PROPAGATION OF PHOTON IN RESTING AND MOVING MEDIUM. J. Zaleśny. Institute of Physics, Technical University of Szczecin, A b s t r a c t

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1 PROPGTION OF PHOTON IN RESTING ND MOVING MEDIUM J Zaleśny Institute of Physis, Tehnial Univesity of Szzein, l Piastów 48, Szzein, Poland b s t a t The popagation of photon in a dieleti may be desibed with the help of the sala and veto potentials of the medium The main novelty of the pape is that the onept of the veto potential (whih is onneted with the veloity of the medium) an be extended to elativisti veloities of the medium The positiondependent photon wave funtion was used to desibe the popagation of the photon The new onepts of the veloity of photon as patile and the photon mass in the dieleti medium wee poposed PCS numbes: 0350De, 0365w

2 INTRODUCTION Conside a photon in a dieleti medium But what photon eally is? In moden physis photon is nothing moe than quantum exitation of the eletomagneti field We have leaned fom quantum eletodynamis, that in the dieleti, in fat, thee ae no photons but polaitons, ie exitations of eletomagneti field oupled to the medium Howeve a diffeent point of view is also possible Remembe the ase of eleton in an extenal eletomagneti field On the fist quantization level the eleton is teated as a quantum objet moving in the lassial field In this pape I develop a simila desiption fo a photon The photon teated as a quantum objet feels the medium as an extenal lassial field To desibe a photon in tems of a onepatile wave funtion, ie, on the fist quantization level, I follow the way pesented in [ 3 ] nothe appoah was poposed in [4] The onept of the positionepesentation photon wave funtion has a long histoy and is still ontovesial Nevetheless I do not want to disuss the question hee The eade inteested in this poblem is efeed to [4] and efeenes theein The pape is oganized in the following way In Setion, I develop the desiption of a photon in medium in tems of the photon wave funtion Some attempts of this kind have been pesented in [ 3 ] What is new in my appoah is to show that the influene of the medium on the photon an be desibed though some potentials Though geneally the idea is not new, see eg,[ 6 5 ] but hee I ealize it within the fomalism of the photon wave funtion On this basis, in Setion 3, the nonzeo mass of photon and the onept of the veloity of photon as a patile appea in a quite natual way The veloity of photon is diffeent fom the phase o goup veloity and, up to my knowledge, is a new onept I show in Setion 4 that the motion of the dieleti an be onneted with the optial analog [ 8 7 ] of the veto potential This idea has been aleady pesented in liteatue, see papes What is new hee is that the onept of the veto potential of the medium an be extended fo

3 elativisti veloities of the medium With the help of the sala and veto potentials of the medium one an define some optial analogs of eleti and magneti fields, and the optial analog of the Loentz foe (ating on the photon in the medium) The potentials ae gauge fields and the analogs of eleti and magneti fields ae gauge invaiant THE SCLR POTENTIL OF MEDIUM In papes [ 3 ] the following fom of the Shödinge equation fo fee photon was poposed ih t F H ff () E (t, ) + i H (t, ) p S, 0 F ; H f E (t, ) i H (t, ) ; 0, p S () p iћ momentum of photon; (S i ) k l i ε i k l spin photon matix (ε i k l antisymmeti LeviCivita symbol ) On the lassial language, the equations ae equivalent to the following Maxwell equations t E H ; t H E ; D E ; B H, (3) desibing fee fields in vauum Sine all the infomation aied by funtion F is ontained in its positive enegy (positive fequeny) pat F (+), following [3], I take this pat as the tue photon wave funtion and denote it as ψ (+) ψ F (4) To beome a omplete set of Maxwell equations, eq(3) must be supplied by divegene onditions E 0, H 0 It is equivalent to the elation p ψ 0 In ode to desibe the popagation of photon in dieleti, one should inlude in Hamiltonian the inteation tem On the miosopi level, suh inteation is athe ompliated, but hee I will take it into aount in a phenomenologial way Let us begin with stationay states of the photon in a homogeneous dieleti Fo a stationay state the wave funtion takes on the fom 3

4 ψ E + i H ϕ ( )exp( it), whee ϕ E i H (5) The popagating photon in evey time and in evey spae point, feels the same oupling with the medium We may ty to desibe the inteation by a single onstant oupling value Ω If some inhomogeneities ae in the dieleti, then the inteation depends on and will be modeled by a funtion Ω () Sometimes, I want to estit onsideations to the nonmagneti media, ie, It means that the medium is oupled only to the eleti pat of the photon wave funtion Geneally, the ouplings of the medium with the eleti and magneti pat of the wave funtion may be diffeent To take it into aount I intodue two eal and symmeti maties γ and η whih split the wave funtion ψ into eleti and magneti pats γ ψ + η ψ ψ, (6) E ih γ ψ and η ψ E, ih η and γ (7) (8) The pojetion opeatos γ, η fulfill the following elations ie: γ γ, η η, γ η 0 Thus the ase of the popagating photon in inhomogeneous nonmagneti dieleti an be modeled in the following way h ψ H fψ Ω γ ψ (9) I intepet the tem Ω () γ as a potential enegy opeato of the photon in dieleti In ode to see what is the meaning of the quantity Ω () in the lassial language, one may tanslate equation (9) into the odinay fom of nonvauum Maxwell equations 4

5 i [ + 4π χ ()] E H ; i H E, (0) whee χ () Ω 4π h Thus Ω () is dietly onneted with the dieleti suseptibility χ () It is easy to genealize this appoah and wite Maxwell equations fo dispesive media in a fom of Shödinge equation If simultaneously many fequenies ae pesent in the medium, then it is easonable to expet (fom the quantum mehanial point of view indistinguishable altenatives) that the inteation of medium with the photon in suh a nonstationay state is desibed by supeposition of singlefequeny inteation tems Thus, if the photon is desibed by a wave paket ψ(t, ), then the inteation of the paket with the medium an be desibed by an integal opeato ˆΩ L in the following way ˆ L Ω ψ ( t, ) Ω ψ ( t, ) d Ω ϕ exp( it) d () The equation of motion of the photon takes on the quasishödinge fom ih ψ H ψ γ ˆΩ ψ () t f The integal fom of the inteation tem makes the elation between fields D and E nonloal in time, ie, D( t, ) E( t, ) + 4π χ( τ ) E( t τ, ) dτ (3) Equation () simplifies in some speial ases Eg in a nondispesive medium Ω ( ) 4π χ( ) h, (4) L whee χ() does not depend on the equation () beomes ih t ( + 4πχ γ ) ψ H f ψ (5) In this ase it is possible to onstut some effetive wave funtion (and Hamiltonian) and [ 3 ] suh effetive fom is used in 5

6 nothe inteesting ase is when Ω () is independent of, then Ω ψ ( t, ) Ω () ψ ( t, ) In this ase the similaity to the ase of eleton in an extenal field is the most appealing Fo simpliity, in the next Setions, I estit disussion to stationay states 3 THE MSS ND VELOCITY OF PHOTON IN THE MEDIUM When the ouplings with the eletial and magneti pats ae taken into aount, the Shödinge equation takes on the fom ˆ L h ψ H fψ Ω γ ψ Γ η ψ (6) Ω (), Γ () have intepetation of potential enegies Γ () is onneted with magneti suseptibility χ m () (/4π) Γ ()/ћ In ode to obtain the onnetion between the total enegy and momentum one may iteate this equation In the ase of a homogeneous medium one obtains [ f ] ψ ( h) ψ ( hω + hγ + Ω Γ ) H, (7) whee the identities H f γ + γ H f H f, H f η + η H f H f have been used The equation (7) is in fat the lassial wave equation It is easy to note that Ω h + Γ + h Ω ( h) Γ ε n, (8) whee ε, ae pemittivity and pemeability of the medium, and n is efative index, and also that ( ) p H f S p ; (p ψ 0) (9) Thus, putting p ih in eq(7) one obtains ψ + n ψ 0 (0) 6

7 The tem n mixes the kineti and potential tems of the Shödinge equation (6) Fom my point of view it is moe natual to intepet the equation (7) in anothe way That is, to put E h and to ewite the equation (7) as a onnetion between E and p in the fom ( Ω Γ ) E p + ( Ω + Γ ) 4 () It is appaent that E is the enegy of a massive elativisti patile in an extenal field Thus the photon in dieleti gains the mass m given by ( Ω Γ ) 4 m 4 () The photon gains the mass, beause of the inteation with the sea of hages in dieleti It eminds the Feynman s emak: mass is inteations The tem U ( Ω + Γ ) (3) is a lassial potential enegy It onfims the pevious intepetation of the quantities Ω and Γ as some potential enegies The equation () pedits that the mass of photon beomes zeo not only in empty spae (when Ω and Γ vanish) but also when Ω Γ (equivalently ε ) Thus the wave equation (7) takes on the fom of the KleinGodon equation ( E U ) ψ ( p + m 4 ) ψ Cetainly, the massive photon in dieleti has enegy, (4) E m + U v (5) Putting E h and using (), (3) one an alulate fom (5) the veloity v of photon in dieleti: 7

8 ( ε ) ( ε + ) v (6) Note that v is veloity of photon as a patile It neve exeeds The emakable featue is that v is equal to not only in empty spae but also if ε (as one should expet beause then the photon mass is zeo) Knowing m and v one an alulate the photon momentum p mv v n h (7) The veloity v is neithe phase no goup veloity The phase veloity v ph of photon is E v ph, (whee p hk ) (8) k p n nd the goup veloity v g is E vg (9) k p n n + In the nondispesive ase (n/ 0) the goup veloity is equal to the phase veloity On the othe hand the goup veloity v g is equal to v in the independent of ase (Ω/ 0, Γ/ 0) I think that one should onside the possibility that v is eally the tue veloity of photon in dieleti Cetainly, the veloity of photon in dieleti medium is not the question of definition The answe an give only an expeiment Note that h plays two distint oles in the above desiption It is the total enegy and apat it is a paamete detemining photonmedium inteation It is the eason why the ight hand side of the equation () depends on t the end of this setion I biefly omment the ase of the wave paket ψ d ϕ ( ) exp( it) d (30) 8

9 Fo evey Fouie omponent of the paket one may wite Shödinge equation (6) and thus KleinGodon equation (4) Beause of diffeent patile veloities v the wave paket dispeses If the dispesion of veloities is v the width of the paket is (in one dimension) ineasing in time as x v t The paket desibes one photon in a nonstationay state (the enegy and the mass of the photon ae not peisely detemined) and x is the egion in whih it is possible to detet it Usually the beam of light ontains many photons It means that all the photons ae in the same onepatile nonstationay state (30) Now, in the egion x you an detet many photons in diffeent points at the same time Pobability is popotional to the ate of detetion, and thus to the enegy density in given point You have a maosopi quantum state 4 THE VECTOR POTENTIL OF THE MEDIUM Developing analogy with the theoy of haged patiles it is inteesting to onstut and examine onsequenes of veto potential of the medium in the ase of photon Replaing p P +, whee p is kinetial and P anonial momentum, the Shödinge equation (6) beomes 0 h ψ ( P + ) S ψ Ω γ ψ Γ η ψ 0 (3) To detemine the situation, I suppose that the light soue emitting photons of enegies ћ ests with espet to the obseve Then, as will be shown, the physial meaning of the veto potential of the medium is dietly onneted with the veloity u of the medium Note, thee is no Dopple shift between the obseve and the soue and theefoe the obseved fequeny is the same as the soue fequeny Expessing the Shödinge equation (3) in the lassial language one finds 9

10 D i H, B i E ; (3) D ε E + a H, B H a E, whee a ( / ) h is a dimensionless veto potential of the medium This may be ompaed with the nonelativisti appoximation of the Minkowski elations dieleti: obliging fo unifomly moving [ 0 9 ] D ε E + (ε ) β H, B H (ε ) β E, (33) whee β u / One finds immediately the onnetion between veto potential a and the veloity u of the medium a (ε ) β (34) The esult is in ageement with papes,[ 8 7 ] whee it has been obtained in anothe way If one wants to examine the puely elativisti veloities ase, the fom of the Shödinge equation (3) must be hanged One eason is that in the nonelativisti veloities ase we assumed that the ouplings with the medium ae the same as in the ase of the esting medium It does not need be tue The seond eason is that the moving medium in fat podues anisotopy of the whole system This is not taken into aount in the nonelativisti veloities ase Theefoe one should admit that the ouplings fo fields pependiula and paallel with espet to the veloity of the medium β may be diffeent Thus one should onside the following Shödinge equation 0 h ψ ( P + ) S 0 ψ + E E Ω Ω ( ) Γ E E ih - ih Γ ih - ih (35) Ou task is to find the elativisti fom of (ћ/) a I look fo the solution in the fom 0

11 a α (β) β, (36) whee α (β) is yet unknown funtion Expessing the Shödinge equation (35) in the lassial language one obtains Maxwell equations with mateial elations in the fom D ε E + ε E + α(β) β H, (37) B H + H + α(β) β E (ε is defined as +Ω /ћ, et) The mateial elations should be ompaed with the elativisti [ 0 9 ] fom of the Minkowski elations D ε E + ε β β ε ε E + ε β β H, (38) B H + ε β β ε H ε β β E, One immediately finds α( β ) ε, (39) ε β and as well the othe paametes as ε, ε, et Fo a unifomly moving homogeneous medium the esult (39) is exat To a good appoximation it an be useful as well in the ase of nonunifomly moving inhomogeneous medium povided that the potentials Ω, Γ and the flow u vay only gadually, ie, do not vay signifiantly ove one optial wave length and one optial yle Fo nonelativisti veloities one an immediately wite onnetion between enegy and momentum simply by substitution in eq() p P + ( Ω Γ ) E ( P + ) + ( Ω + Γ ) 4 (40) It is possible beause (as one an easily hek with the help of Maxwell equations (3)) the divegene ondition D 0, B 0 is equivalent to the ondition p ψ 0 (similaly as it

12 was in the ase of the esting medium) The eq(40) is exat fo unifomly moving medium, and a good appoximation in the ase of nonunifomly moving inhomogeneous medium Note that the mass of photon is the same as it was in esting medium If one wants to iteate the elativisti Shödinge equation (35) it will be advantageous to wite it in the moe suitable fom ( H f Ω R γ Ω R γ Γ R η Γ R η) ψ h ψ, (4) whee pojetion opeatos R and R have been defined in the following way: R V V and R V V (V is a veto, and with espet to the veloity of the medium) The R and R ommute with γ, η The Hamiltonian H f has the same stutue as in the ase of esting medium but now p P + nothe useful elations ae H f γ + γ H f H f, H f η + η H f H f and H f R + R H f H f H f, H f R + R H f H f +H f The Hamiltonian H f diffes fom H f in suh a way that p S is eplaed by p S In geneal the obtained esult of iteation is athe ompliated and I will not wite it down I only examine hee the simplest but physially inteesting ase when the light in fom of plane wave popagates in the same dietion as the medium moves In this ase the esult has exatly the same fom as eq(40) with the only diffeene that Ω is eplaed by Ω and Γ by Γ In patiula, the mass of photon m β in the elativisti ase is given by m β Ω Γ ε β β ε β h m, ε β (4) whee m ε ћ / is the mass of photon in the est medium t fist, it might seem quee that the mass has hanged but it must be so, beause the mass of photon in medium (neve mind esting o moving) is always detemined by the ouplings (Ω and Γ) and these ouplings in the moving medium have hanged The hange of mass does not appea in the nonelativisti veloities ase beause it is only the seond ode effet (with espet to β)

13 3 In the following I estit onsideations to the nonelativistially moving media whih, no doubt, is easonable fom patial point of view It is inteesting that in some limited sense it is possible to develop photodynamis desibing behavio of photon, in lose analogy to the eletodynamis as the theoy of haged patiles In patiula one an detemine a lassial foe F ating on photon P p F & & & + (43) In the ase of nonunifomly moving homogeneous medium fom Hamiltonian (40) one finds v p P U E E & (44) Hee E U and p ae given by (5), (7) and v it is the patile veloity of photon given by (6) The analogs of eleti E ~ and magneti H ~ fields an be defined H E ~ ; ~ t (45) Theefoe the foe F is nothing else but the optial Loentz foe H v E F ~ ~ + (46) In the moe geneal ase of nonunifomly moving inhomogeneous medium one obtains a bit moe ompliated esult Γ + + Ω + ) ( ~ v v t v m U t E, H ~ (47) Note that in inhomogeneous medium the ouplings Ω, Γ depend on plae, theefoe the mass of photon is positiondependent The fields E ~ and H ~ ae gauge invaiant The hange of the potentials, Ω, Γ in the following way

14 + f ; f Ω Ω t ; f Γ Γ t (48) does not hange the fields (47) Hee f is whateve funtion of time and spae 5 SUMMRY I find that the influene of the medium on photon an be desibed by some sala and veto potentials Sala potentials ae dietly onneted with pemittivity and pemeability of the medium, veto potential is onneted with the veloity of the medium The main novelty in the pape is that the notion of veto potential of the medium an be onstuted also fo elativisti veloities of the medium nothe new esults ae the fomulas fo the mass of photon in esting and moving dieleti and the veloity of photon as patile The veloity is diffeent fom phase and goup veloity Quite inteesting is the fat that the photon veloity is equal to not only in vauum but also if ε onsequene of desibing the medium though sala and veto potentials is existene of analogs of eleti and magneti fields, as well as the optial Loentz foe whih desibe the influene of the medium on the photon The same as in the theoy of haged patiles the potentials ae gauge fields and eleti and magneti fields ae gauge invaiant CKNOWLEDGEMENTS I am vey gateful to Pof B Janewiz fo a helpful oespondene and assistene in pepaation of the pape fo publiation 4

15 utho ontat infomation: Jaosław Zaleśny, Institute of Physis, Tehnial Univesity of Szzein, l Piastów 48, Szzein, Poland jaek@aadiatunivszzeinpl 5

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17 [9] L D Landau and E M Lifshitz, Eletodynamis of Continuous Media, (Pegamon, Oxfod 984) [0] S Solimeno, B Cosignani, P DiPoto, Guiding, Diffation, and Confinement of Optial Radiation, (ademi Pess, In, 986) 7