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# Atomic Physics PHYS 602

UT

GPA 3.61

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This 61 page Class Notes was uploaded by Elenor Morissette on Monday October 26, 2015. The Class Notes belongs to PHYS 602 at University of Tennessee - Knoxville taught by Staff in Fall. Since its upload, it has received 47 views. For similar materials see /class/229884/phys-602-university-of-tennessee-knoxville in Physics 2 at University of Tennessee - Knoxville.

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Date Created: 10/26/15

Coherent trapping three states Like in the twostate case Use velocity gaug Two lasers with vector potentials AacEa0a cos0at AbZCEbDb COS0bt9 W1 Assume initial state is Ogt Intuitively one can expect a situation in which all population which is pumped by laser A up from Ogt to 2gt is forced down to 2gt by laser B Result State 2gt empty all population between Ogt and 2gt State 2gt plays a role of a catalyzer for resonant coupling of nonresonant states Ogt and 1gt while 2gt itself stays decoupled Physics 602 Le not populated Krstic UT October 15 2008 We want to find and discuss conditions for this to occur a a W Energy conserving condition a b u H 39 39 for resonant transntlon Ogt to 1gt Each laser has small detuning Awe wa VV209Aab 20 VV21 At exact 01 resonance Amat Awb wawbm0ml waabWV10 0 Ama Amb Allowing that the transition Ogt gt 1gt is not exactly resonant Awa Aw52 AwbAw 62 Physics 602 I 6 ltlt Aw I Krstic UT October 15 2008 Solving three state eigenvalue problem in RWA Total threestate wave function W 11 0 gt e iWo 8t 1 gt e l Wll W 2 gt el39Wzt Do the same as in 2 state case 39Apply L11 In Schrodinger equation with two laser fields Project with states 012 form the left Apply RWA and omit the fast rotating terms note that W1O is not assumed small This results in 3state2 laser RWA equations a iAwt 0 Z 39 a Note ogt is directly coupled with 2gt 3mg EM 1gt is directly coupled with 2gt g em lat w a 2gt is coupled directly with both 2 7 1 ogt and 1gt 139 e a e of the 3state2laser RWA 2 2 equations Physics 602 Krstic UT October 15 2008 1 4M 1 ZAwbt Problem 8 a Show details of derivation Eigenvalue ansatz allows for removal of the residual time dependence from the equation and results in the 3state eigenvalue problem a aoe i8 Aaa l2 z e i8Awa 2Aabl2 O i8Aaal2 7 706 060 80 70 constants or I I t d 39 If slowly varying functions oftime Replaceg t by 8 t Nearlyadiabatic solution Consistent with switching the laser onoff Cubic eigenvalue equation follows from the condition that determinant of the linear system below is zero 8 Aacla0 layO 5 A606 2Awb gt180 70 8 Awa70 1060 Physics 602 Krs IC UT October 15 2008 We consider nearly resonant transition 5 is a Small quantity to develop the SOIUtion of the cubic equation by a perturbation expansion in 6 Express A0 Awb in terms of Ag9 5 and neglecting terms linear and higher a power in 5 s wasz sz a bgt0 Solutions 81 Ag 06 8i 2 iR 06 Generalized Rabi frequency 2 2 2 12 R A0 W 10 Physics602 Krstic UT October 15 2008 For each eigenvalue dressed stateamplitudes a0 9 o 9 70 are obtained from linearly dependent system of equations detO normalization condition 2 2 2 laol l ol l7ol 1 Thus for 51 A0 Up to O LP1 COL 0 gte jW0t 1 1gteiw where normalization gives C 1021 10b Thus amplitude for 2gt vanishes in the limit 5 0 resonance This not affected by the relative laser phases f 2gt is a continuum state LIleill be stable against ionization Any state can be expressed as linear combination of L111 L111 lf spontaneous decay included L111 not affected but LPJ are Physics 602 In result all population goes to P Krstic UT October 15 2008 Ladder sequence in the Three state problem Pulsating sequence A Eb A TWO lasers a and b separately pulsed lEa Effect of pulsating sequence Awe ma VV109Aab I 0 VV21 Physics 602 Krstic UT October 15 2008 Equations for the amplitudes are now in RWA ft lazjelAwat llt b iAabt z e 7 2 39 b iAwt 1 iAat z e b e a 3 2 7 2 where like before we let T at 0 gt e iWO Mt 1 gt e iW1t 9t 2 gt e iW2t Assume exact resonance excitation 02 0a 60 2W20 gtAaa Aab 2A0 Physics 602 Krstio UT Ootober15 2008 Then the cubic eigenvalue equation simplifies to 8 Aw82 R2 O R Am2 020 xlozb 2 Eigenvalues 81 A0 8i 2 iR Obviously like before LP1 Cxlb 0 gt e jWot 2L 2 gt e szt 1 C 2 2 2 12 1061 10b Physics 602 Krstio UT Ootober15 2008 Simplify again assuming exact resonances of both lasers ie Ag 0 Then 1 LP 2 ta O gt equotW0t R 1gt equotW1 1 2 gt equotW2 eXp iRt 2 R 1 2ta 0 gt e WO R 1 gt e W1 Ab 2 gt e W2 eXpz39Rt 2 L zzle i 2 2 Notethatnow R log i Ob t Assuming adiabatic switching of the lasers the only change is Rt J39 Ralf 00 Physics 602 Krstic UT October 15 2008 1 Assume laser a turned in remote past before b is turned on and in the limit of Eat gt oo gt 0 i 1 LIJI gt quot 2gtelet 0a 1 LIJ gt a Ogte W0t 0a LP gt1gtequot39W1t Obviously LP is the initial state In adiabatic limit the system will stay in LP for example if the laser a was turned off After b Everything changes if the laser a which was turned on first is turned of first Physics 602 Krstic UT October 15 2008 Then as t gt 00 Laser b is being turned off and 1 LIJI gt Ogte 0b 1 b lW t IRRoo2 I gt 2 gt e 2 01 LP gt 1 gt e intiRRoo2 Where no RRoo j Rtdt Remarkable LIJJr 2 gt FULL EXCITATION Physics 602 Krstic UT October 15 2008 Solution of problem 7 First need to define the laser filed EU Zoe 02 sin a where r 10 9 104134au 4134au AL 2 80000529Bohr 21512287Bohr gt f i 137 au 0009au AL 1512287 gt a 27rf 0057au 1015 my 26111 E0 if2 EA A Therefore E 017e 413 42 sin 0057 It is useful to graph the field up to 100fs4134 au Laser eld 015 010 100fs 017expx413402sin0057x 010 Eau I I I I I I 4000 2000 0 2000 4000 t au 020 Following notation and gauge fa21 in Section 2 of the MHM book L I t 39 t 39 I t E 6 07 asere ec ron coup Ing ma rIx e emen 0 a a A p10 8 Assume a a 2 P10 398 1au gt A 156 4134 cm Also Au 00001W10 z 00001 0057 57 10 6au which finally yields a 2 A Z 6et4134 38 At t1OO fs this gives shy 10 a Absolute value of amplitude of excitation into exciteq1gt starting from is ground Ogt state is e yZ 26h12 And thus the relevant probability is P01 239 am 2 10 73 z 0 ao1 lt M1 13 gt e z 51037 b The absolute value of the excitation amplitude in this case is la10l2lt0lD gt which gives exactly the same result for t100fs as in a c The frequency has to be chirped adiabatically slow ie any form of the frequency time dependence which satisfies ltlt1 09 will give the satisfactory solution Q Is Rabi frequency which is here approximately equal to Thus a ltlt 0086au Since 0 needs to Change by 52Aco z 10 56114 and Q 5Aw 5t 4 adiabaticity condition for a is here easy to fulfill au reaching excitation probabmty 1 Note that period of 1 field oscillation is 110 au Timedependent Green s function formalism Schrodinger equation for the wave function is homogeneous a e IE HO HLTrt 0 Convenient to introduce corresponding inhomogeneous equation for nonlocal operator G Green function ig HO HLGrI739tt39 517 17950 139 G can be written symbolically G 2 1 15 H0 HL Also can define unperturbed Green function G0 a 1 i H 0 These can be used to formally solve Schrodinger equation for LP in terms of Integral equations For example if before the laserelectron coupling HL is turned on the atom was in the initial state u0l7t satisfying stationary Schr equation a jg HOMOO39J 0u017t 0 gt e iWOt Then LPuOGHLu0 1 Proof 6 1 Replacing G25 and multiplying form left by i H0 HL Iii H0 HL at 1 6 1 u0 a HLu0 gt lg H0 HL P z E HO HL 6 6 IE H0 HLu0 HLu0 lE H0u0 0 Alternatively LI Z 0 GOHLLIJ 2 Which yields explicit representation 1 0 1 GOHL Which directly leads to the perturbation expansion in the laserelectron coupling 1 00 n n m Z1x L11 uo G0HLu0 GOHLGOHLu0 n20 Problem 9 a Using expansion of the full Green s function G 2 G0 GOHLG0 GOHLGOHLG0 derive the perturbation expansion for LP in HLfrom representation 1 b Prove validity of representation 2 How are the time dependent Green s operator applied How does nonlocal Grtr t produce the local L110quot I G EId3r39Idt39GFtF39t39 F39t39 Explicit form of Go GOG t m39 490 t ozi n gt e l39WN t lt n 60 t 39 Is the unit step function 0 for tltt 1 for tgtt its derivative is 51 139 ngt is the complete set of eigenfunctions of HO and therefore 2 1117 gtlt 1117 39 i 517 1739 lg Homo 60 r39gtZi n gt equotWquotltH gt lt n z396ltr r39gt2ltm H0gt n gtlt n WM 5t t39517 I7390 QED quod erat demonstrandum Then I GOHLuO gt n gtlt nl HOL l 0 gt 6 iWnt J dtlcosat ie iW0 Wnt In order to have the convergence of the t integral at lquot 00 IntrOdUCG small 77 gt 0 Through a factor ent39 And then to guaranty an outgoing wave at t gt 00 need also factor e m to cancel the first one at t t This introduces the need to implement a small exponential factor defining the source and sink of the time evolution ie eXp77t f Green s function which evolve form an initial state at t 00 Into an outgoing wave like above carry index ie G On the other hand one can define the Green s function that evolves from a glVen State at t 00 Backward in time with ingoing boundary conditions usually carry index ie G Therefore correct notation of the Green s functions at previous slides is GWGSHW and Gggtt y 490 r 39Z n gt equotltWn 7gtltH gt lt n MULTIMODE LASER FIELD Et ZEk cosakt pk k1 Example 3mode laser E 2 E0 cosm0t 00 E1 cosm1t m1 E2 cosm2t m2 m1 m0 Amm2 2 m0 Am Use trigonometric relation cosx y cos xcos y sin xsin y to transform E in the form E At cos mot Bt sin mot This yields At 2 E0 cos 00 E1 20501 Amt E2 30501 Amt Bt 2 E0 sin 00 E1 si1r1m1 Amt E2 si1r1m1 Amt Use trigonometric relation 2 2 B Acosx Bsmx 2 VA B cosx atanZ To obtain final form Et Ea cosa0t 30 Condition various modes are close to the central one A0 ltlt 00 More details 60 Z Ifcosmor 30 Ema sina0t 30 177t Zsina0t 30 Multimode laser can be presented with a single mode at the central frequency with slowly varying filed amplitude and phase Slow beating A simple example of a 3mode laser sinl0005xquot2cos006x si n00005x 2 cos 006x I l 05 pl 0i012 E1E2E02 E 2E0 sin2 Amt 2 cos wot 00 M 05 l l l l l l l M w l ll ll l l l39 llll l l l l l ll ll ll ll ll ll l l 2000 4000 E vn aul t au l l l l l l l 1 ll l 39l ll lll Hllll l l l lllll l WWW l l l l39 l l l l l l ilji v l l39Hl 39l W l lllllwlill 2000 4000 6000 tau A mt LIAculJPlM u A a w a unu Slow beating clue to coherence M u r Annv J 71 1 7n U mmw w E1E3Enl2 800 nm TiSapphire laser Loong pulse 100 fs ADIABATIC TIME DEPENDENCE General case Laser has to be switched onoff Laser is multimode Frequency could be chirping A70 an cos 1dr 39at 39 00 You had before Two state system in nearly resonant laser RWA treatment Constant parameters Di 2 Gig2i62 uoeimt2 ieiy2 i62 uoe imt2expiVV0 c 1 Sh Aa Z E a A lt u u gt 12 0191006220616 10 p10 olgpjl 1 a 3 C080 2 ASinO 2 Elliptic polarization 77 0 77 7r 2 near circular Assume a special case of adiabatic switching onoff the laser E0 E001 T gt 20 t T T large in comparison to the laser oscillation period Leading solution in 1T t 1 ei ZHQZ uoeiwtZ iei ZiiQZ uleeiwt2explIVO Wt2TJ8tldtl CID i JZchJ 8 81 of the terms which contain u are proportional to small 1T gt O T gtoo Above is the leading term in formal asymptotic expansion in large T Note that 6 1 in phase is replaced by the integral To satisfy initial conditions an tO when filed is switched on form EO this phase is further written in form 315 80dt 39 80t 50 Am 2 a W10 0 Detuning assumed gt0 The twostate nearlyadiabatic solutions takes now the form 1 1 t e2zt92 uoe zWot e ZIW lg WNW exp i J 81 39 80d1 39 Zehy 2 0 1 1 e luZHQZ Moe 1W1Wt eIu2 92 ulelWlteXpi J y v 0 JZchlu Note 10 gt 0u gt 00 for positive detuning and u2 u2 gt1 gt0 6 6 23962 z39W0I 426ml xZchy CD gt e uoe Therefore as f O D ey2 i62 16 in When the filed is turned off after time T again have 1 T lt1 gt 6 qu WVO exp z39E 30 39 800 139 0 l39 l39 1 T l l c1gt gt e m ule W eXpl J8t g0dt 0 States acquired a phase in the field but no transitions occur in the adiabatic limit One more example of a Great Adiabatic Theorem States can deform under the adiabatically slow perturbation to adapt to a new Hamiltonian but once the perturbation is off all are returned in its initial condition In this case upon adiabatically exposing to the laser field the states are oscillating between each other by Rabi frequency 8 After the laser is turned off the states are returned where they were before the laser field application ie amplitude for excitation is zero in the adiabatic limit In order to enable excitation by a laser field some nonadiabatic change of the laser parameters is needed If the detuning Ad ls negative than D gt MO 131 gt MI When the field goes to zero This brings a possibility for excitation lf initially is detuning positive and then while laser is on the detuning switches fast sign to negative chirping laser than DO gt DPCI1 gt DO When the laser is turned off ie the excitation probability is 1 This is known as adiabatic fast passage Note1 The laser perturbation leading to excitation has to be faster than the natural decay time of the excited state 108109 s Note2 More precise adiabatic conditions for the laser field for both frequency and the field strength can be written a E39 ltlt1 ltlt1 we E8 which couples the pulse duration and the laser intensity FREE ELECTRON IN VARIOUS LASER GAUGES Dipole approximation 2 Z a a a a A r pp 2091 pH 1 2 gt gt V r cm MA 2 c up reduced electron mass replace by electron mass m1 MAatom mass replace with nucleus mass M A Is a function of coordinates through expi 5 1717 lt 1 7 gtltlt1 Dipole approximation 6X a Can couple cm motion to the electron and affect motion of atom p 390 as a whole Very interesting consequences to be discussed later Here assume that motion of atom can be neglected and 5 is fixed to the origin ie I5 0 The cm kinetic energy can be then dropped and Schrodinger equation for the atom fixed at origin is 1 Z a 210 H m 2mi1pl c 2 VI7 Only to this form refer the known gauge transformations for the laser interaction 39I39 P e I J LP 1 lg Hj j O Hj eiQHelqj d J Z 2 pz 2 a Choice DVZEIdtA2t HVZVZAtf7i Velocity or 11 11 gauge Class of solutions for a free single electron wave function in laser field VO is known as Volkov solutions 2 I z eXpi F q7texp i java109m A A0 cosat E 9A0 mm E0 mm C 2 z eXpi31 o f 317 expi z39ci 0 a0 swan 2 oo O q n inwt Fourier expansion eXplq r 2 1 J a0 qe Flocke expansion 712 00 Transition amplitude to any freeelectron state I 13 gt lt p I I gt 5c P Requires conservation of momentum so any transition by absorptionemission of a photon by a free electron would not satisfy conservation of energy and momentum simultaneously All photon states in a Volkov state are virtual no transitions until a third body nucleus for example help in conservation laws QED laser approximation in dipole approximation 2O a 61 QED vector potential 0 H p V 121 13 c my QED Hamiltonian for a single electron problem 2 For VO Look for a solution as expansion a a q in states of photon population number lq r 17t 1Nat 110 161 e Z ave N V gt V av 1VJV070 Obtained 61 In gt n1 2 n1 gtan gt n 2 n l gt Laserska approksimacija N V N 1 2n Jn1x Jn1x Jn x Recurrence relation of Bessel functions x Z 2 Z HLZp 2iVZfEt Lengthgauge F E i1 i1 1 t 2 Volkov state i67AC7 EJ 67AC dt39 L Zqe Kramers gauge Z I gt operator Zz I Accelerated system pk 2 AU IA2Z Vdtl l 2 1 1 Hk Vlti 070 050 J dZ39AU39 Electron sees the oscillating nucleus Volkov state in this case is a plane wave Floquet theory Assume atomic system in monochromatic laser field velocity gauge HH0HLtH0Zlofacosat HHzTT2 r a ie the Hamiltonian is periodic in time with period T Then one can apply FloquetShirley theorem The exact wave function of the system can be written in form km r equotW FI7 r where F is a periodic function of time with period T F73 T 11073 F0731 Z E1Fei t I Fz equotW Z Fnl7e mm n oo n oo W is called electron quasienergy and contains nonperiodic response of the system Replace Floquet expansion in the timedependent Schrodinger equation a lg H0 HLLP Z 0 t And use Fourier theorem to equate the same harmonic components Jimquot lg H0 A0219 em 1th 2 F 0 gtWnw HOF7VAF F1 n1 n i Where V Z 14013 and H0 are timeindependent operators which act on A Rdependent functions F r n This is Floquet eigenvalue problem The operator equations can be transformed into linear equations by expanding Fn into a complete basis for example eigenstates of H01 kgt and projecting from left by ltK s W W WK B 2 VA Kk 13131 Bin 1 k where c 14 FM Z Bk l k gt k System infinite in both k and n This is a generalization of the twostate RWA This is a tridiagonal matrix whose elements are infinite block matrices along the main diagonal and two subdiagonals Example System with only two bound states 1gt and 2gt coupled by the dipole interaction VA The above system becomes A J W quotW 110310 VA12B n1 3204 1 W quotW VV2B W VA21Blnl Blvd 1 To further simplify assume only one photon absorption and emission present Then the above system can be written as A1 V 0 B V A0 V 130 0 0 V A1 B Where AV are 2x2 block matrices and B s are vectors 2 W VVI 0 V 0 VA 122 Blj 0 W W2 3 VA21 0 9 B This constitutes an eigenvalue problem in quasienergy W A1W V 0 V A0W V 0 0 V A1W which gives 3x26 eigenvalues ie 6 Floquet states Problem 12 Obtain Rabi frequency of the 2 state problem described by the Floquet system above assuming singlephoton quasiresonance of the states 1gt and 2gt and applying rotatingwave approximation Highfrequency Floquet theory example Kramers gauge T V07 In the electron rest frame the electron sees like nucleus is oscillating with the laser field For example for hydrogen atom V07 5ct 7 IP 061 Apply Floquet theory 00 a a a a 4m V0 at Z Whack nz oo FU ZI Z E1Fe i t t z7t equot39Wt Z Fn7emw By equating the same Fourier components ltWnw T V0ltMo altfgt i momma k oo k n Vnk except nk term and can be dropped for high laser frequencies 0E0 and high laser intensities This yields W nwT KFamp0EF O which is a stationary Schrodinger equation that defines electron in the potential 1 7 d V0 0 W 27 r ap This is a weakened Coulomb potential splits in two squareroot singularities Wave function split in two quothigh intensityhigh frequency dichotomy The energy levels shifted toward continuum edge The atomic states highly adiabatically deformed still the ionization mechanisms is In next order in small n k 0E0 Contraintuitive High intensity stabilization of an atom against ionization Highfrequency is also stabilizing the atom AC Stark shift 130 H HLQG5gtQHL qu 0 Chose P 2 0 gtlt 0 Then F917 tat39 t39enttyZ n n e iWnU l niO Apply Floquet theory and act with the Green s function Time independent diagonal term leads to the eigenvalue problem 1 1 W w Wji77 Wa Wji77 E0 2 WH0 V40j VA390CDW 0 The shift of energy of the state Ogt for example is 1 1 W w W 177 Wa Wj 177 2ltW0 Wjgt WNW w ho AW z f grzo 604 222 305 392 py Dynamical AC Stark shift of WO jiO E a A m2022 paw 2 5W0 wWj5W0 wWj Width of the state 170 0 This is the lowest nonvanishing order of the level dressing Keeping higher powers in GQ contribute to the higher orders of the level shifts lim 77 0 x PVl 175x x Discussion on problem 10 requested Note that a POE H HLQGQ QHLP P 0 And we have Chosen POgtltO with obvious ansatz PT 2 050 O gt e jWot With assumption that there is only one bound state in the model system a a I a a 39W r r39 Gnggtrr gm l 9t t39e quotI I am W gtlt qltgt e A Applying PT and G0Q In the above equation for gives exactly I 23902 i 613qu0 cosatequ m I dt 39AZD cosat 39e Wq 0 7 az 39 Assuming that W W0 a and dropping fast oscillating terms this simplifies q I o l 39 W W l l a Z Id3qu Oe l q0 0 77t 2064 10 0l 77l i If the laser is weak enough so that both a A are SIOW ghangmg In me The time integration can be approximately done QED Thus in case of two resonantly coupled bound states and continuum 1 I i 42011 We can assume ei 2i62 0 gt eimtZ i ei Z iQZ 1gt e imf2eXpl VV0 i POgtltO1gtlt1 while the continuum Green s function has the same form as before When choosing expansion of It is convenient to assume PP2A40 A10 If Ad gt 0 and we are starting form Ogt state than our physical conditions corres ond to evolution of onl p y 13 while A 0 P A40o t A i M 4 1 61 t 1 v 39 Zeh e expz AW 8CliZ 2At TRANSITIONS IN LASER FIELD Smatrix approach Amplitude for transition into a state Induced by a laser pulse um F t 2 n gt WW t that lasted form T to T where T could gt 00 If the system was before application of the laser in state tltT u017 t O gt e jw0t Sn 0 lim 051600 lim lt u L115 gt I gtT I gtT where LIJ 7 t Is the exact solution of the full Schrodinger equation in the 0 9 laser field which evolved from initial state 710031 2 2 1 at H0 H 0 Which can be expanded in the full set of eigenstates of HO wg mr Zaneommr 057600 Tgt 570 S 0 Is called Smatrix element for transition from Ogt to ngt while set of all Sijfor ijcn us called the Smatrix Could be of infinite dimension countable or continuum Therefore it is the transition amplitude at large T when the laser is off HO could have continuum states u 7 0 2 q gt e iWq of momentum q q 3 Then 8 matrix for ionization of Ogt into a state of definite momentum is defined as Sq0 l1i11ltuqLP0 gt Properties of the Smatrix 1 The Smatrix is independent on the gauge chosen for laserelectron interaction Proof Gauge transformations rely on the independence of the QM observables on the overall phase CD of the total wave function CDwCDb Are two different overall phases two gauges which depend on the laser Than CIDaCIb are identical for tgtT when the laser is off Assume that D s are zero when the laser is off Then 610 b I39CD I an t lt u e LPO gtan t lt u la b LPO gt And thus the transition amplitudes are different for various gauges while laser is on 1 7I 0 But when the laser is off Dal t Z T 0 051 T alibT a b Snag Snjo QED 2 Being dependent on gauge the transition amplitudes while the laser is on have no physical sense 3 Timereversed formulation of the Smatrix Sn0 lim ltLPff u0 gt t gt T Where L111 Is the exact wave function of the problem which evolves for initial quotquot0720 starting form tT and evolving backward in time 4 Equivalent practical definition of the Smatrix T d Sn0 11mltun PPS gt cit lt11 PPS gt11m ltun PPS gt I gtT T dl I gt T ltunu0gt0 d jltun PPS gtlt un z39HO PPS gt ltun z39HO HL 1 gt2 t iltunHL Pggt ie T Sm0 1JdtltunHLTO gt T For ionization T T SqO z39jdzltuggtHLLPggtgt ijdzltwggtHLu0gt T T 5 Generalization Sn 0 lim lt u LP gt lim lt Q LP gt t gtT t gtT where 13 H 9 0 lim or u 81 f n t gtT n n Then T T 5710iJdtltQ iH HfiLPg gt iJdtltLIJ l iH Hfi 2ggt T T a la HAD 2 0 llmT 9g 2 L10 and for ionization by n q t t gt Be aware of possible nonorthogonality lim lt Q PP gt72 0 lim lt LPH Q gt72 0 t gt T n 0 9t gtT n 0 MULTIPHOTON IONIZATION IN STRONG LASERS T T 2190 ijdzltoggt lH Hf My gt 4 dtltzq VLPggt gt T T Where we have Chosen for Q The Volkov state ie state of a free electron in laser satisfying 8 25 17 0 Therefore Atomic potential H Hf H HL V for example Coulomb 39W I Or Is not Significantly perturbed by the laser yields The Born approximation for multiphoton ionization Assuming that L115 N uo O gt e T S 0 ijdrltqVuOgt T In the velocity p o A gauge we had the Vqlkov state as 1 Nexpm 7 q7texp i id 39g At 39 c 2 NexpigF a0tsinmt t 2 oo Nexpiztci 17317 Z 1 Jnltao equot nz oo N is normalization const Where a slowly varying function describing switching of the 0 t 2 E00 Laser on and off 0 2 It is equal to the amplitude of oscillation of a free electron in the laser field Note that lim 25 2 ltl gtT 2 NexpiqFiql Which IS not orthogonal to Ho Can assume approximate orthogonality here T Slow 1N Idtlt1 7Vu0gt T 00 T 39 na I 4lmeVvM0gt lenquotlhmuo kwi m T 712 00 Since 060 Is a slow function of time can be taken in front of integral At a plate value of the field T J eKWq nw WON 6 xm WM T Using lim 6x gt lim 2 L oo x iWq nw W0T e iWq na WOT sinWq na WOT Wq na WO sinW na W0T q 2276Wq nw W0 Wq na W0 Finally SW 27riZ 5Wq na W0on where T5 lt 1gtquotNJnltaoltr 2 WWW l 0 gt Is the Tmatrix for ionization of n photons q is defined the by deltafunction In case of hydrogen atom initially in its ground state 7153 1 Jn Id3rei 739fVU I 0 gt 1n1 Jn a0 Id3reiqrcosg l r e 47 r n1 Jn a0 7r 2 iqrcos 1 r 01951116 dre e l39 1 1 M dre r sin qr i1 1 2 g 0 7239 1 q F39 H a ma y n1 1 Jn a0 T i 1 1 0 7r 1q2 Probability of multiphoton ionization Have product of 2 delta functions but note T dteiatfa dt39eiat39fa39 T dtfti em fr tdr z 27 dtf2t6a Therefore the transition rate for ionization is r 2 272 Jd3q5Wq mo W0 T53 2 Where for hydrogen atom in velocity gauge iJ ao39cY T07 2 I q0 I 7Z2 1q22 Probability rate for nphoton ionization r5quot 272 d3q5Wq mo W0 qu g 2 all Do qintegration but not angular and find E angular spectrum J5 gm 2 gt Z j39jn 050 0 d9 7 1qn w W here q 2VVOna nZ 0 0 The nphoton ionization rate is 1 PM ZxEq cch2 a x and finally the total ionization rate 1 F M 2W im aoqnx 722 0 n 1 Discussion Assumption of the previous derivation was a relatively weak laser field so that LIJ z u 0 0 Le perturbation in the initial state is weak Note that for small argument Jn 5 5 Therefore in that limit 0qu lt1 1 E2n 1 PW IJ aoqnx WVO n60 I xzndx I a 1 1 Problem 11 Assume linearly polarized laser of intensity 5x1014Wcm2 and a single photon energy of 2 eV Assume the target of a long laser pulse is hydrogen atom in ground state a Calculate and graph the angular distribution of photoelectrons from 710 and 15 photons ionization b Calculate and draw the nphoton ionization rate as function of a number of absorbed photons n c Calculate the total ionization probability Note The small argument limit is not valid with the laser in this problem

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