arXiv:1604.03869v4 [physics. optics] 30 May 2018

Electromagnetic fields, size, and copy of a single photon

Shan-Liang Liu

Shandong Key Laboratory of Optical Communication Science and Technology, School of Physical Science and Information Engineering, Liaocheng University, Shandong 252059, People’s Republic of China

Photons are almost involved in each field of science and daily life of everyone. However, there are still some fundamental and puzzling questions such as what a photon is. The expressions of electromagnetic fields of a photon are here proposed. On the basis of the present expressions, we calculate the energy, momentum, and spin angular momentum of a photon, derive the relations between the photon size and wavelength, and reveal the differences between a photon and its copy. The results show that the present expressions properly describe the particle characteristics of a photon; the length of a photon is half of the wavelength, and the radius is proportional to square root of the wavelength; a photon can ionize a hydrogen atom at the ground state only if its radius is less than the Bohr radius; a photon and its copy have the phase difference of π and constitute a phase-entangled photon pair; the phase-entangled n-photon train results from the sequential stimulated emissions and belongs to the Fock state. A laser beam is an ensemble of the n-photon trains and belongs to the coherent state. The threshold power of a laser is equal to the power of the n-photon train. These provide a bridge between the wave theory of light and quantum optics and will further advance research and application of the related fields.

duration [13,14], but how to know size of a photon is I. INTRODUCTION still a puzzling question. The wave-like properties of Light is almost involved in each field of science and light or photons are well described by the classical daily life of everyone, and yet light’s true nature has theory of electromagnetic fields. A monochromatic eluded us for centuries. Albert Einstein successfully plane wave indicates that a single photon would occur explained the photoelectric effect in 1905 by the at any place with the same probability, which clearly assumption that light is composed of photons and contradicts the well-known experimental fact that the showed that a photon has constant energy hν and photons out of a single-photon source can be detected momentum h/λ where h is the Planck constant, ν is the only at right time by the detector at right position, and frequency, and λ is the wavelength. The theory of so it is unable to properly describe a photon. A wave relativity tell us that a photon has zero rest mass and packet results from superposition of plane waves with always moves at the speed of c=λν in vacuum. different wave vectors in the classical theory of However, these answers are not satisfactory. Roy J. electromagnetic fields and is obviously incomparable Glauber once jokingly summarized his theory of photo with the well-known fact that a photon has the definite detection by the sentence: “I don’t know anything energy and momentum, and so it is unable to properly about photons, but I know one when I see one” [1]. describe a photon either. How to properly describe the Single photons are essential for the fundamental study electromagnetic fields of a single photon is still a of the quantum mechanics [2,3] and the development fundamental and unresolved question in physics. The of photonic quantum technologies such as optical no-cloning theorem, which says that one cannot make quantum computing [4-6] and quantum communication a perfect cloning of an unknown quantum state [15], is [7-9]. However, there is not still a satisfactory answer considered to be heart of security of quantum crypto- to the problem what a photon is [10]. graphy [16,17]. The stimulated emission is natural way The wave-like properties of microscopic particles of cloning a photon [18]. However, any difference such as an electron and proton are usually described by between a photon and its copy has not been found. wave functions in quantum mechanics, and the The expressions of electromagnetic fields of a particle-like behaviors are governed by the classical or photon are here proposed. On the basis of the present relativistic dynamic equations where they are usually expressions and the known experimental facts, we simplified as a point with mass. Size of an atom or ion calculate the energy, momentum, and spin angular is usually represented by the orbit diameter of the outer momentum of a photon, derive the relations between electron in it, while radius of the atomic nucleus is the photon size and wavelength, and reveal the proportional to cubic root of the mass number. The differences between a photon and its copy. These experiments have indicated that a single photon can provide a bridge between the wave theory of light and locate in very small space [11,12] and very short time quantum optics and will further advance the research and application of the related fields. 1 / 4 arXiv:1604.03869v4 [physics. optics] 30 May 2018

II. ELECTROMAGNETIC FIELDS The calculated result is the same as the known momentum expression of a photon. It follows that the OF A PHOTON phase interval of a photon is π, i.e. the longitudinal Since light is composed of photons and the Maxwell length of a photon is half of the wavelength λ/2 in equations are linear in vacuum, behaviors of a photon spatial domain and half of the cycle T/2 in temporal as well as light are governed by the Maxwell’s domain where T=1/ν. The half-cycle pulses have been equations, and the electromagnetic fields of a photon used to ionize the Rydberg atoms and found to be can be described by the special solutions of the linear polarization and unipolar [19].This means that Maxwell equations the electric fields in a linearly-polarized photon are in the same direction, and sinφ0=±1. , (1) The spin angular momentum (SAM) density of free × , (2) electromagnetic fields in propagation direction is given by ε0E×A where A is the vector potential [20]. SAM of . (3) a photon is given by the integral 香䁕 iδ where φ=kz−ωt, ω=2πν, k=2π/λ, u=excosα+eysinαe , × , (8) α and δ are thepolarizationp∉arameters, E0 is a real where A1 is the vector potential of a photon. Using the 䁪 constant, and f1 is a profile function which describes a 䁕 relation E1=−∂A1/∂t and Eq. (1) gives the vector cylinder with the length z0−z1=λ(φ0−φ1)/2π and radius potential of a photon ρ0. The photon position varies from −∞ to ∞ with time − , (9) t along the motion direction ez. For a linearly-polarized where A0=E0/ω. SubstitutingEq. (9) into Eq. (8) gives photon δ=0 or π, the electric field varies with phase φ −ћ , (10) in magnitude and has the maximum E0. For a where ћ=h/2π is the reduced Planck constant. For a circularly-polarized photon α=π/4 and δ=±π/2, the 䁪 circularly-polarized spihnoton αs=inπ/4,Ls=−ћ at δ=π/2, and electric field has the constant magnitude of E0sin(π/4), Ls=ћ at δ=−π/2. For a linearly-polarized photon δ=0 or and its direction varies with the phase. An elliptically- π, Ls=0. For an elliptically-polarized photon, Ls varies polarized photon corresponds to the other values of α with the polarization parameters α and δ. Equation (10) and δ, and both magnitude and direction of the electric is in good agreement with the known experimental field vary with the phase. results [21]. III. SIZE OF A PHOTON B. Radius of a photon. There are no electromagnetic fields in the region of After the electron in a hydrogen atom meets a ρ>ρ0. The energy density in a photon is photon, the change rate of the electron momentum is independent of ρ where  0 is the vacuum permittivity, governed by the dynamic equation but it depends on the polarization state. The photon − − × , (11) energy is given by the integral 䁕 where fc is the Coulomb force acted on the electron, e 䁕 h . (4) is the elementary charge, and v is the velocity of the electron. Taking scalar product of Eq. (11) and vdt=dr A. Lengtho䁕fa photon. gives ∙ ∙ − ∙ . (12) Integrating the right side of Eq. (4) gives Since v<ionization process is independent䁪 ofcotshe pol䁪ar香izatisoin state, i.e. the − − ∙ . (13) parameters α and δ should not be included in Eq. (5). where W=−2Ui is work of the Coulomb force on the This requires sin∆−=0, i.e. ∆− be a multiple of π. Since 䁕 a photon is the most fundamental quantum of electron in the ionization process, 0 is the electromagnetic fields, ∆− must take the minimum π, ionization energy according to the Bohr’s theory. The i.e. φ1=φ0−π, and Eq. (5) reduces to last term is the work of the electricfieldofthe photon h , (6) on the electron in the absorption process and has the which indicates that the radius ρ0 as well as E0 is maximum h f the photon is linearly polarized and independent of thepolarizaotion state. pe∙E1=−peE1 in the absorption process. The photon A photon has the momentum density , and energy requireid for ionization of pef=0 has the minimum Ui, and Eq. (13) reduces to its momentum is given by the integral h . (7) − ∙ =Ui. (14) 䁕 2 / 4 䁕 arXiv:1604.03869v4 [physics. optics] 30 May 2018

Since v<amplitude of the electric field . (24) − h . (19) The electric fields of a linearly-polarized photon and | | | Substituting Eq. (19) into Eq. (6) gives the photon its copy have the opposite directions. radius B. A laser beam and coherent state , (20) − A light beam is an ensemble of the phase-entangled where re is the classical radius of an electron. Since the o n-photon trains with different wave vectors and can be length of a photon is equal to half of the wavelength described by the state vector and the radius is proportional to square root of the , (25) wavelength, the size and shape of a photon vary with where is the photon number in the i-th photon train the photon energy or wavelength. A photon is in shape |香 |香 like a thin stick if its energy is lower than the rest with the w ave vector ki. The wave vectors are almost 香 energy of an electron and like a plate if its radius is the same in a laser beam and can be simplified as smaller than the classical radius of an electron. For a n. The photon-counting experime nts have showed that photon of hν=13.6 eV, the photon radius is 34.9 pm the photons from a laser we香ll above the threshold and is less than the Bohr radius. This indicates that a obeys the Poisson distribution [22,23] and a laser beam photon can ionize a hydrogen atom at ground state can be described by the state vector only if its radius is less than the Bohr radius. 香 , (26) where is香 the averaged value of n. It is IV. PHOTON TRAIN | 香 香 |香 follows that a laser beam is an ensemble of the A. Copy of a photon and Fock state phase-en香tang|lαed| n-photon trains and belongs to the coherent state, and the probability finding the n-photon An atom jumps from the ground state |a> to the train obeys the Poisson distribution, i.e. excited sate |a′> by absorption of a photon. The excited atom returns to the ground state under stimulation of . (27) 香 香 the photon |1>0 and simultaneously gives off another Since and approaches infinity in a 香 香 香 香 photon |1>1 which is the same as |1>0 except for the laser beam, the relative deviation is very close to zero, phase according to the theory and experiment of and thep香ow香erand香intensity香 of a laser beam are very stimulated emission. So, a photon and its copy stable. constitute a phase-entangled photon pair. The excited The average number of the n-photon trains in any atom |a′> also returns to the ground state under cross section of a laser beam is given by =P/P1 2 stimulation of the photon pair and simultaneously where P is the beam power and P1=2hν is power of gives off another photon |1>2 which is the same as both the n-photon train. The intensity of the laser beam with |1>0 and |1>1 except for the phase. As such, a the radius r0 can be given by phase-entangled n-photon train occurs by the . (28) sequential stimulated emissions and corresponds to the where is the averaged intensity of the wave train in optics, and its electric field can be written n-photon train.Since the beam radius is generally as much laIrger than thephoton radius, i.e. r0>>ρ0, and the , (21) beam intensity is much weaker than the intensity of the n-photon train. A laser beam is usually described by 香 香 , (22) the plane wave with the intensity where 香䁕 香 香 ∉ 香 3 / 4 arXiv:1604.03869v4 [physics. optics] 30 May 2018

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