SUPPLEMENTARY INFORMATION

Transcription

1 do: 0.08/nature09 I. Resonant absorpton of XUV pulses n Kr + usng the reduced densty matrx approach The quantum beats nvestgated n ths paper are the result of nterference between two exctaton paths of sngly charged krypton atoms: 4p and / / 4p n the / / presence of a photoelectron. The ablty of the exctaton paths to nterfere s referred to as coherence and we quantfy the degree of coherence wth the ad of the reduced densty matrx. We model absorpton n the sem-classcal approxmaton where the electromagnetc felds are descrbed classcally whle the response of the matter s modeled based on the Schrödnger equaton. The sngle-on dpole response dt () = Ψ() t Z Ψ () t where Z s the dpole operator allows us to evaluate the polarzaton response of the gas Pt () = n dt () wth the ntal concentraton of atoms AT n AT. Ths polarzaton response then enters the Maxwell equatons whch descrbe the propagaton of the XUV pulse. Here the wavefuncton Ψ () t represents the result of nteracton wth both the NIR and XUV pulses. When these two pulses overlap t leads to a very complex dpole response the transmtted XUV spectrum s senstve to the nteracton of the NIR pulse wth atoms and ons after the XUV pulse. The theoretcal descrpton s sgnfcantly smplfed when the XUV pulse nteracts wth ons prevously prepared by the NIR pulse wth no overlap between the two pulses. The rest of ths secton s concerned wth ths regme. When an XUV pulse exctes an on from a certan ntal state I wth an energy I to a fnal state whch has an energy the polarzaton response at a frequency ω s

2 do: 0.08/nature09 well known to be IZ ZI P( ω) = nat XUV ( ω) I where the fnal state s assumed to decay at a rate and XUV ( ω) denotes the XUV pulse n the frequency doman. Ths expresson can be generalzed to the case where the XUV pulse nteracts wth a coherent superposton of onc states specfed by a reduced densty matrx ρ. When the XUV pulse s so short that the densty matrx neglgbly changes durng the pulse the polarzaton response s approxmately gven by I Z Z I P( ω txuv ) = nat ρii ( txuv ) XUV ( ω+ II ) II ' I where ρ II are the elements of the reduced densty matrx and t XUV s the moment of tme at whch the XUV pulse reaches ts peak ntensty. When the bandwdth of the XUV pulse s much larger than I I χω ( t ) = n ρ ( t ) XUV AT II XUV II ' the lnear susceptblty I Z ZI I accurately descrbes the XUV absorpton. Ths allows us to evaluate the absorpton crosssecton as [ ] ω Im χ( ω) σω ( ) = 4π. c n AT or the absorpton lnes of sngly-charged krypton ons we can re-wrte ths expresson n the followng more explct form where τ s the delay between the NIR and XUV pulses:

3 do: 0.08/nature09 4p/ Z ω 5/ (/ ) (/ ) σ( ωτ ) = 4π Im // // c ρ + ρ 5 5 5/ 4p / / / / / 4p Z 4p Z (/) (/) + ρ// + ρ// / 4p / / (/ ) + 4p/ Z / 4p/ Z / ρ// 0 ( 4p / 4p ) τ+ δ ( / 4p / 4p ) τδ / e e +. / 4p / / 4p / 4 p/ ρ (/ ) // () By analyzng the measured absorpton data wth the ad of ths expresson we obtaned the absolute values of the densty-matrx elements as well as the phaseδ whch depends on the choce of the tme orgnt = 0. or the modelng we used the followng values (obtaned from GRASP calculatons): 5/ 4p = 79.8eV / / 4p = 8.eV / 4 = 80.4eV d/ p/ = 88meV (the decay rate was assumed j -ndependent) d 4 p Z = 0.9 au.. / 5/ 4 p Z = 0.06 au.. / / 4 p Z = au.. and / / 4 p Z 4 p Z =0.096 au.. / / / /

4 do: 0.08/nature09 II. Vsualzaton of the angular hole-densty dynamcs after the NIR laser pulse The hole-densty dstrbuton as functon of tme t n the sphercal coordnates s gven n []. Snce the radal contrbuton of the hole-densty s tme-ndependent the sphercal hole-densty can be expressed as: (/) (/) (/) h( θϕ t) = ρ //sn θ+ ρ // cos θ+ + ρ // + (/ ) t ρ// cos π + δ cos θ ( SO) T () t where π + δ = φ() t ( SO) T s the quantum phase between the 4 / p and 4 p / states as defned n q. () n the man text. Strctly speakng the hole densty defned here depends only on the tme t and on the angleθ whch s the polar angle wth respect to the laser polarzaton axs. III. ttng of the expermental data To retreve the Kr + on quantum state dstrbutons and the degree of coherence of the spn-orbt wavepacket a two-dmensonal ft of the expermental data shown n g. 4c to q. () s performed. Snce q. () rgorously defnes the transent absorpton sgnal only n the absence of the NIR laser pulse the ft s performed over the tme delay nterval of fs only and covers the spectral range of 78.5 to 8 ev. Keepng n mnd that the absorbance of the sample s determned va ts measured transmsson n the experment the transmsson functon computed from q. () s convolved wth a normalzed Gaussan to account for the fnte XUV spectrometer resoluton. Hence the expermental transent absorpton spectrum s ft to the functon 4

5 do: 0.08/nature09 OD ω N ( τ ) log exp[ σ ( ω τ )] * 4ln 4ln exp πδ Δ = ( ω ) where N s the normalzaton constant for the retreved quantum state dstrbuton σ ( ω τ ) s gven by q. () Δ s a WHM quantty that characterzes the spectrometer resoluton (a ft varable n ths case) and ω 0 = 80.5 ev s the central photon energy for the range over whch the ft s performed ( ev). 0 References. Rohrnger N. & Santra R. Multchannel coherence n strong-feld onzaton. Phys. Rev. A (009). 5