Photon Hall Effect in Atomic Hydrogen B.A. van Tiggelen1 and G.L.J.A. Rikken2 1Universit´e Grenoble 1/CNRS, LPMMC UMR 5493, B.P. 166, 38042 Grenoble, France 2LNCMI, UPR 3228 CNRS/INSA/UJF Grenoble 1/UPS, Toulouse & Grenoble, France (Dated: August 17, 2011) 1 1 We predict a photon Hall effect in the optical cross-section of atomic hydrogen, determined by 0 theinterferencebetweenanelectricquadrupoletransitionandanelectricdipoletransitionfromthe 2 ground state to 3D3/2 and 3P3/2, causing a magneto-transverse acceleration comparable to g. The electric dipole transition only generates a Hall effect when theatom is moving. g u A Light scattering exchanges momentum between mat- cross-section of an atom is expressed by the Kramers- 6 ter and radiation. The momentum of light in matter Heisenberg formula [9], 1 has been subject to a number of controversies. One re- cent debate concerns the exchange of momentum with ] dσ ω3 s the quantum vacuum [1]. (ωkε→ω k ε )= α2 s |f (ω)+f (ω)+···|2 c Light scattering is affected by a magnetic field. One dΩ s s s ω3 ED EQ i W(ω ,k ,ε ) t feature,thephotonHalleffect(PHE),wasfirstpredicted × 1+ s s s (1) p o in multiple light scattering [2], and observed shortly af- W0(ωs) . terwards [3] with typical changes in the photon flux of s ∆T /T = 10−5 per Tesla of applied magnetic field. Here, α is the fine structure constant, ω and ωs < ω c PHE are incident and scattered frequency, ε and ε are the i A Mie theory for the PHE [5] agreedquantitatively with s s polarization vectors of incident and scattered radiation. y the experiments. Given the wave number k of the in- The last factor accounts for stimulated enhancement h cident photon flux and the magnetic field B , the PHE p inducesanexchangeofmomentumbetweenscattererand (SE) of the cross-section caused by a radiation density [ W per steradian, per bandwidth, per polarization, with 1 raalodniagtBio×nikn.tAhelimghatgflnuetxoo-tfr1a0n4svWer/sme(2”iunpcwidaerndt”o)ndaire1c0tµiomn W0 = ~ωs3/(2πc0)3 its value for the quantum vacuum. In this work we will ignore SE and leave the study of v particlewitharelativePHEof10−5perTeslaexperiences 9 atransverseforceof10−19N/T,roughlyequivalenttothe ”stimulatedHallscattering”,likelyto become important 8 for W ≈ W , to future work. Finally f(ω) is the com- Lorentz force on a charge e moving with a velocity of 1 0 1 plex scattering amplitude associated with transitions in 3 m/s. the atom, that can be either elastic or inelastic. We will . Atoms are strong light scatterers with elastic opti- 8 concentrate first on the ED transition between the un- cal cross-sections of order of λ2 near optical transitions 0 polarized singlet ground state 1S and the eight 3P 1 and with promising applications in mesoscopic physics 1/2 3/2 hyperfine states in atomic hydrogen, for which 1 [4]. When the typical Zeeman splitting 1ω (ω = 2 c c : eB/m = 17.5 MHz/Gauss is the cyclotron circular fre- v e i quency)equalstheatomiclinewidth(γ ≈100MHz),the X ar onnse[5iipeott]t.ithoiiPcecnTraHsfiol,fEeocglsrrdiewcon,tascatsnaery-dspntoeihocaccetncnaiuzldoElerynrDboftiaosrcmPrukseHaiwpgiEufannorrtirefidianci,aneacolsnfeoretcltwldtbhyreeiGacattw-slaodtyeuemimeprsnesomgd.luaeeNbzptr(eyoEysan,tnDehbodt)eenhtedemtwlroeceaawsaengsnnn--, circular frequency (MHz) -11-5500000000 1 3P3/22 3 4 5 circular frequency (MHz) -11-5050000000 3D13/2 2 3 4 5 considerthescatteringfrompairsofatoms. Foradensity magnetic field (Gauss) magnetic field (Gauss) of n = 1018/m3 88Sr atoms the relative PHE can be as large as a few percent [6]. Can the PHE of a single atom exist at all, and how FIG. 1: Hyperfine structure of the 3P3/2 (left) and 3D3/2 (right) level of atomic hydrogen, as a function of magnetic large will the magneto-transverse momentum transfer field. Equal colors indicate equal values for the hyperfine to the atom be? We will focus on atomic hydrogen, magneticquantumnumberm. Theheight oftheverticalbar whose physics in a magnetic field has been studied in ontherightindicatesthelinewidthγ. Thezeroinfrequency great detail [7], with important applications to the for- ischosenatthefinestructurelevelof3P3/2. Theoneof3D3/2 mationofthestablespin-polarizedphase[8]. Theoptical is 21.07 MHzlower. 2 ω2 8 (r ·ε )(r ·ε) ω2r2 f (ω)= fH s H0 = 3P ×[P (ε ·ε)+[P −P ](ε ·zˆ)(ε·zˆ)+P (ω)iε ·(ε×zˆ)] (2) ED 0 s 1 0 s 2 s c ω−ω −iγ c 0 HX=1 H P 0 Here r = hf|r|Hi is the ED matrix element be- from the product states of orbital momentum, electron fH tween intermediate and final state, and r = 0.517a spin and proton spin with appropriate Clebsch-Gordan 3P 0 the radial matrix element. In a magnetic field of coefficients, that constitute the amplitudes P , together i a few Gauss all hyperfine levels are non-degenerate, with the 8 line profiles f (ω) = 1/(ω−ω −iγ ). We H H P with an (anomalous) Zeeman splitting comparable to choose kˆ = xˆ, Bˆ = ˆz and let Bˆ ×kˆ = yˆ be the Hall the hyperfine splitting (Fig 1a). The eigenfunctions direction. The differential cross-section, averaged over (cid:12)(cid:12)3Pj=32E⊗(cid:12)f =j± 12,m=−f,..,f(cid:11) can be constructed incident polarization, becomes, (cid:12) (cid:12) dσ α2ω4r2 (ωk→ω k ) = 3P × dΩ s s 2c2 0 |P |2(1 − (kˆ ·yˆ)2)+|P |2(1−(kˆ ·ˆz)2)+|P |2(1−(kˆ ·xˆ)2)+2Im(P P∗)(kˆ ·xˆ)(kˆ ·yˆ) (3) 0 s 1 s 2 s 0 2 s s The last term in Eq. (3) affects the current along the induced ”virtual charge’ as, Hall direction though without inducing a net PHE.This will change when the atom moves. Let K be the frame q = 13πα2ω4r34P Im(P0P2∗)I(k) that moves with the atom, and in which Eq. (3) applies, v 10 c4 B 0 andK′ theoneinwhichtheatommoveswithvelocityv. Im(P P∗)(MHz)−2 W The cross-section dσ relates an incident flux ρ(ω,k)/c ≈ 0.0061e 0 2 ∆ω(MHz)∆Ω(rad) 0 B(Gauss) W to an outgoing current ρ (ω ,k )dΩr2/c at a distance 0 s s s 0 (5) r in the far field. The latter is unaffected by a Lorentz transformation in order v/c , and since the radiation 0 Thisisvalidforincidentwavevectorseitheralignedwith density ρ transforms as ρ′ = (ω′3/ω3)ρ [10] it follows or opposed to the velocity. One could thus use oppo- that, site beams to reduce longitudinal kicks. For this virtual charge to be a genuine charge, an external electric field dσ′ ωω′ 3 dσ dΩ′ (ω′k′ →ωs′k′s) ≈ (cid:18)ω′ωss(cid:19) dΩ(ωk→ωsks) ssyhmoumldetirnyduimcepoasfeotrhceeqlivnEeaorneltehcetroatcormossa-tsercetsito.nPtoanbdeoTf We insert kˆ ≈kˆ′ (1+kˆ′ ·v/c )−v/c and ω ≈ (s) (s) (s) 0 0 (s) ω′ (1−v·kˆ′ /c ) and assume for simplicity that the (s) (s) 0 atom moves either parallel or opposite to the incident v x B wavevectork. TheLorentztransformofthelasttermin Eq.(3)generatesaterm(v·xˆ′)/c )[1−2(kˆ′·xˆ′)2](k′·yˆ′), 0 s s withyˆ′ =Bˆ×kˆ′, xˆ′ =kˆ′ (see Fig. 2). ItexhibitsaPHE with a momentum transfer to the atom, v dσ′ 1 F = − d2Ω′~k′ (ω′k′ →ω′k′) I(k′) Z sdΩ′ s s ~ω′ 13πα2ω4r4 Im(P P∗) v = 3P 0 2 I(k) ×B (4) 10 c3 B c 0 0 FIG.2: Polar diagram oftheHallcross-section -last termin with I = 2Wc0∆ω∆Ω the incident flux (W/m2). This Eq.(3)-associated withtheED3P3/2 transitionofamoving PHE-induced momentum transfer is reminiscent of the atom with v=0.4c0 (incident kkv). Solid indicates increase, magneticforce ona chargedparticle. We canidentify an dashedadecrease. Anetphotoncurrentresultsalongv×B. 3 the formF(kˆ·kˆ )E·(kˆ−kˆ ). However,the EDapprox- steradianincidentdivergence. Withthischarge,ahydro- s s imation does not allow any odd sequences of k-vectors. genatomataspeedof10m/sinafieldof1Gausswould The ”induced charge” is thus not a real electromagnetic undergo a transverse deflection of order 4 mm/s2. The charge. effect is thus small, and of course it would be interesting Fig. 3a shows that the ”induced charge” is of order toinvestigateifthiseffectincreaseswhenW ≫W . This 0 10−8e per (rad)MHz line width and per steradian (as- magneto-transverse force vanishes if the incident radia- suming W = W , B = 1 G). When integrated over the tionfieldisisotropic. Inparticular,thequantumvacuum 0 full line profile,we obtaina chargeq =−5·10−8e for0.1 does not induce a nett charge. -9 aition induced charge (10 e) ---------544332211-050505050055 e+l ainsetilcasti c elastic 2 onic Hall cross-section (a)0-202468 elast ic elastic + inelastic -6 )dv/ dv(10HALLlong 1024680 rad --6505-300 -200 -100 0 100 200 300 Phot-4-300 -200 -100 0 100 200 300 -2-300 -200 -100 0 100 200 300 detuning (radMHz) detuning (radMHz) detuning (radMHz) FIG. 3: Left: The virtual charge qv for the ED transitions from the unpolarized ground state to the 3P3/2 hyperfinelevels in a magnetic field of 1 Gauss (detuning = 0 chosen for the fine structure energy level of 3P3/2). Red line represents the contribution of the elastic transition; Black line includes the inelastic transitions to the ground state levels with f = 1. We adopted W = W0, ∆ω = 1 MHz and ∆Ω = 1 sterad. Middle: PHE cross-section from the interference between the ED transition from the spin-singlet ground state 1S to the 3P3/2 level and the EQ transition to the 3D3/2 level for B =5 G. The red line only counts the elastic transition, the black line includes Raman transitions. Right: Magneto-transverse deflection of thehydrogen atom subject toB =5 G. The second mechanism that can create a PHE by a they overlap and allow the interference between the ED single atom is the interference with electric quadrupole transition from the unpolarized ground state 1S to 1/2 (EQ) radiation. Though common in Mie scattering, the 3P and the EQ transition to 3D . This gives rise 3/2 3/2 strong (hyperfine) splitting and the strict selection rules to an interesting ”which-way” event inside the hydrogen toexcitedifferentmultipoles,makesucheventsextremely atom. We adopt the Hall geometry for which k∼x and rarein atomicsystems. The line profilesofthe hydrogen B∼z. For the elastic EQ transition we find, levels 3P and 3D of hydrogen are unique in that 3/2 3/2 1ω2 8 (k ·r )(r ·ε )(k·r )(r ·ε) ω4q2 f (ω)= s fH fH s H0 H0 = 3D × EQ 4 c ω−ω −iγ 75c3 0 HX=1 H D 0 Q(m,i) (kˆ ·ˆr )(ε ·ˆz)+(kˆ ·ˆz)(ε ·ˆr ) (kˆ·ˆr )(ε·ˆz)+ Q(m)(kˆ ·ˆr )(kˆ·ˆr )(ε ·ˆr )(ε·ˆr ) 1 s ± s s s ± ∓ 2 s ± ∓ s ± ∓ m=±X1;i=1,2 h i mX=±2 withγ =32MHz the naturallinewidthofthe3Dlevel states m = ±2 concern only the hyperfine level f = 2, D andq3D =0.867a20. The summationsareoverthe hyper- sothat Q2(m=±2) =1/(ω−ωm−iγQ), whereasthe states fine magnetic quantum numbers;the magnetic hyperfine with m=±1 split into two orthogonalsuperpositions of 4 f =1,2hyperfinelevelsindicatedbytheindexi. Excited section (summed over the 2 outgoing polarizations, and stateswithm=0intheEQamplitudedonotcontribute averaged over the 2 incident polarizations), but not all to the PHE.The interferencewith the EDamplitude (2) cross-terms generate a PHE. Those who contribute to gives rise to many contributions to the differential cross- the elastic Hall cross-section add up to, dσ α2ω6r2 q2 1 (PHE−elastic) = 3P 3D kˆ ·yˆ ×2Im P∗ Q(−1,i)−Q(1,i) 1−(kˆ ·ˆz)2 dΩ 75 c40 (cid:16) s (cid:17)  1 iX=1,2(cid:16) 1 1 (cid:17)2h s i 1  1 + P∗ Q(−2)−Q(2) (kˆ ·ˆz)2+2(kˆ ·xˆ)2 −P∗ Q(−2)+Q(2) 1−2(kˆ ·xˆ)2 (6) 0 (cid:16) 2 2 (cid:17)2h s s i 2 (cid:16) 2 2 (cid:17)2h s i(cid:27) The total photonic Hall cross-sectionσ determines the roughlyfivelinewidths. Ifwearrangeradiationfluxand H total Hall current in the y-direction. In Figure 3a we time slot such that the atom stops accelerating at this show the results for a magnetic field of 5 Gauss. The moment it will have undergone a transverse shift of 0.3 cross-section is of order a2 as expected from multipolar µm some 10 meters further on. With a radiation flux 0 scaling arguments [9]. The PHE normalized to the ED of I = 6 W/m2 inside a line width of 1 radMHz we transversecross-sectionis maximallyof order0.1α2, and avoid stimulated emission and gives us typical time of thusonlysomewhatsmallerthanthetypicalvaluesfound t = v mc /σ (0)I = 300 µs to accomplish this. This k 0 ED for Mie scattering at 1 Tesla [5]. Once excited to either time is larger than the estimated time of 200 µs to ac- 3P or 3D it is possible for the hydrogen atom to complish the EQ transition. 3/2 3/2 decay inelastically to a polarized hyperfine triplet state. In conclusion, we have quantified the magneto- These transitions have been included to obtain Fig. 3b, transverse scattering of light from unpolarized atomic and slightly change the total Hall cross-section. hydrogen. When moving, the electric dipole transition Scattering of light transfers momentum to the atom. gives a contribution of order v/c , when at rest the Hall The dominating ED radiation will accelerate the atom 0 angle is of order α2 per Gauss, with α the fine structure only along the incident radiation. For an incident en- constant. The study of this effect in the spin-polarized ergyfluxI(k)theaccelerationisdv /dt=I(k)σ /mc . k ED 0 S-state of atomic hydrogen [8] is a next step. This work The total elastic ED cross-section is easily seen to be was supported by the ANR contract PHOTONIMPULS (4π/3)(ω4r4 α2/c2) |P |2,withasimilarcontribution 3P 0 i i ANR-09-BLAN-0088-01. from the inelastic trPansitions. The PHE cross-section induces a magneto-transverse acceleration dv /dt = ⊥ I(k)σ (ω)/mc . For a constant flux I = W c dωdΩ Hall 0 0 0 over the line profile we estimate from Fig 3b that dv⊥/dt ≈ I0(k)600a20dΩ/m ≈ 10 m/s2 for 0.1 steradian [1] A. Feigel, Phys. Rev. Lett. 92, 020404 (2004); S. Kawka angular divergence. This equals the gravitational accel- and B.A. van Tiggelen, EPL 89, 11002 (2010). eration. One uncertain element in this approach is the [2] B.A. van Tiggelen, Phys. Rev. Lett. 75, 422 (1995). largetime toaccomplishthe EQtransition,estimatedto [3] G.L.J.A. Rikkenand B.A. van Tiggelen, Nature 381, 54 be of order 1/γ α2 = 200 µs . The acceleration seems (1996). P [4] C.A.Mueller,C.Miniatura,D.Wilkowski,R.Kaiserand instantaneous at time scales of seconds, but a dynamic D. Delande, Phys. Rev. A 72, 053405 (2005). theory should be developed to understand this better. [5] D. Lacoste, B.A. van Tiggelen, G.L.J.A. Rikken,and A. Themagneto-transversedeflectioncanbeexpressedas Sparenberg, J. Opt. Soc. Am.A15, 1636 (1998). dv⊥/dvk = σH(ω)/σED(ω), which is independent on ra- [6] B. Gr´emaud, D. Delande, O. Sigwarth, CH. Miniatura, diationintensity,andisshowninFigure3basafunction Phys. Rev. Lett. 102, 217401 (2009). of detuning. 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