Pulse shape effect of delayed pulse on non-adiabatic rotational excitation

Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 95 (2012) 491–496

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Pulse shape effect of delayed pulse on non-adiabatic rotational excitation Urvashi Arya a, Brijender Dahiya b, Vinod Prasad b,⇑ a b

Department of Physics and Astrophysics, University of Delhi, Delhi 110007, India Department of Physics, Swami Shraddhanand College, University of Delhi, Delhi 110036, India

a r t i c l e

i n f o

Article history: Received 24 January 2012 Received in revised form 20 March 2012 Accepted 7 April 2012 Available online 23 April 2012 Keywords: Delayed pulse Pulse shape NAREX Zero area pulse

a b s t r a c t We examine the time evolution of Non-adiabatic excitation of polar molecule in static field exposed to a combination of delayed pulses. The delayed pulse pair consists of half cycle pulse (HCP) and an another delayed pulse (either ultrashort half cycle pulse or zero area pulse). We describe how Non-adiabatic rotational excitation (NAREX) due to Gaussian HCP pulse alone can be greatly modified by applying ultrashort HCP/zero area pulse. It is also shown that non-adiabatic rotational excitation can be controlled by various laser parameters, out of which pulse shape plays the most significant role for controlling the dynamics. Time dependent Schrödinger equation (TDSE) of NAREX dynamics, are studied using efficient fourth order Runge Kutta method. Ó 2012 Elsevier B.V. All rights reserved.

Introduction Due to fundamentality and significance of Non-adiabatic rotational excitation (NAREX), in various branches of natural sciences, there is long history of experimental [1–3] and theoretical [4–7] investigation of NAREX. NAREX has been utilized in many applications such as non-resonant electronic excitation [10], internal conversion [11,12], fragmentation to neutral products [13], dissociative ionization [14,15], nuclear rearrangement [16], etc. NAREX plays significant role in the strong field electron dynamics of atom, it also plays an even more vital role in molecular excitation. Since atoms are small in size and less complex than molecule, therefore molecule have more intricate and subtle electronic structure. Molecules also have nuclear degree of freedom (rotational and vibrational) and can undergo internal conversion or dissociate. Therefore, the transition probability of molecule due to non-adiabatic excitation is of great interest. NAREX also provides a promising and versatile way to probe and control molecule with an external field. A lot of literature is available for rotational excitation due to static electric field or half cycle pulses alone [17], but NAREX in static electric field and a combination of delayed pulses is scarce. Here, we have used external field as a continuous static electric field and a combination of delayed pulses. The combination of delayed pulses used here consists of a broad Gaussian half cycle pulse (HCP) and second delayed pulse of very short duration as compared to the initial pulse. Delayed pulse used here are of two types i.e. ultrashort pulse

⇑ Corresponding author. Tel.: +91 9871365843. E-mail addresses: [email protected] (U. Arya), [email protected] (V. Prasad). 1386-1425/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.saa.2012.04.031

of negligible area denoted by 0H and zero area pulse. HCP pulse is electrical pulse with duration shorter than the rotational period and their intensity can be up to 200 kV/cm [18]. Zero area pulses has zero temporal area. An illustration of this pulse consisting of two pulses whose phase differ by p. Zero net area corresponds to zero energy or finite energy with the electric field changing sign. We can define zero area pulses as any pulse with zero net area under the electric field envelope and we use the term 0p pulses to denote the subset that represents the zero area pulses that sustain self-induced transparency [19]. Zero area pulse of relatively low amplitude can be produced by linear approximation approach [20] whereas, for high energy zero area pulses one needs recently developed mathematical technique of inverse scattering [21–23]. Zero area pulse has numerous applications in nonlinear optics, laser physics, medicine, telecommunication system and other fields [24]. It is also used to diagnose some of the ultrafast processes and materials. These have been used to coherently control two photon transition [25,26] in atomic system. They have also been employed to control molecular [27] and solid-state quantum system [28]. In addition pulse shaping may become a versatile tool in communication applications due to its ability to realize encoding and decoding devices [29]. The transition probability of the system shows oscillatory behavior depending upon the pulse area of applied pulse. Rotational excitation is usually studied either under adiabatic or non-adiabatic conditions. In adiabatic condition, the laser pulse period is quite high, according to rough estimate if Tpulse > 5h/B, where h, is the Planck’s constant and B, is the rotational constant of the molecule, the molecule behaves as if the field is static at any instant. The states thereby created are the stationary pendular states and in such cases the rotational excitation dynamics follows the laser pulse shape [7], whereas, in general for smaller pulse

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duration (i.e. Tpulse < h/B) the time evolution is said to be non-adiabatic and molecule is supposed to end up in a rotational wave packet. As far as orientation of molecule [30,31] is concerned system needs an asymmetry field that distinguishes up and down. As already known, a variety of methods have been proposed to break the field symmetry, including introductions of weak electrostatic or magnetic or both fields, in addition a off-resonant pulsed laser field [32,33] and the conventional schemes of the coherent excitation with the laser fields of frequencies x and 2x [34]. In this paper, we have studied non-adiabatic case which is much more effective method to orient the polar molecules utilizing the asymmetric electromagnetic half cycle pulse [35,36]. In general molecule oscillates between the different rotational eigenstates and the time average is zero [36–40]. To overcome this problem we have applied delayed zero-area pulse in the combination of HCP pulse and found that the results get drastically modified. Also the variation in the results due to delayed ultrashort HCP in place of zero area pulse is studied to highlight the effect of zero area pulse. We have also calculated the orientation at two different temperatures.

where E3(t) = sin(pt/Tp) for zero area pulse and E3(t) = sin2(pt/Tp) for ultrashort HCP, respectively; 0 < t < Tp, Tp is the pulse duration of the respective pulses. h0 , is polar angle between molecular axis and zero area pulse/ultrashort HCP. In order to ensure that the observed rotational excitation is induced by NAREX process, a quantum dynamical calculation is performed to calculate the population by solving the time dependent Schrödinger equation (TDSE)

Theory

hcosðhÞðtÞi ¼

i

@w ðh; /; tÞ ¼ HðtÞwðh; /; tÞ @t

The above equation is solved numerically using Runge–Kutta method [41], with initial condition taken as the molecule being in the ground rotational J = M = 0 state at t = 0 time. We have taken states up to J = 19,i.e total of 20 lowest rotational states as the field intensities considered here only lowest few rotational states are populated. Although the convergent results are obtained by taking states up to J = 9. The measure of the orientation is taken as the expectation value of cos(h)

Z 2p Z p 0

The laser induced rotational excitation dynamics is studied for LiCl molecule, in simultaneous presence of three electric fields, first there is continuous static field, second HCP with different pulse duration, which delivers its energy during the positive cycle of the pulse and strong field generated by the delayed zero area pulse/ultrashort HCP. The molecule is treated with in rigid-rotor approximation (frozen internal vibrational motion), interacting with the fields. The molecule plus the two fields Hamiltonian, for this model is [7–9,32]

HðtÞ ¼ BJ 2 þ V s þ V E ðh; tÞ þ V Z ðh0 ; tÞ

ð1Þ 2

where, B, is the rotational constant and J is the squared angular momentum operator. The interaction potential Vs = l0E1; where l0 is the permanent electric dipole moment along the internuclear axis. The laser field is defined as E1, is the static field amplitude. The interaction potentials VE(h, t) and VZ(h0 , t) are the interaction potential of the laser field and interaction potential of zero area pulse/ ultrashort HCP with dipole moment along the polarizability, respectively, with VE, defined as

V E ðh; tÞ ¼ l0 E2 ðtÞcosðhÞ

ð2Þ

where h, being the polar angle between the molecular axis and the laser field. It is precisely this angle which defines the orientation of the molecule with respect to laser field. The laser field is defined as

E2 ðtÞ ¼ E0 f ðtÞsinðxtÞ

ð3Þ

where E0, is the electric field amplitude; and f(t) is envelop of the pulse defined as 2

f ðtÞ ¼ sin ðpt=tp Þ; f ðtÞ ¼ 0;

0 < t < tP

otherwise

ð4Þ ð5Þ

In the Eq. (3), x, is the frequency of the applied field of the HCP, which fits the main features of the experimental shape peak amplitude E0 duration Tp (full width at half maximum FWHM). There is probability flow to few of the low lying rotational states due to static field which get modified considerably by the zero area pulse/ultrashort HCP. We have considered the pulse duration of up to 1 ps, which is quite small as compared to rotational periods of the molecule (23.6 ps), hence non-adiabatic interaction are observed. While VZ(h0 , t), is the interaction due to zero area pulse/ultrashort HCP

V Z ðh0 ; tÞ ¼ l0 E3 ðtÞcosðh0 Þ

ð6Þ

ð7Þ

cosðhðtÞÞjwð/; h; tÞj2 sinhdhd/

ð8Þ

0

The orientation parameter hcosh(t)i varies within the interval [1, 1] and the perfect orientation is signified by the extremum value of hcosh(t)i However for non-zero temperatures, the thermal average of the orientation is taken into account as:

hcosðhÞðtÞiT ¼ 1=Z

J max J X X PðJÞ hcosðhÞðtÞiMJ J¼0

ð9Þ

MJ ¼J

Here P(J) = exp[(BJ(J + 1)/KBT)], is the Boltzmann distribution PJmax function associated with the rotational states, Z ¼ J¼0 ð2J þ 1Þ PðJÞ, is the partition function, T is the temperature, KB is the Boltzmann constant, and B=⁄2/2I is the rotational constant of the molecule. Results and discussion Here we have considered the rotational excitation and orientation of LiCl molecule in the presence of collective consequence of continuous static electric field and a combination of delayed pulses. The combination of delayed pulses used here consists of a Gaussian Half cycle pulse (HCP) and delayed pulse of very short duration as compared to the initial pulse. Delayed pulse used here are of two types i.e. ultrashort pulse of negligible area and zero area pulse. The pulse width variation of initial HCP allows the system to be exposed to the electromagnetic field for different durations and the delayed pulse is used for the kick mechanism. Eq. (8) gives only the dynamical behavior of hcosh(t)i. The consideration of LiCl molecule as a sample is obvious, as it has been extensively studied [42–48] and is an ideal case of orientation as it has large value of permanent dipole moment (2.49 a.u.), rotational constant (3.21  106) and polarizability Da = 1.18 a.u.,Da = ak  a\, is the difference between the parallel ak and perpendicular a\ components of polarizability tensor [48]. As we have taken pulses which interact through only permanent dipole moment, hence the intensities and the duration of fields are such that no ionization and dissociation of molecule takes place. Dependence of the orientation parameter and NAREX is presented for the time delay between the pulses and the pulse duration of initial HCP. The peak electric field strength is kept below the ionization threshold 200 kV/cm. The field due to the combination of delayed pulses will exert an abiding force on the permanent dipole moment of the molecule and then the molecule will exchange angular momentum with external field by rotational excitation, purely NAREX, can be described by Rigid Rotor Approximation (RRA). We have consid-

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ered Rigid Rotor Approximation [49], as the frequency of the laser pulses taken here are close to lower rotational transitions hence vibrational and electronic excitation of the diatomic molecule can be safely neglected. An intriguing feature of the rotational wave packet created by a strongly non-adiabatic pulse in the combined fields is revealed when another delayed pulse is sent in, second pulse can either restore, suppress or increase the rotational excitation of the wavepacket that had during the first pulse. The outcome also depends on the delay between the two pulses. Fig. 1 shows the transition probability as the function of time for two different delayed pulses applied at tc/2, where tc is rotational period of molecule (23.6 ps) and tc/2 is the half of the rotational period as per indication given in each panel. The delayed pulse gives the kick mechanism and the rotational states get reshuffled. It is not only the delay between the two pulses which may change the probability of a particular state but also the shape of the delayed pulse plays a vital role. Also the oscillatory behavior of states get suppressed in the HCP, as the tail duration in the zero area pulse is of longer duration than the tail of HCP. The transition from ground state to different rotational states are indicated in the figure, where 0  0 shows the ground state, 0  j0 shows the rotational excitation from ground state to j0 state. In Fig. 2, the transition probability as a function of time is studied only changing the pulse width of initial HCP keeping rest of the parameter same as for Fig. 1 and obtained the effect of pulse duration of initial HCP on different rotational states by comparing Figs. 1 and 2, thus the rotational probability of particular state can also

probabilities

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be controlled by exposure time before the kick mechanism is applied. In Fig. 3; we have studied the effect of static field strength on the time evolution of transition probabilities of different states for two different types of delayed pulse. The pulse width of initial Gaussian HCP is kept constant at 1.0 ps and the delayed pulse is applied at tc/2. By comparing the panels (a, b) and (c, d); we can conclude that with increase in static field strength, the oscillatory behavior of transition probabilities increases and also the transition probability of higher states (i.e. J = 8, 9) increases with Es. Also transition probability of some states is almost independent of static field strength (except their oscillatory behavior) such as states J = 3, 4 for panel (a, b) and state J = 3 for panel (c, d). In Fig. 4; we have studied the effect of initial Gaussian HCP width on the time variation of transition probabilities of different states. keeping the static field strength, delayed pulse type and the delay time constant. The variation of initial HCP duration does small variations of time evolution of transition probabilities of different states if tp = 0.1 ps, but if tp = 1.0 ps, we get dominant variation of some higher J states. i.e. state J = 9 is enhanced remarkably. when pulse duration is 0.1 ps, states J = 4 and 5, shows enhanced population, but as soon as pulse duration is 1.0 ps, state J = 4 get suppressed and population of state J = 8 and 9, get enhanced. In Fig. 5, calculations were carried out for pulse duration of 0.1 ps and 1.0 ps with peak intensity of the ultra-short HCP pulse fixed at 150 W/cm2 and is applied at tc/2, here static field is zero. The results shown in the figure reveals the existence of an optimal

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Fig. 1. Time evolution of the rotational state distribution calculated at static field intensity of 5 kV/cm. zero area pulse for panel a and half cycle pulse for panel (b) is applied at tc/2, respectively. Here the pulse duration (tp) of initial pulse is 0.1 ps, different states transition are indicated in the key.

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Fig. 2. Same as for Fig. 1, except that here the pulse duration of the initial HCP pulse is 0.5 ps, different states transition are indicated in the key.

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tp=1.ps

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Fig. 3. Time dependent rotational population are calculated for delayed pulse applied at tc/2. The results are calculated for different static field intensity (Es), in panels (a) and (c) Es = 0, in panels (b) and(d), Es = 5 kV/cm. First row shows the result for zero area pulse and second row shows result for ultrashort HCP, different states transition are indicated in the key.

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Fig. 4. The effect of pulse width of initial HCP pulse on the time evolution of rotational states is studied in this figure. The time evolution of rotational state is calculated at static field intensity of 10 kV/cm. Here zero area pulse is applied at tc/2. Pulse width of initial HCP is .1 ps and 1 ps in panels (a) and (b), respectively, different states transition are indicated in the key.

J, values for each pulse duration. Furthermore, we observe that the maximum rotational excitation value depends critically on the pulse duration. As the initial pulse duration increases; the time of exposure of the system also increases; due to this rotational probability also increases of states J = 1, 2, 3 . . . in order, but as soon as the delayed pulse ultrashort HCP is applied, this system get kicked and all the rotational states get reshuffled, so that high J states get excited and low J states get suppressed. In Fig. 6, the absolute degree of the orientation is calculated as a function of time for two different temperatures as indicated in panels. We measure hcos(h)i, for two different pulse duration 0.1 ps and 1.0 ps of initial HCP. The delayed ultrashort HCP is

applied at tc/2. At temperature 5 K, the value orientation is much smaller than the value of orientation at 0 K temperature. The reason is that, at higher temperature the population of higher levels decreases due to which orientation also decreases. It is clearly shown in figure, as the pulse duration is increased the value of orientation is decreased. Hence for the small duration of HCP the ionization of the molecule is less likely to occur for low value of pulse duration, but if we apply the HCP for larger duration (tp = 1.0 ps or more), we increase the interaction duration of the pulse. This may destroy the molecule and ionization is much more likely to occur in the molecule. Hence it is concluded that, for better orientation, we need HCP of smaller duration.

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U. Arya et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 95 (2012) 491–496

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Fig. 5. The time dependent rotational population with ultrashort HCP applied at tc/2, calculated at zero static field intensity, here the initial HCP pulse duration is .1 ps and 1 ps in panels (a) and (b), respectively, different states transition are indicated in the key.

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Fig. 6. Time evolution of the degree of orientation hcoshi(t) for LiCl, by using different pulse duration .1 ps and 1 ps in panels (a) and (b), respectively. The UHCP is applied at tc/2. Two different curves in each panel are of two different temperatures i.e. red for 0 K and green for 5 K. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Summary and conclusion We report the simulations of NAREX, between rotational states, driven by laser fields consisting of continuous static field, HCP with different pulse duration and the delayed pulse (zero area pulse/ ultrashort HCP). We have shown that the NAREX due to continuous static field can be enhanced by inserting second delayed pulse. Our further investigation indicates that delay between the two pulses play significant role for enhancement or repression of any rotational state. We find that the delayed pulse of different shapes seems very promising for enhancing rotational excitation of various states and controlling the molecular orientation. This study has various applications in stereodynamics, in chemical reactions, electronic stereodynamics, molecular separation techniques and molecular waveguides based on the contribution of spatial and orientational control. Hence NAREX and nonadiabatic orientation of molecule system to remain a massive area of research in physics and chemistry. Acknowledgments U.A. acknowledges the help from University Grants Commission (UGC) for financial support. V.P. is thankful to DST for financial support. References [1] A.N. Markevitch, D.A. Romanov, S.M. Smith, H.B. Schlegel, M.Y. Ivanov, R.J. Levis, Phys. Rev. A 69 (2004). 013401(1)–013401(13).

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