ICOAM 2015The electromagnetic momentum transferred transferred to scattering particles is proportional to the intensity of the incident fields, however, the momentum of single photons ℏk does not naturally appear in these classical expressions. Here, we discuss an alternative to Maxwell's stress tensor that renders the classical electromagnetic field momentum compatible to the quantum mechanical one. This is achieved through the introduction of the quantum conversion which allows the transformation, including units, of the classical fields to wave-function equivalent fields.Publisher PD

Mazilu, Michael

St Andrews Research Repository

arXiv:1508.01178v2  [physics.optics]  6 Aug 2015Quantum conversionMichael Mazilu∗SUPA, School of Physics and Astronomy,University of St Andrews, St Andrews KY16 9SS, UK(Dated: August 10, 2015)The electromagnetic momentum transferred transfered to scattering particles is proportional tothe intensity of the incident fields, however, the momentum of single photons (~k) does not nat-urally appear in these classical expressions. Here, we discuss an alternative to Maxwell’s stresstensor that renders the classical electromagnetic field momentum compatible to the quantum me-chanical one. This is achieved through the introduction of the quantum conversion which allows thetransformation, including units, of the classical fields to wave-function equivalent fields.PACS numbers: (03.50.De) Classical electromagnetism, (03.65.-w) Quantum mechanics, (42.50.Wk) Mechan-ical effects of lightINTRODUCTIONCalculating the optical forces acting on mesoscopicparticles typically relies on defining a balance equationbetween the momentum of the electromagnetic field andthe momentum of the mesoscopic particle. This can bedone using various definition of the electromagnetic stresstensor such as Chu, Maxwell-Lorentz, Minkowski, Abra-ham, Eistein-Laub[1]. All these approaches deliver thesame total momentum transfer to rigid particles. Fur-ther, in all these cases the momentum is proportionalto the incident intensity. The differences between eachstress tensor definition lies in their precise microscopic in-terpretation. In the following, we consider only Maxwell-Lorentz stress tensor, however all the discussions are ap-plicable to any of the other definitions.To transform classical fields into quantum compatiblewave-function fields, we introduce the quantum conver-sion (QC). The QC is based on two parts, a mathematicaltransformation and a unit conversion. The transforma-tion part corresponds to a convolution with the Fouriertransform of 1/√ν. On the other side, the unit conver-sion ensures the correct units in the physical context.The QC applied to a function f0(t) with a zero-mean∫∞−∞f0(t)dt = 0 defines the function f(t) = (Q ◦ f0)(t)as(Q ◦ f0)(t) = p.v.t∫−∞f0(τ)√hi− 1√t− τ dτ (1)where p.v. denotes the Cauchy principal value. The in-tegral operator defining the QC also includes Planck’sconstant, h, which converts between frequency and en-ergy (intensity). The linear transformation part of theQC operator is similar to the Hilbert transform of f0(t)while normalising its spectral amplitude by√hν whereν is the spectral frequency. Its fundamental property issummarised in the following Fourier space relationshipbetween a function and its QC:√hν(F ◦ f)(ν) = (F ◦ f0)(ν) (2)where F corresponds to the Fourier transform in the timedomain. Using this property and Perseval’s formula itis possible to introduce an integral relationship betweenclassical and quantum space∫ ∞−∞f∗0(t)g0(t)dt =∫ ∞−∞hνfˆ∗(ν)gˆ(ν)dν=∫ ∞−∞f∗(t)(i~ddt)g(t)dt (3)which represents the main purpose of the QC. In ef-fect, the QC introduces a first order time domain deriva-tion whose role is to make the integral intensity mea-sure proportional to the frequency of the field. As wewill see in the following, this will introduce a propor-tionality between the energy/momentum measures andthe frequency/wave-vector of the fields considered. Theproportionality coefficient is Planck’s constant and theQC converts using a different coefficients each spectralcomponent of the field to encode the energy informationonto the frequency of the field. This is possible as eachfrequency component is orthogonal to each other.For completeness, we can define the inverse QC as:f0(t) = (Q−1 ◦ f)(t) = i~ ddt(Q ◦ f)(t) (4)and exemplify the effect of the QC on a harmonic func-tion Q(exp(iωτ))(t) = sgn(ω) i exp(iωt)/√~ω wheresgn(w) corresponds to the sign function.In the following, we consider the QC of the electromag-netic fields solutions of Maxwell’s equations (Gaussianunits) in free space,∇ · E0 = 0,∇ ·B0 = 0,c∇×E0 = −∂tB0,c∇×B0 = ∂tE0, (5)where E0 and B0 are the electric and magnetic vectorfields and where c is the speed of light. We define theQC of the electromagnetic fields as E = Q ◦ E0 andB = Q ◦B0 where E0 and B0 are considered to be zero2mean analytic signal functions i.e. contain only strictlypositive frequencies.Taking property (3) into account we can define theconverted field energy density asρE =18pi(E∗(i~∂t)E+H∗(i~∂t)H) (6)which will is globally identical to the standard defini-tion of energy density i.e. the total time integrated en-ergy in a region is the same for both definitions. Thetime dependence of the fields has change however usingthe normalised fields gives exactly the same total inten-sity/energy of the pulse. What is more this approachmaintains relative intensities which means that throughrelative measures we are not able to distinguish physi-cally between the classic and the quantum version. Theorigin of property is the orthogonality of the differentspectral components of the pulse i.e. the total energycontent of two monochromatic waves is equal to the sumof their individual energies.The flow of energy is described byjE =c8pi(E∗ × (i~∂t)H+ (i~∂t)E×H∗) . (7)Using this definition of the flow allows us to determinethe energy transferred to a volume V surrounded by asurface S asdEdt=∫SjE · n ds (8)where n is the surface normal and E = ∫VρE dv theintegrated energy density in the volume v.Using the quantum converted fields we can redefinethe linear momentum density and its current/flux. Todo this, we notice the parallelism between the definitionof energy and momentum. They only differ in the gen-erating differential operator. What is more, taking intoaccount the dispersion relationship ck = ω we can definethe density of the linear momentum as :ρPi =18pi(E∗(Pi)E+H∗(Pi)H) (9)where Pi is a place holder for differential operators (i~)∂x,(i~)∂y and (i~)∂z corresponding respectively to the linearmomentum in the x−, y− and z−direction.The conservation relationship implies that the transferof momentum (force acting on scattering body) can becalculated by integrating the momentum flux through asurface surrounding the scattering object. Replacing, inequation (10), the energy operator with a momentum op-erator delivers the variation of the momentum in the cho-sen direction. This momentum variation acts as a forceon the scattering object. For example, in the continuouswave case we can calculate the optical period averagedforce acting of the objectFi =c8pi∫S(E∗ × (Pi)H+ (Pi)E×H∗) · n ds. (10)This integral delivers identical average forces asMaxwell’s stress tensor. It shows the equivalence betweenquantum converted fields in conjunction with the oper-ator approach and classical electromagnetic forces. Inboth cases, the observable macroscopic mechanical forceis the same.We further remark that the density defined by the iden-tity operator ρ = (E∗E+H∗H)/(8pi) corresponds to thephoton density. This explains physically the equivalencebetween the momentum flux integral (10) and Maxwell’sstress tensor, which in this context determines the vari-ation of photon flux due to a scattering body.To complete the equivalence picture between the clas-sical fields and the quantum fields we further need totake into account the optical eigenmodes defined eitherover the whole space [2, 3] or over a finite region of in-terest [4]. Indeed, using this approach one can draw adirect equivalence between quantum mechanics and clas-sical electromagnetism where the optical eigenmodes playthe role of photon wavefunctions [5].Further, the quantum conversion can easily be modi-fied to deal with non analytic functions and negative fre-quencies when considering relative energy transfers. Inphysical terms, the quantum conversion corresponds tothe introduction of frequency dependent units or mea-surement metric. This conversion generalises the con-cept of position dependent metric to the Hilbert space.Indeed, each orthogonal basis vector/function can haveits own units just like each point in space can have itsown distance measure. Finally, we note that the inversequantum conversion can be applied to transform quan-tum wave functions into their classical form opening upto the possibility of a new interpretation of quantum me-chanics.∗ michael.mazilu@st-andrews.ac.uk[1] M. Mansuripur, “Electromagnetic force distribution insidematter,” Phys Rev A 88, p. 023826, May 2013.[2] M. Mazilu, “Optical eigenmodes; spin and angular mo-mentum,” J. Opt. 13(6), p. 064009, 2011.[3] M. Mazilu, “Spin and angular momentum operators andtheir conservation,” J. Opt. A 11(9), p. 094005, 2009.[4] M. Mazilu, J. Baumgartl, S. Kosmeier, and K. Dholakia,“Optical Eigenmodes; exploiting the quadratic nature ofthe energy flux and of scattering interactions,” Opt. Ex-press 19(2), pp. 933–945, 2011.[5] I. Bialynicki-Birula, “On the wavefunction of the photon,”Acta Physica Polonica Series B 86, pp. 97–116, Feb. 2003.

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