Photo-Emission of a Single-Electron Wave-Packet in a Strong Laser Field
Justin Peatross, Carsten Müller, Karen Z. Hatsagortsyan, Christoph H. Keitel
AAPS/123-QED
Photo-Emission of a Single-Electron Wave-Packet in a Strong Laser Field
Justin Peatross † , ∗ Carsten M¨uller, Karen Z. Hatsagortsyan, and Christoph H. Keitel
Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany and † Dept. of Physics and Astronomy, Brigham Young University, Provo, UT 84602 (Dated: November 11, 2018)The radiation emitted by a single-electron wave packet in an intense laser field is considered. Arelation between the exact quantum formulation and its classical counterpart is established via theelectron’s Wigner function. In particular we show that the wave packet, even when it spreads to thescale of the wavelength of the driving laser field, cannot be treated as an extended classical chargedistribution but rather behaves as a point-like emitter carrying information on its initial quantumstate. We outline an experimental setup dedicated to put this conclusion to the test.
PACS numbers: 41.60.-m, 42.50.Ct , 42.50.Xa
The availability of super-intense lasers has stimulatedinterest in relativistic electron dynamics in strong drivingfields [1]. Experimenters have observed the effects of pon-deromotive acceleration, the Lorentz drift, and plasmawake-fields through direct detection of electrons ejectedfrom an intense laser focus. Photoemission from rela-tivistically driven plasmas has also been studied.Much theory and computational effort has been de-voted to the dynamics of free-electron wave packetsdriven by intense fields [2, 3, 4] and the associated scat-tered radiation [3, 5, 6]. Coherent emission from manyelectrons can be viewed in the forward direction withthe emerging laser beam. Here, we consider incoherentphotoemission by free electrons out the side of a focusedlaser, as a means of studying electron dynamics.A free electron wave packet with an initial spatialsize on the scale of an atom undergoes natural quan-tum spreading, which eventually reaches the scale of anoptical wavelength, as illustrated in Fig. 1 [4]. More-over, an electron wave packet born through field ioniza-tion is pulled from its parent atom at a finite rate, typi-cally emerging over multiple laser cycles. This, combinedwith the Lorentz drift and sharp ponderomotive gradientsfound in a tight relativistic laser focus, can cause a single-electron wave packet to be strewn throughout a volumeseveral laser wavelengths across [6]; different portions ofthe same wave packet can even be propelled out oppositesides of the laser focus.The question naturally arises as to how a single-electron wave packet radiates, especially when it under-goes such highly non-dipole dynamics, where differentparts of the electron wave packet experience entirely dif-ferent phases of a stimulating laser field. So far, the prob-lem has been treated [5, 6] within an intuitively appealingmodel where the quantum probability current is multi-plied by the electron charge to produce an extended cur-rent distribution used as a source in Maxwell’s equations[7]. The intensity computed classically from the extendedcurrent distribution is then associated with the probabil-ity of measuring a photon. Due to interference, this ap-proach can lead to dramatic suppression of radiation for many directions. We note that semi-classical descriptionshave, in general, proven very useful to understand pro-cesses like above-threshold ionization or high-harmonicgeneration, which arise from intense laser-matter inter-actions [8].In this letter, we provide a fully quantum mechani-cal treatment of photoemission by a single-electron wavepacket in a laser field and relate it to a classical de-scription via the electron’s Wigner function. We showthat no interference occurs between emission from dif-ferent parts of an initially Gaussian wave packet, evenif spatially large. In a plane-wave driving field, this re-sult holds for wave packets of any shape and size. Theradiative response can be mimicked by the incoherentemission of a classical ensemble of point charges. In afocused laser beam, interferences can occur for certaininitial electron states, but these interferences are of adifferent nature than those arising from a semi-classicalcurrent-distribution approach. We outline an experimen-tal arrangement able to probe the single-electron emis-sion behavior by combining methods from strong-fieldphysics and quantum optics.We first examine radiation interferences that arise fromtreating a single electron as an extended charge distribu-tion in a semiclassical picture. Fig. 1 shows an example
FIG. 1: An electron wave packet after natural spreading froman initially Gaussian-shaped size of 1˚A. The spreading takesplaces during 190 cycles in a plane wave with intensity 2 × W / cm and wavelength λ = 800 nm. a r X i v : . [ qu a n t - ph ] D ec of an electron wave packet (probability density), com-puted using the Klein-Gordon equation [4]. In the low-intensity limit, a wave-packet such as shown in Fig. 1can be associated with a classical Gaussian current dis-tribution: J ∼ ˆ zr − e − r /r e iκx , where r characterizesthe spatial extent of the distribution with fixed overallcharge. The distribution is stimulated by a plane wavewith wave four-vector κ = ( ω κ , κ ) traveling in the x -direction and polarized in the z -direction. The intensityof the Thomson scattering from the distribution is I ( θ, φ ) ∼ sin θ e −| κ | r (1 − sin θ cos φ ) , (1)where θ = 0 defines the z-direction, and θ = π/ φ = 0 defines the x-direction. Fig. 2 shows the inten-sity emitted into several directions, as well as the overallemission, as a function of distribution size. The forwardemission does not vary from that of a single point oscilla-tor with equal net charge. In the perpendicular direction,the intensity drops by orders of magnitude as the wavefunction grows to the scale of the wavelength or bigger.This leads to a substantial loss in the overall scattered-light energy [9]. FIG. 2: Efficiency of light scattering into different directionsas a function of size of a driven Gaussian charge distribution.
Measurements of Compton/Thomson scattering pro-vide an indication that electrons do not radiate as ex-tended charge distributions. For example, >
10 keV pho-tons scattered from electrons bound to helium corre-sponds to a scenario where the size of the electron wavepacket is larger than the wavelengths involved. However,the cross section for the scattered photons with energywell below the electron rest energy is known [7, 10] to fol-low Eq. (1) with r = 0 (with φ averaged over all anglesfor unpolarized light). It is interesting to note that A.H. Compton initially proposed a “large electron” modelto explain the decrease in cross section with angle forharder x rays, which he later abandoned when the effectof momentum recoil was understood [11]. Quantum electrodynamics provides the general frame-work to calculate the radiation from a single-electronwave packet. The radiated intensity is proportionalto the expectation value of the correlation function ofthe current density, rather than the expectation valueof the current density of the wave packet [12]. Fromthe definition of the spectral intensity of the emittedradiation dε k = c π (cid:68) ˆ H ( − ) ω ˆ H (+) ω (cid:69) R d Ω dω , with radi-ation magnetic field ˆ H ( ± ) ω = i k × ˆ A ( ± ) ω and vector-potential ˆ A ( ± ) ω = e e ikR cR (cid:82) d x ˆ J ( ± ) ( x )e ikx , together withthe current-density operators ˆ J ( ± ) in the Heisenberg rep-resentation, one derives dε k = e π c (cid:90) d x (cid:90) d x (cid:48) e ik ( x − x (cid:48) ) (cid:68)(cid:16) k × ˆ J ( − ) ( x ) (cid:17) (cid:16) k × ˆ J (+) ( x (cid:48) ) (cid:17)(cid:69) d Ω dω. (2)Here ( ± ) indicates the positive/negative frequency partsof operators, R the distance to the observation pointfrom the coordinate center, k the radiation wave-vector, d Ω the emission solid angle, e the electron charge and c the speed of light. Equation (2) can be represented ina more familiar form via the transition current density J p (cid:48) i ( x ) in the Schr¨odinger picture: dε k = e π c (cid:90) d p (cid:48) (2 π (cid:126) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) d x ( k × J p (cid:48) i ( x )) e ikx (cid:12)(cid:12)(cid:12)(cid:12) d Ω dω (3)where J p (cid:48) i ( x ) = (cid:16) ¯ ψ p (cid:48) ( x ) γ ψ (L) i ( x ) (cid:17) , ψ (L) i ( x ) is the initialelectron wave packet in the laser field, ψ p (cid:48) ( x ) is a com-plete set of free electron states, and γ are the Dirac ma-trices. Equation (3) indicates that the total probabilityof photon emission should be calculated as an incoherentsum over the final momentum states of the electron, eventhough in the experiment the final electron momentumcould be undetected.Although Eq. (3) shows the general way to calculatethe emission intensity, it is difficult to apply in a real ex-perimental situation, as the quantum eigenstates of theelectron in a focused laser beam are usually unknown.It is therefore indeed desirable to mimic the quantumelectrodynamical result of Eq. (3) by means of classi-cal electrodynamical calculations (in the quasi-classicallimit, when the photon energy is much smaller than theelectron rest energy and recoil effects are negligible).We demonstrate how to do this by way of the exam-ple of one-photon Thomson scattering in a focused laserbeam. Choosing the initial wave packet in the form of ψ i ( x ) = (cid:82) d p α ( p ) ψ p ( x ), one can express the quantum-mechanical formula for spectral intensity of Eq.(3) viathe Wigner function ρ w ( r , p ) = (cid:82) d q α ( p + q / α ∗ ( p − q / i qr of the initial electron wave packet as follows: dε k λ = e ω π c (cid:90) d r (cid:90) d p ρ w ( r , p ) M k λ ( r , p ) d Ω dω, (4)where M k λ ( r , p ) = (2 π (cid:126) ) (cid:82) d κ (cid:82) d κ (cid:48) e i ( κ − κ (cid:48) ) r × A ( κ ) A ∗ ( κ (cid:48) ) (cid:61) λ ( p + , p (cid:48) , κ , k ) (cid:61) ∗ λ ( p − , p (cid:48) , κ (cid:48) , k ) δ ( ε p + + (cid:126) ω κ − ε p (cid:48) − (cid:126) ω ) δ ( ε p − + (cid:126) ω κ (cid:48) − ε p (cid:48) − (cid:126) ω ) givesthe radiation by an electron of momentum p ; (cid:61) λ ( p , p (cid:48) , κ , k ) is defined via (cid:82) d x e ikx ( e ∗ λ J p (cid:48) p ) =(2 π (cid:126) ) (cid:82) d κA ( κ ) (cid:61) λ ( p , p (cid:48) , κ , k ) δ (4) ( p + (cid:126) κ − p (cid:48) − (cid:126) k ), ε p = c (cid:112) p + m c , p ± = p ± (cid:126) ( κ (cid:48) − κ ) / p (cid:48) = p + (cid:126) ( κ (cid:48) + κ ) / − k , e λ is the polarizationof the emitted photon, and A ( κ ) the Fourier com-ponent of the focused laser beam. When the Wignerfunction is non-negative (e.g., for a Gaussian wavepacket), it may be interpreted as the initial electrondistribution in phase space. The message of the structureof Eq. (4) is that the total photoemission probability isan incoherent sum over the contributions of each localphase-space element of the electron distribution. If, forexample, the phase-space distribution consists of twoseparate parts: ρ w = ρ (1)w + ρ (2)w , then the intensitiesemitted from each part incoherently add up to yieldthe total radiation intensity. In the quasi-classicallimit, the term M k λ ≈ |M k λ ( r , p ) | , with M k λ ( r , p ) =2 π (cid:126) (cid:82) d κ e i κr A ( κ ) (cid:61) λ ( p , p (cid:48) , κ , k ) δ ( ε p + (cid:126) ω κ − ε p (cid:48) − (cid:126) ω )can be directly related to the classical electrody-namical calculation: in the nonrelativistic limit (cid:61) λ ( p , p (cid:48) , κ , k ) = ( e ∗ λ e κ ) ( e κ is the polarization of the κ -component of the driving field) and Eq. (4) describesthe incoherent average of the radiation intensity overthe initial electron probability distribution in phasespace. The radiation can thus be modelled by a classicalensemble of point emitters, taken individually.A different situation arises when the Wigner function isnegative in some phase-space region, indicating intrinsicquantum behavior. As an example, we consider an elec-tron in a superposition of two momentum states | p > and | p > ; in this case interference in the emitted radi-ation is possible. In fact, the final electron state | p (cid:48) > with emission of a photon of certain momentum k canbe reached by two indistinguishable paths [13]: eitherfrom the state | p > or from | p > by absorption ofdifferent photons κ , = k + p (cid:48) − p , from the exter-nal field, giving rise to interference. This effect is in-cluded in Eq. (4) which cannot be modelled by classicalmeans here. In the case of a Gaussian wave packet in-terferences are suppressed by the continuous spectrumof initial electron momenta which give rise to many in-terfering paths whose contributions largely cancel out.In any case, interferences do not occur in a plane-wavedriving field due to its uniform propagation direction κ . Then Eq. (4) reduces to the incoherent superposition dε k λ = (cid:82) d p | α ( p ) | ε k λ ( p ) d Ω dω of the spectral energies ε k λ ( p ) = e ω π c M k λ (0 , p ) radiated by an electron of mo-mentum p [14].The quantum interference effects above have to beclearly distinguished, though, from those arising in the coherent part of the radiation mentioned in the intro- duction. The latter is calculated employing an ensembleaverage of the current operator < J ( x ) > as a sourcefor the expectation value of the radiated field < E > .In that picture, the spectral component of the coherentradiation intensity | < E k > | involves interference ofthe fields in the classical sense; i.e., all transitions withemission of a photon of certain momentum k interfere,independent of the final electron state. In the exam-ple of two superimposed momentum states, these are thetransitions | p i > → | p j > with i, j ∈ { , } . The coher-ent radiation is analogous to the classical radiation of amodulated charge distribution here.The coherent radiation is qualitatively different fromthe total radiation in Eq. (4). It accounts for the radi-ated field averaged over the quantum ensemble, which ex-cludes the incoherent field with a fluctuating phase [15].In the case of a single electron, the coherent radiationis only a small part of the total radiation, but becomesdominant in the case of N (cid:29) N -atom ensemble [12], the in-tensity of the phase-matched coherent radiation is multi-plied by a factor of N ( N − N, which becomes negligible. Forexample, any experiment on high-harmonic generationfrom (many) atoms measures the coherent emission (see,e.g., [16]). We note that the terminology of a “single-atom response” that is commonly used in this context, istherefore misleading.Finally, we show that the radiation scattered from asingle free electron in a laser is detectable by modern ex-perimental techniques. The feasibility of seeing scatteredlight depends crucially on whether the electron wavepacket radiates with the strength of a classical point-likeelectron, as argued above. If one looks at light near thefrequency of the stimulating light (linear Thomson scat-tering), the rate of emitted photons is proportional to thestimulating intensity. Thus, beam fluence rather thanpeak intensity determines the number of photons scat-tered by an electron in a laser. Nevertheless, relativisticintensities may be helpful for spectral-discrimination pur-poses: Because the Lorentz drift pushes the electron inthe forward direction of the laser, the scattered funda-mental light is typically red shifted from ∼
800 nm tolonger wavelengths, which is a signature that could bedistinguished with bandpass filters and detected with anavalanche photodiode.We envision an electron born in the laser field (e.g. thesecond electron pulled from He at ∼ × W / cm ).Electrons given up by atoms during the leading edge ofthe pulse tend to be pushed out of the focus by the pon-deromotive gradient. The density of donor atoms can bechosen such that on average one electron experiences thehighest intensities (10 − torr–10 − torr, depending onthe focal volume). Free electrons might also be preparedusing a suitable pre pulse. Only an electron experienc-ing the highest intensity receives a substantial forwarddrift and the accompanying red shift for emission in thedirection perpendicular to laser propagation. Keying inon this red-shift signature may be critical for differenti-ating authentic scattered photon events from other noisesources. Fast timing in the photon-detection electron-ics can also suppress false signals, for example, scatteredfrom the walls of the experimental chamber.We computed a representative single classical electrontrajectory in a tightly focused vector laser field with apeak intensity of 10 W / cm , duration 35 fs, and wave-length 800 nm. Fig. 3(a) shows the total radiated energyin the far field emitted from the electron as it is releasedon the rising edge of the pulse. The electron trajectoryeventually exits the side of the focus due to ponderomo-tive gradients. Much of the scattered radiation emergesout the side of the laser focus. The total radiated energy(all angles and frequencies) from the single electron tra-jectory is only ∼ .
24 eV, indicating less than one photonper shot.Fig. 3(b) shows the spatial distribution of light withwavelengths falling between 850 nm and 950 nm. This ac-counts for approximately 20% of the total emitted power.Assuming 10% collection efficiency, this would amount toan average 0.005 eV energy per shot, or one photon per300 shots. This can of course be increased if more elec-trons are used, where the light radiated out the side ofthe focus adds incoherently for a random distribution.The observation of blue-shifted light may also afford anopportunity for discrimination against background. In-tensities above 10 W / cm are not ideal for this experi-ment; a strong Lorentz drift redirects the photoemissioninto the far forward direction.In conclusion, we have studied the amount of light thatan electron scatters out the side of a laser focus. We have (a) (b) x - a x i s y - a x i s z - a x i s x - a x i s y - a x i s FIG. 3: (a) Far-field intensity (at 100 µ m distance) of lightscattered from a single electron trajectory born on axis duringthe early rising edge of an intense (10 W/cm ) laser pulse.Red (blue) indicates regions of high (low) intensity. The laserbeam (mesh) has waist w = 3 λ . The total scattered energy is0.24 eV. (b) Far-field intensity filtered to wavelengths between850 nm and 950 nm with total scattered energy 0.05 eV. shown that individual electrons radiate with the strengthof point emitters. The electron’s initial quantum state isimprinted on the radiation spectrum via its Wigner func-tion, which in general allows for interference of differentelectron momentum components. The latter is qualita-tively distinct from the classical interference in the co-herent radiation of an extended charge distribution. Ourresults can be tested in an experiment that combines forthe first time the sensitive techniques of quantum optics(e.g., single-photon detectors) with the traditionally op-posite and incompatible discipline of high-intensity laserphysics.The authors acknowledge Guido R. Mocken, J¨org Ev-ers, Mikhail Fedorov, and Michael Ware for helpful input. ∗ Electronic address: [email protected][1] Y. I. Salamin et al. , Phys. Rep. , 41 (2006); G. A.Mourou, T. Tajima, and S. V. Bulanov, Rev. Mod. Phys. , 309 (2006).[2] J. San Rom´an, L. Roso, and H. R. Reiss, J. Phys. B ,1869 (2000); J. San Rom´an, L. Plaja, and L. Roso, Phys.Rev. A , 063402 (2001); M. Mahmoudi, Y. I. Salamin,and C. H. Keitel, ibid. , 033402 (2005).[3] G. R. Mocken and C. H. Keitel, Comp. Phys. Comm. , 171 (2005).[4] J. Peatross, C. M¨uller, and C. H. Keitel, Opt. Expr. ,6053 (2007).[5] P. Krekora et al. , Laser Phys. , 455 (2002).[6] E. A. Chowdhury, I. Ghebregziabiher, and B. C. Walker,J. Phys. B , 517 (2005).[7] J. D. Jackson, Classical Electrodynamics (Wiley, NewYork, 1998), 3rd ed., Eq. (14.70); Eq. (14.124).[8] P. B. Corkum, Phys. Rev. Lett. , 1994 (1993).[9] Interferences in the radiation pattern from a classicalcharge distribution as indicated by Fig. 2 imply a cor-responding amount of work exchanged between differentportions of the distribution via the near-field terms. Theimplication of near-field work is problematic in the con-text of a single electron, since one does not write a Hamil-tonian for the interaction between different parts of thesame electron wave function. Note that radiation reactionis negligible in the regime considered.[10] E. O. Wollan, Phys. Rev. , 862 (1931).[11] R. H. Stuewer, The Compton Effect , (Science HistoryPublications, New York, 1975).[12] P. L. Knight and P. W. Milonni, Phys. Rep. , 21(1980); B. Sundaran and P. Milonni, Phys. Rev. A ,6571 (1990); J. H. Eberly and M. V. Fedorov, ibid. ,4706 (1992).[13] The two paths leading to the same final states of the elec-tron and the emitted photon are indistinguishable as theabsorption of photons with different κ , from the exter-nal classical field are not detectable, even in principle,within the uncertainty of the driving coherent state.[14] As a consequence, in a plane-wave field Thomson scat-tering from a Gaussian electron wave packet does notdepend on the arrival time of the applied laser pulse, i.e.,it does not depend on the size of the wave packet which freely spreads in the time preceeding the interaction.[15] D. Marcuse, J. Appl. Phys. , 2255 (1971).[16] A. L’Huillier et al. , Phys. Rev. A , 2778 (1992); W. Becker et al. , ibid.56