Density-matrix description of partially coherent spin-orbit wave packets produced in short-laser-pulse photodetachment
DDensity-matrix description of partially coherent spin-orbit wave packets produced inshort-laser-pulse photodetachment
S. M. K. Law ∗ and G. F. Gribakin † School of Mathematics and Physics, Queen’s University Belfast, Belfast BT7 1NN, United Kingdom (Dated: August 17, 2018)We investigate orbital alignment dynamics within the valence shell of atoms in coherently excited j = / , / − , Cl − and Br − anions. Using Keldysh-type theory, we calculate the density matrix of the residual atoms generated by few-cycle pulses, whose elementsdetermine the populations and coherence among the electronic states. Our calculations demonstrate that thedegree of atomic coherence can be represented by a near universal function of the ratio between the pulseduration ˜ τ p and the beat period τ j (cid:48) j of the atomic system, which allows one to characterize the coherencegenerated in atomic states. The development of femtosecond and attosecond laserpulses has allowed the possibility to generate and observewave-packet dynamics within atoms and molecules. For in-stance, the interaction of molecular systems with short pulsescan initiate coherent rotational wave-packet dynamics, allow-ing the observation of a high degree of alignment along thefield polarization direction in pump-probe experiments [1–9].For laser pulses with duration in the few-fs range, vibrationalwave packets have also been observed [10, 11], while in theattosecond time domain, electron dynamics have been probedin the valence shell of neutral molecules through the process ofhigh-harmonic generation [6, 12–14]. Since the timescale ofmotion is dictated by the energy splitting of the states involved,the observation of coherent dynamics requires excitation by apulse of comparatively short duration (i.e., large bandwidth).It is known that the process of ionization by short pulsescan produce hole states and electronic wave packets with long-lived coherences. Of particular interest is the orbital alignmenteffect characterized by localization of the electron density holealong the polarization axis of the laser pulse [15–17]. Severalstrong-field pump-probe experiments have revealed evidenceof coherent wave packet dynamics within the valence shellof positive ions initiated by pump-pulse ionization of neutralatoms [16–20]. Additionally, experiments have demonstratedwave packet motion within C, Si and Ge open-shell neutralatoms generated by strong-field detachment of the respectivenegative ions [21, 22]. The data were analyzed theoreticallyin [23], which clearly demonstrated a reduction in electroniccoherence when the pump pulse duration is comparable to orexceeds the atomic spin-orbit period. The periodic variationof coherent electron wave packets through the time delay canalso have a significant effect on the probability of sequentialdouble ionization, as demonstrated in the yields of singly-charged cations obtained from pump pulse detachment of Ag − and Al − [24, 25].A proper treatment of coherence within electronic statescan be done by full numerical simulations of the laser in-teraction with the target and calculation of the density ma-trix of the residual atomic system. Existing developments in-clude time-dependent R -matrix methods [26–28], the time-dependent configuration-interaction singles method [29, 30]and a multichannel theory [16]. In particular, the work of [16] applied a reduced density matrix formalism to describe Ne + and Xe + cationic states produced by tunnel ionization, andexamined the hole dynamics and the value of the coherencebetween the doublet fine-structure j = / , / np ground-state configuration.The original Keldysh theory [31] and its variants are com-monly used to study strong field interactions with matter (seee.g. [32–36]). They work particularly well for modelling elec-tron detachment from negative ions where long-range effectsare insignificant. In this paper we use KTA to calculate theamplitudes of photodetachment by a short laser pulse, lead-ing to a neutral atom (F, Cl, or Br) and an electron with agiven momentum in the final state. The residual atomic statespin-orbit manifold P j consists of the j = / m = ± / , ± / j = / m = ± / − forF, Cl, and Br, respectively (corresponding to beat periods be-tween 82.5 and 9.05 fs), the calculations cover a wide range ofparameter space. Our results indicate that the degree of coher-ence is determined by the ratio of the pulse length and the beatperiod, which can be used to predict the subsequent temporalevolution of the coherently excited electronic wave packets.Throughout our analysis we assume that the laser pulseis linearly polarized, with a sine-squared envelope and vec-tor potential A ( t ) = A sin ( ω t / N ) sin ω t , and of total du-ration τ p = π N / ω , where ω and N are the frequency andnumber of optical cycles, respectively. The correspondingpulse duration at full width at half maximum (FWHM) ofthe intensity is ˜ τ p = . τ p . Within the KTA, the detach-ment amplitude for electron transition from an initial state Ψ jm ( r , t ) = ψ jm ( r ) e − E j t into a final (Volkov) state Ψ p ( r , t ) a r X i v : . [ qu a n t - ph ] A ug with asymptotic momentum p , is written in the form A jm p = − i ∫ τ p ∫ Ψ ∗ p ( r , t ) V F ( t ) Ψ jm ( r , t ) d r dt , (1)where F ( t ) = − d A / dt is the electric field and V F ( t ) = r · F ( t ) represents the interaction operator (atomic units are used).The integral over time in Eq. (1) is evaluated using the saddle-point method [32, 35], which replaces it by a sum over a set ofcomplex saddle points t = t µ satisfying S (cid:48) p ( t µ ) =
0, Im t µ > S p ( t ) = ∫ t [ p + A ( t (cid:48) )] dt (cid:48) − E j t , (2)is the classical action of the electron the field, and with initialbound-state energy E j <
0. This gives A jm p = −( π ) / B N + (cid:213) µ = (cid:213) m l m s (±) l C jmlm l sm s Y lm l ( ˆ p µ ) χ sm s × exp [ iS p ( t µ )] (cid:113) − iS (cid:48)(cid:48) p ( t µ ) , (3)where B is the asymptotic normalization constant of the bound-state wave function [37], the alternating sign (±) = ± l = C jmlm l sm s are the Clebsch-Gordan coefficients, Y l , m l denote the spherical harmonics (evaluated at the cor-responding saddle points), and χ s , m s are orthonormal spinfunctions. Within the LS -coupling scheme, the second sum inEq. (3) is over all m l and m s that satisfy m l + m s = m .The elements of the density matrix of the residual atom atthe conclusion of the pulse are evaluated from the transitionamplitude (1) by the formula ρ j (cid:48) m (cid:48) jm = ∫ A j (cid:48) m (cid:48) ∗ p A jm p d p ( π ) , (4)where the integration is performed numerically in sphericalcoordinates, up to the photoelectron energy of 15 ω . The diag-onal elements ρ jmjm ≡ ρ ( j , m ) are probabilities of populatingdifferent final states jm of the atom ( ρ ( j , m ) = ρ ( j , − m ) for linearpolarization), and w = (cid:205) jm ρ ( j , m ) is the total photodetachmentprobability. The complex off-diagonal elements determine thecoherence between the relevant states. Coherent superpositionscan only be formed between atomic states with the same valueof m [16, 17], as confirmed in our calculations where the den-sity matrix elements with m (cid:48) (cid:44) m are found to be numericallysmall (with relative magnitudes < − ).The coherence between the j = / / ρ
32 12 12 12 . The degree ofwave-packet coherence can be characterised by the ratio of themagnitude of the off-diagonal element to the geometric meanof the corresponding diagonal elements [17], g = (cid:12)(cid:12) ρ
32 12 12 12 (cid:12)(cid:12)(cid:113) ρ ( , ) ρ ( , ) . (5) Here g = g = ρ a ( t ) = (cid:213) j (cid:48) jm ρ j (cid:48) mjm e i ( E j − E j (cid:48) ) t | j (cid:48) m (cid:105)(cid:104) jm | , (6)and consider the expectation value (cid:104) ˆ O ( t )(cid:105) = Tr ( ˆ O ˆ ρ a ) for anyobservable ˆ O . Coherence between the spin-orbit componentsresults in a periodic variation of the electron density. The evo-lution of the laser-generated np atomic states can be probedsimilarly to Refs. [21, 22], by applying a laser pulse whichpredominantly ionizes electrons with m l =
0, and by measur-ing the signal for polarizations parallel and perpendicular tothe polarization of the pump pulse, S | | and S ⊥ , respectively.Considering the ratio S ( t ) = ( S ⊥ − S | | )/( S ⊥ + S | | ) [23], wefind that the oscillations in the m l = ω b = E / − E / , S ( t ) = ¯ S + ∆ S cos ( ω b t + β ) , (7)where ¯ S = (cid:16) ρ ( , ) + ρ ( , ) + ρ ( , ) (cid:17) , (8)is the constant alignment offset, and ∆ S = √ (cid:12)(cid:12) ˜ ρ
32 12 12 12 (cid:12)(cid:12) , (9)is the amplitude of the beats, determined by the normalizeddensity matrix elements ˜ ρ j (cid:48) mjm ≡ ρ j (cid:48) mjm / w , and β is an addi-tional phase which may result due to difficulties in determiningthe zero time delay in experiment [21, 22]. Equation (9) showsthat the amplitude of the beats is proportional to the magnitudeof the off-diagonal element describing the coherence of theatomic system. If we assume that the pump pulse is very short,and only detaches m l = g =
1. The populations arethen simply determined by the LS -coupling coefficients,˜ ρ ( , ) = , ˜ ρ ( , ) = , ˜ ρ ( , ) = , (10)with the maximum beat contrast ∆ S / ¯ S = / ≈ . − ,Cl − , and Br − by an eight-cycle pulse with peak intensity1 . × W/cm and wavelength 1800 nm (the FWHM pulseduration is ˜ τ p = . ρ ) density matrix elements.Here the data are presented as a function of the anion-laserinteraction time, where the detachment amplitude Eq. (3) iscomputed by discretizing the range of the pulse-anion interac-tion time t (cid:48) into 2 ( N + ) =
18 saddle-point times Re t (cid:48) µ with0 ≤ Re t (cid:48) µ ≤ τ p , and taking a cumulative partial sum over a j =3/2, m =1/2 j =1/2, m =1/2 j =3/2, m =3/2 -202 Re partIm part00.040.080.12 P r ob a b ilit y -4-2024 C oh e r e n ce ( i n un it s o f - ) Abs. value0 10 20 30 40
Time (fs)
Time (fs) -202 -0.0200.02-0.0200.02 E l ec t r i c f i e l d ( a . u . ) -0.0200.02Electric field F FCl ClBr Br
FIG. 1. Time development of the diagonal (“probability”) and off-diagonal ( ρ
32 12 12 12 , “coherence”) elements of the density matrix forF, Cl and Br (top, middle and bottom, respectively) during in-teraction with an eight-cycle 1800 nm pulse with peak intensity1.3 × W/cm . To guide the eye, values obtained at discrete timeinstances Re t (cid:48) µ (circles, see text) are connected by the dashed blue line( ρ ( , ) and Re ρ
32 12 12 12 ), dot-dashed red line ( ρ ( , ) and Im ρ
32 12 12 12 ),and solid black line ( ρ ( , ) and (cid:12)(cid:12) ρ
32 12 12 12 (cid:12)(cid:12) ), with the electric field ofthe pulse also superimposed on the right-hand-side panels. subset of saddle-point contributions, 0 ≤ Re t µ ≤ Re t (cid:48) µ . Thediscrete time instances Re t (cid:48) µ shown correspond to the electronemission parallel to laser polarization ( θ = p ∼ .
05 a.u. This allows one to visualize the build-up ofthe population and coherence during interaction with the field.Figure 1 shows that the density matrix elements vary rapidlywithin the central time interval of ∼
15 fs. In all cases, themain contribution comes from 6 central saddle points whichcorrespond to three middle cycles of the field. For F, the beatperiod τ b = π / ω b = . ρ
32 12 12 12 and Im ρ
32 12 12 12 ismonotonic. For Cl ( τ b = . ≈ τ p ), one can detect smallnonmonotonic features in both real and imaginary parts of ρ
32 12 12 12 . For Br one observes oscillations on the time scale ofthe spin-orbit period τ b = .
05 fs, which result in a small finalvalue of the coherence. These oscillations are similar to thoseseen in the calculations for neon and xenon [16].At the end of the 8-cycle, ˜ τ p = . g becomes progressively smaller for heavier atoms, in particular,producing near incoherent (classical) ensembles for the caseof Br. The present calculations clearly demonstrate that largecoherence can be observed only for τ b (cid:38) ˜ τ p , confirming thathigher-bandwidth (i.e., shorter) pulses are required to achievecoherent wave packet formation in heavier systems.To quantify the effect of the pulse duration on the spin-orbit-state coherence of the residual atom, we have performedcalculations of the density matrix for a variety of pulse du-rations τ p . Using the KTA, we computed values of g for F, τ ~ p / τ b C oh e r e n ce FClBr
FIG. 2. The degree of coherence g [Eq. (5)] calculated for j = / j = / τ p to the atomic beat period τ b .The laser pulse for each data point consists of an integer number ofcycles N = , , . . . ,
18 for F and Cl, and N = , , . . . , . × W/cm .The solid line is a Gaussian fit to the data, Eq. (11). Cl, and Br atoms, for τ p = π N / ω with increasing numberof cycles N = , , . . . at fixed frequency ω = . . × W/cm (the Keldysh parameter in the calculations is within the range γ = . g are presented in Fig. 2 as afunction of the temporal ratio ˜ τ p / τ b between the FWHM pulseduration and the atomic beat period. The discrete data pointscorrespond to the coherence generated by a laser pulse with N = , , . . . ,
18 cycles for F and Cl, and N = , . . . , τ p / τ b [39]. The solid line in the graph is a curve of best fit to thedata, assuming a Gaussian shape, g = g exp (cid:104) − ζ (cid:0) ˜ τ p / τ b (cid:1) (cid:105) , (11)with two fit parameters, g and ζ . The values for the best fitshown in Fig. 2 are g = .
89 and ζ = . g is a universal function of thescaled pulse length parameter ˜ τ p / τ j (cid:48) j , whose shape is closeto a Gaussian. The value of g progressively decreases whenthe pulse duration ˜ τ p is increased (by increasing the number ofcycles N or decreasing the wave frequency ω ), and drops closeto zero when ˜ τ p exceeds the atomic beat period τ b . In partic-ular, our calculations predict a reduction of g to about 0.25for ˜ τ p / τ b =
1, while near-complete incoherence is reachedfor ˜ τ p / τ b =
2. The latter is observable for Br that possessesthe shortest beat period τ b = .
05 fs, which is exceeded bythe pulse duration ˜ τ p when N >
5. Additional calculationsshow that the values of the coherence are only weakly depen-dent on the laser intensity and wavelength, at least within thetunnelling regime probed by the calculations [39].It is possible to further illustrate the relation between thecoherence g and the ratio ˜ τ p / τ b , by comparing our results withcalculations that employed computationally more demandingmethods to other systems. In Refs. [16, 17] a time-dependentmultichannel theory (TDMT) was used to calculate the densitymatrix and coherences of the P j ( j = / , /
2) spin-orbitstates of the ions Ne + , Kr + , and Xe + , produced in strong-laser-pulse photoionization of the respective noble-gas atoms. Thecalculations for Ne and Xe [16] were for a 4-cycle, constant-amplitude, 800 nm pulse with intensity of 2 . × W/cm (Ne) and 2 . × W/cm (Xe). For Kr [17], 750 nm pulsesof FWHM duration of 3.8 and 7.6 fs and peak intensity of3 × W/cm were employed. Table I displays the degree ofcoherence g TDMT obtained in [16, 17] for Ne + , Xe + and Kr + ,together with the respective values of the pulse duration ˜ τ p , thespin-orbit period τ b and the ratio ˜ τ p / τ b for each calculation.The last column (labelled g G ) shows estimates of the coherenceobtained from the Gaussian fit to our data, Eq. (11). TABLE I. Comparison of the values of the degree of coherence g , Eq. (5), between j = / , / τ p shown in the second column. The ratio of the pulse duration tothe atomic beat period, ˜ τ p / τ b , is given in the third column. g TDMT denotes the coherence values obtained numerically in Ref. [16] forNe + and Xe + , and Ref. [17] for Kr + . g G is the value predicted byEq. (11) with parameters g = . ζ = .
15. CoherenceIon τ b (fs) ˜ τ p (fs) ˜ τ p / τ b g TDMT g G Ne + a + a + + a Total duration of the constant-amplitude pulse (“rectangular envelope”).
From Table I, we see that the spin-orbit-state coherencesfrom the TDMT calculations for Ne + ( ˜ τ p / τ b = .
25) [16] andKr + ( ˜ τ p / τ b = .
61 and 1.23) [17] are in close agreement withour predictions using Eq. (11). This supports the observation inFig. 2 that the coherence follows a universal dependence on thescaled pulse duration ˜ τ p / τ b , with a shape close to a Gaussian.The value g TDMT = .
21 for Xe + ( ˜ τ p / τ b = .
33) [16] is higherthan our prediction g G ∼ − . However, the pulse intensityused for Xe in Ref. [16] is such that photoionization saturateswell before the end of the pulse, so that effectively, the Xe + ions are created on a shorter timescale. Indeed, using the value g = .
21 in Eq. (11), we find ˜ τ p ≈ . τ b = . ∆ S [Eq. (9)] is reduced with the increase in theratio ˜ τ p / τ b , viz. ∆ S (cid:39) ( g / ) ¯ S . (The probability of forming j , m = / , / m = / m l (cid:44) ∆ S / ¯ S can also beaffected by the difference in the binding energy of the j = / j = / P j spin-orbit wave packets generated by short-pulse photode-tachment of F − , Cl − and Br − . By calculating the elements ofthe density matrix using a Keldysh-type approach, we exam-ined the dependence of the degree of coherence g on the pulseduration ˜ τ p and spin-orbit beat period τ b . Calculations of thedensity matrix for a variety of values of τ p reveal that the de-gree of coherence g is a universal function of the ratio ˜ τ p / τ b ,and exhibits a Gaussian-like decrease with the increase of thepulse duration relative to the timescale of motion. Our analysisconfirms that residual atomic states are nearly pure, i.e., arecomprised of highly coherent superpositions, for very shortpulses, but become close to a classical (incoherent) ensem-ble whenever ˜ τ p / τ b >
2. These findings are in accord withprevious experimental [21, 22] and theoretical observations[16, 17]. Our data for the degree of coherence g is also in closeagreement with the results from a numerical multichannel the-ory [16, 17] for the given values of the ratio ˜ τ p / τ b , whichprovides evidence for the accuracy of the Keldysh approach inmodelling the spin-orbit dynamics produced in a short pulse.The calculation of the density matrix and degree of coher-ence presented in this paper can be extended to other sys-tems, e.g., C − , Si − , and Ge − , with np S ground states andthree-level residual atomic states np P J ( J = , , ∗ [email protected] † [email protected] [1] T. Seideman, J. Chem. Phys. , 7887 (1995).[2] M. Machholm and N. E. Henriksen, Phys. Rev. Lett. , 193001(2001).[3] T. Seideman, J. Chem. Phys. , 5965 (2001).[4] H. Stapelfeldt and T. Seideman, Rev. Mod. Phys. , 543 (2003).[5] I. V. Litvinyuk, K. F. Lee, P. W. Dooley, D. M. Rayner, D. M.Villeneuve, and P. B. Corkum, Phys. Rev. Lett. , 233003(2003).[6] J. Itatani, D. Zeidler, J. Levesque, M. Spanner, D. M. Villeneuve,and P. B. Corkum, Phys. Rev. Lett. , 123902 (2005).[7] F. H. M. Faisal, A. Abdurrouf, K. Miyazaki, and G. Miyaji,Phys. Rev. Lett. , 143001 (2007).[8] M. Artamonov and T. Seideman, J. Chem. Phys. , 154313(2008).[9] C. R. Calvert, W. A. Bryan, W. R. Newell, and I. D. Williams,Phys. Rep. , 1 (2010).[10] B. Feuerstein and U. Thumm, Phys. Rev. A , 063408 (2003).[11] W. A. Bryan, C. R. Calvert, R. B. King, J. B. Greenwood, W. R.Newell, and I. D. Williams, Phys. Rep. , 1 (2010).[12] M. Vacher, L. Steinberg, A. J. Jenkins, M. J. Bearpark, andM. A. Robb, Phys. Rev. A , 040502 (2015).[13] P. M. Kraus, S. B. Zhang, A. Gijsbertsen, R. R. Lucchese,N. Rohringer, and H. J. Wörner, Phys. Rev. Lett. , 243005(2013).[14] S. Baker, J. S. Robinson, C. A. Haworth, H. Teng, R. A. Smith,C. C. Chirila, M. Lein, J. W. G. Tisch, and J. P. Marangos,Science , 424 (2006).[15] L. Young, D. A. Arms, E. M. Dufresne, R. W. Dunford, D. L.Ederer, C. Höhr, E. P. Kanter, B. Krässig, E. C. Landahl, E. R.Peterson, J. Rudati, R. Santra, and S. H. Southworth, Phys. Rev.Lett. , 083601 (2006).[16] N. Rohringer and R. Santra, Phys. Rev. A , 053402 (2009).[17] E. Goulielmakis, Z.-H. Loh, A. Wirth, R. Santra, N. Rohringer,V. S. Yakovlev, S. Zherebtsov, T. Pfeifer, A. M. Azzeer, M. F.Kling, S. R. Leone, and F. Krausz, Nature , 739 (2010).[18] A. Fleischer, H. J. Wörner, L. Arissian, L. R. Liu, M. Meckel,A. Rippert, R. Dörner, D. M. Villeneuve, P. B. Corkum, andA. Staudte, Phys. Rev. Lett. , 113003 (2011).[19] L. Argenti and E. Lindroth, Phys. Rev. Lett. , 053002 (2010).[20] H. J. Worner and P. B. Corkum, J. Phys. B , 041001 (2011).[21] H. Hultgren, M. Eklund, D. Hanstorp, and I. Y. Kiyan, Phys.Rev. A , 031404 (2013).[22] M. Eklund, H. Hultgren, D. Hanstorp, and I. Y. Kiyan, Phys.Rev. A , 023423 (2013).[23] S. M. K. Law and G. F. Gribakin, Phys. Rev. A , 053402(2016).[24] J. B. Greenwood, G. F. Collins, J. Pedregosa-Gutierrez,J. McKenna, A. Murphy, and J. T. Costello, J. Phys. B ,L235 (2003).[25] H. W. van der Hart, Phys. Rev. A , 053406 (2006).[26] M. A. Lysaght, P. G. Burke, and H. W. van der Hart, Phys. Rev.Lett. , 193001 (2009).[27] L. R. Moore, M. A. Lysaght, L. A. A. Nikolopoulos, J. S. Parker,H. W. van der Hart, and K. T. Taylor, J. Mod. Opt. , 1132(2011).[28] H. F. Rey and H. W. van der Hart, Phys. Rev. A , 033402(2014).[29] S. Pabst, L. Greenman, P. J. Ho, D. A. Mazziotti, and R. Santra,Phys. Rev. Lett. , 053003 (2011).[30] S. Pabst, M. Lein, and H. J. Wörner, Phys. Rev. A , 023412(2016).[31] L. V. Keldysh, Sov. Phys. JETP , 1307 (1965).[32] G. F. Gribakin and M. Y. Kuchiev, Phys. Rev. A , 3760 (1997).[33] W. Becker, F. Grasbon, R. Kopold, D. B. Milošević, G. G. Paulus, and H. Walther, Adv. At. Mol. Opt. Phys. , 35 (2002).[34] D. B. Milošević, G. G. Paulus, D. Bauer, and W. Becker, J.Phys. B , R203 (2006).[35] S. F. C. Shearer, M. C. Smyth, and G. F. Gribakin, Phys. Rev.A , 033409 (2011).[36] S. V. Popruzhenko, J. Phys. B , 204001 (2014).[37] ψ jm ( r ) (cid:39) Br − exp (− κ j r ) (cid:205) m l m s C jmlm l sm s Y lm l ( ˆ r ) χ sm s , where κ j = (cid:112) − E j , and we use values of B from Ref. [32] and exper-imental energies E j from [40].[38] G. F. Gribakin and S. M. K. Law, Phys. Rev. A , 057401(2016).[39] A very similar dependence was observed for other intensities(7 . × and 1 . × W/cm ) and wavelengths (1300 nm),see S. Law, PhD thesis, Queen’s University Belfast (2017).[40] T. Andersen, H. K. Haugen, and H. Hotop, J. Phys. Chem. Ref.Data28