- Email: [email protected]
Special classes of optical vector vortex beams are Majorana-like photons
Journal Pre-proof Special classes of optical vector vortex beams are Majorana-like photons Sandra Mamani, Daniel A. Nolan, Lingyan Shi, Robert R. Alfano
PII: DOI: Reference:
S0030-4018(20)30098-5 https://doi.org/10.1016/j.optcom.2020.125425 OPTICS 125425
To appear in:
Optics Communications
Received date : 4 August 2019 Revised date : 27 January 2020 Accepted date : 30 January 2020 Please cite this article as: S. Mamani, D.A. Nolan, L. Shi et al., Special classes of optical vector vortex beams are Majorana-like photons, Optics Communications (2020), doi: https://doi.org/10.1016/j.optcom.2020.125425. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.
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Special Classes of Optical Vector Vortex Beams are Majorana-like Photons
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Sandra Mamani1, Daniel A. Nolan2, Lingyan Shi 3, Robert R. Alfano1, *
Institute for Ultrafast Spectroscopy and Lasers, Departments of Physics and electrical Engineering, The City College of the City University of New York, 160 Convent Avenue, New York, NY 10031, USA 2 3
Corning Research and Development Corporation, Sullivan Park, Corning, NY 14830, USA
Department of Bioengineering, University of California San Diego, La Jolla, CA, 92093, USA
Keywords:
Majorana quasi-particle; Boson; Vector vortex; Optical vortex; Orbital angular momentum (OAM); radially polarized; azimuthally polarized
Abstract
Majorana-like photons are introduced in this paper, which are attributed to the polarization
and wavefront of special function class of optical vector vortex beams. A Majorana photon is a
photon that is identical to its anti-photon. It has within itself both chirality, right and left-handed twist in polarization and wavefront. A theory is presented which reveals that certain types of cylindrical vector vortex photons that are spin-orbit coupled beams—radial, and azimuthal
Laguerre-Gaussian, hybrid π-vector beams, and Airy beams—are Majorana-like based on their SAM (polarization) and OAM (wavefront) modes. Majorana-like vector photons may play an important role in free-space fiber communication, propagation, quantum computing, optical
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computing, and imaging in turbid and bio-media as non-separable entangled polarized photons— a quantum mechanical entity.
1. Introduction
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Photons possess novel salient properties such as polarization, wavelength, coherence, speed, and spatial wavefront modes, which play an important role in different active research areas such as condensed matter, transmission, imaging, classical and quantum free optical space, nonlinear optics, self focusing, and fiber communication and information [1-7]. Light can be viewed as a transverse wave. The electric field E can be described by: 𝐸 𝑟, 𝑡
𝐸 𝑟, 𝑡 𝐽 cos 𝑘𝑟
𝜔𝑡 ,
(1)
where k is propagation vector, r is position vector , t is the time , 𝐸 is the field amplitude, and 𝐽 is the polarization vector. Photons can have unique transverse vector pattern properties, which show that certain photons can be separable and non-separable on their wavefront [5]. *Corresponding author: [email protected]
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These spatial non-homogenous vector polarized beams can possess spin angular momentum (SAM) and orbital angular momentum (OAM) coupled mixed states [8,9] in which Jones’ matrices in cylindrical transverse mode patterns can describe the polarization. These polarized
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photons could be involved in both quantum non-local intersystem entanglement processes and classical non-separable local intrasystem entanglement processes at high and low intensity.
In this paper, it is shown theoretically and mathematically that certain cylindrical transverse mode polarized patterns such as vector vortex photons can be their anti-photons defined here as Majorana photons [10]. These Majorana photons are different from the electron and its antielectron–the positron–these pairs are separated and distinct with no charge, like a proton and antiproton. Majorana, the particle and antiparticle are the same. A Majorana-like photon has within itself both directions of time, right and left-handed twists. Most importantly, it follows the main Majorana symbolic characteristic of its state function being equal to its self-transposed, 𝜓 = 𝜓 ∗ in the basis states.
Majorana's solution comes from the negative squared mass problem, the equation that Dirac theorized in 1928, where antiparticles were manifested. Majorana’s work from 1932 [13] and 1937 [14] shows how he was looking at cases of composite systems. He ended up looking at an equation very similar to the Dirac equation for photons. He realized that to solve this problem, bosons would also provide a solution in order to get around it. This solution gave rise to a Tower of Particles, a tower in number that would obey both Bose-Einstein and Fermi-Dirac statistics, which in return would have the precise relationship between spin and mass. This tower or infinite
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spectrum of particles [10] having a one-to-one correspondence between electrons and positrons, proton and anti-proton, became known as the so-called "Majorana Tower." Since this would be something that can be applied to bosons as well, the particles and antiparticles would be distinct from one another. This solution is something that would be only valid for neutral particles, such an elemental particle of photons. Therefore, in this paper, it is recently speculated that a super class of vector photon–a boson–can be a Majorana particle [11,12] based on combination of
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OAM and SAM modes from polarization and spatial wavefront forms, which are generated from various optical components (spatial light modulators, q-plates, and spiral plates). Moreover, Ettore Majorana proposed that Majorana particles could exist not only as fermions but also as bosons composed of particles and antiparticles [10,13-17]. Electrically neutral particles can indeed be their own antiparticles [18]. Some familiar examples are photons (spin 1), neutral pions (spin 0), gravitons (spin 2) and possibly the neutrinos (spin 1/2). However, *Corresponding author: [email protected]
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electrically charged particles are not Majorana because they are different from their antiparticle, as they have opposite charges such as the electron and positron (-e and +e). A Majorana particle involves not only spin angular momentum (SAM) but also orbital angular momentum (OAM)
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coupling as the vector sum of total angular momentum, J =(OAM ℓ) + SAM (𝜎)) ℏ per photon. A notable example is a photon, which has zero rest mass because of the transverse nature of the electric (E) and magnetic (B) fields. If this photon had a longitudinal field, however, then, it may have a small mass (Procar) in order of 10-49 gm [14] and smaller focused spot size. The key Majorana characteristic, the anti-conjugate wave function feature as stated above, is being its antiparticle, where 𝜓 = 𝜓 ∗ , being Hermitian. This Majorana-like of this type is topologically invariant, which means its property remains almost the same regardless of the scattering exchanged and the path taken in the environment as knots and braids of vortex path states.
It is shown here that certain Majorana-like photons possess both chiralities in the form of polarization vector beams. These photons and anti-photons are defined and attributed as Majorana-like boson as they follow the Majorana fingerprint of 𝜓 = 𝜓 ∗ [10] and carry no charge. Remarkably, Majorana showed that Maxwell's equations could be written in the form of this field function, which includes spin one particle in addition to the photon wave equation as manifesting in the form of a Dirac equation. The Electric fields (E) are a key in Quantum Field theory of the E&M field in second quantization theory. Another interesting point discusses that photons with orbital angular momentum could propagate in plasma. This photon interaction with plasmons quasi-particles is very suggestive of what is known as the Majorana Tower.
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In this paper, a theory is presented showing that a super class of polarized transverse vector vortex photon beams patterns are Majorana-like among them are the radial and azimuthal Laguerre-Gaussian, hybrid π-vector beams, and Airy beams, which are consider spin-orbit coupled beams based on OAM and SAM parts of light. This paper also shows that circularly polarized modes are not Majorana. These types of optical beams are known as the solutions to the paraxial Helmholtz wave equation in cylindrical coordinates. Here, the paraxial approximation is
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considered.
2. Theoretical analysis of Majorana and non-Majorana-like states of polarization This section describes mathematically Majorana-like vector Laguerre-Gaussian (LG) beams such as radial (RP) and azimuthal (AP) LG beams. These beams are non-separable (mixed states) in space and polarization. These photon states modes are defined and attributed as Majorana-like
*Corresponding author: [email protected]
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boson with no charge as shown to be 𝜓 = 𝜓 ∗ . This paper also discusses non-Majorana photons such as the circularly polarized LG beam. These Majorana and non-Majorana polarized states are separation of variables technique [19].
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Derived from the paraxial Helmholtz equation in cylindrical coordinates and then using the Laguerre-Gaussian modes are solutions to the paraxial Helmholtz wave equation in cylindrical coordinates (𝑟, 𝜑, 𝑧), which are the radial, azimuthal, and longitudinal components respectively. LG modes have a rotational symmetry along their propagation axis and are known for carrying an intrinsic rotational OAM 𝑖ℏ per photon. The complex electric field amplitude of the LG beam can be written, as in Equation 2 by using the generalized Laguerre polynomials [19]: 𝐿𝐺ℓ, 𝑟, 𝜑, 𝑧 ℓ,
√
𝑒𝑥𝑝
|ℓ|
|ℓ|
𝐿
𝑒𝑥𝑝
𝑖ℓ𝜑 x 𝑒𝑥𝑝
𝑒𝑥 𝑝 𝑖𝜉ℓ, 𝑧 .
(2)
where 𝑟 is the radial distance from the center of the beam’s axis, 𝑘 is the wave number |ℓ|
2𝜋𝑛/𝜆 , where 𝜆 is the wavelength and 𝑛 is the index of refraction. 𝐿
are the generalized
Laguerre polynomials, where ℓ is the azimuthal mode index associated with the helical wave-front of an optical vortex known as topological charge and 𝑝 is the radial mode index 𝑝
0 . These LG polynomials give the beam a vortex-like structure that is described
by a phase 𝑒𝑥𝑝
𝑖ℓ𝜑 , which is the azimuthal phase dependence of an optical vortex beam,
𝜑 is the azimuthal angle in the transverse plane of the beam. The other constants used in
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Equation1 are represented by the following equations: 𝐶ℓ, are the normalized constants given by Equation 3: |ℓ|
𝐶ℓ,
! |ℓ|!
.
(3)
𝑤 𝑧 is the beam width as a function of 𝑧 and is given by Equation 4: 𝑤
1
.
(4)
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𝑤 𝑧
where 𝑧 is the axial distance from the focus point, 𝑤 is the light beam’s waist at 𝑧
0, and 𝑧 is
the Rayleigh length.
𝑅 𝑧 is the radius of curvature of the light beam’s wavefront, which is given by Equation 5: 𝑅 𝑧
𝑧 1
.
(5)
𝜉ℓ, 𝑧 is the Guoy phase of the beam at 𝑧, which is given by Equation 6: *Corresponding author: [email protected]
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𝜉ℓ, 𝑧
|ℓ|
2𝑝
1 tan
.
(6)
Rewriting a simplified version of Equation 2 by taking the azimuthal phase dependence (OAM 𝐿𝐺ℓ 𝜑
𝐴|ℓ |𝑒𝑥𝑝
𝑖ℓ𝜑 ,
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eigenfunction) and spin polarization as a main modes and feature of interest where: (7)
where the radial index is zero (p=0) and A is the radial enveloping amplitude spatial term, which includes the beam width, the beam’s wavefront, Gouy phase, and the Laguerre polynomials .
Equation 8 represents a Laguerre-Gaussian (LG) beam in bra-ket notation with polarization dependence such as with right (RCP) and left (LCP) circular polarization, which are written as states of right-handed (RH) and left-handed (LH): |𝑅𝐶𝑃 ⟩
and |𝐿𝐶𝑃 ⟩
𝐿𝐺ℓ 𝜑
where |𝑅𝐻⟩
and |𝐿𝐻⟩
√
√
𝐿𝐺
ℓ,
𝜑
(8)
are represented using Jones vector notation.
Equation 9 and 10 represent a Laguerre-Gaussian (LG) beam with radial and azimuthal
polarization dependence. These two equations are superposition solutions to the paraxial Helmholtz wave equation and are written in Ket notation as follow: |𝜓
𝐿𝐺ℓ 𝜑
𝐿𝐺
ℓ
𝜑
|𝜓
𝐿𝐺ℓ 𝜑
𝐿𝐺
ℓ
𝜑
(9)
(10)
Rewriting Equation 8 through 10 in terms of the phase and polarization dependence of an optical
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scalar and vector vortex beam, using Dirac bra-ket state notation for basis polarization and wavefront modes [6,20,21]: |𝑅𝐶𝑃 ⟩
𝐴𝑒
ℓ
|𝑅𝐻 and |𝐿𝐶𝑃 ⟩
𝐴𝑒
ℓ
11
|𝐿𝐻 ,
Equation 11 represents pure circularly polarized states RCP and LCP. By taking the complex conjugate of this equation gives the following (Eq. 12): |𝐿𝐶𝑃 ⟩ and |𝐿𝐶𝑃 ⟩∗
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|𝑅𝐶𝑃 ⟩∗
12
|𝑅𝐶𝑃 ⟩,
which shows that RCP and LCP as pure states are not Majorana-like photons. The polarized basis modes in Equation 13 and 14 show the salient feature of a Laguerre-Gaussian beam with radial and azimuthal polarization (non-separable states) with spatial wavefront and polarization by using the wave Dirac Ket notation function: |𝜓
√
𝑒
ℓ
|𝑅𝐻⟩
𝑒
ℓ
13
|𝐿𝐻⟩ .
*Corresponding author: [email protected]
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|𝜓
𝑒
√
ℓ
|𝑅𝐻⟩
ℓ
𝑒
|𝐿𝐻⟩ .
(14)
The ket notation in Eq. 13 and Eq. 14 show that radial and azimuthal polarizations modes are
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locally non-separable states. Both vector beams show to be on a superposition of two orthogonally components, |𝑅𝐻⟩ and |𝐿𝐻⟩ circularly polarized states of ℓ. In the azimuthal
polarization case, the rotation phase has an extra π/2 phase shift; it is represented by the 𝑖 in front. These two equations are also known as cylindrical transverse modes or higher-order transverse
modes, which can be geometrically represented by using a higher-order Poincaré sphere (HOPS) as shown by Milione et al [8].
Equation 13 is an analog representation of the transverse electric mode (TE01), while Equation 14 is an analog representation of the transverse magnetic mode (TM01) as shown by Ndagano et al
[6]. These types of modes occur in optical fibers and in optical resonators. In addition, other types of fiber waveguide modes are the hybrid modes (HE21) known as 𝜋-vector beams [6,8], which have non-zero electric and magnetic fields in the direction of propagation. They are also
represented through a HOPS, which describes various polarizations of optical fiber waveguides [8]. These hybrid-polarized modes are mathematically represented through Equation 15 and Equation 16 as OAM modes [6]: |HE⟩
√
|HE⟩
𝑒
ℓ ℓ
𝑒
√
|𝑅𝐻⟩
𝑒
|𝑅𝐻⟩
ℓ
𝑒
|𝐿𝐻⟩ ℓ
(15)
|𝐿𝐻⟩
.
(16)
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Taking the complex conjugate of Equations 13 and 14, it is observed that a LaguerreGaussian beam with radial or azimuthal polarization is a Majorana-like photon, which is
represented by Equations 17 and 18. These equations show that these vector beams are in an optical spin-orbit state (mixed state of OAM and CP): ∗
and |𝜓
𝑒
√
∗
ℓ
|𝐿𝐻⟩
𝑒
ℓ
|𝑅𝐻⟩
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|𝜓
ℓ
𝑒
√
|𝐿𝐻⟩
𝑒
ℓ
|𝑅𝐻⟩
|𝜓
;
17
|𝜓
.
(18)
Moreover, taking the complex conjugate of the 𝜋-vector beams (HE21), which are optical fiber modes, it is shown that these two equations have indeed a main feature of a Majorana-like photon as shown in Equation 19 and Equation 20: |HE⟩∗
√
𝑒
ℓ
|𝐿𝐻⟩
𝑒
ℓ
|𝑅𝐻⟩
*Corresponding author: [email protected]
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(19)
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|HE⟩∗
𝑒
√
ℓ
|𝐿𝐻⟩
𝑒
ℓ
|𝑅𝐻⟩ .
(20)
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Therefore, vector beams (radial and azimuthal) and hybrid 𝜋-vector beams from Equations 13 through and 16, are defined and attributed as Majorana-like photons from the basis modes,
following the Majorana feature of 𝜓 = 𝜓 ∗ as shown below in Equation 21 and Equation 22: ∗
|𝜓 |HE⟩∗
|𝜓
; |𝜓
∗
|𝜓
; |HE⟩∗
|HE⟩
,
|HE⟩
and
(21)
.
(22)
Figure.1 Representation of a Radially polarized Majorana-like LG beam
Figure 1 is it a cartoon representation of a Majorana-like beam where it is observed the main
feature of a Majorana is composed of a coupled SAM and OAM generating a vector vortex beam with radial polarization.
3. Other special function with polarization and wavefront is Majorana-like
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complex vector vortex beam
Other spatially structured light beams such as Airy beams (AiB)[22] follow the Majorana fingerprint of 𝜓 = 𝜓 ∗ , base modes where the photon and the anti-photon are identical. This type of complex beams is also a solution to the paraxial Helmholtz wave equation expressed in different coordinate systems. Besides, this beam has a special property of non-diffracting and
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self-healing when propagating through an obstruction. Hence, Airy beams have many applications such as in imaging, atmospheric turbulence propagation, and particle transportation and manipulation among others.
The following represents the Airy radially and azimuthally polarized beams based on modes introduced by Wei et al. [22], where sending a Gaussian beam through a LC q-Airy-plate generates vector Airy beams. As a result the following Electric fields are obtained [22]: 𝐸
𝑅𝑃
𝑒
𝑒
𝑒
*Corresponding author: [email protected]
𝑒
𝑒
𝑒 7
,
(23)
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𝐸
𝐴𝑃
𝑒
𝑒
𝑒
ℓ
𝑒
𝑒
𝑒
,
(24)
Equations 23 and 24 represent vector Airy beams written in terms of Jones vector of RCP and LCP, a helical phase front of 𝑒
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𝑦
exp 𝑖 𝑥
, and with the characteristic Airy cubic phase modulation of
, where q is half of the topological charge ℓ ,and 𝜃 is the angle between the
incident polarization direction and the x-axis.
Rewriting Equation 23 and 24 in a simplified manner for the polarization basis modes by using the wave function: |𝜓
√
|𝜓
√
𝑒
ℓ
𝑒
𝑒
|𝑅𝐻
𝑒
ℓ
𝑒
𝑒
|𝑅𝐻⟩
𝑒
ℓ
𝑒
𝑒 |𝐿𝐻⟩ ,
(25)
𝑒
ℓ
𝑒
𝑒 |𝐿𝐻 ,
(26)
where 2q is replaced by the OAM value is ℓ and the RCP and LCP are represented in terms of RH and LH states.
Taking the complex conjugate of Equation 25 and 26, proves that indeed these types of vector vortex polarization beams are Majorana-like photons as shown below in Equation 27 and 28: ∗
|𝜓
√ ∗
|𝜓
√
𝑒
ℓ
𝑒
𝑒 |𝐿𝐻
𝑒
ℓ
𝑒
𝑒
|𝑅𝐻⟩
𝑒
ℓ
𝑒
𝑒 |𝐿𝐻
𝑒
ℓ
𝑒
𝑒
|𝑅𝐻
|𝜓
|𝜓
,
,
(27) (28)
However, not all radially and azimuthally polarized optical beams are Majorana-like; such is the
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example of the Bessel beam [23], where the longitudinal component does not contribute for it to be its own conjugate unless at z=0 (absence of longitudinal component). Hence Bessel beams may lack the necessary condition.
In quantum field theory (QFT), the second quantization formalism in terms of the harmonic creation (𝑎 ) and annihilation (𝑎) operators is useful for the analysis and characterization of the
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quantum optics properties and can be used to describe features of the Majorana vector photons. The Majorana boson creation (𝑏 ) and annihilation (𝑏) operators can be written in terms of harmonic operators (𝑎 ) and (𝑎), where 𝑏 = (𝑎 + 𝑎) /√2 and 𝑏 = (𝑎+ 𝑎 )/√2 [17]. Using the Majorana boson creation (𝑏 ) and annihilation (𝑏) operators, the Majorana boson operator can be represented as follows (Eq. 29): 𝐶
𝑏
𝑏
29
𝐶 ,
*Corresponding author: [email protected]
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where the wave function of the quantum field is its anti-conjugate. Furthermore, the quantization of a system of bosons is implemented using Fock space including Fock states following Majorana fermions [17]. In Fock space, the creation operators used in Equation 30 are used to quantify the
|𝑛⟩
𝑎
^𝑛 /√ 𝑛! |0⟩,
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nth photon state as referenced to the vacuum state. The state |𝑛⟩ is given by:
30
where |0⟩is the vacuum state and n is the number of photons in the state. The set of |𝑛⟩ states are orthogonal to one another meaning the vacuum set combinations to represent the Majorana
bosons have a totally zero inner product [17], and is represented as follows (Eq. 31 and 32): 𝑎|𝑛⟩
√ 𝑛 |𝑛
31
1⟩,
and 𝑎 |𝑛⟩ √ n 1 |𝑛
1⟩.
32
The set of n radially polarized vector photons can be quantized using this formalism and a
separate set can be used to quantize the azimuthally polarized vector photons as another feature of Majorana boson pair. As stated by Nielsen and Ninomiya et al [17] Equations 29 through 32 are part of the Majorana theory, which is extended for Bosons in which a photon is its own antiphoton following the key Majorana fingerprint 𝜓 = 𝜓 ∗ .
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4. Possible applications of Majorana-photon like vector beams in Topological Quantum Computing
The Majorana-like vector beams represented previously in Section 2 are known as the four vector basis states of information [6]. These are more likely to have a topological effect in optical quantum computers due to their property of non-separability. That means their space and
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polarization degree of freedom cannot be written in a Cartesian product. In addition, it known through vector beam modulation that it is possible to obtain 𝑁 simultaneously, which helps to encode 𝑙𝑜𝑔𝑁 bit of information with 𝑁
4 possible states per pulse
2 bits of information per ℓ value, rather than 1
1 possible states as it happens with scalar beams [5].
Moreover, as shown by Kauffman et al [24], Majorana-photon like vector beams can form knots and braids, which are similar to the DNA structure modes. This study has shown that the Majorana operators form a robust representation of the braid group, which is related to knot theory. Although this application is applied to Majorana Fermions, Majorana bosons such as the *Corresponding author: [email protected]
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photon can also follow a similar trend since we previously proved that a Majorana boson like the photon is its own antiparticle, 𝜓 = 𝜓 ∗ . These Majorana quasi-particles are composed of natural braiding operators [24]. They are also used as qubits since they are intrinsically immune to
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decoherence [25].
Besides that, Leach and O’Holleran et al [26,27] have shown that optical vortices could be
combined to form loops, links or knots embedded in the light wave. Optical vortices’ topology is believed to have a nodal line, which may be indefinitely long or form a closed loop. Although the longer the length might affect other vortex lines.
In general, the inhomogeneous polarization and spin-orbit coupling characteristics of a vector beam could be an advantage when forming braids in 3D space, which represent logic gates in a computer. This type of optical quantum computer-based on knots and braids are more likely to conserve stability because a perturbation such as environmental will not change the topology of these knots and braids meaning they will retain input characteristics. A challenge ahead is to develop a method to use the Majorana and OAM photon vortex as braids and knots for topological computation for future computers.
5. Conclusion
In summary, Majorana-like photons arise from radial and zimuthal Laguerre-Gaussian, Airy beams, and 𝝅-vector beams. These belong to a super class of special functions and are
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characterized as cylindrical polarized transverse and hybrid vector vortex photons. These special vector beams are known as eigenmodes of weakly guiding optical fibers. They have identical anti-conjugate state and complex transpose (𝝍 = 𝝍∗ ) in polarization vector, which is not the case for a circularly polarized beam. These Majorana photons–the photons and the anti-photons–are identical, have a zero rest mass and are chargeless. These photons have within themselves locally both right and left-handed chirality and OAM (+L) and (-L). The study of Majorana-like vector
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beams (non-separable states) can play an important role in potentially improving imaging with a higher flux of photons in the classical limits, in contrast to quantum entanglement, which uses a single low photon number non-locally [3]. In addition, various types of polarization and OAM (𝓵) [28] in different parts of a scattering or turbid medium could give interesting results to compare, where photons could interact with different quasi-particles [1,29,30] while transferring charge and ions and forming new types of polaritons about resonances from quasi-particles [30]. Recently Mamani et al. [12], found that these Majorana-vector beams can increase the transmission when traveling in turbid media due to their unique characteristics of inhomogeneity *Corresponding author: [email protected]
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and non-separability. Experimental work with these Majorana beams to improve deep penetration imaging in bio- and condensed media should be undertaken. In general, these special nonseparable Majorana-like photon modes, which form knots and braids from vortex paths, may be at
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the heart of the future optical and topological quantum computers, communication and imaging applications as Majorana quasi photons could increase transmission [12], transfer, and store more information [5] with less scattering and possibly with less environmental interference, maintaining the quantum coherence or just creating small perturbations. Other applications for Majorana-like photons include high-resolution multi photon microscopies (SRS, 2/3 PEF, SHG among others) due to the stronger longitudinal field obtained from the Majorana-radially polarized light [31], fiber optics sensing, free-space communication, turbulence effects [32], and hidden optical transitions in a condensed matter such as in photoexcited oriented carriers in semiconductors such as GaAs in conduction bands [33]. Also, a Majorana can play a key role in spatial collapse in Kerr medium, in multiphoton, and stimulated Raman imaging in bio and condensed media [34,35].
Funding. This work is supported in part by Corning Incorporated Foundation (71198-00 12) and partially by ARO (47351-00-01).
Acknowledgment. We thank Fabrizio Tamburini, David Schmeltzer, Pouyan Ghaemi, Alexander Poltorak, David Grier, Larry Jacobowitz, and Michio Kaku for helpful discussions. We thank the referees for their
Competing Interests
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help in clarifying and improving this manuscript.
The authors declare no competing interests.
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References
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*Corresponding author: [email protected]
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18. F. Wilczek, Majorana returns, Nature Physics 5 (2009) 614-618.
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*Corresponding author: [email protected]
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Washington, DC, 2019), paper JTu4A.120
*Corresponding author: [email protected]
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Author Statement File: Robert R. Alfano: Conceptualization, Proposing project, Project administration, Supervision, Writing- Original Draft, Writing-Review & Editing, Methodology, Validation; Sandra Mamani: Investigation, Experiments, Writing- Original Draft, Writing-Review & Editing, Visualization; Daniel A. Nolan: Writing-Review & Editing, Discussions; Lingyan Shi: Discussions, Writing-Review & Editing.
PII: DOI: Reference:
S0030-4018(20)30098-5 https://doi.org/10.1016/j.optcom.2020.125425 OPTICS 125425
To appear in:
Optics Communications
Received date : 4 August 2019 Revised date : 27 January 2020 Accepted date : 30 January 2020 Please cite this article as: S. Mamani, D.A. Nolan, L. Shi et al., Special classes of optical vector vortex beams are Majorana-like photons, Optics Communications (2020), doi: https://doi.org/10.1016/j.optcom.2020.125425. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.
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Special Classes of Optical Vector Vortex Beams are Majorana-like Photons
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Sandra Mamani1, Daniel A. Nolan2, Lingyan Shi 3, Robert R. Alfano1, *
Institute for Ultrafast Spectroscopy and Lasers, Departments of Physics and electrical Engineering, The City College of the City University of New York, 160 Convent Avenue, New York, NY 10031, USA 2 3
Corning Research and Development Corporation, Sullivan Park, Corning, NY 14830, USA
Department of Bioengineering, University of California San Diego, La Jolla, CA, 92093, USA
Keywords:
Majorana quasi-particle; Boson; Vector vortex; Optical vortex; Orbital angular momentum (OAM); radially polarized; azimuthally polarized
Abstract
Majorana-like photons are introduced in this paper, which are attributed to the polarization
and wavefront of special function class of optical vector vortex beams. A Majorana photon is a
photon that is identical to its anti-photon. It has within itself both chirality, right and left-handed twist in polarization and wavefront. A theory is presented which reveals that certain types of cylindrical vector vortex photons that are spin-orbit coupled beams—radial, and azimuthal
Laguerre-Gaussian, hybrid π-vector beams, and Airy beams—are Majorana-like based on their SAM (polarization) and OAM (wavefront) modes. Majorana-like vector photons may play an important role in free-space fiber communication, propagation, quantum computing, optical
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computing, and imaging in turbid and bio-media as non-separable entangled polarized photons— a quantum mechanical entity.
1. Introduction
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Photons possess novel salient properties such as polarization, wavelength, coherence, speed, and spatial wavefront modes, which play an important role in different active research areas such as condensed matter, transmission, imaging, classical and quantum free optical space, nonlinear optics, self focusing, and fiber communication and information [1-7]. Light can be viewed as a transverse wave. The electric field E can be described by: 𝐸 𝑟, 𝑡
𝐸 𝑟, 𝑡 𝐽 cos 𝑘𝑟
𝜔𝑡 ,
(1)
where k is propagation vector, r is position vector , t is the time , 𝐸 is the field amplitude, and 𝐽 is the polarization vector. Photons can have unique transverse vector pattern properties, which show that certain photons can be separable and non-separable on their wavefront [5]. *Corresponding author: [email protected]
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These spatial non-homogenous vector polarized beams can possess spin angular momentum (SAM) and orbital angular momentum (OAM) coupled mixed states [8,9] in which Jones’ matrices in cylindrical transverse mode patterns can describe the polarization. These polarized
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photons could be involved in both quantum non-local intersystem entanglement processes and classical non-separable local intrasystem entanglement processes at high and low intensity.
In this paper, it is shown theoretically and mathematically that certain cylindrical transverse mode polarized patterns such as vector vortex photons can be their anti-photons defined here as Majorana photons [10]. These Majorana photons are different from the electron and its antielectron–the positron–these pairs are separated and distinct with no charge, like a proton and antiproton. Majorana, the particle and antiparticle are the same. A Majorana-like photon has within itself both directions of time, right and left-handed twists. Most importantly, it follows the main Majorana symbolic characteristic of its state function being equal to its self-transposed, 𝜓 = 𝜓 ∗ in the basis states.
Majorana's solution comes from the negative squared mass problem, the equation that Dirac theorized in 1928, where antiparticles were manifested. Majorana’s work from 1932 [13] and 1937 [14] shows how he was looking at cases of composite systems. He ended up looking at an equation very similar to the Dirac equation for photons. He realized that to solve this problem, bosons would also provide a solution in order to get around it. This solution gave rise to a Tower of Particles, a tower in number that would obey both Bose-Einstein and Fermi-Dirac statistics, which in return would have the precise relationship between spin and mass. This tower or infinite
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spectrum of particles [10] having a one-to-one correspondence between electrons and positrons, proton and anti-proton, became known as the so-called "Majorana Tower." Since this would be something that can be applied to bosons as well, the particles and antiparticles would be distinct from one another. This solution is something that would be only valid for neutral particles, such an elemental particle of photons. Therefore, in this paper, it is recently speculated that a super class of vector photon–a boson–can be a Majorana particle [11,12] based on combination of
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OAM and SAM modes from polarization and spatial wavefront forms, which are generated from various optical components (spatial light modulators, q-plates, and spiral plates). Moreover, Ettore Majorana proposed that Majorana particles could exist not only as fermions but also as bosons composed of particles and antiparticles [10,13-17]. Electrically neutral particles can indeed be their own antiparticles [18]. Some familiar examples are photons (spin 1), neutral pions (spin 0), gravitons (spin 2) and possibly the neutrinos (spin 1/2). However, *Corresponding author: [email protected]
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electrically charged particles are not Majorana because they are different from their antiparticle, as they have opposite charges such as the electron and positron (-e and +e). A Majorana particle involves not only spin angular momentum (SAM) but also orbital angular momentum (OAM)
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coupling as the vector sum of total angular momentum, J =(OAM ℓ) + SAM (𝜎)) ℏ per photon. A notable example is a photon, which has zero rest mass because of the transverse nature of the electric (E) and magnetic (B) fields. If this photon had a longitudinal field, however, then, it may have a small mass (Procar) in order of 10-49 gm [14] and smaller focused spot size. The key Majorana characteristic, the anti-conjugate wave function feature as stated above, is being its antiparticle, where 𝜓 = 𝜓 ∗ , being Hermitian. This Majorana-like of this type is topologically invariant, which means its property remains almost the same regardless of the scattering exchanged and the path taken in the environment as knots and braids of vortex path states.
It is shown here that certain Majorana-like photons possess both chiralities in the form of polarization vector beams. These photons and anti-photons are defined and attributed as Majorana-like boson as they follow the Majorana fingerprint of 𝜓 = 𝜓 ∗ [10] and carry no charge. Remarkably, Majorana showed that Maxwell's equations could be written in the form of this field function, which includes spin one particle in addition to the photon wave equation as manifesting in the form of a Dirac equation. The Electric fields (E) are a key in Quantum Field theory of the E&M field in second quantization theory. Another interesting point discusses that photons with orbital angular momentum could propagate in plasma. This photon interaction with plasmons quasi-particles is very suggestive of what is known as the Majorana Tower.
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In this paper, a theory is presented showing that a super class of polarized transverse vector vortex photon beams patterns are Majorana-like among them are the radial and azimuthal Laguerre-Gaussian, hybrid π-vector beams, and Airy beams, which are consider spin-orbit coupled beams based on OAM and SAM parts of light. This paper also shows that circularly polarized modes are not Majorana. These types of optical beams are known as the solutions to the paraxial Helmholtz wave equation in cylindrical coordinates. Here, the paraxial approximation is
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considered.
2. Theoretical analysis of Majorana and non-Majorana-like states of polarization This section describes mathematically Majorana-like vector Laguerre-Gaussian (LG) beams such as radial (RP) and azimuthal (AP) LG beams. These beams are non-separable (mixed states) in space and polarization. These photon states modes are defined and attributed as Majorana-like
*Corresponding author: [email protected]
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boson with no charge as shown to be 𝜓 = 𝜓 ∗ . This paper also discusses non-Majorana photons such as the circularly polarized LG beam. These Majorana and non-Majorana polarized states are separation of variables technique [19].
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Derived from the paraxial Helmholtz equation in cylindrical coordinates and then using the Laguerre-Gaussian modes are solutions to the paraxial Helmholtz wave equation in cylindrical coordinates (𝑟, 𝜑, 𝑧), which are the radial, azimuthal, and longitudinal components respectively. LG modes have a rotational symmetry along their propagation axis and are known for carrying an intrinsic rotational OAM 𝑖ℏ per photon. The complex electric field amplitude of the LG beam can be written, as in Equation 2 by using the generalized Laguerre polynomials [19]: 𝐿𝐺ℓ, 𝑟, 𝜑, 𝑧 ℓ,
√
𝑒𝑥𝑝
|ℓ|
|ℓ|
𝐿
𝑒𝑥𝑝
𝑖ℓ𝜑 x 𝑒𝑥𝑝
𝑒𝑥 𝑝 𝑖𝜉ℓ, 𝑧 .
(2)
where 𝑟 is the radial distance from the center of the beam’s axis, 𝑘 is the wave number |ℓ|
2𝜋𝑛/𝜆 , where 𝜆 is the wavelength and 𝑛 is the index of refraction. 𝐿
are the generalized
Laguerre polynomials, where ℓ is the azimuthal mode index associated with the helical wave-front of an optical vortex known as topological charge and 𝑝 is the radial mode index 𝑝
0 . These LG polynomials give the beam a vortex-like structure that is described
by a phase 𝑒𝑥𝑝
𝑖ℓ𝜑 , which is the azimuthal phase dependence of an optical vortex beam,
𝜑 is the azimuthal angle in the transverse plane of the beam. The other constants used in
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Equation1 are represented by the following equations: 𝐶ℓ, are the normalized constants given by Equation 3: |ℓ|
𝐶ℓ,
! |ℓ|!
.
(3)
𝑤 𝑧 is the beam width as a function of 𝑧 and is given by Equation 4: 𝑤
1
.
(4)
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𝑤 𝑧
where 𝑧 is the axial distance from the focus point, 𝑤 is the light beam’s waist at 𝑧
0, and 𝑧 is
the Rayleigh length.
𝑅 𝑧 is the radius of curvature of the light beam’s wavefront, which is given by Equation 5: 𝑅 𝑧
𝑧 1
.
(5)
𝜉ℓ, 𝑧 is the Guoy phase of the beam at 𝑧, which is given by Equation 6: *Corresponding author: [email protected]
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𝜉ℓ, 𝑧
|ℓ|
2𝑝
1 tan
.
(6)
Rewriting a simplified version of Equation 2 by taking the azimuthal phase dependence (OAM 𝐿𝐺ℓ 𝜑
𝐴|ℓ |𝑒𝑥𝑝
𝑖ℓ𝜑 ,
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eigenfunction) and spin polarization as a main modes and feature of interest where: (7)
where the radial index is zero (p=0) and A is the radial enveloping amplitude spatial term, which includes the beam width, the beam’s wavefront, Gouy phase, and the Laguerre polynomials .
Equation 8 represents a Laguerre-Gaussian (LG) beam in bra-ket notation with polarization dependence such as with right (RCP) and left (LCP) circular polarization, which are written as states of right-handed (RH) and left-handed (LH): |𝑅𝐶𝑃 ⟩
and |𝐿𝐶𝑃 ⟩
𝐿𝐺ℓ 𝜑
where |𝑅𝐻⟩
and |𝐿𝐻⟩
√
√
𝐿𝐺
ℓ,
𝜑
(8)
are represented using Jones vector notation.
Equation 9 and 10 represent a Laguerre-Gaussian (LG) beam with radial and azimuthal
polarization dependence. These two equations are superposition solutions to the paraxial Helmholtz wave equation and are written in Ket notation as follow: |𝜓
𝐿𝐺ℓ 𝜑
𝐿𝐺
ℓ
𝜑
|𝜓
𝐿𝐺ℓ 𝜑
𝐿𝐺
ℓ
𝜑
(9)
(10)
Rewriting Equation 8 through 10 in terms of the phase and polarization dependence of an optical
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scalar and vector vortex beam, using Dirac bra-ket state notation for basis polarization and wavefront modes [6,20,21]: |𝑅𝐶𝑃 ⟩
𝐴𝑒
ℓ
|𝑅𝐻 and |𝐿𝐶𝑃 ⟩
𝐴𝑒
ℓ
11
|𝐿𝐻 ,
Equation 11 represents pure circularly polarized states RCP and LCP. By taking the complex conjugate of this equation gives the following (Eq. 12): |𝐿𝐶𝑃 ⟩ and |𝐿𝐶𝑃 ⟩∗
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|𝑅𝐶𝑃 ⟩∗
12
|𝑅𝐶𝑃 ⟩,
which shows that RCP and LCP as pure states are not Majorana-like photons. The polarized basis modes in Equation 13 and 14 show the salient feature of a Laguerre-Gaussian beam with radial and azimuthal polarization (non-separable states) with spatial wavefront and polarization by using the wave Dirac Ket notation function: |𝜓
√
𝑒
ℓ
|𝑅𝐻⟩
𝑒
ℓ
13
|𝐿𝐻⟩ .
*Corresponding author: [email protected]
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|𝜓
𝑒
√
ℓ
|𝑅𝐻⟩
ℓ
𝑒
|𝐿𝐻⟩ .
(14)
The ket notation in Eq. 13 and Eq. 14 show that radial and azimuthal polarizations modes are
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locally non-separable states. Both vector beams show to be on a superposition of two orthogonally components, |𝑅𝐻⟩ and |𝐿𝐻⟩ circularly polarized states of ℓ. In the azimuthal
polarization case, the rotation phase has an extra π/2 phase shift; it is represented by the 𝑖 in front. These two equations are also known as cylindrical transverse modes or higher-order transverse
modes, which can be geometrically represented by using a higher-order Poincaré sphere (HOPS) as shown by Milione et al [8].
Equation 13 is an analog representation of the transverse electric mode (TE01), while Equation 14 is an analog representation of the transverse magnetic mode (TM01) as shown by Ndagano et al
[6]. These types of modes occur in optical fibers and in optical resonators. In addition, other types of fiber waveguide modes are the hybrid modes (HE21) known as 𝜋-vector beams [6,8], which have non-zero electric and magnetic fields in the direction of propagation. They are also
represented through a HOPS, which describes various polarizations of optical fiber waveguides [8]. These hybrid-polarized modes are mathematically represented through Equation 15 and Equation 16 as OAM modes [6]: |HE⟩
√
|HE⟩
𝑒
ℓ ℓ
𝑒
√
|𝑅𝐻⟩
𝑒
|𝑅𝐻⟩
ℓ
𝑒
|𝐿𝐻⟩ ℓ
(15)
|𝐿𝐻⟩
.
(16)
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Taking the complex conjugate of Equations 13 and 14, it is observed that a LaguerreGaussian beam with radial or azimuthal polarization is a Majorana-like photon, which is
represented by Equations 17 and 18. These equations show that these vector beams are in an optical spin-orbit state (mixed state of OAM and CP): ∗
and |𝜓
𝑒
√
∗
ℓ
|𝐿𝐻⟩
𝑒
ℓ
|𝑅𝐻⟩
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|𝜓
ℓ
𝑒
√
|𝐿𝐻⟩
𝑒
ℓ
|𝑅𝐻⟩
|𝜓
;
17
|𝜓
.
(18)
Moreover, taking the complex conjugate of the 𝜋-vector beams (HE21), which are optical fiber modes, it is shown that these two equations have indeed a main feature of a Majorana-like photon as shown in Equation 19 and Equation 20: |HE⟩∗
√
𝑒
ℓ
|𝐿𝐻⟩
𝑒
ℓ
|𝑅𝐻⟩
*Corresponding author: [email protected]
6
(19)
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|HE⟩∗
𝑒
√
ℓ
|𝐿𝐻⟩
𝑒
ℓ
|𝑅𝐻⟩ .
(20)
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Therefore, vector beams (radial and azimuthal) and hybrid 𝜋-vector beams from Equations 13 through and 16, are defined and attributed as Majorana-like photons from the basis modes,
following the Majorana feature of 𝜓 = 𝜓 ∗ as shown below in Equation 21 and Equation 22: ∗
|𝜓 |HE⟩∗
|𝜓
; |𝜓
∗
|𝜓
; |HE⟩∗
|HE⟩
,
|HE⟩
and
(21)
.
(22)
Figure.1 Representation of a Radially polarized Majorana-like LG beam
Figure 1 is it a cartoon representation of a Majorana-like beam where it is observed the main
feature of a Majorana is composed of a coupled SAM and OAM generating a vector vortex beam with radial polarization.
3. Other special function with polarization and wavefront is Majorana-like
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complex vector vortex beam
Other spatially structured light beams such as Airy beams (AiB)[22] follow the Majorana fingerprint of 𝜓 = 𝜓 ∗ , base modes where the photon and the anti-photon are identical. This type of complex beams is also a solution to the paraxial Helmholtz wave equation expressed in different coordinate systems. Besides, this beam has a special property of non-diffracting and
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self-healing when propagating through an obstruction. Hence, Airy beams have many applications such as in imaging, atmospheric turbulence propagation, and particle transportation and manipulation among others.
The following represents the Airy radially and azimuthally polarized beams based on modes introduced by Wei et al. [22], where sending a Gaussian beam through a LC q-Airy-plate generates vector Airy beams. As a result the following Electric fields are obtained [22]: 𝐸
𝑅𝑃
𝑒
𝑒
𝑒
*Corresponding author: [email protected]
𝑒
𝑒
𝑒 7
,
(23)
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𝐸
𝐴𝑃
𝑒
𝑒
𝑒
ℓ
𝑒
𝑒
𝑒
,
(24)
Equations 23 and 24 represent vector Airy beams written in terms of Jones vector of RCP and LCP, a helical phase front of 𝑒
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𝑦
exp 𝑖 𝑥
, and with the characteristic Airy cubic phase modulation of
, where q is half of the topological charge ℓ ,and 𝜃 is the angle between the
incident polarization direction and the x-axis.
Rewriting Equation 23 and 24 in a simplified manner for the polarization basis modes by using the wave function: |𝜓
√
|𝜓
√
𝑒
ℓ
𝑒
𝑒
|𝑅𝐻
𝑒
ℓ
𝑒
𝑒
|𝑅𝐻⟩
𝑒
ℓ
𝑒
𝑒 |𝐿𝐻⟩ ,
(25)
𝑒
ℓ
𝑒
𝑒 |𝐿𝐻 ,
(26)
where 2q is replaced by the OAM value is ℓ and the RCP and LCP are represented in terms of RH and LH states.
Taking the complex conjugate of Equation 25 and 26, proves that indeed these types of vector vortex polarization beams are Majorana-like photons as shown below in Equation 27 and 28: ∗
|𝜓
√ ∗
|𝜓
√
𝑒
ℓ
𝑒
𝑒 |𝐿𝐻
𝑒
ℓ
𝑒
𝑒
|𝑅𝐻⟩
𝑒
ℓ
𝑒
𝑒 |𝐿𝐻
𝑒
ℓ
𝑒
𝑒
|𝑅𝐻
|𝜓
|𝜓
,
,
(27) (28)
However, not all radially and azimuthally polarized optical beams are Majorana-like; such is the
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example of the Bessel beam [23], where the longitudinal component does not contribute for it to be its own conjugate unless at z=0 (absence of longitudinal component). Hence Bessel beams may lack the necessary condition.
In quantum field theory (QFT), the second quantization formalism in terms of the harmonic creation (𝑎 ) and annihilation (𝑎) operators is useful for the analysis and characterization of the
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quantum optics properties and can be used to describe features of the Majorana vector photons. The Majorana boson creation (𝑏 ) and annihilation (𝑏) operators can be written in terms of harmonic operators (𝑎 ) and (𝑎), where 𝑏 = (𝑎 + 𝑎) /√2 and 𝑏 = (𝑎+ 𝑎 )/√2 [17]. Using the Majorana boson creation (𝑏 ) and annihilation (𝑏) operators, the Majorana boson operator can be represented as follows (Eq. 29): 𝐶
𝑏
𝑏
29
𝐶 ,
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where the wave function of the quantum field is its anti-conjugate. Furthermore, the quantization of a system of bosons is implemented using Fock space including Fock states following Majorana fermions [17]. In Fock space, the creation operators used in Equation 30 are used to quantify the
|𝑛⟩
𝑎
^𝑛 /√ 𝑛! |0⟩,
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nth photon state as referenced to the vacuum state. The state |𝑛⟩ is given by:
30
where |0⟩is the vacuum state and n is the number of photons in the state. The set of |𝑛⟩ states are orthogonal to one another meaning the vacuum set combinations to represent the Majorana
bosons have a totally zero inner product [17], and is represented as follows (Eq. 31 and 32): 𝑎|𝑛⟩
√ 𝑛 |𝑛
31
1⟩,
and 𝑎 |𝑛⟩ √ n 1 |𝑛
1⟩.
32
The set of n radially polarized vector photons can be quantized using this formalism and a
separate set can be used to quantize the azimuthally polarized vector photons as another feature of Majorana boson pair. As stated by Nielsen and Ninomiya et al [17] Equations 29 through 32 are part of the Majorana theory, which is extended for Bosons in which a photon is its own antiphoton following the key Majorana fingerprint 𝜓 = 𝜓 ∗ .
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4. Possible applications of Majorana-photon like vector beams in Topological Quantum Computing
The Majorana-like vector beams represented previously in Section 2 are known as the four vector basis states of information [6]. These are more likely to have a topological effect in optical quantum computers due to their property of non-separability. That means their space and
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polarization degree of freedom cannot be written in a Cartesian product. In addition, it known through vector beam modulation that it is possible to obtain 𝑁 simultaneously, which helps to encode 𝑙𝑜𝑔𝑁 bit of information with 𝑁
4 possible states per pulse
2 bits of information per ℓ value, rather than 1
1 possible states as it happens with scalar beams [5].
Moreover, as shown by Kauffman et al [24], Majorana-photon like vector beams can form knots and braids, which are similar to the DNA structure modes. This study has shown that the Majorana operators form a robust representation of the braid group, which is related to knot theory. Although this application is applied to Majorana Fermions, Majorana bosons such as the *Corresponding author: [email protected]
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photon can also follow a similar trend since we previously proved that a Majorana boson like the photon is its own antiparticle, 𝜓 = 𝜓 ∗ . These Majorana quasi-particles are composed of natural braiding operators [24]. They are also used as qubits since they are intrinsically immune to
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decoherence [25].
Besides that, Leach and O’Holleran et al [26,27] have shown that optical vortices could be
combined to form loops, links or knots embedded in the light wave. Optical vortices’ topology is believed to have a nodal line, which may be indefinitely long or form a closed loop. Although the longer the length might affect other vortex lines.
In general, the inhomogeneous polarization and spin-orbit coupling characteristics of a vector beam could be an advantage when forming braids in 3D space, which represent logic gates in a computer. This type of optical quantum computer-based on knots and braids are more likely to conserve stability because a perturbation such as environmental will not change the topology of these knots and braids meaning they will retain input characteristics. A challenge ahead is to develop a method to use the Majorana and OAM photon vortex as braids and knots for topological computation for future computers.
5. Conclusion
In summary, Majorana-like photons arise from radial and zimuthal Laguerre-Gaussian, Airy beams, and 𝝅-vector beams. These belong to a super class of special functions and are
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characterized as cylindrical polarized transverse and hybrid vector vortex photons. These special vector beams are known as eigenmodes of weakly guiding optical fibers. They have identical anti-conjugate state and complex transpose (𝝍 = 𝝍∗ ) in polarization vector, which is not the case for a circularly polarized beam. These Majorana photons–the photons and the anti-photons–are identical, have a zero rest mass and are chargeless. These photons have within themselves locally both right and left-handed chirality and OAM (+L) and (-L). The study of Majorana-like vector
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beams (non-separable states) can play an important role in potentially improving imaging with a higher flux of photons in the classical limits, in contrast to quantum entanglement, which uses a single low photon number non-locally [3]. In addition, various types of polarization and OAM (𝓵) [28] in different parts of a scattering or turbid medium could give interesting results to compare, where photons could interact with different quasi-particles [1,29,30] while transferring charge and ions and forming new types of polaritons about resonances from quasi-particles [30]. Recently Mamani et al. [12], found that these Majorana-vector beams can increase the transmission when traveling in turbid media due to their unique characteristics of inhomogeneity *Corresponding author: [email protected]
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and non-separability. Experimental work with these Majorana beams to improve deep penetration imaging in bio- and condensed media should be undertaken. In general, these special nonseparable Majorana-like photon modes, which form knots and braids from vortex paths, may be at
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the heart of the future optical and topological quantum computers, communication and imaging applications as Majorana quasi photons could increase transmission [12], transfer, and store more information [5] with less scattering and possibly with less environmental interference, maintaining the quantum coherence or just creating small perturbations. Other applications for Majorana-like photons include high-resolution multi photon microscopies (SRS, 2/3 PEF, SHG among others) due to the stronger longitudinal field obtained from the Majorana-radially polarized light [31], fiber optics sensing, free-space communication, turbulence effects [32], and hidden optical transitions in a condensed matter such as in photoexcited oriented carriers in semiconductors such as GaAs in conduction bands [33]. Also, a Majorana can play a key role in spatial collapse in Kerr medium, in multiphoton, and stimulated Raman imaging in bio and condensed media [34,35].
Funding. This work is supported in part by Corning Incorporated Foundation (71198-00 12) and partially by ARO (47351-00-01).
Acknowledgment. We thank Fabrizio Tamburini, David Schmeltzer, Pouyan Ghaemi, Alexander Poltorak, David Grier, Larry Jacobowitz, and Michio Kaku for helpful discussions. We thank the referees for their
Competing Interests
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help in clarifying and improving this manuscript.
The authors declare no competing interests.
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Author Statement File: Robert R. Alfano: Conceptualization, Proposing project, Project administration, Supervision, Writing- Original Draft, Writing-Review & Editing, Methodology, Validation; Sandra Mamani: Investigation, Experiments, Writing- Original Draft, Writing-Review & Editing, Visualization; Daniel A. Nolan: Writing-Review & Editing, Discussions; Lingyan Shi: Discussions, Writing-Review & Editing.