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J. Phys. Chem. A 2010, 114, 11202–11209
Coherent Correlation between Nonadiabatic Rotational Excitation and Angle-Dependent Ionization of NO in Intense Laser Fields† Ryuji Itakura,*,‡ Hirokazu Hasegawa,§ Yuzuru Kurosaki,| Atsushi Yokoyama,‡ and Yasuhiro Ohshima§ Quantum Beam Science Directorate, Kansai Photon Science Institute, Japan Atomic Energy Agency, 8-1-7 Umemidai Kizugawa, Kyoto, 619-0215, Japan, Institute for Molecular Science, National Institutes of Natural Sciences, Myodaiji, Okazaki 444-8585, Japan and SOKENDAI (The Graduate UniVersity for AdVanced Studies), Okazaki 444-8585, Japan, and Quantum Beam Science Directorate, Tokai Research and DeVelopment Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan ReceiVed: March 30, 2010; ReVised Manuscript ReceiVed: June 30, 2010
We investigate coherent correlation between nonadiabatic rotational excitation and angle-dependent ionization of NO in intense laser fields in the state-resolved manner. When neutral NO molecules are partly ionized in intense laser fields (I0 > 35 TW/cm2), a hole in the rotational wave packet of the remaining neutral NO is created because of the ionization rate depending on the alignment angle of the molecular axis with respect to the laser polarization direction. Rotational state distributions of NO are experimentally observed, and then the characteristic feature that the population at higher J levels is increased by the ionization can be identified. Numerical calculation for solving time-dependent rotational Schro¨dinger equations including the effect of the ionization is carried out. The numerical results suggest that NO molecules aligned perpendicular to the laser polarization direction are dominantly ionized at the peak intensity of I0 ) 42 TW/cm2, where the multiphoton ionization is preferred rather than the tunneling ionization. 1. Introduction Control of dynamical behavior of molecules with intense laser fields has been intensively investigated for the past decade.1-9 Intense laser pulses induce a variety of dynamical processes such as dynamical alignment,10-17 vibrational nuclear motion on light-induced potential energy surfaces,18,19 tunneling ionization,20-22 and high-order harmonic generation.23,24 These dynamical processes occur simultaneously with a certain extent of coherent correlation, which causes difficulty in controlling the respective processes independently. When the intensity of a femtosecond laser pulse is moderate to be ∼1 TW/cm2 (1012 W/cm2), only nonadiabatic rotational excitation can be induced without suffering from the influence of different dynamical phenomena, resulting in creation of a rotational wave packet after the interaction with the laser pulse.7,14,16 The mechanism of nonadiabatic rotational excitation is well described by the induced dipole interaction, that is, the interaction between the laser electric field and the anisotropic polarizability of molecules. The investigations to observe a rotational wave packet have been performed in frequency domain7,16,25 as well as in time domain.13,14 It has been recently demonstrated that the wave packet interferometry combined with the spectroscopic detection enables us to measure not only the amplitude but also the phase of a rotational wave packet.26 As the laser intensity is increased more, the influence of ionization becomes apparent. Since ionization depends on the alignment angle of the molecular axis with respect to the laser polarization direction, dynamical alignment and ionization must †
Part of the “Klaus Mu¨ller-Dethlefs Festschrift”. * To whom correspondence should be addressed. E-mail: itakura.ryuji@ jaea.go.jp. ‡ Kansai Photon Science Institute, Japan Atomic Energy Agency. § Institute for Molecular Science and SOKENDAI. | Tokai Research and Development Center, Japan Atomic Energy Agency.
be strongly coupled. It is important to understand the mechanism of ionization as a trigger followed by dissociation,27 high-order harmonic generation,28-31 and further ionization to doubly charged ions.32,33 So far, the efforts to disentangle ionization and dynamical alignment have been made using the pump(alignment)-andprobe experiment, where the ionization probability as a function of alignment angle with respect to the polarization direction of the probe pulse was observed.34-36 When an intense linearly polarized laser pulse is used in a probe process, molecules receive torque by dynamical alignment during a laser pulse,37 resulting in coherent correlation between alignment and ionization. The quantum states to be measured would be strongly perturbed by the probe pulse. In the present study, we investigate the coherent correlation between the ionization and the dynamical alignment of NO by extending the spectroscopic approach7,16,25 to the laser intensity range where the ionization probability becomes a few ten percents. In our scheme, the probe process is performed using a weak laser pulse with high frequency-resolution, which does not perturb the quantum states to be measured. We measure rotational state distribution (RSD) of the neutral NO surviving the ionization in intense laser fields and clarify the influence of the ionization, which make a hole in the rotational wave packet of the neutral NO as already reported for a vibrational wave packet38,39 and an electronic wave packet.40,41 In this study, the investigation is done in the state-resolved manner for the quantum mechanical description using the field-free rotational states with the quantum number J. Numerical calculation is also carried out for understanding the coherent correlation between the ionization and the dynamical alignment. Time-dependent Schro¨dinger equation (TDSE) for the nonadiabatic rotational excitation16 of NO is numerically solved with taking account of the influence of the ionization.42
10.1021/jp102840t 2010 American Chemical Society Published on Web 07/21/2010
Correlation between Rotation and Ionization of NO Discussion will be made on the basis of comparisons between the experimental and numerical results. 2. Experimental Section The experimental setup used in the present study is almost the same as the previous one. The details were already described elsewhere.16,25 Briefly, the sample gas of NO (0.5%) seeded in the Ne buffer gas is expanded into a vacuum chamber as a supersonic beam with a pulsed valve at the repetition rate of 40 Hz and the backing pressure of 2 atm. The rotational temperature of NO is estimated to be less than 2 K, where most of NO molecules are populated at the lowest rotational level J ) 0.5 in the X 2Π1/2, V ) 0 state. An output pulse from a femtosecond (fs) Ti:Sapphire multipass amplifier (λ ∼ 810 nm, 1 kHz rep.) is used as a pump pulse for the nonadiabatic rotational excitation and ionization of NO molecules. Its pulse duration is controlled by adjusting the distance between a pair of gratings in the compressor of the amplifier system, that is, changing the chirp rate of the pulses. Autocorrelation measurement with second harmonic generation is used for measuring the pulse duration. The shortest pulse duration we can achieve is 55 fs (the full width at the half-maximum). The laser pulse energy is changed from 0.4 to 1.2 mJ by using a half-wave plate and a thin film polarizer. As a probe pulse, second harmonic generation of nanosecond (ns) dye laser output (40 Hz rep.) is used for the resonance enhanced (1 + 1) photon ionization (RE(1 + 1)PI) of the NO molecules. The polarization direction of the probe pulse is set to be parallel to that of the pump pulse. The wavelength of the second harmonic is scanned in the range of 225.8-226.4 nm for recording ro-vibronic lines in the A2Σ+-X2Π1/2 (0-0) band. The nanosecond probe beam is counter-propagated with respect to the direction of the femtosecond pump beam and tightly focused on the NO/Ne molecular beam with a lens (f ) 170 mm). The pump beam is focused with a lens (f ) 300 nm), and the focal position is shifted by 7 mm from the crossing point with the NO molecular beam in order to make the pump beam area much larger than the focal spot of the probe beam. Therefore, only the spatially central part of the femtosecond pump beam is monitored by the nanosecond probe beam and the spatial variation of the laser intensity can be neglected. The ionized NO molecules are detected with a time-of-flight (TOF) mass spectrometer, which enables us to separate the NO cations generated by the nanosecond probe pulse from those by the femtosecond pump pulse. The delay of the nanosecond probe pulses is set to be around 200 ns, which is long enough to separate the two peaks of the NO cations in the TOF spectra. Normalization of the spectral intensities in the RE(1 + 1)PI spectra is an important procedure in this study, because the total ionization yield has to be evaluated from the reduction of the yield of the neutral NO molecules. The normalization is performed by comparing the signal intensities of the Q11(J ) 0.5) line in the RE(1 + 1)PI spectra with or without a femtosecond pump laser pulse. RSD is obtained in the normalized population ratio. The laser peak intensities are estimated from comparison between the calculated and observed RSDs in the weak intensity regime of less than 15 TW/cm2, where the ionization is negligible. It is known that the theoretical estimation in the weak intensity regime can reproduce the experimentally derived RSDs very well.16 In the strong intensity regime where the ionization occurs significantly, the intensity is estimated by the linear extrapolation from the weak intensity regime because the intensity is proportional to the pulse energy. The validity of
J. Phys. Chem. A, Vol. 114, No. 42, 2010 11203 the above intensity estimation is confirmed by an alternative method on the basis of the measurement of the spatial and temporal pump laser profile. This simple method, which contains large uncertainty, exhibits the same order of magnitude. 3. Numerical Method The time-dependent Schro¨dinger equations (TDSE) are solved in the same manner as the previous study.16 The initial population is considered to be only in the J ) 0.5 level as measured in the experiment. Only the |M| ) 0.5 subspace is taken into account, because of the selection rule of ∆M ) 0, where M is the quantum number of the projection onto the quantized axis, that is, the laser polarization direction. The effect of the population depletion from the X2Π1/2, V ) 0 state by the ionization is included by adding an imaginary potential42 -ipΓ/2 to the alignment Hamiltonian as follows,
ˆ ) BJˆ2 + 1 E(t)2(∆Rcos2 θ + R⊥) - ip Γ(E(t), θ) H 4 2
(1) where B () 1.696 cm-1) is the rotational constant of NO in the X2Π1/2, V ) 0 state;43 E(t) is the temporally varying amplitude (envelop) of the laser electric field; ∆R is equal to R| - R⊥ with R| () 2.25857 Å3) and R⊥ () 1.43295 Å3) being polarizabilities44 parallel and perpendicular to the molecular axes, respectively; θ is the angle between the molecular axis and the direction of the laser polarization; and Γ(E(t), θ) is the ionization rate as a function of E(t) and θ. Time-dependent wave functions, that is, wave packets are expanded by the Hund’s case (a) basis set for field-free rotation.16 When the imaginary potential representing the ionization rate is included, the norm of the rotational wave packet in the X2Π1/2, V ) 0 state is depleted by the ionization. Two types of model functions for Γ(E(t), θ) are assumed in our calculation. The selection rule of ∆M ) 0 is valid for the interaction by these imaginary potentials, because they are cylindrically symmetric with respect to the laser polarization direction. The first model is based on the molecular Ammosov-DeloneKrainov (MO-ADK) thoery,22 which describes tunneling ionization appropriately. The highest occupied molecular orbital (HOMO) is obtained by the unrestricted Hartree-Fock calculation with the basis set of AUG-cc-pVTZ using the Gaussian 03 program.45 In the MO-ADK theory, the asymptotic region of the HOMO is approximated by a linear combination of onecenter basis functions with the spherical harmonics Ylm(θmol, φmol) as follows:
ΨHOMO(r, θmol, φmol) =
∑ ClFl(r)Ylm(θmol, φmol)
(2)
l
where Fl(r) ) r1/(κ-1)e-κr and κ ) (2IP)1/2 with the ionization potential Ip ) 9.26 eV for NO. The three components of l ) 1-3 are included in the linear combination. The polar coordinates of an electron in the molecular-frame are represented by (r, θmol, φmol). Only the basis functions Ylm(θmol, φmol) with |m| ) 1 are taken into account, because the HOMO of NO is a π orbital. The coefficients Cl are obtained through the least-squares fitting of eq 2 to the HOMO at grid points in the asymptotic region and then the ionization rate Γstat(E, θ) in a static electric field E is calculated using these coefficients Cl following the formula derived by Tong et al.22
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Γstat(E, θ) )
Itakura et al.
1 l · (R)Q(l, m′) |m′| ∑ Cl ∑ Dm′1 2 |m′| ! l
m′
1 κ2/κ-1
( ) 2κ3 E
2/κ-|m′|-1
( )
exp -
2κ3 3E
(3)
l where Dm′1 (R) is the rotational matrix, R is the Euler angles between the molecular axis and the direction of the laser polarization, and
Q(l, m′) ) (-1)m′
+ |m′|)! (2l +2(l1)(l- |m′|)!
The second model is a functional form for the multiphoton ionization. The model function used in the present study is expressed as
Γ(E(t),θ) ) AI(t)5cos2p θsin2q θ
(4)
where A is a variable constant and I(t) ) 0cE(t)2/2, with the speed of light c, and the permittivity in vacuum 0. The temporal intensity profile of a femtosecond pump pulse is assumed to be a Gaussian functional form as
{
I(t) ) I0exp -4 ln(2) ·
( )} t
τfwhm
2
(5)
where I0 is the peak intensity, and τfwhm is the full width at the half-maximum of the pulse duration. If the parallel and perpendicular transitions are included p and q times, respectively, in the multiphoton process, Γ(E(t), θ) should be proportional to cos2p θ sin2q θ as described in eq 4. The factor related to the laser intensity I(t) is assumed to be the fifth power of the laser intensity, because the previous experimental study by Talebpour et al.46 reported that the slope of the log-log plot of the intensity dependent ion yield curve was almost 5. The parameters p and q are variable parameters in our calculation. The variable constant A is also adjusted in order that the ionization yield becomes comparable to the experimental results. Although this model might not be so exact in a quantitative sense, the purpose of the model calculation is to extract a qualitative feature of the angle-dependent ionization of NO. 4. Results and Discussion 4.1. Rotational State Distributions and Influence of Ionization. The RE(1 + 1)PI spectra are recorded with the fs pump pulses, whose duration τfwhm are 55, 85, and 155 fs as shown in Figures 1a-c, respectively. As the laser pulse energy is increased, the transitions from the higher J levels in the X2Π1/2 V ) 0 state appear. The RSDs in Figures 2a-c are obtained through the procedure that the peak intensity of each line in the A2Σ+-X2Π1/2 (0-0) band is divided by its transition probability. The ionization probability from each rotational level in the A2Σ+ V ) 0 state is not taken into account, because the transition from the A2Σ+ to the ionization continuum is expected to be saturated. When the pump pulse energy is 0.4 mJ/pulse, the ionization yield is negligibly small. The RSD experimentally obtained by the 55 fs pulse (Figure 2a) can be reproduced by numerically solving the TDSE with no ionization effect at I0 ) 14 TW/cm2 as shown in Figure 2d. The peak intensity I0 for any pulse energy and duration can be estimated from the result that I0 is 14 TW/
Figure 1. Resonance enhanced (1 + 1) photon ionization spectra of the A2Σ+-X2Π1/2 (0,0) band of NO rotationally excited by the femtosecond pump pulses as a function of the pulse duration and energy. The durations (τfwhm) of the pump pulses are (a) 55, (b) 85, and (c) 155 fs. The pulse energies are 0.4, 0.8, and 1.2 mJ from the bottom in the respective graphs. The spectral intensity is normalized for comparison between the different spectra. The spectra recorded with 1.2 mJ pulses are magnified by factors of 3, 2, and 2 in a-c, respectively.
cm2 for the 55 fs pulse with 0.4 mJ, because the peak intensity is proportional to both the pulse energy and the inverse of the pulse duration. Thus, the peak intensities I0 of the 85 and 155 fs pulses with 0.4 mJ/pulse can be estimated to be 9 and 5 TW/ cm2, respectively. It is confirmed that the numerically derived RSDs assuming no ionization at these two peak intensities are in good agreement with the observed RSDs as shown in Figures 2e and 2f. As the pulse energy is increased to 0.8 mJ, the peak intensity I0 is estimated to be 28, 18, and 10 TW/cm2 for τfwhm ) 55, 85, and 155 fs, respectively. The observed RSDs are reproduced well by the numerical calculation assuming no ionization effect. Therefore, the influence of the ionization is considered to be still negligible. This is consistent with the observation that the total yield of the neutral NO is kept unity within the experimental uncertainty as listed in Table 1. It should be noted that if the pump pulse energy is kept a constant value no more than 0.8 mJ, the degree of the rotational excitation is almost the same irrespective of the pulse duration as shown in Figures 2a-c. RSD is governed by the pulse energy as far as the field intensity is low enough to neglect the ionization. This is reasonable because the magnitude of the interaction of the induced dipole moment is proportional to the square of the electric field E(t),
Correlation between Rotation and Ionization of NO
J. Phys. Chem. A, Vol. 114, No. 42, 2010 11205
Figure 3. Rotational state distributions of NO excited by the 55 fs pump pulses at the peak intensities of (a) 35 TW/cm2 and (b) 42 TW/ cm2: The experimental results (solid circles) and the calculated results assuming no ionization (open circles). Figure 2. Rotational state distributions of NO excited by the femtosecond pump pulses in the experiment (a-c) and simulation (d-f). The durations of the pump pulses are 55 fs (a, d), 85 fs (b, e), and 155 fs (c, f). The pulse energies of the pump pulse in the experiment (a-c) are 0.4 mJ (square), 0.8 mJ (triangle), and 1.2 mJ (circle). The corresponding peak intensities in the simulation (d-f) are described in the respective graphs. In the simulation, no ionization effect is taken into account.
TABLE 1: The Experimentally Derived Total Yields of the Neutral NO Molecules after the Interaction with the Femtosecond Pump Pulses As a Function of the Pulse Duration and Energy. 0.4 mJ 0.8 mJ 1.2 mJ
55 fs
85 fs
155 fs
1.08(37) 1.22(47) 0.70(4)
1.02(32) 1.19(21) 0.88(8)
1.13(38) 1.05(21) 0.82(7)
that is, the instantaneous intensity I(t), whose time integral corresponds to the pulse energy. When the pulse energy is increased to 1.2 mJ, the RSDs exhibit the dependence on the pulse duration, suggesting that the population depletion from the X2Π1/2, V ) 0 state by the ionization cannot be neglected. The decrease of the total yield is also confirmed at the pulse energy of 1.2 mJ as listed in Table 1. Particularly, the RSD by the 55 fs pulse is significantly different from those by the 85 and 155 fs pulses. The observed RSD by the 55 fs pulse at I0 ) 42 TW/cm2 (1.2 mJ) is deviated significantly from the numerically derived RSD assuming no ionization, while the observed RSDs by the 85 fs pulse at I0 ) 27 TW/cm2 and by the 155 fs pulse at I0 ) 15 TW/cm2 show almost the same behavior as the numerically derived RSDs assuming no ionization. It is also confirmed that the RSD obtained by the 55 fs pulse at I0 ) 35 TW/cm2 (1.0 mJ) deviates from the numerical result assuming no ionization whereas the degree of deviation at 35 TW/cm2 is smaller than that at 42 TW/cm2. At both the intensities of 35 and 42 TW/cm2, as shown in Figure 3, the
experimentally observed population at the higher J levels becomes larger than the population calculated assuming no ionization, whereas at the lower J levels the magnitude relation is reversed. It should be emphasized that the increase in the population at the higher J levels is evidence of the rearrangement of the rotational quantum states induced by the angle-dependent ionization. In other words, the angle-dependent ionization makes a hole in the rotational wave packet as recently reported for a vibrational wave packet38,39,42 and an electronic wave packet.40,41 4.2. Numerical Simulation on the Basis of MO-ADK Theory. The discrepancy between the observed RSDs and the calculated ones assuming no ionization reflects the population depletion from the X2Π1/2, V ) 0 state and the population transfer between the different J levels in the X2Π1/2, V ) 0 state. To investigate how this discrepancy is caused, we numerically solve the TDSE using two types of models of the ionization probability Γ(E(t), θ), which is substituted into eq 1. First, the MO-ADK theory22 is adopted for the calculation of Γ(E(t), θ). When I(t) ) 42 TW/cm2 (E(t)) 1.8 × 1010 V/m), the ionization rate calculated from the MO-ADK theory is shown in Figure 4a as the average of eq 3 over a half cycle of an AC field, during which the tunneling electron is emitted toward the angle θ. The angular dependence in Figure 4 reflects the shape of the HOMO, which has the larger lobe at the nitrogen site than at the oxygen site. When the laser electric field is directed to the reverse during the second half of the one laser cycle, a tunneling electron is emitted toward the angle π-θ. Γ(E(t), θ) averaged over one cycle of a laser AC field can be obtained as the mean of the ionization rates for a tunneling electron ejected toward θ and π-θ averaged over a half cycle, resulting in the reflection symmetry with respect to the θ ) 90° plane. The TDSE is numerically solved using the one-cycle averaged ionization rate, which is calculated as a function of the temporally varying intensity in each time step. The calculated result indicates that the influence of the ionization is very limited at I0 ) 42 TW/cm2 and τfwhm ) 55 fs in this model. The total yield of the remaining neutral NO after the interaction with the
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Figure 4. Polar plots of the ionization rate calculated on the basis of the MO-ADK theory at I0 ) 42 TW/cm2. This rate is average over a half cycle of an AC field during which the tunneling electron is emitted toward the angle θ along the laser polarization. The θ ) 0 (upward) in the graphs is the direction from the nitrogen to the oxygen in the molecular frame. The rate averaged over one cycle can be obtained as the mean of the ionization rates for a tunneling electron ejected toward θ (solid line) and π-θ (dotted line) during a half cycle. The rate is calculated with (a) the coefficients Cl obtained in this study and (b) those of Tong et al.22
laser pulse is calculated to be 0.995, which is almost unity and much larger than the observed value of 0.70(4). In Figure 4, the ionization rate Γ(E(t), θ) using the coefficients Cl calculated in this study is compared with that using the coefficients Cl reported in the original paper of MO-ADK by Tong et al., where the multiple-scattering method was adopted.22 The ionization rate Γ(E(t), θ) determined by the coefficients Cl of Tong et al. exhibits almost the same angular profile as that by our coefficients Cl. However, the magnitude of the ionization rate of Tong et al. is larger than our value by a factor of 30. When the ionization rate obtained by the coefficients Cl of Tong et al. is used in solving the TDSE, the total yield of the remaining neutral NO after the interaction with the laser pulse (τfwhm ) 55 fs, I0 ) 42 TW/cm2) becomes 0.888. Although the influence of the ionization becomes visible, the remaining neutral yield is still larger than the observed value of 0.70(4). As plotted in Figure 5a, the RSD numerically derived using Γ(E(t), θ) on the basis of the MO-ADK theory with the coefficients Cl of Tong et al. indicates that the population in the entire range of J is reduced from that assuming no ionization. Comparing this calculated RSD with the observed one, it is found that the calculated population at J < 3.5 is larger than the observed one and that the population at J g 3.5 is smaller than the observed one. This qualitative trend is the same as that in the RSD calculated assuming no ionization. As recently pointed out, the HOMO expressed with a Gaussian basis set is not so accurate in the asymptotic region even if the number of included basis functions is increased.47 The amplitude of a Gaussian HOMO in the asymptotic region drops to zero more rapidly than the exact HOMO due to the nature of the Gaussian functional form. The angular profiles of the two ionization rates Γ(E(t), θ) in Figures 4a and 4b exhibit only small difference. If these angular profiles are expected to be close to the exact angular dependence, it would be a good approximation to introduce a scaling factor b for multiplying all the coefficients Cl in order that the calculated ionization yield becomes comparable to the observed one. The ionization rate Γ(E(t), θ) on the basis of the MOADK theory is linearly proportional to b2, the square of this scaling factor. For the ionization rate derived from the Gaussian HOMO, the square of the scaling factor is determined to be b2 ) 80 in order that the remaining neutral yield becomes comparable to the observed neutral yield. For comparison, the TDSE calculation is also performed with b2 ) 3 and the coefficients Cl of
Figure 5. Rotational state distributions of NO excited by the 55 fs pump pulse at I0 ) 42 TW/cm2: (a) The experimental results (solid circles), the calculated results assuming no ionization (open circles), the calculated results on the basis of MO-ADK theory with the Gaussian HOMO (solid triangles) and with the multiple-scattering theory [ref 22] (open triangles). (b) The calculated results by the MO-ADK ionization rate with the Gaussian basis set and the factor of b2 ) 80 (solid squares), and the calculated results by the MO-ADK ionization rate with the multiple-scattering theory and the factor of b2 ) 3 (open diamonds). The experimental results and the calculated ones assuming no ionization are the same as those in panel a.
Tong et al.22 As shown in Figure 5b, the different sets of the coefficients Cl show almost the same RSD. However, they deviate significantly from the experimentally observed RSD. When the scaling factors are included in Γ(E(t), θ), the calculated RSDs exhibit the decrease in the population at all the rotational levels and cannot reproduce the characteristic observed profile that the population at the higher J increases while that at the lower J decreases. Considering that the Keldysh parameter is 1.34 at the intensity of 42 TW/cm2, the multiphoton ionization mechanism might be more appropriate than the tunneling ionization.20 4.3. Numerical Simulation Using Model Functions for Multiphoton Ionization. The ionization rate for the multiphoton ionization is assumed to be a functional form of eq 4, in which the parameters A, p, and q can be varied. The parameter A is adjusted in order that the total ionization yield becomes comparable to the observed yield. The summation of the ionization potential (9.26 eV) and the ponderomotive shift (2.57 eV) at I0 ) 42 TW/cm2 and λ ∼ 810 nm is 11.83 eV, and then at least eight-photon energy is required for the ionization. Assuming p + q ) 8, Γ(E(t), θ) in the form of eq 4 is substituted into the Hamiltonian (eq 1) for solving the TDSE. Irrespective of the q value, the angular dependence of Γ(E(t), θ) exhibits a single peak in the range from θ ) 0 to 90° and the reflection symmetry with respect to the θ ) 90° plane as shown in Figure 6a. As the q value is increased from 0 to 8, the peak positions in the angular probabilities are monotonically shifted from 0 to 90°. The RSDs calculated with these model functions for the eightphoton ionization are shown in Figure 7. Only the angular profile of sin16 θ demonstrates the characteristic behavior of increase in the population at the higher J (J ) 5.5, 6.5) as well as decrease at the lower J (J ) 0.5-2.5) as identified in the
Correlation between Rotation and Ionization of NO
Figure 6. (a) Angular probabilities of the model functions for the eightphoton ionization proportional to cos2p θ sin2q θ, (p + q ) 8): sin16 θ (solid thick line), cos2 θ sin14 θ (short-dashed line), cos4 θ sin12 θ (longdashed line), cos14 θ sin2 θ (dash-dotted line), and cos16 θ (dotted line). The probabilities are normalized so that the peak values will be unity. As a higher-order model, sin50 θ is plotted (solid thin line). (b) Probabilities of the alignment angle of the molecular axis of NO for the respective rotational levels J with |M| ) 0.5. The plotted values are multiplied by sin θ.
Figure 7. Rotational state distributions of NO excited by the 55 fs pump pulse at I0 ) 42 TW/cm2: (a) The experimental results (solid circles) and the calculated results assuming no ionization (open circles). The calculated results using Γ(E(t), θ) proportional to sin16 θ (solid squares) and sin50 θ (open diamonds). (b) The calculated results using Γ(E(t), θ) proportional to sin14 θ cos2 θ (solid triangles) and sin12 θ cos4 θ (open triangles). The experimental results and calculated ones assuming no ionization are the same as panel a.
experiment. If the contribution of only one parallel transition (p ) 1) is included in the multiphoton process, then the calculated populations at the higher J become smaller than those calculated assuming no ionization effect. These results suggest that the ionization probability needs to have a peak at θ ) 90°
J. Phys. Chem. A, Vol. 114, No. 42, 2010 11207 in order to reproduce the experimental results, that is, only the perpendicular transition moments are included in the multiphoton process. Although it is difficult to conclusively explain the mechanism of the present finding that the ionization probability has a peak at θ ) 90°, the perpendicular transition moment between the lowest electronically excited A2Σ+ state and the electronic ground X2Π1/2 state might play an important role in the earlier steps of the multiphoton ionization. It is worthwhile to compare the angular dependence in the ionization and the angular distribution of the molecular axis in the respective rotational levels with the quantum numbers J and |M| ) 0.5. At θ ) 90°, the angular distribution at the J ) 0.5 level exhibits the largest population among all the J levels. When J - 0.5 is even (odd), the angular distribution at the J level shows the maximum (minimum) at θ ) 90° as shown in Figure 6b. As the quantum number J increases, the peak in the angular distribution at the respective J levels is shifted from θ ) 90° toward θ ) 0° and 180°. In the numerical simulation using Γ(E(t), θ) proportional to sin16 θ, the ionization makes a hole in the rotational wave packet around θ ) 90°, leading to not only the decrease of the population at the lower J levels, but also the creation of the population at the higher J levels. Simultaneously, the nonadiabatic rotational excitation takes place, resulting in the population transfer between the different rotational levels obeying the selection rule of ∆J ) (2, (1.16 Considering the large experimental uncertainty, we cannot discuss the detailed profile in the RSDs. In general, however, a sharper angular profile of Γ(E(t), θ) leads to more significant rearrangement of the amplitude and phase of the respective rotational states composing the rotational wave packet. In order to confirm that Γ(E(t), θ) with a shaper peak at θ ) 90° can create larger population at the higher J levels (J ) 3.5-6.5), we numerically solve the TDSE with Γ(E(t), θ) proportional to sin50 θ, which shows a narrower peak at θ ) 90° than sin16 θ as shown Figure 6a. The RSD calculated with sin50 θ becomes closer to the experimental one except for an overestimate at J ) 0.5. This small discrepancy at J ) 0.5 can be improved by adding another functional form such as cos2 θ sin14 θ into Γ(E(t), θ). However, the definite functional form cannot be determined partly due to the large experimental uncertainty. Comparison of our numerical simulations with the experimental data suggests qualitatively that Γ(E(t), θ) exhibiting the peak at θ ) 90° reproduces the characteristic observed feature of the increased population at the higher J levels. Recently, the surprisingly sharp angular profile in Γ(E(t), θ) was experimentally extracted for CO2 by Pavicˇic´ et al.35 Considering that the Keldysh parameter was 1.0 in their experimental condition, the ionization mechanism should be an intermediate between multiphoton and tunneling. Sharp angular dependence in ionization might be a characteristic feature in the intermediate regime between multiphoton and tunneling although the origin of the sharp peak is unclear. Finally, it would be valuable to mention that the relative phase δJ of the respective rotational states with the quantum number J provides the fruitful information on how the respective rotational levels are influenced by the ionization as well as the nonadiabatic rotational excitation. It is derived from the secondorder perturbation theory48 that in the weak-field limit the phase difference between two states coupled by the lowest-order nonresonant Raman process (∆J ) 2) is π/2 for NO X2Π1/2. Assuming the 55 fs pulse with I0 ) 42 TW/cm2 causes no ionization, the TDSE calculation demonstrates in Figure 8 that the phase difference δJ+2 - δJ approaches to π/2 as J increases.
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Itakura et al. 5. Summary
Figure 8. Calculated phase difference between J + 2 and J levels after the interaction with the 55 fs pulse at I0 ) 42 TW/cm2. Different ionization models are adopted in the calculation as follows: no ionization (solid circles), the MOADK ionization rate with the factor of b2 ) 80 (open circles), and Γ(E(t), θ) proportional to sin16 θ (triangles) and sin50 θ. (squares).
This trend was already demonstrated in the previous study on benzene26 while an intuitive explanation for the deviation from π/2 is difficult. When the effect of ionization is included in the TDSE calculation, the phase difference as a function of J deviates from that assuming no ionization as shown in Figure 8. The calculated result on the basis of the MOADK ionization rate multiplied by the factor of b2 ) 80 shows that the phase difference at J ) 0.5 and J > 4.5 deviates slightly from that assuming no ionization. This slight deviation suggests that the coupling between the different rotational states by the imaginary potential is quite small while the decrease in the population of the respective rotation states composing the rotational wave packet takes place significantly. When Γ(E(t), θ) proportional to sin2q θ (q ) 8 or 25) is used in the calculation, the phase differences deviate largely from that assuming no ionization. This large deviation indicates that the creation of the new population at the higher J levels proceeds through the coupling between the different rotational states by the imaginary potential. The angle dependence of Γ(E(t), θ) results in nonzero values in the imaginary part of the off-diagonal matrix elements in the Hamiltonian. The interaction by the induced dipole moment is related to the real part of the off-diagonal matrix elements. The phase of each rotational state is affected by both the real and imaginary parts of the off-diagonal matrix elements, which corresponds to the coupling between different rotational states. In the higher J levels, the coupling with the other J levels is dominantly governed by the imaginary part of the matrix elements related to the ionization, because the simulation assuming no ionization represents the small yield in the higher J levels. It is reasonable that the deviation of the phase difference for the sin50 θ model is larger than that for the sin16 θ model, because Γ(E(t), θ) proportional to sin50 θ can make a shaper hole in the rotational wave packet and the larger contribution from the rotational states with higher J is necessary for the sharper hole. In the present study, only the amplitude of the respective rotational quantum states can be measured experimentally. However, it has been already demonstrated that the retrieve of the phase of the rotational wave packet is feasible using the wave packet interferometry combined with the spectroscopic technique.26 In the near future, the measurement of the phase will be performed for reconstruction of the rotational wave packet depleted by the ionization and a deeper insight into the angle-dependent ionization will be gained.
We have observed the RSDs of the nonadiabatically excited neutral NO influenced by the ionization in intense laser fields. When the laser peak intensity I0 is above 30 TW/cm2, the influence of the ionization can be identified in the RSDs. Comparing the observed RSDs with the numerically simulated RSDs assuming no ionization, it is clearly recognized that the population at the higher J levels is enhanced by the ionization although the total population is reduced. This finding suggests that the amplitude and phase of the respective rotational states composing a rotational wave packet are modulated by the angledependent ionization. The numerical simulations including the different models for the ionization have been carried out. It is suggested that the multiphoton ionization mechanism is more important at I0 e 42 TW/cm2 than the tunneling ionization. When the model functions proportional to sin2q θ, which include only perpendicular transitions in the multiphoton process, are assumed to be the ionization rate Γ(E(t), θ), the calculated RSD is in qualitative agreement with the experimentally observed RSD. It has been also demonstrated numerically that the phase of the respective rotational states with the quantum number J is sensitive to the influence of the ionization. As an additional experimental technique, pumping with a fewcycle pulse would be useful to clearly disentangle the angledependent ionization from the rotational excitation. The shortest pulse duration in the present experiment is 55 fs, during which the rotational excitation is induced as well as the ionization. If the pulse duration is decreased to less than 10 fs, the contribution of the rotational excitation would be reduced considerably,32,33 leading to the clear and simple situation that the RSD is governed dominantly by the ionization. Acknowledgment. This study was supported by the cooperative research program of IMS. The authors thank Dr. T. Yasuike for his valuable discussion on the theoretical treatment of ionization. The Grants-in-Aid from MEXT Japan (Nos. 18244120, 18750020, and 19685003), the Consortium for Photon Science and Technology, and the financial support from the Matsuo Foundation are also acknowledged. References and Notes (1) Laser Control and Manipulation of Molecules; Bandrauk, A. D., Fujimura, Y., Gordon, R. J. , Eds.; American Chemical Society: Washington, DC, 2002; Vol. 821. (2) Yamanouchi, K. Science 2002, 295, 1659–1660. (3) Rabitz, H.; de Vivie-Riedle, R.; Motzkus, M.; Kompa, K. Science 2000, 288, 824–828. (4) Levis, R. J.; Menkir, G. M.; Rabitz, H. Science 2001, 292, 709– 713. (5) Kling, M. F.; Siedschlag, C.; Verhoef, A. J.; Khan, J. I.; Schultze, M.; Uphues, T.; Ni, Y.; Uiberacker, M.; Drescher, M.; Krausz, F.; Vrakking, M. J. J. Science 2006, 312, 246–248. (6) Yazawa, H.; Tanabe, T.; Okamoto, T.; Yamanaka, M.; Kannari, F.; Itakura, R.; Yamanouchi, K. J. Chem. Phys. 2006, 124, 204314. (7) Meijer, A. S.; Zhang, Y.; Parker, D. H.; van der Zande, W. J.; Gijsbertsen, A.; Vrakking, M. J. J. Phys. ReV. A 2007, 76, 023411. (8) Suzuki, T.; Sugawara, Y.; Minemoto, S.; Sakai, H. Phys. ReV. Lett. 2008, 100, 033603. (9) Kitano, K.; Hasegawa, H.; Ohshima, Y. Phys. ReV. Lett. 2009, 103, 223002. (10) Friedrich, B.; Herschbach, D. Phys. ReV. Lett. 1995, 74, 4623– 4626. (11) Larsen, J. J.; Sakai, H.; Safvan, C. P.; Wendt-Larsen, I.; Stapelfeldt, H. J. Chem. Phys. 1999, 111, 7774–7781. (12) Larsen, J. J.; Hald, K.; Bjerre, N.; Stapelfeldt, H.; Seideman, T. Phys. ReV. Lett. 2000, 85, 2470. (13) Rosca-Pruna, F.; Vrakking, M. J. J. Phys. ReV. Lett. 2001, 87, 153902.
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