1}{\sqrt{m!}}{\left({\hat{a}}_b^{†}\right)}^m|00{⟩}_{ab}}$

and the output is

${\displaystyle |{\psi }_{\text{out}}⟩=\frac{1}{\sqrt{n!}}{\left(r_{ac}^{\ast }{\hat{a}}_c^{†}+t_{ad}^{\ast }{\hat{a}}_d^{†}\right)}^n\frac{1}{\sqrt{m!}}{\left(t_{bc}^{\ast }{\hat{a}}_c^{†}+r_{bd}^{\ast }{\hat{a}}_d^{†}\right)}^m|00{⟩}_{cd}.}$

Using the multi-binomial theorem, this can be written

${\displaystyle \begin{array}{rl}|{\psi }_{\text{out}}⟩& =\frac{1}{\sqrt{n!m!}}\sum _{j=0}^n\sum _{k=0}^m(\genfrac{}{}{0ex}{}{n}{j}){\left(r_{ac}^{\ast }{\hat{a}}_c^{†}\right)}^j{\left(t_{ad}^{\ast }{\hat{a}}_d^{†}\right)}^{(n-j)}(\genfrac{}{}{0ex}{}{m}{k}){\left(t_{bc}^{\ast }{\hat{a}}_c^{†}\right)}^k{\left(r_{bd}^{\ast }{\hat{a}}_d^{†}\right)}^{(m-k)}|00{⟩}_{cd}\\ & =\frac{1}{\sqrt{n!m!}}\sum _{N=0}^{n+m}\sum _{j=0}^N(\genfrac{}{}{0ex}{}{n}{j})r_{ac}^{\ast j}{t}_{ad}^{\ast (n-j)}(\genfrac{}{}{0ex}{}{m}{N-j})t_{bc}^{\ast (N-j)}{r}_{bd}^{\ast (m-N+j)}{\left({\hat{a}}_c^{†}\right)}^N{\left({\hat{a}}_d^{†}\right)}^M|00{⟩}_{cd},\\ & =\frac{1}{\sqrt{n!m!}}\sum _{N=0}^{n+m}\sum _{j=0}^N(\genfrac{}{}{0ex}{}{n}{j})(\genfrac{}{}{0ex}{}{m}{N-j})r_{ac}^{\ast j}{t}_{ad}^{\ast (n-j)}{t}_{bc}^{\ast (N-j)}{r}_{bd}^{\ast (m-N+j)}\sqrt{N!M!}|N,M{⟩}_{cd},\end{array}}$

where ${\displaystyle M=n+m-N}$ and the ${\displaystyle (\genfrac{}{}{0ex}{}{n}{j})}$ is a binomial coefficient and it is to be understood that the coefficient is zero if ${\displaystyle j\notin \{0,n\}}$ etc.

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

${\displaystyle r_{ac}^{\ast j}{t}_{ad}^{\ast (n-j)}{t}_{bc}^{\ast (N-j)}{r}_{bd}^{\ast (m-N+j)}=(-1{)}^j{\tan }^{2j}\theta (-\tan \theta {)}^{m-N}{\cos }^{n+m}\theta \exp -i\left[(n+m)({\varphi }_0+{\varphi }_T)-m({\varphi }_R+{\varphi }_T)+N({\varphi }_R-{\varphi }_T)\right].}$

where it can be seen that if the beam splitter is 50:50 then ${\displaystyle \tan \theta =1}$ and the only factor that depends on j is the ${\displaystyle (-1{)}^j}$ term. This factor causes interesting interference cancellations. For example, if ${\displaystyle n=m}$ and the beam splitter is 50:50, then

${\displaystyle \begin{array}{rl}{\left({\hat{a}}_a^{†}\right)}^n{\left({\hat{a}}_b^{†}\right)}^m& \to {\left[{\hat{a}}_a^{†}{\hat{a}}_b^{†}\right]}^n\\ & ={\left[\left(r_{ac}^{\ast }{\hat{a}}_c^{†}+t_{ad}^{\ast }{\hat{a}}_d^{†}\right)\left(t_{bc}^{\ast }{\hat{a}}_c^{†}+r_{bd}^{\ast }{\hat{a}}_d^{†}\right)\right]}^n\\ & ={\left[\frac{e^{-i{\varphi }_0}}{\sqrt{2}}\right]}^{2n}{\left[\left(e^{-i{\varphi }_R}{\hat{a}}_c^{†}+e^{-i{\varphi }_T}{\hat{a}}_d^{†}\right)\left(e^{i{\varphi }_T}{\hat{a}}_c^{†}-e^{i{\varphi }_R}{\hat{a}}_d^{†}\right)\right]}^n\\ & =\frac{e^{-2in{\varphi }_0}}{2^n}{\left[e^{i({\varphi }_T-{\varphi }_R)}{\left({\hat{a}}_c^{†}\right)}^2+e^{-i({\varphi }_T-{\varphi }_R)}{\left({\hat{a}}_d^{†}\right)}^2\right]}^n\end{array}}$

where the ${\displaystyle {\hat{a}}_c^{†}{\hat{a}}_d^{†}}$ 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 ${\displaystyle n=m=1}$, the output is always ${\displaystyle |20{⟩}_{cd}}$ or ${\displaystyle |02{⟩}_{cd}}$, 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 ${\displaystyle {\varphi }_0={\varphi }_T=0,{\varphi }_R=\pi /2}$), and for other phases where the output goes to one arm (e.g. the dielectric beam splitter ${\displaystyle {\varphi }_0={\varphi }_T={\varphi }_R=0}$) 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 ${\displaystyle |n,m⟩}$ known as Poissonian light.

Rigorous derivation is given in the Fearn–Loudon 1987 paper and extended in Ref  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 ${\displaystyle \theta }$ such that

${\displaystyle \{\begin{array}{l}|R|=\sin (\theta )\\ |T|=\cos (\theta )\end{array}}$

where ${\displaystyle R}$ and ${\displaystyle T}$ are the reflection and transmission coefficients. Then the unitary operation associated with the beam-splitter is then

${\displaystyle \hat{U}=e^{i\theta \left({\hat{a}}_a^{†}{\hat{a}}_b+{\hat{a}}_a{\hat{a}}_b^{†}\right)}.}$

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 entanglement between the Fock states.

Similar settings exist for continuous-variable quantum information processing. In fact, it is possible to simulate arbitrary Gaussian (Bogoliubov) transformations of a quantum state of light by means of beam splitters, phase shifters and photodetectors, given 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). The building block of this simulation procedure is the fact that a beam splitter is equivalent to a squeezing transformation under partial time reversal.

Diffractive beam splitter

7x7 matrix using green laser and diffractive beam splitter.

The diffractive beam splitter (also known as multispot beam generator or array beam generator) is a single optical element that divides an input beam into multiple output beams. Each output beam retains the same optical characteristics as the input beam, such as size, polarization and phase. A diffractive beam splitter can generate either a 1-dimensional beam array (1xN) or a 2-dimensional beam matrix (MxN), depending on the diffractive pattern on the element. The diffractive beam splitter is used with monochromatic light such as a laser beam, and is designed for a specific wavelength and angle of separation between output beams.