Controlling Electron Dynamics with Carrier Envelope Phases of a

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A: Kinetics, Dynamics, Photochemistry, and Excited States

Controlling Electron Dynamics with Carrier Envelope Phases of a Laser Pulse Diptesh Dey, Dhiman Ray, and Ashwani Kumar Tiwari J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.9b02870 • Publication Date (Web): 10 May 2019 Downloaded from http://pubs.acs.org on May 11, 2019

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The Journal of Physical Chemistry

Controlling Electron Dynamics with Carrier Envelope Phases of a Laser Pulse Diptesh Dey, Dhiman Ray, and Ashwani K. Tiwari∗ Indian Institute of Science Education and Research Kolkata, Mohanpur 741246, India E-mail: [email protected] Phone: +91 (0)33 66340000. Fax: +91 (0)33 25873020

Abstract A theoretical study on the ionization dynamics of carbon atom irradiated with few cycle, intense laser field is performed within a quasiclassical model to get mechanistic insights of an earlier reported carrier-envelope phase dependency of ionization probabilities of an atom [Phys. Rev. Lett. 2013, 110 , 083602]. The carrier-envelope phase of the laser pulse is found to govern the overall dynamics, reflecting its importance in controlling electronic motion. To understand the origin of this effect, individual trajectories were analyzed at a particular laser intensity. We found that a variation in the carrier-envelope phase affects the angle of ejection of the electrons and subsequently in the attainment of the desired final state.

Introduction Recent generation of extreme ultraviolet laser light with pulse durations in the attosecond regime (1as = 10−18 s) led to the birth of an exciting new era of ultrafast dynamics, namely, ∗

To whom correspondence should be addressed

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ACS Paragon Plus Environment

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attosecond science. 1–6 These new light sources are comprised of only a few optical cycles 7 and are usually highly intense, with optical fields whose magnitude typically match the inter-particle Coulombic potentials. Interaction of these ultrashort, intense electromagnetic fields with isolated atoms or molecules therefore possibly lead to either single or multiple ionization, or rupture of one or more bonds (for molecules). Since the pulse duration of these modern light sources is comparable to the timescale of atomic or molecular processes, with such laser pulses it has become feasible to visualize and control the electronic motion on attosecond time scales. 8–10 With the availability of these ultrashort, intense bursts of electro-magnetic fields as research tool 7,11 the most challenging task is to obtain a complete understanding of the laser-driven correlated motion of electrons and nuclei to achieve the ultimate goal of controlling the reaction outcome with laser light. 8 With these laser pulses comprising of a few wave cycles, the associated ultrafast dynamics becomes even more richer with a proper choice of a carrier-envelope phase (CEP): the temporal offset between the maximum of the optical cycle and the maximum of the pulse envelope. 4,12–15 This is due to the fact that the subsequent dynamics is now governed by the instantaneous magnitude of the laser pulse experienced by the irradiated system and not by the peak intensity of the pulse envelope which is kept fixed. This has been revealed in number of studies (both experimental and theoretical), such as optical field ionization, 16–19 dissociation of molecules, 20 high harmonic generation, 21–24 atomic coherence, population transfer, 25,26 etc. Therefore, CEP of few cycle pulses plays a crucial role in the attosecond electron dynamics 27–29 and offers a new prospect in the strong-field laser science. This constitutes the primary motivation behind this work. Furthermore, recent experiments have shown that ionization probability of an atom strongly depends on the CEP of a few cycle pulse. 14,15 The origin of this effect is yet to be understood at the electron dynamics level. Therefore, in this article we address the question: How laser CEP influences the correlated electron dynamics of a multielectron atom, in relation to its photoionization? To exemplify the present discussion, investigation on fairly simple multielectron (carbon) atom has been

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carried out. Classical models have been found successful to interpret strong-field experiments in many studies. 30,31 Often, some unnoticed processes are revealed by critically analyzing classical trajectories which lead to valuable insight. In this article, a well founded quasi-classical method 32,33 is employed to theoretically model the laser-driven multielectron dynamics. Within this quasi-classical model also referred to as fermionic molecular dynamics (FMD), the electrons are treated as classical point particles; and two non-classical momentumdependent auxiliary repulsive potentials are introduced to mimic the effects of Heisenberg uncertainty principle and Pauli exclusion principle that stabilizes the multielectron system from auto-ionization or collapse. Stable quasi-classical ground state configuration of atoms have been obtained using this FMD model 33 and this method has already been extensively applied in studying various atomic and molecular processes 34 including description of atoms, 31,35–41 and molecules 42–45 in intense laser fields. Moreover, it was successful in achieving reasonable qualitative agreement with experimental results. 43,46 Recently, this FMD model was employed to understand the many-electron response of carbon atom to intense laser fields. 41 The paper is organized as follows. Section II briefly describes the theoretical model implemented in this study. Numerical simulations are presented and analysed in Section III. Section IV summarizes the important findings and concludes.

Theoretical Model Following this FMD model, the laser field-free Hamiltonian for an atom with N number of electrons having nuclear charge Z is given by P2 H= 2M

+

N h 2 X pi i=1

Z f (Qi , ρi , ξ0 ) i − + 2 ρi µρ2i

N 2f (qij , rij , η0 ) i 1 Xh 1 + + δσi σj , 2 2 i,j=1 rij rij i6=j

3


(1)


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where ri , pi , and σi are the position, momentum, and spin of the i-th electron and R, P and M are the position, momentum, and mass of the nucleus, respectively. The following abbreviations: µ = M/(M + 1), Qi = |P − M pi | /(M + 1), ρi = |ri − R|, qij = |pi − pj | /2, and rij = |ri − rj | have been used. The spin σi ∈ {α, β} for each electron is fixed, and the electrons with same spin (α or β) interact via the last term in equation (1). The auxiliary potentials which take into account the classical phase-space constraints are defined as f (q, r, ξ) =

ξ2 16

exp{4[1 − (qr/ξ)4 ]}. 32 It assures that the electrons will be repelled by the

potential if the |q||r| > ξ condition is violated and thereby expel electrons from regions of classical phase-space that are forbidden quantum mechanically. The constant parameters ξ0 and η0 are chosen following reference. 32 Time evolution of the system is governed by the Hamilton’s equations of motion dri ∂H = , dt ∂pi dR ∂H = , dt ∂P

dpi ∂H dri B(t, ri ) =− − E(t, ri ) − × , dt ∂ri dt c dP ∂H dR B(t, R) =− + Z E(t, R) + × , dt ∂R dt c

(2)

where c is the speed of light, and the laser electric field, E(t, r) and magnetic field, B(t, r) are given by E(t, r) = x Ê0 ψ(t − z/c), B(t, r) = ˆ z × E(t, r)

(3)

where   cos(ω0 τ + ϕ)sin2 ( πτ ) T0 ψ(τ ) =  0

if 0 6 τ 6 T0 , otherwise

with the laser field strength E0 , angular frequency ω0 corresponding to a wavelength of 800 nm, CEP ϕ, and pulse length T0 = 2πnc /ω0 ; nc being the number of optical cycles chosen to be 3 for 7.9 fs pulse width similar to Ref. 41 It is important to point out that the Hamiltonian is invariant with respect to separate

4


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rotations of all position and momentum vectors 32 expressed as,

H(r1 , . . . , rN , R, p1 , . . . , pN , P) = H(Ω1 r1 , . . . , Ω1 rN , Ω1 R, Ω2 p1 , . . . , Ω2 pN , Ω2 P),

(4)

where Ω1 and Ω2 denote two different rotation matrices, Ω1 is applied to all position vectors and Ω2 to all momentum vectors. In the simulation, this symmetry operation is performed with random rotation matrices for the sampling of initial conditions. The quasi-classical ground state configuration of the system is obtained by proper minimization of the Hamiltonian following downhill simplex method. 47 Attainment of the global energy minimum is confirmed by repeating trials several times with various initial guesses of the coordinates and momenta. In the minimum energy configuration, the electrons form a shell structure and are located in the xy plane with pairwise arrangement of opposite spin electrons around the nucleus. Field-free trajectories were run and the ground state has been found to be stable against auto-ionization. The positions and momenta of all the particles in the ground state were also found to be consistent with earlier reported results. 41 Under the influence of few cycle, intense laser pulse the electrons start jitter motion. The temporal evolution of the system can be followed by critically studying the particles trajectories. This is conducted by numerical integration of the classical Hamilton’s equations of motion (2) using an adaptive fifth-order Runge-Kutta solver. The simulation covers a time interval of 0 6 t 6 1.5T0 , until the final state of the system is identified. Lastly, the final state analysis is performed by computing the single electron energy (i ). An electron is considered to be detached from the atom if at the end of the simulation i > 0. The ionization probabilities (PN ) were then computed as PN = kN /ktot , where kN and ktot denote the number of ionization events with final charge state N and the total number of trajectories simulated, respectively.

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0.9

0.7

(a)φ=0°

(b)φ=180°

0 1+ 2+ 3+ 4+

0.5

0 1+ 2+ 3+ 4+

0.3

0.1

0.9 Ionization Probability (PN)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.7

(c)φ=−45°

(d)φ=45°

0 1+ 2+ 3+ 4+

0.5

0 1+ 2+ 3+ 4+

0.3

0.1

0.9

0.7

(e)φ=−90°

(f)φ=90°

0 1+ 2+ 3+ 4+

0.5

0 1+ 2+ 3+ 4+

0.3

0.1 1014

1015

1016

1017 Intensity [W/cm2]

1015

1016

1017

Figure 1: Ionization probabilities (zero-, single-, double-, triple-, and quadruple-) versus laser field intensity, at different CEPs.

Results and Discussion With this theoretical model, 105 trajectories were run at various laser intensities ranging from 1014 −1017 W/cm2 for CEP values 0◦ , 180◦ , ±45◦ , and ±90◦ . Figure 1 reports the laser induced ionization probabilities (PN ) as a function of laser intensity, for different CEPs. Figure 1(a), for CEP = 0◦ , indicates that carbon atom stays unionized with 1014 W/cm2 intensity and the probability of zero-ionization decreases almost exponentially with the rise in laser intensity. Single ionization occurs in the intensity range 2 × 1014 − 1016 W/cm2 , with a gradual rise in the ionization probability first, reaches a peak value, and then decays with further increase in the laser intensity. The double ionization probability shows small ups and downs initially

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for intensities less than 1015 W/cm2 , after which it rises steeply and contributes solely in the intensity range 2 × 1015 − 5 × 1016 W/cm2 , followed by a sharp decay at 1017 W/cm2 . Onset of both triple and quadruple ionizations occur at much higher intensities – 4 × 1016 W/cm2 and 7 × 1016 W/cm2 , respectively, which is followed by steep rise afterwards. Overall, Figure 1(a) reveals for intensity below 1015 W/cm2 , the dominant channel is single ionization, double ionization gradually increases after that and contributes largely until 5 × 1016 W/cm2 , and subsequently followed by higher ionization channels. Apparently, a similar behavior is also observed for other CEP values (Figures 1(b)-(f)). However, a closer look indicates variations in the threshold values of origin, attainment of the peak values, alterations in the shape of the plots, etc. Thereby, each of these ionization channels needs to be analyzed with laser intensities kept unaltered. Figure 2 reports the variation of ionization probabilities with laser CEP at certain fixed intensities. In the case of single ionization, with 9 × 1014 W/cm2 intensity (see Figure 2(a)) ionization probability is highest for CEP = 0◦ and for CEPs ±45◦ and ±90◦ the ionization probabilities are symmetrically located with respect to CEP = 0◦ . Whereas, CEP = 180◦ shows nearly same ionization probability value with CEP = 0◦ . On the other hand, moving towards the slightly higher intensity regime (2 × 1015 W/cm2 ) where double ionization starts to play an important role, the plot is no longer symmetric in nature (see Figure 2(b)). For double ionization, the alteration of ionization probabilities with laser CEPs are reported for intensities 2 × 1015 W/cm2 and 8 × 1016 W/cm2 in Figs. 2(c) and 2(d), respectively. In both the cases, the plots are asymmetric with respect to CEP = 0◦ . For CEP = 0◦ , a lowest ionization probability value is noted for lower intensity (Figure 2(c)), whereas, a highest ionization probability value is noted for higher intensity (Figure 2(d)). Furthermore, Figure 2(c) suggests that the range of variation in double ionization probability as we move from CEP = 0◦ to CEP = −45◦ is large (from 0.426 to 0.783) in relation to other CEP values. This is in contrast to Figure 2(b) where a reverse variation in the single ionization probability is noted (from 0.55 to 0.13) at the same intensity. In case of triple ionization, Figure 2(e)

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0.9

(a) I0 = 9 × 1014

1+

(b) I0 = 2 × 1015

1+

(c) I0 = 2 × 1015

2+

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0.7 0.5 0.3 0.1

0.9

(d) I0 = 8 × 1016

2+

(e) I0 = 8 × 1016

3+

0.7 0.5 0.3 0.1

0.9

(f) I0 = 1 × 1017

3+

0.7 0.5 0.3

180°

90°

45°

0°

−45°

−90°

180°

90°

0° CEP

45°

−45°

0.1 −90°

Ionization Probability (PN)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



CEP

Figure 2: Variation of ionization probabilities with laser CEP, at certain fixed intensities. Statistical errors bars are also included to quantify involvement of any possible standard deviation errors. plotted at an intensity of 8 × 1016 W/cm2 shows similar ionization probabilities for CEP = 0◦ and 180◦ , both of which are now lower compared to other CEPs. On the other hand, with an intensity of 1017 W/cm2 (see Figure 2(f)) CEP = 0◦ favors more towards ionization; whereas, contribution from CEP = −45◦ and −90◦ are the least. To estimate the involvement of statistical errors in the results, standard deviation error of the probabilities are computed following Ref. 48 and are reported alongside in Figure 2. The error bars around the CEP values are found to merge into each other, which eliminates the possibility of significant statistical errors. Thereby, from Figure 2 we can conclude that there is definitely a CEP-dependency on the ionization dynamics of an atom, which is in fact quite large for certain CEP values in 8


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agreement with experimental observations. A simple understanding behind the origin of this effect can be obtained by carefully observing the sub-cycle structure of the electric field beyond the intensity envelope. CEP variation at a particular peak intensity results in the alteration of the sub-cycle structure of the electric field and therefore leads to a variation in the instantaneous magnitude of the optical field experienced by the irradiated atom. Control over this instantaneous field strength via laser CEP at a particular peak intensity can also be understood as a possibility of active laser control and Figure 2 demonstrates this in details where the ultrashort dynamics is amenable to control without actually altering the intensity envelope of the laser electric field. Quantum mechanically CEP-dependence was understood as an overlap/interference of atleast two n-photon channel wavepackets which is possible with intense, short laser pulses. 49 Although interference effects are completely inaccessible to quasiclassical models but keeping in mind that the spreading/overlap of a quantum wave packet in space can be explained with the movement of a swarm of trajectories, this CEP dependence for similar ultrashort and intense laser pulses can be correlated with an analogous quantum picture. It is worth pointing out that significant variation of ionization probability with CEP is also noted at laser intensities other than the ones reported in Figure 2. Quadruple ionization also varies with CEP, although the effect is less pronounced within the range of intensity used in this present study. To obtain a better understanding of this CEP influence therefore we critically analyze individual trajectories at fixed laser intensity. In Figure 2, a contrasting behavior with a simple alteration in laser CEP is noted. With 2 × 1015 W/cm2 intensity, carbon atom mostly prefers single ionization for CEP = 0◦ and have least preference for CEP = −45◦ (Figure 2(b)). On the other hand, with the same laser intensity, the exact opposite behavior is observed for double ionization (Figure 2(c)). Therefore, this particular laser intensity appeared to be a good choice for subsequent studies. Thereby, 5 × 103 trajectories were studied at this particular intensity to obtain a detailed understanding of the CEP dependence. In Figure 3, the distribution of time difference between first and second ionization events (Pt ) for doubly ionized trajectories is plotted

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0.35

CEP = -90° CEP = -45° CEP = 0° CEP = 45° CEP = 90° CEP = 180°

0.3

0.25

0.2 Pt

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.15

0.1

0.05

0 0

20

40

60

80

100

120

140

160

180

200

Time difference between first and second ionization (atomic unit)

Figure 3: Distribution of the time delay between first and second ionizations (Pt ) for different CEPs, at 2 × 1015 W/cm2 intensity. for different CEPs. The plot shows mainly three peak structures, one at very early time (between 0-20 a.u.), and the other two at later times (between 40-70 a.u. and 110-125 a.u., respectively). The former can be linked with a sequential double ionization (SDI) mechanism where the laser intensity causes both the electrons to eject simultaneously. Whereas, the later can be attributed to non-sequential double ionization (NSDI) where the first ejected electron recoils back after certain time and collides with the parent atom to eject another electron. With CEP = −45◦ the probability of reaching double ionization state is higher at early times indicating SDI to take place mostly. Whereas, with CEP = 0◦ the probability peaks at later times, indicating a NSDI phenomena. Overall, the change in the shape of the distribution curve indicates the vital role CEP plays in determining the SDI to NSDI ratio in an atom ionization. To get a more mechanistic understanding, in Figure 4 we plotted the distribution of the relative angle between two ejected electrons (Pθ ) as a function of CEP. Earlier, Hu and Collins 50 have shown that the angular distribution depends on CEP in case of a circularly polarized laser. Here, this plot indicates that CEP = 0◦ prefers a lower ejection angle and therefore, there remains a higher chance for one of the ejected electrons to collide back with the parent atom and recombine with it by electron-hole pairing to ultimately result in a single 10


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1

CEP = -90° CEP = -45° CEP = 0° CEP = 45° CEP = 90° CEP = 180°

0.8

0.6 Pθ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.4

0.2

0 0

20

40

60

80

100

120

140

160

180

Angle between ejected electrons (degrees)

Figure 4: CEP-dependency in the angular distribution of the two ejected electrons (Pθ ), at 2 × 1015 W/cm2 intensity. ionized product. On the other hand, apart from the peak at lower angle, the distribution for CEP = −45◦ showed two more peaks at higher angles (between 30 − 45◦ and at 130◦ ) with an overall tendency for the ejection angle to be much higher than the former and, therefore, lessens the chance of electron-hole pairing leading to increased double ionization probability. It also proves that following SDI both the electrons are ejected simultaneously with higher angles of separation between them and this variation is observed merely by a change in CEP. Attainment of convergence is ensured in both Figs. 3 and 4. Here, it is important to mention that the experimental angular distributions are usually wider than our reported distribution. This is due to fact that in our model, electronic motion is restricted in 2D. In a 3D calculation, one would expect a wider distribution as in full dimension electrons will have access to 4π angle for ejection. The plot of the electron-hole pair probability and the average angle against laser CEP (Figure 5) depicts that these two properties are qualitatively anti-correlated. The higher average angle of ejection for CEP = −45◦ ensures that a lower number of electrons recombine with the parent core and, as a result, the product ion remains primarily doubly ionized following an SDI event. Whereas, for CEP = 0◦ , the average angle is the least, and the first ejected electron recoils back and recombines with the divalent ion at later times forming a 11


0.6

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0.6

Probability Average angle

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 -90

Average angle (radians)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Probability of electron-hole pairing


0 -45

0

45 90 CEP (degrees)

180

Figure 5: Variation of electron-hole pair probability and average angle of ejection between the two electrons with CEP. single ionized product, and leading to a lower double ionization probability.

Conclusion To summarize, a theoretical study on the laser induced ionization dynamics of carbon atom is carried out following a well-established quasiclassical model. The influence of CEP of ultrashort few cycle pulses is studied at wide range of intensity values. The ionization dynamics is found to be significantly affected with the variation in laser CEPs. A detailed understanding of this effect is presented for a particular intensity which clearly indicates that CEP influences the angle of ejection of the emitted electrons and thereby continues to offer new prospect in controlling electron dynamics on ultrafast timescales. It will be interesting to see how 3D calculations modify these results. Presently, we are conducting 3D calculations on a smaller system which will be published subsequently.

Acknowledgement The authors would like to thank Dr. P. Manikandan for providing the Runge-Kutta code and Prof. N. E. Henriksen for helpful discussions. D.D. gratefully acknowledges University Grants Commission (UGC), New Delhi, for the Senior Research Fellowship. D.R. is thankful 12


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to Kishore Vaigyanik Protsahan Yojana (KVPY) for a fellowship.

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I0 = 2×10

W/cm

0.9 0.7 0.5 0.3 0.1

2

1+ 2+

-90 -45 0 45 90 CEP (degrees)

180

0.6 0.5

0.6

Probability Angle

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 -90 -45 0 45 90 CEP (degrees)

19

0 180


Average angle (radians)

φ=0 φ=-45

Electron-hole pair probability

Graphical TOC Entry

Ionization Probability

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Controlling Electron Dynamics with Carrier Envelope Phases of a

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