High-Field Laser Physics (2,3) ETH Zürich Spring Semester 2010 H. R. Reiss

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1 Hgh-Feld Laser Physcs (,3) ETH Zürch Sprng Semester 1 H. R. Ress

2 DIPOLE APPROXIMATION (LONG-WAVELENGTH APPROXIMATION) The dpole approxmaton [or electrc-dpole approxmaton (EDA)] s used almost unversally n Atomc, Molecular and Optcal (AMO) Physcs. It s not generally recognzed that ts valdty s lmted by eld ntensty. We start wth the customary hgh-requency lmt on the EDA. A plane wave eld n general s a propagatng eld wth trgonometrc dependence on a phase actor t k r, k kˆ /c kˆ /, where kˆ s the unt vector n the propagaton drecton. The mportant range o r s when the wave overlaps the atom, or r =O(1 a.u.). Dependence o the phase on r can be neglected when kr << 1, or /c << 1, or << 137. (Fne-structure constant = e / ħc n Gaussan unts; so c = 1/ = a.u.) t kr t s called the dpole approxmaton or electrc dpole approxmaton (EDA) or long-wavelength approxmaton (LWA).

3 Dpole approxmaton, EDA, and LWA are not exactly the same thng, but that s usually gnored, and can be gnored here. For more normaton, see: HRR, PRA, 77 (198) HRR, PRA 9, 698 (1984) There s also a low-requency, ntensty-dependent lmt on the dpole approxmaton, rst ponted out long ago [HRR, Prog. Quant. Elex. 16, 1 (199), Sect. 1.5], but gnored untl a more emphatc recent publcaton: HRR, PRL 11, 43 (8); see also Erratum, PRL 11, (E) (8). 3

4 LOW FREQUENCY LIMIT ON THE DIPOLE APPROXIMATION The magnetc eld B s the key to the low requency lmt. In the dpole approxmaton, A(t,r) A(t) B =. Usually, B = s not consdered a problem because o the Lorentz orce expresson: Even though B = F or a plane wave (Gaussan unts), t s usually assumed that v / c << 1 or nonrelatvstc condtons, so the neglect o the magnetc eld doesn t matter. RELATIVISTIC EFFECTS: MAGNETIC EFFECTS: F Lorentz B A(t,r ) qf v B c 1 rel 1 1 v / c magnetc orce v electrc orce c 1 v c... For v/c <<1: v 1 c v c MAGNETIC EFFECTS HAVE AN EARLIER ONSET THAN RELATIVISTIC EFFECTS. 4

5 EFFECT OF THE MAGNETIC FIELD OF A PLANE WAVE For a plane-wave eld o lnear polarzaton, the electrc eld by tsel causes (n the rame o reerence n whch the electron s at rest on average) an oscllaton o ampltude along the drecton o the electrc eld, wth requency. 1/ 1 / F I 1 / z U p The couplng o the electrc and magnetc elds o a lnearly polarzed plane wave produces a gure-8 moton wth the long axs n the electrc eld drecton F and the short axs n the propagaton drecton k. The component n the k drecton has ampltude and requency. I U z p 3 8c c c These expressons are nonrelatvstc. The ully relatvstc results are on the next slde. Recall the dentons: z=u p /, z = U p /mc 1 4 z 1 z 1 z 4 Reerence: Sarachk & Schappert, PRD 1, 738 (197)

6 CLASSICAL FREE ELECTRON IN A PLANE WAVE FIELD Electron ollows a gure-8 trajectory resultng rom the combned acton o the electrc and magnetc elds. α = ampltude along electrc eld drecton β = ampltude along drecton o propagaton k F α z 1 z α 1 z, β z c 1 z, β 1 4 z The gure-8 proportons shown correspond to the lmtng case β / α =

7 CONSEQUENCES OF MAGNETIC FIELDS FOR THE DIPOLE APPROXIMATION I the magnetc-eld-nduced thckness o the gure-8 s o the order o the sze o the atom, then overlaps o the eld-nduced electron moton on the atom wll be altered, and matrxelement evaluatons wll thereby also change. Requre < 1 a.u., or z/c < 1; z= U p / ; U p = I / 4 I < 8c 3 Ths s now a lower bound on the requency or whch the dpole approxmaton s vald. Overall lmts on the dpole approxmaton are shown on the next slde. Taken together wth the act that a tunnelng theory requres not only the dpole approxmaton, but also << E B, tunnelng theores are lmted as shown on the ollowng slde. Reerence: HRR, PRL 11, 43 (8) [Note: tunnelng s nherently a statc-eld or very-low-requency phenomenon; t s explctly or QSE elds. For example, a hgh-requency ( > E B ) PW eld can onze wth a sngle photon; t does not tunnel.] 7

8 Intensty (a.u.) = c, upper lmt o dpole appr Intensty (W/cm ) THE DIPOLE APPROXIMATION HAS UPPER AND LOWER FREQUENCY LIMITS Wavelength (nm) m CO 8 nm T:sapph z = 1 (U p = mc ) =1, lower lmt o dpole appr 1 ev Feld requency (a.u.)

9 Intensty (a.u.) = 1/ Intensty (W/cm ) TUNNELING THEORIES FOR LASER-INDUCED PROCESSES ARE LIMITED TO THE SHADED AREA Wavelength (nm) m CO z = 1 (U p = mc ) = 1 8 nm T:sapph 1 ev Feld requency (a.u.)

10 DERIVATION OF A GENERAL QUANTUM TRANSITION AMPLITUDE We develop a general quantum transton ampltude based only on measurements made n the laboratory. Ths allows us to avod any descrpton o how a state evolves n tme. We need not even menton adabatc decouplng (an artcal procedure used n some ormal S- matrx theores to descrbe how elds stop nteractng at large tmes). All tme evoluton s handled by ncorporatng the equatons o moton (the Schrödnger equaton) drectly n the transton ampltude. What emerges s the standard transton ampltude, but wth a mnmum o needless assumptons. 1

11 BRIEF HISTORY OF S MATRICES Introduced or use n scatterng problems; hence the letter S J. A. Wheeler, Phys. Rev. 5, 117 (1937). W. Hesenberg, Z. Physk 1, 513, 673 (1943). E. C. G. Stückelberg, Helv. Phys. Acta 17, 3 (1943); 18, 1, 195 (1945). In a search or a general nonperturbatve ormalsm to use or strong-eld problems, t was shown that the S-matrx ormalsm can be employed or any quantum process: ree-ree (scatterng) bound-ree (onzaton, photodetachment) ree-bound (recombnaton) bound-bound (exctaton, de-exctaton) HRR, Phys. Rev. A 1, 83 (197). 11

12 RIGOROUS DERIVATION OF AN S MATRIX Basc requrement: The transton-causng nteracton occurs only wthn a doman bounded n space and tme. (Example: transtons n the ocus o a pulsed laser.) Important propertes o an S matrx as derved here: It can be ormulated entrely n terms o quanttes measurable n the laboratory. It does not requre that dynamcs be tracked over tme. Equatons o moton are ncorporated n the S matrx. There s no need or adabatc decouplng. It gves unambguous rules or gauge transormatons. It can be appled to any process as long as the space-tme doman o the nteracton regon s bounded. 1

13 The general problem s evoked o an atomc electron subjected to a pulsed, ocused laser beam. Ths s a convenence, not a requrement. There wll be a complete set o states { n } that satsy the Schrödnger equaton descrbng the atomc electron that may be undsturbed or n nteracton wth a laser beam: t H The outcome o any experment wll be measured by laboratory nstruments that never experence a laser eld. As ar as the laboratory nstruments are concerned, there s a complete set o states { n } that satsy the Schrödnger equaton descrbng an atomc electron that does NOT experence the laser eld: H t 13

14 Algebrac notaton wll be used or Hlbert space quanttes rather than Drac bra-ket notaton. The correspondences and rules are: u u ) v (v, v u v,u v,cu cv,u cv,u c * v,u ; c complexscalar 14

15 By hypothess, the laser pulse s nte, so H t H lm t We can organze the two complete sets o states and states so that they correspond at t - : lm t t t n Ater the laser nteracton has occurred, the only way or the laboratory nstruments to dscover what has happened s to orm overlaps o all possble nal states wth the state that began as a partcular state. Ths s the S matrx: S lm t n, Subtract the ampltude that no transton has occurred: M S 1 lm, lm, t t 15

16 We now have the orm o a perect derental: M dt, t M M M M t t dt,, H H H H dt dt t H,, H H,H, H H dt,h H H I t I An alternatve orm s especally useul or strong-eld problems. Instead o makng a one-to-one correspondence o and states at t -, do t at t + and then look or the probabltes that partcular ntal states could have led to ths nal result. t t I I I 16

17 The end result o ths procedure s the transton matrx element M dt, H I The rst orm above s called the drect-tme S matrx, and the second orm s the tme-reversed S matrx. In general, states are known exactly or can be smulated accurately (or example, by analytcal Hartree-Fock procedures). It s the states that present the problem because they represent condtons or an electron smultaneously subjected to the Coulomb attracton and to the laser eld. No exact solutons are known. I the ntal state s to be approxmated, ths s very dcult because the laser eld (by hypothess) s too strong to be treated perturbatvely, and so s the Coulomb eld because n an ntal bound state the Coulomb eld has a sngularty at the orgn. For the nal state, the electron s unbound, and the laser eld can be assumed to be more mportant than the Coulomb eld. Ths dentes the tme-reversed S matrx as the preerred orm, and t wll be employed exclusvely hereater. 17

18 GAUGE TRANSFORMATIONS Electromagnetc elds can be represented n terms o dervatves o vector potentals A(t,r)and scalar potentals (t,r). These potentals are not unque; wthn specc constrants, one set o potentals can be exchanged or another that produce exactly the same electrc and magnetc elds F, B. The alteraton rom one set o potentals to another descrbng the same elds s called a gauge transormaton. Wthn perturbatve AMO physcs, gauges are not mportant, and are usually (and saely) gnored. It s normally a matter o calculatonal convenence to decde what gauge to employ. In strong-eld physcs, gauges become very mportant; they are no longer a matter o nderence. It wll be necessary to explan why ths occurs, and to denty the consequences o gnorng the eects o a change n gauge. Ths matter has only recently been subjected to detaled study. Much o the AMO communty s ether unaware that gauge s mportant, or else reject the evdence. 18

19 CONCISE OVERVIEW OF GAUGES Every ntermedate or advanced textbook on classcal electrodynamcs gves a treatment o ths subject. Only those matters o drect mportance to these lectures wll be revewed. The elds F and B are related to the potentals A and as 1 A F - -, B A c t When substtuted nto the two source-ree Maxwell equatons, t s ound that the potentals satsy the Lorentz condton 1 A, c t and ths then leads to two uncoupled wave equatons or A and that ollow rom the remanng two Maxwell equatons contanng sources: 1 c 4, t A 1 c A t where s the densty o electrc charge and J s the current densty. 4 J, c 19

20 I there are no sources: = and J =, then these equatons have plane-wave (PW) solutons. Once ormed, plane waves can propagate ndentely wthout sources. A quasstatc electrc (QSE) eld cannot exst wthout sources:. PW and QSE elds are undamentally derent types o elds. Ths derence wll arse repeatedly n later work.

21 GAUGE TRANSFORMATION I s a scalar uncton, then the gradent o can be added to A wthout any change n the magnetc eld: B A,, set A' A B A' I the Lorentz condton s to reman vald, then t s necessary that 1 1 ' rom whch ollows F -' c t c To mantan the wave equatons or the vector and scalar potentals, t s necessary that 1 c t Ths s a gauge transormaton. The new potentals A, gve exactly the same elds as the orgnal A,. The concluson that any set o potentals meetng the constrants lsted are as vald as any other set o potentals s called gauge nvarance. A' t 1

22 EVIDENCE OF INCONSISTENCIES WITH STRICT GAUGE INVARIANCE In support o gauge nvarance, t s noted that Newton s equatons depend drectly on orces, and the Lorentz orce assocated wth electromagnetc elds s gven explctly n terms o the elds themselves: v m r FLorentz qf B c It s generally shown n textbooks on classcal mechancs that alternatve ormulatons o mechancs ner Newton s equatons. The nverse s not true. Newton s equatons do not ner Hamltonan, Lagrangan, etc. orms o classcal mechancs. System unctons lke the Hamltonan or the Lagrangan depend drectly on potentals, and t has never been ound possble to express those potentals n terms o the elds except n a non-local way. The nerence, to be supported wth much more evdence later, s that the potentals contan more normaton than the elds. That, n turn, suggests that the potentals are more undamental than the elds. Ths concluson s n conlct wth what (almost) all o the textbooks say.

23 GAUGE TRANSFORMATION OF THE TRANSITION AMPLITUDE Gauge selecton s unmportant n perturbaton theory, and the general atttude that all gauges are created equal causes no problems. In strong elds, gauge selecton s undamentally mportant. As already shown, n a gauge transormaton: A A' A,, H H', H H unchanged I 1 ' t c H ', ' U where s a scalar generatng uncton that satses the homogeneous wave equaton. It s clear and unambguous that the states { n } and Hamltonan H or the world o the laboratory nstruments have nothng to do wth the laser eld. A change o gauge o the laser eld can have no possble eect. The transton ampltude changes as: M M dt M I, H I M ' dt ', H I ' ' 3

24 EVIDENCE OF MORE PROBLEMS IN THE CONVENTIONAL UNDERSTANDING In numerous papers n the AMO communty and n textbooks, the presumpton s made that a gauge transormaton s a untary transormaton; that s, a transormaton generated by a untary operator U, dened by the propertes: UU = U U = 1, or U = U -1. Quantum states transorm as = U, and quantum operators as O = UOU = UOU -1. The presumpton s that transton matrx elements transorm as That s, t s presumed that all matrx elements are automatcally gauge nvarant. Ths s absolutely not true!,h. 1 H ',H ' ' U, UH U U, I I I I The transormaton just shown s a change o quantum pcture. It s not a gauge transormaton. Were the AMO assumpton correct, there would be no value or purpose n enorcng gauge nvarance. Note: n QED, gauge nvarance s requred, but t s not a unversal gauge change; rather, t s a change to another transverse-eld gauge. 4

25 ANOTHER FUNDAMENTAL FEATURE OF STRONG FIELDS Gauges become mportant n strong elds, but another basc concept rom classcal mechancs has to be altered. In classcal mechancs there s a smple theorem that any uncton o tme alone: (t) can be added to a Hamltonan or Lagrangan uncton wthout alterng the physcal problem n any way. An even smpler example s that a constant can be added to H(q,p,t) or to L(q, consequence. (For example, a constant sht n the zero o energy.) q,t) wthout On these grounds, Davdovch was able to show that the rst correcton term to the SFA should cancel the leadng term (a mathematcal mpossblty: power seres are unque and no term can cancel a precedng term). Mlonn concluded that the ponderomotve energy can be gnored because U p depends on A (t) and so t can be dscarded. However, the ponderomotve energy s physcally very real, and causes channel closngs, among other eects. The smple and amlar prncple about removal o (t) rom the Hamltonan doesn t seem to be vald here. Why not?? 5

26 CHANNEL CLOSING There s a threshold order: n E B + U p condton. mples there s a smallest order n that satses ths As U p ncreases, n n + 1. Ths s a channel closng. Hence reely removng A (t) changes the apparent physcs. What s gong on? In quantum mechancs, the A (t) dependent states are pnned to the A (t) ndependent states. Hence A (t) can NOT be removed n quantum mechancs. Ths s a basc classcal quantum dstncton. It s a undamental result that s not n the lterature. The A (t) term can also be removed by a gauge transormaton. Ths s a derent matter, snce gauge nvarance does protect basc measurables lke total rates. The next slde shows two very smlar-lookng results rom theores wth and wthout removal o A (t). However, physcal nterpretatons are completely derent! 6

27 VELOCITY-GAUGE SFA AT HIGH FREQUENCY Ionzaton o ground-state hydrogen, = a.u. SFA rom: HRR, JOSA B 13, 355 (1996) HFA rom Pont and Gavrla, PRL 65, 36 (199) 7

28 The precedng slde shows many thngs that wll be addressed n more detal later. SFA = Strong-Feld Approxmaton HFA = Hgh-Frequency Approxmaton The act that, as ntensty ncreases, the transton probablty eventually decreases s called stablzaton. Ths appears to be a near-unversal phenomenon theoretcally, but t has never been conclusvely demonstrated n the laboratory. Because o the hgh requency ( = a.u.), n = 1 at low ntenstes. In the SFA, the maxmum occurs when that channel closes, and n. In the HFA, A (t) s removed by a gauge transormaton, and there s no such thng as a channel closng. Stablzaton must be ascrbed to some other cause. Ths example establshes an mportant basc concept: PHYSICAL INTERPRETATIONS ARE GAUGE-DEPENDENT. Not all aspects o a physcal problem are preserved by a gauge transormaton. Ths s another subject that wll requre more attenton later. 8

29 SOME TERMINOLOGY FOR GAUGES COULOMB GAUGE: Ths s the most wdely used gauge n all o physcs. It has a ormal mathematcal denton, but or our purposes a qualtatve descrpton s more useul. The Coulomb gauge s that gauge n whch a longtudnal eld s descrbed by a scalar potental alone, wth no vector potental A; and a transverse eld s descrbed by a vector potental A alone, wth no scalar potental. A longtudnal eld s one where the drecton o propagaton les along the drecton o the electrc eld. It s typed by a quasstatc electrc (QSE) eld, such as that whch exsts between a par o parallel capactor plates wth a slowly varyng potental derence across them. A statc electrc eld (such as arses rom a Coulomb center o attracton) s a longtudnal eld. A transverse eld has the drecton o propagaton perpendcular to the drecton o the electrc eld, so that kf=. A plane-wave (PW) eld s a pure transverse eld. It has the unque property that t can propagate nnte dstances n vacuum wth a loss o ntensty arsng only rom soldangle eects. Lasers produce transverse elds. 9

30 The velocty gauge s a Coulomb gauge n whch the dpole approxmaton s employed. That s, where A(t, r) A(t). It s called the velocty gauge because the Hamltonan contans the quantty p (1/c) A, whch s the knetc momentum (proportonal to velocty ). The length gauge (also called Göppert-Mayer gauge) s only used n the dpole approxmaton, and t expresses even transverse elds by a scalar potental. It s o the orm = rf. It s called the length gauge because o the presence o r n the potental. Reerence: M. Göppert-Mayer, Ann. Phys. (Lepzg) 9, 73 (1931). The length gauge can also be descrbed as that gauge where a PW eld s treated as t were a QSE eld. The Lorentz gauge s that broad class o gauges where vector and scalar potentals satsy the relaton 1 A c t, whch decouples the wave equatons or and A. Almost all gauges used n physcs are Lorentz gauges, except or the length gauge. When the scalar potental s ether zero or ndependent o tme, then a ormal denton o the Coulomb gauge oten encountered s A 3

31 STRONG-FIELD APPROXIMATION WITHIN THE DIPOLE APPROXIMATION Start wth the tme-reversed S matrx: M dt, H I For an onzaton problem, s an ntal bound state. Two mportant acts: 1. can be assumed to be known accurately. Most o the bndng potental dependence n the problem s n s an onzed or photodetached state. For very strong elds, t can then be assumed that the domnant eect on the ree partcle wll be the laser eld. As a rst estmate, ths requres U p >> E B An exact soluton s known or the moton o a ree charged partcle n a very strong plane-wave eld: the Volkov soluton. When s replaced by V, ths gves the Strong-Feld Approxmaton. M SFA dt Reerences: HRR, PRA, 1786 (198); PRA 4, 1476 (199); Prog. Quant. Elect. 16, 1 (199). V, H I 31

32 3 DIPOLE APPROXIMATION VOLKOV SOLUTION Schrödnger equaton: t 1 t A - c 1 The Volkov soluton s explct to the Coulomb gauge; wthn the Coulomb gauge, plane-wave elds are represented by vector potentals wth zero scalar potental. The Coulomb gauge wthn the dpole approxmaton s generally called the velocty gauge. Another generc orm or the Schrödnger equaton: t t A A t c 1 c 1 H, 1 H, H H I I The dpole-approxmaton Volkov soluton: V V t 1 / V p,t H t H p, H d t exp V I I I p - r p

33 Wth the last expresson substtuted: M V p,t SFA dthi, Express the ntal state wave uncton as a statonary state: t,r r exp E t The only spatal dependence n the Volkov uncton s exp( pr) The Hlbert-space nner product s then smply a Fourer transorm, spatal dependence s replaced by momentum dependence, and the Hlbert-space nner product s no longer n evdence. ˆ p expp r, r The matrx element s then M SFA 1 1 / V ˆ p p dt exp E t H p,t exp dh p,. I t I 33

34 INTERACTION HAMILTONIAN AND POLARIZATION STATES To proceed urther requres a knowledge o the nteracton Hamltonan and that, n turn, requres a speccaton o the polarzaton state o the eld. Only crcular and lnear polarzaton wll be consdered. Crcular polarzaton s the smpler case. 34